Transcription of Unifying Foundations for Physics and Mathematics
1 Unifying Foundations for Physics and MathematicsPeter WoitColumbia UniversityMathematics DepartmentParis, Oct. 30, 2021 Peter Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 20211 / 13 Outline1 Unification in Physics2 Unification in Mathematics3 Unifying Math and Physics4 Last ThoughtsNote: These slides ~ Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 20212 / 13 Unification in PhysicsGeneral RelativityThe gravitational force is described by theGeneral theory of Relativity (1915)Dynamical variables: geometry of space-timeClassical dynamical equations: Einstein equations for metricThe Mathematics used is(Pseudo)-Riemannian geometryRiemann (1854).
2 First version of geometry with local curvatureCartan/Ehresmann (1923-1943): general framework for differentialgeometry in terms of connections and curvaturePeter Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 20213 / 13 Unification in PhysicsThe Standard ModelMatter and the other known forces (electromagnetic, weak, strong) aredescribed by:The Standard Model (1973)Dynamical variables: connections, spinor fields, scalar field (Higgs)Classical dynamical equations: Yang-Mills equations, Dirac equation,Wave equationThe Mathematics used isGeometry of connections, curvature and spinorsSame Cartan/Ehresmann (1923-1943) geometry formalism in terms ofconnections and curvature as for (1913), Weyl (1929): spinor geometryActually want quantization of the Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 20214 / 13 Unification in PhysicsThe problem with the unified theoryMain problem with the unified theoryIt s too good.
3 No clear examples of data in conflict with the theoryHuge amount of attention devoted to dark matter as a potentialconflict, and to quantization of gravity as consistency problem, but littleprogress from this since unified theory came together in guidance from experiment, can one do better than currentsituation?Currently popular point of view:No, the unified theory is just anenvironmental artifact of the multiverseA different point of view:The unified theory is a very specialmathematical structure, can we understand the significance of this?Peter Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 20215 / 13 Unification in MathematicsMathematics as separate fieldsConventionally, Mathematics is divided into different fields, each of whichramifies into more specialized theoryThis is still how the subject is subfields share a foundation in set theory /mathematical commonly noticed are Unifying principles.
4 For example: algebra andgeometry are related by thinking of an algebra as some sort of functions onsome sort of geometrical space. In studying a specialized subfield oneoften runs into a connection to another one seemingly far Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 20216 / 13 Unification in MathematicsUnification: the Atiyah-Singer Index TheoremSome Unifying themes in Mathematics :Th Atiyah-Singer index theorem (1963) relatesAnalysis (solutions to differential equations in background fields on amanifold)Topology (of the manifold and data describing the equation andbackground fields)Geometry (curvature of the manifold and background fields)Peter Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 20217 / 13 Unification in MathematicsUnification: Representation TheoryA well-known Unifying principle in Mathematics :SymmetryUsually presented in terms of theory of groups, but more accurately it isabout groups and what they act on.
5 Representation theory is a mathematical field which cuts across the usualsubfields, often providing a Unifying principle tying together unifies with the Atiyah-Singer index theorem in an equivariant versionof index Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 20218 / 13 Unification in MathematicsUnification: the Langlands ProgramThe original Langlands program (1967) unifies number theory andrepresentation theory , relating galois groups of number fields and theirrepresentations with Lie groups and their unification, with geometry, came from the geometric Langlandsprogram (1980s-90s). This is currently a very active field of Frenkel refers to this as A Grand Unified theory ofMathematics.
6 Peter Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 20219 / 13 Unifying Math and PhysicsUnifying the UnificationsEach of the above Unifying themes in Mathematics has significantconnections to the unified theory in Physics :Atiyah-Singer index theoremDifferential operators are characterised by their relation to a generator: theDirac operator. The theorem can then be understood in terms of thephysics of a single elementary spin 1/2 theoryStudy of symmetry in quantum theory is essentially same Mathematics astheory of unitary ProgramGeometric Langlands can be understood in terms of (twisted)N= 4supersymmetric Yang-Mills theory (Witten-Kapustin), similar to Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 202110 / 13 Unifying Math and PhysicsRecent DevelopmentsA very recent major development (Feb.)
7 2021) Unifying number theoryLanglands and geometric Langlands:Fargues-Scholze :Geometrization of the local Langland correspondenceIn number theory , one thinks of the primes as the points of a space (calledSpecZ), and local Langlands is the version of the Langlandsprogram that describes what happens near a primep. Fargues-Scholzeshow this can be thought of as geometric Langlands on a curve, the Fargues-Fontaine can think ofSpecZ as also having a point corresponding not to aprime, but to the real numbers R. The Fargues-Fontaine curve at the realpoint is the so-called twistor P1 : the sphere with its antipodal theory : a compelling way of thinking about four-dimensionalgeometry and conformally invariant Physics in four-dimensions.
8 In twistortheory, a point in four-dimensional space is exactly a twistor this an aspect of deep unification? Is it a random coincidence?Peter Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 202111 / 13 Unifying Math and PhysicsImplicationsPhilosophy of Mathematics : A different perspective on Foundations ofmathematics , which are usually thought of as having nothing to do withphysics. On the Is Mathematics discovered or invented? question, thistakes the discovered (like Physics ) implications for Physics : search for an improved unifiedtheory should concentrate on current ideas that unify deeply withmathematics, together with new ideas with this property.
9 Very unlikely tolead anywhere: the idea that the deep geometry of the Standard Model isjust a long-distance effective artifact of something fundamentally implications for Mathematics : fundamental physicswill remain an ongoing inspiration for new deep mathematical difficult problem: Current doctoral education in theoretical physicsgives no training in the Mathematics needed to appreciate a fundamentalunity of the two subjects. Similarly for doctoral education in researchers get the training needed to effectively use the insights ofboth Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 202112 / 13 Last ThoughtsLast ThoughtsAfter spending many years among both physicists and mathematicians, Ihave some idea what many of them are likely to think of typical physicist s point of view:this is mysticism.
10 Where s thecalculation?A typical mathematician s point of view:this is vague andmeaningless. Where s the theorem ?Some reasons to worry:I get a lot of emails from people who have a new unified re mostly quite clearly oneself by seeing deep connections in unrelated events is acommon human yet, this sort of unity seems to me very real, since early days as astudent, more so the more I ve learnt over the Woit (Columbia University Mathematics Department) Unifying Foundations for Physics and MathematicsOctober 202113 / 13
