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.

For the proof and precise uniqueness statement, see Wigner’s Theorem. The short theorem statement is useful when you need only the assumptions and conclusion.

Symmetry of pure-state rays

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 a surjective linear map that 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 a surjective antilinear map that 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

For a time-independent Hamiltonian, time reversal illustrates why the antiunitary option matters. 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, reversing $t$ would produce $-\Theta H\Theta^{-1}$ as the transformed Hamiltonian. The ordinary time-reversal operation preserves energy, so it uses the antiunitary alternative. A unitary operation that happens to exchange positive and negative eigenvalues of a special finite-dimensional Hamiltonian is a different spectral symmetry.

For a spinless particle with a real scalar potential and no magnetic vector potential, a suitable position-space convention is

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

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

A continuous symmetry connected to the identity is implemented unitarily. A strongly continuous one-parameter subgroup can be lifted to a unitary group of the form

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

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:

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

represent the same transformed ray. For a Hilbert space of dimension at least two, two unitary or antiunitary operators implementing the same ray map differ by one state-independent phase. Assigning arbitrary phases separately to output vectors generally destroys linearity or antilinearity; it is not a new operator implementation.

When unitary 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.

Limits of the conclusion

The theorem assumes a bijection of all pure-state rays preserving every transition probability. Norm preservation alone, or agreement on a few test states, is insufficient. It does not classify arbitrary noisy maps on density matrices.

The theorem also does not test a particular Hamiltonian. That requires the separate condition $SHS^{-1}=H$. Once a group of symmetries is involved, compatible phase choices are a further question, developed in Projective Representations.

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. A map fixes $|0\rangle$ and $|1\rangle$ but sends $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$ to $i|+\rangle$. Can this map be linear? Does it still represent the identity on these three rays?

Solution

Linearity and the first two assignments force $|+\rangle$ to map to itself, so this chosen vector map is not linear. Each output nevertheless lies on the input ray. This separates arbitrary choices of ray representatives from the linear unitary implementation, which here is $I$ (or a single overall phase times $I$).

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.

References

• V. Bargmann, “Note on Wigner’s theorem on symmetry operations,” Journal of Mathematical Physics 5, 862–868 (1964). doi:10.1063/1.1704188.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013. doi:10.1007/978-1-4614-7116-5.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, chapter 4.
• R. Simon, N. Mukunda, S. Chaturvedi, and V. Srinivasan, “Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics,” arXiv:0808.0779 [quant-ph], 2008. Author manuscript.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.