On model theory, non-commutative geometry and physics B. Zilber University of Oxford http://www.maths.ox.ac.uk/ zilber/ 1
Plan I. Generalities on physics, logical hierarchy of structures and Zariski geometries. II. Structural approximation. III. Zariski geometries as noncommutative spaces. IV. Crush course in physics. V. Zariski geometries for quantum Hamiltonian systems. VI. Time evolution and Feynman propagator for some Hamiltonians. 2
First question: in what sense the (mathematical) physics we study reflects the real universe? If our mathematical models are approximations to reality, then in what sense approximation? A plausible answer to the first part of the question: the real universe is a structure, say M, that is a domain (set of points, events, particles,...) with some relations R(x 1,..., x n ) between its elements. We want to distinguish relations defining (topologically) closed sets. 3
First question: in what sense the (mathematical) physics we study reflects the real universe? If our mathematical models are approximations to reality, then in what sense approximation? A plausible answer to the first part of the question: the real universe is a structure, say M, that is a domain (set of points, events, particles,...) with some relations R(x 1,..., x n ) between its elements. We want to distinguish relations defining (topologically) closed sets. Note, this is not the geometer s point of view. 4
What is an idealised model of such a real universe. Another structure M such that (i) they look similar, i.e. many statements true in M also true in M; (ii) close points in M may be indistinguishable in M. 5
What is an idealised model of such a real universe. Another structure M such that (i) they look similar, i.e. many statements true in M also true in M; (ii) close points in M may be indistinguishable in M. A possible formalisation of this: M {M i : i I} and for some ultrafilter U on I there is lim U, a surjective homomorphism (preserves all primitive relations) lim : M i /U M. U i We say in this case that the sequence of structures M i approximates M. 6
Structural approximation generalises algebro - geometric deformation and metric approximation, e.g. Gromov-Hausdorff limit of metric spaces 7
Second question: what property of the real M allows laws of physics? why do we hope that a few laws of physics can describe M? Philosophers call this property algorithmic compressibility. In model theory we have a corresponding notion categoricity in uncountable powers: very large structure M describable uniquely by its concise (countable) first order theory (and the cardinal number measuring its size). 8
Uncountable categoricity of M has strong structural consequences (stability theory). In combination with the topological assumption on M we come to the definition of a Zariski Geometry: 9
Zariski geometries Let M be a structure given with a family of basic relations (subsets of M n ) called closed. We postulate for a Zariski geometry M: 10
Closed subsets form a Noetherian Topology Dimension is assigned to any closed S M n Completeness: Projections of closed are closed Addition formula: dim S = dim pr(s) + for any closed irreducible S. min a pr(s) dim(pr 1 (a) S) Pre-smoothness: For any closed irreducible S 1, S 2 M n, dim S 1 S 2 dim S 1 + dim S 2 dim M n in each component. 11
Known Zariski geometries 1. Smooth complete algebraic varieties over an algebraically closed field, in the natural language (1990). 2. Compact complex manifolds, in the natural language (1993). 3. Solution spaces of well-defined systems of (partial) differential equations. (2001) 4. New (non-commutative) geometries (1991-...). 12
Classification Theorem (Hrushovski, Z. 1993 and its later (Z. 2003-...) extensions. A typical Zariski geometry (of dimension 1) is a noncommutative finite cover of an algebraic variety (of dimension 1). 13
2 M C M 14
2 M C M 15
2 M C M 16
2 M C M 17
2 M C M 18
2 M C M 19
Classification theorem: there may be hidden relations that can not be detected by coordinate functions. 20
Classification theorem: there may be hidden relations that can not be detected by coordinate functions. How to recover the hidden relations? 21
Classification theorem: there may be hidden relations that can not be detected by coordinate functions. How to recover the hidden relations? Use non-commutative coordinate algebra (Heisenberg 1927,...Drinfeld, Connes 1980-...) 22
Theorem(2006) Any quantum algebra A q over an algebraically closed field F at root of unity q gives rise to a Zariski geometry M q, so that A q = A(M q ). Typically, M q is non-classical, that is not an algebro-geometric object over F. In more general cases M q is a (non-noetherian) analytic Zariski geometry. M q A q. 23
Example. The Heisenberg algebra (P, Q) with defining relation QP PQ = ih together with the time evolution operator e ih h t has a full information about the quantum system with the Hamiltonian H = P2 2 + W (Q) W (Q) = 0 W (Q) = aq 2 free particle harmonic oscillator 24
Example continued. Let U = e iq, V = e ip, q = e ih. Then UV = qv U. A q generated by U and V well approximates the Heisenberg algebra. The M q = T 2 q (C) corresponding to A q is a 2- dimensional non-classical Zariski geometry (quantum torus). If q = ɛ = e i2π N then T 2 ɛ (C) is a Noetherian Zariski geometry. 25
u v U-eigenvector with eigenvalue u 26
u V v U-eigenvector with eigenvalue uq U-eigenvector with eigenvalue u 27
u v U-eigenvectors 28
u V -eigenvectors v 29
Theorem. For a sequence T 2 ɛ (C) to approximate T 2 q (C) it is sufficient and necessary that, (a) lim N 2π N = h and (b) given m N, for almost all N (with respect to the ultrafilter) m N. 30
Theorem. For a sequence T 2 ɛ (C) to approximate T 2 q (C) it is sufficient and necessary that, (a) lim N 2π N = h and (b) given m N, for almost all N (with respect to the ultrafilter) m N. Corollary. We may replace h by 2π N m N for all m << N. such that 31
Now we work in an irreducible (U, V )-module generated by u, v. { uq k, v : k = 0, 1... N 1} forms an orthonormal basis of U-eigenvectors. a dual basis. { vq m, u : m = 0, 1... N 1} 32
Now we work in an irreducible (U, V )-module generated by u, v. { uq k, v : k = 0, 1... N 1} forms an orthonormal basis of U-eigenvectors. { vq m, u : m = 0, 1... N 1} a dual basis of V -eigenvectors. vq m, u = 1 N uq k, v = 1 N 0 k<n 0 m<n N = dim = 2π h. q mk uq k, v q km vq m, u 33
What changes if we replace U, V by U a, V b, a, b Q? Then U a V b = q ab V b U a and the dimension N of the irreducible module change From the condition on structural approximation the new Correspondingly dim = N ab. vq abm, u = uq abk, v = ab q abmk uq abk, v N k ab q abkm vq abm, u N m 34
Time evolution for the free particle Choose t = m n evolution) Then and add one more operator (time K t = e ip2 2h t. K t V K t = V and K t UK t = q t 2V t U. We take these identities for an axiomatic definition of K t. 35
Time evolution for the free particle Choose t = m n evolution) Then and add one more operator (time K t = e ip2 2h t. K t V K t = V and K t UK t = q t 2V t U. We take these identities for an axiomatic definition of K t. Lemma K t maps the (orthonormal) system u, v of U-eigenvectors to an orthonormal system of q t 2V t U- eigenvectors. 36
K t u, 1 = c 0 t N q tk2 2 uq tk, 1 0 k< N t c 0 = 1. 37
K t u, 1 = c 0 t N q tk2 2 uq tk, 1 0 k< N t c 0 = 1. Corollary. In terms of Dirac s calculus x 1 K t ht 1 x 2)2 x 2 = c 0 ei(x 2ht. 2π This is the Feynman propagator for the free particle. 38
K t u, 1 = c 0 t N q tk2 2 uq tk, 1 0 k< N t c 0 = 1. Alternatively we can express K t u, 1 in terms of vq tm, u and then use vq tm t, u = q mk uq tk, v. N 0 k<n Thus we get another expression K t u, 1 = t q t(p2 2 pk) uq kt, 1. N k p 39
Comparing the two expressions for x 1 K t x 2, we get Gauss sum p q t(p2 2 pk) = c 0 N t qtk2 2π 2 = c 0 ht known to hold for even integers N t. 1 x 2)2 ei(x 2ht c 0 = 1 + i 2 This corresponds to the physics (non-convergent) integral calculation R e ax2 2 ibx dx = 2π a e b2 2a, a = iht 40
Time evolution for Harmonic oscillator K t = e ith h. Similar strategy can be applied. First we calculate in the usual way that K t H UK t H K t H V K t H t cos t = qsin 2 V sin t U cos t t cos t = q sin 2 V cos t U sin t 41
Time evolution for Harmonic oscillator K t = e ith h. Similar strategy can be applied. First we calculate in the usual way that K t H UK t H K t H V K t H Pick up t such that t cos t = qsin 2 V sin t U cos t t cos t = q sin 2 V cos t U sin t e = sin t, f = cos t g = ef 1, e, f, g Q We work in a (U f, V g )-system, that is the identity U f V g = q e V g U f (use e = fg) 42
Time evolution for Harmonic oscillator K t = e ith h. Similar strategy can be applied. First we calculate in the usual way that K t H UK t H K t H V K t H Pick up t such that t cos t = qsin 2 V sin t U cos t t cos t = q sin 2 V cos t U sin t e = sin t, f = cos t g = ef 1, e, f, g Q We work in a (U f, V g )-system, that is the identity U f V g = q e V g U f (use e = fg) 43
Direct calculation as above yield = c 0 h sin t 2π x 1 K t x 2 = exp i (x2 1 + x2 2 ) cos t 2x 1x 2 2h sin t c 0 = 1. Or, in a different normalisation (if we replace sums by integrals) c 0 1 2πh sin t exp i (x2 1 + x2 2 ) cos t 2x 1x 2 2h sin t Feynman propagator for the Harmonic oscillator. 1 2πh sin t quantum fluctuations, equal, by our calculation, to the dimension of the correspondent irreducible module. 44
References 1. B.Zilber, The noncommutative torus and Dirac calculus (draft). http://people.maths.ox.ac.uk/ zilber/dirac.dvi 2. B.Zilber, Notes on Approximation (draft) http://people.maths.ox.ac.uk/ zilber/approx.dvi 3. B.Zilber, A class of quantum Zariski geometries. In: Model Theory with applications to algebra and analysis, I and II (Z. Chatzidakis, H.D. Macpherson, A. Pillay, A.J. Wilkie editors), Cambridge University Press, Cambridge. 4. B.Zilber, Non-commutative Zariski geometries and their classical limit. arxiv0707.0780[math.qa] June 2007 45