Category Archives: BRST

Quantization and Dirac Cohomology

For many years I’ve been fascinated by the topic of “Dirac cohomology” and its possible relations to various questions about quantization and quantum field theory. At first I was mainly trying to understand the relation to BRST, and wrote some … Continue reading

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BRST and Dirac Cohomology

For the last couple years I’ve been working on the idea of using what mathematicians call “Dirac Cohomology” to replace the standard BRST formalism for handling gauge symmetries. So far this is just in a toy model: gauge theory in … Continue reading

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Notes on BRST IX: Clifford Algebras and Lie Algebras

Note: I’ve started putting together the material from these postings into a proper document, available here, which will be getting updated as time goes on. I’ll be making changes and additions to the text there, not on the blog postings. … Continue reading

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Notes on BRST VIII: Clifford Algebras

Clifford Algebras Clifford algebras are well-known to physicists, in the guise of matrix algebras generated by the [tex]\gamma[/tex] -matrices first used in the Dirac equation. They also have a more abstract formulation, which will be the topic of this posting. … Continue reading

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Notes on BRST VII: The Harish-Chandra Homomorphism

The Casimir element discussed in the last posting of this series is a distinguished quadratic element of the center [tex]Z(\mathfrak g)=U(\mathfrak g)^\mathfrak g[/tex] (note, here [tex]\mathfrak g[/tex] is a complex semi-simple Lie algebra), but there are others, all of which … Continue reading

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Notes on BRST VI: Casimir Operators

For the case of [tex]G=SU(2)[/tex], it is well-known from the discussion of angular momentum in any quantum mechanics textbook that irreducible representations can be labeled either by j, the highest weight (here, highest eigenvalue of [tex]J_3[/tex] ), or by [tex]j(j+1)[/tex], … Continue reading

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I should finish writing the next installment of the Notes on BRST series soon, but thought I’d post here about two pieces of BRST-related news, concerning the “B” and the “T”. The “T” in BRST is I.V. Tyutin, whose Lebedev … Continue reading

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Notes on BRST V: Highest Weight Theory

In the last posting we discussed the Lie algebra cohomology [tex]H^*(\mathfrak g, V)[/tex] for [tex]\mathfrak g[/tex] a semi-simple Lie algebra. Because the invariants functor is exact here, this tells us nothing about the structure of irreducible representations in this case. … Continue reading

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Notes on BRST IV: Lie Algebra Cohomology for Semi-simple Lie Algebras

In this posting I’ll work out some examples of Lie algebra cohomology, still for finite dimensional Lie algebras and representations. If [tex]G[/tex] is a compact, connected Lie group, it can be thought of as a compact manifold, and as such … Continue reading

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Notes on BRST III: Lie Algebra Cohomology

The Invariants Functor The last posting discussed one of the simplest incarnations of BRST cohomology, in a formalism familiar to physicists. This fits into a much more abstract mathematical context, and that’s what we’ll turn to now. Given a Lie … Continue reading

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