6,689 research outputs found
A note on spacelike and timelike compactness
When studying the causal propagation of a field in a globally hyperbolic
spacetime M, one often wants to express the physical intuition that it has
compact support in spacelike directions, or that its support is a spacelike
compact set. We compare a number of logically distinct formulations of this
idea, and of the complementary idea of timelike compactness, and we clarify
their interrelations. E.g., a closed subset A of M has a compact intersection
with all Cauchy surfaces if and only if A is contained in J(K) for some compact
set K. (However, it does not suffice to consider only those Cauchy surfaces
that partake in a given foliation of M.) Similarly, a closed subset A of M is
contained in a region between two Cauchy surfaces if and only if the
intersection of A with J(K) is compact for all compact K. We also treat future
and past compact sets in a similar way
The Proca Field in Curved Spacetimes and its Zero Mass Limit
We investigate the classical and quantum Proca field (a massive vector
potential) of mass in arbitrary globally hyperbolic spacetimes and in the
presence of external sources. We motivate a notion of continuity in the mass
for families of observables and we investigate the
massless limit . Our limiting procedure is local and covariant and it
does not require a choice of reference state. We find that the limit exists
only on a subset of observables, which automatically implements a gauge
equivalence on the massless vector potential. For topologically non-trivial
spacetimes, one may consider several inequivalent choices of gauge equivalence
and our procedure selects the one which is expected from considerations
involving the Aharonov-Bohm effect and Gauss' law. We note that the limiting
theory does not automatically reproduce Maxwell's equation, but it can be
imposed consistently when the external current is conserved. To recover the
correct Maxwell dynamics from the limiting procedure would require an
additional control on limits of states. We illustrate this only in the
classical case, where the dynamics is recovered when the Lorenz constraint
remains well behaved in the limit.Comment: 35 page
Modular nuclearity: A generally covariant perspective
A quantum field theory in its algebraic description may admit many irregular
states. So far, selection criteria to distinguish physically reasonable states
have been restricted to free fields (Hadamard condition) or to flat spacetimes
(e.g. Buchholz-Wichmann nuclearity). We propose instead to use a modular
l^p-condition, which is an extension of a strengthened modular nuclearity
condition to generally covariant theories.
The modular nuclearity condition was previously introduced in Minkowski
space, where it played an important role in constructive two dimensional
algebraic QFT's. We show that our generally covariant extension of this
condition makes sense for a vast range of theories, and that it behaves well
under causal propagation and taking mixtures. In addition we show that our
modular l^p-condition holds for every quasi-free Hadamard state of a free
scalar quantum field (regardless of mass or scalar curvature coupling).
However, our condition is not equivalent to the Hadamard condition.Comment: 42 page
Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss' law
We quantise the massless vector potential A of electromagnetism in the
presence of a classical electromagnetic (background) current, j, in a generally
covariant way on arbitrary globally hyperbolic spacetimes M. By carefully
following general principles and procedures we clarify a number of topological
issues. First we combine the interpretation of A as a connection on a principal
U(1)-bundle with the perspective of general covariance to deduce a physical
gauge equivalence relation, which is intimately related to the Aharonov-Bohm
effect. By Peierls' method we subsequently find a Poisson bracket on the space
of local, affine observables of the theory. This Poisson bracket is in general
degenerate, leading to a quantum theory with non-local behaviour. We show that
this non-local behaviour can be fully explained in terms of Gauss' law. Thus
our analysis establishes a relationship, via the Poisson bracket, between the
Aharonov-Bohm effect and Gauss' law (a relationship which seems to have gone
unnoticed so far). Furthermore, we find a formula for the space of electric
monopole charges in terms of the topology of the underlying spacetime. Because
it costs little extra effort, we emphasise the cohomological perspective and
derive our results for general p-form fields A (p < dim(M)), modulo exact
fields. In conclusion we note that the theory is not locally covariant, in the
sense of Brunetti-Fredenhagen-Verch. It is not possible to obtain such a theory
by dividing out the centre of the algebras, nor is it physically desirable to
do so. Instead we argue that electromagnetism forces us to weaken the axioms of
the framework of local covariance, because the failure of locality is
physically well-understood and should be accommodated.Comment: Minor corrections to Def. 4.3, acknowledgements and typos, in line
with published versio
What can (mathematical) categories tell us about space-time?
It is widely believed that in quantum theories of gravity, the classical description of space-time as a manifold is no longer viable as a fundamental concept. Instead, space-time emerges as an approximation in appropriate regimes. In order to understand what is required to explain this emergence, it is necessary to have a good understanding of the classical structure of space-time.
In this essay I will focus on the concept of space-time as it appears in locally covariant quantum field theory (LCQFT), an axiomatic framework for describing quantum field theories in the presence of gravitational background fields. A key aspect of LCQFT is the way in which it formulates locality and general covariance, using the language of category theory.
I will argue that the use of category theory gives a precise and explicit statement of how space-time acts as an organizing principle in a certain systems view of the world. Along the way I will indicate how physical theories give rise to categories which act as a kind of models for modal logic, and how the categorical view of space-time shifts the emphasis away from the manifold structure. The latter point suggests that the view of space-time as an organizing principle may persist, perhaps in a generalized way, even in a quantum theory of gravity. I will mention some new questions, which this shift in emphasis raises
What can (mathematical) categories tell us about space-time?
It is widely believed that in quantum theories of gravity, the classical description of space-time as a manifold is no longer viable as a fundamental concept. Instead, space-time emerges as an approximation in appropriate regimes. In order to understand what is required to explain this emergence, it is necessary to have a good understanding of the classical structure of space-time.
In this essay I will focus on the concept of space-time as it appears in locally covariant quantum field theory (LCQFT), an axiomatic framework for describing quantum field theories in the presence of gravitational background fields. A key aspect of LCQFT is the way in which it formulates locality and general covariance, using the language of category theory.
I will argue that the use of category theory gives a precise and explicit statement of how space-time acts as an organizing principle in a certain systems view of the world. Along the way I will indicate how physical theories give rise to categories which act as a kind of models for modal logic, and how the categorical view of space-time shifts the emphasis away from the manifold structure. The latter point suggests that the view of space-time as an organizing principle may persist, perhaps in a generalized way, even in a quantum theory of gravity. I will mention some new questions, which this shift in emphasis raises
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