Dynamics of Clifford generators

[tallphil]

Trying to find the correct forum to post this, trying here because it is for a review essay I'm writing..

I've been trying to follow a proof in this paper: http://arxiv.org/abs/0804.4050v2" (page 9).

The relevant build up:
$$c_\mu$$ are Clifford algebra generators, i.e. $$\{c_\mu,c_\nu\}=2\delta_{\mu\nu}I$$

Consider a Hamiltonian of the form:
$$H=i\displaystyle\sum\limits_{\mu\ne\nu=1}^{2n}h_{\mu\nu}c_\mu c_\nu$$
where $$h_{\mu\nu}$$ is a real antisymmetric 2nx2n matrix of coefficients.

Then write $$c_\mu$$ as $$c_\mu(0)$$ and define $$c_\mu(t)=U(t)c_\mu(0)U(t)^\dagger$$ with $$U(t)=e^{iHt}$$.

It can then be shown that $$\frac{dc_\mu(t)}{dt}=i[H,c_\mu(t)]=\displaystyle\sum\limits_\nu 4h_{\mu\nu}c_\nu(t)$$

The part that is not clear to me is then to hence show that
$$c_\mu(t)=\displaystyle\sum\limits_\nu e^{4h_{\mu\nu}t}c_\nu(0)$$
in that I cannot differentiate this expression to get back to the previous line, let alone integrate the previous line to get this one (if that is indeed what is necessary).

Any help would be deeply appreciated, I have wasted far too much time today staring at this.

 

[Dick]

I think the paper intends the expression e^(4ht) to be the matrix exponential. I.e. e^(4ht)=1+4ht+(4ht)^2/2!+(4ht)^3/3!+... where the powers are matrix multiplication. That's not the same as exponentiating each entry in the matrix.