On the other hand, string theory uses also the deformation. In this article we discuss and compare three approaches. Deformation quantization in quantum mechanics and quantum field theory. There is a vast array of further quantum field theory texts, many of them with redeeming features. Deformation quantization and quantum field theory on. Quantum theory, deformation and integrability, volume 186.
Hence the existence of deformation quantization does not guarantee a physically acceptable deformed geometric background. Deformation quantization, which gives a development of quantum mechanics independent of the operator algebra formulation, and quantum groups, which arose from the inverse scattering method and a study of yangbaxter equations, share a common idea abstracted earlier in algebraic deformation theory. Abm also provides an introductory account of the subject. Deformation quantization of fermi fields sciencedirect. Quantum field theory qft is the mathematical and conceptual framework for contemporary elementary particle physics. In the usual approach to quantum mechanics, we reinterpret the classical. Geometrical approaches to the quantization of gauge theories bucker, beatrice, 2004. We discuss the deformation quantization approach for the teaching of quantum mechanics. Pdf perturbative algebraic field theory, and deformation.
Introduction quantization is, most broadly, the process of forming a quantum mechanical system starting from a classical mechanical one. Quantization converts classical fields into operators acting on quantum states of the field theory. Quantum theory, deformation and integrability by r. The algebraic approach to field theory local quantum physics 22 has deepened and enlarged our understanding of.
We propose a new formula for the star product in deformation quantization of poisson structures related in a specific way to a variational problem for a function s, interpreted as the action functional. Algebraic quantum field theory is an approach to relativistic quantum physics, notably the theory of elementary particles, which complements other modern developments in this field. Causal poisson bracket via deformation quantization. Despite its name, deformation quantization does not constitute a successful quantization scheme, namely a method to produce a quantum theory out of a classical one. Pdf quantum groupoids and deformation quantization. This approach has certain conceptual advantages that make its consideration worthwhile. Deformation quantization focuses on the algebras of observables of a physical system hence on the heisenberg picture. Also, we discuss some of the conceptual advantages of dq and how dq may be related to algebraic quantum field theory. Berezin quantization, deformation quantization, poisson manifolds. In the monograph a new approach to deformation quantization on a symplectic manifold is developed. A brief introduction to enriques surfaces dolgachev, igor v. The dissipative scalar field theory by means of the deformation quantization formalism is studied. The reason for quantizing a theory is to deduce properties of materials, objects or particles through the computation of quantum amplitudes, which may be very complicated.
We demonstrate how it can be used to solve specific problems and clarify its relation to conventional quantization and path. What is called strict or algebraic deformation quantization is an attempt to formalize quantization of phase spaces or more generally of poisson manifolds by continuously deforming, in a precise sense, their commutative algebras of functions algebras of observables to noncommutative calgebras whose commutators are. The quantum matrix bialgebra m,2 and quantum plane ki are constructed as preferred deformations of the classical matrix bialgebra and plane, that is. Annals 4f p14ysics 119, 111151 1978 deformation theory and quantization.
A summary is given here of some of the key developments over the past thirty years in the context of physics, from quantum mechanics to quantum field theory. Notes on deformation quantization contents deformation theory. Hirshfeld fachbereich physik, universitat dortmund 44221 dortmund, germany abstract. As a successful theoretical framework today, quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Perturbative quantum field theory a quantum field theory describes the interactions between fundamental particles. Moreover, as has been shown in deformation quantization theory can be applied to quantize the classical bosonic strings in the lightcone gauge. Under favorable circumstances, one can form from a strict c c\astalgebraic deformation quantization given by a continuous field of calgebras over a subset of the interval the differentiation as. Our approach is motivated by perturbative algebraic quantum field theory paqft. One of the ingredients of paqft is formal deformation quantizationthe con. Strict deformation quantization and noncommutative.
We provide a direct combinatorial formula for the star product, and we show that it can be applied to a. In a rather informal sense qft is the extension of quantum mechanics qm, dealing with particles, over to fields, i. We show that the deformation quantization of noncommutative quantum mechanics previously considered by dias and prata weylwigner formulation of noncommutative quantum mechanics, j. It is worth while to note that also perturbation quantum field theory can be given the form of deformation quantization formalism. This approach gives rise to an important invariant, the socalled weyl curvature, which is a formal deformation of the symplectic form. Solvay workshop on higher spin gauge theories, topological. Deformation quantization was mainly applied to quantize classical mechanics. Quantum field theory is a perturbation to quantum mechanics, where you perform a second quantization. Geometry, integrability and quantization june 615, 2002, varna, bulgaria ivailo m. The fuzzy sphere is included as a special case parametrized by. In particular, any regular triangular lie bialgebroid is shown quantizable. Pdf sinesquare deformation and mobius quantization of 2d.
Basics of deformation quantization basic concepts gerstenhaber theory brief panorama of results 2. This formulation of quantum mechanics is useful in. Perturbative algebraic field theory, and deformation quantization. To every graph g in the theory, one associates an amplitude g i gq i. A 352, 309 2006 for quantum mechanics, a star product which contains the dissipative effect for the damped oscillation modes of the field is constructed. We then find that the continuous virasoro algebra of the dipolar quantization can be. The lowest energy state is called the vacuum state. Sinesquare deformation and mobius quantization of 2d. Modern approaches to the quantization of gauge theories bucker, beatrice, 2003.
The following chapters are dedicated to quantum electrodynamics and quantum chromodynamics, followed by the renormalization theory. How ever, it seems to be very interesting to formulate quantum field theory within. Abstract the purpose of this note is to unify quantum groups and starproducts under a general umbrella. In this part, in the rst three chapters i write about scalar elds, elds with spin, and nonabelian elds. Starting with the welldefined product of quantum fields at two spacetime points, we explore an associated poisson structure for classical field theories within the deformation quantization formalism. We end by a short survey of two main avatars of deformation quantization, quantum groups. We provide a direct combinatorial formula for the star product, and we show that it can be applied to a certain. Topological field theory and deformation quantization brussels, february 17 21, 2020 monday 17 february 10. Deformation quantization is one formalization of the general idea of quantization of a classical mechanical systemclassical field theory to a quantum mechanical system quantum field theory. The aim of this paper is to give a basic overview of deformation quantization dq to physicists.
The second part is dedicated to topological field theories. Intuitively classical mechanics is the limit of quantum mechanics when. We demonstrate how it can be used to solve specific problems and clarify its relation to conventional quantization and. When the twocohomology class is not integer valued, the scalar quantum field theory based on the deformation algebra is not unitary. Moyal quantization hamiltonian mechanics moyal star product relation with quantum mechanics the exemple of the harmonic oscillator 2. In section 2 we deal with the second quantization as a deformation quantization of the schrodinger. Chapter 5 involves the whitham equations in various roles mediating between qm and classical behavior. Quantum field theory stanford encyclopedia of philosophy. Deformation quantization of geometric quantum mechanics.
We study the scalar quantum field theory on a generic noncommutative twosphere as a special case of noncommutative curved space, which is described by the deformation quantization algebra obtained from symplectic reduction and parametrized by h 2 s 2,r. It is shown that a quantum groupoid naturally gives rise to a lie bialgebroid as a classical limit. Interactions are represented by feynman graphs, built out of certain types of edges and vertices. Deformation quantization in the teaching of quantum mechanics. Chapter 3 is on quantum integrable systems, quantum groups, and modern deformation quantization. In particular, it sheds new light on the relation between classical and quantum mechanics. A formal deformation of ais an associative kjkalgebra structure on ajk such that aajk as algebras. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory quantum electrodynamics. Weinberg, the quantum theory of fields, vol 1 this is the rst in a three volume series by one of the masters of quantum eld theory. Strict deformation quantization and noncommutative quantum field theories gandalf lechner summer school 2015 there exist several theoretical frameworks attempting to unify quantum physics with gravity, such as string theory, loop quantum gravity, and quantum. Quantization of tensor representations and deformation of. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Fedosov quantization and perturbative quantum field theory. A very short presentation of deformation quantization, some of its. The foundations were laid in 8,1721, and further results concerning fermionic. In the formulation of quantum theory we will pursue, the propagator as probability amplitude for the time evolution of a particle from the point qt to the point q. Abstract let x be our phase space, which we assume to be a possibly in nitedimensional symplectic vector space admitting a unitary structure. Deformation quantization in quantum mechanics and quantum field theory hirshfeld, allen, 2003. In particular, connections to seibergwitten theory arising in n 2 supersymmetric susy yangmills ym theory are discussed and we would. Thought of in terms of physics this describes a quantization of a system of quantum mechanics, as opposed to full quantum field theory. Apr 12, 2002 we discuss the deformation quantization approach for the teaching of quantum mechanics. Deformation quantization and quantum field theory on curved. We also discuss its recent applications in relativistic quantum field theory. A perturbative formulation of algebraic field theory is presented, both for the classical and for the quantum case, and it is shown that the relation between.356 1233 608 1052 1307 1440 1118 766 203 969 1007 732 844 111 1170 977 428 88 1099 1373 627 1131 61 1318 433 737 173 305 531 98 1057