Exponential in Clifford algebra
Hello, I am trying to take the exponential of A+B*i in Clifford algebra (isomorphic to the complex numbers) .
Code is:
sage: R.<A,B> = PolynomialRing(ZZ);
sage: Q = QuadraticForm(R,1,[-1]);
sage: Cl.<i> = CliffordAlgebra(Q);
sage: exp(A+B*i)*exp(A-B*i)
I would expect it to output exp(2*A), but it produces this error:
TypeError: cannot coerce arguments: no canonical coercion from The Clifford algebra of the Quadratic form in 1 variables over Multivariate Polynomial Ring in A, B over Integer Ring with coefficients:
[ 1 ] to Symbolic Ring
What do you expect as a result? Is it a polynomial in
A,B,i?@MaxAlekseyez Good question. What I want, really, is for this
exp(A+B*i)*exp(A-B*i)to giveexp(2*A)(I've edited the question).There are no such things in the polynomial ring.
@MaxAlekseyez Well, I just want A and B to be symbolic variables. In my mind A and B are elements of the reals. Using polynomial ring was suggested by a colleague on this site to make A and B work as symbolic variables in Clifford algebras. I am willing to look at other things. Specifically I would like to set up a Clifford algebra, such that both
(A+B*i)*(A-B*i)=A^2+B^2andexp(A+B*i)*exp(A-B*i)=exp(2*A)are realized. I don't mind playing around with the definitions of the setup of the algebra to make this possible.You can define
AandBas symbolic variables, andexp(2A)would be a legitimate object, howeverexp(B*i)would still not make sense as exponentiation of the generator of the Clifford algebra is not defined.