# Covariant actions and propagators for all spins, masses, and dimensions

Lukas W. Lindwasser<EMAIL_ADDRESS>Mani L. Bhaumik
Institute for Theoretical Physics,
Department of Physics and Astronomy,
University of California, Los Angeles, CA 90095, USA

###### Abstract

The explicit covariant actions and propagators are given for fields describing
particles of all spins and masses, in any spacetime dimension. Massive
particles are realized as “dimensionally reduced” massless particles. To
obtain compact expressions for the propagators, it was useful to introduce an
auxiliary vector coordinate $s^{\mu}$ and consider “hyperfields” that are
functions of space $X^{\mu}$ and $s^{\mu}$. The actions and propagators serve
as a basic starting point for concrete high spin computations amenable to
dimensional regularization, provided that gauge invariant interactions are
introduced.

###### Contents

  1. I Introduction
    1. I.1 Conventions
    2. I.2 Outline
  2. II Hyperfields
    1. II.1 Hyperfield basics
  3. III Massless integer spins
    1. III.1 Transition to hyperfields
    2. III.2 All propagators
    3. III.3 Other gauges
  4. IV Massless half integer spins
    1. IV.1 Transition to hyperfields
    2. IV.2 All propagators
    3. IV.3 Other gauges
  5. V Massive integer spins
    1. V.1 Massive particles from dimensional reduction
    2. V.2 Equations of motion
    3. V.3 All propagators
  6. VI Massive half integer spins
    1. VI.1 Massive particles from dimensional reduction
    2. VI.2 Equations of motion
    3. VI.3 All propagators
  7. VII Discussion

##  I Introduction

The modelling of particles with spin higher than 3/2 has a long history,
dating back to at least the work of Dirac in 1936 [1]. Equations of motion for
free massive particles of arbitrary spins were written down in [2], and it was
noted then that introducing interactions in this framework will generically
not preserve the physical degrees of freedom of the high spin particle. It was
suggested that a way to fix this was to formulate an action principle for the
higher spin field. Subsequently, the first actions for free massive particles
of arbitrary spins were written down in [3, 4], and then their massless
counterparts in [5, 6].

The problem of introducing interactions persisted. For massless higher spin
particles, general arguments [7, 8, 9] were made which essentially invalidate
the existence of spins higher than 2 in asymptotically flat space, assuming
they interact with gravity in a way which obeys the equivalence
principle111There is of course, a notable theory [10] of massless higher spins
in anti-de Sitter space.. We know on the other hand that massive higher spins
exist in nature [11], and so we expect there to be a way to model them and
their interactions.

In spite of this, the task of finding consistent interactions for massive
higher spins remains a challenging one. Along with the aforementioned change
in degrees of freedom issue, it was realized that minimally coupling spins
higher than 3/2 to an electromagnetic background will generically allow
superluminal propagation [12]. Coupling massive higher spins to gravity has
also been shown to violate causality [13, 14]. For spin 3/2, both of these
problems can be solved when it is realized within the framework of
supergravity [15]. For higher spins, the only known solution to both is to
realize them within string theory [16]. It is clear based on the explicit
construction in [16] that a consistent set of interactions with
electromagnetism is by no means unique. A construction of the space of
consistent interactions from first principles is still warranted.

To facilitate this, one must formulate massive higher spins in a way which
makes the introduction of interactions straightforward. The actions [3, 4] are
not suited for this task for at least two reasons. First, these actions are
only consistent in four spacetime dimensions, and based on experience with
string theory, the consistency of interactions appears to depend sensitively
on the spacetime dimension. Thus we should find a way to formulate massive
higher spins in arbitrary spacetime dimensions $d$. Finally, these actions do
not have an organizing principle such as gauge invariance which helps decide
what kinds of interactions will preserve the particle’s degrees of freedom. In
this paper, we will write down covariant actions for all massive spins which
addresses both of these points.

Another important aspect of massive higher spins is the complexity of their
propagators. Explicit propagators for the physical spin $s$ field resulting
from the actions [3, 4] were written in [17]. A different set of propagators
were written in [18, 19], without any reference to an action. They both find
spin $s$ propagators which grow like $\sim p^{-2+2s}$ in momentum space.
Although these works find propagators for the physical spin $s$ field, they do
not find propagators for the various auxiliary fields needed in any action
formulation of massive particles with spin $s\geq 3/2$. These must be included
in your Feynman rules, and are therefore important for doing any concrete
computation of Feynman diagrams. Propagators for spins 1 and higher are not
unique, and we demonstrate this non-uniqueness as a manifestation of gauge
invariance within this formalism. In this paper, we instead find propagators
for all fields needed in the description which grow like $\sim p^{-2}$ and
$\sim p^{-1}$ for all integer and half integer spins, respectively. This
discrepancy is understood as a natural extension of the well known fact that
massive spin 1 propagators can have different high energy behavior depending
on the gauge one picks, e.g. unitarity or $R_{\xi}$ gauges. The improved high
energy behavior is achieved at the cost of introducing unphysical
polarizations into the propagator, but as in Yang-Mills theory, as long as
gauge invariant interactions are present, these unphysical polarizations
should not contribute to observables. Furthermore, because the propagators are
valid in any spacetime dimension $d$, dimensional regularization can be used
readily.

The expressions for the propagators are compactly written in terms of
Gegenbauer polynomials. This is made possible by introducing an auxiliary
vector coordinate $s^{\mu}$, and writing propagators for the fields
$\frac{1}{n!}\phi_{\mu_{1}\cdots\mu_{n}}(X)s^{\mu_{1}}\cdots s^{\mu_{n}}$. In
particular, this allows us to write the propagators so that the spin
$s=n,n+1/2$ is a freely tunable parameter, enabling analytic control over the
large spin asymptotic limit $n\to\infty$. Indeed, using known properties of
Gegenbauer polynomials [20], we find simple asymptotic $n\to\infty$
expressions for the propagators. This may be particularly helpful in the
current research program of using effective field theory techniques to model
black hole binary dynamics with classical values of spin [21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32]. Indeed, a main motivation for this work is to
develop a formalism which allows one to take an $n\to\infty$ limit without
computing a few low spin examples and having to extrapolate.

Finally, by virtue of the gauge symmetry inherent in this formulation, finding
interactions which preserve the higher spin particle’s degrees of freedom from
first principles is equivalent to maintaining some form of gauge invariance.
Formulating massive higher spins in a way that can be used as a starting point
for straightforwardly introducing interactions is a main motivation for the
current work. The formulation we arrive at is similar to previous work [33,
34, 35, 36, 37]. All such formulations, including ours, are presumably related
via field redefinition, yet vary in their complexity.

Previous attempts at introducing consistent interactions to these actions have
restricted to constrained background fields, e.g. [33, 34, 38, 39, 40]. The
difficulty in constructing interactions with general dynamical fields is in
part due to the inherent complexity of high spin actions. One virtue of the
current work which alleviates this difficulty is the elucidation of natural
objects $\mathcal{F}_{\mu_{1}\cdots\mu_{n}}$,
$\mathcal{F}_{\mu_{1}\cdots\mu_{n-1}}$,
$\mathcal{F}_{\mu_{1}\cdots\mu_{n-2}}$, $\mathcal{F}_{\mu_{1}\cdots\mu_{n-3}}$
for the integer spin case, and $\mathcal{S}_{\mu_{1}\cdots\mu_{n}}$,
$\mathcal{S}_{\mu_{1}\cdots\mu_{n-1}}$, $\mathcal{S}_{\mu_{1}\cdots\mu_{n-2}}$
for the half integer spin case, which are gauge invariant and satisfy Bianchi-
like identities, described in section V and section VI. This is possible
because we formulate $d$ dimensional massive particles as “dimensionally
reduced” $d+1$ dimensional massless particles, which themselves have gauge
invariant field strengths satisfying Bianchi-like identities. Dimensional
reduction of massless particles was first discussed in detail for integer
spins in [41], elaborated in [42, 43], and further studied in [33, 34, 44,
45]. We differentiate our work from theirs by organizing the explicit actions
in terms of the natural objects $\mathcal{F}_{\mu_{1}\cdots\mu_{n-i}}$ and
$\mathcal{S}_{\mu_{1}\cdots\mu_{n-i}}$, and also finding the explicit
propagators, including that of the auxiliary fields. Furthermore, the manifest
gauge invariance of the massive higher spin actions we arrive at is guaranteed
by the properties of $\mathcal{F}_{\mu_{1}\cdots\mu_{n-i}}$ and
$\mathcal{S}_{\mu_{1}\cdots\mu_{n-i}}$ discussed above, whereas in previous
formulations it is less obvious. This allows a simpler and more organized
formulation for constructing gauge invariant interactions. We delay a detailed
discussion of interactions within this formalism for future work.

###  I.1 Conventions

Throughout this paper, we use the metric signature
$\eta=\text{diag}(-1,+1,\cdots,+1)$. Subsequently, the standard massless and
massive Feynman propagators for spins 0 and 1/2 are respectively,

$\displaystyle
G_{0}(X-Y)=\int\frac{d^{d}p}{(2\pi)^{d}}\frac{-i}{p^{2}-i\epsilon}e^{ip\cdot(X-Y)},$
$\displaystyle
G_{m}(X-Y)=\int\frac{d^{d}p}{(2\pi)^{d}}\frac{-i}{p^{2}+m^{2}-i\epsilon}e^{ip\cdot(X-Y)}$
(I.1)
$\displaystyle\Delta_{0}(X-Y)=\int\frac{d^{d}p}{(2\pi)^{d}}\frac{-\mathrlap{\\!\not{\phantom{p}}}p}{p^{2}-i\epsilon}e^{ip\cdot(X-Y)},$
$\displaystyle\Delta_{m}(X-Y)=\int\frac{d^{d}p}{(2\pi)^{d}}\frac{-\mathrlap{\\!\not{\phantom{p}}}p-im}{p^{2}+m^{2}-i\epsilon}e^{ip\cdot(X-Y)}$
(I.2)

These will be basic building blocks for their higher spin generalizations. We
will define all correlation functions via analytic continuation of their
Euclidean space versions. Because of this, when writing correlation functions
such as $\langle\mathcal{O}_{1}(X_{1})\cdots\mathcal{O}_{n}(X_{n})\rangle$, we
leave the time ordered product $T$ as implicit.

Frequently used in this paper is the notation
$\mathcal{O}_{(\mu_{1}\cdots\mu_{n})}$, which denotes the total symmetrization
of the indices enclosed by the parentheses. The normalization we use for
instance for the symmetrization of two indices is

$\displaystyle\mathcal{O}_{(\mu_{1}\mu_{2})}\equiv\frac{1}{2}(\mathcal{O}_{\mu_{1}\mu_{2}}+\mathcal{O}_{\mu_{2}\mu_{1}})$
(I.3)

###  I.2 Outline

In section II, we introduce the “hyperfield” formalism, which will enable
compact expressions for the covariant actions, propagators, and interactions
in subsequent sections. Then in section III and section IV, we review the
formulation of free massless higher spins [5, 6], and we compute their
propagators in a special gauge, as well as characterize their gauge
ambiguities. This review importantly sets the stage for section V and section
VI, where we do the same for massive higher spins, defining them via
“dimensional reduction” of their massless counterparts as in [41]. Finally in
section VII, we discuss our results and present some directions for future
research.

##  II Hyperfields

In this section, we introduce the notion of a hyperfield, which greatly
simplifies the analysis of fields with arbitrary spin. Similar techniques have
been used in the high spin literature, including but not limited to [46, 47,
48, 49]. We will find in later sections that covariant actions and propagators
admit particularly simple closed form expressions within this formalism.

Particles with spin are covariantly described by totally symmetric tensors
$\phi_{\mu_{1}\cdots\mu_{n}}(x)$ or spinor tensors
$\psi_{\mu_{1}\cdots\mu_{n}}(x)$. Totally symmetric tensors have
${d+n-1\choose n}$ independent components, and $2^{\cramped{\left\lfloor
d/2\right\rfloor}}{d+n-1\choose n}$ if there is an additional Dirac index,
where $\cramped{\left\lfloor x\right\rfloor}$ is the floor function222In this
paper we consider for simplicity only parity invariant representations.. This
is far in excess of the physical degrees of freedom needed to describe a
particle with spin $n$ or $n+1/2$. To maintain covariance, one must impose
additional constraints on the fields $\phi_{\mu_{1}\cdots\mu_{n}}(x)$,
$\psi_{\mu_{1}\cdots\mu_{n}}(x)$, outlined in later sections, which remove
unnecessary degrees of freedom. This procedure is quite unwieldy for arbitrary
spins, as evidenced by their covariant actions [3, 4, 5, 6], so much so that
many physicists opt to study the effects of high spin particles using
alternative methods [50, 51, 52, 53, 14, 54, 26, 28]. To study the effects of
high spin particles in a fully covariant and off-shell manner, a new formalism
which simplifies calculations is warranted.

In this paper the approach will be to introduce an auxiliary vector coordinate
$s^{\mu}$, and consider “hyperfields”
$\Phi_{n}(X,s)=\frac{1}{n!}i^{-n/2}\phi_{\mu_{1}\cdots\mu_{n}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n}}$, where the factor $i^{-n/2}$ is added for later convenience. If
$\Phi_{n}(X,s)$ has no external Lorentz indices, the component field will be a
totally symmetric rank $n$ tensor, which will describe an integer spin $n$
particle after appropriate constraints are imposed. If the hyperfield has an
external Dirac index, which we will henceforth denote $\Psi_{n}(X,s)$, the
component field will similarly describe a half integer spin $n+1/2$ particle.
More generally, one can consider a hyperfield $\Phi(X,s)$ which is a general
function of $s^{\mu}$, with its formal Taylor expansion generating all totally
symmetric tensors

$\displaystyle\Phi(X,s)=\sum_{n=0}^{\infty}\frac{1}{n!}i^{-n/2}\phi_{\mu_{1}\cdots\mu_{n}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n}}$ (II.1)

Such hyperfields are sufficient to describe the leading Regge trajectory of
string theory. One can further consider introducing $N$ auxiliary vectors
$s_{i}^{\mu}$, and if $N>d$, the corresponding hyperfield
$\Phi(X,\\{s_{i}\\})$ would contain all representations present in string
theory. This is of course precisely how string field theory packages all of
its particle states into a single object [55]. For the purposes of the present
paper, we make no attempt at making contact with string theory, or its
generalizations [56, 57, 58]. The author hopes to return to this possibility
in future work.

###  II.1 Hyperfield basics

In this section we quickly review how to perform basic operations on
$\phi_{\mu_{1}\cdots\mu_{n}}(X)$ relevant for the covariant formulation of
spinning particles, at the level of a hyperfield $\Phi_{n}(X,s)$. For
instance, one often has to take traces of the fields
$\phi^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n}}(X)$. This can be achieved at
the level of $\Phi_{n}(X,s)$ by taking the Laplacian with respect to $s^{\mu}$

$\displaystyle
i\,\partial_{s}^{2}\Phi_{n}(X,s)=\frac{1}{(n-2)!}i^{-(n-2)/2}\phi^{\lambda}_{\;\;\lambda\mu_{1}\cdots\mu_{n-2}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n-2}}$ (II.2) The divergence
$\partial^{\lambda}\phi_{\lambda\mu_{2}\cdots\mu_{n}}(X)$ may also be written
in terms of $\Phi_{n}(X,s)$ $\displaystyle
i^{1/2}\,\partial_{s}\cdot\partial_{X}\Phi_{n}(X,s)=\frac{1}{(n-1)!}i^{-(n-1)/2}\partial^{\lambda}\phi_{\lambda\mu_{1}\cdots\mu_{n-1}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n-1}}$ (II.3) The symmetric derivative
$\partial_{(\mu_{1}}\phi_{\mu_{2}\cdots\mu_{n+1})}(X)$ is written in terms of
$\Phi_{n}(X,s)$ via $\displaystyle
i^{-1/2}s\cdot\partial_{X}\Phi_{n}(X,s)=\frac{1}{n!}i^{-(n+1)/2}\partial_{(\mu_{1}}\phi_{\mu_{2}\cdots\mu_{n+1})}(X)s^{\mu_{1}}\cdots
s^{\mu_{n+1}}$ (II.4) One can also contract indices between two equal rank $n$
hyperfields $A_{n}(X,s)$ and $B_{n}(X,s)$
$\displaystyle\int\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}A_{n}(X,s)B_{n}(X,s^{\prime})=\frac{1}{n!}a_{\mu_{1}\cdots\mu_{n}}b^{\mu_{1}\cdots\mu_{n}}$
(II.5) These four operations are sufficient for the formulation of integer
spin fields, as we will see in section III and section V. For half integer
spin fields, contraction with gamma matrices $\gamma^{\mu}$ will also be
necessary. For a Dirac hyperfield $\Psi_{n}(X,s)$, we have $\displaystyle
i^{1/2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}(X,s)=\frac{1}{(n-1)!}i^{-(n-1)/2}\gamma^{\lambda}\psi_{\lambda\mu_{1}\cdots\mu_{n}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n}}$ (II.6)

The formula (II.5) is particularly interesting, as it suggests introducing a
(pseudo) inner product on the space of hyperfields

$\displaystyle(A_{n},B_{n})=\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}\tilde{A}_{n}(X,s)B_{n}(X,s^{\prime})=\int
d^{d}X\frac{1}{n!}a^{*}_{\mu_{1}\cdots\mu_{n}}b^{\mu_{1}\cdots\mu_{n}}$ (II.7)
$\displaystyle\text{where
}\tilde{A}_{n}(X,s)=\frac{1}{n!}i^{-n/2}a^{*}_{\mu_{1}\cdots\mu_{n}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n}}$ (II.8)

and we may consider the (pseudo) Hilbert space of hyperfields $\Phi(X,s)$ with
finite (pseudo) norm $(\Phi,\Phi)<\infty$333$(A,B)$ is only positive definite
in Euclidean signature. [59]. Note that
$(i^{-1/2}s^{\mu}A,B)=(A,i^{1/2}\partial_{s}^{\mu}B)$, and so for instance,
the divergence and symmetric derivative are anti-Hermitian adjoints of each
other
$(i^{1/2}\partial_{s}\cdot\partial_{X})^{\dagger}=-i^{-1/2}s\cdot\partial_{X}$
in this space.

Crucial operators for the analysis in subsequent sections are the projection
operators $P_{n,\eta}^{d}(s,t)$, $P_{n,\gamma}^{d}(s,t)$ on to the subspace of
$\eta$ traceless rank $n$ hyperfields $i\,\partial_{s}^{2}\Phi_{n}(X,s)=0$ and
$\gamma$ traceless rank $n$ hyperfields
$i^{1/2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}(X,s)=0$,
respectively. We define operators to act on the hyperfields via

$\displaystyle(P\Phi_{n})(X,s)=\int\frac{d^{d}td^{d}t^{\prime}}{(2\pi)^{d}}e^{it\cdot
t^{\prime}}P(s,t)\Phi_{n}(X,t^{\prime})$ (II.9)

The projection operators can be straightforwardly found from the equations
$\partial_{s}^{2}P_{n,\eta}(s,t)=\partial_{t}^{2}P_{n,\eta}(s,t)=0$,
$\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}P_{n,\gamma}(s,t)=P_{n,\gamma}(s,t)\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial}_{t}=0$,
and they have simple expansions in terms of classical Gegenbauer polynomials
$C_{n}^{\alpha}(x)$ [20], they read

$\displaystyle
P_{n,\eta}^{d}(s,t)=\frac{1}{(\frac{d-2}{2})_{n}}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n}C_{n}^{\frac{d-2}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (II.10) $\displaystyle P_{n,\gamma}^{d}(s,t)=$
$\displaystyle\frac{1}{(\frac{d}{2})_{n}}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n}C_{n}^{\frac{d}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}+i\frac{\mathrlap{\\!\not{\phantom{s}}}s\mathrlap{\\!\not{\phantom{t}}}t}{2}\frac{1}{(\frac{d}{2})_{n}}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-1}C_{n-1}^{\frac{d}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (II.11)

where $(x)_{n}=\Gamma(x+n)/\Gamma(x)$ is the Pochhammer symbol.

##  III Massless integer spins

In this section, we will use the hyperfield formalism of section II to review
massless integer spin fields. We will find compact, closed form expressions
for their covariant actions and propagators. First, we review the standard
description of massless integer spin fields $\phi_{\mu_{1}\cdots\mu_{n}}(X)$.

The covariant formulation of massless integer spin fields
$\phi_{\mu_{1}\cdots\mu_{n}}(X)$ was first worked out in [5]. In this
description, Fronsdal found that it was possible to write down a covariant
action for $\phi_{\mu_{1}\cdots\mu_{n}}$ which is quadratic in derivatives,
provided that $\phi_{\mu_{1}\cdots\mu_{n}}$ is a real, symmetric tensor that
satisfies the unusual double traceless condition
$\phi^{\lambda\;\;\omega}_{\;\;\lambda\;\;\omega\mu_{5}\cdots\mu_{n}}=0$. The
equations of motion are

$\displaystyle\mathcal{F}_{\mu_{1}\cdots\mu_{n}}=\partial^{2}\phi_{\mu_{1}\cdots\mu_{n}}-n\partial_{(\mu_{1}}\partial^{\lambda}\phi_{\lambda\mu_{2}\cdots\mu_{n})}+\frac{1}{2}n(n-1)\partial_{(\mu_{1}}\partial_{\mu_{2}}\phi^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n})}=0$
(III.1)

where we will call $\mathcal{F}_{\mu_{1}\cdots\mu_{n}}$ the Fronsdal field
strength, which may be thought of as the spin $n$ generalization of the
linearized Ricci curvature tensor
$R_{\mu\nu}=\partial^{2}h_{\mu\nu}-\partial_{\mu}\partial^{\lambda}h_{\lambda\nu}-\partial_{\nu}\partial^{\lambda}h_{\lambda\mu}+\partial_{\mu}\partial_{\nu}h$
obtained from the metric $g_{\mu\nu}=\eta_{\mu\nu}-2h_{\mu\nu}$.
$\mathcal{F}_{\mu_{1}\cdots\mu_{n}}$ satisfies the spin $n$ generalization of
the second contracted Bianchi identity

$\displaystyle\partial^{\lambda}\mathcal{F}_{\lambda\mu_{1}\cdots\mu_{n-1}}-\frac{1}{2}(n-1)\partial_{(\mu_{1}}\mathcal{F}^{\lambda}_{\;\;\lambda\mu_{2}\cdots\mu_{n-1})}=0$
(III.2)

As with Maxwell theory and linearized gravity, the equations of motion (III.1)
have a gauge redundancy. Under the gauge transformation

$\displaystyle\phi_{\mu_{1}\cdots\mu_{n}}(X)^{\prime}=\phi_{\mu_{1}\cdots\mu_{n}}(X)+n\partial_{(\mu_{1}}\epsilon_{\mu_{2}\cdots\mu_{n})}(X)$
(III.3)

the Fronsdal field strength is invariant provided that
$\epsilon_{\mu_{1}\cdots\mu_{n-1}}$ is a symmetric and traceless, but
otherwise arbitrary function, i.e.

$\displaystyle\delta\mathcal{F}_{\mu_{1}\cdots\mu_{n}}=\frac{1}{2}n(n-1)(n-2)\partial_{(\mu_{1}}\partial_{\mu_{2}}\partial_{\mu_{3}}\epsilon^{\lambda}_{\;\;\lambda\mu_{4}\cdots\mu_{n})}=0$
(III.4)

This gauge redundancy allows us to impose a gauge fixing condition. The
natural choice is the spin $n$ generalization of the Lorenz and de Donder
gauge conditions

$\displaystyle\partial^{\lambda}\phi_{\lambda\mu_{1}\cdots\mu_{n-1}}-\frac{1}{2}(n-1)\partial_{(\mu_{1}}\phi^{\lambda}_{\;\;\lambda\mu_{2}\cdots\mu_{n-1})}=0$
(III.5)

In this gauge, the equations of motion (III.1) reduce to

$\displaystyle\mathcal{F}_{\mu_{1}\cdots\mu_{n}}=\partial^{2}\phi_{\mu_{1}\cdots\mu_{n}}=0$
(III.6)

and so the particles described by $\phi_{\mu_{1}\cdots\mu_{n}}$ are indeed
massless. In this gauge there remains a residual gauge symmetry under a gauge
parameter $\epsilon_{\mu_{1}\cdots\mu_{n-1}}$ satisfying
$\partial^{2}\epsilon_{\mu_{1}\cdots\mu_{n-1}}=0$. In total then, the number
of degrees of freedom removed from $\phi_{\mu_{1}\cdots\mu_{n}}$ is twice that
of $\epsilon_{\mu_{1}\cdots\mu_{n-1}}$. The final degrees of freedom
$N_{0}(d,n)$ of this field in $d$ dimensions is then

$\displaystyle N_{0}(d,n)$ $\displaystyle={d+n-1\choose n}-{d+n-5\choose
n-4}-2\Bigg{(}{d+n-2\choose n-1}-{d+n-4\choose n-3}\Bigg{)}$
$\displaystyle=\frac{2n+d-4}{d-4}{d+n-5\choose n}$ (III.7)

which reduces to the $2$ helicities of a massless spin $n$ particle when
$d=4$.

The Bianchi identity (III.2) implies that some components of (III.1) offer
constraints on $\phi_{\mu_{1}\cdots\mu_{n}}$. Indeed, the following components
of the gauge invariant field strength

$\displaystyle\mathcal{F}_{0j_{1}\cdots
j_{n-1}},\quad\mathcal{F}^{0}_{\;\;0j_{1}\cdots
j_{n-2}}-\mathcal{F}^{i}_{\;\;ij_{1}\cdots j_{n-2}},$ (III.8)

where the $i,j$ indices run over spatial components, are first order in time.
These constraints do not affect the counting of $N_{0}(d,n)$, because in the
gauge (III.5), these components turn into dynamical equations. In other gauges
the counting of these constraints may become relevant.

In the aim of constructing a gauge invariant, quadratic in derivatives action
for $\phi_{\mu_{1}\cdots\mu_{n}}$, one must introduce an object
$\mathcal{G}_{\mu_{1}\cdots\mu_{n}}$ that is linear in
$\mathcal{F}_{\mu_{1}\cdots\mu_{n}}$ whose divergence is a pure trace. The
appropriate object is
$\mathcal{G}_{\mu_{1}\cdots\mu_{n}}=\mathcal{F}_{\mu_{1}\cdots\mu_{n}}-\frac{1}{4}n(n-1)\eta_{(\mu_{1}\mu_{2}}\mathcal{F}^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n})}$.
This object may be thought of as the spin $n$ generalization of the linearized
Einstein tensor $G_{\mu\nu}$. The divergence of
$\mathcal{G}_{\mu_{1}\cdots\mu_{n}}$ is

$\displaystyle\partial^{\lambda}\mathcal{G}_{\lambda\mu_{2}\cdots\mu_{n}}=-\frac{1}{4}n(n-1)(n-2)\eta_{(\mu_{2}\mu_{3}}\partial^{\lambda}\mathcal{F}^{\omega}_{\;\;\omega\lambda\mu_{4}\cdots\mu_{n})}$
(III.9)

This is a direct consequence of (III.2). Using this, one can construct a gauge
invariant action that generates the equations of motion (III.1), which we will
write in terms of $\mathcal{F}_{\mu_{1}\cdots\mu_{n}}$

$\displaystyle S_{n}=\frac{1}{2}\int
d^{d}X\phi^{\mu_{1}\cdots\mu_{n}}\Big{(}\mathcal{F}_{\mu_{1}\cdots\mu_{n}}-\frac{1}{4}n(n-1)\eta_{(\mu_{1}\mu_{2}}\mathcal{F}^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n})}\Big{)}$
(III.10)

And hence the equations of motion derived from this is the spin $n$
generalization of the linearized Einstein field equations

$\displaystyle\mathcal{G}_{\mu_{1}\cdots\mu_{n}}=\mathcal{F}_{\mu_{1}\cdots\mu_{n}}-\frac{1}{4}n(n-1)\eta_{(\mu_{1}\mu_{2}}\mathcal{F}^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n})}=0$
(III.11)

Which is equivalent to (III.1). From (III.10) it is in principle possible to
calculate the propagator
$\langle\phi_{\mu_{1}\cdots\mu_{n}}(X)\phi_{\nu_{1}\cdots\nu_{n}}(Y)\rangle$.
However, the many indices and symmetrizations involved makes such a
computation prohibitive for general $n$.

###  III.1 Transition to hyperfields

The transition to hyperfields is straightforward, and we simply list some of
the various formulas above in terms of $\Phi_{n}(X,s)$.

$\displaystyle\phi^{\lambda\;\;\omega}_{\;\;\lambda\;\;\omega\mu_{5}\cdots\mu_{n}}=0\qquad$
$\displaystyle\longrightarrow\qquad$
$\displaystyle(\partial_{s}^{2})^{2}\Phi_{n}(X,s)=0$ (III.12)
$\displaystyle\delta\phi_{\mu_{1}\cdots\mu_{n}}=n\partial_{(\mu_{1}}\epsilon_{\mu_{2}\cdots\mu_{n})}\qquad$
$\displaystyle\longrightarrow\qquad$
$\displaystyle\delta\Phi_{n}(X,s)=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}(X,s)$
(III.13)
$\displaystyle\epsilon^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n-1}}=0\qquad$
$\displaystyle\longrightarrow\qquad$
$\displaystyle\partial_{s}^{2}\epsilon_{n-1}(X,s)=0$ (III.14)
$\displaystyle\mathcal{F}_{\mu_{1}\cdots\mu_{n}}=\partial^{2}\phi_{\mu_{1}\cdots\mu_{n}}-n\partial_{(\mu_{1}}\partial^{\lambda}$
$\displaystyle\phi_{\lambda\mu_{2}\cdots\mu_{n})}+\frac{1}{2}n(n-1)\partial_{(\mu_{1}}\partial_{\mu_{2}}\phi^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n})}=0$
$\displaystyle\downarrow$ (III.15)
$\displaystyle\mathcal{F}_{n}(X,s)=\partial_{X}^{2}\Phi_{n}(X,s)-s\cdot\partial_{X}$
$\displaystyle\partial_{s}\cdot\partial_{X}\Phi_{n}(X,s)+\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\Phi_{n}(X,s)=0$

The action for $\Phi_{n}(X,s)$ then takes the form

$\displaystyle S_{n}$ $\displaystyle=\frac{1}{2}n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}\Phi_{n}(X,s)\big{(}1-\frac{1}{4}s^{\prime
2}\partial_{s^{\prime}}^{2}\big{)}\mathcal{F}_{n}(X,s^{\prime})$ (III.16)

The messy algebraic combinations of $\phi_{\mu_{1}\cdots\mu_{n}}$ needed to
describe massless particles are now recast as differential operators in the
$2d$ dimensional space $(X^{\mu},s^{\mu})$.

Gauge invariance of (III.16) is guaranteed by the identity (III.2)

$\displaystyle\partial_{s}\cdot\partial_{X}\mathcal{F}_{n}-\frac{1}{2}s\cdot\partial_{X}\partial_{s}^{2}\mathcal{F}_{n}=0$
(III.17)

###  III.2 All propagators

We now compute the propagator $\langle\Phi_{n}(X,s)\Phi_{n}(Y,t)\rangle$ from
(III.16), which will give the expressions for all massless integer spin
propagators
$\langle\phi_{\mu_{1}\cdots\mu_{n}}(X)\phi_{\nu_{1}\cdots\nu_{n}}(Y)\rangle$.

This action is gauge invariant under
$\delta\Phi_{n}=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}$, and so there will
be an ambiguity in how we define the propagator. To see this, consider adding
an external hyperfield source $\mathcal{J}_{n}(X,s)$ to the equations of
motion

$\displaystyle\big{(}1-\frac{1}{4}s^{2}\partial_{s}^{2}\big{)}\mathcal{F}_{n}=-\mathcal{J}_{n}$
(III.18)

The left hand side is double traceless, and satisfies (III.9). Consistent
external sources $\mathcal{J}_{n}$ must be such that these constraints are
maintained. This imposes constraints on $\mathcal{J}_{n}$

$\displaystyle(\partial_{s}^{2})^{2}\mathcal{J}_{n}=0$
$\displaystyle\partial_{s}\cdot\partial_{X}\mathcal{J}_{n}=-i^{1/2}s^{2}\mathcal{K}_{n-3}$
(III.19)

for some traceless hyperfield $\mathcal{K}_{n-3}$. Solving (III.18) for
$\Phi_{n}(X,s)$, we expect a formula of the sort

$\displaystyle\Phi_{n}(X,s)=i\int
d^{d}Y\frac{d^{d}td^{d}t^{\prime}}{(2\pi)^{d}}e^{it\cdot
t^{\prime}}\mathcal{G}(X-Y,s,t)\mathcal{J}_{n}(Y,t^{\prime})$ (III.20)

The transformation
$\delta\mathcal{G}(X-Y,s,t)=i^{-1/2}s\cdot\partial_{X}\Pi(X-Y,s,t)$, where
$\Pi(X-Y,s,t)$ is a rank $n-1$ traceless hyperfield in $s$ and a rank $n$
double traceless hyperfield in $t$, amounts to a gauge transformation on
$\Phi_{n}(X,s)$, so $\mathcal{G}(X-Y,s,t)$ itself has a gauge redundancy. For
the same reason, $\mathcal{G}(X-Y,s,t)$ cannot be interpreted as a Green’s
function for (III.18). Because of the underlying gauge invariance, the kinetic
operator has a zero mode, so it is not possible to solve the equation

$\displaystyle\big{(}1-\frac{1}{4}s^{2}\partial_{s}^{2}\big{)}\big{(}\partial_{X}^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}+\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\mathcal{G}(X-Y,s,t)=\frac{1}{n!}i\delta^{d}(X-Y)P_{n,\eta^{2}}^{d}(s,t)$
(III.21)

where $P_{n,\eta^{2}}^{d}(s,t)$ is the projection operator onto rank $n$
double traceless hyperfields, which is

$\displaystyle
P_{n,\eta^{2}}^{d}(s,t)=\frac{1}{(\frac{d-2}{2})_{n}}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n}\Bigg{(}C_{n}^{\frac{d-2}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}+\Big{(}\frac{d-2}{2}+n-2\Big{)}C_{n-2}^{\frac{d-2}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}\Bigg{)}$ (III.22)

Instead, it is possible to find a $\mathcal{G}(X-Y,s,t)$ that satisfies

$\displaystyle\big{(}1-\frac{1}{4}s^{2}\partial_{s}^{2}\big{)}\big{(}\partial_{X}^{2}-s\cdot\partial_{X}\partial_{s}$
$\displaystyle\cdot\partial_{X}+\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\mathcal{G}(X-Y,s,t)$
$\displaystyle=\frac{1}{n!}i\delta^{d}(X-Y)P_{n,\eta^{2}}^{d}(s,t)+i^{-1/2}t\cdot\partial_{Y}\Omega(X-Y,s,t)$
(III.23)

for some function $\Omega(X-Y,s,t)$ which is a rank $n$ double traceless
hyperfield in $s$ and a rank $n-1$ traceless hyperfield in $t$. The
$\mathcal{G}(X-Y,s,t)$ which solves (III.2) provides a consistent solution to
(III.18) of the form (III.20) because

$\displaystyle\int d^{d}Y\frac{d^{d}td^{d}t^{\prime}}{(2\pi)^{d}}e^{it\cdot
t^{\prime}}i^{-1/2}t\cdot\partial_{Y}\Omega(X-Y,s,t)\mathcal{J}_{n}(Y,t^{\prime})=0$
(III.24)

for a source $\mathcal{J}_{n}$ which satisfies (III.19). In total then,
$\mathcal{G}(X-Y,s,t)$ is ambiguous up to transformations

$\displaystyle\delta\mathcal{G}(X-Y,s,t)=i^{-1/2}s\cdot\partial_{X}\Pi(X-Y,s,t)+i^{-1/2}t\cdot\partial_{Y}\Omega(X-Y,s,t)$
(III.25)

The discussion above holds equally true for the propagator
$\langle\Phi_{n}(X,s)\Phi_{n}(Y,t)\rangle$, i.e. it must satisfy the equation
(III.2) and is ambiguous up to transformations (III.25). The particular form
of $\Omega(X-Y,s,t)$ in (III.2) amounts to a partial gauge choice.

In practice, we can gauge fix $\Phi_{n}(X,s)$ so that the propagator is
unambiguous. For integer spins there is a Lorentz covariant gauge choice which
makes the calculation of the propagator simple. The gauge choice we make is
(III.5)

$\displaystyle\big{(}\partial_{s}\cdot\partial_{X}-\frac{1}{2}(s\cdot\partial_{X})\partial_{s}^{2}\big{)}\Phi_{n}(X,s)=0$
(III.26)

In this gauge, we have $\mathcal{F}_{n}(X,s)=\partial_{X}^{2}\Phi_{n}(X,s)$,
and the action becomes

$\displaystyle S_{n}=\frac{1}{2}n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}\Phi_{n}(X,s)\big{(}1-\frac{1}{4}s^{\prime
2}\partial_{s^{\prime}}^{2}\big{)}\partial_{X}^{2}\Phi_{n}(X,s^{\prime})$
(III.27)

To find the propagator, we must find the inverse of
$(1-\frac{1}{4}s^{2}\partial_{s}^{2})\partial_{X}^{2}$ in the subspace of
double traceless hyperfields $(\partial_{s}^{2})^{2}\Phi_{n}=0$. To make this
simpler, we decompose $\Phi_{n}$ into its traceless components
$\Phi_{n}(X,s)=A_{n}(X,s)+s^{2}B_{n-2}(X,s)$, with
$\partial_{s}^{2}A_{n}=\partial_{s}^{2}B_{n-2}=0$. In this gauge, $A_{n}$ and
$B_{n-2}$ decouple

$\displaystyle S_{n}=\frac{1}{2}n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot s^{\prime}}$
$\displaystyle\Big{(}A_{n}(X,s)\partial_{X}^{2}A_{n}(X,s^{\prime})$
$\displaystyle+B_{n-2}(X,s)(d+2(n-2))(d+2(n-3))\partial_{X}^{2}B_{n-2}(X,s^{\prime})\Big{)}$
(III.28)

The propagators for $A_{n}$ and $B_{n-2}$ satisfy the equations

$\displaystyle\partial_{X}^{2}\langle A_{n}(X,s)A_{n}(Y,t)\rangle$
$\displaystyle=\frac{1}{n!}i\delta^{d}(X-Y)P_{n,\eta}^{d}(s,t)$ (III.29)
$\displaystyle(d+2(n-2))(d+2(n-3))\partial_{X}^{2}\langle
B_{n-2}(X,s)B_{n-2}(Y,t)\rangle$
$\displaystyle=\frac{1}{n!}i\delta^{d}(X-Y)P_{n,\eta}^{d}(s,t)$ (III.30)

where the insertion of $P_{{}_{n}\eta}^{d}$ from (II.10) on the right hand
sides are necessary to maintain tracelessness. These two equations can be
solved, and when combined back into $\Phi_{n}$ gives

$\displaystyle\langle\Phi_{n}(X,s)\Phi_{n}(Y,t)\rangle$
$\displaystyle=G_{0}(X-Y)\frac{1}{n!}P_{n,\eta}^{d-2}(s,t)$ (III.31)
$\displaystyle=G_{0}(X-Y)\frac{1}{n!}\frac{1}{(\frac{d-4}{2})_{n}}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n}C_{n}^{\frac{d-4}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (III.32)

where $G_{0}(X-Y)$ is the standard massless spin $0$ propagator. This formula
follows directly from the recursive relation
$(n+\alpha)C^{\alpha}_{n}=\alpha(C^{\alpha+1}_{n}-C^{\alpha+1}_{n-2})$. This
propagator agrees with previous work [60, 47].

For $d=4$, the Gegenbauer polynomials showing up in the propagator reduce to
Chebyshev polynomials of the first kind $T_{n}(x)$

$\displaystyle\lim_{d\to
4}\frac{1}{(\frac{d-4}{2})_{n}}C_{n}^{\frac{d-4}{2}}(x)=\begin{cases}1&\quad
n=0\\\ \frac{2}{n!}T_{n}(x)&\quad n>0\end{cases}$ (III.33)

The massless spin $n$ propagators
$\langle\phi_{\mu_{1}\cdots\mu_{n}}(X)\phi_{\nu_{1}\cdots\nu_{n}}(Y)\rangle$
from (III.10) can be obtained from $\langle\Phi_{n}(X,s)\Phi_{n}(Y,t)\rangle$
by applying derivatives

$\displaystyle\langle\phi_{\mu_{1}\cdots\mu_{n}}(X)\phi_{\nu_{1}\cdots\nu_{n}}(Y)\rangle=i^{n}\frac{\partial^{n}}{\partial_{s}^{\mu_{1}}\cdots\partial_{s}^{\mu_{n}}}\frac{\partial^{n}}{\partial_{t}^{\nu_{1}}\cdots\partial_{t}^{\nu_{n}}}\langle\Phi_{n}(X,s)\Phi_{n}(Y,t)\rangle$
(III.34)

Since the $(s,t)$ dependence of the propagator is nicely factored out of the
spacetime dependence, we list the index dependence of the first few
propagators

$\displaystyle n=0:$ $\displaystyle 1$ (III.35) $\displaystyle n=1:$
$\displaystyle\eta_{\mu_{1}\nu_{1}}$ (III.36) $\displaystyle n=2:$
$\displaystyle\frac{1}{2}\big{(}\eta_{\mu_{1}\nu_{1}}\eta_{\nu_{2}\mu_{2}}+\eta_{\mu_{1}\nu_{2}}\eta_{\nu_{1}\mu_{2}}\big{)}-\frac{1}{d-2}\eta_{\mu_{1}\mu_{2}}\eta_{\nu_{1}\nu_{2}}$
(III.37) $\displaystyle n=3:$
$\displaystyle\frac{1}{6}\big{(}\eta_{\mu_{1}\nu_{1}}\eta_{\nu_{2}\mu_{2}}\eta_{\mu_{3}\nu_{3}}+\eta_{\mu_{1}\nu_{1}}\eta_{\nu_{2}\mu_{3}}\eta_{\mu_{2}\nu_{3}}+\eta_{\mu_{2}\nu_{1}}\eta_{\nu_{2}\mu_{3}}\eta_{\mu_{1}\nu_{3}}$
$\displaystyle\phantom{+}+\eta_{\mu_{1}\nu_{2}}\eta_{\nu_{1}\mu_{2}}\eta_{\mu_{3}\nu_{3}}+\eta_{\mu_{1}\nu_{2}}\eta_{\nu_{1}\mu_{3}}\eta_{\mu_{2}\nu_{3}}+\eta_{\mu_{2}\nu_{2}}\eta_{\nu_{1}\mu_{3}}\eta_{\mu_{1}\nu_{3}}\big{)}$
$\displaystyle-\frac{1}{3d}\big{(}\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\nu_{1}}\eta_{\nu_{2}\nu_{3}}+\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\nu_{2}}\eta_{\nu_{3}\nu_{1}}+\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\nu_{3}}\eta_{\nu_{1}\nu_{2}}$
$\displaystyle\hskip
28.45274pt+\eta_{\mu_{2}\mu_{3}}\eta_{\mu_{1}\nu_{1}}\eta_{\nu_{2}\nu_{3}}+\eta_{\mu_{2}\mu_{3}}\eta_{\mu_{1}\nu_{2}}\eta_{\nu_{3}\nu_{1}}+\eta_{\mu_{2}\mu_{3}}\eta_{\mu_{1}\nu_{3}}\eta_{\nu_{1}\nu_{2}}$
$\displaystyle\hskip
28.45274pt+\eta_{\mu_{3}\mu_{1}}\eta_{\mu_{2}\nu_{1}}\eta_{\nu_{2}\nu_{3}}+\eta_{\mu_{3}\mu_{1}}\eta_{\mu_{2}\nu_{2}}\eta_{\nu_{3}\nu_{1}}+\eta_{\mu_{3}\mu_{1}}\eta_{\mu_{2}\nu_{3}}\eta_{\nu_{1}\nu_{2}}\big{)}$
(III.38)

The number of terms in the propagator with exposed indices grows factorially
with $n$, making it impractical to do calculations with them. Instead,
calculations can be done strictly in terms of the propagator (III.31), using
the operations (II.2), (II.3), (II.4), and (II.5).

Possibly the most one gains from the expression (III.31) is the ability to
have full control of the large spin limit. Classical Gegenbauer polynomials
have well studied large $n$ asymptotic limits [20], allowing
$\langle\Phi_{n}(X,s)\Phi_{n}(Y,t)\rangle$ to be written in terms of simple
trigonometric functions at large $n$

$\displaystyle\langle\Phi_{n}(X,s)\Phi_{n}(Y,t)\rangle\sim
G_{0}(X-Y)\frac{1}{2^{n}(n!)^{2}}(-i\sqrt{s^{2}t^{2}})^{n}\frac{2\cos\big{(}(n+\frac{d-4}{2})\theta-\frac{\pi}{2}\frac{d-4}{2}\big{)}}{(2\sin\theta)^{\frac{d-4}{2}}}$
(III.39)

as $n\to\infty$, where $\cos\theta=s\cdot t/\sqrt{s^{2}t^{2}}$, for
$0<\theta<\pi$.

###  III.3 Other gauges

Often it is useful to compute observables using propagators in different
gauges as a consistency check. Furthermore, there may be gauges other than
(III.31) which make computations simpler. For instance, using the gauge
transformation (III.25) one may always in momentum space add a term to the
propagator like

$\displaystyle\xi(p^{2})\frac{-i}{p^{2}}\frac{s\cdot p\,t\cdot
p}{p^{2}}\frac{1}{(\frac{d-\alpha-2}{2})_{n-1}}\frac{1}{n!}\Big{(}\frac{1}{2}\sqrt{s\cdot
A\cdot s\,t\cdot A\cdot
t}\Big{)}^{n-1}C_{n-1}^{\frac{d-\alpha-2}{2}}\Big{(}\frac{s\cdot A\cdot
t}{\sqrt{s\cdot A\cdot s\,t\cdot A\cdot t}}\Big{)}$ (III.40)

where $\xi(p^{2})$ is an arbitrary function, and $A_{\mu\nu}(p)$ is a
projection matrix of rank $d-\alpha$. Some covariant choices of
$A_{\mu\nu}(p)$ include e.g. $\eta_{\mu\nu}$, or
$\eta_{\mu\nu}-p_{\mu}p_{\nu}/p^{2}$. One special modification of the
propagator is

$\displaystyle\langle\Phi_{n}(p,s)\Phi_{n}(-p,t)\rangle=$
$\displaystyle\frac{-i}{p^{2}}\frac{1}{n!}\Bigg{(}\frac{1}{(\frac{d-4}{2})_{n}}\Big{(}\frac{1}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n}C_{n}^{\frac{d-4}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}-\frac{s\cdot p\,t\cdot
p}{p^{2}}\frac{1}{(\frac{d-2}{2})_{n-1}}\Big{(}\frac{1}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-1}C_{n-1}^{\frac{d-2}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}\Bigg{)}$ (III.41)

Indeed, this choice may be thought of as the spin $n$ generalization of the
spin 1 Landau gauge, because

$\displaystyle(p\cdot\partial_{s}-\frac{1}{2}s\cdot
p\,\partial_{s}^{2})\langle\Phi_{n}(p,s)\Phi_{n}(-p,t)\rangle=0$ (III.42)

Though because this propagator satisfies the Bianchi identity (III.26), we
refer to this as the Bianchi gauge.

##  IV Massless half integer spins

In this section, we will use the hyperfield formalism of section II to review
massless half integer spin fields. In order to display uniform results for any
dimension, the discussion will be restricted to Dirac spinors, similar results
can be found for other spinor representations. We will again find compact,
closed form expressions for their covariant actions and propagators. First, we
review the standard description of massless half integer spin fields
$\psi_{\mu_{1}\cdots\mu_{n}}(X)$.

The covariant formulation of massless half integer spin fields
$\psi_{\mu_{1}\cdots\mu_{n}}(X)$ was first worked out in [6]. In this
description, Fang and Fronsdal found that it was possible to write down a
covariant action for $\psi_{\mu_{1}\cdots\mu_{n}}$ which is linear in
derivatives, provided that $\psi_{\mu_{1}\cdots\mu_{n}}$ is a symmetric tensor
that satisfies the unusual triple $\gamma$ traceless condition
$\gamma^{\mu_{1}}\gamma^{\mu_{2}}\gamma^{\mu_{3}}\psi_{\mu_{1}\cdots\mu_{n}}=\gamma^{\omega}\psi^{\lambda}_{\;\;\lambda\omega\mu_{4}\cdots\mu_{n}}=0$.
The equations of motion are

$\displaystyle\mathcal{S}_{\mu_{1}\cdots\mu_{n}}=\mathrlap{\\!\not{\phantom{\partial}}}\partial\psi_{\mu_{1}\cdots\mu_{n}}-n\partial_{(\mu_{1}}\gamma^{\lambda}\psi_{\lambda\mu_{2}\cdots\mu_{n})}=0$
(IV.1)

where we will call $\mathcal{S}_{\mu_{1}\cdots\mu_{n}}$ the Fang field
strength. $\mathcal{S}_{\mu_{1}\cdots\mu_{n}}$ satisfies a fermionic analog of
the Bianchi identity (III.2)

$\displaystyle\gamma^{\lambda}\mathrlap{\\!\not{\phantom{\partial}}}\partial\mathcal{S}_{\lambda\mu_{1}\cdots\mu_{n-1}}-(n-1)\partial_{(\mu_{1}}\mathcal{S}^{\lambda}_{\;\;\lambda\mu_{2}\cdots\mu_{n-1})}=0$
(IV.2)

As with Rarita-Schwinger theory, these equations of motion (IV.1) have a gauge
redundancy. Under the gauge transformation

$\displaystyle\psi_{\mu_{1}\cdots\mu_{n}}(X)^{\prime}=\psi_{\mu_{1}\cdots\mu_{n}}(X)+n\partial_{(\mu_{1}}\epsilon_{\mu_{2}\cdots\mu_{n})}(X)$
(IV.3)

the Fang field strength is invariant provided that
$\epsilon_{\mu_{1}\cdots\mu_{n-1}}$ is a symmetric and $\gamma$ traceless, but
otherwise arbitrary function, i.e.

$\displaystyle\delta\mathcal{S}_{\mu_{1}\cdots\mu_{n}}=-n(n-1)\partial_{(\mu_{1}}\partial_{\mu_{2}}\gamma^{\lambda}\epsilon_{\lambda\mu_{3}\cdots\mu_{n})}=0$
(IV.4)

This gauge redundancy allows us to impose a gauge fixing condition. The
natural choice is the fermionic analog of (III.5)

$\displaystyle\gamma^{\lambda}\mathrlap{\\!\not{\phantom{\partial}}}\partial\psi_{\lambda\mu_{1}\cdots\mu_{n-1}}-(n-1)\partial_{(\mu_{1}}\psi^{\lambda}_{\;\;\lambda\mu_{2}\cdots\mu_{n-1})}=0$
(IV.5)

In this gauge, the $\gamma$ trace of the equations of motion (IV.1) reduces to

$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial(\gamma^{\lambda}\psi_{\lambda\mu_{1}\cdots\mu_{n-1}})=0$
(IV.6)

Applying $\mathrlap{\\!\not{\phantom{\partial}}}\partial$ on to
$\mathcal{S}_{\mu_{1}\cdots\mu_{n}}$ in this gauge then implies

$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial\mathcal{S}_{\mu_{1}\cdots\mu_{n}}=\partial^{2}\psi_{\mu_{1}\cdots\mu_{n}}=0$
(IV.7)

and so the particles described by $\psi_{\mu_{1}\cdots\mu_{n}}$ are indeed
massless. Just as in the bosonic case, in this gauge there remains a residual
gauge symmetry under a gauge parameter $\epsilon_{\mu_{1}\cdots\mu_{n-1}}$
satisfying $\partial^{2}\epsilon_{\mu_{1}\cdots\mu_{n-1}}=0$. Finally, one
should consider the constraints on $\psi_{\mu_{1}\cdots\mu_{n}}$ which are a
result of the Bianchi identity (IV.2). The following components of the gauge
invariant field strength

$\displaystyle\gamma^{0}\mathcal{S}_{0j_{1}\cdots
j_{n-1}}-\gamma^{i}\mathcal{S}_{ij_{1}\cdots j_{n-1}}$ (IV.8)

contain only spatial derivatives. In contrast to the bosonic case, these
remain as constraints in the gauge (IV.5) we picked. In total then, the number
of degrees of freedom removed from $\psi_{\mu_{1}\cdots\mu_{n}}$ is thrice
that of $\epsilon_{\mu_{1}\cdots\mu_{n-1}}$. The final degrees of freedom
$N_{0}(d,n+1/2)$ of this field in $d$ dimensions is then

$\displaystyle N_{0}(d,n+1/2)=$ $\displaystyle\,2^{\cramped{\left\lfloor
d/2\right\rfloor}}{d+n-1\choose n}-2^{\cramped{\left\lfloor
d/2\right\rfloor}}{d+n-4\choose n-3}$
$\displaystyle-3\Bigg{(}2^{\cramped{\left\lfloor
d/2\right\rfloor}}{d+n-2\choose n-1}-2^{\cramped{\left\lfloor
d/2\right\rfloor}}{d+n-3\choose n-2}\Bigg{)}$ $\displaystyle=$
$\displaystyle\,2^{\cramped{\left\lfloor d/2\right\rfloor}}{d+n-4\choose n}$
(IV.9)

which reduces to 4, corresponding to the 2 helicities of a massless spin
$n+1/2$ particle and anti-particle when $d=4$.

Again, in order to construct a gauge invariant, linear in derivatives action
for $\psi_{\mu_{1}\cdots\mu_{n}}$, one needs a object
$\mathcal{T}_{\mu_{1}\cdots\mu_{n}}$ that is linear in
$\mathcal{S}_{\mu_{1}\cdots\mu_{n}}$ whose divergence is a pure $\gamma$
trace. The appropriate object is
$\mathcal{T}_{\mu_{1}\cdots\mu_{n}}=\mathcal{S}_{\mu_{1}\cdots\mu_{n}}-\frac{1}{2}n\gamma_{(\mu_{1}}\gamma^{\lambda}\mathcal{S}_{\lambda\mu_{2}\cdots\mu_{n})}-\frac{1}{4}n(n-1)\eta_{(\mu_{1}\mu_{2}}\mathcal{S}^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n})}$,
and its divergence is

$\displaystyle\partial^{\lambda}\mathcal{T}_{\lambda\mu_{2}\cdots\mu_{n}}=-\frac{1}{2}n(n-1)\gamma_{(\mu_{2}}\gamma^{\lambda}\partial^{\omega}\mathcal{S}_{\omega\lambda\mu_{3}\cdots\mu_{n})}-\frac{1}{4}n(n-1)(n-2)\eta_{(\mu_{2}\mu_{3}}\partial^{\lambda}\mathcal{S}^{\omega}_{\;\;\omega\lambda\mu_{4}\cdots\mu_{n})}$
(IV.10)

This is a direct consequence of (IV.2). Using this, one can construct a gauge
invariant action that generates the equations of motion (IV.1), which we will
write in terms of $\mathcal{S}_{\mu_{1}\cdots\mu_{n}}$

$\displaystyle S_{n+1/2}=-\int
d^{d}X\overline{\psi}^{\mu_{1}\cdots\mu_{n}}\Big{(}\mathcal{S}_{\mu_{1}\cdots\mu_{n}}-\frac{1}{2}n\gamma_{(\mu_{1}}\gamma^{\lambda}\mathcal{S}_{\lambda\mu_{2}\cdots\mu_{n})}-\frac{1}{4}n(n-1)\eta_{(\mu_{1}\mu_{2}}\mathcal{S}^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n})}\Big{)}$
(IV.11)

where
$\overline{\psi}^{\mu_{1}\cdots\mu_{n}}=(\psi^{\mu_{1}\cdots\mu_{n}})^{\dagger}i\gamma^{0}$.
The equations of motion derived from this action are therefore

$\displaystyle\mathcal{T}_{\mu_{1}\cdots\mu_{n}}=\mathcal{S}_{\mu_{1}\cdots\mu_{n}}-\frac{1}{2}n\gamma_{(\mu_{1}}\gamma^{\lambda}\mathcal{S}_{\lambda\mu_{2}\cdots\mu_{n})}-\frac{1}{4}n(n-1)\eta_{(\mu_{1}\mu_{2}}\mathcal{S}^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n})}=0$
(IV.12)

which is equivalent to (IV.1).

###  IV.1 Transition to hyperfields

We now transition to hyperfields, listing some of the various formulas above
in terms of $\Psi_{n}(X,s)$.

$\displaystyle\gamma^{\omega}\psi^{\lambda}_{\;\;\lambda\omega\mu_{4}\cdots\mu_{n}}=0\qquad$
$\displaystyle\longrightarrow\qquad$
$\displaystyle(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})^{3}\Psi_{n}(X,s)=0$
(IV.13)
$\displaystyle\delta\psi_{\mu_{1}\cdots\mu_{n}}=n\partial_{(\mu_{1}}\epsilon_{\mu_{2}\cdots\mu_{n})}\qquad$
$\displaystyle\longrightarrow\qquad$
$\displaystyle\delta\Psi_{n}(X,s)=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}(X,s)$
(IV.14)
$\displaystyle\gamma^{\lambda}\epsilon_{\lambda\mu_{2}\cdots\mu_{n-1}}=0\qquad$
$\displaystyle\longrightarrow\qquad$
$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}(X,s)=0$
(IV.15)
$\displaystyle\mathcal{S}_{\mu_{1}\cdots\mu_{n}}=\mathrlap{\\!\not{\phantom{\partial}}}\partial\psi_{\mu_{1}\cdots\mu_{n}}-$
$\displaystyle
n\partial_{(\mu_{1}}\gamma^{\lambda}\psi_{\lambda\mu_{2}\cdots\mu_{n})}=0$
$\displaystyle\downarrow$ (IV.16)
$\displaystyle\mathcal{S}_{n}(X,s)=\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}\Psi_{n}(X,s)$
$\displaystyle-s\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}(X,s)=0$

The action for $\Psi_{n}(X,s)$ then takes the form

$\displaystyle S_{n+1/2}$ $\displaystyle=-n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}\overline{\Psi}_{n}(X,s)\big{(}1-\frac{1}{2}\mathrlap{\\!\not{\phantom{s^{\prime}}}}s^{\prime}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}-\frac{1}{4}s^{\prime
2}\partial^{2}_{s^{\prime}}\big{)}\mathcal{S}_{n}(X,s^{\prime})$ (IV.17)

where
$\overline{\Psi}_{n}(X,s)=\frac{1}{n!}i^{-n/2}\overline{\psi}_{\mu_{1}\cdots\mu_{n}}s^{\mu_{1}}\cdots
s^{\mu_{n}}$. Gauge invariance of (IV.17) is guaranteed by the identity

$\displaystyle(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-s\cdot\partial_{X}\partial_{s}^{2})\mathcal{S}_{n}(X,s)=0$
(IV.18)

###  IV.2 All propagators

We now compute the propagator
$\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle$ from (IV.17). The
discussion in subsection III.2 on the gauge ambiguity of the propagator
equally applies in this case. The propagator
$\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle$ will solve the equation

$\displaystyle\big{(}1-\frac{1}{2}\mathrlap{\\!\not{\phantom{s}}}s\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}-\frac{1}{4}s^{2}\partial_{s}^{2}\big{)}$
$\displaystyle\big{(}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-s\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\big{)}\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle$
$\displaystyle=-\frac{1}{n!}i\delta^{d}(X-Y)P_{n,\gamma^{3}}^{d}(s,t)+i^{-1/2}t\cdot\partial_{Y}\Omega(X-Y,s,t)$
(IV.19)

where $P_{n,\gamma^{3}}^{d}(s,t)$ is the projection operator onto rank $n$
triple $\gamma$ traceless hyperfields, which is

$\displaystyle
P_{n,\gamma^{3}}^{d}(s,t)=\frac{1}{(\frac{d}{2})_{n}}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n}\Bigg{(}C_{n}^{\frac{d}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}+\Big{(}\frac{d}{2}+n-2\Big{)}C_{n-2}^{\frac{d}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}\Bigg{)}$
$\displaystyle+\;i\frac{\mathrlap{\\!\not{\phantom{s}}}s\mathrlap{\\!\not{\phantom{t}}}t}{2}\frac{1}{(\frac{d}{2})_{n-1}}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-1}C_{n-3}^{\frac{d}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (IV.20)

and $\Omega(X-Y,s,t)$ is some function which is a rank $n$ triple $\gamma$
traceless hyperfield in $s$ and a rank $n-1$ $\gamma$ traceless hyperfield in
$t$
$(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})^{3}\Omega,\,\Omega\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial}_{t}=0$.
Because of this, the propagator is ambiguous up to transformations

$\displaystyle\delta\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle=i^{-1/2}s\cdot\partial_{X}\Pi(X-Y,s,t)+i^{-1/2}t\cdot\partial_{Y}\Omega(X-Y,s,t)$
(IV.21)

To find the propagator, one might proceed as in the integer spin case by gauge
fixing $\Psi_{n}(X,s)$. A natural gauge to pick is (IV.5)

$\displaystyle(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-s\cdot\partial_{X}\partial_{s}^{2})\Psi_{n}(X,s)=0$
(IV.22)

In the case of half integer spins however, this choice does not simplify the
analysis, and there does not appear to be a gauge choice which simplifies the
analysis in an analogous manner to the gauge choice for integer spins
(III.26), so we instead proceed directly by finding the simplest solution to
(IV.2). Working in momentum space, one can classify all possible structures
which have a simple pole at $p^{2}=0$, have a numerator linear in momentum,
and are not of the form (IV.21), which are triple $\gamma$ traceless in both
$s$ and $t$, and make an ansatz that
$\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle$ is a linear combination
of these in momentum space. These structures can be obtained from the three
basic structures

$\displaystyle\frac{\mathrlap{\\!\not{\phantom{p}}}p}{p^{2}}(s\cdot
t)^{n},\quad\frac{\mathrlap{\\!\not{\phantom{s}}}s\,\mathrlap{\\!\not{\phantom{p}}}p\,\mathrlap{\\!\not{\phantom{t}}}t}{p^{2}}(s\cdot
t)^{n},\quad s^{2}t^{2}\frac{\mathrlap{\\!\not{\phantom{p}}}p}{p^{2}}(s\cdot
t)^{n}$ (IV.23)

and applying various combinations of $P_{n,\gamma}^{d}$, $P_{n,\eta}^{d}$, and
$P_{n,\gamma^{3}}^{d}$ on either side. A tedious but straighforward
calculation reveals that this ansatz is sufficient and the propagator can be
taken to be

$\displaystyle\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle$
$\displaystyle=\Delta_{0}(X-Y)\frac{1}{n!}\frac{1}{\big{(}\frac{d-2}{2}\big{)}_{n}}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n}C_{n}^{\frac{d-2}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$
$\displaystyle-\frac{i}{2}\mathrlap{\\!\not{\phantom{s}}}s\,\Delta_{0}(X-Y)\,\mathrlap{\\!\not{\phantom{t}}}t\frac{1}{n!}\frac{1}{\big{(}\frac{d-2}{2}\big{)}_{n}}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-1}C_{n-1}^{\frac{d-2}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (IV.24)

where $\Delta_{0}(X-Y)$ is the standard massless spin $1/2$ propagator. This
propagator agrees with previous work [60]. As in the integer spin case, the
massless spin $n+1/2$ propagators
$\langle\psi_{\mu_{1}\cdots\mu_{n}}(X)\overline{\psi}_{\nu_{1}\cdots\nu_{n}}(Y)\rangle$
from (IV.11) can be obtained from
$\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle$ straightforwardly by
applying derivatives

$\displaystyle\langle\psi_{\mu_{1}\cdots\mu_{n}}(X)\overline{\psi}_{\nu_{1}\cdots\nu_{n}}(Y)\rangle=i^{n}\frac{\partial^{n}}{\partial_{s}^{\mu_{1}}\cdots\partial_{s}^{\mu_{n}}}\frac{\partial^{n}}{\partial_{t}^{\nu_{1}}\cdots\partial_{t}^{\nu_{n}}}\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle$
(IV.25)

We list here the numerator of the first few propagators in momentum space

$\displaystyle n+1/2=1/2:$ $\displaystyle\mathrlap{\\!\not{\phantom{p}}}p$
(IV.26) $\displaystyle n+1/2=3/2:$
$\displaystyle\mathrlap{\\!\not{\phantom{p}}}p\,\eta_{\mu_{1}\nu_{1}}+\frac{\gamma_{\mu_{1}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{1}}}{d-2}$
(IV.27) $\displaystyle n+1/2=5/2:$
$\displaystyle\mathrlap{\\!\not{\phantom{p}}}p\,\Big{(}\frac{1}{2}\big{(}\eta_{\mu_{1}\nu_{1}}\eta_{\nu_{2}\mu_{2}}+\eta_{\mu_{1}\nu_{2}}\eta_{\nu_{1}\mu_{2}}\big{)}-\frac{1}{d}\eta_{\mu_{1}\mu_{2}}\eta_{\nu_{1}\nu_{2}}\Big{)}$
$\displaystyle+\frac{1}{2d}\big{(}\eta_{\mu_{1}\nu_{2}}\gamma_{\mu_{2}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{1}}+\eta_{\mu_{2}\nu_{1}}\gamma_{\mu_{1}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{2}}+\eta_{\mu_{1}\nu_{1}}\gamma_{\mu_{2}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{2}}+\eta_{\mu_{2}\nu_{2}}\gamma_{\mu_{1}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{1}}\big{)}$
(IV.28) $\displaystyle n+1/2=7/2:$
$\displaystyle\mathrlap{\\!\not{\phantom{p}}}p\Big{(}\frac{1}{6}\big{(}\eta_{\mu_{1}\nu_{1}}\eta_{\nu_{2}\mu_{2}}\eta_{\mu_{3}\nu_{3}}+\eta_{\mu_{1}\nu_{1}}\eta_{\nu_{2}\mu_{3}}\eta_{\mu_{2}\nu_{3}}+\eta_{\mu_{2}\nu_{1}}\eta_{\nu_{2}\mu_{3}}\eta_{\mu_{1}\nu_{3}}$
$\displaystyle\phantom{+}+\eta_{\mu_{1}\nu_{2}}\eta_{\nu_{1}\mu_{2}}\eta_{\mu_{3}\nu_{3}}+\eta_{\mu_{1}\nu_{2}}\eta_{\nu_{1}\mu_{3}}\eta_{\mu_{2}\nu_{3}}+\eta_{\mu_{2}\nu_{2}}\eta_{\nu_{1}\mu_{3}}\eta_{\mu_{1}\nu_{3}}\big{)}$
$\displaystyle-\frac{1}{3(d+2)}\big{(}\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\nu_{1}}\eta_{\nu_{2}\nu_{3}}+\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\nu_{2}}\eta_{\nu_{3}\nu_{1}}+\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\nu_{3}}\eta_{\nu_{1}\nu_{2}}$
$\displaystyle\hskip
56.9055pt+\eta_{\mu_{2}\mu_{3}}\eta_{\mu_{1}\nu_{1}}\eta_{\nu_{2}\nu_{3}}+\eta_{\mu_{2}\mu_{3}}\eta_{\mu_{1}\nu_{2}}\eta_{\nu_{3}\nu_{1}}+\eta_{\mu_{2}\mu_{3}}\eta_{\mu_{1}\nu_{3}}\eta_{\nu_{1}\nu_{2}}$
$\displaystyle\hskip
56.9055pt+\eta_{\mu_{3}\mu_{1}}\eta_{\mu_{2}\nu_{1}}\eta_{\nu_{2}\nu_{3}}+\eta_{\mu_{3}\mu_{1}}\eta_{\mu_{2}\nu_{2}}\eta_{\nu_{3}\nu_{1}}+\eta_{\mu_{3}\mu_{1}}\eta_{\mu_{2}\nu_{3}}\eta_{\nu_{1}\nu_{2}}\big{)}\Big{)}$
$\displaystyle+\frac{1}{3(d+2)}\Big{(}\big{(}\frac{1}{2}(\eta_{\mu_{2}\nu_{2}}\eta_{\mu_{3}\nu_{3}}+\eta_{\mu_{2}\nu_{3}}\eta_{\mu_{3}\nu_{2}})-\frac{1}{d}\eta_{\mu_{2}\mu_{3}}\eta_{\nu_{2}\nu_{3}}\big{)}\gamma_{\mu_{1}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{1}}$
$\displaystyle\hskip
64.01869pt+\big{(}\frac{1}{2}(\eta_{\mu_{2}\nu_{1}}\eta_{\mu_{3}\nu_{3}}+\eta_{\mu_{3}\nu_{1}}\eta_{\mu_{2}\nu_{3}})-\frac{1}{d}\eta_{\mu_{2}\mu_{3}}\eta_{\nu_{1}\nu_{3}}\big{)}\gamma_{\mu_{1}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{2}}$
$\displaystyle\hskip
64.01869pt+\big{(}\frac{1}{2}(\eta_{\mu_{1}\nu_{2}}\eta_{\mu_{3}\nu_{3}}+\eta_{\mu_{3}\nu_{2}}\eta_{\mu_{1}\nu_{3}})-\frac{1}{d}\eta_{\mu_{1}\mu_{3}}\eta_{\nu_{2}\nu_{3}}\big{)}\gamma_{\mu_{2}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{1}}$
$\displaystyle\hskip
64.01869pt+\big{(}\frac{1}{2}(\eta_{\mu_{2}\nu_{1}}\eta_{\mu_{3}\nu_{2}}+\eta_{\mu_{3}\nu_{1}}\eta_{\mu_{2}\nu_{2}})-\frac{1}{d}\eta_{\mu_{2}\mu_{3}}\eta_{\nu_{1}\nu_{2}}\big{)}\gamma_{\mu_{1}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{3}}$
$\displaystyle\hskip
64.01869pt+\big{(}\frac{1}{2}(\eta_{\mu_{1}\nu_{2}}\eta_{\mu_{2}\nu_{3}}+\eta_{\mu_{2}\nu_{2}}\eta_{\mu_{1}\nu_{3}})-\frac{1}{d}\eta_{\mu_{1}\mu_{2}}\eta_{\nu_{2}\nu_{3}}\big{)}\gamma_{\mu_{3}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{1}}$
$\displaystyle\hskip
64.01869pt+\big{(}\frac{1}{2}(\eta_{\mu_{1}\nu_{1}}\eta_{\mu_{3}\nu_{3}}+\eta_{\mu_{3}\nu_{1}}\eta_{\mu_{1}\nu_{3}})-\frac{1}{d}\eta_{\mu_{1}\mu_{3}}\eta_{\nu_{1}\nu_{3}}\big{)}\gamma_{\mu_{2}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{2}}$
$\displaystyle\hskip
64.01869pt+\big{(}\frac{1}{2}(\eta_{\mu_{1}\nu_{1}}\eta_{\mu_{3}\nu_{2}}+\eta_{\mu_{3}\nu_{1}}\eta_{\mu_{1}\nu_{2}})-\frac{1}{d}\eta_{\mu_{1}\mu_{3}}\eta_{\nu_{1}\nu_{2}}\big{)}\gamma_{\mu_{2}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{3}}$
$\displaystyle\hskip
64.01869pt+\big{(}\frac{1}{2}(\eta_{\mu_{1}\nu_{1}}\eta_{\mu_{2}\nu_{3}}+\eta_{\mu_{2}\nu_{1}}\eta_{\mu_{1}\nu_{3}})-\frac{1}{d}\eta_{\mu_{1}\mu_{2}}\eta_{\nu_{1}\nu_{3}}\big{)}\gamma_{\mu_{3}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{2}}$
$\displaystyle\hskip
64.01869pt+\big{(}\frac{1}{2}(\eta_{\mu_{1}\nu_{1}}\eta_{\mu_{2}\nu_{2}}+\eta_{\mu_{2}\nu_{1}}\eta_{\mu_{1}\nu_{2}})-\frac{1}{d}\eta_{\mu_{1}\mu_{2}}\eta_{\nu_{1}\nu_{2}}\big{)}\gamma_{\mu_{3}}\,\mathrlap{\\!\not{\phantom{p}}}p\,\gamma_{\nu_{3}}\Big{)}$
(IV.29)

As before, this propagator has a well defined large $n$ asymptotic limit

$\displaystyle\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle\sim$
$\displaystyle\Big{(}\Delta_{0}(X-Y)+\frac{s\cdot
t}{s^{2}t^{2}}\,\mathrlap{\\!\not{\phantom{s}}}s\,\Delta_{0}(X-Y)\,\mathrlap{\\!\not{\phantom{t}}}t\Big{)}\frac{1}{2^{n}(n!)^{2}}(-i\sqrt{s^{2}t^{2}})^{n}\frac{2\cos\big{(}(n+\frac{d-2}{2})\theta-\frac{\pi}{2}\frac{d-2}{2}\big{)}}{(2\sin\theta)^{\frac{d-2}{2}}}$
$\displaystyle+\frac{\sqrt{s^{2}t^{2}-(s\cdot
t)^{2}}}{s^{2}t^{2}}\,\mathrlap{\\!\not{\phantom{s}}}s\,\Delta_{0}(X-Y)\,\mathrlap{\\!\not{\phantom{t}}}t\frac{1}{2^{n}(n!)^{2}}(-i\sqrt{s^{2}t^{2}})^{n}\frac{2\sin\big{(}(n+\frac{d-2}{2})\theta-\frac{\pi}{2}\frac{d-2}{2}\big{)}}{(2\sin\theta)^{\frac{d-2}{2}}}$
(IV.30)

as $n\to\infty$, where $\cos\theta=s\cdot t/\sqrt{s^{2}t^{2}}$, for
$0<\theta<\pi$.

###  IV.3 Other gauges

Just as for the integer spin propagator, it may be useful to perform
computations in different gauges as a consistency check. Using the gauge
transformation (IV.21), possible terms one may add to the propagator (IV.2) in
momentum space include

$\displaystyle\frac{\xi(p^{2})}{p^{2}}\frac{1}{(\frac{d-\alpha}{2})_{n-1}}\frac{1}{n!}$
$\displaystyle\Bigg{(}(s\cdot
p\,\tilde{\mathrlap{\\!\not{\phantom{t}}}t}+t\cdot
p\,\tilde{\mathrlap{\\!\not{\phantom{s}}}s})\Big{(}\frac{1}{2}\sqrt{s\cdot
A\cdot s\,t\cdot A\cdot
t}\Big{)}^{n-1}C_{n-1}^{\frac{d-\alpha}{2}}\Big{(}\frac{s\cdot A\cdot
t}{\sqrt{s\cdot A\cdot s\,t\cdot A\cdot t}}\Big{)}$
$\displaystyle-\frac{1}{2}\tilde{\mathrlap{\\!\not{\phantom{s}}}s}(s\cdot
p\,\tilde{\mathrlap{\\!\not{\phantom{t}}}t}+t\cdot
p\,\tilde{\mathrlap{\\!\not{\phantom{s}}}s})\tilde{t}\Big{(}\frac{1}{2}\sqrt{s\cdot
A\cdot s\,t\cdot A\cdot
t}\Big{)}^{n-2}C_{n-2}^{\frac{d-\alpha}{2}}\Big{(}\frac{s\cdot A\cdot
t}{\sqrt{s\cdot A\cdot s\,t\cdot A\cdot t}}\Big{)}\Bigg{)}$ (IV.31)

where $\xi(p^{2})$ is an arbitrary function, $A_{\mu\nu}(p)$ is a projection
matrix of rank $d-\alpha$, and $\tilde{\mathrlap{\\!\not{\phantom{s}}}s}$ for
instance stands for contracting $s^{\mu}$ with $A_{\mu\nu}\gamma^{\nu}$.

The propagator (IV.2) is already in what may be thought of as the fermionic
Bianchi gauge, in the sense that in momentum space

$\displaystyle(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\mathrlap{\\!\not{\phantom{p}}}p-s\cdot
p\partial_{s}^{2})\langle\Psi_{n}(p,s)\overline{\Psi}_{n}(-p,t)\rangle=0$
(IV.32)

##  V Massive integer spins

In this section, we will use the hyperfield formalism of section II to study
massive integer spin fields. We will find compact, closed form expressions for
their covariant actions and propagators.

Historically it was realized that the following system of equations is
sufficient for describing a freely propagating spin $n$ massive particle with
positive definite energy [61, 2]

$\displaystyle(\partial^{2}-m^{2})\phi_{\mu_{1}\cdots\mu_{n}}=0$ (V.1)
$\displaystyle\partial^{\lambda}\phi_{\lambda\mu_{2}\cdots\mu_{n}}=0$ (V.2)
$\displaystyle\phi^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n}}=0$ (V.3)

for a symmetric rank $n$ tensor. We will refer to the system of equations
(V.1)–(V.3) as a Fierz-Pauli system. The transverse and traceless conditions
ensure that the correct number $N_{m}(d,n)$ of degrees of freedom for a spin
$n$ particle in $d$ spacetime dimensions propagate with mass $m$

$\displaystyle N_{m}(d,n)$ $\displaystyle={d+n-1\choose n}-{d+n-3\choose
n-2}-\Bigg{(}{d+n-2\choose n-1}-{d+n-4\choose n-3}\Bigg{)}$
$\displaystyle=\frac{2n+d-3}{d-3}{d+n-4\choose n}$ (V.4)

which reduces to $2n+1$ when $d=4$. Attempts to construct a covariant action
whose equations of motion imposes (V.1)–(V.3) was met with considerable
difficulty for over three decades. It was noted in [2] that it is impossible
to construct an action principle which produces a Fierz-Pauli system out of
just a symmetric and traceless field $\phi_{\mu_{1}\cdots\mu_{n}}$. Additional
auxiliary fields are in general necessary to impose the transverse constraint
(V.2). How these auxiliary fields are introduced is not unique, but there is a
“minimal” set of auxiliary fields necessary in this construction.

A minimal covariant Lagrangian formulation of massive integer spin fields
$\phi_{\mu_{1}\cdots\mu_{n}}$ in four spacetime dimensions was first worked
out in [3]. In it, Singh and Hagen needed to introduce, apart from the
symmetric and traceless rank $n$ tensor, symmetric and traceless tensors of
rank $0,1,2,\dots,n-2$ which intricately couple so that all fields of rank
lower than $n$ vanish on shell, and the rank $n$ tensor made up a Fierz-Pauli
system. Note that this field content is equivalent to using symmetric rank $n$
and $n-3$ tensors, with no tracelessness conditions.

Because this formulation only works in four spacetime dimensions, we will not
present the action, and will opt in the following subsection instead to
construct a formulation which works in any dimension. What will continue to be
same is the minimal number of auxiliary fields needed in the action.
Interestingly, the massless limit of [3] decouples all fields of rank
$0,\dots,n-3$, enabling one to consider only the rank $n$ and $n-2$ fields.
These two fields can be combined into a single, double traceless rank $n$
field $\phi_{\mu_{1}\cdots\mu_{n}}$, resulting in precisely Fronsdal’s
formulation [5], which is valid for any dimension.

###  V.1 Massive particles from dimensional reduction

Instead of constructing a covariant action for a massive spin $n$ from the
ground up, we note as in [41] that one reliable way of getting a massive
particle of mass $m$ in $d$ dimensions is by starting with a massless particle
in $d+1$ dimensions, and compactifying one of the spatial directions to be a
circle of radius $2\pi/m$. Compactifying in this way will generate an infinite
tower of massive particles, corresponding to different windings around the
circle. If the $d+1$ dimensional theory is free as in our case, the particles
with varying masses do not interact, and we may freely throw away all
particles except for the one with mass $m$ without any problems. This
procedure is different from compactification, and so instead we refer to it as
“dimensional reduction”.

As a simple check that dimensional reduction of a massless spin $n$ particle
in $d+1$ dimensions yields a massive spin $n$ particle in $d$ dimensions, we
note that this procedure is equivalent to fixing the magnitude of one
component of the massless particle’s momentum, and that
$N_{0}(d+1,n)=N_{m}(d,n)$. The degrees of freedom of a dimensionally reduced
$d+1$ dimensional massless spin $n$ particle is therefore the same as that of
two $d$ dimensional massive spin $n$ particles, one for each sign of the fixed
momentum component.

The benefit of this starting point is that it is valid for any spacetime
dimension $d$, making it possible to easily write down results as a function
of $d$. Furthermore, the apparent non-uniqueness of the massive description is
manifestly understood to arise from the $d+1$ dimensional massless gauge
symmetry that is inherited. At the level of the $d$ dimensional massive
theory, the gauge symmetry presents itself as a Stückelberg gauge symmetry,
and different gauges amount to a different choice of auxiliary fields.

We therefore start with the massless hyperfield action (III.16) in $d+1$
dimensions, with coordinates $(X^{\mu},X_{d})$ and an auxiliary $d+1$
dimensional vector $(s^{\mu},s_{d})$ for $\mu=0,\dots,d-1$, and enforce that
the real hyperfield $\Phi_{n}(X,X_{d},s,s_{d})$ has the following dependence
on the $X_{d}$ coordinate

$\displaystyle\Phi_{n}(X,X_{d},s,s_{d})=e^{imX_{d}}\Phi_{m}(X,s,s_{d})+e^{-imX_{d}}\tilde{\Phi}_{m}(X,s,s_{d})$
(V.5)

where $\Phi_{m}(X,s,s_{d})$ will generate the fields necessary to covariantly
describe a massive spin $n$ particle in $d$ dimensions. The additional
auxiliary component $s_{d}$ has the effect of generating $n+1$ $d$ dimensional
hyperfields

$\displaystyle\Phi_{m}(X,s,s_{d})=\sum_{k=0}^{n}\frac{1}{k!}i^{-k/2}(s_{d})^{k}\Phi_{n-k}(X,s)$
(V.6)

At the level of an individual $d+1$ dimensional massless rank $n$ field
$\phi_{\mu_{1}\cdots\mu_{n}}$, this is the same as decomposing
$\phi_{\mu_{1}\cdots\mu_{n}}$ into its $d$ dimensional tensors
$\phi_{\mu_{1}\cdots\mu_{n}}$, $\phi_{d\mu_{1}\cdots\mu_{n-1}}$,
$\dots,\,\phi_{dd\cdots d}$. The $d$ dimensional hyperfields $\Phi_{n-k}(X,s)$
are not all independent of each other, because they descend from a $d+1$
dimensional massless hyperfield which satisfies the $d+1$ dimensional double
traceless condition $(\partial_{s}^{2}+\partial_{d}^{2})^{2}\Phi_{n}=0$, where
$\partial_{d}$ is a derivative with respect to $s_{d}$. All $\Phi_{n-k}$ with
$k>3$ can be written in terms of $\Phi_{n}$, $\Phi_{n-1}$, $\Phi_{n-2}$, and
$\Phi_{n-3}$

$\displaystyle\Phi_{m}(X,s,s_{d})$
$\displaystyle=\sum_{k=0}^{\cramped{\left\lfloor
n/2\right\rfloor}}\frac{(-1)^{k}}{(2k)!}(s_{d})^{2k}\Big{(}(1-k)\partial_{s}^{2k}\Phi_{n}+ik\partial_{s}^{2(k-1)}\Phi_{n-2}\Big{)}$
$\displaystyle+i^{-1/2}\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k+1)!}(s_{d})^{2k+1}\Big{(}(1-k)\partial_{s}^{2k}\Phi_{n-1}+ik\partial_{s}^{2(k-1)}\Phi_{n-3}\Big{)}$
(V.7)

where $\Phi_{n},\dots,\Phi_{n-3}$ are unconstrained $d$ dimensional
hyperfields. The gauge symmetry of the massless theory acts on
$\Phi_{m}(X,s,s_{d})$ via

$\displaystyle\delta\Phi_{m}=i^{-1/2}s\cdot\partial_{X}\epsilon_{m}(X,s,s_{d})+i^{1/2}s_{d}m\epsilon_{m}(X,s,s_{d})$
(V.8)

for some $d+1$ dimensional traceless hyperfield gauge parameter
$\epsilon_{m}(X,s,s_{d})$, which may be written in terms of its independent,
unconstrained, $d$ dimensional hyperfield components $\epsilon_{n-1}(X,s)$ and
$\epsilon_{n-2}(X,s)$

$\displaystyle\epsilon_{m}(X,s,s_{d})=\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k)!}(s_{d})^{2k}\partial_{s}^{2k}\epsilon_{n-1}+i^{-1/2}\sum_{k=0}^{\cramped{\left\lfloor(n-2)/2\right\rfloor}}\frac{(-1)^{k}}{(2k+1)!}(s_{d})^{2k+1}\partial_{s}^{2k}\epsilon_{n-2}$
(V.9)

The Stückelberg gauge symmetry may then be expressed in terms of the gauge
parameters $\epsilon_{n-1}$ and $\epsilon_{n-2}$, acting on
$\Phi_{n},\dots,\Phi_{n-3}$

$\displaystyle\delta\Phi_{n}=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}$
$\displaystyle\delta\Phi_{n-1}=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-2}+im\,\epsilon_{n-1}$
$\displaystyle\delta\Phi_{n-2}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\epsilon_{n-1}+2im\,\epsilon_{n-2}$
$\displaystyle\delta\Phi_{n-3}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\epsilon_{n-2}+3m\partial_{s}^{2}\epsilon_{n-1}$
(V.10)

The $d$ dimensional massive action is written in terms of
$\Phi_{n},\dots,\Phi_{n-3}$, and the independent components
$\mathcal{F}_{n},\dots,\mathcal{F}_{n-3}$ of the $d+1$ dimensional double
traceless Fronsdal field strength $\mathcal{F}_{n}(X,X_{d},s,s_{d})$

$\displaystyle S_{n}=\frac{1}{2}n!\int
d^{d+1}X\frac{d^{d+1}sd^{d+1}s^{\prime}}{(2\pi)^{d+1}}e^{is\cdot
s^{\prime}}\Phi_{n}(X,s)\big{(}1-\frac{1}{4}s^{\prime
2}\partial_{s^{\prime}}^{2}\big{)}\mathcal{F}_{n}(X,s^{\prime})$
$\displaystyle S_{n}=n!\int$ $\displaystyle
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot s^{\prime}}\times$
$\displaystyle\Bigg{\\{}$ $\displaystyle\sum_{k=0}^{\cramped{\left\lfloor
n/2\right\rfloor}}\frac{(-1)^{k}}{(2k)!}\Bigg{(}$
$\displaystyle\Big{(}1-\frac{3k}{2}\Big{)}\partial_{s}^{2k}\tilde{\Phi}_{n}\partial_{s^{\prime}}^{2k}\mathcal{F}_{n}+i\frac{k}{2}\partial_{s}^{2k}\tilde{\Phi}_{n}\partial_{s^{\prime}}^{2(k-1)}\mathcal{F}_{n-2}$
$\displaystyle+i\frac{k}{2}\partial_{s}^{2(k-1)}\tilde{\Phi}_{n-2}\partial_{s^{\prime}}^{2k}\mathcal{F}_{n}-\frac{k}{2}\partial_{s}^{2(k-1)}\tilde{\Phi}_{n-2}\partial_{s^{\prime}}^{2(k-1)}\mathcal{F}_{n-2}\Bigg{)}$
$\displaystyle+\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k+1)!}\Bigg{(}$
$\displaystyle\Big{(}1-\frac{5k}{2}\Big{)}\partial_{s}^{2k}\tilde{\Phi}_{n-1}\partial_{s^{\prime}}^{2k}\mathcal{F}_{n-1}+i\frac{3k}{2}\partial_{s}^{2k}\tilde{\Phi}_{n-1}\partial_{s^{\prime}}^{2(k-1)}\mathcal{F}_{n-3}$
$\displaystyle+i\frac{3k}{2}\partial_{s}^{2(k-1)}\tilde{\Phi}_{n-3}\partial_{s^{\prime}}^{2k}\mathcal{F}_{n-1}+\frac{k}{2}\partial_{s}^{2(k-1)}\tilde{\Phi}_{n-3}\partial_{s^{\prime}}^{2(k-1)}\mathcal{F}_{n-3}\Bigg{)}\Bigg{\\}}$
(V.11)

where

$\displaystyle\mathcal{F}_{n}=$
$\displaystyle\big{(}\partial_{X}^{2}-m^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}+\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n}-\frac{i}{2}(s\cdot\partial_{X})^{2}\Phi_{n-2}-i^{1/2}m\,s\cdot\partial_{X}\Phi_{n-1}$
(V.12) $\displaystyle\mathcal{F}_{n-1}=$
$\displaystyle\big{(}\partial_{X}^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}+\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n-1}-\frac{i}{2}(s\cdot\partial_{X})^{2}\Phi_{n-3}$
$\displaystyle+i^{-1/2}m\partial_{s}\cdot\partial_{X}\Phi_{n}-i^{-1/2}m\,s\cdot\partial_{X}\partial_{s}^{2}\Phi_{n}$
(V.13) $\displaystyle\mathcal{F}_{n-2}=$
$\displaystyle\big{(}\partial_{X}^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}-\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n-2}-\frac{i}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{4}\Phi_{n}-im^{2}\partial_{s}^{2}\Phi_{n}$
$\displaystyle+2i^{-1/2}m\partial_{s}\cdot\partial_{X}\Phi_{n-1}-2i^{-1/2}m\,s\cdot\partial_{X}\partial_{s}^{2}\Phi_{n-1}+i^{1/2}m\,s\cdot\partial_{X}\Phi_{n-3}$
(V.14) $\displaystyle\mathcal{F}_{n-3}=$
$\displaystyle\big{(}\partial_{X}^{2}-m^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}-\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n-3}-\frac{i}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{4}\Phi_{n-1}-3im^{2}\partial_{s}^{2}\Phi_{n-1}$
$\displaystyle+2i^{1/2}m\,s\cdot\partial_{X}\partial_{s}^{4}\Phi_{n}+3i^{-1/2}m\partial_{s}\cdot\partial_{X}\Phi_{n-2}+i^{-1/2}m\,s\cdot\partial_{X}\partial_{s}^{2}\Phi_{n-2}$
(V.15)

That (V.1) is invariant under the gauge transformation (V.1) is guaranteed by
the fact that $\mathcal{F}_{n},\dots,\mathcal{F}_{n-3}$ are themselves gauge
invariant, and that they satisfy the identities

$\displaystyle\partial_{s}\cdot\partial_{X}\mathcal{F}_{n}-\frac{1}{2}s\cdot\partial_{X}\partial_{s}^{2}\mathcal{F}_{n}+\frac{i}{2}s\cdot\partial_{X}\mathcal{F}_{n-2}+i^{1/2}m\mathcal{F}_{n-1}=0$
(V.16)
$\displaystyle\partial_{s}\cdot\partial_{X}\mathcal{F}_{n-1}-\frac{1}{2}s\cdot\partial_{X}\partial_{s}^{2}\mathcal{F}_{n-1}+\frac{i}{2}s\cdot\partial_{X}\mathcal{F}_{n-3}+\frac{1}{2}i^{-1/2}m\partial_{s}^{2}\mathcal{F}_{n}+\frac{1}{2}i^{1/2}m\mathcal{F}_{n-2}=0$
(V.17)

which is a direct consequence of (III.17).

The Stückelberg gauge symmetry (V.1) can be used to set for instance
$\Phi_{n-1}$ and $\Phi_{n-2}$ to zero, leaving $\Phi_{n}$ and $\Phi_{n-3}$
unconstrained. This is precisely the minimal field content of [3]. Massive
spin 0, 1, and 2 particles then do not require additional fields to describe
them off-shell. It is only for spins higher than 2 that additional fields are
necessary.

The action (V.1) is for complex massive integer spin particles. For real
massive integer spin particles, it is sufficient to impose the reality
conditions $\tilde{\Phi}_{n}=\Phi_{n}$, $\tilde{\Phi}_{n-1}=-\Phi_{n-1}$,
$\tilde{\Phi}_{n-2}=\Phi_{n-2}$, and $\tilde{\Phi}_{n-3}=-\Phi_{n-3}$.

###  V.2 Equations of motion

By virtue of the fact that the $d+1$ dimensional field strength
$\mathcal{F}_{n}(X,X_{d},s,s_{d})$ is set to zero by the equations of motion,
the $d$ dimensional massive equations of motion are

$\displaystyle\mathcal{F}_{n}=0,\quad\mathcal{F}_{n-1}=0,\quad\mathcal{F}_{n-2}=0,\quad\mathcal{F}_{n-3}=0$
(V.18)

That these equations describe massive integer spin particles amounts to
showing that they imply that $\Phi_{n-1}$, $\Phi_{n-2}$, and $\Phi_{n-3}$ can
always be set to zero, and that $\Phi_{n}$ is a Fierz-Pauli system in
hyperspace

$\displaystyle(\partial_{X}^{2}-m^{2})\Phi_{n}$ $\displaystyle=0$
$\displaystyle\partial_{s}\cdot\partial_{X}\Phi_{n}$ $\displaystyle=0$
$\displaystyle\partial_{s}^{2}\Phi_{n}$ $\displaystyle=0$ (V.19)

The gauge symmetry (V.1) may be used to set $\Phi_{n-1}=0$ and $\Phi_{n-2}=0$
before imposing the equations of motion, so (V.18) becomes

$\displaystyle\mathcal{F}_{n}=$
$\displaystyle\big{(}\partial_{X}^{2}-m^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}+\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n}=0$
(V.20) $\displaystyle\mathcal{F}_{n-1}=$
$\displaystyle-\frac{i}{2}(s\cdot\partial_{X})^{2}\Phi_{n-3}+i^{-1/2}m\partial_{s}\cdot\partial_{X}\Phi_{n}-i^{-1/2}m\,s\cdot\partial_{X}\partial_{s}^{2}\Phi_{n}=0$
(V.21) $\displaystyle\mathcal{F}_{n-2}=$
$\displaystyle-\frac{i}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{4}\Phi_{n}-im^{2}\partial_{s}^{2}\Phi_{n}+i^{1/2}m\,s\cdot\partial_{X}\Phi_{n-3}=0$
(V.22) $\displaystyle\mathcal{F}_{n-3}=$
$\displaystyle\big{(}\partial_{X}^{2}-m^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}-\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n-3}+2i^{1/2}m\,s\cdot\partial_{X}\partial_{s}^{4}\Phi_{n}=0$
(V.23)

A quick inspection of these gauge fixed equations of motion shows that if
$\Phi_{n-3}=0$, then $\Phi_{n}$ satisfies (V.2). What remains is to show that
$\Phi_{n-3}$ may always be set to zero via the residual gauge symmetry of this
system

$\displaystyle\delta\Phi_{n}=\frac{1}{m}(s\cdot\partial_{X})^{2}\epsilon_{n-2},\quad\delta\Phi_{n-3}=2i^{1/2}\partial_{s}\cdot\partial_{X}\epsilon_{n-2}+2i^{1/2}\partial_{s}^{2}(s\cdot\partial_{X}\epsilon_{n-2})$
(V.24)

These transformations keep $\Phi_{n-1}=\Phi_{n-2}=0$ provided that
$\epsilon_{n-2}$ satisfies444Note that this residual gauge symmetry does not
exist when $n\leq 2$.

$\displaystyle
s\cdot\partial_{X}\partial_{s}^{2}(s\cdot\partial_{X}\epsilon_{n-2})-2m^{2}\epsilon_{n-2}=0$
(V.25)

This is indeed the case, as guaranteed by dimensional reduction. To
demonstrate this, we decompose $\Phi_{n}$ into its transverse and traceless
part $\Phi_{n}^{\prime}$, and a part which is not transverse or not traceless
$\Delta\Phi_{n}$

$\displaystyle\Phi_{n}=\Phi_{n}^{\prime}+\Delta\Phi_{n}$ (V.26)

and note that in order for (V.21) to be consistent, $\Delta\Phi_{n}$ cannot
have a term like $A_{n}$ or $s\cdot\partial_{X}B_{n-1}$ for $A_{n}$ and
$B_{n-1}$ transverse
$\partial_{s}\cdot\partial_{X}A_{n}=\partial_{s}\cdot\partial_{X}B_{n-1}=0$.
Hence we may write $\Delta\Phi_{n}$ suggestively as

$\displaystyle\Delta\Phi_{n}=\frac{1}{m}(s\cdot\partial_{X})^{2}\epsilon_{n-2}$
(V.27)

for some hyperfield $\epsilon_{n-2}$. Solving for $\Phi_{n-3}$ in terms of
$\epsilon_{n-2}$ using (V.21) yields

$\displaystyle\Phi_{n-3}=2i^{1/2}\partial_{s}\cdot\partial_{X}\epsilon_{n-2}+2i^{1/2}\partial_{s}^{2}(s\cdot\partial_{X}\epsilon_{n-2})$
(V.28)

Plugging these forms into (V.20), we find
$(\partial_{X}^{2}-m^{2})\Phi_{n}^{\prime}=0$ and an $\epsilon_{n-2}$
satisfying (V.25). $\Delta\Phi_{n}$ and $\Phi_{n-3}$ can therefore be gauged
away, leaving a massive integer spin field $\Phi_{n}^{\prime}$ satisfying
(V.2).

###  V.3 All propagators

We now compute the correlation functions
$\langle\Phi_{n-i}(X,s)\tilde{\Phi}_{n-j}(Y,t)\rangle$, which will give
expressions for all correlation functions
$\langle\phi_{\mu_{1}\cdots\mu_{n-i}}(X)\phi^{*}_{\nu_{1}\cdots\nu_{n-j}}(Y)\rangle$
with $i,j=0,\dots,3$. These 16 correlation functions are needed to fully
specify the physical integer spin $n$ propagator.

First, we detail the ambiguity in the correlation functions arising from the
Stückelberg gauge symmetry (V.1). Coupling $\Phi_{n}$, $\Phi_{n-1}$,
$\Phi_{n-2}$, and $\Phi_{n-3}$ to external sources $\mathcal{J}_{n}$,
$\mathcal{J}_{n-1}$, $\mathcal{J}_{n-2}$, and $\mathcal{J}_{n-3}$,
respectively, will lead to inconsistencies unless it is done in a gauge
invariant way. The gauge invariant coupling of $\Phi_{n-i}$ to
$\mathcal{J}_{n-i}$ is

$\displaystyle\Delta S=n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}\Big{(}\tilde{\Phi}_{n}\mathcal{J}_{n}+\tilde{\Phi}_{n-1}\mathcal{J}_{n-1}+\tilde{\Phi}_{n-2}\mathcal{J}_{n-2}+\tilde{\Phi}_{n-3}\mathcal{J}_{n-3}+\text{c.c.}\Big{)}$
(V.29)

provided that the sources satisfy

$\displaystyle\partial_{s}\cdot\partial_{X}\mathcal{J}_{n}+is^{2}\partial_{s}\cdot\partial_{X}\mathcal{J}_{n-2}+i^{1/2}m\mathcal{J}_{n-1}-3i^{-1/2}ms^{2}\mathcal{J}_{n-3}=0$
(V.30)
$\displaystyle\partial_{s}\cdot\partial_{X}\mathcal{J}_{n-1}+is^{2}\partial_{s}\cdot\partial_{X}\mathcal{J}_{n-3}+2i^{1/2}m\mathcal{J}_{n-2}=0$
(V.31)

We expect the solution to the equations of motion for $\Phi_{n-i}$ when
coupled to this source to be of the form

$\displaystyle\Phi_{n-i}(X,s)=i\int
d^{d}Y\frac{d^{d}td^{d}t^{\prime}}{(2\pi)^{d}}e^{it\cdot
t^{\prime}}\mathcal{G}_{ij}(X-Y,s,t)\mathcal{J}_{n-j}(Y,t^{\prime})$ (V.32)

where the sum over $j$ is implied. The transformation
$\delta\mathcal{G}_{ij}(X-Y,s,t)=\mathcal{N}_{ij}(X-Y,s,t)$, where

$\displaystyle\mathcal{N}_{0j}=i^{-1/2}s\cdot\partial_{X}\Pi_{1j}$
$\displaystyle\mathcal{N}_{1j}=i^{-1/2}s\cdot\partial_{X}\Pi_{2j}+im\,\Pi_{1j}$
$\displaystyle\mathcal{N}_{2j}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\Pi_{1j}+2im\,\Pi_{2j}$
$\displaystyle\mathcal{N}_{3j}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\Pi_{2j}+3m\partial_{s}^{2}\Pi_{1j}$
(V.33)

and $\Pi_{1j}(X-Y,s,t)$, $\Pi_{2j}(X-Y,s,t)$ are arbitrary rank $n-1$ and
$n-2$ hyperfields in $s$, and rank $n-j$ hyperfields in $t$, respectively,
amounts to a Stückelberg gauge transformation on $\Phi_{n-i}(X,s)$, and so
$\mathcal{G}_{ij}(X-Y,s,t)$ itself has a gauge redundancy.
$\mathcal{G}_{ij}(X-Y,s,t)$ is further ambiguous up to transformations
$\delta\mathcal{G}_{ij}(X-Y,s,t)=\mathcal{M}_{ij}$, where

$\displaystyle\mathcal{M}_{i0}=i^{-1/2}t\cdot\partial_{Y}\Omega_{i1}$
$\displaystyle\mathcal{M}_{i1}=i^{-1/2}t\cdot\partial_{Y}\Omega_{i2}-im\,\Omega_{i1}$
$\displaystyle\mathcal{M}_{i2}=-i^{1/2}t\cdot\partial_{Y}\partial_{t}^{2}\Omega_{i1}-2im\,\Omega_{i2}$
$\displaystyle\mathcal{M}_{i3}=-i^{1/2}t\cdot\partial_{Y}\partial_{t}^{2}\Omega_{i2}-3m\partial_{t}^{2}\Omega_{i1}$
(V.34)

for similarly arbitrary hyperfields $\Omega_{i1}(X-Y,s,t)$ and
$\Omega_{i2}(X-Y,s,t)$, precisely because

$\displaystyle i\int d^{d}Y\frac{d^{d}td^{d}t^{\prime}}{(2\pi)^{d}}e^{it\cdot
t^{\prime}}\mathcal{M}_{ij}(X-Y,s,t)\mathcal{J}_{n-j}(Y,t^{\prime})=0$ (V.35)

In total then, $\mathcal{G}_{ij}(X-Y,s,t)$ is ambiguous up to transformations

$\displaystyle\delta\mathcal{G}_{ij}(X-Y,s,t)=\mathcal{N}_{ij}(X-Y,s,t)+\mathcal{M}_{ij}(X-Y,s,t)$
(V.36)

This ambiguity equally applies to the correlation functions
$\langle\Phi_{n-i}(X,s)\tilde{\Phi}_{n-j}(Y,t)\rangle$, i.e. if we change them
by
$\delta\langle\Phi_{n-i}(X,s)\tilde{\Phi}_{n-j}(Y,t)\rangle=\mathcal{N}_{ij}+\mathcal{M}_{ij}$,
they represent the same massive particle.

Instead of pursuing the exact form of the massive propagators directly from
(V.1), we recognize that the massive propagator can be straightforwardly
obtained from the massless propagator (III.31) in $d+1$ dimensions. In the
gauge (III.26), the dimensionally reduced action may be written as

$\displaystyle S_{n}=$ $\displaystyle n!\int
d^{d}X\frac{d^{d+1}sd^{d+1}s^{\prime}}{(2\pi)^{d+1}}e^{is\cdot
s^{\prime}}\tilde{\Phi}_{m}(X,s,s_{d})\big{(}1-\frac{1}{4}(s^{\prime
2}+s_{d}^{\prime
2})(\partial_{s^{\prime}}^{2}+\partial_{d^{\prime}}^{2})\big{)}\big{(}\partial_{X}^{2}-m^{2}\big{)}\Phi_{m}(X,s^{\prime},s_{d}^{\prime})$
(V.37)

The propagator for $\Phi_{m}(X,s,s_{d})$ in this gauge can be obtained in the
same way it was in subsection III.2, it is

$\displaystyle\langle\Phi_{m}(X,s,s_{d})\tilde{\Phi}_{m}(Y,t,t_{d})\rangle=G_{m}(X-Y)\frac{1}{(\frac{d-3}{2})_{n}}\Big{(}-\frac{i}{2}\sqrt{\overline{s}^{2}\overline{t}^{2}}\Big{)}^{n}C_{n}^{\frac{d-3}{2}}\Big{(}\frac{\overline{s}\cdot\overline{t}}{\sqrt{\overline{s}^{2}\overline{t}^{2}}}\Big{)}$
(V.38)

where $G_{m}(X-Y)$ is the standard massive spin 0 propagator, and
$\overline{s}=(s,s_{d})$ and $\overline{t}=(t,t_{d})$ are $d+1$ dimensional
auxiliary vectors. All that remains is to decompose this propagator into its
independent $d$ dimensional components
$\langle\Phi_{n-i}(X,s)\tilde{\Phi}_{n-j}(Y,t)\rangle$ using (V.6)

$\displaystyle\langle\Phi_{n}(X,s)\tilde{\Phi}_{n}(Y,t)\rangle=G_{m}\frac{1}{(\frac{d-3}{2})_{n}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n}C_{n}^{\frac{d-3}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (V.39)
$\displaystyle\langle\Phi_{n}(X,s)\tilde{\Phi}_{n-2}(Y,t)\rangle=iG_{m}\frac{1}{2}s^{2}\frac{1}{(\frac{d-1}{2})_{n-1}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-2}C_{n-2}^{\frac{d-1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (V.40)
$\displaystyle\langle\Phi_{n-2}(X,s)\tilde{\Phi}_{n-2}(Y,t)\rangle=$
$\displaystyle
G_{m}\frac{2}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-2}\Bigg{(}\frac{1}{(\frac{d-1}{2})_{n-1}}\Big{(}\frac{d-6}{2}+n\Big{)}C_{n-2}^{\frac{d-1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}+\frac{3}{2}\frac{1}{(\frac{d+1}{2})_{n-2}}C_{n-4}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}\Bigg{)}$ (V.41)
$\displaystyle\langle\Phi_{n-1}(X,s)\tilde{\Phi}_{n-1}(Y,t)\rangle=G_{m}\frac{1}{(\frac{d-1}{2})_{n-1}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-1}C_{n-1}^{\frac{d-1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (V.42)
$\displaystyle\langle\Phi_{n-1}(X,s)\tilde{\Phi}_{n-3}(Y,t)\rangle=iG_{m}\frac{3}{2}s^{2}\frac{1}{(\frac{d+1}{2})_{n-2}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-3}C_{n-3}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (V.43)
$\displaystyle\langle\Phi_{n-3}(X,s)\tilde{\Phi}_{n-3}(Y,t)\rangle=$
$\displaystyle
G_{m}\frac{6}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-3}\Bigg{(}\frac{1}{(\frac{d+1}{2})_{n-2}}\Big{(}\frac{d-8}{2}+n\Big{)}C_{n-3}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}+\frac{5}{2}\frac{1}{(\frac{d+3}{2})_{n-3}}C_{n-5}^{\frac{d+3}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}\Bigg{)}$ (V.44)

and all other correlation functions are either zero or related via Hermitian
conjugation. One can straightforwardly consider the $n\to\infty$ asymptotic
limit of these expressions, as we did for the massless propagators. We omit
the explicit asymptotic limits for the sake of brevity.

Recall that the hyperfields $\Phi_{n-i}(X,s)$ correspond to fields
$\phi_{\mu_{1}\cdots\mu_{n-i}}$ for $i=0,\dots,3$. In a simplified notation,
the massive spin $n$ action in terms of these fields is

$\displaystyle S_{n}=\int d^{d}X\Bigg{\\{}$
$\displaystyle\sum_{k=0}^{\cramped{\left\lfloor n/2\right\rfloor}}{n\choose
2k}\Bigg{(}$
$\displaystyle\Big{(}1-\frac{3k}{2}\Big{)}\phi_{0}^{*(k)}\cdot\mathcal{F}_{0}^{(k)}-\frac{k}{2}\phi_{0}^{*(k)}\cdot\mathcal{F}_{2}^{(k-1)}$
$\displaystyle-\frac{k}{2}\phi_{2}^{*(k-1)}\cdot\mathcal{F}_{0}^{(k)}+\frac{k}{2}\phi_{2}^{*(k-1)}\cdot\mathcal{F}_{2}^{(k-1)}\Bigg{)}$
$\displaystyle+\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}{n\choose
2k+1}\Bigg{(}$
$\displaystyle\Big{(}1-\frac{5k}{2}\Big{)}\phi_{1}^{*(k)}\cdot\mathcal{F}_{1}^{(k)}-\frac{3k}{2}\phi_{1}^{*(k)}\cdot\mathcal{F}_{3}^{(k-1)}$
$\displaystyle-\frac{3k}{2}\phi_{3}^{*(k-1)}\cdot\mathcal{F}_{1}^{(k)}-\frac{k}{2}\phi_{3}^{*(k-1)}\cdot\mathcal{F}_{3}^{(k-1)}\Bigg{)}\Bigg{\\}}$
(V.45)

where $\mathcal{O}_{i}^{(k)}$ denotes the $k$-fold trace of
$\mathcal{O}_{\mu_{1}\cdots\mu_{n-i}}$, and the dot product $\cdot$ contracts
the remaining indices. The massive spin $n$ correlation functions
$\langle\phi_{\mu_{1}\cdots\mu_{n-i}}(X)\phi^{*}_{\nu_{1}\cdots\nu_{n-j}}(Y)\rangle$
from (V.3) can be obtained from
$\langle\Phi_{n-i}(X,s)\tilde{\Phi}_{n-j}(Y,t)\rangle$ by applying derivatives

$\displaystyle\langle\phi_{\mu_{1}\cdots\mu_{n-i}}(X)\phi^{*}_{\nu_{1}\cdots\nu_{n-j}}(Y)\rangle=i^{n-(i+j)/2}\frac{\partial^{n-i}}{\partial_{s}^{\mu_{1}}\cdots\partial_{s}^{\mu_{n-i}}}\frac{\partial^{n-j}}{\partial_{t}^{\nu_{1}}\cdots\partial_{t}^{\nu_{n-j}}}\langle\Phi_{n-i}(X,s)\tilde{\Phi}_{n-j}(Y,t)\rangle$
(V.46)

##  VI Massive half integer spins

In this section, we will use the hyperfield formalism of section II to study
massive half integer spin fields. We will find compact, closed form
expressions for their covariant actions and propagators.

The half integer spin case was also considered in [61, 2], but an alternative
formulation was given in [62]. In it, Rarita and Schwinger noted that the
following system of equations is sufficient for describing a freely
propagating spin $n+1/2$ massive particle

$\displaystyle(\mathrlap{\\!\not{\phantom{\partial}}}\partial+m)\psi_{\mu_{1}\cdots\mu_{n}}=0$
(VI.1) $\displaystyle\gamma^{\lambda}\psi_{\lambda\mu_{2}\cdots\mu_{n}}=0$
(VI.2)

for a symmetric rank $n$ tensor $\psi_{\mu_{1}\cdots\mu_{n}}$, with an
additional implicit Dirac index. We will refer to the system of equations
(VI.1) and (VI.2) as a Rarita-Schwinger system. Any Rarita-Schwinger system is
also a Fierz-Pauli system. The transverse and $\gamma$ traceless conditions
ensure that the correct number $N_{m}(d,n+1/2)$ of degrees of freedom for a
spin $n+1/2$ particle in $d$ spacetime dimensions propagate with mass $m$

$\displaystyle N_{m}(d,n+1/2)=$ $\displaystyle 2^{\cramped{\left\lfloor
d/2\right\rfloor}}{d+n-1\choose n}-2^{\cramped{\left\lfloor
d/2\right\rfloor}}{d+n-2\choose n-1}$
$\displaystyle-\Bigg{(}2^{\cramped{\left\lfloor
d/2\right\rfloor}}{d+n-2\choose n-1}-2^{\cramped{\left\lfloor
d/2\right\rfloor}}{d+n-3\choose n-2}\Bigg{)}$ $\displaystyle=$ $\displaystyle
2^{\cramped{\left\lfloor d/2\right\rfloor}}{d+n-3\choose n}$ (VI.3)

which reduces to $2(2n+2)$ when $d=4$.

A minimal covariant Lagrangian formulation of massive half integer spin fields
$\psi_{\mu_{1}\cdots\mu_{n}}$ in four spacetime dimensions that imposes
(VI.1)-(VI.2) was first worked out in [4]. In it, Singh and Hagen needed to
introduce, apart from the symmetric and $\gamma$ traceless rank $n$ Dirac
tensor, a symmetric and $\gamma$ traceless rank $n-1$ Dirac tensor, and two of
each symmetric and $\gamma$ traceless Dirac tensors of rank $0,1,2,\dots,n-2$
which couple so that all fields of lower rank than $n$ vanish on shell, and
the rank $n$ field made up a Rarita-Schwinger system. Note that this field
content is equivalent to using symmetric rank $n$ and $n-2$ Dirac tensors,
with no tracelessness conditions.

###  VI.1 Massive particles from dimensional reduction

In this section, we will construct an action for massive half integer spins
which work in any spacetime dimension $d$ using dimensional reduction. The
dimensional reduction procedure will be similar to the integer spin case, but
will differ in some basic ways.

An immediate difference is that $N_{0}(d+1,n+1/2)=N_{m}(d,n+1/2)$ only when
$d$ is even, and so dimensional reduction appears not to work for odd $d$. The
origin of this mismatch is that the spinor representation being used in the
$d$ dimensional theory will always have a $\gamma_{d}$ with $\gamma_{d}^{2}=1$
and $\\{\gamma_{d},\gamma_{\mu}\\}=0$ for $\mu=0,\dots,d-1$. For the Dirac
representation, this is only possible when $d$ is even. When $d$ is odd then,
the $d$ dimensional fields are not in the Dirac representation, but in some
larger one. The mismatch is therefore a difference in choice of spinor
representation, which does not affect the spin of the particle. Shortly, we
will write down a suitable field redefinition which removes all instances of
$\gamma_{d}$ in the formulation, so that even for odd $d$, we may choose
fields in the Dirac representation. As we will see, the result of performing
this field redefinition is a formulation of massive half integer spin fields
whose equations of motion are independent of $d$, and is conveniently well
defined for the Dirac representation in $d$ spacetime dimensions, whether even
or odd. For odd $d$ then, we may replace the larger representation obtained
from dimensional reduction with the irreducible Dirac representation. A
straightforward analysis of the equations of motion in subsection VI.2 will
reveal that this formulation has the correct degrees of freedom for a massive
spin $n+1/2$ Dirac particle in any dimension $d$.

We therefore start with the massless hyperfield action (IV.17) in $d+1$
dimensions, and enforce that the hyperfield $\Psi_{n}(X,X_{d},s,s_{d})$ has
the following dependence on the $X_{d}$ coordinate

$\displaystyle\Psi_{n}(X,X_{d},s,s_{d})=e^{imX_{d}}\Psi_{m}(X,s,s_{d})$ (VI.4)

The additional auxiliary component $s_{d}$ generates $n+1$ $d$ dimensional
hyperfields

$\displaystyle\Psi_{m}(X,s,s_{d})=\sum_{k=0}^{n}\frac{1}{k!}i^{-k/2}(s_{d})^{k}\Psi_{n-k}(X,s)$
(VI.5)

The hyperfields $\Psi_{n-k}(X,s)$ are not all independent of each other,
because they descend from a $d+1$ dimensional massless hyperfield satisfying
the $d+1$ dimensional triple $\gamma$ traceless condition
$(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}+\gamma_{d}\partial_{d})^{3}\Psi_{n}=0$.
All $\Psi_{n-k}$ with $k>2$ can be written in terms of $\Psi_{n}$,
$\Psi_{n-1}$, and $\Psi_{n-2}$

$\displaystyle\Psi_{m}(X,s,s_{d})$
$\displaystyle=\sum_{k=0}^{\cramped{\left\lfloor
n/2\right\rfloor}}\frac{(-1)^{k}}{(2k)!}(s_{d})^{2k}\Big{(}(1-k)\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k}\Psi_{n}+ik\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2(k-1)}\Psi_{n-2}\Big{)}$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k+1)!}(s_{d})^{2k+1}\Big{(}k\gamma_{d}\,\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k+1}\Psi_{n}+i^{-1/2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k}\Psi_{n-1}-ik\gamma_{d}\,\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k-1}\Psi_{n-2}\Big{)}$
(VI.6)

where $\Psi_{n},\Psi_{n-1},\Psi_{n-2}$ are unconstrained $d$ dimensional
hyperfields. We would like to get rid of the $\gamma_{d}$’s in this
expression. This is achieved by making the field redefinitions $\Psi_{n-i}\to
e^{-i\pi\gamma_{d}/4}\Psi_{n-i}$.

$\displaystyle\Psi_{m}(X,s,s_{d})$
$\displaystyle=e^{-i\pi\gamma_{d}/4}\sum_{k=0}^{\cramped{\left\lfloor
n/2\right\rfloor}}\frac{(-1)^{k}}{(2k)!}(s_{d})^{2k}\Big{(}(1-k)\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k}\Psi_{n}+ik\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2(k-1)}\Psi_{n-2}\Big{)}$
$\displaystyle+e^{-i\pi\gamma_{d}/4}$
$\displaystyle\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k+1)!}(s_{d})^{2k+1}\Big{(}ik\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k+1}\Psi_{n}+i^{-1/2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k}\Psi_{n-1}+k\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k-1}\Psi_{n-2}\Big{)}$
(VI.7)

The $d+1$ dimensional $\gamma$ traceless hyperfield gauge parameter
$\epsilon_{m}(X,s,s_{d})$ may also be written in terms of an unconstrained,
$d$ dimensional hyperfield component $\epsilon_{n-1}(X,s)$

$\displaystyle\epsilon_{m}(X,s,s_{d})=$ $\displaystyle
e^{-i\pi\gamma_{d}/4}\Big{(}\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k)!}(s_{d})^{2k}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k}\epsilon_{n-1}-i\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k+1)!}(s_{d})^{2k+1}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k+1}\epsilon_{n-1}\Big{)}$
(VI.8)

The Stückelberg gauge symmetry may then be expressed in terms of the gauge
parameter $\epsilon_{n-1}$ acting on $\Psi_{n},\,\Psi_{n-1},\,\Psi_{n-2}$

$\displaystyle\delta\Psi_{n}=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}$
$\displaystyle\delta\Psi_{n-1}=-is\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}+im\epsilon_{n-1}$
$\displaystyle\delta\Psi_{n-2}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\epsilon_{n-1}+2i^{1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}$
(VI.9)

The $d$ dimensional massive action is written in terms of
$\Psi_{n},\Psi_{n-1},\Psi_{n-2}$, and the independent components
$\mathcal{S}_{n},\,\mathcal{S}_{n-1},\,\mathcal{S}_{n-2}$ of the $d+1$
dimensional Fang field strength $\mathcal{S}_{n}(X,X_{d},s,s_{d})$

$\displaystyle S_{n+1/2}=-n!\int
d^{d+1}X\frac{d^{d+1}sd^{d+1}s^{\prime}}{(2\pi)^{d+1}}e^{is\cdot
s^{\prime}}\overline{\Psi}_{n}(X,s)\big{(}1-\frac{1}{2}\mathrlap{\\!\not{\phantom{s^{\prime}}}}s^{\prime}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}-\frac{1}{4}s^{\prime
2}\partial^{2}_{s^{\prime}}\big{)}\mathcal{S}_{n}(X,s^{\prime})$ (VI.10)
$\displaystyle S_{n+1/2}$ $\displaystyle=-n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot s^{\prime}}\times$
$\displaystyle\Bigg{\\{}$ $\displaystyle\sum_{k=0}^{\cramped{\left\lfloor
n/2\right\rfloor}}\frac{(-1)^{k}}{(2k)!}\Bigg{(}$
$\displaystyle\Big{(}1-\frac{3k}{2}\Big{)}\partial_{s}^{2k}\overline{\Psi}_{n}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}+i\frac{k}{2}\partial_{s}^{2k}\overline{\Psi}_{n}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle+i\frac{k}{2}\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}+\frac{k}{2}\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle-i^{-1/2}k\partial_{s}^{2(k-1)}\overline{\Psi}_{n-1}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}+i^{-1/2}k\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-1}$
$\displaystyle\hskip
78.24507pt+ik\partial_{s}^{2(k-1)}\overline{\Psi}_{n-1}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-1}\Bigg{)}$
$\displaystyle+\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k+1)!}\Bigg{(}$
$\displaystyle-i\Big{(}\frac{1}{2}+\frac{3k}{2}\Big{)}\partial_{s}^{2k}\overline{\Psi}_{n}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}-\frac{k}{2}\partial_{s}^{2k}\overline{\Psi}_{n}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle-\frac{k}{2}\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}+i\frac{k}{2}\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle+\frac{1}{2}i^{-1/2}\partial_{s}^{2k}\overline{\Psi}_{n}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n-1}-\frac{1}{2}i^{-1/2}\partial_{s}^{2k}\overline{\Psi}_{n-1}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}$
$\displaystyle+i^{1/2}k\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n-1}-i^{1/2}k\partial_{s}^{2k}\overline{\Psi}_{n-1}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle\hskip
96.02805pt+\Big{(}\frac{1}{2}-k\Big{)}\partial_{s}^{2k}\overline{\Psi}_{n-1}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n-1}\Bigg{)}\Bigg{\\}}$
(VI.11)

where

$\displaystyle\mathcal{S}_{n}$
$\displaystyle=(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}+m-s\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})\Psi_{n}+i^{1/2}s\cdot\partial_{X}\Psi_{n-1}$
(VI.12) $\displaystyle\mathcal{S}_{n-1}$
$\displaystyle=(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-s\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})\Psi_{n-1}+i^{1/2}s\cdot\partial_{X}\Psi_{n-2}+i^{-1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}$
(VI.13) $\displaystyle\mathcal{S}_{n-2}$
$\displaystyle=(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-m)\Psi_{n-2}+is\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{3}\Psi_{n}+i^{-1/2}s\cdot\partial_{X}\partial_{s}^{2}\Psi_{n-1}+2i^{-1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n-1}$
(VI.14)

That (VI.1) is invariant under the gauge transformation (VI.1) is guaranteed
by the fact that $\mathcal{S}_{n},\,\mathcal{S}_{n-1},\,\mathcal{S}_{n-2}$ are
themselves gauge invariant, and satisfy the identity

$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-m)\mathcal{S}_{n}-s\cdot\partial_{X}\partial_{s}^{2}\mathcal{S}_{n}-i^{1/2}(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-m)\mathcal{S}_{n-1}+i\,s\cdot\partial_{X}\mathcal{S}_{n-2}=0$
(VI.15)

which is a direct consequence of (IV.18). As promised (VI.1) has no instance
of $\gamma_{d}$, making this action valid for the Dirac representation in any
spacetime dimension.

The Stückelberg gauge symmetry (VI.1) can be used to set $\Psi_{n-1}$ to zero,
leaving the fields $\Psi_{n}$ and $\Psi_{n-2}$, consistent with the minimal
field content of [4].

###  VI.2 Equations of motion

The massive equations of motion are equivalent to setting the $d+1$
dimensional field strength $\mathcal{S}_{n}(X,X_{d},s,s_{d})$ to zero

$\displaystyle\mathcal{S}_{n}=0,\quad\mathcal{S}_{n-1}=0,\quad\mathcal{S}_{n-2}=0$
(VI.16)

That these equations describe massive half integer spin particles amounts to
showing that they imply that $\Psi_{n-1}$ and $\Psi_{n-2}$ can always be set
to zero, and that $\Psi_{n}$ is a Rarita-Schwinger system in hyperspace

$\displaystyle(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}+m)\Psi_{n}$
$\displaystyle=0$
$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}$
$\displaystyle=0$ (VI.17)

The gauge symmetry (VI.1) may be used to set $\Psi_{n-1}=0$ before imposing
the equations of motion, so (VI.16) becomes

$\displaystyle\mathcal{S}_{n}=(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}+m-s\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})\Psi_{n}$
(VI.18)
$\displaystyle\mathcal{S}_{n-1}=i^{1/2}s\cdot\partial_{X}\Psi_{n-2}+i^{-1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}$
(VI.19)
$\displaystyle\mathcal{S}_{n-2}=(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-m)\Psi_{n-2}+is\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{3}\Psi_{n}$
(VI.20)

In this gauge, if $\Psi_{n-2}=0$, then $\Psi_{n}$ satisfies (VI.2). What
remains is to show that $\Psi_{n-2}$ may always be set to zero via the
residual gauge symmetry of this system

$\displaystyle\delta\Psi_{n}=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1},\quad\delta\Psi_{n-2}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\epsilon_{n-1}+2i^{1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}$
(VI.21)

These transformations keep $\Psi_{n-1}=0$ provided that $\epsilon_{n-1}$
satisfies555Note that this residual gauge symmetry does not exist when $n\leq
1$.

$\displaystyle
s\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}-m\epsilon_{n-1}=0$
(VI.22)

To show that this is the case, we decompose $\Psi_{n}$ into a transverse and
$\gamma$ traceless part $\Psi_{n}^{\prime}$ and a part $\Delta\Psi_{n}$ which
is not transverse or not $\gamma$ traceless

$\displaystyle\Psi_{n}=\Psi_{n}^{\prime}+\Delta\Psi_{n}$ (VI.23)

We may always write $\Delta\Psi_{n}$ suggestively as
$\Delta\Psi_{n}=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}$, for some hyperfield
$\epsilon_{n-1}$. Plugging this into (VI.18), we find
$(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}+m)\Psi_{n}^{\prime}=0$,
and an $\epsilon_{n-1}$ satisfying (VI.22). Finally, we solve for $\Psi_{n-2}$
using (VI.19) and (VI.22)

$\displaystyle\Psi_{n-2}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\epsilon_{n-1}+2i^{1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}$
(VI.24)

Hence, $\Delta\Psi_{n}$ and $\Psi_{n-2}$ can be gauged away, leaving a massive
half integer spin field $\Psi_{n}^{\prime}$ satisfying (VI.2).

###  VI.3 All propagators

We now compute the correlation functions
$\langle\Psi_{n-i}(X,s)\overline{\Psi}_{n-j}(Y,t)\rangle$, which will give
expressions for all correlation functions
$\langle\psi_{\mu_{1}\cdots\mu_{n-i}}(X)\overline{\psi}_{\nu_{1}\cdots\nu_{n-j}}(Y)\rangle$
with $i,j=0,1,2$. These 9 correlation functions are needed to fully specify
the physical half integer spin $n+1/2$ propagator.

The discussion in subsection V.3 on the gauge ambiguity of the correlation
functions equally applies here. Coupling $\overline{\Psi}_{n}$,
$\overline{\Psi}_{n-1}$, and $\overline{\Psi}_{n-2}$ to external sources
$\mathcal{Q}_{n}$, $\mathcal{Q}_{n-1}$, and $\mathcal{Q}_{n-2}$ via

$\displaystyle\Delta S=n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}\Big{(}\overline{\Psi}_{n}\mathcal{Q}_{n}+\overline{\Psi}_{n-1}\mathcal{Q}_{n-1}+\overline{\Psi}_{n-2}\mathcal{Q}_{n-2}\Big{)}$
(VI.25)

is gauge invariant provided that the sources satisfy

$\displaystyle\partial_{s}\cdot\partial_{X}\mathcal{Q}_{n}-i^{1/2}\mathrlap{\\!\not{\phantom{s}}}s\partial_{s}\cdot\partial_{X}\mathcal{Q}_{n-1}+i^{1/2}m\mathcal{Q}_{n-1}+is^{2}\partial_{s}\cdot\partial_{X}\mathcal{Q}_{n-2}-2im\mathrlap{\\!\not{\phantom{s}}}s\mathcal{Q}_{n-2}=0$
(VI.26)

Because of this, the correlation functions
$\langle\Psi_{n-i}(X,s)\overline{\Psi}_{n-j}(Y,t)\rangle$ are ambiguous up to
the transformations

$\displaystyle\delta\langle\Psi_{n-i}(X,s)\overline{\Psi}_{n-j}(Y,t)\rangle=\mathcal{N}_{ij}(X-Y,s,t)+\mathcal{M}_{ij}(X-Y,s,t)$
(VI.27)

where

$\displaystyle\mathcal{N}_{0j}=i^{-1/2}s\cdot\partial_{X}\Pi_{1j}$
$\displaystyle\mathcal{M}_{i0}=i^{-1/2}t\cdot\partial_{Y}\overline{\Omega}_{i1}$
$\displaystyle\mathcal{N}_{1j}=-i\,s\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Pi_{1j}+im\Pi_{1j}$
$\displaystyle\mathcal{M}_{i1}=-i\,t\cdot\partial_{Y}\overline{\Omega}_{i1}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{t}}-im\overline{\Omega}_{i1}$
$\displaystyle\mathcal{N}_{2j}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\Pi_{1j}+2i^{1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Pi_{1j}$
$\displaystyle\mathcal{M}_{i2}=-i^{1/2}t\cdot\partial_{Y}\partial_{t}^{2}\overline{\Omega}_{i1}-2i^{1/2}m\overline{\Omega}_{i1}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{t}}$
(VI.28)

for arbitrary hyperfields $\Pi_{1j}(X-Y,s,t)$ and
$\overline{\Omega}_{i1}(X-Y,s,t)$.

The massive correlation functions in a specific gauge are obtained
straightforwardly from the $d+1$ dimensional massless propagator (IV.2). After
dimensionally reducing, the $\Psi_{m}(X,s,s_{d})$ propagator is

$\displaystyle\langle\Psi_{m}(X,s,s_{d})\overline{\Psi}_{m}(Y,t,t_{d})\rangle$
$\displaystyle=\tilde{\Delta}_{m}(X-Y)\frac{1}{\big{(}\frac{d-1}{2}\big{)}_{n}}\Big{(}-\frac{i}{2}\sqrt{\overline{s}^{2}\overline{t}^{2}}\Big{)}^{n}C_{n}^{\frac{d-1}{2}}\Big{(}\frac{\overline{s}\cdot\overline{t}}{\sqrt{\overline{s}^{2}\overline{t}^{2}}}\Big{)}$
$\displaystyle-\frac{i}{2}\overline{\mathrlap{\\!\not{\phantom{s}}}s}\,\tilde{\Delta}_{m}(X-Y)\,\overline{\mathrlap{\\!\not{\phantom{t}}}t}\frac{1}{\big{(}\frac{d-1}{2}\big{)}_{n}}\Big{(}-\frac{i}{2}\sqrt{\overline{s}^{2}\overline{t}^{2}}\Big{)}^{n-1}C_{n-1}^{\frac{d-1}{2}}\Big{(}\frac{\overline{s}\cdot\overline{t}}{\sqrt{\overline{s}^{2}\overline{t}^{2}}}\Big{)}$
(VI.29)

where $\tilde{\Delta}_{m}(X-Y)$ is the massive spin 1/2 propagator, whose mass
is proportional to $\gamma_{d}$

$\displaystyle\tilde{\Delta}_{m}(X-Y)=-\int\frac{d^{d}p}{(2\pi)^{d}}\frac{\mathrlap{\\!\not{\phantom{p}}}p+m\gamma_{d}}{p^{2}+m^{2}-i\epsilon}e^{ip\cdot(X-Y)}$
(VI.30)

and $\overline{s}=(s,s_{d})$ and $\overline{t}=(t,t_{d})$ are $d+1$
dimensional auxiliary vectors. We now decompose this propagator into its
independent $d$ dimensional components
$\langle\Psi_{n-i}(X,s)\overline{\Psi}_{n-j}(Y,t)\rangle$, remembering also to
perform the field redefinition $\Psi_{n-i}\to e^{-i\pi\gamma_{d}/4}\Psi_{n-i}$

$\displaystyle\langle\Psi_{n}(X,s)\overline{\Psi}_{n}(Y,t)\rangle$
$\displaystyle=\Delta_{m}\frac{1}{\big{(}\frac{d-1}{2}\big{)}_{n}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n}C_{n}^{\frac{d-1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$
$\displaystyle-\frac{i}{2}\mathrlap{\\!\not{\phantom{s}}}s\,\Delta_{-m}\,\mathrlap{\\!\not{\phantom{t}}}t\frac{1}{\big{(}\frac{d-1}{2}\big{)}_{n}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-1}C_{n-1}^{\frac{d-1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (VI.31)
$\displaystyle\langle\Psi_{n}(X,s)\overline{\Psi}_{n-1}(Y,t)\rangle$
$\displaystyle=\frac{1}{2}i^{1/2}\mathrlap{\\!\not{\phantom{s}}}s\Delta_{-m}\frac{1}{\big{(}\frac{d-1}{2}\big{)}_{n}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-1}C_{n-1}^{\frac{d-1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (VI.32)
$\displaystyle\langle\Psi_{n}(X,s)\overline{\Psi}_{n-2}(Y,t)\rangle$
$\displaystyle=\frac{1}{2}is^{2}\Delta_{m}\frac{1}{\big{(}\frac{d+1}{2}\big{)}_{n-1}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-2}C_{n-2}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$
$\displaystyle+\frac{1}{4}s^{2}\mathrlap{\\!\not{\phantom{s}}}s\Delta_{-m}\mathrlap{\\!\not{\phantom{t}}}t\frac{1}{\big{(}\frac{d+1}{2}\big{)}_{n-1}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-3}C_{n-3}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (VI.33)
$\displaystyle\langle\Psi_{n-1}(X,s)\overline{\Psi}_{n-1}(Y,t)\rangle$
$\displaystyle=\Delta_{m}\frac{1}{\big{(}\frac{d+1}{2}\big{)}_{n-1}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-1}C_{n-1}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$
$\displaystyle-\frac{1}{2}\Delta_{-m}\frac{1}{\big{(}\frac{d-1}{2}\big{)}_{n}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-1}C_{n-1}^{\frac{d-1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$
$\displaystyle-\frac{i}{2}\mathrlap{\\!\not{\phantom{s}}}s\Delta_{-m}\mathrlap{\\!\not{\phantom{t}}}t\frac{1}{\big{(}\frac{d+1}{2}\big{)}_{n-1}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-2}C_{n-2}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (VI.34)
$\displaystyle\langle\Psi_{n-1}(X,s)\overline{\Psi}_{n-2}(Y,t)\rangle$
$\displaystyle=i^{1/2}\mathrlap{\\!\not{\phantom{s}}}s\Delta_{-m}\frac{1}{\big{(}\frac{d+1}{2}\big{)}_{n-1}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-2}C_{n-2}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$
$\displaystyle-\frac{1}{4}i^{-1/2}s^{2}\Delta_{-m}\mathrlap{\\!\not{\phantom{t}}}t\frac{1}{\big{(}\frac{d+1}{2}\big{)}_{n-1}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-3}C_{n-3}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$ (VI.35)
$\displaystyle\langle\Psi_{n-2}(X,s)\overline{\Psi}_{n-2}(Y,t)\rangle=-2\Delta_{-m}\frac{1}{\big{(}\frac{d+1}{2}\big{)}_{n-1}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-2}C_{n-2}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}$
$\displaystyle+\frac{2\Delta_{m}}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-2}$
$\displaystyle\hskip
85.35826pt\times\Bigg{(}\frac{1}{\big{(}\frac{d+1}{2}\big{)}_{n-1}}\Big{(}\frac{d-4}{2}+n\Big{)}C_{n-2}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}+\frac{3}{2}\frac{1}{\big{(}\frac{d+3}{2}\big{)}_{n-2}}C_{n-4}^{\frac{d+3}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}\Bigg{)}$
$\displaystyle-i\frac{\mathrlap{\\!\not{\phantom{s}}}s\Delta_{-m}\mathrlap{\\!\not{\phantom{t}}}t}{n!}\Big{(}-\frac{i}{2}\sqrt{s^{2}t^{2}}\Big{)}^{n-3}$
$\displaystyle\hskip
85.35826pt\times\Bigg{(}\frac{1}{\big{(}\frac{d+1}{2}\big{)}_{n-1}}\Big{(}\frac{d-6}{2}+n\Big{)}C_{n-3}^{\frac{d+1}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}+\frac{3}{2}\frac{1}{\big{(}\frac{d+3}{2}\big{)}_{n-2}}C_{n-5}^{\frac{d+3}{2}}\Big{(}\frac{s\cdot
t}{\sqrt{s^{2}t^{2}}}\Big{)}\Bigg{)}$ (VI.36)

where now $\Delta_{m}(X-Y)$ is the standard massive spin 1/2 propagator,
obtained by replacing $m\gamma_{d}$ with $im$ in (VI.30). All other
correlation functions are related via Hermitian conjugation. Again, we omit
their asymptotic $n\to\infty$ limits for the sake of brevity.

Recall that the hyperfields $\Psi_{n-i}(X,s)$ correspond to fields
$\psi_{\mu_{1}\cdots\mu_{n-i}}$ for $i=0,1,2$. In a simplified notation, the
massive spin $n+1/2$ action in terms of these fields is

$\displaystyle S_{n+1/2}=-\int d^{d}X\Bigg{\\{}$
$\displaystyle\sum_{k=0}^{\cramped{\left\lfloor n/2\right\rfloor}}{n\choose
2k}\Bigg{(}$
$\displaystyle\Big{(}1-\frac{3k}{2}\Big{)}\overline{\psi}_{0}^{(2k)}\cdot\mathcal{S}_{0}^{(2k)}-\frac{k}{2}\overline{\psi}_{0}^{(2k)}\cdot\mathcal{S}_{2}^{(2k-2)}$
$\displaystyle-\frac{k}{2}\overline{\psi}_{2}^{(2k-2)}\cdot\mathcal{S}_{0}^{(2k)}-\frac{k}{2}\overline{\psi}_{2}^{(2k-2)}\cdot\mathcal{S}_{2}^{(2k-2)}$
$\displaystyle-
ik\overline{\psi}_{1}^{(2k-1)}\cdot\mathcal{S}_{2}^{(2k-2)}+ik\overline{\psi}_{2}^{(2k-2)}\cdot\mathcal{S}_{1}^{(2k-1)}$
$\displaystyle\hskip
85.35826pt-k\overline{\psi}_{1}^{(2k-1)}\cdot\mathcal{S}_{1}^{(2k-1)}\Bigg{)}$
$\displaystyle+\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}{n\choose
2k+1}\Bigg{(}$
$\displaystyle-\Big{(}\frac{1}{2}+\frac{3k}{2}\Big{)}\overline{\psi}_{0}^{(2k+1)}\cdot\mathcal{S}_{0}^{(2k+1)}-\frac{k}{2}\overline{\psi}_{0}^{(2k+1)}\cdot\mathcal{S}_{2}^{(2k-1)}$
$\displaystyle-\frac{k}{2}\overline{\psi}_{2}^{(2k-1)}\cdot\mathcal{S}_{0}^{(2k+1)}-\frac{k}{2}\overline{\psi}_{2}^{(2k-1)}\cdot\mathcal{S}_{2}^{(2k-1)}$
$\displaystyle-\frac{i}{2}\overline{\psi}_{0}^{(2k+1)}\cdot\mathcal{S}_{1}^{(2k)}+\frac{i}{2}\overline{\psi}_{1}^{(2k)}\cdot\mathcal{S}_{0}^{(2k+1)}$
$\displaystyle+ik\overline{\psi}_{2}^{(2k-1)}\cdot\mathcal{S}_{1}^{(2k)}-ik\overline{\psi}_{1}^{(2k)}\cdot\mathcal{S}_{2}^{(2k-1)}$
$\displaystyle\hskip
56.9055pt+\Big{(}\frac{1}{2}-k\Big{)}\overline{\psi}_{1}^{(2k)}\cdot\mathcal{S}_{1}^{(2k)}\Bigg{)}\Bigg{\\}}$
(VI.37)

where now $\mathcal{O}^{(k)}_{i}$ denotes the $k$-fold $\gamma$ trace of
$\mathcal{O}_{\mu_{1}\cdots\mu_{n-i}}$ applied from the left, and
$\overline{\mathcal{O}}^{(k)}_{i}$ denotes the $k$-fold $\gamma$ trace of
$\overline{\mathcal{O}}_{\mu_{1}\cdots\mu_{n-i}}$ applied from the right. The
massive spin $n+1/2$ correlation functions
$\langle\psi_{\mu_{1}\cdots\mu_{n-i}}(X)\overline{\psi}_{\nu_{1}\cdots\nu_{n-j}}(Y)\rangle$
from (VI.3) can be obtained from
$\langle\Psi_{n-i}(X,s)\overline{\Psi}_{n-j}(Y,t)\rangle$ by applying
derivatives

$\displaystyle\langle\psi_{\mu_{1}\cdots\mu_{n-i}}(X)\overline{\psi}_{\nu_{1}\cdots\nu_{n-j}}(Y)\rangle=i^{n-(i+j)/2}\frac{\partial^{n-i}}{\partial_{s}^{\mu_{1}}\cdots\partial_{s}^{\mu_{n-i}}}\frac{\partial^{n-j}}{\partial_{t}^{\nu_{1}}\cdots\partial_{t}^{\nu_{n-j}}}\langle\Psi_{n-i}(X,s)\overline{\Psi}_{n-j}(Y,t)\rangle$
(VI.38)

##  VII Discussion

In this paper, we have outlined the covariant formulation of free particles
with any mass $m$ and spin $s$, in any spacetime dimension $d$. For massless
particles with integer spins, their covariant action is (III.10), first
written down in [5], and their propagator is (III.31), in agreement with [47].
For massless particles with half integer spins, their covariant action is
(IV.11), first written down in [6], and their propagator is (IV.2). For
massive particles we arrived at their covariant actions (V.3), (VI.3) and
their corresponding propagators (V.39) – (V.44), (VI.3) – (VI.3) by following
the prescription of dimensional reduction of massless particles in $d+1$
dimensions first described in [41].

The formulation of massive particles exhibits a gauge symmetry (V.1), (VI.1).
Because of this, we were able to arrive at propagators which grow like $\sim
p^{-2}$, and $\sim p^{-1}$, respectively, at the cost of including unphysical
polarizations. Because of the gauge invariance, there must exist a gauge which
only includes physical polarizations. The physical gauges for integer and half
integer spins are obtained by completely exhausting the gauge freedom. An
example of a physical gauge is respectively

Integer spin: $\displaystyle
P^{d}_{\eta}\Big{(}(1-k)\partial_{s}^{2k}\Phi_{n}+ik\partial_{s}^{2(k-1)}\Phi_{n-2}\Big{)}=0$
$\displaystyle k\geq 1$ $\displaystyle
P^{d}_{\eta}\Big{(}(1-k)\partial_{s}^{2k}\Phi_{n-1}+ik\partial_{s}^{2(k-1)}\Phi_{n-3}\Big{)}=0$
$\displaystyle k\geq 0$ (VII.1) Half integer spin: $\displaystyle
P^{d}_{\gamma}\Big{(}(1-k)\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k}\Psi_{n}+ik\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2(k-1)}\Psi_{n-2}\Big{)}=0$
$\displaystyle k\geq 1$ $\displaystyle
P^{d}_{\gamma}\Big{(}ik\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k+1}\Psi_{n}+i^{-1/2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k}\Psi_{n-1}+k\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{2k-1}\Psi_{n-2}\Big{)}=0$
$\displaystyle k\geq 0$ (VII.2)

where $P^{d}_{\eta}$ and $P^{d}_{\gamma}$ are the projection operators onto
traceless and $\gamma$ traceless hyperfields, respectively, defined in
subsection II.1. In these physical gauges, we are left in the action with
traceless rank $n$, $n-2$, $n-3$, $\dots$, 0 hyperfields for integer spins,
and $\gamma$ traceless rank $n$, $n-1$ hyperfields, and two series of $\gamma$
traceless rank $n-2$, $n-3$, $\dots$, 0 hyperfields for half integer spins.
These are precisely the minimal auxiliary field contents used in the Singh-
Hagen actions [3, 4]. The equations of motion in this gauge imply that
$\Phi_{n}$ and $\Psi_{n}$ Fierz-Pauli and Rarita-Schwinger systems (V.2),
(VI.2), respectively, while the remaining hyperfields vanish. Operators which
vanish on-shell make correlation functions vanish after insertions except at
coincident points. The physical gauges are then such that correlations
functions with $\Phi_{n-1}$, $\Phi_{n-2}$, $\Phi_{n-3}$,
$(\partial_{X}^{2}-m^{2})\Phi_{n}$ , $\partial_{s}\cdot\partial_{X}\Phi_{n}$,
$\partial_{s}^{2}\Phi_{n}$, $\Psi_{n-1}$, $\Psi_{n-2}$,
$(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}+m)\Psi_{n}$, or
$\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}$ inserted vanish
except at coincident points. In momentum space, this forces the correlation
functions to be of the form

$\displaystyle\langle\Phi_{n}(p,s)\tilde{\Phi}_{n}(-p,t)\rangle=$
$\displaystyle
G_{m}(p)\frac{1}{\big{(}\frac{d-3}{2}\big{)}_{n}}\frac{1}{n!}\Big{(}-\frac{i}{2}\sqrt{s\cdot
P\cdot s\,t\cdot P\cdot t}\Big{)}^{n}C_{n}^{\frac{d-3}{2}}\Big{(}\frac{s\cdot
P\cdot t}{\sqrt{s\cdot P\cdot s\,t\cdot P\cdot t}}\Big{)}+\text{polynomial}$
(VII.3)
$\displaystyle\langle\Phi_{n-i}(p,s)\tilde{\Phi}_{n-j}(-p,t)\rangle=\text{polynomial}\quad\quad
j=1,2,3$ (VII.4)
$\displaystyle\langle\Psi_{n}(p,s)\overline{\Psi}_{n}(-p,t)\rangle=\frac{1}{\big{(}\frac{d-1}{2}\big{)}_{n}}\frac{1}{n!}\times$
$\displaystyle\Bigg{(}\Delta_{m}(p)\Big{(}-\frac{i}{2}\sqrt{s\cdot P\cdot
s\,t\cdot P\cdot t}\Big{)}^{n}C_{n}^{\frac{d-1}{2}}\Big{(}\frac{s\cdot P\cdot
t}{\sqrt{s\cdot P\cdot s\,t\cdot P\cdot t}}\Big{)}$
$\displaystyle-\frac{i}{2}\big{(}\mathrlap{\\!\not{\phantom{s}}}s-i\frac{s\cdot
p}{m}\big{)}\Delta_{-m}(p)\big{(}\mathrlap{\\!\not{\phantom{t}}}t-i\frac{t\cdot
p}{m}\big{)}\Big{(}-\frac{i}{2}\sqrt{s\cdot P\cdot s\,t\cdot P\cdot
t}\Big{)}^{n-1}C_{n-1}^{\frac{d-1}{2}}\Big{(}\frac{s\cdot P\cdot
t}{\sqrt{s\cdot P\cdot s\,t\cdot P\cdot t}}\Big{)}\Bigg{)}$
$\displaystyle\hskip 341.43306pt+\text{polynomial}$ (VII.5)
$\displaystyle\langle\Psi_{n-i}(p,s)\overline{\Psi}_{n-j}(-p,t)\rangle=\text{polynomial}\quad\quad
j=1,2$ (VII.6)

where $P_{\mu\nu}=\eta_{\mu\nu}+p_{\mu}p_{\nu}/m^{2}$. The physical poles in
(VII), (VII) are uniquely determined by demanding that
$\partial_{s}^{2}\langle\Phi_{n}(p,s)\tilde{\Phi}_{n}(-p,t)\rangle$,
$p\cdot\partial_{s}\langle\Phi_{n}(p,s)\tilde{\Phi}_{n}(-p,t)\rangle$, and
$\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\langle\Psi_{n}(p,s)\overline{\Psi}_{n}(-p,t)\rangle$,
as well as the equivalent expressions in $t$ are polynomials in $p$. The
normalization is determined such that there exists gauge transformations
(V.36), (VI.27) between this gauge and the expressions (V.39) and (VI.3). It
would be interesting to find the specific polynomials which appear in (VII) –
(VII.6).

The exact forms of the propagators in this physical gauge are however not
necessary to have. Indeed, as long as interactions are introduced in a gauge
invariant way, the propagators (V.39) – (V.44), (VI.3) – (VI.3) are
sufficient. In the case of spin 1 particles, gauge invariance implies
generalized Ward identities which guarantee that all Feynman diagrams with
external legs with unphysical polarizations vanish [63, 64]. Thus whenever
perturbation theory is valid, the gauge invariant interacting theory can
continue to describe a spin 1 particle at the level of the $S$-matrix. A
natural expectation is that this general result will continue to hold for
higher spins, although an independent proof of this is necessary. One
complication of such a proof is that when interactions are introduced, one
generally needs to correspondingly deform the gauge symmetry, as is what
happens for instance in Yang-Mills theory and general relativity. The exact
form of the generalized Ward identities in turn depend on the gauge symmetry.
One should therefore first find a consistent set of interactions and
corresponding deformations of the gauge symmetry in the way outlined for
instance in [65]. Previous work in this direction includes [39, 40, 66, 32]. A
particularly important application is finding consistent interactions of
massive higher spins with electromagnetism and gravity to model black hole
binary dynamics [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. An analysis
of introducing interactions within this formulation is postponed for future
work.

Finally, with the explicit expressions for all propagators, this work may be
thought of as “solving” every Poincaré and parity invariant free theory when,
importantly, $d=4$. When $d>5$, there are more irreducible representations of
the little group which are not covered by totally symmetric representations,
corresponding to Young tableaux with more than one row. In the future it would
be useful to do the same for more general representations with mixed symmetry,
along the lines of the formulation of massless particles in [67], to complete
the program of solving every free theory. Having a complete understanding of
such representations, much more control is to be gained when constructing more
general theories with the same field content as string field theory by using a
hyperfield $\Phi(X,\\{s_{i}\\})$ as described in section II, and allow one to
find classes of interactions which result in UV complete amplitudes, and
avoids the causality violations inherent with interacting massive particles
[12, 13, 14].

### Acknowledgements

The author would like to thank Lucile Cangemi, Henrik Johansson, Callum Jones,
Ruslan Metsaev, Mojtaba Najafizadeh, Paolo Pichini, Massimo Porrati, and Radu
Roiban for sharing useful references. The author would also like to thank Amey
Gaikwad, Trevor Scheopner, and E.T. Tomboulis for helpful discussions during
the preparation of this work. This research is supported in part by the Mani
L. Bhaumik Institute for Theoretical Physics.

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