# Talk:Relativistic quantum mechanics

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 To-do list for Relativistic quantum mechanics: Feel free to edit this list if it becomes more logical. Not done How the EM moments are related to the spin quantum number, anomalous magnetic moments. (What are the possible electric multipoles and magnetic multipoles for a given spin? how does this not fix the relativistic wave equation? what happens if the particles have some orbital angular momentum? what about the hypothetical case of magnetic charge?). Yes, probability interpretations are impossible when interactions occur, but these are one of a number of reasons for difficulties in describing interactions in relativistic wave equations that would be good to explain. Pair production Relativistic Heisenberg picture. Dirac himself did work in this lending credit to Heisenberg's formulation (see this paper in the further reading section). Relativistic many-particle states, Fock spaces. Partially done Included but need serious expansion: More on the Lamb shift. More on the Thomas precession, spin-orbit interactions, hyperfine interactions between atomic nuclei and electrons. Certainly more on fine structure of hydrogen atoms: application of the KG and Dirac equation to the Hydrogen atom. Other possibilities (not likely, since the main articles should explain the details) More on helicity and chirality in RQM? Mention plane wave solutions to the KG and Dirac equation? Astrophysical applications? (cosmic rays, solar radiation...)

## Removed redirect

This is just the beginning - feel free to complain and we can improve the article.. Of course by all means change the settings above. Thanks, M∧Ŝc2ħεИτlk 12:17, 9 April 2013 (UTC)

## New history section

You'll find it to be very similar to this, the history timeline at the end was very well outlined, so decided to include something similar. From these notes, I indirectly inferred and discovered the book - and have never ever come across it at all, so please don't ask me for page numbers. Thanks. M∧Ŝc2ħεИτlk 23:11, 13 April 2013 (UTC)

What book?
Are you are suggesting that we use as a reference a book which you have never even seen? If so, I would oppose that. JRSpriggs (talk) 13:33, 14 April 2013 (UTC)
I should have been clearer, the book Quantum by Manjit Kumar. Since then I added extra context so yes citations should be added where appropriate, not just one citing the whole list. That pdf mentioned the Quantum book on the last page (10) in a short timeline of QM/SR events, so I decided to at first while the list was growing. It will be replaced. M∧Ŝc2ħεИτlk 14:50, 14 April 2013 (UTC)
An extensive number of references has now been added to most of the key points (history section) - I don't think we need any more since there are direct links to the main articles where readers should find the main references. The Quantum (book) is in the further reading section, kept in case it's any good, feel free to delete otherwise. M∧Ŝc2ħεИτlk 21:45, 14 April 2013 (UTC)

## Group representations

Apart from (to an extent) the articles Representation theory of the Lorentz group, angular momentum operator and angular momentum, nowhere on WP is there even a half-decent explanation of groups, representations, and generators are in physics (which may be fine if they're so abstract). In any case I tried to include, in as simple terms as possible, such sections under Lorentz group in relativistic quantum mechanics relevant to spinors in RQM. It's probably filled with typos but should hopefully be clear and correct, from what I've gathered... In the literature, generators and groups can be described so vaguely it's painful (even from the best book I've found, which I wrote completely from: E. Abers QM (2004)).

I will no longer contribute until the end of May... M∧Ŝc2ħεИτlk 12:30, 19 April 2013 (UTC)

Naturally, I still edited. Most of the group theory stuff has been cut out now to prevent overloading this article. M∧Ŝc2ħεИτlk 14:50, 17 July 2013 (UTC)

## Suggestions

Hi!

Below are a number of things that needs review. It looks like a load of things, but this is just natural for a brand new article on a huge and difficult field. Some of the below items may be my personal point of view, but most of them are "correct" remarks.

• QM is nowadays considered a framework rather than a single theory. In this view there is one common quantum mechanical ground valid for both non-relativistic and relativistic applications. By this I mean that the postulates (in all attempted "axiomatizations") of QM look the same whether they are applied in a relativistic context or not. This is a tricky point to convey to the reader; States in Hilbert space - yes, Observables represented by Hermitean operators - Yes, but stuff like Probability Densities - No. That is, the QM this article should refer to is QM at the abstract framework level - not things like the nonrelativistic Schrödinger Equation obtained by canonically quantizing
$E = p^2/2m + V(x)$
To put it another way: RQM is parts of what results when one combines the principles of QM (unmodified) with the principles SR (unmodified). (Actually, there are strong arguments that huge parts of QFT results from QM + SR alone.) Thus RQM is not a "version" of QM compatible with SR, it is rather QM and SR applied together.
By contrast, nonrelativistic QM should be regarded as QM + Gallilean Relativity.
• RQM handles massless particles poorly. It does handle things like energy and momentum balances, but it doesn't handle, for instance, the instability of excited atomic states. For this one need to treat the EM field using QFT while one can keep the atomic states within RQM.
• The Schrödinger and Heisenberg pictures have nothing to do with relativistic invariance. They refer to where one places the time dependence - in the states or in the operators. The concepts are valid in RQM as well as in QFT.
• Regarding GR: I actually think it is possible to throw in GR in RQM by using the standard recipe; partial derivative -> covariant derivative, but this is nothing I am certain about. RQM treats spacetime as a background, not as a "full participant". The problems come about in QFT when one attempts to quantize GR:

Combining special relativity and quantum mechanics:

• The solutions to relativistic equations are not in general possible to interpret as probability amplitudes. Even if (as is the case of the Dirac equation) it is possible to construct a positive definite norm and a current, that norm is not conserved when there are interactions. One need not go to QFT to see that; it is clear already in RQM, for instance by considering scattering at a potential barrier. See any book on RQM for this. Some equations (e.g Klein-Gordon) doesn't even permit a positive definite norm.

Space and time:

• "The inclusion of the space and time coordinates of each particle in the probability amplitude, ψ(r1, r2, r3, ..., t1, t2, t3, ... σ1, σ2, σ3 ...), is not a well-posed procedure as the coordinate time of each particle depends on the frame one measures from, so there are actually multiple time dimensions forming a problematic complication. See spacetime: privileged character of 3+1 spacetime for more on this concept. The coordinate time and spatial coordinates of only one particle can easily be handled by SR, limiting RQM to one-particle dynamics. However, the path integral formulation of quantum mechanics, by Feynman in 1948 placed space and time on equal footing,[11] and is a powerful formalism connected to the principle of least action of classical mechanics."

References for the above? When working in a Lorentz frame, there is one and only one time coordinate. We are not considering anything like proper time for each participating particle - or one Lorentz frame per particle. B t w, the section spacetime: privileged character of 3+1 spacetime is in danger of being deleted, see physics project page and spacetime talk page.

Non-relativistic and relativistic Hamiltonians:

• Does not need much work for now.

• The Klein-Gordon equation is a fully satisfactory RQM equation. What is called "probability density" is really to be interpreted as charge density. (Also, see above about probability density in RQM.) The negative energy states plague the Dirac equation too. This is fully resolved in QFT where there are no negative energy states.

Densities and currents:

• Looks good apart from "Then, the amplitude ψ is not a wavefunction at all, but a reinterpreted as a field." What is meant by field here?

Spin and electromagnetically interacting particles:

• The KG equation applies (as mentioned) to bosons - or more correctly to spin 0 particles, e.g. pi-mesons, and is the correct equation for them. But such particles interact strongly as well, so that the EM influence is very small. But this doesn't mean that the KG eqn with minimal coupling predicts the "wrong" energy levels. It just doesn't apply to fermions and is useless for pi-mesons because of the strong interaction.

I'll take a break here. YohanN7 (talk) 12:12, 17 July 2013 (UTC)

To answer a few points (I'll edit later):
• QM and relativity: Will reword.
• GR in RQM? I hardly think it would work so easily. Formulation of quantum theory in curved spacetime alone is not enough. Please give a reference.
• Currents and interactions: Yes, difference between free and interacting particles will be clarified.
• Schro and Heisenberg pictures: The context about the Schro and Heisenberg pictures is worded wrongly in the lead and will be fixed, it is correctly worded in the first section.
• ψ? OK - I used the term "probability amplitude" from section 1. Yes, the probability interpretation is not always possible, I get it already, and that's even written in the article. My question - what would you call the ψ in the KG and Dirac equations, even though the RQM literature still uses the terms "multi-component (spinor) wavefunction", "(probability) amplitude", or when discussing the reinterpretation of the KG eqn the term "scalar field" is used, etc.? Yes - even Wienberg's book on QFT (vol 1) on pages 8 and 9 for example uses the term "wavefunction" in relativistic wave mechanics, and on page 10 ψ is still interpreted as a probability density because it is possible. At the times probability interpretations aren't possible, it can be stated there and then that ψ is not a probability etc etc. But whenever we refer to a relativistic wave equation, ψ is referred to a "wavefunction" or "amplitude" something or other in the literature.
• The multiple time dimensions is in Penrose's book Road to reality, is Penrose unreliable? Nevertheless, that segment can/should be trimmed.
• Energy levels: The KG equation according to Abers's book on QM doesn't predict the energy levels accurately. The equation has nothing to do with spin so fine structure cannot be explained. Why is this wrong?
• Negative energies: This is mentioned in the article, although the solutions to the KG and Dirac eqns have yet to be expanded on to illuminate the context of negative energies.
M∧Ŝc2ħεИτlk 14:50, 17 July 2013 (UTC)
• QM and relativity: Ok
• GR in RQM: At the level of laws of nature (other than GR) formulated as a SR relativistically invariant differential equation it is that simple. It has "almost" postulate status in GR. This is exactly how the EM field equations are treated in GR. See for instance MTW or (better!) L&L Vol 2. It is a cookbook recipe to turn a valid SR equation to one valid in a GR background.
It's slightly more complicated, (partials -> covariant partials not sufficient for multi-component fields, e.g. a spinorial affine connection must be used to define a covariant derivative in the spin 1/2 case) but see your own article;): Dirac equation in curved spacetime. The story is surely similar for other spins. YohanN7 (talk) 17:36, 19 July 2013 (UTC)
• Schro and Heisenberg pictures: Ok
• ψ: Either wavefunction and/or field sounds perfectly ok. Preferably use both to avoid linguistic monotonicity. Probability amplitude is best avoided except for the special cases where it is possible to use it.
• The multiple time dimensions is in Penrose's book Road to reality, is Penrose unreliable? Nevertheless, that segment can/should be trimmed.
Well, that's a reference at least. I haven't read Penroses book, but I have gathered that it is not a scientific publication or a textbook. It is more popular science and speculation with an advanced mathematical twist to it. Is the expression for a relativistic multiparticle wavefunction taken from that book? In QFT you certainly deal with multiparticle states (represented by wavefunctions) with a single time coordinate. I have never seen anything like the expression in the article.
I might just see what you mean. But to draw the conclusion about different time coordinates you would have to bring in general relativity at a much deeper level (and an extremely warped spacetime for it to have any measurable effect). I don't think that this belongs in the article at all. I just got hold of the Penrose book, and it strikes me as being pretty speculative. I think we should stick entirely to SR in this article. YohanN7 (talk) 20:57, 19 July 2013 (UTC)
• Energy levels: The KG equation according to Abers's book on QM doesn't predict the energy levels accurately. The equation has nothing to do with spin so fine structure cannot be explained. Why is this wrong?
You would get the same type of error if you applied the Dirac equation to a boson. The KG equation is the correct equation for spin-0 particles. It follows in the same way the Dirac equation, the Rarita-Schwinger equation, and the and the Bargmann-Wigner equations follow from Lorentz invariance.
In fact, the KG equation is - if any - the fundamental equation of RQM; Every component of every multi-component free field must satisfy the free KG equation. It expresses the relativistic relationship between energy and momentum. YohanN7 (talk) 18:54, 19 July 2013 (UTC)
• Negative energies: This is mentioned in the article, although the solutions to the KG and Dirac eqns have yet to be expanded on to illuminate the context of negative energies.
Agreed. Within RQM one may "solve" the problem by postulating the mentioned "Dirac Sea"/ Hole Theory. But this doesn't hold for integer spin particles. This was historically a long term obstacle to fully accepting the KG equation. A full explanation is available only in QFT. This theoretical point together with the experimental result of the Lamb shift could serve really well as illuminating the limitations of RQM - even without relating to the nonconstancy of particle numbers. Hole theory gives correct results (as far as I know) but the physical interpretation is wrong; at least nobody believes in it anymore. YohanN7 (talk) 14:00, 19 July 2013 (UTC)

Yes, I know the KG eqn alone is for spin-0 bosons as there is nothing in it to describe spin, and that's written in the article. Yes, I know the solutions to the Dirac equation are also solutions to the KG equation by "squaring" the Dirac eqn to return to the KG eqn, and solutions to any RWE (relativistic wave eqn) must satisfy the KG eqn. Considering the first point - the Dirac equation has features the KG equation alone does not have (gamma matrices for spin-1/2 particles and multi-component spinors for particles and antiparticles), similarly for all eqns of higher spin.

How would you apply the KG eqn to a spin s particle, when in the first place the KG eqn only requires one component? Actually, how would you know that ψ is for a spin s particle? Well, the equation is linear, so ψ could be something really fancy beyond tensors or spinors, for all we could guess. You cannot approach RQM with only the KG eqn, this has to be factored or the RWE has to be derived by another route.

You could say the KG eqn is the fundamental eqn of RQM, but it seems to be more of a constraint than a governing eqn. But then, the energy-momentum relation with the Schro operators gives the KG eqn, so that would mean the ultimate constraint is the energy-momentum relation with the quantum operators, although that biases the Schro picture. I'll have to re-read the sources for the status of the KG eqn in RQM, but one ref at the top of my head (Particle physics by Shaw and Martin) agrees that the KG eqn in itself is not enough.

And of course, one should expect inaccuracies by applying a fermionic equation to bosons and vice versa. M∧Ŝc2ħεИτlk 07:17, 20 July 2013 (UTC)

I was just reacting against the article almost putting the KG equation in doubt in the historical way it was put in doubt which is invalid today. (Won't be able to edit for a while now.) Cheers! YohanN7 (talk) 10:34, 20 July 2013 (UTC)
Fair enough, it shouldn't be downplayed/doubted in any way, but the opposite is also true!
BTW, about GR in QM, I didn't write Dirac equation in curved spacetime, it was cut form the Dirac equation article. Even then, how does this curved spacetime formulation include the gravitational interaction between a mass-energy source and the particle described by the Dirac equation? As well as the EM interaction, or more ambitiously the electroweak interaction? M∧Ŝc2ħεИτlk 00:07, 21 July 2013 (UTC)
To Maschen: You left some of your contributions in this section of talk unsigned. Please fix that. JRSpriggs (talk) 23:08, 20 July 2013 (UTC)
Fixed. M∧Ŝc2ħεИτlk 00:07, 21 July 2013 (UTC)
Hi, long time no see. Here is one thing that ought to go into the article: In RQM the concept of a probability density is inconsistent. We have had some arguments about this a few times already, but I intend to convince you. The qualitative reason is this: Pair creation occurs. Therefore, if momentum uncertainty is present, then you will have uncertainty in the number of particles that are present. Moreover, in relativity, time will be uncertain when position is uncertain. The result is that a probability density interpretation makes no rigorous sense. Only in the non-relativistic limit will you get a true probability density. You have an approximate probability density at best.
I can spell it out in greater detail if you want me to.
Dirac didn't even know about pair creation when he found his equation. (Positrons were found later...)
References: Relativistic QM by walter Greiner (page 1) and QED by Landau and Lifshits (pages 1-5).
B t w, those multiple time coordinates... I'll have a look in Penrose when I come home. Cheers! YohanN7 (talk) 12:35, 1 October 2013 (UTC)
Is this a problem in defining the quantity in an invariant way (as seems to be the case for centre of mass (relativistic)), or is it merely a matter of reinterpretation of the expression, such as the "expected particle density" (which would be appropriate if the assumption of a single particle being present is invalid)? — Quondum 15:37, 1 October 2013 (UTC)
It is (as far as I can see) merely a matter of reinterpretation of the expression. If you have an approximately non-relativistic system, like the hydrogen atom, which concerned Dirac the most, then you are close to the non-relativistic limit, and you will only make a conceptual error when talking about probability densities. But the Dirac equation, and other relativistic equations, apply in particular in the ultrarelativistic domain. (It's their purpose after all;)) In this domain, the probability interpretation is wrong in all respects. Massless particles are always ultrarelativistic. In this case, it makes little sense to talk about a photon being located anywhere unless the system under consideration has a large volume. YohanN7 (talk) 12:03, 2 October 2013 (UTC)
There is a standard example where one takes a normalized(=1) electron wave packet and send it towards a potential barrier with a suitable (=pedagogical) heigth or depth. After the "collision" the "norm" of the packet is is in the range 2-3. Pair production may have occured, so the norm of a Dirac wave function is not always conserved if there are interactions.
Mistake on my part above. The norm is still 1(because there is a conserved current). But the conclusion is still valid since the original wave package is spatially split up and can't reasonably describe one particle and its probability amplitude.

YohanN7 (talk) 12:10, 2 October 2013 (UTC)

So? This merely means that a suitable interpretation must be applied to Ψ|Ψ, surely?. — Quondum 17:42, 2 October 2013 (UTC)
Yes. And any attempt to interpret Ψ|Ψ as a probability amplitude density will not stand up to rigorous inspection. In short, a one-particle interpretation of RQM is - in principle - impossible. You will have - at best - an approximate probability density. See the references I gave above.
There is more to it. In ordinary QM an instantateous measurement of a quantum number is - in principle - possible (because the speed of light is infinite). In RQM it is - in principle - impossible (because the speed of light is finite). Thus you can't even define the probability density as the probability of finding the particle in a volume at time t. Add to that the impossibility of speaking of the particle located in a small volume. If the volume is sufficiently small the particle will automatically have companions. YohanN7 (talk) 12:10, 29 October 2013 (UTC)

### ψ = sigh...

Nice to see both of you (YohanN7 and Quondum) on this talk page.

We seem to be cycling in circles on the issue of probability interpretations when there are interactions. I get it already - there are issues when interactions are present. I happen to be actually learning this topic in a module this year so can clarify my own misconceptions and possibly rewrite the article in the future.

For now, I've made plenty of edits to this article, and would welcome anyone else to simply be bold and make improvements themselves. I'll correct my errors as and when. Best regards, M∧Ŝc2ħεИτlk 20:29, 10 October 2013 (UTC)

I have been bold, see edit.
On the issue of what is required for a full description of RQM (i.e. all spins) in terms of wave functions and wave equations, one can probably use the fact that a state is either a tensor, a spinor, or a spinor-tensor (Weinberg vol 1). These states can be built from spin 0 states and spin 1/2 states. Thus the KG eqn + the Dirac eqn together provide the minimum required raw material. But the higher spin equations also have alternative formulations...
I'll prepare a section on probability and relativity. If it comes out to your liking, I'll put it in the article. Main points will be that a one-particle interpretation is - in principle - impossible, and that instantaneous measurements are - in principle - impossible in RQM, making the probability interpretation impossible other than as an approximation. A conserved positive definite norm does not mean the same thing as a probability density, since the Ψ describes an unknown number of particles. The nice thing is that one doesn't need to go to QFT to see this. By the way, the norm (or any other conserved quantity) is conserved, even in the presence of interactions because there is a conserved current associated with it. It is the number of particles that isn't conserved. [I formulated myself incorrectly regarding this way above.]
Multi-particle states: In RQM, there is no problem describing multi-particle states. Such states are (sums of) tensor products of single particle states, and as such they have well-defined LT properties. I simply don't understand from where (or what) Penrose deduces the need for separate time coordinates for each particle. It certainly doesn't follow from the postulates of SR. You shouldn't use "time and space is to be treated on equal footing" as a replacement for the actual second postulate of relativity, which says something more precise; It deals with laws of nature. State vectors are not laws of nature. Besides, Penrose states himself that such ideas haven't lead anywhere. YohanN7 (talk) 17:28, 31 October 2013 (UTC)
Hi Yohan, thanks for any/all of your edits. I'll just cut out any remaining bits that say multiple time coordinates since it's agreeably pointless to mention. I'll review your edits and discussion in more detail later, too busy for now.
Feel free to edit/rewrite the section Relativistic quantum mechanics#Densities and currents, change the heading too if you want.
BTW: yesterday finally saved up enough to buy the one and only Dirac's Principles of Quantum mechanics (4th edition), which could help. M∧Ŝc2ħεИτlk 03:41, 1 November 2013 (UTC)
$E^2 = c^2\mathbf{p}^2 + c^4 M^2$
with M the invariant mass of the whole system (no, it's not the sum of invariant masses for each particle), and:
$E = \sum_{n=1}^N E_n \quad \mathbf{p} = \sum_{n=1}^N \mathbf{p}_n$
are the total energy and momentum of the system in a particular frame, quantizing using the coordinate time t and particle coordinates x1, x2, ... xn still in this frame:
$E = i\hbar \frac{\partial}{\partial t} \quad \mathbf{p} = -i\hbar \sum_{n=1}^N \boldsymbol{\nabla}_n$
where the n in $\boldsymbol{\nabla}_n = \partial/\partial \mathbf{x}_n$ labels the gradient operator for each particle's coordinates (all measured in the same frame), so the KG equation is:
$-\hbar^2 \frac{\partial^2}{\partial t^2}\psi = -\hbar^2 c^2 \left(\sum_{n=1}^N \boldsymbol{\nabla}_n \right)^2 \psi + c^4 M^2\psi$
and would factorize into something like:
$i\hbar \left(\gamma^0 \frac{\partial}{\partial t} + c \boldsymbol{\gamma} \cdot \sum_{n=1}^N \boldsymbol{\nabla}_n \right) \psi = Mc^2 \psi$
for the many-particle Dirac equation? I'll have to look into this later. M∧Ŝc2ħεИτlk 04:09, 1 November 2013 (UTC)
For the record just found this now:
K.S. Viswanathan (1960). "The Dirac equation for many-electron systems".,
which uses the stress-energy tensor. M∧Ŝc2ħεИτlk 04:23, 1 November 2013 (UTC)
The above heuristic is too simple to be correct, so it was crossed out. M∧Ŝc2ħεИτlk 10:36, 2 November 2013 (UTC)

## Klein–Gordon equation and spin-0

This edit (and the associated edit summary: Spin-0: The KG eqn does say something about spin. It says the spin is zero. Hence it cannot a priori be applied to atoms for predictions of energy levels.) leaves one with the impression that the KG equation does not apply to particles of non-zero spin. The inference is however incorrect: the KG equation only implies spin-0 when the wavefunction is assumed to be scalar a priori. Ergo, the wavefunction is spinless because we have constrained it to be so, before the KG equation comes into play. As soon as we allow the wavefunction to represent spin by making it more general than a scalar, voila, its solutions can have spin: any solution to the Dirac equation is also a solution of the KG equation. Thus, one can only say that the KG equation says nothing about the spin of a particle (and is thus not of much use in predicting energy levels in an atom, basically because some of its solutions are nonphysical). — Quondum 17:14, 31 October 2013 (UTC)

Good point. But wrong if we consider irreducible representations of the Lorentz group. This interpretation is the default interpretation. Perhaps the article should clarify this? YohanN7 (talk) 17:33, 31 October 2013 (UTC)
By the way, which solutions of the KG equation are nonphysical? You have the same interpretation in terms of antiparticles as with the Dirac equation, except that there is no "Dirac sea". In modern interpretations there is no Dirac sea for half-integer spin particles either, but one needs to go to QFT for a full explanation.
When talking about nonphysical states; the position eigenstates are nonphysical in RQM (regardless of spin), which is the reason that they aren't used to label states as they are in ordinary QM. (The reason is the lack of a probability density interpretation which manifests itself in the extreme with position eigenstates.) YohanN7 (talk) 18:48, 31 October 2013 (UTC)
Sorry, I should have been more specific about "nonphysical": what I meant was solutions that have no bearing in nature, so let's drop that term to avoid confusion. For example while momentum and energy eigenstates might be nonphysical (they are not normalizable), a linear sum of them may be a valid, physical state. In contrast, I mean solutions to an equation that are extraneous to the problem, that are excluded by something, but the equation allows it as a solution. So, The KG equation has not only the solutions of the Dirac equation as its solutions, it also has solutions that do not correspond to particles (or forms of a particle) in nature in any sense, such as scalar electrons.
A natural algebraic setting to consider the KG equation in is that of the Clifford algebra Cl1,3(R). This is the algebra in which the Dirac equation works, and also in which we can think of the Dirac equation's equation being part of those of the KG equation. This supports scalar particles and spin-1/2 particles in the algebra, within my limited understanding. (Actually, how I understand it is that the spinors of the Dirac equation and the scalars of the scalar-only KG equation are simultaneously represented in the algebra.)
With reference to the antiparticles in the Dirac equation, one needs care interpreting equations. For example, as I understand it, negative-mass particles cannot be distinguished in this context from positive-mass solutions that are the time-reversed in all properties: that is to say, antiparticles. So the Dirac sea is not needed at all to explain this. Which is to say, maybe we should treat that tack as not particularly illuminating in this context.
And finally, though I do not understand the representations of the Lorentz group, in spacetime the KG equation does not bow to "representations". Its solutions are its solutions. And as I've said, in the Clifford algebra (which is natural given the vector nature of the del operator), electrons are solutions to it in addition charged scalar particles. Perhaps (this is only a guess) your irreducible representations are excluding certain cases precisely because they cannot represent them a priori? And it is "default" only because we chose it so? Which comes back to my point of that it is not the KG equation, but our choices in representation which say anything about permitted spin in the solutions. — Quondum 22:21, 31 October 2013 (UTC)
Well, ..., no, no, no, and no again, but I really don't know where to start. You need to explain what the solutions are that "have no bearing in nature". There are plenty of massive particles with zero spin, without or with charge (if the latter is what you mean by scalar electrons). If you want something elementary, then go for the Higgs. It is massive and has spin 0. (But there is no known elementary charged massive spin 0 particle, but this is of no importance here.)
The Clifford Algebra is not a natural setting for consideration of the KG equation. It contains too much. As a representation space, it is the direct sum of a scalar, a 4-vector, a tensor, a pseudo-4-vector and a pseudo-scalar (but no bispinors!). The matrices γμγν - γνγμ is a spin-1/2 representation (as opposed to a representation space) of so(3;1). It is better to regard any representation describing particles as living outside of the Clifford algebra even if there are representation spaces in the algebra (because for instance physical 4-vectors have four components, not 16). Moreover, the elements of the Clifford algebra have interpretations as operators on the space of states. You should check out the Bispinor article for a little more details.
In relativity, all free particle relativistic wave equations are determined by the irreducible representation that is used in deriving them. So, yes, the KG equation "bows" to the representation in question since it, and its solutions, transform in the prescribed way w r t Lorentz transformations from the chosen irreducible representation - in this case the scalar representation. But it also works the other way around when irreducibility is assumed. Solutions of the Dirac equations are, as you say, solutions of the KG equation. But these solutions do not transform irreducibly under the scalar representation. In other words, given irreducible representation (spin) -> equation, and equation -> irreducible representation (spin). If you don't require irreducibility, then the Dirac equation is equally unable to say something conclusive about spin. (Higher spin fermionic wave functions have spin 1/2 subspaces.) (In Diracs original approach, the irreducibility requirement is hidden behind the requirement of minimal dimensionality.) YohanN7
Perhaps it is better explained this way: The KG equation is a scalar equation, because it has no free indices. It thereby uniquely selects the scalar representation of the Lorentz group. It therefore describes spin 0 particles. Likewise, the Dirac equation transforms under (the adjoint action of) the spin representation. Hence, it describes spin 1/2 particles. (talk) 23:55, 31 October 2013 (UTC)
When it comes to nonphysical states, one normally doesn't rule out a state because it can't be normalized. One can still have a sequence of normalizable truly physical states that approaches, say, the momentum eigenstates arbitrarily closely. The position eigenstates are ruled out for other reasons as described above. There is no sequence of physically meaningful states approaching the position eigenstates (due to pair production). YohanN7 (talk) 23:55, 31 October 2013 (UTC)
I wouldn't challenge you about the Lorentz reps, but it would be confusing for many readers if we explain everything this way. Nevertheless feel free to edit that section on Lorentz reps and irreps, which is later in the article after the first introduction to the KG and Dirac equations. M∧Ŝc2ħεИτlk 03:41, 1 November 2013 (UTC)
I don't know what has the highest priority, being correct or being not confusing to the reader. I'd say we need both, but if I must choose, then correctness takes the drivers seat. I have edited the section again. I hope that it is clearer. YohanN7 (talk) 10:01, 1 November 2013 (UTC)
Well obviously technical correctness is more important, but it's not like Lorentz groups and irreps need to be mentioned first thing, they can be forward-referenced to the specialized section (and main article) after introducing what the KG and Dirac equations are. M∧Ŝc2ħεИτlk 12:04, 2 November 2013 (UTC)
YohanN7, I'm not going to argue, or even try to make clear what I was trying to say (you clearly haven't understood what I meant, even though I'm not going make claims of correctness). You seem to be arguing from a particular interpretation that simply isn't adequately clear without a lot of background and a specific formalism, and I'm sure is not the only interpretation. But so be it. — Quondum 04:38, 1 November 2013 (UTC)
I have tried to understand what you mean, and I have asked about specifications of the things I didn't understand. In your posts above you become quite elaborate in spots, making conclusive statements and taking on an explanatory tone. I have to point out where I believe you go wrong because it is my edits that you are discussing. If you consider this impolite, then I apologize, but if you can't discuss, or don't have the required knowledge to discuss, then you shouldn't have posted in the first place claiming that I am wrong.
On the matter of nonphysical solutions to the KG equation: The irreducibility requirement does rinse out nonphysical solutions (as I now think I understand how you perceive them). And yes, you are safe to assume that, in the literature, when a particle is claimed to satisfy the KG equation, it is the one-dimensional irreducible version that is referred to. In other instances, particular mention is made to the effect that the individual components satisfy the KG equation. This is what I mean by irreducibility being the "default". It is not my invention. See new version in the article. Is this better? YohanN7 (talk) 10:01, 1 November 2013 (UTC)
As an aside, I was working a year ago on the Dirac algebra article. The unfinished draft is here: User:YohanN7/Dirac algebra. It is basically a continuation of Bispinor#Derivation of a bispinor representation that is supposed to show exacly how the Lorentz reps sit inside Cl3,1(R). YohanN7 (talk) 12:48, 1 November 2013 (UTC)
I apologize for my excessively strong assertions. You were not being impolite: I just grew brusque upon realizing that we have a long way to go before we synchronize even our underlying concepts and language. Let's just leave it at that I "shouldn't have posted in the first place claiming..."; you're close enough to the mark here. — Quondum 15:09, 1 November 2013 (UTC)
I don't think we have a long way to go synchronizing concepts and language. There is a clash between the physicists lingo and the mathematicians lingo. I take it that you prefer the mathematicians view and terminology. I prefer that myself. In retrospect, what we were discussing was - in a fairly well-defined sense - how to classify particles. This reduces pretty much to how to classify representations of the Lorentz group. Here, there is no clash between the physicists view and the mathematicians. Both will only consider the irreducible representations only (while perhaps for somewhat different reasons). This seems now like the only discrepancy between my view and your view (apart from certain technical matters where I'll not give in any time soon;)).
But then I was utterly wrong writing that you "shouldn't have posted in the first place claiming...". If you, or anyone else, thinks that there is something wrong, of course the error should be pointed out. In case of doubt, even a mildly suspected error should be pointed out. The section in question is much better now thanks to your posts. We just need to read each others posts a bit more carefully before we proceed with the next post. YohanN7 (talk) 16:41, 1 November 2013 (UTC)

## SO(2)

I made a minor edit, essentially quaternion group -> quaternions. Nearby there is mention of applications of the gammas and sigmas to SO(3) and SO(2). The application of the sigmas to SO(3) is clear to me and gammas to SO(3;1) as well. But the other combinations [(gammas, SO(3)), (gammas, SO(2)), (sigmas, SO(2))] elude me. YohanN7 (talk) 11:54, 1 November 2013 (UTC)

## Higher spins

This passage might need some clarifying remarks:

However, for massless particles, spin can still be integer or half integer, but there are only ever two-component spinor fields; one is for the particle in one helicity state and the other for the antiparticle in the opposite helicity state:

$\psi(\mathbf{r},t) = \begin{pmatrix} \psi_{+}(\mathbf{r},t) \\ \psi_{-}(\mathbf{r},t) \end{pmatrix}$"

This may give the impression that there are no nonzero integer-spin massive particles. There are of course, and the Proca equation is the appropriate RWE. Also, an explanation of why massless particles have either "maximum" or "minimum" helicity is needed. YohanN7 (talk) 12:08, 1 November 2013 (UTC)

?? So you're talking about massive particles with s = 1, 2, 3, ... .
Massive particles have 2(2s + 1) component spinors for s = 0, 1/2, 1, 3/2, 2, 5/2, ... integer or half-integer, while massless particles have one component for the particle and a corresponding component for the antiparticle, correct or not? This is in the article, so what's your point?
My point? A rewording of that sentence: "However, for massless particles, spin can still be integer or half integer, ...". For some idiots (like me) it reads like we have just concluded that for massive particles integer or half integer spin is impossible. YohanN7 (talk) 11:13, 2 November 2013 (UTC)
No you're not an idiot. If someone raises a point it means something is not clear to them. I need to not blend different things within the same sentence, and shall try to reword. M∧Ŝc2ħεИτlk 12:04, 2 November 2013 (UTC)
Anyway, I'm not 100% certain on the reasons for this, but apparently, an explanation requires the stuff on Lorentz groups, "little groups", "double covers", "irreps", all of which you understand so well. Presumably it is related to the relativistic relation:
$|\mathbf{p}| = \frac{1}{c}\sqrt{E^2 - (mc^2)^2} \xrightarrow{m=0} |\mathbf{p}| = E/c$
too. M∧Ŝc2ħεИτlk 09:55, 2 November 2013 (UTC)
Yes, it is surely related to LT's, but I don't know exactly how to derive the result. (Maschen, "..., all of which you understand so well...", do I detect a bit of irony here? ) YohanN7 (talk) 11:13, 2 November 2013 (UTC)
No, no irony! Just saying you understand this far better than me. M∧Ŝc2ħεИτlk 12:04, 2 November 2013 (UTC)

At one point I had a section on helicity, but decided to blend it into the discussion on the Weyl/Dirac equations, which was silly because it is out-of-context there. For now, I'll reintroduce helicity in a subsection, which may be merged into the higher-spin section (we can decide this later). Random aside: currently the helicity (particle physics) is a stub which can/should be expanded (in RQM and RQFT), this article is only going to have a concise summary of it in RQM. M∧Ŝc2ħεИτlk 12:15, 2 November 2013 (UTC)

## Velocity operator

The Schrödinger/Pauli velocity operator can be defined for a massive particle using the classical definition p = m v, and substituting quantum operators in the usual way:[1]

$\widehat{\mathbf{v}} = \frac{1}{m}\widehat{\mathbf{p}}$

which has eigenvalues that take any value. In RQM, the Dirac theory, it is:

$\widehat{\mathbf{v}} = \frac{i}{\hbar}\left[\widehat{H},\widehat{\mathbf{r}}\right]$

which must have eigenvalues between ±c. See Foldy–Wouthuysen transformation for more theoretical background.

Interestingly, the square root of the square of this operator (the speed) has exactly the eigenvalues ±c because
$\frac{i}{\hbar}\left[\widehat{H},\widehat{\mathbf{r}}\right] = \hat{\boldsymbol{\alpha}}c,$
and the eigenvalues of the alphas are ±1. See Greiner, RQM section 2.2. This is another manifestation of the problems with position eigenstates and one-particle interpretations. YohanN7 (talk) 12:27, 1 November 2013 (UTC)
I have access to Greiner's book but not right now. Is the equation above for all the alpha (Dirac) matrices? M∧Ŝc2ħεИτlk 09:55, 2 November 2013 (UTC)
Yep. (Boldface greek letters doesn't work well.)
The amusing thing is that according to the formula, all particles (with mass) obeying the Dirac equation move with the speed of light, so the formula itself is of no use, but the discussion around it is. YohanN7 (talk) 11:23, 2 November 2013 (UTC)
That sounds crazily interesting... In any case relativistic velocity operators seem to be in the scheme of the FW transformation (quite complicated), the section as it stands is nothing...
About bold for Greek use \boldsymbol{alpha}.
M∧Ŝc2ħεИτlk 12:04, 2 November 2013 (UTC)

### Zitterbewegung

I found this: Zitterbewegung. I personally don't think it has physical existence. It seems more like a manifestation of the impossibility (with full rigor) of the probability interpretation and one-particle states. From that follows that the position operator, and operators derived from it, have to be handled with care for reliable results. What is truly surprising to me is that one doesn't need to take an ultra-relativistic limit to get the "clean" result above. YohanN7 (talk) 13:17, 2 November 2013 (UTC)
I've heard "Zitterbewegung" numerous times, yet never read enough to understand in detail. For now, I'll add it to the article in the "see also" section.
It seems the main point you are after is to
• expand on the special-relativistic Heisenberg formalism (which is currently badly needed anyway),
• making it easy to find the time-evolution equations of operators,
• with careful descriptions of physical interpretations (including when probability is possible/relevant), and correspondences with experiment.
If so I agree (in addition to some discussion of the FW transform and the SR/classical mechanics limits). M∧Ŝc2ħεИτlk 17:28, 2 November 2013 (UTC)
The "Zitterbewegung" can be found in Greiner as well when you get your hands on it, same section.
While an exposition of the Heisenberg formalism would be nice, my main concern is about the foundational stuff, where RQM differs from ordinary QM (as presented in introductory courses). This includes the probability interpretation, the one-particle interpretation, the "unreliability" of the position operator and its eigenstates, impossibility of instantaneous measurement, Zitterbewegung and more. The common denominator is pair creation. The really foundational thing behind it all is probably (definitly in most cases) the finiteness of the speed of light. Moreover, I think the implication SR->Antiparticles is true, while Spin->SR (i.e. spin only if SR as often claimed elsewhere) is false. I vaguely remember it from Greiners ordinary QM book. I will try to dig up some hard facts. I wish I had Weinbergs fairly new book on QM. It's said to be extremely good. Not surprised.
• A one-page section on the foundational issues would probably be enough.
• Perhaps a section on the transition to QFT, and what foundational stuff differs between RQM and QFT as a last item in the article?
I think the article is in pretty good shape with one exception: The multi-particle states. They aren't suspect. Ar least they aren't viewed with suspicion by most physicists because, at least on the face of it, it would mean that either SR or QM (or both) is fundamentally incorrect. Interacting multi-particle states is another kettle of beans in one respect, namely that one doesn't try to follow the detailed course of events (this is QFT), but it is as far as I know assumed that one never leaves ordinary Fock space where the multi-particle states live (with a single time coordinate). YohanN7 (talk) 19:42, 2 November 2013 (UTC)

If you were to try to observe the trembling motion, you would have to make two very short and precise measurements of position separated by a very short interval of time. In that case, the first measurement would (in accordance with the Heisenberg uncertainty principle) have to strike the particle with very great force causing it to travel at virtually the speed of light until the second measurement. Thus the trembling motion would be observed to exist. JRSpriggs (talk) 09:26, 7 November 2013 (UTC)

Thus the trembling motion would be observed to exist whether it exists or not? Interesting thought.
B t w, very precise measurements of position are in principle impossible with relativity. We'd need to give the particle close to infinite speed with our first measurement, close to infinite momentum isn't enough. Else we wouldn't be able to tell at what time we actually make these measurements, so no conclusive dx/dt can be found because of the error in dt. See Lev Landau and Evgeny Lifshitz, Quantum Mechanics and Quantum Electrodynamics. YohanN7 (talk) 10:46, 8 November 2013 (UTC)
No. It does exist (as much as anything can be said to exist in quantum mechanics) because it is observed. My point is that it is inevitable given the postulates of quantum mechanics.
The duration of the position measurements can be made short relative to the already short interval between them. This requires by HUP that the energy become very uncertain. However, I fail to see how you can conclude that the speed is infinite (i.e. >> c). And why do you keep blaming these problems on the theory of relativity??? JRSpriggs (talk) 11:33, 8 November 2013 (UTC)
I didn't say the speed is infinite. I say that instantaneous measurements are possible in ordinary QM because the difference of the speed of the particle before and after the measurement is allowed to take on infinite values while they are not in RQM. In QM you have ΔxΔp ≈ h/2π. With relativity, you have in addition ΔEΔt ≈ h/2π. So, yes, relativity makes a difference. Uncertainty in time will ruin the measurement of the speed, and uncertainty in energy will ruin the one-particle state. YohanN7 (talk) 12:43, 8 November 2013 (UTC)

### Square 1 again: PIs, interactions, spin

OK, I'll update the "to do list above" (we'll see how more incomplete it becomes!). Feel free to edit.
Take it you are referring to S. Weinberg (2012) Lectures on Quantum Mechanics.
Agreed on Fock spaces in RQM and many-particle states. However, SR + RQM -> spin and antiparticles, surely? (spin and antiparticles are consequential from, not just SR, but the union with QM). M∧Ŝc2ħεИτlk 04:52, 3 November 2013 (UTC)
Yes, you are right. Should have written (SR + QM) -> antiparticles. This will hold if particles exist. But (SR + QM) -> spin is not true. Spin exists if particles with spin exist. To say this more rigorously, It is not true that for every irreducible representation of the Lorentz group, there is an associated particle. It happens to be true for spin 1/2 and some other spins. But SR + QM indeed makes it possible that spin comes in all flavors. But is SR necessary condition for spin? This is what I think I remember that Greiner proves wrong in his QM (not RQM) book. And, to be sure, I have never ever seen a proof that Spin -> (SR + QM) (i.e that spin requires SR), but the claim is often loosely made ("Spin is a consequence of special relativity..."). YohanN7 (talk) 12:28, 3 November 2013 (UTC)
Another (off-topic) remark is that no QM is required for spin. The classical EM field has spin one. Yet another (on-topic) remark is that spin clearly arises due to the rotation subgroup of the Lorentz group. It is a subgroup of the translation-rotation group of Galilean Relativity as well. This is as close to a short proof outline I can think of of the falsehood of "Spin only if SR". In any case (my assertion being right or wrong), particle spin is not a consequence of SR + QM, it is only possible as a consequence of QM + SR. YohanN7 (talk) 14:35, 3 November 2013 (UTC)
And there is the Pauli-Lubanski pseudovector and the related spin tensor (MTW p. 156–159, §5.11.), applicable to SR and GR all of and quantum theory. M∧Ŝc2ħεИτlk 17:13, 3 November 2013 (UTC)
One thing on interactions to clear up (see the papers by Cédric Lorcé (2009) in the article) is to explain how the EM moments arise from spin higher than hbar/2 (what are the possible electric multipoles and magnetic multipoles for a given spin? what happens if the particles have some orbital angular momentum? what about the hypothetical case of magnetic charge?). Yes, probability interpretations are impossible when interactions occur, but these are one of a number of reasons for difficulties in describing interactions in relativistic wave equations that would be good to explain.M∧Ŝc2ħεИτlk 04:52, 3 November 2013 (UTC)
The thing is this: In RQM, rigorous probability interpretations of the wave function are impossible period. You don't need interactions to show that. Suppose for a moment that I am correct about this, just for the sake of reasoning. Can you see why I am so persistent about it? It practically takes forever to convince somebody who knows ordinary QM, but not RQM, that the "probability density" is not a rigorous interpretation of anything in RQM. This is because the Probability Interpretation (PI) works in QM, and it works well there. It is so deeply entrenched that even the statement per se, "There is no PI in RQM", hardly comes through to the destination. It is automatically dismissed by the autopilot since it is "known" that Ψ*Ψ is the probability density.
Most people reading this article will have some QM background, but rarely anything more. The sad thing is that some authors (e.g Greiner) does point out the impossibility of PI, but then goes on to use the same terminology. Other don't (e.g L&L), they use terminology like "particle current density" instead after having dismissed of PI. Most everything in the literature on the Dirac equation, and certainly everything here at Wikipedia, is written with a nineteentwenty-something perspective. Dirac's original rationale was wrong. This is not to degrade his achievement, it is just to point out that they didn't know back then what we know now. I am not going to repeat the arguments for PI being wrong. They are all over the talk page, sometimes with references. YohanN7 (talk) 12:28, 3 November 2013 (UTC)
My point above on multipoles:
"Yes, probability interpretations are impossible when interactions occur, but these are one of a number of reasons for difficulties in describing interactions in relativistic wave equations that would be good to explain."
is not about PIs "in general". I simply meant to describe EM multipole moments and interactions in more detail instead of mentioning in passing. A reader will have no clue why a spin-1 particle will have electric/magnetic dipoles/quadrupoles, and so on for higher spin, and how these relate to interactions and cause internal inconsistencies with the theory, if we don't explain them. Do we need to keep jumping to square 1 every single time and state to death "you cannot have PIs before you even start" and "PIs are ill-defined in RQM" on every topic? Or can we move on? M∧Ŝc2ħεИτlk 17:13, 3 November 2013 (UTC)
You rush on too fast sometimes. I'm still not sure whether you understand that rigorous probability density interpretations are impossible, interactions or no interactions. The way you formulate your point (emphasized above), and the to do list, indicate that you might not. To answer your question: Do we need to keep jumping to square 1 every single time and state to death "you cannot have PIs before you even start" and "PIs are ill-defined in RQM" on every topic? No. Once would be enough near the top of the article. This can be written into the article only after we have agreed on what is right and wrong. YohanN7 (talk) 18:18, 3 November 2013 (UTC)
The issue of multipole moments is far more interesting, and far more advanced. I am of course not objecting to a treatment of these issues. YohanN7 (talk) 18:44, 3 November 2013 (UTC)
That's what I was getting at. Except you simply couldn't help but ignore it and throw in more talk on PIs instead (like we haven't had enough already). M∧Ŝc2ħεИτlk 22:04, 6 November 2013 (UTC)
(Another) Random aside: This PDF paper by Lasenby, Doran & Arcaute on applying GA to EM, gravity, and multiparticle QM states may be useful, may be not (it's freely available and a pointer into the literature in another formalism). M∧Ŝc2ħεИτlk 04:52, 3 November 2013 (UTC)

## Reflist

{{reflist}} is placed here to absorb any/all references from above, and remove the annoying:

Thereare<ref>tagsonthispage,butthereferenceswillnotshowwithoutareflisttemplate(seethehelppage).

beneath the posts of editors in the lowest section. Please do not delete this section. Thanks, M∧Ŝc2ħεИτlk 09:55, 2 November 2013 (UTC)

## Restructuring the article

Restructuring the article according to these pdfs [1][2][3] may help. Of course they are just lecture notes and not reliable sources, but we can add sources to the content written along the lines of these pdfs. M∧Ŝc2ħεИτlk 06:04, 8 November 2013 (UTC)