The Geometric Origin of the Quantum Potential in Complex Spacetime

Bhushan Poojary

doi:doi:10.11648/j.ajmp.20261502.16

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The Geometric Origin of the Quantum Potential in Complex Spacetime

Bhushan Poojary^*

Published in American Journal of Modern Physics (Volume 15, Issue 2)

Received: 18 March 2026 Accepted: 28 March 2026 Published: 13 April 2026

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Abstract

Background: The unification of General Relativity and Quantum Mechanics represents a fundamental challenge in theoretical physics due to their mutually exclusive conceptual foundations: continuous deterministic geometry versus probabilistic wave functions. While the de Broglie-Bohm pilot-wave theory offers a deterministic, trajectory-based alternative to the standard probabilistic framework, it introduces a non-local, ad-hoc "Quantum Potential" (Q) to account for phenomena such as interference and non-locality. The lack of a geometric origin for Q in standard spacetime remains a significant conceptual barrier to a fully unified theory. Purpose: This paper aims to resolve this interpretational crisis by proposing that the Quantum Potential is not a fundamental or mysterious external force, but rather a derived geometric artifact. We postulate that the physical universe is fundamentally a 4-dimensional Complex Manifold (ℂ⁴), extending the standard Riemannian spacetime manifold to include imaginary coordinates. Methods: To establish this geometric basis, we treat the quantum wave function not as an abstract probabilistic amplitude, but as a physical, holomorphic map describing a particle's deterministic trajectory through this complex spacetime. Utilizing the Cauchy-Riemann conditions, we demonstrate that the Bohmian amplitude (R) is strictly coupled to the particle's location in the imaginary dimension. We then analyze the dynamics of particles following geodesics within ℂ⁴. By projecting the complex geodesic equation of motion onto the observable real slice (ℝ⁴), we separate the real and imaginary components of the complex 4-velocity to observe the energy balance. Conclusions: Our derivation reveals that the Quantum Potential (Q) emerges naturally and is mathematically identical to the kinetic energy component associated with a particle's hidden motion in these imaginary dimensions. This formulation successfully recovers the predictions of the Schrödinger equation while removing the ad-hoc nature of Bohmian mechanics. Furthermore, it interprets quantum non-locality—such as the interference observed in the double-slit experiment—as purely local, deterministic geodesic motion around topological singularities in a curved complex manifold. Ultimately, this framework provides a unified geometric description wherein gravity is the curvature of real coordinates, electromagnetism is the torsion of imaginary coordinates, and quantum mechanics is inertial motion through this complex geometry.

Published in	American Journal of Modern Physics (Volume 15, Issue 2)
DOI	10.11648/j.ajmp.20261502.16
Page(s)	55-61
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Bohmian Mechanics, Quantum, Potential, Complex Spacetime, Holomorphic Wave Function, Geodesic Motion, Geometric Unification

References

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[2]	Einstein, A. The Foundation of the General Theory of Relativity. Annals of Physics. 1916, 354(7), 769-822. https://doi.org/10.1002/andp.19163540702
[3]	Newman, E. T. Complex Coordinate Transformations and the Schwarzschild Metric. Journal of Mathematical Physics. 1973, 14, 102. https://doi.org/10.1063/1.1666160
[4]	Poojary, B. Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields. American Journal of Modern Physics. 2025, 14(4), 186-193. https://doi.org/10.11648/j.ajmp.20251404.12
[5]	Holland, P. R. The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press. 1993. https://doi.org/10.1017/CBO9780511622687
[6]	Penrose, R. Twistor Algebra. Journal of Mathematical Physics. 1967, 8(2), 345. https://doi.org/10.1063/1.1705200
[7]	Santamato, E. Statistical interpretation of the Klein-Gordon equation from the geometry of Weyl space. Journal of Mathematical Physics. 1984, 25, 2477. https://doi.org/10.1063/1.526485
[8]	Aharonov, Y., & Bohm, D. Significance of Electromagnetic Potentials in the Quantum Theory. Physical Review. 1959, 115, 485. https://doi.org/10.1103/PhysRev.115.485
[9]	Anza, F., Crutchfield, J. P. Maximum Geometric Quantum Entropy. Entropy. 2024, 26(3), 225. https://doi.org/10.3390/e26030225
[10]	Ahmad, H. A. S., Umair, H., Nurisya, M. S., Chan, K. T. Classical aspect of spin angular momentum in geometric quantum mechanics. Mathematical Modeling and Computing. 2025, 12(1), 49-56. https://doi.org/10.23939/mmc2025.01.049
[11]	Rahmani, F. Gravitational reduction of the wave function through the quantum theory of motion. General Relativity and Gravitation. 2025, 57(2), 24. https://doi.org/10.1007/s10714-025-03324-x
[12]	Haba, Z. Functional Formulation of Quantum Theory of a Scalar Field in a Metric with Lorentzian and Euclidean Signatures. Entropy. 2024, 26(4), 305. https://doi.org/10.3390/e26040305
[13]	Wen, X. Exactly solvable non-unitary time evolution in quantum critical systems I: effect of complex spacetime metrics. Journal of Statistical Mechanics: Theory and Experiment. 2024, 103101. https://doi.org/10.1088/1742-5468/ad7c3d
[14]	Poojary, B. Holographic Address Space: A Framework for Unifying Quantum Mechanics and General Relativity. Journal of Physics & Astronomy. 2024, 12(12), 346. https://doi.org/10.37532/2320-6756.2024.12(12).346
[15]	Bondarenko, S. Dynamical Signature: Complex Manifolds, Gauge Fields and Non-Flat Tangent Space. Universe. 2022, 8(10), 544. https://doi.org/10.3390/universe8100544

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Poojary, B. (2026). The Geometric Origin of the Quantum Potential in Complex Spacetime. American Journal of Modern Physics, 15(2), 55-61. https://doi.org/10.11648/j.ajmp.20261502.16

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Poojary, B. The Geometric Origin of the Quantum Potential in Complex Spacetime. Am. J. Mod. Phys. 2026, 15(2), 55-61. doi: 10.11648/j.ajmp.20261502.16

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Poojary B. The Geometric Origin of the Quantum Potential in Complex Spacetime. Am J Mod Phys. 2026;15(2):55-61. doi: 10.11648/j.ajmp.20261502.16

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@article{10.11648/j.ajmp.20261502.16,
  author = {Bhushan Poojary},
  title = {The Geometric Origin of the Quantum Potential in Complex Spacetime},
  journal = {American Journal of Modern Physics},
  volume = {15},
  number = {2},
  pages = {55-61},
  doi = {10.11648/j.ajmp.20261502.16},
  url = {https://doi.org/10.11648/j.ajmp.20261502.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20261502.16},
  abstract = {Background: The unification of General Relativity and Quantum Mechanics represents a fundamental challenge in theoretical physics due to their mutually exclusive conceptual foundations: continuous deterministic geometry versus probabilistic wave functions. While the de Broglie-Bohm pilot-wave theory offers a deterministic, trajectory-based alternative to the standard probabilistic framework, it introduces a non-local, ad-hoc "Quantum Potential" (Q) to account for phenomena such as interference and non-locality. The lack of a geometric origin for Q in standard spacetime remains a significant conceptual barrier to a fully unified theory. Purpose: This paper aims to resolve this interpretational crisis by proposing that the Quantum Potential is not a fundamental or mysterious external force, but rather a derived geometric artifact. We postulate that the physical universe is fundamentally a 4-dimensional Complex Manifold (ℂ4), extending the standard Riemannian spacetime manifold to include imaginary coordinates. Methods: To establish this geometric basis, we treat the quantum wave function not as an abstract probabilistic amplitude, but as a physical, holomorphic map describing a particle's deterministic trajectory through this complex spacetime. Utilizing the Cauchy-Riemann conditions, we demonstrate that the Bohmian amplitude (R) is strictly coupled to the particle's location in the imaginary dimension. We then analyze the dynamics of particles following geodesics within ℂ4. By projecting the complex geodesic equation of motion onto the observable real slice (ℝ4), we separate the real and imaginary components of the complex 4-velocity to observe the energy balance. Conclusions: Our derivation reveals that the Quantum Potential (Q) emerges naturally and is mathematically identical to the kinetic energy component associated with a particle's hidden motion in these imaginary dimensions. This formulation successfully recovers the predictions of the Schrödinger equation while removing the ad-hoc nature of Bohmian mechanics. Furthermore, it interprets quantum non-locality—such as the interference observed in the double-slit experiment—as purely local, deterministic geodesic motion around topological singularities in a curved complex manifold. Ultimately, this framework provides a unified geometric description wherein gravity is the curvature of real coordinates, electromagnetism is the torsion of imaginary coordinates, and quantum mechanics is inertial motion through this complex geometry.},
 year = {2026}
}

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TY  - JOUR
T1  - The Geometric Origin of the Quantum Potential in Complex Spacetime
AU  - Bhushan Poojary
Y1  - 2026/04/13
PY  - 2026
N1  - https://doi.org/10.11648/j.ajmp.20261502.16
DO  - 10.11648/j.ajmp.20261502.16
T2  - American Journal of Modern Physics
JF  - American Journal of Modern Physics
JO  - American Journal of Modern Physics
SP  - 55
EP  - 61
PB  - Science Publishing Group
SN  - 2326-8891
UR  - https://doi.org/10.11648/j.ajmp.20261502.16
AB  - Background: The unification of General Relativity and Quantum Mechanics represents a fundamental challenge in theoretical physics due to their mutually exclusive conceptual foundations: continuous deterministic geometry versus probabilistic wave functions. While the de Broglie-Bohm pilot-wave theory offers a deterministic, trajectory-based alternative to the standard probabilistic framework, it introduces a non-local, ad-hoc "Quantum Potential" (Q) to account for phenomena such as interference and non-locality. The lack of a geometric origin for Q in standard spacetime remains a significant conceptual barrier to a fully unified theory. Purpose: This paper aims to resolve this interpretational crisis by proposing that the Quantum Potential is not a fundamental or mysterious external force, but rather a derived geometric artifact. We postulate that the physical universe is fundamentally a 4-dimensional Complex Manifold (ℂ4), extending the standard Riemannian spacetime manifold to include imaginary coordinates. Methods: To establish this geometric basis, we treat the quantum wave function not as an abstract probabilistic amplitude, but as a physical, holomorphic map describing a particle's deterministic trajectory through this complex spacetime. Utilizing the Cauchy-Riemann conditions, we demonstrate that the Bohmian amplitude (R) is strictly coupled to the particle's location in the imaginary dimension. We then analyze the dynamics of particles following geodesics within ℂ4. By projecting the complex geodesic equation of motion onto the observable real slice (ℝ4), we separate the real and imaginary components of the complex 4-velocity to observe the energy balance. Conclusions: Our derivation reveals that the Quantum Potential (Q) emerges naturally and is mathematically identical to the kinetic energy component associated with a particle's hidden motion in these imaginary dimensions. This formulation successfully recovers the predictions of the Schrödinger equation while removing the ad-hoc nature of Bohmian mechanics. Furthermore, it interprets quantum non-locality—such as the interference observed in the double-slit experiment—as purely local, deterministic geodesic motion around topological singularities in a curved complex manifold. Ultimately, this framework provides a unified geometric description wherein gravity is the curvature of real coordinates, electromagnetism is the torsion of imaginary coordinates, and quantum mechanics is inertial motion through this complex geometry.
VL  - 15
IS  - 2
ER  -

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Author Information

Bhushan Poojary

Physics, NIMS University, Jaipur, India

Contact Email

http://orcid.org/0000-0002-5122-0236

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Poojary, B. (2026). The Geometric Origin of the Quantum Potential in Complex Spacetime. American Journal of Modern Physics, 15(2), 55-61. https://doi.org/10.11648/j.ajmp.20261502.16

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ACS Style

Poojary, B. The Geometric Origin of the Quantum Potential in Complex Spacetime. Am. J. Mod. Phys. 2026, 15(2), 55-61. doi: 10.11648/j.ajmp.20261502.16

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AMA Style

Poojary B. The Geometric Origin of the Quantum Potential in Complex Spacetime. Am J Mod Phys. 2026;15(2):55-61. doi: 10.11648/j.ajmp.20261502.16

Copy | Download

@article{10.11648/j.ajmp.20261502.16,
  author = {Bhushan Poojary},
  title = {The Geometric Origin of the Quantum Potential in Complex Spacetime},
  journal = {American Journal of Modern Physics},
  volume = {15},
  number = {2},
  pages = {55-61},
  doi = {10.11648/j.ajmp.20261502.16},
  url = {https://doi.org/10.11648/j.ajmp.20261502.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20261502.16},
  abstract = {Background: The unification of General Relativity and Quantum Mechanics represents a fundamental challenge in theoretical physics due to their mutually exclusive conceptual foundations: continuous deterministic geometry versus probabilistic wave functions. While the de Broglie-Bohm pilot-wave theory offers a deterministic, trajectory-based alternative to the standard probabilistic framework, it introduces a non-local, ad-hoc "Quantum Potential" (Q) to account for phenomena such as interference and non-locality. The lack of a geometric origin for Q in standard spacetime remains a significant conceptual barrier to a fully unified theory. Purpose: This paper aims to resolve this interpretational crisis by proposing that the Quantum Potential is not a fundamental or mysterious external force, but rather a derived geometric artifact. We postulate that the physical universe is fundamentally a 4-dimensional Complex Manifold (ℂ4), extending the standard Riemannian spacetime manifold to include imaginary coordinates. Methods: To establish this geometric basis, we treat the quantum wave function not as an abstract probabilistic amplitude, but as a physical, holomorphic map describing a particle's deterministic trajectory through this complex spacetime. Utilizing the Cauchy-Riemann conditions, we demonstrate that the Bohmian amplitude (R) is strictly coupled to the particle's location in the imaginary dimension. We then analyze the dynamics of particles following geodesics within ℂ4. By projecting the complex geodesic equation of motion onto the observable real slice (ℝ4), we separate the real and imaginary components of the complex 4-velocity to observe the energy balance. Conclusions: Our derivation reveals that the Quantum Potential (Q) emerges naturally and is mathematically identical to the kinetic energy component associated with a particle's hidden motion in these imaginary dimensions. This formulation successfully recovers the predictions of the Schrödinger equation while removing the ad-hoc nature of Bohmian mechanics. Furthermore, it interprets quantum non-locality—such as the interference observed in the double-slit experiment—as purely local, deterministic geodesic motion around topological singularities in a curved complex manifold. Ultimately, this framework provides a unified geometric description wherein gravity is the curvature of real coordinates, electromagnetism is the torsion of imaginary coordinates, and quantum mechanics is inertial motion through this complex geometry.},
 year = {2026}
}

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TY  - JOUR
T1  - The Geometric Origin of the Quantum Potential in Complex Spacetime
AU  - Bhushan Poojary
Y1  - 2026/04/13
PY  - 2026
N1  - https://doi.org/10.11648/j.ajmp.20261502.16
DO  - 10.11648/j.ajmp.20261502.16
T2  - American Journal of Modern Physics
JF  - American Journal of Modern Physics
JO  - American Journal of Modern Physics
SP  - 55
EP  - 61
PB  - Science Publishing Group
SN  - 2326-8891
UR  - https://doi.org/10.11648/j.ajmp.20261502.16
AB  - Background: The unification of General Relativity and Quantum Mechanics represents a fundamental challenge in theoretical physics due to their mutually exclusive conceptual foundations: continuous deterministic geometry versus probabilistic wave functions. While the de Broglie-Bohm pilot-wave theory offers a deterministic, trajectory-based alternative to the standard probabilistic framework, it introduces a non-local, ad-hoc "Quantum Potential" (Q) to account for phenomena such as interference and non-locality. The lack of a geometric origin for Q in standard spacetime remains a significant conceptual barrier to a fully unified theory. Purpose: This paper aims to resolve this interpretational crisis by proposing that the Quantum Potential is not a fundamental or mysterious external force, but rather a derived geometric artifact. We postulate that the physical universe is fundamentally a 4-dimensional Complex Manifold (ℂ4), extending the standard Riemannian spacetime manifold to include imaginary coordinates. Methods: To establish this geometric basis, we treat the quantum wave function not as an abstract probabilistic amplitude, but as a physical, holomorphic map describing a particle's deterministic trajectory through this complex spacetime. Utilizing the Cauchy-Riemann conditions, we demonstrate that the Bohmian amplitude (R) is strictly coupled to the particle's location in the imaginary dimension. We then analyze the dynamics of particles following geodesics within ℂ4. By projecting the complex geodesic equation of motion onto the observable real slice (ℝ4), we separate the real and imaginary components of the complex 4-velocity to observe the energy balance. Conclusions: Our derivation reveals that the Quantum Potential (Q) emerges naturally and is mathematically identical to the kinetic energy component associated with a particle's hidden motion in these imaginary dimensions. This formulation successfully recovers the predictions of the Schrödinger equation while removing the ad-hoc nature of Bohmian mechanics. Furthermore, it interprets quantum non-locality—such as the interference observed in the double-slit experiment—as purely local, deterministic geodesic motion around topological singularities in a curved complex manifold. Ultimately, this framework provides a unified geometric description wherein gravity is the curvature of real coordinates, electromagnetism is the torsion of imaginary coordinates, and quantum mechanics is inertial motion through this complex geometry.
VL  - 15
IS  - 2
ER  -

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