On Operator Equations in Particle Physics

Abstract

1 Many problems in Particle Physics and Quantum Field Theory reduces to the problem of finding solutions of certain operator equations. The most popular are operator equations for currents, e.g.,

$$\left[ {J_\ell (x),\,J_k (y)} \right] = iC_{2k}^s \delta _3 \left( {\overline x - \overline y } \right)J_s (\overline x )$$

((1))

or dynamical equations for scalar quantum fields like

$$( + m^2 )\Phi (x) = \lambda \Phi ^n (x)$$

((2))

The commonly used method of solutions of operator equations like (1) is based on Weyl trick which he used for a classification of representations of Heisenberg commutation relations. It consists of the association with the commutation relations (1), the well-defined infinite-dimensional topological group G. Using then the well elaborated global representation theory of G and passing to the infinitesimal representations, one obtains a class of solutions of current commutation relations (1) (cf. Reference 1).

† Presented at the NATO Summer School in Mathematical Physics, Istanbul, 1970.