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Well, first of all, it is important to realize that the integrals (1) and (3) are not merely ordinary double integrals over a single $x$- and a single $p$-variable. Instead they are (Wick-rotated) path integrals containing, heuristically speaking, infinitely many integrations. The path integral derivation of the free particle and the harmonic oscillator ...

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(Not really an answer, but as one should not state such things in comments, I'm putting it here) You commented: "This seems to boil down to the relationship between the phase space and the Hilbert space." That's a deep question. I recommend reading Urs Schreiber's excellent post on how one gets from the phase space to the operators on a Hilbert space in a ...

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This formula follows the usual heuristic discretization rules (here written in 1D): $$\tag{1} \text{discrete var.}\qquad i\in\{1, \ldots,N\}, ~~x_i=i\Delta ~~\longrightarrow~~x~\in~[0,L] \qquad \text{cont. var.},$$ $$\tag{2} \text{sum}\qquad \sum_{i=1}^N ~~\longrightarrow~~ \int_0^L \! \frac{dx}{\Delta} \qquad\text{integral},$$ $$\tag{3} ... 1 Quantum Mechanics and Path Integrals: This is a book every physicist, or student of physics, should study. Here the author describes the principle of action in quantum physics. It is not a minimum action principle, like in classical mechanics: you can, however, derive the classical minimum principle from it, in the classical limit. Why is this important? ... 3 To study the precise mathematical formulation of path integrals, you actually need probabilistic tools. The path integral is a stochastic integral with suitable measures, such as the Wiener measure associated with brownian motion. The ideas used by physicists are very useful, but not always mathematically accurate, and rely more or less on justification by ... 3 I recommend two resources: Feynman's original book called Quantum Mechanics and Path Integrals. This contains most of the prerequisites in the first two chapters, but you will need some maturity to get through them. A. Zee's quantum field theory book Quantum Field Theory in a Nutshell for its friendly chapter on them. 1 Feynman's path integral formulation is closely related to the action principle of classical mechanics, which relies heavily on the calculus of variations. You need to learn, essentially, how to minimize a functional. Prerequisites are pretty much just calculus (multivariable, hopefully), as well some classical mechanics to understand the motivation behind ... 1 I) The path integral reads$$\tag{1} Z~=~\int_{DBC} \!{\cal D}x ~e^{-S},$$with Dirichlet boundary conditions (DBC)$$\tag{2} x(0)~=~0~=~x(T).$$We expand the periodic variable x\in\mathbb{R} in Fourier series:$$\tag{3} x(t) ~=~ \sum_{n\in\mathbb{Z}} c_n e^{i\omega_n t}, \qquad \omega_n~:=~\frac{2\pi n}{T},\qquad c_n^{\ast}~=~c_{-n}.  The DBC (2) ...

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