Monday, March 26, 2018

mathematical physics - What are Grassmann (even/odd) numbers used in superalgebras?


Are Grassmann numbers a concept of graded Lie algebras or is something specific to superalgebras? What are they (i.e: how are they defined, important properties, etc.)? Is there a reasonable introduction to them?



I think that what makes me wonder a little is, since there does not seem to be a sensible constructivist approach to these entities (other than accepting them as the entities that satisfy the required properties) is there nothing that stops someone from going into 'constructing' meta-superalgebras by defining 'numbers' κi, such that, e.g.,


κiκj=θk(Grassmann odd),

κiκjκmκn=θp(Grassmann even).


So I define such numbers as 'square-roots' of grassmann a-numbers. It seems nothing stops this process ad infinitum. Maybe there is some property I'm missing that will allow the algebra to be closed but I don't know what that could be.


Btw, I think this is a great reference Phys.SE question regarding this topic: "Velvet way" to Grassmann numbers.



Answer



Here I will just make a couple of general remarks.


1) Graded algebras usually refer to Z- or N-graded algebras, while superalgebras are Z2-graded algebras.


2) Grassmann numbers are oddly graded supernumbers.


Please click on the links for further information, important properties and references.


References:



1) Bryce De Witt, "Supermanifolds", Cambridge Univ. Press, 1992.


2) Deligne, Pierre and John W. Morgan, "Notes on Supersymmetry (following Joseph Bernstein)". Quantum Fields and Strings: A Course for Mathematicians (1999). American Mathematical Society. pp. 41–97. ISBN 0-8218-2012-5.




Concerning v3 of the question. The κi's correspond to a Z4 grading, and there are indeed research works in that direction. However, many properties of numbers and supernumbers do not generalize easily to Zn-grading with n>2. For instance, I think that already Berezin showed that it is not possible to define a useful notion of (Berezin) determinant of matrices with Zn-graded entries if n>2.


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