Given a system of *N* particles, with a wave function belonging to the Hilbert space, its dynamics satisfies the following properties. During its evolution, the wave function undergoes repeated spontaneous collapse at random times, mathematically described as

$${\psi}_{t}({\mathbf{x}}_{1},{\mathbf{x}}_{2},\mathrm{\dots}{\mathbf{x}}_{N})\to \frac{{L}_{n}\left(\mathbf{x}\right){\psi}_{t}({\mathbf{x}}_{1},{\mathbf{x}}_{2},\mathrm{\dots}{\mathbf{x}}_{N})}{\parallel {L}_{n}\left(\mathbf{x}\right){\psi}_{t}({\mathbf{x}}_{1},{\mathbf{x}}_{2},\mathrm{\dots}{\mathbf{x}}_{N})\parallel}.$$(4)

Here, *L*_{n}( **x**) is the so-called jump operator, which is defined as

$${L}_{n}\left(\mathbf{x}\right)=\frac{1}{{\left(\pi {r}_{C}^{2}\right)}^{3/4}}{e}^{-{\left({\mathbf{q}}_{n}-\mathbf{x}\right)}^{2}/2{r}_{C}^{2}},$$(5)

which localises the *n*th particle to the spatial location **x** in a region of size *r*_{C}, with **q**_{n} being the position operator of the *n*th particle. The probability for this jump to position **x** by the *n*th particle is given by

$${p}_{n}\left(\mathbf{x}\right)\mathit{\hspace{1em}}\equiv \mathit{\hspace{1em}}{\parallel {L}_{n}\left(\mathbf{x}\right){\psi}_{t}({\mathbf{x}}_{1},{\mathbf{x}}_{2},\mathrm{\dots}{\mathbf{x}}_{N})\parallel}^{2}.$$(6)

The jumps are assumed to occur according to a Poisson process, with a frequency *λ*. Thus *λ* and *r*_{C} are the two new parameters of the model. Between any two jumps, the wave function undergoes normal Schrödinger evolution. The mass density of the *n*th particle in physical space is defined as

$$\begin{array}{ccccc}{\rho}_{t}^{\left(n\right)}\left({\mathbf{x}}_{n}\right)\hfill & \equiv {m}_{n}\int {d}^{3}{x}_{1}\mathrm{\dots}{d}^{3}{x}_{n-1}{d}^{3}{x}_{n+1}\mathrm{\dots}\hfill & & & \\ & {d}^{3}{x}_{N}{\left|{\psi}_{t}({\mathbf{x}}_{1},{\mathbf{x}}_{2},\mathrm{\dots}{\mathbf{x}}_{N})\right|}^{2}.\hfill & & & \end{array}$$(7)

These then are the axioms of the model of spontaneous collapse. It is a universal dynamics, to which quantum mechanics and classical mechanics are limiting approximations. There is no need to refer to any classical measuring apparatus, or environment, or measurement, or macroscopic world. Measurement is just a special case of spontaneous collapse.

The beauty of spontaneous collapse is the natural manner in which the amplification mechanism comes about. In a bound system of *N* particles, any one particle undergoing collapse causes the entire system to collapse. Thus the effective collapse rate for the system is *N**λ*, which is an enormous amplification. If the individual particle is a nucleon, then the collapse rate can become enormously high for a macroscopic system, because *N* is very large. Thus macroscopic objects stay effectively localised in space, and this explains their classical behaviour and solves the measurement problem.

Mathematically, spontaneous collapse can be described as a continuous process, through a stochastic non-linear modification of the Schrödinger equation:

$$\begin{array}{ccccc}\text{d}{\psi}_{t}\hfill & =[-\frac{i}{\mathrm{\hslash}}H\text{d}t+\frac{\sqrt{\gamma}}{{m}_{0}}\int \text{d}\mathbf{x}(M\left(\mathbf{x}\right)-{\u27e8M\left(\mathbf{x}\right)\u27e9}_{t})\text{d}{W}_{t}\left(\mathbf{x}\right)\hfill & & & \\ & -\frac{\gamma}{2{m}_{0}^{2}}\int \text{d}\mathbf{x}{(M\left(\mathbf{x}\right)-{\u27e8M\left(\mathbf{x}\right)\u27e9}_{t})}^{2}\text{d}t]\psi {}_{t}.\hfill & & & \end{array}$$

Here, the first term describes the usual Schrödinger evolution, with *H* being the quantum Hamiltonian. The second and third terms are the new terms that cause dynamical collapse. The new terms are non-unitary, yet they maintain the norm-preserving nature of the evolution. *m*_{0} is a reference mass, conventionally chosen to be the mass of the nucleon. *M*( **x**) is the mass density operator

$$M\left(\mathbf{x}\right)=\sum _{j}{m}_{j}{N}_{j}\left(\mathbf{x}\right),$$(8)

$${N}_{j}\left(\mathbf{x}\right)=\int d\mathbf{y}g\left(\mathbf{y}-\mathbf{x}\right){\psi}_{j}^{\u2020}\left(\mathbf{y}\right){\psi}_{j}\left(\mathbf{y}\right),$$(9)

${\psi}_{j}^{\u2020}\left(\mathbf{y}\right)$ and *ψ*_{j}( **y**) are the creation and annihilation operators, respectively, for a particle *j* at the location **y**. The smearing function *g*( **x**) is defined as

$$g\left(\mathbf{x}\right)=\frac{1}{{\left(\sqrt{2\pi}{r}_{\text{C}}\right)}^{3}}{\text{e}}^{-{\mathbf{x}}^{2}/2{r}_{\text{C}}^{2}}.$$(10)

The collapse inducing stochasticity in the CSL model is described by *W*_{t}( **x**), which is an ensemble of independent Wiener processes, one for each point in space. The constant *γ* is related to the rate parameter *λ* as

$$\lambda =\frac{\gamma}{{\left(4\pi {r}_{\text{C}}^{2}\right)}^{3/2}}.$$(11)

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