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arXiv:1603.08427 · AIP conference proceedings

Why square roots of probabilities?

Kevin H. Knuth

Published Jan 1, 2016• 4 citations
arXivHTMLPDF

Abstract

Square roots of probabilities appear in several contexts, which suggests that they are somehow more fundamental than probabilities.Square roots of probabilities appear in expressions of the Fisher-Rao Metric and the Hellinger-Bhattacharyya distance.They also come into play in Quantum Mechanics via the Born rule where probabilities are found by taking the squared modulus of the quantum amplitude.Why should this be the case and why do these square roots not arise in the various formulations of probability theory?In this short, inconclusive exploration, I consider quantifying a logical statement with a vector defined by a set of components each quantifying one of the atomic statements defining the hypothesis space.I show that conditional probabilities (bi-valuations), such as P(x|y), can be written as the dot product of the two vectors quantifying the logical statements x and y each normalized with respect to the vector quantifying the conditional y.The components of the vectors are proportional to the square root of the probability.As a result, this formulation is shown to be consistent with a concept of orthogonality applied to the set of mutually exclusive atomic statements such that the sum rule is represented as the sum of the squares of the square roots of probability.

DOI
10.1063/1.4959042

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Square roots of probabilities appear in several contexts, which suggests that they are somehow more fundamental than probabilities.Square roots of probabilities appear in expressions of the Fisher-Rao Metric and the Hellinger-Bhattacharyya distance.They also come into play in Quantum Mechanics via the Born rule where probabilities are found by taking the squared modulus of the quantum amplitude.Why should this be the case and why do these square roots not arise in the various formulations of probability theory?In this short, inconclusive exploration, I consider quantifying a logical statement with a vector defined by a set of components each quantifying one of the atomic statements defining the hypothesis space.I show that conditional probabilities (bi-valuations), such as P(x|y), can be written as the dot product of the two vectors quantifying the logical statements x and y each normalized with respect to the vector quantifying the conditional y.The components of the vectors are proportional to the square root of the probability.As a result, this formulation is shown to be consistent with a concept of orthogonality applied to the set of mutually exclusive atomic statements such that the sum rule is represented as the sum of the squares of the square roots of probability.
First steps are taken toward a formulation of quantum mechanics which avoids the use of probability amplitudes and is expressed entirely in terms of observable probabilities. Quantum states are represented not by state vectors or density matrices but by “probability tables,” which contain only the probabilities of the outcomes of certain special measurements. The rule for computing transition probabilities, normally given by the squared modulus of the inner product of two state vectors, is re-expressed in terms of probability tables. The new version of the rule is surprisingly simple, especially when one considers that the notion of complex phases, so crucial in the evaluation of inner products, is entirely absent from the representation of states used here.
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.
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.
Quasi-probabilities appear across diverse areas of physics, but their conceptual foundations remain unclear: they are often treated merely as computational tools, and operations like conditioning and Bayes' theorem become ambiguous. We address both issues by developing a principled framework that derives quasi-probabilities and their conditional calculus from structural consistency requirements on how statements are valued across different universes of discourse, understood as finite Boolean algebras of statements.We begin with a universal valuation that assigns definite (possibly complex) values to all statements. The central concept is Syntactic Locality: every universe can be embedded within a larger ambient one, and the universal valuation must behave coherently under such embeddings and restrictions. From a set of structural principles, we prove a representation theorem showing that every admissible valuation can be re-expressed as a finitely additive measure on mutually exclusive statements, mirroring the usual probability sum rule. We call such additive representatives pre-probabilities. This representation is unique up to an additive regraduation freedom. When this freedom can be fixed canonically, pre-probabilities reduce to finite quasi-probabilities, thereby elevating quasi-probability theory from a computational device to a uniquely determined additive representation of universal valuations. Classical finite probabilities arise as the subclass of quasi-probabilities stable under relativisation, i.e., closed under restriction to sub-universes. Finally, the same framework enables us to define a coherent theory of conditionals, yielding a well-defined generalized Bayes' theorem applicable to both pre-probabilities and quasi-probabilities. We conclude by discussing additional regularity conditions, including the role of rational versus irrational probabilities in this setting.
Quantum mechanics is known for its “square root” probability i.e. the existence of a wavefunction W(x) such that P(x) = W*(x)W(x). This begs the question: Why is this the case? In Maxwell’s electromagnetic theory of the photon, a similar situation arises except that one has energy density instead of P(x) which consists of .5C1 Electric field* Electric field + .5c2 Magnetic field*Magnetic field. The “square root” object is a physical observable (Electric field, Magnetic field) which both behave as cos(-Et+px) and so, more accessible than W(x). Alternatively one may use Fermat’s least time principle for light to show that i two dimensional problems (reflection/refraction) d/dx (used in extremization) is really the operator of momentum in that direction i.e. a spatial invariant direction. Thus A=px such that dA/dx=p = x component of momentum which may be generalized to A=-Et+p dot r. (p.r) may be written as -i ln(exp(ip dot r) with exp(i p dot r) representing a “kind of probability”. This probability may be considered a square root probability because for uniform motion (light) all x points along a ray should have equal probability. Thus exp(i p dot r) maps into a constant which happens if one takes the modulus. Both of the above approaches (using continuity at x or a kind of conservation of probability and a momentum) may be used to solve the problem of an incident beam of light moving in the positive x direction hitting a medium and both reflecting and refracting. For example see (1). The goal is to find the two intensity values associated with the reflected and refracted beams assuming the incident intensity is known. In this note, we take a different approach to finding the two unknowns, one that does not use the idea of the electric field proportional to cos(-Et+px) or the probability exp(ipx). We wish to solve the problem using only the idea of information and stress the importance of beginning with energy conservation a priori unlike the two approaches listed above. We postulate that there are two sets of information in this problem. First there is the notion of Energy = p(momentum in a vacuum) n c/n where n is the index of refraction. Thus the incident and reflected light are associated with pc1,c1 and the refracted with pc2, c2 with energy being the same for all. Second there is the idea of conservation of energy i.e. if ten photons with energy E hit the surface and a10 reflect and (1-a)10 refract, then the sum of the refracted and reflected energy equals the initial energy. Energy is then equated with intensity through Intensity/c = energy (1). Using only these two pieces of information we argue one should be able to find the intensity of both the reflected and refracted ray. This leads to an immediate problem because there are two unknowns and only one equation, namely conservation of energy i.e. I(A)/c1 = I(B)/c1 + I(C)/c2 where A is the incident beam, B the reflected and C the refracted. Writing as: I(A)/c1 - I(B)/c1 = I(C)/c2 (i.e. a spatial instead of time grouping) suggests a possible solution might be to employ a difference of squares noting that 1/c = p/E with E being a constant in the problem. Thus one may reduce the problem to A+B=C and p1A-p1B=p2C. This is of the form of a new kind of “square root” type probability, but leads to two equations with two unknowns (A is assumed to be known) which allows the problem to be solved.
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.