Let the line element of Minkowski spacetime, where $P^{\mu }=m{\mathsf{v}}^{\mu }$ is the ‘Four-momentum’, hence $P^{\mu }{\mathsf{v}}^{\nu }\equiv p^0{\mathsf{v}}^0+p^i{\mathsf{v}}^j$ for $i,j=1,2,3$, and $\mu ,\nu =0,1,2,3$, whereas ${\mathsf{v}}^{\mu }$ is the ‘four-velocity’. Be careful that it is $P^{\mu }{\mathsf{v}}^{\nu }\ne p^0{\mathsf{v}}^0-p^i{\mathsf{v}}^j$, because $g_{\mu \nu }$ takes the ‘−’ sign. Also note that $m\ne m_0$ in (3) for the rest mass $m_0$. Thus, (3) gives us an energy-momentum invariant line element as, when $T=\frac{1}{2}m{\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)}^2$ is the kinetic energy [2], that is, we have a new line element as, then, the rearrangement of (4) gives the following equation by using (3) as,

Let us consider the representation of a wave field $\psi (\overrightarrow{r},t)$ by superposition of a free particle (de Broglie wave) for (6) as follows, where $\overrightarrow{P}\to {\overrightarrow{P}}^{\mu }$ and $\overrightarrow{R}\to {\overrightarrow{r}}^{\mu }$, as ${\overrightarrow{r}}^{\mu }=\left({\mathsf{v}}^{\mu }t\right)$, in addition, note it that we have taken here E as total energy for (6). Thus, from (7), we can get the (total) energy operator $\widehat{E}\to i\hslash {\partial }_t$ (it is analogous with, but not exactly the same as, the classical Quantum Mechanics, as it is now the total energy and related to $(3+1)D$ instead of $3D$ due to the presence of $g_{\mu \nu }$ in (7) in either ways), the three momentum operator $\widehat{\mathit{p}}\to -i\hslash {\overrightarrow{\nabla }}_i$, the ‘Four-momentum’ operator, and the mass operator, where, $\left(\frac{\mathrm{d}t}{\mathrm{d}s}\right)$ is evidently relativistic, but $\frac{\partial }{\partial s}$ is no doubt Quantum Mechanical, so as, For constant velocity, we can develop an uncertainty principle describing the intrinsic indeterminacy with which m and s can be determined as,

The mass-energy relation, i.e., $E=m{c}^2$, of (4) yields the Quantum Mechanical definition for the mass operator $\widehat{m}$ of (9) for (8) and (10) as, since $c\left(\frac{\mathrm{d}t}{\mathrm{d}s}\right)=1/\sqrt{1-\frac{{\mathsf{v}}^2}{c^2}}=\gamma$. Thus, (11) tells us that the total energy of a system directly relates to its geometry, more precisely, to $\partial /\partial s$. Additionally, (11) yields the following for (10) and $c=1$, For the second term of $\widehat{m}$ in (9), we can rewrite (11) by squaring its both sides after considering $c=1$ as follows, which clarifies more precisely that ${\left(\mathrm{d}s/\mathrm{d}t\right)}^2$ is evidently relativistic, but ${\partial }^2/\partial s^2$ is definitely Quantum Mechanical. Thus, (13), so as (12), are very peculiar equations where the lhs spacetime is classical relativistic but the rhs spacetime is Quantum Mechanical, and the total energy is directly related to the rhs spacetime. If the lhs spacetime of (13) changes (not more than $(3+1)D$ and not less than $(1+1)D$, unless it is a vacuum state) and the total energy remains fixed (so as time), then the spacetime of rhs should not remain as same as before, but changes inversely against the lhs spacetime. Though, the increment of rhs spacetime should not be observable, i.e., all extra dimensions would have to remain hidden inside the overall observable system, in other words, inside the lhs observable spacetime of (13). We will discuss it below in more details very soon.

For (10), we can rewrite the mass-energy relation (13) as follows, Thus, it is somehow a kind of Klein-Gordon-like (but not an exactly similar) equation of the relativistic waves due to the above quantum scenario derived from (7). (By the way, we will see at the very end of this Subsection that Klein-Gordon equation is a subset of a Second Order Equation of GQG in Semi-Quantum Minkowski Spacetime). This equation may yield the first order equation as, where, ${\gamma }^{\mu }$ are Dirac’s gamma matrices.

The wave field $\psi (\overrightarrow{r},t)$ in (7) must satisfy the eigenfunctions for a discrete Lorentz transformation as, when ${\psi }^{†}$ is the complex conjugate of $\psi$. Then, using summation convention and (10), we can write the joint state of both spacetimes as, But, (15) also intends to, for ${\partial }_{\mu }^2={\widehat{\partial }}_{\mu }^2=\left[\frac{{\partial }^2}{\partial t^2},\frac{{\partial }^2}{\partial x^i\partial x^j}\right]$ and $g_{\mu \nu }={\widehat{g}}_{\mu \nu }=\mathrm{diag}[1,-1,-1,-1]$, where $\widehat{\partial }$ implies ${\partial }_t$’s dependency on ${\psi }_i$ and ${\partial }_x$’s dependency on ${\psi }_0$, respectively. Here, $\rho$ and $\Gamma$ have been chosen arbitrarily. Hence, the complex conjugate of (16) is, But, if we take, $\mathrm{d}{s}^{-2}\phi ={(\mathrm{d}{t}^2-\sum \mathrm{d}{x}^i\mathrm{d}{x}^j)}^{-1}\phi$, where $\phi$ is an operator, we can say that, ${(\mathrm{d}{t}^2-\sum \mathrm{d}{x}^i\mathrm{d}{x}^j)}^{-1}\ne \mathrm{d}{t}^{-2}-\sum {\left(\mathrm{d}{x}^i\mathrm{d}{x}^j\right)}^{-1}$, so, we may assume without any objection that, ${(\mathrm{d}{t}^2-\sum \mathrm{d}{x}^i\mathrm{d}{x}^j)}^{-1}\equiv \frac{{\partial }^2}{\partial t^2}-\sum \frac{{\partial }^2}{\partial x^i\partial x^j}-\Pi$, for some value of $\Pi$. Thus, thus, from (16), if $g^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }\equiv g^{\mu \nu }\left(\rho \right){\partial }_{\mu }{\partial }_{\nu }$, we can write as, this yields, $\frac{{\partial }^2}{\partial s^2}\Psi \left|s_0\equiv \mathrm{d}{s}^{-2}\phi \right.$ if $\Gamma \to -\Pi$, which gives the equivalency of (16) and (18). No more combinations are possible from (16) apart from $\rho$, $\Gamma$ and (15) itself. The arrangement of $\rho$ and $\Gamma$ implies that $g_{\mu \nu }\left(s\right)=\frac{1}{2}\left(g_{\mu \nu }\left(\rho \right)+{\widehat{g}}_{\mu \nu }(\Gamma )\right)$. Thus, (16) should be rewritten by using (15), (16) and additionally replacing $g_{\mu \nu }\left(s\right)$ with $g_{\mu \nu }\left(s\right)=\frac{1}{2}\left(g_{\mu \nu }\left(\rho \right)+{\widehat{g}}_{\mu \nu }(\Gamma )\right)$ as follows, Note that, $\Pi$ exclusively has to depend upon spacetime. Comparing (16) with (18), let us say that, where $\Gamma \to -\Pi$, suppose. Since $g_{\mu \nu }\left(s\right)\ne \frac{1}{2}\left(g_{\mu \nu }\left(\rho \right)-{\widehat{g}}_{\mu \nu }(\Pi )\right)\equiv 0$ as long as $g_{\mu \nu }={\widehat{g}}_{\mu \nu }=\mathrm{diag}[1,-1,-1,-1]$, then it must be as follows, if $g_{\mu \nu }\left(s\right)\ne 0$, It is impossible to decompose (19) and (21) since both of their lhs are only depended upon ∂, thus,

But, the rhs of (21) gives us, where, the swap operator $F|\Psi \rangle =exp\left[i\theta \right]$ for some phase $exp\left[i\theta \right]$, whereas V is a vector space. Then the corresponding eigenspaces are called the symmetric and antisymmetric subspaces and are denoted by the state spaces ${\mathrm{Sym}}^{2}V$ and ${\mathrm{Anti}}^{2}V$, respectively. Note that, we have not intended here that $\rho$ and $\Pi$ individually are two indistinguishable particles for the state spaces ${\mathrm{Sym}}^{2}V$ and ${\mathrm{Anti}}^{2}V$; the above equations are just the generalization forms of their kinds, because $\rho$ and $\Pi$ do not have distinguished (opposite) spins until otherwise they are dissociated as free particles; so, the observable spin is always the spin of $\rho$, since $\Pi$ is an internal hidden property of the overall system and the observable spacetime is always ∂-dependent. Thus, (21) tells us that, if we allow $\Pi$ to be dissociated as a free particle at very high energy, the internal hidden spacetime of $\Pi$ then must be transformed into a fermionic particle, whereas, the overall ∂-dependent system remains bosonic, since, the observable spacetime is always ∂-dependent.

Similarly, (17) yields, this tells us that the internal hidden spacetime of $\Pi$ must be now bosonic, whereas, the overall system is fermionic, since, the observable spacetime is always ∂-dependent. So, whatever (21) and (22) want to tell us is that the ∂-dependent overall system has Supersymmetry and since the spacetimes of $\rho$ and its supersymmetric partner $\Pi$ are not easily dissociative even upto a very high energy scale; thus, $\Pi$ must require extremely high energy to dissociate itself from the overall system as a free particle. Instead of being a free supersymmetric partner, $\Pi$ actually works quite differently inside of the observable spacetime $\rho$, though, at the same time, $\Pi$ is still satisfying all the properties of Supersymmetry. We will show you $\Pi$’s actual purpose very soon in the below. But Supersymmetry needs extra dimensions and we should have to discuss it now.

Before proceeding with anything, we can develop a $\widehat{\partial }$-dependent scenario as follows, and the complex conjugate of $\Psi$ is, It is not important which state spaces satisfy such bosonic or fermionic representations of (23) and (24), here, the most important thing is that the overall system as a free observable particle is must not be baryonic because now only the internal hidden spacetime of $\rho$ has ‘proper’ spacetime arrangement for its ∂-dependency, whereas, the overall (observable) system’s spacetime arrangement is quite ‘improper’ as it is $\widehat{\partial }$-dependent. Despite of $\rho$’s ∂-dependency, here, being a supersymmetric partner, if it is allowed to be free at very high energy, it must not be baryonic either. We should not be confused with it. We will discuss about its property in Section 4 below.

The internal hidden spacetime of $\Pi$ in (21) and (22) also provides us some additional geometry for its ${\widehat{g}}^{\mu \nu }(\Pi ){\widehat{\partial }}_{\mu }{\widehat{\partial }}_{\nu }\left[\pm \Psi \right]$ structures. Suppose, for ${\widehat{g}}^{\mu \nu }(\Gamma ){\widehat{\partial }}_{\mu }{\widehat{\partial }}_{\nu }\left[\Psi \right]$, we have, $\frac{1}{\sqrt{2}}\left\{\frac{{\partial }^2}{\partial t^2}\left[-{\psi }_i\right]-\frac{{\partial }^2}{\partial x^i\partial x^j}\left[{\psi }_0\right]\right\}$, where, neither spacetime arrangements are matched with one another in either combinations within the curvey brackets. These ‘wrong’ arrangements must have a noticeable effect on the acceptable spacetime, i.e., its temporal part must influence over the spatially depended ${\psi }_i$, or its spatial part must influence over the temporally depended ${\psi }_0$, or vice versa. In other words, the acceptable spacetime should not have to be four-dimensional in this case. Let us check it. Suppose, for $\frac{1}{\sqrt{2}}\left\{\frac{{\partial }^2}{\partial t^2}\left[-{\psi }_i\right]-\frac{{\partial }^2}{\partial x^i\partial x^j}\left[{\psi }_0\right]\right\}$, we can consider a dimension function (see [3]; we follow Nagata’s work throughout this paragraph and all the proofs for this paragraph are easily obtainable by using his book [3]), and the space $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ satisfies a normal $T_1$-space. Let $\mathcal{U}$ be a collection in an initially $(3+1)D$ topological spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$, i.e., $\mathit{dim}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)=(3+1)$, which is actually hidden inside an observable $(3+1)D$ spacetime, i.e., $\mathit{dim}\left({\partial }_{[0,i]}^2\otimes {\Psi }_{[0,i]}\right)=(3+1)$ (be careful here about the subscripts, do not confuse observable spacetime with hidden spacetime), and p a point of $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$, then the order of $\mathcal{U}$ at p should be denoted by, $\mathrm{ord}\mathcal{U}=\sup \left\{{\mathrm{ord}}_p\mathcal{U}p\in \left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)\right\}$, where, ${\mathrm{ord}}_p\mathcal{U}$ is the number of members of $\mathcal{U}$ which contain p. If for any finite open covering $\mathcal{U}$ of the topological spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ there exists an open covering $\mathfrak{B}$ such that $\mathfrak{B}<\mathcal{U}$, $\mathrm{ord}\mathfrak{B}\le n+1$, then $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ has covering dimension $\le n$, i.e., $\mathit{dim}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)\le n$. If $\mathcal{U}$ can be decomposed as $\mathcal{U}={\bigcup }_{i=1}^{\infty }{\mathcal{U}}_i$ for locally finite (star-finite, discrete, etc.) collections ${\mathcal{U}}_i$, then $\mathcal{U}$ is called a $\sigma$-locally finite ($\sigma$-star-finite, $\sigma$-discrete, etc.) collection. The topological spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ has strong inductive dimension $-1$, i.e., $\mathrm{Ind}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)=-1$, if $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)=\varnothing$. If for any disjoint closed sets F and G of the topological spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ there exists an open set U such that $F\subset U\subset \left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)-G$, and $\mathrm{Ind}B\left(U\right)\le n-1$, where $B\left(U\right)$ denotes the boundary of U, then $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ has strong inductive dimension $\le n$, i.e., $\mathrm{Ind}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)\le n$. Let V is an open set and $\overline{V}$ is a closed set of $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$. If $\mathrm{Ind}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)\le n$, then there exists a $\sigma$-locally finite open basis $\mathfrak{B}$ of $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ such that, $\mathrm{Ind}B\left(V\right)\le n-1$ for every $V\in \mathfrak{B}$. If a spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ has a $\sigma$-locally finite open basis $\mathfrak{B}$ such that, $B\left(V\right)=\varnothing$ for every $V\in \mathfrak{B}$, then $\mathrm{Ind}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)\le 0$. Again, $\mathrm{Ind}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)\le n$ if and only if there exists a $\sigma$-locally finite open basis $\mathfrak{B}$ such that $\mathrm{Ind}B\left(V\right)\le n-1$ for every $V\in \mathfrak{B}$. For every subset $A=\bigcup \left\{B\left(V\right)V\in \mathfrak{B}\right\}$, for any integer $n\ge 0$, of a spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$, we have, $\mathrm{Ind}A\le \mathrm{Ind}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$. Hence, if and only if $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)={\bigcup }_{i=1}^{n+1}{A}_i$ for some $n+1$ subsets $A_i$ with $\mathrm{Ind}{A}_i\le 0$, $i=1,\dots ,n+l$. For the spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$, we have then $\mathit{dim}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)=\mathrm{Ind}\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$. Let A be a subset of a spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ and I the unit segment. If U is an open set of the topological product $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)\times I$ such that $U\supset A\times I$, then there exists an open set V of $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ such that $A\subset V$, and $V\times I\subset U$. Let F be a closed set of $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ with $\mathit{dim}F\le n$. Let $F_{\alpha }$ and $U_{\alpha }$, $\alpha <\tau$, be closed and open sets, respectively such that $F_{\alpha }\subset U_{\alpha }$, and $\{U_{\alpha }\alpha <\tau \}$ is locally finite. Then there exist open sets $V_{\alpha }$ satisfying $F_{\alpha }\subset V_{\alpha }\subset {\overline{V}}_{\alpha }\subset U_{\alpha }$, and $\mathit{dim}{B}_k\le n-k$, $k=1,\dots ,n+1$, where, $B_k=\{pp\in F,{\mathrm{ord}}_{p}B\left(\mathfrak{B}\right)\ge k\}$, and $\mathfrak{B}=\{V_{\alpha }\alpha <\tau \}$. Let F, $F_{\alpha }$ and $U_{\alpha }$ satisfy the same condition as above, then there exist open sets $V_{\alpha }$, $W_{\alpha }$, $\alpha <\tau$ satisfying, $F_{\alpha }\subset V_{\alpha }\subset {\overline{V}}_{\alpha }\subset W_{\alpha }\subset U_{\alpha }$, and ${\mathrm{ord}}_p\left\{W_{\alpha }-{\overline{V}}_{\alpha }\alpha <\tau \right\}\le n$ for every $p\in F$. Let $G_k$, $k=0,\dots ,n$, be closed sets with $\mathit{dim}{G}_k\le n-k$ of the spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$. Let $\{F_{\alpha }\alpha <\tau \}$ be a closed collection and $\{U_{\alpha }\alpha <\tau \}$ a locally finite open collection such that $F_{\alpha }\subset U_{\alpha }$. Then there exists an open collection, $\mathfrak{B}=\{V_{\alpha }\alpha <\tau \}$, such that, $F_{\alpha }\subset V_{\alpha }\subset {\overline{V}}_{\alpha }\subset U_{\alpha }$, and ${\mathrm{ord}}_{p}B\left(\mathfrak{B}\right)\le n-k$ for every $p\in G_k$. A mapping f of the spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ into a spacetime S is a closed (open) mapping if the image of every closed (open) set of $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ is closed (open) in S. Then the continuous mappings which lower dimensions of the spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ should be defined as follows,

Since, the temporal axis is unaltered in Lorentz transformation, as we have already seen it in (14), we can express the maximal continuous mapping of the $(3+1)D$ spacetime $\left({\widehat{\partial }}_{[0,i]}^2\otimes {\Psi }_{[i,0]}\right)$ onto the spacetime S of (26) as, since i should not be zero in (25), if the considered state is not vacuum; then the spacetime S definitely intends the basic structure of a 2-dimensional worldsheet $X^{\mu }(\tau ,\sigma )$ with the joint states, $\frac{1}{\sqrt{2}}\left\{\frac{{\partial }^2}{\partial {\tau }^2}\left[-{\psi }_{\sigma }\right]-\frac{{\partial }^2}{\partial {\sigma }^2}\left[{\psi }_{\tau }\right]\right\}$ for the spacetime ${\mathfrak{X}}^{\mu }$, where S is a $(1+1)D$ spacetime, but for the space K, we will like to discuss it below in more details in the Theorem 2. Obviously, a string can sweep out the 2-dimensional worldsheet $X^{\mu }(\tau ,\sigma )$ for the spacetime ${\mathfrak{X}}^{\mu }$.