††thanks: Dedicated to Professor T. Dereli on the occasion of his 60th
birthday

# First-order symmetries of Dirac equation in curved background: a unified
dynamical symmetry condition

Ö. Açık 1<EMAIL_ADDRESS>Ü. Ertem 1<EMAIL_ADDRESS>M. Önder 2<EMAIL_ADDRESS>A. Verçin 1<EMAIL_ADDRESS>1
Department of Physics, Ankara University, Faculty of Sciences, 06100,
Tandoğan-Ankara, Turkey
2 Department of Physics Engineering, Hacettepe University, 06800, Beytepe-
Ankara, Turkey.

###### Abstract

It has been shown that, for all dimensions and signatures, the most general
first-order linear symmetry operators for the Dirac equation including
interaction with Maxwell field in curved background are given in terms of
Killing-Yano (KY) forms. As a general gauge invariant condition it is found
that among all KY-forms of the underlying (pseudo) Riemannian manifold, only
those which Clifford commute with the Maxwell field take part in the symmetry
operator. It is also proved that associated with each KY-form taking part in
the symmetry operator, one can define a quadratic function of velocities which
is a geodesic invariant as well as a constant of motion for the classical
trajectory. Some geometrical and physical implications of the existence of KY-
forms are also elucidated.

###### pacs:

04.20.-q, 02.40.-k

##  I INTRODUCTION

In many evolutions taking place in a flat or curved background, isometries of
the underlying space-time metric lead to conservation laws that also have
clear geometrical meanings expressed by means of their local generators,
Killing vector fields. As space-time transformations, flows of these fields
specify the conserved quantities as their flow invariants. However, since the
beginning of the 1970s, it has been recognized that many interesting
properties of a given space-time are intimately related to hidden symmetries
of its metric, which make themselves manifest in higher rank tensorial objects
that also provide additional conservation laws Penrose ; Walker ; Hughston .
The building blocks of the mathematical structure behind these hidden
symmetries and associated conserved quantities are the symmetric Killing
tensors, KY-forms, conformal Killing tensors and conformal KY (CKY)-forms.
Their defining relations are natural generalizations of those of Killing
vector fields and of conformal Killing vector fields. Although these higher
rank objects can be completely determined by the metric itself (see for
instance ozumav ; ozumav1 ), they are not directly related to isometries or
conformal transformations. But they comprise, by definition, their generators
in the rank one sector of their hierarchy. In most of the cases these
additional conserved quantities lead to complete integrability of the
considered problem. This important step was initiated with the works of
Penrose and his collaborators Penrose ; Walker . They have shown that it is
the existence of a second-rank symmetric Killing tensor which can be written
as the “square” of a KY 2-form that leads to Carter’s fourth constant of
motion, which is responsible for the complete integrability of the geodesic
problem in the Kerr geometry Carter . For other research fields utilizing KY
and CKY-forms and for earlier references for these forms we shall refer to
Benn-Kress ; Collinson ; Frulov1 ; Frulov2  and the references therein.

An important place where some or all of these tensorial objects enter the
analysis is the study of symmetry operators for the Dirac-type equations
describing the motion in curved background with or without additional
interactions. By now it has become a well-established fact that while CKY-
forms take part in symmetry operators, via the $R$-commuting argument for the
massless Dirac equation, KY-forms are indispensable in constructing first
order symmetries of the massless as well as massive Dirac equation in a curved
space-time. It is a four-dimensional Lorentzian space-time where most of
applications have taken place. The first seminal studies in this context were
carried out by Carter, McLenaghan and Kamran Carter1 ; Carter2 ; Kamran . The
results of these earlier studies, obtained in four dimensions for the massive
or massless Dirac equation in the absence of electromagnetic interactions,
were recently extended by Benn and his collaborators, to an arbitrary
dimension and signature Benn-Charlton ; Benn-Kress1 ; Benn-Kress2 .

In the absence of additional interactions, all KY or CKY-forms of the space-
time take part in the symmetry operator without any extra restriction.
However, when interactions are included, some additional conditions arise
which restrict the possible forms that can enter into the symmetry operator.
To the best of our knowledge, these restrictions for a four dimensional curved
background were found for the first time by McLenaghan and Spindel McLenaghan
. In searching for the most general symmetry operator commuting with the Dirac
equation in the presence of an electromagnetic field, they found that the
symmetry operator can be constructed from KY-forms of the underlying
background, provided that they separately fulfill some conditions involving
the field itself. These included some conditions found before by Carter and
McLenaghan Carter2 . Some of these conditions were also obtained before by
Hughston et al in a slightly different context: in searching quadratic first
integrals for the charged particle orbits in the charged Kerr background
Hughston .

In this study we first show that the main results of McLenaghan and Spindel,
that is the construction of the symmetry operators out of KY-forms, can be
extended to an arbitrary dimension and signature. We then obtain a unified
condition which allows one to specify which KY-forms can take part in the
symmetry operator, and hence define hidden dynamical symmetry of the problem.
This is an algebraic condition and can be stated as follows; a KY-form of the
curved background enter the symmetry operator if and only if it Clifford
commutes with the force (Maxwell) field. In particular, we solve all the
consistency conditions and find a concise way to choose the gauge in order to
make 0-form component of the symmetry operator constant. Owing to a non-
integrated consistency condition, this point has remained ambiguous in the
literature.

Our results include the results of Benn and his collaborators when the
electromagnetic field is turned off. Finally, we prove that the quadratic
functions of velocities defined in terms of each KY $p$-form entering the
symmetry operator is not only a geodesic invariant but also, for an arbitrary
number of dimension and signature, a constant of motion for the classical
trajectory.

We shall mainly use the notation of Benn-Tucker  and adopt the following
conventions and terminology. The underlying base manifold is supposed to be an
$n-$dimensional pseudo-Riemannian manifold with arbitrary signature. Covariant
derivative of spinors, that is of sections of a bundle carrying an irreducible
representation of the (real or complexified) Clifford algebra, with respect to
the vector field $X$ is denoted by $S_{X}$. Then the Dirac operator on spinors
is $\displaystyle{\not}S=e^{a}S_{X_{a}}$, where the local co-frame
$\\{e^{a}\\}$ is the dual to the tangent frame $\\{X_{a}\\}$. Summation
convention over repeated indices will be used throughout the paper.
Juxtaposing $e^{a}$ and $S_{X_{a}}$, or any other operator, or form will
denote the Clifford multiplication. When acting on forms $S_{X}$ and
$\displaystyle{\not}S$ will be denoted, in terms of the pseudo-Riemannian
connection $\nabla$, as $\nabla_{X}$ and

$\displaystyle\displaystyle{\not}d=e^{a}\nabla_{X_{a}}=d-\delta\;.$

Here $d=e^{a}\wedge\nabla_{X_{a}}$ and $\delta=-i_{X_{a}}\nabla_{X^{a}}$
denote the exterior derivative and the co-derivative written in terms of
covariant derivative operator $\nabla_{X}$. $\wedge$ is the exterior product
and $i_{X}$ will represent the interior derivative with respect to $X$ whose
action on an arbitrary $p$-form $\alpha$ is defined, for all vector fields
$Y_{j}$, by

$\displaystyle(i_{X}\alpha)(Y_{1},\dots,Y_{p-1})=p\alpha(X,Y_{1},\dots,Y_{p-1})\;.$

For dual basis elements we have $i_{X_{b}}e^{a}=e^{a}(X_{b})=\delta^{a}_{b}$.

The rest of the paper is organized as follows. In section II, the general form
of the first order symmetry operator of the Dirac equation with a potential
term is specified. This is achieved by constructing and then by solving all
consistency equations except the equation for the $0$-form component of the
non-derivative term of the symmetry operator. The unified dynamical symmetry
condition announced in the title is established in section III by analyzing
higher degree components of the consistency equations. Special cases of this
condition and the integration of the remaining $0$-form component are also
presented there. In section IV, the correspondence between KY and closed CKY-
forms are studied and Yano vectors are introduced. Implications of the
existence of Yano vectors related to symmetry analysis and to the global
structure of the underlying space-time are given in the same section. In
section V, the first integrals of geodesic equations, constants of motion of
classical trajectories and their connection with the KY-forms and the
mentioned dynamical symmetry condition are considered. Derivation of
consistency equations and the contraction of curvature $2$-forms with a Yano
vector are given in two appendixes. In the Appendix B determination of upper
bounds for the numbers of linearly independent KY-forms is also provided.
Section VI concludes the paper.

##  II FIRST ORDER SYMMETRY OPERATORS OF THE DIRAC EQUATION

We set out to our analysis by considering the Dirac equation

$\displaystyle(\displaystyle{\not}S+{\rm{i}}A)\psi=m\psi$ (1)

for a complex spinor field $\psi$ and propose the following first order linear
symmetry operator

$\displaystyle L=2\omega^{a}S_{X_{a}}+\Omega\;,$ (2)

such that $L$ Clifford commutes with $\displaystyle{\not}S+{\rm{i}}A$.
Equation (1) describes the motion of a massive, charged and spin $1/2$
particle, with unit charge and mass $m$, interacting with the curved
background encoded in the Dirac operator $\displaystyle{\not}S$ and force
field represented by the potential form $A$. By fixing the charge from the
outset, we assume the effective coupling to the spinor field. In the beginning
$A$ is allowed to be an arbitrary inhomogeneous form, and the consistency
conditions are derived in this general context. Later on, $A$ will be taken to
be a 1-form. In that case, the term involving $A$ in equation (1) describes
coupling to the Maxwell field $F=dA$. $A$ may also involve potential terms of
non-electromagnetic origin such as those of conservative forces. In the latter
case, $F$ will be referred to as a force field. In equation (2) $\omega^{a}$
and $\Omega$ are $\mathbb{Z}_{2}$-homogeneous, both even or both odd forms
which act, like $A$, on spinor fields by the Clifford multiplication.

In recent studies Benn-Charlton ; Benn-Kress1 ; Benn-Kress2  it has been
proved that use of the graded Clifford commutator $[,]$ considerably eases the
calculations in analyzing the symmetries of the Dirac-type operators. For a
$p$-form $\alpha$ and an inhomogeneous Clifford form
$\beta=\sum_{q}\beta_{(q)}$, this commutator is defined as

$\displaystyle[\alpha,\beta]=\alpha\beta-\sum_{q}(-1)^{pq}\beta_{(q)}\alpha\;.$

If $\alpha$ is a one-form this transforms, in terms of the main involution
$\eta$ (which leaves the even forms invariant and changes the sign of the odd
forms) of the Clifford algebra, to
$[\alpha,\beta]=\alpha\beta-\beta^{\eta}\alpha$. Let us suppose that

$\displaystyle[\displaystyle{\not}S,L]+{\rm{i}}[A,L]=0\;,$ (3)

is satisfied and let us call $L$ even (odd) when $\omega^{a}$ and $\Omega$ are
both even (both odd). When $L$ is even its graded commutator with the Dirac
equation becomes the usual Clifford commutator $[,]_{-}$ and irrespective of
$n,\;L$ is a symmetry operator. That is, it Clifford commutes with the Dirac
equation and maps a solution to another. When $L$ is odd it anti-commutes with
Dirac equation and fails to be a symmetry operator. However, in such a case if
$n$ is even $Lz$ is a symmetry operator since the volume form $z$ anti-
commutes with odd forms. (In such a case $A$ must be an odd form.)

###  II.1 The Main Consistency Equations

To determine $\omega^{a}$’s and $\Omega$ in equation (2), we equate the
symmetrized coefficients of the covariant derivatives of each order to zero in
equation (3). The second order derivatives come only from the first bracket of
(3) such that

$\displaystyle[\displaystyle{\not}S,L]=2i_{X_{b}}\omega^{a}[S^{2}(X_{a},X^{b})+S^{2}(X^{b},X_{a})]+\cdots\;,$
(4)

where $i_{X}$ is the interior derivative,

$\displaystyle S^{2}(X,Y)=S_{X}S_{Y}-S_{\nabla_{X}Y}\;,$

denotes the second covariant derivative and the ellipsis stands for lower
order terms. Equating the coefficients of the second order derivatives to zero
in equation (3) yields

$\displaystyle i_{X_{b}}\omega^{a}+i_{X^{a}}\omega_{b}=0\;,$ (5)

for all $a,b=1,2,\dots,n$. This is satisfied if and only if
$\omega^{a}=i_{X^{a}}\omega$ where $\omega$ is possibly a
$\mathbb{Z}$-inhomogeneous form. It has been shown in Appendix A that, in view
of (5), by equating the coefficients of equal power of derivatives in equation
(3) we obtain

$\displaystyle\nabla_{X^{a}}\omega$ $\displaystyle=$ $\displaystyle
i_{X^{a}}\varphi-{\rm{i}}[A,i_{X^{a}}\omega]\;,$ (6)
$\displaystyle\nabla^{2}(X_{a},X^{a})\omega$ $\displaystyle=$
$\displaystyle\displaystyle{\not}d\varphi+2{\rm{i}}(i_{X^{a}}\omega)^{\eta}\nabla_{X_{a}}A-{\rm{i}}[A,\Omega]\;,$
(7)

where $\varphi=\displaystyle{\not}d\omega-\Omega$. These constitute two main
sets of the consistency conditions that specify the form of possible symmetry
operators of the Dirac equation, for all dimensions and signatures, in which
$A$ is a general form. Henceforth, we take $A$ to be a 1-form and write, for
an arbitrary form $\alpha$

$\displaystyle[A,\alpha]=2i_{\tilde{A}}\alpha\;.$

Here $\tilde{A}$ is the metric dual of $A$, which in terms of the metric $g$
of the background is defined by $A(X)=g(\tilde{A},X)$ for all $X$.

###  II.2 Solutions: Emergence of KY-Forms

Let us first concentrate on equation (6), which for 1-form $A$ can be
rewritten as

$\displaystyle\nabla_{X^{a}}\omega=i_{X^{a}}(\varphi+2{\rm{i}}i_{\tilde{A}}\omega)\;.$
(8)

Applying $i_{X_{a}}$ to both sides of this equation, we firstly see that
$\delta\omega=0$, that is, $\omega$ must be co-closed. On the other hand, by
applying $e_{a}\wedge$ we obtain

$\displaystyle d\omega=\pi(\varphi+2{\rm{i}}i_{\tilde{A}}\omega)\;,$ (9)

where the linear map $\pi$ scales each form component by its degree :
$\pi(\alpha)=e^{a}\wedge i_{X_{a}}\alpha$. The $(p+1)$-form component of
equation (9) reads, for $p=0,1,\dots,n-2$, as

$\displaystyle\varphi_{(p+1)}=\frac{1}{p+1}d\omega_{(p)}-2{\rm{i}}i_{\tilde{A}}\omega_{(p+2)}\;,$
(10)

and for $p=n-1$, as $\varphi_{(n)}=n^{-1}d\omega_{(n-1)}$. In view of the last
two equations, equation (8) implies that each $p$-form component of $\omega$
must obey

$\displaystyle\nabla_{X_{a}}\omega_{(p)}=\frac{1}{p+1}i_{X_{a}}d\omega_{(p)}\;,$
(11)

which is the well-known KY-equation.

From equation (10) and by the fact that $\omega$ is co-closed we also obtain

$\displaystyle\Omega_{(p+1)}$ $\displaystyle=$
$\displaystyle\frac{p}{p+1}d\omega_{(p)}+2{\rm{i}}i_{\tilde{A}}\omega_{(p+2)}\;,$
(12) $\displaystyle\Omega_{(n)}$ $\displaystyle=$
$\displaystyle(1-\frac{1}{n})d\omega_{(n-1)}\;,$ (13)

for $p=0,1,\dots,n-2$. Note that for $p=0,1,\dots,n-1$ we have

$\displaystyle i_{\tilde{A}}\Omega_{(p+1)}=p\nabla_{\tilde{A}}\omega_{(p)}\;.$
(14)

These provide the non-derivative term, except for its 0-form component, of the
symmetry operator in terms of KY-forms and the potential.

We now turn to the implications of equation (7). Differentiating equation (8)
once more, we obtain

$\displaystyle\nabla^{2}(X_{a},X^{a})\omega=-\delta(\varphi+2{\rm{i}}i_{\tilde{A}}\omega)\;,$
(15)

and by combining this with equation (7), we arrive at

$\displaystyle
i_{\tilde{A}}\Omega-\frac{{\rm{i}}}{2}d\Omega=\delta(i_{\tilde{A}}\omega)+(i_{X^{a}}\omega)^{\eta}\nabla_{X_{a}}A\;.$
(16)

This can be used to obtain the 0-form component of $\Omega$ and possible
relations among its higher degree components. For this purpose we apply the
general relation

$\displaystyle[\delta,i_{X}]_{+}=-i_{X^{a}}i_{\nabla_{X_{a}}X}\;,$

to KY-forms and obtain

$\displaystyle\delta(i_{\tilde{A}}\omega)$ $\displaystyle=$ $\displaystyle-
i_{X^{a}}i_{\nabla_{X_{a}}\tilde{A}}\omega\;.$ (17)

We now make use of

$\displaystyle(i_{X^{a}}\omega)^{\eta}\kappa_{a}=\kappa_{a}\wedge
i_{X^{a}}\omega-i_{\tilde{\kappa}_{a}}i_{X^{a}}\omega,$ (18)

where $\kappa_{a}$’s are 1-forms. This relation is a direct result of the
standard Clifford multiplication rule for a right multiplication of an
arbitrary form by a 1-form. Using (18), by taking
$\kappa_{a}=\nabla_{X_{a}}A$, and (17) in (16) we get

$\displaystyle
i_{\tilde{A}}\Omega-\frac{{\rm{i}}}{2}d\Omega=(\nabla_{X_{a}}A)\wedge
i_{X^{a}}\omega\;.$ (19)

The $1$-form component of this relation reads as

$\displaystyle d\Omega_{(0)}$ $\displaystyle=$
$\displaystyle-2{\rm{i}}(i_{\tilde{A}}\Omega_{(2)}-i_{X^{a}}\omega_{(1)}\nabla_{X_{a}}A)\;,$
(20) $\displaystyle=$
$\displaystyle-2{\rm{i}}(\nabla_{\tilde{A}}\omega_{(1)}-\nabla_{\tilde{\omega}_{(1)}}A)\;.$

In obtaining the second equality we made use of (14). The equation (20) will
be integrated in the next section after the higher degree components of
equation (19) are further analyzed to uncover the symmetry condition stated in
the title.

##  III A Unified Dynamical Symmetry Condition

In this section we first show that equation (16) contains an important
algebraic condition which plays a prominent role in deciding which KY-forms
can take part in the symmetry operator. To see this, we first recall that
$\Omega=d\omega-\varphi$ which implies

$\displaystyle d\Omega$ $\displaystyle=$
$\displaystyle-d\varphi=2{\rm{i}}di_{\tilde{A}}\omega\;,$ $\displaystyle
i_{\tilde{A}}\Omega$ $\displaystyle=$ $\displaystyle i_{\tilde{A}}d\omega-
i_{\tilde{A}}\varphi=\pi i_{\tilde{A}}\varphi\;.$

In the last equality we made use of (9). On substituting these relations into
(16), we get

$\displaystyle\pi(i_{\tilde{A}}\varphi)=-\displaystyle{\not}di_{\tilde{A}}\omega+(i_{X^{a}}\omega)^{\eta}\nabla_{X_{a}}A\;.$
(21)

We now make use of

$\displaystyle[\displaystyle{\not}d,i_{X}]_{+}=\nabla_{X}+e^{a}i_{\nabla_{X_{a}}X}\;,$
(22)

where $[,]_{+}$ denotes the Clifford anti-commutator. Using (18), once again
by taking $\kappa_{a}=\nabla_{X_{a}}A$, and (22) in the equation (21) we
arrive at

$\displaystyle\pi(i_{\tilde{A}}\varphi)-i_{\tilde{A}}\displaystyle{\not}d\omega+\nabla_{\tilde{A}}\omega=-\frac{1}{2}[dA,\omega]_{-}\;,$
(23)

where we have also used the following relation:

$\displaystyle(\kappa_{a}\wedge i_{X^{a}}-e^{a}\wedge
i_{\tilde{\kappa}_{a}})\omega=-\frac{1}{2}[dA,\omega]_{-}\;.$ (24)

It is now easy to verify that, by the KY-equation (11) and by the equation
(10), the left hand side of (23) vanishes and we obtain an important condition
$[dA,\omega]_{-}=0$. On the other hand, as is apparent in equation (24), since
the Clifford commutator of an arbitrary form by a 2-form does not change the
degrees of its components, each p-form component of $\omega$ must Clifford
commute with the force field $F=dA$:

$\displaystyle[F,\omega_{(p)}]_{-}=0\;.$ (25)

This gauge invariant condition means that among the KY-forms admitted by the
underlying space-time, only those which satisfy the above condition take part
in the symmetry operators. Moreover, it must be emphasized that it is the
force-field, not the potential form itself, that plays a selective role in
this regard.

At this point we should note that, in view of equations (10), (14) and
equation (21) derived above, the higher degree components of equation (19)
lead us again to condition (25) and yield nothing new.

In order to integrate equation (20) as well as to obtain more practical
statements resulting from the equation (25), we rewrite it as

$\displaystyle i_{X^{a}}F\wedge i_{X_{a}}\omega_{(p)}=0\;.$ (26)

This is obviously satisfied for $p=0$ and also for $p=n$. Hence the extreme
cases $p=0,n$ do not impose any condition on $F$ and on possible KY-forms that
take part in the symmetry operator $L$. In general, a KY $p$-form
$\omega_{(p)}$ takes part in the symmetry operator if and only if $F$ is in
the kernel of the operator $i_{X_{a}}\omega_{(p)}\wedge i_{X^{a}}$. More
practical refinements of this condition are attained for intermediate values
of $p$. We first should note that the 0-form component $\omega_{(0)}$ of a KY-
form can be any function and $\omega_{(1)}$ is the dual of a Killing vector
field. Moreover, $\omega_{(n)}$ is a constant (parallel), that is, it is a
constant multiple of the volume form: $\omega_{(n)}=kz$. For $p=1$ we can
write $\omega_{(1)}=\tilde{K}$, where $K$ is a Killing vector field. In this
case, equation (26) reduces to

$\displaystyle i_{K}F=0\;,$ (27)

which implies that $F$ remains invariant under the flow generated by $K$. That
is, ${\cal L}_{K}F=0$, where ${\cal L}_{K}$ denotes the Lie derivative in the
direction of $K$. In other words, even the generator of the isometries will
take part in the symmetry operators if they fulfill condition (27).

In terms of the potential $1$-form $A$ the condition (27) reads as
$i_{K}(e^{a}\wedge\nabla_{X_{a}}A)=0$ which, in view of
$i_{\nabla_{X}K}A=i_{\tilde{A}}\nabla_{X}\omega_{(1)}$, can be evaluated to
obtain

$\displaystyle 0=\nabla_{K}A-di_{K}A-\nabla_{\tilde{A}}\omega_{(1)}.$

By comparing this relation with equation (20) and then by integrating we
obtain

$\displaystyle\Omega_{0}=2{\rm{i}}i_{K}A\;,$ (28)

up to a constant. This relation, which was not recognized in the literature
before, provides a concise way of choosing the gauge in order to make
$\Omega_{0}$ constant. In such a case, only duals of Killing vector fields
whose flows preserve the potential can appear in the symmetry operator. An
equivalent but more instructive way of deriving (28) goes as follows.
Recalling the fact that for a 1-form $\beta$ the metric dual of
$\nabla_{X}\beta$ is $\nabla_{X}\tilde{\beta}$, we can rewrite condition (20)
as

$\displaystyle\widetilde{d\Omega}_{(0)}=-2{\rm{i}}[\tilde{A},K]_{L}=2{\rm{i}}{\cal
L}_{K}\tilde{A}\;.$

Here $[,]_{L}$ denotes the usual Lie bracket of vector fields, and we have
also used the zero torsion condition. Since $K$ is a Killing field, the above
relation simply reads as $d\Omega_{(0)}=2{\rm{i}}{\cal L}_{K}A$ and then the
usual action of Lie derivative on differential forms produces the desired
relation in view of (27).

For a KY 2-form $\omega_{(2)}=2^{-1}\omega_{ab}e^{a}\wedge e^{b}$ the
condition (26) reads as $\omega_{a[b}F_{c]}^{\;a}=0$ for all values of $b$ and
$c$ where the square bracket stands for the anti-symmetrization of the
enclosed indices and $F_{cb}$ are the components of $F$ in the same
$\\{e^{a}\\}$ basis. This condition is the tensor-language version Carter2  of
the condition first obtained by Hughston et al Hughston . Our condition (25),
or equivalently (26), are natural generalizations of this and similar
conditions. This reflects the efficiency of the Clifford calculus in this
context.

To discuss condition (26) for higher forms it is convenient to rewrite it, in
terms of $(p+1)$-forms defined by

$\displaystyle\beta_{a}=F\wedge i_{X_{a}}\omega_{(p)}\;,$

as $i_{X^{a}}\beta_{a}=0$. This is also satisfied for $p=0,n$. In the latter
case $\beta_{a}=0$ are identically satisfied, since all $\beta_{a}$’s are
$(n+1)$-forms. For $p=n-1$ we have $\beta_{a}=f_{a}z$ for some set of
functions $f_{a}$ such that equation (26) amounts to $i_{V}z=0$ for
$V=f_{a}X^{a}$. As the map $\varphi_{z}$ defined by $\varphi_{z}(V)=i_{V}z$
between the vector fields and $(n-1)$-forms is an isomorphism, $i_{V}z$
vanishes if and only if $V=0$, that is if and only if all $f_{a}$ vanish. This
is equivalent to $F\wedge i_{X_{a}}\omega_{(n-1)}=0$ for all $X_{a}$, or to

$\displaystyle i_{X_{a}}F\wedge\omega_{(n-1)}=0\;.$ (29)

In the case of $p=n-2,\;\beta_{a}$’s are $(n-1)$-forms and each one can be
written, in terms of uniquely determined $1$-form
$\sigma_{a}=\sigma_{ab}e^{b}$, as $\beta_{a}=^{\ast}\sigma_{a}$. In such a
case equation (26) amounts to $\sigma_{a}\wedge\tilde{X}^{a}=0$, which implies
that $\sigma_{ab}=\sigma_{ba}$. This can succinctly be expressed as

$\displaystyle F\wedge(\tilde{X}_{a}\wedge i_{X_{b}}-\tilde{X}_{b}\wedge
i_{X_{a}})\omega_{(n-2)}=0\;,$

for $a,b=1,\dots,n$. Note that the considered cases $p=0,1,n-2,n-1,n$ exhaust
all possible forms of equation (25) in four dimensions. More appealing
versions of some of these conditions will appear in the next section (see
equation (34) and the remarks followed).

Finally in this section we should note that, when $A$ is zero we recover
exactly the results of Benn and Kress for all dimensions and signatures,
obtained in Benn-Charlton ; Benn-Kress1 ; Benn-Kress2 . In such a case, since
the condition (25) disappears, all KY-forms of the underlying space-time take
part in the symmetry operators and all of the non-derivative terms explicitly
given by equations (12) and (13), except $\Omega_{0}$ which is constant, are
exact.

##  IV CORRESPONDENCE BETWEEN KY-FORMS AND CLOSED CKY-FORMS

The appearance of higher rank KY-forms in the symmetry operator indicates the
presence of dynamical symmetries which are not isometries. We shall now
establish a general one-to-one correspondence between the KY-forms and closed
CKY-forms and, as a particular case, we shall show that for each KY
$(n-1)$-form there exists a uniquely determined conformal transformation. This
transformation is generated by a locally gradient field whose integral curves
are pre-geodesics. In this way we shall see that the higher KY-forms are
related to special CKY-forms and, in the case of $(n-1)$-forms, to the special
conformal transformations rather than isometry transformations.

For the two points mentioned above, let us consider the CKY-equation

$\displaystyle\nabla_{X}\rho_{(p)}=\frac{1}{p+1}i_{X}d\rho_{(p)}-\frac{1}{n-p+1}\tilde{X}\wedge\delta\rho_{(p)}\;,$
(30)

which has the well-established Hodge duality invariance and conformal
covariance. A p-form $\rho_{(p)}$ is called CKY p-form if and only if it
satisfies equation (30) for all vector fields $X$. It is obvious that a
$p$-form is a KY $p$-form if and only if it is a co-closed CKY $p$-form. A
less obvious fact is that a $(n-p)$-form is a KY $(n-p)$-form if and only if
it is the Hodge dual of a closed CKY $p$-form. Indeed, the Hodge dual of (30)
for closed CKY forms yields

$\displaystyle\nabla_{X_{a}}\;^{\ast}\rho_{(p)}=\frac{1}{n-p+1}i_{{X}_{a}}d^{\ast}\rho_{(p)}\;,$

where we have made use of the relation
$i_{X}\;^{\star}\phi=^{\star}(\phi\wedge\tilde{X})$ which can be considered as
the definition of the Hodge map. Since $\rho_{(p)}$ is closed it is locally
exact such that $\rho_{(p)}=d\alpha$, where the $(p-1)$-form $\alpha$ may be
termed as the KY-potential for the KY $(n-p)$-form
$\omega_{(n-p)}=^{\ast}d\alpha$. Note that $0$-forms of KY and CKY coincide
and that any CKY $n$-form is of the form $\rho_{(n)}=fz$, where $f$ can be any
differentiable function. As a result, the Hodge map establishes a vector space
isomorphism between the vector space of all KY $p$-forms and that of all
closed CKY $(n-p)$-forms for all values of $p$.

###  IV.1 KY $(n-1)$-forms and Yano Vectors

In the case of KY $(n-1)$-forms we can write

$\displaystyle\omega_{(n-1)}=i_{Y}z=^{\ast}\tilde{Y}\;,$ (31)

where the vector field $Y$ is the metric dual of the associated closed CKY
1-form $\tilde{Y}$. $Y$ will be referred to, following McLenaghan and Spindel,
as the Yano vector. In that case, connection with the conformal
transformations can be established in a more instructive manner. As the space
of $(n-1)$-forms and that of $1$-forms are isomorphic, exactly as in (31), any
form of the former type can be written as the Hodge dual of a uniquely
determined latter type. Recalling that the divergence of a vector field $U$
with respect to the volume form $z$ is defined by $L_{U}z=(div_{z}U)z$, we can
write

$\displaystyle\nabla_{X_{a}}\omega_{(n-1)}=\frac{1}{n}(div_{z}V)^{\ast}\tilde{X_{a}}$

for the KY $(n-1)$-form $\omega_{(n-1)}=^{\star}\tilde{V}$ (where $V$ need not
be a Yano vector). Equivalently, by the commutativity of the Hodge map and
covariant derivative, we have

$\displaystyle\nabla_{X_{a}}\tilde{V}=\frac{1}{n}(div_{z}V)\tilde{X_{a}}\;.$
(32)

On the other hand, for an arbitrary vector field $U$, $\nabla_{X}\tilde{U}$
can be decomposed as follows:

$\displaystyle\nabla_{X}\tilde{U}=\frac{1}{2}i_{X}d\tilde{U}-\frac{1}{n}\tilde{X}\delta\tilde{U}+\Gamma_{X}(U)\;,$
(33)

(see Benn-Tucker  section 6.13 and Benn-Kress ) where the 1-form
$\Gamma_{X}(U)$ is defined by

$\displaystyle\Gamma_{X}(U)=\frac{1}{2}[(L_{U}g)-\frac{1}{n}tr(L_{U}g)g](X)\;.$

Thus, for each vector field $V$ corresponding to a KY $(n-1)$-form we have, in
view of (32), $i_{X}d\tilde{V}=-2\Gamma_{X}(V)$. That is, $V$ is conformal if
and only if $\tilde{V}$ is closed.

Let us return to equation (31): $\tilde{Y}$ is closed, obeys the relation
$\delta\tilde{Y}=-div_{z}Y$ and generates conformal transformations
$L_{Y}g=2\lambda g$ with the conformal weight

$\displaystyle\lambda=\frac{1}{2n}tr(L_{Y}g)=\frac{1}{2n}(L_{Y}g)(X^{a},X_{a})\;.$

$Y$ is locally a gradient field for $\tilde{Y}$ is closed. From (32), it also
follows that

$\displaystyle\nabla_{Y}Y=\frac{1}{n}(div_{z}Y)Y\;,$

that is, the integral curves of $Y$ are pre-geodesics and may be re-
parameterized to become geodesics. If $Y$ is divergence-free, then it is
covariantly constant. We should note that in some literature, a curve whose
velocity vector $\dot{\gamma}$ satisfies
$\nabla_{\dot{\gamma}}\dot{\gamma}=f\dot{\gamma}$, for some function $f$, is
termed geodesic, which we here call pre-geodesic by adhering to the
nomenclature of Benn-Tucker ; Benn-son . For the Yano vector we have
$f=div_{z}Y/n$.

Substitution of (31) into (29) yields $i_{X_{a}}F\wedge^{\ast}\tilde{Y}=0$,
which in terms of components can be rewritten as

$\displaystyle F_{ab}Y^{b}=0\;.$ (34)

Note that condition (27) can also be rewritten as $F_{ab}K^{b}=0$, where
$K^{b}$’s are the components of the Killing vector $K$. Thus, KY $(n-1)$-forms
and 1-forms take part in the symmetry operator if and only if the
corresponding Yano and Killing vectors are contained in the kernel of the
matrix $F_{ab}$. These imply that if $F_{ab}$ is non-singular, which can
happen only in even dimensions, no KY $(n-1)$-form and 1-form can take part in
the symmetry operator. Our condition for KY 2-forms can also be stated as: a
KY 2-form will take part in the symmetry operator if and only if the
corresponding anti-symmetric matrix ($\omega_{ab}$) commutes with $F_{ab}$.

###  IV.2 Contractions with a Yano Vector

Contractions of curvature 2-forms $R_{ab}$, Ricci 1-forms $P_{a}$ and
conformal (Weyl) 2-forms $C_{ab}$, defined for $n>3$ by (below ${\cal R}$
denotes the scalar curvature)

$\displaystyle C_{ab}=R_{ab}-\frac{1}{n-2}(P_{a}\wedge e_{b}-P_{b}\wedge
e_{a}-\frac{\cal{R}}{n-1}e_{a}\wedge e_{b})\;,$

with a Yano vector enable one to reach decisive statements about the global
structures of the underlying space-times. The mentioned contractions are found
to be as follows:

$\displaystyle i_{Y}R_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{n-1}[(i_{Y}P_{a})e_{b}-(i_{Y}P_{b})e_{a}]$
$\displaystyle Y_{b}i_{Y}P_{a}$ $\displaystyle=$ $\displaystyle
Y_{a}i_{Y}P_{b}\;,$ (35) $\displaystyle i_{Y}C_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{(n-1)(n-2)}[({\cal R}g_{ac}-P_{ac})e_{b}$
$\displaystyle-({\cal
R}g_{bc}-P_{bc})e_{a}]Y^{c}+\frac{1}{n-2}(P_{a}Y_{b}-P_{b}Y_{a})\;,$

where $P_{ac}$ are the components of the Ricci tensor. The second relation
easily follows from a second application of $i_{Y}$ to the first relation and
the last one also follows from the definition of $C_{ab}$ and the first
relation of (35). A derivation of the first relation is given in Appendix B.
Equations (35) are generalizations in the language of differential forms to an
arbitrary number of dimensions and signature of the tensorial relations first
found by McLenaghan and Spindel in the case of four dimensional Lorentzian
space-times.

If $P_{ab}={\cal R}g_{ab}/n$ such that ${\cal R}$ is constant, that is in
Einstein spaces, we have $i_{Y}C_{ab}=0$. In four dimensions, the existence of
a Yano vector on an Einstein space implies that the space is conformally flat
or of Petrov type N. In higher dimensions, the equation $i_{Y}C_{ab}=0$ for
nonzero $C_{ab}$ is necessary but not sufficient condition for being Petrov
type $N$. In such a case the space-time can also be of Petrov type $II$ Coley
. When $Y$ is non-null we can write the second equation of (35) as
$i_{Y}P_{a}=\lambda Y_{a}$, where $\lambda=Y^{b}i_{Y}P_{b}/g(Y,Y)$. In such a
case the Yano vector is an eigenvector of $P_{ab}$ with eigenvalue
$\lambda={\cal R}/n$. When $Y$ is null, it is also null with respect to the
Ricci tensor, in the sense that $Y^{a}i_{Y}P_{a}=0$, in the directions of non-
zero components of the Yano vector. In other words, in such directions
contraction of Ricci forms with a null Yano vector projects it to an
orthogonal direction.

##  V Quadratic Geodesic Invariants and Constants of Motion

The most important physical implication of the existence of KY-forms is the
fact that they provide first integrals of the geodesic equation and may lead
to quadratic functions of momenta that are invariant also for the classical
trajectories. As is shown in the subsection B below, the condition (26) plays
a crucial and unified role in establishment of the second point.

###  V.1 First Integrals of Geodesic Equations

The easiest way to see the first point mentioned above is to consider the
relation $[\nabla_{X},i_{Y}]=i_{\nabla_{X}Y}$ which holds for two arbitrary
vector fields $X,Y$. When $X$ and $Y$ are equal to a velocity field
$\dot{\gamma}$ of a geodesic, that is $\nabla_{\dot{\gamma}}\dot{\gamma}=0$,
the covariant and interior derivatives commute and we obtain, in view of the
defining relation (11), $\nabla_{\dot{\gamma}}i_{\dot{\gamma}}\omega=0$ for
any KY $(p+1)$-form $\omega$. Therefore, the $p$-form
$\alpha=i_{\dot{\gamma}}\omega$ and hence its “length” $|\alpha|^{2}$ defined
by $|\alpha|^{2}=g_{p}(\alpha,\alpha)$ remain constant along the geodesic
$\gamma$. Here $g_{p}$ is the compatible metric induced by $g$ in the space of
$p$-forms and we assume $g_{0}$ to simply multiply the two $0$-forms (see
Benn-Tucker , Chapter 1).

The constancy of $|\alpha|^{2}$ can be considered as a special case of a more
general fact Semmelman1 . To see this, let us consider the symmetric bilinear
form

$\displaystyle
K^{(\beta)}(X,Y)=g_{p}(i_{X}\beta,i_{Y}\beta)=\epsilon^{\ast}(i_{X}\beta\wedge^{\ast}i_{Y}\beta)\;,$
(36)

defined, in terms of a $(p+1)$-form $\beta$ on the cartesian product of the
space of vector fields. Here $\epsilon$ denotes the sign of the determinant of
$g$. In the case of a KY $(p+1)$-form $\omega$, one can easily verify the
cyclic identity:

$\nabla_{X}K^{(\omega)}(Y,Z)+\nabla_{Y}K^{(\omega)}(Z,X)+\nabla_{Z}K^{(\omega)}(X,Y)=0\;.$

That is, the symmetrized covariant derivatives of $K^{(\omega)}$ vanish. This
shows that to any KY $(p+1)$-form $\omega$ is associated a symmetric bilinear
form $K^{(\omega)}$ which is the Killing tensor generalizing the so-called
Stackel-Killing tensor that corresponds to a KY 2-form, first recognized by
Penrose and Floyd (the second tensor of (38) below). Since

$\nabla_{X}[K^{(\omega)}(X,X)]=2K^{(\omega)}(\nabla_{X}X,X)\;,$ (37)

$K^{(\omega)}(\dot{\gamma},\dot{\gamma})$ is constant along the geodesic
$\gamma$.

For KY $1$-form $\omega_{(1)}=K_{a}e^{a}$, $2$-form
$\omega_{(2)}=2^{-1}\omega_{ab}e^{ab}$ and $n$-form $\omega_{(n)}=kz$, where
$k$ is a constant, the components of the corresponding Killing tensor are
found, respectively, to be

$\displaystyle K_{a}K_{b}\;,\;\;-\omega_{ac}\omega_{\;b}^{c}\;,\;\;\epsilon
k^{2}g_{ab}\;$ (38)

In the case of KY $(n-1)$-form ${}^{\ast}\tilde{Y}$, the components of
$K^{(^{\ast}\tilde{Y})}$ can be written in terms of the Yano vector $Y$ as

$\displaystyle K^{(^{\ast}\tilde{Y})}_{ab}$ $\displaystyle=$ $\displaystyle
g_{n-2}(i_{X_{a}}{}^{\ast}\tilde{Y},i_{X_{b}}{}^{\ast}\tilde{Y})=\epsilon
g_{2}(\tilde{Y}\wedge\tilde{X}_{a},\tilde{Y}\wedge\tilde{X}_{b})\;,$ (39)
$\displaystyle=$ $\displaystyle\epsilon[g(Y,Y)g_{ab}-Y_{a}Y_{b}]\;.$

This last Killing tensor, in particular, has remarkable properties worth
mentioning. By definition $K^{(^{\ast}\tilde{Y})}(Y,Y)=0$, that is, Yano
vectors are null with respect to the associated Killing tensor. When $Y$ is
non-null (that is, $g(Y,Y)\neq 0$ ), $K^{(^{\ast}\tilde{Y})}$ has a one
dimensional kernel spanned by $Y$, and therefore it is singular. In that case
every vector orthogonal to $Y$ is an eigenvector with the eigenvalue $\epsilon
g(Y,Y)$ and the trace of $K^{(^{\ast}\tilde{Y})}$ is $\epsilon(n-1)g(Y,Y)$.
Moreover, the properly normalized Killing tensor
$K^{\prime}=K^{(^{\ast}\tilde{Y})}/\epsilon g(Y,Y)$ is an idempotent projector
$K^{\prime}_{ab}K^{\prime b}_{\;\;c}=K^{\prime}_{ac}$ having rank $n-1$. When
$Y$ is null we have $K^{(^{\ast}\tilde{Y})}_{ab}=-\epsilon Y_{a}Y_{b}$ and in
such a case, it is a rank-1 nilpotent
($K^{(^{\ast}\tilde{Y})}_{ab}K^{(^{\ast}\tilde{Y})b}_{\>c}=0$) projector,
projecting vector fields to the direction of Yano vector.

###  V.2 Constants of Motion for Classical Trajectories

It is known that any rank-$p$ symmetric (covariant) Killing tensor provides a
degree-$p$ polynomial of velocity that is a geodesic invariant Benn-son . But,
the second rank symmetric Killing tensors that can be constructed from KY-
forms, as in (36), have a distinguished property of providing a quadratic
invariant of velocity along classical trajectories. In the remaining part of
this section we shall prove this statement in the most general setting of this
paper. To be precise, let us consider the quadratic function

$f^{(\omega)}=K^{(\omega)}(u,u)=K^{(\omega)}_{ab}\,\>u^{a}u^{b}$ (40)

where $u=\dot{C}$ is the world velocity of a charged (charge assumed to be
unit) material particle obeying the classical equation of motion

$\nabla_{u}u=\frac{1}{m}\widetilde{i_{u}F}\;.$ (41)

Here, $u^{a}=dx^{a}/d\tau$ such that $x^{a}$’s are the local coordinates of
the world curve $C$ parameterized by the proper time $\tau$ and $\nabla_{u}u$
represents the acceleration of the particle. Hence

$\displaystyle\frac{d}{d\tau}(f^{(\omega)}\circ C)$ $\displaystyle=$
$\displaystyle
C_{\ast}(\partial_{\tau})f^{(\omega)}=\nabla_{u}[K^{(\omega)}(u,u)]$ (42)
$\displaystyle=$ $\displaystyle 2K^{(\omega)}(\nabla_{u}u,u)\;,$

and in view of (41)

$\displaystyle\frac{d}{d\tau}(f^{(\omega)}\circ
C)=\frac{2}{m}u^{a}u^{b}F_{ac}K^{(\omega)c}_{\;b}.$ (43)

What we are going to prove is the constancy of $f^{(\omega)}$ along the
classical trajectory determined by (41). Before doing that, it would be
illuminating to examine first some special cases. One can easily verify that
the right hand side of (43) vanishes if the components of the Killing tensor
given by (38) are used, provided that KY-forms employed in defining
$f^{(\omega)}$ satisfy the condition (26). In the case of
$K^{(^{\ast}\tilde{Y})}_{ab}$ given by (39), the contraction with the Maxwell
field is, in view of (34)

$F_{ac}K^{(\omega)c}_{\;b}=\epsilon g(Y,Y)F_{ab}\;,$ (44)

which is obviously anti-symmetric and also makes the right hand side of (43)
vanish. So, as a particular result, all quadratic functions constructed from
KY $p$-forms for $p=1,2,n-1,n$ are constant along the classical trajectory.
These exhaust all possible cases in a four dimensional space-time and, to the
best of our knowledge, these are all that can be found in the related
literature.

We shall now prove that $f^{(\omega)}$ is a constant of motion for the
classical trajectories determined by (41) for all dimensions and signatures as
well as for all KY $p$-forms $\omega_{(p)}$ obeying the symmetry condition
(26). For the proof, we first take the Hodge dual of both sides of the
equation (43) and write

$\displaystyle\frac{d}{d\tau}(f^{(\omega)}\circ
C)^{\ast}1=\frac{2}{m}u^{a}u^{b}I_{ab}\;,$ (45)

where the $n$-form $I_{ab}$ is defined as

$I_{ab}=^{\ast}(F_{ac}K^{(\omega)c}_{\;b})=i_{X_{c}}i_{X_{a}}Fi_{X^{c}}\omega\wedge^{\ast}i_{X_{b}}\omega\;.$
(46)

In passing to the second equality of equation (46) we have used (36). Thus,
$f^{(\omega)}$ is constant along the world line $C(\tau)$ if and only if the
right hand side of the equation (45) vanishes. Evidently, the anti-symmetry
condition $I_{ab}=-I_{ba}$ is sufficient (and can also be shown to be
necessary) for the right hand side of (45) to vanish. To show the anti-
symmetry of $I_{ab}$ we rewrite it as follows:

$\displaystyle I_{ab}$ $\displaystyle=$
$\displaystyle-[i_{X_{a}}(i_{X_{c}}F\wedge i_{X^{c}}\omega)+i_{X_{c}}F\wedge
i_{X_{a}}i_{X^{c}}\omega]\wedge^{\ast}i_{X_{b}}\omega\;,$ $\displaystyle=$
$\displaystyle i_{X_{c}}F\wedge
i_{X^{c}}(i_{X_{a}}\omega\wedge^{\ast}i_{X_{b}}\omega)-i_{X_{a}}\omega\wedge
i_{X_{c}}F\wedge i_{X^{c}}\;^{\ast}i_{X_{b}}\omega\;,$ $\displaystyle=$
$\displaystyle-i_{X_{a}}\omega\wedge
i_{X_{c}}F\wedge^{\ast}(i_{X_{b}}\omega\wedge e^{c})\;.$

The first term in the square bracket of the first line vanishes because of
condition (26), and the first term of the second line vanishes since it can be
written as an interior derivative of a $(n+1)$-form. In the third line of (47)
we have made use of the identity
$i_{X^{c}}\;^{\ast}i_{X_{b}}\omega=^{\ast}(i_{X_{b}}\omega\wedge e^{c})$ which
was also used in the previous section. In view of $i_{X_{c}}F=F_{ck}e^{k}$ and
of another Hodge identity $\alpha\wedge^{\ast}\beta=\beta\wedge^{\ast}\alpha$
which holds for any two $p$-forms $\alpha$ and $\beta$, we obtain from (47)

$\displaystyle I_{ab}$ $\displaystyle=$ $\displaystyle-i_{X_{b}}\omega\wedge
F_{ck}e^{c}\wedge^{\ast}(i_{X_{a}}\omega\wedge e^{k})$ $\displaystyle=$
$\displaystyle i_{X_{b}}\omega\wedge
i_{X_{k}}F\wedge^{\ast}(i_{X_{a}}\omega\wedge e^{k})\;.$

By comparing this with third line of (47) we obtain $I_{ab}=-I_{ab}$ which was
what to be demonstrated.

An immediate corollary of the constancy of $f^{(\omega)}$ is, by virtue of
(42), the relation $K^{(\omega)}(\nabla_{u}u,u)=0$ which can be interpreted as
follows. The world velocity and acceleration of a charged material particle
are perpendicular to each other, not only with respect to the metric, but also
with respect to the symmetric Killing tensors associated with each KY-form
satisfying the condition (26).

##  VI Summary and Conclusion

In this study, the most general first-order linear symmetry operators of the
Dirac equation including interaction with Maxwell field in curved background
of arbitrary $n$-dimension and of signature are specified. We have shown that
all coefficients forms $\omega^{a}$’s of the symmetry operator
$L=2\omega^{a}S_{X_{a}}+\Omega$ are given, in terms of an inhomogeneous KY-
form $\omega$, by $\omega^{a}=i_{X^{a}}\omega$. The components of $\Omega$ are
explicitly calculated and are given by the equations (12), (13) and (28). They
depend on the exterior derivative of KY-forms and their contraction with the
potential field $A$. We have also found a unified, gauge invariant dynamical
symmetry condition which states that among all the KY-forms of underlying
curved background only those which Clifford commute with the Maxwell field can
take part in $L$. When $\omega^{a}$ and $\Omega$ are even, $L$ itself, but
when they are odd and $n$ is even $Lz$ is a first order symmetry operator
which Clifford commutes with the Dirac equation and hence maps a solution to
another.

The special cases of the dynamical symmetry condition are also discussed as
they may provide valuable insights in applications. In particular, the KY
$(n-1)$-forms and 1-forms take part in the symmetry operator if and only if
the corresponding Yano and Killing vectors belong to the kernel of the anti-
symmetric matrix $F_{ab}$ corresponding to the components of the Maxwell
field. These imply that if $F_{ab}$ is non-singular, which can happen only in
even dimensions, no KY $(n-1)$-forms and 1-forms can take part in the symmetry
operator. Implications of the existence of Yano vectors in specifying the
global structure of the curved background are also discussed. Finally it has
been proved that, for all KY-forms obeying the dynamical symmetry condition,
there exists a quadratic function of velocity (defined by the equation (40))
which is a constant of motion for the classical motion in an arbitrary
dimension and signature.

All of these results are expected to provide a unified framework for symmetry
analysis and to serve as a firm base in studying symmetry algebras and related
conserved quantities of the Dirac equation for a given specific curved
background and force field.

###### Acknowledgements.

We are grateful to anonymous referees for their useful comments. This work was
supported in part by the Scientific and Technological Research Council of
Turkey (TÜBİTAK).

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##  Appendix A Consistency Conditions

By direct calculations we obtain

$\displaystyle[\displaystyle{\not}S,\omega]=2\omega^{a}S_{X_{a}}+\displaystyle{\not}d\omega=L+\varphi\;,$

where $\omega^{a}=i_{X^{a}}\omega$. It proves convenient to take $L$ as
$L=[\displaystyle{\not}S,\omega]-\varphi$, where
$\Omega=\displaystyle{\not}d\omega-\varphi$. Using the fact that

$\displaystyle[\displaystyle{\not}S,[\displaystyle{\not}S,\omega]]=[\displaystyle{\not}S^{2},\omega]_{-}\;,$

the symmetry condition (3) transforms to

$\displaystyle[\displaystyle{\not}S^{2},\omega]_{-}=[\displaystyle{\not}S,\varphi]-{\rm{i}}[A,L]\;.$
(48)

${\cal R}$ being the scalar curvature of the spinor connection,
$\displaystyle{\not}S^{2}$ acts on the spinor fields as (see Benn-Tucker
chapter 10)

$\displaystyle\displaystyle{\not}S^{2}=S^{2}(X_{a},X^{a})-\frac{1}{4}{\cal
R}\;.$ (49)

Therefore on substituting the relations

$\displaystyle[\displaystyle{\not}S^{2},\omega]_{-}$ $\displaystyle=$
$\displaystyle[S^{2}(X_{a},X^{a}),\omega]_{-}$ $\displaystyle=$ $\displaystyle
2\nabla_{X^{a}}\omega S_{X_{a}}+\nabla^{2}(X_{a},X^{a})\omega\;,$
$\displaystyle\;[\displaystyle{\not}S,\varphi]$ $\displaystyle=$
$\displaystyle 2i_{X^{a}}\varphi S_{X_{a}}+\displaystyle{\not}d\varphi\;,$
(50) $\displaystyle\;[A,L]$ $\displaystyle=$ $\displaystyle
2[A,i_{X^{a}}\omega]S_{X_{a}}-2(i_{X^{a}}\omega)^{\eta}\nabla_{X_{a}}A+[A,\Omega],$

into equation (A1) and then on equating the coefficients of equal power of
derivatives, we obtain the consistency conditions (6) and (7) of the main
text.

##  Appendix B Contraction of Curvatures with a Yano vector

By differentiating the KY equation (11), the action of the Hessian (see the
remark following equation (4)) on a KY $p$-form $\omega_{(p)}$ is found to be

$\nabla^{2}(X_{a},X_{b})\omega_{(p)}=\frac{1}{p+1}i_{X_{b}}\nabla_{X_{a}}d\omega_{(p)}.$
(51)

Since the difference of $\nabla^{2}(X_{a},X_{b})$ and
$\nabla^{2}(X_{b},X_{a})$ is the curvature operator $R(X_{a},X_{b})$, from
(B1) we obtain

$R(X_{a},X_{b})\omega_{(p)}=\frac{1}{p+1}(i_{X_{b}}\nabla_{X_{a}}-i_{X_{a}}\nabla_{X_{b}})d\omega_{(p)}.$
(52)

On the other hand, the action of the curvature operator on any form $\alpha$
is known to be (see Benn-Tucker  equation (8.1.11) and Benn-Kress )

$R(X_{a},X_{b})\alpha=-i_{X^{k}}R_{ab}\wedge i_{X_{k}}\alpha\;.$ (53)

Note that the action on the one form $\tilde{X}$ simply is $-i_{X}R_{ab}$.
Using (B3) in (B2) and then by multiplying the result with $e^{a}\wedge$, we
obtain

$\nabla_{X_{a}}d\omega_{(p)}=\frac{p+1}{p}R^{b}_{\;\;a}\wedge
i_{X_{b}}\omega_{(p)}.$ (54)

Using this in (B2) for $\omega_{(n-1)}=^{\ast}\tilde{Y}$ we find

$R(X_{a},X_{b})^{\ast}\tilde{Y}=\frac{(-1)^{n}}{n-1}[i_{X_{b}}{}^{\ast}\tilde{Y}\wedge
P_{a}-i_{X_{a}}{}^{\ast}\tilde{Y}\wedge P_{b}]$ (55)

where we have used the contracted Bianchi identity
$i_{X_{b}}P_{a}=i_{X_{a}}P_{b}$. Let us first note that

$\displaystyle R(X_{a},X_{b})^{\ast\ast}\tilde{Y}$ $\displaystyle=$
$\displaystyle\epsilon(-1)^{n}i_{Y}R_{ab}\;,$ (56)
${}^{\ast}[^{\ast}(\tilde{Y}\wedge e_{b})\wedge P_{a}]$ $\displaystyle=$
$\displaystyle i_{\tilde{P_{a}}}{}^{\ast\ast}(\tilde{Y}\wedge e_{b})\;,$ (57)
$\displaystyle=$
$\displaystyle\epsilon[(i_{Y}P_{a})e_{b}-\tilde{Y}i_{X_{b}}P_{a}]\;,$

where $\epsilon$ is the sign of $\det g$. In deriving the above relations, we
have made use of (B3) and of the following identities:

${}^{\ast\ast}\alpha_{(p)}=\epsilon(-1)^{p(n-p)}\alpha_{(p)}\;,\quad
i_{X_{a}}{}^{\ast}\tilde{Y}=^{\ast}(\tilde{Y}\wedge e_{a})\;.$ (58)

If we now take the Hodge dual of both sides of (B5) and use (B6), (B7) we
obtain the first relation of (35) of the main text:
$i_{Y}R_{ab}=(n-1)^{-1}[(i_{Y}P_{a})e_{b}-(i_{Y}P_{b})e_{a}]$.

As an aside, it is worth mentioning that if (B4) is substituted into (B1), we
obtain

$\nabla^{2}(X_{a},X_{b})\omega_{(p)}=\frac{1}{p}i_{X_{b}}(R^{c}_{\;a}\wedge
i_{X_{c}}\omega_{(p)}),$ (59)

which shows that the second covariant derivative of any KY $p$-form is
determined by curvature characteristics and the form itself. This implies that
the value of a KY $p$-form at any point is entirely determined by the values
of the form itself and its first covariant derivatives at the same point.
These remarks can be used to determine the upper bound for the numbers of
linearly independent KY $p$-forms to be the Binomial number $C(n+1,p+1)$
Kastor ; ozumav1 .