HEISENBERG'S FUNDAMENTAL EQUATION AND QUANTUM FIELD THEORY WITH A FUNDAMENTAL LENGTH(Micro-Macro Duality in Quantum Analysis)

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HEISENBERG’S

FUNDAMENTAL

EQUATION AND

QUANTUM FIELD

THEORY

WITH

A

FUNDAMENTAL LENGTH

S. NAGAMACHIAND E. BR\"UNING

ABSTRACT. Heisenberg’s fundamental equation of the universe $[8, 9]$ has a

coupling constant $l$ which has the dimension of length [L]. We consider a

model which has acouplingconstant $l$ ofthe samedimension as Heisenberg’s

fundamental equation of the universe, and solved it. The solution is not an

operator-valued tempered distribution but an ultrahyperfunctions. The

con-stant $\ell=l/(\sqrt{2}\pi)$ is the fundamental length in the senseof[1], that is, events

occuning within the distance $\ell$ cannot be distinguished in the

framework of

Ultrahyperfunction Quantum Field Theory which has been developed recently

by the authors. CONTENTS 1. Introduction ₁ 2. Wightman axioms ₃ 3. Fundamental length ₃ 4. Ultrahyperfunction ₄

5. Axioms for ultrahyperfunction quantum

_fields

of mixed type 5

6. Path integral quantization ₆

7. Operator solution ₉

References ₁₁

1. INTRODUCTION

The basic relativistic equation of quantum _{mechanics called Dirac} _equation

$i \frac{\hslash}{c}\gamma_{\mu}\frac{\partial}{\partial x_{\mu}}\psi(x)-m\psi(x)=0,$ _{$x_{0}=ct,$}

$x_{1}=x,x_{2}=y,$$x_{3}=z$ (1.1)

contains

a

constants $c$ (velocityoflight) whiCh is the

fundamental

constant in

rel-ativity _{theory, and Planck’s constant} $h=2\pi\hslash$which is the

fundamental

_constant

in quantum _{mechanics. The dimension of} $c$ is $[LT^{-1}]$ and that of $h$ is _{$[ML^{2}T^{-1}]$}

.

W. Heisenberg thought that

a

fundamentalequation_{of Physics must} _also_contain

a

constant $l$ with the dimension of length [L].

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then the dimensions of any other quantity

can

be expressed in terms of

combi-nations of the basic constants $c,$ $h$ and $l$, e,g., time $[T]=[L]/[LT^{-1}]$,

or mass

as

$[M]=[ML^{2}T^{-1}]/([LT^{-1}][L])$

.

In 1958, Heisenberg and Pauli introduced the equation

$\frac{\hslash}{c}\gamma_{\mu}\frac{\partial}{\partial x_{\mu}}\psi(x)\pm l^{2}\gamma_{\mu}\gamma_{5}\psi(x)\overline{\psi}(x)\gamma^{\mu}\gamma_{5}\psi(x)=0$, (1.2)

which

was

later called the equation

_of

the universe and studied in $[4, 9]$

.

The

constant $l$ has the dimension [L] and is called the fundamental length of the

theory.

Unfortunately, nobody has been able to solve this equation. At present,

even

inthe

more

advanced Ramework of ultra hyperfunction quantum fieldtheory,

we

do not

see

how this equation could be solved. Accordingly

we

study

a

linearized version ofthisequations which enherits the important property of

a

fundamental

length $l$ and which first has been studied by Okubo [13]. This linearized version

is solvable in the

sense

of classical field theory. We write it in the form

$\{\begin{array}{l}\square \phi(x)+(\frac{m}{\hslash})^{2}\phi(x)=0(i\frac{\hslash}{c}\gamma^{\mu}\frac{\partial}{\partial x^{\mu}}-\tilde{m})\psi(x)=-2\gamma^{\mu}l^{2}\psi(x)\phi(x)\frac{\partial\phi(x)}{\partial x^{\mu}}\end{array}$ (1.3)

andpropose tosolve theseequationsby two methods, (I)constructingtheSchwinger functions of the fields $\phi(x)$ and $\psi(x)$, and (ii) constructing directly the

operator-valued generalized functions which satisfy the system of equations (1.3). In the

following

we

will work with the natural units $c=\hslash=1$

.

Then the system of

equations (1.3) reads

$(\square +m^{2})\phi(x)=0$ (1.4)

$(i\gamma^{\mu}\partial_{\mu}-\tilde{m})\psi=-2l^{2}\gamma^{\mu}\psi(x)\phi(x)\partial_{\mu}\phi(x)$

.

(1.5)

and they

are

the field equations of the following Lagrangian density:

$L(x)=L_{Ff}(x)+L_{Fb}(x)+L_{I}(x)$, (1.6)

$L_{Ff}(x)=\overline{\psi}(x)(i\gamma_{\mu}\partial^{\mu}-\tilde{m})\psi(x)$, (1.7)

$L_{Fb}(x)= \frac{1}{2}\{(\partial^{\mu}\phi(x))^{2}-m^{2}\phi(x)^{2}\}$, (1.8)

$L_{I}(x)=2l^{2}(\overline{\psi}(x)\gamma_{\mu}\psi(x))\phi(x)\partial^{\mu}\phi(x)$

.

(1.9)

Equatim (1.5) has

no

solutions in the axiomaticframework of Wightman, that

is, the field $\psi(x)$ ivnot

an

operator-valuedtempered distribution. But,

as we

are

going to show, Equation (1.5) has

a

solution $(\phi, \psi)$

as

operator-valued tempered

ultrahyperfunctionwhich satisfy the conditions of the framework of

a

relativistic

quantum field theory with

a

fundamental length

as

given in [1].

In Sections 2-5,

_we

_{discuss the Wightman axioms for} _the _fields _with

a

fun-damentallength, which

uses

the theory of tempered ultrahyperfunctions. In

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integral method. The Wightman functions

are

tempered ultrahyperfunctions.

In Section 7, the operator-valued tempered ultrahyperfunctions

are

constructed,

whichsatisfy the systemofequations (1.3). The Wightmanfunctions constructed

from the operator solutions coincide with those which

are

constructed by path

integral method. The detailed calculations

are

found in $[11, 2]$

2. WIGHTMAN AXIOMS

Wightman’s set of axioms consist of the following 7 conditions. A special attention is paid to the locality axiom WVI.

W.I (RelativiStic invariance of the state space).

W.II (Spectral property).

W.III (Ebcistence and uniqueness of the vacuum).

W.IV (Fields and temperedness).

W.V (Poincar\’e-covariance of the fields).

W.VI (Locality,

or

microcausality).

Any two field components $\phi_{j}^{(\kappa)}(x)$ and $\phi_{\ell}^{(\kappa’)}(y)$ either commute

or

anti-commute

under

a

spacelike separation of $x$ and$y$:

If $f$ and $g$ have space-like separated supports, then

$\phi_{j}^{(\kappa)}(f)\phi_{\ell}^{(\kappa’)}(g)\Psi\mp\phi_{\ell}^{(\kappa’)}(g)\phi_{j}^{(\kappa)}(f)\Psi=0$

for all $\Psi\in \mathcal{D}$, the

common

domain for all operator $\phi_{j}^{(\kappa)}(f)$

.

We express this by

saying

$\phi_{j}^{(\kappa)}(x)\phi_{\ell}^{(\kappa’)}(y)\Psi\mp\phi_{\ell}^{(\kappa’)}(y)\phi_{j}^{(n)}(x)\Psi=0$ for $(x-y)^{2}<0$

.

W.VII (Cyclicity of the vacuum).

3. FUNDAMENTAL LENGTH

The axiom W.VI says that two events which are space-likely separated

are

inde-pendent. Even If we replace W.VI by

a

weaker axIom

$\phi_{j}^{(\kappa)}(x)\phi_{\ell}^{(\kappa’)}(y)\Psi\mp\phi_{\ell}^{(\kappa’)}(y)\phi_{j}^{(\kappa)}(x)\Psi=0$ for $(x-y)^{2}<-\ell^{2}<0$, (3.1) which says that the two events which are separated by $\ell$

are

independent,

we

can

prove W.VI by usingthe other axioms. It Is not easy to weaken the condition of

locality ifthe field $\phi_{j}^{(\kappa)}(x)$ has the localizationproperty of Schwartz

distributions.

We must introduce generalized functions which have

more

general localization

properties than distrIbutions.

We indicate briefly

a

way in which locallzationpropertiesofgeneralized functions

can

be ‘we&ened’. Denote $T(-\ell, \ell)=\mathbb{R}+i(-l, l)\subset C$, and let $\mathcal{T}(T(-P,l))$ be

the set offunctions $f$ holomorphic in $T(-P, \ell)$

.

Then for $|a|<\ell$,

we

get

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$=f(-a)= \int_{-\infty}^{\infty}\delta(x+a)f(x)dx$.

The above equality implies the following two facts.

(A) $\Delta_{N}(x)=\sum_{n=0}^{N}\frac{a^{n}}{n!}\delta^{(n)}(x)$ converges to _{$\delta(x+a)=\delta_{-a}(x)$} in $\mathcal{T}(T(-\ell,l))’$

as

$Narrow\infty$

.

Clearly, for all _{$N\in N$}, supp$\Delta_{N}=\{0\}$ while for the limit

we

find supp$\delta_{-a}=\{-a\}$

.

(B) If $|a|>l,$ $\Delta_{N}(x)$ does not converge in $\mathcal{T}(T(-\ell, \ell))’$.

(A) and (B)

say:

Elements in $\mathcal{T}(T(-\ell, \ell))’$ do not allow to distinguish between

$\{0\}$ and $\{-a\}$, if $|a|<p$ but if $|a|>\ell$ then elements in $\mathcal{T}(T(-pp))’$

can

be used

to dIstinguish between the locations $\{0\}$ and $\{-a\}$

.

4. ULTRAHYPERFUNCTION

Tempered ultrahyperfunctios

were

first introduced by Hasumi, M. [7] in 1961

and developed by Morimoto, M. [10] in 1975. Here

we

just mention the basic definition. For

a

subset $A$ of$\mathbb{R}^{n}$,

we

denote by _{$T(A)=\mathbb{R}^{n}+iA\subset \mathbb{C}^{\mathfrak{n}}$}the tubular

set with base $A$

.

For

a

convex

compact set $K$ of$\mathbb{R}^{n},$ $\mathcal{T}_{b}(T(K))$ is, by definition,

the space of all continuous functions $f$

on

$T(K)$ which

are

holomorphic In the

interior of$T(K)$ and satisfy

$\Vert f\Vert^{T(K),j}=\sup\{|z^{p}f(z)|;z\in T(K), |p|\leq j\}<\infty,$ _$j=0,1,$ _$\ldots$

where $p=(p_{1}, \ldots,p_{\mathfrak{n}})$ and $z^{p}=l_{1}^{1}\cdots z_{n}^{p_{n}}$

.

$\mathcal{T}_{b}(T(K))$ is a R\’echet space with

the semi-norms $\Vert f\Vert^{T(K),j}$

.

If $K_{1}\subset K_{2}$

are

two compact

convex

sets,

we

have the

canonical injections:

$\mathcal{T}_{b}(T(K_{2}))arrow \mathcal{T}_{b}(T(K_{1}))$.

Let $O$ be a

convex

open set in $\mathbb{R}^{n}$. We define

$\mathcal{T}(T(O))=\lim_{arrow}\mathcal{T}_{b}(T(K_{1}))$,

where$K_{1}$

runs

throughthe

convex

compactsetscontainedin$O$, and theprojective

limit is taken following the restriction mappings.

Definition 4.1. A tempered ultra-hyperfunction is by

_definition

a

continuous

linear

_functional

on

$\mathcal{T}(T(\mathbb{R}^{n}))$

.

Remark 4.2. It

seems

that the space $\mathcal{T}(T(\mathbb{R}^{n}))$ is quite unique, in the

sense

that it is not among the many spaces considered in the book of $I.M$. Gel’fand and $G.E$

.

Shilov [5]. There

we

find function spaces $S^{1,B}$ and _{$S^{1}= \lim_{Barrow\infty}S^{1,B}=$}

$1{\rm Im} \mathcal{T}_{b}(T(K_{1}))$,

$K_{1}arrow\{0\}$

but

no

space $\lim_{0arrow B}S^{1,B}=\lim_{R^{n}arrow K_{1}}\mathcal{T}_{b}(T(K_{1}))=\mathcal{T}(T(\mathbb{R}^{n}))$

.

By the reason explained in Section 3,

we can

formulate relativistic quantum

field theory with a fundamental length by using tempered ultrahyperfunctions,

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5. AXIOMS FOR ULTRAHYPERFUNCTION QUANTUM FIELDS OF MIXED TYPE

Here

we

state $Wigtmans$ axioms for the ultrahyperfunction quantum field

the-ory. For the

case

of neutral scalar fields, _{these axioms have been} presented in [1].

W.I. Relativistic invariance of the state space;

W.II. Spectral property;

W.III. Existence and uniqueness of the vacuum;

W.IV. Fields: Thecomponents $\phi_{j}^{(\kappa)}$ of thequantumfield $\phi^{(\kappa)}$

are

operator-valued

generalizedfunctions$\phi_{j}^{(\kappa)}(x)$

over

the space_{$\mathcal{T}(T(\mathbb{R}^{4}))wIth$}

common

dense

domain

$D;1.e.$, for all $\Psi\in \mathcal{D}$ and all $\Phi\in \mathcal{H}$,

$\mathcal{T}(T(\mathbb{R}^{4}))\ni farrow(\Phi, \phi_{j}^{(\kappa)}(f)\Psi)\in \mathbb{C}$

is

a

tempered _{ultrahyperfunction. It is assumed that the}

_vacuum

_vector $\Phi_{0}$ is

contained in $\mathcal{D}$ and that $\mathcal{D}$ is taken into itself under the

action of the operators

$\phi_{j}^{(\kappa)}(f)$ and $U(a, A)$, i.e., $\phi_{j}^{(\kappa)}(f)D\subset \mathcal{D},$ _{$U(a, A)\mathcal{D}\subset \mathcal{D}$}

.

Moreover it is assumed that there exist indices $\overline{\kappa},\overline{J}$ such that $\phi_{\frac{}{f}}^{(\overline{n})}(\overline{f})\subset\phi_{j}^{(\kappa)}(f)^{*}$ where * indicates the

Hilbert space adjoint of the operator in question.

W.V. Poincar\’e-covariance of the fields;

W.VI. Extended causality

or

extended local commutativity: _{Any two field}

components $\phi_{j}^{(\kappa)}(x)$ and $\phi_{l}^{(\kappa’)}(y)$ either commute

or

anti-commute if the distance

between $x$ and $y$ is greater than $\ell$:

a) The functionals

$\mathcal{T}(T(\mathbb{R}^{4}))\otimes \mathcal{T}(T(\mathbb{R}^{4}))\ni f\otimes garrow(\Phi, \phi_{j}^{(\kappa)}(f)\phi_{l}^{(\kappa’)}(g)\Psi)$ $\bm{t}d$

$\mathcal{T}(T(\mathbb{R}^{4}))\otimes \mathcal{T}(T(\mathbb{R}^{4}))\ni f\otimes garrow(\Phi, \phi_{l}^{(\kappa’)}(g)\phi_{j}^{(\kappa)}(f)\Psi)$

can

be extended continuously to $\mathcal{T}(T(L^{\ell}))$ in

some

Lorentz frame, for arbitrary

elements $\Phi,$ $\Psi$ in the

common

domain $\mathcal{D}$ of the field operators

$\phi_{j}^{\kappa}(f)$, where

$T(L^{\ell})=\{(z_{1}, z_{2})\in \mathbb{C}^{4\cdot 2};|{\rm Im} z_{1}-{\rm Im} z_{2}|_{1}<l\}$, where $|y|_{1}=|y^{0}|+\sqrt{\sum_{i=1}|y_{i}|^{2}}$

.

b) The carrier of the functional

$f\otimes garrow(\Phi, [\phi_{j}^{(\kappa)}(f), \phi_{l}^{(\kappa’)}(g)]_{\mp}\Psi)$

on

$\mathcal{T}(T(\mathbb{R}^{4}))\otimes \mathcal{T}(T(\mathbb{R}^{4}))$ is contained in the set

$W^{\ell}=\{(z_{1}, z_{2})\in \mathbb{C}^{4\cdot 2};z_{1}-z_{2}\in V^{\ell}\}$,

where

$V^{\ell}=\{z\in \mathbb{C}^{4};\exists x\in V, |{\rm Re} z-x|+|{\rm Im} z|_{1}<p\}$

is

a

complex neighborhood of light

cone

$V$, i.e., this functIonal

can

be extended

continuously to $\mathcal{T}(W^{\ell})$.

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Remark 5.1. The condition (3.1) is expressed that the support of the vector-valueddistribution $\phi_{j}^{(\kappa)}(x)\phi_{l}^{(\kappa’)}(y)\Psi\mp\phi_{\ell}^{(\kappa’)}(y)\phi_{j}^{(\kappa)}(x)\Psi$iscontainedIn the set $W_{\ell}=$ $\{(x, y)\in \mathbb{R}^{4\cdots 2};(x-y)^{2}\geq-l^{2}\}$. However, the tempered ultrahyperfunction has

a

carrier but generally

no

support, the smallest carrier, and therefore

we

replace

the condition (3.1) with the condition b) of W.VI, that is, the functional

$f\otimes garrow(\Phi, [\phi_{j}^{(\kappa)}(f), \phi_{i}^{(\kappa’)}(g)]_{\mp}\Psi)$

is continuously extended to $\mathcal{T}(W^{p})$ where $W^{\ell}$ is

a

complex neighborhood of $W_{\ell}$.

6. PATH INTEGRAL QUANTIZATION

We quantizethis model by path integral methods (see [3]). Formally, the

time-ordered two point function is calculated

as

$\int\overline{\psi}_{\alpha}(x_{1})\psi_{\beta}(x_{2})$exp$i \{\int_{\mathbb{R}^{4}}L_{I}(x)dx\}d\mathcal{D}(\psi,\overline{\psi})d\mathcal{G}(\phi)$

$\cross\{\int\exp i\{\int_{R^{4}}L_{I}(x)dx\}d\mathcal{D}(\psi,\overline{\psi})d\mathcal{G}(\phi)\}^{-1}$,

$d \mathcal{G}(\phi)=\exp i\{\int_{R^{4}}L_{Fb}(x)dx\}\prod_{x\in \mathbb{R}^{4}}d\phi(x)$

$d \mathcal{D}(\psi,\overline{\psi})=\exp i\{\int_{R^{4}}L_{Ff}(x)dx\}\prod_{x\in \mathbb{R}^{4}}\prod_{\alpha=1}^{4}d\psi_{a}(x)d\overline{\psi}_{a}(x)$ .

All theseintegralshave

a

rigorous meaningifthecontinuous space-time is replaced

by a lattice. For positive integers $M,$$N$ define $L=MN,$ $\Delta=\sqrt{\pi}/M$ and the

lattice

$\Gamma=\{t=j\Delta;j\in \mathbb{Z}, -L<j\leq L\}=\Delta \mathbb{Z}/(2\sqrt{\pi}N)$

.

The lattice version of the differential operator $-\triangle+m^{2}$ on $\mathbb{R}^{\Gamma^{4}}=\mathbb{R}^{4\cdot 2L}$ is

the

following difference operator

$-\triangle+m^{2}$ : $\mathbb{R}^{\Gamma^{4}}\ni\Phi(x)arrow$

$- \sum_{\mu=0}^{3}\frac{\Phi(x+e_{\mu})+\Phi(x-e_{\mu})-2\Phi(x)}{\Delta^{2}}+m^{2}\Phi(x)\in \mathbb{R}^{\Gamma^{4}}$

.

Let $dG(\Phi)$ be a Gaussian

measure

on $\mathbb{R}^{4\cdot 2L}$ defined by

$dG(\Phi)=C$exp $\{\frac{1}{2}\sum_{y\in\Gamma^{4}}[\sum_{\mu=0}^{3}\frac{\Phi(y+e_{\mu})+\Phi(y-e_{\mu})-2\Phi(y)}{\Delta^{2}}$

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where $C$ is the normalization constant such that _{$\int dG(\Phi)=1$}

.

The exponent

of the

measure

is the (Euclideanized $x^{0}arrow-iy^{0},$ _{$xarrow y$}) discretIzation of

Lagrangian $i \int L_{Fb}(x)dx$

.

Now we

can

calculate the covariance of $dG(\Phi)$

$\int\Phi(y_{1})\Phi(y_{2})dG(\Phi)=2(-\triangle+m)^{-1}(y_{1},y_{2})=2S_{m}(y_{1}-y_{2})$

$S_{m}(y_{1}-y_{2})=(2 \pi)^{-4}\sum_{p\in\overline{\Gamma}^{4}}e^{ip(y_{1}-y_{2})}[\sum_{\mu=0}^{3}$($2-2$

cos

$p_{\mu}\Delta$)$/\Delta^{2}+m^{2}]^{-1}\eta^{4}$,

$\tilde{\Gamma}=\{s=j\eta;j\in \mathbb{Z}, -L<j\leq L, \eta=\sqrt{\pi}/N\}=\eta \mathbb{Z}/(2\sqrt{\pi}M)$.

The followingfact is shown in [11] by using nonstandard analysis: $S_{m}(y_{1}-y_{2})arrow$

$S_{m}(y_{1}-y_{2}),$ $M,$$Narrow\infty$, where

$S_{m}(y_{1}-y_{2})=(2 \pi)^{-4}\int_{R^{4}}e^{ip(y_{1}-y_{2})}[p^{2}+m^{2}]^{-1}d^{4}p$

is the Schwinger function of neutral scalar field of

mass

$m$

.

In order to deal with

the fermion field $\Psi$, we need the

measure

_{$dD(\Psi^{1}, \Psi^{2})$}

on

the

Grassmann

algebra

generated by $\{\Psi_{\alpha}^{1}(y), \Psi_{\alpha}^{2}(y);\alpha=1, \ldots,4, y\in\Gamma^{4}\}$:

$dD( \Psi^{1}, \Psi^{2})=C’\exp\{-\sum_{y\in\Gamma^{4}}\Psi^{2T}(y)[\sum_{\mu=0}^{3}\gamma_{\mu}^{B}\nabla_{\mu}+\tilde{m}]\Psi^{1}(y)\Delta^{4}\}$

$\cross\prod_{y\in\Gamma^{4}}\prod_{\alpha=1}^{4}d\Psi_{\alpha}^{1}(y)d\Psi_{\alpha}^{2}(y)$,

$\Psi^{1}=(\Psi_{1}^{1},$ $\ldots,$

$\Psi_{4}^{1}\rangle^{T},$ $\Psi^{2}=(\Psi_{1}^{2}, \ldots, \Psi_{4}^{2})^{T}$,

$\gamma_{0}^{E}=\gamma_{0}=(\begin{array}{ll}\sigma_{0} 00 -\sigma_{0}\end{array})$ $\gamma_{j}^{B}=-i\gamma_{j}=(\begin{array}{ll}0 -i\sigma_{j}i\sigma_{j} 0\end{array})$ $j=1,2,3$,

$\sigma_{0}=(\begin{array}{ll}1 00 1\end{array})$ $\sigma_{1}=(\begin{array}{ll}0 1l 0\end{array}),$ $\sigma_{2}=(\begin{array}{l}0-ii0\end{array})$ $\sigma_{3}=(\begin{array}{l}100-l\end{array})$

$\nabla_{\mu}\Psi_{k}=\{\begin{array}{ll}\nabla^{+}\Psi_{k}(y)=(\Psi_{k}(y+e_{\mu})-\Psi_{k}(y))/\Delta if k=1,2,\nabla^{-}\Psi_{k}(y)=(\Psi_{k}(y)-\Psi_{k}(y-e_{\mu}))/\Delta if k=3,4.\end{array}$

The ideato replace the partial derivatives in the continuum

case

by the forward-,

respectivelybackward- difference

as

describe above, has originally beendeveloped

in [12] in order to avoid the doubling problem. Using $P\pm=(1\pm\gamma_{0}^{B})/2$ the

interaction Lagrangian is defined

as:

$-L_{I}(y)= \Psi^{2T}(y)e^{il^{2}\Phi(y)^{2}}\sum_{\mu=0}^{3}\gamma_{\mu}^{E}$

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$+P_{-}\Psi^{1}(y-e_{\mu})\{e^{-il^{2}\Phi(y)^{2}}-e^{-il^{2}\Phi(y-e_{\mu})^{2}}\}/\triangle]$,

Now

we

calculate the lattice version of the Schwinger functions of the interacting

fields. The two point Schwinger function is

$\int\Psi_{\alpha}^{1}(y_{1})\Psi_{\beta}^{2}(y_{2})$ exp $( \sum_{y\in\Gamma^{4}}L_{I}(y)\Delta^{4})dD(\Psi^{1}, \Psi^{2})dG(\Phi)$

$\cross\{\int\exp(\sum_{y\in\Gamma^{4}}L_{I}(y)\Delta^{4})dD(\Psi^{1}, \Psi^{2})dG(\Phi)\}^{-1}$

$= \int e^{il^{2}\Phi(y_{1})^{2}1}\Psi’(y_{1})e^{-:l^{2}\Phi(y2)^{2}}\Psi^{\prime 2}(y_{2})dD(\Psi^{\prime 1}, \Psi^{\prime 2})dG(\Phi)$

$= \int\Psi^{\prime 1}(y_{1})\Psi^{r2}(y_{2})dD(\Psi^{\prime 1}, \Psi^{\prime 2})\int)^{2}$

where

we

used the change of variables

$\Psi^{1}(y)=e^{il^{2}\Phi(y)^{2}}\Psi^{\prime 1}(y),$ $\Psi^{2}(y)=e^{-:l^{2}\Phi(y)^{2}}\Psi^{\prime 2}(y)$

.

The covariance

$\int\Psi^{\prime 1}(y_{1})\Psi^{\prime 2}(y_{2})dD(\Psi^{\prime 1}, \Psi^{\prime 2})=\mathcal{R}_{\tilde{m};\alpha,\beta}(y_{1}-y_{2})$

converges to the Schwinger function

$R_{\tilde{m};\alpha,\beta}(y)= \{-\sum_{\mu=0}^{3}\gamma_{\mu}^{E}(\frac{\partial}{\partial y_{\mu}})+\tilde{m}\}_{\alpha,\beta}S_{\tilde{m}}(y)$

ofthe free Dirac field of

mass

$\tilde{m}$

.

The integral

$\int e^{il^{2}\Phi(y_{1})^{2}}e^{-u^{2}\Phi(y_{2})^{2}}dG(\Phi)$

$=[(1-il^{2}S_{m}(0))(1+il^{2}S_{m}(0))-l^{4}S_{m}(y_{1}-y_{2})^{2}]^{-1/2}$ .

contains $S_{m}(0)$ which diverges to $\infty$

as

$N,$ $Marrow\infty$

.

But this divergent quantity

is removed by using the Wick products instead of the ordinary product in the

Lagrangian. It is defined by (see [6]):

: $e^{h\Phi(y)}$ $:= \sum[:(h\Phi(y))^{n} : /n!]=e^{-h^{2}S_{m}(0)}e^{h\Phi(y)}\infty$

.

$n=0$

This

removes

the divergent quantity and the result is

$\int:e^{:l^{2}\Phi(y_{1})^{2}}$ : : $e^{-il^{2}\Phi(y_{2})^{2}}$ : $dG(\Phi)=[1-4l^{4}S_{m}(y_{1}-y_{2})^{2}]^{-1/2}$ ,

and hence the two point Schwinger function of $\psi$ on the lattice

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converges to the continuum

one:

$[1-4l^{4}S_{m}(y_{1}-y_{2})^{2}]^{-1/2}R_{\tilde{m};\alpha,\beta}(y_{1}-y_{2})$

.

Let $D_{m}^{(-)}(x_{0}-i\epsilon, x)=S_{n}(ix_{0}+\epsilon, x)$

.

Then the two point Wightman function

$W_{\alpha,\beta}(x_{0}-i\epsilon, x)$ is

$[1-4l^{4}D_{m}^{(-)}(x_{0}-i\epsilon, x)^{2}]^{-1/2}(i\gamma_{\mu}\partial^{\mu}+\tilde{m})_{\alpha,\beta}D_{\overline{m}}^{(-)}(x_{0}-i\epsilon, x)$

.

It follows ffom the relations

$|D_{m}^{(-)}(x_{0}-i\epsilon,x)|\leq(2\pi\epsilon)^{-2},$ $\epsilon^{2}D_{m}^{(-)}(-i\epsilon, 0)arrow(2\pi)^{-2}(\epsilonarrow 0)$

that if$\epsilon>\sqrt{2}l/(2\pi)$, then $|4l^{4}D_{m}^{(-)}(x_{0}-i\epsilon,x)^{2}|<1$ and

$[1-4l^{4}D_{m}^{(-)}(z_{0},x)^{2}]^{-1/2}(i\gamma_{\mu}\partial^{\mu}+\tilde{m})_{\alpha,\beta}D_{\tilde{m}}^{(-)}(x_{0}-i\epsilon,x)$

is holomorphic for ${\rm Im} z_{0}>\sqrt{2}l/(2\pi)$ and defines

a

ultrahyperfunction $W_{\alpha,\beta}$ by

the fomula

$W_{\alpha,\beta}(f)= \int_{R^{4}}W_{\alpha,\beta}(x_{0}-i\epsilon, x)f(x_{0}-i\epsilon, x)dx$

for all $f\in \mathcal{T}(T(O_{s})),$ $O_{s}=\{x\in \mathbb{R}^{4};||x||<s\}$ for

some

$s>\sqrt{2}t/(2\pi)$

.

$\sqrt{2}l/(2\pi)$

isthe fundamental lengthinthe

sense

ofW.VI, i.e., twoeventswithin the distance

$\sqrt{2}l/(2\pi)$ cannot be distinguished. Thus the parameter $l$ which appears in the

equation (1.5) tums out to be tfe fundamental length. The full sequence of $n$

point Wightman ultrahyperfunctions is calculated in [11].

7. OPERATOR SOLUTION

Recall the system ofequations (1.3) which which we plan to solve

$\{\begin{array}{l}\phi(x)+()^{2}\phi(x)=0(i\frac{\hslash}{c}\gamma_{\mu}\frac{\partial}{\partial x_{\mu}}-\tilde{m})\psi(x)=-2\gamma_{\mu}l^{2}\psi(x)\phi(x)\frac{\partial\phi(x)}{\partial x_{\mu}}\end{array}$

The basic ideas to solve the above system is quite natural: $?hke$ the free

Klein-Gordon field $\phi$ and suppose that we can show that (i)

$\rho(x)=:e^{il^{2}\phi(x)^{2}}$ $:= \sum_{n=0}^{\infty}i^{n}l^{2n}$ : $\phi(x)^{2n}$ $:/n!$

is well-defined as an operator-valued ultrahyperfunction, and that (ii) it satisfies

$\frac{\partial}{\partial x^{\mu}}\rho(x)=2il^{2}$ : _{\mbox{\boldmath$\rho$}.} r\iota &ノ$2 \phi(x)\frac{\partial}{\partial x^{\mu}}\phi(x):=2il^{2}$ : $\rho(x)\phi(x)\frac{\partial}{\partial x^{\mu}}\phi(x):$,

and that (iii) the hee Dirac field $\psi_{0}(x)$ is

a

multiplier for the field $\rho$

.

Then define the field

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and calculate

$(i \gamma_{\mu}\frac{\partial}{\partial x^{\mu}}-\tilde{m})\psi(x)=[(i\gamma_{\mu}\frac{\partial}{\partial x^{\mu}}-\tilde{m})\psi_{0}(x)]\rho(x)+\gamma_{\mu}\psi_{0}(x)\frac{\partial}{\partial x^{\mu}}\rho(x)$

$=-2l^{2} \gamma_{\mu}\psi_{0}(x):\rho(x)\phi(x)\frac{\partial}{\partial x^{\mu}}\phi(x)$ $:=-2l^{2}\gamma_{\mu}$ : $\psi(x)\phi(x)\frac{\partial}{\partial x^{\mu}}\phi(x)$ :.

Thus the operator-valued ultrahyperfunction $\psi(x)$ satisfies the equation (1.5).

While (i) is shown in E. Br\"uning and S. Nagamachi [1]

we

use

the fundamental

fomula of A.S. Wightman and L. Galrding [14]

$(: D^{\alpha^{(1)}}\phi D^{\alpha^{(2)}}\phi\cdots D^{\alpha^{(l)}}\phi:(f)\Phi)^{(n)}(\xi_{1}, \ldots,\xi_{n})$

$= \frac{\pi^{l/2}}{(2\pi)^{2(l-1)}}\sum_{j=0}^{l}[\frac{(n-l+2j)!}{n!}]^{1/2}\int\cdots/(\prod_{k=1}^{j}d\Omega_{m}(\eta_{k}))$

$\cross\sum_{1\leq k_{1}<k_{2}<\ldots<k_{l-j}\leq nP}(j!)^{-1}\sum P((-i\eta_{1})^{\alpha^{(1)}}\cdots(-i\eta_{j})^{\alpha^{0)}}(i\xi_{k_{1}})^{\alpha^{(3+1)}}\cdots(i\xi_{k_{1-f}})^{\alpha^{(1)}}$

$\cross\tilde{f}\sum_{r=1}^{1^{j}}\eta_{r}-\sum_{r=1}^{l-j}\xi_{k_{r}}))\Phi^{(n-l+2j)}(\eta_{1},$$\ldots,\eta_{j},\xi_{1)}$

...,$\hat{\xi}_{k_{1}},$$\ldots,\hat{\xi}_{k_{l-f}},$ $\ldots,\xi_{n}$). to show (ii). Rom this formula

we

get

$(: \phi^{l} : (-\frac{\partial}{\partial x^{\mu}}f)\Phi)^{(n)}=l(:(\frac{\partial}{\partial x^{\mu}}\phi)\phi^{l-1} : (f)\Phi)^{(n)}$,

$\frac{\partial}{\partial x^{\mu}}$ : $\phi(x)^{i}$ : $\Phi=l$ : $( \frac{\partial}{\partial x^{\mu}}\phi(x))\phi^{l-1}(x)$ : $\Phi$

.

$\Phi=\rho(f_{1})\cdots\rho(f_{n})\Phi_{0},$ $f_{j}\in \mathcal{T}(T(\mathbb{R}^{4})),$ $\Phi_{0}$:

vacuum

for $\rho$ (and $\phi$). It follows

$\sum_{l=0}^{\infty}(ig)^{l}$ : $\phi^{2l}$ : $(- \frac{\partial}{\partial x^{\mu}}f)\Phi/l!$

$= \sum_{l=1}^{\infty}(ig)^{l}2:(\frac{\partial}{\partial x^{\mu}}\phi)\phi\phi^{2(l-1)}$ : $(f)\Phi/(l-1)!$

$=2(ig):( \frac{\partial}{\partial x^{\mu}}\phi)\phi\sum_{l=0}^{\infty}(ig)^{l}\phi^{2l}$ : $(f) \Phi/l!=2ig:(\frac{\partial}{\partial x^{\mu}}\phi)\phi\rho:(f)\Phi$.

$\frac{\partial}{\partial x^{\mu}}\rho(x)\Phi=2ig:(\frac{\partial}{\partial x^{\mu}}\phi(x))\phi(x)\rho(x):\Phi$.

(iii): From the system of axioms

one

proves that the vector-valued function

$\rho(z_{1})\cdots\rho(z_{n})\Phi_{0}$ is holomorphic in

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for

some

$p_{j}>P>0$ $(j=1, \ldots , n)$ (see [2]). Let $\Psi_{0}$ is the

vacuum

for $\psi_{0}$. Then

$\psi_{0,\alpha_{1}}(z_{1})\cdots\psi_{0,\alpha_{n}}(z_{n})\Psi_{0}1s$holomorphic in

$\{(z_{1}, \ldots, z_{n})\in \mathbb{C}^{4n};{\rm Im} z_{1}\in V_{+}, {\rm Im}(z_{j}-z_{j-1})\in V_{+}\}$ Therefore, $\psi_{0,\alpha_{1}}(z)\Psi$ for $\Psi=\psi_{0,\alpha_{2}}(g_{2})\cdots\psi_{0,\alpha_{n}}(g_{n})\Psi_{0},$ $g_{j}\in S(\mathbb{R}^{4})$

and $\rho(z)\Phi$ for $\Phi=\rho(f_{2})\cdots\rho(f_{n})\Phi_{0},$ $f_{j}\in \mathcal{T}(T(\mathbb{R}^{4}))$

are

holomorphic in

$\{z\in \mathbb{C}^{4};{\rm Im} z\in V_{+}+(\ell_{1},0)\}$

.

The product $(\psi_{0,\alpha}\rho)(f)$ is defined by

$(\psi_{0,\alpha}\rho)(f)(\Psi\otimes\Phi)=$

$= \int_{\Gamma_{N}}f(z)\psi_{0,\alpha}(z)\Psi\otimes\rho(z)\Phi dz,$ $\Gamma_{N}=\{z\in \mathbb{C}^{4};z=x+i(N, 0)\}$

for suitable $N>0$

.

Moreover,

we

have $( \frac{\partial}{\partial x^{\mu}}(\psi_{0,\alpha}\rho))(f)\Psi\otimes\Phi$

$=( \psi_{0,\alpha}\rho)(-\frac{\partial}{\partial x^{\mu}}f)\Psi\otimes\Phi=\int_{\Gamma_{N}}(-\frac{\partial}{\partial x^{\mu}}f(z))\psi_{0,\alpha}(z)\Psi\otimes\rho(z)\Phi dz$

$= \int_{\Gamma_{N}}f(z)\{(\frac{\partial}{\partial x^{\mu}}\psi_{0,\alpha}(z)\Psi)\otimes\rho(z)\Phi+\psi_{0,\alpha}(z)\Psi\otimes\frac{\partial}{\partial x^{\mu}}\rho(z)\Phi\}dz$

$=(( \frac{\partial}{\partial x^{\mu}}\psi_{0,\alpha})\rho)(f)\Psi\otimes\Phi+(\psi_{0,\alpha}\frac{\partial}{\partial x^{\mu}}\rho)(f)\Psi\otimes\Phi$

.

Thus it follows

$\frac{\partial}{\partial x^{\mu}}(\psi_{0,\alpha}(x)p(x))(\Psi\otimes\Phi)=$

$=( \frac{\partial}{\partial x^{\mu}}\psi_{0,\alpha}(x))\rho(x)\Psi\otimes\Phi+\psi_{0,\alpha}(x)\frac{\partial}{\partial x^{\mu}}\rho(x)\Psi\otimes\Phi$ ,

and

we

deduce that Equation (1.5) holds. The Wightman functions for the field

$\psi(x)=\psi_{0}(x)\rho(x)1s$ calculated

as

$(\Psi_{0}\otimes\Phi_{0}, \psi_{\alpha_{1}}(z_{1})\cdots\psi_{\alpha_{n}}(z_{n})\Psi_{0}\otimes\Phi_{0})$

$=(\Psi_{0}, \psi_{0,\alpha_{1}}(z_{1})\cdots\psi_{0,\alpha_{r\iota}}(z_{n})\Psi_{0})(\Phi_{0}, \rho(z_{1})\cdots\rho(z_{n})\Phi_{0})$

.

These Wightman functIons coincide with those which are calculated by the path

integral method in Section 6. A detailed analysis is found in [2]. REFBRENCES

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(S. Nagamachi) DEPARTMENT OF APPLIED PHYSICS AND MATHEMATICS, FACULTY OF

ENGINEERING, THE UNIVERSITYOF TOKUSHIMA, TOKUSHIMA 770-8506, JAPAN

E-mail address: shigeakiQpm. tokushima-u.ac.jp

(E. Br\"uning) SCHOOL OF MATHEMATICAL $s\circ IENCES$, UNIVERSITY _OF _{$KWAZULU-NATAL$},

PRIVATE BAG X54001, DURBAN 4000, SOUTH AFRICA