# Regularized polysymplectic geometry and first steps towards Floer theory for
covariant field theories

Ronen Brilleslijper and Oliver Fabert

###### Abstract

It is the goal of this paper to present the first steps for defining the
analogue of Hamiltonian Floer theory for covariant field theory, treating time
and space relativistically. While there already exist a number of competing
geometric frameworks for covariant field theory generalizing symplectic
geometry, none of them are readily suitable for variational techniques such as
Hamiltonian Floer theory, since the corresponding action functionals are too
degenerate. Instead, we show how a regularization procedure introduced by
Bridges leads to a new geometric framework for which we can show that the
finite energy $L^{2}$-gradient lines of the corresponding action functional,
called Floer curves, converge asymptotically to space-time periodic solutions.
As a concrete example we prove the existence of Floer curves, and hence also
of space-time periodic solutions, for a class of coupled particle-field
systems defined in this new framework.

###### Contents

  1. 1 Introduction
  2. 2 From classical mechanics to field theory
    1. 2.1 Non-relativistic field theory
    2. 2.2 Relativistic field theory
    3. 2.3 Degeneracy problem for Floer curves
  3. 3 Bridges regularization
    1. 3.1 Degeneracy problem for Floer curves - solved
    2. 3.2 Underlying structure
    3. 3.3 The infinite-dimensional viewpoint
    4. 3.4 The space of solutions
  4. 4 The polysymplectic formalism
    1. 4.1 The polysymplectic Hamiltonian formalism
    2. 4.2 From polysymplectic forms to action functionals
  5. 5 Prime example: coupled particle-field systems
    1. 5.1 Asymptotics of Floer curves
    2. 5.2 Lorentz model
    3. 5.3 Existence of Floer curves

##  1 Introduction

The world of classical physics essentially contains two major areas: classical
mechanics and classical field theory (CFT). The rigorous mathematical
formulation of the former led to the development of Hamiltonian dynamics and
symplectic geometry. As in other fields of geometry, techniques from
variational calculus can provide important tools for proving theorems. In
variational calculus, solutions to equations are found by translating them
into a minimization problem of some functional. In the 1980’s Andreas Floer
discovered that $L^{2}$-gradient lines of the so-called action functional
satisfy a perturbed version of the Cauchy-Riemann equations ([Flo88]). Thus,
to study these gradient lines he could make use of the previously discovered
theory of Gromov on pseudo-holomorphic curves. As these gradient lines
converge to critical points of the action functional, this allowed him to
prove the Arnold conjecture about the number of periodic solutions to
Hamilton’s equations for a large class of symplectic manifolds (see for
example [Flo88, AD14]). The resulting theory, which was named Floer theory
after him, brought a rich family of tools to symplectic geometers. Having all
of this theory for classical mechanics makes one wonder if similar tools can
be developed for CFT. At present there is no analogue of Floer theory in the
setting of CFT. There are however several natural approaches to take when
attempting to formulate such a theory. We will introduce some of them here.

First of all, CFT can be seen as an infinite-dimensional version of classical
mechanics and its equations can be translated to symplectic geometry on
infinite-dimensional manifolds. Thus, one natural approach of defining a Floer
theory would be to develop it on infinite-dimensional symplectic spaces. This
approach is taken by [FL22] and provides a working theory of which we will
review the basics in section 2.1. As explained at the end of that section, the
major disadvantage of this theory is that it is not symmetric in time and
space. As CFT is capable of describing relativistic phenomena such as
electromagnetism, it would be desirable to formulate a theory that is capable
of dealing with coordinate transformations of the space-time manifold
considered. Moreover, a theory that uses the symmetry of the underlying space
instead of breaking it, is also from a mathematical point of view more natural
to develop. Even so, the theory developed in [FL22] is a very good step in the
direction of formulating a Floer theory for CFT and allows us to prove some
existence results for our theory as well (see section 5.3).

When we think of CFT in a covariant way (i.e. not splitting time and space),
there are several methods of formulating the geometry behind it. Different
articles use different methods, such as multisymplectic geometry ([MPS98]),
polysymplectic geometry ([Gün87, McC21]) and others ([RRMRSV11, Kru02]). The
mathematics world has not (yet) decided on one universally standard choice of
language, but in this article we chose to stick with the polysymplectic
formulation. This is mostly due to the inspiration we got from [Gün87] and
[McC21], who both use polysymplectic geometry. As can be seen in the
aforementioned articles by Günther and McClain, they made good progress in
developing a geometry for CFT. However, we believe that it is impossible to
build a Floer theory for their equations. In fact, the polysymplectic
Hamiltonian equations are not suited to be studied by any kind of variational
techniques. The reason for this, which will be explained more in section 2.3,
is that the equations are too degenerate: the differential operator that lies
at the core of the equations has an infinite-dimensional kernel which
obstructs much of the needed analysis. In particular, some of the fundamental
building blocks of Floer theory (see lemma 2.1) can not be reproduced for
Günther’s or McClain’s equations, which provides major obstacles for the
analysis of the Floer equation. Fortunately, this obstacle can be overcome by
enhancing these equations to a richer structure that solves this degeneracy
problem. In section 3, we will explain how an idea from [Bri06] enables us to
augment the polysymplectic form of GüntheriiiIn a follow-up article we will
deal with the global case described by McClain’s equation. and produce a form
that is in a sense more natural and even allows us to connect the covariant
picture with the infinite-dimensional approach. The main purpose of this
article is to convey the message that the polysymplectic form coming from what
we call _Bridges regularization_ provides the correct formalism to describe
CFT covariantly in a way that allows for variational calculus. Apart from the
fact that this form is better suitable for calculus of variations (in
particular Floer theory) and allows for a connection with the infinite-
dimensional approach of [FL22], it also arises more naturally from a geometric
point of view as is explained at the end of section 3.2.

The idea behind Bridges regularization is as follows. In classical mechanics,
the position of a particle can be interpreted as a 0-form and its momentum as
a 1-form. Since the only variable is the time-variable, there are no higher-
degree forms to consider. When we move to field equations, the field values
and momenta can still be interpreted as 0-forms and 1-forms respectively.
However, there are now both time and space variables in the picture, meaning
that there exist higher-degree forms as well. In the existing approaches for
the covariant formulation of CFT only the field values and momenta are taken
into account, whereas the higher-degree forms are ignored. This is the source
of the degeneracy problem in these frameworks. In Bridges regularization these
higher-degree forms are taken into the equation, which resolves the
degeneracy. In particular, the components of the resulting polysymplectic form
are themselves symplectic forms as opposed to Günthers polysymplectic form,
whose components are merely closed 2-forms.

For this article, we deliberately chose to work only with scalar theories on
linear space and attack the problems hands-on before digressing about the
underlying geometry. The reason for this is that we want the reader to be able
to reach the main messages of this article before getting lost into abstract
constructions and computations. However, the basic background of
polysymplectic geometry on linear spaces that is avoided in sections 2 and 3
is treated in section 4. Also, we are in fact able to generalise much of the
theory in this article to field theories on general manifolds taking values in
non-trivial vector bundles. To include these generalisations into this article
would make it too long and distract from the main message. Therefore, we
decided to publish the global geometric picture in a subsequent article. The
article at hand is meant to motivate the use of a regularized polysymplectic
geometry over the existing frameworks and to prove some first results in this
framework. The connection that this framework allows between the covariant and
the infinite-dimensional picture will also enable us to indeed prove the
existence of Floer curves and periodic solutions to certain field equations.

This article is structured as follows. Section 2 starts with a small recap of
classical mechanics and how Floer theory comes into the picture. Lemma 2.1
gives two fundamental lemmas that are crucial for the validity of Floer
theory. They will be used in other sections to test possible approaches to
Floer theory for CFT. Next, field theories are introduced, starting with the
infinite-dimensional approach of [FL22] (section 2.1) and followed by the
polysymplectic formulation (section 2.2). The section ends with an
illustration of the degeneracy problem of the latter (section 2.3). This leads
to the introduction of Bridges regularization in section 3 and we show that
indeed the degeneracy problem is solved (section 3.1). Section 3 starts very
hands-on by manipulating the equations, but the underlying structure of these
alterations is explained later on in section 3.2. We show that this newly
obtained structure also fits into the infinite-dimensional theory of [FL22] in
section 3.3 and still provides the same set of solutions for the field
equation that we started with (section 3.4). In section 4, we explain more
rigorously how polysymplectic geometry works and give some background for
constructions that were used more ad-hoc in the previous sections. In
particular, it is explained how the action functionals of sections 2 and 3
come about and why they have the correct critical points. Finally, section 5
illustrates the theory by applying it to the example of coupled particle-field
systems. This provides an interesting application, both from a physical and
mathematical point of view. Also, section 5.3 shows that indeed our version of
the Floer equation allows us to prove the existence of solutions to field
equations, which is the problem that we started with. Furthermore, section 5.2
illustrates the advantage of our developed theory over [FL22].

##  2 From classical mechanics to field theory

We start with a small recap of classical mechanics and the basic ideas of
Floer theory. The most fundamental equation in classical mechanics is Newton’s
second law:

$\displaystyle\frac{d^{2}}{dt^{2}}q(t)=-V_{t}^{\prime}(q(t)).$ (1)

Here, $q(t)$ describes the position at time $t$ of an object moving in some
potential field given by $V_{t}$. We start by briefly discussing how this
equation can be formulated with the help of a Hamiltonian and symplectic form
and what the role of Floer theory is for finding solutions.

First of all, to transform this equation into a first order ODE, we define the
momentum $p(t)=q^{\prime}(t)$. Now, equation 1 is equivalent to the system of
equations

$\displaystyle\frac{d}{dt}q(t)=p(t)$
$\displaystyle\frac{d}{dt}p(t)=-V_{t}^{\prime}(q).$

This system of equations can be unified into one equation using the
Hamiltonian function $H_{t}(q,p)=\frac{1}{2}p^{2}+V_{t}(q)$. It becomes

$\displaystyle J\frac{d}{dt}u(t)=\nabla H_{t}(u(t)),$ (2)

where $u(t)=(q(t),p(t))$ and $J=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$ is
the standard complex structure on $\mathbb{R}^{2}$. As treated in standard
textbooks on symplectic geometry (for example [DS08, Chapter 18]), the
geometric picture behind this equation lies in the two-form
$\omega=\langle\cdot,J\cdot\rangle=dp\wedge dq$, called the standard
symplectic form on $\mathbb{R}^{2}$. It enables us to write the equation above
as

$\displaystyle(dH_{t})_{u(t)}=\omega(\cdot,\frac{d}{dt}u(t)).$ (3)

In Floer theory, one tries to prove the existence of periodic solutions of the
equation above. First, the essential observation is made that solutions
$u\mathrel{\mathop{\mathchar 58\relax}}S^{1}\to\mathbb{R}^{2}$ to equation 3
are critical points of the action functional

$\displaystyle\mathcal{A}_{H}(u)$
$\displaystyle=\int_{S^{1}}(u^{*}\lambda)_{t}-\int_{0}^{1}H_{t}(u(t))\,dt$
$\displaystyle=\int_{0}^{1}\left(p(t)q^{\prime}(t)-H_{t}(u(t))\right)\,dt.$

Here, $\lambda=p\,dq$ is a primitive for $\omega$ and $H_{t}=H_{t+1}$ is
assumed to be periodic in $t$. Just like in Morse theory, Floer theory proves
the existence of these critical points by studying gradient lines. If
$\mathcal{X}=-\textup{grad}\,\mathcal{A}_{H}$ is defined to be the negative
$L^{2}$-gradient of $\mathcal{A}_{H}$, then its smooth trajectories
$\widetilde{u}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\times
S^{1}\to\mathbb{R}^{2}$, called Floer curves, are characterized by

$\displaystyle\partial_{s}\widetilde{u}(s,t)+J\partial_{t}\widetilde{u}(s,t)-\nabla
H_{t}(\widetilde{u}(s,t))=0.$ (4)

Equation 4 is called the Floer equation and forms the foundation of Floer
theory. It is interesting to note, that when both $\widetilde{u}$ and $H$ are
independent of $t$, the equation reduces to the Morse theory of the function
$H$. An essential observation made by Floer is that equation 4 reduces to the
Cauchy-Riemann equation when $H\equiv 0$. Its analysis builds upon the well-
established theory of Gromov and Witten.iiiiiiSee [MS12] for a detailed
exposition of this theory. We will give two important ingredients for the
theory below. For a rigorous exposition of Floer theory we refer to [AD14].

There are two results in symplectic geometry that are crucial for the validity
of Floer theory. Before restating them, we must define the energy of a Floer
curve:

$\displaystyle
E(\widetilde{u})=-\int_{-\infty}^{+\infty}\frac{d}{ds}\mathcal{A}_{H}(\widetilde{u}(s,\cdot))\,ds=\int_{-\infty}^{+\infty}\left(\int_{S^{1}}|\partial_{s}\widetilde{u}(s,t)|^{2}\,dt\right)\,ds$

From now on we will denote $\widetilde{u}_{s}\mathrel{\mathop{\mathchar
58\relax}}=\widetilde{u}(s,\cdot)$.

######  Lemma 2.1.

Let $\widetilde{u}$ be a Floer curve with finite energy:
$E(\widetilde{u})<+\infty$.

  1. (i)

If $H\equiv 0$ and $\widetilde{u}_{s}$ has mean zero for every $s$ then

$\lim_{s\to\infty}\widetilde{u}_{s}=0$

in $C^{\infty}(S^{1},\mathbb{R}^{2})$.

  2. (ii)

If all critical points of $\mathcal{A}_{H}$ are non-degenerateiiiiiiiiiMeaning
that the Hessian of $\mathcal{A}_{H}$ is invertible., then there is some $u\in
C^{\infty}(S^{1},\mathbb{R}^{2})$ satisfying equation 3 such that

$\lim_{s\to\infty}\widetilde{u}_{s}=u$

in $C^{\infty}(S^{1},\mathbb{R}^{2})$.

For both parts of the lemma we refer to standard textbooks on holomorphic
curves and Floer theory like [AD14, MS12, Sal99]. Lemma 2.1.ii implies that if
there are Floer curves with finite energy, then automatically the existence of
periodic solutions to equation 3 is established. Lemma 2.1.i plays an
important role in proving the validity of Floer theory. As mentioned briefly
above, Floer theory builds on Gromov’s theory of pseudoholomorphic curves and
lemma 2.1.i is needed for establishing so-called ”bubbling off” results. If
lemma 2.1.i fails to hold, the construction of a Floer theory is hopeless,
thus we will use equivalents of lemma 2.1.i to check the validity of various
generalisations of Floer theory in the subsequent text.

###  2.1 Non-relativistic field theory

Our guiding example for field theories will be the free wave equation

$\displaystyle-\partial_{t}^{2}\varphi(t,x)+\partial_{x}^{2}\varphi(t,x)=0,$
(5)

where $\varphi(t,x)$ denotes a scalar field on 2-dimensional space-time. Of
course, the equation becomes interesting only when we add a non-linearity to
the right-hand side. In particular we will be interested in non-linearities
that come from a coupling of the field with a particle (see equation 30). This
provides interesting non-linearities in the field theory, but also influences
the mechanical system of the particle, thereby including symplectic geometry
into the story. However, for the purpose of this chapter, the free wave
equation already suffices. Just like in the previous paragraphs, we want to
describe equation 5 using some geometric framework and examine the existence
of periodic solutions using an analogue of Floer theory. For our first attempt
in doing so, we follow the exact same strategy as above.

We start by defining the momentum $\pi=\partial_{t}\varphi$ in order to get
rid of the second-order derivative in $t$. Equation 5 becomes equivalent to

$\displaystyle\partial_{t}\varphi(t,x)=\pi(t,x)$
$\displaystyle\partial_{t}\pi(t,x)=\partial_{x}^{2}\varphi(t,x).$

Just as in the case of classical mechanics discussed above, this system of
equations can be unified into one equation using a Hamiltonian function.
However, the Hamiltonian has to be defined on an infinite-dimensional space
now. Let $\mathbb{H}=L^{2}(S^{1},\mathbb{R})\oplus L^{2}(S^{1},\mathbb{R})$ be
the space of square-integrable functions in the space-variable. We denote
elements of $\mathbb{H}$ by $\mathscr{Z}$, where
$\mathscr{Z}(x)=(\varphi(x),\pi(x))$. Note that we have taken the domain of
these functions to be the circle instead of $\mathbb{R}$, since eventually we
are interested in periodic solutions anyways. Define
$\mathscr{H}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{H}\to\mathbb{R}$ by

$\mathscr{H}(\mathscr{Z})=\int_{S^{1}}\left[\frac{1}{2}\pi^{2}(x)+\frac{1}{2}\left(\frac{d}{dx}\varphi(x)\right)^{2}\right]\,dx.$

Then

$\textup{grad}\,\mathscr{H}(\mathscr{Z})=\begin{pmatrix}-\frac{d^{2}}{dx^{2}}\varphi\\\
\pi\end{pmatrix},$

so equation 5 becomes equivalent to

$\displaystyle\mathscr{J}\frac{d}{dt}\mathscr{Z}(t)=\textup{grad}\,\mathscr{H}(\mathscr{Z}(t)),$
(6)

where now we write $\mathscr{Z}(t)(x)=\mathscr{Z}(t,x)$ and where
$\mathscr{J}(\varphi,\pi)=(-\pi,\varphi)$ is the standard complex structure on
$\mathbb{H}$.

Define
$\omega_{\mathbb{H}}=\langle\cdot,\mathscr{J}\cdot\rangle_{L^{2}}=d\pi\wedge
d\varphi$ to be the standard symplectic form on $\mathbb{H}$. Then finally we
can write equation 5 as

$\displaystyle(d\mathscr{H})_{\mathscr{Z}(t)}=\omega_{\mathbb{H}}(\cdot,\frac{d}{dt}\mathscr{Z}(t)).$
(7)

One can compare equation 7 to equation 3 to see that the two equations are
very similar. Whereas equation 3 brings the equation into the world of
symplectic geometry, equation 7 translates the wave equation into the realm of
infinite-dimensional symplectic geometry. In a similar way as before, an
action functional can be defined and periodic solutions of the free wave
equation can be found by studying gradient trajectories of this functional.
This approach has been taken by Fabert and Lamoree in [FL22] and leads to an
interesting and working theory. There is one major problem with this theory
though: it breaks the symmetry between time and space. In the free wave
equation time and space play a very similar role. In fact, every linear
coordinate transformation in $O(1,1)$ preserves the equation. On the other
hand, equation 7 is highly asymmetric in time and space and a choice of
coordinates must be made before applying the theory. This is undesirable,
considering that the wave equation describes fundamentally relativistic
phenomena such as electromagnetism and its interaction with charged particles.
Section 5 will treat an example of the theory in which one cannot assume a
fixed splitting of time and space. This example does not fit into the
framework developed by Fabert and Lamoree and therefore illustrates the need
to find a covariant formulation of the theory. However, also from a purely
theoretical point of view, it is desirable to find a treatment of covariant
field theories that uses their symmetry rather than breaking it. The remainder
of this article will deal with finding a theory that realizes this.

###  2.2 Relativistic field theory

In the previous paragraph, the first step in reformulating the wave equation
was to introduce a momentum variable for the derivative with respect to time.
As a start of our symmetric approach, a logical step would be to introduce
momentum variables for both the time and space derivatives;
$\pi_{1}(t,x)=\partial_{t}\varphi(t,x)$ and
$\pi_{2}(t,x)=\partial_{x}\varphi(t,x)$. The free wave equation 5 is then
equivalent to the system of equations

$\displaystyle-$ $\displaystyle\partial_{t}\pi_{1}(t,x)$
$\displaystyle+\partial_{x}\pi_{2}(t,x)$ $\displaystyle=0$
$\displaystyle\partial_{t}\varphi(t,x)$ $\displaystyle=\pi_{1}(t,x)$ (8)
$\displaystyle-\partial_{x}\varphi(t,x)$ $\displaystyle=-\pi_{2}(t,x).$

The reason for the minus signs in the last row will become apparent below.
Notice that the set of solutions to this system of equations is still in one-
to-one correspondence with the set of solutions of equation 5. Just as before,
we want to unify these equations using a type of Hamiltonian function. To this
extent, define $\mathcal{Z}=(\varphi,\pi_{1},\pi_{2})$ and
$\mathcal{S}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{3}\to\mathbb{R}$
given by
$\mathcal{S}(\mathcal{Z})=\frac{1}{2}\pi_{1}^{2}-\frac{1}{2}\pi_{2}^{2}$. Then
the system of equations can be rewritten as

$\displaystyle
K_{1}\partial_{t}\mathcal{Z}(t,x)+K_{2}\partial_{x}\mathcal{Z}(t,x)=\nabla\mathcal{S}(\mathcal{Z}(t,x)),$
(9)

where

$\displaystyle K_{1}=\begin{pmatrix}0&-1&0\\\ 1&0&0\\\ 0&0&0\end{pmatrix}$
$\displaystyle K_{2}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\ -1&0&0\end{pmatrix}.$

Note that equation 9 is somewhat similar to equations 2 and 6, but differs in
a lot of aspects as well. An important difference is that the matrices $K_{1}$
and $K_{2}$ are no longer non-degenerate and therefore do not define complex
structures. However, both matrices are still antisymmetricivivivThis is why we
introduced the extra minus signs on the last line of equation 8. which means
that $\omega^{K_{i}}\mathrel{\mathop{\mathchar
58\relax}}=\langle\cdot,K_{i}\cdot\rangle$ does define a 2-form for $i=1,2$.
Using these 2-forms, we rewrite equation 9 as

$\displaystyle(d\mathcal{S})_{\mathcal{Z}(t,x)}=\omega^{K_{1}}(\cdot,\partial_{t}\mathcal{Z}(t,x))+\omega^{K_{2}}(\cdot,\partial_{x}\mathcal{Z}(t,x)).$
(10)

As $\omega^{K_{1}}=d\pi_{1}\wedge d\varphi$ and
$\omega^{K_{2}}=-d\pi_{2}\wedge d\varphi$ are both degenerate forms on
$\mathbb{R}^{3}$, this equation lives outside the world of symplectic
geometry. However, the equation above does fit in a different geometric
picture, called polysymplectic geometry. We will very briefly discuss here how
this works and leave the more detailed explanation of polysymplectic geometry
to section 4.

#### The polysymplectic form

First of all, define
$\Omega_{0}=\omega^{K_{1}}\otimes\partial_{t}+\omega^{K_{2}}\otimes\partial_{x}$.
This is a closed, non-degenerate $\mathbb{R}^{2}$-valued 2-form on
$\mathbb{R}^{3}$, where $\mathbb{R}^{2}=T_{t}S^{1}\times T_{x}S^{1}$ is
spanned by $\partial_{t}$ and $\partial_{x}$. By definition, $\Omega_{0}$ is
called a polysymplectic form. Note that $(d\mathcal{Z})_{(t,x)}$ defines a
linear map $\mathbb{R}^{2}\to\mathbb{R}^{3}$. Thus, for any vector
$V\in\mathbb{R}^{3}$ the expression
$\Omega_{0}(V,(d\mathcal{Z})_{(t,x)}(\cdot))$ defines a linear automorphism of
$\mathbb{R}^{2}$ and it therefore makes sense to consider the trace of this
map. We denote this trace by $\Omega_{0}^{\sharp}((d\mathcal{Z})_{(t,x)})(V)$.
With this notation, equation 10 is equivalent to

$\displaystyle(d\mathcal{S})_{\mathcal{Z}(t,x)}=\Omega_{0}^{\sharp}((d\mathcal{Z})_{(t,x)}).$
(11)

For a slightly more elaborate dicussion of $\Omega_{0}^{\sharp}$ and an
explanation of why equations 10 and 11 are equivalent, see section 4 and in
particular example 4.4.

Reformulating the wave equation in a covariant way as done above is usually
called the De Donder-Weyl formulation of covariant Hamiltonian field theories.
It is the standard way to alter the non-relativistic approach from the
previous paragraph to a theory that is invariant under linear coordinate
transformations in $O(1,1)$. The polysymplectic formulation of this approach
can be found in many articles, such as [Gün87, Hél01, Kan93].vvvSome of these
articles use multisymplectic instead of polysymplectic geometry to write down
these equations. However, the two theories are very similar.

In order to define a Floer theory for this equation, we must first define the
action functional $\mathcal{A}_{\mathcal{S}}\mathrel{\mathop{\mathchar
58\relax}}C^{\infty}(\mathbb{T}^{2},\mathbb{R}^{3})\to\mathbb{R}$. Note again
that we are interested in periodic solutions, so we view $\mathcal{Z}$ as a
map on $\mathbb{T}^{2}$. Let $dV=dt\wedge dx$ denote the volume form on
$\mathbb{T}^{2}$ and define

$\displaystyle\mathcal{A}_{\mathcal{S}}(\mathcal{Z})=\int_{\mathbb{T}^{2}}\left(\pi_{1}(t,x)\partial_{t}\varphi(t,x)-\pi_{2}(t,x)\partial_{x}\varphi(t,x)-\mathcal{S}(\mathcal{Z}(t,x))\right)dV.$
(12)

In section 4 it will be explained how this action functional is derived from
the polysymplectic form $\Omega_{0}$. For now it is only important to note
that critical points of this functional coincide with periodic solutions of
equation 8 (see lemma 4.8). To find these critical points, we again study
smooth trajectories of the negative $L^{2}$-gradient of
$\mathcal{A}_{\mathcal{S}}$. These trajectories, or Floer curves, are given by
the equation

$\displaystyle\partial_{s}\widetilde{\mathcal{Z}}(s,t,x)+K_{1}\partial_{t}\widetilde{\mathcal{Z}}(s,t,x)+K_{2}\partial_{x}\widetilde{\mathcal{Z}}(s,t,x)-\nabla\mathcal{S}(\widetilde{\mathcal{Z}}(s,t,x))=0,$
(13)

where $\widetilde{\mathcal{Z}}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{T}^{2}\to\mathbb{R}^{3}$. Equation 13 is the
Floer curve corresponding to the system (8).

Note that when $\widetilde{\mathcal{Z}}$ is independent of both $t$ and $x$,
the equation above reduces to the Morse theory of $\mathcal{S}$. One might
expect the equation to reduce to the original Floer equation 4 when
$\widetilde{\mathcal{Z}}$ is independent of $x$, however this is not the case
as $K_{1}$ does not define a complex structure (compare with equation 18).
Also, when $S\equiv 0$ we would hope that the equation would satisfy a
generalization of Gromov-Witten theory. However, the operator
$K_{1}\partial_{t}+K_{2}\partial_{x}$ is too degenerate for some analogue of
this theory to hold true. The next section will elaborate on this observation.

###  2.3 Degeneracy problem for Floer curves

As mentioned in the discussion of Floer theory for classical mechanics, an
analogue of lemma 2.1.i has to hold true in order for there to be any hope
that this Floer theory will work out. As we saw in lemma 2.1.i, when $H\equiv
0$, Gromov’s theory of pseudoholomorphic curves can be used to control the
limit of the Floer curve for $s$ going to infinity. However, in our current
model we will see that for $S\equiv 0$ problems occur.

The energy of a solution to equation 13 is given by

$\displaystyle
E(\widetilde{\mathcal{Z}})=-\int_{-\infty}^{+\infty}\frac{d}{ds}\mathcal{A}_{\mathcal{S}}(\widetilde{\mathcal{Z}}_{s})\,ds=\int_{-\infty}^{+\infty}\left(\int_{\mathbb{T}^{2}}|\partial_{s}\widetilde{\mathcal{Z}}(s,t,x)|^{2}\,dV\right)\,ds.$

Let $\widetilde{\mathcal{Z}}$ be a Floer curve for $\mathcal{S}\equiv 0$ of
finite energy. Assume that the mean of $\widetilde{\mathcal{Z}}_{s}$ is zero
for any $s$. We have to check if it follows that
$\lim_{s\to+\infty}\widetilde{\mathcal{Z}}_{s}=0$.

######  Counterexample 2.2.

Let

$\displaystyle\widetilde{\mathcal{Z}}(s,t,x)=\begin{pmatrix}0\\\ \cos(x)\\\
0\end{pmatrix}.$

Then
$\partial_{s}\widetilde{\mathcal{Z}}\equiv\partial_{t}\widetilde{\mathcal{Z}}\equiv
0$ and $\partial_{x}\widetilde{\mathcal{Z}}\in\ker K_{2}$, so indeed
$\widetilde{\mathcal{Z}}$ is a solution to equation 13 for $S\equiv 0$. Also,
$E(\widetilde{\mathcal{Z}})=0$ as $\partial_{s}\widetilde{\mathcal{Z}}\equiv
0$. The mean of $\widetilde{\mathcal{Z}}_{s}$ is
$\int_{\mathbb{T}^{2}}\widetilde{\mathcal{Z}}_{s}(t,x)\,dV=0$, so
$\widetilde{\mathcal{Z}}$ satisfies our assumptions. However, the limit for
$s$ going to $+\infty$ is clearly not the zero-function.

The reason that lemma 2.1.i does not hold for equation 13, is the fact that
the operator $K_{\partial}=K_{1}\partial_{t}+K_{2}\partial_{x}$ has an
infinite-dimensional kernel. It is given by $\ker
K_{\partial}=\\{(\phi,\partial_{x}\psi,\partial_{t}\psi)\mid\phi\textup{ is
constant, }\psi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{T}^{2}\to\mathbb{R}\\}$. On the contrary, the kernel of the
operator $J\partial_{t}$ from classical mechanics consists only of constant
functions. The covariant Floer equation described above is therefore too
degenerate to provide us with a working theory. That means that the De Donder-
Weyl formulation of the wave equation is not suited for Floer theory, which
explains why no such theory has been developed as of yet. The degeneracy
problem of the operator $K_{\partial}$ has been pointed out already in
[Bri06]. The next section is devoted to solving this problem. By the end of
the section we will have found a new Floer equation that in fact is suitable
for defining a Floer theory.

##  3 Bridges regularization

This section is devoted to solving the degeneracy problem from section 2.3.
Before discussing the general structure of a solution to that problem, we
start hands-on by manipulating the equations in (8). The idea for the
presented solution comes from [Bri06].

The fundamental problem with equation 8 is that the matrices $K_{1}$ and
$K_{2}$ are degenerate. To make them invertible, we add a new variable and a
new equation to the system:

$\displaystyle\begin{split}-\partial_{t}\pi_{1}(t,x)+\partial_{x}\pi_{2}(t,x)&=0\\\
\partial_{t}\varphi(t,x){\color[rgb]{0,0,1}-\partial_{x}o(t,x)}&=\pi_{1}(t,x)\\\
{\color[rgb]{0,0,1}\partial_{t}o(t,x)}-\partial_{x}\varphi(t,x)&=-\pi_{2}(t,x)\\\
{\color[rgb]{0,0,1}-\partial_{t}\pi_{2}(t,x)+\partial_{x}\pi_{1}(t,x)}&={\color[rgb]{0,0,1}0}.\end{split}$
(14)

If $Z=(\varphi,\pi_{1},\pi_{2},o)$ and
$S(Z)=\frac{1}{2}\pi_{1}^{2}+\frac{1}{2}\pi_{2}^{2}$, then equation 14 can be
formulated as

$\displaystyle M_{1}\partial_{t}Z(t,x)+M_{2}\partial_{x}Z(t,x)=\nabla
S(Z(t,x)),$ (15)

where

$\displaystyle M_{1}=\begin{pmatrix}0&-1&0&{\color[rgb]{0,0,1}0}\\\
1&0&0&{\color[rgb]{0,0,1}0}\\\ 0&0&0&{\color[rgb]{0,0,1}1}\\\
{\color[rgb]{0,0,1}0}&{\color[rgb]{0,0,1}0}&{\color[rgb]{0,0,1}-1}&{\color[rgb]{0,0,1}0}\end{pmatrix}$
$\displaystyle M_{2}=\begin{pmatrix}0&0&1&{\color[rgb]{0,0,1}0}\\\
0&0&0&{\color[rgb]{0,0,1}-1}\\\ -1&0&0&{\color[rgb]{0,0,1}0}\\\
{\color[rgb]{0,0,1}0}&{\color[rgb]{0,0,1}1}&{\color[rgb]{0,0,1}0}&{\color[rgb]{0,0,1}0}\end{pmatrix}.$

The last rows and columns are made blue to highlight the fact that the
matrices $K_{1}$ and $K_{2}$ are in some way ”contained” in these new
matrices. Note that the additional terms in equation 14 are chosen in such a
way that $M_{1}$ and $M_{2}$ are both anti-symmetric invertible matrices.
Moreover, they both define complex structures on $\mathbb{R}^{4}$. Thus, we
can define two symplectic forms on $\mathbb{R}^{4}$ by

$\displaystyle\omega^{M_{1}}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\langle\cdot,M_{1}\cdot\rangle=d\pi_{1}\wedge
d\varphi{\color[rgb]{0,0,1}+d\pi_{2}\wedge do}$ $\displaystyle\omega^{M_{2}}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\langle\cdot,M_{2}\cdot\rangle=-d\pi_{2}\wedge
d\varphi{\color[rgb]{0,0,1}-d\pi_{1}\wedge do}.$

Clearly $\Omega_{B}\mathrel{\mathop{\mathchar
58\relax}}=\omega^{M_{1}}\otimes\partial_{t}+\omega^{M_{2}}\otimes\partial_{x}$
defines a closed, non-degenerate $\mathbb{R}^{2}$-valued 2-form on
$\mathbb{R}^{4}$ and thus is a polysymplectic form. Just like in equation 11,
we can reformulate (15) as

$\displaystyle(dS)_{Z(t,x)}=\Omega_{B}^{\sharp}((dZ)_{(t,x)}).$ (16)

We will refer to this equation as the Bridges formulation of the wave
equation. Equation 16 looks very similar to the De Donder-Weyl formulation of
the wave equation, but the polysymplectic form used here has much more
structure, as its separate components consist of symplectic forms. The idea is
that this extra structure will allow us to prove an analogue of lemma 2.1.i
and even of lemma 2.1.ii for this equation.

To find the Floer equation corresponding to equation 16, we have to define an
action functional once again. Again, we are interested in periodic solution,
so we let $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{T}^{2}\to\mathbb{R}^{4}$ and define the action
functionalviviviWe dropped all the arguments $(t,x)$ in the notation here in
order not to make the formula too long.

$\displaystyle\mathcal{A}_{S}(Z)=\int_{\mathbb{T}^{2}}\left(\pi_{1}\partial_{t}\varphi-\pi_{2}\partial_{x}\varphi-o(\partial_{t}\pi_{2}-\partial_{x}\pi_{1})-\mathcal{S}(\mathcal{Z})\right)dV$
(17)

whose critical points are the solutions to equation 16. We refer to section 4
for the derivation of this action functional and the computation of its
critical points. The corresponding Floer equation for this action can be found
once again by studying its negative $L^{2}$-gradient lines. These trajectories
satisfy the equation

$\displaystyle\partial_{s}\widetilde{Z}(s,t,x)+M_{1}\partial_{t}\widetilde{Z}(s,t,x)+M_{2}\partial_{x}\widetilde{Z}(s,t,x)-\nabla
S(\widetilde{Z}(s,t,x))=0,$ (18)

where $\widetilde{Z}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{T}^{2}\to\mathbb{R}^{4}$ is a smooth map.

This new Floer equation is a lot nicer than equation 13 in multiple ways. Some
of the reasons why will be discussed in the following sections. Note that just
as for equation 13, the equation above reduces to Morse theory when
$\widetilde{Z}$ is independent of both $t$ and $x$. However, unlike equation
13, when $\widetilde{Z}$ is just independent of $x$, the original Floer
equation 4 is recovered. This is due to the fact that $M_{1}$ defines a
complex structure on $\mathbb{R}^{4}$. Moreover, as $M_{2}$ defines a complex
structure as well, the same statement holds when $\widetilde{Z}$ is only
independent of $t$. Finally, when $S\equiv 0$ we no longer have the degeneracy
problem mentioned in section 2.3 and the discussion right before. The next
section will show that indeed an analogue of lemma 2.1.i can be proven for
equation 18.

###  3.1 Degeneracy problem for Floer curves - solved

From this section on, we must explicitly define the periodicity conditions
that we are interested in. We will always assume the space-period to be $2\pi$
and the time-period to be some real number $T$. That is, we identify the
2-torus with
$\mathbb{T}^{2}=(\mathbb{R}/T\mathbb{Z})\times(\mathbb{R}/2\pi\mathbb{Z})$. It
must be mentioned that the degeneracy problem depends on the ratio
$\mathcal{R}=\frac{2\pi}{T}$. Note, for example, when $T=2\pi$ that the free
wave equation (5) has an infinite-dimensional space of solutions containing
$\varphi(t,x)=f(t+x)+g(t-x)$ for all $2\pi$-periodic functions $f$ and $g$.
This type of degeneracy is different from the degeneracy problem of section
2.3 in the sense that it is already a part of the free wave equation and does
not depend on the geometric approach that we choose to take. Even in the
infinite-dimensional symplectic approach taken in [FL22] (see section 2.1)
this type of degeneracy is excluded a priori. Thus, we restrict our attention
to the case where $\mathcal{R}$ has irrationality measure equal to 2.

######  Definition 3.1.

The _irrationality measure_ of a real number $r$ is defined to be the infimum
of all $\rho$ for which there exists some constant $c$ such that

$\displaystyle\frac{c}{q^{\rho}}<|r-\frac{p}{q}|$

for all $\frac{p}{q}\in\mathbb{Q}$.

Theorem E.3 from [Bug12] asserts that the set of real number with
irrationality measure equal to 2 has full measure within the set of real
numbers. Thus, the requirement on $\mathcal{R}$ stated above is not too
restrictive. Note that in particular, our assumption implies that
$\mathcal{R}$ is irrational.

Now that we have settled the necessary condition on $\mathcal{R}$, we will
prove an analogue of lemma 2.1.i. First of all, we define the energy of a
solution $\widetilde{Z}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{T}^{2}\to\mathbb{R}^{4}$ of equation 18 as

$\displaystyle E(\widetilde{Z})$
$\displaystyle=-\int_{-\infty}^{+\infty}\frac{d}{ds}\mathcal{A}_{S}(\widetilde{Z}(s))\,ds$
$\displaystyle=\int_{-\infty}^{+\infty}\left(\int_{\mathbb{T}^{2}}\mathinner{\\!\left\lvert\partial_{s}\widetilde{Z}\right\rvert}^{2}dV\right)ds.$

######  Lemma 3.2.

Let $\widetilde{Z}$ be a Floer curve of finite energy for $S\equiv 0$.
Moreover, assume that $\widetilde{Z}_{s}$ has mean zero for every
$s\in\mathbb{R}$. Then

$\displaystyle\lim_{s\to\infty}\widetilde{Z}_{s}=0$

in $C^{\infty}(\mathbb{T}^{2},\mathbb{R}^{4})$.

###### Proof.

We are going to prove the lemma by showing that the $L^{2}$-norms of all
space-time derivatives of $\widetilde{Z}_{s}$ converge to 0. This will imply
convergence in $C^{\infty}(\mathbb{T}^{2},\mathbb{R}^{4})$. As
$\widetilde{Z}_{s}$ is periodic in time and space we can define its Fourier
transform as follows:

$\widetilde{Z}_{s}(t,x)=\sum_{m,k\in\mathbb{Z}}\hat{Z}_{s}(m,k)e^{\frac{2\pi
i}{T}mt}e^{ikx}.$

Note that the assumption that the mean of $\widetilde{Z}_{s}$ is zero for all
$s$, implies that $\hat{Z}_{s}(0,0)=0$. Thus, in what follows we may restrict
our attention to $(m,k)\neq(0,0)$. Applying equation (18) to the Fourier
series gives

$\displaystyle 0$ $\displaystyle=\partial_{s}\hat{Z}_{s}(m,k)+\frac{2\pi
i}{T}mM_{1}\hat{Z}_{s}(m,k)+ikM_{2}\hat{Z}_{s}(m,k)$ (19)
$\displaystyle=\partial_{s}\hat{Z}_{s}(m,k)+A(m,k)\hat{Z}_{s}(m,k),$

where

$\displaystyle A(m,k)$ $\displaystyle=\frac{2\pi i}{T}mM_{1}+ikM_{2}$
$\displaystyle=\begin{pmatrix}0&-\mathcal{R}im&ik&0\\\
\mathcal{R}im&0&0&-ik\\\ -ik&0&0&\mathcal{R}im\\\
0&ik&-\mathcal{R}im&0\end{pmatrix}.$

As $A(m,k)$ is a Hermitian matrix for every $m$ and $k$, its eigenvalues are
real. The eigenvalues
$\\{\lambda^{1}_{+}(m,k),\lambda^{1}_{-}(m,k),\lambda^{2}_{+}(m,k),\lambda^{2}_{-}(m,k)\\}$
are given by

$\displaystyle\lambda^{1}_{+}(m,k)$
$\displaystyle=-\lambda^{1}_{-}(m,k)=\mathinner{\\!\left\lvert
k+m\mathcal{R}\right\rvert}>0$ $\displaystyle\lambda^{2}_{+}(m,k)$
$\displaystyle=-\lambda^{2}_{-}(m,k)=\mathinner{\\!\left\lvert
k-m\mathcal{R}\right\rvert}>0.$

Note that the strict inequalities above come from the assumption that
$\mathcal{R}$ is irrational and $(m,k)\neq(0,0)$.

Equation 19 is a linear ODE and has solution

$\displaystyle\hat{Z}_{s}(m,k)=e^{-A(m,k)s}\hat{Z}_{0}(m,k)$

for some initial condition $\hat{Z}_{0}(m,k)$. Thus,

$\displaystyle\partial_{s}\hat{Z}_{s}(m,k)=-A(m,k)e^{-A(m,k)s}\hat{Z}_{0}(m,k).$
(20)

If $v^{i}_{+}(m,k)$ and $v^{i}_{-}(m,k)$ denote eigenvectors corresponding to
$\lambda^{i}_{+}(m,k)$ and $\lambda^{i}_{-}(m,k)$ respectively, than we can
write

$\hat{Z}_{0}(m,k)=\sum_{i=1}^{2}\left(\alpha^{i}_{+}(m,k)v^{i}_{+}(m,k)+\alpha^{i}_{-}(m,k)v^{i}_{-}(m,k)\right).$

Note that as $A(m,k)$ is Hermitian, the different eigenvectors are orthogonal.
Also

$E(\widetilde{Z})=\int_{-\infty}^{+\infty}\left(\int_{\mathbb{T}^{2}}\mathinner{\\!\left\lvert\partial_{s}\widetilde{Z}_{s}(t,x)\right\rvert}^{2}dV\right)\,ds=\int_{-\infty}^{+\infty}\left(\sum_{m,k}\mathinner{\\!\left\lvert\partial_{s}\hat{Z}_{s}(m,k)\right\rvert}^{2}\right)\,ds<+\infty,$

which implies in particular that there is a sequence $s_{j}\to+\infty$ such
that $\partial_{s}\hat{Z}_{s_{j}}(m,k)\to 0$. Equation 20 shows that this is
only possible when $\alpha^{i}_{-}(m,k)=0$ for $i=1,2$. We see that

$\displaystyle\hat{Z}_{s}(m,k)=e^{-\lambda^{1}_{+}(m,k)s}\alpha^{1}_{+}(m,k)v^{1}_{+}(m,k)+e^{-\lambda^{2}_{+}(m,k)s}\alpha^{2}_{+}(m,k)v^{2}_{+}(m,k).$
(21)

Fix $l_{1},l_{2}\in\mathbb{N}$ and let $\epsilon>0$. We want to prove that for
$s$ big enough it holds that
$||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}\widetilde{Z}_{s}||_{L^{2}}^{2}<\epsilon$.
Note first that the smoothness of $\widetilde{Z}_{0}$ implies that

$\displaystyle||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}\widetilde{Z}_{0}||_{L^{2}}^{2}=\sum_{m,k}\sum_{i=1}^{2}(\mathcal{R}m)^{2l_{1}}k^{2l_{2}}|\alpha^{i}_{+}(m,k)v^{i}_{+}(m,k)|^{2}<+\infty.$

Thus, in particular we can find some $N$ such that

$\displaystyle\sum_{|(m,k)|>N}\sum_{i=1}^{2}(\mathcal{R}m)^{2l_{1}}k^{2l_{2}}|\alpha^{i}_{+}(m,k)v^{i}_{+}(m,k)|^{2}<\frac{\epsilon}{2}.$

As $\lambda^{i}_{+}(m,k)>0$ for all $m,k$, this implies that for any $s>0$

$\displaystyle\sum_{|(m,k)|>N}\sum_{i=1}^{2}(\mathcal{R}m)^{2l_{1}}k^{2l_{2}}e^{-2\lambda^{i}_{+}(m,k)s}|\alpha^{i}_{+}(m,k)v^{i}_{+}(m,k)|^{2}<\frac{\epsilon}{2}.$
(22)

Now, let $\lambda\mathrel{\mathop{\mathchar
58\relax}}=\min\\{\lambda^{i}_{+}(m,k)\mid i=1,2\textup{ and }|(m,k)|\leq
N\\}$ and note $\lambda>0$. Then

$\displaystyle\sum_{|(m,k)|\leq
N}\sum_{i=1}^{2}(\mathcal{R}m)^{2l_{1}}k^{2l_{2}}e^{-2\lambda^{i}_{+}(m,k)s}|\alpha^{i}_{+}(m,k)v^{i}_{+}(m,k)|^{2}\leq
e^{-\lambda
s}||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}\widetilde{Z}_{0}||_{L^{2}}^{2}.$

Thus, if we choose
$s_{0}>\frac{1}{\lambda}\log(2||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}\widetilde{Z}_{0}||_{L^{2}}^{2}/\epsilon)$
(assuming $\epsilon$ is small enough for this to be positive), then for
$s>s_{0}$ we get that

$\displaystyle\sum_{|(m,k)|\leq
N}\sum_{i=1}^{2}(\mathcal{R}m)^{2l_{1}}k^{2l_{2}}e^{-2\lambda^{i}_{+}(m,k)s}|\alpha^{i}_{+}(m,k)v^{i}_{+}(m,k)|^{2}<\frac{\epsilon}{2}.$

Combining this with equation 22 it follows that for $s>s_{0}$

$\displaystyle||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}\widetilde{Z}_{s}||_{L^{2}}^{2}=\sum_{m,k}\sum_{i=1}^{2}(\mathcal{R}m)^{2l_{1}}k^{2l_{2}}e^{-2\lambda^{i}_{+}(m,k)s}|\alpha^{i}_{+}(m,k)v^{i}_{+}(m,k)|^{2}<\epsilon.$

This proves the lemma ∎

This proof shows that equation 18 solves the degeneracy problem from section
2.3. In section 5 we will prove that even an analogue of lemma 2.1.ii can be
proven for the new Floer equation.

###  3.2 Underlying structure

At the start of section 3, a new variable and equation were introduced in
order to improve the Floer equation corresponding to De Donder-Weyl. However,
even though it turns out that indeed we can get an improved Floer equation out
of this, no explanation was given on where this new set of equations comes
from. This section is devoted to explaining the structure behind equation 14.
The ideas from this section are based on [Bri06].

First, we look back at our discussion of classical mechanics in section 2. The
fundamental operator here was $J\frac{d}{dt}$. This operator comes in
naturally also from a geometric point of view, when we look at differential
forms on $\mathbb{R}$. The most fundamental operator on differential forms is
the exterior derivative $d$. For functions $q$ on $\mathbb{R}$ it is given by
$dq=\frac{d}{dt}q\,dt$. Correspondingly, one can also define the
codifferential $\delta$ that takes a 1-form to a 0-form on $\mathbb{R}$. It is
given by $\delta(p\,dt)=-\frac{d}{dt}p$ for functions $p$ on $\mathbb{R}$.
Choosing the basis $\\{1,dt\\}$ for the total exterior algebra
$\bigwedge(T^{*}\mathbb{R})$ identifies it with $\mathbb{R}^{2}$. Under this
identification the operator $J\frac{d}{dt}$ on functions
$u\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\to\mathbb{R}^{2}$
corresponds to the operator

$\displaystyle\begin{pmatrix}o&\delta\\\ d&0\end{pmatrix}$ (23)

on sections $q+p\,dt\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\to\bigwedge(T^{*}\mathbb{R})$.

Moving from classical mechanics to field theory, a space-variable is added to
the system and the functions $\varphi$ are now defined on $\mathbb{R}^{2}$. We
can try to follow the same idea as above to get an operator on differential
forms on $\mathbb{R}^{2}$. To define the codifferential, first a metric has to
be chosen on $\mathbb{R}^{2}$. Since the wave equation comes from the Laplace
operator on Minkowski space, we put a Minkowski metric on $\mathbb{R}^{2}$. We
choose the Minkowski metric of signature $(+-)$, meaning that
$\langle\partial_{t},\partial_{t}\rangle=-\langle\partial_{x},\partial_{x}\rangle=1$.
We will denote $\mathbb{R}^{2}$ with this Minkowski metric by
$\mathbb{R}^{1,1}$. The total exterior algebra bundle
$\Omega(\mathbb{R}^{1,1})=\Gamma(\bigwedge(T^{*}\mathbb{R}^{1,1}))$ of this
space comes equipped with a codifferential $\delta_{k}$ that transforms
$k$-forms into $(k-1)$-forms. It is defined by
$\delta_{k}=(-1)^{k}\star^{-1}d\star$, where $\star$ denotes the Hodge star.
The latter is the linear operator given by

$\displaystyle\star 1$ $\displaystyle=dV$ $\displaystyle\star dt$
$\displaystyle=dx$ $\displaystyle\star dx$ $\displaystyle=dt$
$\displaystyle\star dV$ $\displaystyle=-1,$

A simple calculation shows

$\displaystyle\delta_{1}(\pi_{1}dt+\pi_{2}dx)$
$\displaystyle=-\partial_{t}\pi_{1}+\partial_{x}\pi_{2}$
$\displaystyle\delta_{2}(o\,dV)$
$\displaystyle=-\partial_{t}odx-\partial_{x}odt,$

for functions $\pi_{1}$,$\pi_{2}$ and $o$. Now, analogously to the operator
(23) defined above, Bridges defines an operator $J_{\partial}$ on
$\Omega(\mathbb{R}^{1,1})$ by

$\displaystyle J_{\partial}=\begin{pmatrix}0&\delta_{1}&0\\\
d_{0}&0&\delta_{2}\\\ 0&d_{1}&0\end{pmatrix}.$

The bundle $\Omega(\mathbb{R}^{1,1})$ can be trivialised by choosing the
global frame $\\{1,dt,dx,dV\\}$. In this identification,
$\Omega(\mathbb{R}^{1,1})$ can be seen as the space of maps
$\mathbb{R}^{1,1}\to\mathbb{R}^{2,2}$, where we use the Lorentzian
spaceviiviiviiSometimes we will just denote it by $\mathbb{R}^{4}$, when the
focus does not lie on the metric. $\mathbb{R}^{2,2}$ because the induced
metric on the total exterior algebra is given by

$\displaystyle\mathcal{L}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{pmatrix}.$

in this basis. We can also write out $J_{\partial}$ as

$\displaystyle J_{\partial}=J_{1}\partial_{t}+J_{2}\partial_{x},$

where

$\displaystyle J_{1}=\begin{pmatrix}0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\\\
0&0&1&0\end{pmatrix}$ $\displaystyle J_{2}=\begin{pmatrix}0&0&1&0\\\
0&0&0&-1\\\ 1&0&0&0\\\ 0&-1&0&0\end{pmatrix}.$

Now, Bridges looks at Hamiltonians $S\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{2,2}\to\mathbb{R}$ and considers the equation:

$\displaystyle J_{\partial}Z(t,x)=\nabla_{L}S(Z(t,x)),$ (24)

for $Z\in\Omega(\mathbb{R}^{1,1})$. Here, $\nabla_{L}=\mathcal{L}\nabla$ is
the Lorentzian gradient. After multiplying both sides of the equation with the
Lorentzian metric $\mathcal{L}$, equation (24) can be written as

$\displaystyle M_{1}\partial_{t}Z(t,x)+M_{2}\partial_{x}Z(t,x)=\nabla
S(Z(t,x)),$

where $\nabla$ is the Riemannian gradient. This is exactly equation 15. Note
that it is equivalent to equation 24 as $\mathcal{L}$ is invertible.

The exposition above shows that equation 15 arises naturally from a geometric
point of view as a generalisation of equation 2. Note that if we would have
restricted the operator $J_{\partial}$ to the space of 0-forms and 1-forms, we
would have recovered equation 9. However, when we are working on a
2-dimensional space-time then excluding the 2-forms is somehow less logical
than considering $J_{\partial}$ as an operator on the full space of
differential forms. Even though in this article we focus primarily on
2-dimensional space-time, the construction of $J_{\partial}$ can easily be
generalized to higher dimensions. See [Bri06] for the details.

###  3.3 The infinite-dimensional viewpoint

As discussed above, equation 18 is in some way more natural to consider as a
Floer equation than equation 13. In this section we will discuss another
advantage of equation 18 over equation 13. That is that, after making a fixed
splitting of time and space, equation 18 fits into the infinite-dimensional
symplectic framework of [FL22].

As discussed in section 2.1, [FL22] analyse the wave equation by using
infinite-dimensional symplectic geometry. To translate to their viewpoint, for
the rest of this section we regard $Z$ as a map $S^{1}\to
L^{2}(S^{1},\mathbb{R}^{4})$. By doing this we break the covariance and view
$Z$ as a map from time to a space of functions of the space-variable. In order
to get a symplectic equation, we first need a complex structure on
$L^{2}(S^{1},\mathbb{R}^{4})$. Note that $M_{1}^{2}=-\mathbb{1}$, so it
provides us with this complex structure. As we know from equation 6, the
symplectic equation corresponding to some Hamiltonian
$\mathscr{H}\mathrel{\mathop{\mathchar
58\relax}}L^{2}(S^{1},\mathbb{R}^{4})\to\mathbb{R}$, is

$\displaystyle M_{1}\frac{d}{dt}Z(t)=\textup{grad}\mathscr{H}(Z(t)).$ (25)

In order to recover equation 15, we define

$\mathscr{H}(Z)=-\frac{1}{2}\int_{S^{1}}\langle
M_{2}\partial_{x}Z,Z\rangle\,dx+\int_{S^{1}}S(Z(x))\,dx.$

Then we see that $\textup{grad}\mathscr{H}(Z)=-M_{2}\partial_{x}Z+S\circ Z$,
so that equation 25 becomes the same as equation 15. Note in particular that
the operator $M_{2}\partial_{x}$ that was used to define this Hamiltonian is
self-adjoint with respect to the $L^{2}$-inner product. This ensures the
existence of a complete eigenbasis with real eigenvalues, which is the
assumption that [FL22] starts with (see section 5.3).

We see that indeed the equations coming from the Bridges regularization can be
put into the framework of [FL22]. This was not possible for the original De
Donder-Weyl equation, as neither $K_{1}$ nor $K_{2}$ define a complex
structure. The upshot is that for certain choices of non-linearities, results
from [FL22] may be used to conclude the existence of Floer curves and thus the
existence of periodic solutions of the wave equation. In section 5.3 we will
encounter a natural class of non-linearities that become _$\infty$
-regularizing_ (see definition 5.9) when translated to the infinite-
dimensional framework and thus fit into the work of Fabert and Lamoree.
However, as we will see in section 5, the covariant Floer equation is also
more general in a certain way and we can introduce non-linearities that the
framework of [FL22] cannot deal with.

###  3.4 The space of solutions

Before concluding this section, we want to see that the extra variable $o$ in
equation 14 does not alter the space of solutions that we consider, in the
sense that the space of solutions should still be in one-to-one correspondence
with the solutions to the wave equation. It turns out that this is not
entirely true, but every solution to the wave equation yields a one-
dimensional space of solutions to equation 14. This problem can easily be
fixed. To make the discussion a little more general, we introduce a non-
linearity to the equation now. That is, we consider the wave equation

$\displaystyle-\partial_{t}^{2}\varphi(t,x)+\partial_{x}^{2}\varphi(t,x)=f(\varphi(t,x)),$
(26)

where, for now, $f$ can be any function. The system of equations corresponding
to equation 14, including this non-linearity is

$\displaystyle\begin{split}-\partial_{t}\pi_{1}(t,x)+\partial_{x}\pi_{2}(t,x)&=f(\varphi(t,x))\\\
\partial_{t}\varphi(t,x){-\partial_{x}o(t,x)}&=\pi_{1}(t,x)\\\
{\partial_{t}o(t,x)}-\partial_{x}\varphi(t,x)&=-\pi_{2}(t,x)\\\
{-\partial_{t}\pi_{2}(t,x)+\partial_{x}\pi_{1}(t,x)}&={0}.\end{split}$ (27)

Clearly, when $Z=(\varphi,\pi_{1},\pi_{2},o)$ is a solution to equation 27,
then $\varphi$ solves the wave equation (26). This can be seen by filling in
$\pi_{1}$ and $\pi_{2}$ from the second and third equation into the first
equation in (27). For the converse, note that filling in $\pi_{1}$ and
$\pi_{2}$ into the last equation, yields that $o$ solves the free wave
equation. We will use the following lemma.

######  Lemma 3.3.

The only periodic solutions of the free wave equation

$\displaystyle{-\partial_{t}^{2}o(t,x)+\partial_{x}^{2}o(t,x)}$
$\displaystyle={0}$

are $o\equiv c$ for some constant $c\in\mathbb{R}$, when
$\mathcal{R}=\frac{2\pi}{T}$ is irrational.

###### Proof.

Clearly, all constant maps $o\equiv c$ satisfy the wave equation. Conversely,
if $o$ satisfies the free wave equation, then its Fourier transform $\hat{o}$
satisfies

$\displaystyle\mathcal{R}^{2}m^{2}\hat{o}(m,k)-k^{2}\hat{o}(m,k)=0.$

As $\mathcal{R}^{2}m^{2}-k^{2}$ is zero only when $m=k=0$, it must hold that
$o(m,k)=0$ for $(m,k)\neq(0,0)$. Thus the only non-zero Fourier coefficient is
$o(0,0)$, which proves that $o$ is constant. ∎

Now, let $\varphi$ be any solution to equation 26. Then setting
$\pi_{1}=\partial_{t}\varphi$, $\pi_{2}=\partial_{x}\varphi$ and $o\equiv c$
yields a solution to equation 27. Thus indeed, we find that every solution of
the wave equation corresponds to a one-dimensional space of solutions to
equation 27. We can circumvent this problem by requiring $o$ to have mean
zero.

One might wonder what the actual difference between equation 8 and equation 14
is, now that we know that the extra variable $o$ is constant anyways. The
difference however doesn’t lie per se in the equation itself, but in the Floer
equation it results in. To stress the difference between the two Floer
equations we write them out once more in a different way. Let $\epsilon$ be a
parameter and consider the set of equations

$\displaystyle\partial_{s}\widetilde{\varphi}(s,t,x)-\partial_{t}\widetilde{\pi}_{1}(s,t,x)+\partial_{x}\widetilde{\pi}_{2}(s,t,x)$
$\displaystyle=0$
$\displaystyle\partial_{s}\widetilde{\pi}_{1}(s,t,x)+\partial_{t}\widetilde{\varphi}(s,t,x)-\epsilon\partial_{x}\widetilde{o}(s,t,x)$
$\displaystyle=\widetilde{\pi}_{1}(s,t,x)$
$\displaystyle\partial_{s}\widetilde{\pi}_{2}(s,t,x)+\epsilon\partial_{t}\widetilde{o}(s,t,x)-\partial_{x}\widetilde{\varphi}(s,t,x)$
$\displaystyle=-\widetilde{\pi}_{2}(s,t,x)$
$\displaystyle\epsilon\partial_{s}\widetilde{o}(s,t,x)-\epsilon\partial_{t}\widetilde{\pi}_{2}(s,t,x)+\epsilon\partial_{x}\widetilde{\pi}_{1}(s,t,x)$
$\displaystyle=0.$

For $\epsilon=0$ this gives the Floer equation 13 for de De Donder-Weyl
equation from section 2.2, while for $\epsilon=1$ it describes the Floer
equation 18 corresponding to Bridges’ equations. Even though the $o$ variable
is constant for periodic solutions of equation 14, it doesn’t have to be for
solutions of the Floer equation. As should hopefully be clear by now, the
change from $\epsilon=0$ to $\epsilon=1$ in the equations above, precisely
turns the Floer equation into a workable equation upon which we can build the
theory.

##  4 The polysymplectic formalism

In sections 2 and 3 the language of polysymplectic geometry was briefly
introduced and used to formulate symmetric approaches to field theory. This
section will give the more rigorous background of the material and show
exactly how the polysymplectic equations come about. As a reference for this
section we refer to [Gün87].

###  4.1 The polysymplectic Hamiltonian formalism

Let $M$ be a vector space. Most of the definitions in this section apply to
more general manifolds as well, but as mentioned in the introduction, the
treatment of this will be postponed to a follow-up article.

######  Definition 4.1.

Let $\Omega$ be an $\mathbb{R}^{n}$-valued form on $M$. If $\Omega(V,\cdot)=0$
implies $V=0$ for $V\in TM$, then $\Omega$ is called _non-degenerate_.

######  Definition 4.2.

An $\mathbb{R}^{n}$-valued 2-form $\Omega$ on $M$ is called _polysymplectic_
if it is closed and non-degenerate. The pair $(M,\Omega)$ is called a
_polysymplectic manifold_.

We know that a symplectic form combined with a Hamiltonian function yields an
equation, like the one in equation 3. In a similar way we want to combine
polysymplectic forms with Hamiltonians. We already saw two examples in
equations 11 and 16. Here, the general construction will be explained.

Let $X\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{n}\to TM$ denote any
linear map and $V\in TM$. Then the map $\nu\mapsto\Omega(V,X(\nu))$ is a
linear map from $\mathbb{R}^{n}$ to itself of which we can take the trace.
Thus we get a map $TM\to\mathbb{R}$ given by $V\mapsto\textup{tr
}\Omega(V,X(\cdot))$ and denote this map by $\Omega^{\sharp}(X)$. Thus
$\Omega^{\sharp}\mathrel{\mathop{\mathchar
58\relax}}\textup{Hom}(\mathbb{R}^{n},TM)\to T^{*}M$ is given by

$\Omega^{\sharp}(X)(V)=\textup{tr}\left(\nu\mapsto\Omega(V,X(\nu))\right).$

Now for any map $Z\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{n}\to M$ it
holds that $(dZ)_{\vec{x}}$ is a linear map $\mathbb{R}^{n}\to TM$ for
$\vec{x}\in\mathbb{R}^{n}$. Given a HamiltonianviiiviiiviiiNote that
polysymplectic Hamiltonians are denoted by $S$ in this article, whereas
symplectic Hamiltonians are denoted $H$. $S\mathrel{\mathop{\mathchar
58\relax}}M\to\mathbb{R}$ we get the Hamiltonian equation

$\displaystyle(dS)_{Z(\vec{x})}=\Omega^{\sharp}((dZ)_{\vec{x}})$ (28)

for any $\vec{x}\in\mathbb{R}^{n}$

######  Example 4.3.

First of all, let’s take $n=1$, $M=\mathbb{R}^{2}$ and $\Omega=\omega=dp\wedge
dq$ the standard symplectic form on $\mathbb{R}^{2}$. For a curve
$u\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\to\mathbb{R}^{2}$ we get
that $\omega^{\sharp}((du)_{t})(V)=\textup{tr
}\omega(V,(du)_{t}(\cdot))=\omega(V,\frac{d}{dt}u(t))$. Thus, equation 28
reduces to equation 3 in this case. It follows that the polysymplectic
formalism indeed extends the symplectic formalism.

######  Example 4.4.

Now let $n=2$, $M=\mathbb{R}^{d}$ and choose 2-forms $\omega_{1}$ and
$\omega_{2}$ on $\mathbb{R}^{d}$ such that
$\Omega=\omega_{1}\otimes\partial_{t}+\omega_{2}\otimes\partial_{x}$ is a
polysymplectic form. For $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{2}\to\mathbb{R}^{d}$ the map
$\nu\mapsto\Omega(V,(dZ)_{(t,x)}(\nu))$ is given by

$\displaystyle\begin{pmatrix}\nu_{t}\\\
\nu_{x}\end{pmatrix}\mapsto\begin{pmatrix}\omega_{1}(V,\partial_{t}Z(t,x))&\omega_{1}(V,\partial_{x}Z(t,x))\\\
\omega_{2}(V,\partial_{t}Z(t,x))&\omega_{2}(V,\partial_{x}Z(t,x))\end{pmatrix}\begin{pmatrix}\nu_{t}\\\
\nu_{x}\end{pmatrix}$

for $\nu=(\nu_{t},\nu_{x})\in\mathbb{R}^{2}$. This map has trace
$\omega_{1}(V,\partial_{t}Z(t,x))+\omega_{2}(V,\partial_{x}Z(t,x))$. Thus,

$\Omega^{\sharp}((dZ)_{(t,x)})=\omega_{1}(\cdot,\partial_{t}Z(t,x))+\omega_{2}(\cdot,\partial_{x}Z(t,x)).$

If we put $d=3$ and $\omega_{i}=\omega^{K_{i}}$, then equation 28 becomes
equation 11 and the above computation shows that this is indeed equivalent to
equation 10. For $d=4$ and $\omega_{i}=\omega^{M_{i}}$ equation 16 is
recovered.

######  Remark 4.5.

Note that definition 4.2 of a polysymplectic form states that the form
$\Omega$ has to be non-degenerate. When $\Omega$ can be written as
$\Omega=\sum\omega_{i}\otimes\partial_{x_{i}}$ this does not however imply
that the separate forms $\omega_{i}$ have to be non-degenerate. Indeed, this
is exactly where the difference between $\Omega_{0}$ from section 2.2 and
$\Omega_{B}$ from section 3 lies. The components $\omega^{K_{i}}$ of
$\Omega_{0}$ are degenerate forms on $\mathbb{R}^{3}$, yet $\Omega_{0}$ itself
is non-degenerate. On the other hand, the components $\omega^{M_{i}}$ of
$\Omega_{B}$ are themselves already non-degenerate forms on $\mathbb{R}^{4}$,
from which it follows that $\Omega_{B}$ is non-degenerate as well.

###  4.2 From polysymplectic forms to action functionals

Now that we have established how polysymplectic forms combine with
Hamiltonians to yield field equations, we want to construct the action
functional that will allow us to construct a Floer theory. Thus, given a
polysymplectic form $\Omega$ on $M$ with values in $\mathbb{R}^{n}$ and a
Hamiltonian $S\mathrel{\mathop{\mathchar 58\relax}}M\to\mathbb{R}$, we want to
construct a functional $\mathcal{A}_{S}^{\Omega}\mathrel{\mathop{\mathchar
58\relax}}C^{\infty}(\mathbb{T}^{n},M)\to\mathbb{R}$, such that the critical
points of this functional coincide with the solutions to equation 28. In
symplectic geometry the action is defined by pulling back the symplectic form
along a periodic map and integrating over the circle. As we are working with
$\mathbb{R}^{n}$-valued forms we cannot simply integrate the pullback. First,
we must construct an ordinary differential form out of the polysymplectic
form.

Since we are interested in periodic solutions to equation 28, we will actually
replace $\mathbb{R}^{n}$ by $\mathbb{T}^{n}$ and consider $\Omega$ as a form
with values in $T\mathbb{T}^{n}$ instead of $\mathbb{R}^{n}$. By contracting
with the volume form $dV=dx_{1}\wedge\cdots\wedge dx_{n}$ on $\mathbb{T}^{n}$
it can also be viewed as a $\bigwedge^{n-1}(T^{*}\mathbb{T}^{n})$-valued form.
As $T\mathbb{T}^{n}$ is a trivial bundle, we can write
$\Omega=\sum_{i}\omega_{i}\otimes\partial_{i}$, for closed 2-forms
$\omega_{i}$ on $M$ and where $\partial_{i}\mathrel{\mathop{\mathchar
58\relax}}=\partial_{x_{i}}$. After contraction with $dV$ it becomes
$\sum_{i}(-1)^{i+1}\omega_{i}\otimes
dx_{1}\wedge\cdots\wedge\widehat{dx_{i}}\wedge\cdots\wedge dx_{n}$. Define
$\widetilde{\Omega}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{i}(-1)^{i+1}\omega_{i}\wedge
dx_{1}\wedge\cdots\wedge\widehat{dx_{i}}\wedge\cdots\wedge dx_{n}$ as an
$(n+1)$-form on $\mathbb{T}^{n}\times M$.ixixixThis interchanging of tensor
products and wedge products seems ad hoc, but comes from a bigger geometric
picture. In the language of [FG13], $\Omega$ is the _symbol_ of the horizontal
form $\widetilde{\Omega}$. As we are dealing with linear spaces, the
construction boils down to the simple change of tensor products and wedge
products. We refer to [FG13] for the general construction.

As $M$ is a vector space, all the forms $\omega_{i}$ are exact and we can
write $\omega_{i}=d\theta_{i}$. This implies that
$\widetilde{\Omega}=d\widetilde{\Theta}$ for
$\widetilde{\Theta}=\sum_{i}(-1)^{i+1}\theta_{i}\wedge
dx_{1}\wedge\cdots\wedge\widehat{dx_{i}}\wedge\cdots\wedge dx_{n}$. Note that
every map $Z\mathrel{\mathop{\mathchar 58\relax}}\mathbb{T}^{n}\to M$ can be
seen as a section $Z\mathrel{\mathop{\mathchar
58\relax}}\mathbb{T}^{n}\to\mathbb{T}^{n}\times M$ of the trivial bundle over
$\mathbb{T}^{n}$. Thus, it holds that $Z^{*}\widetilde{\Theta}$ is a well-
defined $n$-form on $\mathbb{T}^{n}$. Therefore, we can define the action
functional $\mathcal{A}_{S}^{\Omega}\mathrel{\mathop{\mathchar
58\relax}}C^{\infty}(\mathbb{T}^{n},M)\to\mathbb{R}$ by

$\displaystyle\mathcal{A}_{S}^{\Omega}(Z)=\int_{\mathbb{T}^{n}}Z^{*}\widetilde{\Theta}-\int_{\mathbb{T}^{n}}S(Z(t,x))\,dV$

######  Remark 4.6.

Note that there can be multiple choices of primitives $\theta_{i}$ leading to
different action functionals. However, if $\widetilde{\Theta}$ and
$\widetilde{\Psi}$ are two choices of primitives
$\widetilde{\Omega}=d\widetilde{\Theta}=d\widetilde{\Psi}$, then
$d(\widetilde{\Theta}-\widetilde{\Psi})=0$ so that
$Z^{*}(\widetilde{\Theta}-\widetilde{\Psi})\in H^{n}_{dR}(\mathbb{T}^{n})$.
Thus $Z^{*}(\widetilde{\Theta}-\widetilde{\Psi})=const\cdot dV$, meaning that
$\mathcal{A}_{S}^{\Omega}$ just shifts by a constant. As we are only
interested in relative values of $\mathcal{A}_{S}^{\Omega}$ this does not
matter.

######  Example 4.7.

If we take $n=2$ and $\Omega$ to be $\Omega_{0}$ from section 2.2, we get
$\widetilde{\Omega}_{0}=d\pi_{1}\wedge d\varphi\wedge dx+d\pi_{2}\wedge
d\varphi\wedge dt$ and $\widetilde{\Theta}_{0}=\pi_{1}d\varphi\wedge
dx+\pi_{2}d\varphi\wedge dt$. Thus,
$Z^{*}\widetilde{\Theta}_{0}=(\pi_{1}\partial_{t}\varphi-\pi_{2}\partial_{x}\varphi)dV$,
so that indeed the action functional $\mathcal{A}_{S}^{\Omega}$ defined above
coincides with equation 12. A similar computation shows that for
$\Omega=\Omega_{B}$ the action functional from equation 17 is recovered.

######  Lemma 4.8.

The critical points of $\mathcal{A}_{S}^{\Omega}$ coincide with the solutions
of equation 28.

###### Proof.

Let $Y$ be a tangent vector to $Z\in C^{\infty}(\mathbb{T}^{n},M)$. We view
$Y$ as a map $\mathbb{T}^{n}\to TM$. Now

$\displaystyle(d\mathcal{A}_{S}^{\Omega})_{Z}(Y)$
$\displaystyle=\int_{\mathbb{T}^{n}}Z^{*}\mathcal{L}_{Y(\vec{x})}\widetilde{\Theta}-\int_{\mathbb{T}^{n}}(dS)_{Z(\vec{x})}(Y(\vec{x}))\,dV.$
(29)

To work out the first term, notice that $\mathbb{T}^{n}$ has no boundary, so
that

$\mathcal{L}_{Y(\vec{x})}\widetilde{\Theta}=\iota_{Y(\vec{x})}d\widetilde{\Theta}=\iota_{Y(\vec{x})}\widetilde{\Omega}.$

So

$Z^{*}\mathcal{L}_{Y(\vec{x})}\widetilde{\Theta}=\sum_{i}\omega_{i}(Y(\vec{x}),\partial_{i}Z(\vec{x}))dV.$

Filling this back into equation 29 yields that $d\mathcal{A}_{S}^{\Omega}=0$
precisely when

$\displaystyle(dS)_{Z(\vec{x})}=\sum_{i}\omega_{i}(\cdot,\partial_{i}Z(\vec{x}))$

for all $\vec{x}\in\mathbb{T}^{n}$. As example 4.4 shows, this is equivalent
to equation 28 for $n=2$. For general $n$ a similar computation applies. ∎

##  5 Prime example: coupled particle-field systems

As an important example of our theory we consider a mechanical system coupled
to a field, i.e. a symplectic system coupled to a polysymplectic one. This
provides an interesting example from a physicists point of viewxxxCoupled
particle-field systems are studied amongst others in [Spo04, BG93, Kun01].,
but also serves to engage the more pure mathematically inclined reader. The
coupling of these two systems provides us with interesting non-linearities in
the theory already when we are working with linear polysymplectic spaces. For
notational simplicity, we consider only scalar fields and 2-dimensional
periodic space-time for now.

Let $q(t)$ denote the position of a particle moving in some external potential
field $V_{t}$ on $S^{1}=\mathbb{R}/2\pi\mathbb{Z}$ and $\varphi(t,x)$ be a
scalar field on
$\mathbb{T}^{2}=(\mathbb{R}/T\mathbb{Z})\times(\mathbb{R}/2\pi\mathbb{Z})$ as
before. We require $V_{t}=V_{t+T}$ to be smooth in $t$ and uniformly bounded
by some constant $\mu$. In order to avoid singularities, we assume the
particle to be of finite size and let $\rho\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\to\mathbb{R}$ model the coupling parameter between the
particle and the field. We assume $\rho$ to be smooth and supported on some
interval $(-r,r)$, where $r<\pi$ can be interpreted as the radius of the
particle and $\rho$ describes the distribution of the charge over the
particle.xixixiNote that as $\rho$ is only supported on
$(-r,r)\subseteq(-\pi,\pi]$, we might also view $\rho$ as a function on
$S^{1}$ sending $x\in S^{1}=\mathbb{R}/2\pi\mathbb{Z}$ to $\rho(a)$, where
$x=a\textup{ mod }2\pi$ and $a\in(-\pi,\pi]$. We will not make a notational
difference between these two functions. The equations of the coupled system
look as follows (see [FL21a]).

$\displaystyle\begin{split}\frac{d^{2}}{dt^{2}}q(t)&=-V^{\prime}_{t}(q(t))-\nabla(\varphi(t,\cdot)*\rho)(q(t))\\\
-\partial_{t}^{2}\varphi(t,x)+\partial_{x}^{2}\varphi(t,x)&=\varphi(t,x)+\rho(q(t)-x)\end{split}$
(30)

Here $f*\rho(q)\mathrel{\mathop{\mathchar
58\relax}}=\int_{S^{1}}f(x)\rho(q-x)\,dx$ for $f\mathrel{\mathop{\mathchar
58\relax}}S^{1}\to\mathbb{R}$. The coupled equations are described by two
Hamiltonians: one for the mechanical system and one for the field. Let
$u=(q,p)$, $Z=(\varphi,\pi_{1},\pi_{2},o)$ and define

$\displaystyle H_{t}(u,Z)$
$\displaystyle=\frac{1}{2}p^{2}+V_{t}(q)+(\varphi(t,\cdot)*\rho)(q)$
$\displaystyle S_{t,x}(u,Z)$
$\displaystyle=\frac{1}{2}\varphi^{2}+\frac{1}{2}\pi_{1}^{2}-\frac{1}{2}\pi_{2}^{2}-\frac{1}{2}o^{2}+\varphi\cdot\rho(q(t)-x).$

Then equation 30 can be reformulated as

$\displaystyle J\frac{d}{dt}u(t)$ $\displaystyle=\nabla_{u}H_{t}(u(t),Z(t,x))$
$\displaystyle M_{1}\partial_{t}Z(t,x)+M_{2}\partial_{x}Z(t,x)$
$\displaystyle=\nabla_{Z}S_{t,x}(u(t),Z(t,x))$

where $\nabla_{u}$ and $\nabla_{Z}$ denote respectively the gradients with
respect to the $u$ and $Z$ coordinates. We see that these equations indeed
combine the symplectic Hamiltonian formalism and the Bridges regularized
equations. Note that $u$ is now a map into $T^{*}S^{1}$ and, as before, $Z$
maps into $\mathbb{R}^{4}$.

To study these equations once again we can look at the Floer equation given by

$\displaystyle\begin{split}\partial_{s}\widetilde{u}(s,t)&=-J\partial_{t}\widetilde{u}(s,t)+\nabla_{u}H_{t}(\widetilde{u}(s,t),\widetilde{Z}(s,t,x))\\\
\partial_{s}\widetilde{Z}(s,t,x)&=-M_{1}\partial_{t}\widetilde{Z}(s,t,x)-M_{2}\partial_{x}\widetilde{Z}(s,t,x)+\nabla_{Z}S_{t,x}(\widetilde{u}(s,t),\widetilde{Z}(s,t,x)).\end{split}$
(31)

###  5.1 Asymptotics of Floer curves

For the Floer curves of the combined system introduced above, we want to
formulate and prove an analogue to lemma 2.1.ii. The energy of a Floer curve
$\widetilde{F}\mathrel{\mathop{\mathchar
58\relax}}=(\widetilde{u},\widetilde{Z})$ is given as the sum of the energies
of the components:

$E(\widetilde{F})=\int_{-\infty}^{+\infty}\left(\int_{S^{1}}|\partial_{s}\widetilde{u}(s,t)|^{2}\,dt+\int_{\mathbb{T}^{2}}|\partial_{s}\widetilde{Z}(s,t,x)|^{2}\,dV\right)ds.$

As we are mostly interested in the field theory part of the equations in this
article, we will assume the particle Floer curve $\widetilde{u}$ to be known.
In a subsequent article the case of an unknown particle curve will be treated.

######  Theorem 5.1.

Let $\widetilde{F}=(\widetilde{u},\widetilde{Z})$ be a solution to equation 31
of finite energy. We assume that there exists some smooth function $u^{+}\in
C^{\infty}(S^{1},T^{*}S^{1})$, such that

$\displaystyle\lim_{s\to+\infty}\widetilde{u}_{s}=u^{+}$
$\displaystyle\lim_{s\to+\infty}\partial_{s}\widetilde{u}_{s}=0$

in the $C^{\infty}$-topology. If
$\mathcal{R}^{-2}=\left(\frac{T}{2\pi}\right)^{2}$ has irrationality
measurexiixiixiiCompare with definition 3.1 and the assumption made in section
3.1. equal to 2, then there exists some $Z^{+}\in
C^{\infty}(\mathbb{T}^{2},\mathbb{R}^{4})$ such that
$\lim_{s\to+\infty}\widetilde{Z}_{s}=Z^{+}$ in the $C^{\infty}$-topology.

To prove theorem 5.1, we first introduce some notation. By $\hat{Z}_{s}$ and
$\hat{\rho}^{q}_{s}$ we denote respectively the Fourier transforms of
$\widetilde{Z}_{s}(t,x)$ and $\rho(\widetilde{q}_{s}(t)-x)$ with respect to
both time and space. Since $\rho$ is smooth and $\widetilde{q}_{s}$ converges
to $q^{+}$ in the $C^{\infty}$-topology, we get that
$\lim_{s\to+\infty}\rho(\widetilde{q}_{s}(t)-x)$ is well defined. The limit of
the Fourier series is denoted by $\hat{\rho}^{q}_{+}\mathrel{\mathop{\mathchar
58\relax}}=\lim_{s\to+\infty}\hat{\rho}^{q}_{s}$. Moreover,
$\partial_{s}\rho(\widetilde{q}_{s}(t)-x)$ converges to 0 in $C^{\infty}$ for
$s$ going to $+\infty$.

From the Floer equation we see that $\hat{Z}_{s}$ satisfies

$\displaystyle\partial_{s}\hat{Z}_{s}(m,k)=B(m,k)\hat{Z}_{s}(m,k)+\hat{\rho}^{q}_{s}(m,k)e_{1},$
(32)

for all $m,k\in\mathbb{Z}$. Here,
$e_{1}=\begin{pmatrix}1&0&0&0\end{pmatrix}^{T}$ is the first basis vector of
$\mathbb{R}^{4}$ and

$\displaystyle B(m,k)=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{pmatrix}-\mathcal{R}imM_{1}-ikM_{2}.$

The four distinct eigenvalues $\lambda_{1}^{\pm}(m,k)$ and
$\lambda_{2}^{\pm}(m,k)$ of the Hermitian matrix $B(m,k)$ are given by

$\displaystyle\lambda_{1}^{+}(m,k)$
$\displaystyle=-\lambda_{1}^{-}(m,k)=\mathcal{R}|m|+\sqrt{1+k^{2}}>1$
$\displaystyle\lambda_{2}^{+}(m,k)$
$\displaystyle=-\lambda_{2}^{-}(m,k)=\left|\mathcal{R}|m|-\sqrt{1+k^{2}}\right|>0.$

######  Proposition 5.2.

There exists some constant $\widetilde{c}>0$ such that
$\lambda_{2}^{+}(m,k)>\frac{\widetilde{c}}{(1+k^{2})^{2}}$ for all
$m,k\in\mathbb{Z}$.

###### Proof.

By assumption $\mathcal{R}^{-2}$ has irrationality measure 2, so for any
$\delta>0$ there exists some $c>0$, such that

$\left|\left(\frac{T}{2\pi}\right)^{2}-\frac{\alpha}{\beta}\right|>\frac{c}{\beta^{2+\delta}}$

for every $\frac{\alpha}{\beta}\in\mathbb{Q}$. In particlular we get for
$\delta=\frac{1}{2}$ that there is some $c>0$ such that

$\displaystyle\left|\frac{T}{2\pi}-\frac{|m|}{\sqrt{1+k^{2}}}\right|\left|\frac{T}{2\pi}+\frac{|m|}{\sqrt{1+k^{2}}}\right|>\frac{c}{(1+k^{2})^{\frac{5}{2}}}$
(33)

for all $m,k\in\mathbb{Z}$ (this follows from letting $\alpha=m^{2}$ and
$\beta=1+k^{2}$). There are two cases to consider.
_Case 1:_ If $\frac{|m|}{\sqrt{1+k^{2}}}\geq 1+\frac{T}{2\pi}$ then

$\displaystyle\lambda_{2}^{+}(m,k)$
$\displaystyle=\frac{2\pi}{T}|m|-\sqrt{1+k^{2}}$
$\displaystyle\geq\frac{2\pi}{T}\sqrt{1+k^{2}}$
$\displaystyle\geq\frac{2\pi}{T}\cdot\frac{1}{(1+k^{2})^{2}}$

where the last step follows from the fact that $1+k^{2}\geq 1$.
_Case 2:_ If $\frac{|m|}{\sqrt{1+k^{2}}}<1+\frac{T}{2\pi}$ then

$\frac{T}{2\pi}+\frac{|m|}{\sqrt{1+k^{2}}}<1+\frac{T}{\pi}=\mathrel{\mathop{\mathchar
58\relax}}C.$

Filling this into equation 33 gives

$\frac{c}{(1+k^{2})^{\frac{5}{2}}}<C\left|\frac{T}{2\pi}-\frac{|m|}{\sqrt{1+k^{2}}}\right|.$

Thus

$\displaystyle\lambda_{2}^{+}(m,k)$
$\displaystyle=\left|\mathcal{R}|m|-\sqrt{1+k^{2}}\right|$
$\displaystyle>\frac{2\pi}{T}\cdot\frac{c}{C}\cdot\frac{1}{(1+k^{2})^{2}}.$

In both cases we see that
$\lambda_{2}^{+}(m,k)>\frac{\widetilde{c}}{(1+k^{2})^{2}}$ when we pick
$0<\widetilde{c}<\textup{max}\\{\frac{2\pi}{T},\frac{2\pi c}{TC}\\}$. ∎

#### Energy argument

By assumption the energy of the Floer curve is finite. This implies the
existence of a sequence of numbers $\\{s_{j}\\}$ diverging to $+\infty$, such
that $\int_{\mathbb{T}^{2}}|\partial_{s}\widetilde{Z}(s_{j},t,x)|^{2}\,dV$
converges to 0 for $j\to+\infty$. Note that
$\int_{\mathbb{T}^{2}}|\partial_{s}\widetilde{Z}(s_{j},t,x)|^{2}\,dV=||\partial_{s}\hat{Z}_{s_{j}}||_{\ell^{2}}$.

######  Lemma 5.3.

We have $\lim_{s\to+\infty}\partial_{s}\widetilde{Z}_{s}=0$ in the
$C^{\infty}$-topology.

###### Proof.

We prove that the $L^{1}$-norm of all space-time derivatives of
$\partial_{s}\widetilde{Z}_{s}$ converge to 0. Let $\epsilon>0$ and $p(m,k)$
be any polynomial with no integer roots. As we know that
$\partial_{s}\rho(\widetilde{q}_{s}(t)-x)$ converges to 0 with all space-time
derivative, there exists some $s^{0}$ such that for $s>s^{0}$ we have

$\sum_{m,k}|\partial_{s}\hat{\rho}^{q}_{s}(m,k)p(m,k)|^{2}<\epsilon^{2}.$

Then for all $m,k\in\mathbb{Z}$ it follows that

$\displaystyle|\partial_{s}\hat{\rho}^{q}_{s}(m,k)|<\frac{\epsilon}{|p(m,k)|}$
(34)

for $s>s^{0}$.

Define $W_{s}\mathrel{\mathop{\mathchar 58\relax}}=\partial_{s}\hat{Z}_{s}$.
First, we want to prove that
$|W_{s}(m,k)|<\frac{2\epsilon}{|p(m,k)|\lambda(m,k)}$ for all $s>s^{0}$ and
$m,k\in\mathbb{Z}$, where $\lambda(m,k)$ is the smallest positive eigenvalue
of $B(m,k)$.xiiixiiixiiiThe idea for this part of the proof is based on
[Fab20, Lemma 7.2]. For the sake of contradiction suppose that there exists
some $s^{1}>s^{0}$ such that
$|W_{s^{1}}(m,k)|\geq\frac{2\epsilon}{|p(m,k)|\lambda(m,k)}$ for some
$m,k\in\mathbb{Z}$. Then, if we write out $W_{s^{1}}(m,k)$ in an eigenbasis of
$B(m,k)$, there is some component, denoted $W^{i}_{s^{1}}(m,k)$, belonging to
eigenvalue $\lambda^{i}(m,k)$ for which

$|W_{s^{1}}^{i}(m,k)|\geq\frac{\epsilon}{|p(m,k)|\lambda(m,k)}.$

Without loss of generality, we may assume $W_{s^{1}}^{i}(m,k)>0$. Then by
taking derivatives with respect to $s$ of equation 32 we see that

$\partial_{s}W^{i}_{s^{1}}(m,k)=\lambda^{i}(m,k)W^{i}_{s^{1}}(m,k)+(\partial_{s}\hat{\rho}^{q}_{s^{1}}(m,k)e_{1})^{i}>0.$

As $W_{s^{1}}^{i}(m,k)>0$ and $W_{s_{j}}(m,k)$ converges to 0, there has to be
a number $s^{2}>s^{1}$ with $\partial_{s}W^{i}_{s^{2}}=0$ and
$\partial_{s}W^{i}_{s}(m,k)>0$ for all $s\in(s^{1},s^{2})$. So

$\displaystyle
W^{i}_{s^{2}}(m,k)>W^{i}_{s^{1}}(m,k)>\frac{\epsilon}{|p(m,k)|\lambda(m,k)}$

and

$\displaystyle
0=\partial_{s}W^{i}_{s^{2}}(m,k)=\lambda^{i}(m,k)W^{i}_{s^{2}}(m,k)+(\partial_{s}\hat{\rho}^{q}_{s^{2}}(m,k)e_{1})^{i},$

contradicting equation 34. This shows that indeed
$|W_{s}(m,k)|<\frac{2\epsilon}{|p(m,k)|\lambda(m,k)}$ for all $s>s^{0}$ and
$m,k\in\mathbb{Z}$.

When $\lambda(m,k)=\lambda_{1}^{+}(m,k)$ then $\lambda(m,k)^{-1}<1$ and when
$\lambda(m,k)=\lambda_{2}^{+}(m,k)$ then
$\lambda(m,k)^{-1}<\widetilde{c}^{-1}(1+k^{2})^{2}$ by proposition 5.2. So
either way we see that $\lambda(m,k)^{-1}$ is at most polynomial in $m,k$. For
all $l_{1},l_{2}\in\mathbb{Z}$ we get that

$\displaystyle||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}W_{s}(m,k)||_{L^{1}}$
$\displaystyle=\sum_{m,k}|W_{s}(m,k)k^{l_{1}}m^{l_{2}}|$
$\displaystyle<2\epsilon\sum_{m,k}\frac{\lambda(m,k)^{-1}k^{l_{1}}m^{l_{2}}}{|p(m,k)|}$
$\displaystyle<\epsilon\cdot\textup{const},$

where the last line follows from the fact that for fixed $l_{1},l_{2}$ we can
choose $p(m,k)$ such that
$\frac{\lambda(m,k)^{-1}k^{l_{1}}m^{l_{2}}}{|p(m,k)|}<\frac{1}{m^{2}k^{2}}$.
This proves the lemma. ∎

#### The limit

Now define $\hat{Z}^{+}(m,k)\mathrel{\mathop{\mathchar
58\relax}}=-B(m,k)^{-1}\hat{\rho}^{q}_{+}(m,k)e_{1}$ and let $Z^{+}$ be the
inverse Fourier transform of $\hat{Z}^{+}$. Note that the formula for
$\hat{Z}^{+}$ makes sense, as $B(m,k)$ never has vanishing eigenvalues. First
of all we would like to see that $Z^{+}$ is indeed a smooth function.

######  Proposition 5.4.

The function $Z^{+}$ defined above is a smooth function on $\mathbb{T}^{2}$.

###### Proof.

We prove that all derivatives of $Z^{+}$ have finite $L^{1}$-norm. This
implies that they are all continuous and thus that $Z^{+}$ is smooth. Let
$l_{1},l_{2}$ be integers. Then

$\displaystyle||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}Z^{+}||_{L^{1}}=\sum_{m,k}|\hat{Z}^{+}(m,k)m^{l_{1}}k^{l_{2}}|$
$\displaystyle\leq\sum_{m,k}\lambda(m,k)^{-1}|m^{l_{1}}k^{l_{2}}\hat{\rho}^{q}_{+}(m,k)|,$

where $\lambda(m,k)$ is again the smallest positive eigenvalue of $B(m,k)$.
Just like before, $\lambda(m,k)^{-1}$ is at most a polynomial expression in
$m$ and $k$ and it follows from smoothness of $\rho(q^{+}(t)-x)$ that
$||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}Z^{+}||_{L^{1}}<+\infty$. ∎

Finally we can finish the proof of theorem 5.1 with the following lemma.

######  Lemma 5.5.

For $Z^{+}$ defined as above it holds that
$\lim_{s\to+\infty}\widetilde{Z}_{s}=Z^{+}$ in the $C^{\infty}$-topology.

###### Proof.

Again, we will prove convergence by proving that the $L^{1}$-norms of all
derivatives converge. Let $V_{s}(m,k)=\hat{Z}_{s}(m,k)-\hat{Z}^{+}(m,k)$. Then

$\displaystyle\partial_{s}V_{s}(m,k)=B(m,k)V_{s}(m,k)+(\hat{\rho}^{q}_{s}(m,k)-\hat{\rho}^{q}_{+}(m,k))e_{1}.$
(35)

Let $\epsilon>0$ and $p(m,k)$ a polynomial with no integer roots. By lemma 5.3
and the fact that $\hat{\rho}^{q}_{s}$ converges to $\hat{\rho}^{q}_{+}$,
there exists some $s^{0}$ such that for $s>s^{0}$

$\displaystyle\sum_{m,k}|\partial_{s}V_{s}(m,k)p(m,k)|^{2}$
$\displaystyle<\epsilon$
$\displaystyle\sum_{m,k}|(\hat{\rho}^{q}_{s}(m,k)-\hat{\rho}^{q}_{+}(m,k))p(m,k)|^{2}$
$\displaystyle<\epsilon.$

For integers $l_{1},l_{2}$, we have that

$\displaystyle||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}(\widetilde{Z}_{s}-Z^{+})||_{L^{1}}$
$\displaystyle=\sum_{m,k}|V_{s}(m,k)m^{l_{1}}k^{l_{2}}|$ (36)
$\displaystyle<\sum_{m,k}|p(m,k)B(m,k)V_{s}(m,k)|\cdot|m^{l_{1}}k^{l_{2}}\frac{\lambda(m,k)^{-1}}{p(m,k)}|.$
(37)

Equation 35 implies that

$|p(m,k)B(m,k)V_{s}(m,k)|\leq|\partial_{s}V_{s}(m,k)p(m,k)|+|(\hat{\rho}^{q}_{s}(m,k)-\hat{\rho}^{q}_{+}(m,k))p(m,k)|,$

which combined with equation 36 and the Cauchy-Schwarz inequality gives

$\displaystyle||\partial_{t}^{l_{1}}\partial_{x}^{l_{2}}(\widetilde{Z}_{s}-Z^{+})||_{L^{1}}$
$\displaystyle<\sum_{m,k}|\partial_{s}V_{s}(m,k)p(m,k)|^{2}\sum_{m,k}\left|m^{l_{1}}k^{l_{2}}\frac{\lambda(m,k)^{-1}}{p(m,k)}\right|^{2}$
$\displaystyle\,+\sum_{m,k}|(\hat{\rho}^{q}_{s}(m,k)-\hat{\rho}^{q}_{+}(m,k))p(m,k)|^{2}\sum_{m,k}\left|m^{l_{1}}k^{l_{2}}\frac{\lambda(m,k)^{-1}}{p(m,k)}\right|^{2}$
$\displaystyle<\epsilon\sum_{m,k}\left|m^{l_{1}}k^{l_{2}}\frac{\lambda(m,k)^{-1}}{p(m,k)}\right|^{2}$

for $s>s^{0}$. Again we see by proposition 5.2 that we can pick $p(m,k)$ such
that the last line is smaller than some constant multiple of $\epsilon$. So
indeed the $L^{1}$ norms of all derivatives of $\widetilde{Z}_{s}-Z^{+}$
converge to 0, proving the lemma. ∎

###  5.2 Lorentz model

Notice that in equation 30 it seems like we broke the symmetry of space and
time. When introducing a particle this is inevitable since, from the viewpoint
of the particle, space and time are split and the particle has a finite size
only in space. However, if the particle would be moving at some different
speed it might experience a different splitting of space and time. The
particle equation would become

$\displaystyle\frac{d^{2}}{dt^{\prime 2}}q(t^{\prime})$
$\displaystyle=-V^{\prime}_{t^{\prime}}(q(t^{\prime}))-\nabla(\varphi(t^{\prime},\cdot)*\rho)(q(t^{\prime})),$

where $t^{\prime}$ is the new time coordinate. How would the field equation
change? According to the Lorentz modelxivxivxivNote that we are dealing with a
simplification of the Lorentz model here in which the speed of the particle is
significantly bigger than its acceleration. This allows us to regard the
reference frame of the particle to be constant. For an explanation of the
Lorentz model in full generality, see [Spo04], explained in [Spo04], the
particle has a finite size in its own reference frame and thus the field
equation becomes

$\displaystyle-\partial_{t}^{2}\varphi(t,x)+\partial_{x}^{2}\varphi(t,x)$
$\displaystyle=\varphi(t,x)+\rho(q(t^{\prime})-x^{\prime}),$

where $t^{\prime}$ and $x^{\prime}$ are the space-time coordinates in the rest
frame of the particle. Note that $t^{\prime}$ and $x^{\prime}$ are functions
of $t$ and $x$ and all the proofs of section 5.1 still hold. The only thing
about the function $\rho(q(t)-x)$ that we used in that section was that it was
smooth and changing the space-time coordinates to $(t^{\prime},x^{\prime})$
does not change that. In fact, we could even have treated two or more
particles moving in different reference frames and all interacting with the
field. For two particles the equations look as follows

$\displaystyle\frac{d^{2}}{dt^{2}}q_{1}(t)$
$\displaystyle=-V^{\prime}_{1,t}(q_{1}(t))-\nabla(\varphi(t,\cdot)*\rho)(q_{1}(t))$
$\displaystyle\frac{d^{2}}{dt^{\prime 2}}q_{2}(t^{\prime})$
$\displaystyle=-V^{\prime}_{2,t^{\prime}}(q_{2}(t^{\prime}))-\nabla(\varphi(t^{\prime},\cdot)*\rho)(q_{2}(t^{\prime}))$
$\displaystyle-\partial_{t}^{2}\varphi(t,x)+\partial_{x}^{2}\varphi(t,x)$
$\displaystyle=\varphi(t,x)+\rho(q_{1}(t)-x)+\rho(q_{2}(t^{\prime})-x^{\prime}).$

Still, all the proofs of section 5.1 hold.

This is where we see a clear advantage of our covariant theory over [FL22,
FL21a]. Fabert and Lamoree need a fixed splitting of time and space to exist
before starting with the analysis of the equations. They demand the non-
linearities in the equations to satisfy conditions that explicitly depend on
the chosen space-time coordinates. We see that in our theory, no such
splitting a priori is required and we can deal with non-linearities coming
from different choices of coordinates simultaneously.

######  Remark 5.6.

In this section we fixed our attention to particles moving in 1-dimensional
periodic space. However, the proofs generalize to higher dimensional spaces.
In fact, for general $n$, we can take $q$ to be a particle constrained to any
submanifold $Q$ of $\mathbb{T}^{n-1}$, where $\mathbb{T}^{n-1}$ denotes
$(n-1)$-dimensional periodic space and all space periods are taken to be equal
to $2\pi$. The field is defined on
$\mathbb{T}^{n}=(\mathbb{R}/T\mathbb{Z})\times\mathbb{T}^{n-1}$. Now, we can
still prove that the eigenvalues of the linearized operator converge to 0 at
most with polynomial speed, so that the same results hold as above. The proofs
for general $n$ will be part of our subsequent article. This remark serves
mostly to show that by looking at particles constrained to any
$Q\subseteq\mathbb{T}^{n-1}$, we can couple our theory of CFT to any
Hamiltonian system on a cotangent bundle.

###  5.3 Existence of Floer curves

We finish this article with the proof that solutions to the particle-field
Floer equation 31 actually exist. This can be done by using results from
Fabert and Lamoree [FL22, FL21a, FL21b]. Recall from section 3.3 that we can
translate the Floer equation into the infinite-dimensional setting used by
Fabert and Lamoree. Define $\mathbb{H}\mathrel{\mathop{\mathchar
58\relax}}=L^{2}_{0}(S^{1},\mathbb{R}^{4})$, the space of square-integrable
functions with mean zero. After endowing $T^{*}S^{1}\times\mathbb{H}$ with the
complex structure $\textup{Diag}(J,M_{1})$, solutions to equation 31 are in
one-to-one correspondence with infinite-dimensional Floer curves coming from
the Hamiltonian

$\displaystyle\mathscr{H}_{t}\mathrel{\mathop{\mathchar
58\relax}}T^{*}S^{1}\times\mathbb{H}$ $\displaystyle\to\mathbb{R}$
$\displaystyle(u,Z)$ $\displaystyle\mapsto\frac{1}{2}\langle
AZ,Z\rangle_{\mathbb{H}}+F_{t}(u,Z),$

where

$\displaystyle A=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{pmatrix}-M_{2}\partial_{x}$ (38)

and $F_{t}(u,Z)=\frac{1}{2}p^{2}+V_{t}(q)+(\varphi*\rho)(q)$. In order to use
[FL22, Theorem 10.4] and conclude that a Floer curve exists, we must check
that $(A,T)$ and $F_{t}$ are admissible according to definitions 2.1 and 2.5
in [FL21b]. First, notice that since $M_{2}\partial_{x}$ and hence $A$ is
self-adjoint, there exists a complete basisxvxvxvThis basis can be found by
looking at the Fourier transform of $\mathbb{H}$. $\\{e_{k}^{\pm}\mid
k\in\mathbb{Z}\backslash\\{0\\}\\}$ of $\mathbb{H}$ consisting of unit
eigenvectors of $A$ such that $M_{1}e_{k}^{\pm}=\mp e_{k}^{\mp}$ and the
eigenvalues are given by $\lambda_{2k}=\sqrt{1+k^{2}}$ and
$\lambda_{2k+1}=-\sqrt{1+k^{2}}$. This allows us to identify $\mathbb{H}$ with
a complex subspace of $\mathbb{H}\otimes_{\mathbb{R}}\mathbb{C}$ given by the
complex span of $\\{z_{k}=(e^{+}_{k}+ie^{-}_{k})/\sqrt{2}\mid
k\in\mathbb{Z}\backslash\\{0\\}\\}$. For any $h$ the space $\mathbb{H}_{h}$
can be defined by requiring that $||z_{k}||_{\mathbb{H}_{h}}=k^{h}$. First of
all we check that $(A,T)$ is admissible.

######  Definition 5.7 (Compare definition 2.1 from [FL21b]).

The pair $(A,T)$ is _admissible_ if there exists $h_{0}$ such that for
$h>h_{0}$ we can find $c_{h}$ such that $|\epsilon_{k}|>c_{h}k^{-h}$. Here,
$\epsilon_{k}\in(-\pi/T,\pi/T]$ is defined by
$\epsilon_{k}\mathrel{\mathop{\mathchar 58\relax}}=\lambda_{k}\mod 2\pi/T$.

######  Lemma 5.8.

The pair $(A,T)$ with $A$ as in equation 38 and $T$ such that
$\left(\frac{T}{2\pi}\right)^{2}$ has irrationality measure 2 is admissible.

###### Proof.

Note that $|\epsilon_{2k}|=\min\\{\lambda_{2}^{+}(m,k)\mid m\in\mathbb{Z}\\}$,
where $\lambda_{2}^{+}(m,k)=|\frac{2\pi}{T}|m|-\sqrt{1+k^{2}}|$ as in section
5.1. As was proven in proposition 5.2, there is some $\widetilde{c}$ such that
$\lambda^{+}_{2}(m,k)>\frac{\widetilde{c}}{(1+k^{2})^{2}}$ for all
$m,k\in\mathbb{Z}$. Left to prove is that there exists $c_{h}$ such that
$\frac{\widetilde{c}}{(1+k^{2})^{2}}\geq c_{h}(2k)^{-h}$. This can always be
done for $h>h_{0}\mathrel{\mathop{\mathchar 58\relax}}=3$. For
$\epsilon_{2k+1}$ the proof is similar. ∎

Secondly, we must check the admissibility of $F_{t}$.

######  Definition 5.9 (Simplified definition 2.5 of [FL21b]).

For a finite-dimensional symplectic manifold $M$, the $T$-periodic non-
linearity $F_{t}\mathrel{\mathop{\mathchar
58\relax}}M\times\mathbb{H}\to\mathbb{R}$ is called _$\infty$ -regularizing_
if for any $h>h_{0}$, the map $F_{t}$ extends to a smooth map
$M\times\mathbb{H}_{-h}\to\mathbb{R}$ which is smooth in the $t$-variable. Any
$\infty$-regularizing non-linearity is called _admissible_.

######  Lemma 5.10.

The nonlinearity $F_{t}$ defined above is $\infty$-regularizing.

###### Proof.

Assume that $Z\in\mathbb{H}_{-h}$ for some $h>h_{0}$ and write
$Z=\sum\hat{Z}(k)z_{k}$. Now by definition of $\mathbb{H}_{-h}$ it follows
that $(\hat{Z}(k)k^{-h})\in\ell^{2}(\mathbb{Z},\mathbb{C}^{2})$. Also, for
$q\in S^{1}$, define
$\underline{\rho^{q}}=\rho(q-\cdot)\begin{pmatrix}1&0&0&0\end{pmatrix}^{T}=\sum\hat{\underline{\rho^{q}}}(k)z_{k}$.
Note that since $\rho(q-\cdot)$ is smooth,
$||\hat{\underline{\rho^{q}}}(k)k^{h}||_{\ell^{2}}<+\infty$ for all $h>0$.
Then

$\displaystyle|(\varphi*\rho)(q)|$
$\displaystyle=\left|\int_{S^{1}}\varphi(x)\rho(q-x)\,dx\right|$
$\displaystyle=\left|\int_{S^{1}}\langle
Z(x),\underline{\rho^{q}}(x)\rangle\,dx\right|$ $\displaystyle=\left|\langle
Z,\underline{\rho^{q}}\rangle_{\mathbb{H}}\right|$
$\displaystyle=\left|\langle\hat{Z}(k),\hat{\underline{\rho^{q}}}(k)\rangle_{\ell^{2}}\right|$
$\displaystyle=\left|\langle\hat{Z}(k)k^{-h},\hat{\underline{\rho^{q}}}(k)k^{h}\rangle_{\ell^{2}}\right|$
$\displaystyle\leq||\hat{Z}(k)k^{-h}||_{\ell^{2}}||\hat{\underline{\rho^{q}}}(k)k^{h}||_{\ell^{2}}<+\infty.$

Here, the last line follows from Cauchy-Schwarz. In particular, it follows
from this computation that $(\varphi*\rho)(q)\in\mathbb{R}$ and thus $F_{t}$
indeed extends to $T^{*}S^{1}\times\mathbb{H}_{-h}$. It is easy to see that
the resulting map is smooth and also smooth in the $t$-variable (compare with
proposition 2.8 from [FL21b]). ∎

The last thing to check is boundedness in the $p$-coordinate. As we are
working with $T^{*}S^{1}$ for our particle which is not compact, it must be
checked that this does not cause any divergence. Since our Hamiltonian is
quadratic in $p$, this does not pose any problems. Conditions (F1) and (F2)
from section 2 of [FL21a] can be easily checked and we refer to the
explanation in that article for the fact that this suffices.

Now by combining the admissibility of $(A,T)$ and $F_{t}$ and the boundedness
in $p$ with the fact that $T^{*}S^{1}$ has non-trivial homology, theorem 10.4
from [FL22] tells us that Floer curves exist for equation 31. Even though the
field equations are linear, this result is still non-trivial because of the
interaction with the particle non-linearity. Note that the translation to the
infinite-dimensional setting of Fabert and Lamoree is possible for equation
31, because the non-linearity has a specific form. In a subsequent article we
will investigate the existence of of Floer curves in our covariant model for a
larger class of polysymplectic Hamiltonians.

### Concluding remarks

It should be clear from the discussion above that the analysis done in [FL22,
FL21b, FL21a] on the infinite-dimensional framework significantly supports the
development of our covariant framework. The goal is to translate the Fredholm
theory and compactness results from the aforementioned articles into the
covariant setting and thus developing a covariant theory that stands on its
own. These two main ingredients, combined with the work done in this article,
will allow us to define a Floer theory that incorporates the symmetries in
time and space, and is capable of dealing with relativistic models, such as
the Lorentz model discussed above.

## References

  * [AD14] Michele Audin and Mihai Damian.  Morse theory and Floer homology.  Springer, 2014.
  * [BG93] Dario Bambusi and Luigi Galgani.  Some rigorous results on the pauli-fierz model of classical electrodynamics.  In Annales de l’IHP Physique théorique, volume 58, pages 155–171, 1993.
  * [Bri06] Thomas J Bridges.  Canonical multi-symplectic structure on the total exterior algebra bundle.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 462(2069):1531–1551, 2006.
  * [Bug12] Yann Bugeaud.  Distribution modulo one and Diophantine approximation, volume 193\.  Cambridge University Press, 2012.
  * [DS08] Ana Cannas Da Silva.  Lectures on symplectic geometry.  Springer, 2008.
  * [Fab20] Oliver Fabert.  Hamiltonian floer theory for nonlinear schrödinger equations and the small divisor problem, 2020.
  * [FG13] Michael Forger and Leandro G Gomes.  Multisymplectic and polysymplectic structures on fiber bundles.  Reviews in Mathematical Physics, 25(09):1350018, 2013.
  * [FL21a] Oliver Fabert and Niek Lamoree.  Cuplength estimates for periodic solutions of hamiltonian particle-field systems.  arXiv preprint 2107.03989, 2021.
  * [FL21b] Oliver Fabert and Niek Lamoree.  Floer homology for hamiltonian pdes: Fredholm theory.  arXiv preprint 2107.14074, 2021.
  * [FL22] Oliver Fabert and Niek Lamoree.  Time-periodic solutions of hamiltonian pdes using pseudoholomorphic curves.  Algebraic and Geometric Topology, 2022.
  * [Flo88] Andreas Floer.  Morse theory for lagrangian intersections.  Journal of differential geometry, 28(3):513–547, 1988.
  * [Gün87] Christian Günther.  The polysymplectic hamiltonian formalism in field theory and calculus of variations. i. the local case.  Journal of differential geometry, 25(1):23–53, 1987.
  * [Hél01] F Hélein.  Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory.  arXiv preprint math-ph/0212036, 2001.
  * [Kan93] Igor V Kanatchikov.  On the canonical structure of the de donder-weyl covariant hamiltonian formulation of field theory i. graded poisson brackets and equations of motion.  arXiv preprint hep-th/9312162, 1993.
  * [Kru02] Olga Krupková.  Hamiltonian field theory.  Journal of Geometry and Physics, 43(2):93–132, 2002.
  * [Kun01] M. Kunze.  On the period of periodic motions of a particle in a scalar wave field.  ZAMM Journal of applied mathematics and mechanics: Zeitschrift für angewandte Mathematik und Mechanik, 81, 01 2001.
  * [McC21] Tom McClain.  A global version of günther’s polysymplectic formalism using vertical projections.  Journal of Geometry and Physics, 161:104065, 2021.
  * [MPS98] Jerrold E Marsden, George W Patrick, and Steve Shkoller.  Multisymplectic geometry, variational integrators, and nonlinear pdes.  Communications in Mathematical Physics, 199(2):351–395, 1998.
  * [MS12] Dusa McDuff and Dietmar Salamon.  J-holomorphic curves and symplectic topology, volume 52.  American Mathematical Soc., 2012.
  * [RRMRSV11] Narciso Román-Roy, Ángel M. Rey, Modesto Salgado, and Silvia Vilariño.  On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories.  Journal of Geometric Mechanics, 3(1):113–137, 2011.
  * [Sal99] Dietmar Salamon.  Lectures on floer homology.  Symplectic geometry and topology (Park City, UT, 1997), 7:143–229, 1999.
  * [Spo04] Herbert Spohn.  Dynamics of charged particles and their radiation field.  Cambridge university press, 2004.