[Paper Review] Differential Calculi on Quantum (Sub-) Groups and Their Classical Limit

Folkert Müller-Hoissen|ArXiv.org|Jan 28, 1994

advanced mathematical theories4 citations

TL;DR

This paper presents a refined classification of bicovariant differential calculi on the two-parameter quantum group GLp,q(2), showing they form a one-parameter family under a new parametrization. It further analyzes calculi on the quantum subgroup SLq(2), revealing that their classical limit does not recover the standard differential calculus on SL(2), suggesting a nonstandard differential structure with connections to stochastic processes and relativistic quantum theories.

ABSTRACT

For the two-parameter matrix quantum group GLp,q(2) all bicovariant differential calculi (with a four-dimensional space of 1-forms) are known. They form a one-parameter family. Here, we give an improved presentation of previous results by using a different parametrization. We also discuss different ways to obtain bicovariant calculi on the quantum subgroup SLq(2). For those calculi, we do not obtain the ordinary differential calculus on SL(2) in the classical limit. The structure which emerges here can be generalized to a nonstandard differential calculus on an arbitrary differentiable manifold and exhibits relations with stochastic calculus and `proper time' relativistic quantum theories.

Motivation & Objective

• To re-express known bicovariant differential calculi on GLp,q(2) using a new parametrization for improved clarity and consistency.
• To systematically investigate the construction of bicovariant differential calculi on the quantum subgroup SLq(2).
• To analyze the classical limit of these calculi on SLq(2) and determine whether they recover the standard differential calculus on SL(2).
• To explore the broader implications of the emerging differential structure, particularly its generalization to arbitrary differentiable manifolds.
• To identify connections between the nonstandard calculus and concepts in stochastic calculus and relativistic quantum theories involving 'proper time'.

Proposed method

• Utilizes a reparametrization of the two-parameter quantum group GLp,q(2) to present its bicovariant differential calculi in a more coherent and systematic form.
• Applies the formalism of bicovariant differential calculi on quantum groups, focusing on the space of one-forms being four-dimensional.
• Applies the same framework to the quantum subgroup SLq(2), deriving its calculi through reduction from GLp,q(2) or independent construction.
• Analyzes the classical limit of the derived calculi on SLq(2) by taking the deformation parameter q → 1, examining the resulting differential structure.
• Identifies structural similarities between the resulting calculus and stochastic differential calculus, particularly in the context of noncommutative geometry.
• Proposes a generalization of the calculus to arbitrary differentiable manifolds, suggesting a nonstandard differential calculus with potential applications in relativistic quantum theories.

Experimental results

Research questions

• RQ1 How can the classification of bicovariant differential calculi on GLp,q(2) be improved through a new parametrization?
• RQ2 What are the distinct bicovariant differential calculi on the quantum group SLq(2), and how do they relate to those on GLp,q(2)?
• RQ3 Does the classical limit of the differential calculi on SLq(2) reproduce the standard differential calculus on the classical group SL(2)?
• RQ4 What geometric and physical structures underlie the nonstandard calculus that emerges in the classical limit?
• RQ5 Can the derived differential calculus be generalized to arbitrary differentiable manifolds, and what are its implications for stochastic or relativistic quantum theories?

Key findings

• The bicovariant differential calculi on GLp,q(2) form a one-parameter family, and the paper provides a clearer presentation using a new parametrization.
• Bicovariant calculi on SLq(2) are derived, but their classical limit does not yield the standard differential calculus on SL(2).
• The classical limit of the SLq(2) calculi results in a nonstandard differential structure, distinct from the usual de Rham calculus.
• The structure of the calculus exhibits formal analogies with stochastic calculus, particularly in the treatment of infinitesimal increments.
• The calculus can be generalized to arbitrary differentiable manifolds, suggesting a broader geometric framework beyond quantum groups.
• The findings point to potential applications in relativistic quantum theories that incorporate 'proper time' as a dynamical variable.