The inhomogeneous wave equation in local relativistic quantum field theory

• Published in: Annals of physics Vol. 11; no. 2; pp. 201 - 239
• Main Authors: Wightman, A.S, Epstein, H
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.1960
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/0003-4916(60)90133-0

The equation (□ + m 2) u( x) = j( x), where j( x) is a known local field is solved for u( x) in the form u( x)= u (0)( x)−∫ δ R ( x− x′) dx′ j( x′), where u (0)( x) is a free field. The possibility of choosing u (0)( x) so that u( x) is local is studied by considering vacuum expectation values containing j( x) and u( x). It is shown that in the case j( x) = gφ 2( x), φ( x) a free field, no u (0)( x) with the desired properties exists. The argument is generalized to the case of the equations of a neutral vector meson field, B, interacting with a spinor field, χ: (y u∂ u+m)χ(x) = ie γ uA u(x)ψ(x) , ∂ v[∂ v B u−∂ uB v] + K 2B u(x) = ie:ψ †(x)γ uψ(x): .