The Geometry of Quantum Energy Levels

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The geometry of quantum energy levels

Aaron Fenyes (IHÉS)

ReNew Quantum Internal Seminar

11 August, 2020 A particle on a curve

Ω̃ 퐶 푦 Ω

Let’s study a particle moving on a real curve 퐶. ∗ 2 For consistency between charts, we need 푞̃ = 푦 푞 푦푧 .

Classical energy levels

푝2 + 푞(푧)̃ 푝2 + 푞(푧)

푇 ∗ℝ 푇 ∗ℝ

푦

Ω̃ ⊂ ℝ Ω ⊂ ℝ

An energy level is a curve in 푇 ∗퐶, cut out locally by

1 푝2 + 푉 (푧) = 퐸 푝2 + 푞(푧) = 0 2 ∗ 2 For consistency between charts, we need 푞̃ = 푦 푞 푦푧 .

Classical energy levels

푝2 + 푞(푧)̃ 푝2 + 푞(푧)

푇 ∗ℝ 푇 ∗ℝ

푦

Ω̃ ⊂ ℝ Ω ⊂ ℝ

An energy level is a curve in 푇 ∗퐶, cut out locally by

1 푝2 + 푉 (푧) = 퐸 푝2 + 푞(푧) = 0 2 Classical energy levels

푝2 + 푞(푧)̃ 푝2 + 푞(푧)

푇 ∗ℝ 푇 ∗ℝ

푦

Ω̃ ⊂ ℝ Ω ⊂ ℝ

An energy level is a curve in 푇 ∗퐶, cut out locally by

1 푝2 + 푉 (푧) = 퐸 푝2 + 푞(푧) = 0 2 ∗ 2 For consistency between charts, we need 푞̃ = 푦 푞 푦푧 . What condition to impose for consistency between charts? Liouville equivalence is a time-honored choice.

2 2 3/2 1/2 ( 휕 ) − 푞̃ = 푦 ∘ [( 휕 ) − 푦∗푞] ∘ 푦 휕푧 푧 휕푦 푧

It can be motivated using an operator-ordering rule.

Quantization 2 2 ( 휕 ) − 푞(푧)̃ ( 휕 ) − 푞(푧) 휕푧 푦 휕푧 Ω̃ ⊂ ℝ Ω ⊂ ℝ

A quantum energy level should be described locally by

2 2 1 (푖 휕 ) + [푉 (푧) − 퐸] ( 휕 ) − 푞(푧) 2 휕푧 휕푧 Hill’s operator What condition to impose for consistency between charts? Liouville equivalence is a time-honored choice.

2 2 3/2 1/2 ( 휕 ) − 푞̃ = 푦 ∘ [( 휕 ) − 푦∗푞] ∘ 푦 휕푧 푧 휕푦 푧

It can be motivated using an operator-ordering rule.

Quantization 2 2 ( 휕 ) − 푞(푧)̃ ( 휕 ) − 푞(푧) 휕푧 푦 휕푧 Ω̃ ⊂ ℝ Ω ⊂ ℝ

A quantum energy level should be described locally by

2 2 1 (푖 휕 ) + [푉 (푧) − 퐸] ( 휕 ) − 푞(푧) 2 휕푧 휕푧 Hill’s operator Liouville equivalence is a time-honored choice.

2 2 3/2 1/2 ( 휕 ) − 푞̃ = 푦 ∘ [( 휕 ) − 푦∗푞] ∘ 푦 휕푧 푧 휕푦 푧

It can be motivated using an operator-ordering rule.

Quantization 2 2 ( 휕 ) − 푞(푧)̃ ( 휕 ) − 푞(푧) 휕푧 푦 휕푧 Ω̃ ⊂ ℝ Ω ⊂ ℝ

A quantum energy level should be described locally by

2 2 1 (푖 휕 ) + [푉 (푧) − 퐸] ( 휕 ) − 푞(푧) 2 휕푧 휕푧 Hill’s operator

What condition to impose for consistency between charts? Quantization 2 2 ( 휕 ) − 푞(푧)̃ ( 휕 ) − 푞(푧) 휕푧 푦 휕푧 Ω̃ ⊂ ℝ Ω ⊂ ℝ

A quantum energy level should be described locally by

2 2 1 (푖 휕 ) + [푉 (푧) − 퐸] ( 휕 ) − 푞(푧) 2 휕푧 휕푧 Hill’s operator

What condition to impose for consistency between charts? Liouville equivalence is a time-honored choice.

2 2 3/2 1/2 ( 휕 ) − 푞̃ = 푦 ∘ [( 휕 ) − 푦∗푞] ∘ 푦 휕푧 푧 휕푦 푧

It can be motivated using an operator-ordering rule. Quantum energy levels

2 ( 휕 ) − 푞(푧)̃ 휕푧 Ω̃ 퐶 Liouville Ω 2 ( 휕 ) − 푞(푧) 휕푧

A quantum energy level on 퐶 consists of: Data An atlas with a Hill’s operator on the image of each chart. Condition Transitions are Liouville equivalences. It’s a geometric structure! Hence, we can describe a quantum energy level using only projective 2 charts, which come with ( 휕 ) on their images. 휕푧

The Liouville self-equivalences of this operator are the Möbius transformations.

Simplification

2 2 ( 휕 ) − 푞(푧) ( 휕 ) 휕푧 휕푧

Kummer’s coordinate

2 Every Hill’s operator is locally Liouville-equivalent to ( 휕 ) . 휕푧 Simplification

2 2 ( 휕 ) − 푞(푧) ( 휕 ) 휕푧 휕푧

Kummer’s coordinate

2 Every Hill’s operator is locally Liouville-equivalent to ( 휕 ) . 휕푧

Hence, we can describe a quantum energy level using only projective 2 charts, which come with ( 휕 ) on their images. 휕푧

The Liouville self-equivalences of this operator are the Möbius transformations. A quantum energy level on 퐶 becomes: Data An atlas of charts to ℝℙ1. Condition Transitions are Möbius transformations. This is known as a real projective structure (see Segal, 1991).

Real projective structures

2 ( 휕 ) 휕푧 Ω̃ 퐶 Möbius Ω 2 ( 휕 ) 휕푧

If we stick to projective charts, we can forget about Hill’s operators. A quantum energy level on 퐶 becomes: Data An atlas of charts to ℝℙ1. Condition Transitions are Möbius transformations. This is known as a real projective structure (see Segal, 1991).

Real projective structures

2 ( 휕 ) 휕푧 Ω̃ 퐶 Möbius Ω 2 ( 휕 ) 휕푧

If we stick to projective charts, we can forget about Hill’s operators. Real projective structures

2 ( 휕 ) 휕푧 Ω̃ 퐶 Möbius Ω 2 ( 휕 ) 휕푧

Summary

1 푝2 + 푞(푧) 2

2 2 ( 휕 ) − 푞(푧) ( 휕 ) 휕푧 휕푧

classical quantum real projective energy level energy level structure ≅ quantize simplify A quantum energy level is a real projective structure.

Summary

1 푝2 + 푞(푧) 2

2 2 ( 휕 ) − 푞(푧) ( 휕 ) 휕푧 휕푧

classical quantum real projective energy level energy level structure ≅ quantize simplify A quantum energy level is a real projective structure.

Summary

1 푝2 + 푞(푧) 2

2 2 ( 휕 ) − 푞(푧) ( 휕 ) 휕푧 휕푧

classical quantum real projective energy level energy level structure ≅ quantize simplify Summary

1 푝2 + 푞(푧) 2

2 2 ( 휕 ) − 푞(푧) ( 휕 ) 휕푧 휕푧

classical quantum real projective energy level energy level structure ≅ quantize simplify

A quantum energy level is a real projective structure. This is a good foothold. Where do we go next? ▶ Vary the energy. ▶ We get a family of real projective structures. ▶ It encodes all the physics of the system. ▶ When 퐶 is a circle, we should recover textbook results about quantum mechanics in a periodic potential. ▶ Complexify the configuration space.

Classification

Topologically, 퐶 is simple: a line or a circle. Its variety of real projective structures is also pretty simple.

▶ For example, the real projective structures on a circle are classified, up to isotopy, by conjugacy classes in PSL̃2ℝ. ▶ Vary the energy. ▶ We get a family of real projective structures. ▶ It encodes all the physics of the system. ▶ When 퐶 is a circle, we should recover textbook results about quantum mechanics in a periodic potential. ▶ Complexify the configuration space.

Classification

Topologically, 퐶 is simple: a line or a circle. Its variety of real projective structures is also pretty simple.

▶ For example, the real projective structures on a circle are classified, up to isotopy, by conjugacy classes in PSL̃2ℝ. This is a good foothold. Where do we go next? ▶ Complexify the configuration space.

Classification

Topologically, 퐶 is simple: a line or a circle. Its variety of real projective structures is also pretty simple.

▶ For example, the real projective structures on a circle are classified, up to isotopy, by conjugacy classes in PSL̃2ℝ. This is a good foothold. Where do we go next? ▶ Vary the energy. ▶ We get a family of real projective structures. ▶ It encodes all the physics of the system. ▶ When 퐶 is a circle, we should recover textbook results about quantum mechanics in a periodic potential. Classification

Topologically, 퐶 is simple: a line or a circle. Its variety of real projective structures is also pretty simple.

1 푧2 + 휆푧4 perturbation theory (Bender–Wu, 1969) 2 휆 1 − + nuclear scattering (Knoll–Schaeffer, 1976) 1 + 푒푧−1 2푧2

Complexification

A complex-valued potential models a particle that can escape. Complexification

A complex-valued potential models a particle that can escape. The potential often extends holomorphically to a complex neighborhood of the path. Famous examples (휆 ∈ ℂ constant):

1 푧2 + 휆푧4 perturbation theory (Bender–Wu, 1969) 2 휆 1 − + nuclear scattering (Knoll–Schaeffer, 1976) 1 + 푒푧−1 2푧2 Complexification

2 ( 휕 ) − 2[푉 (푧) − 퐸] 휕푧

When 푉 (푧) extends holomorphically, the operators describing quantum energy levels can act on holomorphic functions. That action should encode all physically relevant information. For instance, all generalized eigenfunctions∗ can be represented holomorphically (see F., 2020). ∗ For typical choices of rigged Hilbert space. ▶ Classical energy level (spectral curve) ▶ Quantum energy level ▶ Complex projective structure

Now even basic systems have cool energy level geometry.

Complex projective structures