A Philosophical Approach to Quantum Field Theory

"ALL men by nature desire to know,'' states Aristotle in the famous first sentence of his Metaphysics [1]. Knowledge about fundamental particles and interactions, that is, knowledge about the deepest aspects of matter, is certainly high if not top on the priority list, not only for physicists and philosophers. The goal of the present work [2] is to contribute to this knowledge by going beyond the usual presentations of quantum field theory in physics textbooks, both in mathematical approach and by critical reflections inspired by epistemology, that is, by the branch of philosophy also referred to as the theory of knowledge.

This work is particularly influenced by the epistemological ideas of Ludwig Boltzmann: "...it cannot be our task to find an absolutely correct theory but rather a picture that is as simple as possible and that represents phenomena as accurately as possible'' (see p.91 of [3]). This work is an attempt to construct an intuitive and elegant image of the real world of fundamental particles and their interactions. To clarify the word picture or image, the goal could be rephrased as the construction of a genuine mathematical representation of the real world.

Consciously or unconsciously, the construction of any image of the real world relies on personal beliefs. I hence try to identify and justify my own personal beliefs thoroughly and in various ways. Sometimes I rely on philosophical ideas, for example, about space, time, infinity, or irreversibility; as a theoretical physicist, I have a limited understanding of philosophy, but that should not keep me from trying my best to benefit from philosophical ideas. More often I rely on successful physical theories, principles or methods, such as special relativity, quantum theory, gauge invariance or renormalization. Typically I need to do some heuristic mathematical steps to consolidate the various inputs adopted as my personal beliefs. All these efforts ultimately lead to an image of nature, in the sense of a mathematical representation, but they are not part of this image. The final mathematical representation should convince by logical clarity, mathematical rigor, and natural beauty.

Emphasis on the importance of beliefs, even if they are justified by a variety of philosophical and physical ideas, may irritate the physicist. The philosopher, on the other hand, is used to the definition of knowledge as true justified belief. How can one claim truth for one's justified beliefs? This happens by confronting an image of nature with the real world.

According to Pierre Duhem [4], known to thermodynamicists from the Gibbs-Duhem relation, and the analytic philosopher Willard Van Orman Quine [5], only the whole image rather than individual elements or hypotheses can be tested against the real world. The confrontation of a fully developed image with the real world depends on all its background assumptions or an even wider web-of-belief, including the assumed logics (confirmation holism). Following Boltzmann's approach of "deductive representation'' (see p.107 of [3]), the present work makes an attempt to show how such a testable whole image of fundamental particle physics can be constructed within the framework of quantum field theory.

The focus of this work is on conceptual issues, on the clarification of the foundations of quantum field theory, and ultimately even on ontological questions. For our intuitive approach, we choose to go back to the origins of quantum field theory. In view of the many severe problems that had to be overcome on the way to modern quantum field theory, that may seem to be naive to the experts. However, with the deep present-day knowledge and with philosophical guidance, the intuitive origins can nicely be developed into a perfectly consistent image of the real world. On the one hand, there is a price to pay for this: practical calculations, in particular perturbation methods, may be less elegant and more laborious than in other approaches. Symbolic computation is the modern response to this challenge. On the other hand, there is a promising reward: a new stochastic simulation methodology for quantum field theory emerges naturally from our approach [6].

Hopefully, the present work motivates physicists to appreciate philosophical ideas. Epistemology and the philosophy of the evolution of science often seem to lag behind science and to describe the developments a posteriori. As philosophy here has a profound influence on the actual shaping of an image of fundamental particles and their interactions, our development should stimulate the curiosity and imagination of physicists.