Geometric Mechanism of Fundamental Interactions

In this post, we shall give some insights in postulating; see also Article [3]:

Geometric Mechanism of Interactions: The gravitational force is the curved effect of the space-time manifold ${M}$, and the electromagnetic, weak, strong interactions are the twisted effects of the underlying complex vector bundles ${M\otimes_p {\mathbb C}^n}$.

1. Gravity

One of greatest revolutions in sciences is Albert Einstein’s vision on gravity: the gravitational force is caused by the space-time curvature. This geometric mechanism can be viewed as follows.

First, in mathematical terms, the space-time manifold is a 4D Riemannian manifold ${M}$ with its metric ${\{g_{\mu\nu}\}}$:

$\displaystyle ds^2= g_{\mu\nu} dx^\mu dx^\nu,$

representing the gravitational potential. The gravitational effect of a free falling object causes the object moving on geodesics of ${M}$, given by

$\displaystyle \frac{d^2 x^i}{ds^2} + \Gamma^i_{jk} \frac{d x^j}{ds}\frac{dx^k}{ds} =0. \ \ \ \ \ (1)$

Here ${\Gamma^i_{jk}}$ are the Levi-Civita connection, uniquely determined by the gravitational potential ${\{g_{\mu\nu}\}}$:

$\displaystyle \Gamma^i_{jk} = \frac12 g^{il} \left( \frac{\partial g_{jl}}{\partial x^k} + \frac{\partial g_{kl}}{\partial x^j} - \frac{\partial g_{jk}}{\partial x^l} \right). \ \ \ \ \ (2)$

Second, the geometric mechanism of gravity can also be clearly viewed from the gravitational field equations

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\frac{8\pi G}{c^4}T_{\mu\nu} + \nabla_{\mu}\Phi_{\nu}, \ \ \ \ \ (3)$

$\displaystyle \nabla^\mu\left[ \frac{8\pi G}{c^4}T_{\mu\nu} - \nabla_{\mu}\Phi_{\nu} \right] =0, \ \ \ \ \ (4)$

which are derived based only on the principle of general relativity and PID; see the previous post: Dark energy and Dark Matter: Property of Gravity.

The energy-momentum tensor ${T_{\mu\nu}}$ relates to the baryonic matter in the universe. As distribution and dynamics of the baryonic matter such as stars and galaxies change, the space-time curvature changes, leading to changes of the gravitational potential, which, in return, affects the motion of stars and galaxies.

In a nutshell, the distribution and motion of baryonic matter in the universe are dynamically intertwined with the space-time curvature through the gravitational field equations. In short, the theory of gravity represents the geometrization of gravitational fields:

Gravitational force is the curved effect of the space-time manifold ${M}$.

2. Electromagnetic, Weak and Strong Interactions

These three interactions are fundamental interactions describing atomic and sub-atomic particles. As discussed in the previous post, we now understand that these interactions are best described by the gauge field theory: the actions are standard Yang-Mills actions uniquely dictated by the gauge symmetry and Lorentz symmetry, and the field equations are uniquely determined by PID.

The ${SU(N)}$ gauge field theory is also a geometric theory. The geometric “space” is the complex vector bundle ${M\otimes_p {\mathbb C}^n}$ with base manifold ${M}$ and with fibre space ${\mathbb C^n}$, representing the space of Dirac spinors in the fermionic particle case. Note that the geometric space is the space-time manifold ${M}$ for gravity, and is the complex bundle for other three interactions. The reason for the difference is inherited by the de Broglie matter-wave relation of particles, described by wave functions, as Dirac spinors for fermions.

In view of the difference and similarities between gravity and the other three fundamental interactions, the electromagnetic, the weak and the strong interactions are completely determined by how the corresponding complex bundles being twisted. We now explain this viewpoint using the ${SU(N)}$ gauge theory.

First, consider the ${SU(N)}$ gauge theory, consisting of ${N}$ Dirac spinors ${ \Psi =(\psi_1,\cdots ,\psi_N)^T}$, and the ${N^2-1}$ gauge interaction vector fields ${\{G^a_\mu \ | \ a=1, \cdots, N^2-1\}}$. The covariant derivative on the ${N}$ Dirac spinor complex bundle space ${M\otimes_p {\mathbb C}^{4N}}$ is given by

$\displaystyle D_\mu = \partial_\mu + i g G^a_\mu \tau_a, \ \ \ \ \ (5)$

where ${ \{\tau_k\ | \ 1\leq k \leq K\}}$ is a set of ${SU(N)}$ generators. It is then clear that the gauge fields play the same role as the Levi-Civita connection ${\Gamma^i_{jk}}$ in (2) on the space-time manifold ${M}$. We note (see the previous post) that the specific form of the covariant derivative ${D_\mu}$ in (5) is dictated by the gauge invariance.

Second, mathematically, the geometry of the complex vector bundle ${M\otimes_p {\mathbb C}^{4N}}$, the space for the Dirac spinors, is entirely determined by the connection, represented by the gauge fields ${G^a_\mu}$. Hence we have the following mathematical theorem:

Theorem [3]. The complex bundle ${M\otimes_p {\mathbb C}^{4N}}$ is geometrically nontrivial or twisted if and only if ${G=(G^1_\mu, \cdots G^{N^2-1}_\mu) \neq 0}$.

Thanks to this theorem, by Principle of Gauge Invariance, the presence of the electromagnetic, the weak and strong interactions indicates that the complex vector bundle ${M\otimes_p {\mathbb C}^{4N}}$ is twisted.

Third, the interacting gauge fields ${\{G^a_\mu\ | \ a=1, \cdots N^2-1\}}$, their dual interacting scalar fields ${\{\phi_a \ | \ a=1, \cdots N^2-1\}}$ and the Dirac spinor wave functions ${\Psi=(\psi_1, \cdots, \psi_N)^T}$ are determined by the gauge field equations; see the previous post:

$\displaystyle {\mathcal G}_{ab} \left[ \partial^{\nu}F^b_{\nu\mu} - g \lambda^{b}_{cd} g^{\alpha \beta}F^c_{\alpha\mu}A^d_{\beta}\right] - g \bar{\Psi} \gamma_{\mu}\tau_a \Psi = (\partial_\mu + \alpha_b A^b_\mu) \phi_a, \ \ \ \ \ (6)$

$\displaystyle (i\gamma^{\mu}D_{\mu}- m)\Psi =0, \ \ \ \ \ (7)$

derived based on PID with the Lagrangian action dictated by gauge invariance. Again, as in the gravity case, the interaction is entirely manifested through how the complex bundle is twisted. Each of the ${N}$ fermions ${\psi_j}$ carries an interaction charge, ${g}$, coming into play as the coupling constant of the ${SU(N)}$ gauge theory. The ${N}$ fermions with charge ${g}$ generate ${N^2-1}$ interacting fields ${G^a_\mu}$ through equations (6), which dictate how the complex bundle is twisted. In return, the twistedness of the complex bundle affects the dynamical law of the fermions through equations (7) via ${D_\mu}$. Of course these two sides of the story are highly coupled through the gauge field equations (6) and (7).

In summary, what we have demonstrated is that all three interactions, the electromagnetic, the weak and the strong, are simply the manifestations of the twistedness of the corresponding complex bundles. Therefore the geometric mechanism of all four interactions stated in the beginning of this post follows.

3. Yukawa Mechanism

Quantum Field Theory (QFT) adopts the point of view, originated from Hideki Yukawa’s revolutionary work on mesons (1935), that the force is carried by particles and further, that there is a relation between the range of the force and the mass of the force carrying particle. QFT is certainly a useful theory, which offers a great phenomenological understanding of these interactions.

Basically, Yukawa mechanism amounts to saying that these three fundamental forces take place through exchanging intermediate bosons such as photons for the electromagnetic interaction, W${^\pm}$ and Z intermediate vector bosons for the weak interaction, and gluons for the strong interaction. The current standard model, together with QFT, is a phenomenological model based on Yukawa’s mechanism. The endeavor of searching for quantum gravity is also adopting the Yukawa mechanism, leading to the perception that gravity should be quantized.

A radical difference for the two mechanisms is that the Yukawa mechanism is oriented toward to computing the transition probability for particle decays and scatterings, and the Geometric Mechanism is oriented toward to fundamental laws, such as interaction potentials, of the four interactions, and does not involve a quantization process.

Tian Ma & Shouhong Wang