Stochastic Variational Method (SVM) is the generalization of the variation method to the case with stochastic variables. In the series of papers, we investigate the applicability of SVM as an alternative field quantization scheme. Here, we discuss the complex Klein-Gordon equation. In this scheme, the Euler-Lagrangian equation for the stochastic fields leads to the functional Schroedinger equation, which in turn can be interpreted as the ideal fluid equation in the functional space. We show that the Fock state vector is given by the stationary solution of these differential equations and various results in the usual canonical quantization can be reproduced, including the effect of anti-particles. The present formulation is a quantization scheme based on commutable variables, so that there appears no ambiguity associated with the ordering of operators, for example, in the definition of Noether charges.
In this paper we first introduce the famous Klein paradox. Afterwards by proposing the Krein quantization approach and taking the negative modes into account, we will show that the expected and exact current densities, could be achieved without confronting any paradox.
We quantize the linearised Einstein-Klein-Gordon system on arbitrary on-shell backgrounds in a manifestly covariant and gauge-invariant manner. For the special case of perturbations in Inflation, i.e. on-shell backgrounds of Friedmann-Lemaître-Robertson-Walker type, we compare our general quantization construction with the standard approach to the quantum theory of perturbations in Inflation. We find that not all local quantum observables of the linearised Einstein-Klein-Gordon system can be split into local observables of scalar and tensor type as in the standard approach. However, we argue t…
We give an algebraic quantization, in the sense of quantum groups, of the complex Minkowski space, and we examine the real forms corresponding to the signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum metrics and providing an explicit presentation of the quantized coordinate algebras. In particular, we focus on the Kleinian signature $(2,2)$. The quantizations of the complex and real spaces come together with a coaction of the quantizations of the respective symmetry groups. We also extend such quantizations to the $\mathcal{N}=1$ supersetting.
We construct firstly the complete list of five quantum deformations of $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})\cong \mathfrak{o}(3;\mathbb{C})\oplus \mathfrak{o}(3;\mathbb{C})$, describing quantum rotational symmetry of four-dimensional complex space-time, in particular we provide the corresponding universal quantum $R$-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\ma…
Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-${N}$
School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院）；Peking University（北京大学）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Peng Huanwu Center for Fundamental Theory（彭桓武理论物理中心）
We analyse the Klein-Gordon oscillator in a cosmic string space-time and study the effects stemming from the rotating frame and non-commutativity in momentum space. We show that the latter mimics a constant magnetic field, imparting physical interpretation to the setup. The field equation for the scalar field is solved via separations of variables, and we obtain quantization of energy and angular momentum. The space-time metric is non-degenerate as long as the particle is confined within a hard-wall, whose position depends on the rotation frame velocity and the string mass parameter. We invest…
We develop a rigorous method to parametrize complex structures for Klein-Gordon theory in globally hyperbolic spacetimes that satisfy a completeness condition. The complex structures are conserved under time-evolution and implement unitary quantizations. They can be interpreted as corresponding to global choices of vacuum. The main ingredient in our construction is a system of operator differential equations. We provide a number of theorems ensuring that all ingredients and steps in the construction are well-defined. We apply the method to exhibit natural quantizations for certain classes of g…
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…
We apply a type of background independent "polymer" quantization to a free scalar field in a flat spacetime. Using semi-classical states, we find an effective wave equation that is both nonlinear and Lorentz invariance violating. We solve this equation perturbatively for several cases of physical interest, and show that polymer corrections to solutions of the Klein-Gordon equation depend on the amplitude of the field. This leads to an effective dispersion relation that depends on the amplitude, frequency and shape of the wave-packet, and is hence distinct from other modified dispersion relatio…
We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the Kähler case, state spaces arise as spaces of holomorphic functions on linear spaces of classical solutions in neighborhoods of hypersurfaces. Amplitudes arise as integrals of such functions over spaces of classical solutions in regions of spacetime. We prove the validity of the TQFT-type axioms of the general boundary formulation under reasonable assumptions. We also develop the notions of vacuum and cohere…
The quantization rules recently proposed by M. Navarro (and independently I.V. Kanatchikov) for a finite-dimensional formulation of quantum field theory are applied to the Klein-Gordon and the Dirac fields to obtain the quantum equations of motion of both fields. In doing so several problems arise. Solving these difficulties leads us to propose a new classical canonical formalism, which, in turn, leads us to new, improved rules of quantization. We show that the new classical equations of motion and rules of quantization overcome several known unsatisfactory features of the previous formalism. …
We explore a reduced phase space quantization of loop quantum cosmology (LQC) for a spatially flat FLRW universe filled with reference fields and an inflaton field in a Starobinsky inflationary potential. We consider three separate cases in which the reference fields are taken to be the Gaussian dust, the Brown-Kuchař dust, and massless Klein-Gordon scalar reference fields respectively. This is a "two-fluid" model in which reference fields act as global clocks providing a physical time in an inflationary spacetime, and allow bypassing various technical hurdles in conventional quantum cosmologi…
Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein-Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat-Wiles and the Langlands program is indicated. The Beilinson conjecture…
We study the Wheeler-DeWitt quantum cosmology of a spatially flat Friedmann cosmological model with a massless free scalar field. We compare the consistent histories approach with the de Broglie-Bohm theory when applied to this simple model under two different quantization schemes: the Schrödinger-like quantization, which essentially takes the square-root of the resulting Klein-Gordon equation through the restriction to positive frequencies and their associated Newton-Wigner states, or the induced Klein-Gordon quantization, that allows both positive and negative frequencies together. We show t…
A 4-dimensional Lorentzian static space-time is equivalent to 3-dimensional Euclidean gravity coupled to a massless Klein-field. By canonically quantizing the coupling model in the framework of loop quantum gravity, we obtain a quantum theory which actually describes quantized static space-times. The Kinematical Hilbert space is the product of the Hilbert space of gravity with that of imaginary scalar fields. It turns out that the Hamiltonian constraint of the 2+1 model corresponds to a densely defined operator in the underlying Hilbert space, and hence it is finite without renormalization. As…
We examine the challenge of viewing all the fields in supergravity as arising from a Kaluza-Klein like dimensional reduction of some higher-dimensional theory. This gives rise to what is known as exceptional field theory or double field theory. A particular emphasis is placed on following the Kaluza-Klein intuition leading to the identification of charged states and a reinterpretation of the central charges. We further give a description of the novel extended geometry as a generalised phase space and the relationship to string and M-theory theory and the notion of quantization
There exists a two parameter action, the variation of which produces both the geodesic equation and the geodesic deviation equation. In this paper it is shown that this action can be quantized by the canonical method, resulting in equations which generalize the Klein-Gordon equation. The resulting equations might have applications, and also show that entirely unexpected systems can be quantized. The possible applications of quantized geodesic deviation are to: i)the spreading wave packet in quantum theory, ii)and also to the one particle to many particle problem in second quantized quantum fie…
The nonlinear quantization of the domain wall (relativistic membrane of codimension 1) is considered. The membrane dust equation is considered as an analogue of the Hamilton-Jacobi equation, which allows us to construct its quantum analogue. The resulting equation has the form of a nonlinear Klein-Fock-Gordon equation. It can be interpreted as the mean field approximation for a quantum domain wall. Dispersion relations are obtained for small perturbations (in a linear approximation). The group speed of perturbations does not exceed the speed of light. For perturbations propagating along the do…
While the standard construction of the S-matrix fails on Anti-de Sitter (AdS) spacetime, a generalized S-matrix makes sense, based on the hypercylinder geometry induced by the boundary of AdS. In contrast to quantum field theory in Minkowski spacetime, there is not yet a standard way to resolve the quantization ambiguities arising in its construction. These ambiguities are conveniently encoded in the choice of a complex structure. We explore in this paper the space of complex structures for real scalar Klein-Gordon theory based on a number of criteria. These are: invariance under AdS isometrie…
We use the ideas of symplectic quantization for quantizing fields in finite volumes. We consider, as examples, the Klein-Gordon and electromagnetic fields in three dif- ferent boxes. As a second idea we consider the given boundary conditions as primary constrains. Consistency of primary constrains leads to infinite chains of constraints at the boundaries. Without solving the equation of motion, we impose the set of con- straints on suitable expansions of the fields. We show that if the new set of variables, such as Fourier modes, are chosen appropriately, imposing the constraints omits a few n…
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…
The Klein-Gordon equation, the Maxwell equation, and the Dirac equation are presented as partial difference equations in the eight-dimensional covariant discrete phase space. These equations are also furnished as difference-differential equations in the arena of discrete phase space and continuous time. The scalar field and electromagnetic fields are quantized with commutation relations. The spin-1/2 field is quantized with anti-commutation relations. Moreover, the total momentum, energy and charge of these free relativisitic quantized fields in the discrete phase space and continuous time are…
It is well-known that there exist infinitely-many inequivalent representations of the canonical (anti)-commutation relations of Quantum Field Theory (QFT). A way out, suggested by Algebraic QFT, is to instead define the quantum theory as encompassing all possible (abstract) states. In the present paper, we describe a quantization scheme for general linear (aka. free) field theories that can be seen as intermediate between traditional Fock quantization and full Algebraic QFT, in the sense that: * it provides a constructive, explicit description of the resulting space of quantum states; * it doe…
We discuss an application of the method of the angular quantization to reconstruction of form-factors of local fields in massive integrable models. The general formalism is illustrated with examples of the Klein-Gordon, sinh-Gordon and Bullough-Dodd models. For the latter two models the angular quantization approach makes it possible to obtain free field representations for form-factors of exponential operators. We discuss an intriguing relation between the free field representations and deformations of the Virasoro algebra. The deformation associated with the Bullough-Dodd models appears to b…
Usually the only difference between relativistic quantization and standard one is that the Lagrangian of the system under consideration should be Lorentz invariant. The standard approaches are logically incomplete and produce solutions with unpleasant properties: negative-energy, superluminal propagation etc. We propose a two-projections scheme of (special) relativistic quantization. The first projection defines the quantization procedure (e.g. the Berezin-Toeplitz quantization). The second projection defines a casual structure of the relativistic system (e.g. the operator of multiplication by…
We present a manifestly covariant quantization procedure based on the de Donder--Weyl Hamiltonian formulation of classical field theory. This procedure agrees with conventional canonical quantization only if the parameter space is $d=1$ dimensional time. In $d>1$ dimensions, covariant canonical quantization requires a fundamental length scale, and any bosonic field generates a spinorial wave function, leading to the emergence of spinors as a byproduct of quantization. We provide a probabilistic interpretation of the wave functions for the fields, and apply the formalism to a number of simple e…
Given any (Feynman) propagator which is Lorentz and translation invariant, it is possible to construct an action functional for a scalar field such that the quantum field theory, obtained by path integral quantization, leads to this propagator. In general, such a theory will involve derivatives of the field higher than two and can even involve derivatives of infinite order. The poles of the given propagator determine the dispersion relation for the excitations of this field. I show that it is possible to construct field theories in which the dispersion relation is the same as that of standard …
Two sets of modes of a massive free scalar field are quantized in a pair of Poincaré patches of Lorentzian anti-de Sitter (AdS) space, AdS$_{d+1}$ ($d \geq 2$). It is shown that in Poincaré coordinates $(r,t,\vec{x})$, the two boundaries at $r=\pm \infty$ are connected. When the scalar mass $m$ satisfies a condition $0 < ν=\sqrt{(d^2/4)+(m\ell)^2} <1$, there exist two sets of mode solutions to Klein-Gordon equation, with distinct fall-off behaviors at the boundary. By using the fact that the boundaries at $r=\pm \infty$ are connected, a conserved Klein-Gordon norm can be defined for these two …
A system of generalized coherent states for the de Sitter group obeying Klein-Gordon equation and corresponding to the massive spin zero particles over the de Sitter space is considered. This allows us to construct the quantized scalar field by the resolution over these coherent states; the corresponding propagator can be computed by the method of analytic continuation to the complexified de Sitter space and coincides with expressions obtained previously by other methods. We show that this propagator possess the de Sitter-invariance and causality properties.
In this paper we investigate the vanishing of cosmological singularities by quantization. Starting from a 5d Kaluza--Klein approach we quantize, as a first step, the non--spherical metric part and the dilaton field. These fields which are classically singular become smooth after quantization. In addition, we argue that the incorporation of non perturbative quantum corrections form a dilaton potential. Technically, the procedure corresponds to the quantization of 2d dilaton gravity and we discuss several models. From the 4d point of view this procedure is a semiclassical approach where only the…
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）
We study a coupled system that describes the interacting dynamics between a bulk field, confined to a finite region with timelike boundary, and a boundary observable. In our system the dynamics of the boundary observable prescribes dynamical boundary conditions for the bulk field. We cast our classical system in the form of an abstract linear Klein-Gordon equation, in an enlarged Hilbert space for the bulk field and the boundary observable. This makes it possible to apply to our coupled system the general methods of quantization. In particular, we implement the Fock quantization in full detail…
We propose an improved variant of the third-quantization scheme, for the spatially homogeneous and isotropic cosmological models in Einstein gravity coupled with a neutral massless scalar field. Our strategy is to specify a semi-Riemannian structure on the mini-superspace and to consider the quantum Klein-Gordon field on the mini-superspace. Then, the Hilbert space of this quantum system becomes inseparable, which causes the creation of infinite number of universes. To overcome this issue, we introduce a vector bundle structure on the Hilbert space and the connection of the vector bundle. Then…
Canonical quantization of gravitational systems is obstructed by the problem of time. Due to diffeomorphism symmetry the Hamiltonian vanishes: dynamics with respect to a background time parameter appears "frozen." Two strategies towards the quantization of such systems are the identification of a clock degree of freedom before quantization (deparametrization), and quantization on a kinematical Hilbert space which is subject to constraints (Dirac quantization). The usual canonical quantization in quantum field theory is analogous to deparametrization. Here we introduce a frozen formalism and Di…
A general method to construct free quantum fields for massive particles of arbitrary definite spin in a canonical Hamiltonian framework is presented. The main idea of the method is as follows: a multicomponent Klein-Gordon field that satisfies canonical (anti)commutation relations and serves as an auxiliary higher spin field is introduced, and the physical higher spin field is obtained by acting on the auxiliary field by a suitable differential operator. This allows the calculation of the (anti)commutation relations, the Green functions and the Feynman propagators of the higher spin fields. In…
QFT approaches elaborated for treating quantum effects in time-dependent external electric fields are not directly applicable to time-independent nonuniform electric fields that are given by a step potential and their generalization for the such potentials was not sufficiently developed. Such fields can also create particles from the vacuum, the Klein paradox being closely related to this process. We believe that the present work presents a consistent solution of the latter problem. Quantizing the Dirac and scalar fields with time independent backgrounds, we have found in- and out-creation and…
In a recent paper \cite{[Good1]} Good postulated new rules of quantization, one of the major features of which is that the quantum evolution of the wave function is always given by ordinary differential equations. In this paper we analyse the proposal in some detail and discuss its viability and its relationship with the standard quantum theory. As a byproduct, a simple derivation of the `mass spectrum' for the Klein-Gordon field is presented, but it is also shown that there is a complete additional spectrum of negative `masses'. Finally, two major reasons are presented against the viability o…
In curved spacetimes, the lack of criteria for the construction of a unique quantization is a fundamental problem undermining the significance of the predictions of quantum field theory. Inequivalent quantizations lead to different physics. Recently, however, some uniqueness results have been obtained for fields in non-stationary settings. In particular, for vacua that are invariant under the background symmetries, a unitary implementation of the classical evolution suffices to pick up a unique Fock quantization in the case of Klein-Gordon fields with time-dependent mass, propagating in a stat…
It is observed that the shifted Poisson structure (antibracket) on the solution complex of Klein-Gordon and linear Yang-Mills theory on globally hyperbolic Lorentzian manifolds admits retarded/advanced trivializations (analogs of retarded/advanced Green's operators). Quantization of the associated unshifted Poisson structure determines a unique (up to equivalence) homotopy algebraic quantum field theory (AQFT), i.e. a functor that assigns differential graded $\ast$-algebras of observables and fulfills homotopical analogs of the AQFT axioms. For Klein-Gordon theory the construction is equivalen…
A connection between a unitary quantization scheme and para-Fermi statistics of order 2 is considered. An appropriate extension of Green's ansatz is suggested. This extension allows one to transform bilinear and trilinear commutation relations for the annihilation and creation operators of two different para-Fermi fields $ϕ_{a}$ and $ϕ_{b}$ into identity. The way of incorporating para-Grassmann numbers $ξ_{k}$ into a general scheme of uniquantization is also offered. For parastatistics of order 2 a new fact is revealed, namely, the trilinear relations containing both the para-Grassmann variabl…
We construct a family of quantum scalar fields over a $p-$adic spacetime which satisfy $p-$adic analogues of the Gårding--Wightman axioms. Most of the axioms can be formulated the same way in both, the Archimedean and non-Archimedean frameworks; however, the axioms depending on the ordering of the background field must be reformulated, reflecting the acausality of $p-$adic spacetime. The $p-$adic scalar fields satisfy certain $p-$adic Klein-Gordon pseudo-differential equations. The second quantization of the solutions of these Klein-Gordon equations corresponds exactly to the scalar fields int…
Quantum gravity effects modify the Heisenberg's uncertainty principle to a generalized uncertainty principle (GUP). Earlier work showed that the GUP-induced corrections to the Schrödinger equation, when applied to a non-relativistic particle in a one-dimensional box, led to the quantization of length. Similarly, corrections to the Klein-Gordon and the Dirac equations, gave rise to length, area and volume quantizations. These results suggest a fundamental granular structure of space. In this work, it is investigated how spacetime curvature and gravity might influence this discreteness of space.…
We study the canonical quantization of a bosonic string in presence of N twist fields. This generalizes the quantization of the twisted string in two ways: the in and out states are not necessarily twisted and the number of twist fields N can be bigger than 2. In order to quantize the theory we need to find the normal modes. Then we need to define a product between two modes which is conserved. Because of this we need to use the Klein-Gordon product and to separate the string coordinate into the classical and the quantum part. The quantum part has different boundary conditions than the origina…
We study the Fock quantization of scalar fields in (generically) time dependent scenarios, focusing on the case in which the field propagation occurs in --either a background or effective-- spacetime with spatial sections of flat compact topology. The discussion finds important applications in cosmology, like e.g. in the description of test Klein-Gordon fields and scalar perturbations in Friedmann-Robertson-Walker spacetime in the observationally favored flat case. Two types of ambiguities in the quantization are analyzed. First, the infinite ambiguity existing in the choice of a Fock represen…
Universidad de Guanajuato（瓜纳华托大学）；Universidad Autónoma de Zacatecas（萨卡特卡斯自治大学）；Secretaría de Ciencia, Humanidades, Tecnología e Innovación（科学、人文、技术与创新秘书处）；Dual CP Institute of High Energy Physics（双重CP高能物理研究所）
We study the Fock quantization of scalar fields of Klein-Gordon type in nonstationary scenarios propagating in spacetimes with compact spatial sections, allowing for different field descriptions that are related by means of certain nonlocal linear canonical transformations that depend on time. More specifically, we consider transformations that do not mix eigenmodes of the Laplace-Beltrami operator, which are supposed to be dynamically decoupled. In addition, we assume that the canonical transformations admit an asymptotic expansion for large eigenvalues (in norm) of the Laplace-Beltrami opera…
We consider the massive relativistic particle models on fourdimensional Minkowski space extended by $N$ commuting Weyl spinors for N=1 and N=2. The N=1 model is invariant under the most general form of bosonic counterpart of simple D=4 supersymmetry, and provides after quantization the bosonic counterpart of chiral superfields, satisfying Klein--Gordon equation. In massless case these fields do satisfy the Fierz-Pauli equations. For N=2 we obtain after quantization the free massive higher spin fields for arbitrary spin satisfying linear Bargman--Wigner equations. Finally the problem of statist…
A system of generalized coherent states for the de Sitter group obeying the Klein-Gordon equation and corresponding to the massive spin zero particles over the de Sitter space is considered. This allows us to construct the quantized scalar field by the resolution over these coherent states; the corresponding propagator is computed by the method of analytic continuation to the complex de Sitter space and coincides with expressions obtained previously by other methods. Considering the case of spin 1/2 we establish the connection of the invariant Dirac equation over the de Sitter space with irred…
The relation between symmetry reduction before and after quantization of a field theory is discussed using a toy model: the axisymmetric Klein-Gordon field. We consider three possible notions of symmetry at the quantum level: invariance under the group action, and two notions derived from imposing symmetry as a system of constraints a la Dirac, reformulated as a first class system. One of the latter two turns out to be the most appropriate notion of symmetry in the sense that it satisfies a number of physical criteria, including the commutativity of quantization and symmetry reduction. Somewha…
The basic principles of the quantum mechanics in the K-field formalism are stated in the paper. The basic distinction of this theory arises from that the quantum theory equations (including well-known Schrodinger, Klein-Gordon and quadratic Dirac equations) are obtained from de Broglie postulate geometric generalization. Rather, they are obtained as the free wave equations on a manifold metrizing force interactions of particles. Such view on the quantum theory basic equations allows one to use semiclassical models for the quantum system simulation. The quantization principle modifies as well. …
