23 research outputs found
The Minimal Modal Interpretation of Quantum Theory
We introduce a realist, unextravagant interpretation of quantum theory that
builds on the existing physical structure of the theory and allows experiments
to have definite outcomes, but leaves the theory's basic dynamical content
essentially intact. Much as classical systems have specific states that evolve
along definite trajectories through configuration spaces, the traditional
formulation of quantum theory asserts that closed quantum systems have specific
states that evolve unitarily along definite trajectories through Hilbert
spaces, and our interpretation extends this intuitive picture of states and
Hilbert-space trajectories to the case of open quantum systems as well. We
provide independent justification for the partial-trace operation for density
matrices, reformulate wave-function collapse in terms of an underlying
interpolating dynamics, derive the Born rule from deeper principles, resolve
several open questions regarding ontological stability and dynamics, address a
number of familiar no-go theorems, and argue that our interpretation is
ultimately compatible with Lorentz invariance. Along the way, we also
investigate a number of unexplored features of quantum theory, including an
interesting geometrical structure---which we call subsystem space---that we
believe merits further study. We include an appendix that briefly reviews the
traditional Copenhagen interpretation and the measurement problem of quantum
theory, as well as the instrumentalist approach and a collection of
foundational theorems not otherwise discussed in the main text.Comment: 73 pages + references, 9 figures; cosmetic changes, added figure,
updated references, generalized conditional probabilities with attendant
changes to the sections on the EPR-Bohm thought experiment and Lorentz
invariance; for a concise summary, see the companion letter at
arXiv:1405.675
Gauge Invariance for Classical Massless Particles with Spin
Wigner's quantum-mechanical classification of particle-types in terms of
Hilbert spaces that provide irreducible representations of the Poincar\'e group
has a classical analogue, which we review and extend in this letter. In
particular, we study the compactness properties of the resulting phase spaces
at fixed energy, and show that in order for a classical massless particle to be
physically sensible, its phase space must feature an equivalence relation that
is the classical-particle counterpart of gauge invariance. By examining the
connection between massless and massive particles in the massless limit, we
also derive a classical-particle version of the Higgs mechanism.Comment: 5 pages, no figure
Gauge Invariance for Classical Massless Particles with Spin
Wigner's quantum-mechanical classification of particle-types in terms of irreducible representations of the Poincaré group has a classical analogue, which we extend in this paper. We study the compactness properties of the resulting phase spaces at fixed energy, and show that in order for a classical massless particle to be physically sensible, its phase space must feature a classical-particle counterpart of electromagnetic gauge invariance. By examining the connection between massless and massive particles in the massless limit, we also derive a classical-particle version of the Higgs mechanism
The Stochastic-Quantum Theorem
This paper introduces several new classes of mathematical structures that
have close connections with physics and with the theory of dynamical systems.
The most general of these structures, called generalized stochastic systems,
collectively encompass many important kinds of stochastic processes, including
Markov chains and random dynamical systems. This paper then states and proves a
new theorem that establishes a precise correspondence between any generalized
stochastic system and a unitarily evolving quantum system. This theorem
therefore leads to a new formulation of quantum theory, alongside the
Hilbert-space, path-integral, and quasiprobability formulations. The theorem
also provides a first-principles explanation for why quantum systems are based
on the complex numbers, Hilbert spaces, linear-unitary time evolution, and the
Born rule. In addition, the theorem suggests that by selecting a suitable
Hilbert space, together with an appropriate choice of unitary evolution, one
can simulate any generalized stochastic system on a quantum computer, thereby
potentially opening up an extensive set of novel applications for quantum
computing.Comment: 15 pages, no figure
On Magnetic Forces and Work
In this letter, we review a manifestly covariant Lagrangian formulation for
the dynamics of a classical particle with intrinsic spin and elementary
electric and magnetic dipole moments. We then couple the particle to the
electromagnetic field, derive the appropriate generalization of the Lorentz
force law, show that the particle's dipole moments must be collinear with its
spin axis, and argue that the magnetic field does mechanical work on the
particle's permanent magnetic dipole moment. As additional support for this
last claim, we calculate the overall system's energy-momentum and angular
momentum from Noether's theorem and show that their local conservation
equations lead to precisely the same force law. We conclude by computing the
system's Belinfante-Rosenfeld energy-momentum tensor.Comment: 5 pages, no figure
Manifestly Covariant Lagrangians, Classical Particles with Spin, and the Origins of Gauge Invariance
In this paper, we review a general technique for converting the standard Lagrangian description of a classical system into a formulation that puts time on an equal footing with the system's degrees of freedom. We show how the resulting framework anticipates key features of special relativity, including the signature of the Minkowski metric tensor and the special role played by theories that are invariant under a generalized notion of Lorentz transformations. We then use this technique to revisit a classification of classical particle-types that mirrors Wigner's classification of quantum particle-types in terms of irreducible representations of the Poincaré group, including the cases of massive particles, massless particles, and tachyons. Along the way, we see gauge invariance naturally emerge in the context of classical massless particles with nonzero spin, as well as study the massless limit of a massive particle and derive a classical-particle version of the Higgs mechanism
Measurement and Quantum Dynamics in the Minimal Modal Interpretation of Quantum Theory
Any realist interpretation of quantum theory must grapple with the measurement problem and the status of state-vector collapse. In a no-collapse approach, measurement is typically modeled as a dynamical process involving decoherence. We describe how the minimal modal interpretation closes a gap in this dynamical description, leading to a complete and consistent resolution to the measurement problem and an effective form of state collapse. Our interpretation also provides insight into the indivisible nature of measurement—the fact that you can't stop a measurement part-way through and uncover the underlying 'ontic' dynamics of the system in question. Having discussed the hidden dynamics of a system's ontic state during measurement, we turn to more general forms of open-system dynamics and explore the extent to which the details of the underlying ontic behavior of a system can be described. We construct a space of ontic trajectories and describe obstructions to defining a probability measure on this space
On Magnetic Forces and Work
We address a long-standing debate over whether classical magnetic forces can do work, ultimately answering the question in the affirmative. In detail, we couple a classical particle with intrinsic spin and elementary dipole moments to the electromagnetic field, derive the appropriate generalization of the Lorentz force law, show that the particle's dipole moments must be collinear with its spin axis, and argue that the magnetic field does mechanical work on the particle's elementary magnetic dipole moment. As consistency checks, we calculate the overall system's energy-momentum and angular momentum, and show that their local conservation equations lead to the same force law and therefore the same conclusions about magnetic forces and work. We also compute the system's Belinfante-Rosenfeld energy-momentum tensor
Can Magnetic Forces Do Work?
Standard lore holds that magnetic forces are incapable of doing mechanical work. More precisely, the claim is that whenever it appears that a magnetic force is doing work, the work is actually being done by another force, with the magnetic force serving only as an indirect mediator. On the other hand, the most familiar instances of magnetic forces acting in everyday life—bar magnets lifting other bar magnets—appear to present manifest evidence of magnetic forces doing work. These sorts of counterexamples are often dismissed as arising from quantum effects that lie outside the classical regime. In this paper, however, we show that quantum theory is not needed to account for these phenomena, and that classical electromagnetism admits a model of elementary magnetic dipoles on which magnetic forces can indeed do work. In order to develop this model, we revisit the foundational principles of the classical theory of electromagnetism, showcase the importance of constraints from relativity, examine the structure of the multipole expansion, and study the connection between the Lorentz force law and conservation of energy and momentum
Manifestly Covariant Lagrangians, Classical Particles with Spin, and the Origins of Gauge Invariance
In this paper, we review a general technique for converting the standard
Lagrangian description of a classical system into a formulation that puts time
on an equal footing with the system's degrees of freedom. We show how the
resulting framework anticipates key features of special relativity, including
the signature of the Minkowski metric tensor and the special role played by
theories that are invariant under a generalized notion of Lorentz
transformations. We then use this technique to revisit a classification of
classical particle-types that mirrors Wigner's classification of quantum
particle-types in terms of irreducible representations of the Poincar\'e group,
including the cases of massive particles, massless particles, and tachyons.
Along the way, we see gauge invariance naturally emerge in the context of
classical massless particles with nonzero spin, as well as study the massless
limit of a massive particle and derive a classical-particle version of the
Higgs mechanism.Comment: 22 pages, no figure