Possible Extensions on Quantum Field Theory, Loop Quantum Gravity and Statistics

© The Author(s) 2025. Published by AMO Publisher. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https:// creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Possible Extensions on Quantum Field Theory, Loop Quantum Gravity and Statistics Yi-Fang Chang  Department of Physics, Yunnan University, Kunming, 650091, China Article History: Received: 09.10.2025 Revised: 21.10.2025 Accepted: 22.10.2025 Published: 23.10.2025 Abstract Based on the extensive quantum theory, the extensive quantum field theory first is discussed. Second, we propose the extensive loop quantum gravity and nineteen possible ways. Third, we research the extensive quantum statistics and relations with various interactions, in which FD and BE statistics correspond to repulsion and attraction interactions, respectively. Fourth, we study these theoretical applications in biology. Keywords: quantum theory, quantum field, loop quantum gravity, extension, statistics, interaction, biology, application. Suggested citation: Chang, Y.-F. (2025). Possible Extensions on Quantum Field Theory, Loop Quantum Gravity and Statistics. European Journal of Theoretical and Applied Sciences, 3(6), 62-72. https://doi.org/10.59324/ejtas.2025.3(6).05 Introduction Physics and mathematics are two important bases of modern science. Based on a new form ran n=2 of the Titius-Bode law in the solar system, we developed a similar theory with the Bohr atom model, and obtained the quantum constants HaGM=( ) /  1 2 and corresponding Schrödinger equation (Chang, 2002). Further, we proposed the extensive quantum theory and its three laws: 1. Extensive quantum is its element in any system. 2. Its theory is the same with the quantum mechanics and only quantum constant →H, and corresponding basic quantum elements are different (Chang, 2018a). 3. Evolutions of systems may be continuous, but stable states are quantized. Its mathematical base is fractal. It is the same with Feynman’s idea: “There are certain situations in which the peculiarities of quantum mechanics can come out in a special way on large scale.” In a special situation “quantum mechanics will produce its own characteristic effects on a large or ‘macroscopic’ scale” (Feynman, et al. 1966). Using the geometric average method, three different values of the quantum constants of man, cell and macromolecule may be derived for biological, chemical and physical discrete systems with different scales. We researched superconductivity, superfluidity, Bose-Einstein condensation (BEC), and various macroscopic quantum phenomena by this theory (Chang, 2018a,2024a). It is quantum spans from microscopic to astronomical and macroscopic scales, and is consistent with R and 1/R symmetries. In 2025 Nobel Prize in Physics three winner J. Clarke, M.H. Devoret and J.M. Martinis affirmed the existence of macroscopic quantum effects. In this paper, we research the extensive quantum field, the extensive loop quantum gravity, the extensive quantum statistics with various interactions and their some applications.  www.ejtas.com European Journal of Theoretical and Applied Sciences (ISSN 2786-7447) 2025 | Volume 3 | Number 6 63 Extensive Quantum Field Theory There have the extensive quantum theory (Chang, 2002,2018a,2024a), and should have the extensive quantum field theory (QFT). QFT includes from second quantization to quantum electrodynamics (QED) and renormalization, path integration, quantum chromodynamics (QCD) and so on. General formalism of QFT is based on Schrödinger equation (Bjorken, et al. 1965): )( )( tE t t i   =    , (1) and Heisenberg equation (Bjorken, et al. 1965): ][ 1)()( EFFE it tF dt tdF −+   = . (2) The four-dimensional equations of Eq.(1) are:    x ip   −=  . (3) In momentum-energy operator representation there are the quantum equations (Dirac, 1958; Ballentine, 1998; Chang, 2020a):    p ix   = . (4) They include the quantum equation in energy representation:  E iT   −=  . (5) In theory, quantum field theory can describe various scattering cross sections. Energy infinity corresponds to renormalization, and to △t→0 and simultaneity. An important tool of QFT is Feynman diagrams. Based on the combination of Feynman diagrams and the tree-field of graph (Chang, 2014a), we proposed a new development on graph theory, which includes five types of the basic elements: various solid lines, dotted lines, wavy lines and vertices, fields. Then we researched their possible applications in biology, physics and social sciences, etc. In particular, the hypercycle and its matrix representations of graph theory are discussed (2014b). In the extensive quantum field theory the corresponding equations of Eqs.(3) are:    x iHp  −= . (6) In momentum-energy operator representation they are:    p iHx  = . (7) A corresponding Heisenberg equation is: ][ 1)()( EFFE iHt tF dt tdF −+   = . (8) Quantum field corresponds to gravitational field, thus quantum redshift should exist. Extensive Loop Quantum Gravity Theory Starting from general relativity and incorporating certain ideas of QCD, it will include to effort and achieve compatibility with quantum field theory. The result space structure becomes quantum, and formed by interconnected gravitational loops that create a “spin network” system. Different spin networks define distinct quantum www.ejtas.com European Journal of Theoretical and Applied Sciences (ISSN 2786-7447) 2025 | Volume 3 | Number 6 64 states through varying spatial shapes. Their evolution generates spin-foam. It is namely the loop quantum gravity (LQG). LQG theory is a fascinating problem in theoretical physics (Ashtekar, 1986,1987; Ashtekar, et al. 2017; Baggott, 2018). It constitutes a very small discontinuous loop space, and provides a natural embedding of the constraint surface in the phase space of Einstein theory into that of Yang-Mills (YM) gauge theory. Ashtekar (1987) introduced a complex coordinate on the extended phase space, and given a certain complexified SU(2) connection. Based on the canonical quantization, Ashtekar, Rovelli and Smolin (1992) investigated the nonperturbative quantum gravity, and these methods of loop variable have opened up bridges between gravity and other areas in mathematics and physics such knot theory, Chern-Simons theory and YM theory. Then Ashtekar, et al. (1998), introduced a black hole sector of nonperturbative canonical quantum gravity. In the loop quantum gravity the fabric of space is like a weave of tiny threads, and area comes in discrete units: each thread poking through a surface gives it a little bit of area (Baes, 2003). Ashtekar (1987) introduced new variables  i a (the square root of three-metric) and Aa i (the potential for the self-dual part of the curvature). The dynamical equations are: ~ {~ ~ ~ ), [ ] [ ]     a b b a b a D iT T= +  2 (9) = ([ ~ A iT a 1 2  b ab b ab F T F, ] ).   − (10) For the gravity including matter, the new equations for metric gab are: G g ab ab bAA A a A a A A A a A a A A + =  −  +  −  1 2          ' ' ' ' ' [ ( ) ( ) ] −  + + 1 48 8    ab cd c d ab ab k E KG EYM( ) ( ). (11) Here Gab is the Einstein tensor, EKG ab ( ) and EYM ab ( ) are the standard stress-energy tensors of the Klein-Gordon and YM fields. The equations of a massive Dirac spin-1/2 field ( ' ,A A  ) are:    AA a a a A A ikim ' ' ( ) , − = 3 82 (12)    AA a a a A A ikim ' ' ( ) , + = 3 82 (13) Where k i a a AA A A A A = − −2      ' ' ' (), (14) and    A A , are the complex conjugate variables. All equations (9)-(13) are nonlinear. It is very difficult that these equations are solved exactly. We researched the extensive superstring (Chang, 2023a). Such it should have also the extensive loop quantum gravity (ELQG), which can reach the macro network theory. ELQG may be obtained by nineteen possible ways: 1). Dirac canonical quantization combined with geometric dynamics extended to quantum gravity. www.ejtas.com European Journal of Theoretical and Applied Sciences (ISSN 2786-7447) 2025 | Volume 3 | Number 6 65 2). ADM formulation and Wheeler-De Witt cosmological equation of the wave function. 3). Covariant approach to quantum gravity. 4). Wilson-Polyakov-Barbour loop. Loop defines the quantum space. 5). Ashtekar variables and Smolin-Rovelli LQG. 6). From closed string to the knot. The trefoil corresponds to two spins, while Borromean rings correspond to quark non-division (Chang, 2014c). 7). Weave (Ashtekar, et al. 1992). 8). Limited Hamiltonians can be constructed using background-independent area and volume operators. Intersection points serve as nodes in the network, while volume operators at these nodes derive measurable quantities: specific quantum volumes at each node. The force lines correspond to area operators, which are related to the square of the Planck length, and the spectrum of area is proportional to: 2702103)1(8 mjjlp − =+  . (15) 9). In1971 Penrose discussed spin networks and twistor theory, and proposed that collapse might be a gravitational effect caused by spacetime curvature. Gravity leads to decoherence (Penrose, 1990,2016). Rovelli and Smolin (1995a,b) discussed discreteness of area and volume in quantum gravity, and relation of spin networks and quantum gravity. Such loops, knots, links, spin networks and quantized areavolume can all be proved by general relativity and quantum theory. 10). Graphics and topological changes are time, and introducing spin foam (Smolin, 1995b; Wheeler, et al. 1998; Penrose, 2005). 11). A general boundary determines the relations between regions (Oeckl, 2003). The BarberoImmirzi parameter is already present in Ashtekar’s original formulation of the constrained Hamiltonian based on his new variables. 12). Bianchi, et al. (2013), used the method of combining fermions with QCD lattice, and spinfoam tends to spacetime points at the boundary to obtain the fermion action. A quantum theory of gravitational and fermion interactions is established. The endpoints of the open string and the non-closed loop are charges. 13). Wheeler, et al. (1998), searched the unification of gravity and electromagnetism. 14). Combining prequark models, such as Harari-Shupe model, LQG model may be derived (Pfister, et al. 2003). 15). In QG theory the first is the quantum nature and behavior, and spacetime and general relativity are secondary phenomena at low energy approximations. Then it is possible that everything evolves from a set of abstract interactions, whether quantum or spacetime (gravitational field). 16). Loop quantum cosmology (LQC) (Sundance, et al. 2007; Bojowald, 2001; Rovelli, et al. 2014). 17). AdS/CFT duality. AdS is anti-de Sitter space, CFT is conformal field theory, and Gauge/gravity duality. 18). The universe expands, contracts, and bounces back, and form a cycle of the universe (Agullo, et al. 2013). Smolin searched the principles of the open future (Steinhardt, et al. 2007). 19). Smolin (1997,2001,2008,2014) and Rovelli (2007) discussed LQG (Unger, et al. 2015). One of ELQG is an emerged loop, and is related to the hypercycle (Eigen, et al. 1979). Moreover, we may research the extensive membrane and the extensive M theory, etc. Bjorken and Drell (1964) proposed the negativeenergy solutions: 0),()( =+ spvmcp , (16) and the corresponding negative-energy eigenvalue equation: www.ejtas.com European Journal of Theoretical and Applied Sciences (ISSN 2786-7447) 2025 | Volume 3 | Number 6 66  EemeAi−=++−− ])([ . (17) They discussed the negative-energy wave. In 1954 Einstein proposed, one cannot understand why the gravitational masses all have the same sign. Based on Dirac negative energy, and combined Einstein mass-energy relation and principle of equivalence (inertial mass and gravitational mass are equal always), for negative mass Bondi proposed three kinds of mass, and there are four cases. It is a fallacy with contradictions. Since 2007 we proposed and gradually completed the negative matter as the simplest model of unified dark matter and dark energy. Because there is repulsion between positive matter and negative matter, both form two different regions of topological separation, so it is invisible dark matter, and repulsion as dark energy (Chang, 2007,2011,2013b,2020b,2023b,c,2024b,2025b). Extensive Quantum Statistics and Various Interactions Particles described by a symmetric wave function are called bosons with integer spin, while particles by an antisymmetric wave function are called fermions with semi-integer spin. Peyrard and Bishop (1989) investigated the statistical mechanics of a simple nonlinear lattice model for the denaturation of the DNA double helix. This PB model consists of two chains connected by Morse potentials representing the H bonds. Then Peyrard, et al., discussed nonlinear dynamics of DNA, its statistical mechanics, and researched the fundamental properties of nonlinear lattices and their applications in condensed matter and biomolecular physics (Terraneo, et al. 2002). Based on the most basic features whole and nonlinearity of the biology and combining the general nonlinear theory, we proposed the nonlinear whole biology and its four basic hypotheses (Chang, 2012b). The fundamental thought is based on the biological structure and holism. It is well-known that quantum statistics includes Bose-Einstein (BE) and Fermi-Dirac (FD) statistics, both numbers of particle are (Landau, et al. 1980): 1 /)(  =−T e gd dN   . (18) We proposed the possible entropy decrease due to fluctuation magnified and internal interactions in some isolated systems (Chang, 1997,2005,2012c,2013a,2018b, 2020c,d, 2022c,d, 2023d, 2024a). For any isolated system we proposed a generalized formula: ia dSdSdS += , (19) where a dS is an additive part of entropy and is always positive, and i dS is an interacting part of entropy and can be positive or negative. Eq.(13) is similar to a well known formula: SdSddS ei += , (20) in the theory of dissipative structure proposed by Prigogine. Two formulae are applicable for internal or external interactions, respectively. We found that the negative temperature will derive necessarily entropy decrease dS<0 (Chang, 2012c), in Fig.1 this is from S=NklnG to S=0. The negative temperature (Landau, et al. 1980) is contradiction with usual meaning of temperature and with some basic concepts of physics and mathematics. It is a fallacy in thermodynamics. We researched some possible tests of entropy decrease in isolated systems in physics, chemistry and biology, etc. It should be confirmed by many stable states in Nature. www.ejtas.com European Journal of Theoretical and Applied Sciences (ISSN 2786-7447) 2025 | Volume 3 | Number 6 67 Figure 1. Negative Temperature A complete formulation of entropy should be the symmetrical structure:         += += → → . .dddS . ei ia dSdSdS SS decrease increase Entropy (21) Here entropy decrease may be the dissipative structure for an open system, or be the internal interactions for an isolated system. In the statistical system, the general elements have no interaction g=0. It corresponds to the ideal gases, and the absence of interaction between molecules (Landau, et al. 1980). Based on in FD statistics the quantum exchange effects lead to the occurrence of an additional effective repulsion between the particles, and in BE statistics there is an effective attraction between the particles, we further research general relations between the extensive quantum statistics (EQS) and various interactions. For attraction g>0, for repulsion g<0, the entropy increase should be extended. EQS is probably ±1→±n. EQS is generalized to analogies with electromagnetic interactions, and strong and weak interactions, etc. For ordinary charges, Coulomb force is: 2 r eQ F= . (22) So the opposite charges attract, and the same charges repel. If the negative matter is introduced (Chang, 2007,2011,2013,2020b,2022b,2023b,c,2024b, 2025b), 21 2MM r G F−= , (23) it will be the same attracts and the opposite repels. Statistical mechanics corresponds and is extended to interactions. Such quantum statistics may be applied to chemistry, biology, astronomy, geoscience, social sciences and so on. Statistics is applied when interactions exist. In this case, the first law of thermodynamics states that energy is conserved, and it holds universally. When temperature is actually measured, the system remains permanently out of equilibrium. The second law should be revised accordingly. If some interactions cannot apply statistics, they will be a correspondence: The extensive BE statistics because BEC correspond to the attraction, and similar to opposite charges and the interaction of various attractions, such as strong interactions and Van der Waals (VdW) forces. The extensive FD statistics because Pauli exclusion principle (PEP) correspond to the repulsion, and similar to same charges and various incompatible things, such as weak interactions and decay. Both unifications are Schrödinger equation: 0)( 2 2=−+  VE m  . (24) Unification corresponds to ghost particle with spin-0, but is similar to fermion, and it is a complex isospin vector field (Lee, 1981). www.ejtas.com European Journal of Theoretical and Applied Sciences (ISSN 2786-7447) 2025 | Volume 3 | Number 6 68 Interaction as the external field may correspond to the thermal potential E, W, F (Landau, et al. 1980). Pressure is: S V E P)(  −= . (25) PdVTdSdE −= . (26) This is one of the most important relations in thermodynamics. The balance is P constant, dS=dE/T. T>0, so dE>0, dS>0; dE<0, dS<0. For BE statistics the chemical potential is always the negative value (Landau, et al. 1980). Statistical unity corresponds to strong-weak micro unity (Chang, 2020e,2022d), and distance scale unity. Strong interactions correspond to quark confinement r> 13 10− cm, and weak interactions r< 15 10− cm. This is the asymptotic freedom between strong and weak interactions, and corresponds to no interaction. Both electrons and protons are fermions, and carry charges. So e-e and p-p repel each other, and e-p attracts each other. The general interaction is far attraction, near repulsion. VdW forces of average between molecules are attractive. BE statistics should have a similar nuclear magnetic resonance effect. Further, the extensive statistics are extended to general systems, in which opposite attracts, and same repels; and high or low emotional intelligence. There are three types of statistics: 1. Similar BE statistics, mutual attraction. 2. Similar FD statistics, mutual repulsion. 3. It is variable between the two types. The interactions between many macromolecules can be defined as networks (Bilalic, 2017), such as metabolic networks and regulatory networks (Odom, et al. 2004; Zaslaver, et al. 2004) For the reasonable regulation of the network, BE is the connection and FD is the disconnection. In a word, various extensive theories are fractals of different scales. Theoretical Applications in Biology We discussed two new mathematical methods applied in biology. For Cambrian life explosion we proposed a chaotic explanation (Chang, 2025a). Based on the inseparability and correlativity of the biological systems, we proposed the nonlinear whole biology and four basic hypotheses. It may unify reductionism and holism, structuralism and functionalism (Chang, 2012b). We proposed the extensive quantum biology, in which different live quantum may be gene, DNA, RNA, cell, man and any live individual as the smallest live element in various levels, and discussed LQG in biology and general biological string (Chang, 2012a). Since LQG constitutes a very small discontinuous loop space, we discussed that the loop quantum theory may describe qualitatively the protein folding and the structure of lungs, and obtained four approximate conclusions: their structures are quantized, their space regions are finite, various singularities correspond to folding and crossed points, and different types of catastrophe exist. The method may be applied to describe a knot theory, etc. The statistical associations between nucleotides are shortrange, which are some internal interactions in biological systems. Further, we proposed the extensive quantum statistics of DNA and biology, and corresponding quantum equations. We discussed some new research of biology, such as the biological QED and QCD. Various levels in the biological systems have all laws with randomness and statistics, for which the most similar physical theory will be general quantum theory (Chang, 2025a). Biological macromolecules are mainly divided into two categories: proteins and nucleic acids. DNA is double-stranded, and RNA is single-stranded. The two major biological molecules, purine A-G and pyrimidine C-U-T, are similar to diatomic and monatomic molecules. Life is self-organization, which must be related to attraction and repulsion. The principle of www.ejtas.com European Journal of Theoretical and Applied Sciences (ISSN 2786-7447) 2025 | Volume 3 | Number 6 69 competition and exclusion in ecology is a fundamental one. This may be applied to biology, sociobiology, and even the Pauli principle in paramecium. We should define and classify various interactions in biology, including various levels, from the basic electromagnetic interaction and VdW force of biomacromolecules to genes, cells and so on to plants, animals and the Earth ecosystem. They are basically competition or cooperation. Electromagnetic interactions govern both intracellular and extracellular environments. These technologies include electroencephalography (EEG), magnetoencephalography (MEG), nuclear magnetic resonance imaging (NMRI), computed tomography (CAT), positron emission tomography (PET), functional magnetic resonance imaging (fMRI), and related modalities. It is related with the formdevelopment field. Survival of the fittest seems to require statistical variability, and corresponds to variable particle properties. Moreover, we may research the extensive quantum theory in social sciences, which includes the social entangled states and exclusion. The direction of social development must be from exclusion to compatibility and then to harmony. Conclusion Based on the extensive quantum theory, we research the extensive quantum field theory, and propose the extensive loop quantum gravity and nineteen possible ways. We discuss the extensive quantum statistics and relations with various interactions. These theories may be applied to biology and so on. References Agullo, I., Ashtekar, A., & Nelson, W. (2013). The preinflationary dynamics of loop quantum cosmology: Confronting quantum gravity with observations. Classical and Quantum Gravity, 30(8), 085014. https://doi.org/10.1088/02649381/30/8/085014 Ashtekar, A. (1986). New variables for classical and quantum gravity. Physical Review Letters, 57, 2244-2246. https://doi.org/10.1103/PhysRevLett.57.2244 Ashtekar, A. (1987). New Hamiltonian formulation of general relativity. 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