6 research outputs found
Towards a mathematical Theory of the Madelung Equations
Even though the Madelung equations are central to many 'classical' approaches
to the foundations of quantum mechanics such as Bohmian and stochastic
mechanics, no coherent mathematical theory has been developed so far for this
system of partial differential equations. Wallstrom prominently raised
objections against the Madelung equations, aiming to show that no such theory
exists in which the system is well-posed and in which the Schr\"odinger
equation is recovered without the imposition of an additional 'ad hoc
quantization condition'--like the one proposed by Takabayasi. The primary
objective of our work is to clarify in which sense Wallstrom's objections are
justified and in which sense they are not, with a view on the existing
literature. We find that it may be possible to construct a mathematical theory
of the Madelung equations which is satisfactory in the aforementioned sense,
though more mathematical research is required.Comment: 85 pages, 1 figure; keywords: Madelung equations, Schr\"odinger
equation, quantum potential, quantum vortices, stochastic mechanic
Comment on "Born's rule for arbitrary Cauchy surfaces"
A recent article has raised the question of how to generalize the Born rule
from non-relativistic quantum theory to curved spacetimes and claimed to answer
it for the special-relativistic case (Lienert and Tumulka, Lett. Math. Phys.
110, 753 (2019)). The proposed generalization originated in prior works on
`hypersurface Bohm-Dirac models' as well as approaches to relativistic quantum
theory developed by Bohm and Hiley. In this comment, we raise three objections
to the rule and the broader theory in which it is embedded. In particular, to
address the underlying assertion that the Born rule is naturally formulated on
a spacelike hypersurface, we provide an analytic example showing that a
spacelike hypersurface need not remain spacelike under proper time evolution --
even in the absence of curvature. We finish by proposing an alternative `curved
Born rule' for the one-body case on general spacetimes, which overcomes these
objections, and in this instance indeed generalizes the one Lienert and Tumulka
attempted to justify. The respective mathematical theory is almost analogous
for the conservation of charge and mass, being two additional examples of
physical quantities obtained from integrating a scalar field over particular
hypersurfaces. Our approach can also be generalized to the many-body case,
which shall be the subject of a future work.Comment: 12 pages, 3 figures; Keywords: Integral conservation laws, continuity
equation, Born rule, detection probability, multi-time wave function,
spacelike hypersurfac
The Differentiation Lemma and the Reynolds Transport Theorem for Submanifolds with Corners
The Reynolds Transport Theorem, colloquially known as 'differentiation under
the integral sign', is a central tool of applied mathematics, finding
application in a variety of disciplines such as fluid dynamics, quantum
mechanics, and statistical physics. In this work we state and prove
generalizations thereof to submanifolds with corners evolving in a manifold via
the flow of a smooth time-independent or time-dependent vector field. Thereby
we close a practically important gap in the mathematical literature, as related
works require various 'boundedness conditions' on domain or integrand that are
cumbersome to satisfy in common modeling situations. By considering manifolds
with corners, a generalization of manifolds and manifolds with boundary, this
work constitutes a step towards a unified treatment of classical integral
theorems for the 'unbounded case' for which the boundary of the evolving set
can exhibit some irregularity.Comment: 54 pages, 5 figures; Keywords: differentiation under the integral
sign, manifolds with corners, integral conservation laws, Reynolds Transport
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