2,358 research outputs found
Infinite Shannon entropy
Even if a probability distribution is properly normalizable, its associated
Shannon (or von Neumann) entropy can easily be infinite. We carefully analyze
conditions under which this phenomenon can occur. Roughly speaking, this
happens when arbitrarily small amounts of probability are dispersed into an
infinite number of states; we shall quantify this observation and make it
precise. We develop several particularly simple, elementary, and useful bounds,
and also provide some asymptotic estimates, leading to necessary and sufficient
conditions for the occurrence of infinite Shannon entropy. We go to some effort
to keep technical computations as simple and conceptually clear as possible. In
particular, we shall see that large entropies cannot be localized in state
space; large entropies can only be supported on an exponentially large number
of states. We are for the time being interested in single-channel Shannon
entropy in the information theoretic sense, not entropy in a stochastic field
theory or QFT defined over some configuration space, on the grounds that this
simple problem is a necessary precursor to understanding infinite entropy in a
field theoretic context.Comment: 13 pages; V2: 4 references adde
Inertial frames without the relativity principle
Ever since the work of von Ignatowsky circa 1910 it has been known (if not
always widely appreciated) that the relativity principle, combined with the
basic and fundamental physical assumptions of locality, linearity, and
isotropy, leads almost uniquely to either the Lorentz transformations of
special relativity or to Galileo's transformations of classical Newtonian
mechanics. Thus, if one wishes to entertain the possibility of Lorentz symmetry
breaking within the context of the class of local physical theories, then it
seems likely that one will have to abandon (or at the very least grossly
modify) the relativity principle. Working within the framework of local
physics, we reassess the notion of spacetime transformations between inertial
frames in the absence of the relativity principle, arguing that significant and
nontrivial physics can still be extracted as long as the transformations are at
least linear. An interesting technical aspect of the analysis is that the
transformations now form a groupoid/pseudo-group --- it is this technical point
that permits one to evade the von Ignatowsky argument. Even in the absence of a
relativity principle we can nevertheless deduce clear and compelling rules for
the transformation of space and time, rules for the composition of
3-velocities, and rules for the transformation of energy and momentum. As part
of the analysis we identify two particularly elegant and physically compelling
models implementing "minimalist" violations of Lorentz invariance --- in the
first of these minimalist models all Lorentz violations are confined to
carefully delineated particle physics sub-sectors, while the second minimalist
Lorentz-violating model depends on one free function of absolute velocity, but
otherwise preserves as much as possible of standard Lorentz invariant physics.Comment: V1: 42 pages; V2: now 43 pages; added 8 references, added brief
discussion of Carroll kinematics, added brief discussion of
Robertson-Mansouri-Sexl framework, added various minor clarifications. V3:
now 51 pages; added another 34 references; more discussion of DSR and
relative locality; various clarifications and extensions; this version
accepted for publication in JHE
Null Energy Condition violations in bimetric gravity
We consider the effective stress-energy tensors for the foreground and
background sectors in ghost-free bimetric gravity. By considering the
symmetries of the theory, we show that the foreground and background null
energy conditions (NECs) are strongly anti-correlated. In particular, the NECs
can only be simultaneously fulfilled when they saturate, corresponding to
foreground and background cosmological constants. In all other situations,
either the foreground or the background is subject to a NEC-violating
contribution to the total stress-energy.Comment: v1: 16 pages; v2: 2 references adde
Gordon and Kerr-Schild ansatze in massive and bimetric gravity
We develop the "generalized Gordon ansatz" for the ghost-free versions of
both massive and bimetric gravity, an ansatz which is general enough to include
almost all spacetimes commonly considered to be physically interesting, and
restricted enough to greatly simplify calculations. The ansatz allows explicit
calculation of the matrix square root gamma = sqrt{g^{-1} f} appearing as a
central feature of the ghost-free analysis. In particular, this ansatz
automatically allows us to write the effective stress-energy tensor as that
corresponding to a perfect fluid. A qualitatively similar "generalized
Kerr-Schild ansatz" can also be easily considered, now leading to an effective
stress-energy tensor that corresponds to a null fluid. Cosmological
implications are considered, as are consequences for black hole physics.
Finally we have a few words to say concerning the null energy condition in the
framework provided by these ansatze.Comment: 22 page
The information recovery problem
The issue of unitary evolution during creation and evaporation of a black
hole remains controversial. We~argue that some prominent cures are more
troubling than the disease, demonstrate that their central element---forming of
the event horizon before the evaporation begins---is not necessarily true, and
describe a fully coupled matter-gravity system which is manifestly unitary.Comment: 7 pages +1 fig Published versio
The particle interpretation of N = 1 supersymmetric spin foams
We show that N = 1 supersymmetric BF theory in 3d leads to a supersymmetric
spin foam amplitude via a lattice discretisation. Furthermore, by analysing the
supersymmetric quantum amplitudes, we show that they can be re-interpreted as
3d gravity coupled to embedded fermionic Feynman diagrams.Comment: Pages: 16+1
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