Random walk in generalized quantum theory

Martín, Xavier; O’Connor, Denjoe; Sorkin, Rafael D.

doi:10.1103/physrevd.71.024029

Cited by 37 publications

(88 citation statements)

References 21 publications

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“…ordinary unitary quantum mechanics, for example, is strongly positive. Strong positivity is a powerful requirement because it implies in general that there is a Hilbert space associated with the quantum measure, which turns out to be the standard Hilbert space in the case of unitary quantum mechanics [14] [15]. Decoherence functionals which merely satisfy condition (iii) above are termed "weakly positive".…”

Section: Quantum Mechanics As Quantum Measure Theorymentioning

confidence: 99%

See 1 more Smart Citation

A Bell inequality analog in quantum measure theory

Craig¹,

Dowker²,

Henson³

et al. 2006

J. Phys. A: Math. Theor.

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One obtains Bell's inequalities if one posits a hypothetical joint probability distribution, or measure, whose marginals yield the probabilities produced by the spin measurements in question. The existence of a joint measure is in turn equivalent to a certain causality condition known as "screening off". We show that if one assumes, more generally, a joint quantal measure, or "decoherence functional", one obtains instead an analogous inequality weaker by a factor of √ 2. The proof of this "Tsirel'son inequality" is geometrical and rests on the possibility of associating a Hilbert space to any strongly positive quantal measure. These results lead both to a question: "Does a joint measure follow from some quantal analog of 'screening off' ?", and to the observation that non-contextual hidden variables are viable in histories-based quantum mechanics, even if they are excluded classically.

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Section: Quantum Mechanics As Quantum Measure Theorymentioning

confidence: 99%

“…However, in order to demonstrate this, we will need to apply to D a certain basic construction via which any strongly positive decoherence functional gives rise to a Hilbert space [14] [15].…”

Section: Quantal Casementioning

confidence: 99%

A Bell inequality analog in quantum measure theory

Craig¹,

Dowker²,

Henson³

et al. 2006

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…Moreover, there exist quantal measures that yield theories more general than Quantum Mechanics, for example non-unitary theories. [17] As we have said, the feature that distinguishes a quantum theory from a classical theory is interference. This means that the measure will enjoy different formal properties than classically.…”

Section: Quantum Measure Theorymentioning

confidence: 99%

“…In this paper we will work in the framework of Quantum Measure Theory [24,17,8,11,25], which generalizes the mathematical concept of a measure-space so as to allow for quantal interference. When refashioned in this language, Quantum Mechanics appears as a generalized probability theory of a type inspired by the path integral.…”

Section: Introductionmentioning

confidence: 99%

How to Measure the Quantum Measure

Mozota

Sorkin

2017

Int J Theor Phys

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The histories-based framework of Quantum Measure Theory assigns a generalized probability or measure µ(E) to every (suitably regular) set E of histories. Even though µ(E) cannot in general be interpreted as the expectation value of a selfadjoint operator (or POVM), we describe an arrangement which makes it possible to determine µ(E) experimentally for any desired E. Taking, for simplicity, the system in question to be a particle passing through a series of Stern-Gerlach devices or beam-splitters, we show how to couple a set of ancillas to it, and then to perform on them a suitable unitary transformation followed by a final measurement, such that the probability of a final outcome of "yes" is related to µ(E) by a known factor of proportionality. Finally, we discuss in what sense a positive outcome of the final measurement should count as a minimally disturbing verification that the microscopic event E actually happened.

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“…• Does the quantal random walk defined in [11] have, as conjectured, a continuum limit that reproduces the quantum mechanics of the free non-relativistic particle? (This random walk appears to offer one of the few fully defined examples of a genuinely non-unitary quantal process.…”

Section: Further Work and Final Commentsmentioning

confidence: 99%

Quantum dynamics without the wavefunction

Sorkin¹

2007

J. Phys. A: Math. Theor.

168

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When suitably generalized and interpreted, the path-integral offers an alternative to the more familiar quantal formalism based on state-vectors, selfadjoint operators, and external observers. Mathematically one generalizes the path-integral-as-propagator to a quantal measure µ on the space Ω of all "conceivable worlds", and this generalized measure expresses the dynamics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such "histories-based" schemes new, and more "realistic" possibilities open up for resolving the philosophical problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of µ in its sets of measure zero: such sets are to be "precluded". But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of "quantum logic", would accommodate the contradictions by dualizing Ω to a space of "co-events" and effectively identifying reality with an element of this dual space.Reading the literature on "quantum foundations", you could easily get the impression that the problems begin and end with non-relativistic quantum mechanics. Perhaps most authors have limited themselves to this special case because they thought that no essentially new philosophical questions arose in relativistic quantum field theory and quantum gravity, ⋆

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Random walk in generalized quantum theory

Cited by 37 publications

References 21 publications

A Bell inequality analog in quantum measure theory

A Bell inequality analog in quantum measure theory

How to Measure the Quantum Measure

Quantum dynamics without the wavefunction

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