&gco_d.html

Function: Bigebra [`&gco_d`] - dotted Graßmann co-product for a different filtration

Calling Sequence:

t2 := &gco_d(t1,i)
t2 := &gco_d(c1)

Parameters:

t1 : tensor polynoms

i : tensor slot to be acted on

c1 : Clifford polynom

Output:

t1 : a tensor polynom

Global:

for the transition to the regular wedge basis and back to the dotted basis (both represented as eawebw...) the two global matrices F and FT are used.

Description:

The dotted Graßmann co-product is isomorphic to the regular Graßmann co-product on the dotted wedge basis. The function &gco_d(t1,i) computes this product using the original Graßmann co-product but w.r.t. the undotted basis. It is hence the counter part to the function Cli5plus[dwedge] which computes the dotted wedge product in the undotted basis. The dotted and undotted bases arise from different filtrations of the underlying Graßmann algebra. As Graßmann algebras they are isomorphic, but they are not isomorphic as Hopf algebras!

The function &gco_d needs the Cli5plus and it will load it automatically if it was not done previously.

This functionality is simply gained by wrapping the original Graßmann co-product and using internally the two functions `convert/wedge_to_dwedge` and `convert/dwedge_to_wedge` from the Cli5plus package.

If F and FT are antisymmetric arrays which are mutually transposed to each other (negative of each other) this mapping is an isomorphism (F and FT need not be non-singular !). Hence the dotted Graßmann co-product can be computed in the undotted basis by transforming back and forth.

From a physical point of view, it is possible to consider the undotted basis as a fermionic time-ordered product and the dotted basis as a fermionic normal-ordered product. Hence, the dotted Graßmann co-product employs the co-product of normal-ordered fields in the time-ordered basis, see [1].

It should be noted that the loop tangle 'product \circ coproduct' works out completely differently if one mixes products and co-products for the same algebra with different filatration (ordering).

References:
[1] B. Fauser, On the Hopf algebraic origin of Wick normal ordering, J. Math. Phys.: Gen. Math. 34, 2001:105-115

Examples:

> restart:_CLIENV[_SILENT]:=true:with(Bigebra):

Warning, new definition for drop_t

Warning, new definition for gco_d_monom

Warning, new definition for gco_monom

Warning, new definition for init

Define the dimension of the vectorspace V under consideration to be 3, and define F and FT

> dim_V:=3:
F:=array(antisymmetric,1..dim_V,1..dim_V):
F=evalm(F);
FT:=evalm(-1*F);
w_bas:=cbasis(dim_V); ## the wedge basis

Now we invoke for the first time the dotted Graßmann co-product which should also load the Cli5plus package:

> &gco_d(e1we2);

Warning, Need to load Cli5plus.....

Warning, new definition for init

Warning, definitions for type/climon and type/clipolynom now include &C and &C[K]. Type ?cliprod for help.

Note that Cli5plus was loaded. As the Graßmann co-product, &gco_d can act on tensors in the i-th slot. For example, let's act on the second tensor slot of the tensor &t(e1,e2we3,e3) occupied by e2we3:

> &gco_d(&t(e1,e2we3,e3),2);

Now let us show how the co-product acts on dotted and undotted elements

> w_p1:=e1we2; #selection of an element in undotted basis
w_c1:=&gco_d(w_p1); #action of &gco_d on undotted element e1we2
d_p1:=dwedge[F](e1,e2); #transformation of e1we2 to dotted basis
d_p2:=convert(w_p1,wedge_to_dwedge,F); #another way to accomplish the same transformation
d_c1:=&gco_d(d_p1); #action of &gco_d on the image of e1we2 in dotted basis

The following examples compose the dotted co-product with dotted and undotted wedge (acting on a wedge basis!!). First, let's show a dotted basis:

> Grassmann_basis:=cbasis(3); #Grassmann un-dotted basis
dotted_basis:=map(convert,Grassmann_basis,wedge_to_dwedge,F); #dotted basis

We will use the following notation for the dotted basis, e.g., e1We2 = e1we2+F[1,2]*Id, etc.:

> S:={e1we2+F[1,2]*Id=e1We2,e1we3+F[1,3]*Id=e1We3,e2we3+F[2,3]*Id=e2We3,
e1we2we3+F[2,3]*e1-F[1,3]*e2+F[1,2]*e3=e1We2We3}:
subs(S,dotted_basis); #dotted basis in shorter (dotted wedge) notation

Then, we compose dotted co-product with undotted and dotted wedge:

> for i in dotted_basis do
d_p1:=&gco_d(i):
drop_t(&map(d_p1,1,dwedge[F]));
`action_dwedge_o_&gco_d` = 2^maxgrade(%)*subs(S,%/2^maxgrade(%));
d_p2:=convert(%%,dwedge_to_wedge,-F);
print(`****************`);
od;

>

Thus, the above shows that

(wedge o Grassmann co-product)(x) = 2^(grade of x) * x

for any x in the Grassmann basis (Grassmann_basis) shown above and that the same is true, namely,

(dotted wedge o dotted Grassmann co-product)(y) = 2^(grade of y) * y

for any y in the dotted wedge basis (dotted_basis) shown above.

>

See Also: Bigebra [`&gco`] , Bigebra[`&cco`] , Bigebra[`&t`] , Bigebra[drop_t] , Bigebra[`&map`] , Cli5plus[dwedge]