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Journal of Physics A Mathematical and General

Diagonalisation of the quadratic fermion Hamiltonian with a linear part

J.H.P. Colpa

Published Apr 1, 1979• 29 citations
Publisher

Abstract

It is well known that the spectrum of a homogeneous quadratic Hamiltonian in m fermion construction-operator pairs is characterised by m mode energies. In this paper it is proved that the spectrum of a quadratic Hamiltonian with a linear part presents the same feature. Simple methods are given for the calculation of the mode energies and of the ground-state energy of the Hamiltonian. It is shown that, in contrast to the boson case, the diagonalisation cannot be carried out by a linear transformation of the fermion construction operators. Application of such a transformation can only result in a diagonalisation after the introduction of a 'ghost' particle (fermion) and a corresponding pair of Hermitian-conjugate construction operators.

DOI
10.1088/0305-4470/12/4/008

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It is well known that the spectrum of a homogeneous quadratic Hamiltonian in m fermion construction-operator pairs is characterised by m mode energies. In this paper it is proved that the spectrum of a quadratic Hamiltonian with a linear part presents the same feature. Simple methods are given for the calculation of the mode energies and of the ground-state energy of the Hamiltonian. It is shown that, in contrast to the boson case, the diagonalisation cannot be carried out by a linear transformation of the fermion construction operators. Application of such a transformation can only result in a diagonalisation after the introduction of a 'ghost' particle (fermion) and a corresponding pair of Hermitian-conjugate construction operators.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
A self-contained treatment of the Bogoliubov-Valatin transformation for homogeneous fermionic Hamiltonians is presented. The aim is to provide a quick reference that may also serve as supplementary material for a graduate-level course, and that can be understood with quantum mechanics knowledge up to the level of the second quantization's rules. The objective of the transformation is to cast a quadratic Hamiltonian into a diagonal form that resembles the Hamiltonian of a system of non-interacting particles. To obtain this, the first step consists in putting its coefficient matrix into its canonical form; the transformation can always be performed on fermionic Hamiltonians, only some care must be taken when this form is singular. Having explained how to cast a general matrix into its standard form, a complete description of the transformation is provided; a novel procedure is proposed here for the singular matrix case.
Diagonalisation of quadratic multi-mode, time-dependent Hamiltonians can be fraught with difficulties. We set about diagonalising Hamiltonians from quantum optics. The Hamiltonians we choose represent Raman scattering, parametric amplification and the frequency converter. We present a systematic approach to this problem. The process of diagonalisation allows us to find invariants of the Hamiltonians under time evolution, find ground-state energies, apply the theory of Greens functions to find the solution to Schrodinger's equation, examine the conditions for oscillation or growth, and squeezing.
We regard the real and imaginary parts of the Schrodinger wave function as canonical conjugate variables.With this pair of conjugate variables and some other 2n pairs, we construct a quadratic Hamiltonian density. We then show that the Schrodinger Equation follows from the Hamilton's Equation of motion when the Planck frequency is much larger than the characteristic frequencies of the Hamiltonian system. The Hamiltonian and the normal mode solutions coincide, respectively, with the energy expectation value and the energy eigenstates of the corresponding quantum mechanical system.
Abstract The main purpose of this work is to identify invariant quadratic operators associated with Linear Canonical Transformations (LCTs) which could play important roles in physics. In quantum physics, LCTs are the linear transformations which keep invariant the Canonical Commutation Relations (CCRs). In this work, LCTs corresponding to a general pseudo-Euclidian space are considered and related to a phase space representation of quantum theory. Explicit calculations are firstly performed for the monodimensional case to identify the corresponding LCT-invariant quadratic operators then multidimensional generalizations of the obtained results are deduced. The eigenstates of these operators are also identified. A first kind of LCT-invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of the coordinates and momenta operators themselves. It is shown that these statistical variances-covariances can be related with thermodynamic variables. In this context, new quantum corrections to the ideal gas state equation are deduced from correction to the Hamiltonian operator of non-relativistic free quantum particles that is suggested by LCT-covariance. Two other LCT-invariant quadratic operators, which can be considered as the number operators of some quasiparticles, are also identified: the first one is a number operator of bosonic type quasiparticles and the second one corresponds to fermionic type. This fermionic LCT-invariant quadratic operator is directly related to a spin representation of LCTs. It is shown explicitly, in the case of a relativistic pentadimensional theory, that the eigenstates of this operator can be considered as basic quantum states of elementary fermions. A classification of the fundamental fermions, compatible with the Standard Model of particle physics, is established from a classification of these states.