In reading 14.4 of Gregory Moore's notes on abstract group theory, I was left with some questions on the computation he did of the path integral that may be general features.

Let consider a spacetime $M=\Sigma\times[t_0,t_f]$ on which we have a space of fields $C^\infty(M)$. Let $\mathcal E_1$ be the set of fields $\phi\in C^\infty(M)$ such that $\phi|_{\Sigma\times\{t_0\}}=\phi_0$ and $\phi|_{\Sigma\times\{t_f\}}=\phi_f$ for some fixed $\phi_0,\phi_f\in C^\infty(\Sigma)$. The technique employed in the notes above to compute $$\int_{\mathcal E_1}\mathcal{D}\phi\, e^{-\frac{1}{\hbar}S(\phi)},$$ is to first find a solution of the classical equations of motion $\phi_c\in\mathcal E_1$ and then reduce this to an integral $$\propto\int_{\mathcal E_2}\mathcal{D}\phi_q\, e^{-\frac{1}{\hbar}\tilde{S}(\phi_q)},$$ where $\mathcal{E}_2$ is the same as $\mathcal{E}_1$ except that $\phi_0=\phi_f=0$. He the proceeds to compute the integral of $\mathcal{E}_2$ using Gaussian integration. However, the original integral was also Gaussian. Why can't we compute the integral over $\mathcal{E}_1$ using Gaussian integration?

Of course, if one where to compute this integral through Gaussian integration the obvious problem appears of how to incorporate the boundary conditions. But that is at the root of my problem. Namely, what is special about the boundary conditions in $\mathcal{E}_2$ vs. $\mathcal{E}_1$? In usual Gaussian integration the integration of each variable $dx^i$ is on the range $-\infty$ to $\infty$. If one thinks naively of the measure as $\mathcal{D}\phi=\prod_{x\in M}d\phi(x)$, each variable of integration $\phi(x)$ is still being integrated in the range $-\infty$ to $\infty$ except for the ones at the boundaries.

(I posted a similar question yesterday but, after a suggestion by QMechanic, I decided to delete it to focus on just this point).

In reading 14.4 of Gregory Moore's notes on abstract group theory, I was left with some questions on the computation he did of the path integral that may be general features.

Let consider a spacetime $M=\Sigma\times[t_0,t_f]$ on which we have a space of fields $C^\infty(M)$. Let $\mathcal E_1$ be the set of fields $\phi\in C^\infty(M)$ such that $\phi|_{\Sigma\times\{t_0\}}=\phi_0$ and $\phi|_{\Sigma\times\{t_f\}}=\phi_f$ for some fixed $\phi_0,\phi_f\in C^\infty(\Sigma)$. The technique employed in the notes above to compute $$\int_{\mathcal E_1}\mathcal{D}\phi\, e^{-\frac{1}{\hbar}S(\phi)},$$ is to first find a solution of the classical equations of motion $\phi_c\in\mathcal E_1$ and then reduce this to an integral $$\propto\int_{\mathcal E_2}\mathcal{D}\phi_q\, e^{-\frac{1}{\hbar}\tilde{S}(\phi_q)},$$ where $\mathcal{E}_2$ is the same as $\mathcal{E}_1$ except that $\phi_0=\phi_f=0$. He the proceeds to compute the integral of $\mathcal{E}_2$ using Gaussian integration. However, the original integral was also Gaussian. Why can't we compute the integral over $\mathcal{E}_1$ using Gaussian integration?

Of course, if one where to compute this integral through Gaussian integration the obvious problem appears of how to incorporate the boundary conditions. But that is at the root of my problem. Namely, what is special about the boundary conditions in $\mathcal{E}_2$ vs. $\mathcal{E}_1$? In usual Gaussian integration the integration of each variable $dx^i$ is on the range $-\infty$ to $\infty$. If one thinks naively of the measure as $\mathcal{D}\phi=\prod_{x\in M}d\phi(x)$, each variable of integration $\phi(x)$ is still being integrated in the range $-\infty$ to $\infty$ except for the ones at the boundaries.

I posted a similar question yesterday but, after a suggestion by QMechanic, I decided to delete it to focus on just this point. In that question I asked why it was that $S(\phi_c+\phi_q)=S(\phi_c)+S(\phi_q)$ when $\phi_c$ is a solution of the classical eoms. This turns out to be true whenever the theory is free (quadratic) since the second derivative of the action is independent of the fields.