# The hidden Lorentz Covariance of Quantum Mechanics

Partha<EMAIL_ADDRESS>
and
Frederik G<EMAIL_ADDRESS>
Department of Physics,
University of Stellenbosch, Stellenbosch-7600, South Africa

###### Abstract

This paper introduces a systematic algorithm for deriving a new unitary
representation of the Lorentz algebra ($so(1,3)$) and an irreducible unitary
representation of the extended (anti) de-Sitter algebra ($so(2,4)$) on
$\mathcal{L}^{2}(\mathcal{R}^{3},\frac{1}{r})$. This representation is
equivalent to a representation on $\mathcal{L}^{2}(\mathcal{R}^{3})$, and the
corresponding similarity transformation is identified. An explicit
representation in terms of differential operators is given, and it is shown
that the inner product is Lorentz invariant. Ensuring Lorentz covariance
demands a modification of the Heisenberg algebra, recognized as a phase space
algebra at the interface of gravitational and quantum realms (IGQR), which we
consider subordinate to Lorentz covariance. It is also demonstrated that time
evolution can be cast in a manifestly covariant form. Each mass sector of the
Hilbert space carries a representation of the Lorentz algebra, and the (anti)
de-Sitter algebra on each mass sector contracts to the Poincare algebra in the
flat configuration and momentum space limits. Finally, we show that three-
dimensional fuzzy space also carries a unitary representation of these
algebras, algebraically equivalent to the
$\mathcal{L}^{2}(\mathcal{R}^{3},\frac{1}{r})$ representation but not
necessarily equivalent as representations. Several outstanding issues are
identified for future exploration.

##  1 Introduction

Quantum mechanics and general relativity (GR) stand as the twin pillars of our
contemporary understanding of the physical world. The classical theory of
gravity, rooted in deterministic solutions to the Einstein field equations,
employs a commutative formalism and ensures causal time evolution. It excels
in elucidating observations like gravitational waves on cosmic and galactic
scales. In contrast, quantum mechanics thrives in describing the quantum world
at atomic scales. It introduces non-determinism in quantum phenomena, operates
under a non-commutative formalism, and mandates instantaneous collapses during
quantum measurements [1]. However, at very short time and length scales of the
order of the Planck time and length, these two theories seem to be
incompatible, and most likely both are not the correct description of the
world at these scales. Instead, a new, as yet unknown, theory that unites both
of these in a consistent framework seems to be required. This new framework is
commonly referred to as quantum gravity. Clearly, quantum mechanics and GR
must be good approximations of quantum gravity at the appropriate length
scales.

Despite many attempts over the past century, quantum gravity is still alluding
us. Indeed, the absence of a quantum theory of gravity may be the reason why
we struggle to explain such puzzling phenomena as dark matter, dark energy,
and the cosmological constant problem [2, 3]. It may even be at the root of
two closely related fundamental problems in quantum mechanics, namely,
measurement and quantum-to-classical transition problems [4, 5, 6, 7]. At this
stage, there also does not seem to be a convergence of opinion. String theory,
our most promising candidate for quantum gravity [8], is faced with several
serious conceptual problems [9]. At the other extreme, there is even some
scepticism about whether gravity is truly quantum or not [12]. Indeed, it is
not even known whether a gravitational field may exist in a quantum
superposition of states [11]. Clearly, we need observational data to guide us
at this point, which may come from the recent and rapid improvement in
observational precision that may reveal phenomenological signals of quantum
gravity [12, 13].

Conceptually, the quantization of general relativity (GR) can be approached in
two main ways. One method involves quantizing the gravitational field itself,
represented by gravitons, metric fluctuations, or their extensions.
Alternatively, quantization can be applied to the underlying space-time
geometry. Since GR portrays gravity as the curvature of space-time, these
approaches are interconnected. Various attempts to address quantum gravity
highlight a consistent theme: the need for a fundamental reevaluation of
space-time at small length scales, often referred to as quantum space-time.
Despite the multitude of approaches, the quest for quantum space-time lacks
clear guiding principles. Nevertheless, a few indicative directions have
emerged in the pursuit of understanding the nature of quantum space-time. The
first is the realization, emphasised by Doplicher et al [14], that it is
operationally impossible to localize an event with arbitrary accuracy without
encountering a gravitational instability, which provides compelling arguments
for non-commutative space-time as a candidate for quantum space-time. The
second is Lorentz covariance. The Lorentz covariance of physical laws is a
principle deeply embedded in relativity and strongly supported by observation.
Sacrificing it may have unwanted observational consequences. From these two
perspectives, the ideal would be a non-commutative, Lorentz covariant space-
time, and quantum theory [15]. Further compelling arguments in favour of non-
commutative space-time as a candidate for quantum space-time arise from GR and
Born’s reciprocity argument [16]. In GR, it is commonly accepted that space is
curved, which implies non-commutative momentum. Thus, the notion of non-
commutative momentum is part and parcel of GR. The only reason why space is
treated as commutative in GR is because momentum space is not viewed as
curved. However, if one adopts Born’s reciprocity argument, which states that
there is a position-momentum duality of the form $\hat{x}\rightarrow\hat{p}$
and $\hat{p}\rightarrow-\hat{x}$ that leaves the Heisenberg commutation
relation invariant, one would also expect curvature in momentum space if there
is curvature in space, and thus also non-commutativity of coordinates. This
implies a profound connection between curvature and non-commutativity,
suggesting that they are two facets of a unified underlying reality. This
relationship between curved momentum space and non-commutative geometry was
recently explored in [17, 18, 19].

The realization that quantum space-time may be non-commutative drove several
developments in the recent past, most notably non-commutative geometry [20,
21] and non-commutative quantum field theory [22]. The significant motivation
behind these developments is the prospect of non-commutativity serving as a
distinctive feature of quantum gravity [23]. For a comprehensive overview,
refer to the recent review on non-commutative field theory and gauge theory
[24]. Recent studies have also demonstrated that a Moyal-like non-commutative
space-time geometry naturally emerges due to the perturbative aspects of
quantum gravity [25]. However, the most naive model of non-commutative space-
time using constant commutation relations breaks Lorentz covariance [26], and
a deformation of the Lorentz algebra is required to restore it [27, 28, 29].
An unwanted consequence is the mixing of short and long length scales,
commonly referred to as ultraviolet-infrared mixing [30].

On a historical note, the introduction of non-commutative space-time is a very
old concept that was introduced by Snyder[31] to regularize the divergence in
quantum field theory. This uses a non-commutative model of space-time that
does not violate Lorentz covariance. Following that, the Snyder model was
extended to account for a de Siiter background in [32]. This generalization
was motivated by the necessity of including a cosmological constant
($\Lambda$), among the bare parameters of a theory of quantum gravity. This
clearly indicates that the Heisenberg uncertainty relation should be modified
by quantum gravity corrections, which are very much reminiscent of the
Extended Generalized Uncertainty Principle (EGUP) [33]. However, the
commutation relations of the translation generators with the position
variables depend on the specific choice of coordinates on the hyperboloid of
the Snyder de-Sitter background. Usually, the most natural description of the
Snyder de Sitter phase space algebra can be expressed in terms of operators
acting on six-dimensional Minkowski space time [34].

Recent findings hint at a potential reconciliation between the concept of a 4D
flat Lorentzian space-time and the principles of quantum mechanics, especially
in the domain of harmonic oscillators [35]. This challenges the conventional
belief in space-time as a fundamental concept, aligning with the viewpoint
advocated by physicists like Pauli [36]. This raises a fundamental question:
Can quantum mechanics and the Lorentz covariance of physical laws coexist
harmoniously?

The aforementioned general observations lay the groundwork for the present
paper. As may have become evident, our main aim here is to present a
systematic approach to formulate a fully Lorentz covariant version of quantum
mechanics applicable to both commutative and non-commutative spaces and to
investigate the relationships between them. At first glance, this endeavor
might appear challenging. The traditional view of Lorentz covariance requires
space and time to be treated on the same footing, yet in quantum mechanics we
construct the Hilbert space at a fixed time slice, and subsequently the inner
product is not Lorentz invariant [37]. Furthermore, time and space are treated
on different footings in the Schrödinger equation [38]. These were indeed the
main issues faced in the first attempts to formulate relativistic quantum
mechanics on commutative space. This forced Dirac to replace the Schrödinger
equation with a covariant equation of motion, the famous Dirac equation. The
issues regarding the construction of the Hilbert space can be resolved in a
second quantized or quantum field theoretic setting.

Fortunately, it turns out that Lorentz covariance does not require the
treatment of space and time on the same footing. This has clearly not been
appreciated in the literature and is one of the take-away messages of this
paper. The trade-off to achieve this is precisely the necessity of the
deformation of the Heisenberg algebra (notably, it has been previously noted
that relativistic corrections to quantum mechanics, derived from the single-
particle sector of scalar quantum field theory, may lead to a deformation of
the Heisenberg algebra [39]), rather than the Lorentz algebra.

There are two routes to the construction of Lorentz-covariant quantum
mechanics. The first is to construct non-commutative quantum mechanics on
three-dimensional fuzzy space, as was done in [40, 41]. This construction
exhibits puzzling features of a Lorentz covariant theory [42], which suggests
the possibility that it may be a Lorentz covariant theory. A closer inspection
of this theory indeed reveals that the Hilbert space of this quantum theory
carries a representation of the Lorentz $so(1,3)$ and, indeed, the (anti) de
Sitter algebra $so(2,4)$. We briefly review this point of view in Section 6.
This approach is rather ad hoc, as it is based on a seemingly ad hoc choice of
quantum space-time. It will be much more gratifying if we can show directly
that this is indeed the quantum space-time that emerges from the construction
of a Lorentz covariant quantum theory. This is the second route and the one we
prefer to follow here. These two routes lead to completely equivalent
algebraic structures.

Many phenomenological consequences of non-commutative quantum mechanics in
three-dimensional fuzzy space have already been worked out [43, 44, 45, 46].
Even the consequences at large length scales, which indicate a modification of
Newtonian dynamics, have been explored [47]. These are also the most likely
observable phenomena, given the recent advancements in observational
astronomy. The algebraic equivalence between the current construction and non-
commutative quantum mechanics on three-dimensional fuzzy space suggests that
those conclusions may also hold here. However, a more careful analysis is
required here as the two representations are seemingly non-equivalent, but
this falls outside the scope of the current paper.

The ultimate aim of these endeavors is to find clear phenomenological signals
that can support or falsify the proposed structure of quantum space-time. As
the construction proposed here is very tight, suggesting that it may even be
unique, this may provide us with key insights into the potential structure of
a theory of quantum gravity.

The paper is organized as follows: In section 2, we construct a representation
of the Lorentz algebra on $\mathcal{L}^{2}(\mathcal{R}^{3},\frac{1}{r})$
through a Hopf fibration and show its equivalence to a representation on
$\mathcal{L}^{2}(\mathcal{R}^{3})$. In section 3, we show that the Schrödinger
equation can be cast in a covariant form. In section 4, we briefly review the
construction of non-commutative quantum mechanics on three-dimensional fuzzy
space and show its algebraic equivalence with the construction on
$\mathcal{L}^{2}(\mathcal{R}^{3})$ given in section 2. In section 5, we
summarize our results and draw conclusions. Some intermediate results are
collected in an appendix.

##  2 Representation of the Lorentz and (anti) de Sitter algebras on
$\mathcal{L}^{2}(\mathcal{R}^{3})$

The key to the construction of covariant quantum mechanics outlined in the
introduction, is the construction of a representation of the Lorentz and
(anti) de Sitter algebras on $\mathcal{L}^{2}(\mathcal{R}^{3})$, i.e. the
space of square integrable functions on $\mathcal{R}^{3}$. As will immediately
be noted, time does not appear in this representation, which is at the origin
of the claim that it is possible to have Lorentz covariance without treating
space and time on the same footing. The construction is done in a very
systematic way, exploiting a Hopf fibration [48].

###  2.1 Spinor representation of the Lorentz and (anti) de Sitter algebras

Consider a dimensionless complex doublet
$Z=(z_{1}~{}z_{2})^{T}\in\mathcal{C}^{2}$ which transforms as a spinor under
$SU(2)$ transformations and is normalised as follows:

$Z^{\dagger}Z=r.$ (1)

This clearly suggests that the spheres $\mathcal{S}^{3}$ have been foliated to
$\mathcal{R}^{4}$ with radii $\sqrt{r}$.

The embedding $\mathcal{S}^{2}$ with radius $r$ is specified by the
dimensionless Cartesian coordinates of $\mathcal{R}^{3}$ and given by

$x_{1}=r\sin\theta\cos\phi,\;x_{2}=r\sin\theta\sin\phi,\;x_{3}=r\cos\theta.$
(2)

Based on the Hopf fibration these Cartesian coordinates are given by the
$\mathcal{C}^{2}$ coordinates as

$x_{i}=Z^{\dagger}\sigma_{i}Z=(\sigma_{i})_{\alpha\beta}\bar{z}_{\alpha}z_{\beta};~{}~{}~{}~{}~{}~{}~{}~{}~{}i=1,2,3,~{}and~{}\alpha,\beta=1,2.$
(3)

with $\sum_{i}x^{2}_{i}=r^{2}$. This allows to parameterise the complex
doublets above as follows:

$\begin{pmatrix}z_{1}\\\
z_{2}\end{pmatrix}=\sqrt{r}\begin{pmatrix}\cos\frac{\theta}{2}e^{\frac{i(\gamma+\phi)}{2}}\\\
\sin\frac{\theta}{2}e^{\frac{i(\gamma-\phi)}{2}}\end{pmatrix},$ (4)

which is nothing but the $SU(2)$ coadjoint orbits in terms of local
coordinates and the coordinates $(\sqrt{r},\theta,\phi,\gamma)$ are spherical
coordinates on $\mathcal{R}^{4}$ with $r\in[0,\infty)$, $\theta\in[0,\pi]$,
$\phi\in[0,2\pi)$ and $\gamma\in[0,2\pi)$ [49].

Next we introduce the Hilbert space $\mathcal{L}^{2}(\mathcal{C}^{2})$ of
square integrable functions on ${\cal C}^{2}$ with inner product

$(\phi,\psi)=\int\frac{dz_{\alpha}d\bar{z}_{\alpha}}{\pi^{2}}~{}\phi^{*}(z_{\alpha},\bar{z}_{\alpha})\psi(z_{\alpha},\bar{z}_{\alpha})$
(5)

and the finite norm requirement $(\psi,\psi)<\infty$. On this space we can now
construct a pair of bosonic annihilation and creation operators [50] as
follows

$\hat{a}_{\alpha}=z_{\alpha}+\frac{1}{2}\partial_{\bar{z}_{\alpha}};~{}~{}~{}\hat{b}_{\alpha}=z_{\alpha}-\frac{1}{2}\partial_{\bar{z}_{\alpha}},$
(6)
$\hat{a}^{\dagger}_{\alpha}=\bar{z}_{\alpha}-\frac{1}{2}\partial_{z_{\alpha}};~{}~{}~{}\hat{b}^{\dagger}_{\alpha}=\bar{z}_{\alpha}+\frac{1}{2}\partial_{z_{\alpha}},$
(7)

that satisfy the commutation relations (note the unconventional sign)

$[\hat{a}_{\alpha},\hat{a}^{\dagger}_{\beta}]=\delta_{\alpha\beta},~{}~{}[\hat{b}_{\alpha},\hat{b}^{\dagger}_{\beta}]=-\delta_{\alpha\beta}~{}~{}~{}\alpha,\beta=1,2$
(8)

with all other commutators vanishing. Note that the creation and annihilation
operators are adjoints with respect to the inner product (5).

Now we can readily define a 4-component bosonic operator-valued quantum spinor
[51, 52] (transforming under the $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$
representation of $SU(2)\otimes SU(2))$ acting on
$\mathcal{L}^{2}(\mathcal{C}^{2})$ as

$\hat{B}=\begin{pmatrix}\hat{Z}^{+}\\\ \hat{Z}^{-}\end{pmatrix},$ (9)

where $\hat{Z}^{+}=(\hat{a}_{1}~{}~{}\hat{a}_{2})^{T}$ ,
$\hat{Z}^{-}=(\hat{b}_{1}~{}~{}\hat{b}_{2})^{T}$
,$\hat{Z}^{+\dagger}=(\hat{a}^{\dagger}_{1}~{}~{}\hat{a}^{\dagger}_{2})^{T}$
and
$\hat{Z}^{-\dagger}=(\hat{b}^{\dagger}_{1}~{}~{}\hat{b}^{\dagger}_{2})^{T}$are
two component spinors that satisfy the commutation relations

$[\hat{Z}^{\pm}_{\alpha},\hat{Z}^{\pm\dagger}_{\beta}]=\pm\delta_{\alpha\beta};~{}~{}\alpha,\beta=1,2,$
(10)

where $\alpha,\beta$ are the two component spinor indices (note that
$\hat{Z}^{+}$ and $\hat{Z}^{-\dagger}$ commute. The same applies visa versa).

We now construct the following bi-linear forms

$\hat{\Gamma}^{A}=\hat{\bar{B}}\Gamma^{A}\hat{B}$ (11)

with $\bar{\hat{B}}=\hat{B}^{\dagger}\gamma^{0}$ and $\Gamma^{A}$ a general
$4\times 4$ matrix. It is simple to check that these have the property

$[\hat{\Gamma}^{A},\hat{\Gamma}^{B}]=\hat{\bar{B}}[\Gamma^{A},\Gamma^{B}]\hat{B}\equiv\hat{\Gamma}^{[A,B]}.$
(12)

Note that for this to hold the unconventional sign in (8) requires
$\bar{\hat{B}}$ rather than just ${\hat{B}^{\dagger}}$ in the definition of
the bi-linear forms (11). This property has the important consequence that the
operators $\hat{\Gamma}^{A}$ satisfy the same algebra as the $4\times 4$
matrices $\Gamma^{A}$. Since the Lorentz algebra has a well-known
representation in terms of the $4\times 4$ $\gamma$ matrices, we can now
readily construct a representation of the Lorentz algebra on
$\mathcal{L}^{2}(\mathcal{C}^{2})$. Similarly, as we know how the $\gamma$
matrices transform under the Lorentz algebra, we can construct tensor
operators of the Lorentz algebra on $\mathcal{L}^{2}(\mathcal{C}^{2})$. Recall
that the $\gamma$ matrices are explicitly given by ($\sigma_{i}$ denotes Pauli
spin matrices and $I$ the $2\times 2$ identity matrix)

$\gamma^{0}=\left(\begin{array}[]{cc}I&0\\\
0&-I\end{array}\right),\quad\gamma^{i}=\left(\begin{array}[]{cc}0&\sigma_{i}\\\
-\sigma_{i}&0\end{array}\right),\quad\gamma^{5}=\left(\begin{array}[]{cc}0&I\\\
I&0\end{array}\right),$ (13)

and that they satisfy the anti-commutation relations
$[\gamma^{\mu},\gamma^{\nu}]_{+}=2\eta^{\mu\nu}\mathbb{I};$ $\mu,\nu=0,1,2,3$
with $\eta^{\mu\nu}=(1,-1,-1,-1)$ the Minkowski metric. From these we can
construct a basis of the Clifford algebra as
$\Gamma^{A}=\\{\mathbb{I},\gamma^{\mu},\sigma^{\mu\nu},\gamma^{5},\gamma^{5}\gamma^{\mu}\\}$
with $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$, which also
close on the Lorentz algebra.

We can now construct the following bi-linear operators:

$\hat{S}=\frac{1}{2}\hat{\bar{B}}\hat{B},~{}~{}\hat{D}=-\frac{i}{2}\hat{\bar{B}}\gamma^{5}\hat{B},~{}~{}\hat{V}^{\mu}=\frac{1}{2}\hat{\bar{B}}\gamma^{\mu}\hat{B};~{}~{}~{}\hat{A}^{\mu}=\frac{1}{2}\hat{\bar{B}}\gamma^{5}\gamma^{\mu}\hat{B},$
(14)

as well as the two indices anti-symmetric objects

$\hat{M}^{\mu\nu}=\frac{1}{2}\hat{\bar{B}}\sigma^{\mu\nu}\hat{B}.$ (15)

From the discussion above it is clear that the latter satisfy the usual
$so(1,3)$ (Lorentz) algebra:

$[\hat{M^{\mu\nu}},\hat{M}^{\rho\sigma}]=-i(\eta^{[\mu[\rho}\hat{M}^{\nu]\sigma]}),$
(16)

with $\eta_{\mu\nu}$ the inverse Minkowski metric. From the transformation
properties of the $\gamma$ matrices under the Lorentz algebra
$\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$ it is also clear that
$\hat{V}^{\mu}$ and $\hat{A}^{\mu}$ transform like 4-vectors:

$[\hat{V}^{\mu},\hat{M}^{\rho\nu}]=i(\eta^{\mu\rho}\hat{V}^{\nu}-\eta^{\mu\nu}\hat{V}^{\rho}),$
(17)
$[\hat{A}^{\mu},\hat{M}^{\rho\nu}]=(\eta^{\mu\rho}\hat{A}^{\nu}-\eta^{\rho\nu}\hat{A}^{\rho}),$
(18)

while $\hat{S}$ and $\hat{D}$ are scalars:

$[\hat{S},\hat{M}^{\mu\nu}]=[\hat{D},\hat{M}^{\mu\nu}]=0.$ (19)

Note that all these operators are Hermitian on
$\mathcal{L}^{2}(\mathcal{C}^{2})$ so that this is a manifestly unitary
representation of the Lorentz algebra.

The 4-vectors also obey the following commutation relations

$[\hat{V}^{\mu},\hat{V}^{\nu}]=-i\hat{M}^{\mu\nu},~{}[\hat{A}^{\mu},\hat{A}^{\nu}]=i\hat{M}^{\mu\nu}~{}~{}[\hat{V}^{\mu},\hat{A}^{\nu}]=-i\hat{D}\eta^{\mu\nu},~{}[\hat{V}^{\mu},\hat{D}]=i\hat{A}^{\mu},~{}~{}[\hat{A}^{\mu},\hat{D}]=i\hat{V}^{\mu}$
(20)

If we exclude $\hat{S}$ and introduce

$\hat{M}^{5\mu}=\hat{V}^{\mu},~{}~{}~{}~{}\hat{M}^{4\mu}=\hat{A}^{\mu}~{}~{}~{}\hat{M}^{54}=\hat{D},$
(21)

we note that these 15 operators close on the six-dimensional pseudo-orthogonal
algebra with the commutation relations:

$[\hat{M}^{ab},\hat{M}^{cd}]=-i(\eta^{ac}\hat{M}^{bd}-\eta^{bc}\hat{M}^{ad}+\eta^{bd}\hat{M}^{ac}-\eta^{ad}\hat{M}^{bc})$
(22)

where $\eta_{ab}=diag(+,-,-,-,-,+)$, for $a,b=\mu,4,5$.

This is the $so(2,4)$ (anti) de Sitter algebra, which is the isometry group of
this six-dimensional Minkowski space-time in that it preserves a hyperboloid.
In fact, this group can also be regarded as a group that encompasses conformal
transformations of the (3+1)D flat Minkowski space-time [53, 54].

It is useful to have an explicit differential operator representation of these
operators on $\mathcal{L}^{2}(\mathcal{C}^{2})$ :

$\hat{S}=\frac{1}{2}(\bar{z}_{\alpha}\partial_{\bar{z}_{\alpha}}-z_{\alpha}\partial_{z_{\alpha}}-2],~{}~{}~{}\hat{D}=\frac{i}{2}(\bar{z}_{\alpha}\partial_{\bar{z}_{\alpha}}+z_{\alpha}\partial_{z_{\alpha}}+2)$
(23)
$\hat{V}^{0}=[\bar{z}_{\alpha}z_{\alpha}-\frac{1}{4}\partial_{z_{\alpha}}\partial_{\bar{z}_{\alpha}}],~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{V}^{i}=(\sigma^{i})_{\alpha\beta}[\bar{z}_{\alpha}z_{\beta}+\frac{1}{4}\partial_{z_{\alpha}}\partial_{\bar{z}_{\beta}}];$
(24)
$\hat{A}^{0}=-(\bar{z}_{\alpha}z_{\alpha}+\frac{1}{4}\partial_{z_{\alpha}}\partial_{\bar{z}_{\alpha}}),~{}~{}~{}~{}~{}~{}\hat{A}^{i}=-(\sigma^{i})_{\alpha\beta}[\bar{z}_{\alpha}z_{\beta}-\frac{1}{4}\partial_{z_{\alpha}\partial_{\bar{z}_{\beta}}}]$
(25)

and the Lorentz generators can be written as

$\hat{L}_{i}=\frac{1}{2}\epsilon_{ijk}\hat{M}^{jk}=\frac{1}{2}(\sigma_{i})^{\alpha\beta}[\hat{a}^{\dagger}_{\alpha}\hat{a}_{\beta}-\hat{b}^{\dagger}_{\alpha}\hat{b}_{\beta}]=\frac{1}{2}\sigma^{\alpha\beta}_{i}[\bar{z}_{\alpha}\partial_{\bar{z}_{\beta}}-z_{\beta}\partial_{z_{\alpha}}],$
(26)

and

$\hat{K}_{i}=\hat{M}_{oi}=\frac{i}{2}(\sigma_{i})^{\alpha\beta}[\hat{b}^{\dagger}_{\alpha}\hat{a}_{\beta}-\hat{a}^{\dagger}_{\alpha}\hat{b}_{\beta}]=\frac{i}{2}\sigma_{i}^{\alpha\beta}[z_{\beta}\partial_{z_{\alpha}}+\bar{z}_{\alpha}\partial_{\bar{z}_{\beta}}],$
(27)

where the generators ( $\vec{L}$ and $\vec{K}$) obey the usual $so(1,3)$
algebra:

$[\hat{L}_{i},\hat{L}_{j}]=i\epsilon_{ijk}\hat{L}_{k},~{}[\hat{K}_{i},\hat{K}_{j}]=-i\epsilon_{ijk}\hat{L}_{k},~{}[\hat{L}_{i},\hat{K}_{j}]=i\epsilon_{ijk}\hat{K}_{k}.$
(28)

We can introduce a Casimir relation within the framework of $SO(2,4)$ as
follows

$\hat{C}_{2}\equiv-\frac{1}{2}\hat{M}_{ab}\hat{M}^{ab}=-\left(\hat{V}_{\mu}\hat{V}^{\mu}-\hat{A}_{\mu}\hat{A}^{\mu}-\hat{D}^{2}+\frac{1}{2}\hat{M}_{\mu\nu}\hat{M}^{\mu\nu}\right),$
(29)

where we have used the fact that
$D^{2}=(\hat{M}^{54})^{2},~{}\hat{A}_{\mu}=-\hat{M}_{4\mu}$, and
$\hat{X}_{\mu}=\hat{M}_{5\mu}$. It is worth noting that the operator $\hat{S}$
plays a significant role. It demonstrates commutativity with all other
operators, making it another Casimir operator of $SO(2,4)$ and can be used to
label the representations of $SO(2,4)$ carried by
$\mathcal{L}^{2}(\mathcal{C}^{2})$. This will play an important role in what
follows.

###  2.2 Lorentz covariant deformed Heisenberg-Poincare algebra

Similarly to the usual covariant construction in which coordinates and
momentum transform as 4-vectors, it is not only natural but essential for
covariance that we identify the two 4-vectors constructed above as position
and momentum. Of course, these operators do not satisfy the standard
Heisenberg algebra, which implies that it is necessary to deform the
Heisenberg algebra in order to maintain Lorentz covariance. Note that our
perspective here is that Lorentz covariance is more fundamental than the
Heisenberg algebra. We note, though, that there is an ambiguity in this
identification, as any linear combination of the two 4-vectors is again a
4-vector and can be identified with position and momentum. This therefore
allows for different deformations of the Heisenberg algebra, depending on the
choice of position and momentum as a four-vector. We need to consider two
cases, as outlined below:

####  2.2.1 Case 1: Commuting coordinates and momenta

First we introduce

$\hat{V}^{\mu}_{\pm}=\frac{1}{2}(\hat{V}^{\mu}\pm\hat{A}^{\mu})$ (30)

which satisfy the following commutation relations:

$[\hat{V}^{\mu}_{+},\hat{V}^{\nu}_{+}]=0;~{}~{}~{}[\hat{V}^{\mu}_{-},\hat{V}^{\nu}_{-}]=0,~{}~{}~{}~{}[\hat{V}^{\mu}_{-},\hat{V}^{\nu}_{+}]=-\frac{i}{2}(\eta^{\mu\nu}\hat{D}+\hat{M}^{\mu\nu}).$
(31)

It is instructive to introduce the following dimensionfull operators:

$\hat{X}^{\mu}=\lambda\hat{V}^{\mu}_{-};~{}~{}~{}~{}~{}~{}~{}\hat{P}^{\mu}=\frac{2\hbar}{\lambda^{{}^{\prime}}}\hat{V}^{\mu}_{+},~{}~{}~{}~{}\hat{J}^{\mu\nu}=\hbar\hat{M}^{\mu\nu}.$
(32)

Here, $\lambda$ represents the length scale linked to the inverse of the local
curvature within the curved momentum space, while $\lambda^{{}^{\prime}}$
pertains to the length scale associated with the inverse of local curvature in
the coordinate space. These then satisfy the commutation relations

$[\hat{X}^{\mu},\hat{X}^{\nu}]=0;~{}~{}~{}[\hat{P}^{\mu},\hat{P}^{\nu}]=0,~{}~{}~{}~{}[\hat{X}^{\mu},\hat{P}^{\nu}]=-i\hbar\hat{\cal{D}}\eta^{\mu\nu}-i\frac{\lambda}{\lambda^{{}^{\prime}}}\hat{J}^{\mu\nu}.$
(33)

where $\hat{\cal{D}}=\frac{\lambda}{\lambda^{{}^{\prime}}}\hat{D}$.

We note that the coordinates and momenta are commutative as usual, but that
the position-momentum commutators are deformed. Intriguingly, this algebra is
not closed; therefore, in order to ensure the closure of the algebra, the
standard Heisenberg algebra can be extended to a kinematical algebra with the
inclusion of the Lorentz generators ($\hat{J}^{\mu\nu}$) and the central
charge ($\hat{\cal{D}}$) for the Lorentz algebra. Thus, the additional
commutation relations to eqs. (33) are

$[\hat{X}^{\mu},\hat{\cal{D}}]=-i\frac{\lambda}{\lambda^{{}^{\prime}}}\hat{X}^{\mu}~{}~{}~{}[\hat{P}^{\mu},\hat{\cal{D}}]=i\frac{\lambda}{\lambda^{{}^{\prime}}}\hat{P}^{\mu},$
(34)

and

$[\hat{X}^{\mu},\hat{J}^{\rho\nu}]=i\hbar(\eta^{\mu\rho}\hat{X}^{\nu}-\eta^{\rho\nu}\hat{X}^{\mu}),~{}~{}~{}[\hat{P}^{\mu},\hat{J}^{\rho\nu}]=i\hbar(\eta^{\mu\rho}\hat{P}^{\nu}-\eta^{\rho\nu}\hat{P}^{\mu}),\;\;[\hat{\cal{D}},\hat{J}^{\mu\nu}]=0.$
(35)

Finally, let us consider the differential operator representation of these
operators. From (25) these read

$\displaystyle\hat{P}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2\lambda^{\prime}}\tilde{\sigma}^{\mu}_{\alpha\beta}\partial_{z_{\alpha}}\partial_{\bar{z}_{\beta}},$
$\displaystyle\hat{X}^{\mu}$ $\displaystyle=$
$\displaystyle\lambda\sigma^{\mu}_{\alpha\beta}{\bar{z}}_{\alpha}z_{\beta}.$
(36)

where $\sigma^{\mu}=\\{\mathbb{I},\sigma^{i}\\}$ and
$\tilde{\sigma}^{\mu}=\\{-\mathbb{I},\sigma^{i}\\}$. It is clear that the
momentum operators can be simultaneously diagonalized and that they have
continuous spectra. We also note

$\hat{P}^{\mu}\hat{P}_{\mu}=\hat{X}^{\mu}\hat{X}_{\mu}=0.$ (37)

This situation therefore apparently describes lightcone physics with zero
mass.

####  2.2.2 Case 2: Non-commuting coordinates and momenta

In contrast to case 1 above, we can also generate non-commuting coordinates
and momenta by identifying them with 4-vectors that are a linear combinations
of the commuting $\hat{X}^{\mu}$ and $\hat{P}^{\mu}$ above (the analogue of a
Bopp-shift)

$\displaystyle\tilde{X}^{\mu}$ $\displaystyle=$
$\displaystyle[c_{1}\hat{X}^{\mu}+(\frac{\lambda\lambda^{{}^{\prime}}}{2\hbar})d_{1}\hat{P}^{\mu}],$
(38) $\displaystyle\tilde{P}^{\mu}$ $\displaystyle=$
$\displaystyle[c_{2}\hat{P}^{\mu}+(\frac{\hbar}{2\lambda\lambda^{{}^{\prime}}})d_{2}\hat{X}^{\mu}].$
(39)

where $c_{1},c_{2},d_{1},d_{2}$ are dimensionless real free parameters. These
are non-commuting coordinates and momenta with

$\displaystyle[\tilde{X}^{\mu},\tilde{X}^{\nu}]=-i\frac{\lambda^{2}}{\hbar}\beta_{1}\hat{J}^{\mu\nu},\
\ \
[\tilde{P}^{\mu},\tilde{P}^{\nu}]=-i\frac{\hbar}{\lambda^{{}^{\prime}2}}\beta_{2}\hat{J}^{\mu\nu},$
$\displaystyle[\tilde{X}^{\mu},\tilde{P}^{\nu}]=-i\hbar\beta_{0}\hat{\cal{D}}\eta^{\mu\nu}-i\frac{\lambda}{\lambda^{{}^{\prime}}}\beta_{3}\hat{J}^{\mu\nu},\
\ \
[\tilde{X}^{\mu},\hat{\cal{D}}]=i\frac{1}{\beta_{0}}[\frac{\lambda^{2}}{\hbar}\beta_{1}\tilde{P}^{\mu}-\beta_{3}\frac{\lambda}{\lambda^{{}^{\prime}}}\tilde{X}^{\mu}],$
$\displaystyle[\tilde{P}^{\mu},\hat{\cal{D}}]=-i\frac{1}{\beta_{0}}[\frac{\hbar}{\lambda^{{}^{\prime}2}}\beta_{2}\tilde{X}^{\mu}-\beta_{3}\frac{\lambda}{\lambda^{{}^{\prime}}}\tilde{P}^{\mu}],$
$\displaystyle[\tilde{X}^{\mu},\hat{J}^{\rho\nu}]=i\hbar(\eta^{\mu\rho}\tilde{X}^{\nu}-\eta^{\rho\nu}\tilde{X}^{\mu}),$
$\displaystyle[\tilde{P}^{\mu},\hat{J}^{\rho\nu}]=i\hbar(\eta^{\mu\rho}\tilde{P}^{\nu}-\eta^{\rho\nu}\tilde{P}^{\mu}),$
$\displaystyle[\hat{J^{\mu\nu}}\hat{J}^{\rho\sigma}]=-i\hbar(\eta^{\mu\rho}\hat{J}^{\nu\sigma}-\eta^{\nu\rho}\hat{J}^{\mu\sigma}+\eta^{\nu\sigma}\hat{J}^{\mu\rho}-\eta^{\mu\sigma}\hat{J}^{\nu\rho}),$
$\displaystyle[\hat{\cal{D}},\hat{J}^{\mu\nu}]=0.$ (40)

where $\beta_{1}=c_{1}d_{1}$, $\beta_{2}=d_{2}c_{2}$,
$\beta_{3}=\frac{1}{4}(4c_{1}c_{2}+d_{1}d_{2})$, and
$\beta_{0}=\frac{1}{4}(4c_{1}c_{2}-d_{1}d_{2})$. We also require $\beta_{0}$
to be non-zero and finite. The latter requirement is needed to avoid a linear
dependence between $\tilde{X}^{\mu}$ and $\tilde{P}^{\mu}$. This algebra is
nothing but a general deformed Heisenberg Poincare relativistic covariant
algebra involving four parameters, $\beta_{1},\beta_{2},\beta_{3}$, and
$\beta_{0}$, satisfying $\beta^{2}_{3}-\beta^{2}_{0}=\beta_{1}\beta_{2}$. A
four-parameter class of closed algebras is found, but they all exhibit the
same stability [55, 56, 57, 58], as defined by the fact that the isomorphic
set of algebras represented by $so(2,4)$ cannot be interpreted as resulting
from the Inonu-Wigner contraction [59] of some other group. There are two
$so(1,4)$ sub-algebras for coordinate operators $\hat{\tilde{X}}^{\mu}$ and
momentum operators $\hat{\tilde{P}}^{\mu}$, respectively, with a common
$so(1,3)$ Lorentz algebra. It’s evident from (40) that when we make a specific
parameter choice, setting $\beta_{0}=\beta_{1}=-\beta_{2}=1$ and
$\beta_{3}=0$, the standard algebra for the AdS version of the Yang model [60,
61, 62, 63] is effectively reproduced. This standard algebra maintains Lorentz
and translational symmetries within fuzzy space-time due to non-commutativity
and assumes a background manifold with constant curvature. This results in a
duality between the curvature of position and momentum spaces.

These read as differential operators, using (25)

$\displaystyle\tilde{X}^{\mu}$ $\displaystyle=$
$\displaystyle\lambda\left(c_{1}\sigma^{\mu}_{\alpha\beta}{\bar{z}}_{\alpha}z_{\beta}+\frac{d_{1}}{4}\tilde{\sigma}^{\mu}_{\alpha\beta}\partial_{z_{\alpha}}\partial_{\bar{z}_{\beta}}\right),$
$\displaystyle\tilde{P}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2\lambda^{\prime}}\left(d_{2}\sigma^{\mu}_{\alpha\beta}{\bar{z}}_{\alpha}z_{\beta}+c_{2}\tilde{\sigma}^{\mu}_{\alpha\beta}\partial_{z_{\alpha}}\partial_{\bar{z}_{\beta}}\right).$
(41)

In this case we have

$\tilde{P}^{\mu}\tilde{P}_{\mu}=\frac{\beta_{2}\hbar^{2}}{\beta_{1}{\lambda^{2}\lambda^{\prime}}^{2}}\tilde{X}^{\mu}\tilde{X}_{\mu}=\frac{\beta_{2}\hbar^{2}}{{\lambda^{\prime}}^{2}}\left(\hat{S}^{2}+\hat{D}^{2}+2\hat{S}\right)\equiv\tilde{M}^{2}v^{2}$
(42)

where we have introduced the mass operator $\tilde{M}$ and $v$ is some
velocity scale for which the speed of light would be a natural choice. We do
not discuss the spectral properties of these operators further but only note
that the spectra of these operators change from discrete to continuous,
depending on the signs of $\beta_{1}$ and $\beta_{2}$.

###  2.3 Cartesian coordinate representation

Now comes the most important step in this paper, which is the construction of
a representation of the (anti) de Sitter and subsequently Lorentz algebra on
$\mathcal{L}^{2}(\mathcal{R}^{3})$. To do this we transform from the complex
coordinates in (4) to Cartesian coordinates and the coordinate $\gamma$, i.e.
$(x_{1},x_{2},x_{3},\gamma)$. Keeping in mind that the Lorentz scalar
$\hat{S}$ commutes with all the generators of the $so(2,4)$ algebra
constructed above on $\mathcal{L}^{2}(\mathcal{C}^{2})$, we note that we can
use it to construct subspaces of $\mathcal{L}^{2}(\mathcal{C}^{2})$ that are
invariant under the action of these operators and hence will also carry a
representation of $so(2,4)$. It is a straightforward but lengthy exercise to
compute the Cartesian coordinate representation of these operators from the
complex differential operator representation (23)-(27) on
$\mathcal{L}^{2}(\mathcal{C}^{2})$ using (4). For $\hat{S}$, we find

$\hat{S}=\frac{i}{2}\partial_{\gamma}-1.$ (43)

From this, we note that the subspace of functions of
$\mathcal{L}^{2}(\mathcal{C}^{2})$ that depend only on $(x_{1},x_{2},x_{3})$
is the subspace on which $\hat{S}$ takes the value $-1$. We denote these
physical states as $\psi_{\rm ph}(z,\bar{z})$. This space of functions is
actually not $\mathcal{L}^{2}(\mathcal{R}^{3})$, but rather
$\mathcal{L}^{2}(\mathcal{R}^{3},\frac{1}{r})$ as this is the inner product
induced on it by the inner product of $\mathcal{L}^{2}(\mathcal{C}^{2})$, as
we discuss below.

One can start by not imposing the constraint $\hat{S}=-1$, but construct the
Cartesian coordinate representation of the Lorentz and (anti) de Sitter
algebras on the full $\mathcal{L}^{2}(\mathcal{C}^{2})$. As we do not need the
explicit form of these operators, we do not list them here, but for
completeness, they are listed in Appendix A. For our present purposes, it
suffices to note that the generic form of these operators is

$O(a,b,c)=a(x_{i},\partial_{i})+b(x_{i},\partial_{i})\partial_{\gamma}+c(x_{i},\partial_{i})\partial_{\gamma}^{2}$
(44)

so that $O(a,b,c)\psi_{\rm ph}(z,\bar{z})=a(x_{i},\partial_{i})\psi_{\rm
ph}(z,\bar{z})$. We also note that

$\displaystyle[O(a,b,c),O(a^{\prime},b^{\prime},c^{\prime})]\psi_{\rm
ph}(z,\bar{z})=[a(x_{i},\partial_{i}),a^{\prime}(x_{i},\partial_{i})]\psi_{\rm
ph}(z,\bar{z})$ (45)

so that the operators $a(x_{i},\partial_{i})$ must satisfy the same algebra on
the physical subspace as the operators on the full space. This implies that
the operators projected on the physical subspace must themselves provide a
representation of the Lorentz algebra and corresponding 4-vectors. We can
therefore drop all the terms containing $\gamma$ derivatives and focus on the
physical subspace only. For these we find (see Appendix A)

$\displaystyle\hat{L}_{k}$ $\displaystyle=$
$\displaystyle-i\epsilon_{kj\ell}x_{j}\partial_{\ell},$
$\displaystyle\hat{K}_{k}$ $\displaystyle=$ $\displaystyle-ir\partial_{k},$
$\displaystyle\hat{V}^{0}_{+}$ $\displaystyle=$
$\displaystyle-\frac{r}{4}\nabla^{2},$ $\displaystyle\hat{V}^{i}_{+}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(2\partial_{i}-x_{i}\nabla^{2}+2\left(x\cdot\nabla\right)\partial_{i}\right),$
$\displaystyle\hat{V}^{\mu}_{-}$ $\displaystyle=$ $\displaystyle(r,x,y,z),$
$\displaystyle\hat{D}$ $\displaystyle=$ $\displaystyle
i\left(x\cdot\nabla+1\right).$ (46)

It will be noted that these operators are not hermitian with respect to the
normal inner product on $\mathcal{L}^{2}(\mathcal{R}^{3})$, which may look
like an inconsistency given that they are hermitian on
$\mathcal{L}^{2}(\mathcal{C}^{2})$. This is, however, clarified when one
computes from (5) the inner product induced on
$\mathcal{L}^{2}(\mathcal{R}^{3})$:

$\displaystyle(\phi_{\rm ph},\psi_{\rm ph})$ $\displaystyle=$
$\displaystyle\int\frac{dz_{\alpha}d\bar{z}_{\alpha}}{\pi^{2}}~{}\phi^{*}_{\rm
ph}(z_{\alpha},\bar{z}_{\alpha})\psi_{\rm ph}(z_{\alpha},\bar{z}_{\alpha})$
(47) $\displaystyle=$ $\displaystyle\int\frac{d^{3}x}{r\pi}\phi^{*}_{\rm
ph}(x_{i})\psi_{\rm ph}(x_{i})$

where the $\frac{1}{r}$ factor arises from the Jacobian $J=\frac{1}{2r}$. The
operators above are therefore hermitian with respect to this inner product on
$\mathcal{L}^{2}(\mathcal{R}^{3})$ and we have actually arrived at an unitary
representation of the (anti) de Sitter algebra on
$\mathcal{L}^{2}(\mathcal{R}^{3},\frac{1}{r})$. Finally we can transform to
$\mathcal{L}^{2}(\mathcal{R}^{3})$ by performing the similarity transformation
$\psi_{\rm ph}(x_{i})\rightarrow\tilde{\psi}_{\rm
ph}(x_{i})=\frac{1}{\sqrt{r}}\psi_{\rm ph}(x_{i})$ and
$\hat{O}\rightarrow\hat{\tilde{O}}=\frac{1}{\sqrt{r}}\hat{O}\sqrt{r}$ where
$\hat{O}$ represents any of the operators above. Obviously this yields a
completely isomorphic algebra, but note that $\tilde{\psi}_{\rm ph}(x_{i})$
must now be square integrable.

It is immediately clear that $\hat{\tilde{S}}=\hat{S}$. For the other
operators we find

$\displaystyle\hat{\tilde{L}}_{k}$ $\displaystyle=$
$\displaystyle\hat{L}_{k}=-i\epsilon_{kj\ell}x_{j}\partial_{\ell},$
$\displaystyle\hat{\tilde{K}}_{k}$ $\displaystyle=$
$\displaystyle-i\left(r\partial_{k}+\frac{x_{k}}{2r}\right),$
$\displaystyle\hat{\tilde{V}}^{0}_{+}$ $\displaystyle=$
$\displaystyle-\frac{r}{4}\nabla^{2}-\frac{1}{4r}x\cdot\nabla-\frac{3}{16r},$
$\displaystyle\hat{\tilde{V}}^{i}_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(3\partial_{i}-x_{i}\left(\nabla^{2}+\frac{1}{4r^{2}}\right)+2\left(x\cdot\nabla\right)\partial_{i}\right),$
$\displaystyle\hat{V}^{\mu}_{-}$ $\displaystyle=$ $\displaystyle(r,x,y,z),$
$\displaystyle\hat{\tilde{D}}$ $\displaystyle=$ $\displaystyle
i\left(x\cdot\nabla+\frac{3}{2}\right).$ (48)

One can also verify the following relations:

$\displaystyle\hat{\tilde{V}}^{\mu}_{+}\hat{\tilde{V}}_{+\mu}=\hat{\tilde{V}}^{\mu}_{-}\hat{\tilde{V}}_{-\mu}=0,$
$\displaystyle\hat{\tilde{V}}^{\mu}_{+}\hat{\tilde{V}}_{-\mu}+\hat{\tilde{V}}^{\mu}_{-}\hat{\tilde{V}}_{+\mu}=\hat{\tilde{D}}^{2}-1.$
(49)

Using this in ( 29), we can write the second order Casimir as

$\hat{C}_{2}=\hat{\tilde{K}}^{2}-\hat{\tilde{D}}^{2}-\hat{\tilde{L}}^{2}+2=3.$
(50)

This is quite a remarkable result as it implies that we have found a unitary
representation of $so(2,4)$ and the Lorentz algebra on
$\mathcal{L}^{2}(\mathcal{R}^{3})$. The fact that the $so(2,4)$ Casimir is a
constant also implies that this representation is irreducible. Note that time
is completely absent in this representation. The question that faces us now is
what the implications of the existence of this representation are. If we take
Lorentz covariance as a principle that should guide our construction of
physical theories, we must accept that the physical laws should be covariant
under these transformations. In particular this may imply that the Heisenberg
has to be deformed to comply with Lorentz covariance.

In this regard, a first check we should carry out is if the inner product (47)
is Lorentz invariant. This is clearly the case for rotations. Under boosts, we
note from the discussion above on hermiticity that the boost operator
$\hat{B}(a)=e^{ia\cdot\hat{K}}=e^{ra\cdot\nabla}$ is unitary with respect to
this inner product so that it is also invariant under boosts. We can calculate
this also more explicitly by noting that due to rotation invariance, we can,
without loss of generality, choose the $x_{1}$-direction as the boost
direction, i.e. $\hat{B}(\phi)=e^{\phi r\partial_{1}}$. Applying this
transformation to the wave-functions we have

$\hat{B}(\phi)\psi_{\rm ph}(x_{i})=\psi_{\rm ph}(x_{i}^{\prime})$ (51)

with

$x_{1}^{\prime}=x_{1}\cosh\phi+r\sinh\phi,\;x^{\prime}_{2}=x_{2},\;x^{\prime}_{3}=x_{3}.$
(52)

Changing variables to $x_{i}^{\prime}$ in the integral and computing the
Jacobian we find $\frac{d^{3}x}{r}=\frac{d^{3}x^{\prime}}{r^{\prime}}$ and we
conclude

$(B(\phi)\phi_{\rm ph},B(\phi)\psi_{\rm ph})=(\phi_{\rm ph},\psi_{\rm ph}).$
(53)

This result can also be obtained by noting

$\hat{B}^{\dagger}(\phi)\frac{1}{r}B(\phi)=\frac{1}{r}$ (54)

where conjugation is taken w.r.t. the normal inner product on
$\mathcal{L}^{2}(\mathcal{R}^{3})$. A final point to note is that after the
similarity transformation, the unitary boost operator is given by
$\tilde{\hat{B}}(a)=e^{ia\cdot\tilde{\hat{K}}}$ and that, as readily can be
checked, we now have the transformation rule for the wave-functions

$\tilde{\hat{B}}(\phi)\tilde{\psi}_{\rm
ph}(x_{i})=\sqrt{\frac{r^{\prime}}{r}}\tilde{\psi}_{\rm ph}(x_{i}^{\prime}).$
(55)

In this case, the $\mathcal{L}^{2}(\mathcal{R}^{3})$ measure is not invariant,
but the resulting Jacobian cancels the scaling factors on the wave functions
so that the inner product is again Lorentz invariant.

We can again introduce the dimensionful operators of (32). Note from (33) that
position and momentum are commuting operators. We also note from (2.3)

$\displaystyle\hat{X}^{\mu}\hat{X}_{\mu}$ $\displaystyle=$
$\displaystyle\hat{P}^{\mu}\hat{P}_{\mu}=0,$
$\displaystyle\hat{P}^{\mu}\hat{X}_{\mu}+\hat{X}^{\mu}\hat{P}_{\mu}$
$\displaystyle=$
$\displaystyle\frac{2\hbar\lambda}{\lambda^{\prime}}\left(\hat{\tilde{D}}^{2}-1\right).$
(56)

More generally, we can consider the dimensionful non-commuting coordinates and
momenta $\tilde{X}^{\mu}$ and $\tilde{P}^{\mu}$ of (38). As mentioned before,
they are non-commuting and satisfy the algebra (40). We can now compute the
Lorentz invariants $\tilde{X}^{\mu}\tilde{X}_{\mu}$,
$\tilde{P}^{\mu}\tilde{P}_{\mu}$ and
$\tilde{P}^{\mu}\tilde{X}_{\mu}+\tilde{X}^{\mu}\tilde{P}_{\mu}$ using (2.3) to
find

$\displaystyle\tilde{X}^{\mu}\tilde{X}_{\mu}$ $\displaystyle=$
$\displaystyle\beta_{1}\lambda^{2}\left(\hat{\tilde{D}}^{2}-1\right),$ (57)
$\displaystyle\tilde{P}^{\mu}\tilde{P}_{\mu}$ $\displaystyle=$
$\displaystyle\beta_{2}\left(\frac{\hbar}{\lambda^{\prime}}\right)^{2}\left(\hat{\tilde{D}}^{2}-1\right)\equiv\tilde{M}^{2}v^{2}$
$\displaystyle\tilde{P}^{\mu}\tilde{X}_{\mu}+\tilde{X}^{\mu}\tilde{P}_{\mu}$
$\displaystyle=$
$\displaystyle\frac{2\beta_{3}\hbar\lambda}{\lambda^{\prime}}\left(\hat{\tilde{D}}^{2}-1\right).$
(58)

If we make the usual identification of $\tilde{E}=\tilde{P}^{0}$ as the energy
(Hamiltonian), we obtain the usual relativistic dispersion relation

$\tilde{E}=\pm\sqrt{\hat{M}^{2}v^{2}+{\vec{\tilde{P}}}\cdot\vec{\tilde{P}}}$
(59)

###  2.4 The Heisenberg algebra as an Inönu–Wigner contraction of the deformed
Heisenberg Poincare algebra

We now demonstrate that, on a certain sector of the Hilbert space, the
deformed Heisenberg-Poincaré algebra introduced above can be contracted to the
conventional Heisenberg-Poincaré algebra associated with Minkowski space-time.
To do this, we consider a fixed mass sector, i.e.,
$\tilde{M}^{2}|m\rangle=m^{2}|m\rangle$ in the limit of large
$\lambda^{\prime}$, i.e., small curvature in coordinate space, and for
$\beta_{2}>0$. From (57) we then have

$\hat{\tilde{D}}|m\rangle=\sqrt{1+\frac{{\lambda^{\prime}}^{2}m^{2}v^{2}}{\hbar^{2}\beta_{2}}}|m\rangle\approx\frac{{\lambda^{\prime}}mv}{\hbar\sqrt{\beta_{2}}}|m\rangle.$
(60)

From (40), we note that on this sector the position-momentum commutator
assumes the form

$[\tilde{X}^{\mu},\tilde{P}^{\nu}]=-i\frac{\lambda
mv\beta_{0}}{\sqrt{\beta_{2}}}\hat{\mathbb{I}}_{m}\eta^{\mu\nu}-i\frac{\lambda}{\lambda^{{}^{\prime}}}\beta_{3}\hat{J}^{\mu\nu}$
(61)

with $\mathbb{I}_{m}$ the identity (projector) on this sector. This will
reduce to the standard position-momentum commutator on this sector if we set
$\frac{\lambda mv\beta_{0}}{\sqrt{\beta_{2}}}=\hbar$. We then have for small
$\lambda$, i.e. small curvature in momentum space,

$\beta_{3}=\sqrt{\beta_{2}}\sqrt{\beta_{1}+\frac{\hbar^{2}}{\lambda^{2}m^{2}v^{2}}}\approx\frac{\hbar\sqrt{\beta_{2}}}{\lambda
mv}.$ (62)

This turns (40) into

$\displaystyle[\tilde{X}^{\mu},\tilde{X}^{\nu}]=-i\frac{\lambda^{2}}{\hbar}\beta_{1}\hat{J}^{\mu\nu},\
\ \
[\tilde{P}^{\mu},\tilde{P}^{\nu}]=-i\frac{\hbar}{\lambda^{{}^{\prime}2}}\beta_{2}\hat{J}^{\mu\nu},$
$\displaystyle[\tilde{X}^{\mu},\tilde{P}^{\nu}]=-i\hat{\mathbb{I}}_{m}\eta^{\mu\nu}-i\frac{\hbar\sqrt{\beta_{2}}}{\lambda^{\prime}mv}\hat{J}^{\mu\nu},\
\ \ [\tilde{X}^{\mu},\hat{\cal{D}}]=i\frac{\lambda
mv}{\hbar\sqrt{\beta_{2}}}[\frac{\lambda^{2}}{\hbar}\beta_{1}\tilde{P}^{\mu}-\frac{\hbar\sqrt{\beta_{2}}}{mv\lambda^{{}^{\prime}}}\tilde{X}^{\mu}],$
$\displaystyle[\tilde{P}^{\mu},\hat{\cal{D}}]=-i\frac{\lambda
mv}{\hbar\sqrt{\beta_{2}}}[\frac{\hbar}{\lambda^{{}^{\prime}2}}\beta_{2}\tilde{X}^{\mu}-\frac{\hbar\sqrt{\beta_{2}}}{mv\lambda^{{}^{\prime}}}\tilde{P}^{\mu}],$
$\displaystyle[\tilde{X}^{\mu},\hat{J}^{\rho\nu}]=i\hbar(\eta^{\mu\rho}\tilde{X}^{\nu}-\eta^{\rho\nu}\tilde{X}^{\mu}),$
$\displaystyle[\tilde{P}^{\mu},\hat{J}^{\rho\nu}]=i\hbar(\eta^{\mu\rho}\tilde{P}^{\nu}-\eta^{\rho\nu}\tilde{P}^{\mu}),$
$\displaystyle[\hat{J^{\mu\nu}}\hat{J}^{\rho\sigma}]=-i\hbar(\eta^{\mu\rho}\hat{J}^{\nu\sigma}-\eta^{\nu\rho}\hat{J}^{\mu\sigma}+\eta^{\nu\sigma}\hat{J}^{\mu\rho}-\eta^{\mu\sigma}\hat{J}^{\nu\rho}),$
$\displaystyle[\hat{\cal{D}},\hat{J}^{\mu\nu}]=0.$ (63)

Finally, taking the limits $\lambda\rightarrow 0$ and
$\lambda^{\prime}\rightarrow\infty$, we obtain the standard Poincare algebra
on this sector

$\displaystyle[\tilde{X}^{\mu},\tilde{X}^{\nu}]=0,\ \ \
[\tilde{P}^{\mu},\tilde{P}^{\nu}]=0,$
$\displaystyle[\tilde{X}^{\mu},\tilde{P}^{\nu}]=-i\hat{\mathbb{I}}_{m}\eta^{\mu\nu},\
\ \ $
$\displaystyle[\tilde{X}^{\mu},\hat{J}^{\rho\nu}]=i\hbar(\eta^{\mu\rho}\tilde{X}^{\nu}-\eta^{\rho\nu}\tilde{X}^{\mu}),$
$\displaystyle[\tilde{P}^{\mu},\hat{J}^{\rho\nu}]=i\hbar(\eta^{\mu\rho}\tilde{P}^{\nu}-\eta^{\rho\nu}\tilde{P}^{\mu}),$
$\displaystyle[\hat{J^{\mu\nu}}\hat{J}^{\rho\sigma}]=-i\hbar(\eta^{\mu\rho}\hat{J}^{\nu\sigma}-\eta^{\nu\rho}\hat{J}^{\mu\sigma}+\eta^{\nu\sigma}\hat{J}^{\mu\rho}-\eta^{\mu\sigma}\hat{J}^{\nu\rho}).$
(64)

Note that the operator $\hat{\cal{D}}$ contracts to the null operator in this
limit, and we therefore do not list its commutators here as they all vanish.
Essentially, this result implies that in flat space and momentum space, the
coordinates and momenta are close to the standard Heisenberg algebra, and mass
becomes a good quantum number. Obviously, this means that in the presence of
weak curvature, the Heisenberg algebra is a good approximation to the deformed
Heisenberg-Poincare algebra.

##  3 Covariant quantum dynamics

So far, our focus has centered on foundational aspects at the kinematical
level. In this section, we elevate our exploration to introduce Lorentz
covariant dynamics. Consider the representation of the Lorentz algebra on
$\mathcal{L}^{2}(\mathcal{R}^{3})$ with generators $\hat{\tilde{L}}_{i}(t=0)$
and $\hat{\tilde{K}}_{i}(t=0)$ which leave the physical subspace

$\frac{\partial}{\partial\gamma}\psi_{ph}(z,\bar{z})=0$ (65)

invariant. We now define time evolution in the physical subspace as a one-
parameter family of states:

$\psi_{ph}(x_{1},x_{2},x_{3},t)=e^{-it\hat{H}}\psi_{ph}(x_{1},x_{2},x_{3},t=0)\equiv\hat{U}(t)\psi_{ph}(x_{1},x_{2},x_{3},t=0).$
(66)

Here, $\hat{H}$ is for the moment a time-independent Hermitian operator that
commutes with the constraint. A natural choice is the energy
$\hat{H}=\hat{\tilde{P}}^{0}$. This ensures that

$\frac{\partial}{\partial\gamma}\psi_{ph}(x_{1},x_{2},x_{3},t)=0,\quad\forall
t.$ (67)

Clearly,

$[i\partial_{t}-\hat{H}]\psi_{ph}(x_{1},x_{2},x_{3},t)=0.$ (68)

When defining these time-dependent states through the unitary transformation
$\hat{U}(t)$, we also need to transform the differential operator
representation of the Lorentz generators:

$\hat{\tilde{L}}_{i}(t)=\hat{U}(t)\hat{\tilde{L}}_{i}\hat{U}^{\dagger}(t),\quad\hat{\tilde{K}}_{i}(t)=\hat{U}(t)\hat{\tilde{K}}_{i}\hat{U}^{\dagger}(t).$
(69)

It may be noted that

$i\hat{U}(t)\partial_{t}\hat{U}^{\dagger}(t)=i\partial_{t}-\hat{H}.$ (70)

Clearly,

$[\hat{L}_{i}(t=0),\frac{\partial}{\partial
t}]=0,\quad[\hat{K}_{i}(t=0),\frac{\partial}{\partial t}]=0.$ (71)

On applying the unitary transformation, we have

$[\hat{L}_{i}(t),\frac{\partial}{\partial
t}-\hat{H}]=0,\quad[\hat{K}_{i}(t),\frac{\partial}{\partial t}-\hat{H}]=0.$
(72)

Thus, $i\partial_{t}-\hat{H}$ is a Lorentz scalar, and consequently, (68) must
represent a fully covariant equation of motion. Note that the Lorentz
generators now have, as usual, an explicit time dependence.

The argument above can easily be generalised to time-dependent Hamiltonian by
simply replacing the exponential in (66) by a path-ordered exponential. The
rest of the argument remains the same.

##  4 The Fuzzy $\mathcal{R}_{\lambda}^{3}$ representation of the Lorentz and
(anti) de Sitter algebras.

In this section we construct an alternative representation of the Lorentz and
(anti) de Sitter algebras on fuzzy three dimensional space and explore its
connection with the ${\cal L}^{2}({\cal R}^{3})$ construction in the previous
section.

###  4.1 Quantum mechanics on fuzzy ${\cal R}^{3}_{\lambda}$

Three dimensional fuzzy space assumes the fuzzy sphere $su(2)$ commutation
relations for coordinates

$[\hat{x}_{i},\hat{x}_{j}]=2i\lambda\varepsilon_{ijk}\hat{x}_{k}.$ (73)

Here $\lambda$ has units of length and $\varepsilon_{ijk}$ is the standard
completely anti-symmetric tensor. The first step in constructing a quantum
theory on this space is to construct a concrete realisation of the fuzzy
sphere non-commutative coordinates on some Hilbert space that we’ll denote by
${\cal H}_{c}$ . The representation we choose for this purpose is the standard
Schwinger realisation of $su(2)$. Thus, ${\cal H}_{c}$ is a two-mode Fock
space on which the coordinates are realised as

$\hat{x}_{i}=\lambda a^{\dagger}_{\alpha}\sigma^{(i)}_{\alpha\beta}a_{\beta}.$
(74)

Here, a summation over repeated indices is implied. $\alpha,\beta=1,2$,
$\sigma^{(i)}_{\alpha\beta}$, $i=1,2,3$ are the Pauli spin matrices, and
$a_{\alpha}^{\dagger}$, $a_{\alpha}$ are standard boson creation and
annihilation operators. The radius operator is

$\hat{r}^{2}=\hat{x}_{i}\hat{x}_{i}=\lambda^{2}\hat{n}(\hat{n}+2),$ (75)

with $\hat{n}=a_{\alpha}^{\dagger}a_{\alpha}$ the boson number operator. Note
that the radius operator is also the Casimir of $su(2)$ and commutes with the
coordinates. As a measure of the radius we use

$\hat{r}=\lambda(\hat{n}+1),$ (76)

which is to leading order in $\lambda$ the square root of $\hat{r}^{2}$. Note
that this representation contains each $su(2)$ representation, and thus each
quantised radius, exactly once and therefore corresponds to a complete single
covering of ${\cal R}^{3}$, commonly referred to as ${\cal R}^{3}_{\lambda}$
fuzzy space.

The next step is to construct the quantum Hilbert space ${\cal H}_{q}$ in
which the states of the system are to be represented. In analogy with ${\cal
L}^{2}({\cal R}^{3})$ this is now defined as the algebra of operators
generated by the coordinates, i.e. the operators acting on ${\cal H}_{c}$ that
commute with $\hat{r}^{2}$ and have a finite norm with respect to a weighted
Hilbert-Schmidt inner product [41]

$\displaystyle{\cal H}_{q}=\left\\{\psi:[\psi,\hat{r}^{2}]=0\ \ {\rm and}\ \
{\rm tr_{c}}(\psi^{\dagger}\hat{r}\psi)<\infty\right\\}.$ (77)

The inner product on ${\cal H}_{q}$ is

$(\psi|\phi)_{q}=4\pi\lambda^{2}{\rm
tr}_{c}(\psi^{\dagger}\hat{r}\phi)=4\pi\lambda^{3}{\rm
tr}_{c}(\psi^{\dagger}\left(\hat{n}+1\right)\phi)$ (78)

with the trace taken over ${\cal H}_{c}$. We use the standard $|\cdot\rangle$
notation for elements of ${\cal H}_{c}$ and $|\cdot)$ for elements of ${\cal
H}_{q}$. This choice of the inner product is motivated by the observation that
the norm of the operator that projects on the subspace of spheres with radius
$r\leq\lambda(N+1)$, with $N$ large, corresponds to the volume of a sphere in
three dimensional Euclidean space.

Quantum observables are identified with self-adjoint operators acting on
${\cal H}_{q}$. We use capitals to distinguish them from operators acting on
${\cal H}_{c}$. These include the coordinates which act through left
multiplication as

$\hat{X}_{i}|\psi)=|\hat{x}_{i}\psi)$ (79)

and the angular momentum operators which act adjointly according to

$\hat{L}_{i}|\psi)=|\frac{\hbar}{2\lambda}[\hat{x}_{i},\psi])\quad{\rm
with}\quad[\hat{L}_{i},\hat{L}_{j}]=i\hbar\varepsilon_{ijk}\hat{L}_{k}.$ (80)

There is a further important conserved quantity, namely, the operator
$\hat{\Gamma}$, which acts as follows

$\hat{\Gamma}|\psi)=|[a^{\dagger}_{\alpha}a_{\alpha},\psi]).$ (81)

From the defining relation (77) it is clear that this operator tests whether a
state is actually in ${\cal H}_{q}$ and we refer to it as the physicality
condition. Physical observables are required to commute with $\hat{\Gamma}$ in
order to leave ${\cal H}_{q}$ invariant.

###  4.2 The representation of the Lorentz and (anti) de Sitter algebras on
fuzzy ${\cal R}_{\lambda}^{3}$

Let us enlarge ${\cal H}_{q}$ to the Hilbert space ${\cal H}$ defined as
follows:

$\displaystyle{\cal H}=\left\\{\psi:{\rm
tr_{c}}(\psi^{\dagger}\psi)<\infty\right\\},$ (82)

i.e. we dropped the physicality condition and adopted the inner product

$(\psi|\phi)=4\pi\lambda^{2}{\rm tr}_{c}(\psi^{\dagger}\phi).$ (83)

We note that

$(\psi|\phi)=(\psi|\frac{1}{\hat{r}}|\phi)_{q},$ (84)

reminiscent of (47).

On this space we define the following bosonic operators

$\displaystyle A_{\alpha L}^{\dagger}|\psi)=|a_{\alpha}^{\dagger}\psi),\quad
A_{\alpha L}|\psi)=|a_{\alpha}\psi),$ $\displaystyle A_{\alpha
R}^{\dagger}|\psi)=|\psi a_{\alpha}^{\dagger}),\quad A_{\alpha R}|\psi)=|\psi
a_{\alpha}).$ (85)

It is easily checked that these operators are hermitian conjugates with
respect to the inner product (83). Also note that

$[A_{\alpha L},A_{\beta L}^{\dagger}]=\delta_{\alpha\beta},\quad[A_{\alpha
R},A_{\beta R}^{\dagger}]=-\delta_{\alpha\beta}$ (86)

and all other commutators vanish. We now note a complete isomorphism with the
construction in Section 2, under the map

$A_{\alpha L}\rightarrow\hat{a}_{\alpha},\;A^{\dagger}_{\alpha
L}\rightarrow\hat{a}^{\dagger}_{\alpha},\;A_{\alpha
R}\rightarrow\hat{b}_{\alpha},\;A^{\dagger}_{\alpha
R}\rightarrow\hat{b}^{\dagger}_{\alpha}.$ (87)

We can therefore proceed in exactly the same way as there to construct a
representation of the Lorentz and (anti) de Sitter algebras as bilinears of
these creation and annihilation operators acting on $\cal{H}$. It is then easy
to establish that $\hat{S}$ and $\hat{\Gamma}$ play the same role, i.e., the
restriction to the physical subspace ${\cal H}_{q}$ is equivalent to the
restriction to ${\cal L}^{2}({\cal R}^{3})$. The rest of the analysis is the
same as in Section 2.

###  4.3 The link between the ${\cal L}^{2}({\cal R}^{3})$ and fuzzy ${\cal
R}^{3}_{\lambda}$ representations of the Lorentz and (anti) de Sitter algebras

From the above, it should be clear that we now have two representations of the
Lorentz and (anti) de Sitter algebras, one on ${\cal L}^{2}({\cal R}^{3})$ and
one on fuzzy ${\cal R}^{3}_{\lambda}$ that are, by construction, algebraically
equivalent. This equivalence can be made even more explicit by noting the
following: For the noncommutative spatial and momentum sectors of the deformed
Heisenberg algebra, given by:

$[\tilde{X}_{i},\tilde{X}_{j}]=-i\frac{\lambda^{2}}{\hbar}\beta_{1}\hat{J}_{ij},~{}~{}[\tilde{P}_{i},\tilde{P}_{j}]=-i\frac{\hbar}{\lambda^{{}^{\prime}2}}\beta_{2}\hat{J}_{ij},$
(88)

where $\hat{J}_{ij}=\epsilon_{ijk}\hat{L}^{k}$ and
$[\hat{L}_{i},\hat{L}_{j}]=i\hbar\epsilon_{ijk}\hat{L}_{k}$ we can define a
new basis for the spatial sector of the $so(2,4)$ algebra as:

$\hat{x}_{i}=\tilde{X}_{i}+\frac{\lambda\sqrt{\beta_{1}}}{\hbar}\hat{L}_{i},~{}~{}~{}\hat{p}_{i}=\tilde{P}_{i}+\frac{\sqrt{\beta_{2}}}{\lambda^{{}^{\prime}}}\hat{L}_{i}.$
(89)

Then, the spatial sector of the (anti) de Sitter algebra (40) has the
following structure:

$\displaystyle[\hat{x}_{i},\hat{x}_{j}]$ $\displaystyle=$ $\displaystyle
2i\lambda\sqrt{\beta_{1}}\epsilon_{ijk}\hat{x}_{k},\,\,\,\,\,\,[\hat{L}_{i},\hat{x}_{j}]=i\hbar\epsilon_{ijk}\hat{x}_{k},$
(90) $\displaystyle[\hat{p}_{i},\hat{p}_{j}]$ $\displaystyle=$
$\displaystyle\frac{2\hbar\sqrt{\beta_{2}}}{\lambda^{{}^{\prime}}}\epsilon_{ijk}\hat{p}_{k},\,\,\,[\hat{x}_{i},\hat{p}_{j}]=i\hbar\beta_{0}\hat{D}+i\frac{\lambda}{\lambda^{{}^{\prime}}}(\beta_{3}+\sqrt{\beta_{1}\beta_{2}})\epsilon_{ijk}\hat{L}_{k}-i\lambda\sqrt{\beta_{1}}\epsilon_{ijk}\hat{p}_{k}+i\frac{\hbar\sqrt{\beta_{2}}}{\lambda^{{}^{\prime}}}\epsilon_{ijk}\hat{x}_{k},$
(91) $\displaystyle[\hat{L}_{i},\hat{p}_{j}]$ $\displaystyle=$ $\displaystyle
i\hbar\epsilon_{ijk}\hat{p}_{k},\,\,\,\,\,\,\,\,\,\,\,[\hat{x}_{i},\hat{\mathcal{D}}]=i\frac{1}{\beta_{0}}[\frac{\lambda^{2}}{\hbar}\beta_{1}\hat{p}_{i}-\beta_{3}\frac{\lambda}{\lambda^{{}^{\prime}}}\hat{x}_{i}+\frac{\lambda^{2}}{\lambda^{{}^{\prime}}\hbar}(\beta_{3}\sqrt{\beta_{1}}-\beta_{1}\sqrt{\beta_{2}})\hat{L}_{i}],$
(92) $\displaystyle[\hat{p}_{i},\hat{\mathcal{D}}]$ $\displaystyle=$
$\displaystyle-i\frac{1}{\beta_{0}}[\frac{\hbar\beta_{2}}{\lambda^{{}^{\prime}2}}\hat{x}_{i}-\beta_{3}\frac{\lambda}{\lambda^{{}^{\prime}}}\hat{p}_{i}+\frac{\lambda}{\lambda^{{}^{\prime}2}}(\beta_{3}\sqrt{\beta_{2}}-\beta_{2}\sqrt{\beta_{1}})\hat{L}_{i}].$
(93)

where the form of $\hat{\mathcal{D}}$ is already defined through the Casimir
relationship given in equation (29). It is noteworthy that the above structure
bears resemblance to a nonstandard closure of the spatial components of the
deformed Heisenberg algebra. Indeed, it is crucial to note that the
commutation between spatial coordinates, as expressed in equation (93)

$[\hat{x}_{i},\hat{x}_{j}]=2i\lambda\epsilon_{ijk}\hat{x}_{k}$ (94)

illuminates a captivating geometric insight [64]. This commutation relation
unveils a three-dimensional space characterized by a set of concentric fuzzy
spheres of intersecting radii. This intricate spatial structure emerges
prominently when considering specific parameter values, specifically
$\beta_{1}=-\beta_{2}=\beta_{0}=1$ and $\beta_{3}=1$, chosen for brevity and
clarity. This structure has surfaced as a solution to modified Einstein
equations, resembling a black hole solution [65], UV/IR mixing freedom [70],
and quantum gauge theory [71]. Additionally, it has been independently
scrutinized across different quantum mechanical contexts, as referenced, for
instance, in [68].

Although the two representations above are algebraically equivalent, they may
be non-equivalent as representations. For this, one has to establish an
isometry between ${\cal H}_{q}$ and ${\cal L}^{2}({\cal R}^{3})$. This falls
outside the scope of the present paper but will be discussed in a follow-up.

##  5  Conclusion and Future Outlook

Let us summarise, point by point, the new significant findings of the paper,
comparing with existing results found in the literature.

  * •

In special relativity, space and time are treated as coordinates within a
smooth, commutative, flat manifold. This framework provides a unitary space-
time representation of the Lorentz algebra, $so(1,3)$, acting on square-
integrable functions over the space-time manifold. However, when transitioning
to quantum mechanics, there is a tendency to assume the existence of absolute
time. This assumption serves as a reference for describing time as an
evolutionary parameter at the quantum level.

In the Schrodinger picture, the state vectors initially exhibit no time
dependency, and time dependence is introduced to these states through their
evolution. Hence, the time “coordinate” $t$ is simply a parameter susceptible
to arbitrary selection and modification by the observer’s perspective [69].
This prompts the intriguing question: can we introduce the representation of
$so(1,3)$ and the concept of Lorentz four vectors in a “timeless” manner at
the quantum level? The answer is affirmative.

In this paper, a unitary representation of the Lorentz and (anti) de Sitter
algebras has been constructed on ${\cal L}^{2}({\cal R}^{3})$. Notably, time
remains entirely absent within this representation. However, it can be
introduced in a fully covariant manner through a parameter associated with a
time translation operator. This operator can naturally be selected as the
zeroth component of the non-commutative momentum four vector.

  * •

To construct this fully covariant formulation of quantum mechanics, we
required a deformation of the Heisenberg phase space algebra together with
Poincare algebra; however, it appears naturally, which may affect the entire
quantum mechanical framework of fields and particles. Here, we also observe
that the zeroth component of the position four vector does not represent the
time coordinate. In our formulation, Lorentz symmetry holds greater
significance than the Heisenberg phase space algebra at the quantum level.
Essentially, the Heisenberg algebra emerges as a consequence of the undeformed
Lorentz algebra realized on ${\cal L}^{2}({\cal R}^{3})$. This deformed
Heisenberg algebra can be categorized into two equivalent forms, where one
entails commutative coordinates and momentum. Conversely, the other involves a
non-commutative phase space Lie algebra, akin to the one deduced by Mandes
[56] and Yang [60]. Notably, our non-commutative deformed Heisenberg-Poincare
algebra also shares correspondence with a linear (Lie algebraic) rendition of
the non-linear triple special relativity (TSR) phase space algebra proposed by
Gilkman and Smolin [70], following the correct identification of its
generators.

  * •

Moreover, it’s crucial to note that the modification in our deformed phase
space algebra explicitly signifies an inherent curvature in both coordinate
and momentum spaces. This curvature is revealed through the non-commutative
structure of phase space variables [71], wherein the curvature scales in
position and momentum spaces are contingent upon the fundamental length scales
$\lambda$ and $\lambda^{{}^{\prime}}$, respectively. Remarkably, this
curvature persists even in the absence of conventional sources.

The mass of an individual particle is influenced by the momentum space’s
curvature scale $\lambda^{{}^{\prime}}$—an element akin to the cosmological
constant in General Relativity (GR)—and the Planck constant ($\hbar$), which
characterizes the quantum world, encapsulating the wave nature of particles.
We have also demonstrated that our representation of the deformed Heisenberg-
Poincaré algebra on ${\cal L}^{2}({\cal R}^{3})$ can be contracted to the
conventional Heisenberg and Poincaré algebra by taking appropriate flat limits
within fixed mass sectors of the physical Hilbert space. Also note that this
implies the necessity of curvature scales to achieve this unique
representation of the Lorentz algebra in the quantum domain.

In conclusion, unlocking Lorentz covariance at the level of the Schrödinger
evolution in quantum states leads to the recovery of a deformed Heisenberg-
Poincaré algebra. This strongly implies that our deformed Heisenberg-Poincaré
algebra is instrumental in delineating physical phenomena at the interface of
gravitational and quantum realms (IGQR).

  * •

We also demonstrated that the same algebraic construction can be made on fuzzy
${\cal R}^{3}_{\lambda}$, but at this stage, it is not clear whether these
constructions are truly equivalent as representations. This will be explored
elsewhere.

We close this section with some remarks on future developments. The most
pressing issue is how standard, classical (special) relativity emerges from
this picture in light of the different role time plays here. Relativistic
effects are well-tested phenomena, and if they cannot be reproduced, at least
in some approximation, this may invalidate the construction. Another obvious
outstanding issue is the physical content of the current construction. For
this, one would like to explore the spectral properties of the observables in
the two constructions, as these have specific physical consequences analysed
in [43, 44, 45, 46, 47] for fuzzy ${\cal R}^{3}_{\lambda}$. The applicability
of these results in the case of the ${\cal L}^{2}({\cal R}^{3})$
representation hinges critically on the equivalence of these representations
and subsequently the spectral properties of the observables. An important
issue that needs to be explored further is the possible impact of this
construction on cosmic, galactic, and solar length scales. In particular, one
would like to explore whether it may lead to Modified Newton Dynamics (MOND).
One strongly suspects that this may be the case, as this was already confirmed
in [47] for fuzzy ${\cal R}^{3}_{\lambda}$. This may also have a bearing on
the Hubble tension problem [72, 73].

There are also several other possible experimental routes to explore our
proposed top-down approach to Lorentz-covariant quantum mechanics. For
example, recently, there have been many phenomenological studies in atomic
physics, as some experiments in atomic physics are now sensitive to small
frequency shifts below 1 mHz [74]. With such sensitivity, IGQR effects might
be detected at low energies, especially if small energy shifts have a
qualitative impact. Here, such a possibility is analyzed by looking at the
effect of the non-commutative algebra on the spectrum of the quantum Coulomb
problem, even if the corrections turn out to be too small to be measurable in
the near future.

Furthermore, within our formulation, non-commutative momentum signals the
modification of zero-point energy eigenvalues (see [75] for detail), a factor
measurable through precise laboratory experiments involving Planck mass
oscillators and superconducting quantum interference devices (SQUIDs). This is
evident as SQUIDs are known to conduct supercurrents with a tunable
superconducting mass dependent on temperature, potentially providing a
laboratory signature [76] of the IGQR effect.

##  6 Acknowledgements

One of the authors (PN) acknowledges the support from a postdoctoral
fellowship grants from Stellenbosch University, South Africa.

## Appendix A

Transforming from the complex variables to the Cartesian coordinates
$(x_{1},x_{2},x_{3})$ and $\gamma$ we find for the Lorentz generators

$\displaystyle\hat{L}_{k}$ $\displaystyle=$
$\displaystyle\frac{ir\alpha_{k}}{2\left(x_{1}^{2}+x_{2}^{2}\right)}\partial_{\gamma}-i\epsilon_{kj\ell}x_{j}\partial_{\ell}$
(95) $\displaystyle\hat{K}_{k}$ $\displaystyle=$
$\displaystyle\frac{i\beta_{k}}{2\left(x_{1}^{2}+x_{2}^{2}\right)}\partial_{\gamma}-ir\partial_{k}$
(96)

where $\alpha_{k}=\left(x_{1},-x_{2},0\right)$,
$\beta_{k}=\left(x_{2}x_{3},-x_{1}x_{3},0\right)$ and
$r=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$.

Next we consider the Lorentz scalar $\hat{D}$. Since this is a Lorentz scalar,
it commutes with the Lorentz generators and can be used to label the SO(1,3)
representations carried by $\mathcal{L}^{2}(\mathcal{R}^{3})$ . Note though
that this operator does not commute with all the bilinears constructed above.
We find its Cartesian representation to be

$\hat{D}=i\left(x\cdot\nabla+1\right),$ (97)

which is just the dilation operator.

For the commutative 4-vector $\hat{V}^{\mu}_{-}$ we find

$\hat{V}^{\mu}_{-}=(r,x,y,z),$ (98)

which justifies the interpretation of these operators as commuting
coordinates, and for the commutative 4-vector $\hat{V}^{\mu}_{+}$

$\displaystyle\hat{V}^{0}_{+}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\left(x_{1}^{2}+x_{2}^{2}\right)}\left(r\partial_{\gamma}-4x_{2}x_{3}\partial_{1}+4x_{1}x_{3}\partial_{2}\right)\partial_{\gamma}-\frac{r}{4}\nabla^{2}$
$\displaystyle\hat{V}^{1}_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{16\left(x_{1}^{2}+x_{2}^{2}\right)}\left(x_{1}\partial_{\gamma}-4rx_{2}\partial_{3}\right)\partial_{\gamma}+\frac{1}{4}\left(2\partial_{1}-x_{1}\nabla^{2}+2\left(x\cdot\nabla\right)\partial_{1}\right)$
$\displaystyle\hat{V}^{2}_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{16\left(x_{1}^{2}+x_{2}^{2}\right)}\left(x_{2}\partial_{\gamma}+4rx_{1}\partial_{3}\right)\partial_{\gamma}+\frac{1}{4}\left(2\partial_{2}-x_{2}\nabla^{2}+2\left(x\cdot\nabla\right)\partial_{2}\right)$
$\displaystyle\hat{V}^{3}_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{16\left(x_{1}^{2}+x_{2}^{2}\right)}\left(-x_{3}\partial_{\gamma}+4r\left(x_{2}\partial_{1}-x_{1}\partial_{2}\right)\right)\partial_{\gamma}+\frac{1}{4}\left(2\partial_{3}-x_{3}\nabla^{2}+2\left(x\cdot\nabla\right)\partial_{3}\right)$

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