Wigner's Theorem: Unitary and Anti-Unitary Symmetries

The Strategy: A Constructive Proof

The proof of Wigner's theorem is constructive. We begin with a symmetry transformation, which is a map $T$ that acts on the rays of the projective Hilbert space $\mathcal{P}(\mathcal{H})$. Our task is to build a corresponding operator $U$ that acts on the vectors of the Hilbert space $\mathcal{H}$ itself.

The starting point and sole constraint is the definition of symmetry: for any two rays $R_{\psi },R_{\varphi }$, and their images $T(R_{\psi }),T(R_{\varphi })$, the transition probability is preserved. If we pick representative normalized vectors $|\psi ⟩\in R_{\psi }$, $|\varphi ⟩\in R_{\varphi }$, $|{\psi }^{\prime }⟩\in T(R_{\psi })$, and $|{\varphi }^{\prime }⟩\in T(R_{\varphi })$, this means:

Our goal is to show that the operator $U$ which connects the vectors, such that $U|\psi ⟩$ is a representative of the ray $T(R_{\psi })$, must be either linear and unitary or anti-linear and anti-unitary.

A clear, step-by-step outline of this proof, closely following the reasoning developed by Steven Weinberg, is provided in the following paper from the journal Resonance. We will use this as our guide.

Step 1: Preserving the Structure of the Space

The first logical step is to confirm that the symmetry transformation $T$ preserves the fundamental structure of the Hilbert space. Specifically, we must show that a complete orthonormal basis (CONB) is mapped to another complete orthonormal basis.

This is intuitive: if the "scaffolding" of the space were not preserved—if orthogonal states were mapped to non-orthogonal ones, or if the transformed basis was no longer complete—the geometry of the space would be distorted, and it would be difficult to see how all transition probabilities could be maintained.

The proof of this is quite direct. Orthonormality is preserved because the inner product modulus between any two distinct basis vectors is ${\delta }_{kl}$, and this must be preserved under the transformation. Completeness is preserved because if the new basis were not complete, you could find a vector orthogonal to all of its elements, which would imply the existence of a pre-transformation vector orthogonal to all original basis elements—a contradiction.

To see the formal argument, please read Step 1 in the article.

Steps 2 & 3: The Emergence of the Linear/Anti-Linear Dichotomy

Now we come to the heart of the proof. We have a CONB $\{|{\psi }_k⟩\}$ and its image $\{|{\psi }_k^{\prime }⟩\}$, which is also a CONB. We can define an operator $U$ such that $U|{\psi }_k⟩=|{\psi }_k^{\prime }⟩$. But what is the action of $U$ on an arbitrary superposition, $|\psi ⟩={\sum }_k{c}_k|{\psi }_k⟩$?

Let the transformed state be $|{\psi }^{\prime }⟩=U|\psi ⟩$. Since $\{|{\psi }_k^{\prime }⟩\}$ is a basis, we can write $|{\psi }^{\prime }⟩={\sum }_k{c}_k^{\prime }|{\psi }_k^{\prime }⟩$. The central question is: What is the relationship between the new coefficients $c_k^{\prime }$ and the old ones $c_k$?

To find out, we must use our only weapon: the invariance of transition probabilities. By cleverly choosing states and examining their inner products, we can constrain the relationship between the coefficients.

1. The inner product of $|\psi ⟩$ with a basis vector $|{\psi }_k⟩$ gives $⟨{\psi }_k|\psi ⟩=c_k$. The invariance condition $|⟨{\psi }_k^{\prime }|{\psi }^{\prime }⟩{|}^2=|⟨{\psi }_k|\psi ⟩{|}^2$ immediately tells us that $|c_k^{\prime }{|}^2=|c_k{|}^2$. So, the new coefficients must have the same magnitude as the old ones, differing at most by a phase: $c_k^{\prime }=e^{i{\theta }_k}{c}_k$.
2. To fix the relative phases, we consider a more complex state, like $|{\varphi }_k⟩=(|{\psi }_1⟩+|{\psi }_k⟩)/\sqrt{2}$. By enforcing probability conservation for the inner product $⟨{\varphi }_k|\psi ⟩$, we arrive at a powerful geometric constraint on the ratios of coefficients.

The result of this analysis is that for any $k$, the ratio of coefficients $c_k^{\prime }/c_1^{\prime }$ has only two possibilities:

This is the fundamental bifurcation. It arises directly from the algebraic properties of complex numbers. In the previous lesson, we saw an image illustrating this point, which is worth revisiting now.

The details of this derivation are the most technical part of the proof.

Steps 4 & 5: Enforcing Global Consistency

We've found that for any given state $|\psi ⟩=\sum c_k|{\psi }_k⟩$, the transformation of its coefficients is either effectively linear or anti-linear. But could it be that one state, $|{\psi }_A⟩$, transforms linearly while another, $|{\psi }_B⟩$, transforms anti-linearly under the same symmetry $T$?

The next two steps of the proof show that this is impossible. The choice must be global for the entire Hilbert space.

• Step 4 shows that for a single state vector, all of its coefficients must transform in the same way (i.e., you can't have $c_2^{\prime }/c_1^{\prime }=c_2/c_1$ while $c_3^{\prime }/c_1^{\prime }=(c_3/c_1{)}^{\ast }$). This is done by constructing a test state involving three basis vectors and showing that a mixed transformation leads to a contradiction.
• Step 5 extends this to the entire space, proving that if $|{\psi }_A⟩$ transforms linearly, then so must every other state $|{\psi }_B⟩$. This is also a proof by contradiction, showing that if they transformed differently, the transition probability $|⟨{\psi }_A|{\psi }_B⟩{|}^2$ would not be preserved.

Step 6: The Final Connection to Unitarity

At this point, we have successfully constructed an operator $U$ that represents the symmetry $T$, and we have shown it must be either linear or anti-linear for all states. The final step is to show that these properties, combined with the probability preservation rule, imply that $U$ must be unitary or anti-unitary.

Case 1: U is linear.
The operator preserves the modulus of the inner product: $|⟨U\psi |U\varphi ⟩|=|⟨\psi |\varphi ⟩|$.
Since $U$ is linear, $⟨U\psi |U\varphi ⟩=⟨\psi |U^{†}U|\varphi ⟩$. The condition implies that $U^{†}U$ is an operator that scales every inner product by a phase factor, which must be global and can be shown to be 1. Thus, $U^{†}U=I$, which is the definition of a unitary operator.

Case 2: U is anti-linear.
Here, the operator $U$ also preserves the modulus of the inner product. A similar analysis shows that this requires $⟨U\psi |U\varphi ⟩=⟨\varphi |\psi ⟩=⟨\psi |\varphi {⟩}^{\ast }$. This is precisely the definition of an anti-unitary operator.

The distinction between unitary and anti-unitary operators is not merely a mathematical artifact. Continuous symmetries, which can be built up by composing infinitesimal transformations near the identity, must be unitary. The product of two anti-unitary operators is unitary, so an infinitesimal transformation cannot be anti-unitary. The anti-unitary case is reserved for certain discrete symmetries, most famously time reversal, where the complex conjugation is physically necessary to reverse the sign of the $i$ in the Schrödinger equation, $i\hslash \frac{d}{dt}|\psi ⟩=H|\psi ⟩$.

Conclusion

Today we have dissected the proof of Wigner's theorem, revealing the beautiful logical chain that connects a simple physical principle to a profound mathematical constraint.

Key Takeaways:

• The proof of Wigner's theorem is constructive, building an operator $U$ on the Hilbert space from a symmetry map $T$ on the projective space of rays.
• The argument proceeds in steps: first showing that the basis structure is preserved, then using probability conservation to constrain the transformation of expansion coefficients.
• The core dichotomy between linear and anti-linear operators arises directly from the algebraic structure of complex numbers—specifically, that the only continuous field automorphisms are the identity and complex conjugation.
• This choice is shown to be global, applying to all coefficients and all states under a given symmetry transformation.
• Finally, the linear case is shown to correspond to a unitary operator, while the anti-linear case corresponds to an anti-unitary operator.

You now have a much deeper understanding of why symmetries in quantum mechanics are not arbitrary non-linear maps but are restricted to these two elegant forms. This theorem provides the solid mathematical foundation upon which we can build the entire theory of quantum symmetries.

In our next lesson, we will begin to apply this framework. We will construct the operator for spatial translations, identify its generator as the momentum operator, and prove that the invariance of a system under translations leads directly to the conservation of momentum.