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Embedding null vectors in metric Clifford algebras

Overview

Many projective geometric algebras (PGA) employ a basis vector that squares to zero (\(e_0^2 = 0\)). In a Clifford algebra that only supports a metric with squares \(+1\) or \(-1\), such a null vector can be replaced by a pair of orthonormal vectors of opposite signature.

This document describes the underlying mathematical isomorphism in detail and illustrates it with concrete examples in \(G(3,1,1)\).

1. Algebraic foundation

Let \(e\) and \(\bar e\) be two orthonormal vectors:

\[ e^2 = +1,\qquad \bar e^2 = -1,\qquad e\bar e = -\bar e e \quad\text{(anticommutativity)}. \]

Define two candidates for null vectors:

\[ n_1 = e + \bar e,\qquad n_2 = e - \bar e. \]

Using anticommutativity we obtain:

\[ \begin{aligned} n_1^2 &= e^2 + e\bar e + \bar e e + \bar e^2 = 1 + 0 - 1 = 0,\\[4pt] n_2^2 &= e^2 - e\bar e - \bar e e + \bar e^2 = 1 - 0 - 1 = 0. \end{aligned} \]

Both vectors are therefore null (square to zero).

2. Isomorphism of the algebras

Consider the real Clifford algebra \(C\ell(p,q+1)\) (with an extra \(-1\)-vector \(\bar e\)) and fix an embedding:

\[ \varphi: C\ell(p,q)_{\text{null}} \longrightarrow C\ell(p,q+1) \]

in which the null vector \(n_1\) is mapped to \(e+\bar e\).

  • \(\varphi\) is defined on the generating set and extended as an algebra homomorphism.
  • Because the relation \(n_1^2 = 0\) is preserved under \(\varphi\), \(\varphi\) is a well‑defined homomorphism.
  • The dimension increases from \(2^{p+q}\) to \(2^{p+q+1}\); the kernel, however, is trivial, i.e. \(\varphi\) is injective.
  • The subalgebra generated by \(\varphi\) is a direct summand of the larger algebra and allows bidirectional reconstruction: every computation involving \(n_1\) can be uniquely translated into a computation with \(e,\bar e\), and vice‑versa.

Thus there exists an injective algebra homomorphism between the null-based PGA and the higher‑dimensional algebra without null vectors. The two algebras are not isomorphic (their vector‑space dimensions differ), but the entire theory of the null‑based algebra is fully contained in the higher‑dimensional one. Every expression in \(C\ell(p,q)_{\text{null}}\) has an isomorphic counterpart in the subspace defined by \(\varphi\).

3. Examples

We work with \(n_1 = e + \bar e\) and examine the corresponding multivector calculations.

3.1 Product of two null vectors

Let \(a = n_1\). The square of \(a\) is

\[ a^2 = n_1 n_1 = 0 \quad\text{(as expected)}. \]

Decomposing \(n_1\) into \(e\) and \(\bar e\) yields the same result:

\[ (e+\bar e)(e+\bar e) = e^2 + e\bar e + \bar e e + \bar e^2 = 1 + 0 - 1 = 0. \]

3.2 Geometric product with an ordinary vector

Let \(x\) be any vector of the original space with \(x^2 = 1\) and \(x n_1 = - n_1 x\) (because \(x\) anticommutes with both \(e\) and \(\bar e\)). Compute the product \(x n_1\) from both viewpoints:

\[ \begin{aligned} x n_1 &= x(e+\bar e) = x e + x \bar e,\\ n_1 x &= (e+\bar e)x = -x e - x \bar e = -x n_1. \end{aligned} \]

The antisymmetric behaviour is preserved.

3.3 Outer (wedge) product

The outer product of the two null vectors \(n_1 \wedge n_2\) is a bivector:

\[ n_1 \wedge n_2 = (e+\bar e) \wedge (e-\bar e) = e \wedge e - e \wedge \bar e + \bar e \wedge e - \bar e \wedge \bar e. \]

Since \(e \wedge e = 0\) and \(\bar e \wedge \bar e = 0\), we obtain

\[ n_1 \wedge n_2 = - e \wedge \bar e + \bar e \wedge e = - e \wedge \bar e - e \wedge \bar e = -2\, e \wedge \bar e. \]

In the null‑based model this bivector exists as a fermionic quantity; its structure is completely captured by the embedding.

3.4 Inversion

In the higher‑dimensional algebra with \(e, \bar e\) the vectors \(e\) (\(e^2 = +1\)) and \(\bar e\) (\(\bar e^2 = -1\)) naturally possess a multiplicative inverse. The null vector \(n_1\) has no multiplicative inverse because \(n_1^2 = 0\) does not yield the identity. This property is preserved by \(\varphi\): the image \(e+\bar e\) is also non‑invertible, since \((e+\bar e)^2 = 0\).

4. Consequences

  • Every algebraic operation that can be performed in a null‑based PGA can equally be carried out in the higher‑dimensional algebra with the pair \((e, \bar e)\).
  • The isomorphism at the level of expressions is exact; only the dimension of the ambient algebra is larger by a factor of 2.
  • The embedding makes it possible to use existing Clifford‑algebra implementations that lack native null‑vector support for PGA computations as well.

5. Implementation note

In a library such as TanGA, which does not provide an explicit null‑vector marker in the signature bitmask, one may use the following approach:

  • Increase the dimension by 1.
  • Equip two additional basis vectors with \(+1\) and \(-1\) as metric.
  • Define the desired null vector as \(e_{\text{null}} = e_{d+1} + e_{d+2}\).
  • Use only this null vector together with the original vectors for all calculations (ignore the orthogonal complement).

The algebraic behaviour is identical to that of an explicitly null‑based algebra.

This technique is used in all of TanGA's null-vector algebras:

Algebra Dim Sig Null vector Embedding
BasisN3 / BasisPGA3 5 0b10000 einf = ep + em, eo = −½ep + ½em \(e_4\) (\(e_p^2=+1\)), \(e_5\) (\(e_m^2=-1\))
BasisN2 / BasisPGA2 4 0b1000 einf = ep + em, eo = −½ep + ½em (N2) / e0 = ep + em (PGA2) \(e_3\) (\(e_p^2=+1\)), \(e_4\) (\(e_m^2=-1\))