How do you prove the following commutation relation for the lattice QFT
\begin{equation} [\phi(X),\Pi(y)]=\text{i}a^{-d}\delta_{x,y}\mathbb{I}? \end{equation}
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$\begingroup$ I want to know how to derive the commutation relation from the functional-integral formulation of lattice QFT. Also, I would like to know why it is defined in this way in general? I want have a little more insight. $\endgroup$– HeisenbergOct 13, 2020 at 14:19
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$\begingroup$ I mean give that eigenstates of momentum operator are Fourier transform of eigenstates field operator, it should be possible to show the commutation relation. $\endgroup$– HeisenbergOct 13, 2020 at 14:29
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$\begingroup$ One shifts from a lattice discrete-time path integral picture to an operator picture by writing the partition function $Z$ as the trace of a transfer matrix $T$. To pass to the continuous-time Hamitonian picture you have a scale the couplings and the discrete time steps so that $T\to \exp\{-H \delta t\}$. $\endgroup$– mike stoneOct 13, 2020 at 15:14
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$\begingroup$ Can we prove this directly using Fourier transform? $\endgroup$– HeisenbergOct 14, 2020 at 4:30
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$\begingroup$ is it possible to prove it using Fourier transform? $\endgroup$– HeisenbergOct 14, 2020 at 6:34
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