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International Journal of Geometric Methods in Modern Physics (02198878)22(8)
In this paper, we aim to interpret the background gravitational effects appearing in quantum field theory on curved space-time by studying the Brownian motion of quantum states along with the Hamilton–Perelman Ricci flow. It has been shown that the Wiener measure automatically contains the Einstein–Hilbert action and the path-integral formulation of the scalar quantum field theory on curved space-time at the first order of local approximations. This provides a well-defined formulation of the path-integral measure for quantum field theory in the presence of gravity. However, we establish that the emergence of Einstein–Hilbert action is independent of the matter field interactions and is a merely entropic/geometric effect stemming from the nature of the Ricci flow of the universe geometry. We also extract an explicit formula for the cosmological constant in terms of the Ricci flow and Hamilton’s theorem for 3-manifolds. Then, we discuss the cosmological features of the FLRW solution in ΛCDM Model via the derived equations of the Ricci flow. We also argue the correlation between our formulations and the entropic aspects of gravity. Finally, we provide some theoretical evidence that proves the second law of thermodynamics is the basic source of gravity and probably a more fundamental concept. © 2025 World Scientific Publishing Company.
International Journal of Geometric Methods in Modern Physics (02198878)21(12)
This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman's path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman's path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman's path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of nonlocal terms but produces the Klein-Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman's path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure. © 2024 World Scientific Publishing Company.
International Journal of Geometric Methods in Modern Physics (02198878)20(10)
A differential geometric statement of the noncommutative topological index theorem is worked out for covariant star products on noncommutative vector bundles. To start, a noncommutative manifold is considered as a product space X = Y × Z, wherein Y is a closed manifold, and Z is a flat Calabi-Yau m-fold. Also, a semi-conformally flat metric is considered for X which leads to a dynamical noncommutative spacetime from the viewpoint of noncommutative gravity. Based on the Kahler form of Z, the noncommutative star product is defined covariantly on vector bundles over X. This covariant star product leads to the celebrated Groenewold-Moyal product for trivial vector bundles and their flat connections, such as C∞(X). Hereby, the noncommutative characteristic classes are defined properly and the noncommutative Chern-Weil theory is established by considering the covariant star product and the superconnection formalism. Finally, the index of the ?-noncommutative version of elliptic operators is studied and the noncommutative topological index theorem is stated accordingly. © 2023 World Scientific Publishing Company.
International Journal of Geometric Methods in Modern Physics (02198878)19(1)
The entire geometric formulations of the BRST and the anti-BRST structures are worked out in presence of the Nakanishi-Lautrup field. It is shown that in the general form of gauge fixing mechanisms within the Faddeev-Popov quantization approach, the anti-BRST invariance reflects thoroughly the classical symmetry of the Yang-Mills theories with respect to gauge fixing methods. The Nakanishi-Lautrup field is also defined and worked out as a geometric object. This formulation helps us to introduce two absolutely new topological invariants of quantized Yang-Mills theories, so-called the Nakanishi-Lautrup invariants. The cohomological structure of the anti-BRST symmetry is also studied and the anti-BRST topological index is derived accordingly. © 2022 World Scientific Publishing Company.
International Journal of Geometric Methods in Modern Physics (02198878)18(6)
A representation of general translation-invariant star products in the algebra of M(C) =limN∞MN(C) is introduced which results in the Moyal-Weyl-Wigner quantization. It provides a matrix model for general translation-invariant noncommutative quantum field theories in terms of the noncommutative calculus on differential graded algebras. Upon this machinery a cohomology theory, the so-called cohomology, with groups Hk(C), k ≥ 0, is worked out which provides a cohomological framework to formulate general translation-invariant noncommutative quantum field theories based on the achievements for the commutative fields, and is comparable to the Seiberg-Witten map for the Moyal case. Employing the Chern-Weil theory via the integral classes of Hk(a) a noncommutative version of the Chern character is defined as an equivariant form which contains topological information about the corresponding translation-invariant noncommutative Yang-Mills theory. Thereby, we study the mentioned Yang-Mills theories with three types of actions of the gauge fields on the spinors, the ordinary, the inverse, and the adjoint action, and then some exact solutions for their anomalous behaviors are worked out via employing the homotopic correlation on the integral classes of cohomology. Finally, the corresponding consistent anomalies are also derived from this topological Chern character in the cohomology. © 2021 World Scientific Publishing Company.
Reports on Mathematical Physics (00344877)86(2)pp. 157-173
Topological structure of translation-invariant noncommutative Yang–Mills theories are studied by means of a cohomology theory, the so-called ⋆-cohomology, which plays an intermediate role between de Rham and cyclic (co)homology theory for noncommutative algebras and gives rise to a cohomological formulation comparable to Seiberg–Witten map. © 2020 Polish Scientific Publishers
Physical Review D (24700010)97(6)
In this paper, we discuss the noncommutative QED2 in the S-matrix framework. We are interested in perturbatively proving that the exact Schwinger mass μ2=e2π does not receive noncommutative corrections to any order in loop expansion. In this sense, the S-matrix approach is useful since it allows us to work with the effective action Γ[A] (interaction term) to compute the corresponding gauge field 1PI two-point function at higher orders. Furthermore, by means of α∗-cohomology, we generalize the QED2 S-matrix analysis in the Moyal star product to all translation-invariant star products. © 2018 authors. Published by the American Physical Society.
International Journal of Geometric Methods in Modern Physics (02198878)14(11)
It is shown that anti-BRST invariance in quantum gauge theories can be considered as the quantized version of the symmetry of classical gauge theories with respect to different gauge fixing mechanisms. © 2017 World Scientific Publishing Company.
International Journal of Geometric Methods in Modern Physics (02198878)13(9)
It is shown that, anti-BRST symmetry is the quantized counterpart of local axial symmetry in gauge theories. An extended form of descent equations is worked out, which yields a set of modified consistent anomalies. © 2016 World Scientific Publishing Company.
Journal of Geometry and Physics (03930440)83pp. 53-68
Translation-invariant {star operator} products are studied in the setting of α *-cohomology. It is explicitly shown that all quantum behaviors including Green's functions and the scattering matrix of translation-invariant non-commutative quantum field theories are thoroughly characterized by α *-cohomology classes of the star products. © 2014 Elsevier B.V.
Journal of Mathematical Physics (00222488)54(7)
The theory of α*-cohomology is studied thoroughly and it is shown that in each cohomology class there exists a unique 2-cocycle, the harmonic form, which generates a particular Groenewold-Moyal star product. This leads to an algebraic classification of translation-invariant non-commutative structures and shows that any general translation-invariant non-commutative quantum field theory is physically equivalent to a Groenewold-Moyal non-commutative quantum field theory. © 2013 AIP Publishing LLC.
Journal of Mathematical Physics (00222488)53(4)
Translation-invariant noncommutative gauge theories are discussed in the setting of matrix modeled gauge theories. Using the matrix model formulation the explicit form of consistent anomalies and consistent Schwinger terms for translation-invariant noncommutative gauge theories are derived. © 2012 American Institute of Physics.
International Journal of Geometric Methods in Modern Physics (02198878)8(8)pp. 1747-1762
A gauge invariant partition function is defined for gauge theories which leads to the standard quantization. It is shown that the descent equations and consequently the consistent anomalies and Schwinger terms can be extracted from this modified partition function naturally. © 2011 World Scientific Publishing Company.
