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### Geometry and interactions

Or, how do we mathematically describe the interaction of particles?

In my previous post, I addressed some questions concerning the nature of the wavefunction, the most truthful mathematical representation of a particle. Now let us make this simple idea more complete, getting closer to the deep mathematical structure of particle physics. This post is a bit more “mathematical” than the last, and will likely make the most sense to those who have taken a calculus course. But if you bear with me, you may also come to discover that this makes particle interactions even more attractive!

The field theory approach considers wavefunctions as fields. In the same way as the temperature field $$T(x,t)$$ gives the value of the temperature in a room at space $$x$$ and time $$t$$, the wavefunction $$\phi (x,t)$$ quantifies the probability of presence of a particle at space point $$x$$ and time $$t$$.
Cool! But if this sounds too abstract to you, then you should remember what Max Planck said concerning the rise of quantum physics: “The increasing distance between the image of the physical world and our common-sense perception of it simply indicates that we are gradually getting closer to reality”.

Almost all current studies in particle physics focus on interactions and decays of particles. How does the concept of interaction fit into the mathematical scheme?

The mother of all the properties of particles is called the Lagrangian function. Through this object a lot of properties of the theory can be computed. Here let’s consider the Lagrangian function for a complex scalar field without mass (one of the simplest available), representing particles with electric charge and no spin:

$$L(x) = \partial_\mu \phi(x)^* \partial^\mu \phi(x)$$.

Mmm… Is it just a bunch of derivatives of fields? Not really. What do we mean when we read $$\phi(x)$$? Mathematically, we are considering $$\phi$$ as a vector living in a vector space “attached” to the space-time point $$x$$. For the nerds of geometry, we are dealing with fiber bundles, structures that can be represented pictorially in this way:

The important consequence is that, if $$x$$ and $$y$$ are two different space-time points, a field $$\phi(x)$$ lives in a different vector space (fiber) with respect to $$\phi(y)$$! For this reason, we are not allowed to perform operations with them, like taking their sum or difference (it’s like comparing a pear with an apple… either sum two apples or two pears, please). This feature is highly non-trivial, because it changes the way we need to think about derivatives.

In the $$L$$ function we have terms containing derivatives of the field $$\phi(x)$$. Doing this, we are actually taking the difference of the value of the field at two different space-time points. But … we just outlined that we are not allowed to do it! How can we solve this issue?

If we want to compare fields pertaining to the same vector space, we need to slightly modify the notion of derivative introducing the covariant derivative $$D$$:

$$D_\mu = \partial_\mu + ig A_\mu(x)$$.

Here, on top of the derivative $$\partial$$, there is the action of the “connection” $$A(x)$$, a structure which takes care of “moving” all the fields in the same vector space, and eventually allows us to compare apples with apples and pears with pears.
So, a better way to write down the Lagrangian function is:

$$L(x) = D_\mu \phi(x)^* D^\mu \phi(x)$$.

If we expand $$D$$ in terms of the derivative and the connection, $$L$$ reads:

$$L(x) = \partial_\mu \phi(x)^* \partial^\mu \phi(x) +ig A_\mu (\partial^\mu \phi^* \phi – \phi^* \partial^\mu \phi) + g^2 A^2 \phi^* \phi$$.

Do you recognize the role of these three terms? The first one represents the propagation of the field $$\phi$$. The last two are responsible for the interactions between the fields $$\phi, \phi^*$$ and the $$A$$ field, referred to as the “photon” in this context.

This slightly hand-waving argument involving fields and space-time is a simple handle to understand how the interactions among particles emerge as a geometric feature of the theory.

If we consider more sophisticated fields with spin and color charges, the argument doesn’t change. We need to consider a more refined “connection” $$A$$, and we could see the physical interactions among quarks and gluons (namely QCD, Quantum Chromo Dynamics) emerging just from the mathematics.

Probably the professor of geometry in my undergrad course would call this explanation “Spaghetti Mathematics”, but I think it can give you a flavor of the mathematical subtleties involved in the theory of particle physics.