Antiparticle - Quantum Field Theory

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 a_k 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 a_k and b_k shows that one is the annihilation operator for particles and the other for antiparticles. This is the case of a fermion.

This approach is due to Vladimir Fock, Wendell Furry and Robert Oppenheimer. If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator; therefore, real scalar fields describe neutral bosons. Since complex scalar fields admit two different kinds of annihilation operators, which are related by conjugation, such fields describe charged bosons.