Wigner's classification

In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative $~(~E\geq 0~)~$ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (These unitary representations are infinite-dimensional; the group is not semisimple and it does not satisfy Weyl's theorem on complete reducibility.) It was introduced by Eugene Wigner, to classify particles and fields in physics—see the article particle physics and representation theory. It relies on the stabilizer subgroups of that group, dubbed the Wigner little groups of various mass states.

The Casimir invariants of the Poincaré group are $~C_{1}=P^{\mu }\,P_{\mu }~,$ (Einstein notation) where P is the 4-momentum operator, and $~C_{2}=W^{\alpha }\,W_{\alpha }~,$ where W is the Pauli–Lubanski pseudovector. The eigenvalues of these operators serve to label the representations. The first is associated with mass-squared and the second with helicity or spin.

The physically relevant representations may thus be classified according to whether

• $~m>0~;$
• $~m=0~$ but $~P_{0}>0~;\quad$ or whether
• $~m=0~$ with $~P^{\mu }=0~,{\text{ for }}\mu =0,1,2,3~.$

Wigner found that massless particles are fundamentally different from massive particles.

For the first case

Note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with $~P=(m,0,0,0)~$ is a representation of SO(3).

In the ray interpretation, one can go over to Spin(3) instead. So, massive states are classified by an irreducible Spin(3) unitary representation that characterizes their spin, and a positive mass, m.

For the second case

Look at the stabilizer of

$~P=(k,0,0,-k)~.$

This is the double cover of SE(2) (see projective representation). We have two cases, one where irreps are described by an integral multiple of ⁠1/2⁠ called the helicity, and the other called the "continuous spin" representation.

For the third case

The only finite-dimensional unitary solution is the trivial representation called the vacuum.

Massive scalar fields

As an example, let us visualize the irreducible unitary representation with $~m>0~,$ and $~s=0~.$ It corresponds to the space of massive scalar fields.

Let M be the hyperboloid sheet defined by:

$~P_{0}^{2}-P_{1}^{2}-P_{2}^{2}-P_{3}^{2}=m^{2}~,\quad$ $~P_{0}>0~.$

The Minkowski metric restricts to a Riemannian metric on M, giving M the metric structure of a hyperbolic space, in particular it is the hyperboloid model of hyperbolic space, see geometry of Minkowski space for proof. The Poincare group P acts on M because (forgetting the action of the translation subgroup ℝ⁴ with addition inside P) it preserves the Minkowski inner product, and an element x of the translation subgroup ℝ⁴ of the Poincare group acts on $~L^{2}(M)~$ by multiplication by suitable phase multipliers $~\exp \left(-i{\vec {p}}\cdot {\vec {x}}\right)~,$ where $~p\in M~.$ These two actions can be combined in a clever way using induced representations to obtain an action of P acting on $~L^{2}(M)~,$ that combines motions of M and phase multiplication.

This yields an action of the Poincare group on the space of square-integrable functions defined on the hypersurface M in Minkowski space. These may be viewed as measures defined on Minkowski space that are concentrated on the set M defined by

$E^{2}-P_{1}^{2}-P_{2}^{2}-P_{3}^{2}=m^{2}~,\quad$ $~E~\equiv ~P_{0}>0~.$

The Fourier transform (in all four variables) of such measures yields positive-energy, finite-energy solutions of the Klein–Gordon equation defined on Minkowski space, namely

${\frac {\partial ^{2}}{\partial t^{2}}}\psi -\nabla ^{2}\psi +m^{2}\psi =0,$

without physical units. In this way, the $~m>0,\quad s=0~$ irreducible representation of the Poincare group is realized by its action on a suitable space of solutions of a linear wave equation.

The theory of projective representations

Physically, one is interested in irreducible projective unitary representations of the Poincaré group. After all, two vectors in the quantum Hilbert space that differ by multiplication by a constant represent the same physical state. Thus, two unitary operators that differ by a multiple of the identity have the same action on physical states. Therefore the unitary operators that represent Poincaré symmetry are only defined up to a constant—and therefore the group composition law need only hold up to a constant.

According to Bargmann's theorem, every projective unitary representation of the Poincaré group comes from an ordinary unitary representation of its universal cover, which is a double cover. (Bargmann's theorem applies because the double cover of the Poincaré group admits no non-trivial one-dimensional central extensions.)

Passing to the double cover is important because it allows for half-odd-integer spin cases. In the positive mass case, for example, the little group is SU(2) rather than SO(3); the representations of SU(2) then include both integer and half-odd-integer spin cases.

Since the general criterion in Bargmann's theorem was not known when Wigner did his classification, he needed to show by hand (§5 of the paper) that the phases can be chosen in the operators to reflect the composition law in the group, up to a sign, which is then accounted for by passing to the double cover of the Poincaré group.

Further classification

Left out from this classification are tachyonic solutions, solutions with no fixed mass, infraparticles with no fixed mass, etc. Such solutions are of physical importance, when considering virtual states. A celebrated example is the case of deep inelastic scattering, in which a virtual space-like photon is exchanged between the incoming lepton and the incoming hadron. This justifies the introduction of transversely and longitudinally-polarized photons, and of the related concept of transverse and longitudinal structure functions, when considering these virtual states as effective probes of the internal quark and gluon contents of the hadrons. From a mathematical point of view, one considers the SO(2,1) group instead of the usual SO(3) group encountered in the usual massive case discussed above. This explains the occurrence of two transverse polarization vectors $~\epsilon _{T}^{\lambda =1,2}~$ and $~\epsilon _{L}~$ which satisfy $~\epsilon _{T}^{2}=-1~$ and $~\epsilon _{L}^{2}=+1~,$ to be compared with the usual case of a free $~Z_{0}~$ boson which has three polarization vectors $~\epsilon _{T}^{\lambda }{\text{ for }}\lambda =1,2,3~;$ each of them satisfying $~\epsilon _{T}^{2}=-1~.$

See also

• Induced representation
• Representation theory of the diffeomorphism group
• Representation theory of the Galilean group
• Representation theory of the Poincaré group
• System of imprimitivity
• Pauli–Lubanski pseudovector

References

• 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): 211–223. Bibcode:1948PNAS...34..211B. doi:10.1073/pnas.34.5.211. PMC 1079095. PMID 16578292.

• Mackey, George (1978). Unitary Group Representations in Physics, Probability and Number Theory. Mathematics Lecture Notes Series. Vol. 55. The Benjamin/Cummings Publishing Company. ISBN 978-0805367034.

• Sternberg, Shlomo (1994). "§3.9. Wigner classification". Group Theory and Physics. Cambridge University Press. ISBN 978-0521248709.

• Tung, Wu-Ki (1985). "Chapter 10. Representations of the Lorentz group and of the Poincare group; Wigner classification". Group Theory in Physics. World Scientific Publishing Company. ISBN 978-9971966577.

• Weinberg, S. (2002). "Chapter 2. Relativistic quantum mechanics". The Quantum Theory of Fields. Vol. I. Cambridge University Press. ISBN 0-521-55001-7.

• Wigner, E.P. (1939). "On unitary representations of the inhomogeneous Lorentz group". Annals of Mathematics. 40 (1): 149–204. Bibcode:1939AnMat..40..149W. doi:10.2307/1968551. JSTOR 1968551. MR 1503456. S2CID 121773411.