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The correspondence principle in light of holographic correspondence and the law of large numbers

Blind Review

Published Sep 8, 2023• 0 citations
Publisher

Abstract

The main philosophical lesson of holographic gravity is that spatiotemporal descriptions of a field theory may not be respected by observational reality. We apply this lesson to the correspondence principle, and then later justify the lesson (holographic correspondence) without necessitating such spatiotemporal deviations via the law of large numbers in statistics, applied to an average of n quantum measurements. Classical observables are considered to emerge as n approaches infinity. The averaging approach can be interpreted both as a generalization of Bohr's correspondence principle (large number limit) and Planck's correspondence principle (Planck's constant approaching zero). Holographic correspondence and the averaging approach are then re-analyzed in light of preceding works in the correspondence principle. A key consequential lesson is that we should not conflate the nature of fundamental theory observables with that of emergent theory observables, and once this confusion is avoided, the correspondence principle has a solid standing.

DOI
10.31219/osf.io/zn356

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The main philosophical lesson of holographic gravity is that spatiotemporal descriptions of a field theory may not be respected by observational reality. We apply this lesson to the correspondence principle, and then later justify the lesson (holographic correspondence) without necessitating such spatiotemporal deviations via the law of large numbers in statistics, applied to an average of n quantum measurements. Classical observables are considered to emerge as n approaches infinity. The averaging approach can be interpreted both as a generalization of Bohr's correspondence principle (large number limit) and Planck's correspondence principle (Planck's constant approaching zero). Holographic correspondence and the averaging approach are then re-analyzed in light of preceding works in the correspondence principle. A key consequential lesson is that we should not conflate the nature of fundamental theory observables with that of emergent theory observables, and once this confusion is avoided, the correspondence principle has a solid standing.
Planck formulated the correspondence principle between quantum and classical mechanics as the limit in which the Planck constant h goes to zero. Bohr formulated the correspondence principle to be the limit of large quantum numbers. For three common quantum mechanical systems it is shown that in order for eigenvalues of quantum mechanical observables to have meaningful classical limits, it is necessary to take the double limit as both the Planck constant goes to zero and the quantum number goes to infinity, subject to the constraint that their product is equal to an appropriate classical action. This synthesis of the Bohr and Planck formulations of the correspondence principle is also used to show that the quantum mechanical transition frequency between adjacent levels approaches the corresponding classical frequency. The features these systems have in common in their classical limit are explained by general considerations of the classical limit of the Schrödinger equation.
Niels Bohr’s “correspondence principle” is typically believed to be the requirement that in the limit of large quantum numbers (n→∞) there is a statistical agreement between the quantum and classical frequencies. A closer reading of Bohr’s writings on the correspondence principle, however, reveals that this interpretation is mistaken. Specifically, Bohr makes the following three puzzling claims: First, he claims that the correspondence principle applies to small quantum numbers as well as large (while the statistical agreement of frequencies is only for large n); second, he claims that the correspondence principle is a law of quantum theory; and third, Bohr argues that formal apparatus of matrix mechanics (the new quantum theory) can be thought of as a precise formulation of the correspondence principle. With further textual evidence, I offer an alternative interpretation of the correspondence principle in terms of what I call Bohr’s selection rule. I conclude by showing how this new interpretation of the correspondence principle readily makes sense of Bohr’s three puzzling claims.
Abstract Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.
In this paper we suggest a simple mathematical procedure to derive the classical probability density of quantum systems via Bohr’s correspondence principle. Using Fourier expansions for the classical and quantum distributions, we assume that the Fourier coefficients coincide for the case of large quantum number. We illustrate the procedure by analyzing the classical limit for the quantum harmonic oscillator and the particle in a box, although the method is quite general. We find, in an analytical fashion, the classical distribution arising from the quantum one as the zeroth order term in an expansion in powers of Planck’s constant. We interpret the correction terms as residual quantum effects at the microscopic-macroscopic boundary.
The question of Bohr correspondence in quantum field theory is considered from a dynamical point of view. It is shown that the classical description of particle interactions is inapplicable even in the limit of large particles' masses because of finite quantum fluctuations of the fields produced. In particular, it is found that the relative value of the root mean square fluctuation of the Coulomb and Newton potentials of a massive particle is equal to 1/sqrt{2}. It is shown also that in the case of a macroscopic body, the quantum fluctuations are suppressed by a factor 1/sqrt{N}, where N is the number of particles in the body. An adequate macroscopic interpretation of the correspondence principle is given.
We show in this pedagogical review that far from being an apparent law of physics that stands by itself, the holographic principle is a straightforward consequence of the quantum information theory of separable systems. It provides a basis for the theories of measurement, time, and scattering. Utilizing the notion of holographic screens, which are information encoding boundaries between physical subsystems, we demonstrate that the physical interaction is an information exchange during which information is strictly conserved. Then we use generalized holographic principle in order to flesh out a fully-general quantum theory of measurement in which the measurement produces finite-resolution, classical outcomes. Further, we show that the measurements are given meaning by quantum reference frames and sequential measurements induce topological quantum field theories. Finally, we discuss principles equivalent to the holographic principle, including Markov blankets and the free-energy principle in biology, multiple realizability and virtual machines in computer science, and active inference and interface theories in cognitive science. This appearance in multiple disciplines suggests that the holographic principle is not just a fundamental principle of physics, but of all of science. Quanta 2022; 11: 72–96.