Interpretations of Quantum Mechanics
And Superstring Theories

By: Dr. Sam Vaknin

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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.

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