Symmetries and Conservation Laws

Symmetry is the organizing principle of quantum mechanics. Every continuous symmetry of a system corresponds to a unitary operator that commutes with the Hamiltonian, and its Hermitian generator is a conserved observable — the quantum version of Noether's theorem. This single idea explains why momentum, angular momentum, and energy are conserved, why spectra are degenerate, and which transitions are allowed. It is also the template for the gauge symmetries that organize QFT.

1. Symmetries as Unitary Operators (Wigner)

A symmetry is a transformation of states that preserves all transition probabilities, $|⟨{\varphi }^{\prime }|{\psi }^{\prime }⟩{|}^2=|⟨\varphi |\psi ⟩{|}^2$. Wigner's theorem says any such map is implemented by an operator $\hat{U}$ that is either unitary or antiunitary, unique up to phase. Continuous symmetries (connected to the identity) are always unitary; antiunitary operators are reserved for time reversal.

A symmetry of the dynamics is a $\hat{U}$ that commutes with the Hamiltonian:

$$[\hat{U},\hat{H}]=0.$$

2. Generators and Conservation

A continuous symmetry is a one-parameter family $\hat{U}(s)=e^{-is\hat{G}/\hslash }$ with Hermitian generator $\hat{G}$. Invariance $[\hat{U}(s),\hat{H}]=0$ for all $s$ is equivalent to

$$[\hat{G},\hat{H}]=0,$$

and the Heisenberg equation (see heisenberg-picture.md) then gives

$$\frac{d}{dt}⟨\hat{G}⟩=\frac{i}{\hslash }⟨[\hat{H},\hat{G}]⟩=0.$$

$$\boxed{\text{continuous symmetry of }\hat{H}⟺\text{conserved observable }\hat{G}.}$$

This is Noether's theorem in quantum form — cleaner than the classical version, because the generator is the conserved charge and also generates the transformation.

3. The Standard Space-Time Symmetries

That $\hat{\mathbf{p}}$ generates translations follows from $e^{-ia\hat{p}/\hslash }\hat{x}{e}^{ia\hat{p}/\hslash }=\hat{x}+a$, provable from the canonical commutator. Rotational invariance is why angular momentum is conserved and why central-potential spectra carry the $(2\ell +1)$-fold degeneracy.

4. Symmetry and Degeneracy

If $\hat{U}$ commutes with $\hat{H}$ and $|\psi ⟩$ is an eigenstate, so is $\hat{U}|\psi ⟩$ with the same energy. When $\hat{U}|\psi ⟩$ is independent of $|\psi ⟩$, the level is degenerate. Thus:

$$\text{symmetry group of }\hat{H}⟶\text{degeneracies organized into its irreducible representations.}$$

Rotational $SO(3)$ symmetry gives the $m$-multiplets; the larger $SO(4)$ symmetry of the Coulomb problem explains hydrogen's extra $\ell$-degeneracy. Conversely, a perturbation that breaks a symmetry lifts the associated degeneracy — the mechanism behind Zeeman and Stark splittings.

5. The Galilei Group and Its Central Extension

The full symmetry group of non-relativistic space-time is the Galilei group: translations, rotations, and boosts $\mathbf{v}$. A subtlety distinguishes QM from classical mechanics: implementing Galilean boosts unitarily requires a projective (ray) representation — the boost and translation generators fail to commute by a term proportional to the mass,

$$[{\hat{K}}_i,{\hat{P}}_j]=i\hslash m{\delta }_{ij}.$$

This central extension by the mass is not optional; it is why a boosted wavefunction picks up a position- and time-dependent phase $e^{im(\mathbf{v}\cdot \mathbf{x}-\frac{1}{2}{v}^{2}t)/\hslash }$, and why mass behaves as a superselection charge in non-relativistic QM. The relativistic analog is the Poincaré group, whose unitary irreducible representations classify particles by mass and spin.

See also

• Angular momentum — rotations and their generators.
• Discrete symmetries — parity and time reversal.
• Hydrogen atom — degeneracy from hidden symmetry.
• QFT: Modern Foundations — Poincaré symmetry and particle classification.