3,686 research outputs found
The State Space of Perturbative Quantum Field Theory in Curved Spacetimes
The space of continuous states of perturbative interacting quantum field
theories in globally hyperbolic curved spacetimes is determined. Following
Brunetti and Fredenhagen, we first define an abstract algebra of observables
which contains the Wick-polynomials of the free field as well as their
time-ordered products, and hence, by the well-known rules of perturbative
quantum field theory, also the observables (up to finite order) of interest for
the interacting quantum field theory. We then determine the space of continuous
states on this algebra. Our result is that this space consists precisely of
those states whose truncated n-point functions of the free field are smooth for
all n not equal to two, and whose two-point function has the singularity of a
Hadamard fundamental form. A crucial role in our analysis is played by the
positivity property of states. On the technical side, our proof involves
functional analytic methods, in particular the methods of microlocal analysis.Comment: 24 pages, Latex file, no figure
Generalized Hadamard Product and the Derivatives of Spectral Functions
In this work we propose a generalization of the Hadamard product between two
matrices to a tensor-valued, multi-linear product between k matrices for any . A multi-linear dual operator to the generalized Hadamard product is
presented. It is a natural generalization of the Diag x operator, that maps a
vector into the diagonal matrix with x on its main diagonal.
Defining an action of the orthogonal matrices on the space of
k-dimensional tensors, we investigate its interactions with the generalized
Hadamard product and its dual. The research is motivated, as illustrated
throughout the paper, by the apparent suitability of this language to describe
the higher-order derivatives of spectral functions and the tools needed to
compute them. For more on the later we refer the reader to [14] and [15], where
we use the language and properties developed here to study the higher-order
derivatives of spectral functions.Comment: 24 page
Two-point functions in (loop) quantum cosmology
The path-integral formulation of quantum cosmology with a massless scalar
field as a sum-over-histories of volume transitions is discussed, with
particular but non-exclusive reference to loop quantum cosmology. Exploiting
the analogy with the relativistic particle, we give a complete overview of the
possible two-point functions, pointing out the choices involved in their
definitions, deriving their vertex expansions and the composition laws they
satisfy. We clarify the origin and relations of different quantities previously
defined in the literature, in particular the tie between definitions using a
group averaging procedure and those in a deparametrized framework. Finally, we
draw some conclusions about the physics of a single quantum universe (where
there exist superselection rules on positive- and negative-frequency sectors
and different choices of inner product are physically equivalent) and
multiverse field theories where the role of these sectors and the inner product
are reinterpreted.Comment: 29 pages; v2: typos corrected, references adde
On the Hadamard product of Hopf monoids
Combinatorial structures which compose and decompose give rise to Hopf
monoids in Joyal's category of species. The Hadamard product of two Hopf
monoids is another Hopf monoid. We prove two main results regarding freeness of
Hadamard products. The first one states that if one factor is connected and the
other is free as a monoid, their Hadamard product is free (and connected). The
second provides an explicit basis for the Hadamard product when both factors
are free.
The first main result is obtained by showing the existence of a one-parameter
deformation of the comonoid structure and appealing to a rigidity result of
Loday and Ronco which applies when the parameter is set to zero. To obtain the
second result, we introduce an operation on species which is intertwined by the
free monoid functor with the Hadamard product. As an application of the first
result, we deduce that the dimension sequence of a connected Hopf monoid
satisfies the following condition: except for the first, all coefficients of
the reciprocal of its generating function are nonpositive
Permutation Equivalence Classes of Kronecker Products of Unitary Fourier Matrices
Kronecker products of unitary Fourier matrices play important role in solving
multilevel circulant systems by a multidimensional Fast Fourier Transform. They
are also special cases of complex Hadamard (Zeilinger) matrices arising in many
problems of mathematics and theoretical physics. The main result of the paper
is splitting the set of all kronecker products of unitary Fourier matrices into
permutation equivalence classes. The choice of permutation equivalence to
relate the products is motivated by the quantum information theory problem of
constructing maximally entangled bases of finite dimensional quantum systems.
Permutation inequivalent products can be used to construct inequivalent, in a
certain sense, maximally entangled bases.Comment: 26 page
The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes
Quantum fields propagating on a curved spacetime are investigated in terms of
microlocal analysis. We discuss a condition on the wave front set for the
corresponding n-point distributions, called ``microlocal spectrum condition''
(SC). On Minkowski space, this condition is satisfied as a consequence of
the usual spectrum condition. Based on Radzikowski's determination of the wave
front set of the two-point function of a free scalar field, satisfying the
Hadamard condition in the Kay and Wald sense, we construct in the second part
of this paper all Wick polynomials including the energy-momentum tensor for
this field as operator valued distributions on the manifold and prove that they
satisfy our microlocal spectrum condition.Comment: 21 pages, AMS-LaTeX, 2 figures appended as Postscript file
Pinching, Pontrjagin classes, and negatively curved vector bundles
We prove several finiteness results for the class of
-manifolds that have fundamental groups isomorphic to and that can be
given complete Riemannian metrics of sectional curvatures within where
. In particular, if is a closed negatively curved manifold of
dimension at least three, then only finitely many manifolds in the class
are total spaces of vector bundles over . Furthermore,
given a word-hyperbolic group and an integer there exists a positive
such that the tangent bundle of any manifold in the
class has zero rational Pontrjagin classes.Comment: 32 page
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