Reduction and coherent states CMS meeting, Toronto 2019 Alejandro - PowerPoint PPT Presentation

Reduction and coherent states CMS meeting, Toronto 2019 Alejandro Uribe University of Michigan Joint work with J. Rousseva 1

Summary: 1. Symplectic reduction can be used to construct interesting symplectic manifolds 2. There is a quantum analogue of symplectic reduction, perhaps not as well-known or utilized 3. We use quantum reduction to construct interesting wave functions, squeezed coherent states on CP N − 1 4. They have a symbol , a Schwartz function describing them micro-locally 5. We prove that they propagate nicely, as do their symbols. 2

I. Symplectic reduction ( M , ω ) symplectic , L → M pre-quantum line bundle moment map of S 1 action on L → M . µ : M → R X = µ − 1 ( 0 ) / S 1 X = M // S 1 . or X inherits pre-quantization L X → X , L ↓ µ − 1 ( 0 ) M ֒ → ↓ X L X → ahler, L → M holomorphic and S 1 acts by Assume now M is K¨ isometries L X → X is also holomorphic / K¨ ahler. 3

Bargman spaces Let: B M = H 0 ( M , L ) ∩ L 2 ( M , L ) with the natural Hilbert inner product Then S 1 acts linearly on B M , by translations. Define ( B M ) S 1 := the space of invariant vectors 4

[ Q , R ] = 0 (Atiyah, Guillemin-Sternberg) ◮ Similarly, one defines B X . ◮ Quantization commutes with reduction: B X � ( B M ) S 1 , X = M // S 1 . The isomorphism is just restriction–push forward. It will be important to do this for all tensor powers k = 1 , 2 , · · · , � = 1 L k → M , k B ( k ) M = H 0 ( M , L k ) ∩ L 2 ( M , L k ) . Then � S 1 � B ( k ) B ( k ) . � X M 5

II. Quantum reduction By this I mean the operator(s) R k : B ( k ) M → B ( k ) X , which are just the composition � S 1 Π k � R k : B ( k ) B ( k ) � B ( k ) − − → M M X Π k being orthogonal projection (averaging). Theorem: This operator quantizes the canonical relation � ( x , m ) ∈ X × M ; m ∈ µ − 1 ( 0 ) and π ( m ) = x � ⊂ X × M . 6

Example: ω = 1 µ = | z | 2 − 1 M = C N , i dz ∧ d z , e it · z = e − it z . Action of S 1 : � � B ( k ) = ψ = f ( z ) e − k | z | 2 / 2 ; ¯ ∂ f = 0 . Representation of S 1 on B ( k ) : ρ ( e it )( ψ )( z ) = e − ikt ψ ( e it z ) . B ( k ) � S 1 � � ψ = f ( z ) e − k | z | 2 / 2 ; f homog. polyn. degree k � = . 7

Example: = � f | S 2 N − 1 ; f homog. polyn. degree k � . B ( k ) X = CP N − 1 , X L = C N × C ↓ S 2 N − 1 C N ֒ → ↓ CP N − 1 L X → Reduction operator: � 2 π R k ( ψ )( z ) = 1 e − ikt ψ ( e ikt z ) dt ∀ z ∈ S 2 N − 1 2 π 0 8

Lemma: The reduction of ψ ( z ) = f ( z ) e − k | z | 2 / 2 ∈ B ( k ) , is R k ( ψ ) = e − k f k ( z ) , where f k = sum of the terms of degree k in the power series expansion of f . 9

III. Gaussian coherent states on C n The standard coherent state centered at w ∈ C N , e w ∈ B ( k ) , is � N � k e kzw e − k | w | 2 / 2 e − k | z | 2 / 2 e w ( z ) = π � N � k e − k | z − w | 2 / 2 e ik ω ( z , w ) . = π e ( z , w ) := e w ( z ) is the kernel of the orthogonal projection L 2 ( C N ) → B ( k ) . Husimi function: | e w | 2 : � 2 N � k | e w | 2 = e − k | z − w | 2 / 2 π 10

Squeezed Gaussian coherent states: ψ A , w ( z ) := e kQ A ( z − w ) / 2 e w ( z ) Q A ( z ) = zAz T A ∈ D where D := { A ; A is an N × N symmetric matrix such that A ∗ A < I } ψ A , w ( z ) ∈ L 2 . A ∗ A < I ⇒ These are necessary to describe the quantum evolution of standard states. (Among other things.) 11

Figure: Husimi function of a standard coherent state Figure: Husimi function of a squeezed state in N = 1, A = − 1 4 + i 2 12

IV. Reduction of coherent states to CP n − 1 1. Reducing the standard coherent states � N � k e kzw e −| w | 2 / 2 e −| z | 2 / 2 e w ( z ) = π gives R k ( e w )( z ) = Const. ( zw ) k , the standard coherent states of CP N − 1 . 2. Reducing squeezed Gaussian states gives what? 13

Exact formulae, N = 2 Lemma The reduction of the squeezed Gaussian C.S. is � N � k e − k e kQ A ( w ) × Ψ A , w ( z ) = π k � k − ℓ � k ℓ � 1 � 2 ℓ − k . � z ( w − Aw T ) 2 Q A ( z ) ( k − ℓ )!( 2 ℓ − k )! ℓ ≥ k / 2 14

with different notation... Orthonormal basis of B ( k ) CP 1 : 0 ≤ n ≤ k | n � = k k / 2 + 1 1 z n 1 z k − n 2 π � n !( k − n )! then if w = ( 1 , 0 ) the reduction is √ | o , µ, k � ( 1 + O ( 1 / k )) , where c 2 � a � c µ = b − A = 1 + a , c b and � 1 �� 2 ℓ � ℓ | o , µ, k � := k k / 2 + 1 1 � � µ ℓ | k − 2 ℓ � 2 k π � ℓ ( k − 2 ℓ )! 0 ≤ ℓ ≤ k / 2 15

V. Local picture The previous formulae are opaque . What is going on? Introduce the notion of symbol of a coherent state. Easiest to define in adapted coordinates and trivialization, whatever that means. In those coords: Given a center w , define the symbol of a coherent state ϕ w by: � � η σ ( η ) = lim k →∞ ϕ w w + , √ k this is a Schwartz function in the Bargmann space of T w M . It is a well-defined object. ∗ 16

After the construction, results: Theorem: The symbol of the reduction is the reduction of the symbol What “reduction of the symbol” means is an interesting question re: quantization of symplectic vector spaces. Theorem Under a quantum Hamiltonian, the reduced C.S. evolve (to leading order) to C.S. in the same class, and their symbols evolve according to the metaplectic representation. There is a well-defined class of squeezed coherent states on any K¨ ahler-quantized manifold, a special case of “isotropic states”. 17

Thank you for your attention 18