The Spectrum of String States
THE ELEGANT UNIVERSE, Brian Greene, 1999, 2003
```(annotated and with added bold highlights by Epsilon=One)
Chapter 10 - Quantum Geometry
The Spectrum of String States*
* Some of the ideas in this and the next few sections are rather subtle, so don't be put off if you have trouble following every link in the explanatory chain—especially in a single reading.
or The new possibility of wound-string configurations implies that the energy of a string in the Garden-hose universe comes from two sources: vibrational motion and winding energy. (Epsilon=One: The "Garden-hose universe" is ludicrous. Where is a fundamental postulate? Oscillating motion is much more complex than "vibrational"; and exactly what is "winding energy"? All that follows concerning a Garden-hose universe is mental fluff.) From the legacy of Kaluza and Klein, each depends on the geometry of the hose, that is, on the radius of its curled-up circular component, but with a distinctly stringy twist, since point particles cannot wrap around dimensions. Our first task, then, will be to determine precisely how the winding and vibrational contributions to the energy of a string depend on the size of the circular dimension. For this purpose, it proves convenient to separate the vibrational motion of strings into two categories: uniform and ordinary vibrations. Ordinary vibrations refer to the usual oscillations we have discussed repeatedly, such as those illustrated in Figure 6.2; uniform vibrations refer to even simpler motion: the overall motion of string as it slides from one position to another without changing its shape. All string motion is a combination of sliding and oscillating of uniform and ordinary vibrations—but for the present discussion it is easier to separate them in this manner. (Epsilon=One: "Sliding" is considered a form of oscillation. "Strings" have many forms of oscillation—slide, swing, vibration, pulse, et cetera—all of which manifeat as, or within, a Pulsoid.) In fact, the ordinary vibrations will not play a central part in our reasoning, and we will therefore include their effects only after we have finished giving the gist of the argument.
Here are the two essential observations. First, uniform vibrational excitations of a string have energies that are inversely proportional to the radius of the circular dimension. This is a direct consequence of the quantum-mechanical uncertainty principle: A smaller radius more strictly confines a string and therefore, through quantum-mechanical claustrophobia, increases the amount of energy in its motion. So, as the radius of the circular dimension decreases, the energy of motion of the string necessarily increases—the hallmark feature of an inverse proportionality. Second, as found in the preceding section, the winding mode energies are directly—not inversely—proportional to the radius. Remember, this is because the minimum length of wound strings, and hence their minimum energy, is proportional to the radius. These two observations establish that large values of the radius imply large winding energies and small vibration energies, whereas small values of the radius imply small winding energies and large vibration energies.
This leads us to the key fact: For any large circular radius of the Garden-hose universe, there is a corresponding small circular radius for which the winding energies of strings in the former universe equal the vibration energies of strings in the latter, and vibration energies of strings in the former equal winding energies of strings in the latter. As physical properties are sensitive to the total energy of a string configuration—and not to how the energy is divided between vibration and winding contributions—there is no physical distinction between these geometrically distinct forms for the Garden-hose universe. And so, strangely enough, string theory claims that there is no difference whatsoever between a "fat" Garden-hose universe and a "thin" one.
It's a cosmic hedging of bets, somewhat akin to what you, as a smart investor, should do if faced with the following puzzle. Imagine you learn that the fate of two stocks trading on Wall Street—say, a company making fitness machines and a company making heart-bypass valves—are inextricably connected. They each closed trading today valued at one dollar per share, and you are told by a reliable source that if one company's stock goes up the other's will go down, and vice versa. Moreover, your source—who is completely trustworthy (but whose guidance might be crossing over legal boundaries)—tells you that the next day's closing prices of these two companies are absolutely certain to be inversely related to one another. That is, if one stock closes at $2 per share, the other will close at $˝ (50 cents) per share; if one stock closes at $10 per share, the other will close at $1/10 (10 cents) per share, and so on. But the one thing your source can't tell you is which stock will close high and which will close low. What do you do?
Well, you immediately invest all of your money in the stock market, equally divided between the shares of these two companies. As you can easily check by working out a few examples, no matter what happens on the next day, your investment cannot lose value. At worse it can remain the same (if both companies again close at $1), but any movement of share prices—consistent with your insider information—will increase your holdings. For instance, if the fitness company closes at $4 and the heart-valve company closes at $1/4 (25 cents), their combined value is $4.25 (for each pair of shares), compared with $2 the previous day. Furthermore, from the perspective of net worth, it does not matter one bit whether the fitness company closes high and the heart-valve company low, or vice versa. If you care only about the total amount of money, these two distinct circumstances are financially indistinguishable.
The situation in string theory is analogous in that the energy in string configurations comes from two sources—vibrations and windings—whose contributions to the total energy of a string are generally different. But, as we shall see in more detail below, certain pairs of distinct geometrical circumstances—leading to high-winding-energy/low-vibration-energy or low-winding-energy/high-vibration-energy—are physically indistinguishable. And, unlike the financial analogy for which considerations beyond total wealth can distinguish between the two types of stock holdings, there is absolutely no physical distinction between the two string scenarios.
Actually, we shall see that to make the analogy with string theory tighter, we should consider what would happen if you did not divide your money equally between the two companies in your initial investment, but bought, say, 1,000 shares of the fitness company and 3,000 shares of the heart-valve company. Now the total value of your holdings does depend on which company closes high and which closes low. For instance, if the stocks close at $10 (fitness) and 10 cents (heart-valve), your initial investment of $4,000 will now be worth $10,300. If the reverse happens—the stocks close at 10 cents (fitness) and $10 (heart-valve)—your holdings will be worth $30,100—significantly more.
Nevertheless, the inverse relationship between the closing stock prices does ensure the following. If a friend of yours invests exactly "opposite" to you—3,000 shares of the fitness company and 1,000 shares of the heart-valve company—then the value of her holdings will be $10,300 if stocks close valves-high/fitness-low (the same as your holdings in the fitness-high/valves-low closing) and $30,100 if they close with fitness-high/valves-low (again, the same as your holdings in the reciprocal situation). That is, from the point of view of total stock value, interchanging which stock closes high and which closes low is exactly compensated by interchanging the number of shares you own of each company.
Hold this last observation in mind as we now return to string theory and think about the possible string energies in a specific example. Imagine that the radius of the circular Garden-hose dimension is, say, ten times the Planck length. We will write this as R = 10. A string can wrap around this circular dimension one time, two times, three times, and so forth. The number of times a string wraps around the circular dimension is called its winding number. The energy from winding, being determined by the length of wound string, is proportional to the product of the radius and the winding number. Additionally, for any amount of winding, the string can undergo vibrational motion. As the uniform vibrations we are currently focusing on have energies that are inversely dependent on the radius, they are proportional to whole-number multiples of the reciprocal of the radius—1/R—which in this case is one-tenth of the Planck length. We call this whole number multiple the vibration number. 2
As you can see, this situation is very similar to what we encountered on Wall Street, with the winding and vibration numbers being direct analogs of the shares held in the two companies, while R and 1/R are the analogs of the closing prices per share in each. Now, just as you can easily calculate the total value of your investment from the number of shares held in each company and the closing prices, we can calculate the total energy carried by a string in terms of its vibration number, its winding number, and the radius. In Table 10.1 we give a partial list of these total energies for various string configurations, which we specify by their winding and vibration numbers, in a Garden-hose universe with radius R = 10.
A complete table would be infinitely long, since the winding and vibration numbers can take on arbitrary whole-number values, but this representative piece of the table is adequate for our discussion. We see from the table and our remarks that we are in a high-winding-energy/low-vibration-energy situation: Winding energies come in multiples of 10, while vibrational energies come in multiples of the smaller number 1/10.
Now imagine that the radius of the circular dimension shrinks, say, from 10 to 9.2 to 7.1 and on down to 3.4, 2.2, 1.1, .7, all the way to .1 (1/10), where, for our present discussion, it stops. In this geometrically distinct form of the Garden-hose universe we can compile an analogous table of string energies: Winding energies are now multiples of 1/10 while vibration energies are multiples of its reciprocal, 10. The results are shown in Table 10.2.
At first glance, the two tables might appear to be different. But closer inspection reveals that although arranged in a different order, the "total energy" columns of both tables have identical entries. To find the corresponding entry in Table 10.2 for a chosen entry in Table 10.1, one must simply interchange the vibration and winding numbers. That is, vibration and winding contributions play complementary roles when the radius of the circular dimension changes from 10 to 1/10. And so, as far as total string energies go, there is no distinction between these different sizes for the circular dimension. Just as the interchange of fitness-high/valves-low with valves-high/fitness-low is exactly compensated by an interchange of the number of shares held in each company, interchange of radius 10 and radius 1/10 is exactly compensated by the interchange of vibration and winding numbers. Moreover, while for simplicity we have focused on an initial radius of R = 10 and its reciprocal 1/10, the conclusions drawn are the same for any choice of the radius and its reciprocal. 3
Tables 10.1 and 10.2 are incomplete for two reasons. First, as mentioned, we have listed only a few of the infinite possibilities for winding/vibration numbers that a string can assume. This, of course, poses no problem—we could make the tables as long as our patience allows and would find that the relation between them will continue to hold. Second, beyond winding energy, we have so far considered only energy contributions arising from the uniform-vibrational motion of a string. We should now include the ordinary vibrations as well, since these give additional contributions to the string's total energy and also determine the force charges it carries. The important point, however, is that investigations have revealed that these contributions do not depend on the size of the radius. Thus, even if we were to include these more detailed features of string attributes in Tables 10.1 and 10.2, the tables would still correspond exactly, since the ordinary vibrational contributions affect each table identically. We therefore conclude that the masses and the charges of particles in a Garden-hose universe with radius R are completely identical to those in a Garden-hose universe with radius 1/R. And since these masses and force charges govern fundamental physics, there is no way to distinguish physically these two geometrically distinct universes. Any experiment done in one such universe has a corresponding experiment that can be done in the other, leading to exactly the same results.
Table 10.1 Sample vibration and winding configurations of a string moving in a universe shown in Figure 10.3, with radius R = 10. The vibration energies contribute in multiples of 1/10 and the winding energies contribute in multiples of 10, yielding the total energies listed. The energy unit is the Planck energy, so for example, 10.1 in the last column means 10.1 times the Planck energy.
Table 10.2 As in Table 10.1, except that the radius is now taken to be 1/10.