The geometrical structure of quantum theory as a natural - PowerPoint PPT Presentation

The geometrical structure of quantum theory as a natural generalization of information geometry Marcel Reginatto Physikalisch-Technische Bundesanstalt Braunschweig, Germany MaxEnt 2014, Amboise, France, September 21-26, 2014 ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 1 / 20

Preliminary remarks QM has a rich geometrical structure which allows for an equivalent geometrical formulation. ◮ Kibble, Geometrization of Quantum Mechanics (1979); many others. ◮ Detailed but accessible: Ashtekar and Schilling, Geometrical formulation of Quantum Mechanics (1998). The usual approach: ◮ Start from standard QM. ◮ Identify relevant geometrical features. ◮ “Translate” the theory into a geometrical language. Here this procedure is inverted : The geometrical structure of QM is derived from information geometry. ◮ It is a natural generalization of information geometry ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 2 / 20

Preliminary remarks What is the Geometrical formulation of QM? ◮ States are represented by points in a symplectic manifold (which happens to have a compatible metric). ◮ Observables are represented by certain real-valued functions on this space. ◮ The Schrödinger evolution is captured by by the symplectic flow generated by a Hamiltonian formulation. Ashtekar and Schilling, “Geometrical formulation of Quantum Mechanics” (1998) The work presented here relies heavily on, and extends, previous work done in collaboration with M. J. W. Hall (Centre for Quantum Dynamics, Griffith University, Brisbane, Australia). ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 3 / 20

Outline Probabilities, translations, and information geometry 1 Symplectic geometry 2 Kähler geometry 3 Unitary transformations 4 Hilbert space formulation from the geometric approach 5 ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 4 / 20

Translations and the Fisher-Rao metric Consider an n -dimensional configuration space, x ≡ { x 1 , . . . , x n } . d n xP ( x ) = 1. � Probability densities P ( x ) : P ( x ) ≥ 0 and Translation group acting on P ( x ) , T : P ( x ) → P ( x + θ ) . The natural metric on the space of parameters is the Fisher-Rao metric (Rao, 1945), γ jk = α � 1 ∂ P ( x + θ ) ∂ P ( x + θ ) d n x , P ( x + θ ) ∂θ j ∂θ k 2 where α is a constant. ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 5 / 20

The metric in the space of probabilities With a change of integration variables x → x − θ , γ jk = α ∂ P ( x ) ∂ P ( x ) � 1 d n x ∂ x k . 2 P ( x ) ∂ x j The line element of the Fisher-Rao metric induces a line element in the space of probability densities, ds 2 = α � d n x 1 � d n x d n x ′ g PP ( x , x ′ ) δ P x δ P x ′ , δ P x δ P x = 2 P x where P x = P ( x ) , δ P x ≡ ( ∂ P ( x ) /∂ x j )∆ j . We have a Riemannian geometry with metric α g PP ( x , x ′ ) = δ ( x − x ′ ) . 2 P x ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 6 / 20

Information metric: Equal distance contours on P 2 Information metric (left) Euclidean metric (right) For discrete probabilities, g ij = α 2 P i δ ij Figures: Guy Lebanon, Riemannian geometry and statistical machine learning ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 7 / 20

Uniqueness of the information metric g ij = α 2 P i δ ij N. N. ˇ Cencov, based on invariance under “certain probabilistically meaningful transformations” known as congruent embeddings by a Markov mapping . A simpler proof later provided by L. L. Campbell. Markov mappings can be used to map probability spaces of different dimensions; e.g., Figure: Guy Lebanon, Riemannian geometry and statistical machine learning ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 8 / 20

Uniqueness of the information metric: The basic idea Basic idea of the proof: The inner product of any two tangent vectors must be invariant under all Markov mappings. ptb-logo Figures: Guy Lebanon, Riemannian geometry and statistical machine learning Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 9 / 20

Dynamics for P Consider now probabilities P ( x , t ) that evolve in time. Two constraints that must be satisfied at all times t : d n x P = 1 ⇒ I [ P ] a constant of the motion. ◮ I [ P ] = � ◮ P ( x , t ) ≥ 0. The problem of time evolution under these constraints is solved by deriving the equations of motion from an action principle. ◮ A reasonable ansatz : A constants of the motion is often related to the invariance of a Lagrangian or Hamiltonian under a particular symmetry. ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 10 / 20

Hamiltonian dynamics for P Introduce an auxiliary field S canonically conjugate to P , and the Poisson bracket � δ A � � δ B δ S − δ A δ B d n x { A , B } = . δ P δ S δ P The equations of motion for P and S are P = { P , H} = δ H S = { S , H} = − δ H ˙ ˙ δ S , δ P , where H is the ensemble Hamiltonian . Normalization of P is preserved if H does not depend explicitly on S . ◮ Implies gauge invariance under S → S + c , where c is a constant. ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 11 / 20

Symplectic geometry The Poisson bracket can be written as � d n x d n x ′ δ A Ω ab ( x , x ′ ) δ B { A , B } = , δ F a δ F b x x ′ where F a x = ( P x , S x ) . The symplectic structure is � � 0 1 Ω ab ( x , x ′ ) = δ ( x − x ′ ) . − 1 0 We have a symplectic structure and a corresponding symplectic geometry . ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 12 / 20

A more general metric Can we extend the metric g PP ( x , x ′ ) , which is only defined on the subspace of probabilities P , to the space of P and S ? It can be done, but certain conditions which ensure the compatibility of the metric and symplectic structures have to be satisfied. These conditions amount to requiring that the space have a Kähler structure. The natural geometry of the space of probabilities in motion is a Kähler geometry . ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 13 / 20

Kähler geometry A Kähler structure brings together metric, symplectic and complex structures in a harmonious way. The Kähler conditions are g ac J c Ω ab = b , (1) J a c g ab J b = g cd , (2) d J a b J b − δ a = c . (3) c Eq. (1) : compatibility between Ω ab and g ab , Eq. (2) : the metric is Hermitian, Eq. (3) : J a b is a complex structure. ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 14 / 20

General solution to the Kähler conditions Assume g ab ( x , x ′ ) = g ab ( x ) δ ( x − x ′ ) , � � α g PS 2 P x δ ( x − x ′ ) . g ab = g SP g SS The solutions of the Kähler conditions are of the form � � 0 1 Ω ab = δ ( x − x ′ ) , − 1 0 � � α A x 2 P x δ ( x − x ′ ) , g ab = 2 P x α ( 1 + A 2 x ) A x 2 P x α ( 1 + A 2 � � A x x ) J a = δ ( x − x ′ ) . b − α − A x 2 P x But... The functional A is not determined by the Kähler conditions! ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 15 / 20

Solution for A in the case of discrete probabilities In the discrete case, � G A T � g ab = , A ( 1 + A 2 ) G − 1 where A is an n × n matrix and GAG − 1 = A T . To fix A , use the same strategy that leads to the proof of the uniqueness of the information metric. Introduce canonical transformations which generalize Markov mappings. Invariance under these “ generalized Markov mappings” forces A = A 1 , where A is a constant. ◮ A further canonical transformation maps the metric to the particular value A = 0. ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 16 / 20

Kähler structure ( A = 0): Complex coordinates With A x = 0, the Kähler structure is given (up to a product with δ ( x − x ′ ) ) by 2 P � � � � � � 0 1 α 0 0 , J a 2 P Ω ab = , g ab = b = . α 2 P − 1 0 0 − α 0 2 P α √ √ P exp ( iS /α ) , ψ ∗ = Define ψ = P exp ( − iS /α ) . This complex transformation leads to the standard form for a flat Kähler space, � 0 � − i � � � � 0 i α α 0 , J a Ω ab = , g ab = b = . − i α 0 α 0 0 i If α = � , the ψ are precisely the wave functions of QM. This is a remarkable result because it is based on geometrical arguments only. ptb-logo Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 17 / 20