6,876 research outputs found
Pictures of complete positivity in arbitrary dimension
Two fundamental contributions to categorical quantum mechanics are presented.
First, we generalize the CP-construction, that turns any dagger compact
category into one with completely positive maps, to arbitrary dimension.
Second, we axiomatize when a given category is the result of this construction.Comment: Final versio
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
Quantum Picturalism
The quantum mechanical formalism doesn't support our intuition, nor does it
elucidate the key concepts that govern the behaviour of the entities that are
subject to the laws of quantum physics. The arrays of complex numbers are kin
to the arrays of 0s and 1s of the early days of computer programming practice.
In this review we present steps towards a diagrammatic `high-level' alternative
for the Hilbert space formalism, one which appeals to our intuition. It allows
for intuitive reasoning about interacting quantum systems, and trivialises many
otherwise involved and tedious computations. It clearly exposes limitations
such as the no-cloning theorem, and phenomena such as quantum teleportation. As
a logic, it supports `automation'. It allows for a wider variety of underlying
theories, and can be easily modified, having the potential to provide the
required step-stone towards a deeper conceptual understanding of quantum
theory, as well as its unification with other physical theories. Specific
applications discussed here are purely diagrammatic proofs of several quantum
computational schemes, as well as an analysis of the structural origin of
quantum non-locality. The underlying mathematical foundation of this high-level
diagrammatic formalism relies on so-called monoidal categories, a product of a
fairly recent development in mathematics. These monoidal categories do not only
provide a natural foundation for physical theories, but also for proof theory,
logic, programming languages, biology, cooking, ... The challenge is to
discover the necessary additional pieces of structure that allow us to predict
genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures,
some colo
The role of positivity and causality in interactions involving higher spin
It is shown that the recently introduced positivity and causality preserving string-local quantum field theory (SLFT) resolves most No-Go situations in higher spin problems. This includes in particular the Velo–Zwanziger causality problem which turns out to be related in an interesting way to the solution of zero mass Weinberg–Witten issue. In contrast to the indefinite metric and ghosts of gauge theory, SLFT uses only positivity-respecting physical degrees of freedom. The result is a fully Lorentz-covariant and causal string field theory in which light- or space-like linear strings transform covariant under Lorentz transformation.
The cooperation of causality and quantum positivity in the presence of interacting
particles leads to remarkable conceptual changes. It turns out that the presence of H-selfinteractions in the Higgs model is not the result of SSB on a postulated Mexican hat potential, but solely the consequence of the implementation of positivity and causality. These principles (and not the imposed gauge symmetry) account also for the Lie-algebra structure of the leading contributions of selfinteracting vector mesons.
Second order consistency of selfinteracting vector mesons in SLFT requires the presence of H-particles; this, and not SSB, is the raison d'être for H.
The basic conceptual and calculational tool of SLFT is the S-matrix. Its string-independence is a powerful restriction which determines the form of interaction densities in terms of the model-defining particle content and plays a fundamental role in the construction of pl observables and sl interpolating fields
Averaged Null Energy Condition from Causality
Unitary, Lorentz-invariant quantum field theories in flat spacetime obey
microcausality: commutators vanish at spacelike separation. For interacting
theories in more than two dimensions, we show that this implies that the
averaged null energy, , must be positive. This non-local
operator appears in the operator product expansion of local operators in the
lightcone limit, and therefore contributes to -point functions. We derive a
sum rule that isolates this contribution and is manifestly positive. The
argument also applies to certain higher spin operators other than the stress
tensor, generating an infinite family of new constraints of the form . These lead to new inequalities for the coupling
constants of spinning operators in conformal field theory, which include as
special cases (but are generally stronger than) the existing constraints from
the lightcone bootstrap, deep inelastic scattering, conformal collider methods,
and relative entropy. We also comment on the relation to the recent derivation
of the averaged null energy condition from relative entropy, and suggest a more
general connection between causality and information-theoretic inequalities in
QFT.Comment: 31+8 page
Representation Theory of Chern Simons Observables
Recently we suggested a new quantum algebra, the moduli algebra, which was
conjectured to be a quantum algebra of observables of the Hamiltonian Chern
Simons theory. This algebra provides the quantization of the algebra of
functions on the moduli space of flat connections on a 2-dimensional surface.
In this paper we classify unitary representations of this new algebra and
identify the corresponding representation spaces with the spaces of conformal
blocks of the WZW model. The mapping class group of the surface is proved to
act on the moduli algebra by inner automorphisms. The generators of these
automorphisms are unitary elements of the moduli algebra. They are constructed
explicitly and proved to satisfy the relations of the (unique) central
extension of the mapping class group.Comment: 63 pages, late
A pedagogical presentation of a -algebraic approach to quantum tomography
It is now well established that quantum tomography provides an alternative
picture of quantum mechanics. It is common to introduce tomographic concepts
starting with the Schrodinger-Dirac picture of quantum mechanics on Hilbert
spaces. In this picture states are a primary concept and observables are
derived from them. On the other hand, the Heisenberg picture,which has evolved
in the algebraic approach to quantum mechanics, starts with the
algebra of observables and introduce states as a derived concept. The
equivalence between these two pictures amounts essentially, to the
Gelfand-Naimark-Segal construction. In this construction, the abstract algebra is realized as an algebra of operators acting on a constructed
Hilbert space. The representation one defines may be reducible or irreducible,
but in either case it allows to identify an unitary group associated with the
algebra by means of its invertible elements. In this picture both
states and observables are appropriate functions on the group, it follows that
also quantum tomograms are strictly related with appropriate functions
(positive-type)on the group. In this paper we present, by means of very simple
examples, the tomographic description emerging from the set of ideas connected
with the algebra picture of quantum mechanics. In particular, the
tomographic probability distributions are introduced for finite and compact
groups and an autonomous criterion to recognize a given probability
distribution as a tomogram of quantum state is formulated
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