Finding a Klein-Gordon Lagrangian

022 cmpp

0222 cm

xii

2

The Klein-Gordon Equation02

22

cm

or

Provided we can identify the appropriatethis should be derivable by

The Euler-Lagrange Equation

0

)(

i

i

L

L L

2

22

21

2

1

mc

tcL

I claim the expression

2

2

2

1

mc

xx

2

2

2

1

mc

serves this purpose

2

2

002

1

mczzyyxx

0

)(

i

i

L LL

0000

0 2

1

)(

tc

xxxx

x

x

)(2

1

)(

L L

L L

2

222

2

1

cm L

02

mc

You can (and will for homework!) show the Dirac Equation can be derived from:

)( 2mci LDIRAC(r,t)

We might expect a realistic Lagrangian that involves systems of particles

= LK-G + LDIRAC L(r,t)describes

e+e objectsdescribesphotons

)()(),( 2221 mitr

but each termdescribes free

non-interactingparticles

L+ LINT

But what does terms look like?How do we introduce the interactions the experience?

We’ll follow (Jackson) E&M’s lead:

A charge interacts with a field through:

)(INT AJV

L);(

);(

AVA

JJ

AJ

current-field interactions

the fermion(electron)

the boson(photon) field

Ae )(INT L from the Dirac

expression for J

antiparticle(hermitian conjugate)

state

particlestate

What does such a PRODUCT of states mean?

Recall the “state functions”have coefficients that must

satisfy anticommutation relations.They must involve operators!

We introduce operators (p) and †(p) satisfying either of 2 cases:

)()(),( kpkp † )()}(),({ kpkp †

or

)()(),()(),( kpkpkp †† )()}(),({)}(),({ kpkpkp ††

Along with a representation of the (empty) vacuum state: | 0

such that:

†

The Creation Operator

†

pp 0)(† “creates” a free particle of 4-momentum p

pp )(0

10)()(0 pppp †

1)(0 pp

pp)(

The complex conjugate of this equation reads:

The above expression also tells us:

| 0 which we interpret as:

The Annihilation Operator

If we demand, in general, the orthonormal states we’ve assumed:

00 1

p0 0

0p 0

0)(0 p†

0)(0 p

)(0 p†

0)( p

0

0an operation that

makes no contribution to any calculation

)()(),( kpkp † )()}(),({ kpkp †

Then

)()( qpq †(p)|0 =

zero

= (p-q)|0 This is how the annihilation operator works:If a state contains a particle with momentum q, it destroys it. The term simply vanishes (makes no contribution to any calculation) if p q.

|p1 p2 p3 = †(p1)†(p2)†(p3)

|0[ (p-q) †(p)(q) ]

| 0

)()(),( kpkp † )()}(),({ kpkp †

or

)()(),()(),( kpkpkp †† )()}(),({)}(),({ kpkpkp ††

†(p)†(k)|0 = †(k)†(p)|0

|pk = |kp

and if p=k this gives |pp = |pp

This is perfectly OK! These must be symmetric states

BOSONS

†(p)†(k)|0 = †(p)†(k)|0

|pk = |kp

and if p=k this must give 0

These are anti-symmetric states

FERMONS

in contrast

The most general state

2121212

10

),(

)(0

ppdpdpppC

pdppCC

tistisrk

s

ehveguedk

tr

2)2(),(

3

3

s

s sdk3 k g h

{(r,t), †(r´,t)} = 3(r – r)If we insist:that these Dirac

particles are fermions

Recall the most general DIRAC solution:

we can identify (your homework) g as an annihilation operator a(p,s) and h as a creation operator b†(-p,-s)

tistisrk

s

ehveguedk

tr

2)2(),(

3

3

a b†

tistisrk

s

evheugedk

tr

3

3

2)2(

),(

a b†

Similarly for the photon field (vector potential)

tistisrki eCeCedk

trA

213

3

2)2(),(

[A(r,t), A†(r´,t)] = 3(r – r)If we insist: Bosons!

tistisrki eeedk

trA

2)2(),(

3

3d

-s-k

†d

-s-k

Remember here there is no separate anti-particle (but 1 particle with 2 helicities).

Still, both solutions are needed for mathematical completeness.

interactions between Dirac particles (like electrons)and photons appear in the Lagrangian as

Now, since

Ae

It means these interactions involve operator products of

(a† + b ) (a + b† ) (d† + d )

creates anelectron

annihilatesan electron

annihilatesa positron

creates apositron

creates aphoton

annihilatesa photon

giving terms with all these possible combinations:

a†b†d† a†ada†ad†a†ad†

bb†d† bb†d bad† bad

a†b†d† a†ada†ad†a†ad†

bb†d† bb†d bad† bad

What do these mean?

In all computations/calculations we’re interested in,we look for amplitudes/matrix elements like:

inout

ppp132 0|daa†|0

Dressed up by the full integrals tocalculate the probability coefficients

a†b†d†

creates anelectron

creates apositron

creates aphoton

ee

a†ad†

annihilatesan electron

e

e

a†ad†

e

e

time

a†ad

e

e

time

Particle Physicists Awarded the Nobel Prize since 1948

1948 Lord Patrick Maynard Stuart Blackett For development of the Wilson cloud chamber

1949 Hideki Yukawa Prediction of the existence of mesons as the mediators of nuclear force

1950 Cecil Frank Powell Development of photographic emulsions to study mesons

1951 Sir John Douglas Cockcroft Ernest Thomas Walton

Transmutation of nuclei using artificial particle accelerator

1952 Felix Bloch Edward Mills PurcellDevelopment of precision nuclear magnetic measurements

Particle Physicists Awarded the Nobel Prize since 1948

1954 Max Born The statistical interpretation of quantum mechanics wavefunction

Walther Bothe Development of coincident measurement techniques

1955 Eugene Willis Lamb Discovery of the fine structure of the hydrogen spectrum Polykarp Kusch Precision determination of the electron’s magnetic moment

1957 Chen Ning Yang & Tsung-Dao Lee Prediction of violation of Parity in elementary particles

1958 Pavel Alekseyevich ČerenkovIl’ja Mikhailovich Frank Igor Yevgenyevich TammDiscovery and interpretation of the Čerenkov effect

Particle Physicists Awarded the Nobel Prize since 1948

1959 Emilio Gino Segre & Owen Chamberlain Discovery of the antiproton

1960 Donald A. Glaser Invention of the bubble chamber.

1961 Robert Hofstadter Discovery of nuclear structure through electron scattering off atomic nuclei

1965 Sin-Itiro Tomonaga, Julian Schwinger, and Richard P. Feynman

Fundamental work in quantum electrodynamics

1968 Luis W. AlvarezDiscovery of resonance states through bubble chamber analysis techniques

1969 Murray Gell-MannClassification scheme of elementary particles by quark content

Particle Physicists Awarded the Nobel Prize since 1948

1976 Burton Richter and Samuel C. C. Ting Discovery of new heavy flavor (charm) particle

1979 Sheldon L. Glashow, Abdus Salam, andSteven Weinberg

Theory of a unified weak and electromagnetic interaction.

1980 James W. Cronin and Val. L. Fitch Discovery of CP violation in the decay of neutral K-mesons

1984 Carlo Rubbia and Simon Van Der MeerContributions to the discovery of the W and Z field particles.

1988 Leon M. Lederman, Melvin Schwartz, andJack SteinbergerDiscovery of the muon neutrino

Particle Physicists Awarded the Nobel Prize since 19481989 Norman F. Ramsey Work on the hydrogen maser and atomic clocks

(founding president of Universities Research Association, which operates Fermilab)

1990 Jerome I. Friedman, Henry W. Kendall andRichard E. Taylor

Deep inelastic scattering studies supporting the quark model.

1992 Georges Charpak Invention of the multiwire proportional chamber.

1995 Martin L. Perl Discovery of the tau lepton.

Frederick Reine Detection of the neutrino.

1999 Gerardus ‘t Hooft and Martinus J. G. VeltmanRenormalization theories of electroweak interactions

2002 Raymond Davis, Jr. and Masatoshi Koshiba The detection of cosmic neutrinos

David J. Gross H. David Politzer Frank Wilczek

Kavli Institute for Theoretical Physics, University of California Santa Barbara, CA, USA

California Institute of Technology Pasadena, CA, USA

Massachusetts Institute of Technology (MIT) Cambridge, MA, USA

The Nobel Prize in Physics 2004

"for the discovery of asymptotic freedom in the theory of the strong interaction"

b. 1941 b. 1949 b. 1951

In Quantum Electrodynamics (QED)

All physically are ultimately reducible to this elementary 3-branched process.

We can describe/explain ALL electromagnetic processes by patching together copies of this “primitive vertex”

e

e

e

e

p1

p3

p2

p4

Two electrons (in momentum states p1 and p2) enter…

…a is exchanged (one emits/one absorbs)… Our general solution

allows waves traveling in BOTH directions

Calculations will include bothand not distinguish the

contributions from either case.

…two final state electrons exit.

Coulomb repulsion (or “Møller scattering”) Mediated by anexchanged photon!

ee

timebad†

These diagrams can be twisted/turned as long as we preserve the topology(all vertex connections) and describe an equally valid (real, physical) process

What does this describe?

Bhaba Scattering

A few additional notes on ANGULAR MOMENTUM

Combined states of individual j1 , j2 values can be written as a “DIRECT PRODUCT” to represent the new physical state:

| j1 m1 > | j2 m2 >

J 0 I 00 I 0 J

We define operators for such direct product states

A1 B2 | j1 m1> | j2 m2> = (A| j1 m1>)(B | j2 m2>)

then old operators like the MOMENTUM operator take on the new appearance

J = J 1 I 2 + I 1 J 2

+

So for a fixed j1, j2

| j1 m1> | j2 m2>

all possible combinationswhich form the BASIS SET of the matrix

representation of the direct product operators

How many? How big is this basis?

)12)(12(21 jj

)12)(12()12)(12(2121 jjjjGiving us NEW - dimensional operators

)12)(12(21 jjacting on new long column vectors

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

1 0 00 1 00 0 1

J 1 +

0 0 00 0 00 0 00 0 00 0 0

J 2

0 0 00 0 00 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

0 0 0 0 00 0 0 0 0 0 0 0 0 0

2j1+1states

2j2+1 states

We’ve expanded our space into:

Obviously we still satisfy ALL angular momentum commutator relations.

All angular momentum commutator relations still valid. J3 is still diagonal.

OOPS!

RECALL in general the direct product state

is a LINEAR COMBINATION of different final momentum states.

This is the irreducible 1x1 representation for m = j1 + j2.Two eigenstates

give m=j1+j2-1

But

The best that can be done is to block diagonalize the representation

m = j1 + j2 only one possible state (singlet) gives this maximum m-value!

| j1 j1 > | j2 j2 > = | j1+j2 ,j1+j2 >

m = ( j1 + j2 )

either | j1 , j1-1 >| j2, j2 >or | j1, j1 >| j2, j2 -1 >corresponding to states in the irreducible 2 dimensional representation

| j1+j2, j1+j2-1> and | j1+j2-1, j1+j2-1 >

J 2 is no longer diagonal!

This reduces the (2j1+1)(2j2+2) space into sub-spaces you recognize

as spanning the different combinations that result in a particular total m value.

These are the degenerate energy states

corresponding to fixed m values

that quantum mechanicallymix within themselves

but not across the sub-block boundaries.

The raising/lowering operatorsprovide the prescription for filling in entries of the sub-blocks.

The sub-blocks, correspond to fixed m values and can’t mix.They are the separate (lower dimensional) representations of

Angular momentum Space Dimensions Irreducible Subspaces

21

21

211

11

123

23

23

21

23

2 2

4 2

3 2

3 3

4 3

4 4

= 1 2 1

= 1 2 2 1

= 1 2 2 2 1

= 1 2 3 2 1

= 1 2 3 3 2 1

= 1 2 3 4 3 2 1