Uncertainty Principles for Quaternion
Linear Canonical S-Transform

M. Younus Bhat${}^{1},$ and Aamir H. Dar2

Department of Mathematical Sciences, Islamic University of Science and
Technology Awantipora, Pulwama, Jammu and Kashmir 192122, India.
1<EMAIL_ADDRESS>and 2<EMAIL_ADDRESS>
Abstract: In this paper, we introduce the notion of quaternion linear
canonical S-transform(Q-LCST) which is an extension of the linear canonical
S-transform. Firstly, we study the fundamental properties of quaternion linear
canonical S-transform(Q-LCST) and then establish some basic results including
orthogonality relation and reconstruction formula. Finally, we derive the
associated Heisenberg’s uncertainty inequality and the corresponding
logarithmic version for quaternion linear canonical S-transform(Q-LCST).

Keywords: Linear canonical transform; Linear canonical S-transform; Quaternion
linear canonical S-transform; Uncertainty Principle.

Mathematics Subject Classification: 42A38; 46S10; 44A15; 44A35

1\. Introduction

The quaternion linear canonical transform (QLCT), as a generalized form of the
quaternion Fourier transform, is a powerful analyzing tool in image and signal
processing. A nontrivial generalization of the classical real and complex
Fourier transform is the quaternion Fourier transform. This generalization
using quaternion algebra has been of great interest to the mathematicians and
physicists for so many years. Many of the properties of the Fourier transform
hold good in the new setting however others needed to be modified. It was R.
G. Stockwell who constructed the novel Stockwell transform by using wavelet
transform and the classical windowed Fourier transform. For more details we
refer to [1, 2, 3, 9] and references therein.

The well known Heisenberg’s uncertainty principle was first introduced by
German physicist Heisenberg. This principle plays a vital role in signal and
image processing, physics and mathematics and other allied subjects. To be
precise, it states that a signal or function cannot be located in both time
and frequency domains, simultaneously and sharply. while as on the other side,
the linear canonical transform (LCT) is a linear integral transformation with
three free parameters, and it has been widely used in physical optics and
signal processing. Indeed, it is also regarded as a generalization of many
mathematical transforms such as the Fourier transform, the fractional Fourier
transform and others. For further details we suggest [4, 5, 6, 7, 8, 10].

Continuing the studies of the extensions of the linear canonical transform, we
introduced the notion of quaternion linear canonical S-transform(Q-LCST) which
is an extension of the linear canonical S-transform. Firstly, we study the
fundamental properties of quaternion linear canonical S-transform(Q-LCST) and
then establish some basic results including orthogonality relation and
reconstruction formula. Finally, we derive the associated Heisenberg’s
uncertainty inequality and the corresponding logarithmic version for
quaternion linear canonical S-transform(Q-LCST). The rest of the paper is
organised as follows:

In Section 2, we discuss some preliminary results and definitions which are
used in subsequent sections. In Section 3, we introduce the well known
properties of Q-LCST like orthogonality, reconstruction formula and many more.
In Section 4, we establish the well known Heisenberg’s uncertainty inequality
for quaternion linear canonical S-transform.

Methods/Experiments

2\. Preliminaries

In this section we review Linear canonical transform(LCT) , Linear canonical
S-transform(LCST) and Quaternion linear canonical transform(QLCT).

2.1.Linear canonical transform

Let $\mu_{1},\mu_{2},\mu_{3}$ denote the three imaginary units in the
quaternion algebra[26OLCT] then the linear canonical transform (LCT) of any
signal $f\in L^{2}(\mathbb{R})$ is defined as

$\displaystyle\mathcal{L}_{M}[f](u)=\int_{\mathbb{R}}f(x)K^{\mu_{1}}_{M}(x,u)dx$
(2.1)

and its inversion is given by

$\displaystyle f(x)$
$\displaystyle=\mathcal{L}^{-1}_{M}\\{\mathcal{L}_{M}[f]\\}(x)$
$\displaystyle=\int_{\mathbb{R}}\mathcal{L}_{M}[f](u)K^{-\mu_{1}}_{M}(x,u)du$
(2.2)

where $u,x\in\mathbb{R},M=(A,B,C,D)$ is a uni-modular matrix parameter, and
$K^{\mu_{1}}_{M}(x,u)$ is so-called the kernel of the LCT and is given by

$\displaystyle K^{\mu_{1}}_{M}(x,u)=\begin{cases}\frac{1}{\sqrt{2\pi
B}}e^{\frac{\mu_{1}}{2}\left(\frac{A}{B}x^{2}-\frac{2}{B}xu+\frac{D}{B}u^{2}-\frac{\pi}{2}\right)},&B\neq
0\\\ \sqrt{D}e^{\mu_{1}\frac{CD}{2}u^{2}}\delta(x-Du),&B=0\end{cases}$ (2.3)

2.2.Linear Canonical S-Transform

The Linear Canonical S-Transform(ST) is known as a hybrid form of the
S-transform. This transform is an extension of the LCT and ST,which gives a
time-linear canonical domain joint expression. It is defined as:

For any non-zero window function $\psi\in L^{2}(\mathbb{R})$ the LCST of any
signal $f\in L^{2}(\mathbb{R})$ with respect to $\psi$ is defined by

$\displaystyle\mathcal{S}_{\psi}^{M}[f](u,w)$
$\displaystyle=\mathcal{L}_{M}\left\\{f(x)\overline{\psi(u-x,w)}\right\\}(w,u)$
$\displaystyle=\int_{\mathbb{R}}f(x)\overline{\psi(u-x,w)}K^{\mu_{1}}_{M}(x,u)dx$
(2.4)

and its inverse is given by

$\displaystyle f(x)\overline{\psi(u-x,w)}$
$\displaystyle=\mathcal{L}^{-1}_{M}\\{\mathcal{S}_{\psi}^{M}f(u,w)\\}$
$\displaystyle=\int_{\mathbb{R}}\mathcal{S}_{\psi}^{M}f(u,w)K^{-\mu_{1}}_{M}(x,u)dw$
(2.5)

where $K^{\mu_{1}}_{M}(x,u)$ is given by (2.3 ) for $B\neq 0.$

It is worth noting that when $M=(0,1,-1,0),$ the LCST boils down S-transform.

2.3.Quaternion linear canonical transform

The QLCT is a generalization of the LCT in the frame of quaternion algebra.The
QLCT of a signal $f\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ is defined as

$\displaystyle\mathcal{L}_{M_{1},M_{2}}^{\mathbb{H}}\\{f\\}(\mathbf{u})=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},u_{1})f(\mathbf{x})K_{M_{2}}^{\mu_{2}}(x_{2},u_{2})d\mathbf{x},$
(2.6)

and its inverse is given by

$\displaystyle f(\mathbf{x})$
$\displaystyle=\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f]\\}(\mathbf{x})$
$\displaystyle=\int_{\mathbb{R}^{2}}K^{-\mu_{1}}_{M_{1}}(u_{1},x_{1})\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{u})K^{-\mu_{2}}_{M_{2}}(u_{2},x_{2})d\mathbf{u}$
(2.7)

where $\mathbf{u}=(u_{1},u_{2})$,$\mathbf{x}=(x_{1},x_{2})$
,$M_{s}=(A_{s},B_{s},C_{s},D_{s})$ be a matrix parameter satisfying
$det(M_{s})=1$ and the kernel

$\displaystyle\begin{split}K_{M_{s}}^{\mu_{s}}(x_{s},u_{s})=\begin{cases}\frac{1}{\sqrt{2\pi
B_{s}}}e^{\mu_{s}\left(\frac{A_{s}}{2B_{s}}x_{s}^{2}-\frac{x_{s}u_{s}}{B_{s}}+\frac{D_{s}}{2B_{s}}u_{s}^{2}-\frac{\pi}{4}\right)},&B_{s}\neq
0\\\
\sqrt{D_{s}}e^{\mu_{s}\frac{C_{s}D_{s}}{2}u_{s}^{2}}\delta(x_{s}-D_{s}u_{s}),&B_{s}=0\end{cases}\end{split}$
(2.8)

for $s=1,2,\delta(x)$ represents the Dirac function.

Results and Discussion

3.The Quaternion Linear Canonical S-Transform(Q-LCST)

The linear canonical S-transform (LCST) is a linear integral transformation
having more than one free parameters and includes the traditional Stockwell
spectrum and the fractional Stockwell spectrum as its special cases.LCST is
widely used in various fields viz: non-stationary signal representation,
detection, parameters estimation, filter design and etc.This section will
consider generalizing the LCST using the quaternion algebra.This extension is
then called the quaternion linear canonical S-transform which is denoted by
Q-LCST.Moreover, several basic properties of them are investigated.

3.1.Definition of Quaternion linear canonical S-transform

Based on the definition of the linear canonical S-transform(LCST), we obtain a
definition of the quaternion linear canonical S-transform(Q-LCST) by replacing
the kernel of the linear canonical transform(LCT) with the kernel of the
quaternion linear canonical transform(QLCT) in the classical definition of
linear canonical S-transform(LCST) as follows.

Definition 3.1: Let $\Psi\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ be a non zero
quaternion window function satisfying

$\int_{\mathbb{R}^{2}}\Psi(\mathbf{x})d\mathbf{x}=1$ $None$

then 2D quaternion linear canonical S-transform(Q-LCST)of quaternion signal
$f\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ with respect $\Psi$ is defined by

$\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})f(\mathbf{x})\overline{\Psi(\mathbf{u}-\mathbf{x},\mathbf{w})}K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}$
$None$

where
$\mathbf{u}=(u_{1},u_{2})\in{\mathbb{R}^{2}},\mathbf{w}=(w_{1},w_{2})\in{\mathbb{R}^{2}}$,$\mathbf{x}=(x_{1},x_{2})$
,$M_{s}=(A_{s},B_{s},C_{s},D_{s})\in{\mathbb{R}^{2}}$ be a matrix parameter
satisfying $det(M_{s})=1$ and the kernel

$\displaystyle\begin{split}K_{M_{s}}^{\mu_{s}}(x_{s},w_{s})=\begin{cases}\frac{1}{\sqrt{2\pi
B_{s}}}e^{\mu_{s}\left(\frac{A_{s}}{2B_{s}}x_{s}^{2}-\frac{x_{s}w_{s}}{B_{s}}+\frac{D_{s}}{2B_{s}}w_{s}^{2}-\frac{\pi}{2}\right)},&B_{s}\neq
0\\\
\sqrt{D_{s}}e^{\mu_{s}\frac{C_{s}D_{s}}{2}w_{s}^{2}}\delta(x_{s}-D_{s}w_{s}),&B_{s}=0\end{cases}\end{split}$
(3.3)

for $s=1,2,\delta(x)$ represents the Dirac function.

Note that the Q-LCST the case when $B_{s}=0,s=1,2$ is not interesting because
it is essentially a multiplication by a quaternion chirp.Hence, without loss
of generality, we set $B_{s}\neq 0,s=1,2$ in this paper.

It is worth noting that the Q-LCST gives birth to some new time-frequency
transforms which are not yet reported in the open literature:

  * •

For $M_{s}=(cos\theta,sin\theta,-sin\theta,cos\theta),\theta\neq n\pi$ we can
obtain a novel quaternion fractional S-transform.

  * •

For $M_{s}=(1,B,0,1),B\neq 0$ we can obtain a new transform namely the
Quaternion Fresnel S-transform.

  * •

For $M_{s}=(0,1,-1,0)$ the Q-LCST boils down to Quaternion S-transform.

From definition 3.1 we have

$\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})=\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}\left\\{f(\mathbf{x})\overline{\Psi(\mathbf{u}-\mathbf{x},\mathbf{w})}\right\\}(\mathbf{w},\mathbf{u})$
$None$

applying inverse QLCT to(3.4),we have

$\displaystyle f(\mathbf{x})\overline{\Psi(\mathbf{u}-\mathbf{x},\mathbf{w})}$
$\displaystyle=\mathcal{L}^{-1}_{M_{1},M_{2}}\left\\{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})\right\\}$
$\displaystyle=\int_{\mathbb{R}^{2}}K_{M_{1}}^{-\mu_{1}}(x_{1},w_{1})\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})K_{M_{2}}^{-\mu_{2}}(x_{2},w_{2})d\mathbf{w}$
(3.5)

In the sequel, our intention is to study the fundamental properties of the
proposed Q-LCST(3.2).

3.2.Some Properties of Q-LCST

In this subsection, we discuss several basic properties of the Q-LCST. These
properties play important roles in signal representation.Most of the
properties of the quaternion S-transform can be established in the Q-LCST
domain with some modifications in quaternion windowed linear canonical
transform.However, it is clearly visible that the properties of the Q-LCST
like shift,modulation and etc. vary from the corresponding properties of the
quaternion windowed linear canonical transform.

Property 1. (Multiplication operator)
$\left[\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}\right]$ is a multiplication
operator if $\Psi(\mathbf{u-x},\mathbf{w})=\Gamma(\mathbf{w})$ i.e independent
of $\mathbf{x}$.

Proof. From (3.2), we have

$\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})f(\mathbf{x})\overline{\Psi(\mathbf{u}-\mathbf{x},\mathbf{w})}K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}$
$None$

Substituting $\Psi(\mathbf{u-x},\mathbf{w})=\Gamma(\mathbf{w})$ in (3.6),we
obtain

$\displaystyle\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})$
$\displaystyle=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})f(\mathbf{x})\Gamma(\mathbf{w})K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}$
$\displaystyle=\Gamma(\mathbf{w})\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})f(\mathbf{x})K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}$
$\displaystyle=\Gamma(\mathbf{w})\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{w}).$
(3.7)

So $\left[\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}\right]$ is a
multiplication operator.

Example 1. If $\Psi(\mathbf{u-x},\mathbf{w})=\Gamma(\mathbf{w})=1$ then (3.7)
reduces to QLCT

$\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})=\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{w}).$
$None$

Example 2. If $\Psi(\mathbf{u-x},\mathbf{w})=\Gamma(\mathbf{x})$ i.e $\Psi$ is
dependent on $\mathbf{x}$ then from (3.6), we have

$\displaystyle\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})$
$\displaystyle=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})f(\mathbf{x})\Gamma(\mathbf{x})K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}$
$\displaystyle=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})g(\mathbf{x})K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}$
$\displaystyle=\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[g](\mathbf{w}).$ (3.9)

Where $g(\mathbf{x})=f(\mathbf{x})\Gamma(\mathbf{t})$ thus we see Q-LCST boils
down to QLCT.

Property 2. If $\Psi\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ such that
$\int_{\mathbb{R}^{2}}\Psi(\mathbf{u-x},\mathbf{w})d\mathbf{u}=1$ then

$\int_{\mathbb{R}^{2}}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}f(\mathbf{u,w})d\mathbf{u}=\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{w})$
$None$

Proof. From (3.2), we have

$\displaystyle\int_{\mathbb{R}^{2}}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})f(\mathbf{x})\overline{\Psi(\mathbf{u}-\mathbf{x},\mathbf{w})}K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}d\mathbf{x}$
$\displaystyle=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})f(\mathbf{x})K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})\int_{\mathbb{R}^{2}}\overline{\Psi(\mathbf{u}-\mathbf{x},\mathbf{w})}d\mathbf{u}d\mathbf{x}$
$\displaystyle=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})f(\mathbf{x})K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}$
$\displaystyle=\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{w})$

which completes proof.

Property 3.(Linearity) Let $\Psi$ be a non zero quaternion window function in
$L^{2}(\mathbb{R}^{2},\mathbb{H})$ and $f_{n}\in
L^{2}(\mathbb{R}^{2},mathbbH),n\in\mathbb{N}$ then following holds:

$\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}\left\\{\sum_{n\in\mathbb{N}}\alpha_{n}f_{n}\right\\}(\mathbf{u},\mathbf{w})=\sum_{n\in\mathbb{N}}\alpha_{n}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f_{n}](\mathbf{u},\mathbf{w}),\alpha_{n}\in\mathbb{H}$
$None$

Proof. Follows directly from definition (3.2).

Property 4.(Parity) Let $\Psi\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ be a non
zero quaternion window function and $f\in L^{2}(\mathbb{R}^{2},\mathbb{H}),$
then we have

$\mathbb{S}^{\mathbb{H}}_{P\Psi,M_{1},M_{2}}[Pf](\mathbf{u},\mathbf{w})=\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{-u},\mathbf{-w})$
$None$

where $P\Psi(\mathbf{x})=\Psi(\mathbf{-x}).$

Proof. Easy to prove so avoided.

Property 5.(Shift) Let $\Psi\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ be a non
zero quaternion window function and $f\in L^{2}(\mathbb{R}^{2},\mathbb{H}),$
then following property holds

$\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f(\mathbf{x-\alpha})](\mathbf{u},\mathbf{w})=e^{\frac{\mu_{1}}{2B_{1}}(A_{1}\alpha^{2}_{1}-2\alpha_{1}w_{1})}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[\tilde{f}(\mathbf{u-\alpha},\mathbf{w})](\mathbf{.},\mathbf{.})e^{\frac{\mu_{2}}{2B_{2}}(A_{2}\alpha^{2}_{2}-2\alpha_{2}w_{2})}.$
$None$

where
$\tilde{f}(\mathbf{t})=e^{{\mu_{1}\frac{A_{1}t_{1}\alpha_{1}}{B_{1}}}+{\mu_{2}\frac{A_{2}t_{2}\alpha_{2}}{B_{2}}}}f(\mathbf{t})$

Proof.By taking into account of (3.2),we get

$\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f(\mathbf{x-\alpha})](\mathbf{u},\mathbf{w})=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})f(\mathbf{x-\alpha})\overline{\Psi(\mathbf{u}-\mathbf{x},\mathbf{w})}K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}$
$None$

Applying change of variable $\mathbf{t}=\mathbf{x-\alpha}$ in (3.14),we obtain

$\displaystyle\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f(\mathbf{x}-\mathbf{\alpha})](\mathbf{u},\mathbf{w})$
$\displaystyle=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(t_{1}+\alpha_{1},w_{1})f(\mathbf{t})\overline{\Psi(\mathbf{u}-(\mathbf{t+\alpha}),\mathbf{w})}K_{M_{2}}^{\mu_{2}}(t_{2}+\alpha_{2},w_{2})d\mathbf{t}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{\mu_{1}\left(\frac{A_{1}}{2B_{1}}(t_{1}+\alpha_{1})^{2}-\frac{(t_{1}+\alpha_{1})w_{1}}{B_{1}}+\frac{D_{1}}{2B_{1}}w_{1}^{2}-\frac{\pi}{4}\right)}f(\mathbf{t})\overline{\Psi((\mathbf{u-\alpha})-\mathbf{t},\mathbf{w})}$
$\displaystyle\qquad\qquad\times\frac{1}{\sqrt{2\pi
B_{2}}}e^{\mu_{2}\left(\frac{A_{2}}{2B_{2}}(t_{2}+\alpha_{2})^{2}-\frac{(t_{2}+\alpha_{2})w_{2}}{B_{2}}+\frac{D_{2}}{2B_{2}}w_{2}^{2}-\frac{\pi}{4}\right)}d\mathbf{t}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{\mu_{1}\left(\frac{A_{1}}{2B_{1}}(t^{2}_{1}+\alpha^{2}_{1}+2t_{1}\alpha_{1})-\frac{(t_{1}+\alpha_{1})w_{1}}{B_{1}}+\frac{D_{1}}{2B_{1}}w_{1}^{2}-\frac{\pi}{4}\right)}f(\mathbf{t})\overline{\Psi((\mathbf{u-\alpha})-\mathbf{t},\mathbf{w})}$
$\displaystyle\qquad\qquad\times\frac{1}{\sqrt{2\pi
B_{2}}}e^{\mu_{2}\left(\frac{A_{2}}{2B_{2}}(t^{2}_{2}+\alpha^{2}_{2}+t_{2}\alpha_{2})-\frac{(t_{2}+\alpha_{2})w_{2}}{B_{2}}+\frac{D_{2}}{2B_{2}}w_{2}^{2}-\frac{\pi}{4}\right)}d\mathbf{t}$
$\displaystyle=e^{\frac{\mu_{1}}{2B_{1}}(A_{1}\alpha^{2}_{1}-2\alpha_{1}w_{1})}\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{\mu_{1}\left(\frac{A_{1}}{2B_{1}}t^{2}_{1}-\frac{t_{1}w_{1}}{B_{1}}+\frac{D_{1}}{2B_{1}}w_{1}^{2}-\frac{\pi}{4}\right)}$
$\displaystyle\qquad\qquad\times\left\\{e^{{\mu_{1}\frac{A_{1}t_{1}\alpha_{1}}{B_{1}}}+{\mu_{2}\frac{A_{2}t_{2}\alpha_{2}}{B_{2}}}}f(\mathbf{t})\right\\}\overline{\Psi((\mathbf{u-\alpha})-\mathbf{t},\mathbf{w})}$
$\displaystyle\qquad\qquad\qquad\times\frac{1}{\sqrt{2\pi
B_{2}}}e^{\mu_{1}\left(\frac{A_{2}}{2B_{2}}t^{2}_{2}-\frac{t_{2}w_{2}}{B_{2}}+\frac{D_{2}}{2B_{2}}w_{2}^{2}-\frac{\pi}{4}\right)}d\mathbf{t}e^{\frac{\mu_{2}}{B_{2}}(\frac{A_{2}\alpha^{2}_{2}}{2}-\alpha_{2}w_{2})}.$

This completes proof.

Property 6.(Modulation)Let $\Psi\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ be a non
zero quaternion window function and $f\in L^{2}(\mathbb{R}^{2},\mathbb{H}),$
then we have

$\displaystyle\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[\mathcal{M}_{\mathbf{s}}f](\mathbf{u},\mathbf{w})$
$\displaystyle=e^{\mu_{1}\frac{D_{1}}{2}(2w_{1}s_{1}-B_{1}s^{2}_{1})}\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(w_{1}-s_{1}B_{1},x_{1})f(\mathbf{x})\overline{\Psi((\mathbf{u-x}),\mathbf{w})}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times
K_{M_{2}}^{\mu_{2}}(w_{2}-s_{2}B_{2},x_{2})d\mathbf{x}e^{\mu_{2}\frac{D_{2}}{2}(2w_{2}s_{2}-B_{2}s^{2}_{2})},$
(3.15)

where
$\mathcal{M}_{\mathbf{s}}f(\mathbf{x})=e^{\mu_{1}s_{1}x_{1}}f(\mathbf{x})e^{\mu_{2}s_{2}x_{2}}$

Proof. It follows from(3.2) that,

$\displaystyle\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[\mathcal{M}_{\mathbf{s}}f](\mathbf{u},\mathbf{w})$
$\displaystyle=\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(x_{1},w_{1})e^{\mu_{1}s_{1}x_{1}}f(\mathbf{x})e^{\mu_{2}s_{2}x_{2}}\overline{\Psi(\mathbf{u}-\mathbf{x},\mathbf{w})}K_{M_{2}}^{\mu_{2}}(x_{2},w_{2})d\mathbf{x}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{\mu_{1}\left(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}w_{1}}{B_{1}}+\frac{D_{1}}{2B_{1}}w_{1}^{2}+s_{1}x_{1}-\frac{\pi}{4}\right)}f(\mathbf{x})\overline{\Psi((\mathbf{u-x}),\mathbf{w})}$
$\displaystyle\qquad\qquad\times\frac{1}{\sqrt{2\pi
B_{2}}}e^{\mu_{2}\left(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}w_{2}}{B_{2}}+\frac{D_{2}}{2B_{2}}w_{2}^{2}+s_{2}x_{2}-\frac{\pi}{4}\right)}d\mathbf{x}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{\mu_{1}\left(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}(w_{1}-s_{1}B_{1})+\frac{D_{1}}{2B_{1}}w_{1}^{2}+-\frac{\pi}{4}\right)}f(\mathbf{x})\overline{\Psi((\mathbf{u-x}),\mathbf{w})}$
$\displaystyle\qquad\qquad\times\frac{1}{\sqrt{2\pi
B_{2}}}e^{\mu_{2}\left(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}(w_{2}-s_{2}B_{2})+\frac{D_{2}}{2B_{2}}w_{2}^{2}+s_{2}x_{2}-\frac{\pi}{4}\right)}d\mathbf{x}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{\mu_{1}\left(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}(w_{1}-s_{1}B_{1})+\frac{D_{1}}{2B_{1}}((w_{1}-s_{1}B_{1})+s_{1}B_{1})^{2}-\frac{\pi}{4}\right)}f(\mathbf{x})\overline{\Psi((\mathbf{u-x}),\mathbf{w})}$
$\displaystyle\qquad\qquad\times\frac{1}{\sqrt{2\pi
B_{2}}}e^{\mu_{2}\left(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}(w_{2}-s_{2}B_{2})+\frac{D_{2}}{2B_{2}}((w_{2}-s_{2}B_{2})+s_{2}B_{2})^{2}-\frac{\pi}{4}\right)}d\mathbf{x}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{\mu_{1}\left(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}(w_{1}-s_{1}B_{1})+\frac{D_{1}}{2B_{1}}((w_{1}-s_{1}B_{1})^{2}-\frac{\pi}{4}+2(w_{1}-s_{1}B_{1})s_{1}B_{1}+s_{1}B^{2}_{1})\right)}$
$\displaystyle\qquad\times
f(\mathbf{x})\overline{\Psi((\mathbf{u-x}),\mathbf{w})}$
$\displaystyle\qquad\times\frac{1}{\sqrt{2\pi
B_{2}}}e^{\mu_{2}\left(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}(w_{2}-s_{2}B_{2})+\frac{D_{2}}{2B_{2}}((w_{2}-s_{2}B_{2})^{2}-\frac{\pi}{4}+2(w_{2}-s_{2}B_{2})s_{2}B_{2}+s_{2}B^{2}_{2})\right)}d\mathbf{x}$
$\displaystyle=e^{\mu_{1}\frac{D_{1}}{2}(2w_{1}s_{1}-B_{1}s^{2}_{1})}\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{\mu_{1}\left(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}(w_{1}-s_{1}B_{1})+\frac{D_{1}}{2B_{1}}((w_{1}-s_{1}B_{1})^{2}-\frac{\pi}{4})\right)}$
$\displaystyle\qquad\times
f(\mathbf{x})\overline{\Psi((\mathbf{u-x}),\mathbf{w})}$
$\displaystyle\qquad\times\frac{1}{\sqrt{2\pi
B_{2}}}e^{\mu_{2}\left(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}(w_{2}-s_{2}B_{2})+\frac{D_{2}}{2B_{2}}((w_{2}-s_{2}B_{2})^{2}-\frac{\pi}{4})\right)}d\mathbf{x}e^{\mu_{2}\frac{D_{2}}{2}(2w_{2}s_{2}-B_{2}s^{2}_{2})}$
$\displaystyle=e^{\mu_{1}\frac{D_{1}}{2}(2w_{1}s_{1}-B_{1}s^{2}_{1})}\int_{\mathbb{R}^{2}}K_{M_{1}}^{\mu_{1}}(w_{1}-s_{1}B_{1},x_{1})f(\mathbf{x})\overline{\Psi((\mathbf{u-x}),\mathbf{w})}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times
K_{M_{2}}^{\mu_{2}}(w_{2}-s_{2}B_{2},x_{2})d\mathbf{x}e^{\mu_{2}\frac{D_{2}}{2}(2w_{2}s_{2}-B_{2}s^{2}_{2})}.$

Hence completes the proof.

Now we are going to develop the orthogonality relation for Q-LCST and using
this we derive reconstruction formula associated with Q-LCST both of which are
fundamental properties for signal analysis.Before we begin with orthogonality
relation we note that throughout the rest of the paper, we will always assume
that

$\int_{\mathbb{R}^{2}}|\Psi(\mathbf{u,w})|^{2}d\mathbf{u}=\Lambda_{\Psi},\qquad
0<\Lambda_{\Psi}<\infty$ $None$

Theorem 3.1(Orthogonality relation).Let $\Psi\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$ be a non zero quaternion window function and
$f,g\in L^{2}(\mathbb{R}^{2},\mathbb{H}),$ then we have

$\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})\overline{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[g](\mathbf{u},\mathbf{w})}d\mathbf{w}d\mathbf{u}=\left\langle\Lambda_{\Psi}f,g\right\rangle$
$None$

Proof. By definition of Q-LCST,we obtain

$\begin{array}[]{lcr}\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})\overline{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[g](\mathbf{u},\mathbf{w})}d\mathbf{w}d\mathbf{u}&&\\\
\\\
\qquad=\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{-\mu_{1}\left(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}w_{1}}{B_{1}}+\frac{D_{1}}{2B_{1}}w_{1}^{2}-\frac{\pi}{4}\right)}f(\mathbf{x})\overline{\Psi(\mathbf{u-x,w})}&&\\\
\\\ \qquad\qquad\times\frac{1}{\sqrt{2\pi
B_{2}}}e^{-\mu_{2}\left(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}w_{2}}{B_{2}}+\frac{D_{2}}{2B_{2}}w_{2}^{2}-\frac{\pi}{4}\right)}\displaystyle\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{\mu_{1}\left(\frac{A_{1}}{2B_{1}}t^{2}_{1}-\frac{t_{1}w_{1}}{B_{1}}+\frac{D_{1}}{2B_{1}}w_{1}^{2}-\frac{\pi}{4}\right)}\overline{g(\mathbf{t})}&&\\\
\\\ \qquad\qquad\times\Psi(\mathbf{u-t,w})\frac{1}{\sqrt{2\pi
B_{2}}}e^{\mu_{2}\left(\frac{A_{2}}{2B_{2}}t^{2}_{2}-\frac{t_{2}w_{2}}{B_{2}}+\frac{D_{2}}{2B_{2}}w_{2}^{2}-\frac{\pi}{4}\right)}d\mathbf{t}d\mathbf{x}d\mathbf{u}d\mathbf{w}&&\\\
\\\
\qquad=\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{-\mu_{1}\frac{A_{1}}{2B_{1}}\left(x^{2}_{1}-t^{2}_{1}\right)}f(\mathbf{x})\overline{\Psi(\mathbf{u-x,w})}\frac{1}{\sqrt{2\pi
B_{2}}}e^{-\mu_{2}\frac{A_{2}}{2B_{2}}\left(x^{2}_{2}-t^{2}_{2}\right)}&&\\\
\\\ \qquad\qquad\times\frac{1}{\sqrt{2\pi
B_{1}}}e^{\frac{\mu_{1}}{B_{1}}\left(t_{1}-x_{1}\right)}\overline{g(\mathbf{t})}\Psi(\mathbf{u-x,w})\frac{1}{\sqrt{2\pi
B_{2}}}e^{\frac{\mu_{2}}{B_{2}}\left(t_{2}-x_{2}\right)}d\mathbf{t}d\mathbf{x}d\mathbf{u}d\mathbf{w}&&\\\
\\\ \qquad=2\pi
B_{1}\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\frac{1}{2\pi
B_{1}}e^{-\mu_{1}\frac{A_{1}}{2B_{1}}\left(x^{2}_{1}-t^{2}_{1}\right)}f(\mathbf{x})\overline{\Psi(\mathbf{u-x,w})}\frac{1}{2\pi
B_{2}}e^{-\mu_{2}\frac{A_{2}}{2B_{2}}\left(x^{2}_{2}-t^{2}_{2}\right)}&&\\\
\\\
\qquad\qquad\times\overline{g(\mathbf{t})}\Psi(\mathbf{u-t,w})\delta(\mathbf{t-x})d\mathbf{t}d\mathbf{x}d\mathbf{u}.2\pi
B_{2}&&\\\ \\\
\qquad=\displaystyle\int_{\mathbb{R}^{2}}f(\mathbf{x})\overline{g(\mathbf{t})}d\mathbf{t}\displaystyle\int_{\mathbb{R}^{2}}\overline{\Psi(\mathbf{u-t,w})}\Psi(\mathbf{u-t,w})d\mathbf{u}&&\\\
\\\
\qquad=\displaystyle\int_{\mathbb{R}^{2}}f(\mathbf{x})\overline{g(\mathbf{t})}d\mathbf{t}\displaystyle\int_{\mathbb{R}^{2}}|{\Psi(\mathbf{u-t,w})}|^{2}d\mathbf{u}&&\\\
\\\
\qquad=\displaystyle\int_{\mathbb{R}^{2}}\Lambda_{\Psi}f(\mathbf{x})\overline{g(\mathbf{t})}d\mathbf{t}&&\\\
\\\ \qquad=\left\langle\Lambda_{\Psi}f,g\right\rangle.\end{array}$

Thus the proof is completed.

Remark 3.1. If we take $f=g$ in (3.17) Theorem 3.1 takes the form

$\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})|^{2}d\mathbf{w}d\mathbf{u}=\Lambda_{\Psi}\|f\|^{2}.$
$None$

Theorem 3.2(Reconstruction formula).If $f\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$, then f can be reconstructed by the formula

$\displaystyle f(\mathbf{x})$
$\displaystyle=\frac{1}{\Lambda_{\Psi}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{-\mu_{1}(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}w_{1}+\frac{D_{1}}{2B_{1}}w^{2}_{1}-\frac{\pi}{4})}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})$
$\displaystyle\qquad\qquad\qquad\times\Psi(\mathbf{u-x,w})\frac{1}{\sqrt{2\pi
B_{2}}}e^{-\mu_{2}(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}w_{2}+\frac{D_{2}}{2B_{2}}w^{2}_{2}-\frac{\pi}{4})}d\mathbf{w}d\mathbf{u}.$
(3.19)

Where $\Psi\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ is a quaternion window
function that satisfies (3.16)

Proof. From theorem 3.1, we have

$\begin{array}[]{lcr}\left\langle f\Lambda_{\Psi},g\right\rangle&&\\\ \\\
\qquad=\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})\overline{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[g](\mathbf{u},\mathbf{w})}d\mathbf{w}d\mathbf{u}&&\\\
\\\
\qquad=\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})\displaystyle\int_{\mathbb{R}^{2}}e^{-\mu_{1}(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}w_{1}+\frac{D_{1}}{2B_{1}}w^{2}_{1}-\frac{\pi}{4})}\overline{g(\mathbf{x})}&&\\\
\qquad\qquad\qquad\qquad\times\Psi(\mathbf{u-x,w})\frac{1}{\sqrt{2\pi
B_{2}}}e^{-\mu_{2}(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}w_{2}+\frac{D_{2}}{2B_{2}}w^{2}_{2}-\frac{\pi}{4})}d\mathbf{w}d\mathbf{u}d\mathbf{x}&&\\\
\\\
\qquad=\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})e^{-\mu_{1}(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}w_{1}+\frac{D_{1}}{2B_{1}}w^{2}_{1}-\frac{\pi}{4})}&&\\\
\qquad\qquad\qquad\qquad\times\Psi(\mathbf{u-x,w})\frac{1}{\sqrt{2\pi
B_{2}}}e^{-\mu_{2}(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}w_{2}+\frac{D_{2}}{2B_{2}}w^{2}_{2}-\frac{\pi}{4})}\overline{g(\mathbf{x})}d\mathbf{w}d\mathbf{u}d\mathbf{x}&&\\\
\\\
\qquad=\left\langle\displaystyle\int_{\mathbb{R}^{2}}\displaystyle\int_{\mathbb{R}^{2}}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})e^{-\mu_{1}(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}w_{1}+\frac{D_{1}}{2B_{1}}w^{2}_{1}-\frac{\pi}{4})}\right.&&\\\
\qquad\qquad\qquad\times\left.\Psi(\mathbf{u-x,w})\frac{1}{\sqrt{2\pi
B_{2}}}e^{-\mu_{2}(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}w_{2}+\frac{D_{2}}{2B_{2}}w^{2}_{2}-\frac{\pi}{4})}d\mathbf{w}d\mathbf{u},g\right\rangle&&\\\
\\\ \end{array}$

As last equation is true for every $g\in L^{2}(\mathbb{R}^{2},\mathbb{H})$
therefore we have

$\displaystyle f(\mathbf{x})\Lambda_{\Psi}$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{-\mu_{1}(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}w_{1}+\frac{D_{1}}{2B_{1}}w^{2}_{1}-\frac{\pi}{4})}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})$
$\displaystyle\qquad\qquad\qquad\times\Psi(\mathbf{u-x,w})\frac{1}{\sqrt{2\pi
B_{2}}}e^{-\mu_{2}(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}w_{2}+\frac{D_{2}}{2B_{2}}w^{2}_{2}-\frac{\pi}{4})}d\mathbf{w}d\mathbf{u}.$

Which implies

$\displaystyle f(\mathbf{x})$
$\displaystyle=\frac{1}{\Lambda_{\Psi}}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\frac{1}{\sqrt{2\pi
B_{1}}}e^{-\mu_{1}(\frac{A_{1}}{2B_{1}}x^{2}_{1}-\frac{x_{1}}{B_{1}}w_{1}+\frac{D_{1}}{2B_{1}}w^{2}_{1}-\frac{\pi}{4})}\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\Psi(\mathbf{u-x,w})\frac{1}{\sqrt{2\pi
B_{2}}}e^{-\mu_{2}(\frac{A_{2}}{2B_{2}}x^{2}_{2}-\frac{x_{2}}{B_{2}}w_{2}+\frac{D_{2}}{2B_{2}}w^{2}_{2}-\frac{\pi}{4})}d\mathbf{w}d\mathbf{u}.$

Hence completes the proof

4\. Uncertainty Principles for the Q-LCST

The classical uncertainty principle of harmonic analysis states that a non-
trivial function and its Fourier transform cannot both be simultaneously
sharply localized. In quantum mechanics, an uncertainty principle asserts that
one cannot be certain of the position and of the velocity of an electron (or
any particle) at the same time. In other words, increasing the knowledge of
the position decreases the knowledge of the velocity or momentum of an
electron.The uncertainty principles in harmonic analysis are of central
importance as they provide a lower bound for optimal simultaneous resolution
in the time and frequency domains.In this Section, we shall establish an
analogue of the well-known Heisenberg’s uncertainty inequality and the
corresponding logarithmic uncertainty principle for the Q-LCST as defined by
(3.2). First, we prove the following lemma.

Lemma 4.1. Let $\Psi\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ be a non zero
quaternion window function, then for every $f\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$, we have

$\Lambda_{\Psi}\int_{\mathbb{R}^{2}}x^{2}_{s}|f(\mathbf{x})|^{2}d\mathbf{x}=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}x^{2}_{s}|\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u},\mathbf{w})\\}(\mathbf{x})|^{2}d\mathbf{x}d\mathbf{u}.$
$None$

where $s=1,2.$

Proof. We avoided proof as it follows by theorem (3.1) and theorem (3.2).

Now we can establish the Heisenberg-type inequalities for the proposed Q-LCST
as defined by (3.2).

Theorem 4.2. Let $\Psi\in L^{2}(\mathbb{R}^{2},\mathbb{H})$ be a non zero
quaternion window function and $\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f]$
be the Q-LCST of any signal $f\in L^{2}(\mathbb{R}^{2},\mathbb{H}),$ then we
have

$\left(\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{u}d\mathbf{w}\right)^{1/2}\left(\int_{\mathbb{R}^{2}}x^{2}_{s}|f(\mathbf{x})|^{2}d\mathbf{x}\right)^{1/2}\geq\frac{n_{s}\sqrt{\Lambda_{\Psi}}}{2}\|f\|^{2}$
$None$

where $s=1,2.$

Proof. By virtue of the Heisenberg’s inequality for the QLCT [14QWLCT], we can
write

$\int_{\mathbb{R}^{2}}x^{2}_{s}|f(\mathbf{x})|^{2}d\mathbf{x}\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](w)|^{2}d\mathbf{w}\geq\frac{n^{2}_{s}}{4}\left(\int_{\mathbb{R}^{2}}|f|^{2}d\mathbf{x}\right)^{2}.$
$None$

Equation(4.3) can be rewritten as

$\int_{\mathbb{R}^{2}}x^{2}_{s}|\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f]\\}|^{2}d\mathbf{x}\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{w})|^{2}d\mathbf{w}\geq\frac{n^{2}_{s}}{4}\left(\int_{\mathbb{R}^{2}}|f|^{2}d\mathbf{x}\right)^{2}.$
$None$

Now by applying Plancherel’s theorem for the QLCT to the right-hand side of
(4.4), we have

$\int_{\mathbb{R}^{2}}x^{2}_{s}|\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f]\\}|^{2}d\mathbf{x}\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{w})|^{2}d\mathbf{w}\geq\left(\frac{n_{s}}{2}\int_{\mathbb{R}^{2}}|\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{w})|^{2}d\mathbf{w}\right)^{2}.$
$None$

Since $\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f]\in
L^{2}(\mathbb{R}^{2},\mathbb{H})$,therefore we can replace
$\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f]$ by
$\mathbb{S}^{\Psi,\mathbb{H}}_{\Psi,M_{1},M_{2}}[f]$ on the both sides of
(4.5) to get

$\int_{\mathbb{R}^{2}}x^{2}_{s}|\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})\\}|^{2}d\mathbf{x}\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathbb{S}^{\Psi,\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}\geq\left(\frac{n_{s}}{2}\int_{\mathbb{R}^{2}}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f]\mathbf{w}|^{2}d\mathbf{w}\right)^{2}.$
$None$

On taking square root to both sides of (4.6) and integrating with respect
$d\mathbf{u}$,we have

$\displaystyle\int_{\mathbb{R}^{2}}\left\\{\left(\int_{\mathbb{R}^{2}}x^{2}_{s}|\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})\\}|^{2}d\mathbf{x}\right)^{1/2}\left(\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{w})|^{2}d\mathbf{w}\right)^{1/2}\right\\}d\mathbf{u}$
$\displaystyle\qquad\geq\frac{n_{s}}{2}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f]\mathbf{w}|^{2}d\mathbf{w}d\mathbf{u}.$
(4.7)

Now by virtue of Cauchy-Schwarz inequality (4.7) becomes

$\displaystyle\left(\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}x^{2}_{s}|\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})\\}|^{2}d\mathbf{x}d\mathbf{u}\right)^{1/2}\left(\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{w})|^{2}d\mathbf{w}d\mathbf{u}\right)^{1/2}$
$\displaystyle\qquad\geq\frac{n_{s}}{2}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f]\mathbf{w}|^{2}d\mathbf{w}d\mathbf{u}.$
(4.8)

Now by applying Lemma 4.1 on L.H.S and Remark 3.1 on R.H.S of the above
inequality,we have

$\displaystyle\left(\Lambda_{\Psi}\int_{\mathbb{R}^{2}}x^{2}_{s}|f(\mathbf{x})|^{2}d\mathbf{x}\right)^{1/2}\left(\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}d\mathbf{u}\right)^{1/2}$
$\displaystyle\qquad\geq\frac{n_{s}\Lambda_{\Psi}}{2}\|f\|^{2}.$ (4.9)

On further simplifying (4.9),we get

$\left(\Lambda_{\Psi}\int_{\mathbb{R}^{2}}x^{2}_{s}|f(\mathbf{x})|^{2}d\mathbf{x}\right)^{1/2}\left(\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}d\mathbf{u}\right)^{1/2}\geq\frac{n_{s}\Lambda_{\Psi}}{2}\|f\|^{2}.$
$None$

Hence,

$\left(\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}w^{2}_{s}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}d\mathbf{u}\right)^{1/2}\left(\int_{\mathbb{R}^{2}}x^{2}_{s}|f(\mathbf{x})|^{2}d\mathbf{x}\right)^{1/2}\geq\frac{n_{s}\sqrt{\Lambda_{\Psi}}}{2}\|f\|^{2}.$
$None$

Which completes the proof.

We now establish the logarithmic uncertainty principle for the Q-LCST as
defined by (3.2).

Theorem 4.3 For a quaternion window function
$\Psi\in\mathcal{S}(\mathbb{R}^{2},\mathbb{H})$ and for
$f\in\mathcal{S}(\mathbb{R}^{2},\mathbb{H})$ the Q-LCST satisfies the
following logarithmic estimate of the uncertainty inequality:

$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\ln|\mathbf{w}||\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}d\mathbf{u}+\Lambda_{\Psi}\int_{\mathbb{R}^{2}}\ln|\mathbf{x}||f(\mathbf{x})|^{2}d\mathbf{x}\geq\mathcal{D}\Lambda_{\Psi}\|f\|^{2}$
(4.12)

where $\mathcal{D}=\left(\frac{\Gamma^{\prime}(1/2)}{\Gamma(1/2)}-\ln
2\right)$ and $\Gamma$ is a Gamma function.

Proof. By virtue of the logarithmic inequality for the QLCT [14QWLCT], we can
write

$\int_{\mathbb{R}^{2}}\ln|\mathbf{w}||\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{w})|^{2}d\mathbf{w}+\int_{\mathbb{R}^{2}}\ln|\mathbf{x}||f(\mathbf{x})|^{2}d\mathbf{x}\geq\mathcal{D}\int_{\mathbb{R}^{2}}|f(\mathbf{x})|^{2}d\mathbf{x}.$
$None$

Now invoking the inversion formula of QLCT on the L.H.S and Parseval’s formula
for QLCT on R.H.S, we obtain

$\displaystyle\int_{\mathbb{R}^{2}}\ln|\mathbf{x}||\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f]\\}(\mathbf{x})|^{2}d\mathbf{x}+\int_{\mathbb{R}^{2}}\ln|\mathbf{w}||\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{w})|^{2}d\mathbf{w}$
$\displaystyle\qquad\geq\mathcal{D}\int_{\mathbb{R}^{2}}|\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f](\mathbf{x})|^{2}d\mathbf{x}.$
(4.14)

Since $\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f]$ and
$\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f]$ are in
$\mathcal{S}(\mathbb{R}^{2},\mathbb{H})$ thus we can replace
$\mathcal{L}^{\mathbb{H}}_{M_{1},M_{2}}[f]$ by
$\mathbb{S}^{\Psi,\mathbb{H}}_{\Psi,M_{1},M_{2}}[f]$ on the both sides of
(4.14) to get

$\displaystyle\int_{\mathbb{R}^{2}}\ln|\mathbf{w}||\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}+\int_{\mathbb{R}^{2}}\ln|\mathbf{x}||\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})\\}(\mathbf{x})|^{2}d\mathbf{x}$
$\displaystyle\qquad\geq\mathcal{D}\int_{\mathbb{R}^{2}}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}.$
(4.15)

On integrating (4.15) with respect du and then by applying Fubini theorem,we
have

$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\ln|\mathbf{w}||\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}d\mathbf{u}+\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\ln|\mathbf{x}||\mathcal{L}^{-1}_{M_{1},M_{2}}\\{\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})\\}(\mathbf{x})|^{2}d\mathbf{x}d\mathbf{u}$
$\displaystyle\qquad\geq\mathcal{D}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}|\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}d\mathbf{u}.$
(4.16)

Now applying Lemma 4.1 on L.H.S and Remark 3.1 on R.H.S of (4.16), we get

$\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\ln|\mathbf{w}||\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}d\mathbf{u}+\Lambda_{\Psi}\int_{\mathbb{R}^{2}}\ln|\mathbf{x}|f(\mathbf{x})|^{2}d\mathbf{x}\geq\mathcal{D}\Lambda_{\Psi}\|f\|^{2}.$

Which completes the proof.

Remark 4.1 If f is normalized, namely
$\|f\|_{L^{2}(\mathbb{R}^{2},\mathbb{H})}=1$ then (4.12) becomes

$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\ln|\mathbf{w}||\mathbb{S}^{\mathbb{H}}_{\Psi,M_{1},M_{2}}[f](\mathbf{u,w})|^{2}d\mathbf{w}d\mathbf{u}+\Lambda_{\Psi}\int_{\mathbb{R}^{2}}\ln|\mathbf{x}||f(\mathbf{x})|^{2}d\mathbf{x}\geq\mathcal{D}\Lambda_{\Psi}$

Conclusions

We have introduced the notion of quaternion linear canonical
S-transform(Q-LCST) which is an extension of the linear canonical S-transform.
Firstly, we discussed the fundamental properties of quaternion linear
canonical S-transform(Q-LCST) and then establish some basic results including
orthogonality relation and reconstruction formula. Most importantly, we
derived the stronger version of the associated Heisenberg’s uncertainty
inequality and the corresponding logarithmic version for quaternion linear
canonical S-transform(Q-LCST). This will pay way for further findings in this
area. Our research will help in finding more generalised and stronger versions
of the inequality which will revolutionise the signal and image processing.

Declarations

  * •

Availability of data and materials: The data is provided on the request to the
authors.

  * •

Competing interests: The authors have no competing interests.

  * •

Funding: No funding was received for this work

  * •

Author’s contribution: Both the authors equally contributed towards this work.

  * •

Acknowledgements: This work is supported by the UGC-BSR Research Start Up
Grant(No. F.30-498/2019(BSR)) provided by UGC, Govt. of India.

References

## References

  * [1] Akila, L., Roopkumar, R.: Quaternion Stockwell transform, Intg. Trans. Special Func., 2016
  * [2] Bahari, M., Toaha, S., Lande, C.: A generalized S-transform in linear canonical transform, J. Phy. Conf. Series. 1341(2019)
  * [3] Gao, W.B., Li, B.Z.: Quaternion windowed linear canonical transform of two-dimensional quaternionic signals, Adv. Appl. Clifford Algebr. 30 (1) (2020) 1–18
  * [4] Huo, H.: Uncertainty principles for the offset linear canonical transform. Circuits Sys.Signal Process. 38(2019), 395–406
  * [5] Huo, H., Sun, W., Xiao, L.: Uncertainty principles associated with the offset linear canonical transform. Math. Meth. Appl. Sci. 42(2019) 466–474
  * [6] Kamel, B., Tefjeni, E.: Continuous quaternion Stockwell transform and Uncertainty principle. arXiv:1912.11404v1 2019
  * [7] Kou, K.I., Jian, Y.O., Morais, J.: On uncertainty principle for quaternionic linear canonical transform. Abstr. Appl. Anal. (2013), (Article ID 725952). https://doi.org/10.1155/2013/725952
  * [8] Lian, P.: Uncertainty principle for the quaternion Fourier transform. J. Math. Anal. Appl. 467(2018) 1258–1269
  * [9] Moshinsky, M., Quesne, C.: Linear canonical tansformations and their unitary representations. J. Math. Phys. 12(8)(1971) 1772–1880
  * [10] Zhu, X., Zheng, S.: Uncertainty principles for the two-sided quaternion linear canonical transform, Circuits Sys. Signal . Process. 39(2020) 4436-4448


11institutetext:  Fukui Prefectural University, 910-1195 Fukui, JAPAN

Probability theory Kinetic theory Kinetic and transport theory of gases

# Relativistic Equilibrium Distribution by Relative Entropy Maximization

Tadas K. Nakamura

###### Abstract

The equilibrium state of a relativistic gas has been calculated based on the
maximum entropy principle. Though the relativistic equilibrium state was long
believed to be the Jüttner distribution, a number of papers have been
published in recent years proposing alternative equilibrium states. However,
some of these papers do not pay enough attention to the covariance of
distribution functions, resulting confusion in equilibrium states. Starting
from a fully covariant expression to avoid this confusion, it has been shown
in the present paper that the Jüttner distribution is the maximum entropy
state if we assume the Lorentz symmetry.

###### pacs:

02.50.Cw

###### pacs:

05.20.Dd

###### pacs:

51.10.+y

##  1 Introduction

Little after the establishment of the theory of relativity, the equilibrium
particle distribution of a relativistic gas was investigated. The distribution
obtained, which is called Jüttner distribution [1, 2], has been long and
widely believed. However, relatively recent years a number of papers have been
published proposing equilibrium distribution functions other than the Jüttner
distribution ([3, 4, 5, 6, 7] and references therein). Dunkel and coworkers
[7, 8] have examined the discrepancy in the equilibrium distributions as the
maximum entropy state, and showed that the difference comes from the choice of
the reference measure.

The maximum entropy state cannot be uniquely determined when one naively
defines the entropy such as $S=-\int f(\mathbf{x},\mathbf{v})\ln
f(\mathbf{x},\mathbf{v})\,d\mathbf{x}d\mathbf{v}$ (symbols have conventional
meaning in the present paper unless otherwise stated). For instance, the
result would be different if we rewrite distribution function as a function of
momentum $\mathbf{p}$ instead of velocity $\mathbf{v}$. To overcome this
difficulty, it was proposed in Ref [7] to maximize the following relative
entropy

$S=-\int f(\mathbf{x},\mathbf{v})\ln
f(\mathbf{x},\mathbf{v})/\rho(\mathbf{x},\mathbf{v})\,d\mathbf{x}d\mathbf{v}\,,$
(1)

based on a given reference measure $\rho$. In the above expression $f$ is the
phase space distribution of particles and $\rho$ is the reference measure [7].
In this paper we denote a three vector by a bold font (e.g., $\mathbf{x}$) and
a four vector by an upper bar (e.g., $\bar{x}$). Each component of a vector is
represented by a subscript or a superscript (e.g., $x_{\mu}$ or $x^{\mu}$).

The equilibrium distribution is uniquely determined by maximizing the relative
entropy once the reference measure is given. The mathematical procedure in
this approach is essentially the same as the one utilized in Ref [2] to derive
the Jüttner distribution. What is called “a priori probability” in Ref [2]
plays the same role as the reference measure in Ref [7].

Two possibilities for the reference frame were suggested in Ref [7]. One is
the constant distributions a function of momentum, and the Jẗtner distribution
is obtained from this measure. This calculation is essentially the same as the
one in Ref[2]. Another possibility suggested in Ref [7] which is inversely
proportional to the energy. It was argued this measure is derived from the
Lorentz symmetry in Ref [7] and the result is the alternative equilibrium
distribution proposed in recent papers. However, as we will see in the present
paper, there is a confusion on the relativistic phase space density in this
argument. The Lorentz invariant reference measure is the same as the one in
Ref [2], i.e., the constant measure, which gives the Jüttner distribution.

There is a misleading point in defining a phase space density such as a
particle distribution in relativity. When we express a phase space density as
the time evolution of the density in a six (three space + three momentum)
dimensional phase space, it appears to be a Lorentz invariant. Actually, it
can be proved [9] (see also [2, 10]) that
$f(t,\mathbf{x},\mathbf{p})=f(t^{\prime},\mathbf{x^{\prime}},\mathbf{p}^{\prime})$
when the two sets of coordinates $(t,\mathbf{x},\mathbf{p})$ and
$(t^{\prime},\mathbf{x}^{\prime},\mathbf{p}^{\prime})$ are related by the
Lorentz transform, in other words, they are the same point in the spacetime
denoted by different reference coordinates. However, this does not mean
$f(t,\mathbf{x},\mathbf{p})d\mathbf{x}d\mathbf{p}=f(t^{\prime},\mathbf{x^{\prime}},\mathbf{p}^{\prime})d\mathbf{x}^{\prime}d\mathbf{p}^{\prime}$
because $\mathbf{x}$ and $\mathbf{x}^{\prime}$ do not belong to the same
spatial volume. In this sense, phase a space density in the form of
$f(t,\mathbf{x},\mathbf{p})$ is not covariant but frame dependent. It seems
that some of recent papers do not pay enough attention to this fact, resulting
confision in treating Lorentz transfrom.

In the present paper, we examine this confusing point by starting from the
fully covariant distribution function proposed by Hakim [11], and the result
shows the reference measure should be constant to satisfy the full Lorentz
symmetry; the one introduced in Ref [7] is invariant under the Lorentz
transform only in the momentum space. This result means the maximum entropy
state with Lorentz symmetry must be the Jüttner distribution.

##  2 Relativistic Phase Space Density

Let us suppose a relativistic gas as an example. The conservation law of its
particle number is expressed in the form of flux divergence in relativity:

$\frac{\partial}{\partial x_{\mu}}\,J_{\mu}=0\,,$ (2)

where

$J_{\mu}=n_{0}u_{\mu}\,.$ (3)

is the four flux derived from the proper number density $n_{0}$ and the four
velocity of the matter $u_{\mu}$. When we split the spacetime as
$t_{\Sigma}=x_{\Sigma 0}$ and $\mathbf{x_{\Sigma}}=(x_{\Sigma 1,}x_{\Sigma
2},x_{\Sigma 3})$ by choosing a specific reference frame $\Sigma$, the above
conservation is written as

$\frac{\partial}{\partial
t}n_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma})+\nabla\mathbf{J}_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma})=0\,.$
(4)

In the above expression, $n_{\Sigma}=J_{\Sigma 0}$ and
$\mathbf{J}_{\Sigma}=(J_{\Sigma 1},J_{\Sigma 2},J_{\Sigma 3})$ are the number
density and flux in the three dimensional space; the subscript $\Sigma$ is to
explicitly express the frame dependence.

When we decompose the spacetime in another reference frame $\Sigma^{\prime}$,
obviously $n_{\Sigma^{\prime}}$ is different from $n_{\Sigma}$. Moreover,
$n_{\Sigma}$ and $n_{\Sigma^{\prime}}$ cannot be related with a Jacobian
$\partial\mathbf{x}_{\Sigma}/\partial\mathbf{x}_{\Sigma^{\prime}}$ as

$n_{\Sigma}d\mathbf{x}_{\Sigma}=n_{\Sigma^{\prime}}\frac{\partial\mathbf{x}_{\Sigma}}{\partial\mathbf{x}_{\Sigma^{\prime}}}\,d\mathbf{x}_{\Sigma^{\prime}}\,,$
(5)

because $\mathbf{x}_{\Sigma}$ and $\mathbf{x}_{\Sigma}$ belong to different
spacelike volumes. There is no function to relate $\mathbf{x}_{\Sigma}$ and
$\mathbf{x}_{\Sigma^{\prime}}$ as
$\mathbf{x}_{\Sigma}=\mathbf{X}(\mathbf{x}_{\Sigma^{\prime}})$ where
$\mathbf{X}$ that does not depend on the time coordinate.

The above argument on the number density in a three dimensional space is also
valid for phase space densities in a six dimensional space. A phase space
density is often expressed as $f(t,\mathbf{x},\mathbf{p})$ and it should be
denoted in our notation as
$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$ because the
expression is based on a specific choice of the reference frame like
$n_{\Sigma}$ in (4). However, it is generally believed that the phase space
density is unchanged under the Lorentz transform. This is true in the sense
that the value of the phase space density is unchanged [2, 9, 10], but
$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$ is defined
only on a space volume in a specific reference frame, and not directly
applicable to other reference frames.

To correctly treat the phase space density, we derive the frame-dependent
phase space density
$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$ from the
fully covariant expression proposed by Hakim [11]. The relativistic particle
distribution $N(\bar{x},\bar{p})$ is defined such that $\bar{j}$ in the
following expression becomes the particle four-current:

$j_{\mu}(\bar{x})=\int
d_{4}p\,2mu_{\mu}N(\bar{x},\bar{p})\,\theta(p^{0})\delta(p^{\mu}p_{\mu}-m^{2})\,,$
(6)

where $\theta$ and $\delta$ are the theta and delta functions, and $m$ is the
particle rest mass.

In the above expression, $N(\bar{x},\bar{p})$ can be interpreted as the proper
density of the fluid element that has the four velocity $\bar{u}=\bar{p}/m$,
just like $n_{0}$ in (3). Thus its covariant form must be a four vector, which
is expressed as $N(\bar{x},\bar{p})\bar{u}$, like $\bar{J}$ in (3). The delta
function is due to the energy shell and the theta function is to discard the
negative energy solution. Hakim [11] has introduced the above expression for
the distribution of particle number, however, it is generally valid for a
conserved density flowing with the four velocity $\bar{u}$, therefore, it can
be applied to a probability distribution or a reference measure to calculate
entropy in the following.

When we pick up one reference frame $\Sigma$ and denote its unit vectors in
each coordinate direction as $(\bar{e}_{\Sigma t},\bar{e}_{\Sigma
x},\bar{e}_{\Sigma y},\bar{e}_{\Sigma z})$, an arbitrary point in the eight
dimensional phase space $(\bar{x},\bar{p})$ can be represented in this
reference frame as

$t_{\Sigma}=e_{\Sigma t}^{\mu}x_{\mu},~{}~{}\mathbf{x}_{\Sigma}=(e_{\Sigma
x}^{\mu}x_{\mu},e_{\Sigma y}^{\mu}x_{\mu},e_{\Sigma z}^{\mu}x_{\mu})\,,$ (7)

and

$E_{\Sigma}=e_{\Sigma t}^{\mu}p_{\mu},~{}~{}\mathbf{p}_{\Sigma}=(e_{\Sigma
x}^{\mu}p_{\mu},e_{\Sigma y}^{\mu}p_{\mu},e_{\Sigma z}^{\mu}p_{\mu})\,.$ (8)

A frame-dependent phase space density
$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$ is then
calculated from $N(\bar{x},\bar{p})$ as

$\displaystyle f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$
$\displaystyle=$ $\displaystyle 2m\int e_{\Sigma
t}^{\mu}u_{\mu}N(\bar{X},\bar{P})\,\theta(p^{0})\delta(E_{\Sigma}^{2}-\mathbf{p}_{\Sigma}^{2}-m^{2})\,dE_{\Sigma}$
$\displaystyle=$ $\displaystyle\frac{me_{\Sigma
t}^{\mu}u_{\mu}}{E_{\Sigma}}\,N(\bar{X},\bar{P})=N(\bar{X},\bar{P})\,,$ (9)

where $\bar{u}=\bar{p}/m$, and $\bar{X}$ and $\bar{P}$ are the covariant
expression of the four dimensional position and momentum correspond to
$(t,\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma})$,

$\bar{X}=t_{\Sigma}\bar{e}_{\Sigma t}+x_{\Sigma}\bar{e}_{\Sigma
x}+y_{\Sigma}\bar{e}_{\Sigma y}+z_{\Sigma}\bar{e}_{\Sigma z}\,,$ (10)

and

$\bar{P}=\sqrt{\mathbf{p}^{2}+m^{2}}\,\bar{e}_{\Sigma t}+p_{\Sigma
x}\bar{e}_{\Sigma x}+p_{\Sigma y}\bar{e}_{\Sigma y}+p_{\Sigma
z}\bar{e}_{\Sigma z}\,.$ (11)

From (9) van Kampen [9] concluded that $f$ is unchanged under the Lorentz
transform (his derivation is different from ours, but the result is the same).
He considered the above result is purely kinematical. It is true in the sense
that no equation of motion is required for (9), however, it implicitly
includes kinetics in the expression of the energy shell. For example, if the
relativistic kinetics were such that the energy shell is expressed as
$4m^{3}\delta(E_{\Sigma}^{4}-\mathbf{p}_{\Sigma}^{4}-m^{4})$, (9) would be

$f_{\Sigma}(t_{\Sigma},\mathbf{x}_{\Sigma}\mathbf{p}_{\Sigma})=\frac{m^{3}e_{\Sigma
t}^{\mu}u_{\mu}}{E_{\Sigma}^{3}}\,N(\bar{X},\bar{P})=\frac{m^{2}}{E_{\Sigma}^{2}}\,N(\bar{X},\bar{P})\,,$
(12)

which means the value of $f$ changes under the Lorentz transform. This example
demonstrates the fact that $f_{\Sigma}$ is not identical to $N$, but should be
derived from $N$.

##  3 Lorentz Invariant Reference Frame

In (9) we assumed the spatial coordinates $(t_{\Sigma},\mathbf{x}_{\Sigma})$
and the momentum coordinates $(E_{\Sigma},\mathbf{p}_{\Sigma})$ are defined in
the same reference frame $\Sigma$. Mathematically the reference frames to
define spatial and momentum coordinates do not have to be the same; we may
have a phase space density whose spatial coordinates are defined in $\Sigma$
and momentum coordinates are in $\Sigma^{\prime}$ as in the following form:

$\displaystyle
f_{\Sigma\Sigma^{\prime}}(t_{\Sigma},\mathbf{x}_{\Sigma},\mathbf{p}_{\Sigma^{\prime}})$
$\displaystyle=$ $\displaystyle 2m\int e_{\Sigma
t}^{\mu}u_{\mu}N(\bar{X},\bar{P}^{\prime})\,\theta(p^{0})\delta(E_{\Sigma^{\prime}}^{2}-\mathbf{p}_{\Sigma^{\prime}}^{2}-m^{2})\,dE_{\Sigma^{\prime}}$
$\displaystyle=$ $\displaystyle\frac{me_{\Sigma
t}^{\mu}u_{\mu}}{E_{\Sigma^{\prime}}}\,N(\bar{X},\bar{P}^{\prime})\,,$ (13)

with

$\bar{P}^{\prime}=E_{\Sigma^{\prime}}\bar{e}_{\Sigma^{\prime}t}+p_{\Sigma^{\prime}x}\bar{e}_{\Sigma^{\prime}x}+p_{\Sigma^{\prime}y}\bar{e}_{\Sigma^{\prime}y}+p_{\Sigma^{\prime}z}\bar{e}_{\Sigma^{\prime}z}\,.$
(14)

From the above expression we understand that the factor of $e_{\Sigma
t}^{\mu}u_{\mu}$ comes from the spatial Lorentz transform whereas the factor
of $1/E_{\Sigma^{\prime}}$ is due to the transform in the momentum space. They
are canceled out when $\Sigma=\Sigma^{\prime}$ and $f_{\Sigma\Sigma}$ becomes
unchanged as seen in the previous section. This fact also indicates the phase
space density is not a covariant expression; if it were covariant,
$f_{\Sigma\Sigma^{\prime}}$ should be unchanged even when
$\Sigma\neq\Sigma^{\prime}$.

Since $f_{\Sigma\Sigma}$ and $f_{\Sigma\Sigma^{\prime}}$ are the densities
defined on a same spatial volume in $\Sigma$, we can relate them by

$\frac{1}{E_{\Sigma}}f_{\Sigma\Sigma}\,d\mathbf{p}_{\Sigma^{\prime}}d\mathbf{x}_{\Sigma}=\frac{1}{E_{\Sigma^{\prime}}}f_{\Sigma\Sigma^{\prime}}\,d\mathbf{p}_{\Sigma^{\prime}}d\mathbf{x}_{\Sigma}\,.$
(15)

When we apply the above result to the reference measure $\rho$ to calculate
the relative entropy, it has the same meaning as Equation (34) in Ref [7]. If
the measure $\rho$ is to be invariant under the transform of
$\rho_{\Sigma\Sigma}\rightarrow\rho_{\Sigma\Sigma^{\prime}}$, it must be

$\rho(p_{\Sigma})\propto\frac{1}{E_{\Sigma}}\,,$ (16)

which is suggested in Ref [7]. However, as seen from (15), the Lorentz
transform in this context is in the momentum space only and the spatial volume
to define the measure $\rho$ is unchanged.

The present paper proposes that the measure should have the Lorentz symmetry
under the transform both in space and momentum coordinates:
$\rho_{\Sigma\Sigma}\rightarrow\rho_{\Sigma^{\prime}\Sigma^{\prime}}$. Then we
have to choose the phase space density defined by (9) instead of (13) for the
reference measure. As discussed above, two phase space densities with
different reference frames $\Sigma$ and $\Sigma^{\prime}$ is not directly
connected with a equation such as (15). The Lorentz symmetry in this case
means the mathematical expression is unchanged under the transform, and this
is satisfied when $N(\bar{x},\bar{p})$ is constant. Therefore we obtain

$\rho(p_{\Sigma})=\textrm{constant}\,,$ (17)

in the reference frame $\Sigma$ instead of (16). Following the relative
entropy maximization procedure proposed in Ref [7] we obtain the Jüttner
distribution as

$\phi(p_{\Sigma})\propto\exp(-\beta E_{\Sigma})\,.$ (18)

by maximizing the relative entropy in (1).

##  4 Concluding Remarks

It has been shown in the present paper the maximum entropy state based on the
Lorentz symmetry is the Jüttner distribution. Recent years a number of papers
have been published claiming the relativistic equilibrium state is different
from the long believed Jüttner distribution. Dunkel and coworkers [7, 8] have
shed a light to this controversy by pointing the importance of the reference
measure in the maximum entropy approach.. They have shown that the difference
of the reference measure causes the difference of the equilibrium distribution
as the maximum entropy state.

Two typical reference measures were suggested in Ref [7]. One is constant as a
function of $p$ and the other is inversely proportional to the energy. In Ref
[7] it is conjectured the former is derived from the invariance of momentum
transition, and the latter comes from the Lorentz symmetry. However, as we
have seen in the present paper, the reference measure with Lorentz symmetry is
also found to be constant when we correctly formulate the covariance of
relativistic phase space density.

The constant reference measure we derived in this paper corresponds to the
constant “prior probability” employed by Synge [2]. The information theory was
developed long after the days of Synge, therefore, he did not know the modern
concepts such as relative entropy or reference measure. Nevertheless, his
calculation is quite similar to ours, and the result is the same Jüttner
distribution. (The author guesses his basic idea historically comes from the
probabilistic interpretation of the entropy by Boltzmann in his late years
[12].)

Therefore, the argument in the present paper might seem just another
interpretation of Synge’s result with information theory if one believes his
derivation. However, considerable number of papers have been published
recently against the Jüttner distribution and it is important to clarify the
foundation of the maximum entropy process based on information theory.
Moreover, it has become clear in the present paper what causes the confusion
of the reference measure (the difference of $\rho_{\Sigma^{\prime}\Sigma}$ and
$\rho_{\Sigma^{\prime}\Sigma^{\prime}}$).

The result in the present paper strongly suggests that the relativistic
equilibrium state is the Jüttner distribution. There are papers in favor of
the Jüttner distribution in the recent controversy. Debbasch [13] critically
reviewed the theories proposing alternatives to the Jüttner distribution. He
examined the relative entropy approach in Ref [7] and showed the result would
be inconsistent unless the reference measure is constant.

Also there is a result of numerical experiment that supports the Jüttner
distribution [14]. It was aregued in Ref [8] that the distribution measured in
Ref [14] is based on what they call “coordinate-time”, and and the modified
distribution would be obtained if it is defined with “proper-time”. In this
sense, what we examined in the present paper is the one with “coordinate-
time”, in agreement with the numerical experiment.

It has been known, but has not been well recognized, that the a conserved
quantity (energy-momentum, particle number etc.) distributed over a finite
volume is not a Lorentz invariant quantity because it belongs to a different
time slice of the volume’s world tube. Confusions on this point have caused
controversy on the relativistic thermodynamics ([15, 16] and references
therein).

To treat this point correctly any spatial density must be expressed by a flux
four vector as a covariant form. The density in the phase space is no
exception. However, the phase space density in the form of
$f(t,\mathbf{x},\mathbf{p})$ is often regarded as a covariant expression since
the value $f$ is unchanged under the Lorentz transform.

As discussed in Section 2, the expression of $f(t,\mathbf{x},\mathbf{p})$ is
frame dependent since it is defined on a three dimensional space volume in a
specific reference frame. It seems some of recent papers do not pay enough
attention to this point, and treat the phase space density in a confusing way.
In the present paper we start with the fully covariant expression of the phase
space density [11] to avoid this confusion. We have seen that the reference
measure with Lorentz symmetry is constant as a function of momentum.
Consequently the maximum entropy state with the Lorentz symmetry is the
Jüttner distribution.

It is known that the maximum entropy approach used in Ref [7] has the
mathematical structure almost parallel to the traditional ensemble approach
[17]. Therefore, the equilibrium distributions derived from ensemble approach
can be examined with the same basis. This means the result in the present
paper can be applicable to theories with the traditional approach.

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