Introduction, Overview and Special Relativity Flashcards
overview, natural units, special relativity, lorentz transformations,
What is particle physics?
-the combination of quantum mechanics and special relativity
Classifying Particles
Quantum Mechanics
-quantum mechanics is described by a wave function, a function of the positions etc. of all the particles in a system
Classifying Particles
System of Two Particles
-consider a system of two particles with positions x1 and x2 with wavefunction:
Ψ = Ψ(x1,x2)
-if these particles are identical it seems obvious that:
Ψ(x1,x2) = Ψ(x2,x1)
-BUT THIS IS NOT TRUE
Classifying Particles
Observing the Wavefunction
-the wavefunction itself is unobservable, we can only observe |Ψ|
-this means that for identical particles we actually only require:
|Ψ(x1,x2)|² = |Ψ(x2,x1|²
Classifying Particles
Exchange Operator
-define the exchange operator such that:
E Ψ(x1,x2) = Ψ(x2,x1)
-two exchanges must bring us back to the original state
EE Ψ(x1,x2) = Ψ(x1,x2)
=> E² = I (identity)
=> Ψ(x2,x1) = E Ψ(x1,x2) = ±Ψ(x1,x2)
-this gives two classes of particle, some require +, some require -
Classifying Particles
Boson Definition
-if Ψ(x1,x2) = Ψ(x2,x1), then particles are bosons
Classifying Particles
Fermion Definition
-if Ψ(x1,x2) = -Ψ(x2,x1), then particles are fermions
Classifying Particles
Pauli Exclusion Principle
- it is easy to check that fermions (and only fermions) obey the Pauli exclusion principle
- i.e. that two particles cannot be in exactly the same state
The Standard Model
- Bosons
- -fundamental bosons
- –gauge bosons
- —photon
- —gluons (x8)
- —weak bosons (W+,W-,Z0)
- –Higgs
- -hadron bosons
- –mesons (π+,π0,…)
- Fermions
- -fundamental fermions
- –leptons
- –quarks
- -hadron fermions
- –baryons
Spin in the Standard Model
- leptons and quarks are spin 1/2
- higgs are spin 0
- gauge bosons are spin 1
Fermions and Gauge Bosons
-fermions can be thought of as ‘matter particles’ and the gauge bosons as ‘force particles’ that mediate interaction between other particles
Generations of Fermions
- fundamental fermions come in three generations
- mass increases for each generation
- gen I
- -quarks: u,d
- -leptons: νe, e
- gen II
- -quarks: c, s
- -leptons: νμ, μ
- gen III
- -quarks: t, b
- -leptons: ντ, τ
Forces in the Standard Model
- there exist four known fundamental forces; gravity, electromagnetic, weak nuclear and strong nuclear
- in the standard model there are only three, we ignore gravity
- this is OK as the effect of gravity at particle scales is negligible in comparison to the other forces
- the photon carries the EM force, gluons mediate the strong force holding quarks together in hadrons and weak bosons (W+, W-, Z0 mediate the weak force, responsible for β-decay
Feynman Diagrams
Real and Virtual Particles
- external lines in a feynman diagram are observable particles, we call them real particles
- internal lines are not observable since the act of observation would prevent the interaction, these are called virtual particles since we can’t be sure that they exist at all, they may just serve as a useful mathematical tool
Feynman Diagrams
The Mass Shell
-real particles are energy and momentum eigenstates so have well defined energy and momentum and a correct relationship between E and p, i.e.:
E² = p²c² + m²c^4
-where m is the particle mass
-this relation does not hold true for virtual particles
-we say that real particles are ‘on the mass shell’ or ‘on shell’ whereas virtual particles are ‘off shell’
-note that particles that are ‘on shell’ can have negative energy
Natural Units
-from a relativistic perspective we exist in 4D space time, three spatial dimensions and time
-it therefore makes sense to use units which can be standardised in all directions
-set c=1
-as we are also considering quantum mechanics we set ħ=1
-this leaves us with only one unit, typically the MeV is chosen
E, p, m measured in MeV
x, t measured in 1/MeV
L, v unitless
-where L is angular momentum
How do you construct a scalar from vectors?
-multiply a covariant vector by a contravariant vector
Scalar Product in Terms of Kronecker Delta
u_ . v_ = ui δij vj
Transforming Gradient Between Reference Frames
∂iφ* = ∂xj/∂xi* ∂jφ
- where * indicates the transformed version
- this is a covariant vector
Transforming Vectors Between Reference Frames
vi* = ∂xi*/∂xj vj
- where * indicates the transformed version
- this is a contravariant vector
What does the metric do?
- applying the metric to the contravariant vector gives the covariant vector
- the inverse metric turns the covariant into the contravariant
Covariant Vector
Definition
- typically have inverse units
- has components that change oppositely to the coordinates or, equivalently, transform like the reference axes
Contravariant Vector
Definition
- has components that transform ‘as the coordinates do’ under changes of coordinates and so inversely to the transformation of the reference axis
- i.e. if the component axes rotate in one direction, the vector rotates in exactly the opposite
Special Relativity Assumptions
- the speed of light is constant and equal for all observers
- this tells us that both the time and space coordinates of two observers in relative motion are different
Lorentz Transformation
Definition
- a coordinate transformation that mixes up spatial coordinates and time
- so the correct vector space to work in is four dimensional spacetime
Four Vectors
- four component vectors
- indexed by μ=0,1,2,3
- the 0th component is time
How do you use indices to distinguish between covariant and contravariant vectors?
-write in upper position to indicate contravariant vector
Lorentz Transformations
x*
x* = [x-vt] / √[1-v²]
Lorentz Transformations
t*
t* = [t-vx] / √[1-v²]
Lorentz Transformations in Index Notation
xμ* = Λμν xν
-where μ is in the upper position and ν is in the lower position
Spacetime Invariant Quantity
ds² = dt² - dx² - dy² - dz²
Spacetime Metric
- 4x4 matrix
- all zeros except diagonal
- entries on diagonal, 1,-1,-1,-1
Spacetime Contravarient Vector
(t,x,y,z)
Spacetime Covariant Vector
(t,-x,-y,-z)
Spacetime Inverse Metric
-in the case of the spacetime four vector, the inverse metric is the same as the metric
Relationship Between Metric and Inverse Metric
-the metric multiplied by the inverse metric is the identity matrix
Momentum Four Vector
p = (E,px,py,pz) -the invariant quantity is mass, m m² = E² - px² - py² - pz² -or E² = p_² + m² -where p_ is the momentum 3-vector
Potential Four Vector
A = (V, Ax, Ay, Az)
Current Four Vector
J = (ϱ, jx, jy, jz)
-where ϱ is the charge density
Is there a four vector for electric and magnetic fields?
-E_ and B_ transform as an antisymmetric rank two tensor
