How to Measure the Quantum Measure

Abstract

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.

Schwinger’s picture of quantum mechanics III: The statistical interpretation

“…The remarkable formula (15) embodies, in the abstract groupoid formalism ,Sorkin's quantum measure expression for systems described on spaces of histories 16 (see for instance [15, eq. 14]) and explains the quadratic dependence of quantum measures on physical transitions.…”

“…Which requires that Ω is large enough, for instance the fundamental representation of a group is not faithful as Ω consists of just one element 15. In the continuous or infinite case, it will be assumed that Ω carries a probability measure ν, the one used to define L 2 (Ω, ν) and |Ω| = 1.…”

Schwinger’s picture of quantum mechanics II: Algebras and observables

“…"...(selective) Measurements that we have already considered involve the passage of all systems or no systems at all between the two stages, as represented by the multiplicative numbers 1 and 0. More generally, measurements of properties B, performed on a system in a state a that refers to properties incompatible with B, will yield a statistical distribution 11 of possible values. Hence only a determinate fraction of the systems emerging from the first state will be accepted by the second stage.…”

“…Apart from the standard well-known pictures of Quantum Mechanics already discussed in [1], many other settings have been proposed, some of them motivated by the problem of achieving a quantum theoretical description of Gravity. Without pretending to be exhaustive, not even covering all relevant contributions on the subject, we would like to mention here R. Penrose's spin-networks [6], [7], von Weizsacker urs [8], [9], the theory of causalnets developed from R. Sorkin's insight [10,11], C. Isham's categorical foundation of gravity [12], the noncommutative geometry approach to the description of space-time inspired on A. Connes conception of geometry [24], [14], [15], etc. All of them share a notion of "discretness" and "non-commutativity" in Dirac's spirit [16,17] towards the description of fundamental physical theories.…”

“…As it was commented before, the discussion of Schwinger's dynamical principle as well as a detailed description of the probabilistic interpretation of the theory in terms of Sorkin's quantum measures [11], as well as the application to other physical systems of interest, will be left for subsequent works.…”