2,217 research outputs found
On probability, indeterminism and quantum paradoxes
If the quantum mechanical description of reality is not complete and a hidden
variable theory is possible, what arises is the problem to explain where the
rates of the outcomes of statistical experiments come from, as already noticed
by Land\'e and Popper. In this paper this problem is investigated, and a new
"paradigm" about the nature of dynamical and statistical laws is proposed. This
paradigm proposes some concepts which contrast with the usual intuitive view of
evolution and of physical law, such as: initial conditions could play no
privileged role in determining the evolution of the universe; the statistical
distribution of the particles emitted by a source could depend on the future
interactions of the particles; indeterministic trajectories can be defined by
the least action principle.
This paradigm is applied to the analysis of the two-slit experiment and of
the EPR paradox, and a coherent picture for these phenomena is proposed; this
new picture shows how the well known difficulties in completing the quantum
mechanical description of reality could be overcome.Comment: 16 pages, 2 figures, minor change
Could quantum statistical regularities derive from a measure on the boundary conditions of a classical universe?
The problem of defining the boundary conditions for the universe is
considered here in the framework of a classical dynamical theory, pointing out
that a measure on boundary conditions must be included in the theory in order
to explain the statistical regularities of evolution. It is then suggested that
quantum statistical regularities also could derive from this measure. An
explicit definition of such a measure is proposed, using both a simplified
model of the universe based on classical mechanics and the non-relativistic
quantum mechanics formalism. The peculiarity of such a measure is that it does
not apply to the initial conditions of the universe, i.e. to the initial
positions and momenta of particles, but to their initial and final positions,
from which the path is derived by means of the least action principle. This
formulation of the problem is crucial and it is supported by the observation
that it is incorrect to liken the determination of the boundary conditions of
the universe to the preparation of a laboratory system, in which the initial
conditions of the system are obviously determined. Some possible objections to
this theory are then discussed. Specifically, the EPR paradox is discussed, and
it is explained by showing that, in general, a measure on the boundary
conditions of the universe generates preinteractive correlations, and that in
the presence of such correlations Bell's inequality can no longer be proven
true. Finally, it is shown that if one broadens the dynamical scheme of the
theory to encompass phenomena such as particle decay and annihilation, the
least action principle allows for an indeterministic evolution of the system.Comment: 32 pages, LaTe
The permanent spatial decomposition of the wave function
Permanent spatial decomposition (PSD) is the (hypothesized) property of the
wave function of a macroscopic system of decomposing into localized permanently
non-overlapping parts when it spreads over a macroscopic region. The typical
example of this phenomenon is the measurement process, in which the wave
function of the laboratory (quantum system + apparatus + environment)
decomposes into n parts, corresponding to the n outcomes of the measurement:
the parts are non-overlapping, because they represent a macroscopic pointer in
different positions, and they are permanently non-overlapping due to the
irreversible interaction with the environment. PSD is often mentioned in the
literature, but until now no formal definition or systematic study of this
phenomenon has been undertaken. The aim of this paper is to partially fill this
gap by giving a formal definition of PSD and studying its possible connection
with scattering theory. The predictive and explanatory powers of this
phenomenon are also discussed and compared with those of Bohmian mechanics
Origin of which-way information and generalization of the Born rule
The possibility to recover the which-way information, for example in the two
slit experiment, is based on a natural but implicit assumption about the
position of a particle {\it before} a position measurement is performed on it.
This assumption cannot be deduced from the standard postulates of quantum
mechanics. In the present paper this assumption is made explicit and formally
postulated as a new rule, the {\it quantum typicality rule}. This rule
correlates the positions of the particles at two different times, thus defining
their trajectories. Unexpectedly, this rule is also equivalent to the Born rule
with regard to the explanation of the results of statistical experiments. For
this reason it can be considered a generalization of the Born rule. The
existence of the quantum typicality rule strongly suggests the possibility of a
new trajectory-based formulation of quantum mechanics. According to this new
formulation, a closed quantum system is represented as a {\it quantum process},
which corresponds to a canonical stochastic process in which the probability
measure is replaced by the wave function and the usual frequentist
interpretation of probability is replaced by the quantum typicality rule.Comment: 14 pages, 5 figures; acknowledgments adde
Relativistic Bohmian mechanics without a preferred foliation
In non-relativistic Bohmian mechanics the universe is represented by a
probability space whose sample space is composed of the Bohmian trajectories.
In relativistic Bohmian mechanics an entire class of empirically equivalent
probability spaces can be defined, one for every foliation of spacetime. In the
literature the hypothesis has been advanced that a single preferred foliation
is allowed, and that this foliation derives from the universal wave function by
means of a covariant law. In the present paper the opposite hypothesis is
advanced, i.e., no law exists for the foliations and therefore all the
foliations are allowed. The resulting model of the universe is basically the
"union" of all the probability spaces associated with the foliations. This
hypothesis is mainly motivated by the fact that any law defining a preferred
foliation is empirically irrelevant. It is also argued that the absence of a
preferred foliation may reduce the well known conflict between Bohmian
mechanics and Relativity.Comment: 10 pages, minor changes in J. Stat. Phys (2015
Quantum field theory without divergence: the method of the interaction operators
The recently proposed interior boundary conditions approach [S. Teufel and R.
Tumulka: Avoiding Ultraviolet Divergence by Means of Interior Boundary
Conditions, arXiv:1506.00497] is a method for defining Hamiltonians without UV
divergence for quantum field theories. In this approach the interactions
between sectors of the Fock space with different number of particles
(inter-sector interactions) are obtained by extending the domain of the free
Hamiltonian to include functions with singularities. In this paper a similar
but alternative strategy is proposed, in which the inter-sector interactions
are implemented by specific interaction operators. In its simplest form, an
interaction operator is obtained by symmetrizing the asymmetric operator
. The
inter-sector interactions derive from the singularities generated by the
factors enclosing the Laplacian, while the domain
of the interaction operator does not include singular functions. As a
consequence the interaction operators and the free Hamiltonian have a common
dense domain, and they can be added together to form the complete Hamiltonian
with interaction.Comment: 15 pages v2>v3: important improvment
A Newtonian Hidden Variable Theory
A new hidden variable theory is proposed, according to which particles
follows definite trajectories, as in Bohmian Mechanics or Nelson's stochastic
mechanics; in the new theory, however, the trajectories are classical, i.e.
Newtonian. This result is obtained by developing the following concepts: (i)
the essential elements of a hidden variable theory are a set of trajectories
and a measure defined on it; the Newtonian HCT will be defined by giving these
two elements. (ii) The universal wave function has a tree structure, whose
branches are generated by the measurement processes and are spatially
disjoined. (iii) The branches have a classical structure, i.e. classical paths
go along them; this property derives from the fact that the paths close to the
classical ones give the main contribution to the Feynman propagator. (iv)
Classical trajectories can give rise to quantum phenomena, like for instance
the interference phenomena of the two-slit experiment, by violating the so
called Independence Assumption, which is always implicitely made in the
conceptual analysis of these phenomena.Comment: 9 page
Asymptotic velocities in quantum and Bohmian mechanics
In this paper the relations between the asymptotic velocity operators of a
quantum system and the asymptotic velocities of the associated Bohmian
trajectories are studied. In particular it is proved that, under suitable
conditions of asymptotic regularity, the probability distribution of the
asymptotic velocities of the Bohmian trajectories is equal to the one derived
from the asymptotic velocity operators of the associated quantum system. It is
also shown that in the relativistic case the distribution of the asymptotic
velocities of the Bohmian trajectories is covariant, or equivalently, it does
not depend on a preferred foliation (it is well known that this is not the case
for the structure of the Bohmian trajectories or for their spatial distribution
at a finite time). This result allows us to develop a covariant formulation of
relativistic Bohmian mechanics; such a formulation is proposed here merely as a
mathematical possibility, while its empirical adequacy will be discussed
elsewhere.Comment: 14 page
On the preferred-basis problem and its possible solutions
The preferred basis problem is mentioned in the literature in connection with
the measurement problem and with the Many World Interpretation. It is argued
that this problem actually corresponds to two inequivalent problems: (i) the
preferred-decomposition problem, i.e., what singles out a preferred
decomposition of a suitable state vector of a system as the sum of a finite or
countable set of vectors?, and (ii) the preferred-representation problem, i.e.,
what singles out a preferred representation for the Hilbert space of a system?
In this paper the preferred-decomposition problem is addressed and two
processes, namely decoherence and permanent spatial decomposition (PSD), are
examined and compared as possible solutions to this problem. It is shown that,
perhaps contrary to common belief, in realistic situations decoherence is not
sufficient to solve the preferred-decomposition problem. PSD is the
(hypothesized) tendency of the wave function of the universe to decompose into
permanently non-overlapping wave packets. Three phases can be roughly
identified as composing PSD: Microscopic decomposition, amplification, and
interaction with the environment. Decoherence theory considers only the
interaction with the environment and ignores the first two phases. For this
reason PSD is fundamentally different from decoherence and, unlike decoherence,
provides a simple and non-elusive solution to the preferred-decomposition
problem.Comment: 16 pages, 1 figur
Regular Hamiltonians for non-relativistic interacting quantum field theories
In the context of non-relativistic quantum field theory, a method is proposed
for multiplying field operators at the same spatial point and obtaining regular
(i.e. rigorously defined) interaction terms for the Hamiltonian. The basic idea
is to modify the Lebesgue measure of configuration space of many particles by
adding singular measures over the subspaces of configuration space in which the
positions of two or more particles coincide.Comment: 22 + 9 page
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