Thoughts on String Theories

The Multiverse

As a universe tunnels through the landscape (of string theory), from (mathematically modeled) "hill" to "valley", it retains (conserves) the entire information regarding the volume of (mathematically modeled) "space" (or of the space-like volume) of the portion of the landscape that it has traversed. These data are holographically encoded and can be fully captured by specifying the information regarding the universe's (lightlike) boundary (e.g., its gravitational horizon).

As the universe's entropy grows (and energy density falls), it "decays" and its inflation stops. This event determines its nature (its physical constants and laws of Nature). Eternal inflation is, therefore, a feature of the entire landscape of string theory, not of any single "place" or space-time (universe) within it. 

Strings

Strings are described as probabilistic ripples (waves) of spacetime (NOT in a quantum field) propagating through spacetime at the speed of light. From the point of view of an observer in a gravitational field, strings will appear to be point particles (Special Relativity). The same formalism used to describe ripples in quantum fields (i.e., elementary particles) is, therefore, applied.

Strings collapse (are resolved) and "stabilize" as folds, wrinkles, knots, or flaps of spacetime.

The vibrations of strings in string theories are their probabilities in this theory (described in a wave function).

The allowed, netted resonances (vibrations) of the strings are derived from sub-Planck length quantum fluctuations ("quantum foam"). One of these resonances yields the graviton.

Strings probabilistically vibrate in ALL modes at the same time (superposition) and their endpoints are interference patterns.

D-branes are the probability fields of all possible vibrations.

The Universe

A 12 dimensional universe is postulated, with 9 space dimensions and 3 time dimensions.

Every "packet" of 3 spatial dimensions and 1 temporal dimension curls up and creates a Planck length size "curled Universe".

At every point, there are 2 curled up Universes and 1 expressed Universe (=the Universe as we know it).

The theory is symmetric in relation to all curled Universe ("curl-symmetric").

All the dimensions - whether in the expressed Universe (ours) or in the curled ones - are identical. But the curled Universes are the "branches", the worlds in the Many Worlds interpretation of Quantum Mechanics.

Such a 12 dimensional Universe is reducible to an 11 dimensional M Theory and, from there, to 10 dimensional string theories.

In the Appendix we study an alternative approach to Time:

A time quantum field theory is suggested. Time is produced in a non-scalar field by the exchange of a particle ("Chronon").

A. OVERVIEW OF STRING AND SUPERSTRING THEORIES

String theories aim to unify two apparently disparate physical theories: QFT (Quantum Field Theory) and the General Relativity Theory GRT). QFT stipulates the exchange of point-like particles. These exchanges result in the emergence of the four physical forces (weak, strong, electromagnetic and gravity). As the energy of these interactions increases, the forces tend to merge until they become a single, unified force at very high energies. The pursuit of a Grand Unified Theory or, even, a Theory of Everything - is not a new phenomenon. Einstein's Special Theory of Relativity (SRT) (preceded by Maxwell) unified the electromagnetic forces. Glashow, Salam and Weinberg unified the electroweak forces. In the Standard Model (SM), the strong and electroweak forces attain the same values (i.e., are the same) at high energy and gravitation joins in at even higher energies.

GRT and QFT are mathematically interfaced. Macro-objects (dealt with in the GRT) tend to create infinite spacetime curvature when infinitely compressed (to become point particles). The result is a "quantum foam" which really reflects the probabilities of point particles. But relativistic QFT fails to account for gravity. It copes well with elementary particles but only in an environment with a vanishingly weak force of gravity. Some physicists tried to add a "graviton" (gravity force carrying particle) to QFT - and ended up with numerous singularities (particle interactions at a single point and at a zero distance).

Enter the strings. These are 1-dimensional (length) entities (compared to zero-dimensional points). They move across the surface their "worldsheet". They vibrate and each type of vibration is characterized by a number which we otherwise know as a quantum number (such as spin or mass). Thus, reach vibrational modes, with its distinct set of quantum number corresponds to a specific particle.

String theories strive to get rid of infinities and singularities (such as the aforementioned infinite curvature, or the infinities in the Feynman diagrams). They postulate the existence of matter-forming, minuscule, open or closed, strings with a given - and finite - length. The vibrations of these entities yields both the four elementary forces and four corresponding particles. in other words, particles are excitatory modes of these strings, which otherwise only float in spacetime. The string tension being related to its length, strings need to have a Planck length to be able to account for quantum gravity. One of these states of excitation is a particle with zero mass and 2 spin units - known in Quantum Theory of Gravity (QTG) as "graviton". Moreover, strings tend to curl (though, counterintuitively, they are wrapped around space rather than in it - very much like the topological chimeras the Mobius strip, or the Klein bottle). Mathematics dictate an 11-dimensional universe. Four of its dimensions have "opened" and become accessible to us. The other 7 remain curled up in a "Calabi-Yau space" in which strings vibrate. In later version of string theory (like the M-Theory), there is a 7-dimensional, curled up Calabi-Yau space wrapped on every 4-dimensional point in our universe. But Calabi-Yau spaces are not fixed entities. New ones can be created every time space is "torn" and "repairs" itself in a different curvature. Lastly, strings merge when they interact, which is very useful mathematically-speaking. Technically speaking, one of 2 interacting strings "opens up" in an intermediate phase - and then closes up again.

But what is the contribution of this hidden, strange world and of the curling up solution to our understanding of the world?

String theories do not deal with the world as we know it. They apply in the Planck scale (where quantum gravity prevails). On the other hand, to be of any use, even conceptually, they must encompass matter (fermions). Originally, fermions are thought to have been paired with bosons (force conveying particles) in a super-symmetric, superstring world. Supersymmetry broke down and vanished from our expanding Universe. This necessitated the "elimination" of the extra-dimensions and hence their "compactification" (curling up).

Moreover, some string theories describe closed but openable strings - while others describe closed and NON-openable ones. To incorporate Quantum Mechanics (QM) fully, one needs to resort to outlandish 26 dimensional universes, etc.

Still, string theories are both mathematically simpler than anything else we have to offer - and powerfully explanatory.

We use Perturbation Theory (PT) To compute QM amplitudes. We simply add up contributions from all the orders of quantum processes. To be effective, contributions need to get smaller (until they become negligible) the "higher" we climb the order hierarchy. The computation of the first few diagrams should be yield an outcome asymptotic to "reality". This is necessary because in point-like particle field theories, the number of diagrams required to describe higher orders grows exponentially and demands awesome computing power.

Not so in string theories. Holes and "handles" (protrusions) in the worldsheet replace the diagrams. Each PT order has one diagram - the worldsheet. This does not alleviate the mathematical complexity - solving a 2-handle worldsheet is no less excruciating than solving a classic PT diagram. But if we want to obtain complete knowledge about a quantum system, we need a non-perturbative theory. PT is good only as an approximation in certain circumstances (such as weak coupling).

B. MORE ON THE INNER WORKINGS OF STRING THEORIES

String vibrate. In other words, they change shape - but revert to their original form. Closed strings are bound by boundary conditions (such as the period of their vibration). Open strings also succumb to boundary conditions known as the Neumann and Dirichlet boundary conditions. Neumann allowed the end point of a string free movement - but with no loss of momentum to the outside. Dirichlet constrained its movement to one "plane" (or manifold) known as a D-brane or Dp-brane (the "p" stands for the number of spatial dimensions of the manifold). Thus, if a spacetime has 11 dimensions - of which 10 are spatial - it would have a D10 D-brane as its upper limit. p could be negative (-1) if all space and time coordinates are fixed (and "instanton"). When p=0, all the spatial coordinates are fixed, the endpoint is at a single spatial point (i.e., a particle). A D0-brane is what we know as a particle and a D1-brane would be a string. D-branes are mobile and interact with closed strings (and particles). Strings (such as the graviton) may open and "affix" their endpoints on a D2-brane (during the interaction).

But these interactions are confined to bosons. When we add fermions to the cocktail, we get supersymmetry and pairs of fermions and bosons. When we try to construct a "supersymmetric" QFT, we need to add 6 dimensions to the 4 we are acquainted with. This contraption cancel the anomalous results we otherwise obtain. In terms of PT, we get only five consistent string theories: I, IIA, IIB, E8XE8 Heterotic, SO(32) Heterotic. In terms of weakly coupled PT, they appear very different. But, in reality, they are all aspects of a single string theory and are related by "string dualities" (i.e., different formalisms that describe the same physical phenomena).

C. A LITTLE HISTORY

From its very inception in 1987, it was clear one of the gauge groups at the heart of E8XE8 is identical to the gauge group of the Standard Model (SM). Thus, matter in one E8 interacted through all the forces and their particles - and matter in the other E8 interacted only through gravity. This did nothing to explain why the breakdown of supersymmetry - and why the SM is so complex and muti-generational. Six of the 10 dimensions curled up into (non-observable) Planck length and compact 6-d balls attached to every 4-d point in our observable universe. This was a throwback to the neat mathematics of Kaluza-Klein. By compactifying 1 dimension in a 5-d universe, they were able to derive both GRT and electromagnetism (as a U(1) gauge theory of rotation around a circle).

We need to compactify the extra dimensions of (10-d and 11-d alike) superstring theories to get to our familiar universe. Various methods of doing this still leave us with a lot of supersymmetry. A few physicists believe that supersymmetry is likely to emerge - even in our pedestrian 4-d world - at ultra high energies. Thus, in order to preserve a minimum of supersymmetry in our 4-d universe, we use Calabi-Yau (CY) manifolds (on which the extra dimensions are compactified) for low energies. A certain CY manifold even yields the transition from the big bang (10 or 11 dimensional) universe to our dimensions-poorer one.

D. DUALITIES

The various string theories are facets of one underlying theory. Dualities are the "translation mechanisms" that bind them together. The T-duality relates theories with dimensions compactified on a circle with the radius R to theories whose dimensions are compactified on a circle with the radius 1/R. Thus, one's curled dimension is the other's uncurled one. The S-duality relates the coupling limits of the various theories. One's upper (strong coupling) limit becomes another's weak coupling limit. The celebrated M Theory is also a duality, in a way.

M Theory is not a string theory, strictly speaking. It is an 11-d supergravity with membranes and solitons (its 5-branes). Only when  compactified does it yield a 10-d string theory (the IIA version, to be precise). It is not as counterintuitive as it sounds. If the 11th dimension is of finite length, the endpoints of a line segment define 9-dimensional boundaries (the 10th dimension is time). The intersection of an open membrane with these boundaries creates strings. We can safely say that the five string theories, on the one hand, and M Theory on the other hand constitute classical LIMITS. Perturbation theory was used to derive their corresponding quantum theories - but to little effect. the study of non-perturbative attributes (dualities, supersymmetry and so on) yielded much more and led us to the conviction that a unified quantum theory underlies these myriad manifestations.

E. PARTICLES

Every physical theory postulates physical entities, which are really nothing more than conventions of its formalism. The Standard Model (SM) uses fields. The physical properties of these fields (electric, magnetic, etc.) are very reminiscent of the physical properties of the now defunct pre-relativistic ether. Quantized momenta and energy (i.e., elementary particles) are conveyed as ripples in the field. A distinct field is assigned to each particle. Fields are directional. The SM adds scalar fields (=fields without direction) to account for the (directionless) masses of the particles. But scalar fields are as much a field as their non-scalar brethren. Hence the need to assign to them Higgs particles (bosons) as their quanta. SM is, therefore, an isotropy-preserving Quantum Field Theory (QFT).

The problem is that gravity is negligibly weak compared to the enormous energies (masses) of the Higgs, W, Z and Gluon particles. Their interactions with other fields are beyond the coupling strengths (measurement energies) of today's laboratories. The strong and electroweak forces get unified only at 10 to the 16th power GeV. Gravity - at 10 to the 18th power (though some theories suggest a lower limit). This is almost at the Planck scale of energy. There is an enormous gap between the mass of the Higgs particles (200 Gev) and these energies. No one knows why. Supersymmetric and "Technicolor" solutions suggest the existence of additional forces and particles that do not interact with the SM "zoo" at low energies.

But otherwise SM is one of the more successful theories in the history of physics. It renormalized QFT and, thus, re-defined many physical constants. It also eliminated the infinities yielded by QFT calculations. Yet, it failed to renormalize a gravitational QFT.

The result is a schism between the physics of low energies and the physics of high and ultra-high energies. Particle theories look totally disparate depending on the energies of the reactions they study. But, luckily, the reactions of massive particles are negligible in low energies - so renormalizable QFT (e.g., SM) is a fair approximation, althesame. At low energies, the combination of Special Relativity Theory (SRT) and any quantum theory is indistinguishable from a renormalizable QFT. These are the fundaments of a possible unification. Unfortunately, these theories break down at high energy and, though very effective, they are far from being simple or aesthetic (i.e., classic). Too many interactions yielded by the formalism are arbitrarily suppressed below this or that energy threshold. Most of these suppressed interactions are figments of the imagination at the energy scales we are accustomed to or which are attainable in our labs. Not so gravitation - also a non-renormalizable, suppressed (though extremely weak) interaction. Other suppressed reactions threaten to unsettle the whole edifice - yielding such oddities as unstable photons, or neutrinos with masses.

Hence the intuitive appeal of string theories. The vibratory modes of strings appear to us as particles. Gravitation is finally made a part of a finite theory. The drawbacks are the extra-dimensions, which seem to unparsimoniously run contra to Occam's razor - and the outlandishly high energies in which they are supposed to reveal themselves (uncurl). M Theory tries to merge QFT with the classic string theories - but this alleviates only a few marginal issues.

The more philosophically and aesthetically inclined reject the operationalism which characterizes modern physics ("if it works - I am not interested to know WHY it works or even HOW it works"). They demand to know what is the underlying PHYSICAL reality (or at least, physical PRINCIPLE). The great pre-QM (Quantum Mechanics) theories always sprang from such a principle. The general Relativity Theory (GRT) was founded on the principle of the equivalence (i.e., indistinguishability) of gravity and inertia. Even the SM is based on a gauge symmetry. Special Relativity Theory (space-time) constrains QFTs and is, therefore, their "principle". No one is quite sure about string theories.

Arguably, their most important contribution is to have dispensed with Perturbation Theory (PT). PT broke down quantum processes into intermediate stages and generated an "order of complexity". The contributions from simpler phases were computed and added up first, then the same treatment was accorded to the contributions of the more complex phases and so on. It worked with weak forces and many theories which postulate stronger forces (like some string theories) are reducible to PT-solvable theories. But, in general, PT is useless for intermediate and strong forces.

Another possible contribution - though highly theoretical at this stage - is that adding dimensions may act to reduce the energy levels at which grand unification (including gravity) is to be expected. But this is really speculative stuff. No one know how large these extra dimensions are. If too small, particles will be unable to vibrate in them. Admittedly, if sufficiently large, new particles may be discovered as well as new force conveyance modes (including the way gravity is transmitted). But the mathematical fact is that the geometrical form of the curled dimensions determines the possible modes of vibration (i.e., which particle masses and charges are possible).

Strings also constitute a lower limit on quantum fluctuations. This, in due time and with a lot more work (and possibly a new formalism), may explain why our universe is the way it is. Unconstrained quantum fluctuations should have yielded a different universe with a different cosmological constant.

F. THE MICRO AND THE MACRO

Strings have two types of energy states, depending on the shape of space time. If curled (cylindrical) space-time is "fat" (let's say, the whole universe) there will be closely spaced energy states, which correspond to the number of waves (vibrations) of the string and its length, and widely spaced energy states, which correspond to the number of loops a string makes around curled (cylindrical) space-time (winding modes).  If the curled (cylindrical) space time is "thin" (let's say a molecule), a mirror picture emerges. Obviously, in both cases - "fat" space-time and "thin" space-time - the same vibrations and winding states are observed. In other words, the microcosm yields the same physics as the macrocosm.

G. BLACK HOLES

String theory, which is supposed to incorporate quantum gravity, should offer insights regarding black holes. String theories make use of the General Relativity Theory (GRT) formalism and add to it specific matter fields. Thus, many classical black hole solutions satisfy string equations of motion. In an effort to preserve some supersymmetry, superstring theory has devised its own black hole solutions (with D-branes, or "black branes", as the description of certain supersymmetric black holes). A match was even found between types of supersymmetric black holes and supergravity including greybody factors (frequency dependent corrections). String theorists have derived most of Hawking's (and Bekenstein's) work regarding the entropy of black holes from string theories.

This led to novel ways of thinking about strings. What if "open" strings were really closed ones with one part "hidden" behind a black brane? What if intersecting black branes wrapped around seven curled dimensions gave rise to black holes? The vanishing masses of black branes delineate a cosmological evolutionary tree - from a universe with one topology to another, with another topology. Our world may be the "default" universe on the path of least resistance and minimum energy from one universe to another.

H. FROM SUPERGRAVITY TO MEMBRANES - A RECAP

The particles with half integer spins predicted by supersymmetry are nowhere to be found. Either supersymmetry is a wrong idea or the particles are too heavy (or too something) to be detected by us with our current equipment. The latter (particles too heavy) is possible only if supersymmetry has broken down (which is almost the same as saying that it is wrong). Had it existed, it would probably have encompassed gravity (as does the General Theory of Relativity) in the form of "supergravity". The non-supersymmetric equivalent of supergravity can be gravity as we know it. In terms of particles, supersymmetry in an 11-dimensional universe talks about a supersymmetric gravitino and a spin 2 graviton.

Supersymmetric supergravity was supplanted by 10-dimensional superstring theory because it could not account for handedness in nature (i.e., the preference of left or right in spin direction and in other physical phenomena) and for many quantum effects. From there it was a short - and inevitable - way to membrane theories. Branes with "p" dimensions moved in worldvolumes with p+1 dimensions and wrapped around curled dimensions to produce strings. Strings are, therefore, the equivalents of branes. To be more precise, strongly interacting (10-dimensional) strings are the dual equivalent of weakly interacting five-branes (solitons) (Duff, Scientific American, February 1998). Later, a duality between solitonic and fundamental strings in 6 dimensions (the other 4 curled and the five-brane wrapped around them) was established and then dualities between strings from the 5 string theories. Duff's "duality of dualities" states that the T-duality of a solitonic string is the S-duality of the fundamental string and vice versa. In other words, what appears as the charge of one object can also be construed as the inversion of the length of another (and, hence, the size of the dimension). All these insights - pulled together by Witten - led to M Theory in 11 dimensions. Later on, matrix theories replaced traditional coordinates in space time with non-commutable matrices. In other words, in an effort to rigorously define M Theory (that is, merge quantum physics with gravity), space time itself has been "sacrificed" or "quantum theorized".