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Ï/2âC+ = e iâÏ/4âC+ and Cââ = eâiâÏ/2âC+ = eâ iâÏ/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Š 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 eiÎžÂ =Â cosÎž +Â iÂ·sinÎž 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iÎžÂ =Â cos(âÎž) +Â iÂ·sin(âÎž) =Â cosÎž âÂ iÂ·sinÎž, right? The hand of our clock now goes clockwise, so that’s theÂ oppositeÂ direction of our convention for measuring angles. Hence, instead ofÂ eiÎž, we writeÂ eâiÎž, 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 ÎŒÂ 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 eiÎžÂ =Â cosÎž +Â iÂ·sinÎž 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cosÎž âÂ iÂ·sinÎž =Â âeiÎžÂ function. Of course,Â âcosÎž =Â cos(Îž +Â Ï) andÂ âsinÎž =Â sin(Îž +Â Ï) so we can write this as:

âcosÎž âÂ iÂ·sinÎž =Â cos(Îž +Â Ï) +Â iÂ·sinÎž =Â eiÂ·(Îž+Ï)Â =Â eiÏÂ·eiÎžÂ = âeiÎž.

Sweet ! But… Well… First note this isÂ notÂ the complex conjugate:Â eâiÎžÂ =Â cosÎž âÂ iÂ·sinÎžÂ â Â âcosÎž âÂ iÂ·sinÎž =Â âeiÎž. 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eiÎžÂ function instead of aÂ eâiÎžÂ function.

Let us now think about the eiÂ·(Îž+Ï)Â function. It’s the same asÂ âeiÎžÂ but… Well… We walked around theÂ z-axis taking a full 180Â° turn, right? So that’s Ï 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 Î±. So we rotate the reference frame about theÂ z-axis byÂ Î± and… Well… Of course, ourÂ eiÂ·ÎžÂ now becomes anÂ ourÂ eiÂ·(Îž+Î±)Â function, right? We’ve just derived the transformation coefficient for a rotation about theÂ z-axis, didn’t we? It’s equal toÂ eiÂ·Î±, right? We get the transformed wavefunction in the new reference frame by multiplying the old one byÂ eiÂ·Î±, right? It’s equal toÂ eiÂ·Î±Â·eiÂ·ÎžÂ =Â eiÂ·(Îž+Î±), right?

Well…

[…]

No. The answer is: no. TheÂ transformation coefficient is notÂ eiÂ·Î±Â butÂ eiÂ·Î±/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 Î± 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iÂ·Î±/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 eiÂ·Î±Â coefficient describes a rotation of the reference frame. In contrast, theÂ eiÂ·Î±/2Â andÂ eâiÂ·Î±/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 eiÂ·Î±/2Â instead of eiÂ·Î±, he just says the phase shiftâwhich he denotes by Î»âmust be someÂ proportionalÂ to the angle of rotationâwhich he denotes byÂ Ï rather than Î± (so as to avoid confusion with the EulerÂ angleÂ Î±). So he writes:

Î» =Â mÂ·Ï

Initially, he also tries the obvious thing: m should be one, right? SoÂ Î» = Ï, 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 eiÂ·mÂ·2Ï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 eiÂ·ÏC+Â = âC+Â and Cââ =Â eâiÂ·Ï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 eiÂ·2Ï/2C+Â = âC+Â and Cââ =Â eâiÂ·2Ï/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eiÎžÂ function we got when looking at theÂ same experimental set-up from the other side: ourÂ eiÎžÂ and âeiÎžÂ 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eiÎžÂ =Â cosÎž +Â iÂ·sinÎžâ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 eiÎžÂ represents something real, thenÂ eâiÎžÂ 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 eiÎžÂ and eâiÎžÂ 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: Ï(Îži)Â = eiÎžiÂ = aiÂ·(cosÎži + iÂ·sinÎži). In contrast,Â an elementary left-handed wave would be written as: Ï(Îži)Â =Â eâiÎžiÂ = aiÂ·(cosÎži â iÂ·sinÎži).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+iÂ·ÎžÂ 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iÂ·ÎžÂ 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 eiÂ·ÎžÂ andÂ eâiÂ·ÎžÂ function (again: think of aÂ FourierÂ sum, so that’s a waveÂ packet, really) becomes aÂ âeÂ±iÂ·ÎžÂ 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+iÂ·ÎžÂ wave packet, I am still anÂ anÂ e+iÂ·ÎžÂ wave packet now. OrÂ if I was an eâiÂ·ÎžÂ wave packet, then I am still anÂ an eâiÂ·ÎžÂ 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+iÂ·ÎžÂ wave packet, then I’ve become aÂ âe+iÂ·ÎžÂ wave packet now. Not hugely different but… Well… ThatÂ minusÂ sign matters, right? OrÂ If I wasÂ wave packet built up from elementaryÂ aÂ·eâiÂ·ÎžÂ waves, then I’ve become aÂ âeâiÂ·ÎžÂ 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Â±iÂ·ÎžÂ 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cosÎž =Â cos(Îž Â± Ï) andÂ âsinÎž =Â sin(Îž Â± Ï). 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

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here:

Transforming amplitudes for spin-1/2 particles

Pre-script (dated 26 June 2020): This post got mutilated by the removal of some material by the dark force. You should be able to follow the main story line, however. If anything, the lack of illustrations might actually help you to think things through for yourself. In any case, we now have different views on these concepts as part of our realist interpretation of quantum mechanics, so we recommend you read our recent papers instead of these old blog posts.

Original post:

Some say it is not possibleÂ to fullyÂ understandÂ quantum-mechanical spin. Now, I do agree it is difficult, but I do notÂ believe it is impossible. That’s why I wrote so many posts on it. Most of these focused on elaborating how the classical view of how a rotating charge precesses in a magnetic field might translate into the weird world of quantum mechanics. Others were more focused on the corollary of theÂ quantizationÂ of the angular momentum, which is that, in the quantum-mechanical world, the angular momentum is never quite all in one direction onlyâso that explains some of the seemingly inexplicable randomness in particle behavior.

Frankly, I think those explanations help us quite a bit already but… Well… We need to go the extra mile, right? In fact, that’s drives my search for aÂ geometric (orÂ physical)Â interpretation of the wavefunction: the extra mile. đ

Now, in one of these many posts on spin and angular momentum, I advise my readers –Â you, that isÂ – to try to work yourself through Feynman’s 6th Lecture on quantum mechanics, which is highly abstract and, therefore, usually skipped. [Feynman himself told his students to skip it, so I am sure that’s what they did.] However, if we believe theÂ physicalÂ (orÂ geometric) interpretation of the wavefunction that we presented in previous posts is, somehow,Â true, then we need to relate it to the abstract math of these so-calledÂ transformationsÂ between representations.Â That’s what we’re going to try to do here. It’s going to be just a start, and I will probably end up doing several posts on this but… Well… We do have to start somewhere, right? So let’s see where we get today. đ

The thought experiment that Feynman uses throughout his LectureÂ makes use of what Feynman’s refers to as modified or improved Stern-Gerlach apparatuses. They allow us to prepare a pure state or, alternatively, as Feynman puts it, to analyzeÂ a state. In theory, that is. The illustration below present a side and top view of such apparatus. We may already note that the apparatus itselfâor, to be precise, ourÂ perspectiveÂ of itâgives us two directions: (1) theÂ upÂ direction, so that’s the positive direction of the z-axis, and (2) the direction of travel of our particle, which coincides with the positive direction of theÂ y-axis. [This is obvious and, at the same time, not so obvious, but I’ll talk about that in my next post. In this one, we basically need to work ourselves through the math, so we don’t want to think too much about philosophical stuff.]

The kind of questions we want to answer in this post are variants of the following basic one: if a spin-1/2 particle (let’s think of an electron here, even if the Stern-Gerlach experiment is usually done with an atomic beam) was prepared in a given condition by one apparatus S, say the +SÂ state,Â what is the probability (or theÂ amplitude) that it will get through aÂ second apparatus TÂ if that was set to filter out the +TÂ state?

The result will, of course, depend on the angles between the two apparatuses S and T, as illustrated below. [Just to respect copyright, I should explicitly note here that all illustrations are taken from the mentioned Lecture, and that the line of reasoning sticks close to Feynman’s treatment of the matter too.]

We should make a few remarks here. First, this thought experiment assumes our particle doesn’t get lost. That’s obvious but… Well… If you haven’t thought about this possibility, I suspect you will at some point in time. So we do assume that, somehow, this particle makes a turn. It’s an important point because… Well… Feynman’s argumentâwho, remember, represents mainstream physicsâsomehow assumes that doesn’t really matter. It’s the same particle, right? It just took a turn, so it’s going in some other direction. That’s all, right? Hmm… That’s where I part ways with mainstream physics: the transformation matrices for the amplitudes that we’ll find here describe something real, I think. It’s not justÂ perspective: somethingÂ happenedÂ to the electron. That something does not onlyÂ changeÂ the amplitudes but… Well… It describes a different electron. It describes an electron that goes in a different direction now. But… Well… As said, these are reflections I will further develop in my next post. đ Let’s focus on the math here. The philosophy will follow later. đÂ Next remark.

Second, we assume theÂ (a) and (b) illustrations above represent the sameÂ physicalÂ reality because the relative orientation between the two apparatuses, as measured by the angle Î±, is the same. NowÂ thatÂ isÂ obvious, you’ll say, but, as Feynman notes, we can only make that assumption because experiments effectively confirm that spacetime is, effectively, isotropic. In other words, there is noÂ aetherÂ allowing us to establish some sense of absoluteÂ direction. Directions areÂ relativeârelative to the observer, that is… But… Well… Again, in my next post, I’ll argue that it’sÂ notÂ because directions areÂ relativeÂ that they are, somehow,Â notÂ real. Indeed, in my humble opinion, it does matter whether an electron goes here or, alternatively, there. These twoÂ differentÂ directions are not just two different coordinate frames. But… Well… Again. The philosophy will follow later. We need to stay focused on the math here.

Third and final remark. This one is actually very tricky. In his argument, FeynmanÂ also assumes the two set-ups below are, somehow,Â equivalent.

You’ll say: Huh?Â If not, say it!Â Huh? đÂ Yes. Good.Â Huh? Feynman writesÂ equivalentânotÂ the same because… Well… They’re not the same, obviously:

1. In the first set-up (a), TÂ is wide open, so the apparatus is not supposed to do anything with the beam: it just splits and re-combines it.
2. In set-up (b) theÂ TÂ apparatus is, quite simply,Â not there, so… Well… Again. Nothing is supposed to happen with our particles as they come out ofÂ S and travel toÂ U.

TheÂ fundamental idea here is that our spin-1/2 particle (again, think of an electron here) enters apparatus U in the same state as it left apparatus S. In both set-ups, that is!Â Now that is aÂ very tricky assumption, because… Well… While the netÂ turn of our electron is the same, it is quite obvious it has to takeÂ twoÂ turns to get to U in (a), while it only takesÂ oneÂ turn in (b). And so… Well… You can probably think of other differences too.Â So… Yes. And no.Â Same-same but different, right? đ

Right. That isÂ why Feynman goes out of his way to explain the nitty-gritty behind: he actually devotes a full page in small print on this, which I’ll try to summarize in just a few paragraphs here. [And, yes, you should check my summary against Feynman’s actual writing on this.] It’s like this. While traveling through apparatus TÂ in set-up (a), time goes by and, therefore, the amplitude would be different by someÂ phase factorÂ ÎŽ. [Feynman doesn’t say anything about this, but… Well… In the particle’s own frame of reference, this phase factor depend on the energy, the momentum and the time and distance traveled. Think of the argument of the elementary wavefunction here:Â Îž = (Eât âÂ pâx)/Ä§).]Â Now, if we believe that the amplitude is just some mathematical constructâso that’s what mainstream physicists (not me!) believeâthen weÂ couldÂ effectively say that the physics of (a) and (b) are the same, as Feynman does. In fact, let me quote him here:

“TheÂ physicsÂ of set-up (a) and (b) should be the same but the amplitudes could be different by some phase factor without changing the result of any calculation about the real world.”

Hmm… It’s one of those mysterious short passages where we’d all like geniuses like Feynman (or Einstein, or whomever) to be more explicit on their world view: if the amplitudes are different, can theÂ physicsÂ really be the same? I mean…Â ExactlyÂ the same? It all boils down to that unfathomable belief that, somehow, the particle is real but the wavefunction thatÂ describesÂ it, is not.Â Of course, I admit that it’s true that choosing another zero point for the time variable would also change all amplitudes by a common phase factor and… Well… That’s something that I consider to beÂ notÂ real. But… Well… The time and distance traveled in theÂ TÂ apparatus is the time and distance traveled in theÂ TÂ apparatus, right?

Bon…Â I have to stay away from these questions as for nowâwe need to move on with the math hereâbut I will come back to it later. But… Well… Talking math, I should note a very interesting mathematical point here. We have these transformation matrices for amplitudes, right? Well… Not yet. In fact, the coefficient of these matrices are exactly what we’re going to try toÂ derive in this post, but… Well… Let’s assume we know them already. đ So we have a 2-by-2 matrix to go from S to T, from T to U, and then one to go from S to U without going through T, which we can write as RST,Â  RTU,Â  andÂ RSUÂ respectively. Adding the subscripts for theÂ baseÂ states in each representation, theÂ equivalenceÂ between the (a) and (b) situations can then be captured by the following formula:

So we have that phase factor here: the left- and right-hand side of this equation is, effectively, same-same but different, as they would say in Asia. đ Now, Feynman develops a beautiful mathematical argument to show that theÂ eiÎŽÂ factor effectively disappears if weÂ convertÂ our rotation matrices to some rather specialÂ form that is defined as follows:

I won’t copy his argument here, but I’d recommend you go over it because it is wonderfully easy to follow and very intriguing at the same time. [Yes. Simple things can beÂ very intriguing.] Indeed, the calculation below shows that theÂ determinantÂ of theseÂ specialÂ rotation matrices will be equal to 1.

So… Well… So what? You’re right. I am being sidetracked here. The point is that, if we put all of our rotation matrices in this special form, theÂ eiÎŽÂ factor vanishes and the formula above reduces to:

So… Yes. End of excursion.Â Let us remind ourselves of what it is that we are trying to do here. As mentioned above, the kind of questions we want to answer will be variants of the following basic one: if a spin-1/2 particle was prepared in a given condition by one apparatus (S), say the +SÂ state,Â what is the probability (or theÂ amplitude) that it will get through aÂ second apparatus (T) if that was set to filter out the +TÂ state?

We said the result would depend on the angles between the two apparatuses S and T. I wrote: anglesâplural. Why? Because a rotation will generally be described by the three so-calledÂ Euler angles:Â  Î±, ÎČ and Îł. Now, it is easy to make a mistake here, because there is a sequence to these so-calledÂ elemental rotationsâand right-hand rules, of courseâbut I will let you figure that out. đ

The basic idea is the following: if we can work out the transformation matrices for each of theseÂ elementalÂ rotations, then we can combine them and find the transformation matrix forÂ anyÂ rotation. So… Well… That fills most of Feynman’sÂ LectureÂ on this, so we don’t want to copy all that. We’ll limit ourselves to the logic for a rotation about the z-axis, and then… Well… You’ll see. đ

So… TheÂ z-axis… We take that to be the direction along which we are measuring the angular momentum of our electron, so that’s the direction of the (magnetic) field gradient, so that’s theÂ up-axis of the apparatus. In the illustration below, that direction pointsÂ out of the page, so to speak, because it is perpendicular to the direction of the x– and the y-axis that are shown. Note that the y-axis is the initial direction of our beam.

Now, because the (physical) orientation of the fields and the field gradients of S and T is the same, Feynman says thatâdespite the angleâtheÂ probabilityÂ for a particle to beÂ upÂ orÂ downÂ with regard toÂ SÂ andÂ T respectively should be the same. Well… Let’s be fair. He does not onlyÂ sayÂ that: experimentÂ showsÂ it to be true. [Again, I am tempted to interject here that it isÂ notÂ because the probabilities for (a) and (b) are the same, that theÂ realityÂ of (a) and (b) is the same, but… Well… You get me. That’s for the next post. Let’s get back to the lesson here.]Â The probability is, of course, the square of theÂ absolute valueÂ of the amplitude, which we will denote asÂ C+,Â Câ, C’+, andÂ C’âÂ respectively. Hence, we can write the following:

Now, theÂ absolute values (or the magnitudes)Â are the same, but theÂ amplitudes may differ. In fact, theyÂ mustÂ be different by some phase factor because, otherwise, we would not be able to distinguish the two situations, which are obviously different. As Feynman, finally, admits himselfâjokingly or seriously: “There must be some way for a particle to know that it has turned the corner at P1.” [P1Â is the midwayÂ pointÂ betweenÂ SÂ andÂ TÂ in the illustration, of courseânot some probability.]

So… Well… We write:

C’+Â =Â eiÎ»Â Â·C+Â andÂ C’âÂ =Â eiÎŒÂ Â·Câ

Now, Feynman notes that anÂ equal phase change in all amplitudes has no physical consequence (think of re-defining our t0Â = 0 point), so we can add some arbitrary amount to bothÂ Î» and ÎŒ without changing any of the physics. So then we canÂ chooseÂ this amount asÂ â(Î» + ÎŒ)/2. We write:

Now, it shouldn’t you too long to figure out thatÂ Î»’ is equal toÂ Î»’ =Â Î»/2 + ÎŒ/2 =Â âÎŒ’. So… Well… Then we can just adopt the convention thatÂ Î» = âÎŒ. So ourÂ C’+Â =Â eiÎ»Â Â·C+Â andÂ C’âÂ =Â eiÎŒÂ Â·CâÂ equations can now be written as:

C’+Â =Â eiÎ»Â Â·C+Â andÂ C’âÂ =Â eâiÎ»Â·Câ

The absolute values are the same, but theÂ phasesÂ are different. Right. OK. Good move. What’s next?

Well… The next assumption is that the phase shiftÂ Î» is proportional to the angle (Î±) between the two apparatuses. Hence,Â Î» is equal to Î» =Â mÂ·Î±, and we can re-write the above as:

C’+Â =Â eimÎ±Â·C+Â andÂ C’âÂ =Â eâimÎ±Â·Câ

Now, this assumption may or may not seem reasonable. Feynman justifies it with a continuity argument, arguing any rotation can be built up as a sequence of infinitesimal rotations and… Well… Let’s not get into the nitty-gritty here. [If you want it, check Feynman’s Lecture itself.] Back to the main line of reasoning. So we’ll assume weÂ canÂ writeÂ Î» as Î» =Â mÂ·Î±. The next question then is:Â what is the value for m? Now, we obviously do get exactly the same physicsÂ if we rotateÂ TÂ by 360Â°, or 2Ï radians. So weÂ mightÂ conclude that the amplitudes should be the same and, therefore, that eimÎ±Â =Â eimÂ·2ÏÂ has to be equal to one, soÂ C’+Â =Â C+Â andÂ C’âÂ =Â Câ . That’s the case if m is equal to 1. But… Well… No. It’s the same thing again: theÂ probabilities (or theÂ magnitudes)Â have to be the same, but the amplitudes may be different because of some phase factor. In fact, theyÂ should be different. If m = 1/2, then we also get the same physics, even if the amplitudes areÂ notÂ the same. They will be each other’s opposite:

Huh?Â Yes. Think of it. The coefficient of proportionality (m) cannot be equal to 1. If it would be equal to 1, and we’d rotateÂ TÂ by 180Â° only, then we’d also get thoseÂ C’+Â =Â âC+Â andÂ C’âÂ =Â âCâÂ equations, and so these coefficients would, therefore,Â also describeÂ the same physical situation. Now, you will understand,Â intuitively, that a rotation of theÂ TÂ apparatusÂ byÂ 180Â° willÂ notÂ give us the sameÂ physicalÂ situation… So… Well… In case you’d want a more formal argument proving a rotation by 180Â° does not give us the same physical situation, Feynman has one for you. đ

I know that, by now, you’re totally tired and bored, and so you only want the grand conclusion at this point. Well… All of what I wrote above should, hopefully, help you to understand that conclusion, which â I quote Feynman here â is the following:

If we know the amplitudesÂ C+Â andÂ CâÂ of spin one-half particles with respect to a reference frame S, and we then use new base states, defined with respect to a reference frameÂ TÂ which is obtained from S byÂ a rotationÂ Î± around theÂ z-axis, the new amplitudes are given in terms of the old by the following formulas:

[Feynman denotes our angleÂ Î± byÂ phi (Ï) because… He uses the Euler angles a bit differently. But don’t worry: it’s the same angle.]

What about the amplitude to go from theÂ CâÂ to theÂ C’+Â state, and from theÂ C+Â to the C’âÂ state? Well… That amplitude is zero. So the transformation matrix is this one:

Let’s take a moment and think about this. Feynman notes the following, among other things:Â “It is very curious to say that if you turn the apparatus 360Â° you get new amplitudes. [They aren’t really new, though, because the common change of sign doesn’t give any different physics.] But if something has been rotated by a sequence of small rotations whose net result is to return it to the original orientation, then it is possible toÂ defineÂ the idea that it has been rotatedÂ 360Â°âas distinct from zero net rotationâif you have kept track of the whole history.”

This is very deep. It connects space and time into one single geometric space, so to speak. But… Well… I’ll try to explain this rather sweeping statement later. Feynman also notes that a net rotation of 720Â° does give us the same amplitudes and, therefore, cannot be distinguished from the original orientation. Feynman finds that intriguing but… Well… I am not sure if it’s very significant. I do note some symmetries in quantum physics involve 720Â° rotations but… Well… I’ll let you think about this. đ

Note that the determinant of our matrix is equal to aÂ·dÂ â bÂ·c =Â eiÏ/2Â·eâiÏ/2Â = 1. So… Well… Our rotation matrix is, effectively, in that special form! How comes? Well… When equatingÂ Î» = âÎŒ, we are effectively putting the transformation into that special form.Â  Let us also, just for fun, quickly check the normalization condition.Â It requires that the probabilities, in any given representation,Â add to up to one. So… Well… Do they? When they come out ofÂ S, our electrons are equally likely to be in the upÂ orÂ downÂ state. So theÂ amplitudesÂ are 1/â2. [To be precise, they areÂ Â±1/â2 but… Well… It’s the phase factor story once again.] That’s normalized:Â |1/â2|2Â +Â |1/â2|2 = 1. The amplitudes to come out of theÂ TÂ apparatus in the up or down state areÂ eiÏ/2/â2 andÂ eiÏ/2/â2 respectively, so the probabilities add up toÂ |eiÏ/2/â2|2Â +Â |eâiÏ/2/â2|2 = … Well… It’s 1. Check it. đ

Let me add an extra remark here. The normalization condition will result in matrices whose determinant will be equal to some pure imaginary exponential, likeÂ eiÎ±. So is that what we have here? Yes. We can re-write 1 as 1 =Â eiÂ·0Â = e0, soÂ Î± = 0. đ Capito? Probably not, but… Well… Don’t worry about it. Just think about the grand results. As Feynman puts it, this Lecture is really “a sort of cultural excursion.” đ

Let’s do a practical calculation here. Let’s suppose the angle is, effectively, 180Â°. So theÂ eiÏ/2Â and eâiÏ/2/â2Â factors areÂ equal toÂ eiÏ/2Â =Â +i andÂ eâiÏ/2Â = âi, so… Well… What does thatÂ meanâin terms of theÂ geometryÂ of the wavefunction?Â Hmm… We need to do some more thinking about the implications of all this transformation business for ourÂ geometricÂ interpretation of he wavefunction, but so we’ll do that in our next post. Let us first work our way out of this rather hellish transformation logic. đ [See? I do admit it is all quite difficult and abstruse, but… Well… We can do this, right?]

So what’s next? Well… Feynman develops a similar argument (I should sayÂ same-same but differentÂ once more) to derive the coefficients for a rotation ofÂ Â±90Â° around theÂ y-axis. Why 90Â° only? Well… Let me quote Feynman here, as I can’t sum it up more succinctly than he does: “With just two transformationsâ90Â°Â about theÂ y-axis,Â and an arbitrary angle about theÂ z-axis [which we described above]âwe can generate any rotation at all.”

So how does that work? Check the illustration below. In Feynman’s words again: “Suppose that we want the angleÂ Î± around x. We know how to deal with the angleÂ Î±Â Î±Â aroundÂ z, but now we want it aroundÂ x.Â How do we get it? First, we turn the axisÂ zÂ down ontoÂ xâwhich is a rotation ofÂ +90Â°.Â Then we turn through the angleÂ Î±Â aroundÂ xÂ =Â z’. Then we rotateÂ â90Â°Â aboutÂ y”. The net result of the three rotations is the same as turning aroundÂ xÂ by the angleÂ Î±. It is a property of space.”

Besides helping us greatly to derive the transformation matrix forÂ anyÂ rotation, the mentioned property of space is rather mysterious and deep. It sort of reduces theÂ degrees of freedom, so to speak. FeynmanÂ writes the following about this:

“These facts of the combinations of rotations, and what they produce, are hard to grasp intuitively. It is rather strange, because we live in three dimensions, but it is hard for us to appreciate what happens if we turn this way and then that way. Perhaps, if we were fish or birds and had a real appreciation of what happens when we turn somersaults in space, we could more easily appreciate such things.”

In any case, I should limit the number of philosophical interjections. If you go through the motions, then you’ll find the following elemental rotation matrices:

What about the determinants of the Rx(Ï) andÂ Ry(Ï) matrices? They’re also equal toÂ one, so… Yes.Â A pure imaginary exponential, right? 1 =Â eiÂ·0Â = e0. đ

What’s next? Well… We’re done. We can now combine theÂ elementalÂ transformations above in a more general format, using the standardized Euler angles. Again, just go through the motions. The Grand Result is:

Does it give us normalized amplitudes? It should, but it looks like our determinant is going to be a much more complicated complex exponential. đ Hmm… Let’s take some time to mull over this. As promised, I’ll be back with more reflections in my next post.

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here:

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here:

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here:

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here:

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here:

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here:

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here:

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here:

Some content on this page was disabled on June 16, 2020 as a result of a DMCA takedown notice from The California Institute of Technology. You can learn more about the DMCA here: