6,080 research outputs found
Reflection positivity and invertible topological phases
We implement an extended version of reflection positivity (Wick-rotated
unitarity) for invertible topological quantum field theories and compute the
abelian group of deformation classes using stable homotopy theory. We apply
these field theory considerations to lattice systems, assuming the existence
and validity of low energy effective field theory approximations, and thereby
produce a general formula for the group of Symmetry Protected Topological (SPT)
phases in terms of Thom's bordism spectra; the only input is the dimension and
symmetry group. We provide computations for fermionic systems in physically
relevant dimensions. Other topics include symmetry in quantum field theories, a
relativistic 10-fold way, the homotopy theory of relativistic free fermions,
and a topological spin-statistics theorem.Comment: 136 pages, 16 figures; minor changes/corrections in version 2; v3
major revision; v4 minor revision: corrected proof of Lemma 9.55, many small
changes throughout; v5 version for publication in Geometry & Topolog
Quantum Langevin equations for optomechanical systems
We provide a fully quantum description of a mechanical oscillator in the
presence of thermal environmental noise by means of a quantum Langevin
formulation based on quantum stochastic calculus. The system dynamics is
determined by symmetry requirements and equipartition at equilibrium, while the
environment is described by quantum Bose fields in a suitable non-Fock
representation which allows for the introduction of temperature. A generic
spectral density of the environment can be described by introducing its state
trough a suitable P-representation. Including interaction of the mechanical
oscillator with a cavity mode via radiation pressure we obtain a description of
a simple optomechanical system in which, besides the Langevin equations for the
system, one has the exact input-output relations for the quantum noises. The
whole theory is valid at arbitrarily low temperature. This allows the exact
calculation of the stationary value of the mean energy of the mechanical
oscillator, as well as both homodyne and heterodyne spectra. The present
analysis allows in particular to study possible cooling scenarios and to obtain
the exact connection between observed spectra and fluctuation spectra of the
position of the mechanical oscillator.Comment: 37 pages, 2 figures. Major revisions; new reference
Reflection Positivity and Monotonicity
We prove general reflection positivity results for both scalar fields and
Dirac fields on a Riemannian manifold, and comment on applications to quantum
field theory. As another application, we prove the inequality
between Dirichlet and Neumann covariance operators on a manifold with a
reflection.Comment: 11 page
Quantum Spacetime: a Disambiguation
We review an approach to non-commutative geometry, where models are
constructed by quantisation of the coordinates. In particular we focus on the
full DFR model and its irreducible components; the (arbitrary) restriction to a
particular irreducible component is often referred to as the "canonical quantum
spacetime". The aim is to distinguish and compare the approaches under various
points of view, including motivations, prescriptions for quantisation, the
choice of mathematical objects and concepts, approaches to dynamics and to
covariance.Comment: special issue of SIGMA "Noncommutative Spaces and Fields
A Semidefinite Hierarchy for Containment of Spectrahedra
A spectrahedron is the positivity region of a linear matrix pencil and thus
the feasible set of a semidefinite program. We propose and study a hierarchy of
sufficient semidefinite conditions to certify the containment of a
spectrahedron in another one. This approach comes from applying a moment
relaxation to a suitable polynomial optimization formulation. The hierarchical
criterion is stronger than a solitary semidefinite criterion discussed earlier
by Helton, Klep, and McCullough as well as by the authors. Moreover, several
exactness results for the solitary criterion can be brought forward to the
hierarchical approach. The hierarchy also applies to the (equivalent) question
of checking whether a map between matrix (sub-)spaces is positive. In this
context, the solitary criterion checks whether the map is completely positive,
and thus our results provide a hierarchy between positivity and complete
positivity.Comment: 24 pages, 2 figures; minor corrections; to appear in SIAM J. Opti
Stability of Covariant Relativistic Quantum Theory
In this paper we study the relativistic quantum mechanical interpretation of
the solution of the inhomogeneous Euclidean Bethe-Salpeter equation. Our goal
is to determine conditions on the input to the Euclidean Bethe-Salpeter
equation so the solution can be used to construct a model Hilbert space and a
dynamical unitary representation of the Poincar\'e group. We prove three
theorems that relate the stability of this construction to properties of the
kernel and driving term of the Bethe-Salpeter equation. The most interesting
result is that the positivity of the Hilbert space norm in the non-interacting
theory is not stable with respect to Euclidean covariant perturbations defined
by Bethe-Salpeter kernels. The long-term goal of this work is to understand
which model Euclidean Green functions preserve the underlying relativistic
quantum theory of the original field theory. Understanding the constraints
imposed on the Green functions by the existence of an underlying relativistic
quantum theory is an important consideration for formulating field-theory
motivated relativistic quantum models.Comment: 29 pages, Latex, corrected typos, added background section, improved
proof of key resul
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