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Deformation Quantization FFP14 Pierre Bieliavsky (U. Louvain, Belgium) 15 Jully 2014 FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization References Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.; Sternheimer, D.;

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Deformation Quantization

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

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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References

  • Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.;

Sternheimer, D.; Deformation theory and quantization. Ann. Physics (1978)

  • Weinstein, Alan; Deformation quantization. S´

eminaire 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 ;

Deformation Quantization for actions of Kahlerian Lie groups Memoirs of the Amercian Mathematical Society (2014)

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

A := Mn(C)

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

A := Mn(C) µ : A ⊗ A → A : a ⊗ b → a.b matrix multiplication

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

A := Mn(C) µ : A ⊗ A → A : a ⊗ b → a.b matrix multiplication µ(a ⊗ b) =: < K , a ⊗ b > with K ∈ A ⊗ A ⊗ A

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

A := Mn(C) µ : A ⊗ A → A : a ⊗ b → a.b matrix multiplication µ(a ⊗ b) =: < K , a ⊗ b > with K ∈ A ⊗ A ⊗ A

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Consider n points (“configuration space”):

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Consider n points (“configuration space”):

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Consider n points (“configuration space”): Consider the set M of all the arrows between pairs of points (“phase space”):

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Consider n points (“configuration space”): Consider the set M of all the arrows between pairs of points (“phase space”): Note: |M| = n2 = dimC(A) .

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Consider n points (“configuration space”): Consider the set M of all the arrows between pairs of points (“phase space”): Note: |M| = n2 = dimC(A) .

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Triangle: loop constituted by sequence of 3 consecutive arrows.

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Triangle: loop constituted by sequence of 3 consecutive arrows. A triangle:

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Triangle: loop constituted by sequence of 3 consecutive arrows. A triangle: A := Mn(C) is viewed as the space of continuous functions on M (“observables”) : A = C(M) .

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Triangle: loop constituted by sequence of 3 consecutive arrows. A triangle: A := Mn(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: E(→)(x) := 1 if x =→

  • therwise

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Then: K =

  • E(

) ⊗ E( ) ⊗ E( ) .

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Then: K =

  • E(

) ⊗ E( ) ⊗ E( ) . Interpretation: union of edges tensor products of characteristic functions

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Matrices and triangles

Then: K =

  • E(

) ⊗ E( ) ⊗ E( ) . Interpretation: union of edges tensor products of characteristic functions Union = additive operation (on sets)

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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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

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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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??

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

  • Phase space= T ⋆(Rn) = R2n = {x = (q, p) q, p ∈ Rn}

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

  • Phase space= T ⋆(Rn) = R2n = {x = (q, p) q, p ∈ Rn}
  • Poisson bracket =

{f , g} = ωij∂xif ∂xjg ⇔ skewsymmetric tensor field: ω = dα

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

  • Phase space= T ⋆(Rn) = R2n = {x = (q, p) q, p ∈ Rn}
  • Poisson bracket =

{f , g} = ωij∂xif ∂xjg ⇔ skewsymmetric tensor field: ω = dα remind: E( ) ⊗ E( ) ⊗ E( )

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

  • Phase space= T ⋆(Rn) = R2n = {x = (q, p) q, p ∈ Rn}
  • Poisson bracket =

{f , g} = ωij∂xif ∂xjg ⇔ skewsymmetric tensor field: ω = dα remind: E( ) ⊗ E( ) ⊗ E( ) Triangle(x, y, z) → exp

  • µ (
  • Triangle(x,y,z)

ω

  • (µ ∈ C)

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

  • Phase space= T ⋆(Rn) = R2n = {x = (q, p) q, p ∈ Rn}
  • Poisson bracket =

{f , g} = ωij∂xif ∂xjg ⇔ skewsymmetric tensor field: ω = dα remind: E( ) ⊗ E( ) ⊗ E( ) Triangle(x, y, z) → exp

  • µ (
  • Triangle(x,y,z)

ω

  • (µ ∈ C)

i.e E( ) :=

  • α

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

In other words, to observables a, b ∈ C ∞

c (R2n), 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) := ωijxiyj)

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

In other words, to observables a, b ∈ C ∞

c (R2n), 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) := ωijxiyj) Asymptotics: a ⋆ b ∼ a.b +

∞

  • k=1

1 k! 2i k ωi1j1... ωikjk ∂k

i1...ika ∂k j1...jkb

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

In other words, to observables a, b ∈ C ∞

c (R2n), 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) := ωijxiyj) Asymptotics: a ⋆ b ∼ a.b +

∞

  • k=1

1 k! 2i k ωi1j1... ωikjk ∂k

i1...ika ∂k j1...jkb

Theorem On A := C ∞(R2n)[[]], A × A → A : (a, b) → a ⋆ b is associative.

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

Did we know all this already??

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

Did we know all this already?? YES! Theorem[Weyl - von Neumann (1931)] Canonical Schr¨

  • dinger quantization (Weyl ordered):

Polynomials(R2n) − → L(L2(Rn)) : a → Op(a) Op(qj)ϕ(q) = qjϕ(q) Op(p)ϕ(q) = i∂qjϕ(q) Then Op(a) ◦ Op(b) = Op(a ⋆ b) .

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Deformation Quantization

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} := wij∂xif ∂xjg satisfies {f , {g, h}} + {h, {f , g}} + {g, {h, f }} = 0. Examples:

  • canonical phase space: (T ⋆(N), ωLiouville)
  • g = Lie algebra, dual: M = g⋆ with

{f , g}(x) := < x , [dfx , dgx] > ((g⋆)⋆ = g)

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Deformation Quantization

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 kCk(a, b)

(ii) C1(a, b) − C1(b, a) = i {a, b} (iii) Ck= bi-differential operator on C ∞(M) vanishing on constants.

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Deformation Quantization

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

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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Deformation Quantization

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

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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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).

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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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.

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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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.

FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization