# Dimensional analysis and coupling constant

## Main Question or Discussion Point

I'm learning QFT from Srednicki's book. He introduces dimensional analysis in section 12. Coupling constant needs to be dimensionless in order to avoid a number of problems. So phi-cubed theory needs 6 space time dimensions to make sense, but isn't phi-4th-powered theory just right for our 4 space time dimensions? Why to use phi-cubed theory?

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vanhees71
Gold Member
2019 Award
Yes! I never understood why Srednicky deals with $\phi^3$ theory. It's totally useless and physically ill-defined to begin with. Otherwise it's a great book to learn the fundamentals of QFT.

The constraint for renormalizability is that your coupling constants have positive mass dimension, as you get from power counting. $\phi^3$ theory in 1+3 space-time dimensions is superrenormalizable in a formal sense, and you can do formal calculations within perturbation theory. However, it doesn't have a stable ground state and thus is flawed already in the very beginning.

Yes! I never understood why Srednicky deals with $\phi^3$ theory. It's totally useless and physically ill-defined to begin with. Otherwise it's a great book to learn the fundamentals of QFT.

The constraint for renormalizability is that your coupling constants have positive mass dimension, as you get from power counting. $\phi^3$ theory in 1+3 space-time dimensions is superrenormalizable in a formal sense, and you can do formal calculations within perturbation theory. However, it doesn't have a stable ground state and thus is flawed already in the very beginning.
I guess phi-cubed theory is super-renormalizable in 4 spacetime dimensions so the calculations will be easier. Phi-4th-powered theory is well-defined and just right, but it needs especially careful treatments.
Another question is, which L1 best describes the process of a photon creates an electron and a positron? It should be a hermitian field interacts with a non-hermitian field. It's natural to write φχ†χ, but φφχ†χ seems to be better since it has a ground state. However, Feynman's rule implies that in 4-th-powered theories, the total number of incoming and outgoing particles is even. More generally, if want the number of incoming and outgoing particles to be arbitrary, the exponent needs to be odd. But odd exponent L1 will not have a ground state. How to fix it?

vanhees71
Gold Member
2019 Award
Hm, just do QED ;-). Spinor QED is, in my opinion, the simplest theory to deal with. The only problem didactics wise is that it is a U(1) gauge theory and thus you need some more formalism to quantize it properly (best in terms of the path-integral formalism using Feynman-Faddeev-Popov techniques) before you can derive the Feynman rules, compared to a simple toy model like $phi^4$ theory. Also these toy model has its own right in terms of the linear O(N) $\sigma$ model, describing pions and $\sigma$ mesons, employing spontaneous symmetry breaking, etc.

Hm, just do QED ;-). Spinor QED is, in my opinion, the simplest theory to deal with. The only problem didactics wise is that it is a U(1) gauge theory and thus you need some more formalism to quantize it properly (best in terms of the path-integral formalism using Feynman-Faddeev-Popov techniques) before you can derive the Feynman rules, compared to a simple toy model like $phi^4$ theory. Also these toy model has its own right in terms of the linear O(N) $\sigma$ model, describing pions and $\sigma$ mesons, employing spontaneous symmetry breaking, etc.
QED,,,I'm not sure whether Srednicki has introduced QED in his book. He talks a little about photon field and eletro-dynamics. After all I'm learning QFT from his book. Are there any good books on QFT(or StringTheory) other than Srednicki's? I may need more books.

vanhees71
Gold Member
2019 Award
The best books on the subject are

Weinberg, Quantum Theory of Fields, Cambridge University Press

However, that's not good as an introduction. My newest favorite is

M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press

I learned it during my Diploma Thesis from Ryder and Bailin&Love. The latter is particularly nice, concerning the path-integral formalism.

Srednicky is a good book too. I only don't like his engagement with $\phi^3$ theory. His strength in my opinion is the careful treatment of the LSZ reduction formalism. Of course, he also covers QED and also the entire Standard Model to some extent in Part III.

The best books on the subject are

Weinberg, Quantum Theory of Fields, Cambridge University Press

However, that's not good as an introduction. My newest favorite is

M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press

I learned it during my Diploma Thesis from Ryder and Bailin&Love. The latter is particularly nice, concerning the path-integral formalism.

Srednicky is a good book too. I only don't like his engagement with $\phi^3$ theory. His strength in my opinion is the careful treatment of the LSZ reduction formalism. Of course, he also covers QED and also the entire Standard Model to some extent in Part III.
Thanks very much. $\phi^3$ theory is just a toy before going to QED. He actually introduces Feynman rules in Part I, not $\phi^3$ theory. Is it possible to skip some sections and go directly to QED?

vanhees71
It should be possible. Just try it. You can always read the parts on $\phi^3$ theory, if needed.