Its never a bad idea to expose yourself to different school of thoughts, especially in theoretical physics. My experience with QFT steams mainly from Condensed matter theory starting with the idea of second quantization and my rendezvous with relativity was during my early work in gravitational physics. Owing to the beauty of QFT, I would love to expose myself to the school of thoughts in QFT in the high energy physics (HEP) side. As a condensed matter theorist, this is exciting because, you often come across a new mathematics tool (for instance, braid groups or sewing matrices..) but seldom come across mathematical tools that totally alter your way of looking at systems. One such tool is indeed relativistic QFT. My plan is to post a series of blogs while I re-brush my QFT from scratch with a HEP eye. I will be using various books and tools to understand and I may take longer than I anticipated to get a glimpse of this beauty, but nonetheless, here is an humble attempt.

We will start by laying out the basic mathematical frame work and understand the origins/reduction of QFT to Quantum mechanics(QM) in this first of many posts

We will start by first constructing a relativistic theory with hints of QM in it. Given a set of 4 coordinates $x^\nu$ for observer $A$ and $x^{\prime \nu}$ for observer $B$, we have the conditions for their relation as

\[\begin{equation} \eta_{\mu \nu}dx^\nu dx^\mu = \eta_{\rho \sigma}dx^{\prime \sigma} dx^{\prime \rho} \label{eq:1} \end{equation}\]

this gives us

\[\begin{equation} \implies \eta_{\mu \nu} \frac{dx^\nu}{dx^{\prime \sigma}} \frac{dx^\mu}{dx^{\prime \rho}}=\eta_{\sigma\rho} \label{eq:2} \end{equation}\]

This condition above (if one closely thinks) dictates that the transformation above needs to be a linear transformation. And not just any transformation, but rather a poincare transformation, which is given by

\[\begin{equation} x^\mu=\Lambda^\mu_\nu x^\nu + \alpha^\mu \label{eq:3} \end{equation}\]

These transformations $:=(\Lambda,\alpha)$ form a set and in-fact have a structure on the set which forms a group with a trivial addition called the Poincare group ($P_4$). With some work, one can see that the product of the group is defined as

\[\begin{equation} (\Lambda,\alpha)*(\Lambda^\prime,\alpha^\prime)=(\Lambda\Lambda^\prime,\Lambda^\prime\alpha+\alpha^\prime) \label{eq:4} \end{equation}\]

Now to build a theory that is consistent with QM where we generally need to impose symmetries, we need to construct transformations that are either unitary $U$ or anti-unitary $\mathcal{U}$ upon this already existing framework. The first step is to have a Hilbert space and then impose the unitary condition. Thus we have the ingredients

  1. Hilbert space $\mathcal{H}$
  2. To any $(\Lambda,\alpha)\in P_4$; we need to have $U(\Lambda,\alpha):\mathcal{H}\rightarrow\mathcal{H}$

But one cannot choose any $U$ that does the above trick. To be consistent also one needs to have

  1. $U(I,0)=e^{i\phi}I$ where $\phi\in R$
  2. $U(\Lambda^\prime ,a^\prime) U(\Lambda ,a)=U(\Lambda\Lambda^\prime,\Lambda^\prime\alpha+\alpha^\prime)e^{i\phi\left[(\Lambda,a),(\Lambda^\prime,a^\prime)\right]}$

This in fact is where the difficulty of QFT comes in. From out mathematical friends, we know that representations of these $U$ are called “Projective Unitary Representations” and unfortunately, there exists no finite dimensional unitary representations for Poincare group ($P_4$). This leads to the famous known fact that QFT is always infinite dimensional. For those familiar with classical mech, one can clearly understand the beauty of having finite dimensional representation where $SO(3)$, which is the group of 3 dimensional rotations can be represented in finite dimensional matrices. The mathematical term for this ineptness is called compactness. It turns out that $SO(3)$ is compact while $P_4$ is not.

One interesting subgroup of $P_4$ is

\[\begin{equation} V_t=\{(I,(t,0,0,0))|t\in R\} \subset P_4 \label{eq:5} \end{equation}\]

This is nothing but a group of time transformations. If one write $U$ for this sub group of $P_4$, we have $V_t \rightarrow U(I,(t,0,0,0))$, which is a one parameter of unitary family such that

\[\begin{equation} V_s V_t = V_{s+t} \end{equation}\]

which are nothing but solutions of schrodinger’s equation which is given

\[\begin{equation} \frac{V_t}{dt}=i\hat{H}V_t \end{equation}\]

where $\hat{H}$ is a self adjoint operator. One important side note is that, all such unitary transformations of the above subgroup that have 1 particle Hilbert space, can actually be classified and labeled by just 2 numbers! These numbers are nothing but the mass of the particle and the spin of particle (helicity).

But unfortunately, it turns out that one can never model any working system ( will sad truth coming from condensed matter theorist) with a single particle hilbert space because of non-locality. This makes the idea of QFT even more harder as one needs not just infinite dimensional representation, but rather $2^n$ infinite dimensional representation for n spin 1/2 particles.

We will further thus explore the idea of quantum fields in next post.