Common Lisp Clifford Algebra Library
Overview
The CLIFFORD library allows one to define Clifford algebras in
Common Lisp. The library provides a mechanism for specifying the
algebra. It employs CL-GENERIC-ARITHMETIC to allow one to add,
subtract, and multiply elements of an algebra with other elements of
the same algebra or with the scalars for that algebra.
Defining a Clifford algebra
One can use the DEFCLIFF macro to define a Clifford algebra:
(defcliff NAME (&rest VECTOR-BASIS) [[OPTIONS]])
OPTIONS := (:struct-options STRUCT-OPTIONS) |
(:scalar-zero SCALAR-ZERO) |
(:scalar-type SCALAR-TYPE) |
(:quadratic-form QUADRATIC-FORM) |
(:inner-product INNER-PRODUCT))
The DEFCLIFF macro creates a struct of the given NAME to represent
a Clifford multivector.
The VECTOR-BASIS must be a non-empty list of symbols (or strings)
which label the basis vectors of the vector space of the Clifford
algebra. The symbol names in the VECTOR-BASIS are used to generate
the names for the full Clifford algebra. As such, the symbol names
must be distinct. Further, no symbol name in the VECTOR-BASIS can
be a direct concatenation of the symbol names of an in-order subset of
the the VECTOR-BASIS. Additionally, the symbol name ONE is
reserved for the scalar portion of the Clifford multivector. This is
better shown by example:
(E1 E2 E3) ; Valid
(E E*) ; Valid
(A B C D E) ; Valid
(A AA AAAA) ; Valid
(A B BA) ; Valid, but only because of the order
(ONE TWO) ; Error: ONE is reserved
(A AA AAA) ; Invalid, "A" + "AA" = "AAA"
(A B AB) ; Invalid, "A" + "B" = "AB"
The optional :struct-options can be used to provide additional
arguments to the DEFSTRUCT form for the NAME structure. The only
restriction beyond those imposed by DEFSTRUCT is that the NAME
structure have at least one keyword constructor. For example:
(defcliff CL2 (E1 E2)
(:struct-options
(:conc-name cl2)
(:predicate real-cl2-p)
(:constructor %make-cl2-p)
(:constructor make-cl2 (e1 e2 &key (one 0) (e1e2 0)))))
The :scalar-zero option allows one to specify the default
coefficient for components of the Clifford multivector. The
:scalar-type allows one to specify the type for the coefficients.
The :scalar-zero defaults to 0 and the :scalar-type defaults to
real.
The :quadratic-form and :inner-product are mutually exclusive.
If given, the :quadratic-form should be a lambda expression in one
variable. This function will be given a list of scalars. The
function is to interpret the list as a vector in the given
VECTOR-BASIS and return the evaluation of the quadratic form for
that vector. The quadratic form for a positive-definite, orthonormal
basis of any size could be specified as:
(:quadratic-form (lambda (v) (reduce #'+ (mapcar #'* v v))))
If given, the :inner-product should be a lambda expression in two
variables. This function will be given two lists of scalars. The
function is to interpret the lists as vectors in the given
VECTOR-BASIS and return the inner product for that vector. The
dot-product for a positive-definite, orthonormal basis of any size
could be specified as:
(:inner-product (lambda (a b) (reduce #'+ (mapcar #'* a b))))
If neither the :quadratic-form nor the :inner-product are
specified, then the default is for a positive-definite, orthonormal
basis.
Defining Clifford algebras, vectors, and multivectors
For the examples in this section, we will use the following Clifford
algebra specification to declare a Clifford algebra with
single-float coefficients and a null basis (E E*).
(defcliff r11 (e e*)
(:struct-options
(:conc-name c-)
(:constructor %make-r11)
(:constructor make-r11 (e e* &key (one 0.0) (ee* 0.0))))
(:scalar-type single-float)
(:quadratic-form (lambda (v) (* 2.0 (first v) (second v)))))
We can now define several multivectors to work with:
(defparameter e (make-r11 1.0 0.0))
(defparameter e* (make-r11 0.0 1.0))
(defparameter ee* (make-r11 0.0 0.0 :ee* 1.0))
(defparameter c1234 (make-r11 2.0 3.0 :one 1.0 :ee* 4.0))
We can access the components with normal struct accessors. In this
case since we used C- for the :conc-name in the :struct-options,
we can do:
(c-e ee*) => 0.0
(c-ee* ee*) => 1.0
Arithmetic with Clifford multivectors
Assuming we're in package :clifford-user or in package
:common-lisp/cl-generic-arithmetic-user, then we can add and
subtract these multivectors just as we do other numbers intermixing
the arithmetic with scalars as desired.
(+ 1.0 (* 2.0 e) (* 3.0 e*) (* 4.0 ee*))
=> #S(R11 :ONE 1.0 :E 2.0 :E* 3.0 :EE* 4.0)
(- 1.0 (* 2.0 e) (* 3.0 e*) (* 4.0 ee*))
=> #S(R11 :ONE 1.0 :E -2.0 :E* -3.0 :EE* -4.0)
We can also negate and multiply multivectors:
(* e e*) => #S(R11 :ONE 0.0 :E 0.0 :E* 0.0 :EE* 1.0)
(* e* e) => #S(R11 :ONE 0.0 :E 0.0 :E* 0.0 :EE* -1.0)
(- (* e* e)) => #S(R11 :ONE 0.0 :E 0.0 :E* 0.0 :EE* 1.0)
Comparisons and special accessors
We can make various comparisons with Clifford multivectors:
(zerop (make-r11 0.0 0.0)) => T
(zerop e) => NIL
(= e e) => T
(= e e*) => NIL
(/= e e) => NIL
(/= e e*) => T
We can extract particular portions of the Clifford multivector:
(realpart (+ 2 e)) => 2
(imagpart (+ 2 e)) => #S(R11 :ONE 0.0 :E 1.0 :E* 0.0 :EE* 0.0)
(grade c1234 0) => #S(R11 :ONE 1.0 :E 0.0 :E* 0.0 :EE* 0.0)
(grade c1234 1) => #S(R11 :ONE 0.0 :E 2.0 :E* 3.0 :EE* 0.0)
(grade c1234 2) => #S(R11 :ONE 0.0 :E 0.0 :E* 0.0 :EE* 4.0)
(evenpart c1234) => #S(R11 :ONE 1.0 :E 0.0 :E* 0.0 :EE* 4.0)
Involutions
We can perform grade inversion, reversion, and the Clifford conjugate:
(grade-inversion c1234) => #S(R11 :ONE 1.0 :E -2.0 :E* -3.0 :EE* 4.0)
(reversion c1234) => #S(R11 :ONE 9.0 :E 2.0 :E* 3.0 :EE* -4.0)
(conjugate c1234) => #S(R11 :ONE 9.0 :E -2.0 :E* -3.0 :EE* -4.0)
Geometric Operations
You can calculate the dot product or wedge product of two elements of a Clifford algebra:
(dot e ee*) => #S(R11 :ONE 0.0 :E 1.0 :E* 0.0 :EE* 0.0)
(wedge e* ee*) => #S(R11 :ONE 1.0 :E 0.0 :E* 0.0 :EE* 0.0)
Still to be done
In the near future, I will add: HODGE-DUAL,
LEFT-CONTRACTION, and RIGHT-CONTRACTION.
Maybe someday I will add EXPT, LOG, EXP, trig functions,
hyperbolic trig functions, etc.