Quantum theory without classical time: a route to quantum gravity and unification

If you are interested in the current status of the of the octonionic theory, you might find this new review article useful:

Quantum theory without classical time: a route to quantum gravity and unification

https://www.tifr.res.in/~tpsingh/Penrose90singh.pdf

152 pages, 17 figures. Invited review article submitted to the special collection "Celebrating Sir Roger Penrose's Nobel Prize"

[AVS Quantum Science (AIP Publishing and AVS), Guest Editor: Hendrik Ulbricht]

Abstract

There must exist a reformulation of quantum field theory which does not employ classical time to describe evolution, even at low energies. To achieve this goal, we have proposed a pre-quantum, pre-spacetime theory, which is a matrix valued Lagrangian dynamics on an octonionic spacetime. This is a deterministic but non-unitary dynamics in which evolution is described by Connes time, a feature unique to non-commutative geometry. From here, quantum field theory and its indeterminism, as well as classical space-time geometry, are emergent under suitable approximations. In the underlying theory, the algebra of the octonions reveals evidence for the standard model of particle physics, and for its unification with a pre-cursor of gravitation, through extension to the Left-Right symmetric model and the symmetry group $E_6$. When elementary particles are described by spinors made from a Clifford algebra, the exceptional Jordan algebra yields a theoretical derivation of the low energy fine-structure constant, and of the observed mass ratios for charged fermions. We identify the Left-Right symmetry breaking with electroweak symmetry breaking, which also results in separation of emergent four-dimensional Minkowski spacetime from the internal symmetries which describe the standard model. This `compactification without compactification’ is achieved through the Ghirardi-Rimini-Weber mechanism of dynamical wave function collapse, which arises naturally in our theory, because the underlying fundamental Hamiltonian is necessarily non-self-adjoint. Only classical systems live in four dimensions; quantum systems always live in eight octonionic (equivalently ten Minkowski) dimensions. We explain how our theory overcomes the puzzle of quantum non-locality, while maintaining consistency with special relativity. We speculate on the possible connection of our work with twistor spaces and spinorial space-time, and with Modified Newtonian Dynamics (MOND). We point to the promising phenomenology of $E_6$, and mention possible experiments which could test the present proposal. In the end we outline further work that still remains to be done towards completion of this programme.

How does a pre-quantum, pre-spacetime theory know about the low-energy fine structure constant and mass ratios?

Because such a theory is needed both in UV and in IR, and the octonionic theory is such a theory. UV is obvious, but IR could do with some explaining.

Even at low energies, there can be a situation where for a given system, all sub-systems have action of the order \hbar. Then there is no background classical spacetime anymore, and the pre-quantum, pre-spacetime theory is required. e.g. when a massive object is in a quantum superposition of two position states and we want to know what spacetime geometry it produces.

The pre-theory is in principle required also for a more exact description and understanding of the standard model, even at low energies. And the octonionic theory achieves just that, thereby being able to derive the low-energy SM parameters. This is BSM in IR, and has implications for how we plan BSM experiments: these have to be not only towards UV, but also in the IR.

The O-theory has only three fundamental constants, and these happen to be such that both the UV and IR limits can be easily investigated. These constants are Planck length, Planck time and Planck's constant \hbar. Note that, as compared to conventional approaches to quantum gravity, Planck's constant \hbar has been traded for Planck mass/energy. And this is very important:

The pre-theory is in principle required whenever one or more of the following three conditions are satisfied: times scales T of interest are order Planck time, Length scales L of interest are order Planck length, actions S of interest are order \hbar.

If T and L are respectively much larger than Planck time and Planck length, but S is of order \hbar, that requires the pre-theory in IR.

If T is order Planck time, then the energy scale \hbar / T is Planck energy scale. However, \hbar / T is in IR for T >> Planck time, and yet the pre-theory is required (for an exact in-principle description of SM) if all actions are order \hbar.

The BSM physics in IR is achieved by replacing 4D Minkowski spacetime by 8D octonionic non-commutative spacetime. This is the pre-theory analog of flat spacetime - and it has consequences - it predicts the low energy SM parameters, without switching on high-energy interactions in the UV.

Going to high energies is just like in GR. In GR we switch on the gravitational field around Minkowski spacetime and doing so takes us from IR to UV. Same way, in the O-theory we switch on SM interactions and would-be-gravity, *around* the `flat' octonionic spacetime [=10D Minkowski] and this takes us from IR to UV. But unlike in the GR case, we already learn a lot of BSM physics in the IR, because the spacetime is non-commutative. String Theory missed out on this important IR physics, because it continued to work with 10D Minkowski spacetime which is commutative, and from there went to UV. Should have looked at octonionic spacetime and Clifford algebras.

More on MOND and the octonionic theory

In what way might we think of Newtonian gravitation as the square of MOND?

To get some insight, let us write the acceleration a in a circular orbit as

a = [ GM / R^2 ] ( 1 + \beta (R) ]

The function \beta (R) depends on a and goes to zero for a >> a_0, thus recovering Newtonian gravitation. For a in the vicinity of a_0 we approximate the last bracket to a_0 / a, thus yielding MOND. At large cosmic distances, relativistic effects become important. GM/R^2 is replaced by its GR counterpart, and (1+\beta) becomes the MOND induced modification of GR. It is important to ask if dark energy is a manifestation of this MOND induced modification. Were this to be so, we will have a common original cause for flat galaxy rotation curves and cosmic acceleration, without dark matter or dark energy, and because GR and Newtonian gravity are limiting cases of a more general law of gravity. On the clusters scale MOND will need warm dark matter such as sterile neutrinos, or dark baryons.

What then is this more general law of gravity? Which we demand must come from first principles. The Left-Right symmetric octonionic theory proposes SU(3)_g X SU(2)_R X U(1)_g as would-be-gravity, or square-root-gravity, this is the right-hand counterpart of the broken L-R symmetric theory, whose left-handed counterpart is the standard model. The gauge theory of would-be-gravity on an 8D octonionic space-time is proposed as the more general law of gravity, which explains the origin of the critical acceleration a_0 and the emergence of GR, MOND, and Newton as special cases.

Cosmology and the scale a_0: In the L-R symmetric theory, the very early universe undergoes an inflationary expansion having a time-dependent cosmic acceleration a_0(t). This inflationary expansion is halted (and converted to a power law expansion) when significant seeding of density perturbations causes a quantum-to-classical transition, L-R symmetry breaking, and emergence of 4D classical spacetime. The gravitational acceleration in the vicinity of the seeded relativistic density perturbations exceeds the then a_0, and GR emerges from would-be-gravity as its square. MOND is the transition zone between would-be-gravity and Newton/GR. Thus would-be-gravity can seed the scale-invariant matter perturbations whose effect is seen in the CMB (hence relativistic MOND).

In today's universe, away from compact objects, would-be-gravity (relativistic MOND) dominates because accelerations are smaller than a_0 and tend to the current cosmic a_0. In this deep MOND regime there is space-time scale invariance, and the universe tends to de Sitter.

Would-be-gravity when squared yields GR at high accelerations and the condensation of SU(2)_L and SU(2)_R into 4D spacetime geometry is indicated. SU(2)_L mediated on small scales by heavy weak bosons is the weak interaction. The electro-weak symmetry breaking is in reality an L-R symmetry breaking same as

QCD Color + U(1)_em - Grav Color + U(1)_g

breaks from - breaks from

Weak SU(2)_L - SU(2)_R

It appears that if we do cosmology with the L-R symmetric octonionic theory and its emergent approximations, all will be well without cold dark matter and without cosmological constant as dark energy. In this theory the cosmological constant (zero point energy of vacuum) is strictly zero. Cosmic acceleration is caused by U(1)_g