Plan and reuse products

Use the geometric product for \(AB\), the wedge for its highest-grade part, and contractions when one homogeneous factor should reduce the grade of the other. For homogeneous \(A_r\) and \(B_s\), the conventions are

\[ A_r\wedge B_s=\langle A_rB_s\rangle_{r+s},\qquad A_r\mathbin{\lrcorner}B_s=\langle A_rB_s\rangle_{s-r}\quad(r\leq s),\qquad A_r\mathbin{\llcorner}B_s=\langle A_rB_s\rangle_{r-s}\quad(r\geq s). \]

The contractions are zero outside the indicated grade conditions and extend bilinearly across mixed grades. No reversal of either argument is implicit.

import torch
from clifra import make_algebra

algebra = make_algebra(2, dtype=torch.float64)
vectors = algebra.layout((1,))
bivectors = algebra.layout((2,))
even = algebra.layout((0, 2))

e1 = torch.tensor([1., 0.], dtype=torch.float64)
e2 = torch.tensor([0., 1.], dtype=torch.float64)

product = algebra.geometric_product(e1, e2, left=vectors, right=vectors)
torch.testing.assert_close(product, torch.tensor([0., 1.], dtype=e1.dtype))
wedge = algebra.wedge(e1, e2, left=vectors, right=vectors)
torch.testing.assert_close(wedge, torch.tensor([1.], dtype=e1.dtype))
contracted = algebra.left_contraction(e1, wedge, left=vectors, right=bivectors)
torch.testing.assert_close(contracted, e2)
planned = algebra.plan_product(op="geometric_product", left=vectors, right=vectors, output=even)
torch.testing.assert_close(planned(e1, e2), product)

symmetric_product means \((AB+BA)/2\). commutator_product means \(AB-BA\), and anti_commutator_product means \(AB+BA\); the latter two omit the factor of one half. scalar_product selects \(\langle AB\rangle_0\), without reversing either factor. These distinctions matter for mixed-grade coefficients.

An explicit output selects grades of the requested product. For a sequence of products, retain every intermediate grade needed by later factors: projecting \(AB\) before multiplying by \(C\) can change the selected part of \(ABC\). See rotor composition for a three-factor use.