16:00 to 17:00 p.m.

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.

2:15 to 3:15 p.m.

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 $\mathcal{L}_T(K)\subset \mathcal{L}_T(K)$ 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).

AGPhil 9.4: Quantum Foundations 5 - 3:30 to 4 p.m. - JAN/0027

The measurement problem is the most intensely investigated issue at the foundations of the quantum theory. Since what counts as a solution depends on how the problem is defined, a historical investigation of the development that has conditioned the standard formulation of the problem is most needful as a test of its adequacy. 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 new, detailed mathematical and conceptual analysis of the structures of Matrix and Wave Mechanics. It will follow 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 demonstrate. The physical relevance of this relation, stated in exact mathematical terms, 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.

4:45 to 6:15 p.m.

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.

Salón de seminarios A225 - 1 to 2 p.m.

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.

Seminarroom SE 30 - 2 to 3:30 p.m.

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.

4:15 - 5:15 p.m.

11 to 11:40 a.m.