Lecture 7

Reading:

Recall from last lecture:

The particles of the Standard Model:

particle charge flavor

quarks +2/3 u c t

-1/3 d s b

leptons -1 e μ τ

0 ν_e ν_μ <ν_τ

particle	charge	flavor
quarks	+2/3	u	c	t
-1/3	d	s	b
leptons	-1	e	μ	τ
0	ν_e	ν_μ	<ν_τ

The Interactions

Four forces

The known interactions are four:

Interaction mediator symbol spin/parity mass

strong gluon g 1^- massless

electromagnetic photon γ 1^- massless

weak intermediate vector bosons W^±, Z⁰ 1^-, 1⁺ about 90 GeV/c²

gravity graviton G 2⁺ massless

Interaction	mediator	symbol	spin/parity	mass
strong	gluon	g	1^-	massless
electromagnetic	photon	γ	1^-	massless
weak	intermediate vector bosons	W^±, Z⁰	1^-, 1⁺	about 90 GeV/c²
gravity	graviton	G	2⁺	massless

Unification of forces

Unification of forces has been a theme in Physics for centuries. Newton unified the force that caused objects to fall to Earth, and the force that keeps planets in orbit around the Sun with a single force of gravity. Faraday, Maxwell, and others, unified the electric and magnetic forces into one electromagnetic force. In the late 1960's and early 1970's, Weinberg, Glashow, and Salam put forth a theory that unified electromagnetism with the weak interaction of nuclear beta decay.

The combined interaction is called the electroweak interaction. This theory relies on a process of spontaneous symmetry breaking that derives from the BCS theory of superconductors. In the theory, all particles are initially massless and symmetry is unbroken. After symmetry is broken, many particles acquire mass -- we live in this world of broken symmetry.

It is believed that the strong force will combine with the electroweak force at a high energy scale, about 10¹⁶ GeV. This is even predicted by supersymmetry (SUSY) theories (more about SUSY below). Ultimately, many physicists believe that all 4 forces will be seen to derive from a single original force. Such theories are known as grand unified theories or GUT's.

The Sun

All of the above forces are at work in the Sun.

The initial step in the fusion chain:
p p --> d e⁺ ν_e
is governed by the weak interaction. If the weak interaction were a bit stronger, then the Sun would exhaust all its hydrogen fuel in a fraction of the time it now takes. If the weak interaction were a bit weaker, then the Sun would perhaps not shine at all. In either case, life on Earth would be enormously different.
The strong interaction governs other nuclear process within the Sun. For instance
p d --> ³He γ , and
³He ³He --> ⁴He p p
are two more process in the p-p cycle that are governed by the strong force.
The propagation of energy out of the Sun by radiation, and the emission of the solar blackbody spectrum are governed by the electromagnetic interaction.
And of course gravity binds the material of the Sun together and supplied the energy initially required for fusion reaction to commence.

Particle Classification: Fermions and Bosons

Particles are classified as either fermions or bosons depending on their spin.

Fermions are particles with half integral spin (1/2, 3/2, ...).
Bosons are particles with integral spin (0, 1, 2, ...).

The spins are in units of hbar.

Particle exchange symmetry

All particles have are described by a wave function. Combinations of free particles are described by a product of individual particle wave functions. For instance, for two particles, 1 and 2:

ψ = φ₁ χ₂

The spin-statistics theorem of quantum field theory supplies the rule to apply to the interchange of two identical particles, 1 and 2. For bosons, the total wave function remains unchanged under the exchange of particles (it is symmetric in the indices 1 and 2). For fermions, the total wave function changes sign under the exchange of particles (it is antisymmetric in the indices 1 and 2). The above wave function must be modified for pairs of identical particles to satisfy these requirements:

ψ_fermion = φ₁ χ₂ - φ₂ χ₁ , and
ψ_boson = φ₁ χ₂ + φ₂ χ₁ .

Relation to supersymmetry

One of the possible extensions to the Standard Model is something called supersymmetry. A number of symmetries exist in particle physics: left--right symmetry also known as parity; particle--antiparticle symmetry; and time reversal symmetry are the three most common. Supersymmetry postulates a symmetry between fermions and bosons. In supersymmetry, every fermion has a (super-)partner boson, and vice versa. Tests of this hypothesis are ongoing.

Particles and Antiparticles

Free Particle Wave Equations

In scattering and decay experiments, the particles under investigation can be considered free before and after the interaction. This is a great simplification, since then we need deal with only their free particle wave equations, and the messy details of the interaction can be avoided.

The free particle solution from Schrodinger's equation is inadequate since it isn't relativistically invariant. We need something that is a quantum mechanical wave equation and satisfies special relativity. How can we write down a relativistically invariant wave equation?

The free particle Schrodinger equation is arrived at basically from the non-relativistic expression:

E = p²/2m

then replacing E and p by their quantum mechanical operators,

E --> iħ d/dt p --> -iħ d/dx

and supplying them with a wave function to operate on. What if we start with the relativistic expression:

E² = p²c² + m²c⁴

and apply the same idea. What we get is known at the Klein-Gordon wave equation:

δ²ψ/δt² = (del² - m²)ψ

This wave equation is suitable for describing spin 0 bosons.