# The definition of the weaker notion of symmetry in the sense of Wigner's theorem

The weaker notion of symmetry, in the sense of Wigner's theorem, is a transformation on the states that leave all quantum mechanical amplitudes invariant. This tells that such transformations are represented by Unitary operators. See the answer here by @ACuriousMind.

Now, instead of demanding that all quantum mechanical amplitudes are left unchanged, can we not define this same weaker notion of symmetry as those transformations that leave the norm of all states vectors invariant? That too will lead to the unitarity of the transformation operators. Will that be less rigorous/useful or wrong?

• What do you mean for "amplitude", $|\langle \psi|\phi\rangle|^2$ or $\langle \psi|\phi\rangle$? for unit vectors $\psi,\phi$. Wigner theorem deals with the former. Sep 8 '20 at 9:10

It is simply non-physical.

Wigner symmetries deal with rays $$[\psi]$$, i.e., non-vanishing vectors of the Hilbert space up to multiplicative factors.

$$[\psi] := \{a\psi \:|\: a \in \mathbb{C} \setminus \{0\}\}\:, \quad \psi \in H \setminus \{0\}$$ Transition probabilities of couples of rays are defined as $$P([\psi], [\psi']) := \frac{|\langle\psi|\psi'\rangle|^2}{||\psi||^2\: ||\psi'||^2}\:.$$ It is evident that $$P([\psi], [\psi'])$$ does not depend on the choice of the representative in each ray of the couple above.

Wigner Theorem: If a map from rays to rays $$\chi : [\psi] \mapsto [\psi']$$ is bijective and preserves transition probabilities, $$P([\psi], [\phi])= P([\psi'], [\phi'])$$ then it can be produced by working on the vectors of the Hilbert space by means of a unitary or anti unitary map $$U_\chi :H \to H$$ In other words $$[\psi'] = [U_\chi \psi]\:.$$

Remarks (1) Probability transitions and not complex ampitudes are used in the hyptheses of Wigner's theorem. This is a crucial physical fact, since we directly measure probabilities. This is also the door through which anti-unitary transformations enter the theory.

(2) Wigner symmetries are by definition the maps in the hypothesis of the Wigner theorem: bijective maps transforming rays to rays that preserve the transition probabililities.

(3) It is clear that Wigner symmetries do not consider the norms of vectors, since norms are irrelevant when defining the notion of ray.

(4) I think the name "weaker" symmetry is misleading (I just realized that this terminology in your question arises from an ansewr by @ACuriousMind, but that is not the standard terminology). There is only one notion of quantum symmetry according to Wigner (and other notions more or less equivalent according to other persons like Kadison). There is, in fact, a more rigid definition concerning dynamics. In that case the Wigner symmetry is said to be a dynamical (Wigner) symmetry.

A map from the Hilbert space to the Hilbert space that preserves the norms is a much stronger requirement than a Wigner symmetry in a sense, since it deals with generic norms. It sees the norms differently form Wigner symmetries.

And also it is quite non-physical since, in QM, we cannot physically distinguish between two vectors in the Hilbert space that are different just due to a non-vanishing multiplicative constant. We can only handle probabilities and probabilities are independent of the norms of the vectors that represent the states. (That is equivalent to saying that we only use unit-norm vectors up to phases to represent pure quantum states).

In summary, Wigner's definition is disjoint with an apparent alternative/equivalent definition where one requires the preservation of norms. This is not a good definition of symmetry as it deals with unphysical objects.

The weaker notion will also allow for anti-unitary operators, such as time-reversal and charge conjugation:

https://en.wikipedia.org/wiki/T-symmetry#Time_reversal_in_quantum_mechanics

These transform the amplitude to its complex conjugate, keeping the norm the same. They can be very interesting, such as in the CPT theorem in relativistic QFT.