# Foundations of quantum physics I. A critique of the tradition

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Referee this paper: arXiv:1902.10778 by Arnold Neumaier

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Abstract: This paper gives a thorough critique of the foundations of quantum physics in its mainstream interpretation (i.e., treating pure states as primitives, without reference to hidden variables, and without modifications of the quantum laws). This is achieved by cleanly separating a concise version of the (universally accepted) formal core of quantum physics from the (controversial) interpretation issues. The latter are primarily related to measurement, but also to questions of existence and of the meaning of basic concepts like 'state' and 'particle'. The requirements for good foundations of quantum physics are discussed.

Main results:

•  Born's rule cannot be valid universally, and must be considered as a scientific law with a restricted domain of validity.
• If the state of every composite quantum system contains all information that can be known about this system, it cannot be a pure state in general.
requested Mar 1
summarized
paper authored Feb 27 to quant-ph
edited Mar 1

I found a typo on page 7, the Schrödinger equation must contain $\partial\psi/\partial t=\dot{\psi}$.

I do not mind if you or anybody else revisits the QM "foundations" including "interpretations". However I disagree with your "subdivision" of QM into a "core" and "interpretations".

Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with a few meager pointers on how to match it with experimental reality) and an interpretational halo that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics, where many interpretations coexist and compete for the attention of those interested in what quantum physics really means.

...

Measurement should not figure in the foundations of physics...

As to me, I believe that everything in Physics is measurement-based, with all our human imperfections, and only then "idealized" to math, so the core is as questionable/unquestionable as interpretations. Measurements are always implied, explicitly or implicitly.

@VladimirKalitvianski: yes, a dot over the $\psi$ on the left was missing.

As to the core - there is an overwhelming agreement on this, and hardly anybody questions it; this alone justifies the separation. Measurement (though necessary for the development and testing of a theory) is not usually treated in theoretical physics books (which consist almost exclusively of math) but left to the specialized literature.

You are right in your statement - there is a (bad) habit to "formalise" or "axiomatise" theoretical physics, or even to "derive" it from some "principles". But actually it is all done "a posteriori" - after making sure that it "works".

## 1 Review

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The purpose of this first paper of the series about the foundations of quantum physics is to demonstrate the need for new foundations of quantum mechanics and quantum field theory.  This is done by separating the formal axiomatic core of quantum mechanics from controversial interpretational issues.  Among other things, Born’s rule as traditionally applied in quantum mechanics and its applications is claimed to be of restricted validity only.

In the first technical chapter, the axioms of the formal core are chosen in order to highlight the similarities between quantum mechanics and classical mechanics and to work smoothly with statistical mechanics to describe macroscopic measurements. One needs to define a Hilbert space and a trace one positive semidefinite trace class operator, suggestively denoted $\rho$, to describe the state of the system. The evolution of the state is given by von Neumann’s equation, and the Hamiltonian and additional self-adjoint operators provide the quantum observables. Quantum expectation values are calculated by the statistical mechanics formula. Pure states are defined as those that are described by a density operator that maps the Hilbert space to a one-dimensional subspace. For a closed system, Born’s rule is then derived from the axioms of the formal core. The so called universally accepted minimal interpretation (MI),  saying  that measurements on identically prepared systems statistically agree with the definition of quantum expectations, should be replaced by the Thermal Interpretation as described by the second paper of the series.

The outline of the formal core of quantum mechanics is followed by  a criticism of Born’s rule; mostly based on the fact that it deals with theoretical idealized measurements. While the first formulations by Born himself were independent from measurements, the modern version for measuring eigenvalues of self-adjoint Hermitian operators with discrete and continuous spectra has a limited domain of applicability due to the impossibility of exact measurements, lack of knowledge of the spectra of the operators of interest, or for derived and indirect measurements. Also, seemingly FTL propagation for subsequent position measurements can occur. To legitimately apply Born's rule, repeated measurements, measurement errors smaller than the separation of the eigenvalues, and an exact prior knowledge of the spectra are needed. This chapter terminates with defining the state of a system to contain everything that can be said, properties of the subsystems are properties of the whole system,  and the state of a system determines the state of its subsystems. The conventional derivation of the density operator involving subsequent quantum and statistical means is refuted. Finally, algebraic quantum field theory arguments for the non-existence of pure states in QFT and the breaking of the relationship between quantum observables and hermitian self-adjoint linear operators are presented.

The last part of the paper discusses the requirements for better foundations of quantum mechanics. They should not contain measurements but nevertheless provide a formal measurement theory to describe what experimental physicists measure, and reproduce  the  correct interplay between theory and experiment such that  something in real life ’is’ an instance of the theoretical concept if it matches the theoretical description sufficiently well. Measuring individual particles is considered impossible, so-called beables (in quantum optics given by densities, intensities, and correlation functions used to describe optical fields) are considered the only objectively real things. The existence of particles is contested except when they are measured. They are deemed detection events created by the detector and mediated by fields. Photon signals are claimed to be just the response of the detector to the impinging classical field, and the existence of massive particles as measured in bubble chambers are challenged by analogies to cracks propagating in glass after the impact of a projectile.

The biggest difference between the point of view presented in this paper and what one usually learns when studying physics seems to concern the scope of fundamental theoretical quantum physics.  Compared to the usual textbook notion of what (theoretical) quantum mechanics itself is meant to do, the scope of quantum mechanics as outlined in this paper seems unusually broad.  Indeed, the thermal interpretation described later in this series in more detail  is meant to give new foundations that connect quantum physics (including quantum mechanics, statistical mechanics, quantum field theory and their applications) to experiment. One could say that it is meant to be an IOE, an  interpretation of everything. Conventinally, the goal of fundamental physics is among other things to understand things at the smallest scales possible. For example theoretical quantum mechanics is meant to deal  with microscopic situations, the formalism just predicts microscopic probabilities.   How quantum mechanics is used in real macroscopic measurements and even more so in applications is conventionally considered not a task for fundamental theoretical quantum mechanics but for experimental or applied quantum mechanics respectively. Theoretical quantum mechanics without interpretation just deals with idealized measurements for which Born’s rule holds while leaving the real-world measurements of its theoretical observables (including the data analysis)  as a task for experimenters. Basic quantum mechanics is defined to hold for microscopic systems, going to quantum statistical mechanics already involves additional steps towards the macroscopic regime. Statistical and macroscopic physics are usually considered to be rather applied  or derived subfields that do not belong to the theoretical foundations of quantum mechanics themself.

For people who agree with the usual hierarchical organization of physics into subfields there is nothing wrong with Born’s rule and its application to idealized measurements. If one adopts the point of view that those imagined idealized measurements take the same role as Platonic perfect points, lines, circles, planes, etc. in mathematics and what experimenters really measure is an approximation to those, there is no problem in using idealized measurements to define theoretical quantum mechanics. The real-world experimental issues are conventionally not included in the definition of theoretical quantum mechanics.To mention some smaller points, even if it is true that in a statistical situation the two kinds of uncertainties used to derive the density matrix cannot be separated by measurement, it does not necessarily mean that the derivation is wrong mathematically. In the conventional notion of quantum mechanics everything can happen with a corresponding probability. So not everybody will probably consider the interpretation of the squared amplitude $|\psi(x)|^2$,
as probability of finding a particle at position $x$ equally useless as claimed in this paper, because the probability for the issue of FTL propagation in this context is insanely small.

For me personally, it is rather difficult to vote on this paper. While the upcoming coherent quantum mechanics, which will put quantum mechanics and classical mechanics in a single mathematical structure is intriguing enough to me, I generally feel rather satisfied with considering a narrower scope of theoretical quantum mechanics than it is targeted by the thermal interpretation. For me, things are not broken so I don’t need to mend them. However, things such as the failure of Born’s rule if one wants to merge real-world measurements with theoretical quantum physics in a single line of thought and the resulting motivation for a new interpretation are presented well enough and the ideas are also rather original. The only thing which gives me rather a pause is the questioning of the existence of particles. Here it seems the business as usual, such as the standard model to explain three of the four fundamental forces by force mediating particles, works well enough. To give up for example this theoretical framework which is based on the existence of particles, would be a huge paradigm-shift  which should be done only after strong proofs of necessity for it.

reviewed Mar 3 by (5,240 points)
edited Mar 12 by Dilaton

Thanks for the review; here some comments:

Paragraph 2: Born's rule as traditionally stated, always refers to experiment, hence cannot be deduced from the formal core (in which ''experiment'' is undefined). The formal core only allows one to deduce a formal Born rule - (3) on p.8, yet without any interpretation in terms of experiments. The latter is then provided by the interpretation axiom (MI), which, however, does not belong to the formal core and indeed is shown later to be suspect.

Paragraph 4: Photon detection signals may be due to an incident classical or quantum field; the classical case was discussed in more detail only in order to show that the detection signals cannot be interpreted as signals of the quantum nature of the incident light, hence must be solely due to the properties of the detector.

Paragraph 6: If Born's rule would only refer to eigenvalues as idealized (rather than actual) measurement results, these must have an objective existence independent of (and hence before) actual measurement. But assuming this in 2-slit experiments leads to well-known problems.

Paragraph 7: The existence of particles (outside the asymptotic scattering framework) is already questionable in interacting relativistic quantum field theory, as explained in more detail in Part II. The standard model is about interacting fields, not about interacting particles. On this level we already have a relativistic quantum field theory only, not a relativistic quantum particle theory!

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