121 research outputs found
Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon
We prove two theorems which concern difficulties in the formulation of the
quantum theory of a linear scalar field on a spacetime, (M,g_{ab}), with a
compactly generated Cauchy horizon. These theorems demonstrate the breakdown of
the theory at certain `base points' of the Cauchy horizon, which are defined as
`past terminal accumulation points' of the horizon generators. Thus, the
theorems may be interpreted as giving support to Hawking's `Chronology
Protection Conjecture', according to which the laws of physics prevent one from
manufacturing a `time machine'. Specifically, we prove: Theorem 1: There is no
extension to (M,g_{ab}) of the usual field algebra on the initial globally
hyperbolic region which satisfies the condition of F-locality at any base
point. In other words, any extension of the field algebra must, in any globally
hyperbolic neighbourhood of any base point, differ from the algebra one would
define on that neighbourhood according to the rules for globally hyperbolic
spacetimes. Theorem 2: The two-point distribution for any Hadamard state
defined on the initial globally hyperbolic region must (when extended to a
distributional bisolution of the covariant Klein-Gordon equation on the full
spacetime) be singular at every base point x in the sense that the difference
between this two point distribution and a local Hadamard distribution cannot be
given by a bounded function in any neighbourhood (in MXM) of (x,x). Theorem 2
implies quantities such as the renormalized expectation value of \phi^2 or of
the stress-energy tensor are necessarily ill-defined or singular at any base
point. The proofs rely on the `Propagation of Singularities' theorems of
Duistermaat and H\"ormander.Comment: 37 pages, LaTeX, uses latexsym and amsbsy, no figures; updated
version now published in Commun. Math. Phys.; no major revisions from
original versio
Reissner Nordstr\"{o}m Background Metric in Dynamical Co-ordinates: Exceptional Behaviour of Hadamard States
We cast the Reissner Nordstrom solution in a particular co-ordinate system
which shows dynamical evolution from initial data. The initial data for the
case is regular. This procedure enables us to treat the metric as a
collapse to a singularity. It also implies that one may assume Wald axioms to
be valid globally in the Cauchy development, especially when Hadamard states
are chosen. We can thus compare the semiclassical behaviour with spherical dust
case, looking upon the metric as well as state specific information as
evolution from initial data. We first recover the divergence on the Cauchy
horizon obtained earlier. We point out that the semiclassical domain extends
right upto the Cauchy horizon. This is different from the spherical dust case
where the quantum gravity domain sets in before. We also find that the
backreaction is not negligible near the central singularity, unlike the dust
case. Apart from these differences, the Reissner Nordstrom solution has a
similarity with dust in that it is stable over a considerable period of time.
The features appearing dust collapse mentioned above were suggested to be
generally applicable within spherical symmetry. Reissner Nordstrom background
(along with the quantum state) generated from initial data, is shown not to
reproduce them
Infrared problem for the Nelson model on static space-times
We consider the Nelson model with variable coefficients and investigate the
problem of existence of a ground state and the removal of the ultraviolet
cutoff. Nelson models with variable coefficients arise when one replaces in the
usual Nelson model the flat Minkowski metric by a static metric, allowing also
the boson mass to depend on position. A physical example is obtained by
quantizing the Klein-Gordon equation on a static space-time coupled with a
non-relativistic particle. We investigate the existence of a ground state of
the Hamiltonian in the presence of the infrared problem, i.e. assuming that the
boson mass tends to 0 at infinity
Bounds on negative energy densities in flat spacetime
We generalise results of Ford and Roman which place lower bounds -- known as
quantum inequalities -- on the renormalised energy density of a quantum field
averaged against a choice of sampling function. Ford and Roman derived their
results for a specific non-compactly supported sampling function; here we use a
different argument to obtain quantum inequalities for a class of smooth, even
and non-negative sampling functions which are either compactly supported or
decay rapidly at infinity. Our results hold in -dimensional Minkowski space
() for the free real scalar field of mass . We discuss various
features of our bounds in 2 and 4 dimensions. In particular, for massless field
theory in 2-dimensional Minkowski space, we show that our quantum inequality is
weaker than Flanagan's optimal bound by a factor of 3/2.Comment: REVTeX, 13 pages and 2 figures. Minor typos corrected, one reference
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A general worldline quantum inequality
Worldline quantum inequalities provide lower bounds on weighted averages of
the renormalised energy density of a quantum field along the worldline of an
observer. In the context of real, linear scalar field theory on an arbitrary
globally hyperbolic spacetime, we establish a worldline quantum inequality on
the normal ordered energy density, valid for arbitrary smooth timelike
trajectories of the observer, arbitrary smooth compactly supported weight
functions and arbitrary Hadamard quantum states. Normal ordering is performed
relative to an arbitrary choice of Hadamard reference state. The inequality
obtained generalises a previous result derived for static trajectories in a
static spacetime. The underlying argument is straightforward and is made
rigorous using the techniques of microlocal analysis. In particular, an
important role is played by the characterisation of Hadamard states in terms of
the microlocal spectral condition. We also give a compact form of our result
for stationary trajectories in a stationary spacetime.Comment: 19pp, LaTeX2e. The statement of the main result is changed slightly.
Several typos fixed, references added. To appear in Class Quantum Gra
Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime
We prove that the singularity structure of all n-point distributions of a
state of a generalised real free scalar field in curved spacetime can be
estimated if the two-point distribution is of Hadamard form. In particular this
applies to the real free scalar field and the result has applications in
perturbative quantum field theory, showing that the class of all Hadamard
states is the state space of interest. In our proof we assume that the field is
a generalised free field, i.e. that it satisies scalar (c-number) commutation
relations, but it need not satisfy an equation of motion. The same argument
also works for anti-commutation relations and it can be generalised to
vector-valued fields. To indicate the strengths and limitations of our
assumption we also prove the analogues of a theorem by Borchers and Zimmermann
on the self-adjointness of field operators and of a very weak form of the
Jost-Schroer theorem. The original proofs of these results in the Wightman
framework make use of analytic continuation arguments. In our case no
analyticity is assumed, but to some extent the scalar commutation relations can
take its place.Comment: 18 page
Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems
We show in this article that the Reeh-Schlieder property holds for states of
quantum fields on real analytic spacetimes if they satisfy an analytic
microlocal spectrum condition. This result holds in the setting of general
quantum field theory, i.e. without assuming the quantum field to obey a
specific equation of motion. Moreover, quasifree states of the Klein-Gordon
field are further investigated in this work and the (analytic) microlocal
spectrum condition is shown to be equivalent to simpler conditions. We also
prove that any quasifree ground- or KMS-state of the Klein-Gordon field on a
stationary real analytic spacetime fulfills the analytic microlocal spectrum
condition.Comment: 31 pages, latex2
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