Feynman's path integral and uncertainty principle?

Question

The Feynman's path integral representation gives the quantum amplitude to go from point $x$ to point $y$ as an integral over all paths.

1. How is that idea consistent with the uncertainty principle that is considered to be fundamental? That is, having a definite, initial point $x$ is impossible for a physical particle.

2. Probability distributions are allowed but definite values are not. Thus, can we replace definite points with well-localized distributions, together with the finite expectation values of observables, to make more physical/mathematical sense?

3. Can we argue that the Feynman's picture is not "real" but only a way of interpreting the integral-like sum? A related idea might be the Ptolemy's picture of planetary epicycle motion that also gave correct results but for wrong reasons. A similar issue with the Huygens' principle.

Answer 1

I can't claim I understand the question, but, as you point out, the Feynman graphic visualization of Dirac's 1933 construction of QM amplitudes is an invitation for technical algorithms, allowing for creative techniques. Notably, at the Pocono conference, Bohr misunderstood it at first and rebuked RPF on that very UP ground.

1. But... the UP certainly allows for perfectly fine transition probability amplitudes $K(x,t;x',t') = \left \langle x \mid \hat{U}(t,t') \mid x'\right \rangle,$ for perfectly precise x s, just as it does in plain vanilla Hilbert-space QM; it doesn't much care how you compute these. The very same flip-flopping between the x representation and the p representation you see in Hilbert space is utilized in defining the path integral, and proving its equivalence to standard techniques.

2. Well, the output of the path integral is probability amplitudes! The path integral, through the magic of Gaussians, quantifies the localization of distributions, and their spread via $\hbar$ corrections. Feynman expanded on Dirac's crucial observation, and quantified departures from the "classical limit" of stationary actions. The sum over all paths is actually a technique of practically generating these distributions, which, among other things, makes lattice simulations of QFTs like QCD possible and practical!

3. I don't understand your point and I never understood what "real" means, but, yes, the path integral is an umbrella of techniques of deriving amplitudes. Magnificent techniques, when you appreciate the functional equations and insight on ghosts it has lavished on QFT, and the practical simulation answers it makes possible in lattice gauge theory...