Talk:Wigner's classification

Misc

Unbounded continuous operators? Where I come from, continuous operators are bounded.

Charles Matthews 08:36, 5 Feb 2004 (UTC)

also, are we really using this method to find the unitary irreps of Poincare? a noncompact group does not have finite dimensional unitary reps. and Poincare is noncompact, and it looks like these are finite dimensional irreps, thus they cannot be unitary. i am going to delete the word unitary, and the word continuous, according to Charles' comment above._Lethe 21:49, 8 Mar 2004 (UTC)

unitary or not?

at some time in the past, i mentioned on this talk page that I don't think the irreps of Poincaré should be unitary. So I deleted that word from the article. Someone recently added the word back in, and it seems that in Wigner's original paper, he does mention unitary irreps. However, I still think it's true that noncompact groups do not have finite dimensional unitary irreps. So what's the deal? Maybe the idea is just that the induced rep of the little group is unitary? Lethe

I guess it's not clear from the article but the irreps here are infinite dimensional (except for the vacuum). Phys 19:21, 1 Aug 2004 (UTC)

Also, the "little groups" (stabilizers) are spin(3) and the double cover of SE(2). The first is compact (but the resulting FULL Poincaré rep is still infinite dimensional since we'd have to integrate over all energy-momenta on the mass shell) and the latter, while noncompact, still admits finite dimensional unitary reps (the continuous spin rep is infinite dimensional, though)! Phys 19:27, 1 Aug 2004 (UTC)

Physics needs „unitary irreducible ray representations" of the Poincare group in order to leave the defining semi-linear form of quantum mechanics invariant (i.e. probabilities).
By the way, there is a second publication by V. Bargmann and E. Wigner. Can someone include it in the list of references? — Preceding unsigned comment added by 130.133.155.68 (talk) 19:03, 31 October 2012 (UTC)

You mean the joint 1948 one, Bargmann, V., & Wigner, E. P. (1948). Group theoretical discussion of relativistic wave equations. Proceedings of the National Academy of Sciences of the United States of America, 34(5), 211? Why? All right, if you thought this is warranted supplementary material, I could link it, for pedagogy. Cuzkatzimhut (talk) 16:16, 13 June 2014 (UTC)

Nonnegative energy

I am a mathematician, so the definition in terms of "nonnegative energy" makes absolutely no sense to me. Also phrases like "sharp mass eigenvalues". What do they mean mathematically? Surely there is room for both kinds of explanation (physical and mathematical). - 72.57.120.3 05:36, 20 September 2006 (UTC)

Admittedly, these are both standard physics colloquialisms. For physical reasons, the energy, i.e. the eigenvalue of the generator P₀ for self-standing particles must be non-negative in the irrep under scrutiny, even though negative values could get topical, on occasion. "Sharp mass eigenvalues" is perhaps less fortunate: the mass m is the eigenvalue of the Casimir invariant operator defined in the next paragraph, and the sharpness dramatizes the qualitative difference between the m=0 and the m>0 representations characterized by such eigenvalues. It automatically excludes unconventional hybrids like unparticle physics, infraparticles, etc... Cuzkatzimhut (talk) 16:48, 13 June 2014 (UTC)

But how is the "sharpness" defined? Is it a property of the representation? In the introduction, it says:

In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative (E ≥ 0) energy irreducible unitary representations of the Poincaré group which have sharp mass eigenvalues.

It seems that sharpness is being used as a criterion for the representation to be included in the Wigner list. Which representations are excluded by this? If sharpness corresponds to the distinction between the m=0 case and the m>0, would we have to exclude one of these? 178.39.122.125 (talk) 10:26, 16 October 2016 (UTC)

I think that the problem can be solved by replacing "sharp" by "either finite or zero". Stefan Groote (talk) 08:10, 21 September 2022 (UTC)

Regarding nonnegative energy: after Fourier transform, the plane waves are $u_{P}(x):=\exp(-i(Et-x_{1}P_{1}-x_{2}P_{2}-x_{3}P_{3}))$. Here $t\equiv x_{0}$, $E\equiv P_{0}$, and $E=\pm {\sqrt {P_{1}^{2}+P_{2}^{2}+P_{3}^{2}+m^{2}}}$. If $E>0$, it is said to have "positive energy". If $E<0$, it has "negative energy". The span of the positive energy ones (suitably restricted so that it is a Hilbert space) is the space under consideration. They are the ones that come, under Fourier transform, from the positive sheet of the two-sheeted hyperboloid defined by the "dispersion relation" $E^{2}=P_{1}^{2}+P_{2}^{2}+P_{3}^{2}+m^{2}$ 178.39.122.125 (talk) 10:55, 16 October 2016 (UTC)

"Trivial" central extensions of the Poincaré group

It's important to point out that, though the Poincaré has no non-trivial central extension, it still admits (trivial) central extensions -- as do all groups. The importance of this is that a central extension of the Poincaré group exists that has the same Wigner classification as the Poincaré group, has a continuous deformation to the non-trivial central extension of the Galilei group, but is immune to Haag's theorem, the Coleman-Mandula theorem and the Leutwyler Theorem. So "trivial" does not mean "physically insignificant"! — Preceding unsigned comment added by 204.128.235.10 (talk) 20:42, 18 March 2013 (UTC)

It does admit a central extension that is non-trivial. However, it is not immune to any of these no-go theorems. The theorems can all be strengthened to a form that brings the trivially-extended Poincaré under their scope.

The "note" is slightly incorrect

The content of the "note" at the bottom about tachyons and off-shell particles are misleading. Tachyonic solutions and virtual states do not furnish irreps of the Poincare group, and so leaving them out is "correct". The paragraph makes it sound like the classification is incomplete. I suggest removing it entirely, or shortening it.

TriTertButoxy (talk) 12:37, 24 July 2015 (UTC)

Article too brief

Very technical, as it should be, but doesn't define stuff. 84.226.185.221 (talk) 12:27, 10 October 2015 (UTC)

Fuill out formula

These two actions can be combined in a clever way using induced representations to obtain an action of P on $L^{2}(M)$ that combines motions of M and phase multiplication.

I would love to see the explicit formula for this. So far, I have been too lazy to work it out or look it up. 178.39.122.125 (talk) 10:16, 16 October 2016 (UTC)

References

These references really spell it out, very mathematically but with all the physics.

(1) George Mackey, Unitary Group Representations in Physics, Probability and Number Theory, 1978.

(2) Shlomo Sternberg, Group Theory and Physics, 1994, Section 3.9. (Wigner classification)

(3) Wu-Ki Tung, Group Theory in Physics, 1985, Chapter 10. (Representations of the Lorentz group and of the Poincare group; Wigner classification)

178.39.122.125 (talk) 10:37, 16 October 2016 (UTC)

I added those to the article. Thanks for the new section on massive scalar fields. YohanN7 (talk) 11:02, 17 October 2016 (UTC)