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.
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.
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.
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).
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.
In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.
Abstract We develop a quantum field theory based on random nonHermitian actions, which upon quantization lead to stochastic nonlinear Schrödinger dynamics for the state vector. In this framework, Lorentz and spacetime translation symmetries are preserved only in a statistical sense: the probability distribution of the action remains invariant under these transformations. As a result, the theory describes ensembles of quantum-state trajectories whose probability distributions remain invariant under changes of reference frame. As a concrete example, we augment the Dirac action with a purely imaginary term coupling the fermion density operator to a universal colored noise. This noise is constructed by solving the d’Alembert equation with white noise as its source, using a generalized stochastic calculus in 1+3 dimensions. We demonstrate that the colored noise drives stochastic localization of wave packets and derive the localization length analytically. Remarkably, the localization length decreases as the size of the observable universe increases. Our model thus provides a potential framework for relativistic spontaneous wave-function collapse. While establishing consistency with Born’s law remains an open challenge, the present work constitutes a step toward embedding collapse models into a Lorentz-invariant quantum field theory.
Spontaneous collapse models aim to resolve the measurement problem in quantum mechanics by considering wave-function collapse as a physical process. We analyze how these models affect a decaying flavor-oscillating system whose evolution is governed by a phenomenological non-Hermitian Hamiltonian. In turn, we apply two popular collapse models, the Quantum Mechanics with Universal Position Localization and the Continuous Spontaneous Localization models, to a neutral meson system. By using the equivalence between the approaches to the time evolution of decaying systems with a non-Hermitian Hamiltonian and a dissipator of the Lindblad form in an enlarged Hilbert space, we show that spontaneous collapse can induce the decay dynamics in both quantum state and master equations. Moreover, we show that the decay property of a flavor-oscillating system is intimately connected to the time (a)symmetry of the noise field underlying the collapse mechanism. This (a)symmetry, in turn, is related to the definition of the stochastic integral and can provide a physical intuition behind the It\ifmmode \bar{o}\else \={o}\fi{}-Stratonovich dilemma in stochastic calculus.