'abcd products' in G(4)


I am currently studying \mathbb{G}_4 and would like to have a feel for abcd product (i.e. geometric product of four vectors). Is there any algebra software where one could explore these conveniently in symbolic form?

What I mean is that decomposing products into inner and outer products (or dot and wedge products) has been informative, and I would like to continue it. Here are the forms for ab and abc products.

ab = a \cdot b + a \wedge b

abc = (a \cdot b)c - (a \cdot c)b + (b \cdot c)a + a \wedge b \wedge c

It is nice to see that as b is in a special position in the middle of the product it also gets a negative sign in the decomposition, and also illustrative to set c=a to be able to see the reflection aba quite easily (as trivector then vanishes).

For abcd, here is a decomposition which still needs work:
abcd = (a \cdot b)(c \cdot d) - (a \cdot c)(b \cdot d) + (b \cdot c)(a \cdot d) + (a \cdot b)(c \wedge d) - (a \cdot c)(b \wedge d) + (b \cdot c)(a \wedge d) + (a \wedge b \wedge c) \cdot d + a \wedge b \wedge c \wedge d

It is starting to display the scalar, bivector and 4-vector parts (is this correct?), but there is still the (a \wedge b \wedge c) \cdot d term that is in need of decomposing (and I am not even sure about the other terms yet, this was a quick idea), and thus I thought maybe to ask here what are your thoughts on how to most conveniently and efficiently do these kind of explorations.

Hey @cyber, I don’t know, whether there is a software which can do this expansion for you. I did those expansions up to four vectors with pen and paper by separating the vector on the right hand side of the geometric product and inserting the expansion one level below on the left hand side (or vice versa), i.e. abc = (a\cdot b + a\wedge b)c = a(b\cdot c + b \wedge c). I also had difficulties to expand the multiple wedge products together with the “inner product” with one vector. I used the formula
(a\wedge b\wedge c)\cdot d = (a\wedge b)(c\cdot d) - (a\wedge c)(b\cdot d) + (b\wedge c)(a\cdot d) which is a generalisation of (a \wedge b)\cdot c = a (b\cdot c) - b (a\cdot c) = -c\cdot(a\wedge b), see Doran, Lasenby: Geometric Algebra for Physicists; p. 94, eqs. (4.49) and (4.50). I hope this helps you finishing the four-vector expansion. Btw.: I think it is correct that the geometric product of four vectors decomposes into a scalar, a bivector, and a 4-vector part in general.

Thank you for the reference, it helped to finish the decomposition. I guess one printing is:

abcd = \\ (a \cdot b)(c \cdot d) - (a \cdot c)(b \cdot d) + (b \cdot c)(a \cdot d) \\ + (a \cdot b)(c \wedge d) - (a \cdot c)(b \wedge d) + (b \cdot c)(a \wedge d) \\ + (a \wedge b)(c \cdot d) - (a \wedge c)(b \cdot d) + (b \wedge c)(a \cdot d) \\ + a \wedge b \wedge c \wedge d.

It seems that there are many kinds of symmetries in the result, so there are different kinds of partial forms which could highlight different features of the product. The book also seems like a quality book, I may study it, as I am interested in applying GA to novel gravity theories.

1 Like

Hey @cyber! I used this product expansion to calculate the commutator between two 2-blades [a\wedge b, c\wedge d] = \frac{1}{2} ((a \wedge b)(c \wedge d) - (c \wedge d)(a \wedge b)) which is itself a linear combination of 2-blades and corresponds (as far as I understood) to the commutator in Lie algebras [v_i, v_j] = f_{ij}^{\quad k} v_k ,

The book is a very good book and for me the approach to gravitation is very interesting, but as I am a GA noob, it is still very difficult for me to understand :wink:. At the moment I am at a stage of learning GA where I try to write all expressions in terms of basis vectors and components to have the familiar tensor calculus at hand, where I mostly know how to calculate :rofl:. The concepts of the Doran-Lasenby gravitation theory are still not trivial and I ask myself how it corresponds to the Poincaré gauge theory of Hehl et al.