The Dancing Wu Li Masters

by Gary Zukav

  It is as if the top could spin, by some strange law that nobody understands, only at 100 revolutions per minute, 90 revolutions per minute, 80 revolutions per minute, and so on, with absolutely no exceptions in between. If our hypothetical top wants to spin slower than 100 revolutions per minute, it must jump all the way down to the next slower speed of 90 revolutions per minute. This is analogous to the situation with subatomic particles except that (1) particular types of particles forever spin at the same speed, and (2) the spin of subatomic articles is calculated in terms of angular momentum.

  Angular momentum depends upon the mass, size, and rate of rotation of a spinning object. More of any one of these properties increases the angular momentum of the object. In general, angular momentum is the strength of the rotation or, put another way, the effort required to stop the rotation. The more angular momentum an object has, the more effort is required to stop it from spinning. A spinning top does not have much angular momentum, because it is small and it has little mass. A merry-go-round, in comparison, has an enormous angular momentum, not because it rotates very fast, but because it is large and it has so much mass.

  Now that you understand spin, forget everything that you have just learned except the bottom line (angular momentum). Every subatomic particle has a fixed, definite, and known angular momentum, but nothing is spinning! If you don’t understand, don’t worry. Physicists don’t understand these words, either. They just use them. (If you try to understand them, they become a koan).*

  The angular momentum of a subatomic particle is fixed, definite, and known. “But,” wrote Max Born,

  one should not imagine that there is anything in the nature of matter actually rotating.8

  Said another way, the “spin” of a subatomic particle involves “The idea of a spin without the existence of something spinning…”9 Even Born had to admit that this concept is “rather abstruse.”10 (Rather!?) Nonetheless, physicists use this concept because subatomic particles do behave as if they have angular momentum and that angular momentum has been determined to be fixed and definite in each case. Because of this, in fact, “spin” is one of the major characteristics of subatomic particles.

  The angular momentum of a subatomic particle is based upon our old friend, Planck’s constant. Remember that Planck’s constant, which physicists call “the quantum of action,” was the discovery that set into motion the revolution of quantum mechanics. Planck discovered that energy is emitted and absorbed not continuously, but in small packages which he called quanta. Since that initial discovery, Planck’s constant, which represents the quantized nature of energy emission and absorption, has appeared again and again as an essential element in the understanding of subatomic phenomena. Five years after Planck’s discovery, Einstein used Planck’s constant to explain the photoelectric effect, and later still he used it to determine the specific heat of solids, an area far removed from Planck’s original study of black-body radiation. Bohr discovered that the angular momentum of electrons as they orbit atomic nuclei is a function of Planck’s constant, de Broglie used Planck’s constant to calculate the wavelength of matter waves, and it is a central element in Heisenberg’s uncertainty principle.

  As important as Planck’s constant is in the realm of subatomic particles, however, it is entirely unobservable in the world at large. This is because the size of the packages by which energy is emitted and absorbed is so small that energy at our gross level appears to be one continuous flow. Similarly, because the indivisible unit of angular momentum is so small, it, too, cannot be observed in the macroscopic world. A spectator swiveling in his chair at a tennis match has 1000000000000000000000000000000000 (1033) times more angular momentum than an electron. Put another way, a change of one penny in the gross national product of the United States is a disturbance more than a billion billion times greater than a change by one unit of the spectator’s angular momentum.11

  Instead of writing out the actual angular momentum of a subatomic particle, physicists usually indicate the spin of a subatomic particle by comparing it to the spin of a photon, whose spin they call one. This system has revealed yet another unexplainable pattern of subatomic phenomena. Entire families of particles have similar spin characteristics. The entire family of leptons, the light-weight particles, for example, has a spin of ½, which means that they all have an angular momentum which is ½ of a photon’s angular momentum. The same is true for the entire family of baryons, the heavy-weight particles. The mesons also have peculiar spin characteristics. They spin in such a way that their angular momenta is always either 0, 1, 2, 3, etc. in relation to the angular momentum of a photon, but nothing in between (0 = no spin, 1 = the same angular momentum as a photon, 2 = twice the angular momentum of a photon, etc.). The spin characteristics of all of the families and all of the particles are in the table at the back of the book.

  The values of a particle’s charge, spin and other major characteristics are represented by specific numbers. These numbers are called quantum numbers. Every particle has a set of quantum numbers which identify it as a particular type of particle.* Every particle of a particular type has the same set of quantum numbers as every other particle of the same type. Every electron, for example, has the same quantum numbers as every other electron. An electron’s quantum numbers, however, distinguish it from protons, all of which also have the same quantum numbers. Individual particles don’t have much personality. In fact, they don’t have any personality at all.

  When Dirac imposed the requirements of relativity on quantum theory, his formalism indicated the existence of a particular positively charged particle. Since the only positively charged particle known in those days (1928) was the proton, Dirac, and most other physicists, assumed that his theory had accounted (mathematically) for the proton. (His theory even was criticized for yielding the “wrong” proton mass.)

  Upon closer examination, however, it became evident that Dirac’s theory described not the proton, but an entirely different particle. Dirac’s new particle was like an electron except that its charge and some of its other major properties were exactly opposite to those of an electron.

  In 1932, Carl Anderson, at Cal Tech (who hadn’t heard of Dirac’s theory) actually discovered this new particle and called it a positron. Physicists later discovered that every particle has a counterpart which is exactly like it but opposite in several major respects. This new class of particles was called anti-particles. An anti-particle, despite its name, is a particle. (The anti-particle of an anti-particle is another particle.)

  Some particles have other particles as anti-particles (for example, a positive pi meson is the anti-particle of a negative pi meson, and the other way round). A few particles are their own anti-particles (like the photon). All of the particles and their anti-particles are in the table at the back of the book.

  The meeting of a particle and its anti-particle is always spectacular. Whenever a particle and its anti-particle meet, they annihilate each other! When an electron meets a positron, for example, both of them disappear and in their place are two photons which instantly depart the scene at the speed of light. The particle and the anti-particle literally disappear in a puff of light. Conversely, particles and anti-particles can be created out of energy and always in pairs.

  The universe is made of both particles and anti-particles. Our part of it, however, is made almost entirely of regular particles which combine into regular atoms to make regular molecules, which make regular matter, which is what we are made of. Physicists speculate that in other parts of the universe anti-particles combine into anti-atoms to make anti-molecules, which make anti-matter, which is what anti-people would be made of. There are no anti-people in our part of the universe because, if there were, they all long since have disappeared in a flash of light.

  Leptons, mesons, baryons, mass, charge, spin, and anti-particles are some of the concepts that physicists use to categorize subatomic phenomena when they momentarily assume that subatomic particles are real objects tha
t move through space and time. These concepts are useful, but only in a limited context. That context is when physicists, for convenience, pretend, as we all do, that dancers can exist apart from a dance.

  1

  The Dance

  The dance of subatomic particles never ends and it is never the same. However, physicists have found a way to diagram the parts of it that interest them.

  The simplest drawing of any type of movement is a space map. A space map shows the location of things in space. The map, for example (first drawing, next page), shows the positions of San Francisco, California, and Berkeley, California. The vertical axis is the north-south axis, as on any map, and the horizontal axis is the east-west line. The map also shows the path of a helicopter flying between San Francisco and Berkeley and, on a greatly enlarged basis, it shows the path of a proton traveling around the cyclotron at the Lawrence Berkeley Laboratory.

  Like all road maps, this space map is two-dimensional. It shows how far north (the first dimension) and how far east (the second dimension) Berkeley is of San Francisco. It does not show the altitude of the helicopter (the third dimension) and it does not indicate how much time (the fourth dimension) the flight from San Francisco to Berkeley required. If we want to show the time involved in the San Francisco-Berkeley flight we must draw a space-time map.

  A space-time map shows the positions of things in space and it also shows their positions in time. The vertical axis on a space-time map is the time axis. Space-time maps are read from the bottom up because the passage of time is represented by movement up the time axis. The horizontal axis of a space-time map is the space axis which shows the movement of objects in space. The path traced by an object on a space-time map is called its “world line.” For example, the space-time map below shows the same flight from San Francisco to Berkeley.

  Initially the helicopter is sitting on the ground in San Francisco. Its world line is vertical, because, although it is not moving in space, it is moving in time. A to B is the world line of the helicopter while it sits on its pad in San Francisco. When the helicopter takes off for Berkeley it moves forward both in time and space, and its world line traces the path on the space-time map between B and C. When it lands in Berkeley its world line is vertical once again because it no longer moves in space, but, like all things, it continues to move in time (C to D). The arrowheads show which direction and helicopter is moving. It can move backward and forward in space but, of course, it only can move forward in time. The dashed lines show the world lines of San Francisco and Berkeley which do not move in space at all except during California earthquakes.

  Physicists use similar space-time maps to diagram particle interactions. Below is a space-time diagram of an electron emitting a photon.

  Starting at the bottom, an electron moves through space with a certain velocity. At the point in space and time indicated by the dot, it emits a photon. The photon flies off at the speed of light to the right and the electron, its momentum affected by the emission of the photon, alters course and moves off more slowly to the left.

  In 1949, Richard Feynman discovered that space-time maps like these have an exact correspondence with mathematical expressions which give the probabilities of the interactions that they depict. Feynman’s discovery was an extension of Dirac’s 1928 theory and it helped to evolve that theory into the quantum field theory that we know today. Therefore, this type of diagram is sometimes called a Feynman diagram.*

  Here is a Feynman diagram of a particle/anti-particle annihilation. An electron on the left approaches an anti-electron (a positron) which is coming from the right. At their point of contact, indicated by the dot, they mutually annihilate each other and two photons are created which depart in opposite directions at the speed of light.†

  A happening in the subatomic world is called an “event.” Events are indicated in Feynman diagrams by dots. Every subatomic event is marked by the annihilation of the initial particles and the creation of new ones. This is true for every event and not only those involving particles and anti-particles.

  With this in mind, we now can look at the particle diagram again, and see it in a different light. Instead of saying that an electron moving through space emitted a photon which changed its (the electron’s) momentum, we can say as well that an electron moving through space emitted a photon and went out of existence at that point! A new electron was created in this process and it departed the scene with a new momentum. There is no way of knowing if this interpretation is correct or not because all electrons are identical. However, it is simpler and more consistent to assume that the original particle was annihilated and a new particle was created. The indistinguishability of subatomic particles makes this possible.

  On the next page is a Feynman diagram of the process that we discussed.

  A negative pi meson collides with a proton and the two particles are annihilated. Their energy of being (mass) and energy of motion create two new particles, a lambda particle and a neutral K meson. These two new particles are unstable and live less than a billionth of a second before they decay into other particles (actual decay times are in the table at the back of the book). The neutral K meson decays into a positive pi meson and a negative pi meson. The lambda particle, and this is the interesting part, decays into the original two particles (a negative pi meson and a proton)! It is as if we smash two toy automobiles together and instead of shattering into bits and pieces, they come apart into more toy automobiles, some of which are as large as the originals.

  Subatomic particles forever partake of this unceasing dance of annihilation and creation. In fact, subatomic particles are this unceasing dance of annihilation and creation. This twentieth-century discovery, with all its psychedelic implications, is not a new concept. In fact, it is very similar to the way that much of the earth’s population, including the Hindus and the Buddhists, view their reality.

  Hindu mythology is virtually a large-scale projection into the psychological realm of microscopic scientific discoveries. Hindu deities such as Shiva and Vishnu continually dance the creation and destruction of universes while the Buddhist image of the wheel of life symbolizes the unending process of birth, death, and rebirth which is a part of the world of form, which is emptiness, which is form.

  Imagine that a group of young artists have founded a new and revolutionary school of art. Their paintings are so unique that they have come to share them with the curator of an old museum. The curator regards the new paintings, nods his head, and disappears into the vaults of the museum. He returns carrying some very old paintings, which he places beside the new ones. The new art is so similar to the old art that even the young artists are taken aback. The new revolutionaries, in their own time and in their own way, have rediscovered a very old school of painting.

  Let us look again at the Feynman diagram of an electron-positron annihilation. Suppose that we use the arrowhead to indicate which is the particle (the electron) and which is the anti-particle (the positron) by making the arrowheads that point up indicate the particles and the arrowheads that point down indicate the anti-particles. That would make the diagram look like the drawing below.

  Naturally, time, as we experience it, only travels in one direction, forward, and that is up on a space-time diagram. Nonetheless, this simple convention would give us an easy way of telling particles from anti-particles. World lines that appear to move forward in time would belong to particles and world lines that appear to move backward in time would belong to anti-particles. (Photons would not have arrowheads because they are their own anti-particles.)

  Feynman demonstrated in 1949 that this convention is more than an artistic device. He discovered that a positron field propagating forward in time is mathematically the same as an electron field propagating backward in time! In other words, according to quantum field theory, an anti-particle is a particle moving backward in time. An anti-particle does not have to be considered as a particle moving backward in time, but that is the simplest and most symmetric wa
y of viewing anti-particles.

  For example, because the arrowheads distinguish the particles from the anti-particles, we can twist the original Feynman diagram around into any position that we choose and still be able to distinguish the one from the other. Here are some different ways that we can twist the original Feynman diagram.

  Each of these variations is a separate diagram and represents a particle/anti-particle interaction.* By twisting the original diagram completely around we can represent every possible interaction between an electron, a positron, and two photons. The precision, simplicity, and symmetry of Feynman diagrams make them a special type of poetry.

  On the next page is a space-time diagram of two events. A collision between two photons (at B) creates an electron-positron pair and, subsequently, an electron and a positron annihilate each other and create two photons (at A). (The left half of this diagram, the interaction at A, is the same as the electron-positron annihilation.

  Ordinarily we would interpret these events as follows: Two photons collide in the lower right of the diagram producing an electron-positron pair. The electron flies off to the right while the positron flies off to the left where it meets another electron which has entered the diagram from the lower left. There they mutually annihilate and create two photons which depart in opposite directions.

  The preferred interpretation of quantum field theory, however, is much simpler. In it there is only one particle. That particle, an electron, enters the diagram from the lower left and travels forward in time and space until it emits two photons at A. This causes it to reverse its direction in time. Traveling backward in time as a positron it absorbs two photons at B, reverses its direction in time again, and again becomes an electron. Instead of three particles there is only one particle which, moving from left to right, travels first forward in time, then backward in time, and then forward in time again.