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Superstring Theory§

Contents

Classical Theory of String
Quantization
Supersymmetry and Superstring
Compactification
Types of Superstring Theory
M-theory
Problems and Future Development

Classical Theory of String

The first task to construct the mathematical equations was to begin with a string of the right length and tension. Since the theory was supposed to account for, among other things, the quantized force of gravity, the strings could not be longer than the length scale where the granular structure of space-time becomes important, i.e., 10-33 cm - the Planck length. This is the same length scale associated with the gravitational constant when it is expressed in units of = 1 and c = 1. The tension of the string can be estimated from the strength of the force transmitted by the graviton. It is inversely proportional to the strength of the gravitational constant by the formula (c5/G)1/2 = 1.22x1019 Gev (the point particle approximation corresponds to higher tension in the string). However, it is thought that, this colossal energy may be cancelled largely by the vacuum energy. The net result becomes the observed mass of the elementary particles (it is the vibrational pattern that determine the type of particle in the theory of string). The massless graviton can be constructed simply as the lowest string state. But it is not that easy to arrive at the correct mass for massive elementary particles. Moreover, the vacuum energy estimated by the string theory (and quantum field theory for that matter) is too large in comparison with the observed value derived from cosmic acceleration (the cosmological constant). For particle physics, the failure to derive the correct vacuum energy is unsatisfying, but it is fatal for the string theory, which is purported to be a "Theory Of Everything". The superstring theory contains another fault (or virtue depending on point of view) - it gives rise to an infinite number of different configurations (universes) that don't have the same vacuum energy. According to the "anthropic principle" (which simply states: "it is there because we are here"), we are living in just the right one with exceptionally low vacuum energy out of the many others in the multiverse.

The string is an one-dimensional object, which can move in various ways. Its movement sweeps out a two-dimensional sheet, called the world-sheet (see Figure 01a) similar to the 4D space-time world-line. In the diagram, X represents a vector defined in D-dimensional space-time that begins at the origin of a coordinate system and ends at some point along the two-dimensional world-sheet with components X(,), where = 0, 1, 2, ... D-1, the 0th index denotes the time component; the rest are treated as spatial components, is a parameter similar to space, and is a parameter like time. All the real action takes place on the (,) surface - the inner spacetime (worldsheet). The processes are then translated into events occurring in the ordinary spacetime X - the outer spacetime. The string
String, World Sheet degrees of freedom X(,) trace out a curve as varies at fixed . The curve may be open or closed and varies within a range between 0 and as the string is traced out from one end to the other for an open string, or 0 to 2 round the circle for a closed string. The string sweeps out a worldsheet as varies from one instant to another. However, these two parameters have nothing whatsoever to do with the real space and real time. Since nobody has ever seen or detected this artificial space-time, they have to be hidden from the observables in a theory, and thus the requirement of reparametrization invariance. It turns out that such theory can be formulated only with the one-dimensional string within the framework of artificial space-time.

Figure 01a String, World Sheet [view large image]

Construction of the superstring theory starts with the "Action Principle", which confines the movement of the string such that the world sheet sweeps out a minimum area (similar to the shorest distance for the case of point particle). If we demand the action S to be Lorentz covariant in form with conformal and reparametrization invariant, then it can be written as (with = 1, c = 1):

The variations associated with the Poincare (generalized Lorentz transformation), reparametrization (change of coordinates), and conformal invariances (change of artificial space-time) are:

    
where , = (0, 1) = (, ), are coordinates on the world-sheet. The requirement of the invariances largely determines the form of the action. Reparametrization invariance is to ensure that physics doesn't change with relabelling of these coordinates.

Conformal Transformation The Lorentz transformation in special relativity has been generalized to D-dimensions. The requirement of Lorentz covariant is to make sure that the superstring theory behaves the same in all inertial frames. Just as in gauge field theory or general relativity, there are fewer independent dynamical degrees of freedom than appear explicitly in the action. The degrees of freedom in the formulation can be reduced via the requirement of conformal invariance under the conformal transformation (Figure 01b), which rescale the world sheet metric. Such change in topology makes it feasible to evaluate string diagrams. Among other things this makes it possible to compactify the world sheet, closing off the holes corresponding to incoming and outgoing strings. For example, a world sheet with one incoming and one outgoing string (as in (a) of Figure 01b) can be conformally mapped to the plane of (a') with the incoming string appearing at the origin and the outgoing string at infinity (not shown) or to the sphere of (a") with the incoming and outgoing strings appearing at the south and north poles. The external string states in (b) of Figure 01b with four awkward legs are projected to points as indicated in (b'). By a suitable choice of gauge (known as covariant or conformal gauge with in the following case - a gauge is a mathematical device to fix the redundant degrees of freedom), the string equations of motion can be derived by varying X to minimize the action S, thus we obtain:

Figure 01b Conformal Transformations

---------- (2)
which is an one-dimensional wave equation for X. The number of such equation is equal to "D" - the dimension of the coordinate system. The degree of freedom X is independent of each others.

Then by varying the world sheet metric to minimize the action, more equations can be derived in the form:
---------- (3)
where the indices 0 and 1 are used to refer to and respectively. may be interpreted as the energy-momentum tensor for a two-dimensional field theory of "D" free scalar field X.

For a closed string, the general solution of Eq.(2) consistent with the boundary conditions X(+,) = X(,) is:

---------- (4)
where l = (T)-1/2 (sometimes it is taken to be 1), and n can be a positive or negative integer except zero. The solution has been separated into the sum of a "right-mover" part (the first term) and a "left-mover" part (the second term) as they move in opposite directions. Each part consists of two terms corresponding to the motion of the center-of-mass, and a sum of oscillators with associated coefficients designated as n and n. It is the oscillating "wave" that enables the string to imitate the various kind of particles through quantization. This is the simplest structure that can be endowed to a particle in a mathematical theory beyond a structural-less point.

The solution for an open string with the boundary condition
is:
---------- (5)
In the open string case, the left- and right-mover oscillator terms are not independent, having been linked by the boundary condition, and a separation into left and right movers is not particularly useful.

The two examples in the followings serve to visualize the geometry of the closed and open strings. Let us first consider a closed string momentarily at rest in the (X1, X2) plane. According to Eq.(4), the time component and n=1 mode can be expressed as:
Closed String Vibrating String X0 = (l2p0)
X1 = R cos(2)
X2 = R sin(2)
where p0 = iE, and R ~ Rcos(2) near ~ 0 is the radius of the cylinder. The energy-momentum tensor defined in Eq.(3) yields the relation E = 2RT confirming that T is indeed the energy per unit length of the string. Figure 02a shows the world sheet swept out by this closed string for a small interval of in the form of cylindrical surface. In general, the closed string will vibrate in the three dimensional space (X1, X2, X3) in various harmonic modes (see Figure 02b, by Steuard Jensen, Alma College).

Figure 02a Closed String
[view large image]

Figure 02b Vibrating String [view large image]

The solution for open string is given by Eq.(5). The rotating open string in the (X1, X2) plane is expressed by:
Open String X0 = (l2p0)
X1 = (l) cos()cos()
X2 = (l) cos()sin()

The energy-momentum tensor defined in Eq.(3) yields the relation E2 = (/l)2 for this case. As the total angular momentum is given by the formula:
,
it follows that J = 2/2 for this case of spinning string. Thus J / E2 = 1/2T, which can be

Figure 02c Open String
[view large image]

 identified as the Regge slope in Figure 03b. Furthermore, since the velocity of the string V can be computed from the formula:
V = {|X1/X0|2 + |X2/X0|2}1/2 = |cos()| ,
it means that the endpoints (= 0, and ) move at the speed of light (recalling that c = 1 in the unit using here). Figure 02c shows the rotating open string, and the world sheet it sweeps out in the form of helical surface.

This is the original string theory developed in the late 1960s. It is called the bosonic string theory. The name bosonic indicates that all of the vibrational patterns of the bosonic string have spins that are a whole number - there are no fermionic patterns. This led to two problems. Firstly, the theory is incomplete because it misses the world of fermions. Secondly, it was realized that there was one un-physical pattern of vibration in bosonic string theory whose mass-squared was negative - a so-called tachyon (elementary particle that moves faster than light - on the other side of the speed of light barrier). As it will be discussed later, both problems are resolved by introducing super-symmetry into the string theory.

§The official string theory website is at http://www.superstringtheory.com. It is at the level approachable by general audience.

This is an example for the mathematics of conformal transformation on the world sheet. As shown in Figure 01b diagram a, the original world sheet is a single incoming closed string and a single outgoing closed string. In cylindrical coordinates with
- < z < , and 0 < 2 the metric is in the form:
ds2 = dz2 + R2d2,
where R is the radius of the cylinder. If the coordinate z is changed to:
z = 2R ln(tan /2) with 0 < < ,
then ds2 = R2[(sin)-2 d2 + d2].
The conformal invariance allows the rescaling of the metric by a factor:
= sin2.
The metric is now in the form:
ds2 = R2(d2 + sin2 d2),
which is the metric for a 2-sphere of radius R. The initial and final string states at z = -, + correspond to = 0, , i.e. the north and south poles of the 2-sphere, as shown in Figure 01b diagram a".

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