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|From: Peter Morgan <firstname.lastname@example.org>
Title: Regularization by Test Function
Authors: Peter Morgan
Categories: quant-ph hep-th
Comments: 7 pages. Some text adapted from arXiv:1211.2831v2 [math-ph]
Quantum fields are generally taken to be operator-valued distributions, linear functionals of test functions into an algebra of operators; here the effective dynamics of an interacting quantum field is taken to be nonlinearly modified by properties of test functions, in a way that preserves Poincar\'e invariance, microcausality, and the Fock-Hilbert space structure of the free field. The construction can be taken to be a physically comprehensible regularization because we can introduce a sequence that has a limit that is a conventional interacting quantum field, with the usual informal dependence of the effective dynamics on properties of the experimental apparatus made formally explicit as a dependence on the test functions that are used to model the experimental apparatus.
In descriptions of interacting QFT, the "measurement scale", "renormalization scale", etc., of an experiment are repeatedly invoked, but they are always informally related to the experimental apparatus. That informality is unnecessary, as well as contrary to the detailed specificity we usually expect in Physics, insofar as the test functions (generally in a Schwartz space of functions that are smooth both in real space and in fourier space) that are used to describe experimental apparatus in detail can provide formal measures of the "measurement scale" of each individual measurement operator, and hence of the measurement scale of the model of an experimental apparatus as a whole.
The idea that there is a renormalization scale that is independent of the test functions that are used to model an experiment appears to be a consequence of the idealization that the test functions are always point-like, either in real space or in fourier space, whereas detailed discussions of QFT always introduce smooth real-space cutoffs (for example, at T and at -T at very large times), which could instead (and, I would say, could more properly) be subsumed into the test functions that model the experimental apparatus.
Once we take the renormalization scale to be a function of the test functions instead of independent of the test functions, an interacting QFT is (at least weakly) nonlinearly dependent on the test functions. Admitting this gives us a considerable variety of well-defined quantum field theories, which I cannot yet characterize, but for which I can point out a number of examples (but definitely not all examples), some of which can be understood to be a new, formal, and physically comprehensible regularization. A curious interpretation becomes possible, that details of an experimental apparatus "conditions" the dynamics of the interaction in the region of space-time that it prepares and measures. One interest of this approach is that it suggests that we should experimentally determine what the conditioning of the dynamics is for any given test function.