Introduction, Overview and Special Relativity

Question 1

What is particle physics?

Answer 1

-the combination of quantum mechanics and special relativity

Question 2

Classifying Particles

Quantum Mechanics

Answer 2

-quantum mechanics is described by a wave function, a function of the positions etc. of all the particles in a system

Question 3

Classifying Particles

System of Two Particles

Answer 3

-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

Question 4

Classifying Particles

Observing the Wavefunction

Answer 4

-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

Question 5

Classifying Particles

Exchange Operator

Answer 5

-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 -

Question 6

Classifying Particles

Boson Definition

Answer 6

-if Ψ(x1,x2) = Ψ(x2,x1), then particles are bosons

Question 7

Classifying Particles

Fermion Definition

Answer 7

-if Ψ(x1,x2) = -Ψ(x2,x1), then particles are fermions

Question 8

Classifying Particles

Pauli Exclusion Principle

Answer 8

• 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

Question 9

The Standard Model

Answer 9

• 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

Question 10

Spin in the Standard Model

Answer 10

• leptons and quarks are spin 1/2
• higgs are spin 0
• gauge bosons are spin 1

Question 11

Fermions and Gauge Bosons

Answer 11

-fermions can be thought of as ‘matter particles’ and the gauge bosons as ‘force particles’ that mediate interaction between other particles

Question 12

Generations of Fermions

Answer 12

• 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

Question 13

Forces in the Standard Model

Answer 13

• 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

Question 14

Feynman Diagrams

Real and Virtual Particles

Answer 14

• 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

Question 15

Feynman Diagrams

The Mass Shell

Answer 15

-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

Question 16

Natural Units

Answer 16

-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

Question 17

How do you construct a scalar from vectors?

Answer 17

-multiply a covariant vector by a contravariant vector

Question 18

Scalar Product in Terms of Kronecker Delta

Answer 18

u_ . v_ = ui δij vj

Question 19

Transforming Gradient Between Reference Frames

Answer 19

∂iφ* = ∂xj/∂xi* ∂jφ

• where * indicates the transformed version
• this is a covariant vector

Question 20

Transforming Vectors Between Reference Frames

Answer 20

vi* = ∂xi*/∂xj vj

• where * indicates the transformed version
• this is a contravariant vector

Question 21

What does the metric do?

Answer 21

• applying the metric to the contravariant vector gives the covariant vector
• the inverse metric turns the covariant into the contravariant

Question 22

Covariant Vector

Definition

Answer 22

• typically have inverse units

- has components that change oppositely to the coordinates or, equivalently, transform like the reference axes

Question 23

Contravariant Vector

Definition

Answer 23

• 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

Question 24

Special Relativity Assumptions

Answer 24

• 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

Question 25

Lorentz Transformation

Definition

Answer 25

• a coordinate transformation that mixes up spatial coordinates and time
• so the correct vector space to work in is four dimensional spacetime

Question 26

Four Vectors

Answer 26

• four component vectors
• indexed by μ=0,1,2,3
• the 0th component is time

Question 27

How do you use indices to distinguish between covariant and contravariant vectors?

Answer 27

-write in upper position to indicate contravariant vector

Question 28

Lorentz Transformations

x*

Answer 28

x* = [x-vt] / √[1-v²]

Question 29

Lorentz Transformations

t*

Answer 29

t* = [t-vx] / √[1-v²]

Question 30

Lorentz Transformations in Index Notation

Answer 30

xμ* = Λμν xν

-where μ is in the upper position and ν is in the lower position

Question 31

Spacetime Invariant Quantity

Answer 31

ds² = dt² - dx² - dy² - dz²

Question 32

Spacetime Metric

Answer 32

• 4x4 matrix
• all zeros except diagonal
• entries on diagonal, 1,-1,-1,-1

Question 33

Spacetime Contravarient Vector

Answer 33

(t,x,y,z)

Question 34

Spacetime Covariant Vector

Answer 34

(t,-x,-y,-z)

Question 35

Spacetime Inverse Metric

Answer 35

-in the case of the spacetime four vector, the inverse metric is the same as the metric

Question 36

Relationship Between Metric and Inverse Metric

Answer 36

-the metric multiplied by the inverse metric is the identity matrix

Question 37

Momentum Four Vector

Answer 37

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

Question 38

Potential Four Vector

Answer 38

A = (V, Ax, Ay, Az)

Question 39

Current Four Vector

Answer 39

J = (ϱ, jx, jy, jz)

-where ϱ is the charge density

Question 40

Is there a four vector for electric and magnetic fields?

Answer 40

-E_ and B_ transform as an antisymmetric rank two tensor