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 in http://en.wikipedia.org/wiki/Clifford_algebra#As_quantization_of_exterior_algebra 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_Moduleforhow to 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 --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "sympy" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sympy?hl=en -~----------~----~----~----~------~----~------~--~---
