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The Born Rule is a Feature of Superposition

David Ellerman

Published Nov 2, 2023• 0 citations
Publisher

Abstract

Finite probability theory is enriched by introducing the mathematical notion (no physics involved) of a superposition event \(\Sigma S\)--in addition to the usual discrete event \(S\) (subset of the outcome space \(U=\left( u_{1},...,u_{n}\right)\)). Mathematically, the two types of events are distinguished using \(n\times n\) density matrices. The density matrix \(\rho\left( S\right)\) for a discrete event is diagonal and the density matrix \(\rho\left( \Sigma S\right)\) is obtained as an outer product \(\left\vert s\right\rangle \left\langle s\right\vert\) of a normalized vector \(\left\vert s\right\rangle \in\mathbb{R}^{n}\). Probabilities are defined using density matrices as \(\Pr\left( T|\rho\right) =\operatorname*{tr}\left[ P_{T}\rho\right]\) where \(T\subseteq U\) and \(P_{T}\) is the diagonal projection matrix with diagonal entries \(\chi_{T}\left( u_{i}\right)\). Then for the singleton \(\left\{ u_{i}\right\} \subseteq U\), the probability of the outcome \(u_{i}\) conditioned by the _superposition_ event \(\Sigma S\) is \(\Pr\left( \left\{ u_{i}\right\} |\Sigma S\right) =\left\langle u_{i}|s\right\rangle ^{2}\), the Born Rule. Thus the Born Rule arises naturally from the mathematics of superposition when superposition events are added to ordinary finite probability theory. No further explanation is required when the mathematics uses \(\mathbb{C}^{n}\) instead of \(\mathbb{R}^{n}\) except that the square \(\left\langle u_{i}|s\right\rangle ^{2}\) is the absolute square \(\left\vert \left\langle u_{i}|s\right\rangle \right\vert ^{2}\).

DOI
10.32388/ysqjql

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Finite probability theory is enriched by introducing the mathematical notion (no physics involved) of a superposition event (\Sigma S)--in addition to the usual discrete event (S) (subset of the outcome space (U=\left( u_{1},...,u_{n}\right))). Mathematically, the two types of events are distinguished using (n\times n) density matrices. The density matrix (\rho\left( S\right)) for a discrete event is diagonal and the density matrix (\rho\left( \Sigma S\right)) is obtained as an outer product (\left\vert s\right\rangle \left\langle s\right\vert) of a normalized vector (\left\vert s\right\rangle \in\mathbb{R}^{n}). Probabilities are defined using density matrices as (\Pr\left( T|\rho\right) =\operatorname*{tr}\left[ P_{T}\rho\right]) where (T\subseteq U) and (P_{T}) is the diagonal projection matrix with diagonal entries (\chi_{T}\left( u_{i}\right)). Then for the singleton (\left{ u_{i}\right} \subseteq U), the probability of the outcome (u_{i}) conditioned by the superposition event (\Sigma S) is (\Pr\left( \left{ u_{i}\right} |\Sigma S\right) =\left\langle u_{i}|s\right\rangle ^{2}), the Born Rule. Thus the Born Rule arises naturally from the mathematics of superposition when superposition events are added to ordinary finite probability theory. No further explanation is required when the mathematics uses (\mathbb{C}^{n}) instead of (\mathbb{R}^{n}) except that the square (\left\langle u_{i}|s\right\rangle ^{2}) is the absolute square (\left\vert \left\langle u_{i}|s\right\rangle \right\vert ^{2}).
The Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.
The Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.
We deduce the Born rule from a purely statistical take on quantum theory within minimalistic math-setup. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics—a linear, not Hilbert’, vector space—and empirical notion of the Statistical Length of a state. Its statistical nature comes from the lab micro-events (detector-clicks) being formalized into the C -coefficients of quantum superpositions. We also comment that not only has the use not been made of quantum axioms (scalar-product, operators, interpretations , etc.), but that the involving thereof would be, in a sense, inconsistent when deriving the rule. In point of fact, the quadratic character of the statistical length, and even not (the ‘physics’ of) Born’s formula, represents a first step in constructing the mathematical structure we name the Hilbert space of quantum states.
We can use quantum theory to explain an enormous variety of phenomena by showing why they were to be expected and what they depend on. These explanations of probabilistic phenomena involve applications of the Born rule: to accept quantum theory is to let relevant Born probabilities guide one’s credences about presently inaccessible events. We use quantum theory to explain a probabilistic phenomenon by showing how its probabilities follow from a correct application of the Born rule, thereby exhibiting the phenomenon’s dependence on the quantum state to be assigned in circumstances of that type. This is not a causal explanation since a probabilistic phenomenon is not constituted by events that may manifest it: but each of those events does depend causally on events that actually occur in those circumstances. Born probabilities are objective and sui generis, but not all Born probabilities are chances.
One of the most successful, yet most puzzling, structures of quantum mechanics is Born's rule:P(\mathbf{x},t)=|\Psi(\mathbf{x},t)|^2.It establishes, with extraordinarily high experimental accuracy, the connection between the quantum wave function and the statistical distribution of actual measurement outcomes. Yet a more fundamental question remains worth asking: Why is probability given precisely by the modulus squared of the complex wave amplitude? Standard quantum mechanics uses Born's rule as a fundamental rule. This paper instead poses an open research question: Could Born's rule be not an irreducible probabilistic axiom, but a statistical emergence resulting from the combined action of continuous wave-amplitude superposition, quantized energy exchange, and local interaction dynamics? We propose a candidate physical picture:wave-amplitude superposition->spatial interference structure->local interaction->quantized energy exchange->discrete events}->probability distribution The central mathematical question is \Gamma(\mathbf{x})\stackrel{?}{\propto}|\Psi(\mathbf{x})|^2 where Γ(x)\Gamma(\mathbf{x})Γ(x) denotes the occurrence rate of localized discrete detection events. This paper does not claim to have completed such a derivation. Instead, it explicitly formulates the problem as an \textbf{open, computable, testable, and falsifiable research challenge}. We particularly invite young physicists, mathematicians, and researchers skilled in AI and modern computational tools to explore this problem.
Abstract Quantification starts with sum and product rules that express combination and partition. These rules rest on elementary symmetries that have wide applicability, which explains why arithmetical adding up and splitting into proportions are ubiquitous. Specifically, measure theory formalizes addition, and probability theory formalizes inference in terms of proportions. Quantum theory rests on the same simple symmetries, but is formalized in two dimensions, not just one, in order to track an object through its binary interactions with other objects. The symmetries still require sum and product rules (here known as the Feynman rules), but they apply to complex numbers instead of real scalars, with observable probabilities being modulus squared (known as the Born rule). The standard quantum formalism follows. There is no mystery or weirdness, just ordinary probabilistic inference.