- All Group Representation is Equivalent to Unitary Representation

Introduction #

In this article, we will prove that all group representations are equivalent to unitary representations. This is a fundamental result in representation theory with important applications in physics and mathematics.

Definitions and Background #

1. Equivalence of Representations: Two representations $A$ and $B$ are equivalent if and only if there exists an invertible matrix $X$ such that:

   $$A = XBX^{-1}$$

2. Hermitian Operators: A key property we’ll use is that all Hermitian operators are diagonalizable.

Proof of Equivalence to Unitary Representation #

Let’s start by defining an operator $S$ for a representation $D$ of a group $G$:

$$S = \sum_{g \in G} D(g)^{\dagger} D(g)$$

Step 1: Show $S$ is Hermitian and Positive Definite #

Since $S^{\dagger} = S$, the operator $S$ is Hermitian.

When $S|\mu\rangle = \lambda|\mu\rangle$, $\lambda$ is real (because $S$ is Hermitian). Furthermore, since $D(e)=1$ (where $e$ is the identity element), we have:

$$\langle\mu|S|\mu\rangle = \sum_{g \in G} ||D(g)|\mu\rangle||^2 > 0$$

This proves that $S$ is positive definite.

Step 2: Construct a Unitary Representation #

Since $S$ is Hermitian and positive definite, we can write:

$$S^{1/2} = UD^{1/2}U^{-1}$$

Now, define a new representation $D’$ as:

$$D’(g_1) = S^{1/2} D(g_1) S^{-1/2}$$

Step 3: Verify $D’$ is Unitary #

Let’s check if $D’$ is unitary by computing $D’^{\dagger}(g_1)D’(g_1)$:

$$D’^{\dagger}(g_1)D’(g_1) = S^{-1/2} D^{\dagger}(g_1)SD(g_1) S^{-1/2}$$

We need to analyze $D^{\dagger}(g_1)SD(g_1)$:

$$D^{\dagger}(g_1)SD(g_1)$$ $$ = D^{\dagger}(g_1)\left(\sum_{g \in G} D(g)^{\dagger} D(g)\right)D(g_1)$$

Note that: $$D^{\dagger}(g_1)D^{\dagger}(g) = (D(g)D(g_1))^{\dagger}$$ $$ = D^{\dagger}(gg_1) = D^{\dagger}(g’)$$

Therefore:

$$D’^{\dagger}(g_1)D’(g_1)$$ $$ = S^{-1/2}\left(\sum_{g’ \in G} D(g’)^{\dagger} D(g’)\right)S^{-1/2}$$ $$ = S^{-1/2}SS^{-1/2} = I$$

This confirms that $D’(g_1)$ is unitary.

Conclusion #

Since $D(g_1)$ is equivalent to $D’(g_1)$ by construction, and $D’(g_1)$ is unitary, we have proven that any group representation $D(g_1)$ is equivalent to a unitary representation. This result is fundamental in representation theory and has wide-ranging applications in physics, particularly in quantum mechanics.