## Covariant Operator Formalism of Gauge Theories and Quantum Gravity, World Scientific (1990), Chap.5. As for the method of solving quantum field theory in the Heisenberg picture

- N. Nakanishi, I. Ojima
- 1990

- Published 2006

The universe consists of its container, spacetime, and its contents, substance. The substance which we experience daily is very much different from its intrinsic constituents, elementary particles. On the contrary, the spacetime at the microscopic level is not significantly different from the spacetime which we experience daily. Of course, the relativity theory clarified that space and time must be treated not separately but in a unified way as a 4-dimensional manifold, but it did not suggest any essentially new aspect in the microscopic structure of the spacetime. Historically, in Heisenberg’s quantum mechanics, the coordinates of the configuration space was formulated not as real numbers (c-numbers) but as operators (q-numbers). However, in Schrödinger’s wave mechanics, describing the same contents, the spatial coordinates are formulated as c-numbers by representing their conjugate momenta by differntial operators. The concept of the c-number spacetime was carried over to quantum field theory by promoting the wave functions to field operators. But quantum field theory is most naturally formulated in the Heisenberg picture but not in the Schrödinger picture. The most fundamental quantities in quantum field theory are quantum fields, which are the operator-valued (generalized) functions of the c-number spacetime. The present standard theory of elementary particles is completely describable by quantum field theory in the 4-dimensional spacetime. Within this framework, there is no necessity for requiring any change for the concept of the c-number 4-dimensional spacetime. Never-

@inproceedings{Nakanishi2006SpacetimeIT,
title={Spacetime in the Ultimate Theory},
author={N. Nakanishi},
year={2006}
}