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Title:

Elementary proof for asymptotics of large Haar-distributed unitary matrices, unpublished, www.arxiv.org

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We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random n × n unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) k × k submatrix converges in distribution, after multiplying by a normalization factor √ n and as n → ∞, to a matrix of indepen...

We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random n × n unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) k × k submatrix converges in distribution, after multiplying by a normalization factor √ n and as n → ∞, to a matrix of independent complex Gaussian random variables with mean Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-22

Source:

http://arxiv.org/pdf/0705.3146v2.pdf

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text

Language:

en

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Title:

On Spontaneous Wave Function Collapse and Quantum Field Theory

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One way of obtaining a version of quantum mechanics without observers, and thus of solving the paradoxes of quantum mechanics, is to modify the Schrödinger evolution by implementing spontaneous collapses of the wave function. An explicit model of this kind was proposed in 1986 by Ghirardi, Rimini, and Weber (GRW), involving a nonlinear, stochast...

One way of obtaining a version of quantum mechanics without observers, and thus of solving the paradoxes of quantum mechanics, is to modify the Schrödinger evolution by implementing spontaneous collapses of the wave function. An explicit model of this kind was proposed in 1986 by Ghirardi, Rimini, and Weber (GRW), involving a nonlinear, stochastic evolution of the wave function. We point out how, by focussing on the essential mathematical structure of the GRW model and a clear ontology, it can be generalized to (regularized) quantum field theories in a simple and natural way. PACS numbers: 03.65.Ta; 03.70.+k. Key words: quantum field theory without observers; Ghirardi–Rimini–Weber model; identical particles; second quantization. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-12-05

Source:

http://arxiv.org/pdf/quant-ph/0508230v2.pdf

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Document Type:

text

Language:

en

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Title:

A relativistic version of the Ghirardi–Rimini–Weber model

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Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi–Rimini–Weber (GRW) model of spontaneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schrödinger equat...

Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi–Rimini–Weber (GRW) model of spontaneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schrödinger equation. It deviates very little from the Schrödinger equation for microscopic systems but efficiently suppresses, for macroscopic systems, superpositions of macroscopically different states. As suggested by Bell, we take the primitive ontology, or local beables, of our model to be a discrete set of space-time points, at which the collapses are centered. This set is random with distribution determined by the initial wavefunction. Our model is nonlocal and violates Bell’s inequality though it does not make use of a preferred slicing of space-time or any other sort of synchronization of spacelike separated points. Like the GRW model, it reproduces the quantum probabilities in all cases presently testable, though it entails deviations from the quantum formalism that are in principle testable. Our model works in Minkowski space-time as well as in (well-behaved) curved background space-times. PACS numbers: 03.65.Ta; 03.65.Ud; 03.30.+p. Key words: spontaneous wavefunction collapse; relativity; quantum theory without observers. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-12-05

Source:

http://arxiv.org/pdf/quant-ph/0406094v2.pdf

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Document Type:

text

Language:

en

DDC:

115 Time *(computed)*

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Contents

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We develop an extension of Bohmian mechanics by defining Bohm-like trajectories for quantum particles in a curved background space-time containing a spacelike singularity. As an example of such a metric we use the Schwarzschild metric, which contains two spacelike singularities, one in the past and one in the future. Since the particle world lin...

We develop an extension of Bohmian mechanics by defining Bohm-like trajectories for quantum particles in a curved background space-time containing a spacelike singularity. As an example of such a metric we use the Schwarzschild metric, which contains two spacelike singularities, one in the past and one in the future. Since the particle world lines are everywhere timelike or lightlike, particles can be annihilated but not created at a future spacelike singularity, and created but not annihilated at a past spacelike singularity. It is argued that in the presence of future (past) spacelike singularities, there is a unique natural Bohm-like evolution law directed to the future (past). This law differs from the one in nonsingular space-times mainly in two ways: it involves Fock space since the particle number is not conserved, and the wave function is replaced by a density matrix. In particular, we determine the evolution equation for the density matrix, a pureto-mixed evolution equation of a quasi-Lindblad form. We have to leave open whether a curvature cut-off needs to be introduced for this equation to be well Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-08-04

Source:

http://arxiv.org/pdf/0808.3060v1.pdf

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Document Type:

text

Language:

en

DDC:

190 Modern western philosophy *(computed)*

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Title:

Response to Horton and Dewdney

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There are some points in the reply of Horton et al. [3] to my comment [2] on their paper [1] which I cannot let stand without a response. I provide here some clarification of how much I proved about the set of points where their law of motion is ill-defined. In a recent article in J. Phys. A [1], Horton et al. present what they claim is a Bohm-t...

There are some points in the reply of Horton et al. [3] to my comment [2] on their paper [1] which I cannot let stand without a response. I provide here some clarification of how much I proved about the set of points where their law of motion is ill-defined. In a recent article in J. Phys. A [1], Horton et al. present what they claim is a Bohm-type law of motion for point particles, based on a Klein–Gordon wave function and implying (unlike a similar law proposed by de Broglie) timelike world lines. Concerning this claim I pointed out in a comment [2] that the prescription they give is ill-defined in some situations, and underpinned this by a concrete example. In addition, I gave arguments to the effect that the set of “bad ” space-time points, where the law of motion is ill-defined, is a set of positive measure for many wave functions. To this Horton et al. have responded [3], ignoring my arguments, that although bad points may exist, they form a set of lesser dimension and therefore can be dealt with by a limiting procedure. I wish here to point out that the response of Horton et al. is entirely without merit. Here is why: In my comment I pointed out that those space-time points are bad where both vectors W + µ and W − µ that appear in the law of Horton et al. are spacelike, or, equivalently, where W + µ W +µ < 0 and W − µ W −µ < 0. Given that the vector fields Pµ and Sµ (on which the construction of W + µ and W − µ relies) are continuous, the functions W + µ W +µ and W − µ W −µ are continuous, too, and thus their values remain negative in an entire neighborhood of any bad space-time point. Therefore, the bad points form an open set, quite contrary to the picture of “nodal lines ” or even “isolated points ” that Horton et al. suggest in their Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-12-05

Source:

http://arxiv.org/pdf/quant-ph/0210018v1.pdf

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Document Type:

text

Language:

en

DDC:

340 Law *(computed)*

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Title:

Smoothness of Wave Functions in Thermal Equilibrium

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We consider the thermal equilibrium distribution at inverse temperature β, or canonical ensemble, of the wave function Ψ of a quantum system. Since L 2 spaces contain more nondifferentiable than differentiable functions, and since the thermal equilibrium distribution is very spread-out, one might expect that Ψ has probability zero to be differen...

We consider the thermal equilibrium distribution at inverse temperature β, or canonical ensemble, of the wave function Ψ of a quantum system. Since L 2 spaces contain more nondifferentiable than differentiable functions, and since the thermal equilibrium distribution is very spread-out, one might expect that Ψ has probability zero to be differentiable. However, we show that for relevant Hamiltonians the contrary is the case: with probability one, Ψ is infinitely often differentiable and even analytic. We also show that with probability one, Ψ lies in the domain of the Hamiltonian. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-08-24

Source:

http://www.maphy.uni-tuebingen.de/Members/rotu/papers/smooth.pdf

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text

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en

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Title:

Can We Make a Bohmian Electron Reach the Speed of Light, at Least for One Instant?

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In Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron’s instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle’s speed is less than or equal to th...

In Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron’s instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle’s speed is less than or equal to the speed of light, but not necessarily less. That leads us to the question whether situations in which the speed actually reaches the speed of light can be arranged experimentally. Our conclusion is that in practice the probability of the particle reaching the speed of light, even for only one point in time, is zero. PACS: 03.65.Ta. Key words: Bohmian mechanics; Dirac equation; instantaneous velocity; quantum probability current. Introduction. In relativistic classical mechanics, the speed of a particle with positive rest mass (such as an electron) can come arbitrarily close to the speed of light c but cannot actually reach it, not even for a single instant, as this would require an infinite amount of energy. In contrast, Bohm’s law of motion for a single relativistic electron Minimize

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Year of Publication:

2013-08-04

Source:

http://arxiv.org/pdf/0806.4476v1.pdf

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Document Type:

text

Language:

en

DDC:

190 Modern western philosophy *(computed)*

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Title:

Tumulka: Elementary Proof for Asymptotics of Large HaarDistributed Unitary Matrices

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We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random n × n unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) k × k submatrix converges in distribution, after multiplying by a normalization factor √ n and as n → ∞, to a matrix of indepen...

We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random n × n unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) k × k submatrix converges in distribution, after multiplying by a normalization factor √ n and as n → ∞, to a matrix of independent complex Gaussian random variables with mean Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-21

Source:

http://arxiv.org/pdf/0705.3146v1.pdf

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Document Type:

text

Language:

en

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Title:

the Klein–Gordon equation”

Description:

energy-momentum and particle trajectories for

energy-momentum and particle trajectories for Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-12-05

Source:

http://arxiv.org/pdf/quant-ph/0202140v2.pdf

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text

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en

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and

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We present a quantum model for the motion of N point particles, implying nonlocal (i.e., superluminal) influences of external fields on the trajectories, that is nonetheless fully relativistic. In contrast to other models that have been proposed, this one involves no additional space-time structure as would be provided by a (possibly dynamical) ...

We present a quantum model for the motion of N point particles, implying nonlocal (i.e., superluminal) influences of external fields on the trajectories, that is nonetheless fully relativistic. In contrast to other models that have been proposed, this one involves no additional space-time structure as would be provided by a (possibly dynamical) foliation of space-time. This is achieved through the interplay of opposite microcausal and macrocausal (i.e., thermodynamic) arrows of time. PACS numbers 03.65.Ud; 03.65.Ta; 03.30.+p We challenge in this paper a conclusion that is almost universally accepted: that quantum phenomena, relativity, and realism are incompatible. We show that, just as in the case of the no-hidden-variables theorems, this conclusion is hasty. And, as in the hidden variables case, we do so with a counterexample. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-12-05

Source:

http://arxiv.org/pdf/quant-ph/0105040v2.pdf

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Document Type:

text

Language:

en

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