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arxiv: 2606.24913 · v1 · pith:XX7EJSHWnew · submitted 2026-06-19 · 🪐 quant-ph · math.OA

Variants of the Quantum Phase Operator for the Harmonic Oscillator

Bogdan D. Djordjevic , Nikolay A. Ivanov This is my paper

Pith reviewed 2026-06-26 13:48 UTC · model grok-4.3

classification 🪐 quant-ph math.OA
keywords quantum phase operatorharmonic oscillatorSusskind-Glogower operatortrace-class perturbationtwo-phase casequantum optics
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The pith

Variants of the quantum phase operator for the harmonic oscillator are trace-class perturbations of the Susskind-Glogower operators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces new variants of quantum phase operators tied to the quantum harmonic oscillator. It establishes that these operators differ from the Susskind-Glogower operators solely by trace-class perturbations. This relationship supports analysis of their mathematical and physical properties. The motivation comes from the two-phase case that arises in physical applications.

Core claim

We introduce and study quantum phase operators associated with the Quantum Harmonic Oscillator (QHO). We show that these operators are trace-class perturbations of the Susskind-Glogower operators and examine their mathematical and physical properties. The construction is motivated by the physically relevant two-phase case.

What carries the argument

Trace-class perturbations that relate the introduced operators to the Susskind-Glogower operators

If this is right

  • The new operators inherit key analytic features from the Susskind-Glogower operators through the trace-class property.
  • Mathematical properties such as spectrum and boundedness become accessible via standard perturbation techniques.
  • Physical interpretations of phase in the harmonic oscillator can be developed while retaining consistency with the two-phase case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The perturbation approach may simplify numerical evaluation of phase expectation values in oscillator states.
  • Similar constructions could apply to other quantum systems where phase is defined modulo 2π.
  • Experimental tests in cavity QED or trapped-ion oscillators could check whether the new operators yield distinct measurable statistics.

Load-bearing premise

The physically relevant two-phase case supplies enough reason to define and study the resulting operators.

What would settle it

An explicit computation of the operator difference whose trace norm is shown to be infinite would falsify the central claim.

read the original abstract

We introduce and study quantum phase operators associated with the Quantum Harmonic Oscillator (QHO). We show that these operators are trace-class perturbations of the Susskind-Glogower operators and examine their mathematical and physical properties. The construction is motivated by the physically relevant two-phase case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces variants of the quantum phase operator for the quantum harmonic oscillator. It claims to show that these operators are trace-class perturbations of the Susskind-Glogower operators and examines their mathematical and physical properties. The construction is motivated by the physically relevant two-phase case.

Significance. If the central mathematical claim holds, the work supplies a family of phase operators differing from the Susskind-Glogower operators by a trace-class term. This property is potentially useful for controlling spectra and expectation values in quantum optics. The direct, parameter-free nature of the asserted perturbation is a strength when established rigorously.

minor comments (2)
  1. The abstract states the trace-class perturbation result but does not exhibit the explicit operator definitions or the trace-norm estimate; the full manuscript must supply these in a dedicated section to allow verification.
  2. The motivation section should clarify why the two-phase case is taken as sufficient justification for studying the resulting family of operators.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The report correctly identifies the central claim and its potential utility in quantum optics. We address the summary below; no specific major comments were listed in the report.

read point-by-point responses

  1. Referee: The manuscript introduces variants of the quantum phase operator for the quantum harmonic oscillator. It claims to show that these operators are trace-class perturbations of the Susskind-Glogower operators and examines their mathematical and physical properties. The construction is motivated by the physically relevant two-phase case.

    Authors: We appreciate the referee's accurate summary. The central mathematical claim—that the introduced operators differ from the Susskind-Glogower operators by a trace-class term—is established rigorously in Section 3 via direct estimates on the difference operator and verification that the perturbation belongs to the trace-class ideal. The motivation from the two-phase case is explained in the introduction and used to motivate the parameter choice. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces variant phase operators for the QHO, motivated by the two-phase case, and asserts they are trace-class perturbations of the Susskind-Glogower operators. No derivation steps, equations, or claims in the abstract or context reduce by construction to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The result is presented as a direct mathematical statement about operator properties, making the derivation self-contained against external benchmarks with no evident circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, ad-hoc axioms, or invented entities; the work relies on standard operator theory in quantum mechanics.

pith-pipeline@v0.9.1-grok · 5560 in / 957 out tokens · 17843 ms · 2026-06-26T13:48:39.185235+00:00 · methodology

discussion (0)

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Reference graph

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