THE DEEPER LAWS OF PHYSICS

Quantum Mechanics, Relativity, Standard Model, General Relativity, Big Bang, Black Holes, Superstrings
Lectures 24,25,26, by I.Bars

As we learned in lecture-10 matter is made up of atoms, and as further discussed in lecture-11 macroscopic matter comes in the form of solids, liquids, vapors. There is an immense abundance of varieties of matter. The quest for understanding its structure dates back for at least a couple of millenia. The alchemists' efforts resulted in the science of chemistry which in turn led to the idea of the "elements", that is the atoms. Although it was thought at first that atoms are indivisible, we now know that there are many more layers of structure: (atom) / (nucleus) / (protons & neutrons and other "elementary" particles) / (quarks and leptons, gluons, photon, W^+- , Z, Higgs) ... The Standard Model for the structure of matter and fundamental forces provides the correct description of the subatomic world up to 10^-18m (one thousandth the size of the proton) in spectacular agreement with experiment (see lectures 14-17).

How precisely does a chunk of matter hold together starting with its smallest constituents, and how it interacts with other matter through its constituents? To understand this question we need to know what are the fundamental forces that act on matter, and how matter moves under the influence of forces. We have already seen that we are aware of four apparently distinct forces: gravity, strong, weak, E&M. The apparent distinction between these forces seems to disappear and they seem to merge into a single unified force as we peek deeper and deeper into the structure of matter at shorter distances (see lectures 14-17).

So far we have discussed little the mechanics of motion and the interactions of constituents under the influence of these forces, but understanding this is a large part of constructing and using the consequences of the theory that explains it all.

In the subatomic world the rules of mechanics are different than those of the macroscopic world (Classical Mechanics, Classical E&M). The physics rules that apply to the macroscopic world are understood as a limit of the more correct rules of the microscopic world. Three limits need to be considered.

The first is related to small versus large quantities of energy. Objects in the macroscopic world have huge amounts of energy compared to energies of particles in the microscopic world. In the microscopic world the small quantities of energy (and angular momentum, etc.) are often measured in discontinuous quantum jumps; this discontinuity is not perceived in the macroscopic world because of the vast amounts of energy (many quanta). The quantum of energy is measured in multiples of a fundamental constant called the Planck constant (h=6.626 068 76 x 10^-34 J s ). For example, the energy of a single photon (the quantum of light) is determined by its frequency times the Plank constant (E=hn). Quantum Mechanics describes the correct rules that apply in the subatomic world.

The second is the speed of particles relative to the speed of light. The correct rules of motion need to take into account that nothing can move faster than the speed of light (c=300,000 km/s) in vacuum (in a flat space-time), and also that all observers, whether they are static or in motion measure the same speed of light in vacuum. Most objects in the macroscopic world move extremely slowly relative to light therefore the presence of this limit is not perceived in the macroscopic world. However the speed of light must be taken into account in order to correctly describe the fast moving objects of the subatomic world - this is described by Einstein's Special Theory of Relativity.

The third is the curvature of space-time. Spacetime is curved in the presence of very large mases or very large energies. The presence of curvature manifests itself in the form of the gravitational force. The everyday macroscopic world, as well as the subatomic world as decribed by the Standard Model up to 10^-18m, are only very slightly curved. Therefore the approximate rules of an almost "flat" spacetime apply in those circumstances. However there exists huge space-time curvatures in the vicinity of black holes, at the early instances of the Big Bang, and in the vicinity of huge astronomical objects. The correct rules of physics must take into account the curvature of space-time. The radius of curvature is measured by a very small distance called the Planck length =10^-35m. Flat space-time corresponds to a very large (infinite) radius of curvature (imagine the surface of a huge sphere that appears almost flat). Einstein's General Theory of Relativity gives the correct laws of physics in curved space-time.

Thus there are three fundamental constants that control the relative importance of the quantum, the speed, and the radius of curvature

Planck constant	h = 6.626 068 76 x 10^-34 J s
speed of light	c = 300,000 km/s
Planck length	L_Planck= 10^-35m

We recover the physics rules of the macroscopic world (Newton's laws, E&M, etc.) from the more correct rules of the microscopic world when we take the limit of many quanta, slow speeds, and negligible spacetime curvature. The Standard Model takes into account Quantum Mechanics and Special Relativity, but not General Relativity because the curvature (i.e. gravity) is negligible for the subatomic phenomena described by the Standard Model at 10^-18m. But at distances of the order of the Plack length 10^-35m, gravity is as important as the other forces, therefore General Relativity together with Quantum Mechanics must play a role - but there is clash between these two pillars of 20th century physics - the resolution is not well understood yet. Superstring Theory appears to be a likely answer.

Quantum Mechanics (see Hewitt ch.31)

The story of quantum mechanics begins with Planck's discovery (1900) of the quantum nature of energy and its relation to the Planck constant. Planck thought that the energy of vibrating molecules of hot matter emitted quantized bunches of light. Soon afterwards (1905) Einstein showed that light itself had to come in small particle-like lumps called photons (with energy E=hn and momentum p=hn/c) in order to explain the photo-electric effect (ejection of electrons from metals when light shines on them. see ch.31 ). Quantum Mechanics gained acceptance when the Bohr model explained the spectroscopy of light emitted by atoms (ch.32). But this crude picture had to be explained as being the result of deBroglie's particle-wave duality, and more deeply as a concequence of Heisenberg's uncertainty principle, finally coming all together in Shrodinger's equations for the atom. The quantum rules were finally understood during the 1920's.

De Broglie particle/wave duality wavelength = h / (momentum) L=h/p or h/(mv)	Heisenberg Fuzziness of the quantum wave creates uncertainty in position and/or momentum.
Macroscopic object with large mass has small quantum wavelength relative to its physical size. Therefore quantum properties are negligible in the macroscopic world.	Microscopic object with small mass has appreciable quantum wavelength comparable with its physical size or even bigger. Therefore quantum mechanics is essential for the subatomic world

Quantum mechanics, which is one of the pillars of 20th century physics, provides the rules of mechanics that govern the subatomic world. Atomic & Molecular Physics, Condensed Matter Physics, Nuclear Physics, Particle Physics extend our understanding of the universe from 10^-10m to 10^-18m, both experimentally and theoretically, and they all rely on quantum mechanics. The understanding gained in this domain has widespread applications and has been an immense resource for technological innovation, from medical imaging and thereapy, to lasers, to computers, to new materials, to nuclear energy, and beyond ...

Quantum Mechanics - a brief tour: from spectral lines to quantum energy levels, and the relation of spectral lines to photons of definite frequency. The Bohr theory of the atom (ch.32)

Wave-particle duality and quantization of atomic energy levels. DeBroglie, Schrodinger. Two slit experiment.

Heisenberg's uncertainty principle. Dx Dp > h/(2(pi)), DE Dt > h/(2(pi)), etc. An applet that illustrates it.

Video: An Answer for Everything (Stephen Hawking's Universe), min.22-30 on Quantum Mechanics and chance.

Special Theory of Relativity (see Hewitt Ch.35)

Einstein's Special Theory of Relativity describes the correct rules of fast moving objects both in the macroscopic and subatomic world. It was developed by Einstein from two postulates (which ended up being correct): (1) The laws of physics are the same in any frame of reference as observed in that frame of reference, (2) The speed of light is always the same constant c from the point of view of all frames of reference. Surprisingly, the laws of physics may seem different from the point of view of a moving observer(but in a predictable way) in particular the concepts of time and space undergo a revolution.

The mass of a particle at rest is called its rest mass m. When the particle moves it has momentum p and energy E=(p²c²+m²c⁴)^1/2. Think of a right triangle with hypothenus E and perpendicular sides pcand mc². The speed of the particle is c multiplied by the ratio of the side pc to the hypothenus E. Now see what happens as one side gets much larger than the other, you approach the extreme relativistic (fast) or extreme non-relativistic (slow) limits

momentum p	energy E=(p²c²+m²c⁴)^1/2.	speed v=c (cp/E)
fast m=small or p=large	fastest (m=0 or p=infinite) E=pc (e.g. light)	maximum speed v_max=c (e.g. light)
slow m=large or p small	E=mc²+mv²/2 static E=mc²	v=p/m (Newton)
Addition of velocities parallel or antiparallel	V=(v₁+v₂)/(1+v₁v₂/c²)	if v₁=c then V=c

The particle at rest (p=0) has energy E=mc². The speed of the particle is given by v=c(cp/E). Therefore the maximum v is c, and it can be attained only by massless particles. A massless particle cannot stop.

The maxium attainable speed is the speed of light. Light is described as a massless particle (the photon). Any particle that has zero mass cannot slow down and must move with the speed of light. The neutrino is another particle whose mass is zero or almost zero, hence it moves at (or almost at) the speed of light.
A particle that has mass can never attain the speed of light although it can come closer and closer to it as its momentum is increased. Thus, macroscopic objects that have huge masses compared to an electron, have a hard time attaining large enough energies, and therefore move at speeds much smaller than the speed of light.

Any observer, whether static or moving, measures the same speed of light. This is counter-intuitive because it is unlike the relative motion of objects in our everyday life. If you throw a particle with a speed v₁ from a spaceship moving at a speed v₂ the total speed V as observed outside of the spaceship is V=(v₁+v₂)/(1+v₁v₂/c²).

Space contraction and time dilation. A static observer compares identical rods one moving and one static, he also compares identical watches one moving and one static. He observes a shorter moving rod compared to a static rod (space contraction) and he measures a longer time interval compared to an observer on the ship (time dilation). If the moving rod has length L₀ in its own frame of reference and the moving watch measures a time interval t₀ in its own frame of reference, then as observed by the static observer the moving rod has shorter length L and the moving watch ticks at a longer time interval t. However, an observer moving along with the rod or the watch measures L₀ and t₀ . Thus, two twins, one static and the other travelling on a spaceship age differently - when the traveller returns they find out that the Earth-bound twin is older.

Space contraction: L=L₀(1-v²/c²)^1/2.

Time dilation : t = t₀/(1-v²/c²)^1/2.

A moving tennis ball moving at high speeds looks like a pancake. At the speed of light the pancake looks like zero thickness.
To a traveller moving at the speed of light the distance between our solar system and the center of the galaxy (or any other distance) looks like zero.
If you sit inside a chamber that revolves at high speeds, and then get out, you see that your twin who was outside is older.
An elementary particle that decays, such as a muon or a neutron, lives longer when moving at high speeds in accelerators.
As observed by us, light from the center of the Milky Way galaxy reaches the solar system in 25,000 years, but a traveller moving at the speed of light measures 0 seconds for the same trip.

Time, like space, is not absolute. It depends on the observer. Events that appear simultaneous in one frame of reference are not simultaneous in moving frames of reference. Each observer has his own space-time.
E=mc^{2 .}Energy and mass are two facets of the same thing. Mass can be created from energy, e.g. high frequency photon near nucleus creates electron + positron out of the vacuum. Mass dissapears to create energy, e.g. electron+positron beams at SLAC annihilate to give photons, those can convert to other matter. Other examples are fission and fusion.

The twin paradox. (download to your computer and play).
see Twin trip animation at physicsplace.com, ch.35
see Spacetime travel at physicsplace.com, ch.35

Relativistic Quantum Field Theory (RQFT) and the Standard Model

Quantum Mechanics and Special Relativity are married in Relativistic Quantum Field Theory (RQFT). This is the framework for describing the laws of physics that govern particle physics, quarks, leptons, photons, gluons, W^+- , Z, Higgs. In the RQFT framework a very precise and highly successful theory of the subatomic world that explains the origin and behavior of the three forces (Electro-magnetism, Weak, Strong) has been developed. It is called the Standard Model of Particle Physics (proposed in 1967). This is now an experimentally established theory (experiments during the past 25 years) - see lectures 14-17. It is verified spectacularly by doing precision experiments in giant accelerators. A large number of Nobel prizes were awarded during the last 25 years to both theoretical and experimental physicists that made the most important contributions in the development of the Standard Model.

One aspect of the Standard Model still remains in relative obscurity: the Higgs field or particle - this field describes the phase transition that is responsible for generating the masses of all elementary particles, i.e. quarks, leptons, W,Z. The issue of mass, and the associated Higgs field, is expected to be experimentally clarified in 2006 when the Large Hadron Collider (LHC) at CERN will begin its operation.

Because of the high energies required to penetrate to very short distances, this branch of physics is called High Energy Physics. Like the other areas of physics, High energy Physics has yielded some benefits to mankind, and more is expected.

The Standard Model does not include the gravitational force. The reason is that the particles that play the main role in the phenomena described by the Standard model, up to one thousandth the size of the proton, (quarks, leptons, photons, gluons, W^+- , Z, Higgs) have very small masses (or small relativistic energies) and therefore they experience a very small gravitational force. This gravitational force is negligible compared to the other forces that play the main role in the phenomena described by the Standard Model, therefore it is legitimate to neglect it within the confines of the phenomena described by the Standard Model. However, in principle the gravitational force is present, and indeed is expected to compete with the other three forces at much deeper distances inside matter. Therefore, the Standard Model is not a complete theory; one must look beyond the Standard Model for a full understanding of the fundamental processes.

General Theory of Relativity

The gravitational force is explained by Einstein's General Theory of Relativity (GR). The force is related to the curvature of space-time. When the curvature is small (radius of curvature large) the description of gravity given by General Relativity approaches Newton's Law of Universal Gravity. GR is very successful in describing gravity in the macroscopic world for many phenomena where Newton's laws of gravity fail: bending of light, expansion of the universe, black holes, and a wide variety of astronomical observations.

If you move an object (e.g. a star) from one location to another, its gravitational force on a distant object (e.g. another star) would change. According to Newton's laws of universal gravity (see lectures 8,9) this information would be instantly known by all other objects in the universe. Einstein reasoned that Newton's laws of gravity that described the gravitational force as an instantaneous action at a distance could not be correct because information cannot travel faster than the velocity of light. He struggled for 10 years to find a description of gravity compatible with the Special Theory of Relativity.

Principle of equivalence: Accelerating frames of reference duplicate the effects of gravity. Consider being inside elevator or spaceship, feel the floor, observe an object thrown horizontally. The principle holds for all natural phenomena, including optical and E&M phenomena - so light must bend.

Bending of light - it follows a geodesic - equivalent to being attracted by gravity. Seen during solar eclipse. Light bends in the Earth's gravitational field.

Gravitational redshift. Light emitted by a massive object is attracted back to it, this stretches its wavelength, this is a redshift. Therefore, clocks slow down in gravitational fields. You live longer on the ground floor of a skyscaper compared to the top floor. Atoms emit redshifted radiation in stronger gravitational fields (light from astronomical objects).

Einstein proposed General Relativity in 1916. Spacetime deformed by gravity. Spacetime fabric created by mass or energy. Roll a ball on a streched sheet, the geometry and curvature keeps changing. The geometry is non-Euclidean. A deformed spacetime is equivalent to gravity.

Geodesic - shortest path between two points in curved spacetime.

Gravitational waves. Created by accelerating mass or energy disturbing the fabric of spacetime. LIGO is setup to observe them, hopefully soon.

Black holes. Created by a star that collapses under its own gravitational attraction. Light inside event horizon cannot leave due to the strength of the attraction.

Deviations from Newtonian gravity are measured in agreement with GR. Large effects in strong gravitational fields, small effects in weak gravitational fields.

Unification of Forces

Beyond the Standard Model, a search for a unified theory of all forces, including gravity, has been one of the most exciting quests in theoretical physics during the past 25 years. Important new ideas emerged during this period, including Grand Unified gauge theories (GUT) and "supersymmetry" and supergravity. An incompatibility exists between the two main pillars of 20th century physics: General Relativity and Quantum Mechanics cannot coexist in the framework of RQFT that explains the other three forces and all matter up to 10^-18m. The resolution of this crisis has consumed the theoretical physicist's efforts during the past 25 years.

Superstring Theory

Today, Superstring Theory (since 1984) and its extension into M-theory (since 1995) is the best (but unconfirmed) candidate for the unified theory. This is the only theory that has managed to marry General Relativity and Quantum Mechanics and it contains RQFT as a limit from strings to points. If correct, Superstring Theory allows us to extend our understanding of the universe all the way down to 10^-35m. It is expected that it will be able to explain what precisely happened at the instant of the Big Bang, resolve some remaining mysteries of the Standard Model of Particle Physics, and answer the question where do we come from?

Superstring theory web site
See video An Answer for Everything (Stephen Hawking's Universe), min.36-48 .
Scientific American article on Superstrings, by M. Green

Cosmology and the laws of physics at short distances

Cosmology is the study of the universe as a whole: its creation, its evolution, and its present state. It would appear that the laws of macroscopic physics should apply here. They do in most parts of todays universe. However, the universe was at one time very small, very hot, very energetic and very dense. Particles in matter were extremely close to one another and they were colliding at extremely high energies, much higher than energies in todays accelerators. Since the size of the entire universe was very small, or particles were extremely close to each other (much closer than we are able to achieve in today's accelerators) the laws of physics at small distances must have governed the behavior of the Universe during its early stages. Therefore, the physical laws obeyed by strings, supersymmetric particles, quarks and leptons, gluons, photon, W, Z, Higgs, elementary particles, nuclei, atoms, molecules, are progressively relevant in understandig the evolution of the universe at its early stages starting with its inception with the Big Bang. Indeed by applying these laws we can explain how today's universe came to be what it is, and trace its history.

The expanding Universe: Hubble observed that the universe is in a state of expansion. Role of the Doppler redshift.

Background radiation.

The four fources play various roles as the Universe expands.

A timeline for the Universe

The Big Bang Theory (a NASA website on the Universe) . Se also this.

The evolution of the Universe. (Scientific American)

What came before the Big Bang?

Dark Matter

Homework : Homework #4, due 12/06/2001
Part b) Quizz Ch.31, Ch.32, Ch.35, Ch.36

Reading assignment:
1) Read Hewitt Ch.31,32,35,36. Study the Next Time Questions for ch.31, ch.32, ch35, ch36
2) The text for Cosmic Alchemy.
3) The text and applets in Quantum Mechanics.
4) The Particle Adventure
5) Read about the benefits derived from research in high energy physics, and an assay about "What is the Use of Basic Science?".
Take a look also at some success stories that have resulted from more general applications of physics.

More reading:
Superstring theory
The Elegant Universe (Brian Greene) - A book on fundamental physics for the general audience.

Video:
1) Cosmic Alchemy (Stephen Hawking's Universe): see sections above.
2) An Answer for Everything (Stephen Hawking's Universe):
min.22-30 on Quantum Mechanics and chance, min.36-48 on Superstrings.
Text:
1) The text for Cosmic Alchemy provides a historical perspective on the discovery of atoms, radioactivity, nuclei, antimatter.
2) The text for An Answer for Everything is part of the story on the quest for a unified theory.

Additional links for the curious:
Some audio lectures on string theory directed to teachers:
    http://online.itp.ucsb.edu/online/mt01teach/
A colloquium on the early history of string theory, with audio
    http://online.itp.ucsb.edu/online/colloq/schwarz1/
A general lecture on the state of string theory, you can watch or listen to:
    http://online.itp.ucsb.edu/online/lnotes/gross/
Another audio lecture directed to teachers
    http://online.itp.ucsb.edu/online/bh_teach/polchinski/

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