representations. It must be specified separately. The matrix (or a multiple of modulus −1 times it) may be used in the representation. If parity is included with a minus sign (the matrix ) in the representation, it is called a pseudoscalar representation. Time reversal Time reversal , acts similarly on by By explicitly including a representative for , as well as one for , a representation of the full Lorentz group is obtained. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full Poincaré group, four more generators, the , in addition to the and generate the group. These are interpreted as generators of translations. The time-component is the Hamiltonian . The operator satisfies the relation in analogy to the relations above with replaced by the full Poincaré algebra. By just cancelling the 's, the result would imply that for every state with positive energy in a Hilbert space of quantum states with time-reversal invariance, there would be a state with negative energy . Such states do not exist. The operator is therefore chosen antilinear and antiunitary, so that it anticommutes with , resulting in , and its action on Hilbert space likewise becomes antilinear and antiunitary. It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix. This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, is not a representation. When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. The Dirac representation, , is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation. The third discrete symmetry entering in the CPT theorem along with and , charge conjugation symmetry , has nothing directly to do with Lorentz invariance. Action on function spaces If is a vector space of functions of a finite number of variables , then the action on a scalar function given by produces another function . Here is an -dimensional representation, and is a possibly infinite-dimensional representation. A special case of this construction is when is a space of functions defined on the a linear group itself, viewed as a -dimensional manifold embedded in (with the dimension of the