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