I know that the Gaussian integral over complex vectors $\mathbf{u}=(u_1,\dots u_N)$, where $u_i$ are complex valued, is $$\int \mathcal{D}[\mathbf{u}^*,\mathbf{u}] e^{-\mathbf{u}^* A\mathbf{u}+\mathbf{a}^*\cdot\mathbf{u}+\mathbf{u}^*\cdot\mathbf{b}} = \pi^N \mathrm{det}(A^{-1}) e^{-\mathbf{a}^* A^{-1} \mathbf{b}}$$
Now, I am interested in obtaining an effective action for a system where I will only partly integrate out (so that I can have some boundary fields left over), namely
$$\int \mathcal{D}\left[\begin{pmatrix}u_2^*\\u_3^*\\\vdots\\u_{N-1}^*\\u_N^*\end{pmatrix},\begin{pmatrix}u_1\\u_2\\\vdots\\u_{N-2}\\u_{N-1}\end{pmatrix}\right] e^{-\begin{pmatrix}u_2^*\\u_3^*\\\vdots\\u_{N-1}^*\\u_N^*\end{pmatrix} A\begin{pmatrix}u_1\\u_2\\\vdots\\u_{N-2}\\u_{N-1}\end{pmatrix} \,+\, \begin{pmatrix}v_2^*\\v_3^*\\\vdots\\u_{v-1}^*\\v_N^*\end{pmatrix}\cdot\begin{pmatrix}u_1\\u_2\\\vdots\\u_{N-2}\\u_{N-1}\end{pmatrix} \,+\,\begin{pmatrix}u_2^*\\u_3^*\\\vdots\\u_{N-1}^*\\u_N^*\end{pmatrix}\cdot\begin{pmatrix}v_1\\v_2\\\vdots\\v_{N-2}\\v_{N-1}\end{pmatrix}}\ \ =\ ?$$ where I have excluded the $u_1^*$ and the $u_N$ fields from the integration, modified the $\mathbf{u}^*$ and $\mathbf{u}$ so that it is no longer quite Gaussian. Additionally I have written out an explicit expression for $\mathbf{a}^*$ and $\mathbf{b}$, although this isn't necessary for the form of the integral--$v_1,\dots v_N$ are also complex variables.
I have tried:
- excluding the $u_1$ and $u_N^*$ fields from the integral as well, but this gives me extra boundary terms and for the $\mathbf{v}^*\cdot\mathbf{u}$ and $\mathbf{u}^*\cdot\mathbf{v}$ terms these are no longer a dot product
- using that for my case $u_i$ and $u_{i+1}$ are separated by some small time $\delta t$ so that $u_i\approx u_{i+1}$ and using the standard integration formula (first equation above), but I think this is incorrect since the nature of the path integral requires that we integrate over all trajectories not just those with semiclassical trajectories
- additionally I don't want to expand to include $u_1^*$ and $u_N$ since then I lose the boundary conditions I am looking to keep
Thoughts and ideas for how to complete this integral exactly are much appreciated!