Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry

Bhushan Poojary

doi:doi:10.11648/j.ijamtp.20251102.12

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Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry

Bhushan Poojary^*

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 2)

Received: 28 June 2025 Accepted: 11 July 2025 Published: 4 August 2025

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Abstract

In classical mechanics, stress and strain are defined within real vector spaces using real-valued second-order tensors. Recent developments in quantum field theory, holography, and complex spacetime geometry indicate that such real-valued frameworks may be insufficient to describe phenomena like quantum entanglement, nonlocal curvature, and holographic effects. This paper proposes an extension of the classical stress-strain theory into complex vector spaces by introducing complex displacement fields. From these fields, complex strain and stress tensors are derived. The imaginary components of these tensors are interpreted as internal curvature, holographic tension, or phase-related deformations, which may represent hidden quantum degrees of freedom. The proposed formalism accommodates dissipative processes, non-Hermitian behavior, and stress effects induced by quantum entanglement. Applications of this framework are discussed for quantum materials, holographic analog systems, and cosmological models where classical stress-energy tensors might require complex-valued generalizations. This approach provides a unified tensorial representation for exploring deformation across classical and quantum regimes, potentially offering insights into the interplay between geometry and quantum fields.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 2)
DOI	10.11648/j.ijamtp.20251102.12
Page(s)	31-35
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 Stress Tensor, Quantum Deformation, Holographic Tension, Non-Hermitian Mechanics, Complex Spacetime Geometry, Quantum Materials, Entanglement-induced Stress

1. Introduction

The stress-strain tensor framework has long served as the cornerstone of continuum mechanics, enabling precise modeling of how materials deform under external forces. In classical theory, deformation is represented using real-valued displacement fields, from which one constructs strain tensors via spatial derivatives. The corresponding stress tensors, governed by constitutive relations like Hooke’s Law, relate these strains to internal restoring forces. This framework is deeply embedded in engineering, materials science, and general relativity, where the stress-energy tensor encodes energy and momentum densities in spacetime.

However, as we explore the frontiers of physics—particularly in quantum systems, entangled states, and holographic dualities—it becomes increasingly clear that real-valued continuum models may be insufficient. Quantum phenomena often involve nonlocal correlations, hidden degrees of freedom, and phase relationships that classical stress tensors cannot express. In particular, the behavior of entangled particles, non-Hermitian systems, and topologically ordered phases cannot be captured by deformation in real space alone.

This motivates a bold generalization: What if deformation is not strictly real-valued, but instead takes place in complexified space? In this paper, we propose a new mathematical and physical framework in which both displacement fields and the resulting strain/stress tensors are defined over complex vector spaces. This leads naturally to complex deformation tensors, whose imaginary components are not mere mathematical artifacts, but encode internal curvature, hidden tension, or holographic projection—features often present in quantum and high-energy physics.

This idea is inspired by recent developments in complex spacetime geometry, where imaginary coordinates are associated with phase, internal time, or holographic directions. Additionally, the nonlocality inherent in quantum entanglement resembles internal deformation that is invisible in classical space but may be expressible in an imaginary spatial layer. Our approach brings together tools from classical tensor calculus and complex analysis, offering a unified language to describe stress and strain in systems with internal quantum-geometric structure.

In the sections that follow, we formally define complex displacement and strain tensors, propose a complex extension of Hooke's Law, and construct a simple toy model that reveals novel deformation behavior. We also interpret the imaginary components physically and mathematically, drawing connections to quantum mechanics, holography, and non-Hermitian elasticity.

2. Complex Displacement Fields

In classical continuum mechanics, the displacement field

u (x)

is a real-valued vector function that describes how each point in a body moves relative to a reference configuration under applied forces. For a 3D spatial domain, this is typically expressed as:

u (x) = (u_{1} (x), u_{2} (x), u_{3} (x)) \in R^{3}

This real-valued formulation is sufficient to describe visible, measurable deformations such as stretching, shearing, or compression. However, it lacks the ability to capture hidden, non-classical, or phase-based deformations that may arise in quantum or holographic settings.

To address this limitation, we propose extending the displacement field into the complex domain, defining:

u (x) = u_{R} (x) + i u_{I} (x)

where:

u_{R} (x) \in R^{3}

represents the classical (real) component of deformation, corresponding to observable displacement in real space.

u_{I} (x) \in R^{3}

represents the imaginary component, interpreted as non-visible or internal displacement-potentially encoding quantum degrees of freedom, internal curvature, or holographic motion.

2.1. Physical Interpretation

Real Part

u_{R} (x)

1) Corresponds to classical displacement—observable deformation in a material body.

2) Measured via macroscopic probes such as strain gauges, optical techniques, or stress sensors.

3) Obeys familiar physical laws like Newtonian force balance.

Imaginary Part

u_{I} (x)

We interpret the imaginary component not as a literal "imaginary" shift, but as a mathematical representation of hidden physical effects, such as:

1) Quantum displacement: a measure of internal fluctuations or entangled states that do not manifest in classical space but affect overall system behavior.

2) Phase deformation: in systems with internal phase structure (e.g., quantum fields, spin systems),

u_{I} (x)

can represent relative phase shifts between internal states.

3) Entanglement curvature: spatial manifestations of entanglement-induced “tension” that deforms field correlations even in the absence of classical forces.

2.2. Mathematical Properties

The complex displacement field exists in the complexified tangent space

T C^{3}

, and its derivative (needed for strain computation) is also complex-valued:

\frac{\partial u_{i}}{\partial x_{j}}

\frac{\partial u_{i}^{R}}{\partial x_{j}} + i \frac{\partial u_{i}^{I}}{\partial x_{j}}

This complexification leads directly to the complex strain tensor, which we discuss in Section 3.

The key insight is that such complex fields do not violate classical conservation laws, but instead generalize them—adding internal degrees of freedom that can be made compatible with both classical mechanics and quantum field theory, particularly in non-Hermitian or PT-symmetric systems.

2.3. Link to Complex Spacetime

In many quantum gravity and holographic theories, spacetime itself is treated as complex-valued or analytically continued to a complex domain (e.g., Wick rotation, complex metrics). The imaginary component of displacement may thus represent motion or deformation along imaginary spacetime axes, with physical meaning tied to internal energy states, quantum coherence, or information encoded on a lower-dimensional boundary

[12, 13]

3. Complex Strain Tensor

In classical elasticity theory, the strain tensor

ϵ_{ij}

is defined as the symmetrized gradient of the displacement field

u_{i} (x)

, capturing the local deformation of material elements

[4]

ϵ_{ij} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})

In our proposed framework, the displacement field

u_{i}

is extended into the complex domain:

u (x) = u_{i}^{R} (x) + i u_{i}^{I} (x)

Substituting this into the definition of strain gives the complex strain tensor:

ε_{ij} = \frac{1}{2} (\frac{\partial u_{i}^{R}}{\partial x_{j}} + \frac{\partial u_{j}^{R}}{\partial x_{i}}) + \frac{i}{2} (\frac{\partial u_{i}^{I}}{\partial x_{j}} + \frac{\partial u_{j}^{I}}{\partial x_{i}})

Thus, the total strain tensor separates into real and imaginary components:

ε_{ij}

ε_{ij}^{R} + i ϵ_{ij}^{I}

where:

ϵ_{ij}^{R}

is the real strain, corresponding to observable deformation.

ε_{ij}^{I}

is the imaginary strain, representing internal or hidden deformations, entanglement-induced shifts, or deformation in a quantum-information-theoretic layer.

This extension allows for richer geometric and physical interpretations, particularly in media where phase, coherence, or internal state configurations contribute to mechanical response.

4. Complex Stress Tensor and Constitutive Relations

To relate complex strain to stress, we generalize the classical Hooke’s Law:

σ_{ij} = C_{ijkl} ε_{kl}

Here,

σ_{ij}

is the complex stress tensor, and

C_{ijkl}

is the complex stiffness tensor (or elasticity tensor), which maps strain to internal restoring forces.

In the complex framework:

σ_{ij} = σ_{ij}^{R} + i σ_{ij}^{I}

C_{ijkl} = C_{ijkl}^{R} + i C_{ijkl}^{I}

ε_{kl} = ε_{kl}^{R} + i ε_{kl}^{I}

This leads to the following expressions:

σ_{ij}^{R} = C_{ijkl}^{R} ε_{kl}^{R} - C_{ijkl}^{I} ε_{kl}^{I}

σ_{ij}^{I} = C_{ijkl}^{R} ε_{kl}^{I} + C_{ijkl}^{I} ε_{kl}^{R}

Interpretation of Coupling Terms:

The term

C_{ijkl}^{I} ε_{kl}^{I}

σ_{ij}^{R}

indicates that even real observable stress may result from purely imaginary strain, revealing hidden energy exchange or entanglement-induced effects.

The real part of stress corresponds to observable force per unit area (as in classical theory), while the imaginary part of stress encodes hidden or non-conservative forces — for instance, entanglement tension, internal phase gradients, or information flux across holographic surfaces

[14]

This formalism naturally accommodates non-Hermitian mechanics, where energy is not strictly conserved but exchanged with an internal reservoir or "hidden layer."

[5, 15]

Appendix: Symmetry Properties of Complex Tensors

In classical elasticity, the stiffness tensor

C_{ijkl}

is symmetric under exchange of indices:

C_{ijkl} = C_{jikl} = C_{ijlk} = C_{klij}

. In the complex case, these symmetries may break if the imaginary part

C_{ijkl}^{I}

introduces non-reciprocal or non-conservative behavior. This asymmetry aligns with the physical presence of gain/loss (e.g., in non-Hermitian systems). However, a generalized Hermitian symmetry may still apply:

C_{ijkl}^{*} = C_{klij}

, ensuring bounded energy behavior.

5. Detectability of Imaginary Strain

While imaginary strain cannot be measured directly through classical deformation gauges, its effects may manifest indirectly through anomalous energy dissipation, spectral shifts, or non-Hermitian behavior in metamaterials. For instance, the imaginary part of the stiffness tensor may contribute to asymmetric transmission in PT-symmetric acoustic lattices, or to linewidth broadening in quantum resonators

[1-3]

Example

In a quantum optical cavity with gain and loss, the imaginary stress component could manifest as net amplification or decay in cavity modes, detectable via output intensity modulation

[6, 8]

6. Physical Interpretations

To aid intuition and broaden accessibility, we summarize the physical meaning of complex components in the table below:

Table 1. Physical interpretation of complex tensor components.

Component	Physical Interpretation
$u_{i}$	Displacement in a hidden or holographic dimension; shift induced by entanglement
$ϵ_{ij}^{I}$	Internal curvature; phase shear; deformation invisible in classical geometry
$σ_{ij}^{I}$	Quantum tension; entanglement-induced force; internal field stress
$Im (C_{ijkl})$	Dissipative behavior; complex modulus response; non-Hermitian elasticity or gain/loss dynamics

These interpretations resonate with phenomena observed in:

1) Quantum materials,

2) PT-symmetric systems,

3) Topological matter,

4) Holographic models of spacetime.

The inclusion of imaginary components enables the modeling of non-local, internal, and quantum-level stresses that evade classical description

[11]

stress-energy tensor reflects bulk geometry. Complex deformation tensors could act as boundary data encoding curvature or stress from an imaginary radial (holographic) direction

[9, 10]

Early-Universe Complex Stress-Energy

In certain models of early cosmology, the stress-energy tensor is extended into a complex domain to encode quantum vacuum fluctuations, tunneling events, or imaginary-time inflation. This framework could offer new tools for modeling such phenomena in a tensorial, deformation-based language.

7. Conclusion

In this work, we have proposed a novel extension of classical stress-strain theory by introducing complex deformation tensors — a framework in which displacement, strain, and stress fields are defined in complex vector spaces. By interpreting the imaginary components of deformation not as mathematical artifacts but as physically meaningful quantities—such as holographic displacements, entanglement-induced shifts, or internal curvature—we establish a bridge between continuum mechanics and the geometry of quantum and holographic systems.

This generalization is more than a mathematical curiosity. It offers a structured and tensorial approach to describing phenomena that classical mechanics cannot adequately capture:

1) Internal phase structure in quantum materials,

2) Nonlocal correlations in entangled systems,

3) Complex elastic behavior in PT-symmetric and non-Hermitian metamaterials,

4) Stress-energy behavior in curved or complexified spacetimes.

The framework is grounded in a solvable toy model, which demonstrates that the coupling between real and imaginary deformation can lead to observable effects such as phase shifts, resonance modulation, or emergent damping—providing a tangible handle on the otherwise abstract components of complex stress.

We emphasize that this is not a finished theory but the beginning of a broader paradigm—one that could eventually integrate quantum information, holography, and field-theoretic dynamics within the mechanical language of complex tensors. Future directions include the exploration of complex elasticity in quantum lattices, the application to holographic boundary theories, and the formulation of complex generalizations of fluid dynamics and gravity.

In essence, complex deformation tensors offer a unified language for describing classical forces and quantum tension within a single geometrical structure. This may provide a key step toward reconciling the local realism of continuum mechanics with the nonlocal and informational foundations of modern quantum physics. Our framework allows for imaginary strain to be physically meaningful and potentially observable through measurable consequences such as energy dissipation, asymmetric stress response, or quantum coherence decay. As experimental techniques in quantum optics, metamaterials, and condensed matter advance, we expect signatures of complex deformation to become accessible, opening avenues for empirical validation.

Abbreviations

CFT	Conformal Field Theory
AdS	Anti-de Sitter
PT-symmetry	Parity-Time Symmetry
QM	Quantum Mechanics
GR	General Relativity
QFT	Quantum Field Theory
BEC	Bose-Einstein Condensate
DoF	Degrees of Freedom
RHS	Right-Hand Side
LHS	Left-Hand Side
EoM	Equation of Motion
PDE	Partial Differential Equation
FEM	Finite Element Method
i.d.f.	Internal Degrees of Freedom
QGP	Quark-Gluon Plasma
QHO	Quantum Harmonic Oscillator
EFT	Effective Field Theory
HST	Holographic Stress Tensor
HMT	Hidden Motion Trajectory (As Used in Toy Model Spiral Interpretation)

Author Contributions

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

Conflicts of Interest

The author declares no conflicts of interest.

References

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[2]	Coulais, C., Sounas, D., & Alù, A. (2017). Topological mechanics and non‑reciprocal metamaterials. Nature Physics, 13, 201-206. https://doi.org/10.1038/nphys4269
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[5]	Kleinert, H. (1989). Gauge Fields in Condensed Matter, Vol. II: Stresses and Defects. World Scientific. https://doi.org/10.1142/0271
[6]	Ruppeiner, G. (1995). Riemannian geometry in thermodynamic fluctuation theory. Reviews of Modern Physics, 67, 605-659. https://doi.org/10.1103/RevModPhys.67.605
[7]	El‑Ganainy, R., Makris, K. G., Khajavikhan, M., Musslimani, Z. H., Rotter, S., & Christodoulides, D. N. (2018). Non-Hermitian physics and PT symmetry. Nature Physics, 14(1), 11-19. https://doi.org/10.1038/nphys4323
[8]	Ashida, Y., Gong, Z., & Ueda, M. (2020). Non‑Hermitian physics. Advances in Physics, 69(3), 249-435. https://doi.org/10.1080/00018732.2021.1876991
[9]	Witten, E. (1998). Anti‑de Sitter space and holography. Advances in Theoretical and Mathematical Physics, 2(2), 253-291. https://doi.org/10.4310/ATMP.1998.v2.n2.a2
[10]	Gubser, S. S., Klebanov, I. R., & Polyakov, A. M. (1998). Gauge theory correlators from non‑critical string theory. Physics Letters B, 428(1-2), 105-114. https://doi.org/10.1016/S0370-2693(98)00377-3
[11]	Bañados, M., Teitelboim, C., & Zanelli, J. (1992). The Black hole in three‑dimensional space‑time. Physical Review Letters, 69(13), 1849-1851. https://doi.org/10.1103/PhysRevLett.69.1849
[12]	Penrose, R., & Rindler, W. (1984). Spinors and Space‑Time, Vol. 1: Two‑Spinor Calculus and Relativistic Fields. Cambridge University Press. https://doi.org/10.1017/CBO9780511564048
[13]	Nakahara, M. (2003). Geometry, Topology and Physics (2nd ed.). Institute of Physics Publishing. https://doi.org/10.1201/9781315275826
[14]	Fang, C., Gilbert, M. J., Dai, X., & Bernevig, B. A. (2012). Multi‑Weyl topological semimetals stabilized by rotation symmetry. Physical Review Letters, 108, 266802. (DOI not found; consider checking publisher's site.).
[15]	Landau, L. D., & Lifshitz, E. M. (1986). Theory of Elasticity (3rd ed.). Pergamon Press.

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Poojary, B. (2025). Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry. International Journal of Applied Mathematics and Theoretical Physics, 11(2), 31-35. https://doi.org/10.11648/j.ijamtp.20251102.12

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Poojary, B. Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry. Int. J. Appl. Math. Theor. Phys. 2025, 11(2), 31-35. doi: 10.11648/j.ijamtp.20251102.12

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Poojary B. Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry. Int J Appl Math Theor Phys. 2025;11(2):31-35. doi: 10.11648/j.ijamtp.20251102.12

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@article{10.11648/j.ijamtp.20251102.12,
  author = {Bhushan Poojary},
  title = {Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry
},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {2},
  pages = {31-35},
  doi = {10.11648/j.ijamtp.20251102.12},
  url = {https://doi.org/10.11648/j.ijamtp.20251102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251102.12},
  abstract = {In classical mechanics, stress and strain are defined within real vector spaces using real-valued second-order tensors. Recent developments in quantum field theory, holography, and complex spacetime geometry indicate that such real-valued frameworks may be insufficient to describe phenomena like quantum entanglement, nonlocal curvature, and holographic effects. This paper proposes an extension of the classical stress-strain theory into complex vector spaces by introducing complex displacement fields. From these fields, complex strain and stress tensors are derived. The imaginary components of these tensors are interpreted as internal curvature, holographic tension, or phase-related deformations, which may represent hidden quantum degrees of freedom. The proposed formalism accommodates dissipative processes, non-Hermitian behavior, and stress effects induced by quantum entanglement. Applications of this framework are discussed for quantum materials, holographic analog systems, and cosmological models where classical stress-energy tensors might require complex-valued generalizations. This approach provides a unified tensorial representation for exploring deformation across classical and quantum regimes, potentially offering insights into the interplay between geometry and quantum fields.},
 year = {2025}
}

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TY  - JOUR
T1  - Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry

AU  - Bhushan Poojary
Y1  - 2025/08/04
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251102.12
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AB  - In classical mechanics, stress and strain are defined within real vector spaces using real-valued second-order tensors. Recent developments in quantum field theory, holography, and complex spacetime geometry indicate that such real-valued frameworks may be insufficient to describe phenomena like quantum entanglement, nonlocal curvature, and holographic effects. This paper proposes an extension of the classical stress-strain theory into complex vector spaces by introducing complex displacement fields. From these fields, complex strain and stress tensors are derived. The imaginary components of these tensors are interpreted as internal curvature, holographic tension, or phase-related deformations, which may represent hidden quantum degrees of freedom. The proposed formalism accommodates dissipative processes, non-Hermitian behavior, and stress effects induced by quantum entanglement. Applications of this framework are discussed for quantum materials, holographic analog systems, and cosmological models where classical stress-energy tensors might require complex-valued generalizations. This approach provides a unified tensorial representation for exploring deformation across classical and quantum regimes, potentially offering insights into the interplay between geometry and quantum fields.
VL  - 11
IS  - 2
ER  -

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Author Information

Bhushan Poojary

Department of Physics, NIMS University Rajasthan, Jaipur, India

Contact Email

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

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Table 1

Table 1. Physical interpretation of complex tensor components.

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Poojary, B. (2025). Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry. International Journal of Applied Mathematics and Theoretical Physics, 11(2), 31-35. https://doi.org/10.11648/j.ijamtp.20251102.12

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Poojary, B. Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry. Int. J. Appl. Math. Theor. Phys. 2025, 11(2), 31-35. doi: 10.11648/j.ijamtp.20251102.12

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Poojary B. Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry. Int J Appl Math Theor Phys. 2025;11(2):31-35. doi: 10.11648/j.ijamtp.20251102.12

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@article{10.11648/j.ijamtp.20251102.12,
  author = {Bhushan Poojary},
  title = {Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry
},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {2},
  pages = {31-35},
  doi = {10.11648/j.ijamtp.20251102.12},
  url = {https://doi.org/10.11648/j.ijamtp.20251102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251102.12},
  abstract = {In classical mechanics, stress and strain are defined within real vector spaces using real-valued second-order tensors. Recent developments in quantum field theory, holography, and complex spacetime geometry indicate that such real-valued frameworks may be insufficient to describe phenomena like quantum entanglement, nonlocal curvature, and holographic effects. This paper proposes an extension of the classical stress-strain theory into complex vector spaces by introducing complex displacement fields. From these fields, complex strain and stress tensors are derived. The imaginary components of these tensors are interpreted as internal curvature, holographic tension, or phase-related deformations, which may represent hidden quantum degrees of freedom. The proposed formalism accommodates dissipative processes, non-Hermitian behavior, and stress effects induced by quantum entanglement. Applications of this framework are discussed for quantum materials, holographic analog systems, and cosmological models where classical stress-energy tensors might require complex-valued generalizations. This approach provides a unified tensorial representation for exploring deformation across classical and quantum regimes, potentially offering insights into the interplay between geometry and quantum fields.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Complex Deformation Tensors: Extending Stress-Strain Theory into Quantum Geometry

AU  - Bhushan Poojary
Y1  - 2025/08/04
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251102.12
DO  - 10.11648/j.ijamtp.20251102.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 31
EP  - 35
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20251102.12
AB  - In classical mechanics, stress and strain are defined within real vector spaces using real-valued second-order tensors. Recent developments in quantum field theory, holography, and complex spacetime geometry indicate that such real-valued frameworks may be insufficient to describe phenomena like quantum entanglement, nonlocal curvature, and holographic effects. This paper proposes an extension of the classical stress-strain theory into complex vector spaces by introducing complex displacement fields. From these fields, complex strain and stress tensors are derived. The imaginary components of these tensors are interpreted as internal curvature, holographic tension, or phase-related deformations, which may represent hidden quantum degrees of freedom. The proposed formalism accommodates dissipative processes, non-Hermitian behavior, and stress effects induced by quantum entanglement. Applications of this framework are discussed for quantum materials, holographic analog systems, and cosmological models where classical stress-energy tensors might require complex-valued generalizations. This approach provides a unified tensorial representation for exploring deformation across classical and quantum regimes, potentially offering insights into the interplay between geometry and quantum fields.
VL  - 11
IS  - 2
ER  -

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