# Multisymplectic Constraint Analysis of Scalar Field Theories, Chern-Simons
Gravity, and Bosonic String Theory

a Joaquim Gomis , b Arnoldo Guerra IV , c Narciso Román-Roy .
aDepartament de Física Quàntica i Astrofísica and Institut de Ciències del
Cosmos (ICCUB),
Universitat de Barcelona, Barcelona, Spain.
bDepartament de Física Quàntica i Astrofísica, Universitat de Barcelona,
Barcelona, Spain.
dDepartment of Mathematics, Universitat Politècnica de Catalunya, Barcelona,
Spain<EMAIL_ADDRESS>(ORCID:
<EMAIL_ADDRESS>(ORCID:
<EMAIL_ADDRESS>(ORCID: 0000-0003-3663-9861).

###### Abstract

The (pre)multisymplectic geometry of the De Donder–Weyl formalism for field
theories is further developed for a variety of field theories including a
scalar field theory from the canonical Klein-Gordon action, the electric and
magnetic Carrollian scalar field theories, bosonic string theory from the
Nambu-Goto action, and $2+1$ gravity as a Chern-Simons theory. The Lagrangians
for the scalar field theories and for $2+1$ Chern-Simons gravity are found to
be singular in the De Donder–Weyl sense while the Nambu-Goto Lagrangian is
found to be regular. Furthermore, the constraint structure of the
premultisymplectic phase spaces of singular field theories is explained and
applied to these theories. Finally, it is studied how symmetries are developed
on the (pre)multisymplectic phase spaces in the presence of constraints.

Key words: Classical field theories, multisymplectic formulation, Lagrangian
and Hamiltonian formalisms, Symmetries, Scalar field theory, Carrollian
theories, Bosonic String theory, $p$-branes, Chern–Simons gravity.

AMS (2020): Primary: 53D42, 70S05, 70S10. Secondary: 35Q75, 53C15, 53Z05,
81T30, 83D05. PACS (2022): 03.50.-z, 11.10.Ef, 02.40.-k, 04.50.Kd, 04.20.Fy,
11.25.-w, 11.30.-j, 02.30.Jr.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary Differential Geometry
    1. 2.1 Multivector Fields
    2. 2.2 Multisymplectic Lagrangian Field Theory
    3. 2.3 Multisymplectic Hamiltonian Field Theory
    4. 2.4 Geometric Constraint Algorithm
    5. 2.5 Multisymplectic Symmetries and Noether Currents
  3. 3 Scalar Field Theories
    1. 3.1 Klein–Gordon From Geometric Constraints
    2. 3.2 Carrollian Electric Scalar Field Theory
    3. 3.3 Carrollian Magnetic Scalar Field Theory
  4. 4 Bosonic Strings and p-Branes
    1. 4.1 The String Lagrangian Formulation
    2. 4.2 The String Hamiltonian Formulation
    3. 4.3 Symmetries
    4. 4.4 Generalization to p-Branes (p >1)
  5. 5 (2+1)-dimensional Gravity
    1. 5.1 Lagrangian Formulation
    2. 5.2 Hamiltonian Formulation
    3. 5.3 Symmetries
  6. 6 Conclusions and Outlook

##  1 Introduction

Covariant phase space formalisms for field theories have been relevant in
theoretical physics since the development of the variational calculus for
classical field theories. When a field theory under investigation is invariant
under Lorentz transformations, it is possible to develop a manifestly Lorentz
covariant construction of the variational principle and the symmetries of the
Lagrangian. This is evident in the Lagrangian formulation of field theories in
which the fields are, in general, tensors on spacetime and the spacetime
coordinates are treated as indistinguishable independent variables. In the
case of string theory, the fields are scalar fields on the string worldsheet
and covariance is similarly maintained. One appropriate geometrical setting
for constructing covariant phase spaces for field theories in the Lagrangian
formalism is given by jet bundles which are fiber bundle over spacetime whose
fiber coordinates consist of the fields and the spacetime derivatives of the
fields. The covariant phase space manifold in the Lagrangian setting for
first-order field theories (“first-order” in the sense that only first
derivatives appear in the Lagrangian) is called the first-order jet bundle
which will be referred to colloquially here as the multivelocity phase space,
in analogy with the velocity phase space where the Lagrangian formulation of
mechanical systems takes place. Thus, the coordinates on the first-order jet
bundle, whose fibers over spacetime are the spacetime first derivatives of the
fields, are called the multivelocities. In this paper only first-order field
theories will be investigated.

Lorentz covariance in Hamiltonian formulations of Lorentz invariant field
theories is much more subtle. The first attempt to develop a covariant
Hamiltonian formalism was made in the 1930s by Théophile De Donder and Hermann
Weyl [2, 3] who generalized the variational calculus for the Hamiltonian
formalism in mechanics to the multivariable context in field theory, giving
rise to what is today referred to as the De Donder–Weyl approach in which the
variational principle leads to the so-called (Hamilton)–De Donder–Weyl
equations. The phase space manifold on which the De Donder–Weyl formalism for
first-order field theories is constructed is called the dual first-order jet
bundle and is referred to as the multimomentum phase space. The coordinates on
this phase space are called multimomenta and are obtained from the Legendre
map given as the partial derivatives of the Lagrangian with respect to all
multivelocities. The De Donder–Weyl formulation of field theories has become
better understood by learning about the underlying (pre)multisymplectic
geometry [4, 5] of the multivelocity and multimomentum phase spaces.
(Pre)multisymplectic geometry in classical field theory is the geometric
multivariable extension of (pre)symplectic geometry in classical mechanics
where the physical observables of interest (position, velocity, momentum)
depend on a single variable (time). Similarly to how to the simple (single-
variable) variational calculus can be performed on (pre)symplectic phase
spaces as done in classical mechanics, the multiple variational calculus can
be posed geometrically on the (pre)multisymplectic phase spaces.

The De Donder–Weyl formalism, however, was hindered at the time of its
development due to a lack of an understanding of constraints. A constraint
analysis is needed to obtain the full dynamics of a field theory with a
Lagrangian whose Legendre map is singular. In the De Donder–Weyl approach, the
Legendre map is singular when the multi-Hessian constructed from the
multivelocities is singular. The appropriate constraint analysis that is
needed for such singular field theories was not understood at the time when
the De Donder–Weyl approach was first developed. Instead, field theories were
studied using the canonical Hamiltonian formalism which arises from performing
a space+time splitting where, given an initial value Cauchy data surface, one
solves a set of partial differential equations for dynamical functions which
describe the time evolution of the classical fields of interest. This approach
sacrifices the manifest Lorentz covariance exhibited by Lorentz invariant
field theories in the Lagrangian formalism. However, the advantage at the time
was that the (non-covariant) Dirac–Bergmann constraint analysis [6, 7] gave
the full dynamics of singular field theories (“singular” with respect to the
Legendre transform that gives the canonical momenta rather than the
multimomenta used in the De Donder–Weyl approach). Furthermore, the canonical
formalism gave a straightforward method for performing quantization, making
the canonical formalism for field theories exceptionally useful.

The success of the canonical formalism in quantum field theory made it the
standard Hamiltonian field theory formalism in theoretical physics. However,
understanding how to develop the canonical formulation of field theories in a
manifestly Lorentz covariant manner remained a mystery for many years until
Crnković and Witten discovered how to construct a classical covariant phase
space in which the canonical formulation (on-shell) can be carried out without
having to perform a space + time splitting [8]. Soon after, Lee and Wald
generalized this new covariant phase space formalism to also work off-shell in
[9]. The Dirac–Bergmann constraint analysis makes use of Poisson brackets
which are defined on symplectic phase space manifolds; the covariant phase
space manifolds presented by Crnković, Witten, Lee, and Wald are infinite-
dimensional (pre)symplectic manifolds it is therefore possible to carry out
the geometric Gotay–Nester–Hinds constraint algorithm [10] which is equivalent
to the Dirac–Bergmann constraint analysis. In fact, Lee and Wald specify in
[9] how the covariant Poisson brackets of constraints corresponding to a local
symmetry are related to the Lie algebra of the local symmetry under
inspection. For a treatment of boundary terms in the covariant phase space
formalism presented by Lee and Wald, see [11, 12].

The difficulty with performing an analogous constraint analysis a la Dirac on
the multivelocity and multimomentum phase spaces on which the De Donder–Weyl
formalism takes place arises fundamentally from the fact that Poisson brackets
on jet and dual jet bundles are not yet fully understood. A geometric
algorithm for performing the desired covariant constraint analysis in the De
Donder–Weyl formalism was developed in [13] where the premultisymplectic
geometry of the multivelocity and multimomentum phase spaces is used to
describe the constraint submanifolds on which field theories are physically
relevant. This analysis is geometrically analogous to how the presymplectic
geometry of the phase spaces in classical mechanics is used to perform the
geometric constraint analysis of mechanical systems equivalent to the
Dirac–Bergmann constraint procedure for mechanics developed in [10, 14, 15,
16, 17, 18]. For classical field theories constructed on $m$-dimensional
spacetimes, the jet and dual jet bundles, i.e. the multivelocity and
multimomentum phase spaces, are (pre)multisymplectic manifolds equipped with
(pre)multisymplectic $(m+1)$-forms. These differential forms multisymplectic
for regular field theories and are premultisymplectic for singular field
theories; the precise mathematical definitions of these forms will be given
later. Furthermore, the symmetries of a field theory are Lie groups which act
on the phase spaces in such a way that their (pre)multisymplectic forms are
preserved and Noether’s theorem yields conserved currents called covariant
momentum maps which are $(m-1)$-forms on the (pre)multisymplectic phase
spaces. A general overview of (pre)multisymplectic geometry, including new
geometry developed in this work, is provided in Section 2 of this paper. For
reviews of the previously known aspects of the multisymplectic formulation of
field theories see also, for example, [19, 20, 21]. The symplectic form on the
covariant phase space introduced by Crnković and Witten is derived from the
multisymplectic form on the multimomentum phase space in [22]. Furthermore, in
[23, 24] it is shown how performing a space + time splitting of the
multimomentum phase space gives the standard canonical formalism for field
theories along with a canonical constraint analysis (equivalent to the
constraint analysis a la Dirac) in which covariant momentum maps play a
crucial role.

In this paper, the (pre)multisymplectic geometry of the multivelocity and
multimomentum phase spaces is used in a variety of field theories to develop:
the field equations both in the Lagrangian and in the De Donder–Weyl
Hamiltonian formulations, the geometric constraint analysis [13], and the
covariant momentum maps [19] which give the conserved currents associated with
Noether symmetries. The premultisymplectic constraint algorithm originally
formalised in [13] is performed here using a local-coordinate approach which
significantly simplifies the entire constraint algorithm; this exposition of
the constraint algorithm along with some new properties of the constraints are
presented in Section 2.4.

The specific field theories developed in this paper were chosen to point out
certain subtle features of the geometric constraint analysis and the
investigation of symmetries. This paper presents the first premultisymplectic
treatment of Carrollian field theories [25, 26, 27, 28]. It should be noted
that the De Donder–Weyl formalism and its (pre)multisymplectic interpretation
are not restricted to work only for Lorentz invariant theories; in fact, the
Carrollian scalar field theories investigated in the premultisymplectic
formalism in this paper are not Lorentz invariant. Carrollian field theories
arise from taking the limit $c\rightarrow 0$ for the speed of light as first
introduced in [29, 30]. Furthermore, the multisymplectic formulation of string
theory and $p$-branes on the multimomentum phase space found in [31] is
extended in this work by additionally providing the multisymplectic
formulation on the multivelocity phase space and discussing the relevant
symmetries on the multisymplectic phase spaces. The De Donder–Weyl formulation
of string theory (and $p$-branes in general) can be found in [32] without
mention of the underlying multisymplectic structure. Finally, this work
reproduces the results found in the premultisymplectic treatment of
Chern–Simons gravity done in [33] where the primary constraint submanifold of
the multimomentum phase space arising from the De Donder–Weyl Legendre
transform is presented. This work additionally provides the full
premultisymplectic constraint analysis of Chern–Simons gravity, on both the
multimomentum and multivelocity phase spaces, where it is found that the field
equations define the constraint submanifolds of the so-called Second Order
Partial Differential Equation (sopde) type; these constraint submanifolds
arise from imposing the sopde condition which guarantees that the final field
equations are second order partial differential equations. The treatment of
the symmetries of Chern–Simons gravity in the premultisymplectic context is
also given. For additional references on the multisymplectic treatment of
General Relativity, see [34, 35, 36]. The existence of boundaries on the
multisymplectic phase spaces will not be considered in this work. For a
detailed treatment of boundaries in the multisymplectic and other formalisms
of field theories see, for instance [37, 38, 39].

The organization of the paper is as follows: Section 2 provides a general
overview of the (pre)multisymplectic setting used to formulate first-order
classical field theories, including the local description of the constraint
algorithm. Additionally, symmetries and conservation laws in the
(pre)multisymplectic framework are reviewed in this section. Section 3
contains the premultisymplectic analysis of the canonical Klein Gordon
Lagrangian and of the electric and magnetic Carrollian scalar field theories.
All three scalar field theories exhibit different constraint and rigid
symmetry structures. In section 4, the Nambu–Goto Lagrangian for bosonic
strings and $p$-branes is found to be regular in the De Donder–Weyl sense and
the symmetries of the Nambu–Goto action are presented in multisymplectic
framework. Finally, the premultisymplectic treatment of gravity in $2+1$
dimensions as a Chern–Simons theory [40] is presented in Section 5.

All the manifolds are real, second countable and of class ${\rm C}^{\infty}$.
Manifolds and mappings are assumed to be smooth. Sum over crossed repeated
indices is understood.

##  2 Preliminary Differential Geometry

This section reviews the foundations of the multisymplectic formulation of
first-order field theories and basic concepts of symmetries and conservation
laws; the constraint algorithm is also described in the premultisymplectic
context.

###  2.1 Multivector Fields

Let $E$ be an $n$-dimensional manifold and $m<n$; sections of
$\Lambda^{m}({\rm T}E):=\overbrace{{\rm T}E\wedge\ldots\wedge{\rm T}E}^{(m)}$
(the $m$th exterior product of ${\rm T}E$) are called $m$-multivector fields
on $E$ and they are the contravariant skew-symmetric tensor fields of order
$m$ on $E$. (For more details on multivector fields and all the related topics
presented here see, for instance, [41, 42, 43, 44]). The set of
$m$-multivector fields on $E$ is denoted $\mathfrak{X}^{m}(E)$.

For every $m$-multivector field $\textbf{X}\in\mathfrak{X}^{m}(E)$ and $p\in
E$, there exists an open neighborhood $U_{p}\subset Y$ and
$X_{1},...,X_{r}\in\mathfrak{X}(E)$ such that

$\textbf{X}|_{U_{p}}=\sum_{1\leq i_{1}\ldots i_{m}\leq
r}f^{i_{1}...i_{m}}X_{i_{1}}\wedge\ldots\wedge X_{i_{m}},$

where $f^{i_{1}...i_{m}}\in C^{\infty}(U_{p})$ and $m\leq r\leq\text{dim}E$.
Then, $\textbf{X}\in\mathfrak{X}^{m}(E)$ is locally decomposable if, for every
$p\in E$, there is an open neighborhood $U_{p}\subset Y$ and
$X_{1},...X_{m}\in\mathfrak{X}(U_{p})$ such that
$\left.\textbf{E}\right|_{U_{p}}=X_{1}\wedge...\wedge X_{m}$. All multivector
fields used in this paper are assumed to be locally decomposable. The
contraction between multivector fields $\textbf{X}\in\mathfrak{X}^{m}(E)$ and
differentiable $k$-forms $\xi\in{\mit\Omega}^{k}(E)$ is

$i(\textbf{X})\xi|_{U_{p}}=\sum_{1\leq i_{1}\ldots i_{m}\leq
r}f^{i_{1}...i_{m}}i(X_{i_{1}}\wedge...\wedge X_{i_{m}})\xi=\sum_{1\leq
i_{1}\ldots i_{m}\leq r}f^{i_{1}...i_{m}}\,\xi(X_{i_{1}}),\ldots,X_{i_{m}}),$

if $k\geq m$, and $i(\textbf{X})\xi|_{U_{p}}=0$, if $k<m$. A distribution
$\mathpzc{D}$ of rank $k$ on $E$ is a subbundle of ${\rm T}E$ of rank $k$ (and
an $m$-dimensional distribution on $E$ is an $m$-dimensional subbundle of
${\rm T}E$). For every $p\in E$ there is a linear subspace
$\mathpzc{D}_{p}\subseteq{\rm T}_{p}Y$ such that $\mathpzc{D}=\bigcup_{p\in
Y}\mathpzc{D}_{p}$. Given a distribution $\mathpzc{D}\subseteq{\rm T}E$, a
nonempty immersed submanifold $W\subseteq E$ is a an integral manifold of
$\mathpzc{D}$ if ${\rm T}_{p}W=\mathpzc{D}_{p}$. The distribution
$\mathpzc{D}\subseteq{\rm T}E$ is an integrable distribution if at every $p\in
E$ there is an integral manifold of $\mathpzc{D}$ containing $p$. A
multivector field $\textbf{X}\in\mathfrak{X}^{m}(E)$ and an $m$-dimensional
distribution $\mathpzc{D}$ on $E$ are locally associated if there exists an
open set $U\subseteq E$ such that $\left.\textbf{X}\right|_{U}$ is a section
of $\left.\Lambda^{m}\mathpzc{D}\right|_{U}$.

Locally decomposable $m$-multivector fields are locally associated to
$m$-dimensional distributions. Multivector fields associated with the same
distribution form an equivalence class $\\{\textbf{X}\\}$ in
$\mathfrak{X}^{m}(E)$. If $\textbf{X}_{1},\
\textbf{X}_{2}\in\mathfrak{X}^{m}(E)$ are two nonvanishing multivector fields,
both locally associated with the same distribution on the open set $U\subseteq
E$, then there exists $f\in C^{\infty}(U)$ such that
$\textbf{X}_{1}=f\textbf{X}_{2}$ (on $U$), and this is precisely the
equivalence relation with equivalence class denoted $\\{\textbf{X}\\}_{U}$. A
multivector field is integrable if its locally associated distribution is
integrable. An $m$-dimensional submanifold $W\hookrightarrow E$ is an integral
manifold of $\textbf{Z}\in\mathfrak{X}^{m}(E)$ if, and only if, for every
point $p\in W$, the multivector $\textbf{X}_{p}$ spans $\Lambda^{m}{\rm
T}_{p}W$.

Now, let $\pi\colon E\rightarrow M$ be a fiber bundle with $\dim{M}=m$ and
$\dim{E}=m+n$. Vector fields on $E$ are called $\pi$-vertical vector fields
when they are tangent to the fibres of $\pi$. The set of such vector fields
will be denoted as $\mathfrak{X}^{{\rm V}(\pi)}(E)$. A multivector field
$\textbf{Z}\in\mathfrak{X}^{m}(E)$ is $\pi$-transverse if at every point $p\in
Y$, $\left.i(\textbf{X})(\pi^{*}\beta)\right|_{p}\neq 0$ for every
$\beta\in{\mit\Omega}^{m}(Y)$ with $\beta(\pi(p))\neq 0$. Consider a local
section of $\pi$, $\phi\colon U_{x}\subset M\rightarrow E:x\mapsto y=\phi(x)$
where $x\in M$ and $y\in E$; if $\phi(U_{x})$ is the integral manifold of the
multivector field $\textbf{X}\in\mathfrak{X}^{m}(E)$ at $y$, then $\phi$ is
called an integral section of X. If a multivector field
$\textbf{X}\in\mathfrak{X}^{m}(E)$ is integrable, then it is $\pi$-transverse
if, and only if, its integral manifolds are local sections of $\pi$; that is,
the local sections of $\pi$ are integral sections of X. If $M$ is an
orientable manifold, the condition that a multivector field
$\textbf{X}\in\mathfrak{X}^{m}(E)$ is $\pi-$transverse can be written as
$i(\textbf{X})\text{d}^{m}x\neq 0$, where ${\rm d}^{m}x\equiv{\rm
d}x^{1}\wedge\ldots\wedge{\rm d}x^{m}$ is the local coordinate expression
$(x^{\mu})$ on $M$ $(\mu=0,...,m-1)$ of the volume form. In particular, it is
possible to take a representative X in the class of $\pi$-transverse
multivector fields such that $f=1$ so that

$\mathop{i}\nolimits(\textbf{X})\text{d}^{m}x=1\ .$ (1)

If $x\in M$, two sections $\phi,\widetilde{\phi}$ of $\pi$ are _( $1st$-order)
equivalent_ at $x$ if $\phi(x)=\widetilde{\phi}(x)$ and
$\partial_{\mu}\phi|_{x}=\partial_{\mu}\widetilde{\phi}|_{x}$, where
$\displaystyle\partial_{\mu}\phi\equiv\displaystyle\frac{\partial{\phi}}{\partial{x^{\mu}}}$;
i.e. $T_{x}\phi=T_{x}\widetilde{\phi}$. The corresponding equivalence classes
are called the $1$-jets of $\phi$ at $x$, denoted $j^{1}_{x}\phi$. Then, the
first-order jet bundle $J^{1}E$ of $E$ is defined as
$J^{1}E=\\{j^{1}_{x}\phi:x\in M,\phi\in\Gamma(\pi)\\}$. It is a bundle over
$E$ and $M$ whose natural projections are denoted $\pi^{1}\colon
J^{1}E\rightarrow E$ and $\bar{\pi}^{1}\colon J^{1}E\rightarrow M$, and
$\text{dim}J^{1}E=m+n+mn$. The set of sections of $\pi$ and $\bar{\pi}^{1}$
are denoted by $\Gamma(\pi)$ and $\Gamma(\bar{\pi}^{1})$ respectively. A
section $\psi\in\Gamma(\bar{\pi}^{1})$ is said to be a holonomic section if
$\psi=j^{1}\phi$; that is, $\psi$ is is a holonomic section if it is the first
jet prolongation to $J^{1}E$ of a section $\phi\in\Gamma(\pi)$. A multivector
field $\textbf{X}\in\mathfrak{X}^{m}(J^{1}E)$ is said to be a holonomic
multivector field if it is integrable and its integral sections are holonomic
sections of $\bar{\pi}^{1}$. Observe that a multivector field
$\textbf{X}\in\mathfrak{X}^{m}(J^{1}E)$ is a holonomic multivector field if,
and only if, (i) X is integrable, (ii) X is $\bar{\pi}^{1}-$transverse, and
(iii) the integral sections of X are holonomic.

Given coordinates $(x^{\mu},y^{A})$ $(A=1,...,n)$ on $E$ adapted to the bundle
structure, the induced coordinates on $J^{1}E$ are
$(x^{\mu},y^{A},y_{\mu}^{A})$. Then holonomic sections are written as
$\psi=j^{1}\phi(x)=\left(x^{\mu}(x),y^{A}(x),\partial_{\mu}y^{A}(x)\right)$.
Furthermore, the local expression for $\textbf{X}\in\mathfrak{X}^{m}(J^{1}E)$
is

$\textbf{X}=\bigwedge^{m}_{\mu=1}\textbf{X}_{\mu}=\bigwedge^{m}_{\mu=1}f\left(\frac{\partial}{\partial
x^{\mu}}+D_{\mu}^{A}\frac{\partial}{\partial
y^{A}}+H_{\mu\nu}^{A}\frac{\partial}{\partial y_{\nu}^{A}}\right),\ \ \ \ f\in
C^{\infty}(J^{1}E)\ ,$

which defines equivalence classes of multivector fields on $J^{1}E$.

Integral sections of X satisfy
$\displaystyle\displaystyle\frac{\partial{y^{A}}}{\partial{x^{\mu}}}=D_{\mu}^{A}$,
$\displaystyle\displaystyle\frac{\partial{y_{\nu}^{A}}}{\partial{x^{\mu}}}=H^{A}_{\mu\nu}$.
Furthermore, if X is holonomic, then $D_{\mu}^{A}=y_{\mu}^{A}$ and hence
$\displaystyle\frac{\partial^{2}y^{A}}{\partial x^{\mu}\partial
x^{\nu}}=H^{A}_{\mu\nu}$. Holonomic multivector fields are also called Second
Order Partial Differential Equations (sopde).

###  2.2 Multisymplectic Lagrangian Field Theory

Multisymplectic geometry can be viewed as the field theoretic extension of the
symplectic geometrization of classical mechanics (for more details on the
multisymplectic Lagrangian formulation of first-order field theories see, for
instance, [19, 20, 21, 42, 45, 46, 47, 48]). The analysis begins by
considering an orientable $m$-dimensional pseudo-Riemannian manifold $M$. The
configuration manifold is taken to be a fiber bundle $E$ over $M$ with
$n$-dimensional fibers with (surjective) projection map $\pi:E\rightarrow
M:(x^{\mu},y^{A})\mapsto x^{\mu}$. The fields under consideration are denoted
$y^{A}(x)$ and are given by the local sections $\phi:M\rightarrow
E:x^{\mu}\mapsto(x^{\mu},y^{A}(x))$ of $\pi$. The multivelocity phase space on
which the Lagrangian formalism takes place is the first-order jet bundle
$J^{1}E$ of $\pi$ which has natural coordinates $(x^{\mu},y^{A},y_{\mu}^{A})$
and hence $\text{dim}J^{1}E=n+m+nm$; $J^{1}E$ is a fiber bundle over $E$ with
projection map $\pi^{1}:J^{1}E\rightarrow
E:(x^{\mu},y^{A},y_{\mu}^{A})\mapsto(x^{\mu},y^{A})$ and also a fiber bundle
over $M$ with projection $\bar{\pi}^{1}:J^{1}E\rightarrow
M:(x^{\mu},y^{A},y^{A}_{\mu})\mapsto x^{\mu}$.

Densities on $J^{1}E$ can be obtained by lifting the volume form
$\text{d}^{m}x$ on $M$ to $J^{1}E$. The Lagrangian density is then written as
$\textbf{L}=\mathscr{L}(x^{\mu},y^{A},y_{\mu}^{A})\text{d}^{m}x$ and
$\mathscr{L}(x^{\mu},y^{A},y_{\mu}^{A})\in C^{\infty}(J^{1}E)$ is referred to
as the Lagrangian function. The Lagrangian energy $E_{\mathscr{L}}\in
C^{\infty}(J^{1}E)$ is defined as

$E_{\mathscr{L}}\equiv\frac{\partial\mathscr{L}}{\partial
y^{A}_{\mu}}y^{A}_{\mu}-\mathscr{L}(x^{\mu},y^{A},y^{A}_{\mu})\ .$

The Lagrangian is said to be regular when the generalized Hessian matrix

$H_{AB}^{\mu\nu}=\frac{\partial^{2}\mathscr{L}}{\partial y^{A}_{\mu}\partial
y^{B}_{\nu}}$ (2)

is non-singular everywhere. Furthermore, the bundle $J^{1}E$ comes equipped
with an $m$-form called the $m$-Poincaré–Cartan form
$\Theta_{\mathscr{L}}\in{\mit\Omega}^{m}(J^{1}E)$, given by

$\Theta_{\mathscr{L}}=\frac{\partial\mathscr{L}}{\partial
y^{A}_{\mu}}\text{d}y^{A}\wedge\text{d}^{m-1}x_{\mu}-E_{\mathscr{L}}\text{d}^{m}x\
,$

and a closed (and hence locally exact) $(m+1)$-form
$\Omega_{\mathscr{L}}\in{\mit\Omega}^{m+1}(J^{1}E)$, which is called the
$(m+1)$-Poincaré–Cartan form, given by

$\Omega_{\mathscr{L}}=-\text{d}\Theta_{\mathscr{L}}=-\text{d}\left(\frac{\partial\mathscr{L}}{\partial
y^{A}_{\mu}}\right)\wedge\text{d}y^{A}\wedge\text{d}^{m-1}x_{\mu}+\text{d}E_{\mathscr{L}}\wedge\text{d}^{m}x\
,$

where $\text{d}^{m-1}x_{\mu}\equiv
i(\partial_{\mu})\text{d}^{m}x=\displaystyle\frac{1}{(m-1)!}\epsilon_{\mu\mu_{2}\dotsm\mu_{m}}\text{d}x^{\mu_{2}}\wedge\dotsc\wedge\text{d}x^{\mu_{n}}$,
$\text{d}^{m-2}x_{\mu\nu}\equiv
i(\partial_{\nu})i(\partial_{\mu})\text{d}^{m}x=\displaystyle\frac{1}{(m-2)!}\epsilon_{\mu\nu\mu_{3}\dotsm\mu_{m}}\text{d}x^{\mu_{3}}\wedge\dotsc\wedge\text{d}x^{\mu_{m}}$,
and so on.

The form $\Omega_{\mathscr{L}}$ is $j$-nondegenerate ($1\leq j\leq m)$ if, for
every $p\in J^{1}E$ and $\textbf{Y}\in\mathfrak{X}^{j}(J^{1}E)$, it follows
that $\left.i(\textbf{Y})\Omega_{\mathscr{L}}\right|_{p}=0\
\Longleftrightarrow\ \left.\textbf{Y}\right|_{p}=0$. When
$\Omega_{\mathscr{L}}$ is $1$-nondegenerate, it is referred to as being a
multisymplectic form. This occurs when
$\mathscr{L}(x^{\mu},y^{A},y_{\mu}^{A})$ is regular. Otherwise, when
$\mathscr{L}$ is singular, $\Omega_{\mathscr{L}}$ is $1$-degenerate and is
referred to as being a premultisymplectic form (although here it will also be
referred to as a degenerate multisymplectic form when this is the case).

The couple $(J^{1}E,\ \Omega_{\mathscr{L}})$ is called a Lagrangian system.
The field equations for this system are obtained from a variational action
principle posed on $J^{1}E$. The action, denoted

$\displaystyle
S\left[(j^{1}\phi)^{*}x^{\mu},(j^{1}\phi)^{*}y^{A},(j^{1}\phi)^{*}y^{A}_{\mu}\right]=\int_{\Sigma}(j^{1}\phi)^{*}\Theta_{\mathscr{L}}\
,$

is a functional on the set of sections $\Gamma(\pi)$ given by
$\displaystyle\Gamma(\Sigma,E)\rightarrow\mathds{R}:\phi\mapsto\int_{M}(j^{1}\phi)^{*}\Theta_{\mathscr{L}}$.
The variational problem consists of finding critical (stationary) sections of
$\Gamma(\pi)$ which satisfy

$\left.\frac{\text{d}}{\text{d}s}\right|_{s=0}\int_{\Sigma}(j^{1}\phi_{s})^{*}\Theta_{\mathscr{L}}=0\
,$

for all variations $\phi_{s}=\eta_{s}\circ\phi$ of $\phi$. Here $\eta_{s}$ is
the flow of a vertical vector field on $E$ compactly supported on $\Sigma$.
For more details see [42, 47, 48]. Critical sections which are solutions to
this variational problem can be characterized in the following equivalent
ways:

  1. 1.

$(j^{1}\phi)^{*}i(\mathpzc{X})\Omega_{\mathscr{L}}=0$, for every
$\mathpzc{X}\in\mathfrak{X}(J^{1}E)$.

  2. 2.

$j^{1}\phi$ is an integral section of a class of $\bar{\pi}^{1}$-transverse,
locally decomposable, holonomic, multivector fields
$\\{\textbf{X}_{\mathscr{L}}\\}\subset\mathfrak{X}^{m}(J^{1}E)$ which satisfy

$\mathop{i}\nolimits(\textbf{X}_{\mathscr{L}})\Omega_{\mathscr{L}}=0\quad,\quad\mbox{\rm
for every $\textbf{X}_{\mathscr{L}}\in\\{\textbf{X}_{\mathscr{L}}\\}$}\ .$ (3)

Furthermore, it is possible to choose a representative of the class
$\\{\textbf{X}_{\mathscr{L}}\\}$ which satisfies the normalized transverse
condition (1):

$\mathop{i}\nolimits(\textbf{X}_{\mathscr{L}}){\rm d}^{m}x=1\ .$ (4)

  3. 3.

Given a natural system of coordinates $(U;x^{\mu},y^{A},y_{\mu}^{A})$ on
$J^{1}E$, the first jet prolongations
$j^{1}\phi=\left(x^{\mu},y^{A}(x^{\nu}),\partial_{\mu}y^{A}(x^{\nu})\right)\in
U\subset J^{1}E$ satisfy the Euler–Lagrange equations:

$\frac{\partial\mathscr{L}}{\partial y^{A}}\circ
j^{1}\phi-\frac{\partial}{\partial
x^{\mu}}\left(\frac{\partial\mathscr{L}}{\partial y_{\mu}^{A}}\circ
j^{1}\phi\right)=0\ .$

When the Lagrangian is regular, the field equations are compatible (and have
solutions) on all of $J^{1}E$. In the singular case they are, in general, not
compatible on all of $J^{1}E$ and it is therefore necessary to implement a
constraint algorithm in order to find a submanifold of $J^{1}E$ on which the
field equations are compatible where consistent solutions exist. This
constraint algorithm is explained in detail in Section 2.4.

###  2.3 Multisymplectic Hamiltonian Field Theory

For the De Donder–Weyl Hamiltonian formulation of first-order field theories
(see, for instance, [19, 20, 43, 49, 50, 51, 52, 53] for details), let ${\cal
M}\pi\equiv\Lambda_{2}^{m}{\rm T}^{*}E$ be the bundle of $m$-forms on $E$
vanishing by the action of two $\pi$-vertical vector fields. The manifold
${\cal M}\pi$ is called the extended multimomentum bundle and has local
coordinates $(x^{\mu},\,y^{A},p_{A}^{\mu},p)$, hence $\text{dim}{\cal
M}\pi=m+n+mn+1$. Then, consider the quotient bundle $J^{1}E^{*}:={\cal
M}\pi/\Lambda^{m}_{1}({\rm T}^{*}{\rm E})$ (where $\Lambda_{1}^{m}({\rm
T}^{*}{\rm E})$ is the bundle of $\pi$-semibasic $m$-forms in ${\rm E}$),
which is called the restricted multimomentum bundle of $E$. The bundle
$J^{1}E^{*}$ has natural local coordinates $(x^{\mu},y^{A},p^{\mu}_{A})$ and
natural projections

$\displaystyle\tau:J^{1}E^{*}\rightarrow E$ $\displaystyle:$
$\displaystyle(x^{\mu},y^{A},p^{\mu}_{A})\mapsto(x^{\mu},y^{A})\ ,$
$\displaystyle\bar{\tau}:J^{1}E^{*}\rightarrow M$ $\displaystyle:$
$\displaystyle(x^{\mu},y^{A},p^{\mu}_{A})\mapsto x^{\mu}\ ,$
$\displaystyle\sigma:{\cal M}\pi\rightarrow J^{1}E^{*}$ $\displaystyle:$
$\displaystyle(x^{\mu},y^{A},p^{\mu}_{A},p)\mapsto(x^{\mu},y^{A},p^{\mu}_{A})\
.$

As ${\cal M}\pi$ is a subbundle of $\Lambda^{m}{\rm T}^{*}E$ (the
multicotangent bundle of $E$ of order $m$), ${\cal M}\pi$ is endowed with
canonical forms which are known as the multimomentum Liouville $m$ and
$(m+1)$-forms, whose local expressions are

$\Theta=p_{A}^{\alpha}{\rm d}y^{A}\wedge{\rm d}^{m-1}x_{\alpha}+p{\rm
d}^{m}x\quad,\quad\Omega=-{\rm d}p_{A}^{\alpha}\wedge{\rm d}y^{A}\wedge{\rm
d}^{m-1}x_{\alpha}-{\rm d}p\wedge{\rm d}^{m}x\ .$

If $\mathscr{L}\in C^{\infty}(J^{1}E)$ is a Lagrangian function, then the
extended Legendre map, $\widetilde{\mathscr{FL}}\colon J^{1}E\to{\cal M}\pi$,
and the restricted Legendre map,
$\mathscr{FL}:=\sigma\circ\widetilde{\mathscr{FL}}\colon J^{1}E\to
J^{1}E^{*}$, are locally given as

$\begin{array}[]{ccccccc}\widetilde{\mathscr{FL}}^{*}x^{\nu}=x^{\nu}&,&\widetilde{{\mathscr{F}L}}^{*}y^{A}=y^{A}&\quad,&\widetilde{\mathscr{FL}}^{*}p_{A}^{\nu}=\displaystyle\frac{\partial\mathscr{L}}{\partial
y^{A}_{\nu}}&\quad,&\widetilde{\mathscr{FL}}^{*}p=\mathscr{L}-\displaystyle
v^{A}_{\nu}\frac{\partial\mathscr{L}}{\partial y^{A}_{\nu}}\\\
\mathscr{FL}^{*}x^{\nu}=x^{\nu}&\quad,&\mathscr{FL}^{*}y^{A}=y^{A}&\quad,&\mathscr{FL}^{*}p_{A}^{\nu}=\displaystyle\frac{\partial\mathscr{L}}{\partial
y^{A}_{\nu}}\quad.&&\end{array}$

The Lagrangian $\mathscr{L}$ is regular if, and only if, $\mathscr{FL}$ is a
local diffeomorphism (this definition is equivalent to that given above). As a
particular case, $\mathscr{L}$ is hyper-regular if $\mathscr{FL}$ is a global
diffeomorphism. A singular Lagrangian $\mathscr{L}$ is almost-regular if
$P_{0}:=\mathscr{FL}(J^{1}E)$ is a submanifold of $J^{1}E^{*}$ (where
$\jmath_{0}\colon P_{0}\hookrightarrow J^{1}E^{*}$ denotes the natural
embedding), ${\mathscr{FL}}$ is a submersion onto its image, and for every
$p\in J^{1}E$, the fibres ${\mathscr{FL}}^{-1}({\mathscr{FL}}(p))$ are
connected submanifolds of $J^{1}E$.

If $\mathscr{L}$ is a hyper-regular Lagrangian, then the following diagram
illustrates the full structure:

$\begin{array}[]{ccccc}\begin{picture}(15.0,52.0)(0.0,0.0)\put(0.0,0.0){\mbox{$J^{1}E$}}
\end{picture}&\begin{picture}(65.0,52.0)(0.0,0.0)\put(20.0,27.0){\mbox{$\widetilde{\mathscr{FL}}$}}
\put(25.0,6.0){\mbox{${\mathscr{FL}}$}} \put(0.0,7.0){\vector(2,1){65.0}}
\put(0.0,4.0){\vector(1,0){65.0}}
\end{picture}&\begin{picture}(15.0,52.0)(0.0,0.0)\put(0.0,0.0){\mbox{$J^{1}E^{*}$}}
\put(0.0,41.0){\mbox{${\cal M}\pi$}} \put(5.0,38.0){\vector(0,-1){25.0}}
\put(-5.0,22.0){\mbox{$\sigma$}} \put(10.0,13.0){\vector(0,1){25.0}}
\put(15.0,22.0){\mbox{$h$}} \end{picture}\end{array}$

The map constructed as $h:=\widetilde{\mathscr{FL}}\circ\mathscr{FL}^{-1}$ is
called the Hamiltonian section associated with the hyper-regular Lagrangian
$\mathscr{L}$. The differential forms defined as

$\Theta_{\mathscr{H}}:=h^{*}\Theta\in{\mit\Omega}^{m}(J^{1}E^{*}),\qquad\Omega_{\mathscr{H}}:=-{\rm
d}\Theta_{h}=h^{*}\Omega\in{\mit\Omega}^{m+1}(J^{1}E^{*})\ ,$ (5)

are called the Hamilton–Cartan $m$ and $(m+1)$ forms of $J^{1}E^{*}$
associated with the Hamiltonian section $h$. If the form
$\Omega_{\mathscr{H}}$ is $1$-nondegenerate (and hence multisymplectic), then
the Hamiltonian section
$h(x^{\nu},y^{A},p^{\nu}_{A})=(x^{\nu},y^{A},p^{\nu}_{A},p=-{\mathscr{H}}(x^{\gamma},y^{B},p_{B}^{\gamma}))$
is locally specified by the Hamiltonian function
$\mathscr{H}=p^{\nu}_{A}(\mathscr{FL}^{-1})^{*}y_{\nu}^{A}-(\mathscr{FL}^{-1})^{*}\mathscr{L}$
known as the De Donder–Weyl Hamiltonian function and it follows that
$\mathscr{FL}^{*}\mathscr{H}=E_{\mathscr{L}}$. The local expressions for the
Hamilton–Cartan forms defined in (5) are

$\Theta_{\mathscr{H}}=p_{A}^{\nu}{\rm d}y^{A}\wedge{\rm
d}^{m-1}x_{\nu}-\mathscr{H}{\rm d}^{m}x\quad,\quad\Omega_{\mathscr{H}}=-{\rm
d}p_{A}^{\nu}\wedge{\rm d}y^{A}\wedge{\rm d}^{m-1}x_{\nu}+{\rm
d}\mathscr{H}\wedge{\rm d}^{m}x\ ,$

so ${\mathscr{F}}\mathscr{L}^{*}\Theta_{\mathscr{H}}=\Theta_{\mathscr{L}}$ and
${\mathscr{F}}\mathscr{L}^{*}\Omega_{\mathscr{H}}=\Omega_{\mathscr{L}}$. For
regular, but not hyper-regular, Lagrangians the construction is the same, but
on the open sets ${\mathscr{F}}\mathscr{L}(J^{1}E)\subset J^{1}E^{*}$.

The couple $(J^{1}E^{*},\Omega_{\mathscr{H}})$ is called a regular Hamiltonian
system when $\Omega_{\mathscr{H}}$ is multisymplectic and its corresponding
field equations are obtained from a variational action principle on
$J^{1}E^{*}$. The action, denoted

$\displaystyle
S\left[\psi^{*}x^{\mu},\psi^{*}y^{A},\psi^{*}p^{\mu}_{A}\right]=\int_{M}\psi^{*}\Theta_{\mathscr{H}}\
,$

is a functional on the set of sections $\Gamma(\bar{\tau})$ given by
$\displaystyle\Gamma(M,J^{1}E^{*})\rightarrow\mathds{R}:\psi\mapsto\int_{M}\psi^{*}\Theta_{\mathscr{H}}$.
As in the Lagrangian formalism, the objective is to find sections that are
critical (stationary):

$\left.\frac{\text{d}}{\text{d}s}\right|_{s=0}\int_{M}\psi^{*}_{s}\Theta_{\mathscr{H}}=0\
,$

for variations $\psi_{s}=\eta_{s}\circ\psi$ of $\psi$. Critical sections are
characterized in the following equivalent ways:

  1. 1.

$\psi^{*}i(\mathpzc{X})\Omega_{\mathscr{H}}=0$, for every
$\mathpzc{X}\in\mathfrak{X}(J^{1}E^{*})$.

  2. 2.

$\psi$ is an integral section of a class of integrable, locally decomposable,
$\bar{\tau}$-transverse multivector fields
$\\{\textbf{X}_{\mathscr{H}}\\}\subset\mathfrak{X}^{n}(J^{1}E^{*})$ which
satisfy

$i(\textbf{X}_{\mathscr{H}})\Omega_{\mathscr{H}}=0\quad,\quad\mbox{\rm for
every $\textbf{X}_{\mathscr{H}}\in\\{\textbf{X}_{\mathscr{H}}\\}$}\ .$ (6)

Furthermore, we can take as a representative of the class
$\\{\textbf{X}_{\mathscr{L}}\\}$ a normalized $\bar{\tau}^{1}$-transverse
multivector field:

$i(\textbf{X}_{\mathscr{H}}){\rm d}^{m}x=1\ .$ (7)

  3. 3.

If $(U;x^{\mu},y^{A},p_{A}^{\mu})$ is a natural system of coordinates in
$J^{1}E^{*}$, then $\psi\in U\subset J^{1}E^{*}$ satisfies the Hamilton–De
Donder–Weyl equations:

$\frac{\partial(y^{A}\circ\psi)}{\partial
x^{\mu}}=\frac{\partial\mathscr{H}}{\partial
p_{A}^{\mu}}\circ\psi\quad,\quad\frac{\partial(p^{A}_{\mu}\circ\psi)}{\partial
x^{\mu}}=-\frac{\partial\mathscr{H}}{\partial y^{A}}\circ\psi\ .$

For the almost-regular case, $P_{0}$ is a fibre bundle over $E$ and $M$ with
natural projections $\tau_{0}\colon P_{0}\to E$ and
$\bar{\tau}_{0}:=\pi\circ\tau_{0}\colon P_{0}\to M$. The $\sigma$-transverse
submanifold
$\tilde{P}_{0}=\widetilde{\mathscr{F}\mathscr{L}}(J^{1}E)\hookrightarrow{\cal
M}\pi$ with natural embedding
$\tilde{\jmath}_{0}\colon\tilde{P}_{0}\hookrightarrow{\cal M}\pi$ is
diffeomorphic to $P_{0}$ according to the restriction of the projection
$\sigma$ to $\tilde{P}_{0}$ given as $\tilde{\sigma}\colon\tilde{P}_{0}\to
P_{0}$. Taking $\tilde{h}:=\tilde{\sigma}^{-1}$, the Hamilton–Cartan
$(m+1)$-form is defined as
$\Omega^{0}_{\mathscr{H}}=(\tilde{\jmath}_{0}\circ\tilde{h})^{*}\Omega\in{\mit\Omega}^{m+1}(P)$,
which verifies that
${\mathscr{F}}\mathscr{L}_{0}^{*}\Omega^{0}_{\mathscr{H}}=\Omega_{\mathscr{L}}$
where ${\mathscr{F}}\mathscr{L}_{0}$ is the restriction map of
${\mathscr{F}}\mathscr{L}$ onto $P_{0}$ given as
${\mathscr{F}}\mathscr{L}=\jmath_{0}\circ{\mathscr{F}}\mathscr{L}_{0}$.
Furthermore, there exists a De Donder–Weyl Hamiltonian function
$\mathscr{H}_{0}\in{\rm C}^{\infty}(P_{0})$ such that
$E_{\mathcal{L}}=\mathscr{FL}_{0}^{*}\mathscr{H}_{0}$. This case is
illustrated in the following diagram:

$\begin{array}[]{cccc}\begin{picture}(20.0,52.0)(0.0,0.0)\put(0.0,0.0){\mbox{$J^{1}E$}}
\end{picture}&\begin{picture}(65.0,52.0)(0.0,0.0)\put(7.0,28.0){\mbox{$\widetilde{{\mathscr{F}}\mathscr{L}}_{0}$}}
\put(24.0,7.0){\mbox{${\mathscr{F}}\mathscr{L}_{0}$}}
\put(0.0,7.0){\vector(2,1){65.0}} \put(0.0,4.0){\vector(1,0){65.0}}
\end{picture}&\begin{picture}(90.0,52.0)(0.0,0.0)\put(5.0,0.0){\mbox{$P_{0}$}}
\put(5.0,42.0){\mbox{$\tilde{P}_{0}$}} \put(5.0,13.0){\vector(0,1){25.0}}
\put(10.0,38.0){\vector(0,-1){25.0}} \put(-10.0,22.0){\mbox{$\tilde{h}$}}
\put(12.0,22.0){\mbox{$\tilde{\sigma}$}} \put(30.0,45.0){\vector(1,0){55.0}}
\put(30.0,4.0){\vector(1,0){55.0}} \put(48.0,12.0){\mbox{$\jmath_{0}$}}
\put(48.0,33.0){\mbox{$\tilde{\jmath}_{0}$}}
\end{picture}&\begin{picture}(15.0,52.0)(0.0,0.0)\put(0.0,0.0){\mbox{$J^{1}E^{*}$}}
\put(0.0,41.0){\mbox{${\cal M}\pi$}} \put(10.0,38.0){\vector(0,-1){25.0}}
\put(0.0,22.0){\mbox{$\sigma$}} \end{picture}\\\
&&\begin{picture}(90.0,35.0)(0.0,0.0)\put(10.0,35.0){\vector(1,-1){35.0}}
\put(5.0,11.0){\mbox{$\bar{\tau}_{0}$}} \put(90.0,11.0){\mbox{$\bar{\tau}$}}
\put(100.0,35.0){\vector(-1,-1){35.0}} \end{picture}&\\\ &&\qquad
M&\end{array}$

In general, the couple $(P_{0},\Omega_{\mathscr{H}}^{0})$ (where
$\Omega_{\mathscr{H}}^{0}$ is a pre-multisymplectic form) is a non-regular
Hamiltonian system. Now, the variational principle and the field equations are
stated as in the regular case, but for sections $\Gamma(\bar{\tau}_{0})$ and
the analog to the field equations (6) read

$i(\textbf{X}^{0}_{\mathscr{H}})\Omega^{0}_{\mathscr{H}}=0\quad,\quad\mbox{\rm
for every
$\textbf{X}^{0}_{\mathscr{H}}\in\\{\textbf{X}^{0}_{\mathscr{H}}\\}\subset\mathfrak{X}^{m}(P_{0})$}\
.$ (8)

As in the Lagrangian singular case, when $(P_{0},\Omega_{\mathscr{H}}^{0})$ is
a non-regular Hamiltonian system, in the best situations, the Hamiltonian
field equations have consistent solutions only on a submanifold of $P_{0}$
which is obtained by using the constraint algorithm described in the next
section 2.4.

The general geometric structure on which the field theories is developed is
depicted in the diagram below:

$\mathbb{R}\
$$\mathscr{L}$$\mathscr{H}$$J^{1}E$$\mathrm{j^{1}\phi}$$J^{1}E^{*}$$M$$\mathrm{\phi}$$\pi$$\overline{\pi}^{1}$$\tau$$\psi$$E$$\mathscr{FL}$$\left(x^{\mu},y^{A},y^{A}_{\mu}\right)$$\left(x^{\mu},y^{A},p^{\mu}_{A}\right)$$\left(x^{\mu},y^{A}\right)$$x^{\mu}$$\pi^{1}$$\overline{\tau}$$\mathbb{R}\
$

###  2.4 Geometric Constraint Algorithm

As stated in the previous sections, it is compulsory to implement a constraint
algorithm in the Lagrangian and the Hamiltonian formalisms of singular
(almost-regular) field theories in order to find the submanifolds of the
multivelocity and multimomentum phase spaces on which the field equations are
compatible and where consistent solutions exist. The geometric algorithm for
uncovering the intrinsic constraint structure in premultisymplectic systems is
developed in detail in [13] and is a generalization of the algorithms stated
for non-autonomous singular mechanics [14, 15]. The constraint algorithm
presented in [13] uses the characterization of the variational principle and
the resulting field equations via multivector fields as discussed in the
previous sections (i.e. equations (3) and (4) in the Lagrangian case, and (6)
and (7) in the Hamiltonian case), as the multivector field characterization is
the most suitable for its implementation.

This section presents a procedure for obtaining the constraints which define
the submanifolds arising at each step of the constraint algorithm in a more
practical manner than what is given in previous literature. It should be noted
that the procedure described here is done in local charts as opposed to the
intrinsic global description given in [13]. The coordinate description of the
constraint algorithm will be carried out explicitly in each example field
theory studied in this paper.

The general geometrical setting for the Lagrangian and Hamiltonian formalisms
for singular field theories consists of taking a fiber bundle $\kappa\colon
F\to M$, where $\dim\,M=m>1$ and $\dim\,F=N+m$ ($N>m$), and $M$ is an
orientable manifold with a volume form whose pull-back to $F$ is denoted
$\omega\in{\mit\Omega}^{m}(F)$. Let $\Omega\in{\mit\Omega}^{m+1}(F)$ be a
premultisymplectic form on $F$ satisfying the so-called variational condition
given as

$\mathop{i}\nolimits(Y_{1})\mathop{i}\nolimits(Y_{2})\mathop{i}\nolimits(Y_{3})\Omega=0\
,\ \mbox{\rm for every $Y_{1},Y_{2},Y_{3}\in\mathfrak{X}^{{\rm
V}(\kappa)}(F)$}\ .$ (9)

Now, consider the problem of finding a submanifold $\jmath_{C}\colon
C\hookrightarrow F$ and a locally decomposable $m$-vector field ${\bf
X}\in\mathfrak{X}^{m}(F)$ along $C$ such that

$\mathop{i}\nolimits({\bf X})\Omega|_{C}=0\quad,\quad\mathop{i}\nolimits({\bf
X})\omega|_{C}\not=0\ ,$ (10)

where the second equation is the condition that ${\bf X}$ is
$\kappa$-transverse, and can be written simply as $\mathop{i}\nolimits({\bf
X})\omega|_{C}=1$.

The algorithmic procedure consists of the following steps:

  * •

Compatibility conditions: First, look for the conditions called compatibility
constraints which define the set $C_{1}\subset F$ where the field equations
(10) have solutions. This set is assumed to be a submanifold
$C_{1}\hookrightarrow F$. The compatibility constraints can be obtained from
direct inspection of the local expression of the field equations (10).

  * •

Tangency conditions: Now, given locally decomposable multivector fields X
which are solutions to the field equations (10) on $C_{1}$, ensure the
stability of these solutions by imposing the stability or tangency condition
of these multivector fields on $C_{1}$. This is done by requiring that the
components of X are tangent to $C_{1}$; that is, if
$\textbf{X}=\bigwedge_{\mu}X_{\mu}$, for every compatibility constraints
defining $C_{1}$, $\zeta_{j}=\mathop{i}\nolimits(Z_{j})\alpha\in{\rm
C}^{\infty}(F)$, the stability condition is given by

$\mathpzc{L}_{X_{\mu}}\zeta|_{C_{1}}=0\ .$

This procedure may produce a new constraint submanifold $C_{2}\hookrightarrow
C_{1}\hookrightarrow F$; if this is the case, then it is necessary to impose
the tangency of X to $C_{2}$ as well. This procedure is reiterated until, in
the most favorable cases (which include most physical field theories), no new
constraints are produced and hence, a final constraint submanifold
$C\hookrightarrow\ldots C_{1}\hookrightarrow F$ is found such that the
solutions X to the field equations are tangent to $C$. However, note that this
procedure does not in general terminate for all field theories, but these
cases will not be studied in this work.

  * •

Remark: Recall that the variational principle in the Lagrangian formalism
requires that multivector field solutions to the field equations on $J^{1}E$
are holonomic multivector fields. However, multivector fields which solve the
field equations are not, in general, holonomic and it is therefore sometimes
necessary to impose this condition after finding the compatibility constraint
submanifold $C_{1}\hookrightarrow J^{1}E$. Imposing the holonomy (sopde)
condition may give rise to additional constraints which define a new
constraint submanifold $S_{1}\hookrightarrow C_{1}\hookrightarrow J^{1}E$;
such constraints will be referred to here as sopde constraints. Then, as
described above, the tangency condition for the sopde multivector fields
$\textbf{X}_{\mathscr{L}}\in\mathfrak{X}^{m}(J^{1}E)$ is imposed on all the
constraints defining $S_{1}$ and subsequently also on any other resulting
constraints until (in well behaved field theories) a final constraint
submanifold is found.

It is important to note that the sopde condition does not apply in the
Hamiltonian formalism and the algorithm is simply applied as described in the
previous points.

Integrability conditions for ${\bf X}$ are typically imposed after the
constraint analysis, however, this analysis will be unnecessary in the field
theories studied in this work since they are integrable.

Now that the geometric constraint algorithm has been described in detail, it
is possible to establish various properties of constraints by means of a local
analysis. The first of such properties presented here concerns compatibility
constraints. As a consequence of the variational condition (9), the
premultisymplectic form can be expressed as the following splitting:

$\Omega=\widetilde{\Omega}+\alpha\wedge\omega\ ,$ (11)

where $\widetilde{\Omega}\in{\mit\Omega}^{m+1}(F)$ and
$\alpha\in{\mit\Omega}^{1}(F)$ are closed forms. In order to perform such a
splitting globally, it is necessary to use an Ehresmann connection on the
bundle $F\to M$ (although all the results are independent of this choice) [13,
15, 46]; however, since the field theories in this work are studied only
locally, the connection on each local chart $U\subset F$ induced by the
natural connection on $\mathbb{R}^{N}\times\mathbb{R}^{m}\to\mathbb{R}^{m}$
will always be used. It thereby follows that, in the Lagrangian and the
Hamiltonian formulations of classical field theories, the (pre)multisymplectic
forms are split in the following manner:

$\displaystyle\Omega_{\mathscr{L}}=\widetilde{\Omega}_{\mathscr{L}}+\alpha_{\mathscr{L}}\wedge\omega\in{\mit\Omega}^{m+1}(J^{1}E)$
$\displaystyle\Rightarrow$
$\displaystyle\widetilde{\Omega}_{\mathscr{L}}=\displaystyle-\text{d}\left(\frac{\partial\mathscr{L}}{\partial
y^{A}_{\mu}}\right)\wedge\text{d}y^{A}\wedge\text{d}^{m-1}x_{\mu}\ ,$
$\displaystyle\alpha_{\mathscr{L}}\wedge\omega={\rm
d}E_{\mathscr{L}}\wedge{\rm d}^{m}x\ ,$
$\displaystyle\Omega_{\mathscr{H}}=\widetilde{\Omega}_{\mathscr{H}}+\alpha_{\mathscr{H}}\wedge\omega\in{\mit\Omega}^{m+1}(P_{0})$
$\displaystyle\Rightarrow$
$\displaystyle\widetilde{\Omega}_{\mathscr{H}}=-\text{d}p_{A}^{\mu}\wedge\text{d}y^{A}\wedge\text{d}^{m-1}x_{\mu}\
,$ $\displaystyle\alpha_{\mathscr{H}}\wedge\omega={\rm d}\mathscr{H}\wedge{\rm
d}^{m}x\ ,$

where $F=J^{1}E$ is the multivelocity phase space for the Lagrangian
formulation of any generic field theory and $F=P_{0}\subset J^{1}E^{*}$ is the
multimomentum phase space for the De Donder–Weyl Hamiltonian formulation of
singular field theories. Then, as a consequence of splitting the
(pre)multisymplectic forms as shown above, the following lemma and subsequent
propositions hold:

###### Lemma 1

If ${\bf X}\in\mathfrak{X}^{m}(U)$ is a locally decomposable and
$\kappa$-transverse multivector field on $U$, then

$\mathop{i}\nolimits({\bf X})\Omega=\mathop{i}\nolimits({\bf
X})\widetilde{\Omega}+\widetilde{\alpha}_{\bf X}+(-1)^{m}f\,\alpha\ ,$ (12)

where $f=\mathop{i}\nolimits({\bf X})\omega\in{\rm C}^{\infty}(U)$ is a
nonvanishing function and $\widetilde{\alpha}_{\bf X}\in{\mit\Omega}^{1}(U)$
is a $\kappa$-semibasic $1$-form (that is,
$\mathop{i}\nolimits(Y)\widetilde{\alpha}_{\bf X}=0$, for every
$Y\in\mathfrak{X}^{{\rm V}(\kappa)}(F)$).

( Proof ) If ${\bf X}=X_{0}\wedge\ldots\wedge X_{m-1}$ in $U$, it follows from
(11) that

$\displaystyle\mathop{i}\nolimits({\bf X})\Omega$ $\displaystyle=$
$\displaystyle\mathop{i}\nolimits({\bf
X})\widetilde{\Omega}+\mathop{i}\nolimits({\bf X})(\alpha\wedge\omega)$
$\displaystyle=$ $\displaystyle\mathop{i}\nolimits({\bf
X})\widetilde{\Omega}+\sum_{\mu=0}^{m-1}(-1)^{m-1}\mathop{i}\nolimits(X_{\mu})\alpha\,\mathop{i}\nolimits(X_{0}\wedge\ldots\wedge
X_{\mu-1}\wedge X_{\mu+1}\wedge\ldots\wedge
X_{m-1})\omega+(-1)^{m}\alpha\,\mathop{i}\nolimits({\bf X})\omega$
$\displaystyle=$ $\displaystyle\mathop{i}\nolimits({\bf
X})\widetilde{\Omega}+\sum_{\mu=0}^{m-1}(-1)^{m-1}g_{\mu}\,\mathop{i}\nolimits(X_{\mu})\alpha\,{\rm
d}x^{\mu}+(-1)^{m}\alpha\,\mathop{i}\nolimits({\bf X})\omega\,;$

then, denoting $f\equiv\mathop{i}\nolimits({\bf X})\omega$ and

$\widetilde{\alpha}_{\bf
X}=\sum_{\mu=0}^{m-1}(-1)^{m-1}\mathop{i}\nolimits(X_{\mu})\alpha\,\mathop{i}\nolimits(X_{0}\wedge\ldots\wedge
X_{\mu-1}\wedge X_{\mu+1}\wedge\ldots\wedge
X_{m-1})\omega\equiv\sum_{\mu=0}^{m-1}(-1)^{m-1}\mathop{i}\nolimits(X_{\mu})\alpha\,g_{\mu}\,{\rm
d}x^{\mu}\ ,$

where $f,g_{\mu}\in{\rm C}^{\infty}(U)$ are non-vanishing functions and the
result follows.

In particular, taking $\mathop{i}\nolimits({\bf X})\omega=1$, it follows that
$f=1=g_{\mu}$ and $\mathop{i}\nolimits({\bf X})\Omega=\mathop{i}\nolimits({\bf
X})\widetilde{\Omega}+\widetilde{\alpha}_{\bf X}+(-1)^{m}\alpha$, with
$\widetilde{\alpha}_{\bf X}=(-1)^{m-1}\mathop{i}\nolimits(X_{\mu})\alpha\,{\rm
d}x^{\mu}$.

###### Proposition 1

Let $(F,\Omega)$ be a premultisymplectic bundle $F\to M$ where $\Omega$
satisfies the variational condition (9) and is written as the splitting (11).
If $C_{1}\hookrightarrow F$ is the maximal submanifold on which solutions to
the field equations (10) exist (the compatibility constraint submanifold),
then the following condition holds at ${\rm p}\in U\subset C_{1}$:

$[\mathop{i}\nolimits(Z)\alpha]({\rm p})=0,\ \mbox{\rm for every
$Z\in\text{ker}\,\widetilde{\Omega}\cap\text{ker}\,\omega$}\ .$ (13)

( Proof ) As $C_{1}$ is the maximal submanifold where the field equations are
compatible, there exists a locally decomposable multivector ${\bf
X}\in\mathfrak{X}^{m}(U)$ such that $\mathop{i}\nolimits({\bf X})\Omega=0$ and
$\mathop{i}\nolimits({\bf X})\omega\not=0$. Then (12) holds and, as
$\ker\omega=\mathfrak{X}^{{\rm V}(\kappa)}(F)$, if
$Z\in\ker\,\widetilde{\Omega}\cap\ker\,\omega$, then
$\mathop{i}\nolimits(Z)\widetilde{\alpha}_{\bf X}=0$, because
$\widetilde{\alpha}_{\bf X}$ is $\kappa$-semibasic. Therefore, for every
$Z\in\text{ker}\,\widetilde{\Omega}\cap\text{ker}\,\omega$ and for ${\rm p}\in
U$ it follows that

$0=\mathop{i}\nolimits(Z_{\rm p})\mathop{i}\nolimits({\bf X}_{\rm
p})\Omega_{\rm p}=\mathop{i}\nolimits(Z_{\rm p})\left[\mathop{i}\nolimits({\bf
X}_{\rm p})\widetilde{\Omega}_{\rm p}+\widetilde{\alpha}_{{\bf X}{\rm
p}}+(-1)^{m}f\,\alpha_{\rm p}\right]=(-1)^{m}f\,\mathop{i}\nolimits(Z_{\rm
p})\alpha_{\rm p}\ \Longleftrightarrow\ \mathop{i}\nolimits(Z_{\rm
p})\alpha_{\rm p}=0\ .$

Thus, (13) is a necessary condition of compatibility of the field equations
(10). In the particular case where $m=1$ (that is, the non-autonomous
mechanics) this is also a sufficient condition and then (13) is a geometrical
characterization of all the compatibility constraints. Nevertheless, it
remains under investigation to prove the converse of Proposition 1 in the
general case where $m>1$: the compatibility constraint submanifold $C_{1}$ is
completely determined by (13).

As a final remark, in the Lagrangian and Hamiltonian formalisms of field
theories, it is interesting to show the relation between the vector fields
belonging to $\ker{\mathscr{F}}\mathscr{L}_{*}$ and the primary Hamiltonian
constraints arising from the Legendre map. This result is analogous to that
verified for singular dynamical systems [54, 55].

Let $(J^{1}E,\ \Omega_{\mathscr{L}})$ be an almost-regular Lagrangian system.
First observe that in natural coordinates of $J^{1}E$ the map
${\mathscr{F}}\mathscr{L}_{*}$ is represented by the matrix

${\rm T}{{\mathscr{F}}\mathscr{L}}\equiv\left(\begin{matrix}({\rm
Id})_{m\times m}&(0)_{m\times n}&(0)_{m\times nm}\\\ (0)_{n\times m}&({\rm
Id})_{n\times n}&(0)_{n\times nm}\\\ (0)_{nm\times
m}&\left(\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial y^{B}\partial
y^{A}_{\mu}}\right)&\left(\displaystyle\frac{\partial^{2}\mathscr{L}}{\partial
y^{B}_{\nu}\partial y^{A}_{\mu}}\right)\end{matrix}\right)\ .$

It follows that
$Y_{i}=\left(\gamma_{i}\right)_{\mu}^{A}\displaystyle\frac{\partial{}}{\partial{y^{A}_{\mu}}}\in\ker{\mathscr{F}}\mathscr{L}_{*}$
with (local) component functions $\left(\gamma_{i}\right)_{\mu}^{A}\in{\rm
C}^{\infty}(J^{1}E)$ which annihilate the Hessian,
$\displaystyle\left(\frac{\partial^{2}\mathscr{L}}{\partial
y^{B}_{\nu}\partial
y^{A}_{\mu}}\right)(\left(\gamma_{i}\right)_{\mu}^{A})=(0)$, and hence
$\ker{\mathscr{F}}\mathscr{L}_{*}\subset\mathfrak{X}^{{\rm
V}(\pi_{1})}(J^{1}E)$. Furthermore, is worth noting that

$\text{ker}\,\Omega_{\mathscr{L}}\cap\mathfrak{X}^{{\rm
V}(\pi_{1})}(J^{1}E)=\text{ker}\,\mathscr{FL}_{*}=\text{ker}\,\widetilde{\Omega}_{\mathscr{L}}\cap\mathfrak{X}^{{\rm
V}(\pi_{1})}(J^{1}E)\subset\text{ker}\,\widetilde{\Omega}_{\mathscr{L}}\cap\mathfrak{X}^{{\rm
V}(\bar{\pi}_{1})}(J^{1}E)=\text{ker}\,\widetilde{\Omega}_{\mathscr{L}}\cap\text{ker}\,\omega.$

Finally:

###### Proposition 2

For every constraint $\zeta_{i}\in{\rm C}^{\infty}(J^{1}E^{*})$ which locally
defines the (primary Hamiltonian constraint) submanifold
$P_{0}={\mathscr{F}}\mathscr{L}(J^{1}E)\hookrightarrow J^{1}E^{*}$, there
exists a vector field $\displaystyle
Y_{i}=\left(\gamma_{i}\right)_{\mu}^{A}\displaystyle\frac{\partial{}}{\partial{y^{A}_{\mu}}}\in\ker{\mathscr{F}}\mathscr{L}_{*}$
such that

$\left(\gamma_{i}\right)_{\mu}^{A}={\mathscr{F}}\mathscr{L}^{*}\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{p^{\mu}_{A}}}\right)\
.$ (14)

( Proof ) As $\zeta_{i}$ is a primary Hamiltonian constraint it follows that
${\mathscr{F}}\mathscr{L}^{*}\zeta_{i}=0$. Then,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle{\rm
d}{\mathscr{F}}\mathscr{L}^{*}\zeta_{i}={\rm
d}(\zeta_{i}\circ{\mathscr{F}}\mathscr{L})$ $\displaystyle=$
$\displaystyle\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{p^{\mu}_{A}}}\circ{\mathscr{F}}\mathscr{L}\right)\displaystyle\frac{\partial{p^{\mu}_{A}}}{\partial{y^{B}_{\nu}}}\,{\rm
d}y^{B}_{\nu}+\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{p^{\mu}_{A}}}\circ{\mathscr{F}}\mathscr{L}\right)\displaystyle\frac{\partial{p^{\mu}_{A}}}{\partial{y^{B}}}\,{\rm
d}y^{B}+\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{y^{B}}}\circ{\mathscr{F}}\mathscr{L}\right)\,{\rm
d}y^{B}$ $\displaystyle=$
$\displaystyle{\mathscr{F}}\mathscr{L}^{*}\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{p^{\mu}_{A}}}\right)\left(\frac{\partial^{2}\mathscr{L}}{\partial
y^{B}_{\nu}\partial y^{A}_{\mu}}\right)\,{\rm
d}y^{B}_{\nu}+\left[{\mathscr{F}}\mathscr{L}^{*}\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{p^{\mu}_{A}}}\right)\left(\frac{\partial^{2}\mathscr{L}}{\partial
y^{B}\partial
y^{A}_{\mu}}\right)+{\mathscr{F}}\mathscr{L}^{*}\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{y^{B}}}\right)\right]\,{\rm
d}y^{B}$ $\displaystyle\Longleftrightarrow$
$\displaystyle{\mathscr{F}}\mathscr{L}^{*}\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{p^{\mu}_{A}}}\right)\left(\frac{\partial^{2}\mathscr{L}}{\partial
y^{B}_{\nu}\partial y^{A}_{\mu}}\right)=0\ ,\
{\mathscr{F}}\mathscr{L}^{*}\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{p^{\mu}_{A}}}\right)\left(\frac{\partial^{2}\mathscr{L}}{\partial
y^{B}\partial
y^{A}_{\mu}}\right)+{\mathscr{F}}\mathscr{L}^{*}\left(\displaystyle\frac{\partial{\zeta_{i}}}{\partial{y^{B}}}\right)=0\
,$

and equation (14) follows from the first equality above.

###  2.5 Multisymplectic Symmetries and Noether Currents

In the physics literature it is typical to work with the sections of the
various fiber bundles that have been discussed. As mentioned earlier, the
fields $y^{A}(x)$ are given by sections of the configuration bundle
$\pi:E\rightarrow M$. Furthermore, the transformations which are used to
investigate symmetries are also typically constructed on the sections of the
configuration bundle in the physics literature. The Lagrangian function is
typically defined on the jet prolongations $j^{1}\phi:M\rightarrow
J^{1}E:x^{\mu}\mapsto(x^{\mu},y^{A}(x),\partial_{\mu}y^{A}(x))$ and the field
transformations naturally produce transformations of the spacetime derivatives
of the fields. The De Donder–Weyl formalism is similarly constructed on
sections of $J^{1}E^{*}$ (or $P_{0}$). In this paper, the physics is being
studied on the configuration manifold $E$, the jet manifold $J^{1}E$, and the
dual jet manifold $J^{1}E^{*}$ (or $P_{0}$) as opposed to working on the
sections of those manifolds as it is usual in most physical literature. That
is, one transforms $y^{A}$ which is a coordinate on some fiber bundle whose
fibers give the field $y^{A}(x)$.

Noether’s theorem can be formulated on the multivelocity and multimomentum
phase spaces. Noether symmetries are regarded as transformations on the phase
spaces which preserve the (pre)multisymplectic forms [19]. The conserved
quantities on the multivelocity and multimomentum phase spaces associated with
Noether symmetries are called covariant momentum maps which will be discussed
later in 2.5.1. Gauge transformations are transformations on the configuration
manifold $E$ which do not induce transformations on $M$; that is, the vector
fields that generate gauge transformations are vector fields on $E$ which are
$\pi$-vertical. Gauge transformations whose liftings to $J^{1}E$ preserve the
form $\Omega_{\mathscr{L}}$ are called gauge symmetries. Alternatively, lifted
base space diffeomorphisms (such as spacetime diffeomorphisms or worldsheet
diffeomorphisms in the case of string theory) are not referred to as gauge
symmetries in this work.

Since physical symmetries preserve $\Omega_{\mathscr{L}}$, and consequently
$\Omega_{\mathscr{H}}$ as well, it is necessary to discuss how Lie groups act
on the (pre-)multisymplectic multivelocity and multimomentum phase spaces.
Afterwards, it will be shown how gauge transformations and base space
diffeomorphisms are lifted to the multivelocity and multimomentum phase
spaces.

####  2.5.1 Covariant Momentum Maps and Noether Currents

Covariant momentum maps are the conserved quantities which result from the
group action of a Lie group on a multisymplectic manifold. Consider a
multisymplectic phase space $F$ with a multisymplectic $(m+1)$-form
$\Omega=-\text{d}\Theta$ and a Lie group $G$ with group action $\Phi$ on $F$
written as

$\Phi:\mathpzc{G}\times F\rightarrow\text{Diff}(F):g\mapsto\Phi_{g}\quad;\quad
g\in G\ ,$

and denote $\Phi_{g}:F\rightarrow F$ the corresponding diffeomorphisms induced
by this action. If $\mathfrak{g}$ is the Lie algebra of $G$, every
$\xi\in\mathfrak{g}$ induces a vector field on $F$

$\left.\textbf{X}_{\xi}=\frac{\text{d}}{\text{d}\lambda}\right|_{\lambda=0}\Phi^{*}_{exp(\lambda\xi)}\
,$

with the use of some parameter $\lambda$ which is used to write $g\in G$ in
terms of the exponential map

$exp:\textgoth{g}\rightarrow G:\xi\mapsto g=exp(\lambda\xi)\ .$

The group $G$ is a symmetry group of the multisymplectic phase space $F$ if
every $g\in G$ generates a multisymplectomorphism (or a covariant canonical
transformation) on $F$; that is, a diffeomorphism $\Phi_{g}\colon F\to F$ such
that $\Phi^{*}_{g}\Omega=\Omega$, or equivalently:

$\mathpzc{L}_{\textbf{X}_{\xi}}\Omega=0\quad,\quad\mbox{\rm for every
$\xi\in\mathfrak{g}$}\ ,$ (15)

where $\mathpzc{L}_{\textbf{X}_{\xi}}$ denotes the Lie derivative associated
with ${\textbf{X}_{\xi}}$. Furthermore, exact multisymplectomorphisms (or
special covariant canonical transformations) are those which also preserve the
form $\Theta\in{\mit\Omega}^{m}(F)$:

$\Phi^{*}_{g}\Theta=\Theta\Longleftrightarrow\mathpzc{L}_{\textbf{X}_{\xi}}\Theta=0\
.$

Then, using of Cartan’s identity,
$\mathpzc{L}_{\textbf{X}_{\xi}}=i(\textbf{X}_{\xi})\text{d}+\text{d}i(\textbf{X}_{\xi})$,
and as $\text{d}\Omega=0$, it follows from (15) that
$\text{d}i(\textbf{X}_{\xi})\Omega=0$, so (locally) there exists some
$(m-1)$-from $J_{\xi}\in{\mit\Omega}^{m-1}(F)$ such that

$i(\textbf{X}_{\xi})\Omega=-\text{d}J_{\xi}\ .$ (16)

Working with the infinitesimal generators of every $\xi\in\textgoth{g}$
amounts to $J_{\xi}$ to be linear in $\xi$ as will be the case in the theories
that are investigated in this paper. This makes it possible to define a linear
map, $\left.J\right|_{\rm p}$, by considering a point ${\rm p}\in F$ so that
$J_{\rm p}(\xi)\equiv\left.J_{\xi}\right|_{\rm p}\in\Lambda^{m-1}({\rm T}_{\rm
p}F)$ gives $\left.J\right|_{\rm p}:\textgoth{g}\rightarrow\Lambda^{m-1}({\rm
T}_{\rm p}F):\xi\mapsto J_{\rm p}(\xi)$, for every $\xi\in\textgoth{g}$. The
linear map $\left.J\right|_{\rm p}$ is an element of
$\textgoth{g}^{*}\otimes\Lambda^{m-1}({\rm T}_{z}F)$. This structure makes it
possible to construct another map, $J$, using $J({\rm
p})\equiv\left.J\right|_{\rm p}$ so that
$J:F\rightarrow\textgoth{g}^{*}\otimes{\mit\Omega}^{m-1}(F):{\rm p}\mapsto
J({\rm p})$, for every ${\rm p}\in F$. The map $J$ is called the covariant
momentum map, although this terminology is used to refer to both $J$ and
$\left.J\right|_{\xi}$. This setting can be concisely specified through the
natural pairing $\left<\cdot,\cdot\right>$ between $\textgoth{g}^{*}$ and
$\textgoth{g}$ written as

$\left.J_{\xi}\right|_{z}=\left<J(z),\xi\right>\ .$ (17)

Working directly with equation (16) and Cartan’s identity gives

$\text{d}J_{\xi}=-i(\textbf{X}_{\xi})\Omega=i(\textbf{X}_{\xi})\text{d}\Theta=\mathpzc{L}_{\textbf{X}_{\xi}}\Theta-\text{d}i(\textbf{X}_{\xi})\Theta\
,$ (18)

but also

$0=\mathpzc{L}_{\textbf{X}_{\xi}}\text{d}\Theta=\text{d}\mathpzc{L}_{\textbf{X}_{\xi}}\Theta\
,$

so (at least locally)
$\mathpzc{L}_{\textbf{X}_{\xi}}\Theta=\text{d}\alpha_{\xi}$ for some
$\alpha_{\xi}\in{\mit\Omega}^{m-1}(F)$. Then, equation (18) yields

$J_{\xi}=-i(\textbf{X}_{\xi})\Theta+\alpha_{\xi}\ .$

However, equation (16) only determines $J_{\xi}$ up to some exact form
$\text{d}\beta_{\xi}$ with $\beta_{\xi}\in{\mit\Omega}^{m-2}(F)$. That is,
equation (16) is still satisfied with the redefinition $J_{\xi}\rightarrow
J_{\xi}-\text{d}\beta_{\xi}$; so, in order to keep full generality, the
momentum map $J_{\xi}$ takes the form

$J_{\xi}=-i(\textbf{X}_{\xi})\Theta+\alpha_{\xi}+\text{d}\beta_{\xi}\ .$ (19)

It is worth noting that, for symmetries generated by a vector field
$\textbf{X}_{\xi}$ which is a lift from $M$ to $F$, it follows that
$\mathpzc{L}_{\textbf{X}_{\xi}}\Theta=0\Rightarrow\alpha_{\xi}=\text{d}\gamma_{\xi}$
for some $\gamma_{\xi}\in\Omega^{m-2}(F)$. Exact forms such as
$\text{d}\beta_{\xi}$ and $\text{d}\gamma_{\xi}$ will be ignored in this paper
as boundary terms will not be considered.

Furthermore, in the Lagrangian formulation of field theories where $F=J^{1}E$,
the covariant momentum maps on $J^{1}E$ are said to be in the Lagrangian
representation and the Noether current
$j=j^{\mu}\text{d}^{m-1}x_{\mu}\in{\mit\Omega}^{m-1}(M)$ of a symmetry is
obtained by pulling back the associated momentum map
$J_{\xi}\in{\mit\Omega}^{m-1}(J^{1}E)$ to $M$, using the first jet
prolongations $j^{1}\phi$ as follows: $j=j^{1}\phi^{*}J_{\xi}$.

####  2.5.2 Lifting Transformations to $J^{1}E$ and $J^{1}E^{*}$

Consider infinitesimal spacetime diffeomorphisms $M\rightarrow M$ produced by
the coordinate transformation $x^{\prime\mu}=x^{\mu}+\xi^{\mu}(x)$. Such
spacetime diffeomorphisms are generated by the vector field
$\displaystyle\xi=-\xi^{\mu}(x)\frac{\partial}{\partial
x^{\mu}}\in\mathfrak{X}(M)$. Then, the $\pi$-projectable vector field
$\xi_{E}\in\mathfrak{X}(E)$, which generates the corresponding transformations
$(x^{\mu},y^{A})\rightarrow(x^{\mu}+\xi^{\mu}(x),y^{A}+\xi^{A}(x,y))$ on the
configuration manifold $E$ with $\xi^{A}(x,y)\in{\rm C}^{\infty}(E)$ can be
written as

$\xi_{E}=-\xi^{\mu}(x)\frac{\partial}{\partial
x^{\mu}}-\xi^{A}(x,y)\frac{\partial}{\partial y^{A}}\in\mathfrak{X}(E)\ .$
(20)

Furthermore, the canonical lift of $\xi_{E}\in\mathfrak{X}(E)$ to $J^{1}E$ is
given by the following expression [19, 21]:

$X_{\xi}=-\xi^{\mu}\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}-\xi^{A}\frac{\partial}{\partial
y^{A}}-\left(\displaystyle\frac{\partial{\xi^{A}}}{\partial{x^{\mu}}}-y^{A}_{\nu}\displaystyle\frac{\partial{\xi^{\nu}}}{\partial{x^{\mu}}}+y^{B}_{\mu}\frac{\partial\xi^{A}}{\partial
y^{B}}\right)\frac{\partial}{\partial y^{A}_{\mu}}\in\mathfrak{X}(J^{1}E)\ .$
(21)

The field transformations $\delta y^{A}(x)$, which are sometimes referred to
as the local variation of the fields $y^{A}(x)$, are defined as the Lie
derivatives of the fields $y^{A}(x)$ by the vector field $\xi$ on $M$ and take
the form,

$\delta y^{A}(x)\equiv\mathpzc{L}_{\xi}y^{A}(x)=y^{\prime
A}(x)-y^{A}(x)=-\xi^{\mu}(x)\displaystyle\frac{\partial{y^{A}}}{\partial{x^{\mu}}}(x)+\widetilde{\xi}^{A}(x)\
,$ (22)

where $\widetilde{\xi}^{A}(x)=\xi^{A}\circ\phi$. The first term in (22),
$-\xi^{\mu}(x)\displaystyle\frac{\partial{y^{A}}}{\partial{x^{\mu}}}(x)$, is
sometimes referred to as the transport term while the second term,
$\widetilde{\xi}^{A}(x)$, is referred to as the global variation of the fields
$y^{A}(x)$ and is given as

$\widetilde{\xi}^{A}(x)=\Delta_{g}y^{A}(x)\equiv y^{\prime
A}(x^{\prime})-y^{A}(x)\ .$

Since the fields $y^{A}(x)$ are given by the local sections $\phi:M\rightarrow
E$, the field transformations $\delta y^{A}(x)$ in (22) can be interpreted
geometrically as the Lie derivative of the local sections $\phi$ defined as
[19, 21]:

$\mathpzc{L}_{\xi}\phi=T\phi\circ\xi-\xi_{E}\circ\phi\ .$ (23)

The variation of the spacetime derivatives of the fields, $\delta
y^{A}_{\mu}(x)$, can also be characterized by the Lie derivative of local
sections as follows: given holonomic local sections
$\psi=j^{1}\phi:M\rightarrow J^{1}E$, the Lie derivative of $\psi$ with
respect to the vector field
$\displaystyle\xi=-\xi^{\mu}(x)\frac{\partial}{\partial
x^{\mu}}\in\mathfrak{X}(M)$ is defined as

$\mathpzc{L}_{\xi}\psi=T\psi\circ\xi-X_{\xi}\circ\psi,$ (24)

where $X_{\xi}\in\mathfrak{X}(J^{1}E)$ is the canonical lift of
$\xi\in\mathfrak{X}(M)$ to $J^{1}E$ given by (21). It thereby follows that

$\mathpzc{L}_{\xi}\psi^{A}_{\mu}=\delta
y^{A}_{\mu}(x)=-\xi^{\nu}\displaystyle\frac{\partial{y^{A}_{\mu}}}{\partial{x^{\nu}}}-\displaystyle\frac{\partial{\xi^{\nu}}}{\partial{x^{\mu}}}\displaystyle\frac{\partial{y^{A}}}{\partial{x^{\nu}}}+\displaystyle\frac{\partial{\widetilde{\xi}^{A}}}{\partial{x^{\mu}}}+\displaystyle\frac{\partial{y^{B}}}{\partial{x^{\mu}}}\frac{\partial\widetilde{\xi}^{A}}{\partial
y^{B}}\ ,$

where $\widetilde{\xi}^{A}(x)=\xi^{A}\circ\psi$. It is important to point out
that, sometimes, it is only possible to interpret the field variations $\delta
y^{A}(x)$ as Lie derivatives of the jet prolongations $j^{1}\phi$; this may
occur when the vector fields which generate Noether symmetries on the
(pre)multisymplectic phase spaces are not projectable onto $E$ and are not
produced from canonical lifts from $E$ to $J^{1}E$. Such examples will come up
in sections 3.1.3 and 3.3.3.

In the case of a gauge transformation, the vector field $\xi_{E}$ only has the
component along $\partial/\partial y^{A}$,

$\xi_{E}=-\xi^{A}(x,y)\frac{\partial}{\partial y^{A}}\in\mathfrak{X}(E)\ ,$

so the canonical lift to $J^{1}E$ is given by

$X_{\xi}=-\xi^{A}\frac{\partial}{\partial
y^{A}}-\left(\displaystyle\frac{\partial{\xi^{A}}}{\partial{x^{\mu}}}+y^{B}_{\mu}\frac{\partial\xi^{A}}{\partial
y^{B}}\right)\frac{\partial}{\partial y^{A}_{\mu}}\in\mathfrak{X}(J^{1}E)\ .$

Canonical lifts from $E$ to $J^{1}E$ satisfy the property that they preserve
the Lagrangian density if, and only if, they preserve the Poincaré–Cartan
form, i.e. $\mathpzc{L}_{X_{\xi}}\textbf{L}=0\
\Leftrightarrow\mathpzc{L}_{X_{\xi}}\Theta_{\mathscr{L}}=0$ [46, 21]. The
vector fields on $J^{1}E$ used to construct the covariant momentum maps on
$J^{1}E$ are precisely canonical lifts and, for regular Lagrangians (i.e. when
$\mathscr{FL}$ is a diffeomorphism), the vector fields used to construct the
covariant momentum maps on $J^{1}E^{*}$ are the pushforward by the Legendre
map of the canonical lifts:
$Y_{\xi}=\mathscr{FL}_{*}X_{\xi}\in\mathfrak{X}(J^{1}E^{*})$. When the
Lagrangian is singular (in particular, almost-regular), it is sometimes the
case that the vector field $X_{\xi}\in\mathfrak{X}(J^{1}E)$ is only
projectable onto the multimomentum phase space on some constraint submanifold
of the multivelocity phase space: $Y_{\xi}=\mathscr{FL}_{0*}X_{\xi}|_{S\subset
J^{1}E}\in\mathfrak{X}(P_{0})$. This scenario occurs in both the Klein–Gordon
field theory in Section 3.1 and the magnetic Carrollian scalar field theory in
Section 3.3.

##  3 Scalar Field Theories

The scalar field theories presented in this section are formulated on an
$m$-dimensional spacetime $M$ and are constructed from two scalar fields,
$\phi(x)$ and $\pi(x)$, given by the sections of the configuration bundle
$E\rightarrow M:(\phi,\pi)\mapsto x^{\mu}$ with local sections denoted as
$\phi:M\rightarrow E:x^{\mu}\mapsto(x^{\mu},\phi(x),\pi(x))$. Three different
scalar field theories will be constructed on this geometric setup while the
second and third field theories come from Carroll contractions of the
canonical Klein–Gordon Lagrangian. For more details on Carrollian field
theories, see for example [25, 26, 27, 28].

###  3.1 Klein–Gordon From Geometric Constraints

In this section, spacetime, $M$, is taken to be $m-$dimensional Minkowski
space with the Minkowski metric denoted as $\eta_{\mu\nu}$ with signature
$(-+\dotsc+)$.

####  3.1.1 Lagrangian Formulation

The scalar field theory investigated in this section is given by the canonical
Klein–Gordon Lagrangian which originates from the canonical construction of
the Klein–Gordon field on the canonical momentum phase space where the
canonical momentum, $\pi(x)$, is given by the Legendre transform
$\pi(x)=\partial\mathscr{L}(x)/\partial\dot{\phi}(x)$. Conversely, in the
construction of the canonical Klein–Gordon Lagrangian given here, $\pi(x)$ is
proposed simply as a second variational field while the notion of the
canonical Legendre transform is abandoned and the relation that is typically
obtained from the canonical Legendre transform instead shows up as a sopde
constraint. The jet bundle which serves as the multivelocity phase space for
this specific field theory has coordinates
$(x^{\mu},\phi,\pi,\phi_{\mu},\pi_{\mu})$ and it thereby follows that the
first-order jet prolongations of sections $\phi:M\to E$ have the expression
$j^{1}\phi:M\rightarrow
J^{1}E:x^{\mu}\mapsto(x^{\mu},\phi(x),\pi(x),\partial_{\mu}\phi(x),\partial_{\mu}\pi(x))$.

Now, instead of writing the Lagrangian in terms of the fields themselves, the
Lagrangian is written as a function on the multivelocity phase space,

$\mathscr{L}=\pi\phi_{0}-\frac{1}{2}\pi^{2}-\frac{1}{2}\phi_{i}\phi^{i}\in
C^{\infty}(J^{1}E)\ .$ (25)

This Lagrangian is singular as it is directly evident from the Hessian matrix
(with respect to the multivelocities), whose components are

$\frac{\partial^{2}\mathscr{L}}{\partial\phi_{0}\partial\phi_{0}}=0\ ,\
\frac{\partial^{2}\mathscr{L}}{\partial\phi_{0}\partial\phi_{i}}=0\ ,\
\frac{\partial^{2}\mathscr{L}}{\partial\phi_{i}\partial\phi_{j}}=-\delta^{ij}\
,\ \frac{\partial^{2}\mathscr{L}}{\partial\pi_{\mu}\partial\pi_{\nu}}=0\ .$

The null vectors associated with
$\displaystyle\frac{\partial{{}^{2}\mathscr{L}}}{\partial{\phi_{\mu}\partial\phi_{\nu}}}$
and
$\displaystyle\frac{\partial{{}^{2}\mathscr{L}}}{\partial{\pi_{\mu}\partial\pi_{\nu}}}$
are

$\gamma^{(\phi)}_{\mu}=\begin{pmatrix}1\\\ 0\\\ \vdots\\\
0\end{pmatrix}\quad,\quad\gamma^{(\pi)}_{\mu}=\begin{pmatrix}1\\\ 1\\\
\vdots\\\ 1\end{pmatrix}\ ,$ (26)

respectively. The Lagrangian energy is

$E_{\mathscr{L}}=\frac{\partial\mathscr{L}}{\partial\phi_{\mu}}\phi_{\mu}+\frac{\partial\mathscr{L}}{\partial\pi_{\mu}}\pi_{\mu}-\mathscr{L}=\frac{1}{2}(\pi^{2}-\phi_{i}\phi^{i})\in
C^{\infty}(J^{1}E)\ ,$

and the Poincaré–Cartan forms are

$\displaystyle\Theta_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathscr{L}}{\partial\phi_{\mu}}\text{d}\phi\wedge\text{d}^{m-1}x_{\mu}+\frac{\partial\mathscr{L}}{\partial\pi_{\mu}}\text{d}p\wedge\text{d}^{m-1}x_{\mu}-E_{\mathscr{L}}\text{d}^{m}x$
(27) $\displaystyle=$
$\displaystyle\pi\text{d}\phi\wedge\text{d}^{m-1}x_{0}-\phi^{i}\text{d}\phi\wedge\text{d}^{m-1}x_{i}-\frac{1}{2}(\pi^{2}-\phi_{i}\phi^{i})\text{d}^{m}x\in{\mit\Omega}^{m}(J^{1}E)\
,$ $\displaystyle\Omega_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle-\text{d}\Theta_{\mathscr{L}}=\text{d}\phi\wedge\text{d}\pi\wedge\text{d}^{m-1}x_{0}-\text{d}\phi^{i}\wedge\text{d}\phi\wedge\text{d}^{m-1}x_{i}+(\pi\text{d}\pi-\phi^{i}\text{d}\phi_{i})\wedge\text{d}^{m}x$
(28) $\displaystyle\equiv$
$\displaystyle\widetilde{\Omega}_{\mathscr{L}}+\text{d}E_{\mathscr{L}}\wedge\text{d}^{m}x\in{\mit\Omega}^{m+1}(J^{1}E)\
.$

As $\mathscr{L}$ is a singular Lagrangian, ${\Omega}_{\mathscr{L}}$ is a
premultisymplectic form. Now, take a locally decomposable multivector field
$\textbf{X}_{\mathscr{L}}=X_{0}\wedge X_{1}\wedge\dotsc\wedge
X_{m-1}\in\mathfrak{X}^{m}(J^{1}E)$, with components

$X_{\mu}=\frac{\partial}{\partial
x^{\mu}}+A_{\mu}\frac{\partial}{\partial\phi}+B_{\mu}\frac{\partial}{\partial\pi}+C_{\mu\nu}\frac{\partial}{\partial\phi_{\nu}}+D_{\mu\nu}\frac{\partial}{\partial\pi_{\nu}}\
,$

which satisfies $i(\textbf{X}_{\mathscr{L}}){\rm d}^{m}x=1$. Then, the field
equation (3) gives

$\displaystyle 0=i(\textbf{X}_{\mathscr{L}})\Omega_{\mathscr{L}}$
$\displaystyle=$ $\displaystyle-
B_{0}\text{d}\phi+A_{0}\text{d}\pi+C_{i}^{i}\text{d}\phi-A^{i}\text{d}\phi_{i}-\pi\text{d}\pi+\phi^{i}\text{d}\phi_{i}$
$\displaystyle+\left(B_{0}A_{\lambda}-B_{\lambda}A_{0}+C_{\lambda
i}A_{i}-C_{ii}A_{\lambda}+\pi
B_{\lambda}-\phi_{i}C_{\lambda}^{i}\right)\text{d}x^{\lambda}\ ,$

which, by setting differential forms separately equal to zero, produces the
following equations

$\displaystyle C_{i}^{i}-B_{0}=0\ ,$ (29a) $\displaystyle A_{0}-\pi=0\ ,$
(29b) $\displaystyle\phi_{i}-A_{i}=0\ ,$ (29c) $\displaystyle
B_{0}A_{\lambda}-B_{\lambda}A_{0}+C_{\lambda i}A_{i}-C_{ii}A_{\lambda}+\pi
B_{\lambda}-\phi_{i}C_{\lambda}^{i}=0\ .$ (29d)

Equation (29d) is a combination of the first three equations and is thereby
and identity. Equation (29b) determines the coefficient $A_{0}$, equation
(29a) is a relation among the coefficients $B_{0}$ and $C_{i}^{i}$, and
equation (29c) is the spatial part of the holonomic or sopde condition for the
$\partial/\partial\phi$ piece of the multivector field. As these equations
show, no compatibility constraints appear. As it has been proven in
Proposition 1, this can also be analyzed geometrically by enforcing

$i(Z)\text{d}E_{\mathscr{L}}=0\quad,\quad\mbox{\rm for every
$Z\in\text{ker}\,\widetilde{\Omega}_{\mathscr{L}}\cap\text{ker}\,\text{d}^{m}x$}\
,$

where,
$\widetilde{\Omega}_{\mathscr{L}}=\text{d}\phi\wedge\text{d}\pi\wedge\text{d}^{m-1}x_{0}-\text{d}\phi^{i}\wedge\text{d}\phi\wedge\text{d}^{m-1}x_{i}$,
as it is shown in (28). In this case

$\text{ker},\widetilde{\Omega}_{\mathscr{L}}\cap\text{ker}\,\text{d}^{m}x=\left\langle\frac{\partial}{\partial\phi_{0}},\frac{\partial}{\partial\pi_{\mu}}\right\rangle\
,$

and it follows that

$i\left(\frac{\partial}{\partial\phi_{0}}\right)\text{d}E_{\mathscr{L}}=0=i\left(\frac{\partial}{\partial\pi_{\mu}}\right)\text{d}E_{\mathscr{L}}\
,$

which agrees with the fact that no compatibility constraints are produced in
this system.

Now, imposing the sopde condition $A_{\mu}=\phi_{\mu}$, $B_{\mu}=\pi_{\mu}$ on
$\textbf{X}_{\mathscr{L}}$ (i.e. demanding that the multivector fields
$\textbf{X}_{\mathscr{L}}$ are holonomic), it follows that the relevant field
equations left over are

$\displaystyle C_{i}^{i}-\pi_{0}=0\ ,$ (30a) $\displaystyle\phi_{0}-\pi=0\ .$
(30b)

At this stage, the first equation (30a) gives another relation for the
coefficients $C_{i}^{i}$. Equation (30b) is a sopde constraint which defines
the constraint submanifold $S_{1}\hookrightarrow J^{1}E$; this is a constraint
giving a relation between the fields and multivelocities. Notice that the
Lagrangian function (25) on the submanifold $S_{1}$ takes the form of the
standard Klein–Gordon Lagrangian:

$\left.\mathscr{L}\right|_{S_{1}}=-\frac{1}{2}\phi_{\mu}\phi^{\mu}\ .$

As a final step, it is necessary to impose the tangency condition of the
multivector fields ${\bf X}_{\mathscr{L}}$ to the submanifold $S_{1}$:

$\mathpzc{L}_{X_{\mu}}(\phi_{0}-\pi)=C_{\mu 0}-B_{\mu}=0\quad\mbox{\rm(on
$S_{1}$)}\ .$ (31)

These are new relations among the coefficients of ${\bf X}_{\mathscr{L}}$ but
are not constraints, therefore $S_{1}$ is the final constraint submanifold on
which the constraint algorithm terminates.

Now, upon taking $(x^{\mu},\phi(x),\pi(x),\phi_{\nu}(x),\pi_{\nu}(x))$ to be
integral sections of the multivector field $\textbf{X}_{\mathscr{L}}$ by
setting

$A_{\mu}=\displaystyle\frac{\partial{\phi}}{\partial{x^{\mu}}}\quad,\quad
B_{\mu}=\displaystyle\frac{\partial{\pi}}{\partial{x^{\mu}}}\quad,\quad
C_{\mu\nu}=\displaystyle\frac{\partial{\phi_{\nu}}}{\partial{x^{\mu}}}\quad,\quad
D_{\mu\nu}=\displaystyle\frac{\partial{\pi_{\nu}}}{\partial{x^{\mu}}}\ ,$ (32)

the field equations on $S_{1}$ become

$\displaystyle\frac{\partial^{2}\phi}{\partial x^{i}\partial
x_{i}}-\dot{\pi}\equiv\partial_{i}\partial^{i}\phi-\dot{\pi}$
$\displaystyle=0\ ,$ (33a) $\displaystyle\dot{\phi}-\pi$ $\displaystyle=0\ .$
(33b)

The combination of equations (33a) and (33b) gives the Klein–Gordon equation:

$\frac{\partial^{2}\phi}{\partial x^{\mu}\partial
x_{\mu}}\equiv\partial_{\mu}\partial^{\mu}\phi=0\ .$

Furthermore, the integral sections (32) which satisfy the field equations
(33a) and (33b) satisfy the tangency condition on $S_{1}$ given by (31)
particularly as a consequence of (33b).

####  3.1.2 Hamiltonian Formulation

The De Donder–Weyl Hamiltonian formulation takes place on the bundle
$J^{1}E^{*}$ which serves as the multimomentum phase space and has coordinates
$(x^{\mu},\phi,\pi,p_{\phi}^{\mu},p^{\mu}_{\pi})$, where the multimomenta
$p_{\phi}^{\mu}$, $p^{\mu}_{\pi}$ are obtained from the Legendre map
$\mathscr{FL}:J^{1}E\rightarrow J^{1}E^{*}$,

$\mathscr{FL}^{*}p_{\phi}^{i}=\frac{\partial\mathscr{L}}{\partial\phi_{i}}=-\phi^{i}\quad,\quad\mathscr{FL}^{*}p_{\phi}^{0}=\frac{\partial\mathscr{L}}{\partial\phi_{0}}=\pi\quad,\quad\mathscr{FL}^{*}p_{\pi}^{\mu}=\frac{\partial\mathscr{L}}{\partial\pi_{\mu}}=0\
.$ (34)

The first equation above is invertible and gives the relation among the
multimomenta $\pi^{i}$ and the multivelocities $\phi^{i}$ and is therefore not
a constraint. The second and third equations give the primary constraints
$p_{\phi}^{0}-\pi=0$, $p^{\mu}_{\pi}=0$, which define the submanifold
$\jmath_{0}:P_{0}\hookrightarrow J^{1}E^{*}$ with local coordinates
$(x^{\mu},\phi,\pi,p_{\phi}^{i})$. Note that, by Proposition 2, the null
vectors (26) are given as

$\gamma^{(\phi)}_{\mu}=\mathscr{FL}^{*}\displaystyle\frac{\partial{}}{\partial{p^{\mu}_{\phi}}}(p^{0}_{\phi}-\pi)\quad,\quad\gamma^{(\pi)}_{\mu}=\mathscr{FL}^{*}\displaystyle\frac{\partial{}}{\partial{p^{\mu}_{\phi}}}\left(p^{\mu}_{\pi}\right)\
.$

Furthermore, $\mathscr{FL}$ maps onto the primary constraint submanifold
$P_{0}\subset J^{1}E^{*}$ as a consequence of the fact that the Lagrangian
function $\mathscr{L}$ is almost-regular and the restricted Legendre transform
$\mathscr{FL}_{0}$ is given by $\mathscr{FL}=\jmath_{0}\circ\mathscr{FL}_{0}$.

The De Donder–Weyl Hamiltonian $\mathscr{H}_{0}\in{\rm C}^{\infty}(P_{0})$
obtained from $E_{\mathscr{L}}=\mathscr{FL}_{0}^{*}\mathscr{H}_{0}$ is given
as,

$\mathscr{H}_{0}=\frac{1}{2}(\pi^{2}-p_{\phi}^{i}p_{\phi i})\ .$

The Hamilton–Cartan forms obtained from (5) are

$\displaystyle\Theta_{\mathscr{H}}^{0}$ $\displaystyle=$
$\displaystyle\pi\text{d}\phi\wedge\text{d}^{m-1}x_{0}+p_{\pi}^{i}\text{d}\phi\wedge\text{d}^{m-1}x_{i}-\frac{1}{2}(\pi^{2}-p_{\phi}^{i}p_{\phi
i})\text{d}^{m}x\in{\mit\Omega}^{m}(P_{0})\ ,$
$\displaystyle\Omega_{\mathscr{H}}^{0}=-\text{d}\Theta_{\mathscr{H}}^{0}$
$\displaystyle=$
$\displaystyle\text{d}\phi\wedge\text{d}\pi\wedge\text{d}^{m-1}x_{0}+\text{d}\phi\wedge\text{d}p_{\phi}^{i}\wedge\text{d}^{m-1}x_{i}+(\pi\text{d}\pi-
p_{\phi i}\text{d}p_{\phi}^{i})\wedge\text{d}^{m}x$ $\displaystyle\equiv$
$\displaystyle\widetilde{\Omega}_{\mathscr{H}}^{0}+{\rm
d}\mathscr{H}_{0}\wedge\text{d}^{m}x\in{\mit\Omega}^{m+1}(P_{0})\ .$

Now, in order to produce the Hamiltonian field equations, take a locally
decomposable multivector field $\textbf{X}_{\mathscr{H}}^{0}=X_{0}\wedge
X_{1}\wedge\dotsb\wedge X_{m-1}\in\mathfrak{X}^{m}(P_{0})$ satisfying the
normalized transverse condition $i(\textbf{X}_{\mathscr{H}}^{0}){\rm
d}^{m}x=1$, whose components are written as

$X_{\mu}=\frac{\partial}{\partial
x^{\mu}}+A_{\mu}\frac{\partial}{\partial\phi}+B_{\mu}\frac{\partial}{\partial\pi}+C^{i}_{\mu}\frac{\partial}{\partial
p_{\phi}^{i}}\ .$

Then, the field equation
$i(\textbf{X}_{\mathscr{H}}^{0})\Omega_{\mathscr{H}}^{0}=0$ yields

$\displaystyle B_{0}+C^{i}_{i}=0\ ,$ (35a) $\displaystyle A_{0}-\pi=0\ ,$
(35b) $\displaystyle A_{i}+p_{\phi i}=0\ ,$ (35c) $\displaystyle\pi
B_{\lambda}-p_{\phi
i}C_{\lambda}^{i}+A_{\lambda}B_{0}-A_{0}B_{\lambda}+A_{\lambda}C^{i}_{i}-A_{i}C_{\lambda}^{i}=0\
.$ (35d)

Similarly to the Lagrangian formalism, equation (35d) is a combination of the
first three equations and is thereby an identity. Equations (35b) and (35c)
determine the coefficients $A_{0}$ and $A_{i}$ respectively and equation (35a)
is a relation among the coefficients $B_{0}$ and $C_{i}^{i}$. As before, no
compatibility constraints appear; this can also be shown geometrically as in
the Lagrangian formulation by searching for compatibility constraints via the
application of Proposition 1: set $i(Z)\text{d}\mathscr{H}_{0}=0$ for every
$Z\in\text{ker}\,\widetilde{\Omega}_{\mathscr{H}}^{0}\cap\text{ker}\,\text{d}^{m}x$.
However, as

$\text{ker}\,\widetilde{\Omega}_{\mathscr{H}}^{0}\cap\text{ker}\,\text{d}^{m}x=\\{0\\}\
,$ (36)

it follows that condition (13) holds in agreement with the fact that there are
no compatibility constraints (as one would expect from the Lagrangian
formulation).

Now, taking $(x^{\mu},\phi(x),\pi(x),p^{i}_{\phi}(x))$ to be integral sections
of $\textbf{X}_{\mathscr{H}}^{0}$ given by, and hence

$A_{\mu}=\displaystyle\frac{\partial{\phi}}{\partial{x^{\mu}}}\quad,\quad
B_{\mu}=\displaystyle\frac{\partial{\pi}}{\partial{x^{\mu}}}\quad,\quad
C^{i}_{\mu}=\displaystyle\frac{\partial{p^{i}_{\phi}}}{\partial{x^{\mu}}}\ ,$

the field equations above become

$\displaystyle\dot{\pi}+\displaystyle\frac{\partial{p^{i}_{\phi}}}{\partial{x^{i}}}=0\
,$ (37a) $\displaystyle\dot{\phi}-\pi=0\ ,$ (37b)
$\displaystyle\displaystyle\frac{\partial{\phi}}{\partial{x^{i}}}+p_{\phi}^{i}=0\
.$ (37c)

Plugging (37b) and (37c) into (37a) yields

$\ddot{\phi}-\partial_{i}\partial^{i}\phi=0\ ,$

which is again the Klein–Gordon equation as expected and thereby displaying
the equivalence between the Lagrangian and the Hamiltonian formalisms.

####  3.1.3 Symmetries

The Lagrangian density associated with the Lagrangian function (25),

$\textbf{L}=\mathscr{L}\text{d}^{m}x=\left(\pi\phi_{0}-\frac{1}{2}\pi^{2}-\frac{1}{2}\phi_{i}\phi^{i}\right)\text{d}^{m}x\
,$

is invariant under Lorentz transformations $\Lambda^{\mu}_{\ \nu}$:

$x^{\prime\mu}=\Lambda^{\mu}_{\ \nu}x^{\nu}\ ,\
\phi^{\prime}(x^{\prime})=\phi(x)\ ,\ \pi^{\prime}(x^{\prime})=\pi(x)\ ,\
\phi^{\prime}_{\mu}(x^{\prime})=\frac{\partial x^{\nu}}{\partial
x^{\prime\mu}}\phi_{\nu}(x)\ ,\ \pi^{\prime}_{\mu}(x^{\prime})=\frac{\partial
x^{\nu}}{\partial x^{\prime\mu}}\pi_{\nu}(x)\ .$

Let the Lorentz transformations on $M$ be infinitesimal so that the Lorentz
matrices take the form $\Lambda^{\mu}_{\ \nu}=\delta^{\mu}_{\
\nu}+\epsilon^{\mu}_{\ \nu}$, where $\epsilon_{\mu\nu}$ is the infinitesimal
antisymmetric Lorentz matrix. It follows that the spacetime coordinates
transform as $x^{\prime\mu}=x^{\mu}+\xi^{\mu}(x)$ with
$\xi^{\mu}(x)=\epsilon^{\mu}_{\ \nu}x^{\nu}$ and the resulting infinitesimal
Lorentz transformations of the fields and their spacetime derivatives are
given by the Lie derivatives of the first–jet prolongations
$j^{1}\phi:M\rightarrow J^{1}E$ with respect to the vector field
$\xi=-\xi^{\mu}(x)\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}\in\mathfrak{X}(M)$
as defined in (24):

$\displaystyle\begin{split}\delta\phi(x)&\equiv\phi^{\prime}(x)-\phi(x)=-\epsilon^{\mu}_{\
\nu}x^{\nu}\displaystyle\frac{\partial{\phi}}{\partial{x^{\mu}}}\ ,\\\
\delta\pi(x)&\equiv\pi^{\prime}(x)-\pi(x)=-\epsilon^{\mu}_{\
\nu}x^{\nu}\displaystyle\frac{\partial{\pi}}{\partial{x^{\mu}}}-\epsilon^{\alpha}_{\
0}\phi_{\alpha}\ ,\\\
\delta\phi_{\mu}(x)&\equiv\phi^{\prime}_{\mu}(x)-\phi_{\mu}(x)=-\epsilon^{\alpha}_{\
\beta}x^{\beta}\displaystyle\frac{\partial{\phi_{\mu}}}{\partial{x^{\alpha}}}-\epsilon^{\alpha}_{\
\mu}\phi_{\alpha}\ ,\\\
\delta\pi_{\mu}(x)&\equiv\pi^{\prime}_{\mu}(x)-\pi_{\mu}(x)=-\epsilon^{\alpha}_{\
\beta}x^{\beta}\displaystyle\frac{\partial{\pi_{\mu}}}{\partial{x^{\alpha}}}-\epsilon^{\alpha}_{\
\mu}\pi_{\alpha}\ ,\end{split}$ (38)

which give
$\displaystyle\delta\mathscr{L}(x)=-\displaystyle\frac{\partial{}}{\partial{x^{\alpha}}}\left(\epsilon^{\alpha}_{\
\beta}x^{\beta}\mathscr{L}(x)\right)$. The vector field which generates the
Lorentz transformations above on $J^{1}E$ is

$\displaystyle\begin{split}X_{\Lambda}&=-\epsilon^{\mu}_{\
\nu}x^{\nu}\frac{\partial}{\partial x^{\mu}}+\epsilon^{\nu}_{\
0}\phi_{\nu}\frac{\partial}{\partial\pi}+\epsilon^{\nu}_{\
\mu}\phi_{\nu}\frac{\partial}{\partial\phi_{\mu}}+\epsilon^{\nu}_{\
\mu}\pi_{\nu}\frac{\partial}{\partial\pi_{\mu}}\\\ &=-\epsilon^{\mu}_{\
\nu}x^{\nu}\frac{\partial}{\partial x^{\mu}}+\epsilon^{i}_{\
0}\phi_{i}\frac{\partial}{\partial\pi}+\epsilon^{i}_{\
0}\phi_{i}\frac{\partial}{\partial\phi_{0}}+\left(\epsilon^{0}_{\
i}\phi_{0}+\epsilon^{j}_{\
i}\phi_{j}\right)\frac{\partial}{\partial\phi_{i}}+\epsilon^{\nu}_{\
\mu}\pi_{\nu}\frac{\partial}{\partial\pi_{\mu}}\in\mathfrak{X}(J^{1}E)\
.\end{split}$

It follows that $\mathpzc{L}_{X_{\Lambda}}\textbf{L}=0$ holds everywhere on
$J^{1}E$ while

$\mathpzc{L}_{X_{\Lambda}}\Theta_{\mathscr{L}}=\epsilon^{0i}(\pi-\phi_{0})(\text{d}\phi\wedge\text{d}^{m-1}x_{i}-\phi_{i}\text{d}^{m}x)\
.$

Also, observe that $X_{\Lambda}$ is tangent to the constraint submanifold
$S_{1}\subset J^{1}E$ given by the constraint $\phi_{0}-\pi=0$ as desired
since $\mathpzc{L}_{X_{\Lambda}}(\phi_{0}-\pi)=0$. However, $X_{\Lambda}$ is
not projectable onto $E$ as a result of the component attached to
$\displaystyle\frac{\partial{}}{\partial{\pi}}$; it is for this reason that
the transformations (38) can be interpreted as the Lie derivatives of
$j^{1}\phi$ only (as the sections $\phi$ do not provide the term associated
with the second term of $X_{\Lambda}$ that is not projectable onto $E$).
Furthermore, the vector field $X_{\Lambda}\in\mathfrak{X}(J^{1}E)$ is
evidently $\mathscr{FL}_{0}$-projectable only on the constraint submanifold
$S_{1}$ on which

$\left.X_{\Lambda}\right|_{S_{1}}=-\epsilon^{\mu}_{\
\nu}x^{\nu}\frac{\partial}{\partial x^{\mu}}+\epsilon^{i}_{\
0}\phi_{i}\frac{\partial}{\partial\pi}+\left(\epsilon^{0}_{\
i}\pi+\epsilon^{j}_{\
i}\phi_{j}\right)\frac{\partial}{\partial\phi_{i}}+\epsilon^{\nu}_{\
\mu}\pi_{\nu}\frac{\partial}{\partial\pi_{\mu}}\ .$

Now $\mathpzc{L}_{X_{\Lambda}}\Theta_{\mathscr{L}}|_{S_{1}}=0$, so the exact
Noether symmetry exists only on the constraint submanifold $S_{1}$. Then, the
corresponding momentum map
$J_{\mathscr{L}}(X_{\Lambda})=-i(X_{\Lambda})\Theta_{\mathscr{L}}$ also exists
only on the points of $S_{1}$ and is given by

$J_{\mathscr{L}}(X_{\Lambda})|_{S_{1}}=-\epsilon^{i}_{\
\beta}x^{\beta}\pi\text{d}\phi\wedge\text{d}^{m-2}x_{0i}+\epsilon^{j}_{\
\beta}x^{\beta}\phi^{i}\text{d}\phi\wedge\text{d}^{m-2}x_{ij}-\frac{1}{2}\epsilon^{i}_{\
\beta}x^{\beta}(\pi^{2}-\phi_{j}\phi^{j})\text{d}^{m-1}x_{i}\ .$

In the Hamiltonian formalism on $P_{0}\subset J^{1}E^{*}$, the symmetry
transformations are generated by
$Y_{\Lambda}=\mathscr{FL}_{0*}\left.X_{\Lambda}\right|_{S_{1}}\in\mathfrak{X}(P_{0})$
which is given as

$Y_{\Lambda}=-\epsilon^{\mu}_{\ \nu}x^{\nu}\frac{\partial}{\partial
x^{\mu}}-\epsilon_{0i}p_{\phi}^{i}\frac{\partial}{\partial\pi}-\left(\epsilon^{i}_{\
0}\pi+\epsilon^{i}_{\ j}p_{\phi}^{j}\right)\frac{\partial}{\partial
p_{\phi}^{i}}\ .$

The momentum map
$J_{\mathscr{H}_{0}}(Y_{\Lambda})=-i(Y_{\Lambda})\Theta_{\mathscr{H}}^{0}\in{\mit\Omega}^{m-1}(P_{0})$
is given by

$\displaystyle J_{\mathscr{H}_{0}}(Y_{\Lambda})=-\epsilon^{i}_{\
\beta}x^{\beta}\pi\text{d}\phi\wedge\text{d}^{m-2}x_{0i}-\epsilon^{j}_{\
\beta}x^{\beta}p_{\phi}^{i}\text{d}\phi\wedge\text{d}^{m-2}x_{ij}-\frac{1}{2}\epsilon^{i}_{\
\beta}x^{\beta}(\pi^{2}-p_{\phi j}p_{\phi}^{j})\text{d}^{m-1}x_{i}\ .$

###  3.2 Carrollian Electric Scalar Field Theory

The electric Carrollian contraction [27] of the canonical Klein–Gordon
Lagrangian is performed by making the field redefinition given by
$\phi(x)\rightarrow c\phi(x)$, $\pi(x)\rightarrow\frac{1}{c}\pi(x)$, and
taking the limit $c\rightarrow 0$ for the speed of light. It follows that the
Minkowski metric becomes degenerate as

$\text{d}s^{2}=\eta_{\mu\nu}\text{d}x^{\mu}\otimes\text{d}x^{\nu}=-c^{2}\text{d}x^{0}\otimes\text{d}x^{0}+\delta_{ij}\text{d}x^{i}\otimes\text{d}x^{j}\
\longrightarrow\ \delta_{ij}\text{d}x^{i}\otimes\text{d}x^{j}\ .$ (39)

####  3.2.1 Lagrangian Formulation

The Lagrangian function $\mathscr{L}\in C^{\infty}(J^{1}E)$ obtained from the
electric Carrollian contraction of the canonical Klein–Gordon Lagrangian is

$\mathscr{L}=\pi\phi_{0}-\frac{1}{2}\pi^{2}\ ,$ (40)

which is clearly singular since

$\frac{\partial^{2}\mathscr{L}}{\partial\phi_{\mu}\partial\phi_{\nu}}=0\quad,\quad\frac{\partial^{2}\mathscr{L}}{\partial\pi_{\mu}\partial\pi_{\nu}}=0\
,$

whose null vectors are given as

$\gamma^{(\phi)}_{\mu}=\begin{pmatrix}1\\\ 1\\\ \vdots\\\
1\end{pmatrix}\quad,\quad\gamma^{(\pi)}_{\mu}=\begin{pmatrix}1\\\ 1\\\
\vdots\\\ 1\end{pmatrix}\ ,$ (41)

respectively. The Lagrangian energy function and the Poincaré–Cartan forms are

$\displaystyle E_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\pi^{2}\in C^{\infty}(J^{1}E)\ ,$
$\displaystyle\Theta_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle\pi\text{d}\phi\wedge\text{d}^{m-1}x_{0}-\frac{1}{2}\pi^{2}\text{d}^{m}x\in\Omega^{m}(J^{1}E)\
,$ $\displaystyle\Omega_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle\text{d}\phi\wedge\text{d}\pi\wedge\text{d}^{n-1}x_{0}+\pi\text{d}\pi\wedge\text{d}^{m}x\equiv\widetilde{\Omega}_{\mathscr{L}}+\text{d}E_{\mathscr{L}}\wedge\text{d}^{m}x\in\Omega^{m+1(}J^{1}E)\
.$

Since the Lagrangian function (40) is singular, $\Omega_{\mathscr{L}}$ is a
premultisymplectic form. As before, the field equations are obtained by using
locally decomposable multivector fields $\textbf{X}_{\mathscr{L}}=X_{0}\wedge
X_{1}\wedge\dotsb\wedge X_{m-1}\in\mathfrak{X}^{m}(J^{1}E)$ such that
$i(\textbf{X}_{\mathscr{L}}){\rm d}^{m}x=1$, so it follows that their
components are given by

$X_{\mu}=\frac{\partial}{\partial
x^{\mu}}+A_{\mu}\frac{\partial}{\partial\phi}+B_{\mu}\frac{\partial}{\partial\pi}+C_{\mu
0}\frac{\partial}{\partial\phi_{0}}+C_{\mu
i}\frac{\partial}{\partial\phi_{i}}+D_{\mu\nu}\frac{\partial}{\partial\pi_{\nu}}\
.$

Then the field equation is

$0=i(\textbf{X}_{\mathscr{L}})\Omega_{\mathscr{L}}=A_{0}\text{d}\pi-
B_{0}\text{d}\phi-\pi\text{d}\pi+(A_{\lambda}B_{0}-B_{\lambda}A_{0}+\pi
B_{\lambda})\text{d}x^{\lambda}\ .$

which leads to

$\displaystyle-B_{0}=0\ ,$ (42a) $\displaystyle A_{0}-\pi=0\ ,$ (42b)
$\displaystyle A_{\lambda}B_{0}-B_{\lambda}A_{0}+\pi B_{\lambda}=0\ .$ (42c)

Equations (42a) and (42b) determine the coefficients $B_{0}$ and $A_{0}$;
equation (42c) is an identity as a result of the previous equations as usual.
There are no compatibility constraints, which is corroborated by applying
Proposition 1:

$\text{ker}\ \widetilde{\Omega}_{\mathscr{L}}\cap\text{ker}\
\text{d}^{m}x=\left\langle\frac{\partial}{\partial\phi_{\mu}},\frac{\partial}{\partial\pi_{\mu}}\right\rangle,$

and $\displaystyle
i\left(\frac{\partial}{\partial\phi_{\mu}}\right)\text{d}E_{\mathscr{L}}=0=i\left(\frac{\partial}{\partial\pi_{\mu}}\right)\text{d}E_{\mathscr{L}}$
is satisfied a priori without producing any constraints.

Imposing the sopde condition, $A_{\mu}=\phi_{\mu}$, $B_{\mu}=\pi_{\mu}$, the
field equations (42a) and (42b) produce two sopde constraints

$\pi_{0}=0\quad,\quad\phi_{0}-\pi=0\ ,$ (43)

which define the constraint submanifold $S_{1}\hookrightarrow J^{1}E$. The
tangency condition for ${\bf X}_{\mathscr{L}}$ on $S_{1}$ is imposec by

$\mathpzc{L}_{X_{\mu}}(\pi_{0})=D_{\mu
0}=0\quad,\quad\mathpzc{L}_{X_{\mu}}(\phi_{0}-\pi)=C_{\mu
0}-B_{\mu}=0\quad\mbox{\rm(on $S_{1}$)}\ .$ (44)

It is evident above that no new constraints arise, hence, $S_{1}$ is the final
constraint submanifold in the Lagrangian formalism.

Consider now $(x^{\mu},\phi(x),\pi(x),\phi_{\nu}(x),\pi_{\nu}(x))$ which are
integral sections of the multivector fields $\textbf{X}_{\mathscr{L}}$ and
therefore satisfy (32). Then, the sopde constraints (43) yield

$\dot{\pi}=0\quad,\quad\dot{\phi}-\pi=0\ ,$ (45)

which can be combined giving

$\ddot{\phi}=0\ ,$

as in [25]. The tangency conditions (44) lead to

$\displaystyle\frac{\partial{\pi_{0}}}{\partial{x^{\mu}}}\equiv\displaystyle\frac{\partial{\dot{\pi}}}{\partial{x^{\mu}}}=0\quad,\quad\displaystyle\frac{\partial{\pi}}{\partial{x^{\mu}}}=\displaystyle\frac{\partial{\phi_{0}}}{\partial{x^{\mu}}}\equiv\displaystyle\frac{\partial{\dot{\phi}}}{\partial{x^{\mu}}}\quad;\quad\mbox{\rm(on
$S_{1}$)}\ .$

These equations are fulfilled by the solutions to the field equations, in
particular, as a consequence of the second equation in (45).

####  3.2.2 Hamiltonian Formulation

The multimomenta obtained from the Legendre map are

$\mathscr{FL}^{*}p_{\phi}^{0}\equiv\frac{\partial\mathscr{L}}{\partial\phi_{0}}=\pi\quad,\quad\mathscr{FL}^{*}p_{\phi}^{i}=\frac{\partial\mathscr{L}}{\partial\phi_{i}}=0\quad,\quad\mathscr{FL}^{*}p^{\mu}_{\pi}=\frac{\partial\mathscr{L}}{\partial\pi_{\mu}}=0\
,$

and give the primary constraints $p_{\phi}^{0}-\pi=0$, $p^{i}_{\phi}=0$, and
$p^{\mu}_{\pi}=0$, which define the primary constraint submanifold
$\jmath_{0}:P_{0}\hookrightarrow J^{1}E^{*}$ with local coordinates
$(x^{\mu},\phi,\pi)$ and the restricted Legendre transform $\mathscr{FL}_{0}$
is given by $\mathscr{FL}=\jmath_{0}\circ\mathscr{FL}_{0}$ as before. Once
again, the primary constraints can be used to compute the null vectors (41) by
using equation (14) given in Proposition 2. The Lagrangian function
$\mathscr{L}$ is almost-regular and the De Donder–Weyl Hamiltonian
$\mathscr{H}_{0}\in{\rm C}^{\infty}(P_{0})$ obtained from
$E_{\mathscr{L}}=\mathscr{FL}_{0}^{*}\mathscr{H}_{0}$ is

$\mathscr{H}_{0}=\frac{1}{2}\pi^{2}\ ,$

and the Hamilton-Cartan forms are

$\displaystyle\Theta_{\mathscr{H}}^{0}$ $\displaystyle=$
$\displaystyle\pi\text{d}\phi\wedge\text{d}^{m-1}x_{0}-\frac{1}{2}\pi^{2}\text{d}^{m}x\in{\mit\Omega}^{m}(P_{0})\
,$ $\displaystyle\Omega_{\mathscr{H}}^{0}$ $\displaystyle=$
$\displaystyle\text{d}\phi\wedge\text{d}\pi\wedge\text{d}^{m-1}x_{0}+\pi\text{d}\pi\wedge\text{d}^{m}x\equiv\widetilde{\Omega}_{\mathscr{H}}^{0}+{\rm
d}\mathscr{H}_{0}\wedge\text{d}^{m}x\in{\mit\Omega}^{m+1}(P_{0})\ .$

The locally decomposable multivector fields
$\textbf{X}_{\mathscr{H}}^{0}=X_{0}\wedge X_{1}\wedge\dotsb\wedge
X_{m-1}\in\mathfrak{X}^{m}(P_{0})$ satisfying the normalized transverse
condition $i(\textbf{X}_{\mathscr{H}}^{0}){\rm d}^{m}x=1$ used to produce the
field equations have the following components:

$X_{\mu}=\frac{\partial}{\partial
x^{\mu}}+A_{\mu}\frac{\partial}{\partial\phi}+B_{\mu}\frac{\partial}{\partial\pi}\
.$

The field equations obtained from
$i(\textbf{X}_{\mathscr{H}}^{0})\Omega_{\mathscr{H}}^{0}=0$ are

$\displaystyle B_{0}=0\ ,$ (46a) $\displaystyle A_{0}-\pi=0\ ,$ (46b)
$\displaystyle A_{\lambda}B_{0}-A_{0}B_{\lambda}+\pi B_{\lambda}=0\ .$ (46c)

Once again, using (46a) and (46b), the third equation (46c) is an identity.
These equations do not produce any compatibility constraints. This again is in
agreement with Proposition 1 which dictates that compatibility constraints may
be produced by $i(Z)\text{d}\mathscr{H}_{0}=0$, for every
$Z\in\text{ker}\,\widetilde{\Omega}_{\mathscr{H}}^{0}\cap\text{ker}\,\text{d}^{m}x$,
and as

$\text{ker}\,\widetilde{\Omega}_{\mathscr{H}}^{0}\cap\text{ker}\,\text{d}^{m}x=\\{0\\}$
(47)

is obtained computationally, it is confirmed that no compatibility constraints
exist.

Upon working on $(x^{\mu},\phi(x),\pi(x))$ which are integral sections of the
multivector fields $\textbf{X}_{\mathscr{H}}^{0}$, and hence

$A_{\mu}=\displaystyle\frac{\partial{\phi}}{\partial{x^{\mu}}}\quad,\quad
B_{\mu}=\displaystyle\frac{\partial{\pi}}{\partial{x^{\mu}}}\ ,$

it follows that the field equations (46a) and (46b) take the form

$\dot{\pi}=0\quad,\quad\dot{\phi}-\pi=0\ .$

Notice that plugging in the second equation into the first above gives

$\ddot{\phi}=0\ .$

The equivalence between the Lagrangian and the Hamiltonian formalisms again is
evident.

####  3.2.3 Symmetries

The spacetime symmetry of the Carroll spacetime geometry are the Carroll
transformations [27]:

$x^{\prime 0}=x^{0}+b_{k}x^{k}\quad,\quad x^{\prime i}=x^{i}+\epsilon^{i}_{\
j}x^{j}\quad\Rightarrow\quad x^{\prime\mu}=C^{\mu}_{\ \nu}x^{\nu}\ ,$ (48)

where $C^{0}_{\ 0}=1$, $C^{0}_{\ k}=b_{k}$, $C^{k}_{\ 0}=0$, $(C^{i}_{\ j})\in
O(d)$. These transformations can be represented infinitesimally by

$C^{\mu}_{\ \nu}=\delta^{\mu}_{\ \nu}+\epsilon^{\mu}_{\ \nu}\ ,$ (49)

so that now $b_{k}=\epsilon^{0}_{\ k}$ is infinitesimal. Up to linear terms in
$\phi^{\prime}(x^{\prime})=\phi(x)$ and $\pi^{\prime}(x^{\prime})=\pi(x)$, the
infinitesimal transformations of the fields,
$\delta\phi(x)\equiv\phi^{\prime}(x)-\phi(x)$ and
$\delta\pi(x)\equiv\pi^{\prime}(x)-\pi(x)$, are

$\displaystyle\delta\phi(x)$ $\displaystyle=-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\phi}}{\partial{x^{\alpha}}}=-b_{k}x^{k}\displaystyle\frac{\partial{\phi}}{\partial{x^{0}}}-\epsilon^{i}_{\
j}x^{j}\displaystyle\frac{\partial{\phi}}{\partial{x^{i}}}\ ,$
$\displaystyle\delta\pi(x)$ $\displaystyle=-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\pi}}{\partial{x^{\alpha}}}=-b_{k}x^{k}\displaystyle\frac{\partial{\pi}}{\partial{x^{0}}}-\epsilon^{i}_{\
j}x^{j}\displaystyle\frac{\partial{\pi}}{\partial{x^{i}}}\ .$

Furthermore

$\phi^{\prime}_{\mu}(x^{\prime})=\frac{\partial x^{\nu}}{\partial
x^{\prime\mu}}\phi_{\nu}(x)\ \Longleftrightarrow\ \phi^{\prime}_{\mu}(C\cdot
x)=(C^{-1})_{\ \mu}^{\nu}\phi_{\nu}(x)\ ,$

where $C\equiv(C_{\ \nu}^{\mu})$ and $C\cdot x\equiv C_{\ \nu}^{\mu}x^{\nu}$;
hence,

$\displaystyle\delta\phi_{\mu}(x)$
$\displaystyle\equiv\phi^{\prime}_{\mu}(x)-\phi_{\mu}(x)=-\epsilon^{\nu}_{\
\mu}\phi_{\nu}-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\phi_{\mu}}}{\partial{x^{\alpha}}}\
,$ $\displaystyle\delta\pi_{\mu}(x)$
$\displaystyle\equiv\pi^{\prime}_{\mu}(x)-\pi_{\mu}=-\epsilon^{\nu}_{\
\mu}\pi_{\nu}-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\pi_{\mu}}}{\partial{x^{\alpha}}}\
.$

Component-wise,

$\displaystyle\delta\phi_{0}(x)$ $\displaystyle=-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\phi_{0}}}{\partial{x^{\alpha}}}=-b_{k}x^{k}\displaystyle\frac{\partial{\phi_{0}}}{\partial{x^{0}}}-\epsilon^{k}_{\
j}x^{j}\displaystyle\frac{\partial{\phi_{0}}}{\partial{x^{k}}}\ ,$
$\displaystyle\delta\phi_{i}(x)$ $\displaystyle=-\epsilon^{\nu}_{\
i}\phi_{\nu}-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\phi_{i}}}{\partial{x^{\alpha}}}=-b_{i}\phi_{0}-\epsilon^{k}_{\
i}\phi_{k}-b_{k}x^{k}\displaystyle\frac{\partial{\phi_{i}}}{\partial{x^{0}}}-\epsilon^{k}_{\
j}x^{j}\displaystyle\frac{\partial{\phi_{i}}}{\partial{x^{k}}}\ ,$

and similarly for the components of $\delta\pi_{\mu}(x)$. Under the
transformations presented above, the Lagrangian transforms as
$\delta\mathscr{L}(x)=-\partial_{\alpha}\left(\epsilon^{\alpha}_{\
\beta}x^{\beta}\mathscr{L}(x)\right)$. These transformations are produced by
the Lie derivatives of the local sections $\phi:M\rightarrow E$ and
$j^{1}\phi:M\rightarrow J^{1}E$ in the direction of a vector field
$\xi=-\xi^{\mu}(x)\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}\in\mathfrak{X}(M)$
which generates the Carroll transformations, where now
$\xi^{\mu}(x)=\epsilon^{\mu}_{\ \nu}x^{\nu}$ is given by (48) and (49). The
vector field which generates the Carroll transformations on the configuration
bundle $E$, obtained from (23), is

$\xi_{E}=-\epsilon^{\mu}_{\ \nu}x^{\nu}\frac{\partial}{\partial
x^{\mu}}\in\mathfrak{X}(E)\ ,$

while the canonical lift of $\xi_{E}$ to $J^{1}E$ which generates the Carroll
transformations on the multivelocity phase space is given by

$\displaystyle\begin{split}X&=-\epsilon^{\mu}_{\
\nu}x^{\nu}\frac{\partial}{\partial x^{\mu}}+\epsilon^{\nu}_{\
\mu}\phi_{\nu}\frac{\partial}{\partial\phi_{\mu}}+\epsilon^{\nu}_{\
\mu}\pi_{\nu}\frac{\partial}{\partial\pi_{\mu}}\\\
&=-b_{i}x^{i}\frac{\partial}{\partial x^{0}}-\epsilon^{i}_{\
j}x^{j}\frac{\partial}{\partial x^{i}}+\left(b_{i}\phi_{0}+\epsilon_{\
i}^{j}\phi_{j}\right)\frac{\partial}{\partial\phi_{i}}+\left(b_{i}\pi_{0}+\epsilon_{\
i}^{j}\pi_{j}\right)\frac{\partial}{\partial\pi_{i}}\in\mathfrak{X}(J^{1}E)\
.\end{split}$

Evidently $X$ is tangent to the constraint submanifold $S_{1}$ since
$\mathpzc{L}_{X}(\pi-\phi_{0})=0$ and $\mathpzc{L}_{X}\pi_{0}=0$. Finally,
since
$\mathpzc{L}_{X}\textbf{L}=0\Rightarrow\mathpzc{L}_{X}\Theta_{\mathscr{L}}=0$,
the momentum map $J_{\mathscr{L}}(X)$ corresponding to the Carroll spacetime
transformations in the Lagrangian setting is given by

$J_{\mathscr{L}}(X)=-i(X)\Theta_{\mathscr{L}}=-\epsilon^{i}_{\
j}x^{j}\pi\text{d}\phi\wedge\text{d}^{m-2}x_{0i}-\frac{1}{2}\pi^{2}\left(b_{i}x^{i}\text{d}^{m-1}x_{0}+\epsilon^{i}_{\
j}x^{j}\text{d}^{m-1}x_{i}\right)\in{\mit\Omega}^{m-1}(J^{1}E)\ .$

In the Hamiltonian setting, which takes place on $P_{0}\subset J^{1}E^{*}$,
the momentum map $J_{\mathscr{H}_{0}}(Y)\in{\mit\Omega}^{m-1}(P_{0})$ is
constructed using the vector field $Y\in\mathfrak{X}(P_{0})$ given by

$Y=\mathscr{FL}_{0*}X=-\epsilon^{\mu}_{\
\nu}x^{\nu}\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}=-b_{i}x^{i}\frac{\partial}{\partial
x^{0}}-\epsilon^{i}_{\ j}x^{j}\frac{\partial}{\partial x^{i}}\ .$

As in the Lagrangian setting, $\mathpzc{L}_{Y}\Theta^{0}_{\mathscr{H}}=0$, it
follows that $J_{\mathscr{H}_{0}}(Y)$ is given by,

$J_{\mathscr{H}_{0}}(Y)=-i(Y)\Theta_{\mathscr{H}}^{0}=-\epsilon^{i}_{\
j}x^{j}\pi\,\text{d}\phi\wedge\text{d}^{m-2}x_{0i}-\frac{1}{2}\pi^{2}\left(b_{i}x^{i}\text{d}^{m-1}x_{0}+\epsilon^{i}_{\
j}x^{j}\text{d}^{m-1}x_{i}\right)\ .$

###  3.3 Carrollian Magnetic Scalar Field Theory

The magnetic Carrollian contraction [27] of the canonical Klein–Gordon
Lagrangian is performed by reinserting the factors of $c$ (speed of light)
into the Lagrangian (25) and taking the limit $c\rightarrow 0$. The Minkowski
metric (39) becomes degenerate as in the electric Carrollian scalar field
theory.

####  3.3.1 Lagrangian Formulation

The Lagrangian function $\mathscr{L}\in C^{\infty}(J^{1}E)$ obtained from
taking the magnetic Carrollian contraction of the canonical Klein–Gordon
Lagragian (25) is

$\mathscr{L}=\pi\phi_{0}-\frac{1}{2}\phi_{i}\phi^{i}\ .$

This Lagrangian is singular since the components of the Hessian matrix (with
respect to the multivelocities) are

$\frac{\partial^{2}\mathscr{L}}{\partial\phi_{0}\partial\phi_{0}}=0\quad,\quad\frac{\partial^{2}\mathscr{L}}{\partial\phi_{0}\partial\phi_{i}}=0\quad,\quad\frac{\partial^{2}\mathscr{L}}{\partial\phi_{i}\partial\phi_{j}}=-\delta^{ij}\quad,\quad\frac{\partial^{2}\mathscr{L}}{\partial\pi_{\mu}\partial\pi_{\nu}}=0\
.$

The null vectors of the Hessians
$\displaystyle\frac{\partial{{}^{2}\mathscr{L}}}{\partial{\phi_{\mu}\partial\phi_{\nu}}}$
and
$\displaystyle\frac{\partial{{}^{2}\mathscr{L}}}{\partial{\pi_{\mu}\partial\pi_{\nu}}}$
are

$\gamma^{(\phi)}_{\mu}=\begin{pmatrix}1\\\ 0\\\ \vdots\\\
0\end{pmatrix}\quad,\quad\gamma^{(\pi)}_{\mu}=\begin{pmatrix}1\\\ 1\\\
\vdots\\\ 1\end{pmatrix}\ ,$ (50)

respectively. The Lagrangian energy function and the Poincaré–Cartan forms are
now

$\displaystyle E_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\phi_{i}\phi^{i}\in C^{\infty}(J^{1}E)\ ,$
$\displaystyle\Theta_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle\pi\text{d}\phi\wedge\text{d}^{m-1}x_{0}-\phi^{i}\text{d}\phi\wedge\text{d}^{m-1}x_{i}+\frac{1}{2}\phi_{i}\phi^{i}\text{d}^{m}x\in\Omega^{m}(J^{1}E)\
,$ $\displaystyle\Omega_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle\text{d}\phi\wedge\text{d}\pi\wedge\text{d}^{m-1}x_{0}+\text{d}\phi^{i}\wedge\text{d}\phi\wedge\text{d}^{m-1}x_{i}+\phi^{i}\text{d}\phi_{i}\wedge\text{d}^{m}x\equiv\widetilde{\Omega}_{\mathscr{L}}+\text{d}E_{\mathscr{L}}\wedge\text{d}^{m}x\in\Omega^{m+1}(J^{1}E),$

and $\Omega_{\mathscr{L}}$ is a premultisymplectic form. Taking locally
decomposable multivector fields $\textbf{X}_{\mathscr{L}}=X_{0}\wedge
X_{1}\wedge\dotsb\wedge X_{m-1}\in\mathfrak{X}^{m}(J^{1}E)$ satisfying
$i(\textbf{X}_{\mathscr{L}}){\rm d}^{m}x=1$, their components are given as

$X_{\mu}=\frac{\partial}{\partial
x^{\mu}}+A_{\mu}\frac{\partial}{\partial\phi}+B_{\mu}\frac{\partial}{\partial\pi}+C_{\mu
0}\frac{\partial}{\partial\phi_{0}}+C_{\mu
i}\frac{\partial}{\partial\phi_{i}}+D_{\mu\nu}\frac{\partial}{\partial\pi_{\nu}}\
,$

and the field equations obtained from
$i(\textbf{X}_{\mathscr{L}})\Omega_{\mathscr{L}}=0$ are

$\displaystyle C_{i}^{i}-B_{0}=0\ ,$ (51a) $\displaystyle A_{0}=0\ ,$ (51b)
$\displaystyle
A_{\lambda}B_{0}-B_{\lambda}A_{0}+C_{\lambda}^{i}A_{0}+A_{\lambda}C_{0}^{i}+\phi^{i}C_{\lambda
i}=0\ .$ (51c)

Equations (51a) and (51b) are relations for the coefficients of the
multivector fields while equation (42c) is an identity following from (51a)
and (51b) as usual. There are no compatibility constraints in agreement with
the geometric analysis presented in Proposition 1, which shows that

$\text{ker}\,\widetilde{\Omega}_{\mathscr{L}}\cap\text{ker}\,\text{d}^{m}x=\left\langle\frac{\partial}{\partial\phi_{0}},\frac{\partial}{\partial\pi_{\mu}}\right\rangle\
,$

and
$\displaystyle\mathop{i}\nolimits\left(\frac{\partial}{\partial\phi_{0}}\right)\text{d}E_{\mathscr{L}}=0=i\left(\frac{\partial}{\partial\pi_{\mu}}\right)\text{d}E_{\mathscr{L}}$
is satisfied without producing any constraints.

Then, upon imposing the sopde condition $A_{\mu}=\phi_{\mu}$,
$B_{\mu}=\pi_{\mu}$, equations (51a) and (51b) become

$C_{i}^{i}-\pi_{0}=0\quad,\quad\phi_{0}=0\ .$

The first equation gives a relation for the coefficients $C_{i}^{i}$, and the
second one is a sopde-constraint which defines the constraint submanifold
$S_{1}\hookrightarrow J^{1}E$.

Now impose the tangency condition of the multivector fields ${\bf
X}_{\mathscr{L}}$ to $S_{1}$,

$\mathpzc{L}_{X_{\mu}}(\phi_{0})=C_{\mu 0}=0\quad\mbox{\rm(on $S_{1}$)}\ .$
(52)

These are not constraints, but new relations among the coefficient of ${\bf
X}_{\mathscr{L}}$ and hence the final constraint submanifold in the Lagrangian
formalism is $S_{1}$. Then, on
$(x^{\mu},\phi(x),\pi(x),\phi_{\nu}(x),\pi_{\nu}(x))$, the integral sections
of the multivector fields $\textbf{X}_{\mathscr{L}}$ satisfy equations (32)
and the field equations take the form

$\partial_{i}\partial^{i}\phi-\dot{\pi}=0\quad,\quad\dot{\phi}=0\quad;\quad\mbox{\rm(on
$S_{1}$)}\ .$ (53)

For further details on these equations see [25, 27]. Furthermore, from (52) it
follows that, on $S_{1}$,

$\partial_{\mu}\phi_{0}\equiv\partial_{\mu}\dot{\phi}=0\ .$

The equations above are fulfilled by the solutions to the field equations, in
particular, as a consequence of the second equation in (53).

####  3.3.2 Hamiltonian Formulation

The Legendre map $\mathscr{FL}:J^{1}E\rightarrow J^{1}E^{*}$ gives the
multimomenta

$\mathscr{FL}^{*}p_{\phi}^{0}=\frac{\partial\mathscr{L}}{\partial\phi_{0}}=\pi\quad,\quad\mathscr{FL}^{*}p_{\phi}^{i}=\frac{\partial\mathscr{L}}{\partial\phi_{i}}=-\phi^{i}\quad,\quad\mathscr{FL}^{*}p^{\mu}_{\pi}=\frac{\partial\mathscr{L}}{\partial\pi_{\mu}}=0\
.$

It follows that $p_{\phi}^{0}-\pi=0$ and $p_{\pi}^{\mu}=0$ are primary
constraints which define the primary constraint submanifold
$P_{0}\hookrightarrow J^{1}E^{*}$ with local coordinates
$(x^{\mu},\phi,\pi,p_{\phi}^{i})$; consequently, the Lagrangian is again
almost-regular. As above, the null vectors (50) are given by the partial
derivatives of the primary constraints with respect to the multimomenta
according to (14) in Proposition 2. The restricted Legendre transform
$\mathscr{FL}_{0}$ is given by $\mathscr{FL}=\jmath_{0}\circ\mathscr{FL}_{0}$
as usual.

The De Donder–Weyl Hamiltonian $\mathscr{H}_{0}\in C^{\infty}(P_{0})$ is given
by

$\mathscr{H}_{0}=-\frac{1}{2}p_{\phi i}p_{\phi}^{i}\ ,$

and the Hamilton–Cartan forms on $P_{0}$ are

$\displaystyle\Theta_{\mathscr{H}}^{0}$ $\displaystyle=$
$\displaystyle\pi\text{d}\phi\wedge\text{d}^{m-1}x_{0}+p_{\phi}^{i}\text{d}\phi\wedge\text{d}^{m-1}x_{i}+\frac{1}{2}p_{\phi
i}p_{\phi}^{i}\text{d}^{m}x\in{\mit\Omega}^{m}(P_{0})\ ,$
$\displaystyle\Omega_{\mathscr{H}}^{0}$ $\displaystyle=$
$\displaystyle\text{d}\phi\wedge\text{d}\pi\wedge\text{d}^{m-1}x_{0}+\text{d}\phi\wedge\text{d}p_{\phi}^{i}\wedge\text{d}^{m-1}x_{i}-p_{\phi
i}\text{d}p_{\phi}^{i}\wedge\text{d}^{m}x\equiv\widetilde{\Omega}_{\mathscr{H}}^{0}+{\rm
d}\mathscr{H}_{0}\wedge\text{d}^{m}x\in{\mit\Omega}^{m+1}(P_{0})\,.$

Now take a locally decomposable multivector field
$\textbf{X}_{\mathscr{H}}^{0}=X_{0}\wedge X_{1}\wedge\dotsb\wedge
X_{m-1}\in\mathfrak{X}^{m}(P_{0})$ satisfying
$i(\textbf{X}_{\mathscr{H}}^{0}){\rm d}^{m}x=1$, so their components are given
as

$X_{\mu}=\frac{\partial}{\partial
x^{\mu}}+A_{\mu}\frac{\partial}{\partial\phi}+B_{\mu}\frac{\partial}{\partial\pi}+C^{i}_{\mu}\frac{\partial}{\partial
p_{\phi}^{i}}\ .$

The field equations obtained from
$i(\textbf{X}_{\mathscr{H}}^{0})\Omega_{\mathscr{H}}^{0}=0$ are:

$\displaystyle B_{0}+C_{i}^{i}=0\ ,$ (54a) $\displaystyle A_{0}=0\ ,$ (54b)
$\displaystyle A_{i}+p_{\phi i}=0\ ,$ (54c) $\displaystyle
A_{\lambda}B_{0}-A_{0}B_{\lambda}+A_{\lambda}C_{i}^{i}-A_{i}C_{\lambda}^{i}-p_{\phi
i}C_{\lambda}^{i}=0\ .$ (54d)

Equation (54d) is a combination of the equations (54a), (54b), and (54c) which
give relations or determine some coefficients. Then, no compatibility
constraints appear. Again, this is in agreement with the procedure given in
Proposition 1: set $i(Z)\text{d}\mathscr{H}_{0}=0$ for every
$Z\in\text{ker}\,\widetilde{\Omega}_{\mathscr{H}}^{0}\cap\text{ker}\,\text{d}^{m}x$;
however,

$\text{ker}\,\widetilde{\Omega}_{\mathscr{H}}^{0}\cap\text{ker}\
\text{d}^{m}x=\\{0\\}\ ,$

and condition (13) from Proposition 1 holds.

Working on $(x^{\mu},\phi(x),\pi(x),p^{i}_{0}(x))$ taken to be integral
sections of $\textbf{X}_{\mathscr{H}}^{0}$,

$A_{\mu}=\displaystyle\frac{\partial{\phi}}{\partial{x^{\mu}}}\quad,\quad
B_{\mu}=\displaystyle\frac{\partial{\pi}}{\partial{x^{\mu}}}\quad,\quad
C^{i}_{\mu}=\displaystyle\frac{\partial{p_{\phi}^{i}}}{\partial{x^{\mu}}}\ ,$

then (54a), (54b), and (54c) give

$\dot{\pi}+\displaystyle\frac{\partial{p_{\phi}^{i}}}{\partial{x^{i}}}=0\quad,\quad\dot{\phi}=0\quad,\quad\displaystyle\frac{\partial{\phi}}{\partial{x^{i}}}+p_{\phi
i}=0\ .$

Plugging in the third equation into the first equation gives

$\dot{\pi}-\frac{\partial^{2}\phi}{\partial x^{i}\partial
x_{i}}\equiv\dot{\pi}-\partial_{i}\partial^{i}\phi=0\quad,\quad\dot{\phi}=0\
.$

Once again, the equivalence between the Lagrangian and the Hamiltonian
formalisms is evident as the resulting field equations are shown to be
equivalent.

####  3.3.3 Symmetries

The Carroll transformations for the magnetic scalar field theory are written
slightly differently than how they are written for the electric scalar field
3.2. The Carroll transformations which are symmetries of the magnetic scalar
field theory are

$\displaystyle\delta\phi(x)$ $\displaystyle=-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\phi}}{\partial{x^{\alpha}}}=-b_{k}x^{k}\displaystyle\frac{\partial{\phi}}{\partial{x^{0}}}-\epsilon^{i}_{\
j}x^{j}\displaystyle\frac{\partial{\phi}}{\partial{x^{i}}}\ ,$
$\displaystyle\delta\pi(x)$ $\displaystyle=-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\pi}}{\partial{x^{\alpha}}}-b_{i}\phi^{i}=-b_{k}x^{k}\displaystyle\frac{\partial{\pi}}{\partial{x^{0}}}-\epsilon^{i}_{\
j}x^{j}\displaystyle\frac{\partial{\pi}}{\partial{x^{i}}}-b_{i}\phi^{i}\ ,$
$\displaystyle\delta\phi_{0}(x)$ $\displaystyle=-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\phi_{0}}}{\partial{x^{\alpha}}}=-b_{k}x^{k}\displaystyle\frac{\partial{\phi_{0}}}{\partial{x^{0}}}-\epsilon^{k}_{\
j}x^{j}\displaystyle\frac{\partial{\phi_{0}}}{\partial{x^{k}}}\ ,$
$\displaystyle\delta\phi_{i}(x)$ $\displaystyle=-\epsilon^{\nu}_{\
i}\phi_{\nu}-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\phi_{i}}}{\partial{x^{\alpha}}}=-b_{i}\phi_{0}-\epsilon^{k}_{\
i}\phi_{k}-b_{k}x^{k}\displaystyle\frac{\partial{\phi_{i}}}{\partial{x^{0}}}-\epsilon^{k}_{\
j}x^{j}\displaystyle\frac{\partial{\phi_{i}}}{\partial{x^{k}}}\ .$

Under these transformations the Lagrangian transforms as
$\displaystyle\delta\mathscr{L}(x)=-\displaystyle\frac{\partial{}}{\partial{x^{\alpha}}}\Big{(}\epsilon^{\alpha}_{\
\beta}x^{\beta}\mathscr{L}(x)\Big{)}$. Notice that the field $\pi(x)$ no
longer transforms as a Carrollian scalar; instead, $\pi(x)$ transforms under a
Carroll spacetime transformation $-\epsilon_{\
\beta}^{\alpha}x^{\beta}\displaystyle\frac{\partial{\pi}}{\partial{x^{\alpha}}}$
plus a term $-b_{i}\phi^{i}(x)$ which cannot be represented on $E$. The field
transformations behave geometrically as the Lie derivatives of the local
sections $j^{1}\phi:M\rightarrow J^{1}E$. The vector field generating the
Carroll transformations on $J^{1}E$ is

$\displaystyle\begin{split}X&=-\epsilon^{\mu}_{\
\nu}x^{\nu}\frac{\partial}{\partial
x^{\mu}}+b_{i}\phi^{i}\frac{\partial}{\partial\pi}+\epsilon^{\nu}_{\
\mu}\phi_{\nu}\frac{\partial}{\partial\phi_{\mu}}+\epsilon^{\nu}_{\
\mu}\pi_{\nu}\frac{\partial}{\partial\pi_{\mu}}\\\
&=-b_{i}x^{i}\frac{\partial}{\partial x^{0}}-\epsilon^{i}_{\
j}x^{j}\frac{\partial}{\partial
x^{i}}+b_{i}\phi^{i}\frac{\partial}{\partial\pi}+\left(b_{i}\phi_{0}+\epsilon_{\
i}^{j}\phi_{j}\right)\frac{\partial}{\partial\phi_{i}}+\epsilon^{\nu}_{\
\mu}\pi_{\nu}\frac{\partial}{\partial\pi_{\mu}}\in\mathfrak{X}(J^{1}E)\
,\end{split}$

which is tangent to the constraint submanifold $S_{1}\subset J^{1}E$, given by
the constraint $\phi_{0}=0$, since $\mathpzc{L}_{X}\phi_{0}=0$. Furthermore,
although the vector field $X$ leaves the Lagrangian density invariant
($\mathpzc{L}_{X}\textbf{L}=0$), it does not produce an exact Noether symmetry
as

$\mathpzc{L}_{X}\Theta_{\mathscr{L}}=b^{i}\phi_{0}(\phi_{i}\text{d}^{m}x-\text{d}\phi\wedge\text{d}^{m-1}x_{i})\
.$

Moreover, the vector field $X$ as written above on all of $J^{1}E$ is not
projectable onto $P_{0}\subset J^{1}E^{*}$. Instead, $X$ on $S_{1}$ given by

$\left.X\right|_{S_{1}}=-b_{i}x^{i}\frac{\partial}{\partial
x^{0}}-\epsilon^{i}_{\ j}x^{j}\frac{\partial}{\partial
x^{i}}+b_{i}\phi^{i}\frac{\partial}{\partial\pi}+\epsilon_{\
i}^{j}\phi_{j}\frac{\partial}{\partial\phi_{i}}+\epsilon^{\nu}_{\
\mu}\pi_{\nu}\frac{\partial}{\partial\pi_{\mu}}\ ,$ (55)

is $\mathscr{FL}_{0}$-projectable onto $P_{0}$, giving

$Y=\mathscr{FL}_{0*}X|_{S_{1}}=-b_{i}x^{i}\frac{\partial}{\partial
x^{0}}-\epsilon^{i}_{\ j}x^{j}\frac{\partial}{\partial
x^{i}}-b_{i}p_{\phi}^{i}\frac{\partial}{\partial\pi}+\epsilon^{i}_{\
j}p_{\phi}^{j}\frac{\partial}{\partial p_{\phi}^{i}}\in\mathfrak{X}(P_{0})\ .$

Now, letting $\widetilde{X}$ be the local extension of $X|_{S_{1}}$ to
$J^{1}E$ whose coordinate expression is given by (55); then, it follows that
$\mathpzc{L}_{\widetilde{X}}\Theta_{\mathscr{L}}=0$ on $J^{1}E$. It is thereby
possible to define the momentum maps as
$J_{\mathscr{L}}(\widetilde{X})=-i(\widetilde{X})\Theta_{\mathscr{L}}\in{\mit\Omega}^{m-1}(J^{1}E)$
and
$J_{\mathscr{H}_{0}}(Y)=-i(Y)\Theta^{0}_{\mathscr{H}}\in{\mit\Omega}^{m-1}(P_{0})$,
giving:

$\displaystyle J_{\mathscr{L}}(\widetilde{X})$ $\displaystyle=$
$\displaystyle-\left(\epsilon^{i}_{\
j}x^{j}\pi+b_{j}x^{j}\phi^{i}\right)\text{d}\phi\wedge\text{d}^{m-2}x_{0i}+\phi^{i}\epsilon^{j}_{\
k}x^{k}\text{d}\phi\wedge\text{d}^{m-2}x_{ij}$
$\displaystyle+\frac{1}{2}\phi_{i}\phi^{i}\left(b_{k}x^{k}\text{d}^{m-1}x_{0}+\epsilon^{k}_{\
j}x^{j}\text{d}^{m-1}x_{k}\right)\ ,$ $\displaystyle J_{\mathscr{H}_{0}}(Y)$
$\displaystyle=$ $\displaystyle-\left(\epsilon^{i}_{\ j}x^{j}\pi-
b_{j}x^{j}p_{\phi}^{i}\right)\text{d}\phi\wedge\text{d}^{m-2}x_{0i}-p_{\phi}^{i}\epsilon^{j}_{\
k}x^{k}\text{d}\phi\wedge\text{d}^{m-2}x_{ij}$
$\displaystyle+\frac{1}{2}p_{\phi
i}p_{\phi}^{i}\left(b_{k}x^{k}\text{d}^{m-1}x_{0}+\epsilon^{k}_{\
j}x^{j}\text{d}^{m-1}x_{k}\right)\ .$

##  4 Bosonic Strings and p-Branes

In the next sections, the multisymplectic formalism for bosonic $p$-branes is
given by working out explicitly the case for string theory $(p=1)$ and then
discussing the generalization to $p>1$. The De Donder–Weyl Hamiltonian
treatment for $p$-branes can be found in [32].

The fields of interest are the embeddings of a brane worldvolume in spacetime.
Spacetime $M$ is a smooth $D=(d+1)$-dimensional manifold with local
coordinates $x^{\mu}$ ($\mu=0,1,...,d$) and spacetime metric $G_{\mu\nu}$ with
signature $(-+\dotsb+)$. The $p$-brane worldvolume $\Sigma$ is a smooth
manifold with $\text{dim}\Sigma=m=p+1$ and local coordinates $\sigma^{a}$
(with $a=0,1,...,p$). Given the embedding $X:\Sigma\rightarrow
M:\sigma^{a}\mapsto x^{\mu}(\sigma)$, the embedding maps $X^{\mu}(\sigma)$ are
taken to be fields on $\Sigma$. The configuration bundle $E$ over $\Sigma$ is
taken to be the trivial bundle $E=\Sigma\times M$ and comes with a surjective
projection map $\pi:E\rightarrow\Sigma$ as usual and sections of $E$ are given
by $\phi:\Sigma\rightarrow\Sigma\times
M:\sigma^{a}\mapsto(\sigma^{a},x^{\mu}(\sigma))$. The coordinates in $E$ are
$(\sigma^{a},x^{\mu})$, in the jet bundle $J^{1}E$ are
$(\sigma^{a},x^{\mu},x_{a}^{\mu})$, and in $J^{1}E^{*}$ are
$(\sigma^{a},x^{\mu},p^{a}_{\mu})$. It follows that $\text{dim}E=m+D$ and
$\text{dim}J^{1}E=\text{dim}J^{1}E^{*}=m+D+mD$. It is also worth noting that
the first-order jet prolongations are given as $j^{1}\phi:\Sigma\rightarrow
J^{1}E:\sigma^{a}\mapsto\Big{(}\sigma^{a},x^{\mu}(\sigma),\displaystyle\frac{\partial{x^{\mu}}}{\partial{\sigma^{a}}}(\sigma)\Big{)}$.

The Lagrangian density is given by

$\textbf{L}=\mathscr{L}(\sigma^{a},x^{\mu},x^{\mu}_{a})\text{d}^{m}\sigma=-T_{p}\sqrt{-\text{det}g}\
\text{d}^{m}\sigma=-T_{p}\sqrt{-\text{det}(G_{\mu\nu}x^{\mu}_{a}x^{\nu}_{b})}\
\text{d}^{m}\sigma,$

where $T_{p}$ is the $p$-brane tension which has units of
$[T_{p}]=(mass)/(length)^{p}$ and
$g=\frac{1}{2}g_{ab}\text{d}\sigma^{a}\wedge\text{d}\sigma^{b}$ is a 2-form on
$\Sigma$ on $J^{1}E$ whose pullback by jet prolongations give the induced
metric on $\Sigma$:

$j^{1}\phi^{*}g=h\equiv\frac{1}{2}h_{ab}\text{d}\sigma^{a}\wedge\text{d}\sigma^{b}\
,\quad\text{where}\quad h_{ab}=G_{\mu\nu}\frac{\partial
X^{\mu}}{\partial\sigma^{a}}\frac{\partial X^{\nu}}{\partial\sigma^{b}}\ .$
(56)

The 2-form $g$ can also be constructed on $J^{1}E^{*}$ by pushforward of the
Legendre map, that is $\mathscr{FL}_{*}g=g\in\Omega^{2}(J^{1}E^{*})$, so that
$\psi^{*}g=h$. The Legendre map $\mathscr{FL}:J^{1}E\rightarrow
J^{1}E^{*}:(\sigma^{a},x^{\mu},x^{\mu}_{a})\mapsto(\sigma^{a},x^{\mu},p^{a}_{\mu})$
gives

$\mathscr{FL}^{*}p^{a}_{\mu}=\frac{\partial\mathscr{L}}{\partial x^{\mu}_{a}}\
;$

the Hessian matrix

$\frac{\partial^{2}\mathscr{L}}{\partial x_{a}^{\mu}\partial
x_{b}^{\nu}}=-pT_{p}\sqrt{-\text{det}g}G_{\mu\nu}g^{ba}\ ,$ (57)

is non-singular and the Lagrangian is regular thusly.

The full geometric setting is illustrated in the following figure:

$\mathbb{R}\
$$\mathscr{L}$$\mathscr{H}$$J^{1}E$$\mathrm{j^{1}\phi}$$J^{1}E^{*}$$M$$\Sigma$$\mathrm{\phi}$$\pi$$\overline{\pi}^{1}$$\tau$$\psi$$X$$E=\Sigma\times
M$$\mathscr{FL}$$\left(\sigma^{a},x^{\mu},x^{\mu}_{a}\right)$$\left(\sigma^{a},x^{\mu},p^{a}_{\mu}\right)$$\left(\sigma^{a},x^{\mu}\right)$$\sigma^{a}$$x^{\mu}$$\pi^{1}$$\overline{\tau}$$\mathbb{R}\
$

###  4.1 The String Lagrangian Formulation

The Lagrangian for the Nambu–Goto string ($p=1$) is

$\textbf{L}=\mathscr{L}(\sigma^{a},x^{\mu},x^{\mu}_{a})\text{d}^{2}\sigma=-T\sqrt{-\text{det}g}\
\text{d}^{2}\sigma\ ,$

from which it follows that

$\frac{\partial\mathscr{L}}{\partial x_{a}^{\mu}}=-T\sqrt{-\text{det}g}\
G_{\mu\nu}g^{ba}x_{b}^{\nu}=\frac{T}{\sqrt{-\text{det}g}}\
\epsilon^{bd}\epsilon^{ac}g_{dc}G_{\mu\nu}x^{\nu}_{b}\ ,$ (58)

where
$g^{ba}\equiv(g^{-1})^{ba}=\frac{1}{\text{det}g}\epsilon^{bc}\epsilon^{ad}g_{dc}$
and so it follows that the Hessian can be written as

$\frac{\partial^{2}\mathscr{L}}{\partial x_{a}^{\mu}\partial
x_{b}^{\nu}}=T\frac{\epsilon^{bd}\epsilon^{ac}}{\sqrt{-\text{det}g}}\
G_{\mu\nu}g_{dc}\ .$

The Lagrangian energy is

$E_{\mathscr{L}}\equiv\frac{\partial\mathscr{L}}{\partial
x_{a}^{\mu}}x_{a}^{\mu}-\mathscr{L}=-T\sqrt{-\text{det}g}\
(g^{ba}g_{ab}-1)=-T\sqrt{-\text{det}g}\in C^{\infty}(J^{1}E)\ ,$ (59)

where the last equality follows as $g_{ab}$ is a $2\times 2$ matrix in the
case of the string. Using (58) and (59), the Poincaré–Cartan $2$-from and
multisymplectic Poincaré–Cartan $3$-form on $J^{1}E$ are

$\displaystyle\Theta_{\mathscr{L}}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathscr{L}}{\partial x_{a}^{\mu}}\
\text{d}x^{\mu}\wedge\text{d}^{1}\sigma_{a}-E_{\mathscr{L}}\wedge\text{d}^{2}\sigma=\frac{T}{\sqrt{-\text{det}g}}\left[\epsilon^{bd}\epsilon^{ac}g_{dc}G_{\mu\nu}x_{b}^{\nu}\text{d}x^{\mu}\wedge\text{d}^{1}\sigma_{a}-(\text{det}g)\text{d}^{2}\sigma\right]\
,$ $\displaystyle\Omega_{\mathscr{L}}$ $\displaystyle=$ $\displaystyle
T\epsilon^{bd}\epsilon^{ac}\Bigg{\\{}-\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}\text{d}x^{\nu}_{b}\wedge\text{d}x^{\mu}\wedge\text{d}^{1}\sigma_{a}-\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x^{\nu}_{b}\right)\text{d}x^{\rho}\wedge\text{d}x^{\mu}\wedge\text{d}^{1}\sigma_{a}+$
$\displaystyle\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x^{\mu}_{a}\text{d}x^{\nu}_{b}\wedge\text{d}^{2}\sigma+$
$\displaystyle\left[\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}\left(\frac{G_{\rho\nu}g_{dc}}{\sqrt{-\text{det}g}}v^{\nu}_{b}\right)x^{\rho}_{a}-\frac{g_{dc}}{2\sqrt{-\text{det}g}}x^{\alpha}_{a}x^{\beta}_{b}\partial_{\mu}G_{\alpha\beta}+\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\left(\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x^{\nu}_{b}\right)\right]\text{d}x^{\mu}\wedge\text{d}^{2}\sigma\Bigg{\\}}\
.$

The field equations, $i(\textbf{X}_{\mathscr{L}})\Omega_{\mathscr{L}}=0$, can
be computed using a representative of a class of $\bar{\pi}^{1}$-transverse
and locally decomposable multivector fields
$\textbf{X}_{\mathscr{L}}\in\mathfrak{X}^{2}(J^{1}E)$, written as
$\textbf{X}_{\mathscr{L}}=X_{0}\wedge X_{1}$, with components given by the
following local expression

$X_{a}=\frac{\partial}{\partial\sigma^{a}}+B_{a}^{\mu}\frac{\partial}{\partial
x^{\mu}}+H_{ab}^{\nu}\frac{\partial}{\partial x_{b}^{\nu}},$

from which it follows that $i(\textbf{X}_{\mathscr{L}})\text{d}^{2}\sigma=1$.
Then, $i(\textbf{X}_{\mathscr{L}})\Omega_{\mathscr{L}}=0$ is given as

$\displaystyle\begin{split}i(\textbf{X}_{\mathscr{L}})\Omega_{\mathscr{L}}=&T\epsilon^{bd}\epsilon^{ac}\Bigg{\\{}\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}\left(B_{a}^{\mu}-x_{a}^{\mu}\right)\text{d}x_{b}^{\nu}\\\
&+\bigg{[}\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}\left(\frac{G_{\rho\nu}g_{dc}}{\sqrt{-\text{det}g}}x_{b}^{\nu}\right)B^{\rho}_{a}-\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x_{b}^{\nu}\right)B^{\rho}_{a}-\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}H_{ab}^{\nu}\\\
&+\frac{g_{dc}}{2\sqrt{-\text{det}g}}x_{a}^{\alpha}x_{b}^{\beta}\displaystyle\frac{\partial{G_{\alpha\beta}}}{\partial{x^{\mu}}}-\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}\left(\frac{G_{\rho\nu}g_{dc}}{\sqrt{-\text{det}g}}x_{b}^{\nu}\right)x^{\rho}_{a}-\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\left(\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x_{b}^{\nu}\right)\bigg{]}\text{d}x^{\mu}\\\
&+\bigg{[}\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x^{\nu}_{b}\right)x_{a}^{\mu}B_{k}^{\rho}-\frac{g_{dc}}{2\sqrt{-\text{det}g}}x_{a}^{\alpha}x_{b}^{\beta}B_{k}^{\rho}\displaystyle\frac{\partial{G_{\alpha\beta}}}{\partial{x^{\rho}}}+\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\left(\frac{G_{\rho\nu}g_{dc}}{\sqrt{-\text{det}g}}x^{\nu}_{b}\right)B_{k}^{\rho}\\\
&+\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x_{a}^{\mu}H^{\nu}_{kb}-\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}(B_{a}^{\mu}H^{\nu}_{kb}-B_{k}^{\mu}H^{\nu}_{ab})\\\
&-\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\frac{G_{\mu\nu}g_{dc}}{\sqrt{-\text{det}g}}x_{b}^{\nu}\right)(B_{a}^{\mu}B_{k}^{\rho}-B_{k}^{\mu}B_{a}^{\rho})\bigg{]}\text{d}\sigma^{k}\Bigg{\\}}=0.\end{split}$

Setting differential forms separately equal to zero produces the following
equations:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\epsilon^{bd}\epsilon^{ac}\Bigg{[}\displaystyle\frac{\partial{}}{\partial{x^{\mu}}}\left(\frac{g_{dc}G_{\rho\nu}}{\sqrt{-\text{det}g}}x_{b}^{\nu}\right)(B^{\rho}_{a}-x^{\rho}_{a})-\displaystyle\frac{\partial{}}{\partial{x^{\rho}}}\left(\frac{g_{dc}G_{\mu\nu}}{\sqrt{-\text{det}g}}x_{b}^{\nu}\right)B^{\rho}_{a}-\displaystyle\frac{\partial{}}{\partial{\sigma^{a}}}\left(\frac{g_{dc}G_{\mu\nu}}{\sqrt{-\text{det}g}}x_{b}^{\nu}\right)-$
(60)