数理解析研究所講究録
第 1820 巻 2012 年 87-101

87

How to Unify Interactions?
–Independence and Dependence in Physics–*
Izumi Ojima
RIMS, Kyoto University
Kyoto 606-8502, Japan

Abstract
In this article, a new scheme is proposed for unifying the four interactions goveming nature, i.e., strong, weak, electromagnetic andgravitational ones, based on the triangular matrix features of the coupling scheme which constitutes mathematically a composition seriesof the characteristic dynamical laws and their underlying stabilized
hierarchical domains.

1

Introduction: Micro-Macro Duality & Quadrality Scheme

In this article, a new scheme is proposed for unifying the four interactions
governing nature, i.e., strong, weak, electromagnetic and gravitational ones,
based on the triangular matrix structure of the coupling scheme which constitutes mathematically a composition series of the characteristic dynamicallaws and their underlying stabilized hierarchical domains. To explain it, wefirst need the theoretical framework of quadrality scheme [1] based upon
Micro-Macro duality [2, 3].
1 Micro-Macro duality [2, 3]: as a mathematical version of “quantumclassical correpsondence”, Micro-Macro duality describes bi-directional
relations between microscopic sectors as quasi-equivalence $(=$ unitary equivalence up to multiplicity) classes of factor states of observable algebra andmacroscopic inter-sectorial level in terms of geometrical structures on central spectrum Spec :$)$$(\mathcal{Z})$*An invited talk at RIMS workshop, “Mathematical Studies on Independence and
Dependence Structure $-$ Algebra Meets Probability -,, , 19-21 December 2011, RIMS,
Kyoto University organized by Prof. Muraki.

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According to Fourier & Galois dualities, these Micro and Macro are induality, meaning that the data given at Macro level is derived from the structural analysis of Micro and vice versa. This implies also the bi-directionality
between Induction & Deduction. Essence of duality in Fourier transform$( \mathcal{F}f)(\gamma)=\int\overline{\gamma(g)}f(g)dg)(f\in L^{1}(G))$ is formulated by Fourier-Pontryagin
and its dual
duality
between a locally compact abelian group. Via extension to:consisting of all the characters
group
compact cases due to Tannaka & Krein, the most general form can now befound in Tatsuuma-Enock-Schwartz theorem of the duality between a locallycompact non-abelian group and its representation category Rep $(G)$ consisting of “all” the representations. The corresponding version is formulated
by Takesaki (and $/or$ Takai) for dynamical system with (non-commutative)of on in such a form (inalgebraand with an action :-versions, respectively) as
or$G\Leftrightarrow\hat{G}$$G$$Garrow \mathbb{T}$$\hat{G}$$\chi$$G$$a$$\mathcal{F}$$\tau$$\mathcal{F}rightarrow G$$G$$C^{*}-$$\mathcal{F}$$W^{*}$$\mathcal{F}^{G}\rtimes_{\hat{\tau}}\hat{G}=\mathcal{F}$: Recovery of$\mathcal{F}\rtimes_{\tau}G=\mathcal{F}^{G}\otimes \mathcal{K}(L^{2}(G))$

or$\mathcal{F}$

from

$G-$

invar
.

$\dot{u}ants\mathcal{F}^{G}$;$\mathcal{F}^{G}\otimes B(L^{2}(G))$

2 Quadrality scheme [1]: as a combination of two kinds of Micro-Macrodualities in “horizontal” and “vertical” directions, a general methodologicalframework can be formulated for theoretical descriptions of physical phenomena which I call quadrality scheme:$)$

Here, the map $V$ maps modules or representations to the classifying space
specifying the components of spectmm contained in modules and the inverse

89
map assigns the corresponding modules to subsets (or varieties) in theclassifying space of sectors. Moreover, this quadrality scheme is equivalent$I$

to “adjunction” as a categorical formulation of duality! [i.e., Duality
Quadrality]
It is remarkable that this quadrality structure overlaps with the basic
structure controlling the four interactions appearing in particle physics asfollows:$dual\Leftrightarrow$in which the meaning of “unification of four forces” need not be restricted
to such a simple-minded version as converging to single entity, like the fashionable one. Instead, their mutual relations may well be understood alternatively in their integrated organization in nature and the corresponding
theory, where they occupy mutually different places, playing different roles
inherent in each, through which a unified totality of nature and its theoreticalexplanations are achieved. From this viewpoint, the standard simple “unification” pursued in “Geometrization of Physics“ seems to be too narrow toincorporate these non-trivial aspects.
It is also interesting to note that the four kinds of $QP$-independence
can be accommodated in the quadrality scheme, according to their mutual
relations in such a way as:

Here the emergence of monotone independence from tensor type one is due
to Dr. H. Saigo’s “Quantum-Classical Correspondence between Harmonic
Oscillator and Arcsin Law” involving emergence as an essential step: harmonic oscillator as a typical model of (quantum-decomposed”) statisticalemergence
independence
arcsin law CLT of monotone independence. This is, from asimilar to the Bogoliubov transformation:
harmonic oscillator to Poisson distribution (see Accardi-Obata’s textbook).
$arrow$$=$$aarrow a+\sqrt{\lambda},$ $a^{*}arrow a^{*}+\sqrt{\lambda}$

90
bi-modulewith left and right acttions, respectively, of the algebraof microscopic variables and of the observable algebra
and -valued left
equipped with -valued right inner productcan be defined by a subalgebra
The linking algebraonemaps
has such
to . Whenfromof adjointableof the algebra, the fullness ofanda unit element that( “imprimitivity”) can be expressed by theinin as well as of, the former of which can beandexact sequences,
understood as the bundle structure and the latter as a connection definedby a splitting of the former. This construction of a linking algebra
starting from a Hilbert module can be inverted, in a sense, by considering
the Grassmannian Hilbert module as a module version of Grassmannian
manifolds in such forms as$E$$\mathcal{X}$$\mathcal{A}:\mathcal{X}arrow E\wedge \mathcal{A},$$E^{*}E=\mathcal{A}$$\mathcal{A}$$\mathcal{X}=EE^{*}$$\lambda(E)$$E$$\mathcal{L}(E)$$I$$\langle I|E\rangle\cong|E\rangle$$EE^{*}$$\mathcal{A}$$\mathcal{X}$

$E$

$E$

$E^{*}E$$|I\rangle\langle E|\cong\langle E|$$=$$\mathcal{X}$$\mathcal{A}arrow E\mapsto \mathcal{X}$$\mathcal{A}arrow Earrow \mathcal{X}$$\lambda(E)$$E$

The conceptual meaning of these formulae can be understood through suchan example as Prof. Mikio Sato’s formulation of the setof -valued solutions of a system $Pf=0$ of linear, the latpartial differential equations with a differential map
ter of which can be boiled down to specifying a subalgebra $\mathcal{X}=coker(P)=$. In this way, off-diagonal terms in theof a CCR algebra
hnking algebra of a Hilbert bimodules can be reformulated into Grassmannian Hilbert-modules: the-bimodule $F$ plays the role of visualizing the-bimodule $E$
unknown solution of equations in the known and thecan work in the opposite direction to find the equation from the known
solutions in$Hom_{\mathcal{D}}(\mathcal{A}arrow \mathcal{X})=$$Hom(\mathcal{A}\cdotarrow \mathcal{X})^{\mathcal{D}}=F^{\mathcal{D}}$$\mathcal{A}$$P:\mathcal{D}^{m}arrow \mathcal{D}^{n}$$\mathcal{D}^{n}[4]$$\mathcal{D}^{n}/\mathcal{D}^{m}P$$\mathcal{A}-\mathcal{X}$$\mathcal{X}$$\mathcal{A}$$\mathcal{X}-\mathcal{A}$$\mathcal{X}$$\mathcal{A}.$2

Coupling Patterns of Four Interactions

Applying this formulation, we can find an interesting physical meaning of
the above relations in the following new observations for “unification offorces” [IO, in preparation]. The process of “co-emergence” of the algebraof observables from the dynamical flows is seen, in the tablc of coupling
patterns below, to carry such a characteristic feature as a lower triangularmatrix structure of the interactions:91

To incorporate this information into the construction of the relevant dynamical systems corresponding to different kinds of interactions, we consider
a functor of dynamical co-emergence from the category of ordered typesof force $=\{g($ ravity) $arrow w($ eak $)arrow e($ l.mg. $)arrow s($ trong) to a -category
generated by algebras, respectively,of charged leptons,of hadrons,
of $W,$ -bosons,of electromagnetic andofneutrinos$\varphi$$\}$$\mathcal{M}_{h}$$\mathcal{M}_{W}$$Z$$\mathcal{M}_{\nu},of$$v,$$\otimes$$\mathcal{M}_{l}$$\mathcal{M}_{e}$$\mathcal{M}_{g}$gravitational fields:

Objects
for $i=$ g, w, e, h, are respective automorphism groups due
to each force whose acting patterns are indicated above:$\varphi(i)$Morphisms:$\{\begin{array}{l}\varphi(g)arrow \mathcal{M}_{g}\otimes \mathcal{M}_{e}\otimes[\mathcal{M}_{\nu}\otimes \mathcal{M}_{W}\otimes \mathcal{M}_{l}]\otimes \mathcal{M}_{h}\varphi(w)arrow [\mathcal{M}_{v}\otimes \mathcal{M}_{W}\otimes \mathcal{M}_{l}]\otimes \mathcal{M}_{h}\varphi(e)\varphi(s)arrowarrow [\mathcal{M}_{W}\otimes \mathcal{M}_{l}]\otimes_{\mathcal{M}_{h}}\mathcal{M}_{h}\varphi(iarrow j)=[\varphi(i)arrow\varphi(j)]\end{array}$where the morphisms
describe the mutual relations
among different forces and fields due to the off-diagonal terms. The coemergence flows from Dynamics to Algebra are expressed here by the le $ft-$
actions of dynamics on the respective algebras, which can be seen as thetable of coupling pattems in the following upper triangular form:$\varphi(iarrow j)=[\varphi(i)arrow\varphi(j)]$Here the direction from the right bottom to the left top can be regarded as
the histori $cal$ formation of stabilized hierarchical domains with thefirst appearance 2 of stabilized forms of matter and with the associatedrepeatable laws established to run. The opposite arrowfrom the left
top to the right bottom can be viewed as the directions of deepening humanrecognitions and controllability, which always involve the dilations touncover microscopic agents causing unknown $phenomenaarrow$ at the macroscopic
levels. The divergent
and the convergent
can be seen to constitutethe historical paths of nature. The question mark ? above shows different$\nwarrow$$\searrow$$\nwarrow\uparrow$$\nwarrow$

$($

$)$

92
behaviours of electromagnetic field at different scales:$?=\{\begin{array}{lllll}0 behavioursas apurelybehan fi gaugeeldin Macro2 Weinberg- Salam m\ddot{m}ng of(e)arrowarrow(w)in Micro\end{array}\}$$*)$ , the level of hadrons
At the bottom row $(s$ (trong) $|0$
as the only carriers of strong forces shows a sharp contrast between the $ex-$
tremely high instability of individual members and the persistent existence of the level itself as a whole under the influence of all different forces$0$

shown in the rightmost column:$0$$(\begin{array}{l}\frac{hadrons}{*}***\end{array})$$0$. It may be interesting to findthe answer to the question where we have seen similar patterns elsewhere, insuch a form as the level of bacteria in the organic world [5]! This situation$*)$ , showing
should be contrasted with the top line, gravity $|*$
the universality of gravitational attractions.
$*$

$*$

$*$

$($

The above upper triangular block structure,

can be related with a generalized flag manifold:$GM(m_{1}|m_{2}|\cdots|m_{r})$$=U(m_{1}+m_{2}+\cdots+m_{r})/(U(m_{1})\cross U(m_{2})\cross\cdots\cross U(m_{r}))$$:=GM(m1|m_{2}+\cdots+m_{r})\cross GM(m_{2}|m_{3}+\cdots+m_{r})\cross\cdots\cross GM(m_{\tau-1}|m_{r})$$=GL(m1+\cdots+m_{r};\mathbb{C})/\{\begin{array}{llll}GL(m_{1}.\mathbb{C}) M(m_{1},m_{2}\cdot \mathbb{C}) \cdots M(m_{l},m_{r}.\mathbb{C})0 GL(m_{2}\cdot \mathbb{C}) \cdots M(m_{2},m_{r}\cdot \mathbb{C})\vdots \vdots \cdots \vdots 0 0 \cdots M(m_{r-l},m_{r}\cdot \mathbb{C})0 0 0 GL(m_{r}\cdot \mathbb{C})\end{array}\}$whose denominator is a Borel parabolic subalgebra describing a nested
solvability structure of composition series.
Dual to the above $co$ -emergence: $(Alg\wedge Dyn)$ , is the emergence:$(Algrightarrow Dyn)^{*}=(Spec\Leftarrow$ States) which can be seen as follows:

93: upper triangular (with electric charges $Q=I_{3}+(B+S-L)/2$ )Moreover, hadrons are “terminal objects” dual to the spacetime points
as initial objects which is equivalent to “confinement”!! (because, inthis context, “quarks” are nothing but constituents of weak-electro-magnetic
currents
to couple withwithout need for any
andreality!)$x^{\mu}$$J_{\mu},$

2.1$J_{\mu}^{V-A}$$A_{\mu},$$W_{\mu}^{\pm}$$Z_{\mu}$

History of universe vs. repeatable laws

We can interpret the upper off-diagonal terms
as a composite system of
diagonals
andcharacterized by exact sequences, equipped with a structure ofHilbert(withmodules-valued left inner product and-valued right inner product).
can also be regarded as$F_{ij}$$\mathcal{A}_{1)}\cdots,$$\mathcal{A}_{r}$$\mathcal{X}_{1},$$\cdots,$$\mathcal{X}_{r-1}$$1eft-\mathcal{A}_{i}right-\mathcal{X}_{i}$$\mathcal{A}_{i}arrow F_{i,i+1}+arrow \mathcal{X}_{i}$$\mathcal{A}_{i}Fi,i+1\mathcal{X}_{i}$$\mathcal{A}_{i}=F_{i,i+1}F_{i,i+1}^{*}$$F_{i,i+1}$$\mathcal{X}_{i}=F_{i}^{*}{}_{i+1}F_{i,i+1}$

a lefl-adjoint functor. Thus, the upper triangularity with in
the lower left corner implies the presence of $uni$ -directional arrows
which provide a monotone scaling parameter (similarly to Lieb-Yngvason’s
axiomatic derivation of entropy from adiabatic ordering) of what can beinterpreted as historical $uni$ -directionality inherent already at theMicro-level.
To understand the mathematical structure relevant to the mutual relations between micro- and macroscopic levels, what is crucial is to clarify
the processes of emergence of the latter from the former: they can be understood here as condensation of order parameters arising from $sym-$
metry breaking (spontaneous or explicit). Since any transformation
associated with an explicitly broken symmetry changes (by definition) thephysical constants characteristic of a physical system described by a specifictheory, the notion of explicitly broken symmetries has been excluded systematically from the traditional schemes for mathematical and $/or$ theoreti$\mathcal{A}_{i}^{F_{i,i+1}}arrow \mathcal{X}_{i}$$0$$\nwarrow,$

94
cal treatments of a “fixed” physical theory. From a more general viewpoint,
however, nothing prohibits us to consider many different theories at once
and the use of explicitly broken symmetry tansformations can be systematized in such a context into a method of variation of natural constants
to relate different physical theories; in fact, such a method has been usedin [6] in a efficient way to identify (inverse) temperature with an order parameter of explicitly broken scale invariance. In this context we supposea structure of Theory Bundle with a base space consisting of physicalconstants on each point of which fibre is given by a physical theory with
fixed physical constants specified by the base point [7]. In this formulation,such a transformation can easily and consistently be performed as mappinga given physical theory with a fixed set of physical constants into a different
theory with another fixed one. By doing this, we can not only compare
different theories but also pursue the historical formations of stabilized domains such as the worlds of nuclei and of atoms equipped with repeatable
and reproducible physical laws in the evolution processes of the universe.
Along the line of the latter problem, we can discuss the following problem:[Historical formation of Stabilized $Domains$ ]Realization of repeatable laws] duality between [History processes governed by rvnndomness to form hierarchical domains] and [hierarchical domains $=$worlds of necessity controlled by laws].
A simple mathematical model of this situation can be found in the decomposition of broken symmetry group
into a compact subgroup $H$ describing an unbroken symmetry as a kinematically repeatable law and ahomogeneous space $G/H$ describing a stabilized domain consisting of degenerate vacua mutually connected by broken dynamical transitions (due toGoldstone modes) which arises from a historical process of emergence due
to condensations.
$\Leftrightarrow[$

$=$

$=$

$G$

3

Emergence of Relativistic Spacetime

At this point we notice the similarity of the relevant structures involved inthis picture to the basic features of “punctuated equilibria” [8] found in theevolution processes in the biological world. We confirm this viewpoint by examining the typical and, perhaps, most important case of emergence of thespacetime, among transition processess bridging various regimes controlled
by different forces. bom this angle, the etiology of the current fashionable “unification programme” can be found in “Geometrization of Physics”,
whose origin goes back to Einstein’s unsuccessful “Umified Field Theory” tounify electromagnetism and gravity in spacetime geometry: unfortunately,
the physical origin of gravity and spacetime cannot be unveiled from thisviewpoint: for instance, what does spacetime mean when we are faced withtotally indeterminate future??

95
To overcome such conceptual difficulty, we propose here a scenario forderiving gravity and spacetime as epigenetic secondary notions emerging
from microscopic regimes of (microscopic) matter motions. To this end, theessence of the following discussion is just to explain the following diagram
consisting of the structures relevant to the emergence of special- and generalrelativistic spacetimes (see below).
The basic ingredients necessary for this purpose are as follows:i Independence
freely falling frames as “sectors” without gravity
containing only strong, weak & electromagnetic couplings
$=$

$)$

ii) Coupling

defined as Levi-Civita connection
to connect different free-falling fames as “sectors” at the meta-level, and,$=$gravitational force$\Gamma_{\mu\nu}^{\lambda}$iii) Dependence the composite system arising from the above physicalsystems constructed by three kinds of forces (strong, weak & electromagnetic) coupled with each other by the gravitational force.
$=$

By re-examining how general-relativistic spacetime emerges from the physical processes in Micro quantum systems, we clarify here under which condition the notion of “spacetime” can be meaningful from the viewpoint of“Micro-Macro duality”

96
3.1

Symmetry Breaking & Condensed Statesin a stateof a dynamical system
Breakdown of a symmetry
is characterized [2] by non-invariance of the “central extension” ofunder the corresponding G\‘on the centre
is broken, which. In this case, Galois closedness ofaction on
described by a compact Lie subis recovered by dynamical system
accordgroup $H$ of corresponding to unbroken symmetry:
ing to the general method of Galois extension by means of crossed products. Then, the sector structure is determined by the factor spectrum$\mathcal{F}\wedge G$$G$$\omega\in E_{\mathcal{F}}$$\mathcal{Z}_{\pi_{\omega}}(\mathcal{F});=\pi_{\omega}(\mathcal{F})"\cap\pi_{\omega}(\mathcal{F})’$$\omega$$\mathcal{F}^{G}$$\mathcal{Z}_{\pi_{\omega}}(\mathcal{F})$$\mathcal{F}oH$$\mathcal{F}=\mathcal{F}^{H}\rtimes\hat{H}$$G$$[9\wedge 2]$$\mathcal{F}^{H}=Spec(\mathcal{Z}(\mathcal{F}^{H}))=\hat{H}$as a group dual consisting of irreducible unitary

represetations of . On the basis of these key roles played by the symmetry
breaking, we can understand the emergence of macroscopic phenomena ascondensation effects coming from the microscopic dynamics, whose precise mathematical formulation can be given by the application of forcing
method whose common use has benn restricted so far in the contexts offoundations of mathematics (and which also implies the Bom rule [5]).
$H$

For this purpose, we need to extend the standard Doplicher-Roberts
from the observable
method for reconstructing the field algebra$H$
algebrawith the unbroken symmetry described by , to $incorpo_{\wedge}ratethe$situation of a broken symmetry in the following way [2]: withcalled an augmented algebra [2], we have a split bundle exact$\mathcal{F}=\mathcal{F}^{H}\rtimes\hat{H}$$\mathcal{F}^{H}$$\tilde{\mathcal{F}}$$:=\mathcal{F}^{H}\rtimes G=$$\mathcal{F}\rtimes(\overline{H\backslash G})$. In this situation, the minimality ofsequence-central ergodicity, i.e., -ergodicity of the centre
is guaranteedin the representation given by the GNS representation of. Then we have the followingof
induced from the vacuum state
commutativity diagram of algebra extension:$\mathcal{F}^{H_{\langlearrow}^{\overline{m}}}\tilde{\mathcal{F}}\mapsto\overline{\mathcal{F}}/\mathcal{F}^{H}\simeq\hat{G}$$\tilde{\mathcal{F}}$$\vec{b}ytheGarrow$$\mathcal{Z}_{\tilde{\pi}}(\tilde{\mathcal{F}})$$G$$\omega_{0}\circ\tilde{m}$$\tilde{\pi}$$\omega_{0}$$\mathcal{F}^{H}[2]$which is dual to the diagram for sectors:

97

where
denotes the factor spectrum of , etc.
The physical essence of extension
from the G-fixed point$H$
subalgebra
to the -fixed one
can now be understood as the “exten”
sion of a coefficient algebraby (the dual of) $G/H$ whi $ch$ parametrizes
the degenerate vacua:. In
this extension, a part $G/H$ of originally invisible
has become visible through the emergence of degenerate vacua parametrized by $G/H$
due to the condensation of order parameters $\in G/H$ associated with(ponteneous) (ymmetry) $B$ (reaking) of to $H.$
As a result, observables
acquire $G/H$-dependence: $\tilde{A}=(G/H\ni$, which should just be interpreted as an
example of the logical extension [10] transforming a “constant object”
into a “variable object”
having functional
dependence on the universal classifying space $G/H$ for multi-valued
semantics(, as is familiar in non-standard and Boolean-valued analysis).
$\hat{\mathcal{F}}=Spec(\mathcal{Z}(\mathcal{F}))$$\mathcal{F}$$\mathcal{F}^{G}\Rightarrow \mathcal{F}^{H}$$\mathcal{F}^{G}$$\mathcal{F}^{H}$$\mathcal{F}^{G}$$\mathcal{F}^{H}=\tilde{\mathcal{F}}^{G}=[(\mathcal{F}\rtimes(\overline{H\backslash G})]^{G}=\mathcal{F}^{G}\rtimes(\overline{H\backslash G})$

$G$

$S$

$S$

$G$

$A\in \mathcal{A}$$\dot{g}\mapsto\tilde{A}(\dot{g})\in \mathcal{A})\in \mathcal{A}\rtimes(\overline{H\backslash G})$$(\tilde{A}\in \mathcal{A}\rtimes(\overline{H\backslash G}))$$(A\in \mathcal{A})$3.2

Emergence of Spacetime as Symmetry Breaking

By replacing $G/H$ with spacetime, the above situation can be regarded as aprototype for the origin of functional dependence of physical quantities on
spacetime coordinates, due to the physical emergence of spacetime frommicroscopic physical world. Along this line, we prescribe the similar logicalextension procedure on the observable algebra
adding $G/H$-dependence:$\mathcal{F}^{H}$$\mathcal{F}^{H}\rtimes(\overline{H\backslash G})=(\mathcal{F}\rtimes(\overline{H\backslash G}))^{H}=\tilde{\mathcal{F}}^{H}.$The whole sector structure ofits factor spectrum$\tilde{\mathcal{F}}^{H}=(\mathcal{F}^{H}\rtimes(\overline{H\backslash G}))$$\tilde{\mathcal{F}}^{H}\wedge=G_{H}\cross\hat{H}$$\tilde{\mathcal{F}}^{H}=G_{H}\cross\hat{H}arrow G/H$

can be identified with; this constitutes a sector bundle,$\hat{H}\hookrightarrow$, consisting of the classifying space $G/H$ of degeneratevacua, each fibre over which describes the sector structure of unbroken
remaining symmetry $H$ (or, more precisely, the family of conjugated groups$gHg^{-1}$ for the vacuum parametrized by $\dot{g}=gH\in G/H$ ).
Namely, thesector bundle can be seen as the connection or a splitting of bundle exact$\hat{H}$$-$sequence dual to$\mathcal{F}^{H}=\hat{H}arrow\tilde{\mathcal{F}}^{H}=G_{H}\cross\hat{H}rightarrow G/H$of observable triples,$\mathcal{F}^{H}\hookrightarrow\tilde{\mathcal{F}}^{H}=\mathcal{F}^{H}\rtimes(\overline{H\backslash G})arrow(\overline{H\backslash G})!$

Now we apply the above scheme to the situation with the group containing both extemal ( spacetime) and intemal symmetries. For simplicity, the latter component described by a subgroup $H$ of is assumed to be
unbroken, and hence, the broken symmetry described by $G/H$ represents
$G$

$=$

$G$

98
the spacetime structure. It would be convenient (not essential, though) totake $H$ as a normal subgroup of . To be precise, $G/H$ may contain suchnon-commutative components as spatial rotations (and Lorentz boosts) actas
ing on spacetime, but, we simply neglect this aspect to identify
spacetime itself (from which the corresponding transformation group caneasily be recovered).
$G$$G \int H$

Then, by identifying $G/H$ with a spacetime domain , we find an impressive parallelism between the commutative diagram in the previous subsection and the diagram in the Dophcher-Roberts reconstruction [9] of local(without
from the local observable net
field net
the two bottom lines) as follows:$\mathcal{R}$$\mathcal{R}\mapsto \mathcal{A}(\mathcal{R})$$\mathcal{R}\mapsto \mathcal{F}(\mathcal{R})$$\mathcal{F}^{H}\swarrow \mathcal{F}^{H}=\tilde{\mathcal{F}}^{G}\Downarrow\searrow G/H\tilde{\mathcal{F}}^{H}$$\tilde{\mathcal{F}}$$\mathcal{O}_{d^{G}}\swarrow \mathcal{O}_{\rho}=\mathcal{O}_{d}^{G}\Downarrow\searrow \mathcal{R}\mathcal{A}(\mathcal{R})$$\downarrow \mathcal{R}\searrow\searrow$$\Leftrightarrow$$\swarrow H\downarrow$$\mathcal{F}(\mathcal{R})$$\swarrow c$$\downarrow$, where$\downarrow\downarrow G/H\swarrow\searrow\searrow$$\hat{G}\downarrowarrow\swarrow\frac{\Downarrow}{G\cross \mathcal{R}}$$\hat{G}\Downarrow$$\searrow\searrowrightarrow$$\downarrow\hat{\mathcal{R}}$$\mapsto\searrow\searrow_{\frac{\downarrow}{G/H}}$$\hat{H}$$arrow$is the Cuntz algebra of -isometries.
Thus we have arrived at the stage just before gravity to be switched on, toenter General Relativity via Equivolence Principle. This can naturally
be formulated and understood by the above scheme in combination withinduced representation. So, we should recall here the diagram at the endof the previous section, Sec.3. Here, the unbroken symmetry may be eithercan, however, be a bigger
Poincar\’e groupor rotation group $SO$ (3).
group of symmetry transformations formulated as functors in the context of
category.
$d$$\mathcal{O}_{d}$$G$$\mathcal{P}_{+}^{\uparrow}$4

Physical meaning of Equivalence Principle in
General Relativity in he emergence process$t$

We consider processes of spacetime emergence taking place in parallel under
the influence of strong and electro-weak interactions other than gravity,
each of which results in a “fiber” ( sector pure phase) parametrized by. The word “fiber” hcre means a flat tangent
spacetime coordinates
space as a fiber $(T_{x}(M))$ of a tangent bundle $(T(Al))$ on each pointof the base space as a “spacetime manifold” (which would be called $M$ butwhich cannot be recognized yet as such); as its physics is controlled by thethree interactions other than the gravity, this fiber describes a free-falling
frame without any gravitational force (the last of which has not emergedyet). In connection with our discussion up to this point, the word “sector”
should be more appropriate than “fiber”, we accept the latter use in order
to avoid the misunderstanding of what we are concerned with here.
To be precise, what we know up to now is only the simultaneous processes
$=$

$=$

$x^{\mu}$$x^{\mu}$99of spacetime emergence at many “fiber” points
each of which consists ofthe physical world of Poincar\’e covariant quantum fields governed by thestrong and electro-weak interactions in the Minkowski spacetime but we do
not know anything about the mutual relations among different fibers. By
picking up just one specific “fiber”, we focus on the local physics describedby the Poincar\’e covariant QFT developed inside of the “free falling system
without gravity”, which is nothing but the physical contents of “tangential
world” equipped with local Lorentz structure, on (or “in”?) a pointin the emergent “base space”
Now, we pose a question: what does it mean to impose the physicalrequirement of “equivalence principle” between gravitational and inertial
masses, $m_{grav}=m_{inert}$ on the situation after the “individual” processes offree-falling systems arising from the emergence of special-relativistic local
spacetime? While the notion of “inertial mass” already exists in the “standard” physics formulated within the free-falling frames without gravity, it
does not apply to the case of “gravitational mass” before our starting todiscuss the situations governed by the gravitational interaction. It can bemeaningful only in the context where such an attribute is assigned to $a(n$asymptotically) free mass point on the mass-shell, as generating the gravitational force or field as the forth one other than strong and electro-weakforces, through Einstein’s gravitational equation:$x^{\mu}$$(x^{\mu})$$R_{\mu\nu}- \frac{1}{2}g_{\mu v}R=\kappa T_{\mu\nu}.$

When we find the first (or, the 0-th approximated) roles of gravity inregulating the mutual relations among different fibers sectors as free-falling
frames, the proper range of action of the gravitational mass $m_{grav}$ is at thelevel of “inter-fiber $=inter$-sectorial relations”, but, in contrast, that of theinertial one $m_{inert}$ is in the physics within each fiber” (or sector). Therefore,
the equivalence principle qualitatively controls in a bi-directional way themutual duality relation between the inside and the outside of “fibers” (or,sectors : we suppose that the inter-fiber relation of free-falhng frames on
the “neighbouring” points
andis controlled by the connection, as is indicated in the diagram at the beginning, which results
coefficientsin a force propertional to the gravitational mass $m_{grav}$ acting on the inertial$=$$)^{}$$x^{\mu}$$x^{\mu}+\delta x^{\mu}$$\Gamma_{\mu\nu}^{\lambda}$mass$m_{inert}.$Then, the Newtonian equation of motion of the mass point $m_{inert}$ withthe velocity vector$v^{\lambda}$$:= \frac{dx^{\lambda}}{d\tau}$can be written as,$them_{inert}dv^{\lambda}=-v^{\mu}(m_{grav}\Gamma_{\mu\nu}^{\lambda}dx^{\nu})=m_{grav}v^{\mu}\nabla_{\mu}dx^{\lambda}$$\overline{lFrom}$
the viewpointofemergenceasaprocessofphaseseparation,

roles played by
the free-falling frame in each “fiber” and by “base space” can be compared with $H$ and$G/H$ whose duality relation can be seen in the form of “Helgason duality”. In this sense,
the gravitational equivalence principle is analogous to “Helgason duality”.

100
By the requirement of equivalence principle $m_{grav}=m_{inert}$ , this reduces, whose purely geometricto the geodesic equation,
form and independence of the specific mass values ensure the universalityof the mass-point motions. Namely, through the validity of equivalence
acquires its own abprinciple $m_{grav}=m_{inert}$ , the spacetime notion
stract universal meaning, independently of its physical origin in the mutual
relations among different “fibers” of local physics consisting of three interactions, to such an extreme extent that space and “time” exist in themselves,
extending from the past, the present and even the future! Eventually, thephysical motions of mass points are now absorbed into (small) part ofspacetime geometry in the form of geodesic motions, without exhibiting
their individuality [11].
Owing to this mechanism, we can easily forget about the physical $ori-$
gin of spacetime, which can, however, exhibit its existence in the situation
where the validity of equivalence principle is threatened. It is also interesting to note that the above equation of motion can be rewritten in terms of
into
momentum$\frac{dv^{\lambda}}{d\tau}+\Gamma_{\mu\nu}^{\lambda}v^{\mu}v^{\nu}=0$$x^{\mu}$$a$$p^{\lambda}=mv^{\lambda}$$dp^{\lambda}=p^{\mu}\nabla_{\mu}dx^{\lambda},$which explains that mass-point motions as geodesic motion can be absorbed
into the covariance (of physical motions) under the (covariantized) generalcoordinate transformations. If the above discussion is compared with thestandard mathematical treatment of bundle structures in differential geometry, we understand that ours go from physics in the (standard) fiber tothe mathematical structure of tbe bundle and base spaces in the opposite
direction to the latter and that the mathematical essence of the equivalenceprinciple hes in the -structure of the tangent and frame bundles of thespacetime $M$ with being identffied with the Lorentz group [12].
$G$

$G$

References[1] Ojima, I., Meaming of Non-Extensive Entropies in Micro-Macro Duality,
J. Phys.: Conf. Ser. 201012017 (2010); Ojima, I. and Okamura, K.,
Large deviation strategy for inverse problem, to appear in Open Sys.
Info. Dyn. (arXiv:1101.3690).
[2] Ojima, I., A umified scheme for generalized sectors based on selection
criteria-Order parameters of symmetries and of thermal situations andphysical meamings of classifying categorical adjunctions-, Open Sys.
Info. Dyn. 10, 235-279 (2003).

[3] Ojima, I., $Micre\succ$ macro duality in quantum physics, 143-161, Proc. Intern. Conf. “Stochastic Analysis: Classical and Quantum”, World Sci.,

101
2005; Ojima, I. and Takeori, M., How to observe and recover quantum
fields from observational data? -Takesaki duality as a Micro-macro
duality-, Open Sys. Info. Dyn. 14, 307-318 (2007); Harada, R. andOjima, I., A unified scheme of measurement and amplification processes
based on Micro-Macro Duality-Stern-Gerlach experiment as a typicalexample-, Open Sys. Info. Dyn. 16, 55-74 (2009).
[4] See for instance, Tanisaki, T. and Hotta, R., $D$ -modules and Algebraic
Groups (in Japanese), Springer-Verlag Tokyo.

[5] Ojima, I., Space(-Time) Emergence as Symmetry Breaking Effect, Invited talk at International Conference in Quantum Bio-Informatics
Center, Tokyo Unviersity of Sciences, 2010; Dilation and Emergence inPhysical Sciences, Invited talk at International Conference, “Advancesin Quantum Theory” at Linnaeus University, June 2010.
[6] Ojima, I., Temperature as order parameter of broken scale invariance,
Publ. RIMS (Kyoto Univ.) 40, 731-756 (2004).
[7] Ojima, I., Micro-Macro duality and emergence of macroscopic levels,
Quantum Probability and White Noise Analysis, 21, 217-228 (2008).
[8] Eldredge, N. and Gould, S. J., Punctuated equilibria: an alternative
to phyletic gradualism, pp. 82-115 in T.J.M. Schopf, ed., Models in
Paleobiology. San ancisco, Freeman Cooper.
$\mathbb{R}$[9] Dophchcr, S. and Roberts, J.E., Why there is a field algebra with acompact gauge group describing the superselection structure in particle
physics, Comm. Math. Phys. 131 (1990), 51-107.

[10] Ojima, I. and Ozawa, M., Unitary representations of the hyperfinite
Heisenberg group and the logical extension methods in physics, Open
Systems and Information Dynamics 2, 107-128 (1993).
[11] Ojima, I., New Interpretation of Equivalence Principle in General Relativity from the viewpoint of Micro-Macro duality, Invited Talk at International Conference, “ Foundations of Probability and Physics 6” at
Linnaeus University, June 2011.
[12] Dieudonn\’e, J., Treatise on Analysis, Vol.4, Academic Press.