The aim of this blog is to clarify that the fundamental theory of four interactions must contain the following three basic constituents:
- the symmetries,
- the Lagrangian actions, also called functionals, and
- the field equations.
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 
Tian Ma & Shouhong Wang
of a thermodynamic system, the fluctuation of 
is the deviation from 
![\displaystyle \frac{\text{d}w}{\text{d}t}=- [\delta F(u_0+w)-\delta F(u_0)]+\widetilde{f}. \ \ \ \ \ (3)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B%5Ctext%7Bd%7Dw%7D%7B%5Ctext%7Bd%7Dt%7D%3D-+%5B%5Cdelta+F%28u_0%2Bw%29-%5Cdelta+F%28u_0%29%5D%2B%5Cwidetilde%7Bf%7D.+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{\text{d}w}{\text{d}t}=- [\delta F(u_0+w)-\delta F(u_0)]+\widetilde{f}. \ \ \ \ \ (3)")
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 
as given by 
are given by 
be a one-parameter family of 2D divergence-free vector fields with the first-order Taylor expansion with respect to 
at 
: 
be a degenerate singular point of 
, and the Jacobian 
. Then, there are two orthogonal unit vectors 
and 
such that 
is an isolated singular point if 
with 
.
and initial temperature 
. One main result we obtain is the following interior separation equations: 
and the external force 
.
or 
in 
of the fluid satisfies the Navier-Stokes equations (NSEs) or their variations, and 
, 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 ![\displaystyle \frac{\partial u_\tau(x,t)}{\partial n} =\frac{\partial \varphi_\tau}{\partial n} +\int_{0}^{t}[\nu\nabla\times\Delta u+k\nu\Delta u\cdot \tau+\nabla\times f+kf_\tau]\mbox{d}t. \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B%5Cpartial+u_%5Ctau%28x%2Ct%29%7D%7B%5Cpartial+n%7D+%3D%5Cfrac%7B%5Cpartial+%5Cvarphi_%5Ctau%7D%7B%5Cpartial+n%7D+%2B%5Cint_%7B0%7D%5E%7Bt%7D%5B%5Cnu%5Cnabla%5Ctimes%5CDelta+u%2Bk%5Cnu%5CDelta+u%5Ccdot+%5Ctau%2B%5Cnabla%5Ctimes+f%2Bkf_%5Ctau%5D%5Cmbox%7Bd%7Dt.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{\partial u_\tau(x,t)}{\partial n} =\frac{\partial \varphi_\tau}{\partial n} +\int_{0}^{t}[\nu\nabla\times\Delta u+k\nu\Delta u\cdot \tau+\nabla\times f+kf_\tau]\mbox{d}t. \ \ \ \ \ (1)")
![\displaystyle u_\tau(x,t)=\varphi_\tau(x)+\int_{0}^{t} \bigg[\nu\bigg(\frac{\partial^2 u_\tau}{\partial \tau^2}+\frac{\partial^2 u_\tau}{\partial n^2}\bigg)-g_\tau(u)+F_\tau\bigg]\mbox{d}t, \ \ \ \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+u_%5Ctau%28x%2Ct%29%3D%5Cvarphi_%5Ctau%28x%29%2B%5Cint_%7B0%7D%5E%7Bt%7D+%5Cbigg%5B%5Cnu%5Cbigg%28%5Cfrac%7B%5Cpartial%5E2+u_%5Ctau%7D%7B%5Cpartial+%5Ctau%5E2%7D%2B%5Cfrac%7B%5Cpartial%5E2+u_%5Ctau%7D%7B%5Cpartial+n%5E2%7D%5Cbigg%29-g_%5Ctau%28u%29%2BF_%5Ctau%5Cbigg%5D%5Cmbox%7Bd%7Dt%2C+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0 "\displaystyle u_\tau(x,t)=\varphi_\tau(x)+\int_{0}^{t} \bigg[\nu\bigg(\frac{\partial^2 u_\tau}{\partial \tau^2}+\frac{\partial^2 u_\tau}{\partial n^2}\bigg)-g_\tau(u)+F_\tau\bigg]\mbox{d}t, \ \ \ \ \ (2)")
and 
are the divergence-free parts of the external forcing and the nonlinear term 
as defined: 
, the external forcing and the initial velocity field 
exceeds certain threshold 
,vortices near the north Atlantic coast will form. Moreover, the scale (radius) of the vortices is an increasing function of 
;
of 
at the tip point on the east coast of Canada is sufficiently large, and the combined effect of the convexity of the tangential component of the Gulf stream shear flow and the strength of the tangential friction force is positive; and
is the heat effect coefficient.
![\displaystyle \rho\bigg[\frac{\partial u}{\partial t}+(u\cdot\nabla)u\bigg]= \mu\Delta u-\nabla p +\rho_e(\pmb{E}+ u\times \pmb{H})-g\pmb{k}\rho(1-\alpha T), \ \ \ \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Crho%5Cbigg%5B%5Cfrac%7B%5Cpartial+u%7D%7B%5Cpartial+t%7D%2B%28u%5Ccdot%5Cnabla%29u%5Cbigg%5D%3D+%5Cmu%5CDelta+u-%5Cnabla+p+%2B%5Crho_e%28%5Cpmb%7BE%7D%2B+u%5Ctimes+%5Cpmb%7BH%7D%29-g%5Cpmb%7Bk%7D%5Crho%281-%5Calpha+T%29%2C+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0 "\displaystyle \rho\bigg[\frac{\partial u}{\partial t}+(u\cdot\nabla)u\bigg]= \mu\Delta u-\nabla p +\rho_e(\pmb{E}+ u\times \pmb{H})-g\pmb{k}\rho(1-\alpha T), \ \ \ \ \ (2)")
blows up at 
, 
is the solar radius, and 
with the thickness of solar atmosphere 
.
are just 
. Hence, in the neighborhood of 
, where 
is the gas constant and 
is the particle mass, the equation 
direction.
in the 
is the Riemannian metrics of 4-dimensional space-time, representing the gravitational potential, depicting the curved space-time;
is the dual gravitational potential, representing the gravitational field particle and carrying the field energy, which is similar to the electromagnetic interaction field particle: the photon.
, which is regarded as the graviton.
, which we call G-temperature, satisfies a diffusion equation given by 
represents the gravitational source.
in the steady state solution of the model is 
is the constant velocity in the Rubin rotational curve. Then the nondimensional deviation system from the basic solution is given by ![\displaystyle \frac{1}{\text{Pr}}\bigg[\frac{\partial P}{\partial t}+(P\cdot\nabla)P\bigg] =\Delta P -a\frac{\partial P}{\partial x_1}-\frac{1}{\text{Pr}}\nabla P+\sqrt{\text{Ra}}\mathcal{T}\vec{k}, \ \ \ \ \ (4)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7B%5Ctext%7BPr%7D%7D%5Cbigg%5B%5Cfrac%7B%5Cpartial+P%7D%7B%5Cpartial+t%7D%2B%28P%5Ccdot%5Cnabla%29P%5Cbigg%5D+%3D%5CDelta+P+-a%5Cfrac%7B%5Cpartial+P%7D%7B%5Cpartial+x_1%7D-%5Cfrac%7B1%7D%7B%5Ctext%7BPr%7D%7D%5Cnabla+P%2B%5Csqrt%7B%5Ctext%7BRa%7D%7D%5Cmathcal%7BT%7D%5Cvec%7Bk%7D%2C+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{1}{\text{Pr}}\bigg[\frac{\partial P}{\partial t}+(P\cdot\nabla)P\bigg] =\Delta P -a\frac{\partial P}{\partial x_1}-\frac{1}{\text{Pr}}\nabla P+\sqrt{\text{Ra}}\mathcal{T}\vec{k}, \ \ \ \ \ (4)")
is the Rubin number, and 
is the ratio between the disk width and the halo radius. These are non-dimensional parameters given by 
:
or 
and 
of the quantum states;
in the wave function 
.
, 
represents the chemical potential;
. Then mathematically we say that the system undergoes a QPT at a critical 
is different from that of 
.
, an applied magnetic field 
induces spin 
Cooper pairs;
, together with the surface supercurrents, can cancel out the magnetism induced by the applied field 