Beyond the God Particle

by Leon M. Lederman

  We might ask, “If a moving car consumes energy, where did the all that energy go?” If you ask that question, then you have indeed learned our all-important lesson about energy—energy is conserved and cannot be created or destroyed—it therefore must have gone somewhere else. In the case of our car, the kinetic energy is lost through friction between mechanical parts, into heating the engine, through the sound energy the car produces, into the energy content of the air that the car moves around as it travels down the highway, and into the energy of heating and compressing and deforming the tires as they spin around. However, most of the wasted energy goes into heat, increasing the speed of the molecules of water (engine coolant), tires, the road, etc. Since this is chaotic and random molecular motion, it is virtually impossible to usefully recover this energy.

  4. In relativity we find the energy and momentum of the moving particle

  where we have written the Newtonian expressions for comparison. We can see the “Newtonian limit” and the nonzero rest energy implied for zero velocity, v = 0, where the Einstein formulas go into

  In special relativity, we can never get the speed of a massive particle (one with nonzero inertial mass m) to equal the speed of light. As the momentum and the energy become infinite. It therefore would require an infinite energy to accelerate a proton to the speed of light. E = mc2 holds strictly for a particle at rest.

  At the CERN LHC at full design energy we accelerate protons to seven trillion electron volts. The rest mass energy of a proton is about one billion electron volts. Hence, the LHC “boosts” a proton to have a “Lorentz factor” of about 7,000. This means that approximately (v/c)2 = 0.99999998, or that the LHC accelerates protons to 99.999999 percent of the speed of light.

  How, then, can anything travel at the speed of light? We see that if we take and also allow our particle to be massless, then the energy is actually indeterminate, that is, we get, E = 0/0. However, this allows for the possibility that a massless particle, something with no inertial mass, can have finite energy and momentum. If we look at the relationship between energy and momentum, we see that a massless particle must satisfy: . Indeed, this describes the particles of light, the photons. Photons have absolutely no inertial mass, yet they carry energy and momentum as they travel through space. Photons travel forever at the speed of light. They cannot be at rest, or have a finite velocity less than c, for then their energy would be zero.

  5. For a short yet poignant biography of Emmy Noether, see chapter 3 of Symmetry and the Beautiful Universe by Lederman and Hill. Also, the “conservation of energy” and its relationship to the “constancy of the laws of physics in time” are explained by the amusing example of the “ACME Power Company” in chapter 2.

  CHAPTER 5. MASS UNDER THE MICROSCOPE

  1. To measure the mass of a water molecule, we start by looking up the mass of 1 mole of hydrogen, which is listed on the Periodic Table of the Elements as about 1 gram. A mole of oxygen is listed as 16 grams. Therefore, an H2O molecule has 18 grams per mole. A “mole” is a definite number of particles, about 6 × 1023 particles. This is “Avogadro's number,” and determining this number makes the basic connection between aggregate matter and the number of atoms per unit mass comprising it (we're rounding things to one significant digit). So the mass of a single H2O molecule is 18 grams/ (6 × 1023), or 3 × 10–23 grams, hence 3 × 10–26 kilograms.

  In this way, with more care, we can measure the mass of the proton, the neutron, and—with more effort—even the lowly electron, as well as masses of all sorts of atomic and subatomic things. We often quote these masses as an energy, using Einstein's E = mc2. We did this earlier for the electron (about 0.5 million electron volts, or MeV, of mass in equivalent energy units) and our celebrant, the muon (about 105 MeV).

  Sometimes you'll see a mass quoted in MeV/c2, or GeV/c2, which rewrites Einstein's formula as m = E/c2 and installs the factor of /c2. Most physicists simply use units in which c = 1 and Planck's constant h-bar = h/2π = 1, and then use electron volts, eV, as the measure of mass, distance, and time. For example, 1 GeV (Giga electron volts) corresponds to (0.2 × 10–13 cm)–1.

  2. See “Galaxy,” http://en.wikipedia.org/wiki/Galaxy (site last visited 4/1/2013).

  3. See chapter 1, note 13; see also “Superconductivity,” http://en.wikipedia.org/wiki/Superconductivity, “Ginzburg–Landau Theory,” http://en.wikipedia.org/wiki/Ginzburg%E2%80%93Landau_theory, “John Bardeen,” http://en.wikipedia.org/wiki/John_Bardeen, “Leon Cooper,” http://en.wikipedia.org/wiki/Leon_Cooper, “Robert Schrieffer,” http://en.wikipedia.org/wiki/Robert_Schrieffer, “V. Ginzburg,” http://en.wikipedia.org/wiki/Vitaly_Lazarevich_Ginzburg, “Lev Landau” http://en.wikipedia.org/wiki/Lev_Landau (sites last visited 4/1/2013).

  In a superconductor the vibrations of the lattice of atoms that make up the material interact with the electrons. This interaction is very delicate, but at ultra-cold temperatures, it causes the electrons to pair up, forming little particles that are like “two-electron atoms” and that behave quite differently than unpaired electrons. These are called Cooper pairs, and they form a kind of quantum soup inside of the superconductor. This quantum soup is the structure of the vacuum itself, the state of lowest energy, inside of a superconductor. When a photon then enters the superconductor it interacts with the soup. This causes the photon to blend with the soup and behave exactly like a photon that has a mass. The massive photon can be brought to rest in the superconductor, but in fact it is really the original massless photon and soup combination that is brought to rest.

  4. See “Jeffrey Goldstone,” http://en.wikipedia.org/wiki/Jeffrey_Goldstone, “Giovanni Jona-Lasinio,” http://en.wikipedia.org/wiki/Giovanni_Jona-Lasinio, “Yoichiro Nambu,” http://en.wikipedia.org/wiki/Yoichiro_Nambu (sites last visited 4/2/2013).

  5. This is a quantum phenomenon. The rapid flip-flop between L and R chiralities is associated with a rapid oscillating behavior of the muon at rest. This was first realized when people considered the resting state solutions to Dirac's equation that described quantum states of electrons, muons, etc. in a manner consistent with relativity. It was given a fancy German name: “Zitterbewegung.”

  One might be worried that the L-R-L-R oscillation involves a rapid change in momentum from east to west, back to east, etc. For a localized particle, one at rest, this rapid fluctuation in momentum can be viewed as consistent with Heisenberg's uncertainty principle, Δp Δx
  CHAPTER 6. THE WEAK INTERACTIONS AND THE HIGGS BOSON

  1. Leon Lederman, Melvin Schwartz, and Jack Steinberger received the Nobel Prize in Physics in 1988 for their research that revealed the “flavors” of neutrinos. The neutrinos are paired with corresponding charged leptons. There are therefore three charged leptons: electron, muon, and tau, and three associated neutrinos:

  2. Heisenberg's uncertainty principle: The uncertainty principle implies that if we try to localize any particle in space within a very small region of distance, Δx, the uncertainty in position, then the uncertainty in the x-component of the momentum of the particle, Δpx, will grow larger, becoming at least as big as Δpx ≥ h/2πx. Similarly, if we want to localize some event in a system within a tiny time interval, Δt, then we will necessarily disturb the system and cause a range in its energy of ΔE, where ΔEΔt ≥ h/2π, so the smaller we make Δt, the larger becomes ΔE, as ΔE ≥ /Δt. The atomic orbitals of electrons have a typical size in most atoms of roughly Δx ≈ 10–10 meters in any given direction in space. Therefore, electrons must, by the uncertainty principle, have a range of momentum within their orbitals that is as large as Δpx ≥ /Δx, hence, Δpx ≈ 10–24 kilogram-meter/second. Electrons move in their orbitals with velocities that are much less than c (i.e., they are non-relativistic), and the electron mass is known to be me ≈ 9.1 × 10–31 kilograms. Therefore, we can estimate the typical elect
ron kinetic energies to be of order, E ≈ (Δpx)/2me ≈ 6 × 10–19 joules, or about 3.8 electron volts (1 electron volt = 1.6 × 10–19 joules; we have done a lot of “rounding off” to do this “back-of-the-envelope” estimate). The force that holds the electrons in their orbitals must therefore provide a negative potential energy that exceeds, in magnitude, this result. This is the electromagnetic force, and the typical scale of the binding energies of electrons in an atom (the energy we must supply to liberate them) is of this order, ranging over about 0.1 to 10 electron volts. In fact, this is the typical energy scale of all chemical processes, and it contains the typical energies of visible light photons.

  A “quantum fluctuation” is a bit like a “thermal fluctuation.” It is physically possible for a thermal system, like a hot gas in a room, to suddenly fluctuate in density and pressure—even the extreme fluctuation of all of the gas momentarily condensing onto the floor then evaporating back into the room is physically possible, but such a thing is ultra-ultra rare. Note that a top quark is heavy enough that the top can directly decay, converting to a b-quark and a W+, without requiring the uncertainty principle.

  3. In any physical process, when a direction in space (called a vector) becomes correlated with a spin or a magnetic field (which is a “pseudovector”; the mirror image of a pseudovector is opposite that of a vector), then there is parity violation, since in the looking-glass house the correlation would be opposite; i.e., for Leon's experiment, in the looking-glass house the electrons would come out in the opposite direction of the muon spin. In Madame Wu's version of the experiment, the electrons coming out of 60Co decay were aligned with the direction of the magnetic field used to align the spins of the nuclei.

  4. The relationship is simple math, if you have had calculus: the electric and magnetic fields are the particular derivatives of the gauge field in space and time.

  5. Sheldon Glashow, “Partial Symmetries of the Weak Interactions,” Nuclear Physics 22 (1961): 579–88.

  6. Steven Weinberg, “A Model of Leptons,” Physical Review Letters 19 (1967): 1264–66.

  7. Gerard ‘t Hooft and Martinus Veltman, “Regularization and Renormalization of Gauge Fields,” Nuclear Physics B44 (1972): 189–213.

  8. In particular, fermions are spin-1/2, or “half-integer” spins. See Leon M. Lederman and Christopher T. Hill, Quantum Physics for Poets (Amherst, NY: Prometheus Books, 2011) or the discussion of spin in the Appendix.

  9. See “Satyendra Nath Bose,” http://en.wikipedia.org/wiki/Satyendra_Nath_Bose (site last visited 1/23/2013).

  10. This is a complicated phenomenon that involves the interactions of the electrons in the atoms—for most magnetic materials the atoms align but cancel the overall field to zero. Iron prefers a ground state in which there is an exact common alignment, and we get a big magnetic field emanating from the iron.

  CHAPTER 7. MICROSCOPES TO PARTICLE ACCELERATORS

  1. This news bulletin was from CNN Tech: http://articles.cnn.com/2009-11-21/tech/cern.hadron.collider_1_large-hadron-collider-lhc-cern?_s=PM:TECH (site last visited 1/23/2013).

  2. Some early history of microscopes can be found here: http://en.wikipedia.org/wiki/History_of_optics; http://en.wikipedia.org/wiki/Magnifying_glass; http://www.history-of-the-microscope.org/ (sites last visited 6/21/2013).

  3. See “Zacharias Jannsen” and references therein, http://en.wikipedia.org/wiki/Sacharias_Jansen; “History of the Microscope” http://www.history-of-the-microscope.org/hans-and-zacharias-jansen-microscope-history.php (sites last visited 1/23/2013).

  4. Miscellaneous references on the history of microscopes: http://inventors.about.com/od/mstartinventions/a/microscope.htm (site last visited 1/23/2013); R. M. Allen, The Microscope (New York: D. Van Nostrand Company, Inc., 1940); The S. Bradbury, Evolution of the Microscope (Oxford: Pergamon Press, 1967); W. G. Hartly, The Light Microscope (Oxford: Senecio Publishing Company, 1993). See also “Telescope,” http://en.wikipedia.org/wiki/Telescope; Henry C. King and Harold Spencer Jones, The History of the Telescope (Courier Dover Publications, 2003).

  5. See “Antonie van Leeuwenhoek,” http://en.wikipedia.org/wiki/Van_Leeuwenhoek (site last visited 1/23/2013); Alma Smith Payne, The Cleere Observer: A Biography of Antoni van Leeuwenhoek (London: Macmillan, 1970).

  6. Anton van Leeuwenhoek, Letter of June 12, 1716. The letter and short biography can be found here: http://www.ucmp.berkeley.edu/history/leeuwenhoek.html (site last visited 1/23/2013).

  7. See “Robert Hooke” and references therein, http://en.wikipedia.org/wiki/Robert_Hooke (last visited 1/23/2013).

  8. A beam of light bends as it obliquely hits water or glass. This bending of light by transparent materials is called refraction. The amount of refraction is controlled by the “index of refraction” of the medium the light is exiting (e.g., air) and that the light is entering. The index of refraction varies with light wavelength. This is the basis of the phenomenon of a glass prism that splits the white light into its spectral constituents: Red-Orange-Yellow-Green-Blue-Indigo-Violet (ROY G. BIV). White light is therefore composed of equal amounts of the different colors of light. We can take the colors of light and combine them to make white light. The chromatic aberration is mainly the prism effect of the glass lens.

  9. See “Joseph Jackson Lister,” http://en.wikipedia.org/wiki/Joseph_Jackson_Lister; see also “Lens,” under “Optics” heading,” http://en.wikipedia.org/wiki/Lens (sites last visited 1/23/2013). Lister published his work in 1830 in a paper titled “On Some Properties in Achromatic Object-Glasses Applicable to the Improvement of the Microscope,” submitted to the Royal Society.

  10. Lenses and aberrations are thoroughly discussed here: http://en.wikipedia.org/wiki/Lens_%28optics%29#Compound_lenses, and here: http://en.wikipedia.org/wiki/Compound_lens (sites last visited 4/10/2013).

  11. D. Edwards and M. Syphers explain this and much of optics with some simple matrix algebra in an elegant book: An Introduction to the Physics of High Energy Accelerators (Wiley Series in Beam Physics and Accelerator Technology) (Wiley, 1992), pp. 60–65.

  12. Let's review the physics of a wave: Consider a long traveling wave as it moves through space. We can visualize this as a freight train moving by as we are stopped at a railroad crossing. A traveling wave is sometimes called a wave train, with many sequential crests and troughs of the train as it traverses space. Such a wave is described by three quantities: its frequency, its wavelength, and its amplitude. The wavelength is the distance between two neighboring troughs or crests of the wave. The frequency is the number of times per second the wave undulates up and down through complete cycles at any fixed point in space. If we think of the wave as a long freight train, its wavelength is then the length of a boxcar. Its frequency is the number of box cars per second passing in front of us as we patiently wait for the train to pass. The speed of the traveling wave is therefore the length of a boxcar divided by the time it takes to pass, or (speed of wave) = (wavelength) times (frequency). Thus, knowing the speed, the wavelength and frequency are inversely related, or (wavelength) = (speed of wave) divided by (frequency) and (frequency) = (speed of wave) divided by (wavelength). The amplitude of the wave is the height of the crests, or the depth of the troughs, measured from the average. That is, the distance from the top of a crest to the bottom of a trough is twice the amplitude of the wave, and it can be thought of as the height of the boxcars. For an electromagnetic wave, the amplitude is the strength of the electric field in the wave. For a water wave, twice the amplitude is the distance that a boat is lifted from the trough to the crest as the wave passes by. Figure 2.1 says it all. The color of a visible light wave was understood in the nineteenth-century Maxwellian theory of electromagnetism to be determined by the wavelength (and, inversely, the by frequency). If we take the frequency to be small, we correspondingly find that the wavelength becomes large. Longer-wavelength visible light is red, while shorter-wavelength visible light is blue. For graphic display in color of the various wavelengths of light, see http://science-e
du.larc.nasa.gov/EDDOCS/Wavelengths_for_Colors.html#blue (site last visited 3/26/2013).

  13. See “Louis de Broglie,” http://en.wikipedia.org/wiki/Louis_de_Broglie; see also “Davisson–Germer experiment,” http://en.wikipedia.org/wiki/Davisson%E2%80%93Germer_experiment (site last visited 4/10/2013).

  14. According to Newton, the magnitude of the force of gravity exerted upon object a by object b is called Fab and is given by the formula:

  where R is the separation between them. This is an example of an inverse square law force, that is, a force that falls off in magnitude, or strength, with distance, like 1/R2. The electric force between two stationary electric charges is also an inverse square law force.

  In this formula ma is the mass of object a, and mb is the mass of object b. This means that the force of gravity is stronger between two very massive objects than it is between two very low-mass objects. For example, if a is the earth, we substitute ma = mEarth, and if b is the sun, we substitute mb = mSun into the formula. Thus, if we could somehow double the mass of the sun, holding everything else fixed, then the force of gravity that the earth would experience from the sun would become doubled, and the earth's orbit would change, becoming a tighter ellipse with a smaller average distance from the sun. Technically, the force is a vector and must therefore also have a direction. We could write a better formula that illustrates that, but words suffice. Object a experiences the force of gravity, with the magnitude we have written, but the force points as a vector at the direction of object b. And, by symmetry, object b experiences the same magnitude of force, which points in exactly the opposite direction, back to object a.

  The quantity GN in the numerator of the formula is a fundamental constant. Newton had to introduce this factor in order to specify the strength of the gravitational force. We call this Newton's gravitational constant or just Newton's constant, for short. GN is measured from experiment and takes the value GN = 6.673 × 10–11 (meters3) / (kilograms seconds2). We have quoted GN in the meter-kilogram-second system of units. Indeed, we can write, in nonscientific notation, GN = 0.00000000006673 (meters3) / (kilograms seconds2), and we see that GN is a seemingly very small number. Gravity, despite its ubiquitous character in nature, is actually a very feeble force. To get a sense of this, we can estimate that the force of gravitational attraction between two fully loaded oil tankers that are ten miles apart is about the same as the force you feel holding a gallon of milk due to the pull of gravity by the entire earth. For more discussion of gravity, see our book Symmetry and the Beautiful Universe (Amherst, NY: Prometheus Books, 2007).