Chern-Simons理论中的简化Schwinger-Dyson方程及其对称性关系

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在《核物理B》(Nuclear Physics B) 2016年第912期的文章《Schwinger-Dyson函数在Chern-Simons理论中的作用》中，作者E. Guadagnini探讨了摄动性SU(N) Chern-Simons规范理论中的一个关键概念。该理论通常用于研究拓扑量子场论，特别是描述强相互作用中的量子效应。Chern-Simons理论以其独特的拓扑性质而著称，其中最显著的是其与全局规范不变性的联系。文章的核心内容聚焦于Schwinger-Dyson方程在这一理论中的简化形式。在量子场论中，Schwinger-Dyson方程是一组关联着物理量之间的关系，它们是量子修正的Heisenberg方程，有助于揭示场的期望值如何随时间演化以及相互作用的效应。在这个特定的背景下，Guadagnini研究了曲率相关函数的生成函数，这是描述量子场行为的重要工具。在Chern-Simons理论中，规范化后的Schwinger-Dyson函数展现出了一种意想不到的对称性。通过某种形式的对偶变换，作者发现这些函数与规范连接相关函数的生成函数之间存在深层次的联系。这种对偶性可能反映了理论内部的某种基本原理或对称性变换，有助于简化计算并揭示理论的内在结构。值得注意的是，这篇论文是在纪念Raymond Stora的论文集中发表的，Stora是一位著名的理论物理学家，对量子场论尤其是规范场论有重大贡献。此外，该论文是开放获取的，根据Creative Commons BY 4.0许可证发布，这意味着读者可以自由地访问、分享和使用论文的内容，只要给予适当的引用。这篇文章的重要性在于它提供了对Chern-Simons理论中Schwinger-Dyson方程的新理解，并可能开启探索更深层次的理论结构和对称性的新途径。对于研究者来说，这可能引发进一步的研究，如改进计算技术、探索新的物理现象或者发展新的理论模型。

240 E. Guadagnini / Nuclear Physics B 912 (2016) 238–248

In general, equation (2.3) must be integrated with the corrections coming from the gauge-ﬁxing

procedure. This point will be discussed in a while; for the moment let us proceed with the basic

argument. Since the curvature F

μν

transforms covariantly under a local gauge transformation,

the contact terms in the Schwinger–Dyson equations can combine to produce a gauge-invariant

expression. In particular, according to the rule shown in equation (2.3), the 3-points function of

the curvature should be given by the gauge-invariant combination





μαβ

αβ

(x) 

νγδ

γδ

(y) 

τσξ

σξ

(z)



= 16



4π



abc



μντ

(x − y)δ

(z − y) . (2.4)

In the CS theory, the value of the 3-points correlation function of the curvature corresponds to

the gauge-invariant expression (2.4), which is proportional to the structure-constants tensor f

abc

of the gauge group divided by the square of the coupling constant k. As it will be shown below,

the expression appearing in equation (2.4) should have a gauge-independent meaning because

it can also be obtained in the limit of vanishing gauge-ﬁxing. As a check, one can easily verify

the validity of equation (2.4) to lo

west orders of perturbation theory when the CS theory is

formulated in R

In order to proceed with the deri

vation of the properties of the renormalized Schwinger–Dyson

functional in the CS theory, one needs to specify the gauge-ﬁxing procedure and the renormaliza-

tion conditions. Let us consider the perturbative approach to the CS theory in R

. Really, in the

following argument R

can be replaced by a generic 3-manifold M which is a homology sphere

because, in this case, the ﬁeld variables have no zero modes [6] and the standard perturbative

expansion is well deﬁned. In the Landau gauge, the gauge-ﬁxing term [9] is given by

φπ

4π





− B

∂

+ ∂



∂

− f

abc



. (2.5)

Let  be the renormalized effective action of the CS theory; i is given by the sum of the

one-particle-irreducible diagrams with external legs represented by classical ﬁelds.

 can be computed by means of v

arious techniques; one convenient method is the standard

quantum ﬁeld theory procedure which is called the renormalized perturbation theory in the

Peskin–Schroeder book [10]. Of course, any other renormalization method leads to the same

physical conclusions; the use of renormalized perturbation theory is quite instructive because the

basic concepts of the renormalization clearly emer

ge. In renormalized perturbation theory, the

values of all the parameters entering the lagrangian coincide with the renormalized values, and

the so-called local counterterms cancel precisely all the possible contributions to these parame-

ters which are found in the loop expansion. In this wa

y, the normalization conditions [3,10] are

indeed satisﬁed to all orders of perturbation theory, as it must be.

Renormalized perturbation theory represents one of the fundamental constituents of the theory

of quantized ﬁelds [11,12]; this subject wa

s of particular interest for Raymond Stora [13]. So I

will elaborate a bit on this issue in the context of the quantum CS ﬁeld theory. At the beginning

of the years 90’

s, with Raymond we had fruitful discussions on this matter.

The renormalization process of the CS action (2.1) concerns tw

o parameters: the wave func-

tion normalization and the coupling constant. Actually, as in any gauge theory, because of the

gauge invariance one of the normalization conditions is superﬂuous [3]; therefore, in our case,

only one parameter needs to be speciﬁed. The CS normalization conditions can then be e

xpressed

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