# Kaluza, Klein, and the Grand Unified Theory

1. Nov 4, 2011

### PhysicsPost

Extra dimensions have been a mainstay of science fiction for years. Many authors have explored the practical, logical, and comical implications of a universe with more dimensions than we are used to, but this use of artistic license has led to some strange ideas of what makes a dimension. It seems that the common science fiction definition involves something akin a parallel universe, a world similar to the one in which we live, but different in minute or sometimes antithetical ways. Although this type of extra dimension can be an entertaining plot device, it has no bearing on real life. Thankfully modern physics will, once again, save us from the mundane world we perceive. Extra dimensions are a hot topic on the forefront of theoretical physics, and their mathematical implications can be quite stunning.

First, we should discuss just what the term “dimension” means. The scientific definition is much simpler than the fictional concepts that we are used to. A dimension is simply an extent or mode in which something can move. Our universe appears to be four-dimensional. The three spatial dimensions, usually referred to as length, height, and depth, and denoted by the Cartesian coordinates x, y, and z, are the spatial dimensions that we are used to. We can move whichever direction we want in the spatial dimensions. The fourth dimension is time. Many people find it difficult to imagine time as a dimension because we don’t have the same freedom of movement that we do with the spatial dimensions, but that may be a consequence of the way we perceive the universe, rather than the universe itself. Time is a dimension because it is a necessary part of any coordinate system we use to describe the location of an object. If you want to know the location of a planet, not only do you have to know its position in space (its x, y, and z Cartesian coordinates) you also have to know when it will be there (its time coordinate). Special and General Relativity established a direct connection between motion in the three spatial dimensions and time. Four pieces of information are needed to define the position of an object, thus a four dimensional universe.

In 1919 a relatively unknown Polish mathematician named Theodor Kaluza challenged the obvious. He attempted to unify the General Theory of Relativity with Maxwell’s Electromagnetic Field Theory, through a very unique method. Kaluza discovered that when he introduced a fifth dimension into his calculations, he was able to describe both gravity and electromagnetism from the same underlying framework. This was an extremely important step, did not catch on very quickly. Many people simply shrugged off the idea of a fifth dimension because it was “obviously flawed,” after all we can’t see a fifth dimension. Kaluza originally intended the fifth dimension as a mathematical trick to unify the theories rather than a real physical occurrence. Although the fifth dimension was a powerful unification tool, it was eclipsed by the newly developing field of Quantum Mechanics and fell into the shadows of theoretical physics for several years.

Oskar Klein, a Swedish mathematician, proposed a possible solution to the obvious flaw in Kaluza’s theory. Klein suggested that an extra dimension could exist in our universe, provided that it was curled up into a small enough space to escape ordinary detection. The concept of a curled up dimension sounds a bit odd at first, but it’s actually fairly familiar. Picture an ordinary drinking straw. If you view the straw from far away, it looks like a line. All you can see is the length of the straw, it appears to have no width at all. When you get closer to the straw, you can see two distinct dimensions-- the extended dimension of the straw’s length, and the smaller circular dimension of its width. This is an example of a curled dimension. From a distant vantage point, the straw appears to be a one-dimensional line, but up close it appears to be a two-dimensional cylinder. This is not a perfect analogy for our universe, but it is a good place to start. The major importance of Klein’s postulate is that it is now possible for Kaluza’s theory to have real, physical meaning. Kaluza’s fifth dimension could simply be curled up so small that we had never noticed it. Theories involving extra, curled-up dimensions have come to be known as Kaluza-Klein theories. Once again, major developments in Quantum Mechanics pushed these theories into the background.

Unified theories are a bit of a holy grail in physics. A true unified force theory would be able to explain all four of the known fundamental forces (gravity, electromagnetic, weak nuclear, and strong nuclear) under the same mathematical framework. Symmetry and unification are very alluring concepts for scientists. Physicists want to find that all forces are different manifestations of the same thing, and through the last century science has discovered several strong indicators of such an underlying unity. In 1855, James Clerk Maxwell found that electricity and magnetism are not merely related to each other, but are fundamentally the same. This unified force is known as electromagnetism. Maxwell found that electromagnetic radiation traveled in waves, with the electric and magnetic waves lying on planes perpendicular to each other. Years later Sheldon Glashow, Steven Weinberg, and Abdus Salam were awarded the Nobel Prize for discovering a deep connection between the weak nuclear force and electromagnetism. Salam and Glashow were able to show that at high enough energies, electromagnetism and weak nuclear force would merge into what is known as the electroweak force. There have been great advances in the quest to unify the electroweak force with strong nuclear force. Already it seems that there is a connection between the two, but no strong mathematical framework is in place yet. The combination of the electroweak and strong nuclear forces is known as grand unification. Although a grand unified theory seems plausible, it is still beyond the current reaches of science.

Amidst all these developments in unified field theory, one thing remained a mystery—gravity. Newton was the first to describe the effects of gravity. He developed the first mathematical relation of mass to the force of gravity, but was unable to describe what gravity was. Einstein was the first to describe the nature of gravity, and in doing so completely reshaped our ideas about space and time. General Relativity is Einstein’s theory of gravity. It relates the effects of gravity to a warping of spacetime, the multidimensional fabric of the universe. The standard analogy is that of a bowling ball placed on a rubber membrane. The membrane deforms the most at the center of the mass, and the deformity becomes less pronounced as one moves away from the ball. With this analogy, it is easy to see how an object’s gravitational field can influence other objects. They simply follow the shortest path on a warped background. The flaw in this analogy is that the warping of the rubber sheet is caused by earth’s gravity, whereas the warping of spacetime is gravity. The details of General Relativity are unnecessary at this point, but suffice to say it has stood the test of experiment time and time again.

The problem with unification may lie in our ideas of what the forces are and how they interact with matter. Relativity tells us gravity is a warping of spacetime, but the other three forces are different. Electromagnetism, strong nuclear, and weak nuclear forces are typically referred to as fields. A field is simply an area in which a specific type of particle will experience a force. For example, in an electromagnetic field, any charged particle such as an electron will experience a force in accordance with the strength of the field and the particle’s orientation relative to the field. You can picture a field as a region of space that is populated with a sea of messenger particles. These messenger particles interact with matter and tell it how to behave. The specific type of particles involved depends on the force in question. Photons are the messenger particles of the electromagnetic force. The weak nuclear force is mediated by W+, W-, and Z bosons, and it acts on atomic nuclei. Strong nuclear force is mediated by eight varieties of gluons that act on quarks and nucleons.

Many physicists, including Einstein, believed that there should be a link between electromagnetism and gravity. They are the only two forces with an infinite range, and they both obey the inverse square law. This means that their strength is inversely proportional to the square of the distance between the bodies being acted upon. Gravity and electromagnetism have the same fundamental mathematical format, so it is only natural to think they might be related. Oddly enough, this has been one of the most difficult steps to take. Gravity and electromagnetism may share the same basic mathematical formulation, but they behave quite differently from each other. By the early 1980s, progress in the standard model of Quantum Mechanics had started to slow. Physicists began to look new ways to solve old problems, and Kaluza’s idea of a fifth dimension was resurrected.
Just as gravity is warping of spacetime in four dimensions, electromagnetism can be described as warping occurring in the fifth dimension. Gravity in the fifth dimension behaves just like electromagnetism in the standard four dimensions. From here, the natural progression is to add more dimensions and see what happens. At this point, the precise shape of the curled dimension starts to come into play, and herein lies the problem. Nobody knows just what shape it should be. The possibilities have been narrowed down to a type of shape known as a Calabai-Yau space, but that still leaves a huge (possibly infinite) number of specific shapes. The mathematics required to determine the exact shape are just too complex to be solved right now. Although the exact shape is not known, it is possible to make rough estimates of the effects of the extra dimensions. It seems that if there is seven curled dimensions, all four fundamental forces could be accounted for. In the coming few years, we will probably see great advances in the physics of curled dimensions. When the topology of these hidden dimensions is better understood, their true significance will become more clear.

There are currently three main theories that employ curled dimensions, Superstring Theory, Eleven-Dimensional Supergravity, and the most promising of them all, M-Theory. As it stands now, Superstring Theory and Eleven-Dimensional Supergravity appear to be separate parts of M-Theory. When the math behind M-Theory can be worked out further, it will probably add another layer of unification to modern physics. All three theories employ a total of eleven dimensions; three extended, one curled, and one time dimension, to explain the four fundamental forces and the three families of elementary particles. The current theories are very different from the original Kaluza-Klein theory, but they could never exist without the willingness of two scientists to challenge not only the general opinion at the time, but the way our universe appears to us.

In lieu of a final paragraph in which all is revealed and resolved, I am forced to leave you with a bit of uncertainty, which I suppose is strangely appropriate for an article about theoretical physics.