Hyperspace (58 page)

6
. Peter van Nieuwenhuizen, “Supergravity,” in
Supersymmetry and Supergravity
, ed. M. Jacob (Amsterdam: North Holland, 1986), 794.

7.
Quoted in Crease and Mann,
Second Creation
, 419.

1
. Quoted in K. C. Cole, “A Theory of Everything,”
New York Times Magazine
, 18 October 1987, 20.

2
. John Horgan, “The Pied Piper of Superstrings,”
Scientific American
, November 1991, 42, 44.

3.
Quoted in Cole, “Theory of Everything,” 25.

4
. Edward Witten, Interview, in
Superstrings: A Theory of Everything?
ed. Paul Davies and J. Brown (Cambridge: Cambridge University Press, 1988), 90–91.

5.
David Gross, Interview, in
Superstrings
, ed. Davies and Brown, 150.

6.
Witten, Interview, in
Superstrings
, ed. Davies and Brown, 95.

Witten stresses that Einstein was led to postulate the general theory of relativity starting from a physical principle, the equivalence principle (that the gravitational mass and inertial mass of an object are the same, so that all bodies, no matter how large, fall at the same rate on the earth). However, the counterpart of the equivalence principle for string theory has not yet been found.

As Witten points out, “It’s been clear that string theory does, in fact, give a logically consistent framework, encompassing both gravity and quantum mechanics. At the same time, the conceptual framework in which this should be properly understood, analogous to the principle of equivalence that Einstein found in his theory of gravity, hasn’t yet emerged” (ibid., 97).

This is why, at present, Witten is formulating what are called
topological field theories
—that is, theories that are totally independent of the way we measure distances. The hope is that these topological field theories may correspond to some “unbroken phase of string theory”—that is, string theory beyond the Planck length.

7.
Gross, Interview, in
Superstrings
, ed. Davies and Brown, 150.

8.
Horgan, “Pied Piper of Superstrings,” 42.

9
. Let us examine compactification in terms of the full heterotic string, which has two kinds of vibrations: one vibrating in the full 26-dimensional space-time, and the other in the usual ten-dimensional space time. Since 26 − 10 = 16, we now assume that 16 of the 26 dimensions have curled up—that is, “compactified”
into some manifold—leaving us with a ten-dimensional theory. Anyone walking along any of these 16 directions will wind up precisely at the same spot.

It was Peter Freund who suggested that the symmetry group of this 16-dimensional compactified space was the group E(8) X E(8). A quick check shows that this symmetry is vastly larger and includes the symmetry group of the Standard Model, given by SU(3) X SU(2) X U(l).

In summary, the key relation is 26 − 10 = 16, which means that if we compactify 16 of the original 26 dimensions of the heterotic string, we are left with a 16-dimensional compact space with a leftover symmetry called E(8) X E(8). However, in Kaluza-Klein theory, when a particle is forced to live on a compactified space, it must necessarily inherit the symmetry of that space. This means that the vibrations of the string must rearrange themselves according to the symmetry group E(8) X E(8).

As a result, we can conclude that group theory reveals to us that this group is much larger than the symmetry group appearing in the Standard Model, and can thus include the Standard Model as a small subset of the ten-dimensional theory.

10
. Although the supergravity theory is defined in 11 dimensions, the theory is still too small to accommodate all particle interactions. The largest symmetry group for supergravity is 0(8), which is too small to accommodate the Standard Model’s symmetries.

At first, it appears that the 11-dimensional supergravity has more dimensions, and hence more symmetry, than the ten-dimensional superstring. This is an illusion because the heterotic string begins by compactifying 26-dimensional space down to ten-dimensional space, leaving us with 16 compactified dimensions, which yields the group E(8) X E(8). This is more than enough to accommodate the Standard Model.

11.
Witten, Interview, in
Superstrings
, ed. Davies and Brown, 102.

12
. Note that other alternative nonperturbative approaches to string theory have been proposed, but they are not as advanced as string field theory. The most ambitious is “universal moduli space,” which tries to analyze the properties of string surfaces with an infinite number of holes in them. (Unfortunately, no one knows how to calculate with this kind of surface.) Another is the renormalization group method, which can so far reproduce only surfaces without any holes (tree-type diagrams). There is also the matrix models, which so far can be defined only in two dimensions or less.

13
. To understand this mysterious factor of two, consider a light beam that has two physical modes of vibration. Polarized light can vibrate, say, either horizontally or vertically. However, a relativistic Maxwell field
A
_μ
has four components, where μ= 1,2,3,4. We are allowed to subtract two of these four components using the gauge symmetry of Maxwell’s equations. Since 4 − 2 = 2, the original four Maxwell fields have been reduced by two. Similarly, a relativistic string vibrates in 26 dimensions. However, two of these vibratory modes can be
removed when we break the symmetry of the string, leaving us with 24 vibratory modes, which are the ones that appear in the Ramanujan function.

14
. Quoted in Godfrey H. Hardy,
Ramanujan
(Cambridge: Cambridge University Press, 1940), 3.

15
. Quoted in James Newman,
The World of Mathematics
(Redmond, Wash.: Tempus Books, 1988), 1: 363.

16.
Hardy,
Ramanujan
, 9.

17.
Ibid., 10.

18.
Ibid., 11.

19.
Ibid., 12.

20.
Jonathan Borwein and Peter Borwein, “Ramanujan and Pi,”
Scientific American
, February 1988, 112.

1.
David Gross, Interview, in
Superstrings: A Theory of Everything? ed
. Paul Davies and J. Brown (Cambridge: Cambridge University Press, 1988), 147.

2.
Sheldon Glashow,
Interactions
(New York: Warner, 1988), 335.

3.
Ibid., 333.

4.
Ibid., 330.

5
. Steven Weinberg,
Dreams of a Final Theory
(New York: Pantheon, 1992), 218–219.

6
. Quoted in John D. Barrow and Frank J. Tipler,
The Anthropic Cosmological Principle
(Oxford: Oxford University Press, 1986), 327.

7
. Quoted in F. Wilczek and B. Devine,
Longing for the Harmonies
(New York: Norton, 1988), 65.

8
. John Updike, “Cosmic Gall,” in
Telephone Poles and Other Poems
(New York: Knopf, 1960).

9
. Quoted in K. C. Cole, “A Theory of Everything,”
New York Times Magazine
, 18 October 1987, 28.

10
. Quoted in Heinz Pagels,
Perfect Symmetry: The Search for the Beginning of Time
(New York: Bantam, 1985), 11.

11
. Quoted in K. C. Cole,
Sympathetic Vibrations: Reflections on Physics as a Way of Life
(New York: Bantam, 1985), 225.

1
. Quoted in E. Harrison,
Masks of the Universe
(New York: Macmillan, 1985), 211.

2
. Quoted in Corey S. Powell, “The Golden Age of Cosmology,”
Scientific American
, July 1992, 17.

3
. The orbifold theory is actually the creation of several individuals, including L. Dixon, J. Harvey, and Edward Witten of Princeton.

4
. Years ago, mathematicians asked themselves a simple question: Given a curved surface in
N
-dimensional space, how many kinds of vibrations can exist on it? For example, think of pouring sand on a drum. When the drum is vibrated at a certain frequency, the particles of sands dance on the drum surface and form beautiful symmetrical patterns. Different patterns of sand particles correspond to different frequencies allowed on the drum surface. Similarly, mathematicians have calculated the number and kind of resonating vibrations allowed on the surface of a curved
N
-dimensional surface. They even calculated the number and kind of vibrations that an electron could have on such a hypothetical surface. To the mathematicians, this was a cute intellectual exercise. No one thought it could possibly have any physical consequence. After all, electrons, they thought, don’t vibrate on
N
-dimensional surfaces.

This large body of mathematical theorems can now be brought to bear on the problem of GUT families. Each GUT family, if string theory is correct, must be a reflection of some vibration on an orbifold. Since the various kinds of vibrations have been cataloged by mathematicians, all physicists have to do is look in a math book to tell them how many identical families there are! Thus the origin of the family problem is
topology
. If string theory is correct, the origin of these three duplicate families of GUT particles cannot be understood unless we expand our consciousness to ten dimensions.

Once we have curled up the unwanted dimensions into a tiny ball, we can then compare the theory with experimental data. For example, the lowest excitation of the string corresponds to a closed string with a very small radius. The particles that occur in the vibration of a small closed string are precisely those found in supergravity. Thus we retrieve all the good results of supergravity, without the bad results. The symmetry group of this new supergravity is E(8) X E(8), which is much larger than the symmetry of the Standard Model or even the GUT theory. Therefore, the superstring contains both the GUT and the supergravity theory (without many of the bad features of either theory). Instead of wiping out its rivals, the superstring simply eats them up.

The problem with these orbifolds, however, is that we can construct hundreds of thousands of them. We have an embarrassment of riches! Each one of them, in principle, describes a consistent universe. How do we tell which universe is the correct one? Among these thousands of solutions, we find many that predict exactly three generations or families of quarks and leptons. We can also predict thousands of solutions where there are many more than three generations. Thus while GUTs consider three generations to be too many, many solutions of string theory consider three generations to be too few!

5
. David Gross, Interview, in
Superstrings: A Theory of Everything?
ed. Paul Davies and J. Brown (Cambridge: Cambridge University Press, 1988), 142–143.

6
. Ibid.

1
. More precisely, the Pauli exclusion principle states that no two electrons can occupy the same quantum state with the same quantum numbers. This means that a white dwarf can be approximated as a Fermi sea, or a gas of electrons obeying the Pauli principle.

Since electrons cannot be in the same quantum state, a net repulsive force prevents them from being compressed down to a point. In a white dwarf star, it is this repulsive force that ultimately counteracts the gravitational force.

The same logic applies to neutrons in a neutron star, since neutrons also obey the Pauli exclusion principle, although the calculation is more complicated because of other nuclear and general relativistic effects.

2
. John Michell, in
Philosophical Transactions of the Royal Society
74 (1784): 35.

3
. Quoted in Heinz Pagels,
Perfect Symmetry: The Search for the Beginning of Time
(New York: Bantam, 1985), 57.

1. Quoted in Anthony Zee,
Fearful Symmetry
(New York: Macmillan, 1986), 68.

2
. K. Gödel, “An Example of a New Type of Cosmological Solution of Einstein’s Field Equations of Gravitation,”
Reviews of Modern Physics
21 (1949): 447.

3
. F. Tipler, “Causality Violation in Asymptotically Flat Space-Times,”
Physical Review Letters
37 (1976): 979.

4
. M. S. Morris, K. S. Thorne, and U. Yurtsever, ’’Wormholes, Time Machines, and the Weak Energy Condition,”
Physical Review Letters
61 (1988): 1446.

5
. M. S. Morris and K. S. Thorne, “Wormholes in Spacetime and Their Use for Interstellar Travel: A Tool for Teaching General Relativity,”
American Journal of Physics
56 (1988): 411.

6
. Fernando Echeverria, Gunnar Klinkhammer, and Kip S. Thorne, “Billiard Balls in Wormhole Spacetimes with Closed Timelike Curves: Classical Theory,”
Physical Review D
44 (1991): 1079.

7
. Morris, Thorne, and Yurtsever, “Wormholes,” 1447.

1
. Steven Weinberg, “The Cosmological Constant Problem,”
Reviews of Modern Physics 61
(1989): 6.

2
. Heinz Pagels,
Perfect Symmetry: The Search for the Beginning of Time
(New York: Bantam, 1985), 377.

3
. Ibid., 378.

4
. Quoted in Alan Lightman and Roberta Brawer,
Origins: The Lives and
Worlds of Modern Cosmologists
(Cambridge, Mass.: Harvard University Press, 1990), 479.

5
. Richard Feynman, Interview, in
Superstrings: A Theory of Everything?
ed. Paul Davies and J. Brown (Cambridge: Cambridge University Press, 1988), 196.

6
. Weinberg, “Cosmological Constant Problem,” 7.

7
. Quoted in K. C. Cole,
Sympathetic Vibrations: Reflections on Physics as a Way of Life
(New York: Bantam, 1985), 204.