Pauli–Villars ghost fields?

[pines-demon]

TL;DR

How do I think of Pauli–Villars regularization as ghosts?

I understand how to regularize the integrals in a electron vertex function using Pauli–Villars regularization, however in books like Schwartz this is seen as having introduced some "ghosts fields". How do I get that idea? When writing in that vertex function
$$\frac{1}{k^2-m^2}\to\frac{1}{k^2-m^2}-\frac{1}{k^2-M^2}$$
for some large ghost mass $M$. How do I understand the inclusion of these particles?
Is it that the electron is taken as a ghost particle during renormalization? or is the right side some new fermion that can be added/substracted to the propagator somehow? How does the Feynman diagram change? Do they modify the Lagrangian?

 

[Haborix]

It has been awhile since I studied ghosts in a QFT class. My recollection is that ghosts, e.g., Faddeev-Popov ghosts, are normally used in the context of non-abelian gauge theories to account for equivalence of different field configurations related by gauge transformations in the path-integral formalism. I can't really see the parallel here with those ghosts. It seems like a bit of hand-wavey storytelling to relate Pauli-Villars regularization to ghosts. But, like I said, it has been awhile since I thought about these things.

 

[Demystifier]

Pauli-Villars ghost is not a Faddeev-Popov ghost, but they share some similarities. You can think of PV ghost as a new particle, having the same spin as the electron, but quantized as a boson, rather than a fermion. This provides the opposite sign in the propagator, which leads to the desired cancellation. Having the wrong spin-statistics relation, it does not satisfy the assumptions of the spin-statistic theorem, so it cannot be a physical particle. This means that it cannot appear in outer lines of a Feynman diagram, but only in the internal lines. Unlike FP ghost, the PV ghost is not described by a Lagrangian, but only by a modified Feynman rule for a propagator. The Feynman diagram looks the same as without the ghost, but the mathematical formula for writing down the propagator of the internal line is modified. This is an ad hoc modification of Feynman rules, there is no deeper physical meaning of it.

 

[pines-demon]

It has been awhile since I studied ghosts in a QFT class. My recollection is that ghosts, e.g., Faddeev-Popov ghosts, are normally used in the context of non-abelian gauge theories to account for equivalence of different field configurations related by gauge transformations in the path-integral formalism. I can't really see the parallel here with those ghosts. It seems like a bit of hand-wavey storytelling to relate Pauli-Villars regularization to ghosts. But, like I said, it has been awhile since I thought about these things.

Pauli-Villars ghost is not a Faddeev-Popov ghost, but they share some similarities. You can think of PV ghost as a new particle, having the same spin as the electron, but quantized as a boson, rather than a fermion. This provides the opposite sign in the propagator, which leads to the desired cancellation. Having the wrong spin-statistics relation, it does not satisfy the assumptions of the spin-statistic theorem, so it cannot be a physical particle. This means that it cannot appear in outer lines of a Feynman diagram, but only in the internal lines. Unlike FP ghost, the PV ghost is not described by a Lagrangian, but only by a modified Feynman rule for a propagator. The Feynman diagram looks the same as without the ghost, but the mathematical formula for writing down the propagator of the internal line is modified. This is an ad hoc modification of Feynman rules, there is no deeper physical meaning of it.

Well actually Schwartz does introduce a Lagrangian with a PV ghost fermion and a PV ghost photon, but it seems as you say that it is mostly to motivate the "ghostly" appearance even if we can do without it.

 

[dextercioby]

Sorry for the off-topic, but any particular value from looking into this aged topic? I mean MS and MSbar are simpler and can be used throughout the Standard Model.

 

[pines-demon]

Sorry for the off-topic, but any particular value from looking into this aged topic? I mean MS and MSbar are simpler and can be used throughout the Standard Model.

Just reading from the standard books (Peskin). Where can I read about MS/MSbar?

Edit: isn’t MS some kind of renormalization, you still need to regularize right? Did you mean dimensional regularization?

 

[dextercioby]

Dimensional regularization first, then MS or MSbar. I used the brief treatment of Bailin & Love. Another nice option would be P. Ramond.

 

[Demystifier]

Sorry for the off-topic, but any particular value from looking into this aged topic? I mean MS and MSbar are simpler and can be used throughout the Standard Model.

Dimensional regularization first, then MS or MSbar. I used the brief treatment of Bailin & Love. Another nice option would be P. Ramond.

Dimensional regularization is great for practical purposes, but it does not provide a good intuitive conceptual picture of what is really going on. Not many physicists find plausible that the physical quantities are finite because we really live in $4+\epsilon$ dimensions. From that point of view, Pauli-Villars ghosts might look more plausible. But in my opinion, the most plausible picture is provided by the Wilson philosophy of regularization/renormalization. According to this philosophy, our theories are just effective theories valid at long distances, while at smaller distances the nature is described by some unknown completely different theory, so we make a cutoff reflecting the idea that our known theories should not be applied at those smaller distances.

 

[Haborix]

Dimensional regularization is great for practical purposes, but it does not provide a good intuitive conceptual picture of what is really going on. Not many physicists find plausible that the physical quantities are finite because we really live in 4+ϵ dimensions. From that point of view, Pauli-Villars ghosts might look more plausible. But in my opinion, the most plausible picture is provided by the Wilson philosophy of regularization/renormalization. According to this philosophy, our theories are just effective theories valid at long distances, while at smaller distances the nature is described by some unknown completely different theory, so we make a cutoff reflecting the idea that our known theories should not be applied at those smaller distances.

Indeed, so many stories in QFT to help us cope with the lack of rigor that we truly desire, ha!