# Why do we say that in non-relativistic limit we need only two component spinor?

Why do we say that in non-relativistic limit we need only two component spinor? (As in Schrödinger equation, we do not even talk of spinors,... they are one component object) I have read this statement in several books discussing non relativistic limit of Dirac equation.

## 1 Answer

The term "Schrodinger equation" is ambiguous, and can sometimes refer to the abstract equation $H|\psi> = E|\psi>$ and sometimes refer to more specific things such as the spacial portion of the non-relativistic wave-function. The non-relativistic wave-function of a particle with spin does involve spinors, so the general Schrodinger equation applied to it has both a spacial part and a spin part.

It's true that the spinors you need to describe a non-relativistic particle are only 2-component rather than 4-component, as long as the particle's kinetic energy is much less than its rest energy ($\frac{p^2}{2m} << mc^2$)

The easiest way I can think of to explain the difference is in terms of anti-particles. Non-relativistic quantum mechanics only describes particles, there is no such thing as anti-particles. So it makes sense that you only need half the number of degrees of freedom. The same 4-component Dirac spinor in quantum field theory is used to represent both the electron and the positron. Another way to say this is that there are both positive-energy and negative-energy solutions to the Dirac equation. But in non-relativistic quantum mechanics, there is no reason to consider antiparticles or negative-energy solutions. We only care about describing electrons (or other similar particles). If you take the non-relativistic limit of a 4-component Dirac spinor, you just end up with 2 of the components always being the same, and the other 2 components always being the same. So you may as well just discard the duplicated parts and only use 2-components.

In terms of group theory, the spacetime group for relativistic quantum mechanics is the Lorentz group SO(3,1). When you look at all representations of that, you find a trivial scalar representation, some spinor representations, some vector representations, etc. The spinor representations are better thought of as SU(2)xSU(2), which is locally isomorphic to SO(3,1). Since there are 2 copies of SU(2), you have two 2-component Weyl spinors (which can be stacked on top of each other to build a 4-component Dirac spinor). But in non-relativistic quantum mechanics, time is not a dimension, it's just a separate parameter. The group of rotations in 3D space is just SO(3). So you only need one copy of SU(2) to find a spinor representation that's locally isomorphic to that. Adding a second copy would just be redundant. This means you only need a single 2-component spinor.