2.4 Conservation of charge and U(1) symmetry

The way we have defined observable operators in terms of a group representation on $H .$ , the action of these operators has nothing to do with the dynamics. If we start at time $t = 0$ in a state in $H_{q_{j}}$ , with definite numerical value $q_{j}$ for the observable, there is no reason that time evolution should preserve this. Recall from one of our basic axioms that time evolution of states is given by the Schrödinger equation

\frac{d}{d t} ∣ ψ (t)⟩ = - i H ∣ ψ (t)⟩

(we have set $ℏ = 1)$ . We will later more carefully study the relation of this equation to the symmetry of time translation (the Hamiltonian operator $H$ generates an action of the group $R$ of time translations, just as the operator $Q$ generates an action of the group $U (1))$ . For now though, note that for time-independent Hamiltonian operators $H$ , the solution to this equation is given by exponentiating $H .$ , with

∣ ψ (t)⟩ = U (t) ∣ ψ (0)⟩

where

U (t) = e^{- i t H} = 1 - i t H + \frac{( - i t ) ^{2}}{2 !} H^{2} + \dots

The commutator of two operators $O_{1}, O_{2}$ is defined by

[O_{1}, O_{2}] := O_{1} O_{2} - O_{2} O_{1}

and such operators are said to commute if $[O_{1}, O_{2}] = 0$ . If the Hamiltonian operator $H$ and the charge operator $Q$ commute, then $Q$ will also commute with all powers of $H$

[H^{k}, Q] = 0

and thus with the exponential of $H$ , so

[U (t), Q] = 0

This condition

U (t) Q = Q U (t) (2.1)

implies that if a state has a well-defined value $q_{j}$ for the observable $Q$ at time $t = 0$ , it will continue to have the same value at any other time $t,$ since

Q ∣ ψ (t)⟩ = Q U (t) ∣ ψ (0)⟩ = U (t) Q ∣ ψ (0)⟩ = U (t) q_{j} ∣ ψ (0)⟩ = q_{j} ∣ ψ (t)⟩

This will be a general phenomenon: if an observable commutes with the Hamiltonian observable, one gets a conservation law. This conservation law says that if one starts in a state with a well-defined numerical value for the observable (an eigenvector for the observable operator), one will remain in such a state, with the value not changing, i.e., “conserved”.

When $[Q, H] = 0;$ , the group $U (1)$ is said to act as a “symmetry group” of the system, with $π (e^{i θ})$ the “symmetry transformations”. Equation 2.1 implies that

U (t) e^{i Qθ} = e^{i Qθ} U (t)

so the action of the $U (1)$ group on the state space of the system commutes with the time evolution law determined by the choice of Hamiltonian. It is only when a representation determined by $Q$ has this particular property that the action of the representation is properly called an action by symmetry transformations, and that one gets conservation laws. In general $[Q, H] \neq = 0$ , with $Q$ then generating a unitary action on $H$ that does not commute with time evolution and does not imply a conservation law.

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正文：英文 · 原著转录

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原书 PDF · 印刷页 21、22

来源版本：2025-10-20

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2.4 Conservation of charge and U(1) symmetry