# GTR Effects in Quantum Theory as Operator Algebras
## Deformation Cause

Julio Cesar Jaramillo Quiceno, Plamen Neitchev Nechev

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Julio Cesar Jaramillo Quiceno, Plamen Neitchev Nechev. GTR Effects in Quantum Theory as Operator Algebras Deformation Cause. 2025. hal-05144218

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GTR Effects in Quantum Theory as
Operator Algebras Deformation Cause

Julio Cesar Jaramillo Quiceno¹, Plamen Neytchev Nechev²

¹Departamento de Física, Universidad Nacional de Colombia,
Edificio Yu Takeuchi, Bogotá

²Independent Researcher, Strasbourg, France.

¹jcjaramilloq@unal.edu.co, ²pnechev@abv.bg.

**Abstract**

This work explores the deformation of operator algebras arising from general relativistic effects in quantum theory. We present a geometric framework where time-space curvature induces modifications to canonical commutation relations in a *q*-deformed Heisenberg algebra. By approximating the general relativistic manifold with its tangent space at a point, we derive a deformed Minkowski timespace metric that alters the fundamental quantum mechanical operators. The symmetric and antisymmetric components of a covariant 2-form field naturally describe gravitational and electromagnetic interactions, respectively, within this framework. We construct explicit quantization rules demonstrating how time-space curvature parameters modify the standard commutation relations, leading to a generalized harmonic oscillator model. The resulting *q*-deformed algebra differs significantly from conventional treatments, particularly in its failure to satisfy standard *q*-deformation relation. This approach provides new insights into the interface between general relativity and quantum mechanics, suggesting observable consequences for quantum systems in curved timespace.

**Keywords:**

General Relativity, Quantum Mechanics, Operator Algebras, *q*-Deformation, Time-space Curvature, Harmonic Oscillator.
# 1 Introduction

The unification of gravitational and electromagnetic interactions remains one of the most profound challenges in theoretical physics. While General Relativity (GR) successfully describes gravity as the curvature of a differentiable manifold, and Maxwell’s equations elegantly encapsulate electromagnetic phenomena, a consistent theoretical framework integrating both remains elusive. In this context, differential 2-forms provide a geometric language for modeling these classical fields, where the symmetric and antisymmetric parts of a covariant tensor field correspond to gravitational and electromagnetic components, respectively [1–4]. In order to reconcile the apparent incompatibility between curved time-space and quantum mechanics, we approximate the GR manifold locally by its tangent space at a given point, effectively “flattening” the manifold into a “Lorentzian hyperplane” [7]. This approximation facilitates the application of relativistic and non-relativistic quantum mechanics, where position and momentum are represented as operators acting on an equipped Hilbert space [10]. The deformation of Minkowski space by a pseudo-Riemannian metric modifies the canonical commutation relations, reflecting the underlying geometric structure corresponding to gravity. In this work, we develop a $q$-deformed operator algebra arising from this geometric setting. We analyze how the time-space metric influences commutation relations and demonstrate the emergence of a consistent $q$-deformed Heisenberg algebra. A harmonic oscillator model is constructed within this framework, highlighting deviations from conventional treatments and implications for quantization rules and symmetry representations. The paper is organized as follows: Section 2 introduces the geometric modeling of classical fields via differential forms and discusses the deformation of Minkowski timespace and its consequences for quantum mechanical operators. In Section 3, we provide a comprehensive discussion of the results and draw significant conclusions regarding the $q$-deformed operator algebras in curved timespace.

# 2 Differential 2-forms as a description of the gravitational and electromagnetic fields

Let us assume that our time-space in the sense of General Relativity is “modelled” sufficiently rigorously by any of the differentiable manifolds homeomorphic to the standard topological space $\mathbb{R} \times \mathbb{S}^3$. It is well-known that, this topological space has unnumerable differentiable structures [5]. Suppose further that our reference frame choice’s selects the appropriate differentiable structure (each one of those allows the introduction of pseudo-Riemannian metrics [6]; the signature chois is $(+,-,-,-)$ too). Consequently, we can describe the existence of the two classical
fields (i.e. gravitational and electromagnetic) with the sections of the cotangent manifold tensor product (or "square") $T^*(\mathbb{R} \times \mathbb{S}^3) \otimes T^*(\mathbb{R} \times \mathbb{S}^3)$.

For any point $x \in \mathbb{R} \times \mathbb{S}^3$ this section selects a 2-form belonging to the cotangent fibre $T_x^*(\mathbb{R} \times \mathbb{S}^3) \otimes T_x^*(\mathbb{R} \times \mathbb{S}^3)$. At the considered point $x \in \mathbb{R} \times \mathbb{S}^3$ we are located our reference frame's origin. Thus, the symmetric part of above 2-form $g \in T_x^*(\mathbb{R} \times \mathbb{S}^3) \otimes T_x^*(\mathbb{R} \times \mathbb{S}^3)$ [8] leads to the Universal Gravitation Law while the antisymmetric part, which is an element of $T_x^*(\mathbb{R} \times \mathbb{S}^3) \wedge T_x^*(\mathbb{R} \times \mathbb{S}^3)$, involves the Maxwell's equations.

Naturally, the big problems of unifying both fields are common knowledge. One way out of this situation is to "flatten" the continuum by replacing it with the tangent (four-dimensional) plane at point $x$ turning it into a Minkowski time-space. Here, we will not consider the previous problem but will focus on the metric, that is, the symmetric part of the tensor $g$.

Let $X$ ($\approx \mathbb{R} \times \mathbb{S}^3$) be the GR continuum and $x \in X$ the point of consideration. As soon as the pseudo-metric $g \in T_x^*(X) \odot T_x^*(X)$ the Silvester theorem [12] implies that there exists a local map $U_x$ where the above pseudo-metric tensor $g = |a|dx^0 \odot dx^0 - |b|dx^1 \odot dx^1 - |c|dx^2 \odot dx^2 - |d|dx^3 \odot dx^3, abcd \neq 0$; which is equivalent to a Lorentz metric with different units of measurement along the four axes $\chi = \sqrt{|a|}x^0, \xi = \sqrt{|b|}x^1, \eta = \sqrt{|c|}x^2$ and $\zeta = \sqrt{|d|}x^3$:

$$g = d\chi \odot d\chi - d\xi \odot d\xi - d\eta \odot d\eta - d\zeta \odot d\zeta. \quad (1)$$

Under the above conditions, $T_x(X)$ can be considered as a "Lorentz-like" hyper-plane tangent to $x$ that approximates the manifold $X$ in a certain neighborhood of the same point. Let us annotate the group of symmetry of the pseudo-metric (1) by $L_{abcd}$ which, is a subgroup of the diffeomorphisms at the point $x$. Its elements are the matrices:

$$\underline{\Lambda} = \begin{pmatrix} \underline{\Lambda}_0^0 & \underline{\Lambda}_1^0 & \underline{\Lambda}_2^0 & \underline{\Lambda}_3^0 \\ \underline{\Lambda}_0^1 & \underline{\Lambda}_1^1 & \underline{\Lambda}_2^1 & \underline{\Lambda}_3^1 \\ \underline{\Lambda}_0^2 & \underline{\Lambda}_1^2 & \underline{\Lambda}_2^2 & \underline{\Lambda}_3^2 \\ \underline{\Lambda}_0^3 & \underline{\Lambda}_1^3 & \underline{\Lambda}_2^3 & \underline{\Lambda}_3^3 \end{pmatrix},$$

being $det\underline{\Lambda} > 0$ (to do not change the tanged space orientation). To conclude, its respective orthochronous subgroup $L_{abcd}^\dagger$ requires that $\underline{\Lambda}_0^0 \ge 1$ and adding the translations, the elements of the corresponding Poincaré group $\mathcal{P}$ are ($g \neq g$):

$$g = \left\| \begin{array}{cc} \underline{\Lambda} & \underline{t} \\ 0 & 1 \end{array} \right\|, \quad (2)$$

where the block $\underline{t} = \begin{pmatrix} \underline{t}^0 \\ \underline{t}^1 \\ \underline{t}^2 \\ \underline{t}^3 \end{pmatrix}$. Clearly, the Euclidean geodesic lines of a continuum
seen as imbedded into a flat space, are always shorter (in the limiting case equal to) than those of a curved space [11].

## 3. Deformed Minkowski time-space and its implications

In relativistic quantum mechanics the states of the particles are determined by the corresponding bra-vectors (distributions), which, for the exact quantum stats, are eigenfunctional belonging to a respective equipped Hilbert space [10]. For us, the most important now are the positions' and its momentums' operators. Following the fundamental insight and logic of the quantum mechanics, we once more must treat the position operators' actings as multiplications of the respective vectors within an equipped Hilbert space, being $\chi, \xi, \eta$ and $\zeta$ this time the coordinates of an event in a "Minkowski" time-space, deformed by the metric fixed through Eq. (1). It is obvious, its commutations are subject to the following relations:

$$ \hat{\xi}\hat{\eta} - \hat{\eta}\hat{\xi} = 0, \quad \hat{\eta}\hat{\zeta} - \hat{\zeta}\hat{\eta} = 0, \quad \hat{\zeta}\hat{\xi} - \hat{\xi}\hat{\zeta} = 0. \qquad (3) $$

The momentum (and energy) operators are obtained from the as called generators of the translation group's representations (which elements are the matrices Eq. (2) where, the block $\Delta$ is replaced by the unitary minor). From the definition of this representation

$$ R_g \psi(x) = \psi(g^{-1}x), $$

for its generators we obtain

$$ \hat{I}_i \psi(x) = \frac{\partial R_g}{\partial \hat{t}^i} \Big|_{\hat{t}=0} \psi(x). $$

Considering that both, the coordinates $x, y, z$ and the translations $t$ are elements of the Minkowski space, we can write down:

$$ \hat{p}_x = -i\hbar\frac{\partial}{\partial x}, \quad \hat{p}_y = -i\hbar\frac{\partial}{\partial y}, \quad \hat{p}_z = -i\hbar\frac{\partial}{\partial z}. $$

Obviously

$$ \hat{p}_x\hat{p}_y - \hat{p}_y\hat{p}_x = 0, \quad \hat{p}_y\hat{p}_z - \hat{p}_z\hat{p}_y = 0, \quad \hat{p}_z\hat{p}_x - \hat{p}_x\hat{p}_z = 0. \qquad (4) $$

Since the momentum and energy introductions are strictly justified as long as the conditions for their conservation exist, the guesstimate consisting of the treatment of the tangent space as a Minkowski space would be acceptable. Therefore,
in the referential frame obtained by the Silvester method we can calculate the following commutators which establish the quantization rules:

$$ \hat{\xi}\hat{p}_x - \hat{p}_x\hat{\xi} = i\hbar\sqrt{|b|}, \quad \hat{\eta}\hat{p}_y - \hat{p}_y\hat{\eta} = i\hbar\sqrt{|c|}, \quad \hat{\zeta}\hat{p}_z - \hat{p}_z\hat{\zeta} = i\hbar\sqrt{|d|}. \quad (5) $$

Without encountering any problems, we can add the rest of the commutators

$$ \hat{\xi}\hat{p}_y - \hat{p}_y\hat{\xi} = \hat{\xi}\hat{p}_z - \hat{p}_z\hat{\xi} = \hat{\eta}\hat{p}_x - \hat{p}_x\hat{\eta} = \hat{\eta}\hat{p}_z - \hat{p}_z\hat{\eta} = \hat{\zeta}\hat{p}_x - \hat{p}_x\hat{\zeta} = \hat{\zeta}\hat{p}_y - \hat{p}_y\hat{\zeta} = 0, $$

thereby removing the complications associated with momentum deformation

In this order of ideas, using the non-relativistic relationship between energy and impulses ((in the non-relativistic case, no time deformation can be considered since the time is absolute and the energy is expressed through the measurable coordinates, for example the penetration into an emulsion, etc.) we can obtain:

$$ \hat{H} = -\frac{1}{2m} \left[ \left( \hbar \sqrt{|b|} \right)^2 \frac{\partial^2}{\partial \xi^2} + \left( \hbar \sqrt{|c|} \right)^2 \frac{\partial^2}{\partial \eta^2} + \left( \hbar \sqrt{|d|} \right)^2 \frac{\partial^2}{\partial \zeta^2} \right]. \quad (6) $$

If, because the differences among $|b|, |c|$ and $|d|$ are minor or/and, through an appropriate choice of frame $|b| = |c| = |d|$ then, by writing down its square roots as $q$, from Eq. (5) and (6) arrive at the following simplification:

$$ \hat{\zeta}\hat{p}_x - \hat{p}_x\hat{\zeta} = \hat{\eta}\hat{p}_y - \hat{p}_y\hat{\eta} = \hat{\zeta}\hat{p}_z - \hat{p}_z\hat{\zeta} = i\hbar q, \quad (7) $$

and

$$ \hat{H} = -\frac{\hbar^2 q^2}{2m} \left[ \frac{\partial^2}{\partial \xi^2} + \frac{\partial^2}{\partial \eta^2} + \frac{\partial^2}{\partial \zeta^2} \right]. \quad (8) $$

For free motion the impulses $\hat{p}_x, \hat{p}_y$ and $\hat{p}_z$ commute with the Hamiltonian Eq. (8) and according to Eq. (7) the stationary plane wave is:

$$ \psi \sim e^{i \frac{p \cdot x}{\hbar q} \xi}. $$

In order to construct the respective q-Heisenberg operator algebra let us consider a harmonic quantum oscillator with frequency $\omega$. Defining the annihilation and creation operators as follows:

$$ \hat{a} = \frac{1}{\sqrt{2}} \left( \sqrt{\frac{m\omega}{\hbar k}} \hat{\xi} + \frac{i}{\sqrt{m\omega\hbar}} \hat{p}_x \right), \quad \hat{a}^{\dagger} = \frac{1}{\sqrt{2}} \left( \sqrt{\frac{m\omega}{\hbar k}} \hat{\xi} - \frac{i}{\sqrt{m\omega\hbar}} \hat{p}_x \right), \quad (9) $$

and introducing the quantum number operator $\hat{N} = \hat{a}^{\dagger}\hat{a}$ we will have (see Eq. (9))
$$[\hat{a}, \hat{a}] = [\hat{a}^{+}, \hat{a}^{+}] = 0, \quad [\hat{N}, \hat{a}^{+}] = \hat{a}^{+}, \quad [\hat{N}, \hat{a}] = -\hat{a}.$$

Therefore, the q-Heisenberg algebra is defined by the generators $\hat{1}$, $\hat{a}$, $\hat{a}^{+}$ and $\hat{N}$ and unlike [13] it is not subject to relations

$$\hat{a}\hat{a}^{+} - q\hat{a}^{+}\hat{a} = q^{-N},$$

unfourtanelly, the "commutations" as the last, do not show how the sets of operators with common eigenvectors could commute.

## 3 Discussion and conclusions

Although the derived here q-deformed Heisenberg algebra introduces time-space curvature parameters into quantum mechanics, modifying canonical quantization rules, our framework does not satisfy all the conventional q-algebraic deformed relations (for example, the last formula) suggesting a distinct algebraic structure. In any case, it is an approximation because, for a given point, the GR continuum is changed by a deformed tangent hyperplane since a, b, c and d are constants. It is obvious, that the parameter q depends on the flattening site. For example, if on the Earth q has some value > 1, when measured during an experiment carried out on a satellite (or in outer space) q = 1. In other words, in this case the Heisenberg Algebra will not be deformed.

It is clear that, if we want to work on the GR continuum, we must consider the metric coefficients as functions of the coordinates and integrate the symmetric differential form g (as far as possible). In relation to the Dirac equation, we will go into more detail about this aspect in a subsequent article on the subject.

Likewise, one can conclude (following the path of [9]) that for the case of the (non-relativistic) spin we will have the same deformed commutation relations Eqs. (3), (4), (5), (7), etc.

In conclusion, we could assert that the so-called (and mentioned above) q-deformation in writing the canonical quantization equations are better considered as a *legacy* of General Relativity understood from the perspective (because the measured distances could sometimes not be by perfect segments) of Minkowski space.

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