The interaction between spinors and gravity is the most complicated and subtle interaction in the universe, which involves the basic problem to unified quantum theory and general relativity. By means of Clifford Algebra, a unified language and tool to describe the rules of nature, this paper systematically discusses the dynamics and properties of spinor fields in curved space-time, such as the decomposition of the spinor connection, the classical approximation of Dirac equation, the energy momentum tensor of spinors and so on. To split spinor connection into Keller connection $\Upsilon_\mu\in\Lambda^1$ and pseudo-vector potential $\Omega_\mu\in\Lambda^3$ by Clifford algebra not only makes the calculation simpler, but also highlights their different physical meanings. The representation of the new spinor connection is dependent only on the metric, but not on the Dirac matrix. Keller connection only corresponds to geometric calculations, but the potential $\Omega_\mu$ has dynamical effects, which couples with the spin of a spinor and may be the origin of the celestial magnetic field. Only in the new form of connection can we clearly define the classical concepts for the spinor field and then derive its complete classical dynamics, that is, Newton's second law of particles. To study the interaction between space-time and fermion, we need an explicit form of the energy-momentum tensor of spinor fields. However, the energy-momentum tensor is closely related to the tetrad, and the tetrad cannot be uniquely determined by the metric. This uncertainty increases the difficulty of deriving rigorous expression. In this paper, through a specific representation of tetrad, we derive the concrete energy-momentum tensor and its classical approximation. In the derivation of energy-momentum tensor, we obtain a spinor coefficient table $S^{\mu\nu}_{ab}$, which plays an important role in the interaction between spinor and gravity. From this paper we find that, Clifford algebra has irreplaceable advantages in the study of geometry and physics.

Clifford algebra is unified language and efficient tool for geometry and physics. In this paper, we introduce this algebra to derive the integrable conditions for Dirac and Pauli equations. This algebra shows the standard operation procedure and deep insights into the structure of the equations. Usually, the integrable condition is related to the special symmetry of transformation group, which involves some advanced mathematical tools and its availability is limited. In this paper, the integrable conditions are only regarded as algebraic conditions. The commutators expressed by Clifford algebra have a neat and covariant form, which are naturally valid in curvilinear coordinate system and curved space-time. For Pauli and Schr\"odinger equation, many solutions in axisymmetric potential and magnetic field are also integrable. We get the scalar eigen equation in dipole magnetic field. By the virtue of Clifford algebra, the physical researches may be greatly promoted.

In this note we construct explicit complex and real matrix representations for the generators of real Clifford algebra $C\ell_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. We find two classes of representation, the normal representation and exceptional one. The normal representation is a large class of representation which can only be expanded into $4m+1$ dimension, but the exceptional representation can be expanded as generators of the next period. In the cases $p+q=4m$, the representation is unique in equivalent sense. These results are helpful for both theoretical analysis and practical calculation. The generators of Clifford algebra are the faithful basis of $p+q$ dimensional Minkowski space-time or Riemann space, and Clifford algebra converts the complicated relations in geometry into simple and concise algebraic operations, so the Riemann geometry expressed in Clifford algebra will be much simple and clear.

Most fully developed galaxies have vivid spiral structure, but the formation and evolution of spiral structure is still a mystery that is not fully understood in astrophysics. We find that the currently used equations of galactic dynamics contain some unreasonable components. In this paper, the following three working assumptions are introduced to simplify the galactic structural equations. 1. In the research of large-scale structure, the retarded potential of the gravitational field should be taken into account. The propagating time of the gravitational field from center to border is longer than the revolution periods of the stars near the center of galaxy. Newton's gravitational potential is unreasonable for such case, and the weak field and low velocity approximation of Einstein's field equation should be adopted. 2. The stars in a fully developed galaxy should be zero-pressure and inviscid fluid, and the equation of motion is different from that of ordinary continuum mechanics. Stars move along geodesics. 3. The structure of the galaxy is only related to the total mass density distribution. The equation of state of dark halo is different from that of ordinary luminous interstellar matter, so their trajectories are also very different. Dark halo and ordinary matter in galaxy are automatically separated. The total mass density distribution can be presupposed according to the observation data, and then it can be determined by comparing the solution of the equations with the observed data. These assumptions and treatments are supported by theory and observation. The variables of the equations of simplified galactic dynamics are separated from each other, and the equations are well-posed and can be solved according to a definite procedure. Therefore, this simplified dynamic equation system provides a more reasonable and practical framework for the further study of galactic structure, and can solve many practical problems. Besides, it is closely related to the study of dark energy and dark matter.

In cosmology, the cosmic curvature $K$ and the cosmological constant $\Lambda$ are two most important parameters, whose values have strong influence on the behavior of the universe. By analyzing the energy-momentum tensor and equations of state of ideal gas, scalar, spinor and vector potential in detail, we find that the total mass density of all matter is always positive, and the initial total pressure is negative. Under these conditions, by qualitatively analyzing the global behavior of the dynamical equation of cosmological model, we get the following results: (i) $K= 1$, namely, the global spatial structure of the universe should be a 3-dimensional sphere $S^3$. (ii) $0\le\Lambda < 10 ^ {-24} {\rm ly} ^ {-2}$, the cosmological constant should be zero or an infinitesimal. (iii) $a(t)>0$, the initial singularity of the universe is unreachable, and the evolution of universe should be cyclic in time. This means that the initial Big Bang is impossible at all. Since the matter components considered are quite complete and the proof is very elementary and strict, these logical conclusions should be quite reliable. Obviously, these conclusions will be much helpful to correct some popular misconceptions and bring great convenience to further research other problems in cosmology such as property of dark matter and dark energy.

The magnetic field of the earth plays an important role in the ecosystem, and the magnetic field of celestial bodies is also important in the formation of cosmic large-scale structures, but the origin and evolution of the celestial magnetic field is still an unresolved mystery. Many hypotheses to explain the origin have been proposed, but there are some insurmountable difficulties for each one. At present, the theory widely accepted in scientific society is the dynamo model, it says that, the movement of magnetofluid inside celestial bodies, which can overcome the Ohmic dissipative effect and generate persistent weak electric current and macroscopic magnetic field. However, this model needs an initial seed magnetic field, and the true values of many physical parameters inside the celestial body are difficult to obtain, and there is no stable solution to the large range of fluid motion. These are all difficulties for the dynamo model. Furthermore, it is difficult for the dynamo to explain the correlation between the dipole magnetic field and angular momentum of a celestial body. In this paper, by calculating the interaction between spin of particles and gravity of celestial body according to Clifford algebra, we find that a rotational celestial body provides a field $\Omega^\alpha$ for spins, which is similar to the magnetic field of a dipole, and the spins of charged particles within the celestial body are arranged along the flux line of $\Omega^\alpha$, then a macroscopic magnetic field is induced. The calculation shows that the strength of $\Omega^\alpha$ is proportional to the angular momentum of the celestial body, which explains the correlation between the magnetic intensity and angular momentum. The results of this paper suggest that further study of the effects of internal variables such as density, velocity, pressure and temperature of a celestial body on $\Omega^\alpha$ may provide some new insights into the origin and evolution of celestial magnetic field.

In this paper, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area and volume, and unifies the calculus of scalar, spinor, vector and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion and vector algebra, converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.

The description of the microscopic world in quantum mechanics is very different from that in classical physics, and there are some points of view that are contrary to intuition and logic. The first is the loss of reality, the behavior of micro particles shows randomness and hopping. The second is the loss of certainty, the conjugate physical variables of a system cannot be determined synchronously, they satisfy the Heisenberg uncertainty principle. The third is the non-local correlation. The measurement of one particle in the quantum entanglement pair will change the state of the other entangled particle simultaneously. In this paper, some concepts related to quantum entanglement, such as EPR correlation, quantum entanglement correlation function, Bell's inequality and so on, are analyzed in detail. Analysis shows that the mystery and confusion in quantum theory may be caused by the logical problems in its basic framework. Bell's inequality is only a mathematical theorem, but its physical meaning is actually unclear. The Bell state of quantum entangled pair may not satisfy the dynamic equation of quantum theory, so it cannot describe the true state of microscopic particles. In this paper, the correct correlation functions of spin entanglement pair and photonic entanglement pair are strictly derived according to normal logic. Quantum theory is a more fundamental theory than classical mechanics, and they are not parallel relation in logic. However, there are still some unreasonable contents in the framework of quantum theory, which need to be improved. In order to disclose the real relationship between quantum theory and classical mechanics, we propose some experiments which show the wave-particle duality simultaneously and provide intuitionistic teaching materials for the new interpretation of quantum theory.