Collapse Models

This is a review of formalisms and models (nonrelativistic and relativistic) which modify Schrodinger's equation so that it describes wavefunction collapse as a dynamical physical process.Comment: 40 pages, to be published in "Open Systems and Measurement in Relativistic Quantum Theory," F. Petruccione and H. P. Breuer eds. (Springer Verlag, 1999

Apparent wave function collapse caused by scattering

Some experimental implications of the recent progress on wave function collapse are calculated. Exact results are derived for the center-of-mass wave function collapse caused by random scatterings and applied to a range of specific examples. The results show that recently proposed experiments to measure the GRW effect are likely to fail, since the effect of naturally occurring scatterings is of the same form as the GRW effect but generally much stronger. The same goes for attempts to measure the collapse caused by quantum gravity as suggested by Hawking and others. The results also indicate that macroscopic systems tend to be found in states with (Delta-x)(Delta-p) = hbar/sqrt(2), but microscopic systems in highly tiltedly squeezed states with (Delta-x)(Delta-p) >> hbar.Comment: Final published version. 20 pages, Plain TeX, no figures. Online at http://astro.berkeley.edu/~max/collapse.html (faster from the US), from http://www.mpa-garching.mpg.de/~max/collapse.html (faster from Europe) or from [email protected]

The Fermionic Projector, Entanglement, and the Collapse of the Wave Function

After a brief introduction to the fermionic projector approach, we review how entanglement and second quantized bosonic and fermionic fields can be described in this framework. The constructions are discussed with regard to decoherence phenomena and the measurement problem. We propose a mechanism leading to the collapse of the wave function in the quantum mechanical measurement process.Comment: 17 pages, LaTeX, 2 figures, minor changes (published version

Symplectic geometry of entanglement

We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using Kostant-Sternberg theorem, we show that separable states form a unique Kaehler orbit, whereas orbits of entanglement states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement and we show how to calculate it for various multiparticle systems providing also simple criteria of separability. The presented method is general and can be applied also under different additional symmetry conditions stemming, eg. from the indistinguishability of particles.Comment: LaTex, 31 pages, typos correcte

Generalized stochastic Schroedinger equations for state vector collapse

A number of authors have proposed stochastic versions of the Schr\"odinger equation, either as effective evolution equations for open quantum systems or as alternative theories with an intrinsic collapse mechanism. We discuss here two directions for generalization of these equations. First, we study a general class of norm preserving stochastic evolution equations, and show that even after making several specializations, there is an infinity of possible stochastic Schr\"odinger equations for which state vector collapse is provable. Second, we explore the problem of formulating a relativistic stochastic Schr\"odinger equation, using a manifestly covariant equation for a quantum field system based on the interaction picture of Tomonaga and Schwinger. The stochastic noise term in this equation can couple to any local scalar density that commutes with the interaction energy density, and leads to collapse onto spatially localized eigenstates. However, as found in a similar model by Pearle, the equation predicts an infinite rate of energy nonconservation proportional to $\delta^3(\vec 0)$, arising from the local double commutator in the drift term.Comment: 24 pages Plain TeX. Minor changes, some new references. To appear in Journal of Physics

On the Consequences of Retaining the General Validity of Locality in Physical Theory

The empirical validity of the locality (LOC) principle of relativity is used to argue in favour of a local hidden variable theory (HVT) for individual quantum processes. It is shown that such a HVT may reproduce the statistical predictions of quantum mechanics (QM), provided the reproducibility of initial hidden variable states is limited. This means that in a HVT limits should be set to the validity of the notion of counterfactual definiteness (CFD). This is supported by the empirical evidence that past, present, and future are basically distinct. Our argumentation is contrasted with a recent one by Stapp resulting in the opposite conclusion, i.e. nonlocality or the existence of faster-than-light influences. We argue that Stapp's argumentation still depends in an implicit, but crucial, way on both the notions of hidden variables and of CFD. In addition, some implications of our results for the debate between Bohr and Einstein, Podolsky and Rosen are discussed.Comment: revtex, 11 page

The Post-Decoherence Density Matrix Propagator for Quantum Brownian Motion

Using the path integral representation of the density matrix propagator of quantum Brownian motion, we derive its asymptotic form for times greater than the localization time, (\hbar / \gamma k T )^{\half}, where $\gamma$ is the dissipation and $T$ the temperature of the thermal environment. The localization time is typically greater than the decoherence time, but much shorter than the relaxation time, $\gamma^{-1}$. We use this result to show that the reduced density operator rapidly evolves into a state which is approximately diagonal in a set of generalized coherent states. We thus reproduce, using a completely different method, a result we previously obtained using the quantum state diffusion picture (Phys.Rev. D52, 7294 (1995)). We also go beyond this earlier result, in that we derive an explicit expression for the weighting of each phase space localized state in the approximately diagonal density matrix, as a function of the initial state. For sufficiently long times it is equal to the Wigner function, and we confirm that the Wigner function is positive for times greater than the localization time (multiplied by a number of order 1).Comment: 17 pages, plain Tex, submitted to Physical Review

Relativistic Restrictions on the Distinguishability of Orthogonal Quantum States

We analyze the restrictions on the distinguishability of quantum states imposed by special relativity. An explicit expression relating the error probability for distinguishing between two orthogonal single-photon states with the time $T$ elapsed from the start of the measurement procedure until the measurement result is obtained by the observer.Comment: 9 pages, 1 figure (misprints in formulas corrected

Quantum correlations versus Multisimultaneity: an experimental test

Multisimultaneity is a causal model of relativistic quantum physics which assigns a real time ordering to any set of events, much in the spirit of the pilot-wave picture. Contrary to standard quantum mechanics, it predicts a disappearance of the correlations in a Bell-type experiment when both analysers are in relative motion such that, each one in its own inertial reference frame, is first to select the output of the photons. We tested this prediction using acousto-optic modulators as moving beam-splitters and interferometers separated by 55 m. We didn't observe any disappearance of the correlations, thus refuting Multisimultaneity.Comment: 4 pages, 3 figures, RevTex 4 versio

Consistent Resolution of Some Relativistic Quantum Paradoxes

A relativistic version of the (consistent or decoherent) histories approach to quantum theory is developed on the basis of earlier work by Hartle, and used to discuss relativistic forms of the paradoxes of spherical wave packet collapse, Bohm's formulation of Einstein-Podolsky-Rosen, and Hardy's paradox. It is argued that wave function collapse is not needed for introducing probabilities into relativistic quantum mechanics, and in any case should never be thought of as a physical process. Alternative approaches to stochastic time dependence can be used to construct a physical picture of the measurement process that is less misleading than collapse models. In particular, one can employ a coarse-grained but fully quantum mechanical description in which particles move along trajectories, with behavior under Lorentz transformations the same as in classical relativistic physics, and detectors are triggered by particles reaching them along such trajectories. States entangled between spacelike separate regions are also legitimate quantum descriptions, and can be consistently handled by the formalism presented here. The paradoxes in question arise because of using modes of reasoning which, while correct for classical physics, are inconsistent with the mathematical structure of quantum theory, and are resolved (or tamed) by using a proper quantum analysis. In particular, there is no need to invoke, nor any evidence for, mysterious long-range superluminal influences, and thus no incompatibility, at least from this source, between relativity theory and quantum mechanics.Comment: Latex 42 pages, 7 figures in text using PSTrick