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%===================================================================== 
%\noindent{\it file translat.tex, 27-06-1994, Tresguerres}
%\bigskip\bigskip
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\hfill {Preprint IMAFF 98/01}
\bigskip\bigskip
\centerline{\bf{OSTROGRADSKI FORMALISM}}
\centerline{\bf{FOR}}
\centerline{\bf{HIGHER-DERIVATIVE SCALAR FIELD THEORIES}}
\vskip 0.3cm 
\centerline{by}
\vskip 0.7cm
\centerline{F.J. de Urries $^{(*)(\dagger)}$}
\vskip 0.2cm
\centerline{and}
\vskip 0.2cm
\centerline{J.Julve $^{(\dagger)}$}
\vskip 0.2cm
\centerline{$^{(*)}$\it Departamento de F\'\i sica, Universidad de Alcal\'a de %%@
Henares,}
\centerline{\it 28871 Alcal\'a de Henares (Madrid), Spain}
\vskip 0.2cm   
\centerline{$^{(\dagger)}$\it{IMAFF, Consejo Superior de Investigaciones
Cient\'\i ficas,}} 
\centerline{\it{Serrano 113 bis, Madrid 28006, Spain}}

\vskip 0.7cm

\centerline{ABSTRACT}\bigskip 
We carry out the extension of the Ostrogradski method to relativistic field
\break
theories. Higher-derivative Lagrangians reduce to second differential--order %%@
with one explicit independent field for each degree of freedom. We %%@
consider a higher-derivative relativistic theory of a scalar field and %%@
validate a powerful order-reducing covariant procedure by a rigorous phase-space %%@
analysis. The physical and ghost fields appear explicitly. Our results %%@
strongly support the formal covariant methods used in higher-derivative gravity.
 

\bigskip
\centerline{PACS numbers: 11.10.Ef, 11.10.Lm, 04.60} 

\bigskip
\centerline{Keywords: higher-derivative field theories}




\vfill
%\noindent{Corresponding} author:\quad Jaime Julve

%Postal address: given above

%Telephone: 00-34-1-5616800/Ext.3103

%Fax: 00-34-1-5854894

%e-mail:\quad imtjj51@cc.csic.es



\eject








    
 
\sectio{\bf{Introduction}}\bigskip 
Theories with higher order Lagrangians have an old tradition in physics, and %%@
Podolski's Generalized Electrodynamics [1] (later visited as a useful testbed  %%@
[2]), effective gravity and tachyons [3] are examples. The interest in higher %%@
order mechanical systems is alive until today [4].

Theories of gravity with terms of any order in curvatures arise as part of %%@
the low energy effective theories of the strings [5] and from the dynamics of %%@
quantum fields in a curved spacetime background [6]. Theories of second order 
(4--derivative theories in the following) have been studied more closely in %%@
the literature because they are renormalizable [7] in four dimensions. They %%@
greatly affect the effective potential and phase transitions of scalar fields  %%@
in curved space-time, with a wealth of astrophysical and cosmological %%@
properties [8]. In particular a procedure based on the Legendre %%@
transformation was devised [9] to recast them as an equivalent theory of %%@
second differential order. A suitable diagonalization of the resulting theory %%@
was found later [10] that yields the explicit independent fields for the
degrees of freedom (DOF) involved, usually including Weyl ghosts. 

In [11] the simplest example of this procedure was given using a model 
of one scalar field with a massless and a massive DOF.
In an appendix, Barth and Christensen [12] gave the splitting of the
higher-derivative (HD) propagator into quadratic ones for the 4th, 6th and %%@
8th differential-order scalar theories. A scalar 6--derivative theory 
has been considered in [13] as a regularization of the Higgs model, yielding %%@
a finite theory.

Classical treatises [14] study the Lagrangian and Hamiltonian
theories of systems with a finite number of DOF and higher time-derivatives
of the generalized coordinates. Later work has considered the variational %%@
problem of those theories with the tools of the Cartan form, k-jets, %%@
symplectic geometry and Legendre mappings [15].  

However the particular case of relativistic covariant field theories has %%@
complications of its own which are not covered by those general treatments.
Our presentation highlights the Lorentz covariance and the particle aspect of %%@
the theory, with emphasis in the structure of the propagators and the %%@
coupling to other matter sources. We address this issue by using a simplified %%@
model with scalar fields as in [11] and [12], and our extrapolation of the %%@
canonical analysis to these continous systems validates the formal %%@
procedures introduced there. The analysis here presented mostly focuses on %%@
the free part of the Lagrangian, and self-interactions and interactions with %%@
other fields are embodied in a source term.

In Section 2 we briefly review the Ostrogradski method and outline our %%@
extension to the field theories. In Section 3 we study the case of the
4-derivative theory for arbitrary non-degenerate masses, which exemplifies %%@
the use of the Helmholtz Lagrangian and the crucial diagonalization
of the fields. The 8-derivative case and higher 4$N$-derivative cases are %%@
considered in Section 4. For even $N$ the 2$N$-derivative cases present some %%@
peculiarities that deserve the separate discussion of Section 5. Our results    
are summarized in the Conclusions. 

As a general feature, our procedure involves vectors with pure real and %%@
imaginary components as well as symmetric matrices with equally assorted
elements. Diagonalizing symmetric matrices of this kind is a non-standard %%@
task which is detailed in an Appendix.



\vfill
\eject











\sectio{\bf The Ostrogradski's method.}
\bigskip

We consider a HD Lagrangian theory for a system described by
configuration variables \quad$q(t)$\quad. By dropping total derivatives, it
can be always brought to a standard form 

$$ L[q,\dot{q},\ddot{q},...,{\buildrel {(m)}\over{q}}]
                                               \quad , \eqno(\z)$$ 

\noindent{depending} on time derivatives of the lowest possible order. 
The variational principle then yields equations of motion which
are of differential order \quad$2m$\quad at most:

$$  {{\partial L}\over{\partial q}}
   -{d\over dt}{{\partial L}\over{\partial{\dot{q}}}}
   +\cdots
   +(-1)^m {d^m \over dt^m}
    {{\partial L}\over{\partial{\buildrel {(m)}\over{q}}}}
   = 0 \quad .                                               \eqno(\z)$$


\noindent{Hamilton's} equations are obtained by defining $m$ generalized
momenta 

$$\eqalign {p_m &\equiv  
      {{\partial L}\over{\partial{\buildrel {(m)}\over{q}}}}  \cr
            p_i &\equiv 
      {{\partial L}\over{\partial{\buildrel {(i)}\over{q}}}}  
      -{d\over dt}p_{i+1}
      \quad\quad (i=1,...,m\! -\! 1)\quad ,  \cr}  \eqno(\z)$$ 

\noindent{and} the \quad$m$\quad independent variables

$$\eqalign {q_1 &\equiv q  \cr
            q_i &\equiv {\buildrel {(i\!-\!1)}\over{q}}  
            \quad\quad (i=2,...,m)\quad .  \cr}  \eqno(\z)$$ 


\noindent{Then} the Lagrangian may be considered to depend on
the coordinates \quad$q_i$\quad and only on the first time derivative
\quad${\dot q}_m={\buildrel {(m)}\over{q}}$\quad.
A Hamiltonian on the phase space \quad[$q_i,p_i$]\quad may then be found
by working \quad${\dot q}_m$\quad out of the first equation (2.3) as a
function 

$${\bf{\dot q}_m}[q_1,...,q_m;p_m]\quad , \eqno(\z)$$

\noindent{the} remaining velocities \quad${\dot q}_i \;
(i=1,...,m\! -\! 1)$\quad already being expressed in terms of
coordinates, because of (2.4), as

$${\bf {\dot q}_i}=q_{i+1}\quad . \eqno(\z)$$

\noindent{Thus}

$$ H[q_i,p_i]=\sum_{i=1}^{m}\; p_i{\bf{\dot q}_i} - L
               =\sum_{i=1}^{m\!-\!1}\; p_i q_{i+1} + p_m{\bf{\dot q}_m}
                   - L[q_1,...q_m;{\bf{\dot q}_m}]\quad .
                                                      \eqno(\z)$$ 
\noindent{Therefore}

$$\eqalign{\delta H = \sum_{i=1}^{m\!-\!1}&( p_i\delta q_{i+1} + q_{i+1}
\delta p_i)+p_m\delta{\bf {\dot q}_m}+ {\bf {\dot q}_m}\delta p_m  \cr
&-\sum_{i=1}^{m}{{\partial L}\over{\partial q_i}}\delta q_i 
-{{\partial L}\over{\partial {\bf {\dot q}_m}}}\delta{\bf{\dot q}_m}
                                                    \quad,\cr}\eqno(\z)$$ 

\noindent{but} (2.3) can be written as 

$$\eqalign{{\partial L}\over{\partial{\bf{\dot q}_m}} &= p_m  \cr 
      {{\partial L}\over{\partial q_i}}&={\dot p}_i+p_{i-1}
      \quad\quad (i=2,...,m)\quad ,  \cr}  \eqno(\z)$$

\noindent{and} (2.2), because of (2.3), gives

$${{\partial L}\over{\partial q_1}}={{\partial L}\over{\partial q}}
                                           ={\dot p}_1\quad, \eqno(\z)$$

\noindent{so} we get 

$$\delta H= \sum_{i=1}^m(-{\dot p}_i\delta q_i+{\dot q}_i\delta p_i)
                                                       \quad,\eqno(\z)$$


\noindent{and} the canonical equations of motion turn out to be

$$ {\dot q}_i={{\partial H}\over{\partial {p}_i}}\quad ; \quad
    {\dot p}_i=-{{\partial H}\over{\partial {q}_i}}\quad . 
                                                      \eqno(\z)$$

\noindent{Summarizing} we may say that a theory with one
configuration coordinate \quad$q$\quad obeying equations of motion of 
\quad$2m$\quad differential order (stemming from a Lagrangian with quadratic
terms in \quad${\buildrel {(m)}\over{q}}$\quad as its highest derivative
dependence) can be cast as a set of 1st--order canonical equations 
for \quad$2m$\quad phase-space variables \quad$[q_i,p_i]$\quad.

As it is well known, once the differential order has been reduced by 
the Hamiltonian formalism, one may prefer to obtain the same canonical 
equations of motion from a variational principle. Then the canonical %%@
equations (2.12) are the Euler equations of the so-called Helmholtz %%@
Lagrangian

$$  L_H[q_i,{\dot q}_i,p_i] = \sum_{i=1}^{m}\; p_i{\dot q}_i 
                                   - H[q_i,p_i] \eqno(\z)$$
 
\noindent{which} depends on the \quad$2m$\quad coordinates \quad$q_i$\quad %%@
and \quad$p_i$\quad, and 
only on the velocities \quad${\dot q}_i$\quad. This alternative setting will %%@
be adopted later on.

As far as finite--dimensional mechanical systems are considered, only time %%@
derivatives are involved. The generalized momenta above have a mechanical %%@
meaning and the resulting Hamiltonian is the energy of the system up to %%@
problems of positiveness linked to the occurrence of ghost states.
\bigskip


\noindent{\bf Extension to field theories}

Continuous systems with field coordinates \quad$\phi (t,{\bf x})$\quad %%@
usually involve space derivatives as well, chiefly if relativistic covariance %%@
is assumed. We now generalize the previous formalism to this case. A HD field %%@
Lagrangian density will have the general dependence

$${\cal L}[\phi ,{\phi}_{\mu},...,{\phi}_{{\mu}_1\cdots{\mu}_m}]\quad,
                                                              \eqno(\z)$$ 

\noindent{where}\quad ${\phi}_{{\mu}_1\cdots{\mu}_i}\equiv                           %%@
{\partial}_{\mu_1}\cdots{\partial}_{\mu_i}\phi$\quad, with corresponding
equations of motion

$$  {{\partial{\cal L}}\over{\partial\phi}}
   -{\partial_\mu}{{\partial{\cal L}}\over{\partial{\phi_\mu}}}
   +\cdots
   +(-1)^m{\partial}_{\mu_1}\cdots{\partial}_{\mu_m}
    {{\partial{\cal L}}\over{\partial{\phi}_{{\mu}_1\cdots{\mu}_m}}}
   = 0 \quad .                                               \eqno(\z)$$

\noindent{The} generalized momenta now are

$$ \eqalign{\pi^{{\mu}_1\cdots{\mu}_m}&\equiv 
   {{\partial{\cal L}}\over{\partial{\phi}_{{\mu}_1\cdots{\mu}_m}}}\cr
    \pi^{{\mu}_1\cdots{\mu}_i}&\equiv 
   {{\partial{\cal L}}\over{\partial{\phi}_{{\mu}_1\cdots{\mu}_i}}}
   -\partial_{\mu_{i+1}} \pi^{{\mu}_1\cdots{\mu}_i{\mu_{i+1}}}
    \quad\quad (i=1,...,m\!-\!1)\quad.\cr} \eqno(\z)$$
 
\noindent{Though} they have not a direct mechanical meaning of impulses
they still are suitable to perform a Legendre transformation upon. 

Assuming also that the highest derivative can be worked out of the first %%@
equation of 
(2.16) as a function \quad %%@
${\bar\phi}_{{\mu}_1\cdots{\mu}_m}[\phi,\phi_\mu,...,{\phi}_{{\mu}_1\cdots
  \mu_{m-1}};\pi^{{\mu}_1\cdots{\mu}_m}]$\quad, the "Hamiltonian" density 
now is 

$$\eqalign{{\cal H}[\phi,\phi_\mu,...,{\phi}_{{\mu}_1\cdots{\mu}_{m-1}};
                                   \pi^\mu,...,&\pi^{{\mu}_1\cdots{\mu}_m}]
  =\pi^\mu\phi_\mu +\cdots
   + \pi^{{\mu}_1\cdots{\mu}_{m-1}}{\phi}_{{\mu}_1\cdots{\mu}_{m-1}}\cr
   &+\pi^{{\mu}_1\cdots{\mu}_m}{\bar\phi}_{{\mu}_1\cdots{\mu}_m}
   -{\cal L}[\phi,\phi_{\mu},...,{\bar\phi}_{{\mu}_1\cdots{\mu}_m}]\quad.\cr}
                                                            \eqno(\z) $$

\noindent{Then} the canonical equations are

$$\eqalign{\partial_\mu\phi&={{\partial{\cal H}}\over{\partial\pi^\mu}}
                                      \quad,\quad
\partial_\mu\phi_\nu={{\partial {\cal H}}\over{\partial\pi^{\mu\nu}}}
                              \quad,\;...\;,\quad
\partial_\mu{\phi}_{{\mu}_1\cdots{\mu}_{m-1}}=
        {{\partial {\cal H}}\over{\partial\pi^{\mu\mu_1\cdots\mu_{m-1}}}}
                                                          \quad,\cr
\partial_\mu\pi^\mu&=-{{\partial {\cal H}}\over{\partial\phi}} 
                                       \;,\;\;                                                                                                   %%@
\partial_\nu\pi^{\mu\nu}=-{{\partial {\cal H}}\over{\partial\phi_\mu}}
                              \;\;\;,\;...\;,\;\;
\partial_\sigma{\pi}^{{\mu}_1\cdots{\mu}_{m-1}\sigma}=
        -{{\partial {\cal H}}\over{\partial\phi_{\mu_1\cdots\mu_{m-1}}}}
                                                 \;\;.\cr}\eqno(\z)$$



This general setting may be hardly applicable to systems of practical %%@
interest (generally involving internal symmetries and/or fields belonging to %%@
less trivial Lorentz representations) if suitable strategies are not adopted %%@
to refine the method. One crucial observation is that the momenta may be %%@
defined in more useful and general ways than the plain one introduced in %%@
(2.16): instead of differentiating with respect to the simple field %%@
derivatives \quad$\phi_{\mu_1\cdots\mu_i}$\quad one may consider combinations %%@
of field derivatives of different orders belonging to the same Lorentz and %%@
internal group representations. For instance, in HD gravity [9], the Ricci
tensor is a most suited combination of second derivatives of the metric %%@
tensor field. The only condition is that the Lagrangian be regular in the %%@
highest "velocity" so defined. This will be made clear in the following. 

In fact this general Ostrogradski treatment can be significantly simplified %%@
for the Lorentz invariant theory of a scalar field, which is the example we %%@
will consider in this paper. In this case, dropping total derivatives, the %%@
general form (2.14 ) can be expressed in a more convenient way that singles %%@
out the free quadratic part, namely

$$ {\cal L} = -{c\over 2} 
              \,\phi\kg1\gk\kg2\gk\cdots\kg N\gk\phi
                                          -j\,\phi\quad,\eqno(\z)$$

\noindent{where}\quad$\kg i\gk \equiv (\square + m^2_i)\;$, our Minkowski %%@
signature  is \quad$(+,-,-,-)$\quad so that \break %%@
$\square\equiv\partial^2_t-\triangle\;$, and \quad$c$\quad is a dimensional %%@
constant. The masses are ordered such that \quad$m_i > m_j$\quad when 
\quad$ i < j$\quad so that the objects \quad$ \langle ij \rangle \equiv %%@
(m^2_i - m^2_j)$\quad are always positive when \quad$i < j\;$. 

It turns out to be very advantageous to consider only Lorentz 
invariant combinations of  derivatives of the type %%@
\quad${\square\,}^n\phi$\quad and of the \quad$\phi$\quad field itself with %%@
suitable dimensional coefficients. Further, it is even more useful to %%@
consider expressions of the form\quad$\kg i \gk^n\phi\;$. 

\noindent{Thus,} arbitrarily focusing ourselves on \quad$i=1\;$ without loss %%@
of generality, equation (2.19) may be recast as

$${\cal L}={1\over 2}\sum^N_{n=1}c_n\phi\kg 1\gk^n\phi\,-j\phi\quad,
                                                             \eqno(\z)$$

\noindent{where} the \quad$c_n$\quad are redefined constants.

\noindent{Calling} \quad$m={N\over2}$\quad for even \quad$N\;$, and %%@
\quad$m={{N+1}\over2}$\quad for 
odd \quad$N\;$, the motion equation now reads

$$\sum^m_{n=1}\kg 1\gk^n{{\partial{\cal L}}\over{\partial(\kg 1\gk^n\phi)}}
     =\sum^N_{n=1}c_n\kg 1\gk^n\phi = j                    \eqno(\z)$$

\noindent{The} Legendre transform can now be performed upon the simpler set %%@
of {\it generalized momenta}

$$\eqalign{\pi_m &={{\partial{\cal L}}\over{\partial(\kg 1\gk^m\phi)}}  \cr
\pi_{m-1}&={{\partial{\cal L}}\over{\partial(\kg1\gk^{m-1}\phi)}}
                                                        +\kg1\gk\pi_m   \cr
                    \cdots\;\; &\quad\quad\cdots                        \cr
\pi_s &={{\partial{\cal L}}\over{\partial(\kg 1\gk^s\phi)}}+\kg1\gk\pi_{s+1}
                      \quad (s=1,...,m-2)\,. \cr}\eqno(\z)$$

\noindent{The} Hamiltonian will depend on the new phase--space coordinates 
\break $H[\phi_1,...,\phi_m;\pi_1,...,\pi_m]\;$, where 
\quad$\phi_i\equiv\kg1\gk^{i-1}\phi\;$. To this end \quad$\kg1\gk^m\phi$\quad %%@
has been worked out of the 1st (2.22) for even \quad$N$\quad, or of the 2nd  %%@
(2.22) for odd \quad$N$\quad, in terms of these coordinates.

\noindent{The} dynamics of the system is given by the \quad$2m$\quad %%@
equations of second order

$$\eqalign{\kg 1\gk \phi_i &= {{\partial H}\over{\partial\pi_i}}\quad\cr
           \kg 1\gk \pi_i &= {{\partial H}\over{\partial\phi_i}}\quad\cr}
                                          (i=1,...,m)\quad.     \eqno(\z)$$         

\noindent{Notice} that, in comparison with (2.12),(2.16) and (2.18), no %%@
negative sign occurs in both (2.22) and (2.23), because each step now %%@
involves two derivative orders.

As a final comment, the treatment followed above keeps Lorentz invariance %%@
explicitely, and this will turn advantageous later on. The price has been %%@
that neither the \quad$\pi$'s\quad have the meaning of mechanical momenta nor %%@
\quad$H$\quad has to do with the energy of the system. However they are %%@
adequate for providing a set of "canonical" equations that correctly describe %%@
the evolution of the system. Moreover, these equations are Lorentz invariant %%@
and of 2nd differential order, which will lend itself to an almost direct %%@
particle interpretation.
\bigskip

One may however choose to work with the genuine Hamiltonian and mechanical %%@
momenta obtained when the Legendre transformation built-in in the %%@
Ostrogradski method involves only the true "velocities" %%@
\quad$\partial_t^n\phi\;$. The price now is loosing the explicit Lorentz %%@
invariance and  facing more cumbersome calculations, as we will see by an %%@
example in the 2nd part of the next Section. 
\bigskip
\bigskip



\sectio{\bf N=2 theories.}

These theories allow a particularly simple treatment that will be illustrated %%@
in the examples \quad$N=2$\quad and \quad$N=4\;$. The equations (2.23) for
\quad$N=2$\quad will now be obtained from a Helmholtz--like Lagrangian of 2nd %%@
differential order, which is closer to a direct particle interpretation.

Consider the \quad$N=2$\quad Lagrangian

$$ {\cal L}^{4} = -{1 \over 2}{1 \over M}\,\phi\kg 1\gk\kg 2\gk\phi
                                          -j\,\phi\quad.\eqno(\z)$$

\noindent{with} non-degenerate masses \quad$m_1>m_2\;$. Taking the %%@
dimensional constant \break $M=(m_1^2 - m_2^2)\equiv \langle 12 \rangle %%@
>0\;$, equation (3.1) yields the propagator

$$-{{\langle 12 \rangle}\over{\kg 1\gk\kg 2\gk}} 
      = {1 \over{\kg 1\gk}}-{1 \over{\kg 2\gk}}
                                        \quad, \eqno(\z)$$

\noindent{We} thus see that the pole at \quad$m_2$\quad then corresponds to a %%@
physical particle and the one at \quad$m_1$\quad to a negative norm %%@
"poltergeist". The 2nd order Lagrangian we are seeking should describe two %%@
fields with precisely the particle propagators occurring in the r.h.s. of 
(3.2).

The Lagrangian (3.1) can be brought to the form (2.20), namely

 $$ \eqalign{{\cal L}^{4}[\phi,\kg 1\gk\phi]
                    &=-{1 \over 2}{1 \over{\langle 12 \rangle}}
                                      \bigl[\phi\kg 1\gk^2\phi 
             - \langle 12 \rangle \phi\kg 1\gk\phi\bigl]
                                    -j\,\phi       \cr
                    &=-{1 \over 2}{1 \over{\langle 12 \rangle}}
                                      \bigl[(\kg 1\gk\phi)^2 
             - \langle 12 \rangle \phi(\kg 1\gk\phi)\bigl]
                                    -j\,\phi\quad ,\cr}\eqno(\z)$$

\noindent{where} the relationship \quad$\kg 2\gk=\kg 1\gk-\langle 12 \rangle$ %%@
\quad has been used. 

We define one momentum

$$\pi={{\partial{\cal L}}\over{\partial(\kg 1\gk\phi)}}\eqno(\z)$$

\noindent{from} which \quad$\kg 1\gk\phi$\quad is readily worked out, %%@
obtaining

$${\cal H}^4[\phi,\pi]=-{1\over 2}\langle 12 \rangle(-\pi+{1\over 2}\phi)^2
                                                    +j\,\phi\eqno(\z)$$
\noindent{and} the Helmholtz-like Lagrangian is

$${\cal L}^4_H[\phi,\kg 1\gk\phi,\pi]=\pi\kg 1\gk\phi- {\cal H}[\phi,\pi]
                                                        \quad.\eqno(\z)$$

\noindent{It} contains mixed terms \quad$\pi\,\phi$\quad that obscure the %%@
particle contents. The diagonalization is achieved by new fields %%@
\quad$\phi_1\;$ , $\;\phi_2$
 
$$\eqalign{\phi&= \phi_1 + \phi_2 \cr 
            \pi&= {1\over2}(\phi_1 - \phi_2)\cr} \eqno(\z)$$

\eject
\noindent{to} yield

$${\cal L}^{2}={1\over2}\phi_1\kg 1\gk \phi_1
                -{1\over2}\phi_2\kg 2\gk \phi_2
                -j(\phi_1+\phi_2)\quad ,\eqno(\z)$$

\noindent{where} the particle propagators in the r.h.s. of (3.2) are %%@
apparent. This result is physically meaningful: where we had a single field %%@
\quad$\phi\;$, coupled to a source \quad$j\;$, propagating with the quartic %%@
propagator in the l.h.s. of (3.2) as implied by the HD Lagrangian (3.1), we %%@
now have two fields \quad$\phi_1\;$ , $\;\phi_2$\quad describing particles %%@
with quadratic propagators, and the source couples to the sum %%@
\quad$\phi_1+\phi_2\;$.
\bigskip

A deeper insight of the phase-space structure of the theory can be achieved %%@
by the plain use of the Ostrogradski method, eventually confirming the final
form (3.8). In order to explicitely show the velocities, we write (3.1) in %%@
the form of the Lagrangian density

$${\cal L}^{4}=-{1\over 2}{1\over{\langle 12\rangle}}
      \{(\partial^2_t\phi)^2-(\partial_t\phi)S(\partial_t\phi)
                             + \phi P\phi\} -j\,\phi \eqno(\z)$$ 

\noindent{where} \quad$S\equiv M^2_1+M^2_2\;,\;P\equiv M^2_1M^2_2$\quad and
 \quad$M^2_i\equiv m^2_i-\triangle$\quad are operators containing the space %%@
derivatives.

The Ostrogradski formalism yields the Hamiltonian density

$$ {\cal H}^{4}[\phi,\dot\phi;\pi_1,\pi_2]= -{1\over 2}\langle %%@
12\rangle\pi^2_2
   +\pi_1\dot\phi-{1\over 2}{1\over{\langle 12\rangle}}\dot\phi S\dot\phi
   +{1\over 2}{1\over{\langle 12\rangle}}\phi P\phi +j\,\phi    \eqno(\z)$$

\noindent{that} depends on the phase-space coordinates %%@
\quad$\phi\;,\;\dot\phi\;,\;\pi_1\;,\;\pi_2$\quad and on their space %%@
derivatives. The highest-order "velocity" \quad$\partial^2_t\phi$\quad has %%@
been worked out of the momenta

$$\eqalign{\pi_2&\equiv
       {{\partial {\cal L}^4}\over{\partial(\partial^2_t\phi)}}
           =-{1\over{\langle 12\rangle}}\partial^2_t\phi\quad ,\cr
           \pi_1&\equiv
       {{\partial {\cal L}^4}\over{\partial(\partial_t\phi)}}
                                    -\partial_t\pi_2\quad . \cr}\eqno(\z)$$

\noindent{The} canonical equations may be derived from the Helmholtz %%@
Lagrangian

$$\eqalign{ 
   {\cal L}^4_H[\phi,\dot\phi;\pi_1\pi_2;\partial_t\phi,\partial_t\dot\phi]         
   =\pi_2\partial_t\dot\phi+\pi_1\partial_t\phi
    & + {1\over 2}\langle 12\rangle\pi^2_2
    -\pi_1\dot\phi+{1\over 2}{1\over{\langle 12\rangle}}\dot\phi S\dot\phi\cr
    &-{1\over 2}{1\over{\langle 12\rangle}}\phi P\phi -j\,\phi\quad.\cr}                                                                  %%@
\eqno(\z)$$

\noindent{This} is a Lagrangian density of 1st order in time derivatives,
and we express it in matrix form for later convenience:

$$ {\cal L}^4_H = {1\over 2}\Phi^T\mu\,\Sigma\,\partial_t\Phi
                 +{1\over 2}\Phi^T{\cal M}_4\,\Phi-J^T\,\Phi\quad,\eqno(\z)$$

\noindent{where} \quad$\mu$\quad is an arbitrary mass parameter and


$$\eqalign{\Phi&\equiv
      \left(\matrix{\pi_2\cr\mu^{-1}\dot\phi\cr\mu^{-1}\pi_1\cr\phi}\right)
                                               \quad,\quad
 \Sigma\equiv
       \left(\matrix{0&1&0&0\cr-1&0&0&0\cr0&0&0&1\cr0&0&-1&0\cr}\right)
                                                                  \quad,\cr
 {\cal M}_4&\equiv
      \left(\matrix{\langle 12\rangle&0&0&0\cr
                         0&{{\mu^2S}\over{\langle 12\rangle}}&-\mu^2&0\cr
                         0&-\mu^2&0&0\cr
                         0&0&0&-{P\over{\langle 12\rangle}}\cr}\right)
                                              \quad,\quad
     J\equiv\left(\matrix{0\cr0\cr0\cr j}\right)\quad,\cr}\eqno(\z)$$

\noindent{with} mass dimensions \quad$[\Phi]=1\;,\; [{\cal M}_4]=2$\quad and %%@
\quad$[J]=3\;$.

In order to relate (3.13) to (3.8), we have to convert the latter into a 1st %%@
order theory as well. This is readily done by expressing the velocities
\quad$\partial_t\phi_1$\quad and \quad$\partial_t\phi_2$\quad in terms of the %%@
momenta

$$\eqalign{{\tilde\pi}_1&\equiv
       {{\partial {\cal L}^2}\over{\partial(\partial_t\phi_1)}}
           =-\partial_t\phi_1\quad ,\cr
           {\tilde
\pi}_2&\equiv
       {{\partial {\cal L}^2}\over{\partial(\partial_t\phi_2)}}
           =\partial_t\phi_2\quad , \cr}\eqno(\z)$$

\noindent{so} that

$${\cal H}^2[\phi_1,\phi_2,{\tilde\pi}_1,{\tilde\pi}_2]
   = -{1\over2}{\tilde\pi}_1^2+{1\over2}{\tilde\pi}_2^2
      -{1\over2}\phi_1M^2_1\phi_1+{1\over2}\phi_2M^2_2\phi_2
                            +j\,(\phi_1+\phi_2)\quad.\eqno(\z)$$

The Helmholtz Lagrangian that yields the canonical equations now is

$$ {\cal L}^2_H = {1\over 2}\Theta^T\mu\,\Sigma\,\partial_t\Theta
     +{1\over 2}\Theta^T{\cal M}_2\,\Theta-J^T{\cal Z}\,\Theta\eqno(\z)$$
 
\noindent{where}
$$\Theta\equiv
      \left(\matrix{\mu^{-1}{\tilde\pi}_1\cr\phi_1\cr\mu^{-1}{\tilde\pi}_2                                                           %%@
\cr\phi_2}\right)
                                               \quad,\quad
{\cal M}_2\equiv
      \left(\matrix{\mu^2&0&0&0\cr
                     0&M^2_1&0&0\cr
                     0&0&-\mu^2&0\cr
                     0&0&0&-M^2_2\cr}\right)
                                           \quad,\quad \eqno(\z)$$

\noindent{with} mass dimensions \quad$[\Theta]=1$\quad and
\quad$[{\cal M}_2]=2\;$, and \quad${\cal Z}$\quad is any matrix with the %%@
fourth row equal to \quad$(0,1,0,1)\;$.  

The field redefinition analogous to the diagonalizing equations 
(3.7) now is a \quad$4\times4$\quad mixing of fields given by

$$ \Phi= {\cal X}\,\Theta   \eqno(\z)$$

\noindent{where} the invertible matrix

$${\cal X}\equiv \left(\matrix{0&-{{M^2_1}\over{\langle 12\rangle}}&0&
                                      -{{M^2_2}\over{\langle 12\rangle}}\cr
                     -1&0&1&0\cr
                     -{{M^2_2}\over{\langle 12\rangle}}&0&
                      {{M^2_1}\over{\langle 12\rangle}}&0\cr
                      0&1&0&1\cr}\right)                     \eqno(\z)$$

\noindent{verifies}

            $${\cal X}^T\Sigma\,{\cal X}=\,\Sigma    \eqno(\z)$$

            $${\cal X}^T{\cal M}_4{\cal X}=\,{\cal M}_2 \eqno(\z)$$

\noindent{so} we can identify \quad${\cal Z}={\cal X}\;$.

We thus see that (3.19) translates (3.13) into (3.17), and therefore the %%@
Lagrangians (3.9) and (3.8) are again seen to be equivalent. The derivation %%@
of the matrix \quad${\cal X}$\quad is cumbersome but contains interesting %%@
details that worth the Appendix. Notice that the components of %%@
\quad$\Phi$\quad are expressed by (3.19) in terms of the components of %%@
\quad$\Theta$\quad {\it and} of their space derivatives. This is not %%@
surprising as long as \quad$\pi_1\;$, given by (3.11), contains space %%@
derivatives of \quad$\phi$\quad as well.

Though the plain non-covariant Ostrogradski method we have just implemented %%@
eventually shows up the Lorentz invariance, the readiness of the explicitely %%@
covariant procedure  formerly introduced in this Section is apparent. The %%@
non-covariant approach using  the canonical Hamiltonian and mechanical %%@
momenta is rigourous and validates the former, but involves more bulky %%@
diagonalizing matrices with elements that contain space derivatives.     
\bigskip



\sectio{\bf N=4  and higher even N theories}

We treat the \quad$N=4$\quad Theory with the far more practical Lorentz %%@
invariant method of the previous Section. Otherwise one would have to face %%@
the diagonalization of \quad$8\times 8$\quad matrices analogous to %%@
\quad${\hat{\cal M}}_2$\quad and \quad${\hat{\cal M}}_4$\quad in Appendix A. %%@
Our Lagrangian now is

$${\cal L}^8=-{1\over 2}{{\mu^6}\over M}
            \phi\kg 1\gk\kg 2\gk \kg 3\gk\kg 4\gk\phi-j\,\phi \eqno(\z)$$

\noindent{where} the mass dimensions %%@
\quad$[\mu]=[\phi]=1\quad,\quad[M]=12$\quad and \quad$[j]=3$\quad
are such that \quad$[{\cal L}^8]=4\;$. Taking \quad$M=\langle12\rangle %%@
\langle13\rangle
\langle14\rangle \langle23\rangle \langle24\rangle \langle34\rangle\;$, %%@
equation (4.1) treats the masses \quad$m_i\,(i=1,...,4)$\quad on an equal %%@
footing, which is apparent in the propagator

$$-{{\mu^{-6}M}\over{\kg 1\gk\kg 2\gk \kg 3\gk\kg 4\gk}}
     = {{\langle1\rangle}\over{\kg1\gk}}-{{\langle2\rangle}\over{\kg2\gk}}
      +{{\langle3\rangle}\over{\kg3\gk}}-{{\langle4\rangle}\over{\kg4\gk}}
                                                              \eqno(\z)$$

\noindent{where} \quad${\langle i\rangle}\equiv\mu^{-6}M\prod\limits_{j\neq %%@
i}
 {1\over{\langle ij\rangle}}$\quad (remind the ordering convention %%@
\quad$i<j\;$) with
mass dimensions \quad$[\langle i\rangle]=0\;$.

As for (3.2), the propagator expansion (4.2) suggests that the    
lower-derivative equivalent theory should now be 

$$\eqalign{{\cal L}^{2}={1\over2}{1\over{\langle1\rangle}}\phi_1\kg 1\gk %%@
\phi_1
                -{1\over2}{1\over{\langle2\rangle}}\phi_2\kg 2\gk \phi_2
                &+{1\over2}{1\over{\langle3\rangle}}\phi_3\kg 3\gk \phi_3
                -{1\over2}{1\over{\langle4\rangle}}\phi_4\kg 4\gk \phi_4\cr
                &-j(\phi_1+\phi_2+\phi_3+\phi_4)\quad .\cr}\eqno(\z)$$

\noindent{We} derive this Lagrangian from (4.1) in the following. In matrix %%@
form, (4.3) reads

$${\cal L}^{2}={1\over2}\tau^T\kg 1\gk I\tau+{1\over2}\tau^T{\cal M}_2\tau
                   -J^T\,F\tau \quad, \eqno(\z)$$

\noindent{where}

$$\tau\equiv\left(\matrix{\langle1\rangle^{-{1\over 2}}\phi_1\cr
                        -i\langle2\rangle^{-{1\over 2}}\phi_2\cr
                          \langle3\rangle^{-{1\over 2}}\phi_3\cr
                        -i\langle4\rangle^{-{1\over 2}}\phi_4\cr}\right)
                                             \; ,\;
J\equiv\left(\matrix{0\cr0\cr0\cr j}\right)
                                             \; ,\;
{\cal M}_2\equiv\left(\matrix{0&0&0&0\cr
                              0&-\langle12\rangle&0&0\cr 
                              0&0&-\langle13\rangle&0\cr
                              0&0&0&-\langle14\rangle\cr}\right)
                                              \quad ,\eqno(\z)$$

\noindent{$I$}\quad is the \quad$4\times 4$\quad identity, and \quad$F$\quad %%@
is any matrix with the fourth row equal to \break$(\langle1\rangle^{1\over %%@
2},i\langle2\rangle^{1\over 2},
     \langle3\rangle^{1\over 2},i\langle4\rangle^{1\over 2})\;.$

By dropping total derivatives we express (4.1) in a standard form involving %%@
derivatives of the lowest possible order, namely

$$\eqalign{{\cal L}^8[\phi,\kg 1\gk\phi,\kg 1\gk^2\phi]
        =-{1\over 2}{{\mu^6}\over M}
         \{(\kg 1\gk^2\phi)^2&-S(\kg 1\gk\phi)(\kg 1\gk^2\phi)
            +p(\kg 1\gk\phi)^2\cr &-P\phi(\kg 1\gk\phi)\} - j\,\phi\quad,\cr} %%@
\eqno(\z)$$

\noindent{where}\quad$S\equiv\langle12\rangle+\langle13\rangle+\langle14
\rangle \quad,\quad
   p\equiv\langle12\rangle\langle13\rangle+\langle12\rangle\langle14\rangle                                                                                   %%@
+\langle13\rangle\langle14\rangle\;$, and\break  %%@
$P\equiv\langle12\rangle\langle13\rangle\langle14\rangle\;$.

Ostrogradski-like momenta are defined as follows

$$\eqalign{\pi_2&={{\partial{\cal L}^8}\over{\partial(\kg 1\gk^2\phi)}}
          =-{{\mu^6}\over M}(\kg 1\gk^2\phi)+{{\mu^6S}\over 2M}\kg1\gk\phi\cr
           \pi_1&={{\partial{\cal L}^8}\over{\partial(\kg 1\gk\phi)}}
                   +\kg 1\gk\pi_2\quad .\cr} \eqno(\z)$$

From the 1st of (4.7) the highest derivative is worked out, namely

$$ {\bf \kg 1\gk^2\phi}[\pi_2\,,\kg 1\gk\phi]
                 =-{M\over{\mu^6}}\pi_2+{S\over 2}(\kg %%@
1\gk\phi)\quad.\eqno(\z)$$

The "Hamiltonian" functional is

$${\cal H}^8[\psi_1,\psi_2,\pi_1,\pi_2]
                =\pi_2{\bf \kg 1\gk^2\phi}+\pi_1\psi_2
                 -{\cal L}^8[\psi_1,\psi_2,{\bf \kg %%@
1\gk^2\phi}]\quad,\eqno(\z)$$ 

\noindent{where} \quad$\psi_1\equiv \phi$\quad and \quad$\psi_2\equiv
\kg 1\gk\phi\;$. Its canonical equations can be derived from the Lagrangian

$${\cal L}^8_H={1\over 2}\Phi^T\kg 1\gk{\cal K}\Phi
                +{1\over 2}\Phi^T{\cal M}_8\Phi-J^T\Phi\quad, \eqno(\z)$$

\noindent{where}\quad$J$\quad is the same as in (4.5),

$$\eqalign{\Phi&\equiv\left(\matrix{\mu^2\pi_2\cr
                          \mu^{-2}\psi_2\cr
                          \pi_1\cr
                          \psi_1\cr}\right)
                                            \quad ,\quad
{\cal K}\equiv\left(\matrix{0&1&0&0\cr
                            1&0&0&0\cr
                            0&0&0&1\cr
                            0&0&1&0\cr}\right)
                                            \quad {\rm and} \cr
 {\cal M}_8&\equiv\left(\matrix{\mu^{-10}M&-{S\over 2}&0&0\cr
     -{S\over 2}&-{\mu^{-10}\over M}(p-{{S^2}\over 4})&-\mu^2&
                         {{\mu^2}\over{2\langle 1\rangle}}\cr
                                              0&-\mu^2&0&0\cr
                   0&{{\mu^2}\over{2\langle 1\rangle}}&0&0\cr}\right)
                                                 \quad .\cr}\eqno(\z)$$ 

{\it Prior} to its diagonalization we write (4.10) in the form

$${\cal L}^8_H={1\over2}\Omega^T\kg 1\gk I\Omega
              +{1\over2}\Omega^T{\hat{\cal M}}_8\Omega
                  -J^T{\cal D}^T\Omega\quad,  \eqno(\z)$$

\noindent{where} \quad$\Omega\equiv({\cal D}^T)^{-1}\Phi\;$, with

$$ {\cal D}\equiv {1\over{\sqrt{2}}}\left(\matrix{1&1&0&0\cr
                                                 -i&i&0&0\cr
                                                  0&0&1&1\cr
                                                  0&0&-i&i\cr}\right)
                                                              \eqno(\z)$$
\noindent{and}

$$ {\hat{\cal M}}_8\equiv{\cal D}{\cal M}_8{\cal D}^{-1}=
   {1\over 2}\left(\matrix{M_--S&-iM_+&-\mu^21_-&i\mu^21_+\cr
                           -iM_+&-(M_-+S)&-i\mu^21_-&-\mu^21_+\cr
                             -\mu^21_-&-i\mu^21_-&0&0\cr
                             i\mu^21_+ &-\mu^21_+&0&0\cr}\right)
                                                              \eqno(\z)$$

\noindent{with} \quad$M_{\pm}\equiv{M\over{\mu^{10}}}\pm{{\mu^{10}}\over M}
                   (p-{{S^2}\over 4})$ \quad and 
               \quad$1_{\pm}\equiv 1\pm {1\over{2\langle 1\rangle}}\;$.

Now the task is to establish the equivalence of (4.12) and (4.4). One may %%@
first check that the eigenvalues \quad$\lambda_i\,(i=1,...,4)$\quad of %%@
\quad${\hat{\cal M}}_8$\quad
are the diagonal elements of \quad${\cal M}_2$\quad in (4.5). The orthogonal %%@
matrix \quad$T$\quad
that diagonalizes \quad${\hat{\cal M}}_8$\quad is obtained by working out its %%@
orthonormal eigenvectors \quad$\mid\!\lambda_i\rangle$\quad with the suitable %%@
sign, and
arranging them as the columns. These are

$$\eqalign{
\mid\lambda_1\rangle&={{\langle 1\rangle^{1\over 2}}\over{\sqrt 2}}
                         \left(\matrix{0\cr0\cr1_+\cr-i1_-\cr}\right)
                                               \quad ,\cr
\mid\lambda_j\rangle&=
{{i^{(1-\delta_{3j})}\langle j\rangle^{1\over2}}\over
                      {\sqrt 2[-{2\over{\mu^{10}}}M+2\langle 1j\rangle-S]}}
\left(\matrix{{2\over{\mu^{2}}}[-{{\mu^4}\over{\langle 1\rangle}}
          +\langle 1j\rangle(2\langle 1j\rangle-S-M_-)]\cr
          i{2\over{\mu^{2}}}[-{{\mu^4}\over{\langle 1\rangle}}
                                +\langle 1j\rangle M_+]\cr
                 1_-[-2\mu^{-10}M+2\langle 1j\rangle-S]\cr
              -i 1_+[-2\mu^{-10}M+2\langle 1j\rangle-S]\cr}\right)
                                                   \quad ,\cr}\eqno(\z)$$
  

\noindent{where} \quad$j=2,3,4\;.$ If \quad$I$\quad is the identity matrix, %%@
we therefore have

$$ T^T\,I\,T=\,I \quad ,\quad 
                    T^T\,{\hat{\cal M}}_8\,T={\cal M}_2\quad ,\eqno(\z)$$                                        

\noindent{and} the fourth row of \quad${\cal D}^TT$\quad can be seen to be 
\quad$(\langle1\rangle^{1\over 2},i\langle2\rangle^{1\over 2},
  \langle3\rangle^{1\over 2},i\langle4\rangle^{1\over 2}\,)\;$, i.e. it has 
the required form for \quad$F\;$. Then, by taking \quad$\Omega=T\tau\;$, %%@
(4.12) is identical to (4.4).
\bigskip

The general case for even \quad$N\geq6$\quad in the covariant treatment would %%@
involve
\quad${N\over 2}$\quad Ostrogradski--like momenta and the diagonalization of %%@
a \quad$N\times N$\quad mass matrix. The non--covariant Ostrogradski method %%@
introduced in Section 3, which reduces the theory to a 1st differential--%%@
order form,
would now involve \quad$2N\times 2N$\quad matrices. In both treatments the %%@
procedure would follow analogous paths, albeit with the occurrence of %%@
intractable eigenvector and diagonalization problems.  
\eject




\sectio{\bf N=3 and higher odd N theories.}

For \quad$N=3\;$, the HD Lagrangian

$$ {\cal L}^6=-{1\over 2}{{\mu^2}\over M}\phi\kg1\gk\kg2\gk\kg3\gk\phi
                                              -j\,\phi  \quad , \eqno(\z)$$  

\noindent{where} %%@
\quad$M\equiv\langle12\rangle\langle13\rangle\langle23\rangle$\quad
and \quad$[{\cal L}^6]=4\;$, yields the propagator

$$-{{\mu^{-2}M}\over{\kg1\gk\kg2\gk\kg3\gk}}
         =-{{\mu^{-2}\langle23\rangle}\over{\kg1\gk}}
          +{{\mu^{-2}\langle13\rangle}\over{\kg2\gk}}
          -{{\mu^{-2}\langle12\rangle}\over{\kg3\gk}} \quad. \eqno(\z)$$

\noindent{Then}, the expected equivalent 2nd--order theory is 

$${\cal L}^2=-{1\over 2}{{\mu^2}\over{\langle23\rangle}}\phi_1\kg1\gk\phi_1
             +{1\over 2}{{\mu^2}\over{\langle13\rangle}}\phi_2\kg2\gk\phi_2
             -{1\over 2}{{\mu^2}\over{\langle12\rangle}}\phi_3\kg3\gk\phi_3
                                 -j(\phi_1+\phi_2+\phi_3)\,. \eqno(\z)$$

Already for \quad$N=3\;$, the non--covariant Ostrogradski method becomes %%@
exceedingly cumbersome. In fact, it reduces both (5.1) and (5.3) to
1st differential order in time. Proving the equivalence of those theories %%@
then involves the diagonalization of \quad$6\times 6$\quad matrices (the %%@
counterpart
of \quad${\hat{\cal M}}_4$\quad and \quad${\hat{\cal M}}_2$\quad in (A.4)), %%@
although with a reasonable amount of work it can still be checked that both %%@
mass matrices have the same eigenvalues, namely \quad$\pm\mu M_1$\quad, %%@
\quad$\pm\mu M_2$\quad and \quad$\pm\mu M_3\;$.
Finding the eigenvectors and building up the compound diagonalizing %%@
transformation does not worth the effort.

For the odd \quad$N$\quad theories, the covariant method exhibits an %%@
interesting feature. Without loss of generality we again single out the %%@
Klein-Gordon operator \quad$\kg 1\gk$\quad and write (5.1) as

$${\cal L}^6[\phi,\kg1\gk\phi,\kg1\gk^2\phi]
            =-{1\over 2}{{\mu^2}\over M}
              \{(\gk1\gk\phi)(\gk1\gk^2\phi)-S(\gk1\gk\phi)^2
                +P\phi(\gk1\gk\phi)\}-j\,\phi\quad, \eqno(\z)$$

\noindent{where} now \quad$S\equiv\langle12\rangle+\langle13\rangle$\quad and 
\quad$P\equiv\langle12\rangle\langle13\rangle\;$.

The momenta are

$$\eqalign{\pi_2&={{\partial{\cal L}^6}\over{\partial(\kg1\gk^2\phi)}}
                 =-{1\over 2}{{\mu^2}\over M}\kg1\gk\phi  \cr
           \pi_1&={{\partial{\cal L}^6}\over{\partial(\kg1\gk\phi)}}                                                      %%@
+\kg1\gk\;\pi_2 =-{{\mu^2}\over M}\kg1\gk^2\phi+{{\mu^2}\over M}S\kg1\gk\phi %%@
-{1\over 2}{{\mu^2}\over M}P\phi\quad .\cr}                                                                                                                               %%@
\eqno(\z)$$
\eject


\noindent{Unlike} in (4.7), the highest derivative now is worked out of  
\quad$\pi_1$\quad (instead of \quad$\pi_2\;$), namely

$$\kg1\gk^2\phi[\phi,\kg1\gk\phi,\pi_1]=-{M\over{\mu^2}}\pi_1
                            +S\kg1\gk\phi-{1\over 2}P\phi\quad , \eqno(\z)$$   

\noindent{and}, in terms of the coordinates                                      %%@
\quad$\pi_1\;,\pi_2\;,\psi_1\equiv\phi$\quad and
               \quad$\psi_2\equiv\kg1\gk\phi\;$, the "Hamiltonian" reads  

$${\cal H}^6[\psi_1,\psi_2,\pi_1,\pi_2]=\pi_2\kg1\gk^2\phi+\pi_1\psi_2
                 -{\cal L}^6[\psi_1,\psi_2,\kg1\gk^2\phi]\quad.\eqno(\z)$$

\noindent{The} Helmholtz Lagrangian is

$$\eqalign{{\cal %%@
L}^6_H[\psi_1,\psi_2,\pi_1,\pi_2]=\pi_2\kg1\gk\psi_2+\pi_1\kg1\gk\psi_1
          &+{M\over{\mu^2}}\pi_1\pi_2-S\pi_2\psi_2+{1\over 2}P\pi_2\psi_1\cr
          &-{1\over 2}\pi_1\psi_2
              -{{\mu^2}\over 4M}P\psi_1\psi_2-j\psi_1\quad.\cr}\eqno(\z)$$

The distinctive feature of the odd \quad$N$\quad cases is that the 1st of 
(5.5), namely 
\quad$\pi_2=-{1\over 2}{{\mu^2}\over M}\psi_2\;$, is a constraint that %%@
guarantees the relationship \quad$\kg1\gk\psi_1=\psi_2\;$, so one just has %%@
\quad$N$\quad degrees of freedom.  For even \quad$N$\quad it arises directly %%@
as an equation of motion. Moreover, unlike the Dirac Lagrangian for spin-%%@
${1\over2}$ fields or the constraints introduced by means of  multipliers, %%@
the constraint above can be freely imposed on the Lagrangian since it does %%@
not eliminate the dependence on the remaining variables \quad$\psi_1$\quad %%@
and \quad$\pi_1\;$. Thus, (5.8) can be expressed in terms of only the three %%@
fields \quad$\psi_1\;$, $\;\pi_1$\quad and \quad$\pi_2\;$:    

$${\cal L}^6_H[\psi_1,\pi_1,\pi_2]={1\over 2}\Phi^T\kg 1\gk{\cal K}'\Phi
                +{1\over 2}\Phi^T{\cal M}_3\Phi-J^T\Phi\quad ,\eqno(\z)$$

\noindent{where}

$$\eqalign{\Phi&\equiv\left(\matrix{\mu^2\pi_2\cr
                                         \pi_1\cr
                                          \phi\cr}\right) \quad ,\quad
             J\equiv\left(\matrix{0\cr 0\cr j\cr}\right)     \quad ,\cr
{\cal K}'&\equiv\left(\matrix{-4{M\over{\mu^6}}&0&0\cr
                                              0&0&1\cr
                                              0&1&0\cr}\right)\quad , \cr
{\cal M}_3&\equiv
\left(\matrix{4{MS\over{\mu^6}}&2{M\over{\mu^4}}&{P\over{\mu^2}}\cr
                                            2{M\over{\mu^4}}&0&0\cr
                                             {P\over{\mu^2}}&0&0\cr}\right) 
                                                              \quad .\cr}
                                                                \eqno(\z)$$

\noindent{The} Lagrangian (5.9) is expected to be equivalent to (5.3), which %%@
in
matrix form reads

$${\cal L}^2=-{1\over 2}\tau^T\kg 1\gk I\tau 
             +{1\over 2}\tau^T{\cal M}'_2\tau - J^TG\tau\quad , \eqno(\z)$$

\noindent{where} \quad$I$\quad is the \quad$3\times 3$\quad identity matrix,

$$\tau\equiv\left(\matrix{\mu\langle 23\rangle^{-{1\over 2}}\phi_1\cr 
                         i\mu\langle 13\rangle^{-{1\over 2}}\phi_2\cr
                          \mu\langle 12\rangle^{-{1\over 2}}\phi_3\cr}
                                                  \right)\quad , \quad
{\cal M}'_2\equiv\left(\matrix{0&0&0\cr
                             0&\langle 12\rangle&0\cr
                             0&0&\langle 13\rangle\cr}\right)\quad,
                                                          \eqno(\z)$$

\noindent{and} $\;G\;$ is any matrix with the third row given by
                $\;(\mu^{-1}\langle 23\rangle^{1\over 2},
                -i\mu^{-1}\langle 13\rangle^{1\over 2},
                  \mu^{-1}\langle 12\rangle^{1\over 2})\;$.

The transformation of (5.9) into (5.11) is performed by the field %%@
redefinition

$$\Phi={\cal D}'T\tau  \quad ,\eqno(\z)$$

\noindent{where}

$${\cal D}'\equiv{1\over{\sqrt 2}}
           \left(\matrix{{{\mu^3}\over{\sqrt{2M}}}&0&0\cr
                                               0&-i&-1\cr
                                               0&-i&1 \cr}\right)\quad ,
                                                                \eqno(\z)$$

\noindent{and} \quad$T$\quad is an orthogonal matrix built up with the %%@
eigenvectors of \quad${\cal D}'^T{\cal M}_3{\cal D}'\;$, namely

$$T={{\mu}\over{2\sqrt 2}}\langle 23\rangle^{-{1\over 2}}
  \left(\matrix{0&i{{2\sqrt 2}\over{\mu}}\langle 12\rangle^{1\over 2}
                 &-{{2\sqrt 2}\over{\mu}}\langle 13\rangle^{1\over 2}\cr
                  -i{{P_-}\over P}
                 &{{P_+}\over{\sqrt M}}\langle 12\rangle^{-{1\over 2}}
                 &i{{P_+}\over{\sqrt M}}\langle 13\rangle^{-{1\over 2}}\cr
                  {{P_+}\over P}
                 &i{{P_-}\over{\sqrt M}}\langle 12\rangle^{-{1\over 2}}
                 &-{{P_-}\over{\sqrt M}}\langle 13\rangle^{-{1\over 2}}\cr}
                                            \right)\quad ,\eqno(\z)$$

\noindent{with} \quad$P_{\pm}\equiv P\pm\mu^{-2}2M\;$. 

Then \quad${\cal D}'^T{\cal K}'{\cal D}'=-I$\quad and
\quad$T^T{\cal D}'^T{\cal M}_3{\cal D}'T={\cal M}'_2\;$. One may also check %%@
that \quad${\cal D}'T$\quad has the same third row required for \quad$G\;$.

\bigskip

The covariant treatment of the general odd \quad$N\geq 5$\quad case proceeds %%@
along hte same lines. Initially \quad$(N+1)/2$\quad Ostrogradski coordinates %%@
plus the corresponding momenta occur. Again the definition of the highest %%@
momentum yields a constraint with the same meaning as above, while the %%@
highest field derivative is worked out of the next momentum definition. Then %%@
one faces the diagonalization of a Helmholtz Lagrangian depending on just %%@
\quad$N$\quad fields. 

Already in the \quad$N=3$\quad case one might have chosen not to implement %%@
the constraint on the Lagrangian (5.8) and let it to arise in the equations %%@
of motion. These equations are the canonical ones for the Hamiltonian (5.7) %%@
and involve an even number of variables, as required by phase space. Thus one %%@
keeps the dependence of the Lagrangian (5.8) on the four fields
\quad$\psi_1\;,\;\psi_2\;,\;\pi_1$\quad and \quad$\pi_2\;$. Notwithstanding %%@
this enlarged dependence, it may still be diagonalized by new fields 
\quad$\phi_1\;,\;\phi_2\;,\;\phi_3$\quad and \quad$\zeta\;$, the (expected) %%@
surprise being  
that \quad$\zeta$\quad does not couple to the source \quad$j\;$. It is a %%@
spurious field,
which moreover vanishes when the constraint is implemented. We skip here
the details of this derivation. 
\vfill
\eject





\sectio{\bf Conclusions}

We have shown the physical equivalence between relativistic 
HD theories of a scalar field and a reduced 2nd 
differential--order counterpart. The free part of the HD scalar theories can %%@
always be brought to the form (2.19) integrating by parts, and the only %%@
limitation of our procedure is the non-degeneracy of the resulting 
Klein--Gordon masses, i.e. we consider regular Lagrangians. The existence of a %%@
lower--derivative version is already suggested by the algebraic decomposition %%@
of the HD propagator into a sum of second--order pieces showing (physical and %%@
ghost) particle poles. The order--reducing program we have developed relies %%@
on an extension of the Legendre transformation procedure, on the use of the %%@
modified action principle (Helmholtz Lagrangian) and on a suitable %%@
diagonalization. A basic ingredient of this program is the Ostrogradski %%@
formalism, which we have extended to field systems.

Two approaches have been worked out. The first one follows Ostrogradski more %%@
closely by defining generalized momenta and Hamiltonians with a standard %%@
mechanical meaning, at the price of treating time separately and loosing the %%@
explicit Lorentz invariance. It validates a second and more powerful one %%@
which is explicitly Lorentz invariant. The rigorous non-invariant phase-space %%@
analysis strongly backs also the formal covariant methods used in HD gravity, %%@
where \quad$R_{\mu\nu}[g,\partial g,\partial^2g]$\quad and 
\quad$\square\, h_{\mu\nu}$\quad (in the linearized theory) are used in the %%@
Legendre transformation. 

The HD theories of a scalar field we have considered are generalized Klein--%%@
Gordon theories, and hence of \quad$2N$\quad differential order according to %%@
the number \quad$N$\quad of KG operators involved. While the non--invarint %%@
approach treats all the theories on the same footing, the odd \quad$N$\quad %%@
and the even \quad$N$\quad cases feature qualitative differences in the %%@
invariant method. Also the ratio of physical versus ghost fields varies. For %%@
even \quad$N$\quad one finds \quad$N/2$\quad fields of each type. For odd
\quad$N$\quad one has \quad$(N-1)/2$\quad ghost (physical) and 
\quad$(N+1)/2$\quad physical (ghost) fields according to the overall negative %%@
(positive) sign of the free part of the HD Lagrangian. The squared masses may %%@
be shifted by an arbitrary common ammount, since only their differences are %%@
involved in the procedure. Then any of them may be zero (only one in this %%@
case), or tachyonic. 

On the other hand, the non--invariant procedure gets exceedingly cumbersome %%@
already for \quad$N=3\;$, in contrast with the (more compact) invariant one %%@
which remains tractable up to \quad$N=4$\quad at least. Both approaches are %%@
applicable to higher \quad$N$\quad, only at the prize of increasing the %%@
length of the calculations (namely analitically diagonalizing \quad$N\times %%@
N$\quad matrices). An intriguing feature of the odd \quad$N$\quad cases when %%@
treated with the invariant method is the occurrence of a constraint on an %%@
otherwise overabundant set of Ostrogradski--like coordinates and momenta, %%@
together with a less conventional way of working out the highest field %%@
derivative. Ignoring the constraint causes the appearance of a spurious %%@
decoupled scalar field.
        
\bigskip
\bigskip


{\bf Acknowledgements}

We are indebt to Dr. J. Le\'on for the careful reading of the manuscript
and useful suggestions.

\vfill
\eject





{\bf Appendix}

The problem of finding a matrix \quad${\cal X}$\quad with the properties %%@
(3.21) and (3.22) can be brought to the one of diagonalizing a symmetric %%@
\quad$4\times 4$\quad matrix with pure real and imaginary elements. The %%@
procedure is somehow tricky since there is no similarity--like transformation %%@
that brings the symplectic matrix \quad$\Sigma$\quad to the identity matrix, %%@
thus preventing a plain use of the weaponry of orthonormal transformations. %%@
We introduce the diagonal matrices
\quad$f\equiv diag(i,1,1,-i)$\quad and \quad$g\equiv diag(1,i,i,-1)$\quad so %%@
that 

$$\Sigma = gKf \quad,\, {\rm where}\quad
   K\equiv\left(\matrix{0&1&0&0\cr
                        1&0&0&0\cr
                        0&0&0&1\cr
                        0&0&1&0\cr}\right)\quad.$$\hfill (A.1)

\noindent{Taking} \quad$f\neq g$\quad does not compromise the uniqueness of %%@
the transformation \quad$\Phi\rightarrow\Theta$\quad as shown at the end.

Now we transform the symmetric matrix \quad$K$\quad into the \quad$4\times %%@
4$\quad identity by a similarity transformation

$${\cal D}={1\over{\sqrt 2}}\left(\matrix{1&1&0&0\cr
                                          -i&i&0&0\cr
                                          0&0&1&1\cr
                                          0&0&-i&i\cr}\right)
                                                 \quad ,$$\hfill (A.2)

\noindent{so} that

$${\cal D}\,{\cal K}\,{\cal D}^T 
                       ={\cal D}\,g^{-1}\Sigma\,f^{-1}\,{\cal D}^T
                       = I\quad. $$                 \hfill (A.3) 

\noindent{This} same transformation converts \quad${\cal M}_4$\quad and %%@
\quad${\cal M}_2$\quad into

$$\eqalign
     {{\hat{\cal M}}_4&={\cal D}\,g^{-1}{\cal M}_4 \,f^{-1}\,{\cal D}^T\cr
      {\hat{\cal M}}_2&={\cal D}\,g^{-1}{\cal M}_2 \,f^{-1}\,{\cal D}^T                                                                                           %%@
\quad.\cr}$$ \hfill (A.4)

\noindent{Notice} that \quad${\hat{\cal M}}_2$\quad and \quad${\hat{\cal %%@
M}}_4$\quad are symmetric as well. This is a consequence of the vanishing of %%@
some critical elements in both matrices. One then verifies that they have the %%@
same eigenvalues, namely \quad$-i\mu M_1\;,\;i\mu M_1\;,\;i\mu M_2$\quad and
\quad$-i\mu M_2\;$, so that there exist orthogonal matrices \quad$R$\quad and %%@
\quad$T$\quad such that
\eject

$$T^T{\hat{\cal M}}_4T = R^T{\hat{\cal M}}_2R 
                        =i\mu\, diag(-M_1,M_1,M_2,-M_2)\;,$$  \hfill (A.5)

\noindent{while} conserving the euclidean metric \quad$I\;$:

      $$R^T I\,R = T^T I\,T =  I$$                            \hfill (A.6)

\noindent{With} the orthonormal eigenvectors as columns one obtains

$$R={1\over{2\sqrt\mu}}\left(\matrix{-R^+_1&-iR^-_1&0&0\cr
                                     -iR^-_1&R^+_1&0&0 \cr
                                       0&0&R^+_2&iR^-_2\cr 
                                      0&0&iR^-_2&-R^+_2\cr}\right)$$
                                                               \hfill (A.7)

\noindent{where} 
                
$$R^{\pm}_i\equiv{{M_i\pm\mu}\over{\sqrt{M_i}}}\quad,$$        \hfill (A.8)


\noindent{and}

$$T={1\over{2\langle 12\rangle\sqrt\mu}}
                       \left(\matrix{T^+_1&-iT^-_1&-T^-_2&iT^+_2\cr
                                     iT^-_1&T^+_1&-iT^+_2&-T^-_2\cr
                                       P^-_1&iP^+_1&P^+_2&iP^-_2\cr 
                                     iP^+_1&-P^-_1&iP^-_2&-P^+_2\cr}\right)$$
                                                                \hfill (A.9)                                                  

\noindent{where}
 
$$\eqalign{T^{\pm}_i
               &\equiv\sqrt{M_i}(\mu\sqrt{M_i}\pm\langle 12\rangle)\cr
               P^{\pm}_i
               &\equiv{{\langle 12\rangle\sqrt{M_i}}\over{M^2_i}}
                ({P\over{\langle 12\rangle}}\pm\mu M_i)\quad.\cr}$$
                                                                \hfill (A.10)                                            

\noindent{Notice} that one has pure real and imaginary matrix elements and %%@
vector components, and that the norm of a vector, defined as 
\quad$\mid\!V\!\mid\equiv V^TV\;$, may be imaginary as well. Since 
\quad$M^2_i\equiv m^2_i-\triangle\;$, a regularization (the dimensional one, %%@
for instance) is understood such that \quad$R$\quad and \quad$T$\quad have %%@
well defined elements.
\eject


Finally, from (A.4) and (A.5) one gets that

                         $$Y\,{\cal M}_4W={\cal M}_2\quad,$$   \hfill (A.11)

\noindent{where} \quad$W\equiv f^{-1}{\cal D}^TT\,R^T{{\cal D}^{-1}}^Tf$\quad 
and \quad$Y\equiv g\,{\cal D}^{-1}R\,T^T{\cal D}\,g^{-1}\;$. The matrix %%@
\quad$W$\quad has some imaginary elements and the fourth row is not %%@
\quad$(\,0\,1\,0\,1\,)\;$, so that it is not suitable to relate the real %%@
vectors \quad$\Phi$\quad and \quad$\Theta$\quad as in (3.19) yet. Moreover, %%@
\quad$Y\neq W^T\;$. However one may check that

$$ \left(\matrix{i&0&0&0\cr
                 0&i&0&0\cr
                 0&0&1&0\cr
                 0&0&0&1\cr}\right)\,Y = {\cal X}^T\;,\;{\rm where}\quad
  {\cal X}\equiv \;W\;\left(\matrix{-i&0&0&0\cr
                                    0&-i&0&0\cr
                                     0&0&1&0\cr
                                     0&0&0&1\cr}\right) $$      \hfill (A.12)

\noindent{is} the matrix given in (3.20), so that (A.11) writes

       $${\cal X}^T{\cal M}_4{\cal X}={\cal M}_2\quad. $$      \hfill (A.13) 

\noindent{Furthermore}, from (A.3) and (A.6) one has that

            $${\cal X}^T\Sigma\,{\cal X}=\Sigma\quad. $$      \hfill  (A.14)

\noindent{The} fourth row of \quad${\cal X}$\quad has the desired elements %%@
\quad$(\,0\,1\,0\,1\,)$\quad only if suitable signs are chosen for the %%@
eigenvectors that build up \quad$R$\quad and \quad$T\;$, so that the %%@
handedness of the frame is conserved by \quad${\cal X}\;$. We stress that
\quad${\cal X}$\quad is also well--defined as a differential operator, and %%@
that the regularization is needed only for defining the intermediate %%@
operators \quad$T$\quad and \quad$R^T\;$. At the end of the process the %%@
regularization can be put off.


\vfill
\eject

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\vfill
\bye

