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On the Relation Betweeen Gauge and Phase Symmetries Gabriel Catren Laboratoire SPHERE - Sciences, Histoire, Philosophie (UMR 7219) - Université Paris Diderot/CNRS ERC Project Philosophy of Canonical Quantum Gravity elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 1/45 Symmetries & Reduction n ries & Reduction Symmetries assical to Quantum ure nt Map ⇓ Reduction ectic Points eories ... in the amount of (invariant) information... ges to Phases ns & Momenta ... that is necessary to completely describe a system. uge Quantization Quantization se Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 2/45 Symmetries & Reduction n ries & Reduction Symmetries assical to Quantum ure nt Map ⇓ Reduction ectic Points eories ... in the amount of (invariant) information... ges to Phases ns & Momenta ... that is necessary to completely describe a system. uge Quantization Quantization se Quantization • Example: gauge theories (or constrained Hamiltonian systems): 2n degrees of freedom + k first-class constraints ⇓ 2(n -k) physical degrees of freedom elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 2/45 From Classical to Quantum n • The transition from classical to quantum mechanics... ries & Reduction assical to Quantum ure nt Map ... entails a reduction in the number of obs. that are necessary to define a physical state: ectic Points eories 2n classical observables q and p ges to Phases ns & Momenta ⇓ uge Quantization Quantization n quantum observables q or p. se Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 3/45 From Classical to Quantum n • The transition from classical to quantum mechanics... ries & Reduction assical to Quantum ure nt Map ... entails a reduction in the number of obs. that are necessary to define a physical state: ectic Points eories 2n classical observables q and p ges to Phases ns & Momenta ⇓ uge Quantization Quantization n quantum observables q or p. se Quantization • In the simplest case, the phase invariance of |pi i under translations in q q0 · |pi i 7→ e2πiq0 pi |pi i ≈ |pi i can be interpreted by saying that the position q of |pi i is completely “undetermined”. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 3/45 From Classical to Quantum n • The transition from classical to quantum mechanics... ries & Reduction assical to Quantum ure nt Map ... entails a reduction in the number of obs. that are necessary to define a physical state: ectic Points eories 2n classical observables q and p ges to Phases ns & Momenta ⇓ uge Quantization Quantization n quantum observables q or p. se Quantization • In the simplest case, the phase invariance of |pi i under translations in q q0 · |pi i 7→ e2πiq0 pi |pi i ≈ |pi i can be interpreted by saying that the position q of |pi i is completely “undetermined”. • Heisenberg indeterminacy principle generalizes this reduction to more gral. states (e.g. coherent states). elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 3/45 Conjecture n • In analogy to gauge theories, we could try to understand this transition... ries & Reduction assical to Quantum ure nt Map ... as a reduction induced by some form of symmetry. ectic Points eories ges to Phases ns & Momenta uge Quantization Quantization se Quantization elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 4/45 Conjecture n • In analogy to gauge theories, we could try to understand this transition... ries & Reduction assical to Quantum ure nt Map ... as a reduction induced by some form of symmetry. ectic Points eories • Far from being a mere analogy, I will argue that ges to Phases ns & Momenta uge Quantization ... the quantum phase symmetries can be understood... Quantization se Quantization ... as a consequence of the same formalism underlying the gauge symmetries, i.e. the symplectic reduction procedure. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 4/45 Hamiltonian G-manifolds n nt Map nian • Let (M, ω, µ) be a Hamiltonian G-manifold, i.e. a connected symplectic manifold endowed G-manifolds Map g ∗ useful for? Orbit Method Φ∗ gω .with an action Φ : G × M → M of a Lie group G preserving ω (i.e. = ω for all g ∈ G). ectic Points eories .with an equivariant moment map (introduced by J.-M. Souriau) ges to Phases µ : M → g∗ ns & Momenta uge Quantization i.e. a (Poisson) map intertwining the G-action on M and the G-co-adjoint action on g∗ : Quantization se Quantization µ M / g∗ Ad∗−1 g Φg M µ / g∗ . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 5/45 Moment Map • Given Xi ∈ g, the moment map n nt Map nian G-manifolds ∗ Map ∗ g µ:M →g useful for? Orbit Method ectic Points defines a generating function of the group action on M eories ges to Phases fi (m) = hµ(m), Xi i, Xi ∈ g ns & Momenta uge Quantization such that its symplectic gradient Quantization se Quantization vfi = ω −1 dfi is the fundamental vector field that infinitesimally generates the G-action on M . elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 6/45 Moment Map • Given Xi ∈ g, the moment map n nt Map nian G-manifolds ∗ Map ∗ g µ:M →g useful for? Orbit Method ectic Points defines a generating function of the group action on M eories ges to Phases fi (m) = hµ(m), Xi i, Xi ∈ g ns & Momenta uge Quantization such that its symplectic gradient Quantization se Quantization vfi = ω −1 dfi is the fundamental vector field that infinitesimally generates the G-action on M . • The fact of considering M over g∗ implies that there is a privileged family {fi }X ∈g of i observables on M (i.e. the generating functions fi ). elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 6/45 What is g∗ useful for? n nt Map nian • A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action, but also with a map towards g∗ . G-manifolds Map g ∗ useful for? Orbit Method ectic Points eories ges to Phases ns & Momenta uge Quantization Quantization se 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? n nt Map nian • A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action, but also with a map towards g∗ . G-manifolds Map g ∗ useful for? Orbit Method • Now, how can we interpret this relation between M and g∗ ? ectic Points eories ges to Phases ns & Momenta uge Quantization Quantization se 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? n nt Map nian • A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action, but also with a map towards g∗ . G-manifolds Map g ∗ useful for? Orbit Method • Now, how can we interpret this relation between M and g∗ ? ectic Points eories • Roughly speaking, g∗ encodes the unitary representation theory of G. ges to Phases ns & Momenta uge Quantization Quantization se 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? n nt Map nian • A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action, but also with a map towards g∗ . G-manifolds Map g ∗ useful for? Orbit Method • Now, how can we interpret this relation between M and g∗ ? ectic Points eories • Roughly speaking, g∗ encodes the unitary representation theory of G. ges to Phases ns & Momenta • Firstly, g∗ is a Poisson manifold with respect to the so-called Lie-Poisson structure uge Quantization Quantization {f, g} (x) = hx, [df (x), dg(x)]i f, g ∈ C ∞ ∗ (g ) se 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? n nt Map nian • A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action, but also with a map towards g∗ . G-manifolds Map g ∗ useful for? Orbit Method • Now, how can we interpret this relation between M and g∗ ? ectic Points eories • Roughly speaking, g∗ encodes the unitary representation theory of G. ges to Phases ns & Momenta • Firstly, g∗ is a Poisson manifold with respect to the so-called Lie-Poisson structure uge Quantization {f, g} (x) = hx, [df (x), dg(x)]i Quantization se Quantization f, g ∈ C ∞ ∗ (g ) • 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? n nt Map nian • A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action, but also with a map towards g∗ . G-manifolds Map g ∗ useful for? Orbit Method • Now, how can we interpret this relation between M and g∗ ? ectic Points eories • Roughly speaking, g∗ encodes the unitary representation theory of G. ges to Phases ns & Momenta • Firstly, g∗ is a Poisson manifold with respect to the so-called Lie-Poisson structure uge Quantization {f, g} (x) = hx, [df (x), dg(x)]i Quantization se Quantization f, g ∈ C ∞ ∗ (g ) • Secondly, ∗ ∗ Symplective leaves of g = Coadjoint orbits O of G g • The coadjoint orbits O are endowed with a canonical G-invariant symplectic structure ωO . 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 n nt Map nian G-manifolds Map g ∗ useful for? Orbit Method ectic Points eories ges to Phases ns & Momenta uge Quantization ∗ gZ /G ∼ Ĝ, (where Ĝ is the unitary dual of G) given by O HO , where HO is the Hilbert space obtained by applying the geometric quantization procedure to the symplectic manifold O... H or by applying the functor IndG H to the 1-dim unirrep ρξ of H = exp(h) defined by ξ ∈ O where h ⊂ g is a max. subalg. subordinated to ξ: Quantization se Quantization hξ, [h, h]i = 0. 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 n nt Map nian ∗ G-manifolds gZ /G ∼ Ĝ, Map g ∗ useful for? Orbit Method (where Ĝ is the unitary dual of G) given by ectic Points eories ges to Phases ns & Momenta uge Quantization O HO , where HO is the Hilbert space obtained by applying the geometric quantization procedure to the symplectic manifold O... H or by applying the functor IndG H to the 1-dim unirrep ρξ of H = exp(h) defined by ξ ∈ O where h ⊂ g is a max. subalg. subordinated to ξ: Quantization hξ, [h, h]i = 0. se 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: ρξ : G → U (1) eX 7→ e2πihξ,Xi , 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 n nt Map ectic Points Conjecture • Exactly as the G-action on the homogeneous symplectic orbit O ⊂ g∗ is lifted to an irreducible unitary action on HO ... ... we could expect the G-action on M to be lifted to a unitary action on HM . lectic Quotients ction entails a on ould we interpret n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points Conjecture • Exactly as the G-action on the homogeneous symplectic orbit O ⊂ g∗ is lifted to an irreducible unitary action on HO ... ... we could expect the G-action on M to be lifted to a unitary action on HM . lectic Quotients ction entails a on ould we interpret • Since M is not in gral. G-homogeneous, the lifted unitary action will not in gral. be irreducible: n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M HM = M m(O, M )HO , O⊂g∗ . where m(O, M ) = dim(HomG (HO , HM )) is the multiplicity with which the unirrep HO occurs in HM . m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points Conjecture • Exactly as the G-action on the homogeneous symplectic orbit O ⊂ g∗ is lifted to an irreducible unitary action on HO ... ... we could expect the G-action on M to be lifted to a unitary action on HM . lectic Quotients ction entails a on ould we interpret • Since M is not in gral. G-homogeneous, the lifted unitary action will not in gral. be irreducible: n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M HM = M m(O, M )HO , O⊂g∗ . where m(O, M ) = dim(HomG (HO , HM )) is the multiplicity with which the unirrep HO occurs in HM . m Intertwiner Spaces eories • Kirillov’s conjecture: µ tells which unirreps of G occur in HM . ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points Conjecture • Exactly as the G-action on the homogeneous symplectic orbit O ⊂ g∗ is lifted to an irreducible unitary action on HO ... ... we could expect the G-action on M to be lifted to a unitary action on HM . lectic Quotients ction entails a on ould we interpret • Since M is not in gral. G-homogeneous, the lifted unitary action will not in gral. be irreducible: n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M HM = M m(O, M )HO , O⊂g∗ . where m(O, M ) = dim(HomG (HO , HM )) is the multiplicity with which the unirrep HO occurs in HM . m Intertwiner Spaces eories ges to Phases • Kirillov’s conjecture: µ tells which unirreps of G occur in HM . • Guillemin-Sternberg conjecture: µ also gives m(O, M ). ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points Conjecture • Exactly as the G-action on the homogeneous symplectic orbit O ⊂ g∗ is lifted to an irreducible unitary action on HO ... ... we could expect the G-action on M to be lifted to a unitary action on HM . lectic Quotients ction entails a on ould we interpret • Since M is not in gral. G-homogeneous, the lifted unitary action will not in gral. be irreducible: n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M HM = M m(O, M )HO , O⊂g∗ . where m(O, M ) = dim(HomG (HO , HM )) is the multiplicity with which the unirrep HO occurs in HM . m Intertwiner Spaces eories ges to Phases • Kirillov’s conjecture: µ tells which unirreps of G occur in HM . • Guillemin-Sternberg conjecture: µ also gives m(O, M ). ns & Momenta uge Quantization Quantization • Hence, µ encodes the quantization of M over g∗ , i.e. the quantization of M with respect to the observable algebra induced by the G-action on M . se 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 n nt Map • We must learn how to use µ for “pulling-back” the G-unirreps supported by g∗ to M . Let’s consider the case of an abelian G... ectic Points Conjecture lectic Quotients ction entails a on ould we interpret n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map • We must learn how to use µ for “pulling-back” the G-unirreps supported by g∗ to M . Let’s consider the case of an abelian G... ectic Points Conjecture lectic Quotients ction entails a • Since the unirreps Hξ (ξ ∈ g∗ ) are “supported” by ξ, let’s consider the states in M corresponding to a fixed “value” ξ of the “momentum”, that is on ould we interpret µ−1 (ξ). n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map • We must learn how to use µ for “pulling-back” the G-unirreps supported by g∗ to M . Let’s consider the case of an abelian G... ectic Points Conjecture lectic Quotients ction entails a • Since the unirreps Hξ (ξ ∈ g∗ ) are “supported” by ξ, let’s consider the states in M corresponding to a fixed “value” ξ of the “momentum”, that is on ould we interpret µ−1 (ξ). n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” • Now, the preimage µ−1 (ξ) of the (trivial) symp. manifold O is not a symp. manifold. n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map • We must learn how to use µ for “pulling-back” the G-unirreps supported by g∗ to M . Let’s consider the case of an abelian G... ectic Points Conjecture lectic Quotients ction entails a • Since the unirreps Hξ (ξ ∈ g∗ ) are “supported” by ξ, let’s consider the states in M corresponding to a fixed “value” ξ of the “momentum”, that is on ould we interpret µ−1 (ξ). n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” • Now, the preimage µ−1 (ξ) of the (trivial) symp. manifold O is not a symp. manifold. n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases • (Shifted) Mardsen-Weinstein reduction theorem: . Mξ = µ−1 (ξ)/G is a symp. manifold called the ξ-symplectic quotient of M . ns & Momenta uge Quantization Quantization se 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 n nt Map • We must learn how to use µ for “pulling-back” the G-unirreps supported by g∗ to M . Let’s consider the case of an abelian G... ectic Points Conjecture lectic Quotients ction entails a • Since the unirreps Hξ (ξ ∈ g∗ ) are “supported” by ξ, let’s consider the states in M corresponding to a fixed “value” ξ of the “momentum”, that is on ould we interpret µ−1 (ξ). n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” • Now, the preimage µ−1 (ξ) of the (trivial) symp. manifold O is not a symp. manifold. n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases • (Shifted) Mardsen-Weinstein reduction theorem: . Mξ = µ−1 (ξ)/G is a symp. manifold called the ξ-symplectic quotient of M . ns & Momenta uge Quantization • So, Mξ is the symp. counterpart of ξ in M . Quantization se 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 n nt Map • In gauge-theoretic terms, when we fix the “value” of the “momentum” µ to ξ by means of the restriction to the “ξ-constraint surface” ectic Points Conjecture lectic Quotients ction entails a µ−1 (ξ) ⊂ M, on ould we interpret ... the “conjugate coordinate” acted upon by G becomes completely “undetermined”... n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” .... in the sense that it is “gauged out” by means of the projection n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces µ−1 (ξ) ։ Mξ . eories ges to Phases ns & Momenta uge Quantization Quantization se 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 Mξ ? n • We shall argue... nt Map ectic Points 1) that the ξ-symplectic quotient Conjecture lectic Quotients ction entails a on ould we interpret n’s Symplectic Creed y-Theoretical “Points” . −1 Mξ = µ (ξ)/G is the “moduli space” parameterizing the category-theoretical symplectic ξ-points of M. n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M 2) that the notion of symplectic point elicits a category-theoretical interpretation of Heisenberg indeterminacy principle. m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points Conjecture lectic Quotients ction entails a on ould we interpret n’s Symplectic Creed “The Heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum-mechanical particle. This can be rephrased as follows: the smallest subsets of classical phase space in which the presence of a quantum-mechanical particle can be detected are its Lagrangian submanifolds. For this reason it makes sense to regard the Lagrangian submanifolds of phase space as being its true “points”.” y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces V. Guillemin and S. Sternberg, Geometric Quantization and Multiplicities of Group Representations, 1982. lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points Conjecture lectic Quotients ction entails a on ould we interpret n’s Symplectic Creed “The Heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum-mechanical particle. This can be rephrased as follows: the smallest subsets of classical phase space in which the presence of a quantum-mechanical particle can be detected are its Lagrangian submanifolds. For this reason it makes sense to regard the Lagrangian submanifolds of phase space as being its true “points”.” y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces V. Guillemin and S. Sternberg, Geometric Quantization and Multiplicities of Group Representations, 1982. lectic Points of M m Intertwiner Spaces eories ges to Phases • This notion of Lagrangian true “points” acquires a precise category-theoretical meaning in the framework of Weinstein’s symplectic “category”. ns & Momenta uge Quantization Quantization se 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” n • A point x in a manifold M can be identified with the morphism nt Map ectic Points ϕx : {∗} → M Conjecture lectic Quotients ction entails a on ould we interpret given by {∗} 7→ x n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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” n • A point x in a manifold M can be identified with the morphism nt Map ectic Points ϕx : {∗} → M Conjecture lectic Quotients ction entails a on ould we interpret given by {∗} 7→ x n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” • More generally, given two objects A and B in a category, the morphisms n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M B→A m Intertwiner Spaces eories define the so-called B-points of A. ges to Phases ns & Momenta uge Quantization Quantization se 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” n nt Map • Objects: .Symplectic manifolds (M, ω). ectic Points Conjecture lectic Quotients ction entails a on ould we interpret n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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” n nt Map • Objects: .Symplectic manifolds (M, ω). ectic Points Conjecture lectic Quotients • Morphisms (or Lagrangian correspondences) (M2 , ω2 ) → (M1 , ω1 ): ction entails a on ould we interpret HomSymp (M2 , M1 ) = n L2,1 ֒→ M1 × − M2 o n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M where (M2− , −ω2 ) is the dual of (M2 , ω2 ) and (M1 × M2− , π1∗ ω1 − π2∗ ω2 ), is the product symplectic manifold. m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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” n nt Map • Objects: .Symplectic manifolds (M, ω). ectic Points Conjecture lectic Quotients • Morphisms (or Lagrangian correspondences) (M2 , ω2 ) → (M1 , ω1 ): ction entails a on ould we interpret HomSymp (M2 , M1 ) = n L2,1 ֒→ M1 × − M2 o n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M where (M2− , −ω2 ) is the dual of (M2 , ω2 ) and (M1 × M2− , π1∗ ω1 − π2∗ ω2 ), is the product symplectic manifold. m Intertwiner Spaces eories • In particular, a symplectomorphism defines a Lagrangian corresp. ges to Phases ns & Momenta uge Quantization Quantization se 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” n nt Map • Objects: .Symplectic manifolds (M, ω). ectic Points Conjecture lectic Quotients • Morphisms (or Lagrangian correspondences) (M2 , ω2 ) → (M1 , ω1 ): ction entails a on ould we interpret HomSymp (M2 , M1 ) = n L2,1 ֒→ M1 × − M2 o n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M where (M2− , −ω2 ) is the dual of (M2 , ω2 ) and (M1 × M2− , π1∗ ω1 − π2∗ ω2 ), is the product symplectic manifold. m Intertwiner Spaces eories • In particular, a symplectomorphism defines a Lagrangian corresp. ges to Phases ns & Momenta uge Quantization • The symplectic points of (M, ω) is given by the morphisms in HomSymp ((∗, 0), (M, ω)) = {L ֒→ M × {∗} ≃ M } , Quantization i.e. by the Lagrangian submanifolds of M . se 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” n nt Map • Objects: .Hamiltonian G-manifolds (M, ω, µ). ectic Points Conjecture lectic Quotients ction entails a on ould we interpret n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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” n nt Map ectic Points Conjecture • Objects: .Hamiltonian G-manifolds (M, ω, µ). • Morphisms (M2 , ω2 , µ2 ) → (M1 , ω1 , µ1 ): lectic Quotients ction entails a on ould we interpret L2,1 / / M2− M1 ×g∗ M2− n’s Symplectic Creed µ2 y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces M1 µ1 / g∗ lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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” n • Objects: .Hamiltonian G-manifolds (M, ω, µ). nt Map ectic Points Conjecture • Morphisms (M2 , ω2 , µ2 ) → (M1 , ω1 , µ1 ): lectic Quotients ction entails a L2,1 on ould we interpret / / M2− M1 ×g∗ M2− n’s Symplectic Creed µ2 y-Theoretical “Points” n’s Symplectic ry” M1 n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces HomG-Symp (M2 , M1 ) = ges to Phases uge Quantization Quantization se Quantization µ1 • In other terms, eories ns & Momenta / g∗ n −1 L2,1 ֒→ Φ (0) ⊂ M1 × M2− o , where . − ∗ ∗ (M1 × M2 , π1 ω1 − π2 ω2 , Φ = µ1 − µ2 ), is the product Hamiltonian G-manifold. 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 n • It can be shown (♣ ) that L2,1 ⊂ M1 ×g∗ M2− are G-invariant... nt Map ectic Points ... and that there is a bijection Conjecture lectic Quotients ction entails a on ould we interpret HomG-Symp (M2 , M1 ) ≃ n L⊂ o (M1 ×g∗ M2− )/G . n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization ____________________________________________________________________________ ♣ Xu, P. [1994]: “Classical Intertwiner Space and Quantization,” Commun. Math. Phys. 164, 473-488. se 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 n • It can be shown (♣ ) that L2,1 ⊂ M1 ×g∗ M2− are G-invariant... nt Map ectic Points ... and that there is a bijection Conjecture lectic Quotients ction entails a on ould we interpret HomG-Symp (M2 , M1 ) ≃ n L⊂ o (M1 ×g∗ M2− )/G . n’s Symplectic Creed y-Theoretical “Points” n’s Symplectic ry” • Under nice conditions, (M1 ×g∗ M2− )/G is a symplectic manifold... n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M ... whose symplectic points are the classical intertwiners over g∗ between M2 and M1 ... m Intertwiner Spaces eories ... or, in category-theoretical terms, the M2 -sympletic points of M1 . ges to Phases ns & Momenta uge Quantization Quantization ____________________________________________________________________________ ♣ Xu, P. [1994]: “Classical Intertwiner Space and Quantization,” Commun. Math. Phys. 164, 473-488. se 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 n nt Map ectic Points Conjecture • In particular, the morphisms (ξ, 0, µξ : ξ 7→ ξ) → (M, ω, µ) are given by the Lagrang. subman. of − −1 (M ×g∗ ξ )/G = Φ (0)/G, lectic Quotients ction entails a where the twisted moment map is on ould we interpret n’s Symplectic Creed Φ : M × ξ− → g∗ (m, ξ) 7→ µ(m) − ξ. y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points Conjecture • In particular, the morphisms (ξ, 0, µξ : ξ 7→ ξ) → (M, ω, µ) are given by the Lagrang. subman. of − −1 (M ×g∗ ξ )/G = Φ (0)/G, lectic Quotients ction entails a where the twisted moment map is on ould we interpret n’s Symplectic Creed Φ : M × ξ− → g∗ (m, ξ) 7→ µ(m) − ξ. y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces • Now, since Φ−1 (0) ≃ µ−1 (ξ) ⊂ M , then Φ−1 (0)/G ≃ Mξ . lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map • In particular, the morphisms (ξ, 0, µξ : ξ 7→ ξ) → (M, ω, µ) are given by the Lagrang. subman. of ectic Points − −1 (M ×g∗ ξ )/G = Φ Conjecture (0)/G, lectic Quotients ction entails a where the twisted moment map is on ould we interpret n’s Symplectic Creed Φ : M × ξ− → g∗ (m, ξ) 7→ µ(m) − ξ. y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces • Now, since Φ−1 (0) ≃ µ−1 (ξ) ⊂ M , then Φ−1 (0)/G ≃ Mξ . lectic Points of M m Intertwiner Spaces eories • All in all, HomG-Symp (ξ, M ) = {L ⊂ Mξ } ges to Phases ns & Momenta i.e. the symplectic points of Mξ are in correspondence with the ξ-points of M . uge Quantization Quantization Mξ can be interpreted as the moduli space of symplectic ξ-points of M . se 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 n nt Map ectic Points Conjecture • Guillemin and Sternberg (1982) showed (for particular M and G) that the (geometric) quantization of the classical intertwiner space: MO ∼ = HomG-Symp (O, M ) lectic Quotients ction entails a between a coadjoint orbit O and M yields the quantum intertwiner space: on ould we interpret n’s Symplectic Creed HM O ∼ = HomG (HO .HM ), y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points Conjecture • Guillemin and Sternberg (1982) showed (for particular M and G) that the (geometric) quantization of the classical intertwiner space: MO ∼ = HomG-Symp (O, M ) lectic Quotients ction entails a between a coadjoint orbit O and M yields the quantum intertwiner space: on ould we interpret n’s Symplectic Creed HM O ∼ = HomG (HO .HM ), y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories • Whereas MO parameterizes the symplectic G-morphisms O → M, HMO parameterizes the unitary G-intertwiners HO → HM ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points Conjecture • Guillemin and Sternberg (1982) showed (for particular M and G) that the (geometric) quantization of the classical intertwiner space: MO ∼ = HomG-Symp (O, M ) lectic Quotients ction entails a between a coadjoint orbit O and M yields the quantum intertwiner space: on ould we interpret n’s Symplectic Creed HM O ∼ = HomG (HO .HM ), y-Theoretical “Points” n’s Symplectic ry” n’s G-Symplectic ry” l Intertwiner Spaces lectic Points of M m Intertwiner Spaces eories • Whereas MO parameterizes the symplectic G-morphisms O → M, HMO parameterizes the unitary G-intertwiners HO → HM ges to Phases ns & Momenta uge Quantization • Kirillov’s conjecture revisited: The unirrep HO occurs in HM if M has symplectic O-points where the multiplicity is given by Quantization se Quantization m(O, M ) = dim(HMO ). 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 n nt Map ectic Points • A gauge theory is a Ham. G-manifold (M, ω, µ) such that the Ham. equations constraint the solutions to be in the constraint surface Σ=µ −1 (0) ⊂ M. eories n-Weinstein ction ation Commutes with ction ges to Phases ns & Momenta uge Quantization Quantization se 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 n nt Map • A gauge theory is a Ham. G-manifold (M, ω, µ) such that the Ham. equations constraint the solutions to be in the constraint surface ectic Points Σ=µ −1 (0) ⊂ M. eories n-Weinstein ction ation Commutes with • The Mardsen-Weinstein reduction theorem shows that the 0-symplectic quotient ction M0 ≃ µ −1 (0)/G ges to Phases ns & Momenta has a canonical symplectic form ωM0 satisfying uge Quantization π ∗ ω M 0 = ι∗ ω M , Quantization se Quantization where µ−1 M (0) ι / M π . M0 = 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 n • In this case, the Guillemin-Sternberg conjecture is nt Map HM 0 ∼ = HomG (H0 , HM ), ectic Points eories n-Weinstein ction ation Commutes with ction or, in other terms, HM 0 ≃ HG M, where HG M is the space of G-invariant states in HM . ges to Phases ns & Momenta uge Quantization Quantization se 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 n • In this case, the Guillemin-Sternberg conjecture is nt Map HM 0 ∼ = HomG (H0 , HM ), ectic Points eories n-Weinstein ction ation Commutes with ction or, in other terms, HM 0 ≃ HG M, where HG M is the space of G-invariant states in HM . ges to Phases ns & Momenta • Diagramatically, uge Quantization M Quantization se Quantization O 0-symplectic reduction /o /o Quantization o/ /o /o /o /o /o /o / HM O O O M0 O /o /o /o /o /o /o /o / HM0 O G-invariant states ≃ HG 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 n nt Map • A gauge theory is a given by a Ham. G-manifold such that the restriction of the theory to the classical intertwiner space ectic Points eories M0 ≃ Hom(0, M ) 0 ∈ g∗ ges to Phases Theories Groups vs. Phase containing the 0-symplectic points of M ... plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ... implies that the resulting quantum theory only includes G-invariant states. ns & Momenta uge Quantization Quantization se 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 n nt Map • A gauge theory is a given by a Ham. G-manifold such that the restriction of the theory to the classical intertwiner space ectic Points M0 ≃ Hom(0, M ) eories 0 ∈ g∗ ges to Phases Theories Groups vs. Phase containing the 0-symplectic points of M ... plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ns & Momenta ... implies that the resulting quantum theory only includes G-invariant states. • In this case, G is called the gauge group of the theory and the generating functions of the G-action uge Quantization Gi (m) = hµ(m), Xi i Quantization Xi ∈ g se 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 n • We are here interested in ordinary theories defined on a Ham. G-manifolds (M, ω, µ), nt Map ectic Points eories ... where by ordinary we mean that the theories are not constrained to a unique value of µ. ges to Phases Theories Groups vs. Phase plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ns & Momenta uge Quantization Quantization se 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 n • We are here interested in ordinary theories defined on a Ham. G-manifolds (M, ω, µ), nt Map ectic Points eories ... where by ordinary we mean that the theories are not constrained to a unique value of µ. ges to Phases Theories Groups vs. Phase plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries • We shall call G the phase group and the non-constrained generating functions of the G-action fi (m) = hµ(m), Xi i ns & Momenta uge Quantization phase observables. Quantization se 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 n nt Map • Differently from the constraints Ga , the phase observables fi do not select a single unirrep of G. ectic Points eories ges to Phases Theories Groups vs. Phase plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ns & Momenta uge Quantization Quantization se 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 n nt Map ectic Points • Differently from the constraints Ga , the phase observables fi do not select a single unirrep of G. • Therefore, while a gauge group action defines a unique 0-symplectic quotient eories ges to Phases Theories . −1 M0 = µ (0)/G, Groups vs. Phase plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ... associated to the trivial unirrep of G... ... a phase group action defines a different O-symplectic quotient ns & Momenta uge Quantization . −1 MO = µ (O)/G Quantization se Quantization for each unirrep O ⊂ g∗ Z of the phase group G. 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 n nt Map ectic Points • Differently from the constraints Ga , the phase observables fi do not select a single unirrep of G. • Therefore, while a gauge group action defines a unique 0-symplectic quotient eories ges to Phases Theories . −1 M0 = µ (0)/G, Groups vs. Phase plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ... associated to the trivial unirrep of G... ... a phase group action defines a different O-symplectic quotient ns & Momenta uge Quantization . −1 MO = µ (O)/G Quantization se Quantization for each unirrep O ⊂ g∗ Z of the phase group G. • The phase G-action on M entails the existence of a whole set of O-symplectic quotients MO . 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 n nt Map • The Guillemin & Sternberg’s conjecture for a ξ-symplectic quotient with G abelian states that this diagram commutes: ectic Points eories M ges to Phases O Theories Groups vs. Phase ξ-symplectic reduction plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ns & Momenta uge Quantization O O Mξ (G,ξ) .HM /o /o Quantization o/ /o /o /o /o /o /o /o / HM O O (G,ξ)-phase invariant states O /o /o /o /o /o /o /o / HMξ ≃ H(G,ξ) , M is the space of (G, ξ)-phase invariant states in HM ,... ... i.e. the states that are invariant modulo a phase factor given by the 1-dim. unirrep ρG ξ of G defined by ξ: Quantization se Quantization G (G,ξ) → HM (eX , |ξ, ...i) 7→ e2πihξ,Xi |ξ, ...i, ρξ : G × HM (G,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 n nt Map • The (G, ξ)-phase invariance of quantum states is the quantum counterpart of the symplectic reduction with respect to a non-zero ξ ∈ g∗ . ectic Points eories ges to Phases Theories Groups vs. Phase Quantum phase invariance is the generalization... plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ... of the strict gauge invariance... .... to the case of ξ-symplectic reductions with ξ 6= 0. ns & Momenta uge Quantization Quantization se 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 n • We have argued that the existence of (G, ξ)-phase invariant states in HM ... nt Map ectic Points ... results from the existence of ξ-symplectic points in M . eories ges to Phases Theories Groups vs. Phase plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ns & Momenta uge Quantization Quantization se 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 n • We have argued that the existence of (G, ξ)-phase invariant states in HM ... nt Map ectic Points ... results from the existence of ξ-symplectic points in M . eories ges to Phases • Far from being structureless set-theoretic points,... Theories Groups vs. Phase plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ... the ξ-symplectic points of M are non-trivial subman. of M endowed with a G-action. ns & Momenta uge Quantization Quantization se 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 n • We have argued that the existence of (G, ξ)-phase invariant states in HM ... nt Map ectic Points ... results from the existence of ξ-symplectic points in M . eories ges to Phases • Far from being structureless set-theoretic points,... Theories Groups vs. Phase plectic Reductions ation Commutes with ction nvariance vs. Phase ce mplectic to Phase ries ... the ξ-symplectic points of M are non-trivial subman. of M endowed with a G-action. • The (G, ξ)-phase invariance of the states |ξ, ...i,... ns & Momenta ... i.e. the “indeterminacy” in the variable acted upon by G... uge Quantization Quantization ... is the quantum counterpart of the fact... se 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... n • Let’s consider the simplest case of G = R acting on M = T ∗ R by nt Map q0 · (q, p) 7→ (q + q0 , p) ectic Points eories with moment map ges to Phases ns & Momenta ance... Out the Position ∗ µ:T R → R (q, p) 7→ p plectic Localization g the Phase ce Group Actions vs... e Group Actions uge Quantization Quantization se 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... n • Let’s consider the simplest case of G = R acting on M = T ∗ R by nt Map q0 · (q, p) 7→ (q + q0 , p) ectic Points eories with moment map ges to Phases ns & Momenta ance... ∗ Out the Position µ:T R → R (q, p) 7→ p plectic Localization g the Phase ce Group Actions vs... e Group Actions • The pi -symplectic quotient uge Quantization Quantization se Quantization Mpi = µ−1 (pi )/G = {∗} contains the unique symplectic pi -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 n • According to nt Map ectic Points [Quantization, ξ-Reduction] = 0, eories ges to Phases ns & Momenta the (1-dimensional) quantization of Mpi yields the (unique) (G, pi )-phase invariant state |pi i in HM : ance... Out the Position plectic Localization g the Phase ce Group Actions vs... 2πiq0 pi q0 · |pi i 7→ e |pi i ≈ |pi i. e Group Actions uge Quantization Quantization se 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 n • According to nt Map ectic Points [Quantization, ξ-Reduction] = 0, eories ges to Phases ns & Momenta the (1-dimensional) quantization of Mpi yields the (unique) (G, pi )-phase invariant state |pi i in HM : ance... Out the Position plectic Localization g the Phase ce Group Actions vs... 2πiq0 pi q0 · |pi i 7→ e |pi i ≈ |pi i. e Group Actions uge Quantization • The indeterminacy in the position q of the state |pi i is a symptom of the fact... Quantization se Quantization ... that the unique symplectic pi -point of M has an internal symmetry under 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 n nt Map • The category-theoretical notion of symplectic point seems to be the symplectic seed of Heisenberg indeterminacy principle. ectic Points eories ges to Phases ns & Momenta ance... Out the Position plectic Localization g the Phase ce Group Actions vs... e Group Actions uge Quantization Quantization se 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 n nt Map • The category-theoretical notion of symplectic point seems to be the symplectic seed of Heisenberg indeterminacy principle. ectic Points eories ges to Phases • The (im)possibility of sharply localizing quantum states in phase space depends on the notion of point that we are using: ns & Momenta ance... Out the Position plectic Localization g the Phase ce Group Actions vs... ... while a quantum state cannot be sharply localized at the set-theoretic points of M ,... e Group Actions uge Quantization Quantization ... it can be sharply localized at its symplectic point... se 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 n nt Map • The superposition of two G-phase invariant states |pi i and |pj i transforming in different unirreps of G,... ectic Points eories ... is no longer G-phase invariant... ges to Phases ns & Momenta ance... ... since the G-action changes the relative phases between the two terms |pi i + |pj i. Out the Position plectic Localization g the Phase ce Group Actions vs... e Group Actions uge Quantization Quantization se 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 n nt Map • The superposition of two G-phase invariant states |pi i and |pj i transforming in different unirreps of G,... ectic Points eories ... is no longer G-phase invariant... ges to Phases ns & Momenta ance... ... since the G-action changes the relative phases between the two terms |pi i + |pj i. Out the Position plectic Localization g the Phase ce Group Actions vs... e Group Actions • Therefore, by introducing an “indeterminacy” in the value of the variable p that labels the unirreps of G,... uge Quantization Quantization ... we break the G-phase invariance,... se 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... n A gauge theory is restricted to a single value of µ (namely 0) nt Map ectic Points ⇓ eories ges to Phases The quantum theory only contains G-invariant quantum states ns & Momenta ance... Out the Position plectic Localization g the Phase ce Group Actions vs... (i.e. states transforming in the trivial unirrep of G) e Group Actions uge Quantization ⇓ Quantization se Quantization Since we do not have different unirreps to superpose... ⇓ ... 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 n Phase observables are not restricted to a single value of µ nt Map ectic Points ⇓ eories ges to Phases ns & Momenta ance... Out the Position plectic Localization g the Phase ce Group Actions vs... e Group Actions The quantum theory contains (G, ξ)-phase invariant states for all ξ ∈ g∗ Z (i.e. states transforming in different unirreps of G) ⇓ uge Quantization Quantization We can superpose (G, ξ)-phase invariant states defined by different unirreps ξ. se 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 n nt Map The Dirac observables f ∈ C ∞ (M0 ) of a gauge theory can be recovered... ectic Points eories ges to Phases ns & Momenta ... by means of the BRST-cohomological reformulation of the 0-symplectic reduction. uge Quantization logical Reduction Theories in a Diagram Resolution bra Cohomology ohomology Quantization se 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 n nt Map ectic Points eories ges to Phases ns & Momenta uge Quantization logical Reduction Theories in a Diagram Resolution bra Cohomology ohomology Quantization se Quantization . The reduction from M to M0 = Σ/G is a two-step procedure: .Restriction from M to Σ .Projection from Σ to Σ/G g-module ∧g ⊗ C ∞ (M ). Koszul resolution of C ∞ (Σ). Lie algebra cohomology of g with values in the 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 n • We can describe the algebra of observables on Σ nt Map C ∞ (Σ) = C ∞ (M )/hGa i ectic Points eories in homological terms by extending the (co-moment) map ges to Phases ns & Momenta ∞ g → C Xi 7→ fi (m) = hµ(m), Xi i (M ) uge Quantization logical Reduction Theories in a Diagram to the quasi-acyclic complex Resolution bra Cohomology ohomology Quantization se Quantization q ... ∧ g ⊗ C ∞ (M ) → ∧ q−1 g⊗C ∞ (M ) → ... → g ⊗ C ∞ δ 1 (M ) − − → C ∞ δ 0 (M ) − − → 0, where the Koszul differential is defined by δ(Xi ⊗ 1) = 1 ⊗ fi , 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 n • We can describe the algebra of observables on Σ nt Map C ∞ (Σ) = C ∞ (M )/hGa i ectic Points eories in homological terms by extending the (co-moment) map ges to Phases ns & Momenta ∞ g → C Xi 7→ fi (m) = hµ(m), Xi i (M ) uge Quantization logical Reduction Theories in a Diagram to the quasi-acyclic complex Resolution bra Cohomology ohomology Quantization q ... ∧ g ⊗ C ∞ (M ) → ∧ q−1 g⊗C ∞ (M ) → ... → g ⊗ C ∞ δ 1 (M ) − − → C ∞ δ 0 (M ) − − → 0, where the Koszul differential is defined by se Quantization δ(Xi ⊗ 1) = 1 ⊗ fi , δ(1 ⊗ f ) = 0. • Hence, H0δ (∧g ⊗ C ∞ (M )) = Ker(δ0 )/Im(δ1 ) = C ∞ (M )/hGa i = 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 n nt Map • Given the g-module K = ∧g ⊗ C ∞ (M ), we can define the vertical differential (or Chevalley-Eilenberg differential) ectic Points ∗ d : K → g ⊗ K = Hom(g, K) eories given by ges to Phases ns & Momenta (dk)(X) = X · k, X ∈ g, k ∈ K. uge Quantization logical Reduction Theories in a Diagram Resolution bra Cohomology ohomology Quantization se 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 n nt Map • Given the g-module K = ∧g ⊗ C ∞ (M ), we can define the vertical differential (or Chevalley-Eilenberg differential) ectic Points ∗ d : K → g ⊗ K = Hom(g, K) eories given by ges to Phases ns & Momenta (dk)(X) = X · k, X ∈ g, k ∈ K. uge Quantization logical Reduction Theories in a Diagram • This can be extended to a map Resolution bra Cohomology d : ∧q g∗ ⊗ K → ∧q+1 g∗ ⊗ K, ohomology Quantization se Quantization by p d(η ⊗ k) = dη ⊗ k + (−1) η ⊗ dk, p ∗ η∈∧ g . 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 n nt Map • Given the g-module K = ∧g ⊗ C ∞ (M ), we can define the vertical differential (or Chevalley-Eilenberg differential) ectic Points ∗ d : K → g ⊗ K = Hom(g, K) eories given by ges to Phases ns & Momenta (dk)(X) = X · k, X ∈ g, k ∈ K. uge Quantization logical Reduction Theories in a Diagram • This can be extended to a map Resolution bra Cohomology d : ∧q g∗ ⊗ K → ∧q+1 g∗ ⊗ K, ohomology Quantization se Quantization by p d(η ⊗ k) = dη ⊗ k + (−1) η ⊗ dk, p ∗ η∈∧ g . • The 0-Lie algebra cohomology of g with values in the g-module K is given by 0 ∗ g Hd (∧g ⊗ K) = K . 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 n • We can then form the double complex nt Map ectic Points ∧p g∗ ⊗ ∧q g ⊗ C ∞ (M ) δ / ∧p g∗ ⊗ ∧q−1 g ⊗ C ∞ (M ) eories ges to Phases d ns & Momenta ∧ uge Quantization p+1 ∗ g ⊗ ∧ g ⊗ C ∞ (M ) q logical Reduction Theories in a Diagram with Resolution bra Cohomology 2 ohomology Quantization δ =0 2 d =0 δd = dδ such that se Quantization Hd0 (H0δ (∧g∗ ⊗ ∧g ⊗ C ∞ (M ))) = C ∞ (M0 ). 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 n • We can then form the double complex nt Map ectic Points ∧p g∗ ⊗ ∧q g ⊗ C ∞ (M ) δ / ∧p g∗ ⊗ ∧q−1 g ⊗ C ∞ (M ) eories ges to Phases d ns & Momenta ∧ uge Quantization p+1 ∗ g ⊗ ∧ g ⊗ C ∞ (M ) q logical Reduction Theories in a Diagram with Resolution bra Cohomology 2 ohomology Quantization δ =0 2 d =0 δd = dδ such that se Quantization Hd0 (H0δ (∧g∗ ⊗ ∧g ⊗ C ∞ (M ))) = C ∞ (M0 ). • Hence, the Dirac observables in C ∞ (M0 ) can be described as elements f00 that are d-closed modulo δ: 0 1 df0 = −δf1 ≈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 n nt Map • Given a Hamiltonian G-manifold (M, ω, µ) such that ω satisfies the integrality condition (c.f. Kostant, Souriau, Kirillov) according to which [ω] is in the image of the map ectic Points eories HCech (M, Z) → HDe Rahm (M ) ges to Phases ns & Momenta uge Quantization ... there exists a complex line bundle L → M with a Hermitian inner product h·, ·i and a compatible connection ∇ such that Quantization ntum Geometry in a curv(∇) = 2πiω. ions in a Nutshell se 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 n nt Map • Given a Hamiltonian G-manifold (M, ω, µ) such that ω satisfies the integrality condition (c.f. Kostant, Souriau, Kirillov) according to which [ω] is in the image of the map ectic Points eories HCech (M, Z) → HDe Rahm (M ) ges to Phases ns & Momenta uge Quantization ... there exists a complex line bundle L → M with a Hermitian inner product h·, ·i and a compatible connection ∇ such that Quantization ntum Geometry in a curv(∇) = 2πiω. ions in a Nutshell se 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 n nt Map • Given a Hamiltonian G-manifold (M, ω, µ) such that ω satisfies the integrality condition (c.f. Kostant, Souriau, Kirillov) according to which [ω] is in the image of the map ectic Points eories HCech (M, Z) → HDe Rahm (M ) ges to Phases ns & Momenta uge Quantization ... there exists a complex line bundle L → M with a Hermitian inner product h·, ·i and a compatible connection ∇ such that Quantization ntum Geometry in a curv(∇) = 2πiω. ions in a Nutshell se Quantization • The sections ψ ∈ Γ(L) define the so-called pre-quantum states. • The Lie algebra elements Xi ∈ g act on these sections by means of the operators v̂i = −i~∇vf + fi . i 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 n nt Map • Since the pre-quantum states can be sharply localized in M , they do not satisfy Heisenberg indeterminacy principle. ectic Points eories ges to Phases ns & Momenta uge Quantization Quantization ntum Geometry in a ions in a Nutshell se 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 n nt Map • Since the pre-quantum states can be sharply localized in M , they do not satisfy Heisenberg indeterminacy principle. ectic Points eories ges to Phases • The pre-quantum geometry has to be enriched by choosing a polarization P, i.e. an involutive Lagrangian subbundle of T M . ns & Momenta uge Quantization Quantization ntum Geometry in a ions in a Nutshell se 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 n nt Map • Since the pre-quantum states can be sharply localized in M , they do not satisfy Heisenberg indeterminacy principle. ectic Points eories ges to Phases • The pre-quantum geometry has to be enriched by choosing a polarization P, i.e. an involutive Lagrangian subbundle of T M . ns & Momenta uge Quantization Quantization ntum Geometry in a • We can then “cut in half’ the space of pre-quantum states by means of an eq. of the form ions in a Nutshell se Quantization ∇P ψ = 0. 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 n nt Map • Since the pre-quantum states can be sharply localized in M , they do not satisfy Heisenberg indeterminacy principle. ectic Points eories ges to Phases • The pre-quantum geometry has to be enriched by choosing a polarization P, i.e. an involutive Lagrangian subbundle of T M . ns & Momenta uge Quantization Quantization ntum Geometry in a • We can then “cut in half’ the space of pre-quantum states by means of an eq. of the form ions in a Nutshell se Quantization ∇P ψ = 0. • 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 n nt Map • Since the pre-quantum states can be sharply localized in M , they do not satisfy Heisenberg indeterminacy principle. ectic Points eories ges to Phases • The pre-quantum geometry has to be enriched by choosing a polarization P, i.e. an involutive Lagrangian subbundle of T M . ns & Momenta uge Quantization Quantization ntum Geometry in a • We can then “cut in half’ the space of pre-quantum states by means of an eq. of the form ions in a Nutshell se Quantization ∇P ψ = 0. • 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 n nt Map • We do not want to quantize the single 0-symplectic quotient M0 as in gauge theories... ... but rather all the ξ-symplectic quotients Mξ at once. ectic Points eories ges to Phases ns & Momenta uge Quantization Quantization se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” n ons he 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 n nt Map • We do not want to quantize the single 0-symplectic quotient M0 as in gauge theories... ... but rather all the ξ-symplectic quotients Mξ at once. ectic Points eories • Hence, instead of considering a single ξ-constraint surface Σξ = µ−1 (ξ)/G... ges to Phases ns & Momenta uge Quantization Quantization se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” n ons ... we shall consider the universal constraint surface −1 Σ=Φ ∗ (0) ⊂ M × g− , where M × g∗ − is endowed with the product Poisson structure and the shifted moment map ∗ ∗ Φ : M × g− → g (m, ξ) 7→ Φ(m, ξ) = µ(m) − ξ he 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 n nt Map • We do not want to quantize the single 0-symplectic quotient M0 as in gauge theories... ... but rather all the ξ-symplectic quotients Mξ at once. ectic Points eories • Hence, instead of considering a single ξ-constraint surface Σξ = µ−1 (ξ)/G... ges to Phases ns & Momenta uge Quantization Quantization se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” n ons ... we shall consider the universal constraint surface −1 Σ=Φ ∗ (0) ⊂ M × g− , where M × g∗ − is endowed with the product Poisson structure and the shifted moment map ∗ ∗ Φ : M × g− → g (m, ξ) 7→ Φ(m, ξ) = µ(m) − ξ he End • The surface Σ is defined by the common zeros of the involutive universal constraints Gi (m, ξ) = fi (m) − hξ, Xi i, Xi ∈ g k {Gi , Gj } = cij Gk . 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 n • Following Guillemin & Sternberg (1982), we can introduce the shifted bundle . ∗ ∗ ∗ LM ⊠ L∗ g∗ = πM LM ⊗ πg∗ Lg∗ nt Map − ectic Points eories defined by the diagram ges to Phases ns & Momenta LM LM ⊠ L∗ g∗ L∗ g∗ uge Quantization Quantization M se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” n ons he End o πM M× g∗ − πg∗ − / g∗ − and endowed with the vertical differential ∇ acting along the G-orbits M O ∇(m,ξ) = ∇m ⊗ id + id ⊗ ∇ξ , ∗ ξ∈O⊂g which is flat on Σ: F (vi , vj )(m, ξ) = ck ij (fk (m) − hξ, Xk i) ≈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 n nt Map • The BRST construction applied to this setting yields in degree 0 the sections of LM ⊠ L∗ g∗ whose restriction to Σ is g-invariant... ectic Points eories ges to Phases ns & Momenta ... i.e. the sections that are ∇-closed modulo δ: ∇i Ψ(m, ξ) = ϕji Gj (m, ξ) ≈Σ 0. uge Quantization Quantization se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” n ons he 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 n nt Map • The BRST construction applied to this setting yields in degree 0 the sections of LM ⊠ L∗ g∗ whose restriction to Σ is g-invariant... ectic Points eories ges to Phases ns & Momenta uge Quantization ... i.e. the sections that are ∇-closed modulo δ: ∇i Ψ(m, ξ) = ϕji Gj (m, ξ) ≈Σ 0. • If we consider the (distribution) sections whose restrictions to Σ are supported by the elements (µ−1 (ξ), ξ) ∈ Σ for a fixed ξ ∈ g∗ , the cocycle eq. becomes Quantization se Quantization al Constraint Surface ∇M v ψ(m) ≈(µ−1 (ξ),ξ) 0. i Pre-Quantum ry Group-Polarization” n ons he 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 n nt Map • The BRST construction applied to this setting yields in degree 0 the sections of LM ⊠ L∗ g∗ whose restriction to Σ is g-invariant... ectic Points eories ges to Phases ns & Momenta uge Quantization ... i.e. the sections that are ∇-closed modulo δ: ∇i Ψ(m, ξ) = ϕji Gj (m, ξ) ≈Σ 0. • If we consider the (distribution) sections whose restrictions to Σ are supported by the elements (µ−1 (ξ), ξ) ∈ Σ for a fixed ξ ∈ g∗ , the cocycle eq. becomes Quantization se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” n ons he End ∇M v ψ(m) ≈(µ−1 (ξ),ξ) 0. i • By using the pre-quantum operators v̂f = −i~∇vf + f this eq. can be rewritten as an eigenvalue eq. v̂i ψ(m) ≈ fi (m)ψ(m), ≈ hµ(m), Xi i ψ, = hξ, Xi 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 Conclusions n • We have argued that phase symmetries and gauge symmetries... nt Map ectic Points eories ... are different manifestations of the same geom. formalism, i.e. the Mardsen-Weinstein symplectic reduction. ges to Phases ns & Momenta uge Quantization Quantization se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” n ons he End elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 44/45 Conclusions n • We have argued that phase symmetries and gauge symmetries... nt Map ectic Points eories ... are different manifestations of the same geom. formalism, i.e. the Mardsen-Weinstein symplectic reduction. ges to Phases ns & Momenta uge Quantization • From a conceptual viewpoint, this fact suggests a gauge-theoretic interpretation... Quantization se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” ... of the fact that quantum states can be completely described by using half the N of observables required in classical mechanics. ◦ n ons he End elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 44/45 Conclusions n • We have argued that phase symmetries and gauge symmetries... nt Map ectic Points eories ... are different manifestations of the same geom. formalism, i.e. the Mardsen-Weinstein symplectic reduction. ges to Phases ns & Momenta uge Quantization • From a conceptual viewpoint, this fact suggests a gauge-theoretic interpretation... Quantization se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” n ons he End ... of the fact that quantum states can be completely described by using half the N of observables required in classical mechanics. ◦ • From a technical viewpoint, this facts points towards the possibility of a BRST cohomological quantization of ordinary (non-constrained) theories... ... in which the “polarization” of quantum states naturally arises from the condition of g-invariance on the cocycles. elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 44/45 This is the End n nt Map ectic Points eories ges to Phases Many thanks for your kind attention !!! ns & Momenta uge Quantization Quantization se Quantization al Constraint Surface Pre-Quantum ry Group-Polarization” n ons he End elation Betweeen Gauge and Phase Symmetries - Gabriel Catren - XIV International Symposium Frontiers of Fundamental Physics, Université Aix Marseille 15-18 July, 2014 - p. 45/45