It is shown that quantization of the dynamical systems with second class constraints actually can be reduced to quantization of the systems with first class constraints. The motion of the non-relativistic particle along the plane curve and on a surface is considered. The results coincide with those of the "thin layer method". Influence of the non-physical variables on the physical sector is demonstrated.
We propose a method of quantization based on Hamilton-Jacobi theory in the presence of a random constraint due to the fluctuations of a set of hidden random variables. Given a Lagrangian, it reproduces the results of canonical quantization yet with a unique ordering of operators if the Lagrange multiplier that arises in the dynamical system with constraint can only take binary values $\pm\hbar/2$ with equal probability.
A careful reexamination of the quantization of systems with first- and second-class constraints from the point of view of coherent-state phase-space path integration reveals several significant distinctions from more conventional treatments. Most significantly, we emphasize the importance of using path-integral measures for Lagrange multipliers which ensure that the quantum system satisfies the quantum constraint conditions. Our procedures involve no delta-functionals of the classical constraints, no need for gauge fixing of first-class constraints, no need to eliminate second-class constraint…
A practical method is developed to deal with the second quantization of the many-body system containing the composite particles. In our treatment, the modes associated with composite particles are regarded approximately as independent ones compared with those of unbound particles. The field operators of the composite particles thus arise naturally in the second quantization Hamiltonian. To be emphasized, the second quantization Hamiltonian has the regular structures which correspond clearly to different physical processes.
Recent experiments with trapped alkali atoms have drawn enormous interest to the theoretical studies concerning Bose-Einstein condensation. The purpose of this paper is to review one of the approaches to study bosonic matter at zero temperature, namely the Bogoliubov approximation. Review of a necessary tool, the second quantization, will also be made.
Second quantization is an essential topic in senior undergraduate and postgraduate level Quantum Mechanics course. However, it seems that there is a lack of transparent and natural derivation of this formalism from the first-quantization one in most existing textbooks. Without introducing the concept of a simple harmonic oscillator and taking the case of an assembly of identical fermions as an example, we provide a less ad hoc derivation of the second quantization formalism based on the equivalence of the fully antisymmetric many-fermion states and the Fock states.
On the example of stationary states of a system consisting of an atom and a quantized electromagnetic field (the Jaynes-Cummings model in free space), it is shown that the physical characteristics of the system (as the energy and the probability distribution of finding the center of mass of the atom) explicitly depend on the choice of the basis of quantization of the electromagnetic field. Therefore, the secondary quantization procedure is not independent of the choice of the quantization basis in the interaction of the field with the material medium.
Using azimuthally symmetrized cylindrical coordinates, we report the consequences of zero-energy quantal states on the von Roos Hamiltonian. A position-dependent mass M(ρ,ϕ,z)=bz^{j}ρ^{2\u{psion}+1}/2 is used. We show that the zero-energy setting not only offers an additional degree of freedom towards feasible separability for the von Roos Hamiltonian, but also manifestly yields auxiliary quantized ambiguity parametric constraints.
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.
Second quantization has been widely used in quantum mechanics and quantum chemistry, which is trivial and error-prone for researchers. Fortunately it is a good candidate for automatic evaluation with its simple, trivial and intrinsic iterative nature. This article presents an automatic and generic tool for these kinds of evaluations. We believe it could be a helpful tool for scientists, and hope the idea of automatic evaluation could be generalized to other equation derivations in the future.
New measures for the quantization of systems with constraints are discussed and applied to several examples, in particular, examples of alternative but equivalent formulations of given first-class constraints, as well as a comparison of regular and irregular constraints.
We examine two singular Lagrangian systems with constraints which apparently reduce the phase space to a 2-dimensional sphere and a 2-dimensional hyperboloid. Rigorous constraint analysis by Dirac's method, however, gives 2-dimensional open disc and an infinite plane with a hole in the centre respectively as the reduced phase spaces. Upon canonical quantisation the classical constraints show up as restrictions on the Hilbert space.
The conversion of second-class constraints into first-class constraints is used to extend the coordinate-free path integral quantization, achieved by a flat-space Brownian motion regularization of the coherent-state path integral measure, to systems with second-class constraints.
A causal interpretation of the quantum world needs second quantization in order to cover phenomena like creation and annihilation of particles, and this leads to quantum field theory. The causal effects of the second quantization can be described through a new quantum potential. In this article we have dealt with the second quantization of Schrodinger and its effects on the path of a particle. This generalization leads to a modified Schrodinger which affects the particle through a modified quantum potential and a new term in the continuity equation. We have shown that these effects can provide…
By making the second quantization for the Cini Model of quantum measurement without wave function collapse [M. Cini, Nuovo Cimento, B73 27(1983)], the second order quantum decoherence (SOQD) is studied with a two mode boson system interacting with an idealized apparatus composed by two quantum oscillators. In the classical limit that the apparatus is prepared in a Fock state with a very large quantum number, or in a coherent state with average quantum numbers large enough, the SOQD phenomenon appears similar to the first order case of quantum decoherence.
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…
Based on the results of a recent reexamination of the quantization of systems with first-class and second-class constraints from the point of view of coherent-state phase-space path integration, we give additional examples of the quantization procedure for reparameterization invariant Hamiltonians, for systems for which the original set of Lagrange multipliers are elevated to dynamical variables, as well as extend the formalism to include cases of first-class constraints the operator form of which have a spectral gap about the value zero that characterizes the quantum constraint subspace.
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) has been shown to provide a unified origin for the existence of Dirac's Hamiltonian and a self adjoint time operator that circumvents Pauli's objection. As such, this approach restores to Quantum Mechanics (QM) the treatment of space and time on an equivalent footing as that of momentum and energy. Second quantization of the time operator field follows step by step that of the Dirac Hamiltonian field. It introduces the concept of time quanta, in a similar way to the energy quanta …
We continue in this paper our program of rederiving all quantum mechanical formalism from the classical one. We now turn our attention to the derivation of the second quantized equations, both for integral and half-integral spins. We then show that all the quantum results may be derived using our approach and also show the interpretation suggested by this derivation. This paper may be considered as a first approach to the study of the quantum field theory beginning by the same classical ideas we are supporting since the first paper of this series.
Systems under holonomic constraints are classified within the generalized Hamiltonian framework as second-class constraints systems. We show that each system of point particles with holonomic constraints has a hidden gauge symmetry which allows to quantize it in the original phase space as a first-class constraints system. The proposed method is illustrated with quantization of a point particle moving on an $n-1$-dimensional sphere $S^{n-1}$ as well as its field theory analog the O(n) nonlinear sigma model.
The theories defined by Lagrangians containing second time derivative are considered. It is shown that if the second derivatives enter only the terms multiplied by coupling constant one can consistently define the perturbative sector via Dirac procedure. The possibility of introducing standard canonical variables is analysed in detail. The ambiguities in quantization procedure are pointed out.
We present a quantization scheme for optical systems with absorptive losses, based on an expansion in the complete set of scattering solutions to Maxwell's equations. The natural emergence of both absorptive loss and fluctuations without introducing a thermal bath is demonstrated. Our model predicts mechanisms of absorption induced squeezing and dispersion mediated photon conversion.
A method is developed for realizing entanglement and general second quantized fermionic and bosonic fields in the framework of the fermionic projector.
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.
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.
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.
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 …
The quantization of systems with first- and second-class constraints within the coherent-state path-integral approach is extended to quantum systems with fermionic degrees of freedom. As in the bosonic case the importance of path-integral measures for Lagrange multipliers, which in this case are in general expected to be elements of a Grassmann algebra, is emphasized. Several examples with first- and second-class constraints are discussed.
We develop an approximate second quantization method for describing the many-particle systems in the presence of bound states of particles at low energies (the kinetic energy of particles is small in comparison to the binding energy of compound particles). In this approximation the compound and elementary particles are considered on an equal basis. This means that creation and annihilation operators of compound particles can be introduced. The Hamiltonians, which specify the interactions between compound and elementary particles and between compound particles themselves are found in terms of t…
We quantize spherically symmetric vacuum gravity without gauge fixing the diffeomorphism constraint. Through a rescaling, we make the algebra of Hamiltonian constraints Abelian and therefore the constraint algebra is a true Lie algebra. This allows the completion of the Dirac quantization procedure using loop quantum gravity techniques. We can construct explicitly the exact solutions of the physical Hilbert space annihilated by all constraints. New observables living in the bulk appear at the quantum level (analogous to spin in quantum mechanics) that are not present at the classical level and…
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.
The program of quantizing the gravitational field with the help of affine field variables is continued. For completeness, a review of the selection criteria that singles out the affine fields, the alternative treatment of constraints, and the choice of the initial (before imposition of the constraints) ultralocal representation of the field operators is initially presented. As analogous examples demonstrate, the introduction and enforcement of the gravitational constraints will cause sufficient changes in the operator representations so that all vestiges of the initial ultralocal field operato…
The present article is primarily a review of the projection-operator approach to quantize systems with constraints. We study the quantization of systems with general first- and second-class constraints from the point of view of coherent-state, phase-space path integration, and show that all such cases may be treated, within the original classical phase space, by using suitable path-integral measures for the Lagrange multipliers which ensure that the quantum system satisfies the appropriate quantum constraint conditions. Unlike conventional methods, our procedures involve no $δ$-functionals of …
Motivated by certain concepts introduced by the Refined Algebraic Quantization formalism for constrained systems which has been successfully applied within the context of Loop Quantum Gravity, in this paper we propose a phase space implementation of the Dirac quantization formalism to appropriately include systems with constraints. In particular, we propose a physically prescribed Wigner distribution which allows the definition of a well-defined inner product by judiciously introducing a star version of the group averaging of the constraints. This star group averaging procedure is obtained by …
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…
We consider spin system defined on the coadjoint orbit with noncompact symmetry and investigate the quantization. Classical spin with noncompact SU(N,1) symmetry is first formulated as a dynamical system and the constraint analysis is performed to reduce the system from the group space to the coadjoint orbit which is a symplectic manifold with Kahler structure. We achieve this by solving the constraint directly. It is shown that the dynamical variables describing the noncompact spins can be written as functions of canonically conjugate variables and canonical quantization is possible on the re…
