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arXiv:2604.19861

Excitability in quantum field theory

Jacqueline Caminiti, Federico Capeccia, Jonathan Sorce

Published Apr 21, 2026• 0 citations
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Abstract

In quantum field theory, it is not always possible to excite one state out of another using only local operators. This paper establishes abstract algebraic criteria for (local) excitability in general quantum theories, and computes these criteria explicitly for zero-mean Gaussian states in (generalized) free field theories. We find that in this context, due to the special nature of Gaussian states, one-way excitability always implies two-way excitability, and our results generalize the "quasiequivalence theorems" of Powers, Stormer, van Daele, Araki, and Yamagami. A key role in our proof is played by the information-theoretic tool of canonical purification. In appendices, we provide a pedagogical introduction to the algebraic formulation of (generalized) free field theory.

DOI
10.48550/arxiv.2604.19861

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In quantum field theory, it is not always possible to excite one state out of another using only local operators. This paper establishes abstract algebraic criteria for (local) excitability in general quantum theories, and computes these criteria explicitly for zero-mean Gaussian states in (generalized) free field theories. We find that in this context, due to the special nature of Gaussian states, one-way excitability always implies two-way excitability, and our results generalize the "quasiequivalence theorems" of Powers, Stormer, van Daele, Araki, and Yamagami. A key role in our proof is played by the information-theoretic tool of canonical purification. In appendices, we provide a pedagogical introduction to the algebraic formulation of (generalized) free field theory.
We construct and characterize canonical purifications for general algebraic states, extending prior constructions by Woronowicz and by Dutta/Faulkner to general quantum theories. Given a state on a ∗*∗-algebra, the canonical purification is a state on a "doubled" algebra that admits an interpretation in terms of CRT reflection. This interpretation holds for all quantum theories, even in the absence of gravity. We then identify conditions under which canonical purifications are "pure" in the technical sense, compute their modular conjugations, and relate them to GNS and natural-cone purifications in certain settings. In an appendix, we develop a general theory of von Neumann algebras generated by unbounded ∗*∗-algebras. In a forthcoming paper with Caminiti and Capeccia, we provide an application of this general theory to the problem of excitability in quantum field theory.
There has been some recent interest in applying the techniques of Algebraic Quantum Field Theory (AQFT) to entanglement problems in perturbative QFT. In particular, the Hilbert space independence of this formulation makes it particularly interesting in the context of curved spacetimes and the emphasis on the algebra of observables makes the treatment of Bell inequalities in QFT resemble such treatment in non-relativistic Quantum Mechanics. In this work, we present the mathematical structures needed for formulating AQFT in terms of the Haag-Araki-Kastler (HAK) axioms and discuss their implications. Moreover, we discuss the algebraic approach to quantum entanglement in the form of Bell inequalities. We provide an extension of this formulation to general globally hyperbolic spacetimes using the so-called Locally Covariant approach to QFT, which extends the HAK axioms to general spacetimes by means of the Category Theory language.
Let H1{\mathcal H}_1H1​ be a finite dimensional complex Hilbert space. Let ψ↦Z(ψ)ψ\mapsto Z(ψ)ψ↦Z(ψ) be a canonical anti-commutation relations (CAR) field over H1{\mathcal H}_1H1​ acting irreducibly on a Hilbert space K{\mathord{\mathscr K}}K. The ∗*∗-algebra AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ generated by the Z(ψ)Z(ψ)Z(ψ), ψ∈H1ψ\in {\mathcal H}_1ψ∈H1​, is simply all operators on K{\mathscr K}K. However, the CAR field endows AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ with additional structure, and we are concerned with quantum operations acting in harmony with this structure. In particular, there is a "gauge" automorphism group generated by "second quantizing'' ψ↦eitψψ\mapsto e^{it}ψψ↦eitψ. The fixed point algebra of the gauge group, GH1{\mathscr G}_{{\mathcal H}_1}GH1​​, is a sub-algebra of AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ studied by Araki and Wyss. It contains the density matrices of an important class of states, the {\em gauge invariant Gaussian states}, SGIG{\mathfrak S}_{GIG}SGIG​. Our focus is on semigroups {etL}t≥0\{e^{t{\mathscr L}}\}_{t\geq 0}{etL}t≥0​ of quantum operations on AH1{\mathscr A}_{{\mathcal H}_1}AH1​​ that map SGIG{\mathfrak S}_{GIG}SGIG​ into itself. Each etLe^{t{\mathscr L}}etL is one-to-one, and our first main result is a structure theorem for such quantum operations on GH1{\mathscr G}_{{\mathcal H}_1}GH1​​ that map SGIG{\mathfrak S}_{GIG}SGIG​ into itself. We apply this to study semigroups of quantum operations on GH1{\mathscr G}_{{\mathcal H}_1}GH1​​ that map SGIG{\mathfrak S}_{GIG}SGIG​ into itself. Our second main result is a structure theorem showing that they are parameterized by pairs (G,A)(G,A)(G,A) where GGG is a contraction semigroup generator on H1{\mathcal H}_1H1​, and 0≤A≤−G−G∗0 \leq A \leq -G -G^*0≤A≤−G−G∗. We then show that each of these semigroups has a natural extension to the full CAR algebra AH1{\mathscr A}_{{\mathcal H}_1}AH1​​. Further results are obtained under further assumptions on the pair (G,A)(G,A)(G,A).
Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the formalism of algebraic quantum field theory (AQFT) provides a rigorous framework within which to analyze entanglement in the context of a fully relativistic formulation of quantum theory. What emerges from the analysis are new practical and theoretical limitations on an experimenter’s ability to perform operations on a field in one spacetime region that can disentangle its state from the state of the field in other spacelike-separated regions. These limitations show just how deeply entrenched entanglement is in relativistic quantum field theory, and yield a fresh perspective on the ways in which the theory differs conceptually from both standard nonrelativistic quantum theory and classical relativistic field theory. “...despite its conservative way of dealing with physical principles, algebraic QFT leads to a radical change of paradigm. Instead of the Newtonian view of a space-time filled with a material content one enters the reality of Leibniz created by relation (in particular inclusions) between ‘monads ’ (∼ the hyperfinite type III1 local von Neumann factors A(O) which as single algebras are nearly void of physical meaning) ” Schroer (1998, p. 302).
Quantum operations that are perfectly admissible in non-relativistic quantum theory can enable signalling between spacelike separated regions when naively imported into quantum field theory (QFT). Prominent examples of such "impossible measurements" include certain unitary kicks and projective measurements. It is generally accepted that only those quantum operations whose physical implementation arises from a fully relativistically covariant interaction, between the quantum field and a suitable probe, should be regarded as admissible. While this idea has been realised at the level of abstract algebraic QFT, or via particular measurement models, there is still no general set of operational criteria characterising which measurements are physically implementable. In this work we adopt the local S-matrix formalism to describe local operations, and make use of a hierarchy of factorisation conditions that exclude both superluminal signalling and retrocausality, thereby providing such a criterion. Realising the local S-matrices through explicit interactions between smeared field operators and a pointer degree of freedom, we further derive local causality conditions for the induced Kraus operators, which guarantee the absence of signalling in "impossible measurement" scenarios. Finally, we show that the accuracy with which local field observables can be measured is fundamentally limited by the retarded propagator of the field, which plays an essential role in the factorisation identity we prove for the field Kraus operators.
Abstract In this work, the perturbative aspects of quantum mechanics (QM) and quantum field theory (QFT), to a large extent, are studied with functional (path or field) integrals and functional techniques. This physics textbook thus begins with a discussion of algebraic properties of Gaussian measures, and Gaussian expectation values for a finite number of variables. The important role of Gaussian measures is not unrelated to the central limit theorem of probabilities, although the interesting physics is generally hidden in essential deviations from Gaussian distributions. A few algebraic identities about Gaussian expectation values, in particular Wick's theorem are recalled. Integrals over some type of formally complex conjugate variables, directly relevant for boson systems are defined. Fermion systems require the introduction of Grassmann or exterior algebras, and the corresponding generalization of the notions of differentiation and integration. Both for complex and Grassmann integrals, Gaussian integrals, and Gaussian expectation values are calculated, and generalized Wick's theorems proven. The concepts of generating functions and Legendre transformation are recalled.