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I am studying many-body path integral through Altland & Simons's textbook called "Condensed Matter Field Theory," and the book states the completeness of the coherent states as below: $$\int\prod_i\frac{d\bar{\phi_i}d\phi_i}{\pi}e^{-\Sigma_i\bar{\phi_i}\phi_i}\left|\phi\right>\left<\phi\right|=\mathbf{1}_\mathcal{F}\tag{4.7}$$ where $\mathbf{1}_\mathcal{F}$ represents the identity operator in Fock space (I will also use it as a definition of LHS), and the coherent state $\left|\phi\right>$ is chosen as follows: $$\left|\phi\right>=e^{\sum_i\phi_ia_i^\dagger}\left|0\right>\tag{4.1}$$

This statement is proven using Schur's lemma by showing the commutativity of $a_i$ and $\mathbf{1}_\mathcal{F}$ (or $G$-linearity of $\mathbf{1}_\mathcal{F}$), by showing $$a_i\mathbf{1}_\mathcal{F}=\mathbf{1}_\mathcal{F}a_i.$$

Here, I wonder what is the group that is represented by $a_i$ (i.e., what is $G$ in $a: G \rightarrow \mathbf{1}_\mathcal{F}$), and how to show $\{a_i\}_{i\in J}$ irreducibly represents that group.

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Note that Schur's Lemma is both formulated for groups and algebras. Altland & Simons use the Heisenberg Lie algebra generated by creation and annihilation operators and the identity operator. The representation is an infinite-dimensional irreducible Fock space. However, the standard formulation of Schur's Lemma assumes a finite-dimensional representation. Nevertheless, it is straightforward to prove eq. (4.7) directly by converting the coherent states into an eigenbasis for the number operators $\hat{n}_i=\hat{a}^{\dagger}_i\hat{a}_i $.

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  • $\begingroup$ Thanks always!! $\endgroup$
    – Neophyte
    Sep 27, 2022 at 12:29

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