30.1 The Grassmann algebra of polynomials on anticommuting generators

30.1 The Grassmann algebra of polynomials on anticommuting generators

Concept links · terms present in this machine draft; source roles are unverified: vector space · dual space

Given a phase space $M={\mathbf{R}}^{2d}$ , one gets classical observables by taking polynomial functions on $M$. These are generated by the linear functions $q_j,p_j,j=$ $1,\dots ,d,$ which lie in the dual space $\mathcal{M}=M^{\ast }$ One can instead start with a real vector space $V={\mathbf{R}}^n$ with $n$ not necessarily even, and again consider the space $V^{\ast }$ of linear functions on $V_{:}$ , but with a diferent notion of multiplication, one that is anticommutative on elements of $V^{\ast }$ . Using such a multiplication, an anticommuting analog of the algebra of polynomials on $V$ can be generated in the following manner, beginning with a choice of basis elements ${\xi }_j$ of $V^{\ast }$ :

Definition (Grassmann algebra)

The algebra over the real numbers generated

by ${\xi }_j,j=1,\dots ,n$ , satisfying the relations

$${\xi }_j{\xi }_k+{\xi }_k{\xi }_j=0$$

is called the Grassmann algebra.

Note that these relations imply that generators satisfy ${\xi }_j^2=0$ . Also note that sometimes the Grassmann algebra product of ${\xi }_j$ and ${\xi }_k$ is denoted ${\xi }_j\wedge {\xi }_k$ We will not use a diferent symbol for the product in the Grassmann algebra, relying on the notation for generators to keep straight what is a generator of a conventional polynomial algebra $(e.g.,q_j\mathrm{or}{p}_j)$ and what is a generator of a Grassmann algebra $(e.g.,{\xi }_j)$

The Grassmann algebra is the algebra ${\Lambda }^{\ast }(V^{\ast })$ of antisymmetric multilinear forms on $V$ discussed in section 9.6, except that we have chosen a basis of $V$ and have written out the definition in terms of the dual basis ${\xi }_j$ of $V^{\ast }$ . It is sometimes also called the “exterior algebra”. This algebra behaves in many ways like the polynomial algebra on ${\mathbf{R}}^n$ , but it is finite dimensional as a real vector space, with basis

$$1,{\xi }_j,{\xi }_j{\xi }_k,{\xi }_j{\xi }_k{\xi }_l,\dots ,{\xi }_1{\xi }_2\dots {\xi }_n$$

for indices $j<k<l<\cdots$ · taking values $1,2,\dots ,n$ . As with polynomials, monomials are characterized by a degree (number of generators in the product), which in this case takes values from 0 only up to n. $\bar{\mathbf{\Gamma }}{\mathbf{\Lambda }}^k({\mathbf{R}}^n)$ is the subspace of ${\Lambda }^{\ast }({\mathbf{R}}^n)$ of linear combinations of monomials of degree $k$.

Digression (Diferential forms). Readers may have already seen the Grassmann algebra in the context of diferential forms on ${\mathbf{R}}^n$ . These are known to physicists as “antisymmetric tensor ${fields}^{\prime \prime }$ , and given by taking elements of the exterior algebra ${\mathbf{\Lambda }}^{\ast }({\mathbf{R}}^n)$ with coeficients not constants, but functions on ${\mathbf{R}}^n$ This construction is important in the theory of manifolds, where at a point $x$ in a manifold $M$, one has a tangent space $T_{x}M$ and its dual space $(T_{x}M{)}^{\ast }.A$ set of local coordinates $x_j$ on $M$ gives basis elements of $(T_{x}M{)}^{\ast }$ denoted by dx_$j$ and diferential forms locally can be written as sums of terms of the form

$$f(x_1,x_2,\dots ,x_n)d{x}_j\wedge \cdots \wedge d{x}_k\wedge \cdots \wedge d{x}_l$$

where the indices $j,k,l$ satisfy $1\le j<k<l\le n$

A fundamental principle of mathematics is that a good way to understand a space is in terms of the functions on it. What we have done here can be thought of as creating a new kind of space out of ${\mathbf{R}}^n$ , where the algebra of functions on the space is ${\Lambda }^{\ast }({\mathbf{R}}^n)$ , generated by coordinate functions ${\xi }_j$ with respect to a basis of ${\mathbf{R}}^n$ . The enlargement of conventional geometry to include new kinds of spaces such that this makes sense is known as “supergeometry”, but we will not attempt to pursue this subject here. Spaces with this new kind of geometry have functions on them, but do not have conventional points since we have seen that one can’t ask what the value of an anticommuting function at a point is.

Remarkably, an analog of calculus can be defined on such unconventional spaces, introducing analogs of the derivative and integral for anticommuting functions (i.e., elements of the Grassmann algebra). For the case $n=1$ , an arbitrary function is

$$F(\xi )=c_0+c_1\xi$$

and one can take

$$\frac{\partial }{\partial \xi }F=c_1$$

For larger values of $n,$ an arbitrary function can be written as

$$F({\xi }_1,{\xi }_2,\dots ,{\xi }_n)=F_A+{\xi }_j{F}_B$$

where $F_A,F_B$ are functions that do not depend on the chosen ${\xi }_j$ (one gets $F_B$ by using the anticommutation relations to move ${\xi }_j$ all the way to the left). Then one can define

$$\frac{\partial }{\partial {\xi }_j}F=F_B$$

This derivative operator has many of the same properties as the conventional derivative, although there are unconventional signs one must keep track of. An unusual property of this derivative that is easy to see is that one has

$$\frac{\partial }{\partial {\xi }_j}\frac{\partial }{\partial {\xi }_j}=0$$

Taking the derivative of a product one finds this version of the Leibniz rule for monomials $F$ and $G$

$$\frac{\partial }{\partial {\xi }_j}(FG)=\left(\frac{\partial }{\partial {\xi }_j}F\right)G+(-1{)}^{|F|}F\left(\frac{\partial }{\partial {\xi }_j}G\right)$$

where $|F|$ is the degree of the monomial $F.$

A notion of integration (often called the “Berezin integral”) with many of the usual properties of an integral can also be defined. It has the peculiar feature of being the same operation as diferentiation, defined in the $n=1$ case by

$$\int (c_0+c_1\xi )d\xi =c_1$$

and for larger n by

$$\int F\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)d{\xi }_{1}d{\xi }_2\dots d{\xi }_n=\frac{\partial }{\partial {\xi }_n}\frac{\partial }{\partial {\xi }_{n-1}}\dots \frac{\partial }{\partial {\xi }_1}F=c_n$$

where $c_n$ is the coeficient of the basis element ${\xi }_1{\xi }_2\cdot \cdot \cdot {\xi }_n$ in the expression of $F$ in terms of basis elements.

This notion of integration is a linear operator on functions, and it satisfies an analog of integration by parts, since if

$$F=\frac{\partial }{\partial {\xi }_j}G$$

then

$$\int Fd{\xi }_j=\frac{\partial }{\partial {\xi }_j}F=\frac{\partial }{\partial {\xi }_j}\frac{\partial }{\partial {\xi }_j}G=0$$

using the fact that repeated derivatives give zero.

Chapter contents · Next section: 30.2 Pseudo-classical mechanics and the fermionic Poisson bracket · 中文版本

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