Useful Equations and Links

• Relativistic units
• Special relativity (Skidmore text)
• PHYS226 - Sp2015

Table of particles

Simple (basic) particles include fermions (leptons, quarks) and bosons. Composite particles include mesons, baryons and strange quarks.

Simple particles

Wikipedia's simple particle set

Fermions

• Quarks build composite particles more than act independently
  • Operate via strong force
• Antiparticles have opposite charge
• Mass in terms of $$m=\text{MeV}/c^2$$

Table of quarks

Bosons

• These are the force carriers which cover fundamental forces.
• Photons and gluons are their own antiparticles.

Conservations

• Energy, momentum, and angular momentum are all conserved
  • Spin on right can be different by integer 1s, but not by $1/2$.
• Total electric charge
• Lepton number (number of leptons + antileptons) $$L=n_l-n_{-l}$$
• Lepton flavor (electron-ness, muon-ness, tau-ness conserved, except for neutrinos)
  • Opposite neutrino if creation, same neutrino if decay.
  • Whenever neutrinos are produced, weak force
  • Weak force ignores this idea
• Baryon number (quarks + antiquarks in composite particles) $$B=\frac{1}{3}(n_q-n_{\bar{q}})$$
• Quark flavor (conservation of quark type)
  • "Similarly, the only way for the strong or electromagnetic to get rid of a strange quark is to annihilate it with an anti-strange quark"
  • Violated by weak force

Complex particles

Baryons: 3x quarks, $B=+1$ Mesons: 1 quark, 1 antiquark, $B=0$ Antibaryons: 3x antiquarks, $B=-1$

• Leptons have baryon number of zero (not composed of quarks)
• Same with bosons

Which interaction?

• Short lifetimes ($10^{-23}$), only hadrons and composite particles $\to$ likely strong force.
  • Flavor conserved.
• Medium lifetimes $(10^{-17})$, photons involved, likely electromagnetic.
  • Flavor conserved.
• Long lifetimes $(10^{-8})$, results include neutrinos and leptons, likely weak force
  • Flavor violation allowed.

Group theory

In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold.

• $GL(n)$: general linear group, all invertible $n\times n$ matrices
• $U(n)$: unitary group, all matrices such that $U{U}^{†}=1$
• $SU(n)$: belong to $U(n)$ and have $\det =1$
• $O(n)$: belong to $U(n)$, only made with real numbers
• $SO(n)$: belong to $O(n)$ and have unit determinant

Misc

Matrix-vector mult:

Commutator of two operators is $$[\hat{A}\hat{B}]=AB-BA$$

• If zero, share same eigenvalues

$$\beta =\frac{v}{c}\gamma =\frac{1}{\sqrt{1-{\beta }^2}}$$ $$L=\frac{L_0}{\gamma }\Delta t^{\prime }=\gamma \Delta t$$

$L_0$ is the "rest length", $\Delta t$ is the "rest time".

Finding eigenstuff

Eigenvalues: $$\det |A-\lambda I|=0$$

Eigenvectors: $$A|\lambda ⟩=\lambda |\lambda ⟩$$

$\lambda$ eigenvalue, $|\lambda ⟩$ corresponding eigenvector

Quantum stuff

$j$ is the spin (angular momentum) quantum number, $m$ is the spin component quantum number. $$|\frac{1}{2},\frac{1}{2}⟩=|+⟩$$ $$|\frac{1}{2},-\frac{1}{2}⟩=|-⟩$$

Operator

$$J^2|jm⟩=j(j+1)|jm⟩$$ $$J_3|jm⟩=m|jm⟩$$ For example: $$S_z|jm⟩=m\hslash |jm⟩$$