Applied Mathematics & Optimization

, Volume 69, Issue 3, pp 359-392

First online:

Unified Field Theory and Principle of Representation Invariance

  • Tian MaAffiliated withDepartment of Mathematics, Sichuan University
  • , Shouhong WangAffiliated withDepartment of Mathematics, Indiana University Email author 

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


The main objectives of this article are to postulate a new principle of representation invariance (PRI), and to refine the unified field model of four interactions, derived using the principle of interaction dynamics (PID). Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint, and PRI requires that physical laws be independent of representations of the gauge groups. One important outcome of this unified field model is a natural duality between the interacting fields (g,A,W a ,S k ), corresponding to graviton, photon, intermediate vector bosons W ± and Z and gluons, and the adjoint bosonic fields \((\varPhi_{\mu}, \phi^{0}, \phi^{a}_{w}, \phi^{k}_{s})\). This duality predicts two Higgs particles of similar mass with one due to weak interaction and the other due to strong interaction. The unified field model can be naturally decoupled to study individual interactions, leading to (1) modified Einstein equations, giving rise to a unified theory for dark matter and dark energy (Ma and Wang in Discrete Contin. Dyn. Syst., Ser A. 34(2):335–366, 2014), (2) three levels of strong interaction potentials for quark, nucleon/hadron, and atom respectively (Ma and Wang in Duality theory of strong interaction, 2012), and (3) two weak interaction potentials (Ma and Wang in Duality theory of weak interaction, 2012). These potential/force formulas offer a clear mechanism for both quark confinement and asymptotic freedom—a longstanding problem in particle physics (Ma and Wang in Duality theory of strong interaction, 2012).


Principle of Interaction Dynamics (PID) Principle of Representation Invariance (PRI) Unified field equations Duality theory of interactions Quark confinement Asymptotic freedom Higgs mechanism Higgs bosons Quark potential Nucleon potential Atom potential Weak interaction potential Strong interaction force formulas Weak interaction force formula Electroweak theory