The relation between these three components is
Symmetries. The symmetry for gravity is the invariance under general coordinate transformations, which is precisely described by the Einstein principle of general relativity (PGR), and the symmetry for the electromagnetism, the weak and the strong interactions is the gauge symmetry, originally proposed by Herman Weyl.
Uniqueness of Actions. The symmetries determine uniquely the actions (functionals): the PGR uniquely determines the Einstein-Hilbert functional, and the gauge symmetry uniquely dictates the Yang-Mills action.
Of course, the uniqueness is derived under the principle that the law of nature must be simple; simplicity implies stability and beauty.
Field Equations by PID. The principle of interaction dynamics (PID) takes variation of the actions subject to generalized energy-momentum conservation constraints. It is the direct consequence of the presence of dark energy and dark matter, is the requirement of the presence of the Higgs field for the weak interaction, and is the consequence of the quark confinement phenomena for the strong interaction. Hence
PID is the principle for deriving the field equations of fundamental interaction.
Summary. The fundamental theory of four interaction is now complete. The symmetry for gravity is different from the gauge symmetry for the electromagnetism, the weak and the strong interactions, leading to different actions and field equations. In essence, the electromagnetism, the weak and the strong is unified by the gauge field theory. The symmetry for gravity was discovered by Einstein and the action by Einstein and Hilbert. For the electromagnetism, the weak and the strong interactions, the gauge symmetry was discovered by Weyl. The general action was introduced by Yang and Mills.
Tian Ma & Shouhong Wang
]]>This paper is aimed to establish a dynamical law of fluctuations, and to derive the critical exponents based on the standard model with fluctuations, leading to correct critical exponents in agreement with experimental results.
1. For a thermodynamic system, the PDP proposed in
provides the dynamic law for statistical physics:
which offers a complete description of associated phase transitions and transformation of the system from non-equilibrium states to equilibrium states. This dynamic law (1) also describes automatically irreversibility.
In view of (1), we developed a systematic theory in
for deriving explicit expressions of thermodynamic potentials, based on first principles, rather than on the mean-field theoretic expansions.
The dynamic law (1) with expression formulas for the thermodynamic potentials and the dynamic transition theory developed in
Tian Ma & Shouhong Wang, Phase Transition Dynamics, Springer-Verlag, xxii, 555pp., 2013
provide a complete theoretical understanding of phase transitions and critical phenomena for thermodynamic systems. This is the basic theory of the standard model for thermodynamical systems.
2. There is, however, a discrepancy between the theoretical exponents and their experimental values, as in the case of mean-field theoretic approach. We demonstrate in this paper that in reality, there is a critical fluctuation effect, and we show that the discrepancy just mentioned is due entirely to the spontaneous fluctuation.
To have an accurate account of the fluctuations, we need to derive its governing fundamental law, which have to stem from the thermodynamic potential and the dynamic law (1).
In fact, for an equilibrium state of a thermodynamic system, the fluctuation of is the deviation from :
Then the needed dynamic law for fluctuations is given by
where is the fluctuation of the external force. 3. We derive two basic theorems on critical exponents. The first theorem is based on the dynamic law (1).
First Theorem of Critical-Exponents.
For a second-order phase transition, near the critical point, using the dynamical law (1) without fluctuations, we derive the theoretical critical exponents as given by
The second theorem takes into consideration of fluctuations.
Second Theorem of Critical-Exponents.
For a second-order phase transition, near the critical point, using the dynamical law (3) with fluctuations, the fluctuation critical exponents are given by
We now list three groups of exponent data for different thermodynamic systems:
1) experimental exponents,
2) theoretical exponents without taking into consideration of fluctuations, and
3) theoretical exponents using the standard model with fluctuations.
This table shows clearly the strong agreement of the results using the standard model of thermodynamics with fluctuations.
exponent | magnetic system | PVT system | binary system | without fluctuation | with fluctuation |
0.30-0.36 | 0.32-0.35 | 0.30-0.34 | 1/2 | 1/3 | |
4.2-4.8 | 4.6-5.0 | 4.0-5.0 | 3 | 3.0-6.0 | |
0.0-0.2 | 0.1-0.2 | 0.05-0.15 | 0 | 0-2/3 | |
1.2-1.4 | 1.2-1.3 | 1.2-1.4 | 1 | 1-5/3 |
4. We have shown that the theoretical values from the dynamic law (1) do reflect the nature under the ideal assumption that no fluctuations are present in the system. However, the fluctuations are inevitable, and are completely accounted for by the dynamic law of fluctuations (3). In a nutshell,
the standard model (1), together with the dynamic law of fluctuation (3), offers correct information for critical exponents; and
this in return validates the standard model of thermodynamics, which is derived based on first principles.
Ruikuan Liu, Tian Ma, Shouhong Wang and Jiayan Yang
]]>This is the last one in this series of papers on topological phase transitions (TPTs); see I, II, III, and IV for details.
1. Interior separation of fluid flows is a common phenomenon in fluid dynamics, especially in geophysical fluid dynamics, such as the formation of hurricanes, typhoons and tornados, and gyres of oceanic flows. In general, the interior separation refers to sudden appearance of a vortex from the interior of a fluid flow.
It is clear that fluid interior separation is a typical TPT problem, similar to the TPT associated with boundary-layer separations. Also, mathematical, the geometric theory of incompressible flows developed by two of the authors offers the needed mathematical foundation for understanding interior separations, as well as for the quantum phase transitions of the Bose-Einstein condensates, superfluidity and superconductivity. For this geometric theory, see
2. At the kinematic level, a structural bifurcation theorem theorem was proved in [MW05]. Basically, let be a one-parameter family of 2D divergence-free vector fields with the first-order Taylor expansion with respect to at :
Let be a degenerate singular point of , and the Jacobian . Then, there are two orthogonal unit vectors and such that
If is an isolated singular point if with , and satisfies that
then has an interior separation from .
3. The above result is of kinematic in nature. We need to derive a separation equation, which connects the topological structure of the fluid flows to the solutions of the Navier-Stokes equations, which govern the the fluid motion.
For geophysical fluid phenomena such as hurricanes, typhoons, and tornados, the typical interior separation phenomena are caused by external wind-driven forces and by the non-homogenous temperature distributions. Therefore, the crucial factors for the formation of interior separations in the atmospheric and oceanic flows are
Hence the dynamical fluid model for interior separations has to incorporate properly the heat effect.
The Boussinesq equations are mainly for convective flows, and are not suitable for studying interior separations, associated in particular with the such geophysical processes as hurricanes, typhoons, and tornados.
For this purpose, we use the horizontal heat-driven fluid dynamical equations by Yang and Liu [18], which couple the Navier-Stokes equation and the heat diffusion equation with the following equations of state:
4. Consider the solution
of the aforementioned fluid model with initial velocity and initial temperature . One main result we obtain is the following interior separation equations:
These separations include all physical information about the interior separations of the solution for the system, in terms of the initial state and the external force .
5. Theoretical analysis and observations show that interior separation can only occur when
For the case where is a U-flow and is a flat flow, the interior separation equations are expressed as:
Here h.o.s.t referes to higher-order terms with small cofficients. Also we show that an interior separation takes place from :
6. A typical development of a hurricane consists of several stages including an early tropical disturbance, a tropical depression, a tropical storm, and finally a hurricane stage.
Using the U-flow theory as described in Section 5 above, we derive the formation mechanism of tornados and hurricanes, providing precise conditions for their formation and explicit formulas on the time and location where tornados and hurricanes form:
Basically, we demonstrate that the early stage of a hurricane is through the horizontal interior flow separations, and we identify the physical conditions for the formation of the U-flow, corresponding to the tropical disturbance, and the temperature-driven counteracting force needed as the source for tropical depression. Basically, we demonstrate that the early stage of a hurricane is through the horizontal interior flow separations, and we identify the physical conditions for the formation of the U-flow, corresponding to the tropical disturbance, and the temperature-driven counteracting force needed as the source for tropical depression.
Ruikuan Liu, Tian Ma, Shouhong Wang and Jiayan Yang
]]>
This is the fourth of a series of papers on topological phase transitions (TPTs), including
1. A TPT refers to the change in the topological structure in the physical space of the solutions of the governing partial differential equation (PDE) models of the underlying physical problem.
Boundary-layer separation phenomenon is one of the most important processes in fluid flows, and there is a long history of studies which go back at least, if not earlier, to the pioneering work of L. Prandtl in 1904. Basically, in the boundary-layer, the shear flow can detach/separate from the boundary, generating vortices and leading to more complicated turbulent behavior. The fundamental level understanding of this challenge problem boils down to TPTs of fluid flows. Mathematically, the velocity field of the fluid satisfies the Navier-Stokes equations (NSEs) or their variations, and defines its own topological structure in the physical space , where is the physical domain that the fluid occupies. Then the TPT associated with the boundary-layer or interior separations studies transition of the topological structure of as the system control parameter varies.
2. It is clear then the geometric theory of incompressible flows developed by the authors plays a crucial role for the study of TPTs of fluids; and the complete account of this geometric theory is given in the authors’ research monograph
One component of this geometric theory is the necessary and sufficient conditions for structural stability of divergence-free vector fields. Another component of the theory crucial for the study in this paper is the theorems on structural bifurcations. These theorems form the kinematic theory for understanding the topological phase transitions associated with fluid flows.
3. The most difficult and important aspect of TPTs associated with fluid flows is to make connections between the solutions of the NSEs and their structure in the physical space. The first such connection is the separation equation [MW05, Theorem 5.4.1] for the NSEs with the rigid boundary condition:
In this paper, we derive the following separation equation for the NSE with the free-slip boundary condition:
where and are the divergence-free parts of the external forcing and the nonlinear term as defined:
The separation equations (1) and (2) provide necessary and sufficient conditions for the flow separation at a boundary point:
By exploring the leading order terms of the forcing and the initial velocity field (Taylor expansions), more detailed condition, called predicable condition, are derived in [3, 14] for the Dirichlet boundary condition case, and in this paper for the free boundary condition case.
The separation equations (1) and (2), as well as the predicable conditions determine precisely when, where, and how a boundary-layer separation occurs.
4. For example, using the separation equations (1) and (2), we derive
In particular, for the wind-driven north Atlantic circulations, with careful analysis using the separation equations (1) and (2), we derive the following conclusions:
Tian Ma, Shouhong Wang
]]>
The physical and mathematical reasons for solar surface eruptions and sunspots are not satisfactorily understood. This paper is aimed to provide a new theory for the formation of the solar surface eruptions, sunspots and solar prominences, based on the recently developed statistical theory of heat by the authors, and on the theory and notion of topological phase transitions.
1. The most important ingredient of the study is the recently developed theory of heat [T. Ma & S. Wang, Statistical Theory of Heat, 2017, hal-01578634]. We derived the energy level temperature formula, showing that the temperature is essentially the average energy level of system particles. We also obtained the photon number entropy formula, demonstrating that the entropy is the number of photons in the gap between system particles, and the physical carrier of heat is the photons.
Another important component of the theory is the vibratory mechanism of photon absorption and radiation:
a particle can only absorb and radiate photons while experiencing vibratory motion. The higher the frequency of the vibration of the particle, the larger the absorbing and radiating energy. The vibration or irregular motion of particles in a system is caused by collisions between particles and by absorbing and radiating photons.
2. This above mechanism shows immediately that for particles in high speed vibration and irregular motion, the rate of photon emission and absorption increases, leading to the number density of photons to increase, and further causing the particle energy levels to elevate. Hence, the photon absorption and emission induce the concentration of temperature, which we call the anti-diffusive effect of heat:
Due to the higher rate of photon absorption and emission of the particles with higher energy levels, the photon flux will move toward to the higher temperature regions from the lower temperature regions.
By the Stefan-Boltzmann law, the reversed heat flux measuring the anti-diffusive effect is expressed as
where is the heat effect coefficient.
3. Then by the Fourier law, we derive the following law for heat transfer for the solar atmosphere:
Here on the right-hand side, the first term represents the usual diffusion of heat, the last term is the heat source due to the solar electromagnetic fields. Importantly the second term represents the anti-diffusive effect of heat, and it is this anti-diffusive effect that leads to the formation of sunspots, the solar flares and the prominences.
4. The full model governing Sun’s surface plasma fluid combines the fluid dynamical equations, the above new heat equation (1), and the Maxwell equations. They are given as follows:
5. As the anti-diffusive term in (3) counteracts with the diffusion term, we are able to prove a temperature blow-up theorem, which shows that there exist and , such that the temperature blows up at with blow-up time estimated as
where is the initial value of temperature, , is the solar radius, and with the thickness of solar atmosphere .
6. The sunspots can now be clearly explained by the anti-diffusive effect of heat and the temperature blow-up that we just mentioned. We summarize this explanation as follows:
In fact, equation (2) dictates the behavior of mass ejections. When the temperature blows up at , the maximal forces acting on the particles near are just . Hence, in the neighborhood of , (2) can be approximatively expressed as
By the gaseous equation of state: , where is the gas constant and is the particle mass, the equation (10) is written as
The temperature blow-up shows that
Therefore we deduce from (11) that
which represents the high speed gas explosion and particle ejections. The ejection direction is
In view of (12) and (13), the equations (14) and (15) generate very strong electromagnetic radiation in the direction.
which gives rise to violent magnetic loops, perpendicular to the direction , leading to the solar prominences.
This is part of the research program initiated recently by the authors on theory and applications of topological phase transitions, including
Tian Ma, Shouhong Wang
]]>The aim of this paper is to derive a new mechanism for the formation of the galactic spiral patterns.
1. There are three types of galactic structures: the spiral, the elliptical, and the irregular. The existing theory of the formation of the spiral galactic structure is the density wave theory by Chia-Ciao Lin and Frank Shu in 1964. They proposed that the spiral arms, being non material, are caused by the non-homogeneous velocity of stars and nebulae, similar to a traffic jam in a highway. However, the reasons behind the nonhomogeneous velocity of stars and nebulae are still not clear, and the density wave theory is not completely satisfactory.
In essence the physical origin of the density wave theory is not clearly understood. This is resolved in this paper using the recent development by the authors on gravitational field particle and gravitational radiation.
2. Due to the presence of dark matter and dark energy, the Einstein general theory of relativity can be uniquely modified using PID to take into account the effect of dark energy and dark matter phenomena, and to preserve the Einstein’s two fundamental principles: the principle of equivalence and the principle of general relativity. The new field equations are given by
The gravitational field particle is described by the dual field , which is regarded as the graviton.
3. Motivated by the new statistical theory of heat by the authors, we have the following conclusions for the gravitational field particle:
the absorption and radiation of gravitons could generate a gravitational temperature field, representing the average energy level of massive matter, reminiscent of the photons yielding the temperature in thermodynamical systems.
Also, the gravitational temperature , which we call G-temperature, satisfies a diffusion equation given by
where represents the gravitational source.
The new gravitational temperature field provides the needed key source for the formation of different galactic patterns, which was entirely missing in existing theories such as the pioneering work of Lin and Shu.
4. With the G-temparature equation at our disposal, we can derive the momentum form of the astrophysical fluid dynamical model coupling the diffusion equation of G-temperature field. This new astrophysical galactic dynamics model is a dissipative system. Hence the mechanism of the formation of different galactic structures is of characteristic for dissipative systems, in contrast with the density wave theory.
5. The spatial domain of the model is the ring domain in polar coordinates
and we take the transformation
The Rubin rotational curve suggests that the momentum density in the steady state solution of the model is
where is the constant velocity in the Rubin rotational curve. Then the nondimensional deviation system from the basic solution is given by
Here Pr is the Prandtl number, is the Rubin number, and is the ratio between the disk width and the halo radius. These are non-dimensional parameters given by
For this galactic dynamics system, there are three crucial parameters for the formation of different galactic structures:
5. Mathematically, using the dynamical transition theory and the geometric theory of incompressible flows, both developed by the authors, we can demonstrate that both topological phase transition and dynamic phase transition occur at the same critical control parameters. In particular we derive the following conclusions on the formation of galactic structures, dictated by the parameters Ra, :
This paper is part of the research program initiated recently by the authors on theory and applications of topological phase transitions, including
Tian Ma, Shouhong Wang
]]>The aim of the above paper is to provide a systematic theoretical study on quantum phase transitions associated with the Bose-Einstein condensates, the superfluidity and the superconductivity.
I. Based on current developments, we now know that there are only two types of phase transitions:
The authors have developed the dynamic transition theory for dissipative systems; see [Ma & Wang, Phase Transition Dynamics, Springer-Verlag, 2013, 555pp.] and the references therein. This is a new notion of phase transitions, applicable to all dissipative systems, including nonlinear dissipative systems in statistical physics, fluid dynamics, atmospheric and oceanic sciences, biological and chemical systems etc.
A TPT refers to the transition in its topological structure in the physical space of the system, and quantum phase transitions (QPTs) belong to the category of TPTs. The notion of TPTs is originated from the pioneering work by J. Michael Kosterlitz and David J. Thouless (1972) where they identified a completely new type of phase transitions in two-dimensional systems where topological defects play a crucial role. With this work, together with F. Duncan M. Haldane, they received 2016 Nobel prize in physics.
There have been many attempts, but the basic theoretical understanding of TPTs is still largely open.
II. QPTs are TPTs for condensate systems, including the gaseous Bose-Einstein condensates, superconductivity, and superfluidity. The basic physical characteristics of a QPT are as follows:
- A QPT is a transition between quantum states, and the state quantities or the order parameters of the quantum system describing the transition should be the wave functions of the quantum states;
- the control parameters are non-thermal; and
- a QPT is a TPT of a condensate system, rather than a dynamical phase transition. Consequently, the state functions describing a QPT are in the wave function .
III. The field equations governing the condensation are determined by the principle of Hamiltonian dynamics (PHD) or equivalently by the principle of Lagrangian dynamics (PLD):
where is the control parameter. The associated phase transition equations are then the following topological structure equations:
where , , and represents the chemical potential;
IV. Based on the above three characteristics and the topological structure equations (2), the wave function of the condensate is a function of the control parameter , i.e. . Then mathematically we say that the system undergoes a QPT at a critical if the topological structure of is different from that of .
V. We deduce from the topological structure equations (2) that the supercurrents of superconductivity and superfluidity are as follows:
VI. The supercurrents enjoy the divergence–free condition (incompressibility):
The authors have developed a geometric theory for incompressible flows to study the structure and its stability and transitions of incompressible fluid flows in the physical spaces. The complete account of this geometric theory is given in the authors’ research monograph [Ma & Wang, Geometric Theory of Incompressible Flows and Applications to Fluid Dynamics, AMS, 2005]. This geometric theory can then be directly applied to study the transitions of topological structure associated with the quantum phase transitions of BEC, superfluidity and superconductivity. We derive in particular the basic theory for different TPTs, leading to transparent physical pictures of various condensates.
VII. After careful examination of the formation of Cooper pairs in superconductivity, we derive the following microscopic mechanism for the Meissner effect:
- Below the critical temperature , an applied magnetic field induces spin Cooper pairs;
- only the Cooper pairs reversely parallel to the applied are stable, leading to their physical formation; and
- the total magnetic moment of all Cooper pairs with , together with the surface supercurrents, can cancel out the magnetism induced by the applied field in the superconductor, and resists the applied field from entering its body.
Also, this paper is the first one in the series of papers by the authors on TPTs, including the forthcoming papers for the following science problems:
Tian Ma, Shouhong Wang
]]>Ruikuan Liu, Tian Ma, Shouhong Wang, Jiayan Yang, Thermodynamical Potentials of Classical and Quantum Systems, hal-01632278
The aim of the above paper is to systematically introduce thermodynamic potentials for thermodynamic systems and Hamiltonian energies for quantum systems of condensates.
I. In a recent paper [Ma-Wang, Dynamic Law of Physical Motion and Potential-Descending Principle, J. Math. Study, 50:3 (2017), 215-241; see also hal-01558752 and here], we postulated the potential-descending principle (PDP). We have shown the following conclusions:
- PDP leads to the first and second laws of thermodynamics,
- PDP provides the first principle for describing irreversibility, and
- leads to all three distributions: the Maxwell-Boltzmann distribution, the Fermi-Dirac distribution and the Bose-Einstein distribution in statistical physics.
In a nutshell, PDP is the first principle of statistical physics.
II. For a thermodynamic system with thermodynamic potential , order parameters and control parameters , PDP gives rise to the following dynamic equation:
which offers a complete description of associated phase transitions and transformation of the system from non-equilibrium states to equilibrium states.
Consequently an important issue in statistical physics boils down to find a better and more accurate account of the thermodynamic potentials, which justifies the objectives of this paper.
III. The paper studies statistical systems in three categories:
The typical conventional thermodynamic systems include the physical-vapor transport (PVT) systems, the -component systems, and the magnetic and dielectric systems.
There are two cases of condensates. The first is the case where the system is near the critical temperature , the condensation is in its early stage and the condensed particle density is small. At this stage, the system is treated essentially as a thermodynamic system, and it is crucial then to find its thermodynamic potential. Such thermodynamic condensate systems belong to Category 2 above, and include thermodynamic systems of superconductors, superfluids, and the Bose-Einstein condensates.
The second is the case where away from the critical temperature, the system enters a deeper condensate state. In this case, the system is a quantum system and obeys the principle of Hamiltonian dynamics. We need to derive the related Hamiltonian energy. Such systems are quantum systems of condensates, and as classified as Category 3 above, which include superconducting systems, superfluid systems, and the Bose-Einstein condensates.
IV. Our study in this paper is based on
Of course, as mentioned earlier, the study presented in this paper certainly relies on the rich previous work done by pioneers in the related fields. It is worth mentioning that the potentials and Hamiltonians we shall introduce are based on first principles, and no mean-field theoretic expansions are used.
Ruikuan Liu, Tian Ma, Shouhong Wang & Jiayan Yang
]]>Talk at 2017 Midwest Relativity Meeting. October 13, 2017, Ann Arbor, University of Michigan
The above paper addresses radiation and field particles of the four fundamental interactions, demonstrating that each individual interaction possesses two basic attributes:
In this blog, we examine the law of gravity and the gravitational field particle and radiation; see also the talk given in 2017 Midwest Relativity Conference.
The Einstein theory of general relativity is the most profound scientific theory in the recorded human history. The Einstein theory is built on two first principles: the principle of equivalence (PE) and the principle of general relativity (PGR). In essence, PE amounts to saying that space time is a four-dimensional Riemannian manifold with the metric being the gravitational potential.
The PGR is a symmetry principle, and says that the law of gravity is the same (covariant) under all coordinate systems. In other words, the Lagrangian action of gravity, called the Einstein-Hilbert action, is invariant under all coordinate transformations.
The Einstein–Hilbert functional (1) is uniquely dictated by this profound and simple looking symmetry principle, together with simplicity of laws of Nature, and is given as follows:
Indeed, in Riemannian geometry, the invariant quantities satisfying the principle of general relativity and containing the second order derivative terms of is just the scalar curvature , which is unique.
The presence of the dark matter and dark energy phenomena requires the inevitable need for modifying the Einstein general theory of relativity. Such modification needs to preserve the following basic physical requirements:
We have shown that, under these basic requirements, the unique route for altering the Einstein general theory of relativity is through the principle of interaction dynamics (PID), which takes variation of the Einstein-Hilbert action subject of energy-momentum conservation constraint. This leads to the the following new gravitational field equations; see [Tian Ma & Shouhong Wang, Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34:2 (2014), pp. 335-366; see also arXiv:1206.5078]:
Also, we have shown that PID is the direct consequence of the presence of dark energy and dark matter, is the requirement of the presence of the Higgs field for the weak interaction, and is the consequence of the quark confinement phenomena for the strong interaction; see [Tian Ma & Shouhong Wang, Mathematical Principles of Theoretical Physics, Science Press, 524pp., 2015].
The gravitational field particle is described by the dual field in (2), and the governing radiation equations are
where stands for the energy-momentum tensor of the visible matter, and is a massless, spin-1 and electric neutral boson.
In fact, the gravitational field particle represents the dark matter that we have been searching for, and the energy that carries is the dark energy. Equation (3) is the field equations for dark matter and dark energy. The gravitational effect of the field particle is manifested through the mutual coupling and interaction with the gravitational potential , through the field equations (2), leading to both attractive and repulsive behavior of gravity, which is exactly the dark energy and dark matter phenomena.
Gravitational force formula By the gravitational field equations (2), we derive the following approximate gravitational force formula:
Tian Ma & Shouhong Wang, October 13, 2017
]]>In this post, we describe the new interpretation of quantum mechanical wave functions introduced in the above paper.
In classical quantum mechanics, a micro-particle is described by a complex-valued wave function , satisfying such a wave equation as the Schrödinger equation with external interaction potential :
where is the region that the particle occupies, and is the mass of the particle. The Schrödinger equation conserves the energy, and the wave function can be expressed as
where is the energy, and is the time-independent wave function, satisfying
The classical Born statistical interpretation of quantum mechanics amounts to saying that without constraints, the motion of a micro-particle is random and there is no definite trajectory of the motion. Also stands for the probability density of the particle appearing at the particular point . The Born interpretation of the wave function is treated as a fundamental postulate of quantum mechanics. This leads to the classical Einstein-Bohr debates, and is the origin of absurdities associated with the interpretation of quantum mechanics.
The key observation for the new interpretation is that
can be regarded as the velocity field of the particles, and the wave function is the field function for the motion of all particles with the same mass in the same class determined by the external potential . More precisely, we have the following new interpretation of quantum mechanical wave functions:
New Interpretation of Wave Functions
- Under the external potential field , the wave function is the field function for the motion of all particles with the same mass in the same class determined by the external potential . In other words, it is not the wave function of a particular particle in the classical sense;
- When a particle is observed at a particular point , then the motion of the particle is fully determined by the solution of the following motion equation with initial position at :
where is the phase of the wave function in (2);
- With being the field function,
- The energy in (3) represents the average energy level of the particles and can be written as
where in integrand on the right-hand side, the fist term represents the non-uniform distribution potential of particles, the second term is the average kinetic energy, and the third term is the potential energy of the external field. Here is characteristic of quantum mechanics and there is no such a term in classical mechanics.
In summary, our new interpretation says that is the common wave function for all particles in the same class determined by the external potential , represents the distribution density of the particles, and is the velocity field of the particles. The trajectories of the motion of the particles are then dictated by this velocity field. The observed particles are the particles in the same class described by the same wave function, rather than a specific particle in the sense of classical quantum mechanics.
This is an entirely different interpretation from the classical Bohr interpretation. Also this new interpretation of wave functions does not alter the basic theories of quantum mechanics, and instead offers new understanding of quantum mechanics, and plays a fundamental role for the quantum theory of condensed matter physics and quantum physics.
It is worth mentioning that the Landau school of physics was the first who noticed that the relation between the superfluid velocity and the wave function of the condensate is given by (4); see (26.12) on page 106 of [E. Lifshitz and L. Pitaevskii, Statistical physics Part 2, Landau and Lifshitz Course of Theoretical Physics vol. 9, 1980]. However they fail to make an important connection between (4) and the basic interpretation of quantum mechanics.
Tian Ma & Shouhong Wang
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