Following explanation is due to Feynman and Hibbs [25] but I cited here from Albeverio and Hoegh-Krohn [26] .

Let consider the representation formula for the solution of Schrödinger equa-tion on ${ℝ}^d$

$i\hslash \frac{\partial }{\partial t}\psi \left(q,t\right)=-\frac{{\hslash }^2}{2m}\text{Δ}\psi \left(q,t\right)+V\left(q\right)\psi \left(q,t\right),$

with the initial data $\psi \left(q,0\right)=\underset{\_}{\psi }\left(q\right)$. Decomposing

$H=H_0+V\text{with}{H}_0=-\frac{{\hslash }^2}{2m}\text{Δ},$

and assuming that H is selfadjoint in $L^2\left({ℝ}^d\right)$, we have the solution $\psi \left(q,t\right)={\text{e}}^{-i{\hslash }^{-1}tH}\underset{\_}{\psi }\left(q\right)$ by Stone’s theorem¹⁴. On the other hand, Lie-Trotter-Kato’s product formula¹⁵ asserts that even though $\left[H_0,V\right]\ne 0$, we have

${\text{e}}^{-i{\hslash }^{-1}tH}={\text{e}}^{-i{\hslash }^{-1}t\left(H_0+V\right)}=\underset{n\to \infty }{\text{lim}}{\left({\text{e}}^{-i{\hslash }^{-1}\left(t/n\right)V}{\text{e}}^{-i{\hslash }^{-1}\left(t/n\right)H_0}\right)}^n.$

Since

${\text{e}}^{-i{\hslash }^{-1}t{H}_0}u\left(q\right)={\left(2\pi i\hslash t/m\right)}^{-d/2}{\displaystyle {\int }_{{ℝ}^d}\text{d}{q}^{\prime }{\text{e}}^{im{\left(q-q^{\prime }\right)}^2/\left(2\hslash t\right)}u}\left(q^{\prime }\right),$

we have

${\text{e}}^{-i{\hslash }^{-1}\left(t/n\right)V}{\text{e}}^{-i{\hslash }^{-1}\left(t/n\right)H_0}u\left(q\right)={\text{e}}^{-i{\hslash }^{-1}\left(t/n\right)V\left(q\right)}{\displaystyle {\int }_{{ℝ}^d}\text{d}{q}^{\prime }{\text{e}}^{im{\left(q-q^{\prime }\right)}^2/\left(2\hslash t/n\right)}u}\left(q^{\prime }\right),$

we get,

$\begin{array}{l}{\left(2\pi i\hslash t/m\right)}^{-d/2}{\displaystyle {\int }_{{ℝ}^d}\text{d}{q}^{\prime }{\text{e}}^{im{\left(q-q^{\prime }\right)}^2/\left(2\hslash t\right)}u}\left(q^{\prime }\right)\\ ={\left(2\pi i\hslash t/m\right)}^{-dn/2}{\displaystyle {\int }_{{ℝ}^{dn}}\text{d}{q}_0\cdots \text{d}{q}_{n-1}{\text{e}}^{i\hslash S_t^{\text{*}}\left(q_n,\cdots ,q_0\right)}u\left(q_0\right)}\end{array}$

with $q^{\prime }=q_0$ and $q_n=q$, where

$S_t^{\text{*}}\left(q_n,\cdots ,q_0\right)=\underset{j=1}{\overset{n}{{\displaystyle \sum }}}\left[\frac{m}{2}\frac{{\left(q_j-q_{j-1}\right)}^2}{{\left(t/n\right)}^2}-V\left(q_j\right)\right]\frac{t}{n}.$

Taking ${\gamma }^{\text{*}}\left(\tau \right)$ on $\left[0,t\right]$ as a zigzag path from $q_0$ at time 0, passing through

$\gamma \left({\tau }_j\right)=q_j$ $j=0,\cdots ,n$ where ${\tau }_j=j\frac{t}{n}$ and $q_0,\cdots ,q_n$ are given points in ${ℝ}^d$.

Feynman regards this $S_t^{\text{*}}\left(q_n,\cdots ,q_0\right)$ as a Riemann approximation for the classical action $S_t\left(\gamma \right)$ along the path $\gamma \left(\tau \right)$:

$S_t\left(\gamma \right)={\displaystyle {\int }_0^t\text{d}\tau }\left[\frac{m}{2}{\left(\frac{\text{d}\gamma }{\text{d}\tau }\right)}^2-V\left(\gamma \left(\tau \right)\right)\right].$

Assuming ${\gamma }^{\text{*}}\left(\tau \right)$ is an approximation for any classical path belonging to the path space

${\Gamma }_{\left(t,\underset{\_}{q},\bar{q}\right)}=\left\{\gamma \in AC\left(\left[0,t\right]:{ℝ}^d\right);\gamma \left(0\right)=\underset{\_}{q},\gamma \left(t\right)=\bar{q}\right\}$

and using assumed “Lebesgue-like measure” ${\text{d}}_F\gamma$ on ${\Gamma }_{\left(t,\underset{\_}{q},\bar{q}\right)}$, we express its solution as

$\psi \left(\bar{q},t\right)={\displaystyle {\int }_{\gamma \left(t\right)=\bar{q}}{\text{d}}_F\gamma {\text{e}}^{i{\hslash }^{-1}{S}_t\left(\gamma \right)}\underset{\_}{\psi }}\left(\gamma \left(0\right)\right),$

called Feynman’s path integral expression for the solution of Schrödinger equation by Feynman’s time-slicing method.

If we permit this integral representation of the solution with operations under integral sign admitted, when making $\hslash \to 0$, we have the main contribution

stems from the stationary point ${\gamma }_c(\cdot )$, that is, ${\frac{\delta S_t\left(\gamma \right)}{\delta \gamma }|}_{\gamma ={\gamma }_c}=0$. This expression

with above interpretation is persuasive to claim that classical mechanical equation appeared from quantum one when making $\hslash \to 0$.

Using Feynman’s idea rather conversely, Fujiwara [28] [30] constructed a funda-mental solution of Schrödinger equation. From a given Lagrangian $L\left(q,\stackrel{\dot{}}{q}\right)$, he found a classical orbit ${\gamma }_c(\cdot )\in {\Gamma }_{\left(t,\underset{\_}{q},\bar{q}\right)}$ of that Lagrangian mechanics:

$\ddot{q}\left(s\right)-\frac{\delta L}{\delta q}\left(q\left(s\right),\stackrel{\dot{}}{q}\left(s\right)\right)=0\text{with}q\left(0\right)=\underset{\_}{q},\stackrel{\dot{}}{q}\left(0\right)={\underset{\_}{p}}^{\text{*}}$

such that $\bar{q}=q\left(t,\underset{\_}{q},{\underset{\_}{p}}^{\text{*}}\right)$, that is, ${\gamma }_c\left(0\right)=\underset{\_}{q}$, ${\gamma }_c\left(t\right)=\bar{q}$. Then, the action integral corresponding to ${\gamma }_c$ is given

$S_t\left({\gamma }_c\right)=S_t\left(\bar{q},\underset{\_}{q}\right)={\displaystyle {\int }_0^t\text{d}s}L\left({\gamma }_c\left(s\right),{\stackrel{\dot{}}{\gamma }}_c\left(s\right)\right),$

moreover defining van Vleck determinant as

$D\left(t,\bar{q},\underset{\_}{q}\right)=\text{det}\left(\frac{{\partial }^2}{\partial \underset{\_}{q}\partial \bar{q}}{S}_t\left(\bar{q},\underset{\_}{q}\right)\right),$

he finally defined a short time propagator as a FIOp (Fourier Integral Operator)

$T_t\underset{\_}{\psi }\left(\bar{q}\right)={\displaystyle {\int }_{{ℝ}^d}\text{d}\underset{\_}{q}}A\left(t,\bar{q},\underset{\_}{q}\right){\text{e}}^{i{\hslash }^{-1}{S}_t\left(\bar{q},\underset{\_}{q}\right)}\underset{\_}{\psi }\left(\underset{\_}{q}\right)$

where

$A\left(t,\bar{q},\underset{\_}{q}\right)=\sqrt{D\left(t,\bar{q},\underset{\_}{q}\right)}.$

Above Fujiwara’s process is justified when ${\text{sup}}_q\left|{\partial }_q^{\alpha }V\left(q\right)\right|\le C,\left(\left|\alpha \right|\ge 2\right)$. That is, under this condition, not only there exists a unique classical orbit, and above quantities are well-defined, but also $T_t$ defines a bounded linear operator on $L^2\left({ℝ}^d\right)$ with the property

$\Vert T_{t+s}u-T_s{T}_{t}u\Vert \le C\left(t^2+s^2\right)\Vert u\Vert .$

Moreover, following operator $E_t$ is defined by

$\underset{n\to \infty }{\text{lim}}\Vert {\left(T_{t/n}\right)}^{n}u-E_{t}u\Vert =0$

which gives a parametrix of the given initial value problem of Schrödinger equation [28] . As is known by the kernel theorem of Schwartz [29] , it is clear there exists a distribution kernel of the fundamental solution such that

$\left(E_{t}u\right)\left(\bar{q}\right)={\displaystyle {\int }_{{ℝ}^d}\text{d}\underset{\_}{q}}K\left(t,\bar{q},\underset{\_}{q}\right)u\left(\underset{\_}{q}\right).$

How to represent this kernel as concretely as possible? is the next problem, which is attacked by Fujiwara [30] . Finally, not only how to characterize the long-time behavior¹⁶ of $K\left(t,\bar{q},\underset{\_}{q}\right)$ [30] , but also how to use the # products of FIOps by Kumano-go group [32] - [34] are the problems reconsidered.

Remark 2.1. (1) [Feynman-Kac formula, see Simon [35] ]: Stimulated by Feyn-man’s idea, M. Kac represents the solution of

$\frac{\partial }{\partial t}u\left(q,t\right)=\sigma \text{Δ}u\left(q,t\right)-V\left(q\right)u\left(q,t\right),$

using Wiener measure $\text{d}W\left(\gamma \right)$ by

$u\left(q,t\right)={\displaystyle \int \text{d}W}\left(\gamma \right){\text{e}}^{-{\displaystyle {\int }_0^{t}V\left(\gamma \left(s\right)+q\right)\text{d}s}}\underset{\_}{u}\left(\gamma \left(0\right)+\underset{\_}{q}\right).$

(2) [Problem of R/12]: For a given Riemann manifold $\left(M,g_{jk}\right)$, applying Fujiwara’s idea to the heat equation on manifold, that is, purely imaginary time quantization, Inoue and Maeda [36] got the notorious term R/12 where R is the scalar curvature of the Riemann metric $g_{jk}\left(q\right)\text{d}{q}^j\text{d}{q}^k$. Curiously, this term was also derived as the most probable path calculation in probability theory, Takahashi and Watanabe [37] , Fujita and Kotani [38] .

Rather recently the term R/12 appeared in Fukushima [39] , though he tries to

construct a fundamental solution of given equation $i\frac{\partial }{\partial t}-\frac{1}{2}{\text{Δ}}_g+\frac{R}{12}$. That means

the origin of the term R/12 isn’t explained there from the quantization process of $g_{jk}\left(q\right)\text{d}{q}^j\text{d}{q}^k$¹⁷ like physicists arguments, for example, DeWitt [40] or Bastianelli, Corradini and Vassura [41] .

For a given Lagrangian $L\left(q,\stackrel{\dot{}}{q}\right)$, above process is denoted formally as

$K={\displaystyle \int {\text{d}}_F\gamma {\text{e}}^{i{\hslash }^{-1}S\left(\gamma \right)}}\text{or}K\left(t,\bar{q},\underset{\_}{q}\right)={\displaystyle {\int }_{{\Gamma }_{\left(t,\bar{q},\underset{\_}{q}\right)}}{\text{d}}_F\gamma {\text{e}}^{i{\hslash }^{-1}S\left(\gamma \right)}}$

where

$S\left(\gamma \right)={\displaystyle {\int }_0^t\text{d}s}L\left(\gamma \left(s\right),\stackrel{\dot{}}{\gamma }\left(s\right)\right),\text{for}\text{each}\gamma \in {\Gamma }_{\left(t,\bar{q},\underset{\_}{q}\right)}.$

What does it become when Hamiltonian $H\left(q,p\right)$ is given? Here, as is well-known, $H\left(q,p\right)$ is related to $I\left(q,\stackrel{\dot{}}{q}\right)$ by Legendre transformation. How to give meaning to the following formal expression?:

$K={\displaystyle \int {\text{d}}_{F}p{\text{d}}_{F}q{\text{e}}^{i{\hslash }^{-1}{\displaystyle \int \text{d}s}\left(\stackrel{\dot{}}{q}p-H\left(q,p\right)\right)}}.$

Under same assumption as above, there exists a classical orbit $\left(q\left(t,\underset{\_}{q},\underset{\_}{p}\right),p\left(t,\underset{\_}{q},\underset{\_}{p}\right)\right)$ to the Hamilton equation for the given $H\left(q,p\right)$ with initial data $\left(\underset{\_}{q},\underset{\_}{p}\right)$. For any fixed $\underset{\_}{p}$, for the map $\underset{\_}{q}\to \bar{q}=q\left(t,\underset{\_}{q},\underset{\_}{p}\right)$, there exists an inverse map $\underset{\_}{q}=x\left(t,\bar{q},\underset{\_}{p}\right)$ for sufficiently small $\left|t\right|$:

$\bar{q}=q\left(t,x\left(t,\bar{q},\underset{\_}{p}\right),\underset{\_}{p}\right)\text{and}\underset{\_}{q}=x\left(t,q\left(t,\underset{\_}{q},\underset{\_}{p}\right),\underset{\_}{p}\right)$

Using these, we put

$S\left(t,\bar{q},\underset{\_}{p}\right)={\left(\underset{\_}{q}\underset{\_}{p}-S_0\left(t,\underset{\_}{q},\underset{\_}{p}\right)\right)|}_{\underset{\_}{q}=x\left(t,\bar{q},\underset{\_}{p}\right)}$

and define

$D\left(t,\bar{q},\underset{\_}{p}\right)=\text{det}\left({\partial }_{\bar{q}}{\partial }_{\underset{\_}{p}}S\left(t,\bar{q},\underset{\_}{p}\right)\right).$

Finally we define

$T_{t}u\left(\bar{q}\right)={\left(2\pi \hslash \right)}^{-d/2}{\displaystyle {\int }_{{ℝ}^d}\text{d}\underset{\_}{p}}D{\left(t,\bar{q},\underset{\_}{p}\right)}^{1/2}{\text{e}}^{i{\hslash }^{-1}S\left(t,\bar{q},\underset{\_}{p}\right)}\hat{u}\left(\underset{\_}{p}\right).$

This gives a parametrix for Schrödinger equation for short time. More precisely, see, Inoue [8] .

Remark 2.2. (1) In order to consider a classical orbit connecting 2 points $\bar{q}$ and $\underset{\_}{q}$ in t, we need 2nd order time derivatives in corresponding classical mecha-nical equation. Then, how one treats Weyl or Dirac equation? Hint is given in the next section.

(2) [Hamiltonian formulation on manifolds] Above mentioned works containing physicist papers, are formulated in Lagrangian case. Since it needs works to formulate Fourier transformations on manifolds, physicist papers such as Field [42] [43] , claim something interesting without any mathematical estimates. Seemingly to use directly Hamilton formulation in quantization as in [8] , it is indispensable to define Fourier transformation even on general manifold, see Widom [44] .

There are several reasons at least for me except the claim of Manin [48] : (i) As is written in p. 355 of [25] , “spin” has been the object outside Feynman’s procedures at that time:

… path integrals suffer grievously from a serious defect²⁰. They do not permit a discussion of spin operators or other such operators in a simple and lucid way. They find their greatest use in systems for which coordinates and their conjugate momenta are adequate. Nevertheless, spin is a simple and vital part of real quantum-mechanical systems. It is a serious limitation that the half-integral spin of the electron does not find a simple and ready representation. It can be handled if the amplitudes and quantities are considered as quarter-nions instead of ordinary complex numbers, but the lack of commutativity of such numbers is a serious complication.

(ii) Berezin, who invented the second quantization, proposed with Marinov [49] to treat boson and fermion on equal footing. That is, instead of $ℝ$ or $ℂ$, he claims the necessity to construct a scalar-like, non-commutative ground field, where lives electron and photon equally.

(iii) Witten [15] explained the meaning of supersymmetric idea in physics to mathematician by new derivation of Morse inequalities using exterior differential operations. On the other hand, Getzler [16] tries to understand the Atiyah-Singer Index Theorem by extending Weyl’s theorem to that of the heat-type equation on the supermanifolds, but from my point of view, he doesn’t treat new variables on equal footing. But be careful, in many case, physicists usage of supersymmetry only means to use fermion or odd variables, not exactly treating some “symmetry” in ${\Re }^{m|n}$, for example Efetov’s works [17] [18] , Brezin [50] and Fyodorov [51] .

A claim “any Clifford algebra has a representation on Grassmann algebra”²¹ is straight forwardly explained for 2 × 2 matrices case. Though analogous argument works for a set of $2^{\ell }\times 2^{\ell }$ matrices because it has Clifford relation, but for a set of general N × N matrices, we need to find suitable algebra on which we need foundation of analysis. I imagine not only the work of Martin [52] [53] but also the one by Campoamor-Stursberg and Rausch de Traubenberg [54] give some hint on this. Especially $N=3$ will be interesting when we consider vorticity equa-tion for Euler or Navier-Stokes equation which is represented by 3 × 3 matrices. (Recall the construction of the classical solution of Euler equation by Kato [55] using method of characteristics since vorticity equation is reduced to scalar one when the space dimension is 2.)

Recall Pauli matrices ${\left\{{\sigma }_k\right\}}_{k=1}^3$:

${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_2=-i\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$

These matrices satisfy not only Clifford relation

${\sigma }_j{\sigma }_k+{\sigma }_k{\sigma }_j=2{\delta }_{jk},\text{where}j,k=1,2,3,$ (4.1)

but also the following for any $\left(j,k,\ell \right)$, an even permutation of $\left(1,2,3\right)$:

${\sigma }_j{\sigma }_k=i{\sigma }_{\ell }.$ (4.2)

Decompose any 2 × 2 matrix $\mathbb{A}$ by $\left\{{\sigma }_j\right\}$,

$\begin{array}{c}\mathbb{A}=\left(\begin{array}{ll}a\hfill & c\hfill \\ d\hfill & b\hfill \end{array}\right)\\ =\frac{a+b}{2}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+\frac{a-b}{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)+\frac{c+d}{2}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)+\frac{c-d}{2}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\\ =\frac{a+b}{2}{\mathbb{I}}_2+\frac{a-b}{2}{\sigma }_3+\frac{c+d}{2}{\sigma }_1+i\frac{c-d}{2}{\sigma }_2.\end{array}$ (4.3)

That is, a set of all 2 × 2 matrices has the Clifford algebra structure with Pauli matrices $\left\{{\sigma }_k\right\}$ as a basis.

We identify a vector as a function of Grassmann variables. Then, a matrix is regarded as a differential operator acting on a set of functions composed with Grassmann variables, which forms something-like field similar to real or complex number field but non-commutative.

Rather abruptly we prepare²² two odd variables ${\theta }_1,{\theta }_2$ having the following relations:

${\theta }_1\cdot {\theta }_2+{\theta }_2\cdot {\theta }_1=0,{\theta }_j\cdot {\theta }_j=0,\left(j=1,2\right)$ (4.4)

That is, they have product $\cdot$ having Grassmann relations and differentiation such as

$\frac{\partial }{\partial {\theta }_1}1=0,\frac{\partial }{\partial {\theta }_1}{\theta }_1=1,etc.$

Here, when we differentiate, bringing that variable in front, i.e.

$\frac{\partial }{\partial {\theta }_2}{\theta }_1\cdot {\theta }_2=\frac{\partial }{\partial {\theta }_2}\left(-{\theta }_2\cdot {\theta }_1\right)=\frac{\partial }{\partial {\theta }_2}\left({\theta }_2\cdot \left(-{\theta }_1\right)\right)=-{\theta }_1.$

More concretely, for example, taking two variables z₁ and z₂ in ${ℝ}^2$, construct differential forms $\text{d}{z}_1$, $\text{d}{z}_2$ with exterior product $\wedge$ and interior product such that

$\frac{{\partial }^2}{\partial {\theta }_1\partial {\theta }_2}{\theta }_1{\theta }_2={\partial }_{{\theta }_1}{\partial }_{{\theta }_2}{\theta }_1{\theta }_2={\partial }_{{\theta }_1}\left({\partial }_{{\theta }_2}{\theta }_1{\theta }_2\right)={\partial }_{{\theta }_1}\left({\partial }_{{\theta }_2}\left(-{\theta }_2{\theta }_1\right)\right)={\partial }_{{\theta }_1}\left(-{\theta }_1\right)=-1.$

They act on ${\Gamma }_0$ as,

$\begin{array}{l}{\sigma }_1\left(\theta ,{\partial }_{\theta }\right){\sigma }_2\left(\theta ,{\partial }_{\theta }\right)=i{\sigma }_3\left(\theta ,{\partial }_{\theta }\right),\\ {\sigma }_2\left(\theta ,{\partial }_{\theta }\right){\sigma }_3\left(\theta ,{\partial }_{\theta }\right)=i{\sigma }_1\left(\theta ,{\partial }_{\theta }\right),\\ {\sigma }_3\left(\theta ,{\partial }_{\theta }\right){\sigma }_1\left(\theta ,{\partial }_{\theta }\right)=i{\sigma }_2\left(\theta ,{\partial }_{\theta }\right).\end{array}$

Remark 4.1. (1) The action ${\sigma }_1\left(\theta ,{\partial }_{\theta }\right)$ on ${\Gamma }_0$ has a matrix representation

$♭{\sigma }_1\left(\theta ,{\partial }_{\theta }\right)♯\left(\begin{array}{l}{u}_0\hfill \\ u_1\hfill \end{array}\right)=\left(\begin{array}{l}{u}_1\hfill \\ u_0\hfill \end{array}\right),♭{\sigma }_1\left(\theta ,{\partial }_{\theta }\right)♯={\sigma }_1,etc.$

(2) Moreover, putting

${\Gamma }_1=\left\{v\left(\theta \right)=v_1{\theta }_1+v_2{\theta }_2|v_1,v_2\in ℂ\right\}\underset{♯}{\overset{♭}{\rightleftarrows }}{ℂ}^2=\left\{v=\left(\begin{array}{l}{v}_1\hfill \\ v_2\hfill \end{array}\right)|v_1,v_2\in ℂ\right\}$

we have

${\sigma }_j\left(\theta ,{\partial }_{\theta }\right)v\left(\theta \right)=0$ for any $v\in {\Gamma }_1$ and $j=1,2,3$.

We define Fourier transformation²³ for odd variables²⁴. Taking number k in ${ℝ}^{\times }=ℝ\backslash \left\{0\right\}$ or $i{ℝ}^{\times }$ called it as spin constant, we define

$\hat{u}\left(\pi \right)=k{\displaystyle {\int }_{{\Re }^{0|2}}\text{d}\theta }{\text{e}}^{-i{k}^{-1}\langle \theta |\pi \rangle }u\left(\theta \right),u\left(\theta \right)=k{\displaystyle {\int }_{{\Re }^{0|2}}\text{d}\pi }{\text{e}}^{i{k}^{-1}\langle \theta |\pi \rangle }\hat{u}\left(\pi \right).$

In the above, the integral interval ${\Re }^{0|2}$ may be considered only as a symbol, at least for the time being. Here, $\langle \theta |\pi \rangle ={\theta }_1{\pi }_1+{\theta }_2{\pi }_2$ and

$\begin{array}{l}{\langle \theta |\pi \rangle }^2=2{\theta }_1{\pi }_1{\theta }_2{\pi }_2=-2{\theta }_1{\theta }_2{\pi }_1{\pi }_2,\\ {\langle \theta |\pi \rangle }^j=\stackrel{j\text{times}}{\stackrel{︷}{\langle \theta |\pi \rangle \langle \theta |\pi \rangle \cdots \langle \theta |\pi \rangle }}=0\text{if}j\ge 3\end{array}$

we have

${\text{e}}^{a\langle \theta |\pi \rangle }=\underset{\ell =1}{\overset{\infty }{{\displaystyle \sum }}}\frac{a^{\ell }{\langle \theta |\pi \rangle }^{\ell }}{\ell !}=1+a\langle \theta |\pi \rangle -a^2{\theta }_1{\theta }_2{\pi }_1{\pi }_2.$

Remark 4.2. “Integration” should be considered preferable such that (i) all polynomials are integrable, (ii) linear with integrand and (iii) translation invariant, and “Berezin integration” satisfies these properties.

By Fourier transformation, differentiation is regarded as multiplication in dual space²⁵, and theory of pseudo-differential operators treats PDE with variable coefficients controlling error estimates.

Defining Weyl symbols of differential operators ${\left\{{\sigma }_j\left(\theta ,{\partial }_{\theta }\right)\right\}}_{j=1}^3$ as

$\{\begin{array}{l}{\sigma }_1\left(\theta ,\pi \right)={\theta }_1{\theta }_2+k^{-2}{\pi }_1{\pi }_2,\\ {\sigma }_2\left(\theta ,\pi \right)=i\left({\theta }_1{\theta }_2-k^{-2}{\pi }_1{\pi }_2\right),\\ {\sigma }_3\left(\theta ,\pi \right)=-i{k}^{-1}\langle \theta |\pi \rangle =-i{k}^{-1}\left({\theta }_1{\pi }_1+{\theta }_2{\pi }_2\right),\end{array}$ (4.7)

we have, for example,

$\begin{array}{c}\left({\hat{\sigma }}_3^{\text{w}}\left(\theta ,{\partial }_{\theta }\right)u\right)\left(\theta \right)=-ik{\displaystyle {\iint }_{{\Re }^{0|2}\times {\Re }^{0|2}}\text{d}\pi \text{d}{\theta }^{\prime }}{\text{e}}^{i{k}^{-1}\langle \theta -{\theta }^{\prime }|\pi \rangle }{\sigma }_3\left(\frac{\theta +{\theta }^{\prime }}{2},\pi \right)\left(u_0+u_1{{\theta }^{\prime }}_1{{\theta }^{\prime }}_2\right)\\ =u_0-u_1{\theta }_1{\theta }_2={\sigma }_3\left(\theta ,{\partial }_{\theta }\right)\left(u_0+u_1{\theta }_1{\theta }_2\right).etc.\end{array}$

Here, we used

${\displaystyle {\int }_{{\Re }^{0|2}}\text{d}\pi }{\text{e}}^{i{k}^{-1}\langle \theta -{\theta }^{\prime }|\pi \rangle }{\sigma }_3\left(\frac{\theta +{\theta }^{\prime }}{2},\pi \right)=-i{k}^{-1}\left({\theta }_1{\theta }_2-{{\theta }^{\prime }}_1{{\theta }^{\prime }}_2\right).$

Remark 4.3. (i) It takes many time to perceive the meaning of 1 in ${\sigma }_3\left(\theta ,{\partial }_{\theta }\right)$ appeared in (4.6), that is, 1 stems from Weyl quantization!

(ii) Artificially introduced spin constant k gives nice result only when $k{\hslash }^{-1}=1$, that is, the action integral obtained by Jacobi method satisfies Hamilton-Jacobi equation in case $k{\hslash }^{-1}=1$, therefore we assume $k=\hslash$!

Therefore, from (4.3), we have

$\left(♯\mathbb{A}u\right)\left(\theta \right)=\left[\frac{a+b}{2}+\frac{a-b}{2}{\sigma }_3\left(\theta ,{\partial }_{\theta }\right)+\frac{c+d}{2}{\sigma }_1\left(\theta ,{\partial }_{\theta }\right)+i\frac{c-d}{2}{\sigma }_2\left(\theta ,{\partial }_{\theta }\right)\right]u\left(\theta \right),$ (4.8)

whose Weyl symbol is

${\sigma }^{\text{w}}\left(♯\mathbb{A}♭\right)=\frac{a+b}{2}+\frac{a-b}{2}{\sigma }_3^{\text{w}}\left(\theta ,\pi \right)+\frac{c+d}{2}{\sigma }_1\left(\theta ,\pi \right)+i\frac{c-d}{2}{\sigma }_2\left(\theta ,\pi \right).$ (4.9)

Remark 4.4. In the theory of pseudo-differential operators, symbol calculus is the main ingredient where algebra is essentially $ℝ$. Whether analogous procedure works on superspace ${\Re }^{m|n}$? More explicitly, though the symbol of ${\sigma }_1\left(\theta ,{\partial }_{\theta }\right)$ is given as ${\theta }_1{\theta }_2+k^{-2}{\pi }_1{\pi }_2$, but whether the inverse of ${\sigma }_1\left(\theta ,{\partial }_{\theta }\right)$ is calculated di-rectly from ${\theta }_1{\theta }_2+k^{-2}{\pi }_1{\pi }_2$?

Consider the following initial value problem

${\partial }_t^{2}u-L\left(t,q,{\partial }_q\right)u=f\text{with}u\left(0,q\right)={\underset{\_}{u}}_0\left(q\right),u_t\left(0,q\right)={\underset{\_}{u}}_1\left(q\right).$ (4.10)

The following result is a part of Chi [13] in 1958:

Theorem 4.1 (An example of weakly hyperbolic equation) Let

$L\left(t,q,{\partial }_q\right)=t^2{\partial }_q^2+b{\partial }_q.$

Solving the above initial value Problem (4.10) with data $f=0$ and ${\underset{\_}{u}}_1=0$ when $b=4k+1$, $k\in {\mathbb{Z}}_{+}$, we have the solution given by

$u\left(t,q\right)=\underset{\ell =0}{\overset{k}{{\displaystyle \sum }}}\frac{2^{2\ell }k!}{\left(2\ell \right)!\left(k-\ell \right)!}{t}^{2\ell }{\underset{\_}{u}}_0^{\left(\ell \right)}\left(q+\frac{t^2}{2}\right).$

In this section, we derive this result by completely different method from Chi.

Applying change of variables,

$\xi =x+\frac{t^2}{2},\eta =x-\frac{t^2}{2}$

to $u\left(t,x\right)$, Chi reduces above problem to the Euler-Darboux equation for

$\tilde{u}\left(\xi ,\eta \right)=u\left(\sqrt{\xi -\eta },\frac{\xi +\eta }{2}\right)$ yielding,

$\frac{{\partial }^2\tilde{u}}{\partial \xi \partial \eta }-\frac{1-b}{4\left(\xi -\eta \right)}\frac{\partial \tilde{u}}{\partial \xi }+\frac{1+b}{4\left(\xi -\eta \right)}\frac{\partial \tilde{u}}{\partial \eta }=0$ (4.11)

with initial condition

$\tilde{u}\left(\xi ,\xi \right)={\underset{\_}{u}}_0\left(\xi \right),\underset{\xi -\eta \to 0}{\text{lim}}{\left(\xi -\eta \right)}^{1/2}\left(\frac{\partial \tilde{u}}{\partial \xi }-\frac{\partial \tilde{u}}{\partial \eta }\right)={\underset{\_}{u}}_1\left(\xi \right).$

Remark 4.5. For

$E\left(\alpha ,\beta \right)={\partial }_{\xi }{\partial }_{\eta }-\frac{\beta }{\xi -\eta }{\partial }_{\xi }+\frac{\alpha }{\xi -\eta }{\partial }_{\eta },$

with $\alpha =\frac{1+b}{4},\beta =\frac{1-b}{4}$, it gives (4.11). He uses the fact such that if $0<\alpha ,\beta <1$

then the solution of $E\left(\alpha ,\beta \right)u=0$ is expressed as the convergent Euler-Darboux integral

$\begin{array}{c}u\left(\xi ,\eta \right)=\frac{\text{Γ}\left(\alpha +\beta \right)}{\text{Γ}\left(\alpha \right)\text{Γ}\left(\beta \right)}{\displaystyle {\int }_0^1\text{d}t}{u}_0\left(\xi +\left(\eta -\xi \right)t\right)t^{\beta -1}{\left(1-t\right)}^{\alpha -1}\\ +\frac{\text{Γ}\left(1-\alpha -\beta \right)}{2\text{Γ}\left(1-\alpha \right)\text{Γ}\left(1-\beta \right)}{\left(\eta -\xi \right)}^{t-\alpha -\beta }{\displaystyle {\int }_0^1\text{d}t}{u}_1\left(\xi +\left(\eta -\xi \right)t\right)t^{-\alpha }{\left(1-t\right)}^{-\beta }.\end{array}$

Moreover, for other $\alpha ,\beta$, this integral diverges as it stands, he gives these as Riemann-Liouville integral.

Seemingly, Euler-Darboux equation has some relations to special or Painleve’s functions, therefore proceeding inversely, we might dream another perspective to these functions by using superanalysis?

Without reducing to Euler-Darboux equation, we give a simple minded proof (at least as a story) of Chi’s result applying superanalysis (see, a forthcoming paper [14] ).

Putting $u_0=u$, $u_1=u_t$ and assuming for the moment $f=0$, we make (4.10) to a system:

$i\frac{\partial }{\partial t}\left(\begin{array}{l}{u}_0\hfill \\ u_1\hfill \end{array}\right)=i\left(\begin{array}{cc}0& 1\\ L\left(t,{\partial }_q\right)& 0\end{array}\right)\left(\begin{array}{l}{u}_0\hfill \\ u_1\hfill \end{array}\right).$

Here, we multiply i to both sides, rather artificially. Changing a vector repre-sentation to non-commutative scalar one by putting $u\left(t,x,\theta \right)=u_0\left(t,x\right)+u_1\left(t,x\right){\theta }_1{\theta }_2$, we have

$\left(L\left(t,{\partial }_x\right){\theta }_1{\theta }_2-{\partial }_{{\theta }_1}{\partial }_{{\theta }_2}\right)\left(u_0+u_1{\theta }_1{\theta }_2\right)=u_1+L{u}_0{\theta }_1{\theta }_2\sim \left(\begin{array}{cc}0& 1\\ L\left(t,{\partial }_x\right)& 0\end{array}\right)\left(\begin{array}{l}{u}_0\hfill \\ u_1\hfill \end{array}\right).$

Here, $q\in ℝ$ is imbedded in $x\in {\Re }^{1|0}$ such that $X_{\text{B}}=q$. Defining

$\mathscr{H}\left(t,{\partial }_x,\theta ,{\partial }_{\theta }\right)=iL\left(t,{\partial }_x\right){\theta }_1{\theta }_2-i{\partial }_{{\theta }_1}{\partial }_{{\theta }_2}$

whose symbol is given by

$\mathscr{H}\left(t,\xi ,\theta ,\pi \right)=i\left(-t^2{\xi }^2+ib\xi \right){\theta }_1{\theta }_2+i{\pi }_1{\pi }_2$

and corresponding Hamilton-Jacobi equation is given by

${\mathcal{S}}_t+\mathscr{H}\left(t,{\mathcal{S}}_x,\theta ,{\mathcal{S}}_{\theta }\right)=0$ (4.12)

which is solved with the initial condition $\mathcal{S}\left(0,x,\xi ,\theta ,\pi \right)=\langle x|\xi \rangle +\langle \theta |\pi \rangle$.

In the supersmooth category, since we have

$\begin{array}{c}\mathcal{S}\left(t,x,\xi ,\theta ,\pi \right)=S\left(t,x,\xi \right)+X\left(t,x,\xi \right){\theta }_1{\theta }_2+Y\left(t,x,\xi \right){\theta }_1{\pi }_1\\ +\tilde{Y}\left(t,x,\xi \right){\theta }_2{\pi }_2+V\left(t,x,\xi \right){\theta }_1{\pi }_2+\tilde{V}\left(t,x,\xi \right){\theta }_2{\pi }_1\\ +Z\left(t,x,\xi \right){\pi }_1{\pi }_2+W\left(t,x,\xi \right){\theta }_1{\theta }_2{\pi }_1{\pi }_2,\end{array}$ (4.13)