Stochastic metric space and quantum mechanics

The hydrogen atom is the first and simplest system to which QM was successfully used to provide a quantitative structural explanation. In 1947, W E Lamb and R C Retherford reported an energy difference between the states, ${}^{2}{S}_{1/2}$ and ${}^{2}{P}_{1/2}$ [7] that does not appear in the lowest perturbative solution of the Dirac equation. This energy shift is referred to the Lamb shift. Eventually, quantum electrodynamics (QED) would be used as a successful application of renormalization calculus to calculate a precise value of the Lamb shift as a higher order correction. Before renormalization calculus was established, however, in 1948 T A Welton gave an intuitive explanation [8] of the Lamb shift as the mean square amplitude of an electron coupled with the zero-point fluctuations of an electromagnetic field. Welton used the following method to derive the Lamb shift. The potential energy, V(r), is given as an isotropic function of the distance $r=| {\boldsymbol{r}}|$, where ${\boldsymbol{r}}$ is the three-dimensional vector of the electron's position relative to the proton. The potential energy can be Taylor-expanded with respect to the small displacement of the electron position ${\boldsymbol{r}}\to {\boldsymbol{r}}+\delta {\boldsymbol{r}}$ as;

$$\begin{eqnarray*}&&V({\boldsymbol{r}}+\delta {\boldsymbol{r}})=\left(1+\delta {\boldsymbol{r}}\cdot {\rm{\nabla }}+\displaystyle \frac{1}{2}{(\delta {\boldsymbol{r}}\cdot {\rm{\nabla }})}^{2}+\cdots \right)V({\boldsymbol{r}}).\end{eqnarray*}$$

When the displacement is identified as a variance of the electron position arising from the quantum fluctuation, an averaged value with respect to the vacuum state can be estimated as;

$$\begin{eqnarray*}\langle \delta {\boldsymbol{r}}\cdot {\rm{\nabla }}\rangle & = & 0,\\ \langle {(\delta {\boldsymbol{r}}\cdot {\rm{\nabla }})}^{2}\rangle & = & \displaystyle \frac{1}{3}\langle \delta {{\boldsymbol{r}}}^{2}\rangle {{\rm{\nabla }}}^{2},\end{eqnarray*}$$

where the factor of 1/3 in the second relation comes from averaging over the three spatial dimensions. The hydrogen atom has a Coulomb potential energy V(r) = −α/r, where α = 1/137.0359895 is the fine structure constant at the low energy limit. The average of the potential energy fluctuation can be written as

$$\begin{eqnarray*}&&\langle \delta V\rangle =\langle V({\boldsymbol{r}}+\delta {\boldsymbol{r}})-V({\boldsymbol{r}})\rangle =\displaystyle \frac{1}{6}\langle {(\delta {\boldsymbol{r}})}^{2}\rangle {\left\langle {\rm{\Delta }}\left(-\displaystyle \frac{\alpha }{r}\right)\right\rangle }_{H},\end{eqnarray*}$$

where $\langle \bullet {\rangle }_{H}$ is an average over the hydrogen atom. The average of a square fluctuation of the electron's position can be estimated as;

$$\begin{eqnarray*}&&\langle {(\delta {\boldsymbol{r}})}^{2}\rangle =\displaystyle \frac{2{\alpha }^{3}{a}_{B}^{2}}{\pi }{\int }_{{k}_{0}}^{{k}_{1}}\displaystyle \frac{{dk}}{k},\end{eqnarray*}$$

where a_B = 1/(m_eα) is the Bohr radius of the hydrogen atom and m_e is the electron mass. The upper and lower bounds of the integration must be fixed according to quantum field theoretic considerations. The average of the Laplacian on the potential energy over the hydrogen atom can be estimated as

$$\begin{eqnarray*}&&{\left\langle {\rm{\Delta }}\left(-\displaystyle \frac{\alpha }{r}\right)\right\rangle }_{H}=4\pi \alpha {| \psi (0)| }^{2},\end{eqnarray*}$$

where ψ(r) is the wave function of the hydrogen atom. Here well-know relation ${\rm{\Delta }}(1/r)=-4\pi \delta ({\boldsymbol{r}})$ is used. The average value of the quantum fluctuation of the potential energy can be obtained as;

$$\begin{eqnarray*}&&\langle \delta V\rangle =\displaystyle \frac{4}{3}{\alpha }^{4}{a}_{B}^{2}| \psi (0){| }^{2}\mathrm{log}\displaystyle \frac{{k}_{1}}{{k}_{0}}.\end{eqnarray*}$$

The S-wave solution for the hydrogen atom at the origin is then given as;

$$\begin{eqnarray*}&&{\psi }_{n}(0)={[\pi {({{na}}_{B})}^{3}]}^{-1/2},\end{eqnarray*}$$

where n is the main quantum number. From field theoretic considerations, the lower and upper cutoffs of the electron momentum are chosen to be ${k}_{0}={\alpha }^{2}{m}_{e}$ and k₁ = m_e respectively [9]. To enable appropriate setting of these cutoffs, UV and IR divergences are avoided in this method. Finally, the deviation of the potential energy for the S-wave solution with the main quantum number n is obtained as;

$$\begin{eqnarray*}&&\langle \delta V\rangle =\displaystyle \frac{4{\alpha }^{4}}{3\pi {n}^{3}{a}_{B}}\mathrm{log}\displaystyle \frac{1}{{\alpha }^{2}}.\end{eqnarray*}$$

This result is consistent with a more precise result [9] obtained using perturbative QED, which is;

$$\begin{eqnarray*}&&\langle \delta {E}_{{QED}}\rangle =\displaystyle \frac{4{\alpha }^{4}}{3\pi {n}^{3}{a}_{B}}\left(\mathrm{log}\displaystyle \frac{1}{{\alpha }^{2}}+{L}_{n}+\displaystyle \frac{19}{30}\right).\end{eqnarray*}$$

Where the values of the correction term, L_n, are given in table 1.

Table 1.  Numerical values of the correction term, L_n [9].

n	1	2	3	4	$\infty$
Ln	−2.984	−2.812	−2.768	−2.750	−2.721

Welton's method suggests that the quantum effects on the hydrogen energy levels are caused by the variance of the electron position. This scenario can be realized more naturally using the stochastic metric space. As is well known in quantum field theory, the lowest result in the perturbative calculations is the same as the result of classical mechanics, and then a quantum effect will appear starting from one loop corrections. Let us apply a stochastic method on the lowest Bohr solutions of the hydrogen energy levels.

From the stochastic metrical point of view, the distance between two points (the proton and electron positions) is given as a stochastic variable. When distances are given stochastically, their distribution must be Gaussian following the central limit theorem, variance of the distance possibly be proportional to distance. Here, we assume a variance ${\sigma }_{n}^{2}$, given as;

$$\begin{eqnarray}&&{\sigma }_{n}^{2}=\displaystyle \frac{{a}_{B}^{2}}{3n}{\left(\displaystyle \frac{\alpha }{\pi }\right)}^{2},\end{eqnarray} \tag{ 1 }$$

where the factor 1/3 reflects the dimensionality of space, the standard deviation of the Gaussian distribution is taken as proportional to ${a}_{B}/\sqrt{n}$, and a verification of the factor (α/π)² will be given later in this section. Hereafter, the atomic unit a_B = 1 will be used in this section. The statistical fluctuation of the potential energy can be written as;

$$\begin{eqnarray*}&&\delta V=V(r(1+{\delta }_{r}))-V(r),=-\displaystyle \frac{\alpha }{r}{(1+{\delta }_{r})}^{-1}+\displaystyle \frac{\alpha }{r},=-\displaystyle \frac{\alpha }{r}(-{\delta }_{r}+{\delta }_{r}^{2}-{\delta }_{r}^{3}+\cdots ).\end{eqnarray*}$$

The relative deviation is then obtained as

$$\begin{eqnarray*}&&\displaystyle \frac{\delta V}{V}=-{\delta }_{r}+{\delta }_{r}^{2}+{ \mathcal O }({\delta }_{r}^{3}).\end{eqnarray*}$$

When the distance r is given stochastically, the fluctuation δ_r should have a Gaussian distribution. Thus, the average value of the relative deviation of the potential energy is given as;

$$\begin{eqnarray}&&\left\langle \displaystyle \frac{\delta V}{V}\right\rangle =\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{n}}{\displaystyle \int }_{-\infty }^{+\infty }d{\delta }_{r}\displaystyle \frac{\delta V}{V}\exp \left(-\displaystyle \frac{{\delta }_{r}^{2}}{2{\sigma }_{n}^{2}}\right)={\sigma }_{n}^{2}+{ \mathcal O }({\sigma }_{n}^{4}),\end{eqnarray} \tag{ 2 }$$

where σ_n is given by (1). In reality, the Gaussian mean of the reciprocal variable

$$\begin{eqnarray}&&\left\langle \displaystyle \frac{1}{r}\right\rangle =\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{n}}{\int }_{0}^{+\infty }{dr}\displaystyle \frac{1}{r}\exp \left(-\displaystyle \frac{{(r-{\bar{r}}_{n})}^{2}}{2{\sigma }_{n}^{2}}\right),\end{eqnarray} \tag{ 3 }$$

does not exist because the integration on the right-hand side does not converge. Here, ${\bar{r}}_{n}$ is the mean radius of S-wave hydrogen with the main quantum number n, which is considered to correspond to the UV divergence of the QED because this divergence comes from the short distance limit $r\to 0$. In our calculation, the divergence is avoided by looking at variances only around the average value. If the integration in (3) is performed numerically only around the mean value (from which the main contribution must come), results consistent with (2) can be obtained.

We can now compare the numerical results of the stochastic metric (2) with those of QED. Because the S-wave hydrogen energy levels at the Born level are $\langle {E}_{{QED}}\rangle =-\alpha {a}_{B}/2{n}^{2}$, the relative correction on the hydrogen energy can be given as;

$$\begin{eqnarray*}&&{\delta }_{{QED}}=\left|\displaystyle \frac{\langle \delta {E}_{{QED}}\rangle }{\langle {E}_{{QED}}\rangle }\right|=\displaystyle \frac{8{\alpha }^{3}}{3\pi n}\left(\mathrm{log}\displaystyle \frac{1}{{\alpha }^{2}}+{L}_{n}+\displaystyle \frac{19}{30}\right).\end{eqnarray*}$$

The numerical results are summarized in table 2, along with those from SMQ. A relative correction from SMQ, δ_SMQ, is defined as;

$$\begin{eqnarray*}&&{\delta }_{{SMQ}}=\sqrt{2}\left\langle \displaystyle \frac{\delta V}{V}\right\rangle ,\end{eqnarray*}$$

in which the factor $\sqrt{2}$ appears because two independent statistical fluctuations of δ V's with n = 1 and n ≥ 2 are, in principle, involved with δ_SMQ. For the calculation of δ_QED at n = 100, the value of L_n at $n=\infty$ is used. The result at n = 100 is shown to examine the asymptotic behavior of the correction at a large distance. As seen in table 2, the agreement is excellent, which justifies the assumed factor (α/π)² in the variance (1).

Table 2.  Energy corrections of S-wave hydrogen. Numerical values of δ_QED and δ_SM are obtained using QED and SMQ, respectively. As seen in the last row of the table, the agreement between the two methods, ${\delta }_{{SMQ}/{QED}}=({\delta }_{{SMQ}}-{\delta }_{{QED}})/{\delta }_{{QED}}$, is excellent.

n	2	3	4	100
δQED	1.264 × 10−6	8.473 × 10−7	6.369 × 10−7	2.557 × 10−8
${\delta }_{{SMQ}}$	1.272 × 10−6	8.478 × 10−7	6.359 × 10−7	2.543 × 10−8
δSMQ/QED	+0.64%	+0.07%	−0.17%	−0.53%

Here basic definitions of the SM-space are introduced based primarily on S̆erstnev [13], Schweitzer and Sklar [15]. The distribution and triangle functions play essential roles in characterizing the SM-space. A distribution function F, is a non-decreasing function defined as a map such that;

$$\begin{eqnarray*}&&F\,:{\mathbb{R}}\to [0,1]\,:s\mapsto F(s),\end{eqnarray*}$$

with

$$\begin{eqnarray*}&&F(-\infty )=0,\,F(\infty )=1,\,\,\mathrm{and}\,s\lt t\Rightarrow F(s)\leqslant F(t).\end{eqnarray*}$$

The distribution function satisfying F(0) = 0, which is referred to as the distance distribution function, plays the role of a cumulative (probability) distribution function to obtain a distance, s. A set of distance distribution functions is denoted as Δ⁺, and the triangle function, T, is introduced to give a natural extension of the triangle inequality on Euclidean space. T is defined as a map such as;

$$\begin{eqnarray*}&&T\,:{{\rm{\Delta }}}^{+}\otimes {{\rm{\Delta }}}^{+}\to {{\rm{\Delta }}}^{+}\,:\{F,G\}\mapsto T(F,G),\end{eqnarray*}$$

which satisfies the following for any distribution functions F, G, H, K ∈ Δ⁺,

• 1.  
  T(F, G) = T(G, F);
• 2.  
  T(F, G) ≤ T(H, K), whenever F ≤ H, and G ≤ K ;
• 3.  
  T(F, Θ) = F, where Θ(s) is the Heaviside unit function. Note that Θ(s) ∈ Δ⁺ ;
• 4.  
  T(T(F, G), H) = T(F, T(G, H)).

Here, F ≤ H means F(s) ≤ H(s) for any s > 0. The SM-space can be defined as follows using the above two functions.

Definition 3.1 A stochastic metric space is a triple $({ \mathcal S },{\mathscr{F}},T)$ of

• ${ \mathcal S }$: a set of points on the space;
• ${\mathscr{F}}$: a map ${\mathscr{F}}:{ \mathcal S }\otimes { \mathcal S }\to {{\rm{\Delta }}}^{+}$, and;
• T: a triangle function,

which satisfies for any points $p,q,r\in { \mathcal S }$ and $s,s^{\prime} \in {\mathbb{R}};$

• 1.  
  ${\mathscr{F}}(p,p)(s)={\rm{\Theta }}(s);$
• 2.  
  ${\mathscr{F}}(p,q)(s)\ne {\rm{\Theta }}(s)$, if $p\ne q;$
• 3.  
  ${\mathscr{F}}(p,q)(s)={\mathscr{F}}(q,p)(s)$, and;
• 4.  
  ${\mathscr{F}}(p,r)(s+s^{\prime} )\geqslant T({\mathscr{F}}(p,q)(s),{\mathscr{F}}(q,r)(s^{\prime} ))$.

If Requirement 2 is omitted, the space is referred to as a stochastic pseudometric space. Requirement 4 is a type of triangle inequality originally introduced by Menger [10].

To utilize the SM-space for relativistic field theories, the stochastic metric should be introduced on a Minkowski manifold. The Minkowski manifold equipping the stochastic metric is referred to as the stochastic Lorentz metric space (SLM-space) hereafter. A distance between two points on the SLM-space will fluctuate around the geometric distance measured by the Lorentz metric, with the distance measured by the (non-fluctuating) Lorentz metric referred here as the geometrical distance. A distribution function is required to give the geometrical distance as an average value over a two-point ensemble on the SLM-space, with the variance of the distribution function set to be proportional to its geometrical distance. This distribution function makes a null vector (vector with zero length) without any fluctuation, a desirable characteristic for restraining a null photon mass after quantum corrections. Furthermore, to satisfy the causality condition the probability changing a sign of the length of a string stretching between two points must be zero. The SLM-space satisfying above requirements can be introduced as follows:

Definition 3.2 A stochastic Lorentz-metric-space (SLM-space) is a triple of $({ \mathcal M },{F}_{e},T)$ such that:

• 1.  
  ${ \mathcal M }$ is a 4-dimensional Minkowski manifold with a metric tensor ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1).$ The geometrical distance between two points on ${ \mathcal M }$, say ${x}^{\mu }$ and ${y}^{\mu }$ in the local coordinate system is defined as

  $$\begin{eqnarray*}&&{ \mathcal D }(x-y)=\sqrt{| {\eta }_{\mu \nu }{(x-y)}^{\mu }{(x-y)}^{\nu }| },\end{eqnarray*}$$

  using the Lorentz-metric tensor. In this work, the Einstein convention to take the sum over repeated indices is used. The indices run from zero to three.
• 2.  
  ${F}_{e}(s;d)$ is a map of $\{d;s\}\mapsto {F}_{e}(s;d)\in [0,1],$ where ${F}_{e}$ is chosen as the following exponential probability density;

  $$\begin{eqnarray*}{F}_{e}(s;d)=\left\{\begin{array}{ll}0 & s\lt 0,\\ 1-\exp \left(-\tfrac{s}{d}\right) & s\geqslant 0.\end{array}\right.\end{eqnarray*}$$
• 3.  
  $T$ is the triangle function defined as

  $$\begin{eqnarray*}&&T({F}_{e}(s;{ \mathcal D }(x-y)),{F}_{e}(s^{\prime} ;{ \mathcal D }(y-z)))={F}_{e}(s;{ \mathcal D }(x-y)){F}_{e}(s^{\prime} ;{ \mathcal D }(y-z)),\end{eqnarray*}$$

  for any $x,y,z\in { \mathcal M }$ and $s,s^{\prime} \gt 0$.

The function ${F}_{e}(s;{ \mathcal D }(x-y))$ introduced above can be interpreted as the probability of observing a distance less than s when the geometrical distance between x and y is ${ \mathcal D }(x-y)$. In other words, the probability of finding a distance greater than s ≥ 0 is ${\bar{F}}_{e}(s,d)=1-{F}_{e}(s,d)=\exp (-s/d)$ when the geometrical distance is d. The SM-space, whose distribution function ${\mathscr{F}}$ is a function of $s/{ \mathcal D }(x-y)$ only, is called the simple space [16]; the SLM-space is an example of a simple space. A probability measure can be derived from ${F}_{e}(s;{ \mathcal D }(x-y))$ as follows. The exponential probability density of a variable s with a mean value of λ is

$$\begin{eqnarray*}&&{\mu }_{e}(s;\lambda )=\displaystyle \frac{1}{\lambda }\exp \left(-\displaystyle \frac{s}{\lambda }\right){ds}.\end{eqnarray*}$$

Therefore, the probability measure corresponding to ${F}_{e}(s;{ \mathcal D }(x-y))$ in the SLM-space can be written as;

$$\begin{eqnarray*}&&{{dF}}_{e}(s;{ \mathcal D }(x-y))=\displaystyle \frac{1}{{ \mathcal D }(x-y)}\exp \left(-\displaystyle \frac{s}{{ \mathcal D }(x-y)}\right){ds}={\mu }_{e}(s;{ \mathcal D }(x-y)).\end{eqnarray*}$$

An interpretation of the probability measure ${\mu }_{e}(s;{ \mathcal D }(x-y))$ is that it is the probability of obtaining the distance within [s, s + ds] for ${ \mathcal D }(x-y)$. It can be shown that the SLM-space satisfies the above definition of a Stochastic Lorentz metric space with an additional requirement, i.e., it is a stochastic pseudometric space. Thus, we give the following remark.

Remark 3.3 The stochastic Lorentz metric-space $({ \mathcal M },{F}_{e},T)$ is a stochastic pseudometric space.

A proof of the above remark is given in appendix A. The SM-space has a rich structure [16] as follows:

• The SM-space is a topological space with a λ-neighborhood at $p\in { \mathcal M }$ as follows;
  $$\begin{eqnarray*}&&{N}_{p}(\varepsilon ,\lambda )=\{q| {F}_{e}(\varepsilon ;{ \mathcal D }(p-q))\gt 1-\lambda \},\end{eqnarray*}$$
  where ε, λ > 0.
• The SM-space is a Hausdorff space with a triangle function under the above topology.
• On the SM-space with the continuous triangle function, the convergence and continuity of the distribution function ${F}_{e}(s;{ \mathcal D }(p-q))$ is ensured.

These properties ensure that the physical theory can be constructed on the base of the SLM-space following remark 3.3.

In addition to the above properties, it is essential that the physical theory must retain causality. In relativistic theories, causality expressed by the following statement: when the (geometrical) distance between two points $p,q\in { \mathcal M }$ is space like such that $| p-q| \lt 0$, the two-point correlation function must be zero. Our choice of distribution function, ${F}_{e}(s;{ \mathcal D }(p-q))$, does not destroy causality because it does not change the sign of the distance; moreover, the existence of the triangle function ensures that there are no shortcuts between two space time points.

Consider a series of independent random variables ${\{{X}_{i}\}}_{i=1,2,\cdots }$ with a probability measure with P(0 < s < X_i < s + ds) = p(s)ds. A series of random variables ${\{{A}_{j}\}}_{j=0,1,2,\cdots }$ such as; A₀ = 0 and ${A}_{j}={\sum }_{i=1}^{j}{X}_{i}$ will (a.s.) be a the stochastic-process induced by the probability measure p(s)ds. It is known that the exponential measure induces the Poisson process; a detailed explanation of the Poisson process is given in appendix B. The following remark will play an essential role in constructing the SMQ method.

Remark 3.4 Suppose that the Poisson process is induced by the exponential measure ${\mu }_{e}(s,\lambda )$ with a series of random variables, ${\{{X}_{s}\}}_{s=1,2,\cdots }$, and

$$\begin{eqnarray*}&&{Z}_{n}=\displaystyle \frac{1}{\sqrt{n}}\displaystyle \sum _{s=1}^{n}({X}_{s}-\lambda ).\end{eqnarray*}$$

The random variable Z_n is know to show a convergence in law to a normal distribution with a mean value at zero and a variance $\lambda$. In particular, for $-\infty \lt a\lt b\lt \infty$,

$$\begin{eqnarray*}&&\mathop{\mathrm{lim}}\limits_{n\to \infty }P(a\leqslant {Z}_{n}\leqslant b)=\displaystyle \frac{1}{\sqrt{2\pi \lambda }}{\int }_{a}^{b}\exp \left(-\displaystyle \frac{{x}^{2}}{2\lambda }\right){dx},\end{eqnarray*}$$

can be obtained. Brownian motion is induced by this random variable.

The proof of this remark is given in appendix C. To summarize the above derivations, if the position of a force-free mass point at time t can be represented by a series of random variable {X_s}, the mass point can follow a Brownian motion and a set of positions, each at an equal geometrical distance, should be observed as the Poisson process in the SLM-space.

One of the more important calculations in the dynamical theory of a mass point is determining its trajectory. The trajectory may be a continuous smooth line between two space time points as would occur under a classical theory. Such trajectory, called a curvilinear path, is defined geometrically on the Minkowski manifold. In this subsection, a classical Minkowski manifold is assumed and the stochastic aspects of the manifold are suspended for the time being.

The geometrical setup introduced in this subsection is based on reference [17]. A set of bounded functions γ such that

$$\begin{eqnarray}&&\gamma \,:\tau \in [{\tau }_{1},{\tau }_{2}]\mapsto {\boldsymbol{\gamma }}(\tau )=(\tau ,{\gamma }^{1}(\tau ),{\gamma }^{2}(\tau ),{\gamma }^{3}(\tau )),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\gamma }^{i}\,:\tau \mapsto {\gamma }^{i}(\tau ),\,\,{\{{\gamma }^{i}\}}_{i=\mathrm{1,2,3}}\in {C}^{2}({\mathbb{R}}),\end{eqnarray} \tag{ 5 }$$

is called a curvilinear path (or simply path), and a set of paths, denoted by Γ, is called a curvilinear-path space. A parameter τ is an order parameter used to measure the length of a path from the starting point τ₁, which is set to zero without any loss of generality. We set τ₁ = 0 and τ₂ = T, and hereafter only paths whose end points are fixed at $\gamma (0)={\xi }_{1}^{\mu }=(0,{\xi }_{1}^{1},{\xi }_{1}^{2},{\xi }_{1}^{3})=(0,{{\boldsymbol{\xi }}}_{1})$ and $\gamma (T)={\xi }_{2}^{\mu }=(T,{\xi }_{2}^{1},{\xi }_{2}^{2},{\xi }_{2}^{3})=(T,{{\boldsymbol{\xi }}}_{2})$ are considered. The dynamics of a mass point cannot be determined using only information on the path itself; the time-derivative along the path is also necessary. A velocity vector is a tangent vector at a point γ(τ) along the curvilinear path, defined as;

$$\begin{eqnarray}&&\displaystyle \frac{d\gamma }{d\tau }(\tau )=\left(1,\displaystyle \frac{d{\gamma }^{1}}{d\tau }(\tau ),\displaystyle \frac{d{\gamma }^{2}}{d\tau }(\tau ),\displaystyle \frac{d{\gamma }^{3}}{d\tau }(\tau )\right).\end{eqnarray} \tag{ 6 }$$

Hereafter, the velocity vector is written using shorthand notation as $\dot{{\boldsymbol{\gamma }}}(\tau )=(1,{\dot{\gamma }}^{1}(\tau ),{\dot{\gamma }}^{2}(\tau ),{\dot{\gamma }}^{3}(\tau ))$. The velocity vector can be expressed in terms of natural bases on a tangent space T_τγ at γ(τ) as;

$$\begin{eqnarray}&&{\dot{{\boldsymbol{\gamma }}}}^{i}(\tau )=\displaystyle \frac{d{\gamma }^{i}(\theta )}{d\theta }{\left.\displaystyle \frac{\partial }{\partial {\gamma }^{i}}\right|}_{\theta =\tau },\end{eqnarray} \tag{ 7 }$$

where i runs from 1 to 3 and is not summed in the right-hand side of (7). A tangent vector bundle $\dot{{\rm{\Gamma }}}={\bigcup }_{\gamma (\tau )\subset \gamma \in {\rm{\Gamma }}}{T}_{t}\gamma$ is referred to a velocity bundle.

All the information needed to determine the dynamics of the system is contained in the Lagrangian, which can be defined as a map from $\{{\rm{\Gamma }},\dot{{\rm{\Gamma }}}\}$ to a smooth function such that

$$\begin{eqnarray}&&{ \mathcal L }\,:{\rm{\Gamma }}\otimes \dot{{\rm{\Gamma }}}\to {C}^{\infty }\,:\{\gamma ,\dot{\gamma }\}\mapsto { \mathcal L }(\gamma ,\dot{\gamma }).\end{eqnarray} \tag{ 8 }$$

A new dynamical object, the canonical momentum, can be derived from the Lagrangian as;

$$\begin{eqnarray}&&{\pi }^{i}=\displaystyle \frac{\delta { \mathcal L }(\gamma ,\dot{\gamma })}{\delta {\dot{\gamma }}_{i}},\end{eqnarray} \tag{ 9 }$$

where δ denotes the functional derivative. A tuple of ${\mathscr{G}}=\{\gamma ,\pi \}$ is referred to as a phase space.

The Hamiltonian can be derived from the Lagrangian and the canonical momentum as;

$$\begin{eqnarray*}&&{ \mathcal H }\,:{\mathscr{G}}\to {C}^{\infty }\,:\{\gamma ,\pi \}\mapsto { \mathcal H }(\gamma (t),\pi (t)),\end{eqnarray*}$$

where

$$\begin{eqnarray}&&{ \mathcal H }(\gamma (t),\pi (t))={({\pi }_{i}{\dot{\gamma }}^{i}-{ \mathcal L }(\gamma ,\dot{\gamma }))| }_{\dot{\gamma }=\phi }={\pi }_{i}{\phi }^{i}(\gamma ,\pi )-{ \mathcal L }(\gamma ,\phi (\gamma ,\pi )).\end{eqnarray} \tag{ 10 }$$

The function ϕⁱ(γ, π) gives the solutions of

$$\begin{eqnarray}&&{\pi }_{i}-\displaystyle \frac{\delta { \mathcal L }}{\delta {\dot{\gamma }}^{i}}=0,\,\,\,\,i=1,2,3,\end{eqnarray} \tag{ 11 }$$

with respect to the velocity vector of ${\dot{\gamma }}^{i}={\phi }^{i}(\gamma ,\pi )$. Here, the Lagrangian is assumed to be a holomorphic function and therefore a solution ϕⁱ(γ, π) always exists and the Hamiltonian is defined on the phase space. The existence of the inverse Legendre transform is ensured because

$$\begin{eqnarray}&&{ \mathcal L }(\gamma ,\dot{\gamma })={\tilde{\phi }}_{i}(\gamma ,\dot{\gamma }){\dot{\gamma }}^{i}-{ \mathcal H }(\gamma ,\tilde{\phi }(\gamma )),\end{eqnarray} \tag{ 12 }$$

where $\tilde{\phi }$ is a solution of the equation of ${\dot{\gamma }}^{i}-\delta { \mathcal H }/\delta {\pi }_{i}=0$, that is

$$\begin{eqnarray}&&{\dot{\gamma }}^{i}={\left.\displaystyle \frac{\delta { \mathcal H }}{\delta {\pi }_{i}}\right|}_{\pi =\tilde{\phi }}.\end{eqnarray} \tag{ 13 }$$

The action integral can be introduced using the Lagrangian or Hamiltonian as follows: a map ${\mathscr{I}}(\gamma )$ from a path γ to a real number ${\mathbb{R}}$ such as;

$$\begin{eqnarray*}&&{\mathscr{I}}\,:{\rm{\Gamma }}\otimes \dot{{\rm{\Gamma }}}\to {\mathbb{R}}\,:\{\gamma ,\dot{\gamma }\}\mapsto {\mathscr{I}}(\gamma ,\dot{\gamma }),\end{eqnarray*}$$

where

$$\begin{eqnarray}&&{\mathscr{I}}(\gamma ,\dot{\gamma })={\int }_{0}^{T}d\tau \,{ \mathcal L }(\gamma (\tau ),\dot{\gamma }(\tau ))={\left.{\int }_{0}^{T}d\tau ({\pi }_{i}{\dot{\gamma }}^{i}-{ \mathcal H }(\gamma ,\pi ))\right|}_{\pi =\tilde{\phi }}\end{eqnarray} \tag{ 14 }$$

is referred to an action integral or simply an action. As is well known, the curvilinear path that gives $\delta {\mathscr{I}}=0$ satisfies the canonical equations of motion;

$$\begin{eqnarray}\displaystyle \frac{d{\gamma }_{c}^{i}}{d\tau } & = & \displaystyle \frac{\partial { \mathcal H }}{\partial {\pi }_{{ci}}},\,\,\,\,\displaystyle \frac{d{\pi }_{c}^{i}}{d\tau }=-\displaystyle \frac{\partial { \mathcal H }}{\partial {\gamma }_{{ci}}},\\ \displaystyle \frac{d{ \mathcal H }}{d\tau } & = & \displaystyle \frac{\partial { \mathcal H }}{\partial \tau }.\end{eqnarray} \tag{ 15 }$$

A solution of the equation of motion is called the classical path and denoted by γ_c(τ). The classical velocity and momentum can be further defined from the solution of the equation of motion and denoted as ${\dot{\gamma }}_{c}$ and π_c, respectively.

In the Lagrangian formalism of classical mechanics, the principle of least action (Hamilton's principle) is the most fundamental principle. The behavior of a dynamical system can be described by the Euler–Lagrange, or canonical equations of motion; or the Hamilton–Jacobi equation, which can be derived from the Lagrangian, Hamiltonian or action integral, respectively. In the traditional method, QM is introduced only after establishing the classical dynamics of the system by requiring some quantization conditions on the system. However, if QM is the most fundamental theory of nature, a dynamical system should be formulated using QM as a precedential theory to classical mechanics. Following this, some appropriate approximation can be used to derive classical dynamics from the quantum system. Here, we propose a new algorithm establishing a quantum system preceding a classical system. (There is another approach to extract quantum mechanics directly using the principle of least action [18, 19]). As our approach is based on the stochastic property of the metric, the principle of least action is not the most fundamental principle but is instead a theorem that can be derived from a more fundamental stochastic principle. To replace the principle of least action, we propose using the entropy-extremal principle to extract dynamics from the system that equips the Lagrangian.

In the following, we first show that path-integral quantization can be deduced from the entropy-extremal principle. Then, yet another method—Newton–Nelson equation [20]—to extract the quantum equation of motion is given for the SLM-space.

4.2.1. Path integral on the SLM-space

A mass point traveling in the SLM-space will follow a trajectory that, owing to the probabilities of the SLM-space, would be expected to be a stochastic process. Such a trajectory cannot be a smooth curvilinear path as defined in the previous subsection; correspondingly, the instantaneous velocity of the mass point will not exist in classical sense and must be redefined in terms of probability theory.

Let us consider a following integration:

$$\begin{eqnarray}&&{\gamma }_{c}^{i}(\tau )-{\gamma }_{c}^{i}(0)={\int }_{\gamma (0)}^{\gamma (\tau )}d{\gamma }_{c}^{i}={\int }_{0}^{\tau }{\dot{\gamma }}_{c}^{i}(t){dt},\end{eqnarray} \tag{ 16 }$$

where i = 1, 2, 3 is the spatial components of the vectors. The parameter τ is an ordering parameter along the curvilinear path, functioning as, e.g. tick marks along the path. We then consider the integration of a bounded function, f(x) ∈ C² along the classical path;

$$\begin{eqnarray}&&I={\int }_{\gamma (0)}^{\gamma (\tau )}f({\gamma }_{c})| d{\gamma }_{c}| ={\int }_{0}^{\tau }f({\gamma }_{c}(t))| {\dot{\gamma }}_{c}(t)| {dt},\end{eqnarray} \tag{ 17 }$$

where the integration measure is given as;

$$\begin{eqnarray}| d{\gamma }_{c}| & = & \sqrt{| {\eta }_{\mu \nu }d{\gamma }_{c}^{\mu }d{\gamma }_{c}^{\nu }| \,}=\sqrt{| {\eta }_{\mu \nu }{\dot{\gamma }}_{c}^{\mu }(t){\dot{\gamma }}_{c}^{\nu }(t)| \,}\,{dt},\\ & = & | {\dot{\gamma }}_{c}(t)| {dt}.\end{eqnarray} \tag{ 18 }$$

We note that these lengths are also bounded under classical mechanics, and, therefore, that integrations (16) and (17) are well-defined as Riemann-Stieltjes integrations. The classical path and velocity, γ_c(τ) and ${\dot{\gamma }}_{c}(\tau )$, respectively, are functions of τ under the Lagrangian formalism. On the SLM-space, the trajectory of a mass point is cast as a stochastic process induced by the exponential measure; therefore, the motion of a mass point in the SLM-space can be treated as Brownian motion following remark 3.4. We consider, e.g. a mass point stopping at the origin of the coordinate system in classical mechanics. The mass point remains at the same spatial position for a length of time τ. The geometrical distance between the initial and final points is also τ. In the SLM-space, distance is a random variable and has a Gaussian distribution with a standard deviation proportional to $\sqrt{\tau }$ following the stipulation of remark 3.4 that there is no motion other than Brownian motion.

The quantum effects on the mass point appear because of the stochastic treatment for the path. The classical path and velocity are replaced by random variables induced by the exponential measure of the SLM-space. The Itô process, defined in definition D.2 in appendix D, is introduced as

$$\begin{eqnarray}&&{\gamma }_{\omega }^{i}(\tau )={\gamma }_{c}^{i}(0)+{\int }_{0}^{\tau }{\dot{\gamma }}_{\omega }^{i}(t){dt}={\gamma }_{c}^{i}(0)+{\int }_{0}^{\tau }{\sigma }_{k}^{i}(t)\cdot {{dB}}^{k}(t),\end{eqnarray} \tag{ 19 }$$

in analogy with equation (16). A precise definition of (19) can be found in appendix D. We note that the initial point is fixed a.s. at ${\gamma }_{\omega }^{i}(0)={\gamma }_{c}^{i}(0)$. Here, γ_ω(τ) is referred to as the sample path. The sample path is an element of Γ⁰, a set of continuous lines between γⁱ(0) and γⁱ(T) in which the condition C² is relaxed to C⁰ from a definition of the curvilinear path (5). Moreover, ${\gamma }_{\omega }(\tau )$ is unbounded owing to the Brownian motion B^k(s), as will be discussed below. The random variable ${\dot{\gamma }}_{\omega }(\tau )$ must also be treated as a stochastic process. B^k(s) (k = 1, 2, 3) are three independent Brownian motions and ${\sigma }_{k}^{i}(s)={\delta }_{k}^{i}\sigma (s)$ is a square integrable function. The Brownian motion B(s) is differentiable nowhere, and the integration measure;

$$\begin{eqnarray*}&&{{dB}}^{k}(s)=\displaystyle \frac{{{dB}}^{k}(s)}{{ds}}{ds}={\dot{B}}^{k}{ds},\end{eqnarray*}$$

must be understood as the Gaussian white noise (Hida-distribution) introduced by Hida [21]. From definition D.2, the correspondence is then as $\{{\gamma }_{\omega }^{i}(\tau ),{\dot{\gamma }}_{\omega }^{i}(\tau )\}\iff \{{X}_{t}^{i},{\beta }_{s}^{i}\}$. Note that ${\dot{\gamma }}_{\omega }$ and ${\mathop{\gamma }\limits^{\prime\prime}}_{\omega }$ do not denote dγ_ω/dτ and $d{\dot{\gamma }}_{\omega }/d\tau$, respectively, but are random variable independent of γ_ω and ${\dot{\gamma }}_{\omega }$, respectively. The stochastic-space –> stochastic space induced by the random variable γ_ω defined by (19) becomes the Brownian motion, which is ensured in the SLM-space following remark 3.4. Therefore, the phase space $\{{\gamma }_{\omega }^{i}(\tau ),{\dot{\gamma }}_{\omega }^{i}(\tau )\}$ can be understood as a pair of random variables representing a sample trajectory of a mass point on the SLM-space, while integration (19) must be understood as a stochastic integration. The convergence of these integrations are not trivial, unlike the convergence of integration (17).

The stochastic representation of the canonical momentum π_ω is also introduced as;

$$\begin{eqnarray}&&{\pi }_{\omega }={\left.\displaystyle \frac{\delta { \mathcal L }}{\delta {\dot{\gamma }}_{c}}\right|}_{c\to \omega }.\end{eqnarray} \tag{ 20 }$$

Here, $\bullet {| }_{c\to \omega }$ replacing an operation (functional variation) with a corresponding stochastic process. In the SLM-space, the Lagrangian must be understood to be a random variable defined as ${{ \mathcal L }}_{\omega }={ \mathcal L }({\gamma }_{\omega },{\dot{\gamma }}_{\omega })(\tau );$ therefore, the action integral can be considered to be a random variable as follows;

$$\begin{eqnarray}&&{{\mathscr{I}}}_{\omega }={\int }_{0}^{T}d\tau \,{ \mathcal L }({\gamma }_{\omega }(\tau ),{\dot{\gamma }}_{\omega }(\tau )).\end{eqnarray} \tag{ 21 }$$

This integration cannot be a Lebesgue-Stieltjes integration because the stochastic process ${ \mathcal L }({\gamma }_{\omega },{\dot{\gamma }}_{\omega })$ is a continuous but unbounded function. If it exists, the expected value of this random variable might coincide with the classical action. The expected value is formally written using (stochastic) integration over all elements of a set of possible sample paths, which are denoted as ${{\mathscr{B}}}_{{\rm{\Gamma }}}$. The existence of the Lebesgue measure on such an infinite dimensional space cannot be expected in general; a mathematical treatment of the integration on ${{\mathscr{B}}}_{{\rm{\Gamma }}}$ will be discussed later.

The classical path is obtained as the expected values of the stochastic process of the γ_ω as follows:

$$\begin{eqnarray*}&&{{\mathscr{I}}}_{c}={E}_{\omega }[{{\mathscr{I}}}_{\omega }| {{\mathscr{B}}}_{{\rm{\Gamma }}}]\,=\,\displaystyle \sum _{{\gamma }_{\omega }\in {\rm{\Gamma }}}{| \psi ({\gamma }_{\omega })| }^{2}{\mathscr{I}}({\gamma }_{\omega })={\int }_{0}^{T}d\tau \,{ \mathcal L }({\gamma }_{c}(\tau ),{\dot{\gamma }}_{c}(\tau )),\end{eqnarray*}$$

where ψ(γ_ω) is the probability amplitude, which is introduced according to Definition 4.1 in [17]. Under this setup, an additional principle—the extremal entropy principle—can be stated: the quantum probability amplitude and probability measure that extremalize the entropy;

$$\begin{eqnarray}&&S=-\displaystyle \sum _{{\gamma }_{\omega }\in {\rm{\Gamma }}}{| \psi ({\gamma }_{\omega })| }^{2}\mathrm{log}{| \psi ({\gamma }_{\omega })| }^{2},\end{eqnarray} \tag{ 22 }$$

under the constraints,

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\displaystyle \sum }_{{\gamma }_{\omega }\in {\rm{\Gamma }}}{| \psi ({\gamma }_{\omega })| }^{2}{\mathscr{I}}({\gamma }_{\omega })={{\mathscr{I}}}_{c},\\ {\displaystyle \sum }_{{\gamma }_{\omega }\in {\rm{\Gamma }}}{| \psi ({\gamma }_{\omega })| }^{2}=1,\end{array}\right.\end{eqnarray} \tag{ 23 }$$

is given by;

$$\begin{eqnarray}&&\psi ({\gamma }_{\omega })\simeq C\,{e}^{\tfrac{i}{{\hslash }}{\mathscr{I}}({\gamma }_{\omega })},\end{eqnarray} \tag{ 24 }$$

where $C\in {\mathbb{C}}$ is an appropriate normalization constant and ${\mathscr{I}}$ is rendered dimensionless by dividing by the dimensional constant ℏ. This remark is identical to Theorem 4.1 in [17], whose proof can be found in the same reference. Here, ψ gives the transition probability (propagator) from t = 0 to t = T. At this stage, ℏ is simply a constant to adjust the dimension of the argument of the exponential function. When it is taken as the Planck constant (divided by 2π), it becomes a solution of the Shrödinger equation¹ . The sum over possible paths can be interpreted as the path integral and, therefore, the maximum entropy principle has induced the path integral quantization.

On the other hand, in the context of this report, the sum must be understood as a stochastic integration such as;

$$\begin{eqnarray}&&\displaystyle \sum _{{\gamma }_{\omega }\in {\rm{\Gamma }}}\bullet \Rightarrow {E}_{\omega }[\,\bullet \,| {{\mathscr{B}}}_{{\rm{\Gamma }}}].\end{eqnarray} \tag{ 25 }$$

Therefore, the transition amplitude can be obtained from (24) as;

$$\begin{eqnarray}&&\psi (\gamma (T))=C\,{E}_{\omega }[{e}^{\tfrac{i}{{\hslash }}{\mathscr{I}}({\gamma }_{\omega })}| {{\mathscr{B}}}_{{\rm{\Gamma }}}],\end{eqnarray} \tag{ 26 }$$

where the probability that a mass point moves from γ(0) to γ(T) is given as $| \psi (\gamma (T)){| }^{2}$. As this integration is not a functional but a stochastic integration, it is not trivial that the integration (26) is equivalent to the path integral introduced by Feynman [23, 24]. Fortunately, in 1983 Hida and Streit [25, 26] rigorously proved that the path integral can be formulated by means of the Hida-distribution for some classes of potential functions. To do this, they used the following approach [27]. The path integral of a test functional f(γ) can be expressed formally as;

$$\begin{eqnarray}&&{I}_{\mathrm{path}}=C\int f(\gamma ){e}^{\tfrac{i}{{\hslash }}{\mathscr{I}}(\gamma )}{ \mathcal D }\gamma ,\end{eqnarray} \tag{ 27 }$$

where ${ \mathcal D }\gamma$ is the integration 'measure' of the functional integration, C is an appropriate normalization factor and the integration is performed over all possible paths. However, as is well known, the ${ \mathcal D }\gamma$ cannot exist as the Lebesgue measure and, therefore, the integration is not well-defined in general. To treat this functional 'measure' rigorously, they introduced white noise, as is briefly explained in example F.1 in appendix F.1. Using $1=\exp [{\dot{B}}^{2}/2]\exp [-{\dot{B}}^{2}/2]$, integration (27) can be rewritten as;

$$\begin{eqnarray}{I}_{\mathrm{path}} & = & C\displaystyle \int f(\gamma ){e}^{\tfrac{i}{{\hslash }}{\mathscr{I}}(\gamma )+\dot{B}{(\tau )}^{2}/2}{e}^{-\dot{B}{(\tau )}^{2}/2}{ \mathcal D }\gamma ,\\ & = & C^{\prime} {\displaystyle \int }_{{E}^{* }}f({\gamma }_{\omega }){e}^{\tfrac{i}{{\hslash }}{\mathscr{I}}({\gamma }_{\omega })+\dot{B}{(\tau )}^{2}/2}{\mu }_{G}(d{\gamma }_{\omega }),\end{eqnarray} \tag{ 28 }$$

where $\dot{B}(\tau )$ is a white noise defined on the dual space ${E}^{* }$ of $E={{\mathscr{B}}}_{{\rm{\Gamma }}}$ and

$$\begin{eqnarray*}&&{\mu }_{G}(d{\gamma }_{\omega })={e}^{-{\dot{B}}^{2}/2}{ \mathcal D }\gamma \end{eqnarray*}$$

is the Gaussian measure (classical Wiener measure) on ${E}^{* }$. The flattening factor $\exp ({\dot{B}}^{2}/2)$ appears in addition to the action integral. The classical path γ_c(τ) is assumed to be a square-integrable function as;

$$\begin{eqnarray*}&&{\int }_{0}^{T}| {\gamma }_{c}(\tau )| d\tau \lt \infty ,\quad {\gamma }_{c}(\tau )\in {L}_{2}([0,T],d\gamma )\end{eqnarray*}$$

by means of the norm given by (18). Therefore, the Gel'fand triple $E\subset H={L}_{2}([0,T],d\gamma )\subset {E}^{* }$ exists.

Even though the integration measure becomes well-defined as the Hida-distribution, it does not mean this integration converges, and the 'measure' $\exp ({\dot{B}}^{2}/2){\mu }_{G}$ still cannot be understood as the Lebesgue measure. It is shown that the path integral is convergent when the limit

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{{\delta }_{k}\to 0}\exp \left[c\displaystyle \sum _{k}{\left(\displaystyle \frac{{\delta }_{k}B}{{\delta }_{k}}\right)}^{2}{\delta }_{k}\right]\end{eqnarray} \tag{ 29 }$$

exists, where $c\ne 1/2$ [25]. There is a wide class of potential functions known to give convergent results [28–31].

From our standpoint that the SLM-space is the most fundamental object in nature, the stochastic-integration representation of the transition probability (26) has priority over the path integral (27). In other words, the path integral approximates the exact representation of transition amplitude and the flattening procedure is not necessary. In a conclusion, the following remark given.

(Quantum Amplitude).

Remark 4.1 [25] For the Lagrangian of

$$\begin{eqnarray}&&{ \mathcal L }=\displaystyle \frac{m}{2}{\dot{\gamma }}^{2}-V(\gamma ),\end{eqnarray} \tag{ 30 }$$

the trajectory of the mass point is expected to be (19). The quantum probability amplitude is given as;

$$\begin{eqnarray}\psi (\gamma (T)) & = & {E}_{\omega }\left[C\exp \left[\displaystyle \frac{i}{{\hslash }}\displaystyle \frac{m}{2}{\displaystyle \int }_{0}^{T}{\dot{\gamma }}_{\omega }{(t)}^{2}{dt}\right]\exp \left[-\displaystyle \frac{i}{{\hslash }}{\displaystyle \int }_{0}^{T}V({\gamma }_{\omega }(t)){dt}\right]\delta ({\gamma }_{\omega }(T)-{\gamma }_{c}(T))| {{\mathscr{B}}}_{{\rm{\Gamma }}}\right],\\ & = & \exp \left[\displaystyle \frac{i}{2}\displaystyle \frac{m}{{\hslash }}{\displaystyle \int }_{0}^{T}{\dot{\gamma }}_{c}{(t)}^{2}{dt}\right]{E}_{\omega }\left[C\exp \left[{\displaystyle \int }_{0}^{T}\displaystyle \frac{i}{2}{\dot{B}}^{2}{dt}\right]\exp \left[-\displaystyle \frac{i}{{\hslash }}{\displaystyle \int }_{0}^{T}V\left({\gamma }_{c}(t)+\sqrt{\displaystyle \frac{{\hslash }}{m}}\dot{B}\right){dt}\right]\right.\\ & & \left.\times \sqrt{\displaystyle \frac{m}{{\hslash }}}\delta \left.\left(B(t)-\sqrt{\displaystyle \frac{m}{{\hslash }}}({\gamma }_{c}(T)-{\gamma }_{c}(t))\right)\right|{{\mathscr{B}}}_{{\rm{\Gamma }}}\right],\end{eqnarray} \tag{ 31 }$$

where we set $\dot{\gamma }\dot{B}{dt}=0$. Here, Donscker's $\delta$-functional [21], $\delta (B-a)$, is used. Instead, the pinned Brownian motion defined in E.3 gives the same result.

This remark directory follows from the extremal entropy principle and is consistent with that of Hida and Streit [25] except for the omission of the flattening factor. If the factor in front of ${\dot{B}}^{2}$ is replaced as $i/2\to (1+i)/2$, the representation becomes identical to that in [25] completely (an exact definition of ${\dot{B}}^{2}$ is given in appendix F.2.1). This modification is absorbed in the normalization factor C and will give the same probability amplitude. Although the modified form of the flat 'measure' applied to the Gaussian measure might induce some effects on the path integral, it can be confirmed that the path integral with the Gaussian measure gives result consistent with those obtained using the flat measure, as is shown in appendix G. In addition, the integration using the Gaussian measure gives better convergence. The real part of the factor in front of ${\dot{B}}^{2}$ may cause divergent integrations over the path integral; however, using a standard procedure of physics this divergent part can be absorbed into a normalization constant and eliminated by redefining the normalization constant² . On the other hand, the definition of the quantum probability amplitude in (31) can be expressed as a type of stochastic integration;

$$\begin{eqnarray*}&&\int g({\gamma }_{\omega }){e}^{\tfrac{i}{{\hslash }}{\mathscr{I}}({\gamma }_{\omega })}{\mu }_{G}(d{\gamma }_{\omega }).\end{eqnarray*}$$

This integration is simply the Fourier transformation with the Gaussian measure and converges for any functions g(γ_c) if the function belongs to the Schwartz space after multiplying the Gaussian weight.

Before closing this subsection, we would like to discuss in further detail a relation between the results in this work and in [17], in which the author treated QM as a phenomenological theory instead of a fundamental theory without assuming any underlying structure, and then constructed the thermodynamics of particle trajectories that were shown to be equivalent to those under QM. In this work, the underlying structure of particle dynamics is identified as the probabilistic nature of the SLM-space. By contrast, in [17] the function $\psi (\gamma )$ is assumed to be a slow-moving function with respect to variance of γ around the classical solution. This assumption can also be justifies within the framework of this report. A set of sample paths going through a neighborhood of γ_c(τ), where τ is a fixed time in 0 < τ < T, is introduced as;

$$\begin{eqnarray}&&{{\mathscr{B}}}_{\tau }=\{{\gamma }_{\omega }| {\delta }_{\tau }\lt \epsilon \}\,\subset \,{{\mathscr{B}}}_{{\rm{\Gamma }}},\end{eqnarray} \tag{ 32 }$$

where,

$$\begin{eqnarray}&&{\delta }_{\tau }=| {\gamma }_{c}(\tau )-{\gamma }_{\omega }(\tau )| =\sqrt{\displaystyle \sum _{i=1}^{3}{| {\gamma }_{c}^{i}(\tau )-{\gamma }_{\omega }^{i}(\tau )| }^{2}},\end{eqnarray} \tag{ 33 }$$

and $\epsilon \in {\mathbb{R}}$. The expected value of the function ψ(γ), the sample paths of which go through the neighborhood of γ_c can be expressed as;

$$\begin{eqnarray*}&&{E}_{\omega }[\psi ({\gamma }_{\omega })(t)\delta (t-\tau )| {{\mathscr{B}}}_{\tau }]=C{\int }_{-\epsilon }^{+\epsilon }\psi ({\gamma }_{c})(\tau )\exp \left[-\displaystyle \frac{{l}_{\omega }^{2}}{2{\sigma }^{2}}\right]{l}_{\omega }^{2}{{dl}}_{\omega },\end{eqnarray*}$$

where ${l}_{\omega }=| {\gamma }_{c}-{\gamma }_{\omega }| (\tau ),C$ a normalization factor and the integral is understood to be Riemann integration. Here, the Gaussian distribution appears as a result of the stochastic integration over the Brownian motion, as shown in equations (70) and (71) in appendix D.1. Therefore, the derivative can be expressed as;

$$\begin{eqnarray*}&&{\left.\displaystyle \frac{d\psi (\gamma )(\tau )}{d| \gamma (\tau )| }\right|}_{\gamma ={\gamma }_{c}}\sim \displaystyle \frac{{E}_{\omega }[\psi ({\gamma }_{c})(\tau )-\psi ({\gamma }_{\omega })(\tau )| {{\mathscr{B}}}_{{\rm{\Gamma }}}]}{{\delta }_{\tau }}=O(\epsilon )\end{eqnarray*}$$

Therefore, the limitation ${\mathrm{lim}}_{\epsilon \to 0}d\psi ({\gamma }_{c})/d| \gamma | =0$ can be obtained.

We note that, in this procedure QM is formulated earlier than classical mechanics. From this point of view, the reason why the principle of least action can function as the principle for extracting the classical path is that the path obtained from this principle is robust against perturbation form stochastic shaking of the metric. This can be understood from the fact that the probability amplitude around the classical path varies very slowly because the first derivative disappears, as shown above, because of the properties of the Brownian motion of the stochastic process.

4.2.2. Equation of motion

The discussion in a previous section is based on the white noise analysis of the quantum partition function, which is equivalent to the path-integral method. Therefore, the quantum transition function is obtained from the stochastic integration directly instead of through the equation of motion. Here, we propose an independent method to introduce a quantum equation of motion based on the stochastic differential equation (a brief explanation of the stochastic differential equation can be found in appendix E).

The Hamiltonian represents the total energy of the system defined on the classical manifold. When it does not explicitly include a time-variable, it is conserved during the time evolution of the system. This energy conservation must be retained in the SLM-space as a property using the expected value;

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}{E}_{\omega }[{ \mathcal H }({\gamma }_{\omega },{\pi }_{\omega })| {{\mathscr{B}}}_{{\rm{\Gamma }}}](t)={E}_{\omega }[d{ \mathcal H }({\gamma }_{\omega },{\pi }_{\omega })/{dt}| {{\mathscr{B}}}_{{\rm{\Gamma }}}]=0.\end{eqnarray} \tag{ 34 }$$

Here, the stochastic differentiation of the Hamiltonian can be written as;

$$\begin{eqnarray}&&d{ \mathcal H }({\gamma }_{\omega },{\pi }_{\omega })={\left.\displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\gamma }_{c}}\right|}_{c\to \omega }\cdot d{\gamma }_{\omega }+{\left.\displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\pi }_{c}}\right|}_{c\to \omega }\cdot d{\pi }_{\omega }.\end{eqnarray} \tag{ 35 }$$

We take note that π and γ are vectors and '·' denotes the inner product of two three-vectors. The functional variation in (35) can be obtained as;

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\pi }_{{ci}}}\right|}_{c\to \omega }={\dot{\gamma }}_{\omega }^{i},\end{eqnarray} \tag{ 36 }$$

where i = 1, 2, 3 are the components of the spatial coordinate. The preceding is obtained from the classical relation (13) and is not an equation of motion. If solutions to the equation of motion are obtained as γ_ω(t) and π_ω(t), their stochastic differentiations can be respectively written as;

$$\begin{eqnarray*}&&\left\{\begin{array}{l}d{\gamma }_{\omega }^{i}={{\dot{\gamma }}_{c}^{i}(t)| }_{c\to \omega }{dt}+{\displaystyle \sum }_{j=\mathrm{1,2},3}{[{\sigma }_{\gamma }(t)]}_{j}^{i}{{dB}}_{\gamma }^{j}(t),\\ d{\pi }_{\omega }^{i}={{\dot{\pi }}_{c}^{i}(t)| }_{c\to \omega }{dt}+{\displaystyle \sum }_{j=\mathrm{1,2},3}{[{\sigma }_{\pi }(t)]}_{j}^{i}{{dB}}_{\pi }^{j}(t),\end{array}\right.\end{eqnarray*}$$

where ${\sigma }_{\bullet }(t)$ is a 3 × 3 matrix of functions of τ and dB_•(t) is the Gaussian white noise with a mean value of zero and a unit variance. When ${\dot{\pi }}_{\omega }$ and ${\dot{\gamma }}_{\omega }$ are multiplied by the first and second equations above, respectively, the stochastic differential equations can be obtained as;

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\dot{\pi }}_{\omega }\cdot d{\gamma }_{\omega }={{\dot{\pi }}_{c}\cdot {\dot{\gamma }}_{c}| }_{c\to \omega }{dt}+{\dot{\pi }}_{\omega }\cdot {\sigma }_{\gamma }(t)\cdot {{dB}}_{\gamma }(t),\\ {\dot{\gamma }}_{\omega }\cdot d{\pi }_{\omega }={{\dot{\gamma }}_{c}\cdot {\dot{\pi }}_{c}| }_{c\to \omega }{dt}+{\dot{\gamma }}_{\omega }\cdot {\sigma }_{\pi }(t)\cdot {{dB}}_{\pi }(t),\end{array}\right.\end{eqnarray} \tag{ 37 }$$

where ·  indicates either an inner product of two three-vectors or a product of 3 × 3 matrix and three-vector. Here, the Itô-rule of (dt)(dB) = 0 is used, and the relation;

$$\begin{eqnarray*}&&{\dot{\gamma }}_{\omega }\cdot d{\pi }_{\omega }={\dot{\pi }}_{\omega }\cdot d{\gamma }_{\omega }+{\dot{\gamma }}_{\omega }\cdot {\sigma }_{\pi }(t)\cdot {{dB}}_{\pi }(t)+{\dot{\pi }}_{\omega }\cdot {\sigma }_{\gamma }(t)\cdot {{dB}}_{\gamma }(t)\end{eqnarray*}$$

follows from (37). As the Brownian motion is symmetric around its mean value(=0), it can be replaced as ${dB}\to -{dB}$. As a result, the stochastic differentiation of the Hamiltonian can be obtained from (35) as;

$$\begin{eqnarray*}&&d{ \mathcal H }=\left({\left.\displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\gamma }_{c}}\right|}_{c\to \omega }+{\dot{\pi }}_{\omega }\right)\cdot d{\gamma }_{\omega }+{\dot{\gamma }}_{\omega }\cdot {\sigma }_{\pi }(t)\cdot {{dB}}_{\pi }(t)+{\dot{\pi }}_{\omega }\cdot {\sigma }_{\gamma }(t)\cdot {{dB}}_{\gamma }(t),\end{eqnarray*}$$

When the expected value is taken on both sides of equation (38), the conservation relation

$$\begin{eqnarray}&&{E}_{\omega }[d{ \mathcal H }/{dt}| {{\mathscr{B}}}_{{\rm{\Gamma }}}]={E}_{\omega }\left[\left({\left.\displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\gamma }_{c}}\right|}_{c\to \omega }+{\dot{\pi }}_{\omega }\right)\cdot {\dot{\gamma }}_{\omega }| {{\mathscr{B}}}_{{\rm{\Gamma }}}\right]=0,\end{eqnarray} \tag{ 38 }$$

can be obtained because the mean value of the Gaussian noise is zero. Therefore, the stochastic equations of motion such as

$$\begin{eqnarray}&&d{\pi }_{\omega }^{i}={\left.-\displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\gamma }_{{ci}}}\right|}_{c\to \omega }{dt}+\displaystyle \sum _{j=1,2,3}{[{\sigma }_{\pi }(t)]}_{j}^{i}{{dB}}_{\pi }^{j}(t),\end{eqnarray} \tag{ 39 }$$

follow from (38). After solving the above stochastic equation (of motion), further stochastic integration

$$\begin{eqnarray}&&d{\gamma }_{\omega }^{i}={\left.\displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\pi }_{c}^{i}}\right|}_{c\to \omega }{dt}+\displaystyle \sum _{j=1,2,3}{[{\sigma }_{\gamma }(t)]}_{j}^{i}{{dB}}_{\gamma }^{j}(t),\end{eqnarray} \tag{ 40 }$$

from (36) can produce a trajectory of the mass point. These differential equations, including that for Brownian motion, are referred to as the stochastic differential equations (SDE). The existence and uniqueness of the stochastic processes as solutions of the SDEs are ensured mathematically when the stochastic processes satisfy the local Lipschitz condition [33], as is true in this case.

To this point, we have not used the constraint that the Brownian motion is pinned, as the starting and ending points are fixed as in equation (E.3). The set of SDEs above does not have invariance under time reversal; to preserve the time-reversal symmetry, another set of SDEs must be added as follows;

$$\begin{eqnarray}&&\left\{\begin{array}{l}d{\gamma }_{\bar{\omega }}^{i}={\left.-\tfrac{\delta {{ \mathcal H }}_{c}}{\delta {\pi }_{c}^{i}}\right|}_{c\to \bar{\omega }}{dt}+{[{\sigma }_{\bar{\gamma }}(t)]}_{j}^{i}{{dB}}_{\bar{\gamma }}^{j}(t),\\ d{\pi }_{\bar{\omega }}^{i}={\left.\tfrac{\delta {{ \mathcal H }}_{c}}{\delta {\gamma }_{c}^{i}}\right|}_{c\to \bar{\omega }}{dt}+{[{\sigma }_{\bar{\pi }}(t)]}_{j}^{i}{{dB}}_{\bar{\pi }}^{j}(t).\end{array}\right.\end{eqnarray} \tag{ 41 }$$

The sample path indexed by $\bar{\omega }$ is also an element of Γ⁰. The stochastic process ${\gamma }_{\bar{\omega }}(t)$ with Brownian motion ${B}_{\bar{\bullet }}(t)$ can be interpreted as a process evolving from the future to the past, as it obey the equation of motion with time reversal.

This approach can be compared with a stochastic quantization of Nelson [6]. When the functions σ_•(t) are set to be constants as;

$$\begin{eqnarray*}&&\left\{\begin{array}{l}{[{\sigma }_{\gamma }(t)]}_{j}^{i}=\sqrt{{\hslash }/m}\,{\delta }_{j}^{i},\\ {[{\sigma }_{\pi }(t)]}_{j}^{i}=\sqrt{m{\hslash }}\,{\delta }_{j}^{i},\end{array}\right.\end{eqnarray*}$$

equations of (39), (40) and (41) together reduce to the Newton-Nelson equation. For instance, the classical Hamiltonian of ${{ \mathcal H }}_{c}={\pi }_{c}^{2}/2m+V({\gamma }_{c})$ gives

$$\begin{eqnarray*}-\displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\gamma }_{c}^{i}} & = & -\displaystyle \frac{d}{d{\gamma }_{c}^{i}}V({\gamma }_{c})={F}_{i}({\gamma }_{c}),\\ \displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\pi }_{{ci}}} & = & \displaystyle \frac{1}{m}{\pi }_{c}^{i}.\end{eqnarray*}$$

Therefore, the stochastic equation of motion from $d{{ \mathcal H }}_{\omega }=0$ gives a result consistent with stochastic quantisation [20].

As shown above, the dynamics of a mass point in the SLM-space are equivalent to QM under stochastic quantization. If the stochastic quantization is mathematically equivalent to canonical quantization, the canonical commutation relations must be derived from the stochastic properties of the system. Although the relation between the commutation relation and stochastic processes has been discussed by several previous authors [34–36], here we will show the derivation of the commutation relation in our formalism.

First, let us consider the uncertainty relation in terms of stochastic processes. The uncertainty relation between the solutions of equations of motion (39) and (41) can be obtained on the covariance matrix of two Gaussian distributions. When the equations of motion are assumed to be linear SDEs such as;

$$\begin{eqnarray}&&d{\gamma }_{\omega }(t)=({\alpha }_{\gamma }(t)+{\beta }_{\gamma }(t){\gamma }_{\omega }(t)){dt}+{\sigma }_{\gamma }{{dB}}_{\omega },\end{eqnarray} \tag{ 42 }$$

their solutions can be obtained as;

$$\begin{eqnarray}{\gamma }_{\omega }(t) & = & {U}_{\gamma }(t)\left({\gamma }_{0}+{\displaystyle \int }_{0}^{t}{\alpha }_{\gamma }(u){U}_{\gamma }^{-1}(u){du}+{\sigma }_{\gamma }{\displaystyle \int }_{0}^{t}{U}_{\gamma }^{-1}(u){{dB}}_{\omega }(u)\right),\\ {U}_{\gamma }(t) & = & \exp \left[{\displaystyle \int }_{0}^{t}{\beta }_{\gamma }(u){du}\right],\end{eqnarray} \tag{ 43 }$$

as shown in example E.2 in appendix E. Here, {α_γ, β_γ} are integrable functions given by the Hamiltonian. The SDEs and their solutions with respect to the canonical momentum are written by simple replacements of ${\bullet }_{\gamma }\to {\bullet }_{\pi }$ in (42) and (43). The expected values of these solutions coincide with classical solutions as follows:

$$\begin{eqnarray*}{\gamma }_{c}(t) & = & {E}_{\omega }[{\gamma }_{\omega }| {{\mathscr{B}}}_{{\rm{\Gamma }}}],\\ {\pi }_{c}(t) & = & {E}_{\omega }[{\pi }_{\omega }| {{\mathscr{B}}}_{\pi }].\end{eqnarray*}$$

The classical solutions can be obtained as the solutions of a set of classical equations of motion as;

$$\begin{eqnarray}&&\displaystyle \frac{d{\gamma }_{c}(t)}{{dt}}={\alpha }_{\gamma }(t)+{\beta }_{\gamma }(t){\gamma }_{c}(t),\,\,{\gamma }_{c}(0)={\gamma }_{0},\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{\pi }_{c}(t)}{{dt}}={\alpha }_{\pi }(t)+{\beta }_{\pi }(t){\pi }_{c}(t),\,\,{\pi }_{c}(0)={\pi }_{0}.\end{eqnarray} \tag{ 45 }$$

Therefore, the covariance matrix can be obtained as

$$\begin{eqnarray}{\sigma }^{2}(\gamma ,\pi ) & = & {E}_{\omega }[({\gamma }_{\omega }-{\gamma }_{c})({\pi }_{\omega }-{\pi }_{c})| {{\mathscr{B}}}_{{\rm{\Gamma }}}],\\ & = & {\sigma }_{\gamma }{\sigma }_{\pi }{\displaystyle \int }_{0}^{T}{U}_{\gamma }(t){\bar{U}}_{\gamma }^{-1}(t){U}_{\pi }(t){\bar{U}}_{\pi }^{-1}(t){dt},\\ & = & {\sigma }_{\gamma }{\sigma }_{\pi }{\displaystyle \int }_{0}^{T}\left({\displaystyle \int }_{0}^{t}{U}_{\gamma }(t){U}_{\gamma }^{-1}(u){U}_{\pi }(t){U}_{\pi }^{-1}(u){du}\right){dt},\\ & = & C{\hslash },\end{eqnarray} \tag{ 46 }$$

where

$$\begin{eqnarray*}&&{\bar{U}}_{\gamma }^{-1}(t)={\int }_{0}^{t}{U}_{\gamma }^{-1}(s){{dB}}_{\omega }(s),\,\,(\gamma \to \pi ).\end{eqnarray*}$$

The constant C can be obtained when the equation of motion and duration T are fixed. Here we use the Itô rule, ${E}_{\omega }[{{dB}}_{\omega }(t){{dB}}_{\omega }(s)]\Rightarrow \delta (t-s){ds}$, given in appendix D.1, and note that γ_ω and π_ω are induced by common stochastic process. If these processes are independent each other, then it must be true that σ²(γ, π) = 0. If the functions β_•(t) are simply constants, then C = T²/2, and therefore the relation,

$$\begin{eqnarray}&&\displaystyle \frac{1}{{T}^{2}}{\sigma }^{2}(\gamma ,\pi )=\displaystyle \frac{{\hslash }}{2},\end{eqnarray} \tag{ 47 }$$

can be obtained, which is simply Heisenberg's uncertainty relation.

Next, we consider the commutation relation using the same framework and assumptions as above. The commutation relation becomes

$$\begin{eqnarray}[{\pi }_{\omega },{\gamma }_{\omega }](t) & = & ({\pi }_{\omega }\otimes {\gamma }_{\omega }-{\gamma }_{\omega }\otimes {\pi }_{\omega })(t),\\ & = & {\sigma }_{\pi }{\sigma }_{\gamma }\left\{{U}_{\pi }(t){\displaystyle \int }_{0}^{t}{U}_{\pi }^{-1}(u){U}_{\gamma }(u)\left({\displaystyle \int }_{0}^{u}{U}_{\gamma }^{-1}(v){{dB}}_{\omega }(v)\right){{dB}}_{\omega }(u)-(\pi \leftrightarrow \gamma )\right\}.\end{eqnarray} \tag{ 48 }$$

Here, the convolution of two stochastic integrations is defined as

$$\begin{eqnarray*}{X}_{\omega }^{12}(t) & = & ({X}_{\omega }^{1}\otimes {X}_{\omega }^{2})(t)={\displaystyle \int }_{0}^{t}{x}^{1}(u)\left({\displaystyle \int }_{0}^{u}{x}^{2}(v){{dB}}_{\omega }(v)\right){{dB}}_{\omega }(u),\\ {X}_{\omega }^{i}(t) & = & {\displaystyle \int }_{0}^{t}{x}^{i}(u){{dB}}_{\omega }(u).\end{eqnarray*}$$

The convolution induces a new stochastic process ${X}_{\omega }^{12}(t)$, and the stochastic integration appearing in equation (48) can be estimated as

$$\begin{eqnarray}&&{\int }_{0}^{u}{U}_{\gamma }^{-1}(v){{dB}}_{\omega }(v)={B}_{\omega }(u;{\sigma }^{2}),\end{eqnarray} \tag{ 49 }$$

where B_ω(u; σ²) is a Brownian motion with zero mean and a variance of

$$\begin{eqnarray}&&{\sigma }^{2}={\int }_{0}^{u}{({U}_{\gamma }^{-1}(v))}^{2}{dv},\end{eqnarray} \tag{ 50 }$$

as defined in appendix D. It is known that two Brownian motions with different variances will be mutually disjoint, and therefore, the stochastic integration vanishes as

$$\begin{eqnarray}&&{\int }_{0}^{t}{U}_{\pi }^{-1}(u){U}_{\gamma }(u){B}_{\omega }(u;{\sigma }^{2}){{dB}}_{\omega }(u)=0.\end{eqnarray} \tag{ 51 }$$

Therefore, any commutation relation comprising two solutions of the equations of motion, that share the same sample path, ω, will be zero due owing to the same mechanism. Only commutation relations of type $[{\pi }_{{\omega }^{\star }},{\gamma }_{\omega }]$ can be non-zero; in this case, a convolution of two stochastic processes can be defined as;

$$\begin{eqnarray*}{X}_{{\omega }^{\star }\omega }^{12}(t) & = & ({X}_{{\omega }^{\star }}^{1}\otimes {X}_{\omega }^{2})(t)={\displaystyle \int }_{0}^{t}\left[{\displaystyle \int }_{t}^{u}{x}^{1}(v){{dB}}_{{\omega }^{\star }}(v)\right]\left[{\displaystyle \int }_{0}^{u}{x}^{2}(v){{dB}}_{\omega }(v)\right]{du},\\ & = & {\displaystyle \int }_{0}^{t}\left[{\displaystyle \int }_{0}^{t-u}{x}^{1}(v+u){{dB}}_{\omega }(v+u)\right]\left[{\displaystyle \int }_{0}^{u}{x}^{2}(v){{dB}}_{\omega }(v)\right]{du},\end{eqnarray*}$$

where ${{dB}}_{{\omega }^{\star }}$ is a time-reversed Gaussian process with the same sample path, ω. Each stochastic integral may produce a Gaussian process with a different variance, for instance, ${\sigma }_{1}^{2}$, and ${\sigma }_{2}^{2}$, and as the result new stochastic process ${X}_{{\omega }^{\star }\omega }^{12}$ will also be a Gaussian process with a mean value of zero. If the integration region is from $-\infty$ to $+\infty$ instead of 0 to t, the variance becomes ${\sigma }_{12}^{2}={\sigma }_{1}^{2}+{\sigma }_{2}^{2}$. Again, using the same equations (44) and (45) a commutation relation can be obtained as;

$$\begin{eqnarray}[{\pi }_{{\omega }^{\star }},{\gamma }_{\omega }](t) & = & ({\pi }_{{\omega }^{\star }}\otimes {\gamma }_{\omega }-{\gamma }_{\omega }\otimes {\pi }_{{\omega }^{\star }})(t),\\ & = & {\sigma }_{\pi }{\sigma }_{\gamma }{e}^{({\beta }_{\pi }+{\beta }_{\gamma })t}\left\{{\displaystyle \int }_{0}^{t}\left[{\displaystyle \int }_{0}^{t-u}{e}^{-{\beta }_{\pi }(v+u)}{{dB}}_{\omega }(v+u)\right]\left[{\displaystyle \int }_{0}^{u}{e}^{-{\beta }_{\gamma }v}{{dB}}_{\omega }(v)\right]{du}-(\pi \leftrightarrow \gamma )\right\}.\,\,\end{eqnarray} \tag{ 52 }$$

This is a convolution of two Gaussian processes, whose variances are

$$\begin{eqnarray*}{\sigma }_{\pi }^{2} & = & {\displaystyle \int }_{u}^{t}{e}^{-2{\beta }_{\pi }v}{dv}=\displaystyle \frac{{e}^{-2u{\beta }_{\pi }}-{e}^{-2t{\beta }_{\pi }}}{2{\beta }_{\pi }},\\ {\sigma }_{\gamma }^{2} & = & {\displaystyle \int }_{0}^{u}{e}^{-2{\beta }_{\gamma }v}{dv}=\displaystyle \frac{1-{e}^{-2u{\beta }_{\gamma }}}{2{\beta }_{\gamma }},\end{eqnarray*}$$

respectively. The variance of the resulting Gaussian process can be obtained as;

$$\begin{eqnarray*}&&{\sigma }_{\pi \gamma }^{2}={\sigma }_{\pi }^{2}+{\sigma }_{\gamma }^{2}.\end{eqnarray*}$$

When ${\beta }_{\pi }\ne {\beta }_{\gamma }$, it is easily confirmed that ${\sigma }_{\pi \gamma }^{2}\ne {\sigma }_{\gamma \pi }^{2}$, where

$$\begin{eqnarray*}&&{\sigma }_{\gamma \pi }^{2}={\int }_{0}^{u}{e}^{-2{\beta }_{\pi }v}{dv}+{\int }_{u}^{t}{e}^{-2{\beta }_{\gamma }v}{dv}.\end{eqnarray*}$$

Therefore, owing to the reproducing property of the Gaussian distribution, the commutation relation is the following stochastic processes;

$$\begin{eqnarray*}[{\pi }_{{\omega }^{\star }},{\gamma }_{\omega }](t) & = & {\sigma }_{\pi }{\sigma }_{\gamma }{e}^{({\beta }_{\pi }+{\beta }_{\gamma })t}({B}_{\omega }(t;{\sigma }_{\pi \gamma }^{2})-{B}_{\omega }(t;{\sigma }_{\gamma \pi }^{2})),\\ & = & {\sigma }_{\pi }{\sigma }_{\gamma }{e}^{({\beta }_{\pi }+{\beta }_{\gamma })t}{B}_{\omega }(t;{\sigma }_{\pi \gamma }^{2}+{\sigma }_{\gamma \pi }^{2}).\end{eqnarray*}$$

After taking an average over a long time interval, the variance becomes

$$\begin{eqnarray}&&{\bar{\sigma }}^{2}=\mathop{\mathrm{lim}}\limits_{t\to \infty }\displaystyle \frac{{\sigma }_{\pi \gamma }^{2}+{\sigma }_{\gamma \pi }^{2}}{t}=\displaystyle \frac{1}{2}\left|\displaystyle \frac{1}{{\beta }_{\pi }}+\displaystyle \frac{1}{{\beta }_{\gamma }}\right|.\end{eqnarray} \tag{ 53 }$$

Thus, the commutation relation can be expressed as a single Gaussian process with zero mean and the variance derived above. The classical limit of this commutation relation is given as;

$$\begin{eqnarray}&&{E}_{\omega }\left[{\int }_{0}^{T}[{\pi }_{{\omega }^{\star }},{\gamma }_{\omega }](t){dt}| {{\mathscr{B}}}_{\omega }\right]=0.\end{eqnarray} \tag{ 54 }$$

Note the similarity between the commutation relation in (52) and the stochastic area defined in definition D.3. The commutation relation can be understood to be the stochastic area on the phase space. As the average value of the stochastic area on the phase space is zero, the area surrounded by the classical trajectory is conserved (Liouville's theorem). On the other hand, the variance of the stochastic area is not zero, which means that it cannot vanish as a result of quantum (stochastic) effects (the uncertainty principle).

Here, we will discuss the relation between the path integral and stochastic equation of motion methods. In the path-integral method, the meaning of the amplitude, ψ(γ), is given in its definition. By contrast, the meaning of the solutions of the SDE under the stochastic equation of motion needs clarifying. The physical meaning of the solutions can be elucidated by investigating the relation between two methods. First, let us consider an operator such as;

$$\begin{eqnarray*}{\left.-i{\hslash }\displaystyle \frac{\delta \psi ({\gamma }_{c})}{\delta {\gamma }_{c}}\right|}_{c\to \omega } & = & {\left.-\left({\displaystyle \int }_{0}^{T}\displaystyle \frac{\delta {{ \mathcal H }}_{c}}{\delta {\gamma }_{c}}{dt}\right)\psi ({\gamma }_{c})\right|}_{c\to \omega },\\ & = & {\left.\left({\displaystyle \int }_{0}^{T}{\dot{\pi }}_{c}{dt}\right)\psi ({\gamma }_{c})\right|}_{c\to \omega }={\pi }_{\omega }\psi ({\gamma }_{\omega }).\end{eqnarray*}$$

This operator can be interpreted as the momentum as follows;

$$\begin{eqnarray}&&{\left.-i{\hslash }\displaystyle \frac{\delta }{\delta {\gamma }_{c}}\right|}_{c\to \omega }\Rightarrow {\pi }_{\omega }.\end{eqnarray} \tag{ 55 }$$

Next, let us look closely at equation (24) from the point of view of the stochastic equation of motion. After solving the SDE, the transition probability (propagator) becomes;

$$\begin{eqnarray*}&&\psi (\gamma (T))=C{\int }_{\gamma (0)}^{\gamma (T)}\exp \left[\displaystyle \frac{i}{{\hslash }}{\int }_{0}^{T}{ \mathcal L }({\gamma }_{\omega },{\dot{\gamma }}_{\omega })(\tau )d\tau \right]{\mu }_{G}(d{\gamma }_{\omega }).\end{eqnarray*}$$

From equations (39) and (42), the solution of the quantum equation of motion is expected to have the form;

$$\begin{eqnarray}&&{\gamma }_{\omega }={\gamma }_{c}+\sqrt{\displaystyle \frac{{\hslash }}{m}}{B}_{\omega },\end{eqnarray} \tag{ 56 }$$

where ${B}_{\omega }$ is the pinned-Brownian motion given in (E.3). This stochastic integration has been proven to be equivalent to the Feynman path-integration [25, 37] (see also chapter 10.4.2 in [38]). The above results show that the amplitude defined in the path-integral method is consistent with solutions based on the quantum equation of motion.

First, we summarize classical general relativity in terms of a vierbein formalism. The mathematical setups introduced in this subsection are based on [39]; only classical (non-probabilistic) setups of the differential geometry are treated in this subsection.

First, classical general relativity is geometrically re-formulated in terms of a vierbein formalism. Let us introduce a four-dimensional Riemannian manifold ${\mathscr{M}}$ with a metric tensor g_•• of dimension four, which is referred to as the global manifold. A line element on ${\mathscr{M}}$ can be expressed as;

$$\begin{eqnarray}&&{{ds}}^{2}={g}_{{\mu }_{1}{\mu }_{2}}(x){{dx}}^{{\mu }_{1}}\otimes {{dx}}^{{\mu }_{2}}.\end{eqnarray} \tag{ 57 }$$

The covariant derivative for a tensor defined on ${\mathscr{M}}$ can be introduced using the affine connection ${{\rm{\Gamma }}}_{\,\bullet \bullet }^{\bullet }$. Because the affine connection is not a tensor, it is always possible to find a frame in which the affine connection vanishes as ${{\rm{\Gamma }}}_{\,\bullet \bullet }^{\bullet }({x})=0$ at any point x on the global manifold. A local manifold with a vanishing affine connection is required to have the symmetries SO(1, 3) and T⁴ (4-dimensional translation symmetry) and is referred to as the local (Lorenz) manifold and denoted by ${ \mathcal M }$. On ${ \mathcal M }$, a vanishing gravity, ${\partial }_{\bullet }{g}_{\bullet \bullet }=0$, is not required, and a transformation from the global to the local manifold can be achieved using a tensor function³ ${{ \mathcal E }}_{\bullet }^{\bullet }(x)$, such as

$$\begin{eqnarray*}&&{g}_{{\mu }_{1}{\mu }_{2}}(x)\,{{ \mathcal E }}_{a}^{{\mu }_{1}}(x)\,{{ \mathcal E }}_{b}^{{\mu }_{2}}(x)={\eta }_{{ab}},\end{eqnarray*}$$

where η_ab is a local Lorentz metric under a particle physics convention such as ${\eta }_{\bullet \bullet }=\mathrm{diag}(1,-1,-1,-1)$. Here, we employ the convention that indices of Greek and Roman letters represent coordinates on the global and local manifolds, respectively. Both indices run from zero to three. Note that the sign of the determinant of the metric tensor is invariant under the general coordinate transformation. The inverse function is represented as ${({{ \mathcal E }}_{a}^{\mu })}^{-1}={{ \mathcal E }}_{\mu }^{a}$, which satisfies

$$\begin{eqnarray}&&{{ \mathcal E }}_{a}^{\mu }(x){{ \mathcal E }}_{\mu }^{b}(x)={\delta }_{a}^{b},\,\,{{ \mathcal E }}_{a}^{\mu }(x){{ \mathcal E }}_{\nu }^{a}(x)={\delta }_{\nu }^{\mu }.\end{eqnarray} \tag{ 58 }$$

Both ${{ \mathcal E }}_{a}^{\mu }$ and its inverse ${{ \mathcal E }}_{\mu }^{a}$ are called the vierbein. An alternative definition of the vierbein can be given using a set of bi-local functions ξ^a(y, x), defined on a neighborhood $U\in -\gt \subset {\mathscr{M}}$, in which $x,y\subset U$. While ξ^a is a function on the global manifold, it is a vector on the local manifold; it is given as a solution of the differential equation

$$\begin{eqnarray}&&{{ \mathcal E }}_{\mu }^{a}={\left.\displaystyle \frac{\partial {\xi }^{a}(y,x)}{\partial {y}^{\mu }}\right|}_{y=x},\end{eqnarray} \tag{ 59 }$$

and of equations (57) and (58) with the boundary condition ${\xi }^{a}(x,x)=0$. The vierbein (one-)form is introduced as ${{\mathfrak{e}}}^{a}={{ \mathcal E }}_{\mu }^{a}{{dx}}^{\mu }$ and behaves as a local vector under the Lorentz transformation (throughout this report, Fraktur letters indicate differential forms).

The covariant (under the general coordinate transformation) derivative ∇_μ applying on a local tensor can be written using the spin-connection ω as follows:

$$\begin{eqnarray*}&&{{\rm{\nabla }}}_{\mu }{T}_{b}^{a}={\partial }_{\mu }{T}_{b}^{a}+{\omega }_{\mu \,b}^{\,a}{T}_{b}^{c}-{\omega }_{\mu \,b}^{\,c}{T}_{c}^{a},\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&{\omega }_{\mu \,b}^{a}={{ \mathcal E }}_{\nu }^{a}{{\rm{\Gamma }}}_{\mu \rho }^{\nu }\,{{ \mathcal E }}_{b}^{\rho }-({\partial }_{\mu }{{ \mathcal E }}_{\rho }^{a}){{ \mathcal E }}_{b}^{\rho }.\end{eqnarray*}$$

A spin-connection form (or simply spin form) can be defined using the spin connection as

$$\begin{eqnarray*}&&{{\mathfrak{w}}}^{{ab}}={{dx}}^{\mu }{\omega }_{\mu \,b}^{a}\,{\eta }^{{cb}}={{\mathfrak{w}}}_{c}^{a}\,{\eta }^{{cb}},\end{eqnarray*}$$

which satisfies ${{\mathfrak{w}}}^{{ab}}=-{{\mathfrak{w}}}^{{ba}}$. We note that the spin form is a generator of the Lie algebra of ${\mathfrak{so}}(1,3)$ and is not a Lorentz tensor. The torsion and curvature forms can be introduced as;

$$\begin{eqnarray*}{{\mathfrak{T}}}^{a} & = & d{{\mathfrak{e}}}^{a}+{{\mathfrak{w}}}_{b}^{a}\wedge {{\mathfrak{e}}}^{b},\\ {{\mathfrak{R}}}^{{ab}} & = & d{{\mathfrak{w}}}^{{ab}}+{{\mathfrak{w}}}_{c}^{a}\wedge {{\mathfrak{w}}}^{{cb}},\end{eqnarray*}$$

respectively. Any two-forms defined on the local manifold can be expanded using the orthogonal bases of the two-forms on the local manifold, ${{\mathfrak{e}}}^{\bullet }\wedge {{\mathfrak{e}}}^{\bullet }$. The curvature form can be expressed as

$$\begin{eqnarray}&&{{\mathfrak{R}}}^{{a}{b}}=\displaystyle \frac{1}{2}{{R}}_{\,\,{{c}}_{1}{{c}}_{2}}^{{a}{b}}\,{{\mathfrak{e}}}^{{{c}}_{1}}\wedge {{\mathfrak{e}}}^{{{c}}_{2}}.\end{eqnarray} \tag{ 60 }$$

The expansion coefficients ${R}_{\bullet \bullet }^{\bullet \bullet }$ are referred to the Riemann curvature tensor.

The volume form can be expressed using vierbein forms, e.g.

$$\begin{eqnarray}&&{\mathfrak{v}}=\displaystyle \frac{1}{4!}{\epsilon }_{{a}_{1}{a}_{2}{a}_{3}{a}_{4}}{{\mathfrak{e}}}^{{a}_{1}}\wedge {{\mathfrak{e}}}^{{a}_{2}}\wedge {{\mathfrak{e}}}^{{a}_{3}}\wedge {{\mathfrak{e}}}^{{a}_{4}},\end{eqnarray} \tag{ 61 }$$

where ${\epsilon }_{\bullet \bullet \bullet \bullet }$ is a complete anti-symmetric tensor with ₀₁₂₃ = +1. Similarly, three-dimensional volume-and surface-forms are introduced as;

$$\begin{eqnarray*}{{\mathfrak{V}}}_{a} & = & \displaystyle \frac{1}{3!}{\epsilon }_{{{ab}}_{1}{b}_{1}{b}_{2}}{{\mathfrak{e}}}^{{b}_{1}}\wedge {{\mathfrak{e}}}^{{b}_{2}}\wedge {{\mathfrak{e}}}^{{b}_{3}},\\ {{\mathfrak{S}}}_{{ab}} & = & \displaystyle \frac{1}{2!}{\epsilon }_{{{abc}}_{1}{c}_{2}}{{\mathfrak{e}}}^{{c}_{1}}\wedge {{\mathfrak{e}}}^{{c}_{2}}.\end{eqnarray*}$$

The surface form ${{\mathfrak{S}}}_{{ab}}$ is perpendicular to plane spanned by ${{\mathfrak{e}}}^{a}$ and ${{\mathfrak{e}}}^{b}$. Hereafter, dummy Roman indices are omitted in expressions when the index pairing is obvious, with dots placed instead at the index, e.g. $2{{\mathfrak{S}}}_{{ab}}={\epsilon }_{{ab}\cdot \cdot }{{\mathfrak{e}}}^{\cdot }\wedge {{\mathfrak{e}}}^{\cdot }$.

The Lagrangian form and action integral for gravitation can be represented in this terminology as;

$$\begin{eqnarray}&&{{\mathfrak{L}}}_{G}=\displaystyle \frac{1}{2!}{{\mathfrak{R}}}^{\cdot \cdot }({\mathfrak{w}})\wedge {{\mathfrak{S}}}_{\cdot \cdot }-{\rm{\Lambda }}\,{\mathfrak{v}},\end{eqnarray} \tag{ 62 }$$

and

$$\begin{eqnarray*}&&{{\mathscr{I}}}_{G}={\int }_{{\rm{\Sigma }}}{{\mathfrak{L}}}_{G},\end{eqnarray*}$$

respectively, where Λ is the cosmological constant. To complete the general relativity formulation, Lagrangian forms of the matter and gauge fields must be added. The equation of motion can be obtained as the Euler–Lagrange equation by taking variations with respect to ${{\mathfrak{w}}}^{\bullet \bullet }$ and ${{\mathfrak{S}}}_{\bullet \bullet }$. The Einstein equation and the torsion-less condition can be obtained as the Euler–Lagrange equation of motion as;

$$\begin{eqnarray*}&&{\epsilon }_{{\boldsymbol{\cdot }}{\boldsymbol{\cdot }}{\boldsymbol{\cdot }}}\,\left(\displaystyle \frac{1}{2}{{\mathfrak{R}}}^{\cdot \cdot }\wedge {{\mathfrak{e}}}^{\cdot }-\displaystyle \frac{{\rm{\Lambda }}}{3!}{{\mathfrak{e}}}^{\cdot }\wedge {{\mathfrak{e}}}^{\cdot }\wedge {{\mathfrak{e}}}^{\cdot }\right)=\displaystyle \frac{1}{2}{T}_{a{\boldsymbol{\cdot }}}{{\mathfrak{V}}}^{\cdot },\end{eqnarray*}$$

and ${{\mathfrak{T}}}^{\bullet }=0$, respectively, where T_•• is the energy-momentum tensor of the matter and gauge fields.

The general relativity equipping the local SLM-space is constructed in this sub-section. The stochastic aspects of general relativity may be implemented by modifying the vierbein form; through which the effects of these aspect may appear on the spin and surface forms.

Let us start from the definition of the vierbein in (59). The bi-local function ξ^a(y, x) has non-local information of the space time manifold, and the vierbein can be defined through the differentiation of this function. By contrast, the distance between two points on the SLM-space is a random variable, that follows a Gaussian distribution with a mean value and variance given by the geometrical distance between the points, as per remark 3.4. A possible modification of the function ξ^a(y, x) that is consistent with the above requirements is given as follows;

$$\begin{eqnarray}&&{\hat{\xi }}^{a}(y,x)={\xi }^{a}(y,x)+{\sigma }_{\varepsilon }{\int }^{y}{{dB}}^{(a)}(x),\end{eqnarray} \tag{ 63 }$$

where B^(a)(x) (a = 0, 1, 2, 3,) are four independent Brownian motions. Note that the second term of the right-hand side of (63) represents the total fluctuation of the length between x and y owing to the Brownian motion; a constant, ${\sigma }_{\epsilon }$, is prepared to adjust the variance as well as the dimension. The order parameter of the Brownian motion is not simply a single parameter but a four-dimensional coordinate-vector on the local coordinate-patch of the global manifold. Moreover, dB^(a)(x) must be understood as the white noise introduced in appendix F.2. In this definition, the stochastic function still retains the condition ${\hat{\xi }}^{a}(x,x)=0$. Note that B^(a)(x) cannot be considered to be a vector on ${ \mathcal M };$ if it is a Lorentz vector with each component fluctuating independently on some local inertial frame, the components will mix with each other following a Lorentz transformation and therefore will not be independent. In fact, white noise can be treated as an independent Gaussian process in any local coordinate systems (detailed discussions on the SO(1, 3) symmetries of white noise are given in appendix H). From the stochastic function ${\hat{\xi }}^{a}$, the stochastic vierbein is obtained as;

$$\begin{eqnarray*}&&{\hat{{ \mathcal E }}}_{\mu }^{a}(x)={\left.\displaystyle \frac{\partial {\xi }^{a}(y,x)}{\partial {y}^{\mu }}\right|}_{y=x}+{\sigma }_{\varepsilon }\displaystyle \frac{\partial {B}^{(a)}(x)}{\partial {x}^{\mu }},\end{eqnarray*}$$

following (59). Here, the derivative of the Brownian motion is merely a formal expression, as it is differentiable nowhere; an exact definition will be introduced later. As a short hand notation, the Brownian motion is denoted as ${B}_{;\mu }^{(a)}(x)=\partial {B}^{(a)}(x)/\partial {x}^{\mu }$ hereafter.

In the SLM-space, the vierbein form can be a stochastic process formally expressed as;

$$\begin{eqnarray}&&{\hat{{\mathfrak{e}}}}^{a}(x)=({{\mathfrak{e}}}^{a}+{\sigma }_{\varepsilon }{{\mathfrak{B}}}^{(a)})(x)=({{ \mathcal E }}_{\mu }^{a}(x)+{\sigma }_{\varepsilon }{B}_{;\mu }^{(a)}(x)){{dx}}^{\mu },\end{eqnarray} \tag{ 64 }$$

where ${{\mathfrak{B}}}^{(a)}(x)={B}_{;\mu }^{(a)}(x){{dx}}^{\mu }$ (a = 0, 1, 2, 3) can be understood as the local one-form valued white-noise. The ${\sigma }_{\epsilon }{B}_{;\bullet }^{(a)}$ must have the same dimensionality as the vierbein, which is dimensionless; therefore, σ has the dimension of length only.

The precise definition of ${B}_{;\bullet }(x)$ can be stated as follows: consider the probabilistic distribution introduced in appendix D, which will be denoted as ${B}_{;\bullet }(x)$, under the following setups. A probability space, $({E}^{* },{\mathscr{B}},\mu )$, is introduced on the square-integrable functions $H={L}^{2}({{\mathbb{R}}}^{4})$ defined on the global manifold, the Gel'fand triple $E\subset H\subset {E}^{* }$ and the sigma-field ${\mathscr{B}}$ generated by the cylinder subset of ${E}^{* }$.

Definition 5.1 The ${B}_{;\mu }(x)$ is called the vierbein white-noise, when its characteristic functional is

$$\begin{eqnarray*}&&{C}^{\mu }(x)=\exp \left[-\displaystyle \frac{\parallel x{\parallel }^{2}}{2}\right]{e}^{\mu }=\exp \left[-\displaystyle \frac{{ \mathcal D }{(x)}^{2}}{2}\right]{e}^{\mu },\end{eqnarray*}$$

where ${e}^{\mu }$ is the orthonormal base on the local coordinate-patch and ${x}^{\mu }$ is a coordinate vector. ${ \mathcal D }(x)$ is the geometrical distance defined in definition 3.2.

This characteristic functional induces the Gaussian measure uniquely as

$$\begin{eqnarray*}&&{C}^{\lambda }(\mu )={\int }_{{E}^{* }}\exp [{{iB}}_{;\mu }{x}^{\mu }]d\mu ({x}^{\lambda }).\end{eqnarray*}$$

The existence of this measure is ensured by the Bochner–Minlos theorem [21, 40]. The VWN is a stochastic functional with a Gaussian distribution and is pointwise independent. Note that, in the definition of the stochastic vierbein $\hat{{\mathfrak{e}}}$ from equation (64) and definition 5.1, the Brownian motion itself does not appear explicitly. In the VWN, the one-dimensional order parameter, the Morkovian time, does not exist; therefore, the stochastic processes do not allow for simple interpretation such as a sample path of the one-dimensional trajectory of the mass point. One interpretation of the VWN, for instance, is that it gives a sample universe, that equips the sample metric tensor.

To confirm that the VWN can construct the SLM-space properly, let us calculate the line element using the stochastic vierbein form equation (64):

$$\begin{eqnarray*}&&d{\hat{s}}^{2}={\eta }_{{\boldsymbol{\cdot }}{\boldsymbol{\cdot }}}\,{\hat{{\mathfrak{e}}}}^{\cdot }\otimes {\hat{{\mathfrak{e}}}}^{\cdot }={{ds}}^{2}+{\eta }_{\cdot \cdot }(2{\sigma }_{\varepsilon }{{ \mathcal E }}_{{\mu }_{1}}^{\cdot }{B}_{;{\mu }_{2}}^{({\boldsymbol{\cdot }})}+{\sigma }_{\varepsilon }^{2}{B}_{;{\mu }_{1}}^{(\cdot )}{B}_{;{\mu }_{2}}^{({\boldsymbol{\cdot }})}){{dx}}^{{\mu }_{1}}\otimes {{dx}}^{{\mu }_{2}}.\end{eqnarray*}$$

The terms in parentheses in the above equation can be understood as stochastic processes of the Gaussian type. The first term becomes a Gaussian process with a mean value of zero and a variance of;

$$\begin{eqnarray*}&&{{\sigma }}_{{B}}^{2}=4{{\sigma }}_{{\varepsilon }}^{2}\,{{\eta }}_{\cdot \cdot }{{\mathfrak{e}}}^{\cdot }\otimes {{\mathfrak{e}}}^{\cdot }\propto { \mathcal D }{(| {ds}| )}^{2}\end{eqnarray*}$$

as shown in equation (70). When we set σ_ε = 1/2, the variance of the white noise becomes ${\sigma }_{B}^{2}={ \mathcal D }{(| {ds}| )}^{2}$. As mentioned previously, σ_ε should have a length dimension, and therefore the precise representation must be σ_ε = l_p/2, where ${l}_{p}=\sqrt{{\hslash }G/{c}^{3}}$ is the Planck length. Although the third term includes Brownian motion, it is not a stochastic integration but it gives ${dB}(x){dB}(x)={dx}$ from the Itô rule. Therefore, this term gives a fixed correction to the line element, and can be neglected as a higher order effect. Finally, the stochastic line element can be obtained as;

$$\begin{eqnarray*}&&d{\hat{s}}^{2}(x)={{ds}}^{2}(1+B)(x),\end{eqnarray*}$$

where $B(x)={\eta }_{{\boldsymbol{\cdot }}{\boldsymbol{\cdot }}}{{\mathfrak{e}}}^{\cdot }(x)\otimes {{\mathfrak{B}}}^{\cdot }(x)$. This is the representation of the line element on the local SLM-space.

The volume form (61) on the local SLM-space becomes;

$$\begin{eqnarray}\hat{{\mathfrak{v}}} & = & \displaystyle \frac{1}{4!}{\epsilon }_{{a}_{1}{a}_{2}{a}_{3}{a}_{4}}{\hat{{\mathfrak{e}}}}^{{a}_{1}}\wedge {\hat{{\mathfrak{e}}}}^{{a}_{2}}\wedge {\hat{{\mathfrak{e}}}}^{{a}_{3}}\wedge {\hat{{\mathfrak{e}}}}^{{a}_{4}},\\ & = & \displaystyle \frac{1}{4!}{\epsilon }_{{a}_{1}{a}_{2}{a}_{3}{a}_{4}}({{\mathfrak{e}}}^{{a}_{1}}\wedge {{\mathfrak{e}}}^{{a}_{2}}\wedge {{\mathfrak{e}}}^{{a}_{3}}\wedge {{\mathfrak{e}}}^{{a}_{4}}+2{{\mathfrak{B}}}^{({a}_{1})}\wedge {{\mathfrak{e}}}^{{a}_{2}}\wedge {{\mathfrak{e}}}^{{a}_{3}}\wedge {{\mathfrak{e}}}^{{a}_{4}}\\ & & +\displaystyle \frac{3}{2}{{\mathfrak{B}}}^{({a}_{1})}\wedge {{\mathfrak{B}}}^{({a}_{2})}\wedge {{\mathfrak{e}}}^{{a}_{3}}\wedge {{\mathfrak{e}}}^{{a}_{4}}+\displaystyle \frac{1}{2}{{\mathfrak{B}}}^{({a}_{1})}\wedge {{\mathfrak{B}}}^{({a}_{2})}\wedge {{\mathfrak{B}}}^{({a}_{3})}\wedge {{\mathfrak{e}}}^{{a}_{4}}\\ & & +\displaystyle \frac{1}{16}{{\mathfrak{B}}}^{({a}_{1})}\wedge {{\mathfrak{B}}}^{({a}_{2})}\wedge {{\mathfrak{B}}}^{({a}_{3})}\wedge {{\mathfrak{B}}}^{({a}_{4})}),\,\,\end{eqnarray} \tag{ 65 }$$

where (64) and σ_ε = 1/2 are used. As all white noises appearing here are independent, the mean value of the volume form becomes the geometric volume as $E[\hat{{\mathfrak{v}}}| {\mathscr{B}}]={\mathfrak{v}}$, as expected. On the other hand, the variance of the volume form has a finite value, which corresponds to the zero-point vibration of space time. This can be interpreted to mean that the four-dimensional volume cannot be zero owing to the uncertainty principle of quantum general relativity. In cases of a flat global manifold, the vierbein is simply a unit matrix, in which case the variance of the white noise is estimated to be

$$\begin{eqnarray*}&&\delta [{B}_{;\mu }]=E[{({B}_{;\mu })}^{2}| {\mathscr{B}}]-E{[{B}_{;\mu }| {\mathscr{B}}]}^{2}\,=\,{{dx}}^{\mu },\end{eqnarray*}$$

from the Itô rule. Thus, we can obtain $\delta [{{\mathfrak{B}}}^{(a)}]={{\mathfrak{e}}}^{a}$ for the flat space time case, and the variance of the volume form becomes;

$$\begin{eqnarray}&&{\delta }_{{\mathfrak{v}}}=\displaystyle \frac{65}{16}\,{\mathfrak{v}}\end{eqnarray} \tag{ 66 }$$

form (65).

The stochastic effect on other forms can be estimated similarly. For both ${{\mathfrak{V}}}_{a}$ and ${{\mathfrak{S}}}_{{ab}}$, the mean values are the same as the classical values. The variances are obtained as;

$$\begin{eqnarray}&&{\delta }_{{\mathfrak{V}}a}=\displaystyle \frac{19}{8}\int {{\mathfrak{V}}}_{a},\,{\delta }_{{\mathfrak{S}}{ab}}=\displaystyle \frac{5}{4}\int {{\mathfrak{S}}}_{{ab}},\end{eqnarray} \tag{ 67 }$$

for the flat global spacetime case. In general, the variance of a n-dimensional volume form in the flat SLM-space is given as δ_n = (3/2)ⁿ − 1.

The variance of the volume form (66) has the same shape as the cosmological term in the Lagrangian form (62). Therefore, even starting from a universe with ${\rm{\Lambda }}=0$, an effective cosmological term appears because of the fluctuation of the metric. To discuss the quantum effect on the Universe precisely, it is necessary to formulate the quantum Einstein equation in the stochastic metric space and solve it under certain boundary conditions. As these tasks are beyond the scope of this report, we employ a semi-classical approximation of quantum general relativity to estimate the quantum effect from the fluctuations of forms. In this approximation, the effect of the stochastic metric on the vierbein and other forms are considered around the classical solution of the (classical) Einstein equation.

Let us estimate an order of total fluctuation for a flat universe under this approximation. To estimate a four-dimensional volume of the current universe, $\int {\mathfrak{v}}$, we need to know a complete history of the Universe. Here, the simple approximation of a constant expansion of the Universe is assumed. A recent precise measurement gives an age of the Universe to be t₀ = (4.35 ± 0.01) × 10¹⁷ sec [41, 42]. The total four-dimensional volume with a radius of t₀ can be calculated as

$$\begin{eqnarray*}&&{V}_{U}={\int }_{{Univ}}{\mathfrak{v}}=\displaystyle \frac{{\pi }^{2}}{2}{t}_{0}^{4},\end{eqnarray*}$$

for a flat, constantly expanding universe (this is very rough estimation. It is known that the a comoving distance of the Universe is about three times larger than c × t₀). Thus, the variance becomes;

$$\begin{eqnarray*}&&{\delta }_{{\mathfrak{v}}}=\displaystyle \frac{65}{16}{V}_{U}=3.26\times {10}^{167}\,{(\mathrm{GeV})}^{-4}.\end{eqnarray*}$$

The standard deviation averaged over the Universe is

$$\begin{eqnarray*}&&{\bar{\sigma }}_{{\mathfrak{v}}}=\displaystyle \frac{{\delta }_{{\mathfrak{v}}}^{1/2}}{{V}_{U}}=2.13\times {10}^{-84}\,{(\mathrm{GeV})}^{2}.\end{eqnarray*}$$

Interestingly, recent measurements of the dark energy gives [41, 42];

$$\begin{eqnarray*}&&{\rm{\Lambda }}\sim 3{H}_{0}=6.16\times {10}^{-84}\,{(\mathrm{GeV})}^{2}.\end{eqnarray*}$$

This coincidence suggests that the cosmological constant in the classical Hamiltonian of the Universe is zero (Λ = 0) and the origin of the dark energy (the cosmological term) may be a quantum fluctuation of space time arising from the stochastic effect. We note that the unnaturally small values appearing here originated from division by a large number (the total volume of the Universe in Planck units). The fluctuation itself is on the order of σ = l_p/2, where the Planck length l_p is explicitly written to emphasize that the ${\sigma }_{\epsilon }$ has a length dimension. Even though the cosmological constant is zero at the classical level, an effective term corresponding to the cosmological term appears towing to the stochastic (quantum) effect as;

$$\begin{eqnarray}&&{\sigma }_{{\mathfrak{v}}}{\mathfrak{v}}\sim \sqrt{\displaystyle \frac{65}{16}}{V}_{4}^{-1/2}\,{\mathfrak{v}},\end{eqnarray} \tag{ 68 }$$

where V₄ is the four-dimensional volume in question. This effective term may therefore have induced the re-acceleration seen in the Universe [43, 44].

Definition D.1 If a set of Random variable, ${\boldsymbol{B}}(t)=\{B(t),t\in {\mathbb{R}}\}$ satisfies the following properties:

• 1.  
  ${\boldsymbol{B}}(t)$ is a Gaussian system with $E[B(t)]=0$ for every $t$,
• 2.  
  $B(0)=0$,
• 3.  
  $E[{(B(t)-B(s))}^{2}]=| t-s|$

it is called Brownian motion [21].

The covariant matrix of the Brownian motion can be obtained as

$$\begin{eqnarray*}&&{\rm{\Gamma }}[s,t]=E[B(t)B(s)]=\displaystyle \frac{1}{2}(| t| +| s| -| t-s| ),\end{eqnarray*}$$

due to the definition.

Definition D.2 On following setups:

• Functions:
  • 1.  
    $x=\{{x}_{t}^{i}\}{}_{i=1,\cdots ,d}\in {C}^{1}({{\mathbb{R}}}^{d})$ for $t\geqslant 0$,
  • 2.  
    $\forall \varphi =\varphi (x)\in {C}^{2}({\mathbb{R}})$.
• Two stochastic processes:
  • 1.  
    N-independent Brownian motion: ${\boldsymbol{B}}(t)={\{{B}^{k}(t)\}}_{k=1,\cdots ,N}$.
  • 2.  
    stochastic processes: ${\{{A}^{i}(t)\}}_{i=1,\cdots ,d}$, where $\forall i,\omega ,\,\,{A}_{\omega }^{i}(0)=0\wedge {A}_{\omega }^{i}(t)\lt \infty$,
  where ω is a index for sample paths. The index ω will be often omitted.

Itô process is defined as random variables of $\{{X}_{t}^{i}\}{}_{i=1,\cdots ,d},t\geqslant 0$, which follows:

$$\begin{eqnarray*}&&{X}_{t}^{i}={X}_{0}^{i}+{\int }_{0}^{t}{{\boldsymbol{\sigma }}}^{i}\cdot d{\boldsymbol{B}}(s)+{A}^{i}(t),\end{eqnarray*}$$

where ${{\boldsymbol{\sigma }}}_{k}^{i}(s)=\{{\sigma }_{k}^{i}(s)\in {{ \mathcal L }}^{2}\}{}_{k=1,\cdots ,N}$, $({{ \mathcal L }}^{2}$ is a set of square integrable functions). Then the Itô formula [45] is introduced as;

$$\begin{eqnarray*}&&\varphi ({X}_{t})=\varphi ({X}_{0})+{\int }_{0}^{t}\displaystyle \frac{\partial \varphi }{\partial {x}^{i}}({X}_{s}){{\boldsymbol{\sigma }}}^{i}(s)\cdot d{\boldsymbol{B}}(s)+{\int }_{0}^{t}\displaystyle \frac{\partial \varphi }{\partial {x}^{i}}({X}_{s}){{dA}}_{s}^{i}+\displaystyle \frac{1}{2}\displaystyle \frac{{\partial }^{2}\varphi }{\partial {x}^{i}\partial {x}^{j}}({X}_{s}){{\boldsymbol{\sigma }}}^{i}(s)\cdot {{\boldsymbol{\sigma }}}^{j}(s){ds}.\end{eqnarray*}$$

Here, integration with respect to dAⁱ_s must be understood as the Riemann-Stieltjes integration, and the stochastic integration with respect to the Brownian motion ${{\boldsymbol{\sigma }}}^{i}\cdot d{\boldsymbol{B}}(s)$ must be understand as the Itô integral, that is defined as;

$$\begin{eqnarray*}&&{\int }_{0}^{t}{{\boldsymbol{\sigma }}}^{i}(s)\cdot d{\boldsymbol{B}}(s)=\mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \sum _{k=0}^{n-1}{{\boldsymbol{\sigma }}}^{i}({s}_{k})[{{\boldsymbol{B}}}_{k+1}-{{\boldsymbol{B}}}_{k}],\end{eqnarray*}$$

where ${\{{s}_{i}\}}_{i=1,\cdots ,n}$ are the dissection 0 = s₁ < ⋯ < s_n = t, and ${{\boldsymbol{B}}}_{k}={\boldsymbol{B}}({s}_{k})$. Each ${{\boldsymbol{B}}}_{k}$ will be independent Gaussian noise. The Itô integral is the martingale because ${{\boldsymbol{\sigma }}}^{i}({s}_{k})$ is only depends on the process [0, s_k]. Another definition of the stochastic integration can be given as;

$$\begin{eqnarray*}&&{\int }_{0}^{t}{{\boldsymbol{\sigma }}}^{i}(s)\,\circ \,d{\boldsymbol{B}}(s)=\mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \sum _{k=0}^{n-1}{{\boldsymbol{\sigma }}}^{i}({s}_{k+1/2})[{{\boldsymbol{B}}}_{k+1}-{{\boldsymbol{B}}}_{k}],\end{eqnarray*}$$

where ${s}_{k+1/2}=({s}_{k+1}+{s}_{k})/2$. It is known as the Stratonovich integral, which is not martingale, but sub-martingale. We use the Itô convention as the stochastic integral in this work, unless otherwise stated. The Itô formula can be expressed as;

$$\begin{eqnarray*}&&\varphi (X(t))=\varphi ({X}_{0})+{\int }_{0}^{t}\displaystyle \frac{\partial \varphi }{\partial {x}^{i}}(X(s)){{\boldsymbol{\sigma }}}^{i}(s)\cdot d{\boldsymbol{B}}(s)+\left[{\int }_{0}^{t}\displaystyle \frac{\partial \varphi }{\partial {x}^{i}}(X(s)){\beta }^{i}(s)+\displaystyle \frac{1}{2}\displaystyle \frac{{\partial }^{2}\varphi }{\partial {x}^{i}\partial {x}^{j}}(X(s)){{\boldsymbol{\sigma }}}^{i}(s)\cdot {{\boldsymbol{\sigma }}}^{j}(s)\right]{ds},\end{eqnarray*}$$

when the stochastic process Aⁱ(s) is written as;

$$\begin{eqnarray*}&&{A}^{i}(t)-{A}^{i}(0)={\int }_{0}^{t}{\beta }^{i}(s){ds}.\end{eqnarray*}$$

This formula can be understood from a Taylor expansion of φ with simple rules (Itô rules) as

$$\begin{eqnarray*}&&\left\{\begin{array}{l}d{\boldsymbol{B}}(s)\cdot d{\boldsymbol{B}}(s)={\boldsymbol{I}}{ds},\\ d{\boldsymbol{B}}(s)({\beta }^{i}(s){ds})=0,\\ ({\beta }^{i}(s){ds})({\beta }^{j}(s){ds})=0,\end{array}\right.\end{eqnarray*}$$

where ${\boldsymbol{I}}$ is a N × N unit matrix. In other words, they are evaluated as $d{\boldsymbol{B}}(s)\sim \sqrt{{ds}}$ and ${dA}(s)\sim {ds}$ in the expansion and taken up to order ds such as;

$$\begin{eqnarray*}&&d\varphi (X(t))=\displaystyle \frac{\partial \varphi }{\partial {x}^{i}}(X(t)){{dX}}^{i}(t)+\displaystyle \frac{1}{2}\displaystyle \frac{{\partial }^{2}\varphi }{\partial {x}^{i}\partial {x}^{j}}(X(t)){{dX}}^{i}(t){{dX}}^{j}(t)\,+\,\cdots ,\end{eqnarray*}$$

where

$$\begin{eqnarray*}{{dX}}^{i}(t) & = & {{dA}}^{i}(t)+{{\boldsymbol{\sigma }}}^{i}(t)\cdot d{\boldsymbol{B}}(t)={\beta }^{i}(t){dt}+{{\boldsymbol{\sigma }}}^{i}(t)\cdot d{\boldsymbol{B}}(t),\\ {{dX}}^{i}(t){{dX}}^{j}(t) & = & ({{\boldsymbol{\sigma }}}^{i}(t)\cdot d{\boldsymbol{B}}(t))({{\boldsymbol{\sigma }}}^{j}(t)\cdot d{\boldsymbol{B}}(t))=({{\boldsymbol{\sigma }}}^{i}(t)\cdot {{\boldsymbol{\sigma }}}^{j}(t)){dt}.\end{eqnarray*}$$

Then, the Itô formula follows from them. Above expressions must be understood as a shorthand for the integration form, and a following remark for one and two random variables Xⁱ(t) can also be written as;

$$\begin{eqnarray*}d(f(t){X}^{i}(t)) & = & f(t)({{dX}}^{i}(t))+({df}(t)/{dt}){X}^{i}(t){dt},\\ d({X}^{i}(t){X}^{j}(t)) & = & {X}^{i}(t)({{dX}}^{j}(t))+({{dX}}^{i}(t)){X}^{j}(t)+({{dX}}^{i}(t))({{dX}}^{j}(t)),\end{eqnarray*}$$

where f(t) ∈ C². Above formulae are referred as to the Stochastic Leibniz rule. Integration forms of above expressions can be given as:

$$\begin{eqnarray*}f(t){X}^{i}(t)-f(0){X}^{i}(0) & = & {\displaystyle \int }_{0}^{t}f(s){{dX}}^{i}(s)+{\displaystyle \int }_{0}^{t}\displaystyle \frac{{df}(s)}{{ds}}{X}^{i}(s){ds},\\ & = & {\displaystyle \int }_{0}^{t}\left({\beta }^{i}(s)+\displaystyle \frac{{df}(s)}{{ds}}{X}^{i}(s)\right){ds}+{\displaystyle \int }_{0}^{t}{{\boldsymbol{\sigma }}}^{i}(s)\cdot d{\boldsymbol{B}}(s),\\ {X}^{i}(t){X}^{j}(t)-{X}^{i}(0){X}^{j}(0) & = & {\displaystyle \int }_{0}^{t}({X}^{i}(s){\beta }^{j}(s)+{X}^{j}(s){\beta }^{i}(s)){dt}+{\displaystyle \int }_{0}^{t}({X}^{i}(s){{\boldsymbol{\sigma }}}^{j}(s)+{X}^{j}(s){{\boldsymbol{\sigma }}}^{i}(s))\cdot d{\boldsymbol{B}}(s)\\ & & +{\displaystyle \int }_{0}^{t}({\boldsymbol{\sigma }}{(t)}^{i}\cdot {\boldsymbol{\sigma }}{(t)}^{j}){dt}.\end{eqnarray*}$$

The second line is just a special case of the Itô formula with φ(X(t)) = Xⁱ(t) X^j(t).

The stochastic integration of a function f(x) by the Brownian motion is given as;

$$\begin{eqnarray}&&{\int }_{0}^{t}f(s){dB}(s)=B(t;{\sigma }^{2}),\end{eqnarray} \tag{ 70 }$$

where $B(t;{\sigma }^{2})$ is the Gaussian process with a variance of

$$\begin{eqnarray}&&{\sigma }^{2}={\int }_{0}^{t}f{(s)}^{2}{ds},\end{eqnarray} \tag{ 71 }$$

and mean value of zero. One of an important property of the Gaussian distribution is a reproducing property, which means the convolution of two Gaussian distributions make a Gaussian distribution.

First, let us introduce an iterated classical (non-stochastic) integration of one-forms ${\omega }_{i=1,\cdots ,n}$ along the curvilinear path $\{\gamma \in {\rm{\Gamma }}\}$ as defined in section 4.1. A pull of ω_i using the path γ is expressed as ${\gamma }^{* }{\omega }_{i}={f}_{i}(t){dt}$, and then, the iterative integration can be introduced as;

$$\begin{eqnarray*}&&{\int }_{\gamma }{\omega }_{1}\cdots {\omega }_{n}={\int }_{{\delta }_{n}}{\gamma }^{* }{\omega }_{1}\cdots {\gamma }^{* }{\omega }_{n},={\int }_{0}^{T}{{dt}}_{n}\left({f}_{n}({t}_{n}){\int }_{0}^{{t}_{n}}{{dt}}_{n-1}\left(\cdots \left({f}_{2}({t}_{2}){\int }_{0}^{{t}_{2}}{{dt}}_{1}{f}_{1}({t}_{1})\right)\cdots \right)\right),\end{eqnarray*}$$

using simplexes δ_n defined as 0 ≤ t₁ ≤ ⋯ ≤ t_n ≤ T. A natural extension of this iterative integration to the iterative stochastic integrations might be;

$$\begin{eqnarray*}&&{I}_{n}(\{{f}_{i}\})={\int }_{0}^{T}{dB}({t}_{n})\left({f}_{n}({t}_{n}){\int }_{0}^{{t}_{n}}{dB}({t}_{n-1})\left(\cdots \left({f}_{2}({t}_{2}){\int }_{0}^{{t}_{2}}{dB}({t}_{1}){f}_{1}({t}_{1})\right)\cdots \right)\right).\end{eqnarray*}$$

The existence of this multiple integrations can be shown using approximation of f_i being step functions [40]. It is known that the stochastic integration for the unit function can be represented by using the parameterized-Hermite polynomials as follows [40];

$$\begin{eqnarray*}&&{I}_{n}(\{{\bf{1}}\})={\int }_{0}^{T}{dB}({t}_{n})\left({\int }_{0}^{{t}_{n}}{dB}({t}_{n-1})\left(\cdots \left({\int }_{0}^{{t}_{2}}{dB}({t}_{1})\right)\cdots \right)\right)\,=\,{H}_{n}(B(T)-B(0),T-0),\end{eqnarray*}$$

where H_n can be expressed as;

$$\begin{eqnarray*}&&{H}_{n}(x,t)=\displaystyle \frac{{(-t)}^{n}}{n!}\exp \left(\displaystyle \frac{{x}^{2}}{2t}\right)\displaystyle \frac{{d}^{n}}{{{dx}}^{n}}\exp \left(-\displaystyle \frac{{x}^{2}}{2t}\right).\end{eqnarray*}$$

One of interesting examples of the iterated stochastic integration is a stochastic area introduced by Lèvy [46]:

Definition D.3 

$$\begin{eqnarray*}{S}_{\omega }(T) & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{0}^{T}[{{dB}}_{\omega }^{1}(t),{{dB}}_{\omega }^{2}(t)]=\displaystyle \frac{1}{2}\left\{{\displaystyle \int }_{0}^{T}\left({\displaystyle \int }_{0}^{t}{{dB}}_{\omega }^{1}(s)\right){{dB}}_{\omega }^{2}(t)-{\displaystyle \int }_{0}^{T}\left({\displaystyle \int }_{0}^{t}{{dB}}_{\omega }^{2}(s)\right){{dB}}_{\omega }^{1}(t)\right\},\\ & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{0}^{T}({B}_{\omega }^{1}(t){{dB}}_{\omega }^{2}(t)-{B}_{\omega }^{2}(t){{dB}}_{\omega }^{1}(t)),\end{eqnarray*}$$

where ${{\boldsymbol{B}}}_{\omega }(t)=({{dB}}_{\omega }^{1}(t),{{dB}}_{\omega }^{2}(t))$, with $t\geqslant 0$, is a 2-dimensional Brownian motion.

This quantity can be interpreted as the area surrounding by trajectory of ${{\boldsymbol{B}}}_{\omega }$ and straight line connecting between initial and final points of the trajectory.

Let us consider distributions of stochastic processes in the sense of Schwartz. The real Hilbert-space of square integrable functions is taken as a set of test functions as $E\subset H={L}^{2}(T\in {\mathbb{R}})$. A space of distribution, ${E}^{* }$, is a dual space of E, which has a relation of

$$\begin{eqnarray*}&&E\subset H\subset {E}^{* }.\end{eqnarray*}$$

A standard bilinear form is written as $\langle x,\xi \rangle ,$ where $\xi \in {E}^{* },\,x\in E$. When ξ ∈ H, this is nothing but an inner product of two vectors on the Hilbert space. A norm of two elements in the Hilbert space is denoted as $\parallel x{\parallel }^{2}=\langle x,x\rangle$. A set of stochastic processes defined on some probabilistic space is considered to be parameterized by x ∈ E such as $\{X(x)| x\in E\}$. For the probabilistic distribution (measure) μ on ${E}^{* }$, the characteristic functional is defined as

$$\begin{eqnarray*}&&{C}_{X}(x)={\int }_{{E}^{* }}\exp (i\langle \xi ,x\rangle )\mu (d\xi ).\end{eqnarray*}$$

The characteristic functional must have following characteristics:

• 1.  
  C_X is continuous on E,
• 2.  
  C_X is positive definite,
• 3.  
  ${C}_{X}(0)=1$.

The measure μ(dξ) on ${E}^{* }$ is uniquely determined by the characteristic functional, thanks to the Bochner–Minlon theorem.

A vector space of the standard bilinear form, $\langle x,f\rangle ,\,(f\in {L}^{2}({{\mathbb{R}}}^{1}))$ is considered. This vector space becomes the Hilbert space with the inner product as the covariance integral and the subspace of L²(μ). We denote this Hilbert space as ${H}_{1}^{(1)}$ and its dual space as ${H}_{1}^{(-1)}$. A subscript 1 means the Hilbert space defined on a one-parameter space. (±1) is put for future convenience. Isomorphism ${H}_{1}^{(1)}\cong {L}^{2}({{\mathbb{R}}}^{1})$ can be obtained under the isomorphic map $f\leftrightarrow \langle x,f\rangle$. Thus the Gel'fand triple of ${H}_{1}^{(1)}\subset H\subset {H}_{1}^{(-1)}$ is obtained.

Example F.1 The measured space induced by the characteristic function of

$$\begin{eqnarray*}&&C(x;{\sigma }^{2})=\exp \left(-\displaystyle \frac{{\sigma }^{2}}{2}\parallel x{\parallel }^{2}\right)\end{eqnarray*}$$

is named White noise with a variance of ${\sigma }^{2}$, which denoted as $\dot{B}(t)$. Here $\parallel \bullet \parallel$ is a Hilbertian norm.

The probabilistic distribution of white nose is the Gaussian distribution with mean value of zero and variance σ². It is easily confirmed that the white noises with different variances are mutually disjoint.

Example F.2 The characteristic functional of

$$\begin{eqnarray*}&&C^{\prime} (\xi )=\exp \left(-\displaystyle \frac{1}{2}\parallel \xi ^{\prime} {\parallel }^{2}\right)\end{eqnarray*}$$

induces a process of second derivative of the Brownian motion, $\mathop{B}\limits^{ \prime\prime }(t)$ for $-\infty \lt t\lt \infty$.

The measure of this process, μ', is defined on the same space as that of the original white noise, however those are the mutually disjoint each other. If the support of the original measure is X, the measure μ' gives

$$\begin{eqnarray*}&&\mu (X)=1\Rightarrow \mu ^{\prime} (X)=0,\end{eqnarray*}$$

and vice versa.

The definition of Brownian motion and white noise can be extended to a multi-dimensional parameter space.

Definition F.3 If a set of Random variables, $B(x)=\{B(x),x\in {{\mathbb{R}}}^{d}\}$ satisfies the following properties:

• 1.  
  $B(x)$ is a Gaussian system,
• 2.  
  $E[B(x)]=0$,
• 3.  
  Covariant matrix is

  $$\begin{eqnarray*}&&{\rm{\Gamma }}[x,y]=E[B(x)B(y)]=\displaystyle \frac{1}{2}(| x| +| y| -| x-y| ),\end{eqnarray*}$$

it is called Brownian motion.

From the definition, B(o) = 0 can be obtained, where o = (0, 0, ⋯, 0) is the origin of the parameter space. A proof of existence of multi-dimensional Brownian motions can be found in [21].

In order to define the multi-dimensional white noise, Hilbert space, its dual space and Gel'fand triple defined in the previous subsection are extended to ${E}_{d}\subset {H}_{d}\cong {L}^{2}(x\in {{\mathbb{R}}}^{d}),{E}_{d}^{* }$ and ${E}_{d}\subset {H}_{d}\subset {E}_{d}^{* }$. On the dual space ${E}_{d}^{* }$, a completely additive measure space $({E}_{d}^{* },{\mathscr{B}},\mu )$ can be constructed. According to notations for th one-parameter case, we can denote the Gel'fand triple as

$$\begin{eqnarray*}&&{H}_{d}^{(1)}\subset {H}_{d}\subset {H}_{d}^{(-1)}\end{eqnarray*}$$

Existence of the completely additive measure associated with the characteristic function of

$$\begin{eqnarray*}C(\xi ) & = & {\displaystyle \int }_{{E}_{d}^{* }}\exp (i\langle x,\xi \rangle )d\mu (x),\\ & = & \exp \left(-\displaystyle \frac{1}{2}\parallel \xi {\parallel }^{2}\right),\end{eqnarray*}$$

where ξ ∈ E_d and $\parallel \bullet \parallel$ is a Hilbertian norm, is ensured by the Bochner–Minlos theorem.

Definition F.4 A stochastic process induced by this characteristic function of equation (73) is called multi-dimensional white noise. The white noise induced by the multi-dimensional Brownian motion $B(x)$ is denoted as ${B}_{;\mu }(x)$.

F.2.1. Polynomial of white noise

Further extension of the white-noise space for polynomials of the white noise is possible [21, 40]. At first, a n-dimensional white-noise functional space is introduced as

• 1.  
  Test functional space ${H}_{d}^{(n)}$: a symmetric (n + 1)/2 dimensional Sobolev space on ${{\mathbb{R}}}^{n}$.
• 2.  
  Hilbert space H_n: a symmetric tensor product of n square-integrable functions.
• 3.  
  White-noise functional space ${H}_{d}^{(-n)}$: a dual space of the test functional space, which is an isomorphic with a symmetric −(n + 1)/2 dimensional Sobolev space on ${{\mathbb{R}}}^{n}$.

The Gel'fand triple can be obtained as ${H}_{d}^{(n)}\subset {H}_{d}\subset {H}_{d}^{(-n)}.$ Power functionals of the white noise are a element of the dual space (${B}_{;\bullet }^{n}\in {H}_{d}^{(-n)}$).

A polynomial of the white-noise functional can be defined on the space ${\bigotimes }_{n}\,{c}_{n}{H}_{n}^{(-n)}$ with the Gel'fand triple of

$$\begin{eqnarray*}&&\mathop{\displaystyle \bigotimes }\limits_{n}\displaystyle \frac{1}{{c}_{n}}{H}_{n}^{(n)}\subset ({L}^{2})\subset \mathop{\displaystyle \bigotimes }\limits_{n}\,{c}_{n}{H}_{n}^{(-n)},\end{eqnarray*}$$

where c_n is a positive monotonically non-increasing series.