[Paper Review] Path integral measure factorization in path integrals for diffusion of Yang--Mills fields

S. N. Storchak|ArXiv.org|Nov 19, 2007

Numerical methods in inverse problems3 references3 citations

TL;DR

This paper develops a path integral measure factorization method for Yang–Mills field diffusion using nonlinear filtering stochastic differential equations. By transforming the path integral from the full principal fiber bundle to a reduced manifold via the Coulomb gauge, it derives an integral relation between the original and reduced path integrals, revealing a nontrivial Jacobian factor that modifies the effective quantum Hamiltonian.

ABSTRACT

Factorization of the (formal) path integral measure in a Wiener path integrals for Yang--Mills diffusion is studied. Using the nonlinear filtering stochastic differential equation, we perform the transformation of the path integral defined on a total space of the Yang--Mills principal fiber bundle and come to the reduced path integral on a Coulomb gauge surface. Integral relation between the path integral representing the "quantum" evolution given on the original manifold of Yang--Mills fields and the path integral on the reduced manifold defined by the Coulomb gauge is obtained.

Motivation & Objective

• To establish a rigorous measure factorization procedure for path integrals in Euclidean Yang–Mills field theory.
• To reduce the path integral from the full space of connections to a gauge-fixed manifold using the Coulomb gauge.
• To derive an integral relation between the original path integral and the reduced one, accounting for symmetry-induced measure changes.
• To identify the Jacobian factor arising from the transformation, which modifies the effective dynamics of the reduced system.
• To extend finite-dimensional measure factorization techniques to infinite-dimensional Yang–Mills theories using stochastic processes.

Proposed method

• Utilizes nonlinear filtering stochastic differential equations to decompose the path integral measure into components associated with group orbits and orbit space.
• Applies the Faddeev–Popov method and Rossi–Testa approach to handle gauge symmetry and reduce degrees of freedom.
• Implements the Coulomb gauge condition to fix residual time-independent gauge freedom, restricting the domain to a local slice of the principal fiber bundle.
• Derives the projection operator $P_{\bot}$ onto the orthogonal complement of the gauge orbit direction, expressed via differential operators and Green's functions.
• Constructs the inverse of the metric-like matrix $\chi \cdot \chi^\top$ using the Green's function $K(\mathbf{x}-\mathbf{y})$ satisfying $(-\partial^2)K = \delta^3(\mathbf{x}-\mathbf{y})$.
• Symbolically expresses the projection and orthogonal projection operators using differential operators and inverse Laplacians, such as $P_{\bot}^{(\alpha,k,x)}_{(\beta,m,y)} = \delta^\alpha_\beta \left[ \delta^k_m + \partial_m \frac{1}{(-\partial^2)} \partial^k \right] \delta^3(\mathbf{y}-\mathbf{x})$.

Experimental results

Research questions

• RQ1 How can the path integral measure in Yang–Mills diffusion be factorized under gauge symmetry?
• RQ2 What is the precise relation between the original path integral on the full connection space and the reduced path integral on the Coulomb gauge surface?
• RQ3 How does the transformation of the path integral induce a Jacobian factor in the effective quantum Hamiltonian?
• RQ4 What role does the nonlinear filtering stochastic differential equation play in the measure decomposition process?
• RQ5 How can finite-dimensional measure factorization techniques be generalized to infinite-dimensional Yang–Mills theories?

Key findings

• The path integral measure decomposes into a product of measures: one along the gauge orbits and one on the orbit space, with the decomposition governed by the nonlinear filtering SDE.
• The transformation from the full space to the Coulomb gauge surface leads to a nontrivial Jacobian factor that modifies the differential generator of the reduced stochastic process.
• The reduced quantum Hamiltonian acquires an additional term due to the measure Jacobian, which is explicitly derived via the projection operator $P_{\bot}$.
• The projection operator $P_{\bot}$ is expressed as $P_{\bot}^{(\alpha,k,x)}_{(\beta,m,y)} = \delta^\alpha_\beta \left[ \delta^k_m + \partial_m \frac{1}{(-\partial^2)} \partial^k \right] \delta^3(\mathbf{y}-\mathbf{x})$, representing orthogonal projection onto transverse modes.
• The orthogonal projection $N$ satisfies $N^{(\alpha,i,x)}_{(\beta,j,y)} = \left( \delta^\alpha_\beta \delta^i_j - \mathcal{D}^{\alpha i}_\epsilon \left( \frac{1}{\mathcal{D} \partial} \right)_\beta^\epsilon \partial_j \right) \delta^3(\mathbf{x}-\mathbf{y})$, ensuring orthogonality to gauge directions.
• The key identities, such as $P_{\bot} P_{\bot} = P_{\bot}$ and $P_{\bot} \chi^\top = 0$, confirm the consistency of the projection structure in the measure transformation.