# Leptons and the Lorentz Group O(3,3)

• I
This is note about O(3,3) space-time. The related article is:

https://doi.org/10.3390/sym12050817

Here's some background:

In O(3,1) space-time (Minkowski), the six generators of rotations and boosts can form an SU(2) x SU(2) Lie algebra. This algebra is then used generically by all the leptons so that their associated spinors obey special relativity.

In O(3,3) space-time (a mathematical space), the fifteen generators can form a number of SU(2) x SU(2) and related algebras. There are actually enough algebras to allow each lepton in the SM to transform under a unique algebra. This raises two questions, first, is there a one-to-one correspondence between leptons and algebras, and second, if this is true, which leptons are associated with which algebras.

Implications of the article:

The linked article investigates the internal symmetry space of O(3,3). The findings in the article suggest that: mathematical properties and relationships of certain group theory algebras in O(3,3) resemble mathematical properties and relationships of leptons in the SM.

Related Beyond the Standard Model News on Phys.org
Here's a breakdown of O(3,3):

O(3,3) has three O(3,1) subspaces (Minkowski) and three O(1,3) subspaces (associated with one space dimension and three time dimensions). Each of these subspaces is associated with a unique SU(2) x SU(2) subalgebra.

A spinor transforms under the spin 1/2 representation of an SU(2) x SU(2) algebra. In this sense, SU(2) x SU(2) algebras are "proxies" for spinors.

In O(3,1) space, the solutions of the Dirac equation are spinors and describe the time-evolution of a spin 1/2 particle in 3D space. Time is a c-number.

In O(1,3) space (a mathematical space), the solutions of the Dirac equation are spinors and describe the space-evolution in 3D time. Space is a c-number.

The linked article finds that: the units of measure for the conserved quantity due to invariance under rotations in time, are the same as those for the planck constant. This is implies that spinors in O(1,3) space also have spin 1/2.

In the context of O(3,3), this gives us six spinors, split into two families each with three members. That is, spinors associated with O(3,1) subspaces in one family and those associated with O(1,3) subspaces in the other.

jedishrfu
Mentor
@Marty4691 is interested in discussing the mathematical points of his paper. He can explain it better than I So over to @Marty4691

jedi

Thanks jedishrfu.

Just to be clear: O(3,3) space is a mathematical space and the linked article makes no claims about the physics in O(3,3).

As mentioned above the mathematical properties and relationships of some group theory algebras in O(3,3) overlap with the mathematical properties and relationships of leptons. The question is whether this is just a coincidence or warrants further investigation.

Any feedback is appreciated.

jedishrfu
Mentor
@fresh_42 may have something to say here.

bobob
Gold Member
The difficult part is relating the physics to the math and this has been pursued in a number of other papers (in the context of SO(3,3), which eliminates the improper lorentz transforms), so it seems worth investigating. For example, the paper below looks at SO(3,3) and a potential connection of gravity and EM, but interprets the extra dimensions differently.

https://cds.cern.ch/record/688763/files/ext-2003-090.pdf

Marty4691
Thanks bobob.

a number of SU(2) x SU(2) and related algebras. There are actually enough algebras to allow each lepton in the SM to transform under a unique algebra.
But e.g. when you rotate a physical system, there isn't a separate rotation group for each species of lepton. The same spatial rotation acts on everything at once.

There are independent lepton family number symmetries but they are only U(1).

bobob
Gold Member
There are independent lepton family number symmetries but they are only U(1).
That's true in the standard model, but presumably the only point of doing any physics along these lines is to look for something that fails in the standard model. Lepton family number (as per your example) is something that has been looked for experimetally in several different types of experiments and although the limits on violation are really stringent (aside from neutrino oscillations which violate lepton family number for neutrinos) , no one has a good explanation for why it should be so, but it begs an explanation. Also, at least according to what most believe, black holes violate those standard model symmetries.

Most if not all of physics, at some level, seems to come down to finding a larger group to encompass everything else (including gravity) and breaking symmetries. Basically, it comes down to a question of, is the mathematics correct? Can it be made to describe things we measure (in principle) in a simpler way than what we have now? At this point in time, I don't think anyone has a good idea of where there is any new physics, so starting with some mathematics and finding a way to make an experimental prediction about something new that hasn't already been ruled out is pretty much par for the course in HEP. (Obvious examples are minimal SU(5), which was ruled out by the proton lifetime, supersymmetry, which will probably always be "just out of reach of the current accelerators," etc.)

In the case at hand, I think the only question is to what can you apply this symmetry to make an experimentally testable prediction about something we don't know? If the OP does that, great. If not, it will just be a somewhat different attempt to look at almost the same symmetries a number of others have tried.