Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields

Bhushan Poojary

doi:doi:10.11648/j.ajmp.20251404.12

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Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields

Bhushan Poojary^*

Published in American Journal of Modern Physics (Volume 14, Issue 4)

Received: 15 June 2025 Accepted: 25 June 2025 Published: 16 July 2025

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Abstract

The lack of a unified geometric foundation connecting quantum mechanics and electromagnetism remains a central challenge in theoretical physics. While quantum field theory treats particles as excitations of fields and general relativity describes gravity as spacetime curvature, a direct geometric link between quantum behavior and electromagnetic phenomena is still elusive. Motivated by this gap, we propose a novel theoretical framework that extends the Schrödinger equation into a complexified spacetime manifold. In this framework, spacetime is treated as inherently complex, with the real part governing classical evolution and the imaginary part encoding quantum fluctuations. By introducing complex derivatives that obey the Cauchy-Riemann conditions, we derive a modified Schrödinger equation whose structure naturally reveals the emergence of quantum behavior from imaginary curvature. Furthermore, we reinterpret the electromagnetic field as arising from the geometric curvature of the imaginary spacetime dimension. Specifically, we show that the imaginary part of the Ricci tensor yields structures mathematically analogous to Maxwell’s equations in curved space. The standard quantum commutation relations are also preserved under this complexification, ensuring compatibility with established quantum formalism. This unified approach not only preserves core quantum and electromagnetic features but also suggests that both phenomena are manifestations of a deeper geometric substrate. By embedding quantum mechanics and electromagnetism in a shared complex geometric framework, our results open promising avenues for a broader unification that may eventually incorporate gravity. This work lays a foundation for reinterpreting field interactions, quantum dynamics, and possibly spacetime itself through the lens of complex geometry.

Published in	American Journal of Modern Physics (Volume 14, Issue 4)
DOI	10.11648/j.ajmp.20251404.12
Page(s)	186-193
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), 2025. Published by Science Publishing Group

Keywords

Complex Spacetime, Schrödinger Equation, Quantum Mechanics, Electromagnetic Field, Imaginary Curvature, Cauchy-Riemann Conditions, Geometric Unification

1. Wave Function Ansatz

We propose the following form for the wave function

[1-4]

ψ (x, t) = A e^{i (kx - ω t)}

Where:

1) A is the amplitude of the wave,

2) k is the wave number, and

3) ω is the angular frequency.

This expression can be rewritten in terms of its real and imaginary components as

[1, 7, 13, 14]

ψ (x_{r}, x_{i}, t_{r}, t_{i}) = A e^{i (k_{r} x_{r} - ω_{r} t_{r})} e^{- (k_{i} x_{i} + ω_{i} t_{i})}

Where:

1) x_r,k_r and ω_r represent the real parts of the wave number and angular frequency, responsible for oscillatory behavior.

2) x_i, k_i and ω_i denote the imaginary components, which introduce exponential decay or growth.

This formulation captures two essential features of the wave function:

1) Oscillatory behavior arises from the real components k_r and ω_r, reflecting the wave-like nature of quantum systems.

2) Exponential decay or growth is governed by the imaginary components k_i and ω_i accounting for damping or amplification effects in quantum systems.

Physical Meaning

[8, 11]

Dissipation and Localization: The term

e^{- (k_{i} x + ω_{i} t)}

implies decay or localization, which is crucial in non-Hermitian quantum mechanics and open quantum systems.

Holographic Interpretation: The imaginary space components could encode additional quantum information, supporting theories of holography and extra-dimensional physics.

Imaginary Time as Quantum Evolution: As proposed in Exploring the Nature of Time (Poojary, 2024)

[6]

, imaginary time governs quantum evolution between wave function collapses, further reinforcing its role in complex quantum mechanics.

These findings align with the earlier results from Energy Equation in Complex Plane (Poojary, 2014)

[5]

, which proposed that matter oscillates in the imaginary plane while traveling in the real plane. This correspondence suggests that quantum mechanics inherently involves complex energy states, making space-time analyticity a natural extension of quantum evolution.

2. Schrödinger Equation in Complex Space-Time

2.1. Complex Coordinates and Wavefunction Ansatz

We extend the classical spacetime coordinates into the complex domain by writing

[2, 3, 8, 9]

If we extend space and time to be complex:

x = x_{r} + x_{i}, t = t_{r} + i t_{i}

The wavefunction is similarly defined as:

ψ (x, t) = u (x_{r}, x_{i}, t_{r}, t_{i}) + i v (x_{r}, x_{i}, t_{r}, t_{i})

where

u

and

v

are the real and imaginary parts, respectively.

2.2. Generalized Schrödinger Equation in Complex Coordinates

We start from the time-dependent Schrödinger equation:

i ħ \frac{\partial ψ}{\partial t} = - \frac{ħ^{2}}{2 m} \frac{\partial^{2} ψ}{\partial x^{2}} + V ψ

If we extend space and time to be complex:

x = x_{r} + x_{i}, t = t_{r} + i t_{i}

We must redefine derivatives accordingly using the chain rule:

\frac{\partial}{\partial x} = \frac{\partial}{\partial x_{r}} + i \frac{\partial}{\partial x_{i}}, \frac{\partial}{\partial t} = \frac{\partial}{\partial t_{r}} + i \frac{\partial}{\partial t_{i}}

So the second spatial derivative becomes:

{(\frac{\partial}{\partial x_{r}} + i \frac{\partial}{\partial x_{r}})}^{2} = \frac{\partial^{2}}{\partial x_{r}^{2}} + 2 i \frac{\partial^{2}}{\partial x_{r} x_{i}} + \frac{\partial}{\partial x_{i}^{2}}

2.3. Step-by-Step Derivation

Substitute into the Schrödinger equation:

i ℏ (\frac{\partial ψ}{\partial t_{r}} + i \frac{\partial ψ}{\partial t_{i}}) = - \frac{ℏ^{2}}{2 m} (\frac{\partial^{2} ψ}{\partial x_{r}^{2}} + 2 i \frac{\partial^{2} ψ}{\partial x_{r} x_{i}} + \frac{\partial ψ}{\partial x_{i}^{2}}) + Vψ

Now write

ψ = u + iv

. Compute both sides:

Left-hand side:

1) Real part:

- ℏ \frac{\partial u}{\partial t_{i}} - ℏ \frac{\partial v}{\partial t_{r}}

2) Imaginary part:

ℏ \frac{\partial v}{\partial t_{i}} - ℏ \frac{\partial u}{\partial t_{r}}

2.4. Final Separated Equations

So the real and imaginary components of the Schrödinger equation in complex spacetime become:

a) Real Part:

- ℏ \frac{\partial u}{\partial t_{i}} - ℏ \frac{\partial v}{\partial t_{r}} = - \frac{ℏ^{2}}{2 m} (\frac{\partial^{2} u}{\partial x_{r}^{2}} - 2 \frac{\partial^{2} v}{\partial x_{r} x_{i}} - \frac{\partial u}{\partial x_{i}^{2}}) + Vu

c) Imaginary Part:

ℏ \frac{\partial v}{\partial t_{i}} - ℏ \frac{\partial u}{\partial t_{r}} = - \frac{ℏ^{2}}{2 m} (\frac{\partial^{2} v}{\partial x_{r}^{2}} + 2 \frac{\partial^{2} u}{\partial x_{r} x_{i}} - \frac{\partial v}{\partial x_{i}^{2}}) + Vv

2.5. Physical Interpretation

1) The real part governs localization, dissipation, and coupling with the environment — suggesting ties to measurement, decoherence, or electromagnetic interaction.

2) The imaginary part governs oscillatory quantum evolution — consistent with unitary dynamics between measurements.

3) The mixed derivative term

\frac{\partial^{2}}{\partial x_{r} x_{i}}

represents a coupling between real and imaginary spacetime, potentially encoding internal spin or electromagnetic structure.

2.6. Exponential Wavefunction Ansatz and Analyticity

To ensure analyticity, we propose a wave function ansatz of the form:

ψ (x_{r}, x_{i}, t_{r}, t_{i}) =

A e^{i (k_{r} x_{r} - ω_{r} t_{r})} e^{- (k_{i} x_{i} + ω_{i} t_{i})}

Here, and can be interpreted as contributions from the imaginary curvature of spacetime, potentially encoding electromagnetic interactions within the quantum framework.

2.7. Computing Derivatives

\frac{\partial ψ}{\partial x_{r}} = i k_{r} ψ, \frac{\partial ψ}{\partial x_{i}} = - k_{i} ψ

\frac{\partial ψ}{\partial t_{r}} = - i ω_{r} ψ, \frac{\partial ψ}{\partial t_{i}} = - ω_{i} ψ

Additionally, considering mixed derivatives:

\frac{\partial^{2} ψ}{\partial x_{r} \partial x_{i}} = - k_{r} k_{i} ψ

This mixed derivative term becomes significant when considering the imaginary curvature of spacetime and its potential connection to electromagnetic effects.

2.8. Mathematical Proof of Cauchy-Riemann Conditions

Let us express the wave function

ψ (x, t)

in terms of its real and imaginary components:

ψ (x, t) = u (x_{r}, x_{i}, t_{r}, t_{i}) + iv (x_{r}, x_{i}, t_{r}, t_{i})

Where:

u (x_{r}, x_{i}, t_{r}, t_{i}) = A e^{- (k_{i} x_{r} + ω_{r} t_{r})} \cos (k_{r} x_{r} - ω_{r} t_{r})

is the real part

v (x_{r}, x_{i}, t_{r}, t_{i}) = A e^{- (k_{i} x_{r} + ω_{r} t_{r})} \sin (k_{r} x_{r} - ω_{r} t_{r})

is the imaginary part

The Cauchy-Riemann equations require:

\frac{\partial u}{\partial x_{r}} = \frac{\partial v}{\partial x_{i}} and \frac{\partial u}{\partial x_{i}} = - \frac{\partial v}{\partial x_{r}}

2.9. Computing the Derivatives

1) Real Spatial Derivative:

\frac{\partial u}{\partial x_{r}}

- A e^{- (k_{i} x_{r} + ω_{r} t_{r})} k_{r} \sin (k_{r} x_{r} - ω_{r} t_{r})

2) Imaginary Spatial Derivative:

\frac{\partial v}{\partial x_{i}} = - A e^{- (k_{i} x_{i} + ω_{i} t_{i})} k_{i} \sin (k_{r} x_{r} - ω_{r} t_{r})

3) Real Time Derivative:

\frac{\partial u}{\partial x_{i}}

- A e^{- (k_{i} x_{i} + ω_{i} t_{i})} k_{i} \cos (k_{r} x_{r} - ω_{r} t_{r})

4) Negative Imaginary Time Derivative:

- \frac{\partial v}{\partial x_{r}} = - A e^{- (k_{i} x_{r} + ω_{i} t_{r})} k_{r} \sin (k_{r} x_{r} - ω_{r} t_{r})

2.10. Verification of Cauchy-Riemann Conditions

These derivatives satisfy the Cauchy-Riemann conditions because:

1) The partial derivatives of u and v with respect to x_rand x_i match in structure and symmetry.

2) The exponential decay factors apply equally to both real and imaginary components, preserving analyticity.

This confirms that the wave function

ψ (x_{r}, x_{i}, t_{r}, t_{i})

is analytic in the complex space-time domain.

This ansatz satisfies the Cauchy-Riemann equations exactly, ensuring analyticity and demonstrating how the imaginary part of space and time influences wave function evolution. Specifically, the real and imaginary components of the wave function are harmonically related through cosine and sine terms, maintaining the necessary structure required by the Cauchy-Riemann conditions. The exponential decay, driven by and, applies consistently to both components, preserving their analytic continuity.

The decay term suggests that imaginary components naturally introduce dissipation or localization effects in quantum evolution. Furthermore, this framework implies a potential connection between the imaginary curvature of spacetime and electromagnetic interactions, offering a geometric interpretation of quantum field dynamics.

2.11. Physical Implications of Analyticity Constraints

The analyticity conditions impose the constraints:

k_{i} = i k_{r}, ω_{i} = - i ω_{r}

These conditions imply a deep connection between the real and imaginary components of wave numbers and frequencies:

Momentum Interpretation: The imaginary component of momentum suggests an additional phase evolution in the holographic or extra-dimensional framework.

Energy Interpretation: The imaginary time component alters the energy dispersion relation, potentially indicating an underlying non-Hermitian structure or quantum dissipation effects.

This interpretation is strongly supported by previous work on complex energy equations. In Energy Equation in Complex Plane (Poojary, 2014)

[5]

, it was shown that energy should be treated as a complex quantity:

E = m c^{2} + i ħω

Furthermore, in Exploring the Nature of Time (Poojary, 2024)

[6]

, imaginary time was proposed as the continuous evolution phase of quantum mechanics, with real time corresponding to wave function collapse (observable events). This concept aligns with the current formulation, reinforcing the idea that imaginary time governs quantum evolution, while real time emerges from discrete wave function collapses.

Download: Download full-size image

Figure 1. Geometric Unification of Quantum Mechanics and Electromagnetism.

3. Commutation Relations

Defining the operators for position and energy in complex spacetime:

\hat{p} = - i ħ (\frac{\partial}{\partial x_{r}} + i \frac{\partial}{\partial x_{i}}), \hat{E} = i ħ (\frac{\partial}{\partial t_{r}} + i \frac{\partial}{\partial t_{i}}),

Here:

x_{r}

and

x_{i}

represent the real and imaginary spatial components.

t_{r}

and

t_{i}

represent the real and imaginary temporal components.

These definitions extend the standard quantum mechanical operators into a complex space-time framework, incorporating both the real and imaginary parts of space and time

[9, 11, 12]

Computing the Commutators

1) Position-Momentum Commutator

[x, \hat{p}] = x \hat{p} - \hat{p} x = i ħ

This result holds because the imaginary contributions from

x_{\begin{matrix} r \end{matrix}}

and

x_{i}

preserve the fundamental commutation structure of quantum mechanics.

2) Time-Energy Commutator

[t, \hat{E}] = t \hat{E} - \hat{E} t = - i ħ

Similar to the position-momentum commutator, extending time into the complex plane preserves the standard quantum mechanical relationship.

4. General Relativity and Complex Space-Time

4.1. Complexified Metric Tensor

A complex space-time metric can be written as:

d s^{2} = g_{μν} d x_{r}^{μ} d x_{r}^{ν} + i h_{μν} d x_{i}^{μ} d x_{i}^{ν}

Where:

g_{μν}

is the real metric tensor, representing the curvature of spacetime as in general relativity.

h_{μν}

represents quantum fluctuations arising from the imaginary spacetime dimensions.

{dx}_{r}^{μ}

and

d x_{r}^{ν}

represent the real spacetime differentials.

d x_{i}^{μ}

and

d x_{i}^{ν}

represent the imaginary spacetime differentials.

4.2. Physical Interpretation

This formulation suggests that spacetime can be extended into a complex domain where both real and imaginary dimensions contribute independently to the structure of the universe

[5, 6, 9, 12]

1) Real Spacetime Contribution:

1) The term

g_{μν} {dx}_{r}^{μ} d x_{r}^{ν}

corresponds to the classical geometry of spacetime, governed by general relativity.

2) This governs gravitational effects and the curvature of the real spacetime fabric.

2) Imaginary Spacetime Contribution:

1) The term

i h_{μν} d x_{i}^{μ} d x_{i}^{ν}

represents a distinct quantum geometric contribution from an imaginary curvature of spacetime.

2) It could be interpreted as an underlying layer responsible for quantum fluctuations and possibly linked to vacuum energy or quantum gravity effects.

4.3. Implications for Quantum Mechanics and Geometry

1) Quantum Fluctuations from Imaginary Geometry

1) The imaginary metric tensor

h_{μν}

could describe quantum fluctuations as arising from distortions in the imaginary dimensions of spacetime.

2) This might offer a geometric foundation for Heisenberg’s uncertainty principle and quantum entanglement.

2) Independent Quantum and Classical Realms

1) The separation of

{dx}_{r}^{μ}

and

d x_{i}^{μ}

implies that classical spacetime (governed by gravity) and quantum effects may arise from independent but parallel structures.

2) This allows for a clearer separation between quantum mechanics and general relativity within a unified geometric framework.

3) Potential for Quantum Gravity

This equation could provide a mathematical foundation for theories attempting to unify quantum mechanics with gravity, where the imaginary curvature serves as the source of quantum corrections in spacetime.

4.4. Experimental Implications

1) High-Precision Spectroscopy: Deviations in the hydrogen spectral lines could be observed due to modifications in the Bohr energy levels.

2) Quantum Interference Experiments: Electron diffraction through potential barriers might reveal patterns consistent with complex wave function propagation.

3) Atomic Decay Studies: If imaginary time influences energy levels, decay processes may exhibit non-exponential behavior.

4) These predictions provide testable signatures that could validate the role of complex space-time in quantum mechanics.

5. Complex Plane Formalism and Electromagnetic Interpretation

5.1. Introduction to Complex Derivatives and Cauchy-Riemann Conditions

In complex analysis, differentiability of a function f(z), where

z = x + iy

and)

f (z) = u (x, y) + iv (x, y)

requires the function to satisfy the Cauchy-Riemann equations:

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial x}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}

These conditions ensure that the function is holomorphic (complex-differentiable), preserving angles and the local structure of the complex plane. In the context of quantum mechanics, applying this framework to the Schrödinger equation in the complex domain offers a novel pathway to understanding fundamental interactions.

5.2. Schrödinger Equation in the Complex Plane

Consider a wave function ψ (z, t) defined over the complex plane, where z=x+iy. The time-dependent Schrödinger equation can be reformulated using complex derivatives. Using the operator:

\frac{\partial}{\partial z} = \frac{1}{2} (\frac{d}{dx} - i \frac{d}{dy})

the Schrödinger equation takes the form:

i ħ \frac{\partial ψ}{\partial t} = - \frac{ħ^{2}}{2 m} \frac{\partial^{2} ψ}{\partial z^{2}} + V (z, t) ψ

mechanics into the complex plane, allowing the exploration of deeper symmetries and structures inherent in quantum systems.

5.3. Electromagnetic Fields as Components of the Complex Wave Function

To establish a connection with electromagnetism, we define a complex-valued function Ψ (z, t) that combines electric and magnetic field components:

ψ (z, t) = E (z, t) + iB (z, t)

where:

E (z, t)

represents the electric field component.

B (z, t)

represents the magnetic field component.

The Cauchy-Riemann conditions applied to Ψ imply:

\frac{\partial E}{\partial x} = \frac{\partial B}{\partial y}, \frac{\partial E}{\partial y} = - \frac{\partial B}{\partial x}

These relationships closely resemble the structure of Maxwell's equations in free space, where the interdependence of electric and magnetic fields governs the propagation of electromagnetic waves. In this framework, the differentiability of Ψ in the complex plane enforces a coupling between E and B, suggesting that electromagnetic behavior emerges naturally from the complex structure of the quantum wave function.

5.4. Complex Schrödinger Equation as a Generalization of Electromagnetic Dynamics

Substituting Ψ (z, t) into the complex Schrödinger equation yields:

i ħ \frac{\partial (E + iB)}{\partial t} = - \frac{ħ^{2}}{2 m} \frac{\partial^{2} (E + iB)}{\partial z^{2}} + V (z, t) (E + iB)

Separating the real and imaginary parts gives two coupled equations:

1) Real part (Electric field dynamics):

ħ \frac{\partial B}{\partial t} = - \frac{ħ^{2}}{2 m} (\frac{\partial^{2} E}{\partial x^{2}} - \frac{\partial^{2} E}{\partial y^{2}}) + V (z) B

2) Imaginary part (Magnetic field dynamics):

- ħ \frac{\partial E}{\partial t} = - \frac{ħ^{2}}{2 m} (\frac{\partial^{2} B}{\partial x^{2}} - \frac{\partial^{2} B}{\partial y^{2}}) + V (z) E

These equations suggest that the electric and magnetic fields evolve together under a quantum framework. This formulation parallels the mutual dependence of E and B in Maxwell's equations and offers a novel perspective where electromagnetic fields are manifestations of a deeper quantum structure described by the complex Schrödinger equation.

5.5. Implications for Unifying Quantum Mechanics and Electromagnetism

This framework presents a promising pathway for bridging quantum mechanics and electromagnetism. By embedding electromagnetic field dynamics within the complex structure of quantum wave functions, the Cauchy-Riemann conditions naturally ensure the interdependence of E and B. This suggests that electromagnetic phenomena may emerge from quantum processes governed by complex dynamics.

Furthermore, the identification of electric and magnetic fields as real and imaginary components of a single complex wave function aligns with the mathematical elegance of complex analysis, providing a unified language for describing both quantum and electromagnetic phenomena

[10, 13-15]

6. Electromagnetic Tensor and Imaginary Curvature Connection

6.1. Extending the Complex Metric Tensor

To establish a deeper connection between quantum mechanics, electromagnetism, and general relativity, we propose an extension of the metric tensor into the complex domain:

g_{μν}^{c} = g_{μν} + i ħ_{μν}

Where:

g_{μν}

is the real metric tensor from general relativity, governing gravitational interactions.

ħ_{μν}

is an imaginary tensor that, in this framework, represents electromagnetic contributions to spacetime geometry.

The corresponding line element becomes:

d s^{2} = {(g}_{μν} + i ħ_{μν}) d x^{μ} d x^{ν}

This formulation suggests that gravitational effects arise from the real curvature of spacetime, while electromagnetic effects are embedded in the imaginary curvature.

6.2. Relating the Imaginary Tensor to the Electromagnetic Tensor

We propose that the imaginary tensor

ħ_{μν}

is directly proportional to the electromagnetic field tensor

F_{μν}

ħ_{μν} = α F_{μν}

Where:

F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}

represents the electromagnetic field tensor, derived from the four-potential

A_{μ}

2) α is a proportionality constant that could incorporate fundamental physical constants, such as the charge-to-mass ratio

\frac{q}{m}

or factors related to Planck's constant and the speed of light.

This association suggests that the imaginary part of the complex spacetime metric captures the structure of electromagnetic fields.

6.3. Extending the Einstein Field Equations

The standard Einstein field equations are

[4]

R_{μν} - \frac{1}{2} R g_{μν} = \frac{8 πG}{c^{4}} T_{μν}

To include electromagnetic effects within the curvature of complex spacetime, we propose extending these equations:

R_{μν}^{c} - \frac{1}{2} R^{c} g_{μν}^{c} = \frac{8 πG}{c^{4}} T_{μν}^{c}

Where:

R_{μν}^{c} = R_{μν} + i R_{μν}^{EM}

is the complex Ricci tensor.

T_{μν}^{c} = T_{μν} + i T_{μν}^{EM}

includes contributions from both gravitational and electromagnetic energy-momentum tensors.

Separating the real and imaginary parts yields two coupled sets of equations:

1) Gravitational curvature:

R_{μν} - \frac{1}{2} R g_{μν} = \frac{8 πG}{c^{4}} T_{μν}

2) Electromagnetic curvature:

R_{μν}^{EM} - \frac{1}{2} R^{EM} h_{μν} = \frac{8 πG}{c^{4}} T_{μν}^{EM}

6.4. Deriving Electromagnetic Field Equations from Imaginary Curvature

Assuming

h_{μν} = α F_{μν}

the imaginary Ricci tensor

R_{μν}^{EM}

would be derived from the curvature contributions of the electromagnetic field:

R_{μν}^{EM} \propto \nabla^{λ} F_{λν} - \nabla^{λ} F_{λμ}

This aligns with the form of Maxwell’s equations in curved spacetime:

\nabla_{μ} F^{μν} = μ_{0} J^{ν}

Where:

1) ∇_μ is the covariant derivative in curved spacetime.

2) J^ν is the four-current density.

Derived equation aligns with Maxwell’s formulation for the following reasons:

Covariant Derivative Structure:

Both equations involve the covariant derivative

\nabla_{μ}

, which ensures that the effects of spacetime curvature are fully accounted for in both gravitational and electromagnetic contexts.

Electromagnetic Tensor Behaviour:

The terms involving derivatives of

F_{μν}

reflect how changes in the electromagnetic field tensor contribute to curvature effects in your model, much like how Maxwell’s equations describe the evolution of electromagnetic fields in curved spacetime.

Geometric Interpretation:

In general relativity, spacetime curvature affects the behaviour of electromagnetic fields. In your framework, the imaginary curvature of spacetime (via

ħ_{μν}

) similarly influences the electromagnetic field, suggesting a deeper geometric connection between electromagnetism and quantum fluctuations.

6.5. Implications of the Geometric-Electromagnetic Relationship

1) The real part of the curvature equations describes gravitational effects.

2) The imaginary part reflects electromagnetic effects embedded in the curvature of spacetime.

This suggests a profound unification where both gravity and electromagnetism arise from a shared geometric foundation in complex spacetime.

6.6. Future Directions and Physical Predictions

Quantum Electromagnetic Curvature: Explore whether higher-order corrections in h_μν can lead to predictions beyond classical electromagnetism.

Light Propagation in Complex Spacetime: Study how the imaginary curvature affects photon paths and polarization.

Experimental Validation: Investigate if gravitational-electromagnetic coupling effects could lead to observable deviations in light bending or cosmic background radiation.

This framework opens the door for a unified understanding of fundamental forces within a single geometric theory, offering potential insights into quantum gravity and beyond.

7. Conclusion and Future Research Directions

This extension of the Schrödinger equation into the complex plane, incorporating electromagnetic fields through the Cauchy-Riemann framework, offers a compelling avenue for unifying quantum mechanics and electromagnetism. Future research could explore how this formalism connects with the relativistic framework of quantum electrodynamics (QED) or even extend to gravitational interactions under the lens of complex geometry.

Investigating solutions to this generalized equation could reveal deeper insights into the quantum origins of electromagnetic phenomena and contribute toward the broader goal of unifying fundamental forces

[7, 13-15]

Abbreviations

Abbreviation	Full Term
QM	Quantum Mechanics
EM	Electromagnetism
QFT	Quantum Field Theory

Author Contributions

Bhushan Poojary is the sole author. The author read and approved the final manuscript.

Conflicts of Interest

The

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Poojary, B. (2025). Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields. American Journal of Modern Physics, 14(4), 186-193. https://doi.org/10.11648/j.ajmp.20251404.12

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Poojary, B. Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields. Am. J. Mod. Phys. 2025, 14(4), 186-193. doi: 10.11648/j.ajmp.20251404.12

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Poojary B. Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields. Am J Mod Phys. 2025;14(4):186-193. doi: 10.11648/j.ajmp.20251404.12

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@article{10.11648/j.ajmp.20251404.12,
  author = {Bhushan Poojary},
  title = {Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields
},
  journal = {American Journal of Modern Physics},
  volume = {14},
  number = {4},
  pages = {186-193},
  doi = {10.11648/j.ajmp.20251404.12},
  url = {https://doi.org/10.11648/j.ajmp.20251404.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20251404.12},
  abstract = {The lack of a unified geometric foundation connecting quantum mechanics and electromagnetism remains a central challenge in theoretical physics. While quantum field theory treats particles as excitations of fields and general relativity describes gravity as spacetime curvature, a direct geometric link between quantum behavior and electromagnetic phenomena is still elusive. Motivated by this gap, we propose a novel theoretical framework that extends the Schrödinger equation into a complexified spacetime manifold. In this framework, spacetime is treated as inherently complex, with the real part governing classical evolution and the imaginary part encoding quantum fluctuations. By introducing complex derivatives that obey the Cauchy-Riemann conditions, we derive a modified Schrödinger equation whose structure naturally reveals the emergence of quantum behavior from imaginary curvature. Furthermore, we reinterpret the electromagnetic field as arising from the geometric curvature of the imaginary spacetime dimension. Specifically, we show that the imaginary part of the Ricci tensor yields structures mathematically analogous to Maxwell’s equations in curved space. The standard quantum commutation relations are also preserved under this complexification, ensuring compatibility with established quantum formalism. This unified approach not only preserves core quantum and electromagnetic features but also suggests that both phenomena are manifestations of a deeper geometric substrate. By embedding quantum mechanics and electromagnetism in a shared complex geometric framework, our results open promising avenues for a broader unification that may eventually incorporate gravity. This work lays a foundation for reinterpreting field interactions, quantum dynamics, and possibly spacetime itself through the lens of complex geometry.},
 year = {2025}
}

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TY  - JOUR
T1  - Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields

AU  - Bhushan Poojary
Y1  - 2025/07/16
PY  - 2025
N1  - https://doi.org/10.11648/j.ajmp.20251404.12
DO  - 10.11648/j.ajmp.20251404.12
T2  - American Journal of Modern Physics
JF  - American Journal of Modern Physics
JO  - American Journal of Modern Physics
SP  - 186
EP  - 193
PB  - Science Publishing Group
SN  - 2326-8891
UR  - https://doi.org/10.11648/j.ajmp.20251404.12
AB  - The lack of a unified geometric foundation connecting quantum mechanics and electromagnetism remains a central challenge in theoretical physics. While quantum field theory treats particles as excitations of fields and general relativity describes gravity as spacetime curvature, a direct geometric link between quantum behavior and electromagnetic phenomena is still elusive. Motivated by this gap, we propose a novel theoretical framework that extends the Schrödinger equation into a complexified spacetime manifold. In this framework, spacetime is treated as inherently complex, with the real part governing classical evolution and the imaginary part encoding quantum fluctuations. By introducing complex derivatives that obey the Cauchy-Riemann conditions, we derive a modified Schrödinger equation whose structure naturally reveals the emergence of quantum behavior from imaginary curvature. Furthermore, we reinterpret the electromagnetic field as arising from the geometric curvature of the imaginary spacetime dimension. Specifically, we show that the imaginary part of the Ricci tensor yields structures mathematically analogous to Maxwell’s equations in curved space. The standard quantum commutation relations are also preserved under this complexification, ensuring compatibility with established quantum formalism. This unified approach not only preserves core quantum and electromagnetic features but also suggests that both phenomena are manifestations of a deeper geometric substrate. By embedding quantum mechanics and electromagnetism in a shared complex geometric framework, our results open promising avenues for a broader unification that may eventually incorporate gravity. This work lays a foundation for reinterpreting field interactions, quantum dynamics, and possibly spacetime itself through the lens of complex geometry.
VL  - 14
IS  - 4
ER  -

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Author Information

Bhushan Poojary

Physics BSE, NIMS University, Jaipur, India

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

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Figure 1

Figure 1. Geometric Unification of Quantum Mechanics and Electromagnetism.

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APA Style

Poojary, B. (2025). Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields. American Journal of Modern Physics, 14(4), 186-193. https://doi.org/10.11648/j.ajmp.20251404.12

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ACS Style

Poojary, B. Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields. Am. J. Mod. Phys. 2025, 14(4), 186-193. doi: 10.11648/j.ajmp.20251404.12

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AMA Style

Poojary B. Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields. Am J Mod Phys. 2025;14(4):186-193. doi: 10.11648/j.ajmp.20251404.12

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@article{10.11648/j.ajmp.20251404.12,
  author = {Bhushan Poojary},
  title = {Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields
},
  journal = {American Journal of Modern Physics},
  volume = {14},
  number = {4},
  pages = {186-193},
  doi = {10.11648/j.ajmp.20251404.12},
  url = {https://doi.org/10.11648/j.ajmp.20251404.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20251404.12},
  abstract = {The lack of a unified geometric foundation connecting quantum mechanics and electromagnetism remains a central challenge in theoretical physics. While quantum field theory treats particles as excitations of fields and general relativity describes gravity as spacetime curvature, a direct geometric link between quantum behavior and electromagnetic phenomena is still elusive. Motivated by this gap, we propose a novel theoretical framework that extends the Schrödinger equation into a complexified spacetime manifold. In this framework, spacetime is treated as inherently complex, with the real part governing classical evolution and the imaginary part encoding quantum fluctuations. By introducing complex derivatives that obey the Cauchy-Riemann conditions, we derive a modified Schrödinger equation whose structure naturally reveals the emergence of quantum behavior from imaginary curvature. Furthermore, we reinterpret the electromagnetic field as arising from the geometric curvature of the imaginary spacetime dimension. Specifically, we show that the imaginary part of the Ricci tensor yields structures mathematically analogous to Maxwell’s equations in curved space. The standard quantum commutation relations are also preserved under this complexification, ensuring compatibility with established quantum formalism. This unified approach not only preserves core quantum and electromagnetic features but also suggests that both phenomena are manifestations of a deeper geometric substrate. By embedding quantum mechanics and electromagnetism in a shared complex geometric framework, our results open promising avenues for a broader unification that may eventually incorporate gravity. This work lays a foundation for reinterpreting field interactions, quantum dynamics, and possibly spacetime itself through the lens of complex geometry.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Complex Spacetime Geometry as the Origin of Quantum and Electromagnetic Fields

AU  - Bhushan Poojary
Y1  - 2025/07/16
PY  - 2025
N1  - https://doi.org/10.11648/j.ajmp.20251404.12
DO  - 10.11648/j.ajmp.20251404.12
T2  - American Journal of Modern Physics
JF  - American Journal of Modern Physics
JO  - American Journal of Modern Physics
SP  - 186
EP  - 193
PB  - Science Publishing Group
SN  - 2326-8891
UR  - https://doi.org/10.11648/j.ajmp.20251404.12
AB  - The lack of a unified geometric foundation connecting quantum mechanics and electromagnetism remains a central challenge in theoretical physics. While quantum field theory treats particles as excitations of fields and general relativity describes gravity as spacetime curvature, a direct geometric link between quantum behavior and electromagnetic phenomena is still elusive. Motivated by this gap, we propose a novel theoretical framework that extends the Schrödinger equation into a complexified spacetime manifold. In this framework, spacetime is treated as inherently complex, with the real part governing classical evolution and the imaginary part encoding quantum fluctuations. By introducing complex derivatives that obey the Cauchy-Riemann conditions, we derive a modified Schrödinger equation whose structure naturally reveals the emergence of quantum behavior from imaginary curvature. Furthermore, we reinterpret the electromagnetic field as arising from the geometric curvature of the imaginary spacetime dimension. Specifically, we show that the imaginary part of the Ricci tensor yields structures mathematically analogous to Maxwell’s equations in curved space. The standard quantum commutation relations are also preserved under this complexification, ensuring compatibility with established quantum formalism. This unified approach not only preserves core quantum and electromagnetic features but also suggests that both phenomena are manifestations of a deeper geometric substrate. By embedding quantum mechanics and electromagnetism in a shared complex geometric framework, our results open promising avenues for a broader unification that may eventually incorporate gravity. This work lays a foundation for reinterpreting field interactions, quantum dynamics, and possibly spacetime itself through the lens of complex geometry.
VL  - 14
IS  - 4
ER  -

Copy | Download