[Paper Review] Photon fields in a fluctuating spacetime

Jan Naudts, Maciej Kuna+1 more|ArXiv.org|Oct 20, 2002

Noncommutative and Quantum Gravity Theories14 references3 citations

TL;DR

This paper proposes a non-perturbative quantum field model where the electromagnetic field is quantized in a spacetime with intrinsic quantum fluctuations, treating spacetime as a quantum state with non-trivial correlations. It recovers standard photon theory in the classical limit and establishes localized photon field operators as unbounded operators in Hilbert space, enabling a more fundamental description of electromagnetic interactions without the distributional formalism of conventional QFT.

ABSTRACT

We present a model of interacting quantum fields, formulated in a non-perturbative manner. One of the fields is treated semi-classically, the other is the photon field. The model has an interpretation of an electromagnetic field in a fluctuating spacetime. The model is equivalent with the quantization of electromagnetism proposed recently by Czachor. Interesting features are that standard photon theory is recovered as a limiting case, and that localized field operators for the electromagnetic field exist as unbounded operators in Hilbert space.

Motivation & Objective

• To develop a non-perturbative framework for quantum electrodynamics in a spacetime with intrinsic quantum fluctuations.
• To resolve inconsistencies between Fock space quantization and harmonic oscillator ground states in standard photon theory.
• To construct localized photon field operators as genuine Hilbert space operators, avoiding the distributional nature of conventional field operators.
• To demonstrate that standard quantum electrodynamics emerges as a limiting case of the proposed model.
• To maintain Poincaré invariance at large distances despite short-distance spacetime fluctuations.

Proposed method

• Model spacetime as a quantum state |Ω⟩ in Hilbert space, with non-trivial autocorrelation function w(q) = ⟨Ω|U(−q)Ω⟩, encoding spacetime fluctuations.
• Use a covariance system (A, G, I) with Weyl operators W(φ) to define field operators Â(φ) via functional calculus on the state |Ω⟩.
• Introduce a modified Fock space structure with a non-unique vacuum and generalized commutation relations, inspired by Czachor’s noncanonical photon theory.
• Construct field operators as real linear functions of test functions φ via Stone’s theorem applied to the Weyl group representation.
• Ensure positivity and normalization of correlation functions using Schur’s lemma on positive-definite matrices derived from test function overlaps.
• Prove covariance and continuity of correlation functions, confirming the state defines a valid quantum field theory on the covariance system.

Experimental results

Research questions

• RQ1 Can a non-perturbative quantum field theory be formulated where spacetime itself is a quantum fluctuating background, rather than a classical manifold?
• RQ2 How can localized photon field operators be defined as unbounded operators in Hilbert space, avoiding the standard distributional formalism?
• RQ3 Does the model recover standard quantum electrodynamics in an appropriate classical limit?
• RQ4 How is Poincaré invariance preserved despite the presence of a short-distance spacetime cutoff due to fluctuations?
• RQ5 What is the role of non-unique vacuum states and generalized commutation relations in reconciling Fock space with harmonic oscillator ground states?

Key findings

• The model recovers standard photon theory as a limiting case when spacetime fluctuations vanish, confirming consistency with established QED.
• Localized field operators Â(φ) exist as unbounded operators in Hilbert space, enabling a more fundamental description of local interactions.
• The vacuum state is not unique, and field operators satisfy generalized commutation relations, consistent with Czachor’s noncanonical photon theory.
• Correlation functions are covariant under spacetime translations and satisfy positivity and normalization, confirming the state is physically well-defined.
• The field operator Â(φ) is a real linear function of its argument φ, ensuring proper transformation properties under superposition.
• The model maintains Poincaré invariance at large distances, even though short-distance fluctuations break strict Poincaré symmetry, which is restored via the autocorrelation function w(q) ≈ 0 for large |q|.