I have seen two definitions for the norm in the Clifford algebra $\mathrm{Cℓ}_{p, q, r}$.

According to one of them $\Vert x\Vert = ⟨x. x^\dagger⟩_0$, where the dagger stands for the reversal of the order of all Clifford products. That is, if the basis is $e_1, e_2 \ldots$, then $ (e_1 . e_2 . e_3)^\dagger = e_3. e_2 . e_1$. The $⟨x. y⟩_0$ denotes the $0$ grade (i.e. scalar part) of the symmetric inner product $\frac{ x.y + y.x}{2}$.

The other definition is $\Vert x\Vert = x. \hat{x} $, where hat stands for Clifford conjugation, that is the composition of the main involution and the reversal.

If we take $\mathrm{Cℓ}_{0, 1, 0} \simeq \mathbf{C}$, then the Clifford conjugation coincides with the usual complex conjugation, while the reversal doesn't. That is, if we take $x=1+e_1$, then $\hat{x}=1 - e_1 = x^*$. It seems that, if we deal only with vector elements, the definitions are equivalent.

So the question is: which one is more natural definition? And which one holds also for bivectors and multivectors in general?

The literature is confusing.


So the question is which one is more natural definition?

This is entirely a matter of opinion, but my vote would be with the one that matches complex conjugation. A better question would ask where each one is useful. That question would probably be best answered by paying attention to where they are used in the literature.

And which one holds also for bi-vectors and multivectors in general?

They are both defined for all elements of the algebra, and that includes all multivectors, so I'm not sure what you want.

The literature is confusing.

Yes, a lot of it is. Hopefully this will improve over time...

  • $\begingroup$ Thanks for the answer. I am writing a computer algebra package so I was looking for definitions that are simple to code and yet general. My personal preference is also for the conjugation but I was reading the Leo Dorst's book there it is given with reverses. $\endgroup$ – user48672 Feb 1 '15 at 13:00
  • $\begingroup$ @user48672 What language are you writing the package in? $\endgroup$ – rschwieb Feb 1 '15 at 13:23
  • $\begingroup$ The Maxima language. It is a CAS system. $\endgroup$ – user48672 Feb 1 '15 at 13:49
  • $\begingroup$ @user48672 Cool. There is a Python package for geometric algebra which might interest you (or maybe not, but it doesn't hurt.) $\endgroup$ – rschwieb Feb 1 '15 at 15:25

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