Transition Theory and Phase Transition Dynamics

[1] Tian Ma and Shouhong Wang, Phase Transition Dynamics, Springer, pp 555, 2013

This book synthesizes the mathematical and physical theories for a phase transition phenomena, established by the authors. This blog post intends to give a brief introduction to the theories developed in this book, and to point out some main differences between the transition theory and the classical bifurcation theory.

1. Mathematical theory

The mathematical theory is also called dynamic transition theory, and we summarize hereafter the main ingredients of the theory.

First, we have proved a general transition classification theorem, which states that all dynamic transitions are classified into three categories: the Type-I, Type-II and Type-III.

Second, one important tool of the dynamic transition theory is the central manifold reduction, and we have derived for the first time approximate central manifold reduction formulas, which are crucial for many applications of the theory to physical problems. Also, these formulas has been generalized to random dynamical systems by Chekroun et al.

Third, we have systematically developed theorems and criteria for different transition types. For example, for the first time, we introduced the concept of attractor bifurcation, and proved a general attractor bifurcation theorem, which can be used to handle most Type-I transitions. A sequence of theorems are established also for the Type-II, and Type-III transitions. These theorems are easy to use in applications and are crucial in deriving the needed physical theory for the related physical problems.

2. Physical theory

Our physical theory of transition dynamics involves a wide range of scientific fields, including statistical physics, fluid dynamics, atmospheric and ocean physics, and biology and chemistry. Hereafter we present a few physical theory we have derived, and we refer interesting readers to the book [1] for more details and for other physical applications.

Principle of phase transition dynamics: We discovered a general principle of phase transitions for dissipative physical systems, which we call principle of phase transition dynamics. Namely, phase transitions of all dissipative physical systems are classified into three categories: continuous, catastrophic, and random.

General dynamic model for equilibrium phase transitions: We introduced a unified dynamical model for equilibrium phase transitions, based on the Le Chatelier principle and the Ginzburg-Landau mean field theory.

Discovery of third-order phase transition: It is well-known that the gas-liquid coexistence curve terminates at a critical point, also called the Andrews critical point, and gas-liquid transition is of first order before the critical point and of the second-order at the critical point. Going beyond the critical point, physical phenomena indicates that a high-order phase should occur. However, it is a longstanding open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. For the first time, 1) we derived the gas-liquid co-existence curve beyond the Andrews critical point, and 2) we show that the transition is first order before the critical point, second-order at the critical point, and third order beyond the Andrews critical point. This gives rise to the mechanism of the Andrews critical point, and the reason why it is hard to observe the gas-liquid phase transition beyond the Andrews critical point.

Prediction of a new superfluid phase in liquid helium-3: We have derived new dynamical models for liquid helium-3, helium-4 and their mixture, leading to various physical predictions, such as the existence of a new phase {C} for helium-3. Although these predictions need yet to be verified experimentally, they certainly offer new insights to both theoretical and experimental studies for a better understanding of the underlying physical problems.

New mechanism of El Nino Southern Oscillation (ENSO): We discovered a new mechanism of the ENSO, as a self-organizing and self-excitation system, with two highly coupled oscillatory processes: 1) the oscillation between the two metastable warm (El Nino phase) and cold events (La Nina phase), and 2) the spatiotemporal oscillation of the sea surface temperature (SST) field. The interplay between these two processes gives rises the climate variability associated with the ENSO, leads to both the random and deterministic features of the ENSO, and defines a new natural feedback mechanism, which drives the sporadic oscillation of the ENSO.

3. Differences between the dynamic transition theory and classical bifurcation theory

It is important to emphasize the main differences between the dynamical transition theory and the classical bifurcation theory.

First the key difference is that the transition states derived in our dynamic transition theory are physical, and the bifurcation states derived from the classical bifurcation theory may not be physical.

In fact, the classical bifurcation theory first seeks bifurcation solutions and then decides the stability of the bifurcated solutions. The main drawback for this approach is that there is no way to know if the bifurcated solutions represent all transition physical states. In addition, it is always technically difficult to derive the stability of the bifurcated solution.

Instead, our dynamic transition theory finds all physical phase transition states.

Second, our theory indicates that transition always happen at the critical point, but bifurcation may not occur.

Third, it is clear that the general physical principle for phase transitions can only be discovered by using the dynamic transition theory.

4. Summary

The dynamic transition theory can be viewed as a true mathematical representation of a physical theory. The general principle of phase transition dynamics clearly offers guidance to the understanding of dissipative physical systems.

Tian Ma & Shouhong Wang

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Problems in Classical Electroweak Theory

The classical electroweak theory forms the core of the standard model of particle physics. The great success of both the electroweak theory and the standard model include e.g. the prediction of the intermediate vector bosons {W^\pm, Z} and the Higgs boson. In spite of its success, there are a number of issues and difficulties for the classical electroweak theory, which we will address in this blog post.

In the next blog post, we shall introduce the PID electroweak theory, resolving all these difficulties. In particular, the PID approach provides a first principle approach for introducing the Higgs field.

1. Classical electroweak theory

In essence, the electroweak theory is the generalization of the Fermi theory, and provides a useful computational tool for transition probability and amplitudes. It is a {U(1) \times SU(2)} gauge theory incorporating the Higgs field, and its main ingredients include

  • It involves three {SU(2)} gauge potentials, {W^1_\mu, W^2_\mu, W^3_\mu}, and and one {U(1)} potential {B_\mu};
  • The Higgs scalar doublet {\phi=(\phi^+, \phi^0)} was introduced into the Yang-Mills Lagrangian action in order to derive proper mass generation mechanism for the intermediate bosons.
  • With the gauge potentials, the following combinations are introduced to represent the intermediate vector bosons {W^\pm_\mu}, {Z_\mu} and the electromagnetic potential {A_\mu}, respectively:

    \displaystyle W^\pm_\mu =\frac{1}{\sqrt2} ( W^1_\mu \pm i W^2_\mu), \ \ \ \ \ (1)

    \displaystyle Z_{\mu}=\cos\theta_wW^3_{\mu}+\sin\theta_wB_{\mu}, \ \ \ \ \ (2)

    \displaystyle A_{\mu}=-\sin\theta_wW^3_{\mu}+\cos\theta_wB_{\mu}, \ \ \ \ \ (3)

2. Lack of weak force formulas

This problem is that all weak interaction theories have to face, and it is also that all existing theories cannot solve.

In fact, the classical electroweak theory, there are four gauge field potentials:

\displaystyle W^1_{\mu},\ W^2_{\mu},\ W^3_{\mu},\ B_{\mu},

and we don’t know which of these potentials plays the role of weak interaction potential.

3. Violation of Principle of Representation Invariance (PRI)

We have discovered a basic principle, called the principle of representation invariance (PRI), for the {SU(N)} gauge theory, which describes an {interacting} {N} particle system; see the previous post for details about PRI.

Elements in {SU(N)} are expressed as { \Omega =e^{i\theta^a\tau_a}}, where {\{\tau_1, \cdots ,\tau_{N^2-1}\}} is a basis of the set of traceless Hermitian matrices, and plays the role of a coordinate system in this representation. Consequently, an {SU(N)} gauge theory should be invariant under the following global transformation of the representation bases:

\displaystyle \tilde{\tau}_a=x^b_a\tau_b, \ \ \ \ \ (4)

where { X=(x^b_a) } is a a nondegenerate complex matrix. We call such invariance of the {SU(N)} gauge theory the principle of representation invariance (PRI).

PRI is a logic requirement for any gauge theory, and has profound physical consequences. In particular, by PRI, any linear combination of gauge potentials from two different gauge groups are prohibited.

In the classical electroweak theory, a key ingredient is the linear combinations of {W^3_{\mu}} and {B_{\mu}}. By PRI,

\displaystyle W^3_{\mu}\ \text{ is\ the\ third\ component\ of a}\ SU(2)\ \text{ tensor } \{W^a_\mu\},

\displaystyle B_{\mu}\ \text{ is\ the}\ U(1)\ \text{ gauge\ field}.

Hence, for the combinations of two different types of tensors:

\displaystyle Z_{\mu}=\cos\theta_wW^3_{\mu}+\sin\theta_wB_{\mu},

\displaystyle A_{\mu}=-\sin\theta_wW^3_{\mu}+\cos\theta_wB_{\mu},

used in the classical electroweak theory and the standard model of particle physics, violate PRI.

4. Decoupling obstacle

The classical electroweak theory has a difficulty for decoupling the electromagnetic and the weak interactions. In reality, electromagnetism and weak interaction often are independent to each other. Hence, as a unified theory for both interactions, one should be able to decouple the model to study individual interactions. However, the classical electroweak theory manifests a radical decoupling obstacle.

For example, if there is no weak interaction involved, then

\displaystyle W^{\pm}_{\mu}=0,\ \ \ \ Z_{\mu}=0, \ \ \ \ \ (5)

hold true. In this case, the theory should return to the {U(1)} gauge invariant Maxwell equations. But we see that

\displaystyle A_{\mu}=\cos\theta_wB_{\mu}-\sin\theta_wW^3_{\mu},

where {B_{\mu}} is a {U(1)} gauge field, and {W^3_{\mu}} is a component of {SU(2)} gauge field. Therefore, {A_{\mu}} is not independent of {SU(2)} gauge transformation. In particular, the condition (5) means

\displaystyle W^1_{\mu}=0,\ \ \ \ W^2_{\mu}=0,\ \ \ \ W^3_{\mu}=-\tan \theta_wB_{\mu}. \ \ \ \ \ (6)

Now we take the transformation (4) for the generators of {SU(2)}, {W^a_{\mu}} becomes

\displaystyle \left(\tilde{W}^1_{\mu}, \tilde{W}^2_{\mu}, \tilde{W}^3_{\mu} \right)=\left( y^1_3W^3_{\mu}, y^2_3W^3_{\mu}, y^3_3W^3_{\mu} \right),\ \ \ \ (y^b_a)^T=(x^b_a)^{-1}.

It implies that under a transformation (4), a nonzero weak interaction can be generated from a zero weak interaction field of (5)-(6):

\displaystyle \tilde{W}^{\pm}_{\mu}\neq 0,\ \ \ \ \tilde{Z}_{\mu}\neq 0\ \ \ \ \text{ as}\ y^a_3\neq 0\ (1\leq a\leq 3),

and the nonzero electromagnetic field {A_{\mu}\neq 0} will become zero:

\displaystyle \tilde{A}_{\mu}=0\ \ \ \ \text{ as}\ \ \ \ y^3_3=\cot \theta_w.

Obviously, it is not reality.

5. Artificial Higgs mechanism

In the classical electroweak action, the Higgs sector {\mathcal{L}_H} is not based on a first principle, and is artificially added into the action.

6. Presence of a massless and charged boson {\phi^+}

In the WS theory, the Higgs scalar doublet {\phi=(\phi^+, \phi^0)} contains a massless boson {\phi^+} with positive electric charge. Obviously there are no such particles in reality. In particular, the particle {\phi^+} is formally suppressed in the classical electroweak theory by transforming it to zero. However, from a field theoretical point of view, this particle field still represents a particle. This is one of major flaws for the electroweak theory and for the standard model.

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Angular Momentum Rule and Scalar Photons

[1] Tian Ma and Shouhong Wang, Quantum Rule of Angular Momentum, AIMS Mathematics, 1:2(2016), 137-143.

[2] Tian Ma and Shouhong Wang, Mathematical Principles of Theoretical Physics, Science Press, 2015

1. Angular Momentum Rule of Quantum Systems

Quantum physics is the study of the behavior of matter and energy at molecular, atomic, nuclear, and sub-atomic levels. Two most distinct features of quantum mechanics, drastically different from classical mechanics, are the Heisenberg uncertainty relation and the Pauli exclusion principle.

We present a new feature, the angular momentum rule, discovered recently by the authors [1, 2], This new angular momentum rule can be considered as an addition to the Heisenberg uncertainty relation and the Pauli exclusion principle in quantum mechanics.

Quantum Rule of Angular Momentum [1, 2]. Only fermions with spin {J=\frac{1}{2}} and bosons with {J=0} can rotate around a center with zero moment of force, and particles with {J\neq 0,\frac{1}{2}} will move on a straight line unless there is a nonzero moment of force present.

This quantum mechanical rule is important for the structure of atomic and sub-atomic particles. In fact, the rule gives the very reason why the basic constituents of atomic and sub-atomic particles are all spin-{\frac{1}{2}} fermions.

The angular momentum rule provides the theoretical evidence and support of scalar photons, a recent prediction from our unified field theory and the weakton model of elementary particles.

2. Prediction of Scalar Photons

First, we recall that the photon, denoted by {\gamma}, is the mediator of the electromagnetic force. The photon is a massless spin-1 particle, described by a vector field {A_\mu} defined on the space-time manifold, which obeys the Maxwell equations:

\displaystyle \partial^\mu F_{\mu\nu}=0, \qquad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.

Second, the scalar photon, denoted by {\gamma_0}, was first introduced as a natural byproduct of our unified field theory based on the principle interaction dynamics (PID), which we have discussed in the previous posts. The scalar photon {\gamma_0} is a massless, spin-0 particle, described by a scalar field {\phi_0}, satisfying the following Klein-Gordon equation:

\displaystyle \Box \phi_0=0.

Third, the puzzling decay and reaction behavior of subatomic particles suggest that there must be interior structure of charged leptons, quarks and mediators. Careful examinations of subatomic decays/reactions lead us to propose six elementary particles, which we call weaktons, and their anti-particles:

\displaystyle w^*, \quad w_1, \quad w_2, \quad \nu_e, \quad \nu_{\mu}, \quad \nu_{\tau},

\displaystyle \bar{w}^*, \quad \bar{w}_1, \quad \bar{w}_2, \quad \bar{\nu}_e, \quad \bar{\nu}_{\mu}, \quad \bar{\nu}_{\tau},

where {\nu_e,\nu_{\mu},\nu_{\tau}} are the three generation neutrinos, and {w^*,w_1,w_2} are three new particles, which we call {w}-weaktons.

Remarkably, the weakton model offers a perfect explanation for all sub-atomic decays. In particular, all decays are achieved by 1) exchanging weaktons and consequently exchanging newly formed quarks, producing new composite particles, and 2) separating the new composite particles by weak and/or strong forces.

In the weakton model, the constituents of the photon {\gamma} is given as follows:

\displaystyle \gamma =\cos\theta_ww_1\bar{w}_1-\sin\theta_ww_2\bar{w}_2\ (\uparrow \uparrow,\downarrow \downarrow),

and different spin arrangements of the weaktons give rise naturally to the scalar photon {\gamma_0} with the following constituents:

\displaystyle \gamma_0=\cos\theta_ww_1\bar{w}_1-\sin\theta_ww_2\bar{w}_2\ (\downarrow \uparrow,\uparrow \downarrow).

3. Bremsstrahlung as an Experimental Evidence for Scalar Photons

It is known that an electron emits photons as its velocity changes, which is called the bremsstrahlung. The reasons why bremsstrahlung can occur is unknown in classical theories.

In fact, our viewpoint is that the bremsstrahlung suggest that a mediator cloud is present near a naked electron, and the mediator cloud contains photons. The angular momentum rule demonstrates that the photons circling the naked electron must be scalar photons, as free vector photons can only take straight line motion. We refer the interested readers to Section 5.4 of [2] for more detailed discussions.

In summary, bremsstrahlung, together with the angular momentum rule, offers an experimental evidence for scalar photons. Of course, further direct experimental verification and discovery of scalar photons are certainly important feasible.

Tian Ma & Shouhong Wang

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Contradictions between the Black Hole and the Big-Bang Theories, and the Structure of the Universe

Based on the authors’ recent work [1, 2, 3], the main objectives of this post are as follows:

  • First, assuming the Einstein theory of general relativity, we demonstrate that the theory of black holes and the theory of Big-Bang are contradicting to each other.
  • Second, we introduce a theorem, proved recently by the authors, on the structure and geometry of our Universe, assuming the Einstein theory of general relativity and the principle of cosmological principle. This theorem gives rise to a new theory of the Universe, which is consistent with redshift and CMB observations.

1. Conditions for the existence of black holes

Consider a massive body with mass {M} and radius {R}. The Schwarzschild radius is defined by {R_s=2GM/c^2}. Based on the Einstein field equations, in the exterior of the body, the Schwarzschild solution is given by

\displaystyle ds^2= -\left(1-\frac{R_s}{r}\right)c^2dt^2+\left(1-\frac{R_s}{r}\right)^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2 \qquad \text{ for } r > R, \ \ \ \ \ (1)

and in the interior the Tolman-Oppenheimer-Volkoff (TOV) metric is

\displaystyle ds^2= -e^u c^2dt^2+\left(1-\frac{r^2R_s}{R^3}\right)^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2 \qquad \text{ for } r < R. \ \ \ \ \ (2)

The theory of black holes then shows that the massive body is a black hole if and only if {R=R_s}. Here we note that if {R> R_s}, the massive body is not a black hole. Also, it is impossible to have {R< R_s}. Otherwise, the Schwarzschild solution would be valid in the vacuum region {R< r< R_s}, where the factor {1-\frac{R_s}{r}} changes sign and the metric becomes nonphysical. In other words, a black hole must be filled with matter in the entire interior.

The above necessary and sufficient condition implies that a massive body with radius {R} and density {\rho} is a black hole whenever

\displaystyle R^2=\frac{3c^2}{8\pi G \rho}. \ \ \ \ \ (3)

2. Contradictions between Big-Bang and Black-Hole Theories

First, if the Universe began with the big-bang, the energy density of the early universe {\rho} would be so big that the universe would be filled with many black holes of infinitesimal radius {R} defined by (3).

Second, if the Universe were born to a Big-Bang and expanded continuously, then in the expansion process it would generate successively a large number of black holes, whose radii vary as follows:

\displaystyle r=\sqrt{\frac{R}{R_T}}R,\ \ \ \ R_0<R\leq R_T=\frac{2M_TG}{c^2}, \ \ \ \ \ (4)

where {M_T} is the total mass in the universe, {R_0} is the initial radius, {R} is the expanding radius, {r} is the radius of sub-black holes, and {R_T} is the radius of the Universe viewed as a black hole.

To see this, we consider a homogeneous universe with radius {R<R_T}. Then the mass density {\rho} is given by

\displaystyle \rho =\frac{3M_T}{4\pi R^3},

which implies that the mass of a ball {B_r} of radius {r} with this mass density is

\displaystyle M_r=\frac{4\pi}{3}r^3\rho =\frac{r^3}{R^3}M_T=\frac{r^3}{R^3} \frac{c^2 R_T}{2G}, \ \ \ \ \ (5)

Recall that the condition for the ball {B_r} to form a black hole is

\displaystyle M_r =\frac{c^2 r}{2G}. \ \ \ \ \ (6)

The combination of (5) and (6) implies immediately (4).

In summary, (4) demonstrates clearly that if the Universe were born to a Big-Bang and continuously expands, then it would contain many black holes with smaller ones being embedded in the larger ones. This is not what we observed in our Universe.

3. Theorem on Structure of the Universe

The aforementioned contradictions force us to examine the black hole and Big-Bang theories from both the fundamental level and the observational evidences.

It is clear that the large scale structure of our Universe is essentially dictated by the law of gravity, which is based on Einstein’s two principles: the principle of general relativity and the principle of equivalence. Also, strong cosmological observational evidence suggests that the large scale Universe obey the cosmological principle that the Universe is homogeneous and isotropic.

The black theory is a direct consequence of the Einstein theory of general relativity, and is a more trustable theory. Consequently, a careful examination of modern cosmological theories on the structure of the Universe is inevitable. In fact, we have recently shown a theorem on the geometry and structure of our Universe under the assumption of general relativity and the cosmological principle:

Theorem on Structure of our Universe [2, 3]. Assume the Einstein theory of general relativity, and the principle of cosmological principle, then the following assertions hold true:

  1.  our Universe is not originated from a Big-Bang, and is static;
  2. the topological structure of our Universe is the 3D sphere {S^3} such that to each observer, the corresponding equator with the observer at the center of the hemisphere can be viewed as the black hole horizon;
  3. the total mass {M_{\text{total}}} in the Universe includes both the cosmic observable mass {M} and the non-observable mass, regarded as dark matter, due to the space curvature energy; and
  4. a negative pressure is present in our Universe to balance the gravitational attracting force, and is due to the gravitational repelling force, also called dark energy.It is clear that this theorem changes drastically our view on the geometry and the origin of the Universe.

Inevitably, a number of important questions need to be addressed for this scenario of our Universe in the following sections.

 4. Redshift problem

The natural and important question that one has to answer is the consistency with astronomical observations, including the cosmic edge, the flatness, the horizon, the redshift, and the cosmic microwave background (CMB) problems. These problems can now be easily understood based on the structure of the Universe and the blackhole theorem we derived. Hereafter we focus only on the redshift and the CMB problems.

The most fundamental problem is the redshift problem. Observations clearly show that light coming from a remote galaxy is redshifted, and the farther away the galaxy is, the larger the redshift. In modern astronomy and cosmology, it is customary to characterize the redshift by a dimensionless quantity {z} in the formula

\displaystyle 1+z=\frac{\lambda_{\rm observ}}{\lambda_{\rm emit}}, \ \ \ \ \ (7)

where {\lambda_{\rm observ}} and {\lambda_{\rm emit}} represent the observed and emitting wavelenths.

There are three sources of redshifts: the Doppler effect, the cosmological redshift, and the gravitational redshift. If the Universe is not considered as a black hole, then the gravitational redshift and the cosmological redshift are both too small to be significant. Hence, modern astronomers have to think that the large port of the redshift is due to the Doppler effect.

However, due to black hole properties of our Universe, the black hole and cosmological redshifts cannot be ignored. Due to the horizon of the sphere, for an arbitrary point in the spherical Universe, its opposite hemisphere relative to the point is regarded as a black hole. Hence, {g_{00}} can be approximatively taken as the Schwarzschild solution for distant objects as follows

\displaystyle -g_{00}(r)=\alpha(r)\left(1-\frac{R_s}{\tilde r}\right),\qquad \alpha(0)=2, \qquad \alpha(R_s)=1, \qquad \alpha'(r) <0,

where {\tilde{r}=2R_s-r} for {0\leq r<R_s} is the distance from the light source to the opposite radial point, and {r} is the distance from the light source to the point. Then by the gravitational redshift formula:

\displaystyle \lambda_{\rm observ} = \frac{\lambda_{\rm emit}}{\sqrt{-g_{00}(r)}},

we derive the following redshift formula, which is consistent with the observed redshifts:

\displaystyle 1+z=\frac{1}{\sqrt{\alpha(r)(1-\frac{R_s}{\tilde r})}}= \frac{\sqrt{2R_s-r}}{\sqrt{\alpha(r)(R_s-r)}} \qquad \text{for } 0 < r<R_s. \ \ \ \ \ (8)

5. CMB problem

In 1965, two physicists A. Penzias and R. Wilson discovered the low-temperature cosmic microwave background (CMB) radiation, which fills the Universe, and it has been regarded as the smoking gun for the Big-Bang theory. However, based on the unique scenario of our Universe we derived, it is the most natural thing that there exists a CMB, because the Universe has always been there as a black-body, and CMB is a result of blackbody equilibrium radiation.

6. PID-cosmological model

We have demonstrated that the right cosmological model should be derived from the new gravitational field equations [1], taking into consideration the presence of dark matter and dark energy:

\displaystyle R_{\mu\nu}-\frac{1}{2} g_{\mu\nu} R = -\frac{8\pi G}{c^4} T_{\mu\nu} -\nabla_\mu\nabla_\nu \phi. \ \ \ \ \ (9)

In this case, based on the cosmological principle, the the metric of a homogeneous spherical universe is of the form

\displaystyle ds^2=-c^2dt^2+R^2\left[\frac{dr^2}{1-r^2}+r^2(d\theta^2+\sin^2\theta d\varphi^2)\right],

where {R=R(t)} is the cosmic radius. The new gravitational field equations provide then the following PID-cosmological model:

\displaystyle R^{\prime\prime}=-\frac{4\pi G}{3}\left(\rho +\frac{3p}{c^2}+\frac{\varphi}{8\pi G}\right)R \qquad \text{ with } \varphi=\phi'', \ \ \ \ \ (10)

\displaystyle (R^{\prime})^2=\frac{1}{3}(8\pi G\rho +\varphi )R^2-c^2, \ \ \ \ \ (11)

\displaystyle \varphi^{\prime}+\frac{3R^{\prime}}{R}\varphi =-\frac{24\pi G}{c^2}\frac{R^{\prime}}{R}p, \ \ \ \ \ (12)

\displaystyle (R')^2 \phi'=0, \ \ \ \ \ (13)

supplemented with the equation of state:

\displaystyle p=f(\rho ,\varphi ). \ \ \ \ \ (14)

Note that only two equations in (10)-(12) are independent.

Also, the model describing the static Universe is the equation of state (14) together with the stationary equations of (10)-(12), which are equivalent to the form

\displaystyle \varphi =-8\pi G\left(\rho +\frac{3p}{c^2}\right), \ \ \ \ \ (15)

\displaystyle p=-\frac{c^4}{8\pi GR^2}. \ \ \ \ \ (16)

Now we consider a perturbation of the steady state solution, which gives

\displaystyle \rho +\frac{3p}{c^2}+\frac{\varphi}{8\pi G} \rightarrow \varepsilon,

which implies {R''\not=0}, and consequently, {R'\not=0}. By (13), {\phi'=0}, which implies that {\varphi=0}. Then by (12), {p=0}, which contradicts with (16). In other words, this simple argument tells us that the dual field {\varphi} plays a role of balancing, preventing any perturbation of the combined quantity:

\displaystyle \rho +\frac{3p}{c^2}+\frac{\varphi}{8\pi G}.

In a nutshell, we have shown that the steady state solution of (10)-(14) is physically stable.

References

[1] Tian Ma and Shouhong Wang, Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34:2(2014), 335-366; see also arXiv:1206.5078v2.

[2] Tian Ma and Shouhong Wang, Astrophysical dynamics and cosmology, Journal of Mathematical Study, 47:4(2014), 305–378.

[3] Tian Ma and Shouhong Wang,  Mathematical Principles of Theoretical Physics, Science Press, Beijing, 524pp., August, 2015.

Tian Ma and Shouhong Wang

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On Indeterminacy Problem in Quantum Mechanics

Tian Ma & Shouhong Wang, On indeterminacy problem in quantum mechanics, IUISC Preprint #1601

The main objectives of this Note are

  1.  to demonstrate that the indeterminacy problem and its associated absurdities in interpretations of quantum mechanics are due to the wrong fundamental premise that there is no interference of other particles;
  2. to reveal the fact that the Universe is filled with mediators, giving rise to the correct Mediator Sea Premise of quantum mechanics; and
  3. to show that under the Mediator Sea Premise,
    • causality holds true for the quantum-mechanical description of physical reality, removing the absurdities and confusions;
    • the interference of the mediators to a moving particle is reminiscent to Brownian motion; and
    • quantum mechanics is a correct and complete theory.

I. Classical Statistical Interpretation and Indeterminacy Problem

Two dominant views of the interpretation of quantum mechanics are the realistic view, advocated by Albert Einstein, and the orthodox view, also called the Copenhagen interpretation, which was mainly advocated by Niels Bohr and Werner Karl Heisenberg. The main characteristic for the Einstein realistic view is that causality must hold true in the quantum-mechanical description of physical reality, and quantum mechanics is an incomplete theory–the indeterminacy is caused by hidden variables. The key point for the Copenhagen interpretation is non-causality of quantum behavior of particles, leading to various absurdities; see among many others [3, 2, 1, 4].

II. The Fundamental Premise of Indeterminacy Problem

All scientific theories and conclusions are built upon a fundamental premise. If the fundamental premise is true, then we would expect the conclusions are true as well. To understand the confusion caused by the indeterminacy as classically formulated, one needs to examine its fundamental premise.

In fact, the fundamental premise of the indeterminacy in quantum mechanics is that one assumes there is no interference of other particles. Our viewpoint is that such fundamental premise is in fact incorrect. Consequently, under an incorrect premise, confusions and misunderstandings arise, and more importantly

the indeterminacy problem is a wrong question to be asked.

III. Mediator Sea in the Background Space

One natural outcome of our field theory of fundamental interactions and the weakton theory of elementary particles suggests that our Universe is filled with mediators; see [5] and the reference by the authors therein. This clearly shows that the classical fundamental premise of the indeterminacy in the quantum-mechanical description of physical reality is a wrong assumption.

In other words, quantum mechanics should be understood under the following fundamental premise of the quantum-mechanical description of physical reality:

Mediator Sea Premise:

  1.  The entire space is filled with a sea of mediators, including photons, gluons and the $\nu$-particles, as evidenced by the cosmic microwave background;
  2. All mediators carry weak charges, participate in the weak interaction, and consequently will interact with a moving particle in proper ranges.

The interference of the mediator sea towards a moving particle resembles similar features as the Brownian motion.

IV. Causality

In essence, the heart of the debate between Einstein and Bohr is the causality of the quantum-mechanical description of physical reality. We believe what puzzled Einstein was the non-causality conclusion of the Copenhagen interpretation, rather than the randomness in the quantum mechanics.

With the new Mediator-Sea-Premise, it is clear that the principle of causality holds true, as Einstein believed. The randomness is caused by interference of the mediators as a particle moves in the mediator sea, leading to the indeterminacy of the precise position and momentum of the particle. Instead, in the Copenhagen interpretation, randomness is innate with no causality, resulting various spurious paradoxes in the interpretation of quantum mechanics.

Also, as in the Brownian motion, precise physical law expressed by the wave equation for the wave function offers complete information about the system. In other words, quantum mechanics is a correct and complete theory for describing the physical reality under the Mediator Sea Premise.

References

[1] John Bell, On the Einstein-Podolsky-Rosen paradox, Physics, 1, 195-200, 1964.

[2]  Niels Bohr, Can quantum-mechanical description of physical reality be considered complete?, Physical Review 48:8, 696-702, 1935.

[3] A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Physical Review 47:10, 777-780, 1935.

[4] D. Griffiths, Introduction to quantum mechanics, Prentice Hall, 1995.

[5] Tian~Ma and Shouhong~Wang, Mathematical Principles of Theoretical Physics, Science Press, 524 pp, 2015.

Tian Ma and Shouhong Wang

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On Unification of Four Fundamental Interactions

I.

There are four fundamental interactions in Nature–the electromagnetism, the gravity, the strong and the weak interactions. The current route of the unification of the four interactions is under the following assumption:

there is only one interaction of Nature, which, under different energy conditions, degenerates to the four interactions we observe.

Mathematically, this assumption translates to searching for unification through a large symmetry group.

Our first viewpoint is that such an assumption is imaginary and non-falsifiable. We have no reason to believe its validity, and consequently, any unification theory built upon this imaginary assumption may not reflect Nature. In fact, this assumption breaks the principle of representation invariance (PRI), discovered by (Ma-Wang, 2012), which is simply a logic requirement.

II.

Nature tells us that when several interactions are present in a given physical system, each interaction obeys its own symmetry. These interactions are affecting each other in the physical system, and the coupling is inevitably necessary. In other words, Nature suggests us that

the coupling is the essence and a natural route for the unification of the four fundamental interactions.

This route of unification is the reflection of Nature, and is directly built upon observable natural phenomena, instead of on an imaginary assumption in the above traditional route of unification.

This is the route of unification we adopt in the PID unified field theory we have recently developed; see the previous blog  and a recent book.

III.

Now we introduce briefly the main ideas of the PID unified field theory.

  1. Given a physical system, involving all four interactions, the goal of a unified field theory is to derive physical laws of the system:
    • a unified field theory is to derive field equations coupling the interactions and dynamics of the system, and
    • the field equations should be derived based on a few fundamental principles.
  2. Our PID unified field theory is based on the following principles:
  3. The symmetry principles play a decisive role in determining laws of Nature. In other words, symmetry principles dictate the actions of the interactions.With the actions at our disposal, PID gives rise to the field equations of the interacting physical system coupling different interactions and subsystems.The PRI ensures the sources of the interaction, the mass charge for gravity, the electric charge for electromagnetic interaction, the weak charge for the weak interaction and the strong charge for the strong interaction.In deriving the field equations above, the coupling principle of symmetry breaking (PSB) validates the symmetry-breaking of certain subsystems in the coupling.
  4. The unified (coupling) field equations offer solutions to a few challenging problems, which include the dark matter and dark energy phenomena, sources of interactions, asymptotic freedom and the quark confinement, formulas for the weak and strong interaction potentials and forces, and the nucleon interaction potential.

Tian Ma and Shouhong Wang

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Remarks on the four fundamental interactions

An interaction is a force or a potential energy between two different particles. So far, we know that there are four fundamental interactions in Nature– the electromagnetism, the gravity, the strong and the weak interactions. The discovery of the four interactions and the development of scientific theories of these interactions have always been an important endeavor of the mankind.

The most fundamental characteristic of a scientific theory for an interaction is that it can provide the interaction potential and force formulas, in agreement with experiments and observations.  Otherwise, the theory is at least incomplete or incorrect.  This fundamental characteristic suggests the following:

1. the Maxwell theory of electromagnetism, the Einstein theory of general relativity, and our field theory appear to be the only scientific theories of fundamental interactions, and

2. all other theories cannot provide strong and weak interaction potentials and force formulas, and are incomplete to be qualified as a scientific theory for the four fundamental interactions.

Two remarks are now in order.

  1. The Maxwell theory of electromagnetism is a scientific theory, in which the interaction potential is electromagnetic potential {A_\mu}, dictated by the Maxwell equations. Interaction potential and force formulas for an electrically charged particle are the Coulomb formulas, which can be derived from the Maxwell equations.The Einstein theory of gravity, the theory of general relativity (GR), is certainly a scientific theory of the gravitational interaction. In GR, the Riemannian metric represents the basic interaction potentials, and the gravitational force formula including the Newtonian universal law of gravity can be derived from the Einstein gravitational field equations.
  2. As mentioned in the previous posts, we have discovered three fundamental principles: 1) the principle of interaction dynamics (PID),  2) the principle of representation invariance (PRI), and 3) the coupling principle of symmetry-breaking (PSB) for unification.We have then derived a number of experimentally verifiable laws of Nature based only on the Einstein principle of general relativity, the gauge symmetry and the above three principles. In particular, among other implications, our theory offers basic interaction potential and force formulas for the weak and the strong interactions.

Tian Ma & Shouhong Wang

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