Topic
Direct integral
About: Direct integral is a research topic. Over the lifetime, 200 publications have been published within this topic receiving 4826 citations.
Papers published on a yearly basis
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Book•
1 Jan 2002
TL;DR: In this article, the authors present a general overview of Banach Algebras and C*-AlgebrAs, as well as a discussion of their properties and properties.
Abstract: I Fundamentals of Banach Algebras and C*-Algebras.- 0. Introduction.- 1. Banach Algebras.- 2. Spectrum and Functional Calculus.- 3. Gelfand Representation of Abelian Banach Algebras.- 4. Spectrum and Functional Calculus in C*-Algebras.- 5. Continuity of Homomorphisms.- 6. Positive Cones of C*-Algebras.- 7. Approximate Identities in C*-Algebras.- 8. Quotient Algebras of C*-Algebras.- 9. Representations and Positive Linear Functional.- 10. Extreme Points of the Unit Ball of a C*-Algebra.- 11. Finite Dimensional C*-Algebras.- Notes.- Exercises.- II Topologies and Density Theorems in Operator Algebras.- 0. Introduction.- 1. Banach Spaces of Operators on a Hilbert Space.- 2. Locally Convex Topologies in ?(?).- 3. The Double Commutation Theorem of J. von Neumann.- 4. Density Theorems.- Notes.- III Conjugate Spaces.- 0. Introduction.- 1. Abelian Operator Algebras.- 2. The Universal Enveloping von Neumann Algebra of a C*-Algebra.- 3. W*-Algebras.- 4. The Polar Decomposition and the Absolute Value of Functionals.- 5. Topological Properties of the Conjugate Space.- 6. Semicontinuity in the Universal Enveloping von Neumann Algebra*.- Notes.- IV Tensor Products of Operator Algebras and Direct Integrals.- 0. Introduction.- 1. Tensor Product of Hilbert Spaces and Operators.- 2. Tensor Products of Banach Spaces.- 3. Completely Positive Maps.- 4. Tensor Products of C*-Algebras.- 5. Tensor Products of W*-Algebras.- Notes.- 6. Integral Representations of States.- 7. Representation of L2(?,?) ? ?, L1(?,?) ?y? *, and L(?,?) ?? ?.- 8. Direct Integral of Hubert Spaces, Representations, and von Neumann Algebras.- Notes.- V Types of von Neumann Algebras and Traces.- 0. Introduction.- 1. Projections and Types of von Neumann Algebras.- 2. Traces on von Neumann Algebras.- Notes.- 3. Multiplicity of a von Neumann Algebra on a Hilbert Space.- 4. Ergodic Type Theorem for von Neumann Algebras*.- 5. Normality of Separable Representations*.- 6. The Borel Spaces of von Neumann Algebras.- 7. Construction of Factors of Type II and Type III.- Notes.- Appendix Polish Spaces and Standard Borel Spaces.- Monographs.- Papers.- Notation Index.
1,979 citations
TL;DR: The existence of inequivalent representations of the canonical commutation relations which describe a nonrelativistic infinite free Bose gas of uniform density is investigated in this article, with a view to possible applications to statistical mechanics.
Abstract: The existence of inequivalent representations of the canonical commutation relations which describe a nonrelativistic infinite free Bose gas of uniform density is investigated, with a view to possible applications to statistical mechanics. The functional E(f, g)=(Ψ, eiφ(f) eiπ(g)Ψ) is used to describe the inequivalent representations. This functional is calculated for the free Bose gas in a box of volume V, and the limit is then taken as V → ∞. In this way we construct cyclic representations describing an infinite system of particles with a density distribution ρ(k) in momentum space. For a given ρ(k) the operator algebra generated by the φ(f), π(g) is reducible. For the ground‐state representation (all particles in the zero‐momentum state), the representation is a direct integral of irreducible representations (analogous to BCS theory). For finite temperatures the situation is complicated by the occurrence of representations which are not type I. The physical significance of the reducibility of the repre...
455 citations
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TL;DR: In this article, a generalization to l-loop expressing the loops as integrals over the on-shell phase space of exactly l particles is proposed. But the integrand for l > 2 does not involve the forward limit of any physical tree amplitude, except in planar gauge theories.
Abstract: We investigate relations between loop and tree amplitudes in quantum field theory that involve putting on-shell some loop propagators. This generalizes the so-called Feynman tree theorem which is satisfied at 1-loop. Exploiting retarded boundary conditions, we give a generalization to l-loop expressing the loops as integrals over the on-shell phase space of exactly l particles. We argue that the corresponding integrand for l > 2 does not involve the forward limit of any physical tree amplitude, except in planar gauge theories. In that case we explicitly construct the relevant physical amplitude. Beyond the planar limit, abandoning direct integral representations, we propose that loops continue to be determined implicitly by the forward limit of physical connected trees, and we formulate a precise conjecture along this line. Finally, we set up technology to compute forward amplitudes in supersymmetric theories, in which specific simplifications occur.
132 citations
[...]
TL;DR: In this article, a generalization to L-loop expressing the loops as integrals over the on-shell phase space of exactly L particles was proposed, and the corresponding integrand for L>2 does not involve the forward limit of any physical tree amplitude, except in planar gauge theories.
Abstract: We investigate relations between loop and tree amplitudes in quantum field theory that involve putting on-shell some loop propagators. This generalizes the so-called Feynman tree theorem which is satisfied at 1-loop. Exploiting retarded boundary conditions, we give a generalization to L-loop expressing the loops as integrals over the on-shell phase space of exactly L particles. We argue that the corresponding integrand for L>2 does not involve the forward limit of any physical tree amplitude, except in planar gauge theories. In that case we explicitly construct the relevant physical amplitude. Beyond the planar limit, abandoning direct integral representations, we propose that loops continue to be determined implicitly by the forward limit of physical connected trees, and we formulate a precise conjecture along this line. Finally, we set up technology to compute forward amplitudes in supersymmetric theories, in which specific simplifications occur.
85 citations
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| Year | Papers |
|---|---|
| 2021 | 9 |
| 2020 | 3 |
| 2019 | 3 |
| 2018 | 4 |
| 2017 | 3 |
| 2016 | 8 |