11,294 research outputs found
Mapping local Hamiltonians of fermions to local Hamiltonians of spins
We show how to map local fermionic problems onto local spin problems on a
lattice in any dimension. The main idea is to introduce auxiliary degrees of
freedom, represented by Majorana fermions, which allow us to extend the
Jordan-Wigner transformation to dimensions higher than one. We also discuss the
implications of our results in the numerical investigation of fermionic
systems.Comment: Added explicit mappin
Tall tales from de Sitter space II: Field theory dualities
We consider the evolution of massive scalar fields in (asymptotically) de
Sitter spacetimes of arbitrary dimension. Through the proposed dS/CFT
correspondence, our analysis points to the existence of new nonlocal dualities
for the Euclidean conformal field theory. A massless conformally coupled scalar
field provides an example where the analysis is easily explicitly extended to
'tall' background spacetimes.Comment: 31 pages, 2 figure
On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants
We consider Dirichlet-to-Neumann maps associated with (not necessarily
self-adjoint) Schrodinger operators describing nonlocal interactions in
, , where is an open set with a compact,
nonempty boundary satisfying certain regularity conditions. As an application
we describe a reduction of a certain ratio of Fredholm perturbation
determinants associated with operators in to Fredholm
perturbation determinants associated with operators in . This leads to an extension of a variant of a celebrated
formula due to Jost and Pais, which reduces the Fredholm perturbation
determinant associated with a Schr\"odinger operator on the half-line
, in the case of local interactions, to a Wronski determinant of
appropriate distributional solutions of the underlying Schrodinger equation.Comment: 18 page
Entanglement, Holography and Causal Diamonds
We argue that the degrees of freedom in a d-dimensional CFT can be
re-organized in an insightful way by studying observables on the moduli space
of causal diamonds (or equivalently, the space of pairs of timelike separated
points). This 2d-dimensional space naturally captures some of the fundamental
nonlocality and causal structure inherent in the entanglement of CFT states.
For any primary CFT operator, we construct an observable on this space, which
is defined by smearing the associated one-point function over causal diamonds.
Known examples of such quantities are the entanglement entropy of vacuum
excitations and its higher spin generalizations. We show that in holographic
CFTs, these observables are given by suitably defined integrals of dual bulk
fields over the corresponding Ryu-Takayanagi minimal surfaces. Furthermore, we
explain connections to the operator product expansion and the first law of
entanglement entropy from this unifying point of view. We demonstrate that for
small perturbations of the vacuum, our observables obey linear two-derivative
equations of motion on the space of causal diamonds. In two dimensions, the
latter is given by a product of two copies of a two-dimensional de Sitter
space. For a class of universal states, we show that the entanglement entropy
and its spin-three generalization obey nonlinear equations of motion with local
interactions on this moduli space, which can be identified with Liouville and
Toda equations, respectively. This suggests the possibility of extending the
definition of our new observables beyond the linear level more generally and in
such a way that they give rise to new dynamically interacting theories on the
moduli space of causal diamonds. Various challenges one has to face in order to
implement this idea are discussed.Comment: 84 pages, 12 figures; v2: expanded discussion on constraints in
section 7, matches published versio
A non-local vector calculus,non-local volume-constrained problems,and non-local balance laws
A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoints operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The nonlocal calculus gives rise to volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations
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