# Mourre theory and spectral analysis of energy-momentum operators
in relativistic quantum field theory

Janik Kruse

Adam Mickiewicz University in Poznań
Faculty of Mathematics and Computer Science
ul. Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
E-mail<EMAIL_ADDRESS>
(April 8, 2024)

###### Abstract

A central task of theoretical physics is to analyse spectral properties of
quantum mechanical observables. In this endeavour, Mourre’s conjugate operator
method emerged as an effective tool in the spectral theory of Schrödinger
operators. This paper introduces a novel class of examples amenable to
Mourre’s method, focusing on the spectra of energy-momentum operators in
relativistic quantum field theory. Under the assumption of Lorentz or dilation
covariance and the spectrum condition, we provide new proofs of the absolute
continuity of the energy-momentum spectra, showcasing the efficiency of
Mourre’s method in the relativistic setting.

Keywords: spectral theory, Mourre’s conjugate operator method, absence of
singular continuous spectrum, representations of the Poincaré group, dilation-
covariant representations

Mathematics Subject Classification 2020: 81Q10

##  1 Introduction

Understanding spectral properties of quantum mechanical observables is of
fundamental importance in theoretical physics. By analysing the spectrum,
insights into the system’s stability and long-time evolution can be obtained.
Specifically, the spectrum of a self-adjoint operator decomposes into a pure
point, absolutely continuous and singular continuous part, where the pure-
point part corresponds to bound states and the absolutely continuous part to
scattering states. Typically, the singular continuous spectrum, which evades a
simple physical interpretation, is empty. The objective of this paper is to
prove the absence of singular continuous spectrum for relativistic energy-
momentum operators.

Various mathematical techniques have been developed to investigate the
spectrum of a self-adjoint operator. One powerful method in this endeavour is
Mourre’s conjugate operator method, which is based on a strictly positive
commutator estimate. The idea of analysing spectral properties of self-adjoint
operators through commutator identities and estimates can be traced back to
the pioneering works of Putnam [Pu67], Kato [Ka68], and Lavine [La69].
However, it was Mourre [Mo81] who advanced this approach by introducing local
commutator estimates.

Mourre’s method led to substantial progress in the spectral and scattering
theory of Schrödinger operators. In his seminal paper, Mourre [Mo81]
demonstrated the absence of singular continuous spectrum for 2- and 3-body
Schrödinger operators. The applicability of the method to $N$-body systems was
extended by Perry, Sigal, and Simon [PSS81] and by Froese and Herbst [FH82].
Mourre’s method also played a decisive role in establishing asymptotic
completeness of $N$-body Schrödinger operators (see [DG97] for a textbook
exposition). On an abstract level, the mathematical theory underlying the
conjugate operator method was notably improved by Amrein, Boutet de Monvel,
and Georgescu [ABG96].

While originally developed for non-relativistic quantum mechanics, Mourre’s
method has also been extended to other areas. Worthy of note, it has been
applied in non-relativistic quantum electrodynamics (QED). Among many works,
we mention here that a Mourre estimate was proved by Dereziński and Gérard
[DG99] for confined Pauli–Fierz Hamiltonians, by Fröhlich, Griesemer, and
Schlein [FGS04] for Compton scattering, by Fröhlich, Griesemer, and Sigal
[FGS08] for the standard model of non-realtivistic QED, by Chen, Faupin,
Fröhlich, and Sigal [CFFS11] for dressed electrons, and by Møller and
Rasmussen [MR13] for the translation-invariant massive Nelson model.

More recently, Mourre’s method has expanded its scope to the domain of
Haag–Kastler quantum field theory to prove the existence of asymptotic
observables [Kr23]. One of the primary challenges encountered in axiomatic
quantum field theory pertains to the abstract nature of the Hamiltonian.
Scattering theory enabled the application of Mourre’s method by comparing the
abstract dynamics to a more concrete free dynamics.

This paper introduces a novel class of examples amenable to Mourre’s conjugate
operator method. We analyse the spectra associated with spacetime translations
in quantum field theory, that is, the spectra of the energy-momentum operators
$P$. By assuming Lorentz covariance and the spectrum condition, we present a
new proof of the well-known theorem asserting that the energy-momentum spectra
are purely absolutely continuous if translation-invariant vectors are removed.
More precisely, we prove the following proposition, where
$\mathcal{H}_{\mathrm{ac}}(A)$ denotes the absolutely continuous spectral
subspace of a commuting family of self-adjoint operators $A$ and $x\cdot
y=x_{0}y_{0}-\mathbf{x}\cdot\mathbf{y}$, $x,y\in\mathbb{R}^{d}$, denotes the
Minkowski scalar product.

######  Proposition 1.1.

Let $\mathcal{H}$ be a Hilbert space,
$U:\mathcal{P}\to\mathfrak{B}(\mathcal{H})$ a strongly continuous unitary
representation of the Poincaré group
$\mathcal{P}=\mathcal{L}\ltimes\mathbb{R}^{d}$, and
$e\in\mathbb{R}^{d}\backslash\\{0\\}$ a spacetime vector such that $e\cdot
e\neq 0$ if $d=2$. Assume that the generators $P=(P_{0},\mathbf{P})$ of the
translation subgroup $U|_{\mathbb{R}^{d}}$ obey the spectrum condition (see
Assumption 4.1). If $Q_{0}$ denotes the projection onto the subspace of
translation-invariant vectors, then

$\displaystyle\mathcal{H}=Q_{0}\mathcal{H}\oplus\mathcal{H}_{\mathrm{ac}}(e\cdot
P)=Q_{0}\mathcal{H}\oplus\mathcal{H}_{\mathrm{ac}}(\mathbf{P}).$ (1.1)

The first identity in (1.1) was originally proved by Maison [Ma68, Satz 2] for
$d=4$ through the application of Wigner’s theorem. The second identity
(absolute continuity of the joint momentum spectrum) has been proved under
differing assumptions in the realm of quantum field theory. Buchholz and
Fredenhagen [BF82, Proposition 2.2] utilised the locality principle to
establish this identity, and Bachmann, Dybalski, and Naaijkens [BDN16, Lemma
4.16] presented a simplified proof under the additional assumption of the
existence of a vacuum vector.

We employ Mourre’s method to prove Proposition 1.1, finding it to offer a
relatively straightforward proof. As an illustration, we consider the
commutation relation between the momentum operator $P_{j}$ and the generator
of Lorentz boosts $K_{j}$ in the spatial direction $j$:

$\displaystyle[P_{j},\mathop{}\\!\mathrm{i}K_{j}]=P_{0}.$ (1.2)

By virtue of the spectrum condition, this commutator is locally strictly
positive on spectral subspaces of $P_{j}$ that are separated from 0.
Consequently, the absolute continuity of the spectrum associated with $P_{j}$
emerges as a consequence of the limiting absorption principle (see Theorem 3.5
and Corollary 3.6). The only aspect complicating our proof to a certain extent
lies in the regularity requirements of the limiting absorption principle.

An immediate consequence of Proposition 1.1 and the Riemann–Lebesgue lemma is
the following clustering property. If $f,g\in\mathcal{H}$ are two arbitrary
vectors, then

$\displaystyle\lim_{t\to\pm\infty}\langle f,U(te)g\rangle=\langle
f,Q_{0}g\rangle.$ (1.3)

In quantum field theory, the clustering property is well-known for space-like
directions (i.e. $e\cdot e<0$) due to the locality principle. It is somewhat
unexpected that this property is also valid for light-like (i.e. $e\cdot e=0$)
and time-like directions (i.e. $e\cdot e>0$) under the assumption of Lorentz
covariance.

In two spacetime dimensions (i.e. $d=2$), the clustering property may not hold
for light-like directions. This stems from the potential presence of massless
excitations, as elaborated upon in the remark subsequent to Proposition 4.5.

######  Proposition 1.2.

Let $U$ be as in Proposition 1.1 and let $E_{P_{0}\pm P_{1}}$ denote the
spectral measure of $P_{0}\pm P_{1}$. If $d=2$, then

$\displaystyle\mathcal{H}=E_{P_{0}\pm
P_{1}}(\\{0\\})\mathcal{H}\oplus\mathcal{H}_{\mathrm{ac}}(P_{0}\pm P_{1}).$
(1.4)

Another operator of interest in quantum field theory is the mass operator
$M=\sqrt{P_{0}^{2}-|\mathbf{P}|^{2}}$. We prove the absence of singular
continuous spectrum for the mass operator if $U$ satisfies dilation
covariance. A unitary representation
$U:\mathbb{R}^{d}\to\mathfrak{B}(\mathcal{H})$ of the translation group
$\mathbb{R}^{d}$ is dilation-covariant if a self-adjoint operator $D$ exists
such that, for all $t,x\in\mathbb{R}^{d}$,

$\displaystyle\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tD}U(x)\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tD}=U(\mathop{}\\!\mathrm{e}^{-t}x),$
(1.5)

that is, the adjoint action of
$\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tD}$ scales the translation $x$
by the factor $\mathop{}\\!\mathrm{e}^{-t}$.

######  Proposition 1.3.

Let $U:\mathbb{R}^{d}\to\mathfrak{B}(\mathcal{H})$ be a strongly continuous
unitary representation of the translation group $\mathbb{R}^{d}$. Assume that
the generators $P=(P_{0},\mathbf{P})$ of $U$ obey the spectrum condition. If
$U$ is dilation-covariant, then the spectrum of the mass operator
$M=\sqrt{P_{0}^{2}-|\mathbf{P}|^{2}}$ is purely absolutely continuous in
$(0,\infty)$.

Given that the spectrum of the mass operator $M\geq 0$ is contained in the
interval $[0,\infty)$, the proposition implies that the pure point spectrum of
$M$ must be empty or $\\{0\\}$, while the singular continuous spectrum is
empty. Under the assumptions of Proposition 1.1, the mass operator may have
other eigenvalues than 0. The assumption of dilation covariance excludes
additional mass shells within the energy-momentum spectrum.

The structure of the paper is as follows. In Section 2, we define locally
smooth operators for a commuting family of self-adjoint operators, and we
relate the existence of a locally smooth operator to the regularity of the
joint spectrum. In Section 3, we review essential results of Mourre’s
conjugate operator method. We explain how to obtain locally smooth operators
based on a strictly positive commutator estimate. In Section 4, we present the
proofs of Propositions 1.1–1.3.

##  2 Locally smooth operators

Let $U:\mathbb{R}^{n}\to\mathfrak{B}(\mathcal{H})$ be a strongly continuous
unitary representation of $\mathbb{R}^{n}$ on a Hilbert space $\mathcal{H}$
with self-adjoint generators $H=(H_{1},\dots,H_{n})$. We denote by $D(H)$ the
intersection of the domains of $H_{1},\dots,H_{n}$, by $E$ the spectral
measure of the family $H$, and by $\sigma(H)\subset\mathbb{R}^{n}$ its joint
spectrum. We consider $D(H)$ as a Banach space equipped with the graph norm.

######  Definition 2.1.

A continuous operator $T:D(H)\to\mathcal{H}$ is locally $H$-smooth on an open
set $J\subset\mathbb{R}^{n}$ if, for every compact subset $K\subset J$, a
constant $C_{K}$ exists such that, for all $f\in\mathcal{H}$,

$\displaystyle\int_{\mathbb{R}^{n}}\|TU(x)E(K)f\|^{2}\mathop{}\\!\mathrm{d}x\leq
C_{K}\|f\|^{2}.$ (2.1)

In [Kr23, Definition A.9], we defined locally $H$-smooth operators by
demanding that all subfamilies of $H$ are locally $H$-smooth in the sense of
the above definition. In this paper, we prefer the weaker formulation of
Definition 2.1 because it is sufficient to characterise the regularity of the
joint spectrum $\sigma(H)$. The following proposition is a straightforward
generalisation of [RS78, Theorem XIII.23].

######  Proposition 2.2.

If $T$ is locally $H$-smooth on $J$, then
$E(J)\overline{\mathop{}\\!\mathrm{ran}(T^{*})}\subset
E(J)\mathcal{H}_{\mathrm{ac}}(H)$. If, additionally, $\ker(T)=\\{0\\}$, then
the joint spectrum of $H$ is purely absolutely continuous in $J$.

###### Proof.

Let $f\in E(J)\overline{\mathop{}\\!\mathrm{ran}(T^{*})}$. Due to the inner
regularity of the spectral measure $E$, we can approximate $f$ by a sequence
$\\{f_{j}\\}_{j\in\mathbb{N}}$ such that, for every $j\in\mathbb{N}$,
$f_{j}\in E(K_{j})\mathop{}\\!\mathrm{ran}(T^{*})$ for a compact subset
$K_{j}\subset J$. If, for every $j\in\mathbb{N}$, $f_{j}\in
E(J)\mathcal{H}_{\mathrm{ac}}(H)$, then also $f\in
E(J)\mathcal{H}_{\mathrm{ac}}(H)$ because $E(J)\mathcal{H}_{\mathrm{ac}}(H)$
is a closed subspace of $\mathcal{H}$. We write $f_{j}=E(K_{j})T^{*}g_{j}$
with $g_{j}\in D(T^{*})$. Let $\mu_{f_{j}}$ be the spectral measure associated
with $f_{j}$ and $F_{j}$ its inverse Fourier transform:

$\displaystyle
F_{j}(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}p\cdot
x}\mathop{}\\!\mathrm{d}\mu_{f_{j}}(p)=\frac{1}{(2\pi)^{n}}\langle
E(K_{j})T^{*}g_{j},U(x)f_{j}\rangle.$ (2.2)

The function $F_{j}$ is square-integrable because
$|F_{j}(x)|\leq(2\pi)^{-n}\|g_{j}\|\|TU(x)E(K_{j})f_{j}\|$ and $T$ is locally
$H$-smooth on $J$ by assumption. Thus,
$\mathop{}\\!\mathrm{d}\mu_{f_{j}}(p)=\hat{F}_{j}(p)\mathop{}\\!\mathrm{d}p$
is an absolutely continuous measure.

To prove the second statement, we apply the decomposition
$\mathcal{H}=\ker(T)\oplus\overline{\mathop{}\\!\mathrm{ran}(T^{*})}$. If
$\ker(T)=\\{0\\}$, then
$E(J)\mathcal{H}=E(J)\overline{\mathop{}\\!\mathrm{ran}(T^{*})}\subset
E(J)\mathcal{H}_{\mathrm{ac}}(H)$. We conclude
$E(J)\mathcal{H}=E(J)\mathcal{H}_{\mathrm{ac}}(H)$ because
$\mathcal{H}_{\mathrm{ac}}(H)\subset\mathcal{H}$, that is, the joint spectrum
of $H$ is purely absolutely continuous in $J$. ∎

It is possible to determine the optimal constant $C_{K}$ for the bound (2.1),
which is useful to verify the $H$-smoothness of an operator $T$. The following
proposition is proved as in [Kr23, Proposition A.10].

######  Proposition 2.3.

If $T$ is locally $H$-smooth, then the optimal constant for the bound (2.1) is

$\displaystyle
C_{K}^{0}=2^{n}\sup_{\lambda\in\mathbb{R}^{n},\mu\in(0,1)^{n}}\mu_{1}\dots\mu_{n}\|TE(K)R(\lambda+\mathop{}\\!\mathrm{i}\mu)\|^{2},$
(2.3)

where

$\displaystyle
R(\lambda+\mathop{}\\!\mathrm{i}\mu)=\prod_{j=1}^{n}(H_{j}-\lambda_{j}-\mathop{}\\!\mathrm{i}\mu_{j})^{-1}.$
(2.4)

Moreover, a continuous operator $T:D(H)\to\mathcal{H}$ is locally $H$-smooth
on $J$ if $C_{K}^{0}<\infty$ for every compact subset $K\subset J$.

If $n=1$, we can formulate a necessary and sufficient condition for
$C_{K}^{0}<\infty$ in terms of a limiting absorption principle (see Section
3).

######  Proposition 2.4.

Let $n=1$. For every compact subset $K\subset\mathbb{R}$, $C_{K}^{0}<\infty$
if and only if

$\displaystyle\sup_{\lambda\in K,\mu\in(0,1)}\|T\Im
R(\lambda+\mathop{}\\!\mathrm{i}\mu)T^{*}\|<\infty.$ (2.5)

The proof of Proposition 2.4 is contained in the proof of [ABG96, Proposition
7.1.1]; see [Kr23, Theorem A.12] for a generalisation to $n\geq 2$. In the
next section, we apply Proposition 2.4 to obtain locally smooth operators.

##  3 Mourre’s conjugate operator method

Mourre’s conjugate operator method is a mathematical tool to analyse the
spectrum of a self-adjoint operator based on a commutator estimate. In this
section, we outline the framework and state the most important results of the
method, in particular the limiting absorption principle. At the end of this
section, we apply Mourre’s method to construct locally smooth operators.

We introduce the following regularity classes, which are relevant for defining
commutators.

######  Definition 3.1.

Let $A$ be a self-adjoint operator on $\mathcal{H}$ and let
$k\in\mathbb{N}\cup\\{\infty\\}$. We denote by $C^{k}(A)$ the space of all
self-adjoint operators $H$ such that
$t\mapsto\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tA}(H+\mathop{}\\!\mathrm{i})^{-1}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tA}$
is a $C^{k}$-map in the strong operator topology. We denote by
$C_{\mathrm{u}}^{k}(A)$ the subspace of operators $H$ for which the same map
is $C^{k}$ in norm.

We collect some properties of the class $C^{1}(A)$ from [ABG96, Theorem
6.2.10]. If $H\in C^{1}(A)$, then $D(A)\cap D(H)$ is dense in $D(H)$ (equipped
with the graph topology). Moreover, the sesquilinear form defined by the
commutator $HA-AH$ on $D(A)\cap D(H)$ extends to a continuous sesquilinear
form on $D(H)$. We denote the extended sesquilinear form by $[H,A]$. It is
possible to characterise the class $C^{1}(A)$ in terms of the commutator
$[H,A]$.

######  Proposition 3.2 ([ABG96, Theorem 6.2.10 (a)]).

A self-adjoint operator $H$ belongs to the class $C^{1}(A)$ if and only if
there exists $z\in\rho(H)$ such that $\\{f\in D(A)\mid(H-z)^{-1}f\in
D(A)\text{ and }(H-\overline{z})^{-1}f\in D(A)\\}$ is a core for $A$ and, for
all $f\in D(A)\cap D(H)$,

$\displaystyle|\langle Hf,Af\rangle-\langle Af,Hf\rangle|\leq
c(\|Hf\|^{2}+\|f\|^{2}).$ (3.1)

Having formalised the commutator of two self-adjoint unbounded operators, we
are able to define strictly positive commutator estimates.

######  Definition 3.3.

A self-adjoint operator $H$ obeys a Mourre estimate on an open and bounded
subset $J\subset\mathbb{R}$ if a self-adjoint operator $A$ (conjugate
operator) exists such that $H\in C^{1}(A)$ and, for an $a>0$,

$\displaystyle E(J)[H,\mathop{}\\!\mathrm{i}A]E(J)\geq aE(J),$ (3.2)

where $E$ is the spectral measure of $H$.

######  Example.

Let $H$ be $A$-homogeneous, that is, for every $t,x\in\mathbb{R}$,

$\displaystyle\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tA}\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}xH}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tA}=\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}\mathop{}\\!\mathrm{e}^{-t}xH}.$
(3.3)

It is not difficult to show that (3.3) implies $H\in C^{\infty}(A)$ and
$[H,\mathop{}\\!\mathrm{i}A]=H$. Moreover, for every open and bounded interval
$J=(a,b)\subset(0,\infty)$, $0<a<b<\infty$,

$\displaystyle E(J)[H,\mathop{}\\!\mathrm{i}A]E(J)=HE(J)\geq aE(J),$ (3.4)

that is, $H$ obeys a Mourre estimate on $J$ with conjugate operator $A$.
Similarly, for every open and bounded interval $I=(a,b)\subset(-\infty,0)$,
$-\infty<a<b<0$,

$\displaystyle E(I)[H,-\mathop{}\\!\mathrm{i}A]E(I)=-HE(I)\geq-bE(I),$ (3.5)

that is, $H$ obeys a Mourre estimate on $I$ with conjugate operator $-A$.

A consequence of the Mourre estimate on $J$ is that $H$ has no eigenvalues in
$J$. This fact emerges as a direct corollary of the virial theorem [ABG96,
Proposition 7.2.10].

######  Theorem 3.4 (Virial theorem).

Let $H\in C^{1}(A)$. If $f$ is an eigenvector of $H$, then $\langle
f,[H,A]f\rangle=0$.

Next, we state the limiting absorption principle. To formulate the limiting
absorption principle under optimal conditions, we introduce new regularity
classes. Let $p\in[1,\infty]$ and $k\in\mathbb{N}$. For $0<s<k$, we denote by
$\mathcal{C}^{s,p}(A)$ the real interpolation space [ABG96, (5.2.6)]

$\displaystyle\mathcal{C}^{s,p}(A)=(C^{k}_{\mathrm{u}}(A),C^{0}_{\mathrm{u}}(A))_{\theta,p},\
\theta=1-s/k.$ (3.6)

Moreover, we write $H\in C^{k+0}(A)$ [ABG96, p. 204] if $H\in C^{k}(A)$ and if
the operator-valued function
$S(t)=\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tA}\mathrm{ad}_{A}^{k}((H+\mathop{}\\!\mathrm{i})^{-1})\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tA}$
is Dini continuous in norm, that is,

$\displaystyle\int_{|t|<1}\|S(t)-S(0)\|\frac{\mathop{}\\!\mathrm{d}t}{t}<\infty.$
(3.7)

We mention the following chain of inclusions [ABG96, (5.2.14)]:

$\displaystyle C^{k+1}(A)\subset C^{k+0}(A)\subset\mathcal{C}^{k,1}(A)\subset
C^{k}_{\mathrm{u}}(A),\ k\in\mathbb{N}.$ (3.8)

Note that $C^{k+1}(A)\subset C^{k+0}(A)$ as a consequence of the mean value
theorem and the uniform boundedness principle.

We say that $H$ has a spectral gap if $\sigma(H)\neq\mathbb{R}$ and we write
$\langle A\rangle$ for $\sqrt{A^{2}+1}$.

######  Theorem 3.5 (Limiting absorption principle).

Let $H\in\mathcal{C}^{1,1}(A)$ if $H$ has a spectral gap and $H\in C^{1+0}(A)$
if $\sigma(H)=\mathbb{R}$. If $H$ obeys a Mourre estimate on $J$, then, for
every compact subset $K\subset J$ and every $\nu>1/2$,

$\displaystyle\sup_{\lambda\in K,\mu>0}\|\langle
A\rangle^{-\nu}(H-\lambda\mp\mathop{}\\!\mathrm{i}\mu)^{-1}\langle
A\rangle^{-\nu}\|<\infty.$ (3.9)

######  Remark.

The assumption $H\in\mathcal{C}^{1,1}(A)$ is optimal on the Besov scale in the
sense that counterexamples of the limiting absorption principle for operators
possessing less regularity exist [ABG96, Section 7.B]. A proof of the limiting
absorption principle under the assumption $H\in\mathcal{C}^{1,1}(A)$ has been
accomplished in the case that $H$ has a spectral gap [ABG96, Theorem 7.4.1].
Sahbani [Sa97] proved the limiting absorption principle under the more
restrictive assumption $H\in C^{1+0}(A)$. It is an open problem to prove the
limiting absorption principle for $H\in\mathcal{C}^{1,1}(A)$ if
$\sigma(H)=\mathbb{R}$.

According to Proposition 2.2, $H$ has no singular spectrum in $J$ if an
injective locally $H$-smooth operator on $J$ exists. If each $H_{i}$ obeys a
Mourre estimate with conjugate operator $A_{i}$ such that, for $i\neq j$,
$A_{i}$ commutes strongly with $H_{j}$, then, for every $\nu>1/2$, $\langle
A_{1}\rangle^{-\nu}\dots\langle A_{n}\rangle^{-\nu}$ is an injective operator
that is locally $H$-smooth. This is a corollary of the limiting absorption
principle and the results from Section 2.

######  Corollary 3.6.

Consider a commuting family of self-adjoint operators $H=(H_{1},\dots,H_{n})$
and a corresponding family $A=(A_{1},\dots,A_{n})$ of self-adjoint operators
such that, for every $i\in\\{1,\dots,n\\}$, $H_{i}\in\mathcal{C}^{1,1}(A_{i})$
if $H_{i}$ has a spectral gap and $H_{i}\in C^{1+0}(A_{i})$ if
$\sigma(H)=\mathbb{R}$. If $H_{i}$ obeys a Mourre estimate on $J_{i}$ with
conjugate operator $A_{i}$ and, for $i\neq j$, $A_{i}$ commutes strongly with
$H_{j}$, then the joint spectrum of $H$ is purely absolutely continuous in the
product region $J=J_{1}\times\dots\times J_{n}$.

###### Proof.

We demonstrate that the injective operator $T=\langle
A_{1}\rangle^{-\nu}\dots\langle A_{n}\rangle^{-\nu}$ is locally $H$-smooth on
$J$ for every $\nu>1/2$. If this claim is proved, the statement follows from
Proposition 2.2. We show that the optimal constant $C_{K}^{0}$ from
Proposition 2.3 is finite for every compact subset $K\subset J$. It suffices
to consider compact hyperrectangles $K=K_{1}\times\dots\times K_{n}$, where
$K_{i}\subset J_{i}$ are compact intervals because every compact subset
$K\subset J$ can be covered by finitely many compact hyperrectangles in $J$.
If $E_{i}$ denotes the spectral measure of $H_{i}$, then

$\displaystyle E(K)=E_{1}(K_{1})\dots E_{n}(K_{n}).$ (3.10)

By assumption, $A_{i}$ commutes strongly with $H_{j}$ if $i\neq j$; hence,

$\displaystyle C_{K}^{0}$
$\displaystyle=2^{n}\sup_{\lambda\in\mathbb{R}^{n},\mu\in(0,1)^{n}}\mu_{1}\dots\mu_{n}\|\langle
A_{1}\rangle^{-\nu}\dots\langle
A_{n}\rangle^{-\nu}E(K)R(\lambda+\mathop{}\\!\mathrm{i}\mu)\|^{2}$
$\displaystyle\leq\prod_{i=1}^{n}2\sup_{\lambda_{i}\in\mathbb{R},\mu_{i}\in(0,1)}\mu_{i}\|\langle
A_{i}\rangle^{-\nu}E_{i}(K_{i})R_{i}(\lambda_{i}+\mathop{}\\!\mathrm{i}\mu_{i})\|^{2}.$
(3.11)

Each factor on the r.h.s. is finite due to Proposition 2.4 and the limiting
absorption principle (Theorem 3.5). ∎

##  4 Spectral analysis of relativistic energy-momentum operators

We analyse the spectra associated with the generators of spacetime
translations, employing the methods presented above. In the first subsection,
we analyse unitary representations of the Poincaré group. In the second
subsection, we explore dilation-covariant representations of the translation
group.

###  4.1 Representations of the Poincaré group

The Poincaré group is the semi-direct product
$\mathcal{P}=\mathcal{L}\ltimes\mathbb{R}^{d}$ of the Lorentz group
$\mathcal{L}=O(d-1,1)$ and the translation group $\mathbb{R}^{d}$. Its
multiplication law is defined as follows:

$\displaystyle(\lambda_{1},a_{1})\cdot(\lambda_{2},a_{2})=(\lambda_{1}\lambda_{2},a_{1}+\lambda_{1}a_{2}),\
\lambda_{1},\lambda_{2}\in\mathcal{L},a_{1},a_{2}\in\mathbb{R}^{d}.$ (4.1)

Let $U:\mathcal{P}\to\mathfrak{B}(\mathcal{H})$ be a strongly continuous
unitary representation of the Poincaré group on a Hilbert space $\mathcal{H}$.
The generators of the translation subgroup $U|_{\mathbb{R}^{d}}$ are the
energy-momentum operators $P=(P_{0},\mathbf{P})$. We denote by
$\Lambda_{1}(t),\dots,\Lambda_{s}(t)$ the Lorentz boosts in the space
directions $1,\dots,s=d-1$, for example,

$\displaystyle\Lambda_{1}(t)=\begin{pmatrix}\cosh(t)&\sinh(t)&0&\cdots&0\\\
\sinh(t)&\cosh(t)&0&\cdots&0\\\ 0&0&1&&0\\\ \vdots&\vdots&&\ddots&\\\
0&0&0&&1\end{pmatrix},$ (4.2)

and by $K_{1},\dots,K_{s}$ the self-adjoint generators of the Lorentz boosts,
that is,
$U(\Lambda_{j}(t),0)=\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tK_{j}}$,
$j\in\\{1,\dots,s\\}$. We mention the following identities, which are
consequences of the multiplication law ($t,x\in\mathbb{R}$):

$\displaystyle\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tK_{j}}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}xP_{0}}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tK_{j}}$
$\displaystyle=\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}x(\cosh(t)P_{0}-\sinh(t)P_{j})},$
(4.3)
$\displaystyle\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tK_{j}}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}xP_{j}}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tK_{j}}$
$\displaystyle=\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}x(\cosh(t)P_{j}-\sinh(t)P_{0})}.$
(4.4)

Formally, by differentiating in $x=0$ and $t=0$, these identities are
equivalent to the commutation relations
$[P_{0},\mathop{}\\!\mathrm{i}K_{j}]=P_{j}$ and
$[P_{j},\mathop{}\\!\mathrm{i}K_{j}]=P_{0}$.

Utilising Wigner’s theorem, it has been proved in [Ma68, Satz 2] that the
spectra of the energy-momentum operators are absolutely continuous if
translation-invariant vectors are removed. Translation-invariant vectors are
vectors $f\in\mathcal{H}$ for which $U(1,x)f=f$ for all $x\in\mathbb{R}^{d}$.
We present a new proof employing Mourre’s conjugate operator method. Our
analysis crucially depends on the spectrum condition, which will be assumed
for the remainder of this section.

######  Assumption 4.1 (Spectrum condition).

The joint spectrum of the energy-momentum operators is a subset of the closed
forward light cone $V_{+}=\\{p=(p_{0},\mathbf{p})\in\mathbb{R}^{d}\mid
p_{0}\geq|\mathbf{p}|\\}$ (i.e. $\sigma(P)\subset V_{+}$).

The spectrum condition is a quantum field theory axiom in the frameworks of
Wightman and Haag–Kastler. From the spectrum condition, it follows that the
momentum operators are relatively bounded w.r.t. the energy operator $P_{0}$,
that is, $|\mathbf{P}|\leq P_{0}$.

$p_{0}$$\mathbf{p}$$S_{\lambda}$–$\lambda$ Figure 1: The set
$S_{\lambda}\subset V_{+}$ is invariant under Lorentz boosts.

We introduce, for $\lambda>0$, the following Lorentz-invariant sets (see
Figure 1):

$\displaystyle
S_{\lambda}=\\{\Lambda_{1}(t_{1})\dots\Lambda_{s}(t_{s})(p_{0},\mathbf{0})\mid
t_{1},\dots,t_{s}\in\mathbb{R},p_{0}\in[0,\lambda]\\}.$ (4.5)

Clearly, the sets $S_{\lambda}$ cover the forward light cone $V_{+}$ (i.e.
$\bigcup_{\lambda>0}S_{\lambda}=V_{+}$). Hence, by the spectrum condition, the
subspaces $E(S_{\lambda})\mathcal{H}$ cover the Hilbert space $\mathcal{H}$,
where $E$ is the joint spectral measure of the energy-momentum operators $P$.
Moreover,

$\displaystyle\mathcal{H}_{\mathrm{ac}}(\mathbf{P})=\bigcup_{\lambda>0}\mathcal{H}_{\mathrm{ac}}(\mathbf{P}_{\lambda}),\
\mathcal{H}_{\mathrm{ac}}(P_{0})=\bigcup_{\lambda>0}\mathcal{H}_{\mathrm{ac}}(P_{0,\lambda}),$
(4.6)

where $\mathbf{P}_{\lambda}=E(S_{\lambda})\mathbf{P}$ and
$P_{0,\lambda}=E(S_{\lambda})P_{0}$ are operators on
$E(S_{\lambda})\mathcal{H}$. The advantage of introducing the operators
$P_{0,\lambda}$ and $\mathbf{P}_{\lambda}$ is that $P_{0,\lambda}$ is
relatively bounded w.r.t. $P_{j,\lambda}=E(S_{\lambda})P_{j}$.

The following lemmas will be applied to prove absolute continuity of the
energy-momentum spectra.

######  Lemma 4.2.

For every $\lambda>0$ and $j\in\\{1,\dots,s\\}$, $P_{0,\lambda}\in
C^{\infty}(K_{j})$, $P_{j,\lambda}\in C^{\infty}(K_{j})$, and

$\displaystyle[P_{0,\lambda},\mathop{}\\!\mathrm{i}K_{j}]=P_{j,\lambda},$
(4.7)
$\displaystyle[P_{j,\lambda},\mathop{}\\!\mathrm{i}K_{j}]=P_{0,\lambda}.$
(4.8)

###### Proof.

The spectral projection $E(S_{\lambda})$ commutes with $K_{j}$ because
$S_{\lambda}$ is a Lorentz-invariant set. Thus, we can replace $P_{0}$ and
$P_{j}$ in (4.4) with $P_{0,\lambda}$ and $P_{j,\lambda}$:

$\displaystyle\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tK_{j}}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}xP_{j,\lambda}}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tK_{j}}$
$\displaystyle=\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}x(\cosh(t)P_{j,\lambda}-\sinh(t)P_{0,\lambda})}.$
(4.9)

From (4.9), we obtain the following identity:

$\displaystyle\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tK_{j}}(P_{j,\lambda}+\mathop{}\\!\mathrm{i})^{-1}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tK_{j}}$
$\displaystyle=(\cosh(t)P_{j,\lambda}-\sinh(t)P_{0,\lambda}+\mathop{}\\!\mathrm{i})^{-1}.$
(4.10)

The r.h.s. is a smooth function in $t$ in the strong operator topology because
$P_{0,\lambda}$ and $P_{j,\lambda}$ are relatively bounded w.r.t. each other;
hence, $P_{j,\lambda}\in C^{\infty}(K_{j})$. Moreover, by differentiating
(4.9) in $t=0$ and $x=0$, we obtain, for every $f,g\in D(P_{j,\lambda})\cap
D(K_{j})$,

$\displaystyle\langle P_{j,\lambda}f,K_{j}g\rangle-\langle
K_{j}f,P_{j,\lambda}g\rangle=-\mathop{}\\!\mathrm{i}\langle
f,P_{0,\lambda}g\rangle.$ (4.11)

Because $P_{j,\lambda}\in C^{1}(K_{j})$, the sesquilinear form (4.11) has a
unique extension to $D(P_{j,\lambda})=D(P_{0,\lambda})$, yielding the
commutator identity
$[P_{j,\lambda},\mathop{}\\!\mathrm{i}K_{j}]=P_{0,\lambda}$. The proof of
$P_{0,\lambda}\in C^{\infty}(K_{j})$ and
$[P_{0,\lambda},\mathop{}\\!\mathrm{i}K_{j}]=P_{j,\lambda}$ is analogous. ∎

######  Lemma 4.3.

Let $f\in\mathcal{H}$. If $P_{\mu}f=0$ for one $\mu\in\\{0,\dots,s\\}$, then
$P_{\mu}f=0$ for all $\mu\in\\{0,\dots,s\\}$, that is, if a vector is
translation-invariant in one spacetime direction, then it is translation-
invariant in all spacetime directions.

###### Proof.

If $P_{0}f=0$, then $P_{j}f=0$ for all $j\in\\{1,\dots,s\\}$ due to the
spectrum condition. If $P_{j}f=0$ for one $j\in\\{1,\dots,s\\}$, then also
$P_{j,\lambda}f=0$ for all $\lambda>0$. From the commutation relation
$[P_{j,\lambda},\mathop{}\\!\mathrm{i}K_{j}]=P_{0,\lambda}$ and the virial
theorem (Theorem 3.4), it follows that

$\displaystyle\|\sqrt{P_{0,\lambda}}f\|^{2}=\langle
f,[P_{j,\lambda},\mathop{}\\!\mathrm{i}K_{j}]f\rangle=0;$ (4.12)

hence, $P_{0,\lambda}f=0$ for all $\lambda>0$, that is, $P_{0}f=0$. ∎

We split the statements of Proposition 1.1 and Proposition 1.2 into three
separate propositions. First, we prove the absolute continuity of the joint
momentum spectrum, which implies also the absolute continuity of $e\cdot P$
for space-like vectors $e\in\mathbb{R}^{d}$. Subsequently, we prove the
absolute continuity of $e\cdot P$ for light-like and time-like vectors $e$,
respectively. In the latter cases, $e\cdot P$ is unitarily equivalent to a
scalar multiple of $P_{0}-P_{1}$, $P_{0}+P_{1}$, or $P_{0}$, depending on
whether $e$ is light-like or time-like. We denote by $Q_{0}$ the projection
onto the subspace of translation-invariant vectors.

######  Proposition 4.4.

Under the assumptions of Proposition 1.1,
$\mathcal{H}=Q_{0}\mathcal{H}\oplus\mathcal{H}_{\mathrm{ac}}(\mathbf{P})$.

###### Proof.

Let $a>0$ and let $J$ be an open and bounded subset of
$(-\infty,-a]\cup[a,\infty)$. From Lemma 4.2 and the spectrum condition, we
obtain the following Mourre estimate on the Hilbert space
$E(S_{\lambda})\mathcal{H}$:

$\displaystyle
E_{j,\lambda}(J)[P_{j,\lambda},\mathop{}\\!\mathrm{i}K_{j}]E_{j,\lambda}(J)=P_{0,\lambda}E_{j,\lambda}(J)\geq
aE_{j,\lambda}(J),$ (4.13)

where $E_{j,\lambda}$ denotes the spectral measure of $P_{j,\lambda}$. Because
$a>0$ can be arbitrarily small, it follows from Corollary 3.6 that $E(\\{p\in
S_{\lambda}\mid\forall j\in\\{1,\dots,s\\}\colon p_{j}\neq
0\\})\mathcal{H}\subset\mathcal{H}_{\mathrm{ac}}(\mathbf{P}_{\lambda})$. To
apply Corollary 3.6, we used that $P_{j,\lambda}\in C^{\infty}(K_{j})\subset
C^{1+0}(K_{j})$ due to (3.8). By taking the union over $\lambda>0$, we obtain
$E(\\{p\in V_{+}\mid\forall j\in\\{1,\dots,s\\}\colon p_{j}\neq
0\\})\mathcal{H}\subset\mathcal{H}_{\mathrm{ac}}(\mathbf{P})$. By Lemma 4.3,
$E(\\{p\in V_{+}\mid\forall j\in\\{1,\dots,s\\}\colon p_{j}\neq
0\\})\mathcal{H}=E(V_{+}\backslash\\{0\\})\mathcal{H}$. The statement follows
from the decomposition $\mathcal{H}=Q_{0}\mathcal{H}\oplus
E(V_{+}\backslash\\{0\\})\mathcal{H}$. ∎

######  Proposition 4.5.

Under the assumptions of Proposition 1.1,
$\mathcal{H}=Q_{0}\mathcal{H}\oplus\mathcal{H}_{\mathrm{ac}}(P_{0}\pm P_{1})$
if $d\geq 3$ and $\mathcal{H}=E_{P_{0}\pm
P_{1}}(\\{0\\})\mathcal{H}\oplus\mathcal{H}_{\mathrm{ac}}(P_{0}\pm P_{1})$ if
$d=2$.

###### Proof.

From (4.3) and (4.4), we obtain the following identity:

$\displaystyle\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tK_{1}}\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}x(P_{0}\pm
P_{1})}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tK_{1}}=\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}\mathop{}\\!\mathrm{e}^{\mp
t}x(P_{0}\pm P_{1})}.$ (4.14)

In the terminology of the example subsequent to Definition 3.3, the operator
$P_{0}\pm P_{1}$ is $\pm K_{1}$-homogeneous. It follows that $P_{0}\pm P_{1}$
obeys a Mourre estimate on every open and bounded interval that is separated
from 0. Thus, by Corollary 3.6, $E(V_{+}\backslash\\{p_{0}\pm p_{1}\neq
0\\})\mathcal{H}\subset\mathcal{H}_{\mathrm{ac}}(P_{0}\pm P_{1})$. If $d\geq
3$, then $E(V_{+}\backslash\\{p_{0}\pm p_{1}\neq
0\\})\mathcal{H}=E(V_{+}\backslash\\{0\\})\mathcal{H}$. In fact, for
$f\in\mathcal{H}$, $P_{0}f=\pm P_{1}f$ implies $P_{2}f=0$ by the spectrum
condition. And, by Lemma 4.3, $P_{2}f=0$ implies $P_{0}f=P_{1}f=0$. ∎

######  Remark.

Assume that an eigenstate $f\in\mathcal{H}/Q_{0}\mathcal{H}$ with eigenvalue 0
of the mass operator $M=\sqrt{P_{0}^{2}-|\mathbf{P}|^{2}}$ exists. If $d=2$,
then $P_{0}f=|\mathbf{P}|f=|P_{1}|f$ for such a massless excitation. If
$f_{\pm}$ denotes the positive/negative momentum component of $f$, then
$(P_{0}\mp P_{1})f_{\pm}=0$. This illustrates that, in the case $d=2$, the
subspace $E_{P_{0}\pm P_{1}}(\\{0\\})\mathcal{H}$ may be larger than
$Q_{0}\mathcal{H}$.

######  Proposition 4.6.

Under the assumptions of Proposition 1.1,
$\mathcal{H}=Q_{0}\mathcal{H}\oplus\mathcal{H}_{\mathrm{ac}}(P_{0})$.

###### Proof.

Let $\lambda>0$, select $\theta\in C_{c}^{\infty}(\lambda,\infty)$, and set
$F(P_{\lambda})=\theta(P_{0,\lambda})/P_{1,\lambda}$. The operator
$F(P_{\lambda})$ is well-defined because if $p\in S_{\lambda}$ with
$p_{0}\in\mathrm{supp}(\theta)$, then $p_{1}$ is separated from 0. We define
the following operator on the domain $D(A_{\lambda})=D(K_{1})$:

$\displaystyle
A_{\lambda}=\frac{1}{2}(F(P_{\lambda})K_{1}+K_{1}F(P_{\lambda})).$ (4.15)

By Lemma 4.7 below, the operator $A_{\lambda}$ is essentially self-adjoint,
$P_{0,\lambda}\in C^{\infty}(\overline{A_{\lambda}})$, and, for $0<a<b$,

$\displaystyle
E_{0,\lambda}(\lambda+(a,b))[P_{0,\lambda},\mathop{}\\!\mathrm{i}\overline{A_{\lambda}}]E_{0,\lambda}(\lambda+(a,b))=\theta(P_{0,\lambda})E_{0,\lambda}(\lambda+(a,b)),$
(4.16)

where $E_{0,\lambda}$ is the spectral measure of $P_{0,\lambda}$. If we select
$\theta\in C_{c}^{\infty}(\lambda,\infty)$ such that $\theta=1$ on
$\lambda+(a,b)$, then $P_{0,\lambda}$ obeys a Mourre estimate on
$\lambda+(a,b)$. From Corollary 3.6, it follows that $E(\\{p\in
S_{\lambda}\mid
p_{0}\in\lambda+(a,b)\\})\mathcal{H}\subset\mathcal{H}_{\mathrm{ac}}(P_{0,\lambda})$.
We take the union over $\lambda$, $a$, and $b$:

$\displaystyle
E(V_{+}\backslash\\{\mathbf{p}=0\\})\mathcal{H}=\bigcup_{\lambda>0,b>a>0}E(\\{p\in
S_{\lambda}\mid
p_{0}\in\lambda+(a,b)\\})\mathcal{H}\subset\mathcal{H}_{\mathrm{ac}}(P_{0}).$
(4.17)

By Lemma 4.3, the l.h.s. is equal to $E(V_{+}\backslash\\{0\\})\mathcal{H}$. ∎

######  Remark.

The choice of the conjugate operator $\overline{A_{\lambda}}$ is similar as in
[ABG96, Lemma 7.6.4].

######  Lemma 4.7.

Let $\lambda>0$, $\theta\in C_{c}^{\infty}(\lambda,\infty)$, and set
$F(P_{\lambda})=\theta(P_{0,\lambda})/P_{1,\lambda}$. The symmetric operator

$\displaystyle
A_{\lambda}=\frac{1}{2}(F(P_{\lambda})K_{1}+K_{1}F(P_{\lambda}))$ (4.18)

is essentially self-adjoint on $D(K_{1})$. If we denote its self-adjoint
closure by $\overline{A_{\lambda}}$, then $P_{0,\lambda}\in
C^{\infty}(\overline{A_{\lambda}})$ and

$\displaystyle[P_{0,\lambda},\mathop{}\\!\mathrm{i}\overline{A_{\lambda}}]=\theta(P_{0,\lambda}).$
(4.19)

###### Proof.

  1. (i)

If $G\in C_{c}^{\infty}(\mathbb{R}^{d},\mathbb{R})$ and $j\in\\{1,\dots,s\\}$,
then $G(P_{\lambda})\in C^{\infty}(K_{j})$ and the following commutator
identity holds:

$\displaystyle[G(P_{\lambda}),\mathop{}\\!\mathrm{i}K_{j}]$
$\displaystyle=-\partial_{t}\big{|}_{0}\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tK_{j}}G(P_{\lambda})\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tK_{j}}$
$\displaystyle=-\partial_{t}\big{|}_{0}G(\cosh(t)P_{0,\lambda}-\sinh(t)P_{j,\lambda},P_{1,\lambda},\dots,\cosh(t)P_{j,\lambda}-\sinh(t)P_{0,\lambda},\dots,P_{s,\lambda})$
$\displaystyle=\partial_{0}G(P_{\lambda})P_{j,\lambda}+\partial_{j}G(P_{\lambda})P_{0,\lambda}.$
(4.20)

Applying this identity to $F(P_{\lambda})$, we obtain the following
commutator:

$\displaystyle[F(P_{\lambda}),\mathop{}\\!\mathrm{i}K_{1}]=\theta^{\prime}(P_{0,\lambda})-\theta(P_{0,\lambda})\frac{P_{0,\lambda}}{P_{1,\lambda}^{2}}.$
(4.21)

The operator on the r.h.s. is bounded. It follows that $F(P_{\lambda})$ leaves
$D(K_{1})$ invariant. In particular, $A_{\lambda}$ is well-defined on
$D(K_{1})$.

  2. (ii)

We apply Nelson’s commutator theorem [RS75, Theorem X.36] to establish the
essential self-adjointness of $A_{\lambda}$ on $D(K_{1})$. Setting
$N=K_{1}^{2}+1$, we define the integer scale $\mathcal{H}_{k}$,
$k\in\mathbb{Z}$, corresponding to $N$ as the completion of $D(N^{k/2})$ with
respect to the norm $\|f\|_{k}=\|N^{k/2}f\|$. Clearly, $A_{\lambda}$ is a
symmetric bounded operator from $\mathcal{H}_{n}$ to $\mathcal{H}_{-n}$ for
every $n\geq 1$. Moreover, as an operator identity from $\mathcal{H}_{n+2}$ to
$\mathcal{H}_{-n-2}$,

$\displaystyle
NA_{\lambda}-A_{\lambda}N=2K_{1}[K_{1},F(P_{\lambda})]K_{1}+\frac{1}{2}[K_{1},[K_{1},[K_{1},F(P_{\lambda})]]].$
(4.22)

The triple commutator in the second summand is a bounded operator on
$\mathcal{H}$ according to (4.20). It follows that, for every
$f\in\mathcal{H}_{n+2}$,

$\displaystyle\|NA_{\lambda}f-A_{\lambda}Nf\|_{-n}\leq c\|f\|_{n};$ (4.23)

hence, the commutator $NA_{\lambda}-A_{\lambda}N$ extends to a bounded
operator from $\mathcal{H}_{n}$ to $\mathcal{H}_{-n}$. We conclude that
$A_{\lambda}$ is essentially self-adjoint on any core of $N$, particularly on
$D(K_{1}^{2})$. The closure of $A_{\lambda}$ restricted to $D(K_{1}^{2})$
coincides with the closure of $A_{\lambda}$ defined on $D(K_{1})$. Thus,
$A_{\lambda}$ is essentially self-adjoint on $D(K_{1})$.

  3. (iii)

If $\chi\in C_{c}^{\infty}(\mathbb{R})$ is a function such that $\chi=1$ on
$\mathop{}\\!\mathrm{supp}(\theta)$, then
$\overline{A_{\lambda}}=\chi(P_{0,\lambda})\overline{A_{\lambda}}$. In fact,
if $f\in D(K_{1})$, then, by (4.21),

$\displaystyle\overline{A_{\lambda}}f=A_{\lambda}f=F(P_{\lambda})K_{1}f+\frac{1}{2}\left(\theta^{\prime}(P_{0,\lambda})-\theta(P_{0,\lambda})\frac{P_{0,\lambda}}{P_{1,\lambda}^{2}}\right)f=\chi(P_{0,\lambda})A_{\lambda}f=\chi(P_{0,\lambda})\overline{A_{\lambda}}f.$
(4.24)

This identity extends to $f\in D(\overline{A_{\lambda}})$ by approximating $f$
with elements from $D(K_{1})$ in the graph topology of
$\overline{A_{\lambda}}$.

  4. (iv)

We prove $P_{0,\lambda}\in C^{1}(\overline{A_{\lambda}})$. Utilising
$[P_{0,\lambda},\mathop{}\\!\mathrm{i}K_{1}]=P_{1,\lambda}$ (see Lemma 4.2),
it is easy to verify that, for every $f\in D(K_{1})\cap D(P_{0,\lambda})$,

$\displaystyle|\langle
P_{0,\lambda}f,\overline{A_{\lambda}}f\rangle-\langle\overline{A_{\lambda}}f,P_{0,\lambda}f\rangle|=|\langle
f,\theta(P_{0,\lambda})f\rangle|\leq\|\theta(P_{0,\lambda})\|\|f\|^{2}.$
(4.25)

By approximating $f\in D(\overline{A_{\lambda}})$ in the graph topology of
$\overline{A_{\lambda}}$ with elements from $D(K_{1})$, it follows from the
previous step that (4.25) is valid for $f\in D(\overline{A_{\lambda}})$.
Moreover, $\\{f\in
D(\overline{A_{\lambda}})\mid(P_{0,\lambda}\pm\mathop{}\\!\mathrm{i})^{-1}f\in
D(\overline{A_{\lambda}})\\}$ contains the core $D(K_{1})$. In fact, if $f\in
D(K_{1})$, then

$\displaystyle\lim_{t\to
0}\frac{1}{\mathop{}\\!\mathrm{i}t}(\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tK_{1}}-1)(P_{0,\lambda}\pm\mathop{}\\!\mathrm{i})^{-1}f=-\mathop{}\\!\mathrm{i}P_{1,\lambda}(P_{0,\lambda}\pm\mathop{}\\!\mathrm{i})^{-2}f+(P_{0,\lambda}\pm\mathop{}\\!\mathrm{i})^{-1}K_{1}f\in\mathcal{H}.$
(4.26)

We conclude $P_{0,\lambda}\in C^{1}({\overline{A_{\lambda}}})$ by Proposition
3.2.

  5. (v)

From the previous step, it follows that the commutator
$[P_{0,\lambda},\mathop{}\\!\mathrm{i}\overline{A_{\lambda}}]=\theta(P_{0,\lambda})$
is a bounded operator. By similar arguments as in step (iv), we compute the
higher order commutators:

$\displaystyle\mathrm{ad}_{-\mathop{}\\!\mathrm{i}\overline{A_{\lambda}}}^{k}(P_{0,\lambda})=\theta_{k-1}(P_{0,\lambda}),\
\theta_{k}(x)=[\theta(x)\partial_{x}]^{k}\theta(x).$ (4.27)

All commutators are bounded operators; thus, $P_{0,\lambda}\in
C^{\infty}(\overline{A_{\lambda}})$. ∎

###  4.2 Dilation-covariant representations

Let $U:\mathbb{R}^{d}\to\mathfrak{B}(\mathcal{H})$ be a strongly continuous
unitary representation of the translation group, whose generators obey the
spectrum condition (Assumption 4.1). In this subsection, we assume that $U$ is
dilation-covariant, that is, a self-adjoint operator $D$ exists such that, for
every $\mu\in\\{0,\dots,s\\}$ and $t,x\in\mathbb{R}$,

$\displaystyle\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}tD}\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}xP_{\mu}}\mathop{}\\!\mathrm{e}^{-\mathop{}\\!\mathrm{i}tD}=\mathop{}\\!\mathrm{e}^{\mathop{}\\!\mathrm{i}\mathop{}\\!\mathrm{e}^{-t}xP_{\mu}}.$
(4.28)

We analyse the spectrum of the mass operator
$M=\sqrt{P_{0}^{2}-|\mathbf{P}|^{2}}$, which is a well-defined self-adjoint
operator due to the spectrum condition.

###### Proof of Proposition 1.3.

In the terminology of the example subsequent to Definition 3.3, $M$ is
$D$-homogeneous due to (4.28), that is, $M\in C^{\infty}(D)$ and

$\displaystyle[M,\mathop{}\\!\mathrm{i}D]=M.$ (4.29)

Thus, $M$ obeys a Mourre estimate with conjugate operator $D$ on every open
and bounded subset of $[a,\infty)$, $a>0$. Because $a>0$ is arbitrary, it
follows from Corollary 3.6 that the spectrum of $M$ is purely absolutely
continuous in $(0,\infty)$. ∎

### Acknowledgements

I thank Wojciech Dybalski for carefully proofreading the draft of this paper
and for assistance preparing the publication. The research leading to these
results received funding from National Science Centre, Poland, under the grant
‘Sonata Bis’ 2019/34/E/ST1/00053.

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#  $q$-Fibonacci bicomplex quaternions

F. Torunbalcı Aydın<EMAIL_ADDRESS>Yildiz Technical University, Faculty
of Chemical and Metallurgical Engineering, Department of Mathematical
Engineering, Davutpasa Campus, 34220, Esenler, İstanbul, TURKEY

###### Abstract

In the paper, we define the $q$-Fibonacci bicomplex quaternions and the
$q$-Lucas bicomplex quaternions, respectively. Then, we give some algebraic
properties of $q$-Fibonacci bicomplex quaternions and the $q$-Lucas bicomplex
quaternions.

###### keywords:

Bicomplex number; $q$-integer; Fibonacci number; Bicomplex Fibonacci
quaternion; $q$-quaternion; $q$-Fibonacci quaternion.

††journal: Chaos, Solitons $\&$ Fractals

##  1 Introduction

The real quaternions were first described by Irish mathematician William Rowan
Hamilton in 1843. The real quaternions constitute an extension of complex
numbers into a four-dimensional space and can be considered as four-
dimensional vectors, in the same way that complex numbers are considered as
two-dimensional vectors. Hamilton [1] introduced the set of quaternions which
can be represented as

$H=\left\\{\,q={{q}_{0}}+i\,{{q}_{1}}+j\,{{q}_{2}}+k\,{{q}_{3}}\,\left.{}\right|\
\ {{q}_{0}},\,{{q}_{1}},\,{{q}_{2}},\,{q}_{3}\in\mathbb{R}\,\right\\}$ (1)

where

${{i}^{2}}={{j}^{2}}={{k}^{2}}=-1\,,\ \ i\ j=-j\ i=k\,,\quad j\ k=-k\
j=i\,,\quad k\ i=-i\ k=j\,.$ (2)

Horadam [2, 3] defined complex Fibonacci and Lucas quaternions as follows

$Q_{n}=F_{n}+F_{n+1}\,i+F_{n+2}\,j+F_{n+3}\,k$ (3)

and

$K_{n}=L_{n}+L_{n+1}\,i+L_{n+2}\,j+L_{n+3}\,k$ (4)

where $F_{n}$ and $L_{n}$ denote the $n-th$ Fibonacci and Lucas numbers,
respectively. Also, the imaginary quaternion units $i\,j,\,\,k$ have the
following rules

$i^{2}=j^{2}=k^{2}=-1\,,\,\,i\,j=-j\,i=k\,,\quad j\,k=-k\,j=i\,,\quad
k\,i=-i\,k=j$

There are several studies on quaternions as the Fibonacci and Lucas
quaternions and their generalizations, for example,[4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14].
Bicomplex numbers were introduced by Corrado Segre in 1892 [15]. In 1991, G.
Baley Price, the bicomplex numbers are given in his book based on multicomplex
spaces and functions [16]. In recent years, fractal structures of these
numbers have also been studied [17, 18]. The set of bicomplex numbers can be
expressed by the basis $\\{1\,,i\,,j\,,i\,j\,\\}$ (Table 1) as,

$\displaystyle\mathbb{C}_{2}=\\{\,q=q_{1}+i\,q_{2}+j\,q_{3}+i\,j\,q_{4}\
|\,q_{1},q_{2},q_{3},q_{4}\in\mathbb{R}\\}$ (5)

where $i$,$j$ and $i\,j$ satisfy the conditions

$i^{2}=-1,\,\,\,j^{2}=-1,\,\,\,i\,j=j\,i.$ (6)

In 2018, the bicomplex Fibonacci quaternions defined by Aydın Torunbalcı [19]
as follows

$\displaystyle\mathbb{BF}_{n}$
$\displaystyle={F}_{n}+i\,{F}_{n+1}+j\,{F}_{n+2}+i\,j\,{F}_{n+3}$ (7)
$\displaystyle=({F}_{n}+i\,{F}_{n+1})+({F}_{n+2}+i\,{F}_{n+3})\,j$

where $i$, $j$ and $i\,j$ satisfy the conditions (6).
The theory of the quantum $q$-calculus has been extensively studied in many
branches of mathematics as well as in other areas in biology, physics,
electrochemistry, economics, probability theory, and statistics [20, 21]. For
$n\in\mathbb{N}_{0}$ , the $q$-integer ${[{n}]}_{q}$ is defined as follows

$\displaystyle{[{n}]}_{q}$
$\displaystyle=\frac{1-q^{n}}{1-q}=1+q+q^{2}+\,\ldots\,+q^{n-1}.$ (8)

By (8), for all $m,n\in{\mathbb{Z}}$, can be easily obtained
$[{m+n}]_{q}=[{m}]_{q}+q^{m}\,[{n}]_{q}$. For more details related to the
quantum $q$-calculus, we refer to [22, 23].
In 2019, $q$-Fibonacci hybrid and $q$-Lucas hybrid numbers defined by
Kızılateş [24] as follows

$\displaystyle\mathbb{HF}_{n}(\alpha;q)=$
$\displaystyle\alpha^{n-1}[{n}]_{q}+\alpha^{n1}[{n+1}]_{q}\,\mathbb{i}+\alpha^{n+1}[{n+2}]_{q}\,\,\boldsymbol{\varepsilon}$
(9) $\displaystyle+\alpha^{n+2}[{n+3}]_{q}\,\mathbb{h}\,$

and

$\displaystyle\mathbb{HL}_{n}(\alpha;q)=$
$\displaystyle\alpha^{n}\frac{{[{2\,n}]}_{q}}{{[{n}]}_{q}}+\alpha^{n+1}\,\frac{{[{2\,n+2}]}_{q}}{{[{n+1}]}_{q}}\,\mathbb{i}+\alpha^{n+2}\,\frac{{[{2\,n+4}]}_{q}}{{[{n+2}]}_{q}}\,\,\boldsymbol{\varepsilon}$
(10)
$\displaystyle+\alpha^{n+3}\frac{{[{2\,n+6}]}_{q}}{{[{n+3}]}_{q}}\,\mathbb{h}$

where $i$, $\varepsilon$ and $h$ satisfy the conditions

$\mathbb{i^{2}}=-1,\,\,\,\boldsymbol{\varepsilon^{2}}=0,\,\,\,\mathbb{h^{2}}=1,\,\,\mathbb{i}\,\mathbb{h}=\mathbb{h}\,\mathbb{i}=\boldsymbol{\varepsilon}+\mathbb{i}.$
(11)

Also, Kızılateş derived several interesting properties of these numbers such
as Binet-Like formulas, exponential generating functions, summation formulas,
Cassini-like identities, Catalan-like identities and d’Ocagne-like identities
[24].

##  2 $q$-Fibonacci bicomplex quaternions

In this section, we define $q$-Fibonacci bicomplex quaternions and $q$-Lucas
bicomplex quaternions by using the basis $\\{1,\,i\,,j\,,i\,j\\}$, where $i$,
$j$ and $i\,j$ satisfy the conditions (6) as follows

$\begin{array}[]{rl}\mathbb{BF}_{n}(\alpha;q)=&\alpha^{n-1}[{n}]_{q}+\alpha^{n}[{n+1}]_{q}\,\,i+\alpha^{n+1}[{n+2}]_{q}\,\,j+\alpha^{n+2}[{n+3}]_{q}\,i\,j\\\
\\\
=&\alpha^{n}\,(\frac{1-q^{n}}{\alpha-\alpha\,q})+\alpha^{n+1}\,(\frac{1-q^{n+1}}{\alpha-\alpha\,q})\,i\\\
\\\
&+\alpha^{n+2}\,(\frac{1-q^{n+2}}{\alpha-\alpha\,q})\,j+\alpha^{n+3}\,(\frac{1-q^{n+3}}{\alpha-\alpha\,q})\,\,i\,j\\\
\\\
=&\frac{\alpha^{n}}{\alpha-(\alpha\,q)}\,[\,1+\alpha\,i+\alpha^{2}\,j+\alpha^{3}\,i\,j\,]\\\
\\\
&-\frac{(\alpha\,q)^{n}}{\alpha-(\alpha\,q)}\,\,[\,1+(\alpha\,q)\,i+(\alpha\,q)^{2}\,j+(\alpha\,q)^{3}\,i\,j\,]\end{array}$
(12)

and

$\begin{array}[]{rl}\mathbb{BL}_{n}(\alpha;q)=&\alpha^{n}\frac{{[{2\,n}]}_{q}}{{[{n}]}_{q}}+\alpha^{n+1}\,\frac{{[{2\,n+2}]}_{q}}{{[{n+1}]}_{q}}\,i+\alpha^{n+2}\,\frac{{[{2\,n+4}]}_{q}}{{[{n+2}]}_{q}}\,j+\alpha^{n+3}\frac{{[{2\,n+6}]}_{q}}{{[{n+3}]}_{q}}\,i\,j\\\
\\\
=&\alpha^{2n}\,(\frac{1-q^{2n}}{\alpha^{n}-(\alpha\,q)^{n}})+\alpha^{2n+2}\,(\frac{1-q^{2n+2}}{\alpha^{n+1}-(\alpha\,q)^{n+1}})\,i\\\
\\\
&+\alpha^{2n+4}\,(\frac{1-q^{2n+4}}{\alpha^{n+2}-(\alpha\,q)^{n+2}})\,j+\alpha^{2n+6}\,(\frac{1-q^{2n+6}}{\alpha^{n+3}-(\alpha\,q)^{n+3}})\,i\,j\\\
\\\ =&{\alpha^{n}}\,(1+\alpha\,i+\alpha^{2}\,j+\alpha^{3}\,i\,j\,)\\\ \\\
&-(\alpha\,q)^{n}\,(1+(\alpha\,q)\,i+(\alpha\,q)^{2}\,j+(\alpha\,q)^{3}\,i\,j\,)\end{array}$
(13)

For $\alpha=\frac{1+\sqrt{5}}{2}$ and $(\alpha\,q)=\frac{-1}{\alpha}$,
q-Fibonacci bicomplex quaternion $\mathbb{BF}_{n}(\alpha;q)$ become the
bicomplex Fibonacci quaternions $\mathbb{BF}_{n}$.

The addition, substraction and multiplication by real scalars of two
$q$-Fibonacci bicomplex quaternions gives $q$-Fibonacci bicomplex quaternion.
Then, the addition, subtraction and multiplication by scalar of $q$-Fibonacci
bicomplex quaternions are defined by

$\begin{array}[]{rl}\mathbb{BF}_{n}(\alpha;q)\pm\mathbb{BF}_{m}(\alpha;q)=&(\alpha^{n-1}[{n}]_{q}+\alpha^{n}[{n+1}]_{q}\,i+\alpha^{n+1}[{n+2}]_{q}\,j\\\
&+\alpha^{n+2}[{n+3}]_{q}\,i\,j)\\\
&\pm(\alpha^{m-1}[{m}]_{q}+\alpha^{m}[{m+1}]_{q}\,i+\alpha^{m+1}[{m+2}]_{q}\,j\\\
&+\alpha^{m+2}[{m+3}]_{q}\,i\,j)\\\
=&[\alpha^{n}(\frac{1-q^{n}}{\alpha-\alpha\,q})\pm\alpha^{m}(\frac{1-q^{m}}{\alpha-\alpha\,q})]\\\
&+[\alpha^{n+1}(\frac{1-q^{n+1}}{\alpha-\alpha\,q})\pm\alpha^{m+1}(\frac{1-q^{m+1}}{\alpha-\alpha\,q})]\,i\\\
&+[\alpha^{n+2}(\frac{1-q^{n+2}}{\alpha-\alpha\,q})\pm\alpha^{m+2}(\frac{1-q^{m+2}}{\alpha-\alpha\,q})]\,j\\\
&+[\alpha^{n+3}(\frac{1-q^{n+3}}{\alpha-\alpha\,q})\pm\alpha^{m+3}(\frac{1-q^{m+3}}{\alpha-\alpha\,q})]\,i\,j\\\
=&\frac{1}{\alpha-\alpha\,q}\,\\{\,(\alpha^{n}\pm\alpha^{m})(1+\alpha\,i+\alpha^{2}\,j+\alpha^{3}\,i\,j)\\\
&+((\alpha\,q)^{n}\pm(\alpha\,q)^{m}\,)(1+(\alpha\,q)\,i+(\alpha\,q)^{2}\,j\\\
&+(\alpha\,q)^{3}\,i\,j\,)\,\\}.\end{array}$ (14)

The multiplication of $q$-Fibonacci bicomplex quaternion by the real scalar
$\lambda$ is defined as

$\begin{array}[]{rl}{\lambda}\,\mathbb{BF}_{n}(\alpha;q)&=\lambda\,\alpha^{n}(1+\alpha\,i+\alpha^{2}\,j+\alpha^{3}\,i\,j\,)\\\
&+\lambda\,(\alpha\,q)^{n}\,(\,1+(\alpha\,q)\,i+(\alpha\,q)^{2}\,j+(\alpha\,q)^{3}\,i\,j\,).\\\
\end{array}$ (15)

The scalar and the vector part of $\mathbb{BF}_{n}(\alpha;q)$ which is the
$n-th$ term of the $q$-Fibonacci bicomplex quaternion are denoted by

${S}_{\mathbb{BF}_{n}(\alpha;q)}=\alpha^{n-1}[{n}]_{q},\,\,{V}_{\mathbb{BF}_{n}(\alpha;q)}=\alpha^{n}[{n+1}]_{q}\,i+\alpha^{n+1}[{n+2}]_{q}\,j+\alpha^{n+2}[{n+3}]_{q}\,i\,j.$
(16)

Thus, the $q$-Fibonacci bicomplex quaternion $\mathbb{BF}_{n}(\alpha;q)$ is
given by

$\mathbb{BF}_{n}(\alpha;q)={S}_{\mathbb{BF}_{n}(\alpha;q)}+{V}_{\mathbb{BF}_{n}(\alpha;q)}$

.
The multiplication of two $q$-Fibonacci bicomplex quaternions is defined by

$\begin{array}[]{rl}\mathbb{BF}_{n}(\alpha;q)\times\,\mathbb{BF}_{m}(\alpha;q)=&(\alpha^{n-1}[{n}]_{q}+\alpha^{n}[{n+1}]_{q}\,i+\alpha^{n+1}[{n+2}]_{q}\,j\\\
&+\alpha^{n+2}[{n+3}]_{q}\,i\,j)\\\
&\times\,(\alpha^{m-1}[{m}]_{q}+\alpha^{m}[{m+1}]_{q}\,i+\alpha^{m+1}[{m+2}]_{q}\,j\\\
&+\alpha^{m+2}[{m+3}]_{q}\,i\,j)\\\ \\\
=&\frac{1}{\alpha-\alpha\,q}\,(\alpha^{n+m})\,\\{\,(1-\alpha^{2}-\alpha^{4}+\alpha^{6})\\\
&+2\,i\,(\alpha-\alpha^{5})+2\,j\,(\alpha^{2}-\alpha^{4})+4\,i\,j\,(\alpha^{3})\\}\\\
&-q^{m}\,\\{(1-\alpha(\alpha\,q)-\alpha^{2}(\alpha\,q)^{2}+\alpha^{3}(\alpha\,q)^{3})\\\
&+i\,(\alpha+(\alpha\,q)-\alpha^{2}(\alpha\,q)^{3}-\alpha^{3}(\alpha\,q)^{2})\\\
&+j\,(\alpha^{2}+(\alpha\,q)^{2}-\alpha(\alpha\,q)^{3}-\alpha^{3}(\alpha\,q))\\\
&+i\,j\,(\alpha^{3}+(\alpha\,q)^{3}-\alpha(\alpha\,q)^{2}-\alpha^{2}(\alpha\,q))\\}\\\
&-q^{n}\,\\{(1-\alpha(\alpha\,q)-\alpha^{2}(\alpha\,q)^{2}+\alpha^{3}(\alpha\,q)^{3})\\\
&+i\,(\alpha+(\alpha\,q)-\alpha^{2}(\alpha\,q)^{3}-\alpha^{3}(\alpha\,q)^{2})\\\
&+j\,(\alpha^{2}+(\alpha\,q)^{2}-\alpha(\alpha\,q)^{3}-\alpha^{3}(\alpha\,q))\\\
&+i\,j\,(\alpha^{3}+(\alpha\,q)^{3}+\alpha^{2}(\alpha\,q)+\alpha(\alpha\,q)^{2})\\}\\\
&+q^{n+m}\,\\{(1-(\alpha\,q)^{2}-(\alpha\,q)^{4}+(\alpha\,q)^{6})\\\
&+2\,i\,((\alpha\,q)-(\alpha\,q)^{5})+2\,j\,((\alpha\,q)^{2}-(\alpha\,q)^{4})\\\
&+4\,i\,j\,((\alpha\,q)^{3}\,)\\}\\\
=&\mathbb{BF}_{m}(\alpha;q)\times\,\mathbb{BF}_{n}(\alpha;q)\end{array}$ (17)

Here, quaternion multiplication is done using bicomplex quaternionic units
(table 1), and this product is commutative.

Table 1: Multiplication scheme of bicomplex units $x$ | $1$ | $i$ | $j$ | $i\,j$
---|---|---|---|---
$1$ | $1$ | $i$ | $j$ | $i\,j$
$i$ | $i$ | $-1$ | $i\,j$ | $-j$
$j$ | $j$ | $i\,j$ | $-1$ | $-i$
$i\,j$ | $i\,j$ | $-j$ | $-i$ | $1$

Also, the $q$-Fibonacci bicomplex quaternion product may be obtained as
follows

$\begin{array}[]{lr}\mathbb{BF}_{n}(\alpha;q)\times\,\mathbb{BF}_{m}(\alpha;q)=\\\
\\\
\scriptsize{\begin{matrix}&\left(\begin{array}[]{cccc}\alpha^{n-1}[{n}]_{q}&-\alpha^{n}[{n+1}]_{q}&-\alpha^{n+1}[{n+2}]_{q}&\alpha^{n+2}[{n+3}]_{q}\\\
\alpha^{n}[{n+1}]_{q}&-\alpha^{n-1}[{n}]_{q}&-\alpha^{n+2}[{n+3}]_{q}&-\alpha^{n+1}[{n+2}]_{q}\\\
\alpha^{n+1}[{n+2}]_{q}&-\alpha^{n+2}[{n+3}]_{q}&\alpha^{n-1}[{n}]_{q}&-\alpha^{n}[{n+1}]_{q}\\\
\alpha^{n+2}[{n+3}]_{q}&\alpha^{n+1}[{n+2}]_{q}&\alpha^{n}[{n+1}]_{q}&\alpha^{n-1}[{n}]_{q}\end{array}\right)\end{matrix}}.\scriptsize{\begin{matrix}\left(\begin{array}[]{c}\alpha^{m-1}[{m}]_{q}\\\
\alpha^{m}[{m+1}]_{q}\\\ \alpha^{m+1}[{m+2}]_{q}\\\
\alpha^{m+2}[{m+3}]_{q}\end{array}\right)\end{matrix}}\end{array}$ (18)

Three kinds of conjugation can be defined for bicomplex numbers [17, 18].
Therefore, conjugation of the $q$-Fibonacci bicomplex quaternion is defined in
three different ways as follows

$\displaystyle(\mathbb{BF}_{n}(\alpha;q))^{*_{1}}=$
$\displaystyle(\alpha^{n-1}[{n}]_{q}-i\,\alpha^{n}[{n+1}]_{q}+j\,\alpha^{n+1}[{n+2}]_{q}-i\,j\,\alpha^{n+2}[{n+3}]_{q}),$
(19) $\displaystyle(\mathbb{BF}_{n}(\alpha;q))^{*_{2}}=$
$\displaystyle(\alpha^{n-1}[{n}]_{q}+i\,\alpha^{n}[{n+1}]_{q}-j\,\alpha^{n+1}[{n+2}]_{q}-i\,j\,\alpha^{n+2}[{n+3}]_{q}),$
(20) $\displaystyle(\mathbb{BF}_{n}(\alpha;q))^{*_{3}}=$
$\displaystyle(\alpha^{n-1}[{n}]_{q}-i\,\alpha^{n}[{n+1}]_{q}-j\,\alpha^{n+1}[{n+2}]_{q}+i\,j\,\alpha^{n+2}[{n+3}]_{q}).$
(21)

Therefore, the norm of the $q$-Fibonacci bicomplex quaternion
${\,\mathbb{BF}_{n}(\alpha;q)}$ is defined in three different ways as follows

$\begin{array}[]{rl}{N}_{(}\mathbb{BF}_{n}(\alpha;q))^{*_{1}}=&\|(\mathbb{BF}_{n}(\alpha;q))\times\,(\mathbb{BF}_{n}(\alpha;q))^{*_{1}}\|^{2},\end{array}$
(22)
$\begin{array}[]{rl}{N}_{(}\mathbb{BF}_{n}(\alpha;q))^{*_{2}}=&\|(\mathbb{BF}_{n}(\alpha;q))\times\,(\mathbb{BF}_{n}(\alpha;q))^{*_{2}}\|^{2},\end{array}$
(23)
$\begin{array}[]{rl}{N}_{(}\mathbb{BF}_{n}(\alpha;q))^{*_{3}}=&\|(\mathbb{BF}_{n}(\alpha;q))\times\,(\mathbb{BF}_{n}(\alpha;q))^{*_{3}}\|^{2}.\end{array}$
(24)

###### Theorem 2.1

(Binet’s Formula). Let ${\mathbb{BF}_{n}(\alpha;q)}$ and
${\mathbb{BL}_{n}(\alpha;q)}$be the $q$-Fibonacci bicomplex quaternion and the
$q$-Lucas bicomplex quaternion. For $n\geq 1$, Binet’s formula for these
quaternions respectively, is as follows:

$\mathbb{BF}_{n}(\alpha;q)=\frac{\alpha^{n}\,\widehat{\gamma}-(\alpha\,q)^{n}\,\widehat{\delta}}{\alpha-\alpha\,q},$
(25)

and

$\mathbb{BL}_{n}(\alpha;q)={\alpha^{n}\,\widehat{\gamma}+(\alpha\,q)^{n}\,\widehat{\delta}}$
(26)

where

$\begin{array}[]{l}\widehat{\gamma}=1+{\alpha}\,i+{\alpha}^{2}\,j+{\alpha}^{3}\,i\,j,\,\,\,\,\,\alpha=\frac{1+\sqrt{5}}{2}\end{array}$

and

$\begin{array}[]{l}\widehat{\delta}=1+(\alpha\,q)\,i+(\alpha\,q)^{2}\,j+(\alpha\,q)^{3}\,i\,j,\,\,\,\,\,\alpha\,q=\frac{-1}{\alpha}.\end{array}$

###### Proof 1

(25): Using (8) and (12), we find that

$\begin{array}[]{rl}\mathbb{BF}_{n}(\alpha;q)=&\alpha^{n-1}[{n}]_{q}+\alpha^{n}[{n+1}]_{q}\,i+\alpha^{n+1}[{n+2}]_{q}\,j+\alpha^{n+2}[{n+3}]_{q}\,i\,j\\\
\\\
=&\alpha^{n}\,\frac{1-q^{n}}{\alpha-\alpha\,q}+\alpha^{n+1}\,\frac{1-q^{n+1}}{\alpha-\alpha\,q}\,i+\alpha^{n+2}\,\frac{1-q^{n+2}}{\alpha-\alpha\,q}\,j+\alpha^{n+3}\,\frac{1-q^{n+3}}{\alpha-\alpha\,q}\,i\,j\\\
\\\
=&\frac{\alpha^{n}\,[\,1+\alpha\,i+\alpha^{2}\,j+\alpha^{3}\,i\,j\,]-(\alpha\,q)^{n}\,[\,1+(\alpha\,q)\,i+(\alpha\,q)^{2}\,j+(\alpha\,q)^{3}\,i\,j\,]}{\alpha-(\alpha\,q)}\\\
\\\
=&\frac{\alpha^{n}\,\widehat{\gamma}-(\alpha\,q)^{n}\,\widehat{\delta}}{\alpha-\alpha\,q}\end{array}$

In a similar way, equality (26) can be derived as follows

$\begin{array}[]{rl}\mathbb{BL}_{n}(\alpha;q)=&\alpha^{n}\frac{{[{2\,n}]}_{q}}{{[{n}]}_{q}}+\alpha^{n+1}\,\frac{{[{2\,n+2}]}_{q}}{{[{n+1}]}_{q}}\,i+\alpha^{n+2}\,\frac{{[{2\,n+4}]}_{q}}{{[{n+2}]}_{q}}\,j+\alpha^{n+3}\frac{{[{2\,n+6}]}_{q}}{{[{n+3}]}_{q}}\,i\,j\\\
=&\alpha^{2n}\,(\frac{1-q^{2n}}{\alpha^{n}-(\alpha\,q)^{n}})+\alpha^{2n+2}\,(\frac{1-q^{2n+2}}{\alpha^{n+1}-(\alpha\,q)^{n+1}})\,i\\\
\\\
&+\alpha^{2n+4}\,(\frac{1-q^{2n+4}}{\alpha^{n+2}-(\alpha\,q)^{n+2}})\,j+\alpha^{2n+6}\,(\frac{1-q^{2n+6}}{\alpha^{n+3}-(\alpha\,q)^{n+3}})\,i\,j\\\
=&\frac{\alpha^{2n}}{\alpha^{n}}\,(1+\alpha\,i+\alpha^{2}\,j+\alpha^{3}\,i\,j\,)\\\
\\\
&-\frac{(\alpha\,q)^{2n}}{(\alpha\,q)^{n}}\,(1+(\alpha\,q)\,i+(\alpha\,q)^{2}\,j+(\alpha\,q)^{3}\,i\,j\,)\\\
\\\
=&\alpha^{n}\,\widehat{\gamma}-(\alpha\,q)^{n}\,\widehat{\delta}.\end{array}$

where $\widehat{\gamma}=1+{\alpha}\,i+{\alpha}^{2}\,j+{\alpha}^{3}\,i\,j$,
$\widehat{\delta}=1+(\alpha\,q)\,i+(\alpha\,q)^{2}\,j+(\alpha\,q)^{3}\,i\,j$
and $\widehat{\gamma}\,\widehat{\delta}=\widehat{\delta}\,\widehat{\gamma}$.

###### Theorem 2.2

(Exponential generating function)
Let $\mathbb{BF}_{n}(\alpha;q)$ be the $q$-Fibonacci bicomplex quaternion. For
the exponential generating function for these quaternions is as follows:

$\displaystyle g_{\mathbb{BF}_{n}(\alpha;q)}\,(\frac{t^{n}}{n!})=$
$\displaystyle\sum\limits_{n=0}^{\infty}\,{\mathbb{BF}_{n}(\alpha;q)}\,\frac{t^{n}}{n!}=\frac{\widehat{\gamma}\,e^{\alpha\,t}\,-\widehat{\delta}\,e^{(\alpha\,q)\,t}}{\alpha-\alpha\,q}\,$
(27)

###### Proof 2

Using the definition of exponential generating function, we obtain

$\begin{array}[]{rl}\sum\limits_{n=0}^{\infty}\,{\mathbb{BF}_{n}(\alpha;q)}\,\frac{t^{n}}{n!}&=\sum\limits_{n=0}^{\infty}\,(\frac{\alpha^{n}\,\widehat{\gamma}-(\alpha\,q)^{n}\,\widehat{\delta}}{\alpha-\alpha\,q})\,\frac{t^{n}}{n!}\\\
&=\frac{\widehat{\gamma}}{\alpha-\alpha\,q}\,\sum\limits_{n=0}^{\infty}\,\frac{t^{n}}{n!}-\frac{\widehat{\delta}}{\alpha-\alpha\,q}\,\sum\limits_{n=0}^{\infty}\,\frac{(\alpha\,q\,t)^{n}}{n!}\\\
&=\frac{\widehat{\gamma}\,e^{\alpha\,t}\,-\widehat{\delta}\,e^{(\alpha\,q)\,t}}{\alpha-\alpha\,q}.\end{array}$
(28)

Thus, the proof is completed.

###### Theorem 2.3

(Honsberger identity)
For $n,m\geq 0$ the Honsberger identity for the $q$-Fibonacci bicomplex
quaternions ${\mathbb{BF}_{n}(\alpha;q)}$ and ${\mathbb{BF}_{m}(\alpha;q)}$ is
given by

$\begin{array}[]{lr}\mathbb{BF}_{n}(\alpha;q)\,\mathbb{BF}_{m}(\alpha;q)+\mathbb{BF}_{n+1}(\alpha;q)\,\mathbb{BF}_{m+1}(\alpha;q)\\\
\\\
=\frac{\alpha^{n+m}}{\alpha-\alpha\,q}\,\\{\,(1+\alpha^{2})\,\widehat{\gamma}-\widehat{\gamma}\,\delta\,(1+\alpha(\alpha\,q))\,(q^{n}+q^{m})+(1+(\alpha\,q)^{2}\,q^{n+m}\,\widehat{\delta^{2}}\,\\}.\end{array}$
(29)

###### Proof 3

(29): By using (12) and (25) we get,

$\begin{array}[]{lr}\mathbb{BF}_{n}(\alpha;q)\,\mathbb{BF}_{m}(\alpha;q)+\mathbb{BF}_{n+1}(\alpha;q)\,\mathbb{BF}_{m+1}(\alpha;q)\\\
\\\
=(\frac{\alpha^{n}\,\widehat{\gamma}-(\alpha\,q)^{n}\,\widehat{\delta}}{\alpha-\alpha\,q})\,(\frac{\alpha^{m}\,\widehat{\gamma}-(\alpha\,q)^{m}\,\widehat{\delta}}{\alpha-\alpha\,q})+(\frac{\alpha^{n+1}\,\widehat{\gamma}-(\alpha\,q)^{n+1}\,\widehat{\delta}}{\alpha-\alpha\,q})\,(\frac{\alpha^{m+1}\,\widehat{\gamma}-(\alpha\,q)^{m+1}\,\widehat{\delta}}{\alpha-\alpha\,q})\\\
\\\
=\frac{\alpha^{n+m}}{(\alpha-\alpha\,q)^{2}}\,\\{\,(\widehat{\gamma}-q^{n}\,\widehat{\delta})(\widehat{\gamma}-q^{m}\,\widehat{\delta})\\}+\frac{\alpha^{n+m+2}}{(\alpha-\alpha\,q)^{2}}\,\\{\,\widehat{\gamma}-q^{n+1}\,\widehat{\delta})(\widehat{\gamma}-q^{m+1}\,\widehat{\delta})\\}\\\
\\\
=\frac{\alpha^{n+m}}{(\alpha-\alpha\,q)^{2}}\,\\{\,(1+\alpha^{2})\,{\widehat{\gamma}}^{2}-\widehat{\gamma}\,\widehat{\delta}\,(1+\alpha(\alpha\,q))\,(q^{n}+q^{m})+(1+(\alpha\,q)^{2})\,q^{n+m}\,{\widehat{\delta}}^{2}\\}.\end{array}$

where $\widehat{\gamma}\,\widehat{\delta}=\widehat{\delta}\,\widehat{\gamma}$.

###### Theorem 2.4

(d’Ocagne’s identity)
For $n,m\geq 0$ the d’Ocagne’s identity for the $q$-Fibonacci bicomplex
quaternions $\mathbb{BF}_{n}(\alpha;q)$ and $\mathbb{BF}_{m}(\alpha;q)$ is
given by

$\begin{array}[]{lr}\mathbb{BF}_{m}(\alpha;q)\,\mathbb{BF}_{n+1}(\alpha;q)-\mathbb{BF}_{m+1}(\alpha;q)\,\mathbb{BF}_{n}(\alpha;q)=&\frac{\alpha^{n+m-1}(q^{n}-q^{m})\,\widehat{\gamma}\,\widehat{\delta}}{(1-q)}.\end{array}$
(30)

###### Proof 4

(30): By using (12) and (25) we get,

$\begin{array}[]{lr}\mathbb{BF}_{m}(\alpha;q)\,\mathbb{BF}_{n+1}(\alpha;q)-\mathbb{BF}_{m+1}(\alpha;q)\,\mathbb{BF}_{n}(\alpha;q)\\\
\\\
=(\frac{\alpha^{m}\,\widehat{\gamma}-(\alpha\,q)^{m}\,\widehat{\delta}}{\alpha-\alpha\,q})\,(\frac{\alpha^{n+1}\,\widehat{\gamma}-(\alpha\,q)^{n+1}\,\widehat{\delta}}{\alpha-\alpha\,q})-(\frac{\alpha^{m+1}\,\widehat{\gamma}-(\alpha\,q)^{m+1}\,\widehat{\delta}}{\alpha-\alpha\,q})\,(\frac{\alpha^{n}\,\widehat{\gamma}-(\alpha\,q)^{n}\,\widehat{\delta}}{\alpha-\alpha\,q})\\\
\\\
=\frac{\alpha^{n+m+1}}{(\alpha-\alpha\,q)^{2}}\,\\{(1-q)\,(q^{n}-q^{m})\,\widehat{\gamma}\,\widehat{\delta}\,\\}\\\
\\\
=\frac{\alpha^{n+m-1}(q^{n}-q^{m})\,\widehat{\gamma}\,\widehat{\delta}}{(1-q)}.\end{array}$

Here, $\widehat{\gamma}\,\widehat{\delta}=\widehat{\delta}\,\widehat{\gamma}$
is used.

###### Theorem 2.5

(Cassini Identity)
Let $\mathbb{BF}_{n}(\alpha;q)$ be the $q$-Fibonacci bicomplex quaternion. For
$n\geq 1$, Cassini’s identity for $\mathbb{BF}_{n}(\alpha;q)$ is as follows:

$\mathbb{BF}_{n+1}(\alpha;q)\,\mathbb{BF}_{n-1}(\alpha;q)-\mathbb{BF}_{n}(\alpha;q)^{2}=\frac{\alpha^{2n-2}\,q^{n}\,(1-q^{-1})\,\widehat{\gamma}\,\widehat{\delta}}{(1-q)}.$
(31)

###### Proof 5

(31): By using (12) and (25) we get

$\begin{array}[]{rl}\mathbb{BF}_{n+1}(\alpha;q)\,\mathbb{BF}_{n-1}(\alpha;q)-\mathbb{BF}_{n}(\alpha;q)^{2}=&(\frac{\alpha^{n+1}\,\widehat{\gamma}-(\alpha\,q)^{n+1}\,\widehat{\delta}}{\alpha-\alpha\,q})\,(\frac{\alpha^{n-1}\,\widehat{\gamma}-(\alpha\,q)^{n-1}\,\widehat{\delta}}{\alpha-\alpha\,q})\\\
&-(\frac{\alpha^{n}\,\widehat{\gamma}-(\alpha\,q)^{n}\,\widehat{\delta}}{(\alpha-\alpha\,q)})^{2}\\\
=&\frac{\alpha^{2n}\,q^{n}\,(1-q)(1-q^{-1})\,\widehat{\gamma}\,\widehat{\delta}}{(\alpha-\alpha\,q)^{2}}\\\
=&\frac{\alpha^{2n-2}\,q^{n}\,(1-q^{-1})\,\widehat{\gamma}\,\widehat{\delta}}{(1-q)}\,.\end{array}$

Here, $\widehat{\gamma}\,\widehat{\delta}=\widehat{\delta}\,\widehat{\gamma}$
is used.

###### Theorem 2.6

(Catalan’s Identity)
Let $\mathbb{BF}_{n}(\alpha;q)$ be the $q$-Fibonacci bicomplex quaternion. For
$n\geq 1$, Catalan’s identity for $\mathbb{BF}_{n}(\alpha;q)$ is as follows:

$\mathbb{BF}_{n+r}(\alpha;q)\,\mathbb{BF}_{n-r}(\alpha;q)-\mathbb{BF}_{n}(\alpha;q)^{2}=\frac{\alpha^{2n-2}\,q^{n}\,(1-q^{r})(1-q^{-r})\,\widehat{\gamma}\,\widehat{\delta}}{(1-q)^{2}}\,.$
(32)

###### Proof 6

(32): By using (12) and (25) we get

$\begin{array}[]{rl}\mathbb{BF}_{n+r}(\alpha;q)\,\mathbb{BF}_{n-r}(\alpha;q)-\mathbb{BF}_{n}(\alpha;q)^{2}=&(\frac{\alpha^{n+r}\,\widehat{\gamma}-(\alpha\,q)^{n+r}\,\widehat{\delta}}{\alpha-\alpha\,q})\,(\frac{\alpha^{n-r}\,\widehat{\gamma}-(\alpha\,q)^{n-r}\,\widehat{\delta}}{\alpha-\alpha\,q})\\\
&-(\frac{\alpha^{n}\,\widehat{\gamma}-(\alpha\,q)^{n}\,\widehat{\delta}}{(\alpha-\alpha\,q)^{2}}\\\
=&\frac{-\alpha^{2n}\,q^{n-r}\,\widehat{\gamma}\,\widehat{\delta}-\alpha^{2n}\,q^{n+r}\,\widehat{\gamma}\,\widehat{\delta}+2\,\alpha^{2n}\,q^{n}\,\widehat{\gamma}\,\widehat{\delta}}{(\alpha-\alpha\,q)^{2}}\\\
=&-\frac{\alpha^{2n}\,q^{n}\,\widehat{\gamma}\,\widehat{\delta}\,[\,(q^{-r}-1)+(q^{r}-1)\,]}{(\alpha-\alpha\,q)^{2}}\\\
=&\frac{\alpha^{2n-2}\,q^{n}\,\widehat{\gamma}\,\widehat{\delta}\,(1-q^{-r})(1-q^{r})}{(1-q)^{2}}\end{array}$

Here, $\widehat{\gamma}\,\widehat{\delta}=\widehat{\delta}\,\widehat{\gamma}$
is used.

##  3 Conclusion

In this paper, algebraic and analytic properties of the $q$-Fibonacci
bicomplex quaternions are investigated.

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