Book contents
4 - Perspectives
Published online by Cambridge University Press: 21 December 2017
Summary
Let us summarize what we have achieved, which mathematical problems remain to be solved, and where we should go in the future to achieve a faithful image of nature for fundamental particles and all their interactions that fully reflects the state-of-the-art knowledge of quantum field theory. Sections 4.2 and 4.3 of this final chapter may be considered as a program for future work.
The Nature of Quantum Field Theory
Lagrangian quantum field theory, the branch of quantum field theory that has delivered spectacular predictions in fundamental particle physics, is plagued by divergencies and lacks mathematical rigor. We consider these mathematical difficulties to be self-imposed because the field idealization, which implies limiting procedures, is made too easily and too early in the development of the theory. In doing so, the natural physical ultraviolet and infrared cutoffs given by the Planck length and the size of the universe are dauntlessly ignored.
In this book we have elaborated that, when guided by a number of pertinent philosophical considerations, we naturally arrive at a perfectly healthy, intuitive version of quantum field theory. All the crucial ingredients to quantum field theory can be established a priori, not just as an a posteriori reaction to the occurrence of divergencies or other problems associated with the representation of an infinite number of degrees of freedom. In this section, we briefly summarize the key elements of the theory, that is, of our mathematical image of nature, and discuss some implications for the ontology of the most fundamental theory of matter.
The Image
We choose to consider space and time as prerequisites for developing physical theories. Although we clearly need to respect the space–time transformation behavior of special relativity, we can rely on separate concepts of space and time provided that we choose a particular reference frame. Time appears explicitly in the fundamental evolution equation of our approach; spatial dependencies are eliminated by Fourier transformation.
Philosophical doubts about our ability to handle uncountable sets of degrees of freedom lead us to consider finite volumes of increasing size instead of an infinite space. We thus obtain a countable set of momentum vectors providing a priori infrared regularization. Philosophical considerations moreover suggest that irreversibility, or dissipation, occurs naturally.We thus obtain a dynamic smoothing of fields providing a priori ultraviolet regularization.
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- A Philosophical Approach to Quantum Field Theory , pp. 220 - 228Publisher: Cambridge University PressPrint publication year: 2018