AIMS-pp-05.pdf

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Space and time Translations Mass-energy (the rest mass, m) Rotations Intrinsic spin (spin s = 0, i₂ ,i , . . .)

Parity Intrinsic parity (P = ±1)



 Remember how we classified particles:

•  What we want to do now is find out what happens when we

include relativity

•  We will see that the classification above does not change

•  But the form of the wavefunctions does change

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Wavefunction for a

Non-relativistic Free Spinless Particle

Important sign!! Non-relativistic:

where

Suppose we write

Doesn’_{t exist as a solution!}

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Now let’s try to make this relativistic! Let’_{s guess:}

where now

Now let’_{s require this to solve the simplest differential equation we}

can think of:

Rewrite this

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Klein-Gordon Equation

The Klein-Gordon equation is a scalar equation:

scalar differential operator

Problem: Negative-energy solutions

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Not surprising -- a 2nd order D.E. must have two distinct solutions!

Both positive & negative energy wavefunctions are solutions to KG eqn!

But what to do with ˆ

φ

?

Why negative energy?

φ

=

φ

₀

exp

[

−

ip

i

x

]

ˆ

φ

=

φ

₀

exp

[

+

ip

i

x

]

positive energy

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Wrong relationship!

So try:

Guess at an equation linear in the derivatives:

matrix!

And guess:

N-component ``column matrix" identity matrix!

Multiply by

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All other solutions are unitarily

equivalent

anticommutator Simplest non-trivial

solution is

Pauli matrices

So the wavefunction ψ is a 4-component object that obeys

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The Dirac Equation



Want space and time on same footing



Try an equation with one-time and one-space derivative



 Must be a matrix equation



Must also be consistent with p2_=m2

i

γ

µ

∂

_µ

+

m

(

)

ψ

=

0

All other solutions are unitarily

equivalent

γ

µ

,

γ

ν

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Solve the Dirac equation:

(1) Guess:

ξ and χ are each 2-component objects

(2) Insert into Dirac equation:

(3) Break up into 2x2 components and solve for lower part:

(4) Insert result into the upper part:

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A positive energy solution to the Dirac equation!

There are really only two

independent components! (6) Choose the normalization so that the energy term in the

denominator is eliminated:

Consider the rest frame of the wavefunction

Normalize these:

spin-up spin-down

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The Dirac equation should have 4 independent solutions

(since it’s a 4-component coupled 1st_{order DE)}

Now guess:

Repeat the previous argument – this gives:

Two negative

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What are these new solutions?

Antiparticles! ₂_mc2

− e + e 0 = E Empty Filled energy the positron

Positive energy forward in time = Negative energy backward in time

ψ

( )

x = 2m

E + m

2m ξ i

( )

E − m

2m pˆ i  σ

(

)

ξ( )i

⎛ ⎝ ⎜ ⎜ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟⎟

exp

[

−ipi x

]

ψ

( )

x = 2m

E − m

2m pˆ i 

σ

(

)

ξ( )i

E + m

2m ξ

i ( ) ⎛ ⎝ ⎜ ⎜ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟⎟

exp

[

+ipi x

]

positive energy

negative energy

i =↑ spin up

i =↓ spin down

i =↑ spin up

i =↓ spin down

2 solutions:

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•  In general a fermion wavefunction will be a linear combination

of all 4 solutions to the Dirac equation

•  In quantum field theory the coefficients in this linear

combination are quantum operators that create/destroy particles, analogous to the way that the quantum operators in the harmonic oscillator raise/lower energy levels

•  The same approach will work for scalars using the KG equation

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Summary

Klein-Gordon equation

has the complete set of solutions

Spin-up/Spin-down particle _{Spin-up/Spin-down antiparticle}

Dirac equation

has the complete set of solutions

particle

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Not Lorentz-invariant – depends on energy!

spin-up spin-down

(same will be true for the spin down solution)

Is ψ †ψ the probability density? Let's check for spin up:

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Solution – Change the definition of to

f

!

f

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Dirac conjugation:

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Write

f

!

f

=

f L

0

f

Like 0th_{component of a 4-vector!}

How do we interpret

f

!

f

?

f L

W

_f

Also transforms like a 4-vector!

Furthermore

/WYf LW_f _{? = Y/W}_{f L}W_?Y_f _?+ _{f L}W_/W_f = imf f ? imf f = 0

(not obvious: must prove separately)

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So interpret

charge density of wavefunction

electric current density of wavefunction

Use these as sources in Maxwell’_s

equations!

So the Dirac wavefunction represents a charged particle that can generate an electromagnetic field!

But how can an electromagnetic field influence a Dirac wavefunction?

Must modify the Dirac

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Under the transformation

Both and are unchanged

But the Dirac equation does change:

Gauge Transformation

=

α

constant GLOBAL

)

(

x

α

=

LOCAL

So let’_{s modify it by changing the derivative:}

valid for any wavefunction

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Locally Gauge-invariant Dirac Equation:

Equation for the vector potential:

a +b = 0

1) Try

2) Require Gauge-invariance

3) Scale out the irrelevant constant:

or where

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5) Next write

The Source-free Maxwell

Eqs!

6) Include a source:

7) Set

The Maxwell-Dirac Equations

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Gauge Theories



All non-gravitational interactions are founded

on these principles: Lorentz-covariance and

local gauge invariance



As a consequence of the gauge principle the

wavefunction

A

(or gauge field as it is more

commonly called) is massless



The group of gauge transformations depends

on one parameter -- the phase function



symmetry group of the theory is U(1)



Charge is conserved



Gauge theories are renormalizable

µ

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Charge Conservation

Total charge in a volume:

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Photon Wavefunction

If _{solves the source-free Maxwell eqs, so does}

Impose a condition:

Lorentz condition

Maxwell’_{s eqs become:}

Like 4 Klein-Gordon eqs!

Solve these:

where From Maxwell eqs

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2 independent polarizations!

2 spin states for the spin-1 photon!

Further Gauge Transforms are allowed: provided

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Summary

The Maxwell-Dirac Equations

Gauge Derivative

Free Photon wavefunction

Free Fermion wavefunctions

D

_µ

ψ

=

∂

_µ

ψ

+

ieA

_µ

ψ