# On auxiliary fields and Lagrangians
for relativistic wave equations

Alexey Sharapov E-mail<EMAIL_ADDRESS>Centro de Matemática, Computação
e Cognição, Universidade Federal do ABC, Santo André, SP, Brazil Physics
Faculty, Tomsk State University, Lenin ave. 36, Tomsk 634050, Russia  David
Shcherbatov Physics Faculty, Tomsk State University, Lenin ave. 36, Tomsk
634050, Russia

###### Abstract

We address the problem of the existence of a Lagrangian for a given system of
linear PDEs with constant coefficients. As a subtask, this involves bringing
the system into a pre-Lagrangian form, wherein the number of equations matches
the number of unknowns. We introduce a class of overdetermined systems, called
co-flat, and show that they always admit a pre-Lagrangian form, which can be
explicitly constructed by means of auxiliary variables. Moreover, we argue
that such systems enjoy pre-Lagrangian formulations without auxiliary
variables at all. As an application of our method, we construct new pre-
Lagrangian and Lagrangian formulations for free massive fields of arbitrary
integer spin. In contrast to the well-known models of Singh and Hagen, our
Lagrangians involve much fewer auxiliary fields.

##  1 Introduction

Most of the equations encountered in physics, especially in the theory of
fundamental interactions, come from the least action principle. This is not
surprising, since the Lagrangian formulation of classical dynamics is
generally considered as the first step towards quantum theory. Therefore, the
following problem often occurs: given a system of partial differential
equations (PDEs), find a Lagrangian it comes from. This is known as the
inverse problem of the calculus of variations.

A typical example of the above situation is provided by the relativistic wave
equations for higher spin particles. Although at the free level, such
equations can be systematically derived and classified using the
representation theory of the Poincaré group, the Lagrangian formulation for
them had long remained an open problem. The point is that, unlike the
classical inverse problem studied in mathematics, the number of field
equations usually exceeds the number of fields. In other words, the PDE
systems of physical interest are usually overdetermined and finding their
Lagrangian formulations is not reduced to the problem of ‘variational
multiplier’. For instance, a free massive particle of spin $s$ can be
described by a traceless symmetric tensor field $\phi_{\mu_{1}\cdots\mu_{s}}$
subject to the equations of motion

$(\square-m^{2})\phi_{\mu_{1}\cdots\mu_{s}}=0\,,\qquad\partial^{\mu_{1}}\phi_{\mu_{1}\cdots\mu_{s}}=0\,.$
(1)

Here $\square=\partial^{\mu}\partial_{\mu}$ is the D’Alembert operator and
$m\neq 0$ is the particle’s mass. In $4$-dimensional Minkowski space, this
yields $(s+1)^{2}+s^{2}$ equations for $(s+1)^{2}$ field components.

As was first noted by Fierz and Pauli [1], one can correct the imbalance
between the fields and equations by means of the so-called auxiliary fields.
The idea is to introduce a suitable number of additional fields so that they
all vanish due to the equations of motion, whereupon the field equations
become equivalent to Eqs. (1). In particular, a minimal Lagrangian formulation
for the massive spin-2 field was shown to involve a single auxiliary scalar.
Later, developing the ideas of Fierz and Pauli, Singh and Hagen proposed
Lagrangians for free massive fields of arbitrary spin in four dimensions [2,
3]. As auxiliary fields for system (1), they used symmetric and traceless
tensors of ranks $0$, $1$, …, $s-2$ (one of each). The massless limit of the
Singh–Hagen Lagrangians for integer spins was considered by Fronsdal [4]. In
the massless limit, the Lagrangian equations become underdetermined (enjoy
gauge invariance) and all but one of the auxiliary tensor fields decouple.
Thus, both massive and massless particles of higher spins need auxiliary
fields for their Lagrangian formulation.

In this paper, we address the inverse problem of the calculus of variations
for overdetermined systems of PDEs with an eye to applications in relativistic
field theory. Since the inverse problem for such systems is still in its
infancy, we restrict ourselves to linear PDE systems with constant
coefficients. Even in such a restricted formulation, the problem is far from
being trivial and its complete solution is elusive. In general, one can divide
the problem into two parts. At the first stage, it is necessary to extend the
original PDE system with a suitable set of auxiliary fields. The extended
system must satisfy two conditions: (i) be equivalent to the original system
and (ii) be square in size, so that the total number of fields is equal to the
number of field equations. We call such a system pre-Lagrangian. Generally,
the matrix differential operator determining the extended system, being
square, may not be formally self-adjoint, in which case the pre-Lagrangian
equations are not Lagrangian per se. Therefore, at the next stage, one needs
to bring the pre-Lagrangian system into a Lagrangian one by means of a
variational multiplier. Unlike much of the mathematical literature (see e.g.
[5, 6, 7, 8]), we look for multipliers that are square matrices with entries
in differential operators, rather than functions. The variational multiplier
should be chosen in such a way as to make the operator of the pre-Lagrangian
system formally self-adjoint. Furthermore, the matrix of the multiplier must
be unimodular to ensure equivalence. The last two conditions are not easy to
satisfy, and we are unaware of any practical criterion for deciding whether a
given operator admits a variational multiplier. Some necessary conditions are
discussed below in Sec. 2.3. This drawback motivates us to focus on the first
step, i.e., the existence of a pre-Lagrangian form for a given system of
linear PDEs with constant coefficients. In this formulation, the problem can
be systematically studied by methods of homological algebra. All necessary
mathematics is reviewed in the next section.

Using the language of homological algebra, we introduce the notion of a co-
flat system. Loosely speaking, the co-flat systems represent the simplest
class of overdetermined PDE systems. For such systems, a systematic method is
proposed for introducing auxiliary fields that make them pre-Lagrangian.
Despite their simplicity, the co-flat systems cover a wide class of physically
interesting theories. For example, all massive higher spin fields (1) fall
into this class. In that case, the auxiliary fields for spin-$s$ particles, as
offered by our method, constitute a symmetric traceless tensor of rank $s-1$.
This is much less than in the Singh–Hagen models. The price to be paid for
such simplification is that the pre-Lagrangian equations necessarily include
higher derivatives. Direct calculations in particular cases show that these
pre-Lagrangian equations are equivalent to Lagrangian ones so that the
variational multiplier problem has a positive solution for them. This gives
new Lagrangians for higher spin massive theories with fewer auxiliary fields.

It might be tempting to assume that a single Lorentz irreducible tensor
$\psi_{\mu_{1}\cdots\mu_{s-1}}$ provides the minimum number of auxiliary
fields required for spin $s>2$. Surprisingly, this is not true. It follows
from the Quillen–Suslin theorem that any co-flat system admits a pre-
Lagrangian formulation without auxiliary fields at all! Unfortunately, the
theorem does not guarantee that the corresponding equations will be Lorentz
invariant (although their solutions carry a unitary irreducible representation
of the Poincaré group). An examination of simple examples shows that this is
indeed the case. Perhaps this points to some mechanism of spontaneous breaking
of Lorentz symmetry in nonlinear models with higher-spin fields. In any case,
the very existence of relativistic wave equations with broken Lorentz symmetry
looks intriguing and deserves further study.

##  2 Algebraic preliminaries

In this section, we fix our notation and briefly recall some algebraic notions
and constructions that will be needed later. For a systematic exposition of
the subject, we refer the reader to [9, 10, 11, 12].

###  2.1 Free resolutions

Let ${\mathcal{P}}=\mathbb{R}[p_{1},\ldots,p_{d}]$ be the ring of polynomials
in $d\geq 1$ indeterminates over reals. By
${\mathrm{Mat}}_{n,m}({\mathcal{P}})$ we denote the set of all $n\times
m$-matrices with entries in ${\mathcal{P}}$. In this paper, we are mostly
interested in the systems of homogeneous linear equations over the polynomial
ring ${\mathcal{P}}$. Each such system is determined by a matrix
$A\in{\mathrm{Mat}}_{n,m}({\mathcal{P}})$ and has the form $AX=0$ for unknown
$X\in{\mathrm{Mat}}_{m,1}({\mathcal{P}})$. More abstractly, one can think of
the matrix $A$ as defining a homomorphism
$A:{\mathcal{P}}^{m}\rightarrow{\mathcal{P}}^{n}$ of free
${\mathcal{P}}$-modules111We recall that a module is an algebraic structure
defined by the same axioms as for a vector space but where the scalars belong
to a ring (in our case, ${\mathcal{P}}$) rather than a field. of ranks $m$ and
$n$. In the following, we will frequently identify the ${\mathcal{P}}$-module
${\mathrm{Hom}}_{\mathcal{P}}({\mathcal{P}}^{m},{\mathcal{P}}^{n})$ with
${\mathrm{Mat}}_{n,m}({\mathcal{P}})$. Associated with the homomorphism $A$ is
the pair of ${\mathcal{P}}$-modules $\ker A\subset{\mathcal{P}}^{m}$ and
${\mathrm{coker}}A={\mathcal{P}}^{n}/\mathrm{Im}A$. In general, neither module
is free. The former is identified with the module of solutions to the above
system of linear equations. The latter module admits a finite free resolution

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}^{m_{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{N-1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}^{m_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{1}}$$\textstyle{{\mathcal{P}}^{m_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{0}}$$\textstyle{{\mathcal{P}}^{m_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathrm{coker}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\,,}$
(2)

where $\ker A_{i-1}=\mathrm{Im}\,A_{i}$ (exactness) and $N\leq d$. This is the
content of Hilbert’s syzygy theorem. The matrix $A_{0}$ defines a finite
presentation of the module $\mathrm{coker}A$. In particular, one can always
take $m_{1}=m$, $m_{0}=n$, and $A_{0}=A$. Then the exactness of the sequence
(2) implies that any solution to the equation $AX=0$ can be written as
$X=A_{1}Y$ for some $Y\in{\mathrm{Mat}}_{m_{2},1}({\mathcal{P}})$. Thus, the
sequence of matrices $(A_{1},A_{2},\ldots,A_{N-1})$ may be thought of as
parameterizing the ‘general solution’ to the linear system $AX=0$. By abuse of
language, we will sometimes refer to the exact sequence (2) as a (left)
resolution of $A$. The number $N$ is called the length of the resolution.
Neither the resolution (2) nor its length is uniquely determined by the matrix
$A$. The only invariant meaning has the minimum length of a free resolution.
It is called the projective dimension of the module ${\mathrm{coker}}\,A$. The
projective dimension measures how far a ${\mathcal{P}}$-module is from being
free.

We say that the two matrices $A\in{\mathrm{Mat}}_{n,m}({\mathcal{P}})$ and
$B\in{\mathrm{Mat}}_{s,l}({\mathcal{P}})$ are equivalent (in writing $A\sim
B$) if ${\mathrm{coker}}\,A\simeq{\mathrm{coker}}\,B$ as
${\mathcal{P}}$-modules. It is known that the last condition takes place iff
the corresponding free resolutions are homotopy equivalent, that is, there
exist homomorphisms $F_{i}$, $G_{i}$, $h^{A}_{i}$, and $h^{B}_{i}$ fitting the
diagram

$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{2}}$$\textstyle{{\mathcal{P}}^{m_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{1}}$$\scriptstyle{h^{A}_{2}}$$\scriptstyle{G_{2}}$$\textstyle{{\mathcal{P}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\scriptstyle{h^{A}_{1}}$$\scriptstyle{G_{1}}$$\textstyle{{\mathcal{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{h^{A}_{0}}$$\scriptstyle{G_{0}}$$\textstyle{{\mathrm{coker}}\,A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{2}}$$\textstyle{{\mathcal{P}}^{l_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{1}}$$\scriptstyle{h^{B}_{2}}$$\scriptstyle{F_{2}}$$\textstyle{{\mathcal{P}}^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B}$$\scriptstyle{h^{B}_{1}}$$\scriptstyle{F_{1}}$$\textstyle{{\mathcal{P}}^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{h^{B}_{0}}$$\scriptstyle{F_{0}}$$\textstyle{{{\mathrm{coker}}\,B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$

such that

$\begin{array}[]{ll}A_{i}F_{i+1}=F_{i}B_{i}\,,&B_{i}G_{i+1}=G_{i}A_{i}\,,\\\\[8.53581pt]
h^{A}_{i}A_{i}+A_{i+1}h^{A}_{i+1}=1-F_{i+1}G_{i+1}\,,&B_{i+1}h^{B}_{i+1}+h^{B}_{i}B_{i}=1-G_{i+1}F_{i+1}\end{array}$
(3)

for $i=0,1,2,\ldots$ and $A_{0}=A$. If one regards Rels. (3) as equations for
unknown homomorphisms $F_{i}$, $G_{i}$, $h^{A}_{i}$, and $h^{B}_{i}$, then the
existence of a solution to the equations with $i=0$ ensures the existence of
all other homomorphisms, see e.g. [12, Prop. 1.2.8]. The mutually inverse
isomorphisms

$f:{\mathrm{coker}}\,B\rightarrow{\mathrm{coker}}\,A\,,\qquad
g:{\mathrm{coker}}\,A\rightarrow{\mathrm{coker}}\,B$

are induced by the homomorphisms $F_{0}$ and $G_{0}$, respectively.

When doing linear algebra over polynomial rings, it might be useful to extend
the ring ${\mathcal{P}}$ to its field of fractions
${\mathcal{R}}=\mathbb{R}(p_{1},\ldots,p_{d})$. The elements of the field
${\mathcal{R}}$ are just rational functions in $p$’s. Since
${\mathcal{P}}\subset{\mathcal{R}}$, the field ${\mathcal{R}}$ comes with the
natural structure of a ${\mathcal{P}}$-module. This module is known to be
flat, meaning that the covariant functor ${\mathcal{R}}\otimes_{\mathcal{P}}-$
maps exact sequences of ${\mathcal{P}}$-modules to exact. The obvious
isomorphism
${\mathcal{R}}\otimes_{\mathcal{P}}{\mathcal{P}}^{n}\simeq{\mathcal{R}}^{n}$
allows one to pass easily from the free ${\mathcal{P}}$-module
${\mathcal{P}}^{n}$ to the ${\mathcal{R}}$-vector space ${\mathcal{R}}^{n}$.

Recall that a ${\mathcal{P}}$-module ${\mathcal{F}}$ is called injective if
the contravariant functor ${\mathrm{Hom}}_{\mathcal{P}}(-,{\mathcal{F}})$
transforms exact sequences to exact ones. Applying such a functor to the exact
sequence (2) with $A_{0}=A$ and using the obvious isomorphism
${\mathrm{Hom}}_{\mathcal{P}}({\mathcal{P}}^{n},{\mathcal{F}})\simeq{\mathcal{F}}^{n}$,
one obtains the new exact sequence

$\textstyle{0}$$\textstyle{{\mathcal{F}}^{m_{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A^{t}_{N}}$$\textstyle{{\mathcal{F}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A^{t}_{1}}$$\textstyle{{\mathcal{F}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A^{t}}$$\textstyle{{\mathrm{Hom}}_{\mathcal{P}}({\mathrm{coker}}\,A,{\mathcal{F}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\,,}$
(4)

where by $A^{t}$ we denote the matrix transpose to $A$. In particular,

$\ker_{\mathcal{F}}A^{t}\simeq{\mathrm{Hom}}_{\mathcal{P}}({\mathrm{coker}}\,A,{\mathcal{F}})\,.$
(5)

This isomorphism is due to Malgrange [13]. It establishes a relation between
the solutions to the linear system $A^{t}X=0$ for
$X\in{\mathrm{Mat}}_{n,1}({\mathcal{F}})$ and the module
${\mathrm{coker}}\,A$.

An injective ${\mathcal{P}}$-module ${\mathcal{F}}$ is called an injective
cogenerator if ${\mathrm{Hom}}_{\mathcal{P}}(M,{\mathcal{F}})\neq 0$ for any
nontrivial ${\mathcal{P}}$-module $M$. A typical example of an injective
cogenerator for the polynomial ring ${\mathcal{P}}$ is provided by the space
of formal power series
${\mathcal{P}}^{\ast}=\mathbb{R}[[x_{1},\ldots,x_{d}]]$, the dual of the
$k$-vector space ${\mathcal{P}}$. Another example is the smooth real-valued
functions $C^{\infty}(\Omega)$ on an open convex domain
$\Omega\subset\mathbb{R}^{d}$. In both cases the generators of the ring act as
partial derivatives, $p_{i}f={\partial f}/{\partial x^{i}}$. It follows from
the definition that once ${\mathrm{coker}}\,A\neq 0$, the equation $A^{t}X=0$
has a nonzero solution whenever one looks for an $n\times 1$-matrix $X$ with
entries in an injective cogenerator. Roughly speaking, the injective
cogenerator condition on $\cal F$ plays the same role as the condition of an
algebraically closed base field in classical algebraic geometry.

###  2.2 Quillen–Suslin theorem

A matrix $A\in{\mathrm{Mat}}_{n,m}({\mathcal{P}})$ is called unimodular if it
admits either a left or right inverse matrix $B$, that is,

$BA=1_{m}\quad\mbox{or}\quad AB=1_{n}\,.$ (6)

It is clear that the first (resp. second) identity implies $n\geq m$ (resp.
$m\geq n$). We will denote the set of all $n\times m$ unimodular matrices by
$U_{n,m}({\mathcal{P}})$. The unimodular matrices of $U_{n,n}({\mathcal{P}})$,
being invertible, form a group denote by $U_{n}({\mathcal{P}})$. The groups
$U_{n}({\mathcal{P}})$ and $U_{m}({\mathcal{P}})$ naturally act on
${\mathrm{Mat}}_{n,m}({\mathcal{P}})$ from the left and right, and one may
wonder what is the simplest form that every matrix of
${\mathrm{Mat}}_{n,m}({\mathcal{P}})$ can be reduced to by this action. In the
case of unimodular matrices, the answer to this question is given by the
Quillen–Suslin theorem [10].

######  Theorem 2.1.

For any $A\in U_{n,m}({\mathcal{P}})$ with $n\leq m$, there exists a matrix
$U\in U_{m}({\mathcal{P}})$ such that

$AU=\left(\begin{array}[]{ccccccccc}1&0&\cdots&0&0&\cdots&0\\\
0&1&\cdots&0&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&1&0&\cdots&0\end{array}\right)=(1_{n},0)\,.$ (7)

The matrix on the right represents the canonical form of a unimodular matrix.
It follows from the definition of $U$ that

$U^{-1}=\left(\begin{array}[]{c}A\\\ A^{\vee}\end{array}\right)$

for some unimodular $(m-n)\times m$-matrix $A^{\vee}$. We will refer to
$A^{\vee}$ as a unimodular completion of the matrix $A$. This gives an
alternative formulation of the Quillen–Suslin theorem: Every unimodular matrix
can be completed to an invertible one. Writing the matrix $U$ in the block
form $U=(B,B^{\vee})$, where $B^{\vee}$ consists of the last $m-n$ columns of
$U$, one finds that $AB=1_{n}$ and $AB^{\vee}=0$. The columns of the
unimodular matrix $B^{\vee}$ define the basis of $\ker A$ and we arrive at the
short exact sequence

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29.25pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{{\mathcal{P}}^{m-n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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97.34306pt\raise 5.1525pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.1525pt\hbox{$\scriptstyle{A}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 115.05557pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
115.05557pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
154.20575pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
154.20575pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces\,,$ (8)

which can be viewed as the Hilbert resolution of the trivial module
${\mathrm{coker}}\,A=0$. By construction, the sequence splits as
${\mathcal{P}}^{m}=\mathrm{Im}\,B\oplus\mathrm{Im}\,B^{\vee}$.

Finally, a unimodular matrix $A\in U_{n}({\mathcal{P}})$ is called elementary
if it has either the upper or lower triangular form. For $n\neq 2$, the
elementary matrices are known to generate the whole group
$U_{n}({\mathcal{P}})$, see [10, Ch. VI.4.5]. This means that one can bring
every unimodular matrix into the canonical form by a sequence of elementary
column (or row) transformations. The case $n=2$ is special. For example, it is
known that the following unimodular matrix

$\left(\begin{array}[]{cc}1+xy&x^{2}\\\ -y^{2}&1-xy\end{array}\right)$ (9)

cannot be written as a finite product of elementary matrices [10, Ch. I.8].

###  2.3 Resolutions of differential operators with constant coefficients

Let $\mathcal{F}$ denote the ring of smooth functions on an open convex domain
$\Omega\subset\mathbb{R}^{d}$. Then the free $\mathcal{F}$-module
$\mathcal{F}^{n}$ consists of collections of functions
$f=(f_{1},\ldots,f_{n})$ with $f_{a}\in C^{\infty}(\Omega)$. Denote by
$\partial_{i}=\partial/\partial x^{i}$ the partial derivatives w.r.t. the
Cartesian coordinates $(x^{1},\ldots,x^{d})$ in $\mathbb{R}^{d}$. Given a
matrix $A=\big{(}A_{a\alpha}(p)\big{)}$ of
${\mathrm{Mat}}_{n,m}({\mathcal{P}})$, define the differential operator
$\hat{A}:{\mathcal{F}}^{n}\rightarrow{\mathcal{F}}^{m}$ by the rule

$(\hat{A}f)_{\alpha}=\sum_{a=1}^{n}A_{a\alpha}(\partial)f_{a}\,,\qquad\alpha=1,\ldots,m\,,$

where $A_{a\alpha}(\partial)$ is obtained from $A_{a\alpha}(p)$ by the
substitution $p_{i}\rightarrow\partial_{i}$. We will refer to the matrix $A$
as the symbol of a differential operator $\hat{A}$. By differential operators
on $\Omega$ with constant coefficients we understand the operators that are
obtained by the above assignment $A\mapsto\hat{A}$. The $\mathbb{R}$-space of
all differential operators
$\hat{A}:{\mathcal{F}}^{n}\rightarrow{\mathcal{F}}^{m}$ with constant
coefficients will be denoted by ${\mathrm{Op}}_{n,m}({\mathcal{F}})$. If
$\hat{B}$ is an operator of ${\mathrm{Op}}_{m,l}(\mathcal{F})$, then it
follows from the definition that

$\hat{B}\hat{A}=\widehat{AB}\,.$ (10)

(Notice the change of order.)

The space of smooth functions ${\mathcal{F}}$, considered as a
${\mathcal{P}}$-module under the assignment $p_{i}\rightarrow\partial_{i}$, is
known to be injective and the Malgrange isomorphism (5) reduces the study of
linear PDEs $\hat{A}f=0$ to the study of their symbols $A$. In particular, two
equations have isomorphic solution spaces iff their symbols are equivalent.

In this paper, we focus on particular classes of operators. Of most interest
to us are operators of ${\mathrm{Op}}_{n,n}({\mathcal{F}})$, $n=1,2,\ldots$,
which we call pre-Lagrangian. By definition, the symbol of a pre-Lagrangian
operator is a square matrix $A\in{\mathrm{Mat}}_{n,n}({\mathcal{P}})$. The
commutative ring ${\mathcal{P}}$ enjoys the involution $a(p)\mapsto
a^{\dagger}(p)=a(-p)$, which extends to the involution of the
${\mathcal{P}}$-module ${\mathrm{Mat}}_{n,n}({\mathcal{P}})$ as $A(p)\mapsto
A^{\dagger}(p)=A^{t}(-p)$. If one regards
${\mathrm{Mat}}_{n,n}({\mathcal{P}})$ as a ring, then the involution above
defines an anti-isomorphism of order two, i.e.,
$(AB)^{\dagger}=B^{\dagger}A^{\dagger}$.

A pre-Lagrangian operator $\hat{A}\in{\mathrm{Op}}_{n,n}({\mathcal{F}})$ is
called Lagrangian if $A^{\dagger}=A$. The terminology is obvious: If $\hat{A}$
is Lagrangian, then, under suitable boundary conditions, one can obtain the
equation $\hat{A}f=g$ for unknown $f\in{\mathcal{F}}^{n}$ from the variational
principle for the action functional

$S[f]=\int_{\Omega}d^{d}x\Big{(}\frac{1}{2}f^{t}\hat{A}f-f^{t}g\Big{)}\,.$

Notice that whenever (2) defines a resolution of the symbol $A$, the
nonhomogeneous equation $\hat{A}f=g$ implies the equality $\hat{A}_{1}g=0$,
called usually the compatibility condition. Correspondingly, the operator
$\hat{A}_{1}$ is referred to as a compatibility operator.

######  Example 2.2.

Let us add sources to equations (1) to get the nonhomogeneous linear system

$(\square-m^{2})\phi_{\mu_{1}\cdots\mu_{s}}=j^{\prime}_{\mu_{1}\cdots\mu_{s}}\,,\qquad\partial^{\mu_{1}}\phi_{\mu_{1}\cdots\mu_{s}}=j^{\prime\prime}_{\mu_{2}\cdots\mu_{s}}\,.$
(11)

Applying the Klein–Gordon operator to the second equation and subtracting the
divergence of the first one yields the following constraint on the sources:

$(\square-m^{2})j^{\prime\prime}_{\mu_{2}\cdots\mu_{s}}-\partial^{\mu_{1}}j^{\prime}_{\mu_{1}\cdots\mu_{s}}=0\,.$
(12)

The matrix differential operator determining this constraint is the
compatibility operator for the wave operator in (11).

We say that a pre-Lagrangian operator $\hat{A}$ is equivalent to a Lagrangian
one if there exists a unimodular matrix $U$ such that the operator
$\hat{U}\hat{A}$ is Lagrangian, i.e.,

$AU=U^{\dagger}A^{\dagger}\,.$ (13)

Clearly, $AU\sim A$, so that the equations $\hat{U}\hat{A}f=0$ and
$\hat{A}f=0$ have the same solution space for any ${\mathcal{F}}$. In the
context of the inverse problem of the calculus of variations, the operator
$\hat{U}$, making pre-Lagrangian differential equations Lagrangian, is called
a variational multiplier. We do not know any effective criterion for deciding
whether a given pre-Lagrangian operator admits a variational multiplier.
Computing the determinant of the left- and right-hand sides of (13), one
obtains the following necessary condition:

$\det A^{\dagger}=\det A\,.$ (14)

For example, the operator determining the ordinary differential equation
$d^{k}f/dx^{k}=0$ is (equivalent to) Lagrangian iff $k$ is even. To formulate
finer necessary conditions, we need more notation. Let
$A\in{\mathrm{Mat}}_{n,n}({\mathcal{P}})$ and $r\leq n$. An $r$-minor of $A$
is the determinant of any $r\times r$-submatrix of $A$. Let $\Delta_{r}(A)$
denote the set of all $r$-minors of the matrix $A$ and $d_{r}(A)$ denote the
g.c.d. of the polynomials of $\Delta_{r}(A)$. Define the ideals
$\mathcal{I}_{r}(A)=\langle\Delta_{r}(A)\rangle$ and
$\mathcal{J}_{r}(A)=\langle\Delta_{r}(A)/d_{r}(A)\rangle$ generated,
respectively, by all $r$-minors $\Delta_{r}(A)$ and their quotients of
division by $d_{r}(A)$. $\mathcal{I}_{r}(A)$ is known as the $(n-r)$-th
Fitting ideal of the module $\mathrm{coker}A$. It is clear that
$\mathcal{I}_{r}(A)\subset\mathcal{I}_{r-1}(A)$. By applying the generalized
Cauchy–Binet formula one can prove the following

######  Proposition 2.3.

If a unimodular matrix $U\in U_{n}({\mathcal{P}})$ satisfies Eq. (13), then

$\mathcal{I}_{r}(A)=\mathcal{I}_{r}(A^{\dagger})\,,\qquad
d_{r}(A)=d_{r}(A^{\dagger})\,,\qquad\mathcal{J}_{r}(A)=\mathcal{J}_{r}(A^{\dagger})\,,\qquad\forall
r=1,2,\ldots,n\,.$ (15)

For $r=n$, one gets the stronger condition
$\Delta_{n}(A^{\dagger})=\Delta_{n}(A)$. In the absence of effective criteria
for the solvability of equation (13), we confine ourselves to studying a
simpler question about the existence of an equivalent pre-Lagrangian operator.

Following the terminology of Ref. [14], we say that an operator $\hat{A}$ is
co-flat if its symbol admits a one-step resolution

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{1}}$$\textstyle{{\mathcal{P}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{{\mathcal{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{coker}}\,A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\,,}$
(16)

where the matrix $A_{1}$ is unimodular. As we will see in the next section,
co-flat operators are plentiful in relativistic physics. Let us start with the
following simple

######  Proposition 2.4.

Any two matrices $A_{1}$ and $A^{\prime}_{1}$ associated with one-step
resolutions of a co-flat symbol $A$ are related by the equality
$A^{\prime}_{1}=A_{1}U$ for some $U\in U_{l}({\mathcal{P}})$. In particular,
both matrices are unimodular and have the same size.

###### Proof.

Let

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}^{l^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A^{\prime}_{1}}$$\textstyle{{\mathcal{P}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{{\mathcal{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{coker}}\,A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\,,}$
(17)

be another one-step resolution of the matrix $A$. As we know the resolutions
(16) and (17) are homotopy equivalent to each other. Furthermore, we can
satisfy Eqs. (3) with $F_{1}=G_{1}=1_{m}$ and $F_{0}=G_{0}=1_{n}$. Then there
are matrices $F_{2}\in{\mathrm{Mat}}_{ll^{\prime}}({\mathcal{P}})$ and
$G_{2}\in{\mathrm{Mat}}_{l^{\prime}l}({\mathcal{P}})$ such that

$A_{1}F_{2}=A^{\prime}_{1}\,,\qquad A^{\prime}_{1}G_{2}=A_{1}\,.$ (18)

If $l^{\prime}>l$, then $\ker F_{2}\neq 0$. On the other hand, $\ker
A^{\prime}_{1}=0$ and restricting the first equation in (18) onto $\ker
F_{2}$, we arrive at the contradiction. Similarly, the assumption
$l>l^{\prime}$ contradicts the facts that $\ker G_{2}\neq 0$, while $\ker
A_{1}=0$. Hence, $l=l^{\prime}$ and both $F_{2}$ and $G_{2}$ are square
matrices. Let $B$ be a left inverse for the matrix $A_{1}$. Applying $B$ to
both sides of the second equation in (18) yields $BA^{\prime}_{1}G_{2}=1_{l}$.
This means that the square matrix $G_{2}$ is unimodular and we can set
$U=G_{2}^{-1}$. ∎

For a general co-flat operator $\hat{A}$, the ranks of the free modules in
(16) are not related to each other. We say that a differential operator
$\hat{A}\in{\mathrm{Op}}_{n,m}({\mathcal{F}})$ is gauge-free if $\ker
A^{t}=0$. It is clear that $n\geq m$ whenever the operator $\hat{A}$ is gauge-
free. For example, the relativistic wave equations (1) enjoy no gauge symmetry
and the corresponding operator is gauge-free. If $\hat{A}$ is a pre-Lagrangian
gauge-free operator, then $\det A\neq 0$.

######  Proposition 2.5.

Suppose a co-flat operator $\hat{A}\in{\mathrm{Op}}_{n,m}({\mathcal{F}})$ is
gauge-free. Then the ranks of the modules in the one-step resolution (16) are
related by the equality $m=l+n$.

###### Proof.

Applying the functor ${\mathcal{R}}\otimes_{\mathcal{P}}-$ to the exact
sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{1}}$$\textstyle{{\mathcal{P}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{{\mathcal{P}}^{n}}$
(19)

yields the exact sequence of ${\mathcal{R}}$-vector spaces

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{R}}^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{1}}$$\textstyle{{\mathcal{R}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{{\mathcal{R}}^{n}\,.}$
(20)

The requirement for the operator $\hat{A}$ to be gauge-free gives two more
exact sequences

$0\longrightarrow{{\mathcal{P}}}^{n}\stackrel{{\scriptstyle
A^{t}}}{{\longrightarrow}}{{\mathcal{P}}}^{m}\qquad\Rightarrow\qquad
0\longrightarrow\mathcal{R}^{n}\stackrel{{\scriptstyle
A^{t}}}{{\longrightarrow}}\mathcal{R}^{m}\,.$ (21)

Since the dualization of finite-dimensional vector spaces preserves exactness,
we also have the exact sequence

$\mathcal{R}^{m}\stackrel{{\scriptstyle
A}}{{\longrightarrow}}\mathcal{R}^{n}\longrightarrow 0\,.$ (22)

By gluing (20) with (22), we obtain the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{R}}^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{1}}$$\textstyle{{\mathcal{R}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{{\mathcal{R}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\,.}$
(23)

It remains to note that every short exact sequence of finite-dimensional
vector spaces necessary splits,
${\mathcal{R}}^{m}\simeq{\mathcal{R}}^{l}\oplus{\mathcal{R}}^{n}$, so that
$m=l+n$. Another argument: the Euler characteristic of an acyclic complex of
finite-dimensional vector spaces must be zero, i.e., $\chi=l-m+n=0$.

∎

######  Proposition 2.6.

Every co-flat matrix $A$ is equivalent to a matrix $B$ with $\ker B=0$.

###### Proof.

We will present two equivalent matrices $B$ of different size. The first one
comes from the diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{1}}$$\textstyle{{\mathcal{P}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\scriptstyle{h^{A}_{1}}$$\scriptstyle{G_{1}}$$\textstyle{{\mathcal{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{h^{A}_{0}}$$\scriptstyle{G_{0}}$$\textstyle{{\mathrm{coker}}\,A}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{P}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B}$$\scriptstyle{F_{1}}$$\textstyle{{\mathcal{P}}^{l+n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{h^{B}_{0}}$$\scriptstyle{F_{0}}$$\textstyle{{{\mathrm{coker}}\,B}}$

whose homomorphisms are defined by the following matrices:

$\begin{array}[]{llll}G_{0}=\binom{0}{\;1_{n}}\,,&F_{0}=(0,1_{n})\,,&h_{0}^{A}=0\,,&h_{1}^{A}=X\,,\\\\[8.53581pt]
G_{1}=1_{m}-A_{1}X\,,&F_{1}=1_{m}\,,&h_{0}^{B}=(A_{1},0)\,,&B=\binom{X}{A}\,.\\\
\end{array}$ (24)

Here $X$ is a left inverse to $A_{1}$. One can easily check that the matrices
satisfy Eqs. (3), which means $A\sim B$.

To construct an alternative matrix $B$ we use the Quillen–Suslin theorem. Let
$A_{1}^{\vee}$ be a unimodular completion of $A_{1}$ such that the square
matrix $U=(A_{1},A_{1}^{\vee})$ is invertible. Obviously,
${\mathrm{coker}}\,A={\mathrm{coker}}\,AU$ and $A\sim AU$. On the other hand,
$AU=(0,AA_{1}^{\vee})$ and $\mathrm{Im}\,AU=\mathrm{Im}\,AA_{1}^{\vee}$, where
$AA_{1}^{\vee}:{\mathcal{P}}^{m-l}\rightarrow{\mathcal{P}}^{n}$. Therefore,
${\mathrm{coker}}\,AU={\mathrm{coker}}\,AA_{1}^{\vee}$ and we conclude that
$A\sim AA_{1}^{\vee}$. It remains to note that $\ker AA_{1}^{\vee}=0$.
Otherwise, there would exist a nonzero
$X\in{\mathrm{Mat}}_{m-l,1}({\mathcal{P}})$ and
$Y\in{\mathrm{Mat}}_{l,1}({\mathcal{P}})$ such that $A_{1}^{\vee}X=A_{1}Y$.
But this contradicts the invertibility of $U=(A_{1},A_{1}^{\vee})$. ∎

The proposition above is a particular case of a more general statement.

######  Proposition 2.7.

The projective dimension of the module ${\mathrm{coker}}\,A$ is equal to $N$
iff there exists a finite free resolution (2) of length $N$ where the leftmost
matrix $A_{N}$ is not unimodular.

For the proof, see [9, Prop. 5.11]. In Refs. [15] and [14, Prop. 2], an
explicit construction was proposed to reduce the free resolution (2) by one
step in the case when $A_{N}$ is unimodular. All this makes the calculation of
projective dimension a completely algorithmic procedure.

The projective dimension, being an invariant of a module, provides a useful
necessary condition for a given operator $\hat{A}$ to be equivalent to a
(pre-)Lagrangian one.

######  Proposition 2.8.

If an operator $\hat{A}$ is equivalent to a gauge-free pre-Lagrangian
operator, then the projective dimension of the module ${\mathrm{coker}}\,A$ is
either zero or one.

###### Proof.

Every gauge-free pre-Lagrangian operator
$\hat{B}\in{\mathrm{Op}}_{n,n}({\mathcal{F}})$ is determined by a
nondegenerate square matrix $B\in{\mathrm{Mat}}_{n,n}({\mathcal{P}})$. This
gives the short exact sequence
$0\rightarrow{\mathcal{P}}^{n}\stackrel{{\scriptstyle
B}}{{\rightarrow}}{\mathcal{P}}^{n}\rightarrow{\mathrm{coker}}\,B\rightarrow
0$, meaning that the projective dimension of ${\mathrm{coker}}\,B$ is less
than or equal to one. The zero projective dimension implies that the matrices
$B$ and $A$ are unimodular and the differential equation $\hat{A}f=0$ has only
the trivial solution $f=0$. If $A\sim B$, the corresponding cokernels should
have the same projective dimension. ∎

Combining Propositions 2.5 and 2.6, we conclude that every co-flat gauge-free
matrix $A$ is equivalent to a square nondegenerate matrix $B$. In the course
of the proof of Prop. 2.6, two such matrices $B$ were actually constructed:
one given by $\binom{X}{A}$, where $X$ is the left inverse to $A_{1}$, and the
other given by the product $AA_{1}^{\vee}$. The former has size $m\times m$,
while the latter is of dimension $n\times n$. Since $m>n$, we will refer to
the corresponding differential operators as the big and small pre-Lagrangian
forms of the operator $\hat{A}$.

##  3 Free massive fields of integer spin

In this section, we exemplify the above constructions by relativistic wave
equations for free massive particles. As a by-product, we get new free
Lagrangians with fewer auxiliary fields.

###  3.1 Pre-Lagrangian equations for free massive fields

Our starting point is the system of relativistic wave equations of the form

$T^{1}_{\mu_{1}\cdots\mu_{s}}:=(\square-m^{2})\phi_{\mu_{1}\ldots\mu_{s}}(x)=0\,,\qquad
T^{2}_{\mu_{1}\cdots\mu_{s-1}}:=\partial^{\mu_{s}}\phi_{\mu_{1}\ldots\mu_{s}}(x)=0\,.$
(25)

Here, $\phi_{\mu_{1}\ldots\mu_{s}}(x)$ is a fully symmetric and traceless
tensor field in $d$-dimensional Minkowski space with Cartesian coordinates
$x^{\mu}$ and all indices are raised and lowered with the (mostly plus)
Minkowski metric. Equations (25) are known to be gauge-free and describe the
dynamics of free quantum particles of mass $m$. The rank of the tensor $s$
defines the spin of a particle222Of course, the relativistic wave equations
(25) do not cover all free massive particles for $d>4$.. Scalar particles
correspond to $s=0$, in which case the second equation in (25) is absent. For
$s>0$, the system is non-Lagrangian since the equations outnumber the fields.
The symbol of the operator determining Eqs. (25) reads

$A=\big{(}A^{\nu_{1}\cdots\nu_{s}}_{\mu_{1}\cdots\mu_{s}},\,A^{\nu_{1}\cdots\nu_{s}}_{\mu_{1}\cdots\mu_{s-1}}\big{)}=\big{(}(p^{2}-m^{2})\delta_{\mu_{1}}^{\nu_{1}}\cdots\delta_{\mu_{s}}^{\nu_{s}},\,\delta^{\nu_{1}}_{\mu_{1}}\cdots\delta_{\mu_{s-1}}^{\nu_{s-1}}p^{\nu_{s}}\big{)}_{st}\,.$
(26)

The subscript $st$ stands for the symmetric and traceless projection over the
upper and lower indices of the component tensors. The equations satisfy the
obvious identity

$\partial^{\mu_{s}}T^{1}_{\mu_{1}\cdots\mu_{s-1}\mu_{s}}-(\square-m^{2})T^{2}_{\mu_{1}\cdots\mu_{s-1}}\equiv
0\,,$ (27)

from which one can read off the symbol $A_{1}$ of the compatibility operator
for (25). Explicitly,

$A_{1}=\binom{\delta_{\lambda_{1}}^{\mu_{1}}\cdots\delta_{\lambda_{s-1}}^{\mu_{s-1}}p^{\mu_{s}}}{-\delta^{\mu_{1}}_{\lambda_{1}}\cdots\delta^{\mu_{s-1}}_{\lambda_{s-1}}(p^{2}-m^{2})}_{st}\,.$
(28)

It turns out that the matrix $A_{1}$ is unimodular or, what is the same, the
matrix $A$ is co-flat. As explained in the previous section, the last fact
implies that the PDE system (25) is equivalent to a pre-Lagrangian one. To
construct the big pre-Lagrangian form one needs a left inverse matrix to the
matrix $A_{1}$. In view of the tensor structure of the equations of motion
(25) and the identities (27), the left inverse matrix has the block form
$X=({X}_{\mu_{1}\cdots\mu_{s}}^{\nu_{1}\cdots\nu_{s-1}},{X}_{\mu_{1}\cdots\mu_{s-1}}^{\nu_{1}\cdots\nu_{s-1}})$,
the blocks being fully symmetric and traceless tensors in upper and lower
indices. Once the matrix $X$ is identified, one can write the pre-Lagrangian
equations as

${T}^{1}_{\mu_{1}\cdots\mu_{s}}+\hat{X}_{\mu_{1}\cdots\mu_{s}}^{\nu_{1}\cdots\nu_{s-1}}\psi_{\nu_{1}\cdots\nu_{s-1}}=0\,,\qquad
T^{2}_{\mu_{1}\cdots\mu_{s-1}}+\hat{X}_{\mu_{1}\cdots\mu_{s-1}}^{\nu_{1}\cdots\nu_{s-1}}\psi_{\nu_{1}\cdots\nu_{s-1}}=0\,,$
(29)

where $\psi_{\nu_{1}\cdots\nu_{s-1}}$ is a symmetric and traceless tensor
field of rank $s-1$. By construction, the total number of the initial fields
$\phi$’s and the auxiliary fields $\psi$’s is equal to the number of equations
(29). On substituting the extended equations (29) in the identity (27), we get
the differential consequence

$\big{(}\partial^{\mu_{s}}\hat{X}_{\mu_{1}\cdots\mu_{s}}^{\nu_{1}\cdots\nu_{s-1}}-(\square-m^{2})\hat{X}_{\mu_{1}\cdots\mu_{s-1}}^{\nu_{1}\cdots\nu_{s-1}}\big{)}\psi_{\nu_{1}\cdots\nu_{s-1}}=0\,,$
(30)

which involves the auxiliary fields alone. If the matrix $X$ has been chosen
to satisfy $XA_{1}=1$, then the differential operator in (30) defines the
identity map, which means that $\psi_{\nu_{1}\cdots\nu_{s-1}}=0$ and the
extended equations (29) are fully equivalent to the original ones (25). It
should be noted that the left inverse matrix $X$ is not unique. In the case
under consideration, it is natural to require the operator $\hat{X}$ to be
Lorentz invariant. The most general Lorentz-invariant Ansatz for $\hat{X}$
reads

$\begin{array}[]{rcl}\hat{X}_{\mu_{1}\cdots\mu_{s}}^{\nu_{1}\cdots\nu_{s-1}}\psi_{\nu_{1}\cdots\nu_{s-1}}&=&A_{1}\partial_{\\{\mu_{1}}\psi_{\mu_{2}\cdots\mu_{s}\\}}+A_{2}\partial_{\\{\mu_{1}\mu_{2}}(\partial\psi)_{\mu_{3}\cdots\mu_{s}\\}}+\ldots+A_{s}\partial_{\\{\mu_{1}\cdots\mu_{s}\\}}\partial^{s-1}\psi\,,\\\\[8.53581pt]
\hat{X}_{\mu_{1}\cdots\mu_{s-1}}^{\nu_{1}\cdots\nu_{s-1}}\psi_{\nu_{1}\cdots\nu_{s-1}}&=&B_{1}\psi_{\mu_{1}\cdots\mu_{s-1}}+B_{2}\partial_{\\{\mu_{1}}(\partial\psi)_{\mu_{2}\ldots\mu_{s-1}\\}}+\ldots+B_{s}\partial_{\\{\mu_{1}\cdots\mu_{s-1}\\}}\partial^{s-1}\psi\,.\end{array}$
(31)

Here we introduced the following shorthand notation for repeated gradients and
divergences:

$\partial_{\mu_{1}\cdots\mu_{n}}(\partial^{l}\Phi)_{\nu_{l+1}\cdots\nu_{m}}:=\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}\partial^{\nu_{1}}\cdots\partial^{\nu_{l}}\Phi_{\nu_{1}\cdots\nu_{m}}\,.$
(32)

The braces stand for symmetrization and the traceless part of the enclosed
indices, see (63). Finally, the coefficients $A$’s and $B$’s are supposed to
be polynomials in the D’Alembert operator $\square$. Substituting the Ansatz
(31) in Eq. (30) and requiring the wave operator to be unity, we get a set of
equations for determining the $A$’s and $B$’s. The details of the computation
can be found in Appendix A. The final answer is

$\begin{array}[]{ll}\displaystyle A_{1}=\frac{1}{m^{2}},&\displaystyle
B_{1}=\frac{1}{m^{2}},\\\\[14.22636pt] \displaystyle
A_{n}=\frac{1}{m^{2}}\left(-\frac{1}{m^{2}}\frac{2s+d-6}{2s+d-5}\right)^{n-1},&\displaystyle
B_{n}=\frac{1}{m^{2}}\frac{2s+d-5}{2s+d-4}\left(-\frac{1}{m^{2}}\frac{2s+d-6}{2s+d-5}\right)^{n-1},\\\\[14.22636pt]
\displaystyle
A_{s}=-\frac{1}{m^{4}}\frac{d-2}{s+d-3}\left(-\frac{1}{m^{2}}\frac{2s+d-6}{2s+d-5}\right)^{s-2},&\displaystyle
B_{s}=-\frac{1}{m^{4}}\frac{d-2}{2s+d-4}\left(-\frac{1}{m^{2}}\frac{2s+d-6}{2s+d-5}\right)^{s-2},\end{array}$
(33)

where $1<n<s$. As is seen, all coefficients can be chosen constant. Noteworthy
also are the two special cases $(s=1)$ and $(s=2,d=2)$, where only the leading
coefficients $A_{1}=B_{1}=1/m^{2}$ are different from zero. In these cases,
one can exclude the auxiliary field $\psi$ from the extended equations (29) to
get Lagrangian equations in terms of the original field $\phi$:

$(\square-m^{2})\phi_{\mu}-\partial_{\mu}\partial^{\nu}\phi_{\nu}=0\,,\qquad(\square-m^{2})\phi_{\mu\nu}-\partial_{\mu}\partial^{\lambda}\phi_{\lambda\nu}-\partial_{\nu}\partial^{\lambda}\phi_{\lambda\mu}+\eta_{\mu\nu}\partial^{\lambda}\partial^{\gamma}\phi_{\lambda\gamma}=0\,.$
(34)

The first equation is known as the Proca equation for the massive vector
field. As for the second equation, passing to the light-cone coordinates one
can easily see that the wave operator is unimodular and the equation has only
zero solution.

One can regard the relativistic wave equations (34) as the small pre-
Lagrangian form of the corresponding equations (25). What about the small pre-
Lagrangian form for the other values of $s$ and $d$? Although the existence of
such a form was proved in Proposition 2.6, its Lorentz invariance is not
ensured. Counterintuitive as the last statement might seem333It implies that
one can describe unitary irreducible representations of the Poincaré group by
means of PDEs that are not Lorentz invariant!, it is supported by specific
examples. Consider, for instance, the case $s=2$, $d=3$. The small pre-
Lagrangian form is constructed by means of the unimodular completion of the
matrix $A_{1}$ defined by (28). In the case at hand, $A_{1}$ is an $8\times 3$
matrix, which unimodular completion $A_{1}^{\vee}$ must be a matrix of size
$8\times 5$. The most general Lorentz invariant Ansatz for the matrix
$A_{1}^{\vee}$ is given by

$A_{1}^{\vee}=\binom{(A_{1}^{\vee})^{\nu_{1}\nu_{2}}_{\mu_{1}\mu_{2}}}{(A_{1}^{\vee})^{\nu}_{\mu_{1}\mu_{2}}}=\binom{a_{1}\delta^{\nu_{1}}_{\mu_{1}}\delta^{\nu_{2}}_{\mu_{2}}+2a_{2}\delta^{\nu_{1}}_{\mu_{1}}p^{\nu_{2}}p_{\mu_{2}}+a_{3}p^{\nu_{1}}p^{\nu_{2}}p_{\mu_{1}}p_{\mu_{2}}}{2a_{4}\delta^{\nu}_{\mu_{1}}p_{\mu_{2}}+a_{5}p_{\mu_{1}}p_{\mu_{2}}p^{\nu}}_{st}\,,$
(35)

where the coefficients $a_{1},\ldots,a_{5}$ are polynomials in $p^{2}$.
Computing the determinant of the $8\times 8$ matrix $(A_{1},A^{\vee}_{1})$
with Wolfram Mathematica, we find

$\det(A_{1},A^{\vee}_{1})=\frac{1}{3}a_{1}^{2}\Delta^{2}_{1}(p^{2})\Delta_{2}(p^{2})\,,$
(36)

where

$\begin{array}[]{l}\Delta_{1}(t)=a_{1}(t-m^{2})+t(a_{2}(t-m^{2})+a_{4})\,,\\\\[8.53581pt]
\Delta_{2}(t)=3a_{1}(t-m^{2})+2t(2a_{2}(t-m^{2})+t(a_{3}(t-m^{2})+a_{5})+2a_{4})\,.\end{array}$
(37)

Each factor in (36) must be a nonzero constant. Evaluating the derivatives of
$\Delta$’s at zero, we get

$\Delta_{1}^{\prime}(0)=a_{4}-m^{2}a_{2}+a_{1}=0\,,\qquad\Delta^{\prime}_{2}(0)=4(a_{4}-m^{2}a_{2})+3a_{1}=0\,.$
(38)

Therefore

$4\Delta_{1}^{\prime}(0)-\Delta_{2}^{\prime}(0)=a_{1}\neq 0$ (39)

and we arrive at a contradiction.

Despite this somewhat disappointing result, the Quillen–Suslin theorem still
guarantees the existence of a (noninvariant) unimodular completion for the
matrix $A_{1}$ above. Applying the package QuillenSuslin [16] in the Computer
Algebra System MAPLE yields the answer for $A_{1}^{\vee}$ that takes about 30
pages and involves polynomials of degree $20$. It should be noted, however,
that a unimodular completion is not unique and the output largely depends on
the particular algorithm implemented in the package. Whether there is a more
compact expression for $A_{1}^{\vee}$ is not yet clear.

######  Remark 3.1.

That equations without explicit Lorentz invariance can nevertheless describe
irreducible representations of relativistic symmetry groups is nothing new.
The self-dual vector field in $4$-dimensional Euclidean space provides the
simplest example. The self-duality equations read

$T_{\mu\nu}:=F_{\mu\nu}-\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}=0\,,$
(40)

where $F_{\mu\nu}=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}$ is the
gauge invariant strength tensor of the vector-potential $\phi_{\mu}(x)$ and
$\varepsilon_{\mu\nu\alpha\beta}$ is the Levi-Civita symbol. All indices are
raised and lowered with the Euclidean metric $\delta_{\mu\nu}$. The field
equations (40) are explicitly invariant under the action of the isometries of
$ISO(4)$ and constitute an anti-self-dual tensor. Anti-self-duality implies
that only $3$ out of $6$ components of the anti-symmetric tensor $T_{\mu\nu}$
are algebraically independent. Unfortunately, there is no $SO(4)$ invariant
way to extract these three independent equations. A possible way to get rid of
some redundant equations is as follows. Let us fix a constant $4$-vector
$\xi^{\mu}$ of unit length, $\xi^{2}=1$. The contraction of $\xi^{\mu}$ with
(40) gives the system of four equations

$T^{\xi}_{\mu}:=\xi^{\nu}T_{\nu\mu}=0\,.$ (41)

Of these equations, only three are linearly independent because of the
identity $\xi^{\mu}T^{\xi}_{\mu}\equiv 0$. Another easily verified identity,

$T_{\mu\nu}=\xi_{\mu}T^{\xi}_{\nu}-\xi_{\nu}T^{\xi}_{\mu}-\frac{1}{2}\varepsilon_{\mu\nu}{}^{\alpha\beta}(\xi_{\alpha}T^{\xi}_{\beta}-\xi_{\beta}T^{\xi}_{\alpha})\,,$
(42)

shows that Eqs. (40) and (41) are completely equivalent to each other and have
the same solutions. In other words, the solution space of equation (41) is
invariant under the action of $ISO(4)$, even though the presence of a given
$4$-vector $\xi$ violates the explicit $SO(4)$ invariance.

We conclude this subsection with a brief comment on the massless limit of
equations (25). Although the value $m=0$ leads to an unsatisfactory physical
theory, it can be used to illustrate some mathematical concepts of the
previous section. In that case, the matrix of the compatibility operator (28),
being of full rank, is not unimodular any more. The reason is simple: it
vanishes at $p=0$. Therefore, the resulting system is not co-flat. The
corresponding symbol matrices $A$ and $A_{1}$ still define a free resolution
of the module $\mathrm{coker}\,A$. By Proposition 2.7 this module has
projective dimension two. Then it follows from Proposition 2.8 that any
Lagrangian for massless equations (25), if it exists, must necessarily involve
gauge fields.

###  3.2 New Lagrangians for free massive fields

Having brought the relativistic wave equations (25) into a pre-Lagrangian
form, one may wonder about their Lagrangian formulation. Notice that the
solution (31, 33) does not give a Lagrangian system of equation, since the
operator of the extended system (29) is not formally self-adjoint. It should
be remembered that the matrix $X$ – a left inverse to $A_{1}$ – is not unique.
Furthermore, any relativistic wave operator is defined up to a unimodular
factor. Given this ambiguity, we start with the most general Lorentz invariant
Ansatz for the Lagrangian:

$\mathcal{L}=\sum_{n=0}^{s}(-1)^{n}(\partial^{n}\phi)\cdot
A_{n}(\partial^{n}\phi)+\sum_{n=0}^{s-1}(-1)^{n}(\partial^{n}\psi)\cdot
B_{n}(\partial^{n}\psi)+\sum_{n=1}^{s}(-1)^{n}(\partial^{n}\phi)\cdot
C_{n}(\partial^{n-1}\psi)\,.$ (43)

Here the dots stand for the full contraction of tensor indices with the help
of the Minkowski metric and the coefficients $A_{n}$, $B_{n}$, and $C_{n}$ are
polynomials in $\square$ to be determined. The corresponding Euler–Lagrange
equations read

$\begin{array}[]{l}\displaystyle
A_{0}\phi_{\mu_{1}\ldots\mu_{s}}+\sum\limits_{n=1}^{s}\Big{(}A_{n}\partial_{\\{\mu_{1}\cdots\mu_{n}}(\partial^{n}\phi)_{\mu_{n+1}\cdots\mu_{s}\\}}+\frac{1}{2}C_{n}\partial_{\\{\mu_{1}\cdots\mu_{n}}(\partial^{n-1}\psi)_{\mu_{n+1}\cdots\mu_{s}\\}}\Big{)}=0\,,\\\\[14.22636pt]
\displaystyle
B_{0}\psi_{\mu_{1}\ldots\mu_{s-1}}+\sum\limits_{n=1}^{s-1}B_{n}\partial_{\\{\mu_{1}\cdots\mu_{n}}(\partial^{n}\psi)_{\mu_{n+1}\ldots\mu_{s-1}\\}}-\sum\limits_{n=1}^{s}\frac{1}{2}C_{n}\partial_{\\{\mu_{1}\cdots\mu_{n-1}}(\partial^{n}\phi)_{\mu_{n}\cdots\mu_{s-1}\\}}=0\,.\end{array}$
(44)

Our strategy is to fix the free coefficients by requiring the field equations

$(\square-m^{2})\phi_{\mu_{1}\cdots\mu_{s}}=0\,,\qquad\partial^{\mu_{1}}\phi_{\mu_{1}\cdots\mu_{s}}=0\,,\qquad\psi_{\mu_{1}\cdots\mu_{s-1}}=0\,.$
(45)

to be among the differential consequences of (44). As explained in Appendix B,
it is enough to consider only successive divergences of equations (44). This
leads to a system of quadratic equations on the coefficients. Below we present
some explicit solutions for lower spins. In all the cases
$A_{0}=\square-m^{2}$ and the remaining coefficients are constants.

####  Spin $2$:

$\begin{array}[]{lll}\displaystyle A_{1}=-\frac{d}{2(d-1)}\,,&\displaystyle
B_{0}=-\frac{(d-1)}{2(d-2)}C_{1}^{2}\,,&\\\\[14.22636pt] \displaystyle
A_{2}=0\,,&\displaystyle B_{1}=\frac{d-1}{2m^{2}d}C_{1}^{2}\,,&C_{2}=0\,.\\\
\end{array}$ (46)

####  Spin $3$:

$\begin{array}[]{lll}\displaystyle
A_{1}=\frac{d^{4}-6d^{2}-2d+4}{2d^{2}+2d}\,,&\displaystyle
B_{0}=-\frac{m^{4}(d+1)}{2d(d-2)^{2}}C_{2}^{2}\,,&\\\\[14.22636pt]
\displaystyle A_{2}=-\frac{(d-2)^{2}(d+2)}{2m^{2}(d+1)}\,,&\displaystyle
B_{1}=\frac{m^{2}(d+1)}{2(d+2)(d-2)^{2}}C_{2}^{2}\,,&\displaystyle
C_{1}=-\frac{m^{2}(d^{2}-2)}{d(d-2)}C_{2}\,,\\\\[14.22636pt] \displaystyle
A_{3}=\frac{(d-2)^{3}(d+2)}{2m^{4}(d+1)^{2}}\,,&\displaystyle
B_{2}=-\frac{1}{2(d+2)(d-2)}C_{2}^{2}\,,&\displaystyle
C_{3}=-\frac{(d-2)d^{2}}{m^{2}d^{2}(d+1)}C_{2}\,,\end{array}$ (47)

or

$\begin{array}[]{lll}\displaystyle
A_{1}=-\frac{11d^{4}+12d^{3}+8d^{2}+24d-64}{12d(d+1)\left(d^{2}+2\right)}\,,&\displaystyle
B_{0}=-\frac{3m^{4}(d+1)(d^{2}+2)}{4d(d-2)^{2}}C_{2}^{2}\,,&\\\\[14.22636pt]
\displaystyle
A_{2}=-\frac{(d-2)^{2}(d+2)}{3m^{2}(d+1)(d^{2}+2)}\,,&\displaystyle
B_{1}=\frac{3m^{2}(d+1)(d^{2}+2)}{4(d-2)^{2}(d+2)}C_{2}^{2}\,,&\displaystyle
C_{1}=\frac{m^{2}(d^{2}+8)}{2(d-2)d}C_{2}\,,\\\\[14.22636pt] \displaystyle
A_{3}=\frac{3(d-2)^{3}(d+2)}{4m^{4}(d+1)^{2}(d^{2}+2)}\,,&\displaystyle
B_{2}=-\frac{3(d^{2}+2)}{4(d-2)(d+2)}C_{2}^{2}\,,&\displaystyle
C_{3}=-\frac{3(d-2)}{2m^{2}(d+1)}C_{2}\,.\end{array}$ (48)

####  Spin $4$:

$\begin{array}[]{ll}\displaystyle
A_{1}=\frac{d^{4}-40d^{2}-104d-32}{4(d^{3}+9d^{2}+22d+12)}\,,&\displaystyle
B_{0}=-\frac{m^{4}(d^{3}+9d^{2}+22d+12)}{4d^{2}(d+2)^{2}}C_{2}^{2}\,,\\\\[14.22636pt]
\displaystyle
A_{2}=-\frac{d^{2}(d^{2}+6d+8)}{m^{2}(d^{3}+9d^{2}+22d+12)}\,,&\displaystyle
B_{1}=\frac{m^{2}(d^{3}+9d^{2}+22d+12)}{4d^{2}(d+2)(d+4)}C_{2}^{2}\,,\\\\[14.22636pt]
A_{3}=A_{4}=0\,,&B_{2}=B_{3}=0\,,\\\\[14.22636pt] \displaystyle
C_{1}=\frac{m^{2}(d^{2}+2d-4)}{2d(d+2)}C_{2}\,,&C_{3}=C_{4}=0\,.\end{array}$
(49)

The coefficients $C_{1}$ in (46) and $C_{2}$ in (47 – 49) are arbitrary
nonzero constants. It should be noted that the solutions above are by no means
unique. For instance, one can change the coefficients by merely rescaling the
fields: $\phi\rightarrow\alpha\phi$, $\psi\rightarrow\beta\psi$. In [17],
O’Brien considered the special case $s=3,d=4$. Requiring that the
Euler–Lagrange equations be of order $4$, he found another solution wherein
some of the coefficients, besides $A_{0}$, are nonconstant polynomials in
$\square$. Numerical experiments using Wolfram Mathematica indicate that
beyond spin $4$ there are no solutions in which all coefficients except
$A_{0}$ are constant.

###  3.3 Comparison with the Fierz–Pauli theory

Let us consider in more detail the case $s=2$ and $d=4$, studied originally by
Fierz and Pauli [1]. The difference between our model and that of Ref. [1] is
in the auxiliary fields: instead of a single scalar field $\chi$ used by Fierz
and Pauli, we introduce the vector field $\psi_{\mu}$. In order to
systematically relate both models, we augment system (25) for $s=2$ with the
divergence of the second equation. This gives the equivalent system of
equations

$T_{\mu\nu}:=(\square-m^{2})\phi_{\mu\nu}=0\,,\qquad
T_{\nu}:=\partial^{\mu}\phi_{\mu\nu}=0\,,\qquad
T:=\partial^{\mu}\partial^{\nu}\phi_{\mu\nu}=0\,.$ (50)

The augmented system enjoys the obvious identities:

$\partial^{\mu}T_{\mu\nu}-(\square-m^{2})T_{\nu}\equiv
0\,,\qquad\partial^{\nu}T_{\nu}-T\equiv 0\,.$ (51)

It is easy to see that the augmented system (50) is co-flat. By Proposition
2.6 it admits a pre-Lagrangian form. The tensor structure of the identities
(51) suggests that the big pre-Lagrangian form of (50) involves one vector
field $\psi_{\nu}$ and one scalar field $\chi$. One can also think of $\chi$
as a generalized Stueckelberg field in the sense of Ref. [20]. We make the
following choice for the extended system:

$\begin{array}[]{l}(\square-m^{2})\phi_{\mu\nu}+A(\partial_{\mu}\psi_{\nu}+\partial_{\nu}\psi_{\mu}-\frac{1}{2}\eta_{\mu\nu}\partial^{\lambda}\psi_{\lambda})\\\\[8.53581pt]
+B(\partial_{\mu}\partial_{\nu}\partial\psi-\frac{1}{4}\eta_{\mu\nu}\square\partial^{\lambda}\psi_{\lambda})+C(\partial_{\mu}\partial_{\nu}\chi-\frac{1}{4}\eta_{\mu\nu}\square\chi)=0\,,\\\\[8.53581pt]
\partial^{\mu}\phi_{\mu\nu}+D\psi_{\nu}+E\partial_{\nu}\partial^{\mu}\psi_{\mu}+F\partial_{\nu}\chi=0\,,\\\\[8.53581pt]
\partial^{\mu}\partial^{\nu}\phi_{\mu\nu}+G\partial^{\mu}\psi_{\mu}+H\chi=0\,.\end{array}$
(52)

Again, the scalar coefficients $A,B,\ldots,H$ are some polynomials in
$\square$ to be determined from the requirement that the extended system be
equivalent to the original one. On substituting the extended equations (52)
into the identities (51), we obtain the system for the auxiliary fields alone:

$\begin{array}[]{l}\big{(}(A-D)\square+m^{2}D\big{)}\psi_{\nu}+\big{(}\frac{1}{2}A+\frac{3}{4}B\square-E(\square-m^{2})\big{)}\partial_{\nu}\partial^{\mu}\psi_{\mu}+\big{(}\frac{3}{4}C\square-F\square+m^{2}F\big{)}\partial_{\nu}\chi=0\,,\\\\[8.53581pt]
(D+E\square-G)\partial^{\mu}\psi_{\mu}+(F\square-H)\chi=0\,.\end{array}$ (53)

Combining the second equation with the divergence of the first gives the pair
of scalar equations

$\begin{array}[]{l}\Big{(}m^{2}D+\big{(}\frac{3}{2}A-D+m^{2}E\big{)}\square+\big{(}\frac{3}{4}B-E\big{)}\square^{2}\Big{)}\partial^{\mu}\psi_{\mu}+\big{(}\frac{3}{4}C\square-F\square+m^{2}F\big{)}\square\chi=0\,,\\\\[8.53581pt]
(D+E\square-G)\partial^{\mu}\psi_{\mu}+(F\square-H)\chi=0\,.\end{array}$ (54)

At this point, we make a simplifying assumption that $E=0$ and
$H=H_{1}+H_{2}\square$ where $H_{1}$, $H_{2}$ and all other coefficients
$A,\ldots,G$ are constant. If we now want the linear equations (54) to have
only a trivial solution, $\partial^{\mu}\psi_{\mu}=\chi=0$, the operator
determinant of the system

$\begin{array}[]{l}\mathrm{Det}=-m^{2}DH_{1}+\left(DH_{1}-m^{2}DH_{2}+m^{2}FG-\frac{3}{2}AH_{1}\right)\square\\\\[8.53581pt]
+\left(\frac{3}{2}AF+\frac{3}{4}CG-\frac{3}{4}CD-
FG-\frac{3}{4}BH_{1}+DH_{2}-\frac{3}{2}AH_{2}\right)\square^{2}+\frac{3}{4}B\left(F-H_{2}\right)\square^{3}\,,\end{array}$
(55)

must be a nonzero constant. Then the vector equation in (53) simplifies to

$\left((A-D)\square+m^{2}D\right)\psi_{\nu}=0$ (56)

It gives the desired equation $\psi_{\nu}=0$ provided $A=D\neq 0$. This leads
to the following system of equations:

$\begin{array}[]{l}A=D\,,\quad F=H_{2}\,,\quad
DH_{1}-m^{2}DH_{2}+m^{2}FG-\frac{3}{2}AH_{1}=0\,,\\\\[8.53581pt]
\frac{3}{2}AF+\frac{3}{4}CG-\frac{3}{4}CD-
FG-\frac{3}{4}BH_{1}+DH_{2}-\frac{3}{2}AH_{2}=0\,.\end{array}$ (57)

We can satisfy this system by setting

$A=D=\frac{m}{2}\,,\quad B=-\frac{1}{3m}(C+1)\,,\quad
F=H_{2}=-\frac{3}{4}\,,\quad G=m\,,\quad H_{1}=-\frac{3m^{2}}{2}\,,$ (58)

with $C$ being an arbitrary constant. Then the field equations (52) take the
form

$\begin{array}[]{l}(\square-m^{2})\phi_{\mu\nu}+\frac{m}{2}(\partial_{\mu}\psi_{\nu}+\partial_{\nu}\psi_{\mu}-\frac{1}{2}\eta_{\mu\nu}\partial^{\lambda}\psi_{\lambda})\\\\[8.53581pt]
-\frac{1}{3m}(C+1)(\partial_{\mu}\partial_{\nu}\partial\psi-\frac{1}{4}\eta_{\mu\nu}\square\partial^{\lambda}\psi_{\lambda})+C(\partial_{\mu}\partial_{\nu}\chi-\frac{1}{4}\eta_{\mu\nu}\square\chi)=0\,,\\\\[8.53581pt]
\partial^{\mu}\phi_{\mu\nu}+\frac{m}{2}\psi_{\nu}-\frac{3}{4}\partial_{\nu}\chi=0\,,\\\\[8.53581pt]
\partial^{\mu}\partial^{\nu}\phi_{\mu\nu}+m\partial^{\mu}\psi_{\mu}-\frac{3}{4}(\square+2m^{2})\chi=0\,.\end{array}$
(59)

Solving the second equation for $\psi_{\nu}$, we get

$\psi_{\nu}=\frac{3}{2m}\partial_{\nu}\chi-\frac{2}{m}\partial^{\mu}\phi_{\mu\nu}\,.$
(60)

On substituting this $\psi_{\nu}$ into the remaining two equations, we arrive
at the PDE system

$\begin{array}[]{l}(\square-m^{2})\phi_{\mu\nu}-\big{(}\partial_{\mu}(\partial\phi)_{\nu}+\partial_{\nu}(\partial\phi)_{\mu}-\frac{1}{2}\eta_{\mu\nu}\partial^{2}\phi\big{)}+\frac{1}{2}\big{(}\partial_{\mu}\partial_{\nu}\chi-\frac{1}{4}\eta_{\mu\nu}\square\chi\big{)}=0\,,\\\\[8.53581pt]
\partial^{\mu}\partial^{\nu}\phi_{\mu\nu}-\frac{3}{4}(\square-2m^{2})\chi=0\,,\end{array}$
(61)

which is identical to the Fierz–Pauli equations for the massive spin-2 field.
As is seen, the free parameter $C$ drops from the final expressions. For other
Lagrangian formulations of the spin-2 field, see [18, 19, 20, 21].

##  4 Conclusion

Let us summarize the main result of our paper in the following thesis: A
system of PDEs with constant coefficients is equivalent to a pre-Lagrangian
system without gauge symmetry if and only if the dual solution module has
projective dimension one. Of course, the absence of a pre-Lagrangian
formulation implies the absence of a Lagrangian one. The relativistic wave
equations (1) exemplify both possibilities depending on whether $m^{2}>0$ or
$m^{2}=0$.

There are several natural directions for further generalization and
application of our results. In Sec. 3, we restricted our consideration to
fully symmetric tensor fields in $d$-dimensional Minkowski space. For $d>4$,
such fields do not exhaust all unitary irreducible representations of the
Poincaré group. Therefore, it is interesting to apply the above analysis to
mixed-symmetry fields. This does not require any modifications to the method.
Another possible direction would be an extension of the above homological
analysis to gauge invariant systems of PDEs. Even if the original PDE system
enjoys no gauge symmetry it may well be equivalent to a gauge invariant
(pre-)Lagrangian system. Specific examples include frame-like formulations of
massive fields [22, 23, 24]. To cover the case of gauge invariant systems one
needs the concept of ‘two-side resolutions’ of wave operators, introduced and
studied in [25, 26]. Unfortunately, such resolutions have not yet received
much attention in the mathematical literature. Here, we can only mention the
paper [27], which deals with two-side resolutions under very restrictive
assumptions, meaning no physical degrees of freedom. Finally, to consider more
general backgrounds than Minkowski space, we need to generalize our
homological analysis to differential operators with non-constant coefficients.
This brings us immediately to the realm of D-modules and their resolutions.
Although the analysis becomes more complex, it is still doable with modern
computer algebra systems.

## Acknowledgements

The work of A. Sh. was partially supported by the São Paulo Research
Foundation (FAPESP), grant 2022/13596-8. D. Sh. acknowledges the financial
support from the Foundation for the Advancement of Theoretical Physics and
Mathematics “BASIS”. The results of Sec. 3.2 were obtained under the exclusive
support of the Tomsk State University Development Program (Priority–2030).

##  Appendix A Pre-Lagrangian form of free massive theories

We use the notation (32) for repeated gradients and divergences of the tensor
$\psi_{\mu_{2}\cdots\mu_{s}}$ of rank $s-1$. The symmetrization of covariant
indices is defined by the formula

$\partial_{(\mu_{1}\cdots\mu_{r}}(\partial^{r-1}\psi)_{\mu_{r+1}\ldots\mu_{s})}:=\frac{1}{r!(s-r)!}\sum\limits_{\sigma\in
S_{n}}\partial_{\sigma(\mu_{1})\cdots\sigma(\mu_{r})}(\partial^{r-1}\psi)_{\sigma(\mu_{r+1})\ldots\sigma(\mu_{s})}$
(62)

and the projection on the traceless part is denoted by braces:

$\partial_{\\{\mu_{1}\cdots\mu_{r}}(\partial^{r-1}\psi)_{\mu_{r+1}\ldots\mu_{s}\\}}:=\partial_{(\mu_{1}\cdots\mu_{r}}(\partial^{r-1}\psi)_{\mu_{r+1}\ldots\mu_{s})}-\frac{1}{2s+d-4}\eta_{(\mu_{1}\mu_{2}}\Theta_{\mu_{3}\ldots\mu_{s})}\,.$
(63)

Here the tensor $\Theta_{\mu_{3}\ldots\mu_{s}}$ is defined by induction on the
number of gradients as

$\Theta_{\mu_{3}\ldots\mu_{s}}=\square\partial_{\\{\mu_{3}\cdots\mu_{r}}(\partial^{r-1}\psi)_{\mu_{r+1}\ldots\mu_{s}\\}}+2\partial_{\\{\mu_{3}\cdots\mu_{r+1}}(\partial^{r}\psi)_{\mu_{r+2}\ldots\mu_{s}\\}}\,.$
(64)

The base of the induction is

$(\partial^{r-1}\psi)_{\\{\mu_{r+1}\ldots\mu_{s}\\}}=(\partial^{r-1}\psi)_{\mu_{r+1}\ldots\mu_{s}}\,.$

The most general Lorentz invariant Ansatz for the extended system of equations
reads

$\begin{array}[]{l}(\square-m^{2})\phi_{\mu_{1}\ldots\mu_{s}}+A_{1}\partial_{\\{\mu_{1}}\psi_{\mu_{2}\ldots\mu_{s}\\}}+\ldots+A_{s}\partial_{\\{\mu_{1}\mu_{2}\cdots\mu_{s}\\}}(\partial^{s-1}\psi)=0\,,\\\\[8.53581pt]
(\partial\phi)_{\mu_{2}\ldots\mu_{s}}+B_{1}\psi_{\mu_{2}\ldots\mu_{s}}+B_{2}\partial_{\\{\mu_{2}}(\partial\psi)_{\mu_{3}\ldots\mu_{s}\\}}+\ldots+B_{s}\partial_{\\{\mu_{2}\ldots\mu_{s}\\}}(\partial^{s-1}\psi)=0\,.\end{array}$
(65)

To study the differential consequences of these equations we use the following
formulas for divergences:

$\begin{array}[]{rcl}\partial^{\mu_{1}}\partial_{\\{\mu_{1}}\psi_{\mu_{2}\ldots\mu_{s}\\}}&=&\displaystyle\square\psi_{\mu_{2}\ldots\mu_{s}}+\frac{2s+d-6}{2s+d-4}\partial_{\
\mu_{2}}(\partial\psi)_{\mu_{3}\ldots\mu_{s}\\}}\,,\\\\[14.22636pt]
\partial^{\mu_{1}}\partial_{\\{\mu_{1}\cdots\mu_{r}}(\partial^{r-1}\psi)_{\mu_{r+1}\ldots\mu_{s}\\}}&=&\displaystyle\frac{2s+d-5}{2s+d-4}\square\partial_{\\{\mu_{2}\cdots\mu_{r}}(\partial^{r-1}\psi)_{\mu_{r+1}\ldots\mu_{s}\\}}\,,\\\\[14.22636pt]
&+&\displaystyle\frac{2s+d-6}{2s+d-4}\partial_{\\{\mu_{2}\cdots\mu_{r+1}}(\partial^{r}\psi)_{\mu_{r+2}\ldots\mu_{s}\\}}\,,\quad
1<r<s-1\,,\\\\[14.22636pt]
\partial^{\mu_{1}}\partial_{\\{\mu_{1}\cdots\mu_{s-1}}(\partial^{s-2}\psi)_{\mu_{s\\}}}&=&\displaystyle\frac{2s+d-5}{2s+d-4}\square\partial_{\\{\mu_{2}\cdots\mu_{s}-1}(\partial^{s-2}\psi)_{\mu_{s}\\}}\\\\[14.22636pt]
&+&\displaystyle\frac{d-2}{2s+d-4}\partial_{\\{\mu_{2}\cdots\mu_{s}\\}}(\partial^{s-1}\psi)\,,\\\\[14.22636pt]
\partial^{\mu_{1}}\partial_{\\{\mu_{1}\cdots\mu_{s}\\}}(\partial^{s-1}\psi)&=&\displaystyle\frac{s+d-3}{2s+d-4}\square\partial_{\\{\mu_{2}}\ldots\partial_{\mu_{s}\\}}(\partial^{s-1}\psi)\,.\end{array}$
(66)

With these relations, Eq. (30) takes the form

$\begin{array}[]{c}\big{(}m^{2}B_{1}+(A_{1}-B_{1})\square\big{)}\psi_{\mu_{2}\ldots\mu_{s}}\\\\[14.22636pt]
\displaystyle+\sum\limits_{r=2}^{s-1}\Big{(}\frac{2s+d-5}{2s+d-4}A_{r}-B_{r}\Big{)}\square\partial_{\\{\mu_{2}\cdots\mu_{r}}(\partial^{r-1}\psi)_{\mu_{r+1}\ldots\mu_{s}\\}}\\\\[14.22636pt]
\displaystyle+\sum\limits_{r=1}^{s-2}\Big{(}\frac{2s+d-6}{2s+d-4}A_{r}+m^{2}B_{r+1}\Big{)}\partial_{\\{\mu_{2}\cdots\mu_{r+1}}(\partial^{r}\psi)_{\mu_{r+2}\ldots\mu_{s}\\}}\\\\[14.22636pt]
\displaystyle+\Big{(}\frac{d-2}{2s+d-4}A_{s-1}+m^{2}B_{s}\Big{)}\partial_{\\{\mu_{2}\cdots\mu_{s}\\}}(\partial^{s-1}\psi)\\\\[14.22636pt]
\displaystyle+\Big{(}\frac{s+d-3}{2s+d-4}A_{s}-B_{s}\Big{)}\square\partial_{\\{\mu_{2}\cdots\mu_{s}\\}}(\partial^{s-1}\psi)=0\,.\end{array}$
(67)

This leads to the following equations on the coefficients:

$\begin{array}[]{ll}m^{2}B_{1}+(A_{1}-B_{1})\square=1\,,&\\\\[8.53581pt]
\displaystyle\frac{2s+d-5}{2s+d-4}A_{r}-B_{r}=0\,,&1<r<s\,,\\\\[8.53581pt]
\displaystyle\frac{2s+d-6}{2s+d-4}A_{r}+m^{2}B_{r+1}=0\,,&r<s-1\,,\\\\[8.53581pt]
\displaystyle\frac{d-2}{2s+d-4}A_{s-1}+m^{2}B_{s}=0\,,&\displaystyle\frac{s+d-3}{2s+d-4}A_{s}-B_{s}=0\,,\end{array}$
(68)

whence

$\frac{2s+d-5}{2s+d-4}A_{r+1}=B_{r+1}=-\frac{2s+d-6}{m^{2}(2s+d-4)}A_{r}\,,\quad
1<r<s\,.$

The last formula gives the recurrence relations for determining the
coefficients. Setting, for example, $A_{1}={1}/{m^{2}}$, we arrive at the
solution (33).

##  Appendix B Derivation of the Lagrangians

Our starting point is the Lagrangian (43) together with the field equations
(44). By making use of formulas (66) we compute the successive divergences of
equations (44). The differential consequences with $s-n$ indices have the
following structure:

$\begin{array}[]{l}A^{(n)}(\partial^{n}\phi)_{\mu_{1}\ldots\mu_{s-n}}+\frac{1}{2}C^{(n)}(\partial^{n-1}\psi)_{\mu_{1}\ldots\mu_{s-n}}+\ldots=0\,,\\\\[8.53581pt]
B^{(n)}(\partial^{n-1}\psi)_{\mu_{1}\ldots\mu_{s-n}}-\frac{1}{2}D^{(n)}(\partial^{n}\phi)_{\mu_{1}\ldots\mu_{s-n}}+\ldots=0\,.\end{array}$
(69)

Here

$\begin{array}[]{ll}\displaystyle
A^{(n)}=\sum_{k=0}^{n}f_{k}(s,n)A_{k}\square^{k}\,,&\qquad\displaystyle
B^{(n)}=\sum_{k=0}^{n-1}f_{k}(s,n-1)B_{k}\square^{k}\,,\\\\[14.22636pt]
\displaystyle
C^{(n)}=\sum_{k=1}^{n}f_{k}(s,n)C_{k}\square^{k}\,,&\qquad\displaystyle
D^{(n)}=\sum_{k=0}^{n-1}f_{k}(s,n-1)C_{k+1}\square^{k}\,,\end{array}$ (70)

and the coefficients $f_{k}(s,n)$ are defined by the relations

$f_{0}(s,n)=1\,,\qquad f_{1}(s,n)=\frac{(2s+d-3-n)n}{2s+d-4}\,,\qquad
f_{s}(s,s)=\prod_{i=0}^{s-2}\frac{s-i+d-3}{2(s-i)+d-4}\,,$
$f_{k}(s,k)=\prod_{i=0}^{k-2}\frac{2(s-i)+d-5}{2(s-i)+d-4}\qquad 1<k<s\,,$
$f_{k}(s,n)=\frac{2s+d-6}{2s+d-4}f_{k}(s-1,n-1)+\frac{2s+d-5}{2s+d-4}f_{k-1}(s-1,n-1)\qquad
1<k<n\leq s,\quad s-k>1\,,$
$f_{k}(s,n)=\frac{d-3}{2s+d-4}f_{k}(s-1,n-1)+\frac{2s+d-5}{2s+d-4}f_{k-1}(s-1,n-1)\qquad
s=n=k+1\,.$

The dots in (69) stand for the terms involving $\partial^{m}\phi$ and
$\partial^{l}\psi$ with $m>n$ and $l>n-1$. These last terms are supposed to
vanish due to equations with less number of free indices. Under this
assumption, (69) becomes a homogeneous linear system of equations on the
unknowns $\partial^{n}\phi$ and $\partial^{n-1}\psi$. In order for this system
to have only the trivial solution, the operator determinant of the system

$\mathrm{Det}_{n}=A^{(n)}B^{(n)}+\frac{1}{4}C^{(n)}D^{(n)}$ (71)

must be a nonzero constant. Evaluating the determinants for all $n=1,\ldots,s$
gives a system of quadratic equations on the coefficients $A_{k}$, $B_{k}$,
and $C_{k}$. Unfortunately, we were unable to solve these equations explicitly
for an arbitrary value of spin. For some lower spins, the equations run as
follows. In all cases $A_{0}=t-m^{2}$ and

$\mathrm{Det}_{1}=\left|\begin{array}[]{cc}(A_{1}+1)t-m^{2}&\frac{1}{2}C_{1}\\\
-\frac{1}{2}C_{1}&B_{0}\end{array}\right|\,,$ (72)

where $t=\square$. The other determinants depend on the value of spin and
dimension.

#### Spin 2:

$\mathrm{Det}_{2}=\begin{vmatrix}\frac{d-1}{d}(A_{2}t^{2}+2A_{1}t)+t-m^{2}&\frac{d-1}{2d}(C_{2}t^{2}+2C_{1}t)\\\\[8.53581pt]
-\frac{1}{2}(C_{1}+C_{2}t)&B_{0}+B_{1}t\end{vmatrix}\,.$ (73)

#### Spin 3:

$\begin{array}[]{l}\mathrm{Det}_{2}=\begin{vmatrix}\frac{d+1}{d+2}(A_{2}t^{2}+2A_{1}t)-m^{2}+t&\frac{d+1}{2(d+2)}(C_{2}t^{2}+2C_{1}t)\\\\[8.53581pt]
-\frac{1}{2}(C_{1}+C_{2}t)&B_{0}+B_{1}t\end{vmatrix}\,,\\\\[19.91692pt]
\mathrm{Det}_{3}=\begin{vmatrix}\frac{(d-1)A_{3}t^{3}+3(d-1)A_{2}t^{2}+3dA_{1}t}{d+2}+t-m^{2}&\frac{(d-1)C_{3}t^{3}+3(d-1)C_{2}t^{2}+3dC_{1}t}{2(d+2)}\\\\[8.53581pt]
-\frac{d-1}{2d}(C_{3}t^{2}+2C_{2}t)-\frac{1}{2}C_{1}&\frac{d-1}{d}(B_{2}t^{2}+2B_{1}t)+B_{0}\end{vmatrix}\,.\end{array}$
(74)

#### Spin 4:

$\begin{array}[]{l}\mathrm{Det}_{2}=\begin{vmatrix}\frac{d+3}{d+4}(A_{2}t^{2}+2A_{1}t)-m^{2}+t&\frac{d+3}{2(d+4)}(C_{2}t^{2}+2C_{1}t)\\\\[8.53581pt]
-\frac{1}{2}(C_{1}+C_{2}t)&B_{0}+B_{1}t\end{vmatrix}\\\\[19.91692pt]
\mathrm{Det}_{3}=\begin{vmatrix}\frac{\alpha_{3}A_{3}t^{3}+\alpha_{2}A_{2}t^{2}+\alpha_{1}A_{1}t}{(d+2)(d+4)}+t-m^{2}&\frac{\alpha_{3}C_{3}t^{3}+\alpha_{2}C_{2}t^{2}+\alpha_{1}C_{1}t}{2(d+2)(d+4)}\\\\[8.53581pt]
-\frac{1}{2}(\frac{d+1}{d+2}(C_{3}t^{2}+2C_{2}t)+C_{1})&\frac{d+1}{d+2}(B_{2}t^{2}+2B_{1}t)+B_{0}\end{vmatrix}\,,\\\\[19.91692pt]
\mathrm{Det}_{4}=\begin{vmatrix}\frac{\beta_{4}A_{4}t^{4}+\beta_{3}A_{3}t^{3}+\beta_{2}A_{2}t^{2}+\beta_{1}A_{1}t}{(d+2)(d+4)}+t-m^{2}&\frac{\beta_{4}C_{4}t^{4}+\beta_{3}C_{3}t^{3}+\beta_{2}C_{2}t^{2}+\beta_{1}C_{1}t}{(d+2)(d+4)}\\\\[8.53581pt]
\frac{(d-1)B_{3}t^{3}+3(d-1)B_{2}t^{2}+3dB_{1}t}{d+2}+B_{0}&-\frac{1}{2}(\frac{(d-1)C_{4}t^{3}+3(d-1)C_{3}t^{2}+3dC_{2}t}{d+2}+C_{1})\end{vmatrix}\,,\end{array}$
(75)

where

$\begin{array}[]{l}\alpha_{3}=d^{2}+4d+3\,,\quad\alpha_{2}=3d^{2}+11d+8\,,\quad\alpha_{1}=3d^{2}+12d+12\,,\\\\[8.53581pt]
\beta_{4}=d^{2}-1\,,\quad\beta_{3}=4d^{2}+3d-7\,,\quad\beta_{2}=6d^{2}+12d-6\,,\quad\beta_{1}=4d^{2}+12d+8\,.\end{array}$
(76)

In the equations above, all $\mathrm{Det}$’s are required to be nonzero
constants. One can look for the coefficients $A_{k}$, $B_{k}$, and $C_{k}$
that are real numbers. This leads to overdetermined systems of quadratic
equations. Applying Wolfram Mathematica yields the explicit solutions (46–49).

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