Approximate Symmetry

An approximate symmetry is a transformation that is not an exact symmetry of the full Hamiltonian, but whose breaking is small enough to organize states, spectra, selection rules, or dynamics in a specified regime.

The minimal model is

$$H = H_{\mathrm{sym}} + \epsilon V_{\mathrm{break}},$$

where

$$U(g)H_{\mathrm{sym}}U(g)^{-1} = H_{\mathrm{sym}},$$

but

$$U(g)V_{\mathrm{break}}U(g)^{-1} \ne V_{\mathrm{break}}.$$

For $\epsilon\ne0$, the symmetry is not exact. It is approximate only when the effects of $\epsilon V_{\mathrm{break}}$ are small compared with the energy gaps, timescales, experimental resolution, or effective-theory accuracy relevant to the question.

Approximate Does Not Mean Exact

The exact symmetry test is binary:

$$U(g)HU(g)^{-1}=H \qquad \text{or} \qquad U(g)HU(g)^{-1}\ne H.$$

Approximate symmetry is not a third logical option between those two statements. It is a controlled use of a nearby exactly symmetric problem. One first identifies the symmetric Hamiltonian $H_{\mathrm{sym}}$, then asks whether the symmetry-breaking correction can be treated perturbatively or neglected within the desired accuracy.

Thus an approximate symmetry must specify:

• the exact symmetry of the reference Hamiltonian;
• the terms that break it;
• the small dimensionless parameter;
• the energy, time, or length scale where the approximation is being used;
• which conclusions remain reliable and which become only approximate.

Without those details, “approximately symmetric” is only a qualitative description.

Approximate Quantum Numbers

When $H_{\mathrm{sym}}$ commutes with a symmetry generator $G$,

$$[H_{\mathrm{sym}},G]=0,$$

eigenstates of $H_{\mathrm{sym}}$ can be labeled by eigenvalues of $G$ or by irreducible-representation labels of the symmetry group. If the full Hamiltonian is

$$H=H_{\mathrm{sym}}+\epsilon V_{\mathrm{break}},$$

then

$$[H,G] = \epsilon[V_{\mathrm{break}},G].$$

The old labels are no longer exact. They remain useful when the breaking-induced mixing between different symmetry sectors is small. For two unperturbed states $\lvert a\rangle$ and $\lvert b\rangle$ with different symmetry labels, a basic diagnostic is

$$\left| \frac{\epsilon\langle b|V_{\mathrm{break}}|a\rangle} {E_a^{(0)}-E_b^{(0)}} \right| \ll 1.$$

If this ratio is small, the state $\lvert a\rangle$ retains mostly its old symmetry character. If the energy denominator is small, even a tiny breaking term can produce strong mixing.

Approximate Conservation

For a generator $G$ with no explicit time dependence, the Heisenberg equation gives

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

If $G$ is exact for $H_{\mathrm{sym}}$ but not for $V_{\mathrm{break}}$, then

$$\frac{d}{dt}\langle G\rangle = \frac{i\epsilon}{\hbar} \langle[V_{\mathrm{break}},G]\rangle.$$

The right side is small only in the relevant states and units. A quantity can be approximately conserved over short or intermediate times but drift over long times. In closed finite systems the expectation value may also oscillate rather than relax. Approximate conservation is therefore a scale statement, not an absolute law.

Degeneracy Changes the Test

Approximate symmetry is most delicate near degeneracy. Suppose $H_{\mathrm{sym}}$ has a degenerate subspace $\mathcal D$ protected by symmetry. Even if $\epsilon$ is small, the first-order problem inside that subspace is

$$P V_{\mathrm{break}} P,$$

where $P$ projects onto $\mathcal D$. One must diagonalize this matrix. The splitting can be first order in $\epsilon$, and the good states may be completely different linear combinations inside $\mathcal D$.

This is why “small perturbation” and “small change in eigenvectors” are not the same statement. A tiny symmetry-breaking field can strongly choose a basis inside an exactly degenerate multiplet. The spectral version of this issue is degeneracy lifting.

Weakly Violated Selection Rules

Exact symmetry can force a matrix element to vanish:

$$\langle f|O|i\rangle=0.$$

If the symmetry is only approximate, the leading symmetric approximation may still give zero, while the full matrix element is small but nonzero:

$$\langle f_{\mathrm{full}}|O|i_{\mathrm{full}}\rangle = O(\epsilon).$$

This is the clean meaning of a weakly allowed transition. The transition is not forbidden by the full Hamiltonian. It is forbidden by a useful idealized symmetry and appears because symmetry-breaking terms mix small components of the allowed symmetry character into the states or operators.

Spin–orbit mixing can weaken a spin selection rule. A static electric field can mix opposite parities and weaken a parity selection rule. Ordinary electrostatic configuration mixing and central spin–orbit coupling preserve parity, so neither by itself opens a transition forbidden by exact parity. If the first nonzero amplitude is of order $\epsilon$, its squared transition strength is typically of order $\epsilon^2$, unless additional cancellations or small denominators change that scaling.

Spin–Orbit Coupling as a Symmetry Reduction

Consider a central spin-independent Hamiltonian

$$H_0 = \frac{\mathbf P^2}{2m} + V(r).$$

If spin is present but uncoupled, $H_0$ has independent orbital and spin rotations. Orbital and spin quantum numbers can be tracked separately.

A spin–orbit term has the form

$$H_{\mathrm{SO}} = \xi(r)\,\mathbf L\cdot\mathbf S.$$

It does not preserve independent rotations of $\mathbf L$ and $\mathbf S$. Instead, it preserves simultaneous rotations generated by

$$\mathbf J=\mathbf L+\mathbf S.$$

For this one-particle central model, $L^2$ and $S^2$ still commute exactly with $H_{\mathrm{SO}}$. Thus $\ell,s,j,m_j$ are exact labels; the separately oriented components $L_z,S_z$ generally are not. Inside an exactly degenerate uncoupled multiplet, an arbitrarily small spin–orbit term can select coupled $j,m_j$ eigenvectors, so its size compared with unrelated principal-shell gaps does not make $m_\ell,m_s$ good stationary-state labels.

In many-electron atoms, total $L$ and $S$ can instead be approximate LS-coupling labels because more general spin–orbit interactions mix terms. Spin–orbit coupling separates these models.

Isospin as a Preview Example

Isospin is a classic approximate internal symmetry in nuclear and particle physics. In an idealized strong-interaction model, the proton and neutron are treated as two states of a doublet, and the strong interaction is approximately invariant under rotations in this internal space.

The symmetry is not exact. Electromagnetism distinguishes proton from neutron, and the underlying up and down quark masses are not exactly equal. Nevertheless, isospin organizes many nuclear and hadronic patterns well enough to be useful.

Here isospin illustrates an internal approximate symmetry: its usefulness must be judged against electromagnetic and quark-mass effects for the process of interest.

Molecular and Effective Symmetries

Molecular spectra often use idealized symmetries: rigid-rotor rotations, inversion or reflection symmetries, separated electronic and nuclear motion, and approximate angular-momentum couplings. Real molecules have centrifugal distortion, vibration-rotation coupling, spin-rotation coupling, hyperfine interactions, tunneling effects, and environmental perturbations.

One must distinguish an approximate energy formula from an approximate symmetry. A centrifugal-distortion Hamiltonian $H=BJ^2-D(J^2)^2$ still commutes with every component of $\mathbf J$; distortion changes the level spacing without breaking overall rotation symmetry. For an isolated field-free molecule the full angular momentum remains exact, even when separate rotational, vibrational, electronic, and spin angular momenta are coupled. External fields or anisotropic environments can break that rotation symmetry.

Approximate Versus Other Symmetry Notions

Approximate symmetry should be kept distinct from nearby concepts.

Exact symmetry means exact invariance of the stated Hamiltonian or structure. Explicit symmetry breaking means a term in the full equations violates the candidate symmetry. Approximate symmetry is usually explicit breaking that is small enough to use perturbatively.

Spontaneous symmetry breaking is different: the equations can remain symmetric while physically relevant many-body states or phases fail to display the symmetry in an appropriate limit.

Accidental symmetry, hidden symmetry, dynamical symmetry, and emergent symmetry answer different questions: why an unexpected degeneracy exists, which conserved quantities are not geometrically obvious, how spectra are organized, or why an effective low-energy theory has more symmetry than the microscopic model. The relevant distinction is the kind of transformation and the regime in which it is valid.

Common Mistakes

• Calling a symmetry approximate without identifying the breaking term.
• Treating approximate quantum numbers as exact labels in matrix elements.
• Using nondegenerate perturbation theory when the symmetry-breaking term acts inside a degenerate subspace.
• Saying a selection rule is “slightly violated” without naming the mixing mechanism.
• Confusing small coefficient with small physical effect; small denominators can amplify weak breaking.
• Forgetting the timescale in approximate conservation.
• Treating an effective low-energy symmetry as automatically valid at high energies.

Exercises

1. Approximate conservation.

Let

$$H=H_0+\epsilon V, \qquad [H_0,G]=0.$$

Assume $G$ has no explicit time dependence. Derive the exact expression for $d\langle G\rangle/dt$. If $[V,G]$ is bounded, give a state-independent bound on the change over time $T$.

Solution

The Heisenberg equation gives

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

Since

$$[H,G] = [H_0,G]+\epsilon[V,G] = \epsilon[V,G],$$

we obtain

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

For normalized states, the operator norm gives

$$|\langle G\rangle_T-\langle G\rangle_0| \le\frac{|\epsilon|T}{\hbar}\,\|[V,G]\|.$$

This sufficient bound explicitly exhibits the timescale. For unbounded operators one instead needs domain control and bounds in the relevant states. A small coefficient alone gives no universal long-time conservation claim.

2. Small breaking near a two-level gap.

Consider

$$H = \begin{pmatrix} E_1&\epsilon v\\ \epsilon v& E_2 \end{pmatrix}, \qquad E_2-E_1=\Delta>0,$$

with real $v$. What is the basic condition for the old level labels to remain good approximate labels?

Solution

The mixing angle is small when the off-diagonal coupling is small compared with the level spacing. The diagnostic ratio is

$$\left| \frac{\epsilon v}{\Delta} \right| \ll1.$$

If $\Delta$ becomes comparable to $\epsilon v$, the eigenstates are strong mixtures of the old labels even though $\epsilon$ may be numerically small.

3. Spin–orbit coupling and separate rotations.

Why does a term $\xi(r)\mathbf L\cdot\mathbf S$ preserve total rotations but not independent orbital and spin rotations?

Solution

The scalar product is invariant when $\mathbf L$ and $\mathbf S$ are rotated together. That simultaneous rotation is generated by

$$\mathbf J=\mathbf L+\mathbf S.$$

But an independent rotation of only the orbital variables changes $\mathbf L$ while leaving $\mathbf S$ fixed, so the scalar product generally changes. Similarly, rotating only spin changes $\mathbf S$ while leaving $\mathbf L$ fixed. Thus separate orbital and spin rotations are reduced to the diagonal total-rotation symmetry.

4. Weakly allowed transition.

Suppose an exact symmetry of $H_0$ would make $\langle f|O|i\rangle=0$. If a small perturbation mixes

$$|i_{\mathrm{full}}\rangle = |i\rangle+\epsilon|i_1\rangle+O(\epsilon^2),$$

what order of matrix element can appear?

Solution

Insert the perturbed state:

$$\langle f|O|i_{\mathrm{full}}\rangle = \langle f|O|i\rangle + \epsilon\langle f|O|i_1\rangle + O(\epsilon^2).$$

The leading symmetric matrix element vanishes, so the first possible contribution is

$$\epsilon\langle f|O|i_1\rangle.$$

The transition is therefore weakly allowed at order $\epsilon$ if the mixed component has the required symmetry character.

References

• P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, 2nd ed., NRC Research Press, 1998.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• H. F. Jones, Groups, Representations and Physics, 2nd ed., CRC Press, 1998.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• 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.