[Paper Review] Dissipative Quantum Mechanics: The Generalization of the Canonical Quantization and von Neumann Equation

Vasily E. Tarasov|ArXiv.org|Oct 5, 1994

Computational Physics and Python Applications10 references3 citations

TL;DR

This paper proposes a generalized canonical quantization framework for dissipative quantum systems by introducing a nonassociative, non-Lie operator W that extends the Heisenberg algebra. The method preserves canonical commutation relations while allowing compatible quantum equations of motion for dissipative systems, leading to a modified von Neumann equation consistent with classical Liouville dynamics for dissipation.

ABSTRACT

The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.

Motivation & Objective

• To resolve long-standing ambiguities in the quantum description of dissipative systems.
• To extend canonical quantization beyond associative algebras to include nonholonomic degrees of freedom.
• To maintain consistency between quantum equations of motion and generalized commutation relations.
• To derive a modified von Neumann equation compatible with classical dissipative Liouville dynamics.
• To provide a mathematical framework applicable to noncritical strings and curved-space bosonic strings as dissipative systems.

Proposed method

• Introduce a nonassociative, non-Lie operator W to extend the standard Heisenberg algebra of coordinate and momentum operators.
• Modify the canonical commutation relations by adding W, which represents nonholonomic (non-integrable) degrees of freedom.
• Ensure the total time derivative of operator products and commutators no longer obey the Leibniz rule due to nonassociativity.
• Construct a generalized von Neumann equation that remains compatible with the modified commutation relations.
• Establish a connection between the generalized quantum dynamics and the classical Liouville equation for dissipative systems.
• Use the framework to describe dissipative systems such as noncritical strings and bosonic strings in affine-metric curved space.

Experimental results

Research questions

• RQ1 How can canonical quantization be generalized to consistently describe dissipative quantum systems?
• RQ2 What algebraic structure allows the preservation of canonical commutation relations while accommodating dissipation?
• RQ3 How does nonassociativity of the operator W affect the time evolution of quantum observables?
• RQ4 Can a modified von Neumann equation be derived that is compatible with dissipative classical dynamics?
• RQ5 What are the implications of this formalism for string theory models like noncritical or curved-space strings?

Key findings

• The introduction of a nonassociative, non-Lie operator W allows consistent quantization of dissipative systems without violating the standard Heisenberg algebra.
• The time derivative of operator products and commutators fails to satisfy the Leibniz rule due to the nonassociativity of W, enabling compatibility with dissipative dynamics.
• The generalized von Neumann equation is derived and shown to be consistent with the classical Liouville equation for dissipative systems.
• The formalism applies to physical systems such as noncritical strings and bosonic strings in affine-metric curved space, which are identified as dissipative in the coupling constant phase space.
• The framework resolves ambiguities in the quantum description of dissipation by extending the operator algebra beyond associative structures.
• The method provides a mathematically consistent path to quantize systems where standard canonical quantization fails due to broken symmetries or non-conserved energy.