Symmetries and Relativistic Scalar Quantum Fields

Abstract

Just as for non-relativistic quantum fields, the theory of free relativistic scalar quantum fields starts by taking as phase space an infinite dimensional space of solutions of an equation of motion. Quantization of this phase space proceeds by constructing field operators which provide a representation of the corresponding Heisenberg Lie algebra, using an infinite dimensional version of the Bargmann–Fock construction. In both cases, the equation of motion has a representation-theoretical significance: It is an eigenvalue equation for the Casimir operator of a group of space-time symmetries, picking out an irreducible representation of that group. In the non-relativistic case, the Laplacian $\Delta$ was the Casimir operator, the symmetry group was the Euclidean group E(3), and one got an irreducible representation for fixed energy.