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Q1: You use the usual topology on $\mathbb{R}^4$. (If you were to use $\eta$ to naively define a topology, you couldn't separate points on the lightcone $\eta(x,x)=0$, which would not be terribly phys …

If you don't want to discuss any kind of specific model as motivation, you could always argue that the $\phi^4$ theory is the only renormalizable theory that shares the $Z_2$ ($\phi\mapsto-\phi$) symm …

Lattice gauge theory is indeed still very demanding, although the last decade has seen tremendous progress, both in terms of algorithms and of software implementations that make good use of modern com …

The sawtooth function $f$ has Fourier decomposition $$ f(t) = \frac{1}{2}-\frac{1}{\pi}\sum_{n=1}^\infty \frac{1}{n} \sin(n\omega t) $$ Therefore, if $\omega=\frac{\omega_0}{n}$, the $n$-th harmonic o …

I don't think it is possible using current knowledge to perform the overall unification of physical theories that you suggest. However, on a much smaller scope similar things are possible. For example …

Your reasoning is correct in that the discrete Laplacian for periodic boundary conditions has a zero mode. On the space of fields satisfying $\sum_x\phi(x)=0$, its spectrum is, however, strictly posit …

I think the misunderstanding at the heart of this question is about what it takes to specify a quantum field theory. A Hilbert space on which a representation of the Poincaré group acts is not enough …

This construction is somewhat similar to the Martin-Siggia-Rose formalism for writing expectation values over solutions of a stochastic differential equation $$ \dot{p}(t)=f(p,t)+\xi(p,t) $$ with $\xi …

Q1: This is basically correct. For a discussion, cf. the discussion in the first chapter of the book by Montvay and Münster and the references given therein. Q2: This is quite correct. Axiomatic QFT c …

A charitable reading suggests that you are referring to the $\zeta$ function regularization $$ \sum_{i=1}^\infty i = \lim_{s\to -1} \sum_{i=1}^\infty i^{-s}=\zeta(-1)=-\frac{1}{12} $$ which occurs in …

Resonance is so universal because Fourier analysis is so universal, and because physical equations of motion are second-order. Michael Engelhardt already explained why Taylor expansion around stable m …

The "baby version" is actually the regularization of the path integral on a finite Euclidean lattice, which is what is studied by theoretical physicists working in lattice quantum field theory. The us …

To make an interacting (i.e. not purely quadratic) QFT at all meaningful, you have to impose a regulator. The most transparent regulator in many ways, and the only known regulator that allows to addre …

First of all, you're right that by "the Casimirs" the authors mean the eigenvalues of the quadratic Casimir operator on the irreps in question — this is a common phrasing in the physics literature. Fo …

As to question 2, there are certainly plenty of non-trivial discrete models in statistical physics, such as the Ising or Potts models, or lattice gauge theories with discrete gauge groups, that requir …