Transcript First Results from RHIC - University of California, Riverside

How does a theory come to be?
Old “Classical” ideas
e.g. F=ma
Unanswered questions
Puzzling Experimental Facts
New Ideas (might be crazy)
New Theory (A guess)
New Predictions
Experimental Test
No
Yes
A New “Good”
Theory !
Richard Seto
“Once upon a time”

Pre-1900 “The standard model”



Newton, Kepler
1 A
Maxwell B    ,   c t
Pretty good
x, p, E, v, a, F=ma, F=



GMm
r2
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
Richard Seto
The bad guys
e-
p
Discreet shells!



Spectra of Hydrogen atom
Strange – like a merry-go-round which only
went v=1mph, 5mph, 10mph,….
THE WHITE KNIGHT – QUANTUM MECHANICS
Richard Seto
Getting to QM


Classically
p2
E=
+V(r )
2m
To “quantize”




Formally quantize: [ A, B ]  ( AB  BA)
  x, p   i 
where is very small (classically=0)
Particles change to

“wave functions”
One Solution: x  x = x p  p 
i x
E,p,V Change to
check it!
operators
(EH, pp, VV)


x
,
p


x
(

)

( x )
 
i x
i x
But operate on what??


Answer: (x)
 
x
( )  x
( )   x 


 p2

Guess: H ( x, t )  
 V ( r )   ( x, t )
 2m

the “wave function”
 is the “real thing”
i x
i x
 i  It works!
i  x 
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 discrepancies
Relativity
Decay 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?
To Illustrate, let's use a Simple Harmonic Oscillator
1
V ( x)  kx 2
2
k
p 2 m 2 x 2
H

m
2m
2

2

m 2 x 2
  
i
 

  
t t
2m  x 
2
2
m
2
ip 

†
define a 
x

a



m 

†
means hermitian conjugate

xx p
N  a† a
i x
a† n
n 1 a n
n 1
1
n is a state with E=En  (n  )ÿ 
2
m
2
ip 

x

m



|n> is state with E2
e.g. <x|2> is
n=2
Richard Seto
Now skip lots of steps-creation and annihilation operators
Let a † and a be the basic "things" and 0 be the vacuum
Then a † 0
1 i.e. a is an operator which creates 1 quanta
Later we will write a †   † which we call a field operator
An aside: Really the typical way we do this is as follows:
Start from E and M, with the classical radiation field
 Quantize  Field operators
E and M E    B 2  E 2 d 3 r
Remember??
Define P  B, E  and X( B, E )
1 A
really B  Ñ  A and E  
c t
A    ak ,   eikx  ak†,   e-ikx 
k

Pk ,   ic  ak ,  ak†,  Xk , 
1
c
a
k ,
 ak†, 
Richard Seto
Particles and the Vacuum
Now quantize again
 Xk , Pk   iÿ
 a† n
N  a† a N n  n n
n 1
an
n 1
number operator
1

H    N k ,  ÿ 
  ck
2

k  
1

Total _ Energy    number _ of _ photons  Energy _ of _ photon
2
k  
Sakurai: " The quantum mechanical excitation of the radiation field
can be regarded as a particle, the photon, with mass 0, and spin 1.
Field not excited  vacuum 
0
Wiggle (excite) the field  particle(s)  n
Richard Seto
Vacuum has energy???
1

H    N  ÿ 
2
k  
vacuum when N =0
Evacuum

1
   ÿ 
k   2
???
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!
Richard Seto
What tools to we have to build a theory?

To Build theories (an effective Lagrangian) we invoke
fundamental symmetries.

Why? 


Examples




because it works
and it seems right somehow??? “ H. Georgi”
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
d  dL  dL
0
 
dt  dq  dq

d  dL  dL
0
 
dt  dv  dx
e.g . Simple Harmonic Oscillator (A spring)
V=Pot. Energy =
1 2
kx
2
dL
 mv
dv
1 2 1 2
dL
L  mv  kx
 kx
2
2
dx
d
 mv   kx  0 ma  kx F  ma !!!
dt
Richard Seto
Symmetries - Example

L is independent of x (translational invariance)

I.e. physics doesn’t depend on position
1 2
If we choose L  mv  it is independent of position
2
d  dL  dL
d  dL  dL
then from Lagranges eqn.


0

0
 
 
dt  dq  dq
dt  dv  dx
dL
dL
d
dp
0
 mv
0
 mv   0 
dx
dv
dt
dt
i.e. p is constant !, momentum is conserved!
Symmetries  Conservation Laws
space invariance  conservation of p
time invariance  conservation of E
rotational invariance  conservation of angular momentum
Richard Seto
Aside – for electrons
Add relativity
E  p c m c
units c=1, ÿ =1
2
2 2
2 4
E 2  p 2  m2
guess mass like potential energy
guess H  Kinetic _ energy  mass _ term
L  Kinetic _ energy  mass _ term
Guess Lelectron  i  m
Probability = 
mass
Richard Seto
Require other symmetries
Local gauge invariance  Forces!
Lelectron  i  m
     
Probability = 
what about the phase of  ( x) ?? Ans: Its arbitary  '  ei , =constant
Lelectron  L'electron so Lelectron is invariant under (global) phase (guage) transfromation
What about if we made  dependent on the position ??? =(x)
 '  ei ( x ) local gauge transformation
check it
mass term: m  m ei ( x ) ei ( x )  m so mass term is ok
1st term: i  i ei ( x )   ei ( x ) ( x)    i e i ( x ) ei ( x )  ( x)   ei ( x )  
So we have a problem....
Richard Seto
Solution (a crazy one)
Add a new term to the Lagrangian
L New  i   i  A   m
e.g . B  Ñ  A
Local Gauge transformation :
  x   ei x   x 
A  A      x 
note: Ñ  A '  Ñ   A  Ñ   Ñ  A  B so its OK.
now check L New 1st term:
i   i  A   i e  i ( x )   i  A  i       x     ei ( x ) ( x )  
i e i ( x ) ei ( x )   ei ( x )    x    i  A ei ( x )  i    x   ei ( x ) ( x ) 
 i  i  A  i   i  A 
Richard Seto
So What is this thing A?
1 A
B    ,   c t
Ans: Electricity!
What happens if we give mass to A?
L  mass  m A A
Is this term Guage invariant? Try it!
remember A  A      x 
m A A  m  A    ( x)   A    ( x)   m A 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!
Richard Seto
T<TCurie


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 “picophysicist” 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?
A Magnetic Field
down and to the
right is
SACRED
Richard Seto
T>Tcurie

Heat him up to T>TC


restore the symmetry!
Now I understand!
The Vacuum has
broken symmetry!
His environment is
now magnetic-less
Richard Seto
T<Tcurie again
Richard Seto
A Toy Model




Start building a model of the string interactions with the kinetic
energy term of a pair of fermions
L  i
You could think of these as u or  p 
 d
n
 
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
i 
 e 
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
L  i R  R  i L  L

We can then freely mix up the R and L handed flavors independently
and the Lagrangian is still invariant -
 R, L  e
i 
 R, L
Richard Seto
u
 
d

Mass??
Can we add a mass term to this Lagrangian? OK lets try it!
L  i  m

Mass term
Terms break chiral symmetry!
Now what happens to chiral symmetry?
L  i R  R  i L  L  m( R  R   L L   R  L   L  R )


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
RH



LH!
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??
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?

KE of  = u
d
Interaction
of  with 
KE of 
Scalar field 
1
 2
m 
L  i     g            f (T )

2
4
 
2

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
V   , T  Tc
  L,R  e i    L,R 
Richard Seto
2
The Vacuum Gives us Mass?!

What happens as we lower T? [f(T)=-1]
m

Lowest energy state is at

Expand around equilibrium point
L     i    g 

gm


   

m

1
  2 2m 
            
 
2
4
 


V(),T~Tc
V(),T<Tc
2
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!
 
Richard Seto
A simulation of the vacuum
4 dimensional Action Density of the vacuum
Richard Seto
Living in the cold QCD vacuum
The vacuum –
perceived to
be empty by the
general
fish population

It is generally believed in the fish population that
there is an inherent resistance to motion and that
they swim in a “vacuum”.
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
Richard Seto