Point-splitting Regularization of Composite Operators and Anomalies

Abstract

The point-splitting regularization technique for composite operators is discussed in connection with anomaly calculation. We present a pedagogical and self-contained review of the topic with an emphasis on the technical details. We also develop simple algebraic tools to handle the path ordered exponential insertions used within the covariant and non-covariant version of the point-splitting method. The method is then applied to the calculation of the chiral, vector, trace, translation and Lorentz anomalies within diverse versions of the pointsplitting regularization and a connection between the results is described. As an alternative to the standard approach we use the idea of deformed point-split transformation and corresponding Ward-Takahashi identities rather than an application of the equation of motion, which seems to save the complexity of the calculations.

On the quantization of continuous non-ultralocal integrable systems

“…We conclude this section with remarks on the physical meaning of the •-product (3.12). It follows directly from the definition that this product is a point splitting type of regularization, which is frequently used in quantum field theory in order to regularize singular operator products, e.g., in relation to anomaly computations (for an overview and the relation of point-splitting regularization to other methods, see [43,44]). We recall the standard example of the axial fermionic electromagnetic current, regularized via point splitting method [45]:…”

Nonextremal Kerr/CFT on a stretched horizon

“…We assign the coordinates ϕ + and ϕ − as the right and left conformal coordinates, respectively. The two coordinates are slightly separated from the angular coordinate of the corotating frame at the horizon, φ = ϕ − Ω H t. We may consider these two coordinates as two-fold split of the angular coordinate of the corotating frame at the horizon in the spirit of point-splitting [50]. Now, we assume the existence of two Killing vectors, which are functions of either ϕ + or ϕ − , dubbed as the right (+) and left (−) handed Killing vectors.…”

Green’s functions and Hadamard parametrices for vector and tensor fields in general linear covariant gauges

“…In particular, it does not describe the two-point function of any particular preferred state, but instead specifies the singular part that any such state must have. Hadamard parametrices are used to define renormalized composite operators using the point-splitting method [4][5][6], which includes the renormalized stress tensor for scalars, spinors, vectors, gravitons and p-forms [7][8][9][10][11][12][13][14][15][16], in particular its trace anomaly [6,17,18], and the calculation of chiral anomalies [19][20][21][22]. More generally, they play a crucial role in constructing the local and covariant time-ordered products on curved spacetimes [23][24][25][26], which form the basis of renormalized perturbation theory on arbitrary (globally hyperbolic) curved backgrounds.…”