scholarly journals Infinite-dimensional Categorical Quantum Mechanics

2017 ◽  
Vol 236 ◽  
pp. 51-69 ◽  
Author(s):  
Stefano Gogioso ◽  
Fabrizio Genovese
Keyword(s):  
Quantum Mechanics ◽  
Infinite Dimensional ◽  
Categorical Quantum Mechanics

scholarly journals Dagger linear logic for categorical quantum mechanics

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Robin Cockett ◽  
Cole Comfort ◽  
Priyaa Srinivasan
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Physics ◽  
Linear Logic ◽  
Infinite Dimensional ◽  
Graphical Calculus ◽  
Finite Dimensional ◽  
Categorical Quantum Mechanics ◽  
Definition Of ◽  
Categorical Semantics

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.

scholarly journals AN INFINITE DIMENSIONAL VERSION OF THE SCHUR CONVEXITY PROPERTY AND APPLICATIONS

2007 ◽  
Vol 05 (02) ◽  
pp. 123-136 ◽  
Author(s):  
CLAUDE VALLÉE ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Symmetric Matrix ◽  
Liouville Problem ◽  
Convexity Property ◽  
Infinite Dimensional ◽  
Selfadjoint Operators ◽  
Dimensional Version ◽  
Sturm Liouville ◽  
Schur Convexity

We extend to infinite dimensional separable Hilbert spaces the Schur convexity property of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of linear, selfadjoint, compact operators, and that of linear selfadjoint operators that can be approximated by operators of finite rank and having a countable family of eigenvalues. The abstract results of the present paper are illustrated by several examples from mechanics or quantum mechanics, including the Sturm–Liouville problem, the Schrödinger equation, and the harmonic oscillator.

LIE-ALGEBRAIC APPROACH TO THE PROBLEM OF QUASI-EXACT SOLUBILITY IN QUANTUM MECHANICS

1990 ◽  
Vol 05 (23) ◽  
pp. 1891-1899 ◽  
Author(s):  
A. G. USHVERIDZE
Keyword(s):  
Quantum Mechanics ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
New Method ◽  
Exactly Solvable Models ◽  
Solvable Models ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Exactly Solvable

A new method of constructing quasi-exactly solvable models of quantum mechanics is proposed. This method is based on the use of infinite-dimensional representations of simple and semi-simple Lie algebras.

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