Hello Alan,
Thanks for your answer. Apparently all I need from the exterior
algebra
can be done with '^'. If so, I can still use, as you suggest, a
diferent metric in the setup. (I will probably use a euclidien
metric)
and then use the geometric product ('*') to compute the dual: "-I*a".
I will look tomorrow into the Doran and Lasenby.
Regards,
Luis
-----
On Jan 18, 9:32 pm, Alan Bromborsky <[email protected]> wrote:
> luis wrote:
> > Hi,
>
> > Thanks,
>
> > Non, I don't want a Minkowsky metric. I am simply doing exterior
> > algebra in R^4 (or R^n) and in such case I need a null metric
> > to have the wedge product in Lambda(R^n).
>
> > As you can see
> > inhttp://en.wikipedia.org/wiki/Clifford_algebra#As_quantization_of_exte...
> > the exterior algebra Lamda(R^n) is the same as Cl(R^n,Q) with Q a null
> > metric.
>
> > Do you know another way of 'simulating' the wedge (alternating)
> > product of the
> > exterior algebra?
>
> > Luis
>
> > ------
>
> > On Jan 18, 8:19 pm, Alan Bromborsky <[email protected]> wrote:
>
> >> luis wrote:
>
> >>> Hi,
>
> >>> Thanks. Since I want to do exterior algebra, I am using a null
> >>> metric.
> >>> Also, I am not in 3-dim and so I don't have a cross product.
>
> >>> Let's suppose we have the following code:
>
> >>> #######################
> >>> #!/bin/env python
> >>> from sympy.galgebra.GAsympy import *
> >>> set_main(sys.modules[__name__])
> >>> MV.setup('e0 e1 e2 e3', '0 0 0 0, 0 0 0 0, 0 0 0 0, 0 0 0 0')
> >>> a = e0*e1*e2*e3
> >>> ######################
>
> >>> We know that in this case
>
> >>> a = e0^e1^e2^e3 and
> >>> *a = 1 (one)
>
> >>> but I don't know how to obtain this value (one) with sympy,
> >>> starting from 'a'. I suspect it should be trivial, but I
> >>> don't see how to do.
>
> >>> Luis
>
> >>> ---------------------------
>
> >>> On Jan 18, 7:24 pm, Alan Bromborsky <[email protected]> wrote:
>
> >>>> luis wrote:
>
> >>>>> Hi,
>
> >>>>> Is there some way in 'galgebra.GAsympy'
> >>>>> to apply the Hodge-* operator?
>
> >>>>> Thanks,
>
> >>>>> Luis
>
> >>>>> PS. I am using MV.setup(basis,metric) with a null metric.
>
> >>>> Seehttp://en.wikipedia.org/wiki/Geometric_algebra. I think the Hodge
> >>>> dual of a vector, v, is just the geometrics product of the vector and
> >>>> the pseudoscalar I that is
>
> >>>> v* = v*I
>
> >>>> for 3-dimensions (signature(+++)) we have the relation that the cross
> >>>> product of two vectors a and b is
>
> >>>> a x b = -I a ^ b
>
> >>>> where I is the pseudo-scalar and ^ is the outer (exterior) product
>
> >>>> seehttp://wiki.sympy.org/wiki/Geometric_Algebra_Moduleforhowto set
> >>>> up your metric.
>
> >> I think you want (for minkowski metric)
>
> >> #!/bin/env python
> >> from sympy.galgebra.GAsympy import *
> >> set_main(sys.modules[__name__])
> >> MV.setup('e0 e1 e2 e3', '1 0 0 0, 0 -1 0 0, 0 0 -1 0, 0 0 0 -1')
>
> >> since metric is orthogonal
>
> >> a = I = e0*e1*e2*e3
>
> >> or
>
> >> a = I = e0^e1^e2^e3
>
> >> a* = a*I = I*I = +1 or -1 depending on the metric for minkowski metric a*I
> >> = -1,
>
> >> for 4-dimensional euclidian a*I = 1 (MV.setup('e0 e1 e2 e3', '1 0 0 0, 0 1
> >> 0 0, 0 0 1 0, 0 0 0 1'))
>
> >> remember that for vectors a,b
>
> >> a*b = a.b + a^b
>
> >> geometric algebra is a clifford algebra
>
> >> for higher dimension rotations let U = a^b where U**2 = -1 (always true in
> >> a euclidian space) define the rotation
> >> plane and theta the ammount of rotation. The the rotor is
>
> >> R = cos(theta/2)+sin(theta/2)U
>
> >> and the rotation of a vector x->x' is given by
>
> >> x' = R*x*rev(R) (all geometric products)
>
> >> where rev(R) = cos(theta/2)-sin(theta/2)U
>
> >> if U**2 = 1 replace cos and sin with cosh and sinh
>
> The ^ product of geometric algebra is alternating since a^b = -b^a and
> a^a= 0 for vectors regardless of the metric. Note that the ^
> (wedge,outer,exterior) product and the | (dot,inner) product are built
> into GAsympy along with the * (geometric) product. All are defined for
> arbitrary multivectors. If you use ^ or | be sure to use parenthesis
> since the python operator priority for ^ and | is not what we would use
> in a geometric algebra expression. If you have a good library available
> look at "Geometric Algebra for Physicists" by Doran and Lasenby, page
> 218. They show the relationship between geometric calculus and
> differential forms. I haven't dealt with differential forms in 30 years
> so my knowledge about the correspondence with geometric calculus is
> probably negative.
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