# Wavefunctions, perspectives, reference frames, representations and symmetries

Ouff ! This title is quite a mouthful, isn’t it? đ So… What’s the topic of the day? Well… In our previous posts, we developed a few key ideas in regard to a possible physical interpretation of the (elementary) wavefunction. It’s been an interesting excursion, and I summarized it in another pre-publication paper on the open arXiv.org site.

In my humble view, one of the toughest issues to deal with when thinking about geometric (orÂ physical) interpretations of the wavefunction is the fact that a wavefunction does not seem to obey the classical 360Â° symmetry in space. In this post, I want to muse a bit about this and show that… Well… It does and it doesn’t. It’s got to do with what happens when you change from one representational base (orÂ representation, tout court)Â to another which is… Well… Like changing the reference frame but, at the same time, it is also more than just a change of the reference frameâand so that explains the weird stuff (like that 720Â° symmetry of the amplitudes for spin-1/2 particles, for example).

I should warn you before you start reading: I’ll basically just pick up some statements from my paper (and previous posts) and develop some more thoughts on them. As a result, this post may not be very well structured. Hence, you may want to read the mentioned paperÂ first.

### The reality of directions

Huh? TheÂ realityÂ of directions? Yes. I warned you. This post may cause brain damage. đÂ The whole argument revolves around a thoughtÂ experimentâbut one whose results have been verified in zillions of experiments in university student labs so… Well… We do notÂ doubt the results and, therefore, we do not doubt the basic mathematical results: we just want to try to understandÂ them better.

So what is the set-up? Well… In the illustration below (Feynman, III, 6-3), Feynman compares the physics of two situations involving rather special beam splitters. Feynman calls them modified or âimprovedâ Stern-Gerlach apparatuses. The apparatus basically splits and then re-combines the two new beams along theÂ z-axis. It is also possible to block one of the beams, so we filter out only particles with their spinÂ upÂ or, alternatively, with their spinÂ down. Spin (or angular momentum or the magnetic moment) as measured along theÂ z-axis, of courseâI should immediately add: we’re talking theÂ z-axis of the apparatus here.

The two situations involve a different relative orientation of the apparatuses: in (a), the angle is 0Â°, while in (b) we have a (right-handed) rotation of 90Â° about the z-axis. He then provesâusing geometry and logic onlyâthat the probabilities and, therefore, the magnitudes of the amplitudes (denoted byÂ C+ and Câ and Câ+ and Cââ in the S and T representation respectively) must be the same, but the amplitudes must have different phases, notingâin his typical style, mixing academic and colloquial languageâthat âthere must be some way for a particle to tell that it has turned a corner in (b).â

The various interpretations of what actually happens here may shed some light on the heated discussions on the reality of the wavefunctionâand of quantum states. In fact, I should note that Feynman’s argument revolves around quantum states. To be precise, the analysis is focused on two-state systems only, and the wavefunctionâwhich captures a continuum of possible states, so to speakâis introduced only later. However, we may look at the amplitude for a particle to be in theÂ up– or down-state as a wavefunction and, therefore (but do note that’s my humble opinion once more), the analysis is actuallyÂ notÂ all that different.

We know, from theory and experiment, that the amplitudes are different. For example, for the given difference in the relative orientation of the two apparatuses (90Â°), we know that the amplitudes are given by Câ+ = eiâĎ/2âC+ = e iâĎ/4âC+ and Cââ = eâiâĎ/2âC+ = eâ iâĎ/4âCâ respectively (the amplitude to go from the down to the up state, or vice versa, is zero). Hence, yes, weânotÂ the particle, Mr. Feynman!âknowÂ that, in (b), the electron has, effectively, turned a corner.

The more subtle question here is the following: is the reality of the particle in the two setups the same? Feynman, of course, stays away from such philosophical question. He just notes that, while â(a) and (b) are differentâ, âthe probabilities are the sameâ. He refrains from making any statement on the particle itself: is or is it not the same? The common sense answer is obvious: of course, it is! The particle is the same, right? In (b), it just took a turnâso it is just going in some other direction. Thatâs all.

However, common sense is seldom a good guide when thinking about quantum-mechanical realities. Also, from a more philosophical point of view, one may argue that the reality of the particle is not the same: something mightâor mustâhave happened to the electron because, when everything is said and done, the particle did take a turn in (b). It did not in (a). [Note that the difference between âmightâ and âmustâ in the previous phrase may well sum up the difference between a deterministic and a non-deterministic world view but… Well… This discussion is going to be way too philosophical already, so let’s refrain from inserting new language here.]

Let us think this through. The (a) and (b) set-up are, obviously, different but…Â Wait a minute…Â Nothing is obvious in quantum mechanics, right? How can weÂ experimentally confirmÂ thatÂ they are different?

Huh?Â I must be joking, right? You canÂ seeÂ they are different, right? No.Â I am not joking. In physics, two things are different if we get differentÂ measurementÂ results. [That’s a bit of a simplified view of the ontological point of view of mainstream physicists, but you will have to admit I am not far off.] So… Well… We can’t see those amplitudes and so… Well… If we measure the same thingâsame probabilities, remember?âwhy are they different? Think of this: if we look at the two beam splitters as one singleÂ tube (anÂ ST tube, we might say), then all we did in (b) was bend the tube. Pursuing the logic that says our particle is still the sameÂ even when it takes a turn, we could say the tube is still the same, despite us having wrenched it over a 90Â° corner.

Now, I am sure you think I’ve just gone nuts, but just tryÂ to stick with me a little bit longer. Feynman actually acknowledges the same: we need to experimentallyÂ proveÂ (a) and (b) are different. He does so by getting aÂ thirdÂ apparatus in (U), as shown below, whose relative orientation to T is the same in both (a) and (b), so there is no difference there.

Now, the axis ofÂ UÂ is not theÂ z-axis: it is theÂ x-axis in (a), and theÂ y-axis in (b). So what? Well… I will quote Feynman hereânot (only) because his words are more important than mine but also because every word matters here:

“The two apparatuses in (a) and (b) are, in fact, different, as we can see in the following way. Suppose that we put an apparatus in front ofÂ SÂ which produces a pure +xÂ state. Such particles would be split into +z andÂ âz intoÂ beams inÂ S,Â but the two beams would be recombined to give aÂ +xÂ state again at P1âthe exit ofÂ S.Â The same thing happens again inÂ T.Â If we followÂ TÂ by a third apparatusÂ U,Â whose axis is in the +xÂ direction and, as shown in (a), all the particles would go into the +Â beam ofÂ U.Â Now imagine what happens ifÂ TÂ and UÂ are swung aroundÂ togetherÂ by 90Â°Â to the positions shown in (b).Â Again, theÂ TÂ apparatus puts out just what it takes in, so the particles that enterÂ UÂ are in a +xÂ stateÂ with respect toÂ S,Â which is different. By symmetry, we would now expect only one-half of the particles to get through.”

I should note that (b) shows theÂ UÂ apparatus wide open so… Well… I must assume that’s a mistake (and should alert the current editors of the LecturesÂ to it): Feynman’s narrative tells us we should also imagine it with theÂ minus channel shut. InÂ thatÂ case, it should, effectively, filter approximately half of the particles out, while they all get through in (a). So that’s aÂ measurementÂ result which shows the direction, as weÂ seeÂ it, makes a difference.

Now, Feynman would be very angry with meâbecause, as mentioned, he hates philosophersâbut I’d say: this experiment proves that a direction is something real. Of course, the next philosophical question then is: whatÂ isÂ a direction? I could answer this by pointing to the experiment above: a direction is something that alters the probabilities between the STU tube as set up in (a) versus the STU tube in (b). In factâbut, I admit, that would be pretty ridiculousâwe could use the varying probabilities as we wrench this tube over varying angles toÂ define an angle! But… Well… While that’s a perfectly logical argument, I agree it doesn’t sound very sensical.

OK. Next step. What follows may cause brain damage. đ Please abandon all pre-conceived notions and definitions for a while and think through the following logic.

You know this stuff is about transformations of amplitudes (or wavefunctions), right? [And you also want to hear about those special 720Â° symmetry, right? No worries. We’ll get there.] So the questions all revolve around this: what happens to amplitudes (or the wavefunction) when we go from one reference frameâorÂ representation, as it’s referred to in quantum mechanicsâto another?

Well… I should immediately correct myself here: a reference frame and a representation are two different things. They areÂ relatedÂ but… Well… Different… Quite different. Not same-same but different. đ I’ll explain why later. Let’s go for it.

Before talking representations, let us first think about what we reallyÂ mean by changing the reference frame. To change it, we first need to answer the question: what is our reference frame? It is a mathematical notion, of course, but then it is also more than that: it is ourÂ reference frame. We use it to make measurements. That’s obvious, you’ll say, but let me make a more formal statement here:

The reference frame is given by (1) the geometry (or theÂ shape, if that sounds easier to you) of the measurement apparatusÂ (so that’s the experimental set-up) here) and (2) our perspective of it.

If we would want to sound academic, we might refer to Kant and other philosophers here, who told usâ230 years agoâthat the mathematical idea of a three-dimensional reference frame is grounded in our intuitive notions of up and down, and left and right. [If you doubt this, think about the necessity of the various right-hand rules and conventions that we cannot do without in math, and in physics.] But so we do not want to sound academic. Let us be practical. Just think about the following.Â The apparatus gives us two directions:

(1) TheÂ upÂ direction, whichÂ weÂ associate with theÂ positive direction of theÂ z-axis, and

(2) the direction of travel of our particle, whichÂ we associateÂ with the positive direction of theÂ y-axis.

Now, if we have two axes, then the third axis (theÂ x-axis) will be given by the right-hand rule, right? So we may say the apparatus gives us the reference frame. Full stop.Â So… Well… Everything is relative? Is this reference frame relative? Are directions relative? That’s what you’ve been told, but think about this:Â relativeÂ to what?Â Here is where the object meets the subject. What’s relative? What’s absolute?Â Frankly, I’ve started to think that, in this particular situation, we should, perhaps, not use these two terms. I am notÂ saying thatÂ our observation of what physically happens here gives these two directions any absolute character but… Well… You will have to admit they are more than just some mathematical construct: when everything is said and done, we will have to admit that these two directions are real. because… Well… They’re part of theÂ realityÂ that we are observing, right? And the third one… Well… That’s given by our perspectiveâby our right-hand rule, which is… Well… OurÂ right-hand rule.

Of course, now you’ll say: if you think that ârelativeâ and âabsoluteâ are ambiguous terms and that we, therefore, may want to avoid them a bit more, then ârealâ and its opposite (unreal?) are ambiguous terms too, right? WellâŚ Maybe. What language would youÂ suggest? đ Just stick to the story for a while. I am not done yet. So… Yes… WhatÂ isÂ theirÂ reality?Â Let’s think about that in the next section.

### Perspectives, reference frames and symmetries

You’ve done some mental exercises already as you’ve been working your way through the previous section, but you’ll need to do plenty more. In fact, they may become physical exercise too: when I first thought about these things (symmetries and, more importantly, asymmetries in space), I found myself walking around the table with some asymmetrical everyday objects and papers with arrows and clocks and other stuff on itâeffectively analyzing what right-hand screw, thumb or grip rules actuallyÂ mean. đ

So… Well… I want you to distinguishâjust for a whileâbetween the notion of a reference frame (think of the xyz reference frame that comes with the apparatus) and yourÂ perspective on it. What’s our perspective on it? Well… You may be looking from the top, or from the side and, if from the side, from the left-hand side or the right-hand sideâwhich, if you think about it, you can only defineÂ in terms of the various positive and negative directions of the various axes. đÂ If you think this is getting ridiculous… Well… Don’t. Feynman himselfÂ doesn’t think this is ridiculous, because he starts his own “long and abstract side tour” on transformations with a very simple explanation of how the top and side view of the apparatus are related to theÂ axesÂ (i.e. the reference frame) that comes with it. You don’t believe me? This is theÂ very first illustration of hisÂ LectureÂ on this:

He uses it to explain the apparatus (which we don’t do here because you’re supposed to already know how these (modified or improved) Stern-Gerlach apparatuses work). So let’s continue this story. Suppose that we are looking in the positiveÂ y-directionâso thatâs the direction in which our particle is movingâthen we might imagine how it would look like whenÂ weÂ would make a 180Â°Â turn and look at the situation from the other side, so to speak. We do not change the reference frame (i.e. the orientation) of the apparatus here: we just change our perspective on it. Instead of seeing particles going away from us, into the apparatus, we now see particles comingÂ towardsÂ us, out of the apparatus.

What happensâbut that’s not scientific language, of courseâis that left becomes right, and right becomes left. Top still is top, and bottom is bottom. We are looking now in theÂ negativeÂ y-direction, and the positive direction of the x-axisâwhich pointed right when we were looking in the positiveÂ y-directionânow points left. I see you nodding your head nowâbecause you’ve heard about parity inversions, mirror symmetries and what have youâand I hear you say: “That’s the mirror world, right?”

No. It is not. I wrote about this in another post: the world in the mirror is theÂ world in the mirror. We don’t get a mirror image of an object by going around it and looking at its back side. I can’t dwell too much on this (just check that post, and another one who talks about the same), but so don’t try to connect it to the discussions on symmetry-breaking and what have you. Just stick toÂ this story, which is about transformations of amplitudes (or wavefunctions). [If you really want to knowâbut I know this sounds counterintuitiveâthe mirror world doesn’t really switch left for right. Your reflection doesn’t do a 180 degree turn: it is just reversed front to back, with no rotation at all. It’s only your brain which mentallyÂ adds (or subtracts) the 180 degree turn that you assume must have happened from the observed front to back reversal. So the left to right reversal is onlyÂ apparent. It’s a common misconception, and… Well… I’ll let you figure this out yourself. I need to move on.]Â Just note the following:

1. TheÂ xyzÂ reference frame remains a valid right-handed reference frame. Of course it does: it comes with our beam splitter, and we can’t change its reality, right? We’re just looking at it from another angle. OurÂ perspectiveÂ on it has changed.
2. However, if we think of the real and imaginary part of the wavefunction describing the electrons that are going through our apparatus as perpendicular oscillations (as shown below)âa cosine and sine function respectivelyâthen our change in perspectiveÂ might, effectively, mess up our convention for measuring angles.

I am not saying itÂ does. Not now, at least. I am just saying it might. It depends on the plane of the oscillation, as I’ll explain in a few moments. Think of this: we measure angles counterclockwise, right? As shown below… But… Well… If the thing below would be some funny clock going backwardsâyou’ve surely seen them in a bar or so, right?âthen… Well… If they’d be transparent, and you’d go around them, you’d see them as going… Yes… Clockwise. đ [This should remind you of a discussion on real versus pseudo-vectors, or polar versus axial vectors, but… Well… We don’t want to complicate the story here.]

Now, ifÂ we wouldÂ assume this clock represents something realâand, of course, I am thinking of theÂ elementary wavefunctionÂ eiÎ¸Â =Â cosÎ¸ +Â iÂˇsinÎ¸ nowâthen… Well… Then it will look different when we go around it. When going around our backwards clock above and looking at it from… Well… The back, we’d describe it, naively, as… Well…Â Think! What’s your answer? Give me the formula!Â đ

[…]

We’d see it asÂ eâiÎ¸Â =Â cos(âÎ¸) +Â iÂˇsin(âÎ¸) =Â cosÎ¸ âÂ iÂˇsinÎ¸, right? The hand of our clock now goes clockwise, so that’s theÂ oppositeÂ direction of our convention for measuring angles. Hence, instead ofÂ eiÎ¸, we writeÂ eâiÎ¸, right? So that’s the complex conjugate. So we’ve got a differentÂ imageÂ of the same thing here. Not good. Not good at all.

You’ll say: so what? We can fix this thing easily, right?Â YouÂ don’t need the convention for measuring angles or for the imaginary unit (i) here.Â This particle is moving, right? So if you’d want to look at the elementary wavefunction as some sort of circularly polarized beam (which, I admit, is very much what I would like to do, but its polarization is rather particular as I’ll explain in a minute), then you just need to define left- and right-handed angles as per the standard right-hand screw rule (illustrated below).Â To hell with the counterclockwise convention for measuring angles!

You are right. WeÂ couldÂ use the right-hand rule more consistently. We could, in fact, use it as anÂ alternativeÂ convention for measuring angles: we could, effectively, measure them clockwise or counterclockwise depending on the direction of our particle.Â But… Well… The fact is:Â we don’t. We do not use that alternative convention when we talk about the wavefunction. Physicists do use theÂ counterclockwiseÂ convention all of the time and just jot down these complex exponential functions and don’t realize that,Â if they are to represent something real, ourÂ perspectiveÂ on the reference frame matters. To put it differently, theÂ directionÂ in which we are looking at things matters! Hence, the direction is not…Â Well… I am tempted to say… NotÂ relative at all but then… Well… We wanted to avoid that term, right? đ

[…]

I guess that, by now, your brain may suffered from various short-circuits. If not, stick with me a while longer. Let us analyze how our wavefunction model might be impacted by this symmetryâorÂ asymmetry, I should say.

### The flywheel model of an electron

In our previous posts, we offered a model that interprets the real and the imaginary part of the wavefunction as oscillations which each carry half of the total energy of the particle. These oscillations are perpendicular to each other, and the interplay between both is how energy propagates through spacetime. Let us recap the fundamental premises:

1. The dimension of the matter-wave field vector is forceÂ per unit mass (N/kg), as opposed to the force per unit charge (N/C) dimension of the electric field vector. This dimension is an acceleration (m/s2), which is the dimension of the gravitational field.
2. We assume this gravitational disturbance causes our electron (or a charged massÂ in general) to move about some center, combining linear and circular motion. This interpretation reconciles the wave-particle duality: fields interfere but if, at the same time, they do drive a pointlike particle, then we understand why, as Feynman puts it, âwhen you do find the electron some place, the entire charge is there.â Of course, we cannot prove anything here, but our elegant yet simple derivation of the Compton radius of an electron is… Well… Just nice. đ
3. Finally, and most importantly in the context of this discussion, we noted that, in light of the direction of the magnetic moment of an electron in an inhomogeneous magnetic field, the plane which circumscribes the circulatory motion of the electron should also compriseÂ the direction of its linear motion. Hence, unlike an electromagnetic wave, theÂ planeÂ of the two-dimensional oscillation (so that’s the polarization plane, really) cannotÂ be perpendicular to the direction of motion of our electron.

Let’s say some more about the latter point here. The illustrations below (one from Feynman, and the other is just open-source) show what we’re thinking of.Â The direction of the angular momentum (and the magnetic moment) of an electronâor, to be precise, its component as measured in the direction of the (inhomogeneous) magnetic field through which our electron is travelingâcannotÂ be parallel to the direction of motion. On the contrary, it must be perpendicularÂ to the direction of motion. In other words, if we imagine our electron as spinning around some center (see the illustration on the left-hand side), then the disk it circumscribes (i.e. theÂ planeÂ of the polarization)Â has toÂ compriseÂ the direction of motion.

Of course, we need to add another detail here. As my readers will know, we do not really have a precise direction of angular momentum in quantum physics. While there is no fully satisfactory explanation of this, the classical explanationâcombined with the quantization hypothesisâgoes a long way in explaining this: an object with an angular momentumÂ JÂ and a magnetic momentÂ ÎźÂ that is not exactly parallel to some magnetic fieldÂ B, willÂ notÂ line up: it willÂ precessâand, as mentioned, the quantization of angular momentum may well explain the rest.Â [Well… Maybe… We haveÂ detailed our attempts in this regard in various posts on this (just search for spinÂ orÂ angular momentumÂ on this blog, and you’ll get a dozen posts or so), but these attempts are, admittedly, not fully satisfactory. Having said that, they do go a long way in relating angles to spin numbers.]

The thing is: we do assume our electron is spinning around. If we look from theÂ up-direction only, then it will be spinningÂ clockwise if its angular momentum is down (so itsÂ magnetic moment isÂ up). Conversely, it will be spinningÂ counterclockwise if its angular momentum isÂ up. Let us take theÂ up-state. So we have a top view of the apparatus, and we see something like this:I know you are laughing aloud now but think of your amusement as a nice reward for having stuck to the story so far. Thank you. đ And, yes, do check it yourself by doing some drawings on your table or so, and then look at them from various directions as you walk around the table asâI am not ashamed to admit thisâI did when thinking about this. So what do we get when we change the perspective? Let us walk around it, counterclockwise, let’s say, so we’re measuring our angle of rotation as someÂ positiveÂ angle.Â Walking around itâin whatever direction, clockwise or counterclockwiseâdoesn’t change the counterclockwise direction of our… Well… That weird object that mightâjust mightârepresent an electron that has its spin up and that is traveling in the positive y-direction.

When we look in the direction of propagation (so that’s from left to right as you’re looking at this page), and we abstract away from its linear motion, then we could, vaguely, describe this by some wrenchedÂ eiÎ¸Â =Â cosÎ¸ +Â iÂˇsinÎ¸ function, right? The x- andÂ y-axesÂ of the apparatus may be used to measure the cosine and sine components respectively.

Let us keep looking from the top but walk around it, rotating ourselves over a 180Â° angle so we’re looking in theÂ negativeÂ y-direction now. As I explained in one of those posts on symmetries, our mind will want to switch to a new reference frame: we’ll keep theÂ z-axis (up is up, and down is down), but we’ll want the positive direction of the x-axis to… Well… Point right. And we’ll want theÂ y-axis to point away, rather than towards us. In short, we have a transformation of the reference frame here:Â z’ =Â z,Â y’ = âÂ y, andÂ x’ =Â âÂ x. Mind you, this is still a regular right-handed reference frame. [That’s the difference with aÂ mirrorÂ image: aÂ mirroredÂ right-hand reference frame is no longer right-handed.]Â So, in our new reference frame, that we choose to coincide with ourÂ perspective,Â we will now describe the same thing as someÂ âcosÎ¸ âÂ iÂˇsinÎ¸ =Â âeiÎ¸Â function. Of course,Â âcosÎ¸ =Â cos(Î¸ +Â Ď) andÂ âsinÎ¸ =Â sin(Î¸ +Â Ď) so we can write this as:

âcosÎ¸ âÂ iÂˇsinÎ¸ =Â cos(Î¸ +Â Ď) +Â iÂˇsinÎ¸ =Â eiÂˇ(Î¸+Ď)Â =Â eiĎÂˇeiÎ¸Â = âeiÎ¸.

Sweet ! But… Well… First note this isÂ notÂ the complex conjugate:Â eâiÎ¸Â =Â cosÎ¸ âÂ iÂˇsinÎ¸Â â Â âcosÎ¸ âÂ iÂˇsinÎ¸ =Â âeiÎ¸. Why is that? Aren’t we looking at the same clock, but from the back? No. The plane of polarization is different. Our clock is more like those in Dali’s painting: it’s flat. đ And, yes, let me lighten up the discussion with that painting here. đ We need to haveÂ someÂ fun while torturing our brain, right?

So, because we assume the plane of polarization is different, we get anÂ âeiÎ¸Â function instead of aÂ eâiÎ¸Â function.

Let us now think about the eiÂˇ(Î¸+Ď)Â function. It’s the same asÂ âeiÎ¸Â but… Well… We walked around theÂ z-axis taking a full 180Â° turn, right? So that’s Ď in radians. So that’s the phase shiftÂ here. Hey!Â Try the following now. Go back and walk around the apparatus once more, but letÂ the reference frame rotate with us, as shown below. So we start left and look in the direction of propagation, and then we start moving about theÂ z-axis (which points out of this page, toward you, as you are looking at this), let’s say by some small angleÂ Îą. So we rotate the reference frame about theÂ z-axis byÂ Îą and… Well… Of course, ourÂ eiÂˇÎ¸Â now becomes anÂ ourÂ eiÂˇ(Î¸+Îą)Â function, right? We’ve just derived the transformation coefficient for a rotation about theÂ z-axis, didn’t we? It’s equal toÂ eiÂˇÎą, right? We get the transformed wavefunction in the new reference frame by multiplying the old one byÂ eiÂˇÎą, right? It’s equal toÂ eiÂˇÎąÂˇeiÂˇÎ¸Â =Â eiÂˇ(Î¸+Îą), right?

Well…

[…]

No. The answer is: no. TheÂ transformation coefficient is notÂ eiÂˇÎąÂ butÂ eiÂˇÎą/2. So we get an additional 1/2 factor in theÂ phase shift.

Huh?Â Yes.Â That’s what it is: when we change the representation, by rotating our apparatus over some angle Îą about the z-axis, then we will, effectively, get a new wavefunction, which will differ from the old one by a phase shift that is equal to onlyÂ half ofÂ the rotation angle only.

Huh?Â Yes. It’s even weirder than that. For a spin downÂ electron, the transformation coefficient is eâiÂˇÎą/2, so we get an additional minus sign in the argument.

Huh?Â Yes.

I know you are terribly disappointed, but that’s how it is. That’s what hampers an easy geometric interpretation of the wavefunction. Paraphrasing Feynman, I’d say that, somehow, our electron not only knows whether or not it has taken a turn, but it also knows whether or not it is moving away from us or, conversely, towards us.

[…]

But…Â Hey! Wait a minute! That’s it, right?Â

What? Well… That’s it! The electron doesn’t know whether it’s moving away or towards us. That’s nonsense. But… Well… It’s like this:

OurÂ eiÂˇÎąÂ coefficient describes a rotation of the reference frame. In contrast, theÂ eiÂˇÎą/2Â andÂ eâiÂˇÎą/2Â coefficients describe what happens when we rotate the T apparatus! Now thatÂ is a very different proposition.Â

Right! You got it! RepresentationsÂ and reference frames are different things.Â QuiteÂ different, I’d say: representations areÂ real, reference frames aren’tâbut then you don’t like philosophical language, do you? đÂ But think of it. When we just go about theÂ z-axis, a full 180Â°, but we don’t touch thatÂ T-apparatus, we don’t changeÂ reality. When we were looking at the electron while standing left to the apparatus, we watched the electrons going in and moving away from us, and when we go about theÂ z-axis, a full 180Â°, looking at it from the right-hand side, we see the electrons coming out, moving towards us. But it’s still the same reality. We simply change the reference frameâfrom xyz to x’y’z’ to be precise: we doÂ not changeÂ the representation.

In contrast, when we rotate theÂ TÂ apparatus over a full 180Â°, our electron now goes in the opposite direction. And whether that’s away or towards us, that doesn’t matter: it was going in one direction while traveling throughÂ S, and now it goes in the opposite directionârelative to the direction it was going in S, that is.

So what happens,Â really, when weÂ change the representation, rather than the reference frame? Well… Let’s think about that. đ

### Quantum-mechanical weirdness?

The transformation matrix for the amplitude of a system to be in anÂ upÂ orÂ downÂ state (and, hence, presumably, for a wavefunction) for a rotation about theÂ z-axis is the following one:

Feynman derives this matrix in a rather remarkable intellectualÂ tour de forceÂ in the 6th of hisÂ Lectures on Quantum Mechanics. So that’s pretty early on. He’s actually worried about that himself, apparently, and warns his students that “This chapter is a rather long and abstract side tour, and it does not introduce any idea which we will not also come to by a different route in later chapters. You can, therefore, skip over it, and come back later if you are interested.”

Well… That’s howÂ IÂ approached it. I skipped it, and didn’t worry about those transformations for quite a while. But… Well… You can’t avoid them. In some weird way, they are at the heart of the weirdness of quantum mechanics itself. Let us re-visit his argument. Feynman immediately gets that the whole transformation issue here is just a matter of finding an easy formula for that phase shift. Why? He doesn’t tell us. Lesser mortals like us must just assume that’s how the instinct of a genius works, right? đ So… Well… Because heÂ knowsâfrom experimentâthat the coefficient isÂ eiÂˇÎą/2Â instead of eiÂˇÎą, he just says the phase shiftâwhich he denotes by Îťâmust be someÂ proportionalÂ to the angle of rotationâwhich he denotes byÂ Ď rather than Îą (so as to avoid confusion with the EulerÂ angleÂ Îą). So he writes:

Îť =Â mÂˇĎ

Initially, he also tries the obvious thing: m should be one, right? SoÂ Îť = Ď, right? Well… No. It can’t be. Feynman shows why that can’t be the case by adding a third apparatus once again, as shown below.

Let me quote him here, as I can’t explain it any better:

“SupposeÂ TÂ is rotated byÂ 360Â°; then, clearly, it is right back at zero degrees, and we should haveÂ Câ+ = C+Â andÂ Cââ =Â CâÂ or,Â what is the same thing,Â eiÂˇmÂˇ2ĎÂ = 1. We get m =Â 1. [But no!]Â This argument is wrong!Â To see that it is, consider thatÂ TÂ is rotated byÂ 180Â°. If mÂ were equal to 1, we would have Câ+ =Â eiÂˇĎC+Â = âC+Â and Cââ =Â eâiÂˇĎCâÂ =Â âCâ. [Feynman works with statesÂ here, instead of the wavefunction of the particle as a whole. I’ll come back to this.] However, this is just theÂ originalÂ state all over again.Â BothÂ amplitudes are just multiplied byÂ â1Â which gives back the original physical system. (It is again a case of a common phase change.) This means that if the angle betweenÂ TÂ andÂ SÂ is increased to 180Â°, the system would be indistinguishable from the zero-degree situation, and the particles would again go through the (+)Â state of theÂ UÂ apparatus. AtÂ 180Â°, though, the (+)Â state of theÂ UÂ apparatus is theÂ (âx)Â state of the originalÂ SÂ apparatus. So a (+x)Â state would become aÂ (âx)Â state. But we have done nothing toÂ changeÂ the original state; the answer is wrong. We cannot haveÂ m = 1.Â We must have the situation that a rotation byÂ 360Â°, andÂ no smaller angleÂ reproduces the same physical state. This will happen ifÂ m = 1/2.”

The result, of course, is this weird 720Â° symmetry. While we get the same physics after a 360Â° rotation of the T apparatus, we doÂ notÂ get the same amplitudes. We get the opposite (complex) number:Â Câ+ =Â eiÂˇ2Ď/2C+Â = âC+Â and Cââ =Â eâiÂˇ2Ď/2CâÂ =Â âCâ. That’s OK, because… Well… It’s aÂ commonÂ phase shift, so it’s just like changing the origin of time. Nothing more. Nothing less. Same physics. Same reality. But… Well…Â Câ+ â Â âC+Â andÂ Cââ â Â âCâ, right? We only get our original amplitudes back if we rotate theÂ T apparatus two times, so that’s by a full 720 degreesâas opposed to the 360Â° we’d expect.

Now, space is isotropic, right? So this 720Â° business doesn’t make sense, right?

Well… It does and it doesn’t. We shouldn’t dramatize the situation. What’s the actual difference between a complex number and its opposite? It’s like x orÂ âx, or t and ât.Â I’ve said this a couple of times already again, and I’ll keep saying it many times more:Â NatureÂ surely can’t be bothered by how we measure stuff, right? In the positive or the negative directionâthat’s just our choice, right?Â OurÂ convention. So… Well… It’s just like thatÂ âeiÎ¸Â function we got when looking at theÂ same experimental set-up from the other side: ourÂ eiÎ¸Â and âeiÎ¸Â functions didÂ notÂ describe a different reality. We just changed our perspective. TheÂ reference frame. As such, the reference frame isn’tÂ real. The experimental set-up is. AndâI know I will anger mainstream physicists with thisâtheÂ representationÂ is. Yes. Let me say it loud and clear here:

A different representation describes a different reality.

In contrast, a different perspectiveâor a different reference frameâdoes not.

### Conventions

While you might have had a lot of trouble going through all of the weird stuff above, the point is: it isÂ notÂ all that weird. WeÂ canÂ understand quantum mechanics. And in a fairly intuitive way, really. It’s just that… Well… I think some of the conventions in physics hamper such understanding. Well… Let me be precise: one convention in particular, really. It’s that convention for measuring angles. Indeed, Mr. Leonhard Euler, back in the 18th century, might well be “the master of us all” (as Laplace is supposed to have said) but… Well… He couldn’t foresee how his omnipresent formulaâeiÎ¸Â =Â cosÎ¸ +Â iÂˇsinÎ¸âwould, one day, be used to representÂ something real: an electron, or any elementary particle, really. If he wouldÂ have known, I am sure he would have noted what I am noting here:Â NatureÂ can’t be bothered by our conventions. Hence, ifÂ eiÎ¸Â represents something real, thenÂ eâiÎ¸Â must also represent something real. [Coz I admire this genius so much, I can’t resist the temptation. Here’s his portrait. He looks kinda funny here, doesn’t he? :-)]

Frankly, he would probably have understood quantum-mechanical theory as easily and instinctively as Dirac, I think, and I am pretty sure he would have notedâand, if he would have known about circularly polarized waves, probably agreed toâthatÂ alternative convention for measuring angles: we could, effectively, measure angles clockwise or counterclockwise depending on the direction of our particleâas opposed to Euler’s ‘one-size-fits-all’ counterclockwise convention. But so we didÂ notÂ adopt that alternative convention because… Well… We want to keep honoring Euler, I guess. đ

So… Well… If we’re going to keep honoring Euler by sticking to that ‘one-size-fits-all’ counterclockwise convention, then I doÂ believe thatÂ eiÎ¸Â and eâiÎ¸Â represent twoÂ differentÂ realities: spin up versus spin down.

Yes. In our geometric interpretation of the wavefunction, these are, effectively, two different spin directions. And… Well… These are real directions: we seeÂ something different when they go through a Stern-Gerlach apparatus. So it’s not just some convention toÂ countÂ things like 0, 1, 2, etcetera versus 0,Â â1,Â â2 etcetera. It’s the same story again: different but relatedÂ mathematicalÂ notions are (often) related to different but relatedÂ physicalÂ possibilities. So… Well… I think that’s what we’ve got here.Â Think of it. Mainstream quantum math treats all wavefunctions as right-handed but… Well…Â A particle with up spin is a different particle than one withÂ downÂ spin, right? And, again,Â NatureÂ surely cannotÂ be bothered about our convention of measuring phase angles clockwise or counterclockwise, right? So… Well… Kinda obvious, right? đ

Let me spell out my conclusions here:

1. The angular momentum can be positive or, alternatively, negative: J = +Ä§/2 orÂ âÄ§/2. [Let me note that this is not obvious. Or less obvious than it seems, at first. In classical theory, you would expect an electron, or an atomic magnet, to line up with the field. Well… The Stern-Gerlach experiment shows they don’t: they keep their original orientation. Well… If the field is weak enough.]

2. Therefore, we would probably like to think that an actual particleâthink of an electron, or whatever other particle you’d think ofâcomes in twoÂ variants:Â right-handed and left-handed. They will, therefore,Â either consist of (elementary) right-handed waves or,Â else, (elementary) left-handed waves. An elementary right-handed wave would be written as: Ď(Î¸i)Â = eiÎ¸iÂ = aiÂˇ(cosÎ¸i + iÂˇsinÎ¸i). In contrast,Â an elementary left-handed wave would be written as: Ď(Î¸i)Â =Â eâiÎ¸iÂ = aiÂˇ(cosÎ¸i â iÂˇsinÎ¸i).Â So that’s the complex conjugate.

So… Well… Yes, I think complex conjugates are not just someÂ mathematicalÂ notion: I believe they represent something real. It’s the usual thing:Â NatureÂ has shown us that (most) mathematical possibilities correspond to realÂ physical situations so… Well… Here you go. It is reallyÂ just like the left- or right-handed circular polarization of an electromagnetic wave: we can have both for the matter-wave too! [As for the differencesâdifferent polarization plane and dimensions and what have youâI’ve already summed those up, so I won’t repeat myself here.]Â The point is: ifÂ we have two differentÂ physicalÂ situations, we’ll want to have two different functions to describe it. Think of it like this: why would we haveÂ twoâyes, I admit, two relatedâamplitudes to describe the upÂ or downÂ state of the same system, but only one wavefunction for it?Â You tell me.

[…]

Authors like me are looked down upon by the so-called professional class of physicists. The few who bothered to react to my attempts to make sense of Einstein’s basic intuition in regard to the nature of the wavefunction all said pretty much the same thing: “Whatever your geometric (orÂ physical) interpretation of the wavefunction might be, it won’t be compatible with theÂ isotropyÂ of space. You cannot imagineÂ an object with a 720Â° symmetry. That’sÂ geometrically impossible.”

Well… Almost three years ago, I wrote the following on this blog: “As strange as it sounds, aÂ spin-1/2 particle needsÂ twoÂ full rotations (2Ă360Â°=720Â°) until it is again in the same state. Now, in regard to that particularity, youâll often read something like: âThere isÂ nothingÂ in our macroscopic world which has a symmetry like that.â Or, worse, âCommon sense tells us that something like that cannot exist, that it simply is impossible.â [I wonât quote the site from which I took this quotes, because it is, in fact, the site of a very respectable Â research center!]Â Bollocks!Â TheÂ Wikipedia article on spinÂ has this wonderful animation: look at how the spirals flip between clockwise and counterclockwise orientations, and note that itâs only after spinning a full 720 degrees that this âpointâ returns to its original configuration after spinning a full 720 degrees.

So… Well… I am still pursuing my original dream which is… Well… Let me re-phrase what I wrote back in January 2015:

Yes, weÂ canÂ actually imagine spin-1/2 particles, and we actually do not need all that much imagination!

In fact, I am tempted to think that I’ve found a pretty good representation or… Well… A pretty goodÂ image, I should say, because… Well… A representation is something real, remember? đ

Post scriptum (10 December 2017):Â Our flywheel model of an electron makes sense, but also leaves many unanswered questions. The most obvious one question, perhaps, is: why theÂ upÂ andÂ downÂ state only?

I am not so worried about that question, even if I can’t answer it right away because… Well… Our apparatusâthe way weÂ measureÂ realityâis set up to measure the angular momentum (or the magnetic moment, to be precise) in one direction only. If our electron isÂ capturedÂ by someÂ harmonicÂ (or non-harmonic?) oscillation in multiple dimensions, then it should not be all that difficult to show its magnetic moment is going to align, somehow, in the same or, alternatively, the opposite direction of the magnetic field it is forced to travel through.

Of course, the analysis for the spinÂ upÂ situation (magnetic moment down) is quite peculiar: if our electron is aÂ mini-magnet, why would itÂ notÂ line up with the magnetic field? We understand the precession of a spinning top in a gravitational field, but…Â Hey… It’s actually not that different. Try to imagine some spinning top on the ceiling. đ I am sure we can work out the math. đ The electron must be some gyroscope, really: it won’t change direction. In other words, its magnetic moment won’t line up. It will precess, and it can do so in two directions, depending on its state. đ […] At least, that’s why my instinct tells me. I admit I need to work out the math to convince you. đ

The second question is more important. If we just rotate the reference frame over 360Â°, we see the same thing: some rotating object which we, vaguely, describe by someÂ e+iÂˇÎ¸Â functionâto be precise, I should say: by some Fourier sum of such functionsâor, if the rotation is in the other direction, by someÂ eâiÂˇÎ¸Â function (again, you should read: aÂ FourierÂ sum of such functions). Now, the weird thing, as I tried to explain above is the following: if we rotate the object itself, over the sameÂ 360Â°, we get aÂ differentÂ object: ourÂ eiÂˇÎ¸Â andÂ eâiÂˇÎ¸Â function (again: think of aÂ FourierÂ sum, so that’s a waveÂ packet, really) becomes aÂ âeÂąiÂˇÎ¸Â thing. We get aÂ minusÂ sign in front of it.Â So what happened here? What’s the difference, really?

Well… I don’t know. It’s very deep. If I do nothing, and you keep watching me while turning around me, for a fullÂ 360Â°, then you’ll end up where you were when you started and, importantly, you’ll see the same thing.Â ExactlyÂ the same thing: if I was anÂ e+iÂˇÎ¸Â wave packet, I am still anÂ anÂ e+iÂˇÎ¸Â wave packet now. OrÂ if I was an eâiÂˇÎ¸Â wave packet, then I am still anÂ an eâiÂˇÎ¸Â wave packet now. Easy. Logical. Obvious, right?

But so now we try something different:Â IÂ turn around, over a fullÂ 360Â° turn, and youÂ stay where you are. When I am back where I wasâlooking at you again, so to speakâthen… Well… I am not quite the same any more. Or… Well… Perhaps I am but youÂ seeÂ me differently. If I wasÂ e+iÂˇÎ¸Â wave packet, then I’ve become aÂ âe+iÂˇÎ¸Â wave packet now. Not hugely different but… Well… ThatÂ minusÂ sign matters, right? OrÂ If I wasÂ wave packet built up from elementaryÂ aÂˇeâiÂˇÎ¸Â waves, then I’ve become aÂ âeâiÂˇÎ¸Â wave packet now. What happened?

It makes me think of the twin paradox in special relativity. We know it’s aÂ paradoxâso that’s anÂ apparentÂ 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 someÂ force.

Can we relate this to the twin paradox? Maybe. Note that aÂ minusÂ sign in front of theÂ eâÂąiÂˇÎ¸Â 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 180Â°: âcosÎ¸ =Â cos(Î¸ Âą Ď) andÂ âsinÎ¸ =Â sin(Î¸ Âą Ď). Now, adding or subtracting aÂ commonÂ phase factor to/from the argument of the wavefunction amounts toÂ changingÂ the origin of time. So… Well… I do think the twin paradox and this rather weird business of 360Â° and 720Â° symmetries are, effectively, related. đ

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