Pointwise analysis of scalar fields: A nonstandard approach
Abstract
A new nonstandardanalytical approach to quantum fields is presented, which gives a mathematical foundation for manipulating pointwisedefined quantum fields. In our approach, a field operator ϕ (x) is not a standard operatorvalued distribution, but a nonstandard operatorvalued function. Then formal expressions containing, e.g., ϕ (x)^{2} can be understood literally, and shown to be well defined. In the free field cases, we show that the Wightman functions are explicitly calculated with the pointwise field, without any regularization, e.g., Wick product. Our notion of pointwise fields is applied also to the path integral formalisms of scalar fields. We show that some of physicists' naive expressions of Lagrangian path integral formulas can be rigorously justified.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 September 2006
 DOI:
 10.1063/1.2339017
 Bibcode:
 2006JMP....47i2301Y
 Keywords:

 11.10.Cd;
 11.10.Ef;
 Axiomatic approach;
 Lagrangian and Hamiltonian approach