Unitary and antiunitary symmetry operators

Assume a bijection of the rays of a complex Hilbert space preserves all transition probabilities. Wigner's theorem says that it can be represented by a unitary or an antiunitary operator, up to irrelevant phases. A unitary operator is complex linear and preserves inner products. An antiunitary operator is antilinear and conjugates inner products. Both preserve their absolute squares. The theorem concerns the entire state space, not a rule that happens to preserve the norm of one selected preparation.

Antilinearity means that a scalar is conjugated when it passes through the map. In particular, $A(i|\psi\rangle)=-iA|\psi\rangle$. This is different from a linear operator with some negative matrix entries. In a chosen orthonormal basis, any antiunitary map can be written $A=UK$, where $K$ conjugates the components and $U$ is unitary. The matrix describing $U$ and the meaning of $K$ depend on that basis; the antilinear character of $A$ does not.

A continuously connected family starting at the identity stays in the unitary class. For a strongly continuous one-parameter unitary group, a self-adjoint generator describes the family by exponentiation. In finite dimensions this is an ordinary Hermitian matrix; in infinite dimensions its domain matters. The exponent must be dimensionless. A translation parameter measured in metres therefore pairs with momentum divided by Planck's reduced constant, whereas a rotation angle pairs with angular momentum divided by that constant.

The infinitesimal form gives a practical diagnostic. In finite dimensions, expanding the unitary condition to first order forces the generator to equal its adjoint. For unbounded operators the first-order condition instead gives symmetry on the relevant domain; self-adjointness follows from Stone’s theorem for the full strongly continuous group. This does not say that every unitary transformation is a physical symmetry of every Hamiltonian. A spin rotation is a legitimate probability-preserving operation, but a fixed external magnetic field selects a direction and usually prevents that rotation from being a symmetry of the same apparatus. Transforming both the apparatus and the state is a different comparison.

Because states are rays, representative symmetry operators can compose with an extra phase. Such a projective representation is physically consistent: transition probabilities cannot detect the phase on an isolated transformed state. Relative phases between coherently compared operations can still matter. Spin one half supplies an important example, since a full spatial turn returns the ray while changing the sign of a representative spinor. Keeping the vector and ray statements separate avoids an apparent contradiction.

Antiunitarity also has an operational boundary. Ordinary closed-system time evolution generated by a Hamiltonian is unitary, so complex conjugation of an arbitrary unknown input is not another universal Hamiltonian gate. Antiunitary symmetries relate descriptions or possible histories; they need not be directly implementable as deterministic preparation devices. Time reversal is the central physical example. Whether it preserves a particular Hamiltonian requires examining magnetic fields, angular momenta, and all other time-odd quantities rather than only conjugating a displayed column.