### About

### Quantum Theory, Groups and Representations

### Not Even Wrong: The Book

### Subscribe to Blog via Email

Join 666 other subscribers### Recent Comments

- The Crisis in String Theory is Worse Than You Think... 41

Scott Caveny, Matthew Foster, Peter Woit, SDR, Marvin, Low+Math,+Meely+Interacting [...] - Various Items 15

Stephane Dubedat, Peter Woit, Pasquale Di Cesare, Andrew, Mathematician, James [...] - All Langlands, all the time 7

toto, Peter Woit, Jim+Eadon, kitchin, Peter Woit, Quasiparticular [...] - Is Spacetime Unraveling? 19

Peter Woit, Marty, Peter Woit, Bertie, Mathias, Mark Hillery [...] - Richard S. Hamilton 1943-2024 6

David Ross, James Allen Isenberg, Matt Grayson, Jason Starr, Jason Starr, Ian Stopher

- The Crisis in String Theory is Worse Than You Think... 41
### Categories

- abc Conjecture (21)
- Book Reviews (122)
- BRST (13)
- Euclidean Twistor Unification (15)
- Experimental HEP News (153)
- Fake Physics (8)
- Favorite Old Posts (50)
- Film Reviews (15)
- Langlands (52)
- Multiverse Mania (163)
- Not Even Wrong: The Book (27)
- Obituaries (35)
- Quantum Mechanics (23)
- Quantum Theory: The Book (7)
- Strings 2XXX (27)
- Swampland (20)
- This Week's Hype (141)
- Uncategorized (1,290)
- Wormhole Publicity Stunts (15)

### Archives

### Links

### Mathematics Weblogs

- Alex Youcis
- Alexandre Borovik
- Anton Hilado
- Cathy O'Neil
- Daniel Litt
- David Hansen
- David Mumford
- David Roberts
- Emmanuel Kowalski
- Harald Helfgott
- Jesse Johnson
- Johan deJong
- Lieven Le Bruyn
- Mathematics Without Apologies
- Noncommutative Geometry
- Persiflage
- Pieter Belmans
- Qiaochu Yuan
- Quomodocumque
- Secret Blogging Seminar
- Silicon Reckoner
- Terence Tao
- The n-Category Cafe
- Timothy Gowers
- Xena Project

### Physics Weblogs

- Alexey Petrov
- AMVA4NewPhysics
- Angry Physicist
- Capitalist Imperialist Pig
- Chad Orzel
- Clifford Johnson
- Cormac O’Raifeartaigh
- Doug Natelson
- EPMG Blog
- Geoffrey Dixon
- Georg von Hippel
- Jacques Distler
- Jess Riedel
- Jim Baggott
- John Horgan
- Lubos Motl
- Mark Goodsell
- Mark Hanman
- Mateus Araujo
- Matt Strassler
- Matt von Hippel
- Matthew Buckley
- Peter Orland
- Physics World
- Resonaances
- Robert Helling
- Ross McKenzie
- Sabine Hossenfelder
- Scott Aaronson
- Sean Carroll
- Shaun Hotchkiss
- Stacy McGaugh
- Tommaso Dorigo

### Some Web Pages

- Alain Connes
- Arthur Jaffe
- Barry Mazur
- Brian Conrad
- Brian Hall
- Cumrun Vafa
- Dan Freed
- Daniel Bump
- David Ben-Zvi
- David Nadler
- David Vogan
- Dennis Gaitsgory
- Eckhard Meinrenken
- Edward Frenkel
- Frank Wilczek
- Gerard ’t Hooft
- Greg Moore
- Hirosi Ooguri
- Ivan Fesenko
- Jacob Lurie
- John Baez
- José Figueroa-O'Farrill
- Klaas Landsman
- Laurent Fargues
- Laurent Lafforgue
- Nolan Wallach
- Peter Teichner
- Robert Langlands
- Vincent Lafforgue

### Twitter

### Videos

# 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

Posted in BRST
Comments Off on Quantization and Dirac Cohomology

## 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

Posted in BRST
16 Comments

## 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

Posted in BRST
4 Comments

## 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

Posted in BRST
6 Comments

## 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

Posted in BRST
2 Comments

## 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

Posted in BRST
8 Comments

## BRST News

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

Posted in BRST
4 Comments

## 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

Posted in BRST
4 Comments

## 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

Posted in BRST
2 Comments

## 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

Posted in BRST
6 Comments