# Tensor gauge fields and dark matter in general relativity with fermions

Kevin Cahill<EMAIL_ADDRESS>Department of Physics and Astronomy
University of New Mexico
Albuquerque, New Mexico 87106

(August 28, 2024)

###### Abstract

The action of general relativity with fermions has two independent symmetries:
general coordinate invariance and local Lorentz invariance. General coordinate
transformations act on coordinates and tensor indices, while local Lorentz
transformations act on Dirac and Lorentz indices, much like a noncompact
internal symmetry. The internal-symmetry character of local Lorentz invariance
suggests that it might be implemented by tensor gauge fields with their own
Yang-Mills action rather than by the spin connection as in standard
formulations. But because the Lorentz group is noncompact, their Yang-Mills
action must be modified by a neutral vector field whose average value at low
temperatures is timelike. This vector field and the tensor gauge fields are
neutral and interact gravitationally, so they contribute to hot and cold dark
matter. The two independent symmetries of the action are reduced to a single
symmetry of the vacuum, local Lorentz invariance, by the nonzero average
values of the tetrads $c^{a}_{\phantom{a}k}$. The local Lorentz invariance of
general relativity with fermions can be extended to local U(2,2) invariance.
If the contracted squares of the covariant derivatives of the tetrads
multiplied by the square of a mass $M$ are added to the action, then in the
limit $M^{2}\to\infty$, the equation of motion of the tensor gauge fields is
the vanishing of the covariant derivatives of the tetrads, which is Cartan’s
first equation of structure. In the same limit, the tensor gauge fields
approach the spin connection.

##  I Introduction

The action of general relativity with fermions has two independent symmetries:
general coordinate invariance and local Lorentz invariance. These symmetries
are traditionally implemented by Cartan’s tetrads $c^{a}_{\phantom{a}i}$ and
by the spin connection $\omega_{i}$ which is a quartic polynomial in the
tetrads and their first derivatives. In this paper, the spin connection is
replaced by tensor gauge fields with their own Yang-Mills action.

General coordinate invariance is the defining symmetry of Einstein’s general
relativity. A general coordinate transformation $x\to x^{\prime}$ acts on
coordinates $x^{i}$ and on tensor indices $i,k$ but leaves Dirac indices
$\alpha,\beta$ and Lorentz indices $a,b$ unchanged

$\displaystyle\psi^{\prime}_{\alpha}(x^{\prime})={}\psi_{\alpha}(x),\quad
c_{a}^{\prime\phantom{a}i}(x^{\prime})={}\frac{\partial x^{\prime i}}{\partial
x^{k}}\,c_{a}^{\phantom{a}k}(x).$ (1)

General coordinate invariance is implemented by Cartan’s tetrads
$c^{a}_{\phantom{a}k}$ and their derivatives.

Local Lorentz transformations act on Dirac and Lorentz indices but leave
coordinates and tensors unchanged

$\begin{split}\psi^{\prime}_{\alpha}(x)={}&D_{\alpha\beta}(\Lambda(x))\,\psi_{\beta}(x)\\\
(D_{i}\psi)^{\prime}_{\alpha}={}&(\partial_{i}+\omega^{\prime}_{i})^{\alpha\beta}\psi^{\prime}_{\beta}=D_{\alpha\beta}(\Lambda)\,(D_{i}\psi)_{\beta}\\\
c^{\prime
a}_{\phantom{{}^{\prime}a}i}(x)={}&\Lambda^{a}_{\phantom{a}b}(x)\,c^{b}_{\phantom{b}i}(x).\end{split}$
(2)

Local Lorentz invariance is implemented by the spin connection $\omega_{i}$ in
standard formulations [1, 2, 3, 4, 5, 6, 7].

Invariance under general coordinate transformations and invariance under local
Lorentz transformations are both exact and independent symmetries of the
action of general relativity with fermions. But while general coordinate
transformations (1) act on coordinates and tensor indices, local Lorentz
transformations (2) act on Lorentz and Dirac indices leaving coordinates
unchanged. In this respect, local Lorentz invariance is like a noncompact
internal symmetry [8, *Cahill:1979qt, *Cahill:1980, *Cahill:1981rq].

These observations motivated an attempt [12] to implement local Lorentz
invariance by means of tensor gauge fields $L^{ab}_{\phantom{ab}i}$, but the
focus was so exclusively upon fermions and the spin connection, that the
invariance of the action of the tensor gauge fields was neglected and repaired
belatedly in an erratum [13]. In that erratum, a hermitian matrix $h(x)$ was
introduced that under a local Lorentz transformation $\Lambda(x)$ transforms
as $h^{\prime}(x)=D^{\dagger}(\Lambda(x))h(x)D(\Lambda(x))$ in which
$D(\Lambda)$ is Dirac’s $D^{(1/2,0)}\oplus D^{(0,1/2)}$ representation of
$SO(3,1)$.

In the present paper, local Lorentz invariance is implemented by means of
tensor gauge fields $L^{ab}_{\phantom{ab}i}$ and a real vector field $K_{i}$
in terms of which the matrix $h$ is realized as
$h={}i\beta\gamma^{a}c_{a}^{\phantom{a}i}K_{i}=i\beta\gamma^{a}K_{a}$. The
spin connection
$\omega_{i}={}{\textstyle{\frac{1}{8}}}\omega^{ab}_{\phantom{ab}i}\bm{[}\gamma_{a},\gamma_{b}\bm{]}$
is replaced by a “Lorentz connection”
$L_{i}={}{\textstyle{\frac{1}{8}}}L^{ab}_{\phantom{ab}i}\bm{[}\gamma_{a},\gamma_{b}\bm{]}$
with field-strength
$F_{ik}={}\bm{[}\partial_{i}+L_{i},\partial_{k}+L_{k}\bm{]}$ and Yang-Mills-
like action

$S_{L}={}-\frac{1}{4m^{2}\lambda^{2}}\int{\rm
Tr}\left(F^{\dagger}_{ik}\,h\,F^{ik}\,\beta h\beta\right)\sqrt{g}\,d^{4}x$ (3)

in which $\beta=i\gamma^{0}$, $g=|\det(g_{ik})|$, and $\lambda$ is a coupling
constant. The vector field $K_{i}(x)$ makes the trace in the action $S_{L}$
invariant under noncompact Lorentz transformations, but the squares of the
time derivatives $(\dot{L}^{ab}_{\phantom{ab}i})^{2}$ appear in $S_{L}$ with
positive signs only if the average value of $K_{i}$ is timelike.

The action of the vector boson $K_{i}$ is

$\begin{split}S_{K}={}&\int\Big{[}-{\textstyle{\frac{1}{4}}}(D_{i}K_{k}-D_{k}K_{i})(D^{i}K^{k}-D^{k}K^{i})-{\textstyle{\frac{1}{4}}}\xi^{2}(K_{i}K^{i}+m^{2})^{2}\Big{]}\sqrt{g}\,d^{4}x\end{split}$
(4)

in which $\xi>0$ is a positive coupling constant. The potential energy density
${\textstyle{\frac{1}{4}}}\xi^{2}(K_{i}K^{i}+m^{2})^{2}$ makes the average
value $\langle 0|K_{i}(x)|0\rangle$ of $K_{i}(x)$ timelike at low
temperatures.

At low temperatures, $T\ll\xi m$, the vibrations about $\langle
0|K_{i}(x)|0\rangle$ are massless and neutral, and so would contribute to hot
dark matter. But at high temperatures, $T\gg\xi m$, the the field $K_{i}$
radiates particles of mass $m_{K}=\xi m$ which would contribute to cold dark
matter low temperatures.

The tensor gauge fields $L^{ab}_{\phantom{ab}i}$ are neutral and massless and
so like the vector field $K_{i}$ would contribute to hot dark matter.

Since the tensor gauge fields $L^{ab}_{\phantom{ab}i}$ and the vector boson
$K_{i}$ interact with gravitational strength, they would have decoupled much
earlier than the photons, and so would not have been heated by the
annihilations of quarks and leptons. They would have a present temperature
much colder than the 2.7 K of the CMB, which decoupled when the present
universe was 380,000 years old and had a temperature of about 0.26 eV.

In terms of the connection $L_{i}$ and the gauge fields of the standard model
$A_{i}={}iA^{\alpha}_{is}\,t^{\alpha s}$, the covariant derivative of a Dirac
field $\psi$ is defined in this paper as

$D_{i}\psi={}\big{(}\partial_{i}+L_{i}+A_{i}\big{)}\psi$ (5)

rather than in terms of the spin connection as

$D_{i}\psi={}\big{(}\partial_{i}+\omega_{i}+A_{i}\big{)}\psi$ (6)

in which
$\omega_{i}={}{\textstyle{\frac{1}{8}}}\omega^{ab}_{\phantom{ab}i}\bm{[}\gamma_{a},\gamma_{b}\bm{]}$
and [1, 2, 3, 4, 5, 6, 7]

$\displaystyle\omega^{ab}_{\phantom{ab}i}={}$
$\displaystyle{\textstyle{\frac{1}{2}}}c^{aj}\left(\partial_{i}c^{b}_{\phantom{b}j}-\partial_{j}c^{b}_{\phantom{b}i}\right)-{\textstyle{\frac{1}{2}}}c^{bj}\left(\partial_{i}c^{a}_{\phantom{a}j}-\partial_{j}c^{a}_{\phantom{a}i}\right)-{\textstyle{\frac{1}{2}}}c^{ak}c^{b\ell}c^{c}_{\phantom{c}i}\left(\partial_{k}c_{c\ell}-\partial_{\ell}c_{ck}\right).$
(7)

The present formalism has a serious disadvantage: it introduces one vector
boson $K_{i}$ and six tensor gauge fields $L^{ab}_{\phantom{ab}i}$. On the
other hand, it has three advantages:

  1. 1.

It treats local Lorentz invariance as an internal symmetry and gives it a
Yang-Mills action.

  2. 2.

It adds to the usual internal-symmetry gauge fields
$A_{i}={}iA^{\alpha}_{is}\,t^{\alpha s}$ in the Dirac covariant derivative
$D_{i}\psi={}\big{(}\partial_{i}+L_{i}+A_{i}\big{)}\psi$ a linear combination
of gauge fields
$L_{i}={}{\textstyle{\frac{1}{8}}}L^{ab}_{\phantom{ab}i}\bm{[}\gamma_{a},\gamma_{b}\bm{]}$
and not a quartic polynomial $\omega_{i}$ in the tetrads and their
derivatives.

  3. 3.

At high temperatures, the neutral vector field $K_{i}$ radiates particles of
mass $\xi m$ which at low temperatures would contribute to cold dark matter.

The action $S_{L}$ is discussed in Sec. II. The matrix $h$, the vector
$K_{i}$, and the positive signs of squares of the time derivatives
$(\dot{L}^{ab}_{\phantom{ab}i})^{2}$ are discussed in Sec. III. The Dirac
action

$S_{D}={}-\int\bar{\psi}\,\gamma^{a}c_{a}^{\phantom{a}i}D_{i}\psi\,\sqrt{g}\,d^{4}x$
(8)

in which $D_{i}$ is the covariant derivative (5) is discussed in Sec. IV. The
actions $S_{L}$, $S_{K}$ and $S_{D}$ are invariant under local Lorentz
transformations and under independent general coordinate transformations.

Although general coordinate invariance and local Lorentz invariance are
independent symmetries of the action, they are not independent symmetries of
the ground state of the universe because they do not leave invariant the
nonzero average values of Cartan’s tetrads $c^{a}_{\phantom{a}k}$. Their
average values $\langle 0|c^{a}_{\phantom{a}k}|0\rangle$ or ${\rm
Tr}(\rho\,c_{\phantom{a}i}^{a}(x))$ reduce the symmetries of the action —
general coordinate invariance and local Lorentz invariance — to a single
symmetry of the ground state: local Lorentz invariance. Since nonzero average
tetrad values are intrinsic to the theory, this reduction of symmetry is
intrinsic rather than spontaneous. It is discussed in Sec. V.

Local invariance under the Lorentz group SO(3,1) is extended to U(2,2) in
Sec.VI.

If tensor gauge fields do gauge SO(3,1) and if all invariant terms of
dimension $(\text{mass})^{4}$ or less occur in the action (which is not
obvious in a theory of gravity), then the total action would include the
contracted squares

$S_{C}={}-M^{2}\int
D_{i}c^{a}_{\phantom{a}k}\,D^{i}c_{a}^{\phantom{a}k}\,\sqrt{g}\,d^{4}x$ (9)

of the covariant derivatives of the tetrads

$D_{i}c^{a}_{\phantom{a}k}={}\partial_{i}c^{a}_{\phantom{a}k}+L_{i\phantom{a}b}^{\phantom{i}a}\,c^{b}_{\phantom{a}k}-\Gamma^{\ell}_{\phantom{\ell}ki}\,c^{a}_{\phantom{a}\ell}.$
(10)

The factor $M^{2}$ is required because tetrads are dimensionless. The linear
combination
$L_{i\phantom{a}b}^{\phantom{i}a}\,c^{b}_{\phantom{a}k}-\Gamma^{\ell}_{\phantom{\ell}ki}\,c^{a}_{\phantom{a}\ell}$
of the tensor gauge field and the Levi-Civita connection acquires a mass of
order $M$. In the limit $M\to\infty$ the equation of motion of the tensor
gauge fields $L^{ab}_{\phantom{ab}i}$ arises from the term $S_{C}$ and is
$D_{i}c^{a}_{\phantom{a}k}={}0$ which is Cartan’s first equation of structure.
In the same limit, the tensor gauge fields approach the spin connection as
described in Sec. VII.

##  II Action of Tensor Gauge Bosons

The action (3) proposed for the tensor gauge fields is

$S_{L}={}-\frac{1}{4m^{2}\lambda^{2}}\int{\rm
Tr}\left(F^{\dagger}_{ik}\,h\,\,F^{ik}\,\beta h\beta\right)\sqrt{g}\,d^{4}x$
(11)

in which the field strength $F_{ik}$ is

$F_{ik}={}\bm{[}\partial_{i}+L_{i},\partial_{k}+L_{k}\bm{]}\,,$ (12)

the matrix of tensor gauge fields $L_{i}$ is

$\begin{split}L_{i}={}&{\textstyle{\frac{1}{8}}}\,L^{ab}_{\phantom{ab}i}\,\bm{[}\gamma_{a},\gamma_{b}\bm{]}\,,\end{split}$
(13)

$h={}i\beta\gamma^{a}c_{a}^{\phantom{a}i}K_{i}$ is a $4\times 4$ hermitian
matrix, $K_{i}$ is a real vector field, and $\lambda$ is a coupling constant.

The action (11) is real because

$\begin{split}{\rm Tr}\big{(}F^{\dagger}_{ik}\,h\,F^{ik}\,\beta
h\beta\big{)}^{*}={}&{\rm Tr}\big{(}\beta
h\beta\,F^{{\dagger}ik}\,h\,F_{ik}\big{)}={}{\rm
Tr}\big{(}F^{\dagger}_{ik}\,h\,F^{ik}\,\beta h\beta\big{)}.\end{split}$ (14)

The gamma matrices

$\begin{split}\gamma^{0}={}&-i\,\begin{pmatrix}0&1\\\ 1&0\\\
\end{pmatrix},\qquad\qquad\beta={}i\gamma^{0}=\begin{pmatrix}0&1\\\ 1&0\\\
\end{pmatrix},\qquad i\beta\gamma^{0}={}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\\\ {}\gamma^{i}={}&-i\,\begin{pmatrix}0&\sigma^{i}\\\
{}-\sigma^{i}&0\\\ \end{pmatrix},\qquad
i\beta\bm{\gamma}=\begin{pmatrix}-\bm{\sigma}&0\\\
0&\bm{\sigma}\end{pmatrix},\qquad\,\,\,\gamma^{5}=\begin{pmatrix}1&0\\\
0&-1\\\ \end{pmatrix}\end{split}$ (15)

satisfy $\\{\gamma^{a},\gamma^{b}\\}=2\eta^{ab}I$. The commutators
$\bm{[}\gamma_{a},\gamma_{b}\bm{]}$ in
$L_{i}={}-{\textstyle{\frac{1}{8}}}L^{ab}_{\phantom{ab}i}\,\bm{[}\gamma_{a},\gamma_{b}\bm{]}$
are for spatial $a,b,c=1,2,3$

$\bm{[}\gamma_{a},\gamma_{b}\bm{]}=2i\epsilon_{abc}\sigma^{c}I\quad\mbox{and}\quad\bm{[}\gamma_{0},\gamma_{a}\bm{]}=-2\sigma^{a}\gamma^{5}.$
(16)

The gauge fields associated with rotations and boosts are

$\bm{r}^{a}_{\phantom{s}i}\equiv{}{\textstyle{\frac{1}{2}}}\epsilon_{abc}L^{bc}_{\phantom{bc}i}\quad\mbox{and}\quad\bm{b}^{a}_{\phantom{a}i}\equiv{}L^{a0}_{\phantom{a0}i}\,,$
(17)

and the matrix of gauge fields $L_{i}$ is

$\begin{split}L_{i}={}&-i\,{\textstyle{\frac{1}{2}}}\,\bm{r}_{i}\cdot\bm{\sigma}\,I-{\textstyle{\frac{1}{2}}}\,\bm{b}_{i}\cdot\bm{\sigma}\,\gamma^{5}.\end{split}$
(18)

The field strength (12) is then

$\displaystyle F_{ik}={}\bm{[}\partial_{i}+L_{i},\partial_{k}+L_{k}\bm{]}={}$
$\displaystyle-i{\textstyle{\frac{1}{2}}}\big{[}\partial_{i}\bm{r}_{k}-\partial_{k}\bm{r}_{i}+(\bm{r}_{i}\times\bm{r}_{k}-\bm{b}_{i}\times\bm{b}_{k})\big{]}\cdot\bm{\sigma}I$
$\displaystyle-{\textstyle{\frac{1}{2}}}\big{[}\partial_{i}\bm{b}_{k}-\partial_{k}\bm{b}_{i}+(\bm{r}_{i}\times\bm{b}_{k}+\bm{b}_{i}\times\bm{r}_{k})\big{]}\cdot\bm{\sigma}\gamma^{5}.$

Under a local Lorentz transformation $D=D(\Lambda(x))$, these fields transform
as

$\begin{split}L^{\prime
ab}_{\phantom{{}^{\prime}ab}i}={}&\Lambda^{a}_{\phantom{a}c}\Lambda^{b}_{\phantom{b}d}L^{cd}_{\phantom{cd}i}-{}{\textstyle{\frac{1}{2}}}{\rm
Tr}\left(D\partial_{i}D^{-1}\bm{[}\gamma^{a},\gamma^{b}\bm{]}\right)\\\
\partial_{i}+L^{\prime}_{i}={}&D\,\big{(}\partial_{i}+L_{i}\big{)}\,D^{-1}\\\
F^{\prime}_{ik}={}&D\,F_{ik}\,D^{-1}\\\
h^{\prime}={}&D^{-1{\dagger}}\,h\,D^{-1}\\\ (\beta
h\beta)^{\prime}={}&D\,\beta h\beta\,D^{{\dagger}}\\\
\psi^{\prime}={}&D\,\psi\end{split}$ (19)

and so the action density $s_{L}$ of the action $S_{L}$ (11) is invariant

$\displaystyle s^{\prime}_{L}={}$ $\displaystyle{\rm
Tr}\left(F^{\prime{\dagger}}_{ik}\,h^{\prime}\,F^{\prime ik}\,\beta
h^{\prime}\beta\right)={}{\rm
Tr}\left(D^{-1{\dagger}}F^{\dagger}_{ik}D^{{\dagger}}\,D^{-1{\dagger}}hD^{-1}\,DF_{ik}D^{-1}\,D\beta
h\beta D^{{\dagger}}\right)={}{\rm Tr}\left(F^{\dagger}_{ik}\,h\,F^{ik}\,\beta
h\beta\right)=s_{L}$ (20)

under local Lorentz transformations (2) as well as under general coordinate
transformations (1).

The squares of the time derivatives $(\dot{L}^{ab}_{\phantom{ab}i})^{2}$ and
$(\dot{L}^{a0}_{\phantom{a0}i})^{2}$ of the gauge fields must appear with a
positive sign in the action $S_{L}$ (11) if the gauge-field action is to be
bounded below. They will appear with a positive sign at low temperatures if
the vector boson $K_{i}$ in the matrix
$h={}i\beta\gamma^{a}c_{a}^{\phantom{a}i}K_{i}$ has an average value $\langle
0|K_{i}|0\rangle=K_{0i}$ in the low-temperature ground state that is timelike,
$K_{0i}\,K_{0}^{\phantom{0}i}\simeq-m^{2}<0$. The matrix $h$, the vector
$K_{i}$, and the signs of the time derivatives
$(\dot{L}^{ab}_{\phantom{ab}i})^{2}$ and $(\dot{L}^{a0}_{\phantom{a0}i})^{2}$
are discussed in Sec. III.

The tensor gauge fields $L^{ab}_{\phantom{ab}i}$ have spin 2 (not 3) because
they are antisymmetric in $a$ and $b$.

An explicit formula for the matrix $D(\Lambda)$ is given in Appendix A along
with derivations of the identities

$D^{\dagger}\beta D={}\beta\quad\mbox{and}\quad D\beta D^{\dagger}={}\beta.$
(21)

These identities (21) imply that the action $S_{L}$ with $h$ replaced by
$\beta$, i.e, the trace ${\rm Tr}\big{(}F^{\dagger}_{ik}\beta
F^{ik}\beta\big{)}$, is invariant under local Lorentz transformations. But the
choice $h\to\beta$ gives an action in which the squares of the time
derivatives of the tensor gauge fields that gauge boosts and those that gauge
rotations occur with opposite signs. So the trace (11) which uses the matrix
$h={}i\beta\gamma^{a}K_{a}$ may be the only viable choice.

With the abbreviations

$\begin{split}\bm{R}_{ik}={}&\partial_{i}\bm{r}_{k}-\partial_{k}\bm{r}_{i}+\,(\bm{r}_{i}\times\bm{r}_{k}-\bm{b}_{i}\times\bm{b}_{k})\\\
\bm{B}_{ik}=&\partial_{i}\bm{b}_{k}-\partial_{k}\bm{b}_{i}+\,(\bm{r}_{i}\times\bm{b}_{k}+\bm{b}_{i}\times\bm{r}_{k}),\end{split}$
(22)

the field strength $F_{ik}$ is
$F_{ik}={}-i{\textstyle{\frac{1}{2}}}\bm{R}_{ik}\cdot\bm{\sigma}\
I-{\textstyle{\frac{1}{2}}}\bm{B}_{ik}\cdot\bm{\sigma}\,\gamma^{5}$, and the
action $S_{L}$ is

$\displaystyle S_{L}={}$ $\displaystyle-\frac{1}{16m^{2}\lambda^{2}}\int{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I+i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}h\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I-i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\beta
h\beta\Big{]}\sqrt{g}\,d^{4}x.$ (23)

Tensor gauge fields $L^{ab}_{\phantom{ab}i}$ possess at least two other
actions that are invariant under local Lorentz transformations and general
coordinate transformations. One is succinct and linear

$S_{E}={}M^{2}_{E}\int
F^{ab}_{\phantom{ab}ik}\,c_{a}^{\phantom{a}i}\,c_{b}^{\phantom{b}k}\sqrt{g}\,d^{4}\\!x$
(24)

in the coefficients $F^{ab}_{\phantom{ab}ik}$ of the field strength

$\displaystyle F_{ik}={}$ $\displaystyle
F^{ab}_{\phantom{ab}ik}\,\bm{[}\gamma_{a},\gamma_{b}\bm{]}=\partial_{i}L_{k}-\partial_{k}L_{i}+\bm{[}L_{i},L_{k}\bm{]}.$
(25)

These coefficients

$\displaystyle F^{ab}_{\phantom{ab}ik}={}$
$\displaystyle\partial_{i}L^{ab}_{\phantom{ab}k}-\partial_{k}L^{ab}_{\phantom{ab}i}+L^{bc}_{\phantom{bc}\,i}\,L^{a}_{\phantom{a}ck}-L^{ac}_{\phantom{ac}i}\,L^{b}_{\phantom{b}c\,k}$

resemble Riemann’s curvature tensor

$R^{k}_{\phantom{k}i\ell
n}={}\partial_{\ell}\Gamma^{k}_{\phantom{k}in}-\partial_{n}\Gamma^{k}_{\phantom{k}i\ell}+\Gamma^{k}_{\phantom{k}m\ell}\,\Gamma^{m}_{\phantom{m}in}-\Gamma^{k}_{\phantom{k}mn}\,\Gamma^{m}_{\phantom{m}i\ell}.$
(26)

But the action $S_{E}$ (24) does not lead to second-order differential
equations for the gauge fields $L^{ab}_{\phantom{ab}i}$.

A third invariant action is based upon the scalar

$\displaystyle F^{ab}_{\phantom{ab}ik}$ $\displaystyle
F_{ab}^{\phantom{ab}ik}={}\Big{(}\partial_{i}L^{ab}_{\phantom{ab}k}-\partial_{k}L^{ab}_{\phantom{ab}i}+L^{bc}_{\phantom{bc}\,i}\,L^{a}_{\phantom{a}ck}-L^{ac}_{\phantom{ac}i}\,L^{b}_{\phantom{b}c\,k}\Big{)}{}\Big{(}\partial^{i}L_{ab}^{\phantom{ab}k}-\partial^{k}L_{ab}^{\phantom{ab}i}+L_{bd}^{\phantom{bd}\,i}\,L_{a}^{\phantom{a}dk}-L_{ad}^{\phantom{ad}i}\,L_{b}^{\phantom{b}d\,k}\Big{)}.$

It is hermitian and invariant under general coordinate transformations and
local Lorentz transformations, but the squares of its time derivatives occur
with opposite signs.

##  III The Matrix $h$ and the Vector $K$

The action $S_{L}$ of the proposed tensor gauge fields will be invariant under
local Lorentz transformations (19) if the matrix $h$ which appears in the
trace (11) transforms as

$h^{\prime}={}D^{-1{\dagger}}\,h\,D^{-1}$ (27)

where $D=D(\Lambda(x))$, $D^{\dagger}\beta D={}\beta$ and $D\beta
D^{\dagger}={}\beta$ .

The simplest choice is the hermitian matrix

$h={}i\beta\gamma^{a}c_{a}^{\phantom{a}i}K_{i}$ (28)

in which $K_{i}$ is a real vector transforming as

$K^{\prime}_{i}(x^{\prime})={}\frac{\partial x^{k}}{\partial x^{\prime
i}}\,K_{k}(x)$ (29)

under general coordinate transformations. Its action $S_{K}$ (4) is simpler
than it looks since

$D_{i}K_{k}-D_{k}K_{i}={}\partial_{i}K_{k}-\partial_{k}K_{i}.$ (30)

Under a Lorentz transformation $\Lambda$, Dirac’s gamma matrices transform as
a 4-vector

$\begin{split}D(\Lambda)&\gamma^{a}D^{-1}(\Lambda)={}\Lambda_{b}^{\phantom{b}a}\,\gamma^{b}\\\
D^{-1}(\Lambda)&\gamma^{a}D(\Lambda)={}\Lambda^{a}_{\phantom{a}b}\,\gamma^{b}\end{split}$
(31)

where $\Lambda_{b}^{\phantom{b}a}=\Lambda^{-1a}_{\phantom{-1a}b}$. And since
the matrix $D(\Lambda)$ leaves $\beta$ invariant
$D^{-1{\dagger}}(\Lambda)\,\beta={}\beta\,D(\Lambda)$ as seen earlier (21),
the matrices $\beta\gamma^{a}$ also transform as a 4-vector

$\begin{split}D^{-1{\dagger}}(\Lambda)\,\beta&\gamma^{a}\,D^{-1}(\Lambda)={}\Lambda_{b}^{\phantom{b}a}\,\beta\gamma^{b}\\\
D^{{\dagger}}(\Lambda)\,\beta&\gamma^{a}\,D(\Lambda)={}\Lambda^{a}_{\phantom{a}b}\,\beta\gamma^{b}.\end{split}$
(32)

So $\Lambda$ changes $h^{\prime}$ to

$\begin{split}h^{\prime}={}&i\beta\gamma^{a}c^{\prime\phantom{a}i}_{a}K^{\prime}_{i}=i\beta\gamma^{a}\Lambda_{a}^{\phantom{a}b}\,c^{\phantom{b}i}_{b}\,K_{i}={}iD^{-1{\dagger}}\beta\gamma^{b}D^{-1}\,c^{\phantom{b}i}_{b}K_{i}={}D^{-1{\dagger}}\,h\,D^{-1}\end{split}$
(33)

which satisfies (19) as does $\beta h\beta$ since

$\begin{split}(\beta\,h\,\beta)^{\prime}={}&\beta\,D^{-1{\dagger}}hD^{-1}\,\beta=D\,\beta
h\beta\,D^{{\dagger}}.\end{split}$ (34)

The squares of the time derivatives $\dot{L}^{ab}_{\phantom{ab}i}$ of the
gauge fields must appear with a positive sign in the action (3) if the gauge-
field action is to be bounded below. They will appear with a positive sign at
low temperatures if the vector boson $K_{i}$ in the matrix
$h={}i\beta\gamma^{a}c_{a}^{\phantom{a}i}K_{i}$ has an average value
$K_{0i}={}\langle 0|K_{i}|0\rangle$ in the low-temperature vacuum that is
timelike, $K_{0i}\,K^{0i}\simeq-m^{2}<0$. At low temperatures, the average
value $K_{0i}$ is made timelike by the second term
$-{\textstyle{\frac{1}{4}}}(K_{i}K^{i}+m^{2})^{2}$ in its action $S_{K}$ (4)
which due to antisymmetry (30) may be written in the simpler form

$\begin{split}S_{K}={}&\int\Big{[}-{\textstyle{\frac{1}{4}}}(\partial_{i}K_{j}-\partial_{k}K_{i})(\partial^{i}K^{j}-\partial^{k}K^{i})\,-{\textstyle{\frac{1}{4}}}\xi^{2}(K_{i}K^{i}+m^{2})^{2}\Big{]}\sqrt{g}\,d^{4}x.\end{split}$
(35)

At low temperatures and presumably in the rest frame of the CMB, the vector
$K^{i}$ has the average vacuum value $\langle
0|K^{i}|0\rangle={}K^{i}_{0}=m\,\delta^{i}_{0}$, and the average value of the
matrix $h$ (28) is

$\begin{split}h={}&i\beta\gamma^{a}c_{a}^{\phantom{a}i}K_{0i}=i\beta\gamma^{0}c_{0}^{\phantom{a}0}K_{00}={}-mI.\end{split}$
(36)

Then in the rest frame of the CMB and apart from the fluctuations
$k^{i}=K^{i}-K^{i}_{0}$, the action $S_{L}$ (23) is

$\displaystyle S_{L}={}$ $\displaystyle-\frac{1}{16\lambda^{2}}\int{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I+i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I-i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\Big{]}\sqrt{g}\,d^{4}x$
(37) $\displaystyle={}$
$\displaystyle-\frac{1}{4\lambda^{2}}\int\left(\bm{R}_{ik}\cdot\bm{R}^{ik}+\bm{B}_{ik}\cdot\bm{B}^{ik}\right)\sqrt{g}\,d^{4}x$

in which the squares of the time derivatives $(\bm{\dot{r}_{k}})^{2}$ and
$(\bm{\dot{b}_{k}})^{2}$ appear with positive signs as promised. So the action
$S_{L}$ is bounded below in the rest frame of the CMB.

At low temperatures, the vector boson $K_{i}$ fluctuates about its average
value, $K_{i}(x)=K_{0i}+k_{i}(x)$. The fluctuations $k_{i}(x)$ are those of a
massless vector field with $k_{0}(x)=0$ as discussed in Appendix C. The six
gauge fields $L^{ab}_{\phantom{ab}i}$ remain massless despite the nonzero
average value $K^{i}_{0}$ of the vector boson $K^{i}$.

The ground state $|0,\bm{v}\rangle$ of a Lorentz frame moving at velocity
$\bm{v}$ relative to the CMB is related to the ground state of the CMB by a
unitary Lorentz transformation $|0,\bm{v}\rangle=U_{\bm{v}}|0\rangle$ that
represents matched (60) Lorentz and general-coordinate transformations. The
action is invariant under Lorentz and general-coordinate transformations

$U^{-1}_{\bm{v}}\,S_{L}\,U_{\bm{v}}={}S_{L}.$ (38)

So the average value of the action in the state $|0,\bm{v}\rangle$ is the same
as in the state $|0\rangle$ in which the CMB is at rest

$\begin{split}\langle 0,\bm{v}|S_{L}|0,\bm{v}\rangle={}&\langle
0|U^{-1}_{\bm{v}}\,S_{L}\,U_{\bm{v}}|0\rangle=\langle
0|S_{L}|0\rangle.\end{split}$ (39)

Thus the action is bounded below in all Lorentz frames.

The matrix $h$ takes a simpler form in two-component notation as discussed in
Appendix D.

It may be useful here to distinguish different kinds of symmetry. One kind is
an exact symmetry of the action and of the vacuum, like that of the group
$SU_{c}(3)$ of QCD.

A second kind is a symmetry of the action but not of the vacuum, like that of
SU${}_{\ell}(2)\otimes$U(1)Y in which the average value of a component of the
Higgs field in the low-temperature vacuum makes the W and Z bosons massive.
Their masses make their interactions of short range and therefore weak.

A third kind is symmetries of the action that are intrinsically reduced by the
ground state of the actual universe. As described in Sec. V, the average
values of the tetrads $c_{0i}^{a}(x)={}{\rm
Tr}(\rho\,c_{\phantom{a}i}^{a}(x))$ in the ground state of the late, low-
temperature universe reduce the two independent symmetries of general
coordinate and local Lorentz invariance to a single exact symmetry of the
vacuum — local Lorentz invariance. Every local Lorentz transformation
$\Lambda^{a}_{\phantom{a}b}(x)$ must be accompanied by a specific general
coordinate transformation (64)

$\frac{\partial x^{\prime i}}{\partial
x^{k}}={}c_{a}^{\phantom{a}i}(x)\,\Lambda^{a}_{\phantom{a}b}(x)\,c_{\phantom{b}k}^{b}(x)$
(40)

in order to preserve the average values of the tetrads.

In the theory sketched in this paper, the average value of the vector boson
$K_{i}$ in the present low-temperature universe and in the rest frame of the
CMB, $\langle 0|K_{i}|0\rangle={}m\,\delta_{i}^{0}$, gives the action $S_{L}$
(3, 23) of the six gauge fields $L^{ab}_{\phantom{ab}i}$ the approximate and
simpler form (37). The six gauge fields $L^{ab}_{\phantom{ab}i}$ remain
massless, and local Lorentz invariance remains exact — apart from $\langle
0|K_{i}|0\rangle$, the CMB, and the matter and energy of the actual universe.

##  IV Dirac Action

The explicitly hermitian Dirac action is

$\begin{split}S_{D}={}&-{\textstyle{\frac{1}{2}}}\\!\int\Big{[}\overline{\psi}\,\gamma^{a}c_{a}^{\phantom{a}i}D_{i}\psi+\\!\big{(}\overline{\psi}\,\gamma^{a}c_{a}^{\phantom{a}i}D_{i}\psi\big{)}^{\dagger}\Big{]}\sqrt{g}\,d^{4}x\end{split}$
(41)

in which the covariant derivative $D_{i}\psi$ is

$D_{i}\psi={}\big{(}\partial_{i}+{\textstyle{\frac{1}{8}}}\,\,L^{ab}_{\phantom{ab}i}\,\bm{[}\gamma_{a},\gamma_{b}\bm{]}\big{)}\psi.$
(42)

To avoid clutter, I am using a single Dirac field $\psi$ and am suppressing
the gauge bosons $A^{\alpha}_{i}$ of $SU_{c}(3)\times SU_{\ell}(2)\times
U(1)$. To include them, one would replace the single Dirac field $\psi$ by a
vector $\Psi$ whose components $\Psi_{\alpha}$ would be the 6 leptons and 18
quark fields. One would add the 12 gauge bosons $A^{\alpha}_{i}$ of
$SU_{c}(3)\times SU_{\ell}(2)\times U(1)$ and their actions. Then the
covariant derivative of the 96-component Dirac field $\Psi$ would be

$D_{i}\Psi={}\big{(}\partial_{i}+{\textstyle{\frac{1}{8}}}\,\,L^{ab}_{\phantom{ab}i}\,\bm{[}\gamma_{a},\gamma_{b}\bm{]}+A^{\alpha}_{i}t^{\alpha}\big{)}\Psi$
(43)

in which the $t^{\alpha}$’s are the generators of the Lie algebras of
$SU_{c}(3),SU_{\ell}(2)$ and $U(1)_{Y}$.

The simplest choice for $\overline{\psi}$ is Dirac’s choice
$\overline{\psi}={}\psi^{\dagger}\beta$

$\mathcal{L}_{D}={}-\psi^{\dagger}\beta\,c_{b}^{\phantom{b}i}\gamma^{b}D_{i}\psi$
(44)

in which the real $4\times 4$ hermitian symmetric matrix $\beta=i\gamma^{0}$
obeys $D^{{\dagger}}(\Lambda)\,\beta\,D(\Lambda)={}\beta$. The resulting Dirac
action (8) is then invariant under local Lorentz and general coordinate
transformations.

Under a local Lorentz transformation $D{(\Lambda)}={}D(\Lambda(x))$, the Dirac
field $\psi$, its covariant derivative $D_{i}\psi$ and the tetrads
$c_{a}^{\phantom{a}i}$ transform as

$\displaystyle\psi^{\prime}={}$ $\displaystyle D{(\Lambda)}\,\psi$
$\displaystyle\overline{\psi}^{\prime}={}$
$\displaystyle(\psi^{\dagger}\beta)^{\prime}={}\psi^{\dagger}D^{{\dagger}}{(\Lambda)}\beta=\psi^{\dagger}\beta\,D^{-1}{(\Lambda)}$
$\displaystyle(D_{i}\psi)^{\prime}={}$ $\displaystyle D{(\Lambda)}\,D_{i}\psi$
(45) $\displaystyle(c_{a}^{\phantom{a}i})^{\prime}={}$
$\displaystyle\Lambda_{a}^{\phantom{a}b}\,c_{b}^{\phantom{b}i}$

while
$D{(\Lambda)}\,\gamma^{a}\,D^{-1}{(\Lambda)}={}\Lambda_{b}^{\phantom{b}a}\,\gamma^{b}={}\Lambda^{-1a}_{\phantom{-1a}b}\gamma^{b}$.
Thus the action density (8) is invariant under local Lorentz transformations

$\displaystyle\mathcal{L}^{\prime}_{D}={}$
$\displaystyle\left(\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}D_{i}\psi\right)^{\prime}={}\psi^{\dagger}D^{{\dagger}}{(\Lambda)}\,\beta\,\gamma^{a}\,D{(\Lambda)}\,c_{c}^{\phantom{c}i}\Lambda_{a}^{\phantom{a}c}\,D_{i}\psi={}\psi^{\dagger}\beta\,\gamma^{b}\,\Lambda^{a}_{\phantom{a}b}\,\Lambda^{-1c}_{\phantom{-1c}a}\,c_{c}^{\phantom{c}i}D_{i}\psi={}\psi^{\dagger}\beta\,\gamma^{b}\,c_{b}^{\phantom{b}i}D_{i}\psi=\mathcal{L}_{D}$
(46)

as well as under general coordinate transformations.

The explicitly hermitian action density is

$\displaystyle\mathcal{L}_{Dh}={}$ $\displaystyle-\psi^{\dagger}\beta
c_{a}^{\phantom{a}i}\gamma^{a}\partial_{i}\psi-{\textstyle{\frac{1}{2}}}\psi^{\dagger}\,\beta\,\gamma^{a}\,(\partial_{i}c_{a}^{\phantom{a}i})\,\psi\,+\,\psi^{\dagger}{\textstyle{\frac{1}{2}}}\Big{(}c_{0}^{\phantom{0}i}\bm{r}_{i}\cdot\bm{\sigma}\,I-c_{s}^{\phantom{s}i}r_{i}^{\phantom{i}s}\gamma^{5}\Big{)}\psi\,-\,{\textstyle{\frac{1}{2}}}\,\epsilon_{jsk}c_{j}^{\phantom{j}i}b_{i}^{\phantom{i}s}\psi^{\dagger}\sigma^{k}\psi.$
(47)

as shown in Appendix E.

Although the current that generates the rotational field $r^{s}_{i}$ is

$\begin{split}j_{r}^{is}={}&{\textstyle{\frac{1}{2}}}\psi^{\dagger}\Big{(}c_{0}^{\phantom{0}i}\sigma^{s}-c_{s}^{\phantom{s}i}\gamma^{5}\Big{)}\psi,\end{split}$
(48)

the current that generates the boost field $b^{s}_{i}$ has no time component

$j^{is}_{b}={}-{\textstyle{\frac{1}{2}}}\epsilon_{jsk}c_{j}^{\phantom{j}i}\psi^{\dagger}\sigma^{k}\psi.$
(49)

So unless the spatial tetrads are nondiagonal so that
$c_{j}^{\phantom{j}0}\neq 0$ for $j=1,2$ or 3, the time components of the
boost bosons $b_{0}^{\phantom{0}s}$ do not occur in the Dirac action, and do
not generate Coulomb potentials.

These comments apply also to the gauge fields of groups larger than the
Lorentz group: unless the spatial tetrads are nondiagonal,
$c_{a}^{\phantom{a}0}\neq 0$ for $a>0$, the time components of the gauge
bosons of the generators of the noncompact directions do not appear in the
Dirac action and do not generate Coulomb potentials.

In the static limit, the exchange of the three massless tensor gauge fields
$\bm{r}_{i}$ that gauge rotations would imply that two macroscopic bodies of
$F$ and $F^{\prime}$ fermions separated by a distance $r$ would contribute to
the energy a static Coulomb potential

$K_{L}(r)=\frac{3\,FF^{\prime}f^{2}}{4\pi r}.$ (50)

This potential is positive and repulsive (between fermions and between
antifermions). It violates the weak equivalence principle because it depends
upon the number $F$ of fermions as $F=3B+L$ and not upon their masses.

The potential $K_{L}(r)$ changes Newton’s potential to

$K_{NL}(r)={}-G\,\frac{mm^{\prime}}{r}\left(1+\alpha\right)$ (51)

in which the repulsive coupling strength $\alpha$ is

$\alpha={}-\frac{3FF^{\prime}\lambda^{2}}{4\pi
Gmm^{\prime}}={}-\frac{3FF^{\prime}m_{\text{P}}^{2}\lambda^{2}}{4\pi
mm^{\prime}}.$ (52)

This force would increase the need for dark matter and decrease the need for
dark energy. Couplings $\alpha\sim 1$ are of gravitational strength.

Experiments [14, 15, 16, 17, 18, 19, 20, 16, 21, 22, 23, 24, 25, 26, 27, 16,
28, 29, 30, 31, 28, 32, 33, 34, 15, 27, 22, 26, 35, 36, 19, 37] put no upper
limits on the masses of tensor gauge fields and no lower limits on their
coupling $\lambda$.

##  V Intrinsic Reduction of Symmetry

When quantizing a gauge theory, one picks a gauge. For general relativity in
flat space, the usual gauge choice is to set the average value of the metric
$g_{ik}(x)$ equal to the Minkowski metric $\eta$

$\langle 0|g_{ik}(x)|0\rangle={}\eta_{ik}=\begin{pmatrix}-1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}.$ (53)

The average vacuum value of the metric is then quadratic in the average values
of Cartan’s tetrads

$\langle 0|g_{ik}(x)|0\rangle={}\langle
0|c_{\phantom{a}i}^{a}(x)\,\eta_{ab}\,c_{\phantom{b}k}^{b}(x)|0\rangle=\eta_{ik}.$
(54)

A further gauge choice of Lorentz frame sets the average vacuum values of the
tetrads equal to Kronecker deltas

$\langle 0|c_{\phantom{a}i}^{a}(x)|0\rangle={}\delta_{\phantom{a}i}^{a}.$ (55)

The independent symmetry transformations of general coordinate invariance (1)

$c^{\prime b}_{\phantom{{}^{\prime}a}i}(x^{\prime})={}\frac{\partial
x^{k}}{\partial x^{\prime i}}\,c^{b}_{\phantom{b}k}(x)$ (56)

and local Lorentz invariance (2)

$c^{\prime
a}_{\phantom{{}^{\prime}a}i}(x)={}\Lambda^{a}_{\phantom{a}b}(x)\,c^{b}_{\phantom{b}i}(x)$
(57)

map a tetrad $c^{b}_{\phantom{b}k}(x)$ to

$\begin{split}c^{\prime
a}_{\phantom{{}^{\prime}a}i}(x^{\prime})={}&\Lambda^{a}_{\phantom{a}b}(x)\,\frac{\partial
x^{k}}{\partial x^{\prime i}}\,c^{b}_{\phantom{b}k}(x)\end{split}$ (58)

So if the average vacuum values (55) of the tetrads $\langle
0|c_{\phantom{a}i}^{a}(x)|0\rangle={}\delta_{\phantom{a}i}^{a}$ are to be
invariant, then

$\begin{split}\Lambda^{a}_{\phantom{a}b}(x)\,\frac{\partial x^{k}}{\partial
x^{\prime
i}}\,\delta^{b}_{\phantom{b}k}=\Lambda^{a}_{\phantom{a}k}(x)\,\frac{\partial
x^{k}}{\partial x^{\prime i}}=\delta^{a}_{\phantom{a}i}.\end{split}$ (59)

By multiplying this last equation by $\partial x^{\prime i}/\partial
x^{\ell}$, we see that the tetrad values $\langle
0|c_{\phantom{a}i}^{a}(x)|0\rangle={}\delta_{\phantom{a}i}^{a}$ will be
unchanged only if the general coordinate transformation $x\to x^{\prime}$ and
the local Lorentz transformation $\Lambda^{a}_{\phantom{a}\ell}(x)$

$\frac{\partial x^{\prime a}}{\partial
x^{\ell}}={}\Lambda^{a}_{\phantom{a}\ell}(x)\quad\iff\quad\frac{\partial
x^{k}}{\partial x^{\prime i}}=\Lambda^{-1k}_{\phantom{-1k}i}(x)$ (60)

are the same. In that case, we have

$\Lambda^{a}_{\phantom{a}k}\,\frac{\partial x^{k}}{\partial x^{\prime
i}}={}\Lambda^{a}_{\phantom{a}k}\,\Lambda^{-1k}_{\phantom{-1k}i}(x)=\delta^{a}_{\phantom{k}i}$
(61)

which maintains the average values of the tetrads in the vacuum

$\begin{split}\langle 0|c^{\prime
a}_{\phantom{{}^{\prime}a}i}(x^{\prime})|0\rangle={}&\Lambda^{a}_{\phantom{a}b}(x)\,\frac{\partial
x^{k}}{\partial x^{\prime i}}\,\langle
0|c^{b}_{\phantom{b}k}(x)|0\rangle={}\Lambda^{a}_{\phantom{a}b}(x)\,\frac{\partial
x^{k}}{\partial x^{\prime
i}}\,\delta^{b}_{\phantom{b}k}={}\Lambda^{a}_{\phantom{a}k}(x)\,\Lambda^{-1k}_{\phantom{-1k}i}(x)={}\delta^{a}_{\phantom{k}i}.\end{split}$
(62)

In a universe described by a density operator $\rho$, the average values of
the tetrads are traces

$c_{\phantom{a}i}^{a}(x)={}{\rm Tr}(\rho\,c_{\phantom{a}i}^{a}(x)).$ (63)

By using the tetrad identity
$c_{a}^{\phantom{a}k}c^{a}_{\phantom{a}i}=\delta^{k}_{i}$, one may show that
the joint transformations (58) preserve the average values (63) of the tetrads
if the general coordinate transformation $x\to x^{\prime}$ is related to the
local Lorentz transformation $\Lambda(x)$ by two tetrads

$\frac{\partial x^{\prime i}}{\partial
x^{k}}={}c_{a}^{\phantom{a}i}(x)\,\Lambda^{a}_{\phantom{a}b}(x)\,c_{\phantom{b}k}^{b}(x).$
(64)

The nonzero average values of the tetrads reduce the two symmetries of the
action to a single symmetry of the ground state of the universe: local Lorentz
invariance. This reduction of symmetry is intrinsic rather than spontaneous
because tetrads intrinsically have nonzero average values ${\rm
Tr}(\rho\,c_{\phantom{a}i}^{a}(x))$.

The ideas of this section are independent of the existence of the tensor gauge
fields proposed in this paper.

##  VI Is the gauge group U(2,2)?

So far we have been assuming that the gauge group of the Dirac field is the
Lorentz group which acts on the Dirac field as the direct sum

$D(\Lambda)={}D^{(1/2,0)}(\Lambda)\oplus D^{(0,1/2)}(a).$ (65)

Is this a hint of a larger symmetry? These $4\times 4$ matrices
$D(\Lambda)=D(\Lambda(x))$ leave $\beta=i\gamma^{0}$ invariant

$D^{{\dagger}}(\Lambda)\,\beta\,D(\Lambda)={}\beta\quad\mbox{and}\quad
D(\Lambda)\beta D^{\dagger}(\Lambda)={}\beta.$ (66)

as noted earlier (21) and transform Dirac’s gamma matrices as a 4-vector

$D^{{\dagger}}(\Lambda)\,\beta\gamma^{a}\,D(\Lambda)={}\Lambda^{a}_{\phantom{a}b}\,\beta\gamma^{b}$
(67)

as noted earlier (31 and 32).

The gauge group of the Dirac field may be the group of all $4\times 4$ complex
matrices $U$ that leave $\beta$ invariant

$D^{{\dagger}}(U)\,\beta\,D(U)={}\beta.$ (68)

This group has 16 generators and is known as U(2,2) [38] as one may see by
rotating all four of the matrices of this equation of $\beta$ invariance (68)
from the $\beta=i\gamma^{0}$ direction to the $\gamma^{5}$ direction. This
rotation shows that the group must leave $\gamma^{5}$ invariant

$D^{{\dagger}}(U^{\prime})\,\gamma^{5}\,D(U^{\prime})={}\gamma^{5}$ (69)

which is the defining equation of U(2,2).

The $4\times 4$ direct-sum matrices $D(\Lambda)$ (65) leave $\beta$ invariant
(66) and so form a subgroup of U(2,2).

To implement U(2,2) gauge symmetry, we’ll need to extend the 6 generators
$\mathcal{J}^{ab}={}-{\textstyle{\frac{1}{4}}}i[\gamma^{a},\gamma^{b}]$ and 6
gauge fields $L^{ab}_{\phantom{ab}i}$ to 16 generators $G^{A}$ and 16 gauge
fields $L^{A}_{\phantom{A}i}$ so that

$\begin{split}D^{{\dagger}}(U)\beta
G^{A}D(U)={}&\mathscr{D}(U)^{A}_{\phantom{A}B}\,\beta G^{B}\\\
D^{-1}(U)G^{A}D(U)={}&\mathscr{D}(U)^{A}_{\phantom{A}B}\,G^{B}\\\
\psi^{\prime}={}&D(U)\,\psi\\\
D_{i}\psi={}&\big{(}\partial_{i}+{\textstyle{\frac{1}{2}}}i\,L^{A}_{\phantom{A}i}\,G_{A}\big{)}\psi\\\
(D_{i}\psi)^{\prime}={}&D(U)D_{i}\psi\end{split}$ (70)

in which $\mathscr{D}(U)^{A}_{\phantom{A}B}$ is the $16\times 16$ matrix that
represents $U$ in the adjoint representation of $U(2,2)$. The Dirac action
will be invariant under these local U(2,2) transformations (70) if Cartan’s
tetrads are also extended from four 4-vectors to 16 4-vectors transforming as

$\begin{split}e^{\prime
A}_{\phantom{{}^{\prime}A}i}={}&U^{A}_{\phantom{A}B}\,e^{B}_{\phantom{B}i}\end{split}$
(71)

The metric $g_{ik}$ must be invariant under U(2,2)

$g_{ik}={}e^{A}_{\phantom{A}i}e_{Ak}=e^{\prime
A}_{\phantom{A}i}e^{\prime}_{Ak}=g^{\prime}_{ik}.$ (72)

We have seen (71) that $e^{\prime
A}_{\phantom{{}^{\prime}A}i}={}U^{A}_{\phantom{A}B}\,e^{B}_{\phantom{B}i}$,
but we can choose how $e_{Ai}$ transforms. If we choose

$e^{\prime}_{A\,k}={}F_{A}^{\phantom{A}C}\,e_{Ck}$ (73)

then to keep

$\begin{split}g_{ik}={}&e^{A}_{\phantom{A}i}e_{fk}=e^{\prime
f}_{\phantom{f}i}e^{\prime}_{fk}={}e^{h}_{\phantom{\varepsilon}i}\,F^{A}_{\phantom{A}C}\,\,e_{Bk}\,U^{B}_{\phantom{B}A}\end{split}$
(74)

we need

$U^{A}_{\phantom{A}B}\,\,F_{A}^{\phantom{A}C}=\delta^{C}_{B}\quad\mbox{or}\quad(F^{-1})_{B}^{\phantom{B}A}={}U^{A}_{\phantom{A}B}.$
(75)

for then we’d have

$\begin{split}g_{ik}={}&e^{A}_{\phantom{A}i}e_{Ak}=e^{\prime
A}_{\phantom{A}i}e^{\prime}_{Ak}=U^{A}_{\phantom{A}B}\,e^{B}_{\phantom{B}i}\,\,F_{A}^{\phantom{A}C}\,e_{Ck}={}e^{B}_{\phantom{B}i}\,\delta^{C}_{B}\,e_{Ck}={}e^{C}_{\phantom{C}i}\,e_{Ck}=g_{ik}.\end{split}$
(76)

The real Lie algebras su(2,2) and sl(4,R) are not isomorphic, but over the
complex numbers they both belong to $A_{3}$ in the Cartan-Weyl classification,
so their complexifications are isomorphic [39]. It therefore may make sense to
consider the possibility that GL(4,R) or GL(4,C) is the gauge group of the
Dirac field.

##  VII Cartan’s first equation of structure

If we add all invariant terms of dimension $(\text{mass})^{4}$ or less to the
action (which is by no means required in a non-renormalizable theory of
gravity), then the covariant derivatives (10) of the tetrads

$D_{i}c^{a}_{\phantom{a}k}={}\partial_{i}c^{a}_{\phantom{a}k}+\,L_{i\phantom{a}b}^{\phantom{i}a}\,c^{b}_{\phantom{a}k}-\Gamma^{\ell}_{\phantom{\ell}ki}\,c^{a}_{\phantom{a}\ell}$
(77)

would appear in the action squared and contracted

$\begin{split}S_{C}={}&-M^{2}\int
D_{i}c^{a}_{\phantom{a}k}\,D^{i}c_{a}^{\phantom{a}k}\,\sqrt{g}\,d^{4}x\end{split}$
(78)

the coefficient $M^{2}$ being required because tetrads are dimensionless.

If the mass $M$ in the action $S_{C}$ is huge, say of the order of the Planck
mass $M_{P}$, then the equation of motion of the tensor gauge fields would be
approximately the condition that $S_{C}$ be stationary

$\displaystyle\delta S_{C}={}$ $\displaystyle-M^{2}\int\left[\left(\delta
D_{i}c^{a}_{\phantom{a}k}\right)D^{i}c_{a}^{\phantom{a}k}+D_{i}c^{a}_{\phantom{a}k}\delta
D^{i}c_{a}^{\phantom{a}k}\right]\sqrt{g}\,d^{4}x$ $\displaystyle={}$
$\displaystyle-2M^{2}\int\left(\delta
L_{i\phantom{a}b}^{\phantom{i}a}\right)c^{b}_{\phantom{a}k}D^{i}c_{a}^{\phantom{a}k}\sqrt{g}\,d^{4}x=0$
(79)

because the other terms in the action that contain the fields
$L_{i\phantom{a}b}^{\phantom{i}a}$ — namely the action terms $S_{D}$, $S_{L}$
and $S_{K}$ (8, 11, & 4) — lack the huge coefficient $M^{2}$.

Thus in the limit $M^{2}\to\infty$, the equation of motion of the tensor gauge
fields is

$0={}D_{i}c^{a}_{\phantom{a}k}=\partial_{i}c^{a}_{\phantom{a}k}+L_{i\phantom{a}b}^{\phantom{i}a}\,c^{b}_{\phantom{a}k}-\Gamma^{\ell}_{\phantom{\ell}ki}\,c^{a}_{\phantom{a}\ell}$
(80)

which is Cartan’s first equation of structure usually derived from the tetrad
postulate or from the assumption that the torsion vanishes. Multiplying it by
$c^{ck}$, we find that in the $M^{2}\to\infty$ limit, the tensor gauge fields
approach the spin connection

$\begin{split}L_{i}^{\phantom{i}ac}\simeq{}&\Gamma^{\ell}_{\phantom{\ell}ki}\,c^{a}_{\phantom{a}\ell}c^{ck}+c^{a}_{\phantom{a}k}\,\partial_{i}c^{ck}=\omega_{i}^{\phantom{i}ac}.\end{split}$
(81)

##  VIII Conclusions

The action of general relativity with fermions has two independent symmetries:
invariance under general coordinate transformations and invariance under local
Lorentz transformations. The action of local Lorentz transformations on Dirac
and Lorentz indices is similar to the action of noncompact internal-symmetry
transformations on Lie-group indices.

The internal-symmetry character of local Lorentz invariance suggests that it
might be implemented not by the spin connection but by tensor gauge fields
$L^{ab}_{\phantom{ab}i}$ with their own Yang-Mills action. But because the
Lorentz group is noncompact, their Yang-Mills action must be modified by a
neutral vector field $K_{i}(x)$ whose average value at low temperatures is
timelike. This vector boson is massless at low temperatures. The vector gauge
fields $L^{ab}_{\phantom{ab}i}$ are massless at all temperatures.

The particles of the neutral, gravitationally interacting and massless fields
$L^{ab}_{\phantom{ab}i}$ and $K_{i}(x)$ would contribute to the hot dark
matter of the Universe. The massive particles radiated by the vector field
$K_{i}$ at high temperatures would contribute at lower temperatures to cold
dark matter.

The nonzero average values of the tetrads reduce the spacetime symmetries of
the vacuum to local Lorentz invariance which can be extended to local U(2,2)
invariance.

If the contracted squares of the covariant derivatives of the tetrads
multiplied by the square of a mass $M$ are added to the action, then in the
limit $M^{2}\to\infty$, the equation of motion of the tensor gauge fields is
the vanishing of the covariant derivatives of the tetrads, which is Cartan’s
first equation of structure. In the same limit, the tensor gauge fields
approach the spin connection.

###### Acknowledgements.

I must thank Charles Boyer, Dorothy Burnet, Bobby Middleton, Alain Comtet,
Peter van Nieuwenhuizen, Teun van Nuland and Edward Witten for valuable email,
and Michael Grady for helpful conversations.

##  Appendix A The Matrix D($\Lambda$)

The matrix $D(\Lambda(\bm{\theta},\bm{\lambda}))$ is

$\displaystyle D(\Lambda(\bm{\theta},\bm{\lambda}))={}$
$\displaystyle\begin{pmatrix}D^{(1/2,0)}(\bm{\theta},\bm{\lambda})&0\\\
0&D^{(0,1/2)}(\bm{\theta},\bm{\lambda})\end{pmatrix}={}\begin{pmatrix}e^{\mbox{}-\bm{z}\cdot\bm{\sigma}}&0\\\
0&e^{\mbox{}\bm{z}^{*}\cdot\bm{\sigma}}\end{pmatrix}$ (82)

in which $\bm{z}={}{\textstyle{\frac{1}{2}}}(\bm{\lambda}+i\bm{\theta})$ [40].
So

$\displaystyle D\beta={}$
$\displaystyle\begin{pmatrix}e^{\mbox{}-\bm{z}\cdot\bm{\sigma}}&0\\\
0&e^{\mbox{}\bm{z}^{*}\cdot\bm{\sigma}}\end{pmatrix}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}={}\begin{pmatrix}0&e^{-\bm{z}\cdot\bm{\sigma}}\\\
e^{\bm{z}^{*}\cdot\bm{\sigma}}&0\end{pmatrix}={}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\begin{pmatrix}e^{\bm{z}^{*}\cdot\bm{\sigma}}&0\\\
0&e^{-\bm{z}\cdot\bm{\sigma}}\end{pmatrix}=\beta D^{{\dagger}-1}$ (83)

and

$\displaystyle\beta D^{-1}={}$ $\displaystyle\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\begin{pmatrix}e^{\bm{z}\cdot\bm{\sigma}}&0\\\
0&e^{-\bm{z}^{*}\cdot\bm{\sigma}}\end{pmatrix}={}\begin{pmatrix}0&e^{-\bm{z}\cdot\bm{\sigma}}\\\
e^{\bm{z}\cdot\bm{\sigma}}&0\end{pmatrix}={}\begin{pmatrix}e^{-\bm{z}^{*}\cdot\bm{\sigma}}&0\\\
0&e^{\bm{z}\cdot\bm{\sigma}}\end{pmatrix}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}=D^{\dagger}\beta$ (84)

as noted earlier (21).

##  Appendix B Explicit Form of $S_{L}$ in Frame of Earth

The Solar System moves at $v={}368\pm 2$ km/s relative to the CMB. So in the
Lorentz frame of the Earth, the average value of $K^{i}$ is
$K^{i}_{0v}={}\langle 0,v|K^{i}|0,v\rangle={}m(1,-\bm{v})/\sqrt{1-v^{2}}$. The
average value of the matrix $h={}i\beta\gamma^{a}c_{a}^{\phantom{a}i}K_{i}$ is
then

$\langle
0,E|h|0,E\rangle={}m\,\frac{\bm{\sigma}\cdot\bm{v}\gamma^{5}-I}{\sqrt{1-v^{2}}}.$
(85)

Since
$\beta\bm{\sigma}\cdot\bm{v}\gamma^{5}\beta=-\bm{\sigma}\cdot\bm{v}\gamma^{5}$,
the average value of $\beta h\beta$ is

$\langle 0,E|\beta
h\beta|0,E\rangle={}-m\frac{\bm{\sigma}\cdot\bm{v}\gamma^{5}+I}{\sqrt{1-v^{2}}}.$
(86)

So at low temperatures and apart from the fluctuations $k_{i}(x)$, the action
$S_{L}$ (23) is

$\displaystyle S_{L}={}$ $\displaystyle-\frac{1}{16m^{2}\lambda^{2}}\int{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I+i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}h\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I-i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\beta
h\beta\Big{]}\sqrt{g}\,d^{4}x$ $\displaystyle={}$
$\displaystyle\frac{1}{16\lambda^{2}}\int{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I+i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\frac{\bm{\sigma}\cdot\bm{v}\gamma^{5}-I}{1-v^{2}}\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I-i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}(\bm{\sigma}\cdot\bm{v}\gamma^{5}+I)\Big{]}\sqrt{g}\,d^{4}x.$
(87)

Since $v\simeq 10^{-3}$, it’s useful to separate terms according to the number
of powers of $v$

$\displaystyle S_{L}={}$ $\displaystyle-\frac{1}{16\lambda^{2}}\int{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I+i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I-i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\Big{]}\frac{\sqrt{g}\,d^{4}x}{1-v^{2}}$
(88) $\displaystyle-\frac{1}{16\lambda^{2}}\int{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I+i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I-i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\bm{\sigma}\cdot\bm{v}\gamma^{5}\Big{]}\sqrt{g}\,d^{4}x$
(89) $\displaystyle+\frac{1}{16\lambda^{2}}\int{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I+i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\bm{\sigma}\cdot\bm{v}\gamma^{5}\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I-i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\Big{]}\sqrt{g}\,d^{4}x$
(90) $\displaystyle+\frac{1}{16\lambda^{2}}\int{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I+i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\bm{\sigma}\cdot\bm{v}\gamma^{5}\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I-i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\bm{\sigma}\cdot\bm{v}\gamma^{5}\Big{]}$
(91)

in which the first integral contains terms of order zero and two in $v$.

Terms with an odd number of $\gamma^{5}$’s cancel. The first term (88) is thus

$I_{1}={}-\frac{1}{4\lambda^{2}}\int\left(\bm{R}_{ik}\cdot\bm{R}^{ik}+\bm{B}_{ik}\cdot\bm{B}^{ik}\right)\frac{\sqrt{g}\,d^{4}x}{1-v^{2}}.$
(92)

The second (89) and third (90) integrals involve a commutator

$\displaystyle I_{2}+I_{3}={}$ $\displaystyle\frac{1}{16\lambda^{2}}\int{\rm
Tr}\Big{\\{}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I-i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\,,\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I+i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\Big{]}\bm{\sigma}\cdot\bm{v}\gamma^{5}\Big{\\}}\sqrt{g}\,d^{4}x$
$\displaystyle={}$ $\displaystyle\frac{i}{8\lambda^{2}}\int{\rm
Tr}\Big{\\{}\Big{[}\bm{R}_{ik}\cdot\bm{\sigma}I,\bm{B}^{ik}\cdot\bm{\sigma}\Big{]}\,\bm{\sigma}\cdot\bm{v}\Big{\\}}\sqrt{g}\,d^{4}x={}-\frac{1}{\lambda^{2}}\int\bm{R}_{ik}\times\bm{B}^{ik}\cdot\bm{v}\sqrt{g}\,d^{4}x.$
(93)

The fourth term (91) is quadratic in $v$

$\displaystyle I_{4}={}$ $\displaystyle\frac{1}{16\lambda^{2}}\int{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}\cdot\bm{\sigma}I+i\bm{B}_{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\bm{\sigma}\cdot\bm{v}\gamma^{5}\,\big{(}\bm{R}^{ik}\cdot\bm{\sigma}I-i\bm{B}^{ik}\cdot\bm{\sigma}\gamma^{5}\big{)}\bm{\sigma}\cdot\bm{v}\gamma^{5}\Big{]}\sqrt{g}\,d^{4}x$
$\displaystyle={}$ $\displaystyle\frac{1}{16\lambda^{2}}\int{\rm
Tr}\Big{[}\bm{R}_{ik}\cdot\bm{\sigma}\,\bm{\sigma}\cdot\bm{v}\,\bm{R}^{ik}\cdot\bm{\sigma}\,\bm{\sigma}\cdot\bm{v}\,+\,\bm{B}_{ik}\cdot\bm{\sigma}\,\bm{\sigma}\cdot\bm{v}\,\bm{B}^{ik}\cdot\bm{\sigma}\,\bm{\sigma}\cdot\bm{v}\Big{]}\sqrt{g}\,d^{4}x.$
(94) $\displaystyle={}$
$\displaystyle\frac{1}{4\lambda^{2}}\int\Big{[}2\bm{R}_{ik}\cdot\bm{v}\,\bm{R}^{ik}\cdot\bm{v}-\bm{R}_{ik}\cdot\bm{R}^{ik}\,\bm{v}\cdot\bm{v}\,+\,2\bm{B}_{ik}\cdot\bm{v}\,\bm{B}^{ik}\cdot\bm{v}-\bm{B}_{ik}\cdot\bm{B}^{ik}\,\bm{v}\cdot\bm{v}\Big{]}\sqrt{g}\,d^{4}x.$

The action $S_{L}$ in the frame of the Earth is the sum

$S_{L}={}I_{1}+I_{2}+I_{3}+I_{4}.$ (95)

##  Appendix C The Vector Boson $K^{i}$

At high temperatures, the Lagrange density

$\displaystyle L_{K}={}$
$\displaystyle-{\textstyle{\frac{1}{4}}}(\partial_{i}K_{j}-\partial_{j}K_{i})(\partial^{i}K^{j}-\partial^{j}K^{i})\,-\,{\textstyle{\frac{1}{4}}}\xi^{2}m^{4}-{\textstyle{\frac{1}{2}}}\xi^{2}m^{2}K_{i}K^{i}-{\textstyle{\frac{1}{4}}}\xi^{2}K_{i}K^{i}K_{j}K^{j}$

of the action $S_{K}$ (35) describes a spin-one vector boson $K^{i}$ that of
mass $\xi m$. These massive particles of the boson $K_{i}$ would have
contributed to cold dark matter at temperatures $T\ll\xi m$.

At the low temperatures of the present universe, the same Lagrange density
describes a vector boson $K^{i}$ whose average value $\langle
0|K^{i}(x)|0\rangle$ is timelike; in some Lorentz frame, presumably the rest
frame of the CMB, the vector $K^{i}(x)={}m\delta^{i}_{0}+k^{i}(x)$ has average
values $\langle 0|K^{0}(x)|0\rangle={}m$ and $\langle 0|k^{i}(x)|0\rangle=0$
which make $\langle 0|K^{i}(x)|0\rangle$ timelike. Its small fluctuations
$k^{i}(x)$ are those of a massless vector $\bm{k}(x)$ described by the
Lagrange density

$\begin{split}L_{k}={}&-{\textstyle{\frac{1}{4}}}(\partial_{i}k_{j}-\partial_{j}k_{i})(\partial^{i}k^{j}-\partial^{j}k^{i})\,-\,{\textstyle{\frac{1}{4}}}\xi^{2}m^{4}-{\textstyle{\frac{1}{2}}}\xi^{2}m^{2}\big{(}\bm{k}^{2}-(m+k^{0})^{2}\big{)}\,-\,{\textstyle{\frac{1}{4}}}\xi^{2}\big{(}\bm{k}^{2}-(m+k^{0})^{2}\big{)}^{2}\end{split}$
(96)

or

$\displaystyle L_{k}={}$
$\displaystyle-{\textstyle{\frac{1}{4}}}(\partial_{i}k_{j}-\partial_{j}k_{i})(\partial^{i}k^{j}-\partial^{j}k^{i})\,-\,{\textstyle{\frac{1}{4}}}\xi^{2}m^{4}-{\textstyle{\frac{1}{2}}}\xi^{2}m^{2}\big{(}\bm{k}^{2}-m^{2}-2mk^{0}-k^{02}\big{)}$
(97)
$\displaystyle-{\textstyle{\frac{1}{4}}}\xi^{2}(\bm{k}^{2})^{2}+{\textstyle{\frac{1}{2}}}\xi^{2}\bm{k}^{2}(m^{2}+2mk^{0}+k^{02})\,-\,{\textstyle{\frac{1}{4}}}\xi^{2}\big{(}m^{4}+4m^{3}k^{0}+6m^{2}k^{02}+4mk^{03}+k^{04}\big{)}.$

These massless particles would contribute hot dark matter at $T\ll\xi m$.

Combining terms, we get

$\displaystyle L_{k}={}$
$\displaystyle-{\textstyle{\frac{1}{4}}}(\partial_{i}k_{j}-\partial_{j}k_{i})(\partial^{i}k^{j}-\partial^{j}k^{i})\,-\,{\textstyle{\frac{1}{4}}}\xi^{2}m^{4}+{\textstyle{\frac{1}{2}}}\xi^{2}m^{4}-{\textstyle{\frac{1}{4}}}\xi^{2}m^{4}+\xi^{2}m^{3}k^{0}-m^{3}\xi^{2}k^{0}$
(98)
$\displaystyle-{\textstyle{\frac{1}{2}}}\xi^{2}m^{2}\bm{k}^{2}+{\textstyle{\frac{1}{2}}}\xi^{2}m^{2}\bm{k}^{2}+{\textstyle{\frac{1}{2}}}\xi^{2}m^{2}k^{02}-{\textstyle{\frac{3}{2}}}\xi^{2}m^{2}k^{02}\,+\,m\xi^{2}\bm{k}^{2}k^{0}-m\xi^{2}k^{03}-{\textstyle{\frac{1}{4}}}\xi^{2}(\bm{k}^{2})^{2}+{\textstyle{\frac{1}{2}}}\xi^{2}\bm{k}^{2}k^{02}-{\textstyle{\frac{1}{4}}}\xi^{2}k^{04}$

or

$\displaystyle L_{k}={}$
$\displaystyle-{\textstyle{\frac{1}{4}}}(\partial_{i}k_{j}-\partial_{j}k_{i})(\partial^{i}k^{j}-\partial^{j}k^{i})\,-\,\xi^{2}m^{2}k^{02}+\xi^{2}m\bm{k}^{2}k^{0}-\xi^{2}mk^{03}\,-\,{\textstyle{\frac{1}{4}}}\xi^{2}(\bm{k}^{2})^{2}+{\textstyle{\frac{1}{2}}}\xi^{2}\bm{k}^{2}k^{02}-{\textstyle{\frac{1}{4}}}\xi^{2}k^{04}.$
(99)

The quadratic part of $L_{k}$ is

$\begin{split}L_{k2}={}&-{\textstyle{\frac{1}{4}}}(\partial_{i}k_{j}-\partial_{j}k_{i})(\partial^{i}k^{j}-\partial^{j}k^{i})-\xi^{2}m^{2}k^{02}.\end{split}$
(100)

The linear Euler-Lagrange equations for $k^{i}(x)$ are

$\partial_{j}\big{(}\partial^{j}k^{i}-\partial^{i}k^{j}\big{)}={}-2\xi^{2}m^{2}\,k^{0}\,\delta^{i}_{0}$
(101)

or

$\begin{split}\triangle
k^{0}+\nabla\cdot\dot{\bm{k}}={}-2\xi^{2}m^{2}k^{0}\end{split}$ (102)

and

$-\ddot{\bm{k}}+\triangle\bm{k}-\nabla\big{(}\dot{k}^{0}+\nabla\cdot\bm{k}\big{)}={}0.$
(103)

One solution is $-\triangle k^{0}=2m^{2}k^{0}$, $\nabla\cdot\bm{k}=0$,
$\dot{k}^{0}=0$, and $\partial_{j}\partial^{j}\bm{k}={}0$.

##  Appendix D Two-Component Formalism

Dirac’s formalism is economical, but the two-component formalism is better
suited to a discussion of a new interpretation of the matrix $h$.

Since the Dirac-Lorentz matrix (82) is block diagonal

$\displaystyle D(\bm{\theta},\bm{\lambda}){}$
$\displaystyle\begin{pmatrix}D^{(1/2,0)}(\bm{\theta},\bm{\lambda})&0\\\
0&D^{(0,1/2)}(\bm{\theta},\bm{\lambda})\end{pmatrix}={}\begin{pmatrix}e^{\mbox{}-\bm{z}\cdot\bm{\sigma}}&0\\\
0&e^{\mbox{}\bm{z}^{*}\cdot\bm{\sigma}}\end{pmatrix}\equiv\begin{pmatrix}D_{\ell}&0\\\
0&D_{r}\end{pmatrix},$ (105)

the matrix $h$

$h={}\begin{pmatrix}h_{\ell}&0\\\ 0&h_{r}\end{pmatrix}$ (106)

and its transformation law $h^{\prime}=D^{-1{\dagger}}\,h\,D^{-1}$ (27) are
block diagonal

$\begin{split}\begin{pmatrix}h^{\prime}_{\ell}&0\\\
0&h^{\prime}_{r}\end{pmatrix}={}&\begin{pmatrix}D^{-1{\dagger}}_{\ell}h_{\ell}D^{-1}_{\ell}&0\\\
0&D^{-1{\dagger}}_{r}h_{r}D^{-1}_{r}\end{pmatrix}.\end{split}$ (107)

The matrix $\beta h\beta$ is just $h$ with left and right interchanged

$\beta h\beta={}\begin{pmatrix}h_{r}&0\\\ 0&h_{\ell}\end{pmatrix}.$ (108)

The transformation law $\psi^{\prime}={}D\,\psi$ of a Dirac field

$\psi={}\begin{pmatrix}\ell\\\ r\end{pmatrix}$ (109)

is

$\psi^{\prime}={}\begin{pmatrix}\ell^{\prime}\\\
r^{\prime}\end{pmatrix}=\begin{pmatrix}D_{\ell}\,\ell\\\
D_{r}\,r\end{pmatrix}.$ (110)

So the combinations $\ell^{\dagger}h_{\ell}\ell$ and $r^{\dagger}h_{r}r$ are
invariant

$\begin{split}(\ell^{\dagger}h_{\ell}\ell)^{\prime}={}&\ell^{\dagger}D_{\ell}^{\dagger}D^{-1{\dagger}}_{\ell}h_{\ell}D^{-1}_{\ell}D_{\ell}\ell=\ell^{\dagger}h_{\ell}\ell\\\
(r^{\dagger}h_{r}r)^{\prime}={}&r^{\dagger}D_{r}^{\dagger}D^{-1{\dagger}}_{r}h_{r}D^{-1}_{r}D_{r}r=r^{\dagger}h_{r}r\end{split}$
(111)

much as contracted tensors are invariant

$(X^{i}g_{ik}Y^{k})^{\prime}={}X^{\prime i}g^{\prime}_{ik}Y^{\prime
k}=X^{i}g_{ik}Y^{k}.$ (112)

We now see that the $2\times 2$ matrices $h_{\ell}$ and $h_{r}$ do for spinor
indices what $g_{ik}$ does for tensor indices.

Now if $D_{\ell}=e^{-\bm{z}\cdot\bm{\sigma}/2}$, then
$D_{r}=e^{\bm{z}^{*}\cdot\bm{\sigma}/2}$, and so
$D_{r}=D^{{\dagger}-1}_{\ell}$. So we can set
$h_{r}={}h_{\ell}^{-1{\dagger}}$.

Thus in two-component notation, the gauge-field action $S_{L}$ (23) is

$\begin{split}S_{L}={}&-\frac{1}{16m^{2}}\int\Big{\\{}{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}+i\bm{B}_{ik}\big{)}\cdot\bm{\sigma}\,h_{\ell}\,\big{(}\bm{R}^{ik}-i\bm{B}^{ik}\big{)}\cdot\bm{\sigma}\,h_{\ell}^{-1{\dagger}}\Big{]}\\\
{}&\qquad\qquad+{\rm
Tr}\Big{[}\big{(}\bm{R}_{ik}+i\bm{B}_{ik}\big{)}\cdot\bm{\sigma}\,h_{\ell}^{-1{\dagger}}\,\big{(}\bm{R}^{ik}+i\bm{B}^{ik}\big{)}\cdot\bm{\sigma}\,h_{\ell}\Big{]}\Big{\\}}\sqrt{g}\,d^{4}x.\end{split}$
(113)

Since $h^{\prime}_{\ell}={}D^{-1{\dagger}}_{\ell}h_{\ell}D^{-1}_{\ell}$, we
may choose $h_{\ell}$ to be hermitian $h^{\dagger}_{\ell}={}h_{\ell}$, which
implies that the diagonal form of $h_{\ell}$ is just two real numbers. The
most general $2\times 2$ hermitian matrix is a linear combination of the Pauli
matices $\bm{\sigma}$ and the identity matrix $I$. Under a Lorentz
transformation $\Lambda$, the 4-vector $s^{a}_{\ell}\equiv(-I,\bm{\sigma})$
transforms as

$D_{\ell}^{\dagger}(\Lambda)\,s^{a}_{\ell}\,D_{\ell}(\Lambda)=\Lambda^{a}_{\;\;b}\,s^{b}_{\ell}$
(114)

while the 4-vector $s^{a}_{r}\equiv(I,\bm{\sigma})$ transforms as

$D_{r}^{\dagger}(\Lambda)\,s^{a}_{r}\,D_{r}(\Lambda)=\Lambda^{a}_{\;\;b}\,s^{b}_{r}.$
(115)

The $4\times 4$ matrix $h$ (28) is

$h={}i\beta\gamma^{a}c_{a}^{\phantom{a}i}K_{i}=i\beta\gamma^{a}K_{a}=\begin{pmatrix}s^{a}_{\ell}K_{a}&0\\\
0&s^{a}_{r}K_{a}\end{pmatrix}.$ (116)

Under a local Lorentz transformation $\Lambda=\Lambda(x)$, the vector field
$K_{a}(x)$ goes to

$K^{\prime}_{a}(x)={}U(\Lambda)K_{a}(x)U^{-1}(\Lambda)=\Lambda^{\phantom{a}b}_{a}K_{b}(x).$
(117)

So the explicitly
$({\textstyle{\frac{1}{2}}},0)\oplus(0,{\textstyle{\frac{1}{2}}})$ version of
$h$ goes as

$\begin{split}h^{\prime}={}&\begin{pmatrix}s^{a}_{\ell}\Lambda_{a}^{\phantom{a}b}K_{b}&0\\\
0&s^{a}_{r}\Lambda_{a}^{\phantom{a}b}K_{b}\end{pmatrix}=D^{-1{\dagger}}hD^{-1}\,={}\,\begin{pmatrix}D^{-1{\dagger}}_{\ell}s^{b}_{\ell}D^{-1}_{\ell}K_{b}&0\\\
0&D^{-1{\dagger}}_{r}s^{b}_{r}D^{-1}_{r}K_{b}\end{pmatrix}\end{split}$ (118)

since

$\begin{split}D^{-1{\dagger}}_{\ell}s^{b}_{\ell}D^{-1}_{\ell}K_{b}={}&s^{a}_{\ell}\Lambda^{-1b}_{\phantom{-1b}a}K_{b}=s^{a}_{\ell}\Lambda_{a}^{\phantom{a}b}K_{b}\\\
D^{-1{\dagger}}_{r}s^{b}_{r}D^{-1}_{r}K_{b}={}&s^{a}_{r}\Lambda^{-1b}_{\phantom{-1b}a}K_{b}=s^{a}_{r}\Lambda_{a}^{\phantom{a}b}K_{b}.\end{split}$
(119)

##  Appendix E Hermitian Form of the Dirac Action

The fully expanded covariant derivative $D_{i}\psi$ is

$\begin{split}D_{i}\psi={}&\left(\partial_{i}-i\,{\textstyle{\frac{1}{2}}}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I-{\textstyle{\frac{1}{2}}}\,\,\bm{b}_{i}\cdot\bm{\sigma}\,\gamma^{5}\right)\psi.\end{split}$
(120)

If $\overline{\psi}={}\psi^{\dagger}\beta$, then
$-\overline{\psi}\,\gamma^{a}\,c_{a}^{\phantom{a}i}D_{i}\psi$ is

$\displaystyle-\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}D_{i}\psi$
$\displaystyle={}-\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}(\partial_{i}\psi)\,-\,\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}(-i{\textstyle{\frac{1}{2}}}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I)\psi\,-\,\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}(-{\textstyle{\frac{1}{2}}}\,\,\bm{b}_{i}\cdot\bm{\sigma}\,\gamma^{5})\psi$
$\displaystyle={}-\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}(\partial_{i}\psi)\,+\,{\textstyle{\frac{1}{2}}}\,i\,\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I\psi\,+\,{\textstyle{\frac{1}{2}}}\,\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\,\bm{b}_{i}\cdot\bm{\sigma}\,\gamma^{5}\psi.$
(121)

Its adjoint
$[-\overline{\psi}c_{a}^{\phantom{a}i}\gamma^{a}D_{i}\psi]^{\dagger}$ is

$\displaystyle[-\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}D_{i}\psi$
$\displaystyle]^{\dagger}={}\big{[}-\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}(\partial_{i}\psi)\big{]}^{\dagger}\,+\,\big{[}{\textstyle{\frac{1}{2}}}i\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I\psi\big{]}^{\dagger}\,+\,\big{[}{\textstyle{\frac{1}{2}}}\,\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\,\bm{b}_{i}\cdot\bm{\sigma}\,\gamma^{5}\psi\big{]}^{\dagger}$
$\displaystyle\equiv A_{1}+A_{2}+A_{3}.$ (122)

Now $\beta\,\gamma^{a}={}i\gamma^{0}\gamma^{a}$ is antihermitian, so the first
term is

$\begin{split}A_{1}={}&\big{[}-\psi^{\dagger}\beta\,\gamma^{a}c_{a}^{\phantom{a}i}(\partial_{i}\psi)\big{]}^{\dagger}={}(\partial_{i}\psi^{\dagger})\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\psi={}-\psi^{\dagger}\,\beta\,\gamma^{a}\,(\partial_{i}c_{a}^{\phantom{a}i})\,\psi-\psi^{\dagger}\,\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\partial_{i}\psi.\end{split}$
(123)

The second term is

$\begin{split}A_{2}={}&\big{[}{\textstyle{\frac{1}{2}}}i\psi^{\dagger}\,\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I\psi\big{]}^{\dagger}={}\,{\textstyle{\frac{1}{2}}}\,i\,\psi^{\dagger}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I\,\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\psi={}{\textstyle{\frac{1}{2}}}\,i\,\psi^{\dagger}\,\beta\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\psi.\end{split}$
(124)

The third term is

$\displaystyle A_{3}={}$
$\displaystyle\big{[}{\textstyle{\frac{1}{2}}}\,\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\,\bm{b}_{i}\cdot\bm{\sigma}\,\gamma^{5}\psi\big{]}^{\dagger}={}-{\textstyle{\frac{1}{2}}}\psi^{\dagger}\,\gamma^{5}\,\,\bm{b}_{i}\cdot\bm{\sigma}\,\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\psi={}-{\textstyle{\frac{1}{2}}}\psi^{\dagger}\beta\,\bm{b}_{i}\cdot\bm{\sigma}\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\gamma^{5}\,\psi.$
(125)

Since $[\sigma^{s},\gamma^{j}]={}2i\,\epsilon_{sjk}\gamma^{k}$, we have

$\sigma^{s}\,\gamma^{j}={}\gamma^{j}\,\sigma^{s}+2i\,\epsilon_{sjk}\gamma^{k}.$
(126)

So the $\bm{r}$ terms are

$\displaystyle{\textstyle{\frac{1}{4}}}\,i\,\psi^{\dagger}\beta\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I\psi+$
$\displaystyle{\textstyle{\frac{1}{4}}}\,i\,\psi^{\dagger}\,\beta\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\psi={}{\textstyle{\frac{1}{4}}}\,\psi^{\dagger}\Big{(}2c_{0}^{\phantom{0}i}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I+\beta\,c_{j}^{\phantom{j}i}\,\,r_{i}^{\phantom{i}s}\\{i\gamma^{j},\sigma^{s}\\}\Big{)}\psi$
$\displaystyle={}$
$\displaystyle{\textstyle{\frac{1}{2}}}\psi^{\dagger}\Big{(}c_{0}^{\phantom{0}i}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I+\beta\,c_{s}^{\phantom{s}i}\,\,r_{i}^{\phantom{i}s}\gamma^{5}\beta\Big{)}\psi={}{\textstyle{\frac{1}{2}}}\psi^{\dagger}\Big{(}c_{0}^{\phantom{0}i}\,\,\bm{r}_{i}\cdot\bm{\sigma}\,I-c_{s}^{\phantom{s}i}\,\,r_{i}^{\phantom{i}s}\gamma^{5}\Big{)}\psi,$
(127)

while the $\bm{b}$ terms are

$\displaystyle{\textstyle{\frac{1}{2}}}\,\psi^{\dagger}\beta$
$\displaystyle\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\,\bm{b}_{i}\cdot\bm{\sigma}\,\gamma^{5}\psi-{\textstyle{\frac{1}{2}}}\psi^{\dagger}\beta\,\,\bm{b}_{i}\cdot\bm{\sigma}\,\gamma^{a}\,c_{a}^{\phantom{a}i}\,\gamma^{5}\,\psi={}i\,\epsilon_{tsk}\,c_{t}^{\phantom{t}i}\,\,b_{i}^{\phantom{i}s}\,\psi^{\dagger}\,\beta\,\gamma^{k}\,\gamma^{5}\,\psi.$
(128)

The hermitian Dirac action density then is

$\displaystyle\mathcal{L}_{Dh}={}$ $\displaystyle-\psi^{\dagger}\beta
c_{a}^{\phantom{a}i}\gamma^{a}\partial_{i}\psi-{\textstyle{\frac{1}{2}}}\psi^{\dagger}\,\beta\,\gamma^{a}\,(\partial_{i}c_{a}^{\phantom{a}i})\,\psi{}\,+\,\psi^{\dagger}{\textstyle{\frac{1}{2}}}\,\,\Big{(}c_{0}^{\phantom{0}i}\bm{r}_{i}\cdot\bm{\sigma}\,I-c_{s}^{\phantom{s}i}r_{i}^{\phantom{i}s}\gamma^{5}\Big{)}\psi\,-\,{\textstyle{\frac{1}{2}}}\epsilon_{jsk}\,\,c_{j}^{\phantom{j}i}b_{i}^{\phantom{i}s}\psi^{\dagger}\sigma^{k}\psi.$
(129)

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basis vectors is first obtained corresponding to a fixed error, which shows in
the steps $1-13$ of the algorithm. Further, the first $k$ number of basis
vectors out of $\ell$ are used to construct the sparse coefficient vectors
with $k$ non-zero elements. The steps $14-20$ of the algorithm show this
sparsification of coefficient vectors using the MP algorithm.

##  Appendix E  Applicability of RMF algorithms over the CBC parameter space

This section shows the pictorial representation of the applicability of the
various proposed RMF schemes in the different regimes of the parameter space
for the search of GW signal from CBC sources.

Figure 17: The figure shows the applicability of prescribed RMF algorithms
(Type-I, Type-II) for the different parameter space regimes. Figure-18 shows
the duration of the waveform for the various CBC sources with a fixed lower
cut-off frequency. For BNS and NSBH systems, the waveform duration is much
longer than BBH and IMBH systems. Hence, Type-I RMF can be applicable for IMBH
in which the waveforms are shorter than other systems. Similarly, Type-II RMF
is directly relevant to BNS and NSBH systems. However, for the BBH system,
both Type-I and Type-II schemes are applicable.  Figure 18: The figure shows a
two-dimensional template bank on the mass components $m_{1,2}$. The color bar
indicates $10^{4}$ randomly picked template waveforms’ time-duration in the
specific parameter regime, including BNS, BBH, NSBH, and IMBH. The template
bank boundary has been chosen as same as taken as GstLAL pipeline in the third
observing runs ($\texttt{O}3$) [45]. The details of the parameter space are as
follows. (a) BNS region: $1$-$3$ in solar masses, $15$-$1024$ $\mathrm{Hz}$,
aligned or anti-aligned spins to $0.05$, match = $.99$. (b) BBH-low-$q$:
$3$-$100$ in solar masses, $15-1024$ $\mathrm{Hz}$, aligned or anti-aligned
spins to $0.999$, match = $.99$, mass ratios under $10$. (c) NSBH: $1$-$3$ and
$3$-$150$ in solar masses, $15$-$1024$ $\mathrm{Hz}$, aligned or anti-aligned
spins to $0.05$ and $.999$, match = $.97$, mass ratios under $50$. (d) IMBH:
$10$-$400$ in solar masses, $10$-$1024$ $\mathrm{Hz}$, aligned or anti-aligned
spins to $0.99$. (e) Other BBH: $3$-$100$ in solar masses, $15$-$1024$
$\mathrm{Hz}$, aligned or anti-aligned spins to $.999$, match = $0.97$, mass
ratios = $10$-$33$.  Figure 19: The figure shows the evolution of a time-
domain template waveform with increased frequency components. We have
generated the waveform corresponding to a NSBH system, in which the component
masses $m_{1}=1.3\,M_{\odot}$, $m_{2}=20.0\,M_{\odot}$, the lower cut-off
frequency $15$ $\mathrm{Hz}$ are chosen. The TaylorT4 waveform model is used
for the generating of the waveform. From the frequency profile, it is clear
that the early inspiral part can be sampled with $128$ $\mathrm{Hz}$, whereas
for the merger and ringdown parts, we can use a higher sampling frequency such
as $2048$ $\mathrm{Hz}$ and $512$ $\mathrm{Hz}$.

##  Appendix F Optimization of the template bank division

From the sub-section II.2, it is clear that for the template matrix
factorization (SVD, RMF) based matched filtering scheme, one has to make a
balance between the cost of filtering and the cost of reconstruction of the
SNR time-series. There is always a trade-off between the filtering cost and
reconstruction cost depending on the size of a template bank. For a large
template matrix, the reconstruction cost is the dominant cost. Hence, the
current online search pipeline dividing the whole template bank into sub-banks
to optimally reduce the reconstruction cost. This trade-off entirely depending
on the size of the bank, the number of sub-banks. Also, it depends on the
relation between the number of basis vectors obtained from a full bank
factorization and the total number of basis vectors obtained from all sub-
banks factorization for a fixed $\langle{\delta\rho}/\rho\rangle$. It is
expected that the number of basis vectors is small for a full bank
decomposition in comparison to the total number of basis vectors obtained
after the decomposing of all sub-banks. Therefore, in that case, the filtering
cost is high if one has used the sub-banks division setup. Hence, for a large
bank, only considering reducing reconstruction cost by prescribed the division
of the bank into sub-banks may not always be optimal. Hence, it is crucial to
analyze the total time complexity cost theoretically, including the filtering
cost and the reconstruction cost for both the cases of the full bank and the
sub-banks.

Let a full bank contains $N_{T}$ number of templates, and each template has
$N_{s}$ number of sample points. Also, let the whole bank can be divided into
a $p$ number of sub-banks, and each sub-bank contains ${N_{T}}_{i}$ of
templates. For simplicity, we have assumed that each sub-banks includes an
equal number of templates ($b$). Hence $N_{T}=p\,{N_{T}}_{b}$, Let $k_{i}$
represent the number of required basis vectors for the $i^{\texttt{th}}$ sub-
bank corresponding to a fixed $\langle{\delta\rho}/\rho\rangle$.
Reconstruction cost for each of the sub-bank is $k_{i}\,{N_{T}}_{b}\,N_{s}$.
Filtering cost is $k_{i}\,N_{s}\,\log N_{s}$. Therefore the total cost for
each of the sub-bank is $\big{(}{N_{T}}_{b}\,k_{i}\,N_{s}+k_{i}\,N_{s}\,\log
N_{s}\big{)}$. Therefore the total cost considering all the sub-banks is

$\sum_{i=1}^{p}\\{{N_{T}}_{b}\,k_{i}\,N_{s}+k_{i}\,N_{s}\,\log N_{s}\\}\,$
(18)

Whereas, for a full-bank the reconstruction cost is $N_{T}\,k\,N_{s}$. Where
$k$ is the number of the required basis for the whole bank corresponding to
the same $\langle{\delta\rho}/\rho\rangle$ considered for each sub-banks.
Therefore, The filtering cost is $k\,N_{s}\,\log N_{s}$ and the total cost for
a full bank is

$N_{T}\,k\,N_{s}+k\,N_{s}\,\log N_{s}$ (19)

Let $\gamma$ is the ratio between the total time complexity of matched
filtering using full bank and sub-banks. Hence $\gamma$ is

$\gamma=\frac{\sum_{i=1}^{p}\\{{N_{T}}_{b}\,k_{i}\,N_{s}+k_{i}\,N_{s}\,\log
N_{s}\\}}{N_{T}\,k\,N_{s}+k\,N_{s}\,\log N_{s}}$ (20)

For simplicity, assume that all the sub-banks have the same number of basis
vectors corresponding to a fixed $\langle{\delta\rho}/\rho\rangle$. Therefore
the cost for every sub-bank is also the same. Thus The above Eq.18 can be
written as follows:

$p\Big{(}{N_{T}}_{b}\,k_{i}\,N_{s}+k_{i}\,N_{s}\,\log N_{s}\Big{)}$ (21)

The total number of basis considering all the sub-banks is $p\,k_{i}$.
Generally $k\ll p\,k_{i}$. Therefore, we can consider $k$ as a $\beta$
fraction of $pk_{i}$ i.e. $k=\frac{pk_{i}}{\beta}$. Combining Eq.19 $\&$ 21
and incorporating $k=\frac{pk_{i}}{\beta}$, one can redefine the ratio
$\gamma$ as follows:

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{p\Big{(}{N_{T}}_{b}\,k_{i}\,N_{s}+k_{i}\,N_{s}\,\log
N_{s}\Big{)}}{N_{T}\,k\,N_{s}+k\,N_{s}\,\log N_{s}}$ (22) $\displaystyle=$
$\displaystyle\frac{\beta\,\Big{(}N_{T}+p\,\log
N_{s}\Big{)}}{p\,\Big{(}N_{T}+\,\log N_{s}\Big{)}}$

We can formulate three different cases based on the relation between $p$ and
$\beta$.

  1. 1.

Case-I: Let us suppose if $p=\beta$, then For a large $N_{T}$ and small value
of $p$, $\gamma\rightarrow 1$. This implies that the total cost for a sub-bank
division is small compared to the full bank for this scenario. Whereas if $p$
is as large as $N_{T}$ then $\gamma\rightarrow(1+\log N_{s})$. This shows that
if each sub-bank contains a minimal number of templates. But this is not a
feasible case as if $p$ becomes large, then $\beta$ has to be large as the
underline assumption is $p=\beta$. Hence, for this case, either $p$ is small
or large, the total cost for the sub-bank division-based matched filtering
scheme is optimal.

  2. 2.

Case-II: Let $p<\beta:\beta=m\,p$, then from Eq.22,

$\gamma=\frac{m\big{(}N_{T}+p\,\log N_{s}\big{)}}{N_{T}+\log N_{s}}$ (23)

If $N_{T}$ is large, then $\gamma\rightarrow m$ no matter $p$ is small or
large. Hence for this Case, the computational time for a full-bank decomposed-
based matched filtering scheme is optimal.

  3. 3.

Case III: Let $p>\beta:p=n\,q$, then from Eq.22,

$\gamma=\frac{N_{T}+p\,\log N_{s}}{n\big{(}N_{T}+\log N_{s}\big{)}}$ (24)

for large $N_{T}$ and small value of $p$, $\gamma\rightarrow\frac{1}{n}$
whereas for $p=N_{T}$, $\gamma\rightarrow\frac{1+\log N_{s}}{n}$. For a
substantial value of $p$, it seems to be observed that the computational cost
for the full bank-based matched filtering scheme is less in comparison to the
sub-banks-based filtering scheme. However, using the same argument described
in the Case-I, it is clear that practically $p$ can’t be the as same order as
$N_{T}$. Hence, In this Case, the sub-bank division-based matched filtering
scheme is optimal.

Figure 20: The figure shows the relation between the variables $\beta$
(defined in Ref. [6], Figure-1) and $\gamma$. If $\beta=1$, then it is implied
that the number of required basis considering all the sub-banks (w.r.t a fixed
$\langle{\delta\rho}/\rho\rangle$) is the same as the basis obtained from the
full bank. The shaded region shows the possible configuration of $\beta$ and
$\gamma$ in which the filtering cost will be less for the decomposition of the
entire template bank than decompose into sub-banks and split banks together.

After analyzing all possible relations between $p$ and $\beta$, It is clear
that the total computational cost will be less in the case of the full bank in
comparison to the sub-banks if and only if $p<\beta$ (Case-II). For a large
template bank, one can conclude that the ratio $\gamma$ solely depends on the
ratio between $\beta$ and $p$. Therefore, theoretically, it is easy to find
those values of $\beta$ for which the computational cost for the full bank is
less in comparison to the sub bank division. Still, practically, it is hard to
directly find the relation between $p$ and $\beta$ without trials and testing
the decomposition of the template matrix as a whole and into sub-banks. To
provide a notion of this relation, we have consider an example in which, let
us consider $N_{T}=10^{5}$, $N_{s}=10^{6}$ and $p=200$ that means each sub-
banks contain $500$ templates. in this set-up, $\gamma>1$ only when
$\beta>p(=200)$. Figure-20 depicts the same relation. This relation implies
that if the total number of basis vectors after decomposed the full bank is at
least $200$ (= $p$) times less than the total number of basis vectors
considering all the sub-banks, then the only total computational cost will be
less for the full-bank factorized based matched filtering scheme. But for the
other two cases, the reconstruction cost is the dominant cost. Therefore sub-
bank division is the only known optimal solution. Thus, it is important to
address the problem of reducing the reconstruction cost of the SNR time-series
construction. For any matrix factorization-based matched filtering scheme, the
set of bases are used to filter against the data, and the computation of the
approximated SNR time-series depends on the multiplication of the coefficient
vectors with the filter output. Therefore multiplication with the coefficient
matrix is an essential step in this process and can not be ruled out. In
General, the coefficient matrix obtained using SVD or RMF is a dense matrix
that means the weights corresponding to the basis are non-zero. Thus the
reconstruction cost is high. However, the MP algorithm allows us to obtain
sparse coefficients for a set of bases. The use of the sparse coefficient
matrix by replacing the dense one also incurred more approximation error in
the waveform reconstruction and as well as in the SNR time-series
construction. However, the approximation error is small. Therefore the sparse
matrix can be used to construct the SNR time-series. In this way, we can
reduce the reconstruction cost if we use a sparse coefficient matrix obtained
from the MP algorithm in the place of a dense coefficient matrix using SVD or
RMF.

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