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1

We are talking about chiral invariance, right? First of all, the mass term $$m\bar{\psi} \psi$$ breaks the chiral symmetry. So if your professor demands chiral invariance, then we are dealing with massless QED. For massless QED, you can add a chiral symmetric mass dimension 6 term like (NJL 4-fermion interaction)  \Delta \mathcal{L} = g (\bar{\psi}\psi ...

1

It DOES contribute to the renormalized mass, albeit its divergent contribution is absorbed by the mass counter term.

0

This is certainly a semantic question, but I will say that I have both heard and used the term "dangerously irrelevant" in both of those contexts as well as others where your vague initial definition fit. So far I haven't had anyone get angry at me, so I suppose that is a good sign. I'd definitely be interested in hearing about the history of the term (I ...

1

I'm going to attempt a different answer, based on what I think you're trying to get at in the comments, especially since you've been asking how things look when one simulates field theory on a lattice. So let's set up our theory with an explicit cutoff, $\Lambda > 0$, which can be a placeholder for anything you wish (an inverse lattice spacing, a hard ...

2

Chiral Anomaly has already given a good answer. Here is another answer using slightly different words. The $\Theta$-function on the RHS of eq. (12.8) reflects where in momentum space the heavy modes $\hat{\phi}$ live. It is still possible to Fourier transform action terms from position space to momentum space. The $x$-integration becomes part of a Dirac ...

1

The problem is the distinction between a theory being "perturbatively super-renormalizable" (or renormalizable/non-renormalizable), and the non-perturbative renormalization of scaling dimensions which you can see in the "exact" theory. The issue appears here because $\phi^4$ theory is only perturbative in $\lambda$ away from the massless point. For example, ...

21

This problem is nowadays referred to under the names 'self-force' and 'radiation reaction'. In classical electromagnetism it can be solved by noticing that the standard concepts (Maxwell's equations plus the Lorentz force equation) make sense when applied to continuous distributions of charge where there is no infinite charge density (such as a point charge)....

2

Here are a few hints... The factor $\Theta(k)$ is zero whenever $k$ is in the range where $\hat\phi(k)=0$. This enforces the fact that if $\hat\phi(k)=0$, then the integrand is zero, and therefore the integral must also be zero. The term being questioned comes from a term $\int d^dx\ \phi^2 \hat\phi^2$ in the integrand, which in turn comes from expanding ...

1

In the scaling/continuum limit, the microscopic constants should really be taken to $J \rightarrow \infty$, $a \rightarrow 0$, and $g \rightarrow 1$, so one shouldn't refer to the "scaling dimensions" of $a$ and $J$. Once one puts the theory into its scaling form (Eq. (10.25) in Sachdev, which crucially must be independent of the microscopic constants), the ...

0

It should be stressed that the fact that the propagator has a pole at the physical mass $k^2=-m_{\rm ph}^2$ is the very definition of the physical mass, i.e. it does not depend on the renormalization scheme, cf. e.g. Ref. 1. The on-shell (OS) renormalization scheme equates the renormalized mass $m\equiv m_r$ and the physical mass $m_{\rm ph}$. The unit ...

2

This is a very general question, and I am not sure whether there is a simple way to answer it. 1) Fluid dynamics: The need to have scale dependent parameters can be seen purely within fluid dynamics, without making explicit reference to an underlying theory. Fluctuation-dissipation relations require that dissipative fluids have thermal fluctuations (just ...

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