Observable Algebras in the S Matrix Approach

The algebra of observables of an N-body Calogero model is represented on the ${S}_{N}$ -symmetric subspace of the positive-definite Fock space. We discuss some general properties of the algebra and construct four different realizations of the dynamical symmetry algebra of the Calogero model. Using the fact that the minimal algebra of observables is common to the Calogero model and the finite Chern-Simons (CS) matrix model, we extend our analysis to the CS matrix model. We point out the algebraic similarities and distinctions of these models.The algebra of observables of an N-body Calogero model is represented on the ${S}_{N}$-symmetric subspace of the positive-definite Fock space. We discuss some general properties of the algebra and construct four different realizations of the dynamical symmetry algebra of the Calogero model. Using the fact that the minimal algebra of observables is common to the Calogero model and the finite Chern-Simons (CS) matrix model, we extend our analysis to the CS matrix model. We point out the algebraic similarities and distinctions of these models.

arXiv:hep-th/0203245Physical review. B, Condensed matterNov 12, 20026 citations

Observable Algebras and Channels

Elsevier eBooksFeb 15, 20110 citations

The Algebra of Observables

Jean-Louis Basdevant

Graduate texts in physicsJan 1, 20230 citations

Quantum Mechanics and Quantum Field Theory: Algebraic and Geometric Approaches

Igor Frolov, Albert S. Schwarz

This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory, including recent results. It is based on the algebraic approach in which the starting point is a star-algebra and on the geometric approach in which the starting point is a convex set of states. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state). The conventional scattering matrix does not work for quasiparticles (or even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of an inclusive scattering matrix, which is closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied in the geometric approach as well. In particular, this result can be used to provide a simple derivation of the diagram technique for generalized Green functions. The notion of an inclusive scattering matrix makes sense in the geometric approach, although it seems that a definition of the conventional scattering matrix cannot be provided in this situation. The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices are able to measure only a part of the observables.This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory, including recent results. It is based on the algebraic approach in which the starting point is a star-algebra and on the geometric approach in which the starting point is a convex set of states. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state). The conventional scattering matrix does not work for quasiparticles (or even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of an inclusive scattering matrix, which is closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied in the geometric approach as well. In particular, this result can be used to provide a simple derivation of the diagram technique for generalized Green functions. The notion of an inclusive scattering matrix makes sense in the geometric approach, although it seems that a definition of the conventional scattering matrix cannot be provided in this situation. The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices are able to measure only a part of the observables.

arXiv:2301.03804UniverseJul 17, 20232 citations

The Algebra of Observables

M. D. Maia

Springer eBooksJan 1, 20110 citations

TQCにおけるモジュラーS行列をめぐって

Jun Ichi Fujii

The term ‘TQC' means the topological quantum computation, which is one of remarkable theories to realize quantum computers. This theory is related to various fields; theory of quantum groups, Lie algebras, Hopf algebras and conformal field or string theory. But there are few textbooks which readers of other fields can read easily. We wrote a survay[10]easy to observe from the basics of TQC theory to basic quantum gates for quantum computers. But we cannot explain modular S-matrices that is one of the keys in TQC in the preceding survey. So, in this paper, we try to express total basic issue including theories around modular S-matrix. In the many papers for TQC, there are some subtle mistakes and we correct them in this paper. At first we introduce formulae for modular S-matrix in the axiomatic (categorical) approach. Secondly we give an important model for TQC; spin network model based on the Temperly-Lieb-Jones algebra. Finally we mention another important model suk(2). In each levels, Fibannaci anyon is a basic example to explain them.The term ‘TQC' means the topological quantum computation, which is one of remarkable theories to realize quantum computers. This theory is related to various fields; theory of quantum groups, Lie algebras, Hopf algebras and conformal field or string theory. But there are few textbooks which readers of other fields can read easily. We wrote a survay[10]easy to observe from the basics of TQC theory to basic quantum gates for quantum computers. But we cannot explain modular S-matrices that is one of the keys in TQC in the preceding survey. So, in this paper, we try to express total basic issue including theories around modular S-matrix. In the many papers for TQC, there are some subtle mistakes and we correct them in this paper. At first we introduce formulae for modular S-matrix in the axiomatic (categorical) approach. Secondly we give an important model for TQC; spin network model based on the Temperly-Lieb-Jones algebra. Finally we mention another important model suk(2). In each levels, Fibannaci anyon is a basic example to explain them.

Feb 29, 20160 citations

The algebra of observables of an N-body Calogero model is represented on the

{S}_{N}

-symmetric subspace of the positive-definite Fock space. We discuss some general properties of the algebra and construct four different realizations of the dynamical symmetry algebra of the Calogero model. Using the fact that the minimal algebra of observables is common to the Calogero model and the finite Chern-Simons (CS) matrix model, we extend our analysis to the CS matrix model. We point out the algebraic similarities and distinctions of these models.