Spinors play a fundamental role in quantum field theory (QFT) and are used to describe particles with half-integer spin, such as electrons and neutrinos. In this context, spinors are mathematical objects that transform under certain representations of the Lorentz group, which encapsulates the symmetries of spacetime.
To understand the basics of how spinors work in QFT, let's start with a few key concepts:
Lorentz Group: The Lorentz group consists of all possible transformations that leave the spacetime interval invariant. It includes rotations and boosts (transformations between reference frames moving at different velocities). In four dimensions, the Lorentz group is described by SO(3,1) or sometimes denoted as SL(2,C) × SL(2,C).
Lorentz Representations: The Lorentz group has different representations, each describing how different types of physical objects (particles) transform under its symmetries. For spin-0 particles (e.g., scalar fields), the representation is trivial. Spin-1 particles (e.g., vectors) are described by the vector representation. Spinors, on the other hand, are associated with spin-1/2 particles.
Dirac Spinors: The most commonly used spinors in QFT are Dirac spinors, named after physicist Paul Dirac. Dirac spinors describe particles with spin-1/2, such as electrons. In four-dimensional spacetime, Dirac spinors have four components, forming a 4-dimensional complex vector.
Lorentz Transformations of Spinors: Dirac spinors transform under the Lorentz group via a specific matrix representation. These transformations mix the components of the spinor and account for the particle's rotation and boosts in spacetime. The transformation properties of spinors are different from those of vectors or scalars, reflecting their distinct behavior under rotations and boosts.
Spinor Fields: In QFT, spinors are promoted to fields, meaning that instead of describing a single particle, they describe a field that can give rise to multiple particles. Spinor fields are operators that create and annihilate particles with half-integer spin. They can be quantized and treated as quantum operators, with creation and annihilation operators acting on Fock space to create particle states.
Spinor Lagrangians: Lagrangians in QFT describe the dynamics of fields and their interactions. Spinor fields have their own Lagrangians, such as the Dirac Lagrangian, which incorporates the kinetic and interaction terms for spin-1/2 particles. These Lagrangians are used to derive the equations of motion for spinor fields, which determine how they evolve in time and space.
Overall, spinors in QFT provide a mathematical framework for describing and quantizing particles with half-integer spin, such as fermions. They have specific transformation properties under the Lorentz group and are represented by spinor fields in Lagrangian formalism, allowing us to study their dynamics and interactions.