Unified Field Theory

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:

Gravity. The principle of general relativity dictates the following Einstein-Hilbert functional

\displaystyle \mathcal{L}_{EH}(g_{\mu\nu})=R + \frac{8\pi G}{c^4} S, \ \ \ \ \ (1)

where {R} stands for the scalar curvature of {M}, and {S} is the energy-momentum density of baryonic matter fields in the universe.

Electromagnetism. The electromagnetic potential is given in terms of the {U(1)} gauge field {A_{\mu}=(A_0,A_1,A_2,A_3)}, the field strength

\displaystyle A_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},

is the curvature of the Dirac spinor vector bundle, {M\otimes_p \mathbb C^4}, associated with the connection {D_\mu = \partial_{\mu}+ieA_{\mu}}. The Lagrangian action is

\displaystyle \mathcal{L}_{EM}(A_\mu, \psi^E)=-\frac{1}{4}A_{\mu\nu}A^{\mu\nu} + \bar{\psi}^E\left[ i\gamma^{\mu}(\partial_{\mu}+ieA_{\mu})-m_e\right] \psi^E. \ \ \ \ \ (2)

Weak interaction. The weak interacting potentials are the {SU(2)} gauge fields

\displaystyle W^a_{\mu}=(W^a_0,W^a_1,W^a_2,W^a_3) \ \ \ \text{ for } 1\leq a\leq 3,

leading to a connection {D_\mu=\partial_{\mu}+ig_wW^a_{\mu}\sigma_a} on spinor bundle {M\otimes_p (\mathbb C^4)^2}, and the curvature is given by

\displaystyle W^a_{\mu\nu}=\partial_{\mu}W^a_{\nu}-\partial_{\nu}W^a_{\mu}+g_w\lambda^a_{bc}W^b_{\mu}W^c_{\nu} \ \ \text{ for } 1\leq a\leq 3.

The Lagrangian action is dictated by the gauge invariance:

\displaystyle \mathcal{L}_W=-\frac{1}{4}{\mathcal G}^w_{ab}W^a_{\mu\nu}W^{\mu\nu b}+\bar{\psi}^W \left[ i\gamma^{\mu}(\partial_{\mu}+ig_wW^a_{\mu}\sigma_a)-m_W\right] \psi^W. \ \ \ \ \ (3)

Strong interaction: The strong interaction potentials are the {SU(3)} gauge fields

\displaystyle S^k_{\mu}=(S^k_0,S^k_1,S^k_2,S^k_3) \qquad \text{ for } 1\leq k\leq 8,

and the action is

\displaystyle \mathcal{L}_S=-\frac{1}{4} {\mathcal G}^s_{kl}S^k_{\mu\nu}S^{\mu\nu l}+\bar{\psi}^S\left[i\gamma^{\mu}(\partial_{\mu}+ig_sS^k_{\mu}\tau_k)-m_S\right] \psi^S. \ \ \ \ \ (4)


\displaystyle S^k_{\mu\nu}=\partial_{\mu}S^k_{\nu}-\partial_{\nu}S^k_{\mu}+g_s\Lambda^k_{rl}S^r_{\mu}S^l_{\nu} \ \ \ \text{ for } 1\leq k\leq 8

is the curvature tensor on the complex spinor bundle {M\otimes_p (\mathbb C^4)^3}.

2. Lagrangian Action Coupling Four Interactions

For an {SU(N)} gauge theory, the principle of representation invariance (PRI) requires that an {SU(N)} 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 {\alpha A_\mu + \beta W^3_\mu} in the electroweak theory violates PRI, as this term does not represent a gauge potential for any gauge group.

In fact, the physical quantities {W^a_\mu}, {W^a_{\mu\nu}}, {\lambda^a_{bc}} and {\mathcal G^w_{ab}} are {SU(2)}-tensors under the following transformations of representation generators:

\displaystyle \widetilde{\sigma}_a=x^b_a\sigma_b, \ \ \ \ \ (5)

where {X=(x^b_a)} is a nondegenerate complex matrix, and {\{\sigma_a \ | \ a=1, 2, 3\}} and {\{\widetilde{\sigma}_a \ | \ a=1, 2, 3\}} are two sets of generators of {SU(2)}. Hence {W^3_\mu} is simply one particular component of the {SU(2)} tensor {(W^1_\mu, W^2_\mu, W^3_\mu)}. Hence {\alpha A_\mu + \beta W^3_\mu} 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 {\alpha A_\mu + \beta W^3_\mu} 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 {U(1) \times SU(2) \times SU(3)} 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 {U(1), SU(2), SU(3)} Yang-Mills actions for the electromagnetic, weak and strong interactions:

\displaystyle L=\int_M\left[\mathcal{L}_{EH}+\mathcal{L}_{EM}+\mathcal{L}_W+\mathcal{L}_S\right]\sqrt{-g}dx. \ \ \ \ \ (6)

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:

\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R + \frac{8\pi G}{c^4}T_{\mu\nu}= \left[\nabla_{\mu}+\alpha^0A_{\mu}+\alpha^1_bW^b_{\mu}+\alpha^2_kS^k_{\mu}\right] \phi^G_{\nu}, \ \ \ \ \ (7)

\displaystyle \partial^{\mu}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})-e \bar{\psi}^E\gamma_{\nu}\psi^E=\left[\nabla_{\nu}+\beta^0A_{\nu}+\beta^1_bW^b_{\nu}+\beta^2_kS^k_{\nu} \right] \phi^E. \ \ \ \ \ (8)

\displaystyle {\mathcal G}^w_{ab}\left[\partial^{\mu}W^b_{\mu\nu}-g_w\lambda^b_{cd}g^{\alpha\beta}W^c_{\alpha\nu}W^d_{\beta}\right] - g_w\bar{\psi}^w\gamma_{\nu}\sigma_a\psi^w = \left[ \nabla_{\nu}+\gamma^0A_{\nu}+\gamma^1_bW^b_{\nu}+\gamma^2_kS^k_{\nu}-\frac{1}{4}m^2_wx_{\nu} \right]\phi^W_a, \ \ \ \ \ (9)

\displaystyle {\mathcal G}^s_{kj}\left[\partial^{\mu}S^j_{\mu\nu}-g_s\Lambda^j_{cd}g^{\alpha\beta}S^c_{\alpha\nu}S^d_{\beta}\right]-g_s\bar{\psi}^s\gamma_{\nu}\tau_k\psi^s = \left[\nabla_{\nu}+\delta^0A_{\nu}+\delta^1_bW^b_{\nu}+\delta^2_kS^k_{\nu}-\frac{1}{4}m^2_sx_{\nu}\right]\phi^S_k, \ \ \ \ \ (10)

\displaystyle \left[ i\gamma^{\mu}(\partial_{\mu}+ieA_{\mu})-m\right] \psi^E=0, \ \ \ \ \ (11)

\displaystyle \left[ i\gamma^{\mu}(\partial_{\mu}+ig_wW^a_{\mu}\sigma_a)-m_l \right]\psi^w=0, \ \ \ \ \ (12)

\displaystyle \left[ i\gamma^{\mu}(\partial_{\mu}+ig_sS^k_{\mu}\tau_k)-m_g \right] \psi^s=0. \ \ \ \ \ (13)


\displaystyle T_{\mu\nu}=\frac{\delta S}{\delta g_{\mu\nu}}+\frac{c^4g^{\alpha\beta}}{16\pi G}\left[ G^w_{ab}W^a_{\alpha\mu}W^b_{\beta\nu}+G^s_{kl}S^k_{\alpha\mu}S^l_{\beta\nu} +A_{\alpha\mu}A_{\beta\nu} - \mathcal{L}_{EM}-\mathcal{L}_W-\mathcal{L}_S\right].

A few remarks are now in order.

First, the right-hand sides of the field equations (7)-(10) above are due to PID. In other words, PID induces a natural duality between the interaction fields {(g_{\mu\nu},A_{\mu},W^a_{\mu},S^k_{\mu})} and their corresponding dual fields {(\phi^G_{\mu},\phi^E,\phi^W_a,\phi^S_k)}.

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-{k}, then its dual mediator has spin-{(k-1)}. 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 {H^0} (the dual particle of {Z}) had been discovered experimentally by LHC in 2012.

In the weakton model [8], 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 [1] 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.

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5 Responses to Unified Field Theory

  1. Pingback: On Unification of Four Fundamental Interactions | Interplay between Mathematics and Physics

  2. hehuijing says:

    In order to make this excellent theory useful, we need to fix the coupling coefficient appeared in the equations. Once they are fixed, we can derived the accurate inter-molecular potential and inter-atomic potential, these results would be of fundamental importance for molecular dynamics simulation of many interesting physical phenomenon, like phase transition, fracture of solid.


  3. hehuijing says:

    According to Eq. (7), in addition to the scalar potential, all the EM potential, weak potential and strong potential can interact with the gravitation potential Rij, do they have some observable effects? Could we make use of this interaction in the future? One of the most interesting effects is called anti-gravity effect, attracted intensive interest from both scientific community and general public. Could the mysterious phenomenon UFO be explained using this theory?


  4. Ma-Wang says:

    The questions you asked are interesting, but further investigations are needed before we can any positive assertions.


  5. Ma-Wang says:

    Thanks again for your interest and your very insightful questions.

    The coupling coefficients depend on the specific physical system under study. You’re right, one can use the field theory for multi particle systems to derive the inter-molecular potential and inter-atomic potential, which will be useful. See Section 6.5 for preliminary studies in this direction.


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