So far, it is not well known how to deal with dissipative systems. There are many paths of investigation in the literature and none of them present a systematic and general procedure to tackle the problem. On the other hand, it is well known that the fractional formalism is a powerful alternative when treating dissipative problems. In this paper we propose a detailed way of attacking the issue using fractional calculus to construct an extension of the Dirac brackets in order to carry out the quantization of nonconservative theories through the standard canonical way. We believe that using the extended Dirac bracket definition it will be possible to analyze more deeply gauge theories starting with second-class systems.
It is shown that a Dirac bracket algebra is isomorphic to the original Poisson bracket algebra of first class functions subject to first class constraints. The isomorphic image of the Dirac bracket algebra in the star-product commutator algebra is found.
In this article the methods of canonical analysis and quantization that were reviewed in the first part of the series are applied to the case of the Dirac field in the presence of electromagnetic interaction. It is shown that the quantization of electrodynamics, which begins with a given Lagrangian and ends by perturbative calculation of scattering probability amplitudes, can be performed in the way that does not employ Poincaré symmetry of space--time at any stage. Also, the causal structure is not needed.
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…
We study several classes of non-associative algebras as possible candidates for deformation quantization in the direction of a Poisson bracket that does not satisfy Jacobi identities. We show that in fact alternative deformation quantization algebras require the Jacobi identities on the Poisson bracket and, under very general assumptions, are associative. At the same time, flexible deformation quantization algebras exist for any Poisson bracket.
The derivation of the brackets among coordinates and momenta for classical constrained systems is a necessary step toward their quantization. Here we present a new approach for the determination of the classical brackets which does neither require Dirac's formalism nor the symplectic method of Faddeev and Jackiw. This approach is based on the computation of the brackets between the constants of integration of the exact solutions of the equations of motion. From them all brackets of the dynamical variables of the system can be deduced in a straightforward way.
We present an explicit matrix algebra quantization of the algebra of volume-preserving diffeomorphisms of the $n$-torus. That is, we approximate the corresponding classical Nambu brackets using $\mathfrak{sl}(N^{\lceil\frac{n}{2}\rceil},\mathbb{C})$-matrices equipped with the finite bracket given by the completely anti-symmetrized matrix product, such that the classical brackets are retrieved in the $N\rightarrow \infty$ limit. We then apply this approximation to the super $4$-brane in $9$ dimensions and give a regularized action in analogy with the matrix quantization of the supermembrane. Th…
The Nambu Bracket quantization of the Hydrogen atom is worked out as an illustration of the general method. The dynamics of topological open branes is controlled classically by Nambu Brackets. Such branes then may be quantized through the consistent quantization of the underlying Nambu brackets: properly defined, the Quantum Nambu Brackets comprise an associative structure, although the naive derivation property is mooted through operator entwinement. For superintegrable systems, such as the Hydrogen atom, the results coincide with those furnished by Hamiltonian quantization--but the method is…
We present several non-trivial examples of the three-dimensional quantum Nambu bracket which involve square matrices or three-index objects. Our examples satisfy two fundamental properties of the classical Nambu bracket: they are skew-symmetric and they obey the Fundamental Identity. We contrast our approach to the existing literature on the quantum deformations of Nambu mechanics. We also discuss possible applications of our results in M-theory.
Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single valuedness of this eigenfunctio…
Starting with the well-defined product of quantum fields at two spacetime points, we explore an associated Poisson structure for classical field theories within the deformation quantization formalism. We realize that the induced star-product is naturally related to the standard Moyal product through an appropriate causal Green's functions connecting points in the space of classical solutions to the equations of motion. Our results resemble the Peierls-DeWitt bracket analyzed in the multisymplectic context. Once our star-product is defined we are able to apply the Wigner-Weyl map in order to in…
The principle of invariance of the c-number symmetric bracket is used to derive both the quantum operator commutator relation $[\hat q, \hat p]=i\hbar$ and the time-dependent Schrödinger equation. A c-number dynamical equation is found which leads to the second quantized field theory of bosons and fermions.
A detailed Hamiltonian analysis for a five-dimensional St{ü}eckelberg theory with a compact dimension is performed. First, we develop a pure Dirac's analysis of the theory, we show that after performing the compactification, the theory is reduced to four-dimensional St{ü}eckelberg theory plus a tower of Kaluza-Klein modes. We develop a complete analysis of the constraints, we fix the gauge and we show that there are present pseudo-Goldstone bosons. Then we quantize the theory by constructing the Dirac brackets. As complementary work, we perform the Faddeev-Jackiw quantization for the theory un…
We discuss deformation quantization of the covariant, light-cone and conformal gauge-fixed p-brane actions (p>1) which are closely related to the structure of the classical and quantum Nambu brackets. It is known that deformation quantization of the Nambu bracket is not of the usual Moyal type. Yet the Nambu bracket can be quantized using the Zariski deformation quantization (discovered by Dito, Flato, Sternheimer and Takhtajan) which is based on factorization of polynomials in several real variables. We discuss a particular application of the Zariski deformed quantization in M-theory by consi…
A convenient formalism is developed to treat classical dynamical systems involving $(p=2)$ parafermionic and parabosonic dynamical variables. This is achieved via the introduction of a parabracket which summarizes the paracommutation relations of the corresponding Green components in a unified manner. Furthermore, it is shown that Peierls quantization scheme may be applied to such systems provided that one uses the above mentioned parabracket to express the quantum paracommutation relations. Application of the Peierls scheme also provides the form of the parafermionic and parabosonic kinetic t…
Phase space is a framework ideally suited for quantizing superintegrable systems through the use of deformation methods, as illustrated here by applications to de Sitter and chiral particles. Within this framework, Nambu brackets elegantly incorporate the additional quantum invariants of such models. New results are presented for the non-Abelian quantization of these brackets.
Phase Space is the framework best suited for quantizing superintegrable systems, naturally preserving the symmetry algebras of the respective hamiltonian invariants. The power and simplicity of the method is fully illustrated through new applications to nonlinear sigma models, specifically for de Sitter N-spheres and Chiral Models, where the symmetric quantum hamiltonians amount to compact and elegant expressions. Additional power and elegance is provided by the use of Nambu Brackets to incorporate the extra invariants of superintegrable models. Some new classical results are given for these b…
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…
The general procedure of constructing a consistent covariant Dirac-type bracket for models with mixed first and second class constraints is presented. The proposed scheme essentially relies upon explicit separation of the initial constraints into infinitely reducible first and second class ones (by making use of some appropriately constructed covariant projectors). Reducibility of the second class constraints involved manifests itself in weakening some properties of the bracket as compared to the standard Dirac one. In particular, a commutation of any quantity with the second class constraints…
We consider a class of \textit{factorizable} Poisson brackets which includes almost all reasonable Poisson structures. A particular case of the factorizable brackets are those associated with symplectic Lie algebroids. The BRST theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case. This can be viewed as an extension of the Fedosov deformation quantization to a wide class of \textit{irregular} Poisson structures. In a more general case, the factorizable Poisson brackets are shown to be closely conn…
The main goal of this work is to perform a nonolonomic deformation (Fedosov type) quantization of fractional Lagrange geometries. The constructions are provided for a (fractional) almost Kahler model encoding equivalently all data for fractional Euler-Lagrange equations with Caputo fractional derivative. For homogeneous generating Finsler functions, the geometric models contain quantum versions of fractional Finsler spaces. The scheme can be generalized for fractional Hamilton systems and various models of fractional classical and quantum gravity. We conclude that the approach with Caputo frac…
We perform a detailed comparison of the Dirac composite fermion and the recently proposed bimetric theory for a quantum Hall Jain states near half filling. By tuning the composite Fermi liquid to the vicinity of a nematic phase transition, we find that the two theories are equivalent to each other. We verify that the single mode approximation for the response functions and the static structure factor becomes reliable near the phase transition. We show that the dispersion relation of the nematic mode near the phase transition can be obtained from the Dirac brackets between the components of the…
Phase Space is the framework best suited for quantizing superintegrable systems--systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally. We illustrate the power and simplicity of the method through new applications to nonlinear sigma-models, specifically for Chiral Models and de Sitter N-spheres, where the symmetric quantum hamiltonians amount to compact and elegant expressions, in accord with the Groenewold-van Hove theorem. Additional power and elegance is provided by the us…
A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang-Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane.
Free noncommutative fields constitute a natural and interesting example of constrained theories with higher derivatives. The quantization methods involving constraints in the higher derivative formalism can be nicely applied to these systems. We study real and complex free noncommutative scalar fields where momenta have an infinite number of terms. We show that these expressions can be summed in a closed way and lead to a set of Dirac brackets which matches the usual corresponding brackets of the commutative case.
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.
In the space-of-histories approach to gauge fields and their quantization, the Maxwell, Yang--Mills and gravitational field are well known to share the property of being type-I theories, i.e. Lie brackets of the vector fields which leave the action functional invariant are linear combinations of such vector fields, with coefficients of linear combination given by structure constants. The corresponding gauge-field operator in the functional integral for the in-out amplitude is an invertible second-order differential operator. For such an operator, we consider advanced and retarded Green functio…
In the paper "Constraint Quantization of Open String in Background $B$ field and Noncommutative D-brane", it is claimed that the boundary conditions lead to an infinite set of secondary constraints and Dirac brackets result in a non-commutative Poisson structure for D-brain. Here we show that contrary to the arguments in that paper, the set of secondary constraints on the boundary is finite and the non-commutativity algebra can not be obtained by evaluating the Dirac brackets.
We give a simple geometrical picture of the basic structures of the covariant $Sp(2)$ symmetric quantization formalism -- triplectic quantization -- recently suggested by Batalin, Marnelius and Semikhatov. In particular, we show that the appearance of an even Poisson bracket is not a particular property of triplectic quantization. Rather, any solution of the classical master equation generates on a Lagrangian surface of the antibracket an even Poisson bracket. Also other features of triplectic quantization can be identified with aspects of conventional Lagrangian BRST quantization without exte…
We deduce the canonical brackets for a two (1 + 1)-dimensional (2D) free Abelian 1-form gauge theory by exploiting the beauty and strength of the continuous symmetries of a Becchi-Rouet-Stora-Tyutin (BRST) invariant Lagrangian density that respects, in totality, six continuous symmetries. These symmetries entail upon this model to become a field theoretic example of Hodge theory. Taken together, these symmetries enforce the existence of exactly the same canonical brackets amongst the creation and annihilation operators that are found to exist within the standard canonical quantization scheme. …
Deformation quantization is a formal deformation of the algebra of smooth functions on some manifold. In the classical setting, the Poisson bracket serves as an initial conditions, while the associativity allows to proceed to higher orders. Some applications to string theory require deformation in the direction of a quasi-Poisson bracket (that does not satisfy the Jacobi identity). This initial condition is incompatible with associativity, it is quite unclear which restrictions can be imposed on the deformation. We show that for any quasi-Poisson bracket the deformation quantization exists and…
In this paper we consider a connection between the Moyal formuration of quantum mechanics and noncommutative lattices on nonlocal spaces. The Moyal bracket generates SU(N) Lie algebras,and representations of these algebras correspond to Bratteli diagrams on Hilbert spaces of C*-algebras, and also representations of noncommutative lattices on nonlocal spaces correspond to the same diagrams. We will show that the quantization thus generates nonlocal spaces by the intermediation of Bratteli diagrams.The above connection are investigated in details in two definite cases, one is one-dimensional lat…
We show that the external algebra $\cal M$ on $GL(N)$ can be equipped with the graded Poisson brackets compatible with the group action. We prove that there are only two graded Poisson-Lie structures (brackets) on $\cal M$ and we obtain their explicit description. We realize that just these two structures appear as the quasiclassical limit of the bicovariant differential calculi on the quantum linear group $GL_q (N)$.
Phase Space is the framework best suited for quantizing superintegrable systems--systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally, as illustrated on nonlinear $σ$-models, specifically for Chiral Models and de Sitter $N$-spheres. Classically, the dynamics of superintegrable models such as these is automatically also described by Nambu Brackets involving the extra symmetry invariants of them. The phase-space quantization worked out then leads to the quantization of the cor…
We use the light front ``machinery'' to study the behavior of a relativistic free particle and obtain the quantum commutation relations from the classical Poisson brackets. We argue that their usual projection onto the light-front coordinates from the covariant commutation relations show that there is an inconsistency in the expected correlation between canonically conjugate variables ``time'' and ``energy''. Moreover we show that this incompatibility originates from the very definition of the Poisson brackets that is employed and present a simple remedy to this problem and envisages a profoun…
We discuss a manifestly covariant way of arriving at the quantization rules based on causality, with no reference to Poisson or Peierls brackets of any kind.
Adopting a particular approach to fractional calculus, this paper sets out to build up a consistent extension of the Faddeev-Jackiw (or Symplectic) algorithm to carry out the quantization procedure of coarse-grained models in the standard canonical way. In our treatment, we shall work with the Modified Riemman Liouville (MRL) approach for fractional derivatives, where the chain rule is as efficient as it is in the standard differential calculus. We still present a case where we consider the situation of charged particles moving on a plane with velocity $\dot{r}$, subject to an external and int…
We use the extension problem proposed by Caffarelli and Silvestre to study the quantization of a scalar nonlocal quantum field theory built out of the fractional Laplacian. We show that the quantum behavior of such a nonlocal field theory in $d$-dimensions can be described in terms of a local action in $d+1$ dimensions which can be quantized using the canonical operator formalism though giving up local commutativity. In particular, we discuss how to obtain the two-point correlation functions and the vacuum energy density of the nonlocal fractional theory as a brane limit of the bulk correlator…
We introduce a general theory of twisting algebraic structures based on actions of a bialgebra. These twists are closely related to algebraic deformations and also to the theory of quasi-triangular bialgebras. In particular, a deformation produced from a universal deformation formula (UDF) is a special case of a twist. The most familiar example of a deformation produced from a UDF is perhaps the "Moyal product" which (locally) is the canonical quantization of the algebra of functions on a symplectic manifold in the direction of the Poisson bracket. In this case, the derivations comprising the …
A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller than the number of velocities) is proposed. The equations of motion become first-order differential equations, and they coincide with those of multi-time dynamics, if a certain condition is imposed. In a singular theory, this condition is fulfilled in the case of the coincidence of the number of generalized momenta with the rank of the Hessian matrix. The non…
The quantization of two-dimensional Hořava theory of gravity without the projectability condition is considered. Our study of the Hamiltonian structure of the theory shows that there are two first-class and two second-class constraints. Then, following Dirac we quantize the theory by first requiring that the two second-class constraints be strongly equal to zero. This is carried out by replacing the Poisson bracket by the Dirac bracket. The two first-class constraints give rise to the Wheeler-DeWitt equations, which yield uniquely a plane-wave solution for the wavefunction. We also study the c…
We study the quantum theory of two-dimensional electrons in a magnetic field and an electric field generated by a homogeneous background. The dynamics separates into a microscopic and macroscopic mode. The latter is a circular Hall current which is described by a chiral quantum field theory. It is shown how in this second quantized picture a Laughlin-type wave function emerges.
The Peierls bracket quantization scheme is applied to the supersymmetric system corresponding to the twisted spin index theorem. A detailed study of the quantum system is presented, and the Feynman propagator is exactly computed. The Green's function methods provide a direct derivation of the index formula. Note: This is essentially a new SUSY proof of the index theorem.
The Zariski quantization is one of the strong candidates for a quantization of the Nambu-Poisson bracket. In this paper, we apply the Zariski quantization for first quantized field theories, such as superstring and supermembrane theories, and clarify physical meaning of the Zariski quantization. The first quantized field theories need not to possess the Nambu-Poisson structure. First, we construct a natural metric for the spaces on which Zariski product acts in order to apply the Zariski quantization for field theories. This metric is invariant under a gauge transformation generated by the Zar…
We apply the Dirac bracket quantization to open strings attached to branes in the presence of background antisymmetric field and recover an inherent noncommutativity in the internal coordinates of the brane.
We study the Moyal quantization for the constrained system. One of the purposes is to give a proper definition of the Wigner-Weyl(WW) correspondence, which connects the Weyl symbols with the corresponding quantum operators. A Hamiltonian in terms of the Weyl symbols becomes different from the classical Hamiltonian for the constrained system, which is related to the fact that the naively constructed WW correspondence is not one-to-one any more. In the Moyal quantization a geometrical meaning of the constraints is clear. In our proposal, the 2nd class constraints are incorporated into the defini…
In the BRST quantization of gauge theories, the zero locus $Z_Q$ of the BRST differential $Q$ carries an (anti)bracket whose parity is opposite to that of the fundamental bracket. We show that the on-shell BFV/BV gauge symmetries are in a 1:1 correspondence with Hamiltonian vector fields on $Z_Q$, and observables of the BRST theory are in a 1:1 correspondence with characteristic functions of the bracket on $Z_Q$. By reduction to the zero locus, we obtain relations between bracket operations and differentials arising in different complexes (the Gerstenhaber, Schouten, Berezin-Kirillov, and Skly…
A generalization of canonical quantization which maps a dynamical operator to a dynamical superoperator is suggested. Weyl quantization of dynamical operator, which cannot be represented as Poisson bracket with some function, is considered. The usual Weyl quantization of observables is a specific case of suggested quantization. This approach allows to define consistent quantization procedure for non-Hamiltonian and dissipative systems. Examples of the harmonic oscillator with friction (generalized Lorenz-Rossler-Leipnik-Newton equation), the Fokker-Planck-type system and Lorenz-type system are…
The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the ``failure" of the Ehrenfest theorem is clarified in terms of the already defined notion of {\it good} (and {\it bad}) operators. The analysis of ``constrained" Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric Quantization. This study is illustrated with the examples of the free particle on the circumference and …
We use Dirac quantization of flux to study fractional charges and axion angles θin interacting topological insulators with gapless surface modes protected by time-reversal symmetry. In interacting topological insulators, there are two types of fractional axion angle due to conventional odd and nontrivial even flux quantization at the boundary. On even flux quantization in a gapped time reversal invariant system, we show that there is a halved quarter fractional quantum Hall effect on the surface with Hall conductance of p/4q e2/2h with p and q odd integers. The gapless surface modes can be cha…
For redundant second-class constraints the Dirac brackets cannot be defined and new brackets must be introduced. We prove here that the Jacobi identity for the new brackets must hold on the surface of the second-class constraints. In order to illustrate our proof we work out explicitly the cases of a fractional spin particle in 2+1 dimensions and the original Brink-Schwarz massless superparticle in D=10 dimensions in a Lorentz covariant constraints separation.
We apply newly improved Batalin-Fradkin-Tyutin Hamiltonian method to the chiral Schwinger Model in the case of the regularization ambiguity $a>1$. We show that one can systematically construct the first class constraints by the BFT Hamiltonian method, and also show that the well-known Dirac brackets of the original phase space variables are exactly the Poisson brackets of the corresponding modified fields in the extended phase space. Furthermore, we show that the first class Hamiltonian is simply obtained by replacing the original fields in the canonical Hamiltonian by these modified fields. P…
We study the presence of symmetry transformations in the Faddeev-Jackiw approach for constrained systems. Our analysis is based in the case of a particle submitted to a particular potential which depends on an arbitrary function. The method is implemented in a natural way and symmetry generators are identified. These symmetries permit us to obtain the absent elements of the sympletic matrix which complement the set of Dirac brackets of such a theory. The study developed here is applied in two different dual models. First, we discuss the case of a two-dimensional oscillator interacting with an …
Orientifold $p$-planes with $p\le4$ have fractional D$p$-charges, and therefore appear inconsistent with Dirac quantization with respect to D$(6-p)$-branes. We explain in detail how this issue is resolved by taking into account the anomaly of the worldvolume fermions using the $η$ invariants. We also point out relationships to the classification of interacting fermionic symmetry protected topological phases. In an appendix, we point out that the duality group of type IIB string theory is the pin+ version of the double cover of $GL(2,\mathbb{Z})$.
We study a quantization via fractional derivative of a nonminimal derivative coupling cosmological theory, namely, the Fab Four John theory. Its Hamiltonian version presents the issue of fractional powers in the momenta. That problem is solved here by the application of the so-called conformable fractional derivative. This leads to a Wheeler-DeWitt equation of second order, showing that a Bohm-de Broglie interpretation can be constructed. That combination of fractional quantization and Bohmian interpretation provides us a new quantization method, in which the quantum potential is the criterion…
Quantum Groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at $SU(2)$ and $SU(1,1)$, as submanifolds of a 4--dimensional phase space with constraints, and deal with two classes of problems. In the first set of examples we consider some hamiltonian systems associated with Lie-Poisson structures and we investigate the equations of the motion. In the second set of examples …
The dynamics of even topological open membranes relies on Nambu Brackets. Consequently, such 2p-branes can be quantized through the consistent quantization of the underlying Nambu dynamical structures. This is a summary construction relying on the methods detailed in refs hep-th/0205063 and hep-th/0212267.
Using Faddeev-Senjanovic path integral quantization for constrained Hamilton system, we quantize SU(n) N=2 supersymmetric gauge field system with non-abelian Chern-Simons topological term in 2+1 dimensions, and use consistency of a gauge condition naturally to deduce another gauge condition. Further, we get the generating functional of Green function in phase space, deduce the angular momentum based on the global canonical Noether theorem at quantum level, obtain the fractional spin of this supersymmetric system, and show that the total angular momentum has the orbital angular momentum and spi…
