A survey on supersymmetry and supergravity

The schemes for unifying the fundamental interactions are among the most exciting programmes and developments in theoretical physics.

Two fundamental problems in this scheme concern gravitation. On the one hand, how is gravitation related to the other fundamental forces, and on the other hand, no workable theory of gravitation is consistent with the principles of quantum mechanics.

The theory of supergravity suggested a novel approach to unification. Supergravity constitutes an extension of general relativity, allowing to make the same predictions for the classical tests. But at microscopic level, supergravity is different:

Probabilities of quantum-mechanical effects of gravitation are infinite but supergravity allows finite answers to be obtained. The fundamental reason is that supergravity brings together several of the most fundamental concepts in modern physics and hence it achieves more than any earlier quantum theory of gravity.

What are then the fundamental new elements present in supergravity?

It is important to remember that physical theories are guided by principles of symmetry. A symmetry can be either global or local form. Theories with local symmetry (or gauge theories) are much more powerful. Important examples are theory of general relativity and electromagnetism.

If a set of physical laws is invariant, the stronger requirement of invariance under local symmetry can only be met by introducing new fields. This then means new forces. The fields are called gauge fields and correspond to new particles whose exchange gives rise to the corresponding forces. A fundamental example is gravitation. Gravity is the gauge field of local Poincare invariance and the gravitational force results from the requirement that Poincare symmetry be local. Electromagnetism can also be derived from the requirement of invariance from the local symmetry associated with the phase of complex fields.

Supergravity is based on supersymmetry. This symmetry proposes that for every ordinary particle there exists a "superpartner" having similar properties - except for a quantity known as spin. There are two kinds of ordinary particles: basic constituents of matter and those that mediate the forces. Constituents of matter are leptons and quarks. Categorised as fermions, they carry a spin equal to half-integer units. Particles that mediate forces are bosons whose spins are integer units (0, 1, 2 and so on). Fermions are "antisocial" and tend to occupy different energy states; bosons are "gregarious" and tend to clump together in the same energy states.

Hence, supersymmetry relates particles with different spins, namely those with the adjacent spins. Any fermion and boson with adjacent spins can be manifestations of a single "superparticle", like an arrow in an auxiliary space. Supersymmetry transformations result in a change in the orientation of a particle.

The stage for supersymmetry is interesting. The standard model of particle physics is quite sucessfull but physicists may have to look beyond it. Some questions remain to be addressed, which suggest more discoveries will come. Why are the masses of the fundamental particles and the strength of the forces acting between them as they are? Supersymmetry may also provide a mechanism by which a single theory can account for two important energy scales: the energies of the electroweak interaction and the Planck mass. In a supersymmetric theory cancellations occur that allow masses to be many orders of magnitude smaller than the Planck mass. If one wants to develop a theory of fundamental particles and interactions, we should not take refuge on an artificial hierarchy.

In supergravity, supersymmetry is extended from the global level to the local level. But repeated supersymmetry transformations move a particle from one point in space to another. This extension leads automatically to incorporate the gravitational force, suggesting the possibility of unifying gravitation with the other forces. A relation between local supersymmetry and the structure of space- time arises, suggesting the presence of the gravitational force. It is thus remarkable, how supersymmetry relates internal symmetries to Poincare invariance, a connection that allows the construction of the new gravitational theory of supergravity.

In more precise terms, supersymmetry theory incorporates as essential elements numbers which are anticommuting numbers (also known as Grassmann numbers). The "superparticle" has an extra arrow in an imaginary auxiliary space: if the arrow is up then it is a fermion, if it points down it is a boson. Supersymmetry works as follows. Denoting boson and fermion fields respectively by b and f, the transformed fields are b' and f', where f' = f + bE and b' = b + fE. The factor E is a measure of the angle of rotation of the superparticle arrow. Notice that b is an ordinary number, but f and E are now anticommuting numbers.

As repeated above, a repeated supersymmetry application moves a particle from one point to another in space-time. Transformation of position are hence obtained by repeating a supersymmetry transformation. Since local Poincare invariance is the symmetry that gives rise to general relativity, a connection between supersymmetry and gravitation can also be expected. The transition from a global to a local symmetry introduces new gauge fields, which in turn give rise to new forces. When global supersymmetry is promoted to local invariance new gauge fields occur. These will be the spin-2 graviton field and a new spin-3/2 field.

Since the product of two supersymmetry rotations is a shift in space-time (a Poincare transformation) the spin-2 graviton is the appropriate gauge particle. Gravitation appears naturally in the theory and hence local supersymmetry is usually called supergravity. The other gauge field for supersymmetry transformations is a fermion 3/2 fiekd called gravitino. The gravitino is massless in the simplest supergravity theories. The gravitino can acquire a mass through spontaneous symmetry-breaking.

Supergravity will then provide corrections to the general theory of relativity at the quantum level. In supergravity there is an additional contribution from the exchange of spin-3/2 gravitinos. Long-range interactions are unchanged, but new effects are predicted at microscopic scale. In fact, a finite probability for loop diagrams in a quantum theory of gravity is obtained by including gravitinos in the interaction: (infinite) contributions of the gravitinos cancel those of the gravitons.

The following comment is also of importace. Supergravity describes general relativity in the language of quantum field theory. But since general relativity describes forces from a geometrical point of view, supergravity can also be explained in geometric terms in an extended space-time. In such space, every point has not only the four usual space and time coordinates but also an additional set of coordinates identified by anticommuting numbers superspace.

It should be said that there are several supergravity theories (and it is in these that the cancelation of infinities really occur). Extended supergravity theories have a characteristic number N of distinct boson-fermion transformations. There is one spin-2 and N spin-3/2 gravitinos. The number of particles with lower spins is also completely determined. However, the important feature is that these theories have a high degree of symmetry.

Each particle is related to particles with adjacent values of spin by supersymmetry transformations and these supersymmetries are of local form. Thus a graviton can be transformed into a gravitino and a gravitino into a spin-1 particle. In addition, each family of particles with the same spin is related by a new global internal symmetry. Combining the supersymmetry and the internal symmetry the entire group of particles is unified.

Like other physical symmetries, extended supergravity can also be viewed in terms of a "superparticle" with an arrow in an auxiliary space of many dimensions. As the arrow rotates, the particle becomes in turn a graviton, a gravitino, a photon, and so on. When this internal symmetries is made local, the local invariance potentially unify gravitation with the strong, the weak and the electromagnetic forces. This local supersymmetry theory is an elegant unification and passed the testof the the infinities. These cancel only in extended supergravity theories because only in those theories can all particles be transformed into gravitons.

In extended supergravity the strength of the gravitational force is determined by one parameter, all the other forces are determined by another parameter. Ideally, in a unified theory all the forces should be described by a single parameter. A third class of supergravity theories are based on a theory of gravitation that allows the scale by which length and time are measured to achieve a unification of gravity with all the other forces, namely by having such single parameter.

Supergravity significatively offers the hope of solving the unification of the fundamental forces and the elimination of infinities in quantum gravity. However, there are nevertheless important problems that remain to be solved. Only when these are subdued can supergravity theories claim sucess.

One problem is that when the internal symmetry in extended supergravity is made a local one, this introduces a cosmological constant term which is too large.

Another conflict is that the particles are massless. A promising approach is to assume some particles acquire a mass through the mechanism of spontaneous symmetry-breaking.

It is also necessary to show that one of these supergavity theories is finite (no infinities) at all approximations.

Finally, the theory of supergravity is most elegant mathematically when it is formulated in 11 dimensions. This number is supported from the facts that supergravity can be formulated in any number of spacetime dimensions up to 11 but seven is the smallest number of hidden dimensions needed to accommodate the three nongravitational forces into a theory. Moreover, supergravity in 11 dimensions is unique.

Theories with more than 4 space-time dimensions are called generalized Kaluza-Klein theories. There is no difficulty in deriving the bosons from a Kaluza-Klein theory. The higher dimesional gravitational field lead to the bosons in a four-dimensional world. So, it is desirable to have a theory in which the number of fermionic fields and also the number of dimensions are given naturally by the structure of the theory.

11-dimesional supergravity is also such a theory, in which bosons and fermions are treated on an equal footing. It also includes a bosonic field which acts as the source that drives the compactification of hidden dimensions. However, this attractive theory also adds more problems to the above mentioned.

First, there is the chirality problem. Chirality is determined by the sense of spin. Every 11 dimensional structure so far predicts an equal number of left and right handed neutrinos However, only the former are observed in nature.

Second, we again have a cosmological problem. The seven extra dimensions must form a small compact structure which induces the four dimensions of spacetime to became highly curved as well. Astronomical observations suggest zero or close to zero space-time curvature. One may add a cosmological constant to solve this but such freedom to adjust the underlying equations is not present in 11 dimensional supergravity.

Thirdly, there is the quantum problem: infinite quantities occur with no obvious physical interpretation. A possiblity to subdue this issue is through superstring theory. These must be instead constructed in 10 dimensional spacetime and in superstring theory infinite quantities are absent.

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