How does a theory come to be?
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Transcript of How does a theory come to be?
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Richard Seto
New Predictions
How does a theory come to be?
Experimental Test
Old “Classical” idease.g. F=ma
New Ideas (might be crazy)
Unanswered questionsPuzzling Experimental Facts
New Theory (A guess)
A New “Good” Theory !
Yes
No
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Richard Seto
“Once upon a time” Pre-1900 “The standard model”
Newton, Kepler Maxwell Pretty good
Explained orbits of planets Explained static electricity
Maxwell –”We have a complete theory. Physics will be over in 10 years…” But there were a few minor problems
2GMmx, p, E, v, a, F=ma, F= r
1, - ABc t
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Richard Seto
The bad guys
Spectra of Hydrogen atom Strange – like a merry-go-round which only
went v=1mph, 5mph, 10mph,…. THE WHITE KNIGHT – QUANTUM MECHANICS
p
e-
Discreet shells!
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Richard Seto
Classically
To “quantize” Particles change to
“wave functions” E,p,V Change to
operators (EH, pp, VV)
But operate on what??
Answer: (x) the “wave function”
is the “real thing”
Getting to QM
2
[ ,
( , ) ( ) ( , )2
] ( ),
, ( ) ( )
Formally quantize:
where is very s
One Solution: x
G
m
ues
all (classically=0)
s:
chec
k itx
!p
x t r x t
i
i x i x
x
m
i x
i x
A B AB BA
x
x =
pH V
p
x p x x
p
( ) ( )
It orks w !
x xi x i x
i
2pE= +V( )2m
r
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Richard Seto
What does it predict? Energy levels!
BUT! Zero probability!
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Richard Seto
What is ? Answer * is a probability We have lost the normal notion of a
particle with position and velocity Crazy, but it EXPERIMENT tells us it
works E.g. double slit experiment with
electrons
electrons
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Richard Seto
Happiness again? Dirac – “underlying physical laws necessary for a
large part of physics and the whole of chemistry are known”
No! More bad guys!Problems:
Hydrogen atom – small discrepanciesRelativityDecay e.g. n pe
How does a particle spontaneously evolve into several other particles? Quantum Field Theory
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Richard Seto
First- dirac notation E.g. look at energy levels of Hydrogen
E1,E2,E3 with wave functions 1(x), 2(x), 3(x),… n(x) Remember we can specify x or p but not both – Can we
write these as a function of p instead of x?? YES! 1(p), 2(p), 3(p),… n(p)
So generalize – call it |n> and specify n(x)=<x|n> and n(p)=<p|n>
In fact we now want n to be an operator so <x|n>=<0| n(x)|0>
Now |n> is the “real thing” X probability distribution = |<x|n>|2
p probability distribution = |<p|n>|2
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Richard Seto
Can we create or destroy particles?
2 2 22
22 2 2
†
1( )2 2 2
2 2
2 2
To Illustrate, let's use a Simple Harmonic Oscillator
define
means hermitian conjugate
k p m xV x kx Hm m
m xit m x
m ip m ipa x a xm m
x x p
t
†
†
1 1
1( )2n
is a state with E=E
N a ai x
a n n a n n
n n
ÿ
†
|n> is state with E2
e.g. <x|2> is n=2
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Richard Seto
Now skip lots of steps-creation and annihilation operators
0 1
a a
a a
a
Let and be the basic "things" and 0 be the vacuumThen i.e. is an operator which creates 1 quantaLater we will write which we call a field operatorAn aside: Really the typical way we
†
†
† †
2 2 3
,
, ( , )
1
k
E B E d r
B E B E
AB A Ec t
A a
P X
do this is as follows:Start from E and M, with the classical radiation field Quantize Field operatorsE and M Remember??Define and
really and Ñ
,
, , , , , ,
ikx -ikxk
k
i 1k k k k k kc c
e a e
a a a a
P X
†
† †
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Richard Seto
Particles and the Vacuum
,
, 1 1
12
1_ _ _ _ _2
k k
kk
k
i a n n a n n
N a a N n n n
H N ck
Total Energy number of photons Energy of photon
X P
Now quantize again
number operator
Sakurai: " The quantum mechanica
ÿ
ÿ
†
†
0
n
l excitation of the radiation fieldcan be regarded as a particle, the photon, with mass 0, and spin 1.
Field not excited vacuum Wiggle (excite) the field particle(s)
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Richard Seto
Vacuum has energy???
The Vacuum has energy? Experiment – measure Force between two
plates in a vacuum F=dEnergy(vacuum)/dx
Done is last several years – agrees with prediction!
12
12
vacuum when =0
???
k
vacuumk
H N
N
E
ÿ
ÿ
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Richard Seto
What tools to we have to build a theory?
To Build theories (an effective Lagrangian) we invoke fundamental symmetries. Why? -
because it works and it seems right somehow??? “ H. Georgi”
Examples translational invariance momentum conservation rotational invariance angular momentum conservation Local gauge invariance (phase change) EM forces
In the standard theories we essentially always start with a massless theory - for QCD this comes from a requirement of “chiral symmetry” Mass comes about from a breaking of symmetry giving
rise to a complicated vacuum.
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Richard Seto
Lagrangian Formulation Compact, Formal way to get eqns. of motion (F=ma)
Lagrangian L=T-V=Potential E – Kinetic E (Hamiltonian E=H=T+V)
Lagrange’s eqn – just comes from some math
2
2 2
0 0
. .12
1 12 2
0 !!!
Simple Harmonic Oscillator (A spring)
V=Pot. Energy =
d dL dL d dL dLdt dq dq dt dv dxe g
kx
dL dLL mv kx mv kxdv dx
d mv kx ma kx F madt
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Richard Seto
Symmetries - Example L is independent of x (translational
invariance) I.e. physics doesn’t depend on position
212
0 0
0 0 0
If we choose it is independent of position
then from Lagranges eqn.
i.e. p is
L mv
d dL dL d dL dLdt dq dq dt dv dx
dL dL d dpmv mvdx dv dt dt
constant !, momentum is conserved!
Symmetries Conservation Lawsspace invariance conservation of ptime invariance conservation of Erotational invariance conservation of angular momentum
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Richard Seto
Aside – for electrons
2 2 2 2 4
2 2 2
_ __ _
L
Add relativity
units c=1, =1
guess mass like potential energyguess
Guess Probability =
electron
E p c m c
E p m
H Kinetic energy mass termL Kinetic energy mass term
i mmass
ÿ
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Richard Seto
Require other symmetriesLocal gauge invariance Forces!
L
( ) 'L L L
Probability = what about the phase of ?? Ans: Its arbitary , =constant
is invariant under (global)
electron
i
electron electron electron
i m
x eso
'
( )
( ) ( )
'
phase (guage) transfromationWhat about if we made dependent on the position ??? = (x)
local gauge transformationcheck itmass term: so mass term is ok1st ter
i x
i x i x
e
m m e e m
( ) ( ) ( ) ( ) ( )( ) ( )m:
So we have a problem....
i x i x i x i x i xi i e e x i e e x e
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Richard Seto
Solution (a crazy one)
(
. .
'
Add a new term to the Lagrangian
Local Gauge transformation :
note: so its OK.now check 1st term:
New
i x
New
i x
i i A m
e g B A
x e x
A A x
A A A B
i i A i e
Ñ
Ñ Ñ Ñ Ñ
L
L
) ( )
( ) ( ) ( ) ( ) ( )
( )
( )
i x
i x i x i x i x i x
i A i x e x
i e e e x i A e i x e x
i i A i i A
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Richard Seto
So What is this thing A?
The term is NOT gauge invariant So do we throw out gauge invariance?? NO – we set m=0 photons are masses
IS it really electricity ??? – photons? (looks like a duck…)
Agrees with experiment!
( ) ( )
Is this term Guage invariant? Try it!remember
mass m A A
A A x
m A A m A x A x m A A
L
Ans: Electricity!What happens if we give mass to A?
1, - ABc t
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Richard Seto
T<TCurie
A Magnetic Field down and to the
right is SACRED
I used to think Masses were sacred
Example from C. Quigg Think about an infinite
ferromagnet : a crystalline array of spins - magnetic dipoles moments, with nearest neighbor interactions
At T<TC the spins will line up imagine a little “pico-
physicist” who lives in this world.
For him an overall magnetization and its direction would be sacred
How do you prove to him that its not?
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Richard Seto
T>Tcurie
Heat him up to T>TC restore the
symmetry! His environment is
now magnetic-less
Now I understand!The Vacuum has broken symmetry!
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Richard Seto
T<Tcurie again
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Richard Seto
Start building a model of the string interactions with the kinetic energy term of a pair of fermions
You could think of these as
This simple Lagrangian has a symmetry. It is invariant under “isotopic” rotations, I.e. it doesn’t care if we mix up the “flavors” That’s good - the strong interaction doesn’t care about
flavor This Lagrangian also has another symmetry- the chiral symmetry - that of R
and L (we will take this symmetry as sacred) and we can write it as
We can then freely mix up the R and L handed flavors independently and the Lagrangian is still invariant -
A Toy ModelL i
pu or d n
L R R L Li i
, , iR L R Le
ie
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Richard Seto
Mass?? Can we add a mass term to this Lagrangian? OK lets try it!
Now what happens to chiral symmetry?
The mass term connects the R and L handed sectors so we cannot transform R and L handed quarks independently! - the mass term has broken the chiral symmetry!
Easy to see why mass does this- for massive particle
WE CANNOT ADD SUCH A TERM WITHOUT DESTROYING THE CHIRAL SYMMETRY- SO WE WON’T Note: There really is such a term, but m0 is small, on the order of 5 MeV, where as the mass of the
proton is 1 GeV, and the constituent quark is 300 MeV. So chiral symmetry is really an approximate symmetry which is broken by the small quark masses.
So we have a massless world??? - what do we do??
( )L R RR L L R L L RR L Li i m
ud
L mi Mass termTerms break chiral symmetry!
RH LH!
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Richard Seto
Spontaneous symmetry breaking
Which wine glass is yours? Or think of a nail sitting on its head
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Richard Seto
Where Does Mass Come From?
222
42( )1 mf Ti g
L
uKE of = d I nteraction of with KE of Scalar field
Invoke a scalar field with a funny potential energy term For High T: f(T)=+1 and the
lowest energy state is at = 0. is chirally symmetric m is not a mass – just a constant
L,R L,Rie
cV , T T
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Richard Seto
The Vacuum Gives us Mass?!
V(),T~Tc
What happens as we lower T? [f(T)=-1] Lowest energy state is at Expand around equilibrium point
New mass term (chiral symmetry has been broken)
What is ? It’s the condensate a glop of quarks and gluons which make up the vacuum!
m
m
2
2 1 22
4
i g
m
gm
LV(),T<Tc
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Richard Seto
A simulation of the vacuum
4 dimensional Action Density of the vacuum
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Richard Seto
Living in the cold QCD vacuum
It is generally believed in the fish population that there is an inherent resistance to motion and that they swim in a “vacuum”.
The vacuum – perceived to
be empty by the general
fish population
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Richard Seto
A clever (and crazy idea)
One clever young fish is enlightened. “The vacuum is complicated and full of water!” he says –” really there is no resistance to motion!”
“Phooey” say his friends, “we all know the vacuum is empty.”
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Richard Seto
How does he prove it? Answer – he builds a machine to boil
the water into steam – to “melt” the vacuum
In steam his “friends” move freely. OK OK – they die because
They can’t breathe in air They are poached because of the heat
So maybe he boils only what’s in a small bottle as an experiment…
you get the point…
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Richard Seto
- 1 microsecond
after big bang Size of universe ~
10 km size of riverside
At the time of strong interaction phase transition
How do we push back there?
Theories… what do they say? Can we trust them?
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Richard Seto
Early universe Different Vacuum?
Present theories (well tested ones!) indicate that the vacuum is NOT empty but is filled with a quark condensate “goo” This is a very weird idea - “wilder than many crackpot
theories, and more imaginative than most science fiction”-F. Wilczek
Explains why particles (quarks) stick together -”confinement” is the origin of hadronic particle masses (protons, neutrons,
pions etc) In the early universe this Vacuum was very different than it is
now. Particles (hadrons) had different (zero) masses! Study of Heavy ions (RHIC) allows us to study this vacuum
directly