非交换时空中的U（N）Yang-Mills理论与洛伦兹结构

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"非交换时空中的U（N）Yang-Mills理论是量子场论的一个研究领域，探讨了在非交换几何背景下U⋆（N）规范场论的构建与分析。这项工作引入了一种新方法，利用动态场的相空间分析和额外的辅助变量来产生洛伦兹对称性和颜色自由度。这种方法特别关注于平面发散度，通过在有效作用量中计算二次项来求得Yang-Mills耦合常数的β函数。尽管示例中涉及的是二次项的计算，但这种方法具有普适性，能够应用于任意高阶耦合常数的有效作用计算，尤其适用于低能量单环散射振幅的分析。该研究由Naser Ahmadiniaz、Olindo Corradini、James P. Edwards和Pablo Pisanie共同完成，发表在JHEP04(2019)067，并由Springer为SISSA出版。" 本文介绍了一个在非交换空间时间背景下的U（N）Yang-Mills理论的研究方法。非交换几何是指空间或时空的坐标不再满足交换律，这在量子场论中引入了新的数学和物理挑战。U（N）Yang-Mills理论是规范场论的一种，它描述了强相互作用，其中U（N）表示一个包含N×N复矩阵的单位ary群，而Yang-Mills场则负责传递这种强相互作用。在这个研究中，研究人员提出了一种新颖的相空间分析技术，通过引入额外的辅助变量，不仅能够处理动态场，还能在非交换几何中构造出洛伦兹结构，这是经典物理中保持相对论不变性的关键要素。同时，这些辅助变量也帮助定义和处理颜色自由度，这是量子色动力学（QCD）中夸克和胶子的内在属性。研究人员专注于计算平面发散度，这是量子场论中处理无穷大的一种手段。在有效作用量的框架下，他们计算了与Yang-Mills耦合常数相关的二次项，这允许他们推导出耦合常数的β函数。β函数描述了耦合常数如何随着能量尺度的变化而变化，是理解渐近自由等基本物理现象的关键工具。虽然这里只展示了如何计算二次项以获取β函数，但作者指出，这种方法可以推广到更高阶的耦合常数计算。此外，这种方法对于低能量的单环散射振幅分析非常适用，这些振幅通常在粒子碰撞实验中用来验证理论预测。这个研究为非交换时空中的U（N）Yang-Mills理论提供了一个新的分析框架，为理解和计算非交换几何中的规范场理论的复杂性质提供了有价值的工具。

JHEP04(2019)067

where

U = e

iα(x)

= 1 + iα(x) −

α(x) ? α(x) + . . . (2.11)

and α(x) = α

(x) T

is Lie-algebra valued.

An action that is invariant under the symmetry transformation (2.10) is

S[A] =

dx tr (F

µν

? F

µν

) (2.12)

where g

is the bare coupling constant, and the trace is computed in the fundamental repre-

sentation of the gauge group. Note that the ﬁeld strength tensor given in (2.5) means that

the anti-commutator of the gauge group algebra enters the action alongside the commutator

familiar in commutative space. This means that we require the Lie algebra of the genera-

tors in the chosen representation to close under anti-commutation, which is not the case for

SU(N); the simplest extension compatible with this requirement is the U(N) group which

we study for the remainder of this article. For a similar analysis of the Super-Yang-Mills

theory in non-commutative space we refer to the analysis of [36].

2.3 Eﬀective action

Since this quantity will occupy our calculations in the following, we brieﬂy recap the deﬁ-

nition and determination of the eﬀective action in QFT. For a general quantum ﬁeld Φ(x)

described by a classical action S[Φ], the eﬀective action Γ

eﬀ

[Φ] is deﬁned as

−Γ

eﬀ

[

Φ]

D Φ e

−S[

Φ+Φ]+

J(x)Φ(x)

, (2.13)

where

J(x) is the source that generates the arbitrary background ﬁeld

Φ, that is,

J(x) = δ

eﬀ

. Expanding the previous expression around the background ﬁeld

Φ and

recalling that

Φ is the expectation value in the presence of the source

J, one obtains, to

leading order in quantum corrections,

−Γ

eﬀ

[

Φ]

= e

−S[

Φ]

D Φ e

−

Φ δ

S Φ + ...

. (2.14)

Therefore, the eﬀective action can be written as

eﬀ

[

Φ] = S[

Φ] + Γ[

Φ] + O(~

) , (2.15)

where the one-loop corrections, Γ[

Φ], to the eﬀective action are given by

Γ[

Φ] = ±

log Det



Φ]

Φ(x)δ

Φ(x

)



. (2.16)

The ﬁeld Φ is used to collectively denote a set of ﬁelds (real, complex conjugates etc.),

eventually comprising Grassmann components. The upper sign in (2.16) corresponds to

the contribution of each bosonic ﬁeld in Φ; the lower, to each Grassmann component.

Next, we shall calculate the one-loop corrections to the eﬀective action Γ[A] for the

non-commutative Yang-Mills ﬁeld by using the background ﬁeld method, expressing the

resulting functional determinants in terms of heat-traces and passing to the ﬁrst-quantised

(worldline) representation of such heat-traces.

– 6 –

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