What every string theorist should know about physics

It is not possible to quantize gravity without first also unifying it with the forces described by the standard model of particle physics. This is because quantum gravity will be sourced by the energy-momentum of bosons and fermions, and these elementary particles take part also in standard model forces, all of which are stronger than gravity. Therefore, switching on quantum gravity necessitates switching on the other forces as well.

This argument has been used (correctly) in the past to criticize stand alone theories of quantum gravity such as loop quantum gravity. On the other hand, in recent times string theory is being presented as a successful UV-complete theory of quantum gravity (without making any reference to unification). But surely the aforesaid criticism applies equally well to stringy quantum gravity. To this some respond by saying that the different vibrations of the string do include fermions and gauge bosons of  Yang-Mills theories. However, since the standard model has not yet been derived from string theory, the UV-complete stringy quantum gravity cannot by itself be nature’s true/correct quantum gravity theory per se.

Without successful unification, stringy quantum gravity in itself suffers from the above criticism and can be conveniently forgotten. Nonetheless, a vibrating string whose different vibrations are various elementary particles is in itself a very attractive idea. Can the idea be saved? The answer is yes, provided it is arrived at in a foundational manner, as follows.

A string is a quantum entity right from the word go, and so is a collection of strings. Such a collection obeys the quantum superposition principle and can never give rise to a classical space-time. Not even at low sub-Planck energies, nor when gravity is negligible. This is because  a quantum particle can be in more than one place at the same time; hence its resultant gravitation is also in a superposition. It is then a consequence of the Einstein hole argument that the point structure of the underlying classical spacetime is destroyed. Therefore, string theory can never be successfully formulated as a perturbative quantum field theory around Minkowski spacetime. Not even at low energies. Trying to do so is the reason why the theory fails as a theory of unification.

Nor is it a sound starting principle to assume, without justification, that fundamental building blocks of nature are extended objects such as strings. It would be fruitful if extended objects can be motivated from some basic premise. Such a premise exists. And that is to demand that there should exist a reformulation of quantum theory which does not depend on classical spacetime (for reasons outlined above), even at low energies. Such a theory is Stephen Adler’s trace dynamics, which is a pre-quantum theory: it is a matrix-valued Lagrangian dynamics from which quantum field theory is an emergent approximation. Trace dynamics can be transformed into a pre-spacetime theory by replacing every point of spacetime by an octonionic space (more precisely split bioctonionic space). Consistency of equations of motion then demands that the fundamental building blocks of space-time-matter must be extended objects, whose different vibrations are bosons and fermions. There is no supersymmetry. The trace dynamics Lagrangian is assumed to have an E8 x E8 symmetry, and symmetry breaking reveals the standard model forces and gravity, and two new forces are predicted. We also predict three sterile neutrinos, a BSM charged Higgs, and that the mass ratios of the electron, up quark, and down quark is precisely 1:4:9 (in the asymptotic free limit). 

Elementary particle states are described not by complex numbers, but my complex octonions. There are necessarily three and only three fermion generations, and the Dirac equation describing them obeys the exceptional Jordan algebra. The eigenvalues of the characteristic equation of this algebra reveal values of (at least some of) the fundamental constants of the standard model. Space-time is obtained from the squaring of the split bioctonionic space, and via a dynamically induced quantum-to-classical transition. The extra dimensions are not compactified – their extent is of the order of the range of the strong force and the weak force. 

The predictions of our unification theory based on extended objects are not at the Planck scale. The Planck scale gets naturally reset to the TeV scale in our theory. While much remains to be done and tested before claiming a successful theory of unification, we are free of the troubles of string theory (compactification and non-uniqueness, non-predictability, inability to derive the standard model and its fundamental constants). This is possible because from the outset we forego Minkowski spacetime in favour of the non-commutative octonionic space – this is where fermions live. And we forego quantum field theory (which needs classical time) in favour of the pre-quantum theory of trace dynamics.

In this octonionic theory, we have a better theory of unification (based on extended objects) than string theory in its current shape is. It is no longer correct to say that string theory is the only known / leading candidate for quantum gravity and unification.

Does our universe possess a second 4D spacetime, with its own light-cone, which is accessible only to quantum systems, and in which distances are necessarily microscopic?

As we discuss briefly in our recent CKM matrix paper 2305.00668, the answer to all the above questions is yes! Just as gravitation is the geometry of our familiar 4D spacetime, the weak force is the geometry of the second 4D spacetime. Indeed, the weak force is a spacetime symmetry masquerading as an internal symmetry. Symmetry breaking of a 6D spacetime in the very early universe gives rise to these two 4D spacetimes, one curved by gravity and the other curved by the weak force. The size of the universe in the other spacetime is of the order of the range of the weak force. A beam of light going through the universe in this other spacetime would be back to the starting point in just 10^-24 seconds! This could help us understand quantum nonlocality and the EPR paradox. When Einstein said that if quantum nonlocality (assuming QM is complete) is true then special relativity must go, one way to interpret Einstein is to propose that our universe has two 4D spacetimes.

In the twistor picture of our spacetime, null lines are more fundamental than spacetime points. Consider two null lines (t-x) and (t+x). A point arises as the intersection of these two null lines. 4D spacetime can be arrived at by overlaying these two null lines on a complex plane (y + iz). Consider the 2x2 matrix

t-x y+iz

y-iz t+x

Its determinant gives the 4D line element.

To get the second 4D spacetime we overlay these two null lines on an independent complex plane (a+ib) and make a new 2x2 matrix

t-x a+ib

a-ib t+x

The determinant now gives the line-element of the second 4D spacetime.

Between them the two 4D spacetimes are labeled by six real numbers (t, x, y, z, a, b) which can be used to obtain a 6D spacetime.

We can try to visualise the second spacetime by thinking of a torus in which the horizontal circle is much much larger than the vertical circle. The horizontal circle is space of our spacetime, the vertical circle is space of the other spacetime. If we are at location A on the torus then a galaxy at location B on the larger circle is correspondingly at location B' on the smaller circle, and B is identified with B' : the smaller circle is a scaled down version of the bigger circle and in one to one correspondence with it. A photon starting from our location will reach the galaxy B much faster along the smaller circle.

Mathematically, the two spacetimes result because of the group theory relations SL(2, H) ~ SO(1,5) and SL(2,C) ~ SO(1,3) where H are the quaternions. And because the Clifford algebra Cl(3) is the direct sum of two copies of Cl(2). Each Cl(2) generates a 4D Lorentz algebra...one is for our spacetime. The other is for the second spacetime whose three rotations are the weak isospin rotations, and the three boosts give Lorentz transformations along the vertical circle of the torus. Cl(3) is the algebra of complex split biquaternions and one can make a 6D spacetime from it.

If there is indeed a second 4D spacetime in our universe, which obeys the laws of special relativity, has its own light cone, is microscopic and accessible only to quantum systems, it can offer a neat solution to the EPR puzzle. Along the other spacetime, the photon arrives causally at B much before than along A, and this looks nonlocal from our perspective. One could call this a much more believable version of ER=EPR.

The trillion rupee question is: can the second spacetime be used for communication? Can we talk to someone on Andromeda in real time? Perhaps by sending weak force waves, analogous to gravitational waves, along the second spacetime.

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.