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  What Parts of QFT Don't Have Classical Analogs?

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We seem to be living in an era of classical analogs. A few examples:

Question: What remaining parts of QFT meaningfully escape any classical analog?

asked Sep 29, 2023 in Theoretical Physics by Matthew D Cory [ no revision ]

Classical and semiclassical methods only capture certain gross effects of the full quantum treatment.

In particular, discrete spectra need quantum physics for their explanation.

Radiative corrections and parametric downconversion are not described by sclassical tochastic electrodynamics.

Certainly. But I detect a form of circular reasoning because there's no well-defined idea of measurement. To invert the point, how can we explain the continuous spectra of so many quantum phenomena as originating from discrete states? There is a kind of triviality revealed by Nyquist-Shannon sampling theory. For instance, imposing a natural UV cutoff on the spectrum of the Laplacian (or d'Alembertian) operator makes physical fields "bandlimited", allowing them to be perfectly reconstructed from samples on a lattice using sampling theory. This establishes an equivalence between continuous and discrete representations. A signal theory perspective provides a classical explanation for how continuous spectra can turn into discrete spectra while preserving the underlying continuity. The discrete samples arise from limitations on measurement precision and frequency bandwidth, not an inherent quantum discreteness. 

Aside: I come from a math background, looking into an "intentional" setting for proof theory and the reflection principles of ordinal analysis to address the halting problem (hint: the redundant metaphysics of metamathematics got everybody off course). In time, I became a big fan of E.T. Jaynes and his attitude towards measure theory (see the Mycielski-Sierpiński Division Paradox) and the Church of Limitology. I'm actually a variety of formalist ultrafinitist (CAS does any given math and neurons need only GLMs), so I have no hostility to discrete accounts of nature. I have come to see that continuity is the degenerate case of discrete analysis, as Doron Zeilberger astutely remarked. It has nothing to do with "infinite discreteness” but emerges from simplifying symmetries. Like with motion paradoxes, mathematical ideas of continuity really don't say anything definitive about the continuum of nature. I don't pretend to know the ultimate granularity of nature. However, I think insights from signal analysis should make us more reluctant to say we've hit bottom.

Your point about Stokes is insightful. You might remember that the Riemann–Silberstein formulation of electromagnetism was using the Shrodinger equation well before Shrodinger used it for quantum mechanics.

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