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Hi guys. I'm working on a model described by a non-local QFT. I think I got the Feynman rules right, but I get divergences from ##\delta(0)##-like factors.

It's a QFT for a complex scalar field ##\psi(x)=\psi(\mathbf{x},t)## with action $$S= \int dx \dot{\bar{\psi}}\dot{\psi}-m^2\bar{\psi}\psi -\int dx dy \bar{\psi}(x)\psi(y) \Omega(x,y) \bar{\psi}(x)\psi(y)$$

where ##\Omega## is a known, positive function. The space-time of the theory is of the form ##X \times \mathbb{R}##, where ##X## is a measure space. There is no a priori differentiable structure on ##X##, therefore I can't use spatial derivatives of momentum space in this model (maybe just formally), just time derivatives. The problem is to renormalize the theory because loop diagrams are "divergent" in the sense that they are proportional to factors of the form ##\delta(\mathbb{x},\mathbf{x})##, whatever those mean in a general measure space setting.

It's not clear to me whether the Delta function ##\delta(x,y)## makes sense. Maybe just as a Dirac measure, not even as a distribution (there's no topology on ##X##, unless you can build one from the measure). Formally one has the free propagator $$\Delta(x,y)=\int \frac{d\omega}{2 \pi}\frac{i\delta(\mathbf{x},\mathbf{y})}{\omega^2-m^2}e^{-i\omega(t-t')}$$

The Feynman rules for the vertex is, according to my calculations, given by ##\int dz dw \Omega(z,w)##, where one specifies a vertex with two points ##z,w## and joins each one to two lines. This causes loop factors of the form ##\delta(\mathbf{x},\mathbf{x})## which don't really make sense in the general case. In case ##X## is discrete it should be the Kronecker delta so it's just 1. In case ##X=\mathbb{R}^n## it should be the usual Dirac delta, so that's problematic to regularize.

The only cause for divergences is in those factors, and they are in front of every diagram with a loop. This means that if I place a formal cutoff ##\Lambda##, pretending to work in ##d##-dimensional momentum space and write ##\int d^d p \sim \Lambda^d##, subtracting the "divergences" with counterterms cancels every loop diagram, leaving me with tree diagrams that are zero because of the delta functions in the propagators. The only non-zero amplitudes in the theory are diagrams with loops, and my interpretation is that "physical" particles are ##\bar{\psi}\psi## pairs.

Knowing that this form of the propagator can be traced back to the absence of spatial derivatives, I tried to add a formal laplacian term in the action, with a parameter ##\epsilon## in front. Pretending again to be in ##d## dimensions, the basic loop integral evaluates to ##\sim (m^2/\epsilon)^{d/2}## which is again an overall divergent factor. Straight dimensional regularization of ##\int d^d p## gives zero.

I tried to modify the kinetic Lagrangian in a way that could extract a sensible finite part from ##\delta(\mathbf{x},\mathbf{x})##, without success. I tried to treat delta functions and propagators as distributions, as measures, as operators, but in those formalisms it's difficult to define loops. I read something about casual perturbation theory and the R- and W-operations, and applying that scheme to the "squared delta" in this case just gives me an arbitrary constant. I'm supposed to fix it with some renormalization condition, but I don't know what I should do.

## Homework Statement

It's a QFT for a complex scalar field ##\psi(x)=\psi(\mathbf{x},t)## with action $$S= \int dx \dot{\bar{\psi}}\dot{\psi}-m^2\bar{\psi}\psi -\int dx dy \bar{\psi}(x)\psi(y) \Omega(x,y) \bar{\psi}(x)\psi(y)$$

where ##\Omega## is a known, positive function. The space-time of the theory is of the form ##X \times \mathbb{R}##, where ##X## is a measure space. There is no a priori differentiable structure on ##X##, therefore I can't use spatial derivatives of momentum space in this model (maybe just formally), just time derivatives. The problem is to renormalize the theory because loop diagrams are "divergent" in the sense that they are proportional to factors of the form ##\delta(\mathbb{x},\mathbf{x})##, whatever those mean in a general measure space setting.

## Homework Equations

It's not clear to me whether the Delta function ##\delta(x,y)## makes sense. Maybe just as a Dirac measure, not even as a distribution (there's no topology on ##X##, unless you can build one from the measure). Formally one has the free propagator $$\Delta(x,y)=\int \frac{d\omega}{2 \pi}\frac{i\delta(\mathbf{x},\mathbf{y})}{\omega^2-m^2}e^{-i\omega(t-t')}$$

The Feynman rules for the vertex is, according to my calculations, given by ##\int dz dw \Omega(z,w)##, where one specifies a vertex with two points ##z,w## and joins each one to two lines. This causes loop factors of the form ##\delta(\mathbf{x},\mathbf{x})## which don't really make sense in the general case. In case ##X## is discrete it should be the Kronecker delta so it's just 1. In case ##X=\mathbb{R}^n## it should be the usual Dirac delta, so that's problematic to regularize.

## The Attempt at a Solution

The only cause for divergences is in those factors, and they are in front of every diagram with a loop. This means that if I place a formal cutoff ##\Lambda##, pretending to work in ##d##-dimensional momentum space and write ##\int d^d p \sim \Lambda^d##, subtracting the "divergences" with counterterms cancels every loop diagram, leaving me with tree diagrams that are zero because of the delta functions in the propagators. The only non-zero amplitudes in the theory are diagrams with loops, and my interpretation is that "physical" particles are ##\bar{\psi}\psi## pairs.

Knowing that this form of the propagator can be traced back to the absence of spatial derivatives, I tried to add a formal laplacian term in the action, with a parameter ##\epsilon## in front. Pretending again to be in ##d## dimensions, the basic loop integral evaluates to ##\sim (m^2/\epsilon)^{d/2}## which is again an overall divergent factor. Straight dimensional regularization of ##\int d^d p## gives zero.

I tried to modify the kinetic Lagrangian in a way that could extract a sensible finite part from ##\delta(\mathbf{x},\mathbf{x})##, without success. I tried to treat delta functions and propagators as distributions, as measures, as operators, but in those formalisms it's difficult to define loops. I read something about casual perturbation theory and the R- and W-operations, and applying that scheme to the "squared delta" in this case just gives me an arbitrary constant. I'm supposed to fix it with some renormalization condition, but I don't know what I should do.

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