An abstract theory of physical measurements

Abstract: Since its early days, quantum mechanics has forced physicists to consider the interaction between quantum systems and classically described experimental devices — a fundamental tenet for Bohr was that the results of measurements need to be communicated using the language of classical physics.

Several decades of progress have led to improved understanding, but the tension between “quantum” and “classical” persists. Ultimately, how is classical information extracted from a measurement? Is classical information fundamental, as in Wheeler’s “it from bit”? In this talk, which is based on ongoing work [1], I approach the problem mathematically by considering spaces whose points are measurements, abstractly conceived in terms of the classical information they produce. Concretely, measurement spaces are stably Gelfand quantales [2] equipped with a compatible sober topology, but essentially their definition hinges on just two binary operations, called composition and disjunction, whose intuitive meanings are fairly clear. Despite their simplicity, these spaces have interesting mathematical properties. C*-algebras yield measurement spaces of “quantum type,” and Lie groupoids give us spaces of “classical type,” such as those which are associated with a specific experimental apparatus. The latter also yield a connection to Schwinger’s selective measurements, which have been recast in groupoid language by Ciaglia et al. An interaction between the two types, providing a mathematical approach to Bohr’s quantum/classical split, can be described in terms of groupoid (or Fell bundle) C*-algebras as in [3]. I will illustrate the basic ideas with simple examples, such as spin measurements performed with a Stern–Gerlach apparatus.

References

[1] P. Resende, An abstract theory of physical measurements (2021), available at arxiv.org/abs/2102.01712.

[2] P. Resende, The many groupoids of a stably Gelfand quantale, J. Algebra 498 (2018), 197–210, DOI 10.1016/j.jalgebra.2017.11.042.

[3] P. Resende, Quantales and Fell bundles, Adv. Math. 325 (2018), 312–374, DOI 10.1016/j.aim.2017.12.001. MR3742593

mathematical physicsalgebraic topologycategory theoryquantum algebra

Audience: researchers in the topic

Comments: Second part of a double session, followed by a 20 minute discussion period.

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