On how not to emulate Einstein only *partially*

On how not to emulate Einstein only *partially*

Einstein’s quest for developing a mathematically beautiful and physically correct field theory, out of pure thought and nothing else, was successful in the discovery of the general theory of relativity. However, his pure thought based quest failed in the attempt to unify gravitation with electromagnetism. But this failure of Einstein has not prevented us from emulating him – searching for a theory of quantum gravity, and a unified theory of known interactions, based on pure thought, without there being any experimental evidence to support quantization of gravity,  or to support its unification with the other forces. We have tried this for more than half a century, but like Einstein, we have also failed. At least we have not succeeded thus far. And yet, on physical grounds we know that there must be a quantum gravity, as well as unification. 

Could the reason for our failure be that we have been emulating Einstein only `partially’?

In hindsight, we know good reasons why general relativity was successfully discovered. Maxwell’s electrodynamics was relativistic to start with – it in fact propelled the discovery of special relativity. Because Galilean-invariant Newtonian mechanics was inconsistent with electrodynamics. Once special relativity was discovered, Newtonian gravitation had to be made relativistic too; hence general relativity was inevitable. The same cannot be said about unifying electrodynamics and gravitation as a geometric theory. Because the world is not classical. Enter quantum theory. Electrodynamics must be quantized, for it to agree with experiments. There goes the Einstein-style unification. Attempts at unification must take quantum theory into account. However, Einstein was not satisfied with quantum theory, and believed it to be an approximation to a more general  theory. 

In trying to pursue Einstein style thought based unification  program, while at the same time ignoring Einstein’s  concerns about quantum theory, we are emulating him selectively – this could be risky.

Einstein objected to the spooky action at a distance, through the EPR argument on quantum non-locality. He was not saying that quantum theory allows superluminal signaling. Rather, he was saying that there was a quantum influence outside the light cone, which is not causal. And this meant  that either the quantum mechanical description of reality is incomplete, or that special relativity and its related description of space-time structure would have to be modified so as to make it compatible with quantum theory. Since Bell’s theorem rules out local hidden variable theories, and since quantum non-locality has been confirmed by experiments, it is indeed special relativity which needs a rethink, in the quantum context.

Einstein also objected to the occurrence of probabilities in a deterministic mechanical theory: God does not play dice, he famously said. 

Can we then, in our quest for a quantum theory of unification, in the Einstein style of pure thought, also  address his concerns about quantum theory? Why do we pick just one half of Einstein? Maybe emulating him all the way will pay dividends? 

If we decide to emulate the full Einstein, how shall we do it? In true Einstein style, we must look closely at the quantisation procedure. The world of classical dynamics works perfectly, almost, in the macroscopic domain: classical bodies and fields on a classical space-time. If we are to quantise, we must quantise everything in one go: matter, gauge fields, AND space-time degrees of freedom [special and general relativity, not just the latter]. Quantising only matter and gauge fields, and leaving spacetime classical, while very successful, is at the heart of the spooky action at a distance that bothered Einstein. If we quantise everything, we will get a pre-quantum, pre-spacetime theory. This is what removes the incompleteness of quantum mechanics, because it makes pre-spacetime compatible with pre-quantum theory.

Through this generalized quantisation, we have gone pre-. How do we recover the classical world of matter and space-time from the pre-theory? Spontaneous localization is the answer. Classical macroscopic bodies and classical space-time emerge *simultaneously*. We say that space-time arises from the collapse of the wave-function.

Those matter particles and gauge fields which do not undergo spontaneous localization must be described by the pre-quantum pre-spacetime theory. This is the world of elementary particles undergoing standard model interactions. Even if these interactions are at low energies, the pre-theory must be used, if we are to avoid the spooky action and the probabilities of quantum theory. The pre-theory has an IR sector. However, we can to an excellent approximation describe quantum systems by using not the pre-theory, but quantum field theory on a classical space-time background, as we conventionally do. The approximation consists of dropping the very tiny quantum correction to space-time, caused by quantum systems, and assuming  spacetime to be classical. This is our beloved quantum theory, the one Einstein correctly calls incomplete. It has spooky action at a distance. The pre-theory is also deterministic [though non-unitary]. It has no probabilities – these arise only in the approximate description.

Moreover, when we examine the pre-quantum pre-spacetime theory, we find evidence for the standard model symmetries. Maybe the pre-theory can explain things about the standard model which we do not otherwise understand.

Maybe it pays to emulate Einstein fully.

What is Trace Dynamics?

 Trace dynamics is quantisation, without the Heisenberg algebra.

1. Quantisation Step 1 is to raise classical degrees of freedom, the q and p, to the status of operators / matrices. A very reasonable thing to do.

2. Quantisation Step 2 is very unreasonable! Impose the Heisenberg algebra [q, p] = i \hbar Its only claim to fame is that the theory it gives rise to is extremely successful.

In classical dynamics, the initial values of q and p are independently prescribed. There is NO relation between the initial q and p. Once prescribed initially, their evolution is determined by the dynamics. Whereas, in quantum mechanics, a theory supposedly more general than classical mechanics, the initial values of the operators q and p must also obey the constraint [q, p] = i \har. This is highly restrictive!

3. It would be more reasonable if there were to be a dynamics based only on Quantisation Step 1. And then Step 2 emerges from this underlying dynamics in some approximation. This is precisely what Trace Dynamics is. Only step 1 is applied to classical mechanics. q and p are matrices, and the Lagrangian is the trace of a matrix polynomial made from q and its velocity. The matrix valued equations of motion follow from variation of the Lagrangian. They describe dynamics.

4. This matrix valued dynamics, i.e. trace dynamics, is more general than quantum field theory, and assumed to hold at the Planck scale. The Heisenberg algebra is shown to emerge at lower energies, after coarse-graining the trace dynamics over length scales much larger than Planck length scale. Thus, quantum theory is midway between trace dynamics and classical dynamics.

5. The moral of the story is that quantum field theory does not hold at the Planck scale. Trace dynamics does. QFT is emergent.

6. The other assumption one makes at the Planck scale is to replace the 4-D classical spacetime manifold by an 8D octonionic spacetime manifold, so as to obtain a canonical definition of spin. This in turn allows for a Kaluza-Klein type unification of gravity and the standard model. Also, an 8D octonionic spacetime is equivalent to a 10-D Minkowski space-time. It is very rewarding to work with 8D octonionic, rather than 10D Minkowski - the symmetries manifest much more easily.

7. Trace dynamics plus octonionic spacetime together give rise to a highly promising avenue for constructing a theory of quantum gravity, and of unification. 4D classical spacetime obeying GR emerges as an approximation at lower energies, alongside the emergent quantum theory.

8. How is this different from string theory? In many ways it IS like string theory, but *without* the Heisenberg algebra! The gains coming from dropping [q,p]=i\hbar at the Planck scale are enormous. One now has a non-perturbative description of space-time at the Planck scale.

The symmetry principle behind the unification is very beautiful: physical laws are invariant under algebra automorphisms of the octonions. This unifies the internal gauge transformations of the standard model with the 4D spacetime diffeomorphisms of general relativity. The automorphism group of the octonions, the Lie group G2, which is the smallest of the five exceptional Lie groups, contains within itself the symmetries SU(3)xSU(2)xU(1) of the standard model, along with the Lorentz symmetry. The free parameters of the standard model are determined by the characteristic equation of the exceptional Jordan algebra J_3(O), whose automorphism group F4 is the exceptional Lie group after G2.

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.