Deformation Quantization

Transcription

1 FFP14 Pierre Bieliavsky (U. Louvain, Belgium) 15 Jully 2014

2 References Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.; Sternheimer, D.; Deformation theory and quantization. Ann. Physics (1978) Weinstein, Alan; Deformation quantization. Séminaire Bourbaki. Astrisque (1995) Kontsevich, Maxim ; Formality conjecture, in Deformation Theory and Symplectic Geometry, Kluwer Academic Publishers (1997) Cattaneo, Alberto S.; Felder, Giovanni; A path integral approach to the Kontsevich quantization formula. Comm. Math. Phys. (2000) Bieliavsky, Pierre; Gayral, Victor ; for actions of Kahlerian Lie groups Memoirs of the Amercian Mathematical Society (2014)

3 Matrices and triangles

4 Matrices and triangles A := M n (C)

5 Matrices and triangles A := M n (C) µ : A A A : a b a.b matrix multiplication

6 Matrices and triangles A := M n (C) µ : A A A : a b a.b matrix multiplication µ(a b) =: < K, a b > with K A A A

7 Matrices and triangles A := M n (C) µ : A A A : a b a.b matrix multiplication µ(a b) =: < K, a b > with K A A A

8 Matrices and triangles Consider n points ( configuration space ):

9 Matrices and triangles Consider n points ( configuration space ):

10 Matrices and triangles Consider n points ( configuration space ): Consider the set M of all the arrows between pairs of points ( phase space ):

11 Matrices and triangles Consider n points ( configuration space ): Consider the set M of all the arrows between pairs of points ( phase space ): Note: M = n 2 = dim C (A).

12 Matrices and triangles Consider n points ( configuration space ): Consider the set M of all the arrows between pairs of points ( phase space ): Note: M = n 2 = dim C (A).

13 Matrices and triangles Triangle: loop constituted by sequence of 3 consecutive arrows.

14 Matrices and triangles Triangle: loop constituted by sequence of 3 consecutive arrows. A triangle:

15 Matrices and triangles Triangle: loop constituted by sequence of 3 consecutive arrows. A triangle: A := M n (C) is viewed as the space of continuous functions on M ( observables ) : A = C(M).

16 Matrices and triangles Triangle: loop constituted by sequence of 3 consecutive arrows. A triangle: A := M n (C) is viewed as the space of continuous functions on M ( observables ) : A = C(M). A natural basis of A is given by the characteristic functions of arrows: { 1 if x = E( )(x) := 0 otherwise

17 Matrices and triangles Then: K = E( ) E( ) E( ).

18 Matrices and triangles Then: K = E( ) E( ) E( ). Interpretation: union of edges tensor products of characteristic functions

19 Matrices and triangles Then: K = E( ) E( ) E( ). Interpretation: union of edges tensor products of characteristic functions Union = additive operation (on sets)

20 Matrices and triangles Then: K = E( ) E( ) E( ). Interpretation: union of edges tensor products of characteristic functions Union = additive operation (on sets) tensor product = multiplicative operation

21 Matrices and triangles Then: K = E( ) E( ) E( ). Interpretation: union of edges tensor products of characteristic functions Union = additive operation (on sets) tensor product = multiplicative operation Would E( ) E( ) E( ) correspond to an exponential??

22 OK! Let s try! (Weyl-Moyal quantization)

23 OK! Let s try! (Weyl-Moyal quantization) Phase space= T (R n ) = R 2n = {x = (q, p) q, p R n }

24 OK! Let s try! (Weyl-Moyal quantization) Phase space= T (R n ) = R 2n = {x = (q, p) q, p R n } Poisson bracket = {f, g} = ω ij x i f x j g skewsymmetric tensor field: ω = dα

25 OK! Let s try! (Weyl-Moyal quantization) Phase space= T (R n ) = R 2n = {x = (q, p) q, p R n } Poisson bracket = {f, g} = ω ij x i f x j g skewsymmetric tensor field: ω = dα remind: E( ) E( ) E( )

26 OK! Let s try! (Weyl-Moyal quantization) Phase space= T (R n ) = R 2n = {x = (q, p) q, p R n } Poisson bracket = {f, g} = ω ij x i f x j g skewsymmetric tensor field: ω = dα remind: E( ) E( ) E( ) ( ) Triangle(x, y, z) exp µ ( ω Triangle(x,y,z) (µ C)

27 OK! Let s try! (Weyl-Moyal quantization) Phase space= T (R n ) = R 2n = {x = (q, p) q, p R n } Poisson bracket = {f, g} = ω ij x i f x j g skewsymmetric tensor field: ω = dα remind: E( ) E( ) E( ) ( ) Triangle(x, y, z) exp µ ( ω Triangle(x,y,z) (µ C) i.e E( ) := α

28 OK! Let s try! (Weyl-Moyal quantization) In other words, to observables a, b Cc (R 2n ), one associates: a b(x) := 1 2n K (x, y, z) a(y) b(z) dy dz where K (x, y, z) = e i (ω(x,y)+ω(y,z)+ω(z,x)) (ω(x, y) := ω ij x i y j )

29 OK! Let s try! (Weyl-Moyal quantization) In other words, to observables a, b Cc (R 2n ), one associates: a b(x) := 1 2n K (x, y, z) a(y) b(z) dy dz where K (x, y, z) = e i (ω(x,y)+ω(y,z)+ω(z,x)) (ω(x, y) := ω ij x i y j ) Asymptotics: a b a.b + k=1 ( ) 1 k ω i 1j 1... ω i kj k i k k! 2i 1...i k a j k 1...j k b

30 OK! Let s try! (Weyl-Moyal quantization) In other words, to observables a, b Cc (R 2n ), one associates: a b(x) := 1 2n K (x, y, z) a(y) b(z) dy dz where K (x, y, z) = e i (ω(x,y)+ω(y,z)+ω(z,x)) (ω(x, y) := ω ij x i y j ) Asymptotics: a b a.b + k=1 ( ) 1 k ω i 1j 1... ω i kj k i k k! 2i 1...i k a j k 1...j k b Theorem On A := C (R 2n )[[ ]], A A A : (a, b) a b is associative.

31 OK! Let s try! (Weyl-Moyal quantization) Did we know all this already??

32 OK! Let s try! (Weyl-Moyal quantization) Did we know all this already?? YES! Theorem[Weyl - von Neumann (1931)] Canonical Schrödinger quantization (Weyl ordered): Then Polynomials(R 2n ) L(L 2 (R n )) : a Op (a) Op (q j )ϕ(q) = q j ϕ(q) Op (p)ϕ(q) = i q j ϕ(q) Op (a) Op (b) = Op (a b).

33 Definition A Poisson manifold is a smooth manifold M endowed with a skewsymmetric bi-vector field w such that the associated bracket on C (M): {f, g} := w ij x i f x j g satisfies {f, {g, h}} + {h, {f, g}} + {g, {h, f }} = 0. Examples: canonical phase space: (T (N), ω Liouville ) g = Lie algebra, dual: M = g with {f, g}(x) := < x, [df x, dg x ] > ((g ) = g)

34 Definition [Bayen,Flato,Fronsdal, Lichnerowicz, Sternheimer (1977)] A star-product (or deformation quantization) on a Poisson manifold (M, w) is an associative C[[ ]]-bilinear product law on C (M)[[ ]] =: A A A A : (a, b) a b such that (i) a b = a.b + k=1 k C k (a, b) (ii) C 1 (a, b) C 1 (b, a) = i {a, b} (iii) C k = bi-differential operator on C (M) vanishing on constants.

35 Theorem [Kontsevitch (1997)] Every Poisson manifold admits a star-product.

36 Theorem [Kontsevitch (1997)] Every Poisson manifold admits a star-product. Remark: the proof uses (highly sophisticated) flux type methods.

37 Conclusion 1. Deformation quantization of Poisson manifolds is a notion that encompasses Quantization of classical phase spaces within the framework of classical observables (no Hilbert space representation is needed).

38 Conclusion 1. Deformation quantization of Poisson manifolds is a notion that encompasses Quantization of classical phase spaces within the framework of classical observables (no Hilbert space representation is needed). 2. Its mathematical background is Poisson geometry that encompasses both symplectic geometry and Lie theory.

39 Conclusion 1. Deformation quantization of Poisson manifolds is a notion that encompasses Quantization of classical phase spaces within the framework of classical observables (no Hilbert space representation is needed). 2. Its mathematical background is Poisson geometry that encompasses both symplectic geometry and Lie theory.