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The Schemannian Reference

Grassmannian Calculus

"Schemannian" can do some easy Grassmannian calculus. To use that, you want to first include the relavent file.

(require "grassmannian-calculus.rkt")

In the current design, Grassmannian numbers are made by make-grassmannian; however, they add and multiple normal numbers by normal expressions (i.e., it doesn't cover the normal numbers by further tag system).

(make-grassmannian x) → grassmannian?
    x : expression?

Current supported functions include

(simplify-grassmannian exp) → grassmannian?
    exp : grassmannian?

(grassmannian-integrate exp var) → expression?
    exp : expression?
    var : grassmannian?

(grassmannian-deriv exp var) → expression?
    exp : expression?
    var : grassmannian?

in which the exp is basically some superfield, or say, normal expressions with some elements in it are grassmannian numbers.

For example, here is a piece of code about a two-dimensional superfield

(require "grassmannian-calculus.rkt")

(define theta1 (make-grassmannian 'theta1))
(define theta2 (make-grassmannian 'theta2))

(define superfield (make-sum (list 'a
                                   (make-product (list theta1 'b1))
                                   (make-product (list theta2 'b2))
                                   (make-product (list theta1 theta2 'c)))))

(grassmannian-integrate superfield theta1)