24,501 research outputs found
Asymptotic observables, propagation estimates and the problem of asymptotic completeness in algebraic QFT
We review recent results on the existence of asymptotic observables in
algebraic QFT. The problem of asymptotic completeness is discussed from this
perspective.Comment: 5 pages. Contribution to proceedings of QMath1
A complete graphical calculus for Spekkens' toy bit theory
While quantum theory cannot be described by a local hidden variable model, it
is nevertheless possible to construct such models that exhibit features
commonly associated with quantum mechanics. These models are also used to
explore the question of {\psi}-ontic versus {\psi}-epistemic theories for
quantum mechanics. Spekkens' toy theory is one such model. It arises from
classical probabilistic mechanics via a limit on the knowledge an observer may
have about the state of a system. The toy theory for the simplest possible
underlying system closely resembles stabilizer quantum mechanics, a fragment of
quantum theory which is efficiently classically simulable but also non-local.
Further analysis of the similarities and differences between those two theories
can thus yield new insights into what distinguishes quantum theory from
classical theories, and {\psi}-ontic from {\psi}-epistemic theories.
In this paper, we develop a graphical language for Spekkens' toy theory.
Graphical languages offer intuitive and rigorous formalisms for the analysis of
quantum mechanics and similar theories. To compare quantum mechanics and a toy
model, it is useful to have similar formalisms for both. We show that our
language fully describes Spekkens' toy theory and in particular, that it is
complete: meaning any equality that can be derived using other formalisms can
also be derived entirely graphically. Our language is inspired by a similar
graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens'
toy bit theory and stabilizer quantum mechanics can be analysed and compared
using analogous graphical formalisms.Comment: Major revisions for v2. 22+7 page
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On the Logic of Belief and Propositional Quantification
We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is some-thing that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss
The Bisognano-Wichmann Theorem for Massive Theories
The geometric action of modular groups for wedge regions (Bisognano-Wichmann
property) is derived from the principles of local quantum physics for a large
class of Poincare covariant models in d=4. As a consequence, the CPT theorem
holds for this class. The models must have a complete interpretation in terms
of massive particles. The corresponding charges need not be localizable in
compact regions: The most general case is admitted, namely localization in
spacelike cones.Comment: 16 pages; improved and corrected formulation
The Algebras of Large N Matrix Mechanics
Extending early work, we formulate the large N matrix mechanics of general
bosonic, fermionic and supersymmetric matrix models, including Matrix theory:
The Hamiltonian framework of large N matrix mechanics provides a natural
setting in which to study the algebras of the large N limit, including
(reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We
find in particular a broad array of new free algebras which we call symmetric
Cuntz algebras, interacting symmetric Cuntz algebras, symmetric
Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the
role of these algebras in solving the large N theory. Most important, the
interacting Cuntz algebras are associated to a set of new (hidden) local
quantities which are generically conserved only at large N. A number of other
new large N phenomena are also observed, including the intrinsic nonlocality of
the (reduced) trace class operators of the theory and a closely related large N
field identification phenomenon which is associated to another set (this time
nonlocal) of new conserved quantities at large N.Comment: 70 pages, expanded historical remark
Infinite Dimensional Free Algebra and the Forms of the Master Field
We find an infinite dimensional free algebra which lives at large N in any
SU(N)-invariant action or Hamiltonian theory of bosonic matrices. The natural
basis of this algebra is a free-algebraic generalization of Chebyshev
polynomials and the dual basis is closely related to the planar connected
parts. This leads to a number of free-algebraic forms of the master field
including an algebraic derivation of the Gopakumar-Gross form. For action
theories, these forms of the master field immediately give a number of new
free-algebraic packagings of the planar Schwinger-Dyson equations.Comment: 39 pages. Expanded historical remark
Two algebraic properties of thermal quantum field theories
We establish the Schlieder and the Borchers property for thermal field
theories. In addition, we provide some information on the commutation and
localization properties of projection operators.Comment: plain tex, 14 page
Synthesising Graphical Theories
In recent years, diagrammatic languages have been shown to be a powerful and
expressive tool for reasoning about physical, logical, and semantic processes
represented as morphisms in a monoidal category. In particular, categorical
quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of
quantum theory into abstract structural properties, expressed in the form of
diagrammatic identities. One way we search for these properties is to start
with a concrete model (e.g. a set of linear maps or finite relations) and start
composing generators into diagrams and looking for graphical identities.
Naively, we could automate this procedure by enumerating all diagrams up to a
given size and check for equalities, but this is intractable in practice
because it produces far too many equations. Luckily, many of these identities
are not primitive, but rather derivable from simpler ones. In 2010, Johansson,
Dixon, and Bundy developed a technique called conjecture synthesis for
automatically generating conjectured term equations to feed into an inductive
theorem prover. In this extended abstract, we adapt this technique to
diagrammatic theories, expressed as graph rewrite systems, and demonstrate its
application by synthesising a graphical theory for studying entangled quantum
states.Comment: 10 pages, 22 figures. Shortened and one theorem adde
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