The theory of relativistic quantum systems. The origins of quantum field theory are connected with problems of the interaction of matter with radiation and with the attempts to construct a relativistic quantum mechanics (P.A.M. Dirac (1927), W. Heisenberg, W. Pauli, and others). At relativistic (that is, high) energies there cannot be a consistent quantum mechanics of particles, since a relativistic quantum particle is capable of producing new (similar or different) particles and of vanishing itself. As a result one cannot distinguish a definite number of mechanical degrees of freedom connected with this particle, but one has to talk about a system with a variable, generally infinite, number of degrees of freedom. Quantum field theory unifies the description of fields and particles, which in classical physics appear as two distinct entities.

The notion of a quantum field plays a central role in the theory. It is convenient to explain how it is introduced by the example of an electromagnetic field, since this is the only field having a clear content both in the classical and the quantum case.

A classical electromagnetic field satisfies the Maxwell equations. These equations can be rewritten in the form of the Hamilton canonical equations, so that the field potential plays the role of coordinate while its derivative with respect to time is the momentum in the corresponding phase space. The field is represented as a canonical system with an infinite number of degrees of freedom since the potential at each point is an independent coordinate. This system can be quantized in the same way as an ordinary mechanical system. In the quantum situation, the basic concepts are the states, described by vectors in Hilbert space, and the observables, described by self-adjoint operators (cf.Self-adjoint operator) acting on this space. Quantization consists in replacing the canonical coordinates and momenta by operators so that the Poisson brackets are replaced by commutators of the corresponding operators. A quantum field becomes an operator acting on state vectors, and it brings about a transition between states with different numbers of quantum particles (photons), that is, the operator describes the creation and destruction (radiation and absorption) of the quanta of the field.

In a similar way one can associate a quantum field with fundamental particles of any other sort. The equations for the operator of a free field are obtained from the fundamental requirement of relativity theory, namely the conditions of invariance with respect to the Poincaré group. The type of particles is characterized by the rest mass , the spin, that is, the intrinsic angular momentum , which can take integer or half-integer values including zero, and the various charges (electric charge, baryon number, lepton number, etc.). The first two numbers define anirreducible representation of the Poincaré group by which the field is transformed and therefore the equations of the field are transformed as well.