Applied Mathematics & Optimization

, Volume 69, Issue 3, pp 359–392

Unified Field Theory and Principle of Representation Invariance


  • Tian Ma
    • Department of MathematicsSichuan University
    • Department of MathematicsIndiana University

DOI: 10.1007/s00245-013-9226-0

Cite this article as:
Ma, T. & Wang, S. Appl Math Optim (2014) 69: 359. doi:10.1007/s00245-013-9226-0


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,Wa,Sk), 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 equationsDuality theory of interactionsQuark confinementAsymptotic freedomHiggs mechanismHiggs bosonsQuark potentialNucleon potentialAtom potentialWeak interaction potentialStrong interaction force formulasWeak interaction force formulaElectroweak theory

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