The Conjugate Entangled Manifold of Space–Time Induced by the Law of Unity of Contradiction

Abstract

The mechanism of the contradiction between two different objects $u$ and $v$ is attributed to a mechanism that their opposite position information “$x_{u}$” and “$x_{v}$” of u and v are transmitted, respectively, from the initial time $t_{0}$ , at different speeds $\dot{x}_{u} \left( t \right)$ and $\dot{x}_{v} \left( t \right)\,\left( \dot{x}_{v} = -\varsigma\dot{x}_{u} \left( t \right) \right)$, and is meeting at the contradiction point $t = t_{\lambda}$ and $x = x_{\lambda}$. Because the coordinate of contradiction point can be noted by $z_{\lambda} \left(t_{\lambda}, x_{\lambda}\right)$ and $z_{\lambda}^{*} \left(x_{\lambda}, t_{\lambda}\right)$ in two space time Complex Coordinates Systems which origins are $z_{o} \left(0_{t}, 0_{x}\right)$ and $z_{o}^{*} \left(1_{t}, 1_{t}\right)$, respectively, such that the time $t_{\lambda}$ and the position $x_{\lambda}$ of the contradictory points can be expressed as the sum of the complex numbers $z_{\lambda} \left(t_{\lambda}, x_{\lambda}\right)$ and its conjugate $\bar{z}_{\lambda} \left(t_{\lambda}, x_{\lambda}\right): t_{\lambda} = z_{\lambda} \left(t_{\lambda}, x_{\lambda}\right) + \bar{z}_{\lambda} \left(t_{\lambda}, x_{\lambda}\right) = w_{\lambda} \left(z_{\lambda}, \bar{z}_{\lambda}\right)$, and the difference of $z_{\lambda}^{*} \left(x_{\lambda}, t_{\lambda}\right)$ and its conjugate: $\bar z_{\lambda}^{*} \left(x_{\lambda}, t_{\lambda}\right): x_{\lambda} = z_{\lambda}^{*} \left(x_{\lambda}, t_{\lambda}\right) - \bar{z}_{\lambda}^{*} \left(x_{\lambda}, t_{\lambda}\right) = w_{\lambda}^{*} \left(z_{\lambda}^{*} , \bar{z}_{\lambda}^{*}\right)$. By synthesizing the time-space coordinate and the space-me coordinate, such their time axis $\left[0_{t}, 1_{t}\right]$ and the space axis $\left[1_{x}, 0_{x}\right]$ of the two complex coordinate systems are coincide with the intervals $\left[u, v\right]$, respectively, then the contradiction point can be expressed in the synthesis Coordinate System to be a wave function: $\psi\left(w_{\lambda}, w_{\lambda}^{*}\right) = t_{\lambda} - ix_{\lambda} = w_{\lambda} - iw_{\lambda}^{*}$. Because of the varying direction of two information “$x_u$” and “$x_v$” and their increments $\Delta{x}_u \left( \Delta_{u}t\right) = \dot{x}_u \left( \Delta_{u}t\right) \Delta_{u}t$ and $\Delta{x}_v \left( \Delta_{v}t\right) = \dot{x}_v \left( \Delta_{v}t\right) \Delta_{v}t$ with time t and increment $\Delta{t} = t - 0_{t}$ are opposite each other, so the $t_{\lambda}$ of the wave function $\psi$is on the time axis $\left[0_{t}, 1_{t}\right]$ and the $x_{\lambda}$ on the space axis $\left[1_{x}, 0_{x}\right]$, constructed a pair of information transmission streams entangled in opposite directions appear, such that the interval [u, v] constitutes a space-time conjugate entangled manifold. The invariance of the contradiction point or wave function $\psi\left(w_{\lambda}, w_{\lambda}^*\right)$, under the unit scale transformation of time and distance measurement, not only make all points $z\left(t, x\right) \in \left[u, v\right]$ is contradiction point, and makes $\lambda = \frac{1}{2}$ and $\zeta = \frac{\lambda}{1}-{\lambda}$ It is also shown that since λ changes from 0 to $\frac{1}{2}$ is equivalent to the integral for the on $t_{\lambda}$ and $x_{\lambda}$ in wave function ψ from 0 to $\frac{1}{2}$, respectively, by it not only the inner product ψ of the ψ and the time component $t_{\lambda}$, respectively $\psi \left(w, w^{*}\right). t_{\lambda}$, and the outer product of ψ and the spatial component $\psi \left(w, w^{*}\right) \wedge x_{\lambda}$ can be get, but also their sum: $\psi \left(w, w^{*}\right) \cdot \psi_{t} + \psi\left(w, w^{*}\right) \wedge \psi_{x}$ can be gotten too.