This post presents a brief introduction to the field theory of weak interactions, developed recently by the authors, based only on a few fundamental principles:

- the action is the classical Yang-Mills action dictated uniquely by the gauge invariance and the Lorentz invariance; and
- the field equations and the Higgs fields are then derived using the principle of interaction dynamics (PID) and the principle of representation invariance (PRI).

The essence of the new field theory is that the Higgs fields are the natural outcome of the PID, which takes variation under energy-momentum conservation constraints. We call this new field theory the PID weak interaction theory.

This new theory leads to layered weak interaction potential formulas, provides duality between intermediate vector bosons , , and their dual neutral Higgs and two charged Higgs , and offers first principle approach for Higgs mechanism.

**1. PID field equations of the weak interaction**

The new weak interaction theory based on PID and PRI was first discovered by the authors in [1]; see also the new book by the authors.

First the weak interaction obeys the gauge symmetry, which, together with the Lorentz invariance and PRI, dictates the standard Yang-Mills action, as we have explained in our previous post.

Then the field equations of the weak interaction and the Higgs fields are determined by by PID and PRI, and are given by:

where represents the mass of the Higgs particle, are the Pauli matrices, () are the three gauge potentials, are the three Higgs fields, and

**2. Prediction of Charged Higgs**

The right-hand side of (1) is due to PID, leading naturally to the introduction of three scalar dual fields. The left-hand side of (1) represents the intermediate vector bosons and , and the dual fields represent two charged Higgs (to be discovered) and the neutral Higgs , with the later being discovered by LHC in 2012.

It is worth mentioning that the right-hand side of (1), involving the Higgs fields, are non-variational, and can not be generated by directly adding certain terms in the Lagrangian action. This might be the very reason why for a long time one has to use logically-inconsistent electroweak theory, as we explained in the previous post here.

**3. First principle approach to spontaneous gauge symmetry-breaking and mass generation**

PID induces naturally spontaneous symmetry breaking mechanism. By construction, the action obeys the gauge symmetry, the PRI and the Lorentz invariance. Both the Lorentz invariance and PRI are universal principles, and, consequently, the field equations (1) and (2) are covariant under these symmetries.

The gauge symmetry is spontaneously breaking in the field equations (1), due to the presence of the term, , in the right-hand side, derived by PID. This term generates the mass for the vector bosons.

**4. Weak charge and weak potential**

As we mentioned in the previous posts here, elements in are expressed as , where is a basis of the set of traceless Hermitian matrices, and plays the role of a coordinate system in this representation. Consequently, an gauge theory should be independent of choices of the representation basis. This leads to the principle of representation invariance (PRI), and is simply a logic requirement for any gauge theory. This was first discovered by the authors in 2012.

With PRI applied to gauge theory for the weak interaction, two important physical consequences are clear.

*First*, by PRI, the gauge coupling constant plays the role of weak charge, responsible for the weak interaction.

In fact, the weak charge concept can only be properly introduced by using PRI, and it is clear now that the weak charge is the source of the weak interaction.

*Second*, PRI induces an important constant vector . The components of this vector represent the portions distributed to the gauge potentials by the weak charge . Hence the (total) weak interaction potential is given by the following PRI representation invariant

and the weak charge potential is the temporal component of this total weak interaction potential , and weak force is

where is the weak charge of a reference particle with radius .

**5. Layered formulas for the weak interaction potential**

The weak interaction is also layered, and we derive from the field equations (1) and (2) the following

where is the weak force potential of a particle with radius and carrying weak charges , taken as the unit of weak charge for each weakton, is the weakton radius, is a parameter depending on the particles, and represents the force-range of weak interactions.

The layered weak interaction potential formula (5) shows clearly that the weak interaction is short-ranged. Also, it is clear that the weak interaction is repulsive, asymptotically free, and attractive when the distance of two particles increases.

An elegant and natural way to introduce Higgs-Bose field. I think the most difficult part of discovering this new theory is that one first needs to give up the old way of unification, which was try to build a large symmetry group, and realize the functional is simply the sum of different interactions. Another difficulty is that the discovery of principle of interaction dynamics, which takes variation of the functional under the constraint of divergence-free! I am very interested in the discovery of this principle. What phenomenon enlightened you that the variation should be taken under divergence-free constraint? The nature does not prefer completely free variation , but the one with divergence free. The series of important discoveries upgrade our understanding of the universe to a completely new level. Personally I think you and Ma deserve to be awarded the Nobel prize in Physics for more than one time.

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hehuijing: Thanks for your interest and your appreciation. In the case of gravity, the principle of interaction dynamics (PID) is essentially a requirement of the presence of dark matter and dark energy phenomena. Then we realize that PID offers a first principle approach to introducing the Higgs fields. Now we know that the quark confinement require PID as well.

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Thanks for your answer! Nearly all the textbooks in general relativity derive the gravitational equation from Einstein-Hilbert functional under no constraint, but when discussing the properties of energy-momentum tensor Tij, they pointed out that this quantity is divergence free, div(Tij)=0! This is mathematically incorrect, since if Tij is div free, the variation could not be taken freely! I think starting from this point, you and Ma developed the divergence-free constraint variational principle. The scalar potential naturally appeared using this mathematically accurate theory. The unified field theory is a generalization of this idea to the functional of the summation of four interactions.

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You are right that the divergence-free condition on Tij cannot be derived from the variation of the action. In other words, div(Tij)=0 provide additional equations, and physically div(Tij)=0 are not correct as Tij represents only the energy-momentum tensor of the visible matter in the universe.

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