9

The quantum analogue of Noether's theorem in classical physics is the Ward-Takahashi identities, which can be formulated in either the operator formalism or the equivalent path integral formalism.


7

In order to observe a specific path one needs to design an appropriate observable. Clear discussions on this subject are usually appear in the context of the two-slit experiment, although it is not the first thing that comes to mind when one is dwelling on all the mathematical difficulties of the path integrals. Some specific realizations of the two-slit ...


4

It's just very tedious algebra. Expand the exponential function as a power series. Commute all the derivatives to the right where they die against "1", and keep all the terms that can possibly survive the large $M$ limit after doing the $k$ integral (this requires going to higher orders than you might expect) and then do the gamma matrix traces. ...


2

OP's calculation seems to match Zee's calculation; except for the final step. Here OP has made a mistake: $$ \left< k | k \right> = (2 \pi)^4 \delta^{(4)}(0) \neq 1. $$ This is where the factor of $VT$ comes from: $$ \left< k | k \right> = \left<k | 1 | k \right> = \int d^4 x \left< k | x \right> \left< x | k \right> = \int d^4 ...


2

Hint: For a local action functional $S[x]$ the Hessian $ \frac{\delta^2 S[x]}{\delta x(t_1) \delta x(t_2)}$ is proportional to a Dirac delta distribution $\delta(t_1\!-\!t_2)$. See e.g. eq. (7.37) in Ref. 1. See also this related Phys.SE post. References: A. Das, Field Theory: A Path Integral Approach; eq. (7.35).


1

Yes OP is right: The lectures are using Gaussian path integration and the identity $$ \det A~=~e^{{\rm tr}\ln A}\quad \Leftrightarrow \quad \frac{1}{\det A}~=~e^{-{\rm tr}\ln A} \tag{i}$$ in eq. (174). Next the lectures are using the completeness relation $$\int\mathrm{d}^4x\,|x\rangle \langle x|~=~{\bf 1} \tag{ii}$$ and $$ {\rm tr}[A|x\rangle \langle x|]...


1

Yes, you are right -the determinant-logarithim identity has been used above. As for the exponential, the following (from the book Field Quantization by Greiner, Reinhardt, chapter 11, page no. 353) has been used - $ \int {d^D v \exp \big\{ {- \frac{1}{2} v^T A v} \big\}} = \big(2\pi\big)^{\frac{D}{2}} \exp \big\{ {- \frac {1}{2} Tr ln A} \big\} = \big(2\pi)^...


1

There is likely not an exhaustive classification for when pairs of classically equivalent theories are equivalent quantum mechanically. The best one can do is probably to present a relatively short list of known examples. The most famous pairs are square root actions vs. non-square root actions, cf. e.g. Nambu-Goto vs. Polyakov action in string theory or the ...


1

The core of OP's question seems to be the following question: What's the difference between (1) the infinitesimal variations to derive EL equations, and (2) the infinitesimal symmetry variations in Noether's theorem? This is explained in this related Phys.SE post. Concerning the Schwinger-Dyson (SD) equations, they are derived using a vertical translation ...


1

Wick rotation is more than just a change of time coordinate from $t_M\in\mathbb{R}$ to $t_E=it_M\in i\mathbb{R}$, cf. e.g. this related Phys.SE post. It the assumption that we can analytically extend the time coordinate to the complex plane $\mathbb{C}$ (minus possible poles and branch cuts), and use complex function theory and Cauchy's integral theorem to ...


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