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

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.