In some branches of physics, such as condensed matter and quantum optics, one deals exclusively with non-relativistic models. However, there are occasionally advantages to using a relativistic formulation in quantum theory; by embedding a theory in a larger framework, one often obtains new insights. It is therefore useful to be able to take the non-relativistic limit of generally covariant theories, both as an indication of how large or small relativistic effects are and as a cultural bridge between covariant physics and non-relativistic quantum theory.

Particles and anti-particles

There is no unified theory of particles and anti-particles in the non-relativistic field theory. Formally there are two separate theories. When we take the non-relativistic limit of a relativistic theory, it splits into two disjoint theories: one for particles, with only positive definite energies, and one for anti-particles, with only negative definite energies. Thus, a non-relativistic theory cannot describe the interaction between matter and anti-matter.

The Green functions and fields reflect this feature. The positive frequency Wightman function goes into the positive energy particle theory, while the negative frequency Wightman function goes into the negative energy anti-particle theory. The objects which one then refers to as the Wightman functions of the non-relativistic field theory are asymmetrical. In normal Schrödinger field theory for matter, one says that the zero temperature negative frequency Wightman function is zero.

Klein–Gordon field

The free scalar field

We begin by considering the Klein–Gordon action for a real scalar field, since this is the simplest of the cases and can be treated at the level of the action. It also reveals several subtleties in the way quantities are defined and the names various quantities go by.