Octonions, scale invariance, and a CPT symmetric universe

In the octonionic theory, prior to the so-called left-right symmetry breaking, the symmetry group is E8 and the Lagrangian of the theory is scale invariant. There is only one parameter, a length scale, which appears as an overall multiplier of the trace Lagrangian.

Something dramatic happens after the symmetry breaking. Three new parameters emerge, to characterise the fermions:

Electric charge, has two signs, sign change operation C is complex conjugation. Ratios (0, 1/3, 2/3, 1)

Chirality / spin, has two signs, sign change operation P is octonionic conjugation. Ratios (1/2, -1/2)

Square-root of mass, has two signs, sign change operation T is time reversal t --> -t . Ratios (0, 1/3, 2/3, 1)

Thus there are 2x2x2 = 8 types of fermions, based on sign of charge, sqrt mass, spin.

This offers an attractive explanation for the origin of matter-antimatter asymmetry. A CPT symmetric universe. The four types of fermions which have positive sqrt mass become matter, our universe, moving forward in time. The other four types of fermions, which have negative sqrt mass, become anti-matter, a mirror universe moving backward in time ! The forward moving universe and the backward moving universe together restore CPT symmetry. Our universe by itself violates T, and hence also CP. Matter and anti-matter repel each other gravitationally, thus explaining their separation.

Prior to the symmetry breaking, an octonionic inflation [scale invariant, time-dependent in Connes time] precedes the `big bang' creation event, which is the symmetry breaking itself. Freeze out happens when radiation --> matter-antimatter is no longer favorable. Segregation takes place; our matter universe has a one in a billion excess of matter over anti-matter. The backward in time mirror universe has a one in a billion excess of anti-matter over matter.

The maths of complex octonions naturally accounts for the C, P, T operations. Scale invariance is transformed into CPT invariance in the emergent universe.

In an elegant proposal, Turok and Boyle have also recently proposed a CPT symmetric universe [mirror universes]. They, however, did not use the octonions.

A glimpse into unification with E8

In a recent talk that Tevian Dray gave, he identified E8 with SU(3, OxO') where O is an octonion, and O' is a split octonion. This is very helpful, suggesting that this is the right way to construct particle states prior to the L-R symmetry breaking.

Because subsequent to this symmetry breaking we have an SU(3,O) for electro-color, with particle states describing electro-color and made from a Cl(6) and a LH Majorana neutrino. And we have an SU(3,O') for pre-gravity, made from a Cl(6) and a RH Majorana neutrino, with the source charge being square-root of mass.

It remains to be understood what is the Clifford algebra to be associated with SU(3, OxO') prior to the symmetry breaking. However, one can guess that the particle states are `scalar lepto-quarks' such as neutrino-antineutrino, anti-down-quark-electron, upquark-upquark, positron-downquark. These states have no electric charge defined for them. Instead the quantum number for the source charge is electric-charge-square-root mass. This is the source for the unified force which has the symmetry E8 and whose Lagrangian has E8 x E8 symmetry.

Fortunately, the Lagrangian is easy to construct, and in our view, very pretty. The action is

S ~ (1/L^2) \int d\tau \dot{q_1} \dot{q_2}

We are doing special relativity in 4, 6 or 10 dim spacetime, but always defining space first through a division algebra: a quaternion, or a split biquaternion, or an octonion, or a split bioctonion. Depending on which space the above action is defined, four different cases emerge:

quaternion: special relativity and GR in 4D

split biquaternion: special relativity in 6D, gravi-weak unification

octonion: special relativity in 10D and elecro-color interaction and electroweak

split bioctonion: special relativity in (?) D and full unification; electro-color unifies with pre-gravitation

To see symmetry breaking, we expand the dynamical variables as

\dot q_1 = \dot Q_1 + \alpha Q_1

\dot q_2 = \dot Q_2 + \alpha Q_2

where \alpha will be the Yang-Mills coupling constant, which comes into play only after symmetry breaking. The dotted part is over O' and describes gravity; the undotted part is over O and describes electro-color-weak.

Bosons and fermions arise from the expansion

Q_1 = Q_B + \beta_1 Q_F

Q_2 = Q_B + \beta_2 Q_F

\beta_1 and \beta_2 are two unequal Grassmann elements, making the action as one for a 2-brane evolving in Connes time \tau on OxO'. The universe is made of enormously many such 2-branes.

This in essence is all that is there to the theory...details remain to be worked out. The coupling constants are fixed by the algebra of the octonions, or equivalently by the octonionic geometry in which the 2-brane lives.

The interested reader can find a little more related detail in the Appendix D p.47-51 of

https://www.preprints.org/manuscript/202101.0474/v5

The figure below attempts to display the role of division algebras in unification.

The characteristic equation of the exceptional Jordan algebra

The attached 1999 paper by Dray and Manogue will perhaps some day be seen as one of the most important papers in theoretical physics. The eigenvalue problem discussed here seems to play a central role in the determination of the coupling constants of the standard model.

The exceptional Jordan algebra (EJA) is the algebra of 3x3 Hermitean matrices with octonionic entries. The algebraic operation is the Jordan product, which is a symmetrized matrix multiplication:

A * B = (AB + BA)/2

The automorphism group of the EJA is the 26 dimensional exceptional group F_4. The automorphism group of the complexified exceptional Jordan algebra is E_6. This same E_6 is also the symmetry group of the Dirac equation for three fermion generations in 10D spacetime (Dray and Manogue, 0911.2255). Hence, the eigenvalue problem for the EJA is also the eigenvalue problem for the Dirac equation in 10D. Its characteristic equation is an elementary algebraic cubic equation whose solutions depend on the trace and the determinant of the 3x3 matrix.

When the octonionic entries in the matrix are the fermionic states and the diagonal entries are value of the electric charge for a fermion, the resulting eigenvalues [all three are real] appear to determine the coupling constants of the standard model [after relating the Dirac equation to the Lagrangian from which it is derived]. Clearly then, the coupling constants are being fixed in 10D, not in 4D. Undoubtedly, quantum systems even at low energies live in 10D. Only classical systems live in 4D.

The eigenvalues take the simple form

q - sqrt{3/8}, q, q + sqrt{3/8}

where q is the value of the electric charge ratio: q is one of (1/3, 2/3, 1) for down quark, up quark, electron.

Reference: Tevian Dray and Corinne Manogue, https://arxiv.org/abs/math-ph/9910004, The Exceptional Jordan Eigenvalue Problem

Will the E_8 x E_8 heterotic string theory make a comeback?

Possibly yes, but in a new avatar.

In the mid 1980s, E_8 x E_8 and SO(32) string theories made big news as possible candidate symmetries for unification. But it all fizzled out, because of the troubles with compactfication of the higher dimensions.

Why then, more than three decades later, might there now be a (promising, and this time successful) comeback?

The answer lies in the various developments that were taking place in high energy theoretical physics during these intervening thirty years, outside of string theory. The confluence of these disparate developments is what suggests a dramatic revival of a conceptually improved string theory.

The first of these is Stephen Adler's theory of Trace Dynamics : a pre-quantum (though not pre-spacetime) theory from which quantum field theory is an emergent approximation. Also emergent is a dynamical mechanism for explaining the quantum-to-classical transition. The advantage of trace dynamics is that it paves the way for a pre-quantum, pre-spacetime theory from which gravitation, and quantum theory, are emergent.

The second development was the Ghirardi-Rimini-Weber phenomenological theory of spontaneous collapse. This is a stochastic non-unitary modification of quantum theory, with a non-Hermitean Hamiltonian, which explains why macroscopic objects are classical, and do not obey quantum superposition. Adler's trace dynamics provides a theoretical underpinning for the phenomenology of GRW.

The third is the work of Alain Connes and collaborators on non-commutative geometry, and in particular the spectral action principle of Chamseddine and Connes, which allows the Einstein-Hilbert action to be expressed in terms of the eigenvalues of the squared Dirac operator. And also the absolute time of Connes, a feature unique to his non-commutative geometry.

The fourth is the long history of research spread over the last five decades, on relating octonions to the standard model of particle physics. This field has picked up pace over the last decade or so, and includes the pioneering work of many on the exceptional Jordan algebra, and the application of Clifford algebras to construct elementary particle states.

Our own work forms a yet different fifth avenue, which is foundational: to seek a reformulation of quantum (field) theory which makes no reference to classical spacetime. When we made very humble beginnings on this project two decades ago, there was not the slightest hint that we will end up with a revised string theory, or even have anything to do with particle physics. We were only seeking a quantum theory of gravity from a foundational motivation.

Trace dynamics, coupled with the spectral action principle, leads to a highly simple form for the fundamental action principle:

S / \hbar = L_P^2 / L^2 \int d\tau \dot{q_1} \dot {q_2}

q_1 and q_2 are two unequal matrices which together define a 2-brane of area L^2 [an `atom' of space-time-matter, our `string']. \tau is Connes time. Various considerations motivate that the 2-brane lives on the non-commutative coordinate geometry of octonionic space. In other words, q_1 (also q_2) is a sum of eight matrices, one for each of the eight directions of the octonion, and these matrices represent the four forces of nature, and fermions, which curve the octonionic space.

Incidentally, this Lagrangian has the same form as the Bateman oscillator; a pair of coupled oscillators with opposite signs for energy. Giving reason to believe that the cosmological constant is exactly zero in this new theory.

More precisely, the 2-brane lives in OxO', which is a 16-D space made of the octonion O and the split octonion O'. This gives enough freedom to construct chiral fermions. There is a known equivalence bewteen SU(3, OxO') and E_8, and the symmetry group of the space of q_1 is indeed E_8, and that of q_2 is also E_8 and hence the Lagrangian of this theory has E_8 x E_8 symmetry.

Why then is this not the same as string theory? Because:

(i) elementary particle states are defined on octonionic space (a twistor space or a spinor spacetime, equivalently), not on its equivalent 10D Minkowski spacetime. This is the first point of departure from string theory.

(ii) the Hamiltonian of the new theory is not self-adjoint on the Planck scale. This is most essential for obtaining the standard model. The anti-Hermitean part of the Hamiltonian enables a quantum to classical transition dynamically and is responsible for the emergence of classical spacetime.

(iii) This is a higher dimensional theory, but the extra dimensions (which are complex) are never compactified. Only classical systems live in 4D. Quantum systems live in octonionic space (equivalently 10D Minkowski) even at low energies. The extra dimensions are complex-valued - their symmetries are precisely those of the standard model forces. We have a Kaluza-Klein theory in which the extra dimensions are provided by the octonionic directions. We have compactification without compactification, and hence overcome the highly troublesome non-uniqueness of string theory.

(iv) The algebra of the octonions, in conjunction with the above Lagrangian, determines the values of the free parameters of the standard model.

There is a great deal of detail to be filled in, but this is likely the correct approach to unification. We understand why string theory came close to being successful, and also why it did not succeed. Developments outside of string theory over the last three decades now provide completely independent motivation for string theory, but this time without the undesirable features which led to the failure of the original theory.

Why is matter electrically neutral?

When some symmetry breaking mechanism in the early universe separated matter from anti-matter, particles were segregated from their anti-particles. And yet, the sign of the electric charge was not the criterion for deciding who went where. Matter has the positively charged up quark (2/3) and the negatively charged down quark (-1/3) and the electron (-1). Anti-matter has their anti-particles. If sign of electric charge was the deciding criterion for separating matter from anti-matter, all particles in our universe ought to have had the same sign of charge. That is not the case, and yet matter is electrically neutral! How could that have come about?

Even the algebraic proof based on the octonions, which shows quantisation of electric charge, naturally clubs positively charged particles together, when their states are made from a Clifford algebra:

Particles and charge Anti-particles and charge

Neutrino 0 Anti-neutrino 0

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

Up quark 2/3 anti-up -2/3

Positron 1 electron -1

What picks the up quark from the left, and down and electron from the right, and club them as matter, and yet maintain electrical neutrality?

We have proposed that the criterion distinguishing matter from anti-matter is square-root of mass, not electric charge. One can make a new Clifford algebra afresh from the octonions, and show that square-root of mass is quantised:

Matter and square-root mass Anti-matter and sqrt mass

Neutrino 0 Anti-neutrino

Electron 1/3 Positron -1/3

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

Down quark 1 anti-down -1

Let us now calculate the net electric charge of matter, remembering that there are three down quarks (color) and three up quarks (color):

0 + (-1x1) + (3 x 2/3) + (3 x -1/3) = 0

It seems remarkable that the sum of the electric charges of matter (particles with +ve sqrt mass) comes out to be zero. It need not have been so. This demonstration might help understand how matter-antimatter separation preserved electrical neutrality.

Before this separation, the net square-root mass of matter and anti-matter was zero, even though individual sqrt masses were non-zero. In this we differ from the standard gauge-theoretic picture of EW symmetry breaking and mass acquisition. In EW, particles are massless before symmetry breaking, because a mass term in the Lagrangian breaks gauge invariance. However, for us sqrt mass is not zero before the symmetry breaking - its non-zero value was already set at the Planck scale (and cosmological expansion scaled down actual mass values while preserving mass ratios). Indeed it is rather peculiar if prior to the symmetry breaking particles have electric charge but no mass. For us, QFT on a spacetime background (and hence gauge theories) are not valid before the left-right symmetry breaking. In fact spacetime itself, along with gravitation, emerge after this symmetry breaking, as a result of the quantum to classical transition. Spacetime emerges iff classical matter emerges.

Prior to the symmetry breaking, dynamics is described by trace dynamics, there is no spacetime, and we have `atoms' of space-time-matter. The concepts of electric charge and mass are not defined separately....there is only a charge-square-root mass [a hypercharge can also be defined, as for EW] and this is the source for a unified force in octonionic space.

Mass quantisation from a number operator

The masses of the electron, the up quark, and the down quark, are in the ratio 1 : 4 : 9

This simple fact calls for a theoretical explanation.

A few years back Cohl Furey proved the quantisation of electric charge as a consequence of constructing the states for quarks and leptons from the algebra of the octonions [arXiv:1603.04078 Charge quantisation from a number operator]. The complex octonions are used to construct a Clifford algebra Cl(6) which is then used to make states for one generation of quarks and leptons. The automorphism group G_2 of the octonions has a sub-group SU(3) and these particle states have the correct transformation properties as expected if this SU(3) is SU(3)_color of QCD. Further, (one-third of) a number operator made from the Cl(6) generators has the eigenvalues (0, 1/3, 2/3, 1) [with 0 and 1 for the SU(3) singlets and 1/3, 2/3 for the triplets] allowing this to be identified with electric charge. This proves charge quantisation and the U(1) symmetry of the number operator is identified with U(1)_em. Anti-particle states obtained by complex conjugation of particle states are shown to have electric charge (0, -1/3, -2/3, -1). Thus the algebra describes the electro-colour symmetry for the neutrino, down quark, up quark, electron, and their anti-particles. Note that it could instead be the second fermion generation, or the third generation. Each generation has the same charge ratio (0, 1/3, 2/3, 1).

This same analysis can now be used to show that the square-root of the masses of electron, up and down are in ratio 1:2:3 All we have to do is to identify the eigenvalues of the number operator with the square-root of the mass of an elementary particle, instead of its electric charge. And we also get a classification of matter and anti-matter, after noting that complex conjugation now sends matter to anti-matter. as follows:

Matter, sqrt mass Anti-matter, sqrt{mass}

anti- Neutrino 0 Neutrino 0

Electron 1/3 positron -1/3

Up quark 2/3 anti-up -2/3

Down 1 anti-down -1

Compared to the electric charge case above, the electron and down quark have switched places, and we already have our answer to the mass quantisation question asked at the start of this post. There is again an SU(3) and a U(1) but obviously this is no longer QCD and EM. We identify this symmetry with a newly proposed SU(3)_grav x U(1)_grav whose physical implications remain to be unravelled. [GR is supposed to emerge from SU(2)_R this being an analog of the weak force SU(2)_L].

The group E_6 admits a sub-group structure with two copies each of SU(3), SU(2) and U(1). Therefore, one set is identified with the standard model SU(3) x SU(3) x U(1) [electric charge based] and the other with the newly introduced SU(3)_grav x SU(2)_R x U(1)_grav [sqrt{mass} based]. In the early universe, the separation of matter from anti-matter is the separation of particles with positive square-root mass from particles of negative sqrt mass. This separation effectively converts the vector-interaction of pre-gravitation into an attractive only emergent gravitation.

However, the second and third fermion generations do not have the simple mass ratios (0, 1, 4, 9) unlike the electric charge ratios which are same for all three generations. Why so?! Because mass eigenstates are not the same as charge eigenstates. We make our measurements using eigenstates of electric charge; these have strange mass ratios, eg muon is 206 times heavier than the electron. If we were to make our measurements using eigenstates of square-root mass, we would find that all three generations have the mass ratios (0, 1, 4, 9) whereas this time around the electric charge ratios will be strange. There is a perfect duality between electric charge and square-root mass.

A free electron in flight - is it in a charge eigenstate or a mass eigenstate? Neither! It is in a superposition of both, and collapses to one or the other, depending on what we choose to measure. In fact the free electron in flight does not separately have a mass and a charge; it has a quantum number which could be called charge -sqrt mass, which is the quantum number for the unified force. Unification is broken by measurement: if we measure EM effect then we attribue electric charge to the source. if we measure inertia or gravity, we attribute mass to the source. These statements are independent of energy scale. A classical measuring apparatus emerges from its quantum constituents as a consequence of sufficient entanglement: the emergence of such classical apparatus is the prelude to breaking of unification symmetry. In the early universe, sufficient entanglement is impossible above a certain energy [possibly the EW scale] and it appears as if symmetry breaking depends on energy. This is only an indirect dependence. The true dependence of symmetry breaking is on the amount of entanglement. In our current low energy universe we have both low entanglement systems (quantum, unified) and high entanglement systems (classical, unification broken).