Wavefunctions and the twin paradox

My previous post was awfully long, so I must assume many of my readers may have started to read it, but… Well… Gave up halfway or even sooner. ūüôā I added a footnote, though, which is interesting to reflect upon. Also, I know many of my readers aren’t interested in the math‚ÄĒeven if they understand one cannot really appreciate quantum theory without the math. But… Yes. I may have left some readers behind. Let me, therefore, pick up the most interesting bit of all of the stories in my last posts in as easy a language as I can find.

We have that weird 360/720¬į symmetry in quantum physics or‚ÄĒto be precise‚ÄĒwe have it for elementary matter-particles (think of electrons, for example). In order to, hopefully, help you understand what it’s all about, I had to explain the often-confused but substantially different concepts of a¬†reference frame¬†and a representational base¬†(or representation¬†tout court). I won’t repeat that explanation, but think of the following.

If we just rotate the reference frame over 360¬į, we’re just using the same reference frame and so we see the same thing: some object which we, vaguely, describe by some¬†ei¬∑őł¬†function. Think of some spinning object. In its own reference frame, it will just spin around some center or, in ours, it will spin while moving along some axis in its own reference frame or, seen from ours, as moving in some direction while it’s spinning‚ÄĒas illustrated below.

To be precise, I should say that we describe it by some Fourier sum of such functions. Now, if its spin direction is… Well… In the other direction, then we’ll describe it by by some¬†e‚ąíi¬∑őł¬†function (again, you should read: a¬†Fourier¬†sum of such functions). Now, the weird thing is is the following: if we rotate the object itself, over the same¬†360¬į, we get a¬†different¬†object: our¬†ei¬∑őł¬†and¬†e‚ąíi¬∑őł¬†function (again: think of a¬†Fourier¬†sum, so that’s a wave¬†packet, really) becomes a¬†‚ąíe¬Īi¬∑őł¬†thing. We get a¬†minus¬†sign in front of it.¬†So what happened here? What’s the difference, really?

Well… I don’t know. It’s very deep. Think of you and me as two electrons who are watching each other. If I do nothing, and you keep watching me while turning around me, for a full¬†360¬į (so that’s a rotation of your reference frame over 360¬į), then you’ll end up where you were when you started and, importantly, you’ll see the same thing: me. ūüôā I mean… You’ll see¬†exactly¬†the same thing: if I was an¬†e+i¬∑őł¬†wave packet, I am still an¬†an¬†e+i¬∑őł¬†wave packet now. Or¬†if I was an e‚ąíi¬∑őł¬†wave packet, then I am still an¬†an e‚ąíi¬∑őł¬†wave packet now. Easy. Logical. Obvious, right?

But so now we try something different:¬†I¬†turn around, over a full¬†360¬į turn, and you¬†stay where you are and watch me¬†while I am turning around. What happens? Classically, nothing should happen but… Well… This is the weird world of quantum mechanics: when I am back where I was‚ÄĒlooking at you again, so to speak‚ÄĒthen… Well… I am not quite the same any more. Or… Well… Perhaps I am but you¬†see¬†me differently. If I was¬†e+i¬∑őł¬†wave packet, then I’ve become a¬†‚ąíe+i¬∑őł¬†wave packet now.

Not hugely different but… Well… That¬†minus¬†sign matters, right? Or¬†If I was¬†wave packet built up from elementary¬†a¬∑e‚ąíi¬∑őł¬†waves, then I’ve become a¬†‚ąíe‚ąíi¬∑őł¬†wave packet now. What happened?

It makes me think of the twin paradox in special relativity. We know it’s a¬†paradox‚ÄĒso that’s an¬†apparent¬†contradiction only: we know which twin stayed on Earth and which one traveled because of the gravitational forces on the traveling twin. The one who stays on Earth does not experience any acceleration or deceleration. Is it the same here? I mean… The one who’s turning around must experience some¬†force.

Can we relate this to the twin paradox? Maybe. Note that a¬†minus¬†sign in front of the¬†e‚ąí¬Īi¬∑őł¬†functions amounts a minus sign in front of both the sine and cosine components. So… Well… The negative of a sine and cosine is the sine and cosine but with a phase shift of 180¬į: ‚ąícosőł =¬†cos(őł ¬Ī ŌÄ) and¬†‚ąísinőł =¬†sin(őł ¬Ī ŌÄ). Now, adding or subtracting a¬†common¬†phase factor to/from the argument of the wavefunction amounts to¬†changing¬†the origin of time. So… Well… I do think the twin paradox and this rather weird business of 360¬į and 720¬į symmetries are, effectively, related. ūüôā

Post scriptum:¬†Google¬†honors Max Born’s 135th birthday today. ūüôā I think that’s a great coincidence in light of the stuff I’ve been writing about lately (possible interpretations of the wavefunction). ūüôā

Time reversal and CPT symmetry (III)

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‚Äďalways.

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‚Äďjust 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 ‚ÄstNo!¬†‚Äď Feynman’s 1985 book on quantum electrodynamics (QED)¬†‚Äďso that’s five years after¬†Fitch and Cronin got a¬†Nobel Prize for their discovery‚Ästis 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¬†‚Äď despite his age¬†‚Äď 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¬†‚Äď 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‚Äďrepeating the experiment over and over and over again‚Äďto 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‚Äďto some extent at least‚Äď 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‚Äďno matter how small‚Äďis 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¬†‚Äď i.e. time-reversal in the ‘philosophical’ or ‘common-sense’ interpretation ‚Äď 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‚Äďi.e. a world view which connects the arrow of time with real-life forces that cause our world to change‚Äď 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‚Äďfrom 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 (‚Äď) 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¬†‚Äď I admit¬†‚Äď 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)‚Äďwhich is how we can distinguish between ‘left’ and ‘right’ in the first place, in some kind of absolute¬†sense that is‚Äďthe¬†combined¬†CP symmetry implies that right-handed¬†matter¬†and left-handed¬†anti-matter behave just the same‚Äďand, 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‚Äďinstead of the agreed-upon¬†right hand… Then… Well… Don’t touch it. ūüôā

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. ūüôā

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. ūüôā

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‚Äďat 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. ūüôā

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. ūüôā

Time reversal and CPT symmetry (I)

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‚Äôt 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 ‚Äď objects defying gravity ‚Äď is plain impossible because of Newton‚Äôs 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 ‚Äď are also plain impossible because of some other ‚Äėlaw‚Äô: the law of ever increasing entropy.

However, there‚Äôs a distinct difference between the two ‚Äėlaws‚Äô (gravity versus increasing entropy). As one entry on Physics Stack Exchange notes, the entropy law ‚Äď better known as the second law of thermodynamics ‚Äď ‚Äúonly describes what is most likely to happen in macroscopic systems, rather than what has to happen‚ÄĚ, but then the author immediately qualifies this apparent lack of determinism, and rightly so: ‚ÄúIt 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.‚ÄĚ Hence, while one will find some people wondering whether this entropy law is a ‚Äėreal law‚Äô of Nature ‚Äď in the sense that they would question that it’s always true no matter what ‚Äď 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 ‚Äď 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‚Äôs time symmetry in physics? The Wikipedia definition is the following: ‚ÄúT-symmetry is the theoretical symmetry (invariance) of physical laws under a time reversal (T) transformation.‚ÄĚ Huh?

A ‘time reversal transformation’ amounts to inserting ‚Äďt (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’ ‚ÄĒ 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‚Äôs First Law is the principle of inertia: if there‚Äôs no (other) force acting on an object, it will just continue doing what it does‚Äďi.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)‚Äďor further engage with the force it was already engaged with.
  2. Newton‚Äôs 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‚Äôs Third Law is the law of gravitation, which describes the origin, the nature and the strength of the gravitational force. That‚Äôs part of dynamics, i.e. the study of the forces themselves ‚Äď 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‚Äôll discuss dynamics later). Suppose some object ‚Äď perhaps an elementary particle but it could also be a car or a rocket indeed ‚Äď is moving through space with some constant acceleration a (so we can write a(t) = a). This means that v(t) ‚Äď the velocity as a function of time ‚Äď 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 ‚Äď i.e. you and me right here and right now¬†‚Äď 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‚Äôs not the origin of time: it’s just ‘now’‚Äďthe start of our analysis.

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

At some point in the future, let‚Äôs 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‚Äôs 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‚Äôs 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‚Äôll remember that the distance traveled is an infinite sum of infinitesimally small products vőĒt: the velocity at each point of time multiplied by an infinitesimally small interval of time. You‚Äôll remember that we write such infinite sum as an integral:

Eq 1

[In case you wonder why we use the letter ‚Äės‚Äô for distance traveled: it‚Äôs because the ‚Äėd‚Äô symbol is already used to denote a differential and, hence, ‚Äės‚Äô is supposed to stand for ‚Äėspatium‚Äô, 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 ‚ÄďT 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‚Äôs 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 ‚Äď just like the acceleration ‚Äď a constant). In fact, it‚Äôs 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 = ‚Äď2.5. 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 =¬†‚Äď2.5, 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‚ÄĒor the car or the rocket in the example above‚ÄĒto 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‚ÄĒfrom 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)‚ÄĒor, in electrostatics, some positive or negative charge in space‚ÄĒ 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/4ŌÄőĶ0r2. In this equation, őĶ0¬†is¬†the electric constant, which we’ve encountered before, and r is the distance between the charge q and the charge qc¬†causing 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¬†= ‚ÄďGmE/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)¬∑t2¬†curve above, and it’s these dynamics¬†that explain why looking back in time does not make sense‚ÄĒnot 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‚Äďand only one‚Äďlogical 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 ‚Äďt in the domain of f, we find that f(t) = f(‚Äďt).

We also have odd functions, such as the one depicted below. An odd function is a function for which f(-t) = ‚Äďf(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) = ‚Äďkx ‚áĒ¬†d2x/dt2¬†= ‚Äď(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 ‚Äďkx 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 ŌĬ†(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‚ÄĒthe 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‚ÄĒin 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‚ÄĒor 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‚Äďonly 2.2 microseconds on average‚Äďbut 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 ‚Äď 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‚Äďin my humble opinion at least‚Äďis 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. ūüôā

[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.