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MULTISYMPLECTIC GEOMETRY
Within the context of covariant canonical quantization Multisymplectic Geometry MG is a gener-alization of symplectic geometry for eld theory. It allows us to construct a general framework for the calculus of variations with several variables. Historically MG was developed in three distinct steps. Its origins are connected with the names of C. Caratheodory [36] (1929), T. De Donder [57, 56] (1935) on one hand and Weyl [236] (1935) on the other. We make this distinction since the motivations involved were di erent: Caratheodory and later Weyl, were involved with the general-ization of the Hamilton-Jacobi equation to several variables and the line of development stemming from their work is concerned with the solution of variational problems in the setting of the action functional. On the other hand, E. Cartan [39] recognized the crucial importance of developing an invariant language for di erential geometry not dependent on local coordinates. De Donder carried this development further. The two approaches merged in the so-called De Donder-Weyl theory based on the multisymplectic manifold MDW. The second step arose with the work of T. Lepage and P. Dedecker. As was rst noticed by Lepage [159, 160, 161], the De Donder-Weyl setting is a special case of the more general multisymplectic theory. The geometrical tools permitting a fully general treatment were provided by Dedecker [52, 53, 54, 55]. Indeed, for eld theory, we are led to think of variational problems as n-dimensional submanifolds embedded in a (n + k)-dimensional manifold Z0. One observes the key role of the Grassmannian bundle as the analogue of the tangent bundle for variational problems for eld theory.
The nal step was taken by the Polish school in the seventies which further developed the geometric setting. W. Tulczyjew [224, 225], J. Kijowski [141, 142], K. Gawedski [93] and W. Szczyrba [144, 145] all formulated important steps. We nd already in their work the notion of algebraic form, and in the work of Kijowski [141] a corresponding formulation of the notion of dynamical observable emerges. We emphasize, for the full geometrical multisymplectic approach, two fundamental points that will be treated later: the generalized Legendre correspondence – introduced by Lepage and Dedecker – and the issue of observable and Poisson bracket, two cornerstones within the universal Hamiltonian formalism developed by Helein, [111, 113, 114] and Helein and Kouneiher [115, 116, 117, 118]. However, in order to understand the di cult issues surrounding the choice of a good Poisson bracket structure for eld theory in the multisymplectic setting, we will also describe some basic ideas about traditional multisymplectic theory in term of jet bundles and contact structures. The rst part of the Thesis is organized as follow. First we recall some basics of symplectic geometry in the Hamiltonian setting for classical mechanics and the transition to the relativistic case by means of the so-called Hamiltonian constraint. Later we give a basic exposition of MG for the calculus of variations with several variables, the tool for eld theory. Finally we concentrate on the main ideas and motives concerning the treatment of observables found in [111, 113, 114, 115, 116, 117, 118]. Also we emphasize some points related to n-phase space theory.
Symplectic geometry and Hamiltonian dynamics
We refer to the classical textbooks of R. Abraham and J.E. Marsden [1], J.E. Marsden and T. Ratiu [169] for complete geometrical introductions to classical mechanics and symmetry. A great part of our present introduction is inspired by various works and papers of Helein [113] – published and unpublished notes, in particular in Sections (3.1) and (3.2). Here we give a short treatment of his work, and present a synthesis on the symplectic roots of Hamiltonian dynamics.
Lagrangian systems and variational principles
First we focus on the description of non-relativistic dynamical systems. In this context, time is the evolution parameter, hence thought of as an external parametrization. The con guration space is denoted Z – described as a k-dimensional manifold – and we describe the dynamical variables on Z with coordinates fyig1 i k. The evolution of such a classical system is described by a curve : I ! Z : t 7! (t). The tangent bundle T Z is the velocity con guration space described by local coordinates f(yi; vi)g1 i k. A path is lifted to the tangent bundle T Z in the following way: ( ; ) : I ! T Z : t 7!( (t); (t)) and projects down to the con guration space for the same motion. In the case where L = L(y; v) does not explicitly depend on time, we speak of an autonomous dynamical system. In the present development, we are also interested in the so-called non-autonomous case where the setting exhibits a Lagrangian de ned on I T Z. In this case the Lagrangian depends explicitly on time so that L = L(t; y; v). This consideration leads a de nition of the evolution space as I T Z which is the data (t; yi; vi). In such a context, the Lagrangian is de ned on the evolution space L : I T Z ! R associates L(t; y; v) to any (t; y; v) 2 I T Z. The Lagrangian is de ned as the functional: Z t2 L[ ] = L(t; (t); (t))dt = L(t; (t); (t))dt: I t1.
Autonomous Hamiltonian setting and geometrization
In this section, we discuss the Legendre transform for autonomous Hamiltonian case in more detail. A similar construction is possible for the non-autonomous case. However, we will return to this point later. The Hamiltonian picture deals with structure on the cotangent bundle T ?Z by means of a non-degenerate Legendre transform: the map described by (7)(i) is a di eomorphism. Then we are able to de ne its inverse J 1 – (7)(ii). Notice that in this section, and in the following ones, we often abuse notation when specifying coordinates (qi; pi) = ( i(t); i(t)). (i) J : T Z ! T?Z@L (x; v) (ii) J 1 : T ?Z ! T Z (x; v) 7! x; (q; p) 7! (x; v) = (q; V(q; p)) @v.
The Legendre transform hypothesis gives us a a characterization of V(q; p) pi(t) = @L (q; V(q; p)) and Vi(q; @L (q; v)) = vi @vi @v (7) (8).
Hamiltonian dynamics gives the time evolution of coordinates (q; p) on T ?Z, when (q; v) satisfy the Euler-Lagrange equations. From the path perspective, : R ! Z is a solution of the Euler-Lagrange equations if and only if the map z : I ! T ?Z=t 7!z (t) = ( (t); (t)) is a solution of the Hamilton equations. This is possible thanks to the Hamiltonian function, de ned in the autonomous case: : T ?Z ! R. We observe the following de nition: De nition 3.3.1. The Hamiltonian function is de ned H : T ?Z ! R such that10 8(q; p) 2 T ?Z, we have H(q; p) = piVi(q; p) L J 1(q; p).
The Hamiltonian constraint and the presymplectic structure
In the previous treatment of Lagrangian and Hamiltonian classical mechanics, note that whatever the time dependence of the Lagrangian (autonomous or non-autonomous), we are faced with the notion of time as an exterior parameter which ows independently: it corresponds broadly to the Newtonian absolute time as an evolution parameter of the system, much as time does in QM. This is the non-relativistic case. This picture is altered when one treats time as a canonical variable. The key point is that in the previous development, time plays a twofold role. First as a formal integration variable, and secondly as the external evolution parameter in the Hamiltonian H = H(t; q(t); p(t)). However, with the advent of relativistic dynamics comes an understanding of space-time itself as the fundamental dynamical entity of the theory. There is no a priori splitting of space-time into space and time. The fundamental covariance of the theory is related to the lack of such splitting. Then we need the notion of extended phase space, usually introduced in the context of classical mechanics. The subsequent development is twofold. On the one hand, we work on the proper covariant con guration space Zcov with local coordinates fq g0 k. On the other, we choose a speci c time foliation and work with Zext, the extended con guration space equivalently denoted Z = R Z. In this case, coordinates are denoted fq g0 k = f(q ; qi)g1 i k. Note that if we perform such a preferred time foliation we depart from the spirit of a purely covariant theory. Our idea is that in the search for a possible QG it is a fundamental necessity to express the eld equations in a fully covariant way. This is the reason why we should distinguish between the relativistic case where we work on Z = R Z – and the covariant case – where we work with Zcov, without a preferred time direction.
Extended phase space. | The extended phase space – equivalently called extended phase space with a preferred topology – is built on the extended con guration space: the set of points (q ( ); qi( )) 2 Z R = Z . The notation is chosen to emphasize that we add one dimension to the classical con guration space. Hence, the con guration space is no longer spanned by k variables – the position-coordinates qi – but we consider instead the trivial extended con guration space as the data of (k +1) variables treated on an equal footing. It appears that the role of the variable is that of a parametrization variable, whereas the time variable12 t( ) = q ( ) is a variable parametrized 12as well as any coordinates on the extended phase space T ?Z = T ?(R Z) denoted q ( ); qi( ); p ( ); pi( ) by . In this connection we observe that q ( ) = t( ) plays the same role as the other variables. It is no longer viewed in the ambiguous role of integration variable vs time variable. Before further considering this dual aspect of time as parameter vs dynamical variable, we summarize the previous three cases in the following table: Con guration Spaces Phase Spaces R) = T?Z Extended con guration space Z (q ; qi) 1 i k Ext. PS T ?(Z (q ; qi; p ; pi) 1 i k Con guration space Z qi cov Phase space (PS) T ?Z (qi ; pi)
Table of contents :
INTRODUCTION
1 Symmetry, Invariance and Observables
1.1 Symmetry and Invariance
1.2 Dirac constraints and Dirac observables
2 The road ahead
2.1 Symbolic vision and diagram
2.2 Ontologic vs Dynamical
2.3 A glimpse of the journey
MULTISYMPLECTIC GEOMETRY
3 Symplectic geometry and Hamiltonian dynamics
3.1 Lagrangian systems and variational principles
3.2 The Hamiltonian setting and variational principles
3.3 Autonomous Hamiltonian setting and geometrization
3.4 The Hamiltonian constraint and the presymplectic structure
3.5 Geometrical construction for relativistic dynamics
3.6 Covariant Hamiltonian dynamics, preferred topology
3.7 Symplectomorphisms and innitesimal symplectomorphisms
3.8 Locally and Globally Hamiltonian vector elds
3.9 Hamiltonian systems and Hamiltonian symmetry
3.10 Observables, dynamics and Poisson structure
4 Multisymplectic geometry
4.1 Covariant Finite Hamiltonian Field Theories
4.2 Multisymplectic zoology
4.3 Generalized Hamilton equations
4.4 Geometrization and Variational Principles
4.5 De Donder-Weyl multisymplectic theory
4.6 Multisymplectic manifolds: the geometrical setting
4.7 Geometrical construction and Legendre lifts
4.8 Innitesimal symplectomorphism and pseudober
5 Traditional Multisymplectic setting: graded standpoint
5.1 Geometrical spaces for rst order jet De Donder-Weyl theory
5.2 Canonical Forms and contact structure
5.3 Hamiltonian (n 1)-forms and Hamiltonian vector elds
5.4 Hamiltonian (p 1)-forms and their related Hamiltonian vector elds
5.5 A glimpse of Poisson structure in the graded standpoint
6 Observables in multisymplectic geometry
6.1 Ontologic vs Dynamical
6.2 Algebraic observable (n 1)-forms (AOF)
6.3 Observable forms (OF)
6.4 Dynamical vs Ontologic: synthesis on (n 1)-forms
Multisymplectic Geometry and Classical Field Theory 3
6.5 Algebraic copolar and copolar (n 1)-forms
6.6 Frozen or kinematical observable functionals
6.7 Dynamical observable functionals
6.8 Copolarization and observables (p 1)-forms
7 Pre-multisymplectic and covariant phase space
7.1 n-phase space: generalized Hamilton equations
7.2 Observable and pre-multisymplectic case
7.3 Covariant phase space and observable functionals
MULTISYMPLECTIC MAXWELL THEORY
8 Multisymplectic De Donder-Weyl-Maxwell theory
8.1 Multisymplectic De Donder-Weyl-Maxwell theory
8.2 Hamilton-Maxwell equations in the De Donder-Weyl framework
8.3 Maxwell theory as an n-phase space
9 Observables for Maxwell Theory
9.1 Some algebraic observable (n 1)-forms
9.2 Poisson Bracket for algebraic (n 1)-forms
9.3 All algebraic (n 1)-forms
9.4 Dynamical equations
9.5 Algebraic (n 1)-forms for pre-multisymplectic case
9.6 Grassman variables vs copolarization
9.7 Copolarization and canonical variables
10 Lepage-Dedecker for two dimensional Maxwell theory
10.1 Lepage-Dedecker correspondence
10.2 Calculation of the Hamiltonian
10.3 Equations of movement
TOWARDS MULTISYMPLECTIC GRAVITY
11 General Relativity
11.1 Hierarchy of structures beyond mathematical model of space-time
11.2 Ontological being of space-time
11.3 Principles of General Relativity
11.4 Canonical framework and topology
11.5 Basic geometrical features
11.6 Moving frame, holonomic frame
11.7 Covariant derivatives and connections
11.8 Torsion and Curvature Operators
11.9 Levi Civita connection
12 General Relativity vs Gauge theory
12.1 Prolegomena
12.2 Tetrad eld
12.3 Frame bundle standpoint and G-structure
12.4 Coframe eld as a bundle isomorphism
12.5 Palatini formulation of gravity
12.6 Topological terms in gauge gravity
12.7 Cartan geometry as ground area for Gravity
13 Actions for Palatini Gravity and equations of movement
13.1 Derivation of Einstein equation from Palatini gravity
13.2 Equivalence of the Einstein-Hilbert action in the language of forms
13.3 Hodge duality on a vector space
13.4 Einstein-Cartan eld equations, with dierential forms
13.5 Einstein-Cartan and the spinorial area
14 Canonical variable and phase space for canonical gravity
14.1 Phase space and canonical variables
14.2 Connection representation: canonical variables (AI ;E I )
14.3 Holonomyux variables
15 Chern-Simon Gravity
15.1 Chern-Simon Lagrangian
15.2 Chern-Simon multisymplectic manifold
15.3 De Donder-Weyl multisymplectic equations
16 Multisymplectic Palatini-Hamilton equations
16.1 Geometrical setting and notations
16.2 Legendre correspondence
16.3 Palatini multisymplectic manifold
16.4 Multisymplectic Hamilton equations on (Mdeg;!deg)
17 Palatini Gravity as an n-phase space
17.1 Pre-Multisymplectic treatement of 3D-Palatini Gravity
17.2 Pre-multisymplectic treatment of Palatini 4D Gravity
18 Algebraic observable (n 1)-forms and canonical forms
18.1 Prolegomena for the ontologic standpoint
18.2 Innitesimal symplectomorphism sp0MPalatini of (MPalatini;!DW)
18.3 Canonical forms for Gravity
19 Topological hypothesis
19.1 Holst and Nieh-Yan terms
19.2 Lepage-Dedecker transform for rst order gravity
SPACE{TIME{MATTER
20 The Double Duality
20.1 Eye-mirror monad
20.2 Dual Nature of the multisymplectic form
20.3 Ontologic groundstate and background independence
20.4 Dual nature of the Hamiltonian function
PERSPECTIVES
Multisymplectic Geometry and Classical Field Theory 5
NOTATIONS
APPENDIX
A Dierential forms and exterior dierential calculus
B Gauge theory, connections, derivatives and all that
B.1 Gauge elds: Fiber bundle framework
B.2 Vector-valued dierential forms
B.3 Curvature of principal connection
B.4 Connection, exterior derivative and curvature on a vector bundle.
B.5 Horizontal, invariant and Basic forms
B.6 Connection and Curvature on associated bundle
B.7 Group of automorphisms and gauge picture
B.8 Yang-Mills theory
B.9 Maurer-Cartan form
C Jet manifold and contact structure
C.1 Jet manifold
C.2 Contact structure
D Algebraic identities for Palatini framework
E Algebraic computation for innitesimal symplectomorphisms
