Spacetime, vector bundles, and unification: the coming together of ideas which have been around for the last three decades or so

I have attached here Ed Witten's nice summary of `our knowledge of physics' :

The attempts at unification in the style of a Kaluza-Klein theory try to bring the vector bundle and the spacetime into the fold of a higher dimensional spacetime with its own metric, and the theory is then to be quantised.

In spirit, the octonionic theory does the same thing, but the formalism is not directly about `metric and higher dimensional spacetime'. Instead, one can construct the Dirac operator on a curved spacetime and find its eigenvalues; these are alternatives to the metric as observables of general relativity, and these eigenvalues are the entities used in our Kaluza-Klein programme, instead of the metric. We consider the eigenvalues of the Dirac operator on a higher dimensional spacetime. And these not only unify gravitation with the gauge fields of the vector bundle, but also unify the fermions with gauge fields and gravity. One quantum of this unified entity is named an atom of spacetime-matter, or an aikyon. Classical fields are recovered as `condensates' (i.e. macroscopic entanglements) of many aikyons.

Furthermore, we do not work on a higher dimensional curved Minkowski spacetime. We work on the square-root of Minkowski, a spinor spacetime, i.e. a twistor space labelled by the octonions. These define an 8D non-commutative space. We construct the Dirac operator on this octonionic space. Its eigenvalues are the dynamical variables which describe the aikyons. The act of quantisation consists of raising the eigenvalues to the status of matrices/operators. One matrix for every eigenvalue: this is the dynamical variable for the aikyon. Which matrix/operator? The very Dirac operator of which it is the eigenvalue. This is important, because in the quantum-to-classical transition, each matrix collapses to one of the eigenvalues, and from the collection of all the collapsed eigenvalues one again recovers the Dirac operator and hence the classical theory (on a 4D spacetime, because the collapse is a spacetime-dimension-reducing process, with classical gauge fields living on the 4D spacetime).

The matrix D for the aikyon has Grassmann numbers as its entries, and hence can be written as a sum of a bosonic matrix (even grade Grassmann) and a fermionic matrix (odd grade Grassmann). D plays the role of canonical momentum. Let us define a configuration variable Q whose time derivative (and hence velocity) is D. The Lagrangian of the aikyon is simply Trace[D^2] where Trace is matrix trace. The action is simply the time integral (Connes time) of this kinetic energy Tr[D^2]. This is nothing but Newton's free particle with a Ph. D. so to say Newton's 3D absolute Euclidean space has been replaced by curved octonionic space: the matrix-valued coordinate components D_i of D are analogous to metric tensor components, they determine the geometry of octonionic space, which encodes gauge fields of the standard model, gravity, and fermions. Newton's absolute time has been replaced by Connes time, the latter being a property of non-commuative geometries. This matrix-valued Lagrangian dynamics is Adler's trace dynamics: a pre-quantum pre-spacetime theory from which QFT on classical curved spacetime is emergent. The fundamental universe is made of a large number of aikyons.

To incorporate chiral fermions the octonionic space is generalised to split bioctonionic space, which is essentially the doubling of the octonionic space to 16D, with the second half being the parity reverse of the first, hence permitting chiral fermions to be introduced. The velocity Q-dot is used to define a new matrix variable q such that Q-dot = q_dot +q and the Lagrangian is rewritten in terms of q and further rewritten in terms of the bosonic and fermionic parts: q=q_B + q_F. Bosons and chiral fermions emerge. The dotted variables relate to gravity and to right-chiral fermions and are defined over the second half of the 16D space; the undotted ones relate to the standard model forces and to left chiral fermions and are defined over the first half of the 16D space. Left chiral fermions are eigenstates of electric charge; right chiral fermions are eigenstates of square-root mass: +sqrt{m} is matter; -sqrt{m} is antimatter. Our universe has only the former, having separated from antimatter in a breaking of scale-invariance (symmetry) in the very early universe. The Lagrangian prior to symmetry breaking is scale invariant. After the breaking, scale invariance is replaced by C, P, T: our universe violates CP and violates T. Together with the mirror antimatter universe which ours separated from, CPT is preserved. The mirror universe is a CP image and a T image of our universe.

The Lagrangian is assumed to have an E8 x E8 symmetry, with the first E8 symmetry being over the 8D half, and the second E8 over the parity reversed (split part) 8D half. The first 8D space has Euclidean signature and is equivalent to SO(10) space. The second 8D space has Lorentzian signature and is equivalent to SO(1,9) spacetime. We see the coming together of Witten's vector bundle and of spacetime into a unified entity, which is a physical reality. The vector bundle is not merely a mathematical construct; it is reality. The aikyon does not live in spacetime, but in E8 x E8 space: we could call it aikyon space. Only classical objects (these result from entanglement of many aikyons) live in spacetime (4D spacetime). Aikyons live in aikyon space, and the Higgs (implied naturally by the Lagrangian) couples left chiral and right chiral fermions.

Octonions are magical. Not only do they define spacetime and gauge field space, they also define elementary particles (SM fermions and bosons) and determine their properties such as quantisation of charge and mass. The octonionic coordinates and the Lagrangian work hand in hand in all this. Spinors made from Clifford algebras made from octonionic maps define quarks and leptons of the standard model. The first E8 branches as SU(3)_EuclideanSpace x SU(3)_ThreeGensLH x SU(3)_color X SU(2)_L x U(1)_Y. The second E8 branches as SU(3)_spacetime x SU(3)_ThreeGensRH x SU(3)_grav x SU(2)_R x U(1)_g Here SU(2)_R x U(1)_g lead to general relativity in the classical limit, whereas SU(3)_grav is new, and seems related to the conformal gravity modification of GR explicit in the Chamseddine-Connes spectral action principle (the heat kernel expansion of Tr[D^2]).

We recover the standard model and modified gravity in the emergent theory. The trace dynamics equations of motion, when reduced to an eigenvalue problem, give evidence for determining values of free parameters of the standard model.

Note that in unifying the vector bundle and the spacetime we never had to go to high energies. We have simply recast what we already know, onto a spinor spacetime, which when enlarged to higher dimensions, casts gauge fields and gravity into the aikyon space with E8xE8 symmetry. Quantum systems live in this aikyon space even at low energies. The quantum-to-classical transition that we observe around us all the time breaks E8xE8 because macroscopic objects are confined to 4D and their localisation is the very process which in the first place gives rise to the 4D classical spacetime and segregates the vector bundle (which is Euclidean space) from emergent spacetime.

A case for Adler's trace dynamics

Suppose we are asked to `quantise' classical dynamics, and are given the following two choices for how to do it. Which one should we choose, given that the second choice agrees with all experiments done so far, and the first one is untested because it is a Planck scale theory :

1. Trace Dynamics [Stephen Adler, 1996]

Starting from classical dynamics, raise all dynamical variables to the status of matrices / operators, and hence arrive at a Lagrangian which is a matrix polynomial. Take its matrix trace, and use this trace Lagrangian (a scalar) as the new Lagrangian in the action principle. You can now develop a matrix-valued Lagrangian dynamics, derive its matrix-valued equations of motion, and also the corresponding Hamiltonian dynamics. Everything proceeds as in conventional classical dynamics,

(i) except that

The dynamical variables, now being matrices, do not commute with each other. The commutator [q,p] evolves with time and is determined by the dynamics.

(ii) and except that

The new matrix dynamics has a conserved Noether charge, absent in classical dynamics, which is a result of the invariance of the trace Hamiltonian under global unitary transformations. This is coming about because we are now working with matrices and with a Hamiltonian which is a trace over matrices. The conserved charge is

Sum over all degrees of freedom i of the commutators

[q_i, p_i]

That is, whereas each [q_i. p_i] evolves with time, the sum of all such commutators is conserved. It is as if the d.o.f. exchange [q,p] with each other dynamically. This conserved quantity known as the Adler-Millard charge has the dimensions of action. Its existence is what makes trace dynamics into a pre-quantum theory. One never quantises trace dynamics; rather quantum theory emerges from it, as follows.

It is assumed that trace dynamics holds at some energy scale, not yet tested in the laboratory, say the Planck scale. We then ask what is the emergent dynamics at a lower energy scale, such as at the LHC, if one is not observing at the Planck scale. Techniques of statistical thermodynamics are employed to answer this question, and it is shown that in the emergent low energy theory, the Adler-Millard charge is equipartitioned over all d.o.f. As a result, for all coarse-grained d.o.f. the averaged commutator < [q,p] > takes the same value, and it is set equal to i\hbar. This is how one gets [q,p]=\ihbar, the Heisenberg algebra.

The averaged Hamilton's equations of motion of the underlying theory become Heisenberg equations of motion, and quantum field theory is recovered as a low energy emergent approximation to trace dynamics.

2. The second choice: Quantum Theory

Raise all classical dynamical variables to the status of matrices / operators, and impose by hand in an ad hoc way the Heisenberg algebra

[q, p] = i\hbar

The resulting quantum field theory agrees with all experiments done so far. But from a theoretical viewpoint imposing the Heisenberg algebra seems ad hoc. q and p do not commute once they are matrices. Shouldn't the dynamics determine the commutator [q,p] as in trace dynamics, with [q,p]=i\hbar emerging in an approximation?

With trace dynamics as a benchmark, one can now view quantum theory as a special case of trace dynamics. Using trace dynamics along with a spacetime described by the octonions opens up new possibilities for better understanding of the standard model of particle physics, and its unification with general relativity.

So which one do we choose: 1 or 2? Should the Heisenberg algebra be imposed a priori, or allowed to emerge from a more general theory which does not constrain the commutator [q,p] but lets it evolve dynamically?

The origin of the coupling constants

The group E_6 is the symmetry group of the Dirac equation for three fermion generations, in 10D spacetime, when the states of the fermions are defined in terms of complex octonions. E_6 is also the automorphism group of the complexified exceptional Jordan algebra. The group F_4 is the automorphism group of the exceptional Jordan algebra of 3x3 Hermitean matrices with octonionic entries. Hence the EJA defines the eigenvalue problem for the Dirac equation, and the characteristic equation of the EJA, which is a cubic, determines the eigenvalues. One of these eigenvalues, in conjunction with the Lagrangian, fixes the value of the electric charge, and hence the low energy fine structure constant. Between them, the eigenvalues also determine the mass ratios. This is another piece of evidence that quantum systems do not live in 4D classical spacetime, but in a 10D complex spacetime, at all energies. The coupling constants are fixed in 10D.

Pauli's question to the Devil: `what is the meaning of the fine structure constant?' can be answered as follows. The value of the electric charge of the electron is an eigenvalue of the Dirac equation in ten (complex) spacetime dimensions. More precisely put, the eigenvalue in question is the projection of the `square-root-mass-charge' of the electron, onto the LH U(1)_em sector. The RH sector U(1)_grav fixes mass ratios.

Prior to measurement by a classical apparatus, an electron is in 10D. After measurement, it is in 4D. Collapse of the wave function localises the electron not only in space, but in spacetime, and furthermore, it reduces the dimensionality of the occupied spacetime from 10 to 4, with the penetration depth into the extra dimensions becoming less than a Planck length. Clearly, the collapse of the wave function is a real physical phenomenon, which cannot be described by our current formulation of quantum theory. Any claims to the contrary will have to give a derivation of the values of the coupling constants from conventional QFT. One cannot get away by just saying that these are fixed at very high energies, or that there are multiverses and values of constants are not fundamental!

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).

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.