Geometric Algebra Module for SymPy

Author:Alan Bromborsky

Abstract

This document describes the implementation of a geometric algebra module in python that utilizes the sympy symbolic algebra library. The python module GA has been developed for coordinate free calculations using the operations (geometric, outer, and inner products etc.) of geometric algebra. The operations can be defined using a completely arbitrary metric defined by the inner products of a set of arbitrary vectors or the metric can be restricted to enforce orthogonality and signature constraints on the set of vectors. In addition the module includes the geometric, outer (curl) and inner (div) derivatives and the ability to define a curvilinear coordinate system. The module requires the numpy and the sympy modules.

What is Geometric Algebra?

Geometric algebra is the Clifford algebra of a real finite dimensional vector space or the algebra that results when a real finite dimensional vector space is extended with a product of vectors (geometric product) that is associative, left and right distributive, and yields a real number for the square (geometric product) of any vector [Hestenes,Lasenby]. The elements of the geometric algebra are called multivectors and consist of the linear combination of scalars, vectros, and the geometric product of two or more vectors. The additional axioms for the geometric algebra are that for any vectors \(a\), \(b\), and \(c\) in the base vector space:

\[\begin{equation*} \begin{array}{c} a \left( bc \right) = \left( ab \right) c \\ a \left( b+c \right) = ab+ac \\ \left( a + b \right) c = ac+bc \\ aa = a^{2} \in \Re \end{array} \end{equation*}\]

By induction these also apply to any multivectors.

Several software packages for numerical geometric algebra calculations are available from Doran-Lazenby group and the Dorst group. Symbolic packages for Clifford algebra using orthongonal bases such as \(e_{i}e_{j}+e_{j}e_{i} = 2\eta_{ij}\), where \(\eta_{ij}\) is a numeric array are available in Maple and Mathematica. The symbolic algebra module, GA, developed for python does not depend on an orthogonal basis representation, but rather is generated from a set of \(n\) arbitrary symbolic vectors, \(a_{1},a_{2},\dots,a_{n}\) and a symbolic metric tensor \(g_{ij} = a_{i}\cdot a_{j}\).

In order not to reinvent the wheel all scalar symbolic algebra is handled by the python module sympy.

The basic geometic algebra operations will be implemented in python by defining a multivector class, MV, and overloading the python operators in Table 1 where A and B are any two multivectors (In the case of +, -, *, ^, |, <<, and >> the operation is also defined if A or B is a sympy symbol or a sympy real number).

Operation Result
A+B Sum of multivectors
A-B Difference of multivectors
A*B Geometric product
A^B Outer product of multivectors
A|B Inner product of multivectors
A<B or A<<B Left contraction of multivectors
A>B or A>>B Right contraction of multivectors

Table 1. Multivector operations for symbolicGA

The option to use < or << for left contraction and > or >> is given since the < and > operators do not have r-forms (there are no __rlt__() and __rgt__() functions to overload) while << and >> do have r-forms so that x << A and x >> A are allowed where x is a scalar (symbol or integer) and A is a multivector. With < and > we can only have mixed modes (scalars and multivectors) if the multivector is the first operand.

Note

Except for < and > all the multivector operators have r-forms so that as long as one of the operands, left or right, is a multivector the other can be a multivector or a scalar (sympy symbol or integer).

Warning

Note that the operator order precedence is determined by python and is not neccessarily that used by geometric algebra. It is absolutely essential to use parenthesis in multivector expressions containing ^, |, <, >, << and/or >>. As an example let A and B be any two multivectors. Then A + A*B = A +(A*B), but A+A^B = (2*A)^B since in python the ^ operator has a lower precedence than the ‘+’ operator. In geometric algebra the outer and inner products and the left and right contractions have a higher precedence than the geometric product and the geometric product has a higher precedence than addition and subtraction. In python the ^, |, <, >, << and >> all have a lower precedence than + and - while * has a higher precedence than + and -.

Vector Basis and Metric

The two structures that define the MV (multivector) class are the symbolic basis vectors and the symbolic metric. The symbolic basis vectors are input as a string with the symbol name separated by spaces. For example if we are calculating the geometric algebra of a system with three vectors that we wish to denote as a0, a1, and a2 we would define the string variable:

basis = 'a0 a1 a2'

that would be input into the multivector setup function. The next step would be to define the symbolic metric for the geometric algebra of the basis we have defined. The default metric is the most general and is the matrix of the following symbols

\[\begin{equation*} g = \left[ \begin{array}{ccc} a0**2 & (a0.a1) & (a0.a2) \\ (a0.a1) & a1**2 & (a1.a2) \\ (a0.a2) & (a1.a2) & a2**2 \\ \end{array} \right] \end{equation*}\]

where each of the \(g_{ij}\) is a symbol representing all of the dot products of the basis vectors. Note that the symbols are named so that \(g_{ij} = g_{ji}\) since for the symbol function \((a0.a1) \ne (a1.a0)\).

Note that the strings shown in equation 1 are only used when the values of \(g_{ij}\) are output (printed). In the GA module (library) the \(g_{ij}\) symbols are stored in a static member list of the multivector class MV as the double list MV.metric (\(g_{ij}\) = MV.metric[i][j]).

The default definition of \(g\) can be overwritten by specifying a string that will define \(g\). As an example consider a symbolic representation for conformal geometry. Define for a basis

basis = 'a0 a1 a2 n nbar'

and for a metric

metric = '# # # 0 0, # # # 0 0, # # # 0 0, 0 0 0 0 2, 0 0 0 2 0'

then calling \(MV.setup(basis,metric)\) would initialize

\[\begin{equation*} g = \left[ \begin{array}{ccccc} a0**2 & (a0.a1) & (a0.a2) & 0 & 0\\ (a0.a1) & a1**2 & (a1.a2) & 0 & 0\\ (a0.a2) & (a1.a2) & a2**2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 2 & 0 \end{array} \right] \end{equation*}\]

Here we have specified that \(n\) and \(nbar\) are orthonal to all the \(a\)‘s, \(n**2 = nbar**2 = 0\), and \((n.nbar) = 2\). Using \(\#\) in the metric definition string just tells the program to use the default symbol for that value.

When MV.setup is called multivector representations of the basis local to the program are instantiated. For our first example that means that the symbolic vectors named a0, a1, and a2 are created and made available to the programmer for future calculations.

In addition to the basis vectors the \(g_{ij}\) are also made available to the programer with the following convention. If a0 and a1 are basis vectors, then their dot products are denoted by a0sq, a2sq, and a0dota1 for use as python program varibles. If you print a0sq the output would be a0**2 and the output for a0dota1 would be (a0.a1) as shown in equation 1. If the default value are overridden the new values are output by print. For examle if \(g_{00} = 0\) then “print a0sq” would output “0.”

More generally, if metric is not a string, but a list of lists or a two dimension numpy array, it is assumed that each element of metric is symbolic variable so that the \(g_{ij}\) could be defined as symbolic functions as well as variables. For example instead of letting \(g_{01} = (a0.a1)\) we could have \(g_{01} = cos(theta)\) where we use a symbolic \(\cos\) function.

Note

Additionally MV.setup has an option for an othogonal basis where the signature of the metric space is defined by a string. For example if the signature of the vector space is \((1,1,1)\) (Euclidian 3-space) set

metric = '[1,1,1]'

Likewise if the signature is that of spacetime, \((1,-1,-1,-1)\) then define

metric = '[1,-1,-1,-1]'.

Representation and Reduction of Multivector Bases

In our symbolic geometric algebra we assume that all multivectors of interest to us can be obtained from the symbolic basis vectors we have input, via the different operations available to geometric algebra. The first problem we have is representing the general multivector in terms terms of the basis vectors. To do this we form the ordered geometric products of the basis vectors and develop an internal representation of these products in terms of python classes. The ordered geometric products are all multivectors of the form \(a_{i_{1}}a_{i_{2}}\dots a_{i_{r}}\) where \(i_{1}<i_{2}<\dots <i_{r}\) and \(r \le n\). We call these multivectors bases and represent them internally with the list of integers \(\left[ i_{1},i_{2},\dots,i_{r}\right]\). The bases are labeled, for the purpose of output display, with strings that are concatenations of the strings representing the basis vectors. So that in our example [1,2] would be labeled with the string 'a1a2' and represents the geometric product a1*a2. Thus the list [0,1,2] represents a0*a1*a2.For our example the complete set of bases and labels are shown in Table 2

Note

The empty list, [], represents the scalar 1.

MV.basislabel = ['1', ['a0', 'a1', 'a2'], ['a0a1', 'a0a2', 'a1a2'],
                 ['a0a1a2']]
MV.basis      = [[], [[0], [1], [2]], [[0, 1], [0, 2], [1, 2]],
                 [[0, 1, 2]]]

Table 2. Multivector basis labels and internal basis representation.

Since there are \(2^{n}\) bases and the number of bases with equal list lengths is the same as for the grade decomposition of a dimension \(n\) geometric algebra we will call the collections of bases of equal length psuedogrades.

The critical operation in setting up the geometric algebra module is reducing the geomertric product of any two bases to a linear combination of bases so that we can calculate a multiplication table for the bases. First we represent the product as the concatenation of two base lists. For example a1a2*a0a1 is represented by the list [1,2]+[0,1] = [1,2,0,1]. The representation of the product is reduced via two operations, contraction and revision. The state of the reduction is saved in two lists of equal length. The first list contains symbolic scale factors (symbol or numeric types) for the corresponding interger list representing the product of bases. If we wish to reduce \(\left[ i_{1},\dots,i_{r}\right]\) the starting point is the coefficient list \(C = \left[ 1 \right]\) and the bases list \(B = \left[ \left[ i_{1},\dots,i_{r}\right] \right]\). We now operate on each element of the lists as follows:

  • contraction: Consider a basis list \(B\) with element \(B[j] = \left[ i_{1},\dots,i_{l},i_{l+1},\dots,i_{s}\right]\) where \(i_{l} = i_{l+1}\). Then the product of the \(l\) and \(l+1\) terms result in a scalar and \(B[j]\) is replaced by the new list representation \(\left[ i_{1},\dots,i_{l-1},i_{l+2},\dots,i_{r}\right]\) which is of psuedo grade \(r-2\) and \(C[j]\) is replaced by the symbol \(g_{i_{l}i_{l}}C[j]\).
  • revision: Consider a basis list \(B\) with element \(B[j] = \left[ i_{1},\dots,i_{l},i_{l+1},\dots,i_{s}\right]\) where \(i_{l} > i_{l+1}\). Then the \(l\) and \(l+1\) elements must be reversed to be put in normal order, but we have \(a_{i_{l}}a_{i_{l+1}} = 2g_{i_{l}i_{l+1}}-a_{i_{l+1}}a_{i_{l}}\) (From the geometric algebra definition of the dot product of two vectors). Thus we append the list representing the reduced element, \(\left[ i_{1},\dots,i_{l-1},i_{l+2},\dots,i_{s}\right]\), to the pseudo bases list, \(B\), and append \(2g_{i{l}i_{l+1}}C[j]\) to the coefficients list, then we replace \(B[j]\) with \(\left[ i_{1},\dots,i_{l+1},i_{l},\dots,i_{s}\right]\) and \(C[j]\) with \(-C[j]\). Both lists are increased by one element if \(g_{i_{l}i_{l+1}} \ne 0\).

These processes are repeated untill every basis list in \(B\) is in normal (ascending) order with no repeated elements. Then the coefficents of equivalent bases are summed and the bases sorted according to psuedograde and ascending order. We now have a way of calculating the geometric product of any two bases as a symbolic linear combination of all the bases with the coefficients determined by \(g\). The base multiplication table for our simple example of three vectors is given by (the coefficient of each psuedo base is enclosed with {} for clarity):

(1)(1) = 1
(1)(a0) = a0
(1)(a1) = a1
(1)(a2) = a2
(1)(a0a1) = a0a1
(1)(a0a2) = a0a2
(1)(a1a2) = a1a2
(1)(a0a1a2) = a0a1a2

(a0)(1) = a0
(a0)(a0) = {a0**2}1
(a0)(a1) = a0a1
(a0)(a2) = a0a2
(a0)(a0a1) = {a0**2}a1
(a0)(a0a2) = {a0**2}a2
(a0)(a1a2) = a0a1a2
(a0)(a0a1a2) = {a0**2}a1a2

(a1)(1) = a1
(a1)(a0) = {2*(a0.a1)}1-a0a1
(a1)(a1) = {a1**2}1
(a1)(a2) = a1a2
(a1)(a0a1) = {-a1**2}a0+{2*(a0.a1)}a1
(a1)(a0a2) = {2*(a0.a1)}a2-a0a1a2
(a1)(a1a2) = {a1**2}a2
(a1)(a0a1a2) = {-a1**2}a0a2+{2*(a0.a1)}a1a2

(a2)(1) = a2
(a2)(a0) = {2*(a0.a2)}1-a0a2
(a2)(a1) = {2*(a1.a2)}1-a1a2
(a2)(a2) = {a2**2}1
(a2)(a0a1) = {-2*(a1.a2)}a0+{2*(a0.a2)}a1+a0a1a2
(a2)(a0a2) = {-a2**2}a0+{2*(a0.a2)}a2
(a2)(a1a2) = {-a2**2}a1+{2*(a1.a2)}a2
(a2)(a0a1a2) = {a2**2}a0a1+{-2*(a1.a2)}a0a2+{2*(a0.a2)}a1a2

(a0a1)(1) = a0a1
(a0a1)(a0) = {2*(a0.a1)}a0+{-a0**2}a1
(a0a1)(a1) = {a1**2}a0
(a0a1)(a2) = a0a1a2
(a0a1)(a0a1) = {-a0**2*a1**2}1+{2*(a0.a1)}a0a1
(a0a1)(a0a2) = {2*(a0.a1)}a0a2+{-a0**2}a1a2
(a0a1)(a1a2) = {a1**2}a0a2
(a0a1)(a0a1a2) = {-a0**2*a1**2}a2+{2*(a0.a1)}a0a1a2

(a0a2)(1) = a0a2
(a0a2)(a0) = {2*(a0.a2)}a0+{-a0**2}a2
(a0a2)(a1) = {2*(a1.a2)}a0-a0a1a2
(a0a2)(a2) = {a2**2}a0
(a0a2)(a0a1) = {-2*a0**2*(a1.a2)}1+{2*(a0.a2)}a0a1+{a0**2}a1a2
(a0a2)(a0a2) = {-a0**2*a2**2}1+{2*(a0.a2)}a0a2
(a0a2)(a1a2) = {-a2**2}a0a1+{2*(a1.a2)}a0a2
(a0a2)(a0a1a2) = {a0**2*a2**2}a1+{-2*a0**2*(a1.a2)}a2+{2*(a0.a2)}a0a1a2

(a1a2)(1) = a1a2
(a1a2)(a0) = {2*(a0.a2)}a1+{-2*(a0.a1)}a2+a0a1a2
(a1a2)(a1) = {2*(a1.a2)}a1+{-a1**2}a2
(a1a2)(a2) = {a2**2}a1
(a1a2)(a0a1) = {2*a1**2*(a0.a2)-4*(a0.a1)*(a1.a2)}1+{2*(a1.a2)}a0a1+{-a1**2}a0a2
              +{2*(a0.a1)}a1a2
(a1a2)(a0a2) = {-2*a2**2*(a0.a1)}1+{a2**2}a0a1+{2*(a0.a2)}a1a2
(a1a2)(a1a2) = {-a1**2*a2**2}1+{2*(a1.a2)}a1a2
(a1a2)(a0a1a2) = {-a1**2*a2**2}a0+{2*a2**2*(a0.a1)}a1+{2*a1**2*(a0.a2)
                -4*(a0.a1)*(a1.a2)}a2+{2*(a1.a2)}a0a1a2

(a0a1a2)(1) = a0a1a2
(a0a1a2)(a0) = {2*(a0.a2)}a0a1+{-2*(a0.a1)}a0a2+{a0**2}a1a2
(a0a1a2)(a1) = {2*(a1.a2)}a0a1+{-a1**2}a0a2
(a0a1a2)(a2) = {a2**2}a0a1
(a0a1a2)(a0a1) = {2*a1**2*(a0.a2)-4*(a0.a1)*(a1.a2)}a0+{2*a0**2*(a1.a2)}a1
                +{-a0**2*a1**2}a2+{2*(a0.a1)}a0a1a2
(a0a1a2)(a0a2) = {-2*a2**2*(a0.a1)}a0+{a0**2*a2**2}a1+{2*(a0.a2)}a0a1a2
(a0a1a2)(a1a2) = {-a1**2*a2**2}a0+{2*(a1.a2)}a0a1a2
(a0a1a2)(a0a1a2) = {-a0**2*a1**2*a2**2}1+{2*a2**2*(a0.a1)}a0a1+{2*a1**2*(a0.a2)
                  -4*(a0.a1)*(a1.a2)}a0a2+{2*a0**2*(a1.a2)}a1a2

Base Representation of Multivectors

In terms of the bases defined an arbitrary multivector can be represented as a list of arrays (we use the numpy python module to implement arrays). If we have \(n\) basis vectors we initialize the list self.mv = [0,0,...,0] with \(n+1\) integers all zero. Each zero is a placeholder for an array of python objects (in this case the objects will be sympy symbol objects). If self.mv[r] = numpy.array([list of symbol objects]) each entry in the numpy.array will be a coefficient of the corresponding psuedo base. self.mv[r] = 0 indicates that the coefficients of every base of psuedo grade \(r\) are 0. The length of the array self.mv[r] is \(n \choose r\) the binomial coefficient. For example the psuedo basis vector a1 would be represented as a multivector by the list:

a1.mv = [0,numpy.array([numeric(0),numeric(1),numeric(0)]),0,0]

and a0a1a2 by:

a0a1a2.mv = [0,0,0,numpy.array([numeric(1)])]

The array is stuffed with sympy numeric objects instead of python integers so that we can perform symbolically manipulate sympy expressions that consist of scalar algebraic symbols and exact rational numbers which sympy can also represent.

The numpy.array is used because operations of addition, substraction, and multiplication by an object are defined for the array if they are defined for the objects making up the array, which they are by sympy. We call this representation a base type because the r index is not a grade index since the bases we are using are not blades. In a blade representation the structure would be identical, but the bases would be replaced by blades and self.mv[r] would represent the r grade components of the multivector. The first use of the base representation is to store the results of the multiplication tabel for the bases in the class variable MV.mtabel. This variable is a group of nested lists so that the geometric product of the igrade and ibase with the jgrade and jbase is MV.mtabel[igrade][ibase][jgrade][jbase]. We can then use this table to calculate the geometric product of any two multivectors.

Blade Representation of Multivectors

Since we can now calculate the symbolic geometric product of any two multivectors we can also calculate the blades corresponding to the product of the symbolic basis vectors using the formula

\[\begin{equation*} A_{r}\wedge b = \frac{1}{2}\left( A_{r}b-\left( -1 \right)^{r}bA_{r} \right), \end{equation*}\]

where \(A_{r}\) is a multivector of grade \(r\) and \(b\) is a vector. For our example basis the result is shown in Table 3.

1 = 1
a0 = a0
a1 = a1
a2 = a2
a0^a1 = {-(a0.a1)}1+a0a1
a0^a2 = {-(a0.a2)}1+a0a2
a1^a2 = {-(a1.a2)}1+a1a2
a0^a1^a2 = {-(a1.a2)}a0+{(a0.a2)}a1+{-(a0.a1)}a2+a0a1a2

Table 3. Bases blades in terms of bases.

The important thing to notice about Table 3 is that it is a triagonal (lower triangular) system of equations so that using a simple back substitution algorithym we can solve for the psuedo bases in terms of the blades giving Table 4.

1 = 1
a0 = a0
a1 = a1
a2 = a2
a0a1 = {(a0.a1)}1+a0^a1
a0a2 = {(a0.a2)}1+a0^a2
a1a2 = {(a1.a2)}1+a1^a2
a0a1a2 = {(a1.a2)}a0+{-(a0.a2)}a1+{(a0.a1)}a2+a0^a1^a2

Table 4. Bases in terms of basis blades.

Using Table 4 and simple substitution we can convert from a base multivector representation to a blade representation. Likewise, using Table 3 we can convert from blades to bases.

Using the blade representation it becomes simple to program functions that will calculate the grade projection, reverse, even, and odd multivector functions.

Note that in the multivector class MV there is a class variable for each instantiation, self.bladeflg, that is set to zero for a base representation and 1 for a blade representation. One needs to keep track of which representation is in use since various multivector operations require conversion from one representation to the other.

Warning

When the geometric product of two multivectors is calculated the module looks to see if either multivector is in blade representation. If either is the result of the geometric product is converted to a blade representation. One result of this is that if either of the multivectors is a simple vector (which is automatically a blade) the result will be in a blade representation. If a and b are vectors then the result a*b will be (a.b)+a^b or simply a^b if (a.b) = 0.

Outer and Inner Products, Left and Right Contractions

In geometric algebra any general multivector \(A\) can be decomposed into pure grade multivectors (a linear combination of blades of all the same order) so that in a \(n\)-dimensional vector space

\[\begin{equation*} A = \sum_{r = 0}^{n}A_{r} \end{equation*}\]

The geometric product of two pure grade multivectors \(A_{r}\) and \(B_{s}\) has the form

\[\begin{equation*} A_{r}B_{s} = \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|} + \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|+2} + \cdots + \left\langle A_{r}B_{s} \right\rangle_{r+s} \end{equation*}\]

where \(\left\langle \right\rangle_{t}\) projects the \(t\) grade components of the multivector argument. The inner and outer products of \(A_{r}\) and \(B_{s}\) are then defined to be

\[\begin{equation*} A_{r}\cdot B_{s} = \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|} \end{equation*}\]
\[\begin{equation*} A_{r}\wedge B_{s} = \left\langle A_{r}B_{s} \right\rangle_{r+s} \end{equation*}\]

and

\[\begin{equation*} A\cdot B = \sum_{r,s}A_{r}\cdot B_{s} \end{equation*}\]
\[\begin{equation*} A\wedge B = \sum_{r,s}A_{r}\wedge B_{s} \end{equation*}\]

Likewise the right (\(\lfloor\)) and left (\(\rfloor\)) contractions are defined as

\[\begin{equation*} A_{r}\lfloor B_{s} = \left \{ \begin{array}{cc} \left\langle A_{r}B_{s} \right\rangle_{r-s} & r \ge s \\ 0 & r < s \end{array} \right \} \end{equation*}\]
\[\begin{equation*} A_{r}\rfloor B_{s} = \left \{ \begin{array}{cc} \left\langle A_{r}B_{s} \right\rangle_{s-r} & s \ge r \\ 0 & s < r \end{array} \right \} \end{equation*}\]

and

\[\begin{equation*} A\lfloor B = \sum_{r,s}A_{r}\lfloor B_{s} \end{equation*}\]
\[\begin{equation*} A\rfloor B = \sum_{r,s}A_{r}\rfloor B_{s} \end{equation*}\]

The MV class function for the outer product of the multivectors mv1 and mv2 is

@staticmethod
def outer_product(mv1,mv2):
     product = MV()
     product.bladeflg = 1
     mv1.convert_to_blades()
     mv2.convert_to_blades()
     for igrade1 in MV.n1rg:
         if not isint(mv1.mv[igrade1]):
             pg1 = mv1.project(igrade1)
             for igrade2 in MV.n1rg:
                 igrade = igrade1+igrade2
                 if igrade <= MV.n:
                     if not isint(mv2.mv[igrade2]):
                         pg2 = mv2.project(igrade2)
                         pg1pg2 = pg1*pg2
                         product.add_in_place(pg1pg2.project(igrade))
     return(product)

The steps for calculating the outer product are:

  1. Convert mv1 and mv2 to blade representation if they are not already in that form.
  2. Project and loop through each grade mv1.mv[i1] and mv2.mv[i2].
  3. Calculate the geometric product pg1*pg2.
  4. Project the i1+i2 grade from pg1*pg2.
  5. Accumulate the results for each pair of grades in the input multivectors.

Warning

In the MV class we have overloaded the ^ operator to represent the outer product so that instead of calling the outer product function we can write mv1^ mv2. Due to the precedence rules for python it is absolutely essential to enclose outer products in parenthesis.

For the inner product of the multivectors mv1 and mv2 the MV class function is

@staticmethod
def inner_product(mv1,mv2,mode='s'):
    """
    MV.inner_product(mv1,mv2) calculates the inner

    mode = 's' - symmetic (Doran & Lasenby)
    mode = 'l' - left contraction (Dorst)
    mode = 'r' - right contraction (Dorst)
    """
    if isinstance(mv1,MV) and isinstance(mv2,MV):
        product = MV()
        product.bladeflg = 1
        mv1.convert_to_blades()
        mv2.convert_to_blades()
        for igrade1 in range(MV.n1):
            if isinstance(mv1.mv[igrade1],numpy.ndarray):
                pg1 = mv1.project(igrade1)
                for igrade2 in range(MV.n1):
                    igrade = igrade1-igrade2
                    if mode == 's':
                        igrade = igrade.__abs__()
                    else:
                        if mode == 'l':
                            igrade = -igrade
                    if igrade >= 0:
                        if isinstance(mv2.mv[igrade2],numpy.ndarray):
                            pg2 = mv2.project(igrade2)
                            pg1pg2 = pg1*pg2
                            product.add_in_place(pg1pg2.project(igrade))
        return(product)
    else:
        if mode == 's':
            if isinstance(mv1,MV):
                product = mv1.scalar_mul(mv2)
            if isinstance(mv2,MV):
                product = mv2.scalar_mul(mv1)
        else:
            product = None
    return(product)

The inner product is calculated the same way as the outer product except that in step 4, i1+i2 is replaced by abs(i1-i2) or i1-i2 for the right contraction or i2-i1 for the left contraction. If i1-i2 is less than zero there is no contribution to the right contraction. If i2-i1 is less than zero there is no contribution to the left contraction.

Warning

In the MV class we have overloaded the | operator for the inner product, > operator for the right contraction, and < operator for the left contraction. Instead of calling the inner product function we can write mv1|mv2, mv1>mv2, or mv1<mv2 respectively for the inner product, right contraction, or left contraction. Again, due to the precedence rules for python it is absolutely essential to enclose inner products and/or contractions in parenthesis.

Reverse of Multivector

If \(A\) is the geometric product of \(r\) vectors

\[\begin{equation*} A = a_{1}\dots a_{r} \end{equation*}\]

then the reverse of \(A\) designated \(A^{\dagger}\) is defined by

\[\begin{equation*} A^{\dagger} \equiv a_{r}\dots a_{1}. \end{equation*}\]

The reverse is simply the product with the order of terms reversed. The reverse of a sum of products is defined as the sum of the reverses so that for a general multivector A we have

\[\begin{equation*} A^{\dagger} = \sum_{i=0}^{N} \left\langle A \right\rangle_{i}^{\dagger} \end{equation*}\]

but

\[\begin{equation*} \left\langle A \right\rangle_{i}^{\dagger} = \left( -1\right)^{\frac{i\left( i-1\right)}{2}}\left\langle A \right\rangle_{i} \end{equation*}\]

which is proved by expanding the blade bases in terms of orthogonal vectors and showing that equation 2 holds for the geometric product of orthogonal vectors.

The reverse is important in the theory of rotations in \(n\)-dimensions. If \(R\) is the product of an even number of vectors and \(RR^{\dagger} = 1\) then \(RaR^{\dagger}\) is a composition of rotations of the vector \(a\). If \(R\) is the product of two vectors then the plane that \(R\) defines is the plane of the rotation. That is to say that \(RaR^{\dagger}\) rotates the component of \(a\) that is projected into the plane defined by \(a\) and \(b\) where \(R=ab\). \(R\) may be written \(R = e^{\frac{\theta}{2}U}\), where \(\theta\) is the angle of rotation and \(u\) is a unit blade \(\left( u^{2} = \pm 1\right)\) that defines the plane of rotation.

Reciprocal Frames

If we have \(M\) linearly independent vectors (a frame), \(a_{1},\dots,a_{M}\), then the reciprocal frame is \(a^{1},\dots,a^{M}\) where \(a_{i}\cdot a^{j} = \delta_{i}^{j}\), \(\delta_{i}^{j}\) is the Kronecker delta (zero if \(i \ne j\) and one if \(i = j\)). The reciprocal frame is constructed as follows:

\[\begin{equation*} E_{M} = a_{1}\wedge\dots\wedge a_{M} \end{equation*}\]
\[\begin{equation*} E_{M}^{-1} = \displaystyle\frac{E_{M}}{E_{M}^{2}} \end{equation*}\]

Then

\[\begin{equation*} a^{i} = \left( -1\right)^{i-1}\left( a_{1}\wedge\dots\wedge \breve{a}_{i} \wedge\dots\wedge a_{M}\right) E_{M}^{-1} \end{equation*}\]

where \(\breve{a}_{i}\) indicates that \(a_{i}\) is to be deleted from the product. In the standard notation if a vector is denoted with a subscript the reciprocal vector is denoted with a superscript. The multivector setup function MV.setup(basis,metric,rframe) has the argument rframe with a default value of False. If it is set to True the reciprocal frame of the basis vectors is calculated. Additionaly there is the function reciprocal_frame(vlst,names='') external to the MV class that will calculate the reciprocal frame of a list, vlst, of vectors. If the argument names is set to a space delimited string of names for the vectors the reciprocal vectors will be given these names.

Geometric Derivative

If \(F\) is a multivector field that is a function of a vector \(x = x^{i}\boldsymbol{\gamma}_{i}\) (we are using the summation convention that pairs of subscripts and superscripts are summed over the dimension of the vector space) then the geometric derivative \(\nabla F\) is given by (in this section the summation convention is used):

\[\begin{equation*} \nabla F = \boldsymbol{\gamma}^{i}\displaystyle\frac{\partial F}{\partial x^{i}} \end{equation*}\]

If \(F_{R}\) is a grade-\(R\) multivector and \(F_{R} = F_{R}^{i_{1}\dots i_{R}}\boldsymbol{\gamma}_{i_{1}}\wedge\dots\wedge \boldsymbol{\gamma}_{i_{R}}\) then

\[\begin{equation*} \nabla F_{R} = \displaystyle\frac{\partial F_{R}^{i_{1}\dots i_{R}}}{\partial x^{j}}\boldsymbol{\gamma}^{j}\left(\boldsymbol{\gamma}_{i_{1}}\wedge \dots\wedge \boldsymbol{\gamma}_{i_{R}} \right) \end{equation*}\]

Note that \(\boldsymbol{\gamma}^{j}\left(\boldsymbol{\gamma}_{i_{1}}\wedge\dots\wedge \boldsymbol{\gamma}_{i_{R}} \right)\) can only contain grades \(R-1\) and \(R+1\) so that \(\nabla F_{R}\) also can only contain those grades. For a grade-\(R\) multivector \(F_{R}\) the inner (div) and outer (curl) derivatives are defined as

\[\begin{equation*} \nabla\cdot F_{R} = \left < \nabla F_{R}\right >_{R-1} \end{equation*}\]

and

\[\begin{equation*} \nabla\wedge F_{R} = \left < \nabla F_{R}\right >_{R+1} \end{equation*}\]

For a general multivector function \(F\) the inner and outer derivatives are just the sum of the inner and outer dervatives of each grade of the multivector function.

Curvilinear coordinates are derived from a vector function \(x(\boldsymbol{\theta})\) where \(\boldsymbol{\theta} = \left(\theta_{1},\dots,\theta_{N}\right)\) where the number of coordinates is equal to the dimension of the vector space. In the case of 3-dimensional spherical coordinates \(\boldsymbol{\theta} = \left( r,\theta,\phi \right)\) and the coordinate generating function \(x(\boldsymbol{\theta})\) is

\[\begin{equation*} x = r \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+ r \sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+ r \cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}} \end{equation*}\]

A coordinate frame is derived from \(x\) by \(\boldsymbol{e}_{i} = \displaystyle\frac{\partial x}{\partial \theta^{i}}\). The following show the frame for spherical coordinates.

\[\begin{equation*} \boldsymbol{e}_{r} = \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+\cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}} \end{equation*}\]
\[\begin{equation*} \boldsymbol{e}_{{\theta}} = \cos\left({\phi}\right) \cos\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+r \cos\left({\theta}\right) \sin\left({\phi}\right){\boldsymbol{{\gamma}_{y}}} - r \sin\left({\theta}\right){\boldsymbol{{\gamma}_{z}}} \end{equation*}\]
\[\begin{equation*} \boldsymbol{e}_{{\phi}} = - r \sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+r \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}} \end{equation*}\]

The coordinate frame generated in this manner is not necessarily normalized so define a normalized frame by

\[\begin{equation*} \boldsymbol{\hat{e}}_{i} = \displaystyle\frac{\boldsymbol{e}_{i}}{\sqrt{\left| \boldsymbol{e}_{i}^{2} \right|}} = \displaystyle\frac{\boldsymbol{e}_{i}}{\left| \boldsymbol{e}_{i} \right|} \end{equation*}\]

This works for all \(\boldsymbol{e}_{i}^{2} \neq 0\) since we have defined \(\left| \boldsymbol{e}_{i} \right| = \sqrt{\left| \boldsymbol{e}_{i}^{2} \right|}\). For spherical coordinates the normalized frame vectors are

\[\begin{equation*} \boldsymbol{\hat{e}}_{r} = \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+\cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}} \end{equation*}\]
\[\begin{equation*} \boldsymbol{\hat{e}}_{{\theta}} = \cos\left({\phi}\right) \cos\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\cos\left({\theta}\right) \sin\left({\phi}\right){\boldsymbol{{\gamma}_{y}}}- \sin\left({\theta}\right){\boldsymbol{{\gamma}_{z}}} \end{equation*}\]
\[\begin{equation*} \boldsymbol{\hat{e}}_{{\phi}} = - \sin\left({\phi}\right){\boldsymbol{{\gamma}_{x}}}+\cos\left({\phi}\right){\boldsymbol{{\gamma}_{y}}} \end{equation*}\]

The geometric derivative in curvilinear coordinates is given by

\[\begin{align*} \nabla F_{R} & = \boldsymbol{\gamma}^{i}\displaystyle\frac{\partial }{\partial x^{i}}\left( F_{R}^{i_{1}\dots i_{R}} \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right) \\ & = \boldsymbol{e^{j}}\displaystyle\frac{\partial }{\partial \theta^{j}}\left( F_{R}^{i_{1}\dots i_{R}} \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right) \\ & = \left(\displaystyle\frac{\partial }{\partial \theta^{j}} F_{R}^{i_{1}\dots i_{R}}\right) \boldsymbol{e^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)+ F_{R}^{i_{1}\dots i_{R}}\boldsymbol{e^{j}} \displaystyle\frac{\partial }{\partial \theta^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right) \\ & = \left(\displaystyle\frac{\partial }{\partial \theta^{j}} F_{R}^{i_{1}\dots i_{R}}\right) \boldsymbol{e^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)+ F_{R}^{i_{1}\dots i_{R}}C\left\{ \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right\} \end{align*}\]

where

\[\begin{equation*} C\left\{ \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right\} = \boldsymbol{e^{j}}\displaystyle\frac{\partial }{\partial \theta^{j}} \left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right) \end{equation*}\]

are the connection multivectors for the curvilinear coordinate system. For a spherical coordinate system they are

\[\begin{equation*} C\left\{\boldsymbol{\hat{e}}_{r}\right\} = \frac{2}{r} \end{equation*}\]
\[\begin{equation*} C\left\{\boldsymbol{\hat{e}}_{\theta}\right\} = \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)} + \frac{1}{r}\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{\theta} \end{equation*}\]
\[\begin{equation*} C\left\{\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{1}{r}{\boldsymbol{\boldsymbol{\hat{e}}_{r}}}\wedge\boldsymbol{\hat{e}}_{{\phi}}+ \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}\boldsymbol{\hat{e}}_{{\theta}}\wedge\boldsymbol{\hat{e}}_{{\phi}} \end{equation*}\]
\[\begin{equation*} C\left\{\hat{e}_{r}\wedge\hat{e}_{\theta}\right\} = - \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)} \boldsymbol{\hat{e}}_{r}+\frac{1}{r}\boldsymbol{\hat{e}}_{{\theta}} \end{equation*}\]
\[\begin{equation*} C\left\{\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{1}{r}\boldsymbol{\hat{e}}_{{\phi}} - \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{{\theta}}\wedge\boldsymbol{\hat{e}}_{{\phi}} \end{equation*}\]
\[\begin{equation*} C\left\{\boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{2}{r}\boldsymbol{\hat{e}}_{r}\wedge \boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi} \end{equation*}\]
\[\begin{equation*} C\left\{\boldsymbol{\hat{e}}_r\wedge\boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = 0 \end{equation*}\]

Module Components

Initializing Multivector Class

The multivector class is initialized with:

MV.setup(basis, metric='', rframe=False, coords=None, debug=False, offset=0)

The basis and metric parameters were described in section Vector Basis and Metric. If rframe=True the reciprocal frame of the symbolic bases vectors is calculated. If debug=True the data structure required to initialize the MV class are printer out. coords is a list of sympy symbols equal in length to the number of basis vectors. These symbols are used as the arguments of a multivector field as a function of position and for calculating the derivatives of a multivector field (if coords is defined then rframe is automatically set equal to True). offset is an integer that is added to the multivector coefficient index. For example if one wishes to start labeling vector coefficient indexes at one instead of zero then set offset=1. Additionally, MV.setup() calculates the pseudo scalar, \(I\) and its inverse, \(I^{-1}\) and makes them available to the programmer as MV.I and MV.Iinv.

After MV.setup() is run one can reinialize the MV class with curvilinear coordinates using:

MV.rebase(x, coords, base_name, debug=False, debug_level=0)

A typical usage of MV.rebase for generating spherical curvilinear coordinate is:

metric = '1 0 0,0 1 0,0 0 1'
gamma_x,gamma_y,gamma_z = MV.setup('gamma_x gamma_y gamma_z',metric,True)
coords = r,theta,phi = sympy.symbols('r theta phi')
x = r*(sympy.cos(theta)*gamma_z+sympy.sin(theta)*\
    (sympy.cos(phi)*gamma_x+sympy.sin(phi)*gamma_y))
x.set_name('x')
MV.rebase(x,coords,'e',True)

The input parameters for MV.rebase are

  • x: Vector function of coordinates (derivatives define curvilinear basis)

  • coords: List of sympy symbols for curvilinear coordinates

  • debug: If True printout (LaTeX) all quantities required for derivative calculation

  • debug_level: Set to 0,1,2, or 3 to stop curvilinear calculation before all quatities are calculated. This is done when debugging new curvilinear coordinate systems since simplification of expressions is not sufficiently automated to insure success of process of any coordinate system defined by vector function x

    To date MV.rebase works for cylindrical and spherical coordinate systems in any number of dimensions (until the execution time becomes too long). To make it work for these systems required creating some hacks for expression simplification since both trigsimp and simplify were not general enough to perform the required simplification.

MV.set_str_format(str_mode=0)

If str_mode=0 the string representation of the multivector contains no newline characters (prints on one line). If str_mode=1 the string representation of the multivector places a newline after each grade of the multivector (prints one grade per line). If str_mode=2 the string representation of the multivector places a newline after each base of the multivector (prints one base per line). In both cases bases with zero coefficients are not printed.

Note

This function directly affects the way multivectors are printed with the print command since it interacts with the __str__() function for the multivector class which is used by the print command.

str_mode Effect on print
0 One multivector per line
1 One grade per line
2 One base per line

Instantiating a Multivector

Now that grades and bases have been described we can show all the ways that a multivector can be instantiated. As an example assume that the multivector space is initialized with e1,e2,e3 = MV.setup('e1 e2 e3').

So that multivectors could be instantiated with statements such as (a1, a2, and a3 are sympy symbols):

x = a1*e1+a2*e2+a3*e3
y = x*e1*e2
z = x|y
w = x^y

or with the multivector class constructor:

class MV(value='', mvtype='', mvname='', fct=False)

mvname is a string that defines the name of the multivector for output purposes. value and type are defined by the following table and fct is a switch that will convert the symbolic coefficients of a multivector to functions if coordinate variables have been defined when MV.setup() is called:

mvtype value Result
default default Zero multivector
‘basisvector’ int i \(\mbox{i}^{th}\) basis vector
‘basisbivector’ int i \(\mbox{i}^{th}\) basis bivector
‘scalar’ symbol x Symbolic scalar of value x
  string s Symbolic scalar of value Symbol(s)
  int i SymPy integer of value i
‘grade’ [int r, 1-D symbol array A] X.grade(r) = A
  [int r, string s] Symbolic grade r multivector
‘vector’ 1-D symbol array A X.grade(1) = A
  string s Symbolic vector
‘grade2’ 1-D symbol array A X.grade(2) = A
‘pseudo’ symbol x X.grade(n) = x
‘spinor’ string s Symbolic even multivector
     
default string s Symbolic general multivector

If the value argument has the option of being a string s then a general symbolic multivector will be constructed constrained by the value of mvtype. The string s will be the base name of the multivector symbolic coefficients. If coords is not defined in MV.setup() the indices of the multivector bases are appended to the base name with a double underscore (superscript notation). If coords is defined the coordinate names will replace the indices in the coefficient names. For example if the base string is A and the coordinates (x,y,z) then the coefficients of a spinor in 3d space would be A, A__xy, A__xz, and A__yz. If the latex_ex is used to print the multivector the coefficients would print as \(A\), \(A^{xy}\), \(A^{xz}\), and \(A^{yz}\).

If the fct argrument of MV() is set to True and the coords argument in MV.setup() is defined the symbolic coefficients of the multivector are functions of the coordinates.

Basic Multivector Class Functions

__call__(self, igrade=0, ibase=0)

__call__ returns the the igrade, ibase coefficient of the multivector. The defaults return the scalar component of the multivector.

convert_to_blades(self)

Convert multivector from the base representation to the blade representation. If multivector is already in blade representation nothing is done.

convert_from_blades(self)

Convert multivector from the blade representation to the base representation. If multivector is already in base representation nothing is done.

project(self, r)

If r is a integer return the grade-\(r\) components of the multivector. If r is a multivector return the grades of the multivector that correspond to the non-zero grades of r. For example if one is projecting a general multivector and r is a spinor, A.project(r) will return only the even grades of the multivector A since a spinor only has even grades that are non-zero.

even(self)

Return the even grade components of the multivector.

odd(self)

Return the odd grade components of the multivector.

rev(self)

Return the reverse of the multivector. See section Reverse of Multivector.

is_pure(self)

Return true if multivector is pure grade (all grades but one are zero).

diff(self, x)

Return the partial derivative of the multivector function with respect to variable \(x\).

grad(self)

Return the geometric derivative of the multivector function.

grad_ext(self)

Return the outer (curl) derivative of the multivector function. Equivalent to curl().

grad_int(self)

Return the inner (div) derivative of the multivector function. Equivalent to div().

Warning

If A is a vector field in three dimensions \(\nabla\cdot {\bf A}\) = A.grad_int() = A.div(), but \(\nabla\times {\bf A}\) = -MV.I*A.grad_ext() = -MV.I*A.curl(). Note that grad_int() lowers the grade of all blades by one grade and grad_ext() raises the grade of all blades by one.

set_coef(self, grade, base, value)

Set the multivector coefficient of index (grade,base) to value.

SymPy Functions Applied Inplace to Multivector Coefficients

All the following fuctions belong to the MV class and apply the corresponding sympy function to each component of a multivector. All these functions perform the operations inplace (None is returned) on each coefficient. For example if you wished to simplify all the components of the multivector A you would invoke A.simplify(). The argument list for each function is the same as for the corresponding sympy function. The only function that differs in its action from the sympy version is trigsimp() which is applied using the recursive option.

collect(self, lst)
sqrfree(self, lst)
subs(self, *args)
simplify(self)
trigsimp(self)
cancel(self)
expand(self)

Helper Functions

These are functions in GA, but not in the multivector (MV) class.

set_names(var_lst, var_str)

set_names() allows one to name a list, var_lst, of multivectors enmass. The names are in var_str, a blank separated string of names. An error is generated if the number of name is not equal to the length of var_lst.

reciprocal_frame(vlst, names='')

reciprocal_frame() implements the proceedure described in section Reciprocal Frames. vlst is a list of independent vectors that you wish the reciprocal frame calculated for. names is a blank separated string of names for the reciprocal vectors if names are required by you application. The function returns a list containing the reciprocal vectors.

trigsimp(f)

In general sympy.trigsimp() will not catch all the trigonometric simplifications in an sympy expression, but by using the recursive option, more of them are caught. So trigsimp() uses this option by default.

S(x)

S() instanciates a scaler multivector of value x, where x can be a sympy variable or integer. This is just a shorthand method for constructing scalar multivectors and can be used when there is any ambiguity in a multivector expression as to whether a symbol or constant should be treated as a scalar multivector or not.

Examples

Algebra

The following examples of geometric algebra (not calculus) are all in the file testsymbolicGA.py which is included in the sympy distribution examples under the galbebra directory. The section of code in the program for each example is show with the respective output following the code section.

Example Header

This is the header of testsymbolicGA.py that allows access to the required modules and also allow variables and certain multivectors to be broadcast from the GA module to the main program.

import os,sys,sympy
from sympy.galgebra.GA import MV, ZERO, ONE, HALF
from sympy import collect, symbols

def F(x, n, nbar):
        """
        Conformal Mapping Function
        """
        Fx = HALF*((x*x)*n+2*x-nbar)
        return(Fx)

if __name__ == '__main__':

Basic Geometric Algebra Operations

Example of basic geometric algebra operation of geometric, outer, and inner products.

        a,b,c,d,e = MV.setup('a b c d e')
        MV.set_str_format(1)

        print 'e|(a^b) =',e|(a^b)
        print 'e|(a^b^c) =',e|(a^b^c)
        print 'a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b)
        print 'e|(a^b^c^d) =',e|(a^b^c^d)
        print -d*(a^b^c)+c*(a^b^d)-b*(a^c^d)+a*(b^c^d)

        print (a^b)|(c^d)

e|(a^b) = {-(b.e)}a
+{(a.e)}b

e|(a^b^c) = {(c.e)}a^b
+{-(b.e)}a^c
+{(a.e)}b^c

a*(b^c)-b*(a^c)+c*(a^b) = {3}a^b^c

e|(a^b^c^d) = {-(d.e)}a^b^c
+{(c.e)}a^b^d
+{-(b.e)}a^c^d
+{(a.e)}b^c^d

{4}a^b^c^d

{(a.d)*(b.c) - (a.c)*(b.d)}1

Examples of Conformal Geometry

Examples of comformal geometry [Lasenby,Chapter 10]. The examples show that basic geometric entities (lines, circles, planes, and spheres) in three dimensions can be represented by blades in a five dimensional (conformal) space.

        print '\nExample: Conformal representations of circles, lines, spheres, and planes'

        metric = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'

        e0,e1,e2,n,nbar = MV.setup('e0 e1 e2 n nbar',metric,debug=0)
        MV.set_str_format(1)
        e = n+nbar
        #conformal representation of points

        A = F(e0,n,nbar)    # point a = (1,0,0)  A = F(a)
        B = F(e1,n,nbar)    # point b = (0,1,0)  B = F(b)
        C = F(-1*e0,n,nbar) # point c = (-1,0,0) C = F(c)
        D = F(e2,n,nbar)    # point d = (0,0,1)  D = F(d)
        x0,x1,x2 = sympy.symbols('x0 x1 x2')
        X = F(MV([x0,x1,x2],'vector'),n,nbar)

        print 'a = e0, b = e1, c = -e0, and d = e2'
        print 'A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.'
        print 'Circle through a, b, and c'
        print 'Circle: A^B^C^X = 0 =',(A^B^C^X)
        print 'Line through a and b'
        print 'Line  : A^B^n^X = 0 =',(A^B^n^X)
        print 'Sphere through a, b, c, and d'
        print 'Sphere: A^B^C^D^X = 0 =',(A^B^C^D^X)
        print 'Plane through a, b, and d'
        print 'Plane : A^B^n^D^X = 0 =',(A^B^n^D^X)

Example: Conformal representations of circles, lines, spheres, and planes
a = e0, b = e1, c = -e0, and d = e2
A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.
Circle through a, b, and c
Circle: A^B^C^X = 0 = {-x2}e0^e1^e2^n
+{x2}e0^e1^e2^nbar
+{-1/2 + 1/2*x0**2 + 1/2*x1**2 + 1/2*x2**2}e0^e1^n^nbar

Line through a and b
Line  : A^B^n^X = 0 = {-x2}e0^e1^e2^n
+{-1/2 + x0/2 + x1/2}e0^e1^n^nbar
+{x2/2}e0^e2^n^nbar
+{-x2/2}e1^e2^n^nbar

Sphere through a, b, c, and d
Sphere: A^B^C^D^X = 0 = {1/2 - 1/2*x0**2 - 1/2*x1**2 - 1/2*x2**2}e0^e1^e2^n^nbar

Plane through a, b, and d
Plane : A^B^n^D^X = 0 = {1/2 - x0/2 - x1/2 - x2/2}e0^e1^e2^n^nbar

Calculation of Reciprocal Frame

Example shows the calculation of the reciprocal frame for three arbitrary vectors and verifies that the calculated reciprocal vectors have the correct properties.

        e1,e2,e3 = MV.setup('e1 e2 e3')

        print 'Example: Reciprocal Frames e1, e2, and e3 unit vectors.\n\n'

        E = e1^e2^e3
        Esq = (E*E)()
        print 'E =',E
        print 'E^2 =',Esq
        Esq_inv = 1/Esq

        E1 = (e2^e3)*E
        E2 = (-1)*(e1^e3)*E
        E3 = (e1^e2)*E

        print 'E1 = (e2^e3)*E =',E1
        print 'E2 =-(e1^e3)*E =',E2
        print 'E3 = (e1^e2)*E =',E3

        w = (E1|e2)
        w.collect(MV.g)
        w = w().expand()
        print 'E1|e2 =',w

        w = (E1|e3)
        w.collect(MV.g)
        w = w().expand()
        print 'E1|e3 =',w

        w = (E2|e1)
        w.collect(MV.g)
        w = w().expand()
        print 'E2|e1 =',w

        w = (E2|e3)
        w.collect(MV.g)
        w = w().expand()
        print 'E2|e3 =',w

        w = (E3|e1)
        w.collect(MV.g)
        w = w().expand()
        print 'E3|e1 =',w

        w = (E3|e2)
        w.collect(MV.g)
        w = w().expand()
        print 'E3|e2 =',w

        w = (E1|e1)
        w = w().expand()
        Esq = Esq.expand()
        print '(E1|e1)/E^2 =',w/Esq

        w = (E2|e2)
        w = w().expand()
        print '(E2|e2)/E^2 =',w/Esq

        w = (E3|e3)
        w = w().expand()
        print '(E3|e3)/E^2 =',w/Esq

Example: Reciprocal Frames e1, e2, and e3 unit vectors.


E = e1^e2^e3
E^2 = -1 - 2*(e1.e2)*(e1.e3)*(e2.e3) + (e1.e2)**2 + (e1.e3)**2 + (e2.e3)**2
E1 = (e2^e3)*E = {-1 + (e2.e3)**2}e1+{(e1.e2) - (e1.e3)*(e2.e3)}e2+{(e1.e3) - (e1.e2)*(e2.e3)}e3
E2 =-(e1^e3)*E = {(e1.e2) - (e1.e3)*(e2.e3)}e1+{-1 + (e1.e3)**2}e2+{(e2.e3) - (e1.e2)*(e1.e3)}e3
E3 = (e1^e2)*E = {(e1.e3) - (e1.e2)*(e2.e3)}e1+{(e2.e3) - (e1.e2)*(e1.e3)}e2+{-1 + (e1.e2)**2}e3
E1|e2 = 0
E1|e3 = 0
E2|e1 = 0
E2|e3 = 0
E3|e1 = 0
E3|e2 = 0
(E1|e1)/E^2 = 1
(E2|e2)/E^2 = 1
(E3|e3)/E^2 = 1

Hyperbolic Geometry

Examples of calculation of distance in hyperbolic geometry [Lasenby,pp373-375]. This is a good example of the utility of not restricting the basis vector to be orthogonal. Note that most of the calculation is simplifying a scalar expression.

        print 'Example: non-euclidian distance calculation'

        metric = '0 # #,# 0 #,# # 1'
        X,Y,e = MV.setup('X Y e',metric)
        XdotY = sympy.Symbol('(X.Y)')
        Xdote = sympy.Symbol('(X.e)')
        Ydote = sympy.Symbol('(Y.e)')
        MV.set_str_format(1)
        L = X^Y^e
        B = L*e
        Bsq = (B*B)()
        print 'L = X^Y^e is a non-euclidian line'
        print 'B = L*e =',B
        BeBr =B*e*B.rev()
        print 'B*e*B.rev() =',BeBr
        print 'B^2 =',Bsq
        print 'L^2 =',(L*L)()
        s,c,Binv,M,S,C,alpha = symbols('s c Binv M S C alpha')
        Bhat = Binv*B # Normalize translation generator
        R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
        print 's = sinh(alpha/2) and c = cosh(alpha/2)'
        print 'R = exp(alpha*B/(2*|B|)) =',R
        Z = R*X*R.rev()
        Z.expand()
        Z.collect([Binv,s,c,XdotY])
        print 'R*X*R.rev() =',Z
        W = Z|Y
        W.expand()
        W.collect([s*Binv])
        print '(R*X*rev(R)).Y =',W
        M = 1/Bsq
        W.subs(Binv**2,M)
        W.simplify()
        Bmag = sympy.sqrt(XdotY**2-2*XdotY*Xdote*Ydote)
        W.collect([Binv*c*s,XdotY])

        W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2))
        W.subs(2*c*s,S)
        W.subs(c**2,(C+1)/2)
        W.subs(s**2,(C-1)/2)
        W.simplify()
        W.subs(1/Binv,Bmag)
        W = W().expand()
        print '(R*X*R.rev()).Y =',W
        nl = '\n'

        Wd = collect(W,[C,S],exact=True,evaluate=False)
        print 'Wd =',Wd
        Wd_1 = Wd[ONE]
        Wd_C = Wd[C]
        Wd_S = Wd[S]
        print '|B| =',Bmag
        Wd_1 = Wd_1.subs(Bmag,1/Binv)
        Wd_C = Wd_C.subs(Bmag,1/Binv)
        Wd_S = Wd_S.subs(Bmag,1/Binv)
        print 'Wd[ONE] =',Wd_1
        print 'Wd[C] =',Wd_C
        print 'Wd[S] =',Wd_S


        lhs = Wd_1+Wd_C*C
        rhs = -Wd_S*S
        lhs = lhs**2
        rhs = rhs**2
        W = (lhs-rhs).expand()
        W = (W.subs(1/Binv**2,Bmag**2)).expand()
        print 'W =',W
        W = (W.subs(S**2,C**2-1)).expand()
        print 'W =',W
        W = collect(W,[C,C**2],evaluate=False)
        print 'W =',W

        a = W[C**2]
        b = W[C]
        c = W[ONE]

        print 'a =',a
        print 'b =',b
        print 'c =',c

        D = (b**2-4*a*c).expand()
        print 'Setting to 0 and solving for C gives:'
        print 'Descriminant D = b^2-4*a*c =',D
        C = (-b/(2*a)).expand()
        print 'C = cosh(alpha) = -b/(2*a) =',C

Example: non-euclidian distance calculation
L = X^Y^e is a non-euclidian line
B = L*e = X^Y
+{-(Y.e)}X^e
+{(X.e)}Y^e

B*e*B.rev() = {2*(X.Y)*(X.e)*(Y.e) - (X.Y)**2}e

B^2 = -2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2
L^2 = -2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2
s = sinh(alpha/2) and c = cosh(alpha/2)
R = exp(alpha*B/(2*|B|)) = {c}1
+{Binv*s}X^Y
+{-(Y.e)*Binv*s}X^e
+{(X.e)*Binv*s}Y^e

R*X*R.rev() = {Binv*(2*(X.Y)*c*s - 2*(X.e)*(Y.e)*c*s) + Binv**2*((X.Y)**2*s**2
               - 2*(X.Y)*(X.e)*(Y.e)*s**2) + c**2}X
             +{2*Binv*c*s*(X.e)**2}Y
             +{Binv**2*(-2*(X.e)*(X.Y)**2*s**2 + 4*(X.Y)*(Y.e)*(X.e)**2*s**2)
               - 2*(X.Y)*(X.e)*Binv*c*s}e

(R*X*rev(R)).Y = {Binv*s*(-4*(X.Y)*(X.e)*(Y.e)*c + 2*c*(X.Y)**2)
                 + Binv**2*s**2*(-4*(X.e)*(Y.e)*(X.Y)**2 +
                 4*(X.Y)*(X.e)**2*(Y.e)**2 + (X.Y)**3) + (X.Y)*c**2}1

(R*X*R.rev()).Y = S*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2)
                  + (X.Y)*Binv*C*(-2*(X.Y)*(X.e)*(Y.e) +
                  (X.Y)**2)**(1/2) + (X.e)*(Y.e)*Binv*(-2*(X.Y)*(X.e)*(Y.e)
                  + (X.Y)**2)**(1/2) -
                  (X.e)*(Y.e)*Binv*C*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2)
Wd = {1: (X.e)*(Y.e)*Binv*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2),
      S: (-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2),
      C: (X.Y)*Binv*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2) -
         (X.e)*(Y.e)*Binv*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2)}
|B| = (-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2)

Wd[ONE] = (X.e)*(Y.e)
Wd[C] = (X.Y) - (X.e)*(Y.e)
Wd[S] = 1/Binv

W = 2*(X.Y)*(X.e)*(Y.e)*C + (X.Y)**2*C**2 + (X.e)**2*(Y.e)**2
    - (X.Y)**2*S**2 + (X.e)**2*(Y.e)**2*C**2 - 2*C*(X.e)**2*(Y.e)**2
    - 2*(X.Y)*(X.e)*(Y.e)*C**2 + 2*(X.Y)*(X.e)*(Y.e)*S**2
W = -2*(X.Y)*(X.e)*(Y.e) + 2*(X.Y)*(X.e)*(Y.e)*C + (X.Y)**2
    + (X.e)**2*(Y.e)**2 + (X.e)**2*(Y.e)**2*C**2 -
     2*C*(X.e)**2*(Y.e)**2
W = {1: -2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2 + (X.e)**2*(Y.e)**2,
     C**2: (X.e)**2*(Y.e)**2,
     C: 2*(X.Y)*(X.e)*(Y.e) - 2*(X.e)**2*(Y.e)**2}

a = (X.e)**2*(Y.e)**2
b = 2*(X.Y)*(X.e)*(Y.e) - 2*(X.e)**2*(Y.e)**2
c = -2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2 + (X.e)**2*(Y.e)**2

Setting to 0 and solving for C gives:
Descriminant D = b^2-4*a*c = 0
C = cosh(alpha) = -b/(2*a) = 1 - (X.Y)/((X.e)*(Y.e))

Calculus

The calculus examples all use the extened LaTeXoutput module, latex_ex, for clarity.

Maxwell’s Equations

In geometric calculus the equivalent of the electromagnetic tensor is \(F = E\gamma_{0}+IB\gamma_{0}\) where a spacetime vector is given by \(x = x^{0}\gamma_{0}+x^{1}\gamma_{1}+x^{2}\gamma_{2}+x^{3}\gamma_{3}\) where \(x^{0} = ct\), the pseudoscalar \(I = \gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}\), \(E\) and \(B\) are four vectors where the time component is zero and the spatial components equal to the electric and magnetic field components. Then Maxwell’s equations can be all written as \(\nabla F = J\) with \(J\) the four current. This example shows that this equations generates all of Maxwell’s equations correctly (in our units:math:\(c=1\)) [Lasenby,pp229-231].

Begin Program Maxwell.py

from sympy import *
from sympy.galgebra.GA import *
from sympy.galgebra.latex_ex import *

if __name__ == '__main__':

    metric = '1  0  0  0,'+\
             '0 -1  0  0,'+\
             '0  0 -1  0,'+\
             '0  0  0 -1'

    vars = symbols('t x y z')
    gamma_t,gamma_x,gamma_y,gamma_z = MV.setup('gamma_t gamma_x gamma_y gamma_z',metric,True,vars)
    LatexPrinter.format(1,1,1,1)
    I = MV(1,'pseudo')
    print '$I$ Pseudo-Scalar'
    print 'I =',I
    B = MV('B','vector',fct=True)
    E = MV('E','vector',fct=True)
    B.set_coef(1,0,0)
    E.set_coef(1,0,0)
    B *= gamma_t
    E *= gamma_t
    J = MV('J','vector',fct=True)
    F = E+I*B
    print ' '
    print '$B$ Magnetic Field Bi-Vector'
    print 'B = Bvec gamma_0 =',B
    print '$F$ Electric Field Bi-Vector'
    print 'E = Evec gamma_0 =',E
    print '$E+IB$ Electo-Magnetic Field Bi-Vector'
    print 'F = E+IB =',F
    print '$J$ Four Current'
    print 'J =',J
    gradF = F.grad()
    print 'Geometric Derivative of Electo-Magnetic Field Bi-Vector'
    MV_format(3)
    print '\\nabla F =',gradF
    print 'All Maxwell Equations are'
    print '\\nabla F = J'
    print 'Div $E$ and Curl $H$ Equations'
    print '<\\nabla F>_1 -J =',gradF.project(1)-J,' = 0'
    print 'Curl $E$ and Div $B$ equations'
    print '<\\nabla F>_3 =',gradF.project(3),' = 0'
    xdvi(filename='Maxwell.tex')

End Program Maxwell.py

Begin Program Output

\(I\) Pseudo-Scalar

\[\begin{equation*} I = {\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z} \end{equation*}\]

\(B\) Magnetic Field Bi-Vector

\[\begin{equation*} B = - {B^{x}}{\gamma}_{t}{\gamma}_{x}- {B^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{t}{\gamma}_{z} \end{equation*}\]

\(F\) Electric Field Bi-Vector

\[\begin{equation*} E = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z} \end{equation*}\]

\(E+IB\) Electo-Magnetic Field Bi-Vector

\[\begin{equation*} F = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{x}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}+ {B^{y}}{\gamma}_{x}{\gamma}_{z}- {B^{x}}{\gamma}_{y}{\gamma}_{z} \end{equation*}\]

\(J\) Four Current

\[\begin{equation*} J = {J^{t}}{\gamma}_{t}+ {J^{x}}{\gamma}_{x}+ {J^{y}}{\gamma}_{y}+ {J^{z}}{\gamma}_{z} \end{equation*}\]

Geometric Derivative of Electo-Magnetic Field Bi-Vector

\[\begin{align*} \nabla F & = \left( \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\ & + \left(-\partial_{t} {E^{x}} + \partial_{y} {B^{z}} - \partial_{z} {B^{y}}\right){\gamma}_{x} \\ & + \left( \partial_{z} {B^{x}} - \partial_{t} {E^{y}} - \partial_{x} {B^{z}}\right){\gamma}_{y} \\ & + \left(-\partial_{y} {B^{x}} - \partial_{t} {E^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z} \\ & + \left(-\partial_{x} {E^{y}} - \partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\ & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\ & + \left(-\partial_{t} {B^{x}} - \partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\ & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z} \end{align*}\]

All Maxwell Equations are

\[\begin{equation*} \nabla F = J \end{equation*}\]

Div \(E\) and Curl \(H\) Equations

\[\begin{align*} <\nabla F>_1 -J & = \left(-{J^{t}} + \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\ & + \left(-{J^{x}} - \partial_{t} {E^{x}} + \partial_{y} {B^{z}} - \partial_{z} {B^{y}}\right){\gamma}_{x} \\ & + \left( \partial_{z} {B^{x}} - \partial_{t} {E^{y}} -{J^{y}} - \partial_{x} {B^{z}}\right){\gamma}_{y} \\ & + \left(-\partial_{y} {B^{x}} - \partial_{t} {E^{z}} -{J^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z} \end{align*}\]
\[\begin{equation*} = 0 \end{equation*}\]

Curl \(E\) and Div \(B\) equations

\[\begin{align*} <\nabla F>_3 & = \left(-\partial_{x} {E^{y}} - \partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{y} \\ & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{z} \\ & + \left(-\partial_{t} {B^{x}} - \partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}\wedge {\gamma}_{y}\wedge {\gamma}_{z} \\ & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}\wedge {\gamma}_{y}\wedge {\gamma}_{z} \end{align*}\]
\[\begin{equation*} = 0 \end{equation*}\]

End Program Output

Dirac’s Equation

The geometric algebra/calculus allows one to formulate the Dirac equation in real terms (no \(\sqrt{-1}\)). Spinors \(\left(\Psi\right)\) are even multivectors in space time (Minkowski space with signature (1,-1,-1,-1)) and the Dirac equation becomes \(\nabla \Psi I \sigma_{z}-eA\Psi = m\Psi\gamma_{t}\). All the terms in the real equation are explined in Doran and Lasenby [Lasenby,pp281-283].

Begin Program Dirac.py

#!/usr/local/bin/python
#Dirac.py

from sympy.galgebra.GA import *
from sympy.galgebra.latex_ex import *
from sympy import *

if __name__ == '__main__':

    metric = '1  0  0  0,'+\
             '0 -1  0  0,'+\
             '0  0 -1  0,'+\
             '0  0  0 -1'

    vars = symbols('t x y z')
    gamma_t,gamma_x,gamma_y,gamma_z = MV.setup('gamma_t gamma_x gamma_y gamma_z',metric,True,vars)

    m,e = symbols('m e')
    Format('1 1 1 1')
    I = MV(ONE,'pseudo')
    nvars = len(vars)
    psi = MV('psi','spinor',fct=True)
    A = MV('A','vector',fct=True)
    sig_x = gamma_x*gamma_t
    sig_y = gamma_y*gamma_t
    sig_z = gamma_z*gamma_t
    print '$A$ is 4-vector potential'
    print A
    print r'$\bm{\psi}$ is 8-component real spinor (even multi-vector)'
    print psi
    dirac_eq = psi.grad()*I*sig_z-e*A*psi-m*psi*gamma_t
    dirac_eq.simplify()
    print 'Dirac equation in terms of real geometric algebra/calculus '+\
          r'$\lp\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi} = m\bm{\psi}\gamma_{t}\rp$'
    print 'Spin measured with respect to $z$ axis'
    Format('mv=3')
    print r'\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi}-m\bm{\psi}\gamma_{t} = ',dirac_eq,' = 0'
    xdvi(filename='Dirac.tex')

End Program Dirac.py

Begin Program Output

\(A\) is 4-vector potential

\[\begin{equation*} A = {A^{t}}{\gamma}_{t}+ {A^{x}}{\gamma}_{x}+ {A^{y}}{\gamma}_{y}+ {A^{z}}{\gamma}_{z} \end{equation*}\]

\(\boldsymbol{\psi}\) is 8-component real spinor (even multi-vector)

\[\begin{equation*} {}\boldsymbol{{\psi}} = {{\psi}}+ {{\psi}^{tx}}{\gamma}_{t}{\gamma}_{x}+ {{\psi}^{ty}}{\gamma}_{t}{\gamma}_{y}+ {{\psi}^{xy}}{\gamma}_{x}{\gamma}_{y}+ {{\psi}^{tz}}{\gamma}_{t}{\gamma}_{z}+ {{\psi}^{xz}}{\gamma}_{x}{\gamma}_{z}+ {{\psi}^{yz}}{\gamma}_{y}{\gamma}_{z}+ {{\psi}^{txyz}}{\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z} \end{equation*}\]

Dirac equation in terms of real geometric algebra/calculus \(\left(\nabla \boldsymbol{\psi} I \sigma_{z}-eA\boldsymbol{\psi} = m\boldsymbol{\psi}\gamma_{t}\right)\) Spin measured with respect to \(z\) axis

\[\begin{align*} \nabla \boldsymbol{\psi} I \sigma_{z}-eA\boldsymbol{\psi}-m\boldsymbol{\psi}\gamma_{t} & = \left(-e {A^{y}} {{\psi}^{ty}} - m {{\psi}} - \partial_{z} {{\psi}^{txyz}} - \partial_{y} {{\psi}^{tx}} - e {A^{z}} {{\psi}^{tz}} - e {A^{x}} {{\psi}^{tx}} + \partial_{x} {{\psi}^{ty}} - e {A^{t}} {{\psi}} + \partial_{t} {{\psi}^{xy}}\right){\gamma}_{t} \\ & + \left(-e {A^{y}} {{\psi}^{xy}} + \partial_{z} {{\psi}^{yz}} - e {A^{z}} {{\psi}^{xz}} - e {A^{t}} {{\psi}^{tx}} + m {{\psi}^{tx}} - \partial_{x} {{\psi}^{xy}} + \partial_{y} {{\psi}} - e {A^{x}} {{\psi}} - \partial_{t} {{\psi}^{ty}}\right){\gamma}_{x} \\ & + \left(-e {A^{y}} {{\psi}} + e {A^{x}} {{\psi}^{xy}} - e {A^{t}} {{\psi}^{ty}} - \partial_{x} {{\psi}} + \partial_{t} {{\psi}^{tx}} - \partial_{z} {{\psi}^{xz}} - e {A^{z}} {{\psi}^{yz}} + m {{\psi}^{ty}} - \partial_{y} {{\psi}^{xy}}\right){\gamma}_{y} \\ & + \left(-\partial_{z} {{\psi}^{xy}} + e {A^{x}} {{\psi}^{xz}} - e {A^{t}} {{\psi}^{tz}} - \partial_{x} {{\psi}^{yz}} + \partial_{y} {{\psi}^{xz}} + e {A^{y}} {{\psi}^{yz}} + \partial_{t} {{\psi}^{txyz}} + m {{\psi}^{tz}} - e {A^{z}} {{\psi}}\right){\gamma}_{z} \\ & + \left( \partial_{y} {{\psi}^{ty}} - e {A^{y}} {{\psi}^{tx}} + e {A^{x}} {{\psi}^{ty}} + \partial_{z} {{\psi}^{tz}} - e {A^{z}} {{\psi}^{txyz}} + \partial_{x} {{\psi}^{tx}} - e {A^{t}} {{\psi}^{xy}} - \partial_{t} {{\psi}} - m {{\psi}^{xy}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\ & + \left(-\partial_{y} {{\psi}^{tz}} + \partial_{x} {{\psi}^{txyz}} - e {A^{t}} {{\psi}^{xz}} + e {A^{x}} {{\psi}^{tz}} + e {A^{y}} {{\psi}^{txyz}} + \partial_{z} {{\psi}^{ty}} - m {{\psi}^{xz}} - e {A^{z}} {{\psi}^{tx}} - \partial_{t} {{\psi}^{yz}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\ & + \left(-e {A^{t}} {{\psi}^{yz}} - \partial_{z} {{\psi}^{tx}} + \partial_{x} {{\psi}^{tz}} - m {{\psi}^{yz}} + \partial_{y} {{\psi}^{txyz}} + \partial_{t} {{\psi}^{xz}} - e {A^{z}} {{\psi}^{ty}} - e {A^{x}} {{\psi}^{txyz}} + e {A^{y}} {{\psi}^{tz}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\ & + \left(\partial_{z} {{\psi}} + e {A^{y}} {{\psi}^{xz}} + m {{\psi}^{txyz}} -\partial_{t} {{\psi}^{tz}} -\partial_{x} {{\psi}^{xz}} -e {A^{x}} {{\psi}^{yz}} -e {A^{t}} {{\psi}^{txyz}} -\partial_{y} {{\psi}^{yz}} -e {A^{z}} {{\psi}^{xy}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z} \end{align*}\]
\[\begin{equation*} = 0 \end{equation*}\]

End Program Output

Spherical Coordinates

Curvilinear coodinates are implemented as shown in section Geometric Derivative. The gradient of a scalar function and the divergence and curl of a vector function (\(-I\left(\nabla\wedge A\right)\) is the curl in three dimensions in the notation of geometric algebra) to demonstrate the formulas derived in section Geometric Derivative.

Begin Program coords.py

#!/usrlocal/bin/python
#EandM.py

from sympy.galgebra.GA import *
from sympy.galgebra.latex_ex import *
from sympy import *

import sympy,numpy,sys

if __name__ == '__main__':
    metric = '1 0 0,'+\
             '0 1 0,'+\
             '0 0 1'

    gamma_x,gamma_y,gamma_z = MV.setup('gamma_x gamma_y gamma_z',metric,True)
    Format('1 1 1 1')

    coords = r,theta,phi = symbols('r theta phi')
    x = r*(sympy.cos(theta)*gamma_z+sympy.sin(theta)*\
        (sympy.cos(phi)*gamma_x+sympy.sin(phi)*gamma_y))
    x.set_name('x')

    MV.rebase(x,coords,'e',False)

    #psi = MV.scalar_fct('psi')
    psi = MV('psi','scalar',fct=True)
    #psi.name = 'psi'
    dpsi = psi.grad()
    print 'Gradient of Scalar Function $\\psi$'
    print '\\nabla\\psi =',dpsi

    #A = MV.vector_fct('A')
    A = MV('A','vector',fct=True)
    #A.name = 'A'
    print 'Div and Curl of Vector Function $A$'
    print A

    gradA = A.grad()
    I = MV(ONE,'pseudo')
    divA = A.grad_int()
    curlA = -I*A.grad_ext()
    print '\\nabla \\cdot A =',divA
    Format('mv=3')
    print '-I\\lp\\nabla \\W A\\rp =',curlA

    xdvi(filename='coords.tex')

End Program coords.py

Begin Program Output

Gradient of Scalar Function \(\psi\)

\[\begin{equation*} \nabla\psi = \partial_{r} {{\psi}}{}\boldsymbol{e}_{r}+\frac{1 \partial_{{\theta}} {{\psi}}}{r}{}\boldsymbol{e}_{{\theta}}+\frac{1 \partial_{{\phi}} {{\psi}}}{r \operatorname{sin}\left({\theta}\right)}{}\boldsymbol{e}_{{\phi}} \end{equation*}\]

Div and Curl of Vector Function \(A\)

\[\begin{equation*} A = {A^{r}}{}\boldsymbol{e}_{r}+ {A^{{\theta}}}{}\boldsymbol{e}_{{\theta}}+ {A^{{\phi}}}{}\boldsymbol{e}_{{\phi}} \end{equation*}\]
\[\begin{equation*} \nabla \cdot A = \left(\frac{{A^{{\theta}}} \operatorname{cos}\left({\theta}\right)}{r \operatorname{sin}\left({\theta}\right)} + 2 \frac{{A^{r}}}{r} + \partial_{r} {A^{r}} + \frac{\partial_{{\theta}} {A^{{\theta}}}}{r} + \frac{\partial_{{\phi}} {A^{{\phi}}}}{r \operatorname{sin}\left({\theta}\right)}\right) \end{equation*}\]
\[\begin{align*} -I\left(\nabla \wedge A\right) & = \left(\frac{\partial_{{\theta}} {A^{{\phi}}}}{r} + \frac{{A^{{\phi}}} \operatorname{cos}\left({\theta}\right)}{r \operatorname{sin}\left({\theta}\right)} -\frac{\partial_{{\phi}} {A^{{\theta}}}}{r \operatorname{sin}\left({\theta}\right)}\right){}\boldsymbol{e}_{r} \\ & - \left(\frac{{A^{{\phi}}}}{r} + \partial_{r} {A^{{\phi}}} -\frac{\partial_{{\phi}} {A^{r}}}{r \operatorname{sin}\left({\theta}\right)}\right){}\boldsymbol{e}_{{\theta}} \\ & + \left(\frac{{A^{{\theta}}}}{r} + \partial_{r} {A^{{\theta}}} -\frac{\partial_{{\theta}} {A^{r}}}{r}\right){}\boldsymbol{e}_{{\phi}} \end{align*}\]

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See also

Hestenes
Clifford Algebra to Geometric Calculus by D.Hestenes and G. Sobczyk, Kluwer Academic Publishers, 1984.
Lasenby
Geometric Algebra for Physicists by C. Doran and A. Lasenby, Cambridge University Press, 2003.