The lambda phi4 scalar field model can be applied to interpret pion-pion scattering and properties of hadrons. In this work, the mathematical basis, phase transitions and singularities of a (3+1)-dimensional (i.e., (3+1)D) phi4 scalar field model are investigated. It is found that as a specific example of topological quantum field theories, the (3+1)D phi4 scalar field model must be set up on the Jordan-von Neumann-Wigner framework and dealt with the parameter space of complex time (or complex temperature). The use of the time average and the topologic Lorentz transformation representing Reidemeister moves ensure the integrability, which takes into account for the contributions of nontrivial topological structures to physical properties of the many-body interacting system. The ergodic hypothesis is violated at finite temperatures in the (3+1)D phi4 scalar field model. Because the quantum field theories with ultraviolet cutoff can be mapped to the models in statistical mechanics, the (3+1)D phi4 scalar field model with ultraviolet cutoff is studied by inspecting its relation with the three-dimensional (3D) Ising model. Furthermore, the direct relation between the coupling K in the 3D Ising model and the bare coupling lambda0 in the (3+1)D phi4 scalar field model is determined in the strong coupling limit. The results obtained in the present work can be utilized to investigate thermodynamic physical properties and critical phenomena of quantum (scalar) field theories.
The lambda phi4 scalar field model can be applied to interpret pion-pion scattering and properties of hadrons. In this work, the mathematical basis, phase transitions and singularities of a (3+1)-dimensional (i.e., (3+1)D) phi4 scalar field model are investigated. It is found that as a specific example of topological quantum field theories, the (3+1)D phi4 scalar field model must be set up on the Jordan-von Neumann-Wigner framework and dealt with the parameter space of complex time (or complex temperature). The use of the time average and the topologic Lorentz transformation representing Reidemeister moves ensure the integrability, which takes into account for the contributions of nontrivial topological structures to physical properties of the many-body interacting system. The ergodic hypothesis is violated at finite temperatures in the (3+1)D phi4 scalar field model. Because the quantum field theories with ultraviolet cutoff can be mapped to the models in statistical mechanics, the (3+1)D phi4 scalar field model with ultraviolet cutoff is studied by inspecting its relation with the three-dimensional (3D) Ising model. Furthermore, the direct relation between the coupling K in the 3D Ising model and the bare coupling lambda0 in the (3+1)D phi4 scalar field model is determined in the strong coupling limit. The results obtained in the present work can be utilized to investigate thermodynamic physical properties and critical phenomena of quantum (scalar) field theories.
The Ising model describes a many-body interacting spin (or particle) system, which can be utilized to imitate the fundamental forces of nature. Although it is the simplest many-body interacting system of spins (or particles) with Z2 symmetry, the phenomena revealed in Ising systems may afford us lessons for other types of interactions in nature. In this work, we first focus on the mathematical structure of the three-dimensional (3D) Ising model. In the Clifford algebraic representation, many internal factors exist in the transfer matrices of the 3D Ising model, which are ascribed to the topology of the 3D space and the many-body interactions of spins. They result in the nonlocality, the nontrivial topological structure, as well as the long-range entanglement between spins in the 3D Ising model. We review briefly the exact solution of the ferromagnetic 3D Ising model at the zero magnetic field, which was derived in our previous work. Then, the framework of topological quantum statistical mechanics is established, with respect to the mathematical aspects (topology, algebra, and geometry) and physical features (the contribution of topology to physics, Jordan–von Neumann–Wigner framework, time average, ensemble average, and quantum mechanical average). This is accomplished by generalizations of our findings and observations in the 3D Ising models. Finally, the results are generalized to topological quantum field theories, in consideration of relationships between quantum statistical mechanics and quantum field theories. It is found that these theories must be set up within the Jordan–von Neumann–Wigner framework, and the ergodic hypothesis is violated at the finite temperature. It is necessary to account the time average of the ensemble average and the quantum mechanical average in the topological quantum statistical mechanics and to introduce the parameter space of complex time (and complex temperature) in the topological quantum field theories. We find that a topological phase transition occurs near the infinite temperature (or the zero temperature) in models in the topological quantum statistical mechanics and the topological quantum field theories, which visualizes a symmetrical breaking of time inverse symmetry.
An overview of the mathematical structure of the three-dimensional (3D) Ising model is given, from the viewpoints of topologic, algebraic and geometric aspects. By analyzing the relations among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model: 1) The complexified quaternion basis constructed for the 3D Ising model represents naturally the rotation in a (3 + 1) - dimensional space-time, as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function by taking the time average. 2) A unitary transformation with a matrix being a spin representation in 2^(nlo)-space corresponds to a rotation in 2nlo-space, which serves to smooth all the crossings in the transfer matrices and contributes as the non-trivial topologic part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed also by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases Φx, Φy, and Φz. The relation with quantum field and gauge theories, physical significance of weight factors are discussed in details. The conjectured exact solution is compared with numerical results, and singularities at/near infinite temperature are inspected. The analyticity in β = 1/(kB T) of both the hard-core and Ising models has been proved for β > 0, not for β = 0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.
An overview of the mathematical structure of the three-dimensional(3D) Ising model is given from the points of view of topology,algebra,and geometry.By analyzing the relationships among transfer matrices of the 3D Ising model,Reidemeister moves in the knot theory,Yang-Baxter and tetrahedron equations,the following facts are illustrated for the 3D Ising model.1) The complex quaternion basis constructed for the 3D Ising model naturally represents the rotation in a(3+1)-dimensional space-time as a relativistic quantum statistical mechanics model,which is consistent with the 4-fold integrand of the partition function obtained by taking the time average.2) A unitary transformation with a matrix that is a spin representation in 2 n·l·o-space corresponds to a rotation in 2n·l·o-space,which serves to smooth all the crossings in the transfer matrices and contributes the non-trivial topological part of the partition function of the 3D Ising model.3) A tetrahedron relationship would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model,and its existence is guaranteed by the Jordan algebra and the Jordan-von Neumann-Wigner procedures.4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φx,φy,and φz.The relationship with quantum field and gauge theories and the physical significance of the weight factors are discussed in detail.The conjectured exact solution is compared with numerical results,and the singularities at/near infinite temperature are inspected.The analyticity in β=1/(kBT) of both the hard-core and the Ising models has been proved only for β>0,not for β=0.Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.
The three-dimensional (3D) Ising model is mapped into a 3D spinless fermionic model by the Jordan-Wigner transformation. The exact solution of the 3D model for spinless fermions is derived analytically by performing a diagonalization process consisting of the Clifford algebraic approach, the Fourier transformation and the Bogoliubov transformation. The Clifford algebraic approach is the same as that developed for the 3D Ising model, using a time average within the Jordan-von Neumann-Wigner framework, a linearization procedure and a local gauge transformation. The formulas for eigenvalues, partition function, subsequent thermodynamic properties and critical behaviors are presented. The dimensionality and the topological phases are investigated. The present results for many spinless fermions in a 3D lattice are applicable for studying the mechanisms of magnetism, superfluid, superconductors and topological materials.
The phi4 scalar field theory in three dimensions, prototype for the study of phase transitions, is investigated by means of the hierarchical reference theory (HRT) in its smooth cutoff formulation. The critical behavior is described by scaling laws and critical exponents which compare favorably with the known values of the Ising universality class. The inverse susceptibility vanishes identically inside the coexistence curve, providing a first principle implementation of the Maxwell construction, and shows the expected discontinuity across the phase boundary, at variance with the usual sharp cutoff implementation of HRT. The correct description of first and second order phase transitions within a microscopic, nonperturbative approach is thus achieved in the smooth cutoff HRT.
Abstract: This study proposes a unified framework that integrates topological quantum field theories with statistical mechanics to provide a comprehensive understanding of quantum states and dynamics. The problem of reconciling the quantum mechanical description of microscopic systems with macroscopic statistical behavior is addressed by exploring the relationship between topological invariants and thermodynamic properties in quantum systems. The methodology involves the development of a mathematical model that combines the abstract principles of topological quantum field theory with the probabilistic approaches of statistical mechanics, utilizing tensor calculus and advanced mathematical physics tools. Key results indicate that the proposed framework offers new insights into quantum phase transitions, critical phenomena, and the thermodynamic behavior of quantum systems, especially in systems exhibiting topological phases. The framework is further evaluated through a series of theoretical simulations, demonstrating its applicability to quantum information science, condensed matter physics, and quantum technology applications. The implications of this research are significant, providing a foundation for the development of new quantum technologies, such as quantum computing and quantum materials, and contributing to a deeper understanding of the connections between topological properties and statistical mechanics in quantum systems. Keywords: Topological Quantum Field Theory, Statistical Mechanics, Quantum States, Quantum Dynamics, Phase Transitions, Critical Phenomena, Quantum Computing, Quantum Materials, Thermodynamics, Topological Phases, Quantum Information Science.