Pointsplitting

Point-splitting, also called point-splitting regularization, is a method used in quantum field theory and related areas to tame singularities that arise in products of local field operators evaluated at the same spacetime point. By introducing a small separation between the points where the fields are evaluated and then taking the limit as the separation vanishes, one obtains finite, renormalized quantities while preserving important symmetries such as gauge invariance and general covariance.

The basic idea is to replace a product like φ(x)φ(x) with a regulated product φ(x)φ(x′), where x′ is infinitesimally separated from x (often along a prescribed direction in spacetime). In gauge theories, a parallel-transport operator U(x, x′) is inserted to maintain gauge invariance, yielding expressions such as φ(x) U(x, x′) φ(x′). Correlation functions are computed as functions of the separation, and the short-distance singularities are isolated using a Hadamard expansion or related subtraction terms. After subtracting these universal singular parts, one takes the limit as the separation goes to zero to define finite, renormalized composite operators, for example the stress-energy tensor ⟨Tμν⟩.

Historically, point-splitting was developed in the context of quantum field theory in curved spacetime, with foundational work by DeWitt, Christensen, and others in the 1970s and 1980s. It is a covariant and gauge-invariant regularization technique and is closely related to Hadamard renormalization and the use of bitensors and Synge’s world function.

Variants of the method include covariant point-splitting, Hadamard subtraction schemes, and explicit construction of finite parts via the Hadamard parametrix. The approach is a regularization tool: results depend on renormalization conditions chosen, but the procedure provides a consistent, symmetry-preserving way to define renormalized composite operators.