Field Theory Revisited

Following P. A. M. Dirac's critique in " Lectures on Quantum Field Theory " of the usual formalism, I will discuss the role of the time parameter to solve R. Haag's no-go theorem on the non-equivalence of the conventional Schrödinger picture and the Heisenberg picture. This is possible by first defining in a correct way the concept of the vacuum state at a given time in relativity. I will also discuss some consequences such as the spectral condition. We will take as our basic considerations P. A. M. Dirac's beautiful lectures on Quantum Field Theory delivered at the Yeshiva University during the academic year 1963 – 1964 and published by Academic Press in 1966 (1). We will also suppose that the reader knows the abstract Fock space construction as developed along the line introduced by D. Kastler (2). In his lectures cited above Dirac writes [p.6]: We have the kets at one particular time and we may picture them as corresponding to physical states at that time, so we retain the concept of a physical state at a certain time. We do not have the concept of a physical state throughout all time. The latter concept, which lies at the basis of the Schrödinger picture, will not occur in the present formulation. This expresses exactly or point of view, where states are states at the actual time, and the time is a c-number taking a well defined value in any possible state. This is in marked contrast to q-numbers, which have in general no values in any state. In spite of appearances, the noncommutativity of the q-numbers is not the very essence of such physical objects. In fact this depends on the mathematical formalism that you choose and its corresponding interpretation. This can be seen very well in the Wigner representation of p and q. The two objects commute with each other but the usual rules of quantum mechanics are here completely modified and nevertheless in any given physical state p and q have no values. The concept of a physical state throughout all time, a relativistic concept, is not a state at all, it is a trajectory of states labeled by the time, and a solution of a Schrödinger equation (3). But as Dirac demonstrates, such solutions do not exist for the physically justified Hamiltonian for which the interaction is too violent at high frequencies. To exhibit the difficulties …