Symmetries and transformations

In my previous post, I promised to do something on symmetries. Something simple but then… Well… You¬†know how it goes: one question always triggers another one. ūüôā

Look at the situation in the illustration on the left below. We suppose we have something¬†real¬†going on there: something is moving from left to right (so that’s in the 3 o’clock direction), and then something else is going around¬†clockwise (so that’s not¬†the direction in which we measure angles (which also include the argument¬†őł of our wavefunction), because that’s always¬†counter-clockwise, as I note at the bottom of the illustration). To be precise, we should note that the angular momentum here is all about the y-axis, so the angular momentum vector L points in the (positive) y-direction. We get that direction from the familiar right-hand rule, which is illustrated in the top right corner.

mirrorNow, suppose someone else is looking at this from the other side – or just think of yourself going around a full 180¬į to look at the same thing from the back side. You’ll agree you’ll see the same thing going from¬†right¬†to¬†left (so that’s in the¬†9 o’clock direction now – or, if our clock is transparent, the 3 o’clock direction of our reversed clock). Likewise, the thing that’s turning around will now go counter-clockwise.

Note that both observers – so that’s me and that other person (or myself after my walk around this whole thing) – use a regular coordinate system, which implies the following:

  1. We’ve got regular 90¬į degree angles between our coordinates axes.
  2. Our x-axis goes from negative to positive from left to right, and our y-axis does the same going away from us.
  3. We also both define our z-axis using, once again, the ubiquitous right-hand rule, so our z-axis points upwards.

So we have two observers looking at the same reality¬†– some linear¬†as well as some¬†angular¬†momentum – but from opposite sides. And so we’ve got a reversal of both the linear as well as the angular momentum. Not¬†in reality, of course, because we’re looking at the same thing. But we¬†measure¬†it differently. Indeed, if we use the subscripts 1 and 2 to denote the measurements in the two coordinate systems, we find that p2¬†=¬†‚Äďp1.¬†Likewise, we also find that L2¬†=¬†‚ÄďL1.

Now, when you see these two equations, you¬†will probably not worry about¬†that¬†p2¬†=¬†‚Äďp1¬†equation – although you should, because it’s actually only valid for this rather particular orientation of the linear momentum (I’ll come back to that in a moment). It’s the L2¬†=¬†‚ÄďL1¬†equation which should surprise you most. Why? Because you’ve always been told there is a big¬†difference between (1)¬†real¬†vectors (aka polar vectors), like the momentum¬†p, or the velocity¬†v, or the force F,¬†and (2)¬†pseudo-vectors (aka axial vectors), like the angular¬†momentum¬†L. You may also remember how to distinguish between the two:¬†if you change the¬†direction¬†of the axes of your reference frame, polar vectors will change sign too, as opposed to axial vectors: axial vectors do not¬†swap sign if we swap the coordinate signs.

So… Well… How does that work here? In fact, what we should ask ourselves is: why does that not¬†work here? Well… It’s simple, really. We’re not changing the direction of the axes here. Or… Well… Let me be more precise: we’re only swapping the sign of the x– and y-axis. We did¬†not¬†flip the z-axis. So we turned things around, but we didn’t turn them upside down. It makes a huge difference. Note, for example, that if all of the linear momentum would have been in the z-direction only (so our¬†p vector would have been pointing in the z-direction, and in the z-direction only), it would¬†not¬†swap sign. The illustration below shows what really happens with the coordinates of some vector when we’re doing a¬†rotation. It’s, effectively, only the¬†x– and¬†y-coordinates that flip sign.reflection symmetry

It’s easy to see that this¬†rotation about the z-axis here preserves our deep sense of ‘up’ versus ‘down’, but that it swaps ‘left’ for ‘right’, and vice versa. Note that this is not¬†a reflection. We are¬†not¬†looking at some mirror world here. The difference between a reflection (a mirror world) and a rotation (the real world seen from another angle) is illustrated below. It’s quite confusing but, unlike what you might think, a reflection does not swap left for right. It does turn things inside out, but that’s what a rotation does as well: near becomes far, and far becomes near.difference between reflection and rotation

Before we move on, let me say a few things about the¬†mirror world¬†and, more in particular, about the obvious question: could it possibly¬†exist? Well… What do you think? Your first reaction might well be: “Of course! What nonsense question! We just walk around whatever it is that we’re seeing – or, what amounts to the same, we just turn it around – and there it is: that’s the mirror world, right? So of course it exists!” Well… No. That’s¬†not¬†the mirror world. That’s just the¬†real¬†world seen from the opposite direction, and that world… Well… That’s just the real world. ūüôā¬†The mirror world is, literally, the world¬†in the mirror¬†– like the photographer in the illustration below. We don’t swap left for right here: some object going from left to right in the real world is still going from left to right in the mirror world!MirrorOf course, you may now involve the photographer in the picture above and observe – note that you’re now an observer of the observer of the mirror ūüôā – that, if he would move his left arm in the real world, the photographer in the mirror world would be moving his right arm. But… Well… No. You’re saying that because you’re now¬†imaging that you’re the photographer in the mirror world yourself now, who’s looking at the real world from inside, so to speak. So you’ve rotated the perspective in your mind¬†and you’re saying it’s his right arm because you imagine¬†yourself to be the photographer in the mirror.¬†We usually do that because… Well… Because we look in a mirror every day, right? So we’re used to seeing ourselves that way and we always think it’s us we’re seeing. ūüôā However, the illustration above is correct: the mirror¬†world only swaps near for far, and far for near, so it only swaps the sign of the y-axis.

So the question¬†is¬†relevant: could the mirror world actually exist? What we’re really¬†asking here is the following: can we swap the sign of one¬†coordinate axis¬†only in all of our physical laws and equations and… Well… Do we then still get the same laws and equations? Do we get the same Universe – because that’s what those laws and equations describe? If so, our mirror world can exist. If not, then not.

Now,¬†I’ve done a post on that, in which I explain that mirror world can only exist if it would consist of anti-matter. So if our real world and the mirror world would actually meet, they would annihilate each other. ūüôā But that post is quite technical. Here I want to keep it very¬†simple: I basically only want to show what the rotation¬†operation implies for the wavefunction. There is no doubt whatsoever that the rotated¬†world exists. In fact, the rotated world is just our¬†world. We¬†walk around some object, or we turn it around, but so we’re still watching the same object. So we’re not thinking about the mirror world here. We just want to know how things look like when adopting some other perspective.

So, back to the starting point: we just have two observers here, who look at the same thing but from opposite directions. Mathematically, this corresponds to a rotation of our reference frame¬†about¬†the z-axis of 180¬į. Let me spell out – somewhat more precisely – what happens to the linear and angular momentum here:

  1. The direction of the linear momentum in the xy-plane swaps direction.
  2. The angular momentum about the y-axis, as well as about the x-axis, swaps direction too.

Note that the illustration only shows angular momentum about the y-axis, but you can easily verify the statement about the angular momentum about the x-axis. In fact, the angular momentum about any line in the xy-plane will swap direction.

Of course, the¬†x-, y-, z-axes in the other reference frame are different than mine, and so I should give them a subscript, right? Or, at the very least, write something like x’, y’, z’, so we have a primed¬†reference frame here,¬†right? Well… Maybe. Maybe not. Think about it. ūüôā A coordinate system is just a mathematical thing… Only the momentum is real… Linear or angular… Equally real… And then Nature doesn’t care about our position, does it? So… Well… No subscript needed, right? Or… Well… What do you¬†think?¬†ūüôā

It’s just funny, isn’t it? It looks like we can’t really separate reality and perception here. Indeed, note how our¬†p2¬†=¬†‚Äďp1¬†and L2¬†=¬†‚ÄďL1¬†equations already mix reality with how we perceive it. It’s the same thing¬†in reality¬†but the coordinates of p1¬†and L1 are positive, while the coordinates of p2¬†and L2¬†are negative. To be precise, these coordinates will look like this:

  1. p1 = (p, 0, 0) and L1 = (0, L, 0)
  2. p2¬†= (‚ąíp, 0, 0) and L1 =¬†(0, ‚ąíL, 0)

So are they two different things or are they not? ūüôā Think about it. I’ll move on in the meanwhile. ūüôā

Now, you probably know a thing or two about parity¬†symmetry, or¬†P-symmetry: if if we flip the sign of all coordinates, then we’ll still find the same physical laws, like F = m¬∑a and what have you. [It works for all physical laws, including quantum-mechanical laws – except those involving the¬†weak¬†force (read: radioactive decay processes).] But so here we are talking rotational symmetry. That’s not¬†the same as P-symmetry. If we flip the signs of¬†all¬†coordinates, we’re also swapping ‘up’ for ‘down’, so we’re not only turning around, but we’re also getting upside down. The difference between¬†rotational symmetry and P-symmetry is shown below.up and down swap

As mentioned, we’ve talked about P-symmetry at length in other posts, and you can easily¬†google¬†a lot more on that. The question we want to examine here – just as a fun exercise – is the following:

How does that rotational symmetry work for a wavefunction?

The¬†very first illustration in this post gave you the functional form of the¬†elementary¬†wavefunction ¬†eiőł = ei¬∑(E¬∑t p¬∑x)/ńß. We should actually use a bold type¬†x¬†= (x, y, z) in this formula but we’ll assume we’re talking something similar to that p vector: something moving in the x-direction only – or in the xy-plane¬†only. The¬†z-component doesn’t change.¬†Now, you know that we can reduce all¬†actual¬†wavefunctions to some linear combination of such elementary wavefunctions by doing a Fourier¬†decomposition, so it’s fine to look at the elementary¬†wavefunction only – so we don’t make it too complicated here. Now think of the following.

The energy E in the¬†eiőł = ei¬∑(E¬∑t – p¬∑x)/ń߬†function is a scalar, so it doesn’t have any direction and we’ll measure it the same from both sides – as kinetic or potential energy or, more likely, by adding both. But… Well… Writing ei¬∑(E¬∑t – p¬∑x)/ń߬†or ei¬∑(E¬∑t +¬†p¬∑x)/ń߬†is not the same, right? No, it’s not. However, think of it as follows: we won’t be changing the direction of time, right? So it’s OK to not¬†change the sign of E. In fact, we can re-write the two expressions as follows:

  1. ei¬∑(E¬∑t – p¬∑x)/ń߬†= ei¬∑(E/ńß)¬∑t¬∑ei¬∑(p/ńß)¬∑x
  2. ei¬∑(E¬∑t +¬†p¬∑x)/ń߬†= ei¬∑(E/ńß)¬∑t¬∑ei¬∑(p/ńß)¬∑x

The first wavefunction describes some particle going in the positive¬†x-direction, while the second wavefunction describes some particle going in the negative x-direction, so… Well… That’s exactly¬†what we see in those two reference frames, so there is no issue whatsoever. ūüôā It’s just… Well… I just wanted to show the wavefunction¬†does¬†look different too when looking at something from another angle.

So why am I writing about this? Why am I being fussy? Well.. It’s just to show you that those transformations¬†are actually quite natural – just as natural as it is to see some particle go in one direction in one reference frame and see it go in the other in the other. ūüôā It also illustrates another point that I’ve been trying to make: the wavefunction is something¬†real. It’s not just a figment of our imagination. The real and imaginary part of our wavefunction have a precise geometrical meaning – and I explained what that might be in my more speculative posts, which I’ve brought together in the Deep Blue¬†page of this blog. But… Well… I can’t dwell on that here because… Well… You should read that page. ūüôā

The point to note is the following: we¬†do¬†have different wavefunctions in different reference frames, but these wavefunctions describe the same physical reality, and they also¬†do respect the symmetries we’d expect them to respect, except… Well… The¬†laws describing the¬†weak¬†force don’t, but¬†I wrote about that a very¬†long time ago, and it was¬†not¬†in the context of trying to explain the relatively simple basic laws of quantum mechanics. ūüôā If you’re interested, you should check out my post(s) on that or, else, just google¬†a bit. It’s really exciting stuff, but not something that will help you much to understand the basics, which is what we’re trying to do here. ūüôā

The second point to note is that those¬†transformations¬†of the wavefunction – or of quantum-mechanical states¬†–¬†which we go through when rotating our reference frame, for example – are really quite natural. There’s nothing special about them. We had such transformations in classical mechanics too! But… Well… Yes, I admit they do¬†look¬†complicated. But then that’s why you’re so fascinated and why you’re reading this blog, isn’t it? ūüôā

Post scriptum: It’s probably useful to be somewhat more precise on all of this. You’ll remember we visualized the wavefunction in some of our posts using the animation below. It uses a left-handed coordinate system, which is rather unusual but then it may have been made with a software which uses a left-handed coordinate system (like RenderMan, for example). Now the rotating arrow at the center moves with time and gives us the polarization of our wave. Applying our customary right-hand¬†rule,you can see this beam is left-circularly polarized. [I know… It’s quite confusing, but just go through the motions here and be consistent.]AnimationNow, you know that ei¬∑(p/ńß)¬∑x and ei¬∑(p/ńß)¬∑x¬†are each other’s complex conjugate:

  1. ei·k·x = cos(k·x) + i·sin(k·x)
  2. ei¬∑k¬∑x¬†=¬†cos(-k¬∑x) +¬†i¬∑sin(-k¬∑x) = cos(k¬∑x) ‚ąí¬†i¬∑sin(k¬∑x)

Their real part – the cosine function – is the same, but the imaginary part – the sine function – has the opposite sign.¬†So, assuming the direction of propagation is, effectively, the x-direction, then what’s the polarization of the mirror image? Well… The wave will now go from right to left, and its polarization… Hmm…¬†Well… What?¬†

Well… If you can’t figure it out, then just forget about those signs and just imagine you’re effectively looking at the same thing¬†from the backside. In fact, if you have a laptop, you can push the screen down and go around your computer. ūüôā There’s no shame in that. In fact, I did that just to make sure I am not¬†talking nonsense here. ūüôā If you look at this beam from the backside, you’ll effectively see it go from right to left – instead of from what you see on this side, which is a left-to-right direction. And as for its polarization… Well… The angular momentum vector swaps direction too but the beam is still¬†left-circularly polarized. So… Well… That’s consistent with what we wrote above. ūüôā The real world is real, and axial vectors are as real as polar vectors. This real¬†beam will only appear to be¬†right-circularly polarized¬†in a mirror. Now, as mentioned above, that mirror world is not¬†our¬†world. If it would exist – in some other Universe – then it would be made up of anti-matter. ūüôā

So… Well… Might it actually exist? Is there some other world made of anti-matter out there? I don’t know. We need to think about that reversal of ‘near’ and ‘far’ too: as mentioned, a mirror turns things inside out, so to speak. So what’s the implication of that? When we walk around¬†something – or do a¬†rotation¬†– then the reversal between ‘near’ and ‘far’ is something physical: we go near to what was far, and we go away from what was near. But so how would we get into our mirror world, so to speak? We may say that this¬†anti-matter world in the mirror is entirely possible, but then how would we get there? We’d need to turn ourselves, literally, inside out – like short of shrink to the zero point and then come back out of it to do that parity inversion along our line of sight. So… Well… I don’t see that happen, which is why I am a fan of the One World hypothesis. ūüôā So¬†I¬†think¬†the mirror world is just what it is: the mirror world. Nothing real. But… Then… Well… What do¬†you¬†think? ūüôā

Loose ends…

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 Рof course РI 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 ‚Äďj. So the amplitude here will be (‚Äďj)(‚Äďj) =¬†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 ‚Äď1/3 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 ‚Äď0.1. 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 ‚Äď0.1 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,¬†‚Äď0.1 = 0.1eiŌĬ†= 0.1(cosŌÄ + isinŌÄ).]

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 (őĪ). There is a remarkably easy formula for őĪ, which becomes even easier if we fiddle with units to simplify the matter even more. Let me paraphrase Wikipedia on őĪ¬†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¬†őĪ¬†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);¬†ń߬†=¬†h/2ŌÄ is the reduced Planck constant;¬†c¬†is the speed of light (in vacuum);¬†őĶ0¬†is the electric constant (i.e. the so-called¬†permittivity of free space);¬†¬Ķ0¬†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¬†őĶ0¬†,¬†c¬†and¬†ń߬†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 őĪ 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 ‚Äď0.1 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 őĪ under point 1 above), then we get an alpha equal to some 7.297√ó10‚Äď3. In fact, its value will usually be written as 1/őĪ, 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 = hőĹ) 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.

Absorption-1

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 Рfor all forces Рis 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, j3,¬†j4¬†etcetera (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, g3,¬†g4¬†etcetera do not become negligibly small. In fact, they don’t become irrelevant at all. Indeed, if we wrote őĪ for the electromagnetic force as¬†7.297√ó10‚Äď3, then the őĪ 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 őĪ¬†is much less than 1 [in a quantum field theory with a dimensionless coupling constant őĪ], then the theory is said to be¬†weakly coupled. In this case it is well described by an expansion in powers of őĪ¬†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 ¬Ī 40 years ‚Äėonly‚Äô and, hence, measuring how much carbon-14 is left in some organic substance allows us to date it (that‚Äôs what (radio)carbon-dating is about). Now, a¬†beta¬†particle¬†can refer to an electron¬†or¬†a positron, so we can have¬†ő≤‚Ästdecay (e.g. the above-mentioned carbon-14 decay) or¬†ő≤+¬†decay (e.g. magnesium-23 into sodium-23). If we have ő≤‚Ästdecay, then some electron will be flying out in order to make sure the atom as a whole stays electrically neutral. If it’s ő≤+¬†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. ūüôā

Now, all these ‘other particles’¬†are¬†unstable¬†but they survive long enough ‚Äď a¬†muon, for example,¬†disintegrates after¬†2.2 millionths of a second¬†(on average) ‚Äď to deserve the ‚Äėparticle‚Äô title, as opposed to a ‚Äėresonance‚Äô, 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 s¬†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 b¬†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.

Standard_Model_of_Elementary_Particles

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‚Ästand 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‚Äď 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‚Äôt like that for anything that disagrees with an experiment, they cook up an explanation‚Äďa 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.

Gravity

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‚Äď13¬†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‚Äď2¬†approximately. The strength of the weak force as¬†measured by the coupling constant would be smaller with a factor¬†10‚Äď13¬†(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‚Äď38. 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. ūüôā 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:¬†

‚ÄúI don‚Äôt like that they‚Äôre not calculating anything. I don‚Äôt like that they don‚Äôt check their ideas. I don‚Äôt like that for anything that disagrees with an experiment, they cook up an explanation‚Äďa fix-up to say, ‚ÄúWell, it might be true.‚ÄĚ For example, the theory requires ten dimensions. Well, maybe there‚Äôs a way of wrapping up six of the dimensions. Yes, that‚Äôs 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‚Äôs no reason whatsoever in superstring theory that it isn‚Äôt 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‚Äôt produce anything; it has to be excused most of the time. It doesn‚Äôt look right.‚ÄĚ

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. ūüôā