Matter / Anti-matter

How does one define which particles are matter and which are anti-matter? The sign of electric charge is mixed, in both cases.

We can define matter as those particles which have positive sign of square root mass. And anti-particles as those which have negative sign of square-root mass. Square-root mass takes the value (0, 1/3, 2/3, 1) for neutrino, electron, up quark, down quark.

When square-root mass is introduced this way, as a new fundamental quantum number, it immediately implies that (pre-)gravitation must be included in the theory. This requires an extension of the standard model. One possible way to obtain such an extension is a left-right symmetric extension of the standard model, based on the symmetry group E_6. And assign square-root mass as the quantum number for right-handed fermions, corresponding to electric charge which is assigned to left-handed fermions. The Higgs then transfers electric charge from left to right, and square-root mass from right to left, suggesting that the Higgs boson might actually be part of a triplet of Higgs bosons, the other two being electrically charged.

This kind of symmetric relation between electric charge and square-root mass suggests some sort of gauge-gravity duality. Indeed, it is not possible to derive the value of the low-energy fine structure constant without an extension of the standard model so as to include gravitation as well. One can express eigenstates of electric charge as superposition of eigenstates of square-root mass, and vice versa. This expression is key to the derivation of mass ratios.

Perhaps the most significant consequence of introducing square-root mass as a quantum number is that it helps `convert' the vector-interaction of pre-gravitation into an effective spin-2 gravitation, as observed, because of the segregation of matter from anti-matter.

Pre-gravitation [ SU(2)_R x U(1)_grav ] as the right-handed counterpart of the electroweak interaction.

There is a long history of researchers suspecting a connection between the weak force and gravitation. For several reasons; the foremost being that the weak force violates parity - only left-handed particles take part in it. Parity is a space-time symmetry, and if the weak force is exclusively an internal symmetry, how does it know about space-time, which is related to gravity? Also, unlike the strong force (only for quarks) and electrodynamics (only charged particles) the weak force is universal, just as gravity is. Moreover, the coupling constants for the weak force and for gravitation - both are dimensionful, whereas those for QCD and for electromagnetism are both dimensionless. Thus in many ways weak force is more like gravity and less like QCD and ED; maybe the weak force is the `gravity' of extra dimensions which are much smaller in scale than our 4D universe?

Many models have been put forth to unify weak force and gravity, but the going is not easy. Gravity is a spin-2 attractive force; whereas the weak interaction is a spin-one vector interaction unified in the electroweak theory. Trying to quantise gravi-weak will face the difficulties faced by quantum gravity theories.

The octonionic description of interactions provides promising evidence that pre-gravitation is the right-handed counterpart of electroweak. The octonions define the coordinate geometry of a physical space, the geometry of which is related to the four fundamental forces, generalising Einstein's vision of gravitation as geometry of 4D spacetime. The symmetry groups of the octonions [the exceptional Lie groups; particularly E_6 and E_8] exhibit subgroup structures coinciding with the symmetry groups of the standard model.

In particular, one sees subgroups SU(2) x U(1) x SU(2) x U(1) and it's possible to identify the first pair as SU(2)_L x U(1)_Y of the electroweak theory. The second SU(2)xU(1) arises in the description of the Left-Right symmetric extension of the standard model using E_6. Typically the associated vector bosons are interpreted as right-handed ultra-heavy W bosons.

We forego this interpretation, and instead associate the second pair SU(2)xU(1) with pre-gravitation: right handed counterpart SU(2)_R x U(1)_grav of EW. Analogous to how electric charge is defined from a U(1) operator in octonionic physics [U(1)_em --> U(1)_Y] we associate the quantum number sqrt {m} [square-root of mass] with fermions: plus sqrt{m} for matter, and -sqrt{m} for anti-matter. The motivation comes from noting that the mass ratios of electron, up quark and down quark are 1:4:9 whereas their charge ratios are 3:2:1 Invoking square-root of mass as quantum number introduces a L-R symmetry, also a gauge-gravity duality. Square-root of mass is the pre-gravitational charge which mediates pre-gravitation: attractive for matter-matter and for antimatter-antimatter, but repulsive for matter-antimatter.

Whatever was the early universe process that separated matter from antimatter, it leaves behind only attractive pre-gravity in our observed universe. It is known that SU(2)_R chiral gravity [Ashtekar connection gravity] can be mapped to Einstein's general relativity. The Chamseddine-Connes spectral action principle [eigenvalues of the squared Dirac operator as observables for gravity (Landi and Rovelli)] enables the same conclusion. Thus by removing antimatter from the scene, the pre-gravitational vector interaction is FAPP mapped to the attractive gravitational force we are familiar with.

What separated matter and antimatter in the early universe? Possibly a freeze-out accompanied by an effective CP violation, which disallowed further back-conversion of matter-antimatter to radiation. The matter became our universe; the antimatter perhaps went inside primordial black holes - the role of such antimatter PBH in our universe remains to be understood, including whether they are even allowed.

This apart, the octonionic theory gives rise to a decent possibility of convincingly relating the weak force to gravity, once plus-minus square-root mass is introduced as a quantum number mediating pre-gravitation. Weak interaction violates parity because the RH fermions take part in the RH counterpart of the weak force: pre-gravitation. There is evidence that the weak force is the gravity of two of the additional dimensions in octonionic space [ie 6D spacetime]. From the octonionic Lagrangian it is immediately evident that the ED and QCD coupling constants are dimensionless, whereas the other two are not.

It makes a whole lot of sense that all the four forces are vector interactions - unification becomes easier: spin 1/2 fermions and spin one gauge bosons.

The Higgs boson: the Higgs gives mass to the LH bosons, `transferring' it from the RH ones. It appears to be the case that additional Higgs ought to exist, giving electric charge to the RH bosons. This suggests that the Higgs is a triplet; something proposed by other researchers earlier. A triplet Higgs has also been suggested as a possible explanation for the recently claimed W boson mass anomaly [assuming it stands up to further scrutiny].

Why are there two signs for electric charge, whereas there is only one sign for mass?

A possible answer is the following: the analog of electric charge is not mass, but square-root of mass. Given a mass m, its square-root has two signs: + sqrt{m} and - sqrt{m}. The + sign is for matter and the - sign is for anti-matter. Mass has only one sign because our universe is made only of matter. It also follows that whereas matter-matter gravitational interaction is attractive, the matter-antimatter gravitational interaction is repulsive. To our knowledge, this last claim is not ruled out by experiments to date.

These conclusions follow from trying to understand the standard model in the language of the octonions.

Defining elementary particle states using the octonions shows that electric charge is quantised, as observed. The Clifford algebra Cl(6) can be used to deduce the quarks and leptons of the SM:

Particles and electric charge Anti-particles

Neutrino 0 Anti-neutrino 0

Anti-down quark 1/3 Down quark -1/3

Up quark 2/.3 Anti-up quark -2/3

Positron 1 Electron -1

Quantisation of electric charge is deduced from the eigenvalues of a U(1) number operator interpreted as U(1)_em. Anti-particle states are obtained by complex conjugation of particle states and are shown to have an electric charge value opposite to that of the particles.

It turns out that exactly the same construction can alternatively be used to obtain quantisation of square-root mass [a gauge-gravity duality]. There is no reason to confine oneself to eigenstates of electric charge. There is a dual construction of eigenstates of square-root mass, which gives:

Matter and sq. root mass Anti-matter

Neutrino 0 Anti-neutrino 0

Electron 1/3 Positron -1/3

Up quark 2/3 Anti-up -2/3

Down 1 Anti-down -1

This time the complex conjugation maps matter to anti-matter and changes the sign of sqrt m.

Whatever be the reason for the excess of matter over antimatter in our universe, it is obvious that our universe has only particles with plus sign for sqrt m, even though both signs for electric charge are there. The minus sqrt m is with the antimatter.

From the viewpoint of eigenstates of electric charge, the charge is same across generations, for a given family; but mass changes because charge eigenstates are not mass eigenstates.

Equivalently, from the viewpoint of eigenstates of sqrt mass, the sqrt mass is same across generations, for a given family; but electric charge changes because mass eigenstates are not charge eigenstates.

In summary, the gravitational force is only attractive, and not repulsive, because our universe is made only of matter. if we could observe the gravitational interaction of matter and antimatter (which is repulsive) we will see the two signs of sqrt mass in action.

The above also explains how in our approach pre-gravitation is a spin one vector interaction with two signs for sqrt mass, and yet the emergent gravitational interaction is only attractive. It is because matter and antimatter have been separated from each other!

Pre-gravitation is on the same footing as the SM gauge fields: all are spin one vector interactions. gravity appears different because of the matter-antimatter asymmetry in our universe. Whereas SM is SU(3)_c X SU(2)_L X U(1)_Y, pre-gravitation is brought on by a left-right symmetric extension of the SM, and is SU(3)_grav X SU(2)_R X U(1)_grav. Of these, SU(2)_R gets related to GR, whereas SU(3)_grav and U(1)_grav remain to be understood: they might relate to dark matter / MOND and to dark energy.

E_8, octonions, standard model, gravitation, and quantum theory

There is some renewed activity in the field. Dray, Manogue and Wilson have a paper out on the arXiv few days back: Octions, an E_8 description of the standard model. They will give a talk on this coming Monday, you can find out more about it here:

https://www.furey.space

The typical goal here is to show that the standard model gauge symmetries, and the quarks and leptons, are embedded in the larger of the exceptional groups. Most of the time, the attention is focused on a GUTS type scheme, with Lorentz symmetry included, but not gravitation.

I have landed in this octonion space from very different considerations. Those of Quantum Foundations.

There must exist a reformulation of quantum theory which does not make any reference to classical space-time labelled by four real numbers. Such a space-time exists if and only if the universe is dominated by classical macroscopic objects, just as today's low energy universe is. Such classical objects are however a limiting case of quantum systems and therefore, in invoking spacetime, quantum theory depends on its own classical limit. This is approximate, no matter how well it agrees with current experiments. We can in principle imagine a low energy universe devoid of classical objects, and hence devoid of classical spacetime. How to describe the dynamics then?

We can also ask what is the ground state of low energy quantum gravity? It is not 4D classical spacetime. This latter is the ground state of classical general relativity sourced by classical matter fields. When the sources are quantum fermions, they have a non-trivial ground state and a zero point energy - this cannot give rise to a spacetime with a point structure labelled by real numbers. We must resort to a coordinate geometry with non-commuting coordinates. And write the rules of a (pre-) quantum theory on this non-commuting coordinate geometry. On general grounds, the coordinates themselves can be expected to be complex, not real. From here the familiar quantum theory on classical spacetime must emerge as an approximation.

One way to construct the pre-dynamics is to raise the classical dynamical variables to the status of operators/matrices, but not impose the Heisenberg commutation variables. Instead we have a matrix-valued Lagrangian dynamics, and an accompanying Hamiltonian dynamics, from which the Heisenberg algebra and Heisenberg equations of motion are emergent in an approximation.

In this Lagrangian dynamics, it should be possible to define spin as a dynamical variable corresponding to an angular canonical variable. Such an angle clearly cannot be a part of the familiar 4D spacetime nor of its non-commuting version. The doubling of the non-commuting spacetime coordinates from four dimensions to eight is strongly indicated. This is (one of the many reasons) as to why we land up with the octonions. Octonions label the (non-commuting) coordinate geometry of the pre-quantum theory, and all the work being done on octonions and the standard model is seen from this light. The standard model forces as well as gravitation are symmetries of this physical space, which is equivalent to ten dimensional Minkowski spacetime SO(9,1). The octonionic space is like a square-root of this 10D Minkowski spacetime, analogous to tetrads, only now non-commuting. The ground state of quantum gravity is this octonionic space. Quantum gravity and quantum standard model forces are switched on as the curving of this octonion geometry. In the same way that the curving of 4D classical Minkowski spacetime is switched on by classical gravity through the metric and the Riemann tensor.

However, the geometry of octonions has a very fundamental difference from that of Minkowski spacetime. Unlike in the latter, fermions cannot have arbitrary properties in octonion space. Even before the curving of octonion geometry is turned on by switching on interactions, the non-commutative geometry already dictates properties of fermions. There are correct number of quarks and leptons. Electric charge is quantised, just as observed. These properties have been proved earlier by other researchers, but viewed by them in an algebraic context [SM and GUTSs in terms of the algebra of the octonions]. They do not interpret their results as consequences of this new spacetime geometry of the octonions. But such a spacetime interpretation is most essential - it adds dynamics and quantum theory to the algebraic approach. Not only does spacetime tell matter how to move; it also tells matter what to be. This is a natural extension of general relativity, and also of quantum theory - in the latter theory, quantisation of energy levels e.g. is dictated by the dynamics. So it is not a surprise if a non-trivial coordinate geometry dictates particle properties.

The standard model is a mystery when looked at from a 4D Minkowski spacetime labelled by four real numbers. But not when viewed from the `square' root of 10D Minkowski spacetime - the octonionic space.

The extra dimensions are complex and never compactified. Quantum systems probe these extra dimensions, always. Only classical systems live in 4D. The curving of the higher dimensions represents the standard model forces - equivalence principle is not obeyed because the curving is proportional to electric charge, not to mass.

String theory and the octonionic theory, compared.

String theory does not predict the values of the free parameters of the standard model, not yet. Assuming that these parameters take unique values which should be derivable from an underlying theory, string theory is not true, not yet. In fact the theory gives very many different possible values for these parameters, and it is not clear what criteria might favour the observed values. In contrast, the octonionic theory makes unique predictions for these parameters, which are derivable from an underlying theory. The action of the theory, and its symmetries and dynamics, are well defined. 

 It is also quite clear why the octonionic theory succeeds where string theory fails .. actually the two theories are quite similar but have crucial differences, making it quite clear which of the assumptions of string theory are wrong. Flat 10D Minkowski space-time is not the correct ground state on which to define the Fock space. Rather, the correct choice is the equivalent octonionic space SL(2, O) where O is an octonion. The Hamiltonian of the theory is not self-adjoint on the Planck scale. The extra spacetime dimensions are complex... they define symmetries of the standard model. These extra dimensions are never compactified. Only classical systems live in 4D space-time. Quantum systems live in octonionic spacetime, even at low energies. These few basic but key changes set string theory on the right track...that being the octonionic theory.