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# Supercomputers against strong coupling in gravity with curvature and torsion

W.E.V. Barker<EMAIL_ADDRESS>Astrophysics Group, Cavendish Laboratory, JJ
Thomson Avenue, Cambridge CB3 0HE, UK Kavli Institute for Cosmology, Madingley
Road, Cambridge CB3 0HA, UK

###### Abstract

Many theories of gravity are spoiled by strongly coupled modes: the high
computational cost of Hamiltonian analysis can obstruct the identification of
these modes. A computer algebra implementation of the Hamiltonian constraint
algorithm for curvature and torsion theories is presented. These non-
Riemannian or Poincaré gauge theories suffer notoriously from strong coupling.
The implementation forms a package (the ‘Hamiltonian Gauge Gravity Surveyor’ –
_HiGGS_) for the _xAct_ tensor manipulation suite in _Mathematica_. Poisson
brackets can be evaluated in parallel, meaning that Hamiltonian analysis can
be done on silicon, and at scale. Accordingly _HiGGS_ is designed to survey
the whole Lagrangian space with high-performance computing resources (clusters
and supercomputers). To demonstrate this, the space of ‘outlawed’ Poincaré
gauge theories is surveyed, in which a massive parity-even/odd vector or
parity-odd tensor torsion particle accompanies the usual graviton. The survey
spans possible configurations of teleparallel-style multiplier fields which
might be used to kill-off the strongly coupled modes, with the results to be
analysed in subsequent work. All brackets between the known primary and
secondary constraints of all theories are made available for future study.
Demonstrations are also given for using _HiGGS_ – on a desktop computer – to
run the Dirac–Bergmann algorithm on specific theories, such as Einstein–Cartan
theory and its minimal extensions.

###### pacs:

04.50.Kd, 04.60.-m, 04.20.Fy, 02.70.-c, 07.05.Bx

###### Contents

  1. I Introduction
    1. I.1 Conventions
  2. II Implementation
    1. II.1 Geometric setup
    2. II.2 Irreducible decompositions
      1. II.2.1 Lorentz group
      2. II.2.2 Rotation group
    3. II.3 Derivatives
    4. II.4 Low-level functions
      1. II.4.1 Module: ‘ToBasicForm’
      2. II.4.2 Module: ‘ToNesterForm’
      3. II.4.3 Module: ‘PoissonBracket’
    5. II.5 High-level functions
      1. II.5.1 Module: ‘DefTheory’
      2. II.5.2 Module: ‘StudyTheory’ and ‘Velocity’
      3. II.5.3 Module: ‘ViewTheory’
    6. II.6 Parallelisation and HPC
  3. III Examples
    1. III.1 Preparing a science session
    2. III.2 Yo–Nester unit tests
    3. III.3 Simple HPC survey
      1. III.3.1 Scope of survey
      2. III.3.2 Results and prospects for new physics
  4. IV Conclusions
  5. A Linear velocities of the simple spin-$1^{+}$ case

##  I Introduction

Figure 1:  On a supercomputer, 448 processors obtain simplified, covariant
expressions for all nonlinear Poisson brackets, among all discovered primary
and secondary constraints, for 192 modified gravity theories. These theories
are based on Einstein–Cartan gravity with an extra, massive spin-parity
$1^{+}$, $1^{-}$ or $2^{+}$ torsion particle, in which various parts
($2^{3}\times 2^{3}$ choices) of the curvature and torsion are disabled by
multipliers. Each colour is a different theory, black/gray is initialisation.
The objective is eventually to discover whether multipliers can be used to
suppress the strongly-coupled $1^{-}$, $1^{+}$ or $2^{-}$ modes, which
respectively ruin these theories. Ready access to the Poisson bracket
structure is vital for this analysis.

The modern frontier in the search for alternatives to general relativity (GR)
is characterised by a very large number of competing models [1, 2, 3]. These
models are problematic by dint of their heterogeneity: differences in the
mathematical formulation hinder comparison between different classes (e.g.
mimetic [4, 4] and MOND [5, 6] theories). This seems intractable, but we
expect the number of classes to grow only in proportion to the community. A
more serious problem presents when a class contains a large number of
parameters, e.g. the couplings in a Lagrangian. Critical theoretical
properties, such as the number and health of propagating degrees of freedom
(d.o.f) may be sensitive to these parameters in ways which are hard not only
to characterise in general, but even to calculate in detail for a given
parameter choice.

This scenario commonly arises in the Hamiltonian analysis, used routinely to
determine the d.o.fs of a proposed theory [7, 8, 9, 10, 11]. If the coupling
parameters elimniate some velocity $\dot{\psi}$ from the motivated Lagrangian
$\mathcal{L}$, then the total Hamiltonian $\mathcal{H}_{\text{T}}$ must be
modified to express the constraint
$\pi_{\psi}\sim\partial_{\dot{\psi}}\mathcal{L}\approx 0$. That constraint is
only preserved if
$\dot{\pi}_{\psi}\sim\\{\pi_{\psi},\mathcal{H}_{\text{T}}\\}\approx 0$, which
either vanishes identically or constitutes a _new_ constraint, ad infinitum.
Each constraint subtracts d.o.fs from the countable fields $\psi$ naïvely
present in $\mathcal{L}$. The full chain of constraints is systematically
elucidated by means of the _Dirac–Bergmann_ algorithm [12, 10, 7], the
fundamental computational unit of which is the Poisson bracket. In theories
constructed from the higher-spin representations of the Lorentz group,
including most tensor theories of gravity, a single covariant bracket can be
surprisingly cumbersome for manual evaluation [13, 14]; less so, as we will
find, for computer assistance (see Fig. 1). Through the algorithm, the d.o.f
and symmetry structure depend not only on eliminated velocities, but also on
all brackets between all constraints, which themselves may be contingent on
the couplings in ways that are, ab initio, unknowable.

The _strong coupling problem_ is commonly diagnosed in the Hamiltonian
picture. It is usually preferred that two gravitational d.o.f propagate [15,
16]. Additional d.o.f may be tolerated unless ghostly or tachyonic111Perhaps
with some interesting exceptions [17, 18]., but they must kept under close
theoretical and phenomenological control (e.g. by large [19] – or small [20,
21] – masses, screening [22] or other measures). Frequently we start with many
fields $\psi$ (ten in the case of GR), so that reduction to two d.o.f is an
achivement in the linearised Hamiltonian picture: strongly coupled modes may
_increase_ this number the nonlinear analysis [23]. Strong coupling can be
imagined as a finely-tuned suppression of the kinetic coefficients222Where
permitted, a canonical normalisation recasts this as a divergent mass. in a
mode’s linear wave equation [24]. The nonlinear operators are generally still
present, however, and describe a non-perturbative dynamics which may be
unacceptable, having for example an elliptic or parabolic character. The
linearisation in this case refers to some motivated exact vacuum solution.
This solution need not be Minkowskian: for GR in higher (odd) dimensions, a
fine-tuned admixture of the Gauss–Bonnet invariant strongly couples the whole
graviton on a maximally symmetric but curved background [25]. In many versions
of Hořava gravity [26], the ‘detailed balance’ which defines the Lorentz-
asymmetric theory has a similar effect [27, 28].

There are some prominent scenarios where strong coupling seems desirable. The
linearised, massive graviton of Fierz and Pauli [29] appears troubled even in
the massless limit by its fifth d.o.f — the helicity-0 mode or van
Dam–Veltman–Zakharov (vDVZ) scalar [30, 31] — which couples to the matter
trace. However, nonlinear completions [32, 33] of Fierz–Pauli theory revealed
that rather than persisting as a light d.o.f, the vDVZ scalar instead becomes
strongly coupled, i.e. Vainshtein-screened [34, 35]. Unfortunately, these same
theories were also plagued by a sixth d.o.f — the nonlinear Boulware–Deser
ghost [32] — whose origin was apparently connected back to the same strong
coupling effect [36]. These matters remain strongly contested [37], and
massive gravity continues to evolve [38, 39].

Strong coupling in massive gravity had another (equally contested [37])
association with superluminality [40, 41]. Both properties are sometimes
conflated with acausality, and ill-posedness of the Cauchy–Kovalevskaya
problem. While systems might exist in which these are all mutually implicated
pathologies, it is by no means general. In each case one should study the
characteristic surfaces and sift for coordinate artifacts [42, 43]. Strongly
coupled or _singular_ surfaces in the Hamiltonian phase space, where the rank
of the matrix of constraint brackets changes, have also been accused of
causality violation in the _Poincaré gauge theory_ (PGT) of gravity [44, 45,
23]. PGT is the general theory which may be constructed from spacetime
curvature and torsion. Whether or not the same caveats resolve the causal
question here [13], the proclivity for strong coupling in this class remains,
and has yet to be seriously addressed. To this end, we target the Poincaré
gauge theory class in this paper.

The Poincaré gauge theory encompasses a large sector of the general category
of _non-Riemannian_ theories, whose current popularity warrants a quick
introduction. Geometrically interpreted, we retain for non-Riemannian gravity
the system of clocks and rulers in the gravitational metric potential
$\tensor{g}{{}_{\mu\nu}}$. In GR, however, the connection
$\tensor{\Gamma}{{}^{\mu}_{\nu\sigma}}$ is fixed to the Levi–Civita form
$\tensor{*}{C}{{}^{\mu}_{\nu\sigma}}\equiv\tensor{*}{C}{{}^{\mu}_{(\nu\sigma)}}\equiv\frac{1}{2}\tensor{g}{{}^{\mu\rho}}(\tensor{\partial}{{}_{\nu}}\tensor{g}{{}_{\rho\sigma}}+\tensor{\partial}{{}_{\sigma}}\tensor{g}{{}_{\rho\nu}}-\tensor{\partial}{{}_{\rho}}\tensor{g}{{}_{\nu\sigma}})$
– i.e. the non-tensorial Christoffel symbol – by assumption. By relaxing this
convention, the geometry of the manifold can be extended to include not only
the curvature $\tensor{R}{{}^{\mu}_{\nu\sigma\rho}}\equiv
2\tensor{\partial}{{}_{[\sigma|}}\tensor{\Gamma}{{}^{\mu}_{|\rho]\nu}}+\dots$,
but also the _torsion_ $\tensor{T}{{}^{\mu}_{\nu\sigma}}\equiv
2\tensor{\Gamma}{{}^{\mu}_{[\nu\sigma]}}$ and _non-metricity_
$\tensor{Q}{{}_{\mu\nu\sigma}}\equiv\tensor{\nabla}{{}_{\mu}}\tensor{g}{{}_{\nu\sigma}}$.
In this broader context, what is usually meant by ‘GR’ is reached by the
Einstein–Hilbert-like action
$S_{\text{G}}\equiv\int\mathrm{d}^{4}x\sqrt{-g}L_{\text{G}}$, with

$L_{\text{G}}=-\frac{1}{2\kappa}R+\frac{1}{\kappa}\tensor{\lambda}{{}_{\mu}^{\rho\sigma}}\tensor{T}{{}^{\mu}_{\rho\sigma}}+\frac{1}{\kappa}\tensor{\hat{\lambda}}{{}_{\mu}^{\rho\sigma}}\tensor{Q}{{}^{\mu}_{\rho\sigma}}.$
(1)

The multiplers333Note some differences in our convention, regarding the
dimensionality and density status of these [46].
$\tensor{\lambda}{{}_{\mu}^{\rho\sigma}}$ and
$\tensor{\hat{\lambda}}{{}_{\mu}^{\rho\sigma}}$ constrain the geometry to be
_torsion-free_ and _metric_ , while the dynamics are those of _curvature_. At
the time of writing, the attention of the community is drawn to the remarkable
non-uniqueness of (1) as a realisation of GR in this broader, non-Riemannian
context. By cycling multipliers onto different pairs in the
curvature–torsion–non-metricity triad, and identifying suitable dynamical
terms, one can construct the flat, metric _teleparallel_ equivalent of GR
(TEGR) from pure torsion [47], and the flat, torsion-free _symmetric_
alternative (STEGR) from pure non-metricity [48]. With (1), these points in
the space of non-Riemannian theories define the _geometrical trinity of
gravity_ [46].

There is now activity to determine a preferred vertex of the trinity, and the
viability of the surrounding non-Riemannian landscape. Various considerations
must be balanced in a very large parameter space [49, 50, 51, 52]. As we
mentioned previously, the PGT may be thought of as the sector of the landscape
in which $\tensor{Q}{{}^{\mu}_{\nu\sigma}}$ is suppressed, e.g. by a
multiplier as in (1). PGT is a convenient sector to study — without prejudice
to the ultimate rôle of non-metricity in constructing a viable gravity theory
— since it has a relatively self-contained history stemming from the
Einstein–Cartan model [53], and the gauge-theoretic interpretation pioneered
by many authors [54, 8, 55, 56], beginning with Kibble [57], Utiyama [58] and
also Sciama [59].

The linear Hamiltonian structure of PGT is particularly well developed [8, 60,
61, 62]. In the nonlinear structure, strong coupling phenomena are know to be
_abundant_ , and in this regard PGT is feared to be representative of the
broader non-Riemannian landscape. Critically however, the challenge of the
Hamiltonian analysis has prevented this structure from being mapped in any
comprehensive detail. A few islands in the landscape were probed at the turn
of the millenium [23]: minimal extensions to the Einstein–Cartan theory in
which a single extra massive spin-parity ($J^{P}$) $1^{+}$, $1^{-}$ or $2^{-}$
torsion particle is present. In each case respectively, $1^{-}$, $1^{+}$ and
$2^{+}$ modes were suggested to be strongly coupled. Since then, the PGT
sector of the landscape has been treated with a sense of ‘ _hic sunt dracones_
’ – i.e. avoided as potentially dangerous – with apparently the only safe
configurations being exclusive activation of the $0^{+}$ or $0^{-}$ scalar
torsion particles [45]. In light of our comments above, one can even afford to
remain agnostic on the pathology of this strong coupling444It is important to
understand that the main historical objection to the strongly coupled modes of
the PGT has been their ghostly character, as inferred by an inspection of the
signs of squared momenta in the Hamiltonian. These signs are fixed by the
unitarity requirements of the desired, linearly active modes in each case, and
in each case they are _negative_ [23]. This ‘ _catch-22_ ’ is particular to
the PGT, but, as cogently explained in [63], there are principled reasons to
be suspicious of stongly coupled surfaces, which appeal to neither the non-
perturbative dynamics, nor the causality arguments mentioned above.
Generically, a background which is strongly coupled had better not be one
which is also seen in nature, since it cannot have been reached by any smooth
trajectory through the phase space.. It would seem, however, given the recent,
promising developments in the non-Riemannian approach [46, 49, 50, 51, 52],
that the extent of the phenomenon should still be understood, and general
tools be developed to that end.

In this paper we present a computer algebra implementation of the nonlinear
Hamiltonian analysis for a _generalisation_ of the full PGT, in which one may
covariantly disable arbitrary irreps of the torsion and curvature by means of
Lagrangian multiplier fields. This generalisation was put forward as an anti-
strong-coupling measure in [14], and its Hamiltonian structure is elucidated
in the companion paper [64]. In the conventions of (1), we may write
generalised PGT as

$\displaystyle\tensor{L}{{}_{\text{G}}}$
$\displaystyle=-\frac{\tensor{\hat{\alpha}}{{}_{0}}}{2\kappa}R+\sum_{I=1}^{6}\Big{(}\tensor{\hat{\alpha}}{{}_{I}}\tensor{R}{{}^{\mu\nu}_{\sigma\rho}}+\tensor{\bar{\alpha}}{{}_{I}}\tensor{\lambda}{{}^{\mu\nu}_{\sigma\rho}}\Big{)}{\tensor{[}^{I}]{\hat{P}}{{}_{\mu\nu}^{\sigma\rho}{}_{\kappa\pi}^{\xi\zeta}}}\tensor{R}{{}^{\kappa\pi}_{\xi\zeta}}$
$\displaystyle\ \
+\frac{1}{\kappa}\sum_{M=1}^{3}\Big{(}\tensor{\hat{\beta}}{{}_{M}}\tensor{T}{{}^{\mu}_{\nu\sigma}}+\tensor{\bar{\beta}}{{}_{M}}\tensor{\lambda}{{}^{\mu}_{\nu\sigma}}\Big{)}{\tensor{[}^{M}]{\hat{P}}{{}_{\mu}^{\nu\sigma}{}_{\pi}^{\xi\zeta}}}\tensor{T}{{}^{\pi}_{\xi\zeta}}$
$\displaystyle\ \
+\frac{1}{\kappa}\tensor{\hat{\lambda}}{{}_{\mu}^{\rho\sigma}}\tensor{Q}{{}^{\mu}_{\rho\sigma}},$
(2)

where the usual ‘quadratic’ PGT is spanned by the ten parameters
$\tensor{\hat{\alpha}}{{}_{0}}$, $\\{\tensor{\hat{\alpha}}{{}_{I}}\\}$,
$\\{\tensor{\hat{\beta}}{{}_{M}}\\}$. The projections
${\tensor{[}^{I}]{\hat{P}}{{}_{\mu\nu}^{\sigma\rho}{}_{\kappa\pi}^{\xi\zeta}}}$
and ${\tensor{[}^{M}]{\hat{P}}{{}_{\mu}^{\nu\sigma}{}_{\pi}^{\xi\zeta}}}$
extract the ${\mathrm{SO}^{+}(1,3)}$ field strength irreps. The core of the
implementation is (version 1.0.0 of [65]) the _Hamiltonian Gauge Gravity
Surveyor_ (_HiGGS_), a _Mathematica_ package grounded in the popular open-
source _xAct_ tensor manipulation suite [66, 67, 68, 69, 70, 71].

The _HiGGS_ package is suitable for targeted use on a desktop computer. Since
it is parallelised over Poisson brackets, _HiGGS_ also scales to clusters and
supercomputers. Development in this direction is with the aim of surveying the
constraint structure of the non-Riemannian landscape at scale. Modules from
_HiGGS_ , in particular those concerned with bracket evaluation, can be used
as a back-end in searching the parameter space for desirable canonical
features — as such features become better understood with time.

For the moment, we perform the brute-force ‘calibration’ survey illustrated in
Fig. 1. In this run, which takes a little over $1\text{\,}\mathrm{h}$, the
$1^{+}$, $1^{-}$ and $2^{-}$ Einstein–Cartan extensions are modified with all
possible configurations of curvature- and torsion-disabling multipliers: 192
generalised Poincaré gauge theories in total. During the run _HiGGS_ obtains,
for every theory, all the primary and secondary constraints which can be
inferred from a knowledge of the literature, and then computes simple
covariant expressions for the nonlinear Poisson brackets between all
constraint pairs. All brackets identified in this survey can be found in the
supplemental materials [72].

Examples are also provided of how to use _HiGGS_ to calculate constraint
velocities when implementing the Dirac–Bergmann algorithm. Focus is on the
minimal Einstein–Cartan extensions with strong coupling, and the viable
$0^{+}$ and $0^{-}$ extensions. The unmodified Einstein–Cartan theory is also
studied, and _HiGGS_ is used to show that it propagates only the two graviton
polarisations. Finally, we will use the results of the initial survey to show
how multipliers might conceivably be used to suppress strongly coupled fields.
Note that the major undertaking of fully analysing the results, isolating and
confirming any viable multiplier configurations, is left to future work.

The remainder of this paper is structured as follows. In Section I.1 we
briefly set out our conventions for PGT, using the non-geometric, gauge-
theoretic formulation. In Section II we describe the _HiGGS_ implementation,
including general tools for the canonical manipulation curvature and torsion,
up to Poisson brackets and higher-level functionality for the theory-specific
calculation of constraints and velocities. Solutions for scaling the
Hamiltonian analysis to high-performance computing (HPC) resources are also
described. In Section III we present examples of the algorithm, and the
results of our initial survey in Fig. 1. Conclusions follow in Section IV.

###  I.1 Conventions

Before proceeding, we introduce the gauge-theoretic formulation of Poincaré
gauge theory, as used in [8, 73, 74, 75, 13, 14, 64]. The gauge picture does
not offer any special advantage over the geometric setup in (1), but it is
more consistent with previous Hamiltonian analyses in [8, 60, 61, 76, 45, 23,
62].

The geometric covariant derivative appearing in the definition
$\tensor{Q}{{}_{\mu\nu\sigma}}\equiv\tensor{\nabla}{{}_{\mu}}\tensor{g}{{}_{\nu\sigma}}$,
as it acts on a vector $\tensor{V}{{}^{\mu}}$, is written

$\tensor{\nabla}{{}_{\nu}}\tensor{V}{{}^{\mu}}\equiv\tensor{\partial}{{}_{\nu}}\tensor{V}{{}^{\mu}}+\tensor{\Gamma}{{}^{\mu}_{\lambda\nu}}\tensor{V}{{}^{\lambda}}.$
(3)

In the Poincaré gauge theory we enforce $\tensor{Q}{{}_{\mu\nu\sigma}}=0$ by
assumption, though a multiplier could also be used. The general non-Riemannian
connection

$\tensor{\Gamma}{{}^{\mu}_{\nu\sigma}}=\tensor{*}{C}{{}^{\mu}_{\nu\sigma}}+\tensor{K}{{}^{\mu}_{\nu\sigma}}+\tensor{L}{{}^{\mu}_{\nu\sigma}},$
(4)

thus loses its _disformation_ part
$\tensor{L}{{}^{\mu}_{\nu\sigma}}\equiv\frac{1}{2}\tensor{Q}{{}^{\mu}_{\nu\sigma}}-\tensor{Q}{{}_{(\nu|}^{\mu}{}_{|\sigma)}}=0$,
but still conveys torsion through the _contorsion_ d.o.fs
$\tensor{K}{{}^{\mu}_{\nu\sigma}}\equiv\frac{1}{2}\tensor{T}{{}^{\mu}_{\nu\sigma}}+\tensor{T}{{}_{(\nu|}^{\mu}{}_{|\sigma)}}$.
Aside from the contorsion, the metric d.o.fs which source the Levi–Civita part
are defined, as usual, by tangents to the coordinate curves
${\tensor{g}{{}_{\mu\nu}}\equiv\tensor{\bm{e}}{{}_{\mu}}\cdot{\bm{e}}{{}_{\nu}}}$.

We now move away from this geometric picture, to a ‘particle physics’ setup
where the underlying manifold is always flat Minkowski space
$\check{\mathcal{M}}$. The metric associated with (curvillinear) coorinate
tangents is then
${\tensor{\gamma}{{}_{\mu\nu}}\equiv\tensor{\bm{e}}{{}_{\mu}}\cdot{\bm{e}}{{}_{\nu}}}$,
and this metric is _flat_. The coordinate basis is accompanied by a Lorentz
basis, whose dot products give the Minkowskian metric components
${\tensor{\eta}{{}_{ij}}\equiv\tensor{\hat{\bm{e}}}{{}_{i}}\cdot{\hat{\bm{e}}}{{}_{j}}}$.
The Lorentz basis can rotate locally under the proper, orthochronous Lorentz
rotations, and is non-holonomic. Following on from (3), a vector
$\tensor{\mathcal{V}}{{}^{i}}$ with Lorentz indices has a covariant derivative

$\tensor{\mathcal{D}}{{}_{j}}\tensor{\mathcal{V}}{{}^{i}}\equiv\tensor{h}{{}_{j}^{\mu}}(\tensor{\partial}{{}_{\mu}}\tensor{\mathcal{V}}{{}^{i}}+\tensor{A}{{}^{i}_{k\mu}}\tensor{\mathcal{V}}{{}^{k}}),$
(5)

where we define the (inverse of the) translational gauge field
$\tensor{h}{{}_{i}^{\mu}}$ and the rotational gauge field
${\tensor{A}{{}^{ij}_{\mu}}\equiv\tensor{A}{{}^{[ij]}_{\mu}}}$ — we will
return to this derivative in Section II.3. These fields guarantee invariance
under the general coordinate transformations (GCTs or passive diffeomorphisms)
on $\check{\mathcal{M}}$, and Lorentz rotations, and together they _gauge_ the
Poincaré group ${\mathbb{R}^{1,3}\rtimes\mathrm{SO}^{+}(1,3)}$.

How to connect back to the geometric picture? The Minkowskian metric
components can be recovered via the identities
${\tensor{b}{{}^{i}_{\mu}}\tensor{h}{{}_{i}^{\nu}}\equiv\tensor{*}{\delta}{{}_{\mu}^{\nu}}}$
and
${\tensor{b}{{}^{i}_{\mu}}\tensor{h}{{}_{j}^{\mu}}\equiv\tensor{*}{\delta}{{}_{j}^{i}}}$.
On the other hand, the usual system of clocks and rulers can be recovered
using
${\tensor{g}{{}_{\mu\nu}}\equiv\tensor{\eta}{{}_{ij}}\tensor{b}{{}^{i}_{\mu}}\tensor{b}{{}^{j}_{\nu}}}$
and
${\tensor{g}{{}^{\mu\nu}}\equiv\tensor{\eta}{{}^{ij}}\tensor{h}{{}_{i}^{\mu}}\tensor{h}{{}_{j}^{\nu}}}$.
The covariant measures on $\mathcal{M}$ and $\check{\mathcal{M}}$ are
respectively $\sqrt{-g}$, i.e. the conventional
${g\equiv\det\tensor{g}{{}_{\mu\nu}}}$, and ${b\equiv
h^{-1}\equiv\det\tensor{b}{{}^{i}_{\mu}}}$. Finally, the geometric field
strength tensors – referred to as the curvature and the torsion – are provided
by the formulae

$\displaystyle\tensor{\mathcal{R}}{{}^{ij}_{kl}}$ $\displaystyle\equiv
2\tensor{h}{{}_{k}^{\mu}}\tensor{h}{{}_{l}^{\nu}}\big{(}\tensor{\partial}{{}_{[\mu|}}\tensor{A}{{}^{ij}_{|\nu]}}+\tensor{A}{{}^{i}_{m[\mu|}}\tensor{A}{{}^{mj}_{|\nu]}}\big{)},$
(6a) $\displaystyle\tensor{\mathcal{T}}{{}^{i}_{kl}}$ $\displaystyle\equiv
2\tensor{h}{{}_{k}^{\mu}}\tensor{h}{{}_{l}^{\nu}}\big{(}\tensor{\partial}{{}_{[\mu|}}\tensor{b}{{}^{i}_{|\nu]}}+\tensor{A}{{}^{i}_{m[\mu|}}\tensor{b}{{}^{m}_{|\nu]}}\big{)}.$
(6b)

The conversion of these back to the (numerical values of) the geometric
components, is done by contraction with the translational gauge fields. The
theory (2) meanwhile becomes

$\displaystyle\tensor{L}{{}_{\text{G}}}$
$\displaystyle=-\frac{1}{2}\tensor{\hat{\alpha}}{{}_{0}}{m_{\text{p}}}^{2}\mathcal{R}$
$\displaystyle\ \
+\sum_{I=1}^{6}\Big{(}\tensor{\hat{\alpha}}{{}_{I}}\tensor{\mathcal{R}}{{}^{ij}_{kl}}+\tensor{\bar{\alpha}}{{}_{I}}\tensor{\lambda}{{}^{ij}_{kl}}\Big{)}{\tensor{[}^{I}]{\hat{\mathcal{P}}}{{}_{ij}^{kl}{}_{nm}^{pq}}}\tensor{\mathcal{R}}{{}^{nm}_{pq}}$
$\displaystyle\ \
+{m_{\text{p}}}^{2}\sum_{M=1}^{3}\Big{(}\tensor{\hat{\beta}}{{}_{M}}\tensor{\mathcal{T}}{{}^{i}_{jk}}+\tensor{\bar{\beta}}{{}_{M}}\tensor{\lambda}{{}^{i}_{jk}}\Big{)}{\tensor{[}^{M}]{\hat{\mathcal{P}}}{{}_{i}^{jk}{}_{l}^{nm}}}\tensor{\mathcal{T}}{{}^{l}_{nm}},$
(7)

where our metricity assumption dispenses with the need for the final term in
(2), and we use the Planck mass rather than the Einstein constant
${m_{\text{p}}}\equiv 1/\sqrt{\kappa}$. Once again, we reiterate that our use
of the gauge theory setup over geometric alternatives (including those which
use the tetrad and spin-connection over the metric and contorsion) is merely a
matter of convenience in the present work.

In this article we will follow [77] by using the following syntax highlighting
for code listings: keywords for _Mathematica_ are typeset in green, for _xAct_
in blue and for _HiGGS_ in red. This paper uses the ‘West coast’ signature
$(+,-,-,-)$.

##  II Implementation

In this section the implementation of the Hamiltonian analysis in the _HiGGS_
package is described. Many aspects of our particular approach will be
inefficient: the package is monolithic and expensive in terms of memory and
maintenance. At the time of writing, however, HPC is a _cheap_ resource. By
carefully tuning only a few aspects of the implementation it is therefore
possible to produce a product which surveys the theory space in a matter of
hours. It is important to emphasise that these ‘high-level’ features of
_HiGGS_ are currently limited to the generalised Poincaré gauge theory in (7).
The question of scaling to arbitrary theories will be addressed in Section IV,
where we will find that such a scaling is actually likely to _reduce_ the
complexity of the package, so long as the PoissonBracket[] function is
upgraded so as to be ‘aware of’ Leibniz’s rule. Moreover, the ‘low-level’
functionality, which is not specific to a Lagrangian, will be of standalone
utility. In what follows, we assume a basic familiarity with the _xAct_ suite
and Wolfram Language.

###  II.1 Geometric setup

We begin by describing the way in which the Riemann–Cartan geometry is
implemented in _HiGGS_. Whilst _xAct_ is perfectly capable of accommodating
not only a Riemann–Cartan curvature, but also a torsion tensor, we prefer to
adhere to the ‘particle physics’ picture of gravitational gauge theories [78,
8, 56, 14], and set up all the physics on a _flat_ spacetime. The following
equivalent _xAct_ commands are issued when the _HiGGS_ environment is
initially built (see Section III.1) with the command BuildHiGGS[]

⬇

In[]:= DefManifold[M4, 4, IndexRange[{a, z}]];

DefMetric[-1, G[-m, -n], CD, {",", "\\[PartialD]"}, PrintAs $\to$ "\\[Gamma]",
FlatMetric $\to$ True, SymCovDQ $\to$ True];

This sets up a $D=4$ manifold M4 to represent $\check{\mathcal{M}}$, with a
_flat_ negative-signature metric G[-m,-n] to represent
$\tensor{\gamma}{{}_{\mu\nu}}$, and _flat_ covariant derivative
$\tensor{\check{\nabla}}{{}_{\mu}}$ represented by CD[-m][]. This offers
several advantages, and foremost among these is an easy comparison with the
very substantial body of literature on the Hamiltonian structure as mentioned
in Section I.1. We will also not be limited by the fact that the non-
Riemannian features of _xAct_ are (presently) less flexible and comprehensive
than those of a simple Minkowskian setup. This is to be expected, given that
GR is the preferred effective theory of gravitation.

An essential feature of this setup, which goes beyond the particle picture of
the literature, is the conflation of holonomic and non-holonomic tangent
spaces: a single collection of indices a, b, c, etc. is ascribed to TangentM4,
and these represent both the Lorentz indices $i$, $j$, $k$ and coordinate
indices $\mu$, $\nu$, $\sigma$. This practice might be anathema to the field
of differential geometry, but it is acceptable for pragmatic, computational
purposes. The coordinates are assumed implicitly to be Cartesian, so that the
components of the flat metric $\tensor{\gamma}{{}_{\mu\nu}}$ are Minkowskian
(equal to those of $\tensor{\eta}{{}_{ij}}$), and moreover that a rotational
gauge is chosen in which
$\tensor{\bm{e}}{{}_{0}}=\tensor{\hat{\bm{e}}}{{}_{0}}$,
$\tensor{\bm{e}}{{}_{1}}=\tensor{\hat{\bm{e}}}{{}_{1}}$,
$\tensor{\bm{e}}{{}_{2}}=\tensor{\hat{\bm{e}}}{{}_{2}}$ and
$\tensor{\bm{e}}{{}_{3}}=\tensor{\hat{\bm{e}}}{{}_{3}}$. As a consequence of
this gauge-fixed setup, the covariance of quantities is not guaranteed
internally, as it would be if all the features of _xAct_ (such as user-defined
connections) were fully exploited in _HiGGS_. It is instead possible to check
the covariance by visually inspecting the final results for explicit gauge
fields: in practice this turns out to be very easy, and the elimination of
bare gauge fields in _HiGGS_ is reliable.

It is important to note that whilst the gauge is fixed to conveniently
overload the indices of TangentM4, the tensor structure of PGT is wholly
preserved. This is in contrast, for example, with the seminal paper [60] in
which the _time gauge_ imposes $\tensor{h}{{}_{a}^{0}}=0$, thereby massively
simplifying the various algebraic expressions. On the contrary, the algebraic
expressions produced from _HiGGS_ are valid for _any gauge_ , once the various
shared indices are interpreted as being either Greek or Roman. Care is taken
in the definitions to avoid any ambiguity over this division of indices.

Following on from our discussion of indices, we note that the 1, 2, 3
spacelike indices $a$, $b$, $c$ and $\alpha$, $\beta$, $\gamma$ are also
subsumed into a, b, c. These may be extracted by means of the projection
operator G3[a,b], which represents $\tensor{\gamma}{{}^{\alpha\beta}}$, and
lies at the heart of the Arnowitt–Deser–Misner (ADM) split. The ADM or $3+1$
split uses a spacelike foliation which is characterised by timelike unit
vector $\tensor{n}{{}_{k}}$, defined as

$\tensor{n}{{}_{k}}\equiv\tensor{h}{{}_{k}^{0}}/\sqrt{\tensor{g}{{}^{00}}},$
(8)

where we recall that the (gravitational) metric components are recovered by
$\tensor{g}{{}^{\mu\nu}}\equiv\tensor{h}{{}_{i}^{\mu}}\tensor{h}{{}_{j}^{\nu}}\tensor{\eta}{{}^{ij}}$.
Any vector with local Lorentz indices may then be decomposed into
perpendicular and parallel components
${\tensor{\mathcal{V}}{{}^{i}}=\tensor{\mathcal{V}}{{}^{\perp}}\tensor{n}{{}^{i}}+\tensor{\mathcal{V}}{{}^{\overline{i}}}}$.
An overbar is used to denote the parallel indices. There are then some
identities
$\smash{\tensor{b}{{}^{\overline{k}}_{\alpha}}\tensor{h}{{}_{\overline{l}}^{\alpha}}\equiv\tensor{*}{\delta}{{}_{\overline{l}}^{\overline{k}}}}$
and
$\smash{\tensor{b}{{}^{\overline{k}}_{\alpha}}\tensor{h}{{}_{\overline{k}}^{\beta}}\equiv\tensor{*}{\delta}{{}_{\alpha}^{\beta}}}$,
which follow from (8). The gauge fields $\tensor{h}{{}_{i}^{\mu}}$,
$\tensor{b}{{}^{i}_{\mu}}$ and $\tensor{A}{{}^{ij}_{\mu}}$ are denoted by
H[-a,b], B[a,-b] and A[a,b,-c]. The timelike vector $\tensor{n}{{}^{i}}$ is
represented by V[a], and we see that a variety of identities are then implied
within the built _HiGGS_ environment

⬇

In[]:= quantity = {H[-a, b], B[a, -b], A[a, b, -c], V[a], V[a]*V[-a], A[a, -a,
-c], A[a, b, -c] + A[b, a, -c], H[-a, i]*B[a, -j], H[-a, i]*B[c, -i], G3[-a,
-b]*G3[b, -d], G3[-a, a], B[a, -b]*G3[b, -c]*V[-a], CD[-a][G3[-c, b]]}

Out[]=

In[]:= ToCanonical[quantity]

Out[]=

Note that the covariant derivative CD[-i][] represents, in our Cartesian gauge
choice, the basic coordinate derivative
$\tensor{\check{\nabla}}{{}_{\mu}}\to\tensor{\partial}{{}_{\mu}}$. The ADM
_lapse_ function, and a _shift_ vector are next defined, according to

$N\equiv\tensor{n}{{}_{k}}\tensor{b}{{}^{k}_{0}},\quad\tensor{N}{{}^{\alpha}}\equiv\tensor{h}{{}_{\overline{k}}^{\alpha}}\tensor{b}{{}^{\overline{k}}_{0}}.$
(9)

These functions carry information about the part
$\tensor{b}{{}^{\overline{k}}_{0}}$ of the translational gauge field (which,
as we will shortly see, is _non-physical_), wheres $\tensor{n}{{}^{i}}$ is
independent of this quantity. In _HiGGS_ the lapse is Lapse[] and we use also
the spatial measure J[] for the quantity $J\equiv b/N$, where
$b\equiv\det\tensor{b}{{}^{i}_{\mu}}$. Some further identities are then

$\begin{gathered}\frac{\partial\tensor{n}{{}_{l}}}{\partial\tensor{b}{{}^{k}_{\mu}}}\equiv-\tensor{n}{{}_{k}}\tensor{h}{{}_{\overline{l}}^{\mu}},\quad\frac{\partial\tensor{h}{{}_{l}^{\nu}}}{\partial\tensor{b}{{}^{k}_{\mu}}}\equiv-\tensor{h}{{}_{k}^{\nu}}\tensor{h}{{}_{l}^{\mu}},\quad\frac{\partial
b}{\partial\tensor{b}{{}^{k}_{\nu}}}\equiv b\tensor{h}{{}_{k}^{\nu}},\\\
\frac{\partial J}{\partial\tensor{b}{{}^{k}_{\nu}}}\equiv
J\tensor{h}{{}_{\overline{k}}^{\nu}},\quad\frac{\partial
N}{\partial\tensor{b}{{}^{k}_{\nu}}}\equiv
N\tensor{n}{{}_{k}}\tensor{h}{{}_{\perp}^{\nu}}.\end{gathered}$ (10)

These identities are also incorporated into the _HiGGS_ environment, which
prefers to extract – from all derivatives of quantities dependent on the
translational gauge field – the form CD[-a][B[-b,-c]]. It is practical to
replace all instances of CD[-a][H[-b,-c]] accordingly, since the momentum
BPi[-a,-b] is defined according to B[-a,-b], and so we find

⬇

In[]:= quantity = ScreenDollarIndices[{J[], Lapse[], CD[-a][V[-j]],
CD[-a][H[-j, n]], CD[-a][J[]], CD[-a][Lapse[]]}]

Out[]=

Now that the geometric setup is in place, we introduce the canonical setup [8,
7, 79]. The _canonical_ momenta are

$\tensor{\pi}{{}_{i}^{\mu}}\equiv\frac{\partial
bL_{\text{G}}}{\partial(\partial_{0}\tensor{b}{{}^{i}_{\mu}})},\quad\tensor{\pi}{{}_{ij}^{\mu}}\equiv\frac{\partial
bL_{\text{G}}}{\partial(\partial_{0}\tensor{A}{{}^{ij}_{\mu}})}.$ (11)

As with our earlier work [14, 64, 13], we will _neglect_ the matter Lagrangian
$L_{\text{M}}$. The field strengths in (6a) and (6b) are independent of the
velocities for $\tensor{b}{{}^{k}_{0}}$ and $\tensor{A}{{}^{ij}_{0}}$, so (11)
imply $10$ primary constraints of the form

$\tensor{\varphi}{{}_{k}^{0}}\equiv\tensor{\pi}{{}_{k}^{0}}\approx
0,\quad\tensor{\varphi}{{}_{ij}^{0}}\equiv\tensor{\pi}{{}_{ij}^{0}}\approx 0.$
(12)

From (12) we arrive at the result mentioned above, that the conjugate field
$\tensor{b}{{}^{k}_{0}}$ is non-physical; the same applies to
$\tensor{A}{{}^{ij}_{0}}$. By ($\approx$), the _weak_ equality is denoted,
i.e. not an approximation. The constraints (12) are referred to as the ‘sure’
primary, first class (sPFC) constraints: they are a consequence of Poincaré
symmetry, and they apply for all choices of the
$\\{\tensor{\hat{\alpha}}{{}_{I}}\\}$, $\\{\tensor{\hat{\beta}}{{}_{M}}\\}$,
$\\{\tensor{\bar{\alpha}}{{}_{I}}\\}$, $\\{\tensor{\bar{\beta}}{{}_{M}}\\}$.
According to (11), _HiGGS_ must support a field momentum for both B[a,b] and
A[a,b,c]. The $\tensor{\pi}{{}_{i}^{\mu}}$, $\tensor{\pi}{{}_{ij}^{\mu}}$ and
the ‘parallel’ $\tensor{\hat{\pi}}{{}_{i}^{\overline{j}}}$,
$\tensor{\hat{\pi}}{{}_{ij}^{\overline{k}}}$ are defined as follows

⬇

In[]:= quantity = {BPi[-a, b], APi[-a, -b, c], BPiP[-a, -b], APiP[-a, -b, -c],
BPiP[-a, -b]*V[b], APiP[-a, -b, -c]*V[c]}

Out[]=

In[]:= ScreenDollarIndices[ToCanonical[quantity /. PiPToPi]]

Out[]=

The purpose of the parallel momenta is seen above: they are the physical parts
not touched by the sPFCs, and defined according to
$\smash{\tensor{\hat{\pi}}{{}_{i}^{\overline{k}}}\equiv\tensor{\pi}{{}_{i}^{\alpha}}\tensor{b}{{}^{k}_{\alpha}}}$
and
$\smash{\tensor{\hat{\pi}}{{}_{ij}^{\overline{k}}}\equiv\tensor{\pi}{{}_{ij}^{\alpha}}\tensor{b}{{}^{k}_{\alpha}}}$.

We will see in Section II.4 that the specific internal rule PiPToPi should not
often need to be used in practice, and has a more general alternative in the
ToBasicForm[] command which is provided officially by the package. In general,
parallel and perpendicular quantities can be accessed with some projections

⬇

In[]:= quantity = {PPerp[-a, b], PPara[-a, b]}

Out[]=

In[]:= ScreenDollarIndices[ToCanonical[quantity /. PADMActivate]]

Out[]=

The flat manifold of course has vanishing Riemannian curvature
$\tensor{R}{{}^{\mu}_{\nu\sigma\lambda}}=0$, and so we must be careful not to
use the _xAct_ quantities RiemannCD[-a,-b,-c,-d], RicciCD[-c,-d] or
RicciScalarCD[]. Instead, the field strengths
$\tensor{\mathcal{R}}{{}_{ijkl}}$ and $\tensor{\mathcal{T}}{{}_{ijk}}$ are
given by their own tensors, and these expand to give the definitions in Eqs.
6a and 6b

⬇

In[]:= quantity = {R[-a, -b, -c, -d], R[a, -a, -c, -d], R[-a, -b, -c, c], T[a,
-b, -c], T[a, -b, b]}

Out[]=

In[]:= ScreenDollarIndices[ToCanonical[quantity /. ExpandStrengths]]

Out[]=

The final set of fields which we introduce are the multipliers
$\tensor{\lambda}{{}_{ijkl}}$ and $\tensor{\lambda}{{}_{ijk}}$ – these are
precisely the same shape as $\tensor{\mathcal{R}}{{}_{ijkl}}$ and
$\tensor{\mathcal{T}}{{}_{ijk}}$, and we write them

⬇

In[]:= quantity = {RLambda[-a, -b, -c, -d], RLambda[a, -a, -c, -d],
RLambda[-a, -b, -c, c], TLambda[a, -b, -c], TLambda[a, -b, b]}

Out[]=

In[]:= ScreenDollarIndices[ToCanonical[quantity]]

Out[]=

It should be emphasised again that rules such as ExpandStrengths, PADMActivate
and PiPToPi should not often be needed, and are subsumed under the
ToBasicForm[] command.

###  II.2 Irreducible decompositions

The Hamiltonian analysis invites decomposition into irresducible
representations of the Lorentz group ${\mathrm{SO}^{+}(1,3)}$, and (through
the ADM split) the special orthogonal group ${\mathrm{SO}(3)}$. The former is
useful also in the Lagrangian picture, since the Lagrangian formulation of a
theory is typically Lorentz-covariant, it is useful to split field
representations into blocks which transform only among themselves. In the
Hamiltonian case, the choice of slicing introduces a ‘preferred’ timelike
vector – this is not unique, so covariance is not ultimately lost – but the
symmetry in the context of the slicing is reduced to the spatial rotations
${\mathrm{SO}(3)}$. In the case of rotations, the irreps of tensor fields
correspond to states of definite spin and parity, allowing us to designate the
parts as $J^{P}$.

####  II.2.1 Lorentz group

We do not often encounter the Lorentz decomposition in the course of
Hamiltonian calculations, but it is implemented across the higher-rank
tensors, for example the ‘human-readable’ decomposition into familiar irreps
such as the _Weyl_

$\displaystyle\tensor{\mathcal{W}}{{}_{ijkl}}$
$\displaystyle\equiv\tensor{\mathcal{R}}{{}_{ijkl}}-\frac{1}{2}\big{(}\tensor{\eta}{{}_{ik}}\tensor{\mathcal{R}}{{}_{jl}}-\tensor{\eta}{{}_{il}}\tensor{\mathcal{R}}{{}_{jk}}-\tensor{\eta}{{}_{jk}}\tensor{\mathcal{R}}{{}_{il}}$
$\displaystyle\ \
+\tensor{\eta}{{}_{jl}}\tensor{\mathcal{R}}{{}_{ik}}\big{)}+\frac{1}{6}\mathcal{R},$
(13)

the _Ricci_ and the _Ricci scalar_

$\tensor{\mathcal{R}}{{}_{ij}}\equiv\tensor{\mathcal{R}}{{}^{l}_{ilj}},\quad\tensor{\mathcal{R}}{}\equiv\tensor{\mathcal{R}}{{}^{l}_{l}},$
(14)

and the _tensor_ torsion, defined as the remainder of
$\tensor{\mathcal{T}}{{}^{i}_{jk}}$ after removing the _vector_ (i.e. torsion
contraction) and _pseudovector_ components

$\tensor{\mathcal{T}}{{}_{i}}\equiv\tensor{\mathcal{T}}{{}^{l}_{il}},\quad\tensor{[}^{*}]{\mathcal{T}}{{}_{i}}\equiv\tensor{\epsilon}{{}_{ijkl}}\tensor{\mathcal{T}}{{}^{jkl}},$
(15)

– see e.g. [80, 45, 23]. These components may be accessed as follows

⬇

In[]:= ScreenDollarIndices[ToCanonical[R[-a, -b, -c, -d] /.
StrengthSO13Activate]]

Out[]=

In[]:= ScreenDollarIndices[ToCanonical[TLambda[-a, -b, -c] /.
StrengthLambdaSO13Activate]]

Out[]=

More commonly, we might need access to the complete, orthonormal
${\mathrm{SO}^{+}(1,3)}$ operators which appear in (7). These may be
associated with the couplings $\\{\tensor{\hat{\alpha}}{{}_{I}}\\}$,
$\\{\tensor{\hat{\beta}}{{}_{M}}\\}$, $\\{\tensor{\bar{\alpha}}{{}_{I}}\\}$,
$\\{\tensor{\bar{\beta}}{{}_{M}}\\}$ as follows, for the purposes of
constructing a Lagrangian555Note that _HiGGS_ does not define a Planck mass
${m_{\text{p}}}$: all scales are absorbed into the couplings Bet1, Bet2, Bet3,
which are really ${m_{\text{p}}}^{2}\tensor{\hat{\beta}}{{}_{1}}$, etc., and
likewise for the multiplier coefficients and Alp0.

⬇

In[]:= quantity = (Alp1*PR1[-i, -k, -l, -m, a, b, c, d] + Alp2*PR2[-i, -k, -l,
-m, a, b, c, d] + Alp3*PR3[-i, -k, -l, -m, a, b, c, d] + Alp4*PR4[-i, -k, -l,
-m, a, b, c, d] + Alp5*PR5[-i, -k, -l, -m, a, b, c, d] + Alp6*PR6[-i, -k, -l,
-m, a, b, c, d])*R[-a, -b, -c, -d]

Out[]=

In[]:= quantity = ScreenDollarIndices[ToCanonical[quantity /. PActivate /.
{Alp1 $\to$ 1, Alp2 $\to$ 1, Alp3 $\to$ 1, Alp4 $\to$ 1, Alp5 $\to$ 1, Alp6
$\to$ 1}]]

Out[]=

In[]:= ScreenDollarIndices[ContractMetric[ToCanonical[T[i, k, l]*(Bet1*PT1[-i,
-k, -l, a, b, c])*T[-a, -b, -c] + cAlp6*R[i, k, l, m]*PR6[-i, -k, -l, -m, a,
b, c, d]*R[-a, -b, -c, -d] + T[i, k, l]*(cBet2*PT2[-i, -k, -l, a, b, c] +
cBet3*PT3[-i, -k, -l, a, b, c])*TLambda[-a, -b, -c] /. PActivate]]]

Out[]=

From this we see how to access the
${\tensor{[}^{I}]{\hat{\mathcal{P}}}{{}_{ij}^{kl}{}_{nm}^{pq}}}$ and
${\tensor{[}^{M}]{\hat{\mathcal{P}}}{{}_{j}^{kl}{}_{m}^{pq}}}$ projections.

####  II.2.2 Rotation group

The ${\mathrm{SO}(3)}$ projections are similarly defined, but of course refer
to the vector $\tensor{n}{{}_{i}}$, not just the Lorentzian metric
$\tensor{\eta}{{}_{ij}}$. The parallel momenta can be decomposed in this way:
for the translational case

$\displaystyle\tensor{\hat{\pi}}{{}_{k\overline{l}}}$
$\displaystyle\tensor{\hat{\pi}}{{}_{\overline{kl}}}$
$\displaystyle=\frac{1}{3}\tensor{\eta}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}\IfEqCase{B0p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:placeholder}{}]+\IfEqCase{B1p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{kl}}{}]+\IfEqCase{B2p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{kl}}{}],$
(16a)

where the second term in (LABEL:traper) is the $1^{-}$ vector mode, and the
terms in (16a) are respectively the $0^{+}$ scalar, skew-symmetric $1^{+}$
vector and symmetric-traceless $2^{+}$ tensor modes. The rotational case is
similarly decomposed as

$\displaystyle\tensor{\hat{\pi}}{{}_{kl\overline{m}}}$
$\displaystyle=\tensor{\hat{\pi}}{{}_{\overline{klm}}}+2\tensor{n}{{}_{[k}}\tensor{\hat{\pi}}{{}_{\perp\overline{l}]\overline{m}}},$
(16qa) $\displaystyle\tensor{\hat{\pi}}{{}_{\perp\overline{kl}}}$
$\displaystyle=\frac{1}{3}\tensor{\eta}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}\IfEqCase{A0p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:placeholder}{}]+\IfEqCase{A1p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{kl}}{}]+\IfEqCase{A2p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{kl}}{}],$
(16qb) $\displaystyle\tensor{\hat{\pi}}{{}_{\overline{klm}}}$
$\displaystyle=\frac{1}{6}\tensor{\epsilon}{{}_{\IfEqCase{\overline{klm}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{klm}]}{}_{\perp}}\IfEqCase{A0m}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:placeholder}{}]+\IfEqCase{A1m}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{[\overline{k}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[[\overline{k}]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{[\overline{k}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[[\overline{k}]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{[\overline{k}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[[\overline{k}]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{[\overline{k}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[[\overline{k}]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{[\overline{k}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[[\overline{k}]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{[\overline{k}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[[\overline{k}]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{[\overline{k}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[[\overline{k}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:[\overline{k}}{}]\tensor{\eta}{{}_{\IfEqCase{{\overline{l}]\overline{m}}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[{\overline{l}]\overline{m}}]}}+\frac{4}{3}\IfEqCase{A2m}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{\overline{klm}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{klm}]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{\overline{klm}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{klm}]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{\overline{klm}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{klm}]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{klm}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{klm}]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{\overline{klm}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{klm}]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{\overline{klm}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{klm}]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{\overline{klm}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{klm}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{klm}}{}],$
(16qc)

where (16qb) are the $0^{+}$, $1^{+}$ and $2^{+}$ modes and (16qc) are the
$0^{-}$, $1^{-}$ and $2^{-}$ modes. Accordingly, we access these as follows

⬇

In[]:= ScreenDollarIndices[ToCanonical[{APiP[-a, -b, -c], BPiP[-a, -b]} /.
PiPToPiPO3]]

Out[]=

The parallel parts of the field strengths can also be decomposed, since they
are canonical, but we neglect the perpendicular parts which depend on the
unphysical fields $\tensor{b}{{}^{i}_{0}}$ and $\tensor{A}{{}^{ij}_{0}}$, and
also on non-canonical velocities. The field strength decomposition is

$\displaystyle\tensor{\mathcal{R}}{{}_{ijkl}}$
$\displaystyle=\tensor{\mathcal{R}}{{}_{ij\overline{kl}}}+2\tensor{n}{{}_{[k|}}\tensor{\mathcal{R}}{{}_{ij\perp|\overline{l}]}},$
(16ra) $\displaystyle\tensor{\mathcal{T}}{{}_{ikl}}$
$\displaystyle=\tensor{\mathcal{T}}{{}_{i\overline{kl}}}+2\tensor{n}{{}_{[k|}}\tensor{\mathcal{T}}{{}_{i\perp|\overline{l}]}}.$
(16rb)

We notice that $\tensor{\mathcal{R}}{{}_{ij\overline{kl}}}$ and
$\tensor{\mathcal{R}}{{}_{ij\perp\overline{l}}}$ both share all six $J^{P}$
representations present in the rotational momentum. In the parallel case,
these are denoted by the $0^{+}$ scalar
$\IfEqCase{A0p}{{A0p}{\tensor{\underline{\mathcal{R}}}{}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\perp\circ}}}{A1p}{\tensor{\underline{\mathcal{R}}}{{}_{[\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]]}}}{A1m}{\tensor{\mathcal{R}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2p}{\tensor{\underline{\mathcal{R}}}{{}_{\langle\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\rangle}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]$,
$1^{+}$ dual vector
$\IfEqCase{A1p}{{A0p}{\tensor{\underline{\mathcal{R}}}{}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\perp\circ}}}{A1p}{\tensor{\underline{\mathcal{R}}}{{}_{[\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]]}}}{A1m}{\tensor{\mathcal{R}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A2p}{\tensor{\underline{\mathcal{R}}}{{}_{\langle\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\rangle}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ij}}{}]$,
$2^{+}$ symmetric-traceless tensor
$\IfEqCase{A2p}{{A0p}{\tensor{\underline{\mathcal{R}}}{}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\perp\circ}}}{A1p}{\tensor{\underline{\mathcal{R}}}{{}_{[\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]]}}}{A1m}{\tensor{\mathcal{R}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A2p}{\tensor{\underline{\mathcal{R}}}{{}_{\langle\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\rangle}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ij}}{}]$,
$0^{-}$ pseudoscalar
$\IfEqCase{A0m}{{A0p}{\tensor{\underline{\mathcal{R}}}{}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\perp\circ}}}{A1p}{\tensor{\underline{\mathcal{R}}}{{}_{[\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]]}}}{A1m}{\tensor{\mathcal{R}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2p}{\tensor{\underline{\mathcal{R}}}{{}_{\langle\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\rangle}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]$,
$1^{-}$ vector
$\IfEqCase{A1m}{{A0p}{\tensor{\underline{\mathcal{R}}}{}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\perp\circ}}}{A1p}{\tensor{\underline{\mathcal{R}}}{{}_{[\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]]}}}{A1m}{\tensor{\mathcal{R}}{{}_{\perp\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]}}}{A2p}{\tensor{\underline{\mathcal{R}}}{{}_{\langle\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]\rangle}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\perp\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{i}}{}]$
and $2^{-}$ tensor
$\IfEqCase{A2m}{{A0p}{\tensor{\underline{\mathcal{R}}}{}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\perp\circ}}}{A1p}{\tensor{\underline{\mathcal{R}}}{{}_{[\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]]}}}{A1m}{\tensor{\mathcal{R}}{{}_{\perp\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]}}}{A2p}{\tensor{\underline{\mathcal{R}}}{{}_{\langle\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]\rangle}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\perp\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ijk}}{}]$.
In general, the ($\langle\cdot\rangle$) brackets indicate the symmetric-
traceless operation. The perpendicular irreps are denoted as the $0^{+}$
scalar
$\IfEqCase{A0p}{{A0p}{\tensor{\mathcal{R}}{{}_{\perp\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\circ\perp}}}{A1p}{\tensor{\mathcal{R}}{{}_{\perp[\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]]\perp}}}{A1m}{\tensor{\mathcal{R}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\perp}}}{A2p}{\tensor{\mathcal{R}}{{}_{\perp\langle\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\rangle\perp}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]$,
$1^{+}$ dual vector
$\IfEqCase{A1p}{{A0p}{\tensor{\mathcal{R}}{{}_{\perp\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\circ\perp}}}{A1p}{\tensor{\mathcal{R}}{{}_{\perp[\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]]\perp}}}{A1m}{\tensor{\mathcal{R}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\perp}}}{A2p}{\tensor{\mathcal{R}}{{}_{\perp\langle\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\rangle\perp}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ij}}{}]$,
$2^{+}$ symmetric-traceless tensor
$\IfEqCase{A2p}{{A0p}{\tensor{\mathcal{R}}{{}_{\perp\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\circ\perp}}}{A1p}{\tensor{\mathcal{R}}{{}_{\perp[\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]]\perp}}}{A1m}{\tensor{\mathcal{R}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\perp}}}{A2p}{\tensor{\mathcal{R}}{{}_{\perp\langle\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\rangle\perp}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ij}}{}]$,
$0^{-}$ pseudoscalar
$\IfEqCase{A0m}{{A0p}{\tensor{\mathcal{R}}{{}_{\perp\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\circ\perp}}}{A1p}{\tensor{\mathcal{R}}{{}_{\perp[\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]]\perp}}}{A1m}{\tensor{\mathcal{R}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\perp}}}{A2p}{\tensor{\mathcal{R}}{{}_{\perp\langle\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\rangle\perp}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]$,
$1^{-}$ vector
$\IfEqCase{A1m}{{A0p}{\tensor{\mathcal{R}}{{}_{\perp\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\circ\perp}}}{A1p}{\tensor{\mathcal{R}}{{}_{\perp[\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]]\perp}}}{A1m}{\tensor{\mathcal{R}}{{}_{\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]\perp}}}{A2p}{\tensor{\mathcal{R}}{{}_{\perp\langle\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]\rangle\perp}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{i}}{}]$
and $2^{-}$ tensor
$\IfEqCase{A2m}{{A0p}{\tensor{\mathcal{R}}{{}_{\perp\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\circ\perp}}}{A1p}{\tensor{\mathcal{R}}{{}_{\perp[\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]]\perp}}}{A1m}{\tensor{\mathcal{R}}{{}_{\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]\perp}}}{A2p}{\tensor{\mathcal{R}}{{}_{\perp\langle\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]\rangle\perp}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ijk}}{}]$.
The situation for the torsion is slightly different, because of the reduced
number of components. The parallel $\tensor{\mathcal{T}}{{}_{i\overline{kl}}}$
contains the $0^{-}$ pseudoscalar
$\IfEqCase{A0m}{{B1p}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{B1m}{\tensor{\overset{\rightharpoonup}{\mathcal{T}}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{T}}{}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{T}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]$,
$1^{+}$ dual vector
$\IfEqCase{B1p}{{B1p}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{B1m}{\tensor{\overset{\rightharpoonup}{\mathcal{T}}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{T}}{}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{T}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ij}}{}]$,
$1^{-}$ vector
$\IfEqCase{B1m}{{B1p}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]}}}{B1m}{\tensor{\overset{\rightharpoonup}{\mathcal{T}}}{{}_{\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{T}}{}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{T}}{{}_{\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{i}}{}]$
and $2^{-}$ tensor
$\IfEqCase{A2m}{{B1p}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]}}}{B1m}{\tensor{\overset{\rightharpoonup}{\mathcal{T}}}{{}_{\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{T}}{}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{T}}{{}_{\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ijk}}{}]$.
The perpendicular $\tensor{\mathcal{T}}{{}_{i\perp\overline{l}}}$ contains the
$0^{+}$ scalar
$\IfEqCase{B0p}{{B0p}{\tensor{\mathcal{T}}{{}^{\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}_{\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\perp}}}{B1p}{\tensor{\mathcal{T}}{{}_{[\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]]\perp}}}{B1m}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\perp}}}{B2p}{\tensor{\mathcal{T}}{{}_{\langle\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\rangle\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]$,
$1^{+}$ dual vector
$\IfEqCase{B1p}{{B0p}{\tensor{\mathcal{T}}{{}^{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\perp}}}{B1p}{\tensor{\mathcal{T}}{{}_{[\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]]\perp}}}{B1m}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\perp}}}{B2p}{\tensor{\mathcal{T}}{{}_{\langle\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\rangle\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ij}}{}]$,
$1^{-}$ vector
$\IfEqCase{B1m}{{B0p}{\tensor{\mathcal{T}}{{}^{\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]}_{\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]\perp}}}{B1p}{\tensor{\mathcal{T}}{{}_{[\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]]\perp}}}{B1m}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]\perp}}}{B2p}{\tensor{\mathcal{T}}{{}_{\langle\IfEqCase{\overline{i}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{i}]\rangle\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{i}}{}]$
and $2^{+}$ tensor
$\IfEqCase{B2p}{{B0p}{\tensor{\mathcal{T}}{{}^{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\perp}}}{B1p}{\tensor{\mathcal{T}}{{}_{[\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]]\perp}}}{B1m}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\perp}}}{B2p}{\tensor{\mathcal{T}}{{}_{\langle\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]\rangle\perp}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ij}}{}]$
– just as with the translational momentum.

In _HiGGS_ , only the canonical, parallel parts of these tensors are
decomposed. The multipliers share the same tensor structure, and both their
parallel and perpendicular parts are decomposed: this is because all
multiplier fields are assumed to be canonical. The resultant decomposition is

⬇

In[]:= quantity = ScreenDollarIndices[ToCanonical[{R[-a, -b, -c, -d],
RLambda[-a, -b, -c, -d], T[-a, -b, -c], TLambda[-a, -b, -c]} /.
StrengthDecompose /. StrengthLambdaDecompose]]

Out[]=

In[]:= Simplify[ScreenDollarIndices[ContractMetric[ToCanonical[quantity[[1]]
/. StrengthPToStrengthPO3]]]]

Out[]=

In[]:= Simplify[ScreenDollarIndices[ContractMetric[ToCanonical[quantity[[4]]
/. StrengthLambdaPToStrengthLambdaPO3 /.
StrengthLambdaPerpToStrengthLambdaPerpO3]]]]

Out[]=

Note that our typeset convention for the ${\mathrm{SO}(3)}$ multiplier irreps
will be simply to recycle the field strength expressions above, substituting
the symbols $\mathcal{R},\ \mathcal{T}\to\lambda$. This is a general rule
unless stated otherwise: i.e., tensors whose (Lorentz-invariant) index
structure are identical will use the same ${\mathrm{SO}(3)}$ irrep notation,
but a different symbol.

It is possible, if needed, to recover the explicit ‘human-readable’ projection
operators used in the above calculations, such as appear in
$\tensor{[}^{A}]{\hat{\pi}}{{}_{\acute{l}}}\equiv{\tensor{[}^{A}]{\smash{\check{\mathcal{P}}}}{{}_{\acute{l}}^{ij}{}_{\overline{k}}}}\tensor{\hat{\pi}}{{}_{ij}^{\overline{k}}}$
and
$\tensor{[}^{E}]{\hat{\pi}}{{}_{\acute{l}}}\equiv{\tensor{[}^{E}]{\smash{\check{\mathcal{P}}}}{{}_{\acute{l}}^{i}{}_{\overline{k}}}}\tensor{\hat{\pi}}{{}_{i}^{\overline{k}}}$,
where the $J^{P}$ sectors are represented by $A$ and $E$ indices according to
our conventions in [64]; the $\acute{l}$ accent denotes variable indices, and
these projections are defined as obtaining precisely the irreps in
LABEL:traper, 16a, 16qb and 16qc. To reflect the various conventions used in
the literature, these projections are sometimes defined differently for spin
representations contained within parts of the field strengths and momenta
whose tensor structures are identical – for example
$\tensor{\hat{\pi}}{{}_{\overline{ijk}}}$ and
$\tensor{\mathcal{T}}{{}_{\overline{ijk}}}$. We therefore use different symbol
names for momenta and field strength projections (or those of multipliers),
and find e.g. for the selection

⬇

In[]:= quantity = ScreenDollarIndices[ContractMetric[ToCanonical[{PB0p[e,
f]*PBPara[-e, -f, a, c]*BPiP[-a, -c], PB1m[-n, f]*PBPerp[-f, a, c]*BPiP[-a,
-c], PA2p[-n, -m, e, f]*PAPerp[-e, -f, a, b, c]*APiP[-a, -b, -c], PR0p[e, f,
g, h]*RP[-e, -f, -g, -h], PR0m[e, f, g]*RPerp[-e, -f, -g], PR1p[-n, -m, e, f,
g, h]*RLambdaP[-e, -f, -g, -h], PT1p[-a, -b, c, d]*TPerp[-c, -d], PT2m[-a, -b,
-c, d, e, f]*TP[-d, -e, -f]} /. PADMPiActivate]]]

Out[]=

In[]:= quantity = quantity /. PO3PiActivate /. PO3TActivate /. PO3RActivate /.
PiPToPiPO3 /. StrengthPToStrengthPO3 /. StrengthPerpToStrengthPerpO3 /.
StrengthLambdaPToStrengthLambdaPO3 /.
StrengthLambdaPerpToStrengthLambdaPerpO3; quantity =
Simplify[ScreenDollarIndices[ContractMetric[ToCanonical[quantity]]]]; quantity
= Simplify[ScreenDollarIndices[ContractMetric[ToCanonical[quantity]]]];

Out[]=

Note above that it is sometimes necessary to repeatedly apply routines in
order to recover the desired form. In prinicple, the _HiGGS_ commands are
constructed so as to be idempotent, but broadly in _Mathematica_ and _xAct_
repeated commands are sometimes helpful when multiple functions are nested.

###  II.3 Derivatives

While the gauge-fixed Minkowskian setup initially makes for easy development,
it sometimes means that we have to ‘reinvent the wheel’ in order to access
machinery for which there is already a very sophisticated implementation in
_xAct_. A clear example of this is given by the way in which _HiGGS_ handles
gauge-covariant derivatives in the Poincaré gauge theory. Generalising from
(5), we recall from [73, 14, 74, 8] that for some matter field $\varphi$ we
have

$\tensor{\mathcal{D}}{{}_{i}}\varphi\equiv\tensor{h}{{}_{i}^{\mu}}\Big{(}\tensor{\partial}{{}_{\mu}}+\frac{1}{2}\tensor{A}{{}^{kl}_{\mu}}\tensor{\Sigma}{{}_{kl}}\cdot\Big{)}\varphi,\quad\tensor{D}{{}_{\mu}}\varphi\equiv\tensor{b}{{}^{i}_{\mu}}\tensor{\mathcal{D}}{{}_{i}}\varphi,$
(16s)

where $\tensor{\Sigma}{{}_{ij}}$ are the Lorentz group generators specific to
the representation of $\varphi$. This construction could be implemented using
the _xAct_ command DefCovD[], in such a way that the
$\tensor{\Sigma}{{}_{ij}}$ generators are automatically calculated for
tensorial $\varphi$ representations of the kind that arise in the Hamiltonian
analysis of the matter-free theory. In particular, the derivative
$\tensor{D}{{}_{\mu}}\varphi$ is geometrically interpreted as
$\tensor{\nabla}{{}_{\mu}}\varphi$ as it appears in (3). Rather than following
this route, _HiGGS_ defines a pair of first derivatives (broadly corresponding
to the two definitions in (16s)) for every canonical quantity which might be
of interest. This process is not very efficient or flexible, but it is
sufficient when explicit covariant derivatives are scarce. Indeed, the most
common occurance of the gauge covariant derivative is through its commutator

$2\tensor{\mathcal{D}}{{}_{[i}}\tensor{\mathcal{D}}{{}_{j]}}\varphi=\Big{(}\frac{1}{2}\tensor{\mathcal{R}}{{}^{kl}_{ij}}\tensor{\Sigma}{{}_{kl}}\cdot-\tensor{\mathcal{T}}{{}^{k}_{ij}}\tensor{\mathcal{D}}{{}_{k}}\Big{)}\varphi,$
(16t)

in the form of the field strengths: as we saw in Section II.1, these have a
_separate_ implementation. Explicit gradients will tend to arise mostly at the
_end_ of the calculations for which _HiGGS_ is designed. Gradients of the
field strengths cannot usually be fed back into the algorithm anyway, since
they would require an implementation of the second order Euler–Lagrange
equations in order to be processed.

Based on the understanding that we are always dealing with arbitrarily-indexed
(possibly mixed) tensorial representations, _HiGGS_ ’ two preferred forms of
the derivative are
$\tensor{D}{{}_{\mu}}\tensor{\varphi}{{}_{\acute{u}\acute{\mu}}}$ and
$\tensor{*}{\delta}{{}^{\acute{\overline{u}}}_{\acute{\overline{v}}}}\tensor{*}{\delta}{{}^{\acute{\overline{w}}}_{\acute{\overline{s}}}}\tensor{h}{{}_{\acute{w}}^{\acute{\mu}}}\tensor{\mathcal{D}}{{}_{\overline{k}}}\tensor{\varphi}{{}_{\acute{u}\acute{\mu}}}$.
The former is straightforward666Note that we use an accent to indicate an
arbitrary number of indices, following from [81, 64]., and the latter is the
parallel projection of the gradient on all indices. Picking a couple of
${\mathrm{SO}(3)}$ irreps at random, we find for example

⬇

In[]:= quantity = {DPiPA2m[-z, -a, -b, -c], CD[-z][PiPA2m[-a, -b, -c]],
DRP2p[-z, -a, -b], CD[-z][RP2p[-a, -b]]}

Out[]=

In[]:= quantity = ScreenDollarIndices[ContractMetric[ToCanonical[quantity /.
DRPDeactivate /. DPiPDeactivate]]]

Out[]=

In[]:= quantity = ScreenDollarIndices[ContractMetric[ToCanonical[quantity /.
DRPActivate /. DPiPActivate]]]

Out[]=

One cannot generally pass from
$\tensor{D}{{}_{\mu}}\tensor{\varphi}{{}_{\acute{u}\acute{\mu}}}$ to a
parallel derivative, unless only the spacelike indices are involved, e.g.
$\tensor{D}{{}_{\alpha}}\tensor{\varphi}{{}_{\acute{u}\acute{\mu}}}$. Since
_HiGGS_ tries to achive parallel derivatives wherever possible, this serves as
one of the internal checks on the canonical status of a quantity, making sure
that any unacceptable time derivatives would appear explicitally in the
output. Accordingly, continuing from above

⬇

In[]:= quantity = ScreenDollarIndices[ContractMetric[ToCanonical[(G3[-y, z]*#1
& ) /@ quantity[[{1, 3}]]]]]

Out[]=

In[]:= quantity = ScreenDollarIndices[ContractMetric[ToCanonical[quantity /.
DpRPActivate /. DpPiPActivate]]]

Out[]=

In[]:= quantity = ScreenDollarIndices[ContractMetric[ToCanonical[quantity /.
DpRPDeactivate /. DpPiPDeactivate]]]

Out[]=

We see in the penultimate expression above the two parallel derivatives. If an
expression containing gradients is covariant, velocity-independent and
parallel, then the various other terms should cancel among themselves. As we
have mentioned, covariance also requires that gauge fields do not explicitally
appear, but are implicit in covariant quantities. Overall, the work of
packaging an expression into a canonical, covariant form, parallel if possible
and free from unphysical fields, is done by the ToNesterForm[] command, which
we introduce in Section II.4.

As a final comment on the use of derivatives, we recall that the formulae Eqs.
6a and 6b imply a pair of Bianchi identities [80, 8] as follows

$\displaystyle\tensor{\epsilon}{{}^{\rho\mu\lambda\nu}}\tensor{D}{{}_{\mu}}\left(\tensor{b}{{}^{i}_{\lambda}}\tensor{b}{{}^{j}_{\nu}}\tensor{\mathcal{T}}{{}^{s}_{ij}}\right)$
$\displaystyle\equiv\tensor{\epsilon}{{}^{\rho\mu\lambda\nu}}\tensor{b}{{}^{k}_{\mu}}\tensor{b}{{}^{i}_{\lambda}}\tensor{b}{{}^{j}_{\nu}}\tensor{\mathcal{R}}{{}^{s}_{kij}},$
(16ua)
$\displaystyle\tensor{\epsilon}{{}^{\rho\mu\lambda\nu}}\tensor{D}{{}_{\lambda}}\left(\tensor{b}{{}^{k}_{\mu}}\tensor{b}{{}^{l}_{\nu}}\tensor{\mathcal{R}}{{}^{ij}_{kl}}\right)$
$\displaystyle\equiv 0.$ (16ub)

These can be verified using the tools already introduced, but the process will
be easier with the ToBasicForm[] and ToNesterForm[] commands.

###  II.4 Low-level functions

Throughout Sections II.1 and II.2 we have used many rules, such as
DpRPDeactivate, DPiPActivate, etc., which — whilst they are not confined to
xAct‘HiGGS‘Private‘ and so are available to the user — should not often be
needed within a _HiGGS_ science session. Instead these rules are wrapped into
two ‘official’ functions: ToNesterForm[], which strives to collect
expressions777The chosen naming refers to the fact that the irrep conventions
and notation of collected expressions tends to align most closely with those
of a collection of articles – very useful during development – for which
Nester is a common author, see e.g. [76, 54, 45, 23, 82, 83, 84, 85, 86, 87,
44]., and ToBasicForm[], which expands them. Naturally ToNesterForm[] is the
more complicated of the two. It is, in some sense, the extension of the
ToCanonical[] command from _xAct_ into _HiGGS_.

####  II.4.1 Module: ‘ToBasicForm’

We begin by breaking some expressions which we know to be covariant, using
ToBasicForm[]

⬇

In[]:= quantity = ToBasicForm[{T[i, -j, -k], PiPB0p[], PiPA1p[-i, -j]}]

Out[]=

We see that the results are expressed in terms of the bare gauge fields, non-
parallel momenta, independent ADM quantities and coordinate derivatives. If a
Poisson bracket were to be manually evaluated between the fields in quantity,
it would first be necessary to perform the expansion provided by
ToBasicForm[], before calculating variational derivatives and multiplying. For
this reason, the PoissonBracket[] command which we introduce shortly relies on
ToBasicForm[] as an initial step. In the case above, all the brackets would be
relatively straightforward to evaluate by hand: to see why the strong coupling
problem demands an implementation such as _HiGGS_ , let us break open a simple
momentum gradient

⬇

In[]:= ToBasicForm[DpPiPB1m[-k, -i]]

Out[]=

The gradient of a more complicated irrep such as DpPiPA2m[-i,-j,-k] would span
many pages after an application of ToBasicForm[], rendering manual evaluation
of brackets impractical, yet we will see in Section III that _there is nothing
to stop these terms arising in the analysis_.

Now is a convenient time to verify the second Bianchi identity in (16ub).
Setting up the derivative, and using the gauge-fixed
$\tensor{\epsilon}{{}^{\mu\nu\sigma\lambda}}$ which follows from
$\tensor{\gamma}{{}_{\mu\nu}}$ – epsilonG[a,b,c,d] – we find

⬇

In[]:= quantity2 = ToBasicForm[epsilonG[r, m, l, n]*(CD[-l][B[k, -m]*B[q,
-n]*R[i, j, -k, -q]] + A[i, -x, -l]*B[k, -m]*B[q, -n]*R[x, j, -k, -q] + A[j,
-x, -l]*B[k, -m]*B[q, -n]*R[i, x, -k, -q])];

Out[]=

Evidently, this quantity vanishes, and so what we are seeing is a side-effect
of the _HiGGS_ geometric setup: _xAct_ does not know that CD[-i][] is the
partial derivative, so we need a final step

⬇

In[]:= quantity2 = (ToCanonical[(1/2)*(#1 + CommuteCovDs[#1, CD, {l, m}])] &
)[quantity2];

Out[]=

####  II.4.2 Module: ‘ToNesterForm’

Let us now undo the breaking, using ToNesterForm[]. In general, much of the
functionality of ToNesterForm[] has to do with fully incorporating the known
primary and secondary constraints of the theory during the course of
simplification – so as to leave no extra conditions for the user to worry
about. Before a theory has been defined (see Section II.5) using DefTheory[],
we will have to suppress this activity by passing the option
"ToShell"$\to$False, yielding

⬇

In[]:= quantity = ToNesterForm[quantity, "ToShell" $\to$ False]

Out[]=

Note that whilst the results _are_ covariant, we do not get back the same form
from ToNesterForm[] as that which was provided to ToBasicForm[], instead we
recover the ${\mathrm{SO}(3)}$ expansion. The reason for this is that
reducible quantities such as T[-a,-b,-c] are usually quickly broken up during
the course of the Hamiltonian analysis: the output of ToNesterForm[] is tuned
so as to be useful in that context. Moreover, we note that this operation will
not now work in reverse, since input such as TP2m[-a,-b,-c]//ToBasicForm will
not return a broken expression. There is no special reason behind this:
ToBasicForm[] is not a sophisticated function, it essentially imposes a list
of internal rules on its argument, and the expansion of ${\mathrm{SO}(3)}$
field strength irreps, as with those of momenta, would be perfectly
straightforward to implement in the source if and when needed.

When ToNesterForm[] is passed a non-covarint quantity, it is unable to return
a covariant result. Nonetheless, it tries, returning for the innocuous
gradient $\tensor{\partial}{{}_{\mu}}\tensor{b}{{}^{i}_{\nu}}$

⬇

In[]:= ToNesterForm[CD[-i][B[-a, -b]], "ToShell" $\to$ False]

Out[]=

As with this case, it is usually easy to identify nonphysical expressions
which indicate a human error, through the appearance of bare gauge fields. If
needed, this covariance check is easy to automate through an output search
with the _Mathematica_ Head[] function for an unwanted _HiGGS_ quantity, such
as A or B. We can also observe in the output above part of the route taken by
ToNesterForm[]. Gauge field gradients are converted, where possible, to field
strengths and covariant derivatives. The residual spin connection terms are
then extensively manipulated in an attempt to cancel them. Leftover asymmetric
derivatives of the translational gauge field can sometimes be ascribed to
covariant quantities through (10), for example the following gradient cannot
be expressed through the torsion alone

⬇

In[]:= ToNesterForm[PPara[-y, v]*H[-v, w]*G3[-w, m]*PPara[-q, b1]*H[-b1,
a1]*G3[-a1, l]*(G3[-l, n]*V[-k]*CD[-m][B[k, -n]] + G3[-l, n]*B[j, -n]*A[k, -j,
-m]*V[-k]) /. PADMActivate, "ToShell" $\to$ False]

Out[]=

Note that the above result could equally be written as the single term
-DpV[-q,-y], but ToNesterForm[] takes the opportunity to separate out the
antisymmetric part.

Before moving on, we return to verify the first Bianchi identity in (16ub). We
can access the canonical (i.e. velocity-independent) part of this identity by
projecting Eq. 16ub with $\tensor{b}{{}^{\perp}_{\mu}}$, or equivalently using
in place of $\tensor{\epsilon}{{}^{\mu\nu\lambda\sigma}}$ the foliation
equivalent
$\tensor{\epsilon}{{}^{\IfEqCase{\overline{ijk}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ijk}]}{}^{\perp}}$,
which is given in _HiGGS_ by Eps[i,j,k],

⬇

In[]:= quantity = ToBasicForm[Eps[u, v, w]*H[-u, m]*H[-v, l]*H[-w,
n]*(CD[-m][B[i, -l]*B[j, -n]*T[s, -i, -j]] + A[s, -x, -m]*B[i, -l]*B[j,
-n]*T[x, -i, -j]) /. PADMActivate];

Out[]=

In[]:= quantity = (ToCanonical[V[-r]*(1/2)*(#1 + CommuteCovDs[#1, CD, {j,
l}])] & )[quantity];

Out[]=

In[]:= quantity = ToNesterForm[quantity, "ToShell" $\to$ False];

Out[]=

The output here is then equal to the ${\mathrm{SO}(3)}$ irreps contained
within
$\tensor{\epsilon}{{}^{\IfEqCase{\overline{kij}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kij}]}{}^{\perp}}\tensor{\mathcal{R}}{{}^{s}_{\overline{kij}}}$,
as expected.

####  II.4.3 Module: ‘PoissonBracket’

The PoissonBracket[] command is the third ‘official’ function provided by the
_HiGGS_ package. The Poisson bracket appearing in this article is defined for
general functionals $\mathcal{A}$ and $\mathcal{B}$ of the gravitational
fields and their conjugate momenta

$\displaystyle\Big{\\{}\mathcal{A},\mathcal{B}\Big{\\}}\equiv\int\mathrm{d}^{3}x\Bigg{[}\frac{\delta\mathcal{A}}{\delta\tensor{b}{{}^{i}_{\mu}}}\frac{\delta\mathcal{B}}{\delta\tensor{\pi}{{}_{i}^{\mu}}}+\frac{\delta\mathcal{A}}{\delta\tensor{A}{{}^{ij}_{\mu}}}\frac{\delta\mathcal{B}}{\delta\tensor{\pi}{{}_{ij}^{\mu}}}$
(16v)
$\displaystyle-\frac{\delta\mathcal{A}}{\delta\tensor{\pi}{{}_{i}^{\mu}}}\frac{\delta\mathcal{B}}{\delta\tensor{b}{{}^{i}_{\mu}}}-\frac{\delta\mathcal{A}}{\delta\tensor{\pi}{{}_{ij}^{\mu}}}\frac{\delta\mathcal{B}}{\delta\tensor{A}{{}^{ij}_{\mu}}}\Bigg{]},$

with a natural extension of the formula when multiplier fields are admitted.
The formula (16v) may appear no more daunting than a commonplace action
variation, but in practice $\mathcal{A}$ and $\mathcal{B}$ are frequently
_local tensors_ rather than nonlocal scalars. Locality signifies that the
underlying functionals contain Dirac distribuions, themselves subject to the
total derivatives of the generalised Euler–Lagrange equations. The full
ramifications of covariantly removing these Dirac gradients are detailed in
[14, 64], and some special cases are discussed in electrodynamics on the
lightcone [88] and noncritical string theory [89]. Currently, _HiGGS_ is able
to accommodate the _first order_ Euler–Lagrange formalism in PoissonBracket[].
First order brackets, evaluated by inserting the spatial dependence into
(16v), produce four terms of the form

$\displaystyle\Big{\\{}\mathcal{A}(\bm{x}_{1}),$ $\displaystyle\
\mathcal{B}(\bm{x}_{2})\Big{\\}}\equiv$
$\displaystyle\int\mathrm{d}^{3}x\big{[}$ $\displaystyle
J_{1}(\bm{x})\delta^{3}(\bm{x}-\bm{x}_{1})\delta^{3}(\bm{x}-\bm{x}_{2})$
$\displaystyle+\tensor{J}{{}_{2}^{\alpha}}(\bm{x})\delta^{3}(\bm{x}-\bm{x}_{1})\tensor{\partial}{{}_{\alpha}}\delta^{3}(\bm{x}-\bm{x}_{2})$
$\displaystyle+\tensor{J}{{}_{3}^{\alpha}}(\bm{x})\tensor{\partial}{{}_{\alpha}}\delta^{3}(\bm{x}-\bm{x}_{1})\delta^{3}(\bm{x}-\bm{x}_{2})$
$\displaystyle+\tensor{J}{{}_{4}^{\alpha\beta}}(\bm{x})\tensor{\partial}{{}_{\alpha}}\delta^{3}(\bm{x}-\bm{x}_{1})\tensor{\partial}{{}_{\beta}}\delta^{3}(\bm{x}-\bm{x}_{2})\big{]},$
(16w)

where the $J_{1}$, $\tensor{J}{{}_{2}^{\alpha}}$,
$\tensor{J}{{}_{3}^{\alpha}}$ and $\tensor{J}{{}_{4}^{\alpha\beta}}$ can be
determined by certain formulae. Note that by our conventions in [13, 14, 64],
we will in future denote by $\delta^{3}$ the equal-time Dirac function
${\delta^{3}(\bm{x}_{1}-\bm{x}_{2})}$. Without any special instructions,
PoissonBracket[] returns a List of the four Dirac coefficients in (16w). This
behaviour is tied into calls to PoissonBracket[] from within Velocity[], which
we introduce in Section II.5. Note that PoissonBracket[] contains calls to
ToNesterForm[], and so works to exhaust transformations which can be applied
to the output by virtue of the known primary and secondary constrints. In this
sense, it depends on the theory introduced by DefTheory[] and so for the time
being we must again pass the option "ToShell"$\to$False. We begin with a very
simple bracket, whose output can be understood in terms of our previous
ToNesterForm[] result for DpV[-a,-b]

⬇

In[]:= PoissonBracket[PiPB2p[-a, -b], TP1p[-c, -d], "ToShell" $\to$ False];

Out[]=

The bracket is not ‘surficial’, in the sense that the latter three entries
vanish and the nonvanishing part of the bracket is a compact, covariant
expression. If the latter three coefficients are nonvanishing however,
explicit covariance in this expression will be lost, as we can see by
modifying the previous example

⬇

In[]:= PoissonBracket[PiPB2p[-a, -b], TP1m[-c], "ToShell" $\to$ False];

Out[]=

In such ‘surficial’ cases the default output of PoissonBracket[] will not be
helpful for visual inspection. To resolve this we can recall, again from [14],
that (16w) can be alternatively expressed as

$\displaystyle\int$
$\displaystyle\mathrm{d}^{3}x_{2}\big{\\{}{\tensor{\mathcal{A}}{{}_{\acute{u}}}}(\bm{x}_{1}),{\tensor{\mathcal{B}}{{}_{\acute{v}}}}(\bm{x}_{2})\big{\\}}{\tensor{\mathcal{C}}{{}^{\acute{v}}}}(\bm{x}_{2})\equiv\tensor{\mathcal{J}}{{}_{1}{}_{\acute{v}}}(\bm{x}_{1}){\tensor{\mathcal{C}}{{}^{\acute{v}}}}(\bm{x}_{1})$
(16x)
$\displaystyle+\tensor{\mathcal{J}}{{}_{2}{}_{\acute{v}}^{\alpha}}(\bm{x}_{1})\tensor{D}{{}_{\alpha}}{\tensor{\mathcal{C}}{{}^{\acute{v}}}}(\bm{x}_{1})+\tensor{\mathcal{J}}{{}_{3}{}_{\acute{v}}^{\alpha\beta}}(\bm{x}_{1})\tensor{D}{{}_{\alpha}}\tensor{D}{{}_{\beta}}{\tensor{\mathcal{C}}{{}^{\acute{v}}}}(\bm{x}_{1}).$

The three-component List output corresponding to (16x) can be produced by
passing the option "Surficial"$\to$True. Trying again with this option, we
obtain

⬇

In[]:= PoissonBracket[PiPB2p[-i, -j], TP1m[-l], "ToShell" $\to$ False,
"Surficial" $\to$ True];

Out[]=

We see that the result is indeed covariant, and fairly simple. We will come
back to this ‘surficial’ case in Section III.2, where we attempt to recover
the historical results in [45, 23].

###  II.5 High-level functions

As we mentioned in Section II.4, much of the work done by ToNesterForm[] has
to do with the imposition of the theory shell so as to simplify the argument.
The particular shell used is not specifically that of primary vs secondary
constraints, but it is restricted to the constraints of which we have prior
knowledge from the literature, and does not include new constraints discovered
in the course of a _HiGGS_ session. In particular, we rely on the so-called
_if-constraint_ structure which was discovered by Blagojević and Nikolić in
[60]. Depending on the Lagrangian parameters in (7), the number and type of
primary constraints may be radically different: these contingent primaries are
called primary if-constraints (PiC). There are similarly secondary (SiC) and
tertiary (TiC) quantities, etc. Returning to [64], the PiCs of the theory (7)
take the form

$\displaystyle\tensor{[}^{A}]{\varphi}{{}_{\acute{v}}}$
$\displaystyle\equiv\frac{1}{J}\tensor{[}^{A}]{\hat{\pi}}{{}_{\acute{v}}}+2\tensor{\hat{\alpha}}{{}_{0}}{m_{\text{p}}}^{2}{\tensor{[}^{A}]{\smash{\check{\mathcal{P}}}}{{}_{\acute{v}\perp\overline{k}}^{\overline{k}}}}-8\tensor{*}{\bar{\alpha}}{{}_{A}^{\perp\perp}}{\tensor{[}^{A}]{\smash{\check{\mathcal{P}}}}{{}_{\acute{v}}{}_{jk}^{\overline{m}}}}\tensor{\lambda}{{}^{jk}_{\perp\overline{m}}}$
$\displaystyle\ \ \
-4{\tensor{[}^{A}]{\smash{\check{\mathcal{P}}}}{{}_{\acute{v}}{}_{jk}^{\overline{lm}}}}\Big{(}\tensor{*}{\bar{\alpha}}{{}_{A}^{\perp\parallel}}\tensor{\lambda}{{}^{jk}_{\overline{lm}}}+2{\tensor{*}{\hat{\alpha}}{{}_{A}^{\perp\parallel}}}\tensor{\mathcal{R}}{{}^{jk}_{\overline{lm}}}\Big{)},$
(16ya) $\displaystyle\tensor{[}^{E}]{\varphi}{{}_{\acute{v}}}$
$\displaystyle\equiv\frac{1}{J}\tensor{[}^{E}]{\hat{\pi}}{{}_{\acute{v}}}-4{m_{\text{p}}}^{2}\tensor{*}{\bar{\beta}}{{}_{E}^{\perp\perp}}{\tensor{[}^{E}]{\smash{\check{\mathcal{P}}}}{{}_{\acute{v}}{}_{j}^{\overline{m}}}}\tensor{\lambda}{{}^{j}_{\perp\overline{m}}}$
$\displaystyle\ \ \
-2{m_{\text{p}}}^{2}{\tensor{[}^{E}]{\smash{\check{\mathcal{P}}}}{{}_{\acute{v}}{}_{j}^{\overline{lm}}}}\Big{(}\tensor{*}{\bar{\beta}}{{}_{E}^{\perp\parallel}}\tensor{\lambda}{{}^{j}_{\overline{lm}}}+2{\tensor{*}{\hat{\beta}}{{}_{E}^{\perp\parallel}}}\tensor{\mathcal{T}}{{}^{j}_{\overline{lm}}}\Big{)},$
(16yb)

where we defined e.g.
${{\tensor{*}{\hat{\alpha}}{{}_{A}^{\parallel\parallel}}}\equiv\sum_{I}{\tensor{*}{M}{{}_{AI}^{\parallel\parallel}}}\tensor{\hat{\alpha}}{{}_{I}}}$,
with the matrix
${\tensor{[}^{A}]{\smash{\check{\mathcal{P}}}}{{}_{\acute{p}}^{lm}{}_{\overline{nq}}}}{\tensor{[}^{I}]{\hat{\mathcal{P}}}{{}_{lm}^{\overline{nq}}{}_{ij}^{\overline{rk}}}}\equiv{\tensor{*}{M}{{}_{AI}^{\parallel\parallel}}}{\tensor{[}^{A}]{\smash{\check{\mathcal{P}}}}{{}_{\acute{p}}{}_{ij}^{\overline{rk}}}}$.
In the handling of PiCs, it is extremely useful to refer to the functions

$\displaystyle\mu(x)\equiv\left\\{\begin{array}[]{lr}x^{-1},&\text{for }x\neq
0\\\ 0,&\text{for }x=0,\end{array}\right.\quad\nu(x)\equiv
1-|\text{sgn}(\mu(x))|,$ (16ab)

and we note that the PiC functions defined in Eqs. 16ya and 16yb are only
constrained when ${\nu({\tensor{*}{\hat{\alpha}}{{}_{A}^{\perp\perp}}})=1}$ or
${\nu({\tensor{*}{\hat{\beta}}{{}_{E}^{\perp\perp}}})=1}$. The structure of
Eqs. 16ya and 16yb is quite useful in that it allows momenta to substituted
for parallel field strengths. These substitutions are among those performed
every time ToNesterForm[] is called, unless the option "ToShell"$\to$False is
passed as above in Section II.4.

####  II.5.1 Module: ‘DefTheory’

In order to discover these PiCs, we must first set up the shell using
DefTheory[]. Let us consider the simple example of Einstein–Cartan theory,
i.e. the substantial restriction of (7) to the simple Einstein–Hilbert term

$L_{\text{G}}=-\frac{1}{2}\tensor{\hat{\alpha}}{{}_{0}}{m_{\text{p}}}^{2}\mathcal{R}.$
(16ac)

We implement the theory (16ac) by passing to the DefTheory[] command the
system of equations which deactivates all the
$\\{\tensor{\hat{\alpha}}{{}_{A}}\\}$, $\\{\tensor{\hat{\beta}}{{}_{E}}\\}$,
$\\{\tensor{\bar{\alpha}}{{}_{A}}\\}$, $\\{\tensor{\bar{\beta}}{{}_{E}}\\}$,
while leaving $\tensor{\hat{\alpha}}{{}_{0}}$ untouched. We want to store our
knowledge of the shell once it has been obtained, and so we pass the label
"Export"$\to$"EinsteinCartan", which will be used to construct a filename. The
input is

⬇

In[]:= DefTheory[{Alp1 == 0, Alp2 == 0, Alp3 == 0, Alp4 == 0, Alp5 == 0, Alp6
== 0, Bet1 == 0, Bet2 == 0, Bet3 == 0, cAlp1 == 0, cAlp2 == 0, cAlp3 == 0,
cAlp4 == 0, cAlp5 == 0, cAlp6 == 0, cBet1 == 0, cBet2 == 0, cBet3 == 0},
"Export" $\to$ "EinsteinCartan", "Order" $\to$ Infinity];

Out[]=

** DefTheory: Found the following primary if-constraints:

** DefTheory: Found the following secondary perpendicular if-constraints:

** DefTheory: Found the following secondary parallel if-constraints:

** DefTheory: Found the following secondary singular if-constraints:

** DefTheory: The super-Hamiltonian is:

** DefTheory: The linear super-momentum is:

** DefTheory: The 1- part of the angular super-momentum is:

** DefTheory: The 1+ part of the angular super-momentum is:

** DefTheory: Exporting the binary at svy/EinsteinCartan.thr.mx

In the output above we can see the listing of the PiCs. It is clear from
(16ac) and also from the form of the constraints in Eqs. 16ya and 16yb, why we
get this specific list. All the coefficients associated with the
${\mathrm{SO}^{+}(1,3)}$ irreps of $\tensor{\mathcal{R}}{{}_{ijkl}}$ and
$\tensor{\mathcal{T}}{{}_{ijk}}$ are vanishing, and so all the PiC functions
become constraints. Moreover, these constraints are very simple in their form:
they are all pure momenta, with the single exception being in the case of the
$0^{+}$ roton constraint, which contains a constant

$\IfEqCase{A0p}{{B0p}{\varphi}{B1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{B1m}{\tensor{\varphi}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{B2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A0p}{\tensor{\varphi}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\varphi}{}}{A1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\varphi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2m}{\tensor{[}^{\text{T}}]{\varphi}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]=3\tensor{\hat{\alpha}}{{}_{0}}{m_{\text{p}}}^{2}+\frac{1}{J}\IfEqCase{A0p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]\approx
0.$ (16ad)

Following this listing of the PiCs, there are references to perpendicular,
parallel and singular SiCs, none of which are present in the Einstein–Cartan
theory. We will return to these in Section III.3, but for now we note that
they follow from the imposition of multipliers in (7), and their presence is
fully understood in [64], just as the presence of the PiCs of the basic
Poincaré gauge theory is understood in [60, 62]. This is the intended scope of
DefTheory[]: to elucidate not only the primary constraints, but all of that
part of the constraint structure which is already known from the literature.
Every if-constraint identified by DefTheory[] is used during the DefTheory[]
call in the construction of very large internal rule sets
$StrengthPShellToStrengthPO3, $PiPShellToPiPPO3, $TheoryCDPiPToCDPiPO3 and
$TheoryPiPToPiPO3. These rules are applied – by default – during ToNesterForm
calls. Most of the if-constraints that can arise, as detailed in [64], are of
the form $\frac{1}{J}\tensor{[}^{A}]{\hat{\pi}}{{}_{\acute{u}}}+\cdots\approx
0$ or $\frac{1}{J}\tensor{[}^{E}]{\hat{\pi}}{{}_{\acute{u}}}+\cdots\approx 0$,
and so the shell is defined by _replacing_ all possible instances of the
momenta in favor of other quantities. There are also the ‘parallel’ SiCs,
which deactivate irreps within the canonical parts of the field strength
tensors in Eqs. 16ra and 16rb – these are straightforwardly implemented.

Following the listing of the if-constraints, the output above details the
structure of the canonical Hamiltonian, which from [64] is written

$\tensor{\mathcal{H}}{{}_{\text{C}}}\equiv
N\tensor{\mathcal{H}}{{}_{\perp}}+\tensor{N}{{}^{\alpha}}\tensor{\mathcal{H}}{{}_{\alpha}}-\frac{1}{2}\tensor{A}{{}^{ij}_{0}}\tensor{\mathcal{H}}{{}_{ij}}+\tensor{\partial}{{}_{\alpha}}\tensor{\mathscr{D}}{{}^{\alpha}}.$
(16ae)

This _Dirac_ form [61, 90] is useful because it expresses part of the
Hamiltonian as a linear combination of the nonphysical fields $N$,
$\tensor{N}{{}^{\alpha}}$ and $\tensor{A}{{}^{ij}_{0}}$. The physical (and
canonical) coefficients of these undetermined fields are888Note that
$\tensor{\mathscr{D}}{{}^{\alpha}}\equiv\tensor{b}{{}^{i}_{0}}\tensor{\pi}{{}_{i}^{\alpha}}+\frac{1}{2}\tensor{A}{{}^{ij}_{0}}\tensor{\pi}{{}_{ij}^{\alpha}}$.

$\displaystyle\tensor{\mathcal{H}}{{}_{\perp}}$
$\displaystyle\equiv\tensor{\hat{\pi}}{{}_{i}^{\overline{k}}}\tensor{\mathcal{T}}{{}^{i}_{\perp\overline{k}}}+\frac{1}{2}\tensor{\hat{\pi}}{{}_{ij}^{\overline{k}}}\tensor{\mathcal{R}}{{}^{ij}_{\perp\overline{k}}}-JL_{\text{G}}-\tensor{n}{{}^{k}}\tensor{D}{{}_{\alpha}}\tensor{\pi}{{}_{k}^{\alpha}},$
(16afa) $\displaystyle\tensor{\mathcal{H}}{{}_{\alpha}}$
$\displaystyle\equiv\tensor{\pi}{{}_{i}^{\beta}}\tensor{T}{{}^{i}_{\alpha\beta}}+\frac{1}{2}\tensor{\pi}{{}_{ij}^{\beta}}\tensor{R}{{}^{ij}_{\alpha\beta}}-\tensor{b}{{}^{k}_{\alpha}}\tensor{D}{{}_{\beta}}\tensor{\pi}{{}_{k}^{\beta}},$
(16afb) $\displaystyle\tensor{\mathcal{H}}{{}_{ij}}$ $\displaystyle\equiv
2\tensor{\pi}{{}_{[i}^{\alpha}}\tensor{b}{{}_{j]\alpha}}+\tensor{D}{{}_{\alpha}}\tensor{\pi}{{}_{ij}^{\alpha}},$
(16afc)

and they form the ‘sure’ secondary FC constraints (sSFCs)

$\tensor{\mathcal{H}}{{}_{\perp}}\approx
0,\quad\tensor{\mathcal{H}}{{}_{\alpha}}\approx
0,\quad\tensor{\mathcal{H}}{{}_{ij}}\approx 0.$ (16ag)

We see that these 10 constraints are divided up under ${\mathrm{SO}(3)}$ to
give the final four entries in the output above, with
$\tensor{\mathcal{H}}{{}_{ij}}=\tensor{\mathcal{H}}{{}_{\overline{ij}}}+2\tensor{n}{{}_{{[}i|}}\tensor{\mathcal{H}}{{}_{\perp|\overline{j}{]}}}$.
The (linearised) values returned for $\tensor{\mathcal{H}}{{}_{\perp}}$ and
$\quad\tensor{\mathcal{H}}{{}_{\alpha}}$ seem sensible in the context of the
constraint in (16ad) – we can see from this that _HiGGS_ has begun imposing
the PiC shell. What about the angular super-momentum? Let us verify by hand
that the answer is correct. The first term in (16afc) vanishes on the PiC
shell, because _HiGGS_ has told us that
$\IfEqCase{B0p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:placeholder}{}]\approx\IfEqCase{B1p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:placeholder}{}]\approx\IfEqCase{B1m}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:placeholder}{}]\approx\IfEqCase{B2p}{{B0p}{\hat{\pi}}{B1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B1m}{\tensor{\hat{\pi}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{B2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A0p}{\tensor{\hat{\pi}}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\hat{\pi}}{}}{A1p}{\tensor{\overset{\wedge}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\hat{\pi}}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2p}{\tensor{\overset{\sim}{\hat{\pi}}}{{}_{\perp\IfEqCase{placeholder}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}{A2m}{\tensor{[}^{\text{T}}]{\hat{\pi}}{{}_{\IfEqCase{placeholder}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[placeholder]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:placeholder}{}]\approx
0$. For the second term, we find from (16ad) after a few lines that

$\tensor{\pi}{{}_{kl}^{\beta}}\approx-2\tensor{\hat{\alpha}}{{}_{0}}{m_{\text{p}}}^{2}J\tensor{n}{{}_{{[}k}}\tensor{h}{{}_{\overline{l}{]}}^{\beta}},$
(16ah)

from which we determine

$\displaystyle\tensor{\mathcal{H}}{{}_{kl}}$
$\displaystyle\approx-2\tensor{\hat{\alpha}}{{}_{0}}{m_{\text{p}}}^{2}J\bigg{[}\tensor{h}{{}_{\overline{m}}^{\beta}}\left(\tensor{\partial}{{}_{\alpha}}\tensor{b}{{}^{m}_{\alpha}}\right)\tensor{n}{{}_{{[}k}}\tensor{h}{{}_{\overline{l}{]}}^{\alpha}}$
(16ai) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\tensor{D}{{}_{\alpha}}\left(\tensor{n}{{}_{{[}k}}\tensor{h}{{}_{\overline{l}{]}}^{\alpha}}\right)\bigg{]}$
$\displaystyle=-2\tensor{\hat{\alpha}}{{}_{0}}{m_{\text{p}}}^{2}J\bigg{(}\IfEqCase{B1p}{{B1p}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{B1m}{\tensor{\overset{\rightharpoonup}{\mathcal{T}}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{T}}{}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{T}}{{}_{\IfEqCase{\overline{kl}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{kl}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{kl}}{}]+2\tensor{n}{{}_{{[}k}}\IfEqCase{B1m}{{B1p}{\tensor{\mathcal{T}}{{}_{\perp\IfEqCase{\overline{l}{]}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{l}{]}]}}}{B1m}{\tensor{\overset{\rightharpoonup}{\mathcal{T}}}{{}_{\IfEqCase{\overline{l}{]}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{l}{]}]}}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{T}}{}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{T}}{{}_{\IfEqCase{\overline{l}{]}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{l}{]}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{l}{]}}{}]\bigg{)}.$

These are indeed the $1^{+}$ and $1^{-}$ parts of the torsion, as _HiGGS_ is
claiming.

####  II.5.2 Module: ‘StudyTheory’ and ‘Velocity’

The greater part of the analysis – i.e. that which is not necessarily encoded
in the literature – is requested by means of the StudyTheory[] command. We
will see in Section II.6 that this command is just a high-level wrapper for
parallelising Poisson matrices and constraint velocities, and also for
DefTheory[] as discussed in Section II.5.1. We will see how to use
StudyTheory[] for a batch of theories in Section III.2. For now we note that
having called DefTheory[] above, we can pass the option "Import"$\to$True to
the command

⬇

In[]:= JobsBatch = {{"EinsteinCartan", {Alp1 == 0, Alp2 == 0, Alp3 == 0, Alp4
== 0, Alp5 == 0, Alp6 == 0, Bet1 == 0, Bet2 == 0, Bet3 == 0, cAlp1 == 0, cAlp2
== 0, cAlp3 == 0, cAlp4 == 0, cAlp5 == 0, cAlp6 == 0, cBet1 == 0, cBet2 == 0,
cBet3 == 0}}};

JobsBatch~StudyTheory~("Import" $\to$ True);

and – since the calculation is expensive – run it through a Wolfram Language
package file. The evaluation of the Poisson matrix is fairly straightforward,
and made up from calls to PoissonBracket[] as discussed in Section II.4.3. It
is better then to turn to Velocity[].

The linearised velocity of (e.g.) some PiC
$\tensor{[}^{B}]{\varphi}{{}_{\acute{v}}}$ is calculated using the formula

$\tensor{[}^{B}]{\dot{\varphi}}{{}_{\acute{v}}}(\bm{x}_{1})=\int\mathrm{d}^{3}x_{2}\left\\{\tensor{[}^{B}]{\varphi}{{}_{\acute{v}}}(\bm{x}_{1}),\tensor{\mathcal{H}}{{}_{\text{T}}}(\bm{x}_{2})\right\\},$
(16aj)

where from (16ae) we need evaluate only the commutator with
$\mathcal{H}_{\perp}$, which is re-expressed in [64] on the PiC shell as

$\displaystyle\tensor{\mathcal{H}}{{}_{\perp}}$
$\displaystyle\equiv\frac{J}{64}\sum_{A}{\tensor{*}{c}{{}_{A}^{\perp}}}\mu({\tensor{*}{\hat{\alpha}}{{}_{A}^{\perp\perp}}})\tensor{[}^{A}]{\varphi}{{}_{\acute{v}}}\tensor{[}^{A}]{\varphi}{{}^{\acute{v}}}$
(16ak) $\displaystyle\ \ \
+\frac{J}{16{m_{\text{p}}}}\sum_{E}{\tensor{*}{c}{{}_{E}^{\perp}}}\mu({\tensor{*}{\hat{\beta}}{{}_{E}^{\perp\perp}}})\tensor{[}^{E}]{\varphi}{{}_{\acute{v}}}\tensor{[}^{E}]{\varphi}{{}^{\acute{v}}}+\frac{1}{2}\tensor{\hat{\alpha}}{{}_{0}}{m_{\text{p}}}^{2}\IfEqCase{A0p}{{A0p}{\tensor{\underline{\mathcal{R}}}{}}{A0m}{\tensor{[}^{\text{P}}]{\mathcal{R}}{{}_{\perp\circ}}}{A1p}{\tensor{\underline{\mathcal{R}}}{{}_{[\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]]}}}{A1m}{\tensor{\mathcal{R}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2p}{\tensor{\underline{\mathcal{R}}}{{}_{\langle\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]\rangle}}}{A2m}{\tensor{[}^{\text{T}}]{\mathcal{R}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]$
$\displaystyle\ \ \
-J\sum_{I}\Big{(}\tensor{\hat{\alpha}}{{}_{I}}\tensor{\mathcal{R}}{{}^{ij}_{\overline{kl}}}+\tensor{\bar{\alpha}}{{}_{I}}\tensor{\lambda}{{}^{ij}_{kl}}\Big{)}{\tensor{[}^{I}]{\hat{\mathcal{P}}}{{}_{ij}^{kl}{}_{nm}^{\overline{pq}}}}\tensor{\mathcal{R}}{{}^{nm}_{\overline{pq}}}$
$\displaystyle\ \ \
-J{m_{\text{p}}}^{2}\sum_{M}\Big{(}\tensor{\hat{\beta}}{{}_{M}}\tensor{\mathcal{T}}{{}^{i}_{\overline{kl}}}+\tensor{\bar{\beta}}{{}_{M}}\tensor{\lambda}{{}^{i}_{kl}}\Big{)}{\tensor{[}^{M}]{\hat{\mathcal{P}}}{{}_{i}^{kl}{}_{n}^{\overline{pq}}}}\tensor{\mathcal{T}}{{}^{n}_{\overline{pq}}}$
$\displaystyle\ \ \
-\tensor{n}{{}^{k}}\tensor{D}{{}_{\alpha}}\tensor{\pi}{{}_{k}^{\alpha}}.$

A particular problem now is that (16ak) has a _quadratic_ structure. If we
apply PoissonBracket[] to evaluate a velocity for some PiC
$\\{\tensor{[}^{B}]{\varphi}{{}_{\acute{v}}},\mathcal{H}_{\perp}\\}$, we will
actually be calling ToBasicForm[] directly on $\mathcal{H}_{\perp}$. This will
generally result in a very expensive computation, which we avoid in Velocity[]
by manually applying the Leibniz rule. Accordingly in the _HiGGS_ source there
_is_ a collection of prepared formulae for generic expressions such as

$\displaystyle\left\\{\tensor{[}^{B}]{\varphi}{{}_{\acute{v}}},\mathcal{H}_{\perp}\right\\}$
$\displaystyle=\frac{J}{32}\sum_{A}{\tensor{*}{c}{{}_{A}^{\perp}}}\mu({\tensor{*}{\hat{\alpha}}{{}_{A}^{\perp\perp}}})\left\\{\tensor{[}^{B}]{\varphi}{{}_{\acute{v}}},\tensor{[}^{A}]{\varphi}{{}_{\acute{u}}}\right\\}\tensor{[}^{A}]{\varphi}{{}^{\acute{u}}}+...,$
(16al)
$\displaystyle=\frac{1}{32}\sum_{A}{\tensor{*}{c}{{}_{A}^{\perp}}}\mu({\tensor{*}{\hat{\alpha}}{{}_{A}^{\perp\perp}}})\Bigg{[}J\tensor{J}{{}_{1\acute{v}\acute{u}}}\tensor{[}^{A}]{\varphi}{{}^{\acute{u}}}+$
$\displaystyle\ \ \ \ \ \
+\tensor{J}{{}_{2\acute{v}\acute{u}}^{\alpha}}\tensor{\partial}{{}_{\alpha}}\left(J\tensor{[}^{A}]{\varphi}{{}^{\acute{u}}}\right)-\tensor{\partial}{{}_{\alpha}}\left(J\tensor{J}{{}_{3\acute{v}\acute{u}}^{\alpha}}\tensor{[}^{A}]{\varphi}{{}^{\acute{u}}}\right)$
$\displaystyle\ \ \ \ \ \ \ \
-\tensor{\partial}{{}_{\alpha}}\left(\tensor{J}{{}_{4\acute{v}\acute{u}}^{\alpha\beta}}\tensor{\partial}{{}_{\beta}}\left(J\tensor{[}^{A}]{\varphi}{{}^{\acute{u}}}\right)\right)\Bigg{]}+...,$

which collectively act as a template for each velocity. For each
${\mathrm{SO}(3)}$ index $A$ shown in the sum in the second equality in
(16al), we need the four portions of a single Poisson bracket, as given in
(16w) and returned as a List by a call to PoissonBracket[]. Accordingly, the
whole of (16ak) is broken into _blocks_ , each of four terms, which follow
from a single bracket, and these are evaluated in parallel as discussed in
Section II.6.

As we mentioned in Section II.4.3, the formula (16w) is not explicitally
covariant, and hence the appearance in (16al) of partial derivatives.
Ultimately, a call to ToNesterForm[] is needed to restore explicit covariance
to the quadratic expression (16al). This setup is not so good: the use of
block formulae restricts us to calculating velocities over the given
$\mathcal{H}_{\perp}$ – it is also susceptible to human error, and concludes
with an expensive simplification process. We propose that any future
iterations of _HiGGS_ implement velocities based two extensions to
PoissonBracket[]:

  1. 1.

The operands should be expressed as sums of products of covariant quantities,
and the Leibniz rule be used to distribute the operation.

  2. 2.

The operation should return the covariant form given in (16x), so that
covariance is maintained at all steps.

An improvement of PoissonBracket[] along these lines should be straightforward
to implement, and would also be equally emenable to parallelisation: we defer
its development to future work.

####  II.5.3 Module: ‘ViewTheory’

The final module ViewTheory[] is used to produce a human-readable summary of
the results from StudyTheory[]. Since we already have a summary of the
literature constraint structure from the DefTheory[] call in Section II.5.1,
we pass for brevity our earlier theory name with the option
"Literature"$\to$False, producing

⬇

In[]:= ViewTheory["EinsteinCartanVelocities", "Literature" $\to$ False];

Out[]=

** DefTheory: Incorporating the binary at svy/EinsteinCartanVelocities.thr.mx

We see how the results are stored in the theory binary at
svy/EinsteinCartan.thr.mx. There are _four_ nonvanishing Poisson brackets, all
between the translational and rotational pairs of the same ${\mathrm{SO}(3)}$
irrep,
$\IfEqCase{B0p}{{B0p}{\varphi}{B1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{B1m}{\tensor{\varphi}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{B2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A0p}{\tensor{\varphi}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\varphi}{}}{A1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\varphi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2m}{\tensor{[}^{\text{T}}]{\varphi}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]$
vs
$\IfEqCase{A0p}{{B0p}{\varphi}{B1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{B1m}{\tensor{\varphi}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{B2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A0p}{\tensor{\varphi}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\varphi}{}}{A1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\varphi}}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\perp\IfEqCase{}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}{A2m}{\tensor{[}^{\text{T}}]{\varphi}{{}_{\IfEqCase{}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:}{}]$,
$\IfEqCase{B1p}{{B0p}{\varphi}{B1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{B1m}{\tensor{\varphi}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{B2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A0p}{\tensor{\varphi}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\varphi}{}}{A1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\varphi}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A2m}{\tensor{[}^{\text{T}}]{\varphi}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ij}}{}]$
vs
$\IfEqCase{A0p}{{B0p}{\varphi}{B1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{B1m}{\tensor{\varphi}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{B2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A0p}{\tensor{\varphi}{{}_{\perp}}}{A0m}{\tensor{[}^{\text{P}}]{\varphi}{}}{A1p}{\tensor{\overset{\wedge}{\varphi}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A1m}{\tensor{\overset{\rightharpoonup}{\varphi}}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{1}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A2p}{\tensor{\overset{\sim}{\varphi}}{{}_{\perp\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{2}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}{A2m}{\tensor{[}^{\text{T}}]{\varphi}{{}_{\IfEqCase{\overline{ij}}{{placeholder}{\IfEqCase{3}{{1}{\overline{k}}{2}{\overline{kl}}{3}{\overline{klm}}}}}[\overline{ij}]}}}}[\packageError{cosmicclass}{UnidentifiedCriticalCase:\overline{ij}}{}]$,