Alternate View Column AV-06
Keywords: dimensions, Kaluza-Klein, compactified, string theory
Published in the April-1985 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 10/1/84 and is copyrighted © 1984, John G. Cramer. All rights reserved.
No part may be reproduced in any form without the explicit permission of the author.
Does our universe have other dimensions? We know about the three space dimensions (in which we move) and the time dimension (along which we seem to be hurtling without control). But are there hidden dimensions not accessible to us, dimensions in which we could go adventuring, dimensions within which malevolent hyper-dimensional aliens may be lurking, ready to pierce our flimsy paper-thin three-space bodies with their terrible hyper-sharp claws? The notion of other dimensions has been a recurrent theme in science fiction from the time of Wells and Lovecraft to the present. Recent examples are Heinlein's Number of the Beast, in which two extra time dimensions (tau and teh) give access to alternate universes, and Hogan's Genesis Machine, in which two extra spatial dimensions (hi-space) are exploited for instant communications, gravity control, etc.
This Alternate View column is about hints from particle physics that our universe may have other dimensions ... lots of them. In this issue of Analog you may already have encountered "The Lost Dimensions of Reality" by John Gribbin, a science-fact article on this subject. Dr. Gribbin has laid some excellent groundwork, permitting me to go deeper into some of the fascinating aspects of extra-dimensionality. The reader is advised, however, to read Gribbin's article before joining me on the extra-dimensional playground.
Gribbin's article describes progressive attempts by theoretical physicists to provide an overall theory including the three strongest natural forces (the strong, electromagnetic and weak forces) along with gravity (the other known force). Their work suggests that the space in which we live and work is not four-dimensional but eleven-dimensional; that in addition to the familiar three spatial dimensions and one time dimension there may be seven (or more) "lost" dimensions, similar to ordinary space dimensions except that they are curled up into tiny little loops. This is the Kaluza-Klein (K-K) theory of extra dimensions.
You may well ask why we are suddenly talking about dimensions instead of forces. What can extra dimensions have to do with forces? We will start with the simplest application of K-K theory which deals with electric charge.
Every electron spins on its axis in rough analogy to the daily rotation of the Earth. But while the Earth's spin could in principle be slowed or stopped (by a suitable application of Planet Engineering), only the direction of the electron's spin can be changed; it must always have the same magnitude because all electrons have the same amount of spin. Further, when any atomic or nuclear system rotates, its angular momentum (a measure of speed of rotation) can have only a value which is a unit multiple of the electron's spin. The electron's spin is said to be quantized, with an irreducible constant value.
The electron also has another quantized characteristic, its electric charge. All particles found in nature or produced in the laboratory have charges with magnitudes which are integer multiples of the electron charge. Since spin is quantized and charge is quantized, are the two somehow related?
An early suggestion that this may be so comes from nuclear physics. It was discovered in the 1940's that mathematical machinery used to describe the geometrical rotations of particle spin in normal 3-space could also be applied to describe the behavior of electrical charges by inventing a spin-like quantity called "isospin". Isospin is not a normal 3-space vector; it is a vector in some fictitious extra-dimensional space which projects only one dimension into our universe. It is a useful concept because there are forces acting within a nucleus which can rotate or "flip" this vector: neutrons can be "flipped" into protons or electrons into neutrinos by an extra-dimensional reorientation of isospin.
The Kaluza-Klein theory, relegated for six decades to obscurity as a curious but untestable variant of Einstein's general theory of relativity, has suddenly emerged as one of the hottest "new" theories around because it provides a way of relating forces to extra dimensions. The basic gimmick of the original K-K theory is that the electromagnetic force can be incorporated into the framework of general relativity by adding an extra spatial dimension, one which is curled up in a tiny loop. Each point of normal space becomes an extremely small loop of this extra-dimensional space.
A particle moving in this extra dimension travels around the loop and is soon back where it started. What we call electric charge, according to K-K theory, is actually motion in this dimension. A charged particle is travelling repeatedly around the K-K loop, even if it is at rest in normal space. If it moves, say, clockwise around the loop then it has a positive charge, while counterclockwise motion gives it a negative charge. Spin and similar rotations in normal space are quantized into unit angular momentum chunks because a single (or multiple) rotational flip through 360o cannot be distinguished from no rotation at all. In the same way, motion completely around a K-K loop brings you back to where you started, and this analogously leads to quantization of electric charge. The size of the unit charge and the strength of the electric force are inversely proportional to the distance around the loop: the smaller the loop, the larger is a unit charge.
There are also other connections embedded in the theory. Newton's 3rd law of motion (action=reaction) applied along the K-K dimension is equivalent to the law of conservation of electric charge. And the famous CPT theorem (see my article "Antimatter in the Universe", Analog, 8/79) connecting the reversal of electrical charge with reversals in space and time directions is given a simple geometrical interpretation by this geometrization of charge. The K-K theory is "powerful" because it reveals connections between seemingly unrelated physical laws.
But the modern version of K-K theory is more ambitious than this; it seeks to incorporate not only the electromagnetic force but also the weak and strong forces into the framework of general relativity. This is done by adding enough extra dimensions to take care of all known forces. Quark properties like flavor and color become orbital dances in multi-dimensional K-K loops. This is not a simple change to K-K theory because the weak and strong forces are more complicated than electromagnetism. While electromagnetism can be viewed as the exchange (swapping) of a single particle (the photon), the weak interaction needs three different exchanged particles (W+, W-, and Zo) and the strong interaction requires eight different exchanged particles (gluons). One might suspect that twelve extra K-K dimensions (1+3+8) were needed, but careful analysis shows that only seven K-K dimensions plus normal space-time can accommodate all of the known natural forces.
However, there are already reasons for thinking that there might an even larger number of collapsed K-K dimensions. A previous Alternate View column (September, 1984) discussed the inflationary scenario of Big Bang cosmology, the idea that the early universe went through a phase of very rapid expansion in which its volume expanded by 1088 before settling down to the more leisurely expansion rate of today. A problem with this scenario is that to specify the initial conditions of the early universe would require about 1088 fundamental parameters. This is considered to be an absurdly large number of initial conditions. The early expansion can be specified more compactly by considering the initial universe to be a randomly disordered space of many dimensions. As the universe expands, most of these extra dimensions collapse into small K-K loops, transferring their disorder to the three "normal" dimensions of our world. Calculations show that about 40 extra K-K dimensions are needed for this process to match what we know about the early Big Bang.
And so the use of K-K theory to explain various physical phenomena has led us
from one extra dimension for electromagnetism, to seven extra dimensions for
all of the forces, and finally to about 40 extra dimensions to inflate the
early universe. Applying K-K theory in this way is a very new idea. The full
implications and quantitative application may need years of development. And
it is possible that the whole line of development is completely wrong. But 40
extra dimensions ... What a lovely idea from the standpoint of science
fiction!! Even Hogan and Heinlein would probably have been a bit reluctant to
suggest that there were 40 extra dimensions to our universe, but
theoretical physicists are not so modest and have gone ahead and done it.
So let's start from the assumption that our universe has extra K-K dimensions and entertain some questions:
Q: Could one "enter" these other dimensions? For the seven dimensions needed to explain the natural forces, we are already inextricably embedded in them. Your body is made of 1029 or so point-like particles: electrons, up-quarks, and down-quarks. These particles differ from one another principally because each kind is doing a different dance-step as it loops around in the extra dimensions of K-K space. One could imagine superimposing a hyper-rotation to change one dance-step to another, but this would have an enormous energy cost (and you wouldn't like the result).
Q: What about K-K dimensions beyond the first seven? If there are more than seven K-K dimensions, there are two reasons why we may not have noticed them: (a) the force associated with the dimension is so strong (small K-K loop) that all particles in our universe are constrained by available energy to be neutral and "at rest" within its loop, or (b) the force is so weak (big K-K loop) that up to now experiments have not been sensitive enough to show its effects. In fact the time-reversal asymmetry observed in the decay of the K2o. meson has been attributed to a hypothetical "super-weak" force, possible evidence for a new K-K dimension.
Q: If there are other dimensions, could these lead to other universes? The present version of K-K theory does not have much to say on this point, so let me speculate a bit. If all of the particles of our known world are in some particular state of rest or motion involving K-K dimension loops, then similar particles in other states of motion might be non-interactive and invisible, with only gravity to connect the two particle states. Our universe does have some invisible mass (see my 2/85 Alternate View column) so perhaps another universe of particles might be superimposed on ours and without our being aware of it.
Q: What might be learned from a fully developed K-K theory? We have yet to appreciate the full implications of K-K theory as it applies to seven extra dimensions, but it offers the possibility of presenting a unified picture of gravity, electromagnetism, and nuclear/quark forces. No one can really guess what might come from such understanding; this is always the way with new knowledge. But it does seem that we may be on the borders of a new "dimensional frontier", that we may live in a universe of many dimensions while perceiving only 3+1, that we may be taking the first toddling steps toward understanding the other dimensions of the universe in which we live.
A. Salam and J. Strathdee, Annals of Physics 141, 316 (1982).
Inflation and K-K Theory:
R. B. Abbott, S. M. Barr, and S. D. Ellis, Physical Review D30, 720 (1984).
SF Novels by John Cramer: my two hard SF novels, Twistor and Einstein's Bridge, are newly released as eBooks by Book View Cafe and are available at : http://bookviewcafe.com/bookstore/?s=Cramer .
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