Stochastic Quantization of an Abelian Gauge Theory

Shen, Hao

doi:10.1007/s00220-021-04114-x

Cited by 10 publications

(7 citation statements)

References 71 publications

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“…Another closely related work was recently carried out in [She21]. It was shown there that the lattice gauge covariant Langevin dynamic of the scalar Higgs model (the Lagrangian of which is given by (1.10) without the | | 4 term and with an abelian Lie algebra) in d = 2 can be appropriately modified by a DeTurck-Zwanziger term and renormalised to yield local-in-time solutions in the continuum limit.…”

Section: Relation To Previous Workmentioning

confidence: 99%

Langevin dynamic for the 2D Yang–Mills measure

et al. 2022

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We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions.To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric.To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations.Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.

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Section: Relation To Previous Workmentioning

confidence: 99%

Langevin dynamic for the 2D Yang–Mills measure

et al. 2022

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“…We also mention that the first work to study the stochastic quantisation of a gauge theory using regularity structures is [She21] (using a lattice regularisation of T 2 with G = U (1) and a Higgs field), and the first work to give a representation of the YM measure on T 2 as a random variable taking values in a (linear) state space of distributional connections for which certain Wilson loops are defined pathwise is [Che19]. The state space in [Che19] served as the basis for that in [CCHS20], and part of the definition of S in the present work is inspired by these works (see Sections 2.3 and 2.5).…”

Section: Related Work and Open Problemsmentioning

confidence: 99%

Stochastic quantisation of Yang-Mills-Higgs in 3D

Chandra¹,

Chevyrev²,

Hairer³

et al. 2022

Preprint

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We define a state space and a Markov process associated to the stochastic quantisation equation of Yang-Mills-Higgs (YMH) theories. The state space 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 ∼ to S and thus define a quotient space of "gauge orbits" 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 O (up to a potential finite time blow-up) associated to the stochastic YMH flow.

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“…As more far-reaching goals, it would be interesting to study large N problems beyond Φ 4 type models, for either invariant measures or observables or the associated stochastic dynamics. For instance, the coupled KPZ systems [FH17], random loops in N dimensional manifolds [BGHZ22, Hai16, RWZZ20, CWZZ21] and Yang-Mills type models [CCHS22a,CCHS22b,She21,Che22] where the dimension of the Lie group or its representation space tends to infinity. In the last case, the Yang-Mills measure in 2D is known to converge to a deterministic limit called the master field [Lév17, DN20, DL22b, DL22a] which satisfies the Makeenko-Migdal equations; on lattice much more results can be proved, see [Cha19,CJ16] and dynamic approach [SSZ22,SZZ23].…”

Section: Respectively It Was Shown That With Decomposition φmentioning

confidence: 99%

Large N limit of the O(N) linear sigma model via stochastic quantization

et al. 2022

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In this paper we continue the study of large N problems for the Wick renormalized linear sigma model, i.e. N -component Φ 4 model, in two spatial dimensions, using stochastic quantization methods and Dyson-Schwinger equations. We identify the large N limiting law of a collection of Wick renormalized O(N ) invariant observables. In particular, under a suitable scaling, the quadratic observables converge in the large N limit to a mean-zero (singular) Gaussian field denoted by Q with an explicit covariance; and the observables which are 2n-th renormalized powers of the fields converge in the large N limit to suitably renormalized n-th powers of Q. The quartic interaction term of the model has no effect on the large N limit of the field Φ, but has nontrivial contributions to the limiting law of the observables, and the renormalization of the n-th powers of Q in the limit has an interesting finite shift from the standard one.Furthermore, we derive the 1/N asymtotic expansion for the k-point functions of the quadratic observables by employing graph representations and analyzing the order of each graph from Dyson-Schwinger equations. Finally, turning to the stationary solutions to the stochastic quantization equations, with the Ornstein-Uhlenbeck process being the large N limiting dynamic, we derive here its next order correction in stationarity, as described by an SPDE with the right-hand side having explicit fixed-time marginal law which involves the above field Q. Contents√ N :Φ 2 : 29 6. Next order stationary dynamic 41 Appendix A. Besov spaces 48 References 52

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Stochastic Quantization of an Abelian Gauge Theory

Cited by 10 publications

References 71 publications

Langevin dynamic for the 2D Yang–Mills measure

Langevin dynamic for the 2D Yang–Mills measure

Stochastic quantisation of Yang-Mills-Higgs in 3D

Large N limit of the O(N) linear sigma model via stochastic quantization

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