wednesday, mar. 5, 2003phys 5326, spring 2003 jae yu 1 phys 5326 – lecture #13 wednesday, mar. 5,...

Wednesday, Mar. 5, 2003 PHYS 5326, Spring 2003Jae Yu

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PHYS 5326 – Lecture #13Wednesday, Mar. 5, 2003

Dr. Jae Yu

Local Gauge Invariance and Introduction of Massless Vector Gauge Field


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Announcements• Remember the mid-term exam next Friday, Mar.

14, between 1-3pm in room 200– Written exam

• Mostly on concepts to gauge the level of your understanding on the subjects

• Some simple computations might be necessary– Constitutes 20% of the total credit, if final exam will be

administered otherwise it will be 30% of the total• Strongly urge you to go into the colloquium today.


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Prologue• Motion of a particle is express in terms of the position of

the particle at any given time in classical mechanics.• A state (or a motion) of particle is expressed in terms of

wave functions that represent probability of the particle occupying certain position at any given time in Quantum mechanics. Operators provide means for obtaining observables, such as momentum, energy, etc

• A state or motion in relativistic quantum field theory is expressed in space and time.

• Equation of motion in any framework starts with Lagrangians.


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Non-relativistic Equation of Motion for Spin 0 ParticleEnergy-momentum relation in classical mechanics give

EVm

2

2p

Provides non-relativistic equation of motion for field, , Schrodinger Equation

Quantum prescriptions; .t

iEi

,p

tiV

m

2

2

2

2 represents the probability of finding the pacticle of mass m at the position (x,y,z)


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Relativistic Equation of Motion for Spin 0 ParticleRelativistic energy-momentum relation

02242222 cmppcmcE p

With four vector notation of quantum prescriptions;

Relativistic equation of motion for field, , Klein-Gordon Equation

, , ,1

; ; 3210 zyxtcxip where

0222 cm

2

22

2

2

1

mc

tc2nd order in time


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Relativistic Equation of Motion (Direct Equation) for Spin 1/2 Particle

To avoid 2nd order time derivative term, Direct attempted to factor relativistic energy-momentum relation 022 cmpp

This works for the case with 0 three momentum

0002220 mcpmcpcmp

But not for the case with non-0 three momentum 2222 cmpmcppmcpmcpcmpp k

kkk

kk

k

The terms linear to momentum should disappear, so kk

To make it work, we must find coefficients k to satisfy:

pppp k

k

Terms CrossOther 30

033020

022010

0110

2323222221212020

23222120

pppppp

pppp

pppp


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Dirac Equation Continued…The coefficients like 0=1 and 1= 2= 3=i do not work

since they do not eliminate the cross terms.It would work if these coefficients are matrices that satisfy the conditions

Using Pauli matrix as components in coefficient matrices whose smallest size is 4x4

022 mcpmcpcmpp kk

The energy-momentum relation can be factored

when 0

1 ,123222120

g2, Or using Minkowski metric, g

w/ a solution

0 mcp

By applying quantum prescription of momentum

ip

Acting it on a wave function , we obtain Dirac equation 0 mci k


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Euler-Lagrange EquationFor conservative force, it can be expressed as the gradient

of a scalar potential, U, as

UF Therefore the Newton’s law can be written .U

dt

vdm

ii

q

L

q

L

dt

d

The Euler-Lagrange fundamental equation of motion Starting from Lagrangian UmvUTL 2

2

1

x

U

q

L

mvdv

dT

q

Lx

x

1

1In 1D Cartesian

Coordinate system


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Euler-Lagrange equation in QFT

Unlike particles, field occupies regions of space. Therefore in field theory motion is expressed in space and time.

Euler-Larange equation for relativistic fields is, therefore,

ii

LL

Note the four vector form


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Klein-Gordon Largangian for scalar (S=0) FieldFor a single, scalar field variable , the lagrangian is

22

2

1

2

1

mc

L

02

mc

From the Euler-Largange equation, we obtain

Since ii

L

and ii

mc

2

L

This equation is the Klein-Gordon equation describing a free, scalar particle (spin 0) of mass m.


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Dirac Largangian for Spinor (S=1/2) FieldFor a spinor field , the lagrangian

2mcci L

0

mc

i

From the Euler-Largange equation for , we obtain

Since 0

Land

2mcci

L

Dirac equation for a particle of spin ½ and mass m.How’s Euler Lagrangian equation looks like for


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Proca Largangian for Vector (S=1) FieldFor a vector field A, the lagrangian

AAmc

FF

AAmc

AAAA

2

2

8

1

16

1

8

1

16

1

L

022

Amc

FAmc

AA

From the Euler-Largange equation for A, we obtain

Since

AA

A

4

1Land

A

mc

A

2

4

1

L

For m=0, this equation is for an electromagnetic field.

Proca equation for a particle of spin 1 and mass m.


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Local Gauge Invariance - IDirac Lagrangian for free particle of spin ½ and mass m;

2mcci L

Is invariant under a global phase transformation (global gauge transformation) since . ie ie

However, if the phase, , varies as a function of space-time coordinate, x, is L still invariant under the local gauge transformation, ? xie

No, because it adds an extra term from derivative of .


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Local Gauge Invariance - IIThe derivative becomes

2mcci L

xixi eeiSo the Lagrangian becomes

cmcci

mceeici xixi

2

2'L

Since the original L is

L’ is cLL'

Thus, this Lagrangian is not invariant under local gauge transformation!!


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Local Gauge Invariance - IIIDefining a local gauge phase, (x), as

where q is the charge of the particle involved, L becomes

Under the local gauge transformation:

xq

cx

qLL'

cxiqe /


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Local Gauge Invariance - IVRequiring the complete Lagrangian be invariant under (x) local gauge transformation will require additional terms to free Dirac Lagrangian to cancel the extra term

Where A is a new vector gauge field that transforms under local gauge transformation as follows:

AA

Aqmcci 2L

Addition of this vector field to L keeps L invariant under local gauge transformation, but…


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Local Gauge Invariance - VThe new vector field couples with spinor through the last term. In addition, the full Lagrangian must include a “free” term for the gauge field. Thus, Proca Largangian needs to be added.

AAThis Lagrangian is not invariant under the local gauge transformation, , because

AA

cmFF A

2

8

1

16

1

L

AAAA

AAAA


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Local Gauge Invariance - VIThe requirement of local gauge invariance forces the introduction of massless vector field into the free Dirac Lagrangian.

AqFF

mcci

16

1

2L


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Homework• Prove that the new Dirac Lagrangian with an

addition of a vector field A, as shown on page 12, is invariant under local gauge transformation.

• Describe the reason why the local gauge invariance forces the vector field to be massless.

wednesday, mar. 5, 2003phys 5326, spring 2003 jae yu 1 phys 5326 – lecture #13 wednesday, mar. 5,...

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