Quantum Symmetries

A quantum symmetry is a transformation of physical states that preserves physical predictions. Symmetries usually come in groups because transformations can be composed and undone, and a group action specifies what those transformations act on. For pure states, the essential prediction-preserving condition is preservation of transition probabilities:

$$\lvert\langle\phi|\psi\rangle\rvert^2 \quad \text{is unchanged}.$$

In ordinary Hilbert-space quantum mechanics, Wigner’s theorem says that such transformations are represented on state vectors by unitary or antiunitary operators, up to physically irrelevant phases. This page explains how that statement is used, not the full proof.

Physical States Are Rays

Pure states are rays rather than individual vectors. If

$$\lvert\psi'\rangle=e^{i\alpha}\lvert\psi\rangle,$$

then $\lvert\psi'\rangle$ and $\lvert\psi\rangle$ represent the same physical pure state. A quantum symmetry therefore acts first on rays. A representative vector is chosen only after that.

This is why phases appear naturally in symmetry theory. A transformation can act exactly on physical rays while acting on chosen vectors only up to phases.

Preserving Transition Probabilities

For normalized pure states, the transition probability from $\lvert\psi\rangle$ to $\lvert\phi\rangle$ is

$$P_{\psi\to\phi} = \lvert\langle\phi|\psi\rangle\rvert^2.$$

A symmetry $S$ should preserve this number:

$$\lvert\langle S\phi|S\psi\rangle\rvert^2 = \lvert\langle\phi|\psi\rangle\rvert^2.$$

Unitary transformations preserve the inner product itself. Antiunitary transformations preserve its complex conjugate. Both preserve its absolute square.

Transformations of States

For a unitary symmetry $U$,

$$\lvert\psi\rangle \longmapsto U\lvert\psi\rangle.$$

For an antiunitary symmetry $T$, the same expression is often written formally,

$$\lvert\psi\rangle \longmapsto T\lvert\psi\rangle,$$

but $T$ is antilinear:

$$T(a\lvert\psi\rangle+b\lvert\phi\rangle) = a^*T\lvert\psi\rangle+b^*T\lvert\phi\rangle.$$

This distinction is essential for time reversal and for any operation involving complex phases.

Transformations of Observables

For a unitary transformation, observables transform as

$$A\longmapsto UAU^\dagger.$$

For a general unitary or antiunitary symmetry $S$, the compact expression is

$$A\longmapsto SAS^{-1}.$$

This preserves expectation values when states and observables are transformed consistently:

$$\langle S\psi|SAS^{-1}|S\psi\rangle = \langle\psi|A|\psi\rangle,$$

with the antiunitary case understood using antilinearity.

Transformations of Hamiltonians

A transformation is a symmetry of a time-independent Hamiltonian when

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

For a unitary $U$, this is equivalent to

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

For a continuous unitary family $U(\alpha)=e^{-i\alpha G/\hbar}$, the infinitesimal version is

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

This is the bridge from symmetry to conserved quantities.

Symmetry Versus Change of Representation

Do not confuse a physical symmetry with a mere change of basis. A basis change rewrites the same state and operators in different coordinates. A physical symmetry compares a system with a transformed system and asks whether the physical predictions are preserved.

The formulas can look similar because both use unitary transformations. The interpretation is different. Good pages say which is being done.

Examples

Parity in one dimension acts on a wavefunction by

$$(P\psi)(x)=\psi(-x).$$

It is a symmetry of

$$H=\frac{p^2}{2m}+V(x)$$

only if $V(-x)=V(x)$.

Spatial translations are represented by

$$U(a)=e^{-iaP/\hbar}.$$

They are symmetries of a free particle, but not of a particle in a generic position-dependent potential.

Spin rotations act on spin-$1/2$ states by

$$U(\hat{\mathbf n},\theta) = \exp\left( -\frac{i}{2}\theta\,\hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

They are symmetries only when the Hamiltonian respects the corresponding rotational invariance.

Common Mistakes

• Calling a transformation a symmetry before checking the Hamiltonian or measurement structure.
• Ignoring the ray nature of pure states.
• Treating antiunitary symmetries as if they were ordinary unitary matrices.
• Confusing a basis change with an active physical transformation.
• Forgetting that a symmetry may preserve probabilities while changing phases.

Cross-Links

• Symmetry Principles
• Rays and Global Phase
• Probability Amplitudes
• Active and Passive Transformations
• States, Observables, and Hamiltonians
• Symmetry Groups and Representations
• Wigner’s Theorem Preview
• Unitary Symmetries
• Antiunitary Symmetries
• Projective Representations
• Symmetry Constraints on Hamiltonians
• Exact Symmetry
• Superselection Sectors Preview
• Groups
• Group Actions
• Representations
• Unitary Representations
• Antiunitary Symmetries, First Look

References

• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• V. Bargmann, “Note on Wigner’s theorem on symmetry operations,” Journal of Mathematical Physics 5, 862–868, 1964.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.

Exercises

1. Why is preserving $\lvert\langle\phi\rvert\psi\rangle\rvert^2$ more physically direct than preserving $\langle\phi\rvert\psi\rangle$ itself?

Solution

The absolute square is the transition probability. The inner product itself depends on phase choices for the two ray representatives. Since global phases of state vectors are not physical, the probability is the direct invariant.

2. For parity in one dimension, show that $P^2=I$ when $(P\psi)(x)=\psi(-x)$.

Solution

Apply parity twice:

$$(P^2\psi)(x) = (P\psi)(-x) = \psi(x).$$

Thus $P^2$ acts as the identity on wavefunctions.