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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.; 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

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

Matrices and triangles A := M n ( C ) FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

Matrices and triangles A := M n ( C ) µ : A ⊗ A → A : a ⊗ b �→ a . b matrix multiplication FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

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

Matrices and triangles Consider n points (“configuration space”): FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

Matrices and triangles Consider n points (“configuration space”): FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

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

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

Matrices and triangles Triangle: loop constituted by sequence of 3 consecutive arrows. FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

Matrices and triangles Triangle: loop constituted by sequence of 3 consecutive arrows. A triangle: FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

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

Matrices and triangles Then: � K = E ( ) ⊗ E ( ) ⊗ E ( ) . FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

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

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

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

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

OK! Let’s try! (Weyl-Moyal quantization) FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) • Phase space= T ⋆ ( R n ) = R 2 n = { x = ( q , p ) q , p ∈ R n } FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) • Phase space= T ⋆ ( R n ) = R 2 n = { x = ( q , p ) q , p ∈ R n } • Poisson bracket = { f , g } = ω ij ∂ x i f ∂ x j g ⇔ skewsymmetric tensor field: ω = d α FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) • Phase space= T ⋆ ( R n ) = R 2 n = { x = ( q , p ) q , p ∈ R n } • Poisson bracket = { f , g } = ω ij ∂ x i f ∂ x j g ⇔ skewsymmetric tensor field: ω = d α remind: � E ( ) ⊗ E ( ) ⊗ E ( ) FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) • Phase space= T ⋆ ( R n ) = R 2 n = { x = ( q , p ) q , p ∈ R n } • Poisson bracket = { f , g } = ω ij ∂ x i f ∂ x j g ⇔ skewsymmetric tensor field: ω = d α remind: � E ( ) ⊗ E ( ) ⊗ E ( ) � � � Triangle( x , y , z ) �→ exp µ ( ω ( µ ∈ C ) Triangle ( x , y , z ) FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) • Phase space= T ⋆ ( R n ) = R 2 n = { x = ( q , p ) q , p ∈ R n } • Poisson bracket = { f , g } = ω ij ∂ x i f ∂ x j g ⇔ skewsymmetric tensor field: ω = d α remind: � E ( ) ⊗ E ( ) ⊗ E ( ) � � � Triangle( x , y , z ) �→ exp µ ( ω ( µ ∈ C ) Triangle ( x , y , z ) � i.e E ( ) := α FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) c ( R 2 n ), one associates: In other words, to observables a , b ∈ C ∞ 1 � � a ⋆ � b ( x ) := K � ( x , y , z ) a ( y ) b ( z ) d y d z � 2 n where i � ( ω ( x , y )+ ω ( y , z )+ ω ( z , x )) ( ω ( x , y ) := ω ij x i y j ) K � ( x , y , z ) = e FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) c ( R 2 n ), one associates: In other words, to observables a , b ∈ C ∞ 1 � � a ⋆ � b ( x ) := K � ( x , y , z ) a ( y ) b ( z ) d y d z � 2 n where i � ( ω ( x , y )+ ω ( y , z )+ ω ( z , x )) ( ω ( x , y ) := ω ij x i y j ) K � ( x , y , z ) = e Asymptotics: � � � k ∞ 1 � ω i 1 j 1 ... ω i k j k ∂ k i 1 ... i k a ∂ k a ⋆ � b ∼ a . b + j 1 ... j k b k ! 2 i k =1 FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) c ( R 2 n ), one associates: In other words, to observables a , b ∈ C ∞ 1 � � a ⋆ � b ( x ) := K � ( x , y , z ) a ( y ) b ( z ) d y d z � 2 n where i � ( ω ( x , y )+ ω ( y , z )+ ω ( z , x )) ( ω ( x , y ) := ω ij x i y j ) K � ( x , y , z ) = e Asymptotics: � � � k ∞ 1 � ω i 1 j 1 ... ω i k j k ∂ k i 1 ... i k a ∂ k a ⋆ � b ∼ a . b + j 1 ... j k b k ! 2 i k =1 Theorem On A � := C ∞ ( R 2 n )[[ � ]], A � × A � → A � : ( a , b ) �→ a ⋆ � b is associative. FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) Did we know all this already?? FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization

OK! Let’s try! (Weyl-Moyal quantization) Did we know all this already?? YES! Theorem[Weyl - von Neumann (1931)] Canonical Schr¨ odinger quantization (Weyl ordered): Polynomials( R 2 n ) − → 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 ) Then Op � ( a ) ◦ Op � ( b ) = Op � ( a ⋆ � b ) . FFP14 Pierre Bieliavsky (U. Louvain, Belgium) Deformation Quantization