Ricardo Correa da Silva
Mathematical Physicist
Assistant professor at University of São Paulo
Postdoctoral researcher at Friedrich-Alexander-Universität Erlangen-Nürnberg
Mathematical Physicist
Assistant professor at University of São Paulo
Postdoctoral researcher at Friedrich-Alexander-Universität Erlangen-Nürnberg
My research is mainly focused on operator algebras and their physical applications. More specifically, I'm interested in algebraic quantum field theory, KMS states, Tomita-Takesaki modular theory, noncommutative Lp-spaces and related topics.
In addition, I'm also interested in foundations of quantum theory and philosophy of physics.
Standard subspaces are closed real subspaces of a complex Hilbert space that naturally arise in the context of von Neumann algebras as a consequence of the existence of a cyclic and separating vector. Moreover, inclusions of von Neumann algebras sharing a common cyclic and separating vector, such as those encountered in Algebraic Quantum Field Theory, lead to inclusions of such standard subspaces. In this talk, we aim to present several perspectives on the construction and characterisation of inclusions between standard subspaces.
In this talk we will start from an inclusion of standard subspaces, resembling subfactors but consisting of real linear closed subspaces of a complex Hilbert space. The aim is to discuss how such an inclusion can be upgraded to a subfactor or more generally an inclusion of von Neumann algebras. This upgrade is based on a twist operator on the tensor square. In case the twist is compatible with the subspace inclusion in a way that involves crossing symmetry (which is largely analogous to the subfactor Fourier transform), then a natural state vector becomes cyclic and separating and allows us to transfer Tomita-Takesaki modular information from the linear to the algebraic picture. This construction generalizes many known constructions, such as Araki-Woods second quantization factors, q-deformed von Neumann algebras, free group factors, and has applications in quantum field theory.
We will introduce the family \(\mathcal{L}_T(H)\) of von Neumann algebras with respect to the standard subspace \(H\) and the twist \(T\in B(\mathcal{H} \otimes \mathcal{H})\) known as Araki-Woods algebras, which are interesting in Algebraic Quantum Field Theory. These algebras encode localization properties in the standard subspace and provide a general framework of the Bose and Fermi second quantization, the S-symmetric Fock spaces, and the full Fock spaces from free probability. Under the assumption of compatibility between \(T\) and \(H\), we are going to present the equivalence between \(T\) satisfying a standard subspace version of crossing symmetry, and the Yang-Baxter equation (braid equation) and the Fock vacuum being cyclic and separating for \(\mathcal{L}_T(H)\).
Under the same assumptions above, we also determine the Tomita-Takesaki modular data for Araki-Woods algebra and the Fock vacuum, and study the inclusions \(\mathcal{L}_T(K)\subset \mathcal{L}_T(H)\) of such algebras and their relative commutants for standard subspaces \(K\subset H\).
This talk is based on a joint work with Gandalf Lechner (arXiv: 2212.02298).
Finding models for local nets of von Neumann algebras and understanding the relative commutant \(\mathcal{M}\cap \mathcal{N}'\) for the inclusion \(\mathcal{N} \subset\mathcal{M}\) is a central problem in Algebraic Quantum Field Theory.
In this talk, a family of von Neumann algebras \(\mathcal{L}_T(H)\) with respect to a twist \(T\) and a standard subspace \(H\) will be introduced and it will be discussed that the Fock vacuum is separating for these algebras if, and only if, the twist \(T\) satisfies two physically motivated conditions: crossing-symmetry and the Young-Baxter equation. Furthermore, some properties of the relative commutant of the inclusion \(\mathcal{L}_T(K) \subset \mathcal{L}_T(H)\) will be presented.
This talk aims to introduce the "crossing map", a transformation of operators in Hilbert spaces defined in terms of modular theory and inspired by "crossing symmetry" from elementary particle physics, and discuss the strong connection between crossing-symmetric twists and endomorphisms of standard subspaces. We will also mention the connection of the crossing symmetry with T-twisted Araki-Woods algebras and q-systems.
This seminar aims to introduce the "crossing map", a transformation of operators in Hilbert spaces defined in terms of modular theory and inspired by "crossing symmetry" from elementary particle physics, and discuss the strong connection between crossing-symmetric twists and endomorphisms of standard subspaces. Crossing symmetry has many interesting connections, including T-twisted Araki-Woods algebras, q-Systems, and algebraic Fourier transforms.
This is joint work with Gandalf Lechner and Luca Giorgetti.
We will introduce the family \( \mathcal{L}_T(H) \) of von Neumann algebras with respect to the standard subspace \(H\) and the twist \(T\in B(\mathcal{H} \otimes \mathcal{H})\) known as Araki-Woods algebras. These algebras generalize the construction of the Bosonic and Fermionic Fock spaces and provide a general framework of the Bose and Fermi second quantization, the S-symmetric Fock spaces, and the full Fock spaces from free probability.
Under the assumption of compatibility between \(T\) and \(H\), we are going to present the equivalence between \(T\) satisfying a standard subspace version of crossing symmetry, and the Yang-Baxter equation (braid equation) and the Fock vacuum being cyclic and separating for \(\mathcal{L}_T(H)\). Under the same assumptions, we also determine the Tomita-Takesaki modular data for Araki-Woods algebra and the Fock vacuum. Finally, the inclusions \(\mathcal{L}_T(K)\subset \mathcal{L}_T(H)\) of such algebras and their relative commutants for standard subspaces \(K\subset H\) will be discussed.
This is joint work with Gandalf Lechner (arXiv: 2212.02298).
The measurement problem is the most intractable, most intensely investigated issue at the foundations of the quantum theory. Meaningfully one should relate it to the physical hypotheses upon which the theory was built—and lest our analysis not be impaired by half- understood concepts, or worse, by prejudices, this task cannot be undertaken without a detailed historical investigation. Quantum Mechanics is unique in the history of science in that it resulted from the axiomatized merging of two rival—yet putatively equivalent— theories, namely Matrix Mechanics and Wave Mechanics. In this talk, we shall present a mathematical and conceptual analysis of the structures of Matrix and Wave Mechanics that reveals facts hitherto overlooked by the literature. The analysis will serve its purpose by enabling us to show that the measurement problem is a logical consequence of constructing Quantum Mechanics over a fabricated—and therefore fictitious—equivalence. Matrix and Wave Mechanics are not equivalent quantum theories, but their structures are related, in a way we shall indicate in exact mathematical terms. The physical relevance of this relation is that it gives us new insight into the nature of the measurement problem, enabling us to state it in a different, more general setting than it has been done heretofore, opening new paths in our search for solutions.
Similar to the construction of Bosonic and Fermionic Fock spaces, we will introduce the family \(\mathcal{L}_T(H)\) of von Neumann algebras with respect to the standard subspace \(H\) and the twist \(T\in B(\mathcal{H} \otimes \mathcal{H})\) known as Araki-Woods algebras. These algebras provide a general framework of Bose and Fermi second quantization, S-symmetric Fock spaces, and full Fock spaces from free probability.
Under a compatibility assumption on \(T\) and \(H\), we are going to present the equivalence between the Fock vacuum being cyclic and separating for \(\mathcal{L}_T(H)\), and \(T\) satisfying a standard subspace version of crossing symmetry and the Yang-Baxter equation (braid equation). In this case, we also determine the Tomita-Takesaki modular data. Finally, inclusions \(\mathcal{L}_T(K)\subset \mathcal{L}_T(K)\) of such algebras and their relative commutants, for standard subspaces \(K\subset H\), will be discussed as well as applications in QFT.
The measurement problem is the most intractable, most intensely investigated issue at the foundations of the quantum theory. Meaningfully one should relate it to the physical hypotheses upon which the theory was built—and lest our analysis not be impaired by half- understood concepts, or worse, by prejudices, this task cannot be undertaken without a detailed historical investigation. Quantum Mechanics is unique in the history of science in that it resulted from the axiomatized merging of two rival—yet putatively equivalent— theories, namely Matrix Mechanics and Wave Mechanics. In this talk, we shall present a mathematical and conceptual analysis of the structures of Matrix and Wave Mechanics that reveals facts hitherto overlooked by the literature. The analysis will serve its purpose by enabling us to show that the measurement problem is a logical consequence of constructing Quantum Mechanics over a fabricated—and therefore fictitious—equivalence. Matrix and Wave Mechanics are not equivalent quantum theories, but their structures are related, in a way we shall indicate in exact mathematical terms. The physical relevance of this relation is that it gives us new insight into the nature of the measurement problem, enabling us to state it in a different, more general setting than it has been done heretofore, opening new paths in our search for solutions.
Starting from the familiar Weyl algebra on a Bose Fock space over a Hilbert space \(\mathcal{H}\), we introduce a family of von Neumann algebras \(\mathcal{L}_T (H)\) that can be thought of as deformations of the von Neumann algebra generated by Weyl operators \(W(h)\), \(h \in H\). Here \(H \subset \mathcal{H}\) is a real (standard) subspace of \(H\), and \(T\) a "twist" (or deformation), namely a selfadjoint operator on \(\mathcal{H} \otimes \mathcal{H}\) satisfying a positivity condition.
These algebras are called twisted Araki-Woods algebras. They naturally arise in representations of Wick algebras and provide a general framework in which many special cases such as the algebras underlying free Bose fields (on Bose Fock space), free Fermi fields (on Fermi Fock space), integrable QFT models (on S-symmetric Fock space), but also free group factors (on full Fock space) can be discussed in a unified manner.
We will explain the modular theory of these algebras which is closely linked to T being braided and crossing-symmetric, and consider the dependence of LT (H) on the twist T (which is quite discontinuous) and the standard subspace \(H\). This naturally leads to inclusions \(\mathcal{L}_T(K) \subset \mathcal{L}_T (H)\) and applications in QFT.
No deep background in QFT or von Neumann algebras is assumed, only a working knowledge of Hilbert spaces and functional analysis.
The measurement problem is perhaps the most intensely studied issue at the foundations of quantum theory. It is, therefore, remarkable that the question of its mathematical and conceptual origins has been largely ignored; thinking this a lack serious enough to remedy, and without asserting any exhaustiveness, this talk is an attempt to meet this need. By going back to the foundational papers we try to make very explicit the conceptual platforms to which matrix mechanics and wave mechanics were originally fixed—John von Neumann presented his Mathematical Foundations of Quantum Mechanics, let us recall, as a mathematically rigorous “synthesis” of the “equivalent” matrix and wave mechanics—and we show that the measurement problem follows straightforwardly from the peculiar mathematical fusion of quantum theories that were built from the ground up from physically incompatible assumptions.
The measurement problem is perhaps the most intensely studied issue at the foundations of quantum theory. It is, therefore, remarkable that the question of its mathematical and conceptual origins has been largely ignored; thinking this a lack serious enough to remedy, and without asserting any exhaustiveness, this talk is an attempt to meet this need. By going back to the foundational papers we try to make very explicit the conceptual platforms to which matrix mechanics and wave mechanics were originally fixed—John von Neumann presented his Mathematical Foundations of Quantum Mechanics, let us recall, as a mathematically rigorous “synthesis” of the “equivalent” matrix and wave mechanics—and we show that the measurement problem follows straightforwardly from the peculiar mathematical fusion of quantum theories that were built from the ground up from physically incompatible assumptions.
KMS states are very important in quantum statistical mechanics since they describe thermal equilibrium states. It is also important the theory of perturbations of these states, mostly developed by Araki. Unfortunately, Araki's theory is restricted to bounded perturbations.
We will present our progress in extending the theory of perturbations of KMS states using noncommutative Lp-spaces. We will also discuss certain stability property of the domain of the Modular Operator associated to a \(\|.\|_p\)-continuous state that allows us to define an analytic multiple-time KMS condition and to obtain its analyticity together with some bounds to its norm. Finally, we will discuss how it can be used to extend perturbations to a class of unbounded perturbations.
KMS states are very important in quantum statistical mechanics since they describe thermal equilibrium states. It is also important the theory of perturbations of these states, mostly developed by Araki. Unfortunately, Araki's theory is restricted to bounded perturbations.
We will present our progress in extending the theory of perturbations of KMS states using noncommutative Lp-spaces. We will also discuss certain stability property of the domain of the Modular Operator associated to a \(\|.\|_p\)-continuous state that allows us to define an analytic multiple-time KMS condition and to obtain its analyticity together with some bounds to its norm. Finally, we will discuss how it can be used to extend perturbations to a class of unbounded perturbations.
Communications in Mathematical Physics, 406, p. 283 (2025)
Standard subspaces are closed real subspaces of a complex Hilbert space that appear naturally in Tomita-Takesaki modular theory and its applications to quantum field theory. In this article, inclusions of standard subspaces are studied independently of von Neumann algebras. Several new methods for their investigation are developed, related to polarizers, Gelfand triples defined by modular data, and extensions of modular operators. A particular class of examples that arises from the fundamental irreducible building block of a conformal field theory on the line is analyzed in detail.
Studies in History and Philosophy of Science, 109, p. 89–105 (2025)
This paper combines mathematical, philosophical, and historical analyses in a comprehensive investigation of the dynamical foundations of the formalism of orthodox quantum mechanics. The results obtained include: (i) A deduction of the canonical commutation relations (CCR) from the tenets of Matrix Mechanics; (ii) A discussion of the meaning of Schrödinger's first derivation of the wave equation that not only improves on Joas and Lehner's 2009 investigation on the subject, but also demonstrates that the CCR follow of necessity from Schrödinger's first derivation of the wave equation, thus correcting the common misconception that the CCR were only posited by Schrödinger to pursue equivalence with Matrix Mechanics; (iii) A discussion of the mathematical facts and requirements involved in the equivalence of Matrix and Wave Mechanics that improves on F. A. Muller's classical treatment of the subject; (iv) A proof that the equivalence of Matrix and Wave Mechanics is necessitated by the formal requirements of a dual action functional from which both the dynamical postulates of orthodox quantum mechanics, von Neumann's process 1 and process 2, follow; (v) A critical assessment, based on (iii) and (iv), of von Neumann's construction of unified quantum mechanics over Hilbert space. Point (iv) is our main result. It brings to the open the important, but hitherto ignored, fact that orthodox quantum mechanics is no exception to the golden rule of physics that the dynamics of a physical theory must follow from the action functional. If orthodox quantum mechanics, based as it is on the assumption of the equivalence of Matrix and Wave Mechanics, has this "peculiar dual dynamics," as von Neumann called it, then this is so because by assuming the equivalence, one has been assuming a peculiar dual action.
Annales Henri Poincaré, 26, p. 4109–4139 (2025)
KMS states on \(\mathbb{Z}_2\)-crossed products of unital \(C^*\)-algebras \(\mathcal{A}\) are characterized in terms of KMS states and twisted KMS functionals of \(\mathcal{A}\). These functionals are shown to describe the extensions of KMS states \(\omega\) on \(\mathcal{A}\) to the crossed product \(\mathcal{A} \rtimes \mathbb{Z}_2\) and can also be characterized by the twisted center of the von Neumann algebra generated by the GNS representation corresponding to \(\omega\).
As a particular class of examples, KMS states on \(\mathbb{Z}_2\)-crossed products of CAR algebras with dynamics and grading given by Bogoliubov automorphisms are analyzed in detail. In this case, one or two extremal KMS states are found depending on a Gibbs type condition involving the odd part of the absolute value of the Hamiltonian.
As an application in mathematical physics, the extended field algebra of the Ising QFT is shown to be a \(\mathbb{Z}_2\)-crossed product of a CAR algebra which has a unique KMS state.
Reviews in Mathematical Physics, p. 2461005 (2024)
We introduce and study the crossing map, a closed linear map acting on operators on the tensor square of a given Hilbert space that is inspired by the crossing property of quantum field theory. This map turns out to be closely connected to Tomita-Takesaki modular theory. In particular, crossing symmetric operators, namely those operators that are mapped to their adjoints by the crossing map, define endomorphisms of standard subspaces. Conversely, such endomorphisms can be integrated to crossing symmetric operators. We also investigate the relation between crossing symmetry and natural compatibility conditions with respect to unitary representations of certain symmetry groups, and furthermore introduce a generalized crossing map defined by a real object in an abstract \(C^*\)-tensor category, not necessarily consisting of Hilbert spaces and linear maps. This latter crossing map turns out to be closely related to the (unshaded, finite-index) subfactor theoretical Fourier transform. Lastly, we provide families of solutions of the crossing symmetry equation, solving in addition the categorical Yang-Baxter equation, associated with an arbitrary Q-system.
Communications in Mathematical Physics, 402, p. 2339–2386 (2023)
In the general setting of twisted second quantization (including Bose/Fermi second quantization, \(S\)-symmetric Fock spaces, and full Fock spaces from free probability as special cases), von Neumann algebras on twisted Fock spaces are analyzed. These twisted Araki-Woods algebras \(\mathcal{L}_{T}(H)\) depend on the twist operator \(T\) and a standard subspace \(H\) in the one-particle space. Under a compatibility assumption on \(T\) and \(H\), it is proven that the Fock vacuum is cyclic and separating for \(\mathcal{L}_{T}(H)\) if and only if \(T\) satisfies a standard subspace version of crossing symmetry and the Yang-Baxter equation (braid equation). In this case, the Tomita-Takesaki modular data are explicitly determined.
Inclusions \(\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)\) of twisted Araki-Woods algebras are analyzed in two cases: If the inclusion is half-sided modular and the twist satisfies a norm bound, it is shown to be singular. If the inclusion of underlying standard subspaces \(K\subset H\) satisfies an \(L^2\)-nuclearity condition, \(\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)\) has type III relative commutant for suitable twists \(T\).
Applications of these results to localization of observables in algebraic quantum field theory are discussed.
Brazilian Journal of Physics, Vol. 51 (2021)
We present a review on the notion of pure states and mixtures as mathematical concepts that apply for both classical and quantum physical theories, as well as for any other theory depending on statistical description. Here, states will be presented as expectation values on suitable algebras of observables, in a manner intended for the non-specialist reader; accordingly, basic literature on the subject will be provided. Examples will be exposed together with a discussion on their meanings and implications. An example will be shown where a pure quantum state converges to a classical mixture of particles as Planck's constant tends to zero.
Journal of Mathematical Physics, Vol. 60 (8) (2019)
We extend the theory of perturbations of KMS states to a class of unbounded perturbations using noncommutative \(L_p\)-spaces. We also prove certain stability of the domain of the Modular Operator associated with a \(\|\cdot\|_p\)-continuous state. This allows us to define an analytic multiple-time KMS condition and to obtain its analyticity together with some bounds to its norm.
PhD Thesis, arXiv: 1806.03488 (2018)
We extend the theory of perturbations of KMS states to some class of unbounded perturbations using noncommutative \(L_p\)-spaces. We also prove certain stability of the domain of the Modular Operator associated to a \(\|\cdot\|_p\)-continuous state. This allows us to define an analytic multiple-time KMS condition and to obtain its analyticity together with some bounds to its norm. The main results are Theorem 5.1.15, Theorem 5.1.16 and Corollary 5.1.18.
Apart from that, this work contains a detailed review, with minor contributions due to the author, starting with the description of \(C^\ast\)-algebras and von Neumann algebras followed by weights and representations, a whole chapter is devoted to the study of KMS states and its physical interpretation as the states of thermal equilibrium, then the Tomita-Takesaki Modular Theory is presented, furthermore, we study analytical properties of the modular operator automorphism group, positive cones and bounded perturbations of states, and finally we start presenting multiple versions of noncommutative \(L_p\)-spaces.
arXiv: 1803.02390 (2018)
Master's Thesis (2013)
The purpose of this work is the study of \(p\)-compact operators and the \(p\)-approximation property. These concepts are connected with important results by A. Gröthendieck about compactness and approximation property that were generalized in [21] and studied in [3], [6] and [7].
2024 - Present
Department of Mathematical Physics, University of São Paulo
2021 - Present
Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg
2021
Institute of Mathematics and Statistics, University of São Paulo
2019
Department of Mathematics, Federal University of São Carlos
2018 - 2020
Department of Mathematical Physics, University of São Paulo
2013 - 2028
Department of Mathematical Physics, University of São Paulo
2011 - 2023
Institute of Mathematics and Statistics, University of São Paulo
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ricardo.correa.silva(at)fau.de
ricardo.correa.silva(at)usp.br
+49 9131 85-67078