Transcript Klein Gordan Lagrangian: Interaction Terms

Finding a Klein-Gordon Lagrangian
p  p  m2c 2  0

i    i
x
2

  2     m2c 2  0
2 2
m
c

or     
 0
2

Provided we can identify the appropriate
this should be derivable by
The Euler-Lagrange Equation
L
  L  L

 
0
  (  )  
 i 
i

The Klein-Gordon Equation
I claim the expression
2
2

1 1   
 mc  2 
  2           
2  c  t 
  

L
2

1    mc  2 
  
   
2  x x   

2


1
mc


           2 
2 
  

  
serves this purpose
2

1
 mc  2 
  0  0   x  x   y  y   z  z     
2 
  

L
L
L
L
 
 

 
0
  (  )  
 i 
i

L


1

  0   0    0   0
 ( 0 )
   2


 ct 
L
L


1

 ( x   x )   x   x
 ( x )
   2
 
 x 

1
m2c 2 
   2 2 
2
 2 
 
mc
L
 
       0
  

You can (and will for homework!) show
the Dirac Equation can be derived from:
L

2


(
i



mc
)

DIRAC(r,t)
We might expect a realistic Lagrangian that involves systems of particles
L(r,t) = L
K-G
describes
photons
+
L
but each term
describes free
non-interacting
particles
DIRAC
describes
e+e objects

1
( r , t )  (      2 2 )   (i     m)
L
2
+
L
INT
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  ( V  J  A)
L
  J  A

J  (; J )


A  (V ; A)

current-field interactions
the fermion
(electron)
the boson
(photon) field
LINT  (e   ) A
from the Dirac
expression for J
particle
state
antiparticle
(hermitian conjugate)
state
Recall the “state functions”
have coefficients that must
What does such a PRODUCT of states mean? satisfy anticommutation relations.
They must involve operators!
We introduce operators (p) and †(p) satisfying either of 2 cases:
 ( p),  † (k )   ( p  k )
{ ( p ),  † (k )}   ( p  k )
or
 ( p),  (k )   † ( p),  † (k )   ( p  k ) { ( p),  (k )}  { † ( p),  † (k )}   ( p  k )
Along with a representation of the (empty) vacuum state: | 0 
such that:
The Creation Operator
†
†
 † ( p) 0  p
“creates” a free particle of 4-momentum p
The complex conjugate of this equation reads:
0  ( p)  p
0  ( p)  † ( p) 0  p p  1
The above expression also tells us:
0  ( p) p  1
 ( p) p  | 0 
which we interpret as:
The Annihilation Operator
If we demand, in general,
the orthonormal states we’ve assumed:
00  1
0 p  0
0  † ( p) 0
0  † ( p)  0
p0  0
0  ( p) 0
 ( p) 0  0
an operation that
makes no contribution
to any calculation
 ( p),  † (k )   ( p  k )
{ ( p ),  † (k )}   ( p  k )
Then
 ( q) p   ( q) †(p)|0 = [ (p-q)  †(p)(q) |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),  † (k )   ( p  k )
{ ( p ),  † (k )}   ( p  k )
or
 ( p),  (k )   † ( p),  † (k )   ( p  k ) { ( p),  (k )}  { † ( p),  † (k )}   ( p  k )
in contrast
†(p)†(k)|0 = †(k)†(p)|0
†(p)†(k)|0 = †(p)†(k)|0
|pk = |kp
|pk = |kp
and if p=k this gives |pp = |pp
and if p=k this must give
This is perfectly OK!
These must be symmetric states
These are anti-symmetric states
BOSONS
FERMONS
0
The most general state

 C0 0   C1 ( p )dp p
   C2 ( p1 , p2 )dp1dp2 p1 p2  
Recall the most general DIRAC solution:
3

3

dk
k
k r
dk
s 
(2 )3 2
{(r,t),
†(r´,t)}
 (r , t )   
s
If we insist:
e
s
gu
g e
s it
= 3(r
h
hv ss e it

that these Dirac
– r) particles are fermions
we can identify (your homework) g as an annihilation
operator a(p,s) and h as a creation operator b†(-p,-s)

 (r , t )   
s

 (r , t )   
s
dk 3
(2 ) 2
3
dk
e
3
(2 ) 2
3
e

 k r
gu
a e

 k r
gau e
s it
†
s i t
† s it
bhv e

s i t
 hb v e

Similarly for the photon field (vector potential)
 
A(r , t )  
If we insist:
 
A( r , t )  
dk
3
e
(2 ) 2
3

ik r
C  e
s  i t
1
s i t
 C2 e
[A(r,t), A†(r´,t)] = 3(r – r)
dk 3
(2 )3 2

ik r
e

-s†
d -k
s i t
 e

Bosons!

s i t
 e
-s
d -k
Remember here there is no separate anti-particle
(but 1 particle with 2 helicities).
Still, both solutions are needed for mathematical completeness.

Now, since interactions between Dirac particles (like electrons)
and photons appear in the Lagrangian as

e  A
It means these interactions involve operator products of
(a†
+b
creates an
electron
) (a
+ b†
annihilates
an electron
annihilates
a positron
) (d†
creates a
photon
creates a
positron
giving terms with all these possible combinations:
a†b†d† a†ad† a†ad†
a†ad
bb†d†
bb†d
bad†
bad
+d
annihilates
a photon
)
What do these mean?
a†b†d† a†ad† a†ad†
a†ad
bb†d†
bb†d
bad†
bad
In all computations/calculations we’re interested in,
we look for amplitudes/matrix elements like:
p2 p3 p1
out
Dressed up by the full integrals to
calculate the probability coefficients
creates a
positron
e
e
a†b†d†
creates an
electron
in
 0|daa†|0
e


a†ad†
creates a
photon
annihilates
an electron
e
e

a†ad†
time
e
e
a†ad
time
e

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 Purcell
Development 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 Čerenkov
Il’ja Mikhailovich Frank
Igor Yevgenyevich Tamm
Discovery 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. Alvarez
Discovery of resonance states through bubble chamber analysis techniques
1969
Murray Gell-Mann
Classification 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, and
Steven 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 Meer
Contributions to the discovery of the W and Z field particles.
1988 Leon M. Lederman, Melvin Schwartz, and
Jack Steinberger
Discovery of the muon neutrino
Particle Physicists Awarded the Nobel Prize since 1948
1989 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 and
Richard 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. Veltman
Renormalization theories of electroweak interactions
2002 Raymond Davis, Jr. and Masatoshi Koshiba
The detection of cosmic neutrinos
The Nobel Prize in Physics 2004
"for the discovery of asymptotic
freedom in the theory of the strong
interaction"
David J. Gross
H. David Politzer
Frank Wilczek
Kavli Institute for
Theoretical Physics,
University of
California
Santa Barbara, CA,
USA
b. 1941
California Institute of
Technology
Pasadena, CA, USA
Massachusetts
Institute of Technology
(MIT)
Cambridge, MA, USA
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”
p3
p4
e
e
…two final state electrons exit.

…a  is exchanged (one emits/one absorbs)…
Our general solution


e
e
allows waves traveling
p1
in BOTH directions
Calculations will include both
p2
and not distinguish the
contributions from either case.
Two electrons (in momentum states p1 and p2) enter…
Coulomb repulsion (or “Møller scattering”)
Mediated by an
exchanged photon!
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?
bad†
e
time
e
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 >
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 = J1  I2 + I1  J2
J
0
0
I
+
I 0
0 J
So for a fixed j1, j2
| j1 m1> | j2 m2>
all possible combinations
which form the BASIS SET of the matrix
representation of the direct product operators
How many? How big is this basis?
( 2 j1  1)( 2 j2  1)
Giving us NEW (2 j1  1)( 2 j2  1)  (2 j1  1)( 2 j2  1)- dimensional operators
acting on new
( 2 j1  1)( 2 j2  1)
long column vectors
2j1+1
states
We’ve expanded our space into:
0
0
J1 0
0
0
0 0 0 0 0 1
0 0 0 0 0 0
0 0 0 0 0 0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
+
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
J2
2j2+1 states
Obviously we still satisfy ALL
angular momentum commutator relations.
All angular momentum commutator relations still valid. J3 is still diagonal.
But OOPS! 2
J is no longer diagonal!
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 >
Two eigenstates
give m=j1+j2-1
either | j1 , j1-1 >| j2, j2 >
or
| j1, j1 >| j2, j2 -1 >
corresponding to states in the
irreducible 2 dimensional representation
This is the irreducible 1x1
representation for m = j1 + j2.
| j1+j2, j1+j2-1> and | j1+j2-1, j1+j2-1 >
RECALL in general
the direct product state
is a LINEAR COMBINATION
of different final momentum states.
m = ( j1 + j2 )
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 mechanically
mix within themselves
but not across
the sub-block boundaries.
The raising/lowering operators
provide 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
1

2
2
22
=
121
1 12
32
=
1221
1
2 2
42
=
12221
1 1
33
=
12321
3
2 1
43
=
123321
3
3
2 2
44
=1234321
1
3