REVIEW
Quantum Mechanics as Quantum Measure Theory
General Relativity and Quantum Cosmology · Submitted 7 Jan 1994 · 11 references
Pith's verdict: The math core is sound and checkable; the interpretive half is a promissory note to a paper that doesn't exist yet.
T0 · AI review
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Referee report
The paper
The paper claims
Sorkin argues that quantum mechanics is the third-weakest rule in a ladder of probability sum-rules, so the Born rule follows from a single algebraic condition rather than being postulated.
The reading
What carries the argument
The interference functions $I_n$ and the recursion lemma relating $I_{n+1}$ to $I_n$. These carry the whole argument: the lemma simultaneously proves that the sum rules are ordered by strength (each implies its successor) and that each rule is equivalent to multi-additivity of the function measuring the failure of the next stronger rule. At $n=2$ that multi-additivity is bi-additivity, and bi-additivity plus the normalization on equal arguments is exactly what forces the quadratic amplitude-probability relation. The three-slit experiment is the physical face of the same object: the alternating-sign superposition of all eight open/blocked diffraction patterns vanishes for any $I_3=0$ measure, just as it does for classical particles.
Extended reading notes
Core claim
The paper defines a sequence of symmetric set functions $I_n$ that measure the failure of successive additivity rules: $I_1(A)=|A|$, $I_2(A,B)=|A+B|-|A|-|B|$, $I_3(A,B,C)=|A+B+C|-|A+B|-|B+C|-|A+C|+|A|+|B|+|C|$, and so on. A lemma, stated with its inductive proof deferred elsewhere, gives $I_{n+1}(A_0,A_1,...,A_n)=I_n(A_0+A_1,A_2,...,A_n)-I_n(A_0,A_2,...,A_n)-I_n(A_1,A_2,...,A_n)$, from which it follows immediately that setting $I_n$ to zero entails setting $I_{n+1}$ to zero. The rules therefore form a nested hierarchy: $I_2=0$ is ordinary Kolmogorov additivity, appropriate to Brownian-motion-style stochastic processes, and $I_3=0$ is the next weaker rule, which the paper identifies as the level of quantum mechanics. From $I_3=0$ the lemma yields bi-additivity of $I_2$ on disjoint arguments, and once $I$ is extended to arbitrary arguments with $I(X,X)=2|X|$, bi-additivity fixes it completely, giving $I(A,B)=|A cup B|+|A cap B|-|A setminus B|-|B setminus A|$ and hence $|X|=I(X,X)/2$. This is the paper's central result: any measure obeying the three-alternative sum rule can be written this way, so the postulate that probabilities descend from a bi-additive object can be replaced by the sum rule itself. The author notes the resulting $I$ is what the decoherence-functional literature would call twice the real part of that functional, and sketches how ordinary non-relativistic particle quantum mechanics is recovered by taking $I(x,y)=delta(x(T),y(T)) e^{-iS(x)}e^{iS(y)}$ plus its complex conjugate, with a truncation time $T$ to the future of the properties defining the set.
Load-bearing premise
The characterization rests on extending $I(A,B)$ from disjoint arguments to arbitrary overlapping ones with the value $I(X,X)=2|X|$ fixed on equal arguments; the paper says only that a short computation to appear elsewhere confirms this extension is self-consistent, so the uniqueness and well-definedness of that extension is asserted rather than shown here.
Original abstract
The additivity of classical probabilities is only the first in a hierarchy of possible sum-rules, each of which implies its successor. The first and most restrictive sum-rule of the hierarchy yields measure-theory in the Kolmogorov sense, which physically is appropriate for the description of stochastic processes such as Brownian motion. The next weaker sum-rule defines a {\it generalized measure theory} which includes quantum mechanics as a special case. The fact that quantum probabilities can be expressed ``as the squares of quantum amplitudes'' is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. The present paper presents these relationships in the context of a ``realistic'' interpretation of quantum mechanics.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The additivity of classical probabilities is only the first in a hierarchy of possible sum-rules, each of which implies its successor. The first and most restrictive sum-rule of the hierarchy yields measure-theory in the Kolmogorov sense, which physically is appropriate for the description of stochastic processes such as Brownian motion. The next weaker sum-rule defines a {\it generalized measure theory} which includes quantum mechanics as a special case. The fact that quantum probabilities can be expressed ``as the squares of quantum amplitudes'' is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. The present paper presents these relationships in the context of a ``realistic'' interpretation of quantum mechanics.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The additivity of classical probabilities is only the first in a hierarchy of possible sum-rules, each of which implies its successor. | background 0.78 | 0.22 | 0.17 | asserted 0.99 |
| s1 | The first and most restrictive sum-rule of the hierarchy yields measure-theory in the Kolmogorov sense, which physically is appropriate for the description of stochastic processes such as Brownian motion. | background 0.70 | 0.18 | 0.08 | asserted 0.98 |
| s2 | The next weaker sum-rule defines a {\it generalized measure theory} which includes quantum mechanics as a special case. | result 0.51 | 0.62 | 0.47 | asserted 1.00 |
| s3 | The fact that quantum probabilities can be expressed ``as the squares of quantum amplitudes'' is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. | result 0.81 | 0.87 | 0.29 | asserted 1.00 |
| s4 | Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. | result 0.55 | 0.79 | 0.15 | asserted 0.99 |
| s5 | The present paper presents these relationships in the context of a ``realistic'' interpretation of quantum mechanics. | contribution 0.51 | 0.81 | 0.11 | asserted 0.94 |
What was examined, and what was not
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References
How the literature uses it
41 arXiv papers cite this in 78 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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Following Sorkin, we introduce a generalized measure theory based on a hierarchy of ``sum-rules.'' The first sum-rule yields classical probability theory, and the second yields a generalized probability theory that includes quantum mechanics as a special case.
Some Identities for the Quantum Measure and its Generalizations · 0.88
Papers citing it · 48 sentences without a confident label
Compares with it 2 papers
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This well-known fact, representing the “heart” of quantum mechanics, is to be contrasted with Sorkin's results for the following, so-called “second order sum rule” [ref] : [equation] This result is remarkable insofar as it can be inferred that interference terms theoretically always originate from pairings of paths, but never from triples etc.
Born's Rule as Signature of a Superclassical Current Algebra · Three-slit interference and Born's rule · 0.81
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Sorkin's [ref] original definition, and the discussion in [ref] , used the $M=3$ case as their definition of third-order interference, but the two can straightforwardly if somewhat tediously be shown to be equivalent.
Higher-order interference and single-system postulates characterizing quantum theory · Third-order interference · 0.82
Cites it as background 21 papers
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In the following, we first review Sorkin's definition of multi-path interference and the classification of theories based on their presence or absence [ref] .
Neutrino Oscillations at JUNO, the Born Rule, and Sorkin's Triple Path Interference · Introduction · 0.92
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From the theoretical point of view interference-type phenomena have been analyzed thoroughly [ref] , for the cases of only one open slit up to $𝑁$ open slits.
Born's Rule as Signature of a Superclassical Current Algebra · Three-slit interference and Born's rule · 0.96
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It is this measure (or the corresponding probability) that enters the sum rules we are concerned with.Details of the quantum measure theory following a sum over histories approach can be found in [ref] .
Testing Born's Rule in Quantum Mechanics with a Triple Slit Experiment · The 3-slit experiment · 0.97
Discussion
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