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Has anyone written about the graded generalization of the Moyal–Weyl product/star product, that is, where the original algebra is already graded? Is it just a matter of signs?

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Yes, it's just putting signs correctly. Martin Bordemann has a preprint from the 90s where he adapted Fedosov's construction in the graded setting. If you are only interested in the flat situation things are even much easier. The Grassmann part then get deformed into a Clifford-like algebra. You can find this in a recent preprint of mine with a lot of related analytical discussion. The algebraic core however seems to be folklore.

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Here's a reference that explicitly writes down the Grassman version of the integral formula for the Wigner-Weyl-Moyal star-product (Eq. (54)):

I. Galaviz, H. Garcia-Compean, M. Przanowski, F.J. Turrubiates Weyl-Wigner-Moyal Formalism for Fermi Classical Systems Annals Phys. 323 267-290 (2008) (arXiv, doi)

There is a number of references to earlier related work, including some to the older Russian literature, where some of these formulas may have appeared earlier.

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Many thanks to all - nothing beats the collective's memory! – Jim Stasheff Jul 29 '14 at 14:07

Graded generalizations of the Moyal–Weyl product must undoubtedly have been known already early-on to F.A. Berezin and his students, see e.g. Refs. 1-2.

Graded versions (where both Grassmann-even and Grassmann-odd are present) can e.g. be found in the works of E.S. Fradkin and his students, see e.g. Ref. 3.


  1. F.A. Berezin, The Method of Second Quantization, 1966.

  2. F.A. Berezin and M.S. Marinov, Ann. Physics (NY) 104 (1977) 336.

  3. I. A. Batalin and E. S. Fradkin, Ann. Inst Henri Poincare A49 (1988) 145; Section 4.A.

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