# Quantization of measure in gravitation

Vladimir Dzhunushaliev<EMAIL_ADDRESS>Department of Theoretical
and Nuclear Physics, Al-Farabi Kazakh National University, Almaty 050040,
Kazakhstan  Institute of Experimental and Theoretical Physics, Al-Farabi
Kazakh National University, Almaty 050040, Kazakhstan  Academician J. Jeenbaev
Institute of Physics of the NAS of the Kyrgyz Republic, 265 a, Chui Street,
Bishkek 720071, Kyrgyzstan  Vladimir Folomeev<EMAIL_ADDRESS>Institute of
Experimental and Theoretical Physics, Al-Farabi Kazakh National University,
Almaty 050040, Kazakhstan  Academician J. Jeenbaev Institute of Physics of the
NAS of the Kyrgyz Republic, 265 a, Chui Street, Bishkek 720071, Kyrgyzstan

###### Abstract

Using the Radon-Nikodym theorem concerning the relation between any two
measures, as well as the methods employed in loop quantum gravity, it is shown
that, in gravitation, one can quantize any measure which is not even
associated with metric. We have considered the simplest case where the
proportionality coefficient between two operators of measure (the Radon-
Nikodym derivative) is some function. The result is that in the right-hand
side of the commutation relations for the measure the fundamental length
becomes a variable quantity, and this can lead to smoothing the singularity.
The case where the Radon-Nikodym derivative is an operator is also under
discussion. Classical and quantum theories on the background of space endowed
with quantum measure are under consideration.

measure, quantization, field theory, background

##  I Introduction

In differential geometry, there are only three geometric quantities which can
be defined on a manifold – the measure, connection, and metric. In general
relativity, the first two quantities are associated with metric: the measure
is defined by the square root of the determinant of a metric tensor and the
connection is the Christoffel symbols. But there is the following difference
between the measure and connection. The connection associated with metric
arises naturally in general relativity when one uses the Palatini approach. By
varying the Einstein-Hilbert action with respect to the metric and connection,
one gets equations for the connection which ensure that the connection becomes
associated with metric. It is interesting that such a procedure for the
measure is absent. This means that: (a) in general relativity, the measure is
merely postulated as $d\mu=\sqrt{-g}\,dx^{0}dx^{1}dx^{2}dx^{3}$; and (b) for
the measure, there are no any field equations, in contrast to the connection
for which there are the corresponding equations appearing when one uses the
Palatini method. It is easily explained by the fact that between the measure
on the one hand, and the metric and connection on the other, there is the
following difference: the measure is a nonlocal quantity, whereas the metric
and connection are local ones.

An entirely different situation arises in loop quantum gravity (LQG) where,
when quantizing the Ashtekar variables, there emerges the quantization of the
volume, area, and length (see, e.g., the textbooks [1, 2]), i.e., the
quantization of the measure. This remarkable fact suggests that, apparently,
in quantum gravity the measure becomes a dynamical variable and is defined
while quantising, although there are no any dynamical equations for the
measure.

One of the fruitful approaches in quantum gravity is LQG [3, 4], where the
spectra of such geometric quantities like the area [5] and volume [6] have
been calculated. In LQG, the SU(2) Ashtekar-Barbieri connection is introduced;
this enables one to quantize the operators of area and volume. For a review of
main results obtained in LQG, see Ref. [7].

The purpose of the present paper is to apply the methods developed in LQG for
quantizing an arbitrary measure. We use the results obtained to investigate
the dynamics of arbitrary classical and quantized fields on the background of
space endowed with quantum measure. In conclusion, we discuss questions
arising in quantizing the measure.

##  II Quantization of measure in LQG

In the present paper we consider the question of the quantization of measure
in gravitation. This implies that the measure of any region $\mathcal{A}$ in
space, for example, of volume, area or length, becomes an operator:

$\widehat{\mu(\mathcal{A})}=\int\limits_{\mathcal{A}}\widehat{d\mu},$

where $\mathcal{A}$ can be a volume, area or length. Of course, the main
question here is how one can define an operator algebra
$\widehat{\mu(\mathcal{A})}$? One of the main problems here is that such an
operator is nonlocal, since it is defined not at the point but in some region
$\mathcal{A}$. If we compare this situation with quantum field theory, we see
that there fields are described by local operators and one knows how they can
be quantized; this is the difference of principle between quantum field theory
and the problem considered in the present paper when we quantize a nonlocal
operator of the measure $\widehat{\mu(\mathcal{A})}$.

In quantum field theory, the question of quantization of the given theory is
resolved using the fact that there is classical field theory with the
corresponding field equations, and this theory should be quantized by
introducing field operators. To quantize the measure, such an approach is
excluded, since, in classical gravity, there are no field equations for the
measure. In general relativity, the measure is merely postulated as

$d\mu_{GR}=\sqrt{-g}\,d\Omega,$

where $d\Omega=dx^{0}dx^{1}dx^{2}dx^{3}$.

When quantizing the measure, there arise problems which are absent in local
quantum field theory. For example, one has to take into account that in
calculating the two-point Green function

$G\left(\widehat{\mu(\mathcal{A}_{1})},\widehat{\mu(\mathcal{A}_{2})}\right)=\left\langle
Q\left|\widehat{\mu(\mathcal{A}_{1})}\widehat{\mu(\mathcal{A}_{2})}\right|Q\right\rangle$

the regions $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ may intersect or not, they
may have different dimensions, etc. The essence of these problems is due to
the fact that the measure is a nonlocal object, whereas in standard quantum
field theory all physical quantities are local.

The remarkable fact is that the problems of quantization of such a nonlocal
quantity as the measure had been solved in LQG. But for this purpose, in LQG,
some new extra structure is introduced, which is absent in classical general
relativity – a triangulation $\Delta$ of our spacetime $\mathcal{M}$, i.e.,
the spacetime is divided into four-simplices. Every four-simplex is bounded by
five tetrahedra. In LQG, the dual $\Delta^{*}$ of the triangulation with
vertices, edges, and faces are used. The relation between $\Delta$ and
$\Delta^{*}$ is illustrated as follows:

$\begin{matrix}\text{Bulk triangulation }\Delta&&\text{Dual
complex}\Delta^{*}\\\ \text{4-simplex }(v)&\leftrightarrow&\text{vertex
}(\mathrm{v})\\\ \text{tetrahedron }(\tau)&\leftrightarrow&\text{edge
}(\mathrm{e})\\\ \text{triangle }(t)&\leftrightarrow&\text{face
}(\mathrm{f})\\\ \text{segment }(s)&&\\\ \text{point }(p)&&\end{matrix}$ (1)

That is, instead of the triangulation $\Delta$, one can employ the graph
$\Delta^{*}$ with vertices, edges, and faces.

This extra structure $\Delta^{*}$ enables one to resolve the problem of
quantization in the following manner. One introduces the operator (here, we
follow the textbook [1], chapter 7)

$\hat{\vec{E}}_{\ell}=8\pi\gamma G\hbar\hat{\vec{L}}_{\ell},$

where $\gamma$ is the Barbero-Immirzi constant. The operator
$\hat{\vec{L}}_{\ell}$ is the area operator of the faces of the tetrahedron
punctured by the link $\ell$, and, before the quantization (in the classical
case), it is defined as follows:

$L^{i}_{S}=\frac{1}{2\gamma}\int
dS^{i}=\frac{1}{2\gamma}\epsilon^{i}_{jk}\int\limits_{S}e^{j}\wedge e^{k},$

with the tetrad $e^{a}_{\mu}$ chosen in the time gauge where $e^{0}=dt$,
$e^{i}=e^{i}_{\mu}dx^{\mu}$ and $i,j,k=1,2,3$; $S$ is some two-dimensional
surface punctured by the link $\ell$. The area of this surface is defined as

$S=\left|\vec{E}_{\ell}\right|.$

After the quantization, the quantity $L^{i}_{\mathcal{S}}$ becomes an operator
with the following commutation relations that can be obtained from the
hamiltonian analysis of general relativity, by promoting Poisson brackets to
operators, or can be postulated [8]:

$\left[\hat{E}^{i}_{S_{1}},\hat{E}^{j}_{S_{2}}\right]=i\Delta_{S_{1}S_{2}}l_{0}^{2}\epsilon^{ij}_{\phantom{ij}k}\hat{E}_{S_{1}}^{k},$
(2)

where $l_{0}^{2}=8\pi\gamma G\hbar$ is a constant proportional $\hbar$ and
with the dimensions of an area; $\Delta_{S_{1}S_{2}}$ is the Dirac delta
functional defined as

$\Delta_{S_{1}S_{2}}=\begin{cases}1\text{ if }S_{1}\text{ coincides with
}S_{2},\\\ 0\text{ otherwise }\end{cases}.$

The eigenvalues of the square of this operator form a discrete spectrum

$S_{j}=l_{0}^{2}\sqrt{j(j+1)}.$ (3)

From the point of view of quantization of measure, the operator
$\frac{l_{0}^{2}}{2\gamma}\epsilon^{i}_{jk}\hat{e}^{j}\wedge\hat{e}^{k}$ is
the operator of the measure $\widehat{d\mu^{i}_{g}}$ of the area spanned on
the vectors $\hat{e}^{j},\hat{e}^{k}$:

$\widehat{d\mu_{g}^{i}}=\frac{l_{0}^{2}}{2\gamma}\epsilon^{i}_{jk}\hat{e}^{j}\wedge\hat{e}^{k}.$

Here, we use the index $g$ in the definition of the element of the measure
$\widehat{d\mu_{g}^{i}}$ in order to emphasize that this measure is associated
with the metric $g$. Using this definition of the operator of the element of
the measure, one can rewrite the commutation relations (2) in the
infinitesimal form,

$\left[\widehat{d\mu_{g}^{i}},\widehat{d\mu_{g}^{j}}\right]=i\Delta_{dS_{1}dS_{2}}l_{0}^{2}\epsilon^{ij}_{\phantom{ij}k}\widehat{d\mu_{g}^{k}}.$
(4)

Using this notion of the measure $\mu$, one can also rewrite the commutation
relations (2) in the following form:

$\left[\widehat{\mu_{g}}\left({S^{i}_{1}}\right),\widehat{\mu_{g}}\left({S^{j}_{2}}\right)\right]=i\Delta_{S_{1}S_{2}}l_{0}^{2}\epsilon^{ij}_{\phantom{ij}k}\widehat{\mu_{g}}\left({S^{k}_{1}}\right).$
(5)

A complete description of quantization in LQG, including the quantization of a
volume, can be found, for instance, in the textbooks [1, 2]; we will not go
into this subject here.

##  III Quantization of measure in the general case

In this section we suggest a method of quantization of measure in the general
case, analogous to what was done in LQG when quantizing an area. The main idea
is as follows:

  * •

the triangulation of spacetime is introduced, as in LQG;

  * •

the measure $\mu$ is introduced, which in general is not associated with
metric;

  * •

the operator of this measure $\hat{\mu}$ is introduced, for which the
commutation relations are imposed, analogous to what was done for the operator
of area in LQG.

  * •

we use the Radon-Nikodym theorem, which relates two measures $d\mu$ and
$d\mu_{g}$.

Thus we have the measure $\mu$ related to the measure $\mu_{g}$ associated
with the metric $g$ as follows:

$\mu(\mathcal{A})=\int\limits_{\mathcal{A}}fd\mu_{g}.$

This can be rewritten in the infinitesimal form as

$d\mu=fd\mu_{g},$

where $f$ is some function, called the Radon-Nikodym derivative. After the
quantization, we arrive at the following operator equality:

$\widehat{\mu(\mathcal{A})}=\int\limits_{\mathcal{A}}\hat{f}\,\widehat{d\mu_{g}}.$

Formally, in the infinitesimal form, this can be rewritten as

$\widehat{d\mu}=\hat{f}\,\widehat{d\mu_{g}}.$

The restriction of this measure to the area labelled by the link $\ell_{i}$
has the following form:

$\widehat{\mu(S^{i})}=\int\limits_{S^{i}}\widehat{d\mu^{i}}=\int\limits_{S^{i}}\hat{f}\widehat{d\mu_{g}^{i}}.$

Now we are dealing with the very difficult question of defining the properties
of the operator $\widehat{\mu(S^{i})}$. The challenge we face is the presence
of the operator $\hat{f}$; this does not allow us to generalize the
commutation relations (5) in a simple way.

The main problem here is the determination of the properties of the operator
$\hat{f}$. Of course, in the general case this operator can be an arbitrary
quantity, but it is only of interest to consider the case where the operator
$\hat{f}$ is defined using some equations. In this case we have only one
possibility: $\hat{f}$ is a function of some quantized field whose properties
can be defined using the corresponding quantization rules for the given field.
For example, this can be a positively defined Lagrangian of such a field,
$\hat{f}=\mathcal{L}(\hat{\Phi}^{A})$, where the index $A$ corresponds to all
possible indices of the field $\Phi^{A}$. The positiveness of the Lagrangian
is imposed due to the theorem A.1. As such positively defined Lagrangian, one
can take, for example, a Lagrangian in Euclidean space with the field
potential bounded below.

As another possibility, one can consider an approach with the so-called
dynamical measure introduced in Refs. [9, 10, 11]. In these works, it is
assumed that, in the gravitational action, instead of the measure $d\mu_{g}$
associated with metric, one uses a nonmetric measure $d\phi^{1}\wedge
d\phi^{2}\wedge d\phi^{3}\wedge d\phi^{4}$ with the scalar fields
$\phi^{i},i=1,2,3,4$, which are determined using some dynamical equations.

As one more possibility, one can consider modified bimetric gravity [12] where
two measures corresponding to the metrics $g_{1}$ and $g_{2}$ are used in the
gravitational action to integrate the Ricci scalars $R\left(g_{2}\right)$ and
$R\left(g_{1}\right)$, respectively, as follows:

$\mathcal{L}\propto\sqrt{-g_{1}}R\left(g_{2}\right)+\sqrt{-g_{2}}R\left(g_{1}\right).$

This enables one to use in each term the measure which is not associated with
metric in the corresponding scalar curvature. Nevertheless, for each measure,
there are its own dynamical differential Euler-Lagrange equations. Within this
approach, as the $\Phi^{A}$-fields for the metric $g_{1}$, it will be the
metric $g_{2}$ and vice versa.

In the aforementioned case when the operator $\hat{f}$ is taken to be
$\hat{f}=\mathcal{L}(\hat{\Phi}^{A})$, the algebra of operators of the measure
$\widehat{\mu(S)}$ becomes much more complicated compared with the algebra of
operators of the measure $\widehat{\mu_{g}(S)}$ associated with metric, since
it is necessary to take into account the algebra of operators of the quantized
field $\hat{\Phi}^{A}$. In the general case the properties of the operator
$\widehat{\mu}$ will be determined by all Green functions

$G_{n}=\left\langle\Psi_{\hat{\mu}}\right|\hat{\mu}\left(S_{1}\right),\hat{\mu}\left(S_{2}\right),\dots,\hat{\mu}\left(S_{n}\right)\left|\Psi_{\hat{\mu}}\right\rangle,$
(6)

where $\left|\Psi_{\hat{\mu}}\right\rangle$ is a quantum state describing both
the measure $\widehat{\mu_{g}}$ and the quantized field $\hat{\Phi^{A}}$. It
is important to note here that the properties of the operator
$\widehat{\mu(S)}$ are not determined by the commutation relations similar to
(5), but should be determined by all Green functions (6).

The situation is simplified considerably if the operator $\hat{f}$ is just a
function of $\hat{f}=f$. In this case one can write the following equality
that generalizes the corresponding theorem from the mathematical analysis:

$\widehat{\mu(S)}=\int\limits_{S}f\,\widehat{d\mu_{g}}=f(\xi)\int\limits_{S}\widehat{d\mu_{g}},\quad\xi\in
S.$

Then, using the commutation relations (5), one can derive the following
expression:

$\begin{split}\left[\widehat{\mu(S^{i}_{1})},\widehat{\mu(S^{j}_{2})}\right]=&i\Delta_{S_{1}S_{2}}l_{0}^{2}\frac{f\left(\xi_{1}\right)f\left(\xi_{2}\right)}{f\left(\xi_{3}\right)}\epsilon^{ij}_{\phantom{ij}k}\widehat{\mu}\left({S^{k}_{1}}\right)=i\Delta_{S_{1}S_{2}}l_{0}^{2}F\left(\xi_{1,2,3}\right)\epsilon^{ij}_{\phantom{ij}k}\widehat{\mu}\left({S^{k}_{1}}\right)\\\
=&i\Delta_{S_{1}S_{2}}l^{2}\left(\xi_{1,2,3}\right)\epsilon^{ij}_{\phantom{ij}k}\widehat{\mu}\left({S^{k}_{1}}\right),\end{split}$
(7)

where $\xi_{1}\in S^{i}_{1}$, $\xi_{2}\in S^{j}_{2}$, and $\xi_{3}\in
S^{k}_{1}$, and a nonlocal quantity $l\left(\xi_{1,2,3}\right)$ with the
dimensions of length occurs. Formally, in the infinitesimal form, this
expression takes the form

$\left[\widehat{d\mu^{i}},\widehat{d\mu^{j}}\right]=i\Delta_{dS_{1}dS_{2}}l_{0}^{2}f(x)\epsilon^{ij}_{\phantom{ij}k}\widehat{d\mu^{k}}=i\Delta_{dS_{1}dS_{2}}l^{2}(x)\epsilon^{ij}_{\phantom{ij}k}\widehat{d\mu^{k}},$

since in the infinitesimal form $\xi_{1}=\xi_{2}=\xi_{3}=x\in dS_{1}$ and one
can introduce the function $l(x)$ with the dimensions of length.

The appearance of the function $l\left(\xi_{1,2,3}\right)$ instead of the
fundamental length $l_{0}$ has the result that the discrete spectrum (3)
contains now the length $l$ which varies in space; this leads to interesting
consequences. For example, near the singularity, $f\left(x\rightarrow
x_{\text{sing}}\right)\rightarrow\infty$, and hence the length
$l\left(\xi_{1,2,3}\rightarrow x_{\text{sing}}\right)\rightarrow\infty$
becomes infinitely large as well. That is, the size of an elementary cell in
the spin foam becomes infinitely large. But since the singularity occurs at
the point and the size of the elementary cell becomes infinitely large while
approaching the singularity (at that, one cannot speak about the space inside
the cell), this means that while approaching the singularity an increase of
the function $f$ is regulated such that it remains finite inside the cell.
This means that the singularity is regularized when the quantum measure is
taken into account.

Using the same methods as those employed in LQG (see, for example, Ref. [1],
chapters 5 and 8), one can quantize the measure $\widehat{d\mu}$ restricted to
the volume and length.

##  IV Physical theories on the background of space with quantum measure

LQG suggests that our space is a quantum object with fluctuating measure, and,
as we clarified here, the measure need not be associated with metric.
Therefore, the question arises as to how one can define the action in
classical theory or quantize fields when a quantum measure is taken into
account?

The next interesting question is whether it is possible to make somehow a
particular choice of the function $f$ in the classical case or of the operator
$\hat{f}$ in the quantum case? If we compare this situation with classical
physics, we see that there are fundamental equations coming from the principle
of least action which determine the fields, and in quantum theory there are
quantization rules determining the properties of field operators. Thus we are
interested in the following question: how one can assign equations determining
the function $f$ or the operator $\hat{f}$?

We think that the only way of doing this is as follows: in the classical case,
$f$ is a function of some fields, i.e., it is a Lagrangian density; in the
quantum case, $\hat{f}$ is defined by operators of some quantum field whose
algebra must be defined independently. Then in the classical case the Euler-
Lagrange equations determine the fields $\rightarrow$ the Lagrangian
$\rightarrow$ the function $f$ $\rightarrow$ the operator of the measure
$f\widehat{d\mu_{g}}$. In the quantum case, one defines the algebra of field
operators $\rightarrow$ the operator $f$ $\rightarrow$ the operator of the
measure $\hat{f}\widehat{d\mu_{g}}$.

###  IV.1 Classical physics on the background of spacetime with quantum
measure

Consider now the question of how one can assign equations of motion in
classical physics on the background of spacetime with quantum measure. One may
assume that this can be done as follows. The quantum states in LQG are
described by the spin network states
$\psi_{j_{\ell},v_{n}}(\mathcal{U}_{\ell})=\left|\Gamma,j_{\ell},v_{n}\right\rangle,$
where $\Gamma$ is the graph, $v$ are the eigenvalues, $\left|v\right\rangle$
are eigenstates of the volume in each Hilbert space $\mathcal{K}_{j_{1}\ldots
j_{4}}$ (for more details concerning these quantities, see Ref. [1], chapter
7). In each such state, one can write an action for any classical field and
then, using the principle of least action, calculate the field distribution
ensuring the minimum of the corresponding action. In practice, this means that
one obtains a solution to the Euler-Lagrange equations for the given classical
field. Thus we have the following solution to the classical Euler-Lagrange
equations:

$\frac{\partial}{\partial
x^{\mu}}\left(\frac{\partial\mathcal{L}}{\partial\Phi^{A}_{,\mu}}\right)-\frac{\partial\mathcal{L}}{\partial\Phi^{A}_{\mu}}=0\text{
on the background of the spin network state
}\left|\Gamma,j_{\ell},v_{n}\right\rangle,$ (8)

where the index $A$ in the field $\Phi^{A}$ is a generalized index labelling
all indices of the given field. As a result, we have a solution of the Euler-
Lagrange equations (8) $\phi^{A}_{\left|\Gamma,j_{\ell},v_{n}\right\rangle}$
on the background of the spin network state
$\left|\Gamma,j_{\ell},v_{n}\right\rangle$. Physically, this means that for a
pure quantum state $\left|\Gamma,j_{\ell},v_{n}\right\rangle$ the spatial
distribution of the classical field $\Phi^{A}$ is
$\phi^{A}_{\left|\Gamma,j_{\ell},v_{n}\right\rangle}$. If we have a quantum
superposition

$\Psi=\sum_{\left|\Gamma,j_{\ell},v_{n}\right\rangle}C_{j_{\ell},v_{n}}\left|\Gamma,j_{\ell},v_{n}\right\rangle,$

the mean field $\Phi^{A}$ may be determined in a standard manner to be

$\left\langle\Phi^{A}\right\rangle=\sum_{\left|\Gamma,j_{\ell},v_{n}\right\rangle}\left|C_{j_{\ell},v_{n}}\right|^{2}\Phi^{A}_{\left|\Gamma,j_{\ell},v_{n}\right\rangle}.$

Similarly, one can determine the dispersion
$\left\langle\left(\left\langle\Phi^{A}\right\rangle-\Phi^{A}_{\left|\Gamma,j_{\ell},v_{n}\right\rangle}\right)^{2}\right\rangle$
and all other average values.

###  IV.2 Quantum field theory on the background of space with quantum measure

In this subsection we consider the question of how one can define quantum
field theory on the background of space with quantum measure. This means that
we have a space endowed with quantum measure which is associated with metric
and given by the commutation relations in the infinitesimal form (4) or in the
finite form (5), and also the quantized field $\hat{\Phi^{A}}$.

In this case a wave function describing such a system will be the product of
the wave function describing the quantum state of the field $\Phi^{A}$,
$\left|\Psi_{\Phi^{A}}\right\rangle$, and the spin network state,
$\psi_{j_{\ell},v_{n}}(\mathcal{U}_{\ell})=\left|\Gamma,j_{\ell},v_{n}\right\rangle$,

$\left|\Psi_{\hat{\mu},\Phi^{A}}\right\rangle=\left|\Gamma,j_{\ell},v_{n}\right\rangle\left|\Psi_{\Phi^{A}}\right\rangle.$

The presence of this wave function enables us, analogous to what was done in
Sec. III, to define the Green function

$G_{n}\left(x_{1},x_{2},\dots,x_{n};S_{1},S_{2},\dots,S_{n}\right)=\left\langle\Psi_{\hat{\mu}}\right|\Phi^{A}\left(x_{1}\right)\widehat{\mu_{g}}\left(S_{1}\right),\Phi^{A}\left(x_{2}\right)\widehat{\mu_{g}}\left(S_{2}\right),\dots,\Phi^{A}\left(x_{n}\right)\widehat{\mu_{g}}\left(S_{n}\right)\left|\Psi_{\hat{\mu}}\right\rangle.$
(9)

As one can see, the structure of these Green functions is extremely
complicated since they contain both the local arguments $x_{i}$ and nonlocal
ones: the restriction of the measure $\widehat{\mu_{g}}$ to two-dimensional
areas $S_{i}$.

But this is not all. Using the techniques developed in LQG, it is necessary to
define the restriction of the operator of the measure $\widehat{\mu_{g}}$ to
the one- and three-dimensional regions (volumes and lengths). Then, using
these operators and making a generalization, as in Eq. (9), one can define the
corresponding Green functions

$G_{n}\left(x_{1},x_{2},\dots,x_{n};O_{1},O_{2},\dots,O_{n}\right)=\left\langle\Psi_{\hat{\mu}}\right|\Phi^{A}\left(x_{1}\right)\widehat{\mu_{g}}\left(O_{1}\right),\Phi^{A}\left(x_{2}\right)\widehat{\mu_{g}}\left(O_{2}\right),\dots,\Phi^{A}\left(x_{n}\right)\widehat{\mu_{g}}\left(O_{n}\right)\left|\Psi_{\hat{\mu}}\right\rangle,$

where $O_{i}$ are one-, two-, and three-dimensional regions. These Green
functions will completely determine the properties of the operator
$\widehat{\mu}=\hat{f}\widehat{\mu_{g}}$.

##  V Physical discussion and conclusions

One of remarkable results obtained in LQG is the quantization of length, area,
and volume; this implies the possibility of quantization of measure which is
associated with metric. Using the Radon-Nikodym theorem relating any two
measures, we have shown that in gravitation one can quantize any measure which
is not even associated with metric. In this case the properties of the
operator of a measure which is not associated with metric differ from the
properties of the operator of a measure associated with metric. In the
simplest case, if the Radon-Nikodym derivative (the proportionality
coefficient between two measures in the Radon-Nikodym theorem) is some
function, the fundamental length appearing in the right-hand side of the
corresponding commutation relations becomes a variable quantity. For example,
near a singularity where the Radon-Nikodym derivative goes to infinity, the
size of an elementary cell in the spin foam will tend to infinity, and this
can lead to smoothing the singularity. In the case where the Radon-Nikodym
derivative is an operator, the properties of the operator of a measure change
even more stronger, and they are determined by all Green functions of the
operator of a measure.

For such an approach in quantizing the measure, the following interesting
question arises. We have shown that, based on the results obtained within LQG,
one may quantize any measure which can be even not associated with metric. The
quantization of the measure which is not associated with metric is carried out
through Green functions (6) using the quantization of the measure associated
with metric in the form (5) or in the infinitesimal form (4). Here, it is of
great importance that the measure is a nonlocal object and hence nonlocal
objects are present in the commutation relations. The regions where the
nonlocal objects are defined can intersect one another or not, and also they
can have different dimensions. All this greatly complicates the problem of
defining the commutation relations for such nonlocal objects. To solve this
extremely complicated problem, in LQG, in order to make the problem as simple
as possible, one introduces a triangulation of space. In practice, the
triangulation enables one to assign commutation relations only for regions
having identical dimensions. Therefore, a natural question arises as to
whether the triangulation is an auxiliary trick introduced only to quantize
nonlocal objects? If it is true, the space remains continuous (i.e., not
discrete), and the triangulation is just a convenient trick, allowing one to
avoid all problems related to the quantization of nonlocal objects. This
resembles somewhat the situation in general relativity where one can use any
coordinate systems, but there are invariant quantities, e.g., the scalar
curvature.

Next, we have considered classical and quantum theories on the background of
space endowed with quantum measure. In the first case, we have shown that for
any quantum state of the spin foam there is its own solution of the Euler-
Lagrange field equations. This means that in the general case there is a
quantum superposition of solutions of the Euler-Lagrange equations, for which
one can evaluate the average solution, its dispersion, etc. In the second
case, the given quantum system is determined by quantum states of the spin
foam and quantized field.

The question of quantization of measure is the question of quantization of a
nonlocal quantity. The literature in the field discusses the question of
nonlocal theories for a long time (see, e.g., Refs. [13, 14, 15]), and
therefore, in the context of the present paper, the question arises as to
whether there exists any relation between the quantization of measure and
nonlocal quantum field theories? This interesting question requires a special
and careful consideration.

A special and separate question arises when one studies the quantization of
metric and measure. In LQG, it is assumed that the approach being suggested
there will enable one to quantize gravity as well, that is, a metric and
connection. If this is not so, then the question arises concerning the
quantization of two independent quantities – the measure and the metric. The
main challenge appears when one takes into account their mutual influence on
each other, when both quantities are quantum objects. In the classical case
this problem is resolved in such a way that the relation
$d\mu_{g}=\sqrt{-g}d^{4}x$ between them is postulated. An additional
difficulty is that, in the classical case, there are no Euler-Lagrange
equations for the measure which give the relation between the measure and
metric $d\mu_{g}=\sqrt{-g}d^{4}x$, similar to the relation between the metric
and connection within the Palatini approach.

In conclusion, we would like to note that in the present paper we have
discussed positively defined measures, but, using the theorem A.2, one can
generalize the results obtained by us to signed and complex measures.

## Acknowledgments

The work was supported by the program No. AP14869140 (The study of QCD effects
in non-QCD theories) of the Ministry of Education and Science of the Republic
of Kazakhstan. We are also grateful to the Research Group Linkage Programme of
the Alexander von Humboldt Foundation for the support of this research.

##  Appendix A Minimum necessary insight into measure theory

Definition of measure:

######  Definition A.1.

Let $X$ be a set and $\Sigma$ a $\sigma$-algebra over $X$. A function $\mu$
from $\Sigma$ to the extended real number line is called a measure if it
satisfies the following properties:

  * •

Non-negativity: For all $E$ in $\Sigma$, we have $\mu(E)\geq 0$.

  * •

Null empty set: $\mu(\varnothing)=0$.

  * •

Countable additivity (or $\sigma$-additivity): For all countable collections
$\left\\{E_{k}\right\\}_{k=1}^{\infty}$ of pairwise disjoint sets in $\Sigma$,

$\mu\left(\bigcup_{k=1}^{\infty}E_{k}\right)=\sum_{k=1}^{\infty}\mu(E_{k}).$

The pair $\left(X,\Sigma\right)$ is called a measurable space. A triple
$\left(X,\Sigma,\mu\right)$ is called a measure space. The Radon-Nikodym
theorem involves a measurable space $(X,\Sigma)$ on which two $\sigma$-finite
measures are defined, $\mu$ and $\nu$.

######  Theorem A.1 (the Radon-Nikodym theorem [16]).

if $\nu\ll\mu$ (that is, if $\nu$ is absolutely continuous with respect to
$\mu$), then there exists a $\Sigma$-measurable function $f:X\to[0,\infty)$,
such that for any measurable set $\mathcal{A}\subseteq X$,

$\nu(\mathcal{A})=\int\limits_{\mathcal{A}}f\,d\mu.$ (10)

The function $f$ is called the Radon-Nikodym derivative and is denoted as
$\frac{d\nu}{d\mu}$. The meaning of this theorem is that any two measure are
interrelated by some function.

A similar theorem can be proven for signed and complex measures:

######  Theorem A.2 (the Radon-Nikodym theorem).

if $\mu$ is a nonnegative $\sigma$-finite measure, and $\nu$ is a finite-
valued signed or complex measure such that $\nu\ll\mu$, that is, $\nu$ is
absolutely continuous with respect to $\mu$, then there is a $\mu$-integrable
real- or complex-valued function $g$ on $X$ such that for every measurable set
$\mathcal{A}\subseteq X$

$\nu(\mathcal{A})=\int\limits_{\mathcal{A}}g\,d\mu.$ (11)

## References

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  * [9] E. I. Guendelman and A. B. Kaganovich, The Principle of nongravitating vacuum energy and some of its consequences, Phys. Rev. D 53, 7020 (1996).
  * [10] E. I. Guendelman and A. B. Kaganovich, Gravitational theory without the cosmological constant problem Phys. Rev. D 55, 5970 (1997).
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  * [13] G. V. Efimov, Non-local quantum theory of the scalar field Commun. Math. Phys. 5, 42 (1967).
  * [14] J. W. Moffat, Finite nonlocal gauge field theory Phys. Rev. D 41, 1177 (1990).
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  * [16] W. Rudin, Real and Complex Analysis (New York: McGraw-Hill, pp. 116-134, 1986).


# Group analysis for
generalized reaction-diffusion convection equation

M. Nadjafikhah<EMAIL_ADDRESS>S. Dodangeh
<EMAIL_ADDRESS>School of Mathematics, Iran University of
Science and Technology, Narmak, Tehran 1684613114, Iran.

###### Abstract

In this paper we discuss about group classification for non-linear generalized
reaction-diffusion convection equation:
$u_{t}=(f(x,u)u_{x})_{x}+h(x,u)u_{x}+k(x,u)$, by using Lie-classical symmetry
method. For this, first we find its symmetry group and then we find
differential invariants for resulted group by using infinitesimal criterion
method and at the end reduce modeling equations by using resulted invariants.
We present application of this group classification in group classification
and obtaining related similarity solution of KPP equation, too.

###### keywords:

symmetry, group classification, differential invariants, Lie-classical
method,infinitesimal criterion method, RDC equation, KPP equation, similarity
solutions.

††thanks:  Corresponding author: Tel. +9821-73913426. Fax +9821-77240472.

, ,

##  1 Introduction

This paper devoted to group classification of Generalized Reaction-Diffusion
Convection (G-RDC) equation, by using Lie-classical method.

$\displaystyle\Delta\;:\;u_{t}=(f(x,u)u_{x})_{x}+h(x,u)u_{x}+k(x,u),$ (1.1)

Where $u(x,t)$ is unknown function and $f(x,u),h(x,u),k(x,u)$ are arbitrary
functions. The equation (1.1) generalizes a number of the well known second-
order evolution equations, describing various process in physics, chemistry
and biology. Symmetry group method plays an important role in the analysis of
differential equations. The history of group classification methods goes back
to Sophus Lie [9]. (See [2, 4, 7, 6]). His work devoted to finding symmetry
groups, differential invariants and linearized or reduced equation for given
model. There are several approach to group classification of differential
equation, we apply infinitesimal method (See [2, 4, 10]) for this. There are
another useful articles and accounts about group classification for similar
equations of (1.1) via other methods, (See [11, 12, 13]). In this paper we
generalize RDC equation to G-RDC equation and applay Lie-classical symmetry
method via applied approach. we hope this work be useful to applied and
theoretical readers.

##  2 Group classification for modeling equation

Let following one-parameter group

$\displaystyle\overline{x}=x+\varepsilon\xi(x,t,u)+O(\varepsilon^{2}),\qquad\overline{t}=t+\varepsilon\eta(x,t,u)+O(\varepsilon^{2}),\qquad\overline{u}=u+\varepsilon\varphi(x,t,u)+O(\varepsilon^{2}),$
(2.2)

be symmetry group of modeling equation $\Delta$. We can obtain $\xi$, $\eta$
and $\varphi$, by using infinitesimal method.

Consider the vector field
$X:=\xi\partial_{x}+\eta\partial_{t}+\varphi\partial_{u}$ in total space
$M=(x,t,u)$ with $p=2$ and $q=1$. If this vector field be an infinitesimal
generator of $\Delta$’s symmetry group, then

$\displaystyle X^{(2)}\Delta=0,\qquad\textrm{whenever}\qquad\Delta=0.$ (2.3)

Where $X^{(2)}$ is second prolong of $X$ and has following form:

$\displaystyle
X^{(2)}=X+\varphi^{x}\partial_{u_{x}}+\varphi^{t}\partial_{u_{t}}+\varphi^{xx}\partial_{xx}+\varphi^{xt}\partial_{u_{xt}}+\varphi^{tt}\partial_{u_{tt}},$
(2.4)

Where $\varphi^{x},\varphi^{t},\varphi^{xx},\varphi^{xt}$ and $\varphi^{tt}$
are respectively:

$\displaystyle\varphi^{x}=D_{x}Q+\xi u_{xx}+\eta u_{xt},\hskip
45.52458pt\varphi^{t}=D_{t}Q+\xi u_{xt}+\eta u_{tt},$
$\displaystyle\varphi^{xx}=D_{xx}Q+\xi u_{xxx}+\eta
u_{xxt},\qquad\varphi^{xt}=D_{xt}Q+\xi u_{xxt}+\eta
u_{xtt},\qquad\varphi^{tt}=D_{tt}Q+\xi u_{xtt}+\eta u_{ttt}.$

Where $D_{x}$, $D_{t}$ are total derivative with respect to specified
variables, $D_{xx}=D_{x}D_{x}$, $D_{xt}=D_{x}D_{t}$ and $D_{tt}=D_{t}D_{t}$,
and $Q=\varphi-\xi u_{x}-\eta u_{t}$ is the corresponding characteristic of
$X$ (See [2, 4, 7, 6]). By using criterion (2.4), we find:

$\displaystyle\varphi^{t}=(f_{xx}u_{x}+f_{xu}u_{x}^{2}+f_{x}u_{xx}+k_{x}+h_{x}u_{x})\xi+(f_{xu}u_{x}+f_{uu}u_{x}^{2}+f_{u}u_{xx}+h_{u}u_{x}+k_{u})\varphi+$
$\displaystyle\hskip
25.6073pt+(f_{x}+2f_{u}u_{x}+h)\varphi^{x}+f\varphi^{xx}.$ (2.5)

By substituting $\varphi^{x},\varphi^{t},\varphi^{xx},\varphi^{xt}$ and
$\varphi^{tt}$ in (2.5), we have following results:

coefficient monomial $1$
$\varphi_{t}-f_{x}\varphi_{x}-f\varphi_{xx}-h\varphi_{x}-k_{x}\xi-
k_{u}\varphi$ $u_{x}$
$\xi_{t}+f_{xx}\xi+f_{xu}\varphi+f_{x}(\varphi_{u}-\xi_{x})+2f_{u}\varphi_{x}+f(2\varphi_{xu}-\xi_{xx})+h(\varphi_{u}-\xi_{x})+h_{x}\xi+h_{u}\varphi$
$u_{t}$ $\varphi_{u}-\eta_{t}+f_{x}\eta_{x}+f\eta_{xx}+h\eta_{x}$ $u_{x}u_{t}$
$\xi_{u}-f_{x}\eta_{u}-2f_{u}\eta_{x}-f\eta_{xu}-h\eta_{u}$ $u_{t}^{2}$
$\eta_{u}$ $u_{x}^{2}$
$-f_{x}\xi_{u}+f_{xu}\xi+f_{uu}\varphi+2f_{u}(\varphi_{u}-\xi_{x})-h\xi_{u}+f(\varphi_{uu}-2\xi_{xu})$
$u_{x}^{3}$ $2f_{u}\xi_{u}+f\xi_{uu}$ $u_{x}^{2}u_{t}$
$2f_{u}\eta_{u}-f\eta_{uu}$ $u_{xx}$
$f_{x}\xi+f_{u}\varphi+f(\varphi_{u}-2\xi_{x})$ $u_{x}u_{xt}$ $2f\eta_{x}$
$u_{x}u_{xx}$ $3f\xi_{u}$ $u_{t}u_{xx}$ $f\eta_{u}$ $u_{xx}u_{x}^{2}$
$2f\eta_{u}$ (Table 1)

By simplifying above equations we obtain:

$\displaystyle\eta=\eta(t),\qquad\xi=\xi(x,t),\qquad\varphi_{u}-\eta_{t}=0,\qquad
f_{x}\xi+f_{u}\varphi+f(\varphi_{u}-2\xi_{x})=0,$
$\displaystyle\varphi_{t}-f_{x}\varphi_{x}-f\varphi_{xx}-h\varphi_{x}-k_{x}\xi-
k_{u}\varphi=0,\qquad f_{xu}\xi+f_{uu}\varphi+2f_{u}(\varphi_{u}-\xi_{x})=0,$
(2.6)
$\displaystyle\xi_{t}+f_{xx}\xi+f_{xu}\varphi+(f_{x}+h)(\varphi_{u}-\xi_{x})+2f_{u}\varphi_{x}-f\xi_{xx}+h_{x}\xi+h_{u}\varphi=0.$

##  3 Group classification in special cases

In this section we consider four special case of modeling equation and obtain
differential invariants for them by using (2) and infinitesimal criterion
method.

#### A:

$f(x,u)=xu^{-1}$, $h(x,u)=-2/u$, $k(x,u)=au+b$, where $a,b$ are constant real
numbers. In this case we have: $\eta=\frac{1}{a}e^{at}.e^{ac_{1}}+c_{2}$,
$\xi=c_{3}\sqrt{x}$, and $\varphi=u.e^{at}.e^{ac_{1}}$. As a result we find 3
independent vector fields: $X_{1}=e^{at}\partial_{t}+ae^{at}\partial_{u}$,
$X_{2}=\partial_{t}$, $X_{3}=\sqrt{x}\partial_{x}$.

#### B:

$f(x,u)=ax^{4}u$, $h(x,u)={\frac{bx}{u}}$, $k(x,u)=xu$, where $a\neq 0,b$ are
real numbers. In this case we have: $\eta=-c_{1}t+c_{2}$, $\xi=c_{1}x$,
$\varphi=-c_{1}u$. As a result we find 2 independent vector fields:
$X_{1}=-t\partial_{t}+x\partial_{x}-u\partial_{u}$, $X_{2}=\partial_{t}$.

#### C:

$f(x,u)=ax\exp{(-u/b)}$, $h(x,u)=xu$, $k(x,u)=c-bu$, where $a\neq,b\neq 0,c$
are real constant numbers. In this case we have: $\eta=c_{1}$,
$\xi=-\frac{c_{2}}{c}x\exp{(bt)}$, $\varphi=c_{2}\exp{(bt)}$. As a result we
find 2 independent vector fields:
$X_{1}=-\frac{1}{b}x\exp{(bt)}\partial_{x}+\exp{(bt)}\partial_{u},$
$X_{2}=\partial_{t}$.

#### D:

$f(x,u)=ax^{2}u$, $h(x,u)=xu$, $k(x,u)=u$, where $a$ is real nonzero constant.
In this case we have: $\eta=c_{1}$, $\xi=c_{2}x$, $\varphi=0$. As a result we
find 2 independent vector fields: $X_{1}=\partial_{t}$, $X_{2}=x\partial_{x}$.

Similar to above, the reader can use above procedure for finding her or him
interested modeling equation where has form (1.1), with interested $f,h$ and
$k$.

##  4 Resulted differential invariants

In this section we obtain differential invariants for above resulted symmetry
groups in several major and complicated cases. For example we compute
differential equation for B, $X_{1}$ and C, $X_{1}$.

#### B,

$X_{1}$: In this case we have following determination equation:
$\frac{dx}{x}=\frac{dt}{-t}=\frac{du}{-u}$, and by solving this equation we
find: $xt=c_{1}$, $xu=c_{2}$, $u/t=c_{3}$; and we choose $r=xt$ and $w=xu$ as
independent invariants. (we note $u/x=w/r$ and as a result obtain from $r,w$.)

#### C,

$X_{1}$: In this case we have following determination equation:
$\frac{bdx}{x\exp{(bt)}}=\frac{dt}{0}=\frac{du}{\exp{(bt)}}$, and by solving
this equation we find : $t=c_{1}$, $c_{2}=u+b\ln{x}$; and we choose $r=t$ and
$w=u+b\ln{x}$ as independent invariants.

case interested vector field differential invariants A $X_{1}$ $r=x$, $w=u-at$
$X_{2}$ $r=x$, $w=u$ $X_{3}$ $r=t$, $w=u$ B $X_{1}$ $r=xt$, $w=xu$ $X_{2}$
$r=x$, $w=u$ C $X_{1}$ $r=t$, $w=u+b\ln{x}$ $X_{2}$ $r=x$, $w=u$ D $X_{1}$
$r=x$, $w=u$ $X_{2}$ $r=t$, $w=u$ (Table 2)

In the above table, $F$ is an arbitrary function.

In the sequel, we obtain reduced equation respect to specified group symmetry
with infinitesimal generator $X$, (solution of this reduced equation called
$X$-invariants solution of original equation) by using resulted differential
invariants in the above table in two case.

For example, consider $u_{t}=(ax^{4}u)_{x}+bx/uu_{x}+xu$ (case B). By
considering $w=w(r)$, we find: $u_{t}=w_{r}$, $u_{x}=(xtw_{r}-w)/x^{2}$ and
$u_{xx}=(x^{2}(tw_{r}+xt^{2}w_{rr}-tw_{r})-2x(xtw_{r}-w))/x^{4}$. By
substituting this values in the given equation, we find following
$X_{1}$-reduced equation:

$\displaystyle
w_{r}=b+(1-4a)w+(6a+ar^{3})w^{2}+(4ar-2aw+awr-2arw)w_{r}+awr(r-a)w_{rr}$

As and second example, consider
$u_{t}=\big{(}{\frac{ax}{\exp{(u/b)}}}\big{)}_{x}+xuu_{x}+c-bu$, By
considering $w=w(r)$, we find: $u_{t}=w_{r}$, $u_{x}=-1/x$ and
$u_{xx}=1/x^{2}$. By substituting this values in the given equation, we find
following $X_{1}$-reduced equation:

$\displaystyle w_{r}=c+ab-2a,$

##  5 Some Applications

The Kolomogorov-Petrovskii-Piskonov (KPP) equation, (See [1, 8])

$\displaystyle E(u)\equiv bu_{t}-u_{xx}+\gamma uu_{x}+f(u),$ (5.7)

with ($b$,$\gamma$) real numbers, is encountered in reaction-diffusion systems
and prey-predator models. The optional convection term $uu_{x}$ [1, 4]) is
quite important in physical applications to prey-predator models.

###  5.1 Classical symmetries and Differential invaiants

If we let $b\neq 0$, then we have following equation:

$\displaystyle u_{t}=\frac{1}{b}(u_{xx}-\gamma uu_{x}-f(u)),$ (5.8)

By substituting this value in (2), we have following results.

#### Case I:

$b={\frac{\alpha\gamma}{\exp(\beta\alpha)}},f(u)={\frac{(1/2)\gamma\kappa\alpha
u}{\exp(\alpha\beta)}}+s$; Where $\alpha,\beta,\kappa$ and s are arbitrary
constants.

In this case we have:

$\displaystyle\xi=\frac{\exp(\alpha
t)\exp(\alpha\beta)}{\alpha}+c_{2},\qquad\eta=c_{1},\qquad\varphi=\kappa\exp(\alpha
t),$ (5.9)

For symmetry algebra we find:

$\displaystyle X_{1}=\partial_{t}\qquad X_{2}=\partial_{x},$ (5.10)

#### Case II:

$b\neq{\frac{\alpha\gamma}{\exp(\beta\alpha)}},f(u)\neq{\frac{(1/2)\gamma\kappa\alpha
u}{\exp(\alpha\beta)}}+s$; Where $\alpha,\beta,\kappa$ and s are arbitrary
constants.

In this case we have:

$\displaystyle\xi=c_{1}\qquad\eta=c_{2},\qquad\varphi=0,$ (5.11)

For symmetry algebra we find:

$\displaystyle X_{1}=\partial_{t}$ $\displaystyle\qquad X_{2}=\partial_{x},$
(5.12)

As a result we have following theorem:

######  Theorem 1.

Some exact solutions for modeling equation (5.7) invariant under a translation
group respect to $x$ and some solutions of this equation invariant under
translation respect to $t$.

###  5.2 Similarity solutions

In this subsection we find similarity solution of equation (5.8) by using
above resulted symmetry algebra.

#### similarity solution respect to $X=\partial_{t}$.

In this case we have following equation as $X$-reduced equation:

$\displaystyle w_{rr}-\gamma ww_{r}-f(w)=0,$ (5.13)

If we solve equation (5.13) with MAPLE, then we find: $w(x)=c$. Where

$\displaystyle\frac{d}{dc}{F(c)}F(c)-\gamma(cF(c))-f(c)=0\hskip
14.22636pt\mbox{or}\hskip 14.22636pt\frac{d}{dr}w(r)=F(c),\hskip
14.22636pt\mbox{or}\hskip 14.22636ptr=\int\frac{1}{F(c)}dc+C$ (5.14)

Where $F$ is arbitrary function with specified arguments and $c,C$ are
arbitrary constants.

#### similarity solution respect to $Y=\partial_{x}$.

In this case we have following equation as $Y$-reduced equation:

$\displaystyle w_{r}+\frac{1}{b}f(w)=0,$ (5.15)

If we solve equation (5.15) with MAPLE, then we find following solution:

$\displaystyle x-\int^{w(x)}\frac{b}{f(c_{1})}dc_{1}+c_{2}=0,$ (5.16)

Where $c_{1}$ and $c_{2}$ are arbitrary constants.

## Conclusion

In this paper first we find system of equations to finding symmetry group and
symmetry algebra for (G-RDC) equation, then obtain these symmetry groups in
several special cases and at the end we establish symmetry classification for
KPP equation by using group classification of (G-RDC) equation and we find its
similarity solution respect to resulted symmetry algebra.

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