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Particles
Dr. Jay Maron


Fundamental particles
Helium atom
Particle       Charge

Proton           +1       Composed of 2 up quarks, 1 down quark,  and gluons
Neutron           0       Composed of 1 up quark,  2 down quarks, and gluons
Electron         -1
Up quark        +2/3
Down quark      -1/3
Photon            0       Carries the electromagnetic force and binds electrons to the nucleus
Gluon             0       Carries the strong force and binds quarks, protons, and neutrons

History of physics

Particle masses

In this plot, the diameter of each particle proportional to Mass1/3. This is what the particles would look like if they were uniform-density spheres.

The electron is exaggerated otherwise it would be invisible.

The blue particles represent the heaviest particle that can be produced by each accelerator.

At this scale, a Planck-mass particle has a diameter of 10 km.

Photons, Gluons, and Gravitons are massless.


Electron Volts

Particle masses can be expressed in terms of electron Volts.

1 Electron Volt (eV) =  The energy gained by an electron upon falling down a potential of 1 Volt.
                     =  1.602⋅10-19 Joules

M = Particle rest mass
C = Speed of light
E = particle rest energy
  = M C2

Electron mass  =  9.11⋅10-31 kilograms
               =  511000 eV

1 keV = 103 eV
1 MeV = 106 eV
1 GeV = 109 eV

                 Mass (GeV)
Electron              .00051
Proton                .9383
Neutron               .9396
SLAC limit          45         Heaviest particle that can be produced by the Stanford Linear Accelerator
Higgs Boson        125         Discovered at the LHC and Fermilab
Fermilab limit     200         Heaviest particle that can be produced by Fermilab
LHC limit          700         Heaviest particle that can be produced by the Large Hadron Collider
Cosmic rays       1012         Highest-energy events observed
Big bang energy   1019         Energy of particles at the time of the big bang

Big bang energy = Planck energy
                = 1.22⋅1019 GeV
                = 1.956⋅109 Joules
                = 2.2⋅10-8 kg C2

Forces

Two electrons interact by exchanging a photon
Force           Relative    Force-carrying    Felt by
                strength    particle

Gravity          10-40      Graviton          All particles
Weak             10-5       W & Z bosons      All particles except gluons
Electromagnetic    1        Photon            All particles with electric charge
Strong            100       Gluon             Quarks and gluons

Unification
1687  Earth gravity     +  Planetary motion    →  Newtonian gravity         Newton

1752  Electric charge   +  Lightening                                       Franklin

1820  Electric currents +  Magnets             →  Electromagnetism          Oersted

1862  Electric force    +  Magnetic force      →  Maxwell's equations       Maxwell

1864  Electromagnetism  +  Light               →  Elecromagnetic waves      Maxwell

1905  Electromagnetism  +  Lorentz transform   →  Special relativity        Einstein, Poincare, Lorentz

1915  Newtonian Gravity +  Special relativity  →  General relativity        Einstein

1928  Quantum mechanics +  Special relativity  →  Quantum electrodynamics   Dirac, Feynman, Dyson

1967  Electromagnetism  +  Weak force          →  Electroweak force         Salam, Glashow, Weinberg

   ?  Electroweak force +  Strong force        →  Grand Unified Theory (GUT)

   ?  GUT               +  General Relativity  →  Quantum Gravity

"LHC limit" is the heaviest particle that can be produced by the LHC collider.

Weak unification:       The electromagnetic force unifies with the weak force
Strong unification:     The electroweak force unifies with the strong force
Gravity unification:    Gravity unifies with the electroweak and strong forces

Standard Model of particle physics

The "Standard Model" describes:
The strong force (quantum chromodynamics)
The electroweak force
The Higgs particle and the mechanism for generating mass
Dark matter, dark energy, and matter-antimatter asymmetry are examples of things that are not explained by the Standard Model.
String theory is an attempt to construct a theory of quantum gravity.


Photons
1803  Young discovers the diffraction of light, suggesting that light is a wave
1861  Maxwell develops the "Maxwell's equations", unifying electricity and magnetism
1864  Maxwell finds that light is an electromagnetic wave
1900  Planck solves the blackbody problem by assuming that photon energy is quantized as
      E = h F
1905  Einstein publishes the "photoelectric effect" experiment, providing the first
      direct measurement of photon energy and momentum.
1905  Theory of special relativity completed.
      Einstein publishes the photoelectric experiment
1924  de Broglie postulates that for particles with mass,
      Momentum * Wavelength = PlanckConstant
1927  Davisson and Germer experimentally verify the de Broglie relation for electrons.


F  =  Photon frequency
W  =  Photon wavelength
C  =  Photon speed
   =  Speed of light
   =  3.00-34 Joules/second
   =  F W                      (Wave equation)
E  =  Photon energy
Q  =  Photon momentum
h  =  Planck constant
   =  6.63⋅10-34 Joule seconds
Einstein's 1905 photoelectric experiment showed that photons have energy and momentum given by
E = h F        (Planck equation)
E = Q C
Using the wave equation for photons (C=FW) we have
Q W = h        (de Broglie equation)
Photons have zero rest mass. In 1924 de Broglie found that this equation also applies to particles with finite rest mass. W is the "quantum-mechanical wavelength" of a particle.
Particles with finite rest mass

For non-relativistic electrons (V << C), the energy and momentum are

Q  =  m V
E  =  ½ m V2
The relationship between E and Q is different for photons and electrons.
Photon:      E = Q C
Electron:    E = Q2 / (2M)
When electrons are relativistic (V ~ C) they behave similar to photons.
Particle energy and momentum
F  =  Frequency
W  =  Wavelength
V  =  Speed
C  =  Speed of light
   =  3.00-34 Joules/second
   =  F W                     (Wave equation)
E  =  Energy
Q  =  Momentum
h  =  Planck constant
   =  6.63⋅10-34 Joule seconds
m  =  Rest mass
Z  =  Lorentz gamma           (only defined for particles with m > 0)
   =  (1-V2/C2)
M  =  Relativistic mass
   =  Z m                     (for particles with positive rest mass)
   =  E/C2                    (for particles with zero rest mass)
The following equations apply for all particles, whether the rest mass is zero or positive.
Q W  =  h
E    =  (mC2)2 + (QC)2
If the particle has zero rest mass (m=0), such as a photon, then
V  =  C
m  =  0
E  =  Q C
M  =  E / C2
If the particle has positive rest mass then
M  =  Relativistic mass
   =  Z m
E  =  M C2
Q  =  M V
In many cases you can guess the relativistic formula by replacing "m" with "M". For example, in CGS units,
Force           = M * Acceleration
Electric force  = Charge  *  Electric field
Magnetic force  = Charge  *  Velocity  *  Magnetic field
If you try to accelerate a particle to the speed of light,
V/C           ->  1
M             ->  Infinity
Acceleration  ->  0
The particle never reaches the speed of light.
Regimes of special relativity

We can define 4 regimes.

Classical:          V/C << 1
Relativistic:       V/C not close to zero and not close to 1
Ultrarelativistic:  V/C close to 1 but not equal to 1
Light:              V/C = 1
For particles with positive rest mass, the relativistic formulae work for all values of
0  <=  V/C  <  1
In the classical and ultrarelativistic regimes the relativistic formulae can be simplified.
Classical regime

When the formulas of special relativity are taken in the limit
V/C -> 0
they become the formulas of classical physics.
V/C      →    0
Z        →    1 + 1/2 V2/C2
E        →    E = m C2 + ½ m V2              A rest energy plus a kinetic energy
Q        →    m V

Relativistic regime

This is the most complex case. For the classical and ultrarelativistic regimes the formulas simplify and for the relativistic regime they don't.
V/C < 1
E2  = (mC2)2 + (QC)2
E   =  Z m C2
    =  M C2
Q   =  Z m V
    =  M V

Ultrarelativistic

In the limit of V/C -> 1,
Z        ->   Infinity
E        ->     Q C
For particles with V/C << 1,    QC << E.
For particles with V/C almost equal to 1, QC is almost equal to E.
For particles with V/C = 1,    QC = E.

Ultrarelativistic particles behave similar to massless particles in that QC ~ E. In particle colliders, particles are almost always ultrarelativistic.


Photon regime

For particles with zero rest mass, such as photons,

E = Q C
  = F h

Particle wavelength

The following table shows example values for the properties of particles.

                         Rest        Kinetic       V/C        Wavelength    Compton radius
                         mass        energy                   (angstroms)   (amgstroms)
                         (MeV)        (MeV)
Infrared photon            0            .0000001   1           107000       107000
Green photon               0            .0000023   1             5550         5550
X ray photon               0            .01        1                1.28         1.28
Gamma ray photon           0           5           1                 .0026          .0026
Electron                    .511       0           0                 -              .0243
Electron from vacuum tube   .511        .00001      .00626         12.3             .0243       10 electron Volts
Electron from beta decay    .511       5            .9957            .0025          .0243
Electron at SLAC            .511   45000            .999999999936    .00000028      .0243
Proton                   938           0           0                 -              .0000132
Proton in nucleus        938           5            .103             .00013         .0000132
Proton at the LHC        938     7000000            .9999999910      .0000000018    .0000132
Cosmic ray proton        938         e15           nearly 1         1e-17           .0000132
Higgs boson           125000           0           0                 -              .0000000099
Baseball pitch           e29         e15           .00000015        1e-24           e-24         146 grams, 45 m/s
F-22 Raptor, Mach 2.3    e36        3e22           .0000023         5e-31           e-41         20 tons, 700 m/s
Planck particle          e22         e22           nearly 1          e-25           e-25


E  =  Energy
C  =  Speed of light
m  =  Rest mass
Q  =  Momentum
h  =  Planck constant
Particle energy and momentum are related by
E2 = (mC2)2 + (QC)2

If    Q  <<  mC       the particle is nonrelativistic
If    Q   =  mC       the particle is at the boundary between nonrelativistic and ultrarelativistic
If    Q  >>  mC       the particle is ultrarlativistic
If the particle is on the boundary (Q=mC) then its wavelength is called the "Compton wavelength". For force-carrying particles this sets the range of the force.
Compton wavelength  =  h / Q  =  h / (mC)
A F-22 Raptor has the same kinetic energy as a Planck particle.

A baseball pitch has the same Compton wavelength as a Planck particle.


Creating particles

The energy of a particle can be expressed as a rest energy plus a kinetic energy

E  =  Gamma m C2
   =  m C2  +  (Gamma - 1) m C2
   =  m C2  +  1/2 m V2                    in the limit of V/C << 1
If Gamma < 2, a particle's rest energy is larger than its kinetic enegy.
If Gamma > 2, a particle's kinetic energy is larger than its rest energy.

In the Stanford Linear Collider (SLAC), electrons are accelerated to an energy of 45 GeV and collided head-on with antielectrons with the same energy.

Rest energy of electron           = .0005 GeV
Kinetic energy of SLAC electron   =   45  GeV
SLAC electrons are highly relativistic. When a SLAC electron is collided with a SLAC antielectron, they annihilate and new particles are created. The energy available to create new particles is 90 GeV. This is how a high-energy collison can produce particles that have a larger rest energy than the colliding particles.

For example,

Electron with 45 GeV  +  Antielectron with 45 GeV  ->  Annihilate
                                                   ->  Charm Quark + Anti Charm Quark + 87.4 GeV of kinetic energy

                 Energy in GeV
Electron             .0005
Proton              1.0
Neutron             1.0
Charm Quark         1.3
Higgs Boson       125
Big Bang particle  1019 GeV     Energy required for quantum gravity

Physics regimes
                         Quantum   Relativistic  Strong   Planck   Example
                        phenomena     speeds     gravity  energy

Classical physics                                                  Rugby
Special relativity                      *                          Interstellar spaceships
General relativity                      *          *               Black holes
Quantum mechanics           *                                      Atomic sizes & smaller
Quantum field theory (QFT)  *           *                          Particle colliders
Quantum gravity             *           *          *        *      Big bang
Quantum mechanics is relevant if the scale is equal to or less than the quantum wavelength. If an object is larger than the quantum wavelength it behaves classically.

In atoms, the size scale is small enough for quantum phenomena and the characteristic speeds of electrons are nonrelativistic, and so this is the regime of "Quantum mechanics".

Q  = Momentum
h  = Planck Constant               =  6.62e-34  Joule second
W  = Wavelength                    =  h/Q       meter
K  = Electric force constant       =  8.988e9   Newton meter2 / Coulomb2
q  = Proton charge                 =  1.602e-19 Coulombs
G  = Gravity constant              =  6.67e-11  Newton meter2 / kg2
V  = Velocity
C  = Speed of light
R  = Distance from the object
Re = Object event horizon radius   =  2 G M / C2


If  V << C       Special relativity may be neglected
If  V > 0.1 C    Special relativity applies
If  W << R       Quantum mechanics doesn't apply
If  W >= R       Quantum mechanics applies
If  R >> Re      Spacetime is flat enough for general relativity to be neglected
If  R < 10 Re    Gravity warps spacetime and general relativity applies
If gravity is strong enough to make particles relativistic and if the scale is small enough for quantum mechanics, quantum gravity is relevant. This is the "Planck scale". The space near an event horizon can make particles relativistic but a solar mass black hole is too large for quantum mechanics. Quantum gravity occurs at the centers of black holes and during the big bang.

As the mass of a particle increases, its event horizon radius increases and its quantum-mechanical wavelength decreases. When they become equal, quantum-gravity applies. THis is the "Planck scale".

For an ultrarelativistic particle,

E = Q C = h C / W
If the event horizon radius equals the quantum wavelength,
W = (G h / C3)½
  = 4.1*10-35 meters                This is the Planck scale

Physical scales

SI fundamental units:
Distance      Meter
Time          Second
Mass          kg
Charge        Coulomb
SI derived units:
Velocity          Meter  / Second
Momentum       kg Meter  / Second
Energy         kg Meter2 / Second2  =  Joules
Force          kg Meter  / Second2  =  Newtons  =  Joules/meter
Physical constants:
C  =  Speed of light   =  2.998e8    Meters/Second
G  =  Gravity constant =  6.67e-11   Newton (Meter/Kg)2
h  =  Planck constant  =  6.626e-34  Joule Seconds
e  =  Electron charge  =  1.602e-19  Coulombs
We can define time in terms of distance using C.

Define a "Light meter" as the amount of time it takes for light to travel 1 meter.

1 Light meter = 3.34 Nanoseconds

With the "light meter" we no longer need the "second".

We can also define distance in terms of time.

Define a "light second" as the distance light travels in one second.

1 light second = 2.998e8 meters

With this we no longer need the meter. Distance can be defined in terms of light travel time.

_______________________________________

We can define energy in terms of mass using the speed of light.

Energy = Mass C2

1 "kilogram" of energy is equal to C2 = 8.99e16 Joules.

Particle masses are usually expressed in terms of energy. For example,

M  =  Proton rest mass
   =  1.673e-27 kg
E  =  Proton rest energy
   =  M C2
   =  1.673e-27 • 2.998e82
   =  1.504e-10  Joules
If a proton and antiproton annihilate they produce 3e-10 Joules.

_______________________________________

Using the gravity constant we can define mass in terms of distance.

G Mass2 / Distance  =  Energy  =  Mass C2

Mass  =  C2 Distance / G

1 "Gravity mass"  =  C2/G    kilograms
                  =  1.34e27 kilograms
Mass can be expressed in terms of "gravity masses".

An object with this mass has an event horizon radius of 1 meter.

_______________________________________

X = Distance
T = Time
M = Mass

Using the physical constant C we can express either:
X in terms of T
T in terms of X

Using the physical constants {C,G} we can express either:

T and M in terms of X
X and M in terms of T
X and T in terms of M

However we cannot define a universal value for X, T, or M.  If we add Planck's
constant we can.

Planck units

In 1899 Planck hypothesized that photon energy is quantized, and in 1905 Einstein's photoelectric experiment confirmed it.
E  =  Photon energy
f  =  Photon frequency
h  =  Planck constant

E = h f
Planck used the Planck constant to obtain a universal scale for space, time and mass which are called the "Planck scales".
X  =  Planck distance
T  =  Planck time
M  =  Planck mass
Physical constants:
Speed of light         =  C  =  2.998⋅108   Meters/Second
Gravity constant       =  G  =  6.674⋅10-11 Newton (Meter/Kg)2
Electron charge        =  e  =  1.602⋅10-19 Coulombs
Planck constant        =  h  =  6.626⋅10-34 Joule Seconds
Reduced Planck constant=  ℏ  =  h (2π)-1  =  1.055⋅10-34  Joule Seconds
When discussing Planck units, ℏ is used instead of h.

Speed of light:         X = C T
Gravitational energy:   E = G M2 / X     ->     C2 = G M / X
Energy quantization:  E T = ℏ         ->     M C2 T = ℏ
Solving for {X, T, M},
Planck length  =  X  =  (G ℏ / C3)½  =  1.62⋅10-35 meters

Planck time    =  T  =  (G ℏ / C5)½  =  5.39⋅10-44 seconds

Planck mass    =  M  =  (ℏ C / G)½   =  2.18⋅10-8  kg  =  1.22⋅1019 GeV
These are the scales for particles at the time of the big bang.
Planck charge
R  =  Distance between two particles
M  =  Planck mass in kg  =  (ℏ C / G)½
m  =  Proton mass in kg  =  1.673⋅10-27 kg
Q  =  Planck charge in Coulombs
q  =  Charge on the proton
   =  1.602e-19 Coulombs
Fg =  Gravity force     =  G M2 / R2
Fe =  Electric force    =  K Q2 / R2
G  =  Gravity constant  =  6.67e-11   Newton Meter2 / Kg2
K  =  Electric constant =  8.988e9    Newton Meter2 / Coulomb2
The Planck charge is defined such that
Electric force between two Planck charges  =  Gravitational force between two Planck masses

G M2   =  K Q2

Q      =  (ℏ C / K)½
       =  1.876⋅10-18  Coulombs

q/Q    =  Proton Charge / Planck charge  =  .0854

m/M    =  Proton mass   / Planck mass    =  7.68⋅10-20

For two protons,
Gravity force / Electric force
   =  G m m / (K q q)
   =  (m/M)2 / (q/Q)2
   =  8.087⋅10-37
Gravity is vastly weaker than the electric force. This is because the proton mass is much less than the Planck mass, while the proton charge is similar to the Planck charge.
Fine structure constant

We can define a measure of the strength of the electric force "Z" by using photons. Suppose two electrons are a distance R apart and a photon has wavelength W = 2 π R. The photon energy is E = hf = hC/W.

R  =  Distance between two electrons
f  =  Photon frequency
W  =  Photon wavelength
   =  2 π R
h  =  Planck constant
E  =  Photon energy
   =  h f
   =  h C / W
   =  ℏ C / R
q  =  electron charge
Ee =  Electric energy between the electrons
   =  K q2 / R

α  =  Electric energy / Photon energy
   =    (K q2 / R)   /   (hC/W)
   =  2 π K q2 / (hC)
   =  .007297
The dimensionless number α appears everywhere in physics and is called the "fine structure constant". For example,
Z  =  Proton charge / Planck charge
   =  q K½ (ℏ C)
   =  .0854  =  α½

R_classical / R_compton          =  α   / (2Pi)

R_classical / R_quantum          =  α2 / (2π)2

Black hole event horizon
G = Gravitational force constant
  = 6.67*10-11 Newton Meter2 / Kilogram2
C = Speed of light
M = Mass of a black hole
m = Rest mass of a particle
R = Distance between the black hole and the particle
F = Force on a particle from the black hole
E = Potential energy of the particle with respect to the black hole
Re= Black hole event horizon radius
If the particle is far from the black hole then the Newtonian formulas for force and energy apply
F =  G M m / R2

E = -G M m / R
As the particle falls toward the black hole its velocity approaches C and the gravitational energy approaches the rest energy. General relativity becomes important here, and the characteristic scale R for this to happen is determined by C, G, and M. The only combination of these variables that gives units of length is
R  =  DimensionlessConstant * C2 / (GM)
This is an estimate for the radius of a black hole event horizon. Order-of-magnitude estimates like this tend to give the right exponents (C2, G-1, M-1) but they can't give the dimensionless constant in front. Using general relativity, the exact formula for the event horizon radius is
Re = 2 C2 / (GM)
The dimensionless constant turns out to be "2".

_______________________________________

We can also estimate the event horizon radius by setting the gravitational energy equal to the rest energy.

G M m / R = .5 m C2

R = 2 G M / C2
This is only an estimate because the Newtonian formulas break down near the black hole.

_______________________________________

Plugging in the values for C and G,

Event horizon radius  =  2  *  6.67e-11  /  (3*108 m/s)2  *  M
                      =  1.49e-27     *   M
                      =  3000 meters  *  (M / Mass of sun)
If the sun were a black hole the event horizon radius would be 3000 meters. To make the Earth a black hole you would have to squash it to a radius of 7 millimeters.

Neutron stars have a mass between 2 and 4 solar masses and a radius slightly larger than the event horizon radius.

The black hole at the center of the galaxy has a mass of 4 million solar masses and an event horizon radius of 12 million km. This is 1/12 the distance from the Earth to the sun.


Classical electron radius

In classical physics, an accelerating charge emits synchrotron photons and loses energy. If an electron orbits a proton then the emitted photons cause the electron to inspiral into the proton in 10-15 seconds. This is an example where a theory predicts a phenomenon that breaks the theory, and this usually points the way to a more fundamental theory. The thing that stops the electron from crashing into the proton is quantum mechanics.

Suppose an electron is on a circular orbit around a proton.

R  =  Distance between the electron and the proton
V  =  Velocity of the electron
C  =  Speed of light
   =  3.00e8 m/s
M  =  Electron rest mass
   =  5.68e-31 kg
Z  =  Electric charge on an electron
   =  1.602e-19 Coulombs
K  =  Electric force constant
   =  8.988*109  Newton Meter2 / Coulomb2
F  =  Force between the proton and electron
   =  -K Z2 / R2
E  =  Electron kinetic energy
   =  .5 M V2
Ee =  Electric energy between the proton and electron
   =  Integral (Force dR)
   =  -K Z2 / R
Q  =  Electron momentum
   =  M V
h  =  Planck Constant
   =  6.62 * 10-34 Joule seconds
W  =  Electron quantum-mechanical wavelength
   =  h / Q
Balance the electric and the centripetal forces:
K Z2 / R2  =  M V2 / R
The kinetic energy is
E  =  .5 M V2
   =  .5 K Z2 / R
The electron becomes relativistic when E ~ M C^2. Define
R_classical =  Classical radius of the electron
            =  Radius of a circular orbit for which the kinetic energy equals the rest energy
            =  K Z2 / (M C2)
            =  2.818e-15 meters
If R < R_classical, classical physics is guaranteed to fail and so some new physics has to appear. A similar example is a black hole where Newtonian gravity breaks down and general relativity takes over.

The Schwarzschild radius of a black hole is the characteristic distance where infalling matter becomes relativistic.

M        =  Mass of an electron
M_hole   =  Mass of a black hole
E_grav   =  - G M_hole M / R2
R        =  Distance of an electron from a black hole
R_schwarz=  Schwarzschild radius of a black hole, the closest distance from which
            light can escape.
         =  2 G M / C2
Setting the gravitational energy equal to the rest energy gives the gravitational radius of a black hole, which is proportional to the Schwarzschild radius.
Electron quantum radius

As R->0, the scale where quantum mechanics becomes important is the "quantum radius of the electron".

To derive this scale we calculate the electron wavelength as a function of R. We assume that the electron is nonrelativistic and we assume the electron is on a circular orbit around the proton. The balance of electric and centripetal force is

W  =  Quantum-mechanical wavelength of the electrom
R  =  Orbital radius of the electron
The parameter that characterizes the importance of quantum mechanics is W/R.
If  W/R > 1  Quantum mechanics is important
If  W/R < 1  Quantum mechanics is unimportant and classical physics can be used

W/R  =  (h/Q) * Q2 / (K q2 m)
     =  h Q / (K q2 m)
     =  Constant * R

As R-> Infinity    W/R -> 0           Classical physics applies
As R->     0       W/R -> Infinity    Quantum mechanics applies

The radius where W/R=1 is the quantum radius of the electron.

R_quantum   =  h2 / (K q2 m)
            =  2.086e-9 meters
We can compare the quantum radius to the classical radius.
R_quantum / R_classical  =  h2 C2 / (K2 q4)
                         =  740000

Because R_quantum / R_classical >> 1, quantum mechanics becomes important
before relativity.
R_quantum sets the size of atoms. If you calculate the electron orbital radius in a hydrogen atom using the Bohr theory,
R_bohr  =  R_quantum / (2 Pi)
        =  5.29e-11 meters
This is the radius of the S=1 orbital in a hydrogen atom.
Electron Compton radius

If a particle is nonrelativistic (V << C),

Q  =  Momentum  =  m V
E  =  Energy    =  ½ m V2

Q2 = 2 m E
If a particle is relativistic (V~C),
E ~ Q C
For relativistic particles we can define an equivalence between space and time by setting V=C.
Space = C * Time
We can also define an equivalence between space and energy by setting the space scale equal to the quantum mechanical wavelength.
E = Q C = h C / W
If we set E equal to the particle's rest energy, we call the resulting wavelength the "Compton wavelength". This particle is "barely relativistic".
m C2 = h C / R_compton

R_compton  =  h / (m C)
All particles with finite mass have a Compton wavelength.

Suppose a photon has an energy equal to a particle's rest energy E.

f = Photon frequency
W = Photon wavelength
E = h f = h C / W
W = h C / E
  = h / (M C)
The wavelength of the photon is equal to the particle's Compton wavelength. If a photon has the same energy as a particle's rest energy the wavelength of the photon is equal to the Compton wavelength.

For an electron,

R_compton  =  2.426e-12 meters

Virtual particles

In a vacuum, virtual particles continuously appear and disappear.
m  =  Particle rest mass
E  =  Particle rest energy
   =  m C2
T  =  Particle lifetime
X  =  Characteristic distance a virtual particle travels during its lifetime
h  =  Planck constant
Q  =  Particle momentum
C  =  Speed of light
From the Heisenberg uncertainty principle, a virtual particle has a characteristic lifetime such that
E T = h
Virtual particles are relativistic and so we assume V ~ C.

The distance the virtual particle travels in its lifetime is

X  =  T C  =  h C / E
If we set E equal to the rest energy mC2,
X  =  h / (m C)
This is the "Compton wavelength" of a particle.
Strength of forces
Force         Force carrying   Mass of force       Compton radius of
                 particle      carrying particle   force-carrying particle
                                 (GeV)                  (meters)
Gravity           Graviton        0                     Infinite
Weak                 W           80                     1.5e-17
                     Z           91                     1.4e-17
Electromagnetic    Photon         0                     Infinite
Strong              Pion         .135                   9.2e-15
Technically the strong force is carried by gluons. For protons and neutrons in a nucleus the strong force can be considered to be carried by pions.

If a force is carried by a massless particle then the force is given by

Force  =  Constant / R2
such a force is said to have "infinite range".

If the force is carried by a massive particle then the force is said to have "finite range". The range of the force is equal to the carrier's Compton wavelength and beyond this range the force is essentially zero.

The size scale of the nucleus is determined by the pion's Compton wavelength.

The force-carrying particles for the weak force are heavy and hence the weak force has short range, 100 times shorter than the size of a nucleus. This is why the weak force is weak.

The fine structure constant characterizes the strength of the electromagnetic force.

For two electrons, the gravitational force is vastly weaker than the electric force.

Force of gravity / Electric force
= G m2 / K q2
= 2.40e-43

Higgs crisis

Quantum mechanics resolved the electron crisis. The next crisis is the Higgs crisis.

Higgs mass = 125.3 GeV

                           meters
Electron quantum radius    2.09e-9    Limit for classical physics
Atomic scale               2e-10
Electron Compton radius    2.43e-12   Limit for non-relativistic quantum mechanics
Electron classical radius  2.82e-15
Nuclear scale              1e-15
Higgs Compton radius       9.9e-18

Suppose we consider a length scale R.
If  R > Electric quantum radius     Classical physics applies
If  R < Electron quantum radius     Quantum mechanics apples
If  R > Electron Compton radius     Non-relativistic quantum mechanics may be used
If  R < Electron Compton radius     Relativistic quantum mechanics must be used
If  R > Higgs Compton radius        The Standard Model applies
If  R < Higgs Compton radius        The Standard Model breaks down

The Standard Model breaks down for scales below the Higgs scale and some new theory must take over for smaller scales. The most promising candidates are "Supersymmetry" and "The Multiverse". The Large Hadron Collider can at present explore down to the Higgs scale but no further. When it is upgraded to higher energies in 2015 it will be able to go beyond the Higgs scale.

There is much drama in the world of supersymmetry. It is predicted that the LHC should already have detected supersymmetric particles and thus far none have been found.


Particle families


Particles
                Charge  Spin  Rest mass  Strong  Weak   Lifetime
                                (GeV)    force   force  (seconds)
Up quark         +2/3   1/2     ~.0024     *      *     882      (Decay of Neutron)
Charm quark      +2/3   1/2    ~1.25       *      *     1e-12    (Decay of D+ meson)
Top quark        +2/3   1/2  ~171          *      *     5e-25
Down quark       -1/3   1/2     ~.0048     *      *
Strange quark    -1/3   1/2     ~.1        *      *     1.2e-8   (Decay of Kaon+ meson)
Bottom quark     -1/3   1/2    ~4.2        *      *     1.6e-12  (Decay of B+ meson)
Electron          -1    1/2      .000511          *
Muon              -1    1/2      .1057            *     2.20e-6
Tau               -1    1/2     1.777             *     2.9e-13
Electron neutrino  0    1/2     <2.2e-9           *     Mixes
Muon neutrino      0    1/2     <.00017           *     Mixes
Tau neutrio        0    1/2     <.016             *     Mixes
Photon             0     1      0                 *
Gluon              0     1      0          *            1e-23  Timescale of strong force
W boson           -1     1     80.4               *     3e-25
Z boson            0     1     91.2               *     3e-25
Graviton           0     2      0
Higgs Boson        0     0    125.3               *     1.6e-22
Proton            +1    1/2      .9383     *      *
Neutron            0    1/2      .9396     *      *     882
All of the above particles are fundamental (not composed of smaller particles) except for the proton and neutron.

All particles with charge feel the electromagnetic force.
All particles feel the gravitational force.

Hadron:  Particle composed of quarks, such as a proton, muon, and pion
Baryon:  Particle composed of 3 quarks, such as a proton and neutron
Meson:   Particle composed of a quark and an antiquark, such as a pion
Lepton:  Spin 1/2 particle that does not feel the strong force
Bosom:   Integer spin, such as {0, 1, 2, ...}
         Does not obey the Pauli exclusion principle
Fermion: Half-integer spin, such as {1/2, 3/2, ...}
         Obeys the Pauli exclusion principle
The masses of the quarks are not accurately known.

If no halflife is given, the particle is stable (as far as we know). As neutrinos propagate they can change to other kinds of neutrinos.

The length scale of the strong force is 10-15 meters. The timescale of the strong force is the length scale divided by the speed of light, which is on the order of 1e-23 seconds.

Quarks and gluons are never found in isolation. They are always bound together in the form of a meson (2 quarks) or a baryon (3 quarks), and their identities are continuously mixing.

Gluons interact with themselves and so they have a lifetime equal to the characteristic strong force timescale, 10-23 seconds.

Some theories predict that the proton is unstable, with timescales in the range of 1036 years. Experiments have found that the proton has a lifetime of at least 1034 years.


Particle interactions

A blue line indicates that an interaction exists between the given particles. For example, an electron interacts with a photon. The Higgs interacts with all particles with positive rest mass.


Feynman diagrams

Feynman diagrams are a way of illustrating the possible particle interactions.


Density of the universe
                 kg/m3
Planck density   5   ⋅1096   =    PlanckMass / PlanckLength^3
Solar system     2   ⋅10-8   =    Mass of sun / (30 AU)^3
Milky Way        3   ⋅10-21  =    1.2e12 solar masses / (100000 lightyears)^3
Matter            .12⋅10-27       Mean density of protons & electrons in the universe
Dark matter       .66⋅10-27       Mean density of dark matter in the universe
Dark energy      1.67⋅10-27       Mean density of dark energy in the universe
As the universe expands the matter and dark matter density decrease and the dark energy density is constant.

In the early universe the dark matter density was vastly greater than the dark energy density. In the future dark energy will overwhelm dark matter and the universe will expand unchecked.


Big bang timeline
0 seconds:
Big bang
Planck Epoch

10-43 seconds:
Grand unification epoch
The temperature of the universe is 1015 GeV or 1027 K
The strong, electromagnetic and weak forces are unified as the electronuclear force
At this time, gravity separates from the electronuclear force

10-36 seconds:
Strong and electroweak forces separate
Beginning of the inflationary epoch

10-32 seconds:
End of the inflationary epoch
The inflationary phase transition heats the universe into a quark-gluon plasma
The universe is hot enough for W and Z bosons (< 100 GeV)
Physics is highly speculative before this epoch

10-12 seconds:
End of the electroweak epoch and beginning of the quark epoch
The electroweak force separates into the electromagnetic and weak forces
The universe is too cool for W, Z and Higgs bosons
The universe is filled with quarks, leptons, and their antiparticles
The universe is too hot for baryons or mesons

10-6 seconds:
End of the quark epoch and beginning of the hadron epoch
The universe has equal amounts of hadrons and anti-hadrons

1 second:
End of the hadron epoch and beginning of the lepton epoch
Hadrons and anti-hadrons have annihilated, leaving behind only matter
Leptons dominate the mass of the universe
The universe has equal amounts of leptons and anti-leptons

2 seconds:
Neutrinos decouple from other matter and form the cosmic neutrino background

10 seconds:
End of the lepton epoch and beginning of the photon epoch
Leptons and anti-leptons annihilate, leaving behind only leptons
Photons dominate the energy of the universe
Nucleosynthesis occurs

3 minutes:
Nucleosynthesis ends
The photon epoch continues
The universe is a plasma of free electrons and ions
The universe is opaque to photons

300,000 years:
The cosmic "Dark age"
The universe cools enough for nuclei to combine with electrons to
form neutral atoms
The universe becomes transparent to photons
The photons generated at this time become the cosmic background radiation
For the next billion years, the universe is a cold and dark gas

1 Billion years:
End of the cosmic dark age. Stars and Galaxies form.
Supernovae enrich the universe with heavy elements.

9.2 billion years:
Formation of the Sun and the Earth.

9.9 billion years:
Uranus and Neptune change places, causing the Earth to be bombarded with comets.

12 billion years:
Oxygen appears in the Earth's atmosphere from photosynthesis.

13.1 billion years:
Oxygen becomes a major constituent of the atmosphere.
First appearance of complex multicellular life.

13.5 billion years:
An anoxic extinction event wipes out 95 percent of species.
Dinosaurs dominate hereafter.

13.6 billion years:
The dinosaurs are wiped out by a 10 kilometer meteor.
Mammals emerge hereafter.

10 thousand years ago:
Emergence of civilization on the Earth

13.7 billion years:
The present.

1 billion years from now:
The sun increases in brightness and the Earth's oceans evaporate.

4 billion years from now:
The Milky Way and Andromeda galaxies will collide.

5 billion years from now:
The sun expands in a nova and consumes the Earth.

Several billion years from now:
Dark energy causes the universe to expand.

Trillions of years from now:
The last stars burn out. The universe is dark hereafter.
The only remaining objects are white dwarfs, neutrons stars, black holes
and cold planets orbiting dead stars.
The Big Chill.

1040 years:
Prootns decay.

1070 years:
Black holes explode.

Fate of the universe
Tangible matter  =  Stuff that interacts by the strong and/or electromagnetic force,
                    such as protons, neutrons, electrons, photons.
                    These particles can be stopped by a meter of lead.
Dark matter      =  Stuff that does not interact by the strong or electromagnetic
                    force but interacts by the weak force and gravity.
                    These particles easily pass through the Earth.
                    Examples include neutrinos.
                    Most of the dark matter in the universe consists of particles
                    that have not yet been discovered.
Dark energy      =  An energy density that has negative pressure.
Tangible matter and dark matter have positive pressure and dark energy has negative pressure. All three have positive energy density.

Densities:

                 kg/m3
Planck density   5   *1096   =    PlanckMass / PlanckLength^3
Black hole       1.8 *1019   =    Density of a 1 solar mass black hole
Neutron star     1   *1018   =    Upper range for the density at the core
Nuclear matter   2.3 *1017   =    Density of a nucleus
White dwarf      1   *109    =    White dwarf density
Osmium          22.6 *103    =    Densest element
Water            1   *103
Air              1.22*100         At sea level
Solar system     2   *10-8   =    Mass of sun / (30 AU)^3
Milky Way        3   *10-21  =    1.2e12 solar masses / (100000 lightyears)^3
Ordinary matter   .12*10-27  =    Mean density of protons, neutrons, & electrons in the universe
Dark matter       .66*10-27  =    Mean density of dark matter in the universe
Dark energy      1.67*10-27  =    Mean density of dark energy in the universe
Sum              2.45*10-27  =    Total density of matter, dark matter, and dark energy
As the universe expands the matter and dark matter density decrease and the dark energy density is constant.

In the early universe the dark matter density was vastly greater than the dark energy density. In the future dark energy will overwhelm dark matter and the universe will expand unchecked.

The Earth's escape velocity is Ve=11.2 km/s. Suppose the Earth had no atmosphere and you launched a cannonball upward with velocity V.


V < Ve   Elliptic     The cannonball falls back to the Earth
            
V > Ve   Hyperbolic   The cannonball escapes from the Earth and asymptotes to a
                      positive velocity
V = Ve   Parabolic    The cannonball is on the boundary between escape and falling back.
                      It never returns to the Earth and it asymptotes to zero velocity.
If the universe consisted entirely of ordinary matter and dark matter and no dark energy, then there is a critical value of the density such that the expansion of the universe is parabolic. This value is 2.45e-27 kg/m3.
d             =  Density of a parabolic universe  =  2.45e-27 kg/m3
D    =  1.0   =  Density of all matter and dark energy in the universe / d
Dom  =   .049 =  Fraction of ordinary matter in the universe           / d
Ddm  =   .27  =  Fraction of dark matter in the universe               / d
Dde  =   .68  =  Fraction of dark energy in the universe               / d
D, Dom, Ddm, and Dde are scaled relative to the parabolic density d.
If Dde = 0 then

If  D > 1   The density of the universe is large enough to reverse the expansion from
            the big bang and the universe collapses in a Big Crunch.
            The Hubble constant goes from positive to zero to negative.
If  D > 1   The universe expands forever, ending with a positive Hubble constant
If  D = 1   The universe stops expanding and the Hubble constant goes to zero.
            The universe ends in a "Big Chill".
If Dde > 0 then dark energy trumps all of the above. If the universe survives long enough to avoid a Big Crunch then dark energy causes the universe to expand unchecked regardless of the matter density. The universe ends in a Big Chill.

Previous to 2005 we knew that the value of D was close to 1 and we couldn't tell if it was larger or smaller than 1. The value of the dark energy density was unknown and the fate of the universe was unknown. The theory of "Inflation" was developed to explain why D is close to 1.

In 2005 measurements of distant supernovae showed that Dde > 0, implying that the universe will end in a Big Chill.

In 2010 the Planck spacecraft measured the precise values of Dom, Ddm, and Dde.

In the plot,
Omega_M      = Dom + Ddm
Omega_Lambda = Dde

You can illustrate the concept of escape velocity with the "My Solar System" simulation at phet.colorado.edu.

         Mass     Position     Velocity
                    X    Y     X   Y
Body 1    100.      0    0     0    0      Sun
Body 2      1.    100    0     Vx   Vy     Planet
If Vx=100 and Vy=0 the planet orbits the sun on a circular orbit. What do the orbits look like if you vary Vx?

The escape velocity from the sun at X=100 is Ve=100*Squareroot(2).

If  Vx=0  and  Vy > Ve   the planet escapes.

If  Vx=0  and  Vy < Ve   the planet crashes into the sun.

The Multiverse

Naturalness
 Scale    Constituents of matter
(meters)
   1      Materials, gases, chemicals, pizza
10-9      Molecules
10-10     Elements           (hydrogen, helium, ...)
10-14     Nuclei & electrons
10-15     Protons, neutrons, pions, electrons, photons, neutrinos, dark matter
10-16     Quarks, electrons, photons, gluons, neutrinos, W, Z, Higgs, dark matter

Parameters of the universe
Number of     Parameter
parameters
    6         Quark masses.  Up, down, charm, strange, top, bottom
    3         Lepton masses.  Electron, muon, tau
    3         Neutrino masses.  Electron neutrino, muon neutrino, tau neutrino
    1         Z mass.  The W mass is determined by the Z mass.
    1         Higgs mass
    1         Electric force constant
    1         Gravitational force constant
    1         Strong force constant
    1         Weak force constant
    4         3 Neutrino mixing angles and one phase
    6         Cosmological parameters.  These are:
                 Density of tangible matter in the universe (nuclei, electrons, etc)
                 Density of dark matter in the universe
                 Dark energy density (cosmological constant)
                 Scalar spectral index of the universe
                 Curvature fluctuation amplitude of the universe
                 reionization optical depth of the universe
Dark matter is likely a source of new parameters, such as the masses of the dark matter particles.

The masses of the proton and neutron can be calculated from the masses of the up and down quark.

At present the origin of these parameters is unknown. Ideally, a future physics theory will explain the origin of the parameters based on a compact set of principles, and the number of parameters will decrease. This was the hope of Einstein. Since special relativity and general relativity can be generated from compact principles it was hoped that particle physics could as well, but at present no successful principles have been found.

An alternative to principles is The Multiverse, where there are multiple universes, each with different parameters, and we live in a universe where the parameters allow for the existence of intelligent life. This is similar to the Anthropic Principle.

If you change the parameters of the universe there are extreme consequences. For example,

If the electron mass increases, protons will consume the electrons to produce neutrons, leaving behind a boring universe with no nuclei.

If the dark energy density is increased, the universe expands too fast for galaxies to form.

If you increase the electric force relative to the strong force then nuclei can't form.


Multiverse scenarios

Possibilities for the laws of physics:

Natural             The parameters of the universe will be found to originate
                    from compact principles and no fine tuning is required for life.
Fine tuned          The parameters of physics require fine tuning to be
                    amenable to life.
Intelligent design  The parameters of the universe were designed to be amenable
                    to life.
Multiverse          There are multiple universes with different laws of physics,
                    most of them dull and lifeless, but the probability of one of
                    them accommodating life is unity.
Fortuitous          There is only one universe.  The laws of physics require fine
                    tuning and we are lucky that they are amenable to life.
Matrix              The universe is a computer simulation.
Darwin universe     Universes beget universes and the laws of physics evolve
                    by natural selection.

Drake equation

An analogue of The Multiverse is the Drake equation.

P  =  Probability that a star has a planet with intelligent life
N  =  Number of stars in the universe.

If:
P ~ 1                  Life is natural
P << 1                 Life requires fine tuning
P << 1  and  PN >> 1   Life requires fine tuning but life is probable in the universe.
                       This is the "Anthropic principle" or the "Multiverse" scenario.
P << 1  and  PN << 1   Life is improbable in the universe.
                       Either life is "Lucky" or we live in The Matrix or the
                       Earth was intelligently designed.

Soccer vs. Football

The rules of soccer are "natural" in the sense that they flow from a single premise (don't use your arms) and they lead to a rich game. The rules of football are "unnatural" in that the rulebook is thick and it takes a squad of referees to enforce.


The fortuitousness of the 12-tone scale

          Red: equal temperament           Orange: just intonation

The notes for 12-tone equal temperament coincide well with the note of just intonaton.

The most resonant notes in the 12-tone equal temperament scale are the fourth and the fifth and these are particularly close to their just-intonation counterparts.

The frequency ratio between a fourth and a fifth in just-temperament is

R  =  (3/2) / (4/3)  =  9/8  =  1.125
In a 12-tone equal-tempered scale the frequency ratio of a whole step is
R  =  2(2/12)  =  1.122
which is nearly the same as the ratio between a fourth and a fifth. This is why the 12-tone scale works so well. If you try any number other than 12 it doesn't work. This is why the 12-tone scale is the most useful for writing harmony.

Tunings exist that use numbers different from 12, such as for Indian, Thai, and Arabic music. These tunings can generate exotic melodic structure but they are less useful for harmony than the 12-tone scale.

The 12-tone scale is natural in the sense that it doesn't have any "free parameters". The choice of the number "12" emerged naturally from the positions of the resonant notes. It is also "fortuitous" in that the values of Z are so small.

Soccer is an example of a "natural sport". The rules are simple and if you change the parameters (such as field size, number of players, etc) the game is essentially the same.

American football requires "fine tuning". In order for the sport to make sense you need a large rulebook. It also has lots of "free parameters" because there are many different ways the rules could be constructed.

The chess player Edward Lasker once said:

"While the Baroque rules of Chess could only have been created by humans, the rules of Go are so elegant, organic, and rigorously logical that if intelligent life forms exist elsewhere in the universe, they almost certainly play Go."

The rules of chess are an example of "fine tuning" and there are lots of free parameters (the moves allowed by each piece).


Aliens

Timeline of the universe

An alien planet could conceivably have formed as early as 1 billion years after the big bang, meaning that there are likely aliens with a head start on us by billions of years.

An alien civilization could easily build a fission or fusion rocket that travels at 1/10 the speed of light, which would take 1 million years to cross the galaxy. The aliens have plenty of time to get here.

                Millions of years ago

Big bang             13700
First planets formed 13000
Earth formed          4500
Photosynthesis        3000
Oxygen atmosphere      600
Multicellular life     600
Vertebrates            480
Tetrapod vertebrates   400       Mammals, birds, and reptiles are all tetrapods
Mammals                170
Dinosaur extinction     66
Cats                    25
Cheetahs                 6       Fastest land animal
Tigers                   1.8
Humans                   1
Lions                     .9
Agriculture               .01
Civilization              .005
Calculus                  .0004
Smartphones               .00001

Starship

We presently possess the technology to build a fission and fusion rocket, each of which can reach a speed of .1 times the speed of light, and such a rocket can cross the Milky Way galaxy in a time that is a small fraction of the age of the universe. If aliens had built such a rocket they could easily have already colonized the galaxy.

Speed of light                       =  C
Speed of a fission or fusion rocket  =  V  =  .1 C
Diameter of the Milky Way            =  X           =     .1 million light years
Time to cross the galaxy             =  T  =  X/V   =      1 million years
Age of the universe                                 =  13800 million light years

Divinity

The divinity hypothesis becomes persuasive if there is a physical mechanism allowing it to happen, a mechanism that obeys the known laws of physics. Such a mechanism exists. An advanced alien civilization is equivalent to a diety.


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