[Paper Review] Path integral quantization of Yang-Mills theory

Sami I. Muslih|ArXiv.org|Nov 14, 2000

Particle Accelerators and Free-Electron Lasers3 citations

TL;DR

This paper presents a gauge-fixing-free path integral quantization of Yang-Mills theory using a canonical path integral formulation based on Hamilton-Jacobi partial differential equations. By ensuring integrability of the equations of motion through consistency conditions, the method directly constructs the path integral over canonical phase space variables without introducing Lagrange multipliers, delta functions, or gauge-fixing conditions, offering a consistent alternative to Faddeev-Popov quantization.

ABSTRACT

Path integral formulation based on the canonical method is discussed. Path integral for Yang-Mills theory is obtained by this procedure. It is shown that gauge fixing which is essential procedure to quantize singular systems by Faddeev's and Popov's method is not necessary if the canonical path integral formulation is used.

Motivation & Objective

• To develop a path integral quantization of Yang-Mills theory without relying on gauge-fixing procedures.
• To demonstrate that canonical path integral formulation naturally yields a well-defined path integral for singular systems with first-class constraints.
• To show that the standard Faddeev-Popov method is unnecessary when using the canonical path integral approach.
• To establish a direct link between integrability of equations of motion and the construction of a consistent path integral over canonical variables.

Proposed method

• The method starts from the Hamilton-Jacobi partial differential equations (HJPDE) derived from the canonical structure of the constrained system.
• It uses the condition dH′α = 0 for integrability, ensuring the existence of canonical phase space coordinates q_a and p_a as functions of time parameters t_α.
• The canonical action is derived directly from the equations of motion via the total differential form dz = (−H_α + p_a ∂H′_α/∂p_a) dt_α.
• The path integral is constructed as a functional integral over canonical variables A_i^a and π_i^a with the derived action, without gauge-fixing terms.
• The method avoids introducing auxiliary fields, Lagrange multipliers, or determinant ambiguities common in Faddeev-Popov quantization.
• Integrability is verified by checking the closure of constraints through successive variations of H′_α and F_1^a, ensuring consistency.

Experimental results

Research questions

• RQ1 Can Yang-Mills theory be consistently quantized without gauge-fixing using a canonical path integral formulation?
• RQ2 Does the canonical path integral method produce a unitary and well-defined quantum theory for gauge theories with first-class constraints?
• RQ3 How does the integrability of the Hamilton-Jacobi system relate to the absence of gauge-fixing in the path integral?
• RQ4 What is the structure of the path integral action when derived directly from equations of motion in the canonical framework?

Key findings

• The path integral for Yang-Mills theory is derived directly as ∫∏DA_i^a Dπ_i^a exp[i∫(−¼F_ij^aF_a^ij + ½π_a^iπ_i^a + D_iπ_a^i A_0^a − ∂_i(π_a^i A_0^a) + π_a^i ∂_0 A_i^a) d⁴x], without gauge-fixing.
• The method avoids the need for Faddeev-Popov ghosts, determinant Jacobians, and auxiliary constraints by ensuring integrability of the HJPDE system.
• The canonical variables A_i^a and π_i^a are obtained as functions of time parameters t and A_0^a through consistent integration of the total differential equations.
• The condition dH′_0 = 0 and dH′_a = 0 is satisfied only after introducing F_1^a = −D_iπ_a^i as a constraint, which is then consistently closed.
• The resulting path integral is manifestly non-singular and defined over the full canonical phase space, unlike the standard Faddeev-Popov approach.
• The canonical path integral formulation provides a consistent alternative to traditional quantization, bypassing the need for gauge-fixing while preserving unitarity and positivity of the metric.