SU(3) as spacetime structure

Difficulty: ★★★★★ research

Quarks have “colour” charge, which has symmetry group $SU(3)$ . This is normally understood as an “internal symmetry”. But Hestenes has a remarkable proposal to interpret it geometrically in spacetime itself, using tangent vectors and bivectors formed from them, in a 1982 paper research paper §8.

To begin, $SU(3)$ acts on a 3-dimensional complex vector space $\mathbb C^3$ . So we seek an analogue of this structure in Minkowski spacetime. Choose an orthonormal basis: $\mathbf t$ etc. Hestenes defines the bivectors $R := \mathbf{tx} + \mathbf {yz}$ , $G := \mathbf{ty} + \mathbf{zx}$ and $B := \mathbf{tz} + \mathbf{xy}$ . This notation uses the geometric product, but in this case the vectors are orthogonal, so the result is just wedge products, e.g. $\mathbf{tx} = \mathbf t\wedge\mathbf x$ . Together, the scalar and pseudoscalar parts from the geometric algebra form a subalgebra which is isomorphic to $\mathbb C$ , the underlying field for the vector space $\mathbb C^3$ . You can define the equivalent of complex conjugation. I completed an answer on Physics StackExchange today which has more details on this bit.

For intuition behind the bivectors, recall bivectors generate rotations. Here my use of the word “rotation” is contextual: the metric signature is Lorentzian, so I mean Lorentz transformations $SO(1,3)$ (or maybe some subgroup), hence these spacetime “rotations” include both boosts and spatial rotations. Then $R$ encodes a boost in the $x$ -direction, combined with a rotation about the $x$ -axis. This suggests to me a screw or helix picture… but with some strong cautions: the screws are not rigid bodies under many $SU(3)$ transformations, and some transformations even swap handedness.

Now $SU(3)$ consists of “complex rotations”. This may be interpreted as a subgroup of $SO(6)$ , the rotations on a 6-dimensional real space. Specifically, those which preserve the structure of the complex axes, relative to one another. However in our case these are not spacetime rotations, but act on the space of bivectors. This is quite abstract, and I envision making illustrations in future work. But just as a 2D rotation changes the $x$ and $y$ -components of a vector, these $SO(6)$ operators change the components of a bivector (the coefficients when decomposed in the 6 basis bivectors $\mathbf{tx}$ and so on). Returning to $SU(3)) specifically, one example is rotating$ R $and$ G $around (or "into") one another. Another example is changing the "phase" of$ R $in one direction, while changing$ G $the opposite way. And in turn, changing the phase of$ R $means keeping its overall (complex) magnitude the same, while redistributing the magnitude of the timelike and spacelike blades (the boost and spatial rotation). This suggests <em>quark colour is a bivector</em>. Or as Hestenes proposes, "we associate quark states with even multivectors". I am unconcerned with the distinction, for the purposes of this brief sketch. (Though I do lean towards Hestenes tbh.) Both characterise rotations, with even multivectors including an extra phase term, essentially. The gluon potential field may be interpreted as a map from vectors to$ \mathfrak{su}(3) $. This is the Lie algebra of$ SU(3) $, so its elements generate the$ SU(3) $transformations, so we realise them as certain bivectors. But these operators can themselves be characterised as bivectors, since in geometric algebra elements can also act as operators. (I am little concerned about the difference between rotations and rotation generators for the purposes of this current article, which is conceptual and a brief sketch.) The intuition is that as you move in a given direction, there is a "rotation" of colour space. This is very similar to one visualisation of a connection. Similarly the gluon field strength can be taken as a map from bivectors to$ \mathfrak{su}(3) $. As you rotate around a small loop (specified by the bivector), the output of the map describes the resulting rotation in colour space. This is very similar to one visualisation of Riemann curvature. The colour current is also a map from vectors to$ \mathfrak{su}(3) $. What is the interpretation of a "white" or colour-neutral quark combination? Apparently the totally antisymmetric tensor$ \epsilon $appears in the definition, which suggests a rewrite as a wedge product and possibly a Hodge dual. It is simply required the result be nonzero, for example$ R\wedge_{\mathbb C}G\wedge_{\mathbb C}B$ is valid. The wedge here is not the one acting on tangent vectors, but “complex vectors” which for us are basically bivectors. Intuitively, the “complex trivector” fills all directions of (real) bivector space. Just like the 4-volume element formed from vectors fills all spacetime directions (within a tangent space at a point, that is).

I have plenty of questions myself, and topics to ponder on:

Quarks also satisfy the Dirac equation, which leads to a 4-velocity vector. But a quark’s colour bivector would also seem to give rise to a 4-velocity vector. It does not make sense for one particle to have two different velocities! Perhaps we can split them into two separate fields/particles, or else just force the two velocities to coincide, although I’ll assume for now that both those wild ideas are wrong. [I will take the conservative (cautious) approach for now, and treat Hestenes’ idea as just a reformulation. But certainly I have hopes it will be more than that!]
Since the quarks (within a single hadron) are separate particles, they should be housed in different copies of space
Individual quarks are unobservable
Hestenes’ construction relies on choice of a timelike vector, which suggests the meaning of red/green/blue is frame-dependent. It would be very interesting to boost between frames.

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