[Paper Review] Measurement and self-adjoint operators

G.E. Johnson|arXiv (Cornell University)|May 28, 2014

Quantum Mechanics and Applications6 references3 citations

TL;DR

This paper challenges the canonical quantization paradigm by arguing that not all classical dynamical variables correspond to self-adjoint operators in Hilbert space, proposing instead that quantum mechanics should be grounded in states as Hilbert space elements with the Everett-Wheeler-Graham (EWG) relative state interpretation. It demonstrates that quantum mechanics can consistently describe physical systems—including interacting quantum field theories—without requiring Hermitian field operators, showing that classical limits can emerge even when key observables lack self-adjoint representations.

ABSTRACT

The approximations of classical mechanics resulting from quantum mechanics are richer than a correspondence of classical dynamical variables with self-adjoint Hilbert space operators. Assertion that classical dynamic variables correspond to self-adjoint Hilbert space operators is disputable and sets unnatural limits on quantum mechanics. Well known examples of classical dynamical variables not associated with self-adjoint Hilbert space operators are discussed as a motivation for the realizations of quantum field theory that lack Hermitian field operators but exhibit interaction.

Motivation & Objective

• To challenge the canonical quantization assumption that classical dynamical variables must correspond to self-adjoint Hilbert space operators.
• To demonstrate that quantum field theories with interactions can be consistently formulated without Hermitian field operators.
• To argue that the Everett-Wheeler-Graham relative state interpretation provides a more complete and consistent foundation for quantum mechanics than collapse-based interpretations.
• To show that classical limits of quantum systems, such as the harmonic oscillator with observable $x^3p$, can be accurately described even when the observable is not self-adjoint.
• To resolve foundational issues in relativistic quantum mechanics by freeing the formalism from reliance on classical concepts and wave function collapse.

Proposed method

• Analyzes the role of Hilbert space states as the fundamental entities in quantum mechanics, rather than self-adjoint operators.
• Applies the Everett-Wheeler-Graham (EWG) relative state interpretation to eliminate the need for wave function collapse and classical measurement domains.
• Examines specific examples—such as the harmonic oscillator observable $x^3p$—to show that classical dynamics can be approximated without self-adjointness.
• Uses the EPR paradox to illustrate that entangled states cannot be described by classical idealizations, supporting the need for a relational quantum description.
• Contrasts canonical quantization with a state-centric formulation, emphasizing unitary time evolution and the absence of measurement-induced collapse.
• Evaluates the limitations of rigged Hilbert spaces and the role of generalized functions in representing non-self-adjoint observables.

Experimental results

Research questions

• RQ1 Can quantum mechanics be consistently formulated without requiring all classical dynamical variables to be represented by self-adjoint operators?
• RQ2 To what extent does the Everett-Wheeler-Graham interpretation eliminate the need for wave function collapse and classical measurement postulates?
• RQ3 How can classical limits of quantum systems be accurately described when the corresponding observable is not self-adjoint?
• RQ4 Why do standard Lagrangian quantum field theories face foundational difficulties, and can these be traced to the assumption of Hermitian field operators?
• RQ5 Can interacting quantum field theories be realized without Hermitian field operators while still reproducing correct classical dynamics?

Key findings

• The observable $x^3p$ in the linear harmonic oscillator is not associated with a self-adjoint operator in Hilbert space, yet its classical limit is accurately reproduced by quantum mechanics.
• Measurement processes in entangled systems, such as those in the EPR paradox, do not require collapse to eigenstates of the measured observable, as shown by unitary evolution and correlation with observer states.
• The Everett-Wheeler-Graham interpretation allows for a consistent description of quantum mechanics without invoking a classical domain or wave function collapse.
• Quantum field theories with interactions can be realized without Hermitian field operators, challenging the foundational assumption that Hermiticity is necessary for physical observables.
• The correspondence between classical and quantum dynamical variables is richer than canonical quantization suggests, as classical limits emerge even when the observable is not self-adjoint.
• Failure of self-adjointness for an operator does not preclude the physical characterization of a quantity, as long as the state evolves unitarily and correlations are preserved.