Wigner's Theorem: Why Quantum Symmetries Are Unitary

Wigner's theorem (1931) is one of the foundational insights in quantum physics. Starting from only the observable properties of quantum states (their projectivity and the Born rule), it proves that any symmetry of a quantum system must be implemented by a unitary or antiunitary operator on Hilbert space. This is how representation theory enters quantum mechanics.

Prerequisites: familiarity with algebra basics (in particular operators, inner products, and Hilbert spaces) is recommended. See also: Noether's Theorem.

1. Definitions

To describe quantum mechanics, one works in a complex Hilbert space ℋ with inner product ⟨·|·⟩ (see algebra basics for the definitions of inner product and Hilbert space).

1.1 Unitary and Antiunitary Operators

Remember that an operator on ℋ is a map ℋ → ℋ (see algebra basics).

A unitary operator U: ℋ → ℋ is a linear operator satisfying ⟨Uψ|Uφ⟩ = ⟨ψ|φ⟩ for all ψ, φ ∈ ℋ.

which means that a unitary operator preserves lengths, angles (, and probabilities!) under its inner product. It preserves the full structure of the state space. Note that in superposition states (e.g. neutrino oscillations), unitary time evolution changes the relative phases between components, which does change the probabilities of individual measurement outcomes over time. What is preserved is the total inner product structure, not the probability of each specific outcome.

An antiunitary operator U: ℋ → ℋ is an antilinear operator (meaning U(cψ) = c*U(ψ), it conjugates scalars) satisfying ⟨Uψ|Uφ⟩ = ⟨ψ|φ⟩* for all ψ, φ ∈ ℋ.

An antiunitary operator also preserves transition probabilities (since |⟨Uψ|Uφ⟩|² = |⟨ψ|φ⟩*|² = |⟨ψ|φ⟩|²), but it complex-conjugates the inner product. Time reversal for example. Antilinearity is briefly brushed in representation theory where we mentioned complex conjugation z → z* to be antilinear. Note that this meant it can not be a complex linear representation.

1.2 Rays and the Born Rule

In quantum mechanics we often describe pure states as vectors ψ ∈ ℋ. These are not the classical vectors over ℝ³: we operate in the Hilbert space and its vectors are wavefunctions describing probabilities of quantum states. The universe itself is inherently undecided about what state a quantum object is in, and according to the Born interpretation of quantum mechanics it only decides what state something truly is in to the absolute minimum of information you force it to reveal by measuring it. When we measure, the state collapses to an eigenstate of the measured observable (still a vector in ℋ), and the measurement result is a real number (the corresponding eigenvalue).

For the Born rule we observe that the global phase of a vector (wavefunction) leaves the physical meaning of the state unchanged (multiplying the entire state by e^iθ changes no measurement outcome). Note: relative phases between components of a superposition do matter and affect probabilities, but the overall global phase does not. We call this group of underlying vectors an equivalence class because they express an equivalence relation. We call these classes rays:

A ray is the equivalence class [ψ] := {c · ψ | c ∈ U(1)}, where U(1) = {e^iθ : θ ∈ ℝ} is the set of complex numbers with absolute value 1. Two vectors ψ and e^iθψ represent the same physical state.

The set of all rays forms the projective Hilbert space Pℋ. What is concretely observable are transition probabilities between rays:

The transition probability between two rays is: p([ψ₁], [ψ₂]) = ⟨ψ₁|ψ₂⟩ ⟨ψ₂|ψ₁⟩ / (⟨ψ₁|ψ₁⟩ ⟨ψ₂|ψ₂⟩)

For normalized states this simplifies to p([ψ₁], [ψ₂]) = |⟨ψ₁|ψ₂⟩|². Note the absolute value squared: this makes the constant phase shifts of rays physically irrelevant for the actual states. The product ⟨ψ₁|ψ₂⟩ ⟨ψ₂|ψ₁⟩ = |⟨ψ₁|ψ₂⟩|² because ⟨ψ₂|ψ₁⟩ = ⟨ψ₁|ψ₂⟩* (because of the conjugate symmetry of the inner product).

2. Quantum Symmetries

A quantum symmetry is a bijection S: Pℋ → Pℋ on the projective space that preserves transition probabilities: p(S(·), S(·)) = p(·, ·).

The · is a placeholder for any input ray. This is a bijection (see algebra basics, section 5) on rays, not on vectors. It preserves only the absolute value squared of the inner product, not the inner product itself. This is a weaker condition than unitarity. Wigner's theorem is that despite this weakness, the only maps that satisfy this condition are projections of unitary or antiunitary operators.

2.1 Example: Unitary Operators Induce Quantum Symmetries

Any unitary operator U: ℋ → ℋ induces a quantum symmetry via projection: [U]: Pℋ → Pℋ, [ψ] ↦ [U(ψ)]. This preserves transition probabilities since |⟨Uψ|Uφ⟩|² = |⟨ψ|φ⟩|².

2.2 Example: Why Antiunitary Operators Also Work

If U is antiunitary, the projection [U] still preserves transition probabilities: |⟨Uψ|Uφ⟩|² = |⟨ψ|φ⟩*|² = |⟨ψ|φ⟩|². We find that there are more quantum symmetries than just those from unitary operators.

3. Wigner's Theorem

Wigner's Theorem Every quantum symmetry S: Pℋ → Pℋ is the projection [U] of a map U: ℋ → ℋ which is either:

Unitary: U is complex-linear and ⟨Uψ|Uφ⟩ = ⟨ψ|φ⟩ for all ψ, φ ∈ ℋ, or
Antiunitary: U is conjugate-linear and ⟨Uψ|Uφ⟩ = ⟨ψ|φ⟩* for all ψ, φ ∈ ℋ.

The operator U is unique up to a phase factor: if U' also implements S, then U' = e^iθU for some θ ∈ ℝ.

The phase ambiguity reflects the ray structure: the same ray can be represented by any vector in it, so operators that differ only by a phase implement the same transformation on rays.

A geometric proof via the Fubini-Study metric on Pℋ is given in Freed 2012. A generalization to non-pure (mixed) quantum states is given in Moretti 2017.

4. The Group of Quantum Symmetries

Now take T to be a real structure on ℋ (a complex-antilinear involution, meaning T² = id and T is antilinear). Write U(ℋ) for the unitary group and U_anti(ℋ) for the set of antiunitary operators.

The semidirect product (see algebra basics, section 1 for groups) of the unitary group with C₂(t) = {id, T} gives: U(ℋ) ⋊ C₂(t) ≅ U(ℋ) ∪ U_anti(ℋ) via (U, id) ↦ U and (U, T) ↦ U ∘ T.

This passes to projective groups by excluding the phase ambiguity U(1)={e^iθ : θ ∈ ℝ}: PU(ℋ) := U(ℋ) / U(1), PU_anti(ℋ) := U_anti(ℋ) / U(1). The full group of quantum symmetries is PU(ℋ) ⋊ C₂(t).

5. Projective Representations and the Phase Ambiguity

The operator U is implementing a symmetry only determined up to a phase. The symmetry group G therefore is acting on a quantum system which gives rise to a projective representation rather than an ordinary representation (see representation theory). This means the composition law holds only up to a phase:

A projective representation of a group G on ℋ is a map U: G → U(ℋ) satisfying U(g₁) U(g₂) = ω(g₁, g₂) U(g₁g₂), where ω(g₁, g₂) ∈ U(1) is a phase factor (called a 2-cocycle).

This is a homomorphism into the projective unitary group (!): G → PU(ℋ) and not into the unitary group G → U(ℋ).

A bit more advanced: For many groups (e.g., simply connected Lie groups), the cocycle can be removed by passing to a central extension of G. For example, the projective representations of SO(3) (the rotation group) correspond to ordinary representations of its double cover SU(2). This is the mathematical origin of half-integer spin: SO(3) only has integer-spin representations, but SU(2) can also have half-integer spins (spins are just quantum numbers defining quantum states).

6. Unitary vs Antiunitary: When Does Each Occur?

For continuous symmetries (Lie groups), the connected component of the identity always gives unitary operators. This is because a continuous path from the identity to an antiunitary operator would have to cross a discontinuity in linearity type (linear vs antilinear), which is impossible by continuity.

Antiunitary operators arise only from:

Discrete symmetries (like time reversal T)
Disconnected components of symmetry groups

Example: time reversal must be antiunitary because it must reverse the sign of momentum (p → -p) while keeping position unchanged (x → x). The canonical commutation relation [x, p] = iℏ must be preserved (see newtonian mechanics, Hamiltonian formalism). Under time reversal T: T [x, p] T^-1 = [Tx, Tp] = [x, -p] = -[x, p] = -iℏ. A unitary operator gives T(iℏ)T^-1 = iℏ (preserving i), which is contradicting -iℏ. An antiunitary operator gives T(iℏ)T^-1 = -iℏ (conjugating i to -i), which fits the requirements.

7. CPT Symmetries and the 10-Fold Way

In systems with a particle/antiparticle structure, the full set of discrete quantum symmetries involves three involutions:

C (charge conjugation): antilinear, C² = ± id
P (parity): unitary, exchanges particle and antiparticle sectors, P² = id
T (time reversal): antiunitary, T² = ± id

The group of CPT symmetries is C₂(t) × C₂(c) = {id, C, P, T}. NOTE: This group forces the following relation: CP = T

Some call it CPT, some PCT, some TPC symmetry group, etc. it's all the same - this is just up to convention.

To make these symmetries concrete, we work on a graded Hilbert space ℋ_gr = ℋ ⊕ ℋ: two copies of the same Hilbert space stacked together, one for particles, one for antiparticles. A state looks like (ψ₊, ψ_-) where ψ₊ is the particle sector and ψ_- is the antiparticle sector. "Graded" means there is a ℤ₂ labeling (particle/antiparticle) distinguishing the two sectors, and the parity operator P swaps them.

An antiparticle is defined as the CP-conjugate of a particle: apply charge conjugation C and parity P. Since CP = T, this is equivalent to the original particle reversed in time. This is the physical content of the CPT theorem: a particle moving forward in time is equivalent to its antiparticle moving backward in time.

A CPT quantum symmetry is then a lift: you start with an abstract symmetry (e.g. "time reversal exists and squares to -1") and ask whether you can find a concrete operator on ℋ_gr that implements it. Not every abstract symmetry can be lifted, because the signs T² = ± id and C² = ± id constrain which ones work. The classification of which lifts are possible gives the 10-fold way (google Altland-Zirnbauer classification).

The key constraint comes from the (anti)linearity type: for the antiunitary operators T and C, the square T² or C² must be ± id (not just any phase). This is because rescaling an antilinear operator T by a phase does not change T²: (e^iθT)² = e^iθ T e^iθ T = e^iθ e^-iθ T² = T² (since T conjugates the phase). So T² = ω id with ω* = ω, hence ω ∈ {+1, -1}.

The classification of which subgroups of CPT can be lifted, and with which signs for T² and C², yields exactly 10 classes-the 10-fold way (Altland-Zirnbauer classification). These 10 classes correspond to:

The 10 types of random matrix ensembles in condensed matter physics
The 10 flavors of topological K-theory (KU, KO, KR in various degrees)
The 10-fold classification of topological phases of matter
Cartan's 1926 classification of symmetric spaces

The traditional labels for these 10 classes are: A, AI, AII, AIII, BDI, CII, D, C, DIII, CI (following Cartan's notation for symmetric spaces).

8. From Wigner to Particles: The Wigner Classification

Wigner himself used his theorem as the starting point for classifying elementary particles. The logic: in special relativity, the symmetry group of spacetime is the Poincare group (translations, rotations, and Lorentz boosts). The connected component of the Poincare group is a Lie group, so by section 6, its representations are unitary. By Wigner's theorem, particles in a relativistic quantum theory must furnish projective unitary representations of this group.

Wigner's classification: the irreducible unitary representations of the Poincare group are labeled by two numbers:

Mass m ≥ 0: a continuous parameter
Spin s = 0, 1/2, 1, 3/2, ...: a discrete parameter (half-integer because we pass to the double cover SL(2, ℂ))

A particle IS an irreducible unitary representation. The electron is the representation with m = m_e, s = 1/2. The photon is the massless representation with s = 1 (technically helicity ±1). This is representation theory at its most powerful: the abstract classification of irreducible representations directly tells you what kinds of particles can exist. This was used to predict many particles before they were experimentally confirmed.

Antiunitarity now enters not in the classification itself, but in the connection between particles and antiparticles. As described in section 7, antiparticles are CP-conjugates, which because of CP = T is an antiunitary operation. In short: unitary representations classify particles, antiunitary operations (time reversal, CPT) relate them to their antiparticles.

9. Summary: Why Wigner's Theorem Matters

Wigner's theorem is the reason representation theory enters physics. The logical chain:

Quantum states are rays, not vectors (Born rule)
Symmetries must preserve transition probabilities between rays
Wigner: such symmetries are necessarily (anti)unitary operators, unique up to phase
Therefore: symmetry groups act by projective (anti)unitary representations on the Hilbert space
Classifying these representations = classifying the physics (particles, phases of matter, selection rules)

Wigner's theorem introduces particles into particle physics: unitary representations of continuous symmetries classify them, antiunitary operations from discrete symmetries connect them to their antiparticles.