Quantum Field Theory 1

It's never a bad idea to expose yourself to different schools of thought, especially in theoretical physics. My experience with QFT stems 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.

We will start by laying out the basic mathematical framework and understand the origins and reduction of QFT to Quantum Mechanics (QM).

The Poincare Group

We start by constructing a relativistic theory with hints of QM. Given a set of 4 coordinates for observer A and for observer B, the conditions for their relation are

This dictates that the transformation needs to be a linear transformation, specifically a Poincare transformation:

These transformations form the Poincare group , with product defined as

Unitary Representations and Hilbert Space

To build a theory consistent with QM where we need to impose symmetries, we need to construct transformations that are either unitary or anti-unitary upon this framework. We need:

1. Hilbert space

2. For any , we need

From our mathematical friends, we know that representations of these are called "Projective Unitary Representations" and unfortunately, there exist no finite dimensional unitary representations for the Poincare group. This leads to the famous fact that QFT is always infinite dimensional.

Connection to Quantum Mechanics

One interesting subgroup of is

This is nothing but a group of time transformations. Writing for this subgroup gives us a one-parameter unitary family such that , which are nothing but solutions of Schrodinger's equation:

where is a self-adjoint operator. All such unitary transformations of this subgroup with a one-particle Hilbert space can be classified and labeled by just 2 numbers: the mass of the particle and its spin (helicity).

Unfortunately, one can never model any working system with a single particle Hilbert space because of non-locality. This makes QFT even harder as one needs not just infinite dimensional representation, but rather infinite dimensional representation for spin-1/2 particles.

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