Operator regularization is one of the excellent prescriptions for studying gauge theories. Among the many regularization prescriptions Dimensional regularization and Pre-regularization are the best methods for evaluating loop diagrams perturbatively. On the other hand Operator regularization can also be said one of the best methods for studying gauge theories because of its twofold use. With this prescription one can adopt pathintegral method with the combination of background field quantization and Schwinger expansion to find the result of the required problem without considering any Feynman diagrams. Also from this prescription one can consider Feynman diagrams and evaluating these diagrams using the Operator regularization prescription. In this paper we have shown how one can use both the options of Operator regularization method to evaluate Feynman diagrams in QED in (3+1) dimensional space-time.

Quantum field theory can be understood through gauge theories. It is already established that the gauge theories can be studied either perturbatively or non-perturbatively. Perturbative means using Feynman diagrams and non-perturbative means using Path-integral method. Operator regularization (OR) is one of the exceptional methods to study gauge theories because of its twofold prescriptions. That means in OR two types of prescriptions have been introduced, which gives us the opportunity to check the result in self consistent way. In an earlier paper, we have evaluated basic QED loop diagrams in (3 + 1) dimensions using the both methods of OR and Dimensional regularization (DR). Then all three results have been compared. It is seen that the finite part of the result is almost same. In this paper, we are interested to evaluate the same basic loop diagrams in (2 + 1) space-time dimensions, because of two reasons: the main reason in (2 + 1) space-time dimensions, these loops diagrams are finite, on other hand, there are divergences in (3 + 1) space-time dimensions and the other reason is to see validity of using OR to evaluate Feynman loop diagrams in all dimensions. Here we have used both prescriptions of OR and DR to evaluate the basic loop diagrams and compared the results. Interestingly the results are almost same in all cases.