# Coherent state basis of (relativistic) particle Fock space

For a neutral scalar bosonic particle of mass $$m$$, I consider a Fock space with an orthonormal basis of momenta eigenstates $$$$\label{Fock-p-states} \left|p_1p_2\cdots p_n\right\rangle=\frac{1}{n!}\sum_\sigma\left|p_{\sigma(1)}\right\rangle\otimes\left|p_{\sigma(2)}\right\rangle\otimes \cdots\otimes\left|p_{\sigma(n)}\right\rangle\,,$$$$ with a definite number of particles running from 1 to $$\infty$$, together with the vacuum state $$|0\rangle$$; the sum is over all permutations $$\sigma$$ of the particles. The normalization must be of the form $$\langle p'|p\rangle=2E\delta(\vec{p}-\vec{p}')$$, where $$E=\sqrt{m^2+\vec{p}^2}$$, for $$\langle p'|p\rangle$$ to be Lorentz invariant.

I define the creation operator $$\hat{a}^\dagger(p)$$ as $$\begin{equation*} \hat{a}^\dagger(p)|p_1p_2\cdots p_n\rangle=\sqrt{n+1}|p_1p_2\cdots p_np\rangle\,, \end{equation*}$$ and a coherent state $$|a\rangle$$ as an eigenstate of the annihilation operator $$\hat{a}(p)$$ with eigenvalue $$a(p)$$ for every possible $$p$$: $$\begin{equation*} \hat{a}(p)|a\rangle=a(p)|a\rangle\:\:\:\forall p\,. \end{equation*}$$ The set $$\{|a\rangle\}$$ of all coherent states is thus constructed through the functional ($$\langle 0|a\rangle\equiv a_0$$ is fixed by normalization) $$\begin{equation*} a(p)\mapsto|a\rangle=a_0|0\rangle +\frac{a_0}{\sqrt{n!}}\sum_{n=1}^\infty\int\!\!\bar{d}^3\!p_1\cdots\bar{d}^3\!p_n a(p_1)\cdots a(p_n)|p_1\cdots p_n\rangle\,,\hspace{5pt} \int\!\!\bar{d}^3\!p\,|a(p)|^2<\infty\,, \end{equation*}$$ with a coherent state for each element of the set $$A$$ of all modulus-square-integrable complex functions $$a(p)$$; and $$$$\label{square-integrable} \langle b|a\rangle=b^*_0a_0\exp\!\int\!\!\bar{d}^3\!p\,b^*(p)a(p)\,.$$$$ Integrations are performed with the Lorentz invariant momentum element $$\begin{equation*} \bar{d}^3\!p_i\equiv\frac{d^3\!p_i}{2\sqrt{\vec p_i^2+m^2}} =\frac{d^3\!p_i}{2E_i}\,. \end{equation*}$$

Now, the question is if $$\{|a\rangle\}$$ is a basis, an overcomplete basis of the Fock space; more precisely, if there is a suitable definition of the measure $$\mathcal{D}a$$ of a modulus-square-integrable complex function $$a(p)$$ giving a functional integral $$$$\label{identity-alpha} \int_A\!\!\mathcal{D}a|a\rangle\langle a|=\hat{1}\,,$$$$ with $$|a\rangle$$ normalized to $$1$$, i.e., $$|a_0|^2\exp\int\!\!\bar{d}^3\!p\,|a(p)|^2=1$$. In the momenta basis, this is translated into $$$$\label{identity-alpha-p} \delta_{\!mn}\sum_\sigma\prod_i2E_i\delta(\vec{p}_i-\vec{p}'_{\sigma(i)}) =\int\!\!\mathcal{D}a\,e^{-\int\!\bar{d}^3\!p\,|a(p)|^2}a(p_1)\cdots a(p_n)a^*(p'_1)\cdots a^*(p'_m)\,;$$$$ $$$$\label{identity-alpha-0} 1=\int\!\!\mathcal{D}a\,e^{-\int\!\bar{d}^3\!p\,|a(p)|^2}\,,\:\:m=n=0\,.$$$$