15,425 research outputs found
(Never) Mind your p's and q's: Von Neumann versus Jordan on the Foundations of Quantum Theory
In two papers entitled "On a new foundation [Neue Begr\"undung] of quantum
mechanics," Pascual Jordan (1927b,g) presented his version of what came to be
known as the Dirac-Jordan statistical transformation theory. As an alternative
that avoids the mathematical difficulties facing the approach of Jordan and
Paul A. M. Dirac (1927), John von Neumann (1927a) developed the modern Hilbert
space formalism of quantum mechanics. In this paper, we focus on Jordan and von
Neumann. Central to the formalisms of both are expressions for conditional
probabilities of finding some value for one quantity given the value of
another. Beyond that Jordan and von Neumann had very different views about the
appropriate formulation of problems in quantum mechanics. For Jordan, unable to
let go of the analogy to classical mechanics, the solution of such problems
required the identication of sets of canonically conjugate variables, i.e., p's
and q's. For von Neumann, not constrained by the analogy to classical
mechanics, it required only the identication of a maximal set of commuting
operators with simultaneous eigenstates. He had no need for p's and q's. Jordan
and von Neumann also stated the characteristic new rules for probabilities in
quantum mechanics somewhat differently. Jordan (1927b) was the first to state
those rules in full generality. Von Neumann (1927a) rephrased them and, in a
subsequent paper (von Neumann, 1927b), sought to derive them from more basic
considerations. In this paper we reconstruct the central arguments of these
1927 papers by Jordan and von Neumann and of a paper on Jordan's approach by
Hilbert, von Neumann, and Nordheim (1928). We highlight those elements in these
papers that bring out the gradual loosening of the ties between the new quantum
formalism and classical mechanics.Comment: New version. The main difference with the old version is that the
introduction has been rewritten. Sec. 1 (pp. 2-12) in the old version has
been replaced by Secs. 1.1-1.4 (pp. 2-31) in the new version. The paper has
been accepted for publication in European Physical Journal
Assumptions that imply quantum dynamics is linear
A basic linearity of quantum dynamics, that density matrices are mapped
linearly to density matrices, is proved very simply for a system that does not
interact with anything else. It is assumed that at each time the physical
quantities and states are described by the usual linear structures of quantum
mechanics. Beyond that, the proof assumes only that the dynamics does not
depend on anything outside the system but must allow the system to be described
as part of a larger system. The basic linearity is linked with previously
established results to complete a simple derivation of the linear Schrodinger
equation. For this it is assumed that density matrices are mapped one-to-one
onto density matrices. An alternative is to assume that pure states are mapped
one-to-one onto pure states and that entropy does not decrease.Comment: 10 pages. Added references. Improved discussion of equations of
motion for mean values. Expanded Introductio
Recommended from our members
Climate and Land-Use Controls on Surface Water Diversions in the Central Valley, California
Californiaâs Central Valley (CV) is one of the most productive agricultural regions in the world, enabled by the conjunctive use of surface water and groundwater. We investigated variations in the CVâs managed surface water diversions relative to climate variability. Using a historical record (1979â2010) of diversions from 531 sites, we found diversions are largest in the wetter Sacramento basin to the north, but most variable in the drier Tulare basin to the south. A rotated empirical orthogonal function (REOF) analysis finds 72% of the variance of diversions is captured by the first three REOFs. The leading REOF (35% of variance) exhibited strong positive loadings in the Tulare basin, and the corresponding principal component time-series (RPC1) was strongly correlated (Ï >â0.9) with contemporaneous hydrologic variability. This pattern indicates larger than average diversions in the south, with neutral or slightly less than average diversions to the north during wet years, with the opposite true for dry years. The second and third REOFs (20% and 17% of variance, respectively), were strongest in the Sacramento basin and San Francisco BayâDelta. RPC2 and RPC3 were associated with variations in agricultural- and municipal-bound diversions, respectively. RPC2 and RPC3 were also moderately correlated with 7-year cumulative precipitation based on lagged correlation analysis, indicating that diversions in the north and central portions of the CV respond to longer-term hydrologic variations. The results illustrate a dichotomy of regimes wherein diversions in the more arid Tulare are governed by year-to-year hydrologic variability, while those in wetter northern basins reflect land-use patterns and low-frequency hydrologic variations
Quantum Discord and Quantum Computing - An Appraisal
We discuss models of computing that are beyond classical. The primary
motivation is to unearth the cause of nonclassical advantages in computation.
Completeness results from computational complexity theory lead to the
identification of very disparate problems, and offer a kaleidoscopic view into
the realm of quantum enhancements in computation. Emphasis is placed on the
`power of one qubit' model, and the boundary between quantum and classical
correlations as delineated by quantum discord. A recent result by Eastin on the
role of this boundary in the efficient classical simulation of quantum
computation is discussed. Perceived drawbacks in the interpretation of quantum
discord as a relevant certificate of quantum enhancements are addressed.Comment: To be published in the Special Issue of the International Journal of
Quantum Information on "Quantum Correlations: entanglement and beyond." 11
pages, 4 figure
Einstein and Jordan frames reconciled: a frame-invariant approach to scalar-tensor cosmology
Scalar-Tensor theories of gravity can be formulated in different frames, most
notably, the Einstein and the Jordan one. While some debate still persists in
the literature on the physical status of the different frames, a frame
transformation in Scalar-Tensor theories amounts to a local redefinition of the
metric, and then should not affect physical results. We analyze the issue in a
cosmological context. In particular, we define all the relevant observables
(redshift, distances, cross-sections, ...) in terms of frame-independent
quantities. Then, we give a frame-independent formulation of the Boltzmann
equation, and outline its use in relevant examples such as particle freeze-out
and the evolution of the CMB photon distribution function. Finally, we derive
the gravitational equations for the frame-independent quantities at first order
in perturbation theory. From a practical point of view, the present approach
allows the simultaneous implementation of the good aspects of the two frames in
a clear and straightforward way.Comment: 15 pages, matches version to be published on Phys. Rev.
Thermal correlators of anyons in two dimensions
The anyon fields have trivial -commutator for not integer.
For integer the commutators become temperature-dependent operator
valued distributions. The -point functions do not factorize as for quasifree
states.Comment: 14 pages, LaTeX (misprints corrected, a reference added
Fast quantum algorithm for numerical gradient estimation
Given a blackbox for f, a smooth real scalar function of d real variables,
one wants to estimate the gradient of f at a given point with n bits of
precision. On a classical computer this requires a minimum of d+1 blackbox
queries, whereas on a quantum computer it requires only one query regardless of
d. The number of bits of precision to which f must be evaluated matches the
classical requirement in the limit of large n.Comment: additional references and minor clarifications and corrections to
version
Role of interactions in ferrofluid thermal ratchets
Orientational fluctuations of colloidal particles with magnetic moments may
be rectified with the help of external magnetic fields with suitably chosen
time dependence. As a result a noise-driven rotation of particles occurs giving
rise to a macroscopic torque per volume of the carrier liquid. We investigate
the influence of mutual interactions between the particles on this ratchet
effect by studying a model system with mean-field interactions. The stochastic
dynamics may be described by a nonlinear Fokker-Planck equation for the
collective orientation of the particles which we solve approximately by using
the effective field method. We determine an interval for the ratio between
coupling strength and noise intensity for which a self-sustained rectification
of fluctuations becomes possible. The ratchet effect then operates under
conditions for which it were impossible in the absence of interactions.Comment: 18 pages, 10 figure
Density perturbations in Kaluza--Klein theories during a de Sitter phase
In the context of Kaluza-Klein theories, we consider a model in which the
universe is filled with a perfect fluid described by a barotropic equation of
state. An analysis of density perturbations employing the synchronous gauge
shows that there are cases where these perturbations have an exponential growth
during a de Sitter phase evolution in the external space.Comment: LaTex file, 10 pages. To be published in Classical and Quantum
Gravit
- âŠ