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Consolidating from comments: The definitions given in the question are a bit unusual, so it is not straightforward to interpret them. However, best as I understand the question, it is not about whet …

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 $n$-point functions for $n\ge 3$ are required if you are interested in how your system responds to an external probe $\phi$ (e.g. an electromagnetic field). You have to couple this to your system …

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 …

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 …

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 …

I take it that your question is about why the symplectic Euler method is symplectic, while the explicit Euler method is not. The point is that for a Hamiltonian of the form $H(p,q)=T(p)+V(q)$, the sym …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …