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 $^{74}\mathrm{Ge}$ nucleus.
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 nucleus.
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 s 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.
Abstract CSL so far is a general framework to describe dynamical collapse of a statevector to one or another joint eigenstate of a set of mutually commuting collapse-generating operators. This chapter makes the final selection of the collapse generating operators. First, the choice of mass-density operators, A(x)= D(x), is discussed. This leads to an infinite rate of energy increase for particles, which is ameliorated by a slightly different choice, “smeared” mass-density operators, A(x)=∫dx’f(x-x’)D(x’), where f is a Gaussian of width a, f(z) exp-z2/2a2. This introduces a second parameter to the theory (in addition to the collapse rate λ), the width parameter a. This completes construction of non-relativistic CSL, the theory proposed to describe Nature. The chapter goes on to discuss the density matrix, followed by two important calculations, the first consequences of the theory. First, for an initial state that is an object in a superposition of two different places, is the rate to collapse to one or another place, and second, the rate of energy increase of an object due to the wave function narrowing inherent in collapse.
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) (mass m) is discussed. A complete solution for the density matrix is given, for an initial state ∣ψ,0⟩=21[∣L⟩+∣R⟩] when the Hamiltonian H^ is set equal to 0, and when H^ is the free field Hamiltonian. Here ∣L⟩,∣R⟩ are coherent states which represent clumps of particles, with mean particle number density Nχi2(x), where $χ_{1}(x),χ_{1}(x) $ are gaussians of width σ>>m−1 with mean positions separated by distance >>σ. It is shown that, with high probability, the solution for H^=0 (identical to the short time solution for H^=0) favors collapse toward eigenstates of the scalar field whose eigenvalues are close to ∼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.
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.
The continuous spontaneous localization (CSL) theory of dynamical wave-function collapse is an experimentally testable alternative to nonrelativistic quantum mechanics. In it, collapse occurs because particles interact with a classical random field. However, particles gain energy from this field, i.e., particle energy is not conserved. Recently, it has been shown how to construct a theory dubbed ``completely quantized collapse'' (CQC) which is predictively equivalent to CSL. In CQC, a quantized random field is introduced, and CSL's classical random field becomes its eigenvalue. In CQC, energy is conserved, which allows one to understand that energy is conserved in CSL, as the particle's energy gain is compensated by the random field's energy loss. Since the random field has energy, it should have gravitational consequences. For that, one needs to know the random field's energy density. In this paper, it is shown how to construct a symmetric, conserved, energy-momentum-stress-density tensor associated with the quantized random field, even though this field obeys no dynamical equation and has no Lagrangian. Then, three examples are given involving the random field's energy density. One considers interacting particles, the second treats a ``cosmological'' particle creation model, the third involves the gravity of the random field.
Spontaneous collapse models provide a possible, testable solution to the quantum measurement problem. While experiments are providing increasingly stronger bounds on their parameters, a full-fledged relativistic extension is still missing. Previous attempts have encountered different obstacles, such as violation of microcausality, infinite energy rate, and particle production from vacuum. Here, we propose a generalization of the collapse master equation that is characterized by a local field collapse operator and a non-Markovian noise with a Lorentz invariant correlation. Our construction is able to overcome previously encountered problems and has the desirable properties in the nonrelativistic limit. A specific choice of the noise correlation function is also introduced and discussed.