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@menu
* Introduction to atensor::
* Definitions for atensor::
@end menu
@node Introduction to atensor, Definitions for atensor, atensor, atensor
@section Introduction to atensor
@code{atensor} is an algebraic tensor manipulation package. To use @code{atensor},
type @code{load(atensor)}, followed by a call to the @code{init_atensor}
function.
The essence of @code{atensor} is a set of simplification rules for the
noncommutative (dot) product operator ("@code{.}"). @code{atensor} recognizes
several algebra types; the corresponding simplification rules are put
into effect when the @code{init_atensor} function is called.
The capabilities of @code{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.:
@example
i = v j = v k = v . v
1 2 1 2
@end example
Although the @code{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:
@example
(%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 ]
@end example
@code{atensor} recognizes as base vectors indexed symbols, where the symbol
is that stored in @code{asymbol} and the index runs between 1 and @code{adim}.
For indexed symbols, and indexed symbols only, the bilinear forms
@code{sf}, @code{af}, and @code{av} are evaluated. The evaluation
substitutes the value of @code{aform[i,j]} in place of @code{fun(v[i],v[j])}
where @code{v} represents the value of @code{asymbol} and @code{fun} is
either @code{af} or @code{sf}; or, it substitutes @code{v[aform[i,j]]}
in place of @code{av(v[i],v[j])}.
Needless to say, the functions @code{sf}, @code{af} and @code{av}
can be redefined.
When the @code{atensor} package is loaded, the following flags are set:
@example
dotscrules:true;
dotdistrib:true;
dotexptsimp:false;
@end example
If you wish to experiment with a nonassociative algebra, you may also
consider setting @code{dotassoc} to @code{false}. In this case, however,
@code{atensimp} will not always be able to obtain the desired
simplifications.
@c end concepts atensor
@node Definitions for atensor, , Introduction to atensor, atensor
@section Definitions for atensor
@deffn {Function} init_atensor (@var{alg_type}, @var{opt_dims})
@deffnx {Function} init_atensor (@var{alg_type})
Initializes the @code{atensor} package with the specified algebra type. @var{alg_type}
can be one of the following:
@code{universal}: The universal algebra has no commutation rules.
@code{grassmann}: The Grassman algebra is defined by the commutation
relation @code{u.v+v.u=0}.
@code{clifford}: The Clifford algebra is defined by the commutation
relation @code{u.v+v.u=-2*sf(u,v)} where @code{sf} is a symmetric
scalar-valued function. For this algebra, @var{opt_dims} can be up
to three nonnegative integers, representing the number of positive,
degenerate, and negative dimensions of the algebra, respectively. If
any @var{opt_dims} values are supplied, @code{atensor} will configure the
values of @code{adim} and @code{aform} appropriately. Otherwise,
@code{adim} will default to 0 and @code{aform} will not be defined.
@code{symmetric}: The symmetric algebra is defined by the commutation
relation @code{u.v-v.u=0}.
@code{symplectic}: The symplectic algebra is defined by the commutation
relation @code{u.v-v.u=2*af(u,v)} where @code{af} is an antisymmetric
scalar-valued function. For the symplectic algebra, @var{opt_dims} can
be up to two nonnegative integers, representing the nondegenerate and
degenerate dimensions, respectively. If any @var{opt_dims} values are
supplied, @code{atensor} will configure the values of @code{adim} and @code{aform}
appropriately. Otherwise, @code{adim} will default to 0 and @code{aform}
will not be defined.
@code{lie_envelop}: The algebra of the Lie envelope is defined by the
commutation relation @code{u.v-v.u=2*av(u,v)} where @code{av} is
an antisymmetric function.
The @code{init_atensor} function also recognizes several predefined
algebra types:
@code{complex} implements the algebra of complex numbers as the
Clifford algebra Cl(0,1). The call @code{init_atensor(complex)} is
equivalent to @code{init_atensor(clifford,0,0,1)}.
@code{quaternion} implements the algebra of quaternions. The call
@code{init_atensor(quaternion)} is equivalent to
@code{init_atensor(clifford,0,0,2)}.
@code{pauli} implements the algebra of Pauli-spinors as the Clifford-algebra
Cl(3,0). A call to @code{init_atensor(pauli)} is equivalent to
@code{init_atensor(clifford,3)}.
@code{dirac} implements the algebra of Dirac-spinors as the Clifford-algebra
Cl(3,1). A call to @code{init_atensor(dirac)} is equivalent to
@code{init_atensor(clifford,3,0,1)}.
@end deffn
@deffn {Function} atensimp (@var{expr})
Simplifies an algebraic tensor expression @var{expr} according to the rules
configured by a call to @code{init_atensor}. Simplification includes
recursive application of commutation relations and resolving calls
to @code{sf}, @code{af}, and @code{av} where applicable. A
safeguard is used to ensure that the function always terminates, even
for complex expressions.
@end deffn
@deffn {Function} alg_type
The algebra type. Valid values are @code{universal}, @code{grassmann},
@code{clifford}, @code{symmetric}, @code{symplectic} and @code{lie_envelop}.
@end deffn
@defvr {Variable} adim
Default value: 0
The dimensionality of the algebra. @code{atensor} uses the value of @code{adim}
to determine if an indexed object is a valid base vector. See @code{abasep}.
@end defvr
@defvr {Variable} aform
Default value: @code{ident(3)}
Default values for the bilinear forms @code{sf}, @code{af}, and
@code{av}. The default is the identity matrix @code{ident(3)}.
@end defvr
@defvr {Variable} asymbol
Default value: @code{v}
The symbol for base vectors..
@end defvr
@deffn {Function} sf (@var{u}, @var{v})
A symmetric scalar function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using @code{abasep} and if that is the case, substitutes the
corresponding value from the matrix @code{aform}.
@end deffn
@deffn {Function} af (@var{u}, @var{v})
An antisymmetric scalar function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using @code{abasep} and if that is the case, substitutes the
corresponding value from the matrix @code{aform}.
@end deffn
@deffn {Function} av (@var{u}, @var{v})
An antisymmetric function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using @code{abasep} and if that is the case, substitutes the
corresponding value from the matrix @code{aform}.
For instance:
@example
(%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
@end example
@end deffn
@deffn {Function} abasep (@var{v})
Checks if its argument is an @code{atensor} base vector. That is, if it is
an indexed symbol, with the symbol being the same as the value of
@code{asymbol}, and the index having a numeric value between 1
and @code{adim}.
@end deffn
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