[Paper Review] Stochastic quantisation of Yang-Mills-Higgs in 3D

Ajay Chandra, Ilya Chevyrev+2 more|arXiv (Cornell University)|Jan 10, 2022

Black Holes and Theoretical Physics88 references4 citations

TL;DR

This paper establishes a rigorous Markov process for the stochastic quantisation of Yang-Mills-Higgs theory in three spatial dimensions by constructing a nonlinear metric space of distributions as a state space. Using regularity structures and gauge covariance, it proves local-in-time solutions to the renormalised stochastic YMH flow and identifies a unique renormalisation scheme ensuring gauge invariance in law, enabling a canonical Markov process on the quotient space of gauge orbits.

ABSTRACT

We define a state space and a Markov process associated to the stochastic quantisation equation of Yang-Mills-Higgs (YMH) theories. The state space $\mathcal{S}$ is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence $\sim$ to $\mathcal{S}$ and thus define a quotient space of "gauge orbits" $\mathfrak{O}$. We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on $\mathfrak{O}$ (up to a potential finite time blow-up) associated to the stochastic YMH flow.

Motivation & Objective

• To define a well-posed state space for the stochastic quantisation of Yang-Mills-Higgs theories in 3D, accommodating distributional fields with good continuity properties.
• To extend gauge equivalence to the space of distributions and construct a quotient space of gauge orbits to capture physical degrees of freedom.
• To establish local-in-time solutions to the renormalised stochastic Yang-Mills-Higgs flow using the theory of regularity structures.
• To identify a unique renormalisation counterterm scheme ensuring gauge covariance in law of the solutions in the small noise limit.
• To define a canonical Markov process on the quotient space of gauge orbits, up to potential finite-time blow-up.

Proposed method

• Construct a nonlinear metric space of distributions, denoted $\mathcal{S}$, as the state space for initial conditions of the YMH flow.
• Leverage gauge covariance of the deterministic YMH flow to extend the gauge equivalence relation $\sim$ to $\mathcal{S}$, forming the quotient space $\mathfrak{O}$ of gauge orbits.
• Apply the theory of regularity structures to prove existence and uniqueness of local-in-time solutions to the renormalised stochastic YMH equation.
• Use symmetry and small noise analysis to identify the unique renormalisation counterterms that preserve gauge covariance in law of the solution process.
• Construct a Markov process on $\mathfrak{O}$ by lifting the solution dynamics from $\mathcal{S}$, ensuring consistency with gauge invariance.

Experimental results

Research questions

• RQ1 How can one define a suitable state space of distributional fields for the stochastic quantisation of 3D Yang-Mills-Higgs theories?
• RQ2 In what sense can gauge equivalence be extended from classical fields to distributional configurations in the stochastic setting?
• RQ3 What is the role of renormalisation in ensuring gauge covariance in law for the stochastic YMH process?
• RQ4 Can a canonical Markov process be constructed on the space of gauge orbits for the stochastic YMH flow?
• RQ5 What conditions ensure the existence and uniqueness of local-in-time solutions to the renormalised stochastic YMH equation in 3D?

Key findings

• A nonlinear metric space $\mathcal{S}$ of distributions is constructed, providing a well-behaved state space for initial data of the stochastic YMH flow with strong continuity properties.
• Gauge equivalence is extended to $\mathcal{S}$, allowing the definition of a quotient space $\mathfrak{O}$ of gauge orbits, which captures the physical configuration space of the theory.
• Local-in-time solutions to the renormalised stochastic YMH equation are established using the theory of regularity structures, valid for small initial data in $\mathcal{S}$.
• In the small noise limit, there exists a unique choice of renormalisation counterterms such that the law of the solution is gauge covariant, ensuring physical consistency.
• A canonical Markov process is defined on the quotient space $\mathfrak{O}$, representing the stochastic dynamics of the Yang-Mills-Higgs theory up to potential finite-time blow-up.
• The construction ensures that the resulting Markov process respects the gauge symmetry of the underlying theory, providing a mathematically rigorous framework for stochastic quantisation in 3D.