*This is only true for events that are space-like separated. For time-like separated events the relation earlier-later is preserved for all observers. Time-like separated events can never appear simultaneous in any frame of reference moving with a velocity less than c. (Space-like separation is explained later.)
*The Pythagorean theorem is c2 = a2 + b2. The equation for the space-time interval in the special theory of relativity is s2 = t2 - x2. The Pythagorean theorem describes properties in Euclidean space. The equation for the space-time interval describes properties in Minkowski’s flat space-time (Euclidean and non-Euclidean space are discussed in the next chapter). There are other differences as well, but the fundamental relationship between space, time, and the space-time interval is very similar to the relationship expressed in the Pythagorean theorem between the three sides of a right triangle.
*Thanks to Guy Murchie who drew the original version of this drawing in his fine book, Music of the Spheres, New York, Dover, 1961.
*As its hydrogen becomes exhausted, a star begins to fuse the helium at its core. Helium fusion is hotter than hydrogen fusion and produces heavier elements, such as neon, oxygen, and carbon, which, in turn, becomes the solar fuel as its helium becomes exhausted.
*The view presented here is not that geometry comes from the mind. There are many possible geometries (as Riemann and Lobachevsky showed before Einstein), but the actual geometry that we have is determined by the physics. For example, Euclid considered geometry to be closely related to experience (he defined congruence by moving triangles about in space) and he considered his parallel axiom to be not self-evident, i.e., not a product purely of the mind.
The view presented here is that idealizations abstracted from experience (like Euclidean geometry) form a rigid structure of such durability that, when subsequent sensory experience contradicts it, we question the validity of the sensory data rather than the validity of the idealized abstractions. Once such a set of idealized abstractions is erected (verified) in the mind, we thereafter superimpose it upon all subsequent actual and projected sense data (i.e., upon the entire universe as we picture it according to this set of abstractions), whether it fits or not.
*The special theory deals with the unaccelerated (uniform) motion of co-ordinate systems. The special theory can be used to describe the accelerated (non-uniform) motion of objects as long as the co-ordinate system from which the object is being observed is itself in uniform motion.
*Some physicists think that general relativity will be useful on the microscale of high-energy physics (where the effects of gravity usually are ignored), e.g., strong fluctuations of the gravitational field have been detected at very short distances (10-14 cm).
*Eddington expressed this concept most concisely: “A field of force represents the discrepancy between the natural geometry of a co-ordinate system and the abstract geometry arbitrarily ascribed to it.” (Arthur Eddington, The Mathematical Theory of Relativity, Cambridge, England, Cambridge University Press, 1923, pp. 37–38. Italics in the original.)
*This distance, of course, is “invariant,” i.e., the same for all co-ordinate systems. The invariance is the absolute objective aspect of Einstein’s theory that complements the subjective arbitrary choice of co-ordinate system.
†The space-time continuum is not only curved, it also has topological properties, i.e., it can be connected in crazy ways, e.g., like a donut . It also can twist (i.e., torsion).
*This phenomenon was theorized by Pierre-Simon La Place in 1795 using Newtonian physics. Finkelstein was the first physicist to formulate it from the modern point of view, i.e., relativity theory. This modern formulation triggered the current theories on the black hole.
†The very first modern paper on black holes was done by J. R. Oppenheimer and S. Snyder in 1939. The current theories of the black hole, i.e., black hole singularities which are beyond space-time, were developed independently by R. Penrose and S. W. Hawking.
*To a first approximation. Physicists currently theorize that black holes actually shine due to photons and other particles quantum-tunneling out of the one-way membranes.
*The present state of high-energy theory is similar to Ptolemaic astronomy before its collapse under the pressure of the new Copernican world view. The discovery of new particles and new quantum numbers, e.g., charm (to be discussed later) is analogous to the addition of epicycles piled on an already unwieldy theoretical structure.
*The dark-adapted eye can detect single photons. All of the other subatomic particles must be detected indirectly.
†In addition to bubble-chamber physics there is emulsion (photographic plate) physics, counter physics, etc. However, the bubble chamber is probably the most commonly used detection device in particle physics.
*The mass/energy dualism of our ordinary conceptualizations does not exist in the formalism of relativity or quantum theory. According to Einstein’s E = mc2, mass does not change into energy or vice versa: Energy is mass. Wherever energy, E, is present, mass, m, is present and the amount of mass, m, is given by E = mc2. The total amount of energy, E, is conserved, and hence the total amount of mass, m, also is conserved. This mass, m, is defined by the fact that it is a source of the gravitational field.
*S-Matrix theory merges quantum theory and relativity, but it provides limited information on the details of subatomic phenomena and it currently is restricted to hadron interactions (S-Matrix theory is discussed in the next chapter).
*The language of quantum theory is precise but tricky. Quantum theory does not state that something—like light, for example—can be wave-like and particle-like at the same time. According to Bohr’s complementarity (page 103), light reveals either a particle-like aspect or a wave-like aspect depending upon the context, i.e., the experiment. It is not possible to observe both the wave-like aspect and the particle-like aspect in the same situation. However, both of these mutually exclusive (complementary) aspects are needed to understand “light.” In this sense, light is both particle-like and wave-like.
*One of the finest popular books on particle physics is The World of Elementary Particles, by Kenneth Ford, New York, Blaisdell, 1965.
*Einstein’s formula E = mc2 says that mass is energy: energy is mass. Therefore, strictly speaking, mass is not a particular form of energy. Every form of energy is mass. Kinetic energy, for example, is mass. As we speed up a particle, and hence give it energy, ΔE, it gains mass, Δm, in exactly the amount required: ΔE = (Δm)c2. Wherever energy goes, mass goes.
*Physicists no longer use the terms leptons, mesons, and baryons to refer to particle mass alone. These terms now refer to classes of particles which are defined by several properties in addition to mass. For example, the tau particle (τ), which was discovered by a joint team from the Stanford Linear Accelerator Center (SLAC) and the Lawrence Berkeley Laboratory (LBL) in 1975, seems to be a lepton even though it has more mass than the heaviest baryon! Similarly, the D particles, also discovered by a joint SLAC/LBL team (in 1976) are mesons even though they have more mass than the tau particle.
*This peculiar aspect of electrical charge appears to be connected to the unknown properties of quarks and/or magnetic monopoles.
*The quantitative (mathematical) description of particle spin is not any more understandable than the nonquantitative description. Dr. Felix Smith, Head of Molecular Physics, Stanford Research Institute, once related to me the true story of a physicist friend who worked at Los Alamos after World War II. Seeking help on a difficult problem, he went to the great Hungarian mathematician, John von Neumann, who was at Los Alamos as a consultant.
“Simple,” said von Neumann. “This can be solved by using the method of characteristics.”
After the explanation, the physicist said, “I’m afraid I don’t understand the method of characteristics.”
“Young man,” said von Neumann, “in mathematics you don’t understand things, you just get used to them.”
*The basic quantum numbers are spin, isotopic spin,
charge, strangeness, charm, baryon number, and lepton number.
*Original diagrams of this sort were space-time diagrams. However, Feynman also discovered that momentum-energy space descriptions, which are complementary to space-time descriptions, more closely approximate the actual conditions of a collision experiment. The basic concept of space-time descriptions and momentum-energy space descriptions is the same except that momentum-energy space descriptions deal with the momenta and energies of the particles involved instead of their space-time co-ordinates. The diagrams of both space-time descriptions and momentum-energy space descriptions are similar except that diagrams depicting momentum-energy space descriptions can be rotated, as we shall see. Accurately speaking, the remaining Feynman diagrams in this book depict momentum-energy space descriptions unless they are specifically identified as space-time diagrams.
†A detailed analysis of the dot in this particular diagram would reveal a two-step process in which first one photon and then the other is emitted. Technically, diagrams with more than three lines connected to the same vertex are called Mandelsten diagrams.
*These three interactions are: left, a photon and an electron annihilate and create a photon and an electron (electron-photon scattering); middle, two photons annihilate to create a positron and an electron (positron-electron pair creation); and, right, a positron and a photon annihilate to create a positron and a photon (positron-photon scattering).
*The Hagedorn theory of very-high-energy collisions uses the second law of thermodynamics.
†Time reversability exists in potential, i.e., while the particles are represented by propagating wave functions. Time irreversability is an artifact of the measurement process.
*From one point of view virtual photons differ from real ones in that their rest mass is not zero: only zero rest mass photons can escape. There are two ways of looking at virtual photons mathematically. In the first (old-fashioned perturbation theory), the mass of a virtual particle is the same as the mass of a real particle, but energy is not conserved. In the second (Feynman perturbation theory) energy-momentum is exactly conserved, but the virtual particles do not have physical mass.
†In a typical atomic process; high-energy virtual photons have even shorter lifetimes.
*There are other virtual particles in the cloud of virtual particles surrounding an electron, but photons are the most common among them.
*However, the essence of quantum mechanics seems to demand a nondynamic “action-at-a-distance” operating faster than light. A good example of this is the Pauli exclusion principle, which indicates a correlation between the motions of two electrons over and above the exchange of virtual “signal” photons. (Other examples—EPR and Bell’s theorem—are discussed in a later chapter.)
*In fact, there usually is an infinite series of Feynman diagrams for every interaction.
*When the muon was discovered in 1936, it looked like Yukawa’s predicted particle. Gradually, however, it became evident that the muon’s properties were not those of the particle in Yukawa’s theory. Another eleven years passed before Yukawa’s theory was confirmed.
*Recent evidence gives growing credence to the Weinberg-Salam theory that electromagnetic and weak forces are actually different manifestations of the same force, operating at different distances between particles.
†E.g., supergravity theories that use both spin 2 and spin 3/2 virtual exchange particles.
*Paul Schilpp (ed.), Albert Einstein, Philosopher-Scientist, vol. 1, New York, Harper & Row, 1949, has some good essays on this theme.
*From discussions with Prof. John Blofeld, a Buddhist and Taoist scholar, I believe that there are even better illustrations of this concept in The Flower Garland Sutra than the metaphor of Indra’s net. (The Flower Garland Sutra, which also is called the Hua Yen Sutra [Chinese] and the Avatamsaka Sutra [Sanskrit], is extremely long. A complete translation with commentary would be about 150 volumes.) At the time of this printing there is no complete English translation of The Flower Garland Sutra, although one is in progress by the Buddhist Text Translation Society, City of Ten Thousand Buddhas, Talmage, California 95481.
†G. F. Chew’s bootstrap theory may be a physical analog to the Buddhist theory of interdependent originations.
*Brian Josephson, Jack Sarfatti, and Nick Herbert independently have speculated that human sensory systems might detect the zero-point vacuum fluctuations of the dance of virtual particles in empty space predicted by the uncertainty principle. If this is so, such detections might be part of the mechanism of mystic knowing.
†Prajna (Sanskrit) means “wisdom,” but it is a special kind of wisdom which cannot be learned through studying books. Paramita (literally “to cross over”) means “bringing something to perfection.”
*Conservation laws impose absolute checks, but the probability laws can effectively exclude much of what the conservation laws would permit. They impose a great deal of structure.
†The strong interactions are restrained by all twelve conservation laws: energy, momentum, angular momentum, charge, electron-family number, muon-family number, baryon-family number, time reversal (T), combined space inversion and charge conjugation (PC), space inversion alone (P) and charge conjugation alone (C), strangeness, and isotopic spin.
Electromagnetic interactions, next down on the ladder of strengths, lose isotopic spin conservation. Weak interactions, one rung lower, lose strangeness conservation, parity conservation, and charge conjugation invariance as well (but the combination PC remains valid). The last step down the ladder to gravitational interactions at the microscale has not been taken.
*S-Matrix theory is concerned with events in the sense of overall results of a process rather than in the sense of individual things happening during the collision process. There are well-defined entities in the input and output channels (or the S Matrix would not be defined), but in the interaction region itself (inside the circle) everything is blurred and unspecified. “S-Matrix philosophy,” according to Brian Josephson at Cambridge University, “is a statement of unanalyzability in detail.”
†A lay discussion of S-Matrix theory is contained in The Tao of Physics by F. Capra, Berkeley, Shambhala, 1975, pp. 261–76.
*Laser fusion and the search for quarks already have become the partial domain of experimental physics. The new frontiers of theoretical physics appear to be solitons and unified gauge theories.
*There are several interpretations of the formalism of quantum mechanics. Von Neumann thought that only ensembles, i.e., groups of photons, have wave functions and not single particles. A few physicists still agree with this point of view, although most physicists do not.
*It was necessary for me to decide between including the polarizers mentioned in the text and keeping the price of this book within the reach of every person. I chose to omit the polarizers. However, even though it is impossible for words to convey experience, I have kept the text as I originally wrote it to convey the flavor of the demonstration. (Small sheets of polarized plastic are very inexpensive and can be purchased through most popular scientific catalogues.)
*As seen from a particular co-ordinate system. We must be careful about using words like “instantaneous.” Einstein’s special theory of relativity shows that whether one event appears to occur simultaneously with, before, or after another event depends upon the frame of reference from which the observation is made. Accurately speaking, this type of communication is called “space-like” (see next page). Space-like transfers do not always appear to be instantaneous. In fact, they only appear instantaneous from special frames of reference.
*Relativity permits the hypothetical existence of particles called tachyons (tak’i ons) which come into existence already traveling faster than light. In the formalism of the special theory of relativity, tachyons have an imaginary rest mass. Unfortunately, no one knows what an “imaginary rest mass” means in physical terms, or what the interaction forces would be between tachyons and the ordinary particles of real rest
mass out of which we are made.
*The EPR argument for the incompleteness of quantum theory rests squarely on the assumption that the real factual situation in one region cannot depend on what an experimenter does in a far-away region (the principle of local causes).
Einstein, Podolsky, and Rosen point out that we could have chosen to place the axis of the magnet in area A either in the vertical position or in the horizontal position, and that in each case we would have observed a definite result—either up or down in the vertical case, or right or left in the horizontal case. They also assert that what we do (choose to measure or observe) in area A cannot affect the real factual situation in area B. Thus they conclude there must exist simultaneously in area B a definite spin, up or down, and also a definite spin, right or left, to account for all the possible results that we can get by orienting the magnet in area A one way or the other.