These notes provide an introduction to Haagerup noncommutative $L^p$ spaces for quantum field theorists. The aim is to explain how the familiar calculus of density matrices, fractional powers, and Schatten norms extends to local observable algebras of type~III, where an intrinsic trace is unavailable. We develop the construction first for a matrix algebra, making the crossed product, its trace, and the dual action explicit, before passing to general von Neumann algebras. Particular attention is paid to the representation of normal states by Haagerup densities and to the distinction between the crossed-product trace and the canonical Haagerup functional used to compute expectation values. As a concrete application, we calculate the sandwiched Rényi divergence for a coherent excitation of a chiral current, verifying monotonicity under restriction and the Rényi quantum null energy condition at every real order $α>1$. The presentation assumes familiarity with quantum mechanics and elementary quantum field theory, while introducing the operator-algebraic concepts needed for the construction and calculation.
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The paper explains how density-matrix operations used in quantum information extend to type-III local observable algebras, which have no intrinsic trace. It is primarily an introduction to existing Haagerup noncommutative $L^p$ theory, organized around explicit matrix and continuum-QFT calculations.
For matrix algebras, it constructs the modular crossed product, dual action and semifinite trace, identifying Schatten matrices with homogeneous operator fields $J_p(a)(t)=e^{-t/p}a$. It then presents the general von Neumann algebra construction. A central distinction is between the crossed-product trace $\tau$ and the canonical Haagerup functional $\operatorname{tr}$: normalized state densities satisfy $\operatorname{tr}(h_\psi)=1$ but $\tau(h_\psi)=\infty$. Above Rényi order one, finiteness is formulated through a positive-power factorization rather than uncontrolled inverse density powers.
The worked application concerns a coherent excitation of a chiral abelian current restricted to advancing half-lines. For a specified preparation with tail $f_0(u)=q_0\ell/(u+\ell)$, it obtains $\widetilde D_\alpha(V)=\pi\alpha(\alpha+1)q_0^2\ell^2/[12(V+\ell)^2]$ for every finite real $\alpha>1$ and $V\geq0$. Gaussian covariance and BCH identities construct the factorization; expressing its factor as a scalar times a normalized density’s fractional power evaluates its norm. Cut derivatives verify monotonicity and the Rényi QNEC for this excitation. The $\alpha\to1$ limit gives $D_1''(V)=2\pi\Delta\langle T(V)\rangle$.
This is not a general real-order Rényi-QNEC proof. Extensions to noncoherent states, physical finiteness criteria and regions with nongeometric modular flow remain open directions; interpolation theory is not developed.
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These notes provide an introduction to Haagerup noncommutative Lp spaces for quantum field theorists. The aim is to explain how the familiar calculus of density matrices, fractional powers, and Schatten norms extends to local observable algebras of type~III, where an intrinsic trace is unavailable. We develop the construction first for a matrix algebra, making the crossed product, its trace, and the dual action explicit, before passing to general von Neumann algebras. Particular attention is paid to the representation of normal states by Haagerup densities and to the distinction between the crossed-product trace and the canonical Haagerup functional used to compute expectation values. As a concrete application, we calculate the sandwiched Rényi divergence for a coherent excitation of a chiral current, verifying monotonicity under restriction and the Rényi quantum null energy condition at every real order α>1. The presentation assumes familiarity with quantum mechanics and elementary quantum field theory, while introducing the operator-algebraic concepts needed for the construction and calculation.
The paper explains how density-matrix operations used in quantum information extend to type-III local observable algebras, which have no intrinsic trace. It is primarily an introduction to existing Haagerup noncommutative Lp theory, organized around explicit matrix and continuum-QFT calculations.
For matrix algebras, it constructs the modular crossed product, dual action and semifinite trace, identifying Schatten matrices with homogeneous operator fields Jp(a)(t)=e−t/pa. It then presents the general von Neumann algebra construction. A central distinction is between the crossed-product trace τ and the canonical Haagerup functional tr: normalized state densities satisfy tr(hψ)=1 but τ(hψ)=∞. Above Rényi order one, finiteness is formulated through a positive-power factorization rather than uncontrolled inverse density powers.
The worked application concerns a coherent excitation of a chiral abelian current restricted to advancing half-lines. For a specified preparation with tail f0(u)=q0ℓ/(u+ℓ), it obtains Dα(V)=πα(α+1)q02ℓ2/[12(V+ℓ)2] for every finite real α>1 and V≥0. Gaussian covariance and BCH identities construct the factorization; expressing its factor as a scalar times a normalized density’s fractional power evaluates its norm. Cut derivatives verify monotonicity and the Rényi QNEC for this excitation. The α→1 limit gives D1′′(V)=2πΔ⟨T(V)⟩.
This is not a general real-order Rényi-QNEC proof. Extensions to noncoherent states, physical finiteness criteria and regions with nongeometric modular flow remain open directions; interpolation theory is not developed.
This paper provides an explanation of the type classification of von Neumann algebras, which has made many appearances in the recent work on entanglement in quantum field theory and quantum gravity. The goal is to bridge a gap in the literature between resources that are too technical for the non-expert reader, and resources that seek to explain the broad intuition of the theory without giving precise definitions. Reading this paper will provide you with: (i) an argument for why “factors” are the fundamental von Neumann algebras that one needs to study; (ii) an intuitive explanation of the type classification of factors in terms of renormalization schemes that turn unnormalizable positive operators into “effective density matrices”; (iii) a mathematical explanation of the different types of renormalization schemes in terms of the allowed traces on a factor; (iv) an intuitive characterization of type I and type II factors in terms of their “standard forms”; and (v) a list of some interesting connections between type classification and modular theory, including the argument for why type III 1 factors are believed to be the relevant ones in quantum field theory. None of the material is new, but the pedagogy is different from other sources the author has read; it is most similar in spirit to the recent work on gravity and the crossed product by Chandrasekaran, Longo, Penington, and Witten.
Couples are wholes and not wholes, what agrees disagrees, the concordant is discordant. From all things one and from one all things. ( Heraclitus , On the Universe 59.) This chapter introduces the notion of joint (Wigner) density of random variables, for future use in quantum Malliavin calculus. For this we will rely on functional calculus on general Lie algebras, starting with the Heisenberg– Weyl algebra. We also consider some applications to quantum optics and timefrequency analysis. Joint moments of noncommuting random variables The notion of moment ⟨ ϕ , P n ψ ⟩ of order n of P in the state ⟨ ϕ , · ψ ⟩ is well understood as the noncommutative analogue of E [ X n ]. Indeed, in the single variable case, various functional calculi are available for elements of an algebra. In particular, • polynomials can be applied to any element in an algebra; • holomorphic functions can be applied to elements of a Banach algebra provided the function is holomorphic on a neighborhood of the spectrum of the element; • continuous functions can be applied to normal elements of C*-algebras provided they are continuous on the spectrum of the element; • measurable ( i.e. , Borel) functions can be applied to normal elements in a von Neumann algebra ( e.g. , B (h ) ), provided they are measurable on the spectrum of the operator. In addition, the Borel functional calculus can be extended to unbounded normal operators (where normal means that XX ∗ and X ∗ X have the same domain and coincide on this domain). All the above functional calculi work due to a restriction to a commutative subalgebra. In that sense, Weyl calculus is fundamentally different because it applies to functions of noncommuting operators. We now need to focus on the meaning of joint moments. For example, which is the noncommutative analogue of E [ XY ]? A possibility is to choose ⟨ ϕ , PQψ ⟩ as the the noncommutative analogue of E [ XY ], however this would lead to ⟨ ϕ , QPψ ⟩ as the the noncommutative analogue of E [ YX ] and we have E [ XY ] = E [ YX ] while due to noncommutativity. A way to solve this contradiction can be to define the first joint moment as Next, how can we define the analogue of E [ XY 2 ]? Proceeding similarly, a natural extension could be Then what is the correct way to construct E[ X 2 Y 3 ]? Clearly there is a combinatorial pattern which should be made explicit.
An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III _{1} 1 , in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II _∞ ∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.
Quantum Field Theory (QFT) represents a vast generalization of Quantum Mechanics (QM), as it deals with systems that have an infinite number of degrees of freedom. The Stone-von Neumann theorem, which establishes the equivalence of irreducible representations of the canonical commutation relations (CCR) in QM, does not extend to QFT. Consequently, QFT admits multiple inequivalent irreducible representations, leading to a much richer algebraic structure. This essay aims to explore the physics of QFT from the operator algebra perspective, particularly focusing on entanglement entropy. We discuss the role of von Neumann algebras of different types in QFT, describe the local operator algebra approach to QFT, and explain how entanglement entropy can be defined in terms of the algebra of observables. Additionally, we explore the benefits of this approach in concrete applications, specifically in quantum field theory on curved spacetime.
Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.
A note on Meyer's note.- Stochastic integration.- Quantum stochastic parallel transport on non-commutative vector bundles.- Input and output channels in quantum systems and quantum stochastic differential equations.- Some noncommutative Radon-Nikodym Theorems for von Neumann algebras.- Multidimensional quantum diffusions.- An application of de finetti's theorem.- The quantum Langevin equation from the independent-oscillator model.- Quantum poisson processes: Physical motivations and applications.- A noncommutative generalization of conditionally positive definite functions.- Contraction semigroups in L2 over a von neumann algebra.- Survey on a theory of non-commutative stationary markov processes.- Dynamical entropy for quantum systems.- An integral kernel approach to noise.- A note on shifts and cocycles.- Local measures in fock space stochastic calculus and a generalized ito-tanaka formula.- Representation of a class of quantum martingales II.- Conditional expectation in quantum probability.- Mutual quadratic variation and ito's table in quantum stochastic calculus.- On mixing properties of automorphisms of von neumann algebras related to measure space transformations.- First exit time: A theory of stopping times in quantum processes.- A central limit theorem on the free lie group.- Entropy, observability and the generalised second law of thermodynamics.- Remarks on asymptotic properties of groups of the bogoliubov transformations of car c*-algebras.- Linear and non-linear stochastic processes.- Detailed balance and critical slowing down.- Quantum martingales and stochastic integrals.