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How about path integrals? The probability that a system evolves between state $|\phi_1\rangle$ and $|\phi_2\rangle$ is $$\langle \phi_2|\phi_1 \rangle =\int_{\phi_1}^{\phi_2}\mathcal{D}\phi \exp \left(\frac{i}{\hbar}S(\phi)\right)$$ where the measure $\mathcal{D}\phi$ is suitably defined and the action $S(\phi)$ is the integral of the Lagrangian (over ...

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Author gives a clue on the transition: Let us assume that $\delta\vec{A}$ vanishes at infinity and integrate (formula (1)) by parts... This is the usual step in the Lagrangian theory of field (actually, of anything). At first, we have the variation of action written in an awkward form: $$\delta S=\int_{\substack{\text{domain of least}\\\text{action ... 3 The Lagrangian should not only be independent of the direction of \vec{v} but it should also change correctly under a Galilean transformation. For instance, if K and K' are two frames of reference with a relative velocity \vec{V} then the two Lagrangians L and L' should differ only by a total time derivative. If L is a function of fourth power ... 3 This is quite a standard textbook example. Good treatments can, for example, be found in Altland (Condensed Matter Field Theory) or Nagaosa (Quantum Field Theory in Condensed Matter Physics). However, the basic reasoning can be understood quite easily: Consider the change in angle produced as we perform a closed loop around some point. Naively we obtain ... 1 The Lagrangian density for a Dirac field is$$ \mathcal{L} = i\bar\psi\gamma^\mu\partial_\mu\psi -m \bar\psi\psi $$The Euler-Lagrange equation reads$$ \frac{\partial\mathcal{L}}{\partial\psi} - \frac{\partial}{\partial x^\mu}\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\right] = 0  We treat $\psi$ and $\bar\psi$ as independent dynamical ...

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I am not sure if this is what you are up to (it is related to what Xiao-Qi Sun said) to but I'll give it a try too ... At the beginning of Chapter V.2 of his QFT Nutshell, Anthony Zee explains how classical statistical mechanics (characterized by the corresponding partition function involving the Hamilton function) in $d$- dimensional space is related to ...

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