Foundations of the Theory of Dynamical Systems of Infinitely Many Degrees of Freedom, II

Abstract

The notion of quantum field remains at this time still rather elusive from a rigorous standpoint. In conventional physical theory such a field is defined in essentially the same way as in the original work of Heisenberg and Pauli (1) by a function ϕ(x, y, z, t) on space-time whose values are operators. It was recognized very early, however, by Bohr and Rosenfeld (2) that, even in the case of a free field, no physical meaning could be attached to the values of the field at a particular point—only the suitably smoothed averages over finite space-time regions had such a meaning. This physical result has a mathematical counterpart in the impossibility of formulating ϕ(x, y, z, t) as a bona fide operator for even the simplest fields (in any fashion satisfying the most elementary non-trivial theoretical desiderata), while on the other hand for suitable functions f, the integral ∫ϕ(x, y, zy t)f(x, y, z, t)dxdydzdt could be so formulated.