I suggest a simple alteration of my CSL (Continuous Spontaneous Localization) theory, replacing the mass density collapse-generating operators by relativistic energy density operators. Some consequences of the density matrix evolution equation are explored. First, the expression for the mean energy increase of free particles is calculated (which, in the non-relativistic limit, agrees with the usual result). Then, the density matrix evolution is applied to photons. The mean rate of loss of photon number from a laser beam pulse, the momentum distribution of the photons "excited" out of the laser beam pulse, and the alteration of the cosmic blackbody spectrum are all treated to first order in the collapse rate parameter $λ$. Associated possible experimental limits on $λ$ are discussed.
I suggest a simple alteration of my CSL (Continuous Spontaneous Localization) theory, replacing the mass density collapse-generating operators by relativistic energy density operators. Some consequences of the density matrix evolution equation are explored. First, the expression for the mean energy increase of free particles is calculated (which, in the non-relativistic limit, agrees with the usual result). Then, the density matrix evolution is applied to photons. The mean rate of loss of photon number from a laser beam pulse, the momentum distribution of the photons "excited" out of the laser beam pulse, and the alteration of the cosmic blackbody spectrum are all treated to first order in the collapse rate parameter λ. Associated possible experimental limits on λ are discussed.
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 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.
A natural generalization of the CSL (Continuous Spontaneous Localization) theory of dynamical collapse is applied to a relativistic quantum scalar field ϕ(x,t). It is shown that the modified Schrödinger equation is relativistically invariant, that the probabilities associated to all possible values of the classical scalar random field w(x,t) (which determines the eventual state of collapse) add up to 1, that there is no energy production out of the vacuum and, in the limit of large time, the collapse is toward eigenstates of ϕ(x,0).
Abstract In this work we consider a wide variety of alternatives opened when applying the continuous spontaneous localization (CSL) dynamical collapse theory to the inflationary era. The definitive resolution of many of the issues discussed here will have to await, not only for a general relativistic CSL theory, but for a fully workable theory of quantum gravity. Our concern here is to explore these issues, and to warn against premature conclusions. This exploration includes: two different approaches to deal with quantum field theory and gravitation, the identification of the collapse-generating operator and the general nature and values of the parameters of the CSL theory. All the choices connected with these issues have the potential to dramatically alter the conclusions one can draw. We also argue that the incompatibilities found in a recent paper, between the CSL parameter values and the cosmic microwave background observational data, are associated with specific choices made for the extrapolation to the cosmological context of the CSL theory (as it is known to work in non-relativistic laboratory situations) which do not represent the most natural ones.
Abstract Currently there is not a satisfactory relativistic spontaneous collapse model. Here we show the impossibility of a simple generalization of the continuous spontaneous collapse (CSL) model to the relativistic framework. We consider a mass coupled model as in the non-relativistic limit this gives the CSL model. We show that a Lorentz covariant collapse equation cannot simultaneously satisfy the following conditions: (i) to avoid a divergent rate of energy, (ii) to prevent superluminal signalling.