The Operator Approach

In Chap. 14, we have described the approach to quantum field theory which can be traced back to Feynman’s approach in the 1940s based on the Feynman rules for Feynman diagrams and the representation of propagators by functional integrals. Typically, this approach does not use operators in Hilbert spaces, that is, the methods of functional analysis do not play any role. Historically, in the 1920s quantum mechanics was first based on operator theory by Heisenberg, Born, Jordan, Dirac, Pauli, and von Neumann. In order to understand Feynman’s very effective approach, Dyson related this to operator theory via the magic Dyson formula for the S-matrix. Conceptually, the advantage of operator theory is that the duality between particles and waves is formulated in a very transparent manner.

• The waves appear as solutions of classical wave equations. These equations arise as equations of motion from the classical principle of critical action.

• The particles appear after introducing creation and annihilation operators.

• The free quantum field is a linear combination of creation and annihilation operators where the coefficients are classical wave functions (that is, solutions of the free equations of motion).

The disadvantage of operator theory is the fact that there arise serious mathematical difficulties in applying the rigorous theory of functional analysis to quantum electrodynamics and the Standard Model in particle physics. These difficulties are caused by the interactions which are related to nonlinearities.