Domenico Fiorenza Introduction

Introduction

This wiki is meant to be a dictionary between physics and mathematics for the standard model. The standard model is a theory of three of the four fundamental interactions: electro-magnetic, weak and strong. As in every “quantum” theory, the particles fields are vectors $\Psi$ of a certain Hilbert space $H$, satisfying covariant differential equations (example: Kleing-Gordon, Dirac, Proca, etc.). The basic mechanism of the model is that a particle, posseding several quantum parameters (charge, isospin and color charge), interacts with another particle via a so-said intermediary boson (example: photon, W, Z, gluon), with a certain intensity according to quantum numbers. Remark that the theory has been confirmed by every experiment in the 20th century and each particle predicted by the theory, except the Higgs boson, has been observed. First we want to summarize a physical approach to the theory:

1. Since the masses are intended to be generated by the Higgs mechanism, one considers mass-less quantum fields.

2. It is convenient to group together similar quantum fields in a so-said n-plet (example: in electroweak theory neutrino and electron form a 2-plet)

3. One can now write a Lagrangian $L$ for the n-plet and then study the group $G$ of “global” symmetries that conserve $L$. Remark that these symmetries of $L$ are often broken somehow by mass terms in $L$ (example: consider a lagrangian $L$ for the 2-plet neutrino-electron, both mass-less. One verifies that symmetry group of $L$ is $U(1) \times SU(2)$. If the mass term for the electron is taken into account the symmetry is violated).

4. A theory of field with global symmetries has no actually physical meaning: since in Minkowski space there are points causally not connected, it is more fitting a theory with local symmetries. By Young-Mills theory one can write a lagrangian invariant under local transformation. Essentially the task is performed by replacing the derivative with a covariant derivative. The connections associated to the new derivative are the “Gauge Fields”.

5. The only piece missing in the picture are the masses. We already remarked that the mass terms are symmetry-breakers. It could sound a little pretentious generating masses still preserving symmetries. But here it comes the beauty of the Higgs mechanism: by adding to the theory a brand new boson, one can generate the mass term missing for both gauge fields and particles, without losing any symmetry in $L$.

While this approach furnish a “ready-to-use” theory, that could be easily verified with actual experiments, it is not clear what kind of mathematical tools one should use in this process. The first step in creating a dictionary physics-mathematics should be understanding what is in the free case a quantum number: mass, spin, charge etc..

Recall that $\mathbb{R}^4$ equipped with a real metric $g=diag(+1,-1,-1,-1)$ is the Minkowski space. We define the Poincaré group $P$, the group of transformation that preserves the metric. Thus

where $O(3,1)$ is the Lorentz group. We already remarked that system states in quantum mechanics are vector of a Hilbert space $H$. If a state $\Psi$ is a solution of a certain relativistic wave function, a relation occurs between $\Psi$ in different frames connected by a Poincaré transformation: if ${\Psi}^\prime$ is $\Psi$ in a new frame then:

where $U$ is a unitary representation of $P$. For $\Psi$ and $\Psi^\prime$ are both states for the system, then $\forall (\Lambda,a) \in P, U(\Lambda,a)\Psi \in H$. In this behavior classifying particles is the same than classifying irreducible representations (from now on irreps) of $P$. This is done using the induced representation method: one can verify that irreps $V$ for $P$ are parametrized by two values, say $(m,j)$ such that $m \in \mathbb{R}$ and $j \in \mathbb{Z}/2$. We call $m$ mass and $j$ spin. It is now clear that a particle of “spin” $j$ and “mass” $m$ is a vector of the irrep $V_{m,j}$. The other quantum numbers presented so far (charge, isospin and color charge) are still missing. Proceeding in analogy to the case of mass and spin, we want to present them as indexes for irreps of a certain group $G$. To get this done we need a typical result borrowed by representation theory:

Proposition: Let $G_1$ and $G_2$ be two groups. Then every irreducible representation of the direct product $G_1 \times G_2$ is a tensor product $V_1 \otimes V_2$ with $V_i$ irrep for $G_i$. (citation needed)

Given a group $G$, we can now decompose a rep $V$ for $P \times G$ as follwing:

where $\lambda$ is parametrizing irreps of $G$. Finally, assigning $G$ for each type of interaction, we end labeling each $\Psi$ with a complete set of quantum numbers.

1. $G=U(1)$ $\rightarrow$ electro-magnetic case $\rightarrow$ quantum number: charge.

2. $G=SU(2)$ $\rightarrow$ weak interaction $\rightarrow$ quantum number: isospin.

3. $G=SU(3)$ $\rightarrow$ strong interaction $\rightarrow$ quantum number: color charge.

This is an algebraic point of view: so far there is no trace of any covariant different equation. This has been solved by Wigner: there is a “recipe” for, given the representation, writing a covariant differential equation (see Boulanger-Beckaert - Unitary representations of the Poincaré group in any spacetime dimension ).

We have achieved a nice background for the theory: on one side we have the irreps of $P \times G$, on the other side we have a set of covariant differential equations. Since for each covariant equation we can assign at least one lagrangian $L$, whenever we perform some change to the lagrangian, such that the corresponding covariant differential equation changes as well, we want to know exactly what representation we result with and vice versa. This should be an overall strategy in writing a physics-mathematics dictionary.

interaction: TODO