TY - JOUR

T1 - The collapse problem as a consistency problem. Is the quantum measurement hypothesis consistent with conservation laws?

AU - Sen, R. N.

N1 - Funding Information:
I would like to thank Professors Helmut Reeh, Hansjörg Roos, Geoffrey Sewell and Ted Eisenberg for correspondence on earlier versions of this manuscript. I would also like to thank the referee for several useful suggestions – which have been incorporated in the body of this text – and for drawing my attention to Sneed's paper.
Publisher Copyright:
© 2023 The Author(s)

PY - 2023/11/1

Y1 - 2023/11/1

N2 - In his book Mathematical Foundations of Quantum Mechanics von Neumann asserted that the state vector must collapse upon measurement of any self-adjoint operator. This assertion was based on his reading of the Compton-Simon experiment. However, comparing von Neumann's account with the Compton-Simon paper itself, we find that von Neumann had completely misinterpreted the experiment. Compton and Simon had measured two angles on the same photographic plate; in von Neumann's reading it became two successive measurements which gave identical results, where the experimenter could choose which measurement to perform first. In short, von Neumann's case for collapse simply did not exist! In quantum mechanics, superpositions of different eigenvalues of conserved observables are freely admissible, but conservation laws require successive measurements of a conserved quantity upon the same object to return the same value (if the object is isolated and moves under a Schrödinger equation). This can be guaranteed only if the first measurement has caused the state vector to collapse. Thus collapse is a consistency condition which the theory must satisfy. Consistency problems are solved by displaying examples, or models, which fulfil the conditions in question. We therefore have to construct a mathematical model of the measurement process. This requires taking a stand on whether the measuring instrument does (Bohr) or does not (von Neumann–Wigner) have a classical description. If it does not, the ‘quantum measurement problem’ is insoluble. If it does, the problem – for conserved quantities – does have a solution, given by Sewell in 2005. The model displays collapse while evolving under the Schrö- dinger equation with a time-dependent hamiltonian; it does not have to call upon the observer's ‘conscious ego’. Its existence shows that additive conservation laws are consistent with the superposition principle and Schrödinger evolution. Measurement of a conserved observable returns an eigenvalue of the observable; measurement of a non-conserved observable returns a diagonal matrix element of the observable in the basis of energy eigenvectors.

AB - In his book Mathematical Foundations of Quantum Mechanics von Neumann asserted that the state vector must collapse upon measurement of any self-adjoint operator. This assertion was based on his reading of the Compton-Simon experiment. However, comparing von Neumann's account with the Compton-Simon paper itself, we find that von Neumann had completely misinterpreted the experiment. Compton and Simon had measured two angles on the same photographic plate; in von Neumann's reading it became two successive measurements which gave identical results, where the experimenter could choose which measurement to perform first. In short, von Neumann's case for collapse simply did not exist! In quantum mechanics, superpositions of different eigenvalues of conserved observables are freely admissible, but conservation laws require successive measurements of a conserved quantity upon the same object to return the same value (if the object is isolated and moves under a Schrödinger equation). This can be guaranteed only if the first measurement has caused the state vector to collapse. Thus collapse is a consistency condition which the theory must satisfy. Consistency problems are solved by displaying examples, or models, which fulfil the conditions in question. We therefore have to construct a mathematical model of the measurement process. This requires taking a stand on whether the measuring instrument does (Bohr) or does not (von Neumann–Wigner) have a classical description. If it does not, the ‘quantum measurement problem’ is insoluble. If it does, the problem – for conserved quantities – does have a solution, given by Sewell in 2005. The model displays collapse while evolving under the Schrö- dinger equation with a time-dependent hamiltonian; it does not have to call upon the observer's ‘conscious ego’. Its existence shows that additive conservation laws are consistent with the superposition principle and Schrödinger evolution. Measurement of a conserved observable returns an eigenvalue of the observable; measurement of a non-conserved observable returns a diagonal matrix element of the observable in the basis of energy eigenvectors.

KW - Collapse hypothesis

KW - Measurement theory

KW - Superposition principle and conservation laws

UR - http://www.scopus.com/inward/record.url?scp=85171785631&partnerID=8YFLogxK

U2 - 10.1016/j.aop.2023.169462

DO - 10.1016/j.aop.2023.169462

M3 - Article

AN - SCOPUS:85171785631

SN - 0003-4916

VL - 458

JO - Annals of Physics

JF - Annals of Physics

M1 - 169462

ER -