Statistics of the Composite System

The space-like asymptotic limit of the bilocal composite field of the state consisting of a nucleus and an electron is studied. It is shown that the resulting local field of an atom satisfies the proper commutation relations in the sub-Fock-space of the atom. ∗) E-mail address: [email protected] 1 typeset using PTPTEX.sty More than forty years ago, composite systems in the quantum field theory was investigated . One starts from a local scalar field, say A(x), and assumes the existence of discrete eigenvalues m and M of the 4-momentum squared P , where m is the mass of the original field A. If 〈A(x)A(y) | P 2 = M〉 6 = 0 there may be a composite(bound) state of the mass M . Then, one defines a bilocal field B(x, ε) = TA(x+ ε)A(x− ε) (1) representing this state, where T denotes the time-ordered product. Zimmermann showed, under some mathematical assumption, that the asymptotic(t → ±∞) field of B satisfies the proper commutation relation in the limit ε → 0, if it is suitablly normalized. One can infer, from this construction, that composite systems can be described by the local field operators and there are no differences between elementary and composite particles in constructing the S matrix elements. An idea of the boot strapping(nuclear democracy) emerged from this observation. However, the successes of the gauge theories have changed drastically the framework of the elementary particle theory. The quantum theory of field has revived and one gradually recognized the hierarchy structure of the gauge interactions. A field theory now becomes an effective theory for each class of the hierarchy. On the other hand, there is another hierarchy of compositeness in nature, which was revealed through success of the composite models of elementary particles. The hierarchy here consists of the classes of quark, hadron, neucleus, atom and so on, although we are not sure if the class of quark is the deepest. The level of a class is specified by the energy scale indicating the limit of applicability of the theory governing it. We, further, believe that the theory is a quantum field theory and there exist elementary fields for each class, which are constructed from the elementary fields of the deeper class. The most instructive example of this interpolating mechanism may be provided by considering the class of atom. An atom is a composite system which consists of a nucleus field and an electron field interacting through the photon field. Then, we construct the atom field as a composite field of the elementary fields of the deeper class. In this respect, we are specially interested in the statistic property of the composite system, since we are hardly convinced of it by the particle quantum mechanics. For definiteness and simplicity, we restrict ourselves to the hydrogen atom in the following. There is also another reason to take up it: The nucleus of it is a Fermi particle and it changes to the Bose particle by acquiring another Fermion. This is mysterious from the view point of the particle quantum mechanics. ∗) The energy scale can be replaced by the length scale in some case.