Sunday special: Conversing with Grok about the Clifford algebra Cl(2,2)

Over the past few days, I’ve engaged in an extensive dialogue with Grok, the AI developed by xAI, about the Clifford algebra ( Cl(2,2) ). 

Note: For those new to the topic, the Clifford algebra Cl(2,2)  is a mathematical structure used to describe geometric and physical properties of a 4-dimensional space with a metric signature (+,+,−,−). It’s particularly relevant in physics for modeling spacetime symmetries and spinors, offering a rich framework for connecting geometry to quantum mechanics.

Grok was remarkably helpful, anticipating my questions

Grok was remarkably helpful, anticipating my questions, assisting with Mathematica calculations, and discussing relevant publications on Cl(2,2)  and related topics. However, Grok also made a series of mathematical errors, from proposing incorrect generators for Cl(2,2)  to suggesting flawed minimal left ideals. At one point, Grok even claimed that (−1)×(−1)=−1! Each time I caught these mistakes, Grok graciously apologized. This led me to hypothesize that Grok might be programmed to test the mathematical acumen of its interlocutors, subtly checking whether I truly understand the subject! Despite the errors, the experience was both fun and enlightening. So, here’s my new blog post on Cl(2,2). I take full responsibility for its content, and if there are any mistakes, I invite my readers to point them out so I can make corrections. After all, I might be making errors even more frequently than Grok!

There is a clash between what is natural or just convenient for mathematics and what is natural or convenient for physics. We have to deal with this clash somehow, and it may create a confusion. So, let us meet this clash without fears. For space-time we have chosen coordinates x¹,x²,x³,x⁴ and metric signature (+++-). Then we add extra coordinates x5,x6 with signature (+,-). So we end up with coordinates x¹,...,x⁶ and signature  (+++-+-) . We have total signature (4,2), but not in mathematically preferred natural order (++++--). The isometry group of our extended space-time is isomorphic to O(4,2), but not exactly O(4,2). It is not a big problem, it is just an annoyance that may lead to a confusion here and there. Since we will be dealing with mathematics a lot, and physics will come only later, I will choose signature (++++--). At first we will discuss the toy model with suppressed two space dimensions. So, we will have signature (++--), with isometry group O(2,2). To avoid unnecessary confusion I will use coordinates y¹,y²,y³,y⁴. The relation with previously discussed extended space-time coordinates is

y¹ = x¹, y³ = x⁴, y² = x⁵, y⁴ = x⁶.

In particular the conformal infinity condition  x5 = x6, which arises in physical contexts like twistor theory, that we have discussed in the previous post, becomes y² = y⁴ in this framework, preserving the geometric structure.

Clifford algebra Cl(2,2)

We have already had some experience with Clifford algebras in the Spin Chronicles series of posts, where we were discussing Minkowski space spinors. Minkowski space metric has signature (3,1). Now we have signature (2,2).  Our discussion will be in parts similar, but but we will discover new features. For Minkowski space we ended with complex structures and biquaternions. Now everything will be real.  No complex numbers needed. In the beginning I will follow the way Cl(2,2) Clifford algebra is discussed in D.J.H. Garling, "Clifford Algebras: An Introduction",  London Mathematical Society, CUP 2011. But Garling has a opposite sign convention in the definition of Clifford algebras, so I will adapt the discussion from his book to our needs.

A Clifford algebra is a mathematical structure that encodes the geometry of a space through generators that satisfy specific multiplication rules, reflecting the space’s metric. For Cl(2,2) we need four mutually anticommuting generators e₁,...,e₄, with e₁² = e₂² = 1, e₃² = e₄² = -1. For Cl(2,2), the generators e₁,e₂,e₃,e₄ correspond to an orthonormal basis of a 4-dimensional space with signature (+,+,−,−). While we could proceed in purely algebraic way and search for minimal left ideals, we will take an easy way and start with a particular matrix realization of the algebra. Of course a curious Reader would ask: how this particular realization can be derived?  Well, it can, like we have derived Pauli matrices in Spin Chronicles, but it would take us unnecessarily long time. It is much easier to work with a particular realization while remembering that it is not necessarily the best for all purposes, and that there are infinitely many other realizations. This matrix representation, where the generators are 4x4 real matrices, leverages the isomorphism Cl(2,2) ≈ Mat(4,R).  While other representations exist (e.g., via different matrix forms), this choice simplifies our exploration of the algebra’s structure and its spin group.

So here are the four matrices representing the orthonormal basis, listed row by row.

e₁ = {{0, 0, 0, 1}, {0, 0, 1, 0}, {0, 1, 0, 0}, {1, 0, 0, 0}}
e₂ = {{0, 0, 1, 0}, {0, 0, 0, -1}, {1, 0, 0, 0}, {0, -1, 0, 0}}
e₃ = {{0, 0, -1, 0}, {0, 0, 0, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}}
e₄ = {{0, 0, 0, -1}, {0, 0, 1, 0}, {0, -1, 0, 0}, {1, 0, 0, 0}}

Matrices e₁ and e₂ are symmetric, e₃ and e₄ are anti-symmetric. Written in a block matrix form, with 2x2 blocks, e₁  and e₂ are of the form

0	X
X	0

while e₃ and e₄ are of the form

0	X
-X	0

Let us examine the volume element ω = e₁e₂e₃e₄. It is the diagonal matrix, with (1,1,-1,-1) on the diagonal:

ω = diag(1,1,-1,-1).

For the Clifford algebra of space Cl(3) it was commuting with all elements of the C(3), and its square was -1. This gave us the complex structure akin to the imaginary unit ( i ), which was crucial for spinor representations in 3D space. Now ω² = 1, and ω anti-commutes with all e_i, which implies a real structure, aligning with the split signature ((2,2)) and leading to real spinors, which we’ll explore in the context of Spin(2,2). We have an essentially  different situation now. But there is a hope! In the case of Cl(3) we were discussing the whole Clifford algebra. Now we are interested mainly in the automorphisms of our extended mini space-time, that is in the group Spin(2,2), which consists of elements of the even Clifford algebra CL⁺(2,2).

The whole algebra splits into

1) scalars, span of 1 - real numbers (1-dimensional subspace)
2) vectors - span of e_i (4-dimensional subspace)
3) bivectors - span of e_ie_j - e_je_i (i<j) (6-dimensional subspace)
4) 3-vectors - span of ωe_i (4-dimensional subspace)
5) pseudo-scalars - span of ω (1-dimensional subspace)

Together 1+4+6+4+1 = 16 = 2⁴ dimensions. It spans the whole algebra of real 4x4 matrices Mat(4,R).

The even Clifford algebra Cl⁺(2,2)

The even subalgebra Cl⁺(2,2) is crucial because it contains the Spin group Spin(2,2), the double cover of the isometry group SO(2,2). This group describes transformations that preserve the (2,2) metric, and its elements are key to understanding spinors and symmetries in our toy model.The even Clifford algebra Cl⁺(2,2) is the span of scalars, bivectors, and pseudo-scalars. It is 8-dimensional. While the generators ei anti-commute with ω, products of even number of generators all commute with ω. We can see it from the block form of generators: they will always be of the block-diagonal form

X	0
0	Y

This is, in fact, a general form of elements in Cl⁺(2,2) , where X and Y can be arbitrary real 2x2 matrices:

Cl⁺(2,2) = Mat(2,R) ⨁ Mat(2,R).

Since Mat(2,R) is 4-dimensional, we get 4+4 = 8 dimensions, as needed for Cl⁺(2,2).

The next step will be, as we have dome it before for Cl(3), to identify the three involutions of Cl(2,2). We will do it in the next post. In the next post, we’ll explore the three main involutions of Cl(2,2) —grade involution, reversion, and Clifford conjugation—and how they act on our matrix representation. These involutions are essential for defining Spin(2,2) and understanding spinor transformations, paving the way for applications in physics.

This is, in fact, a general form of elements in Cl⁺(2,2) , where X and Y can be arbitrary real 2x2 matrices:

Cl⁺(2,2) = Mat(2,R) ⨁ Mat(2,R).

Since Mat(2,R) is 4-dimensional, we get 4+4 = 8 dimensions, as needed for Cl⁺(2,2).

The next step will be, as we have dome it before for Cl(3), to identify the three involutions of Cl(2,2). We will do it in the next post. In the next post, we’ll explore the three main involutions of Cl(2,2) —grade involution, reversion, and Clifford conjugation—and how they act on our matrix representation. These involutions are essential for defining Spin(2,2)
 and understanding spinor transformations, paving the way for applications in physics.