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 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 >> > > > > 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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