Post on 21-Dec-2015
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