The road to a Theory of Everything
At this current time of writing, string or M-theory, the landscape of possible theories involving strings in a high dimensional space, is…
At this current time of writing, string or M-theory, the landscape of possible theories involving strings in a high dimensional space, is the leading candidate for a Theory Of Everything (TOE). No other theory come close to its sweeping claims on all matter and forces. (Pure group-based theories are mathematical curiosities at best.) Yet, increasingly, experiments have failed to find evidence for strings, and theorists have failed to prove that it leads correctly to a quantum theory of gravity in the required four dimensions.
It is a theory of “shattered dreams”, yet no other candidate has arisen to take its place. This shows how difficult theories of everything are.
String theory was introduced specifically to try to marry Einstein’s theory of general relativity, governing the large scale dynamics of the universe, and quantum Standard Model physics that governs the very small behavior of particles. Yet it is much more. There are many, many theories of quantum gravity but few viable theories of everything.
A theory of everything must explain all forces and matter along with gravity in a unified framework of a single field or entity.
String theory is a particle physicist’s approach to a theory of everything. That makes sense since everything we know about is a particle or has a particle form. Strings, likewise, are particles that have a one dimensional rather than zero dimensional structure.
If strings are what particle physicists dream of, what theories of everything would other physicists come up with?
A gravitational physicist might come up with a theory in which all forces and matter could be reduced to some high dimensional theory of gravity. Perhaps every force is part of the gravitational field in some way?
High dimensional theories of gravity, called Kaluza-Klein theories, were once a promising avenue for theories of everything. Kaluza’s original theory, dating back to the 1920s, showed how you can merge gravity and electromagnetism into one 5 dimensional theory. This result was astonishing at the time, wowing even Einstein, but in the hindsight of group theory, which would arise in its modern form in the 1950s with Yang and Mills, it was not that surprising.
The standard model contains three forces other than gravity: electromagnetic, weak, and strong forces. Electromagnetism is fairly simple. It is “Abelian” which essentially means that the force carriers, photons, do not themselves carry charge and do not self interact. The force carriers for the “non-Abelian” weak and strong forces, on the other hand, (W and Z bosons and gluons) do carry “charge” (flavor or color). They do self-interact.
That is fine, however, because gravity is similar to non-Abelian theories in that gravitons carry energy and self-interact. As far back as 1968, Kaluza-Klein was extended to non-Abelian forces. When you combine all the forces, you need at least seven dimensions in addition to our four, so you get 11 dimensions.
So far so good, but you still have to deal with matter. Matter is made up of fermions, spin-1/2 particles. And these are fundamentally different beasts from gravity. They are left and right handed. Force carriers are all bosons, with integer spin, so the gravitational field can represent them. Fermions are hard to fit into geometry.
Quantization is another problem. Gravity is a classical theory, meaning it does not account for quantum physics at all. No universally accepted quantum theory of gravity exists, and incorporating all forces and fields into gravity means that everything inherits the problems quantum gravity has. Hardly the direction we want to go in.
Seminal papers by Edward Witten, one of the most important figures in the development of string theory, suggested that Kaluza-Klein was unstable and could not account for matter. From then on it was seen as a dead end and that matter and forces should be represented as coupling to geometry but not an aspect of geometry. Rather, gravity was an aspect of strings, acting against some unexplained background geometry.
The majority of physicists interested in TOE dropped Kaluza-Klein although some continued to work on it. For example, a theory of higher dimensional gravity based only on spin-1/2 fields was worked out called Spinor Gravity. This was subsequently incorporated into Loop Quantum Gravity a decade ago as a kind of “spin” off.
I think that Spinor Gravity is a very promising direction for a TOE in the conventional sense and ultimately is more qualified to be called one than string theory which still includes two different types of entities: strings and background geometry. Spinor Gravity only has one type entity: spinors.
Yet, my own take is that there is a fundamental problem with TOEs when we don’t know what “everything” means. What we really mean when we propose a TOE is something that can incorporate all the physics we know about. This may only be a small sliver of all the physics that is possible.
A mathematical physicist, like myself, doesn’t see the world in terms of a specific collection of physical entities but as a collection of principles into which entities can be inserted. Those entities likewise may simply be processes that arise from those principles and have no essential existence.
That was the theory of everything that David Bohm, a prominent quantum theorist and philosopher, proposed. Bohm is probably best known for his quantum theory of Bohmian mechanics developed in 1952, which I’ve written about elsewhere. Another idea he proposed, however, is his theory of implicate and explicate order. The idea isn’t a physical theory but a metaphysics that can be applied to physics and how physical objects can manifest from underlying reality.
Bohm proposed that everything we observe, even spacetime geometry, and certainly particles, is a temporary “explicate order” that arises from a deeper, implicate process. In this sense, reality is a process of becoming while being is deeper and hidden. (Buddhism has espoused this idea for millennia.)
The deeper we go into physical entities, the more we find that they are composed of other entities, and the reason that the entities we know about have the properties they do has to do with the way those smaller entities behave as much as for their intrinsic properties. Elements in the periodic table, for example, have their properties from the configuration of electrons, protons, and neutrons in them, not necessarily from the intrinsic nature of those subatomic particles.
Bohm’s idea is that the deeper we go, the more we find out that the properties of objects arise from the configuration and motion of their underlying components. Change how those components are configured and the “thing”, which is the explicate order, vanishes. At some point, we find that there is nothing at all but configuration itself, pure motion.
In quantum field theory, we know that particles are simply excitations of fields. Thus, fields represent the potential for something to exist. All that is needed is the right energy in the right place to make it appear. This is something like what Bohm means, yet we can go beyond fields to talk about the underlying structure of physical principles.
One way to approach this idea is through the use of a concept called a phase space. A phase space has as many dimensions as are required to describe the complete state of an object. For a rock that may be six dimensions, three position and three velocity numbers in a three dimensional space. For a field, it may be infinite.
Yet, what determines how those descriptions move in the phase space? In classical mechanics, there are two more or less equivalent forms: the principle of least action and Hamilton’s equations.
In quantum physics, there are likewise two descriptions: the Hamiltonian description governed by Schroedinger’s equation and the path integral form which is based on the action. In quantum physics, the phase space is the space of all possible wavefunctions. Schroedinger’s or its equivalent forms (Heisenberg’s and Dirac’s) describe how the wavefunction moves in phase space.
One of the beautiful things about the Hamiltonian description is that it connects to a geometric description of phase spaces called symplectic geometry. Essentially, the Hamiltonian of a system, whether classical or quantum, describes its symplectic geometry. This is not a geometry that you can see or touch, but rather a geometry of a process.
I have wondered if symplectic geometry connects to Bohm’s idea of implicate order at a deeper level than fields.
Consider that we don’t know where Hamiltonians or actions come from. They are descriptions of physical objects, but there is no theory that explains why they have precisely those descriptions. Even a conventional TOE would have some Hamiltonian, some action, and some description as an entity.
A Hamiltonian is some kind of operator or matrix, but it is also a model for the landscape of the phase space. Could it be that the Hamiltonians arise from some symplectic landscape that underlies all of reality rather than vice versa? That is, is a Hamiltonian simply a map of that landscape? In that case, all physical descriptions that we know about, even space and time, are simply descriptions of the particular region of that infinite dimensional geometry we inhabit, a local description of valley in which we live, and not some overarching description of the universe.
Note that symplectic geometry is not a “thing” per se like space and time. It is a description of a process or configuration. The points within a symplectic landscape are not things either. They only relate to our understanding of how to describe the physical motion of things. Could it be that all reality could be reduced to motion in symplectic space?
To me that would be a real theory of everything, not only describing everything we can see or touch but the very principles upon which reality is based.
R. Kerner, “Generalization of the Kaluza-Klein theory for an arbitrary non-abelian gauge group”, Ann. Inst. Henri Poincare, 9, 143–152 (1968)
Witten, Edward. “Search for a realistic Kaluza-Klein theory.” Nuclear Physics B 186.3 (1981): 412–428.
Witten, Edward. “Instability of the Kaluza-Klein vacuum.” Nuclear Physics B 195.3 (1982): 481–492.
Witten, Edward. “Fermion quantum numbers in Kaluza-Klein theory.” et al: Modern Kaluza-Klein Theories (1983): 438–511.
Hebecker, Arthur, and Christof Wetterich. “Spinor gravity.” Physics Letters B 574.3–4 (2003): 269–275.
Livine, Etera, and Johannes Tambornino. “Spinor representation for loop quantum gravity.” Journal of mathematical physics 53.1 (2012): 012503.