Foundations of quantum physics III. Measurement

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Referee this paper: arXiv:1902.10782 by Arnold Neumaier

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

Abstract: This paper presents the measurement problem from the point of view of the thermal interpretation of quantum physics introduced in Part II. Unlike most work on the foundations of quantum mechanics, the present paper involves a multitude of connections to the actual practice of quantum theory and quantum measurement.

The measurement of a Hermitian quantity $A$ is regarded as giving an uncertain value approximating the q-expectation $\langle A\rangle$ rather than (as tradition wanted to have it) as an exact revelation of an eigenvalue of $A$. Single observations of microscopic systems are (except under special circumstances) very uncertain measurements only. The thermal interpretation

• treats detection events as a statistical measurement of particle beam intensity;
• claims that the particle concept is only asymptotically valid, under conditions where particles are essentially free.
• claims that the unmodeled environment influences the results enough to cause all randomness in quantum physics.
• implies that part of Born's rule holds exactly: Whenever a quantity $A$ is measured exactly, its value is an eigenvalue of $A$.
• allows one to derive Born's rule for scattering and in the limit of ideal measurements;
• has no explicit collapse -- the latter emerges approximately in non-isolated subsystems;
• gives a valid interpretation of systems modeled by a quantum-classical dynamics;
• explains the peculiar features of the Copenhagen interpretation (lacking realism between measurements) and the minimal statistical interpretation (lacking realism for the single case) where these interpretations apply -- in the microscopic domain.

The thermal interpretation is an interpretation of quantum physics that is in principle refutable by theoretical arguments leading to a negative answer to a number of open issues collected at the end of the paper, since there is plenty of experimental evidence for each of the points  mentioned there.

requested Mar 1
summarized
paper authored Feb 27 to quant-ph
edited Mar 1

My paper Relative Measurement Theory Measurement, February 2018 appears to resolve the issues discussed in this paper. See https://www.sciencedirect.com/science/article/pii/S0263224117306887

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The third paper of the series about the foundations of quantum physics explains how measurements are seen from the point of view of the thermal interpretation. Unlike in traditional interpretations, single, non reproducible observations do not count as measurements since this would violate the reproducibility of measurements. This difference is most obvious in the interpretation of single discrete microscopic events.

Generally, the measurement principle says that a macroscopic quantum device qualifies as an instrument for approximately, with uncertainty $\Delta a$, measuring a Hermitian quantity $A$ of a system with density operator $\rho$, if it satisfies the conditions of uncertainty bounded from below and reproducibility. There are statistical and deterministic measurements with the corresponding nature of the relationship between the state of the system and the state of the measurement instrument. An event-based instrument is a statistical instrument measuring the probability of events modeled by a discrete (classical or quantum) statistical process. The correspondence between eigenvalues and observables is only perfect for sharp observed quantities with zero uncertainty.

Mathematically, elementary particles are causal unitary irreducible representations of the Poincare group. In an interacting quantum field theory, only asymptotically free particles exist. Specifically, the particle concept is only valid if the semiclassical approximation holds and geometric optics can be applied. Otherwise, the S-matix calculated from renormalized quantum field theory is needed. From the quantum optics point of view, the photoelectric effect can be described as the measurement of a free classical or quantum electromagnetic field by means of a photomultiplier. Stokes’ description of a partially polarized quasi monochromatic beam of classical light behaves exactly like a modern quantum bit, including Born’s rule (Malus law) and Schrödinger’s equation.

In the thermal interpretation of statistical physics, the state of the system is described by the density operator of Gibbs states from statistical mechanics and the usual hierarchy of approximations from nonequilibrium statistical mechanics to thermal equilibrium exists. The Koopmann representation can then be used to unify classical and quantum mechanics, such that states of classical mechanics are represented by diagonal density operators and states of quantum mechanics contain off-diagonal elements. The functions of expectation values follow Hamiltonian dynamics given by Poisson brackets and the classical limit can be explained by the law of large numbers and coarse graining. The information ignored by coarse-graining leads to the intrinsic quantum probability. Empirical randomness is taken to be an emergent feature of deterministic chaos implicit in the deterministic dynamics of the Ehrenfest picture. To describe applications between the classical and quantum regime, quantum-classical models based on the Liouville and Hamilton equations are often used. This results in Ehrenfest equations where the l.h.s. is replaced by classical variables. The Lie algebra of the quantities is the direct product of the Lie algebra of the classical sub-system and the Lie algebra of the quantum subsystem, which results in non-linearities responsible for chaotic behavior of the system.

Compared to other interpretations, the Copenhagen and statistical interpretation can be seen as limits of the thermal interpretation. In the thermal interpretation, the Heisenberg cut is replaced by the choice of relevant variables that describe the coarse grained deterministic system. Particles appear only as detection events created by the detector and mediated by fields and the “collapse” is a result of coarse graining. In a statistical interpretation, all statements claimed about single quantum systems are non-minimal. In particular, the minimal interpretation does not address the foundational problems posed by the ensembles of equilibrium thermodynamics.

As Arnold Neumaier explicitly says, the thermal interpretation is inspired by what physicists actually do rather than what they say. It is therefore the interpretation that people actually work with in the applications (as contrasted to work on the foundations themselves), rather than only paying lip service to it. Accordingly, this  third paper in the series features some really nice discussions of real-world measurements and interesting thoughts about real-world measurements from a wide range of applications.
However, refuting single measurements on microscopic systems and their scientific legitimacy is probably not something everybody agrees with. Even if in particle physics experiments, the interpretation could be changed to avoid particles, I personally see no strong need for doing so. Also, quantum field theory is rather seen as a framework about fields and their excitations than about beams as in geometric optics for example. There is nothing wrong with anti-particles showing up in the formalism of relativistic quantum field theory.
Stokes' description of the quantum bit by means of polarized light in classical optics is rather interesting. And modeling the discrete outcomes of measurements by multistable systems is a nice idea worthwhile to follow up. However, the notion explicitly stated that the thermal interpretation gives a deterministic picture of QM, God does not play dice but moves fast to produce randomness, makes me feel rather uncomfortable. At a fundamental level, the world is surely not quantum-classical but quantum, even though it might, for computational physics purposes for example, be possible to model quantum uncertainty by deterministic chaos.

If the thermal interpretation were rather intended to be an interpretation of statistical mechanics and a unified framework to describe experimental practice in a wide range of applications than to revise and change microscopic theoretical quantum mechanics, I could more fully embrace it. The presented need to change some well established views and concepts, about things such as the existence of particles, the inherent randomness of quantum mechanics, or the observability of single microscopic systems, is for me personally not strong or urgent enough.

reviewed Mar 12 by (5,240 points)
edited Mar 15 by Dilaton

Thanks for the review; here some comments:

1. Sacrificing the particle picture as not being fundamental is not really revolutionary; for example, it is well-known that already nuclei are better described in hydromechanical terms rather than in terms resembling point particles in some reasonable way. References [27] and [34] of Part III show that the same holds on the level of single hadrons.

2. For someone like (apparently) you who just uses quantum mechanics, the Copenhagen interpretation or the statistical interpretation, each combined with a dose of common sense, is indeed good enough. But for those who inquire about the inner consistency of quantum mechanics and the absence of a borderline where quantum mechanics turns into classical mechanics, the traditional interpretations, if taken at face value, leave certain things unexplained and indeed, lead to nonsense when inquiring enough of the detailed implications and nonimplications. This includes in particular the application of quantum mechanics to large objects that cannot be observed from the outside,

This is the reason why the interpretation problem was never settled, and why even today, Nobel prize winners such as Steven Weinberg and Gerard t'Hooft (see the introduction to Section 5 of Part III), are unsatisfied with the foundations. They felt the need for changes, though they verbalized their concerns only after retirement - being more practically oriented before.

The thermal interpretations addresses precisely these questions while leaving everything else intact.

Dear Arnold, can you describe the classical mechanics (CM) and its interpretation, if any, including the measurement problem. It is another, but very interesting topic, because our difficulties stem from its being wrongly taken on, in my opinion. Thanks.

@VladimirKalitvianski: In classical mechanics, the measurement problem is - as in quantum mechanics - the question, how (and how accurately) one part of the universe can obtain reliable (reproducible) information about another part of the universe, assuming only the interactions defined by an appropriate mathematical model of the (relevant part of the) universe. To my knowledge, virtually nothing has been done on the classical side. If you want to discuss this further, please ask a question in the Q&A section; not here.

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