Elementary Particles and the World of Planck Scale

Unifications, A Brief History of Physics

			The history of physics is closely linked to the unification of seemingly disparate phenomena (Figure 15-05a). Each stage of unification in turn advanced a new theoretical framework, which provides a deeper understanding of nature (Figure 15-05b). Figure 15-05c shows an in-precise time line with milestones for each important advance. The TOE in the image stands for "Theory Of Everything", which attempts to
Figure 15-05a Unifications	Figure 15-05b Theories [view large image]	Figure 15-05c History of Physics [view large image]	develop a theory encompassing everything in physics instead of merging (unifying) the rather disparate formulations.

• Gravity is the first discipline modernized under the banner of physics mainly because its effects can be visualized (Figure 15-05c1). Around 1680, Isaac Newton asserted that the force of "terrestrial" gravity (which makes apples fall to the ground, and which in Newton's view was a universal force) was the same as "celestial" gravity (the force which keeps planets in motion around the Sun, see Figure 15-05c2). Such a force is long-range. Its effects can be felt at any distance, though attenuated by the square of the distance between the two "gravitating" objects concerned. Newton introduced a new fundamental constant of nature, G, which characterizes the strength of the gravitational force. Gravity is always attractive in contrast to other forces of nature. Newtonian mechanics works in a three dimensional Euclid space (flat space). Time is independent of the system of reference -- it is just a

		running variable. An invariance in such space is the length. It is the same in any inertial frames of reference. Other consequences derived from Newton's formalism include action at a distance, i.e., there is no time delay for the two "gravitating" objects to interact; and determinism, which assumes that events are entirely determined by other, earlier events.
Figure 15-05c1 History of Gravity [view large image]	Figure 15-05c2 Newtonian Mechanics [view large image]

The electric and magnetic fields (Figure 15-05d) were described in terms of the Coulomb's Law (about charge and electric field), the Ampere's Law (about current and magnetic field), and the Faraday's Law (a relationship between electric and magnetic fields). Faraday's observation has inspired J. C. Maxwell to assemble these laws into a consistent set of equations in 1865 and is now known as Maxwell's equations. The disturbance of the electromagnetic fields was subsequently identified as the light wave in optics.

Figure 15-05d Electromag-netism [view large image]

See "Electromagnetism" for details.

Thermodynamics is a branch of physcis developed in the 19th century. It began with the invention of the steam engine (Figure 15-05e) at the beginning of the industrial revolution. It was driven by the need to have a better source and more efficient use of energy than the competitors (among English, French, and German). It was a case where technology drove basic research rather than vice versa. Thermodynamics provides a macroscopic description of matter and energy. Today better insight is obtained by linking the subject with the statistical behaviour of microscopic particles.

Figure 15-05e Steam Engine [view large image]

• In 1887 Michelson and Morley showed conclusively that the velocity of light is constant in all inertial¹ systems of reference. This observation led Einstein to propose the principle of relativity in 1905. This is also known as the Special Theory of Relativity, which treats space and time on an equal footing such that the velocity of light is constant in this four dimensional space-time. It implies that space and time can transform

among each other in different inertial systems of reference. In particular, time ticks slower and length becomes shorter in a fast moving reference frame (with respect to the observer). Another consequence is the finite propagation of interaction, the signal can travel from the past to object A only with a speed less than or equal to the velocity of light. The light cones in Figure 15-05f define the space within which communication is possible. They form the event horizons for a particular time in the past or future. These ideas are in complete variance with the Newtonian Mechanics. However, classical mechanics is still a good approximation for phenomena involving low velocity (in comparing to the velocity of light).

Figure 15-05f Light Cones [view large image]

Starting from the principle of equivalence (Figure 15-05g), which states that the properties of the motion in a free falling non-inertial (accelerating) system in the presence of a gravitational field is the same as those in an inertial (non-accelerating) system , Einstein proposed the General Theory of Relativity in 1915. In general relativity, it is postulated that the curvature of space-time determines gravity. The mass-energy generates the curvature of space-time and particles moving along the geodesic in this four dimensional curved space. The geodesic is the shortest distance between two points. It is a straight line only in Euclidean space (flat space); it would be different in the curved space (Riemann space). [view large image]

Figure 15-05g Principle of Equivalence

Quantum theory was developed in 1925-1926 by Erwin Schrodinger, Werner Heisenberg, and others in response to many inconsistences between expermental data and classical theories. The first difference between classical and quantum theories is that whereas classical theory always deals with continuously varying quantities, quantum theory must also deals with discontinuous or divisible processes. The second difference is that whereas classical theory completely determines the relationship between variables at an earlier time and those at a later time, quantum laws determine only probabilities (Figure 15-05h) of future events in terms of given conditions in the past. By the early 1930s the application of quantum theory to problems involving nuclei, atoms, molecules, and matter in the solid state made it possible to understand a vast body of otherwise-puzzling data and led to predictions of remarkable accuracy.

Figure 15-05h Electron Cloud [view large image]

• Beta decay was a main topic of research in the first decades of the 20^th century. It was found that the electrons produced in the decay has energy always less than the 2.0 m_e available when the neutron transmutes into a proton. It was Wolfgang Pauli who came up with the answer in 1930. He surmised that there must be another particle running away with the "missing

energy". The particle required to do the job must have zero mass and no electric charge to escape detection by the experimenters. In 1933 Enrico Fermi took up Pauli's idea and put it on a respectable footing by introducing a new force called "weak" interaction (manifested by long mean lives of decay in the order of minutes). The hypothetical particle is called neutrino. He proposed that when a neutron changes into a proton it emits a mediating boson called W-, which carries off the negative electric charge and excess energy, while the neutron changes into a proton and recoils. The W- boson then quickly decays into an electron and an anti-neutrino (see Figure 15-05i). Evidence for the existence of the neutrino came in 1953. The W- boson was discovered in 1983.

Figure 15-05i Weak Interaction

• It was well aware in the 1930s that the force that hold the neutrons and protons together in the nucleus has to be short-range, extending only over about the diameter of a nucleus, otherwise it would pull all the other nuclei together. In 1935 Hideki Yukawa suggested that the strong

	nuclear force (Figure 15-05j) must be mediated by the exchange of another kind of force-carrying particle, which became known as pion. His explanation for the short range of the force is related to the uncertainty principle. If the pion has mass, then its virtual existence can last only for a short time. He estimated the mass of the pion
Figure 15-05j Nuclear Energy [view large image]	to be 150 Mev. The actual mass is 140 Mev when it was discovered in 1947. It is now known that the pion is a composite boson, it is not truly a fundamental force carrier.



• While the quantization of particles is adequte for low energy phemonena in molecules and atoms, the quantization of fields is more appropriate for high energy particles involving creation and annihilation. Such theoretical frame work is called quantum field theory, in which matters are represented by fermion fields and the interactions are mediated by gauge bosons. The interaction of electromagnetic and electron fields was the first treated by such kind of formulism, and known as quantum electrodynamics (QED). It was developed by Richard Feynman, Julian

  Schwinger and others (Figure 15-05k1) in the late 1940s. QED makes predictions about the scattering of photons and electrons and other charged particles that agree with experiment to an accuracy of eleven decimal places. All the computations are perfromed by a procedure called perturbation theory, which seems to be very promising for small coupling constant. Each term in the perturbative series can be represented by a graph known as Feynman diagram (Figure 15-05k2). The K+ and factors in the mathematical expression represent the probobility amplitude of the process between points (s56 is the distance travels by the photon). At the vertex (intersection), the likelihood that an electron would emit or absorb a photon is e, where e is the electron's charge and a vector called Dirac matrices to keep track of the electron's spin. Conversely, a process can be computed by drawing graphs and then applying the Feynman rules. The trouble with Feynman's method is that it always leads to infinite

  Figure 15-05k1 A Mini-Conference in QED

  expressions for the loop diagrams such as those in the fourth order electron-electron scattering (diagrams a - i in Figure 15-05k2). However, Feynman and others discovered that these are related only to quantities involving

  mass and charge. It is then figured out that if one simply corsses out these infinite answeres wherever they appear, and substitutes the right, finite answer, all the calculations become sensible again. This procedure is called renormalization. When it works for a theory, that theory is said to be renormalizable. The electroweak interaction and quantum chromo-dynamics are the other examples of renormalizable theory. Unfortunately, quantum gravity is not renoramlizable. New theories are being developed to allow the merger of quantum theroy and the gravitational force.

  Figure 15-05k2 Feynman Diagrams [view large image]

  See some "animated Feyman Diagrams".

• Parity is the operation P that reverses the coordinate (x,y,z) to (-x,-y,-z) of a system (e.g., a particle or collection of particles). It is equivalent to a mirror reflection followed by a rotation through 180^o. The parity can be +1 or -1 for a system depending on whether the corresponding field remains the same or flips a sign, i.e., in simple mathematical expression: P(x,t) = (-x,t). The overall parity is given by the multiplication of its parts, e.g. (+1)x(-1)x(+1) = (-1) etc. It has been long held that parity is conserved before

and after all particle interactions. The processes in which parity is not conserved would look different in the mirror image (+ 180o rotation) world. With some hints from experimental results, T. D. Lee and C. N. Yang pointed out that conservation of parity may be violated in weak interaction. A test was arranged by C. S. Wu to observe the beta decay of cobalt-60 in a magnetic field. It shows a preferred direction for the emitting electrons (the left-handed electrons) and thus validates the hypothesis of parity violation for weak interaction -- the mirror world behaves differently from the real world (see Figure 15-05l). The three Chinese physicists shared the 1957 Nobel Prize for their efforts in identifying this peculiar behaviour in weak interaction.

Figure 15-05l Parity Violation [view large image]

Note that parity will be conserved if there are equal number of electrons in both directions. It is a useful tool to predict permissible process when parity is conserved.

• One of the greatest triumphs of theoretical physics in the second half of the 20^th century was the discovery, made independently by Abdus Salam (1964) and by Steven Weinberg (1967), of a way to describe the weak interaction and the electromagnetic interaction

in one mathematical formalism, as a single force - the electroweak interaction (based in part on works developed previously by Sheldon Glashow and others). The theory requires three intermediate vector bosons with mass to explain the weak interaction. The predicted masses of these bosons were duly observed in experiments at CERN in the early 1980s. A scalar field called the Higgs field is introduced in this formalism to endow mass to the gauge bosons Zo, W+, and W-. The lower limit on the mass of the Higgs boson is estimated to be 113 Gev. At present, there is no experimental evidence in favor of a Higgs boson, nor is there any against (see LHC updates, also "Discovery of Higgs" in July, 2012; and the subsequent Nobel Prize award in 2013). Figure 15-05m shows the Feynman diagrams for interactions in the Standard model including the electroweak and strong interactions. Development of the Standard Model was in limbo for many years until 1971 when Gerard 't Hooft showed that the theory is

Figure 15-05m Interactions in Standard Model [view large image]

renormalizable by employing mathematical techniques such as path integral, gauge swapping, dimensional regularization, and numerical computation.




A similar formalism was developed for the strong interaction in the 1970s. The story starts with three doublets of quark flavors (u, d), (c, s) and (t, b) (See Figure 15-03). It turns out that there are not just six but eighteen distinct quarks, distinguished from each other by colour charge. Each quark comes in three colour charges -- red, green, and blue (they are just labels, not the colour we see). The gauge symmetry between these three colour charges give rise to eight gauge bosons -- called gluons, which remain massless (no Higgs field interaction is necessary). Detailed tests have supported the idea of the strong interaction being mediated by the gluons. This formalism is called Quantum Chromodynamics (QCD). For example, the neutron is a combination of two down quarks and one up quark bound by

Figure 15-05n QCD [view large image]

gluons to form a white composite particle (Figure 15-05n). Together, the electroweak theory and QCD constitute what has become known as the "Standard Model" of elementary particles.



• General relativity and quantum theory describe two extreme of physics - the former is on the very large scale up to the whole universe, while the latter is on the smallest possible scale down to the size of elementary particles. However, there are many instances in which both general relativity and quantum theory are equally important and a common framework is essential. One obvious such instance is in the very early universe, immediately after the Big Bang, when the size is very small and the curvature of space-time were nearly infinite. Much of the difficulty in merging these theories comes from the radically different assumptions that these theories make on how the universe works. Quantum field theory depends on particle fields embedded in a fixed flat space-time, while general relativity models gravity as space-time curvature that changes as mass moves. The most obvious way of

  combining the two (such as treating gravity as simply another particle field in Quantum Gravity) runs quickly into problem of infinite Feynman diagrams. Figure 15-05o summarizes the steps in the evolution of the theory of gravitation. Each step in this chart builds on the successes of the previous one. Newton thought gravity was a force that acted instantly over a distance. Einstein proposed that gravity is just the manifestion of spacetime curvature. Quantum gravity assumes that gravitation is caused by the exchange of particle-like gravitons. Superstring theory identifies gravitation as the exchange of closed strings. According to Lee Smolin, there are three roads leading to the domain of qunatum gravity:

  Figure 15-05o Theories of Gravity [view large image]

  1. Hawking Radiation - By combining the uncertainty principle in quantum theroy and the general relativity concept of black holes, Stephen Hawking showed that black holes are not completely black, but would actually radiate. As a bonus, he also found new connections between gravity and thermodynamics, i.e., entropy (and hence information) is equal to 1/4 of the area (in Planck units) of the black hole's event horizon. Thus, the amount of information that can be contained in that region is finite. This implies that the world must be discrete on the Planck scale, for were it continuous any region could contain an infinite amount of information. See Hawking Radiation, Relativity in the appendix for more details.
  2. Superstring theory - It started out as a generalization of quantum field theory where instead of point particles, string-like objects (with size ~ the Planck length = 10^-33 cm) propagate in a fixed spacetime background. Although string theory had its origins in the study of quark confinement and not of quantum gravity, it was soon discovered that the string spectrum contains the graviton, and that "condensation" of certain vibration modes of strings is equivalent to a modification of the original background. It is suggested that in the "yet to be formulated" fundamental superstring theory such as the M theory, the background dependent apporach should be replaced by a scheme involving dynamic space time. See section on "Superstrings" for more details.
  3. Loop Quantum Gravity - One reason why it has taken so long to construct a quantum theory of gravity is that all previous quantum theories were background dependent. The problem of how to build a quantum theoretic description of a world in which space and time are nothing but networks of relationships was solved only 20 years ago in the 1980s. The relational theory that resulted is called loop quantum gravity. According to this theory, space is made of discrete units in size of the Planck length. In contrast to ordinary geometry, a given region cannot have a volume which is arbitrarily big or small - instead, the volume must be one of a finite set of numbers (similar to some of the quantized quantities in quantum theory). Similarly, time is also made of discrete units called the Planck time (10^-43 sec). The theory offers more than a picture: it makes precise predictions about what would be observed were it possible to probe the geometry of space at distances as short as the Planck scale. See section on "Quantum Foam and Loop Quantum Gravity" for more details.

A few of the other approaches to quantum gravity may turn out to play significant roles in the final synthesis. Among them will be the twistor theory and the non-commutative geometry. They will provide essential insights into the nature of the quantum geometry of space-time. Quantum gravity will emerge as a more fundamental theory since it will possess more explanatory and predictive powers. Figure 15-05p shows the relationship between quantum gravity and the other branches of physics at the limit of the various universal constants, where the gravitational constant G is associated with gravity, the Planck constant is for quantum, and the velocity of light c comes with special relativity. In quantum gravity, all the fundamental units are expressed in terms of G, , and c: Planck length = (G/c3)1/2 = 1.62x10-33 cm, Planck time = (G/c5)1/2 = 5.39x10-44 sec, Planck mass = (c/G)1/2 = 2.17x10-5 gm, Planck energy = (c5/G)1/2 = 1.22x1019 Gev, and Planck temperature = (c5/GkB2)1/2 = 1.42x1032 oK, where kB is the Boltzmann's constant, which relates energy to absolute temperature on the Kelvin scale.

Figure 15-05p Quantum Gravity [view large image]

Figure 15-05q is a group photo for the physicists of yesterday. It was taken in 1927 at the Solvay Conference on Quantum Mechanics, Belgium. Most of the physicists are European males with 2 exceptions from the U.S. and one lady (Madam M. Curie), whoes entry was secured by the fame of pioneering the investigations into radioactivity. Table 15-01b lists all the participants with their nationality, field(s) of study, and the year when the Nobel prize was conferred (if any). Most of their names are linked to various kinds of theory, equation, and formula. It is very difficult to avoid them in a text book for physics. The year 1927 was within a relatively quiet period between the end of World War I (1919) and the onset of Great Depression in 1929. It was the heyday for the developments of quantum theory and relativity. However, all was not well.

Figure 15-05q 1927 Solvay Conference [view large image]

Hitler was on the way to seize power in Germany. Nightmare would soon begin in 1933 when he became Chancellor of the Reich.

Table 15-01b Physicists of Yesterday

Year	Author	Discipline	Subject	Equation(s)
1687	Isaac Newton	Classical Mechanics	Motion of Partilce
1865	J. C. Maxwell	Electrodynamics	Electricity and Magnetism
1872	L. Boltzmann	Thermodynamics	Tendency toward Disorder
1905	A. Einstein	Special Relativity	Constant Velocity of Light
1915	A. Einstein	General Relativity	Gravity - Warpped Spacetime
1927	W. Heisenerg	Quantum Theory	Microscopic Particle
1928	P. A. M. Dirac	Quantum Field Theory	Free Field Equation for Fermion
1973	GSW	Quantum Field Theory	A Model of Elementary Particles

Table 15-01c The Eight Most Important Equations in Physics

		"Property Space". The "Source" is a specific property of matter responsible for generating the "Property Space". The "Metric Generator" is the rule (the mathematical equations - the distorting medium) for its production. The "Mediating Paricle" is the messenger to implement the rule. The "Property Space" is the final product, which could show up in various configurations or states to direct the movement of other matter with similar "Source" in
Figure 15-05r Property Space [view large image]	Figure 15-05s Distorted Image [view large image]	that particular "Property Space".

Interaction	Sources / Fermions	Metric Generator	Mediating Particle(s)	Property Space
Gravity	Energy-Momentum	General Relativity	Graviton (?)	Space-time with coordinates x, y, z, t
Electromagnetism	Electric charges / e-1	Quantum Electrodynamics	Photon	U(1) symmetry with one gauge and state u(p,s)
Weak Interaction	Hypercharge, 3rd Isospin Component / , e-1	Weinberg-Salam Model	Z0, W mesons	U(1)SU(2) symmetry with a total of 4 gauges and isosingle, isodublet states
Strong Interaction	Color charges / 6 left-handed ur,b,g , dr,b,g + 6 right-handed ur,b,g , dr,b,g	Quantum Chromodynamics	8 Gluons	SU(3) symmetry with 8 gauges and triplet state

Table 15-01d Unification of the Four Interactions

Followings are some comments on this new paradigm about unification of the fundamental interactions :


• The gravity property space is the space-time, which has the special property of providing a continuous scaffold to support the other property spaces. The space-time coordinates (x,y,z,t) are the gauges in general relativity. It is the local change of gauge that creates the various interactions. Global change of gauge generates the various conservation laws.


• The trajectory of a "test" particle in space-time is the geodesic (shortest distance between two points).


• The property space for electromagnetic interaction has U(1) symmetry with a single gauge. The complex state u(p,s) in there is similar to the test particle in space-time. It can be resolved into a real part for the electron with negative charge and an imaginary part for the positron with positive charge. The particle could be in a stationary (definite) state, or in mixture if perturbed.


• The state u(p,s) is a function of spin s and 4-momentum p, which may divert the movement of the "test" particle in space-time (from the geodesic).


• In weak interaction, two property spaces are splice together with U(1) and SU(2) symmetry respectively to account for the fact that there is no right-handed neutrino in this world. There are three gauges associated with the SU(2) symmetry with three corresponding mediating particles W and Z⁰.


• The strong interaction property space has SU(3) symmetry with eight gauges. There are eight corresponding meditation particles (the gluons) - two of them mediate the strong interaction while the rest go around to swap color charges. Figure 15-05t depicts the

  	eight gluon states together with a diagram to illustrate the swapping process (exchange of the blue and green color charges). The numerical factor of 1/ signifies 50% probability for each of the possible state.
  Figure 15-05t Gluon States [view large image]

  
  
• The weak interactions between quarks is similar to the lepton's with the massless neutrino replaced by the up quark u (which has mass), and the electron replaced by the down quark d.



Figure 15-05u depicts a summary of all the sources and their fermion carriers using r (red), g (green), b (blue) for the color charges associated with the u and d quarks in strong interaction. The 3rd component isospin in weak interaction is represented by y (yellow) and p (purple) respectively (double dots denotes both strong and weak interactions). The electric charge in electromagnetic interaction is blank. There are altogether 16 fermions including the non-existing right-handed neutrino. The "L" and "R" in the diagram stand for the "Left-handed" and "Right-handed" sectors of the SU(2)U(1) group; while the subscript Y = -(r+g+b)/3 + (y+p)/2 is the generalized hypercharge. The right-handed neutrino ()R is the odd-man/woman out in the standard model. It is there to provide a possible mechanism for the mass of the neutrinos. It does not involve in any of the interactions - that's why it has not been observed (smelling like dark matter ?).

Figure 15-05u Fermion Table [view large image]

The hypercharge was invented to explain the state of the electron and neutrino prior to Spontaneous Symmetry Breaking (SSB), which endows mass and charge Q to these particles. After SSB, Q = Y + t3. For example, t3 = (1/2, -1/2) for (, e)L and (u, d)L while t3 = 0 for (e)R.



• The "Metric Generator" is obtained by demanding invariance of the equations even though the gauge in the property space is different at each point in space-time, i.e., the change of the gauge is at local level (not global). The mediating particle emerges naturally from such formulation. This is the "Beauty" (about symmetry) referred to in the book on "A Beautiful Question".



This new paradigm on unification of the fundamental interactions allures to a central entity which is viewed through various distorting media resulting in different interactions. It turns out that some entity with SO(10) symmetry would accommodate all the 16 elementary fermions. Within this unified group, all the members are equivalent and they are supposed to interact under one kind of force. The task now is to provide detail of the underlying dynamics and comes up with some theoretical predictions to validate the hypothesis. Regrettably, gravity is not included in this scheme. Anyway, the situation is now likened to a dodecahedron (corresponding to the central entity) broken into

Figure 15-05v Dodecahedron Analogy [view large image]

un-recognizable pieces (Figure 15-05v) - the various property spaces.

also tried to show that all the interactions would merge in the tiny region (corresponding to high probing energy) within which the "central entity" resides. The attempt is not very successful (see left diagram of Figure 15-05w). However, the improvement is drastic (including gravity as shown in the right diagram of Figure 15-05w) if supersymmetry (SUSY) is introduced into the consideration. But none of the partner particles predicted by SUSY has ever been detected despite the deployment of many modern technologies (see dark matter as SUSY/WIMP or LHC updates).

Figure 15-05w Domain of Unification

MSSM = Minimal Supersymmetric Standard Model, and see more beauty in "Conservation Rules".

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