1-loop renormalization of a scalar theory in QFT - part 1

In this video we renormalize the theory for a scalar field with quartic interactions. There will be more videos on this in the following days.
The video is an excerpt from the course on Advanced Quantum Field Theory.
#qft #quantum #quantummechanics #quantumfieldtheory #calculus #integralcalculus #contourintegration #mathematics #mathematician #lovemath #mathlover #physics #physicslover #quantumphysics #theoreticalphysics #mathchallenge #physicschallenge

[@GeoffryGifari]

this is really helpful to understand what the inner loop diagrams do to a bare propagator

[@GeoffryGifari]

I sometimes forgot that there is a stack of diagrams under the banner of "1PI", and then those 1PIs also get stacked together to form the final correction

[@rootjay]

🎉🎉 🏅🏅

[@GeoffryGifari]

Can i say that the λ/4 𝜙^4 term only affect 3-point function and higher n-point functions (when vertices are involved)?
also, why are we evaluating the smallness of the correction in terms of its effect on exp(-S)? is it the path integral formula?

[@math.physics]

Sorry, I had missed this comment. If understand your first question, it does not seem to be completely right, because the lambda term also causes some higher-order corrections to the two-point correlation function (that we calculated in this lecture).
Also, I don't know if I understand your second question, but what I meant in the lecture is that the lambda correction to the propagator arises from the expansion of exp(-S) in the path integral (the previous videos on this channel show this, but the calculations are heavy...)

[@GeoffryGifari]

Do you remember why we can't have mixed terms like 𝜙∂𝜙 in the lagrangian?

[@math.physics]

That's easy if you think about it, that cross term would break Lorentz invariance ! In fact, the derivative operator has a Lorentz index, which must be contracted with another index to obtain a scalar.
However, one could argue that you could write other (Lorentz invariant) cross terms (containing higher derivatives, etc). The theory of the Renormalization group tells us that these operators are "irrelevant" as we change the scale of our theory appropriately.

[@GeoffryGifari]

@@math.physics Ah makes sense! in that case, if our theory has a vector field ψᵘ coupled to a scalar φ, can the Lagrangian contain Lorentz-invariant terms
- ∂ᵤψᵘ (4-divergence of ψᵘ, only 1 power of the field)
- (∂φ)ᵤψᵘ (contraction of a field with the derivative of another field)
?
thanks!

[@math.physics]

@@GeoffryGifari Yes! On the grounds of Lorentz invariance, those terms can appear. In interacting field theory, by imposing certain gauge transformations, terms which are similar to those may naturally appear.