Spectral representation in connection with Schwinger terms

Abstract

The Lehmann-Källen representation shows that Schwinger terms must appear in the commutation relations between spatial and time components of any four-vector operator with four hermitian components. These Schwinger terms disappear from the expectation values of the commutator [ O μ( x), O ν( Y)] when the time component of the fourvector is anti-hermitian while the spatial components are hermitian for the ordinary scalar product. An application of these considerations concerns the free electromagnetic field theory, where such hermiticity conditions are used in the Gupta-Bleuler method.