Representation theory of the Poincaré group

Lie groups

Classical groups General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n)
Simple Lie groups List of simple Lie groups Infinite simple Lie groups: A_n, B_n, C_n, D_n, Exceptional simple Lie groups: G₂ F₄ E₆ E₇ E₈
Other Lie groups Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group
Lie algebras Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry
Structure of semi-simple Lie groups Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Representation of a Lie group Representation of a Lie algebra
Lie groups in Physics Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the Galilean group

In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics.

In a physical theory having Minkowski space as the underlying spacetime, the space of physical states is typically a representation of the Poincaré group. (More generally, it may be a projective representation, which amounts to a representation of the double cover of the group.)

In a classical field theory, the physical states are sections of a Poincaré-equivariant vector bundle over Minkowski space. The equivariance condition means that the group acts on the total space of the vector bundle, and the projection to Minkowski space is an equivariant map. Therefore the Poincaré group also acts on the space of sections. Representations arising in this way (and their subquotients) are called covariant field representations, and are not usually unitary.

For a discussion of such unitary representations, see Wigner's classification.

In quantum mechanics, the state of the system is determined by the Schrödinger equation, which is only invariant under Galilean transformations. Quantum field theory is the relativistic extension of quantum mechanics, where relativistic (Lorentz/Poincaré invariant] wave equations are solved, "quantized", and act on a Hilbert space composed of Fock states; eigenstates of the theory's Hamiltonian which are states with a definite number of particles with individual 4-momentum. There are no finite unitary representations of the full Lorentz (and thus Poincaré) transformations due to the nature of Lorentz boosts (rotations in Minkowski space along a space and time axis).

Representation theory of the Poincaré group

See also