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

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

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!

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

Richard Seto

What does it predict? Energy levels!

BUT! Zero probability!

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

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

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

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

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

†

† †

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)

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

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.

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

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

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

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

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

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

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?

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!

Richard Seto

T<Tcurie again

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

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!

Richard Seto

Spontaneous symmetry breaking

Which wine glass is yours? Or think of a nail sitting on its head

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

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

Richard Seto

A simulation of the vacuum

4 dimensional Action Density of the vacuum

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

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.”

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…

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?

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