Trace dynamics, and a ground state in
spontaneous quantum gravity

Abhinash Kumar Roya1, Anmol Sahub2 and Tejinder P. Singhc3

aIndian Institute of Science Education and Research Kolkata 741246, India

bIndian Institute of Science Education and Research, Pune 411008, India

cTata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005,
India

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Published in Mod. Phys. Lett. (2020) 2150019

DOI: https://doi.org/10.1142/S021773232150019X

ABSTRACT

We have recently proposed a Lagrangian in trace dynamics, to describe a
possible unification of gravity, Yang-Mills fields, and fermions, at the
Planck scale. This Lagrangian for the unified entity - called the aikyon - is
invariant under global unitary transformations, and as a result possesses a
novel conserved charge, known as the Adler-Millard charge. In the present
paper, we derive an eigenvalue equation, analogous to the time-independent
Schrödinger equation, for the Hamiltonian of the theory. We show that in the
emergent quantum theory, the energy eigenvalues of the aikyon are
characterised in terms of a fundamental frequency times Planck’s constant. The
eigenvalues of this equation can, in principle, determine the values of the
parameters of the standard model. We also report a ground state, in this
theory of spontaneous quantum gravity, which could characterise a non-singular
initial epoch in quantum cosmology.

##  I An eigenvalue equation for the hamiltonian of an aikyon

We have recently proposed MPSingh ; Singhspin  the following Lagrangian and
action in trace dynamics Adler:04 ; Adler:94 ; AdlerMillard:1996 , to describe
the unification of gravity, Yang-Mills fields, and fermions:

$\displaystyle\frac{S}{C_{0}}=\frac{1}{2}\int\frac{d\tau}{\tau_{Pl}}\;Tr\biggl{[}\biggr{.}\dfrac{L_{p}^{2}}{L^{4}}\biggl{\\{}\biggr{.}i\alpha\biggl{(}\biggr{.}q_{B}+\dfrac{L_{p}^{2}}{L^{2}}\beta_{1}q_{F}\biggl{.}\biggr{)}+L\biggl{(}\biggr{.}\dot{q}_{B}+\dfrac{L_{p}^{2}}{L^{2}}\beta_{1}\dot{q}_{F}\biggl{.}\biggr{)}\biggl{.}\biggr{\\}}\times$
$\displaystyle\biggl{\\{}\biggr{.}i\alpha\biggl{(}\biggr{.}q_{B}+\dfrac{L_{p}^{2}}{L^{2}}\beta_{2}q_{F}\biggl{.}\biggr{)}+L\biggl{(}\biggr{.}\dot{q}_{B}+\dfrac{L_{p}^{2}}{L^{2}}\beta_{2}\dot{q}_{F}\biggl{.}\biggr{)}\biggl{.}\biggr{\\}}\biggl{.}\biggr{]}$
(1)

Here, the action is made dimensionless by scaling with respect to $C_{0}$, a
real constant with dimensions of action, which is subsequently identified as
being proportional to Planck’s constant (the proportionality constant being
chosen so as to recover the correct numerical factor in the emergent
Einstein’s equations). $q_{B}$ and $q_{F}$ are respectively even-grade (hence
bosonic) and odd-grade (hence fermionic) Grassmann matrices, which describe
Yang-Mills fields and their fermionic current source. Their time derivatives
with respect to the Connes time $\tau$, i.e. $\dot{q}_{B}$ and $\dot{q}_{F}$,
represent gravity and its fermionic mass source, respectively. Together, these
four dynamical variables describe an ‘atom’ of space-time matter, or an
‘aikyon’, which has an associated length scale $L$. $\beta_{1}$ and
$\beta_{2}$ are constant fermionic matrices, and $\alpha$ is the gauge
coupling constant. At the Planck scale, the universe is populated by many such
aikyons, with the total action for the theory being the sum of their
individual actions. At scales below Planck scale, quantum field theory and
classical general relativity are emergent from the Planck scale trace dynamics
of these aikyons. These aspects have been studied in detail in our recent
papers MPSingh ; Singhspin ; Singh2020DA . In particular, we have investigated
the potential for the above Lagrangian to describe the unification of gravity
with the standard model. The reader is referred to these two papers for
details of these investigations.

In the present paper we derive an eigenvalue equation for the Hamiltonian of
the aikyon, and suggest that this equation could in principle predict values
of the standard model parameters. This equation turns out to be an analogue of
the time-independent Schrödinger equation in a potential, for four dynamical
variables. In order to arrive at this equation, we first define two bosonic
variables $q_{1}$ and $q_{2}$ as follows:

$q_{1}=q_{B}+a_{0}\beta_{1}q_{F},\qquad q_{2}=q_{B}+a_{0}\beta_{2}q_{F}$ (2)

where $a_{0}\equiv L_{P}^{2}/L^{2}$. In terms of these two variables, the
above trace Lagrangian for the aikyon can be written as

$Tr{\cal
L}=\frac{1}{2}a_{1}a_{0}Tr\left[\dot{q}_{1}\dot{q}_{2}-\frac{\alpha^{2}c^{2}}{L^{2}}q_{1}q_{2}+i\frac{\alpha
c}{L}\left(\dot{q}_{1}q_{2}+q_{1}\dot{q}_{2}\right)\right]$ (3)

where $a_{1}\equiv C_{0}/cL_{P}$, which would be proportional to Planck mass
once $C_{0}$ is identified to be proportional to Planck’s constant $\hbar$.
[The constant of proportionality is chosen to ensure the correct numerical
factors in the emergent Einstein equations. It is understood that the action
principle is now being written as $S=\int d\tau\;{\cal L}$]. In writing this
expression for the Lagrangian a $c$ factor has been pulled out from the
original definition of the ‘dot’ time derivative. Henceforth, the time
derivative of $q$ has dimensions of velocity, and since $a_{1}$ has dimensions
of mass, the trace Lagrangian has the correct dimensions of energy. The last
term in the trace Lagrangian is a total time derivative, and hence does not
contribute to the equations of motion, and is henceforth dropped, so that we
work with the Lagrangian:

$Tr{\cal
L}=\frac{1}{2}a_{1}a_{0}Tr\left[\dot{q}_{1}\dot{q}_{2}-\frac{\alpha^{2}c^{2}}{L^{2}}q_{1}q_{2}\right]$
(4)

The canonical momenta are given by

$p_{1}=\frac{\delta Tr{\cal
L}}{\delta\dot{q}_{1}}=\frac{1}{2}a_{1}a_{0}\dot{q}_{2};\qquad
p_{2}=\frac{\delta Tr{\cal
L}}{\delta\dot{q}_{2}}=\frac{1}{2}a_{1}a_{0}\dot{q}_{1}$ (5)

The Euler-Lagrange equations of motion are

$\ddot{q}_{1}=-\frac{\alpha^{2}c^{2}}{L^{2}}q_{1};\qquad\ddot{q}_{2}=-\frac{\alpha^{2}c^{2}}{L^{2}}q_{2}$
(6)

In terms of these two complex variables, the aikyon behaves like two
independent complex-valued oscillators. However, the degrees of freedom of the
aikyon couple with each other when expressed in terms of the self-adjoint
variables $q_{B}$ and $q_{F}$. This is because $q_{1}$ and $q_{2}$ both depend
on $q_{B}$ and $q_{F}$, the difference being that $q_{1}$ depends on
$\beta_{1}$ and $q_{2}$ depends on $\beta_{2}$.

The trace Hamiltonian for an aikyon is

$Tr{\cal H}=Tr[p_{1}\dot{q}_{1}+p_{2}\dot{q}_{2}-Tr{\cal
L}]=\frac{a_{1}a_{0}}{2}Tr\left[\frac{4}{a^{2}_{1}a^{2}_{0}}p_{1}p_{2}+\frac{\alpha^{2}c^{2}}{L^{2}}q_{1}q_{2}\right]$
(7)

and Hamilton’s equations of motion are

$\dot{q}_{1}=\frac{2}{a_{1}a_{0}}p_{2}\qquad\dot{q}_{2}=\frac{2}{a_{1}a_{0}}p_{1};\qquad\dot{p}_{1}=-\frac{a_{1}a_{0}\alpha^{2}c^{2}}{2L^{2}}q_{2};\qquad\dot{p}_{2}=-\frac{a_{1}a_{0}\alpha^{2}c^{2}}{2L^{2}}q_{1}$
(8)

Because the trace Lagrangian is invariant under global unitary
transformations, it possesses a novel conserved charge, known as the Adler-
Millard charge, and defined as follows:

$\tilde{C}=\sum_{i}[q_{Bi},p_{Bi}]-\sum_{i}\\{q_{Fi,}p_{Fi}\\}$ (9)

which is the sum, over all bosonic degrees of freedom, of the commutator
$[q_{B},p_{B}]$ minus the sum, over all fermionic degrees of freedom, of the
anti-commutator $\\{q_{F},p_{F}\\}$. This charge plays an important role in
the theory, as we see below.

In our theory, the Dirac operator $D_{B}$ is defined as
$D_{B}\equiv(1/L)dq_{B}/d\tau$, and this operator coincides with the standard
Dirac operator on a Riemannian manifold when space-time emerges in the
classical limit of the theory. The fermionic Dirac operator $D_{F}$ is defined
as time derivative of $q_{F}$ so that together we have $D\equiv D_{B}+D_{F}$
where

$D_{B}\equiv\frac{1}{Lc}\;\frac{dq_{B}}{d\tau};\qquad
D_{F}\equiv\frac{L_{P}^{2}}{L^{2}}\frac{\beta_{1}+\beta_{2}}{2Lc}\;\frac{dq_{F}}{d\tau}$
(10)

When Yang-Mills fields and their fermionic charge source are included in the
theory, we define the generalised Dirac operator $D_{newi}\equiv
D_{Bnewi}+D_{Fnewi}$ where

$\displaystyle D_{Bnewi}=\dfrac{1}{Lc}\dot{\widetilde{Q}}_{B}\qquad and\qquad
D_{Fnewi}=\frac{L_{P}^{2}}{L^{2}}\frac{\beta_{1}+\beta_{2}}{2Lc}\dot{\widetilde{Q}}_{F}$
(11)

and

${\dot{\widetilde{Q}}_{B}}=(i\frac{\alpha
c}{L}q_{B}+\dot{q}_{B});\qquad{\dot{\widetilde{Q}}_{F}}=(i\frac{\alpha
c}{L}q_{F}+\dot{q}_{F});$ (12)

Having transformed to the variables $q_{1}$ and $q_{2}$ it is straightforward
to show that $D_{newi}=D_{1}+D_{2}$ where

$D_{1}=\frac{1}{2Lc}\bigg{(}\dot{q}_{1}+i\frac{\alpha c}{L}q_{1}\bigg{)}\
;\qquad D_{2}=\frac{1}{2Lc}\bigg{(}\dot{q}_{2}+i\frac{\alpha
c}{L}q_{2}\bigg{)}$ (13)

In order to arrive at the desired Schrödinger equation, if we divide $q_{1}$
and $q_{2}$ into their self-adjoint and anti-self-adjoint parts as
$q_{1}=q_{1}^{s}+q_{1}^{as}$, $q_{2}=q_{2}^{s}+q_{2}^{as}$, then we can use
the self-adjoint and anti-self-adjoint parts as new dynamical variables:

$q_{1}^{s}=\frac{1}{2}(q_{1}+q_{1}^{\dagger});\qquad
q_{1}^{as}=\frac{1}{2}(q_{1}-q_{1}^{\dagger});\qquad
q_{2}^{s}=\frac{1}{2}(q_{2}+q_{2}^{\dagger});\qquad
q_{2}^{as}=\frac{1}{2}(q_{2}-q_{2}^{\dagger})$ (14)

The Lagrangian then becomes

${\cal
L}=\frac{a_{1}a_{0}}{2}Tr\bigg{[}\frac{4}{a^{2}_{1}a^{2}_{0}}\bigg{\\{}\dot{q}_{2}^{s}\dot{q}_{1}^{s}+\dot{q}_{2}^{as}\dot{q}_{1}^{as}\bigg{\\}}-\frac{\alpha^{2}c^{2}}{L^{2}}\bigg{\\{}q_{1}^{s}q_{2}^{s}+q_{1}^{as}q_{2}^{as}\bigg{\\}}+\frac{4}{a^{2}_{1}a^{2}_{0}}\bigg{\\{}\dot{q}_{2}^{s}\dot{q}_{1}^{as}+\dot{q}_{2}^{as}\dot{q}_{1}^{s}\bigg{\\}}-\frac{\alpha^{2}c^{2}}{L^{2}}\bigg{\\{}q_{1}^{s}q_{2}^{as}+q_{1}^{as}q_{2}^{s}\bigg{\\}}\bigg{]}$
(15)

In terms of these new variables, and in terms of the self-adjoint and anti-
self-adjoint parts of the momenta $p^{s}$ and $p^{as}$, the real and imaginary
parts of the trace Hamiltonian are given as follows:

$TrH^{s}=\frac{a_{1}a_{0}}{2}Tr\bigg{[}\frac{4}{a^{2}_{1}a^{2}_{0}}\bigg{\\{}p_{1}^{s}p_{2}^{s}+p_{1}^{as}p_{2}^{as}\bigg{\\}}+\frac{\alpha^{2}c^{2}}{L^{2}}\bigg{\\{}q_{1}^{s}q_{2}^{s}+q_{1}^{as}q_{2}^{as}\bigg{\\}}\bigg{]}$
(16)
$TrH^{as}=\frac{a_{1}a_{0}}{2}Tr\bigg{[}\frac{4}{a^{2}_{1}a^{2}_{0}}\bigg{\\{}p_{1}^{s}p_{2}^{as}+p_{1}^{as}p_{2}^{s}\bigg{\\}}+\frac{\alpha^{2}c^{2}}{L^{2}}\bigg{\\{}q_{1}^{s}q_{2}^{as}+q_{1}^{as}q_{2}^{s}\bigg{\\}}\bigg{]}$
(17)

The conserved Adler-Millard charge, and its self-adjoint and anti-self-adjoint
components are

$\begin{split}&\tilde{C}=[q_{1},p_{1}]+[q_{2},p_{2}]\\\
&\tilde{C}^{as}=[q_{1}^{s},p_{1}^{s}]+[q_{1}^{as},p_{1}^{as}]+[q_{2}^{s},p_{2}^{s}]+[q_{2}^{as},p_{2}^{as}]\\\
&\tilde{C}^{s}=[q_{1}^{s},p_{1}^{as}]+[q_{1}^{as},p_{1}^{s}]+[q_{2}^{s},p_{2}^{as}]+[q_{2}^{as},p_{2}^{s}]\end{split}$
(18)

In the theory of trace dynamics, at energies below Planck scale, quantum
theory (without a background spacetime) emerges. This is achieved by coarse-
graining the theory over many Planck time scales, and by applying the methods
of statistical thermodynamics to arrive at the emergent quantum theory. This
happens provided the self-adjoint part of the Adler-Millard charge can be
neglected, and the anti-self-adjoint part of the Hamiltonian can be neglected.
When that happens, the Adler-Millard charge gets equipartitioned over the four
degrees of freedom, the equipartitioned value is identified with Planck’s
constant, and quantum commutation relations emerge, for the statistically
averaged dynamical variables at statistical equilibrium:

$[q_{1}^{s},p_{1}^{s}]=i\hbar,\qquad[q_{1}^{as},p_{1}^{as}]=i\hbar,\qquad[q_{2}^{s},p_{2}^{s}]=i\hbar,\qquad[q_{2}^{as},p_{2}^{as}]=i\hbar$
(19)

The averaged dynamical variables obey Heisenberg equations of motion. The
above commutation relations allow us to write the momenta as displacement
operators conjugate to their corresponding configuration variables, enabling
us to move to the position representation

$p_{1}^{s}=-i\hbar\frac{\partial\ }{\partial q_{1}^{s}},\qquad
p_{1}^{as}=-i\hbar\frac{\partial\ }{\partial q_{1}^{as}},\qquad
p_{2}^{s}=-i\hbar\frac{\partial\ }{\partial q_{2}^{s}},\qquad
p_{2}^{as}=-i\hbar\frac{\partial\ }{\partial q_{2}^{as}},$ (20)

Using these, and the form of the self-adjoint part of the Hamiltonian
operator, we arrive at the following sought after eigenvalue equation
$H^{s}\psi=\lambda\psi$ for the operator Hamiltonian for these four dynamical
variables [with the knowledge that in the emergent quantum theory, the
configuration variables are treated as c-numbers, in the position
representation]:

$-\frac{2\hbar^{2}}{a_{1}a_{0}}\left(\frac{\partial^{2}\ }{\partial
q_{1}^{s}\;\partial q_{2}^{s}}+\frac{\partial^{2}\ }{\partial
q_{1}^{as}\;\partial
q_{2}^{as}}\right)\psi+\frac{\alpha^{2}c^{2}a_{1}a_{0}}{2L^{2}}(q_{1}^{s}q_{2}^{s}+q_{1}^{as}q_{2}^{as})\psi=\lambda\psi$
(21)

This fundamental equation is a time-independent Schrödinger equation for these
four variables, in the potential shown in the equation. It is the analog of
the Wheeler-deWitt equation in our theory, and predicts the quantum
gravitational states of the aikyon. The ground state solution of this equation
will tell us about the non-singular state of the universe at the big bang
epoch. Hopefully, this equation also carries information about the parameters
of the standard model, considering that it depends on the gauge coupling
constant $\alpha$, and also on the length parameter $L$ from which the
fermionic mass $m=\hbar/Lc$ is defined. These investigations are currently in
progress.

Here, we include a few preliminary observations about this equation and its
solutions. The system has the characteristics of an oscillator, considering
the bi-linear nature of the potential. That suggests we could think of the
coefficient of the potential term as one-half of a ‘spring constant’ k, hence
$k=\alpha^{2}c^{2}a_{1}a_{0}/L^{2}$. Also, from the kinetic energy term, the
‘mass’ $m$ can be read off as $m=a_{1}a_{0}/4$ which is proportional to
$m_{Pl}L_{P}^{2}/L^{2}$. This would imply that the ‘frequency’
$\omega=\sqrt{k/m}$ is given by $\omega=2\alpha c/L$. The most remarkable
thing here is that the Planck scale has dropped out from the estimate of the
frequency! We comment below on the possible implications of this result. A
scale analogous to this frequency is actually seen in the solution of the
above Schrödinger equation, though with a twist.

To see this, we start by noting that the above equation for the two self-
adjoint and two anti-self adjoint variables can be separated into two
equations, by writing the wave-function
$\psi(q_{1}^{s},q_{2}^{s},q_{1}^{as},q_{2}^{as})$ as a product state:

$\psi(q_{1}^{s},q_{2}^{s},q_{1}^{as},q_{2}^{as})\equiv\psi_{ss}(q_{1}^{s},q_{2}^{s})\;\psi_{aa}(q_{1}^{as}q_{2}^{as})$
(22)

Substituting this in the above Schrödinger equation yields two decoupled
Schrödinger equations:

$-\frac{2\hbar^{2}}{a_{1}a_{0}}\left(\frac{\partial^{2}\ }{\partial
q_{1}^{s}\;\partial
q_{2}^{s}}\right)\psi_{ss}+\frac{\alpha^{2}c^{2}a_{1}a_{0}}{2L^{2}}(q_{1}^{s}q_{2}^{s})\psi_{ss}=\lambda_{ss}\psi_{ss}$
(23) $-\frac{2\hbar^{2}}{a_{1}a_{0}}\left(\frac{\partial^{2}\ }{\partial
q_{1}^{as}\;\partial
q_{2}^{as}}\right)\psi_{aa}+\frac{\alpha^{2}c^{2}a_{1}a_{0}}{2L^{2}}(q_{1}^{as}q_{2}^{as})\psi_{aa}=\lambda_{aa}\psi_{aa}$
(24)

with $\lambda=\lambda_{aa}+\lambda_{ss}$. It turns out that the solutions are
not oscillatory, but exponentially decaying [expected, for the ‘spatial part’
of a system bounded by a potential, the oscillatory time-dependence in Connes
time is in any case there for this stationary solution] or exponentially
growing. We will find the eigenvalue spectrum and the corresponding
eigenstates in the next section.

The imaginary part of the trace Hamiltonian is given by Eqn. (17) above. It
has contributions from cross-terms which couple the self-adjoint parts of
position and momenta to their anti-self-adjoint parts. This anti-self-adjoint
part of the Hamiltonian is expected to become significant when sufficiently
many aikyons get entangled with each other. When that happens, superpositions
of position states become extremely short-lived and spontaneous localisation
results, in the spirit of the Ghirardi-Rimini-Weber-Pearle theory of objective
collapse Ghirardi:86 ; Pearle:89 ; Ghirardi2:90 ; Bassi:03 ; RMP:2012 . Hence
we expect to be able to derive collapse models, including the collapse rate
parameter, and the spectrum of the collapse noise, starting from the
Lagrangian of our theory. This work is currently in progress. One can expect
the collapse rate parameter to be related to $\alpha c/L$, the frequency that
arises naturally in our theory.

Next, we present the exact spectrum of the eigenvalues of the above
Schrödinger equation, which confirms the above preliminary observations.

###  I.1 Spectrum of eigenvalues for the energy eigenstates of the aikyon

The Schrödinger equation (23) is a partial differential equation in two
variables, of the form

$\frac{\partial^{2}\psi}{\partial x\partial y}+K_{1}xy\psi=K_{2}\psi$ (25)

where

$K_{1}=-\frac{a_{1}^{2}a_{0}^{2}\alpha^{2}c^{2}}{4L^{2}\hbar^{2}};\qquad
K_{2}=-\frac{\lambda a_{1}a_{0}}{2\hbar^{2}}$ (26)

where $\lambda$ is the eigenvalue of the Schrödinger equation, and we will
assume $xy\geq 0$ so that the potential is bounded from below.

Let us consider the following ansatz for the solution to this equation:

$\psi(x,y)\sim f(x)g(y)\exp[-mxy]$ (27)

where $f(x)$ is an arbitrary function of $x$ and $g(y)$ is an arbitrary
function of $y$, and $m>0$ so that the wave-function is bounded.

By differentiating the wave function in this ansatz first with respect to $y$
and then with respect to $x$, and substituting this second order derivative in
(25) we get the following equation:

$\begin{split}f^{\prime}(x)g^{\prime}(y)-&myf(x)g^{\prime}(y)-mxf^{\prime}(x)g(y)-mf(x)g(y)+m^{2}xyf(x)g(y)\\\
&+K_{1}xyf(x)g(y)=K_{2}f(x)g(y)\end{split}$ (28)

Now comparing both sides we get two conditions:

$\qquad m^{2}+K_{1}=0\implies m=\pm(-K_{1})^{1/2}\implies
m=\frac{a_{1}a_{0}\alpha c}{2L\hbar}$ (29)

where we have ignored the negative root because we have assumed $m>0$. Second
condition is

$\quad\frac{f^{\prime}(x)g^{\prime}(y)}{f(x)g(y)}-m\bigg{[}\frac{yg^{\prime}(y)}{g(y)}+\frac{xf^{\prime}(x)}{f(x)}+1\bigg{]}=K_{2}$
(30)

Since $K_{2}$ determines the eigenvalues, LHS of this equation must be
independent of $x$ and $y$. For this to be satisfied we must have the
following;

$f^{\prime}(x)g^{\prime}(y)=0$ (31)

which implies that both $f^{\prime}(x)$ and $g^{\prime}(y)$ cannot be non zero
simultaneously.We basically have two cases for Eqn.(30).

${\rm Case\ I}\mathrel{\mathop{\ordinarycolon}}\qquad g^{\prime}(y)=0\implies
g(y)={\rm cons.}\qquad{\rm and}\qquad\frac{xf^{\prime}(x)}{f(x)}=r\implies
f(x)=x^{r}$ (32)

where we require $r\geq 0$ for wavefunction to be physical.

${\rm Case\ II}\mathrel{\mathop{\ordinarycolon}}\qquad f^{\prime}(x)=0\implies
f(x)={\rm cons.}\qquad{\rm and}\qquad\frac{yg^{\prime}(y)}{g(y)}=s\implies
g(y)=y^{s}$ (33)

where $s\geq 0$. Therefore the the general form of eigenstates for Eqn. (25)
will be

$\psi_{rs}=Cx^{r}y^{s}\exp\bigg{[}-\frac{a_{1}a_{0}\alpha
c}{2L\hbar}xy\bigg{]}$ (34)

with eigenvalue

$\lambda=\frac{\hbar\alpha c}{L}\;(1+r+s)$ (35)

where one of $r$ and $s$ must be zero. The ground state is given by

$\psi_{00}=C_{0}\exp\bigg{[}-\frac{a_{1}a_{0}\alpha
c}{2L\hbar}xy\bigg{]};\qquad\lambda=\frac{\hbar\alpha c}{L}$ (36)

It has the character of a non-singular ground state, with a ‘minimum area
spread’ given by $L\hbar/a_{1}a_{0}\alpha c=L^{3}/6\pi\alpha L_{P}$. [We have
taken $a_{1}=6\pi m_{Pl}$, this coefficient being the correct one for
recovering Einstein equations with correct numerical factors].

We do not yet have reason to assume that $r$ and $s$ take only discrete
values. Therefore we have a continuous spectrum of energy eigenstates. Since
one of $r$ and $s$ must be zero, for each $r\geq 0$ we have doubly degenerate
excited states each having an eigenvalue $(1+r)\hbar\alpha c/L$:

$\psi_{r0}=C_{r}\;x^{r}\exp\bigg{[}-\frac{a_{1}a_{0}\alpha
c}{2L\hbar}xy\bigg{]};\qquad\psi_{0r}=C_{r}\;y^{r}\exp\bigg{[}-\frac{a_{1}a_{0}\alpha
c}{2L\hbar}xy\bigg{]}$ (37)

For the original Schrödinger Equation (21) one can now easily deduce, assuming
that $q_{1}^{as}q_{2}^{as}$ is also positive, the ground state is given by

$\psi_{0}=C_{0}\exp\bigg{[}-\frac{a_{1}a_{0}\alpha
c}{2L\hbar}(q_{1}^{s}q_{2}^{s}+q_{1}^{as}q_{2}^{as})\bigg{]}$ (38)

and has the energy eigenvalue $2\hbar\alpha c/L$. This is the ground state of
an aikyon in our theory. If the aikyon is an electron, this state non-
perturbatively describes the electron, together with its mass aspect, and with
all the fields that it produces! We believe this is the first ever result of
its kind. Again there is a minimum area spread given by $L^{3}/6\pi\alpha
L_{P}$, and suggests a non-singular initial state of the universe at the epoch
of its beginning.

The excited states of the aikyon can be written as

$\psi=C\psi_{r1,s1}(q_{1}^{s},q_{2}^{s})\;\psi_{r2,s2}(q_{1}^{as}q_{2}^{as})$
(39)

and have the eigenvalue

$\lambda=\frac{\hbar\alpha c}{L}(2+r_{1}+s_{1}+r_{2}+s_{2})$ (40)

where one of $r_{1}$ or $s_{1}$ must be zero, and one of $r_{2}$ or $s_{2}$
must be zero.

It remains to be investigated if these solutions are relevant in the quantum
cosmological context, and if they can describe an expanding universe at the
big bang epoch and address the issue of singularity avoidance. What is highly
significant is that the energy eigenvalue is independent of Planck scale. We
recall that in our theory the underlying trace dynamics is assumed to take
place at the Planck scale. Whereas the quantum gravity theory whose solutions
we are describing here emerges after coarse-graining over many Planck time
scales. However, in our theory there is nothing special about Planck energy
scale. We could have chosen any other sufficiently high energy scale at which
quantum field theory has not been tested, and at which scale we assume the
laws of quantum field theory to be replaced by those of trace dynamics. This
suggests that the ratio $\alpha/L$ which appears in the energy eigenvalue
remains unchanged during coarse-graining. Since coarse-graining is a mechanism
for coming to lower energy scales from higher energy scales, and since the
gauge coupling constant $\alpha$ runs with energy, a change in $\alpha$ would
imply a change in $L$. And hence a change in bare mass, since $L$ is
identified with Compton wavelength; thus it might be possible to estimate
lepton and quark bare masses as well as standard model parameters, from a more
careful and detailed analysis of this theory. This will be attempted during
forthcoming investigations, and we hope that other researchers will also find
it interesting to explore this avenue.

##  II An aikyon in octonionic space

It is possible that the energy eigenvalues that we have found above for the
aikyon, relate to the standard model in a novel way. In our recent work
Singhspin  we have proposed that the aikyon lives and evolves in an 8-D
octonionic space. If this were to be so, the fundamental Lagrangian (1),
reproduced below,

$\displaystyle\frac{S}{C_{0}}=\frac{1}{2}\int\frac{d\tau}{\tau_{Pl}}\;Tr\biggl{[}\biggr{.}\dfrac{L_{p}^{2}}{L^{4}}\biggl{\\{}\biggr{.}i\alpha\biggl{(}\biggr{.}q_{B}+\dfrac{L_{p}^{2}}{L^{2}}\beta_{1}q_{F}\biggl{.}\biggr{)}+L\biggl{(}\biggr{.}\dot{q}_{B}+\dfrac{L_{p}^{2}}{L^{2}}\beta_{1}\dot{q}_{F}\biggl{.}\biggr{)}\biggl{.}\biggr{\\}}\times$
$\displaystyle\biggl{\\{}\biggr{.}i\alpha\biggl{(}\biggr{.}q_{B}+\dfrac{L_{p}^{2}}{L^{2}}\beta_{2}q_{F}\biggl{.}\biggr{)}+L\biggl{(}\biggr{.}\dot{q}_{B}+\dfrac{L_{p}^{2}}{L^{2}}\beta_{2}\dot{q}_{F}\biggl{.}\biggr{)}\biggl{.}\biggr{\\}}\biggl{.}\biggr{]}$
(41)

could describe the standard model of particle physics, and its unification
with gravity. This trace Lagrangian is invariant under global unitary
transformations, and in the octonionic space these are generated by the
algebra of automorphisms which form the group $G_{2}$. The symmetries are
described by the Clifford algebra $Cl(2)$ generated from the algebra of
complex quaternions, and the Clifford algebra $Cl(6)$ generated from the
algebra of complex octonions f1 ; f3 . In Lagrangian (41), the objects $q_{B}$
and $\dot{q}_{B}$ together form an octonion, and similarly $q_{F}$ and
$\dot{q}_{F}$ form another octonion. Every one of
$q_{B},q_{F},\dot{q}_{B},\dot{q}_{F}$ carries four indices, with the time
derivative terms carrying the octonionic indices $(1,e_{1},e_{2},e_{3})$ and
$q_{B}$ and $q_{F}$ each carrying the other four octonionic indices
$(e_{4},e_{5},e_{6},e_{7})$. Hence we have

$\begin{split}q_{B}=(q_{Be4},q_{Be5},q_{Be6},q_{Be7});\qquad\dot{q}_{B}=(\dot{q}_{B1},\dot{q}_{Be1},\dot{q}_{Be2},\dot{q}_{Be3})\\\
q_{F}=(q_{Fe4},q_{Fe5},q_{Fe6},q_{Fe7});\qquad\dot{q}_{F}=(\dot{q}_{F1},\dot{q}_{Fe1},\dot{q}_{Fe2},\dot{q}_{Fe3})\end{split}$
(42)

Thus, when the brackets are opened up, the Lagrangian has 256 terms (some of
these will be removed as they are total time derivatives) and these could
incorporate the standard model and gravity Singhspin . $q_{B}$ and $q_{F}$
respectively describe would-be-Yang-Mills and its fermionic charged source.
$\dot{q}_{B}$ and $\dot{q}_{F}$ describe would-be-gravity and its fermionic
mass source. Together these four entities describe the unification of gravity
and Yang-Mills, with each other and with the fermionic source, so as to form
the aikyon, whose Lagrangian is the one written above.

The structure of the Lagrangian (41) in our theory then suggests it can
describe one generation of fermions.This can be inferred by noting that the
space-time symmetries of the charge aspect of the fermions arise from the
cross-terms $i\alpha\beta_{1}q_{F}\times L\dot{q}_{B}$ and $L\dot{q}_{B}\times
i\alpha\beta_{2}q_{F}$. In the octonionic space spanned by the directions
$(1,e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7})$ the would-be space-time
directions are $(1,e_{1},e_{2},e_{3})$ and the other four directions are
internal directions. The Dirac operator $D_{B}\propto\dot{q}_{B}$ has four
space-time components and these are $[D_{B1},D_{Be1},D_{Be2},D_{Be3}]$. The
fermionic charge aspect has the four components
$[q_{Fe4},q_{Fe5},q_{Fe6},q_{Fe7}]$. Therefore, the cross terms, between them,
have $4\times 4+4\times 4=32$ components. Suppressing the labels $D_{F}$ and
$q_{F}$ these thirty-two components can be represented as

$\beta_{1}\;[e_{4},e_{5},e_{6},e_{7}]\times[1,e_{1},e_{2},e_{3}]\quad+\quad[1,e_{1},e_{2},e_{3}]\times\beta_{2}\;[e_{4},e_{5},e_{6},e_{7}]$
(43)

Every one of the four directions $(e_{4},e_{5},e_{6},e_{7})$ gives rise to
eight terms, as can be easily read off from this previous expression. Thus
these thirty-two components can also be equivalently written as four sets of
eight terms each, as follows:

$\begin{split}[\beta_{1}e_{4}\times 1,\beta_{1}e_{4}\times
e_{1},\beta_{1}e_{4}\times e_{2},\beta_{1}e_{4}\times
e_{3}]\quad&+\quad[1\times\beta_{2}e_{4},e_{1}\times\beta_{2}e_{4},e_{2}\times\beta_{2}e_{4},e_{3}\times\beta_{2}e_{4}]\\\
[\beta_{1}e_{5}\times 1,\beta_{1}e_{5}\times e_{1},\beta_{1}e_{5}\times
e_{2},\beta_{1}e_{5}\times
e_{3}]\quad&+\quad[1\times\beta_{2}e_{5},e_{1}\times\beta_{2}e_{5},e_{2}\times\beta_{2}e_{5},e_{3}\times\beta_{2}e_{5}]\\\
[\beta_{1}e_{6}\times 1,\beta_{1}e_{6}\times e_{1},\beta_{1}e_{6}\times
e_{2},\beta_{1}e_{6}\times
e_{3}]\quad&+\quad[1\times\beta_{2}e_{6},e_{1}\times\beta_{2}e_{6},e_{2}\times\beta_{2}e_{6},e_{3}\times\beta_{2}e_{6}]\\\
[\beta_{1}e_{7}\times 1,\beta_{1}e_{7}\times e_{1},\beta_{1}e_{7}\times
e_{2},\beta_{1}e_{7}\times
e_{3}]\quad&+\quad[1\times\beta_{2}e_{7},e_{1}\times\beta_{2}e_{7},e_{2}\times\beta_{2}e_{7},e_{3}\times\beta_{2}e_{7}]\end{split}$
(44)

The eight terms in any one row represent the coupling of one internal
direction to the four would-be-spacetime directions. It appears that there are
two leptons (as desired), these being the components of the type
$(\beta_{1}e_{4}\times 1,1\times\beta_{2}e_{4})$, and the other twenty-four
components possibly represent the six quarks. It could be that a more direct
interpretation arises in terms of the $q_{1}$ and $q_{2}$ variables studied in
the previous section.

The vector bosons come from the square term $-\alpha^{2}q_{B}^{2}$ in the
Lagrangian, which lives in the internal space. So $q_{B}$ carries the four
octonionic indices $(e_{4},e_{5},e_{6},e_{7})$. Hence there are sixteen
components to this square term, $-\alpha^{2}q_{B}^{2}$. We expect that the
twelve off-diagonal components represent the twelve massless vector bosons,
whereas the four diagonal components presumably represent the four Higgs
bosons (prior to spontaneous symmetry breaking). Hence there are a total of
sixteen bosons eight fermions of one generation the theory.

Where do the heavier fermion generations come from? It is possible that there
arises in the theory an eigenvalue equation for the fermion masses, whose
unequal roots give rise to different fermion masses. Possibly, this
information is already contained in the analysis of the previous section.
Further developments in this direction are reported in Singh2020DA .

###  II.1 Understanding the Lagrangian in octonionic space

We will assume for the sake of further investigation that the Grassmann matrix
$q$ is defined over the field of octonions. The space in which it resides is
8-D, and labelled conventionally by the directions
$(1,e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7})$. The first direction is the
real line and the other seven are imaginary directions whose algebra is
described by the multiplication rules of the Fano plane. The following index
allocation is assumed for the variables $q_{B},\dot{q}_{B},q_{F}$ and
$\dot{q}_{F}$. $\dot{q}_{B}$ has quaternionic indices $(1,e_{1},e_{2},e_{3})$
where $(e_{1},e_{2},e_{3})$ is a quaternionic triplet, so that these four
directions form a closed algebra. Therefore $\dot{q}_{B}$ is a quaternion, and
the term $\dot{q}_{B}^{2}$ in the trace Lagrangian describes would-be emergent
spacetime. This term has 16 components. The term $iq_{B}$ carries the
octonionic indices $(e_{4},e_{5},e_{6},e_{7})$. Its square has 16 components
and it describes the standard model gauge fields and the four Higgs bosons.
One can think of the four spacetime directions [could be thought of as
‘horizontal’ Singhspin ] as being orthogonal to the four internal gauge
directions [could be thought of as ‘vertical’].

The Grassmann matrix $\dot{q}_{F}$ is assumed to have the quaternionic indices
$(1,e_{1},e_{2},e_{3})$ and describes the mass aspect of the fermionic degrees
of freedom. It’s square term is order $L_{P}^{6}$ and does not contribute in
the classical limit, which comes from terminating the heat kernel expansion at
order $L_{P}^{4}$. This term provides quantum corrections to the classical
theory, and we need not consider these in the classical limit. However, for
reasons to be mentioned below, we retain them for now. Further, in the trace
Lagrangian, cross terms involving $\dot{q}_{F}$ contribute, but only cross
terms with $\dot{q}_{B}$ contribute: there are 32 such terms. Other cross-
terms involving $\dot{q}_{F}$ contribute to the total time derivative in the
Lagrangian, and are hence dropped [more on this below]. We emphasize that
$\dot{q}_{B}$ and $\dot{q}_{F}$ carry identical quaternionic indices: together
they form the ‘horizontal’ gravity-mass quaternionic ‘plane’ in octonioniic
space; it is the domain of emergent 4-D space-time.

The Grassmann matrix $iq_{F}$ is assumed to have the the octonionic indices
$(e_{4},e_{5},e_{6},e_{7})$; this represents fermionic charge aspect and
together with $q_{B}$, which has the same indices, it forms the ‘vertical’
gauge-charge ‘plane’ in the octonionic space. Note that these four directions
do not involve the real line (which is what makes them different from
spacetime) and they do not form a closed algebra. Thus one can have space-
time-mass without gauge-charge, but one cannot have gauge-charge without
space-time. The contribution of $q_{F}$ to the Lagrangian, from its square, is
order $L_{P}^{6}$ and is retained. Its contribution from product with
$\dot{q}_{B}$ and $\dot{q}_{F}$ is dropped, being part of total time-
derivative. Thus it only contributes from terms of the form $q_{F}q_{B}$ and
$q_{B}q_{F}$, there being a total of 32 such terms.

One can easily check from the Lagrangian (1) and (3) that a total of 128
terms, out of the original 256, contribute to the total time derivatives, and
are hence removed. There are 32 terms of order $L_{P}^{6}$ which are kept.
That gives 128 terms which are as follows: 16 terms in $\dot{q}_{B}^{2}$ which
describe gravity, 16 terms in $q_{B}^{2}$ which describe gauge-fields, 32
terms which describe gravity-mass interaction, 32 terms which describe gauge-
charge interaction, and 32 fermion-squared terms of order $L_{P}^{6}$ whose
curious status we comment on below.

The reduced trace Lagrangian with 128 terms can hence be written as

$\begin{split}\frac{S}{C_{0}}=\frac{1}{2}\int\frac{d\tau}{\tau_{Pl}}\;Tr\bigg{[}\biggr{.}\frac{L_{P}^{2}}{L^{2}}\bigg{\\{}\biggr{.}\dot{q}_{B}^{2}+\frac{L_{P}^{2}}{L^{2}}\dot{q}_{B}\beta_{2}\dot{q}_{F}+\frac{L_{P}^{2}}{L^{2}}\beta_{1}\dot{q}_{F}\dot{q}_{B}+\frac{L_{P}^{4}}{L^{4}}\beta_{1}\dot{q}_{F}\beta_{2}\dot{q}_{F}\\\
-\frac{\alpha^{2}}{L^{2}}\bigg{(}q_{B}^{2}+\frac{L_{P}^{2}}{L^{2}}q_{B}\beta_{2}q_{F}+\frac{L_{P}^{2}}{L^{2}}\beta_{1}q_{F}q_{B}+\frac{L_{P}^{4}}{L^{4}}\beta_{1}q_{F}\beta_{2}q_{F}\bigg{)}\biggl{.}\bigg{\\}}\biggl{.}\bigg{]}\end{split}$
(45)

Each of the eight terms has sixteen components, giving a total of 128, same as
in the Lagrangian in Eqn. (4), there being 64 terms of kinetic energy type,
and 64 terms of potential energy type. Although the $i$ factors have
disappeared, there are anti-self-adjoint terms in the Lagrangian, as expected.
The harmony between gravity and gauge-fields is noteworthy!

If we set $\alpha=0$, half of the terms vanish, and we are in the ‘horizontal’
gravity-mass plane, which is quaternionic and has a closed algebra. This
emerges as spacetime-gravity-mass in the classical limit [will have gauge-
charge contributions if $\alpha$ is not zero]. The Lagrangian is

$\frac{S}{C_{0}}=\frac{1}{2}\int\frac{d\tau}{\tau_{Pl}}\;Tr\bigg{[}\frac{L_{P}^{2}}{L^{2}}\bigg{\\{}\dot{q}_{B}^{2}+\frac{L_{P}^{2}}{L^{2}}\dot{q}_{B}\beta_{2}\dot{q}_{F}+\frac{L_{P}^{2}}{L^{2}}\beta_{1}\dot{q}_{F}\dot{q}_{B}+\frac{L_{P}^{4}}{L^{4}}\beta_{1}\dot{q}_{F}\beta_{2}\dot{q}_{F}\bigg{\\}}\bigg{]}$
(46)

This is the pure gravity case that we studied in maithresh2019 . If the matter
part is ignored and if we consider only the first term $\dot{q}_{B}^{2}$, it
is the complex quaternion part studied by Furey f1  to describe the Lorentz
symmetry $Cl(2)$. When matter is included, it appears to suggest the Clifford
algebra $Cl(4)$. When the full Lagrangian (45) is considered, the associated
symmetry is described by $Cl(6)$ and this is the Lagrangian for one generation
of fermions in the standard model, unified with gravity.

We now illustrate possible adjointness properties of this Lagrangian, after
making some assumptions on the $\beta$ matrices. This choice is not unique but
only illustrative. We assume $q_{B}$ and $q_{F}$ to be self-adjoint, and
$\beta_{1}q_{F}$ and $\beta_{2}q_{F}$ to be each anti-self-adjoint. With these
conditions we have

$\begin{split}&q_{1}^{s}=q_{B},\ q_{1}^{as}=\beta_{1}\ q_{F},\
q_{2}^{s}=q_{B},\ q_{2}^{as}=\beta_{2}q_{F},\\\
&p_{1}^{s}=\frac{1}{2}a_{1}a_{0}\dot{q}_{B},\
p_{2}^{s}=\frac{1}{2}a_{1}a_{0}\dot{q}_{B},\
p_{1}^{as}=\frac{1}{2}a_{1}a_{0}\beta_{2}\dot{q}_{F},\
p_{2}^{as}=\frac{1}{2}a_{1}a_{0}\beta_{1}\dot{q}_{F}\end{split}$ (47)

In terms of these variables, the Lagrangian in (45) becomes

${\cal
L}=\frac{a_{1}a_{0}}{2}Tr\bigg{[}\frac{4}{a^{2}_{1}a^{2}_{0}}\bigg{\\{}\dot{q}_{2}^{s}\dot{q}_{1}^{s}+\dot{q}_{2}^{as}\dot{q}_{1}^{as}\bigg{\\}}-\frac{\alpha^{2}c^{2}}{L^{2}}\bigg{\\{}q_{1}^{s}q_{2}^{s}+q_{1}^{as}q_{2}^{as}\bigg{\\}}+\frac{4}{a^{2}_{1}a^{2}_{0}}\bigg{\\{}\dot{q}_{2}^{s}\dot{q}_{1}^{as}+\dot{q}_{2}^{as}\dot{q}_{1}^{s}\bigg{\\}}-\frac{\alpha^{2}c^{2}}{L^{2}}\bigg{\\{}q_{1}^{s}q_{2}^{as}+q_{1}^{as}q_{2}^{s}\bigg{\\}}\bigg{]}$
(48)

which of course matches with (15) obtained from the Lagrangian (4), as
expected. The first half is the self-adjoint part and the second half is the
anti-self-adjoint part. Each of the eight terms in the Lagrangian has sixteen
components, which is true also for the Hamiltonian in Eqns. (16) and (17). If
we were to drop the 32 terms of order $L_{P}^{6}$ terms from the Lagrangian,
the self-adjoint terms $p_{1}^{as}p_{2}^{as}$ and $q_{1}^{as}q_{2}^{as}$ would
disappear from the above Lagrangian, and also alter the eigen-solutions we
have found for the Hamiltonian operator. The curious role of these terms,
which do not survive in the leading order classical limit, remains to be
understood.

##  III Concluding Remarks

Since the Hamiltonian has an anti-self-adjoint component, it implies that
under the circumstances when this component becomes significant, unitary
evolution is lost, and spontaneous localisation takes place. Spontaneous
localisation then acts as a means of suppressing four of the octonionic
dimensions in the classical limit: As we have noted, the aikyon lives in an
8-D octonionic space. Four of these constitute would-be-classical spacetime.
Which four? This is very interesting indeed. The octonionic space has one real
direction and seven imaginary directions. The rules of multiplication amongst
the vectors along the imaginary direction are given by the Fano plane. The
seven imaginary vectors criss-cross in triplets of three vectors such that any
three in the same triplet form a closed algebra, often called quaternionic
triplets. Pick a triplet - along with the real direction: these four form
would-be-spacetime! It is a closed algebra. Would-be-gravity lives only in
these four directions. The Yang-Mills fields are associated with the other
four directions in the octonionic space; though more strictly they lie all
over the octonionic space, because the other four directions do not form a
closed algebra. There is leakage from these directions onto the would-be-
spacetime directions.

Aikyons evolve in this 8-D octonionic space. Perhaps we can talk of
‘collisions’ in this space as a means of interaction. We can surely talk of
entanglement of aikyon states. When sufficiently many of them get entangled,
spontaneous localisation results [imaginary part of the Hamiltonian becomes
significant], and entanglement breaks down. The entangled system localises to
one of the states in the original superposition. Now, the real part of the
Hamiltonian (to which the localisation takes place) depends only on the four
directions which involve the real line and a quaternionic triplet [the would-
be-spacetime]. Spontaneous localisation hence results in the emergence of a
classical spacetime, with classical matter fields and classical gauge fields
and gravity maithresh2019 ; stcw .

In essence, spontaneous localisation suppresses dimensions. The classical
system lives in 4-D spacetime. Whereas a quantum system lives in 8-D
octonionic space. So there is no need to compactify the extra dimensions -
classical systems never go there. Whereas quantum systems do, and that is
perfectly fine. That is what makes them different from classical systems.

Acknowledgements: It is our pleasure to thank Sourabh Magare, Abhishek Sharma
and Varun Srivastava for discussions and valuable help on the eigenvalue
problem for the Schrödinger equation discussed in this paper.

REFERENCES

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