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| 27.1 Introduction to atensor | ||
| 27.2 Functions and Variables 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.
Categories: Tensors Share packages Package atensor
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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).
Categories: Package atensor
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.
Categories: Package atensor Simplification functions
The algebra type. Valid values are universal, grassmann,
clifford, symmetric, symplectic and lie_envelop.
Categories: Package atensor
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.
Categories: Package atensor Global variables
Default value: ident(3)
Default values for the bilinear forms sf, af, and
av. The default is the identity matrix ident(3).
Categories: Package atensor Global variables
Default value: v
The symbol for base vectors.
Categories: Package atensor Global variables
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.
Categories: Package atensor
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.
Categories: Package atensor
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
Categories: Package atensor
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.
Categories: Package atensor Predicate functions
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