Extreme state. Coleman

Although the concept of an extreme state, Coleman [15], is well known in quantum chemistry, it is not fully acknowledged in physics. In condensed matter physics the custom is often to proceed directly to the thermodynamic limit. Nevertheless, ODLRO is of central importance in super-conductivity/ fluidity and it will also play a central role here. In connection with this development Coleman [15] identified the precise condition for the so-called extreme state which in certain cases could develop ODLRO. 

There are several inter-connections to be mentioned here. The first one concerns the extreme state. If h is the set of two particle determinants and the AGP wave function is constructed from gt, see Coleman [27] for the exact condition for the extreme state, the two-matrix (save the tail contribution from the remaining pair configurations) can be expressed as... 

In Appendix F, we have derived a simplified version of Coleman s extreme state [107] as well as indicated the onset of ODLRO [106]. Our first examples of applications within disordered condensed matter concern the discovery of high-temperature superconductivity. From appendix, Eqs. (F.4)-(F.7), we will use the relation below [note the quadratic expression in the occupation number p in the large eigenvalue A.L to be used in Eq. (82), for more details see Refs. [7,103] and references therein]... 

We will here review some aspects of Yang s famous concept of ODLRO [10]. The formulation has been adapted to several problems of complex phenomena in condensed matter [9, 15]. In this development, we have employed Coleman s notion of an extreme state [18] as a precursor for the onset of the strongly correlated (condensed) phase and/or of a coherent dissipative state, for more details see Refs. [9, 19] and references therein. We will first supply the necessary machinery before proceeding to the conversion of a black hole type entity into the organization of ODLRO. 

In passing we note that the functions in the set g are completely delocalized over the region of sites defined by the localized particle-antiparticle basis h, while the f-basis contains all possible phase-shifted contributions from each site in accordance with Eqs. (56) and (57) above. Some interrelationships can be recognized here. The first connection concerns Coleman s so-called extreme state [18], cf. the theories of superconductivity and superfluidity based on ODLRO. The second observation relates to the identification of the present finite dimensional representation as a precursor for possible condensations, developing correlations and coherences that may extend over macroscopic dimensions. If h is a set of two-particle determinants and the iV-particle fermionic wave function is constructed from an AGP, antisymmetrized geminal power, based on i, see Eq. (57), then the reduced density matrix can be represented as... 

F.6), there appears the possibility to consider the latter to be the reduction of a many-body fermionic pure state to an N-representable two-matrix. Since the density matrix above, if adapted appropriately, consequently is essentially N-representable through its relation to Coleman s extreme case [107], one might, via appropriate projections, completely recover the proper information, cf. corresponding partitioning procedures depicted in Appendix A. The structure described here is also of fundamental importance in connection with the phenomena of superconductivity and superfluidity through its intimate connections with Yang s concept of ODLRO [106], see more under Section 3.2. 

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