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CAUSALITY, MEASUREMENT THEORY AND
THE DIFFERENTIABLE STRUCTURE
OF SPACE-TIME
Introducing graduate students and researchers to mathematical physics, this
book discusses two recent developments: the demonstration that causality can
be defined on discrete space-times; and Sewell’s measurement theory, in which
the wave packet is reduced without recourse to the observer’s conscious ego,
nonlinearities or interaction with the rest of the universe.
Thedefinitionofcausalityonadiscretespace-timeassumesthatspace-timeis
made up of geometrical points. Using Sewell’s measurement theory, the author
concludes that the notion of geometrical points is as meaningful in quantum
mechanicsasitisinclassicalmechanics,andthatitisimpossibletotellwhether
the differential calculus is a discovery or an invention. This mathematical dis-
course on the relation between theoretical and experimental physics includes
detailed accounts of the mathematically difficult measurement theories of von
Neumann and Sewell.
R.N.Sen was a Professor in the Department of Mathematics at Ben-Gurion
University, Beer-Sheva, Israel, and is now retired. His main research inter-
ests were the theory of symmetry of infinite quantum-mechanical systems and
mathematicalinvestigationsintotherelationbetweenmathematicsandphysics,
particularly the origins of the differentiable structure of space-time. He has
taught a broad spectrum of courses on physics and mathematics, as well as
demography. A Life Member of Clare Hall, Cambridge, he has been a Gauss
ProfessorinGo¨ttingenandisalsoamemberoftheInternationalAssociationfor
Mathematical Physics and of the Israel Mathematical Union.
CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS
General Editors: P. V. Landshoff, D. R. Nelson, S Weinberg
S.J.AarsethGravitationalN-BodySimulations:ToolsandAlgorithms
J.Ambjørn,B.DurhuusandT.JonssonQuantumGeometry:AStatisticalFieldTheory
Approach
A.M.AnileRelativisticFluidsandMagneto-fluids:WithApplicationsinAstrophysicsand
PlasmaPhysics
J.A.deAzc´arragaandJ.M.IzquierdoLieGroups,LieAlgebras,CohomologyandSome
ApplicationsinPhysics†
O.Babelon,D.BernardandM.TalonIntroductiontoClassicalIntegrableSystems†
F.BastianelliandP.vanNieuwenhuizenPathIntegralsandAnomaliesinCurvedSpace
V.BelinskiandE.VerdaguerGravitationalSolitons
J.BernsteinKineticTheoryintheExpandingUniverse
G.F.BertschandR.A.BrogliaOscillationsinFiniteQuantumSystems
N.D.BirrellandP.C.W.DaviesQuantumFieldsinCurvedSpace†
K.Bolejko,A.Krasin´ski,C.HellabyandM-N.C´el´erierStructuresintheUniversebyExact
Methods:Formation,Evolution,Interactions
D.M.BrinkSemi-ClassicalMethodsforNucleus-NucleusScattering†
M.BurgessClassicalCovariantFields
E.A.CalzettaandB.-L.B.HuNonequilibriumQuantumFieldTheory
S.CarlipQuantumGravityin2+1Dimensions†
P.CartierandC.DeWitt-MoretteFunctionalIntegration:ActionandSymmetries
J.C.CollinsRenormalization:AnIntroductiontoRenormalization,theRenormalizationGroup
andtheOperator-ProductExpansion†
P.D.B.CollinsAnIntroductiontoReggeTheoryandHighEnergyPhysics†
M.CreutzQuarks,GluonsandLattices†
P.D.D’EathSupersymmetricQuantumCosmology
F.deFeliceandC.J.SClarkeRelativityonCurvedManifolds
B.DeWittSupermanifolds,2ndedition†
P.G.OFreundIntroductiontoSupersymmetry†
Y.FrishmanandJ.SonnenscheinNon-PerturbativeFieldTheory:FromTwoDimensional
ConformalFieldTheorytoQCDinFourDimensions
J.A.FuchsAffineLieAlgebrasandQuantumGroups:AnIntroduction,withApplicationsin
ConformalFieldTheory†
J.FuchsandC.SchweigertSymmetries,LieAlgebrasandRepresentations:AGraduateCourse
forPhysicists†
Y.FujiiandK.MaedaTheScalar-TensorTheoryofGravitation
J.A.H.Futterman,F.A.Handler,R.A.MatznerScatteringfromBlackHoles†
A.S.Galperin,E.A.Ivanov,V.I.OrievetskyandE.S.SokatchevHarmonicSuperspace
R.GambiniandJ.PullinLoops,Knots,GaugeTheoriesandQuantumGravity†
T.GannonMoonshinebeyondtheMonster:TheBridgeConnectingAlgebra,ModularFormsand
Physics
M.G¨ockelerandT.Schu¨ckerDifferentialGeometry,GaugeTheoriesandGravity†
C.G´omez,M.Ruiz-AltabaandG.SierraQuantumGroupsinTwo-DimensionalPhysics
M.B.Green,J.H.SchwarzandE.WittenSuperstringTheoryVolume1:Introduction†
M.B.Green,J.H.SchwarzandE.WittenSuperstringTheoryVolume2:LoopAmplitudes,
AnomaliesandPhenomenology†
V.N.GribovTheTheoryofComplexAngularMomenta:GribovLecturesonTheoreticalPhysics
J.B.GriffithsandJ.Podolsky´ExactSpace-TimesinEinstein’sGeneralRelativity
S.W.HawkingandG.F.R.EllisTheLargeScaleStructureofSpace-Time†
F.IachelloandA.ArimaTheInteractingBosonModel†
F.IachelloandP.vanIsackerTheInteractingBoson–FermionModel
C.ItzyksonandJ.M.DrouffeStatisticalFieldTheoryVolume1:FromBrownianMotionto
RenormalizationandLatticeGaugeTheory†
C.ItzyksonandJ.M.DrouffeStatisticalFieldTheoryVolume2:StrongCoupling,MonteCarlo
Methods,ConformalFieldTheoryandRandomSystems†
C.V.JohnsonD-Branes†
P.S.JoshiGravitationalCollapseandSpacetimeSingularities
J.I.KapustaandC.GaleFinite-TemperatureFieldTheory:PrinciplesandApplications,2nd
edition
V.E.Korepin,N.M.BogoliubovandA.G.IzerginQuantumInverseScatteringMethodand
CorrelationFunctions†
M.LeBellacThermalFieldTheory†
Y.MakeenkoMethodsofContemporaryGaugeTheory
N.MantonandP.SutcliffeTopologicalSolitons†
N.H.MarchLiquidMetals:ConceptsandTheory
I.MontvayandG.Mu¨nsterQuantumFieldsonaLattice†
L.O’RaifeartaighGroupStructureofGaugeTheories†
T.Ort´ınGravityandStrings†
A.M.OzoriodeAlmeidaHamiltonianSystems:ChaosandQuantization†
L.ParkerandD.J.TomsQuantumFieldTheoryinCurvedSpacetime:QuantizedFieldsand
Gravity
R.PenroseandW.RindlerSpinorsandSpace-TimeVolume1:Two-SpinorCalculusand
RelativisticFields†
R.PenroseandW.RindlerSpinorsandSpace-TimeVolume2:SpinorandTwistorMethodsin
Space-TimeGeometry†
S.PokorskiGaugeFieldTheories,2ndedition†
J.PolchinskiStringTheoryVolume1:AnIntroductiontotheBosonicString
J.PolchinskiStringTheoryVolume2:SuperstringTheoryandBeyond
V.N.PopovFunctionalIntegralsandCollectiveExcitations†
R.J.RiversPathIntegralMethodsinQuantumFieldTheory†
R.G.RobertsTheStructureoftheProton:DeepInelasticScattering†
C.RovelliQuantumGravity†
W.C.SaslawGravitationalPhysicsofStellarandGalacticSystems†
R.N.SenCausality,MeasurementTheoryandtheDifferentiableStructureofSpace-Time
M.ShifmanandA.YungSupersymmetricSolitons
H.Stephani,D.Kramer,M.MacCallum,C.HoenselaersandE.HerltExactSolutionsof
Einstein’sFieldEquations,2ndedition
J.StewartAdvancedGeneralRelativity†
T.ThiemannModernCanonicalQuantumGeneralRelativity
D.J.TomsTheSchwingerActionPrincipleandEffectiveAction
A.VilenkinandE.P.S.ShellardCosmicStringsandOtherTopologicalDefects†
R.S.WardandR.O.Wells,JrTwistor GeometryandFieldTheory†
J.R.WilsonandG.J.MathewsRelativisticNumericalHydrodynamics
† Issuedasapaperback
Causality, Measurement Theory and
the Differentiable Structure
of Space-Time
RATHINDRA NATH SEN
Ben-Gurion University, Beer-Sheva
CAMBRIDGEUNIVERSITYPRESS
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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Information on this title: www.cambridge.org/9780521880541
© R. Sen 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2010
ISBN-13 978-0-511-67354-2 eBook (EBL)
ISBN-13 978-0-521-88054-1 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
To the memory of my father, mother, brother and sister
Upendra Nath Sen
Mukul Sen
Tejendra Nath Sen
Aditi Datta Chowdhury
