New Book: Mathematical Principles of Theoretical Physics

Tian Ma & Shouhong Wang, Mathematical Principles of Theoretical Physics, Science Press, Beijing, 524pp., August 2015, ISBN: 9787030452894.

[The book can now be ordered in amazon.cn. Note:  In order to order at amazon.cn, one needs to sign up for an account there, and it does ship to the US. The total cost (book + shipping to Indiana, USA) is about \$35.]

This book is devoted to our discoveries in general relativity, particle physics, cosmology and astrophysics over the last few years. We have derived experimentally verifiable laws of Nature based only on a few fundamental principles, and have established a unified field theory, as Albert Einstein hoped. Our work solves a number of challenging problems, which include

• the dark matter and dark energy phenomena,
• the structure of black holes,
• the structure and origin of our Universe,
• the quark confinement,
• first principle approach to Higgs fields, and
• mechanism of supernovae explosion and active galactic nucleus jets (AGN).

The current discoveries began in 2012, when we made a breakthrough both on the Einstein law of gravity and on the dark matter and dark energy phenomena – two of the greatest unsolved mysteries in physics. Both dark matter and dark energy are not accounted for in the Einstein equations. We rigorously proved that the presence of dark matter and dark energy requires that the variation of the Einstein-Hilbert action be taken under energy-momentum conservation constraint. The Einstein equations, however, are derived as the variation of the Einstein-Hilbert action under no constraint. This gives rise to a new set of field equations, altering the Einstein equations with a new term analytically derived from the constraints:

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

The new law of gravity we established shows that gravity behaves like the Einstein gravity in the solar system, and it has more attraction in the galactic scale (dark matter), and it becomes repulsive over very large scale (dark energy).

Since 2012, we have discovered that the same principle of taking variation of the actions under the energy-momentum conservation constraints is required by the quark confinement and the Higgs field, and is valid for all four forces–the gravity, the electromagnetism, the nuclear weak and the strong forces. We call it the principle of interaction dynamics (PID); see the previous post here. 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 our PID. This has led to new insights and solutions to a number of unsolved challenging problems of theoretical physics and cosmology since 1950s, and led to the unified field theory as Einstein hoped.

Most attempts in modern physics and cosmology focus on artificially modifying the actions such as the Einstein-Hilbert action. This is the primary reason behind the difficulties and challenges that modern physics has faced for many decades. What distinguishes our work is that the actions are uniquely dictated by basic principles, and the laws of Nature are then derived using PID. This leads to dual fields, given for example by the new term in our gravitational field equations. The dual fields are hardly achievable by any other means.

Another discovery we made in 2012 is the principle of representation invariance (PRI); see the previous post. PRI requires that the gauge theory be independent of the choices of the representation generators. These representation generators play the same role as coordinates, and in this sense, PRI is a coordinate-free invariance/covariance, reminiscent of the Einstein principle of general relativity. In other words, PRI is purely a logic requirement for the gauge theory.

PRI suggests the introduction of  the principle of symmetry-breaking (PSB) for unification. The three sets of symmetries — the general relativistic invariance, the Lorentz and gauge invariances, as well as the Galileo invariance — are mutually independent and dictate in part the physical laws in different levels of Nature. PSB assets that for a system coupling different levels of physical laws, part of these symmetries must be broken.

These three new principles–PID, PRI and PSB– have profound physical consequences, and, in particular, provide a new route of unification for the four interactions, different from the Einstein unification route which uses large symmetry group:

• the general relativity and the gauge symmetries dictate the Lagrangian;
• the coupling of the four interactions is achieved through PID and PRI in the field equations, which obey the PGR and PRI, but break spontaneously the gauge symmetry;
• the unified field model can be easily decoupled to study individual interaction, when the other interactions are negligible; and
• the unified field model coupling the matter fields using PSB.

Tian Ma & Shouhong Wang