Three Constituents of the Fundamental Theory for the Four Interactions

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

$\displaystyle \text{\it symmetries} \Longrightarrow \text{\it {\bf unique} actions (functionals)} \Longrightarrow \text{\it field equations}$

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 ${SU(N)}$ action was introduced by Yang and Mills.

Tian Ma & Shouhong Wang