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.

  • $\begingroup$ 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 '18 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. $\endgroup$ – Thomas Bakx May 12 '18 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 '18 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! $\endgroup$ – Thomas Bakx May 12 '18 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 '18 at 0:06

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)

  • $\begingroup$ 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. $\endgroup$ – Thomas Bakx May 29 '18 at 16:24

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