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In Schwarz's QFT he introduces in chapter 27 the Spin-Helicity formalism as a way of calculating gluon-gluon interactions much easier than going through all the Feynman calculus from the beginning to the end. It seems so amazing, but I am not sure I understand what is the fundamental difference between the 2 approaches that makes one a lot easier than the other. The main difference (on which spin-helicity formalism is actually based) is the fact that momentum is treated like a bi-spinor and not a vector. Why is this approach so much simpler? Can someone give me some intuition to it? Thank you!

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The most practical traditional method of calculating scattering amplitudes comes from Feynman diagrams, where one draws all the relevant diagrams up to the desired order for a given theory, then using the set of Feynman rules relevant to the theory, one can assign a mathematical quantity to each diagram and hence sum everything nicely. The problem is that while manageable for small problems, the number of diagrams and intermediate calculations rapidly grows and quickly becomes too difficult to do by hand.

The spinor-helicity formalism is the starting point for studying the structure of amplitudes. By expressing the spinor brackets and using this formalism, we can uncover newer methods which expose the simplicity; cancellations are abundant and it is much easier to do with things, particularly teamed with recursion relations.

You mention that 'the spinor formalism uses bi-spinors rather than vectors'. Even doing Feynman diagram calculations uses polarisation vectors and Dirac spinors (in QCD,QED diagrams etc). It is naturally to keep track of quantum numbers such as spin or helicity (and colour in QCD), and it is much easier to deal with in SH-formalism.

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