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.
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 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.
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.
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.
Abstract This chapter introduces the two equations that completely define CSL, for the simplest case, that is, a state vector in a two-dimensional Hilbert space. One equation is the modified evolution equation for the state vector which depends upon a Brownian motion function of time, B(t), a Hermitian operator A called the “collapse-generating operator,” and a collapse rate parameter λ. The state vector evolution is linear, but not unitary, so its norm changes with time. The second equation is called the probability rule. It says the probability a particular B(t) occurs in Nature is proportional to the squared norm of the state vector, that is, high norm state vectors are most likely to occur. The chapter shows how an initial state vector that is a superposition of the eigenstates of A evolves to one eigenstate, with the Born probability, as predicted by standard quantum theory. Finally, it discusses the density matrix description of the ensemble of evolutions under all possible B(t).
Abstract Since the Hilbert space for state vectors describing Nature is, of course, of higher dimension than two, generalization of the state vector evolution equation given in the previous chapter is needed. It is explained why the Brownian motion function needs to be replaced by its time derivative, that is, white noise w(t). The collapse-generating operator A is allowed to be arbitrary. As an example, the case of a free particle moving in one dimension, where A=X (the particle’s position operator), is completely solved. Then, further generalizing, the single collapse-generating operator A is replaced by a family Ai, a complete commuting set of operators, one of whose joint eigenstates is the end result of collapse. Each operator is linked to its own white noise function wi(t). The chapter then discussed time-ordering, which comes into play in the generalized evolution equation. The chapter also covers the associated density matrix. The chapter concludes with a final generalization where the discrete index i is replaced by the continuous index x, so the operators A(x) (x is a point in space) and the white noise field w(x,t) appear in the evolution equation.
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.