On the Relation Betweeen Gauge and Phase Symmetries Gabriel Catren - PowerPoint PPT Presentation

What is g ∗ useful for? • A Hamiltonian G -manifold ( M, ω, µ ) is not only endowed with a symplectic G -action, ion but also with a map towards g ∗ . ent Map nian G -manifolds t Map g ∗ useful for? • Now, how can we interpret this relation between M and g ∗ ? ’s Orbit Method lectic Points • Roughly speaking, g ∗ encodes the unitary representation theory of G . heories uges to Phases • Firstly, g ∗ is a Poisson manifold with respect to the so-called Lie-Poisson structure ons & Momenta uge Quantization f, g ∈ C ∞ ( g ∗ ) { f, g } ( x ) = � x, [ d f ( x ) , dg ( x )] � ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 7/45

What is g ∗ useful for? • A Hamiltonian G -manifold ( M, ω, µ ) is not only endowed with a symplectic G -action, ion but also with a map towards g ∗ . ent Map nian G -manifolds t Map g ∗ useful for? • Now, how can we interpret this relation between M and g ∗ ? ’s Orbit Method lectic Points • Roughly speaking, g ∗ encodes the unitary representation theory of G . heories uges to Phases • Firstly, g ∗ is a Poisson manifold with respect to the so-called Lie-Poisson structure ons & Momenta uge Quantization f, g ∈ C ∞ ( g ∗ ) { f, g } ( x ) = � x, [ d f ( x ) , dg ( x )] � ic Quantization ase Quantization • Secondly, Symplective leaves of g ∗ = Coadjoint orbits O of G � g ∗ elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 7/45

What is g ∗ useful for? • A Hamiltonian G -manifold ( M, ω, µ ) is not only endowed with a symplectic G -action, ion but also with a map towards g ∗ . ent Map nian G -manifolds t Map g ∗ useful for? • Now, how can we interpret this relation between M and g ∗ ? ’s Orbit Method lectic Points • Roughly speaking, g ∗ encodes the unitary representation theory of G . heories uges to Phases • Firstly, g ∗ is a Poisson manifold with respect to the so-called Lie-Poisson structure ons & Momenta uge Quantization f, g ∈ C ∞ ( g ∗ ) { f, g } ( x ) = � x, [ d f ( x ) , dg ( x )] � ic Quantization ase Quantization • Secondly, Symplective leaves of g ∗ = Coadjoint orbits O of G � g ∗ • The coadjoint orbits O are endowed with a canonical G -invariant symplectic structure ω O . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 7/45

Kirillov’s Orbit Method • For certain G , Kirillov’s orbit method establishes a correspondence ion ent Map g ∗ Z /G ∼ ˆ nian G -manifolds G, t Map g ∗ useful for? (where ˆ G is the unitary dual of G ) given by ’s Orbit Method lectic Points O � H O , heories where H O is the Hilbert space obtained by applying the geometric quantization uges to Phases procedure to the symplectic manifold O ... ons & Momenta or by applying the functor Ind G H to the 1 -dim unirrep ρ H ξ of H = exp ( h ) defined by uge Quantization ξ ∈ O where h ⊂ g is a max. subalg. subordinated to ξ : ic Quantization � ξ, [ h , h ] � = 0 . ase Quantization G-Homogeneous Symplectic Manifolds in g ∗ � Irreducible Unitary Representations of G elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 8/45

Kirillov’s Orbit Method • For certain G , Kirillov’s orbit method establishes a correspondence ion ent Map g ∗ Z /G ∼ ˆ nian G -manifolds G, t Map g ∗ useful for? (where ˆ G is the unitary dual of G ) given by ’s Orbit Method lectic Points O � H O , heories where H O is the Hilbert space obtained by applying the geometric quantization uges to Phases procedure to the symplectic manifold O ... ons & Momenta or by applying the functor Ind G H to the 1 -dim unirrep ρ H ξ of H = exp ( h ) defined by uge Quantization ξ ∈ O where h ⊂ g is a max. subalg. subordinated to ξ : ic Quantization � ξ, [ h , h ] � = 0 . ase Quantization G-Homogeneous Symplectic Manifolds in g ∗ � Irreducible Unitary Representations of G • If G is abelian, each ξ ∈ g ∗ Z is a coadjoint orbit defining a 1 -dim. unirrep of G : ρ ξ : G → U (1) e 2 πi � ξ,X � , e X �→ X ∈ g . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 8/45

Kirillov’s Conjecture • Exactly as the G -action on the homogeneous symplectic orbit O ⊂ g ∗ is lifted to an ion irreducible unitary action on H O ... ent Map lectic Points ... we could expect the G -action on M to be lifted to a unitary action on H M . ’s Conjecture plectic Quotients riction entails a ion ould we interpret ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 9/45

Kirillov’s Conjecture • Exactly as the G -action on the homogeneous symplectic orbit O ⊂ g ∗ is lifted to an ion irreducible unitary action on H O ... ent Map lectic Points ... we could expect the G -action on M to be lifted to a unitary action on H M . ’s Conjecture plectic Quotients riction entails a • Since M is not in gral. G -homogeneous, the lifted unitary action will not in gral. be ion ould we interpret irreducible: ein’s Symplectic Creed ry-Theoretical “Points” � H M = m ( O , M ) H O , ein’s Symplectic O⊂ g ∗ ory” ein’s G -Symplectic where m ( O , M ) . ory” = dim ( Hom G ( H O , H M )) is the multiplicity with which the unirrep al Intertwiner Spaces H O occurs in H M . plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 9/45

Kirillov’s Conjecture • Exactly as the G -action on the homogeneous symplectic orbit O ⊂ g ∗ is lifted to an ion irreducible unitary action on H O ... ent Map lectic Points ... we could expect the G -action on M to be lifted to a unitary action on H M . ’s Conjecture plectic Quotients riction entails a • Since M is not in gral. G -homogeneous, the lifted unitary action will not in gral. be ion ould we interpret irreducible: ein’s Symplectic Creed ry-Theoretical “Points” � H M = m ( O , M ) H O , ein’s Symplectic O⊂ g ∗ ory” ein’s G -Symplectic where m ( O , M ) . ory” = dim ( Hom G ( H O , H M )) is the multiplicity with which the unirrep al Intertwiner Spaces H O occurs in H M . plectic Points of M m Intertwiner Spaces • Kirillov’s conjecture : µ tells which unirreps of G occur in H M . heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 9/45

Kirillov’s Conjecture • Exactly as the G -action on the homogeneous symplectic orbit O ⊂ g ∗ is lifted to an ion irreducible unitary action on H O ... ent Map lectic Points ... we could expect the G -action on M to be lifted to a unitary action on H M . ’s Conjecture plectic Quotients riction entails a • Since M is not in gral. G -homogeneous, the lifted unitary action will not in gral. be ion ould we interpret irreducible: ein’s Symplectic Creed ry-Theoretical “Points” � H M = m ( O , M ) H O , ein’s Symplectic O⊂ g ∗ ory” ein’s G -Symplectic where m ( O , M ) . ory” = dim ( Hom G ( H O , H M )) is the multiplicity with which the unirrep al Intertwiner Spaces H O occurs in H M . plectic Points of M m Intertwiner Spaces • Kirillov’s conjecture : µ tells which unirreps of G occur in H M . heories uges to Phases • Guillemin-Sternberg conjecture : µ also gives m ( O , M ) . ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 9/45

Kirillov’s Conjecture • Exactly as the G -action on the homogeneous symplectic orbit O ⊂ g ∗ is lifted to an ion irreducible unitary action on H O ... ent Map lectic Points ... we could expect the G -action on M to be lifted to a unitary action on H M . ’s Conjecture plectic Quotients riction entails a • Since M is not in gral. G -homogeneous, the lifted unitary action will not in gral. be ion ould we interpret irreducible: ein’s Symplectic Creed ry-Theoretical “Points” � H M = m ( O , M ) H O , ein’s Symplectic O⊂ g ∗ ory” ein’s G -Symplectic where m ( O , M ) . ory” = dim ( Hom G ( H O , H M )) is the multiplicity with which the unirrep al Intertwiner Spaces H O occurs in H M . plectic Points of M m Intertwiner Spaces • Kirillov’s conjecture : µ tells which unirreps of G occur in H M . heories uges to Phases • Guillemin-Sternberg conjecture : µ also gives m ( O , M ) . ons & Momenta • Hence, µ encodes the quantization of M over g ∗ , i.e. the quantization of M with uge Quantization respect to the observable algebra induced by the G -action on M . ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 9/45

ξ -Symplectic Quotients • We must learn how to use µ for “pulling-back” the G -unirreps supported by g ∗ to M . ion Let’s consider the case of an abelian G ... ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion ould we interpret ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 10/45

ξ -Symplectic Quotients • We must learn how to use µ for “pulling-back” the G -unirreps supported by g ∗ to M . ion Let’s consider the case of an abelian G ... ent Map lectic Points • Since the unirreps H ξ ( ξ ∈ g ∗ ) are “supported” by ξ , let’s consider the states in M ’s Conjecture plectic Quotients corresponding to a fixed “value” ξ of the “momentum”, that is riction entails a ion ould we interpret µ − 1 ( ξ ) . ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 10/45

ξ -Symplectic Quotients • We must learn how to use µ for “pulling-back” the G -unirreps supported by g ∗ to M . ion Let’s consider the case of an abelian G ... ent Map lectic Points • Since the unirreps H ξ ( ξ ∈ g ∗ ) are “supported” by ξ , let’s consider the states in M ’s Conjecture plectic Quotients corresponding to a fixed “value” ξ of the “momentum”, that is riction entails a ion ould we interpret µ − 1 ( ξ ) . ein’s Symplectic Creed ry-Theoretical “Points” • Now, the preimage µ − 1 ( ξ ) of the (trivial) symp. manifold O is not a symp. manifold. ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 10/45

ξ -Symplectic Quotients • We must learn how to use µ for “pulling-back” the G -unirreps supported by g ∗ to M . ion Let’s consider the case of an abelian G ... ent Map lectic Points • Since the unirreps H ξ ( ξ ∈ g ∗ ) are “supported” by ξ , let’s consider the states in M ’s Conjecture plectic Quotients corresponding to a fixed “value” ξ of the “momentum”, that is riction entails a ion ould we interpret µ − 1 ( ξ ) . ein’s Symplectic Creed ry-Theoretical “Points” • Now, the preimage µ − 1 ( ξ ) of the (trivial) symp. manifold O is not a symp. manifold. ein’s Symplectic ory” ein’s G -Symplectic ory” • (Shifted) Mardsen-Weinstein reduction theorem : al Intertwiner Spaces plectic Points of M m Intertwiner Spaces M ξ . = µ − 1 ( ξ ) /G heories uges to Phases is a symp. manifold called the ξ -symplectic quotient of M . ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 10/45

ξ -Symplectic Quotients • We must learn how to use µ for “pulling-back” the G -unirreps supported by g ∗ to M . ion Let’s consider the case of an abelian G ... ent Map lectic Points • Since the unirreps H ξ ( ξ ∈ g ∗ ) are “supported” by ξ , let’s consider the states in M ’s Conjecture plectic Quotients corresponding to a fixed “value” ξ of the “momentum”, that is riction entails a ion ould we interpret µ − 1 ( ξ ) . ein’s Symplectic Creed ry-Theoretical “Points” • Now, the preimage µ − 1 ( ξ ) of the (trivial) symp. manifold O is not a symp. manifold. ein’s Symplectic ory” ein’s G -Symplectic ory” • (Shifted) Mardsen-Weinstein reduction theorem : al Intertwiner Spaces plectic Points of M m Intertwiner Spaces M ξ . = µ − 1 ( ξ ) /G heories uges to Phases is a symp. manifold called the ξ -symplectic quotient of M . ons & Momenta uge Quantization • So, M ξ is the symp. counterpart of ξ in M . ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 10/45

A Restriction entails a Projection • In gauge-theoretic terms, when we fix the “value” of the “momentum” µ to ξ by means ion of the restriction to the “ ξ -constraint surface” ent Map lectic Points ’s Conjecture µ − 1 ( ξ ) ⊂ M, plectic Quotients riction entails a ion ould we interpret ... the “conjugate coordinate” acted upon by G becomes completely “ undetermined ”... ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic .... in the sense that it is “ gauged out ” by means of the projection ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M µ − 1 ( ξ ) ։ M ξ . m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 11/45

How should we interpret M ξ ? • We shall argue... ion ent Map 1) that the ξ -symplectic quotient lectic Points ’s Conjecture plectic Quotients M ξ . riction entails a = µ − 1 ( ξ ) /G ion ould we interpret is the “ moduli space ” parameterizing the category-theoretical symplectic ξ -points of ein’s Symplectic Creed M . ry-Theoretical “Points” ein’s Symplectic ory” ein’s G -Symplectic 2) that the notion of symplectic point elicits a category-theoretical ory” al Intertwiner Spaces interpretation of Heisenberg indeterminacy principle. plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 12/45

Weinstein’s Symplectic Creed ion “The Heisenberg uncertainty principle says that it is impossible to ent Map determine simultaneously the position and momentum of a lectic Points quantum-mechanical particle. This can be rephrased as follows: the ’s Conjecture smallest subsets of classical phase space in which the presence of a plectic Quotients riction entails a quantum-mechanical particle can be detected are its Lagrangian ion ould we interpret submanifolds. For this reason it makes sense to regard the Lagrangian submanifolds of phase space as being its true “points”.” ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic V. Guillemin and S. Sternberg, Geometric Quantization and Multiplicities of ory” Group Representations , 1982. ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 13/45

Weinstein’s Symplectic Creed ion “The Heisenberg uncertainty principle says that it is impossible to ent Map determine simultaneously the position and momentum of a lectic Points quantum-mechanical particle. This can be rephrased as follows: the ’s Conjecture smallest subsets of classical phase space in which the presence of a plectic Quotients riction entails a quantum-mechanical particle can be detected are its Lagrangian ion ould we interpret submanifolds. For this reason it makes sense to regard the Lagrangian submanifolds of phase space as being its true “points”.” ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic V. Guillemin and S. Sternberg, Geometric Quantization and Multiplicities of ory” Group Representations , 1982. ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories • This notion of Lagrangian true “points” acquires a precise category-theoretical uges to Phases meaning in the framework of Weinstein’s symplectic “category”. ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 13/45

Category-Theoretical “Points” • A point x in a manifold M can be identified with the morphism ion ent Map lectic Points ϕ x : {∗} → M ’s Conjecture plectic Quotients riction entails a given by ion ould we interpret {∗} �→ x ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 14/45

Category-Theoretical “Points” • A point x in a manifold M can be identified with the morphism ion ent Map lectic Points ϕ x : {∗} → M ’s Conjecture plectic Quotients riction entails a given by ion ould we interpret {∗} �→ x ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic • More generally, given two objects A and B in a category, the morphisms ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M B → A m Intertwiner Spaces heories define the so-called B -points of A . uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 14/45

Weinstein’s Symplectic “Category” • Objects : ion ent Map .Symplectic manifolds ( M, ω ) . lectic Points ’s Conjecture plectic Quotients riction entails a ion ould we interpret ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 15/45

Weinstein’s Symplectic “Category” • Objects : ion ent Map .Symplectic manifolds ( M, ω ) . lectic Points • Morphisms (or Lagrangian correspondences ) ( M 2 , ω 2 ) → ( M 1 , ω 1 ) : ’s Conjecture plectic Quotients riction entails a ion � � → M 1 × M − Hom Symp ( M 2 , M 1 ) = L 2 , 1 ֒ ould we interpret 2 ein’s Symplectic Creed 2 , − ω 2 ) is the dual of ( M 2 , ω 2 ) and where ( M − ry-Theoretical “Points” ein’s Symplectic ory” ( M 1 × M − 2 , π ∗ 1 ω 1 − π ∗ 2 ω 2 ) , ein’s G -Symplectic ory” al Intertwiner Spaces is the product symplectic manifold. plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 15/45

Weinstein’s Symplectic “Category” • Objects : ion ent Map .Symplectic manifolds ( M, ω ) . lectic Points • Morphisms (or Lagrangian correspondences ) ( M 2 , ω 2 ) → ( M 1 , ω 1 ) : ’s Conjecture plectic Quotients riction entails a ion � � → M 1 × M − Hom Symp ( M 2 , M 1 ) = L 2 , 1 ֒ ould we interpret 2 ein’s Symplectic Creed 2 , − ω 2 ) is the dual of ( M 2 , ω 2 ) and where ( M − ry-Theoretical “Points” ein’s Symplectic ory” ( M 1 × M − 2 , π ∗ 1 ω 1 − π ∗ 2 ω 2 ) , ein’s G -Symplectic ory” al Intertwiner Spaces is the product symplectic manifold. plectic Points of M m Intertwiner Spaces • In particular, a symplectomorphism defines a Lagrangian corresp. heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 15/45

Weinstein’s Symplectic “Category” • Objects : ion ent Map .Symplectic manifolds ( M, ω ) . lectic Points • Morphisms (or Lagrangian correspondences ) ( M 2 , ω 2 ) → ( M 1 , ω 1 ) : ’s Conjecture plectic Quotients riction entails a ion � � → M 1 × M − Hom Symp ( M 2 , M 1 ) = L 2 , 1 ֒ ould we interpret 2 ein’s Symplectic Creed 2 , − ω 2 ) is the dual of ( M 2 , ω 2 ) and where ( M − ry-Theoretical “Points” ein’s Symplectic ory” ( M 1 × M − 2 , π ∗ 1 ω 1 − π ∗ 2 ω 2 ) , ein’s G -Symplectic ory” al Intertwiner Spaces is the product symplectic manifold. plectic Points of M m Intertwiner Spaces • In particular, a symplectomorphism defines a Lagrangian corresp. heories uges to Phases • The symplectic points of ( M, ω ) is given by the morphisms in ons & Momenta uge Quantization Hom Symp (( ∗ , 0) , ( M, ω )) = { L ֒ → M × {∗} ≃ M } , ic Quantization i.e. by the Lagrangian submanifolds of M . ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 15/45

Weinstein’s G -Symplectic “Category” • Objects : ion .Hamiltonian G -manifolds ( M, ω, µ ) . ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion ould we interpret ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 16/45

� � Weinstein’s G -Symplectic “Category” • Objects : ion .Hamiltonian G -manifolds ( M, ω, µ ) . ent Map lectic Points • Morphisms ( M 2 , ω 2 , µ 2 ) → ( M 1 , ω 1 , µ 1 ) : ’s Conjecture plectic Quotients riction entails a ion L 2 , 1 � � � M 1 × g ∗ M − � M − ould we interpret 2 2 ein’s Symplectic Creed µ 2 ry-Theoretical “Points” ein’s Symplectic ory” � g ∗ M 1 ein’s G -Symplectic µ 1 ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 16/45

� � Weinstein’s G -Symplectic “Category” • Objects : ion .Hamiltonian G -manifolds ( M, ω, µ ) . ent Map lectic Points • Morphisms ( M 2 , ω 2 , µ 2 ) → ( M 1 , ω 1 , µ 1 ) : ’s Conjecture plectic Quotients riction entails a ion L 2 , 1 � � � M 1 × g ∗ M − � M − ould we interpret 2 2 ein’s Symplectic Creed µ 2 ry-Theoretical “Points” ein’s Symplectic ory” � g ∗ M 1 ein’s G -Symplectic µ 1 ory” al Intertwiner Spaces plectic Points of M • In other terms, m Intertwiner Spaces heories � � → Φ − 1 (0) ⊂ M 1 × M − Hom G - Symp ( M 2 , M 1 ) = L 2 , 1 ֒ , 2 uges to Phases where ons & Momenta uge Quantization 2 ω 2 , Φ . ( M 1 × M − 2 , π ∗ 1 ω 1 − π ∗ = µ 1 − µ 2 ) , ic Quantization is the product Hamiltonian G -manifold. ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 16/45

Classical Intertwiner Spaces • It can be shown ( ♣ ) that L 2 , 1 ⊂ M 1 × g ∗ M − 2 are G -invariant... ion ent Map ... and that there is a bijection lectic Points ’s Conjecture plectic Quotients � � L ⊂ ( M 1 × g ∗ M − Hom G - Symp ( M 2 , M 1 ) ≃ 2 ) /G . riction entails a ion ould we interpret ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta ____________________________________________________________________________ uge Quantization ♣ Xu, P . [1994]: “Classical Intertwiner Space and Quantization,” Commun. Math. Phys. 164, 473-488. ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 17/45

Classical Intertwiner Spaces • It can be shown ( ♣ ) that L 2 , 1 ⊂ M 1 × g ∗ M − 2 are G -invariant... ion ent Map ... and that there is a bijection lectic Points ’s Conjecture plectic Quotients � � L ⊂ ( M 1 × g ∗ M − Hom G - Symp ( M 2 , M 1 ) ≃ 2 ) /G . riction entails a ion ould we interpret ein’s Symplectic Creed ry-Theoretical “Points” • Under nice conditions, ( M 1 × g ∗ M − 2 ) /G is a symplectic manifold... ein’s Symplectic ory” ein’s G -Symplectic ... whose symplectic points are the classical intertwiners over g ∗ between M 2 and ory” al Intertwiner Spaces M 1 ... plectic Points of M m Intertwiner Spaces heories ... or, in category-theoretical terms, the M 2 -sympletic points of M 1 . uges to Phases ons & Momenta ____________________________________________________________________________ uge Quantization ♣ Xu, P . [1994]: “Classical Intertwiner Space and Quantization,” Commun. Math. Phys. 164, 473-488. ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 17/45

ξ -Symplectic Points of M • In particular, the morphisms ( ξ, 0 , µ ξ : ξ �→ ξ ) → ( M, ω, µ ) are given by the ion Lagrang. subman. of ent Map lectic Points ( M × g ∗ ξ − ) /G = Φ − 1 (0) /G, ’s Conjecture plectic Quotients where the twisted moment map is riction entails a ion ould we interpret Φ : M × ξ − g ∗ → ein’s Symplectic Creed ry-Theoretical “Points” ( m, ξ ) �→ µ ( m ) − ξ. ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 18/45

ξ -Symplectic Points of M • In particular, the morphisms ( ξ, 0 , µ ξ : ξ �→ ξ ) → ( M, ω, µ ) are given by the ion Lagrang. subman. of ent Map lectic Points ( M × g ∗ ξ − ) /G = Φ − 1 (0) /G, ’s Conjecture plectic Quotients where the twisted moment map is riction entails a ion ould we interpret Φ : M × ξ − g ∗ → ein’s Symplectic Creed ry-Theoretical “Points” ( m, ξ ) �→ µ ( m ) − ξ. ein’s Symplectic ory” ein’s G -Symplectic ory” • Now, since Φ − 1 (0) ≃ µ − 1 ( ξ ) ⊂ M , then Φ − 1 (0) /G ≃ M ξ . al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 18/45

ξ -Symplectic Points of M • In particular, the morphisms ( ξ, 0 , µ ξ : ξ �→ ξ ) → ( M, ω, µ ) are given by the ion Lagrang. subman. of ent Map lectic Points ( M × g ∗ ξ − ) /G = Φ − 1 (0) /G, ’s Conjecture plectic Quotients where the twisted moment map is riction entails a ion ould we interpret Φ : M × ξ − g ∗ → ein’s Symplectic Creed ry-Theoretical “Points” ( m, ξ ) �→ µ ( m ) − ξ. ein’s Symplectic ory” ein’s G -Symplectic ory” • Now, since Φ − 1 (0) ≃ µ − 1 ( ξ ) ⊂ M , then Φ − 1 (0) /G ≃ M ξ . al Intertwiner Spaces plectic Points of M m Intertwiner Spaces • All in all, heories Hom G - Symp ( ξ, M ) = { L ⊂ M ξ } uges to Phases ons & Momenta i.e. the symplectic points of M ξ are in correspondence with the ξ -points of M . uge Quantization M ξ can be interpreted as the moduli space of symplectic ξ -points of M . ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 18/45

Quantum Intertwiner Spaces • Guillemin and Sternberg (1982) showed (for particular M and G ) that the (geometric) ion quantization of the classical intertwiner space: ent Map lectic Points M O ∼ = Hom G - Symp ( O , M ) ’s Conjecture plectic Quotients between a coadjoint orbit O and M yields the quantum intertwiner space: riction entails a ion ould we interpret H M O ∼ = Hom G ( H O . H M ) , ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic ory” ein’s G -Symplectic ory” al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 19/45

Quantum Intertwiner Spaces • Guillemin and Sternberg (1982) showed (for particular M and G ) that the (geometric) ion quantization of the classical intertwiner space: ent Map lectic Points M O ∼ = Hom G - Symp ( O , M ) ’s Conjecture plectic Quotients between a coadjoint orbit O and M yields the quantum intertwiner space: riction entails a ion ould we interpret H M O ∼ = Hom G ( H O . H M ) , ein’s Symplectic Creed ry-Theoretical “Points” • Whereas M O parameterizes the symplectic G -morphisms ein’s Symplectic ory” ein’s G -Symplectic O → M, ory” al Intertwiner Spaces plectic Points of M H M O parameterizes the unitary G -intertwiners m Intertwiner Spaces H O → H M heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 19/45

Quantum Intertwiner Spaces • Guillemin and Sternberg (1982) showed (for particular M and G ) that the (geometric) ion quantization of the classical intertwiner space: ent Map lectic Points M O ∼ = Hom G - Symp ( O , M ) ’s Conjecture plectic Quotients between a coadjoint orbit O and M yields the quantum intertwiner space: riction entails a ion ould we interpret H M O ∼ = Hom G ( H O . H M ) , ein’s Symplectic Creed ry-Theoretical “Points” • Whereas M O parameterizes the symplectic G -morphisms ein’s Symplectic ory” ein’s G -Symplectic O → M, ory” al Intertwiner Spaces plectic Points of M H M O parameterizes the unitary G -intertwiners m Intertwiner Spaces H O → H M heories uges to Phases ons & Momenta • Kirillov’s conjecture revisited : The unirrep H O occurs in H M if M has symplectic uge Quantization O -points where the multiplicity is given by ic Quantization m ( O , M ) = dim ( H M O ) . ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 19/45

Mardsen-Weinstein 0 -Reduction • A gauge theory is a Ham. G -manifold ( M, ω, µ ) such that the Ham. equations ion constraint the solutions to be in the constraint surface ent Map Σ = µ − 1 (0) ⊂ M. lectic Points heories n-Weinstein uction zation Commutes with uction uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 20/45

�� � Mardsen-Weinstein 0 -Reduction • A gauge theory is a Ham. G -manifold ( M, ω, µ ) such that the Ham. equations ion constraint the solutions to be in the constraint surface ent Map Σ = µ − 1 (0) ⊂ M. lectic Points heories n-Weinstein • The Mardsen-Weinstein reduction theorem shows that the 0 -symplectic quotient uction zation Commutes with M 0 ≃ µ − 1 (0) /G uction uges to Phases has a canonical symplectic form ω M 0 satisfying ons & Momenta uge Quantization π ∗ ω M 0 = ι ∗ ω M , ic Quantization where ase Quantization M (0) � � ι µ − 1 M π M 0 . = µ − 1 (0) /G. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 20/45

Quantization Commutes with 0 -Reduction • In this case, the Guillemin-Sternberg conjecture is ion ent Map H M 0 ∼ = Hom G ( H 0 , H M ) , lectic Points or, in other terms, heories H M 0 ≃ H G n-Weinstein M , uction zation Commutes with where H G M is the space of G -invariant states in H M . uction uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 21/45

� � � � � � � � � � ��� � ��� � � � � � � � � � Quantization Commutes with 0 -Reduction • In this case, the Guillemin-Sternberg conjecture is ion ent Map H M 0 ∼ = Hom G ( H 0 , H M ) , lectic Points or, in other terms, heories H M 0 ≃ H G n-Weinstein M , uction zation Commutes with where H G M is the space of G -invariant states in H M . uction uges to Phases • Diagramatically, ons & Momenta Quantization uge Quantization H M M ic Quantization 0 -symplectic reduction G -invariant states ase Quantization H M 0 ≃ H G M 0 M , Quantum G -invariance � Classical 0 -symplectic reduction. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 21/45

Gauge Theories • A gauge theory is a given by a Ham. G -manifold such that the restriction of the theory ion to the classical intertwiner space ent Map lectic Points 0 ∈ g ∗ M 0 ≃ Hom (0 , M ) heories uges to Phases Theories containing the 0 -symplectic points of M ... Groups vs. Phase mplectic Reductions ... implies that the resulting quantum theory only includes G -invariant states. zation Commutes with uction Invariance vs. Phase nce ymplectic to Phase tries ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 22/45

Gauge Theories • A gauge theory is a given by a Ham. G -manifold such that the restriction of the theory ion to the classical intertwiner space ent Map lectic Points 0 ∈ g ∗ M 0 ≃ Hom (0 , M ) heories uges to Phases Theories containing the 0 -symplectic points of M ... Groups vs. Phase mplectic Reductions ... implies that the resulting quantum theory only includes G -invariant states. zation Commutes with uction Invariance vs. Phase nce ymplectic to Phase • In this case, G is called the gauge group of the theory and the generating functions of tries the G -action ons & Momenta uge Quantization G i ( m ) = � µ ( m ) , X i � X i ∈ g ic Quantization ase Quantization are called constraints . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 22/45

Gauge Groups vs. Phase Groups • We are here interested in ordinary theories defined on a Ham. G -manifolds ( M, ω, µ ) , ion ent Map lectic Points ... where by ordinary we mean that the theories are not constrained to a unique heories value of µ . uges to Phases Theories Groups vs. Phase mplectic Reductions zation Commutes with uction Invariance vs. Phase nce ymplectic to Phase tries ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 23/45

Gauge Groups vs. Phase Groups • We are here interested in ordinary theories defined on a Ham. G -manifolds ( M, ω, µ ) , ion ent Map lectic Points ... where by ordinary we mean that the theories are not constrained to a unique heories value of µ . uges to Phases Theories Groups vs. Phase • We shall call G the phase group and the non-constrained generating functions of the mplectic Reductions zation Commutes with G -action uction Invariance vs. Phase nce ymplectic to Phase f i ( m ) = � µ ( m ) , X i � tries ons & Momenta phase observables . uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 23/45

O -Symplectic Reductions • Differently from the constraints G a , the phase observables f i do not select a single ion unirrep of G . ent Map lectic Points heories uges to Phases Theories Groups vs. Phase mplectic Reductions zation Commutes with uction Invariance vs. Phase nce ymplectic to Phase tries ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 24/45

O -Symplectic Reductions • Differently from the constraints G a , the phase observables f i do not select a single ion unirrep of G . ent Map lectic Points • Therefore, while a gauge group action defines a unique 0 -symplectic quotient heories M 0 . = µ − 1 (0) /G, uges to Phases Theories Groups vs. Phase mplectic Reductions ... associated to the trivial unirrep of G ... zation Commutes with uction Invariance vs. Phase nce ymplectic to Phase ... a phase group action defines a different O -symplectic quotient tries ons & Momenta M O . = µ − 1 ( O ) /G uge Quantization ic Quantization for each unirrep O ⊂ g ∗ Z of the phase group G . ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 24/45

O -Symplectic Reductions • Differently from the constraints G a , the phase observables f i do not select a single ion unirrep of G . ent Map lectic Points • Therefore, while a gauge group action defines a unique 0 -symplectic quotient heories M 0 . = µ − 1 (0) /G, uges to Phases Theories Groups vs. Phase mplectic Reductions ... associated to the trivial unirrep of G ... zation Commutes with uction Invariance vs. Phase nce ymplectic to Phase ... a phase group action defines a different O -symplectic quotient tries ons & Momenta M O . = µ − 1 ( O ) /G uge Quantization ic Quantization for each unirrep O ⊂ g ∗ Z of the phase group G . ase Quantization • The phase G -action on M entails the existence of a whole set of O -symplectic quotients M O . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 24/45

� � � � � � � � � � � ��� � ��� � � � � � � � � � Quantization Commutes with ξ -Reduction • The Guillemin & Sternberg’s conjecture for a ξ -symplectic quotient with G abelian ion states that this diagram commutes: ent Map lectic Points Quantization heories H M M uges to Phases Theories ξ -symplectic reduction ( G,ξ ) -phase invariant states Groups vs. Phase mplectic Reductions H Mξ ≃ H ( G,ξ ) M ξ , zation Commutes with M uction Invariance vs. Phase nce . H ( G,ξ ) is the space of ( G, ξ ) -phase invariant states in H M ,... ymplectic to Phase M tries ons & Momenta ... i.e. the states that are invariant modulo a phase factor given by the 1 -dim. unirrep ρ G ξ of G defined by ξ : uge Quantization ic Quantization ase Quantization ξ : G × H ( G,ξ ) H ( G,ξ ) ρ G → M M ( e X , | ξ, ... � ) e 2 πi � ξ,X � | ξ, ... � , �→ elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 25/45

Gauge Invariance vs. Phase Invariance • The ( G, ξ ) -phase invariance of quantum states is the quantum counterpart of the ion symplectic reduction with respect to a non-zero ξ ∈ g ∗ . ent Map lectic Points heories uges to Phases Theories Quantum phase invariance is the generalization... Groups vs. Phase mplectic Reductions zation Commutes with ... of the strict gauge invariance... uction Invariance vs. Phase nce ymplectic to Phase .... to the case of ξ -symplectic reductions with ξ � = 0 . tries ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 26/45

From Symplectic to Phase Symmetries • We have argued that the existence of ( G, ξ ) -phase invariant states in H M ... ion ent Map ... results from the existence of ξ -symplectic points in M . lectic Points heories uges to Phases Theories Groups vs. Phase mplectic Reductions zation Commutes with uction Invariance vs. Phase nce ymplectic to Phase tries ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 27/45

From Symplectic to Phase Symmetries • We have argued that the existence of ( G, ξ ) -phase invariant states in H M ... ion ent Map ... results from the existence of ξ -symplectic points in M . lectic Points heories • Far from being structureless set-theoretic points,... uges to Phases Theories Groups vs. Phase ... the ξ -symplectic points of M are non-trivial subman. of M endowed with a mplectic Reductions G -action. zation Commutes with uction Invariance vs. Phase nce ymplectic to Phase tries ons & Momenta uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 27/45

From Symplectic to Phase Symmetries • We have argued that the existence of ( G, ξ ) -phase invariant states in H M ... ion ent Map ... results from the existence of ξ -symplectic points in M . lectic Points heories • Far from being structureless set-theoretic points,... uges to Phases Theories Groups vs. Phase ... the ξ -symplectic points of M are non-trivial subman. of M endowed with a mplectic Reductions G -action. zation Commutes with uction Invariance vs. Phase nce ymplectic to Phase • The ( G, ξ ) -phase invariance of the states | ξ, ... � ,... tries ons & Momenta ... i.e. the “indeterminacy” in the variable acted upon by G ... uge Quantization ic Quantization ... is the quantum counterpart of the fact... ase Quantization ... that the corresponding ξ -symplectic points of M have an internal G -symmetry. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 27/45

For Instance... • Let’s consider the simplest case of G = R acting on M = T ∗ R by ion ent Map q 0 · ( q, p ) �→ ( q + q 0 , p ) lectic Points heories with moment map uges to Phases ons & Momenta tance... µ : T ∗ R → R g Out the Position plectic Localization ( q, p ) �→ p g the Phase nce Group Actions vs... se Group Actions uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 28/45

For Instance... • Let’s consider the simplest case of G = R acting on M = T ∗ R by ion ent Map q 0 · ( q, p ) �→ ( q + q 0 , p ) lectic Points heories with moment map uges to Phases ons & Momenta tance... µ : T ∗ R → R g Out the Position plectic Localization ( q, p ) �→ p g the Phase nce Group Actions vs... se Group Actions • The p i -symplectic quotient uge Quantization ic Quantization M pi = µ − 1 ( p i ) /G = {∗} ase Quantization contains the unique symplectic p i -point of M . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 28/45

Phasing Out the Position • According to ion ent Map lectic Points [ Quantization , ξ -Reduction ] = 0 , heories uges to Phases the ( 1 -dimensional) quantization of M pi yields the (unique) ( G, p i ) -phase invariant state | p i � in H M : ons & Momenta tance... g Out the Position plectic Localization q 0 · | p i � �→ e 2 πiq 0 pi | p i � ≈ | p i � . g the Phase nce Group Actions vs... se Group Actions uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 29/45

Phasing Out the Position • According to ion ent Map lectic Points [ Quantization , ξ -Reduction ] = 0 , heories uges to Phases the ( 1 -dimensional) quantization of M pi yields the (unique) ( G, p i ) -phase invariant state | p i � in H M : ons & Momenta tance... g Out the Position plectic Localization q 0 · | p i � �→ e 2 πiq 0 pi | p i � ≈ | p i � . g the Phase nce Group Actions vs... se Group Actions • The indeterminacy in the position q of the state | p i � is a symptom of the fact... uge Quantization ic Quantization ... that the unique symplectic p i -point of M has an internal symmetry under ase Quantization translations in q . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 29/45

On Symplectic Localization • The category-theoretical notion of symplectic point seems to be the symplectic seed of ion Heisenberg indeterminacy principle. ent Map lectic Points heories uges to Phases ons & Momenta tance... g Out the Position plectic Localization g the Phase nce Group Actions vs... se Group Actions uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 30/45

On Symplectic Localization • The category-theoretical notion of symplectic point seems to be the symplectic seed of ion Heisenberg indeterminacy principle. ent Map lectic Points heories • The (im)possibility of sharply localizing quantum states in phase space depends on the uges to Phases notion of point that we are using: ons & Momenta tance... g Out the Position plectic Localization ... while a quantum state cannot be sharply localized at the set-theoretic points of g the Phase M ,... nce Group Actions vs... se Group Actions uge Quantization ... it can be sharply localized at its symplectic point... ic Quantization ase Quantization ... given that the symplectic points “internalize” the unsharp variables. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 30/45

Breaking the Phase Invariance • The superposition of two G -phase invariant states | p i � and | p j � transforming in ion different unirreps of G ,... ent Map lectic Points ... is no longer G -phase invariant... heories uges to Phases ... since the G -action changes the relative phases between the two terms ons & Momenta | p i � + | p j � . tance... g Out the Position plectic Localization g the Phase nce Group Actions vs... se Group Actions uge Quantization ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 31/45

Breaking the Phase Invariance • The superposition of two G -phase invariant states | p i � and | p j � transforming in ion different unirreps of G ,... ent Map lectic Points ... is no longer G -phase invariant... heories uges to Phases ... since the G -action changes the relative phases between the two terms ons & Momenta | p i � + | p j � . tance... g Out the Position plectic Localization g the Phase nce • Therefore, by introducing an “indeterminacy” in the value of the variable p that Group Actions vs... se Group Actions labels the unirreps of G ,... uge Quantization ... we break the G -phase invariance ,... ic Quantization ase Quantization ... i.e. the complete indeterminacy in the variable q acted upon by G . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 31/45

Gauge Group Actions vs... ion A gauge theory is restricted to a single value of µ (namely 0 ) ent Map lectic Points ⇓ heories uges to Phases The quantum theory only contains G -invariant quantum states ons & Momenta tance... g Out the Position plectic Localization (i.e. states transforming in the trivial unirrep of G ) g the Phase nce Group Actions vs... se Group Actions ⇓ uge Quantization ic Quantization Since we do not have different unirreps to superpose... ase Quantization ⇓ ... the gauge invariance cannot be broken. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 32/45

... Phase Group Actions ion Phase observables are not restricted to a single value of µ ent Map lectic Points ⇓ heories uges to Phases The quantum theory contains ( G, ξ ) -phase invariant states for all ξ ∈ g ∗ ons & Momenta Z tance... g Out the Position (i.e. states transforming in different unirreps of G ) plectic Localization g the Phase nce Group Actions vs... ⇓ se Group Actions uge Quantization ic Quantization We can superpose ( G, ξ ) -phase invariant states defined by different unirreps ξ . ase Quantization ⇓ The G -phase invariance is broken for such superposed states. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 33/45

Cohomological Reduction ion The Dirac observables f ∈ C ∞ ( M 0 ) of a gauge theory can be recovered... ent Map lectic Points heories uges to Phases ... by means of the BRST-cohomological reformulation of the 0 -symplectic reduction. ons & Momenta uge Quantization ological Reduction Theories in a Diagram Resolution ebra Cohomology Cohomology ic Quantization ase Quantization ____________________________________________________________________________ ♣ Kostant, B. & Sternberg, S. [1987]: “Symplectic Reduction, BRS Cohomology, and Infinite Dimensional Clifford Algebras,” Annals of Physics 176, 49-113. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 34/45

Gauge Theories in a Diagram ion ent Map lectic Points heories uges to Phases ons & Momenta uge Quantization ological Reduction Theories in a Diagram Resolution ebra Cohomology Cohomology ic Quantization ase Quantization The reduction from M to M 0 . = Σ /G is a two-step procedure: . Restriction from M to Σ � Koszul resolution of C ∞ (Σ) . . Projection from Σ to Σ /G � Lie algebra cohomology of g with values in the g -module ∧ g ⊗ C ∞ ( M ) . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 35/45

Koszul Resolution • We can describe the algebra of observables on Σ ion ent Map C ∞ (Σ) = C ∞ ( M ) / � G a � lectic Points in homological terms by extending the (co-moment) map heories C ∞ ( M ) uges to Phases → g ons & Momenta X i �→ f i ( m ) = � µ ( m ) , X i � uge Quantization ological Reduction to the quasi-acyclic complex Theories in a Diagram Resolution δ 1 δ 0 ... ∧ q g ⊗ C ∞ ( M ) → ∧ q − 1 g ⊗ C ∞ ( M ) → ... → g ⊗ C ∞ ( M ) → C ∞ ( M ) ebra Cohomology − − − − → 0 , Cohomology where the Koszul differential is defined by ic Quantization ase Quantization δ ( X i ⊗ 1) = 1 ⊗ f i , δ (1 ⊗ f ) = 0 . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 36/45

Koszul Resolution • We can describe the algebra of observables on Σ ion ent Map C ∞ (Σ) = C ∞ ( M ) / � G a � lectic Points in homological terms by extending the (co-moment) map heories C ∞ ( M ) uges to Phases → g ons & Momenta X i �→ f i ( m ) = � µ ( m ) , X i � uge Quantization ological Reduction to the quasi-acyclic complex Theories in a Diagram Resolution δ 1 δ 0 ... ∧ q g ⊗ C ∞ ( M ) → ∧ q − 1 g ⊗ C ∞ ( M ) → ... → g ⊗ C ∞ ( M ) → C ∞ ( M ) ebra Cohomology − − − − → 0 , Cohomology where the Koszul differential is defined by ic Quantization ase Quantization δ ( X i ⊗ 1) = 1 ⊗ f i , δ (1 ⊗ f ) = 0 . • Hence, H δ 0 ( ∧ g ⊗ C ∞ ( M )) = Ker ( δ 0 ) /Im ( δ 1 ) C ∞ ( M ) / � G a � = C ∞ (Σ) . = elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 36/45

Lie Algebra Cohomology • Given the g -module K = ∧ g ⊗ C ∞ ( M ) , we can define the vertical differential (or ion Chevalley-Eilenberg differential ) ent Map lectic Points d : K → g ∗ ⊗ K = Hom ( g , K ) heories given by uges to Phases ons & Momenta ( dk )( X ) = X · k, X ∈ g , k ∈ K. uge Quantization ological Reduction Theories in a Diagram Resolution ebra Cohomology Cohomology ic Quantization ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 37/45

Lie Algebra Cohomology • Given the g -module K = ∧ g ⊗ C ∞ ( M ) , we can define the vertical differential (or ion Chevalley-Eilenberg differential ) ent Map lectic Points d : K → g ∗ ⊗ K = Hom ( g , K ) heories given by uges to Phases ons & Momenta ( dk )( X ) = X · k, X ∈ g , k ∈ K. uge Quantization ological Reduction • This can be extended to a map Theories in a Diagram Resolution d : ∧ q g ∗ ⊗ K → ∧ q +1 g ∗ ⊗ K, ebra Cohomology Cohomology by ic Quantization d ( η ⊗ k ) = dη ⊗ k + ( − 1) p η ⊗ dk, η ∈ ∧ p g ∗ . ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 37/45

Lie Algebra Cohomology • Given the g -module K = ∧ g ⊗ C ∞ ( M ) , we can define the vertical differential (or ion Chevalley-Eilenberg differential ) ent Map lectic Points d : K → g ∗ ⊗ K = Hom ( g , K ) heories given by uges to Phases ons & Momenta ( dk )( X ) = X · k, X ∈ g , k ∈ K. uge Quantization ological Reduction • This can be extended to a map Theories in a Diagram Resolution d : ∧ q g ∗ ⊗ K → ∧ q +1 g ∗ ⊗ K, ebra Cohomology Cohomology by ic Quantization d ( η ⊗ k ) = dη ⊗ k + ( − 1) p η ⊗ dk, η ∈ ∧ p g ∗ . ase Quantization • The 0 -Lie algebra cohomology of g with values in the g -module K is given by d ( ∧ g ∗ ⊗ K ) = K g . H 0 elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 37/45

� � BRST Cohomology • We can then form the double complex ion ent Map lectic Points ∧ p g ∗ ⊗ ∧ q g ⊗ C ∞ ( M ) δ ∧ p g ∗ ⊗ ∧ q − 1 g ⊗ C ∞ ( M ) heories uges to Phases d ons & Momenta ∧ p +1 g ∗ ⊗ ∧ q g ⊗ C ∞ ( M ) uge Quantization ological Reduction with Theories in a Diagram Resolution ebra Cohomology δ 2 = 0 d 2 = 0 Cohomology δd = dδ ic Quantization such that ase Quantization 0 ( ∧ g ∗ ⊗ ∧ g ⊗ C ∞ ( M ))) = C ∞ ( M 0 ) . H 0 d ( H δ elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 38/45

� � BRST Cohomology • We can then form the double complex ion ent Map lectic Points ∧ p g ∗ ⊗ ∧ q g ⊗ C ∞ ( M ) δ ∧ p g ∗ ⊗ ∧ q − 1 g ⊗ C ∞ ( M ) heories uges to Phases d ons & Momenta ∧ p +1 g ∗ ⊗ ∧ q g ⊗ C ∞ ( M ) uge Quantization ological Reduction with Theories in a Diagram Resolution ebra Cohomology δ 2 = 0 d 2 = 0 Cohomology δd = dδ ic Quantization such that ase Quantization 0 ( ∧ g ∗ ⊗ ∧ g ⊗ C ∞ ( M ))) = C ∞ ( M 0 ) . H 0 d ( H δ • Hence, the Dirac observables in C ∞ ( M 0 ) can be described as elements f 0 0 that are d -closed modulo δ : f 0 0 = − δf 1 d 1 ≈ Σ 0 . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 38/45

Pre-Quantum Geometry in a Nutshell • Given a Hamiltonian G -manifold ( M, ω, µ ) such that ω satisfies the integrality ion condition (c.f. Kostant, Souriau, Kirillov) according to which [ ω ] is in the image of the ent Map map lectic Points heories H Cech ( M, Z ) → H De Rahm ( M ) uges to Phases ... there exists a complex line bundle L → M with a Hermitian inner product �· , ·� and ons & Momenta a compatible connection ∇ such that uge Quantization ic Quantization antum Geometry in a curv ( ∇ ) = 2 πiω. ll ations in a Nutshell ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 39/45

Pre-Quantum Geometry in a Nutshell • Given a Hamiltonian G -manifold ( M, ω, µ ) such that ω satisfies the integrality ion condition (c.f. Kostant, Souriau, Kirillov) according to which [ ω ] is in the image of the ent Map map lectic Points heories H Cech ( M, Z ) → H De Rahm ( M ) uges to Phases ... there exists a complex line bundle L → M with a Hermitian inner product �· , ·� and ons & Momenta a compatible connection ∇ such that uge Quantization ic Quantization antum Geometry in a curv ( ∇ ) = 2 πiω. ll ations in a Nutshell ase Quantization • The sections ψ ∈ Γ( L ) define the so-called pre-quantum states . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 39/45

Pre-Quantum Geometry in a Nutshell • Given a Hamiltonian G -manifold ( M, ω, µ ) such that ω satisfies the integrality ion condition (c.f. Kostant, Souriau, Kirillov) according to which [ ω ] is in the image of the ent Map map lectic Points heories H Cech ( M, Z ) → H De Rahm ( M ) uges to Phases ... there exists a complex line bundle L → M with a Hermitian inner product �· , ·� and ons & Momenta a compatible connection ∇ such that uge Quantization ic Quantization antum Geometry in a curv ( ∇ ) = 2 πiω. ll ations in a Nutshell ase Quantization • The sections ψ ∈ Γ( L ) define the so-called pre-quantum states . • The Lie algebra elements X i ∈ g act on these sections by means of the operators v i = − i � ∇ vfi + f i . ˆ elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 39/45

Polarizations in a Nutshell • Since the pre-quantum states can be sharply localized in M , they do not satisfy ion Heisenberg indeterminacy principle. ent Map lectic Points heories uges to Phases ons & Momenta uge Quantization ic Quantization antum Geometry in a ll ations in a Nutshell ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 40/45

Polarizations in a Nutshell • Since the pre-quantum states can be sharply localized in M , they do not satisfy ion Heisenberg indeterminacy principle. ent Map lectic Points heories • The pre-quantum geometry has to be enriched by choosing a polarization P , i.e. an involutive Lagrangian subbundle of T M . uges to Phases ons & Momenta uge Quantization ic Quantization antum Geometry in a ll ations in a Nutshell ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 40/45

Polarizations in a Nutshell • Since the pre-quantum states can be sharply localized in M , they do not satisfy ion Heisenberg indeterminacy principle. ent Map lectic Points heories • The pre-quantum geometry has to be enriched by choosing a polarization P , i.e. an involutive Lagrangian subbundle of T M . uges to Phases ons & Momenta uge Quantization • We can then “cut in half’ the space of pre-quantum states by means of an eq. of the ic Quantization form antum Geometry in a ll ations in a Nutshell ∇ P ψ = 0 . ase Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 40/45

Polarizations in a Nutshell • Since the pre-quantum states can be sharply localized in M , they do not satisfy ion Heisenberg indeterminacy principle. ent Map lectic Points heories • The pre-quantum geometry has to be enriched by choosing a polarization P , i.e. an involutive Lagrangian subbundle of T M . uges to Phases ons & Momenta uge Quantization • We can then “cut in half’ the space of pre-quantum states by means of an eq. of the ic Quantization form antum Geometry in a ll ations in a Nutshell ∇ P ψ = 0 . ase Quantization • Now, we have argued that the reduction in the N ◦ of observables that we need in order to define a state might be understood as a consequence of the G -action. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 40/45

Polarizations in a Nutshell • Since the pre-quantum states can be sharply localized in M , they do not satisfy ion Heisenberg indeterminacy principle. ent Map lectic Points heories • The pre-quantum geometry has to be enriched by choosing a polarization P , i.e. an involutive Lagrangian subbundle of T M . uges to Phases ons & Momenta uge Quantization • We can then “cut in half’ the space of pre-quantum states by means of an eq. of the ic Quantization form antum Geometry in a ll ations in a Nutshell ∇ P ψ = 0 . ase Quantization • Now, we have argued that the reduction in the N ◦ of observables that we need in order to define a state might be understood as a consequence of the G -action. • Hence, we can conjecture that the group action induces a sort of natural “group-polarization” of the pre-quantum states. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 40/45

Universal Constraint Surface • We do not want to quantize the single 0 -symplectic quotient M 0 as in gauge theories... ion ent Map ... but rather all the ξ -symplectic quotients M ξ at once. lectic Points heories uges to Phases ons & Momenta uge Quantization ic Quantization ase Quantization sal Constraint Surface Pre-Quantum try Group-Polarization” ion sions the End elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 41/45

Universal Constraint Surface • We do not want to quantize the single 0 -symplectic quotient M 0 as in gauge theories... ion ent Map ... but rather all the ξ -symplectic quotients M ξ at once. lectic Points • Hence, instead of considering a single ξ -constraint surface Σ ξ = µ − 1 ( ξ ) /G ... heories uges to Phases ... we shall consider the universal constraint surface ons & Momenta Σ = Φ − 1 (0) ⊂ M × g ∗ uge Quantization − , ic Quantization where M × g ∗ − is endowed with the product Poisson structure and the shifted moment ase Quantization map sal Constraint Surface Pre-Quantum Φ : M × g ∗ g ∗ try → − Group-Polarization” ion ( m, ξ ) �→ Φ( m, ξ ) = µ ( m ) − ξ sions the End elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 41/45

Universal Constraint Surface • We do not want to quantize the single 0 -symplectic quotient M 0 as in gauge theories... ion ent Map ... but rather all the ξ -symplectic quotients M ξ at once. lectic Points • Hence, instead of considering a single ξ -constraint surface Σ ξ = µ − 1 ( ξ ) /G ... heories uges to Phases ... we shall consider the universal constraint surface ons & Momenta Σ = Φ − 1 (0) ⊂ M × g ∗ uge Quantization − , ic Quantization where M × g ∗ − is endowed with the product Poisson structure and the shifted moment ase Quantization map sal Constraint Surface Pre-Quantum Φ : M × g ∗ g ∗ try → − Group-Polarization” ion ( m, ξ ) �→ Φ( m, ξ ) = µ ( m ) − ξ sions the End • The surface Σ is defined by the common zeros of the involutive universal constraints G i ( m, ξ ) = f i ( m ) − � ξ, X i � , X i ∈ g { G i , G j } = c k ij G k . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 41/45

� � � � Shifted Pre-Quantum Geometry • Following Guillemin & Sternberg (1982), we can introduce the shifted bundle ion g ∗ . ent Map L M ⊠ L ∗ = π ∗ M L M ⊗ π ∗ − L ∗ g ∗ g ∗ lectic Points defined by the diagram heories uges to Phases L M ⊠ L ∗ L ∗ L M g ∗ g ∗ ons & Momenta uge Quantization ic Quantization M × g ∗ � g ∗ M − − ase Quantization πM π g ∗ − sal Constraint Surface Pre-Quantum try and endowed with the vertical differential ∇ acting along the G -orbits Group-Polarization” ion sions ∇ ( m,ξ ) = ∇ M m ⊗ id + id ⊗ ∇ O ξ ∈ O ⊂ g ∗ ξ , the End which is flat on Σ : F ( v i , v j )( m, ξ ) = c k ij ( f k ( m ) − � ξ, X k � ) ≈ Σ 0 . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 42/45

Weak “Group-Polarization” Condition • The BRST construction applied to this setting yields in degree 0 the sections of ion L M ⊠ L ∗ g ∗ whose restriction to Σ is g -invariant... ent Map lectic Points ... i.e. the sections that are ∇ -closed modulo δ : heories uges to Phases ∇ i Ψ( m, ξ ) = ϕ j i G j ( m, ξ ) ≈ Σ 0 . ons & Momenta uge Quantization ic Quantization ase Quantization sal Constraint Surface Pre-Quantum try Group-Polarization” ion sions the End elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 43/45

Weak “Group-Polarization” Condition • The BRST construction applied to this setting yields in degree 0 the sections of ion L M ⊠ L ∗ g ∗ whose restriction to Σ is g -invariant... ent Map lectic Points ... i.e. the sections that are ∇ -closed modulo δ : heories uges to Phases ∇ i Ψ( m, ξ ) = ϕ j i G j ( m, ξ ) ≈ Σ 0 . ons & Momenta • If we consider the (distribution) sections whose restrictions to Σ are supported by the uge Quantization elements ( µ − 1 ( ξ ) , ξ ) ∈ Σ for a fixed ξ ∈ g ∗ , the cocycle eq. becomes ic Quantization ∇ M vi ψ ( m ) ≈ ( µ − 1( ξ ) ,ξ ) 0 . ase Quantization sal Constraint Surface Pre-Quantum try Group-Polarization” ion sions the End elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 43/45

Weak “Group-Polarization” Condition • The BRST construction applied to this setting yields in degree 0 the sections of ion L M ⊠ L ∗ g ∗ whose restriction to Σ is g -invariant... ent Map lectic Points ... i.e. the sections that are ∇ -closed modulo δ : heories uges to Phases ∇ i Ψ( m, ξ ) = ϕ j i G j ( m, ξ ) ≈ Σ 0 . ons & Momenta • If we consider the (distribution) sections whose restrictions to Σ are supported by the uge Quantization elements ( µ − 1 ( ξ ) , ξ ) ∈ Σ for a fixed ξ ∈ g ∗ , the cocycle eq. becomes ic Quantization ∇ M vi ψ ( m ) ≈ ( µ − 1( ξ ) ,ξ ) 0 . ase Quantization sal Constraint Surface Pre-Quantum • By using the pre-quantum operators try Group-Polarization” ion v f = − i � ∇ vf + f ˆ sions the End this eq. can be rewritten as an eigenvalue eq. ˆ v i ψ ( m ) ≈ f i ( m ) ψ ( m ) , ≈ � µ ( m ) , X i � ψ, = � ξ, X i � ψ. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics , Université Aix Marseille 15-18 July, 2014 - p. 43/45