n this past Century, many wonderful things were discovered in physics and cosmology. We learned how stars shine for billions of years and how the elements were formed. Our new technology of lasers and semiconductors is based on Quantum Mechanics, while our understanding of the Cosmos is largely based on General Relativity.
espite its many triumphs,
the Standard Model is based on the acceptance of paradox, singularities, and the infinite energy of the vacuum. A paradox is a form of self-contradiction. Goedel proved that from a logical system which contains a contradiction, absolutely any proposition may be proven. The average nut on the Internet doesn't know Goedel's theorem, but he or she does know that if physicists can accept paradox and obvious absurdities without blinking an eye, so can they! So on the Net we find every sort of crazy idea derived from quantum mechanics, or from String Theory, which hasn't got a single testable consequence! This is the disappearance of physics as we know it. Or as it was known to Newton, Maxwell and Einstein. And of course, the inevitable fall of Western Civilization! :-) Of course, I can fix all that, without changing the equations. Then we will talk about the graviton.
singularity used to be a place where a variable goes to plus or minus infinity, and that is the meaning I use. (In chaos theory, the meaning of "singularity" has been expanded a bit.) We know there can be no actual infinities, because they would gobble up the rest of the universe. We know that the vacuum cannot have infinite energy, because energy is mass, and infinite mass would have folded up the universe long before the CMBR was released, long before atoms were formed, long before the first quasar or galaxy. Yet just such views have been accepted as part of the Standard Model in the 20th Century. Let us hope the 21st Century can do a little better, or rationality will simply dissolve.
ll is not lost. It is not the equations that are wrong, but in some cases our use of them, for instance, applying Heisenberg's uncertainty rules to the vacuum, which results in the infinite energy of the vacuum. And in some cases, the problem is not the equations, but our interpretation of them. That is the problem with basic quantum mechanics.
non-paradoxical interpretation of quantum mechanics is possible if we accept the fundamental reality of the de Broglie wave. All we have to do is show why particles sometimes have a wave-like behavior (while remaining particles) and why vibrations sometimes act like particles, (while remaining vibrations in a field). A thing cannot be simultaneously a particle and a wave. The scale of things doesn't change logic. Fortunately, we can avoid the wave-particle paradox with the de Broglie wave. It was Prince Louis de Broglie, a French aristocrat, who discovered this wave in 1923. His Ph.D. was in history, and he went on to a career as a civil servant. According to Fred Alan Wolf, in Taking the Quantum Leap, de Broglie regarded his wave as either a pilot wave or a matter wave. My view, however, is that it is neither. It is something real in and of itself, which we may call the de Broglie wave, and it is a possibility / probability wave. It shows us what is possible, and their probabilities.
he properties of the de Broglie wave are described by two equations:
he ideas of de Broglie excited immediate interest, because he could use his formula to calculate the orbits of atoms. Each orbit is determined by the standing waves (or resonances) of the de Broglie vibration. The ground state orbit will fit precisely one wavelength, the second orbit two wavelengths, and so forth to higher and higher overtones. These standing waves can themselves move, which indicates movement of their associated particle. This movement of the standing wave is never faster than the speed of light. Prince de Broglie's ideas are very similar to the theory of musical instruments, as he well knew.
here are actually several ways of deriving the orbits of atoms. Before de Broglie, Bohr, noticing that Planck's constant, h, has the units of angular momentum, found that he could calculate the orbits of the hydrogen atom by taking Planck's constant as the unit of angular momentum. The first orbit had an angular momentum of h, the second 2h and so on. And from this, he could calculate the energy levels of each orbit, and more importantly, the difference in energy given up in dropping from a higher orbit down to a lower orbit. Since 1905, scientists had known that for the photon, E=hf, to know the energy lost in a given transition is to know the resulting photon's frequency and thus its wavelength (since C=L*f). Much to Bohr's delight, his calculations agreed exactly with the Balmer series of the spectral lines emitted or absorbed by hydrogen.
standing wave is sometimes called an Eigenstate, with a quantum number N=0,1,2,3..., with an Eigenvalue E(N) associated with each value of the quantum number. In a musical instrument, the standing wave is the note produced by the instrument. In music, the Eigenvalue would be frequency and the quantum number would indicate the fundamental tone and its first, second, third, etc., overtones. This leads to the mature form of the de Broglie wave equation. If the de Broglie function is described by F(r,t), and H is the Hamiltonian differential operator, then H[F(r,t)] becomes the left half of a differential equation. We let E(n) be the eigenvalues, which for the Hamiltonian operator will be energy states, a different one in most cases for each value of n=0,1,2,3... So the mature forms of the equation resemble:
It was Erwin Schroedinger who took de Broglie's formulas and plugged them into the partial differential equation for a wave. He then imposed severe boundary conditions, which in effect made each solution of the equation a probability function. In this way, the 2-dimensional picture of orbits is transformed into a 3-dimensional picture of orbitals. There are s, p, d and f orbitals. All the s orbitals are spherical. The other orbitals can assume more complicated, multi-lobed shapes. There are 3 such shapes for the p orbitals, 5 for the d orbitals and 7 for the f orbitals. Knowing the size, shape, and energy level of each orbital, chemists can understand the periodic table of elements. This, in turn, provides an understanding of ionic bonds and valence bonds and the general behavior of all of the elements, alone or in combination. See The Periodic Kingdom, by P.W. Atkins, especially pages 112 and 113.
f you have trouble understanding resonances, think of musical instruments. Louis de Broglie's theory is similar to the theory of a musical instrument, as he well knew. For a trumpet with a given length of tubing, there is a lowest note, one where exactly one wavelength of sound will resonate in the tube. It is possible to make higher notes on a trumpet, by producing overtones, but impossible to make a lower note. The same is true of electrons in orbit around a nucleus, although this analogy is only a teaching device and doesn't really explain the behavior of atoms.
o far so good. This picture even allows us to see why the first orbit of electrons around a nucleus can only hold two electrons, one somewhere under the maxima of the de Broglie resonance, and the other opposite to it under the minima. These two electrons have slightly different energy, because in the up electron, its magnetic pole aligns with that of the nucleus, while the down electron has its magnetic pole alignment opposite to that of the nucleus. The picture even allows us to understand why we can never predict exactly where the electron is. The de Broglie wave is a probability wave. It shows us where the probability is greatest, but some of the time the electron will be anywhere its probability is non-zero.
hat we must add to this picture are Heisenberg's uncertainty rules, and relativity, which gives us electron spin and anti-particles. I won't go into that. If Z is some observable, then we will indicate the spread of its probability function (its Heisenberg uncertainty) by putting a "d" in front of it, dZ. The Heisenberg rules describe a curious coupling of position with momentum, and energy with time. Heisenberg's rules are:
onsider an electron in an excited orbit in the hydrogen atom. It will remain in that state for a certain duration of time, t, and when it falls back into a lower orbit, it will release a quantum of electro-magnetic energy, where E=hf (f being the frequency). However, if we have a large number of hydrogen atoms making the same transition, both the energy E and the time t will vary a little. That is the "spread." There is a little probability curve that goes with each variable. The product of those spreads is Planck's constant h. Nature is fundamentally probabilistic on an atomic or sub-atomic level. Notice that the product of the spreads is an extremely small number, far smaller than experimental error. The real importance of Heisenberg's laws is that it is from them that we get virtual particles and the infinite energy of the vacuum. More on that later. Incidentally, Heisenberg's rules provide yet a third way of calculating the orbits of the electrons in an atom.
he paradoxical interpretation of QM arose at the Fifth Solvay conference in 1927, where the principals were Bohr, Einstein, de Broglie, Born, and Schroedinger. Schroedinger had plugged in de Broglie's equation for the wavelength into the standard partial differential equation for moving particles, and produced the Schroedinger equation for calculating wave packets. The problem is that these Schroedinger packets steadily spread out in time, unlike the de Broglie wave. Heisenberg suggested that the act of observation collapsed the Schroedinger wave function, so it was once again localized. Thus, reality is created by our observation of it. Naturally mystics liked that idea. Schroedinger was something of a mystic, even before he produced his equation. So it became part of the standard interpretation of QM which is still accepted.
ears later, in 1952, David Bohm argued that the underlying assumptions of Heisenberg's uncertainty rules could be contradicted by an unknown underlying level of reality. This is known as the hidden variables theory. In the 1960s, John Bell proved that hidden variables would be non-local. In other words, a change in a hidden variable might simultaneously and instantaneously change an observable variable light-years away, if the two events were "intangled." Bell's theorem has become famous. It does not exclude hidden variables. But they must be non-local, which would be an even stranger idea than quantum mechanics.
End of historical interlude.
should emphasize that I do not believe in hidden variables. I don't accept Bohm's objections to quantum mechanics, nor those of Einstein. It doesn't bother me in the slightest that sub-atomic reality is fundamentally probabilistic, and is in many ways different from the macroscopic reality we directly observe. All I am saying is that our theories about sub-atomic reality cannot be paradoxical or allow logical impossibilities.
f we assume that the de Broglie wave is not a mathematical fiction, we can make all the quantum paradoxes and all the quantum weirdness go away. The electron does not go through both slits in the famous two-slit interferometer experiment. But its de Broglie wave does, and produces the diffraction patterns on the far side. The electron goes through one slit or the other; we just don't know which, since the de Broglie wave intensity is equal at the two slits. The electron is always in one place or another. It does not have a ghostly presence in each of the places it could be. Thus, observing an electron does not "collapse the wave function," nor does it pick out one among an infinity of universes.
he de Broglie functions describe the experiment, such as the 2-slit interferometer. Remember, it describes what is possible, and their probabilities. For the detector on the 2-slit interferometer, we use wave theory and positive and negative interference to produce a curve of the intensity of the de Broglie wave at the plane of the detector. This will be a curve with two humps. Now suppose we feed our electrons through one at a time. It might land on the right, or it might land on the left. What has collapsed? Nothing. The function for this apparatus remains the same. And if we keep on feeding electrons through it, the results will more and more closely match the two humped probability wave of de Broglie. We do not change it by observing it.
ncidentally, the 2-slit experiment has now been done with atoms, molecules, and even a 60 carbon atom bucky-ball (Arndt, M. et al. (1999)
Letters to Nature, vol 401, October 14, 1999 pg 680). This implies that one can calculate the de Broglie wave of an entire object, such as the Bucky-ball, as if it were a single simple thing having a particular mass, velocity, and location. This is somewhat like the Center of Mass theorem in Newtonian physics.
Of course, someone is sure to repeat the mantra of QM, which is that the position and velocity of an object cannot be known simultaneously. But this is not what the Heisenberg Uncertainty relationship says. It says we can only know each of these quantities to 13 significant digits! I am sure any experimenter would be very happy with that.
he de Broglie vibration explains the sometimes wavelike behavior of electrons. When a beam of electrons is reflected off a crystal, it forms diffraction patterns. This is because the de Broglie wave associated with the electron goes before, (since its velocity is always greater than that of the electron), and bounces off each atom on the surface of the crystal. The result is a whole series of reflected de Broglie waves, which add or subtract, producing a diffraction pattern. The probability of an electron hitting the detector screen in a particular place is proportional to the intensity of its de Broglie wave there.
ecall that the velocity of the de Broglie wave for an object traveling at velocity V is
he problem of photons is quite different from the problem of atoms. In atoms, we have resonances of the de Broglie wave, which determine the orbits of the electrons. With EM radiation, the de Broglie wave does not form a resonance. It just spreads out like ripples in a pond, in three-dimensions, and has positive and negative interference with all the other de Broglie waves from other photons in the neighborhood. As usual, where the de Broglie wave is strongest, that is where you are most likely to find a photon. What does a photon look like? A useless question.
So what is a photon? It is a quantum state of the EM field. That is why its energy can be transferred instantly to an electron, even if the wavelength of the photon is miles long. This would be impossible in classical physics.
should remind everybody that there is no new physics here. I'm not changing the equations. This is just an interpretation of the equations, i.e., a word picture and a mental picture. This interpretation avoids the weirdness of later quantum mechanics, such as the wave-particle paradox, multiple universes, instantaneous action at a distance, the collapse of the wave-function and the entanglement of observer with observed, which crept in with Schroedinger, Heisenberg, and Bohr. This non-paradoxical interpretation is consistent with observation. We cannot ask more of an interpretation. I would make a stronger statement. Physics cannot simply accept paradox, any more than it can just accept singularities. To do so is the end of physics as a rational enterprise, because absolutely any proposition can be derived from a system of ideas which allows logical impossibilities.
he infinite energy of the vacuum (which would curl the whole universe up faster than I can type this sentence) arises from the theory of virtual particles. The entire theory of virtual particles arises from applying Heisenberg's rule of dE*dt=h to the vacuum. Remember, dt is the spread in duration, and dE is the spread in energy. If we make dt very sharp, dE becomes very broad, so broad in fact, that the formation of a particle and an anti-particle has non-zero probability, although this pair will exist for the minute fraction of time allowed it by dt. This is the origin of the theory of virtual particles, which has the unfortunate consequence that the energy of the vacuum is infinite.
t is possible to prevent infinity by cutting off the possible wavelengths when they are small enough to enter the realm of quantum gravity. But that ad hoc device still gives us a vacuum energy 120 orders of magnitude greater than the energy contained in all the matter in the universe! According to Lawrence Krauss, a well-respected neutrino physicist, "[This] discrepancy between theory and observation is the most perplexing quantitative puzzle in physics today (Scientific American, Jan. 1999, "Cosmological Antigravity," p. 55)." I am glad that Lawrence Krauss agrees with me. Some would say that if the vacuum has any energy density, it would be infinite if the universe is spatially infinite. What is wrong with that? Only that the universe has been expanding from a Hawking no-boundary singularity for a finite time, about 15 billion years. So it cannot be spatially infinite, though it could still be unbounded.
ortunately, it is possible to get rid of virtual particles, simply by saying that it is appropriate to apply Heisenberg's rules only when we can calculate a de Broglie wave. There is no de Broglie wave for the vacuum. So how then do we explain Casimir's force and other apparent confirmations of this idea? By applying de Broglie theory to the Electro-Magnetic field, which extends through all of space. As we bring two plates closer together, we begin limiting the wavelengths of photons that can exist between them. This draws the plates together. At least, that is one idea. Better than accepting an absurdity such as infinite energy for the vacuum. If there are no virtual particles, then the vacuum goes back to what it was in Newton's time. Nothing. It is just empty space. Those people who are planning space-ships which will extract energy from the ZPE are just wasting their time.
have now eliminated the paradox from Quantum Mechanics. I would like to begin my discussion of cosmology with an observation about anti-particles.
t was Richard Feynman who suggested that anti-particles are like ordinary particles moving backwards in time. If that is true, anti-particles should have anti-gravity, a conjecture which has never been tested.
ewsweek (Newsweek, May 12, 1997, "Fountain of Annhilation") and Discover magazines have reported a fountain of anti-electrons spouting from the center of our own galaxy, which is suspected of harboring an old quasar, also known as a giant black hole. Giant jets of particles and energy have been seen erupting from many galactic centers. Wouldn't it be interesting if all such jets start out as anti-particles? This is exactly what we would expect if anti-particles have anti-gravity. When particle and anti-particle pairs are formed inside the intense gravitational field of a black hole or quasar, the anti-particle seems to shoot out one pole or the other, producing the jet. This implies that quasars and black holes gradually evaporate, which is why there are no quasars left in recent times. Ordinary stellar black holes could very well have a limited lifetime as well.
hink about this and you see that there is no singularity inside a black hole. First, we must assume that all black holes are spinning, and the unseen particles inside the black hole are orbiting the center as well. For a moment, let us suppose there is a singularity. Any particle plunging down its throat would steadily pick up energy, and continuously split into particles and anti-particles. The anti-particles zoom away with enormous acceleration out the poles. Why? Because in any other direction, it will meet particles, and annihilate. This process would continue until all the energy is drained out of the black hole. This doesn't happen, therefore there is no singularity. For the same reason, the universe could not collapse into a singularity, nor begin with a singularity.
ecent observations show that the universe is not only expanding, it is accelerating. The evaporation of black holes and quasars could explain this, since it produces anti-matter. The proportion of anti-matter to matter should increase (up to a point), and so should the repulsive force produced by anti-matter. The force of repulsion will decline as quasars and black holes disappear, since the anti-matter will gradually be annhilated. The acceleration phase may already be passing, since there have been no quasars in the last billion years. There has been just enough acceleration to make the universe old enough to hold the oldest stars. The universe is closed, and repeatedly returns to the Hartle-Hawking no-boundary and explodes outward again, all its laws of nature unchanged. There is no need for Guth's Inflation theory. Fiddling with Einstein's Cosmic Constant is unnecessary. It is zero, and to make it anything else would be hopelessly ad hoc. Quintessence is not required. There is no ZPE. And there is no need for string theory, since the rationale for developing it is to avoid singularities. Once Quantum Gravity has been developed, there will be nothing left to explain in physics. The laws of the universe are what they are, because they have always been so.
f anti-particles have anti-gravity, they would not clump. Indeed, they would produce the soap bubble large scale structure of the universe that we in fact observe. The anti-matter particles would try to stay as far away from every other particle as possible. Thus, we must imagine the voids inside the soap-bubbles filled with anti-matter, pushing out the walls of matter (both normal and "dark") until the walls collide. It is along these collisions between bubbles that we see matter become dense enough to form galaxies, clusters and super-clusters. Anti-matter still has inertia, in other words, positive mass. The mass density of the universe has on average been close to the critical value needed to make the universe flat, and anti-particles with anti-gravity filling the vast voids between super-clusters of galaxies must comprise a large, but variable, component.
irst, an analogy. Existing quantum theory was developed entirely by observation of the interaction between photons and electrons. The theory of the photon could not be derived from the field theory of Electro-Magnetism, i.e. Maxwell's equations. It required new evidence. So it shall be with the graviton. Thus, quantum gravity will be the theory of the graviton, created by observing the absorption and emission of gravitons.
e have been looking at the absorption and emission of gravitons for 30 years, but not recognizing it as such.
This is based on a little noted experiment, reported in Scientific American back in 1970 by Mansinha and Smylie, which shows that the Earth experiences abrupt changes of spin vector on the order of at most 10 milli-arc-seconds per second. These abrupt changes look exactly like the absorption or emission of gravitational quanta, and thus could be the beginning of a theory of quantum gravity.
he field theory of gravity (Einstein's general theory of relativity) gives us misleading advice about the graviton, just as the field theory of EM gave misleading advice about the photon. Einstein's theory predicts that a graviton carries an extremely small amount of energy. So that is the kind of graviton being looked for, and not being found. And it is assumed in classical gravitational theory that a relatively small object can absorb a graviton, which is also not true. It takes a planetary sized object to absorb or emit a graviton.
illi-arc-second jumps in direction and speed of the planetary spin vector have been observed by D.E. Smylie and L. Mansinha. See "The Rotation of the Earth," vol. 225, #6, December 1971, Scientific American, pp. 80-88. There is nothing in geology that could explain this.
agma movements are too slow, and the flow of currents in the liquid metal outer core of the earth cause continuous rather than discontinuous movements in the magnetic pole, with no associated change in the spin vector. The jumps in the spin vector are not caused by earthquakes. Abrupt changes of as much as ten milliseconds in sidereal time have been observed. Thus, Mansinha & Smylie's observations are a mystery...unless they represent the absorption or emission of gravitons of enormous energy. Add to this Bode's Law and we have the beginning of a theory.
ode's law takes the series 0,3,6,12,24, each time doubling the previous number, adds 4 and divides by 10. The result is the mean distance of each planet's orbit expressed in units of AU (which is one for the Earth). Bode's law very accurately describes the orbits of all the planets (and the asteroid belt) except the outer two, Neptune and Pluto. And we know that Pluto is not really a planet, but just a planetesimal captured by Neptune. We also know that in the early years of the formation of the Solar System, the outermost planets would be busy throwing out planetesimals, and each time, moving in a little closer to the sun. So it is easy to imagine that Neptune formed at 38.8 AU, but gradually moved inward to 30.1 AU as a result of tossing out planetesimals. No one has ever come up with an explanation of Bode's law. I suggest it makes our solar system resemble an atom, and we know that an atom's orbits are determined by the theory of photons. Likewise, it seems reasonable that a solar system should be explained by the theory of gravitons, with a touch of chaos thrown in. By the way, this means our Solar System is typical, and the ones currently being found in the year 2001 are atypical. This increases the odds of finding life and intelligence in the universe.
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