Our understanding of the physical universe which is a shadow of the greater Cosmos consisting of spiritual, mental and physical realms is very limited. Our scientific knowledge of the physical universe is growing at an exponential pace but we have only scratched the surface so far. We have made great progress but we still don’t know what electric charge and particle spin is and we cannot explain the masses of elementary particles. We know that the Higgs field which pervades the entire universe gives the elementary particles their masses but we have no idea why a particular elementary particle has the mass it has. We cannot explain the differences in masses of elementary particles.
We don’t know what an electron is but we can predict its motion in a probabilistic way. This allows us to manipulate the electrons. Our modern age is based on the manipulation of electrons.
We have theories of gravitation such as the Newton’s theory and the Einstein’s theory known as General Relativity but these theories simply describe the attractive force between the masses; they don’t explain gravitation. They give us formulas to predict the motion of falling bodies. These formulas, especially the formula given by the theory of General Relativity is very precise. We can predict the motion of astronomical objects with precision but we still don’t know what gravitation is.
Sometimes, geometrical description is advertised as explanation. For example, according to General Relativity the mass distorts the surrounding space-time creating a curvature in space-time. The smaller objects in the vicinity of this distortion are attracted towards the more massive object because of the curvature of space-time in that vicinity. This is an ingenious approach but there is no explanation regarding the space-time distortion itself. Why does the mass cause a distortion in the surrounding space-time? We don’t know!
Even though the geometric description of physics is not really an explanation we love this approach because it is psychologically most satisfying. Geometric explanations allow us to visualize the forces.
In the geometrical approach to physics the primary strategy is to extend the definition of space-time. If the space-time has additional degrees-of-freedom (dimensions) at the microscopic scale then certain characteristics of elementary particles might be explained by this microscopic structure. That’s the idea. That’s the hope. Two problems:
- So far there is no experimental evidence for this hypothesis.
- Even if we find evidence it will be very difficult to determine the shape and size of these microscopic dimensions.
Below is a quick summary of the proposals none of which has been experimentally confirmed. The sad fact is that all these proposals have severe theoretical difficulties as well. The theoretical difficulties and the lack of experimental confirmation raise doubts about this approach.
Can we really explain the mass differences, electric charge, spin, and the free parameters of the Standard Model of particle physics and discover new physics by working out the consequences of extending space-time in some way?
Thousands of physicists dedicated their lifetimes to work out the details of these proposals. It is so sad that after so much mathematical effort we have made so little progress in terms of understanding physics.
In 1921 Theodor Kaluza published a theory attempting to unify electromagnetism with Einstein’s General Relativity theory. This unification was facilitated by the assumption of an invisible extra spatial dimension. In 1926 Oskar Klein suggested that Kaluza’s extra spatial dimension must be curled up and that’s why it is invisible. Klein theorized that each point in space is a compact circle.
The word “compact” here means infinitesimally small, cm which one tenth of a millionth of a trillionth of a trillionth of a centimeter. The size of this compact circle is also known as the Planck length which is about 24 orders of magnitude smaller than an atom and 19 orders of magnitude smaller than a proton.  The term “order” means 10 times. The point is that this compact circle is supposed to be infinitesimally small.
Klein’s extension of Kaluza’s idea is very appealing because it hints at the possibility of explaining electric charge and mass differences of elementary particles in geometric terms. Klein suggested that the electric charge could be understood as quantized momentum in this compact circle. Similarly, one can imagine standing waves in this compact circle and the quantized frequencies can be interpreted as particle masses. But, unfortunately, this approach does not explain the mass differences of elementary particles and cannot explain the electric charge fully either.
Superspace of Supersymmetry
In the simplest supersymmetric theory, each point in space has additional degrees of freedom (coordinates) that can be represented by a 2-component Weyl spinor. The motivation, of course, is to incorporate the physical property known as the spin into the definition of space itself thereby escaping the need to explain the physics of spin.
When you work out the physics implications of the supersymmetry hypothesis you end up with sparticles. Supersymmetry predicts that each of the particles in the Standard Model has a partner (sparticle) with a spin that differs by half of a unit. The sparticle’s spin will be 1/2 units less than its partner. Electron is a spin=1/2 particle. The selectron (supersymmetric partner of electron) will be a spin=0 particle. Photon is a spin=1 particle. The photino (supersymmetric partner of photon) will be a spin=1/2 particle.
The picture above is kind of misleading in the sense that the so-called “SUSY force carriers” would not really be force carriers because they have spin=1/2. The spin=1/2 particles cannot be force carriers. Most physicists think that supersymmetry simplifies physics. I don’t think so! Supersymmetry may simplify the mathematics of proposed theories but it adds conceptual complexity.
Newly discovered Higgs particle is the only known spin=0 particle in nature. Spin=0 particles are very special. Having another type of spin=0 particle (selectron) would be very interesting.
Space in Superstrings
The idea of strings is that the most elementary manifestation of physicality is in the form of a vibrating string. The term “string” is used in a broader context to include vibrating membranes. String is not necessarily a 1-dimensional object.
The supersymmetry type extension as well as the Kaluza-Klein type extension of space have been incorporated into strings. In other words, strings move in a space where each point in space can have a Kaluza-Klein type internal structure and also have additional degrees of freedom represented by a Weyl spinor (supersymmetry).
The shape and dimensionality of these Kaluza-Klein type structures can be very complicated. There is no principle yet to fix the shape and dimensionality of extra dimensions. One class of shapes known as Calabi-Yau is promising.
The initial hope was that the vibrational modes of the strings can be identified as mass, charge, spin, etc. But, unfortunately, string theorists have not been successful in establishing these connections. I have never seen clear statements about what vibrational mode corresponds to what physical property. They are still working on the theory.
Comment on Background Independence
In his book “Trouble with Physics” Lee Smolin emphasizes the background independence property of General Relativity which treats space-time as a dynamic object. The equations of General Relativity do not specify the geometry (shape, structure, mathematical form) of the space-time. In General Relativity the mathematical form of space-time is dependent on the mass-energy distribution.
The Kaluza-Klein type as well as the supersymmetry type extension of space, on the other hand, specify a mathematical form for space. According to Kaluza-Klein type and supersymmetry type proposals physical laws depend on the mathematical form of space. Otherwise we loose the ability to explain physical quantities such as charge and spin. General Relativity is different in this regard.
Background independence is not really a strength because we loose explanatory power. But, it is a good principle to have because that way we don’t make unnecessary assumptions about the mathematical form of the space-time.
The idea of twistor-space was proposed and developed by Roger Penrose who is one of my heroes. According to this proposal the light-rays are fundamental and space-time is just an aspect of relations among them.
Twistor-space is smooth (not discrete or granular). In this sense it differs fundamentally from some of the proposals mentioned below. Twistor-space has no concept of fuzzy point. In this regard as well it differs from the quantum space-time proposal mentioned below.
Twistor-space concept is finding new applications. Andrew Hodges (“Theory with a twistor”, Nature Physics, Vol 9, April 2013) explains the recent developments regarding twistor space as follows:
“Twistors, introduced by Roger Penrose in the 1960s, extend such spinor geometry into a new picture of space-time. While spinors give a sort of square root of vectors, and so of directions, twistors give a sort of square root for space-time itself. In 1972, Penrose wrote down and evaluated the first twistor diagrams for particle scattering. They had those same white and dark vertices (a duality which expresses this splitting into two) that now appear in the new work.
This journey has taken many years. The critical juncture was in 2003, when Ed Witten brought together insights from string theory, knowledge of twistor geometry, and an appreciation of the practical needs of scattering calculation. With this stimulus, Penrose’s twistor diagrams enjoyed a renaissance.
In 2005, I showed how they fitted into Witten’s new theory, and then in 2009, Arkani-Hamed and colleagues built on this to create a new calculus for scattering theory, in which a central role is taken by the classical geometrical concept of the ‘Grassmannian’, a space that parameterizes all linear subspaces of a vector space.”
Branes are lower-dimensional surfaces embedded in a higher-dimensional space. Branes come in different shapes, forms, and sizes.
Andy Strominger considered the so-called p-branes that wrap around a tiny curled-up region of space and showed that p-branes would act like particles. Later, Joe Polchinski showed that the D-branes (the type of branes to which the open strings attach themselves) morph into p-branes at high energies.
Cosmological Branes are also possible. The Brane-world hypothesis suggests that our universe is a 3-dimensional (big) Brane floating in a multi-dimensional sea (Bulk). Brane-world hypothesis also suggests that photons, electrons and all the rest of the matter as well as the forces other than gravity are restricted to a particular Brane but gravity is not. Gravity is free to extend into extra dimensions and to other Branes.
The Brane-world hyptothesis replaces the space-time with the concept of multiple Branes embedded in a higher dimensional Bulk. The number of possibilities in a Multiverse consisting of multiple Branes (parallel or intersecting) would be infinite.
Large extra dimensions
The sizes of extra curled dimensions of various string theories are typically at Planck length ( cm). In this radical 1998 model proposed by Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali the extra dimensions could have mm size. Results from the Large Hadron Collider experiments, however, do not appear to support the model thus far.
In “Prometheus and Chronos” I mentioned that space, time and matter emerge from the primordial field simultaneously and they are inter-dependent on each other. In a different piece titled “Matter (Mass) is not bottled-up energy” I used the term “space-time-matter.” At that time I did not know about the theory developed by the STM consortium http://www.5dstm.org/ which has similarities to my view. They explain their approach as follows:
“Like Kaluza, we write down Einstein’s field equations in more than four spacetime dimensions, with no explicit higher-dimensional source terms (i.e., we do not put in any matter or energy “by hand”). Unlike Klein and many others since, we avoid overly restrictive assumptions about the physical dimension, scale or topology of the extra coordinates. Dimensional reduction then leaves us with Einstein’s field equations as usual, along with extra terms arising solely from the geometry of the higher-dimensional manifold. We identify these extra terms with matter and energy in the four-dimensional world.”
Biquaternionic extension of space-time
In the Biquaternionic formulation of relativistic quantum mechanics each point in space-time is represented by a biquaternion .
Algebrodynamics over complex space
Biquaternions also form the core of the “algebrodynamics over complex space” paradigm discussed by V.V.Kassandrov  and the references therein.
Algebraic design of physics
Another major effort pointing out the “algebraic design” of physics was developed by G.M.Dixon . In the “algebraic design of physics” space-time is not primary.
Quantum space-time is also known as the non-commutative geometry of space-time. According to this proposal the space-time is not a continuum.
The term “non-commutative” refers to the quantum nature where the position of a particle cannot be known with certainty. The uncertainty is not due to experimental limitations or measurement errors. Quantum uncertainty is intrinsic. The quantum space-time proposal is essentially saying that a point in quantum space-time is a fuzzy point.
Discrete (granular) space-time
One example of a proposal where space-time is discrete (granular) is “Causal Dynamical Triangulations” approach where space-time is approximated as a mosaic of triangles . Obviously, these triangles would be very small otherwise we would have seen the effect in the speed of light in vacuum.
All experiments and astrophysical measurements have shown that the speed of light in vacuum does not depend on its frequency. If space-time was discrete (granular) then light would have a different speed in vacuum depending on its frequency. This has not been observed yet.
Atoms of space
In his book “Trouble with Physics” Lee Smolin explains this idea as follows:
“…smoothness of space is not real and that space emerges as an approximation of something consisting of building blocks that we can count. In some approaches, it is just assumed that space is made of discrete “atoms”; in others this assumption is rigorously derived by combining the principles of general relativity and quantum theory.”
Space-time is emergent from spin-network (spin-foam)
Spin networks were first introduced by Roger Penrose as a purely combinatorial description of the geometry of space-time. Later, Rovelli and Smolin discovered that spin networks can be used to describe states in loop quantum gravity. A spin network is a mathematical graph. Nodes of the graph correspond to quanta of volume (atoms of space). Edges of the graph correspond to quanta of area. Spin networks evolve in time. The main strength of this proposal is its background independence property. Spin-network proposal combines nicely the “discrete space-time” and the “quantum-space-time” proposals and shows how “quantum space-time” can emerge from the spin-network. The weakness, however, is the difficulty to show how “classical space-time” emerges from a spin-network.
This is a huge topic related to quantum nature and the discussions between Niels Bohr and Albert Einstein. In the literature this topic is normally discussed under the subjects of “hidden variables” theories of Quantum Mechanics. I will be very brief here. Einstein never accepted Quantum Mechanics as a complete theory and believed that someday the quantum uncertainty would be explained by other (hidden) variables. After decades of discussion and experiments we now know that Quantum Mechanics with hidden variables can only be constructed if and only if you assume that space is non-local. David Bohm constructed a non-local hidden variables theory of quantum physics. It produces the same results as the standard version known as Quantum Mechanics but his space is weird in the sense that large distances have no meaning. According to Bohm’s quantum theory events on the other side of the galaxy are influencing the events here. If you do not accept this then you have to accept the intrinsic uncertainty of Quantum Mechanics. Debate goes on.
Extra temporal dimensions
Physicists investigated the consequences of having more than 1 time dimension. Extra temporal dimensions imply that there are tachyons which are particles moving faster than light in vacuum. No tachyons have been observed experimentally. According to Einstein’s Special Relativity theory no particle and no information packet can move faster than light in vacuum.
Fabric of space consisting of knots and links
Dmitri Kozlov has a very interesting paper titled “Knots and Links As Form-Generating Structures“. Knots and links is another proposal for the conceptual basis of space-time fabric. In this context you should check out a paper by Atiyah titled “The Geometry and Physics of Knots.”
Events are fundamental, space-time is emergent (causal sets)
I mentioned Kevin H. Knuth and his collaborators’ work in the “No space, no time, just events” piece. In the abstract of “A Derivation of Special Relativity from Causal Sets” Kevin H. Knuth and Newshaw Bahrenyi explain the idea as follows:
“We present a novel derivation of special relativity based on the information physics of events comprising a causal set. We postulate that events are fundamental, and that some events have the potential to receive information about other events, but not vice versa. This leads to the concept of a partially-ordered set of events, which is called a causal set. Quantification proceeds by selecting two chains of coordinated events, each of which represents an observer, and assigning a valuation to each chain. Events can be projected onto each chain by identifying the earliest event on the chain that can be informed about the event. In this way, each event can be quantified by a pair of numbers, referred to a pair, that derives from the valuations on the chains. Pairs can be decomposed into a sum of symmetric and antisymmetric pairs, which correspond to time-like and space-like coordinates. From this pair, we derive a scalar measure and show that this is the Minkowski metric. The Lorentz transformations follow, as well as the fact that speed is a relevant quantity relating two inertial frames, and that there exists a maximal speed, which is invariant in all inertial frames. All results follow directly from the Event Postulate and the adopted quantification scheme.”
The conceptual problems do not go away by demonstrating that events are fundamental. If that’s the case then we have to explain what an “event” is. I wrote few words on this in “What is an event? ”
There is much more
There are many more proposals. Chapter 33 of Penrose’s book “Road to Reality” mentions those other concepts that extend or replace space-time. Space time fabric as lattice is mentioned. Quaternionic and Octonionic space-time are mentioned as well. There is still more.
There is no experimental evidence for any proposal mentioned here or mentioned elsewhere that tries to extend the space-time concept used in relativity theory. To make progress, we need additional degrees-of-freedom to explain the observed physics. The question is: are those additional degrees-of-freedom necessarily geometric or algebraic? If not geometric or algebraic what then? Algorithmic? Rule-based reactionary? Fundamental-DNA type design to physics perhaps?
 Lisa Randall, “Warped Passages”, Harper Perennial (2005)
 Katsusada Morita, “Quaternions, Lorentz Group and the Dirac Theory”, Progress of Theoretical Physics, Vol.117, No.3 (2007)
 Vladimir V. Kassandrov, “Algebrodynamics over Complex Space and Phase Extension of the Minkowski Geometry”, arXiv:gr-qc/0602088. (2006)
 Geoffrey M.Dixon, “Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics”, Springer (1994), Kluwer Academic Publishers (2002), ISBN: 0792328906 / 9780792328902
 Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, “The Self-Organizing Quantum Universe”, Scientific American, July 2008