Parametrized theories: Making EM even ``gaugier

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Parametrized theories:
Making EM even “gaugier”
Juan Margalef Bentabol
Joint work with F. Barbero (CSIC) and Eduardo J.S. Villaseñor (UC3M)
21st International Conference on
General Relativity and Gravitation
July 2016 - New York
Motivation: parametrized
(M ∼
= Σ × I, g) being
Σ 3-manifold
I = [t0 , t1 ]
(M ∼
= Σ × I, g) being
Σ 3-manifold
I = [t0 , t1 ]
(M ∼
= Σ × I, g) being
Σ 3-manifold
I = [t0 , t1 ]
(M ∼
= Σ × I, g) being
Σ 3-manifold
I = [t0 , t1 ]
(M ∼
= Σ × I, g) being
Z1
Σ 3-manifold
I = [t0 , t1 ]
Z2
(M ∼
= Σ × I, g) being
Z1
+
Diffsp (M ) :=
Σ 3-manifold
I = [t0 , t1 ]
Z2
T Z.∂t
future timelike
Z : M → M diff. /
Z(Σ × {t}) spacelike
Our model
J. Margalef Bentabol
Parametrized EM Field Action
SEM : Ω1 (M ) −→ R
Z
SEM [A] =
(dA)∧(?g dA)
M
d ?g (dA) = 0
2
SEM : Ω1 (M ) −→ R
Z
SEM [A] =
(dA)∧(?g dA)
M
d ?g (dA) = 0
P
+
SEM
: Ω1 (M ) × Diffsp
(M ) −→ R
Z
P
SEM
[A, Z] :=
(dA)∧(?Z ∗g dA)
M
d ?Z ∗g (dA) = 0
2
SEM : Ω1 (M ) −→ R
Z
SEM [A] =
(dA)∧(?g dA)
M
d ?g (dA) = 0
P
+
SEM
(M ) −→ R
Z
P
SEM
[A, Z] :=
(dA)∧(?Z ∗g dA)
M
d ?Z ∗g (dA) = 0
Properties
If (A, Z) is a solution, then so is (Y ∗ A, Z ◦ Y ).
2
SEM : Ω1 (M ) −→ R
Z
SEM [A] =
(dA)∧(?g dA)
M
d ?g (dA) = 0
P
+
SEM
(M ) −→ R
Z
P
SEM
[A, Z] :=
(dA)∧(?Z ∗g dA)
M
d ?Z ∗g (dA) = 0
Properties
If (A, Z) is a solution, then so is (Y ∗ A, Z ◦ Y ).
P
P
SEM
[Y ∗ A, Z ◦ Y ] = SEM
[A, Z]
2
Hamiltonian Framework
P
SEM =
Z
M
(dA) ∧ (?Z ∗g dA)
M∼
=Σ×I
3
P
SEM =
Z
M
(dA) ∧ (?Z ∗g dA)
M∼
=Σ×I
3+1
P
=⇒ SEM =
Z
Z
dt
I
(something)t
Σ
3
Z
P
SEM =
M
(dA) ∧ (?Z ∗g dA)
M∼
=Σ×I
3+1
P
=⇒ SEM =
Z
Z
dt
I
(something)t
Σ
Lagrangian LPEM : D ⊂ T Q → R
3
Z
P
SEM =
M
(dA) ∧ (?Z ∗g dA)
M∼
=Σ×I
3+1
P
=⇒ SEM =
Z
Z
dt
I
(something)t
Σ
Fiber derivative F LPEM : D ⊂ T Q → T ∗ Q
3
Z
P
SEM =
M
(dA) ∧ (?Z ∗g dA)
M∼
=Σ×I
3+1
P
=⇒ SEM =
Z
Z
dt
I
(something)t
Σ
P
Hamiltonian HEM
: F LPEM (D) → R
3
Z
P
SEM =
M
(dA) ∧ (?Z ∗g dA)
M∼
=Σ×I
3+1
P
=⇒ SEM =
Z
Z
dt
I
(something)t
Σ
P
Hamiltonian HEM
: F LPEM (D) → R
Important things to retain
P
LPEM homogeneous of degree 1, hence HEM
= 0.
3
Z
P
SEM =
M
(dA) ∧ (?Z ∗g dA)
M∼
=Σ×I
3+1
P
=⇒ SEM =
Z
Z
dt
I
(something)t
Σ
P
Hamiltonian HEM
: F LPEM (D) → R
Important things to retain
P
LPEM homogeneous of degree 1, hence HEM
= 0.
In fact, this is true for any parametrized theory.
3
The geometric arena
Configuration space
+
Ω1 (M ) × Diffsp
(M )
q (4) : Σ × I → T ∗ (Σ × I)
Z :Σ×I →Σ×I
(Σ × I, g)
Z
4
Configuration space
+
Ω1 (M ) × Diffsp
(M )
q (4) : Σ × I → T ∗ (Σ × I)
Z :Σ×I →Σ×I
Xt : Σ ,→ Σ × I
(Σ × I, g)
Z
Xt
it
(Σ, γXt = Xt∗ g)
4
Configuration space
+
Ω1 (M ) × Diffsp
(M )
q (4) : Σ × I → T ∗ (Σ × I)
(q⊥ )t : Σ → R
Z :Σ×I →Σ×I
Xt : Σ ,→ Σ × I
(Σ × I, g)
Z
Xt
it
4
Configuration space
+
Ω1 (M ) × Diffsp
(M )
q (4) : Σ × I → T ∗ (Σ × I)
Z :Σ×I →Σ×I
(q⊥ )t : Σ → R
qt : Σ → T ∗ Σ
Xt : Σ ,→ Σ × I
(Σ × I, g)
Z
Xt
it
4
Configuration space
+
Ω1 (M ) × Diffsp
(M )
QEM = C ∞ (Σ) × Ω1 (Σ) × Emb(Σ, M )sp
q (4) : Σ × I → T ∗ (Σ × I)
(q⊥ )t : Σ → R
qt : Σ → T ∗ Σ
Z :Σ×I →Σ×I
Xt : Σ ,→ Σ × I
(Σ × I, g)
Z
Xt
it
4
Configuration space
+
Ω1 (M ) × Diffsp
(M )
QEM = C ∞ (Σ) × Ω1 (Σ) × Emb(Σ, M )sp
q (4) : Σ × I → T ∗ (Σ × I)
(q⊥ )t : Σ → R
qt : Σ → T ∗ Σ
Z :Σ×I →Σ×I
Xt : Σ ,→ Σ × I
(Σ × I, g)
Z
Xt
it
4
Configuration space
+
Ω1 (M ) × Diffsp
(M )
QEM = C ∞ (Σ) × Ω1 (Σ) × Embsp (Σ, M )
q (4) : Σ × I → T ∗ (Σ × I)
(q⊥ )t : Σ → R
qt : Σ → T ∗ Σ
Z :Σ×I →Σ×I
Xt : Σ ,→ Σ × I
(Σ × I, g)
Z
Xt
it
4
Configuration space
QEM = C ∞ (Σ) × Ω1 (Σ) × Embsp (Σ, M )
T QEM = T C ∞ (Σ) × T Ω1 (Σ) × T Embsp (Σ, M )
(Σ × I, g)
Z
Xt
it
4
Configuration space
QEM = C ∞ (Σ) × Ω1 (Σ) × Embsp (Σ, M )
(Σ × I, g)
Z
Xt
it
4
Configuration space
QEM = C ∞ (Σ) × Ω1 (Σ) × Embsp (Σ, M )
∼
=
C ∞ (Σ)×C ∞ (Σ)
(Σ × I, g)
Z
Xt
it
4
Configuration space
QEM = C ∞ (Σ) × Ω1 (Σ) × Embsp (Σ, M )
∼
=
C ∞ (Σ)×C ∞ (Σ)
(Σ × I, g)
Z
Xt
it
4
Configuration space
QEM = C ∞ (Σ) × Ω1 (Σ) × Embsp (Σ, M )
∼
=
∼
=
C ∞ (Σ)×C ∞ (Σ)
Ω1 (Σ)×Ω1 (Σ)
(Σ × I, g)
Z
Xt
it
4
Configuration space
QEM = C ∞ (Σ) × Ω1 (Σ) × Embsp (Σ, M )
∼
=
∼
=
C ∞ (Σ)×C ∞ (Σ)
Ω1 (Σ)×Ω1 (Σ)
(Σ × I, g)
Z
Xt
it
4
T Emb(Σ, M )
TM
VX
Σ
X
M
(Σ × I, g)
VX ∈ TX Emb(Σ, M )
{Xλ } /
X0 = X
Ẋ0 = VX
σ1
b
X0
b
b
X0 (Σ)
b
Σ
σ2
VX = Ẋ0
5
Back to the Hamiltonian
5
Hamiltonian vector field
Symplectic form
j F LPEM (D), j ∗ Ω ,−→ T ∗ QEM , Ω
~ X , Zp )
If Z ∈ X(F LPEM (D)) then Z = q⊥ , q, X, p; Zq⊥ , Zq , Z
6
Symplectic form
~ X , Zp )
ıZ j ∗ Ω = dH = 0
6
Symplectic form
~ X , Zp )
ıZ j ∗ Ω = dH = 0
pa
• (Zq )a = LZ T qa + √ ZX⊥ − d q⊥ ZX⊥
X
γX
√
• Zpa = LZ T pa + γX ∇b ZX⊥ (dq)ba
X
⊥
b
• ZX (∇b p ) = 0
• ZXT arbitrary
• Zq⊥ − LZ T q⊥ + qa ∇a ZX⊥ ∇b pb = 0
X
6
Symplectic form
~ X , Zp )
ıZ j ∗ Ω = dH = 0
pa
• (Zq )a = LZ T qa + √ ZX⊥ − d q⊥ ZX⊥
X
γX
√
• Zpa = LZ T pa + γX ∇b ZX⊥ (dq)ba
X
⊥
b
• ZX (∇b p ) = 0
• ZXT arbitrary
• Zq⊥ − LZ T q⊥ + qa ∇a ZX⊥ ∇b pb = 0
X
6
Symplectic form
~ X , Zp )
ıZ j ∗ Ω = dH = 0
• (Zq )a = LZ T qa
X
• Zpa = LZ T pa
X
⊥
• ZXT arbitrary
• ZX = 0
• Z q ⊥ = L Z T q⊥
X
6
M. Bauer, P. Harms, P.W. Michor, Almost local metrics on shape space
of hypersurfaces in n-space, SIAM J. Imaging Sci. (2012) 5.1.
J.F. Barbero, J. Margalef-Bentabol, E.J.S. Villaseñor, Hamiltonian
dynamics of the parametrized electromagnetic field, Classical and
Quantum Gravity (2016) 33 [arXiv: 1511.00826]
K. Kuchař, Geometry of hyperspace. I, J. Math. Phys. (1976) 17.
Thanks for your attention
M. Bauer, P. Harms, P.W. Michor, Almost local metrics on shape space
of hypersurfaces in n-space, SIAM J. Imaging Sci. (2012) 5.1.
K. Kuchař, Geometry of hyperspace. I, J. Math. Phys. (1976) 17.

Parametrized theories: Making EM even ``gaugier

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