General Element of U(4)

Question

Relating back to previous question about how to write a general element of $U(2)$, I am now wondering about how to write a general element of $U(4)$.

Define $\Gamma_{(i,j)}:=\sigma_i\otimes\sigma_j$ where $\sigma_0$ is the unit $2\times2$ matrix and $\sigma_1$, $\sigma_2$, $\sigma_3$ are the three Pauli spin-1/2 matrices.

$(\Gamma_{(i,j)})_{(i,j)\in\{0,1,2,3\}^2}$ spans the space of $4\times4$ Hermitian matrices, with sixteen linearly independent elements.

Question: Is it true that a general element of $U(4)$ may be written as $\exp\left(i\sum_{(i,j)\in\{0,1,2,3\}^2}d_{(i,j)}\Gamma_{(i,j)}\right)$ where $(d_{(i,j)})_{(i,j)\in\{0,1,2,3\}^2}\in\mathbb{R}^{16}$?

If not, what is a general way to write an element of $U(4)$ parametrized by 16 real parameters?

Answer 1

It is true.

Proof:

Note that I write the Hermitian conjugate of a matrix $A$ as $A^\dagger$

1. Every matrix of the form $U = \exp(\mathrm iH)$ with $H$ Hermitian is unitary: $$U^\dagger U = \exp(-\mathrm iH^\dagger)\exp(\mathrm iH) \stackrel{(1)}= \exp(-\mathrm iH)\exp(\mathrm iH) \stackrel{(2)}= \exp(-\mathrm iH+\mathrm iH) = I$$ where (1) uses the fact that $H$ is Hermitian, and (2) that of course $\mathrm iH$ commutes with $-\mathrm iH$.

   The proof for $UU^\dagger$ obviously works the same.

2. Every unitary matrix $U$ can be written in the form $\exp(\mathrm iH)$.

   Unitary matrices are diagonizable, that is, there exists an unitary matrix $V$ so that $VUV\dagger = D$ where $D$ is diagonal. Of course, as product of unitary matrices, $D$ is also diagonal. Now a diagonal unitary matrix is always of the form $$\begin{align} D &= \operatorname{diag}(\exp(\mathrm i\phi_1),\ldots,\exp(\mathrm i\phi_n))\\ &= \exp(\mathrm i\underbrace{\operatorname{diag}(\phi_1,\ldots,\phi_n)}_{=:T}) \end{align}$$ Clearly $T$ is a Hermitian diagonal matrix. Now $$U = V^\dagger DV = \exp(\mathrm i V^\dagger T V)$$ Since $T$ is Hermitian, so is $V^\dagger TV$.

Now since your exponent is just a parametrization of a general Hermitian $4\times 4$ matrix, your formula gives the general unitary $4\times 4$ matrix.