Why a quantum theory of gravity is needed at all energy scales; not just at the Planck energy scale?

 

Why a quantum theory of gravity is needed at all energy scales, and not just at the Planck energy scale?

and

How that leads us to partially redefine what is meant by Planck scale: Replace Energy by Action.

We have argued earlier that there must exist a formulation of quantum theory which does not refer to classical time. Such a formulation must in principle exist at all energy scales, not just at the Planck energy scale. For instance, in today's universe, if all classical objects were to be separated out into elementary particles, there would be no classical space-time and we would need such a formulation. Even though the universe today is a low energy universe, not a Planck energy universe.

Such a formulation is inevitably also a quantum theory of gravity. Arrived at, not by quantising gravity, but by removing classical gravity from quantum theory. We can also call such a formulation pure quantum theory, in which there are no classical elements: classical space-time has been removed from quantum theory. We also call it a pre-quantum, pre-spacetime theory.

What is meant by Planck scale, in this pre-theory?

Conventionally, a phenomenon is called Planck scale if: the time scale T of interest is of the order Planck time TP; and/or length scale L of interest is of the order of Planck length LP; and/or energy scale E of interest is of the order Planck energy EP. According to this definition of Planck scale, a Planck scale phenomenon is quantum gravitational in nature.

Since the pre-theory is quantum gravitational, but not necessarily at the Planck energy scale, we must partially revise the above criterion, when going to the pre-theory: replace the criterion on energy E by a criterion on something else. This something else being the action of the system!

In the pre-theory, a phenomenon is called Planck scale if: the time scale T of interest is of the order Planck time TP; and/or length scale L of interest is of the order of Planck length LP; and/or the action S of interest is of the order Planck constant \hbar. According to this definition of Planck scale, a Planck scale phenomenon is quantum gravitational in nature.

Why does this latter criterion make sense? If every degree of freedom has an associated action of order \hbar, together the many degrees of freedom cannot give rise to a classical spacetime. Hence, even if the time scale T of interest and length scale L of interest are NOT Planck scale, the system is quantum gravitational in nature. The associated energy scale \hbar / T for each degree of freedom is much smaller than Planck scale energy EP. Hence in the pre-theory the criterion for a system to be quantum gravitational is DIFFERENT from conventional approaches to quantum gravity. And this makes all the difference to the formulation and interpretation of the theory. e.g. the low energy fine structure constant 1/137 is a Planck scale phenomenon [according to the new definition] because the square of the electric charge is order unity in the units \hbar c = \hbar LP / TP.

In our pre-theory, there are three, and only three, fundamental constants: Planck length LP, Planck time TP and Planck action \hbar. Every other parameter, such as electric charge, Newton's gravitational constant, standard model coupling constants, and masses of elementary particles, are defined and derived in terms of these three constants: \hbar, LP and TP.

In the pre-theory the universe is an 8D octonionic universe, as shown in the attached figure: the octonion. The origin e_0=1 stands in for the real part of the octonion [coordinate time] and the other seven vertices stand in for the seven imaginary directions. A degree of freedom [i.e. `particle' or an atom of space-time-matter (STM)] is described by a matrix q which resides on the octonionic space: q has eight coordinate components q_i where each q_i is a matrix. We have replaced a four-vector in Minkowski space-time by an eight-matrix in octonionic space: and this describes the particle / STM atom. The STM atom evolves in Connes time, this time being over and above the eight octonionic coordinates. Its action is that of a free particle in this same: time integral of kinetic energy, the latter being the square of velocity q-dot, where dot is Connes time. Eight octonionic coordinates are equivalent to ten Minkowski coordinates, because of SL(2,O) ~ Spin(9,1).

The symmetries of this space are the symmetries of the (complexified) octonionic algebra: they contain within them the symmetries of the standard model, including the Lorentz symmetry.

The classical 4D Minkowski universe is one of the three planes (quaternions) intersecting at the origin e_0 = 1. Incidentally the three lines originating from e_0 represent complex numbers. The four imaginary directions not connected to the origin represent directions along which the standard model forces lie (internal symmetries). Classical systems live on the 4D quaternionic plane. Quantum systems (irrespective of whether they are at Planck energy scale) live on the entire 8D octonion. Their dynamics is the sought for quantum theory without classical time. This dynamics is oblivious to what is happening on the 4D classical plane. QFT as we know it is this pre-theory projected to the 4D Minkowski space-time. The present universe has arisen as a result of a symmetry breaking in the 8D octonionic universe: the electroweak symmetry breaking. Which in this theory is actually the color-electro -- weak-Lorentz symmetry breaking. Classical systems condense on to the 4D Minkowski plane as a result of spontaneous localisation, which precipitates the electro-weak symmetry breaking in the first place. The fact that weak is part of weak-lorentz should help understand why the weak interaction violates parity, whereas electro-color does not. Hopefully the theory will shed some light also on the strong-CP problem.