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Consider an $N$-dimensional quantum system with the Hamiltonian $$ H = \left[ \begin{matrix} E_{1} & 0 & \cdots & 0 \\ 0 & E_{2} & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & E_{N} \end{matrix} \right] \ . $$ $H$ generates translations in time $t$, such that $|\Psi(t_2)\rangle = e^{- i H (t_2-t_{1}) } |\Psi(t_1) \rangle$. This is a bit of strange question but does there exist an action $S$ corresponding to the above system? In field theory the action is a scalar, so this seems unlikely to me (I don't know how to build scalars out matrices)...

In the case that there is not, is there a way to see this system as a limit of a some $S$ (possibly built out of fields)?

Sahand Tabatabaei
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QuantumEyedea
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  • Related: https://physics.stackexchange.com/q/3500/ – fqq Dec 16 '19 at 18:43
  • Of course there is, by comparison of infinitesimal propagators; but working out the L is not always practical, requiring reconstruction of the potential from your energy eigenvalues. Dirac's breathtaking 1933 paper reminds you that $\langle x | \exp (-i\Delta t H/\hbar)|x'\rangle \sim \langle x| \exp(-i\Delta t p^2/2m\hbar)\exp (-i\Delta t V(x)/\hbar )|x'\rangle\approx \exp ( i(m(x-x')^2/2\Delta t -\Delta t V(x) )/\hbar )$. – Cosmas Zachos Dec 18 '19 at 15:50

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