aainstitutetext: Department of Physics and Astronomy, Seoul National
University, Seoul 08826, Koreabbinstitutetext: Centre for Research in String
Theory, School of Physics and Astronomy,
Queen Mary University of London, Mile End Road, London E1 4NS, United
Kingdomccinstitutetext: Center for Theoretical Physics, Seoul National
University, Seoul 08826, Koreaddinstitutetext: College of Liberal Studies,
Seoul National University, Seoul 08826, Korea

# The Relativistic Spherical Top as a Massive Twistor

Joon-Hwi Kim b Jung-Wook Kim a,c,d Sangmin Lee<EMAIL_ADDRESS>jung-
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We prove the equivalence between two traditional approaches to the classical
mechanics of a massive spinning particle in special relativity. One is the
spherical top model of Hanson and Regge, recast in a Hamiltonian formulation
with improved treatment of covariant spin constraints. The other is the
massive twistor model, slightly generalized to incorporate the Regge
trajectory relating the mass to the total spin angular momentum. We establish
the equivalence by computing the Dirac brackets of the physical phase space
carrying three translation and three rotation degrees of freedom. Lorentz
covariance and little group covariance uniquely determine the structure of the
physical phase space. The Regge trajectory does not affect the phase space but
enters the equations of motion. Upon quantization, the twistor model produces
a spectrum that agrees perfectly with the massive spinor-helicity description
proposed by Arkani-Hamed, Huang and Huang for scattering amplitudes for all
masses and spins.

QMUL-PH-21-08

##  1 Introduction

The Lagrangian for a free massive particle in special relativity is a rare
gem; being simpler than its Newtonian predecessor, it cries out for a leap to
general relativity. Unfortunately, this extreme simplicity is quickly lost as
soon as the particle starts spinning, even if the particle remains free. All
textbooks explain that the quantum mechanics of an elementary particle of
minimal nonzero spin is governed by the Dirac equation. As for the classical
mechanics of a relativistic particle with arbitrary (possibly macroscopic)
spin, there seems to be no such consensus.

A Lorentz covariant description of a spinning particle inevitably introduces
constraints. The mass-shell constraint ($p^{2}+m^{2}=0$) reveals that only
three of the four translation degrees of freedom are independent. Identifying
and imposing the constraints for the rotation degrees of freedom is a major
source of complication in any model of a spinning particle. Different choices
are made with not-so-obvious advantages or disadvantages.

To simplify the discussion somewhat, we restrict our attention to the
relativistic analog of the “spherical top”; in the rest frame of the particle,
the model should retain the SO$(3)$ rotation symmetry. By a slight abuse of
language, we refer to this SO$(3)$ as the “little group”. Over many decades, a
variety of models for a massive spinning particle have been proposed. It is
not always straightforward to tell whether two models are equivalent or not;
see e.g. Frydryszak:1996mu ; Rivas:2002 ; Deriglazov:2017jub  for reviews with
a comprehensive bibliography.

From a Hamiltonian point of view, ignoring the translation temporarily and
focusing on the rotation, we can distinguish three types of phase spaces with
a manifest SO$(3)$ isometry: $S^{2}$, $T^{*}(S^{2})$, $T^{*}(\mathrm{SO}(3))$,
where $T^{*}(M)$ denotes the cotangent bundle of a manifold $M$. But, even if
two models appear to have phase spaces of different dimensions, there remains
a possibility that imposing an extra constraint on one model reproduces the
other.

The main goal of this paper is to reduce the catalog of competing models to
point out a hitherto unnoticed equivalence between two famous models of a
massive spinning particle. One of them is the spherical top model of Hanson
and Regge Hanson:1974qy . It uses the Lorentz covariant generalization of the
Euler angles. Before constraints are imposed, the rotation degrees of freedom
are described by an element of the Lorentz group SO$(1,3)$ such that the spin
angular momentum becomes an anti-symmetric tensor with six components. The
other model is the massive twistor (a.k.a. two-twistor or bi-twistor) model.
The dynamical variables consist of a pair of usual twistors, amounting to
eight complex variables in total.

We should clarify what we mean by the two models at the technical level.
Hanson and Regge Hanson:1974qy  began with a Lagrangian formulation and moved
on to a constrained Hamiltonian system. They showed that the mass can be an
arbitrary function of the total spin angular momentum. We will call the
function “Regge trajectory”. For a systematic analysis of constraints, it is
more convenient to start directly from a Hamiltonian formulation, where the
choice of Regge trajectory replaces the choice of Lagrangian in the original
formulation. We begin with a recent proposal for constraints for spins by
Steinhoff Steinhoff:2015ksa  and gradually deviate from it. On the twistor
side, while the key ideas for a massive twistor model had been available since
the beginning of the twistor theory Penrose:1972ia ; Newman:1974fr ;
Tod:1976sk ; Hughston:1979pg ; Huggett:1986fs , a particular version of the
model suitable for our purposes was constructed rather recently. Our twistor
model is nearly identical to the one of Fedoruk and Lukierski Fedoruk:2014vqa
, except that we incorporate the Regge trajectory.

Having specified the two models, we give an overview of how to prove the
equivalence. Before embarking on a technical discussion, let us count the
dimensions of the physical phase spaces. In the spherical top model, among the
six rotation degrees of freedom, three (loosely corresponding to boosts) are
removed by constraints. Thus, the physical phase space becomes 12 dimensional
(three coordinate-momenta pairs for translation and three pairs for rotation).
In the massive twistor model, two complex-valued functions of the twistor
variables are set to zero, leaving behind six complex (12 real) variables.

Choosing and imposing constraints in the two models is by no means a new
subject. Our choices differ from previous ones in a few aspects. In the
spherical top model, we begin with a slight variation of the “spin-gauge”
generators of Steinhoff Steinhoff:2015ksa . The three generators form first-
class constraints. We pair them with three “gauge-fixing conditions” to form a
set of second-class constraints. Including the mass-shell constraints and the
corresponding gauge-fixing condition, we have four pairs of constraints in
total. It is straightforward to compute the Dirac brackets taking account of
all constraints, which reveal the physical phase space without leaving the
original coordinates.

The massive twistor space is equipped with a large symmetry:
$\mathrm{SU}(2,2)\times\mathrm{SU}(2)\times\mathrm{U}(1)$. The SU$(2,2)$ is
identified with the SO$(2,4)$ conformal symmetry of the Minkowski space, to be
broken to the Lorentz group by the mass of the particle. The SU$(2)$ is to be
identified with the SO$(3)$ little group. To emphasize this point, following
Fedoruk:2014vqa  and differing from some earlier works, we use a manifestly
SU$(2)$ covariant notation throughout. The U$(1)$ may be associated with
electric charge Newman:1974fr , but we focus on neutral particles only. Again,
the constraints come in pairs: gauge generators and gauge-fixing conditions.
The gauge generators are twistor counterpart of the mass-shell constraints.
The real part of the gauge-fixing conditions corresponds to the dilatation
symmetry and the imaginary part to the internal U$(1)$ symmetry. Computing the
Dirac brackets on the constrained twistor space is straightforward.

To establish the equivalence between the two models, a map between the two
phase spaces are in order. We propose and verify the map between the two
models in two equivalent ways. One is to compare the Dirac brackets and the
other is to compare the invariant coordinates that solve the constraints in
the respective models. The map is determined, to a large extent, by Lorentz
symmetry and little group symmetry. Remarkably, the 12-dimensional physical
phase space turns out to be a copy of $T^{*}(\mathrm{SO}(1,3))$ and is
insensitive to the Regge trajectory. The Regge trajectory contributes to the
dynamics only through the equations of motion.

Section 2 on the spherical top model and section 3 on the massive twistor
model constitute the main body of this paper. In section 2, we review and
slightly modify Steinhoff’s proposal for spin-gauge generators and pair them
with covariant gauge-fixing conditions. After giving an explicit description
of the gauge orbits, we analyze the fully constrained physical phase space. We
apply two equivalent methods. One is the Dirac bracket in the original
coordinates, the other is to find new coordinates for the physical phase
space. We also use the constraints to derive the covariant equations of motion
and confirm that the solutions indeed describe a free spinning particle. In
section 3, we take parallel steps in the massive twistor model. Although we
are not the first to construct the model, we motivate each step in a self-
contained manner in such a way that the map to the spherical top model is
forced upon us under mild assumptions. We spell out the constraints, compute
the Dirac brackets, and derive the equations of motion. The equivalence with
the spherical top model is established quite explicitly. Upon quantization,
the spectrum of the massive twistor model coincides precisely with the
spectrum of massive states expressed in terms of the “massive helicity-
spinors” Arkani-Hamed:2017jhn  in the context of scattering amplitudes of
quantum fields. Section 4 discusses a few possible applications of our
results.

##  2 Spherical top

We revisit the celebrated spherical top model by Hanson and Regge
Hanson:1974qy  from a Hamiltonian perspective, taking hints from a recent work
by Steinhoff Steinhoff:2015ksa .

###  2.1 Spin-gauge symmetry

The spherical top model has manifest Lorentz symmetry. Its translation is
described by the position 4-vector $x^{\mu}$, and its orientation is described
by an $\mathrm{SO}(1,3)$ matrix $\Lambda^{\mu}{}_{A}$ which defines a “body-
attached orthonormal frame”:

$\displaystyle\eta_{\mu\nu}\Lambda^{\mu}{}_{A}\Lambda^{\nu}{}_{B}=\eta_{AB}\,,\qquad\Lambda^{\mu}{}_{A}\Lambda^{\nu}{}_{B}\eta^{AB}=\eta^{\mu\nu}\,,$
(1)

where the Greek and uppercase Latin indices lie in the range
$\mu,A=0,\cdots,3$. We work in the mostly plus signature. The canonical
momenta of $x^{\mu}$ and $\Lambda^{\mu}{}_{A}$ are denoted by $p_{\mu}$ and
$S_{\mu\nu}$, respectively. The non-vanishing components of the elementary
Poisson bracket for the phase space coordinates are

$\displaystyle\begin{split}\\{x^{\mu},p_{\nu}\\}&=\delta^{\mu}{}_{\nu}\,,\\\
\\{\Lambda^{\rho}{}_{A},S_{\mu\nu}\\}&=-(\delta^{\rho}{}_{\mu}\Lambda_{\nu
A}-\delta^{\rho}{}_{\nu}\Lambda_{\mu A})\,,\\\
\\{S_{\mu\nu},S_{\rho\sigma}\\}&=-(\eta_{\nu\rho}S_{\mu\sigma}-\eta_{\mu\rho}S_{\nu\sigma}-\eta_{\nu\sigma}S_{\mu\rho}+\eta_{\mu\sigma}S_{\nu\rho})\,.\end{split}$
(2)

The spin satisfies the same relation as the orbital angular momentum
$L_{\mu\nu}:=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}$. We denote the total angular
momentum by $J_{\mu\nu}=L_{\mu\nu}+S_{\mu\nu}$. The Poisson bracket is
equivalent to the symplectic form,

$\displaystyle\begin{split}\omega=d\theta&=dp_{\mu}\wedge
dx^{\mu}+\frac{1}{2}dS_{\mu\nu}\wedge\Theta^{\mu\nu}-\frac{1}{2}S_{\mu\nu}\hskip
1.00006pt\Theta^{\mu\rho}\wedge\Theta_{\rho}{}^{\nu}\,,\\\
\theta&=p_{\mu}dx^{\mu}+\frac{1}{2}S_{\mu\nu}\Theta^{\mu\nu}\,,\qquad\Theta^{\mu\nu}=\Lambda^{\mu
A}d\Lambda^{\nu}{}_{A}\,.\end{split}$ (3)

The symplectic form appears to be invariant under two Lorentz symmetries, one
acting on the space-time index $\mu$ and the other acting on the body frame
index $A$. As we will see shortly, the body frame symmetry is broken to the
little group SO$(3)$, so that the time-like ($A=0$) component and the space-
like ($A=a=1,2,3$) components do not have to be correlated. Nevertheless, as a
bookkeeping device, we will often combine a little group scalar and a little
group vector as if they form a “4-vector.”

The Hamiltonian mechanics of the spherical top model is nontrivial because the
phase space coordinates in (2) are constrained. Even for a spin-less particle,
incorporating the mass-shell constraint,

$\displaystyle p^{2}+m^{2}=0\quad(p^{2}:=p^{\mu}p_{\mu})\,,$ (4)

in a manifestly covariant way requires some thoughts. Constraining the
rotation degrees of freedom requires extra effort. Intuitively, among the six
apparent degrees of freedom associated with
$(\Lambda_{\mu}{}^{A},S_{\mu\nu})$, the “boost-like” three should be removed
and only the three genuine rotation degrees of freedom should be retained.

Our discussion of the spin constraints begins with a recent proposal by
Steinhoff Steinhoff:2015ksa  who introduced the “spin-gauge generators”: 111
This choice is closely related to the Pryce-Newton-Wigner choice Pryce:1948pf
; Newton:1949cq . A nearly identical set of first-class constraints appeared
earlier in Nielsen:1995cy . Refs. Levi:2015msa ; Levi:2018nxp  discuss the
importance of spin-gauge symmetry in effective field theories of post-
Newtonian gravity.

$\displaystyle\mathcal{C}_{\mu}=(\hat{p}^{\rho}+\Lambda^{\rho}{}_{0})S_{\rho\mu}\,,\quad\hat{p}^{\rho}=p^{\rho}/|p|\,,\quad|p|=\sqrt{-p^{2}}\,.$
(5)

They form a set of first-class constraints:

$\displaystyle\\{\mathcal{C}_{\mu},\mathcal{C}_{\nu}\\}=\mathcal{C}_{\mu}\hat{p}_{\nu}-\mathcal{C}_{\nu}\hat{p}_{\mu}\,.$
(6)

As such, they qualify as “generators” of the spin-“gauge” symmetry in the
standard nomenclature for constrained Hamiltonian systems; see appendix A for
a quick review.

Our first step away from Steinhoff’s proposal is to rewrite the constraints
adapted to the body frame:

$\displaystyle\phi_{a}:=\frac{1}{2}(\hat{p}^{\mu}+\Lambda^{\mu}{}_{0})S_{\mu\nu}\Lambda^{\nu}{}_{a}\quad(a=1,2,3)\,.$
(7)

This rewriting makes it clear that only three components of
$\mathcal{C}_{\mu}$ are independent. Another advantage of the rewriting is
that the three generators commute under the Poisson bracket:

$\displaystyle\\{\phi_{a},\phi_{b}\\}=0\,.$ (8)

In other words, the gauge algebra is abelian.

As emphasized by Steinhoff, when we have a set of gauge generators which forms
a closed algebra under Poisson bracket, we are free to choose any one of
infinitely many gauge-fixing conditions. We will focus on the covariant gauge
in the rest of this paper.

The implications of the covariant gauge include

$\displaystyle\mbox{both}\quad\Lambda^{\mu}{}_{0}=\hat{p}^{\mu}\quad\mbox{and}\quad
S_{\mu\nu}p^{\nu}=0\,.$ (9)

An attempt to use $S_{\mu\nu}p^{\nu}=0$ as first-class constraints as an
alternative to (5) may cause difficulties discussed in e.g. Deriglazov:2015bqa
. We circumvent the difficulty by pairing the spin-gauge generators (5) with
the following “canonically conjugate” gauge-fixing conditions,

$\displaystyle\begin{split}\chi^{a}=\hat{p}_{\mu}\Lambda^{\mu
a}\,.\end{split}$ (10)

Clearly, imposing both $\phi_{a}=0$ and $\chi^{a}=0$ (and a choice of sign)
implies (9).

Sometimes, it is useful to introduce a conjugate constraint for the mass-shell
constraint. All together, we have four pairs of constraints:

$\displaystyle\begin{split}&\phi_{0}=\frac{1}{2}(p^{2}+m^{2})\,,\quad\phi_{a}=\frac{1}{2}(\hat{p}^{\mu}+\Lambda^{\mu}{}_{0})S_{\mu\nu}\Lambda^{\nu}{}_{a}\,,\\\
&\chi^{0}=\frac{1}{p^{2}}x^{\mu}p_{\mu}\,,\qquad\quad\chi^{a}=\hat{p}_{\mu}\Lambda^{\mu
a}\,.\end{split}$ (11)

We have adjusted the constant coefficients such that

$\displaystyle\\{\chi^{0},\phi_{0}\\}\approx
1\,,\quad\\{\chi^{a},\phi_{b}\\}\approx\delta^{a}{}_{b}\,,$ (12)

with all the other Poisson brackets weakly vanishing. The weak equality
$(\approx)$ means that the equality holds on the constraint surface
$\phi_{A}=0=\chi^{A}$. In the rest of this paper, we will not explicitly
distinguish the weak equality from the strong equality, hoping that the
context will clearly indicate which one is being used.

#### Regge trajectory

Following Steinhoff, we turn on the Regge trajectory using the spin-gauge
invariant combination of the spin tensor,

$\displaystyle
m=m(\tilde{S}^{2})\,,\quad\tilde{S}^{2}:=\frac{1}{2}\tilde{S}^{\mu\nu}\tilde{S}_{\mu\nu}\,,\quad\tilde{S}_{\mu\nu}:=\hat{\eta}_{\mu\rho}S^{\rho\sigma}\hat{\eta}_{\sigma\nu}\,,\quad\hat{\eta}_{\mu\nu}:=\eta_{\mu\nu}+\hat{p}_{\mu}\hat{p}_{\nu}\,.$
(13)

It is easy to verify that $\\{\phi_{a},\tilde{S}_{\mu\nu}\\}=0$, which implies
that $\\{\phi_{0},\phi_{a}\\}=0$ even after the Regge trajectory is turned on.
Physically, $\tilde{S}_{\mu\nu}$ is the projection of $S_{\mu\nu}$ onto the
space-like three dimensions transverse to the momentum vector. Compared with
the normalized Pauli-Lubanski vector
$W_{\mu}=-\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}J^{\nu\rho}\hat{p}^{\sigma}$
$(\varepsilon_{0123}=+1)$, we have $W^{2}=\tilde{S}^{2}$.

###  2.2 Gauge orbit in the phase space

The four conjugate pairs of constraints (11) define the gauge orbits in the
unconstrained phase space. The dynamics of the spherical top depends only on
the physical phase space defined by our choice of gauge slice:
$\phi_{A}=0=\chi^{A}$.

In this subsection, we study the geometry of the gauge orbits. We focus on the
spin-gauge symmetry $(\phi_{a},\chi^{b})$ here and leave $(\phi_{0},\chi^{0})$
for a later subsection, since the latter pair governs the time-evolution of
the spherical top.

#### Flow equations

Recall our definition of spin-gauge generators:

$\displaystyle\phi_{a}=\frac{1}{2}(\hat{p}^{\mu}+\Lambda^{\mu}{}_{0})S_{\mu\nu}\Lambda^{\nu}{}_{a}\,.$

We showed earlier that the algebra is abelian, so the gauge orbit should look
like $\mathbb{R}^{3}$ locally. How is the gauge orbit embedded in the
unconstrained phase space?

Let $y^{a}$ be the coordinates for the $\mathbb{R}^{3}$. With hindsight, we
rescale the coordinates slightly such that the flow equations for the gauge
orbits are given by

$\displaystyle\frac{\partial X}{\partial y^{a}}=2\\{X,\phi_{a}\\}\,,$ (14)

where $X$ is any one of the phase space coordinates. Explicitly,

$\displaystyle\begin{split}\partial_{a}p_{\mu}&=0\,,\\\
\partial_{a}\Lambda^{\mu}{}_{B}&=\Lambda^{\mu}{}_{a}[(\hat{p}+\Lambda_{0})^{\rho}\Lambda_{\rho
B}]-(\hat{p}+\Lambda_{0})^{\mu}\eta_{aB}\,,\\\
\partial_{a}S_{\mu\nu}&=(\hat{p}_{\mu}S_{\nu\rho}-\hat{p}_{\nu}S_{\mu\rho})\Lambda^{\rho}{}_{a}\,,\\\
\partial_{a}x_{\mu}&=\frac{1}{|p|}\hat{\eta}_{\mu\nu}S^{\nu}{}_{\rho}\Lambda^{\rho}{}_{a}\,.\end{split}$
(15)

To set the initial condition for the equation, we demand the covariant gauge
conditions $(\phi_{a}=0=\chi^{a})$ at the origin of the $y$-space. As we move
away from the origin, $\phi_{a}$ stay zero but $\chi^{a}$ develop non-zero
values. Since $\\{\chi^{a},\phi_{b}\\}=\delta^{a}{}_{b}$ at the origin, for a
small departure from the origin, $\chi^{a}=2y^{a}+\mathcal{O}(y^{2})$ should
hold.

#### Spin-gauge invariants

Steinhoff Steinhoff:2015ksa  suggests a number of gauge invariant combinations
that are conserved along the flow. In addition to $p_{\mu}$ and $\phi_{a}$,
the list of invariants includes

$\displaystyle\begin{split}&\tilde{\Lambda}^{\mu}{}_{0}=\hat{p}^{\mu}\,,\quad\tilde{\Lambda}^{\mu}{}_{a}=\Lambda^{\mu}{}_{a}-\frac{(\hat{p}+\Lambda_{0})^{\mu}(\hat{p}\cdot\Lambda_{a})}{\hat{p}\cdot(\hat{p}+\Lambda_{0})}\,,\\\
&\tilde{S}_{\mu\nu}=\hat{\eta}_{\mu\rho}S^{\rho\sigma}\hat{\eta}_{\sigma\nu}\,,\quad\tilde{x}^{\mu}=x^{\mu}-S^{\mu\nu}\frac{p_{\nu}}{|p|^{2}}\,.\end{split}$
(16)

The invariance of $\tilde{x}^{\mu}$ and $\tilde{S}_{\mu\nu}$ can easily be
checked against the flow equation. The change of $x^{\mu}$ and $S_{\mu\nu}$
are such that $J_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}+S_{\mu\nu}$ remains
invariant.

The invariance of $\tilde{\Lambda}$ has a nice geometric explanation
Steinhoff:2015ksa , which serves as a motivation behind the original spin-
gauge generators $\mathcal{C}_{\mu}$ in (5). For any pair of time-like, unit-
normalized, future-pointing 4-vectors $u^{\mu}$ and $v^{\mu}$, there exists a
unique “minimal boost” mapping one to the other:

$\displaystyle u^{\mu}=B(u,v)^{\mu}{}_{\nu}v^{\nu}\,,\quad
B(u,v)^{\mu}{}_{\nu}=\delta^{\mu}{}_{\nu}-\frac{(u+v)^{\mu}(u+v)_{\nu}}{1+u\cdot
v}+2u^{\mu}v_{\nu}\,.$ (17)

As emphasized in Steinhoff:2015ksa , the relation between the two frames
$\tilde{\Lambda}$ and $\Lambda$ in (16) is nothing but the minimal boost
between $\Lambda^{\mu}{}_{0}$ and $\tilde{\Lambda}^{\mu}{}_{0}=\hat{p}^{\mu}$:

$\displaystyle\Lambda^{\mu}{}_{A}=B(\Lambda_{0},\tilde{\Lambda}_{0})^{\mu}{}_{\nu}\tilde{\Lambda}^{\nu}{}_{A}\quad\Longleftrightarrow\quad\tilde{\Lambda}^{\mu}{}_{A}=B(\tilde{\Lambda}_{0},\Lambda_{0})^{\mu}{}_{\nu}\Lambda^{\nu}{}_{A}\,.$
(18)

Hence, if we solve the flow equation for $\Lambda^{\mu}{}_{0}$, the solution
for $\Lambda^{\mu}{}_{a}$ will follow automatically from the minimal boost.

####  $\Lambda$ orbit

The equations for $\Lambda$ do not depend on $S_{\mu\nu}$ or $x^{\mu}$. So, we
may solve them first and then feed the result into the other equations. The
relation in (16) implies

$\displaystyle\Lambda^{\mu}{}_{0}\tilde{\Lambda}_{\mu
a}=-\tilde{\Lambda}^{\mu}{}_{0}\Lambda_{\mu
a}=-\chi_{a}\,\,\implies\,\,\Lambda^{\mu}{}_{0}=\sqrt{1+\vec{\chi}^{2}}\,\hat{p}^{\mu}-\chi^{a}\tilde{\Lambda}^{\mu}{}_{a}\quad(\vec{\chi}^{2}=\chi^{a}\chi_{a})\,.$
(19)

Thus, the solution for $\chi^{a}(y)$ completely determines
$\Lambda^{\mu}{}_{0}$. The flow equation for $\chi^{a}$ away from the origin
of the $y$-space is

$\displaystyle\partial_{a}\chi_{b}=2\\{\chi_{b},\phi_{a}\\}=\chi_{a}\chi_{b}+(1+\sqrt{1+\vec{\chi}^{2}})\delta_{ab}\,.$
(20)

Noting that $\partial_{a}\chi_{b}-\partial_{b}\chi_{a}=0$, we look for an
SO$(3)$-covariant solution of the form

$\displaystyle\chi_{a}=\partial_{a}\left[F(\vec{y}^{2})\right]=2y_{a}F^{\prime}(\vec{y}^{2})\qquad(\vec{y}^{2}=y^{a}y_{a})\,.$
(21)

The equations admit a unique solution: $F^{\prime}=(1-\vec{y}^{2})^{-1}$. In
summary, the $\Lambda$ orbit is

$\displaystyle\Lambda^{\mu}{}_{B}(y)=\tilde{\Lambda}^{\mu}{}_{A}R^{A}{}_{B}(y)\,,\quad
R^{A}{}_{B}(y)=\frac{1}{1-\vec{y}^{2}}\begin{pmatrix}1+\vec{y}^{2}&-2y_{b}\\\
-2y^{a}&(1-\vec{y}^{2})\delta^{a}{}_{b}+2y^{a}y_{b}\end{pmatrix}\,.$ (22)

####  $S$ and $x$ orbits

Intuitively, the invariant spin $\tilde{S}_{\mu\nu}$ accounts for the true
rotation of the body, while the spin-gauge dependent part is induced by the
boost (18). In the gauge invariant frame, we split the components of the spin
tensor as

$\displaystyle
S_{\mu\nu}(y)=\tilde{\Lambda}_{\mu}{}^{a}\tilde{\Lambda}_{\nu}{}^{b}\tilde{S}_{ab}+2\tilde{\Lambda}_{[\mu}{}^{0}\tilde{\Lambda}_{\nu]}{}^{a}U_{a}(y)\,.$
(23)

The flow equation for $U_{a}$ and its solution are simply

$\displaystyle\partial_{a}U_{b}=\tilde{S}_{ab}\quad\implies\quad
U_{a}(y)=-\tilde{S}_{ab}y^{b}\,.$ (24)

Finally, feeding this result into a relation in (16), we obtain the solution
for $x^{\mu}$:

$\displaystyle
x^{\mu}(y)=\tilde{x}^{\mu}+S^{\mu\nu}(y)\frac{p^{\nu}}{|p|^{2}}=\tilde{x}^{\mu}+\frac{1}{|p|}\tilde{\Lambda}^{\mu
a}\tilde{S}_{ab}y^{b}\,.$ (25)

###  2.3 Physical phase space

The goal of this subsection is to study the geometry of the 12-dimensional
“physical” or “fully constrained” phase space. We use two slightly different
but equivalent methods. First, we compute the Dirac brackets accounting for
the four pairs of constraints. An advantage of the Dirac bracket is that we
can keep using the original coordinates of the unconstrained phase space.
Second, we choose new coordinates for the physical subspace and spell out the
inclusion map into the unconstrained phase space. The spin-gauge invariant
combinations (16) strongly suggest an ideal choice of coordinates. Remarkably,
we find that the physical phase space is a copy of $T^{*}(\mathrm{SO}(1,3))$
which is independent of the Regge trajectory.

#### Dirac bracket

Appendix A gives a short review of Dirac brackets and related concepts.
Applying the general methods to our model, we compute the “fully constrained”
Dirac brackets,

$\displaystyle\\{f,g\\}_{*}:=\\{f,g\\}-\\{f,\phi_{A}\\}\\{\chi^{A},g\\}+\\{f,\chi^{A}\\}\\{\phi_{A},g\\}\,.$
(26)

The complete list of non-vanishing Dirac brackets is as follows:

$\displaystyle\begin{split}\\{x^{\mu},x^{\nu}\\}_{*}&=\frac{1}{|p|^{2}}J^{\mu\nu}\,,\\\
\\{x^{\mu},p_{\nu}\\}_{*}&=\hat{\delta}^{\mu}{}_{\nu}\,,\\\
\\{x^{\mu},\Lambda_{\nu}{}^{a}\\}_{*}&=\frac{1}{|p|}\left(\hat{p}_{\nu}\Lambda^{\mu
a}-2m^{\prime}x^{\mu}\tilde{S}_{\nu\rho}\Lambda^{\rho a}\right)\,,\\\
\\{p_{\mu},\Lambda_{\nu}{}^{a}\\}_{*}&=2m^{\prime}\hat{p}_{\mu}\tilde{S}_{\nu\rho}\Lambda^{\rho
a}\,,\\\
\\{x^{\rho},S_{\mu\nu}\\}_{*}&=-\frac{1}{|p|}\left(\hat{p}_{\mu}S_{\nu}{}^{\rho}-\hat{p}_{\nu}S_{\mu}{}^{\rho}\right)\,,\\\
\\{\Lambda_{\rho}{}^{a},S_{\mu\nu}\\}_{*}&=-\left(\hat{\eta}_{\rho\mu}\Lambda_{\nu}{}^{a}-\hat{\eta}_{\rho\nu}\Lambda_{\mu}{}^{a}\right)\,,\\\
\\{S_{\mu\nu},S_{\rho\sigma}\\}_{*}&=-\left(\hat{\eta}_{\nu\rho}S_{\mu\sigma}-\hat{\eta}_{\mu\rho}S_{\nu\sigma}+\hat{\eta}_{\nu\sigma}S_{\rho\mu}-\hat{\eta}_{\mu\sigma}S_{\rho\nu}\right)\,.\end{split}$
(27)

It is straightforward to confirm that the Dirac brackets satisfy the Leibniz
rule and the Jacobi identity on the physical phase space. By construction, the
Dirac brackets also satisfy

$\displaystyle\\{\phi_{A},f\\}_{*}=0=\\{\chi^{A},f\\}_{*}\,,$ (28)

where $f$ is any function of the phase space coordinates.

Finally, we note that the difference between the Poisson bracket and the Dirac
bracket may vanish completely for some. For example, since all constraints are
Lorentz invariant,

$\displaystyle\\{J_{\mu\nu},f\\}_{*}=\\{J_{\mu\nu},f\\}\,.$ (29)

If both $f$ and $g$ are fully gauge invariant
($\\{f,\phi_{A}\\}=0=\\{g,\phi_{A}\\}$), then

$\displaystyle\\{f,g\\}_{*}=\\{f,g\\}\,.$ (30)

#### Minimal coordinates

The Dirac bracket (27) completely specifies the symplectic geometry of the
physical phase space, but it may look opaque to an untrained eye. Here, we
wish to introduce new coordinates on the physical phase space and spell out
the inclusion map that solves the constraints explicitly.

We demand that amongst the new coordinates the Dirac bracket and the Poisson
bracket should coincide. We have seen some hints in this direction earlier. We
recognize $\Lambda_{\mu}{}^{A}$ and $J_{\mu\nu}$ as good candidates with $6+6$
components:

$\displaystyle\begin{split}\\{\Lambda_{\mu}{}^{A},\Lambda_{\nu}{}^{B}\\}_{*}&=0\,,\\\
\\{\Lambda_{\rho}{}^{A},J_{\mu\nu}\\}_{*}&=-(\eta_{\rho\mu}\Lambda_{\nu}{}^{A}-\eta_{\rho\nu}\Lambda_{\mu}{}^{A})\,,\\\
\\{J_{\mu\nu},J_{\rho\sigma}\\}_{*}&=-(\eta_{\nu\rho}J_{\mu\sigma}-\eta_{\mu\rho}J_{\nu\sigma}+\eta_{\nu\sigma}J_{\rho\mu}-\eta_{\mu\sigma}J_{\rho\nu})\,.\end{split}$
(31)

Since we are focusing on the physical phase space, $\Lambda$ here may be
understood as the gauge invariant $\tilde{\Lambda}$ discussed in section 2.2.

The covariant spin-gauge condition enforces
$\Lambda^{\mu}{}_{0}=\hat{p}^{\mu}$. To see how this condition is implemented
in the new coordinates, it is useful to define a pair of 4-vectors:

$\displaystyle V_{\mu}=J_{\mu\nu}\hat{p}^{\nu}\,,\quad
W_{\mu}=-(*J)_{\mu\nu}\hat{p}^{\nu}=-\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}J^{\rho\sigma}\hat{p}^{\nu}\,.$
(32)

We recognize $W_{\mu}$ as the normalized Pauli-Lubanski vector. Substituting
$J_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}+S_{\mu\nu}$ into (32) produces many
terms, but most of them vanish on the physical phase space. In terms of the
body frame coordinates $V_{A}=V_{\mu}\Lambda^{\mu}{}_{A}$,
$W_{A}=W_{\mu}\Lambda^{\mu}{}_{A}$, we have

$\displaystyle\begin{split}V_{A=0}=0\,,&\quad V_{a}=-|p|x^{\mu}\Lambda_{\mu
a}\,,\\\ W_{A=0}=0\,,&\quad
W_{a}=\frac{1}{2}\varepsilon_{a}{}^{bc}(S_{\mu\nu}\Lambda^{\mu}{}_{b}\Lambda^{\nu}{}_{c\,})\,.\end{split}$
(33)

These expressions show how the three translation and three rotation degrees of
freedom reside in the new coordinates. To clarify the physical meaning of
$(\Lambda^{\mu}{}_{A},J_{\mu\nu})$ under constraints, we propose to use an
equivalent but suggestive form
$(\Lambda^{\mu}{}_{0},\Lambda^{\mu}{}_{a},V_{a},W_{a})$. The two sets are
related by $\Lambda^{\mu}{}_{0}=\hat{p}^{\mu}$ and

$\displaystyle\begin{split}J_{a0}=L_{a0}=V_{a}=-m(W^{2})x_{a}\,,\quad
J_{ab}=S_{ab}=\varepsilon^{c}{}_{ab}W_{c}\,.\end{split}$ (34)

In other words, the new coordinates encode the physical degrees of freedom as

$\displaystyle\begin{split}&\Lambda_{\mu}{}^{a}\;:\;\mbox{(transverse
rotation)}\,,\qquad J_{ab}=S_{ab}\;:\;\mbox{(spin angular momentum)}\,,\\\
&\Lambda_{\mu}{}^{0}\;:\;\mbox{(momentum
direction)}\,,\quad\;J_{0a}=L_{0a}\;:\;\mbox{(transverse
position)}\,.\end{split}$

The inclusion map of the new coordinates
$(\Lambda_{\mu}{}^{0},\Lambda_{\mu}{}^{a},V_{a},W_{a})$ into the original
coordinates $(x^{\mu},p_{\mu},\Lambda_{\mu}{}^{A},S_{\mu\nu})$ is simply given
by

$\displaystyle x^{\mu}=x_{a}\Lambda^{\mu a}=-\frac{V_{a}\Lambda^{\mu
a}}{m(W^{2})}\,,\quad p_{\mu}=m(W^{2})\Lambda_{\mu 0}\,,\quad
S_{\mu\nu}=\Lambda_{\mu}{}^{a}\Lambda_{\nu}{}^{b}\varepsilon^{c}{}_{ab}W_{c}\,.$
(35)

#### Symplectic geometry

The above discussion can be rephrased in the language of symplectic geometry.
The space-time components and body frame components are related in the usual
way:

$\displaystyle\begin{split}&p_{A}=p_{\mu}\Lambda^{\mu}{}_{A}\,,\quad
x^{A}=x^{\mu}\Lambda_{\mu}{}^{A}\,,\quad
S_{AB}=S_{\mu\nu}\Lambda^{\mu}{}_{A}\Lambda^{\nu}{}_{B}\,,\quad\\\
&\Theta^{AB}=\Theta^{\mu\nu}\Lambda_{\mu}{}^{A}\Lambda_{\nu}{}^{B}=(\Lambda^{\mu
C}d\Lambda^{\nu}{}_{C})\Lambda_{\mu}{}^{A}\Lambda_{\nu}{}^{B}=d\Lambda^{\nu
A}\Lambda_{\nu}{}^{B}\,.\end{split}$ (36)

The symplectic potential (3) can be alternatively expressed as

$\displaystyle\theta$
$\displaystyle=p_{\mu}dx^{\mu}+\frac{1}{2}S_{\mu\nu}\hskip
1.00006pt\Theta^{\mu\nu}=p_{A}dx^{A}+\frac{1}{2}J_{AB}\Theta^{AB}\,.$ (37)

These two guises of $\theta$ naturally arise from a geometrical perspective;
see (139) and explanations thereafter. Since the constraints break the
fictitious Lorentz symmetry acting on the body frame indices, we may write

$\displaystyle\theta$
$\displaystyle=p_{0}dx^{0}+p_{a}dx^{a}+(L_{0a}+S_{0a})\Theta^{0a}+\frac{1}{2}(L_{ab}+S_{ab})\Theta^{ab}\,.$
(38)

The conditions $\chi^{a}=0$ set $p_{a}=0$ and aligns the time axis of the body
frame as $\Lambda_{\mu 0}=\hat{p}_{\mu}$. The mass-shell constraint $\phi_{0}$
fixes the value of $p_{A=0}$ according to the Regge trajectory. The condition
$\chi^{0}=0$ sets $x^{A=0}=0$. Finally, the conditions $\phi_{a}=0$ set
$S_{0a}=0$.

Imposing the constraints amounts to taking the pull-back of the symplectic
potential onto the fully constrained phase space:

$\displaystyle\theta_{*}$
$\displaystyle=L_{0a}\Theta^{0a}+\frac{1}{2}S_{ab}\Theta^{ab}\,.$ (39)

Clearly, the second term carries the rotation degrees of freedom in the
transverse 3-space. As for the first term, we note that

$\displaystyle
L_{0a}=-p_{0}x_{a}=|p|x_{a}=m(\tilde{S}^{2})x_{a}\,,\quad\Theta^{0a}=d\Lambda^{\nu
0}\Lambda_{\nu}{}^{a}=-d(\hat{p}^{\nu})\Lambda_{\nu}{}^{a}\,.$ (40)

Mathematically, $L_{0a}\Theta^{0a}$ describes the boost part of the Lorentz
transformation. Physically, it actually describes the translation degrees of
freedom. To be concrete, $L_{0a}$ represents the transverse body frame
components of the position $x^{\mu}$, and $\Theta^{0a}$ measures the variation
of the unit vector $\hat{p}$ in the body frame.

In terms of $(\Lambda_{\mu}{}^{0},\Lambda_{\mu}{}^{a},V_{a},W_{a})$, the
symplectic potential (39) on the physical phase space can also be written as

$\displaystyle\theta_{*}=-V_{a}\hskip
1.00006pt\Theta^{0a}+W_{a}\left(\textstyle\frac{1}{2}\varepsilon^{a}{}_{bc}\Theta^{bc}\right)\,.$
(41)

The dependence on the Regge trajectory is nowhere to be seen. This should come
as no surprise since this symplectic potential carries precisely the same
information as the Dirac bracket algebra (31) describing
$T^{*}(\mathrm{SO}(1,3))$. Each point on the physical phase space labels a
distinct trajectory (solution to the equations of motion). The “shape” of the
trajectory is not visible until time evolution, generated by $\phi_{0}$,
begins to unfold. Only then will the Regge trajectory begin to play a crucial
role.

#### Complex combinations

For later purposes, we recombine $V_{a}$ and $W_{a}$ slightly. They implement
$\mathfrak{so}(1,3)$ while keeping only the $\mathfrak{so}(3)$ covariance
manifest:

$\displaystyle\\{V_{a},V_{b}\\}_{*}=\varepsilon^{c}{}_{ab}W_{c}\,,\quad\\{V_{a},W_{b}\\}_{*}=-\varepsilon^{c}{}_{ab}V_{c}\,,\quad\\{W_{a},W_{b}\\}_{*}=-\varepsilon^{c}{}_{ab}W_{c}\,.$
(42)

Taking the standard complex combinations,

$\displaystyle
J_{a}:=\frac{1}{2}(W_{a}-iV_{a})\,,\quad\bar{J}_{a}:=\frac{1}{2}(W_{a}+iV_{a})\,,$
(43)

we may rewrite (31) as

$\displaystyle\begin{split}&\\{J_{a},J_{b}\\}_{*}=-\varepsilon^{c}{}_{ab}J_{c}\,,\\\
&\\{\Lambda_{\mu}{}^{0},J_{c}\\}_{*}=-\frac{1}{2i}\Lambda_{\mu
c}\,,\quad\\{\Lambda_{\mu}{}^{a},J_{c}\\}_{*}=-\frac{1}{2i}\delta^{a}{}_{c}\Lambda_{\mu}{}^{0}-\frac{1}{2}\varepsilon^{a}{}_{cb}\Lambda_{\mu}{}^{b}\,,\end{split}$
(44)

and their complex conjugates. Combining the Dirac bracket (44) and the
inclusion map (35) offers perhaps the most intuitive way to understand the
Dirac bracket computed directly in the original coordinates (27).

#### From vector to spinor

Anticipating the comparison with the massive twistor model in the next
section, we convert the vector indices into spinor indices. The Lorentz
indices $(\mu,\nu,\ldots)$ are converted to SL$(2,\mathbb{C})$ bi-spinor
indices $(\alpha,\beta,\ldots;{\dot{\alpha}},{\dot{\beta}},\ldots)$. A body
frame vector $X_{A}$ is converted to a little group $\mathfrak{su}(2)$ bi-
spinor $X_{IJ}$ such that the anti-symmetric part $X_{[IJ]}$ carries the
$X_{A=0}$ component and the symmetric part $X_{(IJ)}$ carries the
$X_{a=1,2,3}$ components. See appendix B for a detailed summary of our
conventions for spinors and twistors, and the precise conversion rules between
vector indices and spinor indices.

We may apply the conversion rules either to the body frame coordinates
$(V_{a},W_{a},\Lambda_{\mu}{}^{A})$ parametrizing the constrained subspace or
to the original coordinates $(x^{\mu},p_{\mu},\Lambda_{\mu}{}^{A},S_{\mu\nu})$
parametrizing the unconstrained phase space. We convert the common part first,

$\displaystyle\Lambda^{IJ}_{\alpha{\dot{\alpha}}}=(\sigma^{\mu})_{\alpha{\dot{\alpha}}}\Lambda_{\mu}{}^{A}(\sigma_{A})^{I}{}_{J}\,.$
(45)

The conversion rule for the body frame vectors $V$ and $W$ is (see (148))

$\displaystyle X_{IJ}=\sigma^{A}_{IJ}X_{A}\,.$ (46)

In the spinor notation, the Dirac bracket (44) can be rewritten as

$\displaystyle\begin{split}\\{J^{I}{}_{J},J^{K}{}_{L}\\}_{*}&=-i(\delta^{I}{}_{L}J^{K}{}_{J}-\delta^{K}{}_{J}J^{I}{}_{L})\,,\\\
\\{(\Lambda_{\alpha{\dot{\alpha}}})^{I}{}_{J},J_{KL}\\}_{*}&=+i(\Lambda_{\alpha{\dot{\alpha}}})^{I}{}_{(K}\epsilon_{L)J}\,.\end{split}$
(47)

The conversion rules for $(x,p,S)$ are as follows:

$\displaystyle
p_{\alpha{\dot{\alpha}}}=(\sigma^{\mu})_{\alpha{\dot{\alpha}}}p_{\mu}\,,\quad
x^{{\dot{\alpha}}\alpha}=(\bar{\sigma}_{\mu})^{{\dot{\alpha}}\alpha}\left(\textstyle{\frac{1}{2}}x^{\mu}\right)\,,\quad\textstyle{\frac{1}{2}}S_{\mu\nu}\sigma^{[\mu}_{\alpha{\dot{\alpha}}}\sigma^{\nu]}_{\beta{\dot{\beta}}}=S_{\alpha\beta}\bar{\epsilon}_{{\dot{\alpha}}{\dot{\beta}}}+\epsilon_{\alpha\beta}\bar{S}_{{\dot{\alpha}}{\dot{\beta}}}\,.$
(48)

The reality conditions are recast as

$\displaystyle[p_{\alpha{\dot{\beta}}}]^{*}=p_{\beta{\dot{\alpha}}}\,,\quad[x^{{\dot{\alpha}}\beta}]^{*}=x^{{\dot{\beta}}\alpha}\,,\quad[\Lambda^{IJ}_{\alpha\dot{\beta}}]^{*}=-\Lambda_{JI,\beta\dot{\alpha}}\,,\quad[S_{\alpha\beta}]^{*}=\bar{S}_{{\dot{\alpha}}{\dot{\beta}}}\,.$
(49)

The non-vanishing Dirac brackets in (27) are rewritten as

$\displaystyle\begin{split}\\{x_{\alpha{\dot{\alpha}}},x_{\beta{\dot{\beta}}}\\}_{*}&=\frac{1}{2|p|^{2}}\left(J_{\alpha\beta}{\bar{\epsilon}}_{{\dot{\alpha}}{\dot{\beta}}}+\epsilon_{\alpha\beta}\bar{J}_{{\dot{\alpha}}{\dot{\beta}}}\right)\,,\\\
\\{x_{\alpha{\dot{\alpha}}},p_{\beta{\dot{\beta}}}\\}_{*}&=-\left(\epsilon_{\alpha\beta}{\bar{\epsilon}}_{{\dot{\alpha}}{\dot{\beta}}}-\hat{p}_{\alpha\dot{\alpha}}\hat{p}_{\beta\dot{\beta}}\right)\,,\\\
\\{x_{\alpha{\dot{\alpha}}},\Lambda^{IJ}_{\beta{\dot{\beta}}}\\}_{*}&=\frac{1}{2|p|}\left(\hat{p}_{\beta{\dot{\beta}}}\Lambda^{IJ}_{\alpha{\dot{\alpha}}}+4m^{\prime}x_{\alpha{\dot{\alpha}}}\big{(}S_{\beta{\gamma}}\epsilon^{{\gamma}{\delta}}\Lambda_{{\delta}{\dot{\beta}}}^{IJ}+\bar{S}_{{\dot{\beta}}{\dot{{\gamma}}}}{\bar{\epsilon}}^{{\dot{{\gamma}}}{\dot{{\delta}}}}\Lambda_{\beta{\dot{{\delta}}}}^{IJ}\big{)}\right)\,,\\\
\\{p_{\alpha{\dot{\alpha}}},\Lambda^{IJ}_{\beta{\dot{\beta}}}\\}_{*}&=-2m^{\prime}\hat{p}_{\alpha{\dot{\alpha}}}\big{(}S_{\beta{\gamma}}\epsilon^{{\gamma}{\delta}}\Lambda_{{\delta}{\dot{\beta}}}^{IJ}+\bar{S}_{{\dot{\beta}}{\dot{{\gamma}}}}{\bar{\epsilon}}^{{\dot{{\gamma}}}{\dot{{\delta}}}}\Lambda_{\beta{\dot{{\delta}}}}^{IJ}\big{)}\,,\\\
\\{x_{\gamma{\dot{\gamma}}},S_{\alpha\beta}\\}_{*}&=\frac{1}{4|p|}\big{(}\hat{p}_{\alpha{\dot{\gamma}}}S_{\beta\gamma}+\hat{p}_{\beta{\dot{\gamma}}}S_{\alpha\gamma}-\hat{p}_{\gamma{\dot{\gamma}}}S_{\alpha\beta}\big{)}\,,\\\
\\{\Lambda^{IJ}_{\gamma{\dot{\gamma}}},S_{\alpha\beta}\\}_{*}&=\frac{1}{2}\left(\epsilon_{\alpha\gamma}\Lambda^{IJ}_{\beta{\dot{\gamma}}}+\epsilon_{\beta\gamma}\Lambda^{IJ}_{\alpha{\dot{\gamma}}}-\frac{1}{2}\hat{p}_{\gamma{\dot{\gamma}}}\big{(}\Lambda^{IJ}_{\alpha{\dot{\alpha}}}\hat{p}_{\beta{\dot{\beta}}}+\Lambda^{IJ}_{\beta{\dot{\alpha}}}\hat{p}_{\alpha{\dot{\beta}}}\big{)}{\bar{\epsilon}}^{{\dot{\alpha}}{\dot{\beta}}}\right)\,,\\\
\\{S_{\alpha\beta},S_{\gamma{\delta}}\\}_{*}&=-\frac{1}{2}\big{(}\epsilon_{\alpha\gamma}S_{\beta{\delta}}+\epsilon_{\beta\gamma}S_{\alpha{\delta}}+\epsilon_{\alpha{\delta}}S_{\beta\gamma}+\epsilon_{\beta{\delta}}S_{\alpha\gamma}\big{)}\,,\\\
\\{S_{\alpha\beta},\bar{S}_{{\dot{\gamma}}{\dot{{\delta}}}}\\}_{*}&=-\frac{1}{2}\left(\hat{p}_{\alpha{\dot{\gamma}}}(S\cdot\hat{p})_{\beta{\dot{{\delta}}}}+\hat{p}_{\beta{\dot{\gamma}}}(S\cdot\hat{p})_{\alpha{\dot{{\delta}}}}+\hat{p}_{\alpha{\dot{{\delta}}}}(S\cdot\hat{p})_{\beta{\dot{\gamma}}}+\hat{p}_{\beta{\dot{{\delta}}}}(S\cdot\hat{p})_{\alpha{\dot{\gamma}}}\right)\,,\end{split}$
(50)

while the covariant gauge condition (9) is rephrased as

$\displaystyle S_{\mu\nu}p^{\nu}=0\quad\Longleftrightarrow\quad
S_{\alpha\beta}\epsilon^{\beta\gamma}p_{\gamma{\dot{\alpha}}}=p_{\alpha{\dot{\beta}}}\bar{\epsilon}^{{\dot{\beta}}{\dot{\gamma}}}\bar{S}_{{\dot{\gamma}}{\dot{\alpha}}}\,,$
(51)

i.e., the bi-spinor $(S\cdot
p)_{\alpha{\dot{\alpha}}}:=S_{\alpha\beta}\epsilon^{\beta\gamma}p_{\gamma{\dot{\alpha}}}$
as a $(2\times 2)$ matrix should be anti-hermitian.

###  2.4 Equation of motion

We study the equations of motion of the spherical top model. A covariant
Hamiltonian description of a relativistic particle is somewhat subtle due to
the fact the time evolution generator $\phi_{0}$ is a gauge generator at the
same time. We circumvent the subtlety by treating $(\phi_{0},\chi^{0})$
differently from $(\phi_{a},\chi^{a})$, although it is possible to put all
gauge generators on an equal footing; see e.g. ref. Barandes:2019oas .

#### Without spin

Let us quickly review the equation of motion of a spin-less particle from a
Hamiltonian point of view. The first-order action is

$\displaystyle\mathcal{S}=\int
d\sigma\left(p_{\mu}\dot{x}^{\mu}-\frac{\kappa}{2}(p^{2}+m^{2})\right)\,,$
(52)

where $\sigma$ is a world-line parameter and $\dot{x}=dx/d\sigma$. The mass-
shell constraint is imposed by the Lagrange multiplier $\kappa$. The
variations of the action give

$\displaystyle\begin{split}\delta_{x}\mathcal{S}=0\quad&\Longrightarrow\quad\dot{p}_{\mu}=0\,,\\\
\delta_{p}\mathcal{S}=0\quad&\Longrightarrow\quad\dot{x}^{\mu}-\kappa
p^{\mu}=0\,,\\\ \delta_{\kappa}\mathcal{S}=0\quad&\Longrightarrow\quad
p^{2}+m^{2}=0\,.\end{split}$ (53)

Combining the equations, we can determine $\kappa$:

$\displaystyle\kappa=\frac{1}{m}\frac{d\tau}{d\sigma}\,,\quad
d\tau=\sqrt{-dx^{\mu}dx_{\mu}}\,.$ (54)

Note that the constraint $\phi_{0}$ in the action plays the role of the
Hamiltonian generating the time evolution, even though its numerical value
stays zero.

#### With spin

Since $\phi_{a}$ form first-class constraints, we can use either of the two
equivalent ways to derive the equation of motion under the constraints. One is
to add a linear combination of $\phi_{a}$ in the action using Lagrange
multipliers. The other is to use the Dirac bracket constructed from the pair
$(\phi_{a},\chi^{a})$. For illustrative purposes, we examine both approaches
in some detail.

For the first approach, we begin with the action:

$\displaystyle\begin{split}\mathcal{S}&=\int
d\sigma\left(p_{\mu}\dot{x}^{\mu}+\frac{1}{2}S_{\mu\nu}\Omega^{\mu\nu}-\kappa\phi_{0}-\kappa^{a}\phi_{a}\right)\\\
&=\int
d\sigma\left(p_{\mu}\dot{x}^{\mu}+\frac{1}{2}S_{\mu\nu}\Omega^{\mu\nu}-\frac{\kappa}{2}(p^{2}+m^{2})-\frac{\kappa^{a}}{2}(\hat{p}^{\mu}+\Lambda^{\mu}{}_{0})S_{\mu\nu}\Lambda^{\nu}{}_{a}\right)\,.\end{split}$
(55)

The angular velocity is defined as the pull back of the one-form
$\Theta^{\mu\nu}$ onto the world-line:

$\displaystyle\Omega^{\mu\nu}=\Lambda^{\mu
A}\frac{d\Lambda^{\nu}{}_{A}}{d\sigma}\,.$ (56)

The variations of the Lagrange multipliers $\kappa^{0}$, $\kappa^{a}$ force
the gauge generators $\phi_{0}$, $\phi_{a}$ to vanish. The variations of the
dynamical variables yield the equations of motion:

$\displaystyle\begin{split}\delta_{x}\mathcal{S}=0\quad&\Longrightarrow\quad\dot{p}_{\mu}=0\,,\\\
\delta_{p}\mathcal{S}=0\quad&\Longrightarrow\quad\dot{x}_{\mu}-\kappa\left(p_{\mu}-\frac{2mm^{\prime}}{|p|^{2}}\tilde{S}_{\mu\nu}S^{\nu\rho}p_{\rho}\right)-\frac{\kappa^{a}}{2|p|}\hat{\eta}_{\mu\nu}S^{\nu}{}_{\rho}\Lambda^{\rho}{}_{a}=0\,,\\\
\delta_{S}\mathcal{S}=0\quad&\Longrightarrow\quad\Omega^{\mu\nu}-2\kappa
mm^{\prime}\tilde{S}^{\mu\nu}-\kappa^{a}(\hat{p}+\Lambda_{0})^{[\mu}\Lambda^{\nu]}{}_{a}=0\,.\end{split}$
(57)

The variation of $\Lambda_{A}{}^{\mu}$ takes values in $\mathfrak{so}(1,3)$:

$\displaystyle\delta\Lambda^{\mu}{}_{A}=-\delta{\theta}^{\mu}{}_{\nu}\Lambda^{\nu}{}_{A}\,,\quad\delta\theta^{\nu\mu}=-\delta{\theta}^{\mu\nu}\,.$
(58)

The intermediate steps include

$\displaystyle\begin{split}\delta_{\Lambda}\left(S_{\mu\nu}\Omega^{\mu\nu}\right)&=-\delta\theta^{\mu\nu}\left(\dot{S}_{\mu\nu}-2S_{\rho[\mu}\Omega_{\nu]}{}^{\rho}\right)\,,\\\
\delta_{\Lambda}\left[(\hat{p}+\Lambda_{0})^{\mu}S_{\mu\nu}\Lambda^{\nu}{}_{a}\right]&=-\delta\theta^{\mu\nu}\left((\hat{p}+\Lambda_{0})^{\rho}S_{\rho[\mu}\Lambda_{\nu]a}-\Lambda_{0[\mu}S_{\nu]\rho}\Lambda^{\rho}{}_{a}\right)\,.\end{split}$
(59)

Combining all terms, we find an equation for $\dot{S}$:

$\displaystyle\dot{S}_{\mu\nu}=2S_{\rho[\mu}\Omega_{\nu]}{}^{\rho}+\kappa^{a}(\hat{p}+\Lambda_{0})^{\rho}S_{\rho[\mu}\Lambda_{\nu]a}-\kappa^{a}\Lambda_{0[\mu}S_{\nu]\rho}\Lambda^{\rho}{}_{a}\,.$
(60)

Before taking the next step, let us collect all the equations of motion in one
place:

$\displaystyle\begin{split}\dot{p}_{\mu}&=0\,,\\\
\dot{x}_{\mu}&=\kappa\left(p_{\mu}-\frac{2mm^{\prime}}{|p|^{2}}\tilde{S}_{\mu\nu}S^{\nu\rho}p_{\rho}\right)+\frac{\kappa^{a}}{2|p|}\hat{\eta}_{\mu\nu}S^{\nu}{}_{\rho}\Lambda^{\rho}{}_{a}\,,\\\
\Omega^{\mu\nu}&=2\kappa
mm^{\prime}\tilde{S}^{\mu\nu}+\kappa^{a}(\hat{p}+\Lambda_{0})^{[\mu}\Lambda^{\nu]}{}_{a}\,,\\\
\dot{S}_{\mu\nu}&=2S_{\rho[\mu}\Omega_{\nu]}{}^{\rho}+\kappa^{a}(\hat{p}+\Lambda_{0})^{\rho}S_{\rho[\mu}\Lambda_{\nu]a}-\kappa^{a}\Lambda_{0[\mu}S_{\nu]\rho}\Lambda^{\rho}{}_{a}\,.\end{split}$
(61)

#### Lagrange multipliers

As shown in (54), The Lagrange multiplier $\kappa$ for the mass-shell
constraint is related to the world-line reparametrization invariance. The
multipliers $\kappa^{a}$ for the spin-gauge symmetry are uniquely fixed by the
gauge-fixing conditions; see appendix A for the relation between the
multipliers, gauge-fixing conditions and Dirac bracket.

Our favorite gauge-fixing condition is the covariant one introduced in (10).
Requiring that the gauge-fixing condition is preserved by the equations of
motion, we find

$\displaystyle\begin{split}0&=\frac{d}{d\sigma}\left(\hat{p}_{\nu}\Lambda^{\nu}{}_{a}\right)\\\
&=-\hat{p}_{\mu}\Omega^{\mu\nu}\Lambda_{\nu a}\\\
&=-\hat{p}_{\mu}\left(2\kappa
mm^{\prime}\tilde{S}^{\mu\nu}+\kappa^{b}(\hat{p}+\Lambda_{0})^{[\mu}\Lambda^{\nu]}{}_{b}\right)\Lambda_{\nu
a}\\\
&\approx-\frac{1}{2}\kappa^{b}\hat{p}_{\mu}\left((\hat{p}+\Lambda_{0})^{\mu}\Lambda^{\nu}{}_{b}-(\hat{p}+\Lambda_{0})^{\nu}\Lambda^{\mu}{}_{b}\right)\Lambda_{\nu
a}=\eta_{ab}\kappa^{b}\,.\end{split}$ (62)

In the last line, we took the liberty to impose both $\phi_{A}=0$ and
$\chi^{A}=0$.

We reach the expected conclusion that, for a free particle, the multipliers
$\kappa^{a}$ all vanish. The final form of the equations of motion describes a
motion with constant (proper) linear velocity and angular velocity.

$\displaystyle\begin{split}&\frac{dp_{\mu}}{d\tau}=0\,,\quad
p^{\mu}=m\frac{dx^{\mu}}{d\tau}\,,\\\
&\frac{dS_{\mu\nu}}{d\tau}=2S_{\rho[\mu}\bar{\Omega}_{\nu]}{}^{\rho}=0\,,\quad
S^{\mu\nu}=\frac{1}{2m^{\prime}}\bar{\Omega}^{\mu\nu}\,,\quad\left(\bar{\Omega}^{\mu\nu}:=\Lambda^{\mu
A}\frac{d\Lambda^{\nu}{}_{A}}{d\tau}\right)\,,\\\
&S_{\mu\nu}p^{\nu}=0\,,\quad\Lambda^{\nu}{}_{0}=\hat{p}^{\nu}\,.\end{split}$
(63)

In the Newtonian limit, the Regge trajectory can be approximated by

$\displaystyle\begin{split}&m(\tilde{S}^{2})=m_{0}+\frac{1}{2I}\tilde{S}^{2}+\cdots\quad\left(m^{\prime}(\tilde{S}^{2})=\frac{1}{2I}+\cdots\right)\\\
\implies\quad&p^{0}=\sqrt{\vec{p}^{2}+m^{2}}=m_{0}+\frac{1}{2m_{0}}{\vec{p}\hskip
1.00006pt}^{2}+\frac{1}{2I}\tilde{S}^{2}+\cdots\,,\end{split}$ (64)

such that $S^{\mu\nu}=(1/2m^{\prime})\bar{\Omega}^{\mu\nu}$ in (63) reduces to
the familiar $\vec{S}=I\,\vec{\Omega}$.

#### Dirac bracket revisited

When first-class constraints $\phi_{a}$ are paired with conjugate constraints
$\chi^{a}$ to form a set of second-class constraints, Dirac brackets can be
used to derive the equations of motion without writing out the full action
with Lagrange multipliers. To apply the method to our model, we construct the
“partly constrained” Dirac brackets,

$\displaystyle\\{f,g\\}_{\diamond}:=\\{f,g\\}-\\{f,\phi_{a}\\}\\{\chi^{a},g\\}+\\{f,\chi^{a}\\}\\{\phi_{a},g\\}\,.$
(65)

Here, we say “partly constrained” to emphasize the fact that we left out the
mass-shell constraints $(\chi^{0},\phi_{0})$. The complete list of non-
vanishing Dirac brackets among phase space coordinates is as follows:

$\displaystyle\begin{split}\\{x^{\mu},x^{\nu}\\}_{\diamond}&=\frac{1}{|p|^{2}}S^{\mu\nu}\,,\\\
\\{x^{\mu},p_{\nu}\\}_{\diamond}&=\delta^{\mu}{}_{\nu}\,,\\\
\\{x^{\mu},\Lambda_{\nu}{}^{a}\\}_{\diamond}&=\frac{1}{|p|^{2}}p_{\nu}\Lambda^{\mu
a}\,,\\\
\\{x^{\rho},S_{\mu\nu}\\}_{\diamond}&=-\frac{1}{|p|^{2}}\left(p_{\mu}S_{\nu}{}^{\rho}-p_{\nu}S_{\mu}{}^{\rho}\right)\,,\\\
\\{\Lambda_{\rho}{}^{a},S_{\mu\nu}\\}_{\diamond}&=-(\hat{\eta}_{\rho\mu}\Lambda_{\nu}{}^{a}-\hat{\eta}_{\rho\nu}\Lambda_{\mu}{}^{a})\,,\\\
\\{S_{\mu\nu},S_{\rho\sigma}\\}_{\diamond}&=-(\hat{\eta}_{\nu\rho}S_{\mu\sigma}-\hat{\eta}_{\mu\rho}S_{\nu\sigma}+\hat{\eta}_{\nu\sigma}S_{\rho\mu}-\hat{\eta}_{\mu\sigma}S_{\rho\nu})\,.\end{split}$
(66)

We did not include the time-like frame vector ($\Lambda^{\mu}{}_{0}$) in the
list, because the covariant gauge condition enforces
$\Lambda^{\mu}{}_{0}=\hat{p}^{\mu}$, which in turn implies
$\\{\Lambda^{\mu}{}_{0},X\\}_{\diamond}=\\{\hat{p}^{\mu},X\\}_{\diamond}$ for
any phase space coordinate $X$.

The Dirac bracket guarantees that both the gauge generators and the gauge-
fixing conditions are preserved throughout the Hamiltonian time evolution.

$\displaystyle\begin{split}\\{\phi_{a},h\\}_{\diamond}&=\\{\phi_{a},h\\}-\\{\phi_{a},\phi_{b}\\}\\{\chi^{b},h\\}+\\{\phi_{a},\chi^{b}\\}\\{\phi_{b},h\\}\\\
&=\\{\phi_{a},h\\}-0-\delta_{a}{}^{b}\\{\phi_{b},h\\}=0\,,\\\
\\{\chi^{a},h\\}_{\diamond}&=\\{\chi^{a},h\\}-\\{\chi^{a},\phi_{b}\\}\\{\chi^{b},h\\}+\\{\chi^{a},\chi^{b}\\}\\{\phi_{b},h\\}\\\
&=\\{\chi^{a},h\\}-\delta^{a}{}_{b}\\{\chi^{b},h\\}+0=0\,.\end{split}$ (67)

Using $\phi_{0}$ as an effective Hamiltonian for the time evolution, we arrive
at a general prescription for the equation of motion:

$\displaystyle\frac{df}{d\sigma}=\kappa^{0}\\{f,\phi_{0}\\}_{\diamond}\qquad\mbox{or}\qquad\frac{df}{d\tau}=\frac{1}{m}\\{f,\phi_{0}\\}_{\diamond}\,.$
(68)

Inserting $x$, $p$ and $S$, we immediately recover

$\displaystyle\frac{dx^{\mu}}{d\tau}=\frac{1}{m}p^{\mu}\,,\quad\frac{dp^{\mu}}{d\tau}=0\,,\quad\frac{dS_{\mu\nu}}{d\tau}=0\,.$
(69)

Inserting $\Lambda$, we find

$\displaystyle\frac{d\Lambda^{\nu}{}_{a}}{d\tau}=2m^{\prime}\tilde{S}^{\rho\sigma}\delta_{[\sigma}^{\nu}\Lambda_{\rho]a}\quad\implies\quad\bar{\Omega}^{\mu\nu}=2m^{\prime}\hskip
1.00006ptS^{\mu\nu}\,.$ (70)

Thus, the Dirac bracket method reproduces all equations of motion in (63) as
expected.

##  3 Massive twistor

We introduce the massive twistor model Fedoruk:2014vqa  in such a way that its
equivalence to the spherical top model appears almost inevitable. In the
previous section, we showed that the physical space of the spherical top model
is a copy of $T^{*}(\mathrm{SO}(1,3))$ and that the Regge trajectory carries
all the dynamical information not dictated by the symmetries.

Our goal in this section is clear from the beginning. First, we should show
that the physical phase space of the twistor model is also a copy of
$T^{*}(\mathrm{SO}(1,3))$. Second, we should show how to incorporate the Regge
trajectory into the twistor model. Finally, the symmetries and the Regge
trajectory should be compatible such that the solutions to the equations of
motion agree perfectly between the two models.

###  3.1 Massive twistor with constraints

We assume that the readers are somewhat familiar with the massless twistor
space. Our conventions for the twistor space and its symmetry algebra are
summarized in the second half of section B. We follow the same conventions
whenever applicable and focus on the differences between the massive case and
the massless case.

#### Twistor space and its symmetry

The massive twistor consists of a doublet of massless twistors:
$Z_{\mathrm{A}}{}^{I}$ ($\mathrm{A}=1,2,3,4$, $I=1,2$). The Lie algebra of the
natural symmetry group acting on this space is
$\mathfrak{su}(2,2)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$. The mass of
the particle partially breaks the conformal algebra $\mathfrak{su}(2,2)$ and
leave behind the Lorentz subalgebra $\mathfrak{sl}(2,\mathbb{C})$. The
“internal” $\mathfrak{su}(2)$ is to be identified with the little group
symmetry of the spherical top model. We will not discuss the possibility of
associating the $\mathfrak{u}(1)$ with an electric charge.

The twistor coordinates and their conjugations with respect to the SU$(2,2)$
invariant tensor are denoted by

$\displaystyle Z_{\mathrm{A}}{}^{I}=\begin{pmatrix}\lambda_{\alpha}{}^{I}\\\
i\mu^{{\dot{\alpha}}I}\end{pmatrix}\,,\quad\bar{Z}_{I}{}^{\mathrm{B}}:=(Z_{\mathrm{A}}{}^{I})^{\dagger}A^{\bar{\mathrm{A}}\mathrm{B}}=\begin{pmatrix}-i\bar{\mu}_{I}{}^{\beta}&\bar{\lambda}_{I{\dot{\beta}}}\end{pmatrix}\,.$
(71)

The symplectic form invariant under the
$\mathfrak{su}(2,2)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ is given by

$\displaystyle\omega=i(d\bar{Z}_{I}{}^{\mathrm{A}}\wedge
dZ_{\mathrm{A}}{}^{I})=d\bar{\mu}_{I}{}^{\alpha}\wedge
d\lambda_{\alpha}{}^{I}-d\bar{\lambda}_{I\dot{\alpha}}\wedge
d\mu^{\dot{\alpha}I}\,.$ (72)

The Poisson brackets are

$\displaystyle\\{Z_{\mathrm{A}}{}^{I},\bar{Z}_{J}{}^{\mathrm{B}}\\}=-i\,\delta_{\mathrm{A}}{}^{\mathrm{B}}\delta^{I}{}_{J}\,\,\implies\,\,\\{\lambda_{\alpha}{}^{I},\bar{\mu}_{J}{}^{\beta}\\}=\delta_{\alpha}{}^{\beta}\delta^{I}{}_{J}\,,\quad\\{\bar{\lambda}_{I\dot{\alpha}},\mu^{\dot{\beta}J}\\}=\delta_{\dot{\alpha}}{}^{\dot{\beta}}\delta_{I}{}^{J}\,.$
(73)

The generators of Hamiltonian flows for $\mathfrak{su}(2,2)$ are

$\displaystyle
G_{\mathrm{A}}{}^{\mathrm{B}}:=Z_{\mathrm{A}}{}^{I}\bar{Z}_{I}{}^{\mathrm{B}}-\frac{1}{4}\delta_{\mathrm{A}}{}^{\mathrm{B}}(\bar{Z}_{I}{}^{\mathrm{C}}Z_{\mathrm{C}}{}^{I})\,.$
(74)

They are identified with the conformal generators as

$\displaystyle\begin{split}G_{\mathrm{A}}{}^{\mathrm{B}}=\begin{pmatrix}-iJ_{\alpha}{}^{\beta}-\textstyle{\frac{i}{2}}\delta_{\alpha}{}^{\beta}D&-P_{\alpha{\dot{\beta}}}\\\
-K^{{\dot{\alpha}}\beta}&+i\bar{J}^{\dot{\alpha}}{}_{{\dot{\beta}}}+\textstyle{\frac{i}{2}}\delta^{\dot{\alpha}}{}_{{\dot{\beta}}}D\end{pmatrix}\,,\end{split}$
(75)

such that

$\displaystyle\begin{split}J_{\alpha\beta}=\bar{\mu}_{I(\alpha}\lambda_{\beta)}{}^{I}\,,\quad&D=\frac{1}{2}\left(\langle{\bar{\mu}}\lambda\rangle+[{\bar{\lambda}}{\mu}]\right)\,,\\\
P_{\alpha{\dot{\alpha}}}=-\lambda_{\alpha}{}^{I}\bar{\lambda}_{I{\dot{\alpha}}}\,,\quad&K^{{\dot{\alpha}}\alpha}=-\mu^{{\dot{\alpha}}I}\bar{\mu}_{I}{}^{\alpha}\,,\end{split}$
(76)

where we defined

$\displaystyle\langle{\bar{\mu}}\lambda\rangle:={\bar{\mu}}_{I}{}^{\alpha}\lambda_{\alpha}{}^{I}\,,\quad[{\bar{\lambda}}{\mu}]:={\bar{\lambda}}_{I{\dot{\alpha}}}{\mu}^{{\dot{\alpha}}I}\,.$
(77)

The generators of Hamiltonian flows for
$\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ are

$\displaystyle
W^{I}{}_{J}=Z_{\mathrm{A}}{}^{I}\bar{Z}_{J}{}^{\mathrm{A}}\,\,\implies\,\,\\{W^{I}{}_{J},W^{K}{}_{L}\\}={-}i(\delta^{I}{}_{L}W^{K}{}_{J}-\delta^{K}{}_{J}W^{I}{}_{L})\,.$
(78)

#### Constraints on the twistor space

Before imposing constraints, the massive twistor space carries 8 real (4
complex) degrees of freedom. To establish the equivalence between the two
models, at the level of the fully constrained physical phase space with 6
physical degrees of freedom, we should introduce 2 real (1 complex) gauge
generators and the corresponding gauge-fixing conditions.

The twistor model has no analogue of the spin-gauge symmetry of the spherical
top model. In a rough sense, the twistor model offers a solution to the
covariant spin condition. So, we only need to consider the twistor incarnation
of the mass-shell constraint. Recall the mass-shell constraint and its
conjugate from the spherical top model:

$\displaystyle\phi_{0}=\frac{1}{2}\left(p^{2}+m^{2}(\tilde{S}^{2})\right)\,,\quad\chi^{0}=\frac{1}{p^{2}}x^{\mu}p_{\mu}\,.$

We propose analogous pairs of constraints in the twistor model:

$\displaystyle\begin{split}\phi&=-\frac{1}{2}\left({\rm
det}(\lambda)-m(\tilde{S}^{2})\right)\,,\quad\bar{\chi}=\frac{1}{{\rm
det}(\lambda)}\langle{\bar{\mu}}\lambda\rangle\,,\\\
\bar{\phi}&=-\frac{1}{2}\left({\rm
det}(\bar{\lambda})-m(\tilde{S}^{2})\right)\,,\quad\chi=\frac{1}{{\rm
det}({\bar{\lambda}})}[{\bar{\lambda}}{\mu}]\,.\end{split}$ (79)

This proposal is natural in many ways. First, the condition ${\rm
det}(\lambda)=m$ takes a “square-root” of the mass shell condition
$-p^{2}=m^{2}$. A slight difference is that the twistor version fixes not only
the magnitude but also the phase of ${\rm det}(\lambda)$. Second, just as
$\phi_{0}$ in the spherical top model generates the “time gauge orbit’ in the
real Minkowski space, $x^{\mu}\mapsto x^{\mu}+kp^{\mu}$, $(\phi,\bar{\phi})$
in the twistor model generates the time gauge orbit in the complexified
Minkowski space, $z^{\mu}\mapsto z^{\mu}+\kappa p^{\mu}$. Third, while
$\chi^{0}$ in the spherical top model projects the $x$-space onto the real
three-dimensional subspace transverse to $p^{\mu}$, $(\bar{\chi},\chi)$ in the
twistor model projects the $z$-space onto the complex three-dimensional
transverse subspace. Finally, the constraints have been unit-normalized:

$\\{\bar{\chi},\phi\\}=1\,,\quad\\{\chi,\bar{\phi}\\}=1\,.$ (80)

Readers unfamiliar with the use of complexified Minkowski space in the twistor
model might have been puzzled by some of the statements above. More
information will be given.

#### Map to the spherical top model

Strictly speaking, the map between the two models is 2-to-1 for the same
reason as $\mathrm{SO}(3)=\mathrm{SU}(2)/\mathbb{Z}_{2}$. We will ignore this
double cover issue in most of what follows.

The map from the twistor model to the spherical top model can be phrased
either in terms of the original coordinates
$(x^{\mu},p_{\mu},\Lambda^{\mu}{}_{A},S_{\mu\nu})$ or in terms of the physical
coordinates $(\Lambda^{\mu}{}_{0},\Lambda^{\mu}{}_{a},V_{a},W_{a})$. We find
it instructive to spell out both descriptions, beginning with the common
factor: $\Lambda^{\mu}{}_{A}\in\mathrm{SO}(1,3)$. The twistor counterpart of
$\Lambda^{\mu}{}_{A}$ involves only $(\lambda,\bar{\lambda})$ and not
$(\mu,\bar{\mu})$. This is natural in that $\Lambda$ and
$(\lambda,\bar{\lambda})$ span Lagrangian submanifolds of their respective
phase spaces both before and after imposing constraints.

#### Momentum and body frame

The map for $p_{\mu}$ is waiting for us in (76):

$\displaystyle
p_{\alpha\dot{\alpha}}=-\lambda_{\alpha}{}^{I}\bar{\lambda}_{I\dot{\alpha}}\,.$
(81)

The overall minus sign is due to the fact that $p_{0}<0$ in our convention.
The reality of $p_{\mu}$ is consistent with the fact that $\lambda$ and
$\bar{\lambda}$ are hermitian conjugates.

In the covariant gauge, $\Lambda^{\mu}{}_{A}$ is aligned such that
$\Lambda^{\mu}{}_{0}=\hat{p}^{\mu}$, while the magnitudes of the momentum
enter the Regge trajectory in a Lorentz invariant way. In an analogous way, we
can separate the magnitudes of the twistor variables by

$\displaystyle{\rm det}(\lambda)\hskip
1.00006pt\epsilon^{IJ}:=\epsilon^{\alpha\beta}\lambda_{\alpha}{}^{I}\lambda_{\beta}{}^{J}\,,\quad{\rm
det}(\bar{\lambda}):=[{\rm det}(\lambda)]^{*}\,,$ (82)

and define the normalized variables as

$\displaystyle\begin{split}\hat{\lambda}_{\alpha}{}^{I}:=({\rm
det}(\lambda))^{-1/2}\lambda_{\alpha}{}^{I}&\,,\quad\hat{{\mu}}^{{\dot{\alpha}}I}:=({\rm
det}(\lambda))^{1/2}{\mu}^{{\dot{\alpha}}I}\,,\\\
\hat{{\bar{\lambda}}}_{I{\dot{\alpha}}}:=({\rm
det}({\bar{\lambda}}))^{-1/2}{\bar{\lambda}}_{I{\dot{\alpha}}}&\,,\quad\hat{{\bar{\mu}}}_{I}{}^{\alpha}:=({\rm
det}({\bar{\lambda}}))^{1/2}{\bar{\mu}}_{I}{}^{\alpha}\,.\\\ \end{split}$ (83)

The map from $(\hat{\lambda},\hat{\bar{\lambda}})$ to $\Lambda$, which
includes the normalized version of (81), is self-evident, because
$(\hat{\lambda},\hat{\bar{\lambda}})$ by definition parametrize the group
manifold SL$(2,\mathbb{C})$. Explicitly, using the spinor notation for
$\Lambda$ in (45), we write the map as

$\displaystyle(\Lambda_{\alpha{\dot{\alpha}}})^{I}{}_{J}=(\sigma^{\mu})_{\alpha{\dot{\alpha}}}\Lambda_{\mu}{}^{A}(\sigma_{A})^{I}{}_{J}=2\hat{\lambda}_{\alpha}{}^{I}\hat{{\bar{\lambda}}}_{J{\dot{\alpha}}}\,.$
(84)

The normalization condition is recast as

$\displaystyle-\frac{1}{2}\epsilon^{\alpha\beta}{\bar{\epsilon}}^{{\dot{\alpha}}{\dot{\beta}}}\Lambda^{IJ}_{\alpha{\dot{\alpha}}}\Lambda_{KL,\beta{\dot{\beta}}}=-2\delta{}^{(I}{}_{K}\delta^{J)}{}_{L}\,.$
(85)

#### Position and spin

Having mapped the “base” part of $T^{*}(\mathrm{SO}(1,3))$ between the two
models, we move on to the “fiber” part. We can try to find the expressions for
$(x^{\mu},S_{\mu\nu})$ directly via the incidence relation of the twistor
model, or by going through the body frame vectors $(V_{a},W_{a})$ at an
intermediate stage. We explain both approaches, one after another.

In the massless twistor theory, it is well known that the incidence relation
with complexified Minkowski coordinates conveniently account for the helicity
of the particle Penrose:1972ia . Essentially the same is true for the massive
twistors, provided that the internal SU$(2)$ of the twistor space is
identified with the little group of the massive particle.

The incidence relation in our convention is given by

$\displaystyle\mu^{{\dot{\alpha}}I}=-z^{\dot{\alpha}\beta}\lambda_{\beta}{}^{I}\,.$
(86)

Since $\lambda_{\alpha}{}^{I}$ is an invertible matrix, unlike in the massless
case, we can explicitly solve the relation for $z^{{\dot{\alpha}}\beta}$.
Taking the real and imaginary parts of $z^{\mu}=x^{\mu}+iy^{\mu}$, we obtain

$\displaystyle\begin{split}x^{{\dot{\alpha}}\alpha}&=\frac{1}{2}\left(\frac{1}{{\rm
det}(\lambda)}\mu^{{\dot{\alpha}}I}\lambda^{\alpha
J}\epsilon_{IJ}+\frac{1}{{\rm
det}(\bar{\lambda})}\epsilon^{IJ}\bar{\lambda}_{I}{}^{\dot{\alpha}}\bar{\mu}_{J}{}^{\alpha}\right)\,,\\\
y^{{\dot{\alpha}}\alpha}&=\frac{1}{2i}\left(\frac{1}{{\rm
det}(\lambda)}\mu^{{\dot{\alpha}}I}\lambda^{\alpha
J}\epsilon_{IJ}-\frac{1}{{\rm
det}(\bar{\lambda})}\epsilon^{IJ}\bar{\lambda}_{I}{}^{\dot{\alpha}}\bar{\mu}_{J}{}^{\alpha}\right)\,.\end{split}$
(87)

The $y^{\mu}$ coordinate does not appear explicitly in the spherical top
model. Instead, it contributes to the spin variable through the relation,

$\displaystyle
S_{\mu\nu}=*(y_{\mu}p_{\nu}-y_{\nu}p_{\mu})=\varepsilon_{\mu\nu}{}^{\rho\sigma}y_{\rho}p_{\sigma}\,.$
(88)

When translated to the spinor notation, it gives

$\displaystyle
S_{\alpha\beta}=\frac{1}{2}\left(\bar{\mu}_{I(\alpha}\lambda_{\beta)}{}^{I}-\frac{1}{{\rm
det}(\lambda)}\lambda_{(\alpha}{}^{I}\lambda_{\beta)}{}^{J}({\mu}^{{\dot{{\gamma}}}}{}_{I}{\bar{\lambda}}_{J{\dot{{\gamma}}}})\right)\,.$
(89)

The orbital angular momentum in the spinor notation gives

$\displaystyle
L_{\alpha\beta}=\frac{1}{2}\left(\bar{\mu}_{I(\alpha}\lambda_{\beta)}{}^{I}+\frac{1}{{\rm
det}(\lambda)}\lambda_{(\alpha}{}^{I}\lambda_{\beta)}{}^{J}({\mu}^{{\dot{{\gamma}}}}{}_{I}{\bar{\lambda}}_{J{\dot{{\gamma}}}})\right)\,.$
(90)

The orbital and spin parts of the angular momentum add up to $J_{\alpha\beta}$
in (76) as expected. In summary, the map from the massive twistor space to the
phase space of the spherical top model is

$\displaystyle\begin{split}p_{\alpha{\dot{\alpha}}}=-\lambda_{\alpha}{}^{I}\bar{\lambda}_{I{\dot{\alpha}}}\,,&\quad
x^{{\dot{\alpha}}\alpha}=\frac{1}{2}\left(\frac{1}{{\rm
det}(\lambda)}\mu^{{\dot{\alpha}}I}\lambda^{\alpha
J}\epsilon_{IJ}+\frac{1}{{\rm
det}(\bar{\lambda})}\epsilon^{IJ}\bar{\lambda}_{I}{}^{\dot{\alpha}}\bar{\mu}_{J}{}^{\alpha}\right)\,,\\\
(\Lambda_{\alpha\dot{\alpha}})^{I}{}_{J}=\frac{2\lambda_{\alpha}{}^{I}\bar{\lambda}_{J{\dot{\alpha}}}}{|{\rm
det}(\lambda)|}\,,&\quad
S_{\alpha\beta}=\frac{1}{2}\left(\bar{\mu}_{I(\alpha}\lambda_{\beta)}{}^{I}-\frac{1}{{\rm
det}(\lambda)}\lambda_{(\alpha}{}^{I}\lambda_{\beta)}{}^{J}({\mu}^{{\dot{{\gamma}}}}{}_{I}{\bar{\lambda}}_{J{\dot{{\gamma}}}})\right)\,.\end{split}$
(91)

#### Minimal coordinates

In the previous section, we combined $V_{a}$ and $W_{a}$ into $J_{a}$,
$\bar{J}_{a}$ and then translated them into the spinor notation. The resulting
Dirac brackets were summarized in (47):

$\displaystyle\begin{split}\\{J^{I}{}_{J},J^{K}{}_{L}\\}_{*}&=-i(\delta^{I}{}_{L}J^{K}{}_{J}-\delta^{K}{}_{J}J^{I}{}_{L})\,,\\\
\\{(\Lambda_{\alpha{\dot{\alpha}}})^{I}{}_{J},J_{KL}\\}_{*}&=+i(\Lambda_{\alpha{\dot{\alpha}}})^{I}{}_{(K}\epsilon_{L)J}\,.\end{split}$

We are interested in the twistor expressions for $J$, $\bar{J}$.

Just as we used $\Lambda^{\mu}{}_{A}$ to switch between the space-time frame
and the body frame in the spherical top model, we may use
$(\lambda,\bar{\lambda})$ to switch between an SL$(2,\mathbb{C})$ index and
the internal SU$(2)$ index. We may call the SU$(2)$-indexed variables “body
frame components”.

Converting the space-time expression for $J_{\alpha\beta}$ in (76) to the body
frame, we find

$\bar{J}_{{\dot{\alpha}}{\dot{\beta}}}=i\hskip 1.00006pt\frac{1}{{\rm
det}({\bar{\lambda}})}{\bar{\lambda}}^{I}{}_{{\dot{\alpha}}}{\bar{\lambda}}^{J}{}_{{\dot{\beta}}}J_{(IJ)}\,,\quad
J_{\alpha\beta}=i\hskip 1.00006pt\frac{1}{{\rm
det}(\lambda)}\lambda_{\alpha}{}^{I}\lambda_{\beta}{}^{J}\bar{J}_{(IJ)}\,,$
(92)

where the body frame components are

$\displaystyle\begin{split}2iJ^{I}{}_{J}=-2{\mu}^{{\dot{\alpha}}I}{\bar{\lambda}}_{J{\dot{\alpha}}}=V^{I}{}_{J}+iW^{I}{}_{J}\,,\\\
-2i\bar{J}^{J}{}_{I}=-2{\bar{\mu}}_{I}{}^{\alpha}{}\lambda_{\alpha}{}^{J}=V^{I}{}_{J}-iW^{J}{}_{I}\,.\end{split}$
(93)

In fact, these expressions contain not only the $\mathfrak{su}(2)$ part but
also the $\mathfrak{u}(1)$ part:

$\displaystyle
V_{(IJ)}=-\left({\mu}^{{\dot{\alpha}}}{}_{(I}{\bar{\lambda}}_{J){\dot{\alpha}}}+\lambda_{\alpha}{}_{(I}{\bar{\mu}}_{J)}{}^{\alpha}\right)$
$\displaystyle\,,\quad
V_{0}=\frac{1}{2}\left([{\bar{\lambda}}{\mu}]+\langle{\bar{\mu}}\lambda\rangle\right)=D\,,$
(94) $\displaystyle
W_{(IJ)}=i\left({\mu}^{{\dot{\alpha}}}{}_{(I}{\bar{\lambda}}_{J){\dot{\alpha}}}-\lambda_{\alpha}{}_{(I}{\bar{\mu}}_{J)}{}^{\alpha}\right)$
$\displaystyle\,,\quad
W_{0}=\frac{1}{2i}\left([{\bar{\lambda}}{\mu}]-\langle{\bar{\mu}}\lambda\rangle\right)\,.$
(95)

To see the fate of the $\mathfrak{u}(1)$ part, we note that

$V^{A}=-|{\rm det}(\lambda)|(x\cdot\Lambda^{A})\,,\quad W^{A}=-|{\rm
det}(\lambda)|(y\cdot\Lambda^{A})\,,$ (96)

where $\Lambda$ should be understood as in (84) and $x$ and $y$ as in (87).
These are the twistor counterpart of (33). Since $V^{0}=0=W^{0}$ in the
spherical top model, we should require the same in the twistor model. The
requirement is fulfilled by the gauge-fixing conditions $\chi$ in (79) which
is equal to $-(V^{0}+iW^{0})/{\rm det}(\bar{\lambda})$, or the time-like body
frame component of $z^{\mu}=x^{\mu}+iy^{\mu}$.

In summary, we have shown that the vectors $V$ and $W$ in the two models
coincide perfectly on the physical subspace. We have also reconfirmed the
identification of the internal SU$(2)$ as the little group; see (78).
Separating the orbital and spin parts of (92), we can rewrite (89) and (90) as

$L_{\alpha\beta}=-\frac{1}{2\hskip 1.00006pt{\rm
det}(\lambda)}\lambda_{\alpha}{}^{I}\lambda_{\beta}{}^{J}V_{(IJ)}\,,\quad
S_{\alpha\beta}=i\frac{1}{2\hskip 1.00006pt{\rm
det}(\lambda)}\lambda_{\alpha}{}^{I}\lambda_{\beta}{}^{J}W_{(IJ)}\,.$ (97)

It implies the transverse spin squared appearing in $m(\tilde{S}^{2})$ can be
expressed as

$\displaystyle\tilde{S}^{2}:=-\frac{1}{2}W^{(IJ)}W_{(IJ)}=W^{a}W_{a}\,.$ (98)

###  3.2 Physical phase space

#### Dirac brackets

Our goal is to check whether the massive twistor phase space equipped with the
constraints (79) reproduce the Dirac bracket (50). The Dirac bracket is
constructed in the usual manner:

$\displaystyle\begin{split}\\{f,g\\}_{*}=\\{f,g\\}&-\big{(}\\{f,\phi\\}\\{\bar{\chi},g\\}-\\{f,\bar{\chi}\\}\\{\phi,g\\}\big{)}\\\
&-\big{(}\\{f,\bar{\phi}\\}\\{\chi,g\\}-\\{f,\chi\\}\\{\bar{\phi},g\\}\big{)}\,.\end{split}$
(99)

On the constraint surface, the non-vanishing terms are

$\displaystyle\\{\lambda_{\alpha}{}^{I},\lambda_{\beta}{}^{J}\\}_{*}$
$\displaystyle=-\frac{m^{\prime}}{2im}\left[\big{(}\lambda_{\alpha}{}^{K}{W}_{K}{}^{I}\big{)}\lambda_{\beta}{}^{J}-\lambda_{\alpha}{}^{I}\big{(}\lambda_{\beta}{}^{K}{W}_{K}{}^{J}\big{)}\right]\,,$
$\displaystyle\\{\lambda_{\alpha}{}^{I},{\bar{\lambda}}_{J{\dot{\beta}}}\\}_{*}$
$\displaystyle=-\frac{m^{\prime}}{2im}\left[\big{(}\lambda_{\alpha}{}^{K}{(W}_{K}{}^{I}\big{)}{\bar{\lambda}}_{J{\dot{\beta}}}+\lambda_{\alpha}{}^{I}\big{(}{W}_{J}{}^{K}{\bar{\lambda}}_{K{\dot{\beta}}}\big{)}\right]\,,$
$\displaystyle\\{\lambda_{\alpha}{}^{I},{\bar{\mu}}_{J}{}^{\beta}\\}_{*}$
$\displaystyle=\left(\delta_{\alpha}{}^{\beta}\delta^{I}{}_{J}+\frac{1}{2m}\lambda_{\alpha}{}^{I}\lambda^{\beta}{}_{J}\right)+\frac{m^{\prime}}{2im}\left[\big{(}\lambda_{\alpha}{}^{K}{W}_{K}{}^{I}\big{)}{\bar{\mu}}_{J}{}^{\beta}-\lambda_{\alpha}{}^{I}\big{(}{W}_{J}{}^{K}{\bar{\mu}}_{K}{}^{\beta}\big{)}\right]\,,$
$\displaystyle\\{\lambda_{\alpha}{}^{I},{\mu}^{{\dot{\beta}}J}\\}_{*}$
$\displaystyle=\frac{m^{\prime}}{2im}\left[\big{(}\lambda_{\alpha}{}^{K}{W}_{K}{}^{I}\big{)}{\mu}^{{\dot{\beta}}J}+\lambda_{\alpha}{}^{I}\big{(}{\mu}^{{\dot{\beta}}K}{W}_{K}{}^{J}\big{)}\right]\,,$
$\displaystyle\\{{\mu}^{{\dot{\alpha}}I},{\mu}^{{\dot{\beta}}J}\\}_{*}$
$\displaystyle=\frac{1}{2m}\left({\mu}^{{\dot{\alpha}}I}{\bar{\lambda}}^{J{\dot{\beta}}}-{\mu}^{{\dot{\beta}}J}{\bar{\lambda}}^{I{\dot{\alpha}}}\right)+\frac{m^{\prime}}{2im}\left[\big{(}\mu^{{\dot{\alpha}}K}{W}_{K}{}^{I}\big{)}{\mu}^{{\dot{\beta}}J}-{\mu}^{{\dot{\alpha}}I}\big{(}{\mu}^{{\dot{\beta}}K}{W}_{K}{}^{I}\big{)}\right]\,,$
$\displaystyle\\{{\mu}^{{\dot{\alpha}}I},{\bar{\mu}}_{J}{}^{\beta}\\}_{*}$
$\displaystyle=\frac{m^{\prime}}{2im}\left[\big{(}{\mu}^{{\dot{\alpha}}K}{W}_{K}{}^{I}\big{)}{\bar{\mu}}_{J}{}^{\beta}+{\mu}^{{\dot{\alpha}}I}\big{(}{W}_{J}{}^{K}{\bar{\mu}}_{K}{}^{\beta}\big{)}\right]\,,$
(100)

and their complex conjugates. The algebra may look unwieldy, but its actual
content is rather simple. If we focus on the normalized variables defined in
(83),

$\displaystyle\begin{split}\hat{\lambda}_{\alpha}{}^{I}:=({\rm
det}(\lambda))^{-1/2}\lambda_{\alpha}{}^{I}&\,,\quad\hat{{\mu}}^{{\dot{\alpha}}I}:=({\rm
det}(\lambda))^{1/2}{\mu}^{{\dot{\alpha}}I}\,,\\\
\hat{{\bar{\lambda}}}_{I{\dot{\alpha}}}:=({\rm
det}({\bar{\lambda}}))^{-1/2}{\bar{\lambda}}_{I{\dot{\alpha}}}&\,,\quad\hat{{\bar{\mu}}}_{I}{}^{\alpha}:=({\rm
det}({\bar{\lambda}}))^{1/2}{\bar{\mu}}_{I}{}^{\alpha}\,,\\\ \end{split}$

the “Regge slope” $m^{\prime}(\tilde{S}^{2})$ terms disappear completely from
the Dirac brackets:

$\displaystyle\begin{split}&\\{\hat{\lambda}_{\alpha}{}^{I},\hat{\lambda}_{\beta}{}^{J}\\}_{*}=0\,,\quad\\{\hat{\lambda}_{\alpha}{}^{I},\hat{{\bar{\lambda}}}_{J{\dot{\beta}}}\\}_{*}=0\,,\\\
&\\{\hat{\lambda}_{\alpha}{}^{I},\hat{{\bar{\mu}}}_{J}{}^{\beta}\\}_{*}=\delta_{\alpha}{}^{\beta}\delta^{I}{}_{J}+{\textstyle\frac{1}{2}}\,\hat{\lambda}_{\alpha}{}^{I}\hat{\lambda}^{\beta}{}_{J}\,,\quad\\{\hat{\lambda}_{\alpha}{}^{I},\hat{{\mu}}^{{\dot{\beta}}J}\\}_{*}=0\,,\\\
&\\{\hat{{\mu}}^{{\dot{\alpha}}I},\hat{{\mu}}^{{\dot{\beta}}J}\\}_{*}=\frac{1}{2}\left(\hat{{\mu}}^{{\dot{\alpha}}I}\hat{{\bar{\lambda}}}^{J{\dot{\beta}}}-\hat{{\mu}}^{{\dot{\beta}}J}\hat{{\bar{\lambda}}}^{I{\dot{\alpha}}}\right)\,,\quad\\{\hat{{\mu}}^{{\dot{\alpha}}I},\hat{{\bar{\mu}}}_{J}{}^{\beta}\\}_{*}=0\,.\end{split}$
(101)

This observation is consistent with the fact that the Dirac bracket algebra of
$(\Lambda_{\alpha{\dot{\alpha}}})^{I}{}_{J}$ and $J^{I}{}_{J}$ is isomorphic
to the Poison bracket algebra of $T^{*}(\mathrm{SO}(1,3))$ and is independent
of the Regge trajectory. Indeed, expressing
$(\Lambda_{\alpha{\dot{\alpha}}})^{I}{}_{J}$ and $J^{I}{}_{J}$ in terms of the
normalized twistor variables, we reproduce the fully constrained Dirac
brackets of the spherical top model in the minimal coordinates in (47).

In switching back to the original coordinates, the only non-trivial step is to
multiply each variable by appropriate factors of ${\rm
det}(\lambda)=m(W^{2})$. Using the explicit map (91), we can eventually
reproduce the Dirac bracket algebra of the spherical top model (50).

###  3.3 Equation of motion

#### First-order action

The symplectic potential on the twistor space can be chosen to be

$\displaystyle\begin{split}\theta&=\frac{i}{2}\left(\bar{Z}_{I}{}^{\mathrm{A}}dZ_{\mathrm{A}}{}^{I}-Z_{\mathrm{A}}{}^{I}d\bar{Z}_{I}{}^{\mathrm{A}}\right)\\\
&=p_{\mu}dx^{\mu}-iy^{{\dot{\alpha}}\alpha}\left(\lambda_{\alpha}{}^{I}d{\bar{\lambda}}_{I{\dot{\alpha}}}-d\lambda_{\alpha}{}^{I}{\bar{\lambda}}_{I{\dot{\alpha}}}\right)\,.\end{split}$
(102)

In the second line, $p_{\mu}$, $x^{\mu}$, and $y^{{\dot{\alpha}}\alpha}$
should be understood as the twistor expressions discussed earlier. The term
involving the $y^{\mu}$ coordinate encodes the spinning dynamics of the
spherical top. The first-order action is

$\mathcal{S}=\int
d\sigma\left(\frac{i}{2}\big{(}\bar{Z}_{I}{}^{\mathrm{A}}\dot{Z}_{\mathrm{A}}{}^{I}-Z_{\mathrm{A}}{}^{I}\dot{\bar{Z}}_{I}{}^{\mathrm{A}}\big{)}-\bar{\kappa}\phi-\kappa\bar{\phi}\right)\,,$
(103)

where $\phi$ and $\bar{\phi}$ are the mass-shell constraints in (79). Although
the symplectic potential respects conformal symmetry, the gauge generators
break the dilatation symmetry (this can be made explicit by introducing
infinity twistors). The variations of the action give

$\begin{split}\delta_{{\bar{\mu}}}\mathcal{S}=0&\,\,\implies\,\,\dot{\lambda}_{\alpha}{}^{I}=-i({\operatorname{Re}}\,\kappa)m^{\prime}\lambda_{\alpha}{}^{J}W_{J}{}^{I}\,,\\\
\delta_{{\bar{\lambda}}}\mathcal{S}=0&\,\,\implies\,\,\dot{{\mu}}^{{\dot{\alpha}}I}=-i({\operatorname{Re}}\,\kappa)m^{\prime}{\mu}^{{\dot{\alpha}}}{}^{J}W_{J}{}^{I}+\frac{\kappa}{2}\hskip
1.00006pt{\bar{\lambda}}{}^{{\dot{\alpha}}I}\,,\end{split}$ (104)

and their complex conjugates.

#### Lagrange multipliers

Requiring that the gauge-fixing condition $W^{0}=0$ is preserved by the
equation of motion, we find that the Lagrange multipliers are subject to the
condition,

$\displaystyle 0=2\dot{W}^{0}={\operatorname{Im}}\big{[}\kappa\hskip
1.00006pt{\rm det}({\bar{\lambda}})\big{]}\,.$ (105)

Then, $\phi=0$ and $\bar{\phi}=0$ imply that
$({\operatorname{Im}}\,\kappa)=0$. Rather than imposing another gauge-fixing
condition, we solve for $({\operatorname{Re}}\,\kappa)$ from the resulting
equations of motion as we did in section 2.4. From the complexified incidence
relation (86),

$\displaystyle\dot{z}^{{\dot{\alpha}}\alpha}=\frac{\kappa}{2m}p^{{\dot{\alpha}}\alpha}\,\,\implies\,\,\dot{x}^{\mu}=({\operatorname{Re}}\,\kappa)\frac{1}{m}p^{\mu}\,,\quad\dot{y}^{\mu}=0\,.$
(106)

Hence,

$\displaystyle({\operatorname{Re}}\,\kappa)=\frac{d\tau}{d\sigma}\,,\quad
d\tau=\sqrt{-dx_{\mu}\hskip 1.00006ptdx^{\mu}}\,.$ (107)

Additional calculations based on the map (91) give the final equations of
motion:

$\displaystyle
p^{\mu}=m\frac{dx^{\mu}}{d\tau}\,,\quad\frac{dp_{\mu}}{d\tau}=0\,,\quad\frac{d\Lambda_{\mu}{}^{a}}{d\tau}=-\varepsilon^{a}{}_{bc}(2m^{\prime}W^{b})\Lambda_{\mu}{}^{c}\,,\quad\frac{dW_{a}}{d\tau}=0\,.$
(108)

The particle propagates with constant velocity and spin angular momentum,
while the body frame rotates with a constant three-angular velocity
$\omega^{a}:=2m^{\prime}W^{a}$. The minus sign of $(-\varepsilon^{a}{}_{bc})$
accounts correctly for the fact that the angular velocity $\omega^{a}$ is
measured with respect to the body frame. It is easy to verify that (108)
conforms to the result (63) of section 2.4.

In terms of the constant of motion $\omega^{a}$, (104) can be integrated to
give

$\displaystyle\lambda_{\alpha}{}^{I}(\tau)=\lambda_{\alpha}{}^{J}(0)\hskip
1.00006ptU_{J}{}^{I}(\tau)\,,\quad\mu^{{\dot{\alpha}}I}(\tau)=\frac{\tau}{2}\hskip
1.00006pt{\bar{\lambda}}^{I{\dot{\alpha}}}(\tau)+\mu^{{\dot{\alpha}}J}(0)\hskip
1.00006ptU_{J}{}^{I}(\tau)\,,$ (109)

where $U(\tau)=\exp[\hskip 1.00006pt\omega^{a}\tau\,(\sigma_{a})/2i\hskip
1.00006pt]$. This translates to

$\displaystyle\begin{split}p_{\mu}(\tau)=p_{\mu}(0)&\,,\quad\Lambda_{\mu}^{a}(\tau)=(U^{-1}(\tau))^{a}{}_{b}\Lambda_{\mu}^{b}(0)\,,\\\
x^{\mu}(\tau)=x^{\mu}(0)+\Lambda^{\mu}{}_{0}(0)\hskip 1.00006pt\tau&\,,\quad
S_{\mu\nu}(\tau)=S_{\mu\nu}(0)\,,\end{split}$ (110)

where $(U^{-1}\sigma^{a}U):=U^{a}{}_{b}(\sigma^{b})$. The position and the
spin can be conveniently extracted from the $J^{I}{}_{J}$ matrix (93) or the
complexified position (87).

###  3.4 Quantization

We review the quantization of the massive twistor model iteFedoruk:2014vqa in
our notation. We quantize the unconstrained phase space first and then impose
the constraints. The quantization is straightforward in the
$(\lambda,\bar{\lambda})$ polarization, where the conjugate variables act as
derivatives:

$\displaystyle\bar{\mu}_{I}{}^{\alpha}=-i\frac{\partial}{\partial\lambda_{\alpha}{}^{I}}\,,\quad\mu^{{\dot{\alpha}}I}=-i\frac{\partial}{\partial\bar{\lambda}_{I{\dot{\alpha}}}}\,.$
(111)

Ignoring the SL$(2,\mathbb{C})$ indices for a moment and focusing on the
SU$(2)$ indices, for an integer or half-integer $s$, any totally symmetrized
polynomial $P^{(2s)}(\lambda,\bar{\lambda})$ of degree $2s$ is an eigenstate
of the transverse spin-squared:

$\displaystyle
W^{2}P^{(2s)}(\lambda,\bar{\lambda})=s(s+1)P^{(2s)}(\lambda,\bar{\lambda})\,,$
(112)

where $W^{2}$ should be understood as a differential operator through (95),
(98) and (111). The mass spectrum of the particle follows from the Regge
trajectory as

$\displaystyle(-p^{2})P^{(2s)}(\lambda,{\bar{\lambda}})=m^{2}(W^{2})P^{(2s)}(\lambda,{\bar{\lambda}})=m^{2}(s(s+1))P^{(2s)}(\lambda,{\bar{\lambda}})\,.$
(113)

In the quantum theory, once we impose the constraints $\phi=0=\bar{\phi}$, the
wave-function is forced to be independent of the conjugate variables
$(\bar{\chi},\chi)$, just as $p=0$ for a classical trajectory translates to
$\partial_{x}\psi(x)=0$ for the corresponding quantum wave-function.

#### On-shell fields

Restoring the SL$(2,\mathbb{C})$ indices, we may distinguish $(2s+1)$ types of
polynomials, each of which can be mapped to an on-shell space-time field
Fedoruk:2014vqa  as

$\displaystyle\begin{split}\Phi^{(2s,0)}_{\alpha_{1}\alpha_{2}\ldots\alpha_{2s}}(x)&=c_{I_{1}\ldots
I_{2s}}\int
d[\lambda,\bar{\lambda}]\,\lambda_{\alpha_{1}}^{I_{1}}\lambda_{\alpha_{2}}^{I_{2}}\cdots\lambda_{\alpha_{2s}}^{I_{2s}}\,\exp\left(ix^{{\dot{\beta}}\beta}\bar{\lambda}_{{\dot{\beta}}J}\lambda_{\beta}^{J}\right)\,,\\\
\Phi^{(2s-1,1)}_{{\dot{\alpha}}_{1}|\alpha_{2}\ldots\alpha_{2s}}(x)&=c_{I_{1}\ldots
I_{2s}}\int
d[\lambda,\bar{\lambda}]\,\bar{\lambda}_{{\dot{\alpha}}_{1}}^{I_{1}}\lambda_{\alpha_{2}}^{I_{2}}\cdots\lambda_{\alpha_{2s}}^{I_{2s}}\,\exp\left(ix^{{\dot{\beta}}\beta}\bar{\lambda}_{{\dot{\beta}}J}\lambda_{\beta}^{J}\right)\,,\\\
\vdots\qquad\,\,\,&\qquad\,\,\,\vdots\\\
\Phi^{(0,2s)}_{{\dot{\alpha}}_{1}{\dot{\alpha}}_{2}\ldots{\dot{\alpha}}_{2s}}(x)&=c_{I_{1}\ldots
I_{2s}}\int
d[\lambda,\bar{\lambda}]\,\bar{\lambda}_{{\dot{\alpha}}_{1}}^{I_{1}}\bar{\lambda}_{{\dot{\alpha}}_{2}}^{I_{2}}\cdots\bar{\lambda}_{{\dot{\alpha}}_{2s}}^{I_{2s}}\,\exp\left(ix^{{\dot{\beta}}\beta}\bar{\lambda}_{{\dot{\beta}}J}\lambda_{\beta}^{J}\right)\,,\end{split}$
(114)

where $c_{I_{1}\ldots I_{2s}}$ is a totally symmetrized SU$(2)$ tensor, and
the integration measure is

$\displaystyle d[\lambda,\bar{\lambda}]=d^{4}\lambda
d^{4}\bar{\lambda}\left[\delta({\rm det}(\lambda)-m)\delta({\rm
det}(\bar{\lambda})-m)\right]\,.$ (115)

The on-shell fields satisfy the coupled, first-order, Dirac-Fierz-Pauli-
Bargmann-Wigner equations, which implies that each field satisfies the Klein-
Gordon equation.

We close this subsection with a few comments. First, we note that the spectrum
of the quantum states shown in (114) takes precisely the same form as the one
based on massive spinor-helicity variables of Arkani-Hamed, Huang and Huang
Arkani-Hamed:2017jhn . Second, it is well known that a twistor description
reveals many interesting properties of scattering amplitudes of massless
particles Witten:2003nn . It would be interesting to see, to what extent, the
twistor point of view would become useful in understanding scattering
amplitudes of massive particles. Technically, the “half-Fourier-transform”
from the $(\lambda,\bar{\lambda})$ polarization to the twistor $(\lambda,\mu)$
polarization may face complications due to the mass-shell constraint.

##  4 Discussion

We have established the equivalence between the spherical top model and the
massive twistor model. Undoubtedly, this is only the beginning of a longer
story.

An immediate question is whether the equivalence persists in an interacting
theory. We could try to couple the spinning particle to an external
electromagnetic field and see if we recover the Bargmann-Michel-Telegdi (BMT)
equation Bargmann:1959gz , or try to couple it to an external gravitational
field and see if we recover the Mathisson–Papapetrou–Dixon (MPD) equation
Mathisson:1937zz ; Papapetrou:1951pa ; Dixon:1970zza . Coupling to an external
field may not be a straightforward exercise as it requires generalizing the
spin-gauge constraints in a way consistent with the U(1) gauge invariance of
electromagnetism or the general covariance of gravity.

For the electromagnetic coupling, the internal U(1) of the twistor model can
be associated with the electric charge Newman:1974fr . An intriguing
application of this idea is Bette’s derivation of a “square-root” of the BMT
equation Bette:1989zt . It would be interesting to identify the spherical top
counterpart of the equation and confirm the classical value of the $g$-factor:
$g=2$.

While the twistor model conforms to the covariant spin-gauge by construction,
the spherical top model allows for other gauge choices. We may keep the gauge
generators $\phi_{a}$ but use gauge-fixing conditions different from
$\chi^{a}$ used in this paper. For instance, when describing a binary system
of two spinning particles weakly interacting through gravity, we may choose a
gauge in which the center-of-momentum frame plays a prominent role.

A version of the spherical top model constitutes the free part of an effective
field theory (EFT) approach Goldberger:2004jt  to the post-Newtonian (PN)
gravity of spinning objects Levi:2015msa ; Levi:2017kzq . In intermediate
steps of the EFT computation, some higher time-derivative terms appear in the
Lagrangian, but they can be eliminated systematically in the final Hamiltonian
of the interacting binary system. It would be interesting to experiment with
different gauge choices and see if one can reduce the appearance of higher
derivative terms.

Recently, some spin-dependent terms of the EFT Hamiltonian have been compared
with a post-Minkowskian (PM) Hamiltonian derived from a method based on
scattering amplitudes Chung:2018kqs ; Chung:2020rrz  in the overlapping regime
of validity. The comparison of the final results required non-trivial
canonical transformation between phase space coordinates in the two
computational frameworks. Since we showed that the spherical top model is
equivalent to the massive twistor model, and that the latter upon quantization
reproduces the on-shell states in terms of the massive spinor-helicity
variables, a closer look at the gauge choices of the spherical top model may
reveal a more direct connection between the EFT-based PN Hamiltonian and the
amplitude-based PM Hamiltonian. This possibility was one of the main
motivations for the authors to examine the spherical top model in the first
place.

Ever since the work of Witten Witten:2003nn , theories based on (or inspired
by) massless twistors have made tremendous contributions to scattering
amplitudes of massless particles. It is tempting to conjecture that a similar
degree of success with massive twistors is on the horizon.

###### Acknowledgements.

We are grateful to Yu-tin Huang, Michèle Levi, Rodolfo Russo, Congkao Wen,
Chris White for discussions and comments on the manuscript. The work of JHK
and SL was supported in part by the National Research Foundation of Korea
grant NRF-2019R1A2C2084608. JWK was supported by the Science and Technology
Facilities Council (STFC) Consolidated Grant ST/T000686/1 _“Amplitudes,
Strings and Duality”_.

##  Appendix A Hamiltonian mechanics under constraints

We give a minimal review of the Hamiltonian mechanics of constrained systems.

#### Basic conventions

Here are our conventions for symplectic form, Poisson bracket, etc. For a flat
phase space $\mathbb{R}^{2n}$ with coordinates $(q^{i},p_{i})$, we orient the
symplectic form $\omega$ as

$\displaystyle\omega=dp_{i}\wedge dq^{i}\,.$ (116)

When a Hamiltonian $H$ generates a time flow $V=d/dt$, Hamilton’s equation
reads

$\displaystyle dH=-i_{V}\omega\,.$ (117)

Short computations,

$\displaystyle dH=\left(\frac{\partial H}{\partial
q^{i}}\right)dq^{i}+\left(\frac{\partial H}{\partial
p_{i}}\right)dp_{i}\,,\quad V=\frac{d}{dt}=\dot{q}^{i}\frac{\partial}{\partial
q^{i}}+\dot{p}_{i}\frac{\partial}{\partial p_{i}}\,,$ (118)

verify that the geometric form of the equation (117) agrees with the
conventional form,

$\displaystyle\dot{p_{i}}=-\left(\frac{\partial H}{\partial
q^{i}}\right)\,,\quad\dot{q}^{i}=\left(\frac{\partial H}{\partial
p_{i}}\right)\,.$ (119)

Geometrically, the Poisson bracket is defined as

$\displaystyle\\{f,g\\}=-\omega(X_{f},X_{g})\,,\;\;\mbox{where}\;\;df=-i_{X_{f}}\omega\,,\;\;dg=-i_{X_{g}}\omega\quad\Longrightarrow\quad\frac{df}{dt}=\\{f,H\\}\,.$
(120)

For $\mathbb{R}^{2n}(q^{i},p_{i})$, since

$\displaystyle df=\left(\frac{\partial f}{\partial
q^{i}}\right)dq^{i}+\left(\frac{\partial f}{\partial
p_{i}}\right)dp_{i}\,,\quad X_{f}=-\left(\frac{\partial f}{\partial
q^{i}}\right)\frac{\partial}{\partial p_{i}}+\left(\frac{\partial f}{\partial
p_{i}}\right)\frac{\partial}{\partial q^{i}}\,,$ (121)

and similarly for $g$, the bracket computes

$\displaystyle\\{f,g\\}=\frac{\partial f}{\partial q^{i}}\frac{\partial
g}{\partial p_{i}}-\frac{\partial g}{\partial q^{i}}\frac{\partial f}{\partial
p_{i}}\,.$ (122)

Equivalently, we can begin with specifying the elementary Poisson bracket,

$\displaystyle\\{q^{i},q^{j}\\}=0=\\{p_{i},p_{j}\\}\,,\quad\\{q^{i},p_{j}\\}=\delta^{i}{}_{j}\,,$
(123)

and require antisymmetry, Leibniz rule and Jacobi identity to recover (122).

Our conventions can be used for general symplectic manifolds with little
modifications:

$\displaystyle\omega=\frac{1}{2}\omega_{mn}dy^{m}\wedge
dy^{n}\,,\quad\\{f,g\\}=\omega^{mn}\frac{\partial f}{\partial
y^{m}}\frac{\partial g}{\partial
y^{n}}\,,\quad\omega^{mk}\omega_{kn}=\delta^{m}{}_{n}\,.$ (124)

When momentum and position can be distinguished, we orient the symplectic form
such that the momentum precedes the position. It is often convenient to
introduce the symplectic potential satisfying $\omega=d\theta$. While $\omega$
is required to be globally well-defined, $\theta$ may be only locally well-
defined.

#### Constrained system and Dirac bracket

We give a brief review of the Hamiltonian mechanics of a constrained system,
focusing on the case when the constraints satisfy a few simplifying
properties. We start with $n$ position-momentum pairs and let the constraints
remove $k$ pairs. The constraints are divided into ‘gauge generators’
$\phi_{A}$ and ‘gauge-fixing conditions’ $\chi^{A}$ $(A=1,\ldots,k)$. We
require that their Poisson brackets take the form

$\displaystyle\\{\phi_{A},\phi_{B}\\}\approx
0\approx\\{\chi^{A},\chi^{B}\\}\,,\quad
C^{A}{}_{B}:=\\{\chi^{A},\phi_{B}\\}\,,\quad{\rm det}(C)\neq 0\,.$ (125)

The weak equality ‘$\approx$’ means that the equality holds on the constraint
surface. The gauge generators among themselves are said to form first-class
constraints. The gauge generators and the gauge-fixing conditions together are
said to form second-class constraints. Finally, to qualify as ‘symmetry’
generators, $\phi_{A}$ should commute with the Hamiltonian $H$.

These constraints provide an ideal setup to introduce the Dirac bracket. We
motivate the Dirac bracket starting from the Hamiltonian action principle.
Consider the action

$\displaystyle\mathcal{S}=\int
dt\left(p_{i}\dot{q}^{i}-H-\kappa^{A}\phi_{A}\right)\,.$ (126)

The linear combination of constraints, $\kappa^{A}\phi_{A}$, serves two
purposes. First, it imposes the constraints $\phi_{A}=0$ through the Lagrange
multipliers $\kappa_{A}$:

$\displaystyle\delta_{\kappa}\mathcal{S}=0\quad\Longrightarrow\quad\phi_{A}=0\,.$
(127)

Second, it generates a flow along the gauge orbits to accompany the
Hamiltonian time flow:

$\displaystyle\delta_{p,q}\mathcal{S}=0\quad\Longrightarrow\quad\frac{df}{dt}=\\{f,H\\}+\kappa^{A}\\{f,\phi_{A}\\}\,.$
(128)

The two types of equations are mutually consistent since

$\displaystyle
0=\frac{d\phi_{A}}{dt}=\\{\phi_{A},H\\}+\kappa^{B}\\{\phi_{A},\phi_{B}\\}\approx
0\,.$ (129)

Now, if we impose the conditions $\chi^{A}=0$, the multipliers $\kappa^{A}$
should be tuned such that the trajectory does not deviate from the gauge
slice. Consistency of the procedure requires

$\displaystyle\begin{split}0&=\frac{d\chi^{A}}{dt}=\\{\chi^{A},H\\}+\kappa^{B}\\{\chi^{A},\phi_{B}\\}=\\{\chi^{A},H\\}+\kappa^{B}C^{A}{}_{B}\approx
0\\\
&\Longrightarrow\quad\kappa^{A}=-(C^{-1})^{A}{}_{B}\\{\chi^{B},H\\}\,.\end{split}$
(130)

Substituting it into (128), we find

$\displaystyle\begin{split}\frac{df}{dt}&=\\{f,H\\}-\\{f,\phi_{A}\\}(C^{-1})^{A}{}_{B}\\{\chi^{B},H\\}\\\
&=\\{f,H\\}-(C^{-1})^{A}{}_{B}\left(\\{f,\phi_{A}\\}\\{\chi^{B},H\\}-\\{f,\chi^{B}\\}\\{\phi_{A},H\\}\right)\approx\\{f,H\\}_{*}\,,\end{split}$
(131)

where we inserted a vanishing term ($\\{\phi_{A},H\\}\approx 0$) to make
contact with the Dirac bracket.

#### Hamiltonian mechanics on a group manifold

In a Hamiltonian system, a continuous symmetry comes with a generator $Q$ and
the associated vector flow $X_{Q}$ satisfying

$\displaystyle\\{H,Q\\}=0\,,\quad dQ=-i_{X_{Q}}\omega\,,\quad
X_{Q}[F]=\\{F,Q\\}\,.$ (132)

When there are two or more continuous symmetries, a Lie algebra structure
arises:

$\displaystyle\\{Q_{a},Q_{b}\\}=+f^{c}{}_{ab}Q_{c}\,,\quad[X_{a},X_{b}]=-f^{c}{}_{ab}X_{c}\,.$
(133)

The “structure function” $f^{c}{}_{ab}$ should satisfy the Jacobi identity at
every point.

Now we specialize to the Hamiltonian mechanics of a particle on a group
manifold. Let $G$ be a Lie group and $\mathfrak{g}\cong T_{e}(G)$ be its Lie
algebra. The phase space is the cotangent bundle $T^{*}(G)$ whose fiber is
$\mathfrak{g}^{*}$. The continuous isometry group of $G$ is $(G\times G)/Z$,
where the two $G$ factors denote the left and right actions, and $Z$ denotes
the center. We choose to use the right action to push-forward the Lie algebra
of $\mathfrak{g}$ to the group manifold $G$, and then up-lift it to the phase
space $T^{*}(G)$. The structure functions (133) become constants.

We summarize a few crucial results on the extension of $\mathfrak{g}$ to
$T^{*}(G)$. Let $\Theta=dg\hskip 1.00006ptg^{-1}$ be the right-invariant
Maurer-Cartan one-form. In a suitably chosen basis, the Maurer-Cartan equation
reads

$\displaystyle
d\Theta^{c}=\frac{1}{2}f^{c}{}_{ab}\Theta^{a}\wedge\Theta^{b}\,.$ (134)

The right-invariant symplectic potential and symplectic form are

$\displaystyle\theta=p_{c}\Theta^{c}\,,\quad\omega=d\theta=dp_{c}\wedge\Theta^{c}+p_{c}\left(\textstyle{\frac{1}{2}}f^{c}{}_{ab}\Theta^{a}\wedge\Theta^{b}\right)\,.$
(135)

Let $\Xi_{a}$ be the vector fields dual to the one-forms $\Theta^{a}$:
$\Theta^{a}(\Xi_{b})=\delta^{a}_{b}$. Its uplift to $T^{*}(G)$ is

$\displaystyle
X_{a}[F]=\\{F,p_{a}\\}=\left(\Xi_{a}-p_{c}f^{c}{}_{ab}\frac{\partial}{\partial
p_{b}}\right)F\,,$ (136)

where $F$ is any function on $T^{*}(G)$. The Lie algebra takes the same form
in the two incarnations,

$\displaystyle[\Xi_{a},\Xi_{b}]=-f^{c}{}_{ab}\Xi_{c}\quad\mapsto\quad[X_{a},X_{b}]=-f^{c}{}_{ab}X_{c}\,.$
(137)

In terms of the momenta $p_{a}$, the Lie algebra reads

$\displaystyle\\{g,p_{a}\\}=\Xi_{a}g\,,\quad\\{p_{a},p_{b}\\}=+f^{c}{}_{ab}p_{c}\,.$
(138)

In the first equation, a group element $g$ is treated as a matrix-valued
function on $G$.

Identifying the relativistic spherical top as the right-invariant Hamiltonian
system on the cotangent bundle of the Poincaré group uniquely determines the
symplectic structure in (2) and (3):
$\theta=\frac{1}{2}J_{\mu\nu}\Theta^{\mu\nu}+p_{\mu}\Theta^{\mu}$. The right-
invariant one-forms $\Theta^{a}$ are

$\displaystyle\begin{split}\Theta^{\mu\nu}&=-d\Lambda^{\mu}{}_{A}\hskip
1.00006pt\Lambda^{\nu A}=\Lambda^{\mu}{}_{A}\hskip 1.00006ptd\Lambda^{\nu
A}\,,\\\
\Theta^{\mu}&=dx^{\mu}+\Theta^{\mu}{}_{\nu}x^{\nu}=\Lambda^{\mu}{}_{A}\hskip
1.00006ptd(\Lambda_{\rho}{}^{A}x^{\rho})\,,\end{split}$ (139)

where an element of the Poincaré group is described by the pair
$(\Lambda^{\mu}{}_{A},x^{\mu})$. As the mixed index types indicate, the “top
parametrization” $\Lambda^{\mu}{}_{A}$ distinguishes the left and right
actions. Note that, if the Poincaré group is regarded as the semidirect
product $\mathrm{SO}(1,3)\ltimes\mathbb{R}^{1,3}$, the symplectic potential
can also be written as
$\theta=\frac{1}{2}S_{\mu\nu}\Theta^{\mu\nu}+p_{\mu}dx^{\mu}$. The two
expressions are equal, provided that
$J_{\mu\nu}-S_{\mu\nu}=2x_{[\mu}p_{\nu]}$.

##  Appendix B Spinors and twistors

####  $\mathrm{SU}(2)$ spinors

The $\epsilon^{IJ}$ of $\mathrm{SU}(2)$ denotes the antisymmetric rank-2
tensor normalized such that $\epsilon^{12}{\;\;\dot{=}\;\;}+1$. Its inverse is
the lower-index epsilon tensor:

$\epsilon^{IK}\epsilon_{KJ}=\delta^{I}{}_{J}=\delta_{J}{}^{I}\,.$ (140)

The spinor indices ($I,J,\cdots$) carry the fundamental representation of
$\mathrm{SU}(2)$ and are raised and lowered by the epsilon tensors with the
convention

$\psi^{I}=\epsilon^{IJ}\psi_{J}\,,\quad\psi_{I}=\epsilon_{IJ}\psi^{J}\,.$
(141)

The representation is pseudo-real; complex conjugation exchanges upper and
lower indices as $[\psi_{I}]^{*}=\bar{\psi}^{I}$. Note also that

$[\epsilon^{IJ}]^{*}=-\epsilon_{IJ}.$ (142)

The three Hermitian generators of $\mathrm{SU}(2)$ are half of the Pauli
matrices $(\sigma_{a})_{I}{}^{J}$. The invariance of the epsilon tensors under
$\mathrm{SU}(2)$ transformations implies that

$(\sigma_{a})_{IJ}=\epsilon_{JK}(\sigma_{a})_{I}{}^{K}\,,\quad(\sigma_{a})^{IJ}=\epsilon^{IK}(\sigma_{a})_{K}{}^{J}$
(143)

are symmetric in their two spinor indices.

An arbitrary spin-$s$ irreducible representation of $\mathrm{SU}(2)$ is
described by a totally symmetric $\mathrm{SU}(2)$ tensor with $2s$ indices:

$\psi_{I_{1}I_{2}\cdots I_{2s}}=\psi_{(I_{1}I_{2}\cdots I_{2s})}\,.$ (144)

Specifically, for $s=1$, $(\sigma^{a})_{IJ}$ and $(\sigma_{a})^{IJ}$ serve as
the Clebsch-Gordan coefficients. Consider

$(\sigma_{a})^{IJ}(\sigma^{a})_{KL}=-2\delta^{I}_{(K}\delta^{J}_{L)}\,,\quad(\sigma^{a})_{IJ}(\sigma_{b})^{IJ}=-2\delta^{a}_{b}\,.$
(145)

These identities ensure completeness and orthogonality for the symmetric
(triplet) subspace of rank-two $\mathrm{SU}(2)$ tensors. For the antisymmetric
(singlet) subspace,

$(\sigma_{0})^{IJ}(\sigma^{0})_{KL}=-2\delta^{I}_{[K}\delta^{J}_{L]}\,,\quad(\sigma^{0})_{IJ}(\sigma_{0})^{IJ}=-2\,,$
(146)

where the Clebsch-Gordan coefficients are given by
$(\sigma^{0})_{IJ}=-\epsilon_{IJ}$ and $(\sigma_{0})^{IJ}=-\epsilon^{IJ}$.
Then, the matrices $(\sigma_{0})_{I}{}^{J}$ and $(\sigma_{a})_{I}{}^{J}$
together serve as the Hermitian generators of $\mathrm{U}(2)$:

$\displaystyle\sigma^{A}=(I,\sigma^{a})\,,\quad\sigma_{A}=(-I,\sigma_{a})=\eta_{AB}\sigma^{B}\,.$
(147)

Finally, the singlet-triplet decomposition of a rank-two $\mathrm{SU}(2)$
tensor is defined as

$X_{IJ}=(\sigma^{A})_{IJ}X_{A}\,,\quad
X_{0}=-\frac{1}{2}(\sigma_{0})^{IJ}X_{IJ}=\frac{1}{2}X_{I}{}^{I}\,,\quad
X_{a}=-\frac{1}{2}(\sigma_{a})^{IJ}X_{IJ}\,.$ (148)

For complex $X_{A}$, the behavior under complex conjugation is

$[(\sigma_{A})_{I}{}^{J}]^{*}=(\sigma_{A})_{J}{}^{I}\,\,\implies\,\,[(\sigma^{A})_{IJ}]^{*}=-(\sigma^{A})^{JI}\,,\quad[X_{IJ}]^{*}=-\bar{X}^{A}(\sigma_{A})^{JI}:=-\bar{X}^{JI}\,.$
(149)

####  $\mathrm{SL}(2,\mathbb{C})$ spinors

Our convention is such that the epsilon tensors of different types coincide
numerically:

$\displaystyle\epsilon^{\alpha\beta}{\;\;\dot{=}\;\;}\bar{\epsilon}^{{\dot{\alpha}}{\dot{\beta}}}{\;\;\dot{=}\;\;}\epsilon^{IJ}\,,\quad\epsilon_{\alpha\beta}{\;\;\dot{=}\;\;}\bar{\epsilon}_{{\dot{\alpha}}{\dot{\beta}}}{\;\;\dot{=}\;\;}\epsilon_{IJ}\,.$
(150)

Then, the rules for raising and lowering $\mathrm{SL}(2,\mathbb{C})$ spinor
indices ($\alpha,\beta,\cdots$ and ${\dot{\alpha}},{\dot{\beta}},\cdots$) are
the same as that of $\mathrm{SU}(2)$ spinor indices.

We work in the mostly-plus metric signature
$\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1)$ and use the widely-adopted
convention for gamma matrices:

$\displaystyle\gamma^{\mu}=\begin{pmatrix}0&(\sigma^{\mu})_{\alpha\dot{\beta}}\\\
(\bar{\sigma}^{\mu})^{\dot{\alpha}\beta}&0\end{pmatrix}\,,\quad\sigma^{\mu}=(I,\vec{\sigma})\,,\quad\bar{\sigma}^{\mu}=(I,-\vec{\sigma})\,.$
(151)

The Clifford algebra takes the form
$\\{\gamma^{\mu},\gamma^{\nu}\\}=-2\eta^{\mu\nu}$. Note also that

$\displaystyle(\bar{\sigma}^{\mu})^{{\dot{\alpha}}\alpha}=\epsilon^{\alpha\beta}\bar{\epsilon}^{{\dot{\alpha}}{\dot{\beta}}}\sigma^{\mu}_{\beta{\dot{\beta}}}\,,\quad\eta_{\mu\nu}\sigma^{\mu}_{\alpha{\dot{\alpha}}}\sigma^{\nu}_{\beta{\dot{\beta}}}=-2\hskip
1.00006pt\epsilon_{\alpha\beta}\bar{\epsilon}_{{\dot{\alpha}}{\dot{\beta}}}\,.$
(152)

Vectors are converted to mixed bi-spinors and vice versa by the following
rule:

$V_{\alpha\dot{\alpha}}=(\sigma^{\mu})_{\alpha\dot{\alpha}}V_{\mu}\,,\quad
V_{\mu}=-\frac{1}{2}(\bar{\sigma}_{\mu})^{\dot{\alpha}\alpha}V_{\alpha\dot{\alpha}}\,.$
(153)

For anti-symmetric rank 2 tensors, we choose a particular convention,

$\displaystyle\frac{1}{2}\gamma^{[\mu}\gamma^{\nu]}=\begin{pmatrix}(-\Sigma^{\mu\nu})_{\alpha}{}^{\beta}&0\\\
0&(\bar{\Sigma}^{\mu\nu})^{\dot{\alpha}}{}_{\dot{\beta}}&\end{pmatrix}\,,\quad(\Sigma^{\mu\nu})=-\frac{1}{2}\sigma^{[\mu}\bar{\sigma}^{\nu]}\,,\quad(\bar{\Sigma}^{\mu\nu})=\frac{1}{2}\bar{\sigma}^{[\mu}\sigma^{\nu]}\,,$
(154)

while $\frac{i}{2}\gamma_{[\mu}\gamma_{\nu]}$ are the Hermitian generators of
Lorentz symmetry. Contracted with the epsilon tensors, the $(2\times 2)$
blocks of $\frac{1}{2}\gamma^{[\mu}\gamma^{\nu]}$ give symmetric matrices in
spinor indices:

$\displaystyle(\Sigma^{\mu\nu})_{\alpha\beta}:=\epsilon_{\beta\gamma}(\Sigma^{\mu\nu})_{\alpha}{}^{\gamma}\,,\,\,(\bar{\Sigma}^{\mu\nu})_{{\dot{\alpha}}{\dot{\beta}}}:=\bar{\epsilon}_{{\dot{\alpha}}{\dot{\gamma}}}(\bar{\Sigma}^{\mu\nu})^{{\dot{\gamma}}}{}_{\dot{\beta}}\;\;\implies\;\;[(\Sigma^{\mu\nu})_{\alpha\beta}]^{*}=(\bar{\Sigma}^{\mu\nu})_{{\dot{\alpha}}{\dot{\beta}}}\,.$
(155)

An anti-symmetric tensor is converted to unmixed bi-spinors as

$\displaystyle
F_{\alpha\beta}\,\bar{\epsilon}_{{\dot{\alpha}}{\dot{\beta}}}+\epsilon_{\alpha\beta}\,\bar{F}_{{\dot{\alpha}}{\dot{\beta}}}:=\frac{1}{2}F_{\mu\nu}\left[(\Sigma^{\mu\nu})_{\alpha\beta}\,\bar{\epsilon}_{{\dot{\alpha}}{\dot{\beta}}}+\epsilon_{\alpha\beta}\,(\bar{\Sigma}^{\mu\nu})_{{\dot{\alpha}}{\dot{\beta}}}\right]=\frac{1}{2}F_{\mu\nu}\sigma^{[\mu}_{\alpha{\dot{\alpha}}}\sigma^{\nu]}_{\beta{\dot{\beta}}}\,.$
(156)

The spinor notation splits the self-dual and anti-self-dual parts of an anti-
symmetric tensor:

$\displaystyle\frac{1}{2}\varepsilon^{\mu\nu}{}_{\rho\sigma}\Sigma^{\rho\sigma}=-i\Sigma^{\mu\nu}\,,\quad\frac{1}{2}\varepsilon^{\mu\nu}{}_{\rho\sigma}\bar{\Sigma}^{\rho\sigma}=+i\bar{\Sigma}^{\mu\nu}\,,\quad\varepsilon_{0123}=+1\,.$
(157)

#### Massless twistors

The massless twistor space consists of four complex variables. It transforms
in the fundamental representation of an $\mathrm{SU}(2,2)$ group, to be
identified with the $\mathrm{SO}(2,4)$ conformal group of the Minkowski space.
Luckily, the twistor space admits a symplectic structure, opening the
possibility of interpreting it as the phase space of a massless spinning
particle. We follow a physicist’s convention for twistor variables similar
(but not identical) to the one of Witten:2003nn . We denote the twistor
coordinates by $Z_{\mathrm{A}}$, their complex conjugates by
$\bar{Z}_{\bar{\mathrm{A}}}$, and the $\mathrm{SU}(2,2)$ action by
$Z_{\mathrm{A}}\mapsto(g)_{\mathrm{A}}{}^{\mathrm{B}}Z_{\mathrm{B}}$. The
indefinite signature of $\mathrm{SU}(2,2)$ is encoded in a “metric”
$A^{\bar{\mathrm{A}}\mathrm{B}}$:

$\displaystyle(g^{\dagger})^{\bar{\mathrm{A}}}{}_{\bar{\mathrm{C}}}A^{\bar{\mathrm{C}}\mathrm{D}}(g)_{\mathrm{D}}{}^{\mathrm{B}}=A^{\bar{\mathrm{A}}\mathrm{B}}\,.$
(158)

The metric defines the invariant “norm” of the twistor:

$\displaystyle\bar{Z}^{\mathrm{B}}Z_{\mathrm{B}}:=A^{\bar{\mathrm{A}}\mathrm{B}}\bar{Z}_{\bar{\mathrm{A}}}Z_{\mathrm{B}}\,.$
(159)

In relating the twistor space to the Minkowski space, it is customary to
decompose $Z_{\mathrm{A}}$ into two $\mathrm{SL}(2,\mathbb{C})$ spinors such
that

$Z_{\mathrm{A}}=\begin{pmatrix}\lambda_{\alpha}\\\
i\mu^{\dot{\alpha}}\end{pmatrix}\,,\quad
A^{\bar{\mathrm{A}}\mathrm{B}}=\begin{pmatrix}0&\delta^{{\dot{\alpha}}}{}_{{\dot{\beta}}}\\\
\delta_{\alpha}{}^{\beta}&0\end{pmatrix}\,,\quad\bar{Z}^{B}=\begin{pmatrix}-i\bar{\mu}^{\beta}&\bar{\lambda}_{\dot{\beta}}\end{pmatrix}\,.$
(160)

The factors of $(\pm i)$ in front of $\mu$ and $\bar{\mu}$ have been inserted
for later convenience.

The $\mathrm{SU}(2,2)$-invariant symplectic structure defined on the twistor
space is

$\displaystyle\omega$ $\displaystyle=i(d\bar{Z}^{\mathrm{A}}\wedge
dZ_{\mathrm{A}})=d\bar{\mu}^{\alpha}\wedge
d\lambda_{\alpha}+d\mu^{\dot{\alpha}}\wedge d\bar{\lambda}_{\dot{\alpha}}\,.$
(161)

The non-vanishing Poisson brackets are

$\displaystyle\\{Z_{\mathrm{A}},\bar{Z}^{\mathrm{B}}\\}=-i\hskip
1.00006pt\delta_{\mathrm{A}}{}^{\mathrm{B}}\quad\implies\quad\\{\lambda_{\alpha},\bar{\mu}^{\beta}\\}=\delta_{\alpha}{}^{\beta}\,,\quad\\{\bar{\lambda}_{\dot{\alpha}},\mu^{\dot{\beta}}\\}=\delta_{\dot{\alpha}}{}^{\dot{\beta}}\,.$
(162)

The Lie algebra $\mathfrak{su}(2,2)$ manifests itself as a set of Hamiltonian
flows satisfying $dG=-\mathrm{i}_{V}\omega$, where $G$ is a function and $V$
is a vector field. A convenient choice of basis for $G$ is

$\displaystyle
G_{\mathrm{A}}{}^{\mathrm{B}}:=Z_{\mathrm{A}}\bar{Z}^{\mathrm{B}}-\frac{1}{4}\delta_{\mathrm{A}}{}^{\mathrm{B}}(\bar{Z}^{\mathrm{C}}Z_{\mathrm{C}})\,.$
(163)

Their Poisson brackets are

$\displaystyle\\{G_{\mathrm{A}}{}^{\mathrm{B}},G_{\mathrm{C}}{}^{\mathrm{D}}\\}=-i(\delta_{\mathrm{A}}{}^{\mathrm{D}}G_{\mathrm{C}}{}^{\mathrm{B}}-\delta_{\mathrm{C}}{}^{\mathrm{B}}G_{\mathrm{A}}{}^{\mathrm{D}})\,.$
(164)

To connect the twistor space to the Minkowski space further, we rewrite the
conformal generators of the latter in the spinor notation using (153) with one
and only exception: for position variables,

$x^{{\dot{\alpha}}\alpha}:=({\bar{\sigma}}_{\mu})^{{\dot{\alpha}}\alpha}\left(\frac{1}{2}\,x^{\mu}\right)\,.$
(165)

This rescaling of $x^{{\dot{\alpha}}\alpha}$ reduces unwarranted factors of 2
in many equations, starting from the fundamental Poisson bracket in the
Minkowski space:

$\displaystyle\\{x^{\mu},p_{\nu}\\}=\delta^{\mu}{}_{\nu}\quad\implies\quad\\{x^{{\dot{\alpha}}\alpha},p_{\beta{\dot{\beta}}}\\}=-\delta^{\alpha}_{\beta}\delta^{\dot{\alpha}}_{\dot{\beta}}\,.$
(166)

The conformal algebra boils down to

$\displaystyle\begin{split}\\{J_{\alpha\beta},J_{\gamma\delta}\\}&=-\frac{1}{2}\left(\epsilon_{\alpha\gamma}J_{\beta\delta}+\epsilon_{\beta\gamma}J_{\alpha\delta}+\epsilon_{\alpha\delta}J_{\beta\gamma}+\epsilon_{\beta\delta}J_{\alpha\gamma}\right)\,,\\\
\\{J_{\alpha\beta},P_{\gamma{\dot{\gamma}}}\\}&=-\frac{1}{2}\left(\epsilon_{\alpha\gamma}P_{\beta{\dot{\gamma}}}+\epsilon_{\beta\gamma}P_{\alpha{\dot{\gamma}}}\right)\,,\quad\,\,\\{D,P_{\alpha{\dot{\alpha}}}\\}=-P_{\alpha{\dot{\alpha}}}\,,\\\
\\{J_{\alpha\beta},K_{\gamma{\dot{\gamma}}}\\}&=-\frac{1}{2}\left(\epsilon_{\alpha\gamma}K_{\beta{\dot{\gamma}}}+\epsilon_{\beta\gamma}K_{\alpha{\dot{\gamma}}}\right)\,,\quad\\{D,K_{\alpha{\dot{\alpha}}}\\}=+K_{\alpha{\dot{\alpha}}}\,,\\\
\\{P_{\alpha{\dot{\alpha}}},K_{\beta{\dot{\beta}}}\\}&=-\left(\epsilon_{\alpha\beta}\bar{J}_{{\dot{\alpha}}{\dot{\beta}}}+\bar{\epsilon}_{{\dot{\alpha}}{\dot{\beta}}}J_{\alpha\beta}-\epsilon_{\alpha\beta}\bar{\epsilon}_{{\dot{\alpha}}{\dot{\beta}}}D\right)\,.\end{split}$
(167)

The Poincaré generators $P_{\mu}=p_{\mu}$ and
$J_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}$ constitute the conformal algebra
together with the dilatation generator $D=-x^{\mu}p_{\mu}$ and the special
conformal generator
$K^{{\dot{\alpha}}\alpha}=x^{{\dot{\alpha}}\beta}p_{\beta{\dot{\beta}}}x^{{\dot{\beta}}\alpha}$.
The normalization of
$K^{{\dot{\alpha}}\alpha}=({\bar{\sigma}}_{\mu})^{{\dot{\alpha}}\alpha}K^{\mu}$
is such that $K_{\mu}=IP_{\mu}I$ holds with the inversion map
$I:x^{\mu}/2\mapsto 2\,{x^{\mu}/x^{2}}$.

Now, we identify the $\mathfrak{su}(2,2)$ generators in (163) with the
conformal generators as

$\displaystyle\begin{split}G_{A}{}^{B}=\begin{pmatrix}-iJ_{\alpha}{}^{\beta}-\textstyle{\frac{i}{2}}\delta_{\alpha}{}^{\beta}D&-P_{\alpha{\dot{\beta}}}\\\
-K^{{\dot{\alpha}}\beta}&+i\bar{J}^{\dot{\alpha}}{}_{{\dot{\beta}}}+\textstyle{\frac{i}{2}}\delta^{\dot{\alpha}}{}_{{\dot{\beta}}}D\end{pmatrix}\,.\end{split}$
(168)

More explicitly,

$\displaystyle\begin{split}J_{\alpha\beta}=\lambda_{(\alpha}\bar{\mu}_{\beta)}\,,\quad&D=\frac{1}{2}\left(\bar{\mu}^{\alpha}\lambda_{\alpha}+\bar{\lambda}_{\dot{\alpha}}\mu^{\dot{\alpha}}\right)\,,\\\
P_{\alpha{\dot{\alpha}}}=-\lambda_{\alpha}\bar{\lambda}_{\dot{\alpha}}\,,\quad&K^{{\dot{\alpha}}\alpha}=-\mu^{\dot{\alpha}}\bar{\mu}^{\alpha}\,.\end{split}$
(169)

It is straightforward to verify that (164) and (168) reproduce (167).

We finally turn to the (complexified) incidence relation,

$\mu^{\dot{\alpha}}=-z^{\dot{\alpha}\beta}\lambda_{\beta}\,.$ (170)

Substituting $(-\bar{z}^{\dot{\alpha}\beta}{\lambda}_{\beta})$ for
$\mu^{\dot{\alpha}}$ into
$\bar{J}_{\dot{\alpha}\dot{\beta}}=\bar{\lambda}_{({\dot{\alpha}}}\mu_{{\dot{\beta}})}$
and using
$p_{\alpha{\dot{\alpha}}}=-\lambda_{\alpha}\bar{\lambda}_{\dot{\alpha}}$, we
find

$\displaystyle\bar{J}_{\dot{\alpha}\dot{\beta}}=-\bar{\lambda}_{(\dot{\alpha}\vphantom{{\dot{\beta}}}}z_{\dot{\beta})}{}^{\gamma}\lambda_{\gamma}=-z_{\gamma(\dot{\alpha}\vphantom{{\dot{\beta}}}}p_{\dot{\beta})}^{\gamma}\,.$
(171)

In the vector notation with $z^{\mu}=x^{\mu}+iy^{\mu}$, it means
$J_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}+*(y_{\mu}p_{\nu}-y_{\nu}p_{\mu})$,
where $*$ denotes the Hodge dual. The real part of $z^{\mu}$ clearly
parametrizes the position in the Minkowski space. The imaginary part can be
used to account for the helicity of a massless particle. One way to see it is
that when $\mathfrak{sl}(2,\mathbb{C})$ is regarded as the complexification of
$\mathfrak{su}(2)$, the orbital and spin angular momenta take the real and
imaginary parts, respectively. To see it in another way, we split the real and
imaginary parts of

$\displaystyle\begin{split}z^{{\dot{\alpha}}\alpha}p_{\alpha{\dot{\alpha}}}&=-\bar{\lambda}_{\dot{\alpha}}z^{{\dot{\alpha}}\alpha}\lambda_{\alpha}=\bar{\lambda}_{\dot{\alpha}}{\mu}^{\dot{\alpha}}\,,\\\
\bar{z}^{\alpha{\dot{\alpha}}}p_{\alpha{\dot{\alpha}}}&=-\lambda_{\alpha}\bar{z}^{\alpha{\dot{\alpha}}}\bar{\lambda}_{\dot{\alpha}}=\lambda_{\alpha}{\bar{\mu}}^{\alpha}\,,\end{split}$
(172)

to find

$\displaystyle\begin{split}x^{\alpha{\dot{\alpha}}}p_{{\dot{\alpha}}\alpha}&=\frac{1}{2}\left(\bar{\lambda}_{\dot{\alpha}}{\mu}^{\dot{\alpha}}+\lambda_{\alpha}{\bar{\mu}}^{\alpha}\right)=D\,,\\\
y^{{\dot{\alpha}}\alpha}p_{\alpha{\dot{\alpha}}}&=\frac{1}{2i}\left(\bar{\lambda}_{\dot{\alpha}}{\mu}^{\dot{\alpha}}-\lambda_{\alpha}{\bar{\mu}}^{\alpha}\right)=-\frac{1}{2}\bar{Z}^{\mathrm{A}}Z_{\mathrm{A}}\,.\end{split}$
(173)

The first line is consistent with the map (168). The second line relates the
helicity with the generator of the overall U$(1)$ commuting with SU$(2,2)$.
The same generator appears as the proportionality constant between the Pauli-
Lubanski vector $-(*J)_{\mu\nu}P^{\nu}$ and the momentum $P_{\mu}$.
Interpreting this U$(1)$ as the helicity of a massless particle also plays a
crucial role in the modern application of twistor theory to scattering
amplitudes.

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