Functional Integral Approach to $C^*$ -algebraic Quantum Mechanics II: Symplectic Quantum MechanicsFunctional Integral Approach to $C^*$-algebraic Quantum Mechanics II: Symplectic Quantum Mechanics

J. LaChapelle

We propose $Sp\,(8,\R)$ and its Langlands dual $SO(9,\R)$ as dynamical groups for closed quantum systems. Restricting here to the non-compact group $Sp\,(8,\R)$ , the quantum theory is constructed and investigated. The functional Mellin transform plays a prominent role in defining the quantum theory. It provides a bridge between the quantum algebra of observables and the algebra of operators on Hilbert spaces furnishing unitary representations that are induced from a distinguished parabolic subgroup of $Sp\,(8,\C)$. As well, the parabolic subgroup renders a fiber bundle construction that models what can be described as a matrix quantum gauge theory. The formulation is strictly quantum mechanics: no \emph{a priori} space-time is assumed and the only geometrical input comes indirectly from the group manifold. But what appears on the surface to be a fairly simple-minded model turns out to have a capacious structure suggesting some compelling physical interpretations regarding space-time and fundamental interactions.We propose $Sp\,(8,\R)$ and its Langlands dual $SO(9,\R)$ as dynamical groups for closed quantum systems. Restricting here to the non-compact group $Sp\,(8,\R)$, the quantum theory is constructed and investigated. The functional Mellin transform plays a prominent role in defining the quantum theory. It provides a bridge between the quantum algebra of observables and the algebra of operators on Hilbert spaces furnishing unitary representations that are induced from a distinguished parabolic subgroup of $Sp\,(8,\C)$. As well, the parabolic subgroup renders a fiber bundle construction that models what can be described as a matrix quantum gauge theory. The formulation is strictly quantum mechanics: no \emph{a priori} space-time is assumed and the only geometrical input comes indirectly from the group manifold. But what appears on the surface to be a fairly simple-minded model turns out to have a capacious structure suggesting some compelling physical interpretations regarding space-time and fundamental interactions.

arXiv:1506.02985Jun 9, 20150 citations

S -Matrix Interpretation of Quantum Theory

Henry Pierce Stapp

The usual interpretation of quantum theory is clouded by the following points. (1) Invalid classical concepts are ascribed fundamental status. (2) The process of measurement is not describable within the framework of the theory. (3) The subject-object distinction is blurred. (4) The observed system is required to be isolated in order to be defined, yet interacting in order to be observed. These problems are resolved in a straightforward way by the unequivocal acceptance of a thoroughly pragmatic $S$ -matrix viewpoint. This interpretation is described, and it is explained how and why the full physical content of quantum theory resides in the $S$ matrix.The usual interpretation of quantum theory is clouded by the following points. (1) Invalid classical concepts are ascribed fundamental status. (2) The process of measurement is not describable within the framework of the theory. (3) The subject-object distinction is blurred. (4) The observed system is required to be isolated in order to be defined, yet interacting in order to be observed. These problems are resolved in a straightforward way by the unequivocal acceptance of a thoroughly pragmatic $S$-matrix viewpoint. This interpretation is described, and it is explained how and why the full physical content of quantum theory resides in the $S$ matrix.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsMar 15, 1971205 citations

Semigroups and the density matrix formulation of quantum mechanics

Herbert S. Green

Abstract A formulation of quantum mechanics is presented based on the theory of semigroups and the associated enveloping algebras of functions defined on countable subsemigroups. The existence of a unique *‐involution is not assumed. The fundamental elements of a semigroup are identified with experimental precedures for the separation of subensembles from a given ensemble of experimental systems. Observables are represented as elements of enveloping algebras, and ensembles as density matrices within an enveloping algebra. The statistical properties of ensembles are expressed in terms of traces defined on the semigroup and its enveloping algebras. The elements and generators of the Poincaré group can be defined and interpreted in the usual way. A variety of applications is described, in which the theory of the density matrix plays an essential or effective role. Advantages associated with the resulting freedom from the limitations of Hilbert space are illustrated.Abstract A formulation of quantum mechanics is presented based on the theory of semigroups and the associated enveloping algebras of functions defined on countable subsemigroups. The existence of a unique *‐involution is not assumed. The fundamental elements of a semigroup are identified with experimental precedures for the separation of subensembles from a given ensemble of experimental systems. Observables are represented as elements of enveloping algebras, and ensembles as density matrices within an enveloping algebra. The statistical properties of ensembles are expressed in terms of traces defined on the semigroup and its enveloping algebras. The elements and generators of the Poincaré group can be defined and interpreted in the usual way. A variety of applications is described, in which the theory of the density matrix plays an essential or effective role. Advantages associated with the resulting freedom from the limitations of Hilbert space are illustrated.

International Journal of Quantum ChemistryJan 1, 19802 citations

Matrix elements of the operator TT¯ in integrable quantum field theory

Gesualdo Delfino A, Giuliano Niccoli

arXiv:hep-th/0407142Nuclear Physics BDec 13, 200426 citations

A Convergent S -Matrix Formalism with Correspondence to Ordinary Quantum Mechanics

P. Kristensen, C. Møller

Physical ReviewMar 1, 19520 citations

Functional Integral Approach to

C^*

-algebraic Quantum Mechanics II: Symplectic Quantum Mechanics

We propose

Sp\,(8,\R)

and its Langlands dual

SO(9,\R)

as dynamical groups for closed quantum systems. Restricting here to the non-compact group

Sp\,(8,\R)

, the quantum theory is constructed and investigated. The functional Mellin transform plays a prominent role in defining the quantum theory. It provides a bridge between the quantum algebra of observables and the algebra of operators on Hilbert spaces furnishing unitary representations that are induced from a distinguished parabolic subgroup of $Sp\,(8,\C)$. As well, the parabolic subgroup renders a fiber bundle construction that models what can be described as a matrix quantum gauge theory. The formulation is strictly quantum mechanics: no \emph{a priori} space-time is assumed and the only geometrical input comes indirectly from the group manifold. But what appears on the surface to be a fairly simple-minded model turns out to have a capacious structure suggesting some compelling physical interpretations regarding space-time and fundamental interactions.

S

-matrix viewpoint. This interpretation is described, and it is explained how and why the full physical content of quantum theory resides in the

S

matrix.