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I have been reading and writing a lot about the Wigner Classification of irreducible unitaries of (the universal cover of) the Poincaré group lately, both from a physicist's and a mathematician's perspective. I am trying to connect the mathematical treatment (using Mackey's work on systems of imprimitivity) to the physical treatment typically seen in textbooks on quantum field theory (like Weinberg). My question is the following:

In the mathematical treatment, one computes the orbit structure of the action of the universal covering $SL(2,\mathbb{C})$ on $\mathbb{R}^{1,3}$ using level sets of the Lorentzian form $\beta$: for each orbit $O$ and every two points $x,y \in O \subset \mathbb{R}^{1,3}$, the invariant lengths of the vectors are the same, i.e. $\beta(x,x) = \beta(y,y)$. This common value is typically denoted $m^2$.

In the physical treatment, one typically argues via the Casimir operators of the Lie algebra of the Poincaré group, one of which is $P_\mu P ^\mu$, the 'total 4-momentum'. By irreducibility of a representation corresponding to an elementary particle, it must act by a scalar, which is denoted $m^2$ as well. In this case, I understand why this value should be interpreted as the mass of the particle (see Rest Mass and Wigner's Classification where a nice answer is given by Arnold Neumaier).

My question is how the first 'definition' of $m^2$ is related to the second. That is:

Why is the length of a vector in an $SL(2,\mathbb{C})$-orbit equal to the eigenvalue of the Casimir $P_\mu P^\mu$?

I understand that this is a very specific question, and any comments or thoughts (even those not directly attempting to answer my question) are welcome.

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  • $\begingroup$ Each vector in $\mathbb{R}^{1,3}$ can be considered to be an eigenvector of $P_\mu$. So a vector is an eigenvector of $P_\mu P^\mu$ with an eigenvalue equal to the length of the vector. $\endgroup$
    – octonion
    May 12, 2018 at 16:56
  • $\begingroup$ Can you elaborate on the first statement? How exactly does $P_\mu$ act on $\mathbb{R}^{1,3}$ itself (by the identity even, you seem to state)? In the representation associated with the particle, the operator $P_\mu P^{\mu}$ acts on the Hilbert space (because the Poincaré group does, so we can pass to the Lie algebra). I think I am missing something. $\endgroup$
    – user_phys
    May 12, 2018 at 17:39
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    $\begingroup$ You have Weinberg's book right? From a physicist's point of view there is a family of states that are simultaneous eigenvectors of the four $P_\mu$ operators. In Weinberg's notation he calls these states $\Psi_p$ where $P^\mu \Psi_p=p^\mu \Psi_p$. Now this label $p$ that tells you the 4 eigenvalues is exactly the mapping to $\mathbb{R}^{1,3}$ in "the mathematician's perspective." In Weinberg's book we are thinking about orbits of these states under the Lorentz group too. $\endgroup$
    – octonion
    May 12, 2018 at 19:27
  • $\begingroup$ Thanks, that may help. Can you point to the specific pages maybe? If you can turn your comments into an answer it'll have my upvote! $\endgroup$
    – user_phys
    May 12, 2018 at 21:23
  • $\begingroup$ It's all in chapter 2.5 starting on page 62. In particular note 2.5.3 which says a unitary transformation on the state is equivalent to a Lorentz transformation of the eigenvalues. Your $\mathbb{R}^{1,3}$ is just the space of 4-momentum eigenvalues $\endgroup$
    – octonion
    May 13, 2018 at 0:06

2 Answers 2

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The correspondence between group orbits and representations is a very general and fruitful principle which has a multitude of applications in physics.

To be precise, the correspondence is between coadjoint orbits and unitary irreducible representations. This correspondence was discovered by A.A. Kirillov; please see his review article on the subject.

The correspondence is not perfect (not always 1:1). It is perfect for nilpotent Lie groups; it is almost perfect for compact Lie groups. For non-compact Lie groups it is not perfect, but nevertheless, it includes many of the physically important cases. For non-compact Lie groups of the Poincaré type having the structure of a semidirect product of an Abelian and a semisimple Lie group, the correspondence is perfect.

The coadjoint orbits are the orbits of the coadjoint action of a Lie group on the dual of its Lie algebra. In the case of the Poincare group it is a $10$ dimensional Poisson space. The orbits themselves are symplectic (in particular even dimensional) subspaces. In the case of the Poincare group in $3+1$ dimensions, they are the level sets of the mass squared and the magnitude of the Pauli- Lubański vector are of the form $j(j+1)$. When the orbits are quantized according to the rules of geometric quantization; the action of the group on the resulting Hilbert space is via an irreducible unitary representation. The value of the angular momentum becom quantized after geometric quantization. Likewise, for the massless orbits, the value of the Helicity becomes quantized after quantization (both to half integers).

For the Poincaré group in $3+1$ dimensions, everything is performed explicitly in the following two articles by Cariñena, Gracia-Bondía1, Lizzi, Marmo Várilly and Vitale. The codjoint action is given explicitly in tables 1, 2 respectively. Please see also the following article by Cushman and van der Kallen. Please see also my answer to the following PSE question which treats coadjoint orbits in general.

For a modern and extensive review of the theory of coadjoint orbits and their quantization please see the following review by Oblak (Chapter 5)

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  • $\begingroup$ Thanks, I will definitely have a look at that too! In the semidirect product case I figured out a way to 'derive' it explicitly from the induced representation, but I am not completely sure if this is rigorous. I will share it later today. $\endgroup$
    – user_phys
    May 29, 2018 at 16:24
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The process is carried out at a pedestrian level in a way that spans the divide between relativistic and non-relativistic theory, even throwing in the "Carrollian" universe (called therein: the "Archimedean" case).

A Unified Framework For The Symplectic Wigner Classification
https://fdocuments.net/document/a-unified-framework-for-symplectic-wigner-classification.html

This could be generalized to cover all the kinematic groups in the Bacry Levi-Leblond (BLL) classification - in a uniform manner. The reason for the uniformity is that the Poisson manifolds associated with each of the Lie groups in this classification can all be combined into a single Poisson manifold. The BLL classification is, in fact, a three-parameter family of deformations of the Static Group. So, the corresponding unified Poisson manifold is obtained by just throwing in the parameters as three extra coordinates. So, the classification consists of the symplectic leaves of the combined Poisson manifold.

(This is in contrast to what happens if you create the classification by starting with the Static Group and finding all of its deformations. The resulting family is quite a bit larger than the BLL family, and not uniform at all.)

Thus, you can speak of continuously transitioning between the different kinematic groups. This supersedes the method of Lie Group contractions, and the results obtained therein.

A key point worth making here: uniformity can only be achieved after centrally extending all the kinematic groups. For the Galilei group, this yields the Bargmann group, which is more correctly regarded as the symmetry group for non-relativistic theory. What this deforms into is not the Poincaré group, but its (trivial) central extension. All the members in the BLL family have trivial central extensions, except those that have absolute simultaneity (that includes: Galilei, Static and the two Newton-Hooke groups). Both the Static and Carroll (or "Archimedean") group have the same central extension, even though the central extension is non-trivial for the Static group.

This has a material bearing on the classification question for Relativity as follows: our current understanding of the Bargmann group being the correct symmetry group for non-relativistic theory arose only late in the game, some time in the 1950's, well past the time that non-relativistic theory was superseded by relativity. So, it represents a retro-update to Newtonian physics. Before that, it was the Galilei group that was understood to be the symmetry group for non-relativistic theory.

But Relativity, itself, was formulated with a Correspondence Principle that connected to the older understanding of non-relativistic theory. The retro-update of the latter entails a revision of the Correspondence Principle, too: it should now retarget a theory that has been lifted, but the other end of the "Correspondence Principle" arrow, on the Relativity side, has not been lifted to match this retargeting. It also needs to be lifted. The result adds in an 11th generator, for the trivial central extension. Doing this also happens to clarify other seemingly obscure issues, like why the Dirac algebra needs to be complexified or why/how the "zero of energy is relative" (as is often asserted in the setting of quantum field theory, but is not justified from the vantage point of Poincaré group representations).

The resulting classification is actually the same as it is for the Poincaré group, but an extra degree of freedom appears in each subclass, arising from the extra generator. For the vacuum class, the extra degree of freedom corresponds to "vacuum energy".

Moreover, BLL family, in its entirety, admits a uniform (but generally non-linear) five-dimensional coordinate representation that includes the Bargmann geometry of non-relativistic theory as one of its cases. The centrally extended Poincaré group is associated with the geometry that has $dx^2 + dy^2 + dz^2 + 2 dt du + (1/c)^2 du^2$ and $ds = dt + (1/c)^2 du$ as its invariants, the latter invariant being the differential for proper time $s$, itself. Minkowski geometry is obtained by setting the former invariant to 0 (i.e. as a light cone in a 4+1 dimensional geometry). For non-relativistic theory, the corresponding geometry is obtained by replacing each of the $(1/c)^2$ factors by 0.

Time permitting, it might get written up and updated to a form suitable for for JMP. But, it's not a high-priority issue. But it probably should be TeX'ed and cleaned up and submitted, though I prefer an IoP journal.

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