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.