We study simple two-dimensional models with massless and massive fermions in the Hamiltonian framework. While our ultimate goal is to gain a deeper insight to structural differences between the usual (“spacelike” – SL) and light-front (LF) forms of the relativistic dynamics, an attempt is also made to clarify a few conceptual problems of quantum field theory. We point out that contrary to the assumption of canonical quantization, interacting Heisenberg fields do not always reduce to free fields at t=0. We also show that by incorporating operator solutions of the field equations to the canonical formalism, SL and LF Hamiltonians of the derivative-coupling model as well as of the Federbush model acquire an equivalent structure. In the usual canonical treatment, physical predictions in the two schemes disagree – the SL Hamiltonians contain interaction terms while their LF counterparts do not. Using a Bogoliubov transformation, the physical vacuum of the Thirring model is then derived for the first time. It has a form of a coherent state quadratic in composite boson operators which, after bosonization of the vector current, are present in the (nondiagonal) interaction Hamiltonian. To find the vacuum of the Federbush model by an analogous Bogoliubov transformation, we propose a massive version of Klaiberʼs current bosonization and demonstrate advantages of the LF treatment of the model.
We study simple two-dimensional models with massless and massive fermions in the Hamiltonian framework. While our ultimate goal is to gain a deeper insight to structural differences between the usual (“spacelike” – SL) and light-front (LF) forms of the relativistic dynamics, an attempt is also made to clarify a few conceptual problems of quantum field theory. We point out that contrary to the assumption of canonical quantization, interacting Heisenberg fields do not always reduce to free fields at t=0. We also show that by incorporating operator solutions of the field equations to the canonical formalism, SL and LF Hamiltonians of the derivative-coupling model as well as of the Federbush model acquire an equivalent structure. In the usual canonical treatment, physical predictions in the two schemes disagree – the SL Hamiltonians contain interaction terms while their LF counterparts do not. Using a Bogoliubov transformation, the physical vacuum of the Thirring model is then derived for the first time. It has a form of a coherent state quadratic in composite boson operators which, after bosonization of the vector current, are present in the (nondiagonal) interaction Hamiltonian. To find the vacuum of the Federbush model by an analogous Bogoliubov transformation, we propose a massive version of Klaiberʼs current bosonization and demonstrate advantages of the LF treatment of the model.
We study a massive Thirring-like model in 2-dimensional space-time, which contains two different species of fermions. This model is a field theoretical version of the quantum mechanical model originally proposed by Glöckle, Nogami and Fukui, where different fermions interact with each other through δ-function potentials. We derive a corresponding boson model by the bosonization technique in the path integral formulation. This is a simple but non-trivial extension of the freedom of the bosonization technique. Operator correspondences between fermion and boson fields are given. One of these could not be realistically expected from the naive correspondence of the original single-species models. It is essential for this point that in our model fermions of the same kind do not interact with each other directly. We find that for a specific value of the coupling constant, one boson field becomes free while the other is a Sine-Gordon field. For this case, therefore, our two-species model is equivalent to the ordinary Sine-Gordon model of a single boson field.
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
It is shown that exact solvability of the finite temperature massless Thirring model, as well as of its zero temperature case, in canonical quantization scheme originates from the intrinsic hidden exact linearizability of Heisenberg equations by means of dynamical mapping onto the Schrödinger physical fields. The normal forms of different one- and two- parametric (thermo) field's solutions are obtained. They are connected with each other by making use of generalized conformal shift transformations. The sequential use of bosonic canonical transformations provides a correct renormalization, anticommutation and symmetry properties of these solutions. The dynamical role of inequivalent representations of 1+1-D free massless Dirac fields, that are induced by inequivalent representations of 1+1-D free massless (pseudo) scalar field, and the appearance of Schwinger terms are elucidated. The inequivalent vacuum is established as coherent state for SU(1,1) group. A new alternative sources of superselection rules are shown. A generalization of Ojima tilde conjugation rules is suggested, which reveals the properties of coherent state for SU(2) group for the fermionic thermal vacuum state and is useful for the thermofield bosonization. The notions of "hot" and "cold" thermofields are introduced to distinguish different thermofield representations giving the correct normal form of thermofield solution. The weak sense of definition of zero and finite temperature operator bosonization rules in the framework of thermofield dynamics is demonstrated.
At the conceptual level, the quantization procedure replaces the pure particle description of classical mechanics by a particle-wave treatment satisfying the requirements of the fundamental wave–particle dualism. Technically, one can always start from a generalized wave equation and use it as a basis to discuss wave-front propagation in the form of rays following classical particle trajectories, see Sections 2.2.1 and 3.1. Consequently, any wave theory inherently contains that aspect of the wave–particle dualism. Especially, Maxwell's equations of electrodynamic theory include both wave and particle aspects of electromagnetic fields. Hence, it could be that there are no additional “quantum,” i.e., wave–particle dualistic aspects. However, as we discuss in this chapter, Maxwell's equations can be expressed in the framework of Hamilton's formulation of particle mechanics. This then suggests that in addition to the approximate ray-like behavior, electromagnetic fields do have a supplementary level of particle-like properties. Due to the fundamental axiom of wave–particle dualism, the additional particle-like aspects of electromagnetic fields must then also have a wave counterpart that can be axiomatically introduced via the canonical quantization scheme described in Section 3.2.3. Since the spatial dependency in Maxwell's equation contains wave–particle dualism already at the classical level, it is clear that the new level of quantization cannot be a simple real-space feature but has to involve some other space describing the structure of the electromagnetic fields. In fact, this quantization yields several very concrete implications such as quantization of the light energy, elementary fluctuations in the light-field amplitude and intensity, and more.
A simple prescription for quantizing fields is developed and applied to the sytsem of interacting spins in an electromagnetic field. The classical equations of motion are written in the standard form $\ensuremath{-}i(\frac{d}{\mathrm{dt}})|V〉+\mathfrak{M}|V〉=0$, where the state vector ∣V〉 has the state variables, such as the generalized coordinates and momenta, as its components, and the time development operator M contains derivative, integral, and matrix operators in general. By diagonalizing M and replacing the resulting classical variables with operators obeying boson commutation relations, the system is quantized, and the Hamiltonian and Heisenberg equations of motion are automatically cast into the harmonic-oscillator form. Thus the quantization is reduced to the simple problem of diagonalizing a matrix. This avoids the laborious process of finding a Lagrangian density (by trial and error in general), determining the canonical variables, forming the Hamiltonian density, and inventing transformations to creation and annihilation operators. Problems with zero-frequency modes, which can be annoying in the standard treatment, are automatically eliminated by the prescription. The bookkeeping of forward and backward waves, different branches of dispersion curves, etc., is simplified. In addition to affording a straightforward and simple method of quantizing fields, the method also gives the necessary and sufficient conditions for a Hamiltonian to be transformable into harmonic-oscillator form, gives a direct relation between the classical equations of motion and the Heisenberg quantum-mechanical equations, and gives a simple method of determining the constants of the motion.
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.