75 research outputs found
On Locality in Quantum General Relativity and Quantum Gravity
The physical concept of locality is first analyzed in the special
relativistic quantum regime, and compared with that of microcausality and the
local commutativity of quantum fields. Its extrapolation to quantum general
relativity on quantum bundles over curved spacetime is then described. It is
shown that the resulting formulation of quantum-geometric locality based on the
concept of local quantum frame incorporating a fundamental length embodies the
key geometric and topological aspects of this concept. Taken in conjunction
with the strong equivalence principle and the path-integral formulation of
quantum propagation, quantum-geometric locality leads in a natural manner to
the formulation of quantum-geometric propagation in curved spacetime. Its
extrapolation to geometric quantum gravity formulated over quantum spacetime is
described and analyzed.Comment: Mac-Word file translated to postscript for submission. The author may
be reached at: [email protected] To appear in Found. Phys. vol. 27,
199
CPT-conserving Hamiltonians and their nonlinear supersymmetrization using differential charge-operators C
A brief overview is given of recent developments and fresh ideas at the
intersection of PT and/or CPT-symmetric quantum mechanics with supersymmetric
quantum mechanics (SUSY QM). We study the consequences of the assumption that
the "charge" operator C is represented in a differential-operator form. Besides
the freedom allowed by the Hermiticity constraint for the operator CP,
encouraging results are obtained in the second-order case. The integrability of
intertwining relations proves to match the closure of nonlinear SUSY algebra.
In an illustration, our CPT-symmetric SUSY QM leads to non-Hermitian polynomial
oscillators with real spectrum which turn out to be PT-asymmetric.Comment: 25 page
Complete Characterization of Pure Quantum Measurements and Quantum Channels
We give a complete characterization for pure quantum measurements, i.e., for
POVMs which are extremals in the convex set of all POVMs. Such measurements are
free from classical noise. The characterization is valid both in discrete and
continuous cases, and also in the case of an infinite Hilbert space. We show
that sharp measurements are clean, i.e. they cannot be irreversibly connected
to another POVMs via quantum channels and thus they are free from any
additional quantum noise. We exhibit an example which demonstrates that this
result could also be approximately true for pure measurements.Comment: 5 page
Positive-Operator-Valued Time Observable in Quantum Mechanics
We examine the longstanding problem of introducing a time observable in
Quantum Mechanics; using the formalism of positive-operator-valued measures we
show how to define such an observable in a natural way and we discuss some
consequences.Comment: 13 pages, LaTeX, no figures. Some minor changes, expanded the
bibliography (now it is bigger than the one in the published version),
changed the title and the style for publication on the International Journal
of Theoretical Physic
The modern tools of quantum mechanics (A tutorial on quantum states, measurements, and operations)
This tutorial is devoted to review the modern tools of quantum mechanics,
which are suitable to describe states, measurements, and operations of
realistic, not isolated, systems in interaction with their environment, and
with any kind of measuring and processing devices. We underline the central
role of the Born rule and and illustrate how the notion of density operator
naturally emerges, together the concept of purification of a mixed state. In
reexamining the postulates of standard quantum measurement theory, we
investigate how they may formally generalized, going beyond the description in
terms of selfadjoint operators and projective measurements, and how this leads
to the introduction of generalized measurements, probability operator-valued
measures (POVM) and detection operators. We then state and prove the Naimark
theorem, which elucidates the connections between generalized and standard
measurements and illustrates how a generalized measurement may be physically
implemented. The "impossibility" of a joint measurement of two non commuting
observables is revisited and its canonical implementations as a generalized
measurement is described in some details. Finally, we address the basic
properties, usually captured by the request of unitarity, that a map
transforming quantum states into quantum states should satisfy to be physically
admissible, and introduce the notion of complete positivity (CP). We then state
and prove the Stinespring/Kraus-Choi-Sudarshan dilation theorem and elucidate
the connections between the CP-maps description of quantum operations, together
with their operator-sum representation, and the customary unitary description
of quantum evolution. We also address transposition as an example of positive
map which is not completely positive, and provide some examples of generalized
measurements and quantum operations.Comment: Tutorial. 26 pages, 1 figure. Published in a special issue of EPJ -
ST devoted to the memory of Federico Casagrand
Propagation of a string across the cosmological singularity
Our results concern the transition of a quantum string through the
singularity of the compactified Milne (CM) space. We restrict our analysis to
the string winding around the compact dimension (CD) of spacetime. The CD
undergoes contraction to a point followed by re-expansion. We demonstrate that
both classical and quantum dynamics of considered string are well defined. Most
of presently available calculations strongly suggest that the singularity of a
time dependent orbifold is useless as a model of the cosmological singularity.
We believe that our results bring, to some extent, this claim into question.Comment: 9 pages, 2 figures, revtex4; version accepted for publication in
Class. Quantum Gra
Quantum theory of successive projective measurements
We show that a quantum state may be represented as the sum of a joint
probability and a complex quantum modification term. The joint probability and
the modification term can both be observed in successive projective
measurements. The complex modification term is a measure of measurement
disturbance. A selective phase rotation is needed to obtain the imaginary part.
This leads to a complex quasiprobability, the Kirkwood distribution. We show
that the Kirkwood distribution contains full information about the state if the
two observables are maximal and complementary. The Kirkwood distribution gives
a new picture of state reduction. In a nonselective measurement, the
modification term vanishes. A selective measurement leads to a quantum state as
a nonnegative conditional probability. We demonstrate the special significance
of the Schwinger basis.Comment: 6 page
On Hilbert-Schmidt operator formulation of noncommutative quantum mechanics
This work gives value to the importance of Hilbert-Schmidt operators in the
formulation of a noncommutative quantum theory. A system of charged particle in
a constant magnetic field is investigated in this framework
Minimal Informationally Complete Measurements for Pure States
We consider measurements, described by a positive-operator-valued measure
(POVM), whose outcome probabilities determine an arbitrary pure state of a
D-dimensional quantum system. We call such a measurement a pure-state
informationally complete (PSI-complete) POVM. We show that a measurement with
2D-1 outcomes cannot be PSI-complete, and then we construct a POVM with 2D
outcomes that suffices, thus showing that a minimal PSI-complete POVM has 2D
outcomes. We also consider PSI-complete POVMs that have only rank-one POVM
elements and construct an example with 3D-2 outcomes, which is a generalization
of the tetrahedral measurement for a qubit. The question of the minimal number
of elements in a rank-one PSI-complete POVM is left open.Comment: 2 figures, submitted for the Asher Peres festschrif
Dirac quantization of membrane in time dependent orbifold
We present quantum theory of a membrane propagating in the vicinity of a time
dependent orbifold singularity. The dynamics of a membrane, with the parameters
space topology of a torus, winding uniformly around compact dimension of the
embedding spacetime is mathematically equivalent to the dynamics of a closed
string in a flat FRW spacetime. The construction of the physical Hilbert space
of a membrane makes use of the kernel space of self-adjoint constraint
operators. It is a subspace of the representation space of the constraints
algebra. There exist non-trivial quantum states of a membrane evolving across
the singularity.Comment: 16 pages, no figures, version accepted for publication in Journal of
High Energy Physic
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