Download Particle Physics

The World of
Elementary Particle Physics
Demystified
Eduardo Pontón - IFT & ICTP/SAIFR
IFT - Perimeter - ICTP/SAIFR Journeys
into Theoretical Physics, July 18-23
Plan
1. What are these lectures about
2. How do we know what we know
3. Field Theory and Symmetries
4. Hidden Symmetry and the Higgs Boson
Part I
Our Subject Matter
Particle Physics
Perseus Galaxy Cluster
What do YOU mean??
Gold Dust?
Installation inspired by the search for the Higgs boson
(by Becs Andrews)
Atoms of Niobium (41) and Selenium (34)
Particle Physics typically understood
as the realm of subatomic particles
Quantum mechanical effects an essential component!
It will be useful to recall some facts in the
context of the birth of Quantum Mechanics!
Planck's law for black-body radiation:
For radiation at temperature T, average energy in volume V with frequencies between ! and
~!
V ~ ! 3 d!
= dN (!) ~!
hdE(!)i = 2 3 ~!
e
1
⇡ c e
1
Number of modes in
V and [!, ! + d!]
! + d! :
Planck, 1900
V ! 2 d!
dN (!) =
⇡ 2 c3
Einstein’s formula for mean energy fluctuations in a black body:
2
hdE(!)i
h( E)2 i =
+ ~!hdE(!)i
dN (!)
Einstein, 1909
To derive it, just need to recall a bit of statistical mechanics:
Canonical Partition Function:
Z=
X
e
En
n
Obtain average energy by differentiating w.r.t.
1 X
hEi ⌘
En e
Z n
Differentiating w.r.t.
:
En
=
@ log Z
@
again, we can obtain the mean square energy fluctuation:
2
2
h( E) i = hE i
2
hEi =
@hEi
@
Upshot: can easily get the energy fluctuation, if we know the energy as a function
of temperature!
Einstein just applied this to Planck’s law:
hdE(!)i = dN (!)
~!
e
~!
~! ⌧ 1
2
h( E) i =
dN (!)
Rayleigh-Jeans law
1
~!
1
dN (!) ~! e
and for any frequency:
2
hdE(!)i
h( E) i =
+ ~!hdE(!)i
dN (!)
2
@hEi
@
Then he proceeded to interpret the result…
~!
Wien’s formula
2
hdE(!)i
2
h( E) i =
+ ~!hdE(!)i
dN (!)
To interpret the linear term
The quadratic term is exactly what
is obtained for classical waves
Recall that for an ideal gas, i.e.
N non-interacting particles:
3
3N
hEi = N kB T =
2
2
(Homework!)
Hence:
2
h( E) i =
@hEi
= kB T hEi
@
The concept of wave-particle duality was born:
“the effects of the two causes of fluctuations [waves and particles] act like fluctuations
from mutually independent causes (additivity of the two terms)” — Einstein (1909)
Attempts at obtaining this from dynamics (as time averages) could only give one or
the other term…
vs
In our typical “particle physics” experiments, we see the events as
particles: they resemble the electron-by-electron setup!
But we are interested in the underlying distribution (in different variables)
In addition, the experimental conditions we are focusing on are such that
the particles are moving close to the speed of light!
Theoretician perspective: need to be able to compute quantum mechanical
amplitudes that reflect the properties of special relativity:
• No inertial frame is special (laws appear the same in any)
• Nothing travels faster than the speed of light (causality structure)
• Amplitudes must allow a probabilistic interpretation (unitarity)
Quantum Mechanics + Special Relativity = Quantum Field Theory
If we want to be more precise: “Quantum Field Theory of Point Particles”
A “point particle” type specified by: mass & spin
(or helicity,
if massless)
What if the fundamental constituents were string-like, or even more complex?
A matter of scales: if our probe cannot
resolve the string/brane, we will be able
to describe its physics by QFT
The lesson is more general:
• Protons/nuclei can look point-like under many experimental conditions
• Atoms/molecules can look point-like to a typical human
QFT can be used to describe any such system…
… it has nothing to do with the system being “fundamental”
But QFT becomes essential when the energies are such that particles
can be produced… the realm we want to explore here!
The Quantum Harmonic Oscillator
(if you need a review)
Quadratic Hamiltonian:
1 2
2 2
H=
p̂ + ! x̂
2
r
~
x̂ =
a† + a
2!
r
Define creation and annihilation operators via:
~! †
p̂ = i
a
a
2
✓
◆
1
~!
Which diagonalize the Hamiltonian: H = N +
N = a† a
2
n-quanta/particle states obeying Bose-Einstein statistics:
†
|1i = a |0i
1
|ni = p (a† )n |0i
n!
[a, a† ] = 1
[a, a] = [a† , a† ] = 0
Let us go back to the wave-particle puzzle
2
hdE(!)i
h( E)2 i =
+ ~!hdE(!)i
dN (!)
(Pasqual Jordan in
the Dreimännerarbeit)
which had to wait until 1925 for a satisfactory solution
Since the E&M field (in a box) is a set of decoupled simple harmonic oscillators, we can quantize
each one, following the quantum mechanical treatment known to apply to a single oscillator.
Considering a 1D case:
(x, t) =
(as Jordan did)
X
n
†
[an , an0 ]
r
=
box of size L and wavevectors
~
an e
2!n
n,n0
kn = ⇡n/L = !n
i!n t+kn x
[an , an0 ] =
+ h.c.
†
†
[an , an0 ]
=0
Considering a 1D case:
(x, t) =
box of size L and wavevectors
X
n
†
[an , an0 ]
r
=
~
an e
2!n
kn = ⇡n/L = !n
i!n t+kn x
[an , an0 ] =
n,n0
+ h.c.
†
†
[an , an0 ]
=0
The energy density is found to be
1
H=
2L
Z
L
⇥
2
dx (@t ) + (@x )
0
⇤
2
=
X
n
1
(Nn + )~!n
2
By putting the oscillators in a thermal bath at temperature T, and computing the mean
square energy fluctuations in a small segment a << L, the two terms were recovered:
2
hdE(!)i
2
h( E) i =
+ ~!hdE(!)i
dN (!)
For a detailed derivation see:
Duncan & Janssen, arXiv:0709.3812
These considerations, together with the seminal papers by Dirac (1926,1927), gave birth
to Quantum Field Theory…
Upshot: the canonical quantization of a field
(continuous) incorporates automatically the
concept of a particle (discreteness)
In addition: the formalism allows to describe the
creation and destruction of quanta (particles)
(provided energy and momentum can be conserved)
Note also that the quantum mechanical treatment
of the E&M field had to be intrinsically relativistic
Dimensional Analysis
The only difference between a rut and a grave are the dimensions — Ellen Glasgow
• Existence of (maximum) universal speed: c = 299 792 458 m/s
• Universal unit of action: ~ = 6.582 119 514(40) ⇥ 10
16
eV · s
• Universal Boltzmann constant: kB = 8.617 3324 (78) eV /K
Since we are interested in quantum, relativistic systems, better to measure
velocities and action in units of c and ~ !
Natural units:
c = ~ = kB = 1
Then any mass/energy scale is associated with a length scale through the
Compton wavelength:
o = ~/(mc)
and to a time scale through c.
“Everything is energy and that’s all there is to it.
Match the frequency of the reality you want and
you cannot help but get that reality.
It can be no other way. This is physics.”
Pseudo-scientific quote, probably from a
“channeler“ named Darryl Anka who has
assigned the words to an entity named Bashar!!
Dimensional Analysis
Some useful reference scales:
• Most of the mass we see around us comes from the mass of protons/neutrons
Nucleon mass:
m p ⇠ 1 GeV = 109 eV ⇠ 10
27
kg
15
Associated length scale: o p ⇠ (1/5) ⇥ 10 m = (1/5) Fermi
Note: the “charge radius” of the proton is about 1 Fermi
Associated time scale: ⇠ 10
24
s
• Sometimes dimensionless constants have a crucial effect on the physics.
Compare to the hydrogen atom. The “underlying” mass scale is:
Relevant energy scale is
and the length scale is
E0 = 12 ↵2 me ⇠ 10 eV
a0 = oe /↵
⇢
me ⇠ 0.5 MeV = 0.5 ⇥ 106 eV ⇠ m p /2000
oe ⇠ 2000 ⇥ o p
A big, non-relativistic
system because
↵ ⇠ 1/137 ⌧ 1
The Standard Model
Mendeleev’s Table
Particles and Particles
A Paradigm
Rutherford at his Lab.
Seeing Particles
~
⌦B
P.A.M. Dirac
(1931)
Discovery of the
Positron (1932)
C. Anderson
Cloud chamber’s picture of cosmic radiation
“Simpler” Times
Cloud Chamber
(muons, alpha and beta radiation)
Alpha Particles
(Polonium)
16 GeV ⇡ beam entering a liquid- H2 bubble chamber at CERN (~1970)
The discovery of neutral currents by the Gargamelle bubble chamber (1973)
... A Long Way
Find an LHC Guide/FAQ with lots of
interesting information at:
http://cds.cern.ch/record/1165534/files/CERNBrochure-2009-003-Eng.pdf