nLab relativistic quantum field theory

Idea

In the context of the physics of fields one speaks of relativistic (quantum) field theory to indicate that the underlying spaces considered are Lorentzian spacetimes that model configurations of the field of gravity according to the theory of “general relativity”, whence the name, and that the equations of motion respect this structure, in that they respect tangent space-wise Lorentz invariance (i.e. the Cartan geometry strucure, see at first-order formulation of gravity).

This is in contrast to other variants of field theory such as for instance

• non-relativistic field theory? typically used in solid state physics, where certainly effects of general relativity are irrelevant, but where even effects of special relativity may often be ignored;

• Euclidean field theory where there is no time direction at all (really: statistical mechanics), which however, in good situations, may be related to relativistic field theory by Wick rotation. In particular, this provides, under suitable conditions, a definition of thermal quantum field theory, see there for more.

Mathematically, the hallmark of relativistic field theory is the key role played by causal locality, notably in the rigorous mathematical formulation of relativistic perturbative quantum field theory via causal perturbation theory (also called perturbative AQFT).

This marks also the contrast to “Euclidean field theory”, where the underlying “spacetimes” are Riemannian manifolds (with Euclidean instead of Lorentzian signature, whence the name). While in good situations relativistic and Euclidean field theory may be related by Wick rotation (Osterwalder-Schrader theorem) generally they are very different.

Related concepts

• relativistic particle

• causal locality

• AQFT on curved spacetimes

• locally covariant perturbative algebraic quantum field theory