There is a new beer commercial currently airing on the Belgian YouTube. The brand seems to have shifted the focus of their adverts from hard-working manly manual laborers to good-looking geeks. Which might tell something on the kind of persons they are willing to attract. In this commercial, two university students are pictured struggling on difficult mathematics on the blackboard of a deserted lecture hall. They finally succeed to crack the problem up and consequently celebrate the result of their hard work with a rewarding draft.

I have always loved when general media feature supposedly difficult mathematics on a blackboard. When this happens, I can’t help but pause the video and work out everything written, trying to identify with what specific domain of mathematics it is concerned and if it is all consistent.

Here is what I found on this occasion. Don’t try to click on the play button from the still below…

The upper-left panel displays the action of an element of the symmetry group $U(1)$ on a scalar $\phi$. The following lines show the Lagrangian for the dynamics of $\phi$ with mass $\mu$ coupled to the boson field associated with $U(1)$ through the inclusion of the term proportional to the square of the field strength tensor $F_{\mu\nu}^2$ and the vector potential $A_\mu$ shown in the expression of the covariant derivative $D_\mu$. The second line of the panel explicits $\mu^2$ as a function of the temperature $T$. If this goes below a critical temperature $T_c$, the value of the scalar potential is minimum at a non-zero value of $\phi$ called the vacuum expectation value (vev, for short) around which the perturbations have to be computed. The shape of the resulting potential is drawn on the bottom-right black board panel has the recognisable Mexican hat potential. This is a very well known process in quantum field theory called spontaneous symmetry breaking used to model phase transitions in condensed matter.

In high energy physics, this same phenomenon is responsible for the Higgs mechanism when it is applied to the more extended Yang-Mills symmetry group $SU(2)\times U(1)_{YM}$ breaking to $U(1)_{EM}$ of electromagnetism. In which case the vev of the scalar field $\phi$ provides mass to the gauge bosons of electroweak interaction $W^+$, $W^-$ and $Z^0$. This is the mechanism exposed on the upper-right panel.

The bottom-left panel shows a simpler example of this mechanism in action when the simple $U(1)$ group is spontaneously broken with a term of the shape $\frac{m^2}{2}\eta^2$ appearing in the final expression of the Lagrangian. The other resulting terms giving the relative strength of the interactions with the gauge fields.

All of this is really consistent and well tied together. This is the subject of highly specialised theoretical physics and it is great that this is shown here without any mistakes even though this can hardly be considered as bleeding-edge research and is more textbook level. Still, a very nice touch !

Interestingly enough The final panel of the board has nothing to do with the rest and is simply an exercise of computing the value of the Euler Gamma function at a point. This is probably the will of the makers of the commercial to come up with a complicated computation with 1966 as result. This being the year of the introduction of that particular beer on the Belgian market. The result is only approximate though. I get

But I think it is close enough.

One cute touch is the little drawing plot of the Gamma function on the real line at the bottom-right for extra geekiness.

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