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We seem to be living in an era of classical analogs. A few examples:
There exist classical analogs to Hawking and Unruh radiation.
As pointed out by Arnold Neumaier, George Stokes, in 1852, "described all the modern quantum phenomena of a single qubit, explaining them in classical terms," including spin-1/2 systems. It appears that all quantum systems can be simulated by classical electromagnetic waves.
Sean Carroll rightly pointed out that "[t]he Klein-Gordon and Dirac equations are actually not quantum at all — they are classical field equations, just like Maxwell’s equations are for electromagnetism and Einstein’s equation is for the metric tensor of gravity."
Quantum collapse can be explained by nonlinearity of particle creation/annihilation within QFT. The nonlinearity is rooted in existing physics of field interactions. Note: Collapse is not instantaneous and neither are Bohr jumps.
Lazaro and Link describe spin-1/2 systems with a quaternionic mechanical system.
Quantum tunneling can be described in a classical way, when objects have extent.
Special Relativity is fairly trivial from a field perspective, when considering classical wave mechanics.
There are a number of classical analogs of quantum entanglement that violate Bell inequalities, such as with Brownian motion, chaotic balls, Ising models, classical electromagnetism and water waves (HQFT). Supercorrelation suggests nonlocal correlations can be found conditioning the quantum state on a background field. This can be derived with classical random field models. See The Straw Man of Quantum Physics, Bell Inequalities for Random Fields, Bell Test in a Classical Pilot-Wave System, Shifting the Quantum-Classical Boundary, Brownian Entanglement, Noncommutative Probability in Classical Systems, Classical Electrodynamics Can Violate Bell-type Inequalities, Apparent Violations of Bell-Boole Inequalities in Elastic Collision Experiments, The Chaotic Ball: An Intuitive Analogy for EPR Experiments, Violation of the Bell-Inequality in Supercorrelated Systems, Contextuality, Complementarity, Signaling, and Bell Tests, Comment on "Loophole-free Bell Inequality Violation"
Considering stochastic QFT, SED provides a classical explanation for a number of different quantum effects.
Quantum teleportation falls into the entanglement phenomena but people make it sound mysterious because they forget that they have to copy the computational basis.
GR is already classical. Scharf, following Einstein and Weinberg, gave a non-geometrical version of gravity that mainly acts as a classical field, although derived from a quantum gauge field. When attempting non-geometrical field theory unifications, few even consider Bondi radar. Obviously, GR is not my concern here.
Question: What remaining parts of QFT meaningfully escape any classical analog?
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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