May 29, 2021
Why there must exist a formulation of quantum theory which does not refer to classical time?
and
Why such a formulation must exist at all energy scales, not just at the Planck energy scale.
Classical time, on which quantum systems depend for a description of their evolution, is part of a classical space-time. Such a space-time - the manifold as well as the metric that overlies it - is produced by macroscopic bodies. These macroscopic bodies are a limiting case of quantum systems. In principle one can imagine a universe in which there are no macroscopic bodies, but only microscopic quantum systems. And this need not be just at the Planck energy scale.
As a thought experiment, consider an electron in a double slit interference experiment, having crossed the slits, and not yet reached the screen. It is in a superposed state, as if it has passed through both the slits. We want to know,
non-perturbatively, what is the spacetime geometry produced by the electron? Furthermore, we imagine that every macroscopic object in the universe is suddenly separated into its quantum, microscopic, elementary particle units. We have hence lost classical space-time! And yet we must be able to describe what gravitational effect the electron in the superposed state is producing. This is the sought for quantum theory without classical time! And the quantum system is at low non-Planckian energies, and is even non-relativistic.
This is the sought for formulation we have developed, assuming only three fundamental constants a priori: Planck length L_P, Planck time t_P, and Planck's constant \hbar. Every other dimensionful constant, e.g. electric charge, and particle masses, is expressed in terms of these three. This new theory is a pre-quantum, pre-spacetime theory, needed even at low energies.
A system will be said to be a Planck scale system if any dimensionful quantity describing the system and made from these three constants, is order unity. Thus if time scales of interest to the system are order t_Pl = 10^-43 s, the system is Planckian. If length scales of interest are order L_P = 10^-33 cm, the system is Planckian. If speeds of interest are of the order L_P/t_P = c = 3x10^8 cm/s then the system is Planckian. If the energy of the system is of the order \hbar / t_P = 10^19 GeV, the system is Planckian. If the action of the system is of the order \hbar, the system is Planckian. If the charge-squared is of the order \hbar c, the system is Planckian. Thus in our concepts, the value 1/137 for the fine structure constant, being order unity in the units \hbar c, is Planckian. This explains why this pre-quantum, pre-spacetime theory knows the low energy fine structure constant.
A quantum system on a classical space-time background is hugely non-Planckian. Because the classical space-time is being produced by macroscopic bodies each of which has an action much larger than \hbar. The quantum system treated in isolation is Planckian, but that is strictly speaking a very approximate description. The spacetime background cannot be ignored - only when the background is removed from the description, the system is Planckian. This is the pre-quantum, pre-spacetime theory.
It is generally assumed that the development of quantum mechanics, started by Planck in 1900, was completed in the 1920s, followed by generalisation to relativistic quantum field theory. This assumption, that the development of quantum mechanics is complete, is not correct - quantisation is not complete until the last of the classical elements - this being classical space-time - has been removed from its formulation.
The pre-quantum, pre-spacetime theory achieves that, giving also an anticipated theory of quantum gravity. What was not anticipated was that removing classical space-time from quantum theory will also lead to unification of gravity with the standard model. And yield an understanding of where the standard model parameters come from. It is clear that the sought for theory is not just a high energy BSM theory. It is needed even at currently accessible energies, so at to give a truly quantum formulation of quantum field theory. Namely, remove classical time from quantum theory, irrespective of the energy scale. Surprisingly, in doing so, we gain answers to unsolved aspects of the standard model and of gravitation.
The process of quantisation works very successfully for non-gravitational interactions, because they are not concerned with space-time geometry. However, it is not correct to apply this quantisation process to spacetime geometry. Because the rules of quantum theory have been written by assuming a priori that classical time exists. How then can we apply these quantisation rules to classical time itself? Doing so leads to the notorious problem of time in quantum gravity - time is lost, understandably.
We do not quantise gravity. We remove classical space-time / gravity from quantum [field] theory. Space-time and gravity emerge as approximations from the pre-theory, concurrent with the emergence of classical macroscopic bodies. In this emergent universe, those systems which have not become macroscopic, are described by the beloved quantum theory we know - namely quantum theory on a classical spacetime background. This is an approximation to the pre-theory: in this approximation, the contribution of the said quantum system to the background spacetime is [justifiably] neglected.
