Table Of ContentPREFACE
ThisvolumecontainstheproceedingoftheNationalResearchSymposium
on Geometric Analysis and Applications held at the Centre for Mathematics
and its Applications, Australian National University, Canberra, from June
26 – 30, 2000. The Symposium celebrated the many significant contributions
of Professor Derek W. Robinson to mathematics, on the occasion of his 65th
birthday. The first day, Monday June 26, in particular was devoted to Derek;
speakers with a particularly close connection to Derek, including A. Carey,
M. Cowling, D. Evans, P. Jorgensen and T. ter Elst recalled and elaborated
on important aspects of Derek’s work, and the day ended with a banquet in
Derek’s honour.
The Symposium brought together researchers working in harmonic anal-
ysis, linear and nonlinear partial differential equations, quantum mechanics
and mathematical physics, and included researchers from North America,
Europe and Asia as well as Australasia.
We gratefullyacknowledge the supportof the contributorstothis volume.
We would like especially thank to Derek W. Robinson for his participation.
A short synopsis of Derek’s career, and a full list of his publications to date
are included in this volume.
Contributions for the proceedings were sought from all participants and
all papers received were carefully refereed by peer referees.
Alexander Isaev, Andrew Hassell,
Alan McIntosh, Adam Sikora
(Editors)
i
At the end of 2000 Derek Robinson retired from his position as Professor
of Mathematics in the Institute of Advanced Studies at ANU, a position he
had held for 19 years. Robinson obtained his early training at Oxford with
a Bachelor’s degree in mathematics in 1957 and a Doctorate of Philosophy
in theoretical nuclear physics in 1960. Subsequently he held various post-
doctoral positions in Switzerland, the United States, Germany and France
before being appointed Professor of Mathematical Physics at the University
of Aix–Marseille in 1968. He remained in this position until 1978 at which
pointheacceptedappointmentasProfessorofMathematicsattheUniversity
of New South Wales. Soon after he moved to Australia he became a Fellow
of the Australian Academy of Science.
In his early years Robinson worked in a wide range of areas of mathemat-
ical physics, quantum field theory, statistical mechanics, operator algebras,
etc. Operator algebras were introduced into quantum field theory in the
1960s as a means to describe the macroscopic observables but their most
fruitful application was to the characterization of the equilibrium states of
statistical mechanics. The observables of classical statistical mechanics form
anabelianC∗-algebraandRobinsonrealizedthatthecorrespondingquantum
algebra was ‘asymptotically abelian’. This observation was of fundamental
importancesincethestatesofasymptoticallyabelianalgebrascouldbeshown
to share many of the good properties of abelian algebras. Therefore extremal
states of the algebra could be identified with pure phases of the system and
decomposition theory of the states could be applied to describe the separa-
tion of the phases. Although these results were of a general abstract nature
Robinson showed that they could be applied to a broad class of realistic mod-
els of quantum spin systems. In particular he demonstrated that a number
of these systems exhibited a phase transition at low temperatures and the
pure phases were indeed described by extremal invariant states. Much of this
is described in the two volume monograph Operator Algebras and Quantum
Statistical Mechanics which Derek Robinson co-authored with Ola Bratteli.
Both volumes of this book appeared in a second edition and continue to be
used by research workers in these areas twenty years later.
The majority of applications of operator algebras to quantum statistical
mechanics concerned the equilibrium theory. Robinson and Bratteli realized
that the description of non-equilibrium phenomena required a dynamical
theory and this in turn required a theory of unbounded derivations of the
algebras. The foundations of this theory were laid in a series of joint papers
which are also described in their book. As a direct result of this latter work
Robinson’s interests then developed to evolution equations and semigroup
ii
theory in a much broader context. This change of direction coincided with
his move to Australia and most of his subsequent work has been on evolution
equations, in particular equations involving elliptic operators on Lie groups.
One of the issues at that time was the integrability of a Lie algebra rep-
resentation. Together with Ola Bratteli, George Elliott and Palle Jørgensen,
Robinson gave a sufficient condition in terms of a dissipativity condition and
an estimate on the semigroup generated by a Laplacian. Using Lipschitz
spaces Robinson could not only weaken the assumptions in the above men-
tioned paper, but he could also prove that the analytic vectors associated
with any representation of a Lie group coincide with the analytic elements of
the Poisson semigroup. By this stage Robinson became more interested in
elliptic operators on Lie groups and decided to write the monograph Elliptic
Operators and Lie Groups. After completion of the book Robinson began
a collaboration with Tom ter Elst and their continuing work can be divided
into three parts.
First, they developed the theory of complex, weighted, higher order sub-
coercive operators on Lie groups, in particular proving Gaussian bounds for
the kernel, and its derivatives, of the semigroup generated by such an op-
erator. They established that the Gaussian bounds are equivalent with a
subcoercivity condition on the operator, under weak additional assumptions.
Secondly, they studied second-order divergence form subelliptic operators on
Lie groups with complex bounded measurable coefficients and proved, under
a variety of conditions, optimal smoothness properties of the kernel. Thirdly,
in joint work with Adam Sikora or Nick Dungey, they studied asymptotic
properties of the semigroup and its kernel. They showed that on a Lie group
with polynomial growth the second-order Riesz transforms associated with
the Laplacian are bounded if, and only if, the group is a direct product of
a compact and a nilpotent Lie group. Their analysis of the asymptotics of
higher order operators continues.
Robinson has maintained an active involvement in University affairs, hav-
ing been Chairman of the Board of the Institute for Advanced Studies at
ANU (1988-1992), and a member of ANU Council (1997-2000), where he
never resiled from robust and provocative debate. He has also been an ac-
tive member of the mathematical community, in particular being president
of the Australian Mathematical Society (1994-1996) and continuing as Vice-
president. He also served four years as a member of the Research Training
and Careers Committee of the ARC (1996-2000)and as Chair of the National
Committee for Mathematics (1997-2001).
iii
LIST OF PUBLICATIONS
Books
[1] TheThermodynamicPressureinQuantumStatisticalMechanics-SpringerVerlagLecture
NotesinPhysics9,(1971)115pages.
[2] Operator Algebras and Quantum Statistical Mechanics Vol I (with O Bratteli) 500 pages,
Springer-Verlag (1979), 2nd edition 505 pages, Springer-Verlag (1979). Russian translation
offirstedition(1981).
[3] Operator Algebras and Quantum Statistical Mechanics Vol II (with O Bratteli) 501 pages,
Springer-Verlag(1981),2ndedition517pages,Springer-Verlag(1996).
[4] BasicSemigroupTheoryProceedingsoftheCentreforMathematicalAnalysis,ANU,Can-
berraVolume2,(1983)138pages.
[5] EllipticOperatorsandLieGroups556pages,OxfordUniversityPress(1991).
Research papers
[1] MultipleCoulombExcitationsofVibrationalNucleiNuclearPhysics45(1961)459–470.
[2] Zero-MassRepresentationsoftheInhomogeneousLorentzGroupHelvPhysActa35(1962)
98–112.
[3] SupportofaFieldinMomentumSpaceHelvPhysActa35(1962)403–413.
[4] MultipleCoulombExcitationsofDeformedNucleiHelvPhysActa36(1963)140–154.
[5] OnaSolubleModelofRelativisticFieldTheoryPhysicsLetters9(1964)189–191.
[6] ATheoremConcerningthePositiveMetricCommunMathPhys1(1965)89–95.
[7] TheGroundStateoftheBoseGasCommunMathPhys1(1965)159–174.
[8] ConservedCurrentsandAssociated Symmetries (withDKastler andASwieca) Commun
MathPhys2(1966)108–120.
[9] Covariance Algebras in Field Theory (with S Doplicher and D Kastler) Commun Math
Phys3(1966)1–28.
[10] Invariant States in Statistical Mechanics (with D Kastler) Commun Math Phys 3 (1966)
151–180.
[11] ExtremalInvariantStates(withDRuelle)AnnInstHenriPoincar´e6(1967)299-310.
[12] MeanEntropyofStatesofClassicalStatisticalMechanics(withDRuelle)CommunMath
Phys5(1967)288–300.
[13] Mean Entropy of States of Quantum Statistical Mechanics (with O E Lanford) J Math
Phys9(1968)1120–1125.
[14] StatisticalMechanicsofQuantumSpinSystemsCommunMathPhys6(1967)151–160.
[15] AsymptoticAbelianSystems(withSDoplicher,RKadisonandDKastler)CommunMath
Phys6(1967)101–120.
[16] Analyticity Properties of a Lattice Gas (with G Gallavotti and S Miracle-Sole) Physics
Letter25A(1967)443–444.
[17] StatisticalMechanicsofQuantumSpinSystemsIICommunMathPhys7(1968)337–346.
[18] Statistical Mechanics of Quantum Spin Systems III (with O E Lanford) Commun Math
Phys9(1968)327–338.
[19] Analyticity Properties of the Anisotropic Heisenberg Model (with G Gallavotti and S
Miracle-Sole)CommunMathPhys10(1968)311–324.
[20] ProofoftheExistenceofPhaseTransitionsintheAnisotropicHeisenbergModelCommun
MathPhys14(1969)196–204.
[21] Physical States of Fermi Systems (with S Miracle-Sole) Commun Math Phys 14 (1969)
235–270.
[22] StatisticalMechanicsofQuantumMechanicalParticleswithHardCoresI(withSMiracle-
Sole)CommunMathPhys16(1970)290–309.
[23] StatisticalMechanicsofQuantumMechanicalParticleswithHardCoresII(withSMiracle-
Sole)CommunMathPhys19(1970)204–218.
[24] NormalandLocallyNormalStatesCommunMathPhys19(1970)219–234.
[25] Normal States and Representations of the Canonical Commutation Relations (with M
CourbageandSMiracle-Sole)AnnInstHenriPoincareXIV(1971)171–178.
iv
[26] Approach to Equilibrium of Free Quantum Systems (with O E Lanford) Commun Math
Phys24(1972)193–210.
[27] TheFiniteGroupVelocityofQuantumSpinSystems(withEHLieb)CommunMathPhys
28(1972)193–210.
[28] ReturntoEquilibriumCommunMathPhys31(1973)171-189.
[29] ScatteringTheorywithSingularPotentialsIAnnInstHenriPoincareXXI(1974)185–215.
[30] Scattering Theory with Singular Potentials II (with P Ferrero and O Depazzis) Ann Inst
HenriPoincareXXI(1974)216–231.
[31] ACharacterisationofClusteringStatesCommunMathPhys41(1975)79–88.
[32] DynamicalStabilityandPureThermodynamicPhases(withHNarnhofer)CommunMath
Phys41(1975)89–97.
[33] UnboundedDerivationsofC∗-AlgebrasI(withOBratteli)CommunMathPhys42(1975)
253–268.
[34] UnboundedDerivationsofC∗-AlgebrasII(withOBratteli)CommunMathPhys46(1976)
11–30.
[35] Unbounded Derivations of Von Neumann Algebras (with O Bratteli) Ann Inst Henri
PoincareXXV(1976)139–164.
[36] UnboundedDerivationsandInvariantTraceStates(withOBratteli)CommunMathPhys
46(1976)31–35.
[37] TheApproximationofFlowsJFunctAnal24(1977)280–290.
[38] Green’sFunctions,HamiltoniansandModularAutomorphisms(withOBratteli)Commun
MathPhys50(1976)53–59.
[39] Bose-EinsteinCondensationwithAttractiveBoundaryConditionsCommunMathPhys50
(1976)53–59.
[40] PropertiesofPropagationofQuantumSpinSystemsJAustMathSocVolXIX(SeriesB)
19(1976)387–399.
[41] QuasiAnalyticVectorsandDerivationsofOperatorAlgebras(withOBratteliandRHer-
man)MathScan39(1976)371–38.
[42] Perturbations of Flows on Banach Spaces and Operator Algebras (with O Bratteli and R
Herman)CommunMathPhys59(1978)167–196.
[43] StabilityPropertiesandtheKMSCondition(withAKishimotoandOBratteli)Commun
MathPhys61(1978)209–238.
[44] Ground States of Quantum Spin Systems (with A Kishimoto and O Bratteli) Commun
MathPhys64(1978)41–48.
[45] PropagationPropertiesinScatteringTheoryJAusMathSoc(SeriesB)21(1979)474–485.
[46] LieandJordanStructureinOperatorAlgebras(withEStormer)JAusMathSoc(Series
A)29(1980)129–142.
[47] EquilibriumStatesofaBoseGaswithrepulsiveinteractions(withOBratteli)JAusMath
Soc(SeriesB)22(1980)129–147.
[48] PositivityandMonotonicityPropertiesofC0-SemigroupsI(withOBratteliandAKishi-
moto)CommunMathPhys75(1980)67–84.
[49] PositivityandMonotonicityPropertiesofC0-SemigroupsII(withAKishimoto)Commun
MathPhys75(1980)85–101.
[50] SubordinateSemigroupsandOrderProperties(withAKishimoto)JAusMathSoc(Series
A)31(1981)69–76.
[51] PositiveC0-SemigroupsonC∗-Algebras(withOBratteli)MathScand49(1981)259–274.
[52] Strictly Positive and Strongly Positive Semigroups (with A Majewski) J Aus Math Soc
(SeriesA)34(1983)36–48.
[53] StronglyPositiveSemigroupsandFaithfulInvariantStatesCommunMathPhys85(1982)
129–142.
[54] OnUnboundedDerivationsCommutingwithaCompactGroupof∗-Automorphisms(with
AKishimoto)PublRIMSKyotoUniv18(1982)1121–1136.
[55] Positive Semigroups on Ordered Banach Spaces (with O Bratteli and T Digernes) J Op
Theor9(1983)371–400.
[56] AdditionofanIdentitytoanOrderedBanachSpace(withSYamamuro)JAusMathSoc
(SeriesA)35(1983)200–210.
[57] HereditaryCones,OrderIdeals,andHalf-norms(withSYamamuro)PacJMath110(1984)
335–343.
v
[58] The Jordan Decomposition and Half-norms (with S Yamamuro) Pac J Math 110 (1984)
345–353.
[59] The Canonical Half-norm, Dual Half-norms, and Monotonic Norms (with S Yamamuro)
TohokuJMath35(1983)375–386.
[60] ContinuousSemigroupsonOrderedBanachSpacesJFunctAnal5(1983)268–284.
[61] OnPositiveSemigroupsPublRIMSKyotoUniv20(1984)213–224.
[62] Extending Derivations (with C J K Batty, A Carey and D E Evans) Publ RIMS Kyoto
Univ20(1984)119–139.
[63] Positive One-parameter Semigroups on Ordered Banach Spaces (with C J K Batty) Acta
ApplMath2(1984)221–296.
[64] AC∗-algebraicSchoenbergTheorem(withOBratteli,PJørgensenandAKishimoto)Ann
InstFourier(Grenoble)XXXIV(1984)155–187.
[65] RelativeLocalityofDerivations(withOBratteliandTDigernes)JFunctAnal59(1984)
12–40.
[66] Derivations,DynamicalSystems,andSpectralRestrictions(withAKishimoto)MathScand
56(1985)83–95.
[67] Dissipations,Derivations,DynamicalSystemsandAsymptoticAbelianness(withAKishi-
moto)JOpTheor13(1985)237–253.
[68] DerivationsofSimpleC∗-algebrasTangentialtoCompactAutomorphismGroups(withE
StormerandMTakesaki)JOpTheor13(1985)189–200.
[69] Strong Topological Transitivity and C∗-dynamical Systems (with O Bratteli and G A El-
liott)JMathSocJapan37(1985)115–133.
[70] TheCharacterisationofDifferentialOperatorsbyLocality;ClassicalFlows(withOBratteli
andGAElliott)CompMath58(1985)279–319.
[71] The Characterisation of Differential Operators by Locality; Abstract Derivations (with C
JKBatty)ErgTheorandDynSyst5(1985)171–183.
[72] The Characterisation of Differential Operators by Locality; Dissipations and Ellipticity
(withOBratteliandGAElliott)PublRIMSKyotoUniv21(1985)1031–1049.
[73] TheCharacterizationofDifferentialOperatorsbyLocality: C∗-algebrasofTypeI(withO
BratteliandGAElliott)JOpTheor16(1986)213–233.
[74] IntegrationinAbelianC∗-dynamicalSystems(withOBratteli,TDigernesandFMGood-
man)PublRIMSKyotoUniv,21(1985)1001–1030.
[75] SmoothCoresofLipschitzFlowsPublRIMSKyotoUniv22(1986)659–669.
[76] SmoothDerivationsonAbelianC∗-dynamicalSystemsJAusMathSoc(SeriesA)42(1987)
247–264.
[77] CommutatorsandGenerators(withCJKBatty)MathScand62(1988)303–326.
[78] CommutatorsandGeneratorsIIMathScand64(1989)87–108.
[79] ReturntoEquilibriumintheX−Y Model(withLRHume)JStatPhys44(1986)829–848.
[80] Embedding Product Type Actions into C∗-dynamical Systems (with O Bratteli and A
Kishimoto)JFunctAnal75(1987)188–210.
[81] FractionalPowersofGeneratorsofEquicontinuousSemigroupsandFractionalDerivatives
(withOELanford)JAusMathSoc(SeriesA)46(1989)1–32.
[82] CommutatorTheoryonHilbertSpaceCanJMath5(1987)1235–1280.
[83] TheDifferentialandIntegralStructureofContinuousRepresentationsofLieGroupsJOp
Theor19(1988)95–128.
[84] TheHeatSemigroupandIntegrabilityofLieAlgebras(withOBratteli,FMGoodmanand
PETJørgensen)JFunctAnal79(1988)351–397.
[85] An Index for Continuous Semigroups of ∗-endomorphisms of B(H) (with R T Powers) J
FunctAnal83(1989)1–12.
[86] LieGroupsandLipschitzSpacesDukeMathJour57(1988)357–395.
[87] The Heat Semigroup, Derivations, and Reynold’s Identity (with O Bratteli and C J K
Batty) Operator algebras and applications. London Math Soc Lecture Notes Series 136
(1988)22–47.
[88] The Heat Semigroup and Integrability of Lie Algebras: Lipschitz Spaces and Smoothness
PropertiesCommunMathPhys132(1990)217–243.
[89] LipschitzOperatorsJFunctAnal85(1989)179–211.
[90] Comparison of Commuting One-parameter Groups of Isometries (with O Bratteli and H
Kurose)TransAmerMathSoc320(1990)677–694.
vi
[91] Unitary Representations of Lie Groups and G˚arding’s Inequality (with O Bratteli, F M
GoodmanandPETJørgensen)ProcAmerMathSoc107(1989)627–632.
[92] Positive Semigroups Generated by Elliptic Operators on Lie Groups (with W Arendt and
CJKBatty).JOpTheor23(1990)369–407.
[93] EllipticDifferentialOperatorsonLieGroupsJFunctAnal97(1991)373–402.
[94] 2ndorderEllipticOperatorsandHeatKernelsonLieGroups(withOBratteli)TransAmer
MathSoc325(1991)683–713.
[95] SubellipticOperatorsonLieGroups: Regularity(withAFMterElst)JAusMathSoc57
(1994)179–229.
[96] Subelliptic Operators on Lie Groups: Variable Coefficients (with O Bratteli) Acta Appl
Math42(1996)1–104.
[97] Subcoercivity and Subelliptic Operators on Lie Groups I: Free Nilpotent Groups (with A
FMterElst)PotAn3(1994)283–387.
[98] SubcoercivityandSubellipticOperatorsonLieGroupsII:TheGeneralCase(withAFM
terElst)PotAn4(1995)205–243.
[99] Subcoercive and Subelliptic Operators on Lie Groups: Variable Coefficients (with A F M
terElst)PublRIMSKyotoUniv5(1993)745–801.
[100] Lp-regularityofSubellipticOperatorsonLieGroups(withRJBurnsandAFMterElst)
JOpTheor31(1994)165–187.
[101] Functional Analysis of Subelliptic Operators on Lie Groups (with A F M ter Elst) J Op
Theor31(1994)277–301.
[102] Asymptotics of Periodic Subelliptic Operators (with C J K Batty, O Bratteli and P E T
Jørgensen)JGeomAnal5(1995)427–443.
[103] Weighted Strongly Elliptic Operators on Lie Groups (with A F M ter Elst) J Funct Anal
125(1994)548–603.
[104] OnPositiveRocklandOperators(withPAuscherandAFMterElst)ColloqMathLXVII
(1994)197–216.
[105] SpectralEstimatesforPositiveRocklandOperators(withAFMterElst)Algebraicgroups
and Lie groups (Ed G I Lehrer), Aus Math Soc Lecture Series 9. Cambridge Univ Press
(1997).
[106] ReducedHeatKernelsonNilpotentLieGroups(withAFMterElst)CommunMathPhys
173(1995)475–511.
[107] Semigroup Kernels, Poisson bounds and Holomorphic Functional Calculus (with X T
Duong)JFunctAn142(1996)89–129.
[108] Weighted Subcoercive Operators on Lie Groups (with A F M ter Elst) J Funct Anal 156
(1998)88–163.
[109] OnKato’sSquareRootProblem(withAFMterElst)HokkaidoMathJ26(1997)1–12.
[110] AnalyticElementsofLieGroups(withAFMterElst)HelvPhysActa69(1996)655–678.
[111] AbundanceofInvariantandAlmostInvariantPureStatesofC∗-dynamicalSystems(with
OBratteliandAKishimoto)CommunMathPhys187(1997)491–507.
[112] Second-orderSubellipticOperatorsonLieGroupsI:ComplexUniformlyContinuousPrin-
cipalCoefficients(withAFMterElst)ActaApplMath59(1999)299–331.
[113] HighOrderDivergence-formEllipticOperatorsonLieGroups(withAFMterElst)Bull
AusMathSoc55(1997)335-348.
[114] Second-order Subelliptic Operators on Lie Groups II: Real Measurable Coefficients (with
A F M ter Elst) Progess in Nonlinear Differential Equations 42 Birkh¨auser Verlag (2000)
103–124.
[115] Second-order Strongly Elliptic Operators on Lie Groups with H¨older Continuous Coeffi-
cients.(withAFMterElst)JAusMathSoc(SeriesA)63 (1997)297–363.
[116] HeatKernelsandRieszTransformsonNilpotentLieGroups.(withAFMterElstandA
Sikora)ColloqMathLXXIV(1997)191–218.
[117] Second-orderSubellipticOperatorsonLieGroupsIII:H¨olderContinuousCoefficients(with
AFMterElst)CalcVarPDE8(1999),327–363.
[118] SpectralAsymptoticsofPeriodicEllipticOperators(withOBratteliandPETJørgensen)
MathZ232(1999),621–650.
[119] LocalLowerBoundsonHeatKernels(withAFMterElst)Positivity2(1998)123–151.
[120] Riesz Transforms and Lie Groups of Polynomial Growth. (with A F M ter Elst and A
Sikora)JFunctAnal162(1999),14–51.
vii
[121] AsymptoticsofSemigroupsonNilpotentLieGroups(withNDungey,AFMterElstand
ASikora)JOpTheor(toappear)
[122] AsymptoticsofSumsofSubcoerciveOperators(withNDungeyandAFMterElst)Colloq
Math82(1999)231–260.
[123] Onsecond-orderPeriodicEllipticOperatorsinDivergenceForm(withAFMterElstand
AdamSikora)MathZ(toappear)
[124] On Anomalous Asymptotics of Heat Kernels (with A F M ter Elst) Evolution equations
and their applications to Physical and Life Sciences, Lecture Notes in Pure and Applied
Mathematics,Vol.215,(2001)89–103,Eds.GLumer,LWeis,MarcelDekker,NewYork
[125] OnSecond-orderAlmost-periodicEllipticOperators(withNDungeyandAFMterElst)
JLondonMathSoc(submitted)
[126] On Anomalous Asymptotics of Heat Kernels on Groups of Polynomial Growth (with N
DungeyandAFMterElst)CompMath(submitted)
Lecture notes, reviews and conference papers
[1] Algebraic Aspects of Relativistic Field Theory - Brandeis Lectures, Gordon and Breach,
NewYork,Vol1(1966).
[2] Symmetries, Broken Symmetries, Charges and Currents - Istanbul Lecture, Freeman and
Co,SanFrancisco,(1967)463–513.
[3] ConservedCurrentsandBrokenSymmetries-ColloqueduCNRS(1966),EdsduCNRS15
(1968)131–135.
[4] Physical States and Finiteness Restrictions - Proceeding of the CNRS Conference at Gif-
sur-Yvette,CNRS(1969).
[5] ExistenceTheoremsinQuantumStatisticalMechanics-ProceedingsoftheCargeseSummer
School(notesbyRLimaandSMiracle)GordonandBreach(1970).
[6] Approach and Return to Equilibrium of Free Fermi States - Colloquium of the American
MathSoc1971-SiamAMSProceeding555(1972).
[7] C∗-Algebras and Quantum Statistical Mechanics - Int School of Physics Enrico fermi
Varenna-EditriceCompBologna(1975)235–252.
[8] Time Dependent Scattering Theory III - Int Colloq on Group Theoretical Methods, Mar-
seille(1974).
[9] Unbounded Derivations of C∗-Algebras - Int Symp on Math Problems in Theor Phys,
Springer-Verlag(1975).
[10] Dynamics in Quantum Statistical Mechanics - Int Summer School Bielefeld 1976, Plenum
Press(1978).
[11] ReviewofDerivations-IntConferenceonMathematicalPhysics,Lausanne1979,Springer-
Verlag(1980).
[12] Commutator Theory and Partial Differential Operators on Hilbert Space - Proceedings of
theCentreofMathematicalAnalysis,ANU,Canberra14(1986)295–302.
[13] Differential Operators on C∗-algebras - Contemporary Mathematics, Amer Math Soc 62
(1987)367–384.
[14] C∗-algebras and a Single Operator - Surveys of some recent results in operator theory II
PitmanResearch171(1988)235–266.
[15] Integration of Lie Algebras - Proceedings of the Centre of Mathematical Analysis, ANU,
Canberra15(1987)255–278.
[16] Schauder Estimates on Lie Groups- Proceedings of the Centre of Mathematical Analysis,
ANU,Canberra24(1989)1–9.
[17] SubellipticOperatorsonLieGroups(withAFMterElst)-ProceedingsoftheCentreof
MathematicalAnalysis,ANU,Canberra29(1992)63–72.
[18] Strongly Elliptic and Subelliptic Operators on Lie Groups - Quantum and Non-
CommutativeAnalysis,EdsHArakietal.,KluwerAcademicPublishers(1993)435–453
[19] Basic Semigroup Theory - Proceedings of the Centre of Mathematical Analysis, ANU,
Canberra34–PartIII(1996)1–33.
[20] EllipticOperatorsonLieGroups(withAFMterElst)ActaAppl.Math.44(1996)133–
150.
[21] C∗-AlgebrasandMathematicalPhysics-FieldsInstituteMonographvolume,EdsRBhat,
GElliottandPFillmore,AmericanMathematicalSociety,Providence,RhodeIsland.
viii
ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO
NONLINEAR PDE
HILARY BOOTH
Abstract. TheMaxwell-Diracequationsgiveamodelofanelec-
tron in an electromagnetic (e-m) (cid:12)eld, in which neither the Dirac
or the e-m (cid:12)elds are quantized. The two equations are coupled
via the Dirac current which acts as a source in the Maxwell equa-
tion,resultinginanonlinearsystemofpartialdi(cid:11)erentialequations
(PDE’s). In this way the self-(cid:12)eld of the electron is included.
We reviewourresults todateand givethefourrealconsistency
conditions (one of which is conservation of charge) which apply
to the components of the wavefunction and its (cid:12)rst derivatives.
These must be met by any solutions to the Dirac equation. These
conditions prove to be invaluable in the analysis of the nonlinear
system, and generalizable to higher dimensional supersymmetric
matter.
Inearlierpapers,wehaveshownanalyticallythatinanisolated
stationarysystem, thesurroundingelecton(cid:12)eldmustbeequaland
opposite to the central (external) (cid:12)eld. The nonlinearity forces
electric neutrality, at least in the static case. We illustrate these
properties with a numerical family of orbits which occur in the
(static) spherical and cylindrical ODE cases. These solutions are
highly localized and die o(cid:11) exponentially with increasing distance
from the central charge.
1. The Maxwell-Dirac Equations and QED
The coupled Maxwell-Dirac equations can be written as follows:
(cid:11)
(cid:13) (@(cid:11) (cid:0)ieA(cid:11)) +im = 0 where (cid:11) = 0;::: ;3
F(cid:11)(cid:12) = A(cid:12);(cid:11) (cid:0)A(cid:11);(cid:12)
(cid:11)
@ F(cid:11)(cid:12) = (cid:0)4(cid:25)ej(cid:12)
(1) where j(cid:11) = (cid:13)(cid:11) :
4
Note that 2 C is the Dirac wave-function or 4-spinor and the
(cid:11)
Dirac conjugate. These are acted upon by (cid:13) which are the usual
gamma matrices (representations of a Cli(cid:11)ord algebra), and A(cid:11) is the
4-potential.
These equations model an electron in an electromagnetic (cid:12)eld. The
two equations are coupled via the Dirac current j(cid:11) i.e. we include
1
2 HILARY BOOTH
the nonlinearity of the self-(cid:12)eld, see Equation (1). These equations
form the foundation of quantum electrodynamics (QED), the theory
of electrons interacting with (cid:12)elds. QED is one of the most successful
physical theories, explaining the Lamb shift and the anomalous mag-
netic moment of the electron. The calculationsof QED are acheived by
quantizing the (cid:12)elds and using perturbation theory. In so doing, well-
known mathematical problems occur, which have yet to be resolved at
the fundamental level.
It is possible that the full nonlinearized equations must be analysed
rigorously before we can hope to resolve these deep problems. For
example, in [23], Lieb and Loss observe that the stability of matter re-
quires that the electron be de(cid:12)ned with a Dirac operator with the mag-
netic vector potential A(x), instead of the free Dirac operator (without
A(x)). That perturbation theory must start from the dressed electrons
(including their own self-(cid:12)eld) might be \fundamentally important in
a non-perturbative QED".
This view is shared by many analysts working on this problem in-
cluding Flato, Simon and Ta(cid:13)in who established global existence for
the M-D equations as recently as 1997 [18], following many years of
sustained interest [21] [9] [20] [19] in the problem. In [18], Flato et al
showed that the nonlinear representation is integrable to a global non-
linear representation of the Poincar(cid:19)e group on a di(cid:11)erential manifold,
U1 of small initial conditions. This established the existence of global
solutions for initial data in U1 at t = 0. They go on to show that the
asymptotic representations are also nonlinear and draw conclusions for
the infrared tail of the electron. Their results show that \in the clas-
sical case (also), one obtains infrared divergencies, if one requires free
asymptotic (cid:12)elds as it is needed in QED". In other words, the (cid:12)elds
must remain coupled via the self-(cid:12)eld if we are to resolve the infrared
problem.
In the case where we assume that the system isstatic and/or station-
ary (see Section 2), we can make some simpli(cid:12)cations. The system is
ellipticinthe stationarycase (notimedependence). Esteban, Georgiev
and S(cid:19)er(cid:19)e showed existence of soliton-like solutions (that is, solutions
which are spatially localized) in this case [16]. Furthermore, the wave-
function, together with all its derivatives decreases exponentially at
in(cid:12)nity.
2. Some simplified versions of the problem
The full 4-dimensional nonlinear problem is somewhat intractable
{ as stated in Section 1, global existence has only been established
recently [18] (and references therein). If we want to get some idea of
