Projection Operators and Group Theory

1. How Projection Operators Relate to Group Theory

Projection operators appear in quantum mechanics whenever we have symmetries described by a group \( G \). They help decompose Hilbert spaces into irreducible representations of these groups.

Key Properties:

  • Idempotency: \( P^2 = P \) (applying twice is the same as once).
  • Orthogonality (for distinct eigenvalues): \( P_i P_j = 0 \) if \( i \neq j \).
  • Completeness: The sum of all projectors over a complete basis gives the identity:
    \[
    \sum_i P_i = I.
    \]

2. Projection Operators in Representation Theory

If a quantum system has a symmetry group \( G \), then its Hilbert space can be decomposed into irreducible representations (irreps). The projection operators onto these representations are:

\[
P_\lambda = \frac{d_\lambda}{|G|} \sum_{g \in G} \chi_\lambda(g)^* U(g)
\]

where:

  • \( d_\lambda \) is the dimension of the irrep labeled by \( \lambda \).
  • \( \chi_\lambda(g) \) is the character of \( g \) in the representation.
  • \( U(g) \) is the unitary representation matrix of \( g \).

Visual Representation:

        Group Symmetry in Quantum Mechanics
        --------------------------------------
        | Irrep 1 | Irrep 2 | Irrep 3 | ...
        --------------------------------------
        |   P₁    |   P₂    |   P₃    | ...
        --------------------------------------
            ⬇         ⬇         ⬇
        |ψ⟩ = c₁P₁ + c₂P₂ + c₃P₃

3. Projection Operators in Angular Momentum (SU(2) Symmetry)

In quantum mechanics, the rotation group SO(3) and its double cover SU(2) play a key role.

Decomposing Angular Momentum:

The total angular momentum operator \( J^2 \) commutes with all rotations, meaning its eigenspaces define invariant subspaces. The projection operator onto a definite angular momentum \( j \) is:

\[
P_j = \sum_{m=-j}^{j} |j, m\rangle \langle j, m|
\]

Spin-1 Representation of SU(2):

            Angular Momentum Subspaces
        ---------------------------------
        | J=1, m=1 | J=1, m=0 | J=1, m=-1 |
        ---------------------------------
        |    P₊    |    P₀    |    P₋    |
        ---------------------------------
            ⬇          ⬇         ⬇
        |ψ⟩ = aP₊ + bP₀ + cP₋

4. Projection Operators in Parity Symmetry (Z₂ Group)

In systems with parity symmetry, the parity operator \( P \) has eigenvalues \( \pm 1 \). The projection operators are:

\[
P_{\pm} = \frac{1}{2} (I \pm P).
\]

Even/Odd Parity States:

        Classical Parity Transformation
        ---------------------------------
        | Even States  (P=+1)  | Odd States  (P=-1) |
        ---------------------------------
        |       P₊        |       P₋        |
        ---------------------------------
            ⬇                    ⬇
        |ψ⟩ = c₊ P₊ + c₋ P₋

5. Projection Operators in Quantum Measurement

In quantum measurement, projection operators describe observable eigenstates and their probabilities follow the Born rule:

\[
P_i = |i\rangle \langle i|,
\]
\[
\rho’ = \sum_i P_i \rho P_i.
\]

Measurement and Decoherence:

        Quantum Measurement and Decoherence
        ---------------------------------------
        | State Before Measurement:  |ψ⟩        |
        ---------------------------------------
        | Projectors:   P₁, P₂, P₃,...            |
        ---------------------------------------
        | Probabilities: P₁⟨ψ|P₁|ψ⟩, P₂⟨ψ|P₂|ψ⟩,... |
        ---------------------------------------

6. Conclusion

  • Projection operators decompose Hilbert spaces into irreducible representations.
  • They appear in angular momentum (SU(2)), parity symmetry (Z₂), and representation theory.
  • They ensure that quantum measurements follow the Born rule and describe state decoherence.