Wigner's classification
In mathematics and theoretical physics, Wigner's classification is a classification of the
The
The physically relevant representations may thus be classified according to whether
- but or whether
- with
Wigner found that massless particles are fundamentally different from massive particles.
- For the first case
- Note that the generalized eigenspaces of unbounded operators) associated with is aSO(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 stabilizerof
This is the
- 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 and It corresponds to the space of massive scalar fields.
Let M be the hyperboloid sheet defined by:
The Minkowski metric restricts to a
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
The Fourier transform (in all four variables) of such measures yields positive-energy,[clarification needed] finite-energy solutions of the Klein–Gordon equation defined on Minkowski space, namely
without physical units. In this way, the 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
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
- PMID 16578292.
- ISBN 978-0805367034.
- 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. ISBN 978-9971966577.
- ISBN 0-521-55001-7.
- S2CID 121773411.