LDirac=iħcmc2

Look at the FREE PARTICLE Dirac Lagrangian

because ei and in all pairings this added phase cancels!

Dirac matrices Dirac spinors(Iso-vectors, hypercharge)

This is just an SU(1) transformation, sometimescalled a “GLOBAL GAUGE TRANSFORMATION.”

Which is OBVIOUSLY invariant under the transformation

ei (a simple phase change)

What if we GENERALIZE this? Introduce more flexibility to the transformation? Extend to:

but still enforce UNITARITY?ei(x)

LOCAL GAUGE TRANSFORMATION

Is the Lagrangian still invariant?

(ei(x)) =

LDirac=iħcmc2

So:

L'Dirac = ħc((x))

iħcei(x)( )ei(x)mc2

i((x)) + ei(x)()

L'Dirac =

ħc((x))iħc( )mc2LDirac

For convenience (and to make subsequent steps obvious) define:

(x) (x)ħc q

L'Dirac = q()LDirac

then this is re-written as

recognize this????

cqe /

the current of the charge carrying particle described by as it appears in our current-field interaction term

L=[iħcmc2qA

L'Dirac = q()LDirac

If we are going to demand the complete Lagrangian be invariant under even such a LOCAL gauge transformation,

AAand that defines its transformation

under the same local gauge transformation

i.e., we must assume the full LagrangianHAS TO include a current-field interaction:

something that can ABSORB (account for) that extra term,it forces us to ADD to the “free” Dirac Lagrangian

L=[iħcmc2qA

•We introduced the same interaction term 3 weeks back following electrodynamic arguments (Jackson)

A ) that “couples” to If we chose to allow gauge invariance, it forces to introduce a vector field (here that means

The search for a “new” conserved quantum number shows that for an SU(1)-invariant Lagrangian, the free Dirac Lagrangain is “INCOMPLETE.”

A' = A + is exactly (check your notes!) the rule for GAUGE TRANSFORMATIONS already introduced in e&m!

•the transformation rule

•the form of the current density is correctly reproduced

The FULL Lagrangian also needs a term describing the free particles of the GAUGE FIELD (the photon we demand the electron interact with).

Of course NOW we want the Lagrangian term that recreates that!

Furthermore we now demand that now be in a form that is both Lorentz and SU(3) invariant!

titis

rki eCeCedk

trA 213

3

2)2(),(

We’ve already introduced the Klein-Gordon equation for a massless particle, the result, the solution

A = 0 was the photon field, A

We will find it convenient to express this term in terms of the ANTI-SYMMETRIC electromagnetic field tensor

More ELECTRODYNAMICS: The Electromagnetic Field Tensor

• E, B do not form 4-vectors

• E, B are expressible in terms of and A

but A=(V,A) and J=(c, vx, vy, vz) do!

the energy of em-fields is expressed in terms of E2, B2

• F = AA transforms as a Lorentz tensor!

xV

txA

AAF

)(

011001 = Ex since tAVE /

yxA

xyA

AAF

)()(

122112= Bz since AB

In general

xA

t

xAxxx AAF

0000

= Ex

0000 xt

xAx

Axx FAA

yxxy FF

00 xx FF

= Bz

etcFF xx 00 =

Actually thedefinition youfirst learned:

0

0

0

0

xyz

xzy

yzx

zyx

BBE

BBE

BBE

EEE

Fik = Fki =

While vectors, like J transform as “tensors” simply transform as

JJ

FF

zyxxxx

, , , ; 00

zyxxxx

g

,,,;00

x' = x or

x = x'

Under Lorentz transformations

xx )(

xx 1

dxxd

xddx 1

xxx

x

x

So, simply by the chain rule:

xx

1

and similarly:

xddx

dxxd

1

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 )(

2

12

L

AAAAFF

2

1

2

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

A = ggA

= ggA

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

AAggAAgg

8

1

AAgg

4

1 where since this tensor is anti-symmetric!