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

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

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

ion tries & Reduction lassical to Quantum ture ent Map lectic Points heories uges to Phases

• ns & 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. 2/45

Symmetries & Reduction

Symmetries ⇓ Reduction ... in the amount of (invariant) information... ... that is necessary to completely describe a system.

ion tries & Reduction lassical to Quantum ture ent Map lectic Points heories uges to Phases

• ns & 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. 2/45

Symmetries & Reduction

Symmetries ⇓ Reduction ... in the amount of (invariant) information... ... that is necessary to completely describe a system.

• Example: gauge theories (or constrained Hamiltonian systems):

2n degrees of freedom + k first-class constraints ⇓ 2(n -k) physical degrees of freedom

ion tries & Reduction lassical to Quantum ture ent Map lectic Points heories uges to Phases

• ns & 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. 3/45

From Classical to Quantum

• The transition from classical to quantum mechanics...

... entails a reduction in the number of obs. that are necessary to define a physical state: 2n classical observables q and p ⇓ n quantum observables q or p.

ion tries & Reduction lassical to Quantum ture ent Map lectic Points heories uges to Phases

• ns & 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. 3/45

From Classical to Quantum

• The transition from classical to quantum mechanics...

... entails a reduction in the number of obs. that are necessary to define a physical state: 2n classical observables q and p ⇓ n quantum observables q or p.

• In the simplest case, the phase invariance of |pi under translations in q

q0 · |pi → e2πiq0pi|pi ≈ |pi can be interpreted by saying that the position q of |pi is completely “undetermined”.

ion tries & Reduction lassical to Quantum ture ent Map lectic Points heories uges to Phases

• ns & 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. 3/45

From Classical to Quantum

• The transition from classical to quantum mechanics...

... entails a reduction in the number of obs. that are necessary to define a physical state: 2n classical observables q and p ⇓ n quantum observables q or p.

• In the simplest case, the phase invariance of |pi under translations in q

q0 · |pi → e2πiq0pi|pi ≈ |pi can be interpreted by saying that the position q of |pi is completely “undetermined”.

• Heisenberg indeterminacy principle generalizes this reduction to more gral. states

(e.g. coherent states).

ion tries & Reduction lassical to Quantum ture ent Map lectic Points heories uges to Phases

• ns & 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. 4/45

Conjecture

• In analogy to gauge theories, we could try to understand this transition...

... as a reduction induced by some form of symmetry.

ion tries & Reduction lassical to Quantum ture ent Map lectic Points heories uges to Phases

• ns & 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. 4/45

Conjecture

• In analogy to gauge theories, we could try to understand this transition...

... as a reduction induced by some form of symmetry.

• Far from being a mere analogy, I will argue that

... the quantum phase symmetries can be understood... ... as a consequence of the same formalism underlying the gauge symmetries, i.e. the symplectic reduction procedure.

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 5/45

Hamiltonian G-manifolds

• Let (M, ω, µ) be a Hamiltonian G-manifold, i.e. a connected symplectic manifold

endowed .with an action Φ : G × M → M of a Lie group G preserving ω (i.e. Φ∗

gω = ω for all g ∈ G).

.with an equivariant moment map (introduced by J.-M. Souriau) µ : M → g∗ i.e. a (Poisson) map intertwining the G-action on M and the G-co-adjoint action on g∗: M

µ

• Φg
• g∗

Ad∗ g−1

• M

µ

g∗.

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 6/45

Moment Map

• Given Xi ∈ g, the moment map

µ : M → g∗ defines a generating function of the group action on M fi(m) = µ(m), Xi, Xi ∈ g such that its symplectic gradient vfi = ω−1d fi is the fundamental vector field that infinitesimally generates the G-action on M.

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 6/45

Moment Map

• Given Xi ∈ g, the moment map

µ : M → g∗ defines a generating function of the group action on M fi(m) = µ(m), Xi, Xi ∈ g such that its symplectic gradient vfi = ω−1d fi 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}Xi∈g of
• bservables on M (i.e. the generating functions fi).

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 7/45

What is g∗ useful for?

• A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action,

but also with a map towards g∗.

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 7/45

What is g∗ useful for?

• A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action,

but also with a map towards g∗.

• Now, how can we interpret this relation between M and g∗?

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 7/45

What is g∗ useful for?

• A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action,

but also with a map towards g∗.

• Now, how can we interpret this relation between M and g∗?
• Roughly speaking, g∗ encodes the unitary representation theory of G.

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 7/45

What is g∗ useful for?

• A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action,

but also with a map towards g∗.

• Now, how can we interpret this relation between M and g∗?
• Roughly speaking, g∗ encodes the unitary representation theory of G.
• Firstly, g∗ is a Poisson manifold with respect to the so-called Lie-Poisson structure

{f, g} (x) = x, [d f(x), dg(x)] f, g ∈ C∞(g∗)

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 7/45

What is g∗ useful for?

• A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action,

but also with a map towards g∗.

• Now, how can we interpret this relation between M and g∗?
• Roughly speaking, g∗ encodes the unitary representation theory of G.
• Firstly, g∗ is a Poisson manifold with respect to the so-called Lie-Poisson structure

{f, g} (x) = x, [d f(x), dg(x)] f, g ∈ C∞(g∗)

• Secondly,

Symplective leaves of g∗ = Coadjoint orbits O of G g∗

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 7/45

What is g∗ useful for?

• A Hamiltonian G-manifold (M, ω, µ) is not only endowed with a symplectic G-action,

but also with a map towards g∗.

• Now, how can we interpret this relation between M and g∗?
• Roughly speaking, g∗ encodes the unitary representation theory of G.
• Firstly, g∗ is a Poisson manifold with respect to the so-called Lie-Poisson structure

{f, g} (x) = x, [d f(x), dg(x)] 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.

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 8/45

Kirillov’s Orbit Method

• For certain G, Kirillov’s orbit method establishes a correspondence

g∗

Z/G ∼ ˆ

G, (where ˆ G 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...

• r by applying the functor IndG

H to the 1-dim unirrep ρH ξ of H = exp(h) defined by

ξ ∈ O where h ⊂ g is a max. subalg. subordinated to ξ: ξ, [h, h] = 0. G-Homogeneous Symplectic Manifolds in g∗ Irreducible Unitary Representations of G

ion ent Map nian G-manifolds t Map g∗ useful for? ’s Orbit Method lectic Points heories uges to Phases

• ns & 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. 8/45

Kirillov’s Orbit Method

• For certain G, Kirillov’s orbit method establishes a correspondence

g∗

Z/G ∼ ˆ

G, (where ˆ G 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...

• r by applying the functor IndG

H to the 1-dim unirrep ρH ξ of H = exp(h) defined by

ξ ∈ O where h ⊂ g is a max. subalg. subordinated to ξ: ξ, [h, h] = 0. 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 → e2πiξ,X, X ∈ g.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

irreducible unitary action on HO... ... we could expect the G-action on M to be lifted to a unitary action on HM .

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

irreducible unitary action on HO... ... we could expect the G-action on M to be lifted to a unitary action on HM .

• Since M is not in gral. G-homogeneous, the lifted unitary action will not in gral. be

irreducible: HM =

• O⊂g∗

m(O, M)HO, where m(O, M) . = dim(HomG(HO, HM)) is the multiplicity with which the unirrep HO occurs in HM.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

irreducible unitary action on HO... ... we could expect the G-action on M to be lifted to a unitary action on HM .

• Since M is not in gral. G-homogeneous, the lifted unitary action will not in gral. be

irreducible: HM =

• O⊂g∗

m(O, M)HO, where m(O, M) . = dim(HomG(HO, HM)) is the multiplicity with which the unirrep HO occurs in HM.

• Kirillov’s conjecture: µ tells which unirreps of G occur in HM .

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

irreducible unitary action on HO... ... we could expect the G-action on M to be lifted to a unitary action on HM .

• Since M is not in gral. G-homogeneous, the lifted unitary action will not in gral. be

irreducible: HM =

• O⊂g∗

m(O, M)HO, where m(O, M) . = dim(HomG(HO, HM)) is the multiplicity with which the unirrep HO occurs in HM.

• Kirillov’s conjecture: µ tells which unirreps of G occur in HM .
• Guillemin-Sternberg conjecture: µ also gives m(O, M).

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

irreducible unitary action on HO... ... we could expect the G-action on M to be lifted to a unitary action on HM .

• Since M is not in gral. G-homogeneous, the lifted unitary action will not in gral. be

irreducible: HM =

• O⊂g∗

m(O, M)HO, where m(O, M) . = dim(HomG(HO, HM)) is the multiplicity with which the unirrep HO occurs in HM.

• Kirillov’s conjecture: µ tells which unirreps of G occur in HM .
• Guillemin-Sternberg conjecture: µ also gives m(O, M).
• 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.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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.

Let’s consider the case of an abelian G...

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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.

Let’s consider the case of an abelian G...

• 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 µ−1(ξ).

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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.

Let’s consider the case of an abelian G...

• 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 µ−1(ξ).

• Now, the preimage µ−1(ξ) of the (trivial) symp. manifold O is not a symp. manifold.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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.

Let’s consider the case of an abelian G...

• 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 µ−1(ξ).

• Now, the preimage µ−1(ξ) of the (trivial) symp. manifold O is not a symp. manifold.
• (Shifted) Mardsen-Weinstein reduction theorem:

Mξ . = µ−1(ξ)/G is a symp. manifold called the ξ-symplectic quotient of M.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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.

Let’s consider the case of an abelian G...

• 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 µ−1(ξ).

• Now, the preimage µ−1(ξ) of the (trivial) symp. manifold O is not a symp. manifold.
• (Shifted) Mardsen-Weinstein reduction theorem:

Mξ . = µ−1(ξ)/G is a symp. manifold called the ξ-symplectic quotient of M.

• So, Mξ is the symp. counterpart of ξ in M.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

A Restriction entails a Projection

• In gauge-theoretic terms, when we fix the “value” of the “momentum” µ to ξ by means
• f the restriction to the “ξ-constraint surface”

µ−1(ξ) ⊂ M, ... the “conjugate coordinate” acted upon by G becomes completely “undetermined”... .... in the sense that it is “gauged out” by means of the projection µ−1(ξ) ։ Mξ.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

How should we interpret Mξ?

• We shall argue...

1) that the ξ-symplectic quotient Mξ . = µ−1(ξ)/G is the “moduli space” parameterizing the category-theoretical symplectic ξ-points of M. 2) that the notion of symplectic point elicits a category-theoretical interpretation of Heisenberg indeterminacy principle.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

“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”.”

• V. Guillemin and S. Sternberg, Geometric Quantization and Multiplicities of

Group Representations, 1982.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

“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”.”

• V. Guillemin and S. Sternberg, Geometric Quantization and Multiplicities of

Group Representations, 1982.

• This notion of Lagrangian true “points” acquires a precise category-theoretical

meaning in the framework of Weinstein’s symplectic “category”.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

ϕx : {∗} → M given by {∗} → x

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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

ϕx : {∗} → M given by {∗} → x

• More generally, given two objects A and B in a category, the morphisms

B → A define the so-called B-points of A.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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:

.Symplectic manifolds (M, ω).

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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:

.Symplectic manifolds (M, ω).

• Morphisms (or Lagrangian correspondences) (M2, ω2) → (M1, ω1):

HomSymp(M2, M1) =

• L2,1 ֒

→ M1 × M−

2

• where (M−

2 , −ω2) is the dual of (M2, ω2) and

(M1 × M−

2 , π∗ 1ω1 − π∗ 2ω2),

is the product symplectic manifold.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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:

.Symplectic manifolds (M, ω).

• Morphisms (or Lagrangian correspondences) (M2, ω2) → (M1, ω1):

HomSymp(M2, M1) =

• L2,1 ֒

→ M1 × M−

2

• where (M−

2 , −ω2) is the dual of (M2, ω2) and

(M1 × M−

2 , π∗ 1ω1 − π∗ 2ω2),

is the product symplectic manifold.

• In particular, a symplectomorphism defines a Lagrangian corresp.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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:

.Symplectic manifolds (M, ω).

• Morphisms (or Lagrangian correspondences) (M2, ω2) → (M1, ω1):

HomSymp(M2, M1) =

• L2,1 ֒

→ M1 × M−

2

• where (M−

2 , −ω2) is the dual of (M2, ω2) and

(M1 × M−

2 , π∗ 1ω1 − π∗ 2ω2),

is the product symplectic manifold.

• In particular, a symplectomorphism defines a Lagrangian corresp.
• The symplectic points of (M, ω) is given by the morphisms in

HomSymp((∗, 0), (M, ω)) = {L ֒ → M × {∗} ≃ M} , i.e. by the Lagrangian submanifolds of M.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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:

.Hamiltonian G-manifolds (M, ω, µ).

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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:

.Hamiltonian G-manifolds (M, ω, µ).

• Morphisms (M2, ω2, µ2) → (M1, ω1, µ1):

L2,1

M1 ×g∗ M−

2

• M−

2 µ2

• M1

µ1

g∗

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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:

.Hamiltonian G-manifolds (M, ω, µ).

• Morphisms (M2, ω2, µ2) → (M1, ω1, µ1):

L2,1

M1 ×g∗ M−

2

• M−

2 µ2

• M1

µ1

g∗

• In other terms,

HomG-Symp(M2, M1) =

• L2,1 ֒

→ Φ−1(0) ⊂ M1 × M−

2

• ,

where (M1 × M−

2 , π∗ 1ω1 − π∗ 2ω2, Φ .

= µ1 − µ2), is the product Hamiltonian G-manifold.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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. 17/45

Classical Intertwiner Spaces

• It can be shown (♣) that L2,1 ⊂ M1 ×g∗ M−

2 are G-invariant...

... and that there is a bijection HomG-Symp(M2, M1) ≃

• L ⊂ (M1 ×g∗ M−

2 )/G

• .

____________________________________________________________________________

♣ Xu, P . [1994]: “Classical Intertwiner Space and Quantization,” Commun. Math. Phys. 164, 473-488.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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. 17/45

Classical Intertwiner Spaces

• It can be shown (♣) that L2,1 ⊂ M1 ×g∗ M−

2 are G-invariant...

... and that there is a bijection HomG-Symp(M2, M1) ≃

• L ⊂ (M1 ×g∗ M−

2 )/G

• .
• Under nice conditions, (M1 ×g∗ M−

2 )/G is a symplectic manifold...

... whose symplectic points are the classical intertwiners over g∗ between M2 and M1... ... or, in category-theoretical terms, the M2-sympletic points of M1. ____________________________________________________________________________

♣ Xu, P . [1994]: “Classical Intertwiner Space and Quantization,” Commun. Math. Phys. 164, 473-488.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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, µξ : ξ → ξ) → (M, ω, µ) are given by the
• Lagrang. subman. of

(M ×g∗ ξ−)/G = Φ−1(0)/G, where the twisted moment map is Φ : M × ξ− → g∗ (m, ξ) → µ(m) − ξ.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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, µξ : ξ → ξ) → (M, ω, µ) are given by the
• Lagrang. subman. of

(M ×g∗ ξ−)/G = Φ−1(0)/G, where the twisted moment map is Φ : M × ξ− → g∗ (m, ξ) → µ(m) − ξ.

• Now, since Φ−1(0) ≃ µ−1(ξ) ⊂ M, then Φ−1(0)/G ≃ Mξ.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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, µξ : ξ → ξ) → (M, ω, µ) are given by the
• Lagrang. subman. of

(M ×g∗ ξ−)/G = Φ−1(0)/G, where the twisted moment map is Φ : M × ξ− → g∗ (m, ξ) → µ(m) − ξ.

• Now, since Φ−1(0) ≃ µ−1(ξ) ⊂ M, then Φ−1(0)/G ≃ Mξ.
• All in all,

HomG-Symp(ξ, M) = {L ⊂ Mξ} i.e. the symplectic points of Mξ are in correspondence with the ξ-points of M. Mξ can be interpreted as the moduli space of symplectic ξ-points of M.

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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)

quantization of the classical intertwiner space: MO ∼ = HomG-Symp(O, M) between a coadjoint orbit O and M yields the quantum intertwiner space: HMO ∼ = HomG(HO.HM),

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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)

quantization of the classical intertwiner space: MO ∼ = HomG-Symp(O, M) between a coadjoint orbit O and M yields the quantum intertwiner space: HMO ∼ = HomG(HO.HM),

• Whereas MO parameterizes the symplectic G-morphisms

O → M, HMO parameterizes the unitary G-intertwiners HO → HM

ion ent Map lectic Points ’s Conjecture plectic Quotients riction entails a ion

• uld we interpret

ein’s Symplectic Creed ry-Theoretical “Points” ein’s Symplectic

• ry”

ein’s G-Symplectic

• ry”

al Intertwiner Spaces plectic Points of M m Intertwiner Spaces heories uges to Phases

• ns & 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)

quantization of the classical intertwiner space: MO ∼ = HomG-Symp(O, M) between a coadjoint orbit O and M yields the quantum intertwiner space: HMO ∼ = HomG(HO.HM),

• Whereas MO parameterizes the symplectic G-morphisms

O → M, HMO parameterizes the unitary G-intertwiners HO → HM

• Kirillov’s conjecture revisited: The unirrep HO occurs in HM if M has symplectic

O-points where the multiplicity is given by m(O, M) = dim(HMO ).

ion ent Map lectic Points heories n-Weinstein uction zation Commutes with uction uges to Phases

• ns & 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

constraint the solutions to be in the constraint surface Σ = µ−1(0) ⊂ M.

ion ent Map lectic Points heories n-Weinstein uction zation Commutes with uction uges to Phases

• ns & 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

constraint the solutions to be in the constraint surface Σ = µ−1(0) ⊂ M.

• The Mardsen-Weinstein reduction theorem shows that the 0-symplectic quotient

M0 ≃ µ−1(0)/G has a canonical symplectic form ωM0 satisfying π∗ωM0 = ι∗ωM, where µ−1

M (0) ι

• π
• M

M0 . = µ−1(0)/G.

ion ent Map lectic Points heories n-Weinstein uction zation Commutes with uction uges to Phases

• ns & 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

HM0 ∼ = HomG(H0, HM ),

• r, in other terms,

HM0 ≃ HG

M ,

where HG

M is the space of G-invariant states in HM.

ion ent Map lectic Points heories n-Weinstein uction zation Commutes with uction uges to Phases

• ns & 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

HM0 ∼ = HomG(H0, HM ),

• r, in other terms,

HM0 ≃ HG

M ,

where HG

M is the space of G-invariant states in HM.

• Diagramatically,

M Quantization

• 0-symplectic reduction
• HM

G-invariant states

• M0
• HM0 ≃ HG

M ,

Quantum G-invariance Classical 0-symplectic reduction.

ion 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

• ns & 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

to the classical intertwiner space M0 ≃ Hom(0, M) 0 ∈ g∗ containing the 0-symplectic points of M... ... implies that the resulting quantum theory only includes G-invariant states.

ion 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

• ns & 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

to the classical intertwiner space M0 ≃ Hom(0, M) 0 ∈ g∗ containing the 0-symplectic points of M... ... 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 Gi(m) = µ(m), Xi Xi ∈ g are called constraints.

ion 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

• ns & 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),

... where by ordinary we mean that the theories are not constrained to a unique value of µ.

ion 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

• ns & 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),

... where by ordinary we mean that the theories are not constrained to a unique value of µ.

• We shall call G the phase group and the non-constrained generating functions of the

G-action fi(m) = µ(m), Xi phase observables.

ion 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

• ns & 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 Ga, the phase observables fi do not select a single

unirrep of G.

ion 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

• ns & 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 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

M0 . = µ−1(0)/G, ... associated to the trivial unirrep of G... ... a phase group action defines a different O-symplectic quotient MO . = µ−1(O)/G for each unirrep O ⊂ g∗

Z of the phase group G.

ion 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

• ns & 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 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

M0 . = µ−1(0)/G, ... associated to the trivial unirrep of G... ... a phase group action defines a different O-symplectic quotient MO . = µ−1(O)/G 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.

ion 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

• ns & 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. 25/45

Quantization Commutes with ξ-Reduction

• The Guillemin & Sternberg’s conjecture for a ξ-symplectic quotient with G abelian

states that this diagram commutes: M Quantization

• ξ-symplectic reduction
• HM

(G,ξ)-phase invariant states

• Mξ
• HMξ ≃ H(G,ξ)

M

, .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:

ρG

ξ : G × H(G,ξ) M

→ H(G,ξ)

M

(eX, |ξ, ...) → e2πiξ,X|ξ, ...,

ion 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

• ns & 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

Gauge Invariance vs. Phase Invariance

• The (G, ξ)-phase invariance of quantum states is the quantum counterpart of the

symplectic reduction with respect to a non-zero ξ ∈ g∗. Quantum phase invariance is the generalization... ... of the strict gauge invariance... .... to the case of ξ-symplectic reductions with ξ = 0.

ion 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

• ns & 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 HM ...

... results from the existence of ξ-symplectic points in M.

ion 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

• ns & 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 HM ...

... results from the existence of ξ-symplectic points in M.

• Far from being structureless set-theoretic points,...

... the ξ-symplectic points of M are non-trivial subman. of M endowed with a G-action.

ion 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

• ns & 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 HM ...

... results from the existence of ξ-symplectic points in M.

• Far from being structureless set-theoretic points,...

... 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.e. the “indeterminacy” in the variable acted upon by G... ... is the quantum counterpart of the fact... ... that the corresponding ξ-symplectic points of M have an internal G-symmetry.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 28/45

For Instance...

• Let’s consider the simplest case of G = R acting on M = T ∗R by

q0 · (q, p) → (q + q0, p) with moment map µ : T ∗R → R (q, p) → p

ion ent Map lectic Points heories uges to Phases

• ns & 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. 28/45

For Instance...

• Let’s consider the simplest case of G = R acting on M = T ∗R by

q0 · (q, p) → (q + q0, p) with moment map µ : T ∗R → R (q, p) → p

• The pi-symplectic quotient

Mpi = µ−1(pi)/G = {∗} contains the unique symplectic pi-point of M.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 29/45

Phasing Out the Position

• According to

[Quantization, ξ-Reduction] = 0, the (1-dimensional) quantization of Mpi yields the (unique) (G, pi)-phase invariant state |pi in HM: q0 · |pi → e2πiq0pi|pi ≈ |pi.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 29/45

Phasing Out the Position

• According to

[Quantization, ξ-Reduction] = 0, the (1-dimensional) quantization of Mpi yields the (unique) (G, pi)-phase invariant state |pi in HM: q0 · |pi → e2πiq0pi|pi ≈ |pi.

• The indeterminacy in the position q of the state |pi is a symptom of the fact...

... that the unique symplectic pi-point of M has an internal symmetry under translations in q.

ion ent Map lectic Points heories uges to Phases

• ns & 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

Heisenberg indeterminacy principle.

ion ent Map lectic Points heories uges to Phases

• ns & 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

Heisenberg indeterminacy principle.

• The (im)possibility of sharply localizing quantum states in phase space depends on the

notion of point that we are using: ... while a quantum state cannot be sharply localized at the set-theoretic points of M,... ... it can be sharply localized at its symplectic point... ... given that the symplectic points “internalize” the unsharp variables.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 31/45

Breaking the Phase Invariance

• The superposition of two G-phase invariant states |pi and |pj transforming in

different unirreps of G,... ... is no longer G-phase invariant... ... since the G-action changes the relative phases between the two terms |pi + |pj.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 31/45

Breaking the Phase Invariance

• The superposition of two G-phase invariant states |pi and |pj transforming in

different unirreps of G,... ... is no longer G-phase invariant... ... since the G-action changes the relative phases between the two terms |pi + |pj.

• Therefore, by introducing an “indeterminacy” in the value of the variable p that

labels the unirreps of G,... ... we break the G-phase invariance,... ... i.e. the complete indeterminacy in the variable q acted upon by G.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 32/45

Gauge Group Actions vs...

A gauge theory is restricted to a single value of µ (namely 0) ⇓ The quantum theory only contains G-invariant quantum states (i.e. states transforming in the trivial unirrep of G) ⇓ Since we do not have different unirreps to superpose... ⇓ ... the gauge invariance cannot be broken.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 33/45

... Phase Group Actions

Phase observables are not restricted to a single value of µ ⇓ The quantum theory contains (G, ξ)-phase invariant states for all ξ ∈ g∗

Z

(i.e. states transforming in different unirreps of G) ⇓ We can superpose (G, ξ)-phase invariant states defined by different unirreps ξ. ⇓ The G-phase invariance is broken for such superposed states.

ion ent Map lectic Points heories uges to Phases

• ns & Momenta

uge Quantization

• logical 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. 34/45

Cohomological Reduction

The Dirac observables f ∈ C∞(M0) of a gauge theory can be recovered... ... by means of the BRST-cohomological reformulation of the 0-symplectic reduction. ____________________________________________________________________________

♣ Kostant, B. & Sternberg, S. [1987]: “Symplectic Reduction, BRS Cohomology, and Infinite Dimensional Clifford Algebras,” Annals of Physics 176, 49-113.

ion ent Map lectic Points heories uges to Phases

• ns & Momenta

uge Quantization

• logical 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. 35/45

Gauge Theories in a Diagram

The reduction from M to M0 . = 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).

ion ent Map lectic Points heories uges to Phases

• ns & Momenta

uge Quantization

• logical 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. 36/45

Koszul Resolution

• We can describe the algebra of observables on Σ

C∞(Σ) = C∞(M)/Ga in homological terms by extending the (co-moment) map g → C∞(M) Xi → fi(m) = µ(m), Xi to the quasi-acyclic complex ... ∧q g ⊗ C∞(M) → ∧q−1g ⊗ C∞(M) → ... → g ⊗ C∞(M)

δ1

− − → C∞(M)

δ0

− − → 0, where the Koszul differential is defined by δ(Xi ⊗ 1) = 1 ⊗ fi, δ(1 ⊗ f) = 0.

ion ent Map lectic Points heories uges to Phases

• ns & Momenta

uge Quantization

• logical 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. 36/45

Koszul Resolution

• We can describe the algebra of observables on Σ

C∞(Σ) = C∞(M)/Ga in homological terms by extending the (co-moment) map g → C∞(M) Xi → fi(m) = µ(m), Xi to the quasi-acyclic complex ... ∧q g ⊗ C∞(M) → ∧q−1g ⊗ C∞(M) → ... → g ⊗ C∞(M)

δ1

− − → C∞(M)

δ0

− − → 0, where the Koszul differential is defined by δ(Xi ⊗ 1) = 1 ⊗ fi, δ(1 ⊗ f) = 0.

• Hence,

Hδ

0 (∧g ⊗ C∞(M))

= Ker(δ0)/Im(δ1) = C∞(M)/Ga = C∞(Σ).

ion ent Map lectic Points heories uges to Phases

• ns & Momenta

uge Quantization

• logical 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

Chevalley-Eilenberg differential) d : K → g∗ ⊗ K = Hom(g, K) given by (dk)(X) = X · k, X ∈ g, k ∈ K.

ion ent Map lectic Points heories uges to Phases

• ns & Momenta

uge Quantization

• logical 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

Chevalley-Eilenberg differential) d : K → g∗ ⊗ K = Hom(g, K) given by (dk)(X) = X · k, X ∈ g, k ∈ K.

• This can be extended to a map

d : ∧qg∗ ⊗ K → ∧q+1g∗ ⊗ K, by d(η ⊗ k) = dη ⊗ k + (−1)pη ⊗ dk, η ∈ ∧pg∗.

ion ent Map lectic Points heories uges to Phases

• ns & Momenta

uge Quantization

• logical 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

Chevalley-Eilenberg differential) d : K → g∗ ⊗ K = Hom(g, K) given by (dk)(X) = X · k, X ∈ g, k ∈ K.

• This can be extended to a map

d : ∧qg∗ ⊗ K → ∧q+1g∗ ⊗ K, by d(η ⊗ k) = dη ⊗ k + (−1)pη ⊗ dk, η ∈ ∧pg∗.

• The 0-Lie algebra cohomology of g with values in the g-module K is given by

H0

d(∧g∗ ⊗ K) = Kg.

ion ent Map lectic Points heories uges to Phases

• ns & Momenta

uge Quantization

• logical 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. 38/45

BRST Cohomology

• We can then form the double complex

∧pg∗ ⊗ ∧qg ⊗ C∞(M)

δ

• d
• ∧pg∗ ⊗ ∧q−1g ⊗ C∞(M)

∧p+1g∗ ⊗ ∧qg ⊗ C∞(M) with δ2 = 0 d2 = 0 δd = dδ such that H0

d(Hδ 0(∧g∗ ⊗ ∧g ⊗ C∞(M))) = C∞(M0).

ion ent Map lectic Points heories uges to Phases

• ns & Momenta

uge Quantization

• logical 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. 38/45

BRST Cohomology

• We can then form the double complex

∧pg∗ ⊗ ∧qg ⊗ C∞(M)

δ

• d
• ∧pg∗ ⊗ ∧q−1g ⊗ C∞(M)

∧p+1g∗ ⊗ ∧qg ⊗ C∞(M) with δ2 = 0 d2 = 0 δd = dδ such that H0

d(Hδ 0(∧g∗ ⊗ ∧g ⊗ C∞(M))) = C∞(M0).

• Hence, the Dirac observables in C∞(M0) can be described as elements f 0

0 that are

d-closed modulo δ: d f 0

0 = −δf 1 1 ≈Σ 0.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 39/45

Pre-Quantum Geometry in a Nutshell

• 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 HCech(M, Z) → HDe Rahm(M) ... there exists a complex line bundle L → M with a Hermitian inner product ·, · and a compatible connection ∇ such that curv(∇) = 2πiω.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 39/45

Pre-Quantum Geometry in a Nutshell

• 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 HCech(M, Z) → HDe Rahm(M) ... there exists a complex line bundle L → M with a Hermitian inner product ·, · and a compatible connection ∇ such that curv(∇) = 2πiω.

• The sections ψ ∈ Γ(L) define the so-called pre-quantum states.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 39/45

Pre-Quantum Geometry in a Nutshell

• 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 HCech(M, Z) → HDe Rahm(M) ... there exists a complex line bundle L → M with a Hermitian inner product ·, · and a compatible connection ∇ such that curv(∇) = 2πiω.

• 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

ˆ vi = −i∇vfi + fi.

ion ent Map lectic Points heories uges to Phases

• ns & 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

Heisenberg indeterminacy principle.

ion ent Map lectic Points heories uges to Phases

• ns & 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

Heisenberg indeterminacy principle.

• The pre-quantum geometry has to be enriched by choosing a polarization P, i.e. an

involutive Lagrangian subbundle of T M.

ion ent Map lectic Points heories uges to Phases

• ns & 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

Heisenberg indeterminacy principle.

• The pre-quantum geometry has to be enriched by choosing a polarization P, i.e. an

involutive Lagrangian subbundle of T M.

• We can then “cut in half’ the space of pre-quantum states by means of an eq. of the

form ∇Pψ = 0.

ion ent Map lectic Points heories uges to Phases

• ns & 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

Heisenberg indeterminacy principle.

• The pre-quantum geometry has to be enriched by choosing a polarization P, i.e. an

involutive Lagrangian subbundle of T M.

• We can then “cut in half’ the space of pre-quantum states by means of an eq. of the

form ∇Pψ = 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.

ion ent Map lectic Points heories uges to Phases

• ns & 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

Heisenberg indeterminacy principle.

• The pre-quantum geometry has to be enriched by choosing a polarization P, i.e. an

involutive Lagrangian subbundle of T M.

• We can then “cut in half’ the space of pre-quantum states by means of an eq. of the

form ∇Pψ = 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.

ion ent Map lectic Points heories uges to Phases

• ns & 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 M0 as in gauge theories...

... but rather all the ξ-symplectic quotients Mξ at once.

ion ent Map lectic Points heories uges to Phases

• ns & 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 M0 as in gauge theories...

... but rather all the ξ-symplectic quotients Mξ at once.

• Hence, instead of considering a single ξ-constraint surface Σξ = µ−1(ξ)/G...

... we shall consider the universal constraint surface Σ = Φ−1(0) ⊂ M × g∗

−,

where M × g∗

− is endowed with the product Poisson structure and the shifted moment

map Φ : M × g∗

−

→ g∗ (m, ξ) → Φ(m, ξ) = µ(m) − ξ

ion ent Map lectic Points heories uges to Phases

• ns & 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 M0 as in gauge theories...

... but rather all the ξ-symplectic quotients Mξ at once.

• Hence, instead of considering a single ξ-constraint surface Σξ = µ−1(ξ)/G...

... we shall consider the universal constraint surface Σ = Φ−1(0) ⊂ M × g∗

−,

where M × g∗

− is endowed with the product Poisson structure and the shifted moment

map Φ : M × g∗

−

→ g∗ (m, ξ) → Φ(m, ξ) = µ(m) − ξ

• The surface Σ is defined by the common zeros of the involutive universal constraints

Gi(m, ξ) = fi(m) − ξ, Xi, Xi ∈ g {Gi, Gj} = ck

ijGk.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 42/45

Shifted Pre-Quantum Geometry

• Following Guillemin & Sternberg (1982), we can introduce the shifted bundle

LM ⊠ L∗

g∗ .

= π∗

MLM ⊗ π∗ g∗ −L∗ g∗

defined by the diagram LM

• LM ⊠ L∗

g∗

• L∗

g∗

• M

M × g∗

− πM

• πg∗

−

g∗

−

and endowed with the vertical differential ∇ acting along the G-orbits ∇(m,ξ) = ∇M

m ⊗ id + id ⊗ ∇O ξ ,

ξ ∈ O ⊂ g∗ which is flat on Σ: F (vi, vj)(m, ξ) = ck

ij(fk(m) − ξ, Xk) ≈Σ 0.

ion ent Map lectic Points heories uges to Phases

• ns & 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

LM ⊠ L∗

g∗ whose restriction to Σ is g-invariant...

... i.e. the sections that are ∇-closed modulo δ: ∇iΨ(m, ξ) = ϕj

i Gj(m, ξ) ≈Σ 0.

ion ent Map lectic Points heories uges to Phases

• ns & 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

LM ⊠ L∗

g∗ whose restriction to Σ is g-invariant...

... i.e. the sections that are ∇-closed modulo δ: ∇iΨ(m, ξ) = ϕj

i 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 ∇M

vi ψ(m) ≈(µ−1(ξ),ξ) 0.

ion ent Map lectic Points heories uges to Phases

• ns & 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

LM ⊠ L∗

g∗ whose restriction to Σ is g-invariant...

... i.e. the sections that are ∇-closed modulo δ: ∇iΨ(m, ξ) = ϕj

i 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 ∇M

vi ψ(m) ≈(µ−1(ξ),ξ) 0.

• By using the pre-quantum operators

ˆ vf = −i∇vf + f this eq. can be rewritten as an eigenvalue eq. ˆ viψ(m) ≈ fi(m)ψ(m), ≈ µ(m), Xi ψ, = ξ, Xi ψ.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 44/45

Conclusions

• We have argued that phase symmetries and gauge symmetries...

... are different manifestations of the same geom. formalism, i.e. the Mardsen-Weinstein symplectic reduction.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 44/45

Conclusions

• We have argued that phase symmetries and gauge symmetries...

... are different manifestations of the same geom. formalism, i.e. the Mardsen-Weinstein symplectic reduction.

• From a conceptual viewpoint, this fact suggests a gauge-theoretic interpretation...

... of the fact that quantum states can be completely described by using half the N◦ of observables required in classical mechanics.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 44/45

Conclusions

• We have argued that phase symmetries and gauge symmetries...

... are different manifestations of the same geom. formalism, i.e. the Mardsen-Weinstein symplectic reduction.

• From a conceptual viewpoint, this fact suggests a gauge-theoretic interpretation...

... 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

• f g-invariance on the cocycles.

ion ent Map lectic Points heories uges to Phases

• ns & 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. 45/45

This is the End

Many thanks for your kind attention !!!