45.4 The geometric significance of the connection

45.4 The geometric significance of the connection

The information contained in a connection $A_{\mu }(x)$ can be put in a diferent form, using it to define a phase for any curve $\gamma$ between two points in $M^4$ :

Definition (Path-dependent phase factor)

Given a connection $A_{\mu }(x)$ , one can define for any curve $\gamma$ parametrized by $\tau \in [0,1]$ , with position at time $\tau$ given by $x(\tau )$, the path-dependent phase factor

$${\int }_{\gamma }A\equiv {\int }_0^1\sum _{\mu =1}^4{A}_{\mu }(x)\frac{d{x}_{\mu }}{d\tau }d\tau$$

The efect of a gauge transformation $\phi$ is

$${\int }_{\gamma }A\to {\int }_0^1\sum _{\mu =1}^4\left(A_{\mu }(x)+\frac{\partial \phi }{\partial x_{\mu }}\right)\frac{d{x}_{\mu }}{d\tau }d\tau ={\int }_{\gamma }A+\phi (\gamma (1))-\phi (\gamma (0))$$

Note that if $\gamma$ is a closed curve, with $\gamma (1)=\gamma (0)$ , then the path-dependent phase factor $\dot{{\int }_{\gamma }A}$ is gauge invariant.

Digression. For readers familiar with diferential forms, $A$ can be thought of as an element of ${\Omega }^1(M^4)$ , the space of 1-forms on space-time $M^4$ . The pathdependent phase ${\int }_{\gamma }\dot{A}$ is then the standard integral of a 1-form along a curve $\gamma$ . The curvature of $A$ is simply the 2-form $F=dA$ , where $d$ is the de Rham diferential. The gauge group acts on connections by

$$A\to A+d\phi$$

and the curvature is gauge invariant since

$$F\to F+d(d\phi )$$

and $d$ satisfies $d^2=0$

Stokes theorem for diferential forms implies that $if\gamma$ is a closed curve, and $\gamma$ is the boundary of a surface $S\left(\gamma =\partial S\right)$ , then

$${\int }_{\gamma }A={\int }_{S}F$$

Note that if $F=0$ , then ${\int }_{\gamma }A=0$ for any closed curve $\gamma ,$ , and this can be used to show that path-dependent phase factors do not depend on the path. To see this, consider any two paths ${\gamma }_1$ and ${\gamma }_2$ from $\gamma (0)$ to $\gamma (1)$ , and $\gamma ={\gamma }_1-{\gamma }_2$ the closed curve that goes from $\gamma (0)$ to $\gamma (1)$ along ${\gamma }_1$ , and then back $to\gamma (0)$ along ${\gamma }_2$ . Then

$${\int }_{S}F=0⟹{\int }_{\gamma }A={\int }_{{\gamma }_1}A-{\int }_{{\gamma }_2}A=0$$

so

$${\int }_{{\gamma }_1}A={\int }_{{\gamma }_2}A$$

The path-dependent phase factors ${\int }_{\gamma }A$ allow comparison of the values of the complex field $\psi$ at diferent points in a gauge invariant manner. To compare the value of a field $\psi$ at $\gamma (0)$ to that of the field at $\gamma (1)$ in a gauge invariant manner, we just need to consider the path-dependent quantity

$$e^{ie{\int }_{\gamma }A}\psi (\gamma (0))$$

where $\gamma$ is a curve from $\gamma (0)$ to $\gamma (1)$ . Under a gauge transformation this will change as

$$e^{ie{\int }_{\gamma }A}\psi (\gamma (0))\to e^{ie({\int }_{\gamma }A+\phi (\gamma (1))-\phi (\gamma (0)))}{e}^{ie\phi (\gamma (0))}\psi (\gamma (0))=e^{ie\phi (\gamma (1))}{e}^{ie{\int }_{\gamma }A}\psi (\gamma (0))$$

which is the same transformation property as that of $\psi (\gamma (1))$ .

Figure 45.1: Comparing a complex field at two points in a gauge invariant manner.

In the path integral formalism (see section 35.3), the minimal coupling of a single particle to a background electromagnetic field described by a vector potential $A_{\mu }$ can be introduced by weighting the integral over paths by the path-dependent phase factor. This changes the formal path integral by

$$\int \mathcal{D}\gamma e^{\frac{i}{\hslash }S[\gamma ]}\to \int \mathcal{D}\gamma e^{\frac{i}{\hslash }S[\gamma ]}{e}^{\frac{ie}{\hslash }{\int }_{\gamma }A}$$

and a path integral with such a weighting of paths then must be sensibly defined. This method only works for the single-particle theory, with minimal coupling for a quantum theory of fields given by the replacement of derivatives by covariant derivatives described earlier.

Previous section: 45.3 Field equations with background electromagnetic fields · Next section: 45.5 The non-Abelian case · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 484、485、486、487、488、489、490、491、492、493

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：d3d285500af2024670a515c59ee5503cd8bd26667c0e1e839c22ee489820b440

OCR 产物 SHA-256：d3d285500af2024670a515c59ee5503cd8bd26667c0e1e839c22ee489820b440