A microscopic theory is proposed to describe not only the pairing rotation, the pairing vibrations and the independent-particle motions, but also their mutual couplings in a consistent manner. With the aid of a BCS-type wave packet and a quasi-particle coherent state, the quantal system is translated into a corresponding classical one. First, we develop a canonical form in a classical image. This classical system obeys certain constraints, by which the double counting in the degrees of freedom is avoided. Then, the quantization is performed with the use of the Dirac theory of a canonical system with constraints. The original fermion operators themselves are expressed in terms of three types of degrees of freedom, which correspond to the pairing rotation, the pairing vibrations and the independent-quasi-particle motions, respectively.
A classical theory for many-fermion system is presented in the canonical form by introducing the ordinary and the Grassmann variables, the number of which exceeds “apparently” the original one. The basic idea is a translation of a theory proposed by the present authors for the case of the particle-hole pairs to the case of the Hartree-Bogoliubov type pairs. Certain constraints governing the variables are introduced and the double counting in the number of the degrees of freedom concerning the extra variables can be avoided. The quantization is performed with the aid of the Dirac bracket to take account of the constraints. It is shown that final result is completely identical with the compact form of the boson expansion given by Marshalek.
With the use of the ordinary and the Grassmann variables, a classical theory for many-fermion system with particle-hole correlations is presented in a canonical form. This system involves many valence particles/holes in addition to many particle-hole pairs. The particle-hole pairs are assigned as the collective degree of freedom and the others as the independent-particle one. In order to avoid the double counting in the number of the degrees of freedom due to the introduction of the collective variables, certain constraints are introduced. The quantization is performed with the aid of the Dirac theory for canonical system with constraints. It is shown that the result is completely equivalent to that obtained with the use of the boson expansion method by Marshalek.
A microscopic theory of collective and independent-particle motions in many-fermion system is developed in the framework of Dirac's canonical theory with constraints.The basic idea is an extension of the conventional time-dependent Hartree-Fock theory based on the use of the Grassmann variables and, in certain sense, it supplements incomplete parts of the paper presented by the present authors.Canonicity condition to certify the canonical property of the variables, equation to determine collective submanifold and constraints to govern the collective and the Grassmann variables are given.
A many•body technique called the dynamical nuclear field theory is developed by taking the example of pair rotation.The time-dependent one-body Hamiltonian contains the collective coordinates; the magnitude and phase aIigleof the energy gap.Effective forms of the original Hamiltonian and other operators are derived for a direct-product space of the collective states and a restricted set of the particle states defined by the driving Hamiltonian in the intrinsic frame.Pair rotations of even-and odd-A systems are treated in the lowest order approximation.The particle-rotor picture and the cranking model are shown to be equivalent.*) Brief report on DNFT in the Tamm-Dancoff approximation can be found in Ref. 13).
In Newtonian spacetime, the canonical description of a classical
N-particle system requires
3N degrees of freedom. Not all of these are physical, however, since the conservation of the net momentum implies that only
3(N−1) accelerations are independent. Hence, three constraints can be used to eliminate the unphysical centre-of-mass variables, at the level of the Lagrangian, leaving only the subset of observable displacements and momenta, which are relational. Imposing the constraints does not change the dynamics of these variables, at the classical level, and is analogous to a gauge-fixing procedure, which removes redundancy in the description of the system. In classical physics, therefore, the number of physical degrees of freedom equals the number of independent relational degrees of freedom. Here, we show that this is not the case in quantum mechanics. While an operator-analogue of the classical net momentum exists, it cannot be used to impose constraints that restrict the degrees of freedom in the theory, without a loss of physical information. This means that all
3N canonical degrees of freedom are physical, even though only
3(N−1) of them are relational. We explore the physical consequences of the non-relational variables and show that they give rise to generalised uncertainty relations (GURs), for the relational quantities that define the quantum reference frame (QRF). Hence, it is shown that the non-relational degrees of freedom refer to the frame itself and that the non-Heisenberg terms in the GURs define its Galilean-invariant spreads, in both real space and momentum space. The implications of this result for recent work on relational models, including the ``perspective neutral'' framework for QRFs, are discussed. Its implications for the wider relational program, and, in particular, the relevance of the latter to quantum gravity research, are also critically assessed