Talk:Bargmann–Wigner equations

For now not perfect...

but at least a start. The literature is completely confusing so this is an attempt to simplify bit by bit. Feel free to complain but I can't write much more than what's in the article for now, if you're an expert in QFT just make the changes. Maschen (talk) 18:23, 17 September 2012 (UTC)

Before anyone complains about how badly written the history section is, I'm going to read the cited paper in that section before making it more coherent. M∧Ŝc²ħεИ_τlk 14:28, 16 December 2012 (UTC)

Next steps

"To do" list (mainly for me... don't take it personally... feel free to edit this list if it becomes more logical)

In (approximate) order: Define the generalized gamma matrix $\gamma ^{\mu _{1}\mu _{2}\cdots \mu _{2s}}$ in the Joos-Weinberg equation. Unfortunately in the literature, it's either rare, or potentially dubious, or difficult to understand... Rewrite history section, better reasons and motivations the numerous wave equations werere developed. more on Bargmann, Wigner, and how their equations came about. This is where much of the "readable" physics behind all the relativistic equations may end up... Add more mathematical background behind the equations in descriptive terms, outlining/summarizing what the papers say, such as group representations. At the same time FAR, FAR, far more on the physical interpretations of the mathematics are obvious to add... Add the Galilean relativity formulation. More context on relativistic Hamiltonians. Make clear what component-minimization of the wavefunction actually is and means. Theoretical predictions of experiments (doesn't seem easy to find in the literature, in spite of extensive searches... Maybe the BW eqns are just for theoretical interest?...) Add more context to the Lagrangian section. Find the first name of H. Joos (No really, it's very hard!... A Google search all over the place didn't work well... everyone EXCEPT the H. Joos in this article shows up; when it's the one to look for, and in papers, they tend to only write the first initial "H"... which doesn't help!...)

Currently easier said than done... M∧Ŝc²ħεИ_τlk 18:16, 18 December 2012 (UTC)

The current history section is about relativistic wave equations in general, so I'll move this section to that article (and sort out the equation numbering in this article). M∧Ŝc²ħεИ_τlk 11:18, 19 December 2012 (UTC)

Heisenberg picture

I think we need to adopt a broader view of what the BW are in the context of QFT. In QFT one uses either the Heisenberg picture or some version of the interaction picture almost all of the time. This means that all of the (interesting) dynamics lie in the operators, while the states are constant in time. The JW equations are, as originally defined, equations relating the operators, not the states. Even the Dirac equation is by default an operator equation in QFT, (it is Heisenbergs equation of motion) while in RQM it is perhaps by default a wave equation in the Schrödinger picture. The equations often look formally about the same, but sometimes they don't. I don't know about the present case.

There is a difference also in the group theoretical aspect. If Ψ→DΨ where Ψ is a state, then DΨD^-1 for operators on the states (same D). One can still without error write Ψ→DΨ for operator, but then another D is implied.

One solution that would make it easier for the reader is to use operator hats, even though it isn't standard in the literature. Some authors in QFT use it though, e.g Walter Greiner in "Field Quantization". YohanN7 (talk) 04:06, 16 February 2013 (UTC)

Ok - I'll introduce hats for the operators. Also the transformation DΨD^-1 is important to mention, we'll get to that.

About your proposed inclusion it looks a bit too heavy for a typical reader, but it can always be simplified later (and difficulty/unfamiliarity can't be helped). Feel free to add. M∧Ŝc²ħεИ_τlk 08:51, 16 February 2013 (UTC)

Forgot to mention on the DΨD^-1 transformation, are you thinking along the lines of this? (link is in the external links section). Not really a "reliable source" since it's one of Prof. Radovan Dermisek's class notes (Indiana University), but certainly a relevant external link. M∧Ŝc²ħεИ_τlk 11:38, 16 February 2013 (UTC)

The DΨD^-1 transformation is general. If V is a vector space and if Π_V is a representation of a group G acting on V, then Π__End(V) given by Π__End(V)(g):End(V) -> End(V); A->Π(g)AΠ^-1(g) for A∈End(V) is always a representation on End(V).

For Lie algebra reps, the corresponding promotion is π__End(V) is given by A -> [π(X), A] = π(X)A - Aπ(X)

See also Representation theory of the Lorentz group#Induced representations YohanN7 (talk) 18:07, 16 February 2013 (UTC)

The hats look good. At least in the JW equation, the Ψ itself needs one too, since it is a field operator, i.e. a Fourier superposition of creation and annihilation operators (whose coefficients may be regarded as momentum space wave functions), but this can probably wait. I'v begun reading the Tóth paper. I must say that his use of notation and terminology in group theory is unfamiliar to me, especially with regard to what constitutes a "complex" resp. "real" representation and what constitutes an "irreducible" representation. Moreover, he treats the 3-dimensional complex Lie algebra sl(2,C) as a six-dimensional real Lie algebra. This latter thing is very helpful though since the isomorphism with the 6-dimensional real Lie algebra so(3;1) is transparent. [SL(2;C is the universal covering group of the Lorentz group, see Weinberg section 2.7, so that so(3;1) ≈ sl(2C)]. Also, the m, and the n does not refer to to the same thing as for the Lorentz group. What I wrote in the math ref desk was probably correct, but that was, I realize now, pure luck. YohanN7 (talk) 20:41, 17 February 2013 (UTC)

I have not managed to figure out what the basic building blocks (1/2, 0) and (0, 1/2) are in the Tóth formalism are. He doesn't actually say. (1/2, 0) may be the standard representation on C² and (0, 1/2) may simply the complex conjugate matrices of the matrices of sl(2,1) acting on C² as well. Do you know? Found it. ((A.18) and (A.19).)

B t w, I meant that my proposed inclusion to go into Representation theory of the Lorentz group, not here. YohanN7 (talk) 21:13, 17 February 2013 (UTC)

Ok (about your work). I left out the hats - since when are fields hatted in the literature? It may be technically correct but it would just clutter the notation, which is never nice... Creation and annihilation operators to act on the BW fields can definitely wait until the article has more structure, as can Fourier transforms between position and momentum spaces (well-known properties of wavefunctions anyway). M∧Ŝc²ħεИ_τlk 00:40, 18 February 2013 (UTC)

Quantum fields are sometimes hatted, especially in introductory QFT texts, e.g. Field Quantization by Walter Greiner, to emphasize that they are oparator fields, not fields representing states, i.e. not wavefunctions representing particles. The JW equation is a differential equation for a field operator, so that the Ψ there is not a wavefunction describing a particle. It is an operator that can act on Hilbert space to create or destroy a particle. The QFT Dirac equation describe a field operator as well, and the Ψ entering in that one is not the wavefunction referred to in RQM (i.e. not the original one in the sense of Dirac). If you have Weinberg at hand, check out the beginning of chapter 14, where he derives the Dirac original wave equation. Terminology is severely abused in QFT, so an author or lecturer may mutter the word "operator" in "field operator" a couple of times, but then consistently say field when he means field operator. When he really means wavefunctions he may say coefficient or coefficient function.

Typically there are annihilation fields

$\psi _{l}^{+}=\sum \int u_{l}(x;p,\sigma ,n)a(p,\sigma ,n)d^{3}p,$

creation fields

$\psi _{l}^{-}=\sum \int v_{l}(x;p,\sigma ,n)a^{\dagger }(p,\sigma ,n)d^{3}p,$

and fields

$\psi =\kappa \psi ^{+}+\lambda \psi ^{-}.$

The u and the v are the wave functions, here in momentum space. (the a and a-dagger are annihilation and creation operators) It is a ψ of this type that enters in the JW equation, not a wavefunction. This is not to say that there aren't wave equations corresponding to the JW equations. What I am saying is that the setting in QFT is the Heisenberg picture (or the interaction picture) where the dynamics lie in the operators. As I mentioned, before, I don't know in which picture the BW equations are formulated, but we must certainly adopt a broader view than relativistic wave equations when the JW equations are treated. YohanN7 (talk) 01:50, 18 February 2013 (UTC)

To clarify what I mean, in the case of the Dirac equation in QFT,

$(\gamma ^{\mu }\partial _{\mu }-m)\psi =0.$

holds for the field (operator) and

$(ip^{\mu }\gamma _{\mu }+m)u(p,\sigma )=0,$

$(-ip^{\mu }\gamma _{\mu }+m)v(p,\sigma )=0$

holds for the coefficient functions in momentum pace.

None of these is the free field Dirac equation in the original sense. It may be something similar in the case of the JW equation, i.e. that the wave equations are formally similar to the operator equations. I think the article should be clear on these things.

I'll try to get time to read the original BW paper soon. It seems, at least according to the title, to be formulated in RQM. YohanN7 (talk) 02:29, 18 February 2013 (UTC)

We can of course discuss the operator aspect of the "field/wavefuntion", but for now a field/wave interpretation would be easier. M∧Ŝc²ħεИ_τlk 16:17, 20 February 2013 (UTC)

I think you misunderstand me.

I don't want to blindly endow each and every Ψ with an operator hat. Only those which are operators should have a hat. In QFT, you must interpret "field" as "being spacetime dependent" when reading the literature, nothing more. The terminology (and the notation) in the literature does not in the slightest way indicate if we are talking about a state/wavefunction or an operator. This obviously leads to initial confusion. To further confuse the issue, in some cases, the differential equation governing an operator is formally identical to the equation governing the states. This is what happens with the Dirac equation for the quantum field (an operator). It becomes the classic Dirac wave equation when it is projected onto the Hilbert space.This is the reason that I vote for operator hats where operator hats are due.

At any rate, the JW equation as formulated in the original paper is an equation for a field operator. Moreover it's transformation rule under LT is Ψ → DΨD^-1 according to equation (7.2) in http://theory.fi.infn.it/becattini/files/weinberg1.pdf, which (if nothing else) makes it clear that Ψ is an operator in that case. The wave interpretation would is easier, but only provided it is the correct interpretation. The master equation in QFT is the Heisenberg equation of motion. The wave equations for the states are always derived from it, never the other way around. If we are lucky, then the wave equations and the field operator equations are formally identical. They may very well be. Even so we need to make the distinction, otherwise inexperienced readers will be confused when looking up the references.

Another thing, the operator assignment

${\hat {P}}_{\mu }=i\hbar \partial _{\mu }$

appears to miss a minus sign, see momentum operator. YohanN7 (talk) 17:48, 20 February 2013 (UTC)

Is this explained in the note to reference #2? YohanN7 (talk) 19:16, 20 February 2013 (UTC)

Well much of the misunderstanding is because I have yet to understand all this in detail... As much as I really appreciate your detailed explanations - I can't take it all in yet. This article was written from the papers (and yes, that's a bad idea and have already been told not to write about stuff not understood, but was still ignorant to make a start anyway...).

The momentum operator is correct, and the matrix operator in the Dirac equation is given in ref 2. If it was -P the equation would be

$(\gamma ^{\mu }{\hat {P}}_{\mu }+mc)\Psi =0\,,$

which is the Dirac equation for particles of negative mass (that's referenced by The Road to Reality, Penrose). Btw - I inserted the 4-momentum operator section in the momentum operator article from a depreciated version of the Dirac equation article. M∧Ŝc²ħεИ_τlk 20:01, 20 February 2013 (UTC)

Forgot to mention, this paper by D. Shay (1968) (ref 14) is a good one for the Lagrangians of the JW equation, one for spin-half integer and spin-integer, with a bit more on the generalized gamma. M∧Ŝc²ħεИ_τlk 15:42, 21 February 2013 (UTC)

History

A history sections should probably begin with a reference to http://courses.theophys.kth.se/SI2390/wigner_1939.pdf, which is Wigner's classification. This essentially partitions Hilbert space into various representations of the Lorentz group. If I understand correctly, the results in Jeffreys paper and in Weinbergs first paper indicate that there is further another partitioning ,one according to particle spin, that goes across the group representation partitioning. Does it sound obscure? YohanN7 (talk) 19:55, 20 February 2013 (UTC)

Not obscure - I have come across that paper before and meant to add it (it was left out temporarily at first thought as something marginally relevant). I'll insert it now. M∧Ŝc²ħεИ_τlk 20:01, 20 February 2013 (UTC)

Added to article. M∧Ŝc²ħεИ_τlk 20:10, 20 February 2013 (UTC)

...charge conjugation, time reversal symmetry, and parity...

There is the sentence:

"It is suitable when the symmetries of charge conjugation, time reversal symmetry, and parity are good.

but this means that "C, P, T conjugations/inversions are each fulfilled (not violated)"?

Next is the statement:

"The representations D^{(j, 0)} and D^{(0, j)} can each separately represent particles of spin j. A state or quantum field in such a representation would satisfy no field equation except the Klein-Gordon equation."

is it correct to interpret the D^{(j, 0)} representation corresponding to the particle spinor transformation and D^{(0, j)} rep corresponding to the antiparticle spinor transformation, as inferred from the (⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠) representation for the Dirac spinor transformation? M∧Ŝc²ħεИ_τlk 15:42, 21 February 2013 (UTC)

Yes, spot on. Weinbergs paper starts wih D^{(j, 0)} (D^{(0, j)}) and then proceeds to D^{(j, 0)} ⊕ D^{(0, j)}. My sentence about the discrete symmetries C, P, and T is murky and full of jargon. Needs work. YohanN7 (talk) 12:03, 22 February 2013 (UTC)

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Transfer of content from Bargmann–Wigner equations to Joos–Weinberg equation

See this (BW article) and this (JW article). They are different equations, and can be sourced separately. M∧Ŝc²ħεИ_τlk 18:58, 28 December 2016 (UTC)

They are two faces of the same coin (QFT a la Weinberg vs RQM, also there are equivalent forms of e.g. the BW equations). The whole idea is to take one Lorentz group rep containing the appropriate spin (generally, more spins occur in a rep), then to project the chosen spin out using differential constraints. One also typically sets up CPT constraints, while this might be problematic. See references given in the below reference (primarily a 1968 Tung paper I can't recall the title of)

The reference

• Tung, Wu-Ki (1985). Group Theory in Physics (1st ed.). New Jersey·London·Singapore·Hong Kong: World Scientific. ISBN 978-9971966577. {{cite book}}: Invalid |ref=harv (help)

actually treats it a bit at the undergraduate level. Separate articles are still probably motivated. YohanN7 (talk) 13:19, 29 December 2016 (UTC)

Arguably, this article (BW eqns) need not exist, and all high spin RWEs (such as the Dirac–Fierz–Pauli equations and Bhaba(?) equations and likely others) could be in the relativistic wave equations article. Also relativistic wave equation may as well be included in relativistic quantum mechanics.

Just because they are two faces of the same coin, it doesn't mean separate articles can't be written. This article isn't focused enough (most of the irrelevancies were mine back in 2012/2013, and will be cut out). It would be clearer if this article was about the $D^{\mathrm {BW} }=\bigotimes _{r=1}^{2j}\left[D_{r}^{(1/2,0)}\oplus D_{r}^{(0,1/2)}\right]$ representation while the other about $D^{\mathrm {JW} }=D^{(j,0)}\oplus D^{(0,j)}$. M∧Ŝc²ħεИ_τlk 16:27, 29 December 2016 (UTC)

I definitely think this article should exist, but perhaps under the name of Bargmann–Wigner programme, and it should also provide some historical and technical background. It started 1939 with Wigner's monumental paper, his most important work (according to himself, naturally suggested to W 10 years before that by Dirac), #2 in the ref list (the year is wrong there). Much background technical could be placed in Representation theory of the Poincaré group, which at present is meager. YohanN7 (talk) 11:10, 30 December 2016 (UTC)

OK, feel free to rename (move article) when you want, and I'll move Wigner's 1937 reference to a history section. M∧Ŝc²ħεИ_τlk 11:21, 30 December 2016 (UTC)