Superstring Theory

Problems and Future Development

• Mass Scale - As mentioned in the very beginning, the tensions of the string is estimated to be an immense 10³⁹ tons, which corresponds to a mass of 10¹⁹ Gev. Thus, only the massless state particle in the string theory is consistent with the real world. In fact, the observed mass of all the elementary particles is negligible in comparison to the Planck mass of 10¹⁹ Gev. It is thought that at the very high temperatures during the creation of the universe, it was totally symmetric with all particles in the lowest string state having zero mass. It is though symmetry breaking that the particles acquire their masses. Working out the fine details of all this is a current problem in phenomenology, which builds models to fit the observational data. At the moment, such details and the precise values of the particle masses cannot be obtained from the superstring theory. What is needed is some new insight, some deeper principle that would explain the mechanisms of symmetry breaking and allows particle masses to be calculated accurately. Attempt has been made to compute the low energy mass by cancellation with the negative energy from vacuum fluctuation. A noticeable

  success is the prefect cancellation for graviton. Cancellation to such a high level of precision is generally beyond theoretical capability at present. Figure 12 shows the large gap between the Planck mass and the mass of known particles. There is "nothing" in this enormous region labelled "energy desert". Note that the mass/energy is referred to binding energy for some composite systems such as molecules, atoms, and nuclei.

  Figure 12 Mass/Energy Scale [view large image]

  
  
• Fractional Charge - In QCD, the quarks are required to have 1/3 or 2/3 of the electric charge of the electron by the symmetry of the theory. In strings, the same result is achieved by the way strings wrap around the extra dimensions. Other wrappings would produce particles with 1/5, 1/7, or 1/11 of the electron's charge. Searchs for fractionally charged particles have been unsuccessful so far.



Large Dimensions - Since the Planck length is the natural scale of strings, the most probable universe would be the one in which all of the dimensions are comparable to the Planck length. Why are three spatial dimensions so large in our universe? Why not four, five, six, ...? A novel explanation suggests that if a dimension has a small circumference, strings can actually wrap around the circle as shown in Figure 13. Like rubber bands around a rolled-up newspaper, these wrapped strings tend to keep the universe from expanding in that dimension. When strings collide, though, they can unwrap and the dimension expand rapidly, like the inflationary model. String theorists discovered that these collisions were likely to happen if (at most) three spatial dimensions were involved. The sudden expansion of the three spatial dimensions is interpreted as the moment of the Big Bang. There are other string-base scenarios on the origin of the universe. A better understanding of the structure of string theory must be developed before these issues can be resolved.

Figure 13 Large Dimensions [view large image]



• Calabi-Yau Manifolds - As mentioned earlier there is a lot of difficulties to figure out the Calabi-Yau shape that agrees with the observed physical properties. A sensible start is to focus only on those Calabi-Yau shapes that yield three families. This cuts down the list of viable choices considerably, although many still remain. There are a few entries in the Calabi-Yau catalog that are closely akin to the particles of the standard model. If many of the Calabi-Yau shapes were in rough agreement with experiment, the link between a specific choice and the physics we observe would be less compelling. On the other hand, if none of the Calabi-Yau shapes came even remotely close to yielding observed physical properties, it would seem that string theory could have no relevance for our universe. Then finding a small number of Calabi-Yau shapes appear to be an extremely encouraging outcome. Because of such impasse of selecting an unique CalabiYau manifold that is fine tuned to our world, some string theorists now turn to the last resort - the "Anthropic Principle". It suggests that we just happen to live in one of the 10⁵⁰⁰ manifolds, which permits life to evolve to its present form. They insist that this is a matter of probability; it is not a "cope out" (see Manifold, Vacuum Energy, and Multiverse).


• Light Cone Gauge - Superstrings were created to be space-time covariant as manifested by the form of its "action". However, the actual calculations are always made using a particular frame of reference called the light cone gauge, which destroys the covariance. The light cone gauge could be thought of as a frame of reference that moves through space-time at the speed of light. It violates the full relativistic covariance that is inherent in superstring theory and decrees that the details of the theory should not really depend on the particular choice of gauge. By obscuring this underlying symmetry, physicists are really working with the shadow of a much deeper theory. It also turns out that using this single gauge is intimately connected to perturbation theory along with all its limitations. One of the important tasks is to rewrite string theory so that it is free from any particular choice of gauge or frame of reference. In this way, it may be possible to penetrate much deeper into the theory and resolve some of its present difficulties.


• Non-perturbation Theory - Interactions in superstring theory are expressed in terms of the splitting and joining of strings (see Figure 14 and 15). The approach used to calculate all the quantities of interest in the theory remains old-fashioned perturbation theory, which assumes a reasonable starting point and then tries to home in on the correct result by adding an infinite series of corrections. In quantum electrodynamics, these perturbation corrections were represented by Feynman diagrams, and, on summing up infinite numbers of terms, results of surprising accuracy were achieved. On the other hand, when it came to gravity, the perturbation series failed totally to represent a curved space-time by adding a finite number of corrections to an initially flat space-time. Perturbation theory is often plagued with infinity, and infinite sum (when the coupling constant is not small). Superstrings are supposed to be a theory about gravity and matter. Yet physicists continue to treat them as if they exist in a flat background space-time, a picture they hope will be corrected by using a perturbation series. Clearly this whole approach is inadequate and obscures the power of the superstrings themselves. M-theory and

  		duality were introduced to overcome the difficulty. Although this approach have yet to give us definitive information about four-dimensional vacua, they have already clarified much of the nonperturbative nature of string theory in 10, 8, and even 6 dimensions, giving us a complex web of dualities between different string compactifications.
  Figure 14 String Inter- actions [view large image]	Figure 15 Sum of Interactions [view large image]

  
  
• Superstring Field Theory - Only the first quantization has been applied to the superstring theory up to now. Classical strings are created according to an action principle that is made manifestly covariant in order to produce minimal surfaces in space-time. The various modes of vibration of these minimal surfaces are then quantized to produce string wave function. Following the development of quantum theory, the next step would be to generate, out of the wave functions for individual string vibrations, a super wave function for the string field. It can be shown that the equation of motion for the superstring wave function is:
  
  where the field functional is defined as:
  
  It has been discretize into a series of points along the string. At the limit N , it becomes a continuous function of X . Thus while in the first quantization the basic object is X_i, which represents just one possible configuration of the string; the string functional is simultaneously a fucntion of every point along the string. Now the Hamiltonian H takes the form:
  
  where and P_i = are the canonical momenta conjugate to Xⁱ.
  Although superstrings are formulated in a ten-dimensional space-time, in another sense a string field is also a theory about two-dimensional surfaces. A string field theory is therefore about the quantum properties of two-dimensional surfaces. Considerable research is now being directed toward this approach. It has been pointed out that the program can proceed in two directions. On one hand, physicists can study the topological properties of these surfaces and in this way produce powerful insights. But it is also possible to consider the geometry of these surfaces in term of algebra. It turns out that physicists had already for some time been looking at these branches of algebra (the sorts of algebras first discovered in the 19th century). Their idea was to discover ways of probing deeper into the quantum theory and freeing it from its attachment to an underlying space-time. String field theory is currently too difficult to solve with the non-perturbative approach.



• Space-time Background Dependence - Superstrings have resolved many of the problems that faced earlier attempts to explain the elementary particles; they are free from infinities and associated with just the right symmetry groups. Yet all these theories are fundamentally flawed because they still regard strings as moving in a fixed, background space-time. Such an approach just has to be wrong; the superstrings in a proper string theory have to interact with the space in which they move and are inseparable from it. We need a theory to describe an universe, which evolves from a more primary state to the fabric of space, time, and by association, dimensions. The graviton in string theory does suggest an idea. It shows that a gravitational field is composed of an enormous number of gravitons all executing some vibrational pattern. Gravitational fields, in turn, are encoded in the warping of the spacetime fabric, and hence we are led to identify the fabric of spacetime itself with a colossal number of strings all undergoing the same, orderly, graviton pattern of vibration. Another approach is to replace ordinary geometry by something known as noncommutative geometry. In this geometrical framework, the conventional notions of space and of distance between points melt away, leaving us in a vastly different conceptual landscape. Nevertheless, as we focus our attention on scales larger than the Planck length, physicists have shown that our conventional notion of space does re-emerge. It seems that string theories work very well above the size of Planck length. The more fundamental theory is buried below this size.


• Testing - The Superstring Theory is often criticized for the lack of testable predictions. By 2007 string theorists have come up with at least 4 ways to put their models to test (Figure 16, - evidence for theory, - against theory):

1. It is found that inflation can stretch the tiny string to astronomical size (length-wise) and very high density (see cosmic string). It would act as gravitational lens to produce two identical images of a light source from behind.
2. If the same cosmic string is located in between a pulsar and Earth, then it would disturb the otherwise precise timing of the pulses arriving here.

3. It is suggested that string theory has already been tested by the RHIC, which produced liquid quark-gluon plasma as predicted by the string theory. According to the holographic conjecture, such plasma in three dimension should correspond exactly to a black hole in the higher dimensional space. It has been shown that the property of the liquid plasma is described by the same equations as a higher dimensional black hole. 4. Inflation generated ripples through space-time with imprint of these primordial gravitational waves in the CMBR (Cosmic Microwave Background Radiation). String theory sets a limit on the strength of such waves. If it is too energetic, the six curled-up dimensions posited by string theory would have unfurled and grown just as large as the three we see around us. This is a test that can falsify the string

Figure 16 Tests for String Theory [view large image]

theory if strong gravitational waves are detected in CMBR. On the other hand, the lack of CMBR gravitational wave can also falsify the theory of inflation.


In addition, the LHC (Large Hadron Collider, to be operational in April, 2008) will provide evidence or the lack of it for higher dimensions (in string theory) and superpartners (in the theory of supersymmetry).

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