This post presents a brief introduction to the unified field theory of four fundamental interactions, developed recently by the authors [3, 4], based only on a few fundamental principles and symmetries. The main breakthrough is the introduction of two basic principles, which we call the principle of interaction dynamics (PID) and the principle of representation invariance (PRI). The field theory leads to formulas for weak and strong interaction potentials, provides first-principle based Higgs fields and mass generation mechanism, and offers explanations of the quark confinement, asymptotic freedom, and the dark energy and dark matter phenomena.
The key ingredients of the unified field model are as follows:
- The Lagrangian actions of the four individual interactions are fully dictated by the principle of general relativity, the principle of gauge invariance, and the principle of Lorentz invariance.
- The principle of representation invariance (PRI) demonstrates that the Lagrangian action coupling the four interactions is the natural combination of the actions for individual interactions.
- The unified field equations are derived based on the principle of interaction dynamics (PID).
1. Symmetry Dictates Lagrangian Actions
Fundamental laws of Nature are universal, and their validity is independent of the space-time location and directions of experiments and observations. The universality of laws of Nature implies that the Lagrangian actions are invariant and the differential equations are covariant under certain symmetry.
In fact, following the simplicity principle of laws of Nature, the three basic symmetries—the Einstein general relativity, the Lorentz invariance and the gauge invariance—uniquely determine the interaction fields and their Lagrangian actions for the four interactions. We list these actions as follows:
Electromagnetism. The electromagnetic potential is given in terms of the gauge field , the field strength
Weak interaction. The weak interacting potentials are the gauge fields
leading to a connection on spinor bundle , and the curvature is given by
Strong interaction: The strong interaction potentials are the gauge fields
is the curvature tensor on the complex spinor bundle .
2. Lagrangian Action Coupling Four Interactions
For an gauge theory, the principle of representation invariance (PRI) requires that an 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 in the electroweak theory violates PRI, as this term does not represent a gauge potential for any gauge group.
where is a nondegenerate complex matrix, and and are two sets of generators of . Hence is simply one particular component of the tensor . Hence will have no meaning if we perform a transformation (5), as it combines one component of a tensor with another component of an entirely different tensor with respect to the transformations of representation generators as given by (5).
We recall that the need for the combination in the classical electroweak theory and in the standard model is due to the particular way of coupling the Higgs fields and the gauge fields. Also, the same difficulty appears in the Einstein route of unification by embedding into a larger Lie group.
Consequently, PRI dictates that the Lagrangian action coupling the four fundamental interactions must be the natural combination of the Einstein-Hilbert functional, the Yang-Mills actions for the electromagnetic, weak and strong interactions:
3. Unified Field Equations Based on PID
One most important ingredient for the unification for four fundamental interactions is the principle of interaction dynamics (PID), discovered by (Ma-Wang, 2012); see also the previous post.
Basically, PID takes the variation of the Lagrangian action under energy-momentum conservation constraint. For gravity, PID is the direct consequence of the presence of dark energy and dark matter; see the previous post: Dark Matter and Dark Energy: a Property of Gravity.
The main breakthrough comes with the natural introduction of the Higgs fields due to PID, the variation of the Lagrangian action under energy-momentum conservation constraint. Namely, by PID and PRI, we derive the following system of field equations coupling the four fundamental interactions:
A few remarks are now in order.
This duality relation can be regarded as a duality between field particles for each interaction. It is clear that each interaction mediator possesses a dual field particle, called the dual mediator, and if the mediator has spin-, then its dual mediator has spin-. Hence the dual field particles consist of spin-1 dual graviton, spin-0 dual photon, spin-0 charged Higgs and neutral Higgs fields, and spin-0 dual gluons. The neutral Higgs (the dual particle of ) had been discovered experimentally by LHC in 2012.
In the weakton model , we realize that these dual particles possess the same weakton constituents, but different spins, as the mediators.
Second, this duality can be considered as a duality of interacting forces: Each interaction generates both attracting and repelling forces. Moreover, for the corresponding pair of dual fields, the even-spin field generates an attracting force, and the odd-spin field generates a repelling force. For the first time, we discovered such attracting and repelling property of each interaction derived from the field model. Such property plays crucial role in the stability of matter in the Universe. For example, repulsive behavior of gravity on a very large scale we discovered in  explains dark energy phenomena.
Third, we have demonstrated that the unification/coupling of the four interactions are achieved through the right-hand sides of the field equations based on PID. It is also clear that the model can be decoupled to study individual interactions when other interactions are negligible.
Fourth, the terms in the right-hand sides of (7)-(10) spontaneously break the gauge symmetries. In other words, the Lagrangian action (6) obeys the gauge gauge invariance, the Lorentz invariance, PRI and the principle of general relativity. The field equations obey all the symmetries except the gauge symmetry.