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arXiv:quant-ph/9902046 · Physical Review A

Relativistic collapse model with tachyonic features

Philip Pearle

Published Jan 1, 1999• 90 citations
arXivHTMLPDF

Abstract

A finite relativistic model for free particles, which describes the collapse of the state vector, is presented. The interaction of particles with a classical fluctuating field $w(x)$ causes collapse, as in the (so far) physically satisfactory nonrelativistic continuous spontaneous localization (CSL) model. In previous relativistic models, where $w(x)$ has a white-noise spectrum, collapse is accompanied by spontaneous creation of particles with all energies out of the vacuum [amounting to infinite energy/(sec-volume)]. In this paper we explore one way of eliminating such vacuum excitation. We present a formalism which allows an arbitrary spectrum for $w(x).$ We point out, in lowest order of perturbation theory, that it is timelike components of this spectrum which are responsible for vacuum excitation but (appropriate, it seems, to a fundamentally nonlocal phenomenon) it is spacelike components which are responsible for collapse. We restrict the spectrum of $w(x)$ to that of a tachyon of mass $\ensuremath{\mu}=\ensuremath{\Elzxh}/ac\ensuremath{\approx}1 \mathrm{eV},$ where $a\ensuremath{\approx}{10}^{\ensuremath{-}5} \mathrm{cm}\mathrm{}$ is the Ghirardi-Rimini-Weber (GRW) collapse model scale parameter used in CSL. However, in higher orders than the first, there is still vacuum excitation due to the energy-momentum supplied by particle propagators. We opt to explore a simple means of eliminating this vacuum excitation: by removing the time-ordering operation from the state vector evolution so that particle propagators are on the mass shell. The result is a finite theory: there is no vacuum excitation (and no need for renormalization) as each vertex describes a finite physical process, the spontaneous emission or absorption of a real tachyon by a particle (pair creation or annihilation is not permitted by energy-momentum conservation). This process may be thought of as analogous to the GRW ``hitting'' description of collapse. The cost of no time ordering is that at each vertex a mass M particle's wave function can spread by as much as $\ensuremath{\approx}cT(\ensuremath{\mu}/M)$ in time T. This has the result that, beginning at perturbation order $2M/\ensuremath{\mu},$ there is a nonvanishing probability for the particle to be found outside its light cone: for electrons and nucleons, the probability of this spacelike transport is negligibly small. Apart from this effect, the density matrix of particles in one region evolves independently of particles in another spacelike separated region. We thus obtain a finite, reasonably sensible, relativistic collapse model for free particles.

DOI
10.1103/physreva.59.80

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A finite relativistic model for free particles, which describes the collapse of the state vector, is presented. The interaction of particles with a classical fluctuating field w(x)w(x)w(x) causes collapse, as in the (so far) physically satisfactory nonrelativistic continuous spontaneous localization (CSL) model. In previous relativistic models, where w(x)w(x)w(x) has a white-noise spectrum, collapse is accompanied by spontaneous creation of particles with all energies out of the vacuum [amounting to infinite energy/(sec-volume)]. In this paper we explore one way of eliminating such vacuum excitation. We present a formalism which allows an arbitrary spectrum for w(x).w(x).w(x). We point out, in lowest order of perturbation theory, that it is timelike components of this spectrum which are responsible for vacuum excitation but (appropriate, it seems, to a fundamentally nonlocal phenomenon) it is spacelike components which are responsible for collapse. We restrict the spectrum of w(x)w(x)w(x) to that of a tachyon of mass \ensuremathμ=\ensuremath\Elzxh/ac\ensuremath≈1eV,\ensuremath{\mu}=\ensuremath{\Elzxh}/ac\ensuremath{\approx}1 \mathrm{eV},\ensuremathμ=\ensuremath\Elzxh/ac\ensuremath≈1eV, where a\ensuremath≈10\ensuremath−5cma\ensuremath{\approx}{10}^{\ensuremath{-}5} \mathrm{cm}\mathrm{}a\ensuremath≈10\ensuremath−5cm is the Ghirardi-Rimini-Weber (GRW) collapse model scale parameter used in CSL. However, in higher orders than the first, there is still vacuum excitation due to the energy-momentum supplied by particle propagators. We opt to explore a simple means of eliminating this vacuum excitation: by removing the time-ordering operation from the state vector evolution so that particle propagators are on the mass shell. The result is a finite theory: there is no vacuum excitation (and no need for renormalization) as each vertex describes a finite physical process, the spontaneous emission or absorption of a real tachyon by a particle (pair creation or annihilation is not permitted by energy-momentum conservation). This process may be thought of as analogous to the GRW ``hitting'' description of collapse. The cost of no time ordering is that at each vertex a mass M particle's wave function can spread by as much as \ensuremath≈cT(\ensuremathμ/M)\ensuremath{\approx}cT(\ensuremath{\mu}/M)\ensuremath≈cT(\ensuremathμ/M) in time T. This has the result that, beginning at perturbation order 2M/\ensuremathμ,2M/\ensuremath{\mu},2M/\ensuremathμ, there is a nonvanishing probability for the particle to be found outside its light cone: for electrons and nucleons, the probability of this spacelike transport is negligibly small. Apart from this effect, the density matrix of particles in one region evolves independently of particles in another spacelike separated region. We thus obtain a finite, reasonably sensible, relativistic collapse model for free particles.
A model is discussed where all operators are constructed from a quantum scalar field whose energy spectrum takes on all real values. The Schr\"odinger picture wave function depends upon space and time coordinates for each particle, as well as an inexorably increasing evolution parameter sss which labels a foliation of spacelike hypersurfaces. The model is constructed to be manifestly Lorentz invariant in the interaction picture. Free particle states and interactions are discussed in this framework. Then, the formalism of the continuous spontaneous localization (CSL) theory of dynamical collapse is applied. The collapse-generating operator is chosen to be the particle number space-time density. Unlike previous relativistically invariant models, the vacuum state is not excited. The collapse dynamics depends upon two parameters, a parameter $\mathrm{\ensuremath{\Lambda}}$ which represents the collapse rate/volume and a scale factor $\ensuremath{\ell}$. A common example of collapse dynamics, involving a clump of matter in a superposition of two locations, is analyzed. The collapse rate is shown to be identical to that of nonrelativistic CSL when the GRW-CSL choice of $\ensuremath{\ell}=a=1{0}^{\ensuremath{-}5}\text{ }\text{ }\mathrm{cm}$, is made, along with $\mathrm{\ensuremath{\Lambda}}=\ensuremath{\lambda}/{a}^{3}$ (GRW-CSL choice $\ensuremath{\lambda}=1{0}^{\ensuremath{-}16}{s}^{\ensuremath{-}1}$). The collapse rate is also satisfactory with the choice $\ensuremath{\ell}$ as the size of the Universe, with $\mathrm{\ensuremath{\Lambda}}=\ensuremath{\lambda}/\ensuremath{\ell}{a}^{2}$. Because the collapse narrows wave functions in space and time, it increases a particle's momentum and energy, altering its mass. It is shown that, with $\ensuremath{\ell}=a$, the change of mass of a nucleon is unacceptably large but, when $\ensuremath{\ell}$ is the size of the Universe, the change of mass over the age of the Universe is acceptably small.
A Markovian wave-function collapse model is presented where the collapse-inducing operator, constructed from quantum fields, is a manifestly covariant generalization of the mass-density operator utilized in the nonrelativistic continuous spontaneous localization (CSL) wave-function collapse model. However, the model is not Lorentz invariant because two such operators do not commute at spacelike separation, i.e., the time-ordering operation in one Lorentz frame, the ``preferred'' frame, is not the time-ordering operation in another frame. However, the characteristic spacelike distance over which the commutator decays is the particle's Compton wavelength so, since the commutator rapidly gets quite small, the model is ``almost'' relativistic. This quasirelativistic CSL (QRCSL) model is completely finite: unlike previous, relativistic, models, it has no (infinite) energy production from the vacuum state. QRCSL calculations are given of the collapse rate for a single free particle in a superposition of spatially separated packets, and of the energy production rate for any number of free particles: these reduce to the CSL rates if the particle's Compton wavelength is small compared to the model's distance parameter. One motivation for QRCSL is the realization that previous relativistic models entail excitation of nuclear states which exceeds that of experiment, whereas QRCSL does not; an example is given involving quadrupole excitation of the 74Ge^{74}\mathrm{Ge}74Ge nucleus.
The CSL dynamical collapse structure, adapted to the relativistically invariant model where the collapse-generating operator is a one-dimensional scalar field ϕ^(x,t)\hatϕ(x,t)ϕ^​(x,t) (mass mmm) is discussed. A complete solution for the density matrix is given, for an initial state ∣ψ,0⟩=12[∣L⟩+∣R⟩]|ψ,0\rangle=\frac{1}{\sqrt{2}}[|L\rangle+|R\rangle]∣ψ,0⟩=2​1​[∣L⟩+∣R⟩] when the Hamiltonian H^\hat HH^ is set equal to 0, and when H^\hat HH^ is the free field Hamiltonian. Here ∣L⟩,∣R⟩|L\rangle, |R\rangle∣L⟩,∣R⟩ are coherent states which represent clumps of particles, with mean particle number density Nχi2(x)Nχ_{i}^{2}(x)Nχi2​(x), where $χ_{1}(x),χ_{1}(x) $ are gaussians of width σ>>m−1σ>>m^{-1}σ>>m−1 with mean positions separated by distance >>σ>>σ>>σ. It is shown that, with high probability, the solution for H^=0\hat H=0H^=0 (identical to the short time solution for H^≠0\hat H\neq 0H^=0) favors collapse toward eigenstates of the scalar field whose eigenvalues are close to ∼χi(x)\simχ_{i}(x)∼χi​(x). Thus, this collapse dynamics results in essentially one clump of particles. However, eventually particle production dominates the density matrix since, as is well known, the collapse generates energy/sec-volume of every particle momentum in equal amounts. Because of the particle production, this is not an experimentally viable physical theory but, as is emphasized by the discussion, it is a sound relativistic collapse model, with sensible collapse behavior.
Spontaneous collapse models provide a possible, testable solution to the quantum measurement problem, but constructing a relativistic version applicable to quantum field theory remains an important open challenge. Challenges in constructing relativistic collapse models include avoiding superluminal signaling and foliation dependence, controlling energy production, and ensuring vacuum stability. Here, we present a relativistic collapse framework that addresses these problems at the cost of introducing a new space of physical states, whose complete characterization is left open. A modified, history-dependent Aharonov--Albert prescription describes instantaneous nonlocal state updates on arbitrary spacelike foliations while preserving foliation independence. To ensure vacuum stability, we modify the standard unitary evolution through a prescription that we name dynamical vacuum subtraction. The resulting dynamical map is linear and trace-preserving but not positive on the full set of standard density operators. We establish no-signaling by proving microscopic parameter independence, and identify conditions under which standard nonrelativistic collapse dynamics are recovered. Finally, we apply our framework to a free massive scalar field, obtaining finite particle-number and energy-increase rates at leading order in the long-time regime, together with further promising evidence for the consistency of the framework.
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 spon-taneous 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 sug-gested 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 wave-function collapse; relativity; quantum theory without observers. 1
A relativistic collapse model for distinguishable particles is presented. Position and time, for each particle, are the fundamental operators of the theory. The Schrödinger equation is of the continuous spontaneous localization form, with a Hermitian Hamiltonian and an anti-Hermitian, white-noise dependent, Hamiltonian. It generates state vector evolution parametrized by an “evolution parameter.” It is shown how this can be interpreted as an evolving state in spacetime with collapses satisfying Born rule probabilities, and how certain choices of collapse generating operators lead to states of definite mass and definite configuration in spacetime. The model is Poincaré covariant and conserves energy in expectation.