Wigner's Theorem and Quantum Symmetries

Physical States as Rays in Hilbert Space

Before we can define a transformation of states, we must be precise about what a "state" is. In our previous work, we've used kets like $|\psi ⟩$ to represent states. However, a crucial subtlety of quantum mechanics is that not every distinct vector in Hilbert space corresponds to a distinct physical state.

All observable information is extracted from probabilities, which are calculated using the Born rule. The transition probability between a state represented by $|\psi ⟩$ and one represented by $|\varphi ⟩$ is:

Notice what happens if we replace $|\psi ⟩$ with $c|\psi ⟩$, where $c$ is any non-zero complex number. The numerator becomes $|c⟨\varphi |\psi ⟩{|}^2=|c{|}^2|⟨\varphi |\psi ⟩{|}^2$, and the denominator acquires a factor of $⟨c\psi |c\psi ⟩=|c{|}^2⟨\psi |\psi ⟩$. The factors of $|c{|}^2$ cancel, leaving the probability unchanged.

This means that the vectors $|\psi ⟩$ and $c|\psi ⟩$ are physically indistinguishable. They represent the same physical state. This leads us to the proper definition of a quantum state:

A physical state corresponds not to a single vector, but to a ray in Hilbert space. A ray is the set of all vectors that are non-zero scalar multiples of each other: $R_{\psi }=\{c|\psi ⟩\mid c\in \mathbb{C},c\ne 0\}$.

The space of all such rays is called the projective Hilbert space, denoted $\mathcal{P}(\mathcal{H})$. Each point in this space corresponds to a unique physical state.

To delve deeper into this crucial concept, the following discussion from Physics Stack Exchange provides an excellent and clear explanation.

Defining Quantum Symmetry

With a clear understanding of physical states as rays, we can now define a symmetry. In classical physics, a symmetry is a transformation of the state space (e.g., phase space) that leaves the laws of physics, and therefore the results of all possible experiments, invariant. The quantum analogue is a transformation on the projective Hilbert space that preserves all possible transition probabilities.

A quantum symmetry transformation is a mapping $T:\mathcal{P}(\mathcal{H})\to \mathcal{P}(\mathcal{H})$ that is a bijection (i.e., it's one-to-one and onto, so it's invertible) and preserves transition probabilities.

This means that if a ray $R_{\psi }$ is mapped to $T(R_{\psi })$ and another ray $R_{\varphi }$ is mapped to $T(R_{\varphi })$, then for any choice of representative vectors $|\psi ⟩\in R_{\psi }$, $|\varphi ⟩\in R_{\varphi }$, $|{\psi }^{\prime }⟩\in T(R_{\psi })$, and $|{\varphi }^{\prime }⟩\in T(R_{\varphi })$, the following equality must hold:

(Assuming normalized vectors for simplicity). This definition is purely physical; it demands that the probability of a system in state $\psi$ being measured to be in state $\varphi$ is the same as the probability of the transformed system being measured in the transformed state.

The following video provides a concise statement of this definition.

Wigner's Theorem

The definition of a symmetry transformation is very general. It does not assume the transformation $T$ is linear. A priori, $T$ could be a highly complex, non-linear map on the Hilbert space. This is where the profound power of Wigner's theorem comes in. It takes this simple, physical requirement of preserving probabilities and imposes an incredibly strong constraint on the mathematical nature of the operator that implements the symmetry on the Hilbert space itself.

Wigner's Theorem: Every symmetry transformation $T$ on the projective Hilbert space $\mathcal{P}(\mathcal{H})$ can be implemented by an operator $U$ acting on the Hilbert space $\mathcal{H}$ such that for any $|\psi ⟩\in R_{\psi }$, the vector $U|\psi ⟩$ is a representative of the transformed ray $T(R_{\psi })$. Furthermore, this operator $U$ must be either:

1. Unitary and linear, or
2. Anti-unitary and anti-linear.

Let's break down these terms:

• A linear operator $U$ satisfies $U(\alpha |\psi ⟩+\beta |\varphi ⟩)=\alpha U|\psi ⟩+\beta U|\varphi ⟩$.
• An anti-linear operator $A$ satisfies $A(\alpha |\psi ⟩+\beta |\varphi ⟩)={\alpha }^{\ast }A|\psi ⟩+{\beta }^{\ast }A|\varphi ⟩$, where ${\alpha }^{\ast },{\beta }^{\ast }$ are the complex conjugates.
• A unitary operator $U$ preserves the inner product: $⟨U\psi |U\varphi ⟩=⟨\psi |\varphi ⟩$. This implies $U^{†}U=I$.
• An anti-unitary operator $A$ satisfies $⟨A\psi |A\varphi ⟩=⟨\psi |\varphi {⟩}^{\ast }=⟨\varphi |\psi ⟩$. An anti-unitary operator can always be written as $A=UK$, where $U$ is a unitary operator and $K$ is the complex conjugation operator (which takes the complex conjugate of all coefficients in a given basis).

The theorem is astonishing because it tells us that the vast landscape of possible non-linear transformations is completely ruled out. Symmetries in quantum mechanics must be fundamentally "simple" in this mathematical sense.

The following video segment provides an excellent statement of the theorem and emphasizes its significance.

The Two Classes of Symmetries

Wigner's theorem gives us two possibilities. When does each one occur?

1. Unitary Symmetries

Most symmetries familiar from classical mechanics, especially continuous symmetries, fall into this category. These include:

• Spatial translations (shifting the system in space)
• Spatial rotations (rotating the system)
• Time translations (letting the system evolve in time)

The reason continuous symmetries must be unitary is intuitive. Any such symmetry can be built up from infinitesimal transformations close to the identity. An infinitesimal transformation must be unitary. Since the composition of two unitary operators is unitary, any finite transformation built from these infinitesimal steps must also be unitary. (Conversely, the composition of two anti-unitary operators is unitary, which would lead to contradictions for transformations near the identity). We will use this fact extensively in upcoming lessons to find the generators of these symmetries.

2. Anti-Unitary Symmetries

The anti-unitary case is rarer and is typically associated with discrete symmetries that involve reversing the direction of time. The paradigmatic example is time reversal.

Consider the time-dependent Schrödinger equation: $i\hslash \frac{d}{dt}|\psi (t)⟩=H|\psi (t)⟩$. A time-reversal symmetry operator $\mathcal{T}$ should transform a state evolving forward in time, $|\psi (t)⟩$, into a state that appears to be evolving backward in time. For this to be a symmetry, the transformed state $|{\psi }^{\prime }(t)⟩=\mathcal{T}|\psi (-t)⟩$ must also satisfy the Schrödinger equation. The anti-unitary nature of $\mathcal{T}$ is crucial here because it complex-conjugates the $i$ on the left-hand side of the equation, effectively flipping the direction of time evolution. We will explore this in detail in a later module.

The image below, taken from a derivation of Wigner's theorem, highlights the mathematical root of this linear/anti-linear dichotomy. It stems from the fact that the only field automorphisms of the complex numbers $\mathbb{C}$ are the identity and complex conjugation.

Conclusion

Today we have laid the formal groundwork for our study of symmetries. By starting from a purely physical principle—the invariance of measurement outcomes—we have arrived at a powerful mathematical classification.

Key Takeaways:

• A physical state in quantum mechanics is not a vector but a ray in Hilbert space—an equivalence class of vectors differing by a non-zero scalar multiple. The space of physical states is the projective Hilbert space.
• A quantum symmetry is a transformation of the projective Hilbert space that preserves the transition probabilities between all pairs of states.
• Wigner's theorem is a fundamental result stating that any such symmetry must be represented on the Hilbert space by an operator that is either unitary and linear or anti-unitary and anti-linear.
• Continuous symmetries (like translations and rotations) are represented by unitary operators, while some discrete symmetries (like time reversal) are represented by anti-unitary operators.

This theorem is not just a mathematical curiosity; it is the gatekeeper for all symmetry operations in quantum theory. It assures us that we can represent symmetries with well-behaved operators, whose properties we can then study. In our next lesson, we will delve into the logic behind the proof of Wigner's theorem to understand why this powerful conclusion holds.