Some more on symmetries…

In our previous post, we talked a lot about symmetries in space – in a rather playful way. Let’s try to take it further here by doing some more thinking on symmetries in spacetime. This post will pick up some older stuff – from my posts on states and the related quantum math in November 2015, for example – but that shouldn’t trouble you too much. On the contrary, I actually hope to tie up some loose ends here.

Let’s first review some obvious ideas. Think about the direction of time. On a time axis, time goes from left to right. It will usually be measured from some zero point – like when we started our experiment or something 🙂 – to some +point but we may also think of some point in time before our zero point, so the minus (−t) points – the left side of the axis – make sense as well. So the direction of time is clear and intuitive. Now, what does it mean to reverse the direction of time? We need to distinguish two things here: the convention, and… Well… Reality. If we would suddenly decide to reverse the direction in which we measure time, then that’s just another convention. We don’t change reality: trees and kids would still grow the way they always did. 🙂 We would just have to change the numbers on our clocks or, alternatively, the direction of rotation of the hand(s) of our clock, as shown below. [I only showed the hour hand because… Well… I don’t want to complicate things by introducing two time units. But adding the minute hand doesn’t make any difference.]

clock problemNow, imagine you’re the dictator who decided to change our time measuring convention. How would you go about it? Would you change the numbers on the clock or the direction of rotation? Personally, I’d be in favor of changing the direction of rotation. Why? Well… First, we wouldn’t have to change expressions such as: “If you are looking north right now, then west is in the 9 o’clock direction, so go there.” 🙂 More importantly, it would align our clocks with the way we’re measuring angles. On the other hand, it would not align our clocks with the way the argument (θ) of our elementary wavefunction ψ = a·eiθ = ei·(E·t – p·x)/ħ is measured, because that’s… Well… Clockwise.

So… What are the implications here? We would need to change t for −t in our wavefunction as well, right? Yep. Good point. So that’s another convention that would change: we should write our elementary wavefunction now as ψ = a·ei·(E·t – p·x)/ħ. So we would have to re-define θ as θ = –E·t + p·x = p·x –E·t. So… Well… Done!

So… Well… What’s next? Nothing. Note that we’re not changing reality here. We’re just adapting our formulas to a new dictatorial convention according to which we should count time from positive to negative – like 2, 1, 0, -1, -2 etcetera, as shown below. Fortunately, we can fix all of our laws and formulas in physics by swapping for -t. So that’s great. No sweat. time reversal

Is that all? Yes. We don’t need to do anything else. We’ll still measure the argument of our wavefunction as an angle, so that’s… Well… After changing our convention, it’s now clockwise. 🙂 Whatever you want to call it: it’s still the same direction. Our dictator can’t change physical reality 🙂

Hmm… But so we are obviously interested in changing physical reality. I mean… Anyone can become a dictator, right? In contrast, we – enlightened scientists – want to really change the world, don’t we? 🙂 So what’s a time reversal in reality? Well… I don’t know… You tell me. 🙂 We may imagine some movie being played backwards, or trees and kids shrinking instead of growing, or some bird flying backwards – and I am not talking the hummingbird here. 🙂

Hey! The latter illustration – that bird flying backwards – is probably the better one: if we reverse the direction of time – in reality, that is – then we should also reverse all directions in space. But… Well… What does that mean, really? We need to think in terms of force fields here. A stone that’d be falling must now go back up. Two opposite charges that were going towards each other, should now move away from each other. But… My God! Such world cannot exist, can it?

No. It cannot. And we don’t need to invoke the second law of thermodynamics for that. 🙂 None of what happens in a movie that’s played backwards makes sense: a heavy stone does not suddenly fly up and decelerate upwards. So it is not like the anti-matter world we described in our previous post. No. We can effectively imagine some world in which all charges have been replaced by their opposite: we’d have positive electrons (positrons) around negatively charged nuclei consisting of antiprotons and antineutrons and, somehow, negative masses. But Coulomb’s law would still tell us two opposite charges – q1 and –q2 , for example – don’t repel but attract each other, with a force that’s proportional to the product of their charges, i.e. q1·(-q2) = –q1·q2. Likewise, Newton’s law of gravitation would still tell us that two masses m1 and m2 – negative or positive – will attract each other with a force that’s proportional to the product of their masses, i.e. m1·m= (-m1)·(-m2). If you’d make a movie in the antimatter world, it would look just like any other movie. It would definitely not look like a movie being played backwards.

In fact, the latter formula – m1·m= (-m1)·(-m2) – tells us why: we’re not changing anything by putting a minus sign in front of all of our variables, which are time (t), position (x), mass (m) and charge (q). [Did I forget one? I don’t think so.] Hence, the famous CPT Theorem – which tells us that a world in which (1) time is reversed, (2) all charges have been conjugated (i.e. all particles have been replaced by their antiparticles), and (3) all spatial coordinates now have the opposite sign, is entirely possible (because it would obey the same Laws of Nature that we, in our world, have discovered over the past few hundred years) – is actually nothing but a tautology. Now, I mean that literally: a tautology is a statement that is true by necessity or by virtue of its logical form. Well… That’s the case here: if we flip the signs of all of our variables, we basically just agreed to count or measure everything from positive to negative. That’s it. Full stop. Such exotic convention is… Well… Exotic, but it cannot change the real world. Full stop.

Of course, this leaves the more intriguing questions entirely open. Partial symmetries. Like time reversal only. 🙂 Or charge conjugation only. 🙂 So let’s think about that.

We know that the world that we see in a mirror must be made of anti-matter but, apart from that particularity, that world makes sense: if we drop a stone in front of the mirror, the stone in the mirror will drop down too. Two like charges will be seen as repelling each other in the mirror too, and concepts such as kinetic or potential energy look just the same. So time just seems to tick away in both worlds – no time reversal here! – and… Well… We’ve got two CP-symmetrical worlds here, don’t we? We only flipped the sign of the coordinate frame and of the charges. Both are possible, right? And what’s possible must exist, right? Well… Maybe. That’s the next step. Let’s first see if both are possible. 🙂

Now, when you’ve read my previous post, you’ll note that I did not flip the z-coordinate when reflecting my world in the mirror. That’s true. But… Well… That’s entirely beside the point. We could flip the z-axis too and so then we’d have a full parity inversion. [Or parity transformation – sounds more serious, doesn’t it? But it’s only a simple inversion, really.] It really doesn’t matter. The point is: axial vectors have the opposite sign in the mirror world, and so it’s not only about whether or not an antimatter world is possible (it should be, right?): it’s about whether or not the sign reversal of all of those axial vectors makes sense in each and every situation. The illustration below, for example, shows how a left-handed neutrino should be a right-handed antineutrino in the mirror world.right-handed antineutrinoI hope you understand the left- versus right-handed thing. Think, for example, of how the left-circularly polarized wavefunction below would look like in the mirror. Just apply the customary right-hand rule to determine the direction of the angular momentum vector. You’ll agree it will be right-circularly polarized in the mirror, right? That’s why we need the charge conjugation: think of the magnetic moment of a circulating charge! So… Well… I can’t dwell on this too much but – if Maxwell’s equations are to hold – then that world in the mirror must be made of antimatter.animation

Now, we know that some processes – in our world – are not entirely CP-symmetrical. I wrote about this at length in previous posts, so I won’t dwell on these experiments here. The point is: these experiments – which are not easy to understand – lead physicists, philosophers, bloggers and what have you to solemnly state that the world in the mirror cannot really exist. And… Well… They’re right. However, I think their observations are beside the point. Literally.

So… Well… I would just like to make a very fundamental philosophical remark about all those discussions. My point is quite simple:

We should realize that the mirror world and our world are effectively separated by the mirror. So we should not be looking at stuff in the mirror from our perspective, because that perspective is well… Outside of the mirror. A different world. 🙂 In my humble opinion, the valid point of reference would be the observer in the mirror, like the photographer in the image below. Now note the following: if the real photographer, on this side of the mirror, would have a left-circularly polarized beam in front of him, then the imaginary photographer, on the other side of the mirror, would see the mirror image of this left-circularly polarized beam as a left-circularly polarized beam too. 🙂 I know that sounds complicated but re-read it a couple of times and – I hope – you’ll see the point. If you don’t… Well… Let me try to rephrase it: the point is that the observer in the mirror would be seeing our world – just the same laws and what have you, all makes sense! – but he would see our world in his world, so he’d see it in the mirror world. 🙂


Capito? If you would actually be living in the mirror world, then all the things you would see in the mirror world would make perfectly sense. But you would be living in the mirror world. You would not look at it from outside, i.e. from the other side of the mirror. In short, I actually think the mirror world does exist – but in the mirror only. 🙂 […] I am, obviously, joking here. Let me be explicit: our world is our world, and I think those CP violations in Nature are telling us that it’s the only real world. The other worlds exist in our mind only – or in some mirror. 🙂

Post scriptum: I know the Die Hard philosophers among you will now have an immediate rapid-backfire question. [Hey – I just invented a new word, didn’t I? A rapid-backfire question. Neat.] How would the photographer in the mirror look at our world? The answer to that question is simple: symmetry! He (or she) would think it’s a mirror world only. His world and our world would be separated by the same mirror. So… What are the implications here?

Well… That mirror is only a piece of glass with a coating. We made it. Or… Well… Some man-made company made it. 🙂 So… Well… If you think that observer in the mirror – I am talking about that image of the photographer in that picture above now – would actually exist, then… Well… Then you need to be aware of the consequences: the corollary of his existence is that you do not exist. 🙂 And… Well… No. I won’t say more. If you’re reading stuff like this, then you’re smart enough to figure it out for yourself. We live in one world. Quantum mechanics tells us the perspective on that world matters very much – amplitudes are different in different reference frames – but… Well… Quantum mechanics – or physics in general – does not give us many degrees of freedoms. None, really. It basically tells us the world we live in is the only world that’s possible, really. But… Then… Well… That’s just because physics… Well… When everything is said and done, it’s just mankind’s drive to ensure our perception of the Universe lines up with… Well… What we perceive it to be. 😦 or 🙂 Whatever your appreciation of it. Those Great Minds did an incredible job. 🙂

Time reversal and CPT symmetry (III)

Pre-scriptum (dated 26 June 2020): While my posts on symmetries (and why they may or may be broken) are somewhat mutilated (removal of illustrations and other material) as a result of an attack by the dark force, I am happy to see a lot of it survived more or less intact. While my views on the true nature of light, matter and the force or forces that act on them – all of the stuff that explains symmetries or symmetry-breaking, in other words – have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics, I think most (if not all) of the analysis in this post remains valid and fun to read. 🙂

Original post:

Although I concluded my previous post by saying that I would not write anything more about CPT symmetry, I feel like I have done an injustice to Val Fitch, James Cronin, and all those other researchers who spent many man-years to painstakingly demonstrate how the weak force does not always respect the combined charge-parity (C-P) symmetry. Indeed, I did not want to denigrate their efforts when I noted that:

  1. These decaying kaons (i.e. the particles that are used to demonstrate the CP symmetry-breaking phenomenon) are rather exotic and very short-lived particles; and
  2. Researchers have not been able to find many other traces of non-respect of CP symmetry, except when studying a heavier version of these kaons (the so-called B- and D-mesons) as soon as these could be produced in higher volumes in newer (read: higher-energy) particle colliders (so that’s in the last ten or fifteen years only), but so these B- and D-mesons are even more rare and even less stable.

CP violation is CP violation: it’s plain weird, especially when Fermilab and CERN experiments observed direct CP violation in kaon decay processes. [Remember that the original 1964 Fitch-Cronin experiment could not directly observe CP violation: in their experiment, CP violation in neutral kaon decay processes could only be deduced from other (unexpected) decay processes.]

Why? When one reverses all of the charges and other variables (such as parity which – let me remind you – has to do with ‘left-handedness’ and ‘right-handedness’ of particles), then the process should go in the other direction in an exactly symmetric way. Full stop. If not, there’s some kind of ‘leakage’ so to say, and such ‘leakage’ would be ‘kind-of-OK’ when we’d be talking some kind of chemical or biological process, but it’s obviously not ‘kind-of-OK’ when we’re talking one of the fundamental forces. It’s just not ‘logical’.

Feynman versus ‘t Hooft: pro and contra CP-symmetry breaking

A remark that is much more relevant than the two comments above is that one of the most brilliant physicists of the 20th century, Richard Feynman, seemed to have refused to entertain the idea of CP-symmetry breaking. Indeed, while, in his 1965 Lectures, he devotes quite a bit of attention to Chien-Shiung Wu’s 1956 experiment with decaying cobalt-60 nuclei (i.e. the experiment which first demonstrated parity violation, i.e. the breaking of P-symmetry), he does not mention the 1964 Fitch-Cronin experiment, and all of his writing in these Lectures makes it very clear that he not only strongly believes that the combined CP symmetry holds, but that it’s also the only ‘symmetry’ that matters really, and the only one that Nature truly respects–always.

So Feynman was wrong. Of course, these Lectures were published less than a year after the 1964 Fitch-Cronin experiment and, hence, you might think he would have changed his ideas on the possibility of Nature not respecting CP-symmetry–just like Wolfgang Pauli, who could only accept the reality of Nature not respecting reflection symmetry (P-symmetry) after repeated experiments re-confirmed the results of Wu’s original 1956 experiment.

But – No! – Feynman’s 1985 book on quantum electrodynamics (QED) –so that’s five years after Fitch and Cronin got a Nobel Prize for their discovery– is equally skeptical on this point: he basically states that the weak force is “not well understood” and that he hopes that “a more beautiful and, hence, more accurate understanding” of things will emerge.

OK, you will say, but Feynman passed away shortly after (he died from a rare form of cancer in 1988) and, hence, we should now listen to the current generation of physicists.

You’re obviously right, so let’s look around. Hmm… Gerard ‘t Hooft? Yes ! He is 67 now but – despite his age – it is obvious that he surely qualifies as a ‘next-generation’ physicist. He got his Nobel Prize for “elucidating the quantum structure of electroweak interactions” (read: for clarifying how the weak force actually works) and he is also very enthusiastic about all these Grand Unified Theories (most notably string and superstring theory) and so, yes, he should surely know, shouldn’t he?

I guess so. However, even ‘t Hooft writes that these experiments with these ‘crazy kaons’ – as he calls them – show ‘violation’ indeed, but that it’s marginal: the very same experiments also show near-symmetry. What’s near-symmetry? Well… Just what the term says: the weak force is almost symmetrical. Hence, CP-symmetry is the norm and CP-asymmetry is only a marginal phenomenon. That being said, it’s there and, hence, it should be explained. How?

‘t Hooft himself writes that one could actually try to interpret the results of the experiment by adding some kind of ‘fifth’ force to our world view – a “super-weak force” as he calls it, which would interfere with the weak force only.

To be fair, he immediately adds that introducing such ‘fifth force’ doesn’t really solve the “mystery” of CP asymmetry, because, while we’d restore the principle of CP symmetry for the weak force interactions, we would then have to explain why this ‘super-weak’ force does not respect it. In short, we cannot just reason the problem away. Hence, ‘t Hooft’s conclusion in his 1996 book on The Ultimate Building Blocks of the universe is quite humble: “The deeper cause [of CP asymmetry] is likely to remain a mystery.” (‘t Hooft, 1996, Chapter 7: The crazy kaons)

What about other explanations? For example, you might be tempted to think these two or three exceptions to a thousand cases respecting the general rule must have something to do with quantum-mechanical uncertainty: when everything is said and done, we’re dealing with probabilities in quantum mechanics, aren’t we? Hence, exceptions do occur and are actually expected to occur.

No. Quantum indeterminism is not applicable here. While working with probability amplitudes and probabilities is effectively equivalent to stating some general rules involving some average or mean value and then some standard deviation from that average, we’ve got something else going on here: Fitch and Cronin took a full six months indeed–repeating the experiment over and over and over again–to firmly establish a statistically significant bias away from the theoretical average. Hence, even if the bias is only 0.2% or 0.3%, it is a statistically significant difference between the probability of a process going one way, and the probability of that very same process going the other way.

So what? There are so many non-reversible processes and asymmetries in this world: why don’t we just accept this?Well… I’ll just refer to my previous post on this one: we’re talking a fundamental force here – not some chemical reaction – and, hence, if we reverse all of the relevant charges (including things such as left-handed or right-handed spin), the reaction should go the other way, and with exactly the same probability. If it doesn’t, it’s plain weird. Full stop.

OK. […] But… Perhaps there is some external phenomenon affecting these likelihoods, like these omnipresent solar neutrinos indeed, which I mentioned in a previous post and which are all left-handed. So perhaps we should allow these to enter the equation as well. […] Well… I already said that would make sense–to some extent at least– because there is some flimsy evidence of solar flares affecting radioactive decay rates (solar flares and neutrino outbursts are closely related, so if solar flares impact radioactive decay, we could or should expect them to meddle with any beta decay process really). That being said, it would not make sense from other, more conventional, points of view: we cannot just ‘add’ neutrinos to the equation because then we’d be in trouble with the conservation laws, first and foremost the energy conservation law! So, even if we would be able to work out some kind of theoretical mechanism involving these left-handed solar neutrinos (which are literally all over the place, bombarding us constantly even if they’re very hard to detect), thus explaining the observed P-asymmetry, we would then have to explain why it violates the energy conservation law! Well… Good luck with that, I’d say!

So it is a conundrum really. Let me sum up the above discussion in two bullet points:

  1. While kaons are short-lived particles because of the presence of the second-generation (and, hence, unstable) s-quark, they are real particles (so they are not some resonance or some so-called virtual particle). Hence, studying their behavior in interactions with any force field (and, most notably, their behavior in regard to the weak force) is extremely relevant, and the observed CP asymmetry–no matter how small–is something which should really grab our attention.
  2. The philosophical implications of any form of non-respect of the combined CP symmetry for our common-sense notion of time are truly profound and, therefore, the Fitch-Cronin experiment rightly deserves a lot of accolades.

So let’s analyze these ‘philosophical implications’ (which is just a somewhat ‘charged’ term for the linkage between CP- and time-symmetry which I want to discuss here) somewhat more in detail.

Time reversal and CPT symmetry

In the previous posts, I said it’s probably useful to distinguish (a) time-reversal as a (loosely defined) philosophical concept from (b) the mathematical definition of time-reversal, which is much more precise and unambiguous. It’s the latter which is generally used in physics, and it amounts to putting a minus sign in front of all time variables in any equation describing some situation, process or system in physics. That’s it really. Nothing more.

The point that I wanted to make is that true time reversal – i.e. time-reversal in the ‘philosophical’ or ‘common-sense’ interpretation – also involves a reversal of the forces, and that’s done through reversing all charges causing those forces. I used the example of the movie as a metaphor: most movies, when played backwards, do not make sense, unless we reverse the forces. For example, seeing an object ‘fall back’ to where it was (before it started falling) in a movie playing backwards makes sense only if we would assume that masses repel, instead of attract, each other. Likewise, any static or dynamic electromagnetic phenomena we would see in that backwards playing movie would make sense only if we would assume that the charges of the protons and electrons causing the electromagnetic fields involved would be reversed. How? Well… I don’t know. Just imagine some magic.

In such world view–i.e. a world view which connects the arrow of time with real-life forces that cause our world to change– I also looked at the left- and right-handedness of particles as some kind of ‘charge’, because it co-determines how the weak force plays out. Hence, any phenomenon in the movie having to do with the weak force (such as beta decay) could also be time-reversed by making left-handed particles right-handed, and right-handed particles left-handed. In short, I said that, when it comes to time reversal, only a full CPT-transformation makes sense–from a philosophical point of view that is.

Now, reversing left- and right-handedness amounts to a P-transformation (and don’t interrupt me now by asking why physicists use this rather awkward word ‘parity’ for what’s left- and right-handedness really), just like a C-transformation amounts to reversing electric and ‘color’ charges (‘color’ charges are the charges involved in the strong nuclear force).

Now, if only a full CPT transformation makes sense, then CP-reversal should also mean T-reversal, and vice versa. Feynman’s story about “the guy in the ‘other’ universe” (see my previous post) was quite instructive in that regard, and so let’s look at the finer points of that story once again.

Is ‘another’ world possible at all?

Feynman’s assumption was that we’ve made contact (don’t ask how: somehow) with some other intelligent being living in some ‘other’ world somewhere ‘out there’, and that there are no visual or other common references. That’s all rather vague, you’ll say, but just hang in there and try to see where we’re going with this story. Most notably, the other intelligent being – but let’s call ‘it’ a she instead of ‘a guy’ or ‘a Martian’ – cannot see the universe as we see it: we can’t describe, for instance, the Big and Small Dipper and explain to her what ‘left’ and ‘right’ is referring to such constellations, because she’s sealed off somehow from it (so she lives in a totally different corner of the universe really).

In contrast, we would be able, most probably, to explain and share the concept of ‘upward’ and ‘downwards’ by assuming that she is also attracted by some center of gravity nearby, just like we are attracted downwards by our Earth. Then, after many more hours and days, weeks, months or even years of tedious ‘discussions’, we would probably be able to describe electric currents and explain electromagnetic phenomena, and then, hopefully, she would find out that the laws in her corner of the universe are exactly the same, and so we could thus explain and share the notion of a ‘positive’ and a ‘negative’ charge, and the notion of a magnetic ‘north’ and ‘south’ pole.

However, at this point the story becomes somewhat more complicated, because – as I tried to explain in my previous post – her ‘positive’ electric charge (+) and her magnetic ‘north’ might well be our ‘negative’ electric charge (–) and our magnetic ‘south’. Why? It’s simple: the electromagnetic force does respect charge and also parity symmetry and so there is no way of defining any absolute sense of ‘left’ and ‘right’ or (magnetic) ‘north’ and (magnetic) ‘south’ with reference to the electromagnetic force alone. [If you don’t believe, just look at my previous post and study the examples.]

Talking about the strong force wouldn’t help either, because it also fully respects charge symmetry.

Huh? Yes. Just go through my previous post which – I admit – was probably quite confusing but made the point that a ‘mirror-image’ world would work just as well… except when it comes to the weak force. Indeed, atomic decay processes (beta decay) do distinguish between ‘left-handed’ and ‘right-handed’ particles (as measured by their spin) in an absolute sense that is (see the illustration of decaying muons and their mirror-image in the previous post) and, hence, it’s simple: in order to make sure her ‘left’ and her ‘right’ is the same as ours, we should just ask her to perform those beta decay experiments demonstrating that parity (or P-symmetry) is not being conserved and, then, based on our common definition of what’s ‘up’ and ‘down’ (the commonality of these notions being based on the effects of gravity which, we assume, are the same in both worlds), we could agree that ‘right’ is ‘right’ indeed, and that ‘left’ is ‘left’ indeed.

Now, you will remember there was one ‘catch’ here: if ever we would want to set up an actual meeting with her (just assume that we’ve finally figured out where she is and so we (or she) are on our way to meet each other), we would have to ask her to respect protocol and put out her right hand to greet us, not her left. The reason is the following: while ‘right-handed’ and ‘left-handed’ matter behave differently when it comes to weak force interactions (read: atomic decay processes)–which is how we can distinguish between ‘left’ and ‘right’ in the first place, in some kind of absolute sense that is–the combined CP symmetry implies that right-handed matter and left-handed anti-matter behave just the same–and, of course, the same goes for ‘left-handed’ matter and ‘right-handed’ anti-matter. Hence, after we would have had a painstakingly long exchange on broken P-symmetry to ensure we are talking about the same thing, we would still not know for sure: she might be living in a world of anti-matter indeed, in which case her ‘right’ would actually be ‘left’ for us, and her ‘left’ would be ‘right’.

Hence, if, after all that talk on P-symmetry and doing all those experiments involving P-asymmetry, she actually would put out her left hand when meeting us physically–instead of the agreed-upon right hand… Then… Well… Don’t touch it. 🙂

There is a way out of course. And, who knows, perhaps she was just trying to be humorous and so perhaps she smiled and apologized for the confusion in the meanwhile. But then… […] Hmm… I am not sure if such bad joke would make for a good start of a relationship, even if it would obviously demonstrate superior intelligence. 🙂

Indeed, the Fitch-Cronin experiment brings an additional twist to this potentially romantic story between two intelligent beings from two ‘different’ worlds. In fact, the Fitch-Cronin experiment actually rules out this theoretical possibility of mutual destruction and, therefore, the possibility of two ‘different’ worlds.

The argument goes straight to the heart of our philosophical discussion on time reversal. Indeed, whatever you may or may not have understood from this and my previous posts on CPT symmetry, the key point is that the combined CPT symmetry cannot be violated.

Why? Well… That’s plain logic: the real world does not care about our conventions, so reversing all of our conventions, i.e.

  1. Changing all particles to antiparticles by reversing all charges (C),
  2. Turning all right-handed particles into left-handed particles and vice versa (P), and
  3. Changing the sign of time (T),

describes a world truly going back in time.

Now, ‘her’ world is not going back in time. Why? Well… Because we can actually talk to her, it is obvious that her ‘arrow of time’ points in the same direction as ours, so she is not living in a world that is going back in time. Full stop. Therefore, any experiment involving a combined CP asymmetry (i.e. C-P violation) should yield the same results and, hence, she should find the same bias, i.e. a bias going in the very same direction of the equation, i.e. from left to right, or from right to left – whatever (what we label it, depends on our conventions, which we ‘re-set’ as we talked to her, and, hence, which we share, based on the results of all these beta decay experiments we did to ensure we’re really talking about the ‘same’ direction, and not its opposite).

Is this confusing? It sure is. But let me rephrase the logic. Perhaps it helps.

  1. Combined CPT symmetry implies that if the combined CP-symmetry is broken, then T-symmetry is also broken. Hence, the experimentally established fact of broken CP symmetry (even if it’s only 2 or 3 times per thousand) ensures that the ‘arrow of time’ points in one direction, and in one direction only. To put it simply: we cannot reverse time in a world which does not (fully) respect the principle of CP symmetry.
  2. Now, if you and I can exchange meaningful signals (i.e. communicate), then your and my ‘arrow of time’ obviously point in the same direction. To put it simply, we’re actors in the same movie, and whether or not it is being played backwards doesn’t matter anymore: the point is that the two of us share the same arrow of time. In other words, God did not do any combined CPT-transformation trick on your world as compared to mine, and vice versa.
  3. Hence, ‘your’ world is ‘my’ world and vice versa. So we live in the same world with the very same symmetries and asymmetries.

Now apply this logic to our imaginary new friend (‘she’) and (I hope) you’ll get the point.

To make a long story short, and also to conclude our philosophical digressions here on a pleasant (romantic) note: the fact that we would be able to communicate with her, implies that she’d be living in the same world as ours. We know that now, for sure, because of the broken CP symmetry: indeed, if her ‘time arrow’ points in the same direction, then CP symmetry will be broken in just the very same way in ‘her’ world (i.e. the ‘bias’ will have the same direction, in an absolute sense) as it it is broken in ‘our’ world.

In short, there are only two possible worlds: (1) this world and (2) one and only one ‘other’ world. This ‘other’ world is our world under a full CPT-transformation: the whole movie played backwards in other words, but with all ‘charges’ affecting forces – in whatever form and shape they come (electric charge, color charge, spin, and what have you) reversed or – using that awful mathematical term – ‘negated’.

In case you’d wonder (1): I consider the many-worlds interpretation of quantum mechanics as… Well… Nonsense. CPT symmetry allows for two worlds only. Maximum two. 🙂

In case you’d wonder (2): An oscillating-universe theory, or some kind of cyclic thing (so Big Bangs followed by Big Crunches) are not incompatible with my ‘two-possible-worlds’ view of things. However, this ‘oscillations’ would all take place in the same world really, because the arrow of time isn’t being reversed really, as Big Bangs and Big Crunches do not reverse charges and parities–at least not to my knowledge.

But, of course, who knows?


1. You may wonder what ‘other’ asymmetries I am hinting at in this post here. It’s quite simple. It’s everything you see around you, including the works of the increasing entropy law. However, if I would have to choose one asymmetry in this world (the real world), as an example of a very striking and/or meaningful asymmetry, it’s the the preponderance of matter over anti-matter, including the preponderance of (left-handed) neutrinos over (right-handed) antineutrinos. Indeed, I can’t shake off that feeling that neutrino physics is going to spring some surprises in the coming decades.

[When you’d google a bit in order to get some more detail on neutrinos (and solar neutrinos in particular, which are the kind of neutrinos that are affecting us right now and right here), you’ll probably get confused by a phenomenon referred to as neutrino oscillation (which refers to a process in which neutrinos change ‘flavor’) but so the basic output of the Sun’s nuclear reactor is neutrinos, not anti-neutrinos. Indeed, the (general) reaction involves two protons combining to form one (heavy) hydrogen atom (i.e. deuterium, which consists of one neutron, one proton and one electron), thereby ejecting one positron (e+) and one (electron) neutrino (ve). In any case, this is not the place to develop the point. I’ll leave that for my next post.]

2. Whether or not you like the story about ‘her’ above, you should have noticed something that we could loosely refer to as ‘degrees of freedom’ is playing some role:

  1. We know that T-symmetry has not been broken: ‘her’ arrow of time points in the same direction.
  2. Therefore, the combined CP-symmetry of ‘her’ world is broken in the same way as in our world.
  3. If the combined CP-symmetry in ‘her’ world is broken in the same way as in ‘our’ world, the individual C and P symmetries have to be broken in the very same way. In other words, it’s the same world indeed. Not some anti-matter world.

As I am neither a physicist nor a mathematician, and not a philosopher either, please do feel free to correct any logical errors you may identify in this piece. Personally, I feel the logic connecting CP violation and individual C- and P-violation needs further ‘flesh on the bones’, but the core argument is pretty solid I think. 🙂

3. What about the increasing entropy law in this story? What happens to it if we reverse time, charge and parity? Well… Nothing. It will remain valid, as always. So that’s why an actual movie being played backwards with charges and parities reversed will still not make any sense to us: things that are broken don’t repair themselves and, hence, at the system level, there’s another type of irreducible ‘arrow of time’ it seems. But you’ll have to admit that the character of that entropy ‘law’ is very different from these ‘fundamental’ force laws. And then just think about it, isn’t it extremely improbable how we human beings have evolved in this universe? And how we are seemingly capable to understand ourselves and this universe? We don’t violate the entropy law obviously (on the contrary: we’re obviously messing up our planet), but I feel we do negate it in a way that escapes the kind of logical thinking that underpins the story I wrote above. But such remarks have nothing to do with math or physics and, hence, I will refrain from them.

4. Finally, for those who’d feel like some kind of ‘feminist’ remark on my use of ‘us’ and ‘her’, I think the use of ‘her’ is explained to underline the idea of ‘other’ and, hence, as a male writer, using ‘her’ to underscore the ‘other’ dimension comes naturally and shouldn’t be criticized. The element which could/should bother a female reader of such ‘through experiments’ is that we seem to assume that the ‘other’ intelligent being is actually somewhat ‘dumber’ than us, because the story above assumes we are actually explaining the experiments of the Wu and Fitch-Cronin team to ‘her’, instead of the other way around. That’s why I inserted the possibility of ‘her’ pulling a practical joke on us by offering us her left hand: if ‘she’ is equally or even more intelligent than us, then she’d surely have figured out that there’s no need to be worried about the ‘other’ being made of anti-matter. 🙂

Time reversal and CPT symmetry (II)

Pre-scriptum (dated 26 June 2020): While my posts on symmetries (and why they may or may be broken) are somewhat mutilated (removal of illustrations and other material) as a result of an attack by the dark force, I am happy to see a lot of it survived more or less intact. While my views on the true nature of light, matter and the force or forces that act on them – all of the stuff that explains symmetries or symmetry-breaking, in other words – have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics, I think most (if not all) of the analysis in this post remains valid and fun to read. 🙂

Original post:

My previous post touched on many topics and, hence, I feel I was not quite able to exhaust the topic of parity violation (let’s just call it mirror asymmetry: that’s more intuitive). Indeed, I was rather casual in stating that:

  1. We have ‘right-handed’ and ‘left-handed’ matter, and they behave differently–at least with respect to the weak force–and, hence, we have some kind of absolute distinction between left and right in the real world.
  2. If ‘right-handed’ matter and ‘left-handed’ matter are not the same, then ‘right-handed’ antimatter and ‘left-handed’ antimatter are not the same either.
  3. CP symmetry connects the two: right-handed matter behaves just like left-handed antimatter, and right-handed antimatter behaves just like left-handed matter.

There are at least two problems with this:

  1. In previous posts, I mentioned the so-called Fitch-Cronin experiment which, back in 1964, provided evidence that ‘Nature’ also violated the combined CP-symmetry. In fact, I should be precise here and say the weak force, instead of ‘Nature’, because all these experiments investigate the behavior of the weak force only. Having said that, it’s true I mentioned this experiment in a very light-hearted manner–too casual really: I just referred to my simple diagrams illustrating what true time reversal entails (a reversal of the forces and, hence, of the charges causing those forces) and that was how I sort of shrugged it all of.
  2. In such simplistic world view, the question is not so much why the weak force violates mirror symmetry, but why gravity, electromagnetism and the strong force actually respect it!

Indeed, you don’t get a Nobel Prize for stating the obvious and, hence, if Val Fitch and James Cronin got one for that CP-violation experiment, C/P or CP violation cannot be trivial matters.

P-symmetry revisited

So let’s have another look at mirror symmetry–also known as reflection symmetry– by following Feynman’s example: let us actually build a ‘left-hand’ clock, and let’s do it meticulously, as Feynman describes it: “Every time there is a screw with a right-hand thread in one, we use a screw with a left-hand thread in the corresponding place of the other; where one is marked ‘IV’ on the face, we mark a ‘VI’ on the face of the other; each coiled spring is twisted one way in one clock and the other way in the mirror-image clock; when we are all finished, we have two clocks, both physical, which bear to each other the relation of an object and its mirror image, although they are both actual, material objects. Now the question is: If the two clocks are started in the same condition, the springs wound to corresponding tightnesses, will the two clocks tick and go around, forever after, as exact mirror images?”

The answer seems to be obvious: of course they will! Indeed, we do observe that P symmetry is being respected, as shown below:

P symmetry

You may wonder why we have to go through the trouble of building another clock. Why can’t we just take one of these transparent ‘mystery clocks’ and just go around it and watch its hand(s) move standing behind it? The answer is simple: that’s not what mirror symmetry is about. As Feynman puts its: a mirror reflection “turns the whole space inside out.” So it’s not like a simple translation or a rotation of space. Indeed, when we would move around the clock to watch it from behind, then all we do is rotating our reference frame (with a rotation angle equal to 180 degrees). That’s all. So we just change the orientation of the clock (and, hence, we watch it from behind indeed), but we are not changing left for right and right for left.

Rotational symmetry is a symmetry as well, and the fact that the laws of Nature are invariant under rotation is actually less obvious than you may think (because you’re used to the idea). However, that’s not the point here: rotational symmetry is something else than reflection (mirror) symmetry. Let me make that clear by showing how the clock might run when it would not respect P-symmetry.

P asymmetry

You’ll say: “That’s nonsense.” If we build that mirror-image clock and also wind it up in the ‘other’ direction (‘other’ as compared to our original clock), then the mirror-image clock can’t run that way. Is that nonsense? Nonsensical is actually the word that Wolfgang Pauli used when he heard about Chien-Shiung Wu’s 1956 experiment (i.e. the first experiment that provided solid evidence for the fact that the weak force – in beta decay for instance – does not respect P-symmetry), but so he had to retract his words when repeated beta decay experiments confirmed Wu’s findings.

Of course, the mirror-image clock above (i.e. the one running clockwise) breaks P-symmetry in a very ‘symmetric’ way. In fact, you’ll agree that the hands of that mirror-image clock might actually turn ‘clockwise’ if its machinery would be completely reversible, so we could wind up its springs in the same way as the original clock. But that’s cheating obviously. However, it’s a relevant point and, hence, to be somewhat more precise I should add that Wu’s experiment (and the other beta decay experiments which followed after hers) actually only found a strong bias in the direction of decay: not all of the beta rays (beta rays consist of electrons really – check the illustration in my previous post for more details) went ‘up’ (or ‘down’ in the mirror-reversed arrangement), but most of them did. 


OK. We got that. Now how do we explain it? The key to explaining the phenomenon observed by Wu and her team, is the spin of the cobalt-60 nuclei or, in the muon decay experiment described in my previous post, the spin of the muons. It’s the spin of these particles that makes them ‘left-handed’ or ‘right-handed’ and the decay direction is (mostly) in the direction of the axial vector that’s associated with the spin direction (this axial vector is the thick black arrow in the illustration below).

Axial vector

Hmm… But we’ve got spinning things in (mechanical) clocks as well, don’t we? Yes. We have flywheels and balance wheels and lots of other spinning stuff in a mechanical clock, but these wheels are not equivalent to spinning muons or other elementary particles: the wheels in a clock preserve and transfer angular momentum.

OK… But… […] But isn’t that what we are talking about here? Angular momentum?

No. Electrons spinning around a nucleus have angular momentum as well – referred to as orbital angular momentum – but it’s not the same thing as spin which, somewhat confusingly, is often referred to as intrinsic angular momentum. In short, we could make a detailed analysis of how our clock and its mirror image actually work, and we would find that all of the axial vectors associated with flywheels, balance wheels and springs in a clock would effectively be reversed in the mirror-image clock but, in contrast with the weak decay example, their reversed directions would actually explain why the mirror-image clock is turning counter-clockwise (from our point of view that is), just like the image of the original clock in the mirror does, and, therefore, why a ‘left-handed’ mechanical clock actually respects P-symmetry, instead of breaking it.

Axial and polar vectors in physics

In physics, we encounter such axial vectors everywhere. They show the axis of spin, and their direction is determined by the direction of spin through one of two conventions: the ‘right-hand screw rule’, or the ‘left-hand screw rule’. Physicists have settled on the former, so let’s work with that for the time being.

The other type of vector is a polar vector. That’s an ‘honest’ vector as Feynman calls it–depicting ‘real’ things such as, for example, a step in space, or some force acting in some direction. The figures below (which I took from Feynman’s Lectures) illustrate the idea (and please do note the care with which Feynman reversed the direction of the arrows above the r and ω in the mirror image):

  1. When mirrored, a polar vector “changes its head, just as the whole space turns inside out.”
  2. An axial vector behaves differently when mirrored. It changes too, but in a very different way: it is usually reversed in respect to the geometry of the whole space, as illustrated in the muon decay image above. However, in the illustration below, that is not the case, because the angular velocity ‘vector’ is not reversed when mirrored. So it’s all quite subtle and one has to carefully watch what’s going on really when we do such mirror reflections.

Axial vectors

What’s the third figure about? Well… While it’s not that difficult to visualize all of the axial vectors in a mechanical clock, it’s a different matter when discussing electromagnetic forces, and then to explain why these electromagnetic forces also respect mirror symmetry, just like the mechanical clock. But let’s me try.

When an electric current goes through a solenoid, the solenoid becomes a magnet, especially when wrapped around an iron core. The direction and strength of the magnetic field is given by the magnetic field vector B, and the force on an electrically charged particle moving through such magnetic field will be equal to F = qv×B. That’s a so-called vector cross product and we’ve seen it before: a×b = na││b│sinθ, so we take (1) the magnitudes of a and b, (2) the sinus of the angle between them, and (3) the unit vector (n) perpendicular to (the plane containing) a and b; multiply it all; and there we are: that’s the result. But – Hey! Wait a minute! – there are two unit vectors perpendicular to a and b. So how does that work out?

Well… As you might have guessed, there is another right-hand rule here, as shown below.


Now how does that work out for our magnetic field? If we mirror the set-up and let an electron move through the field? Well… Let’s do the math for an electron moving into this screen, so in the direction that you are watching.

In the first set-up, the B vector points upwards and, hence, the electron will deviate in the direction given by that cross product above: qv×B. In other words, it will move sideways as it moves away from you, into the field. In which direction? Well… Just turn that hand above about 90 degrees and you have the answer: right. Oh… No. It’s left, because q is negative. Right.

In the mirror-image set-up, we have a B’ vector pointing in the opposite direction so… Hey ! Mirror symmetry is not being respected, is it?

Well… No. Remember that we must change everything, including our conventions, so the ‘right-hand rules’ above becomes ‘left-hand rules’, as shown below for example. Surely you’re joking, Mr. Feynman!

P-parity for screw rules

Well… No. F and v are polar vectors and, hence, “their head might change, just as the whole space turns inside out”, but that’s not the case now, because they’re parallel to the mirror. In short, the force F on the electron will still be the same: it will deviate leftwards. I tried to draw that below, but it’s hard to make that red line look like it’s a line going away from you.


But that can’t be true, you’ll say. The field lines go from north to south, and so we have that B’ vector pointing downwards now.

No, we don’t. Or… Well… Yes. It all depends on our conventions. 🙂  

Feynman’s switch to ‘left-hand rules’ also involves renaming the magnetic poles, so all magnetic north poles are now referred to as ‘south’ poles, and all magnetic south poles are now referred to as ‘north’ poles, and so that’s why he has a B’ vector pointing downwards. Hence, he does not change the convention that magnetic field lines go from north to south, but his ‘north’ pole (in the mirror-image drawing) is actually a ‘south’ pole. Capito? 🙂

[…] OK. Let me try to explain it once again. In reality, it does not matter whether or not a solenoid is wound clockwise or counterclockwise (or, to use the terminology introduced above, whether our solenoid is left-handed or right-handed). The important thing is that the current through the solenoid flows from the top to the bottom. We can only reverse the poles – in reality – if we reverse the electric current, but so we don’t do that in our mirror-image set-up. Therefore, the force F on our charged particle will not change, and B’ is an axial vector alright but this axial vector does not represent the actual magnetic field.

[…] But… If we change these conventions, it should represent the magnetic field, shouldn’t it? And how do we calculate that force then?

OK. If you insist. Here we go:

  1. So we change ‘right’ to ‘left’ and ‘left’ to ‘right’, and our cross-product rule becomes a ‘left-hand’ rule.
  2. But our electrons still go from ‘top’ to ‘bottom’. Hence, the (magnetic) force on a charged particle won’t change.
  3. But if the result has to be the same, then B needs to become –B, or so that’s B’ in our ‘left-handed’ coordinate system.
  4. We can now calculate F using the ‘left-handed’ cross product rule and – because we did not change the convention that field lines go from north to south, we’ll also rename our poles.
  5. Yippee ! All comes out all right: our electron goes left. Sorry. Right. Huh? Yes. Because we’ve agreed to replace ‘left’ by ‘right’, remember? 🙂


If you didn’t get anything of this, don’t worry. There is actually a much more comprehensible illustration of the mirror symmetry of electromagnetic forces. If we would hang two wires next to each other, as below, and we send a current through them, they will attract if the two currents are in the same direction, and they will repel when the currents are opposite. However, it doesn’t matter if the current goes from left to right or from right to left. As long as the two currents have the same direction (left or right), it’s fine: there will be attraction. That’s all it takes to demonstrate P-symmetry for electromagnetism.

Wires attracting

The Fitch-Cronin experiment

I guess I caused an awful lot of confusion above. Just forget about it all and take one single message home: the electromagnetic force does not care about the axial vector of spinning particles, but the weak force does.

Is that shocking?

No. There are plenty of examples in the real world showing that the direction of ‘spin’ does matter. For instance, to unlock a right-hinged door, you turn the key to the right (i.e. clockwise). The other direction doesn’t work. While I am sure physicists won’t like such simplistic statements, I think that accepting that Nature has similar ‘left-handed’ and ‘right-handed’ mechanisms is not the kind of theoretical disaster that Wolfgang Pauli thought it was. If anything, we just should marvel at the fact that gravity, electromagnetism and the strong force are P- and C-symmetric indeed, and further investigate why the weak force does not have such nice symmetries. Indeed, it respects the combined CPT symmetry, but that amounts to saying that our world sort of makes sense, so that ain’t much.

In short, our understanding of that weak force is probably messy and, as Feynman points out: “At the present level of understanding, you can still see the “seams” in the theories; they have not yet been smoothed out so that the connection becomes more beautiful and, therefore, probably more correct.” (QED, 1985, p. 142). However, let’s stop complaining about our ‘limited understanding’ and so let’s work with what we do understand right now. Hence, let’s have a look at that Fitch-Cronin experiment now and see how ‘weird’ or, on the contrary, how ‘understandable’ it actually is.

To situate the Fitch-Cronin experiment, we first need to say something more about that larger family of mesons, of which the kaons are just one of the branches. In fact, in case you’d not be interested in this story as such, then I’d suggest you just read it as a very short introduction to the Standard Model as such, as it gives a nice short overview of all matter-particles–which is always useful I’d think.

Hadrons, mesons and baryons

You may or may not remember that mesons are unstable particles consisting of one quark and one anti-quark (so mesons consist of two quarks, but one of them should be an anti-quark). As such, mesons are to be distinguished from the ‘other’ group within the larger group of hadrons, i.e. the baryons, which are made of three quarks. [The term ‘hadrons’ itself is nothing but a catch-all for all particles consisting of quarks.]

The most prominent representatives of the baryon family are the (stable) neutron and proton, i.e. the nucleons, which consist of u and d quarks. However, there are unstable baryons as well. These unstable baryons involve the heavier (second-generation) or quarks, or the super-heavy (third-generation) b quark. [As for the top quark (t), that’s so high-energy (and, hence, so short-lived) that baryons made of a t quark (so-called ‘top-baryons’) are not expected to exist but, then, who knows really?]

But kaons are mesons, and so I won’t say anything more about baryons The two illustrations below should be sufficient to situate the discussion.



Kaons are just one branch of the meson family. There are, for instance, heavier versions of the kaons, referred to as B- and D-mesons. Let me quickly introduce these:

  1. The ‘B’ in ‘B-meson’ refers to the fact that one of the quarks in a B-meson is a b-quark: a b (bottom) quark is a much heavier (third-generation) version of the (second-generation) s-quark.
  2. As for the ‘D’ in D-meson, I have no idea. D-mesons will always consist of a c-quark or anti-quark, combined with a lighter d, u or s (anti-)quark, but so there’s no obvious relationship between a D-meson and a d-quark. Sorry.
  3. If you look at the quark table above, you’ll wonder whether there are any top-mesons, i.e. mesons consisting of a t quark or anti-quark. The answer to that question seems to be negative: t quarks disintegrate too fast, it is said. [So that resembles the remark on the possiblity of t-baryons.] If you’d google a bit on this, you’ll find that, in essence, we haven’t found any t-mesons as yet but their potential existence should not be excluded.

Anything else? Yes. There’s a lot more around actually. Besides (1) kaons, (2) B-mesons and (3) D-mesons, we also have (4) pions (i.e. a combination of a u and a d, or their anti-matter counterpart), (5) rho-mesons (ρ-mesons can be thought of as excited (higher-energy) pions(6) eta-mesons (η-mesons a rapidly decaying mixture of ud and s quarks or their anti-matter counterparts), as well as a whole bunch of (temporary) particles consisting of a quark and its own anti-matter counterpart, notably the (7) phi (a φ consists of a s and an anti-s), psi (a ψ consists of an c and an anti-c) and upsilon (a φ consists of a b and an anti-b) particles (so all these particles are their own anti-particles).

So it’s quite a zoo indeed, but let’s zoom in on those ‘crazy’ kaons. [‘Crazy kaons’ is the epithet that Gerard ‘t Hooft reserved for them in his In Search of the Ultimate Building Blocks (1996).] What are they really? 

Crazy kaons

Kaons, also know as K-mesons, are, first of all, mesons, i.e. particles made of one quark and one anti-quark (as opposed to baryons, which are made of three quarks, e.g. protons and neutrons). All mesons are unstable: at best, they last a few hundredths of a microsecond, but kaons have much shorter lifetimes than that. Where do we find them? We usually create them in those particle colliders and other sophisticated machinery (the experiment used kaon beams) but we can also find them as a decay product in (secondary) cosmic rays (cosmic rays consist of very high-energy particles and they produce ‘showers’ of secondary particles as they hit our atmosphere).

They come in three varieties: neutral and positively or negatively charged, so we have a K0, a K+, and a K, in principle that is (the story will become more complicated later). What they have in common is that one of the quarks is the rather heavy s-quark (s stands for ‘strange’ but you know what Feynman – and others – think of that name: it’s just a strange name indeed, and so don’t worry too much about it). An s-quark is a so-called second-generation matter-particle and that’s why the kaon is unstable: all second-generation matter-particles are unstable. The second quark is just an ordinary u- or d-quark, i.e. the type of quark you’d find in the (stable) proton or neutron.

But what about the electric charge? Well… I should be complete. The quarks might be anti-quarks as well. That’s nothing to worry about as you’ll remember: anti-matter is just matter but with the charges reversed. So a Kconsists of an s quark and an anti-d quark or –and this is the key to understanding the experiment actually– a K0 can also consist of an anti-s quark and a (normal) d-quark. Note that the s and d quarks have a charge of 1/3 and so the total charge comes out alright. [As for the other kaons, a Kconsists of a u and anti-s quark (the u quark has charge 2/3 and so we have +1 as the total charge), and the K– consists of an anti-u and an s quark (and, hence, we have –1 as the charge), but we actually don’t need them any more for our story.]

So that’s simple enough. Well… No. Unfortunately, the story is, indeed, more complicated than that. The actual kaons in a neutral kaon beam come in two varieties that are a mix of the two above-mentioned neutral K states: a K-long (KL) has a lifetime of about 9×10–11 s, while a K-short (KS) has a lifetime of about 5.2×10–8 s. Hence, at the end of the beam, we’re sure to find Kkaons only.

Huh? mix of two particle states… You’re talking superposition here? Well… Yes. Sort of. In fact, as for what KL and Kactually are, that’s a long and complex story involving what is referred to as a neutral particle oscillation process. In essence, neutral particle oscillation occurs when a (neutral) particle and its antiparticle are different but decay into the same final state. It is then possible for the decay and its time reversed process to contribute to oscillations indeed, that turn the one into the other, and vice versa, so we can write A → Δ → B → Δ → A → etcetera, where A is the particle, B is the antiparticle, and Δ is the common set of particles into which both can decay. So there’s an oscillation phenomenon from one state to the other here, and all the things I noted about interference obviously come into play.

In any case, to make a very long and complicated story short, I’ll summarize it as follows: if CP symmetry holds, then one can show that this oscillation process should result in a very clear-cut situation: a mixed beam of long-lived and short-lived kaons, i.e. a mix of KL and KS. Both decay differently: a K-short particle decays into two pions only, while a K-long particle decays into three pions.

That is illustrated below: at the end of the 17.4 m beam, one should only see three-pion decay events. However, that’s not what Fitch and Cronin measured: the actually saw a one two-pion decay event into every 500 (on average that is)! [I have introduced the pion species in the more general discussion on mesons: you’ll remember they consist of first-generation quarks only, but so don’t worry about it: just note the K-long and K-short particles decay differently. Don’t be confused by the π notation below: it has nothing to do with a circle or so, so 2π just means two pions.]

Kaon beam

That means that the kaon decay processes involved do not observe the assumed CP symmetry and, because it’s the weak force that’s causing those decays, it means that the weak force itself does not respect CP symmetry.

Why is that so?

You may object that these lifetimes are just averages and, hence, perhaps we see these two-pion decays at the end of the beam because some of the K-short particles actually survived much longer !

No. That’s to be ruled out. The short-lived particle cannot be observable more than a few centimeters down the beam line. To show that, one can calculate the time required to drop to 1/500 of the original population of K-short particles. With the stated lifetime (9×10–11 s), the half-life calculation gives a time of 5.5 x 10-10 seconds. At nearly the speed of light, this would give a distance of about 17 centimeters, and so that’s only 1/100 the length of Cronin and Fitch’s beam tube.

But what about the fact that particles live longer when they’re going fast? You are right: the number above ignores relativistic time dilation: the lifetime as seen in the laboratory frame is ‘dilated’ indeed by the relativity factor γ. At 0.98c (i.e. the speed of these kaons, γ =5, and, hence, this “time dilation effect” is very substantial. However, re-calculating the distance gives a revised distance equal to 17γ cm, i.e. 85 cm. Hence, even with kaons speeding at 0.98c, the population would be down by a factor of 500 by the time they got a meter down the beam tube. So for any particle velocity really, all of these K-short particles should have decayed long before they get to the end of the beam line.

Fitch and Cronin did not see that, however: they saw one two-pion decay event for every 500 decay events, so that’s two per thousand (0.2%) and, hence, that is very significant. While the reasoning is complex (these oscillations and the quantum-mechanical calculations involved are not easy to understand), the results clearly shows the kaon decay process does not observe CP symmetry.

OK. So what? How does this violate charge and parity symmetry? Well… That’s a complicated story which involves a deeper understanding of how initial and final states of such processes incorporate CP values, and then showing how these values differ. That’s a story that requires a master’s degree in physics, I must assume, and so I don’t have that. But I can sort of sense the point and I would suggest we just accept it here. [To be precise, the Fitch-Cronin experiment is an indirect ‘proof’ of CP violation only: as mentioned below, only in 1999 would experiments be able to demonstrate direct CP violation.]

OK. So what? Do we see it somewhere else? Well… Fitch and Cronin got a Nobel Prize for this only sixteen years later, i.e. in 1980, and then it took researchers another twenty years to find CP violation in some other process. To be very precise, only in 1999 (i.e. 35 years after the Fitch-Cronin findings), Fermilab and CERN could conclude a series of experiments demonstrating direct CP violation in (neutral) kaon decay processes (as mentioned above, the Fitch-Cronin experiment only shows indirect CP violation), and that then set the stage for a ‘new’ generation of experiments involving B-mesons and D-mesons, i.e. mesons consisting of even heavier quarks (c or b quarks)–so these are things that are even less stable than kaons. So… Well… Perhaps you’re right. There’s not all that many examples really.

Aha ! So what?

Well… Nothing. That’s it. These ‘broken symmetries’ exist, without any doubt, but–you’re right–they are a marginal phenomenon in Nature it seems. I’ll just conclude with quoting Feynman once again (Vol. I-52-9):

“The marvelous thing about it all is that for such a wide range of important phenomena–nuclear forces, electrical phenomena, and gravitation–over a tremendous range of physics, all the laws for these seem to be symmetrical. On the other hand, this little extra piece says, “No, the laws are not symmetrical!” How is it that Nature can be almost symmetrical, but not perfectly symmetrical? […] No one has any idea why. […] Perhaps God made the laws only nearly symmetrical so that we should not be jealous of His perfection.”

Hmm… That’s the last line of the first volume of his Lectures (there are three of them), and so that should end the story really.

However, I would personally not like to involve God in such discussions. When everything is said and done, we are talking atomic decay processes here. Now, I’ve already said that I am not a physicist (my only ambition is to understand some of what they are investigating), but I cannot accept that these decay processes are entirely random. I am not saying there are some ‘inner variables’ here. No. That would amount to challenging the Copenhagen interpretation of quantum mechanics, which I won’t.

But when it comes to the weak force, I’ve got a feeling that neutrino physics may provide the answer: the Earth is being bombarded with neutrinos, and their ‘intrinsic parity’ is all the same: all of them are left-handed. In fact, that’s why weak interactions which emit neutrinos or antineutrinos violate P-symmetry! It’s a very primitive statement – and not backed up by anything I have read so far – but I’ve got a feeling that the weak force does not only involve emission of neutrinos or antineutrinos: I think they enter the equation as well.

That’s preposterous and totally random statement, you’ll say.

Yes. […] But I feel I am onto something and I’ll explore it as good as I can–if only to find out why I am so damn wrong. I can only say that, if and when neutrino physics would allow us to tentatively confirm this random and completely uninformed hypothesis, then we would have an explanation which would be much more in line with the answers that astrophysicists give to questions related to other observable asymmetries such as, for example, the imbalance between matter and anti-matter.

However, I know that I am just babbling now, and that nobody takes this seriously anyway and, hence, I will conclude my series on CPT symmetry right here and now. 🙂

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Time reversal and CPT symmetry (I)

Pre-scriptum (dated 26 June 2020): While my posts on symmetries (and why they may or may be broken) are somewhat mutilated (removal of illustrations and other material) as a result of an attack by the dark force, I am happy to see a lot of it survived more or less intact. While my views on the true nature of light, matter and the force or forces that act on them – all of the stuff that explains symmetries or symmetry-breaking, in other words – have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics, I think most (if not all) of the analysis in this post remains valid and fun to read. 🙂

Original post:

In my previous posts, I introduced the concept of time symmetry, and parity and charge symmetry as well. However, let’s try to explore T-symmetry first. It’s not an easy concept – contrary to what one might think at first.

The arrow of time

Let me start with a very ‘common-sense’ introduction. What do we see when we play a movie backwards? […]

We reverse time. When playing some movie backwards, we look at where things are coming from. And we see phenomena that don’t make sense, such as: (i) cars racing backwards, (ii) old people becoming younger (and dead people coming back to life), (iii) shattered glass assembling itself back into some man-made shape, and (iv) falling objects defying gravity to get back to where they were. Let’s briefly say something about these unlikely or even impossible phenomena before a more formal treatment of the matter:

  1. The first phenomenon – cars racing backwards – is unlikely to happen in real life but quite possible, and some crazies actually do organize such races.
  2. The last example – objects defying gravity – is plain impossible because of Newton’s universal law of gravitation.
  3. The other examples – the old becoming young (and the dead coming back to life), and glass shards coming back together into one piece – are also plain impossible because of some other ‘law’: the law of ever increasing entropy.

However, there’s a distinct difference between the two ‘laws’ (gravity versus increasing entropy). As one entry on Physics Stack Exchange notes, the entropy law – better known as the second law of thermodynamics – “only describes what is most likely to happen in macroscopic systems, rather than what has to happen”, but then the author immediately qualifies this apparent lack of determinism, and rightly so: “It is true that a system may spontaneously decrease its entropy over some time period, with a small but non-zero probability. However, the probability of this happening over and over again tends to zero over long times, so is completely impossible in the limit of very long times.” Hence, while one will find some people wondering whether this entropy law is a ‘real law’ of Nature – in the sense that they would question that it’s always true no matter what – there is actually no room for such doubts.

That being said, the character of the entropy law and the universal law of gravitation is obviously somewhat different – because they describe different realities: the entropy law is a law at the level of a system (a room full of air, for example), while the law of gravitation describes one of the four fundamental forces.

I will now be a bit more formal. What’s time symmetry in physics? The Wikipedia definition is the following: “T-symmetry is the theoretical symmetry (invariance) of physical laws under a time reversal (T) transformation.” Huh?

A ‘time reversal transformation’ amounts to inserting –t (minus t) instead of t in all of our equations describing trajectories or physical laws. Such transformation is illustrated below. The blue curve might represent a car or a rocket accelerating (in this particular example, we have a constant acceleration a = 2). The vertical axis measures the displacement (x) as a function of time (t). , and the red curve is its T-transformation. The two curves are each other’s mirror image, with the vertical axis (i.e. the axis measuring the displacement x) as the mirror axis.

Time reversal 2

This view of things is quite static and, hence, somewhat primitive I should say. However, we can make a number of remarks already. For example, we can see that the slope (of the tangent) of the red curve is negative. This slope is the velocity (v) of the particle: v = dx/dt. Hence, a T-transformation is said to negate the velocity variable (in classical physics that is), just like it negates the time variable. [The verb ‘to negate’ is used here in its mathematical sense: it means ‘to take the additive inverse of a number’ — but you’ll agree that’s too lengthy to be useful as an expression.]

Note that velocity (and mass) determines (linear and angular) momentum and, hence, a T-transformation will also negate p and l, i.e. the linear and angular momentum of a particle.

Such variables – i.e. variables that are negated by the T-transformation – are referred to as odd variables, as opposed to even variables, which are not impacted by the T-transformation: the position of the particle or object (x) is an example of an even variable, and the force acting on a particle (F) is not being negated either: it just remains what it is, i.e. an external force acting on some mass or some charge. The acceleration itself is another ‘even’ variable.

This all makes sense: why would the force or acceleration change? When we put a minus sign in front of the time variable, we are basically just changing the direction of an axis measuring an independent variable. In a way, the only thing that we are doing is introducing some non-standard way of measuring time, isn’t it? Instead of counting from 0 to T, we count from 0 to minus T.

Well… No. In this post, I want to discuss actual time reversal. Can we go back in time? Can we put a genie back into a bottle? Can we reverse all processes in Nature and, if not, why not?

Time reversal and time symmetry are two different things: doing a T-transformation is a mathematical operation; trying to reverse time is something real. Let’s take an example from kinematics to illustrate the matter.


Kinematics can be summed up in one equation, best known as Newton’s Second Law: F = ma = m(dv/dt) = d(mv)/dt.  In words: the time-rate-of-change of a quantity called momentum (mv) is proportional to the force on an object. In other words: the acceleration (a) of an object is proportional to the force (F), and the factor of proportionality is the mass of the object (m). Hence, the mass of an object is nothing but a measure of its inertia.

The numbering of laws (first, second, etcetera) – usually combining some name of a scientist – is often quite arbitrary but, in this case (Newton’s Laws), one can really learn something from listing and discussing them in the right order:

  1. Newton’s First Law is the principle of inertia: if there’s no (other) force acting on an object, it will just continue doing what it does–i.e. nothing or, else, move in some straight line according to the direction of its momentum (i.e. the product of its mass and its velocity)–or further engage with the force it was already engaged with.
  2. Newton’s Second Law is the law of kinematics. In kinematics, we analyze the motion of an object without caring about the origin of the force causing the motion. So we just describe how some force impacts the motion of the object on which it is acting without asking any questions about the force itself. We’ve written this law above: F = ma.
  3. Finally, Newton’s Third Law is the law of gravitation, which describes the origin, the nature and the strength of the gravitational force. That’s part of dynamics, i.e. the study of the forces themselves – as opposed to kinematics, which only looks at the motion caused by those forces.

With these definitions and clarifications, we are now well armed to tackle the subject of T-symmetry in kinematics (we’ll discuss dynamics later). Suppose some object – perhaps an elementary particle but it could also be a car or a rocket indeed – is moving through space with some constant acceleration a (so we can write a(t) = a). This means that v(t) – the velocity as a function of time – will not be constant: v(t) = at. [Note that we make abstraction of the direction here and, hence, our notation does not use any bold letters (which would denote vector quantities): v(t) and a(t) are just simple scalar quantities in this example.]

Of course, when we – i.e. you and me right here and right now – are talking time reversal, we obviously do it from some kind of vantage point. That vantage point will usually be the “now” (and quite often also the “here”), and so let’s use that as our reference frame indeed and we will refer to it as the zero time point: t = 0. So it’s not the origin of time: it’s just ‘now’–the start of our analysis.

Now, the idea of going back in time also implies the idea of looking forward – and vice versa. Let’s first do what we’re used to do and so that’s to look forward.

At some point in the future, let’s call it t = T, the velocity of our object will be equal to v(T) = v(0) + aT. Why the v(0)? Well… We defined the zero time point (t = 0) in a totally random way and, hence, our object is unlikely to stop for that. On the contrary: it is likely to already have some velocity and so that’s why we’re adding this v(0) here. As for the space coordinate, our object may also not be at the exact same spot as we are (we don’t want to be to close to a departing rocket I would assume), so we can also not assume that x(0) = 0 and so we will also incorporate that term somehow. It’s not essential to the analysis though.

OK. Now we are ready to calculate the distance that our object will have traveled at point T. Indeed, you’ll remember that the distance traveled is an infinite sum of infinitesimally small products vΔt: the velocity at each point of time multiplied by an infinitesimally small interval of time. You’ll remember that we write such infinite sum as an integral:

Eq 1

[In case you wonder why we use the letter ‘s’ for distance traveled: it’s because the ‘d’ symbol is already used to denote a differential and, hence, ‘s’ is supposed to stand for ‘spatium’, which is the Latin word for distance or space. As for the integral sign, you know that’s an elongated S really, don’t you? So its stands for an infinite sum indeed. But lets go back to the main story.]

We have a functional form for v(t), namely v(t) = v(0) + at, and so we can easily work out this integral to find s as a function of time. We get the following equation:

Eq 2

When we re-arrange this equation, we get the position of our object as a function of time:

Eq 3

Let us now reverse time by inserting –T everywhere:

Eq 4

Does that still make sense? Yes, of course, because we get the same result when doing our integral:

Eq 5

So that ‘makes sense’. However, I am not talking mathematical consistency when I am asking if it still ‘makes sense’. Let us interpret all of this by looking at what’s happening with the velocity. At t = 0, the velocity of the object is v(0), but T seconds ago, i.e. at point t = -T, the velocity of the object was v(-T) = v(0) – aT. This velocity is less than v(0) and, depending on the value of -T, it might actually be negative. Hence, when we’re looking back in time, we see the object decelerating (and we should immediately add that the deceleration is – just like the acceleration – a constant). In fact, it’s the very same constant a which determines when the velocity becomes zero and then, when going even further back in time, when it becomes negative.

Huh? Negative velocity? Here’s the difference with the movie: in that movie that we are playing backwards, our car, our rocket, or the glass falling from a table or a pedestal would come to rest at some point back in time. We can calculate that point from our velocity equation v(t) = v(0) + at. In the example below, our object started accelerating 2.5 seconds ago, at point t = –2.5. But, unlike what we would see happening in our backwards-playing movie, we see that object not only stopping but also reversing its direction, to go in the same direction as we saw it going when we’re watching the movie before we hit the ‘Play Backwards’ button. So, yes, the velocity of our object changes sign as it starts following the trajectory on the left side of the graph.

time reversal

What’s going on here? Well… Rest assured: it’s actually quite simple: because the car or that rocket in our movie are real-life objects which were actually at rest before t = –2.5, the left side of the graph above is – quite simply – not relevant: it’s just a mathematical thing. So it does not depict the real-life trajectory of an accelerating car or rocket. The real-life trajectory of that car or rocket is depicted below.

real-life car

So we also have a ‘left side’ here: a horizontal line representing no movement at all. Our movie may or may not have included this status quo. If it did, you should note that we would not be able to distinguish whether or not it would be playing forward or backwards. In fact, we wouldn’t be able to tell whether the movie was playing at all: we might just as well have hit the ‘pause’ button and stare at a frozen screenshot.

Does that make sense? Yes. There are no forces acting on this object here and, hence, there is no arrow of time.


The numerical example above is confusing because our mind is not only thinking about the trajectory as such but also about the force causing the particle—or the car or the rocket in the example above—to move in this or that direction. When it’s a rocket, we know it ignited its boosters 2.5 seconds ago (because that’s what we saw – in reality or in a movie of the event) and, hence, seeing that same rocket move backwards – both in time as well as in space – while its boosters operate at full thrust does not make sense to us. Likewise, an obstacle escaping gravity with no other forces acting on it does not make sense either.

That being said, reversing the trajectory and, hence, actually reversing the effects of time, should not be a problem—from a purely theoretical point at least: we should just apply twice the force produced by the boosters to give that rocket the same acceleration in the reverse direction. That would obviously means we would force it to crash back into the Earth. Because that would be rather complicated (we’d need twice as many boosters but mounted in the opposite direction), and because it would also be somewhat evil from a moral point of view, let us consider some less destructive examples.

Let’s take gravity, or electrostatic attraction or repulsion. These two forces also cause uniform acceleration or deceleration on objects. Indeed, one can describe the force field of a large mass (e.g. the Earth)—or, in electrostatics, some positive or negative charge in space— using field vectors. The field vectors for the electric field are denoted by E, and, in his famous Lectures on Physics, Feynman uses a C for the gravitational field. The forces on some other mass m and on some other charge q can then be written as F = mC and F = qE respectively. The similarity with the F = ma equation – Newton’s Second Law in other words – is obvious, except that F = mC and F = qE are an expression of the origin, the nature and the strength of the force:

  1. In the case of the electrostatic force (remember that likes repel and opposites attract), the magnitude of E is equal to E = qc/4πε0r2. In this equation, εis the electric constant, which we’ve encountered before, and r is the distance between the charge q and the charge qcausing the field).
  2. For the gravitational field we have something similar, except that there’s only attraction between masses, no repulsion. The magnitude of C will be equal to C = –GmE/r2, with mE the mass causing the gravitational field (e.g. the mass of the Earth) and G the universal gravitational constant. [Note that the minus sign makes the direction of the force come out alright taking the existing conventions: indeed, it’s repulsion that gets the positive sign – but that should be of no concern to us here.]

So now we’ve explained the dynamics behind that x(t) = x(0) + v(0)·t + (a/2)·tcurve above, and it’s these dynamics that explain why looking back in time does not make sense—not in a mathematical way but in philosophical way. Indeed, it’s the nature of the force that gives time (or the direction of motion, which is the very same ‘arrow of time’) one–and only one–logical direction.

OK… But so what is time reversibility then – or time symmetry as it’s referred to? Let me defer an answer to this question by first introducing another topic.

Even and odd functions

I already introduced the concept of even and odd variables above. It’s obviously linked to some symmetry/asymmetry. The x(t) curve above is symmetric. It is obvious that, if we would change our coordinate system to let x(0) equal x(0) = 0, and also choose the origin of time such that v(0) = 0, then we’d have a nice symmetry with respect to the vertical axis. The graph of the quadratic function below illustrates such symmetry.

Even functionFunctions with a graph such as the one above are called even functions. A (real-valued) function f(t) of a (real) variable t is defined as even if, for all t and –t in the domain of f, we find that f(t) = f(–t).

We also have odd functions, such as the one depicted below. An odd function is a function for which f(-t) = –f(t).

Odd function

The function below gives the velocity as a function of time, and it’s clear that this would be an odd function if we would choose the zero time point such that v(0) = 0. In that case, we’d have a line through the origin and the graph would show an odd function. So that’s why we refer to v as an odd variable under time reversal.

Velocity curve

A very particular and very interesting example of an even function is the cosine function – as illustrated below.

Cosine functionNow, we said that the left side of the graph of the trajectory of our car or our rocket (i.e. the side with a negative slope and, hence, negative velocity) did not make much sense, because – as we play our movie backwards – it would depict a car or a rocket accelerating in the absence of a force. But let’s look at another situation here: a cosine function like the one above could actually represent the trajectory of a mass oscillating on a spring, as illustrated below.

oscillating springIn the case of a spring, the force causing the oscillation pulls back when the spring is stretched, and it pushes back when it’s compressed, so the mechanism is such that the direction of the force is being reversed continually. According to Hooke’s Law, this force is proportional to the amount of stretch. If x is the displacement of the mass m, and k that factor of proportionality, then the following equality must hold at all times:

F = ma = m(d2x/dt2) = –kx ⇔ d2x/dt= –(k/m)x

Is there also a logical arrow of time here? Look at the illustration below. If we follow the green arrow, we can readily imagine what’s happening: the spring gets stretched and, hence, the mass on the spring (at maximum speed as it passes the equilibrium position) encounters resistance: the spring pulls it back and, hence, it slows down and then reverses direction. In the reverse direction – i.e. the direction of the red arrow – we have the reverse logic: the spring gets compressed (x is negative), the mass slows down (as evidence by the curvature of the graph), and – at some point – it also reverses its direction of movement. [I could note that the force equation above is actually a second-order linear differential equation, and that the cosine function is its solution, but that’s a rather pedantic and, hence, totally superfluous remark here.]


What’s important is that, in this case, the ‘arrow of time’ could point either way, and both make sense. In other words, when we would make a movie of this oscillating movement, we could play it backwards and it would still make sense. 

Huh? Yes. Just in case you would wonder whether this conclusion depends on our starting point, it doesn’t. Just look at the illustration below, in which I assume we are starting to watch that movie (which is being played backwards without us knowing it is being played backwards) of the oscillating spring when the mass is not in the equilibrium position. It makes perfect sense: the spring is stretched, and we see the mass accelerating to the equilibrium position, as it should.


What’s going on here? Why can we reverse the arrow of time in the case of the spring, and why can’t we do that in the case of that particle being attracted or repelled by another? Are there two realities here? No. There’s only. I’ve been playing a trick on you. Just think about what is actually happening and then think about that so-called ‘time reversal’:

  1. At point A, the spring is still being stretched further, in reality that is, and so the mass is moving away from the equilibrium position. Hence, in reality, it will not move to point B but further away from the equilibrium position.
  2. However, we could imagine it moving from point A to B if we would reverse the direction of the force. Indeed, the force is equal to –kx and reversing its direction is equivalent to flipping our graph around the horizontal axis (i.e. the time axis), or to shifting the time axis left or right with an amount equal to π (note that the ‘time’ axis is actually represented by the phase, but that’s a minor technical detail and it does not change the analysis: we just measure time in radians here instead of seconds).

It’s a visual trick. There is no ‘real’ symmetry. The flipped graph corresponds to another situation (i.e. some other spring that started oscillating a bit earlier or later than ours here). Hence, our conclusion that it is the force that gives time direction, still holds.

Hmm… Let’s think about this. What makes our ‘trick’ work is that the force is allowed to change direction. Well… If we go back to our previous example of an object falling towards the center of some gravitational field, or a charge being attracted by some other (opposite) charge, then you’ll note that we can make sense of the ‘left side’ of the graph if we would change the sign of the force.

Huh? Yes, I know. This is getting complicated. But think about it. The graph below might represent a charged particle being repelled by another (stationary) particle: that’s the green arrow. We can then go back in time (i.e. we reverse the green arrow) if we reverse the direction of the force from repulsion to attraction. Now, that would usually lead to a dramatic event—the end of the story to be precise. Indeed, once the two particles get together, they’re glued together and so we’d have to draw another horizontal line going in the minus t direction (i.e. to the left side of our time axis) representing the status quo. Indeed, if the two particles sit right on top of each other, or if they would literally fuse or annihilate each other (like a particle and an anti-particle), then there’s no force or anything left at all… except ifwe would alter the direction of the force once again, in which case the two particles would fly apart again (OK. OK. You’re right in noting that’s not true in the annihilation case – but that’s a minor detail).

arrow of time

Is this story getting too complicated? It shouldn’t. The point to note is that reversibility – i.e. time reversal in the philosophical meaning of the word (not that mathematical business of inserting negative variables instead of positive ones) – is all about changing the direction of the force: going back in time implies that we reverse the effects of time, and reversing the effects of time, requires forces acting in the opposite direction.

Now, when it’s only kinetic energy that is involved, then it should be easy but when charges are involved, which is the case for all fundamental forces, then it’s not so easy. That’s when charge (C) and parity (P) symmetry come into the picture.

CP symmetry

Hooke’s ‘Law’ – i.e. the law describing the force on a mass on a stretched or compressed spring – is not a fundamental law: eventually the spring will stop. Yes. It will stop even if when it’s in a horizontal position and with the mass moving on a frictionless surface, as assumed above: the forces between the atoms and/or molecules in the spring give the spring the elasticity which causes the mass to oscillate around some equilibrium position, but some of the energy of that continuous movement gets lost in heat energy (yes, an oscillating spring does actually get warmer!) and, hence, eventually the movement will peter out and stop.

Nevertheless, the lesson we learned above is a valuable one: when it comes to the fundamental forces, we can reverse the arrow of time and still make sense of it all if we also reverse the ‘charges’. The term ‘charges’ encompasses anything measuring a propensity to interact through one of the four fundamental forces here. That’s where CPT symmetry comes in: if we reverse time, we should also reverse the charges.

But how can we change the ‘sign’ of mass: mass is always positive, isn’t it? And what about the P-symmetry – this thing about left-handed and right-handed neutrinos?

Well… I don’t know. That’s the kind of stuff I am currently exploring in my quest. I’ll just note the following:

1. Gravity might be a so-called pseudo force – because it’s proportional to mass. I won’t go into the details of that – if only because I don’t master them as yet – but Einstein’s gut instinct that gravity is not a ‘real’ fundamental force (we just have to adjust our reference frame and work with curved spacetime) – and, hence, that ‘mass’ is not like the other force ‘charges’ – is something I want to further explore. [Apart from being a measure for inertia, you’ll remember that (rest) mass can also be looked at as equivalent to a very dense chunk of energy, as evidenced by Einstein’s energy-mass equivalence formula: E = mc2.]

As for now, I can only note that the particles in an ‘anti-world’ would have the same mass. In that sense, anti-matter is not ‘anti’-matter: it just carries opposite electromagnetic, strong and weak charges. Hence, our C-world (so the world we get when applying a charge transformation) would have all ‘charges’ reversed, but mass would still be mass.

2. As for parity symmetry (i.e. left- and right-handedness, aka as mirror symmetry), I note that it’s raised primarily in relation to the so-called weak force and, hence, it’s also a ‘charge’ of sorts—in my primitive view of the world at least. The illustration below shows what P symmetry is all about really and may or may not help you to appreciate the point.

muon decay

OK. What is this? Let’s just go step by step here.

The ‘cylinder’ (both in (a), the upper part of the illustration, and in (b), the lower part) represents a muon—or a bunch of muons actually. A muon is an unstable particle in the lepton family. Think of it as a very heavy electron for all practical purposes: it’s about 200 times the mass of an electron indeed. Its lifetime is fairly short from our (human) point of view–only 2.2 microseconds on average–but that’s actually an eternity when compared to other unstable particles.

In any case, the point to note is that it usually decays into (i) two neutrinos (one muon-neutrino and one electron-antineutrino to be precise) and – importantly – (ii) one electron, so electric charge is preserved (indeed, neutrinos got the name they have because they carry no electric charge).

Now, we have left- and right-handed muons, and we can actually line them up in one of these two directions. I would need to check how that’s done, but muons do have a magnetic moment (just like electrons) and so I must assume it’s done in the same way as in Wu’s cobalt-60 experiment: through a uniform magnetic field. In other words, we know their spin directions in an experiment like this.

Now, if the weak force would respect mirror symmetry (but we already know it doesn’t), we would not be able to distinguish (i) the muon decay process in the ‘mirror world’ (i.e. the reflection of what’s going on in the (imaginary) mirror in the illustration above) from (ii) the decay process in ‘our’ (real) world. So that would be situation (a): the number of decay electrons being emitted in an upward direction would be the same (more or less) as the amount of decay electrons being emitted in a downward direction.

However, the actual laboratory experiments show that situation (b) is actually the case: most of the electrons are being emitted in only one direction (i.e. the upward direction in the illustration above) and, hence, the weak force does not respect mirror symmetry.

So what? Is that a problem?

For eminent physicists such as Feynman, it is. As he writes in his concluding Lecture on mechanics, radiation and heat (Vol. I, Chapter 52: Symmetry in Physical Laws): “It’s like seeing small hairs growing on the north pole of a magnet but not on its south pole.” [He means it allows us to distinguish the north and the south pole of a magnet in some absolute sense. Indeed, if we’re not able to tell right from left, we’re also not able to tell north from south – in any absolute sense that is. But so the experiment shows we actually can distinguish the two in some kind of absolute sense.]

I should also note that Wolfgang Pauli, one of the pioneers of quantum mechanics, said that it was “total nonsense” when he was informed about Wu’s experimental results, and that repeated experiments were needed to actually convince him that we cannot just create a mirror world out of ours. 

For me, it is not a problem.I like to think of left- and right-handedness as some charge itself, and of the combined CPT symmetry as the only symmetry that matters really. That should be evident from my rather intuitive introduction on time symmetry above.

Consider it and decide for yourself how logical or illogical it is. We could define what Feynman refers to as an axial vector: watching that muon ‘from below’, we see that its spin is clockwise, and let’s use that fact to define an axial vector pointing in the same direction as the thick black arrow (it’s the so-called ‘right-hand screw rule’ really), as shown below.

Axial vector

Now, let’s suppose that mirror world actually exists, in some corner in the universe, and that a guy living in that ‘mirror world’ would use that very same ‘right-hand-screw rule’: his axial vector when doing this experiment would point in the opposite direction (see the thick black arrow in the mirror, which points in the opposite direction indeed). So what’s wrong with that?

Nothing – in my modest view at least. Left- and right-handedness can just be looked at as any other ‘charge’ – I think – and, hence, if we would be able to communicate with that guy in the ‘mirror world’, the two experiments would come out the same. So the other guy would also notice that the weak force does not respect mirror symmetry but so there’s nothing wrong with that: he and I should just get over it and continue to do business as usual, wouldn’t you agree?

After all, there could be a zillion reasons for the experiment giving the results it does: perhaps the ‘right-handed’ spin of the muon is sort of transferred to the electron as the muon decays, thereby giving it the same type of magnetic moment as the one that made the muon line up in the first place. Or – in a much wilder hypothesis which no serious physicist would accept – perhaps we actually do not yet understand everything of the weak decay process: perhaps we’ve got all these solar neutrinos (which all share the same spin direction) interfering in the process.

Whatever it is: Nature knows the difference between left and right, and I think there’s nothing wrong with that. Full stop.

But then what is ‘left’ and ‘right’ really? As the experiment pointed out, we can actually distinguish between the two in some kind of absolute sense. It’s not just a convention. As Feynman notes, we could decide to label ‘right’ as ‘left’, and ‘left’ as ‘right’ right here and right now – and impose the new convention everywhere – but then these physics experiments will always yield the same physical results, regardless of our conventions. So, while we’d put different stickers on the results, the laws of physics would continue to distinguish between left and right in the same absolute sense as Wu’s cobalt-60 decay experiment did back in 1956.

The really interesting thing in this rather lengthy discussion–in my humble opinion at least–is that imaginary ‘guy in the mirror world’. Could such mirror world exist? Why not? Let’s suppose it does really exist and that we can establish some conversation with that guy (or whatever other intelligent life form inhabiting that world).

We could then use these beta decay processes to make sure his ‘left’ and ‘right’ definitions are equal to our ‘left’ and ‘right’ definitions. Indeed, we would tell him that the muons can be left- or right-handed, and we would ask him to check his definition of ‘right-handed’ by asking him to repeat Wu’s experiment. And, then, when finally inviting him over and preparing to physically meet with him, we should tell him he should use his “right” hand to greet us. Yes. We should really do that.

Why? Well… As Feynman notes, he (or she or whatever) might actually be living in an anti-matter world, i.e. a world in which all charges are reversed, i.e. a world in which protons carry negative charge and electrons carry positive charge, and in which the quarks have opposite color charge. In that case, we would have been updating each other on all kinds of things in a zillion exchanges, and we would have been trying hard to assure each other that our worlds are not all that different (including that crucial experiment to make sure his left and right are the same as ours), but – if he would happen to live in an anti-matter world – then he would put out his left hand – not his right – when getting out of his spaceship. Touching it would not be wise. 🙂

[Let me be much more pedantic than Feynman is and just point out that his spaceship would obviously have been annihilated by ‘our’ matter long before he would have gotten to the meeting place. As soon as he’d get out of his ‘anti-matter’ world, we’d see a big flash of light and that would be it.]

Symmetries and conservation laws

A final remark should be made on the relation between all those symmetries and conservation laws. When everything is said and done, all that we’ve got is some nice graphs and then some axis or plane of symmetry (in two and three dimensions respectively). Is there anything more to it? There is.

There’s a “deep connection”, it seems, between all these symmetries and the various ‘laws of conservation’. In our examples of ‘time symmetry’, we basically illustrated the law of energy conservation:

  1. When describing a particle traveling through an electrostatic or gravitation field, we basically just made the case that potential energy is converted into kinetic energy, or vice versa.
  2. When describing an oscillating mass on a spring, we basically looked at the spring as a reservoir of energy – releasing and absorbing kinetic energy as the mass oscillates around its zero energy point – but, once again, all we described was a system in which the total amount of energy – kinetic and elastic – remained the same.

In fact, the whole discussion on CPT symmetry above has been quite simplistic and can be summarized as follows:

Energy is being conserved. Therefore, if you want to reverse time, you’ll need to reverse the forces as well. And reversing the forces implies a change of sign of the charges causing those forces.

In short, one should not be fascinated by T-symmetry alone. Combined CPT symmetry is much more intuitive as a concept and, hence, much more interesting. So, what’s left?

Quite a lot. I know you have many more questions at this point. At least I do:

  1. What does it mean in quantum mechanics? How does the Uncertainty Principle come into play?
  2. How does it work exactly for the strong force, or for the weak force? [I guess I’d need to find out more about neutrino physics here…]
  3. What about the other ‘conservation laws’ (such as the conservation of linear or angular momentum, for example)? How are they related to these ‘symmetries’.

Well… That’s complicated business it seems, and even Feynman doesn’t explore these topics in the above-mentioned final Lecture on (classical) mechanics. In any case, this post has become much too long already so I’ll just say goodbye for the moment. I promise I’ll get back to you on all of this.

Post scriptum:

If you have read my previous post (The Weird Force), you’ll wonder why – in the example of how a mirror world would relate to ours – I assume that the combined CP symmetry holds. Indeed, when discussing the ‘weird force’ (i.e. the weak force), I mentioned that it does not respect any of the symmetries, except for the combined CPT symmetry. So it does not respect (i) C symmetry, (ii) P symmetry and – importantly – it also does not respect the combined CP symmetry. This is a deep philosophical point which I’ll talk about in my next post. However, I needed this post as an ‘introduction’ to the next one.

The weird force

Pre-scriptum (dated 26 June 2020): While one of the illustrations in this post was removed as a result of an attack by the dark force, I am happy to see it still survived more or less intact. While my views on the true nature of light, matter and the force or forces that act on them have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics, I think some of the analysis in this post remains fun to read. 🙂

Original post:

In my previous post (Loose Ends), I mentioned the weak force as the weird force. Indeed, unlike photons or gluons (i.e. the presumed carriers of the electromagnetic and strong force respectively), the weak force carriers (W bosons) have (1) mass and (2) electric charge:

  1. W bosons are very massive. The equivalent mass of a W+ and W– boson is some 86.3 atomic mass units (amu): that’s about the same as a rubidium or strontium atom. The mass of a Z boson is even larger: roughly equivalent to the mass of a molybdenium atom (98 amu). That is extremely heavy: just compare with iron or silver, which have a mass of  about 56 amu and 108 amu respectively. Because they are so massive, W bosons cannot travel very far before disintegrating (they actually go (almost) nowhere), which explains why the weak force is very short-range only, and so that’s yet another fundamental difference as compared to the other fundamental forces.
  2. The electric charge of W and Z bosons explains why we have a trio of weak force carriers rather than just one: W+, W– and Z0. Feynman calls them “the three W’s”.

The electric charge of W and Z bosons is what it is: an electric charge – just like protons and electrons. Hence, one has to distinguish it from the the weak charge as such: the weak charge (or, to be correct, I should say the weak isospin number) of a particle (such as a proton or a neutron for example) is related to the propensity of that particle to interact through the weak force — just like the electric charge is related to the propensity of a particle to interact through the electromagnetic force (think about Coulomb’s law for example: likes repel and opposites attract), and just like the so-called color charge (or the (strong) isospin number I should say) is related to the propensity of quarks (and gluons) to interact with each other through the strong force.

In short, as compared to the electromagnetic force and the strong force, the weak force (or Fermi’s interaction as it’s often called) is indeed the odd one out: these W bosons seem to mix just about everything: mass, charge and whatever else. In his 1985 Lectures on Quantum Electrodynamics, Feynman writes the following about this:

“The observed coupling constant for W’s is much the same as that for the photon. Therefore, the possibility exists that the three W’s and the photon are all different aspects of the same thing. Stephen Weinberg and Abdus Salam tried to combine quantum electrodynamics with what’s called ‘the weak interactions’ into one quantum theory, and they did it. But if you look at the results they get, you can see the glue—so to speak. It’s very clear that the photon and the three W’s are interconnected somehow, but at the present level of understanding, the connection is difficult to see clearly—you can still see the ‘seams’ in the theories; they have not yet been smoothed out so that the connection becomes more beautiful and, therefore, probably more correct.” (Feynman, 1985, p. 142)

Well… That says it all, I think. And from what I can see, the (tentative) confirmation of the existence of the Higgs field has not made these ‘seams’ any less visible. However, before criticizing eminent scientists such as Weinberg and Salam, we should obviously first have a closer look at those W bosons without any prejudice.

Alpha decay, potential wells and quantum tunneling

The weak force is usually explained as the force behind a process referred to as beta decay. However, because beta decay is just one form of radioactive decay, I need to say something about alpha decay too. [There is also gamma decay but that’s like a by-product of alpha and beta decay: when a nucleus emits an α or β particle (i.e. when we have alpha or beta decay), the nucleus will usually be left in an excited state, and so it can then move to a lower energy state by emitting a gamma ray photon (gamma radiation is very hard (i.e. very high-energy) radiation) – in the same way that an atomic electron can jump to a lower energy state by emitting a (soft) light ray photon. But so I won’t talk about gamma decay.]

Atomic decay, in general, is a loss of energy accompanying a transformation of the nucleus of the atom. Alpha decay occurs when the nucleus ejects an alpha particle: an α-particle consist of two protons and two neutrons bound together and, hence, it’s identical to a helium nucleus. Alpha particles are commonly emitted by all of the larger radioactive nuclei, such as uranium (which becomes thorium as a result of the decay process), or radium (which becomes radon gas). However, alpha decay is explained by a mechanism not involving the weak force: the electromagnetic force and the nuclear force (i.e. the strong force) will do. The reasoning is as follows: the alpha particle can be looked at as a stable but somewhat separate particle inside the nucleus. Because of their charge (both positive), the alpha particle inside of the nucleus and ‘the rest of the nucleus’ are subject to strong repulsive electromagnetic forces between them. However, these strong repulsive electromagnetic forces are not as strong as the strong force between the quarks that make up matter and, hence, that’s what keeps them together – most of the time that is.

Let me be fully complete here. The so-called nuclear force between composite particles such as protons and neutrons – or between clusters of protons and neutrons in this case – is actually the residual effect of the strong force. The strong force itself is between quarks – and between them only – and so that’s what binds them together in protons and neutrons (so that’s the next level of aggregation you might say). Now, the strong force is mostly neutralized within those protons and neutrons, but there is some residual force, and so that’s what keeps a nucleus together and what is referred to as the nuclear force.

There is a very helpful analogy here: the electromagnetic forces between neutral atoms (and/or molecules)—referred to as van der Waals forces (that’s what explains the liquid shape of water, among other things)— are also the residual of the (much stronger) electromagnetic forces that tie the electrons to the nucleus.

Now, that residual strong force – i.e. the nuclear force – diminishes in strength with distance but, within a certain distance, that residual force is strong enough to do what it does, and that’s to keep the nucleus together. This stable situation is usually depicted by what is referred to as a potential well:

Potential wellThe name is obvious: a well is a hole in the ground from which you can get water (or oil or gas or whatever). Now, the sea level might actually be lower than the bottom of a well, but the water would still stay in the well. In the illustration above, we are not depicting water levels but energy levels, but it’s equally obvious it would require some energy to kick a particle out of this well: if it would be water, we’d require a pump to get it out but, of course, it would be happy to flow to the sea once it’s out. Indeed, once a charged particle would be out (I am talking our alpha particle now), it will obviously stay out because of the repulsive electromagnetic forces coming into play (positive charges reject each other).

But so how can it escape the nuclear force and go up on the side of the well? [A potential pond or lake would have been a better term – but then that doesn’t sound quite right, does it? :-)]

Well, the energy may come from outside – that’s what’s referred to as induced radioactive decay (just Google it and you will tons of articles on experiments involving laser-induced accelerated alpha decay) – or, and that’s much more intriguing, the Uncertainty Principle comes into play.

Huh? Yes. According to the Uncertainty Principle, the energy of our alpha particle inside of the nucleus wiggles around some mean value but our alpha particle would also have an amplitude to have some higher energy level. That results not only in a theoretical probability for it to escape out of the well but also into something actually happening if we wait long enough: the amplitude (and, hence, the probability) is tiny, but it’s what explains the decay process – and what gives U-232 a half-life of 68.9 years, and also what gives the more common U-238 a much more comfortable 4.47 billion years as the half-life period.


Now that we’re talking about wells and all that, we should also mention that this phenomenon of getting out of the well is referred to as quantum tunneling. You can easily see why: it’s like the particle dug its way out. However, it didn’t: instead of digging under the sidewall, it sort of ‘climbed over’ it. Think of it being stuck and trying and trying and trying – a zillion times – to escape, until it finally did. So now you understand this fancy word: quantum tunneling. However, this post is about the weak force and so let’s discuss beta decay now.

Beta decay and intermediate vector bosons

Beta decay also involves transmutation of nuclei, but not by the emission of an α-particle but by a β-particle. A beta particle is just a different name for an electron (β) and/or its anti-matter counterpart: the positron (β+). [Physicists usually simplify stuff but in this case, they obviously didn’t: why don’t they just write e and ehere?]

An example of β decay is the decay of carbon-14 (C-14) into nitrogen-14 (N-14), and an example of β+ decay is the decay of magnesium-23 into sodium-23. C-14 and N-14 have the same mass but they are different atoms. The decay process is described by the equations below:

Beta decay

You’ll remember these formulas from your high-school days: beta decay does not change the mass number (carbon and nitrogen have the same mass: 14 units) but it does change the atomic (or proton) number: nitrogen has an extra proton. So one of the neutrons became a proton ! [The second equation shows the opposite: a proton became a neutron.] In order to do that, the carbon atom had to eject a negative charge: that’s the electron you see in the equation above.

In addition, there is also the ejection of a anti-neutrino (that’s what the bar above the ve symbol stands for: antimatter). You’ll wonder what an antineutrino could possibly be. Don’t worry about it: it’s not any spookier than the neutrino. Neutrinos and anti-neutrinos have no electric charge and so you cannot distinguish them on that account (electric charge). However, all antineutrinos have right-handed helicity (i.e. they come in only one of the two possible spin states), while the neutrinos are all left-handed. That’s why beta-decay is said to not respect parity symmetry, aka as mirror symmetry. Hence, in the case of beta decay, Nature does distinguish between the world and the mirror world ! I’ll come back on that but let me first lighten up the discussion somewhat with a graphical illustration of that neutron-proton transformation.


As for magnesium-sodium transformation, we’d have something similar but so we’d just have a positron instead of an electron (a positron is just an electron with a positive charge for all practical purposes) and a regular neutrino. So we’d just have the anti-matter counterparts of the electron and the neutrino. [Don’t be put off by the term ‘anti-matter’: anti-matter is really just like regular matter – except that the charges have opposite sign. For example, the anti-matter counterpart of a blue quark is an anti-blue quark, and the anti-matter counterpart of neutrino has right-handed helicity – or spin – as opposed to the ‘left-handed’ ‘ordinary’ neutrinos.]

Now, you surely will have several serious questions. The most obvious question is what happens with the electron and the neutrino? Well… Those spooky neutrinos are gone before you know it and so don’t worry about them. As for the electron, the carbon had only six electrons but the nitrogen needs seven to be electrically neutral… So you might think the new atom will take care of it. Well… No. Sorry. Because of its kinetic energy, the electron is likely to just explore the world and crash into something else, and so we’re left with a positively charged nitrogen ion indeed. So I should have added a little + sign next to the N in the formula above. Of course, one cannot exclude the possibility that this ion will pick up the electron later – but don’t bet on it: the ion might have to absorb another electron, or not find any free electrons !

As for the positron (in a β+ decay), that will just grab the nearest electron around and auto-destruct—thereby generating two high-energy photons (so that’s a little light flash). The net result is that we do not have an ion but a neutral sodium atom. Because the nearest electron will usually be found on some shell around the nucleus (the K or L shell for example), such process is often described as electron capture, and the ‘transformation equation’ can then be written p + e– → n + ve (with p and n denoting a proton and a neutron respectively).

The more important question is: where are the W and Z bosons in this story?

Ah ! Yes! Sorry I forgot about them. The Feynman diagram below shows how it really works—and why the name of intermediate vector bosons for these three strange ‘particles’ (W+, W, and Z0) is so apt. These W bosons are just a short trace of ‘something’ indeed: their half-life is about 3×10−25 s, and so that’s the same order of magnitude (or minitude I should say) as the mean lifetime of other resonances observed in particle collisions. 

Feynman diagram beta decay

Indeed, you’ll notice that, in this so-called Feynman diagram, there’s no space axis. That’s because the distances involved are so tiny that we have to distort the scale—so we are not using equivalent time and distance units here, as Feynman diagrams should. That’s in line with a more prosaic description of what may be happening: W bosons mediate the weak force by seemingly absorbing an awful lot of momentum, spin, and whatever other energy related to all of the qubits describing the particles involved, to then eject an electron (or positron) and a neutrino (or an anti-neutrino).

Hmm… That’s not a standard description of a W boson as a force carrying particle, you’ll say. You’re right. This is more the description of a Z boson. What’s the Z boson again? Well… I haven’t explained it yet. It’s not involved in beta decay. There’s a process called elastic scattering of neutrinos. Elastic scattering means that some momentum is exchanged but neither the target (an electron or a nucleus) nor the incident particle (the neutrino) are affected as such (so there’s no break-up of the nucleus for example). In other words, things bounce back and/or get deflected but there’s no destruction and/or creation of particles, which is what you would have with inelastic collisions. Let’s examine what happens here.

W and Z bosons in neutrino scattering experiments

It’s easy to generate neutrino beams: remember their existence was confirmed in 1956 because nuclear reactors create a huge flux of them ! So it’s easy to send lots of high-energy neutrinos into a cloud or bubble chamber and see what happens. Cloud and bubble chambers are prehistoric devices which were built and used to detect electrically charged particles moving through it. I won’t go into too much detail but I can’t resist inserting a few historic pictures here.

The first two pictures below document the first experimental confirmation of the existence of positrons by Carl Anderson, back in 1932 (and, no, he’s not Danish but American), for which he got a Nobel Prize. The magnetic field which gives the positron some curvature—the trace of which can be seen in the image on the right—is generated by the coils around the chamber. Note the opening in the coils, which allows for taking a picture when the supersaturated vapor is suddenly being decompressed – and so the charged particle that goes through it leaves a trace of ionized atoms behind that act as ‘nucleation centers’ around which the vapor condenses, thereby forming tiny droplets. Quite incredible, isn’t it? One can only admire the perseverance of these early pioneers.

Carl Anderson Positron

The picture below is another historical first: it’s the first detection of a neutrino in a bubble chamber. It’s fun to analyze what happens here: we have a mu-meson – aka as a muon – coming out of the collision here (that’s just a heavier version of the electron) and then a pion – which should (also) be electrically charged because the muon carries electric charge… But I will let you figure this one out. I need to move on with the main story. 🙂


The point to note is that these spooky neutrinos collide with other matter particles. In the image above, it’s a proton, but so when you’re shooting neutrino beams through a bubble chamber, a few of these neutrinos can also knock electrons out of orbit, and so that electron will seemingly appear out of nowhere in the image and move some distance with some kinetic energy (which can all be measured because magnetic fields around it will give the electron some curvature indeed, and so we can calculate its momentum and all that).

Of course, they will tend to move in the same direction – more or less at least – as the neutrinos that knocked them loose. So it’s like the Compton scattering which we discussed earlier (from which we could calculate the so-called classical radius of the electron – or its size if you will)—but with one key difference: the electrons get knocked loose not by photons, but by neutrinos.

But… How can they do that? Photons carry the electromagnetic field so the interaction between them and the electrons is electromagnetic too. But neutrinos? Last time I checked, they were matter particles, not bosons. And they carry no charge. So what makes them scatter electrons?

You’ll say that’s a stupid question: it’s the neutrino, dummy ! Yes, but how? Well, you’ll say, they collide—don’t they? Yes. But we are not talking tiny billiard balls here: if particles scatter, one of the fundamental forces of Nature must be involved, and usually it’s the electromagnetic force: it’s the electron density around nuclei indeed that explains why atoms will push each other away if they meet each other and, as explained above, it’s also the electromagnetic force that explains Compton scattering. So billiard balls bounce back because of the electromagnetic force too and…

OK-OK-OK. I got it ! So here it must be the strong force or something. Well… No. Neutrinos are not made of quarks. You’ll immediately ask what they are made of – but the answer is simple: they are what they are – one of the four matter particles in the Standard Model – and so they are not made of anything else. Capito?

OK-OK-OK. I got it ! It must be gravity, no? Perhaps these neutrinos don’t really hit the electron: perhaps they skim near it and sort of drag it along as they pass? No. It’s not gravity either. It can’t be. We have no exact measurement of the mass of a neutrino but it’s damn close to zero – and, hence, way too small to exert any such influence on an electron. It’s just not consistent with those traces.

OK-OK-OK. I got it ! It’s that weak force, isn’t it? YES ! The Feynman diagrams below show the mechanism involved. As far as terminology goes (remember Feynman’s complaints about the up, down, strange, charm, beauty and truths quarks?), I think this is even worse. The interaction is described as a current, and when the neutral Z boson is involved, it’s called a neutral current – as opposed to…  Well… Charged currents. Neutral and charged currents? That sounds like sweet and sour candy, isn’t it? But isn’t candy supposed to be sweet? Well… No. Sour candy is pretty common too. And so neutral currents are pretty common too.


You obviously don’t believe a word of what I am saying and you’ll wonder what the difference is between these charged and neutral currents. The end result is the same in the first two pictures: an electron and a neutrino interact, and they exchange momentum. So why is one current neutral and the other charged? In fact, when you ask that question, you are actually wondering whether we need that neutral Z boson. W bosons should be enough, no?

No. The first and second picture are “the same but different”—and you know what that means in physics: it means it’s not the same. It’s different. Full stop. In the second picture, there is electron absorption (only for a very brief moment obviously, but so that’s what it is, and you don’t have that in the first diagram) and then electron emission, and there’s also neutrino absorption and emission. […] I can sense your skepticism – and I actually share it – but that’s what I understand of it !

[…] So what’s the third picture? Well… That’s actually beta decay: a neutron becomes a proton, and there’s emission of an electron and… Hey ! Wait a minute ! This is interesting: this is not what we wrote above: we have an incoming neutrino instead of an outgoing anti-neutrino here. So what’s this?

Well… I got this illustration from a blog on physics (Galileo’s Pendulum – The Flavor of Neutrinos) which, in turn, mentions Physics Today as its source. The incoming neutrino has nothing to do with the usual representation of an anti-matter particle as a particle traveling backwards in time. It’s something different, and it triggers a very interesting question: could beta decay possibly be ‘triggered’ by neutrinos? Who knows?

I googled it, and there seems to be some evidence supporting such thesis. However, this ‘evidence’ is flimsy (the only real ‘clue’ is that the activity of the Sun, as measured by the intensity of solar flares, seems to be having some (tiny) impact on the rate of decay of radioactive elements on Earth) and, hence, most ‘serious’ scientists seem to reject that possibility. I wonder why: it would make the ‘weird force’ somewhat less weird in my view. So… What to say? Well… Nothing much at this moment. Let me move on and examine the question a bit more in detail in a Post Scriptum.

The odd one out

You may wonder if neutrino-electron interaction always involve the weak force. The answer to that question is simple: Yes ! Because they do not carry any electric charge, and because they are not quarks, neutrinos are only affected by the weak force. However, as evidenced by all the stuff I wrote on beta decay, you cannot turn this statement on its head: the weak force is relevant not only for neutrinos but for electrons and quarks as well ! That gives us the following connection between forces and matter:

forces and matter

[Specialists reading this post may say they’ve not seen this diagram before. That might be true. I made it myself – for a change – but I am sure it’s around somewhere.]

It is a weird asymmetry: almost massless particles (neutrinos) interact with other particles through massive bosons, and these massive ‘things’ are supposed to be ‘bosons’, i.e. force carrying particles ! These physicists must be joking, right? These bosons can hardly carry themselves – as evidenced by the fact they peter out just like all of those other ‘resonances’ !

Hmm… Not sure what to say. It’s true that their honorific title – ‘intermediate vectors’ – seems to be quite apt: they are very intermediate indeed: they only appear as a short-lived stage in between the initial and final state of the system. Again, it leads one to think that these W bosons may just reflect some kind of energy blob caused by some neutrino – or anti-neutrino – crashing into another matter particle (a quark or an electron). Whatever it is, this weak force is surely the odd one out.

Odd one out

In my previous post, I mentioned other asymmetries as well. Let’s revisit them.

Time irreversibility

In Nature, uranium is usually found as uranium-238. Indeed, that’s the most abundant isotope of uranium: about 99.3% of all uranium is U-238. There’s also some uranium-235 out there: some 0.7%. And there are also trace amounts of U-234. And that’s it really. So where is the U-232 we introduced above when talking about alpha decay? Well… We said it has a half-life of 68.9 years only and so it’s rather normal U-232 cannot be found in Nature. What? Yes: 68.9 years is nothing compared to the half-life of U-238 (4.47 billion years) or U-235 (704 million years), and so it’s all gone. In fact, the tiny proportion of U-235 on this Earth is what allows us to date the Earth. The math and physics involved resemble the math and physics involved in carbon-dating but so carbon-dating is used for organic materials only, because the carbon-14 that’s used also has a fairly short half-time: 5,730 years—so that’s like a thousand times more than U-232 but… Well… Not like millions or billions of years. [You’ll immediately ask why this C-14 is still around if it’s got such a short life-time. The answer to that is easy: C-14 is continually being produced in the atmosphere and, hence, unlike U-232, it doesn’t just disappear.]

Hmm… Interesting. Radioactive decay suggests time irreversibility. Indeed, it’s wonderful and amazing – but sad at the same time:

  1. There’s so much diversity – a truly incredible range of chemical elements making life what it is.
  2. But so all these chemical elements have been produced through a process of nuclear fusion in stars (stellar nucleosynthesis), which were then blasted into space by supernovae, and so they then coagulated into planets like ours.
  3. However, all of the heavier atoms will decay back into some lighter element because of radioactive decay, as shown in the graph below.
  4. So we are doomed !

Overview of decay modes

In fact, some of the GUT theorists think that there is no such thing as ‘stable nuclides’ (that’s the black line in the graph above): they claim that all atomic species will decay because – according to their line of reasoning – the proton itself is NOT stable.

WHAT? Yeah ! That’s what Feynman complained about too: he obviously doesn’t like these GUT theorists either. Of course, there is an expensive experiment trying to prove spontaneous proton decay: the so-called Super-K under Mount Kamioka in Japan. It’s basically a huge tank of ultra-pure water with a lot of machinery around it… Just google it. It’s fascinating. If, one day, it would be able to prove that there’s proton decay, our Standard Model would be in very serious problems – because it doesn’t cater for unstable protons. That being said, I am happy that has not happened so far – because it would mean our world would really be doomed.   

What do I mean with that? We’re all doomed, aren’t we? If only because of the Second Law of Thermodynamics. Huh? Yes. That ‘law’ just expresses a universal principle: all kinetic and potential energy observable in nature will, in the end, dissipate: differences in temperature, pressure, and chemical potential will even out. Entropy increases. Time is NOT reversible: it points in the direction of increasing entropy – till all is the same once again. Sorry? 

Don’t worry about it. When everything is said and done, we humans – or life in general – are an amazing negation of the Second Law of Thermodynamics: temperature, pressure, chemical potential and what have you – it’s all super-organized and super-focused in our body ! But it’s temporary indeed – and we actually don’t negate the Second Law of Thermodynamics: we create order by creating disorder. In any case, I don’t want to dwell on this point. Time reversibility in physics usually refers to something else: time reversibility would mean that all basic laws of physics (and with ‘basic’, I am excluding this higher-level Second Law of Thermodynamics) would be time-reversible: if we’d put in minus t (–t) instead of t, all formulas would still make sense, wouldn’t they? So we could – theoretically – reverse our clock and stopwatches and go back in time.

Can we do that?

Well… We can reverse a lot. For example, U-232 decays into a lot of other stuff BUT we can produce U-232 from scratch once again—from thorium to be precise. In fact, that’s how we got it in the first place: as mentioned above, any natural U-232 that might have been produced in those stellar nuclear fusion reactors is gone. But so that means that alpha decay is reversible: we’re producing stable stuff – U-232 lasts for dozens of years – that probably existed long time ago but so it decayed and now we’re reversing the arrow of time using our nuclear science and technology.

Now, you may object that you don’t see Nature spontaneously assemble the nuclear technology we’re using to produce U-232, except if Nature would go for that Big Crunch everyone’s predicting so it can repeat the Big Bang once again (so that’s the oscillating Universe scenario)—and you’re obviously right in that assessment. That being said, from some kind of weird existential-philosophical point of view, it’s kind of nice to know that – in theory at least – there is time reversibility indeed (or T symmetry as it’s called by scientists). 


What? That’s right. For beta decay, we don’t have T symmetry. The weak force breaks all kinds of symmetries, and time symmetry is only one of them. I talked about these in my previous post (Loose Ends) – so please have a look at that, and let me just repeat the basics:

  1. Parity (P) symmetry or mirror symmetry revolves around the notion that Nature should not distinguish between right- and left-handedness, so everything that works in our world, should also work in the mirror world. Now, the weak force does not respect P symmetry: we need right-handed neutrinos for β decay, and we’d also need right-handed neutrinos to reverse the process – which actually happens: so, yes, beta decay might be time-reversible but so it doesn’t work with left-handed neutrinos – which is what our ‘right-handed’ neutrinos would be in the ‘mirror world’. Full stop. Our world is different from the mirror world because the weak force knows the difference between left and right – and some stuff only works with left-handed stuff (and then some other stuff only works with right-handed stuff). In short, the weak force doesn’t work the same in the mirror world. In the mirror world, we’d need to throw in left-handed neutrinos for β decay. Not impossible but a bit of a nuisance, you’ll agree.  
  2. Charge conjugation or charge (C) symmetry revolves around the notion that a world in which we reverse all (electric) charge signs. Now, the weak force also does not respect C symmetry. I’ll let you go through the reasoning for that, but it’s the same really. Just reversing all signs would not make the weak force ‘work’ in the mirror world: we’d have to ‘keep’ some of the signs – notably those of our W bosons !
  3. Initially, it was thought that the weak force respected the combined CP symmetry (and, therefore, that the principle of P and C symmetry could be substituted by a combined CP symmetry principle) but two experimenters – Val Fitch and James Cronin – got a Nobel Prize when they proved that this was not the case. To be precise, the spontaneous decay of neutral kaons (which is a type of decay mediated by the weak force) does not respect CP symmetry. Now, that was the death blow to time reversibility (T symmetry). Why? Can’t we just make a film of those experiments not respecting P, C or CP symmetry, and then just press the ‘reverse’ button? We could but one can show that the relativistic invariance in Einstein’s relativity theory implies a combined CPT symmetry. Hence, if CP is a broken symmetry, then the T symmetry is also broken. So we could play that film, but the laws of physics would not make sense ! In other words, the weak force does not respect T symmetry either !

To summarize this rather lengthy philosophical digression: a full CPT sequence of operations would work. So we could – in sequence – (1) change all particles to antiparticles (C), (2) reflect the system in a mirror (P), and (3) change the sign of time (T), and we’d have a ‘working’ anti-world that would be just as real as ours. HOWEVER, we do not live in a mirror world. We live in OUR world – and so left-handed is left-handed, and right-handed is right-handed, and positive is positive and negative is negative, and so THERE IS NO TIME REVERSIBILITY: the weak force does not respect T symmetry.

Do you understand now why I call the weak force the weird force? Penrose devotes a whole chapter to time reversibility in his Road to Reality, but he does not focus on the weak force. I wonder why. All that rambling on the Second Law of Thermodynamics is great – but one should relate that ‘principle’ to the fundamental forces and, most notably, to the weak force.

Post scriptum 1:

In one of my previous posts, I complained about not finding any good image of the Higgs particle. The problem is that these super-duper particle accelerators don’t use bubble chambers anymore. The scales involved have become incredibly small and so all that we have is electronic data, it seems, and that is then re-assembled into some kind of digital image but – when everything is said and done – these images are only simulations. Not the real thing. I guess I am just an old grumpy guy – a 45-year old economist: what do you expect? – but I’ll admit that those black-and-white pictures above make my heart race a bit more than those colorful simulations. But so I found a good simulation. It’s the cover image of Wikipedia’s Physics beyond the Standard Model (I should have looked there in the first place, I guess). So here it is: the “simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson (produced by colliding protons) decaying into hadron jets and electrons.”

CMS_Higgs-event (1)

So that’s what gives mass to our massive W bosons. The Higgs particle is a massive particle itself: an estimated 125-126 GeV/c2, so that’s about 1.5 times the mass of the W bosons. I tried to look into decay widths and all that, but it’s all quite confusing. In short, I have no doubt that the Higgs theory is correct – the data is all what we have and then, when everything is said and done, we have an honorable Nobel Prize Committee thinking the evidence is good enough (which – in light of their rather conservative approach (which I fully subscribe too: don’t get me wrong !) – usually means that it’s more than good enough !) – but I can’t help thinking this is a theory which has been designed to match experiment. 

Wikipedia writes the following about the Higgs field:

“The Higgs field consists of four components, two neutral ones and two charged component fields. Both of the charged components and one of the neutral fields are Goldstone bosons, which act as the longitudinal third-polarization components of the massive W+, W– and Z bosons. The quantum of the remaining neutral component corresponds to (and is theoretically realized as) the massive Higgs boson.”

Hmm… So we assign some qubits to W bosons (sorry for the jargon: I am talking these ‘longitudinal third-polarization components’ here), and to W bosons only, and then we find that the Higgs field gives mass to these bosons only? I might be mistaken – I truly hope so (I’ll find out when I am somewhat stronger in quantum-mechanical math) – but, as for now, it all smells somewhat fishy to me. It’s all consistent, yes – and I am even more skeptical about GUT stuff ! – but it does look somewhat artificial.

But then I guess this rather negative appreciation of the mathematical beauty (or lack of it) of the Standard Model is really what is driving all these GUT theories – and so I shouldn’t be so skeptical about them ! 🙂

Oh… And as I’ve inserted some images of collisions already, let me insert some more. The ones below document the discovery of quarks. They come out of the above-mentioned coffee table book of Lederman and Schramm (1989). The accompanying texts speak for themselves.

Quark - 1

Quark - 2

Quark - 3


Post scriptum 2:

I checked the source of that third diagram showing how an incoming neutrino could possibly cause a neutron to become a proton. It comes out of the August 2001 issue of Physics Today indeed, and it describes a very particular type of beta decay. This is the original illustration:

inverse beta decay

The article (and the illustration above) describes how solar neutrinos traveling through heavy water – also known as deuterium – can interact with the deuterium nucleus – which is referred to as deuteron, and which we’ll represent by the symbol d in the process descriptions below. The nucleus of deuterium – which is an isotope of hydrogen – consists of one proton and one neutron, as opposed to the much more common protium isotope of hydrogen, which has just one proton in the nucleus. Deuterium occurs naturally (0.0156% of all hydrogen atoms in the Earth’s oceans is deuterium), but it can also be produced industrially – for use in heavy-water nuclear reactors for example. In any case, the point is that deuteron can respond to solar neutrinos by breaking up in one of two ways:

  1. Quasi-elastically: ve + d → ve + p + n. So, in this case, the deuteron just breaks up in its two components: one proton and one neutron. That seems to happen pretty frequently because the nuclear forces holding the proton and the neutron together are pretty weak it seems.
  2. Alternatively, the solar neutrino can turn a deuteron’s neutron into a second proton, and so that’s what’s depicted in the third diagram above: ve + d → e + p + p. So what happens really is ve + n → e + p.

The author of this article – which basically presents the basics of how a new neutrino detector – the Sudbury Neutrino Observatory – is supposed to work – refers to the second process as inverse beta decay – but that’s a rather generic and imprecise term it seems. The conclusion is that the weak force seems to have myriad ways of expressing itself. However, the connection between neutrinos and the weak force seems to need further exploring. As for myself, I’d like to know why the hypothesis that any form of beta decay – or, for that matter, any other expression of the weak force – is actually being triggered by these tiny neutrinos crashing into (other) matter particles would not be reasonable.

In such scenario, the W bosons would be reduced to a (very) temporary messy ‘blob’ of energy, combining kinetic, electromagnetic as well as the strong binding energy between quarks if protons and neutrons are involved. Could this ‘odd one out’ be nothing but a pseudo-force? I am no doubt being very simplistic here – but then it’s an interesting possibility, isn’t it? In order to firmly deny it, I’ll need to learn a lot more about neutrinos no doubt – and about how the results of all these collisions in particle accelerators are actually being analyzed and interpreted.

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Loose ends…

Pre-scriptum (dated 26 June 2020): My views on the true nature of light, matter and the force or forces that act on them have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics. If you are reading this, then you are probably looking for not-to-difficult reading. In that case, I would suggest you read my re-write of Feynman’s introductory lecture to QM. If you want something shorter, you can also read my paper on what I believe to be the true Principles of Physics. Having said that, I still think there are a few good quotes and thoughts in this post too. 🙂

Original post:

It looks like I am getting ready for my next plunge into Roger Penrose’s Road to Reality. I still need to learn more about those Hamiltonian operators and all that, but I can sort of ‘see’ what they are supposed to do now. However, before I venture off on another series of posts on math instead of physics, I thought I’d briefly present what Feynman identified as ‘loose ends’ in his 1985 Lectures on Quantum Electrodynamics – a few years before his untimely death – and then see if any of those ‘loose ends’ appears less loose today, i.e. some thirty years later.

The three-forces model and coupling constants

All three forces in the Standard Model (the electromagnetic force, the weak force and the strong force) are mediated by force carrying particles: bosons. [Let me talk about the Higgs field later and – of course – I leave out the gravitational force, for which we do not have a quantum field theory.]

Indeed, the electromagnetic force is mediated by the photon; the strong force is mediated by gluons; and the weak force is mediated by W and/or Z bosons. The mechanism is more or less the same for all. There is a so-called coupling (or a junction) between a matter particle (i.e. a fermion) and a force-carrying particle (i.e. the boson), and the amplitude for this coupling to happen is given by a number that is related to a so-called coupling constant

Let’s give an example straight away – and let’s do it for the electromagnetic force, which is the only force we have been talking about so far. The illustration below shows three possible ways for two electrons moving in spacetime to exchange a photon. This involves two couplings: one emission, and one absorption. The amplitude for an emission or an absorption is the same: it’s –j. So the amplitude here will be (–j)(j) = j2. Note that the two electrons repel each other as they exchange a photon, which reflects the electromagnetic force between them from a quantum-mechanical point of view !

Photon exchangeWe will have a number like this for all three forces. Feynman writes the coupling constant for the electromagnetic force as  j and the coupling constant for the strong force (i.e. the amplitude for a gluon to be emitted or absorbed by a quark) as g. [As for the weak force, he is rather short on that and actually doesn’t bother to introduce a symbol for it. I’ll come back on that later.]

The coupling constant is a dimensionless number and one can interpret it as the unit of ‘charge’ for the electromagnetic and strong force respectively. So the ‘charge’ q of a particle should be read as q times the coupling constant. Of course, we can argue about that unit. The elementary charge for electromagnetism was or is – historically – the charge of the proton (q = +1), but now the proton is no longer elementary: it consists of quarks with charge –1/3 and +2/3 (for the d and u quark) respectively (a proton consists of two u quarks and one d quark, so you can write it as uud). So what’s j then? Feynman doesn’t give its precise value but uses an approximate value of –0.1. It is an amplitude so it should be interpreted as a complex number to be added or multiplied with other complex numbers representing amplitudes – so –0.1 is “a shrink to about one-tenth, and half a turn.” [In these 1985 Lectures on QED, which he wrote for a lay audience, he calls amplitudes ‘arrows’, to be combined with other ‘arrows.’ In complex notation, –0.1 = 0.1eiπ = 0.1(cosπ + isinπ).]

Let me give a precise number. The coupling constant for the electromagnetic force is the so-called fine-structure constant, and it’s usually denoted by the alpha symbol (α). There is a remarkably easy formula for α, which becomes even easier if we fiddle with units to simplify the matter even more. Let me paraphrase Wikipedia on α here, because I have no better way of summarizing it (the summary is also nice as it shows how changing units – replacing the SI units by so-called natural units – can simplify equations):

1. There are three equivalent definitions of α in terms of other fundamental physical constants:

\alpha = \frac{k_\mathrm{e} e^2}{\hbar c} = \frac{1}{(4 \pi \varepsilon_0)} \frac{e^2}{\hbar c} = \frac{e^2 c \mu_0}{2 h}
where e is the elementary charge (so that’s the electric charge of the proton); ħ = h/2π is the reduced Planck constant; c is the speed of light (in vacuum); ε0 is the electric constant (i.e. the so-called permittivity of free space); µ0 is the magnetic constant (i.e. the so-called permeability of free space); and ke is the Coulomb constant.

2. In the old centimeter-gram-second variant of the metric system (cgs), the unit of electric charge is chosen such that the Coulomb constant (or the permittivity factor) equals 1. Then the expression of the fine-structure constant just becomes:

\alpha = \frac{e^2}{\hbar c}

3. When using so-called natural units, we equate ε0 , c and ħ to 1. [That does not mean they are the same, but they just become the unit for measurement for whatever is measured in them. :-)] The value of the fine-structure constant then becomes:

 \alpha = \frac{e^2}{4 \pi}.

Of course, then it just becomes a matter of choosing a value for e. Indeed, we still haven’t answered the question as to what we should choose as ‘elementary’: 1 or 1/3? If we take 1, then α is just a bit smaller than 0.08 (around 0.0795775 to be somewhat more precise). If we take 1/3 (the value for a quark), then we get a much smaller value: about 0.008842 (I won’t bother too much about the rest of the decimals here). Feynman’s (very) rough approximation of –0.1 obviously uses the historic proton charge, so e = +1.

The coupling constant for the strong force is much bigger. In fact, if we use the SI units (i.e. one of the three formulas for α under point 1 above), then we get an alpha equal to some 7.297×10–3. In fact, its value will usually be written as 1/α, and so we get a value of (roughly) 1/137. In this scheme of things, the coupling constant for the strong force is 1, so that’s 137 times bigger.

Coupling constants, interactions, and Feynman diagrams

So how does it work? The Wikipedia article on coupling constants makes an extremely useful distinction between the kinetic part and the proper interaction part of an ‘interaction’. Indeed, before we just blindly associate qubits with particles, it’s probably useful to not only look at how photon absorption and/or emission works, but also at how a process as common as photon scattering works (so we’re talking Compton scattering here – discovered in 1923, and it earned Compton a Nobel Prize !).

The illustration below separates the kinetic and interaction part properly: the photon and the electron are both deflected (i.e. the magnitude and/or direction of their momentum (p) changes) – that’s the kinetic part – but, in addition, the frequency of the photon (and, hence, its energy – cf. E = hν) is also affected – so that’s the interaction part I’d say.

Compton scattering

With an absorption or an emission, the situation is different, but it also involves frequencies (and, hence, energy levels), as show below: an electron absorbing a higher-energy photon will jump two or more levels as it absorbs the energy by moving to a higher energy level (i.e. a so-called excited state), and when it re-emits the energy, the emitted photon will have higher energy and, hence, higher frequency.


This business of frequencies and energy levels may not be so obvious when looking at those Feynman diagrams, but I should add that these Feynman diagrams are not just sketchy drawings: the time and space axis is precisely defined (time and distance are measured in equivalent units) and so the direction of travel of particles (photons, electrons, or whatever particle is depicted) does reflect the direction of travel and, hence, conveys precious information about both the direction as well as the magnitude of the momentum of those particles. That being said, a Feynman diagram does not care about a photon’s frequency and, hence, its energy (its velocity will always be c, and it has no mass, so we can’t get any information from its trajectory).

Let’s look at these Feynman diagrams now, and the underlying force model, which I refer to as the boson exchange model.

The boson exchange model

The quantum field model – for all forces – is a boson exchange model. In this model, electrons, for example, are kept in orbit through the continuous exchange of (virtual) photons between the proton and the electron, as shown below.

Electron-protonNow, I should say a few words about these ‘virtual’ photons. The most important thing is that you should look at them as being ‘real’. They may be derided as being only temporary disturbances of the electromagnetic field but they are very real force carriers in the quantum field theory of electromagnetism. They may carry very low energy as compared to ‘real’ photons, but they do conserve energy and momentum – in quite a strange way obviously: while it is easy to imagine a photon pushing an electron away, it is a bit more difficult to imagine it pulling it closer, which is what it does here. Nevertheless, that’s how forces are being mediated by virtual particles in quantum mechanics: we have matter particles carrying charge but neutral bosons taking care of the exchange between those charges.

In fact, note how Feynman actually cares about the possibility of one of those ‘virtual’ photons briefly disintegrating into an electron-positron pair, which underscores the ‘reality’ of photons mediating the electromagnetic force between a proton and an electron,  thereby keeping them close together. There is probably no better illustration to explain the difference between quantum field theory and the classical view of forces, such as the classical view on gravity: there are no gravitons doing for gravity what photons are doing for electromagnetic attraction (or repulsion).

Pandora’s Box

I cannot resist a small digression here. The ‘Box of Pandora’ to which Feynman refers in the caption of the illustration above is the problem of calculating the coupling constants. Indeed, j is the coupling constant for an ‘ideal’ electron to couple with some kind of ‘ideal’ photon, but how do we calculate that when we actually know that all possible paths in spacetime have to be considered and that we have all of these ‘virtual’ mess going on? Indeed, in experiments, we can only observe probabilities for real electrons to couple with real photons.

In the ‘Chapter 4’ to which the caption makes a reference, he briefly explains the mathematical procedure, which he invented and for which he got a Nobel Prize. He calls it a ‘shell game’. It’s basically an application of ‘perturbation theory’, which I haven’t studied yet. However, he does so with skepticism about its mathematical consistency – skepticism which I mentioned and explored somewhat in previous posts, so I won’t repeat that here. Here, I’ll just note that the issue of ‘mathematical consistency’ is much more of an issue for the strong force, because the coupling constant is so big.

Indeed, terms with j2, j3jetcetera (i.e. the terms involved in adding amplitudes for all possible paths and all possible ways in which an event can happen) quickly become very small as the exponent increases, but terms with g2, g3getcetera do not become negligibly small. In fact, they don’t become irrelevant at all. Indeed, if we wrote α for the electromagnetic force as 7.297×10–3, then the α for the strong force is one, and so none of these terms becomes vanishingly small. I won’t dwell on this, but just quote Wikipedia’s very succinct appraisal of the situation: “If α is much less than 1 [in a quantum field theory with a dimensionless coupling constant α], then the theory is said to be weakly coupled. In this case it is well described by an expansion in powers of α called perturbation theory. [However] If the coupling constant is of order one or larger, the theory is said to be strongly coupled. An example of the latter [the only example as far as I am aware: we don’t have like a dozen different forces out there !] is the hadronic theory of strong interactions, which is why it is called strong in the first place. [Hadrons is just a difficult word for particles composed of quarks – so don’t worry about it: you understand what is being said here.] In such a case non-perturbative methods have to be used to investigate the theory.”

Hmm… If Feynman thought his technique for calculating weak coupling constants was fishy, then his skepticism about whether or not physicists actually know what they are doing when calculating stuff using the strong coupling constant is probably justified. But let’s come back on that later. With all that we know here, we’re ready to present a picture of the ‘first-generation world’.

The first-generation world

The first-generation is our world, excluding all that goes in those particle accelerators, where they discovered so-called second- and third-generation matter – but I’ll come back to that. Our world consists of only four matter particles, collectively referred to as (first-generation) fermions: two quarks (a u and a d type), the electron, and the neutrino. This is what is shown below.

first-generation matter

Indeed, u and d quarks make up protons and neutrons (a proton consists of two u quarks and one d quark, and a neutron must be neutral, so it’s two d quarks and one u quark), and then there’s electrons circling around them and so that’s our atoms. And from atoms, we make molecules and then you know the rest of the story. Genesis ! 

Oh… But why do we need the neutrino? [Damn – you’re smart ! You see everything, don’t you? :-)] Well… There’s something referred to as beta decay: this allows a neutron to become a proton (and vice versa). Beta decay explains why carbon-14 will spontaneously decay into nitrogen-14. Indeed, carbon-12 is the (very) stable isotope, while carbon-14 has a life-time of 5,730 ± 40 years ‘only’ and, hence, measuring how much carbon-14 is left in some organic substance allows us to date it (that’s what (radio)carbon-dating is about). Now, a beta particle can refer to an electron or a positron, so we can have β decay (e.g. the above-mentioned carbon-14 decay) or β+ decay (e.g. magnesium-23 into sodium-23). If we have β decay, then some electron will be flying out in order to make sure the atom as a whole stays electrically neutral. If it’s β+ decay, then emitting a positron will do the job (I forgot to mention that each of the particles above also has a anti-matter counterpart – but don’t think I tried to hide anything else: the fermion picture above is pretty complete). That being said, Wolfgang Pauli, one of those geniuses who invented quantum theory, noted, in 1930 already, that some momentum and energy was missing, and so he predicted the emission of this mysterious neutrinos as well. Guess what? These things are very spooky (relatively high-energy neutrinos produced by stars (our Sun in the first place) are going through your and my my body, right now and right here, at a rate of some hundred trillion per second) but, because they are so hard to detect, the first actual trace of their existence was found in 1956 only. [Neutrino detection is fairly standard business now, however.] But back to quarks now.

Quarks are held together by gluons – as you probably know. Quarks come in flavors (u and d), but gluons come in ‘colors’. It’s a bit of a stupid name but the analogy works great. Quarks exchange gluons all of the time and so that’s what ‘glues’ them so strongly together. Indeed, the so-called ‘mass’ that gets converted into energy when a nuclear bomb explodes is not the mass of quarks (their mass is only 2.4 and 4.8 MeV/c2. Nuclear power is binding energy between quarks that gets converted into heat and radiation and kinetic energy and whatever else a nuclear explosion unleashes. That binding energy is reflected in the difference between the mass of a proton (or a neutron) – around 938 MeV/c2 – and the mass figure you get when you add two u‘s and one d, which is them 9.6 MeV/c2 only. This ratio – a factor of one hundred – illustrates once again the strength of the strong force: 99% of the ‘mass’ of a proton or an electron is due to the strong force.    

But I am digressing too much, and I haven’t even started to talk about the bosons associated with the weak force. Well… I won’t just now. I’ll just move on the second- and third-generation world.

Second- and third-generation matter

When physicists started to look for those quarks in their particle accelerators, Nature had already confused them by producing lots of other particles in these accelerators: in the 1960s, there were more than four hundred of them. Yes. Too much. But they couldn’t get them back in the box. 🙂

Now, all these ‘other particles’ are unstable but they survive long enough – a muon, for example, disintegrates after 2.2 millionths of a second (on average) – to deserve the ‘particle’ title, as opposed to a ‘resonance’, whose lifetime can be as short as a billionth of a trillionth of a second. And so, yes, the physicists had to explain them too. So the guys who devised the quark-gluon model (the model is usually associated with Murray Gell-Mann but – as usual with great ideas – some others worked hard on it as well) had already included heavier versions of their quarks to explain (some of) these other particles. And so we do not only have heavier quarks, but also a heavier version of the electron (that’s the muon I mentioned) as well as a heavier version of the neutrino (the so-called muon neutrino). The two new ‘flavors’ of quarks were called s and c. [Feynman hates these names but let me give them: u stands for up, d for down, s for strange and c for charm. Why? Well… According to Feynman: “For no reason whatsoever.”]

Traces of the second-generation and c quarks were found in experiments in 1968 and 1974 respectively (it took six years to boost the particle accelerators sufficiently), and the third-generation quark (for beauty or bottom – whatever) popped up in Fermilab‘s particle accelerator in 1978. To be fully complete, it then took 17 years to detect the super-heavy t quark – which stands for truth.  [Of all the quarks, this name is probably the nicest: “If beauty, then truth” – as Lederman and Schramm write in their 1989 history of all of this.]

What’s next? Will there be a fourth or even fifth generation? Back in 1985, Feynman didn’t exclude it (and actually seemed to expect it), but current assessments are more prosaic. Indeed, Wikipedia writes that, According to the results of the statistical analysis by researchers from CERN and the Humboldt University of Berlin, the existence of further fermions can be excluded with a probability of 99.99999% (5.3 sigma).” If you want to know why… Well… Read the rest of the Wikipedia article. It’s got to do with the Higgs particle.

So the complete model of reality is the one I already inserted in a previous post and, if you find it complicated, remember that the first generation of matter is the one that matters and, among the bosons, it’s the photons and gluons. If you focus on these only, it’s not complicated at all – and surely a huge improvement over those 400+ particles no one understood in the 1960s.


As for the interactions, quarks stick together – and rather firmly so – by interchanging gluons. They thereby ‘change color’ (which is the same as saying there is some exchange of ‘charge’). I copy Feynman’s original illustration hereunder (not because there’s no better illustration: the stuff you can find on Wikipedia has actual colors !) but just because it’s reflects the other illustrations above (and, perhaps, maybe I also want to make sure – with this black-and-white thing – that you don’t think there’s something like ‘real’ color inside of a nucleus).

quark gluon exchange

So what are the loose ends then? The problem of ‘mathematical consistency’ associated with the techniques used to calculate (or estimate) these coupling constants – which Feynman identifies as a key defect in 1985 – is is a form of skepticism about the Standard Model that is not shared by others. It’s more about the other forces. So let’s now talk about these.

The weak force as the weird force: about symmetry breaking

I included the weak force in the title of one of the sub-sections above (“The three-forces model”) and then talked about the other two forces only. The W, W and Z bosons – usually referred to, as a group, as the W bosons, or the ‘intermediate vector bosons’ – are an odd bunch. First, note that they are the only ones that do not only have a (rest) mass (and not just a little bit: they’re almost 100 times heavier than a the proton or neutron – or a hydrogen atom !) but, on top of that, they also have electric charge (except for the Z boson). They are really the odd ones out.  Feynman does not doubt their existence (a Fermilab team produced them in 1983, and they got a Nobel Prize for it, so little room for doubts here !), but it is obvious he finds the weak force interaction model rather weird.

He’s not the only one: in a wonderful publication designed to make a case for more powerful particle accelerators (probably successful, because the Large Hadron Collider came through – and discovered credible traces of the Higgs field, which is involved in the story that is about to follow), Leon Lederman and David Schramm look at the asymmety involved in having massive W bosons and massless photons and gluons, as just one of the many asymmetries associated with the weak force. Let me develop this point.

We like symmetries. They are aesthetic. But so I am talking something else here: in classical physics, characterized by strict causality and determinism, we can – in theory – reverse the arrow of time. In practice, we can’t – because of entropy – but, in theory, so-called reversible machines are not a problem. However, in quantum mechanics we cannot reverse time for reasons that have nothing to do with thermodynamics. In fact, there are several types of symmetries in physics:

  1. Parity (P) symmetry revolves around the notion that Nature should not distinguish between right- and left-handedness, so everything that works in our world, should also work in the mirror world. Now, the weak force does not respect P symmetry. That was shown by experiments on the decay of pions, muons and radioactive cobalt-60 in 1956 and 1957 already.
  2. Charge conjugation or charge (C) symmetry revolves around the notion that a world in which we reverse all (electric) charge signs (so protons would have minus one as charge, and electrons have plus one) would also just work the same. The same 1957 experiments showed that the weak force does also not respect C symmetry.
  3. Initially, smart theorists noted that the combined operation of CP was respected by these 1957 experiments (hence, the principle of P and C symmetry could be substituted by a combined CP symmetry principle) but, then, in 1964, Val Fitch and James Cronin, proved that the spontaneous decay of neutral kaons (don’t worry if you don’t know what particle this is: you can look it up) into pairs of pions did not respect CP symmetry. In other words, it was – again – the weak force not respecting symmetry. [Fitch and Cronin got a Nobel Prize for this, so you can imagine it did mean something !]
  4. We mentioned time reversal (T) symmetry: how is that being broken? In theory, we can imagine a film being made of those events not respecting P, C or CP symmetry and then just pressing the ‘reverse’ button, can’t we? Well… I must admit I do not master the details of what I am going to write now, but let me just quote Lederman (another Nobel Prize physicist) and Schramm (an astrophysicist): “Years before this, [Wolfgang] Pauli [Remember him from his neutrino prediction?] had pointed out that a sequence of operations like CPT could be imagined and studied; that is, in sequence, change all particles to antiparticles, reflect the system in a mirror, and change the sign of time. Pauli’s theorem was that all nature respected the CPT operation and, in fact, that this was closely connected to the relativistic invariance of Einstein’s equations. There is a consensus that CPT invariance cannot be broken – at least not at energy scales below 1019 GeV [i.e. the Planck scale]. However, if CPT is a valid symmetry, then, when Fitch and Cronin showed that CP is a broken symmetry, they also showed that T symmetry must be similarly broken.” (Lederman and Schramm, 1989, From Quarks to the Cosmos, p. 122-123)

So the weak force doesn’t care about symmetries. Not at all. That being said, there is an obvious difference between the asymmetries mentioned above, and the asymmetry involved in W bosons having mass and other bosons not having mass. That’s true. Especially because now we have that Higgs field to explain why W bosons have mass – and not only W bosons but also the matter particles (i.e. the three generations of leptons and quarks discussed above). The diagram shows what interacts with what.

2000px-Elementary_particle_interactions.svgBut so the Higgs field does not interact with photons and gluons. Why? Well… I am not sure. Let me copy the Wikipedia explanation: “The Higgs field consists of four components, two neutral ones and two charged component fields. Both of the charged components and one of the neutral fields are Goldstone bosons, which act as the longitudinal third-polarization components of the massive W+, W– and Z bosons. The quantum of the remaining neutral component corresponds to (and is theoretically realized as) the massive Higgs boson.”

Huh? […] This ‘answer’ probably doesn’t answer your question. What I understand from the explanation above, is that the Higgs field only interacts with W bosons because its (theoretical) structure is such that it only interacts with W bosons. Now, you’ll remember Feynman’s oft-quoted criticism of string theory: I don’t like that for anything that disagrees with an experiment, they cook up an explanation–a fix-up to say.” Is the Higgs theory such cooked-up explanation? No. That kind of criticism would not apply here, in light of the fact that – some 50 years after the theory – there is (some) experimental confirmation at least !

But you’ll admit it does all look ‘somewhat ugly.’ However, while that’s a ‘loose end’ of the Standard Model, it’s not a fundamental defect or so. The argument is more about aesthetics, but then different people have different views on aesthetics – especially when it comes to mathematical attractiveness or unattractiveness.

So… No real loose end here I’d say.


The other ‘loose end’ that Feynman mentions in his 1985 summary is obviously still very relevant today (much more than his worries about the weak force I’d say). It is the lack of a quantum theory of gravity. There is none. Of course, the obvious question is: why would we need one? We’ve got Einstein’s theory, don’t we? What’s wrong with it?

The short answer to the last question is: nothing’s wrong with it – on the contrary ! It’s just that it is – well… – classical physics. No uncertainty. As such, the formalism of quantum field theory cannot be applied to gravity. That’s it. What’s Feynman’s take on this? [Sorry I refer to him all the time, but I made it clear in the introduction of this post that I would be discussing ‘his’ loose ends indeed.] Well… He makes two points – a practical one and a theoretical one:

1. “Because the gravitation force is so much weaker than any of the other interactions, it is impossible at the present time to make any experiment that is sufficiently delicate to measure any effect that requires the precision of a quantum theory to explain it.”

Feynman is surely right about gravity being ‘so much weaker’. Indeed, you should note that, at a scale of 10–13 cm (that’s the picometer scale – so that’s the relevant scale indeed at the sub-atomic level), the coupling constants compare as follows: if the coupling constant of the strong force is 1, the coupling constant of the electromagnetic force is approximately 1/137, so that’s a factor of 10–2 approximately. The strength of the weak force as measured by the coupling constant would be smaller with a factor 10–13 (so that’s 1/10000000000000 smaller). Incredibly small, but so we do have a quantum field theory for the weak force ! However, the coupling constant for the gravitational force involves a factor 10–38. Let’s face it: this is unimaginably small.

However, Feynman wrote this in 1985 (i.e. thirty years ago) and scientists wouldn’t be scientists if they would not at least try to set up some kind of experiment. So there it is: LIGO. Let me quote Wikipedia on it:

LIGO, which stands for the Laser Interferometer Gravitational-Wave Observatory, is a large-scale physics experiment aiming to directly detect gravitation waves. […] At the cost of $365 million (in 2002 USD), it is the largest and most ambitious project ever funded by the NSF. Observations at LIGO began in 2002 and ended in 2010; no unambiguous detections of gravitational waves have been reported. The original detectors were disassembled and are currently being replaced by improved versions known as “Advanced LIGO”.

So, let’s see what comes out of that. I won’t put my money on it just yet. 🙂 Let’s go to the theoretical problem now.

2. “Even though there is no way to test them, there are, nevertheless, quantum theories of gravity that involve ‘gravitons’ (which would appear under a new category of polarizations, called spin “2”) and other fundamental particles (some with spin 3/2). The best of these theories is not able to include the particles that we do find, and invents a lot of particles that we don’t find. [In addition] The quantum theories of gravity also have infinities in the terms with couplings [Feynman does not refer to a coupling constant but to a factor n appearing in the so-called propagator for an electron – don’t worry about it: just note it’s a problem with one of those constants actually being larger than one !], but the “dippy process” that is successful in getting rid of the infinities in quantum electrodynamics doesn’t get rid of them in gravitation. So not only have we no experiments with which to check a quantum theory of gravitation, we also have no reasonable theory.”

Phew ! After reading that, you wouldn’t apply for a job at that LIGO facility, would you? That being said, the fact that there is a LIGO experiment would seem to undermine Feynman’s practical argument. But then is his theoretical criticism still relevant today? I am not an expert, but it would seem to be the case according to Wikipedia’s update on it:

“Although a quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, difficulties arise when one attempts to apply the usual prescriptions of quantum field theory. From a technical point of view, the problem is that the theory one gets in this way is not renormalizable and therefore cannot be used to make meaningful physical predictions. As a result, theorists have taken up more radical approaches to the problem of quantum gravity, the most popular approaches being string theory and loop quantum gravity.”

Hmm… String theory and loop quantum gravity? That’s the stuff that Penrose is exploring. However, I’d suspect that for these (string theory and loop quantum gravity), Feynman’s criticism probably still rings true – to some extent at least: 

I don’t like that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation–a fix-up to say, “Well, it might be true.” For example, the theory requires ten dimensions. Well, maybe there’s a way of wrapping up six of the dimensions. Yes, that’s all possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there’s no reason whatsoever in superstring theory that it isn’t eight out of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn’t produce anything; it has to be excused most of the time. It doesn’t look right.”

What to say by way of conclusion? Not sure. I think my personal “research agenda” is reasonably simple: I just want to try to understand all of the above somewhat better and then, perhaps, I might be able to understand some of what Roger Penrose is writing. 🙂