This is a collection of lectures given at the University of Heidelberg, especially but not exclusively for people who want to learn something about the canonical approach to quantum gravity, which is however not included in these lectures. They are about Dirac's general method to construct a quantum theory out of a classical theory, which has to be defined in terms of a Lagrangian. The classical Hamiltonian formalism is reviewed, with emphasis on the relation between constraints and gauge symmetries, and quantization is carried out without any kind of gauge fixing. The method is applied to three examples: the free electro-magnetic field, the relativistic point particle, and the very first steps of string theory are carried out.
By requiring unambiguous symmetric quantization leading to the Dirac equation in a curved space, we obtain a special representation of the spin connections in terms of the Dirac gamma matrices and their space-time derivatives. We also require that squaring the equation give the Klein-Gordon equation in a curved space in its canonical from (without spinor components coupling and with no first order derivatives). These requirements result in matrix operator algebra for the Dirac gamma matrices that involves a universal curvature constant. We obtain exact solutions of the Dirac and Klein-Gordon e…
The purpose of this paper is to show that, under certain restrictions, we can take a Dirac-Aharonov-Bohm potential as a pure gauge field. We argue that a modified quantization condition comes out for the electric charge that may open up the way for the understanding of fractional charges. One does not need any longer to rely on the existence of a magnetic monopole to justify electric charge quantization.
In the framework of Dirac quantization with second class constraints, a free particle moving on the surface of a $(d-1)-$dimensional sphere has an ambiguity in the energy spectrum due to the arbitrary shift of canonical momenta. We explicitly show that this spectrum obtained by the Dirac method can be consistent with the result of the Batalin-Fradkin-Tyutin formalism, which is an improved Dirac method, at the level of the first-class constraint by fixing the ambiguity, and discuss its physical consequences.
Dirac-Bergmann algorithm and canonical quantization of $k$-essence cosmology
Universidad de La Frontera（拉弗龙特拉大学）；Durban University of Technology（德班理工大学）；North-West University（西北大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）；Universidad Católica del Norte（北方天主教大学）
The aim of present work is to extend the application of Weitzeböck Induced Matter Theory (WIMT) to a dyonic Reissner-Nordström Black Hole (RNBH), by proposing a condition compatible with a quantization relation between gravitational mass and both magnetic and electric charges from a geometric product defined as an invariant in 5D.
The Dirac equation for massive free electrically neutral spin 1/2 particles in a gravitation field is considered. The secondary quantization procedure is applied to it and the Hilbert space of multiparticle quantum states is constructed.
We give an explicit operator realization of Dirac quantization of free particle motion on a surface of codimension 1. It is shown that the Dirac recipe is ambiguous and a natural way of fixing this problem is proposed. We also introduce a modification of Dirac procedure which yields zero quantum potential. Some problems of abelian conversion quantization are pointed out.
A toy model (suggested by Klauder) is analyzed from the perspective of First Class and Second Class Dirac constrained systems. The comparison is made by turning a First Class into a Second Class system with the introduction of suitable auxiliary conditions. The links between Dirac's system of constraints, the Faddeev-Popov canonical functional integral method and the Maskawa-Nakajima procedure to reduced phase space are explicitly illustrated. The model reveals stark contrasts and physically distinguishable results between First and Second class routes. Physically relevant systems such as the …
An elementary presentation of the methods for the canonical quantization of constraint systems with Fermi variables is given. The emphasis is on the subtleties of the construction of an appropriate classical bracket that could be consistently replaced by commutators or anti--commutators of operators, as required by canonical quantization procedure for bosonic and fermionic degrees of freedom respectively. I present a consequent canonical quantization of the Dirac field, in which the role of Poincaré invariance is made marginal. This simple example provides an introduction to the Poincaré--free…
Dirac's quantization of the Maxwell theory on non-commutative spaces has been considered. First class constraints were found which are the same as in classical electrodynamics. The gauge covariant quantization of the non-linear equations of electromagnetic fields on non-commutative spaces were studied. We have found the extended Hamiltonian which leads to equations of motion in the most general gauge covariant form. As a special case, the gauge fixing approach on the basis of Dirac's brackets has been investigated. The problem of the construction of the wave function and physical observables h…
This work addresses certain ambiguities in the Dirac approach to constrained systems. Specifically, we investigate the space of so-called ``rigging maps'' associated with Refined Algebraic Quantization, a particular realization of the Dirac scheme. Our main result is to provide a condition under which the rigging map is unique, in which case we also show that it is given by group averaging techniques. Our results comprise all cases where the gauge group is a finite-dimensional Lie group.
The main goal of this work is to study the Dirac oscillator as a quantum field using the canonical formalism of quantum field theory and to develop the canonical quantization procedure for this system in $(1+1)$ and $(3+1)$ dimensions. This is possible because the Dirac oscillator is characterized by the absence of the Klein paradox and by the completeness of its eigenfunctions. We show that the Dirac oscillator field can be seen as constituted by infinite degrees of freedom which are identified as decoupled quantum linear harmonic oscillators. We observe that while for the free Dirac field th…
In this article, we carry out the Hamiltonization in the axial gauge, of the t'Hooft-Polyakov monopole field outside the localized region, which represents the monopole's core. One feature of the treatment here, is using the Higgs vacuum condition as both strong and weak equation instead of using it in the degree of freedom reduction.
We present an exact quantization condition for the time independent solutions (energy eigenstates) of the one-dimensional Dirac equation with a scalar potential well that gives only two `effective' turning points (defined by the roots of $V(x)+mc^2=\pm E$) for a given energy $E$ and satisfies $\min V(x)+mc^2\geq 0$. This result generalizes the previously known non-relativistic quantization formula and preserves many physically desirable symmetries, besides, attaining the correct non-relativistic limit. Numerical calculations demonstrate the utility of the formula for computing accurate energy …
Dirac in 1931 gave a beautiful argument for the quantization of electric charge, which required only the existence in the universe of one magnetic monopole, because gauge invariance of the interaction between the pole and any charge could hold only if the product of the charge and the pole strength were quantized in half-integer multiples of the reduced Planck constant. However, if the photon had a nonzero mass, implying exponential decrease of the flux out of an electric charge, then Dirac's argument might seem to fail. We demonstrate that the result still should hold. The key point is that m…
We show that an unambiguous and correct quantization of the second-class constrained system of a free particle on a sphere in $D$ dimensions is possible only by converting the constraints to abelian gauge constraints, which are of first class in Dirac's classification scheme. The energy spectrum is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere. A quantization of Dirac's modified Poisson brackets for second-class constraints is also possible and unique, but must be rejected since the resulting energy spectrum is physically…
A fundamental problem regarding the Dirac quantization of a free particle on an $N-1$ curved hypersurface embedded in $N$($\geq 2$) flat space is the impossibility to give the same form of the curvature-induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin-layer sandwiching the surface. We resolve this problem by means of previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator-ordering-free se…
We extend to a non-Hermitian fermionic quantum field theory with PT symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [arXiv:2006.06656]. For illustration, we consider a prototype model containing a single Dirac fermion with a parity-odd, anti-Hermitian mass term. In the phase of unbroken PT symmetry, this Dirac fermion model is equivalent to a Hermitian theory under a similarity transformation, with the non-Hermitian nature of the model residing only in the spinor structure, whereas the algebra of the creati…
In this paper, we present an approach to quantize singular systems. This is an extension of the constant integration method (Belhadi et al. (2014)) which is applicable only for the case of exactly solvable systems. In our approach, we determine Dirac brackets at the initial instant with the help of Taylor expansion, and using their covariance, we deduce the brackets at any time. We also illustrate our approach by studying two systems: the self-dual model and the Dirac spinorial field. Keywords : quantization, singular systems, integration constants, Liouville operator, Taylor expansion.
We addressed quantization phenomena in transport and vortex/precession-motion of low-dimensional systems, stationary quantization of confined motion in phase space due to oscillatory dynamics or compacti fication of space and time for steady-state systems (e.g., particle in a box or torus, Brillouin zone, and Matsubara time zone or Matsubara quantized frequencies), and the quantization of sources. We discuss how the self-consistent Bohr-Sommerfeld quantization condition permeates the relationships between the quantization of integer Hall effect, fractional quantum Hall effect, the Berezenskii-…
The structure of Pais-Uhlenbeck oscillator in the equal-frequency limit has been recently studied by Mannheim and Davidson [Phys.Rev. A71 (2005), 042110]. It appears that taking this limit, as presented in the above paper, is quite subtle and the resulting structure of space of states - involved. In order to clarify the situation we present here the proper way of taking the equal-frequency limit, first under the assumption that the scalar product in the space of states is positive defined. We discuss also the case of indefinite metric space of states. We show that, irrespective of the way the …
Dirac's identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac's canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets betwee…
The problem of ultraviolet divergences is analysed in the quantum field theory. It was found that it has common roots with the problem of cosmological singularity. In the context of fibre bundles the second quantization method is represented as a procedure of the quantization for vector bundle cross-section. It is shown to be quite a different quantization way called as a fibre quantization which leads to an idea on existence of the non-standard dynamical system, i.e. the relativistic be-Hamiltonian system. It takes place on supersmall distances and is well described by the mathematical appara…
We consider the problem of constrained motion along a conic path under a given external potential function. The model is described as a second-class system capturing the behavior of a certain class of specific quantum field theories. By exhibiting a suitable integration factor, we obtain the general solution for the associated non-linear differential equations. We perform the canonical quantization in a consistent way in terms of the corresponding Dirac brackets. We apply the Dirac-Bergmann algorithm to unravel and classify the whole internal constraints structure inherent to its dynamical Ham…
Quantization Mapping on Dirac Dynamics via Voltage-Driven Charge Density in Monolayer Graphene: A Klein Paradox and Entropy-Ruled Wavevector Mechanics Study
KPR Institute of Engineering and Technology（KPR工程技术学院）
This is an overview of Erlangen Programme at Large. Study of objects and properties, which are invariant under a group action, is very fruitful far beyond the traditional geometry. In this paper we demonstrate this on the example of the group SL(2,R). Starting from the conformal geometry we develop analytic functions and apply these to functional calculus. Finally we link this to quantum mechanics and conclude by a list of open problems. Keywords: Special linear group, Hardy space, Clifford algebra, elliptic, parabolic, hyperbolic, complex numbers, dual numbers, double numbers, split-complex n…
A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation.
Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac quantization procedures.
It is well known that the single particle Dirac equation is gauge invariant. This means that observable quantities, such as the current density, are not affected by a gauge transformation. However what happens when the method of second quantization is applied to convert a single particle theory into a field theory? In this case it will be shown that the theory is no longer gauge invariant. This will be shown by considering the second quantization of a zero mass Dirac field in 1+1 dimensions and examining the change of the current density operator due to a gauge transformation.
Within the context of infinite-dimensional representations of the rotation group the Dirac monopole problem is studied in details. Irreducible infinite-dimensional representations, being realized in the indefinite metric Hilbert space, are given by linear unbounded operators in infinite-dimensional topological spaces, supplied with a weak topology and associated weak convergence. We argue that an arbitrary magnetic charge is allowed, and the Dirac quantization condition can be replaced by a generalized quantization rule yielding a new quantum number, the so-called topological spin, which is re…
The non-trivial geometry encoded in the Quantum Mechanical wavefunctions has important consequences for its single-particle as well as many-body dynamics. Yet, our understanding of how the geometry of the single-particle eigenstates are manifest in the characteristics of a many-particle system is still incomplete. Here, we demonstrate how the single-particle Berry curvature modifies the fluxoid quantization of a two dimensional Bardeen-Cooper-Schrieffer (BCS) superconductor, and discuss the experimental scenarios where this anomalous quantization is expected to be realized.
