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1

A good discussion of regularity properties of the Wiener measure is in section II.5 of the book "Functional Integration and Quantum Physics" by Barry Simon. He gives the proofs of most of the relevant theorems except the borderline case $\alpha=\frac{1}{2}$, namely Levy's Theorem showing that $\frac{|x(\tau_1)-x(\tau_2)|}{\sqrt|\tau_1-\tau_2|}$ diverges ...


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You can refer to chap 9 of "An introduction to quantum field theory" of Peskin & Scroeder, which includes a detailed calculation of path integral using the original physical definition of path integral. After the brutal treatment, they will show you more modern treatment using generating functional.


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The Fermat principle does not say light ray follows the fastest path, it says when there is a light ray, the optical path (length divided by index of refraction) is stationary with respect to small variations in the shape of the ray that preserve the position of the boundary points. It is not as if light got everywhere the fastest way possible; it goes ...


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I guess your prof might have been referring to the "spirit" of the derivation, which proceeds by splitting up time into discrete chunks and considering a stroboscopic evolution over short time intervals $\Delta t$. Then by taking the limit $\Delta t \to 0$, the desired expression is obtained. A similar approach is used in the derivation of the Feynman path ...


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The answer is Yes. Define function $g(q):= \frac{1}{f(q)}$ for later convenience. Then the classical Hamiltonian reads $$2h~=~g(q)p^2.$$ One may show that the Weyl-ordered Hamiltonian reads $$2H_W~=~ (g(q)p^2)_W ~=~ \frac{1}{4}P^2 g(Q)+\frac{1}{2} Pg(Q)P+\frac{1}{4} g(Q)P^2$$ $$~=~ Pg(Q)P - \frac{1}{4}\hbar^2g^{\prime\prime}(Q),$$ see e.g. Ref. 1 and this ...


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Yes, eq. (2.193) is a classical formula, and the symmetry of the (Hilbert) stress-energy-momentum tensor $T^{\mu\nu}$ is only valid classically. Quantum mechanically, the symmetry of $T^{\mu\nu}(x)$ is broken by the presence of other fields in positions $x_1,x_2,\ldots$ in the (time-ordered) correlator $$\langle T\left\{ (\hat{T}^{\mu \nu}(x) - \hat{T}^{\nu ...


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Technically no, because if something went faster than c it would be classified as a Tachyon. If they did however, they would only be a Tachyon by definition, physically they would still be Photons.


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since $\phi(x)$ is real, $\phi^*(q) = \phi(-q)$ implies Re$(\phi(q))$ is even and Im$(\phi(q))$ is odd, but nonetheless independent unlike the real and imaginary parts of $\phi(x)$. So in the new measure it must be understood that you are summing only over the even/odd functions. There for ...


2

Yes, the path integral measure has units but they are mostly irrelevant because physically well-defined objects tend to be ratios of path integrals in which the basic portion cancels, along with its units. So the overall normalization factor in front of the path integral (including its units) drops out. But the path integral measure should be assigned the ...



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