We uncover a method of calculation that proceeds at every step without fixing the gauge or specifying details of the regularisation scheme. Results are obtained by iterated use of integration by parts and gauge invariance identities. Calculations can be performed almost entirely diagrammatically. The method is formulated within the framework of an exact renormalisation group for QED. We demonstrate the technique with a calculation of the one-loop beta function, achieving a manifestly universal result, and without gauge fixing.

/pdf/manifestly-gauge-invariant-qcd-4vpeebf9oq.pdf

Manifestly gauge-invariant QCD

We uncover a method of calculation that proceeds at every step without fixing the gauge or specifying details of the regularisation scheme. Results are obtained by iterated use of integration by parts and gauge invariance identities. The initial stages can even be computed diagrammatically. The method is formulated within the framework of an exact renormalization group for SU(N) Yang-Mills gauge theory, incorporating an effective cutoff through a manifest spontaneously broken SU(N|N) gauge invariance. We demonstrate the technique with a compact calculation of the one-loop beta function, achieving a manifestly universal result, and without gauge fixing, for the first time at finite N.

A proposal for a manifestly gauge invariant and universal calculus in Yang-Mills theory

We construct a gauge-invariant regularisation scheme for pure SU(N) Yang–Mills theory in dimension four or less (for N = ∞ in all dimensions), with a physical cutoff scale Λ, by using covariant higher derivatives and spontaneously broken SU(N|N) supergauge invariance. Providing their powers are within certain ranges, the covariant higher derivatives cure the superficial divergence of all but a set of one-loop graphs. The finiteness of these latter graphs is ensured by properties of the supergroup and gauge invariance. In the limit Λ → ∞, all the regulator fields decouple and unitarity is recovered in the renormalized pure SU(N) Yang–Mills theory. By demonstrating these properties, we prove that the regularisation works to all orders in perturbation theory.

Gauge invariant regularisation via SU(N|N)

Manifestly gauge invariant QED

A formulation of supersymmetric quantum mechanics in terms of sesquilinear forms is presented. Otherwise occurring rather complicated domain questions are solved in a simple and natural way. The authors formulate the axioms and describe a number of models. Supersymmetry transformations are implemented as superunitary transformations in a space defined over a Grassmann algebra, generalised Bogoliubov transformations acting in that space are also formulated.

http://iopscience.iop.org/article/10.1088/0305-4470/20/13/030/pdf

Supersymmetric quantum mechanics defined as sesquilinear forms

to Supersymmetry.- Superfields.- Grand Unification With and Without Supersymmetry.- Yang-Mills Theories with Global and Local Supersymmetry - Higgs and Superhiggs Effect in Unified Field Theories.- Supergroups and Their Representations.

L. Urrutia

Papers

Introduction to supersymmetry in particle and nuclear physics