Wigner's theorem

Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT act on the Hilbert space of states.

According to the theorem, any symmetry acts as an unitary or antiunitary transformation in the Hilbert space. More precisely, it states that a surjective (not necessarily linear) map $T:H\rightarrow H$ on a complex Hilbert space $H$ that satisfies

$|\langle Tx,Ty\rangle|=|\langle x,y\rangle|$

for all $x,y \in H$ has the form $Tx=\varphi(x)Ux$ for all $x\in H$, where $\varphi:H\rightarrow \mathbb{C}$ has modulus one and $U:H\rightarrow H$ is either unitary or antiunitary.

Symmetry in quantum mechanics

In quantum mechanics and quantum field theory, the quantum state that characterizes one or more particles or fields is a vector (ket) in a complex Hilbert space. Any symmetry operation, for example "translate all particles and fields forward in time by five seconds", or "Lorentz transform all particles and fields by a 5 m/s boost in the x direction", corresponds to an operator T on that Hilbert space. This operator T must be bijective because every quantum state must have a unique corresponding transformed state and vice-versa. Also, the probability of finding a system in state $x$ when it is initially in state $y$ is given by $|\langle x,y\rangle|^2$. Since T is a symmetry operation, the probability of finding the system in state $Tx$ when it is initially in state $Ty$ must be the same; therefore $|\langle Tx,Ty\rangle|^2=|\langle x,y\rangle|^2$. It follows that T satisfies the hypotheses of Wigner's theorem.

Thus, according to Wigner's theorem, T is either unitary or anti-unitary. In the two examples above (time translations and Lorentz boosts), T corresponds to a unitary symmetry operator. The time-reversal symmetry operator is a famous example of an anti-unitary symmetry operator.

References

• Bargmann, V. "Note on Wigner's Theorem on Symmetry Operations". Journal of Mathematical Physics Vol 5, no. 7, Jul 1964.
• Molnar, Lajos. "An Algebraic Approach to Wigner's Unitary-Antiunitary Theorem". arXiv:math/9808033

See also

• Particle physics and representation theory