Transcript Document

Symmetries and Conservation Laws
1. Costituents of Matter
2. Fundamental Forces
3. Particle Detectors
4. Symmetries and Conservation Laws
5. Relativistic Kinematics
6. The Static Quark Model
7. The Weak Interaction
8. Introduction to the Standard Model
9. CP Violation in the Standard Model (N. Neri)
“Five Deities Mandala
Tibet, XVIIth Century
The word is used also to indicate a circular diagram,
basically made by the association of different geometric
figures (the most used being the dot, the triangle, the
circle and the square). The drawing has spiritual and
ritual meaning in both Buddhism and Hinduism.
“Mandala” from
it.wikipedia.org
1
A classification of
symmetries in particle
physics
Class
Invariance
Conserved quantity
Proper orthochronous
Lorentz symmetry
translation in time
(homogeneity)
energy
translation in space
(homogeneity)
linear momentum
rotation in space
(isotropy)
angular momentum
P, coordinate inversion
spatial parity
C, charge conjugation
charge parity
T, time reversal
CPT
time parity
product of parities
U(1) gauge transformation
electric charge
U(1) gauge transformation
lepton generation number
U(1) gauge transformation
hypercharge
U(1)Y gauge transformation
weak hypercharge
U(2) [U(1) × SU(2)]
electroweak force
Discrete symmetry
Wikipedia:
Internal symmetry
(independent of
spacetime coordinates)
SU(2) gauge transformation Isospin
SU(2)L gauge transformation weak isospin
P × SU(2)
G-parity
SU(3) "winding number"
baryon number
SU(3) gauge transformation quark color
SU(3) (approximate)
quark flavor
S(U(2) × U(3))
[ U(1) × SU(2) × SU(3)]
Standard Model
2
Symmetries of a physical system:
Classical
system
Lagrangian
Formalism
Hamiltonian
Formalism
Invariance of Equations of Motion
Quantum
system
Lagrangian
Formalism
Hamiltonian
Formalism
•Invariance of dynamical equations
•Invariance of commutation relations
(Invariance of probability)
E. Noether’s Theorem (valid for any lagrangian theory, classical or quantum)
relates symmetries to conserved quantities of a physical system
3
A “classical” example :
T 

1 2 1
m1r1  m2 r22
2
2
 
r1  r2
 
V  V (r1  r2 )

 


m1r1    V (r1  r2 )
r1

 


m2 r2    V (r1  r2 )
r2

'  
Let us do a translation :
ri  ri  ri  a
 
   
 
V (r1  r2 )  V (r1  a  r2  a) V (r1  r2 )
'
 


mi ri    ' V (r1  r2 )
ri
The equations of
motion are translation
Invariant !
4
If one calculates the forces acting on 1 and 2:

 
 
 
 
 




FTOT  F1  F2    V (r1  r2 )   V (r1  r2 )   V (r1  r2 )   V (r1  r2 )  0
r1
r2
r2
r2


dPTOT
 FTOT  0
dt
In the classical Lagrangian formalism :
L  L(qi , qi )
d L L

0
dt qi qi
L
pi 
qi
L invariant with respect to q
dpi
L

dt
qi
p conserved
5
In the Hamiltonian formalism
qi   qi , H 
p i   pi , H 
d (qi , pi )
  , H 
dt
Possible conservation of a
dinamical quantity
Commutation (Poisson
brackets) with the
Hamiltonian  conservation
of dynamical quantities
Possible symmetry
This formalism can easily be extended to Quantum Mechanics
In Quantum Mechanics, starting from the Schroedinger Equation :

i   s (t )  H  S (t )
t
 s (t )  exp i(t  t0 ) H /  S (t0 )
T (t , t0 ) Time evolution (unitary operator)
6
Schroedinger and Heisenberg Pictures:
Heisenberg
Schroedinger
Q   s (t0 )* Q(t ) S (t0 ) dV   S (t )* Q0  S (t ) dV
 S (t0 )* Q(t ) S (t0 )   S (t )* Q0  S (t )
 S (t0 )* Q(t ) S (t0 )   S (t0 )* T 1Q0 T S (t0 )
Q(t )  T 1Q0 T
for the operators in the Heisenberg picture
Taking the derivatives:
d
dT 1
dT
i Q(t )  i
Q0 T  iT 1Q0
  HT 1Q0T  T 1Q0TH
dt
dt
dt
i
d
Q(t )   HQ  QH  Q, H 
dt
d
Q
i Q(t )  i
 Q, H 
dt
t
Conserved quantities:
commute with H
In the case when there is an explicit time
dependence (non-isolated systems)
7
Translational invariance: a continuous spacetime symmetry
 
 
 (r  r )  (r )  r
 1  r
  D
r 
r 
 i

D ( r )  1  p  r 
 

The translation operator is naturally
associated to the linear momentum
For a finite translation :
 i

 i r 
i

D (r )  lim 1  p  r   lim1  p   exp p r 
n
 
 n  n 


n
unitary
n
( r  n  r )
Self-adjoint: the generator of space translations
If H does not depend on coordinates
D, H  0
 p, H  0
The momentum is conserved
8
Rotational invariance: a continuous spacetime symmetry

   i

R ( )  1      1  J z  
   


The rotation operator is naturally
associated with the angular momentum
 


J z   i   x  y    i
x 

 y
Angular momentum
(z-component)
operator (angle phi)
Self-adjoint: rotation generator
A finite rotation
unitary
n
 i

i

R ( )  lim 1  J z    exp J z  
n
 



  n
If H does not depend on the rotation angle φ around the z-axis
R, H  0
J z , H  0
The angular momentum is conserved
9
Time invariance (a continuous symmetry)
The generator of time translation is
actually the energy!
 s (t )  expi(t t0 ) H /  S (t0 )
Using the equation of motion of the operators :
i
d
Q
H (t )  i
 Q, H 
dt
t
i
d
H
H (t )  i
 H , H 
dt
t
If H does not dipend from t, the energy is conserved
The continous spacetime symmetries:
Space translation
Space rotation
Time translation
Linear momenum
Angular momentum
Energy
10
Continuous symmetries and groups: the case of SU(2)
Combination of two transformations: the result depends on the commutation
rules of the group generators. For instance, in the case of space translations :
 pi , p j   0
Commutative (Abelian) algebra of translations
Translation operator along x:
i

Dx ( )  exp p x  


i

i

i

i

Di ( ) D j (  )  exp pi   exp p j    exp p j   exp pi   D j (  ) Di ( )








(two translations commute). Moreover:
i

i

i

Dx ( ) Dx (  )  exp px   exp px    exp p j (   )   Dx (   )






and clearly :
i

Dx (0)  exp px 0   1


11
In the case of rotations :
Commutation rules for the generators:
i

Rz ( )  exp Lz  


L , L  i 
j
k
jkl
Ll
A non-commutative algebra
Rotations about different axes do not commute
This is also the case of SU(2), which we are now going to introduce
Why SU(2) ?
In the case of a two level quantum system, the relevant internal symmetries are described by
the SU(2) group, having algebraic structure similar to O(3).
SU(2) finds an application in the Electroweak Theory.
SU(3) can instead be applied to QCD, the candidate theory of Strong Interactions.
12
Isospin Symmetry, SU(2)
Let us consider a two-state quantum system (the original idea of this came from
the neutron and the proton, considered to be degenerate states of the nuclear
force). Since they were considered degenerate, they could be redefined :
H p  E p
 p   'p   p   n
H 'p  E 'p
H n  E n
 n  n'   p   n
H n'  E n'
Degeneration
Redefinition
Double degeneracy, similar to what happens in s=1/2 spin systems.
The degeneration can be removed by a magnetic field
One can introduce a two-components spinor :

(1 / 2 )
 p 
    p  1p/ 2   n  n1/ 2
 n 

1/ 2
p
1 
  
 0

1/ 2
n
 0
  
1 
13
 (1/ 2)   (1/ 2) '  U  (1/ 2)
The ridefinition now becomes :
A symmetry for the Strong Interactions (broken by electromagnetism)
U U  1
det U   1
U 1 i
SU(2) is a Lie group
Properties can be deduced from infinitesimal transformations
  
(1  i )(1  i )  1

Tr   0
det U   1

Which can be written in a general form :

 
2
Pauli matrices
0
1  
1
1
0
0  i 
2  

i
0


1 0
3  

0

1


 i  j 
k
 ,   i  ijk
2
2 2
14
1
S 2  (p1/ 2 )   2  (p1/ 2)  s ( s  1)  (p1/ 2)
4
1
1
S z  (p1/ 2)   3  (p1/ 2 )  s z  (p1/ 2 )   (p1/ 2 )
2
2
1
S 2  n(1/ 2 )   2  n(1/ 2)  s ( s  1)  n(1/ 2)
4
1
 1 (1/ 2 )
S z  n(1/ 2)   3  n(1/ 2)  s z  n(1/ 2) 
p
2
2
 
 'p  
 
   1  i    p 
 
 '  
2

 n
 n
Infinitesimal rotation of the p-n doublet :
A finite rotation in SU(2):
  n

 

U  lim 1  i
  expi / 2
n
n 2

• Generalization of a global
phase transformation
• Three phase angles
• Non-commuting operators
(Non-abelian phase invariance)
example

(1/ 2) '

 expi / 2 (1/ 2)
 
 'p  cos 2
 
 '   
 n
sin
 2

 sin  
2  p 
 
   n 
cos
2 
15
The two nucleon system

A one-nucleon state can be described
with the base of the nuclear spinors
1/ 2
p
1 
  
 0

1/ 2
n
 0
  
1 
A two nucleon state can be constructed by combining the Isospin states :
N ) I  1 / 2, I 3   1 / 2,  1 / 2
n
1 / 2 1 / 2 
I , I3 :
N ) I  1 / 2, I 3   1 / 2,  1 / 2
p
n
I 1
I 3  1
0 1
I 0
I3 
0
1,1 , 1,0 , 1,1
p
0,0
16
1,1
 pp
Isospin triplet
1
 pn  np
1,0 
2
1,1  nn

1
 pn  np
2

0,0

In an Isospin rotation:
These states transform into one
another in Isospin rotations, similar to
a 3-d vector for ordinary rotations
Isospin singlet (scalar)
0,0  0,0
Isospin Invariance means: there are two amplitudes, I=0 e I=1
I=1 states cannot be internally distinguished by the Strong Interactions
Strong Interactions conserves I and I3 (full I-spin invariance)
Electromagnetic interaction conserves I but not I3 (the charge is different in
different states of the same I-spin multiplet, e.g. the proton and neutron)
17
Isospin invariance at work: pion-nucleon reactions
 N  N
Let us consider the (strong) diffusion of pions and nucleons. The diffusion
amplitudes are the element of the S matrix:
 , m ' ; N , l ' S  , m; N , l
which can be written as combinations of the total isospin of the system :
 , N ; I ' , I 3' S  , N ; I , I 3   II  I I S ( I , I 3 )
'
'
3 3
taking into account the conservation of I and I3
Note, however, that S must commute also with I1 and I2 which implies (Schur’s
Lemma) that S is S=(I). Therefore, the amplitude can be expressed as a
function of the allowed total isospin amplitudes.
S ( I  1 / 2)
S ( I  3 / 2)
18
 ) I  1, I 3   1, 0,  1
N ) I  1 / 2, I 3   1 / 2,  1 / 2
 0 
n
I  1/ 2
1 1 / 2 
I  3/ 2
I3 
p
1/ 2  1/ 2
I 3   3 / 2 1/ 2  1/ 2  3 / 2
Using the two invariant amplitudes (1/2 and 3/2), one can write the amplitudes of
the following processes :
  p  3 / 2,3 / 2
p
0p 
2
3 / 2,1 / 2 
3
1
1 / 2,1 / 2
3
 p  p
 n 
1
3 / 2,1 / 2 
3
2
1 / 2,1 / 2
3
p 
1
3 / 2,1 / 2 
3
2
1 / 2,1 / 2
3
 0n 
2
3 / 2,1 / 2 
3
1
1 / 2,1 / 2
3


p 

 p  0 n
19
  p  3 / 2,3 / 2
0p 
2
3 / 2,1 / 2 
3
1
1 / 2,1 / 2
3
 n 
1
3 / 2,1 / 2 
3
2
1 / 2,1 / 2
3
p 
1
3 / 2,1 / 2 
3
2
1 / 2,1 / 2
3
2
3 / 2,1 / 2 
3
1
1 / 2,1 / 2
3
 0n 
Some Isospin amplitudes that one
can study by making Pion-Nucleon
diffusion experiments
π
π
N
N
A(  p    p)  3 / 2,3 / 2 S 3 / 2,3 / 2  S (3 / 2)
1
2
1
2
3 / 2,1 / 2 S 3 / 2,1 / 2  1 / 2,1 / 2 S 1 / 2,1 / 2  S (3 / 2)  S (1 / 2)
3
3
3
3
2
2
A(  p   0 n) 
S (3 / 2) 
S (1 / 2)
3
3
A(  p    p) 
Let us make the Pion-Nucleon experiment at an energy of the Pion of about 200
MeV. This generates a clear resonant peak in the π+p cross section, indicating the
presence of a dominant I=3/2 resonance (the Δ++, m = 1232 MeV)
20
At the energy when the cross section resonates with the Δ++ state, the 3/2
amplitude is dominant. Therefore S(1/2) can be neglected at that energy and we
have a prediction of relative cross sections of :
 (  p    p)
p π+ cross section dominates
at the energy of the Delta
resonance
√s GeV
A(  p    p )  S (3 / 2)
A 1
1
2
1
A(  p    p )  S (3 / 2)  S (1 / 2)  S (3 / 2)
3
3
3
2
2
2
A(  p   0 n) 
S (3 / 2) 
S (1 / 2) 
S (3 / 2)
3
3
3
A 
2
1
9
2
2
A 
9
2
In agreement with the experimental observation of
 (  p    p)  3    p    p    p   0n 
21
A little preview of a Lagrangian for a system of particles: Nucleon, Pions and some
Interaction (Yukawa style) between them.
This effective Lagrangian could describe reasonably well hadronic physics at
energies below ~2 GeV (assuming the proton and the pion to be elementary)




1
L  N i    M N     i    i   i2 mi2  g i N i i 5 N  i
2

Will produce the Dirac Equation
A free non interacting Nucleon
(Kinetic Energy and Mass)
Will produce the Klein-Gordon Equation. This term represents
free Pions (Kinetic Energy and Mass terms for each Pion)
A Yukawa interaction term (interaction term in the simplest form, taking into
account pion parity through the Gamma-5 matrix)
22
Nucleons and Quarks
I3
particle
antiparticle
particle
+1/2
p
u
-1/2
n
n
p
d
antiparticle
d
u
An Isospin triplet : the Pion
Wave Function
I
1
1
I3
Q/e
1
u d   
1
-1
u d  
-1
1
0
0
o
1
uu  dd    0
2
1
dd  uu   
2
o
0
23
Building up strongly interacting particles (hadrons) using Quarks as
building blocks
24
Will need to reconsider later on (Static Quark Model section)
25
 (1,1)  p (1) p ( 2)
More on the two nucleon state:
S
Isospin part
The total wave function :
A
 (tot)   (space)  (spin)  (isospin)
1
 p(1)n(2)  n(1) p(2)
2
 (1,1)  n(1) n( 2)
1
 p(1)n(2)  n(1) p(2)
 (0,0) 
2
 (1,0) 
(non relativistic decomposition)
The Deuton case:
 As it is known the deuton spin = 1  α symmetric
 φ has (-)l symm. It is known that the two nucleons are l=0 or l=2, φ is symm.
 Then  must be antisymmetric.
 This is because ψ tot must be antisymmetric in nucleon exchange.
I 0
The Deuton is an Isospin singlet
Now, since the Deuton has I=0 and the Pion has I=1, considering the reactions :
(i) p  p  d   
Isospin I
1
1
The reaction can proceed only via I=1
(ii) p  n  d   0
0,1
1
 ( pn  d 0 ) /  ( pp  d  )  1/ 2
26
Gauge symmetries (global and local)
Gauge symmetries are continuous symmetries (a continuous symmetry
group). They can be global or local.
Global symmetries: conserved quantities (electric charge)
Local symmetries: new fields and their transformation laws (Gauge theories)
Let us consider the Schoedinger
equation
 2  2
  

 
  V (r )  (r )  E (r )
 2m

Let us consider a global phase transformation: the
change in phase is the same everywhere


i
 (r )  e  (r )
The Schroedinger equation is invariant for this transformation. This invariance ia
associated (E. Noether’s Theorem) to electric charge conservation.
But then what happens if we consider a
local gauge tranformation ?

  k     (r )
27


i
 (r )  e  (r )



i ( r )
 (r )  e  (r )
How does one realize a local gauge invariance ?



i ( r )
 (r )  e  (r )

 2  2
  i ( r ) 

i ( r )
 
  V (r )  e
 (r )  E e  (r )
 2m

Non invariant! And this is because :


 

 i ( r ) 


 

i
i ( r )
 e  (r )  e i      e
 (r )
extra term !
28
To solve the problem (and stick to the invariance) we can introduce a new
field. The field would compensate for the extra term.
The new field would need to have an appropriate transformation law.
Since the free Schroedinger Equation is not invariant under:
 
 2

 i
 
 ( x , t )  i   ( x , t )
2m
 t 
Let us modify the Equation :

Compensating fields

Transformation laws:

i q  ( x ,t )
   e
'

  2

 i  G

 
 ( x , t )  i   R  ( x , t )
2m
 t

 ' 

G G  G  q
R  R '  R  iq


t
29
This allows us to restore the invariance


Gq A
To give the fields physical meaning :
R  iqV

It is invariant, indeed
iq
   e 
'

 2


 i  qA

 
 ( x , t )  i   iqV  ( x , t )
2m
 t


'  
A  A  A  
V  V ' V   t 

'

 i  qA
2m
  ( x, t )  i    iqV  ( x, t )
2
'
'
 t
'

The local gauge U(1) invariance of the free Schroedinger field
• Requires the presence of the Electromagnetic Field
• Dictates the field transformation law
30
This gauge symmetry is the U(1) gauge symmetry related to phase
invariance:

   '  ei q  ( x ,t ) 
There are, of course, other possibilities.
In physics, a gauge principle specifies a procedure for obtaining an
interaction term from a free Lagrangian which is symmetric with respect to a
continuous symmetry -- the results of localizing (or gauging) the global
symmetry group must be accompanied by the inclusion of additional fields
(such as the electromagnetic field), with appropriate kinetic and interaction
terms in the action, in such a way that the extended Lagrangian is covariant
with respect to a new extended group of local transformations.
FREE
INTERACTING
 
 2

 i
 
 ( x, t )  i   ( x, t )
2m
 t 
 2


 i  qA

 
 ( x , t )  i   iqV  ( x , t )
2m
 t



31
The U(1) Gauge Invariance and the Dirac field
Dirac Equation (1928): a quantum-mechanical description of spin ½ elementary
particles, compatible with Special Relativity


i    i    m 
t
 
0  
i    i 
 i 0 
i 
 




  m   0
4 component spinor
 ( x)   ( x)  0
Conjugate spinor




1
0
  

0  1 
 0 
Matrices 4x4 (gamma)
Dirac probability current
j     
 j  0
32
L  i c     mc  

The Dirac Lagrangian:
2
Features a global gauge invariance:
  ei 
(phase transformations)
Let us require this global invariance to hold locally.
The invariance is now a dynamical principle
i ( x )
 e
 e
 iq ( x )
c

Now the gauge transformation depends on the
spacetime points
Let us see how L behaves :
Since
 iq
 iq
 iqc 
iq c
c
   e    e    e     
c


33
L  L'  i c e
 iq
c
 iq



c
     e    mc 2  L  q     




This lagrangian is NOT gauge-invariant.
If we want a gauge-invariant L, one has to introduce a compensating field with a
suitable transformation law :
L  i c     mc2  (q  ) A
A  A   
This new lagrangian is locally gauge-invariant.
This was made possible by the introduction of a new field (the E.M. field).
34
L  i c     mc2  (q  ) A
  ei ( x )  e
 iq ( x )
c

A  A   
iq
c
 iqc 
L  i c e     e    mc 2   (q  ) ( A     ) 


 i ce
iq
c


 e
 iq
c




  q       mc 2   (q  ) A  q       L
The gauge field A must however include a (gauge-invariant) free-field term.
This will be the E.M. free field term (more on this in the Standard Model lecture)
L  ic (     mc 2 )  (q   ) A 
1
F  F
16
35
Discrete symmetries: P,C,T
Discrete symmetries describe non-continuous changes in a system. They
cannot be obtained by integrating infinitesimal transformations.
These transformations are associated to discrete symmetry groups
Parity P
Inversion of all space coordinates :
 x   x 
  

P:  y     y 
 z   z 
  



P (r )  (r )
The determinant of this transform is -1. In the case of rotations, that would be +1



PP (r )  P (r )  (r )
PP 1 P P
PP 1
A unitary operator.
Eigenvalues: +1, -1 (if definite-parity states)
Eigenstates: definite parity states
36
H , P 0
Parity is conserved in a system when
The case of the central potential:



PH (r )  H (r )  H (r )  H (r )
Bound states of a system with radial symmetry have definite parity
Example: the hydrogen atom
Hydrogen atom: wavefunction (no spin)
 (r, ,)   (r) Yl m ( ,)
Radial part
The effect of parity on the state is :
      

P :    
      
Angular part
P: Yl m ( , ) (1)l Yl m ( ,)
Electric Dipole Transition ∆l = ± 1. This implies a
change of parity. Since, however, e.m. interactions
conserve parity, this indicates the need to attribute to
the photon an intrinsic (negative) parity.
P ( )  1
37
The general parity of a quantum state


P ( x,t ) Pa ( x,t )
Let us consider a single particle a.
The intrinsic parity can be represented by a phase


P ( x,t )  expi / 2 ( x,t )
Intrinsic
Spatial


P P ( x,t ) ( x,t )
Which is the physical meaning of the intrinsic parity ?
For instance, in a plane wave (momentum eigenstates) representation




P p ( x,t )Pexpi( px  Et)Pa p (x,t )Pa  p ( x,t )
If one then lets go the momentum to zero, one can see that the intrinsic parity
has the meaning of a parity in the p=0 system
38
The parity of the photon from a classical analogy
A classical E field obeys :
Let us take the P:
To keep the Poisson Equation
invariant, we need to have the
following law for E :
 

E( x,t )  ( x,t )


 

() PE( x,t )  (x,t )


And the parity operation would give :
 
 
PE( x,t )   E( x,t )


 
A
A
E ( x ,t )   

t
t

 
 
A( x,t )  N  (k )exp i(kx  Et)
 
 
A( x,t )  P A( x,t )
In order to make it consistent with the
electric field transformation :
P  1
On the other hand, in vacuum
(no charges):


39
The action of parity on relevant physical quantities
Position:


P: r  r
Momentum:
Angular Momentum:
P: t  t




dr
dr
P: pm
   p
dt
dt
  

 
P : L  r  p  (r )  ( p)  L
Time:
Charge:
E field:
P:

P: E
B field:

P: B  k
Current:
Spin:
P: q  q




J  v  v  J



r
r
 kq 3  kq 3  E
r
r
 



s r
( s )  (r )
I  k
I B
2
2
r
r
P:   
40
Symmetry at work: the parity of the Pion
Parity of a complex system: overall parity times the product of the intrinsic
parities of the parts of the system. Let us illustrate this for the Pion, using the

reaction:
  d n  n
This is a Strong Interaction process  Parity is conserved
L( d )  0 (known) P(d )  1 (known)
P(  d )  (1) L P( ) P(d )  (1)0 P( )1 P( ) P(nn)  (1) L P(n)P(n)  (1) L
P( )  (1) L ( L of the final state)
P conservation
s  0, sd  1  s1  1/ 2, s2  1/ 2 (already known)
J  L  S 1
initial state
  d n  n

P( )  (1)1   1
( L of the final state )
L  S 1 final state J conservation
(1) LS 1   1
L  1, S  1
Pauli symmetry
between neutrons
L  S even with J  1
41
Final considerations on parity :
Some intrinsic parities cannot be observed. For instance (p, n). They are
conventionally chosen to be +1. Because of the Baryon Number conservation the
actual P value is not important as it cancels out in any reaction.
Neutral Pion Parity.
It is deduced from the photon pair distribution in :
P( 0 )  P(space) (1) (1)  P (space)  1
 0  
(study of the relative polarization of the two photons)
Particles can be (and are) classified
using J and P (and C)
Transformation properties for rotations
and space reflections. Spin-parity
J P  0  : pseudoscalar ( )
J P  0  : scalar
J P 1 : vector
J P 1 : axial vector
The intrinsic parities of Particles and Antiparticles
P(particle) = - P (antiparticle)
FERMIONS
P(particle) = P (antiparticle)
BOSONS
42
Time Inversion T : preliminary considerations
Time is a parameter characterizing the evolution of a system
Classical Physics
An absolute Cosmic Flow : just one
Clock for the whole Universe
Special and General Relativity
The time flow depends on :
1. The motional status of the clock
2. The spacetime structure
Is there a Time symmetry? Is there a possible Arrow of Time based on asymmetry
between past and future? (A. Eddington , 1928)
43
How can we define a time symmetry ? What do we really need to run a movie
backward? In Classical Physics :
State A
Dynamical Law
State B
State - A
Dynamical Law
State - B
1. The Initial and Final Conditions can be inverted (State  - State)
2. The Dynamical Law is time-neutral : t -t leaves the law unchanged
Here is a time-neutral dynamical law:
d2
V ( x (t ))
m 2 x(t )  
dt
x
t  t
d2
V ( x (t ))
m 2 x (t )  
dt
x
d2
V Tx (t )
m 2 Tx(t )  
dt
x
44
Time Arrow in the MACROSCOPIC world
Conventional wisdom says that physical processes at the MICROSCOPIC level are timesymmetric  if the time arrow is changed in sign, the theoretical description would not change.
On the other hand, in the MACROSCOPIC world, one can appreciate the existence of a
preferential direction of time : from the past to the future.
Building up the Arrow of Time using the Second Law of Thermodynamic. Entropy can
only increase.
Ink dissolved in water in the left hand side of
the vessel. One has a «trivial» time sequence
corresponding to an increase of Entropy.
A
B
C
D
The only possible sequence is ABCD
,even if – from the MICROSCOPIC viewpoint
– the opposite sequence is possible as well
(and does not violate any other physical law).
There is no conflict between the MICRO
(T-reversible) laws and the MACRO (Tirreversibile) behaviour.
The key point is that the macrostate D
corresponds to a much bigger number of
microstates with respect to A
45
What is the origin of the MACROSCOPIC Time Arrow?
Boltzmann’s suggestion: thermodynamic time can arise from cosmic time.
If the Universe had begun from an equilibrium condition, then (possibly) we would
not have an Entropy concept as we know it.
A
B
Analogy
C
D
Since all started from a relatively ordered state, a very peculiar state indeed (the
first primi istanti del Big Bang), it is then natural (emergent) a tendency to
maximum Entropy  it is naturally emergent a concept of macroscopic cosmic
time
46
Let us suppose a different
Demiurgical Creation
If the Universe had begun from an
equilibrium condition (see below)
then (likely) we would not have a
concept of Entropy like this. Likely,
this means no macroscopic time.
Creazione di Adamo – Michelangelo Buonarroti
(Musei Vaticani - La Cappella Sistina)
Two species of perfect gases (the blue
and the red one). They do not interact
except for perfectly elastic scatterings.
Time arrow undefined
47
Time Inversion T in Particle Physics
It changes the time arrow
T
Classical dynamical
equations are invariant
because of second
order in time
t t
 
r r




dr
dr
pm  m  p
dt
dt
   


L  r  p  r  ( p)   L
Classical microscopic systems : T invariance is fully respected.
Classical macroscopic systems: time arrow selected statistically (defined as
the direction of non decrease of entropy)
In the quantum case, the Schroedinger equation :
Is not invariant for

i   (t )  H  (t )
t
 T? 
 (r , t )  (r ,t )
T

i    H
t

i
 (t )  H  (t )
t
48
Let us now start from the Conjugate Schroedinger Equation :
i



 (r , t )  H  (r , t )
t
i

 * 
 (r , t )  H  * (r , t )
t
 T * 
 (r , t )  (r ,t )
And define a T-inversion operator :



 i   * (r , t )  H  * (r , t )
t
T
So, with this definition of T operator, we have:
 * 
* 
i   (r ,t )  H  (r ,t )
t



i  T (r ,t ) H T (r ,t )
t
The operator representing T is an antilinear operator.
The square modulus of transition amplitudes is conserved
49

i   (t )  H  (t )
t
 *
i
 (t )  H  * (t )
t
T
i
Let us take the complex conjugate
And we have the required result in
terms of the conjugate wave function :
i

 (t )  H  (t )
t




T * (t )  H T * (t )
t

Wigner Theorem on Quantum Systems
Any symmetry of a quantum
system is given by:
•
 '  U
U U  1
U a   b 
  aU 
 bU 
either a unitary
• or an antiunitary operator
 '  W
W W  1
W a   b 
 aW
*
 b*W 
50
x , p  i 
The T operator in Quantum Mechanics
Action on
position and
momentum
xi T  T xi
pi T   T pi
i
T  xi T  xi
T  pi T   pi
j
ij
T  xi  xi T 
T  pi   pi T 
Then considering:
xi , pi  i
and calculating
T  xi , pi T  T  i T
T  xi piT  T  pi xi T  T i T
xi T  piT  pi T  xi T  T i T
 xi pi  pi xi  T  i T
The operator has to be antilinear
 i T  i T
51
Non-interacting particles
in the initial state
Particle Physics : what do we really measure ?
p1
Quantum interference.
Unitary evolution.
Non-interacting particles
in the final state
p3
p2
S
p4
Performing a momentum measurement of a final states involves (in general) a selection process
that traces away the other quantum mechanical degrees of freedom (exception: Dalitz Plots).
 p x  
 p vt  
p

c
200 MeV fm 200 MeV fm
200 MeV fm





p
pvt pvct
E vt
E c t
E  3 1023 ( fm / s)  t
A measurement on a particle can
be done during a relatively long
time  a momentum eigenstate
can be built with (almost)
arbitrary accuracy :
p
s
 21 MeV
12 MeV ns
 10
10
p
E t
E t
52
We see the consequence of T-invariance at the microscopic level by analizing the
the transition amplitudes when no interference is present between i and f. One
single transition amplitude has to be involved. Otherwise the measurement
process would generate phase cancellation and irreversibility !
The selection is made on momentum eigenstate of the final state.
If these conditions are met, one can use the detailed balance principle on the initial
and final momentum eigenstates :
M i  f  M f i
Note: the detailed balance DOES NOT imply the equality of the reaction rates:
M i  f  M f i
Wi  f
2
2
2
2

M i f  f 
M f i  i  W f i


A “classical” test, the study of the (Strong) reaction
p  27Al   24Mg
T is violated at the microscopic level il the Weak Nuclear Interactions
Physical Review Letters 109 (2012) 211801. BaBar experiment at SLAC
0
Comparing the reactions:

B B
53
Charge Conjugation C
C
An internal discrete symmetry
q q
It changes the sign of the charges (and magnetic moments)




r
r
C : E  k q 3  k (q) 3   E
r
r
 
 


s r
s r
C : B  k 2 I  k 2 ( I )   B
r
r
In the case of a quantum state
C


 (q, r , t )   (q, r , t )
C  C 
The C eigenstates are the neutral states
C ( )   1
For the photon case
54
The charge conjugation of the photon from a classical analogy
A classical E field obeys :
Let us take the C:
 

E( x,t )  ( x,t )


 


 CE( x,t ) C ( x,t )    ( x,t )
To keep the Poisson Equation
invariant, we need to have the
following law for E :

On the other hand, for a system of
charges :

 
A
E ( x ,t )   
 
t

 
 
CE( x,t )   E( x,t )
And the charge conjugation operation
would give :


 ( x,t ) C  ( x,t )
In order to make it consistent with the
electric field transformation :
C  1
55
The C-parity reverses the signs of charges and magnetic moments.
The electromagnetic interaction is not affected.
For interactions that are C-invariant :
C, H  0
a a
Let us distinguish between particles that have antiparticles :
And particles that don’t :
a    , K  ,e
 
   , 0 ,...
The action of the C operator :
Spin, momentum
Mass, type
C a,  a ,
C  ,  C  ,
C phase
C  C 1  CC 1  C  1
56
The C-parity of a state can be calculated for a neutral state if we know the
wave function of the state. It is the product between the parities of the total
wavefunction and the parities of the constituents.
Since charge conjugation of two particles of opposite charge, swaps the
identify of the particles, one has to account for the proper quantum statistics
C    , L  (1) L   
True also in general for spin zero particles
For a couple of femions, instead :
In the pi-zero decay
This decay in 3 photons
 0 
C f f , L, S  (1) L S f f
C ( 0 )  C( )C( )  (1) (1) 1
 0 
Is forbidden if C is conserved in
electromagnetic interactions. In fact :
 0 
7

10
 0 
57
Action of C,P,T

r
t

 md r
p
dt
  
L r  p
q
C

r
t

p

L
q



J   v   v   J

qr (  q ) r   E
E 3
r3
r
 
 s  r
s r

B 2 I
(I )  B
2
r
r


P
T

r
t

r
t

md (  r )

p
dt

md r

p
d ( t )



( r )  (  p)  L



r  ( p)   L
q
r

2

I  B

L

J

E


 ( v )   J


q(r )


E
r3

r
t

p
q
q


 ( v )   J


(s )  (r )
CPT


qr

E
r3


(s )  r
r

2

I  B

B

58
Positronium
Similar to the H atom. Actually, the «true» atom.
We require the total wavefunction to be
antysimmetric, considering the electron and the
positron as different C-states of the same particle
   (space)  (spin)  (C )
The space part
 Yml ( ,  )
The spin part
C  (1)
S 1
C  (1)l
 (1,1)  1 (1 / 2) 2 (1 / 2)
Triplet
Singlet
The C conjugation for the Ps state :
1
1 (1 / 2) 2 (1 / 2)  1 (1 / 2) 2 (1 / 2)
2
 (1,1)  1 (1 / 2) 2 (1 / 2)
 (1,0) 
 (0,0) 
1
1 (1 / 2) 2 (1 / 2)  1 (1 / 2) 2 (1 / 2)
2
K  (1)l (1) S 1 C
59
K  (1)l (1) S 1 C
Positronium in the l=0 (fundamental) state
Let us now calculate C
e e  2 
J  0, l  0, S  0
C  1
This state has C=+1 since in decays in 2 γ’s
e  e   3
J  1, l  0, S  1
C  1
This state has C= - 1 since in decays in 3 γ’s
Singlet:
K  (1)l (1) S 1 C  (1)0 (1)1 (1)  1
Triplet: K  (1) (1)
l
S 1
C  (1)0 (1)2 (1)  1
Antisymmetry by
electron/positron
exchange
C-parity conservation determines the Ps decay modes :
Singlet:
 (2 )  1.2521010 s
Triplet:
 (3 )  1.374107 s
See the vertices in the relevant
Feynman diagrams
60
Photons, Spin, Helicity
  
B  A



1 A
E    
c t
Gauge - invariant
  
A  A  
1 
  
c t
Coulomb Gauge

A  0
Free propagation:

2 1  A
 A 2 2 0
c t

 
A  e A0 expi(kr   t )
2


A  0  e k  0
For instance:
ex2  ey2 1
Plane wave solution
Transversality condition in the Coulomb Gauge

 0
k  (0,0, k )
Ax  ex A0 expi(kz   t   )
Plane polarization
Ay  e y A0 expi (kz   t )
Circular polarization
   / 2 ex  e y
61
Circular polarization
  /2
1
eR 
(ex  ie y )
2
1
eL 
(ex  ie y )
2
ex  ey
Which can be expressed by using the rotating vectors:
The polarization vectors can be associated to the photon spin states :
If the wave propagates along z:
Lz  xpy  ypx  0
Let us make a rotation around the z axis :
ex'  ex cos  e y sin 
e 'y  ex sin   e y cos
ez'  ez
1 '
(ex  ie'y ) 
2
1 '
eL' 
(ex  ie'y ) 
2
ez'
eR' 
Transversality: the Jz=0 state does not exist:
Photons with Jz =0 are virtual (longitudinal
photons). They have: m≠0
Jz only due to spin
R  exp(i J z )
1
(ex  ie y ) exp(i )
2
1
(ex  iey ) exp(i )
2
 1 Eigenstates
of Jz with
eigenvalues
1
+1,-1,0
0
e
e
k
J z  1 R
k
J z  1 L
62
Helicity
Projection of the spin in the momentum direction
right handed H  1
left handed H   1
scalar
H0

p

 E
An approximate quantum number for massive particles
So much better inasmuch the particle is relativistic
Exact for photons
The advantages of a description


p,  over a description p, m :
• The Helicity is unchanged by rotation
•
 
• Since  p  J p , the helicity can be defined in a relativistic context
Invariance laws in action:
An example: E.M. interactions conserve Parity
One can then build a Parity-violating quantity, like:

 

P :  p   ( p)    p
Then, in E.M. interactions this quantity must be zero. And one can test this!
In E.M. Interactions right-handed and left-handed photons appear with equal
amplitudes. In this way they compensate to the result and conserve Parity.
63
The Neutrino
C, P are violated in Weak Nuclear Interactions
Neutrinos takes part only in Weak Nuclear Interactions
J z   1/ 2, 1 / 2
In the massless neutrino approximation:
Experimental evidence indicates that in Weak Interactions :
Neutrinos are always left-handed. Antineutrinos are always right-handed !

p
P
L
R
C

p
L

p
CP

R
p
To a very good approximation, Weak interactions conserve CP (not C, not P)
64
The CPT Theorem
In a local, Lorentz-invariant quantum field theory, the interaction (Hamiltonian)
is invariant with respect to the combined action of C,P,T
(Pauli, Luders, Villars, 1957)
A few consequences :
1) Mass of the particle = Mass of the antiparticle
2) (Magnetic moment of the particle) = -- (Magnetic moment of antiparticle)
3) Lifetime of particle = Lifetime of antiparticle
Proton
Protons,
electrons
Antiproton
Electron
Positron
Q
+e
-e
-e
+e
B o L(e)
+1
-1
+1
-1
μ
 2.79(e / 2Mc)
 2.79(e / 2Mc)
 e / 2mc
 e / 2mc
σ
/2
/2
/2
/2
65
CPT Theorem (wikipedia)
In quantum field theory the CPT theorem states that any canonical (that is, local
and Lorentz-covariant) quantum field theory is invariant under the CPT operation,
which is a combination of three discrete transformations: charge conjugation C,
parity transformation P, and time reversal T. It was first proved by G.Lüders,
W.Pauli and J.Bell in the framework of Lagrangian field theory.
At present, CPT is the sole combination of C, P, T observed as an exact symmetry of
nature at the fundamental level.
CPT and the Fundamental Interactions
Concern systems for which a quantum field theory has been developed (i.e.
systems subject to Strong, Electromagnetic and Weak Interactions). The relative
Lagrangian is CPT-invariant.
Note
CPT is a flat-spacetime theorem (it does not concern Gravity)
66
Invariance of Physical Laws and Invariance of Physical Systems
Let us consider the quantity

E

L
L
It is an Electric Dipole Moment (EDM)
How this quantity P,T transforms ?

 

T :  E   E   E

 

P : E   ( E )   E

Now let us consider a system bound by a P conserving interaction, like the
Electromagnetic Interaction. Since P is conserved, one can expect that this
quantity should be zero.
H
N
We will consider the Ammonia Molecule :
d  51030 C m
H
H
67
H
N
This system of minimum energy does not display the
full symmetry of the Electromagnetic Interaction
H
H
There are in fact two degenerate states :
The full ammonia wavefunction has the
form :
  1  2
1
2
Which displays the symmetry of the
underlying interaction.
This is an example of a Broken Symmetry
The minimum energy configuration does
not display the symmetries of the theory
Other examples :
• A ferromagnet (when T is below the Curie temperature)
• The Higgs potential
68
A recipe to search for violation of fundamental laws
Consider a system S that obeys some interaction I.
Build up a quantity Q that is violated by said interaction. Q is violated by I.
Find a (non-degenerate) s state of the system S.
Check that Q is zero on the state s.
Example : the neutron Electric Dipole Moment (EDM)
The magnetic moment (like a spin) changes sign
with the T inversion
The EDM changes sign with the P inversion
Current best limit on neutron EDM:
|dn| < 10–32e·cm
"The Neutron EDM in the SM : A
Review". arXiv:hep-ph/0008248
69
Particle Numbers: baryonic, flavor, and leptonic
Flavor :
The flavor is the quark content of a hadron
Massa
(MeV)
Quark
U
D
S
C
B
p
938
uud
+2
+1
0
0
0
n
940
udd
+1
+2
0
0
0
Λ
1116
uds
+1
+1
-1
0
0
Λc
2285
udc
+1
+1
0
+1
0
π+
140
u-dbar
+1
-1
0
0
0
K-
494
s-ubar
-1
0
-1
0
0
D-
1869
d-cbar
0
+1
0
-1
0
Ds+
1970
c-sbar
0
0
+1
+1
0
B-
5279
b-ubar
-1
0
0
0
-1
Υ
9460
b-bbar
0
0
0
0
0
70
Favor quantum numbers refer to quark content of hadrons
They are conserved in Strong and Electromagnetic Interactions
They are violated in Weak Interaction


U  N (u)  N (u ) D  N (d )  N (d )
S  N (s)  N (s )
Strangeness

C  N (c)  N (c ) B   N (b)  N (b )
Charm

T  N (t )  N (t )
Beauty
In a Stong Nuclear (or E.M.) process,
all flavors are conserved:
In Weak Interactions instead :
Top
p 
p

n 
p  
(uud)  (uud)  (udd)  (uud)  (ud )
n

p  e   e
(udd)  (uud)
Baryon Number:
1
B  U  D  S  C  B T 
3
71
Baryon Number
1
B  U  D  S  C  B  T 
3
The Baryon Number is equivalent to :
1
B  (nQ  nQ )
3
Baryons have B=1 while Antibaryons have B = -1
Mesons have B = 0
This law follows from the conservation of the Quark Number.
Quarks transform into each other. They disappear (or appear) in pairs.
Flavor quantum numbers refer to the
identity of quarks :
Violated in Weak Interactions
(Isospin: +1/2 o -1/2 in doublets)
Strangeness: -1 for the s quark
Charm: +1 for the c quark
Bottom: -1 for the b quark
Top: +1 for the t quark
72
The Leptonic Numbers:
Ne  N (e )  N (e )  N ( e )  N ( e )
Electronic Lepton Number
N  N (  )  N (  )  N (  )  N (  )
Muonic Lepton Number
N  N (  )  N (  )  N ( )  N ( )
Numero leptonico tauonico
The Leptonic Numbers are conserved in any known interaction WITH THE
EXCEPTION OF Neutrino Oscillations. In Neutrino Oscillations, they are violated.
This process does not involve neutrino oscillations :
 e
So, it does not take place .
However, we can define a total lepton number :
Nl  Ne  N  N
To the best of our knowledge the Total Lepton Number (sum of the three
leptonic numbers) is conserved in every interaction.
73
74