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.

Why there must exist a formulation of quantum theory which does not refer to classical time? : Towards quantum gravity and unification

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.

Towards unification of the four fundamental forces: The Aikyon Theory

SCEST21: Schrodinger's Cat, and Einstein's Space-time, in the 21st Century

A blogspot for discussing the connection between quantum foundations and quantum gravity

Managed by: Tejinder Pal Singh, Physicist, Tata Institute of Fundamental Research, Mumbai

If you are a professional researcher / student researching on these topics, and would like to post an article here with you as author, you are welcome to do so. Please e-mail your write-up to tpsingh@tifr.res.in and it will be uploaded here.


Keywords: Quantum foundations; Quantum gravity; Schrodinger's cat; Spontaneous collapse theory; Trace dynamics; Non-commutative geometry; Spontaneous quantum gravity; Classical general relativity; black holes, gyromagnetic ratio 

Saturday, September 26, 2020

Towards unification of the four fundamental forces

https://arxiv.org/abs/2009.05574

https://www.youtube.com/watch?v=uxdvergYNrg&ab_channel=TejinderSingh

The Aikyon Theory

[The word Aikyon derives from `Aikya' in Sanskrit, which means `oneness’. To not make a distinction between space-time and matter].

At the Planck scale, there is no distinction between space-time symmetry and internal symmetry. Physical space is eight dimensional non-commutative octonionic space. One can imagine it as a 2-D complex plane, where the real axis represents 4-D to-be-spacetime, and the imaginary axis represents 4-D to be internal symmetries. The aikyon is an elementary particle, say an electron, *along with* the fields it produces. We do not make a distinction  between the particle and the fields it produces. This is evident from the form of the action for an aikyon, shown below: variables with subscript B stand for the four known forces, and those with subscript F for any of the 24 known fermions of the three generations of the standard model. The Lagrangian is unchanged if B and F variables are interchanged. This is super-symmetry. And since the B-variables include both gravity and gauge-fields, there is a gauge-gravity duality.

The aikyon evolves in this 8-D space in Connes time. The aikyon is a 2D object, as if a membrane [2-brane]. Motion along the real axis is caused by gravity, along vertical axis by electro-colour force, and from real to imaginary by the weak force. Or we can just say, the aikyon moves in the 8D space under the influence of the unified force, given by the  B-variable in the action. 

There is one such action term for every aikyon in this space. Different aikyons interact by `colliding' with each other. The coordinates of this 8D space are the eight components of an octonion. Algebra automorphisms transform one coordinate system to another. These are the analog of general coordinate transformations of general relativity and internal gauge symmetries of gauge theories, and hence unify those concepts. The theory is invariant under 8D algebra automorphisms. And because the laws of motion are those of trace dynamics, this is already a quantum theory.

Why `quantising' a classical system is like going halfway and then stopping!

SCEST21: Schrodinger's Cat, and Einstein's Space-time, in the 21st Century

A blogspot for discussing the connection between quantum foundations and quantum gravity

Managed by: Tejinder Pal Singh, Physicist, Tata Institute of Fundamental Research, Mumbai

If you are a professional researcher / student researching on these topics, and would like to post an article here with you as author, you are welcome to do so. Please e-mail your write-up to tpsingh@tifr.res.in and it will be uploaded here.


Keywords: Quantum foundations; Quantum gravity; Schrodinger's cat; Spontaneous collapse theory; Trace dynamics; Non-commutative geometry; Spontaneous quantum gravity; Classical general relativity; black holes, gyromagnetic ratio 
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Why `quantising' a classical system is like going halfway and then stopping!

When we quantise a classical system, we raise the canonical variables, q and p, to the status of operators, and assign quantum commutation relations to them: [q, p] = ih

How to recover the classical limit back? It is impossible! One cannot un-quantise a quantised system so as to make it classical again. Because the quantum system obeys linear superposition, and it does so irrespective of the size of the quantum system. Even large quantum systems are demanded by theory to obey superposition, even though they are in fact observed to be classical (no superposition).

The way out of this mess is not that we invent some `interpretation' of quantum mechanics!! The way out is to re-examine how we could have done things differently in the first place.

One very promising way is: ok let us make q and p into operators, but not impose *quantum* commutation relations. The commutation relations are arbitrary, and evolve with time, and are determined by dynamical laws. The dynamical laws are similar to those of classical dynamics, but now adapted to operator variables. This is the theory of Trace Dynamics.

It turns out that for microscopic systems, trace dynamics reduces to quantum theory, quantum commutation relations emerge, and the quantum superposition principle holds. But for large systems, trace dynamics reduces to classical mechanics, and superpositions do not hold [because of the mechanism known as spontaneous localisation, which we discussed earlier].

So, instead of quantising a classical system, one should `operatorise' it: make the commutators arbitrary. Quantisation is like going only half the distance. Trace dynamics is the full story. Thus

Trace Dynamics = Quantum Theory + Spontaneous Localisation.

For small systems, the last bit is negligible. For large systems it is very important.

It is possible to include gravity also in trace dynamics. This leads to a quantum theory of gravity:

Trace Dynamics + Trace Gravity = Quantum gravity + Spontaneous Localisation

For large systems, the last bit is important, and responsible for the emergence of classical space-time and laws of general relativity, from quantum gravity.

Spontaneous Localisation is what un-quantises a quantum system and makes it classical again. Experimentalists are carrying out experiments to find out if spontaneous localisation occurs in nature.

Why does a charged rotating black hole have the same gyromagnetic ratio as an electron?!

SCEST21: Schrodinger's Cat, and Einstein's Space-time, in the 21st Century

A blogspot for discussing the connection between quantum foundations and quantum gravity

Managed by: Tejinder Pal Singh, Physicist, Tata Institute of Fundamental Research, Mumbai

If you are a professional researcher / student researching on these topics, and would like to post an article here with you as author, you are welcome to do so. Please e-mail your write-up to tpsingh@tifr.res.in and it will be uploaded here.


Keywords: Quantum foundations; Quantum gravity; Schrodinger's cat; Spontaneous collapse theory; 
Trace dynamics; Non-commutative geometry; Spontaneous quantum gravity; Classical general relativity 
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January 11, 2020


Why does a charged rotating black hole have the same gyromagnetic ratio as an electron?!


We know that a charge moving in a magnetic field experiences a force. A moving charge is of course a current. So, a current carrying wire experiences a force in a magnetic field. The same would be true if the wire forms a closed loop, say for simplicity a circular loop with some radius, in which a charge is going around with some angular velocity. The response of such a charge to an external magnetic field depends on the value of the charge, the angular speed, and the area of the loop. These quantities combine to define the magnetic moment of the current loop, which is simply current times the area. The response of a current in a loop depends on the magnetic moment, and is essentially magnetic moment times the external magnetic field.

It is easy to show that the magnetic moment is proportional to the orbital angular momentum of the charge in the loop. The ratio of the magnetic moment to the angular momentum is a constant q/2m, independent of the parameters of the orbit, where q is the magnitude of the charge, and m the mass of the particle. This ratio, q/2m, known as the gyromagnetic ratio, is the object of our interest in today’s post.

So if we have an electron with a charge e and mass m going around in a loop, it will have a g-ratio of e/2m. Now, an electron also has a spin, I.e. an intrinsic angular momentum, distinct from orbital angular momentum. It is a purely quantum mechanical effect, arising out of the Dirac equation, and the spin of an electron is one-half the Planck constant. The electron also has an intrinsic magnetic moment because it is a charge with angular momentum. Again, this magnetic moment is proportional to intrinsic angular momentum, but the g-ratio is now e/m, not e/2m. This can be proved from the Dirac equation. The g-ratio of a Dirac fermion is twice the classical value - this can be proved from the Dirac equation. Nothing terribly surprising if the quantum value is twice the classical value. The surprise comes now, when we talk of black holes.

Black holes are solutions of Einstein equations, and their properties are determined by the laws of classical general relativity. These laws are *classical*, not quantum, and black holes are classical objects. A black hole is described by at most three parameters - its mass, charge, and angular momentum, let’s label them M, Q and J respectively. The composition of this mass is not relevant for describing a black hole. Already we see something striking - an electron is also described by its mass, charge, and (spin) angular momentum! A black hole with no charge or angular momentum is described only by its mass M, and is called the Schwarzschild black hole. A Reissner-Nordstrom black hole has mass and charge, but no angular momentum. The most general black hole is the Kerr-Newman black hole, where M, Q and J are all non-zero. This is the charged rotating black hole. It has a magnetic moment, and a g-ratio, which can be calculated from Einstein equations. The g-ratio turns out to be Q/M, not Q/2M, agreeing with the Dirac quantum value for the electron, and disagreeing with the classical value! Even though black holes are classical objects.

This should set alarm bells ringing. In my view, this is a great hint that, despite appearances. Einstein equations have something to do with the Dirac equation, and black holes have something to do with Dirac fermions. Thus, since classical general relativity (with matter) is an approximation to a quantum gravity theory (along with matter), the latter theory must explain why a black hole has a g-ratio twice the classical value. This should be a requirement for a quantum gravity theory to be viable.

There is already considerable evidence in known physics that Einstein equations are deeply connected with the Dirac equation. Firstly, if we ask how to describe the dynamics of a relativistic particle of some mass m, the answer is ambiguous. We could describe it by the Dirac equation, which claims to hold for arbitrary values of mass; or we could describe it by Einstein equations, which also claim to hold for arbitrary values of the mass. Obviously, both descriptions cannot be correct. From experience we know that Dirac equation works for small particles, which are quantum in nature, and Einstein equations work for large objects, which are classical in nature. But how small is small, and how large is large? Neither the Dirac equation nor Einstein equations have  a mass scale. Obviously then, there ought to exist an underlying theory with a mass scale (or equivalently a length scale), such that for masses much smaller than this scale, the theory reduces to Dirac equation for quantum systems. And for masses much larger than this scale, the theory reduces to Einstein equations for classical systems.

The next big hint that Einstein equations are connected with Dirac equations, comes from geometry - something we discussed in an earlier post. Given  a Riemannian manifold, and the standard Dirac operator on it, the sum of the eigenvalues of the square of the Dirac operator is proportional to the Einstein-Hilbert action on this manifold. This remarkable result again suggests an Einstein-Dirac connection, provided we figure out a way to include Dirac fermions in the theory.

This is precisely what has been done in our recently proposed theory of spontaneous quantum gravity. An atom of space-time-matter has an associated length scale L, which is interpreted as its Compton wave length. If L is much larger than Planck length the theory reduces to quantum theory and the Dirac equation for a fermion. If L is much smaller than Planck length, the equations describing the STM atoms reduce to Einstein equations of general relativity, and the collection of STM atoms can be shown to be a classical black hole! One way to understand this is to note that given L, one can define another length from it, namely the square of Planck length divided by L. This is a quantity equal to the Schwarzschild radius, and it exceeds Planck length when L goes below Planck length. Thus when L crosses from a value larger than Planck length to a value smaller than Planck length, there is a cross-over from the Dirac fermion phase to the Einstein black hole phase. The net mass in the system crosses over from less than Planck mass to greater than Planck mass. A collection of entangled STM atoms behaves quantum mechanically, obeying the Dirac equation, if the total mass is less than Planck mass. The collection behaves like a black hole if he total mass exceeds Planck mass.

There in fact is a duality which maps a Dirac fermion to a black hole. It can be shown that if a solution describes an STM atom with Compton wavelength L, the adjoint of this solution describes a black hole with Schwarzschild radius L and Compton length L’=L_P^2/L. Now, if we associate an electric charge e with the Dirac fermion, we can associate a dipole moment eL with the STM atom. There is strong evidence that this dipole moment remains unchanged under the said duality map, which maps the Dirac fermion to a black hole with charge Q and mass M=1/L’ in such a way that eL = QL’. But this product is nothing but the gyromagnetic ratio. Hence this duality between Dirac fermions and black holes explains why a charged rotating black hole has the same gyromagnetic ratio as the electron. Black holes and electrons are simply two different states of atoms of space-time-matter. We consider this to be compelling evidence that spontaneous quantum gravity is a viable theory of quantum gravity.