A method of field quantization is investigated which is more general than the usual methods of quantization in accordance with Bose of Fermi statistics, though these are included in the scheme. The commutation properties and matrix representations of the quantized field amplitudes are determined, and the energy levels of the field are derived in the usual way. It is shown that spin-half fields can be quantized in such a way that an arbitrary finite number of particles can exist in each eigenstate. With the generalized statistics, the interchange of two particles of the same kind may or may not be physically significant, according to the type of interaction by means of which they are created or annihilated. Physical consequences of the assumption that there are particles which obey the generalized statistics are briefly examined.
A method of field quantization is investigated which is more general than the usual methods of quantization in accordance with Bose of Fermi statistics, though these are included in the scheme. The commutation properties and matrix representations of the quantized field amplitudes are determined, and the energy levels of the field are derived in the usual way. It is shown that spin-half fields can be quantized in such a way that an arbitrary finite number of particles can exist in each eigenstate. With the generalized statistics, the interchange of two particles of the same kind may or may not be physically significant, according to the type of interaction by means of which they are created or annihilated. Physical consequences of the assumption that there are particles which obey the generalized statistics are briefly examined.
ABSTRACT The Feynman probability amplitudes for interactions between spin ½ fields and fields of integral spin quantized according to the generalized method of Green are investigated. It is shown that the scattering amplitudes are independent of the method of quantization though the amplitudes of the vacuum fluctuations are not. The question of distinguishing particles satisfying the generalized statistics, if they exist in nature, from Fermi or Bose particles is considered. It is shown that the statistical thermodynamics of the particles depends on the statistics, but as this requires a large number of particles in thermodynamical equilibrium, it offers no means of distinction for particles with short lifetimes. In practice, an examination of particles in bound states would identify particles obeying generalized statistics.
A new generalization of quantum statistics is described, which is different from parastatistics, though it includes Fermi statistics and parafermi statistics of order two. It can be applied to the quantization of nonlinear field theories, without violating the correspondence principle. Like parastatistics, it allows the occupation of a given dynamical state by more than one particle of half-odd-integral spin. A special feature is that quark-like particles are naturally associated in modules, which have many of the characteristics of baryons and mesons. The group theoretical properties of the new statistics, and the implied classification of states, are briefly examined.
The formalism of the second quantization is developed entirely within the framework of the Dirac-Jordan representation theory. The equivalence of this formalism and the methods of ordinary coordinate-spin space is shown, and the equations of "connection" are fully developed. It is shown that the "creation" and "annihilation" operators, and indeed all operators which do not commute with the quantized Hamiltonian and hence whose representatives cannot be explicitly given in a configuration space of a fixed number of dimensions are easily treated in terms of the general representation theory and arise quite naturally. Both types of statistics are considered, and the Fock-Dirac density matrix and self-consistent field are treated as a simple illustration.
Synopsis: e connection between a particle’s spin and the statistics it obeys is given. Bose-Einstein and Fermi-Dirac statistics are explained and illustrated with products of wave functions. A new type of particle basis state is described and two possibilities for the commutation properties of the associated raising (“creation”) and lowering (“annihilation”) ladder operators are examined, giving rise to the Bose-Einstein and Fermi-Dirac cases. e relation between particle occupation number and energy is inferred from detailed balance, and cubical enclosures of particles are studied. Two physics mysteries encountered in the earlier chapters (“zweideutigkeit” and the vanishing electronic component of specific heat) are explained. e Hartree-Fock theoretical framework for the approximate treatment of many-particle atomic systems is constructed.
We state in the following a structure theory of the density matrices from the standpoint of secind quantization. The difficulties involved in the quantum-mechanical many body problem have so far proved to be so tremendous that as yet some have been obliged to stop before exact treatments and others have retired to phenomenological model theories. There are two main tools to approach the problem, the theory of second quantization and the theory of density matrix. The former, in spite of its generality and compactness, has never revealed us its adaptability to the practical problems of quantum statistical mechanics. In fact, about the exchange effect, namely the statistical correlation of particles, the latter informs us the detailed features of it by application of the group theory, while the former lacks, at present, such an exposition. For a long time a more general and simpler formulation than the current group theoretic method has been desired in the theoretical researches of spectroscopy. A distinguished contribution has been made by P. Jordan (1934) and H. Ostertag (1937) from the standpoint of second quantization. H. Ostertag made use of the theory of hypercomplex ring and introdeced the original concept of the elementary algebra. (Part II) Since the latter part of his formulation, however, seems to be complicated, it will be better, for clearness' sake, to avoid this part. Accordingly the concepts of the elementary algebra and the symmetric algebra will be introduced as a remedy for it. Then we shall complete our theory to include the whole theory of symmetric group. These concepts will serve to clarify and generalize the previous formulations and to unify the several group theoretic standpoints of the spectroscopy. In §1 and §2 the representation of the quantized density of Bose or Fermi particles will reveal itself as the density matrix of Bose or Fermi particles or as the density matrix of many electrons with a certain term of spin multiplets. Indeed, the theory of the statistical operator, or the density matrix of J. von Neumann and P.A.M. Dirac, the equilibrium properties of which have been fully examined by one of the authors (1940), is the natural formalism of dealing with quantum gases. A contribution has been made by W. Kofink (1938) to introduce the intrinsic degrees of freedom of particles. This attempt has resulted in a fair seccess. His formulation consists in making use of the initial stages of the theory of spectroscopy, which will be reexamined in the second section and Part II. In §3, as an example at zero temperature, the Hartree-Fock equation for a prescribed orbital and spin configuration will be derived, and in §4 some properties of the reduced density matrix will be illustrated.
We state in the following a structure theory of the density matrices from the standpoint of secind quantization. The difficulties involved in the quantum-mechanical many body problem have so far proved to be so tremendous that as yet some have been obliged to stop before exact treatments and others have retired to phenomenological model theories. There are two main tools to approach the problem, the theory of second quantization and the theory of density matrix. The former, in spite of its generality and compactness, has never revealed us its adaptability to the practical problems of quantum statistical mechanics. In fact, about the exchange effect, namely the statistical correlation of particles, the latter informs us the detailed features of it by application of the group theory, while the former lacks, at present, such an exposition. For a long time a more general and simpler formulation than the current group theoretic method has been desired in the theoretical researches of spectroscopy. A distinguished contribution has been made by P. Jordan (1934) and H. Ostertag (1937) from the standpoint of second quantization. H. Ostertag made use of the theory of hypercomplex ring and introdeced the original concept of the elementary algebra. (Part II) Since the latter part of his formulation, however, seems to be complicated, it will be better, for clearness' sake, to avoid this part. Accordingly the concepts of the elementary algebra and the symmetric algebra will be introduced as a remedy for it. Then we shall complete our theory to include the whole theory of symmetric group. These concepts will serve to clarify and generalize the previous formulations and to unify the several group theoretic standpoints of the spectroscopy. In §1 and §2 the representation of the quantized density of Bose or Fermi particles will reveal itself as the density matrix of Bose or Fermi particles or as the density matrix of many electrons with a certain term of spin multiplets. Indeed, the theory of the statistical operator, or the density matrix of J. von Neumann and P.A.M. Dirac, the equilibrium properties of which have been fully examined by one of the authors (1940), is the natural formalism of dealing with quantum gases. A contribution has been made by W. Kofink (1938) to introduce the intrinsic degrees of freedom of particles. This attempt has resulted in a fair seccess. His formulation consists in making use of the initial stages of the theory of spectroscopy, which will be reexamined in the second section and Part II. In §3, as an example at zero temperature, the Hartree-Fock equation for a prescribed orbital and spin configuration will be derived, and in §4 some properties of the reduced density matrix will be illustrated.