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MOMENTUM MAPS & CLASSICAL FIELDS

1. Introduction

MARK J. GOTAY

MARK J. GOTAY (PIMS, UBC) MOMENTUM MAPS & CLASSICAL FIELDS 1. Introduction

Olomou c, August, 2009 1 / 28
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Joint work over the years 19792009 with:

Jim Isenberg (Eugene)

Jerry MARSDEN (Pasadena)

Richard Montgomery (Santa Cruz)

Jedrzej Sniatycki (Calgary)

Phil Yasskin (College Station)




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Goals:

Study the Lagrangian and Hamiltonian structures ofclassical eld theories (CFTs) with constraints


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Goals:


Explore connections between initial value constraints &gauge transformations

MARK J. GOTAY (PIMS, UBC) MOMENTUM MAPS & CLASSICAL FIELDS 1. IntroductionOlomou c, August, 2009 3 / 28
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Goals:



Tie together & understand many different and apparentlyunrelated facets of CFTs

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Goals:



Tie together & understand many different and apparentlyunrelated facets of CFTs

Focus on roles of gauge symmetry and momentum maps

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Methods & Tools:

calculus of variations

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Methods & Tools:


initial value analysis, Dirac constraint theory

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Methods & Tools:



multisymplectic geometry, multimomentum maps

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Methods & Tools:




Noethers theorem



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Methods & Tools:




Noethers theorem

energy-momentum maps



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Generic Properties of Classical Field Theories

Gleaned from extensive study of standard examples:

electromagnetism, YangMills

gravity

strings

relativistic uids

topological eld theories . . .

Pioneers: Choquet-Bruhat, Lichnerowicz, DiracBergmann, andArnowitDeserMisner (ADM).



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1. The EulerLagrange equations are underdetermined : there are not

enough evolution equations to propagate all eld components.



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This is because the theory has gauge freedom .

The corresponding gauge group is known at the outset.

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This is because the theory has gauge freedom .

The corresponding gauge group is known at the outset.

Kinematic elds have no signicance.

Dynamic elds , conjugate momenta have physical meaning.



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2. The Euler Lagrange equations are overdetermined : they include constraints

i (, ) = 0 (1)

on the choice of initial data.



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i (, ) = 0 (1)


So initial data ((0), (0)) cannot be freely specied

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i (, ) = 0 (1)


So initial data ((0), (0)) cannot be freely specied Elliptic system

Assume all constraints are rst class in the sense of Dirac

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3. The i generate gauge transformations of (, ) via the canonical symplectic structure on the space of Cauchy data.

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3. The i generate gauge transformations of (, ) via the canonical symplectic structure on the space of Cauchy data.

So the presence of constraints gauge freedom

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4. The Hamiltonian (with respect to a slicing) has the form

H =

i

i i (, ) d

depending linearly on the atlas elds i

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4. The Hamiltonian (with respect to a slicing) has the form

H =

i

i i (, ) d

depending linearly on the atlas elds i

Atlas elds:

closely related to kinematic elds

arbitrarily speciable

drive the entire gauge ambiguity of the CFT

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5. The evolution equations for the dynamic elds (, ) take the adjoint form

d d

=

J

i

D i (( ), ( ))

i . (2)

MARK J. GOTAY (PIMS, UBC) MOMENTUM MAPS & CLASSICAL FIELDS 1. IntroductionOlomouc , August, 2009 10 / 28
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d d

=

J

i

D i (( ), ( ))

i . (2)

is a slicing parameter (time)

J is a compatible almost complex structure; * an L2-adjoint

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d d

=

J

i

D i (( ), ( ))

i . (2)

is a slicing parameter (time)

J is a compatible almost complex structure; * an L2-adjoint

hyperbolic system (typically)

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Adjoint form displays, in the clearest and most concise way, theinterrelations between the

dynamics

initial value constraints, and

gauge ambiguity of a theory

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6. The EulerLagrange equations are equivalent, modulo gauge transformations, to the combined evolution equations (2) and constraint equations (1).

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6. The EulerLagrange equations are equivalent, modulo gauge transformations, to the combined evolution equations (2) and constraint equations (1).

The constraints are preserved by the evolution equations

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7. The space of solutions of the eld equations is not necessarily smooth. It may have quadratic singularities occurring at symmetric solutions.

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7. The space of solutions of the eld equations is not necessarily smooth. It may have quadratic singularities occurring at symmetric solutions.

symplectic reduction

linearization stability

quantization


Philosophy
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Philosophy

Noethers theorem and the Dirac analysis of constraints do much topredict and explain features 16.


Philosophy
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Philosophy

Noethers theorem and the Dirac analysis of constraints do much topredict and explain features 16.

I wish to go further and provide (realistic) sufcient conditionswhich guarantee that they must occur in a CFT.

I provide such criteria for 16 and lay the groundwork for 7.

A key objective is thus to derive the adjoint formalism for CFTs.


Example: (Abelian) ChernSimons Theory
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p ( ) y

A topological eld theory on 3-dimensional spacetime X = R .


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p ( ) y


Fields are 1-forms A = ( A0 , A ) on X .




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p ( ) y



Lagrangian density is L = dA A.


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p ( ) y




Gauge group is Diff (X ): A = A


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Gauge group is Diff (X ): A = A

EL equations: dA = 0.


Ki i ld A
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Kinematic elds: A0


Ki i ld A
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Kinematic elds: A0

Dynamic elds: A


Ki ti ld A
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Kinematic elds: A0

Dynamic elds: A

IV constraint: (A , ) = ( dA)12 = 0


Kinematic elds: A
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Kinematic elds: A0

Dynamic elds: A

IV constraint: (A

, ) = ( dA)12 = 0 Gauge generators: (D i a )

Ai


Kinematic elds: A
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Kinematic elds: A0

Dynamic elds: A

IV constraint: (A


Ai

Slicing of 1(X ) X generated by:

d d

= x

A ,

A


Kinematic elds: A0
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Kinematic elds: A0

Dynamic elds: A

IV constraint: (A


Ai

Slicing of 1(X ) X generated by:

d d

= x

A ,

A

Hamiltonian: H = 2

(dA)12 ( A ) d

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Atlas eld: A

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Atlas eld: A

Evolution equations in adjoint form reduce to:

d d

Ai

i =

D i ( A )0ij D j ( A )

This is equivalent to (dA)0i = 0.

N.B. We also have primary constraints 0 = 0 and i = 0ij A j .


Traditional Approaches to CFT
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Group-theoretical




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Group-theoretical

concerned with the gauge covariance of a CFT

Lagrangian-oriented

covariant

based on Noethers theorem

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Canonical

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Canonical

initial value analysis

Hamiltonian-oriented

not covariantbased on space + time decomposition

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Connections:

These two aspects of a mechanical system are linked by themomentum map .

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Connections:

These two aspects of a mechanical system are linked by themomentum map .

One would like to have an analogous connection in CFT relatinggauge symmetries to initial value constraints.

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Caveat!

The standard notion of a momentum map associated to a symplecticgroup action usually cannot be carried over to spacetime covarianteld theory, because:

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Caveat!


spacetime diffeomorphisms move Cauchy surfaces,

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Caveat!


spacetime diffeomorphisms move Cauchy surfaces, and the Hamiltonian formalism is only dened relative to a xed

Cauchy surface


Prime Example : Einsteins theory of vacuum gravity
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the gauge group is the spacetime diffeomorphism group


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the only remnants of this group on the instantaneous (i.e., space +time split) level are the superhamiltonian H and supermomenta J


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the only remnants of this group on the instantaneous (i.e., space +time split) level are the superhamiltonian H and supermomenta J

interpreted as the generators of temporal and spatial deformationsof a Cauchy surface

these deformations do not form a group

nor are H and J components of a momentum map.

This circumstance forces us to work on the covariant level.


The Way Out: Multisymplectic Field Theory
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We must construct a covariant counterpart to the instantaneousHamiltonian formalism.

In the spacetime covariant (or multisymplectic ) framework we developherean extension and renement of the formalism of Kijowski andSzczyrba the gauge group does act.

So we can dene a covariant (or multi-) momentum map on thecorresponding covariant (or multi-) phase space .


The Energy-Momentum Map
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Key fact : The covariant momentum map induces an energy-

momentum map on the instantaneous phase space.

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Bridges the covariant & instantaneous formalisms

is the crucial object reecting the gauge transformation

covariance of a CFT in the instantaneous picture.


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Bridges the covariant & instantaneous formalisms

is the crucial object reecting the gauge transformation

covariance of a CFT in the instantaneous picture.

In ADM gravity, = (H , J ), so that the superhamiltonian andsupermomenta are the components of the energy-momentummap.


A recurrent theme is that the energy momentum map encodes
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A recurrent theme is that the energy-momentum map encodesessentially all the dynamical information carried by a CFT : its

Hamiltonian (Item 4)


A recurrent theme is that the energy-momentum map encodes
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A recurrent theme is that the energy-momentum map encodesessentially all the dynamical information carried by a CFT : its


gauge freedom (Item 3)


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A recurrent theme is that the energy momentum map encodesessentially all the dynamical information carried by a CFT : its



initial value constraints (Item 2)


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A recurrent theme is that the energy momentum map encodesessentially all the dynamical information carried by a CFT : its



initial value constraints (Item 2)

stress-energy-momentum tensor

. . .

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Indeed:

Energy-Momentum TheoremThe constraints ( 1) are given by the vanishing of the energy-momentum map associated to the gauge group of the theory.

thus synthesizes the group-theoretical and canonical approaches toCFT.


The Plan

The theoretical underpinnings of the adjoint formalism consist of:
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I. a covariant analysis of CFTs;


The Plan

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I. a covariant analysis of CFTs;II. a space + time decomposition of the covariant formalism followed

by an initial value analysis;


The Plan

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III. a study of the relations between gauge symmetries and initialvalue constraints


The Plan

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III. a study of the relations between gauge symmetries and initialvalue constraints

IV. the derivation of the adjoint formalism.

We will focus on IIII.


Carry along an example:
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bosonic string

Mimics gravity to a signicant extenta rst class rst order theory

Exercise: Work through abelian ChernSimons!

Other sidelights might include:

parametrization theory ( la Kucha r)

covariantization theory ( la YangMills)

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