The Planar Thirring Model
with Kähler-Dirac Fermions

Simon Hands

Department of Physics,
Faculty of Science & Engineering, Swansea University,
Singleton Park, Swansea SA2 8PP, United Kingdom.

###### Abstract

Kähler’s geometric approach in which relativistic fermion fields are treated
as differential forms is applied in three spacetime dimensions. It is shown
that the resulting continuum theory is invariant under global
U($N)\otimes$U($N)$ field transformations, and has a parity-invariant mass
term, both symmetries shared in common with staggered lattice fermions. The
formalism is used to construct a version of the Thirring model with contact
interactions between conserved Noether currents. Under reasonable assumptions
about field rescaling after quantum corrections, a more general interaction
term is derived, sharing the same symmetries but now including terms which
entangle spin and taste degrees of freedom, which exactly coincides with the
leading terms in the staggered lattice Thirring model in the long-wavelength
limit. Finally truncated versions of the theory are explored; it is found that
excluding scalar and pseudoscalar components leads to a theory of six-
component fermion fields describing particles with spin 1, with fermion and
antifermion corresponding to states with definite circular polarisation. In
the UV limit only transverse states with just four non-vanishing components
propagate. Implications for the description of dynamics at a strongly
interacting renormalisation-group fixed point are discussed.

##  1 Introduction

This paper concerns relativistic fermions interacting strongly in three
spacetime dimensions, in the context of a field theory known as the Thirring
model with Lagrangian density

${\cal L}=\bar{\psi}_{i}(\partial\\!\\!\\!/\,+m)\psi_{i}+{g^{2}\over
2N}(\bar{\psi}_{i}\gamma_{\mu}\psi_{i})^{2}.$ (1)

Here the fields $\psi_{i},\bar{\psi}_{i}$ are reducible spinors, so the Dirac
matrices $\gamma_{\mu}$ are $4\times 4$. In Euclidean metric they obey
$\gamma_{\mu}=\gamma_{\mu}^{\dagger}$ and
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2\delta_{\mu\nu}$. The index $i=1,\dots,N$
runs over $N$ distinct fermion species. The contact interaction between
conserved fermion currents $\bar{\psi}\gamma_{\mu}\psi$ results in a repulsive
force between fermions, but attraction between fermion and antifermion. A
question of interest is then whether in the massless limit $m\to 0$ a bilinear
condensate $\langle\bar{\psi}\psi\rangle\not=0$ forms as a result of strong
interactions, leading to the dynamical generation of fermion mass.

It is natural to analyse bilinear condensation in terms of symmetry breaking.
In three dimensions there are two elements of the reducible Dirac algebra
$\gamma_{4}$ and $\gamma_{5}$ which anticommute with the kinetic term of (1).
Accordingly for $m\to 0$ (1) is invariant under the following field rotations:

$\displaystyle\psi\mapsto
e^{i\alpha_{1}}\psi;\;\bar{\psi}\mapsto\bar{\psi}e^{-i\alpha_{1}}\;\;$
$\displaystyle:$ $\displaystyle\psi\mapsto
e^{\alpha_{45}\gamma_{4}\gamma_{5}}\psi;\;\bar{\psi}\mapsto\bar{\psi}e^{-\alpha_{45}\gamma_{4}\gamma_{5}};$
(2) $\displaystyle\psi\mapsto
e^{i\alpha_{4}\gamma_{4}}\psi;\;\bar{\psi}\mapsto\bar{\psi}e^{i\alpha_{4}\gamma_{4}}\;\;$
$\displaystyle:$ $\displaystyle\psi\mapsto
e^{i\alpha_{5}\gamma_{5}}\psi;\;\bar{\psi}\mapsto\bar{\psi}e^{i\alpha_{5}\gamma_{5}}.$
(3)

Together these rotations generate a U(2$N$) global symmetry which can be
broken either explicitly by $m\not=0$ or spontaneously by
$\langle\bar{\psi}\psi\rangle\not=0$ to U($N)\otimes$U($N$), when the
rotations (3) no longer leave the ground state invariant. Goldstone’s theorem
implies spontaneous symmetry breaking results in $2N^{2}$ massless bosons in
the theory’s spectrum.

It is suspected that symmetry breaking occurs for sufficiently large
interaction strength $g^{2}$ and sufficiently small $N$; it is even possible
that the resulting quantum critical point observed at $g_{c}^{2}(N)$ might be
a UV-stable fixed point of the renormalisation group, implying that a
continuum limit at this point is possible. The fixed-point theory is expected
to display universal features of the strongly-interacting dynamics
characterised by the pattern of symmetry breaking. However, there are no small
parameters to enable a systematic investigation of this phenomenon by analytic
means. Determination of the critical exponents, and even the critical flavor
number $N_{c}$ below which symmetry breaking can occur, are essentially non-
perturbative problems.

A natural approach employs numerical simulations of lattice field theory (a
recent review can be found in [1]). Most recent work uses a lattice fermion
formulation which seeks to respect the U($2N$) symmetry such as the SLAC
derivative [2, 3], or domain wall fermions [4, 5]. However, there is also a
substantial body of earlier simulations [6] employing the more primitive
staggered formulation, in which fermion fields are represented by single-
component Grassmann objects $\chi,\bar{\chi}$ located on the sites of a cubic
lattice. As well as U($N$) flavor rotations, staggered fermions also enjoy a
second U($N$) global symmetry protecting them from acquiring mass, of the form

$\chi\mapsto e^{i\beta\varepsilon_{x}}\chi;\;\bar{\chi}\mapsto
e^{i\beta\varepsilon_{x}}\bar{\chi},$ (4)

where $\varepsilon_{x}=(-1)^{x_{1}+x_{2}+x_{3}}$ is an alternating sign in
effect partitioning the sites $x$ into distinct odd and even sublattices. This
time, therefore, bilinear condensation drives a symmetry breaking
U($N)\otimes$U($N)\to$U($N$), resulting in just $N^{2}$ Goldstones. For a
strongly-interacting system, therefore, we expect distinct fixed-point
behaviour, and indeed simulations of the staggered model [7] support a
critical $N_{c}\approx 3.3$ significantly larger than that found for the
U($2N$)-symmetric variants [3, 4, 5]. Moreover, simulation studies of the
minimal staggered model with $N=1$111It is very common in the literature to
designate $N$ staggered flavors in 3$d$ as describing $N_{f}=2N$ “continuum
flavors” [8]. [9] find critical indices indistinguishable from those of the
Gross-Neveu model having the same global symmetries [10], even though in the
latter case symmetry breaking can be described analytically using a $1/N$
expansion.

Despite these apparent shortcomings the staggered Thirring model does exhibit
interesting behaviour; in particular, the critical exponents characterising
the fixed-point are particularly sensitive to the value of $N<N_{c}$. Could
there exist a continuum-based description of the corresponding fixed-point
theories? One question which needs addressing is the significance of $N$ – in
a weak-coupling long-wavelength limit it is natural to interpret staggered
femions in terms of $N_{f}=2N$ autonomous flavors [8], or in modern parlance,
each staggered flavor describes two continuum “tastes”. However, even in early
staggered Thirring studies [6] the factorisation of interaction currents into
mutually-distinct taste sectors was not manifest, and there is no reason a
priori to require this in a strongly-coupled setting. In what follows we will
refer to the difficulty in separating taste and spin components as “spin/taste
entanglement”. A related question is how to engineer the U($N)\otimes$U($N$)
symmetry in the continuum where we have no lattice partition to help recover
(4).

This paper will answer such questions using a framework introduced into
lattice field theory by Becher and Joos in 1982 [11], who found that a version
of the Dirac equation rooted in concepts of differential geometry originally
noted by Kähler [12] in 1962 is in fact the formal continuum limit of
staggered lattice fermions. As set out in the next few sections, in the
Kähler-Dirac approach fermions are not spinor fields but rather are complexes
of $p$-forms, where $p=0,1,\ldots,d$, with $d$ the dimension of spacetime.
This is a natural way to prepare for transcribing continuum fields to a
lattice [13]. Each $p$-form has ${}_{d}C_{p}$ components. In the four
dimensional case analysed in [11] a fermion field has thus $1+4+6+4+1=16$
components, which are recast as 4 independent tastes of 4-component spinor
fields. For the case $d=3$ to be developed in what follows, the corresponding
field has 8 components recast as 2 tastes of reducible 4-component spinor. The
algebraic details very closely mirror the assignment of spin/taste degrees of
freedom to staggered lattice fermions in three spacetime dimensions originally
set out by Burden and Burkitt [8].

The remainder is organised as follows. Sec. 2 is a brief but hopefully self-
contained introduction to the differential geometry machinery required.
Readers who are already expert will find our notations and conventions set
out; those less familiar might also benefit from the helpful Appendix of [11],
or a textbook such as [14]. Sec. 3 derives the equivalence between the free
Kähler-Dirac equation, which with suitable notation assumes the same form in
any dimension, and a continuum Dirac equation in three Euclidean dimensions
describing two tastes of reducible spinor. The same framework is used in Sec.
4, following the introduction of a generalised scalar product between
$p$-forms, to identify the fermion current that will be used in building the
Thirring interaction term. Sec. 5 at last introduces the Thirring model action
in the Kähler-Dirac language, and identifies both the U($N)\otimes$U($N$)
global symmetry and also an important parity symmetry shared in common with
staggered lattice fermions. The Euclidean path integral is introduced
permitting an explicit derivation of the Noether current associated with the
symmetry corresponding to (4).

In Sec. 6 we begin to take the geometrical form of the theory more seriously
by exploring the idea that in a suitably-regularised interacting theory the
renormalisation of the field components should depend on $p$: the Thirring
interaction term is modified in order to accommodate this possibility, and it
is shown that the resulting terms when recast in a spinor basis exhibit
spin/taste entanglement, and are in exact correspondence with the interaction
derived from the staggered Thirring model [6] using the formalism of [8]. This
demonstrates that the proposed $p$-dependent field rescaling is perfectly
consistent with a properly-regularised lattice model, and also that spin/taste
entanglement is not a lattice artifact but rather in fact a feature of an
interacting continuum field theory. Finally in Sec. 7 the idea is taken a step
further with the exploration of truncated actions resulting from retaining
just field components with two consecutive values of $p$. The most interesting
case corresponds to keeping just $p=1,2$, resulting in a theory of six-
component spin-1 fermions, whose physical states are transverse, and for which
fermion and antifermion are states of opposite polarisation. Sec. 8 summarises
the paper’s findings and speculates on the applicability of the exotic
scenario of Sec. 7 to the physics of a putative renormalisation-group fixed
point at strong coupling.

##  2 Mathematical Preliminaries

The theory to be developed uses the language of differential forms in three-
dimensional Euclidean spacetime. We will follow the presentation and notation
of [11] closely. In this approach all physical quantities are viewed as
$p$-forms defined in some vector space ${}^{p}\Lambda$, with $p=0,1,\ldots,3$.
A suitable basis for ${}^{p}\Lambda$ is given by
$dx^{H}=dx^{\mu_{1}}\wedge\ldots\wedge dx^{\mu_{p}}$ with the exterior product
satisfying

$\wedge:\;^{p}\Lambda\times^{q}\Lambda\mapsto\,^{p+q}\Lambda,\;\;\;dx^{H}\wedge
dx^{K}=\begin{cases}\rho_{H,K}dx^{H\cup K}&\mbox{if}\;H\cap K=\emptyset;\\\
0&\mbox{otherwise,}\end{cases}$ (5)

where the sign factor $\rho_{H,K}=(-1)^{s}$, with $s$ the number of pairs
$\mu,\nu\in H\times K$ with $\mu>\nu$, and
$\rho_{\emptyset,H}=\rho_{H,\emptyset}=1$. With this in place any function
$\Phi$ can be expanded as follows:

$\displaystyle\Phi(x)$ $\displaystyle=$
$\displaystyle\varphi_{\emptyset}(x)+\varphi_{\mu}(x)dx^{\mu}+{1\over{2!}}\varphi_{\mu\nu}(x)dx^{\mu}\wedge
dx^{\nu}+\varphi_{123}(x)dx^{1}\wedge dx^{2}\wedge dx^{3}$ (6)
$\displaystyle\equiv$ $\displaystyle\sum_{H}\varphi(x,H)dx^{H}.$

The convention is that repeated indices are summed over, and no special
significance is attached to whether an index is super- or subscript. In
dealing with quantities defined on the whole space
$\Lambda=\bigoplus_{p=0}^{3}{{}^{p}\Lambda}$, it is convenient to define the
main automorphism

${\cal A}:\;\Lambda\mapsto\Lambda,\;\;\;{\cal
A}\Phi=\sum_{H}(-1)^{p(H)}\varphi(x,H)dx^{H},$ (7)

and the main antiautomorphism

${\cal B}:\;\Lambda\mapsto\Lambda,\;\;\;{\cal
B}\Phi=\sum_{H}(-1)^{{}_{p(H)}C_{2}}\varphi(x,H)dx^{H}.$ (8)

where for $p\in\\{0,\ldots,3\\}$ the combinatoric factor ${}_{p}C_{2}$ takes
values $\\{0,0,1,3\\}$.

Three key operations are then:

  * •

Exterior derivative

$d:\;^{p}\Lambda\mapsto\,^{p+1}\Lambda,\;\;\;d\Phi=dx^{\mu}\wedge\partial_{\mu}\Phi$
(9)

Since $\partial_{\mu}\partial_{\nu}=\partial_{\nu}\partial_{\mu}$ it
immediately follows from (5) that $d^{2}=0$.

  * •

Hodge Star

$\star:\;^{p}\Lambda\mapsto\,^{3-p}\Lambda,\;\;\;\star dx^{H}=\rho_{H,{\cal
C}H}dx^{{\cal C}H}$ (10)

where ${\cal C}H$ is the complement of $H$. In odd-dimensional Euclidean
spacetimes $\star\star=1$.

  * •

Co-derivative

$\delta:\;^{p}\Lambda\mapsto\,^{p-1}\Lambda,\;\;\;\delta=\star{\cal
B}\,d\star\\!{\cal B}=\star d\\!\star\\!{\cal A}.$ (11)

and it immediately follows from $d^{2}=0$, $\star\star=1$ that $\delta^{2}=0$.
The co-derivative’s sign depends in general on $p$, $d$ and the signature of
the metric [14], which in (11) is captured by the use of the automorphisms
(7,8). A convenient representation for its action is

$\delta\Phi=-e^{\mu}\lrcorner\partial_{\mu}\Phi,$ (12)

where the contraction operator enabling differentiation with respect to a
differential is defined by

$e^{K}\lrcorner dx^{H}=\begin{cases}\rho_{K,H\backslash K}dx^{H\backslash
K}&\mbox{if}\;K\subset H;\\\ 0&\mbox{otherwise.}\end{cases}$ (13)

##  3 The Kähler-Dirac Equation

The starting point is the observation that $(d-\delta)^{2}=-(d\delta+\delta
d)=\partial_{\mu}\partial_{\mu}=\Delta$, the Laplacian operator. Hence
$d-\delta$ is in effect the square-root of the Laplacian, and therefore linear
in momentum, while still local. It is thus a candidate for incorporating in a
relativistic wave equation, as first written by Kähler [12]:

$(d-\delta+m)\Phi=0.$ (14)

The Kähler-Dirac equation (KDE) takes the same form in any spacetime
dimension. The scalar parameter $m$ is the fermion mass. Note that since $d$
and $\delta$ implement $\Delta p=\pm 1$, the equation only makes sense if
$\Phi\in\Lambda$, ie. $\Phi$ admits an expansion of the form (6), with
components $\varphi(x,H)$ having mass dimension 1 in three spacetime
dimensions.

It is helpful to define the Clifford product between differential forms:

$\vee:\;\Lambda\times\Lambda\mapsto\Lambda,\;\Phi\vee\Xi=\sum_{p}{{(-1)^{{}_{p}C_{2}}}\over
p!}({\cal A}^{p}e^{\mu_{1}}\lrcorner\ldots
e^{\mu_{p}}\lrcorner\Phi)\wedge(e^{\mu_{1}}\lrcorner\ldots
e^{\mu_{p}}\lrcorner\Xi),$ (15)

with particular instances

$dx^{\mu}\vee\Phi=dx^{\mu}\wedge\Phi+e^{\mu}\lrcorner\Phi;\;\;\;\Phi\vee
dx^{\mu}=\Phi\wedge dx^{\mu}-e^{\mu}\lrcorner{\cal A}\Phi.$ (16)

It immediately follows from (9) and (12) that the KDE can be rewritten

$(dx^{\mu}\vee\partial_{\mu}+m)\Phi=0.$ (17)

Now, the identity

$dx^{\mu}\vee dx^{\nu}\vee+dx^{\nu}\vee dx^{\mu}\vee=2\delta^{\mu\nu}$ (18)

is strongly reminiscent of the defining relation
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2\delta_{\mu\nu}$ for Dirac matrices in
Euclidean metric, and suggests the operation $dx^{\mu}\vee$ furnishes a
representation of the Dirac algebra in the 8-dimensional space spanned by
$dx^{H}$. The appropriate representation of the algebra in 3 spacetime
dimensions was identified in [8] in a study of the staggered lattice fermion
operator. It is the direct sum $\sigma_{H}\oplus\tau_{H}$ of two inequivalent
irreducible 2-dimensional representations generated by the Pauli matrices
$\sigma_{\mu}$ ($\mu=1,2,3$), and by $\tau_{\mu}=-\sigma_{\mu}$. The Pauli
matrices have the property $\sigma_{\mu}^{*}=\sigma_{\mu}^{T}$, where *
denotes complex conjugation and $T$ the matrix transpose. Analysis proceeds by
identifying a new basis

$\displaystyle\Sigma\oplus T$ $\displaystyle=$ $\displaystyle
1+(\sigma_{\mu}^{T}\oplus\tau_{\mu}^{T})dx^{\mu}+{1\over
2!}(\sigma_{\mu}^{T}\sigma_{\nu}^{T}\oplus\tau_{\mu}^{T}\tau_{\nu}^{T})dx^{\mu}\wedge
dx^{\nu}$ $\displaystyle+$
$\displaystyle(\sigma_{1}^{T}\sigma_{2}^{T}\sigma_{3}^{T}\oplus\tau_{1}^{T}\tau_{2}^{T}\tau_{3}^{T})dx^{1}\wedge
dx^{2}\wedge dx^{3}=\sum_{H}{\cal B}(\sigma\oplus\tau)_{H}^{T}dx^{H}$

The key result is now

$dx^{\alpha}\vee(\Sigma\oplus
T)=(\sigma^{\alpha}\oplus\tau^{\alpha})^{T}(\Sigma\oplus
T)=\sum_{c}(\sigma^{\alpha}\oplus\tau^{\alpha})^{T}_{ac}(\Sigma\oplus
T)_{cb},$ (20)

where Roman indices $a,b,c=1,2$. The derivation of (20) makes repeated use of
$\\{(\sigma\oplus\tau)_{\mu},(\sigma\oplus\tau)_{\nu}\\}=2\delta_{\mu\nu}\otimes\mathbb{1}_{2\times
2}$.

In order to express the KDE in the basis (3) we need the orthogonality
relations

$\sum_{H}\sigma_{ab}^{H}\sigma_{cd}^{H*}=\sum_{H}\tau_{ab}^{H}\tau_{cd}^{H*}=4\delta_{ac}\delta_{bd};\;\;\;\sum_{H}\sigma_{ab}^{H}\tau_{cd}^{H*}=0,$
(21)

implying

$dx^{H}={1\over
4}(-1)^{{}_{p}C_{2}}\mbox{tr}(\sigma_{H}\oplus\tau_{H})^{*}(\Sigma\oplus T).$
(22)

Using (6) we then define

$\Phi(x)=\sum_{H}\varphi(x,H)dx^{H}=\sum_{a,b}u_{a}^{b}(x)\Sigma_{a}\oplus
d_{a}^{b}(x)T_{a}$ (23)

where we have introduced fields $u$, $d$ whose lower index $a=1,2$ will turn
out to be associated with spinor degrees of freedom in the non-interacting
case, and whose upper index $b=1,2$ will be associated with taste. The field
transformations between bases are then:

$\displaystyle\varphi(x,H)$ $\displaystyle=$
$\displaystyle\sum_{a,b}u_{a}^{b}(x)(\sigma_{H}^{T})_{ab}\oplus
d_{a}^{b}(x)(\tau_{H}^{T})_{ab};$ (24) $\displaystyle(u_{a}^{b}\oplus
d_{a}^{b})(x)$ $\displaystyle=$ $\displaystyle{1\over
4}\sum_{H}\varphi(x,H)(\sigma\oplus\tau)^{H}_{ab}.$ (25)

Combining the result (20) with the KDE equation (17) we deduce

$(\gamma_{\mu}\partial_{\mu}+m)\psi^{b}(x)=0,$ (26)

ie. the free Dirac equation for a two-taste four-component spinor field
$\psi=u\oplus d$, with Euclidean Dirac matrices defined

$\gamma_{\mu}=\left(\begin{matrix}\sigma_{\mu}&\cr&\tau_{\mu}\end{matrix}\right).$
(27)

We will refer to this familar form as the free KDE in the $\psi$-basis.

##  4 Interaction Current

In order to develop an interacting theory we will need a definition of a
current in the Kähler-Dirac formalism. This requires the definition of a
generalised scalar product
$(,)_{p}:\;\Lambda\times\Lambda\mapsto^{3-p}\\!\Lambda$ [12, 11]. The two
cases we will need have $p=0$:

$(\Phi,\Xi)_{0}=({\cal B}\Phi\vee\Xi)\wedge\varepsilon$ (28)

and $p=1$:

$(\Phi,\Xi)_{1}=e_{\mu}\lrcorner(dx^{\mu}\vee\Phi,\Xi)_{0}=e_{\mu}\lrcorner[(dx^{\mu}\vee\Phi\vee{\cal
B}\Xi)\wedge\varepsilon],$ (29)

where $\varepsilon$ is the volume 3-form $dx^{1}\wedge dx^{2}\wedge dx^{3}$.
In components these are expressed

$(\Phi,\Xi)_{0}=\left[\varphi_{\emptyset}\xi_{\emptyset}+\varphi_{\mu}\xi_{\mu}+{1\over
2!}\varphi_{\mu\nu}\xi_{\mu\nu}+\varphi_{123}\xi_{123}\right]\varepsilon=\sum_{H}\varphi(x,H)\xi(x,H)\varepsilon;$
(30)

and

$\displaystyle(\Phi,\Xi)_{1}$ $\displaystyle=$ $\displaystyle{1\over
2!}\Bigl{[}\varphi_{\emptyset}\xi_{\alpha}+\varphi_{\alpha}\xi_{\emptyset}+\varphi_{\mu}\xi_{\alpha\mu}+\varphi_{\alpha\mu}\xi_{\mu}$
$\displaystyle+$ $\displaystyle{\textstyle{1\over
2!}}(\varphi_{\mu\nu}\xi_{\alpha\mu\nu}+\varphi_{\alpha\mu\nu}\xi_{\mu\nu})\Bigr{]}\epsilon_{\alpha\mu\nu}dx^{\mu}\wedge
dx^{\nu}$

The following Green’s formula identity is useful [12]:

$d(\Phi,\Xi)_{1}=(\Phi,(d-\delta)\Xi)_{0}+((d-\delta)\Phi,\Xi)_{0}.$ (32)

Next define $\bar{\Phi}={\cal A}\Phi^{*}$ as the solution of the adjoint KDE:

$(d-\delta-m)\bar{\Phi}=0.$ (33)

A current 1-form is then given by

$j=j_{\mu}dx^{\mu}=-\star{i\over 4}(\bar{\Phi},\Phi)_{1}.$ (34)

Current conservation follows using (32,14,33):

$\displaystyle\delta j=-\star d\star j$ $\displaystyle=$
$\displaystyle\star{i\over 4}d(\bar{\Phi},\Phi)_{1}=\star{i\over
4}[(\bar{\Phi},(d-\delta)\Phi)_{0}+((d-\delta)\bar{\Phi},\Phi)_{0}]$
$\displaystyle=$ $\displaystyle-\star{i\over 4}(\bar{\Phi},\Phi)_{0}(m-m)=0,$

ie. $\partial_{\mu}j_{\mu}=0$. Now use
$(\Phi,\Xi)_{p}=(-1)^{{}_{p}C_{2}}(\Xi,\Phi)_{p}$ and (20) to write

$(\bar{\Phi},\Phi)_{1}=e_{\mu}\lrcorner(\bar{\Phi},dx^{\mu}\vee\Phi)_{0}=e_{\mu}\lrcorner(\bar{u}\Sigma^{*}\oplus\bar{d}T^{*},(\sigma^{\mu}u)\Sigma\oplus(\tau^{\mu}d)T)_{0}.$
(36)

In the $\psi$-basis the current 1-form thus reads

$j=i\sum_{b}\bar{\psi}^{b}(x)\gamma_{\mu}\psi^{b}(x)dx^{\mu}.$ (37)

##  5 Action and Symmetries

Now we have enough equipment to define the action and hence the Euclidean path
intgeral. The action for free fields is

$S_{0}={1\over
4}\int(\bar{\Phi},(d-\delta+m)\Phi)_{0}=\sum_{b=1,2}\int\bar{\psi}^{b}(\gamma_{\mu}\partial_{\mu}+m)\psi^{b}\varepsilon.$
(38)

For the Thirring model this is supplemented by a contact interaction of the
form $-{g^{2}\over 4}j_{\mu}j_{\mu}$, where the normalisation of the coupling
strength, which has mass dimension -1, is somewhat conventional. In the
language of forms this reads

$-{g^{2}\over 4}\int(j,j)_{0}=-{g^{2}\over 4}\int(j\vee
j)\wedge\varepsilon={g^{2}\over
64}\int(\star(\bar{\Phi},\Phi)_{1}\vee\star(\bar{\Phi},\Phi)_{1})\wedge\varepsilon.$
(39)

Using $(\star\Phi,\star\Xi)_{0}=(\Phi,\Xi)_{0}$ we arrive at the Thirring
model action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int{1\over
4}(\bar{\Phi},(d-\delta+m)\Phi)_{0}+{g^{2}\over
64}((\bar{\Phi},\Phi)_{1}\vee(\bar{\Phi},\Phi)_{1})\wedge\varepsilon$ (40)
$\displaystyle=$
$\displaystyle\int\Bigl{[}\bar{\psi}^{b}(\gamma_{\mu}\partial_{\mu}+m)\psi^{b}+{g^{2}\over
4}(\bar{\psi}^{b}\gamma_{\mu}\psi^{b})(\bar{\psi}^{c}\gamma_{\mu}\psi^{c})\Bigr{]}\varepsilon.$
(41)

As a consequence of its construction from $(\bar{\Phi},\Phi)$ bilinears the
action (40) has two manifest global symmetries. First:

$\Phi\mapsto e^{i\theta}\Phi;\;\;\;\bar{\Phi}\mapsto e^{-i\theta}\bar{\Phi}.$
(42)

This symmetry correponds to the conservation of fermion charge, and the
corresponding Noether current is given by (34). Second, in the limit $m\to 0$:

$\Phi\mapsto e^{i\omega{\cal A}}\Phi;\;\;\;\bar{\Phi}\mapsto e^{i\omega{\cal
A}}\bar{\Phi},$ (43)

which follows because $d,\delta$ both yield $\Delta p=\pm 1$, and by
inspection of the component expansion of $(\bar{\Phi},\Phi)_{1}$ (4). This is
analogous to the chiral symmetry protecting fermions from additive mass
renormalisation in $d=4$. The corresponding Noether current is

$j_{\cal A}=-\star{i\over 4}(\bar{\Phi},{\cal A}\Phi)_{1}.$ (44)

In order to translate to the $\psi$-basis, observe that the action of ${\cal
A}$ in effect exchanges $\sigma_{H}$ and $\tau_{H}$ in (25). It then follows
straightforwardly that

$j_{{\cal A}\mu}=i\bar{\psi}^{b}\gamma_{\mu}\gamma_{5}\psi^{b}$ (45)

where we introduce two new hermitian $\gamma$-matrices obeying
$\\{\gamma_{4},\gamma_{\mu}\\}=\\{\gamma_{5},\gamma_{\mu}\\}=\\{\gamma_{4},\gamma_{5}\\}=0$:

$\gamma_{4}=\left(\begin{matrix}&-i\mathbb{1}\cr
i\mathbb{1}&\end{matrix}\right);\;\;\;\gamma_{5}=\left(\begin{matrix}&\mathbb{1}\cr\mathbb{1}&\end{matrix}\right).$
(46)

From here it is straightforward to extend the model by introducing $N$ Kähler-
Dirac fermion flavors $\Phi^{i}$, $i=1,\ldots,N$. The flavor index $i$ is
distinct from the indices $b,c=1,2$ in (41), which run over taste degrees of
freedom. The two U(1) rotation symmetries (42,43) are trivially extended to
U$(N)\otimes$U${}_{\cal A}(N)$, broken to U($N$) either explicitly by
$m\not=0$, or spontaneously by dynamical generation of a non-vanishing
condensate $\langle(\bar{\Phi}^{i},\Phi^{i})_{0}\rangle$.

Finally, consider discrete parity inversion. In odd spacetime dimensions this
is conveniently represented by inversion of all spacetime axes:
$x_{\mu}\mapsto-x_{\mu}$, $\partial_{\mu}\mapsto-\partial_{\mu}$. The action
(40,41) is invariant provided

$\Phi(x)\mapsto{\cal A}\Phi(-x);\;\bar{\Phi}(x)\mapsto{\cal
A}\bar{\Phi}(-x)\Rightarrow\psi(x)\mapsto\gamma_{5}\psi(-x);\;\bar{\psi}(x)\mapsto\bar{\psi}(-x)\gamma_{5}.$
(47)

Note that the Noether currents (34,45), along with all bilinears of the form
$(\Phi,\Xi)_{1}$, are parity-odd.

The Euclidean path integral is defined by

${\cal Z}=\int D\Phi D\bar{\Phi}\exp(-S[\Phi,\bar{\Phi}])$ (48)

where $\Phi,\bar{\Phi}$ are now Grassmann-valued and $\bar{\Phi}$ is
considered independent of $\Phi$. We illustrate its use via a derivation of
the Ward Identity for the divergence of the current $j_{\cal A}$; for
simplicity we consider only the free action (38). Consider the impact of the
field transformation (43) where $\omega(x)$ is infinitesimal but now
spacetime-dependent.

$\displaystyle S_{0}\mapsto S_{0}^{\prime}$ $\displaystyle=$
$\displaystyle{1\over 4}\int\left(e^{i\omega(x){\cal
A}}\bar{\Phi},(d-\delta+m)e^{i\omega(x){\cal A}}\Phi\right)_{0}$
$\displaystyle=$ $\displaystyle{1\over 4}\int\left(e^{i\omega{\cal
A}}\bar{\Phi},dx^{\mu}\vee\partial_{\mu}e^{i\omega{\cal
A}}\Phi+me^{i\omega{\cal A}}\Phi\right)_{0}$ $\displaystyle=$ $\displaystyle
S_{0}+{1\over 4}\int i\partial_{\mu}\omega\left(\bar{\Phi},dx^{\mu}\vee{\cal
A}\Phi\right)_{0}+im\omega\left((\bar{\Phi},{\cal A}\Phi)_{0}+({\cal
A}\bar{\Phi},\Phi)_{0}\right).$

Now use (29) together with $(\Phi,\Xi)_{p}=(-1)^{{}_{p}C_{2}}(\Xi,\Phi)_{p}$
and the definition (44) to write

$\displaystyle S_{0}^{\prime}-S_{0}$ $\displaystyle=$
$\displaystyle-\int(\partial_{\mu}\omega)dx^{\mu}\wedge\star j_{\cal
A}+i{m\over 2}\int\omega(\bar{\Phi},{\cal A}\Phi)_{0}$ $\displaystyle=$
$\displaystyle\int\omega\left(d\star j_{\cal A}+i{m\over 2}(\bar{\Phi},{\cal
A}\Phi)_{0}\right),$

where in the second step we have integrated the first term by parts. Since the
path-integral measure $D\Phi
D\bar{\Phi}=\prod_{x,H}d\varphi(x,H)d\bar{\varphi}(x,H)$ is formally invariant
under the field transformation, the change of variables has no impact on the
path integral, and we conclude

$\left\langle\int\omega\left[-\star\delta j_{\cal A}+i{m\over
2}(\bar{\Phi},{\cal A}\Phi)_{0}\right]\right\rangle=0.$ (51)

Since (51) holds for any $\omega(x)$, we conclude the expectation value of the
3-form in square brackets is identically zero, which is the Ward Identity. In
the $\psi$-basis it has the familiar form

$\left\langle\partial_{\mu}\bar{\psi}\gamma_{\mu}\gamma_{5}\psi-2m\bar{\psi}\gamma_{5}\psi\right\rangle=0.$
(52)

##  6 Impact of Quantum Corrections

Our treatment up to this point has been either classical or formal. In any
application to a genuine interacting quantum field theory, it is inevitable
that the theory will need to be regularised somehow in order to control the
calculation of quantum corrections. As a concrete example, we have already
discussed the close parallels between the KDE continuum formalism and
staggered lattice fermions, and will assume without further discussion that
the proof of [11] that the KDE is the formal continuum limit of staggered
fermions continues to apply in 3 dimensions.

Regularisation is essentially some kind of truncation of the degrees of
freedom present in the classical field theory, and inevitably violates some of
the symmetries of the classical theory. In many cases this leads to the
requirement of renormalisation of both the fields and the coupling parameters
in the theory, which depends on some physical scale. As a concrete example,
consider the Thirring action in the $\psi$-basis (41), where the rotations
(42,43) take the form

$\psi\mapsto
e^{i\theta}\psi;\;\;\;\bar{\psi}\mapsto\bar{\psi}e^{-i\theta}:\;\;\;\;\;\psi\mapsto
e^{i\theta\gamma_{5}}\psi;\;\;\;\bar{\psi}\mapsto\bar{\psi}e^{i\theta\gamma_{5}}$
(53)

Eqn. (41) also looks to be invariant under a U(2) rotation among the tastes
indexed by $b,c$. Beyond that, in the limit $m\to 0$ there is an additional
symmetry

$\psi\mapsto
e^{i\theta\gamma_{4}}\psi;\;\;\;\bar{\psi}\mapsto\bar{\psi}e^{i\theta\gamma_{4}}$
(54)

as well as

$\psi\mapsto
e^{\theta\gamma_{4}\gamma_{5}}\psi;\;\;\;\bar{\psi}\mapsto\bar{\psi}e^{-\theta\gamma_{4}\gamma_{5}},$
(55)

valid for any $m$. Rotations (54,55) combined with (53) and the taste
rotations would generate a U(4$N$) global symmetry broken to
U($2N)\otimes$U($2N)$ by a fermion mass. Our viewpoint is that this symmetry
is not fundamental and can only be recovered in certain limits, such as long
wavelength or weak coupling.

We will proceed on the assumption that the geometric description employed in
the KDE is more natural, so that after quantum corrections the field expansion
of eqn. (6) is modified:

$\Phi_{r}(x)=\sum_{H}Z_{p(H)}\varphi(x,H)dx^{H}.$ (56)

Here a renormalised field $\Phi_{r}$ is defined in terms of bare components
$\varphi(x,H)$ via wavefunction renormalisation constants $Z_{p}$ which depend
on the interaction strength, the renormalisation scale and, crucially in this
context, on the form degree $p$. This correction is covariant, in the sense
that $Z_{p}$ is insensitive to rotations acting on the spacetime indices
specific to $\varphi(x,H)$, and the key symmetries (42) and (43) continue to
be respected by $\Phi_{r}$ even with $Z_{p}\not=1$.

The form of (56) motivates a more general exploration of possible interaction
currrents. In $d=3$ the space of bilinear currents consistent with the four
renormalisation constants $Z_{p}$ is spanned by $\star(\bar{\Phi},\Phi)_{1}$,
$\star(\bar{\Phi},{\cal A}\Phi)_{1}$, $\star({\cal B}\bar{\Phi},\Phi)_{1}$ and
$\star({\cal B}\bar{\Phi},{\cal A}\Phi)_{1}$. Transcription to the
$\psi$-basis for the first two of these is given in (34,45), and eg.

$\displaystyle({\cal B}\bar{\Phi},\Phi)_{1}$ $\displaystyle=$ $\displaystyle
e_{\mu}\lrcorner({\cal B}\bar{\Phi},dx^{\mu}\vee\Phi)_{0}$ $\displaystyle=$
$\displaystyle
e_{\mu}\lrcorner\varepsilon\sum_{H}(-1)^{{}_{p}C_{2}}\bigl{[}\bar{u}\sigma_{H}\oplus\bar{d}\tau_{H}\bigr{]}\bigl{[}(\sigma_{\mu}u)\sigma_{H}^{*}\oplus(\tau_{\mu}d)\tau_{H}^{*}\bigr{]}$

Now observe the following identities for the components of $\sigma_{H}$:

$\displaystyle\sum_{\rho}\sigma_{\rho}\mathbb{1}\sigma_{\rho}$
$\displaystyle=$ $\displaystyle
3;\;\;\;\sum_{\rho}\sigma_{\rho}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{\rho}=3\sigma_{1}\sigma_{2}\sigma_{3};$
$\displaystyle\sum_{\rho}\sigma_{\rho}\sigma_{\mu}\sigma_{\rho}$
$\displaystyle=$
$\displaystyle-\sigma_{\mu};\;\;\sum_{\rho}\sigma_{\rho}\sigma_{\mu}\sigma_{\nu}\sigma_{\rho}=-\sigma_{\mu}\sigma_{\nu}.$
(58)

Recalling $\tau_{H}=(-1)^{p}\sigma_{H}$, we deduce a particularly convenient
combination:

$\displaystyle-\star(\bar{\Phi}\\!\\!\\!$ $\displaystyle+$
$\displaystyle\\!\\!\\!\\!2{\cal B}\bar{\Phi},\Phi)_{1}$ $\displaystyle=$
$\displaystyle
dx^{\mu}\sum_{H}\sum_{\rho}\bigl{[}\bar{u}\sigma_{H}\oplus\bar{d}\tau_{H}\bigr{]}\bigl{[}(\tau_{\rho}\tau_{\mu}d\tau_{\rho}^{*})^{b}_{a}(\sigma_{H}^{*})_{ab}\oplus(\sigma_{\rho}\sigma_{\mu}u\sigma_{\rho}^{*})^{b}_{a}(\tau_{H}^{*})_{ab}\bigr{]}$
$\displaystyle=$ $\displaystyle
4\sum_{\rho}(\bar{u},\bar{d})\left(\begin{matrix}&-\sigma_{\rho}\sigma_{\mu}\otimes\sigma_{\rho}^{*}\\\
+\sigma_{\rho}\sigma_{\mu}\otimes\sigma_{\rho}^{*}&\end{matrix}\right)\left(\begin{matrix}u\\\
d\end{matrix}\right)dx^{\mu}$ $\displaystyle=$ $\displaystyle
4\sum_{\rho}\bar{\psi}(i\gamma_{4}\gamma_{\rho}\gamma_{\mu}\otimes\tau_{\rho}^{*})\psi
dx^{\mu}.$

Here the second component of the tensor product is a $2\times 2$ matrix acting
on taste indices. Similarly,

$-\star(\bar{\Phi}+2{\cal B}\bar{\Phi},{\cal
A}\Phi)_{1}=-4\sum_{\rho}\bar{\psi}(i\gamma_{4}\gamma_{5}\gamma_{\rho}\gamma_{\mu}\otimes\tau_{\rho}^{*})\psi
dx^{\mu}.$ (60)

In either case what emerges is an interaction current which although parity-
odd and respecting the U(1)$\otimes$UA(1) symmetries (42,43) no longer treats
fermion tastes as independent degrees of freedom but rather entangles taste
and spacetime rotations, contrary to what is expected for particle flavor
degrees of freedom. Remarkably, the currents $\star(\bar{\Phi},\Phi_{1})$,
$\star(\bar{\Phi},{\cal A}\Phi)_{1}$, $\star([1+2{\cal
B}]\bar{\Phi},\Phi)_{1}$ and $\star([1+2{\cal B}]\bar{\Phi},{\cal A}\Phi)_{1}$
all feature in equal weight contact interactions in the Thirring model
formulated with staggered fermions on a 3$d$ cubic lattice as derived in a
basis with explicit spinor and taste indices using the formalism of [8], and
given in Eqn. (2.12) of [6]. In view of the equivalence [11] between Kähler-
Dirac fermions and the formal continuum limit of staggered lattice fermions,
this result is not surprising.

It is now clear the interactions survive the long-wavelength $a\to 0$ limit,
where the lattice spacing $a$ furnishes an explicit UV cutoff. Other terms
entangling spinor and taste degrees of freedom which formally vanish as $O(a)$
are also present in the lattice formulation [6]. The current analysis
demonstrates that spin/taste entanglement is not a lattice artifact, but is
rooted in a continuum action of the form (40) with U($N)\otimes$U${}_{\cal
A}(N)$ symmetry. However, it is significant that such terms also emerge from a
well-defined regularisation capable of exploring strongly-interacting
dynamics.

##  7 Reduced Kähler-Dirac Fermions

Kähler-Dirac fermions offer a new language with which to discuss relativistic
fermion dynamics. To quote Becher and Joos, “This differential geometric
description of fermions might be a basis for the construction of different
finds of field theoretic model,” [11]. Once the differential geometric
scaffolding has been removed, what kind of stories will we be able to tell?
With motivation coming from a desire to understand novel structures at
strongly-interacting fixed points, in this section we will hazard some
speculations.

Let’s start by expressing the free action (38) in the $\varphi$-basis (6),
with Lagrangian density

$\displaystyle\star{1\over 4}(\bar{\Phi},(d-\delta)\Phi)_{0}$ $\displaystyle=$
$\displaystyle{1\over
4}\biggl{[}\bar{\varphi}_{\emptyset}\partial_{\mu}\varphi_{\mu}+\bar{\varphi}_{\mu}(\partial_{\mu}\varphi_{\emptyset}+\partial_{\nu}\varphi_{\nu\mu})$
(61) $\displaystyle+$
$\displaystyle\bar{\varphi}_{\mu\nu}\left(\partial_{\mu}\varphi_{\nu}+{1\over
2!}\partial_{\lambda}\varphi_{\lambda\mu\nu}\right)+{1\over
2!}\bar{\varphi}_{\mu\nu\lambda}\partial_{\mu}\varphi_{\nu\lambda}\biggr{]}.$

Each term in (61) is separately invariant under U(1)$\otimes$U${}_{\cal A}(1)$
and parity (47). Now consider a reduced action containing just a subset of the
$p$-form fields $\varphi(x,H)$. The motivation comes from Eqn. (56), where we
envisage a partition of $\\{0,1,2,3\\}$ into sets $P,Q$ with $Z_{p\in P}\gg
Z_{p\in Q}\approx 0$ arising, say, as a consequence of large anomalous scaling
dimensions at a renormalisation group fixed point. Clearly only cases
retaining consecutive values of $p$ will result in propagating states. We
consider two examples.

###  7.1 $P=\\{0,1\\}$

If we truncate the field content to just $p=0,1$ there are four components
$\phi\equiv(\varphi_{\emptyset},\varphi_{\mu})^{T}$ to keep track of. The
Lagrangian density is

${\cal L}_{01}={1\over
4}[\bar{\varphi}_{\emptyset}\partial_{\mu}\varphi_{\mu}+\bar{\varphi}_{\mu}\partial_{\mu}\varphi_{\emptyset}+m\bar{\varphi}_{\emptyset}\varphi_{\emptyset}+m\bar{\varphi}_{\mu}\varphi_{\mu}]\equiv{1\over
4}\bar{\phi}M\phi.$ (62)

The $4\times 4$ matrix $M$ has

$\mbox{det}M=m^{2}(m^{2}-\Delta),$ (63)

which therefore vanishes identically for massless fermions. The propagator
$M^{-1}$ has components

$\displaystyle\langle\varphi_{\emptyset}\bar{\varphi}_{\emptyset}\rangle$
$\displaystyle=$
$\displaystyle{m\over{m^{2}-\Delta}};\;\;\langle\varphi_{\mu}\bar{\varphi}_{\nu}\rangle={m\delta_{\mu\nu}\over{m^{2}-\Delta}}-{{{\cal
P}_{\mu\nu}\Delta}\over{m(m^{2}-\Delta)}};$
$\displaystyle\langle\varphi_{\emptyset}\bar{\varphi}_{\mu}\rangle$
$\displaystyle=$
$\displaystyle\langle\varphi_{\mu}\bar{\varphi}_{\emptyset}\rangle=-{\partial_{\mu}\over{m^{2}-\Delta}},$
(64)

where the transverse projector

${\cal P}_{\mu\nu}=\delta_{\mu\nu}-{\partial_{\mu}\partial_{\nu}\over\Delta}$
(65)

such that ${\cal P}_{\mu\nu}\partial_{\mu}=0$, ${\cal P}_{\mu\lambda}{\cal
P}_{\lambda\nu}={\cal P}_{\mu\nu}$, and $\mbox{tr}{\cal P}=2$. In momentum
space, all components manifest a particle pole at $k^{2}=-m^{2}$, but
asymptotically scale differently: $\emptyset\emptyset\sim k^{-2}$; $\mu\nu\sim
k^{-2}+k^{0}$; $\emptyset\mu\sim k^{-1}$. We conclude that ${\cal L}_{01}$
describes particles of mass $m$, and that the resulting effective theory is
well-behaved in the IR regime $k^{2}\lesssim m^{2}$. The singular part in the
$m\to 0$ limit has vanishing longitudinal component.

Note also that following a field redefinition
$\varphi_{\mu\nu}=\epsilon_{\mu\nu\lambda}\xi_{\lambda}$;
$\varphi_{\mu\nu\lambda}=\epsilon_{\mu\nu\lambda}\xi_{\emptyset}$, the action
${\cal L}_{23}[\xi_{\emptyset,\mu},\bar{\xi}_{\emptyset,\mu}]$ yields an
action identical in form to (62), the only difference being that in contrast
to (62) the field $\xi_{\emptyset}$ has negative intrinsic parity and
$\xi_{\mu}$ positive.

###  7.2 $P=\\{1,2\\}$

In this case there are 6 field components $\varphi_{\mu},\varphi_{\mu\nu}$
with Lagrangian density

${\cal L}_{12}={1\over
4}\left[\bar{\varphi}_{\mu}\partial_{\nu}\varphi_{\nu\mu}+\bar{\varphi}_{\mu\nu}\partial_{\mu}\varphi_{\nu}+m\bar{\varphi}_{\mu}\varphi_{\mu}+{m\over
2!}\bar{\varphi}_{\mu\nu}\varphi_{\mu\nu}\right].$ (66)

After a field redefinition

$\chi_{\mu}=\varphi_{\mu};\;\;\;\bar{\chi}_{\mu}={1\over
2!}\epsilon_{\mu\nu\lambda}\bar{\varphi}_{\nu\lambda};\;\;\bar{\xi}_{\mu}=\bar{\varphi}_{\mu};\;\;\;\xi_{\mu}={1\over
2!}\epsilon_{\mu\nu\lambda}\varphi_{\nu\lambda},$ (67)

${\cal L}_{12}$ can be rewritten

$\displaystyle{\cal L}_{12}$ $\displaystyle=$ $\displaystyle{1\over
4}\Bigl{[}\epsilon_{\mu\nu\lambda}\left[\bar{\chi}_{\mu}\partial_{\nu}\chi_{\lambda}-\bar{\xi}_{\mu}\partial_{\nu}\xi_{\lambda}\right]+m(\bar{\chi}_{\mu}\xi_{\mu}+\bar{\xi}_{\mu}\chi_{\mu})\Bigr{]}$
(68) $\displaystyle=$ $\displaystyle{1\over
4}(\bar{\chi},\bar{\xi})\left(\begin{matrix}\partial_{\mu}\lambda^{\mu}&m\mathbb{1}_{3\times
3}\\\ m\mathbb{1}_{3\times
3}&-\partial_{\mu}\lambda^{\mu}\end{matrix}\right)\left(\begin{matrix}\chi\\\
\xi\end{matrix}\right)\equiv{1\over 4}\bar{\Upsilon}M\Upsilon,$ (69)

where we have implicitly defined 6 component fermion fields
$\Upsilon,\bar{\Upsilon}$. The individual kinetic terms for the three-
component objects $\chi,\xi$ in (68) superficially resemble the Chern-Simons
action for gauge boson fields in $d=3$, and in (69) the $3\times 3$ matrices
$\lambda^{\mu}$, $\mu=1,\ldots,3$ are antihermitian generators of the spin-1
representation of SU(2), ie. obeying
$[\lambda_{\mu},\lambda_{\nu}]=-\epsilon_{\mu\nu\rho}\lambda_{\rho}$, and each
with eigenvalues $i\lambda=0,\pm 1$. Fields respond to rotations in the
$\rho\sigma$ plane via
$(\chi,\xi)^{T}_{\mu}\mapsto(\Lambda_{1})_{\mu\nu}(\chi,\xi)^{T}_{\nu}$ with

$\Lambda_{1}(\theta_{\rho\sigma})=\exp\left(-{{\theta_{\rho\sigma}}\over
2!}[\lambda^{\rho},\lambda^{\sigma}]\right).$ (70)

Unlike Dirac matrices the $\lambda^{\mu}$ don’t obey a Clifford algebra, so
the $6\times 6$ matrix $M$ in (69) is less straightforward to invert than a
conventional Dirac operator. We start by checking its determinant, introducing
the notation $\partial_{\mu}\lambda^{\mu}\equiv\partial\cdot\lambda$:

$\displaystyle\mbox{det}M$ $\displaystyle=$
$\displaystyle-\mbox{det}\left(m^{2}\mathbb{1}_{3\times
3}+(\partial\cdot\lambda)^{2}\right)$ $\displaystyle=$
$\displaystyle-\exp\left[\mbox{tr}(\ln
m^{2})+\mbox{tr}\left(+{{(\partial\cdot\lambda)^{2}}\over
m^{2}}-{{(\partial\cdot\lambda)^{4}}\over
2m^{4}}+{{(\partial\cdot\lambda)^{6}}\over 3m^{6}}-\cdots\right)\right]$

Now use $(\partial\cdot\lambda)^{2}=-\Delta{\cal P}$ and $\mbox{tr}{\cal P}=2$
to write

$\mbox{det}M=-\exp\left(3\ln m^{2}+2\ln(1-{\Delta\over
m^{2}})\right)=-m^{2}(m^{2}-\Delta)^{2}.$ (72)

Again, the determinant vanishes if $m=0$. The propagator exists for $m\not=0$
and is given by

$\langle\Upsilon\bar{\Upsilon}\rangle={1\over{m^{2}-\Delta}}\left(\begin{matrix}\partial\cdot\lambda&m-m^{-1}\partial_{\mu}\partial_{\nu}\\\
m-m^{-1}\partial_{\mu}\partial_{\nu}&-\partial\cdot\lambda\end{matrix}\right).$
(73)

Also note that

$\displaystyle{-1\over{\Delta(m^{2}-\Delta)}}\left[\begin{matrix}(\partial\cdot\lambda)^{3}&m(\partial\cdot\lambda)^{2}\\\
m(\partial\cdot\lambda)^{2}&-(\partial\cdot\lambda)^{3}\end{matrix}\right]M$
$\displaystyle=$
$\displaystyle{1\over{m^{2}-\Delta}}\left[\begin{matrix}\partial\cdot\lambda&m\\\
m&-\partial\cdot\lambda\end{matrix}\right]{\cal P}M$ (74) $\displaystyle=$
$\displaystyle{\cal P}_{\mu\nu}\otimes\mathbb{1}_{2\times 2}$

The fact that $M$ in the massless limit is invertible when acting on a
transverse subspace is reminiscent of gauge theories, where the same issue
occurs due to the redundancy of the field description as a consequence of an
underlying invariance of the action under local gauge transformations of the
form $A_{\mu}\mapsto A_{\mu}+\partial_{\mu}\Lambda$. We can trace this to the
invariance of (68), after integration by parts, under

$\chi_{\mu}\mapsto\chi_{\mu}+\partial_{\mu}\vartheta_{\chi};\;\;\bar{\chi}_{\mu}\mapsto\bar{\chi}_{\mu}+\partial_{\mu}\vartheta_{\bar{\chi}};\;\;\xi_{\mu}\mapsto\xi_{\mu}+\partial_{\mu}\vartheta_{\xi};\;\;\bar{\xi}_{\mu}\mapsto\bar{\xi}_{\mu}+\partial_{\mu}\vartheta_{\bar{\chi}}.$
(75)

Here the $\vartheta(x)$ are Grassmann-valued fields, and the subscripts
emphasise that independent shifts are applied to each fermi field. For this
reason the mass term is not in general invariant under (75), consistent with
the fact that $M$ is invertible once $m\not=0$. Further note that the textbook
solution to defining a gauge-field propagator, namely to fix a gauge by adding
a covariant term of the form $\zeta^{-1}(\partial_{\mu}A_{\mu})^{2}$ to the
action, would in this case yield terms of the form , eg.
$\sim(\epsilon_{\mu\nu\lambda}\partial_{\mu}\bar{\varphi}_{\nu\lambda})(\partial_{\rho}\varphi_{\rho})$,
consistent with U(1)A but violating parity. Rather, it makes more sense to
regard the term $m(\bar{\chi}\xi+\bar{\xi}\chi)$ as the “gauge-fixing term”.

We conclude that ${\cal L}_{12}$ describes a fermion field transforming in the
spin-1 representation of the rotation group, with some features reminiscent of
a gauge field, namely that in the UV limit the only remaining degrees of
freedom are transverse, ie. helicity eigenstates, so that six components are
reduced to four. The Noether currents corresponding to symmetries (42,43) are
given by

$j_{\mu},\,j_{{\cal A}\mu}=-{i\over
4}\left[\bar{\chi}\lambda_{\mu}\chi\mp\bar{\xi}\lambda_{\mu}\xi\right].$ (76)

Assigning 3 as the timelike direction, we identify a fermion charge operator
$-i\lambda^{3}\otimes\sigma_{3}$ with $\pm$ restframe eigenstates
$F,\bar{F}=(1,\mp i,0,1,\pm i,0)^{T}$, ie. fermions (antifermions) correspond
to left(right)- and right(left)-handed circularly polarised
$\chi$($\xi$)-states, which remain transverse under SO(3) rotations.
Asymptotically the propagator scales as $k^{-1}$, as expected for a
relativistic fermion. The propagator pole at $k^{2}=-m^{2}$ again corresponds
to a physical particle.

Finally, we can use the fermion current of (76) to write the Lagrangian for
the Thirring model based on ${\cal L}_{12}$, using the $\Upsilon$-basis:

${\cal L}_{\rm rThir}={1\over
4}\bar{\Upsilon}(\partial\cdot\lambda\otimes\sigma_{3}+m\mathbb{1}\otimes\sigma_{1})\Upsilon+{g^{2}\over
64}\left(\bar{\Upsilon}\lambda_{\mu}\otimes\sigma_{3}\Upsilon\right)^{2}.$
(77)

In the same basis the invariances (42,43) read

$\displaystyle\Upsilon\mapsto e^{i\alpha}\Upsilon$ ;
$\displaystyle\;\;\;\bar{\Upsilon}\mapsto\bar{\Upsilon}e^{-i\alpha}$ (78)
$\displaystyle\Upsilon\mapsto
e^{i(\mathbb{1}\otimes\sigma_{3})\alpha}\Upsilon$ ;
$\displaystyle\;\;\;\bar{\Upsilon}\mapsto\bar{\Upsilon}e^{-i(\mathbb{1}\otimes\sigma_{3})\alpha}$
(79)

while parity is

$\Upsilon(x)\mapsto-(\mathbb{1}\otimes\sigma_{3})\Upsilon(-x);\;\;\;\bar{\Upsilon}(x)\mapsto\bar{\Upsilon}(-x)(\mathbb{1}\otimes\sigma_{3}).$
(80)

##  8 Discussion

This paper has developed the description of relativistic fermions in the
language of differential geometry, originally set out in [12], to three
spacetime dimensions. The principal result is the specification of a continuum
field theory sharing the same parity and global U($N)\otimes$U($N$)
invariances as the “staggered Thirring model” originally studied numerically
using lattice field theory simulations in [6]. In our view this puts the
staggered Thirring model on a firm footing as an interacting quantum field
theory distinct from the U($2N$)-invariant version based on the continuum
action (1), which is the focus of much recent numerical work. This result is
entirely consistent with Becher and Joos’ demonstration that Kähler-Dirac
fermions are the correct continuum limit for staggered lattice fermions [11].
Beyond the weak-coupling and long-wavelength limits, we’ve seen that
spin/taste entanglement is not merely a lattice artifact, but a genuine
feature of an interacting continuum field theory: tastes are not the same as
flavors.

An important consequence of regarding the $\varphi$-basis as more fundamental
than the more familiar $\psi$-basis is the response to quantum corrections
encapsulated in the proposed relation (56) relating renormalised to bare
fields, in which multiplicative renormlisation depends solely on $p$,
consistent with U($N)\otimes$U($N$) symmetry. This was demonstrated explicitly
in Sec. 6 through the recovery of interaction currents entangling spin and
taste originally found in the staggered Thirring model. However, a more
spectacular, if speculative consequence was worked out in Sec. 7, where the
assumption of a strong hierarchy of the $Z_{p}$ arising due to large anomalous
scaling dimensions in the vicinity of a renormalisation-group fixed point
motivated the investigation of truncated actions retaining just two
$p$-values. In particular the Lagrangian ${\cal L}_{\rm rThir}$ (77) was found
to be particularly compelling, describing six-component spin-1 fermions, with
fermions/antifermions being states of well-defined polarisation, and dynamics
dominated by the four components lying in the transverse subspace in the UV
limit. Could this exotica form the basis for a description of strongly
interacting fixed-point dynamics? The answer must await a controlled non-
perturbative investigation.

We conclude with a brief discussion of spin and statistics. The Lagrangian
(77) describes spin-1 fermions which in the canonical approach to field
quantisation would be represented by field operators with anticommutator
$\\{\Upsilon_{\alpha}(\vec{x},t),\Upsilon_{\beta}^{\dagger}(\vec{x}^{\prime},t)\\}=\delta^{2}(\vec{x}-\vec{x}^{\prime})\delta_{\alpha\beta}$.
An immediate concern is the apparent contradiction with the spin-statistics
theorem requiring Lorentz-invariant theories of anti-commuting fields to be
quantised with half-integer spin representations of the Lorentz group. A
symptom of the problem is revealed through the ground state expectation of the
anticommutator of fields at arbitrary spacetime separation [15]:

$\langle
0|\\{\Upsilon(x),\bar{\Upsilon}(x^{\prime})\\}|0\rangle=i(i\partial\cdot\tilde{\lambda}\otimes\sigma_{3}+m\mathbb{1}\otimes\sigma_{1})\Delta_{\rm
sym}(x^{\prime}-x).$ (81)

Here $\tilde{\lambda}_{\mu}$ are Minkowski space versions of the
$\lambda$-matrices, we have assumed that all states are defined in the
transverse subspace, and for field quantisation with the “wrong” statistics
the $PCT$ theorem dictates the appearance on the RHS of the symmetric solution
of the Klein-Gordon equation (or its generalisation):

$\Delta_{\rm sym}(x)=\int{{d^{2}\vec{k}}\over{(2\pi)^{2}}}{\cos(k\cdot
x)\over\omega(k)},$ (82)

where for free fields $\omega(k)=\sqrt{k^{2}+m^{2}}$. Now specialise to the
case of a spacelike interval $\vec{x}=x\hat{x}$ with $|\hat{x}|=1$. We find

$\langle 0|\\{\Upsilon(0),\bar{\Upsilon}(\vec{x})\\}|0\rangle={im^{3\over
2}\over{(2\pi x)^{1\over
2}}}{1\over\pi}\left[(i\hat{x}\cdot\tilde{\lambda}\otimes\sigma_{3})K_{3\over
2}(mx)+(\mathbb{1}\otimes\sigma_{1})K_{1\over 2}(mx)\right].$ (83)

The non-vanishing of the RHS of (81) outside the lightcone signals a violation
of microcausality. This is a general result independent of the detailed form
of the dispersion $\omega(k)$. For free fields the asymptotic properties of
the modified Bessel functions in (83) can be used to to find

$\lim_{x\to\infty}\langle
0|\\{\Upsilon(0),\bar{\Upsilon}(\vec{x}))\\}|0\rangle={i\over{2\pi}}{m\over
x}e^{-mx}[i\hat{x}\cdot\tilde{\lambda}\otimes\sigma_{3}+\mathbb{1}\otimes\sigma_{1}],$
(84)

and

$\lim_{x\to 0}\langle
0|\\{\Upsilon(0),\bar{\Upsilon}(\vec{x}))\\}|0\rangle={i\over{2\pi
x^{2}}}(i\hat{x}\cdot\tilde{\lambda}\otimes\sigma_{3})+{im\over{2\pi
x}}(\mathbb{1}\otimes\sigma_{1});$ (85)

that is, the causality violation is localised to within roughly a Compton
wavelength of the lightcone, but diverges as $x\to 0$, although less severely
than the $x^{-3}$ behaviour of 3+1$d$ [15].

Since microcausality is a desirable property for a fundamental theory, the
correct relation between spin and statistics is a necessary ingredient of a
complete quantum field theory. By hypothesis, however, the spin-1 action (77)
serves only as an effective description of the dynamics near a UV fixed point,
in the deep Euclidean regime $k^{2}\to\infty$ very far from the lightcone. The
question of whether the spin-statistics linkage compromises the fixed-point
description remains open.

## Acknowledgements

This work was supported by STFC Consolidated Grant ST/T000813/1.

## References

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11institutetext:  Department of Physics, The Applied Math Program, and
Department of Astronomy,
The University of Arizona, AZ 85721, USA

Inflation Cosmology Relativistic Astrophysics

# Model Selection with Baryonic Acoustic Oscillations
in the Lyman-$\alpha$ Forest

F. Melia

###### Abstract

The recent release of the final, complete survey of Lyman-$\alpha$ baryonic
acoustic oscillation measurements provides the most significant and accurate
data base for studying cosmic geometry at an effective redshift $z_{\rm
eff}=2.334$, which is inaccessible to other sources. In this Letter, we use
these data to select among four distinct cosmologies: Planck $\Lambda$CDM, the
$R_{\rm h}=ct$ universe, the Milne universe and Einstein-de Sitter. Given the
breadth and depth of the Lyman-$\alpha$ study, this BAO measurement alone
provides a strong model comparison, complementary to previous studies that
combined Lyman-$\alpha$ data with measurements at lower redshifts. Though both
approaches are useful, the latter tends to dilute the disparity between model
predictions and the observations. We therefore examine how the models compare
to each other strictly based on the BAO scale measured in the Lyman-$\alpha$
forest and background quasars. We find that Milne and Einstein-de Sitter are
strongly ruled out by these data. There is also strong evidence disfavoring
the standard model. The Lyman-$\alpha$ measurements are completely consistent
with the cosmic geometry predicted by $R_{\rm h}=ct$. As such, evidence
continues to grow that the zero active mass condition from general relativity
ought to be an essential ingredient in $\Lambda$CDM.

###### pacs:

98.80.Cq

###### pacs:

98.80.-k

###### pacs:

98.80.Jk

##  1 Introduction

The angular power spectrum in the cosmic microwave background (CMB) is
characterized by an angular scale set at the surface of last scattering, when
the radiation began dissociating from baryonic matter. Believed to represent a
sonic horizon, $r_{d}$, this comoving distance is apparently associated with
sound waves created at or near the Planck regime, which then propagated across
the baryon-photon fluid until the CMB was produced [1, 2, 3].

In a Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology, in which all proper
(or physical) distances grow in proportion to a universal scale factor,
$a(t)$, while comoving distances remain constant, the scale $r_{d}$ is
expected to have remained unchanged, reappearing subsequently as a
characteristic length in the matter correlation function. In this context, the
sonic scale is more commonly inferred from baryonic acoustic oscillations
(BAO) seen in the large-scale structure, the first measurement of which was
carried out using the auto-correlation of galaxy positions [4] at $z\sim
0.35$, and the galaxy power spectrum [5] at $z\sim 0.1$. More generally, the
BAO scale at $z\lesssim 2$ has been studied using a variety of discrete
tracers, such as galaxy clusters [6], and quasars [7], in addition to the
aforementioned galaxies [8, 9, 10, 11, 12, 13, 14, 15].

Beyond $z\sim 2$, however, a measurement of the BAO scale must to be handled
differently because the number density of observable dicrete tracers is too
low for high precision clustering studies. The method of choice for measuring
$r_{d}$ in the high-redshift Universe instead rests on the observation of
opacity fluctuations in the Lyman-$\alpha$ forest irradiated by background
quasars. The first such studies, focusing on the Lyman-$\alpha$ auto-
correlation function, were carried out by refs. [16], [17], [18], [19], [20]
and [21]. Complementary results based on the Lyman-$\alpha$ and quasar cross-
correlation function have also been reported by refs. [22], [23] and [24].

The BAO feature is a powerful diagnostic in cosmology because it yields both
angular-diameter distances and the expansion rate normalized to the sound
horizon $r_{d}$. And while the luminosity distance to type Ia supernovae has
already provided strong evidence for the existence of dark energy [25, 26,
27], these transient events are difficult to observe at redshifts $z\gtrsim
1.8$.

The BAO measurements in the Lyman-$\alpha$ forest, at an effective $z\sim
2.334$, may therefore be used in several distinct tests of the geometry of the
Universe at intermediate redshifts not accessible to local surveys focusing on
supernovae ($z\lesssim 1.8$) and instruments designed to study the CMB at
$z\gg 1$.

A common approach is to combine all of the BAO data, those from the galaxy
surveys at low redshifts and those from the Lyman-$\alpha$ forest farther
away, under the assumption that $r_{d}$ is independent of $z$. This can be
done either in the context of $\Lambda$CDM, where one includes the predicted
value of $r_{d}$ in this model to extract the redshift-dependent angular-
diameter distance and expansion rate, or in model selection by examining which
cosmology is preferred by the BAO measurements. For the latter, one considers
$r_{d}$ to be ‘unanchored,’ since its value may not be the same from one model
to the next. In this case, one either optimizes $r_{d}$ along with the other
model parameters individually for each cosmology being tested, or avoids it
altogether by considering ratios of the angular-diameter and Hubble distances,
both of which are proportional to the sonic horizon (see Eqs. 3 and 4 below).

In previous work [28, 29, 30], we have followed the latter approach using
older, less complete BAO catalogs than those available now to compare the
standard model with one of its principal competitors known as the $R_{\rm
h}=ct$ universe [31, 32]. Interestingly, the aggregated BAO data have tended
to favour the latter model rather than $\Lambda$CDM. But the BAO measurements
using the lower redshift galaxy surveys are less discerning than their higher
redshift counterparts, so while the outcome of these studies has been
suggestive, it has not necessarily been compelling.

In view of the significantly more complete eBOSS catalog available now (see
section labeled ‘Data’ below), however, the BAO data at $z=2.334$ by
themselves should constitute an important constraint on the geometry of the
Universe sampled solely by the quasars in this catalog (distributed at
$z\gtrsim 1.77$) and their Lyman-$\alpha$ forests. Since it is well known that
a comparison of the model predictions and the BAO measurements tends to become
more discordant with increasing redshift, it is desirable to use the
Lyman-$\alpha$ data by themselves for model selection purposes, in parallel to
the already completed studies based on the BAO observations at various
redshifts.

In this Letter, we therefore complement our previous model selection analysis
by restricting the comparison to just the Lyman-$\alpha$ measurements. As
noted, we do this for two principal reasons. First, the model contrast at
$z\gtrsim 2$ is significantly greater than that at $z\sim 0$. Second, we now
have the final BAO constraints from the Lyman-$\alpha$ auto-correlation and
Lyman-$\alpha$-quasar cross-correlation functions produced from the sixteenth
(DR16) and final release [33] of the fourth generation Sloan Digital Sky
Survey (SDSS-IV), containing all of the clustering and Lyman-$\alpha$ data
from the completed ‘extended’ Baryonic Oscillation Spectroscopic Survey
(eBOSS) [34]. This catalog of quasars and Lyman-$\alpha$ profiles is so large
(see ‘Data’ below) that the BAO scale measured from it at an effective
redshift $\sim 2.334$ constitutes a crucial probe of the cosmology on its own
merit. As we shall see, these data alone provide an important comparison of
cosmological models and their predictions without having to pre-assume $r_{d}$
and $H_{0}$. The latter is especially desirable in view of the growing
disparity between the measurements of $H_{0}$ at low and high redshifts,
creating a $\sim 4\sigma$ uncertainty in its value [35].

We shall find that the depth and extent of the final Lyman-$\alpha$ BAO
analysis [36], along with the option of altogether avoiding the use of $r_{d}$
and $H_{0}$, provides us with a very clean and compelling test of the various
FLRW cosmologies.

##  2 Data

The analysis in this Letter utilizes the baryonic acoustic oscillations (BAO)
measured in the Lyman-$\alpha$ absorption and background quasars, with an
effective (enemble) redshift $z=2.334$. The data are taken from the complete
extended Baryonic Oscillation Spectroscopic Survey (eBOSS) [36], which
includes the Lyman-$\alpha$ absorption profiles of 210,005 background quasars
distributed at $z_{q}>2.10$. The BAO scale has been measured in both the auto-
correlation of the Lyman-$\alpha$ absorbers and their cross-correlation with
341,468 quasars at $z_{q}>1.77$. This data release represents several advances
over previously published Lyman-$\alpha$ BAO measurements, including improved
statistics from a larger quasar catalog and deeper observations, and a more
accurate modeling of the systematics.

##  3 Model Comparisons

The Lyman-$\alpha$ BAO survey measures the BAO scale in the Lyman-$\alpha$
forest of absorption of light from distant quasars, both via the forest-forest
correlation function [19], and in the forest-quasar cross-correlation [22].
The scale is measured along the line-of-sight,

$\Delta z={r_{d}\over(1+z)\,d_{H}(z)}\;,$ (1)

and in the transverse direction,

$\Delta\theta={r_{d}\over d_{M}(z)}$ (2)

where, as previously noted, $r_{d}$ is the length corresponding to the peak of
the matter two-point function in comoving coordinates. In addition, the
quantity

$d_{H}(z)\equiv{c\over H(z)}$ (3)

is the Hubble scale at $z$, while $d_{M}(z)$ is given in terms of the angular-
diameter distance via the relation

$d_{M}(z)\equiv(1+z)\,d_{A}(z)\;.$ (4)

Table 1: Model comparison using the Lyman-$\alpha$ BAO data Data/Model |  ${d_{M}/d_{H}}\quad\;\,$ (P-value)
---|---
Data | $4.17\pm 0.18$
1\. $R_{\rm h}=ct$ |  $4.02\qquad\quad\;\;\;$ ($0.39$)
2\. Planck-$\Lambda$CDM |  $4.56\qquad\quad\;$ ($0.03$)
3\. Milne |  $5.058\quad$ ($<0.00001$)
4\. Einstein-de Sitter |  $5.507\quad$ ($<0.00001$)

We mention here that surveys often also define the distance

$d_{V}(z)\equiv z^{1/3}d_{M}^{2/3}(z)\,d_{H}^{1/3}(z)\;,$ (5)

an angle-weighted average of $d_{M}(z)$ and $d_{H}(z)$, but this quantity was
not derived independently of $d_{M}$ and $d_{H}$ for these data [36]. Thus,
though it tends to be the best determined scale in measurements of BAO from
galaxy surveys, we will not find it useful for our analysis in this particular
instance, and we shall instead focus our attention on $d_{M}$ and $d_{H}$
themselves.

Given that these BAO observables are proportional to $r_{d}H_{0}$ (see below
for the model-dependent functional form of $H[z]$), the ratios of the various
BAO scales are completely independent of the sonic horizon and Hubble
constant. Indeed, as long as we restrict our comparisons to these ratios,
three of the models we compare here have no free parameters at all, while the
standard model, $\Lambda$CDM (see Eq. 11 below), has just one: the scaled
matter density $\Omega_{\rm m}$. But to avoid unduly ‘punishing’ this
cosmology by treating $\Omega_{\rm m}$ as an unknown variable when assessing
its likelihood, we shall simply adopt its Planck optimized value, $\Omega_{\rm
m}=0.315\pm 0.007$, with a spatial curvature constant $k=0$ [3].

We shall compare fits to the Lyman-$\alpha$ data using four different
cosmological models, each with its own prediction of the angular-diameter and
Hubble distances. The quantity $\Omega_{i}$ is the energy density of species
$i$, scaled to the critical density

$\rho_{\rm c}\equiv{3c^{2}H_{0}^{2}\over 8\pi G}\;,$ (6)

in terms of the Hubble constant, $H_{0}=67.4\pm 0.5$ km s-1 Mpc-1. In
principle, we could completely avoid any reliance on the Planck measurements
by selecting the value of $\Omega_{\rm m}$ that optimizes $\Lambda$CDM’s fit
to the Lyman-$\alpha$ BAO data, but the improvement is too small to justify
the introduction of an additional free parameter. The four models we compare
are:

  1. 1.

The $R_{\rm h}=ct$ universe, a Friedmann-Lemaître-Robertson-Walker cosmology
with zero active mass, $\rho+3p=0$, in terms of the total energy density
($\rho$) and pressure ($p$) in the cosmic fluid [31, 32]. In this case,

$d^{\,(1)}_{M}(z)=\frac{c}{H_{0}}\ln(1+z)\;,$ (7)

and

$d^{\,(1)}_{H}(z)=\frac{c}{H_{0}(1+z)}\;.$ (8)

  2. 2.

Flat Planck-$\Lambda$CDM, with $\Omega_{\Lambda}=1-\Omega_{\rm m}$ and a dark-
energy equation of state parameter $w_{\rm de}=-1$. For this model,

$d^{\,(2)}_{M}(z)=\frac{c}{H_{0}}\int_{0}^{\,z}\frac{du}{E(u)}\;,$ (9)

and

$d^{\,(2)}_{H}(z)=\frac{c}{H_{0}E(z)}\;,$ (10)

where

$E(z)\equiv\left[\Omega_{\rm m}(1+z)^{3}+\Omega_{\Lambda}\right]^{1/2}\;.$
(11)

At these redshifts, the contribution to $E(z)$ from radiation is negligible,
so we have ignored it for the calculation of $d_{H}(z)$ and $d_{M}(z)$.

  3. 3.

The Milne universe. This well studied solution is also a Friedmann-Lemaître-
Robertson-Walker cosmology, but its energy density, pressure and cosmological
constant are all zero. Its expansion instead derives from a non-zero spatial
curvature, with $k=-1$. Like the $R_{\rm h}=ct$ universe, the Milne scale
factor, $a(t)$, is also linear in time [37, 38], but we include it here
primarily because the observable signatures in these two models are very
different. In the Milne universe,

$d^{\,(3)}_{M}(z)=\frac{c}{H_{0}}\sinh\left[\ln(1+z)\right]\;,$ (12)

and

$d^{\,(3)}_{H}(z)=\frac{c}{H_{0}(1+z)}\;.$ (13)

  4. 4.

Einstein-de Sitter (i.e., Eqs. 9 and 10 with $\Omega_{\rm m}=1$ and
$\Omega_{\Lambda}=0$):

$d^{\,(4)}_{M}(z)=\frac{2c}{H_{0}}\left(1-\frac{1}{\sqrt{1+z}}\right)\;,$ (14)

and

$d^{\,(4)}_{H}(z)=\frac{c}{H_{0}(1+z)^{3/2}}\;.$ (15)

This cosmology is already heavily disfavoured by many other observations, but
we include it here because of its relevance as a former ‘standard’ model.

The combined BAO measurements from the auto- and cross-correlation in the
final, complete eBOSS release [36] yield the following constraints:

$\displaystyle d_{H}(z=2.334)/r_{d}$ $\displaystyle=$ $\displaystyle 8.99\pm
0.19$ $\displaystyle d_{M}(z=2.334)/r_{d}$ $\displaystyle=$ $\displaystyle
37.5\pm 1.1\;.$ (16)

Though a ‘fiducial’ cosmology is employed in the calculation of these
distances, the ratios shown in Equation (16) are model independent, as studied
in detail by ref. [39] in the context of galaxy correlations, and confirmed
for the Lyman-$\alpha$ observations by ref. [36].

The model predictions are compared with these data in Table 1, prioritized in
terms of the p-values estimated from the various fits. Given a model’s
prediction for $R^{\rm th}\equiv d_{M}/d_{H}$ and the standard deviation (in
this case, $\sigma_{R}=0.18$) of the measurement, this p-value represents the
probability of observing a difference $|R^{\rm obs}-R^{\rm th}|$ greater than
$|4.17-R^{\rm th}|$ under the assumption that the null hypothesis is true and
that the distribution of $R^{\rm obs}$ is normal. The error quoted for the
measured value of $d_{M}/d_{H}$ includes the correlation between $d_{M}/r_{d}$
and $d_{H}/r_{d}$, characterized by the correlation coefficient
$C(d_{H},d_{M})=-0.45$ [36]. Small p-values provide evidence against the
assumed model, with the evidence getting stronger as the p-value approaches
zero. Typical guidelines suggest the following hierarchy: ($p>0.10$) weak or
no evidence; ($0.05<p\leq 0.10$) moderate evidence; ($0.01<p\leq 0.05$) strong
evidence; ($p\leq 0.01$) very strong evidence.

On the basis of these comparisons, it is clear that the Lyman-$\alpha$ BAO
measurements very strongly rule out the Milne and Einstein-de Sitter
cosmologies, affirming an already established conclusion drawn from many other
comparative tests [32]. There is also strong evidence disfavoring the standard
model, an outcome confirmed by several previous studies, including those
reported in refs. [36] and [40]. But the principal result of this work is that
the Lyman-$\alpha$ BAO measurements are completely consistent with the
geometry of the cosmos predicted by the $R_{\rm h}=ct$ universe.

Pursuing this head-to-head comparison further, we note that if $\Omega_{\rm
m}$ were to differ from its Planck value by $3\sigma$, i.e., if $\Omega_{\rm
m}=0.294$, then the p-value for $\Lambda$CDM would improve somewhat to
$0.065$, so the evidence against the standard model in that case would be
‘moderate’ instead of ‘strong.’ Nevertheless, to make the standard model
equally likely with $R_{\rm h}=ct$, i.e., to improve its p-value to $0.39$,
$\Omega_{\rm m}$ would have to be smaller than $\sim 0.236$, a value different
by many standard deviations from that inferred by Planck.

It is also straightforward to perform a Bayesian analysis [41, 42] of the
head-to-head comparison between $\Lambda$CDM and $R_{\rm h}=ct$ based on the
Lyman-$\alpha$ data at $z=2.334$, specifically, the measurement of
$d_{M}/d_{H}=4.17\pm 0.18$ shown in Table 1. Under the assumption that this
ratio follows a normal distribution, and adopting a point model for both the
null and alternative hypotheses (remember that we are fixing the value of
$\Omega_{\rm m}$ at the Planck measurement to avoid unduly ‘punishing’ the
standard model for having an additional free parameter), one infers a Bayes
factor of 7.39. The marginal likelihoods at the measured value of
$d_{M}/d_{H}$, from which the Bayes factor is estimated, are shown in Figure
1.

Figure 1: Estimation of the Bayes factor from the marginal prediction of the
$R_{\rm h}=ct$ and $\Lambda$CDM models versus the measurement of the ratio
$d_{M}/d_{H}$ from the Lyman-$\alpha$ forest at an effective redshift
$z=2.334$.

A Bayes factor between 3 and 10 (as we have here) indicates ‘moderate’ or
‘substantial’ evidence in favor of the alternative hypothesis [43, 44, 45]. As
such, the Bayesian analysis complements and confirms our earlier conclusion,
derived from the p-values, that the Lyman-$\alpha$ BAO data at $z=2.334$ favor
the $R_{\rm h}=ct$ cosmology over the current standard model.

##  4 Conclusion

The strong rejection of the Milne and Einstein-de Sitter cosmologies by the
final Lyman-$\alpha$ BAO data release is hardly surprising in view of their
similarly strong rejection by other observations. Our main conclusion from
this study instead refocuses our attention on the fact that the data tend to
favour the $R_{\rm h}=ct$ cosmology over the current standard model.

Indeed, the BAO measurements offer a new perspective on this comparison,
following an earlier examination by ref. [40], who attempted to identify the
reason behind the BAO anomaly in the context of $\Lambda$CDM. Their analysis
showed that the BAO data at $z>0.43$ are in tension with the standard model,
whether or not the Planck optimized parameters (e.g., for $\Omega_{\rm m}$)
are assumed. They concluded that this tension arises not from the $\Lambda$CDM
parameters, but instead from the dark energy evolution at $0.57<z<2.334$. If
one further sets $r_{d}$ equal to the acoustic scale measured in the CMB, a
cosmological constant for dark energy is firmly rejected.

The $R_{\rm h}=ct$ cosmology is essentially $\Lambda$CDM, though with the
crucial additional constraint of zero active mass from general relativity
[32]. This constraint features an equation-of-state, $\rho+3p=0$, in terms of
the total energy density $\rho$ and pressure $p$ in the cosmic fluid. Dark
energy is therefore dynamic in this model, evolving along with the other
constituents, presumably an extension to the standard model of particle
physics.

The current standard model suffers from several major conflicts and
inconsistencies that continue to defy serious attempts at mitigation [46].
Together with the continued success of $R_{\rm h}=ct$ to account for the data
better than $\Lambda$CDM [47], as affirmed by the work reported in this
Letter, the prospects for further development of this alternative FLRW
cosmology look very promising. An especially exciting future observation to
anticipate over the coming decade is the real-time measurement of redshift
drift [48, 49], which should provide an unambiguous yes/no answer to the
question of whether or not the cosmic fluid is in fact driven by a zero active
mass equation-of-state.

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