Zeno Toffano
email: zeno.toffano@centralesupelec.fr

François Dubois
email: francois.dubois@cnam.fr

Patrick Canipet

Hanaé Carrié

Guillaume Dubert

Benoit Dufumier

Elvire De

Mathieu La Fresnaye

Philippe Houdebine

Thibault Mantrand

Victoire Marcilhacy

Caroline Pansard

Noémie Perrot

Quantum Observables for Binary, Multi-Valued and Fuzzy Logic : Eigenlogic

We propose a linear algebraic method, named Eigenlogic [1], for two-, many-valued and fuzzy logic using observables in Hilbert space. All logical connectives are represented by observables where the truth values correspond to eigenvalues and the atomic input propositional cases, i.e. the "models" of a propositional system, to the respective eigenvectors. In this way propositional logic can be formalized by using combinations of tensored elementary quantum observables. The outcome of a "measurement" of a logical observable will give the truth value of the associated logical proposition, and becomes "interpretable" when applied to vectors of its eigenspace, leading to an original insight into the quantum measurement postulate. Recently the concept of "quantum predicate" has been proposed in [2] leading to similar concepts.

We develop logical observables for binary logic and extend them to many-valued logic.

For binary logic and truth values {0, 1} logical observables are commuting projector operators [1].

For truth values {+1, -1} the logical observables are isometries formally equivalent to the ones of a composite quantum spin 1 2 system, these observables are reversible quantum logic gates. The analogy of many-valued logic with quantum angular momentum is then established using a general algebraic method, based on classical interpolation framework suggested by the finite-element method. Logical observables for three-valued logic are formulated using the orbital angular momentum observable L z with = 1. The representative 3-valued 2-argument logical observables for the ternary threshold logical connectives Min and Max are then explicitly obtained [3].

Also in this approach fuzzy logic arises naturally when considering vectors outside the eigensystem. The fuzzy membership function [4] is obtained by the quantum mean value (Born rule) of the logical projector observable and turns out to be a probability measure. Fuzziness arises because of the quantum superposition of atomic propositional cases, the truth of a proposition being in this case a probabilistic value ranging from completely false to completely true. This method could be employed for developing algorithms in high-dimensional vector spaces for example in modern semantic theories, such as distributional semantics or in connectionist models of cognition [START_REF] Busemeyer | Quantum models of cognition and decision[END_REF]. For practical implementation, due to the exponential growth of the vector space dimension, adapted logical reduction methods must then be used. LSA (Latent Semantic Analysis) algorithms are often used in quantum-like approaches, this was done in [START_REF] Barros | Contextual Query Using Bell Tests[END_REF] using the HAL (Hyperspace Analogue to Language) algorithm.

Our approach is also of interest for quantum computation because several of the observables in Eigenlogic are well-known quantum gates (for example CONTROL-Z) and other ones can be derived by unitary transformations. Ternary-logic quantum gates using qu-trits lead to less complex circuits, our formulation of multi-valued logical observables could help the development of new multi-valued quantum gate architectures.

Eigenlogic [1] We propose a linear algebraic model for two-and many-valued logic using families of observables in Hilbert space, the eigenvalues give the truth values of logical propositions where the input propositional cases (models for the logical proposition) are represented by the respective eigenvectors. The outcome of a measurement on a logical observable will give the truth value of the associated logical proposition.

In classical logic, predicates are used to describe properties of individuals or systems. A natural idea is that a quantum predicate [2] should be a physical observable.

NOR F F F T I -A -B + AB (1/2) ( + I -U -V -UV)

A B F F T F B -AB (1/2) ( + I -U + V+UV)

¬A F F T T I -A -U

A B F T F F A -AB (1/2) ( + I + U -V + UV)

¬B F T F T I -B -V

XOR ; A B F T T F A + B -2 AB U V = Z Z

NAND F T T T I -AB (1/2) ( -I -U -V + UV)

AND ; A B T F F F AB = Π Π (1/2) ( + I + U + V -UV)

A ≡ B T F F T I -A -B + 2 AB -UV

B T F T F B = I Π V = I Z

A B T F T T I -A + AB (1/2) ( -I -U + V -UV)

A T T F F A = Π I U = Z I

A B T T F T I -B + AB (1/2) ( -I + U -V -UV)

OR ; A B T T T F A + B -AB (1/2) ( -I + U + V + UV)

True T T T T T I -I

Projector Binary Logical Observables

J. von Neumann considered projectors as propositions (quantum logic).

Two-dimensional rank-1 projector (qubit)

; eigenvalue 1. ; eigenvalue 0.

Logical observable as a development in

Here the elements of the development are the rank-1 projectors and the truth values and :

The two-argument logical observables can be developed on the corresponding four rank-1 projectors:

The conjunction is here: All two-argument connectives are derived.

Isometric Binary Logical Observables

There is a linear bijection (isomorphism) from the projector to a reversible observable (isometry) using the Householder transform:

both observables commute and have the same system of eigenvectors. The eigenvalues that correspond to the truth values, respectively "false" and "true", are now and . The seed observable is Pauli-Z:

The logical dictators and are expressed as a function of the seed observable : For the isometric observables negation is obtained by multiplying by . The exclusive disjunction (XOR) :

The logical conjunction (AND) : which is the quantum Control-Z gate

Tri-Valued Logical Observables : Min and Max

In tri-valued logic, popular connectives are Min and Max, defined by their tables:

The corresponding logical observables are given function of the dictators:

The total number of logical connectives for a system of values and arguments is the combinatorial number . For a two-argument system taking values the total number of connectives will be 19683.

Tri-Valued Logical Observables : Min and Max

In tri-valued logic, g , popular p p connectives are Min and Max, defined by their tables:

The corresponding logical observables are given function of the dictators:

The total number of logical connectives for a system of values and arguments is the combinatorial number . For a two-argument system taking values the total number of connectives will be 19683.

Min U \\ V F N T F ≡ +1 +1 +1 +1 N ≡ 0 +1 0 0 T ≡ -1 +1 0 -1 Max U \\ V F N T F ≡ +1 +1 0 -1 N ≡ 0 0 0 -1 T ≡ -1 -1 -1 -1

Discussion and Potential Applications

• Several observables in Eigenlogic are well-known quantum gates. e.g. (Control-Z) corresponds here to the conjunction logical observable .

• Operational correspondence between control logic (Deutsch's paradigm) and propositional logic. A goal could be to express the diagonal forms of observables in Eigenlogic as products of known quantum gates.

• Questions of how could Eigenlogic be used for quantum computing?: measurements give truth values, but how to reinject them in a circuit?

• Ternary-logic quantum gates using qu-trits lead to less complex circuits (e.g. Adder). Our formulation of multi-valued logical observables could help the development of new multi-valued quantum gate architectures.

• The fuzzy membership function is obtained by the quantum mean value (Born rule) of the logical projector observable.

• This method could be employed for developing algorithms in highdimensional vector spaces for example in modern semantic theories, such as distributional semantics or in connectionist models of cognition [4].

Eigenlogic [1] We propose a linear algebraic model for two-and many-valued logic using families of observables in Hilbert space, the eigenvalues give the truth values of logical propositions where the input propositional cases (models for the logical proposition) are represented by the respective eigenvectors. The outcome of a measurement on a logical observable will give the truth value of the associated logical proposition.

In classical logic, predicates are used to describe properties of individuals or systems. A natural idea is that a quantum predicate [2] should be a physical observable.

Fuzzy Logic Interpretation

Fuzzy logic deals with propositions that can lie between "completely true" and "completely false". When the quantum state is not an eigenstate of the logical system one can always express a state vector as a superposition, for example:

when more than one coefficient is non-zero we are in the case of a fuzzy proposition, that can be interpreted as a quantum superposition of propositional models. Mean values are fuzzy membership functions [3].

Using the Born rule for the state:

For a compound quantum state:

We can give the "measure" of the conjunction, , and the disjunction, using the seed projector observable:

The

Multivalued Eigenlogic

Identification with quantum orbital angular moment (OAM), The component of the OAM gives the observable :

The eigenvalues, , , are the logical values:

The logical observables can be expressed as spectral decompositions on the rank-1 projectors of the three-dimensional vector space:

We can define the dictators and simply by the rule of composition:

The logical observables are in this case represented by 9x9 matrices.

Multivalued Eigenlogic

Identification with quantum orbital angular moment nt (OAM) (OAM), The The component of the OAM gives the observable :

The eigenvalues, , , are the logical values:

The logical observables can be expressed as spectral decompositions on the rank-1 projectors of the three-dimensional vector space:

We can define the dictators and simply by the rule of composition:

The logical observables are in this case represented by 9x9 matrices.


A connected history of the expansion

Main focus of my research project:

• The expansive nature of foreign seLlements • The evolu/on of Chinese na/onalism • The maintenance of public order Chronological and spaEal framework:

• Cross-analysis of three major treaty ports: Tianjin, Shanghai, and Hankou.

• From the establishment of the first concession, the Bri/sh Concession in Shanghai in 1846 to the retrocession of the last concessions in 1946.

Research quesEons:

• What did land management issues disclose about the nature of these territories?

• How did the different concep/ons of borders and jurisdic/ons coexist and conflict with each other in the treaty ports? • How did mul/-faceted protests emerging from sec/ons of Chinese society against foreign expansion reflect paLerns of na/onalist aspira/ons in a context of par/al coloniza/on?

A connected approach to reveal connec/ons between processes of expansion in each city and between protest movements sparked by Chinese urban ci/zens.

Sources:

• Western or Chinese language diploma/c archives, including maps, and local press related to each expansion project. • Other historical "voices" of individuals like missionaries, foreign companies, or ins/tu/ons (for example, police and military archives, personal photographic and film materials etc.).

1) RESEARCH QUESTIONS AND AIMS OF THE PROJECT 2) DIGITAL METHODOLOGY

Why? To organize and classify historical sources with a spa/al-based approach.

How? In gathering considerable amount of data (census of popula/on, real estate inves/ng, road construc/on, geographical alloca/on of police forces, etc.) related to foreign seLlements and adjacent Chinese districts over which foreign authori/es were seeking control.

Expected findings: Precise loca/on of these specific data can illuminate urban and social transforma/ons associated with processes of territorial expansion and their evolu/on over /me. What is Actoz?

PracEcal applicaEon:

• Filemaker-based applica/on designed to develop a database for historical analysis.

• and developed by Jean-Pierre Dedieu, Senior Research Fellow at the French Na/onal Center for Scien/fic Research (CNRS).

Purposes and advantages:

• Process, record, and analyze many kinds of historical data and sources, such as data on individuals, genealogical data, contextual data, especially ins/tu/onal data, data on the sources and bibliography as extracted from the primary or secondary literature, data on objects related to actors etc. • Establish rela/onships between the diverse units of informa/on gathered on actors, ac/ons, objects, which the human mind could not possibly handle. • Collabora/ve work.

ContribuEons:

• Make connec/ons and strengthen our grasp of the historical processes and configura/ons in which actors were involved. • Build an innova/ve knowledge base on treaty ports in the nineteenth and twen/eth centuries, with special focus on urban elite networks, interac/ons between polices, militaries, protest movements, etc.

Fleur

  Example of this Chinese map, which shows the evolu/on of geographical alloca/on of French police forces and the installa/on of power lines in an adjacent district of the French Concession of Tianjin. => Used to trace the expansion steps of the French Municipality into this district. CollaboraEve project between the Ins/tute of History of Present Time (IHTP) and the Lyon Ins/tute of East Asian Studies: digital analysis of photographic and film materials through an historical visual narraEve. More than 400 photographs and 24 films taken by André Bontemps (1888-1969) who was a member of the French military colonial troops assigned to Tianjin from 1931 to 1935. PublicaEon in March 2016: h[p://scalar.usc.edu/works/Eanjin_bontemps/index Why Scalar? • Wri/ng and publishing plahorm suited to born-digital works, which incorporate visual materials, such as visual media, graphics, and charts. • Ability to organize and annotate photographic and film materials and juxtapose them with my own wri/ng in a variety of ways. • Ability to display pages in mul/ple views.Lecturer in Chinese History and Language, Na/onal Ins/tute for Oriental Languages and Civiliza/ons


perspective with the behavioral mainstream which was dominant during the 1950s and 1960s in political science. Voters, lobbyists and political entrepreneurs are supposed to accept and understand rules which are immutable and forms of state leadership that are legitimate. Culture determines public and political life, not societal structures and dynamics (Almond and Verba, 1981). Institutional order or stability of the game conditions research and action designs. Institutions are defined as being in a state of equilibrium, unless crises or disruptive periods occur and induce spot rapid and visible changes. Institutions per se do not generate change in an endogenous way. Their transformation is the consequence of exogenous sources such as economic growth, revolutions or coups d'état.

2. Institutions as scientific issues in changing societies : new institutionalist agendas .

Neo-institutionalism defines a perspective which is usually associated with the idea that institutional background matter because they shape public administration realities and processes. Specific research questions renew the agenda of public administration. What roles do institutions play in providing social and economic outcomes to markets, polities and societies? Do welfare states produce and allocate goods and services in an efficient and effective way? Is it possible to fight bureaucracies and to develop democracies in a quick and voluntary way? Neo-institutionalism mixes the will to understand reality through scientifically grounded knowledge and the hope to supply decision criteria for practice.

Public administration defined as a specific field gave birth to alternative theories and approaches elaborated by or borrowed from disciplines such as history, economics and sociology of organizations. In fact the (re)discovery of institutions as independent variables or as explicit causes reflects a general trend which goes beyond the sole domain of public administration. The late 1970s and the early 1980s coincide with growing dissatisfaction about behavioralism and structuro-functionalism. Public administration institutions had been marginalized as an issue for too many years by political scientists. Some failures of the welfare state such as increasing costs, corporatism, urban revolts or technocratic arrogance generated growing debates. Public agencies were used as favorite scapegoats. The capacity of state and local governments to manage society and public policies became increasingly questioned in terms of efficiency and social justice by all kinds of social movements as well as by neo-liberal and communitarian ideologies. Such doubts and critiques were massively expressed in the Anglo-Saxon world.

To a large extent, the notion of neo-institutionalism does not define a unified theory. It covers a rather loose set of specific schools of thought. Some have developed without any reference to the others or even come to contradictory conclusions, as it is the case with the possibility to enact radical changes. Three of them are really relevant as far as their impacts upon the ideological debates and the research paradigms in public administration have been important : rational choice institutionalism, historical institutionalism, sociological institutionalism (Peters, 1998).

2.1.Rational choice institutionalism.

Rational choice institutionalism studies political phenomena in a specific way. It applies concepts and theories developed by the new economy of the organization (Shepsle, 1989).

It stresses the importance of handling transaction costs when collective action is needed (Williamson, 1975). Such is the case when considering the functioning of the Federal Congress in the USA. Contrary to what classic rational choice in economics would expect to happen, majorities are rather stable from one legislature to another. The reason is linked to the fact that procedural rules exist which lower transaction costs legislators usually face. They structure choices and information available to individual members of the Congress in such a way that agreements can be reached which allow a rather fair distribution of benefits among them (Ferejohn and Fiorina, 1975 ;Riker, 1980). Institutions generate equilibria.

At least three different versions exist inside the rational choice inspired new institutionalism.

A first one emphasizes the agency or the principal-agent model. The problem it addresses is the control of the bureaucrats by the elected politicians. Public organizations tend to overspend even in democracies (Downs, 1967;Niskanen, 1971). Some analytic framework (which is neither hierarchical authority nor clientelism or partisan allegiance) is provided by rational choice theorists. It is supposed to allow the principal (the parliament or the political cabinet) to make the agent (a public agency) act in a way the principal would like the latter to behave. Specific incentives and punishments are set up around bilateral contracts in order to generate compliance.

Such an approach has been used in practice. New Zealand has reorganized its public sector organizations along these lines (Boston, 1991). France has set up autonomous regulatory agencies which structure their relationships with the regulated economic industries in terms quite similar to the principal-agents ideal type. On the other hand, public administration offers quite few examples in which one specific public agency is dependant from one single principal. Most if not all the general directorates of the European Commission are dependant from multiple principals. Rational choice institutionalism tends to oversimplify real life cases.

Game theory provides another set of choice models about types of institutions (Calvert, 1995).

The reasoning here again supposes that two or more actors act in a free or non prestructured setting and are involved in a game, in an interdependence relationship around some common task or problem. The model assumes that each of them wants to maximize its own utility in such a context. But none af the player is able to predict the behavior of the partner-opponent. What cooperation scheme to choose when the game is repetitive among the same players and when defection becomes too costly for them, such as is the case in budgetary processes in democracies (Wildavsky, 1992). Formal game theory offers some solutions for institutional arrangements which may satisfy such prerequisites: linking competitors who are simultaneously interdependent and selfish.

A third version of rational choice institutionalism defines institutions as providers of rules for choice and action (Ostrom, 1990).

Rules are considered as the best mechanism by which administrative behaviors can be influenced so as to exploit resources in the most appropriate way. Sanctions for not obeying such rules provide the main and most effective vehicle for conformity. The model is applied to situations in which some common property resources have to be allocated or consumed.

The rational choice institutionalism as such offers a prescriptive perspective. It also pretends to explain actual public administration contexts. For instance, the Federal Congress in the USA acts as a principal who is able to control and supervise in a rational way many autonomous administrative agencies with the help of some specific institutional procedures (Pratt and Zeckhauser, 1991;Milgrom and Roberts, 1992). The growth of international organizations is explained as the consequence of rational choices made by nation-states which maximize the satisfaction of their own preferences, who have a clear perception of their preferences and act in a selfish way (Oye, 1993).

Institutions are the outcomes of a world based around generalized individualism. They get designed and have an influence upon individual behaviors of their members because the latter get in return more than what they loose. Would the individuals change their preferences, the institution would collapse. Public institutions survive and develop because exchanges and interactions satisfy their member preference function.

2.2.Historical institutionalism

Historical institutionalism as a theoretical stream was born in the early 1980s (Hall, 1986)and labeled as such later on (Steinmo, Thelen and Longstreth, 1992).

This perspective defines public administration as a part of political life. The essence of politics is made out of competition processes for scarce resources between rival groups. The State does not act as a neutral agent between competing interests, but as a complex set of differentiated institutions, as underlined by neomarxist (Katzenstein, 1978;Evans, 1985), neocorporatist (Anderson, 1979) or mainstream scholars (Dupuy and Thoenig, 1985). The British Treasury is fragmented into several policy communities, each of them gathering public servants and private associations who share common views or are involved in common problem handling (Heclo and Wildavsky, 1974).

Why are empirically such resources and power allocated in an unequal way by the public sector? Why do old issues and solutions of the past influence in such a massive way the present agenda? A key hypothesis tested by historical institutionalism is that the current outcomes of public policies do not reflect the mere preferences or interests of the current strongest competitors, but that they are channeled by the existing and past institutional arrangements in which competition takes place. It explores how policy choices made in the past shape choices made today. Institutions such as political and administrative organizations, or conventions and procedures regulating the relationships economic actors and the state, are path-dependent. Modes of conflict-cooperation and the structure of outcomes are to a large degree persistently identical throughout time.

Radical changes in public administration are a rather hopeless endeavor in such contexts. Existing institutions structure the design and the content of the decisions themselves. Future action reflects past experience. Such a perspective explains for instance how and why trade unions carry so divergent views of the world as it is the case in the USA and in Great Britain. Different institutional contexts between countries, as characterized by the real power of the judiciary, model divergent preferences and interpretations of action by the labor movement organizations (Hattam, 1993).

Historical institutionalists favor comparative approaches between different countries, combining in depth study and longitudinal research.. They bring political conflict and social dissent back in, studying a variety of settings in which collective action implies interactions between the public sector and society at large. They criticize the idea that the state functions as a single and hands-off agent who would elaborate compromises between competing social and economic interests. Understood as a complex set of specific organizations and loosely coupled procedures which may contradict or conflict, the public sector is structured around non symetric power relationships. Other social and political institutions such as trade unions, economic associations of employers or of farmers also structure social interactions and generate public order and political legitimacy (Rose and Davies, 1994). Certain groups or coalitions consistently win while others consistently lose.

Public institutions influence administrative and socio-political actors in two ways. They offer some degree of predictability about the behaviors and stakes of the actors. They also define models of behaviors and sets of protocoles that are rather stereotyped and ready for immediate use. Institutions such as public agencies provide moral and cognitive frameworks that allow third parties and their own members to make sense of events and to act in specific circumstances. They supply information. They also shape the identity, the image of self and the preferences of administrative behaviors.

Institutional designs themselves do not reflect intentionality. Criteria used when public policies and organizations were initially designed rapidly vanish. Political rationalities and coalitions take over and determine outcomes. A model of "punctuated equilibrium" posits that institutions simply respond to changes in the power balance within society (Krasner, 1984). Therefore they are epiphenomenal, and the pressures for change are external to them. While old institutionalism postulates that institutions shape policies and politics, new institutionalists underline the fact that politics shape institutions.

Public institutions may also be taken for granted and provide the infrastructure for collective action elaboration. Because they acquire the status of social conventions, they are never questioned. In any case, as social constructs, they resist any incremental change or any reform made by one single actor (Graftstein, 1992).

2.3.Sociological institutionalism

Sociologists provide a third major perspective which influences rather strongly public administration as a body of grounded knowledge. It has been carried mainly by American theorists of organization who question the validity of the distinction made between rationality -bureaucracy supposedly providing as such and in a universalistic way the ideal type of ends/means linkaging -and culture -social practices being influenced by norms and values reflecting the specificities of local contexts.

The historic origins of such a perspective go back to the late 1940s when Philip Selznick made a pioneering theoretical contribution around his study of the Tennessee Valley Authority (Selznick, 1948 and1949). He favoured a natural system approach, reacting to what he described as the inadequacy of the rational system perspective developed by scholars like Herbert Simon (Simon, 1945). He defined an organization such as a public agency no longer as been a collectivity deliberately constructed in order to achieve specific goals, but as a collectivity in and of itself.

As field observation consistently suggests, incongruities exist between the declared ends and those that the public organization actually achieves or seeks to achieve. It pursues selfsupport and self-maintenance goals, as well as productive ends. It may turn into a polymorphous system whose struggle to survive may induce it to neglect or to distort its goals. Organizations may be passive instruments, they also possess a live of their own and become active entrepreneurs. People who participate are complete wholes, they do not act solely in accordance with the roles assigned to them.

The issue for public management is not highly formalized structures designed to regulate performance and to coordinate behaviors, but the way these are influenced, transformed and completed by informal structures. Therefore what happens at the bottom of the hierarchy of public organizations, in grass roots units in charge of operational tasks and implementation of national policies, matters a lot, in some cases as much as what happens at the top level.

An agency like TVA must cope with the constraints and pressures applied by the outside social context in which it operates. Therefore it develops its own organization character and acquires values that go beyond the technical requirements of organizational tasks. This is the institutionalization process of the organization, a process that no organization of any duration is completely free of. "To institutionalize is to infuse with value beyond the technical requirements of the task at hand" ( Selznick, 1957 : 17). It takes place for example by selective recruiting of personnel, by establishing strong ties or alliances with outside groups through processes such as cooptation, etc. Thick institutionalization is achieved when some rules or procedures are sanctified, when some units or members of the public agency have become semi-autonomous centers of power and develop their own vested interests, or when administrative rituals symbols, and ideologies exist. Expectations, behaviors, beliefs are channeled and stabilized. Moral communities become set up.

TVA develops as a public institution in a gradual manner, without any explicit design forces behind. It becomes valued by some of its members and by outside vested interests such as local farmers associations for the special place it holds in the larger social system.

The perspective set by Selznick came to be neglected during the 1970s. Sociology of organizations had become dominated by approaches such as contingency theory, resource dependence and population ecology. They had rehabilitated an instrumental and a technicist view of organizations. Efficiency and survival in unstable and unpredictable environments were defined as being incompatible with certain types of formal structures and procedures such as the bureaucratic model. Empirical research attention was also mainly allocated to firms by business school academics. Public administration became a rather neglected topic. Sociological institutionalism arises in the late 1970s, basically in the USA and around departments of sociology. Theorists of organization develop institutional analysis or thinking as a new and promising field (Meyer and Scott, 1983). Public organizations are primarily analyzed such as art museums (DiMaggio, 1983), city administration (Tolbert and Zucker, 1991) private and public elementary schools or healthcare programs (Scott and Meyer, 1994).

This perspective borrows from Selznick but adds novelty and breadth. It studies the process of institutionalization, or, put in other terms, the socially legitimized activities which in the long run come to characterize certain aspects of social life (Scott, 1995) One of the pioneering contributions deals for instance with formal structure as myth and ceremony (Meyer-Rowan, 1977). While Selznick emphasized political processes such as group conflict and intentional cooptation of external constituencies, the new generation of institutional sociologists downplays their importance while it emphasizes the importance of other sources of constraints such as conformity pressures or legitimacy imperatives. It also locates irrationality in the formal structure itself, not only in informal interactions such as influence patterns. Institutionalization is fundamentally a process of a cognitive nature (Powell and DiMaggio, 1991).

The focus in not so much on a single organization with its specific context relegated to the background, but on the individual organization defined as a consequence of this context. More than identifying mere pressures exerted locally on the organization, institutional sociologists study what they define as organizational fields (DiMaggio, 1983), which are formed by bodies ranging from public institutions (hospitals, aging centers, etc.) to professional activities (doctors, teachers, etc.). The organizational field of a public system is examined as a whole, as an activity making rules, supervising or surveilling. In fact it defines an institutional context within which each specific single organization has to plot its courses of action.

Organizational reality is theoretically framed as symbolic construction. New institutionalism emphasizes the cultural messages transmitted by specific institutional systems or the explanatory significance of institutionalized myths. These norms and values shape organizational structurings through interactions among institutional logics made up of socially accepted beliefs and the organizational practices associated with them.

The key research question refers to an empirical observation. Why do so many organizations, whether public or private, adopt formal structures, procedures and institutional symbols which are so identical? a few years, most cities in a given country drop patronage systems and adopt human resource methods to run their public agencies. The ministries in charge of education in most countries around the world adopt rather identical formal structures and modes of functioning. Institutional sociologists study diffusion processes characterized by institutional isomorphic change (DiMaggio and Powell, 1983). Innovation is not adopted mainly because it provides added efficiency value. Other mechanisms are at work such as coercive isomorphism -change results from pressures exerted by political influence and by other organizations considered as legitimate -, mimetic isomorphism -uncertainty and ambiguity about goals or technology increases the adoption of imitation conducts -and normative isomorphism -the existence across different organizations of individuals belonging to the same profession or having followed the same educational processes accelerates similarities of various kinds.

Compared to historical and rational choice neo-institutionalisms, the sociological perspective defines what institutional means in a much wider and global way. Beside formal rules and procedures, it includes symbols, moral models and cognitive schemes. Institutions provide the frames of meaning which guide human action. They are similar to culture or cultural systems. They influence the conducts of public administrators not only by stating what they have to do when where and how, but also by shaping the ways of reasoning and the imagination of the actors about alternatives and solutions in new contexts. Such a macro-sociological definition may lead to a view of public administrations as systems of action that operate without actual specific actors. Society shapes the acts and non-acts, the structures and the values of the public sector.

While both the rational choice and historical schools study the creation of new institutions from scratch, sociological neo-institutionalists underline the fact that the public sector is already crowded with existing organizations which provide references and models for the new ones. Designing institutions radically different from the existing ones becomes an illusion in a world which constraints autonomy of choice and limits action oriented imagination. Why do public organizations adopt new institutional practices? In the short term, pressures toward isomorphism are strong. Reinforcing their political legitimacy or the improving the social image of their members is a major reason why public organizations as well as firms conform. Values that are recognized by their environment drive transformation much more than instrumental rationalities increasing their efficiency or effectiveness.

Sociologists dealing with institutional analysis provide evidence about change processes. In the mid term, more diversity or competition between alternative organizational models is possible and isomorphic pressures are weaker within a given institutional framework (Kondra and Hinings, 1998). The concept of archetype is used to describe operating organizational transformation. Archetype refers to a configuration of structures and systems of organizing with a common orientation or underlying interpretative scheme. Evolutionary change occurs slowly and gradually, as a fine-tuning process within the parameters of an existing archetype (Greenwood and Hinings, 1996). Organizational transformation or revolutionary change is theorized to occur with a change in the dominant archetype. It happens swiftly and affects all the parts of the organization simultaneously. It is associated with interactions between exogenous dynamics (institutional contexts) and endogenous dynamics of interests, values and power dependencies. Pressures for change are precipitated under two conditions. Inside the agency, group dissatisfaction with the way heterogeneous interests are made compatible within the existing template for organizing is coupled with a value commitment. Outside the public agency, exogenous dynamics exist that also push for an alternative template. Situations are also identified that generate de-institutionalization processes (Oliver, 1992). Some institutionalized organizational activity or practice gets eroded or is facing discontinuity when not rejection over time.

3. Institutions as determinants of political life : a normative theory.

New institutionalism as an explicit notion or as a structured school of thought finds its origins in a seminal paper published in the early 1980s by James March and Johan Olsen (March and Olsen, 1984). A scientific ambition should be pursued by public administration. Principles should be dropped and substituted by laws as criteria of knowledge. Much more than sociological institutionalism, it is action oriented and actionable. It is also normative. A general theoretical question drives such a perspective. What are the foundations of democracy in contemporary societies? Public institutions should be considered as key factors. Government is in the business of forming its environments, not adapting to it. Public administrators are driven by societal visions and political projects. Therefore organizations which handle public affairs should be 'conceptualized as institutions rather than as instruments' (Brunsson and Olsen, 1997: 20). They generate and implement rules which define how the rules of the game have to be played, who is legitimate to participate, what are the acceptable agendas, which sanctions to apply in case of deviations, as well as the process by which changes should occur. The way people think, interpret facts, act and cope with conflicts are influenced and simplified by public administration. Any relevant answer to the question of democracy implies a good understanding of the properties of political institutions, of their construction processes and their real modes of functioning. Political science has to address the core issue of their governance. Academics should observe empirically whether public administration reforms match societal needs and widen democratic participation. Is the very idea that it is possible to reform and control public organizations a relevant rule for action? The answer provided by the normative theory tends to be that it is a rather dangerous and inefficient value.

A crucial task for social science research is to make explicit the less than convincing axioms or hypotheses that existing theories carry about organizations and institutions. Contextualism stipulates that politics is a component of society which is the mere product of factors such as social classes, culture or demography but is itself not at all a cause of such factor. Reductionism postulates that political phenomena are mere consequences of individual behaviors, for instance that the functioning of a public agency is explainable by the behavior model of the single bureaucrat. Economic utilitarism implies that conducts of individuals are basically driven by their own selfish interest. Functionalist approaches adopt darwinian view points: historical evolution selects the organizational forms which fit the environmental requirements. An instrumental perspective claims that the core role political life fulfills is to allocate scarce resources.

The founding fathers of self-proclaimed new institutionalism claim that alternative ways should be tested. Public institutions may experience a large degree of autonomy and follow a logic of their own, independently of outside influences or requirements. The historical process happens to select organizational forms which are non efficient. Symbols, myths and rituals have more impact upon political and administrative events than immediate material issues.

To a large extent, new institutionalism opens political science and public administration to organizational theory. More precisely, it is strongly influenced by one particular research tradition: the concepts, models and tool bags which originated out of Carnegie school. Compared with other institutionalism schools, the March and Olsen approach offers at least two distinctive advantages. Administrative behavior is tackled with powerful analytical tools. The real functioning of organizations defines the research agenda.

The logic of consequentiality is questionable. Action in organizations is not very much intrumentally oriented. In fact, absolute or pure rationality is impossible to enact. Only bounded rationality is available. (Simon, 1945).

Public administrators such as city park managers make investment decisions according to some criterion of satisficing that expresses a compromise they make between the content of the problem they address and the local context of uncertainty they face. Behavioral theories supply the best vehicle in order to understand policy making processes and functioning mechanisms inside economic organizations (Cyert and March, 1963). Three concepts are fundamental: the goals the various units pursue, the way information, opportunities and support are built and elaborated, the choice or decisions processes. Four main mechanisms are identified: conflict avoidance, uncertainty reduction, problem solving as solution seeking initiator, organizational learning through former experience and rules of attention allocation. Organizations such as firms are political coalition or arenas, meaning that power as an issue and as a dynamics plays a major role for the administrative behavior (March, 1962). Collective goals do not exist as far as they provide common references subsuming individual goals or particularistic preferences. Therefore institutional devices are needed in firms in order to channel opportunistic behaviors and ensure some collective stability.

Another contribution questions how organizations as such produce decisions (Cohen, March and Olsen, 1972). The garbage can model of choice that is presented as a relevant explanation of what happens in real life is inspired by observations made on universities. These latter institutions look like organized anarchies. Their goals are loosely defined, their technologies such as education are ambiguous (ambiguity meaning that it is not easy to explain why student succeed or not), and the participation of their members to administration and decision processes is fluid. Therefore choices are made in a quasi-random way, resulting from interactions between fluxes of problems, solutions, choice opportunities and participants. In other terms, as it is the case with garbage cans, the piling up of elements is a juxtaposition which sequence or order depends from outside events. Based upon electronic simulations, such a model does not explain universities as specific organizations, it simply underlines two properties of organizations which are organized anarchies. No consensus exists about what a decision is, about its basic ingredients (content, time, arena, etc.). And decisions are seldom purposive or consequence driven. Only a minority of decisions are problem solving directed. Many are not because the choice happens at a time when the problem is already linked to other choices. In a high proportion of cases, a third situation is visible: no decision is made until the problem has migrated toward other more attractive choices. Such a perspective could generate a paradox for public administration. The garbage can model brings the deconstruction of the notion of preference to an extreme limit. If ambiguity and foolishness (March, 1971) are so widely diffused, when random processes structure organizational order and cause-consequence relationships are difficult to establish, how is it possible to manage collective action? Are organizations still a relevant and thick level of reality and coordination? Two basic socialization mechanisms make behaviors predictable and channel the potential risk factor they represent: organizational routines and institutions.

Actors select their conducts according to a logic of appropriateness or conformism (March and Olsen, 1989). Routines or legacies from the past are powerful sources. So are also cognitive patterns and values that are diffused by institutionalization processes. Action mobilizes cultural elements. Actors fulfill identities by following rules that they imagine as being appropriate the situations. A theory of learning in ambiguous environments provides a framework which explains how individuals and organizations try to reach some degree of understanding and intelligence of the contexts which they face (March and Olsen, 1975), how they allocate their attention to a particular subject at a given time, how information is collected and exploited, etc. (March and Olsen, 1976).

New institutionalism questions how organized action can be managed and how some public order can be governed in pluralistic societies. It offers simultaneously a descriptive and a prescriptive perspective to public administration. The platform designed by James March and Johan Olsen gave birth in 1988 to a research consortium involving around the Stanford University campus American and Scandinavian scholars (SCANCOR, Scandinavian Center of Organizational Research). More than thirty field studies were conducted on public sector organizations, specially in Sweden and Norway (for the latter country, see a review by Christensen a nd Laegreid, 1998b). They observed reforms of various kinds: introducing corporate strategic planning in the relationships between the national government and state agencies, running a public rail company in a decentralized way and with a strong market orientation, introducing a three-year budgeting methodology into national government administration, setting up active and participative county councils (Brunsson and Olsen, 1997). They also mobilized a new generation of social scientists who kept an interest in public administration phenomena such as national administrative reform policy (Christensen and Laegreid, 1998a), complex public building projects (Sahlin-Andersson, 1998), decentralization policies in municipalities (Czarniawska and Joerges, 1998), constitutive reforms of the European Union (Blichner and Sangold, 1998), municipal accounting reforms (Bergevaern, Mellenvik and Olson, 1998) or central government officials (Egeberg and Soetren, 1999).

Such a stream provides ample empirical material and alternative ways of reasoning about the adoption of new formal structures and processes by the public sector at a time when in most OECD countries many governments assumed that major surgery was needed in order to modernize governance. They put on comprehensive administrative reform on the agenda. Policies were formulated to reshape the public sector and to redraw the boundary between the state and the market Actual administrative reform looks more like a process of organized attention (Laegreid and Roness, 1999). Time, energy and attention are scarce resources for political leaders. They influence reform policies in an indirect way, maneuvering the scene by steering the actors' role through procedure more than substance, using various degrees of hierarchy, specialization and random control.

Public management is in fact the consequence of human activities, and not the result of applied techniques. Administrative reforms inspired by New Public Management are determined to a large extent by cultural rules and institutional values. . Voluntary reformism based upon free rational calculation postulates that leaders are in control and organizations act in a passive manner. Four common attributes are present in most contemporary administrative reforms : simplicity and clarity (principles and theories rather than descriptions and perceptions), normativity (bringing order into chaotic reality), one-sidedness (a single set of values is legitimate), promises about the future (no instant result production). The most important illusions or errors are the belief that intentions are non-ambiguous, and the very idea that public administration organizations may be controlled and reformed from the top through pure thought. Reformers are prisoners of walls which are mental. It is easy to initiate administrative reforms, but few are completed (Brunsson and Olsen, 1993). Reforms generate more reforms and less changes. Organizational forgetfulness is supplied which allows to accelerate reforms and helps people accepting them. Reforms have become routines. Therefore top-down reforms should be avoided because their relationship with change outcomes is problematic. They paradoxically contribute to stability and prevent change from occurring.

While actual organizational changes are not the result of planned modernization reforms, observation suggests that they are abundant. Public administrations as such are not innovation adverse. They may follow a sequence of transformations which reflect outside factors such as labor market dynamics or inside initiatives informally taken by low ranking units. Real big changes in public administration when they happen take place without much prior discussion. It is also easier to generate them when reforms are undertaken in non controversial areas. In controversial contexts, attempts at reform make for stability. Hotly debated issues are not subjected to any great change.

Normative institutionalism suggests two main prescriptions for public administration change to occur. There should be a match between rules, identity and situations : succesful reforms are culturally sensitive. Context matters because contexts vary.

Institutions and co-constructed systems of action .

The new institutional theory as applied to public administration has become a very visible school of thought in the Anglo-Saxon countries. In other parts of the world, its influence is much less important. In some European countries such as France, Germany or Italy, contributions inspired by the new institutional schools are not many (Gherardi and Lippi, 2000). Several factors may explain such a phenomenon.

In continental Europe, academic markets are much less sensitive to high segmentationdifferentiation processes than in the USA where many specialized schools of thought compete and also experience rather short life cycles. This has clearly been the case in the field of organizational theory during the 1980s, when the Anglo-Saxon research scene was nurturing more than thirty different approaches or so called paradigms. A high level of skepticism across the ruling public elites about fads and fashions such as the New Public Management offers another reason. Administrative reformers find that the value premises on which supposedly rational public management schemes are based lack credibility and are caricatural. A third and decisive reason has to do with the fact that in countries such as France or Germany a recognized tradition of research existed on public organizations as such well before the 1980s.

In fact, public administration approaches were, in Europe, taken over by organization theory as a specific field of inquiry much sooner than in the USA or in the UK, and not by the same academic tribes. When Anglo-Saxon political scientists began challenging the hegemony of behavioralism and rational choice, Continental European social scientists, who basically were sociologists, had already won their battle against two other enemies: macro theories such as sociology of the State, which was mainly inspired by Marxist perspectives, public law inspired science administrative. The scientific and ideological controversies that fueled the emergence of the new institutional theory in the Anglo-Saxon world were considered as less relevant in France or in Germany. To argue that the doctrine of rational choice has to be rejected by public administration, to recognize that administrative life in the public sector has a thickness of its own and that institutions should be considered as independent variables, was not relevant news.

The French sociologie des organisations considers institutional phenomena from an agnostic angle. Institutions should be treated analytically as both independent and dependent variables.

Michel Crozier, its founding father, made pioneering in depth studies of several public agencies (Crozier, 1963). As causes or determinants of bureaucracy, the national education system, the cultural norms assigning how to behave in interpersonal interactions, and social stratification within society shape the way a public agency like the postal office or a public sector firm like the tobacco factory are structured and managed. Later studies made on administrative elites showed that, as consequences or as outcomes of technocratic practices and values, institutional configurations such as criteria of efficiency and effectiveness in the sectors of urban development and housing, or such as the cognitive content defining general interest, are interpreted as social constructs. Public servants who are members of the famous Grands Corps trained in exclusive elitist schools such as ENA and Ecole Polytechnique may control the public agenda of a whole country in a specific set of policies for a certain period of time (Thoenig, 1987). In other terms, analytical schemes used for research purposes should keep the theoretical options open about the status allocated to institutions and institutionalization processes. Such a scientific tradition originated around a specific concern for organizational change. While Anglo-Saxon rulers and intellectuals were debating about the marketization and the economization of public choices, French social scientists questioned the ambitions expressed by politicians and public servants who thought that, using economic planning, social indicators or PPBS, the central state could manage the whole polity, integrate society and guide the economy. Planned change projects defined from the top and implemented through hierarchical designs would make a difference. The argument made by public administration scholars was that redesigning formal structures would remain a pure cosmetic game, given the vicious circles of bureaucratic change within which ministries and public agencies were caught. A series of long lasting field research projects was launched in the mid 1960s and explored various institutional facets and many ministries.

Two main contributions deal specifically with institutions and institutionalization inside the public sector.

Cooptation mechanisms are widely diffused across the public sector. Norms circulate and get shared between interdependent partners. Mutual socialization occurs. It allows the field agencies of the national ministries -in France, 95 % of the central State two and a half million public servants are located in such field agencies across the country -to be embedded in micro-communities, to get legitimacy from their environment (specially from local elected politicians), and to acquire quite a large autonomy in their relationship with their respective headquarters located in Paris (Grémion, 1976).

A breakthrough provided by the French sociologie des organisations refers to the idea that cooptation processes as such structure informal but stable organized systems linking state public agencies to specific local environments (Crozier and Thoenig, 1976). The concept of cross-regulation allows to identify why and how interdependence relationships develop between partners who otherwise perceive each other as even antagonistic. It identifies social sources that give birth and legitimacy to implicit rules of exchange and stable coalitions. A politico-administrative system linking State field agencies and local political and economic leaders, such as prefects and mayors, supplies the backbone of the French public sector. Political centralization as a formal structure as well as a set of values is balanced by the logic of such local arrangements. The machinery of the central State looks like a piecemeal fragmented organizational fabric. Its various subparts cooperate less than each of them cooperates with its local environment leaders. The unified reference framework headquarters in Paris impose to their respective field agencies is in fact appropriated by the local front-line units in an active way. Formal rigid rules of behavior decided at the top are substituted by informal and more flexible norms of implementation that are induced by bargaining and quasi-negotiation practices between partners within each subsystem (Dupuy et Thoenig, 1985). A social norm of flexibility, which content is different according to time and space, and which is perceived as legitimate by social groups, prevails over a rule of formal conformism and of rigid equality of treatment. Public agencies themselves help generating exceptions that become local norms. They also reinforce differentiation of outcome mechanisms and externalize the costs of their mode of functioning. Free riders activists benefit while third parties, unable to adopt voice strategies, pay for the errors or for the consequences. Local polities and politics are shaped by public administration in two ways. Easy access to the public sector allocates advantages to certain social categories. For instance, a local politician gets a kind of rent she/he may use to favor her/his constituency interests. And bureaucratic ways of doing things influence the cognitions and the expectations of social groups.

Beside considering a public agency as a single organization, the French perspective also treats it as one actor among many who intervene in handling public affairs. In such a case, analyses take sectorial public policies as units of research. Freight transportation (Dupuy and Thoenig, 1979) and agricultural affairs (Jobert and Muller, 1988) have offered pioneering examples. Studies of how governmental agendas are set up, how political and bureaucratic decisions are elaborated, and how they are implemented and evaluated, recurrently suggest that for each policy a specific system of organized action, which has a logic of its own, defines the problem as public and governs its management. Even when some state ministry or some regional authority may play a hegemonic role, its acts and non acts remain to some extent dependant from the presence other public agencies, of firms or of not for profit associations. Public affairs are not governed in a hegemonic manner by state public bureaucracies. The latter are institutionally and functionally embedded within specific social fabrics.

Longitudinal observation shows that public administrations have undergone a dramatic change since the end of the 1960s (Duran and Thoenig, 1996). The central state agencies do not any longer play a dominant role, governing national as well as local public affairs through the allocation of subsidies and the elaboration of detailed technical procedures, and relying upon cross regulation mechanisms as a way to build an alliance with local elites around policy implementation issues. A new political and administrative system has emerged. Massive decentralization has been allocated to regional and local authorities. New private, associative or public players such as the European Commission get a role in policy making. Public issues as such coincide less and less with the way sub-national territories are subdivided and administrative jurisdictions defined. Collective problems are more and more of a horizontal nature and are addressed with uncertain solutions.

In such a context, the state administrative machinery may become marginalized. Its main tool in order to keep a relevant role is to use constitutive policies, to diffuse or help emerging new cognitions and new patterns of behavior among the numerous partners involved. In other terms, it creates and provides all kinds of institutional frameworks which are expected to allow multiple partners to coordinate their views, to speak a common language and to share a common perception about what to do how when and for whom. Institutionalization and institutional design become weapons used by a weak public actor who is facing a polity which is complex, active and non consensual by nature.

A rigourous and strong analytical framework characterizes the French academic approach. Intensive field work and in depth observation produce thick empirical evidence. Interdependance phenomena and relational games are interpreted as results of strategic behaviors of actors operating in power settings (Crozier and Friedberg, 1980). Social regulation -how pluralist sets of actors may find normative arrangements in order to achieve some degree of compatibility between their respective logics of action -and systemic implications of such social co-constructions are explored. They help understand inner modes of functioning public agencies conform to, their consequences in terms of outcome production, and identify actual change coalitions and opportunities when they exist (Thoenig, 1997).

By comparison, while scholars belonging to this tradition consider the new institutional theory as a source of massive conceptual inventiveness and as an ambitious theoretical agenda, they express some doubts about its ability to deliver convincing and strong empirical study demonstrations (Friedberg, 1998). For instance, the link between discourses and behaviors is not made in a convincing way. Public agents may adopt some norms or cognitions, but not use them in relational settings and in decision-making processes. Institutions may remain words without consequences on acts. Last but not least, French sociologists adopt a vision of politics which is more cynical or machiavellian than the vision of democratic order the normative institutional theory is favoring and which considers responsibility as a consequence of the institution of the individual (Brunsson and Olsen, 1997), citizens as free, equal and discipline oriented agents, governance as enlightened and rule-constrained (Olsen, 1998).

Collective public action is a key entry in order to understand and predict how institutions and institutionalization processes operate. The other idea is that public organizations and complex systems of organized action have to be considered specific local social orders. They are meso or intermediary social configurations, which are neither passive nor intentional but are constantly co-constructed in terms of social regulation norms and of membership.

A good illustration is provided by education policies. Anglo-Saxon sociologists (see section 2.3. above) have emphasized the intrinsic homogeneity of institutional aspects, the fact that, either formally or in terms of discourses, schools look alike across the territory within a same country, or even across different countries. Observation nevertheless suggests that, below the surface, schools may differ quite much in terms of their real daily mode of functioning. This raises one question never addressed by the new institutional theory: how is it possible that local orders exist which create discontinuities and heterogeneities?

The irony with this research perspective on institutions is that it shows quite much sensitivity to differentiation and local specificities although its members belong to a nation-state, the French Republic, which founding values incorporate the ideals of unity and equality, and which enforcement has been handled in a centralized-authoritarian manner. The French approach is reluctant to institutional theory as a macro perspective that may discover some global logics shaping social regimes. It is also quite skeptical when action contexts such as public administration are studied excluding actors. From this point of view, while, U.S. sociological institutional theory tends to negate the importance of actors, historical institutional theory, with its focus upon path dependaencies, allocates much attention to actors, therefore is welcome. Public institutions considered as specific social arrangements are fragile constructs because they are the non-intentional outcomes of permanent collective tinkering (or bricolage). Discontinuities characterize the essence of public administration in particular and of societal order in general.

Another influential school has also developed in Germany under the influence of Renate Mayntz.

The similarities with the French perspective are quite many. Well before the 1980s, this sociologist had written a comprehensive book on public administration which was influenced by theory of organizations (Mayntz, 1978). After having led a major program of policy implementation studies in her country, and within the framework of the Max-Planck Institute, she and Fritz Scharpf have set up a perspective actor-centered institutionalism (Mayntz and Scharpf, 1995), which supplies less a theoretical framework than a heuristic tool bag for research on the governement of public affairs.


On a Singular Solution in Higgs Field (Ⅲ) -The Phase Transition and Crystallization of SM Higgs Boson.

Kazuyoshi KITAZAWA, Mitsui Chemicals, Inc. FAX: 03-6253-4244, E-mail: Kazuyoshi.Kitazawa@mitsui-chem.co.jp

The phase transition and crystallization of SM Higgs boson are discussed by considering a naïve relativistic energy equation with Cornell potential and then Bethe-Salpeter equation (BS) with Goldstein approximation for tightly bound fermion (t) -antifermion (t_bar) coupling which exchanges with vector particle. We consider that the well known continuous spectrum solution of BS should not be excluded but be interpreted as it closely relates to propagator of the exchanged gluon. Then SM Higgs boson will be constructed from a number of mesons each of which is composed of two gluons (glueball), after emitting a virtual photon. A glueball mass value calculated of grand state is around at 502.55 MeV/c 2 which is expected as f 0 (600) (σ meson) mass. Moreover the mass values of its resonant states are consistent with the known result of lattice simulation and within upper-f 0 mesons' masses from experiment. And we predict that they finally crystallize into certain fullerene structures, as SM Higgs boson, through their color 'valence' of components. So the phase transition of SM Higgs boson is completed to transit to a state of truncated-Octahedron (tr-O) structure by receiving 'melting heat' from another SM Higgs boson before crystallization near itself.

1.Introduction

In preceding paper [START_REF] Kitazawa | On a Singular Solution in Higgs Field[END_REF] the author discussed the mass and the basic structure of SM Higgs boson with reference to bound top quark-pair. Where we saw an intimate relation between them, and we also derived a smaller mass of SM Higgs boson than the predicted one by the dynamical strong coupling theory of top quark condensation. We consider that the mass deference comes from different state of each Higgs boson. Therefore we shall hereafter treat the phase transitions of SM Higgs boson to start with studying a relativistic energy equation, and then applying Bethe-Salpeter equation [START_REF] Salpeter | A Relativistic equation for bound state problems[END_REF] for the states of the system.

( ) / . V r r r     e (1)
r q○----------• q Fig. 1 Quenched Wilson action SU(3) potential [START_REF] Eichten | The Spectrum of Charmonium[END_REF] , normalised to V (r 0 ) = 0. Hence we may write for a tightly bound energy as

  0 22 . H , 2 t bound photon t V M c M c     (2)
because we will treat the situation in next subsection that total mass of the system is zero, we here set its tightly bound (total) energy to equalize to the one with SM Higgs mass 1) which will be produced at second stage from massless vector particle. Then by substituting eq.( 2) into eq.( 1), we have a relativistic energy equation for the system as [START_REF] Eichten | The Spectrum of Charmonium[END_REF] . Here we assumed that bound energy between two gluons in a glueball is small. And, since we consider that there would be some 'latent heat' between the molecular-like state and solidification state of SM Higgs boson (namely, we think, the former would not be in tightly bound state), we will choose for a tightly bound energy with left-hand side of eq.( 3). Then we could describe the diagram of phase transition into SM Higgs boson as shown in Fig. 2. It is very interesting that the rate of outgoing energy from the system to the space (= +Q) by deficit of mass is fairly large at this point of time, as eq.( 4). We will later return to the diagram of Fig. 2.

  2 2. / Total photon t M c r r      e (3)

   

Bethe-Salpeter Equation with Goldstein Approximation

Tightly bound fermion-antifermion coupling which exchanges with vector particle by Bethe-Salpeter equation 2) has long been investigated. [START_REF] Goldstein | Properties of the Salpeter-Bethe Two Nucleon Equation[END_REF] Firstly, Salpeter and Bethe constructed the relativistic equation for two interacted nucleons. Goldstein studied its solution by ladder approximation and discovered the continuous spectral solution with relevant discrete ones. [START_REF] Goldstein | Properties of the Salpeter-Bethe Two Nucleon Equation[END_REF] There Goldstein argued the lack of physical interpretation for the continuous solution of highly singular behavior at the origin of coordinate space. Later, Kummer; Higashijima and Nishimura; Fukui and Seto; and others discussed the continuous spectral solution in the fermion-antifermion or in the spinor-spinor interactions. Then BS for fermion-antifermion bound state with total four momentum P  is given explicitly [START_REF] Kummer | Exact Solution of the Bethe-Salpeter Equation for Fermions[END_REF] in the Bjorken-Drell metric,

        1 1 4 4 ; , 1 1 , 2 2
,; 2 , qP q P q P q P q q P S S dq K

x

x                                    (8)   4 1 1 , 2 2
where 0

.

qP P d x T x x x                            (9)
Hence we will have the Goldstein equation for abelian vector gluon model 4), 5) with the ladder approximation after putting

0 P   and       5 ,0 q F q x     ,         22 2 2 4 . 4
dq qq i qq mq i FF            (10)
After the Wick rotation and then the Fourier transform regarding eq.( 10), we will see that it has the continuous spectrum solution for 0   , putting K  as modified Bessel function of -th order

      1 . f mr r mr K    (11)
On the other hand, recently Iritani et al calculated gluon propagator's functional forms [START_REF] Iritani | Gluon-propagator functional form in the Landau gauge in SU(3) lattice QCD: Yukawa-type gluon propagator and anomalous gluon spectral function[END_REF] in the Landau gauge in SU(3), which fit the result of lattice QCD, one of whose candidates has similar form of eq.( 11). However they abandoned this form because of the deviation from the data of lattice QCD near 0 r  . We consider in this case, it should rather be adopted that

Crystallization as SM Higgs Boson

In lattice QCD it is now believed that there might be several scalar mesons of f 0 (1370), f 0 (1500), f 0 (1710) all of which are supposed to have some contents of glueball (GB) of grand state. So we shall write down them, relating to the glueball mass and SM Higgs boson mass as 0 1.

H 1 2 3 1 2 3 , G G G N N N M M M M                      (13)
      (14)
As the number of the kind for colored gluon is 8, the glueball's color valence should be 4 (cf. Fig. 3), which is same as carbon. After setting N 1 , N 2 N 3 as the fullerene number of f 0 (1370), f 0 (1500), f 0 (1710), under the consideration of similar structure to the carbon fullerenes of C90, C80 and C70 respectively, we put

2.Formulation and the Result 2 . 1 A

 21 Naïve Relativistic Energy Equation for the System Before we later apply Bethe-Salpeter equation for a tightly bound top quark-pair * () tt , we at first make a naïve relativistic energy equation for * () tt with considering Cornell potential () Vr of quarkonium in lattice QCD of Wilson loop 3) :



Fig. 2

 2 Fig. 2 Diagram of Phase Transition into SM Higgs boson

  [START_REF] Kummer | Exact Solution of the Bethe-Salpeter Equation for Fermions[END_REF] They excluded it from the reason of each difficulty of interpreting physical meaning, except that Higashijima and Nishimura considered it as a renormalized vertex function of the solution for the homogeneous BS. Thus we shall hereafter apply BS for tightly bound fermion (top quark)-antifermion (anti-top quark) coupling which exchanges with vector particle, and reconsider the physical meaning of continuous solution. The general form of BS is[START_REF] Nakanishi | General Survey of the Theory of the Bethe-Salpeter Equation[END_REF]

  according to the continuous solution of Goldstein for BS. We prefer from the result of lattice QCD that 1

Fig. 3

 3 Fig. 3 Color valence of glueball <Details are to be explained at talk> References 1) K.-Y. Kitazawa: On a Singular Solution in Higgs Field, Theoretical and Applied Mechanics Japan, 57, pp.217-225, 2009; ditto (II), ibid, 58, pp.61-70, 2010. 2) E.E. Salpeter and H.A. Bethe: A Relativistic equation for bound state problems, Phys. Rev., 84, pp.1232-1242, 1951. 3) E. Eichten et al.: The Spectrum of Charmonium, Phys. Rev. Lett., 34, pp.369-372, 1975; Erratum,ibid,36, p.1276,1976. Gunnar S. Bali, QCD forces and heavy quark bound states, Phys. Rep., 343, pp.1-136, 2001. 4) J.S. Goldstein: Properties of the Salpeter-Bethe Two Nucleon Equation, Phys. Rev., 91, pp.1516-1524, 1953. 5) W. Kummer: Exact Solution of the Bethe-Salpeter Equation for Fermions, Nuovo Cimento, XXXI, pp.219-246,1964. K. Higashijima and A. Nishimura: A Solution to the Goldstein Problem and the Possibility of Dynamical Chiral Symmetry Breaking, Nucl. Phys., B113, pp.173-188, 1976. A. Nishimura and K. Higashijima: Exact Solution of the Spinor-Spinor Bethe-Salpeter Equation and their Gauge Dependence, Prog. Theor. Phys., 58, pp.908-918, 1976. I. Fukui and N. Seto: On the Structure of the Continuous Spectra in the Spinor-Spinor Bethe-Salpeter Equation, Prog. Theor. Phys., 65, pp.1026-1040, 1981. 6) N. Nakanishi: General Survey of the Theory of the Bethe-Salpeter Equation, Suppl. Prog. Theor. Phys., 43, pp.1-81, 1969. 7) T. Iritani et al: Gluon-propagator functional form in the Landau gauge in SU(3) lattice QCD: Yukawa-type gluon propagator and anomalous gluon spectral function, Phys. Rev., D80, 114505, 2009.