(also xyz yzx zxy) both can be re-written with (with the same for x y z) all 4 statements can be...
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4 E
JBct
Ec
41
cEEEc
zzyyxx 4 xc
xyzzy JEBB 40 (also xyzyzxzxy)
both can be re-written with
04000
000 JFFFFc
zz
yy
xx
xc
xzxz
yxy
xxx
JFFFF 400
(with the same for xyz)
All 4 statements can be summarized in
JF
c)(4 zyx ,,,0
The remaining 2 Maxwell Equations: 0 B
01
tB
cE
are summarized by
0ijkkijjki
FFF ijk = xyz, xz0, z0x, 0xy
Where here I have used the “covariant form”
0
0
0
0
xyz
xzy
yzx
zyx
BBE
BBE
BBE
EEE
F
= g g F =
To include the energy of em-fields(carried by the virtual photons)
in our Lagrangian we write:
L=[iħcmc2F FqA
12
But need to check: is this still invariant under the SU(1) transformation?
(A+) (A +)
= A+ A
=
L
L
AqFFmcci )(
1612
AqFFmcci )(
2
12
“The Fundamental Particles and Their Interactions”, Rolnick (Addison-Wesley, 1994)
“Introduction to Elementary Particles”, Griffiths (John Wiley & Sons, 1987)
Jt
EB
E
12
116
Jct
E
cB
E
41
4
Gaussian cgs units
Heaviside-Lorentz
units
AqFF
mcci )(
16
12
L
AAAAFF
16
1
16
1with
(and summing over , )
0)
( L L 0
)
AA
(
L LTheprescriptions
and
give two independent equations OR summing over ALL variables (fields) gives the full equation WITH interactions
Starting from
)) AA
(
(
L FFLet’s look atthe new term:
)(
AFF
AAAA
AA
)(16
1
)(
FF
summing over ,
survive when =, = and when =, =
AAAAAAAA
A
)(16
1
[-( A-A)][-( A-A)]
AAAA
A
2
)(16
1
, now fixed, not summed
1
sum over(but non-zero onlywhen =, = )
AA
4
1
)(
)(
)(
)(
8
1
A
AAAAA
A
A
AAAAgg
8
1
AA 2
8
1 where since this tensor is anti-symmetric!
A = ggA
= ggA
)) AA
(
(
L FF
So with
AA
4
1
0)
AA
(
L L
004
1
AA
A
AAAA
A
))((
16
1 FF = 0
AA
02 AA
and next
we get
0 in theLorentz Gauge
Note:
The Klein Gordon equation for massless photons!
/ = e+iq/ħcU(1) :
+iqq = e for electronsq = +e for positronsq = +2e/3 for up quarks
/ = e+iJ · /ħ
The ± sign is just a convention, as in rotations:
SO(3) :
We can generalize our procedures into a PRESCRIPTION to be followed,noting the difference between LOCAL and GLOBAL transformations
are due to derivatives:
c
qi
for U(1) this is a1×1 unitary matrix
(just a number)
/ = [e+iq/ħc] = e+iq/ħc the extra term
that gets introduced
If we replace every derivative in the original free particle Lagrangianwith the “co-variant derivative”
D= + i A
għc
then the gauge transformation of A will cancel the term that appears through
(D )/ = e-iq/ħcD restores the invariance of Li.e.
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
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.
The conserved quantity “discovered” was ELECTRIC CHARGE.
The particle coupling to CHARGE was interpreted as the PHOTON.
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?
The theorists Yang & Mills extended the U(1) formalism that explained e&m forces in an effort to explore ISOSPIN.
Is there some kind of fundamental ISOSPIN FORCE?
Imagine 2 possible states: the flip sides of some spin-½ (2-component) propertySpin, ISO-spin, or even something NEW
112
111 cmci L 22
2222
cmci Sum of 2 Dirac lagrangians
Applying the Euler-Lagrange equations results in 2 independent Dirac equations
1 satisfying one, 2 satisfying the other.
Written more compactly as spinors
2
1
21 and its adjoint
2Mcci L
2
1
0
0
m
mMwhere “mass
matrix”
If m1=m2 then M=mI and
2mcci L which “looks” just like the
1-particle Dirac Lagrangian
Note: 1 and 2 each already 4-component Dirac spinors
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 obviousglobal phase transformation to a LOCAL invariance, writing
U = ei ( /2)· għc
U = ei (x)qħccompare to the U(1) transformation:
where (x)
Like before, the Dirac Lagrangian (as it stands) is NOT invariant under this transformation
)()( )2/(
igeU = (U) + U()
))2/(()2/(
igeig
The fix again is to replace by a “covariant derivative”
))2/(( Gig D
3 vector fieldsare needed to span the space
of this transformation operator
Then, assuming an appropriate GAUGE transformation of the G fields is possible:
DDD )2/()( ige
so that the (D)' = D term remains invariant
To figure out the necessary transformation property of the Gauge fields
we’ll use the fact that )2/(ige
)2/(ige
then
))()2/((
))2/((
)2/()2/(
)2/(
igig
ig
eGige
Gige
D
))()2/(( )2/()2/(
igig eGigeD
))(( )2/(
2
)2/(
2
igig eGigeig '
D'in other words the transformed
2
)2/(2
)2/( )(
)2/(
igig eGeig
Gig
which means in particular
222
GG U U †
222
GG U U †
if could pull through U or U† this would just be
)(22
GG
which would look similar to thegauge transformations under U(1)
Why can’t we?
Let’s focus on this term:iig
i
iigigig GeeeGe
)2/(
2)2/()2/(
2)2/( )(
OKto commute!Not OK!
which we can just write as
U U†Gi
i
2
You will show for homework that
U U†i
2= RT
(=g/2)
i
2a 3-dim space(-like) rotation
applied to the i/2 matrices
recalling that2
sin)(2
cos)2/( g
ig
eig
SoiT
ijigiiig GReGe j
2
)2/(2
)2/( )(
i,j count over the iso-space generators(Pauli matrices 1,2,3)
counts over the spacetime coordinates(ct, x, y, z)
222
GG U U †Since
kk
jk
Tjk
ii
GRG
Now remember the i are linearly independent
matching like terms we find:
ijTji
i GRG
ijijGR
RG
fields are“rotated” …and shifted by a gradient
(a gauge shift)
The resulting Lagrangian (so far)
L=iħcDmc2
=iħcmc2g·G
2
where we’ve introduced 3 new vector fields3 separate4-vector
fields(like A)
G = (G1, G2
, G3 )
each with its own free Lagrangian (kinetic energy) term
3341
2241
1141 FFFFFF = F ·F
14
but not quite the same as before iii
GGF ?
since THIS is not an invariant
F i' = Gi'G
i' = (RijG
j
i (RijG j
i
= ( Rij G j +Rij (G
j
i
( Rij G j Rij(G
j
i
= Rij (G j G
j ( Rij Rij G
j
RR((x)) for a local transformation
F i
Actually with 3 vector fields there IS another anti-symmetric term possible
G×G
and, with it, the more general
F i = GiGi
G×G
restores invariance!
2għc
L=[iħcmc2F Fg·G
14
So NOW for our newly proposed SU(2) theory we have
2
describing two equal mass Dirac particle states in interaction with
3 massless vector fields G
Think of something like the -fields, A
This followed just by insisting on local SU(2) invariance!
In the Quantum Mechanical view:•Dirac particles generate 3 currents, J = (g
)2
•These particles carry a “charge” g, which is the source for the 3 “gauge” fields
Furthermore with:
FF
FieldGauge
~L
)2()2(
GGgGGGGgGG
)2()2(
kj
ijk
ii
i
kj
ijk
ii GGgGGGGgGG 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 particleswith themselves!!)
field-current interaction
3 like this:
one for each Gi
plus “self-interaction” terms:
These gauge particles (“force carriers”) are NOT neutral!(like s 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 charm topdown strange 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
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