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30.1 Introduction to atensor | ||

30.2 Definitions for atensor |

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`atensor`

is an algebraic tensor manipulation package. To use `atensor`

,
type `load(atensor)`

, followed by a call to the `init_atensor`

function.

The essence of `atensor`

is a set of simplification rules for the
noncommutative (dot) product operator ("`.`

"). `atensor`

recognizes
several algebra types; the corresponding simplification rules are put
into effect when the `init_atensor`

function is called.

The capabilities of `atensor`

can be demonstrated by defining the
algebra of quaternions as a Clifford-algebra Cl(0,2) with two basis
vectors. The three quaternionic imaginary units are then the two
basis vectors and their product, i.e.:

i = v j = v k = v . v 1 2 1 2 |

Although the `atensor`

package has a built-in definition for the
quaternion algebra, it is not used in this example, in which we
endeavour to build the quaternion multiplication table as a matrix:

(%i1) load(atensor); (%o1) /share/tensor/atensor.mac (%i2) init_atensor(clifford,0,0,2); (%o2) done (%i3) atensimp(v[1].v[1]); (%o3) - 1 (%i4) atensimp((v[1].v[2]).(v[1].v[2])); (%o4) - 1 (%i5) q:zeromatrix(4,4); [ 0 0 0 0 ] [ ] [ 0 0 0 0 ] (%o5) [ ] [ 0 0 0 0 ] [ ] [ 0 0 0 0 ] (%i6) q[1,1]:1; (%o6) 1 (%i7) for i thru adim do q[1,i+1]:q[i+1,1]:v[i]; (%o7) done (%i8) q[1,4]:q[4,1]:v[1].v[2]; (%o8) v . v 1 2 (%i9) for i from 2 thru 4 do for j from 2 thru 4 do q[i,j]:atensimp(q[i,1].q[1,j]); (%o9) done (%i10) q; [ 1 v v v . v ] [ 1 2 1 2 ] [ ] [ v - 1 v . v - v ] [ 1 1 2 2 ] (%o10) [ ] [ v - v . v - 1 v ] [ 2 1 2 1 ] [ ] [ v . v v - v - 1 ] [ 1 2 2 1 ] |

`atensor`

recognizes as base vectors indexed symbols, where the symbol
is that stored in `asymbol`

and the index runs between 1 and `adim`

.
For indexed symbols, and indexed symbols only, the bilinear forms
`sf`

, `af`

, and `av`

are evaluated. The evaluation
substitutes the value of `aform[i,j]`

in place of `fun(v[i],v[j])`

where `v`

represents the value of `asymbol`

and `fun`

is
either `af`

or `sf`

; or, it substitutes `v[aform[i,j]]`

in place of `av(v[i],v[j])`

.

Needless to say, the functions `sf`

, `af`

and `av`

can be redefined.

When the `atensor`

package is loaded, the following flags are set:

dotscrules:true; dotdistrib:true; dotexptsimp:false; |

If you wish to experiment with a nonassociative algebra, you may also
consider setting `dotassoc`

to `false`

. In this case, however,
`atensimp`

will not always be able to obtain the desired
simplifications.

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__Function:__**init_atensor***(*`alg_type`,`opt_dims`)__Function:__**init_atensor***(*`alg_type`)Initializes the

`atensor`

package with the specified algebra type.`alg_type`can be one of the following:`universal`

: The universal algebra has no commutation rules.`grassmann`

: The Grassman algebra is defined by the commutation relation`u.v+v.u=0`

.`clifford`

: The Clifford algebra is defined by the commutation relation`u.v+v.u=-2*sf(u,v)`

where`sf`

is a symmetric scalar-valued function. For this algebra,`opt_dims`can be up to three nonnegative integers, representing the number of positive, degenerate, and negative dimensions of the algebra, respectively. If any`opt_dims`values are supplied,`atensor`

will configure the values of`adim`

and`aform`

appropriately. Otherwise,`adim`

will default to 0 and`aform`

will not be defined.`symmetric`

: The symmetric algebra is defined by the commutation relation`u.v-v.u=0`

.`symplectic`

: The symplectic algebra is defined by the commutation relation`u.v-v.u=2*af(u,v)`

where`af`

is an antisymmetric scalar-valued function. For the symplectic algebra,`opt_dims`can be up to two nonnegative integers, representing the nondegenerate and degenerate dimensions, respectively. If any`opt_dims`values are supplied,`atensor`

will configure the values of`adim`

and`aform`

appropriately. Otherwise,`adim`

will default to 0 and`aform`

will not be defined.`lie_envelop`

: The algebra of the Lie envelope is defined by the commutation relation`u.v-v.u=2*av(u,v)`

where`av`

is an antisymmetric function.The

`init_atensor`

function also recognizes several predefined algebra types:`complex`

implements the algebra of complex numbers as the Clifford algebra Cl(0,1). The call`init_atensor(complex)`

is equivalent to`init_atensor(clifford,0,0,1)`

.`quaternion`

implements the algebra of quaternions. The call`init_atensor(quaternion)`

is equivalent to`init_atensor(clifford,0,0,2)`

.`pauli`

implements the algebra of Pauli-spinors as the Clifford-algebra Cl(3,0). A call to`init_atensor(pauli)`

is equivalent to`init_atensor(clifford,3)`

.`dirac`

implements the algebra of Dirac-spinors as the Clifford-algebra Cl(3,1). A call to`init_atensor(dirac)`

is equivalent to`init_atensor(clifford,3,0,1)`

.

__Function:__**atensimp***(*`expr`)Simplifies an algebraic tensor expression

`expr`according to the rules configured by a call to`init_atensor`

. Simplification includes recursive application of commutation relations and resolving calls to`sf`

,`af`

, and`av`

where applicable. A safeguard is used to ensure that the function always terminates, even for complex expressions.

__Function:__**alg_type**The algebra type. Valid values are

`universal`

,`grassmann`

,`clifford`

,`symmetric`

,`symplectic`

and`lie_envelop`

.

__Variable:__**adim**Default value: 0

The dimensionality of the algebra.

`atensor`

uses the value of`adim`

to determine if an indexed object is a valid base vector. See`abasep`

.

__Variable:__**aform**Default value:

`ident(3)`

Default values for the bilinear forms

`sf`

,`af`

, and`av`

. The default is the identity matrix`ident(3)`

.

__Variable:__**asymbol**Default value:

`v`

The symbol for base vectors..

__Function:__**sf***(*`u`,`v`)A symmetric scalar function that is used in commutation relations. The default implementation checks if both arguments are base vectors using

`abasep`

and if that is the case, substitutes the corresponding value from the matrix`aform`

.

__Function:__**af***(*`u`,`v`)An antisymmetric scalar function that is used in commutation relations. The default implementation checks if both arguments are base vectors using

`abasep`

and if that is the case, substitutes the corresponding value from the matrix`aform`

.

__Function:__**av***(*`u`,`v`)An antisymmetric function that is used in commutation relations. The default implementation checks if both arguments are base vectors using

`abasep`

and if that is the case, substitutes the corresponding value from the matrix`aform`

.For instance:

(%i1) load(atensor); (%o1) /share/tensor/atensor.mac (%i2) adim:3; (%o2) 3 (%i3) aform:matrix([0,3,-2],[-3,0,1],[2,-1,0]); [ 0 3 - 2 ] [ ] (%o3) [ - 3 0 1 ] [ ] [ 2 - 1 0 ] (%i4) asymbol:x; (%o4) x (%i5) av(x[1],x[2]); (%o5) x 3

__Function:__**abasep***(*`v`)Checks if its argument is an

`atensor`

base vector. That is, if it is an indexed symbol, with the symbol being the same as the value of`asymbol`

, and the index having a numeric value between 1 and`adim`

.

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