interchanged by the combined application of charge conjugation , parity and time reversal . and are linear, unitary operators, is antilinear and antiunitary, . If denotes the quantum state of a particle with momentum and spin whose component in the z-direction is , then one has where denotes the charge conjugate state, that is, the antiparticle. In particular a massive particle and its antiparticle transform under the same irreducible representation of the Poincaré group which means the antiparticle has the same mass and the same spin. If , and can be defined separately on the particles and antiparticles, then where the proportionality sign indicates that there might be a phase on the right hand side. As anticommutes with the charges, , particle and antiparticle have opposite electric charges q and -q. Quantum field theory This section draws upon the ideas, language and notation of canonical quantization of a quantum field theory. One may try to quantize an electron field without mixing the annihilation and creation operators by writing where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E(k), and ak denotes the corresponding annihilation operators. Of course, since we are dealing with fermions, we have to have the operators satisfy canonical anti-commutation relations. However, if one now writes down the Hamiltonian then one sees immediately that the expectation value of H need not be positive. This is because E(k) can have any sign whatsoever, and the combination of creation and annihilation operators has expectation value 1 or 0. So one has to introduce the charge conjugate antiparticle field, with its own creation and annihilation operators satisfying the relations where k has the same p, and opposite σ and sign of the energy. Then one can rewrite the field in the form where the first sum is over positive energy states and the second over those of negative energy. The energy becomes where E0 is an infinite negative constant. The vacuum state is defined as the state with no particle or antiparticle, i.e., and . Then the energy of the vacuum is exactly E0. Since all energies are measured relative to the vacuum, H is positive definite. Analysis of the properties of ak and bk shows that one is the annihilation operator for particles and the other for antiparticles. This is the case of a fermion.