mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the representation theory of the Poincaré group is an example of the

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

of a

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

that is neither a

compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...

nor a semisimple group. It is fundamental in

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

. In a physical theory having

Minkowski space In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...

as the underlying

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

, the space of physical states is typically a representation of the Poincaré group. (More generally, it may be a

projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where G ...

, which amounts to a representation of the double cover of the group.) In a

classical field theory A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called qua ...

, the physical states are sections of a Poincaré-equivariant

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...

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 In mathematics, equivariance is a form of symmetry for function (mathematics), functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are Group action ( ...

. 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 representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in the ca ...

s, see

Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (These u ...

. In quantum mechanics, the state of the system is determined by the Schrödinger equation, which is 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. There are no finite unitary representations of the full Lorentz (and thus Poincaré) transformations due to the non-compact nature of Lorentz boosts (rotations in Minkowski space along a space and time axis). However, there are finite non-unitary indecomposable representations of the Poincaré algebra, which may be used for modelling of unstable particles. In case of spin 1/2 particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product preserved by this representation by associating a 4-component

Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain comb ...

\psi

with each particle. These spinors transform under Lorentz transformations generated by the

gamma matrices In mathematical physics, the gamma matrices, \ \left\\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra \ \mathr ...

(

\gamma_

). It can be shown that the scalar product :

\langle\psi, \phi\rangle = \bar\phi = \psi^\gamma_0\phi

is preserved. It is not, however, positive definite, so the representation is not unitary.

References

* * * * *.

References

Notes

See also