Quantization, after Souriau Prequantization Quantization? Group - PowerPoint PPT Presentation

Quantization, Group algebra. States after Souriau • C [G] : = {finitely supported functions G → C } ∋ c = � g ∈ G c g δ g Souriau ( δ g ) ∗ = δ g − 1 (and a G-module) δ g · δ h = δ gh , Prequantization is a ∗ -algebra: Quantization? 〈 m , c 〉 = � c g m ( g ) • C [G] ′ ∼ = C G = {all functions m : G → C }: Group algebra • G-invariant sesquilinear forms on C [G] write ( c , d ) �→ 〈 m , c ∗ · d 〉 Classical ( δ e , g δ e ) �→ m ( g ) Quantum Nilpotent Reductive Definition, Theorem (GNS, L. Schwartz) E(3) Call m a state of G if positive definite: 〈 m , c ∗ · c 〉 � 0, and m ( e ) = 1. • Then C [G] / C [G] ⊥ is a unitary G-module, realizable in C [G] ′ as φ ∈ C G such that � φ � 2 : = sup c ∈ C [G] | 〈 φ , c 〉 | 2 � � GNS m = 〈 m , c ∗ · c 〉 < ∞ . • m is cyclic in GNS m (its G-orbit has dense span). • Any unitary G-module with a cyclic unit vector φ is GNS ( φ , · φ ) . 4 / 18

Quantization, Group algebra. States after Souriau • C [G] : = {finitely supported functions G → C } ∋ c = � g ∈ G c g δ g Souriau ( δ g ) ∗ = δ g − 1 (and a G-module) δ g · δ h = δ gh , Prequantization is a ∗ -algebra: Quantization? 〈 m , c 〉 = � c g m ( g ) • C [G] ′ ∼ = C G = {all functions m : G → C }: Group algebra • G-invariant sesquilinear forms on C [G] write ( c , d ) �→ 〈 m , c ∗ · d 〉 Classical ( δ e , g δ e ) �→ m ( g ) Quantum Nilpotent Reductive Definition, Theorem (GNS, L. Schwartz) E(3) Call m a state of G if positive definite: 〈 m , c ∗ · c 〉 � 0, and m ( e ) = 1. • Then C [G] / C [G] ⊥ is a unitary G-module, realizable in C [G] ′ as φ ∈ C G such that � φ � 2 : = sup c ∈ C [G] | 〈 φ , c 〉 | 2 � � GNS m = 〈 m , c ∗ · c 〉 < ∞ . • m is cyclic in GNS m (its G-orbit has dense span). • Any unitary G-module with a cyclic unit vector φ is GNS ( φ , · φ ) . 4 / 18

Quantization, Group algebra. States after Souriau • C [G] : = {finitely supported functions G → C } ∋ c = � g ∈ G c g δ g Souriau ( δ g ) ∗ = δ g − 1 (and a G-module) δ g · δ h = δ gh , Prequantization is a ∗ -algebra: Quantization? 〈 m , c 〉 = � c g m ( g ) • C [G] ′ ∼ = C G = {all functions m : G → C }: Group algebra • G-invariant sesquilinear forms on C [G] write ( c , d ) �→ 〈 m , c ∗ · d 〉 Classical ( δ e , g δ e ) �→ m ( g ) Quantum Nilpotent Reductive Definition, Theorem (GNS, L. Schwartz) E(3) Call m a state of G if positive definite: 〈 m , c ∗ · c 〉 � 0, and m ( e ) = 1. • Then C [G] / C [G] ⊥ is a unitary G-module, realizable in C [G] ′ as φ ∈ C G such that � φ � 2 : = sup c ∈ C [G] | 〈 φ , c 〉 | 2 � � GNS m = 〈 m , c ∗ · c 〉 < ∞ . • m is cyclic in GNS m (its G-orbit has dense span). • Any unitary G-module with a cyclic unit vector φ is GNS ( φ , · φ ) . 4 / 18

Quantization, Group algebra. States after Souriau • C [G] : = {finitely supported functions G → C } ∋ c = � g ∈ G c g δ g Souriau ( δ g ) ∗ = δ g − 1 (and a G-module) δ g · δ h = δ gh , Prequantization is a ∗ -algebra: Quantization? 〈 m , c 〉 = � c g m ( g ) • C [G] ′ ∼ = C G = {all functions m : G → C }: Group algebra • G-invariant sesquilinear forms on C [G] write ( c , d ) �→ 〈 m , c ∗ · d 〉 Classical ( δ e , g δ e ) �→ m ( g ) Quantum Nilpotent Reductive Definition, Theorem (GNS, L. Schwartz) E(3) Call m a state of G if positive definite: 〈 m , c ∗ · c 〉 � 0, and m ( e ) = 1. • Then C [G] / C [G] ⊥ is a unitary G-module, realizable in C [G] ′ as φ ∈ C G such that � φ � 2 : = sup c ∈ C [G] | 〈 φ , c 〉 | 2 � � GNS m = 〈 m , c ∗ · c 〉 < ∞ . • m is cyclic in GNS m (its G-orbit has dense span). • Any unitary G-module with a cyclic unit vector φ is GNS ( φ , · φ ) . 4 / 18

Quantization, Group algebra. States after Souriau • C [G] : = {finitely supported functions G → C } ∋ c = � g ∈ G c g δ g Souriau ( δ g ) ∗ = δ g − 1 (and a G-module) δ g · δ h = δ gh , Prequantization is a ∗ -algebra: Quantization? 〈 m , c 〉 = � c g m ( g ) • C [G] ′ ∼ = C G = {all functions m : G → C }: Group algebra • G-invariant sesquilinear forms on C [G] write ( c , d ) �→ 〈 m , c ∗ · d 〉 Classical ( δ e , g δ e ) �→ m ( g ) Quantum Nilpotent Reductive Definition, Theorem (GNS, L. Schwartz) E(3) Call m a state of G if positive definite: 〈 m , c ∗ · c 〉 � 0, and m ( e ) = 1. • Then C [G] / C [G] ⊥ is a unitary G-module, realizable in C [G] ′ as φ ∈ C G such that � φ � 2 : = sup c ∈ C [G] | 〈 φ , c 〉 | 2 � � GNS m = 〈 m , c ∗ · c 〉 < ∞ . • m is cyclic in GNS m (its G-orbit has dense span). • Any unitary G-module with a cyclic unit vector φ is GNS ( φ , · φ ) . 4 / 18

Quantization, Group algebra. States after Souriau • C [G] : = {finitely supported functions G → C } ∋ c = � g ∈ G c g δ g Souriau ( δ g ) ∗ = δ g − 1 (and a G-module) δ g · δ h = δ gh , Prequantization is a ∗ -algebra: Quantization? 〈 m , c 〉 = � c g m ( g ) • C [G] ′ ∼ = C G = {all functions m : G → C }: Group algebra • G-invariant sesquilinear forms on C [G] write ( c , d ) �→ 〈 m , c ∗ · d 〉 Classical ( δ e , g δ e ) �→ m ( g ) Quantum Nilpotent Reductive Definition, Theorem (GNS, L. Schwartz) E(3) Call m a state of G if positive definite: 〈 m , c ∗ · c 〉 � 0, and m ( e ) = 1. • Then C [G] / C [G] ⊥ is a unitary G-module, realizable in C [G] ′ as φ ∈ C G such that � φ � 2 : = sup c ∈ C [G] | 〈 φ , c 〉 | 2 � � GNS m = 〈 m , c ∗ · c 〉 < ∞ . • m is cyclic in GNS m (its G-orbit has dense span). • Any unitary G-module with a cyclic unit vector φ is GNS ( φ , · φ ) . 4 / 18

Quantization, Group algebra. States after Souriau • C [G] : = {finitely supported functions G → C } ∋ c = � g ∈ G c g δ g Souriau ( δ g ) ∗ = δ g − 1 (and a G-module) δ g · δ h = δ gh , Prequantization is a ∗ -algebra: Quantization? 〈 m , c 〉 = � c g m ( g ) • C [G] ′ ∼ = C G = {all functions m : G → C }: Group algebra • G-invariant sesquilinear forms on C [G] write ( c , d ) �→ 〈 m , c ∗ · d 〉 Classical ( δ e , g δ e ) �→ m ( g ) Quantum Nilpotent Reductive Definition, Theorem (GNS, L. Schwartz) E(3) Call m a state of G if positive definite: 〈 m , c ∗ · c 〉 � 0, and m ( e ) = 1. • Then C [G] / C [G] ⊥ is a unitary G-module, realizable in C [G] ′ as φ ∈ C G such that � φ � 2 : = sup c ∈ C [G] | 〈 φ , c 〉 | 2 � � GNS m = 〈 m , c ∗ · c 〉 < ∞ . • m is cyclic in GNS m (its G-orbit has dense span). • Any unitary G-module with a cyclic unit vector φ is GNS ( φ , · φ ) . 4 / 18

Quantization, Group algebra. States after Souriau • C [G] : = {finitely supported functions G → C } ∋ c = � g ∈ G c g δ g Souriau ( δ g ) ∗ = δ g − 1 (and a G-module) δ g · δ h = δ gh , Prequantization is a ∗ -algebra: Quantization? 〈 m , c 〉 = � c g m ( g ) • C [G] ′ ∼ = C G = {all functions m : G → C }: Group algebra • G-invariant sesquilinear forms on C [G] write ( c , d ) �→ 〈 m , c ∗ · d 〉 Classical ( δ e , g δ e ) �→ m ( g ) Quantum Nilpotent Reductive Definition, Theorem (GNS, L. Schwartz) E(3) Call m a state of G if positive definite: 〈 m , c ∗ · c 〉 � 0, and m ( e ) = 1. • Then C [G] / C [G] ⊥ is a unitary G-module, realizable in C [G] ′ as φ ∈ C G such that � φ � 2 : = sup c ∈ C [G] | 〈 φ , c 〉 | 2 � � GNS m = 〈 m , c ∗ · c 〉 < ∞ . • m is cyclic in GNS m (its G-orbit has dense span). • Any unitary G-module with a cyclic unit vector φ is GNS ( φ , · φ ) . 4 / 18

Quantization, Group algebra. States after Souriau Souriau Example 1: Characters Prequantization If χ : G → U(1) is a character, then χ is a state and Quantization? Group algebra GNS χ = C χ Classical Quantum ( = C where G acts by χ ). Nilpotent Reductive Example 2: Discrete induction (Blattner 1963) E(3) � n ( g ) if g ∈ H, If n is a state of a subgroup H ⊂ G and m ( g ) = 0 otherwise, then m is a state and GNS m = ind G H GNS n (lower case “i” for discrete a.k.a. ℓ 2 induction). 5 / 18

Quantization, Group algebra. States after Souriau Souriau Example 1: Characters Prequantization If χ : G → U(1) is a character, then χ is a state and Quantization? Group algebra GNS χ = C χ Classical Quantum ( = C where G acts by χ ). Nilpotent Reductive Example 2: Discrete induction (Blattner 1963) E(3) � n ( g ) if g ∈ H, If n is a state of a subgroup H ⊂ G and m ( g ) = 0 otherwise, then m is a state and GNS m = ind G H GNS n (lower case “i” for discrete a.k.a. ℓ 2 induction). 5 / 18

Quantization, Group algebra. States after Souriau Souriau Example 1: Characters Prequantization If χ : G → U(1) is a character, then χ is a state and Quantization? Group algebra GNS χ = C χ Classical Quantum ( = C where G acts by χ ). Nilpotent Reductive Example 2: Discrete induction (Blattner 1963) E(3) � n ( g ) if g ∈ H, If n is a state of a subgroup H ⊂ G and m ( g ) = 0 otherwise, then m is a state and GNS m = ind G H GNS n (lower case “i” for discrete a.k.a. ℓ 2 induction). 5 / 18

Quantization, Group algebra. States after Souriau Souriau Example 1: Characters Prequantization If χ : G → U(1) is a character, then χ is a state and Quantization? Group algebra GNS χ = C χ Classical Quantum ( = C where G acts by χ ). Nilpotent Reductive Example 2: Discrete induction (Blattner 1963) E(3) � n ( g ) if g ∈ H, If n is a state of a subgroup H ⊂ G and m ( g ) = 0 otherwise, then m is a state and GNS m = ind G H GNS n (lower case “i” for discrete a.k.a. ℓ 2 induction). 5 / 18

Quantization, Group algebra. States after Souriau Souriau Example 1: Characters Prequantization If χ : G → U(1) is a character, then χ is a state and Quantization? Group algebra GNS χ = C χ Classical Quantum ( = C where G acts by χ ). Nilpotent Reductive Example 2: Discrete induction (Blattner 1963) E(3) � n ( g ) if g ∈ H, If n is a state of a subgroup H ⊂ G and m ( g ) = 0 otherwise, then m is a state and GNS m = ind G H GNS n (lower case “i” for discrete a.k.a. ℓ 2 induction). 5 / 18

Quantization, Group algebra. States after Souriau Souriau Example 1: Characters Prequantization If χ : G → U(1) is a character, then χ is a state and Quantization? Group algebra GNS χ = C χ Classical Quantum ( = C where G acts by χ ). Nilpotent Reductive Example 2: Discrete induction (Blattner 1963) E(3) � n ( g ) if g ∈ H, If n is a state of a subgroup H ⊂ G and m ( g ) = 0 otherwise, then m is a state and GNS m = ind G H GNS n (lower case “i” for discrete a.k.a. ℓ 2 induction). 5 / 18

Quantization, Group algebra. States after Souriau Souriau Example 1: Characters Prequantization If χ : G → U(1) is a character, then χ is a state and Quantization? Group algebra GNS χ = C χ Classical Quantum ( = C where G acts by χ ). Nilpotent Reductive Example 2: Discrete induction (Blattner 1963) E(3) � n ( g ) if g ∈ H, If n is a state of a subgroup H ⊂ G and m ( g ) = 0 otherwise, then m is a state and GNS m = ind G H GNS n (lower case “i” for discrete a.k.a. ℓ 2 induction). 5 / 18

Quantization, Classical (statistical) states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Continuous states m Souriau of ( g , + ) correspond to probability measures μ on g ∗ (Bochner): Prequantization Quantization? � g ∗ e i 〈 x ,Z 〉 d μ ( x ). Group algebra m (Z) = (1) Classical Quantum Nilpotent Definition Reductive A statistical state for X is a state m of g which is concentrated on X, E(3) in the sense that its spectral measure ( μ above) is. This works even without assuming continuity of m : in (1), make g discrete and hence replace g ∗ by its Bohr compactification ˆ g = { all characters of g }, in which X ⊂ g ∗ embeds by x �→ e i 〈 x , · 〉 . Notation: b X = closure of X in ˆ g (“Bohr closure”). 6 / 18

Quantization, Classical (statistical) states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Continuous states m Souriau of ( g , + ) correspond to probability measures μ on g ∗ (Bochner): Prequantization Quantization? � g ∗ e i 〈 x ,Z 〉 d μ ( x ). Group algebra m (Z) = (1) Classical Quantum Nilpotent Definition Reductive A statistical state for X is a state m of g which is concentrated on X, E(3) in the sense that its spectral measure ( μ above) is. This works even without assuming continuity of m : in (1), make g discrete and hence replace g ∗ by its Bohr compactification ˆ g = { all characters of g }, in which X ⊂ g ∗ embeds by x �→ e i 〈 x , · 〉 . Notation: b X = closure of X in ˆ g (“Bohr closure”). 6 / 18

Quantization, Classical (statistical) states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Continuous states m Souriau of ( g , + ) correspond to probability measures μ on g ∗ (Bochner): Prequantization Quantization? � g ∗ e i 〈 x ,Z 〉 d μ ( x ). Group algebra m (Z) = (1) Classical Quantum Nilpotent Definition Reductive A statistical state for X is a state m of g which is concentrated on X, E(3) in the sense that its spectral measure ( μ above) is. This works even without assuming continuity of m : in (1), make g discrete and hence replace g ∗ by its Bohr compactification ˆ g = { all characters of g }, in which X ⊂ g ∗ embeds by x �→ e i 〈 x , · 〉 . Notation: b X = closure of X in ˆ g (“Bohr closure”). 6 / 18

Quantization, Classical (statistical) states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Continuous states m Souriau of ( g , + ) correspond to probability measures μ on g ∗ (Bochner): Prequantization Quantization? � g ∗ e i 〈 x ,Z 〉 d μ ( x ). Group algebra m (Z) = (1) Classical Quantum Nilpotent Definition Reductive A statistical state for X is a state m of g which is concentrated on X, E(3) in the sense that its spectral measure ( μ above) is. This works even without assuming continuity of m : in (1), make g discrete and hence replace g ∗ by its Bohr compactification ˆ g = { all characters of g }, in which X ⊂ g ∗ embeds by x �→ e i 〈 x , · 〉 . Notation: b X = closure of X in ˆ g (“Bohr closure”). 6 / 18

Quantization, Classical (statistical) states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Continuous states m Souriau of ( g , + ) correspond to probability measures μ on g ∗ (Bochner): Prequantization Quantization? � g ∗ e i 〈 x ,Z 〉 d μ ( x ). Group algebra m (Z) = (1) Classical Quantum Nilpotent Definition Reductive A statistical state for X is a state m of g which is concentrated on X, E(3) in the sense that its spectral measure ( μ above) is. This works even without assuming continuity of m : in (1), make g discrete and hence replace g ∗ by its Bohr compactification ˆ g = { all characters of g }, in which X ⊂ g ∗ embeds by x �→ e i 〈 x , · 〉 . Notation: b X = closure of X in ˆ g (“Bohr closure”). 6 / 18

Quantization, Quantum states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Souriau Prequantization Definition ( equivalent to Souriau’s) Quantization? Group algebra A quantum state for X is a state m of G, such that for every abelian Classical subalgebra a of g , the state m ◦ exp | a of a is concentrated on b X | a . Quantum Nilpotent Reductive E(3) a ˆ X ˆ g Statistical interpretation: the spectral measure of m ◦ exp | a gives the probability distribution of x | a (or “joint probability” of the Poisson commuting functions 〈 · , Z j 〉 for Z j in a basis of a ). 7 / 18

Quantization, Quantum states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Souriau Prequantization Definition ( equivalent to Souriau’s) Quantization? Group algebra A quantum state for X is a state m of G, such that for every abelian Classical subalgebra a of g , the state m ◦ exp | a of a is concentrated on b X | a . Quantum Nilpotent Reductive E(3) a ˆ X ˆ g Statistical interpretation: the spectral measure of m ◦ exp | a gives the probability distribution of x | a (or “joint probability” of the Poisson commuting functions 〈 · , Z j 〉 for Z j in a basis of a ). 7 / 18

Quantization, Quantum states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Souriau Prequantization Definition ( equivalent to Souriau’s) Quantization? Group algebra A quantum state for X is a state m of G, such that for every abelian Classical subalgebra a of g , the state m ◦ exp | a of a is concentrated on b X | a . Quantum Nilpotent Reductive E(3) a ˆ X ˆ g Statistical interpretation: the spectral measure of m ◦ exp | a gives the probability distribution of x | a (or “joint probability” of the Poisson commuting functions 〈 · , Z j 〉 for Z j in a basis of a ). 7 / 18

Quantization, Quantum states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Souriau Prequantization Definition ( equivalent to Souriau’s) Quantization? Group algebra A quantum state for X is a state m of G, such that for every abelian Classical subalgebra a of g , the state m ◦ exp | a of a is concentrated on b X | a . Quantum Nilpotent Reductive E(3) a ˆ X ˆ g Statistical interpretation: the spectral measure of m ◦ exp | a gives the probability distribution of x | a (or “joint probability” of the Poisson commuting functions 〈 · , Z j 〉 for Z j in a basis of a ). 7 / 18

Quantization, Quantum states after Souriau Let X be a coadjoint orbit of G (say a Lie group). Souriau Prequantization Definition ( equivalent to Souriau’s) Quantization? Group algebra A quantum state for X is a state m of G, such that for every abelian Classical subalgebra a of g , the state m ◦ exp | a of a is concentrated on b X | a . Quantum Nilpotent Reductive E(3) a ˆ X ˆ g Statistical interpretation: the spectral measure of m ◦ exp | a gives the probability distribution of x | a (or “joint probability” of the Poisson commuting functions 〈 · , Z j 〉 for Z j in a basis of a ). 7 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau If V = GNS m , then ( φ , · φ ) is a quantum state for X for all unit φ ∈ V. Souriau Prequantization Definition Quantization? G-modules V with this property are quantum representations for X. Group algebra Classical They need not be continuous, nor irreducible on transitive subgroups. Quantum Nilpotent Example 1: Point-orbits Reductive Suppose a state n of a connected Lie group H is quantum for a point- E(3) orbit { y } ⊂ ( h ∗ ) H . Then y is integral , and n is the character such that n (exp(Z)) = e i 〈 y ,Z 〉 . (2) A representation of H is quantum for { y } iff it is a multiple of this n . We will call states of G ⊃ H that restrict to (2) eigenstates belonging to y ∈ ( h ∗ ) H — or by abuse, to the (generically coisotropic ) preimage of y in some X ⊂ g ∗ . Weinstein (1982) called attaching waves to lagrangian submanifolds the FUNDAMENTAL QUANTIZATION PROBLEM . 8 / 18

Quantization, Quantum states after Souriau Souriau Example 2: Prequantization is not quantum Prequantization Let L be the prequantization line bundle over X = ( R 2 , dp ∧ dq ). Quantization? The resulting representation of Aut(L) in L 2 (X) is not quantum for X. Group algebra Classical Sketch of proof: Quantum It represents the flow of the bounded hamiltonian H( p , q ) = sin p by a Nilpotent 1-parameter group whose self-adjoint generator is unbounded — it’s Reductive equivalent to multiplication by sin p + ( k − p ) cos p in L 2 ( R 2 , dp dk ). E(3) Remark We are rejecting this representation for spectral reasons. Unlike van Hove who rejected it for being reducible on the Heisenberg subgroup, we can still hope that Aut(L) has a representation quantizing X. (Of course, this remains purely verbal until someone finds it! ) 9 / 18

Quantization, Quantum states after Souriau Souriau Example 2: Prequantization is not quantum Prequantization Let L be the prequantization line bundle over X = ( R 2 , dp ∧ dq ). Quantization? The resulting representation of Aut(L) in L 2 (X) is not quantum for X. Group algebra Classical Sketch of proof: Quantum It represents the flow of the bounded hamiltonian H( p , q ) = sin p by a Nilpotent 1-parameter group whose self-adjoint generator is unbounded — it’s Reductive equivalent to multiplication by sin p + ( k − p ) cos p in L 2 ( R 2 , dp dk ). E(3) Remark We are rejecting this representation for spectral reasons. Unlike van Hove who rejected it for being reducible on the Heisenberg subgroup, we can still hope that Aut(L) has a representation quantizing X. (Of course, this remains purely verbal until someone finds it! ) 9 / 18

Quantization, Quantum states after Souriau Souriau Example 2: Prequantization is not quantum Prequantization Let L be the prequantization line bundle over X = ( R 2 , dp ∧ dq ). Quantization? The resulting representation of Aut(L) in L 2 (X) is not quantum for X. Group algebra Classical Sketch of proof: Quantum It represents the flow of the bounded hamiltonian H( p , q ) = sin p by a Nilpotent 1-parameter group whose self-adjoint generator is unbounded — it’s Reductive equivalent to multiplication by sin p + ( k − p ) cos p in L 2 ( R 2 , dp dk ). E(3) Remark We are rejecting this representation for spectral reasons. Unlike van Hove who rejected it for being reducible on the Heisenberg subgroup, we can still hope that Aut(L) has a representation quantizing X. (Of course, this remains purely verbal until someone finds it! ) 9 / 18

Quantization, Quantum states after Souriau Souriau Example 2: Prequantization is not quantum Prequantization Let L be the prequantization line bundle over X = ( R 2 , dp ∧ dq ). Quantization? The resulting representation of Aut(L) in L 2 (X) is not quantum for X. Group algebra Classical Sketch of proof: Quantum It represents the flow of the bounded hamiltonian H( p , q ) = sin p by a Nilpotent 1-parameter group whose self-adjoint generator is unbounded — it’s Reductive equivalent to multiplication by sin p + ( k − p ) cos p in L 2 ( R 2 , dp dk ). E(3) Remark We are rejecting this representation for spectral reasons. Unlike van Hove who rejected it for being reducible on the Heisenberg subgroup, we can still hope that Aut(L) has a representation quantizing X. (Of course, this remains purely verbal until someone finds it! ) 9 / 18

Quantization, Quantum states after Souriau Souriau Example 2: Prequantization is not quantum Prequantization Let L be the prequantization line bundle over X = ( R 2 , dp ∧ dq ). Quantization? The resulting representation of Aut(L) in L 2 (X) is not quantum for X. Group algebra Classical Sketch of proof: Quantum It represents the flow of the bounded hamiltonian H( p , q ) = sin p by a Nilpotent 1-parameter group whose self-adjoint generator is unbounded — it’s Reductive equivalent to multiplication by sin p + ( k − p ) cos p in L 2 ( R 2 , dp dk ). E(3) Remark We are rejecting this representation for spectral reasons. Unlike van Hove who rejected it for being reducible on the Heisenberg subgroup, we can still hope that Aut(L) has a representation quantizing X. (Of course, this remains purely verbal until someone finds it! ) 9 / 18

Quantization, Quantum states after Souriau On the other hand. . . Souriau Prequantization Theorem (Howe-Z., Ergodic Theory Dynam. Systems 2015) Quantization? Group algebra • G noncompact simple: every nonzero coadjoint orbit has b X = b g ∗ . Classical • G connected nilpotent: every coadjoint orbit has the same Bohr Quantum Nilpotent closure as its affine hull. Reductive E(3) Corollary • G noncompact simple: every unitary representation is quantum for every nonzero coadjoint orbit. • G simply connected nilpotent: a unitary representation is quantum for X iff the center of G / exp(ann(X)) acts by the correct character. 10 / 18

Quantization, Quantum states after Souriau On the other hand. . . Souriau Prequantization Theorem (Howe-Z., Ergodic Theory Dynam. Systems 2015) Quantization? Group algebra • G noncompact simple: every nonzero coadjoint orbit has b X = b g ∗ . Classical • G connected nilpotent: every coadjoint orbit has the same Bohr Quantum Nilpotent closure as its affine hull. Reductive E(3) Corollary • G noncompact simple: every unitary representation is quantum for every nonzero coadjoint orbit. • G simply connected nilpotent: a unitary representation is quantum for X iff the center of G / exp(ann(X)) acts by the correct character. 10 / 18

Quantization, Quantum states after Souriau On the other hand. . . Souriau Prequantization Theorem (Howe-Z., Ergodic Theory Dynam. Systems 2015) Quantization? Group algebra • G noncompact simple: every nonzero coadjoint orbit has b X = b g ∗ . Classical • G connected nilpotent: every coadjoint orbit has the same Bohr Quantum Nilpotent closure as its affine hull. Reductive E(3) Corollary • G noncompact simple: every unitary representation is quantum for every nonzero coadjoint orbit. • G simply connected nilpotent: a unitary representation is quantum for X iff the center of G / exp(ann(X)) acts by the correct character. 10 / 18

Quantization, Quantum states after Souriau On the other hand. . . Souriau Prequantization Theorem (Howe-Z., Ergodic Theory Dynam. Systems 2015) Quantization? Group algebra • G noncompact simple: every nonzero coadjoint orbit has b X = b g ∗ . Classical • G connected nilpotent: every coadjoint orbit has the same Bohr Quantum Nilpotent closure as its affine hull. Reductive E(3) Corollary • G noncompact simple: every unitary representation is quantum for every nonzero coadjoint orbit. • G simply connected nilpotent: a unitary representation is quantum for X iff the center of G / exp(ann(X)) acts by the correct character. 10 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Souriau Prequantization Quantization? Group algebra Classical Quantum Nilpotent Reductive E(3) 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive E(3) 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive E(3) 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive E(3) 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive E(3) 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive E(3) 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive E(3) 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = 0 otherwise. Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = 0 otherwise. Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = 0 otherwise. Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = 0 otherwise. Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = 0 otherwise. Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . a ⊂ h ⇒ x | a certain ; 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau G : connected, simply connected nilpotent Lie group, Souriau X : coadjoint orbit of G, Prequantization x : chosen point in X. Quantization? A connected subgroup H ⊂ G is subordinate to x if, equivalently, Group algebra Classical • { x | h } is a point-orbit of H in h ∗ Quantum • 〈 x , [ h , h ] 〉 = 0 Nilpotent • e i x ◦ log | H is a character of H. Reductive Theorem E(3) Let H ⊂ G be maximal subordinate to x ∈ X. Then there is a unique quantum eigenstate for X belonging to { x | h } ⊂ h ∗ , namely � e i x ◦ log ( g ) if g ∈ H, m ( g ) = 0 otherwise. Moreover GNS m = ind( x , H) : = ind G H e i x ◦ log | H ( discrete induction ) . a ⊂ h ⇒ x | a certain ; a ⋔ h ⇒ x | a equidistributed in ˆ a . 11 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Remark Souriau Prequantization Kirillov (1962) used Ind( x , H) : = Ind G H e i x ◦ log | H (usual induction). Quantization? This is Group algebra (a) irreducible ⇔ H is a polarization at x (: subordinate subgroup Classical such that the bound dim(G / H) � 1 2 dim(X) is attained); Quantum (b) equivalent to Ind( x , H ′ ) if H � = H ′ are two polarizations at x . Nilpotent Reductive E(3) 12 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Remark Souriau Prequantization Kirillov (1962) used Ind( x , H) : = Ind G H e i x ◦ log | H (usual induction). Quantization? This is Group algebra (a) irreducible ⇔ H is a polarization at x (: subordinate subgroup Classical such that the bound dim(G / H) � 1 2 dim(X) is attained); Quantum (b) equivalent to Ind( x , H ′ ) if H � = H ′ are two polarizations at x . Nilpotent Reductive E(3) 12 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Remark Souriau Prequantization Kirillov (1962) used Ind( x , H) : = Ind G H e i x ◦ log | H (usual induction). Quantization? This is Group algebra (a) irreducible ⇔ H is a polarization at x (: subordinate subgroup Classical such that the bound dim(G / H) � 1 2 dim(X) is attained); Quantum (b) equivalent to Ind( x , H ′ ) if H � = H ′ are two polarizations at x . Nilpotent Reductive E(3) 12 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Remark Souriau Prequantization Kirillov (1962) used Ind( x , H) : = Ind G H e i x ◦ log | H (usual induction). Quantization? This is Group algebra (a) irreducible ⇔ H is a polarization at x (: subordinate subgroup Classical such that the bound dim(G / H) � 1 2 dim(X) is attained); Quantum (b) equivalent to Ind( x , H ′ ) if H � = H ′ are two polarizations at x . Nilpotent Reductive E(3) 12 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Remark Souriau Prequantization Kirillov (1962) used Ind( x , H) : = Ind G H e i x ◦ log | H (usual induction). Quantization? This is Group algebra (a) irreducible ⇔ H is a polarization at x (: subordinate subgroup Classical such that the bound dim(G / H) � 1 2 dim(X) is attained); Quantum (b) equivalent to Ind( x , H ′ ) if H � = H ′ are two polarizations at x . Nilpotent Reductive E(3) In contrast: Theorem Let H ⊂ G be subordinate to x . Then ind( x , H) : = ind G H e i x ◦ log | H is (a) irreducible ⇔ H is maximal subordinate to x ; (b) inequivalent to ind( x , H ′ ) if H � = H ′ are two polarizations at x . 12 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Remark Souriau Prequantization Kirillov (1962) used Ind( x , H) : = Ind G H e i x ◦ log | H (usual induction). Quantization? This is Group algebra (a) irreducible ⇔ H is a polarization at x (: subordinate subgroup Classical such that the bound dim(G / H) � 1 2 dim(X) is attained); Quantum (b) equivalent to Ind( x , H ′ ) if H � = H ′ are two polarizations at x . Nilpotent Reductive E(3) In contrast: Theorem Let H ⊂ G be subordinate to x . Then ind( x , H) : = ind G H e i x ◦ log | H is (a) irreducible ⇔ H is maximal subordinate to x ; (b) inequivalent to ind( x , H ′ ) if H � = H ′ are two polarizations at x . 12 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Remark Souriau Prequantization Kirillov (1962) used Ind( x , H) : = Ind G H e i x ◦ log | H (usual induction). Quantization? This is Group algebra (a) irreducible ⇔ H is a polarization at x (: subordinate subgroup Classical such that the bound dim(G / H) � 1 2 dim(X) is attained); Quantum (b) equivalent to Ind( x , H ′ ) if H � = H ′ are two polarizations at x . Nilpotent Reductive E(3) In contrast: Theorem Let H ⊂ G be subordinate to x . Then ind( x , H) : = ind G H e i x ◦ log | H is (a) irreducible ⇔ H is maximal subordinate to x ; (b) inequivalent to ind( x , H ′ ) if H � = H ′ are two polarizations at x . 12 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Remark Souriau Prequantization Kirillov (1962) used Ind( x , H) : = Ind G H e i x ◦ log | H (usual induction). Quantization? This is Group algebra (a) irreducible ⇔ H is a polarization at x (: subordinate subgroup Classical such that the bound dim(G / H) � 1 2 dim(X) is attained); Quantum (b) equivalent to Ind( x , H ′ ) if H � = H ′ are two polarizations at x . Nilpotent Reductive E(3) In contrast: Theorem Let H ⊂ G be subordinate to x . Then ind( x , H) : = ind G H e i x ◦ log | H is (a) irreducible ⇔ H is maximal subordinate to x ; (b) inequivalent to ind( x , H ′ ) if H � = H ′ are two polarizations at x . 12 / 18

Quantization, Eigenstates in nilpotent groups after Souriau Remark Souriau Prequantization Kirillov (1962) used Ind( x , H) : = Ind G H e i x ◦ log | H (usual induction). Quantization? This is Group algebra (a) irreducible ⇔ H is a polarization at x (: subordinate subgroup Classical such that the bound dim(G / H) � 1 2 dim(X) is attained); Quantum (b) equivalent to Ind( x , H ′ ) if H � = H ′ are two polarizations at x . Nilpotent Reductive E(3) In contrast: Theorem Let H ⊂ G be subordinate to x . Then ind( x , H) : = ind G H e i x ◦ log | H is (a) irreducible ⇔ H is maximal subordinate to x ; (b) inequivalent to ind( x , H ′ ) if H � = H ′ are two polarizations at x . 12 / 18

Quantization, Eigenstates in reductive groups after Souriau Souriau Prequantization Quantization? Group algebra Classical Quantum Nilpotent Reductive E(3) 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 0 otherwise. 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 0 otherwise. It is a quantum eigenstate for X belonging to { x | q }, and GNS m = ind G Q χ . 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 0 otherwise. It is a quantum eigenstate for X belonging to { x | q }, and GNS m = ind G Q χ . 13 / 18

Quantization, Eigenstates in reductive groups after Souriau G : linear reductive Lie group (: ⊂ GL n ( R ), stable under transpose) Souriau g ∗ : identified with g by means of the trace form 〈 Z, Z ′ 〉 = Tr(ZZ ′ ) Prequantization x : hyperbolic element of g ∗ (: diagonalizable with real eigenvalues) Quantization? u : sum of the eigenspaces belonging to positive eigenvalues of ad( x ) Group algebra χ : a character of the parabolic Q = G x exp( u ) with differential i x | q . Classical Quantum Remark: The coadjoint orbit G( x ) = Ind G Q { x | q } (symplectic induction). Nilpotent Reductive Conjecture E(3) There is a unique state m of G that extends χ , namely � χ ( g ) if g ∈ Q, m ( g ) = 0 otherwise. It is a quantum eigenstate for X belonging to { x | q }, and GNS m = ind G Q χ . Theorem: The conjecture is true for G = SL 2 ( R ) or SL 3 ( R ), Q Borel. 13 / 18

� A c : A ∈ SO (3) � � � Quantization, Euclid’s group G = g = after Souriau c ∈ R 3 0 1 Souriau Example: TS 2 Prequantization Quantization? G acts naturally and symplectically on the manifold X ≃ TS 2 of oriented lines Group algebra Classical (a.k.a. light rays) in R 3 . 2-form k , s : Quantum ω = k d 〈 u , d r 〉 + s Area S 2 . Nilpotent r Reductive u The moment map E(3) � r × k u + s u � Φ ( u , r ) = k u makes X into a coadjoint orbit of G. 14 / 18

� A c : A ∈ SO (3) � � � Quantization, Euclid’s group G = g = after Souriau c ∈ R 3 0 1 Souriau Example: TS 2 Prequantization Quantization? G acts naturally and symplectically on the manifold X ≃ TS 2 of oriented lines Group algebra Classical (a.k.a. light rays) in R 3 . 2-form k , s : Quantum ω = k d 〈 u , d r 〉 + s Area S 2 . Nilpotent r Reductive u The moment map E(3) � r × k u + s u � Φ ( u , r ) = k u makes X into a coadjoint orbit of G. 14 / 18