Wigner’s Theorem Preview

Wigner’s theorem explains why quantum symmetries are represented on Hilbert space by unitary or antiunitary operators. The input is not a preferred basis, a Hamiltonian, or a demand that inner products themselves be fixed. The input is physical: a symmetry of pure states should preserve transition probabilities between rays.

The theorem is one of the structural reasons quantum mechanics has the symmetry language it does. Ordinary rotations, translations, parity, and internal phase symmetries are represented unitarily. Time reversal is represented antiunitarily. Projective phases appear because the physical states are rays rather than phase-chosen vectors.

The full theorem statement, proof architecture, uniqueness result, and projective-group boundary are in Wigner’s Theorem; the Wigner card is the compact lookup projection. This page retains the physical first encounter needed throughout symmetry and spin.

Informal Statement

Pure states are rays in a complex Hilbert space $\mathcal H$. A ray is written $[\psi]$, meaning all nonzero scalar multiples of $\lvert\psi\rangle$ represent the same physical pure state.

For normalized representatives, the transition probability between two pure states is

$$P(\phi,\psi) = \lvert\langle\phi|\psi\rangle\rvert^2.$$

Wigner’s theorem says, roughly:

If a bijective transformation of pure-state rays preserves all transition probabilities, then it can be implemented on Hilbert-space vectors by either a unitary or an antiunitary operator, up to physically irrelevant phase choices.

In symbols, if a ray map $S$ satisfies

$$P(S\phi,S\psi) = P(\phi,\psi)$$

for all pure states, then one can choose a Hilbert-space operator $\widehat S$ such that

$$S[\psi] = [\widehat S\psi],$$

where $\widehat S$ is either unitary or antiunitary. Different choices of vector representatives can change $\widehat S\psi$ by phases without changing the ray transformation.

Why Transition Probabilities Matter

The transition probability is directly physical. It is the probability that a system prepared in $\lvert\psi\rangle$ is found in the one-dimensional projector onto $\lvert\phi\rangle$:

$$\operatorname{Prob}(\psi\to\phi) = \lvert\langle\phi|\psi\rangle\rvert^2.$$

This number is insensitive to independent phase choices:

$$\lvert \langle e^{i\beta}\phi|e^{i\alpha}\psi\rangle \rvert^2 = \lvert e^{i(\alpha-\beta)} \langle\phi|\psi\rangle \rvert^2 = \lvert\langle\phi|\psi\rangle\rvert^2.$$

The inner product $\langle\phi|\psi\rangle$ itself is not a ray-level observable, because it changes under phase conventions. Its absolute square is the natural ray-level invariant.

This point is why the theorem starts from rays and transition probabilities. Starting instead from vectors and inner products would put in extra convention-dependent structure by hand.

Unitary and Antiunitary Alternatives

A unitary operator $U$ is linear and preserves inner products:

$$\langle U\phi|U\psi\rangle = \langle\phi|\psi\rangle.$$

Therefore it preserves transition probabilities:

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

An antiunitary operator $A$ is antilinear and conjugates inner products:

$$A(a\lvert\psi\rangle+b\lvert\phi\rangle) = a^*A\lvert\psi\rangle+b^*A\lvert\phi\rangle,$$

and

$$\langle A\phi|A\psi\rangle = \langle\phi|\psi\rangle^*.$$

It also preserves transition probabilities:

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

Thus the two possibilities differ in what they do to phases, not in what they do to transition probabilities.

Why Time Reversal Is Antiunitary

Time reversal is the basic reason the antiunitary option is not a mathematical curiosity. The Schrödinger equation contains $i$:

$$i\hbar \frac{d}{dt} \lvert\psi(t)\rangle = H\lvert\psi(t)\rangle.$$

If time reversal maps a solution to another solution by

$$\lvert\psi_T(t)\rangle = \Theta\lvert\psi(-t)\rangle,$$

then the transformed state should satisfy a Schrödinger equation with Hamiltonian $\Theta H\Theta^{-1}$. This works naturally when $\Theta$ is antiunitary, because it conjugates $i$:

$$\Theta i\Theta^{-1} = -i.$$

Using the original equation at time $-t$, one finds

$$i\hbar \frac{d}{dt} \lvert\psi_T(t)\rangle = \Theta H\Theta^{-1} \lvert\psi_T(t)\rangle.$$

If $\Theta$ were complex-linear instead, the sign structure would come out wrong. This is why time reversal cannot be represented by an ordinary unitary operator in the usual complex Hilbert-space formulation.

For a spinless particle in the position representation, one often has

$$(\Theta\psi)(x) = \psi(x)^*.$$

For spin-$1/2$, time reversal also rotates the spinor:

$$\Theta = -i\sigma_y K$$

in a common $\sigma_z$-basis convention. The details and the result $\Theta^2=-I$ are worked out in Time Reversal for Spin-1/2 Particles.

Continuous Symmetries

Most familiar continuous symmetries connected to the identity are represented unitarily:

$$U(\alpha) = \exp \left( -\frac{i\alpha G}{\hbar} \right).$$

The reason is structural. The identity operation is represented by the identity operator, which is linear. A continuous path of symmetry operations starting at the identity cannot jump into the antiunitary component without losing continuity in the usual operator topology. Antiunitary transformations therefore appear most naturally as discrete symmetries or as disconnected components of a larger symmetry group.

This is why translations, rotations connected to the identity, and time evolution are unitary, while time reversal is antiunitary.

Rays and Projective Phases

Wigner’s theorem gives a lift from a ray transformation to a Hilbert-space operator, but the lift is not unique. Multiplying a representative by a phase leaves the ray unchanged:

$$\widehat S\lvert\psi\rangle \quad \text{and} \quad e^{i\alpha(\psi)} \widehat S\lvert\psi\rangle$$

represent the same transformed ray.

When the symmetry transformations form a group $G$, this phase freedom leads naturally to projective representations. Instead of

$$U(g)U(h) = U(gh),$$

one may have

$$U(g)U(h) = e^{i\omega(g,h)} U(gh).$$

The phase factor does not change any single ray, but it can encode real representation-theoretic information. Spin-$1/2$ under rotations is the central elementary example. The quantum-mechanical motivation is developed in Projective Representations.

The bridge from projective phases to, but not identification with, field-theoretic anomalies is From Projective Representations to Anomalies Preview.

What the Theorem Does Not Say

Wigner’s theorem is powerful, but its scope is precise.

• It starts with transformations of pure-state rays, not arbitrary maps on density matrices.
• It assumes preservation of transition probabilities, not merely preservation of norms for a few vectors.
• It identifies unitary or antiunitary implementations up to phases; it does not by itself choose a unique phase convention.
• It does not say every unitary operator is a symmetry of a particular Hamiltonian.
• It does not replace the separate question of whether a Hamiltonian satisfies $SHS^{-1}=H$.
• It does not remove the need to discuss projective phases when a whole symmetry group is represented.

The full mathematical proof uses ray geometry and Hilbert-space assumptions more carefully than a working physics page needs. For most physics applications, the important result is the dichotomy: probability-preserving pure-state symmetries lift to unitary or antiunitary transformations.

Common Mistakes

• Requiring symmetries to preserve $\langle\phi|\psi\rangle$ instead of $\lvert\langle\phi|\psi\rangle\rvert^2$.
• Forgetting that pure states are rays, so vector phases are not physical.
• Ignoring the antiunitary alternative and then treating time reversal incorrectly.
• Treating antiunitary operators as ordinary matrices.
• Assuming Wigner’s theorem alone proves $[H,U]=0$; that is a separate Hamiltonian-symmetry condition.
• Forgetting that group multiplication may be implemented projectively on vectors.

Cross-Links

• Wigner Theorem gives the compact theorem card.
• Quantum Symmetries explains the broader symmetry vocabulary.
• Rays and Global Phase explains why phase-chosen vectors are not the primary pure states.
• Unitary Symmetries develops the ordinary linear case.
• Antiunitary Symmetries develops antilinearity and time reversal.
• Antiunitary Symmetries, First Look gives the mathematical first pass.
• Projective Representations explains phase factors in group composition.
• Time Reversal and Time Reversal for Spin-1/2 Particles are the main antiunitary applications.

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.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Why is the transition probability ray-invariant?

Show that $\lvert\langle\phi|\psi\rangle\rvert^2$ is unchanged under $\lvert\psi\rangle\mapsto e^{i\alpha}\lvert\psi\rangle$ and $\lvert\phi\rangle\mapsto e^{i\beta}\lvert\phi\rangle$.

Solution

Under the phase changes,

$$\langle e^{i\beta}\phi|e^{i\alpha}\psi\rangle = e^{i(\alpha-\beta)} \langle\phi|\psi\rangle.$$

Taking the absolute square removes the phase:

$$\left| e^{i(\alpha-\beta)} \langle\phi|\psi\rangle \right|^2 = \lvert\langle\phi|\psi\rangle\rvert^2.$$

2. Show that antiunitary transformations preserve transition probabilities.

Assume $A$ is antiunitary and

$$\langle A\phi|A\psi\rangle = \langle\phi|\psi\rangle^*.$$

Show that the transition probability is unchanged.

Solution

Compute

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

For any complex number $z$, $\lvert z^*\rvert=\lvert z\rvert$. Therefore

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

3. Why does time reversal conjugate $i$?

Explain in one or two equations why an antiunitary $\Theta$ has $\Theta i\Theta^{-1}=-i$.

Solution

Antilinearity means scalar coefficients are conjugated:

$$\Theta(c\lvert\psi\rangle) = c^*\Theta\lvert\psi\rangle.$$

With $c=i$,

$$\Theta(i\lvert\psi\rangle) = -i\Theta\lvert\psi\rangle.$$

Equivalently, as an operator identity on vectors,

$$\Theta i\Theta^{-1} = -i.$$

4. Separate theorem implementation from Hamiltonian symmetry.

Suppose a ray transformation is implemented by a unitary operator $U$. What extra condition makes it a symmetry of a time-independent Hamiltonian $H$?

Solution

Wigner’s theorem supplies the unitary or antiunitary implementation of a transition-probability-preserving ray map. For $U$ to be a symmetry of the specific Hamiltonian $H$, one also needs

$$UHU^\dagger = H,$$

or equivalently $[U,H]=0$ for a unitary $U$. This is a dynamical condition, not part of the theorem by itself.