45.1 U(1) gauge symmetry

45.1 U(1) gauge symmetry

Concept links · terms present in this machine draft; source roles are unverified: vector space · eigenvalue · Lie algebra · group action · U(1) · Lie bracket

In sections 38.2.1 and 44.1 we saw that the existence of a $U(1)$ group action by overall phase transformations on the complex field values led to the existence of an operator $\hat{Q}$ , which commuted with the Hamiltonian and acted with integral eigenvalues on the space of states. Instead of acting on fields by multiplication by a constant phase $e^{i\phi }$ , one can imagine multiplying by a phase that varies with the coordinates $x.$ . Such phase transformations are called “gauge transformations” and form an infinite dimensional group under pointwise multiplication:

Definition (Gauge group)

The group $\mathcal{G}$ of functions on $M^4$ with values in the unit circle $U(1)$ , with group law given by point-wise multiplication

$$e^{ie{\phi }_1(x)}\cdot e^{ie{\phi }_2(x)}=e^{ie({\phi }_1(x)+{\phi }_2(x))}$$

is called the $U(1)$ gauge group, or group of $U(1)$ gauge transformations.

Here $x=(x_0,\mathbf{x})\in M^4$ , $e$ is a constant, and $\phi$ is a real-valued function

$$\phi (x):M^4\to \mathbf{R}$$

The vector space of such functions is the Lie algebra $Lie\mathcal{G}.$ with a trivial Lie bracket. The constant e determines the normalization of the Lie algebra-valued $\phi (x)$ , with its appearance here a standard convention (such that it does not appear in the Hamiltonian, Poisson brackets, or equations of motion).

The group $\mathcal{G}$ acts on complex functions $\psi$ of space-time as

$$\psi (x)\to e^{ie\phi (x)}\psi (x),\overline{\psi }(x)\to e^{-ie\phi (x)}\overline{\psi }(x)$$

Note that, in quantum mechanics, this is a group action on the wavefunctions, and it does not correspond to any group action on the finite dimensional phase space of coordinates and momenta, so has no classical interpretation. In quantum field theory though, where these wavefunctions make up the phase space to be quantized, this is a group action on the phase space, preserving the symplectic structure.

Terms in the Hamiltonian that just involve $|\psi (x){|}^2=\overline{\psi }(x)\psi (x)$ will be invariant under the group $\mathcal{G}$ , but terms with derivatives such as

$$|\nabla \psi {|}^2$$

will not, since when

$$\psi \to e^{ie\phi (x)}\psi (x)$$

one has the inhomogeneous behavior

$$\frac{\partial \psi (x)}{\partial x_{\mu }}\to \frac{\partial }{\partial x_{\mu }}(e^{ie\phi (x)}\psi (x))=e^{ie\phi (x)}\left(ie\frac{\partial \phi (x)}{\partial x_{\mu }}+\frac{\partial }{\partial x_{\mu }}\right)\psi (x)$$

To deal with this problem, one introduces a new kind of field:

Definition (Connection or vector potential)

A $U(1)$ connection (mathematician’s terminology) or vector potential (physicist’s terminology) is a function $A$ on space-time $M^4$ taking values in ${\mathbf{R}}^4$ , with its components denoted

$$A_{\mu }(x)=(A_0(t,\mathbf{x}),\mathbf{A}(t,\mathbf{x}))$$

The gauge group $\mathcal{G}$ acts on the space of $U(1)$ connections by

$$A_{\mu }(t,\mathbf{x})\to A_{\mu }^{\phi }(t,\mathbf{x})\equiv A_{\mu }(t,\mathbf{x})+\frac{\partial \phi (t,\mathbf{x})}{\partial x_{\mu }}\text{\hspace{1em}(45.1)}$$

The vector potential allows one to define a new kind of derivative, such that the derivative of the field $\psi$ has the same homogeneous transformation properties under $\mathcal{G}$ as $\psi$ itself:

Definition (Covariant derivative)

Given a connection $A$, the associated covariant derivative in the $\mu$ direction is the operator

$$D_{\mu }^A=\frac{\partial }{\partial x_{\mu }}-ie{A}_{\mu }(x)$$

With this definition, the efect of a gauge transformation is

$$\begin{array}{rl}{D}_{\mu }^A\psi & \to \left(D_{\mu }^A-ie\frac{\partial \phi }{\partial x_{\mu }}\right)e^{ie\phi (x)}\psi =e^{ie\phi (x)}\left(D_{\mu }^A+ie\frac{\partial }{\partial x_{\mu }}-ie\frac{\partial }{\partial x_{\mu }}\right)\psi \\ & =e^{ie\phi (x)}{D}_{\mu }^A\psi \end{array}$$

If one replaces derivatives by covariant derivatives, terms in a Hamiltonian such as

$$|\nabla \psi {|}^2=\sum _{j=1}^3\frac{\overline{\partial \psi }}{\partial x_j}\frac{\partial \psi }{\partial x_j}$$

will become

$$\sum _{j=1}^3(\overline{D_j^A\psi })(D_j^A\psi )$$

which will be invariant under the infinite dimensional group $G$. The procedure of starting with a theory of complex fields, then introducing a connection while changing derivatives to covariant derivatives in the equations of motion is called the “minimal coupling prescription.” It determines how a theory of complex free fields describing charged particles can be turned into a theory of fields coupled to a background electromagnetic field, in the simplest or “minimal” way.

Chapter contents · Next section: 45.2 Curvature, electric and magnetic fields · 中文版本

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

来源版本：2025-10-20

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