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tallphil
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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.
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.
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