# Unified Theories

# UNIFIED THEORIES

The quest for unification has been a perennial theme of modern physics, although it dates back many millennia. The belief that all physical phenomena can be reduced to simple elements and explained by a small number of natural laws is the central tenet of physics, indeed of all science. One of the first unifying scientific principles was the atomic hypothesis, beautifully expressed by Democritus (Presocratics, Fragment 125) in 400 B.C.E.:

By convention there is color, by convention sweetness, by convention bitterness, but in reality there are atoms and space.

Separate laws do not dictate the nature of gases, liquids, and solids; rather, these are different phases of the same matter that obeys the same laws. Both life and superconductivity are consequences of the electric forces between the same kinds of atoms.

The reductionist program of reducing complex phenomena to simpler physical processes, which in turn can be reduced to even simpler and more encompassing laws of nature, is at the heart of the search for unified theories. In this pursuit, physics has been extremely successful. Physicists believe that the turbulent motion of fluids can be understood by the laws of classical mechanics, which are a good approximation to the quantum mechanical laws that govern the interaction of atoms, that the structure of atoms can be explained by the laws of interaction of nuclei and electrons, that the structure of nuclei can explained by the theory of quarks and gluons, and finally that these are consequences of an even more comprehensive unified theory to be developed.

Each stage in this development has required exploring natural phenomena at shorter and shorter distances, explaining complex macroscopic phenomena in terms of simple microscopic constituents. Ordinary matter is made of atoms, atoms are made of nuclei and electrons, and nuclei are made of quarks. At each stage, the laws of physics are more unified, explaining a larger class of phenomena and reducing to the laws of the previous in appropriate circumstances. Typically, the more unified theories exhibit more symmetry, are more predictive, and are less arbitrary.

Isaac Newton's theory of gravity was the first grand success of this saga, unifying in a precise mathematical framework the laws that governed the motion of apples and planets. It spurred the search for similar unifying theories that would explain ordinary matter.

James Clerk Maxwell's theory unifying electricity and magnetism was the next major step in the quest for unification. Maxwell's theory is the very paradigm of a unified theory. Its new mathematical formalism was based on new concepts (local fields) and exhibited new symmetries of nature (Lorenz or relativistic invariance, as well as local gauge symmetry). It explained the many electric and magnetic phenomena that had been discovered over the years as manifestations of a single entity, the electromagnetic field. It also had many new consequences. The most dramatic of these was the prediction of the existence of electromagnetic waves and the demonstration (by calculating the velocity of the waves) that light was such a wave. Thus, optics was unified with electromagnetism.

Albert Einstein, after the successful formulation of his general theory of relativity, which explained gravity as the consequence of the dynamics of the metric field of space-time, and unifying the structure of the geometry of space-time with gravitation, dreamed of a unified theory of all the forces of nature and of all forms of matter. Since general relativity was a nonlinear theory, he hoped that its solutions could behave as localized lumps of matter and that these might even imitate quantum mechanics. Although it is now believed that Einstein's quest to explain quantum mechanics in a classical field theory was in vain, many believe that the goal of unification is achievable. Indeed, Einstein's belief that "nature is constituted so that it is possible to lay down such strongly determined laws that within these laws only rationally completely determined constants appear, not constants therefore that could be changed without completely destroying the theory" (Einstein 1949, p.63) beautifully expresses the view that a theory containing arbitrary parameters that cannot be calculated from first principles is incomplete and is to be superceded by a more unified and predictive theory.

The development of quantum mechanics in the 1920s and throughout the twentieth century enabled the completion of the atomic program, unifying chemistry and atomic physics. All the properties of ordinary matter, in all of its variety of forms, can be explained in terms of atoms and the electromagnetic forces between them, realizing Democritus's vision. Quantum mechanics also provides for a theory of the structure of atoms in terms of the electromagnetic forces between the atomic nucleus and the electrons orbiting them.

In the latter half of the twentieth century, a comprehensive theory of the constituents of matter and of the forces of nature was completed—the Standard Model of elementary particle physics. This quantum theory of fields identifies the basic constituents of matter. They are the quarks that make up the nuclei at the center of atoms and the leptons (such as the electron) that revolve about the nuclei. The Standard Model also explains the forces that act on the elementary particles (the electromagnetic, the weak, and the strong or nuclear forces) as consequences of local gauge symmetries.

An essential part of the Standard Model is the unification of the electromagnetic and weak forces in a combined electroweak theory. Much as electricity and magnetism were seen as different phenomena before Maxwell's theory, the electromagnetic and weak forces originally were seen as very different in nature. The electromagnetic forces are long-range and the quanta of the electromagnetic fields (photons) are massless particles. The weak forces are short-ranged and their quanta (the *W* and *Z* bosons) are massive. In the Standard Model, both are consequences of a unified gauge symmetry, and the differences between them are a consequence of the fact that this symmetry is spontaneously broken. Indeed, the theory predicts that if one were to heat the universe to very high temperatures (a circumstance that did occur at very early cosmological times), the symmetry would be restored, and all apparent differences between the forces would disappear.

The Standard Model provides many hints that further unification is required. First, there is remarkable similarity, at the fundamental level, between the various forces and particles. The electroweak and strong forces are both consequences of local gauge invariance and differ only in the specific group responsible for each (SU(2) × U(1) and SU(3), respectively). The fundamental particles, the leptons and the quarks, are very similar in their properties. Indeed, it is quite easy and natural to unify these theories in a way whereby the quarks and leptons fit naturally into larger patterns of symmetry.

At first sight it, might appear that the disparity between the strength of the electroweak force (which is rather weakly coupled at low energies) and the strength of the strong nuclear force (that is strongly coupled at low energies) would argue against unification. However, the strength of these forces varies with energy due to the dynamical properties of the fluctuating quantum mechanical vacuum. The strong force decreases at a short distance or for high-energy processes (asymptotic freedom), whereas the electric and weak forces grow stronger. Precise measurements and theoretical calculations enable us to extrapolate these forces to very high energies, where they coincide in strength at the very-high-energy scale of 10^{18} GeV. (This agreement is greatly improved if one also assumes that a new symmetry, supersymmetry, and many new associated particles are present at energies of approximately 10^{4} GeV.)

This extrapolation provides a compelling hint that unification of the forces of nature might occur at energies of ∼10^{18} GeV. Furthermore, since the force of gravity (which is an extraordinarily weak force at low energy) becomes equally strong at this energy, it is suggested that the next stage of unification should include gravity.

Finally, the many unanswered questions raised by the Standard Model, and the many parameters that are incalculable within the Standard Model, suggest that further unification within a more symmetric and predictive theory is necessary.

Currently, the best hope for a unified theory is based on string theory. String theory is largely based on the notion that the elementary constituents of matter are extended one-dimensional objects, not the pointlike particles of the Standard Model. This is a unifying concept. Since a string has infinitely many shapes, it can describe in one entity many elementary particles. Indeed, each vibrational mode of a string will behave as an elementary pointlike particle. One of the most alluring features of string theory is that when certain special solutions of the theory are analyzed, they contain precisely the spectrum of elementary particles found in nature—the quarks and leptons of the Standard Model, as well as the quanta of the gauge theories that provide for all of the observed forces.

String theories are inherently theories of gravity. Unlike the situation in ordinary quantum field theory, one does not have the option in string theory of turning off gravity. The gravitational, or closed string, sector of the theory must always be present for consistency, even if one starts by considering only open strings, since these can join at their ends to form closed strings. One of string theory's greatest successes is that it is a mathematically consistent quantum theory of gravity, free of the infinities that plagued field theoretic attempts to quantize gravity. Thus, string theory appears to provide the framework to unify all the forces of nature including gravity into a single, tightly woven pattern.

String theories, as is appropriate for unified theories of physics, are incredibly unique. In principle, they contain no freely adjustable parameters, and all physical quantities should be calculable in terms of the fundamental dimensional units of nature: the velocity of light *c,* Planck's constant of action, and Newton's constant of gravity. However physicists' understanding of the structure of string theory is still quite primitive, and thus in practice, they are not yet in the position to exploit such enormous predictive power.

Will the quest for unification ever end? Will a theory of everything, a theory that unifies all the phenomena of physics once and for all, a final theory, ever be formulated? There is no known reason why this is impossible. Experience teaches that each stage of unification leaves many questions unanswered and reveals new mysteries that only find their explanation at the next stage of unification. But, this might end.

Time will tell.

*See also:*Grand Unification; Particle Physics, Elementary; Planck Scale; String Theory

## Bibliography

Einstein, A., and Infeld, L. *The Evolution of Physics* (Simon and Schuster, New York, 1938).

Einstein, A., and Schilpp, P. A., ed. *Albert Einstein: Philosopher-Scientist* (Harper & Brothers, New York, 1949).

Greene, B. *The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory* (W. W. Norton, New York, 1999).

Weinberg, S. *Dreams of a Final Theory* (Pantheon Books, New York, 1992).

*David Gross*

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