Robert's Mistakes

about Grassmann variables

Sep 24, 2013

Grassmann variables were first used by the german mathematician and linguist Hermann Günther Graßmann (1809-1877). Together with the numbers they provide a minimalistic model for an anticommutative algebra and capture the essence of what is so different about anticommutative spaces. How does it work?

First, we introduce symbols for denoting Grassmann variables. Traditionally the greek letter theta is used θ_1, θ_2, ...

For our purposes, two symbols will do and i'll call them t and T because i love ascii. Again, traditionally one would use θ and θ-bar here, drawing an actual bar above the θ. Those symbols should anticommute with each other, that is, the following must hold:

tT = -Tt, sometimes written as tT + Tt = 0

contrasting commutativity, where we have:

tT = Tt, or tT - Tt = 0

One important thing to notice is that t*t = 0, easily to be seen by looking at the second formulation for anticommutativity. This simplifies things a lot. We can drop all and each term containing the same symbol twice.

As i'm not deep into this i'm not sure if it is true that every noncommutative algebra can be split into a commutative part and an anticommutative one. [ Update: No, this holds only for supercommutative algebras. Thanks to +John Baez and see the comments below for more ] - But it sure works for the physics application we have in mind here.

What does it mean? Extra dimensions?? - Fermions (e.g. electrons) have an anticommutative symmetry, swapping two fermions gives a minus sign. It's best not to interpret each theta as extra physical dimension, Susskind describes them as bookkeeping devices for tracking minus signs. See him working with thetas in his "Supersymmetry & Grand Unification: Lecture 5":

Supersymmetry & Grand Unification: Lecture 5

He goes through some calculations for you and demonstrates that working with thetas is much simpler than with numbers. You'll also learn that expressions with thetas are called odd, ones without even. And that integration and differentiation are equivalent when relating to theta. I'm not sure what to make of the rest of the series but there are some insights buried worth finding. At one point he remarks that supersymmetry is a contrived collection of symmetries just giving the right results. But it's not actually supposed to feel simple or natural.

This is how far i'll explain things here but i'd still like to fast forward you a bit towards understanding the #amplituhedron , as promised.

So how do Grassmann variables relate to grassmannians? About the latter wikipedia has to say (emphasis by me):

In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr(1, V) is the space of lines through the origin in V , so it is the same as the projective space P(V).

When looking for Grassmann algebra instead we learn that the Grassmannian is related to a construction using Grassmann algebra:

The Grassmannian of k-dimensional subspaces of V, denoted Gr_k(V), can be naturally identified with an algebraic subvariety of the projective space P(Λ^k V)

You might have heard about Λ under the name wedge product. It's the operator of the exterior algebra - also known as... Grassmann algebra! And it is anticommutative and what that means i just explained.

So next time, when you think of the amplituhedron as a polytope in many dimensions, you'll know that there is also some funny anticommutative stuff going on.