Symmetries, Generators, and Conservation Laws

A conservation law in classical mechanics comes from a symmetry of the action: translation invariance conserves momentum, rotational invariance conserves angular momentum, time-translation invariance conserves energy. This is Noether's theorem. Quantum mechanics realizes the same correspondence more directly. A symmetry is a unitary operator, that operator is the exponential of a Hermitian generator, and the generator is conserved exactly when it commutes with the Hamiltonian.¹ The conserved quantities of the theory and the transformations that leave it invariant are two views of one algebraic object.

Symmetries are unitary

A physical symmetry is a mapping of states that preserves every measurable prediction, which means it preserves inner-product magnitudes: $|⟨{\varphi }^{\prime }|{\psi }^{\prime }⟩|=|⟨\varphi |\psi ⟩|$ for the transformed states. Wigner's theorem identifies the operators that can do this.

A one-parameter family of symmetries $\hat{U}(\epsilon )$ with $\hat{U}(0)=1$ is fixed by its behavior near the identity. Expanding to first order,

and unitarity ${\hat{U}}^{†}\hat{U}=1$ to first order forces $\hat{G}={\hat{G}}^{†}$: the generator is Hermitian, hence an observable. Composing $N$ infinitesimal steps of size $\epsilon /N$ and taking $N\to \infty$ exponentiates the generator,

The factor of $\hslash$ makes the exponent dimensionless and gives $\hat{G}$ the units of the conjugate observable: an action per unit $\epsilon$.

Momentum generates translations

Take the spatial translation $\hat{T}(a)$ that shifts a wave packet by $a$, $\hat{T}(a)\psi (x)=\psi (x-a)$. For an infinitesimal shift $\delta$, Taylor expansion gives

Substituting $\partial /\partial x=(i/\hslash )\hat{p}$ from $\hat{p}=-i\hslash \partial /\partial x$ turns this into the generator form $\hat{T}(\delta )=1-(i/\hslash )\delta \hat{p}$. The generator of translations is momentum, and finite translations exponentiate it:

Two immediate consequences follow. The commutator of the generator with position, $[\hat{x},\hat{p}]=i\hslash$, is the statement that $\hat{p}$ moves $\hat{x}$: ${\hat{T}}^{†}(a)\hat{x}\hat{T}(a)=\hat{x}+a$. And a Hamiltonian invariant under all translations, ${\hat{T}}^{†}(a)\hat{H}\hat{T}(a)=\hat{H}$, must commute with the generator, $[\hat{p},\hat{H}]=0$ — momentum is conserved. A free particle is the archetype; adding any position-dependent potential $V(\hat{x})$ breaks translation invariance and $\hat{p}$ is no longer conserved.

Angular momentum generates rotations

The identical construction in the angular variable gives rotations. A rotation by angle $\varphi$ about the axis $\hat{n}$ is generated by the component of orbital angular momentum along that axis,

Because rotations about different axes do not commute, their generators do not commute either. Working out $[{\hat{L}}_i,{\hat{L}}_j]$ from $\hat{\vec{L}}=\hat{\vec{r}}\times \hat{\vec{p}}$ and $[{\hat{x}}_i,{\hat{p}}_j]=i\hslash {\delta }_{ij}$ reproduces the rotation group's structure:

This non-abelian algebra is the entire content of angular-momentum quantization: the eigenvalues and multiplet structure follow from these commutators alone, exactly as the oscillator spectrum followed from $[\hat{a},{\hat{a}}^{†}]=1$. A Hamiltonian invariant under all rotations, such as any central potential $V(r)$, commutes with every ${\hat{L}}_i$, so all three components of angular momentum are conserved.

Conservation from commuting with the Hamiltonian

The link between symmetry and conservation is the Heisenberg equation of motion. For any observable $\hat{G}$ with no explicit time dependence,

If $\hat{G}$ generates a symmetry of the Hamiltonian, then ${\hat{U}}^{†}(\epsilon )\hat{H}\hat{U}(\epsilon )=\hat{H}$ for all $\epsilon$; differentiating at $\epsilon =0$ gives $[\hat{G},\hat{H}]=0$. The right side vanishes, and $⟨\hat{G}⟩$ is constant — as is the entire probability distribution of $\hat{G}$, since $[\hat{G},\hat{H}]=0$ makes $\hat{U}(\epsilon )$ commute with the evolution operator $e^{-i\hat{H}t/\hslash }$.

Time evolution itself fits the same pattern. The evolution operator $e^{-i\hat{H}t/\hslash }$ is generated by the Hamiltonian, so energy is the conserved quantity associated with time-translation invariance, provided $\hat{H}$ has no explicit time dependence.

Symmetry forces degeneracy

A symmetry does more than conserve a quantity; it organizes the spectrum. Suppose $[\hat{U},\hat{H}]=0$ and $|\psi ⟩$ is an energy eigenstate, $\hat{H}|\psi ⟩=E|\psi ⟩$. Then

so $\hat{U}|\psi ⟩$ is another eigenstate at the same energy. If it is not simply a phase times $|\psi ⟩$, the level is degenerate. A single generator that commutes with $\hat{H}$ but whose transformations rotate one state into a genuinely different one produces a whole family of states sharing an energy.

The size of the degenerate multiplet is fixed by the symmetry group. For a central potential the three components ${\hat{L}}_x,{\hat{L}}_y,{\hat{L}}_z$ all commute with $\hat{H}$ but not with each other, and this non-abelian structure forces the $2\ell +1$ states of fixed $\ell$ to be degenerate: the ${\hat{L}}_{\pm }={\hat{L}}_x\pm i{\hat{L}}_y$ ladder maps any one onto the others without changing the energy. An abelian symmetry (a single commuting generator, like translation) does not force degeneracy; a non-abelian one does. A degeneracy not explained by the obvious geometric symmetry signals a hidden symmetry with extra commuting generators, as for the $1/r$ Coulomb potential and the isotropic oscillator.

The dictionary of symmetries and conserved quantities

The construction is uniform: read off the generator of a symmetry, and it is the conserved observable when the Hamiltonian respects that symmetry.

Each row is one instance of the same theorem. The continuous symmetries treated here are connected to the identity and represented by unitary operators. The remaining symmetries — parity and time reversal — are discrete, cannot be reached by exponentiating a generator, and one of them requires the antiunitary case of Wigner's theorem. Those are the subject of the next lesson.