56 research outputs found
Deformations of quantum field theories and integrable models
Deformations of quantum field theories which preserve Poincar\'e 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 field theories
to integrable models with non-trivial S-matrix.Comment: 36 pages, no figures: Minor changes and corrections in Section 3.
Added new Section 5 on von Neumann algebraic formulation, and modular
structur
Algebraic constructive quantum field theory: Integrable models and deformation techniques
Several related operator-algebraic constructions for quantum field theory
models on Minkowski spacetime are reviewed. The common theme of these
constructions is that of a Borchers triple, capturing the structure of
observables localized in a Rindler wedge. After reviewing the abstract setting,
we discuss in this framework i) the construction of free field theories from
standard pairs, ii) the inverse scattering construction of integrable QFT
models on two-dimensional Minkowski space, and iii) the warped convolution
deformation of QFT models in arbitrary dimension, inspired from non-commutative
Minkowski space.Comment: Review article, 57 pages, 3 figure
Localization in Nets of Standard Spaces
Starting from a real standard subspace of a Hilbert space and a
representation of the translation group with natural properties, we construct
and analyze for each endomorphism of this pair a local, translationally
covariant net of standard subspaces, on the lightray and on two-dimensional
Minkowski space. These nets share many features with low-dimensional quantum
field theory, described by corresponding nets of von Neumann algebras.
Generalizing a result of Longo and Witten to two dimensions and massive
multiplicity free representations, we characterize these endomorphisms in terms
of specific analytic functions. Such a characterization then allows us to
analyze the corresponding nets of standard spaces, and in particular compute
their minimal localization length. The analogies and differences to the von
Neumann algebraic situation are discussed.Comment: 34 pages, 1 figur
Linear hyperbolic PDEs with non-commutative time
Motivated by wave or Dirac equations on noncommutative deformations of
Minkowski space, linear integro-differential equations of the form are studied, where is a normal or prenormal hyperbolic differential
operator on , is a coupling constant, and
is a regular integral operator with compactly supported kernel. In
particular, can be non-local in time, so that a Hamiltonian formulation is
not possible. It is shown that for sufficiently small , the
hyperbolic character of is essentially preserved. Unique advanced/retarded
fundamental solutions are constructed by means of a convergent expansion in
, and the solution spaces are analyzed. It is shown that the acausal
behavior of the solutions is well-controlled, but the Cauchy problem is
ill-posed in general. Nonetheless, a scattering operator can be calculated
which describes the effect of on the space of solutions of .
It is also described how these structures occur in the context of
noncommutative Minkowski space, and how the results obtained here can be used
for the analysis of classical and quantum field theories on such spaces.Comment: 33 pages, 5 figures. V2: Slight reformulation
Towards an operator-algebraic construction of integrable global gauge theories
The recent construction of integrable quantum field theories on
two-dimensional Minkowski space by operator-algebraic methods is extended to
models with a richer particle spectrum, including finitely many massive
particle species transforming under a global gauge group. Starting from a
two-particle S-matrix satisfying the usual requirements (unitarity, Yang-Baxter
equation, Poincar\'e and gauge invariance, crossing symmetry, ...), a pair of
relatively wedge-local quantum fields is constructed which determines the field
net of the model. Although the verification of the modular nuclearity condition
as a criterion for the existence of local fields is not carried out in this
paper, arguments are presented that suggest it holds in typical examples such
as nonlinear O(N) sigma-models. It is also shown that for all models complying
with this condition, the presented construction solves the inverse scattering
problem by recovering the S-matrix from the model via Haag-Ruelle scattering
theory, and a proof of asymptotic completeness is given.Comment: 27 pages. Corrected a few minor typos and added a paragraph in the
conclusions to comply with published versio
Thermal equilibrium states for quantum fields on non-commutative spacetimes
Fully Poincar\'e covariant quantum field theories on non-commutative Moyal
Minkowski spacetime so far have been considered in their vacuum
representations, i.e. at zero temperature. Here we report on work in progress
regarding their thermal representations, corresponding to physical states at
non-zero temperature, which turn out to be markedly different from both,
thermal representations of quantum field theory on commutative Minkowski
spacetime, and such representations of non-covariant quantum field theory on
Moyal Minkowski space with a fixed deformation matrix.Comment: 20 pages. Contribution to the proceedings of the conference 'Quantum
Mathematical Physics', Regensburg, 29.09.-02.10.201
Modular nuclearity: A generally covariant perspective
A quantum field theory in its algebraic description may admit many irregular
states. So far, selection criteria to distinguish physically reasonable states
have been restricted to free fields (Hadamard condition) or to flat spacetimes
(e.g. Buchholz-Wichmann nuclearity). We propose instead to use a modular
l^p-condition, which is an extension of a strengthened modular nuclearity
condition to generally covariant theories.
The modular nuclearity condition was previously introduced in Minkowski
space, where it played an important role in constructive two dimensional
algebraic QFT's. We show that our generally covariant extension of this
condition makes sense for a vast range of theories, and that it behaves well
under causal propagation and taking mixtures. In addition we show that our
modular l^p-condition holds for every quasi-free Hadamard state of a free
scalar quantum field (regardless of mass or scalar curvature coupling).
However, our condition is not equivalent to the Hadamard condition.Comment: 42 page
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