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

Is MOND (Modified Newtonian Dynamics) related to the Left-Right symmetric octonionic theory of the standard model and gravitation?

Is MOND (Modified Newtonian Dynamics) related to the octonionic theory? Possibly yes. At least this question is worth investigating further, for the reasons explained below.

MOND is an alternative to dark matter. Instead of proposing that the dynamical anomaly of galactic rotation curves is due to the presence of additional invisible matter, it is proposed that on sufficiently large distance scales, where gravitational acceleration falls below a critical value a_0, the law of gravitation departs from Newton's.

Roughly put, in MOND, the circular orbital acceleration v^2 / R outside a mass M is given as usual by Newton's GM/R^2, so long as the acceleration exceeds a_0. This gives the well-known Keplerian fall-off V^2 ~ 1/R, which is contradicted by the flat galaxy rotation curves.

As is known from observations, whenever the observed orbital acceleration falls below a_0, the velocity curve stays flat thereafter.

In 1983, Milgrom proposed that in the deep MOND regime, where the acceleration is much below a_0, the law of gravitation changes, so that the orbital acceleration is now given by

v^2 / R ~ (GM a_0)^1/2 / R

which explains the flatness of the rotation curve. This phenomenological modification of Newtonian gravitation at large distances is quite successful in explaining observations. Clearly a modification to general relativity is implied at large distances, including at the cosmological Hubble scale. Curiously, a_0 is numerically of the order of the cosmic acceleration c/H_0. A mysterious coincidence or a pointer to a relativistic theory underlying MOND? Until recently, the main and justified criticism of MOND was that there is no relativistic extension of the theory which can account for structure formation and in particular the CMB anisotropies. Something at which cold dark matter is highly successful. This may have changed last year, when two Czech physicists constructed RMOND, an action principle based phenomenological relativistic extension of MOND, which explains CMB data.

What could be the fundamental origin of MOND? This is where the octonionic theory comes in. What caught my attention is the acceleration being proportional to square-root of M, instead of M, in the MOND regime. In the O-theory as well, the would-be-gravity theory SU(3)_g x SU(2)_R x U(1)_g has as its charge square-root of m, rather than m, where m is the mass of the elementary particle. In the unbroken L-R symmetric regime, the interaction strength goes as sqrt{m}. When L-R symmetry is broken, squaring of would-be-gravity is enabled, GR and Newtonian gravitation emerge, and the interaction strength goes as m.

Where and how does a_0 enter the picture? We will identify a_0 with cosmic acceleration at the corresponding cosmic epoch (making it epoch dependent!). The L-R symmetry breaking [same as electro-weak symmetry breaking] is caused by spontaneous localisation of classical matter perturbations (primordial black holes??) as a result of which the emergent gravitational acceleration in the vicinity of compact objects exceeds the (pre symmetry breaking) cosmic acceleration a_0. This would be the origin of MOND. In the vicinity of compact objects, where acceleration exceeds a_0, the square of would-be-gravity, i.e. GR and Newton, hold. As for instance in the solar system and near stars and black holes. However, in low density regions, where acceleration is below the cosmic acceleration a_0, the unbroken would-be-gravity law holds, where acceleration is proportional to square root mass, not mass.

If this line of thinking were to be correct, the octonionic theory could explain the fundamental origin of MOND ! The critical acceleration a_0 then serves as the order parameter for a phase transition: the L-R symmetry breaking. Could it be that the U(1)_g of would-be-gravity is the sought after dark energy. i.e. dark photons? In a universe made only of matter, all particles have like charge root(m), and the U(1) vector interaction is repulsive.

MOND would then be more fundamental than Newtonian gravitation, with the latter becoming square of MOND! MOND is then same as would-be-gravity.

Elementary particles, and the space-time in which they live.

We are accustomed to the fact that space-time is four dimensional, and its coordinates are labeled by four real numbers (t, x, y, z). All material objects, such as the electron, and the fields they produce, are supposed to live in this 4D spacetime.

But what if this description of space-time is only an approximation? In Newton's world, material bodies as well as space and time, are described by real numbers. However, in quantum mechanics, material particles are described not by real numbers, but by non-commuting matrices, q-hat and p-hat, the position and momentum operators. Eigenvalues of these matrices correspond to the classical Newtonian values of position and momentum.

We assume in quantum mechanics that we can continue to describe space-time by real numbers, which commute, even when position and momenta have been made operators, and these latter do not commute. What if this is only an approximation, and the truth is that when q and p are made operators, spacetime should also be labeled by non-commuting coordinates? Could it be that when the space-time is having non-commuting coordinates, this very property of non-commutativity determines properties of elementary particles? Such as, why is the electric charge of the down quark one third that of the electron? Quantum theory and the standard model have no answer to this question. However, when we replace 4D spacetime by an 8D spacetime labeled by the octonions, we are able to prove this relation theoretically!

So then, what are our choices for non-commuting coordinates? Turns out, not many! If we choose to generalise the real number system and yet retain the property of division (i.e. every element should have an inverse) there are only three other possible number systems.

The first of the three are complex numbers (x + i y)

The next are the quaternions (a + b i-hat + c j-hat + d k-hat)

Here, a, b, c, d are real numbers. i, j, k are three imaginary units each of which square to minus one, but they do not commute with each other: ij = - ji, jk = -kj, ki = - ik, ij=k, jk=i, ki=j.

The last of the four division algebras are the octonions. An octonion is denoted as

a_0 + a_1 e_1 + a_2 e_2 + ... + a_7 e_7

The eight a-s are real numbers, the seven e_i are imaginary units each squaring to minus one, these anti-commute with each other, and have a multiplication table known as the Fano plane. Octonion multiplication is non-associative, besides being non-commutative.

An eight dimensional spacetime labeled by the octonions as their coordinates is known as an octonionic space-time. The usual 4D spacetime is a special case of the 8D spacetime.

When we put fermions on this spacetime, interesting things happen. Only eight types of fermions and their eight anti-particles are allowed [and exactly three generations]. Electric charge is quantised in units of 0, 1/3, 2/3 ad 1. The 1/3 and 2/3 are SU(3) triplets and identified as down and up quark. The 0 and 1 are SU(3) singlets and identified as the anti-neutrino and the positron. This way the standard model fermions arise, and only those ones are allowed.

Quantum theory on an octonionic spacetime is the exact quantum theory.

Why do elementary particles have such strange looking mass ratios?

Why do elementary particles have such strange looking mass ratios?

Elementary fermions of four different values of electric charge are observed in nature. In units of the charge of the positron, these values are 0, 1/3, 2/3, 1. The positron has charge 1, the up quark has charge 2/3, the anti-down quark has charge 1/3, and the anti-neutrino has charge 0. Their respective anti-particles have same absolute value of the electric charge, but of the opposite sign [electron, anti-up, down, neutrino]. We hence say that charge is quantised in these discrete units? But why these and only these, and why is it quantised in the first place?

Each of these four particles has a corresponding second generation particle, and a third generation particle. The second and third generation copies have the same electric charge as its first generation relative. Thus the counterparts of the electron are the muon and the tau-lepton, each of them having charge minus one. The down quark has as its counterparts the strange quark and bottom quark, all having charge 1/3. The up quark's copies are called charm and top, and they all have charge 2/3. We see that the value of electric charge does not change across generations.

The only known difference between the three copies of  a particle is their mass. The electron has a mass of 0.5 MeV. The muon at about 105 MeV is some 200 times heavier than the electron, The tau lepton at 1777 MeV is about 3500 times heavier than the electron. 

The up quark, charm and top have respective masses of 2.3 MeV, 1275 MeV and 173210 MeV. The down, strange and bottom quark have respective masses of 4.7 MeV, 95 MeV and 4180 MeV.

Neutrinos are known to have a non-zero mass, but much smaller than the mass of the charged fermions, and the actual value of their masses is unknown. Let us leave them out of the discussion for now.

It is obvious that unlike electric charge, mass ratios appear strange and random, and show no apparent pattern. If we decide say to compare the various masses with respect to the up quark, and for simplicity take the square root of the ratio, we get the numbers 1, 1.4, 0.47, 6.4, 6.8, 23.5, 42, 274.3, 27.8 The elegant simplicity of quantised values of electric charge is lost. Is there a simple pattern to these mass ratios, or not, and how are we to find the pattern, in case there is one? This profound question has remained unanswered for decades after all these particles were found and their masses measured experimentally.

An answer may now have been found, and it comes from a very surprising and unexpected quarter. We started in a very different realm: addressing a foundational problem in quantum theory. Quantum mechanics is formulated on an external classical space-time but such classical elements should strictly not be a part of quantum theory. We should be able to describe quantum phenomena without referring to classical time. In one specific approach to finding such a description, it has been found that the theory must be  formulated in eight dimensions, which are labelled not by real numbers, but by eight dimensional numbers known as the octonions.

When we try to place fermions in such an octonionic space-time, we are in for a surprise. We are not allowed to assign arbitrary properties and quantum numbers to these particles. The space-time dictates that the electric charge must be quantised, precisely in the units 0, 1/3, 2/3 and 1, as observed. And the space-time also dictates that particles have anti-particles of opposite charge, and come in three generations. 

If the space-time dictates there are three generations, and if the only difference between the three generations is mass, the octonionic space-time must also determine mass ratios, just as it determines ratios of electric charge. Happily, it does. Just as there is a charge value associated with every particle, there is a mass number associated with every particle, called its Jordan eigenvalue. These Jordan mass numbers are shown in the table below (Ignore the numerical entry in the first column). The entries in the next three columns are the respective mass numbers.

These mass numbers are very simple and pretty. For the three generational copies of a particle, the middle value is its electric charge, and the other two values are symmetrically placed about the middle value, always departing from the middle by a factor of square root of 3/8. 

The theory dictates that mass ratios will be determined by these Jordan values, and these mass ratios are shown in the table below, scaled with respect to the down quark mass which is set as one. It can be easily verified that these simple fractions reproduce the strange pattern of mass ratios observed in nature! Mass is quantised, just as electric charge is, and the ratios are quite simply given, though not as simple as the charge ratios 1/3, 2/3 and 1. 

We believe that the mystery of the strange mass ratios has finally been solved. At the heart of the resolution lies the realisation that elementary particles are described by the octonions, as long suspected by several physicists.

Also:

Mass ratios and Majorana neutrinos:

Getting the correct mass ratios from the octonion algebra requires us to assume that the neutrino is a Majorana particle [i.e. its own anti-particle]. Assuming the neutrino to be a Dirac particle gives wrong mass ratios. We hence predict that Neutrinoless Double Beta Decay does occur in nature and will indeed be observed once adequate sensitivity is achieved in experiments.

Reference: https://www.preprints.org/manuscript/202101.0474/v4

What do the interpretational problems of quantum mechanics have to do with the failed unification programme of string theory?

What do the interpretational problems  of quantum mechanics  have to do with the failed unification programme of string theory?

******************

The context of my present post is this beautiful video chat:

Steven Weinberg and Andrew Strominger in conversation [April, 2021] YouTube video https://youtu.be/PFJ46G8BflQ

Towards the end of this very watchable video, a member of the audience put a question to both of them:

In your opinion, is there some problem in our understanding of quantum mechanics [i.e. interpretational issues, measurement problem]. And if so, could this unsolved problem be holding up progress in theoretical high energy physics etc. ?

Weinberg: Yes there is a problem. Quantum mechanics gives a great importance to observers. It pre-assumes a quantum-classical divide so as to be able to make sense of the theory [classical apparatus, measurement]. A reductionist theory must not have to depend on its own limit for us to understand the theory. Rather, working from bottom up, the theory should be able to explain the classical properties of the measuring apparatus as a consequence of the theory itself, instead of having to assume them a priori without proof. Essentially, Weinberg is dissatisfied with Bohr's Copenhagen interpretation. Then he correctly points out that Everett had the same dissatisfaction and therefore came up with what became known as the Everett interpretation. Wave functions never collapse during measurements, and the universe is forever in a state of quantum superpositions of everything. Weinberg says in the conversation that he does not agree with / like the Everett interpretation either. Hence, according to him there is something missing in our understanding of quantum theory.

Strominger: No. There is no problem. We can calculate marvellously with quantum theory. The Lamb shift has been calculated to an unprecedented accuracy. No experiment has ever disagreed with quantum mechanics. As for the interpretational issues, these are just words. It is not physics. You could side with Bohr, or you could side with Everett. It does not make a difference. You can still do your excellent calculations with quantum field theory and predict the world. In other words, Strominger is saying: Shut up and calculate.

***************

These contrasting responses by Prof. Weinberg and by Prof. Strominger are noteworthy, and have a deep connection with the current status of string theory.

Fast forward to the present: "Strominger: It does not make a difference'. ?? Interestingly, it does!! And I try to say this as clearly as possible: The reason string theory has failed in its unification programme, and fails to predict the standard model despite being almost there, is because string theory adheres to the Everett interpretation of quantum mechanics. If string theory is modified a little bit so as to allow for [a dynamical implementation of] Bohr's Copenhagen interpretation and wave function collapse, it can predict the standard model, uniquely. So Bohr vs. Everett makes a huge difference. As big as the difference between success and failure. As I now try to explain.

It is well-known that string theory can be consistently formulated as a unified theory in a higher dimensional space-time [not four]. Ten space-time dimensions, to be precise. (Eleven, for M-theory). So far so good. However the universe we live in is four dimensional, not ten. Why do we not see the remaining six spatial dimensions? The answer proposed in string theory is that the extra dimensions are curled up, compactified, too small to be seen, say as small as the Planck length scale. This proposed solution ruins the theory. For it turns out there are a very large number of inequivalent ways of compactifying the extra dimensions, all of which produce different particle physics theories in four dimensions. String theory loses predictive power. Which compactification to use? In fact it is not clear whether the standard model is even there as one of the compactifications. As a consequence, as Strominger notes, the ambitious unification programme of string theory was over in the 1980s itself, in a couple of years after the excitement set in.

There is a different mechanism, other than compactification, for recovering a four dimensional space-time from a ten dimensional space-time. It requires us to preferentially and deliberately pick Bohr over Everett, and modify quantum mechanics a little bit [while still remaining consistent with all lab tests of quantum theory] and allow for a dynamically induced rapid collapse of the wave function in macroscopic systems. This is known as the Ghirardi-Rimini-Weber (GRW) mechanism of spontaneous localisation, and would happen very naturally in string theory too, provided we remove the restriction that at the Planck scale as well, the Hamiltonian of the theory must be self-adjoint. Instead, allow for the possibility that under suitable circumstances, evolution at the Planck scale can be non-unitary. Such a possibility is certainly not ruled out by current experiments.

How does this help with the compactification problem in string theory? When we say that the universe is four dimensional, what we mean is that classical objects in the universe live and evolve in four spacetime dimensions. Nobody can claim that quantum systems live in a four dimensional space-time!! A quantum system can well be thought of as living in ten space-time dimensions [as string theory does] even in today's universe, provided the support of its wave-function is non-vanishing only over microscopic distances. Here, microscopic does not mean Planck length. Microscopic can be as large as a micron, roughly before the classical-to-quantum transition takes place around an Angstrom, into the world of atoms, which are of course quantum.

Consider then a string theory type quantum field theoretic system living in a ten dimensional spacetime. Except that the dynamics is now given by the GRW modified quantum theory. and the Hamiltonian possesses an anti-self-adjoint part. When sufficiently many degrees of freedom living in 10D get entangled, the GRW mechanism of spontaneous localisation sets in, and the entangled system becomes classical. And now is the key point, in becoming classical, the entangled system descends from ten to four spacetime dimensions. The support of its wave-function over the extra six spatial dimensions is vanishingly small, smaller than Planck length - this has actually been proved. While the size of their extent in the 4D spacetime remains large. This way we have achieved effective dynamical compactification, or so to say, compactification without compactification. Quantum systems, including those in today's universe, continue to live in ten dimensions. The forces that curve the extra six spatial dimensions are precisely the internal symmetries of the standard model.

We have developed a unification theory in ten spacetime dimensions, very similar to string theory. Except that the dynamics is modified quantum dynamics. The theory has a very promising potential to unify gravity and the standard model, and to predict the values of the free parameters of the standard model (work in progress).

So dear Prof. Strominger , it matters: Bohr or Everett. These are not mere words; foundational questions of quantum mechanics are important. And now they are important in your own backyard Prof. Weinberg is absolutely right on this count; sadly he is no longer with us to witness the unfolding of this story.

We can hence have a failed string theory and an unmodified quantum mechanics. Or we can have a successful string theory and a modified quantum mechanics. It confounds me that string theorists, extremely smart physicists though they are, do not get this. Why do they not consider that the problem is not with strings, but with quantum theory? I sincerely hope they change their mind.