Exact Symmetry

An exact symmetry is a transformation that leaves the physical structure of a quantum system invariant without approximation. The chapter guide compares this baseline with explicit, approximate, spontaneous, hidden, dynamical, emergent, and protected structures. For a time-independent closed system, the most common statement is:

$$SHS^{-1}=H,$$

where $H$ is the Hamiltonian and $S$ is the unitary or antiunitary operator representing the transformation. If the symmetry is unitary, this is equivalently

$$[S,H]=0.$$

For a continuous unitary symmetry generated by a Hermitian operator $G$,

$$S(\alpha) = e^{-i\alpha G/\hbar},$$

exact invariance for all $\alpha$ implies

$$[G,H]=0.$$

The word “exact” matters. It means the symmetry is a property of the model or physical setup as stated, not a rough simplification, a low-energy approximation, a perturbative limit, or a pattern visible only in one specially chosen state.

The Object Being Symmetric

A symmetry statement must say what object is invariant. In quantum mechanics, different statements have different meanings:

• a Hamiltonian may be invariant, $SHS^{-1}=H$;
• a state may be invariant, $S\lvert\psi\rangle=e^{i\phi}\lvert\psi\rangle$;
• a set of states may be invariant as a subspace;
• an observable algebra may be invariant, $A\mapsto SAS^{-1}$;
• a measurement setup may or may not transform along with the system.

The Hamiltonian statement is the main one for dynamics. It says that applying the transformation and then evolving gives the same result as evolving and then applying the transformation.

For unitary time evolution

$$U(t) = e^{-iHt/\hbar},$$

an exact unitary symmetry $S$ with $[S,H]=0$ also commutes with the time-evolution operator:

$$SU(t) = U(t)S.$$

Thus symmetry is not merely a label on eigenstates. It is a compatibility condition between transformations and dynamics.

Exact Symmetry of a Hamiltonian

For a unitary symmetry, the Hamiltonian condition can be written in three equivalent ways:

$$\begin{aligned} SHS^{-1}&=H,\\ SH&=HS,\\ [S,H]&=0. \end{aligned}$$

The first form emphasizes invariance under transformation. The second says that the order of symmetry operation and Hamiltonian action does not matter. The third is the commutator test.

For antiunitary symmetries, such as time reversal, the compact condition remains

$$SHS^{-1}=H,$$

but one must remember that $S$ is antilinear. It is usually safer not to manipulate antiunitary operators as ordinary matrices unless the complex conjugation action is explicit.

Continuous Symmetries and Conserved Quantities

Suppose

$$S(\alpha) = e^{-i\alpha G/\hbar}$$

is an exact symmetry for every value of $\alpha$. Then

$$S(\alpha)HS(\alpha)^\dagger=H.$$

Differentiate at $\alpha=0$:

$$\frac{d}{d\alpha} \left. S(\alpha)HS(\alpha)^\dagger \right|_{\alpha=0} = -\frac{i}{\hbar}GH + \frac{i}{\hbar}HG = 0.$$

Therefore

$$[G,H]=0.$$

If $G$ has no explicit time dependence, it is conserved under closed-system evolution:

$$\frac{d}{dt}\langle G\rangle = \frac{i}{\hbar}\langle[H,G]\rangle = 0.$$

This is the quantum version of the symmetry-conservation link.

Discrete Symmetries

Discrete symmetries do not require infinitesimal generators. Parity in one dimension is the simplest example:

$$\mathsf P X\mathsf P^{-1}=-X, \qquad \mathsf P^2=I.$$ $$\mathsf P P_x\mathsf P^{-1}=-P_x.$$

For

$$H = \frac{P_x^2}{2m}+V(X),$$

parity is exact when

$$V(-X)=V(X).$$

Then

$$[\mathsf P,H]=0.$$

Energy eigenstates can be chosen with definite parity, and parity-odd perturbations have vanishing diagonal matrix elements in nondegenerate parity eigenstates.

Rotational Symmetry and Reduced Symmetry

For a spinless particle in a central potential,

$$H = \frac{\mathbf p^2}{2m} + V(r), \qquad r=\lvert\mathbf r\rvert,$$

the Hamiltonian is invariant under all spatial rotations. Equivalently,

$$[H,L_i]=0 \qquad i=x,y,z.$$

The exact symmetry group is rotational symmetry. Energy eigenstates can be organized by angular momentum quantum numbers.

If an external magnetic field chooses the $z$ direction, full rotational symmetry is generally reduced. For example,

$$H = H_0 - \gamma B S_z$$

is invariant under rotations about the $z$ axis if $H_0$ is rotationally invariant, but it is not invariant under arbitrary rotations unless the field is transformed as part of the physical setup. With the field held fixed, the surviving exact continuous symmetry is axial:

$$[H,S_z]=0, \qquad [H,S_x]\ne0, \qquad [H,S_y]\ne0$$

for a generic nonzero $B$.

This distinction is common in applications. A term can reduce a larger symmetry to a subgroup rather than destroy all symmetry.

Consequences of Exact Symmetry

Exact symmetry has several practical consequences.

First, it produces conserved quantities for continuous symmetries. Translational invariance gives momentum conservation; rotational invariance gives angular momentum conservation; time-translation invariance gives energy conservation.

Second, it organizes Hilbert space into invariant sectors. If

$$[S,H]=0,$$

then $H$ maps each eigenspace of $S$ into itself, assuming the relevant spectral decomposition is well-defined. This lets one block-diagonalize the Hamiltonian by symmetry labels.

Third, it constrains matrix elements. If states and operators transform incompatibly under a symmetry, the corresponding matrix element must vanish. This is the logic behind parity selection rules and the more systematic Wigner-Eckart theorem.

Fourth, it can protect degeneracies. If symmetry maps a state to a linearly independent state with the same energy, then degeneracy follows. However, not every exact symmetry forces degeneracy: one-dimensional representations can label nondegenerate states.

Symmetry of a State Is Different

A Hamiltonian can have an exact symmetry even if a particular state does not look symmetric. For example, an even potential has parity symmetry, but a wave packet localized on one side of the origin is not itself a parity eigenstate.

Conversely, a particular state can be invariant under a transformation that is not a symmetry of the Hamiltonian. A spin state aligned with $z$ is an eigenstate of $S_z$, but that alone does not imply the Hamiltonian is invariant under rotations about $z$.

The symmetry of the dynamics and the symmetry of a state answer different questions:

$$\begin{array}{ccl} SHS^{-1}=H &\text{asks}& \text{whether the dynamics has the symmetry},\\ S\lvert\psi\rangle=e^{i\phi}\lvert\psi\rangle &\text{asks}& \text{whether the state carries a definite symmetry label}. \end{array}$$

Keeping these separate prevents many false conclusions about symmetry breaking.

Exact Versus Approximate or Broken

Exact symmetry is the baseline against which other notions are defined.

An explicitly broken symmetry is not a symmetry of the Hamiltonian after all terms are included. For instance, adding a small odd potential to an even one breaks parity explicitly.

An approximate symmetry is not exact in the full Hamiltonian, but the symmetry-breaking terms are small in a useful regime. Approximate symmetries can explain near-degeneracies and weak selection-rule violations.

A hidden symmetry or accidental symmetry may explain degeneracy beyond an obvious geometric symmetry. The hydrogen atom’s Coulomb degeneracy is the standard example.

Spontaneous symmetry breaking is a more subtle notion: the equations or Hamiltonian have a symmetry, while physically relevant states or phases fail to display it in an appropriate many-degree-of-freedom limit. It is not simply the observation that one finite-system state lacks a symmetry.

Those topics have their own canonical pages. This page only fixes the reference notion: exact symmetry means exact invariance of the specified quantum structure.

Common Mistakes

• Calling a transformation a symmetry because it is mathematically available, without checking $SHS^{-1}=H$.
• Confusing symmetry of the Hamiltonian with symmetry of one state.
• Forgetting that external fields must either transform as part of the setup or be treated as fixed backgrounds.
• Assuming every exact symmetry creates degeneracy.
• Treating a weakly broken symmetry as exact when deriving selection rules.
• Using a continuous-symmetry generator after the corresponding symmetry-breaking perturbation has been added.
• Handling antiunitary symmetries as if they were ordinary unitary matrices.

Cross-Links

• Symmetry Breaking and Emergence
• Why Symmetry Matters
• Broken Symmetry Preview
• Explicit Symmetry Breaking
• Spontaneous Symmetry Breaking Preview
• Approximate Symmetry
• Accidental Symmetry
• Hidden Symmetry
• Quantum Symmetries
• Unitary Symmetries
• Antiunitary Symmetries
• Symmetry Constraints on Hamiltonians
• Commutators and Conservation Laws
• Parity
• Central Potentials
• Selection Rules
• Commutators
• Conservation Laws
• Commutator Table

References

• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• H. F. Jones, Groups, Representations and Physics, 2nd ed., CRC Press, 1998.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003.

Exercises

1. Parity of a one-dimensional Hamiltonian.

Let

$$H = \frac{P_x^2}{2m} + V(X),$$

with $\mathsf P X\mathsf P^{-1}=-X$ and $\mathsf P P_x\mathsf P^{-1}=-P_x$. Show that parity is an exact symmetry precisely when $V(-X)=V(X)$.

Solution

Transform the Hamiltonian:

$$\mathsf P H\mathsf P^{-1} = \frac{(\mathsf P P_x\mathsf P^{-1})^2}{2m} + V(\mathsf P X\mathsf P^{-1}) = \frac{P_x^2}{2m} + V(-X).$$

Thus $\mathsf P H\mathsf P^{-1}=H$ exactly when

$$V(-X)=V(X).$$

That is the condition that the potential be even under parity.

2. Continuous symmetry and conservation.

Suppose $S(\alpha)=e^{-i\alpha G/\hbar}$ is an exact unitary symmetry of $H$ for all $\alpha$. Show that $G$ is conserved if it has no explicit time dependence.

Solution

Exact invariance gives

$$S(\alpha)HS(\alpha)^\dagger=H.$$

Differentiating at $\alpha=0$ yields

$$[G,H]=0.$$

The expectation-value equation for an observable with no explicit time dependence is

$$\frac{d}{dt}\langle G\rangle = \frac{i}{\hbar}\langle[H,G]\rangle.$$

Since $[H,G]=0$, the expectation value is constant in time.

3. Axial symmetry of a spin Hamiltonian.

Let

$$H = aI+bS_z.$$

Which spin rotations are exact symmetries?

Solution

Rotations about the $z$ axis are generated by $S_z$:

$$U_z(\theta)=e^{-i\theta S_z/\hbar}.$$

Since $H$ is a function of $S_z$,

$$[H,S_z]=0,$$

so rotations about $z$ are exact symmetries. For nonzero $b$, the Hamiltonian does not generally commute with $S_x$ or $S_y$, because

$$[S_z,S_x]=i\hbar S_y, \qquad [S_z,S_y]=-i\hbar S_x.$$

Thus full rotational symmetry is reduced to axial symmetry.

4. State symmetry versus Hamiltonian symmetry.

Give an example where the Hamiltonian has parity symmetry but a state does not.

Solution

Take an even potential, such as a harmonic oscillator potential $V(X)=m\omega^2X^2/2$. The Hamiltonian commutes with parity.

A localized wave packet centered at $x_0\ne0$ is not generally a parity eigenstate, because parity maps it to a packet centered near $-x_0$. The Hamiltonian has parity symmetry, but that particular state does not.