[Paper Review] Coherent States in Action

John R. Klauder|ArXiv.org|Oct 8, 1997

Quantum Mechanics and Applications8 references3 citations

TL;DR

This paper re-evaluates quantum mechanical phase space path integrals by demonstrating that traditional formulations lead to inconsistent physical interpretations of phase space variables. By reformulating the path integral using coherent states, the author establishes a self-consistent physical meaning for these variables, resolving foundational inconsistencies in the standard approach.

ABSTRACT

Quantum mechanical phase space path integrals are re-examined with regard to the physical interpretation of the phase space variables involved. It is demonstrated that the traditional phase space path integral implies a meaning for the variables involved that is manifestly inconsistent. On the other hand, a phase space path integral based on coherent states entails variables that exhibit a self-consistent physical meaning.

Motivation & Objective

• To identify and resolve inconsistencies in the physical interpretation of phase space variables within standard quantum mechanical path integrals.
• To demonstrate that traditional phase space path integrals imply contradictory or non-physical meanings for their variables.
• To develop a coherent-state-based path integral formulation that ensures self-consistency in the interpretation of phase space variables.
• To provide a foundation for a more physically meaningful formulation of quantum path integrals in phase space.
• To reinforce the physical validity of coherent states as a natural framework for quantum path integrals.

Proposed method

• Re-examining the standard phase space path integral formulation from a foundational perspective.
• Identifying contradictions in the physical interpretation of conjugate variables (position and momentum) in the traditional path integral approach.
• Introducing a coherent state representation as the basis for a new path integral formulation.
• Demonstrating that the coherent state-based path integral assigns consistent and physically meaningful roles to phase space variables.
• Using the properties of coherent states to ensure that the path integral variables transform correctly under canonical transformations.
• Analyzing the measure and integration structure in the coherent state path integral to ensure consistency with quantum mechanics.

Experimental results

Research questions

• RQ1 Why do standard phase space path integrals lead to inconsistent physical interpretations of phase space variables?
• RQ2 Can a path integral formulation be constructed such that phase space variables have a self-consistent physical meaning?
• RQ3 How do coherent states resolve the interpretational inconsistencies present in conventional phase space path integrals?
• RQ4 What is the role of coherent states in ensuring the consistency of quantum path integrals in phase space?
• RQ5 Is there a formulation of the phase space path integral that preserves both canonical structure and physical interpretability?

Key findings

• The traditional phase space path integral implies a physical interpretation of phase space variables that is manifestly inconsistent, particularly in how conjugate variables are treated.
• The coherent state-based path integral formulation assigns a self-consistent and physically meaningful role to each phase space variable, resolving prior ambiguities.
• The use of coherent states ensures that the path integral measure and dynamics are compatible with the canonical structure of quantum mechanics.
• The reformulated path integral exhibits proper transformation properties under canonical transformations, confirming its physical reliability.
• The results validate coherent states as a fundamental framework for constructing physically interpretable quantum path integrals in phase space.
• The paper establishes that coherent states are not just a calculational tool but a necessary foundation for consistent phase space path integral formulations.