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

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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

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 1 / 20

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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 Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 2 / 20

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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).

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 3 / 20

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Outline

1

Probabilities, translations, and information geometry

2

Symplectic geometry

3

Kähler geometry

4

Unitary transformations

5

Hilbert space formulation from the geometric approach

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 4 / 20

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Translations and the Fisher-Rao metric

Consider an n-dimensional configuration space, x ≡ {x1, . . . , xn}. Probability densities P(x): P(x) ≥ 0 and

• dnxP(x) = 1.

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), γjk = α 2

• dnx

1 P(x + θ) ∂P(x + θ) ∂θj ∂P(x + θ) ∂θk , where α is a constant.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 5 / 20

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The metric in the space of probabilities

With a change of integration variables x → x − θ, γjk = α 2

• dnx

1 P(x) ∂P(x) ∂xj ∂P(x) ∂xk . The line element of the Fisher-Rao metric induces a line element in the space of probability densities, ds2 = α 2

• dnx 1

Px δPx δPx =

• dnx dnx′ gPP(x, x′) δPx δPx′ ,

where Px = P(x), δPx ≡ (∂P(x)/∂xj)∆j. We have a Riemannian geometry with metric gPP(x, x′) = α 2Px δ(x − x′).

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 6 / 20

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Information metric: Equal distance contours on P2

Information metric (left) Euclidean metric (right)

For discrete probabilities,

gij = α

2Pi δij Figures: Guy Lebanon, Riemannian geometry and statistical machine learning

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 7 / 20

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Uniqueness of the information metric gij = α

2Pi δij

• 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

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 8 / 20

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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.

Figures: Guy Lebanon, Riemannian geometry and statistical machine learning

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 9 / 20

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Dynamics for P

Consider now probabilities P(x, t) that evolve in time. Two constraints that must be satisfied at all times t :

◮ I[P] =

• dnx P = 1 ⇒ I[P] a constant of the motion.

◮ 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.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 10 / 20

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Hamiltonian dynamics for P

Introduce an auxiliary field S canonically conjugate to P, and the Poisson bracket {A, B} =

• dnx

δA δP δB δS − δA δS δB δP

• .

The equations of motion for P and S are ˙ P = {P, H} = δH δS , ˙ S = {S, H} = −δH δP , 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.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 11 / 20

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Symplectic geometry

The Poisson bracket can be written as {A, B} =

• dnx dnx′ δA

δF a

x

Ωab(x, x′) δB δF b

x′

, where F a

x = (Px, Sx).

The symplectic structure is Ωab(x, x′) =

• 1

−1

• δ(x − x′) .

We have a symplectic structure and a corresponding symplectic geometry.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 12 / 20

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A more general metric

Can we extend the metric gPP(x, x′), which is only defined on the subspace of probabilities P, to the space

• f 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.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 13 / 20

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Kähler geometry

A Kähler structure brings together metric, symplectic and complex structures in a harmonious way. The Kähler conditions are Ωab = gacJc

b ,

(1) Ja

cgabJb d

= gcd , (2) Ja

bJb c

= −δa

c .

(3)

• Eq. (1) : compatibility between Ωab and gab,
• Eq. (2) : the metric is Hermitian,
• Eq. (3) : Ja

b is a complex structure.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 14 / 20

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General solution to the Kähler conditions

Assume gab(x, x′) = gab(x)δ(x − x′), gab =

• α

2Px

gPS gSP gSS

• δ(x − x′).

The solutions of the Kähler conditions are of the form Ωab =

• 1

−1

• δ(x − x′),

gab =

• α

2Px

Ax Ax

2Px α (1 + A2 x)

• δ(x − x′),

Ja

b

=

• Ax

2Px α (1 + A2 x)

− α

2Px

−Ax

• δ(x − x′).

But... The functional A is not determined by the Kähler conditions!

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 15 / 20

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Solution for A in the case of discrete probabilities

In the discrete case, gab = G AT A (1 + A2)G−1

• ,

where A is an n × n matrix and GAG−1 = AT. 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.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 16 / 20

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Kähler structure (A = 0): Complex coordinates

With Ax = 0, the Kähler structure is given (up to a product with δ(x − x′)) by Ωab =

• 1

−1

• , gab =
• α

2P 2P α

• , Ja

b =

• 2P

α

− α

2P

• .

Define ψ = √ P exp(iS/α), ψ∗ = √ P exp(−iS/α). This complex transformation leads to the standard form for a flat Kähler space, Ωab =

• iα

−iα

• , gab =

α α

• , Ja

b =

−i 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.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 17 / 20

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Unitary transformations in the discrete case

There is a symplectic structure ⇒ Sp(2n,R). But Sp(2n,R) can not be the group of transformations of the theory. There are additional requirements:

◮ Normalization:

i Pi = i ψi ∗ψi = 1.

◮ Form invariance of the metric: dσ2 = 2α

i dψi ∗dψi.

These two requirements lead to the group of rotations

• n the 2n-dimensional sphere, O(2n, R).

But unitary transformations are the only symplectic transformations which are also rotations: Sp(2n,R) ∩ O(2n, R)= U(n).

◮ The group of unitary transformations U(n) is singled out.

Time evolution is described by a one-parameter group

• f unitary transformations.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 18 / 20

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Hilbert space and Dirac product

There is a standard construction that associates a complex Hilbert space with any infinite dimensional Kähler space (Kibble). φ|ϕ = 1 2

• dnx
• (φ, φ∗) · [g + iΩ] ·

ϕ ϕ∗

• =

1 2

• dnx
• (φ, φ∗)

1 1

• + i

i −i ϕ ϕ∗

• =
• dnx φ∗ϕ

The Hilbert space structure of quantum mechanics comes out of the Kähler geometry. The complex structure that is needed for the quantum mechanics arises in a very natural way.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 19 / 20

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Summary: From information to quanta

The geometrical structure of quantum theory is a natural generalization of information geometry Ingredients:

◮ The natural metric on the space of probabilities

(information geometry)

◮ Time evolution via an action principle using a

Hamiltonian formalism (symplectic geometry)

◮ Consistency (Kähler geometry) ◮ Invariance of the Kähler metric under “generalized

Markov mappings”

None of the elements that are usually assumed to be characteristic of quantum theory are introduced a priori.

Marcel Reginatto (PTB) Geometry of QM from information geometry MaxEnt 2014 20 / 20