## Blog: Interplay between Mathematics and Physics

This blog explores the symbiotic interplay between 1) the mathematical formulation of nature phenomena and the  development of new mathematics motivated by physical problems, and 2) the discovery of new laws and insights of Nature. Our posts focus on the field theory of four fundamental interactions and elementary particles, cosmology, climate dynamics and phase transitions.  -Tian Ma & Shouhong Wang

## Field Theory and Elementary Particles (Ma & Wang)

We have developed a field theory of four fundamental interactions based only on a few fundamental principles and symmetries. The main ingredient is the introduction of two basic principles, which we call the principle of interaction dynamics (PID) and the principle of representation invariance (PRI).

Principle of Interaction Dynamics (PID) [3]. PID takes the variation of the Lagrangian action under energy-momentum conservation constrain.  For gravity, we have shown that PID is the direct implication of the presence of dark matter and dark energy, together with the principle of general relativity. PID offers a natural and simple way to introduce the Higgs fields, leading to spontaneous gauge symmetry breaking.

Principle of Representation Invariance (PRI) [4]PRI requires that an ${SU(N)}$ gauge theory be invariant under the transformations of different sets of generators. One profound consequence of PRI is that any linear combination of gauge potentials from two different gauge groups are prohibited by PRI. For example, the term ${\alpha A_\mu + \beta W^3_\mu}$ in the electroweak theory violates PRI, as this term does not represent a gauge potential for any gauge group.

Law of Gravity [1, 2]. Einstein’s principle of equivalence amounts to saying that the space-time is a 4D Riemannian manifold ${M}$, with the metric ${\{g_{\mu\nu}\}}$ being the gravitational potential. Einstein’s principle of general relativity requires that the Lagrangian action be invariant and the field equations be covariant under general coordinate transformations, and dictates that the action is given by the Einstein-Hilbert functional. By PID, we derive the gravitational field equations; see [1]:

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\frac{8\pi G}{c^4}T_{\mu\nu} + \nabla_{\mu}\Phi_{\nu},$

supplemented by the energy-momentum conservation:

$\displaystyle \nabla^\mu\left[ \frac{8\pi G}{c^4}T_{\mu\nu} - \nabla_{\mu}\Phi_{\nu} \right] =0.$

Here ${\Phi_\nu}$ is a vector field defined on the 4D space-time manifold ${M}$, and needs to be solved together with the Riemannian metric ${g_{\mu\nu}}$, and ${T_{\mu\nu}}$ is the energy-momentum of the baryonic matter in the universe.

Theory of Dark Matter and Dark Energy [1, 2]. The field equations establish a natural duality between the gravitational field ${g_{\mu\nu}}$ and its dual vector field ${\Phi_\mu}$.  This duality gives rise also to both attractive and repulsive forces. In fact, we have shown that it is the duality between the attracting gravitational field ${\{g_{\mu\nu}\}}$ and the repulsive dual vector field ${\{\Phi_\mu\}}$, together with the nonlinear interaction of these two fields through the field equations, that give rise to gravity, and in particular the gravitational effect of dark energy and dark matter.  In fact, consider a central matter field with total mass ${M}$ and with spherical symmetry. Using the above new gravitational field equations, we can derive an approximate gravitational force formula:

$\displaystyle F=mMG\left[-\frac{1}{r^2} -\frac{k_0}{r} + k_1 r \right], \qquad k_0=4 \times 10^{-18} km^{-1}, \qquad k_1=10^{-57} km^{-3}.$

Here the first term represents the Newton gravitation, the attracting second term stands for dark matter and the repelling third term is the dark energy.