Download slides - Frontiers of Fundamental Physics (FFP14)

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