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The usual way to compute such a path integral is by writing the fields (in your case: the paths) as "classical configuration" (the straight line) plus "quantum fluctuations". So if you write your paths as $\gamma(\tau) = a + \tau b + \hat\gamma(\tau)$ (with $\tau\in[0,1]$ a parameter describing the path), then $\hat\gamma$ will be the "quantum fluctuation" ...

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Definitions differ, but cardinal numbers are either specific sets (e.g. the ordinals $\omega$ where any ordinal $\alpha$ isomorphic to an initial segment of $\omega$ cannot be bijectively mapped to $\omega$), or (proper) equivalence classes (e.g. the cardinality of a set $X$ is the proper equivalence class of all and exactly those sets $Y$ such that there is ...

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In the textbook of TASI 2009, section "Introduction to extra dimension" i can find the answer as follows. They state that $5D$ or higher dimensional gauge coupling has a negative mass dimension, so the 5d or higher dimensional gauge theory is non-renormalizable.

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If you are thinking to apply renormalization to gravity, I would suggest to look at the explanation from "String Theory" of Kevin Wray: the problem is that we need more and more parameters to absorb the infinities that occur in the theory. String theory solve this particular problem because a string has finite extent lp, the divergent integral is cutoff at ...

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Firstly, renormalisation isn't really inherently to do with the UV divergences you get: it's more to do with the idea that interactions with other fields change a particle's energy from its inherent 'bare' mass, so the measured mass isn't the same as the value that appears in the quadratic part of the action. You play a renormalisation game in perturbation ...

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What I think one needs to internalize conceptually is that the program of renormalization is always favourable (and almost always required) in physical theories, be they fundamental or effective phenomenological ones (including condensed matter field theories), be there infinities or not. I think the last point is by far the most important. Yes, ...

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You seem to be confusing regularization with renormalization. Regularization is the process of removing (or, more properly, parameterizing) infinities in loop integrals. Often in elementary texts a "cutoff" representing an energy scale above which the theory is assumed to be invalid is discussed, and counterterms are added to the Lagrangian in order to make ...

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