atensor

abasep (v) — Function

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.

adim — Variable

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.

af (u, v) — Function

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.

aform — Variable

Default value: ident(3)

Default values for the bilinear forms sf, af, and av. The default is the identity matrix ident(3).

Function: alg_type

The algebra type. Valid values are universal, grassmann, clifford, symmetric, symplectic and lie_envelop.

asymbol — Variable

Default value: v

The symbol for base vectors.

atensimp (expr) — Function

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.

av (u, v) — Function

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

init_atensor (alg_type, opt_dims) — Function

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).

sf (u, v) — Function

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.