Braided Field Quantization from Quantum Poincare Covariance
Jerzy Lukierski, Mariusz Woronowicz
备注：
14 pages; based on a talk presented at the meeting devoted to the Scientific and Human Legacy of Julius Wess, Serbia, August 2011, published also in electronic form in IJMP Conf. Series vol. 13 (2012) 127-140
We demonstrate that the covariance of the algebra of quantum NC fields under quantum-deformed Poincare symmetries implies the appearence of braided algebra of fields and the notion of braided locality in NC QFT. We briefly recall the historical development of NC QFT which was firstly formulated in the framework using classical relativistic symmetries but further it was described as generated by the quantum-deformed symmetries. We argue that consistent covariant quantum-deformed formalism requires "braiding all the way", in particular braided commutator of deformed field oscillators as well as the braid between the field oscillators and noncommutative Fourier exponentials. As example of braided quantum-deformed NC QFT we describe the NC scalar free fields on noncommutative canonical (Moyal-Weyl) space-time with braided c-number field commutator which implies braided locality.
We discuss the obstruction to the construction of a multiparticle field theory on a $κ$-Minkowski noncommutative spacetime: the existence of multilocal functions which respect the deformed symmetries of the problem. This construction is only possible for a light-like version of the commutation relations, if one requires invariance of the tensor product algebra under the coaction of the $κ$-Poincaré group. This necessitates a braided tensor product. We study the representations of this product, and prove that $κ$-Poincaré-invariant N-point functions belong to an Abelian subalgebra, and are ther…
We introduce the free quantum noncommutative fields as described by braided tensor products. The multiplication of such fields is decomposed into three operations, describing the multiplication in the algebra M of functions on noncommutative space-time, the product in the algebra H of deformed field oscillators, and the braiding by factor Psi_{M,H} between algebras M and H. For noncommutativity generated by the twist factor we shall employ the star-product realizations of the algebra M in terms of functions on standard Minkowski space. The covariance of single noncommutative quantum fields und…
The coadjoint orbit method is applied to the construction of Hamiltonian dynamics of massless particles of arbitrary helicity. The unusual transformation properties of canonical variables are interpreted in terms of nonlinear realizations of Poincare group. The action principle is formulated in terms of new space-time variables with standard transformation properties.
A fully Poincare' covariant model is constructed out of the k-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincare' group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincare' covariance is realised a` la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of "Poincare covariance".
We present the action and transformation rules of Poincare supergravity coupled to chiral multiplets $(z^α, χ^α, h^α)$ with off-shell auxiliary fields. Starting from the geometric formulation of the superconformal theory with auxiliary fields, we derive the Poincare counterpart by gauge-fixing the Weyl and chiral symmetry and S-supersymmetry. We show how this transition is facilitated by retaining explicit target-space covariance. Our results form a convenient starting point to study models with constrained superfields, including general matter-coupled de Sitter supergravity.
A spinless covariant field $ϕ$ on Minkowski spacetime $\M^{d+1}$ obeys the relation $U(a,Λ)ϕ(x)U(a,Λ)^{-1}=ϕ(Λx+a)$ where $(a,Λ)$ is an element of the Poincaré group $\Pg$ and $U:(a,Λ)\to U(a,Λ)$ is its unitary representation on quantum vector states. It expresses the fact that Poincaré transformations are being unitary implemented. It has a classical analogy where field covariance shows that Poincaré transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covaria…
We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the …
In higher dimensional quantum field theory, irreducible representations of the Poincare group are associated with particles. Their counterpart in two-dimensional massless models are "waves" introduced by Buchholz. In this paper we show that waves do not interact in two-dimensional Moebius covariant theories and in- and out-asymptotic fields coincide. We identify the set of the collision states of waves with the subspace generated by the chiral components of the Moebius covariant net from the vacuum. It is also shown that Bisognano-Wichmann property, dilation covariance and asymptotic completen…
We extend the notion of general coordinate invariance to many-body, not necessarily relativistic, systems. As an application, we investigate nonrelativistic general covariance in Galilei-invariant systems. The peculiar transformation rules for the background metric and gauge fields, first introduced by Son and Wingate in 2005 and refined in subsequent works, follow naturally from our framework. Our approach makes it clear that Galilei or Poincare symmetry is by no means a necessary prerequisite for making the theory invariant under coordinate diffeomorphisms. General covariance merely expresse…
We explore the quantization of a $(1+1)$-dimensional inhomogeneous scalar field theory in which Poincaré symmetry is explicitly broken. We show the `classical equivalence' between a scalar field theory on curved spacetime background and its corresponding inhomogeneous scalar field theory. This implies that a hidden connection may exist among some inhomogeneous field theories, which corresponds to general covariance in field theory on curved spacetime. Based on the classical equivalence, we propose how to quantize a specific field theory with broken Poincaré symmetry inspired by standard field …
Deformations of quantum field theories which preserve Poincaré covariance and localization in wedges are a novel tool in the analysis and construction of model theories. Here a general scenario for such deformations is discussed, and an infinite class of explicit examples is constructed on the Borchers-Uhlmann algebra underlying Wightman quantum field theory. These deformations exist independently of the space-time dimension, and contain the recently studied warped convolution deformation as a special case. In the special case of two-dimensional Minkowski space, they can be used to deform free…
A universal C*-algebra of the electromagnetic field is constructed. It is represented in any quantum field theory which incorporates electromagnetism and expresses basic features of this field such as Maxwell's equations, Poincaré covariance and Einstein causality. Moreover, topological properties of the field resulting from Maxwell's equations are encoded in the algebra, leading to commutation relations with values in its center. The representation theory of the algebra is discussed with focus on vacuum representations, fixing the dynamics of the field.
Gravity theory based on current algebra is formulated. The gauge principle rather than the general covariance combined with the equivalence principle plays the pivotal role in the formalism, and the latter principles are derived as a consequence of the theory. In this approach, it turns out that gauging the Poincaré algebra is not appropriate but gauging the $SO(N,M)$ algebra gives a consistent theory. This makes it possible to have Anti-de Sitter and de Sitter space-time by adopting a relation between the spin connection and the tetrad field. The Einstein equation is a part of our basic equat…
The scattering equivalence of quantum field theories formulated with light-front and instant-form kinematic subgroups is established using non-perturbative methods. The difficulty with field theoretic formulations of Dirac's forms of dynamics is that the free and interacting unitary representations of the Poincaré group are defined on inequivalent representations of the Hilbert space, which means that the concept of kinematic transformations must be modified on the Hilbert space of the field theory. This work addresses this problem by assuming the existence of a field theory with the expected …
Under the hypotheses of analyticity, locality, Lorentz covariance, and Poincare invariance of the deformations, combined with the requirement that the interaction vertices contain at most two space-time derivatives of the fields, we investigate the consistent cross-couplings between two collections of tensor fields with the mixed symmetries of the type (3,1) and (2,2). The computations are done with the help of the deformation theory based on a cohomological approach in the context of the antifield-BRST formalism. Our results can be synthesized in: 1. there appear consistent cross-couplings be…
A noncommutative space-time admitting dilation symmetry was briefly mentioned in the seminal work of Doplicher, Fredenhagen and Roberts. In this paper we explicitly construct the model in details and carry out an in-depth analysis. The C*-algebra that describes this quantum space-time is determined, and it is shown that it admits an action by *-automorphisms of the dilation group, along with the expected Poincaré covariance. In order to study the main physical properties of this scale-covariant model, a free scalar neutral field is introduced as a investigation tool. Our key results are then t…
Foundations of Relational Quantum Field Theory I: Scalars
University of Cambridge（剑桥大学）；Basic Research Community for Physics, Leipzig（莱比锡基础物理社区）；Institute for Quantum Optics and Quantum Information, Vienna（维也纳量子光学与量子信息研究所）；Department of Computer Science, University of Oxford（牛津大学计算机系）
