A question from Srednicki's QFT textbook I have a question in Srednicki's QFT textbook.
In order to compute the vacuum to vacuum transition amplitude given by :
$$\left \langle 0|0 \right \rangle_{J}~=~\int \left [ d\varphi  \right ]e^{i\int dt ~L\left ( \varphi ,\partial _{\mu }\varphi  \right )},$$
one has to Fourier transform the field to the momentum space 
$$\varphi \left ( x \right )~=~ \int d^{4}k~e^{ikx}\phi \left ( k \right ).$$
But what is the motivation to do so? And what other methods that could be used to compute $\left \langle 0|0 \right \rangle_{J}$ in the path integral formalism? I feel that although I understand the derivation, many steps are done in many places in this book but I don't understand what are the motivations for taking them.
 A: Sredniecki probably has his eye on perturbation theory, where you start with a collection of free fields, and then add interactions through formal power series.  
The Fourier basis is useful for treating free fields because a) it diagonalizes the free field Hamiltonian, and because b) the Fourier modes have zero coupling to one another.  In path integrals, this means that the path integral measure factorizes into a product of one-dimensional integrals, which makes it possible to evaluate it explicitly.
A: Fourier transform is usually used to get rid of derivative terms. In the exponential the Lagrangian contains derivatives of field $\varphi$ w.r.t. time as well as space. After Fourier transformation you will get an algebraic expression in terms of $\varphi(k)'s$. This will simplify matters for now the expression that you have to integrate is just (exponential of) some algebraic expression in the variables over which you integrate. After all these formal calculations you will get to some diagrammatic rules for computing vacuum to vacuum transition amplitude. One can also use Hamiltonian point of view to get to the same diagrammatic rules as for example is done in Peskin-Schroeder and some other books. However the point that I would like to stress is that most modern QFT textbooks aim to teach QFT only at a practical level, rather than trying to go very deep into clarification of the arguments they use. In fact (interacting) QFT is mathematically not a well defined subject and so most (though not all:-) of arguments presented in textbooks are only at heuristic and formal level. So don't worry if you do not find these arguments very convincing or do not see much motivation behind them. It only means that you are beginning to understand QFT :-) 
A: The essential motivation to transform to momentum space is that the translation invariance of the dynamics and of the vacuum state requires the 2-point Vacuum Expectation Value to be diagonal in the momentum variable. Thus, for the free field we have
$$\left<0\right|\hat\phi(x)\hat\phi(y)\left|0\right>=G(x-y),$$ for some function $G(x-y)$ of the separation $x-y$, where the Fourier transform of this is $$\tilde G(k)=2\pi\delta(k^2-m^2)\theta(k_0).$$
$G$ has to be a function of $x-y$, not of $x$ and $y$ separately, for it to be translation invariant.
In contrast, in real space,
$$G(x-y)=\int 2\pi\delta(k^2-m^2)\theta(k_0)\mathrm{e}^{-\mathrm{i}k\cdot(x-y)}\frac{\mathrm{d}^4k}{(2\pi)^4},$$
which is manifestly not diagonal in the $x$ and $y$ "indices".
If one considers two general "vectors" in the space of operators, $\hat\phi_f=\int\hat\phi(x)f(x)\mathrm{d}^4x$ and a similarly defined $\hat\phi_g=\int\hat\phi(y)g(y)\mathrm{d}^4y$, in the $f$ and $g$ "directions", one obtains
$$\left<0\right|\hat\phi_f^\dagger\hat\phi_g\left|0\right>=\int f^*(x) G(x-y)g(y)\mathrm{d}^4x\mathrm{d}^4y=\int \tilde f^*(k)\tilde G(k)\tilde g(k)\frac{\mathrm{d}^4k}{(2\pi)^4},$$
which hopefully makes clear the Hermitian structure of the inner product this gives us.
Another fact that is vitally important to the Hilbert space structure is manifest in the Fourier space presentation, that the function $G(x-y)$ is positive semi-definite in the Fourier co-ordinates (and hence it is positive semi-definite, but not diagonal, in all coordinates).
Working in the appropriate co-ordinates as always makes computations more compact, and keeps the computations relatively closer to the underlying conceptual structure, but one can use whatever coordinates one likes, or, better, that one can justify using.
