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 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). 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 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. ▶ 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. ▶ Complexify the configuration space. 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 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 Make 퐶 a complex curve. 2 2 휕 2휕 2 휕 휕 ( 푝2()+−푞(푧))̃ 푞(푧)̃ ( 푝2()+− 푞(푧)) 푞(푧) Carry over definitions: 휕푧 휕푧 휕푧 휕푧 LiouvilleMöbius ▶ Quantum energy level ▶ Complex projective structure Now even basic systems have cool energy level geometry. Complex projective structures Make 퐶 a complex curve. 2 2 휕 2휕 2 휕 휕 ( 푝2()+−푞(푧))̃ 푞(푧)̃ ( 푝2()+− 푞(푧)) 푞(푧) Carry over definitions: 휕푧 휕푧 휕푧 휕푧 ▶ Classical energy level (spectral curve) LiouvilleMöbius ▶ Complex projective structure Now even basic systems have cool energy level geometry.