Transcript Slide 1

 1 E 4 
 B 
 J

  E  4
xE x   yE y  zE z 
4
c
c t
c
c
 y B z   z B y  0E x 
4
c
(also xyzyzxzxy)
both can be re-written with
 x F x 0   y F y 0   z F z 0   0 F 00 
xF
xx
  yF
yx
 zF
zx
 0F
0x

4
c
J0
4
c
Jx
(with the same for xyz)
All 4 statements can be summarized in
  F   (
4
)J
c
  0, x, y, z
Jx

 B  0
 1 B
 E 
0
c t
The remaining 2 Maxwell Equations:
are summarized by
i F jk   j Fki  k Fij  0
ijk = xyz, xz0, z0x, 0xy
Where here I have used the “covariant form”
0
F = g g
F =
 Ex
 Ey
 Ez
Ex
Ey
Ez
0
 Bz
Bz
0
 Bx
Bx
0
 By
By
To include the energy of em-fields
(carried by the virtual photons)
in our Lagrangian we write:
L=[iħcg

 ] F F(qg )A
mc2
1
2
But need to check: is this still invariant under the SU(1) transformation?
  (A+) (A +)
= A+   A  
= 
“The Fundamental Particles and Their Interactions”, Rolnick (Addison-Wesley, 1994)
L 

]
11 

 ic g    mc    F F  (q g  ) A
22
2

Heaviside   E  

-Lorentz
 E 
 B 
J
units
t
“Introduction to Elementary Particles”, Griffiths (John Wiley & Sons, 1987)

L

]
1
1
 ic g    mc   
F  F  (q g  ) A
16
16
2
Gaussian
cgs units

  E  4

 1 E 4 
 B 

J
c t
c

L

L
L


L




The
 
 
 
 0 and  
0
prescriptions  (   )  
 (  A )  A
give two independent equations OR
summing over ALL variables (fields) gives the full equation WITH interactions
Starting from
L
2

 1 
mc

 ic g   
    F F  (q g  ) A


 2
(and summing over ,  )
(
)(
1 
1  
 
with  F F    A   A   A   A
2
2
Let’s look at
the new term:
L

 FF

(  A ) (  A )
)
(
summing over , 
)]
)(

1


 A    A  A    A
(  A )
16 (  A )
survive when =, = and when =, = 
FF
(
)(
)(
(
)(
)]
)(
)
)]
)(
1


  A   A   A   A   A    A  A    A
16 (  A )
,  now fixed, not summed
[-( A-A)][-( A-A)]

1

2   A   A   A   A
16 (  A )
(
1
 
(  A )
 1  ( A )
 
 

  A   A   A   A
8 (  A )
(  A )

 FF
 1  

g g
 (  A ) 8
 1  

g g
4

(
(
(
A = ggA
= ggA
sum over
) g ( A   A )]
where  since this
A   A ) tensor is anti-symmetric!
A   A  g
 
 
1  
 
 A  A
4
 
)
 
 


(but non-zero only
when =, = )
So with
L
(

 FF
1  
 


 A  A
(  A ) (  A )
4
and next
 
)
 
FF
1  ( A   A )(  A    A )

A
16
A
we get
L
=0
L
 
 
 
 
0
 (  A )  A
)
(
1
 
 

  A     A  0  0
4


  A 
Note:
 


  A    A
0 in the
2
A0
Lorentz Gauge
U(1) :
 / = e+iq/ħc
+iq
q = -e for electrons
q = +e for positrons
q = +2e/3 for up quarks
The ± sign is just a convention, as in rotations:
SO(3) :
 / = e+iJ · /ħ
We can generalize our procedures into a PRESCRIPTION to be followed,
noting the difference between LOCAL and GLOBAL transformations
are due to derivatives:
  = 
/
[e+iq/ħc]
for U(1) this is a
1×1 unitary matrix
(just a number)

+iq

/ħc

=e
(
)
q


   i c    
the extra term
that gets introduced
If we replace every derivative  in the original free particle Lagrangian
with the “co-variant derivative”
g
= + i ħc A
D
then the gauge transformation of A will
cancel the term that appears through 
i.e.
(D )/ = e-iq/ħcD restores the invariance of L
The free particle Dirac Lagrangian
can be made U(1) invariant only by
•introducing a charged current
•introducing a VECTOR FIELD particle
•which couples to that charge
The conserved quantity “discovered”
was ELECTRIC CHARGE.
The particle coupling to CHARGE
was interpreted as the PHOTON.
CHARGE is the source of the PHOTON FIELDS through
which Dirac particles interact. This is believed to be the
underlying principal of the fundamental electromagnetic
force: VECTOR PARTICLES mediate interaction forces.
Are there HIGHER symmetries?
SU(2) spin-multiplets
just one of many ANGULAR
MOMENTUM representations
Dirac matrices and Dirac spinors
already keep this space separate.
SU(2) Iso-spin multiplets
already expanded into SU(3) and higher as
we generalized isospin to include concepts
like hypercharge, strangeness, and charm.
Are these all some kind of charge?
Is UP or DOWN some kind of CHARGE
that generates fields?
that couples to a force carrying vector particle?
Is there some kind of fundamental ISOSPIN FORCE?
The theorists Yang & Mills extended the U(1) formalism
that explained e&m forces in an effort to explore ISOSPIN.
Imagine 2 possible states: the flip sides of some spin-½ (2-component) property
Spin, ISO-spin, or even something NEW
L
 ic 1g    1  m1c 2 1 1  ic 2g    2  m2 c 2 2 2
Sum of 2 Dirac lagrangians
Applying the Euler-Lagrange equations results in 2 independent Dirac equations
1 satisfying one, 2 satisfying the other.
 1 
Written more compactly as spinors   
  and its adjoint   (1  2 )
 2
Note: 1 and 2 each already 4-component Dirac spinors
L

 ic g    Mc  
2
where
 m1 0 
M  

 0 m2 
“mass
matrix”
If m1=m2 then M=mI and
L
 ic g     mc 2 
which “looks” just like the
1-particle Dirac Lagrangian
But NOW  represents a 2-element column vector and we can explore
an additional invariance under    U
The most general SU(2) matrix is of course U

= ei( /2)·

where   Pauli matrices
Following the success of U(1) Yang-Mills promoted the obvious
global phase transformation to a LOCAL invariance, writing

g 
i ħc ( /2)·
U=e

where   (x)

compare to the U(1) transformation:
U=
ei ħc (x)
q
Like before, the Dirac Lagrangian (as it stands)
is NOT invariant under this transformation
(  )   U    (e
e


ig ( / 2 )


ig ( / 2 )
 ) = (U) + U()


(   ig ( / 2)    )
The fix again is to replace  by a “covariant derivative”

D  (  ig( / 2)  G )

3 vector fields
are needed to span the space
of this transformation operator
Then,
assuming an appropriate GAUGE transformation of the G fields is possible:


ig ( / 2 )
  (  )  e



D
D
D
so that the (D)' = D
term remains invariant
To figure out the necessary transformation
 property of the Gauge
fields


ig ( / 2)
we’ll use the fact that    e
   eig ( / 2) 
then

D    e
(   ig ( / 2)  G )






 eig ( / 2) (   ig ( / 2)  G )(e ig ( / 2) )


ig ( / 2 )

D   e


ig ( / 2 )



ig ( / 2 )
(   ig ( / 2)  G )(e
 )





ig ( / 2 ) 
ig ( / 2 )
 '
 (   ig    ige
(  G )e
)



2
 
2
D'
in other words the transformed


   ig ( / 2)  G

    ig e
 ig ( / 2 ) 

(  G )e
   
 

ig ( / 2 ) 
2
2
which means in particular


2




 

 G  U  GU †    
2
2
]


2
 





 G  U  GU †    
2
2
if could pull through U or U† this would just be




 

 G   G   
2
2
(
)
which would look similar to the
gauge transformations under U(1)
Why can’t we?
Let’s focus on this term:
 



ig ( / 2)  
ig ( / 2)
e
(  G )e
2
which we can just write as


ig ( / 2)
e
i
U U†Gi
2
 i ig ( / 2) i
e
G
2
i
OK
to commute!
Not OK!

recalling that e


ig ( / 2 )
 
g
g
 
 cos
 i ( ) sin

2
2
You will show for homework that
i
U 2 U† =
RT
i
(=g/2) 2
a 3-dim space(-like) rotation
applied to the i/2 matrices
 
ig ( / 2) i
e
(
2

So
T j i
G )e
 Rij G
2
i,j count over the iso-space generators
(Pauli matrices 1,2,3)
i


ig ( / 2)
 counts over the spacetime coordinates
(ct, x, y, z)


Since 2
 





 G  U  GU †    
2

 iG 
i
2
T
j
R jk k G
  k  
k
Now remember the i are linearly independent
matching like terms we find:
i

T j
G  R jiG
j
 RijG

  
i
  
i

RG
fields are
“rotated” …and shifted by a gradient
(a gauge shift)
The resulting Lagrangian (so far)
L=iħcgDmc2
=iħcg


mc2
 




(gg )·G
2
where we’ve introduced 3 new vector fields

G =
(G1, G2, G3 )

3 separate
4-vector
fields
(like A)
each with its own free Lagrangian (kinetic energy) term


1 
1 
1 
 F1 F1
4

 F2 F2
4

 F3 F3
but not quite the same as before Fi

4
 ?
 
=  F ·F
1
4


  Gi   Gi
since THIS is not an invariant
Fi' = Gi'Gi'
=  (RijG j i )   (RijG j i )
= ( Rij )G j +Rij (G j)   i
 ( Rij )G j  Rij(G j)    i
RR((x)) for a local transformation
= Rij (G j  G j )  ( Rij   Rij )G j
Fi
Actually with 3 vector fields
there IS another anti-symmetric term possible


G×G
and, with it, the more general
F = GiGi 
i

2g 
G ×G
ħc 
restores invariance!
So NOW for our newly proposed SU(2) theory we have
L=[iħcg
2 ]


mc


1 
 F
F

4




(gg )·G
2
describing two equal mass Dirac particle states in interaction with
3 massless vector fields
Think of something like the g-fields, A
G

This followed just by insisting on local SU(2) invariance!
In the Quantum Mechanical view:





•Dirac particles generate 3 currents, J = (gg  )
•These particles carry a “charge” g, which
is the source for the 3 “gauge” fields
2


Furthermore with:
  
LGauge ~ F  F
Field


  




 
 
 ( G   G  2 gG  G )  (  G   G  2 gG  G )
i
i
j
k
  ( G   G  2 g ijkG G )  (  G i   G i   2 g ijkG j  G k )


i
linear
linear
quadratic

The full product has nothing smaller than quadratic G terms
(KE terms of free particles)
plus cubic and quartic terms
(interaction terms describing
VERTICES of gauge particles
with themselves!!)
field-current
interaction
3 like this:
one for each Gi
plus “self-interaction” terms:
These gauge particles (“force carriers”) are NOT neutral!
(like gs are with respect to electric charge)
In general NON-ABELAIN GAUGE THEORIES:
•introduce more interactions (vertices)
•for SU(2) we saw both 3 and 4 particle interaction vertices
•have (still) massless gauge particles (like the photon!)
•the gauge field particles posses “charge” just like the
fundamental Dirac states
•not electric charge - we’re trying to think of NEW forces
The YANG-MILLS was built on the premise that
there existed
•2 elementary Dirac (spin-½) particles of ~equal mass
•serving as sources for the force fields through which they interact
NO SUCH PAIRS EXIST
proton/neutron isospin states were the inspiration, but
•there is NO massless vector (spin 1) iso-triplet
(isospin 1) of known particle states
• -mesons? 770 MeV/c2
•p,n, now recognized as COMPOSITE particles
•isospin of up,dn quarks generalized into SU(3)  SU(4)
The strong force must be
independent of FLAVOR
up
down
charm
strange
top
bottom
i.e., the strong force does not couple to flavors.
SO WHERE DOES THE STRONG FORCE COME FROM?
We WILL find these ideas resurrected in:
SU(3) color symmerty of strong interactions
SU(2) electro-weak symmetry