A Quantum Route to Hamilton-Jacobi Theory: Considerations on the - PowerPoint PPT Presentation

A Quantum Route to Hamilton-Jacobi Theory: Considerations on the Quantum-to-Classical Transition Collaborators : J. F. Cari˜ nena, A. Ibort, G. Morandi N. Mukunda, G. Sudarshan, G. Esposito, J. Clemente-Gallardo X. Gr` acia, E. Mart´ ınez, M. Mu˜ noz, N. Rom´ an J. Grabowski, M. Kus

Two quotations from Dirac The Method of Classical Analogy “... The value of classical analogy in the developments of quantum mechanics depends on the fact that classical mechanics provides a valid description of dynamical sys- tems...” “... Classical mechanics must therefore be a limiting case of quantum mechanics...”

Two quotations from Dirac The Hamiltonian form of field dynamics “..In classical dynamics, one has usually supposed that, when one has solved the equation of motion, one has done everything worth doing...” Talking about the family of solutions of Hamilton equations which fill a Lagrangian submanifold transversal to the fibers of the cotangent bundle, Dirac says: “... The family does not have any importance from the point of view of Newtonian mechanics; but it is a family which corresponds to one state of motion in the quantum theory, so presumable the family has some deep signifi- cance in nature, not yet properly understood...”

General settings q = ∂ H p = − ∂ H • ˙ ˙ Hamilton equations ∂ p , ∂ q � � q , ∂ S = E = ∂ S • H Hamilton-Jacobi ∂ q ∂ t ∇ ) = i � ∂ q , − i � � • H ( � Schr¨ odinger ∂ t

Linear versus non-linear Lie-Scheffers ideology: superposition rules Γ = a j ( t ) X j , [ X j , X k ] = c l jk X l H , R ( H ) , U ( H ) TU ( H ) ≡ [ U ( H )] C , T u ( H ) ⇋ T ∗ u ( H ) ⇋ T ∗ u ∗ ( H ) T u ( H ) is a C*-algebra GNS-construction: from T u ( H ) to H . With any C*-algebra on any manifold M , we construct Hilbert spaces.

Second order dynamics on U ( H ) Quantum systems on R ( H ) � | ψ �� ψ | � d = A | ψ �� ψ | � ψ | ψ � − 1 | ψ �� ψ | � ψ | ψ � A � ψ | ψ � dt i � i � R ( H ) homogeneous space of U ( H ) T R ( H ) homogeneous space of TU ( H ) ≃ GL ( H ) Quantum systems on R ( H ) ≡ complete solution of H.J. associated with � ψ | ( U − 1 ˙ U ) 2 | ψ � L = 1 U ) 2 ) = 1 2tr( ρ ψ ( U − 1 ˙ 2 � ψ | ψ �

Dirac interaction picture On the group L = 1 E.L. equations d 2tr( U − 1 ˙ dt ( U − 1 ˙ U ) 2 − → U ) = 0 Family of generalized solutions of the H.J. on the group θ L = tr( U − 1 ˙ U )( U − 1 dU ) � � ( U − 1 ˙ U ) ∧ ( U − 1 dU ) d ( U − 1 U ) = 0 ω L = tr d , U − 1 ˙ i X ( U − 1 dU ) = iA , U = iA , A Hermitian X ∗ ( θ L ) = i tr( AU − 1 dU )

Unfolding nonlinear classical dynamics Quantization: A procedure to associate linear equations with non-linear ones. More specifically: Write the equations of motion as a Lie-Scheffers system. � z 1 � � H 11 � � z 1 � i � d ξ = z 1 H 12 = ⇒ , z 2 H 21 H 22 z 2 dt z 2 ⇒ i � d dt ξ = H 12 + ( H 11 − H 22 ) ξ − H 21 ξ 2 From an equation on the group we obtain a linear equation on a vector space for any representation of the group.

More on Schr¨ odinger equation and Hamilton- Jacobi equation ∂ t ψ = − � 2 i � ∂ iS / � 2 m ∆ ψ + V ψ, ψ = Ae We obtain ∂ A ∂ t = − 1 2 m ( A ∆ S + 2 grad A · grad S ) � (grad S ) 2 � r ) − � 2 ∆ A ∂ S ∂ t = − + V ( � 2 m 2 m A � � � 2 1 − � 2 u = 1 ∆ A ∆ ρ A 2 = ψ ∗ ψ = ρ, Setting � m grad S , = − 2 m A 2 m ρ ∂ρ ∂ t + div( ρ� u ) = 0 Hamilton-Jacobi ⇐ � 2 ∆ A � = 0 2 m A R 3 ρ d 3 x = 1

Field theoretical aspect � � ∂ S � ∂ t + 1 d 3 x dt , • L = ρ 2 m ∇ S · ∇ S + V δ S | ∂ V = 0 , δρ | ∂ V = 0 V � � � � � � 2 1 ∂ S ∂ t + 1 1 d 3 x dt • L = ρ 2 m ∇ S · ∇ S + ρ 2 ∇ ρ · ∇ ρ + V 2 2 m V Fixed end-point variation with respect to S leads to the continuity equation � � ∂ρ ρ 1 ∂ t + ∇ m ∇ S = 0 Fixed end-point variation with respect to ρ leads respectively to • ∂ S ∂ t + 1 2 m ∇ S · ∇ S + V = 0 � � � 2 � 1 � • ∂ S ∂ t + 1 ρ 2 ∇ ρ · ∇ ρ − 2 1 ρ ∇ 2 ρ 2 m ∇ S · ∇ S + V + = 0 2 2 m

The fate of the continuity equation ∂ρ ∂ t + div( ρ� u ) = 0 Geometric interpretation: dS : Q × Λ → T ∗ Q ∂ 2 S dS ∗ ( θ 0 ) = d λ j ∧ dq k ∂λ j ∂ q k A symplectic structure. Liouville theorem on Q × Λ × R gives the continuity equations.

Generalized Coherent States Embedding “classical manifolds” into R ( H ). M a manifold with a volume form Ω. m �→ | ψ ( m ) � ∈ H ⇒�→ | ψ ( m ) �� ψ ( m ) | � ψ ( m ) | ψ ( m ) � Pull-back of Hermitian tensor fields � d ψ ( m ) | d ψ ( m ) � → Riemannian + skew-symmetric (2,0) tensors on M Fisher-Rao quantum information metric. Tranformations T : | ψ ( m ) � → | ψ ( m ′ ) � defines φ T : M → M , m �→ m ′

Generalized Coherent States Examples: Quantizers and Dequantizers W : m �→ W ( m ) ∈ u ( H ) | 0 � ∈ H a fiducial state | ψ ( m ) � = W ( m ) | 0 � O P ( H ) → F ( M ) A �→ tr( A W ( m )) = f A ( m ) , ( f A ∗ f B )( m ) = tr( A B W ( m )) Remark: When M is a group and W a representation, the ∗ -product becomes the convolution product on the group algebra.

If D : M → u ( H ) is another association such that tr D ( m ) W ( m ′ ) = δ ( m , m ′ ) , we define a “quantizer” map � A = f A ( m ) D ( m ) d µ Ω M M = T ∗ R n ( q , p ) �→ D ( q , p ) Weyl system, projective unitary representation ( f A ∗ f B )( q , p ) Moyal product Locality versus non-locality of the product.

“Quantum Hamilton-Jacobi” Replace Q × Λ with operators p = ∂ P = − ∂ q , � � q , � � q S ( � λ, t ) , S ( � λ, t ) ∂ � ∂ � λ � � q , ∂ S + ∂ q , � � ∂ t S ( � λ, t ) = 0 H q , t ∂ � “Well ordering” � q , � q , t ) g α ( � S ( � λ, t ) = f α ( � λ, t ) α Main ingredient q , � � q | S ( � λ, t ) | λ � = S ( q , λ, t ) � q | λ � �� ∂ � � 2 − i � ∂ 2 1 + V ( q ) + ∂ ∂ q S ′ ( q , λ, t ) ⇒ ∂ q 2 S ( q , λ, t ) ∂ t S ( q , λ, t ) = 0 2 m i � S ( q ,λ, t ) ψ ( q , λ, t ) = e

Summarizing Nonlinear equations − → linear equations (classical) (quantum) Γ = a j X j Lie Scheffers: g − 1 ˙ g = a j X j U − 1 ( g ) ˙ U ( g ) = a j A j , U ( g ) ∈ u ( H )