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.
Promotion of quantum theory from a theory of measurement to a theory of reality requires an unambiguous specification of the ensemble of realizable states (and each state's probability of realization). Although not yet achieved within the framework of standard quantum theory, it has been achieved within the framework of the continuous spontaneous localization (CSL) wave-function collapse model. In CSL, a classical random field
w(x,t) interacts with quantum particles. The state vector corresponding to each
w(x,t) is a realizable state. In this paper, I consider a previously presented model, which is predictively equivalent to CSL. In this completely quantized collapse (CQC) model, the classical random field is quantized. It is represented by the operator
W(x,t) which satisfies $[W(\mathbf{x},t),W({\mathbf{x}}^{\ensuremath{'}},{t}^{\ensuremath{'}})]=0$. The ensemble of realizable states is described by a single state vector, the ``ensemble vector.'' Each superposed state which comprises the ensemble vector at time
t is the direct product of an eigenstate of $W(\mathbf{x},{t}^{\ensuremath{'}})$, for all
x and for $0\ensuremath{\leqslant}{t}^{\ensuremath{'}}\ensuremath{\leqslant}t$, and the CSL state corresponding to that eigenvalue. These states never interfere (they satisfy a superselection rule at any time), they only branch, so the ensemble vector may be considered to be, as Schr\"odinger put it, a ``catalog'' of the realizable states. In this context, many different interpretations (e.g., many worlds, environmental decoherence, consistent histories, modal interpretation) may be satisfactorily applied. Using this description, a long-standing problem is resolved, where the energy comes from the particles gain due to the narrowing of their wave packets by the collapse mechanism. It is shown how to define the energy of the random field and its energy of interaction with particles so that total energy is conserved for the ensemble of realizable states. As a by-product, since the random-field energy spectrum is unbounded, its canonical conjugate, a self-adjoint time operator, can be discussed. Finally, CSL is a phenomenological description, whose connection to, or derivation from, more conventional physics has not yet appeared. We suggest that, because CQC is fully quantized, it is a natural framework for replacement of the classical field
w(x,t) of CSL by a quantized physical entity. Two illustrative examples are given.
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.
We present a self-consistent stochastic semiclassical framework in which objective localization (collapse) of quantum states induces localized fluctuations in the stress–energy distribution that source causal gravitational wave perturbations via an Einstein–Langevin type equation. The model combines a Lindblad-averaged continuous spontaneous localization (CSL)-like dynamics with a local collapse bookkeeping field that enforces local energy–momentum conservation. Linearized analysis yields a causal wave equation for metric perturbations sourced by a stochastic tensor whose statistics are computable from collapse parameters. We derive scaling laws for expected strain statistics from a localized mass superposition, provide order-of-magnitude estimates and outline experimental strategies and model limitations. The construction avoids standard no-go objections (superluminal signaling and Page–Geilker constraints) at the ensemble level while yielding concrete, testable predictions for future high-sensitivity detectors. This model offers a phenomenological approach that can guide future experiments, assuming advancements in preparing mesoscopic superpositions and detecting subtle gravitational effects.
Continuous Spontaneous Localization (CSL) is one possible explanation for dynamically induced collapse of the wave-function during a quantum measurement. The collapse is mediated by a stochastic non-linear modification of the Schrödinger equation. A consequence of the CSL mechanism is an extremely tiny violation of energy-momentum conservation, which can, in principle, be detected in the laboratory via the random diffusion of a particle induced by the stochastic collapse mechanism. In a paper in 2003, Collett and Pearle investigated the translational CSL diffusion of a sphere, and the rotational CSL diffusion of a disc, and showed that this effect dominates over the ambient environmental noise at low temperatures and extremely low pressures (about ten-thousandth of a pico-Torr). In the present paper, we revisit their analysis and argue that this stringent condition on pressure can be relaxed, and that the CSL effect can be seen at the pressure of about a pico-Torr. A similar analysis is provided for diffusion produced by gravity-induced decoherence, where the effect is typically much weaker than CSL. We also discuss the CSL induced random displacement of a quantum oscillator. Lastly, we propose possible experimental set-ups justifying that CSL diffusion is indeed measurable with the current technology.
Continuous Spontaneous Localization (CSL) is one possible explanation for dynamically induced collapse of the wave-function during a quantum measurement. The collapse is mediated by a stochastic non-linear modification of the Schrödinger equation. A consequence of the CSL mechanism is an extremely tiny violation of energy-momentum conservation, which can, in principle, be detected in the laboratory via the random diffusion of a particle induced by the stochastic collapse mechanism. In a paper in 2003, Collett and Pearle investigated the translational CSL diffusion of a sphere, and the rotational CSL diffusion of a disc, and showed that this effect dominates over the ambient environmental noise at low temperatures and extremely low pressures (about ten-thousandth of a pico-Torr). In the present paper, we revisit their analysis and argue that this stringent condition on pressure can be relaxed, and that the CSL effect can be seen at the pressure of about a pico-Torr. A similar analysis is provided for diffusion produced by gravity-induced decoherence, where the effect is typically much weaker than CSL. We also discuss the CSL induced random displacement of a quantum oscillator. Lastly, we propose possible experimental set-ups justifying that CSL diffusion is indeed measurable with the current technology.
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.