数理解析研究所講究録
第 1819 巻 2012 年 148-161

148

Independence, Coupling and Dependence from theviewpoint of Micro-Macro duality
Izumi OJIMARIMS, Kyoto University

June 2011
Abstract

We explain the basic notions pertaining to the scattering processes
in terms of asymptotic fields and of -matri as a kind of central limit
theorem. These notions can be used for understanding the highly dy$S$$X$

namical behaviours of strongly interacting hadrons, from the viewpointof the duality involving independence, couphng and dependence, which
may have some interesting relations with the monotone and $/or$ free independences.

1

Hadrons and Bacteria as “Unsung Heros” behind Nature

Introduction;

In all the physical nature, the hadronic world is characterized by its $ex-$
treme activity and its longest history of existence (of the level asa whole); we cannot imagine and verify the possibility of historical period
without its activities and existence; for instance, the history of universe withevolution of stars starts from protons as the most typical hadrons. Similarlevels in the biosituation can be found in the roles played by the bacte
logical context, as was emphasized by Stephen Jay Gould in his book, “FullHouse–The Spread of Excellence from Plato to Darwin” (Harmony Books,
1996)1. This kind of aspects will be seen to be crucial and indispensablefor our satisfactory understanding of the consistency between repeatablelaws and their histori $cal$ developments without repetitions, as seenin the cosmological and biological evolutions. At the end, we try to examine this problem in the realm of quantum fields and hadrons, from theviewpoint of Micro-Macro duality [1], which will hopefully be useful for
unified understanding of nature according to the longitudinal axis of its historical processes and to the transverse ones of coexisting network structures$7\dot{\tau}al$lThis interesting book was brought to my attention by Prof. I. Yamato at Tokyo
University of Science, to whom the present author expresses his deep gratitude.

149
spanned by various bridges among different hierarchical regimes in nature.
After explaining Micro-Macro duality, we recollect the formulation of scattering amplitudes (-matrix functional) in terms of quantum fields, on
$=$

$S$

the basis of which basic features of hadrons are examined.
While there are no strict boundaries between micro- and macroscopic
levels in nature, it is important to specify such a boundary in a scientificdiscussion of a given restricted domain, for the purpose of which the notionof “sectors” plays a crucial role. The essence of the sector structures foundin various areas in nature can be summarized in the context of Micro-Macroduality as follows, where a $sector^{i}$ ’ is interpreted as quasi-equivalence classof factor states [1]:or, in a little more elaborated form:

A good physical example of the mathematical notion of statistical independence can be found in the form of asymptotic fields arising in the
scattering theory of relativistic quantum fields through the asymptotic condition, which is nothing but a version of Central Limit Theorem [3] inthe physical context. Once the independent objects are successfully identified, the cssence of the most important tasks in mathematical and physicaldescriptions of natural phenomena can be found in the problem concerning
the gaps between idealized ( approximate) world of independence
and realistic interacting world of dependence, which are to be filled
up by the scheme of coupling $=$ interactions [3]. In the case of scat$=$150
tering theory of QFT, the basic scheme can be understood in the following
diagram:

The Precise meaning of the “Central Limit Theorem” can be seen in “MicroMacro Duality” in $QFT$ in the following sense:or,(Remark: there is another local-net version of independence based on
the so-called nuclearity condition in Algebraic QFT. )It is the aim of the present article to clarify the precise meaning of thenotions and the diagrams appearing above.

2

What does Einstein’s Formula $\ll E=mc^{2}\gg$ Mean?:“Unit” of Independence $=$ Free Particles

In Quantum Probability, several versions of independence generalizing
bosonic tensor type have been proposed, developed and classified with interesting results [2]. Here my naive questions are: on which physical grounddo they emerge and what physical meaning do they have? For Gaussian( bosonic CCR & fermionic CAR) case(s), the following is my partial answer in the context of relativistic QFT [3]: emergence of independence
$=$

$=$

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a “Central Limit Theorem” via asymptotic condition,$\varphi_{H}(x)x^{0}=t\mp\infty\vec{arrow}$, from non-independent interacting Heisenberg fields
to independent free asymptotic fields $\phi^{as}=\phi^{in/out}fy$ asymptotic states. To
formulate this problem in a clear-cut way, the notion of the “particles”
characterized by the mass-shell condition plays essential roles whose familar
version can be found in Einstein’s famous formula $E=mc^{2}.$Owing to such serious and actual consequences as atomic bombs andnuclear power plants, Einstein’s famous equality $\ll E=mc^{2}\gg$ of energy & mass has always been regarded as one of the most fundamental
notions of the special theory of relativity. Properly speaking, however, thisis a simple and trivial sort of misunderstanding, because this formula ismeaningful only for asymptotic fields/states as the “on-shell condition” to extract 1-particle modes !! from the interacting Heisenberg
fields: if it were not for the interactions of Heisenberg fields
any kind of nuclear reactions as the sources of radioactivity cannot takeplace, and hence, the formula $\ll E=mc^{2}\gg$ itself yields no actual events,
good or bad!! Its genuine theoretical meaning is simply the condition todefine independent $=free=$ non-interacting asymptotic fields/states,$p^{2}=p_{\mu}p^{\mu}=m^{2}$ containing independent
free $=non$-interacting particles.
The resulting asymptotic fields, provide a vocabulary for describing state changes taking place in the scattering processes: [asymptotic
matrixin-states
out-states]. For lack of interactions, however, on-shell
asymptotic fields
by themselves can not ignite scattering processes, andhence, we need to introduce off-shell interacting Heisenberg fields
which violate Einstein’s formula $\ll E=mc^{2}\gg!$$\phi^{in/out}(x)$$\varphi_{H}$

$($

$)$

$\varphi_{H}$$\varphi_{H},$$=$$\phi^{as}$$s-\Rightarrow$$\phi^{as}$$\varphi_{H},$In fact, taking $m$ as “moving mass”$m= \frac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}$, we have$E=mc^{2}= \frac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}c^{2}$$\Rightarrow(m_{0}c)^{2}=(\frac{E}{c})^{2}(1-v^{2}/c^{2})=(\frac{E}{c})^{2}-(\frac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}\vec{v})^{2}$$=( \frac{E}{c})^{2}-(p\neg)^{2}$$\Rightarrow p^{2}=p_{\mu}p^{\mu}=(m_{0}c)^{2},$where$\frac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}\vec{v}=:\vec{p}$is the relativistic momentum and$p^{\mu}=( \frac{E}{c},p\neg)$isthe -mementum. The actual meaning of
can be seen as follows:i mass-shell (or, on-shell) condition to characterize a mass hyperboloid in $F$-space of 4-momenta
carried by free 1-particle states with$p^{2}=p_{\mu}p^{\mu}=( \frac{E}{c})^{2}-(p\neg)^{2}=(m_{0}c)^{2}$$)$$p_{\mu}\in\hat{\mathbb{R}}^{4}$

152
rest mass$m_{0}.$

By this condition an orbit family $p^{2}=m^{2}>0$ can be picked up among
the four: $p^{2}=<>0,p_{\mu}=0$ (: vacua), of Poincar\’e groupin the Wigner’s construction of unitary representations induced from “little
groups” $(SU(2), E(2), SU(1,1))$ ;$\mathcal{P}1=\mathbb{R}^{4}\rtimes SL(2, \mathbb{C})$ii) through “first quantization”, we have$p_{\mu} arrow iW_{\mu}=i\hslash(\frac{1}{c}\frac{\partial}{\partial t},\vec{\nabla})$of a free scalar
the Klein-Gordon equation
with rest mass
fieldiii) The existence of positive/negative energy solutions $E=\pm\sqrt{(\vec{p}c)^{2}+(m_{0}c^{2})^{2}}$leads to the creation & annihilation operators, particleof
and $PCT$ invari ance.
antiparticle pairs, time reversal
Thus, the famous equivalence $E=mc^{2}$ between energy $E$ and mass
$m$ gives only partial information for dynamical descriptions of relativistic
quantum fields, with off-shell apects being neglected in spite of their vital
importance for non-trivial scattering processes, particle decays and productions, etc., etc.!
$[\hslash^{2}\partial_{\mu}\partial^{\mu}+(m_{0}c)^{2}]\phi(x)=0$$\phi(x)$$m_{0}.$$( \frac{E}{c})^{2}-(\vec{p})^{2}=m_{0}^{2}c^{2}$$T$

Free independent vs. interacting $=$ non-independent

2.1

$=$

It is also remarkable that the free asymptotic fields can be decomposed. Namely, free
into the sum of creation and annihilation operators
quantum field
as quantized solution of Klein-Gordon equation$m^{2})\phi=0$ describes “particle pictures” in terms of creation and annihilation:
operators:
creation and annihilation operators$\phi$$a(\vec{p)}.a^{*}(q\neg)$$(\square +$$\phi(x)$$a(\overline{p}).a^{*}(q\neg)$$\phi(x)\Leftrightarrow$$\phi(x)=\int\frac{d^{3}p}{\sqrt{(2\pi)^{3}2\omega_{\vec{p}}}}(a(p^{-})\exp(-ip_{\mu}x^{\mu})+h.c.)$,$a^{*}(f) :=i \int\emptyset(x)\Re_{f(x)d^{3_{X}}}=\int a^{*}(parrow)\tilde{f}(p\neg)d^{3}p$$=[a(f)]^{*},$$[a(f).a^{*}(g)]= \int\overline{\tilde{f}(\overline{p})}\tilde{g}(\vec{p})d^{3}p=\langle\tilde{f},\tilde{g}\rangle,$$[ \phi(x).\phi(y)]=\int\frac{d^{4}p}{(2\pi)^{3}}\epsilon(p^{0})\delta(p^{2}-m^{2})\exp(-ip(x-y))$$=:i\Delta(x-y;m^{2})$,in the “natural unit system” with $\hslash=c=1$ (restis denoted by $m$ , henceforth).
mass
with
It is customary for most physicists to regard quantum fields
as sufficient objects for describing wave-particle dualism inherent
in elementary particles. Perpetual creation and annihilation processes ofparticles, however, require interactions among elementary particles, whichwith$\omega_{\vec{p}}:=\sqrt{p^{T}+m^{2}}$$m_{0}$$\phi(x)$$a^{*}(\vec{p}),$$a(\overline{p})$153
is not consistent with the linearity of free field equation. In fact, the contentsof the famous Haag theorem is that Poincar\’e (or even, Galilei) covariant
quantum fields related to free fields by a unitary transformation are only freefields, which means that it is meaningless to formulate interacting Heisenberg
fields by means of a unitary transformation of free fields (as is common in
perturbative approaches). Note that this is in sharp contrast to qunatum
systems with finite degrees of freedom.
On the other hand, to describe relativistic scattering processes of elementary particles, we need the following three items: Poincar\’e-covariant
quantum fields/their interactions/free fields. Free fields are necessary because it provide us with indispensable vocabulary for the description of
scattering processes, where an initial state with incoming free particles ischanged int a final one with outgoing particles. According to the above
Haag theorem, however, we cannot discuss directly the relation between
interacting Heisenberg and free fields. Instead, the unitary -matrix $ap-$
pears between two free asymptotic fields,in theandform of a basis change
between in-state basis
and out-state basis
:
$-$

$0$

$S$

$\phi^{in}(x)$$S_{\beta,\alpha}$$|\beta,$$:=\langle\beta,$$\phi^{out}(x)$$out|\alpha,$ $in\rangle$$|\alpha,$$in\rangle$$out\rangle$

To treat Heisenberg fields, we recapitulate briefly the essence ofWightman axioms for relativistic quantum fields (in the vacuum representationin the form of relativistic covariance, local commutativity,
cyclicity or ergodicity of vacuum vector and spectral condition:a [Heisenberg fields]
operator-valued distributions(unbounded)
being
with values
closable operators acting on Hilbert
space is defined on the 4-dimensional Minkowski spacetime, where$:=x\cdot
y=x^{0}y^{0}-\vec{x}\cdot\vec{y}$$x=(x^{0},\vec{x})$is the Minkowski metric: $\eta(x, y)$.
forb [relativistic covariance]: local net:
o -algebras
generated by local fields
withandtheir polynomials defined on the net of double cones in the Minkowski
spacetime constitute a non-commutative covariant dynamical system,$\varphi_{H}(x)$$(\mathcal{P}, \mathfrak{H}, U, \Omega))$$\mathcal{D}(\mathbb{R}^{4})\ni f\mapsto$

$=$

$)$

$\varphi_{H}^{i}(f)$$(\mathbb{R}^{4}, \eta)$$\mathfrak{H}$$\eta$$)$$\mathcal{K}\ni \mathcal{O}\mapsto \mathcal{P}(\mathcal{O})$$\mathcal{P}$$\varphi_{H}^{i}(f)=\int\varphi_{H}^{i}(x)f(x)d^{4}x$$\mathcal{P}(\mathcal{O})$$\mathcal{K}$$f^{*}$$f\in \mathcal{D}(\mathcal{O})$$\mathcal{O}$$\alpha_{a,\Lambda}(\varphi_{H}^{i}(x))=U(a, \Lambda)\varphi_{H}^{i}(x)U(a, \Lambda)^{-1}$$=s(\Lambda)_{j}^{i}\varphi_{H}^{i}(\Lambda^{-1}(x-a))$$\alpha_{a,\Lambda}(\mathcal{P}(\mathcal{O}))=\mathcal{P}(\Lambda \mathcal{O}+a)$under the action$\mathcal{P}_{+}^{\uparrow}=\mathbb{R}^{4}\rtimes L_{+}^{\uparrow}$$\alpha,$,,, of Poincar\’e group) defined by the$\mathcal{P}_{+}^{\uparrow}\ni(a, \Lambda)\mapsto\alpha_{a,\Lambda}\in Aut(\mathcal{P}(\mathbb{R}^{4}))$$(or, its$ universal covering$\mathbb{R}^{4}\rtimes SL(2, \mathbb{C})$154
and (proper) Lorentz
semi-direct product of spacetime translation group(or its
group.
universal coveringonis a unitary representation ofis a finite-dimensional representation of Lorentz group, andassociated with each field multiplet
c [local commutaitivity]: absence of propagation of physical effects exceeding the light velocity due to Einstein causality, implies the local com:mutativity of Heiscnberg fields$\mathbb{R}^{4}$$L_{+}^{\uparrow};=\{\Lambda=(\Lambda_{\nu}^{\mu});\Lambda x\cdot\Lambda y=x\cdot y, \Lambda_{0}^{0}>0, \det(\Lambda)=+1\}$$SL(2, \mathbb{C}))$$\mathcal{P}_{+}^{\uparrow}$$\mathcal{P}_{+}^{\uparrow}\ni(a, \Lambda)\mapsto U(a, \Lambda)\in u(\mathfrak{H})$$L_{+}^{\uparrow}$$s(\Lambda)_{j}^{i}$$\mathfrak{H}$$(\varphi_{H}^{i}(x))_{i}.$$)$$\varphi_{H}^{i}(f)$if (supp ) $\cross(suppf_{2})$$[\varphi_{H}^{i}(f_{1}),\dot{\psi}_{H}(f_{2})]=0$$f_{1}$

are spacelikemeans that any pair of points
whereseparated: $(x-y)^{2}<0.$
Remark: By this condition, the Fourier transform of a Wightman funcin the vacuum
as a correlation function oftion
defined in the next d) admits an analytic continuation
state
into a holomorphic function in the complex energy-momentum space, fromwhich dispersion relations follow.
d [vacuum state and spectrum condition]:of spacetime translationsd-i) energy-momentum spectrum
in prealized on is within the forward light cone,, and the lowest energy is realized by eigenvalue of the vacuum
space
vector $\Omega:U(x)$ $:=U(x, 1)= \int_{p\in\overline{V_{+}}}\exp(ipx)dE(p);U(x)\Omega=\Omega.$
Remark: Similarly to p–analyticity due to local commutativity, -space
follows from specanalyticity of a Wightman function
trum condition, which provides powerful tools for structural analysis.
ofd-ii) The equivalence holds among cyclicity
uniqueness of vacuum $(: U(x)\Psi=\Psi\Rightarrow\Psi\propto\Omega)$
ducibility ofcluster property:$x\in \mathcal{O}_{1},y\in \mathcal{O}_{2}$$\mathcal{O}_{1}\cross \mathcal{O}_{2}$$\omega_{0}(\varphi_{H}^{i_{1}}(x_{1})\cdots\varphi_{H}^{i_{r}}(x_{r}))$$\varphi_{H}^{i}$$\omega_{0}(\cdot)=\langle\Omega|(\cdot)\Omega\rangle$$)$$Sp(U(\mathbb{R}^{4}))$$Sp(U(\mathbb{R}^{4}))\subset\overline{V+}$$\mathbb{R}^{4}$$\mathfrak{H}$$\hat{\mathbb{R}^{4}}$

$0$

$x$

$\omega_{0}(\varphi_{H}^{i_{1}}(x_{1})\cdots\varphi_{H}^{i_{r}}(x_{r}))$$\mathcal{P}(\mathbb{R}^{4})\Omega=\mathfrak{H}$$\Omega\Leftrightarrow irre-$$\mathcal{P}(\mathbb{R}^{4})\Leftrightarrow$$\Leftrightarrow$$|\omega_{0}(A(x)B(y))-\omega_{0}(A)\omega_{0}(B)|arrow 0$

as$(\vec{x}-\vec{y})^{2}arrow\infty.$

where $A(x);=\alpha_{x}(A)=U(x)AU(x)^{*},$ $B(y):=\alpha_{y}(B)$ are the spacetime, respectively. This
by $x,$
translates of local observables $A,$
condition follows from partition of unity due to spectral resolution of space$B\in \mathcal{P}(\mathcal{O})$$y\in \mathbb{R}^{4}$

time translations $U(x)$ :( -particle singularities on mass-shell$1$$1=| \Omega\rangle\langle\Omega|+\sum_{i}$$+$(absolutely continuous$1\succ$$p^{2}=m_{i}^{2}$)

spectra)and is equivalent to the validity of the ergodicity of a unique vacuum vector
invariant under spacetime translations $U(x)$ in combination with the local
commutativity.
$\Omega$155
3

Asymptotic Condition & Yang-Felman Equation

From the above cluster property combined with local commutativity:$\langle\Omega|A\alpha_{\vec{x}}(B)\Omega\rangle^{\vec{x}}\vec{arrow}\langle\Omega|A\Omega\rangle\langle\Omega|B\Omega\rangle\infty,$

the asymptotic condition(as weak convergence) follows. In sharp contrast to this cluster property for interacting
Heisenberg fields as asymptotic factorization valid only in the asymptotic
limit, the asymptotic fields
materialize the kinematical factorization $(=$
independence) of correlations without taking asymptotic limit, which is justequivalent to the validity of “Wick theorem” :$\varphi_{H}(x)x^{0}=t\mp\infty\vec{arrow}\phi^{in/out}(x)$$\phi^{as}$$\omega_{0}(\phi^{as}\phi^{as}\cdots\phi^{as})=\sum\omega_{0}(\phi^{as}\phi^{as})\cdots\omega_{0}(\phi^{as}\phi^{as})$,as the expansion of -point functions into the sum of products of 2-point
functions. This is nothing but the “quasi-freeness” of
w.r.t.
constituting the contents of independence of Gaussian type.
It is also remarkable that
contains creation and annihilation operators
as infinite number of conserved quantities: for any$f(x)$
solutionof the Klein-Gordon equation $(\square +m^{2})f=0$ , we have$n$$\omega_{0}$$\phi^{as}$$\phi^{as}$$a(\vec{p}),$ $a^{*}(\vec{q})$$J_{\mu}(f):=i\phi(x)5_{\mu}^{+}f(x)$;$\partial^{\mu}J_{\mu}(f)=i\partial^{\mu}[\phi(X)5_{\mu}^{+}]\ni,$among which we find $a( \vec{p})=\int dS^{\mu}J_{\mu}(f),$for $f(x)$ $:=$.
Thus, the independence embodied by asymptotic fields
is seento emerge from interacting Heisenberg fields
via asymptotic condition
as a kind of central limit theorem. In this context, what corresponds to“Langevin equation” is the Yang-Feldman equation to connect Heisenberg field
and asymptotic field:$a^{*}( \vec{p})=\int dS^{\mu}J_{\mu}(\overline{f})$$\exp(-ip_{\mu}x^{\mu})$$\phi^{as}$$\varphi_{H}$$\varphi_{H}(x)$$\phi^{as}(x)$$\varphi_{H}(x)=\int\triangle_{ret}(x-y;m^{2})J_{H}(y)d^{4}y+\phi^{in}(x)$$=[\triangle_{ret}*J_{H}+\phi^{in}](x)$$= \int\triangle_{adv}(x-y;m^{2})J_{H}(y)d^{4}y+\phi^{mt}(x)$$=[\triangle_{adv}*J_{H}+\phi^{\sigma ut}](x)$.

where $J_{H}=(\square +m^{2})\varphi_{H}$ : Heisenberg source current, $\triangle_{ret/adv}(x-$$y;m^{2})$ :retarded/advanced Green’s functions (i.e., principal solutions) of
Klein-Gordon equation defined by$(\square _{x}+m^{2})\Delta_{ret/adv}(x-y;m^{2})=\delta(x-y)$$\Delta_{ret/adv}(x-y;m^{2})=0$

for,$x_{0}\lessgtr y_{0}.$156
and the HeisenIn the Yang-Feldman equation, the asymptotic fields
appear, respectively, as residue and quotient in theberg source current
gives the residues. More important is thatbydivision of$\phi^{as}(x)$$J_{H}$$\varphi_{H}$$J_{H}$$\triangle_{ret/adv}$

at the on-shell pole$\frac{1}{p^{2}-m^{2}}$to determine matrix elements of scattering

amplitudes.

“Central Limit Theorem” as Micro-Macro Du-

4

ality
Along this line, we can now find the natural meaning of “central limit” inthe universality guaranteed by the Haag-GLZ expansion, which is similar
to the “Fock expansion” in WNA. Now, the mutual relations between the(:(: Micro) and the asymptotic fields
interacting Heisenberg fields
Macro) are just described by “Micro-Macro duality” controlled by theK- operator $W$ which is given by$\phi^{as}$$\varphi_{H}$$T$$W^{d}=^{ef}:T(\exp(iJ_{H}\otimes\phi^{in})$:$= \sum_{n}\int d^{4}x_{1}\cdots\int d^{4}x_{n}\frac{i^{n}}{n!}T(J_{H}(x_{1})\cdots J_{H}(x_{n}))$$\otimes:\phi^{in}(x_{1})\cdots\phi^{in}(x_{n})$:and is characterized by the pentagonal relation $W_{12}(W^{as})_{23}=(W^{as})_{23}W_{13}W_{12}$corresponding to a regular oep(withbeing K- operator of CCR
resentation). Rom this, the -matrix and the Haag-GLZ-Fock expansion [4, 5] are derived:$W^{as}$$T$$\phi^{as}$

$S$

$S$

$S :=(\omega_{0}\otimes id)(W)=:(\omega_{0}\otimes id)(T\exp(iJ_{H}\otimes\phi^{in})$$=:(\omega_{0}\otimes id)(T\exp(iJ_{H}\otimes\phi^{out})$::$=: \exp(\phi^{in}(\square +m^{2})\frac{\delta}{\delta J}):\omega_{0}(T(\exp(iJ\varphi_{H}))r_{J=0}$$B=S^{-1}$: $(\omega_{0}\otimes id)(T[B\otimes 1]\exp(iJ_{H}\otimes\phi^{in}))$ :$=:(\omega_{0}\otimes id)(T[B\otimes 1]\exp(iJ_{H}\otimes\phi^{out}):S^{-1},$

and the -matrix describing the state changes in the scattering processes
is an intenwiner between two freeon the vacuum state:
fields, in-coming
and out-going
$S$

$S$

$\omega_{0}=\langle\Omega|\cdots\Omega\rangle$$\phi^{out}$$\phi^{in}$$\phi^{in}(x)S=S\phi^{\sigma ut}(x)$.

is seen inThus the essence of Micro-Macro duality between
andcondition,by the asymptotic
isand thatthatis derived from
Haag-GLZ-Fockexpansion:
by
the
from
reconstructed$\varphi_{H}$$\phi^{as}$$\phi^{as}$$\varphi_{H}$$\varphi_{H}$$\phi^{as}$asymp. cond.
$\phi^{as} arrowarrow \varphi_{H}.$Haag-GLZ-Fock157
In this way, the Micro-Macro duality aspect involved in the “Central Limit
Theorem” can be formulated in QFT in terms of the -matrix functional
and the Haag-GLZ expansion formula.
Moreover, the relevance of harmonic-analytic duality is evident from the
Lie-algebraic structure of Heisenberg source currents$S$$J_{H}^{i}(x)=( \square +m^{2})\varphi_{H}^{i}=S^{-1}\frac{\delta}{i\delta\phi_{i}^{in}(x)}S$;$\frac{\delta J_{H}^{i}(x)}{\delta\psi(y)}-\frac{\delta J_{H}^{j}(y)}{\delta\phi^{i}(x)}=i[J_{H}^{i}(x), J_{H}^{j}(y)].$What is closely related here is the relations among (weak) local commutativity, PCT invariance, -matrix and Borchers classes of mutually local$S$

fields:From the definition of PCT transformation,(within combination with the local commutativity of, the vacuum
is seen to be invariant under, which implies the existenceof anti-unitary
to implement
andThen the irreducibility of(following from the assumption of asymptotic
completeness) implies$\theta(\varphi_{H}(x))=\gamma\varphi_{H}(-x)^{*}$$\gamma\in \mathbb{T})$$\omega_{0}$$\varphi_{H}$$\theta:\omega_{0}\circ\theta=\omega_{0}$$\Theta\Omega=\Omega.$$\theta:\theta(\varphi_{H}(x))=\Theta\varphi_{H}(x)\Theta$$\Theta$$\phi^{as}$$S=\Theta^{in}\Theta=\Theta\Theta^{out},$

ffom such relations ae$S\phi^{out}(x)S^{-1}=\phi^{in}(x)=\Theta\gamma^{-1}\phi^{out}(-x)^{*}\Theta=\Theta\Theta^{out}\phi^{out}(x)\Theta^{out}\Theta.$

Thus, the quantum fields sharing the same $PCT$ operator
shares thesame -matrix: this explains the “ambiguities” of interpolating Heisenberg fields having the same -matrix and is related with
the notion of Borchers classess of mutually local fields sharing the same
PCT operator . This kind of consideration is relevant to the “inverse problem” to reconstruct interacting Heisenberg fields
from the knowledge ofasymptotic fields
and the -matrix intertwining them.
$\Theta$$S$$S=\Theta^{in}\Theta=\Theta\Theta^{out}$$S$$\Theta$$\varphi_{H}$$\phi^{as}$

5

$S$

$S$

“Micro-Micro Duality” between Resonances andRegge Poles in Hadronic World of “Dependence$=$ Coupling”

While the above scheme provides a good description of the scattering processes of quantum fields in terms of the asymptotic fields and the -matrix,
$S$

158
its pertinence highly depends on the validity of the asymptotic condition
to be interpreted as a central limit theorem and on the smallness of thedeviation of the Heisenberg fields from the corresponding asymptotic fields.
If it were not for the notion of -matrix based on the asymptotic fields, thefollowing considerations on the hadronic world certainly could not have beenformulated. However, the conditions crucial for the above discussions areevidently violated in the world of strongly interacting hadrons:$S$

1 almost all hadrons are in highly unstable resonance states, appearing temporarily only in the intermediate states of the scattering processes of
low-lying stable (or almost stable) hadrons. This is just the essential featuresof dependence whose sector structure can be understood as follows:$)$

2 It is remarkable that the basic statistical features of the resonance$)$states can be found in the Cauchy distributions$| \frac{1}{E-E_{i}-i\Gamma_{i}/2}|^{2}$$\propto\frac{1}{(E-E_{i})^{2}+(\Gamma_{i}/2)^{2}}=$

w.r.t. the energy variable which correspond to the expo-

nential decays of unstable particle states $\propto|\exp(-it(E-E_{i}-i\Gamma_{i}/2))|^{2}\propto$
in time. Because of this instability dependence caused byinteractions,strongthethe controllable stable states can be found only inhadrons with the lowest masses among those with the same (internal)
quantum numbers such as the proton, neutron, and pions and so on.
$\exp(-t\Gamma_{i})$$=$3 The word ’‘dual’ in the above diagram means the duality inside a
hadronic sector Regge trajectory of hadron poles with energy-dependent
angular momenta
between its resonance poles & Reggepoles, the former of which appear in the time-like region of the -matrix(usually called -channel) and the latter in the spacelike ones called or$)$$(\simeq$$\alpha(s)=\alpha_{0}+\cdot\alpha’\mathcal{S})$

$S$

$t-$

$s$

$\backslash$$u$-channels:$/arrowarrow$$/arrowarrow\backslash$$/\backslash ^{/}--$$–$$/ \backslash / \backslash / \backslash$

The former

159
describes particle-like unstable modes ( $=pseudo$ -independence) whose
lowest level members only such as proton, neutron, pions, etc., can exist
in (meta-)stable ways, and the latter yields the interaction terms between
hadrons reflccting the aspect of dependence.
4 The meaning of duality here should properly be understood in thefollowing two kinds of contrasts to other contexts: first, the basic common
features found in the mixed moments in all kinds of independences is thecoexistence of all the above three terms of types, s-, t-, -channels. In sharp
contrast to it, these three-type diagrams are here mutually transformed
from one to another without $co$ -existing. Perhaps, this can be interpreted
as one of the most essential features of dependence inherent in the hadronic
processes.
Next, in contrast to the usual kinds of dualities valid between objects living at different levels, the duality appearing here holds at one and the samelevel of strongly interacting hadrons, connecting the aspect of independent
objects and that of dynamical processes as coupling and dependence.
Thus, this duality can be seen as Micro-Micro duality.
$)$

$u$

5 In relation with the Wigner’s construction of representations of the
Poincar\’e group, we can understood this last duality (which motivated the investigation of dual resonanceand string models) as the interchange between the little group $SU(2)$ (or$)$$\mathcal{P}_{+}^{\uparrow}=\mathcal{H}_{2}(\mathbb{C})\underline{\triangleleft}SL(2, \mathbb{C})$$(or, \mathbb{R}^{4}\underline{\triangleleft}SO(1,3))$(3) at the timelike momentum $p,p^{2}>0$ and that $SU(1,1)$ $(or SL(2, \mathbb{R}))$at the spacelike momentum $p,p^{2}<0.$
Reformulating this Wigner-Mackey machinery of induction based on little groups $H$ of a semi-direct product group, we can see theclose relation of the present hadronic duality with the (spontaneous) symmetry breakdown, degenerate vacua unified into the notion of augmented
algebras [1] and with the spacetime emergence [6] as follows: the regular-group algebra
of $G$ is isomorphic to the crossed product( $=C^{*}$ -group algebra of abelian group $N$ ) by the action of onof$N$ , which isalso related with the covariant representations of the dynamicalsystem. For each characterof $N$ , we can consider the
little groupof a pure stateofas the isotropy subgroup of
the latter, which can be viewed as the group of the remaining unbroken
symmetry in the pure state
In the case of the Poincar\’e group, the existence of the wellknown four types of the orbits $p^{2}>0,$ $=0,$ $<0,p_{\mu}\equiv 0$ having little groups$SU(2),$ $E(2),$ $SU(1,1)$ and, respectively, can be interpreted as akind of phase transitions taking place in the process of space-time emergence, whose different phases can be mutually connected through the analytic continuation of the dynamical system
:
$SO$

$)$

$G=N\underline{\triangleleft}L$$C^{*}$$C_{r}^{*}(G)$$C_{0}(\hat{N})\rtimes L$$C_{0}(\hat{N})$$L$$C_{0}(\hat{N})\backslash L$$\chi\in\hat{N}$$H_{\chi}$$\delta_{\chi}$$C_{0}(\hat{N})$$\delta_{\chi}.$$\mathcal{P}_{+}^{\uparrow}=G$$SL(2, \mathbb{C})$$C_{0}(\hat{N})\wedge L$

160$E(2)$

at $p^{2}=0$: parabolic$SU(2)atp^{2}>0$$SU(1,1)atp^{2}<0$: elliptic: hyperbolic$SL(2, \mathbb{C})$at$p\equiv 0$

It would be useful and interesting to review the hadronic dual-resonanceaspects encoded in the string model, from the viewpoint of quantum probits complex analysis, in close relationship with the indepenability
dence, coupling and dependence.
$\mathcal{B}$6 In view \‘of the dominant contributions to the -matrix comingfrom the low-lying hadrons with lightest masses, it would be interesting to
examine the relevance of monotone independence to this context offactorization of dominant components w.r. . the (inverse of) energy
variable $(or 1/s)$ as “time parameter” in the quantum probability theoryof monotone independence [7]. Along this line and also taking account of akind of duality relation between monotone and free independences (as was
informed to me by Dr.Saigo and Mr.Hasebe), we may be attracted by thepossible relations of monotone $and/or$ free independences with theenergy-level statistics of nuclei (among dominant figures of low-lying
hadrons) formulated by random matrices which are closely related with
the quantum chaos and also with the free probability with Wign er’ssemi-circle law as its CLT. In this special situation, the mutual relation
between the shell model and the liquid one of nuclei could possibly beunderstood as a kind of duality, similarly to that between resonance andRegge poles of hadrons.
$S$

$)$

$t$

$s$

Acknowledgments
I would like to express my sincere thanks to Prof. N. Muraki for his kind invitation to this interesting workshop at RIMS on quantum probability withsuch an inspiring title as “Mathematics of Independence and Dependence” :without this title, it would have been impossible for me to consider the
problem discussed here from the present viewpoint. am also very grateful to Dr.Saigo and Mr.Hasebe for instructive discussions on the relevanceof quantum probabilistic notions of independence to the hadronic and/or
nuclear physics from a renewed viewpoint.
$I$

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