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\begin{document}
\hfill DSFT93/23
\begin{center}
\vskip 2.5cm
{\Large \bf EIKONAL TYPE EQUATIONS FOR GEOMETRICAL SINGULARITIES OF SOLUTIONS
IN FIELD THEORY}
\vskip 1.0cm
{\Large F.~Lizzi, G.~Marmo, G.~Sparano \footnote{\scriptsize
LIZZI@NA.INFN.IT, GIMARMO@NA.INFN.IT, SPARANO@NA.INFN.IT.
}}
\\
{\large Dipartimento di Scienze Fisiche and I.N.F.N., Sez.\ di Napoli\\
Mostra d'Oltremare Pad.~19, 80125 Napoli, Italy,}
\\
and
\\
{\Large A. M. Vinogradov}
\\
{\large Istituto di Matematica, Universit\`a di Salerno, 84000 Salerno, Italy
\\and \\
Erwin Schr\"odinger Institute, Pasteurgasse 4/7, A1090 Wien, Austria.}
\vskip 0.4cm
\end{center}
\begin{abstract}
We discuss several aspects of singularities of the solutions of the partial
differential equations of KleinGordon, Schr\"odinger and Dirac. In
particular we analyze the fold type singularity, of the first and higher
orders, and the related characteristic equations. We also consider the
field equations as reduction of homogenous equations in higher dimensions,
and discuss how singularities of the solution are reduced.
\end{abstract}
\vfill
\begin{center}
July 1993
\end{center}
\newpage
\setcounter{footnote}{1}
\section{Introduction}
The Lagrangian formalism plays a major role in the description of
evolutionary system in Physics. Among other things, it allows for
manifestly covariant theories, N\"other's theorem and locality.
Many relevant Lagrangians for physics (gauge theories, gravitation,
relativistic particles) give rise to dynamical systems in implicit form,
i.e.\ they do not give rise to {\em vector fields}. They only determine a
submanifold of the relevant carrier space, and this submanifold need not be
a section of the appropriate bundle. For these systems one usually deals
with constrained formalism, as elaborated by Dirac and Bergman. However,
this procedure does not appear to be a natural approach to these equations,
for one is forced to deal with the inverse of a matrix which may change
rank from point to point.
This equations are instances of implicit differential equations and their
solutions may exhibit singularities.
Another familiar example of partial differential equations (PDEs) arising in
implicit form in physics is provided by the HamiltonJacobi form of
dynamics. Here the equations for a function $S$ on the configuration space
$Q$ has the form
$$
H\left( q,\frac{\del S}{\del q} \right)  \frac{\del S}{\del t} = 0\ \ \ .
$$
The Hamiltonian function $H=H(q,p,t)$ is in general non linear and gives
rise to an implicit differential equation for $S$. In more geometrical
terms, the equation for $S$ is replaced by an equation for Lagrangian
submanifolds which are not necessarily sections of the cotangent bundle
$T^*(Q) \stackrel{\pi}{\longrightarrow} Q$. To simply illustrate the
situation we restrict to $Q=R^n, \ \ T^*(Q)=R^{2n}$. Given a Hamiltonian
function $H$, the associated generalized version of the HamiltonJacobi
form of the dynamics can be given along the following lines.
First find the embedding $\Phi$ :
$$
\begin{tabular}{ccc}
$R^n$ & $\stackrel{\Phi}{\longrightarrow}$ & $T^* R^n$\\
& & $\downarrow$\\
& & $R^n$
\end{tabular}
$$
with the property $$
\Phi^* H = E, \ \ \Phi^* \omega_0 = 0
$$
where $\omega_0 = dp_i \wedge d q^i$ is the canonical symplectic structure.
Of course if $\Phi(R^n)$ is transverse to the fibers of $T^*R^n$, we can
find locally a function $S$ such that $dS(R^n)=\Phi(R^n)$ in the
appropriate neighborhood. In many cases solutions $\Phi$ will fail to be
transverse to the fibers, caustics arise in this way. Other singularities
also show up in this respect.
When the symplectic structure is replaced by the contact structure on
$R^{2n+1}$, the 1jets of functions on $R^n$, we have Legendre rather then
Lagrange embeddings. The projection of this submanifold on the base manifold
is called its front. The set where there is lack of tranversality is the
wave front. The connection between HamiltonJacobi theory and the
Schr\"odinger equation shows that the analysis of this singularities is very
important in the WKB approximation of quasiclassical asymptotics of the
solutions of Schr\"odinger equation. One can hypotize that the geometric
background found by V.~Maslov \cite{maslov} for quasi classical asymptotic
solutions gives rise to a similar theory at the level of exact solution.
The study of these singularities is centered around the
subsidiary equations, describing all possible forms of a prescribed type of
singularities admitted by a given system of PDE's.
Therefore it is necessary to
develop a theory of singular solutions of PDE's. Two steps are needed:
\begin{enumerate}
\item{ The first step is to
formalize the concept of singularity for solutions of PDE's, and to
classify them.}
\item{ The second step requires that we
develop a formal procedure to associate, with a given system
of PDE's ${\cal Y}$ and a given singularity type $\Sigma$, the
subsidiary equations (${\cal Y}_\Sigma$) mentioned above.}
\end{enumerate}
Also central in this approach is the reconstruction problem, that is, given
the system of equations ${\cal Y}_\Sigma$, and the singularity type
$\Sigma$ (to which they correspond), find the original system ${\cal
Y}$. The quantization procedure, as well as the problem of the sources
of the fields, are of this kind. It also seems to be very important for the
mechanics of continuous media, as it gives regular methods to deduce the
equations governing the behaviour of the medium from the propagation of
singularities in it.
With this paper we would like to start a systematic investigation of the
correspondence
${\cal Y}\longleftrightarrow {\cal Y}_\Sigma$ for some
fundamental equations of mathematical physics. Our aim here is to deduce and
to discuss the equations ${\cal Y}_\Sigma$ for some well known equations,
supposing $\Sigma$ to be the geometric folding type singularities described
in \cite{av52,av55}.
In section 2 we recall the general feature of fold type singularities, and
the subsidiary equations associated to them. We recognize in some of them
the analog of the classical characteristic equations, we call them
$k$characteristic equations.
In section 3 we study the $k$characteristic equations for some classical
field theory equations: KleinGordon,
Schr\"odinger and Dirac. Here we will see that, since
the $k$characteristic equations depend on the symbol only, the singularities
will not be sensitive to the mass, in the first and third case, or the time
derivative term and potential in the second case. We find, however,
that for the KleinGordon equation, 1 and 2 characteristics describe,
respectively, the propagation of massless pointlike and one dimensional
objects.
In section 4 we reconsider the same equations as reduction of homogeneous
equations in an extended space. Once this is done, all the terms in the
equation contribute to the symbol, and the folds then yield
the correct equations of motion of the corresponding propagating objects.
In section 5 we find the remaining subsidiary equations (for 1singularity,
called complementary equations)
for KleinGordon, Schr\"odinger, Dirac and Maxwell equations.
As for now, our paper is a sort of {\em `phenomenological'} paper, i.e.\
we discuss several aspects of singularities for relevant equations even
though at the moment some equations do not allow for a clear cut physical
interpretation.
\section{Generalities}
We recall that geometric singularities are singularities of multivalued
solutions of PDE's \cite{av43,av49,av17,av28}.
To make more precise these concepts some preliminaries are to be done.
Let $E$ be a $(m+n)$dimensional manifold (the manifold of all dependent and
independent variables). Given two $n$dimensional submanifolds, $L_1$ and
$L_2$ of
$E$ we say that they have the same $k$th order jet at a point $\alpha\in
L_1\cap L_2$ iff they are tangent to each other with order $k$. So, a $k$th
order jet at $\alpha\in E$ is an equivalence class of $n$dimensional
submanifolds of $E$ passing through $\alpha$. The set of all such $k$th order jet
admits in a natural way a smooth manifold structure which is called the $k$th
order jet space of $n$dimensional submanifolds of $E$ and is denoted by $J^k =
J^k(E,n)$. Projections $J_{k,l} : J^k\rightarrow J^l$ are defined in a natural
way. Let $(x,u)$ with $x = (x^1,\dots,x^n)$, $u = (u^1,\dots,x^m)$ be a
divided local chart on $E$, that is a local chart on $E$ where some of the
coordinate functions are proclaimed `independent' variables and the remaining
ones `dependent' variables. Such a divided chart on $E$ generates a local
chart on $J^k(E,n)$ composed of the variables
\be
x^\mu,u^i,\dots,u^i_\sigma,\dots ~~~~~~~~\sigma\leq k
\ee
with $1\leq \mu\leq n$, $1\leq i\leq m$ and $\sigma = (i_1,\dots,i_n)$ being a
multiindex, $\sigma = i_1 +\dots +i_n$.
The Cartan distribution on $J^k(E,n)$, also called the $k$th order contact
structure, is defined as a distribution of tangent subspaces of $E$ given by
the system of Pfaff equations
\be
du^i_\sigma  u^i_{(\sigma + 1)\mu} dx^\mu = 0 ,\label{cart}
\ee
with $1\leq i\leq m$, $1\leq\mu\leq n$, $\sigma< k$. Every $n$dimensional
submanifold $L$ of $E$ given in the form
\be
u^i = f_i(x^1,\dots,x^n), ~~~~~~~1\leq i\leq m ,
\ee
can be lifted canonically on $J^k(E,n)$. This lifted submanifold
$L_{(k)}\subset J^k(E,n)$ is given by the equations
\be
u^i_\sigma = {\partial^{\sigma}f_i\over\partial x_\sigma} ~~~~~~~~~ 1\leq
i\leq m , ~~~0\leq \sigma\leq k
\ee
where ${\partial^{\sigma}\over\partial x_\sigma}$ stands for
${\partial^{\sigma}\over\partial x^{i_1}_1\dots \partial x^{i_n}_n}$
supposing
that $\sigma = (i_1,\dots ,i_n)$. A submanifold $N\subset J^k$ is called
integral if it satisfies \eqn{cart}. Note that all the manifolds of the form
$L_{(k)}$ are integral. An $n$dimensional submanifold $N\subset J^k$ is called
$R$manifold if for almost every point $\theta\in N$ there exists a
neighborhood of $\theta$ in $N$ which is of the form $L_{(k)}$ for an $L\subset
E$. Here `almost every' means excluding a subset $Y$ of $N$ with dim$Y< n$. It
can be proved that this subset $Y$ coincides with the singular set of the
projection $\pi_{k,k1} : J^k\longrightarrow J^{k1}$ restricted to $N$.
Because of this reason $Y$ is denoted by $sing_N\subset N$. Let now $\theta$ be
a point of $sing_N$, $N$ being an $R$manifold, and $T_\theta N$ be the tangent
space of $N$ at $\theta$. The kernel of the projection of $T_\theta N$ along
$\pi_{k,k1}$ is called the label of $\theta$. These labels can be classified
naturally with respect to the group of contact diffeomorphisms of $J^k$. recall
that contact diffeomorphisms are those diffeomorphisms that preserve the
Cartan distribution of $J^k$. The result of this classification (see
\cite{av52,av55}) tells us that the label
equivalence classes can be labeled by the finitedimensional
commutative $R$algebras (in fact this result was formulated in \cite{av52}
in slightly different terms).
As it is well known a finitedimensional commutative $R$ algebra
splits in an essentially unique way into a direct sum of algebras $F_{(k)}$,
with $ F = R, C$, and $F_{(k)}$ stands for the $F$algebra generated by an element,
say $\xi$, subjected to the conditions $\xi^k = 0, \xi^{k1}\neq 0 $.
In this paper we are concerned with solution singularities corresponding to
$R_{(k)}$label type which we will call folds.
These singularities can be paralleled with the
ThomBoardman ones of the standard singularity theory, commonly denoted
by $\Sigma_{(k)}$.
Recall, finally, that a $k$th order system of PDE imposed on a $n$dimensional
submanifold of $E$ can be represented as a submanifold ${\cal I}\subset
J^k(E,n)$. In fact, local equations of ${\cal I}$ are obviously of the form
\be
F_j(\ldots x^\mu \ldots u^i \ldots u^i_\sigma \ldots)=0 \ \ \ j=1,\ldots,l
\label{Fj}
\ee
It is easily seen now that the functions $u^i=f_i(x),\ i=1,\ldots,m$ give us a
solution of \eqn{Fj} iff $L_{(k)}\subset N$ where $L\subset E$ is the
submanifold of $E$ given by the equations $u^i=f_i(x)$. This motivates the
following concept which is crucial for what follows:\\ an $R$manifold is
called a multivalued solution of \eqn{Fj} iff $N\subset{\cal I}$.
We stress that the concept of $R$manifold allows one to generalize the notion
of solution for an arbitrary nonlinear solution system of PDE essentially in
the same way as the concept of lagrangian submanifold in $T^*M$ does for the
HamiltonJacobi equation.
Roughly speaking, equations ${\cal Y}_\Sigma$, as mentioned in the
introduction, describe possible shapes
of singular submanifolds of ${\rm sing}_L$ formed by all $\Sigma$type singular
points. The system ${\cal Y}_\Sigma$ for $\Sigma = R_{(k)}$ will be
called the $k$singularity system associated to ${\cal Y}$. This is a
(generally undetermined) system of partial differential
equations on $nk$ independent
variables, which contains a specific equation which we call $k$characteristic;
$1$characteristic equations coincide with classical characteristic equations
introduced by Hadamard when studying the uniqueness of the Cauchy problem.
Note that the eikonal equation is the characteristic equation for a number
of fundamental equations of mathematical physics. So, $k$characteristic
equations for $k>1$ describe, in particular, `wave front' propagation for
`extended objects'. We call complementary equations those which have to be
added to the characteristic ones to get the full $k$singularity system.
Remembering that the characteristic equations describe the spacetime form
of solution singularities it is natural to think that complementary equations
describe behaviours of internal structures of singularities giving a more
intrinsic description.
It is worth to emphasize that the study of asymptotic
solutions of a differential equation leads to the theory of lagrangian
submanifold on $T^*M$. From his point of view one can treat lagrangian
submanifolds as the asymptotic counterpart of $R$ manifolds. A physical
interpretation of these complementary equations depend obviously on the
physical nature of the original equation in question. We hope to present some
examples of this kind in a future publication.
In the following two sections we deduce both $k$characteristic equations
and complementary equations for fundamental equations of mathematical
physics. The necessary computational algorithms, extracted from the geometrical
description of ${\cal Y}_\Sigma$ given in \cite{av55}are presented here without
proof.
\section{$k$characteristic equations.}
{\Large\em Characteristic equations for a differential equation }
The simplest, but not trivial, case in which $k$characteristic equations
appear is that of second order scalar differential equations.
$k$characteristic equations for them can be found as follows.
Let $x = (x_1,\dots ,x_n)$ be independent variables. The general second
order scalar differential equation is of the form:
\be
F(x,u,u_i,u_{ij}) = 0 \label{eqgeneral}
\ee
where $u_i=\partial u / \partial{x_i}$ etc. The corresponding
characteristic matrix is then of the form:
\be
M= \left( {\partial F\over \partial u_{ij}} \right)\ . \label{matrixm}
\ee
With this matrix we can associate a bilinear pairing on 1forms on the space of
independent variables, namely
\be
_F = {\partial F\over\partial u_{ij}} u_iw_j .
\ee
This pairing can be extended to the full exterior algebra. For instance on
two forms
$$
_F =
_F_F 
$$
\be
 _F
_F.
\ee
The $k$ characteristic equations express the fact that the
$(nk)$vector tangent to $N$ is isotropic with
respect to the metric on $\Lambda^{nk}TM$ induced naturally by the metric on
$TM$ which is in its turn dual to the metric $M_{ij} , \ \ M_{ij} = {\del
F\over\del u_{ij}}$ on $T^*M$. Equivalently these equations state that the
dual $k$covector is isotropic
with respect to the metric on $\Lambda^{k}T^*M$ coming from the just mentioned
metric on $T^*M$. The fact that a decomposable $k$covector $ \theta^*_1\wedge\dots\wedge
\theta^*_k\in \Lambda^{k}T^*_aM\ , where \ \theta^*_1,\dots,
\theta^*_k\in T^*_aM $, is isotropic means that the $K$dimensional subspace $L$
generated by the $\theta$'s is tangent to the characteristic cone
$K^*_a\subset T^*_aM$ given by the equation
\be
\sum_{ij} M_{ij}(x)p_ip_j = 0 \ \ {\rm for} \ x = a .
\ee
Similarly, the dual $(nk)$ vector $ \theta_1\wedge\dots\wedge
\theta_{nk}\in \Lambda^{nk}T_aM\ ,\ \theta_1,dots ,
\theta_{nk}\in T_aM $, being isotropic is tangent to the dual cone $K_a\subset
T_aM$ given by
\be
\sum M^{ij}(x)v_iv_j = 0
\ee
where $M^{ij}$ is the $n1$order minor of the matrix $M_{ij}$ which is the
complement of the element $M_{ij}$. It results that a
solution of equation \eqn{eqcara} is an $(nk)$dimensional submanifold $
N$ of $M$ which is tangent to the cone $K_a$ at each point $a$.
The lines along which $N$ is tangent to the cones $K_a$ form a field of
directions ($=1$dimensional distribution) on $N$. Integral curves of
this distribution are exactly those along which $R_k$singularities propagate.
For $k=1$ they are classical bicharacteristics of the original equations
\eqn{eqgeneral}.
To find explicitly the $k$characteristic equations
divide the variables $x$ into two parts, say $\tau= (\tau_1,\dots,\tau_{nk})$
and $y = (y_1,\dots,y_{k})$,
where for instance
\bea
\tau_i & = & x_{k+i} \ , 1\leq i\leq nk\nonumber\\
y_i & = & x_{i} \ , 1\leq i\leq k\ . \label{choice}
\eea
Suppose that the projection of the
$k$singularity (which lies in $J^2$) on the $x$ space is of the form
$$
{\tilde\Phi}_i = y_i  \phi_i(\tau) = 0, \ \ \ i=1,\dots,k\ .
$$
The $k$form
\be
d{\tilde\Phi}_1\wedge d{\tilde\Phi}_2\wedge\cdots\wedge d{\tilde\phi}_k
\ee
defines the $k$characteristic equations by setting
\be
= 0.\label{kch}
\ee
Using the coordinates introduced above these equations can be written in the
following way: consider the $(nk)\times n$matrix
\be
\Phi =
\frac{\del x_i(\tau)}{\del\tau_j} \label{eqgeneralphi}
\ee
with $i=1,\ldots n,\ j=1\ldots nk$. Indicating with
$\phi_{k,l}=\frac{\del\phi_k}{\del \tau_l}$, with the above choice \eqn{choice}
we have
\be
\Phi=
\left(
\begin{array}{cccccccc}
\phi_{1,1} & \phi_{2,1} & \ldots & \phi_{k,1} & 1 & 0 & \ldots & 0\\
\phi_{1,2} & \phi_{2,2} & \ldots & \phi_{k,2} & 0 & 1 & \ldots & 0\\
\dots &\dots &\dots &\dots &\dots &\dots &\dots &\dots \\
\phi_{1,nk} & \phi_{2,nk} & \ldots & \phi_{k,nk} & 0 & 0 & \ldots & 1\\
\end{array}
\right). \label{expphi}
\ee
Indicating by
$\Phi_{i_1,\dots,i_{nk}},\ 1\leq i_1 m$), $1$ and $2$characteristic
equations are \eqn{1char} and \eqn{2char} imposed on all characteristic determinants.
\section{Examples}
We now discuss in detail some examples, first the case of the
KleinGordon equation. Here the eikonal equation we find for 1folds. For
2folds we find an equation describing a null two dimensional surface, to be
interpreted as an analog of the `wave front' propagation.
The situation is analogous for 3folds, were we will observe the
$n, (nk)$fold duality.
Then we discuss the
Schr\"odinger equation were we find that, since the symbol of the differential
operator does not contain any information, not only on the potential
$V$, but also on the time derivatives, the solutions of the characteristic equation
are `spacelike', that is transverse with respect to time. We will discuss
and interpret these results. finally we will consider $1$ and $2$
characteristic equations for the Dirac equation.
{\Large\em KleinGordon equation}
The KleinGordon equation is:
\be
(\del_t^2 \vec\nabla^2 +m^2)u=0\
\ee
The matrix $M$ and the differential equation on $J_2$ are respectively
\be
M=
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{array}
\right)
\ee
and
\be
F = u_{00}  \Sigma u_{ii} + m^2 u = 0 .
\ee
The parametrizations of the singularities and the corresponding characteristic
equations for 1,2 and 3folds are:
{\large\em 1Folds}
We write the singularity in the form:
\be
t=\phi(x_1,x_2,x_3)\ ,
\ee
therefore the $ 3 \times 4$ matrix $\Phi$ is
\be
\Phi=\left(
\begin{array}{cccc}
\phi_1 & 1 & 0 & 0 \\
\phi_2 & 0 & 1 & 0 \\
\phi_3 & 0 & 0 & 1
\end{array}
\right)\ .
\ee
Using equation \eqn{eqcara}, or equation \eqn{kch}
\be
= 0
\ee
where
\be
{\tilde\Phi} = \phi(x_1,x_2,x_3)  t = 0,
\ee
we obtain
\be
\sum_{i=1}^3 \phi_i^2 = (\vec\nabla\phi)^2 = 1\ . \label{kgic}
\ee
This is the eikonal equation. Observe however
that an interpretation of this equation (more precisely of its characteristics)
in terms of particles associated to the
fields is correct only for $m=0$ as
the surfaces move at the speed of light. The meaning of \eqn{kgic} for the
massive case has been discussed by Racah, \cite{Racah}, in the context of the
Dirac equation. He observed that if the wave function of a particle
is different from zero in a finite region, then the
eikonal equation describes the motion of the boundary of such a region.
Since the Fourier expansion of the
wave function will have components with all possible wave numbers,
this surface will move at the speed of light, even if the particle does not.
This problem of interpretation will obviously be present for all $k$folds, as
well as for the Schr\"odinger case below,
the next section is in
fact dedicated to a discussion of this problem.
{\large\em 2Folds}
In this case it is convenient to parametrize the singularity as follows:
\bea
x_2 & = & \phi_2(t,x_1) \\
x_3 & = & \phi_3(t,x_1)\ .
\eea
As this is not the choice made in \eqn{choice}, the form of $\Phi$ will be
slightly different, using equation \eqn{eqgeneralphi} we obtain:
\be
\Phi = \left(
\begin{array}{cccc}
1 & 0 & \phi_{2,0} & \phi_{3,0}\\
0 & 1 & \phi_{2,1} & \phi_{3,1}
\end{array}
\right)\ .
\ee
And the equation is
\be
(\phi_{2,0}\phi_{3,1}\phi_{2,1}\phi_{3,0})^2
+(\phi_{3,0})^2 +(\phi_{2,0})^2 (\phi_{3,1})^2 (\phi_{2,1})^2
1 = 0
\ee
This equation describes a two dimensional null submanifold, that is a surface
which is everywhere tangent to a null cone. Notice that the world surfaces of
null (tensionless) strings \cite{nullstring} are two dimensional null
submanifolds.
{\large\em 3Folds}
Again here it is convenient to use $t$ in the parametrization of the
singularity:
\bea
x_1 & = & \phi_1(t)\\
x_2 & = & \phi_2(t)\\
x_3 & = & \phi_3(t)\ ,
\eea
thus
\be
\Phi= (1\ \phi_{1,0}\ \phi_{2,0}\ \phi_{3,0})\ .
\ee
And the characteristic equation is
\be
\sum_{i=1}^3 \phi_{i,0}^2 = 1
\ee
This equation describes a null curve.
In this equation we immediately recognize the lagrangian of a free particle,
thus showing the above mentioned duality.
{\Large\em Schr\"odinger Equation}
The Schr\"odinger equation is:
\be
\left( i \hbar {\del\over\del t}  {\hbar^2 \nabla^2\over 2m} +
V(\vec x)\right) u(\vec x,t) =0
\ee
with $t=x_0$. Written in terms of the coordinates on $J_2$ this equation becomes
\be
F=i \hbar u_0  \hm \sum_{i=1}^3 u_{ii} + V u=0 .
\ee
>From equation \eqn{matrixm} we obtain for the matrix $M$:
\be
M=
\left(
\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & \hm & 0 & 0\\
0 & 0 & \hm & 0\\
0 & 0 & 0 & \hm
\end{array}
\right)
\ee
{\large\em 1Folds}
The singularity is of the form:
\be
t=\phi(x_1,x_2,x_3)
\ee
therefore the $ 3 \times 4$ matrix $\Phi$ is
\be
\Phi=\left(
\begin{array}{cccc}
\phi_1 & 1 & 0 & 0 \\
\phi_2 & 0 & 1 & 0 \\
\phi_3 & 0 & 0 & 1
\end{array}
\right)
\ee
and using equation \eqn{eqcara} we obtain
\be
\sum_{i=1}^3 \phi_i^2=0\
\ee
Which has as solution
\be
\phi= t=\mbox{\rm const.}
\ee
That is the fold is transverse with respect to time, this is in agreement
with what we said about the eikonal equation for the massive KleinGordon
equation. Here in fact, being the theory non relativistic, the
surface bounding the region in which the wave function
is different from zero moves with infinite speed
{\large\em 2Folds}
For 2folds instead the equation for the form of the singularity and of the
matrix $\Phi$ are:
\bea
t & = & \phi_0(x_2,x_3)\\
x_1 & = & \phi_1(x_2,x_3)\\
\Phi & =& \left(
\begin{array}{cccc}
\phi_{0,2} & \phi_{1,2} & 1 & 0\\
\phi_{0,3} & \phi_{1,3} & 0 & 1
\end{array}
\right)
\eea
and the equation is
\be
(\phi_{0,2}\phi_{1,3}\phi_{1,2}\phi_{0,3})^2 + \phi_{0,2}^2 +
\phi_{0,3}^2 =0\ .
\ee
The solutions are :
\bea
t& =& \mbox{\rm constant} \nonumber\\
\phi_1 & = & \phi_1(x_2,x_3)
\eea
where $\phi_1(x_2,x_3)$ is an arbitrary function.
They describe two dimensional surface in space at a fixed time.
Even in this case the singularities are transverse with respect to time.
{\large\em 3Folds}
We parametrize the 3Folds as follows:
\bea
t & = & \phi_0(x_3)\\
x_1 & = &\phi_1(x_3)\\
x_2 & = &\phi_2(x_3)
\eea
thus
\be
\Phi=(\phi_{0,3} \ \phi_{1,3} \ \phi_{2,3} \ 1 )
\ee
and the equation is
\be
(\phi_{0,3})^2=0
\ee
which again has solution $\phi_0=t=$const, the other $\phi$'s being arbitrary.
This describes a
onedimensional curve in space at a given time. And therefore such a curve
cannot be considered a worldline.
{\Large\em Dirac Equation}
We finish this section with a brief discussion of Dirac equation, or rather
Dirac equations, as it is a system of four equations, one for each
component of the spinor. The system is
\be
(i \not\!\del m) u = 0
\ee
which written explicitly is:
\be
F_\alpha(x^\mu,u^\alpha,u^\alpha_\mu) =
i \gamma^\mu_{\alpha\beta}u_\mu^\alpha 
m \delta_{\alpha\beta}u^{\beta} = 0
\ee
{\large\em 1Folds}
If the singularity is defined by
\be
f(x^\mu) = 0,
\ee
The equation characteristic is:
\be
\det \left( \frac{\del F_\alpha}{\del u^\beta_\nu} f_\nu \right) = 0
\ee
Which gives:
\be
f_\nu f^\nu = 0
\ee
{\large\em 2Folds}
Parametrizing the fold by
\bea
x_2 & = & \phi_2(x_0,x_1) \\
x_3 & = & \phi_3(x_0,x_1)
\eea
and following the procedure of section (3)
we find the $2$characteristic equation:
\be
\sum_i \phi_{2,i}\phi_{3,i} + \sum_{i,j} \phi_{2,i}^2 \phi_{3,i}^2
\phi_{2,j}^2 +\sum_{i\neq j} \phi_{2,i}^2 \phi_{3,j}^2 = 0
\ee
\section{Extended Equations}
In the case of Maxwell equations, or massless KleinGordon, the characteristic
equation describes the classical motion of the particles associated to the
fields (photons or scalar massless particles). In the Schr\"odinger or massive
KleinGordon case this does not happen. The reason is that the characteristic
equation is sensitive only to the symbol and therefore potential and time
derivative do not appear in the former case, while all information about the
mass is absent from the latter, for which wave fronts move in fact at the
speed of light.
We expect that other singularities, sensitive not only to the symbol, will
provide the particle trajectories even in this case.
We observe however that it is possible to write
the equations above as reduction of homogenous equations in an extended space.
If this is done, all the terms in the equation will contribute to the symbol,
and the folds will then yield equations of motion of the particles even in
these cases. We will describe briefly the reduction procedure and then consider
again the fold singularities and how they get reduced.
Consider a second order differential equation of the kind
\be
(A^{\mu\nu}\del_\mu\del_\nu + B^\mu\del_\mu +C) u=0 \label{eqorig}
\ee
where the coefficients $A^{\mu\nu},B^\mu,C$ are functions, and
$\mu,\nu=0\ldots,n1$. Introducing an extra variable $x_{1}$, this equation
can be obtained as reduction of the following equation homogenous in the second
derivatives:
\be
g^{ab}\del_a\del_b \tilde u=0 \ a,b=1,\ldots,3
\ee
The $n+1 \times n+1$ metric $g^{ab}$, written in terms of the matrix
$A=(A_{\mu\nu})$,
the vector $B=(B_\mu)$ and the scalar $C$, has the form:
\be
g^{ab}=
\left(
\begin{array}{cc}
C & B\\
B^T & A
\end{array}
\right)
\label{extmet}
\ee
If we consider the space with an additional variable as a principal $R$bundle,
functions $\tilde u$ on the total space are simply $R$equivariant functions.
The new operator $\tilde D$ on the total space and the operator $D$ on the base
manifold are related by
\be
g^*(\pi^*(Du)) + \tilde D(g^*\pi^*u).
\ee
The reduction is obtained restricting the dependence of the functions
$\tilde u(x_a)$ on the $x_{1}$ as follows:
\be
\tilde u(x_a) =e^{x_{1}} u(x_\mu) \label{utilde}
\ee
and setting $x_{1}$ equal to a constant. In our examples, in order to have
that the extended metric has signature $(+,,\ldots,)$, we will use
equation \eqn{extmet} with $C$ replaced by $C$ and equation \eqn{utilde}
with $e^{x_{1}}$ replaced by $e^{ix_{1}}$.
The reduction for the singularity is obtained by fixing the value od the
variable $x_{1}$.
The introduction of this reduction procedure \cite{reduction} here may
seem artificial, it is nonetheless interesting because it shows that
already the simplest (fold) singularities have several non trivial
features and are able to capture relevant aspects of the dynamics of
particles associated to fields. In the following we will find convenient
to put some constants in the exponent of $\tilde u$ in \eqn{utilde} rather
than in the metric $g^{ab}$.
{\Large\em Extended KleinGordon case}
The extended KleinGordon equation is:
\be
g^{ab}\tilde u_{ab}=0
\ee
with the metric
\be
g^{ab}=
\left(
\begin{array}{ccccc}
m^2 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 1
\end{array}
\right)
\ee
and the reduction follows from:
\be
\tilde u = e^{ix_{1}} u \label{redk}
\ee
{\large\em 1Folds}
Parametrised the singularity by:
\be
x_{1}=\phi(x_\mu) \ \ \ \ \ \mu=0,\ldots,3 \label{x_=phi}
\ee
and $\Phi$ is
\be
\Phi=\left(
\begin{array}{ccccc}
\phi_0 & 1 & 0 & 0 & 0\\
\phi_1 & 0 & 1 & 0 & 0\\
\phi_2 & 0 & 0 & 1 & 0\\
\phi_3 & 0 & 0 & 0 & 1
\end{array}
\right)\ ,
\ee
the characteristic equation then is:
\be
g^{\mu\nu}\phi_\mu\phi_\num^2=0
\ee
which after reduction, namely setting $x_{1}= const.$ in \eqn{x_=phi}
now correctly describes free particles with arbitrary
masses.
{\large\em 2Folds}
The singularity and $\Phi$ are:
\bea
x_{1} & = &\phi_{1}(x_i)\\
x_{0} & = &\phi_{0}(x_i)\ \ \ \ \ \ i=1,2,3\\
\Phi & = &
\left(
\begin{array}{ccccc}
\phi_{1,1} & \phi_{0,1} & 1 & 0 & 0\\
\phi_{1,2} & \phi_{0,2} & 0 & 1 & 0\\
\phi_{1,3} & \phi_{0,3} & 0 & 0 & 1
\end{array}
\right)
\eea
and the characteristic equation is:
\be
\sum_i(\sum_{jk} \varepsilon_{ijk} \phi_{1,j}\phi_{0,k})^2
+ \sum_i \left(m^2(\phi_{0,i})^2  (\phi_{1,i})^2 \right) m^2 =0
\ee
{\large\em 3Folds}
The singularity is parametrized by
\bea
x_{1} & = &\phi_{1}(x_i)\\
x_{0} & = &\phi_{0}(x_i)\\
x_{1} & = &\phi_{1}(x_i)\ \ \ \ \ \ i=2,3,\\
\Phi & = &
\left(
\begin{array}{ccccc}
\phi_{1,2} & \phi_{0,2} & \phi_{1,2} & 1 & 0\\
\phi_{1,3} & \phi_{0,3} & \phi_{1,3} & 0 & 1
\end{array}
\right)
\eea
The characteristic equation is:
$$
(\phi_{1,2}\phi_{0,3}\phi_{0,2}\phi_{1,3})^2
+(\phi_{1,2}\phi_{1,3}\phi_{1,2}\phi_{1,3})^2
m^2(\phi_{0,2}\phi_{1,3}\phi_{1,2}\phi_{0,3})^2
$$
\be
+(\phi_{1,3})^2 +(\phi_{1,2})^2
+m^2\left( (\phi_{1,3})^2 +(\phi_{1,2})^2
(\phi_{0,3})^2 (\phi_{0,2})^2 + 1 \right) =0
\ee
{\large\em 4Folds}
In this case we parametrize the singularity with $t$:
\bea
x_{1}& =& \phi_{1}(t)\\
x_i & = &\phi_i(t) \ \ \ \ \ i=1,2,3\\
\Phi & = & (\phi_{1,0} \ \ 1 \ \ \phi_{1,0} \ \ \phi_{2,0} \ \ \phi_{3,0})
\eea
and the characteristic equation is:
\be
(\phi_{1,0})^2 = m^2\left( 1 \sum_{i=1}^3(\phi_{i,0})^2 \right)
\ee
Which after reduction gives the square of the usual lagrangian
$m\sqrt{x_\mu x^\mu}$
We now list the parametrization of the singularities and the corresponding
characteristic equations for $k$folds in the Schr\"odinger
case.
{\Large\em Extended Schr\"odinger case}
The extended Schr\"odinger equation is:
\be
g^{ab}\tilde u_{ab}=0\ \ \ \ \ \ a,b=1,0,1,2,3
\ee
with
\be
g^{ab}=
\left(
\begin{array}{ccccc}
V & 1/2 & 0 & 0 & 0\\
1/2 & 0 & 0 & 0 & 0\\
0 & 0 & {1\over 2m} & 0 & 0\\
0 & 0 & 0 & {1\over 2m} & 0\\
0 & 0 & 0 & 0 & {1\over 2m}
\end{array}
\right)
\ee
The expression for $\tilde u$ is
\be
\tilde u(x) = e^{i{x_{1}\over\hbar}}u(x^\mu) \label{reds}
\ee
{\large\em 1Folds}
\be
x_{1}=\phi(x_\mu)
\ee
\be
{1\over 2m}(\nabla \phi)^2 + \phi_0 V = 0
\ee
Reducing $\phi$ one finds that this can be interpreted in four dimensions
as the HamiltonJacobi equation.
{\large\em 2Folds}
\bea
x_{1} & = & \phi_{1}(x_i)\\
x_{0} & = & \phi_0(x_i) \ \ \ i=1,2,3
\eea
\be
({1\over 2m})^2\sum_{ijk}(\epsilon_{ijk}\phi_{1,j}\phi_{0,k})^2 
{1\over 2m}\sum_i
\phi_{1,i}\phi_{0,i} + V{1\over 2m}\sum_i(\phi_{0,i})^2  {1\over 4} = 0
\ee
{\large\em 3Folds}
\bea
x_{1} & =& \phi_{1}(x_2,x_3)\\
x_{0} & =& \phi_0(x_2,x_3)\\
x_{1} & =& \phi_1(x_2,x_3)
\eea
$$
({1\over 2m})^3
\left(
\phi_{1,2} \phi_{0,3}  \phi_{0,2} \phi_{1,3}
\right)^2
$$
$$
+({1\over 2m})^2
\left( \left(
\phi_{0,2} \phi_{1,3}
\phi_{1,2} \phi_{0,3}
\right)
\left(
\phi_{1,2} \phi_{1,3}
\phi_{1,2} \phi_{1,3}
\right)+
\phi_{1,3} \phi_{0,3}
+\phi_{1,2} \phi_{0,2}
\right)
$$
\be

V ({1\over 2m})^2
\left(
\left(
\phi_{0,2} \phi_{1,3}
\phi_{1,2} \phi_{0,3}
\right)^2
+\left(
\phi_{0,3}
\right)^2
+\left(
\phi_{0,2}
\right)^2
\right)
{1\over 8 m}
\left(
\left(
\phi_{1,3}
\right)^2
+
\left(
\phi_{1,2}
\right)^2
+1
\right) = 0
\ee
\pagebreak[2]
{\large\em 4Folds}
\bea
x_{1} &=&\phi_{1}(t)\\
x_i & = & \phi_i(t) \ \ \ i=1\ldots 3
\eea
\be
{1\over 2} m \left( \sum_{i=1}^3(\phi_{i,0})^2 \right) 
\phi_{1,0} + V =0
\ee
After reduction this equation gives the lagrangian of a particle in the
potential $V$.
\section{Complementary equations for $1$folds}
In this last section we discuss the complementary equations which, as we said,
are necessary for a complete description of the singularities.
The procedure for finding the complementary equations for $1$folds in the
case of 1singularity equations for a scalar second order differential equation
is as follows.
Let us consider a singularity described by the equation
\be
x_n\phi(x_i)=0\ \ \ \ i=1,\ldots,n1\ , \label{sing}
\ee
a basis of vector fields tangent to the projection of the fold on the base is
\be
X_i=\del_i+\phi_i\del_n \label{tang}
\ee
We also have the initial data
\bea
uh & = & 0\\
u_ng & = & 0
\eea
Initial data on a singularity are subject to constraints, which can be
described in terms of a set of equations, which turn out to be the
complementary equations. To find them let us proceed as follows.
Acting with the vector fields $X_i$ on the initial data, after some
manipulations and including the differential equations $F=0$, we obtain a
system in the unknowns $u_{\mu \nu}$:
\bea
u_{in} +\phi_iu_{nn} & = & g_i \nonumber\\
u_{ij}\phi_i\phi_j u_{nn} & = & h_{ij}  \phi_{ij} g  (\phi_i g_j + \phi_j
g_i) \nonumber\\
F(x_\mu,u,u_\mu,u_{\mu \nu}) & = & 0 \label{compsys}
\eea
There are $n1+{n (n1)\over 2} +1 = {n (n+1)\over 2}$ equations in the same
number of unknowns.
Using the 1characteristic equation (i.e.\ along the singularity) the system
becomes degenerate, that is the determinant of the matrix of the
coefficients of the $u_{\mu \nu}$'s vanishes identically when the characteristic
equation is substituted into it. Writing the system in matricial form we can express it as
\be
M\cdot U = C
\ee
Where $U$ is the vector of the unknowns, $M$ is the matrix of the coefficients,
and $C$ is the vector of the known factors of the system \eqn{compsys}.
If we indicate by $Y_i$ a basis of vectors of the left kernel of $M$,
\be
Y_i^\dagger
M = 0,
\ee
the complementary equations are
\be
Y_i^\dagger C = 0
\ee
Obviously if we change the choice of the coordinate $x_n$ in \eqn{sing}, the
equations (\ref{tang}\ref{compsys}) will change accordingly.
{\Large\em Examples}
{\large\em KleinGordon case}
The singularities are described by the equation
\be
t\phi(x_i)=0\ ,\ \ i=1\ldots,3\ \ . \label{kgsing}
\ee
and the initial data are:
\bea
u  h &=& 0 \\
u_0  g &=& 0
\eea
The system \eqn{compsys} becomes:
\bea
u_{i0} + u_{00}\phi_i & = & g_i \nonumber\\
u_{ij}  \phi_i \phi_j u_{00} & = & h_{ij}  \phi_{ij} g  (\phi_i g_j
+ g_j \phi_i)\nonumber\\
u_{00}  \sum_{i=1}^3 u_{ii} & = & m^2 h
\eea
The characteristic equation is
\be
(\vec\nabla \phi)^2 = 1\ .
\ee
The complementary equation is
\be
\nabla^2 h + m^2 h  g  (\nabla^2 \phi) g
2\vec \nabla \phi \cdot \vec \nabla g = 0
\ee
{\large\em Schr\"odinger case}
With the same parametrization of the singularity,
the system \eqn{compsys} becomes:
\bea
u_{i0} + u_{00}\phi_i & = & g_i \nonumber\\
u_{ij}  \phi_i \phi_j u_{00} & = & h_{ij}  \phi_{ij} g  (\phi_i g_j
+ g_j \phi_i)\nonumber\\
\hm \sum_{i=1}^3 u_{ii} & = &  V u + i \hbar g
\eea
The characteristic equation is
\be
\sum_{i=1}^3 \phi_i^2 = 0\ ,
\ee
whose solution is
\be
\phi_i = 0\ .
\ee
The complementary equation is:
\be
\hm \nabla^2 h + V u  i\hbar g = 0
\ee
{\large\em Extended KleinGordon}
Parametrising the singularity as follows:
\be
x_{1}\phi(x_\mu)=0\ \ \mu=0,\ldots,3 \ ,
\ee
the complementary equation is:
\be
\Box \tilde h  \Box\phi \tilde g 2\del_\mu\phi\del^\mu \tilde g = 0
\ee
The reduced equation is obtained using for $\tilde h$ and $\tilde g$ the ansatz
in eq. \eqn{redk}:
\bea
\tilde h(x) &=& e^{ix_{1}}h(x^\mu) \\
\tilde g(x) &=& \del_{1} \tilde h(x) = i e{ix_{1}}h(x^\mu).
\eea
The result is
\be
\Box h  \Box\phi h 2i\del_\mu\phi\del^\mu h = 0.
\ee
{\large\em Extended Schr\"odinger }
Using the same parametrisation of the previous example, the complementary
equation is:
\be
\hm(\nabla^2 \tilde h  (\nabla^2 \phi) \tilde g
2 \nabla \phi \cdot \nabla \tilde g)\tilde g_0 = 0.
\ee
According to eq.\eqn{reds}, we set
\bea
\tilde h(x) &=& e^{i{x_{1}\over \hbar}}h(x^\mu) \\
\tilde g(x) &=& \del_{1} \tilde h(x) = {i\over \hbar}
e{i{x_{1}\over \hbar}}h(x^\mu).
\eea
So that the reduced equation is
\be
\hm(\nabla^2 h  {i\over \hbar}(\nabla^2 \phi) h
 {i\over \hbar}2 \nabla \phi \cdot \nabla h)  {i\over \hbar}h_0 = 0
\ee
{\large\em Maxwell Case}
Now we deduce the complementary equations for Maxwell equations.
This time the system of differential equations is degenerate of the first
order, and the fields are vectors, therefore the fields $u^\alpha$ will have
an extra (upper) index, to avoid confusions, we will use for it the first
letters of the greek alphabet. The differential equations are represented on
$J_1$ by $F_a(x_\mu,u^\alpha,u^\alpha_\mu)=0$.
The characteristic equations can be obtained by equating to 0
highest order minors of the characteristic matrix:
$
({\del F_a}/{\del u^\alpha_\mu}) f_\mu
$
where the $f_\mu(x)=0$ is the equation for the singularity.
In the Maxwell case the fields are 6, we identify $u^i=E^i, u^{i+3}=B^i,\
i=1,2,3$, the 8 Maxwell equations are:
\bea
u_i^i & = & 0 \nonumber\\
u_i^{i+3} & = & 0 \nonumber\\
\varepsilon ^{ijk}u^j_i & = & u_0^{k+3} \nonumber\\
\varepsilon ^{ijk}u^{j+3}_i & = & u_0^{k}\epsilon\mu \label{max}
\eea
The characteristic matrix is rectangular $ 6 \times 8$, and its minors of order
6 have determinants of the form
\be
\epsilon\mu f_0^2  \sum_{i=0}^3 f_i^2 = 0.
\ee
So that the only solution is, as expected, the eikonal
equation.
We now look for the complementary equations. To this purpose let us consider a
singularity described by equations
\be
t  \phi(x^i) = 0 ,
\ee
with tangent fields spanned
by
\be
X_i=\del_i+\phi_i\del_0\ .
\ee
This time the initial data will be of the kind:
\be
u^\alpha  h^\alpha = 0
\ee
Proceeding in analogy with the scalar case we obtain:
\be
u^\alpha_i + \phi_i u^\alpha_0 = h^\alpha_i
\ee
which together with the \eqn{max} form a system on $J_1$.
To find the complementary equations we can proceed as before, even if the
system is overdetermined. In this case the matrix of the coefficients is $26
\times 24$ and, using the eikonal, the left kernel has dimension four.
The equations one obtains can be written as:
\bea
\epsilon\mu\nabla\cdot h_E + \nabla f \cdot (\nabla \wedge h_B)
& = & 0\\
\epsilon\mu\nabla\wedge h_E + \nabla f (\nabla\cdot h_B)
\nabla f \wedge (\nabla \wedge h_B) & = & 0
\eea
where
\bean
h_E & = & (h_1,h_2,h_3)\\
h_B & = & (h_4,h_5,h_6)\ .
\eean
{\large\em Dirac equation}
Here we find four complementary equations:
\be
f_\mu\gamma^\mu_{\nu\alpha} (i\gamma^j_{\alpha\beta}u^\beta_j  u^\alpha) = 0
\ee
where
$
j = 1,\ldots , 3 \ \
\alpha,\beta,\mu,\nu = 1,\ldots,4 \ ,
$
and $f = 0$ is the equation of the singularity.
\section{Conclusions}
>From the mathematical point of view the theory of singularities of the
generalized solutions of PDE's is a generalization of the standard
singularity (or `catastrophe') theory. In fact the latter can be viewed as
the part of the former dealing with solution of zeroorder differential
equations. Many interesting aspects appear in this generalization, and we
discussed but a few of them in this paper. Therefore there is no doubt that
this generalized theory of singularities is worth to be developed as a
new branch of pure mathematics to a much larger extent. For the state of
the art see \cite{av49,av17,av28,av52,av27}. A possible important role of
this theory is discussed in \cite{av52,avpep}
The `phenomenology' presented in this article indicates a number of more
concrete problems of interest. Among them there is the systematic development
of the theory of bicharacteristic of $k$characteristic equations. Some
sort of duality between $k$characteristic and $(nk)$characteristic
equations emerges from this paper, this lead to the hypothesis of an
analogue of the Legendre transform, and a natural extension of the
classical Lagrangian formalism. Apart from these argument we can expect a
generalization of the standard Hamiltonian formalism which, with respect to
$\Sigma$characteristic equations, would play the same role the
standard one plays with the usual characteristics. It is very likely that such
a generalization is in the spirit of \cite{michor,avcabras}.
A problem that remains open is that of a systematical physical
interpretation of the new equations presented here, as well as the search
of alternative singularity types. In particular the extended equations
presented in section 5, which at this moment can seem a mere trick, have to
be understood more conceptually. Another open question of possible physical
relevance is that of putting the old problem of the field sources
\cite{levi,luneburg} in the framework of the theory presented here.
\ \\
{\bf Acknowledgments}\\
Part of this work has been performed while we were in Vienna at the
{\sl E.~Schr\"odinger International Institute for Mathematical Physics}, we
would like to thank Profs. W.~Thirring and P.~Michor for the kind hospitality.
% A macro to raise things. Used in math and journal macros.
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%journal references
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\newcommand{\ijmpa}[3]{{\sl Int. J. Mod. Phys. }{\bf A#1} \up(19#2\up) #3}
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\begin{thebibliography}{99}
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\end{document}