Wigner's Theorem and Quantum Symmetries

1. What is a Symmetry? From Physical Principles to a Mathematical Group

Before diving into the quantum specifics, let's build the concept of a symmetry from the ground up. A symmetry is an operation that can be performed on a system that leaves it, in some essential way, unchanged. For a set of such operations to be a useful and self-consistent description of a system's symmetries, it should satisfy a few common-sense requirements.

The following video provides an excellent motivation for how these physical requirements naturally lead to the mathematical structure of a group.

As the video explains, any collection of symmetry operations on a physical system must form a group. The labels for these operations—$g_1,g_2,\dots$—form an abstract group $G$. For example, the group could be the real numbers under addition, $(\mathbb{R},+)$, which describes continuous time translation, or a finite group like $O_h$ for an octahedral crystal.

The question now becomes: how do these abstract group elements $g\in G$ act on the quantum states $|\psi ⟩$ in a Hilbert space $\mathcal{H}$?

2. Symmetry in Quantum Mechanics: Preserving Probabilities

In quantum mechanics, all physically measurable information about a system is encoded in probabilities. The most fundamental of these are transition probabilities. The probability that a system prepared in state $|\varphi ⟩$ will be found in state $|\psi ⟩$ is given by $|⟨\psi |\varphi ⟩{|}^2$.

A symmetry transformation is a change in our point of view (e.g., rotating our apparatus) that does not change the results of any possible experiment. Therefore, the core physical requirement for a symmetry in quantum mechanics is that it must preserve all transition probabilities.

Furthermore, we must remember that a physical state is not represented by a single vector in Hilbert space, but by a ray—an equivalence class of vectors $\{e^{i\alpha }|\psi ⟩\}$ where $|\psi ⟩$ is a normalized vector and $\alpha$ is any real number. All vectors in a ray describe the same physical state.

This leads us to a precise definition of a symmetry transformation.

To summarize the key definition from the reading:
A symmetry transformation is a bijection $T$ on the set of rays in a Hilbert space such that for any two rays $R_1,R_2$ and any choice of normalized vectors $|{\psi }_1⟩\in R_1$ and $|{\psi }_2⟩\in R_2$, the transition probability is preserved:

where $|{\psi }_1^{\prime }⟩\in T(R_1)$ and $|{\psi }_2^{\prime }⟩\in T(R_2)$.

At this point, we've placed no other constraints on $T$. It could, in principle, be a very complicated, non-linear map on the Hilbert space. The set of all invertible maps on a Hilbert space is enormous. Is nature really this complicated?

3. Wigner's Theorem: From Infinite Possibilities to Two Choices

This is where Eugene Wigner's profound insight comes into play. In 1931, he proved a theorem that dramatically narrows the possibilities for what a symmetry transformation can be.

As explained in the video, Wigner's theorem states:

Wigner's Theorem: Any symmetry transformation $T$ (a map on rays that preserves transition probabilities) can be represented by an operator $U$ acting on the Hilbert space that is either linear and unitary or anti-linear and anti-unitary.

Let's formally define these terms, which should be familiar from your studies in linear algebra and quantum mechanics. An operator $U$ is:

• Linear and Unitary if for any vectors $|\psi ⟩,|\varphi ⟩$ and complex scalars $\alpha ,\beta$:

  • Linearity: $U(\alpha |\psi ⟩+\beta |\varphi ⟩)=\alpha U|\psi ⟩+\beta U|\varphi ⟩$
  • Unitarity: $⟨U\psi |U\varphi ⟩=⟨\psi |\varphi ⟩$

• Anti-linear and Anti-unitary if for any vectors $|\psi ⟩,|\varphi ⟩$ and complex scalars $\alpha ,\beta$:

  • Anti-linearity: $U(\alpha |\psi ⟩+\beta |\varphi ⟩)={\alpha }^{\ast }U|\psi ⟩+{\beta }^{\ast }U|\varphi ⟩$
  • Anti-unitarity: $⟨U\psi |U\varphi ⟩=⟨\varphi |\psi ⟩=⟨\psi |\varphi {⟩}^{\ast }$

This result is truly remarkable. The simple, physical requirement of preserving probabilities forces symmetry operations to be represented by these highly structured operators. All the wild, non-linear possibilities are ruled out.

Given your mathematical background, you might be interested in the proof of this theorem. It's an elegant piece of mathematical physics. The article by Rajeev (resource LINK, Section 3) and another excellent text by K. Schoutens (resource LINK, Section 3.2) both provide detailed, step-by-step proofs following Weinberg's classic argument. I encourage you to review one of them at your convenience; it's a very rewarding exercise.

4. The Significance of Wigner's Theorem

Wigner's theorem is not just a mathematical curiosity; it is the foundation upon which the entire theory of symmetry in quantum mechanics is built.

a) Justifying Unitary Representations

The theorem provides the fundamental justification for why we represent symmetry groups with unitary representations. When we say a quantum system "has a symmetry group $G$," what we formally mean is that there exists a homomorphism $\rho$ from the abstract group $G$ into the group of unitary (and anti-unitary) operators acting on the system's Hilbert space $\mathcal{H}$.

This map, $\rho$, is the representation. It assigns a specific unitary operator $\rho (g)$ to each abstract symmetry element $g\in G$. Wigner's theorem guarantees that such a representation is the only way to realize the symmetry while respecting the probabilistic nature of quantum mechanics.

This next clip makes this connection explicit.

b) The Role of Anti-unitary Operators

What about the anti-unitary case? This is not a mere mathematical footnote. It is physically essential. While most symmetries, particularly continuous ones like rotations and translations, are represented by unitary operators, certain discrete symmetries are not. The most famous example is time reversal.

Why must time reversal be anti-unitary? Consider the time-dependent Schrödinger equation: $i\hslash \frac{d}{dt}|\psi ⟩=H|\psi ⟩$. A time-reversal operator $\mathcal{T}$ should change $t\to -t$. For the equation to remain covariant, $\mathcal{T}$ must be anti-linear to flip the sign of $i$. We will explore this in detail in a future lesson, but Wigner's theorem is what opens the door for this possibility. The paper by Rajeev (resource LINK, Section 5) gives a concise argument showing that time-reversal must be anti-unitary to preserve the Hamiltonian.

Conclusion

Today, we've established the fundamental rulebook for symmetries in quantum mechanics. Starting from the intuitive physical idea that symmetries must preserve the outcomes of experiments (i.e., transition probabilities), we arrived at a powerful and restrictive conclusion.

Key Takeaways:

• A symmetry transformation in quantum mechanics is a mapping on the space of physical states (rays) that preserves all transition probabilities.
• Wigner's Theorem states that any such transformation must be represented by an operator on the Hilbert space that is either linear and unitary or anti-linear and anti-unitary.
• This theorem is profoundly significant because it justifies the use of unitary representations to describe symmetry groups in quantum mechanics. A system has a symmetry $G$ if there is a homomorphism from $G$ into the group of unitary operators $U(\mathcal{H})$.
• The anti-unitary case is not a mathematical artifact but is physically necessary to describe symmetries like time reversal.

With Wigner's theorem as our foundation, we now have a solid understanding of why unitary operators are the language of symmetry in the quantum world. In our next lesson, we will begin to explore the direct consequences of this, starting with the deep connection between the symmetry group of a Hamiltonian and the degeneracy of its energy levels.