Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2511.12311

The conditional probabilities and the empirical laws in a free scalar QFT in curved spacetime

Hideyasu Yamashita

math-phgr-qchep-thquant-phSubmitted Nov 15, 2025• Announced Nov 18, 2025• 0 citations
arXivHTMLPDF

Abstract

Unlike QFT in Minkowski spacetime (QFTM), QFT in curved spacetime (QFTCS) suffers from a conceptual obscurity on the empirical (experimentally verifiable/falsifiable) laws. We propose to employ the notion of prior conditional probabilities to describe a part of the empirical laws of QFTCS. This is interpreted as a quantum conditional probability without no information on the initial state. Hence this notion is expected to be free from the inevitable vagueness of the empirical meaning of quantum states in QFTCS. More generally in quantum physics, this notion seems free from the conceptual problems on state reductions. We confine ourselves to the probabilistic laws of the free scalar fields (Klein-Gordon fields) in curved spacetime, which require some reconsideration on the empirical meaning of the canonical commutation relation (CCR). We give some examples of empirical laws in terms of prior conditional probabilities, concerning the CCR and the free scalar QFTCS.

Comments
22 pages

Related papers

The empirical laws for Majorana fields in a curved spacetime

Hideyasu Yamashita

This article is a sequel to our previous paper (arXiv:2511.12311), where we considered the conceptual problem on the empirical laws for the Klein\textendash Gordon quantum field theory in curved spacetime (QFTCS), and we will consider the similar problems for the Majorana field on curved spacetime here. A ``law'' in theoretical physics is said to be observable or empirical only if it can be verified/falsified by some experimental procedure. The notion of empiricality/observability becomes far more unclear in QFTCS, than in QFT in Minkowski (flat) spacetime (QFTM), mainly because QFTCS lacks the notion of vacuum. This could potentially undermine the status of QFTCS as a physical (not only mathematical) theory. We consider this problem for the Majorana field in curved spacetime, and examine some examples of the empirical laws.

gr-qcCross-listedarXiv:2602.16907•Feb 20, 2026

A skepticism on the concept of quantum state related to quantum field theory on curved spacetime

Hideyasu Yamashita

Some skeptical arguments on the physical reality of quantum states are given. First, I argue that the algebraic formalism of quantum field theory in curved spacetime (algebraic QFTCS, AQFTCS) leads to such a skepticism. Of course we have the purely mathematical notion of states on a $C^{*}$-algebra $\mathfrak{A}$, but usually in non-relativistic quantum mechanics and quantum field theory in Minkowski spacetime (QFTM), not all of them are considered to be physically real; Some of them are physically real (or realizable) states, but others are non-physical ``fictional'' states. Only the states which can be expressed as a density matrix on a fixed ``physical Hilbert space'' (the GNS representation space of $\mathfrak{A}$ w.r.t. the vacuum) are viewed to be physically real. On the other hand, in QFTCS, there is no distinguished physical Hilbert space; no distinguished vacuum state. Thus we cannot distinguish physically real states from fictional states. The second part of my argument is a counterargument to what I call ``pragmatic realism on quantum states'', which insists as follows: ``We are permitted to regard a quantum state as a physical reality, because the concept of quantum state is indispensable in quantum physics.'' I argue that the concept of quantum state is indeed dispensable in non-relativistic QM, and hence this pragmatic realist thesis is vacuous there. I give a conjecture that it is also dispensable in QFTM and QFTCS, and some preliminary considerations on it.

quant-pharXiv:2604.11830•Apr 14, 2026•0 citations

Quantum field theory for classical fields

Christof Wetterich

For classical field theories with probabilistic initial conditions the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to employ observables based on fluctuating fields. In terms of these "statistical observables" the probabilistic classical field theory becomes a quantum field theory. Non-commuting operators are associated to observables. The quantum rules follow from the laws for classical probabilities. The ``quantum'' is part of the ``classical''. A regularized functional integral guarantees the unitarity of the quantum field theory. We discuss in detail the classical relativistic Klein-Gordon equation with interactions.

quant-pharXiv:2603.05061•APS Open Science•Mar 6, 2026•0 citations

Extension of quantum information theory to curved spacetimes

Daniel R. Terno

The representation of measurements by positive operator valued measures and the description of the most general state transformations by means of completely positive maps are two basic concepts of quantum information theory. These concepts can be trivially extended to field theories in curved spacetime if all the representations of canonical commutation or anticommutation relations are unitarily equivalent. We show that both concepts can be applied even when there is no such unitary equivalence.

arXiv:quant-ph/0111144•Nov 27, 2001•1 citation

Towards a probabilistic foundation of relativistic quantum theory: the one-body Born rule in curved spacetime

Maik Reddiger, Bill Poirier

Abstract In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.

arXiv:2012.05212•Quantum Studies Mathematics and Foundations•Oct 30, 2024•4 citations

A Pattern-Based Interpretation of Quantum Fields and Spacetime Information

Abhinav gupta Gupta

All known physical systems are described by quantum fields, whose observable properties arise from measurable correlations among their states. In this interpretation, what we call information is not an abstract entity but the physical pattern formed by these correlations. Energy, momentum, and spacetime curvature are already known to be expressible through expectation values of quantum operators, and recent results in quantum information theory show that variations in entanglement correspond directly to variations in energy. From this perspective, macroscopic structures—including matter and spacetime geometry—can be understood as stable, evolving patterns within quantum fields. Apparent non-local correlations, such as quantum entanglement, do not transmit signals faster than light but reflect pre-existing informational structure consistent with relativistic causality. Gravitational effects traditionally attributed to unseen components may, in principle, also be described as consequences of information distribution within known fields, without invoking new particles or interactions. This interpretation does not modify established physical laws or introduce new constants. It remains fully compatible with quantum field theory and general relativity, and is intended as a conceptual framework for organizing existing results rather than a new predictive theory. Its empirical relevance lies in re-expressing known phenomena in terms of information patterns that are, in principle, measurable through correlations and entanglement.

•Feb 3, 2026•0 citations
Unlike QFT in Minkowski spacetime (QFTM), QFT in curved spacetime (QFTCS) suffers from a conceptual obscurity on the empirical (experimentally verifiable/falsifiable) laws. We propose to employ the notion of prior conditional probabilities to describe a part of the empirical laws of QFTCS. This is interpreted as a quantum conditional probability without no information on the initial state. Hence this notion is expected to be free from the inevitable vagueness of the empirical meaning of quantum states in QFTCS. More generally in quantum physics, this notion seems free from the conceptual problems on state reductions. We confine ourselves to the probabilistic laws of the free scalar fields (Klein-Gordon fields) in curved spacetime, which require some reconsideration on the empirical meaning of the canonical commutation relation (CCR). We give some examples of empirical laws in terms of prior conditional probabilities, concerning the CCR and the free scalar QFTCS.
This article is a sequel to our previous paper (arXiv:2511.12311), where we considered the conceptual problem on the empirical laws for the Klein\textendash Gordon quantum field theory in curved spacetime (QFTCS), and we will consider the similar problems for the Majorana field on curved spacetime here. A ``law'' in theoretical physics is said to be observable or empirical only if it can be verified/falsified by some experimental procedure. The notion of empiricality/observability becomes far more unclear in QFTCS, than in QFT in Minkowski (flat) spacetime (QFTM), mainly because QFTCS lacks the notion of vacuum. This could potentially undermine the status of QFTCS as a physical (not only mathematical) theory. We consider this problem for the Majorana field in curved spacetime, and examine some examples of the empirical laws.
Some skeptical arguments on the physical reality of quantum states are given. First, I argue that the algebraic formalism of quantum field theory in curved spacetime (algebraic QFTCS, AQFTCS) leads to such a skepticism. Of course we have the purely mathematical notion of states on a C∗C^{*}C∗-algebra A\mathfrak{A}A, but usually in non-relativistic quantum mechanics and quantum field theory in Minkowski spacetime (QFTM), not all of them are considered to be physically real; Some of them are physically real (or realizable) states, but others are non-physical ``fictional'' states. Only the states which can be expressed as a density matrix on a fixed ``physical Hilbert space'' (the GNS representation space of A\mathfrak{A}A w.r.t. the vacuum) are viewed to be physically real. On the other hand, in QFTCS, there is no distinguished physical Hilbert space; no distinguished vacuum state. Thus we cannot distinguish physically real states from fictional states. The second part of my argument is a counterargument to what I call ``pragmatic realism on quantum states'', which insists as follows: ``We are permitted to regard a quantum state as a physical reality, because the concept of quantum state is indispensable in quantum physics.'' I argue that the concept of quantum state is indeed dispensable in non-relativistic QM, and hence this pragmatic realist thesis is vacuous there. I give a conjecture that it is also dispensable in QFTM and QFTCS, and some preliminary considerations on it.
For classical field theories with probabilistic initial conditions the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to employ observables based on fluctuating fields. In terms of these "statistical observables" the probabilistic classical field theory becomes a quantum field theory. Non-commuting operators are associated to observables. The quantum rules follow from the laws for classical probabilities. The ``quantum'' is part of the ``classical''. A regularized functional integral guarantees the unitarity of the quantum field theory. We discuss in detail the classical relativistic Klein-Gordon equation with interactions.
Abstract In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.
All known physical systems are described by quantum fields, whose observable properties arise from measurable correlations among their states. In this interpretation, what we call information is not an abstract entity but the physical pattern formed by these correlations. Energy, momentum, and spacetime curvature are already known to be expressible through expectation values of quantum operators, and recent results in quantum information theory show that variations in entanglement correspond directly to variations in energy. From this perspective, macroscopic structures—including matter and spacetime geometry—can be understood as stable, evolving patterns within quantum fields. Apparent non-local correlations, such as quantum entanglement, do not transmit signals faster than light but reflect pre-existing informational structure consistent with relativistic causality. Gravitational effects traditionally attributed to unseen components may, in principle, also be described as consequences of information distribution within known fields, without invoking new particles or interactions. This interpretation does not modify established physical laws or introduce new constants. It remains fully compatible with quantum field theory and general relativity, and is intended as a conceptual framework for organizing existing results rather than a new predictive theory. Its empirical relevance lies in re-expressing known phenomena in terms of information patterns that are, in principle, measurable through correlations and entanglement.