The geometrical semantics of algebraic quantum mechanics

Transcription

1 The geometrical semantics of algebraic quantum mechanics B. Zilber University of Oxford November 29, 2016 B.Zilber, The semantics of the canonical commutation relations arxiv.org/abs/ The geometrical semantics of algebraic quantum mechanics 1 / 20

2 Geometric dualities Affine commutative C-algebra R = C[X 1,..., X n ]/I Commutative unital C -algebra A Affine reduced k-algebra R = k[x 1,..., X n ]/I... Complex algebraic variety V R with relations given by polynomial equations. Compact topological space V A with continuous functions into C corresponding to elements of A The geometry of k-definable points, curves etc of an algebraic variety V R... The geometrical semantics of algebraic quantum mechanics 2 / 20

3 Relation to Logic These are syntax semantics dualities. The geometrical semantics of algebraic quantum mechanics 3 / 20

4 Relation to Logic These are syntax semantics dualities. In general the syntax may come with a topology! (as in C -algebras). The geometrical semantics of algebraic quantum mechanics 3 / 20

5 What is a geometric semantic? The geometrical semantics of algebraic quantum mechanics 4 / 20

6 What is a geometric semantic? A Suggestion. The structures on the right hand side of dualities must be model-theoretically stable (Morley, Shelah and others, in a reasonably generalised sense). The geometrical semantics of algebraic quantum mechanics 4 / 20

7 What is a geometric semantic? A Suggestion. The structures on the right hand side of dualities must be model-theoretically stable (Morley, Shelah and others, in a reasonably generalised sense). There also should be a topology associated in a natural way with the syntax. The geometrical semantics of algebraic quantum mechanics 4 / 20

8 What is a geometric semantic? A Suggestion. The structures on the right hand side of dualities must be model-theoretically stable (Morley, Shelah and others, in a reasonably generalised sense). There also should be a topology associated in a natural way with the syntax. Perhaps in the spirit of continuous or/and positive model theory (Keisler, Ben Yaacov, Usviatsov and others) The geometrical semantics of algebraic quantum mechanics 4 / 20

9 What is a geometric semantic? A Suggestion. The structures on the right hand side of dualities must be model-theoretically stable (Morley, Shelah and others, in a reasonably generalised sense). There also should be a topology associated in a natural way with the syntax. Perhaps in the spirit of continuous or/and positive model theory (Keisler, Ben Yaacov, Usviatsov and others) The initial setting can be provided by the notion of Zariski geometry (Hrushovski and Z. 1993). The geometrical semantics of algebraic quantum mechanics 4 / 20

10 Zariski geometries. The structure V = (V ; L) (a universe V and a language L) with a topology on its cartesian powers is said to be Zariski if it satisfies Closed subsets of V n are exactly those which are L-positive-quantifier-free definable. The projection of a closed set is constructible (positive quantifier-elimination). A good dimension notion on closed subsets is given.... The geometrical semantics of algebraic quantum mechanics 5 / 20

11 Zariski geometries. The structure V = (V ; L) (a universe V and a language L) with a topology on its cartesian powers is said to be Zariski if it satisfies Closed subsets of V n are exactly those which are L-positive-quantifier-free definable. The projection of a closed set is constructible (positive quantifier-elimination). A good dimension notion on closed subsets is given.... Theorem (1994) Noetherian Zariski geometries are stable of finite Morley rank. The geometrical semantics of algebraic quantum mechanics 5 / 20

12 A Noncommutative Duality Theorem The classical duality can be extended to non-commutative geometry at root of unity ( algebras with large Azumaya locus). A V V A. N = dimension of the maximal indecomposable module = order of the root of unity. The geometrical semantics of algebraic quantum mechanics 6 / 20

13 A non-commutative example at root of unity Non-commutative 2-torus V A at ɛ = e 2πi m N has co-ordinate ring A = U, V : U = U 1, V = V 1, UV = ɛvu The geometrical semantics of algebraic quantum mechanics 7 / 20

14 A non-commutative example at root of unity A Non-commutative 2-torus V A at ɛ = e 2πi m N has co-ordinate ring A = U, V : U = U 1, V = V 1, UV = ɛvu Points α on the torus have structure of an N-dim Hilbert space V α with a distinguished system of canonical orthonormal bases The geometrical semantics of algebraic quantum mechanics 7 / 20

15 Further geometric dualities Affine commutative C-algebra R Commutative C -algebra A k-scheme of finite type S C -algebra A at roots of unity Complex algebraic variety V R Compact topological space V A The k-definable structure on a Zariski geometry V S (?) Zariski geometry V A Weyl-Heisenberg algebra Q, P : QP PQ = i The integers Z The geometrical semantics of algebraic quantum mechanics 8 / 20

16 Further geometric dualities Affine commutative C-algebra R Commutative C -algebra A k-scheme of finite type S C -algebra A at roots of unity Weyl-Heisenberg algebra Q, P : QP PQ = i The integers Z Complex algebraic variety V R Compact topological space V A The k-definable structure on a Zariski geometry V S (?) Zariski geometry V A?? The geometrical semantics of algebraic quantum mechanics 8 / 20

17 QP PQ = i This canonical commutation relation unfolds into the full-fledged quantum mechanics. The geometrical semantics of algebraic quantum mechanics 9 / 20

18 QP PQ = i This canonical commutation relation unfolds into the full-fledged quantum mechanics. Remark. This does not allow the C -algebra (Banach algebra) setting. The geometrical semantics of algebraic quantum mechanics 9 / 20

19 QP PQ = i On suggestion of Herman Weyl and following the Stone von Neumann Theorem replace the Weyl-Heisenberg algebra by the category of Weyl -algebras A a,b = U a, V b : U a V b = e iab V b U a, a, b R, U a = e iaq, V b = e ibp. where it is also assumed that U a and V b are unitary. The geometrical semantics of algebraic quantum mechanics 10 / 20

20 QP PQ = i On suggestion of Herman Weyl and following the Stone von Neumann Theorem replace the Weyl-Heisenberg algebra by the category of Weyl -algebras A a,b = U a, V b : U a V b = e iab V b U a, a, b R, U a = e iaq, V b = e ibp. where it is also assumed that U a and V b are unitary. We may assume that 2π Q and so, when a, b Q the algebra A a,b is at root of unity. We call such algebras rational Weyl algebras. The geometrical semantics of algebraic quantum mechanics 10 / 20

21 Étale sheaf of Zariski geometries over the category of rational Weyl algebras The category A fin has objects A a,b, rational Weyl algebras, and morphisms = embeddings: A a,b A c,d The geometrical semantics of algebraic quantum mechanics 11 / 20

22 Étale sheaf of Zariski geometries over the category of rational Weyl algebras The category A fin has objects A a,b, rational Weyl algebras, and morphisms = embeddings: A a,b A c,d This corresponds to surjective morphism in the dual category V fin of Zariski geometries V Aa,b V Ac,d. The geometrical semantics of algebraic quantum mechanics 11 / 20

23 Étale sheaf of Zariski geometries over the category of rational Weyl algebras The category A fin has objects A a,b, rational Weyl algebras, and morphisms = embeddings: A a,b A c,d This corresponds to surjective morphism in the dual category V fin of Zariski geometries V Aa,b V Ac,d. Note, that A a,b A c,d can be rephrased in terms of divisibility between numerators and denominators of a, b, c, d. The geometrical semantics of algebraic quantum mechanics 11 / 20

24 Limit objects of A fin and V fin. Determining the limits on the geometric side is the main difficulty of the project. The geometrical semantics of algebraic quantum mechanics 12 / 20

25 Limit objects of A fin and V fin. Determining the limits on the geometric side is the main difficulty of the project. An analogy: define R from Q. We use structural approximation, The geometrical semantics of algebraic quantum mechanics 12 / 20

26 Limit objects of A fin and V fin. Determining the limits on the geometric side is the main difficulty of the project. An analogy: define R from Q. We use structural approximation, a construction within postive (continuos) model theory. The nonstandard universal object A of A fin is the algebra A := A 1 µ, 1 ν = U 1 µ, V 1 ν, µ, ν Z, µ and ν nonstandard integers divisible by any standard ones. The geometrical semantics of algebraic quantum mechanics 12 / 20

27 Limit objects of A fin and V fin. Determining the limits on the geometric side is the main difficulty of the project. An analogy: define R from Q. We use structural approximation, a construction within postive (continuos) model theory. The nonstandard universal object A of A fin is the algebra A := A 1 µ, 1 ν = U 1 µ, V 1 ν, µ, ν Z, µ and ν nonstandard integers divisible by any standard ones. Lemma. A A a,b for all a, b Q iff µ and ν divisible by all standard integers. The geometrical semantics of algebraic quantum mechanics 12 / 20

28 More precisely we choose µ and ν so that the Planck constant. µ ν h = 2π, The geometrical semantics of algebraic quantum mechanics 13 / 20

29 More precisely we choose µ and ν so that the Planck constant. µ ν h = 2π, The non-standard projective object of V fin is the nonstandard geometric object V A, corresponding to the algebra A. The geometrical semantics of algebraic quantum mechanics 13 / 20

30 More precisely we choose µ and ν so that the Planck constant. µ ν h = 2π, The non-standard projective object of V fin is the nonstandard geometric object V A, corresponding to the algebra A. Lemma. For every A a,b with a, b Q, there is a surjective morphism V A V Aa,b. The geometrical semantics of algebraic quantum mechanics 13 / 20

31 Noncommutative V A deforms into V A A A The geometrical semantics of algebraic quantum mechanics 14 / 20

32 The space of states The space of states S is defined as a factor of an observable part of V A (defined algebraically) by an equivalence relation x 1 x 2 meaning the two elements are indiscernible. E.g. two eigenvectors of norm 1 with infinitesimally close eigenvalues are indiscernible. The geometrical semantics of algebraic quantum mechanics 15 / 20

33 The space of states The space of states S is defined as a factor of an observable part of V A (defined algebraically) by an equivalence relation x 1 x 2 meaning the two elements are indiscernible. E.g. two eigenvectors of norm 1 with infinitesimally close eigenvalues are indiscernible. The language (relations) on S is determined by operators from A which survive the passage to S. These turn our to coincide with observables of physics. The geometrical semantics of algebraic quantum mechanics 15 / 20

34 The space of states S is described in terms of standard real and complex numbers and can effectively be identified with the conventional representation of CCR over a symplectic space with (quadratic) time evolution operators acting as symplectomorphism. The geometrical semantics of algebraic quantum mechanics 16 / 20

35 Example. Time evolution operator for the quantum harmonic oscillator. K t = K t QHOsc := e i P2 +Q 2 t, t R. The geometrical semantics of algebraic quantum mechanics 17 / 20

36 Example. Time evolution operator for the quantum harmonic oscillator. K t = K t QHOsc := e i P2 +Q 2 t, t R. The geometrical semantics of algebraic quantum mechanics 17 / 20

37 Example. Time evolution operator for the quantum harmonic oscillator. K t = K t QHOsc := e i P2 +Q 2 t, t R. The matrix element on row x 1 and column x 2 ( kernel of the Feynman propagator) is calculated as 1 x 1 K t x 2 = 2πi sin t exp i (x x 2 2) cos t 2x 1x 2. 2 sin t The trace of K t, Tr(K t ) = R x K t x = 1 sin t. 2 The geometrical semantics of algebraic quantum mechanics 17 / 20

38 Note that in terms of conventional mathematical physics we have calculated Tr(K t ) = a non-convergent infinite sum. e it(n+ 1 2 ), n=0 The geometrical semantics of algebraic quantum mechanics 18 / 20

39 Summary The semantics of the algebraic quantum mechanics presents the conventional space of states as a limit of finite spaces (and in fact suggests that the true space of states is a huge finite space); It explains the mathematical nature of physics observables; It provides a mathematical interpretation to quantum probabilities as the measure of symmetry. It replaces calculations by integrals with calculations by finite sums (such as Gauss quadratic sums). The geometrical semantics of algebraic quantum mechanics 19 / 20