Ghirardi–Rimini–Weber theory

Quantum mechanics has two fundamentally different dynamical principles: the linear and deterministic Schrödinger equation, and the nonlinear and stochastic wave packet reduction postulate. The orthodox interpretation, or Copenhagen interpretation of quantum mechanics, posits a wave function collapse every time an observer performs a measurement. One thus faces the problem of defining what an “observer” and a “measurement” are. Another issue of quantum mechanics is that it forecasts superpositions of macroscopic objects, which are not observed in nature (see Schrödinger's cat paradox). The theory does not tell where the threshold between the microscopic and macroscopic worlds is, that is when quantum mechanics should leave space to classical mechanics. The aforementioned issues constitute the measurement problem in quantum mechanics.

Collapse theories avoid the measurement problem by merging the two dynamical principles of quantum mechanics in a unique dynamical description. The physical idea that underlies collapse theories is that particles undergo spontaneous wave-function collapses, which occur randomly both in time (at a given average rate), and in space (according to the Born rule). The imprecise “observer” and “measurement” that plague the orthodox interpretation are thus avoided because the wave function collapses spontaneously. Furthermore, thanks to a so-called “amplification mechanism” (later discussed), collapse theories recover both quantum mechanics for microscopic objects, and classical mechanics for macroscopic ones.

The GRW is the first spontaneous collapse theory that was devised. In the following years several different models were proposed. Among these are

• the continuous spontaneous localization model (CSL model),{{Cite journal|last1=Ghirardi|first1=Gian Carlo|last2=Pearle|first2=Philip|last3=Rimini|first3=Alberto|date=1990-07-01|title=Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles|journal=Physical Review A|volume=42|issue=1|pages=78–89|doi=10.1103/PhysRevA.42.78|pmid=9903779|bibcode=1990PhRvA..42...78G }} which is formulated in terms of identical particles;
• the Diósi–Penrose model,{{Cite journal|last=Diósi|first=L.|date=1989-08-01|title=Models for universal reduction of macroscopic quantum fluctuations|journal=Physical Review A|language=en|volume=40|issue=3|pages=1165–1174|doi=10.1103/PhysRevA.40.1165|pmid=9902248|bibcode=1989PhRvA..40.1165D |issn=0556-2791}}{{Cite journal|last=Penrose|first=Roger|date=May 1996|title=On Gravity's role in Quantum State Reduction|journal=General Relativity and Gravitation|language=en|volume=28|issue=5|pages=581–600|doi=10.1007/BF02105068|bibcode=1996GReGr..28..581P |s2cid=44038399|issn=0001-7701}} which relates the spontaneous collapse to gravity;
• the quantum mechanics with universal position localization (QMUPL) model,{{Cite journal|last=Bassi|first=Angelo|date=2005-04-08|title=Collapse models: analysis of the free particle dynamics|journal=Journal of Physics A: Mathematical and General|volume=38|issue=14|pages=3173–3192|doi=10.1088/0305-4470/38/14/008|issn=0305-4470|arxiv=quant-ph/0410222|s2cid=37142667}} which proves important mathematical results on collapse theories; and the coloured QMUPL model,{{Cite journal|last1=Bassi|first1=Angelo|last2=Ferialdi|first2=Luca|date=2009-07-31|title=Non-Markovian dynamics for a free quantum particle subject to spontaneous collapse in space: General solution and main properties|journal=Physical Review A|language=en|volume=80|issue=1|pages=012116|doi=10.1103/PhysRevA.80.012116|arxiv=0901.1254|bibcode=2009PhRvA..80a2116B |s2cid=119297164|issn=1050-2947}}{{Cite journal|last1=Bassi|first1=Angelo|last2=Ferialdi|first2=Luca|date=2009-07-28|title=Non-Markovian Quantum Trajectories: An Exact Result|journal=Physical Review Letters|language=en|volume=103|issue=5|pages=050403|doi=10.1103/PhysRevLett.103.050403|pmid=19792469|arxiv=0907.1615|bibcode=2009PhRvL.103e0403B |s2cid=25021141|issn=0031-9007}}{{Cite journal|last1=Ferialdi|first1=Luca|last2=Bassi|first2=Angelo|date=2012-08-08|title=Dissipative collapse models with nonwhite noises|journal=Physical Review A|language=en|volume=86|issue=2|pages=022108|doi=10.1103/PhysRevA.86.022108|arxiv=1112.5065|bibcode=2012PhRvA..86b2108F |s2cid=119216571|issn=1050-2947}}{{Cite journal|last1=Ferialdi|first1=Luca|last2=Bassi|first2=Angelo|date=2012-04-26|title=Exact Solution for a Non-Markovian Dissipative Quantum Dynamics|journal=Physical Review Letters|language=en|volume=108|issue=17|pages=170404|doi=10.1103/PhysRevLett.108.170404|pmid=22680843|arxiv=1204.4348|bibcode=2012PhRvL.108q0404F |s2cid=16746767|issn=0031-9007}} which is the only collapse model involving coloured stochastic processes for which the exact solution is known.

The first assumption of the GRW theory is that the wave function (or state vector) represents the most accurate possible specification of the state of a physical system. This is a feature that the GRW theory shares with the standard Interpretations of quantum mechanics, and distinguishes it from hidden variable theories, like the de Broglie–Bohm theory, according to which the wave function does not give a complete description of a physical system. The GRW theory differs from standard quantum mechanics for the dynamical principles according to which the wave function evolves.{{Cite journal|last1=Bassi|first1=Angelo|last2=Ghirardi|first2=GianCarlo|date=June 2003|title=Dynamical reduction models|journal=Physics Reports|language=en|volume=379|issue=5–6|pages=257–426|doi=10.1016/S0370-1573(03)00103-0|arxiv=quant-ph/0302164|bibcode=2003PhR...379..257B |s2cid=119076099}}{{Cite journal|last1=Bassi|first1=Angelo|last2=Lochan|first2=Kinjalk|last3=Satin|first3=Seema|last4=Singh|first4=Tejinder P.|last5=Ulbricht|first5=Hendrik|date=2013-04-02|title=Models of wave-function collapse, underlying theories, and experimental tests|journal=Reviews of Modern Physics|language=en|volume=85|issue=2|pages=471–527|doi=10.1103/RevModPhys.85.471|arxiv=1204.4325 |bibcode=2013RvMP...85..471B |s2cid=119261020|issn=0034-6861|url=https://eprints.soton.ac.uk/367141/1/RevModPhys.85.471}} More philosophical issues related to the GRW theory and to collapse theories in general one have been discussed by Ghirardi and Bassi.{{Citation|last1=Ghirardi|first1=Giancarlo|title=Collapse Theories|date=2020|url=https://plato.stanford.edu/archives/sum2020/entries/qm-collapse/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2020|publisher=Metaphysics Research Lab, Stanford University|access-date=2020-05-26|last2=Bassi|first2=Angelo}}

• Each particle of a system described by the multi-particle wave function $|\backslash psi\backslash rangle$ independently undergoes a spontaneous localization process (or jump):

$|\backslash psi\backslash rangle\backslash rightarrow\backslash frac\{|\backslash psi\_x^i\backslash rangle\}\{\backslash sqrt\{\backslash langle\backslash psi\_x^i|\backslash psi\_x^i\backslash rangle\}\}$ ,

where $|\backslash psi\_x^i\backslash rangle=\backslash hat\{L\}\_x^i|\backslash psi\backslash rangle$ is the state after the operator $\backslash hat\{L\}\_x^i$ has localized the $i$-th particle around the position $x$.

• The localization process is random both in space and time. The jumps are Poisson distributed in time, with mean rate $\backslash lambda$; the probability density for a jump to occur at position $x$ is $P\_i(x)=\backslash langle\backslash psi\_x^i|\backslash psi\_x^i\backslash rangle$.
• The localization operator has a Gaussian form:

$\backslash hat\{L\}\_x^i=\backslash left(\backslash frac\{1\}\{\backslash pi\; r\_C^2\}\backslash right)^\{\backslash frac\{3\}\{4\}\}e^\{-\backslash frac\{(\backslash hat\{q\}\_i-x)^2\}\{2r\_C^2\}\}$ ,

where $\backslash hat\{q\}\_i$ is the position operator of the $i$-th particle, and $r\_C$ is the localization distance.

• In between two localization processes, the wave function evolves according to the Schrödinger equation.

These principles can be expressed in a more compact way with the statistical operator formalism. Since the localization process is Poissonian, in a time interval $dt$ there is a probability $\backslash lambda\; dt$ that a collapse occurs, i.e. that the pure state $\backslash rho=|\backslash psi\backslash rangle\backslash langle\backslash psi|$ is transformed into the statistical mixture

$\backslash hat\{T\}\_i[\backslash rho]\backslash equiv\backslash int\; dx\; \backslash ,\backslash hat\{L\}\_x^i\; |\backslash psi\backslash rangle\backslash langle\backslash psi|\backslash hat\{L\}\_x^i$ .

In the same time interval, there is a probability $1-\backslash lambda\; dt$ that the system keeps evolving according to the Schrödinger equation. Accordingly, the GRW master equation for $N$ particles reads

$\backslash frac\{d\}\{dt\}\backslash rho(t)=-\backslash frac\{i\}\{\backslash hbar\}[\backslash hat\{H\},\backslash rho(t)]-\backslash sum\_\{i=1\}^N\backslash lambda\_i\backslash left(\backslash rho(t)-\backslash hat\{T\}\_i[\backslash rho]\backslash right)$ ,

where $\backslash hat\{H\}$ is the Hamiltonian of the system, and the square brackets denote a commutator.

Two new parameters are introduced by the GRW theory, namely the collapse rate $\backslash lambda$ and the localization distance $r\_C$. These are phenomenological parameters, whose values are not fixed by any principle and should be understood as new constants of Nature. Comparison of the model's predictions with experimental data permits bounding of the values of the parameters (see CSL model). The collapse rate should be such that microscopic object are almost never localized, thus effectively recovering standard quantum mechanics. The value originally proposed was $\backslash lambda=10^\{-16\}\backslash mathrm\{s\}^\{-1\}$, while more recently Stephen L. Adler proposed that the value $\backslash lambda=10^\{-8\}\backslash mathrm\{s\}^\{-1\}$ (with an uncertainty of two orders of magnitude) is more adequate.{{Cite journal|last=Adler|first=Stephen L|date=2007-03-07|title=Lower and upper bounds on CSL parameters from latent image formation and IGM heating|journal=Journal of Physics A: Mathematical and Theoretical|volume=40|issue=12|pages=2935–2957|doi=10.1088/1751-8113/40/12/s03|arxiv=quant-ph/0605072 |bibcode=2007JPhA...40.2935A |issn=1751-8113|url=https://cds.cern.ch/record/954625}} There is a general consensus on the value $r\_C=10^\{-7\}\backslash mathrm\{m\}$ for the localization distance. This is a mesoscopic distance, such that microscopic superpositions are left unaltered, while macroscopic ones are collapsed.

When the wave function is hit by a sudden jump, the action of the localization operator essentially results in the multiplication of the wave function by the collapse Gaussian.

Let us consider a Gaussian wave function with spread $\backslash sigma$, centered at $x=a$, and let us assume that this undergoes a localization process at the position $x=a$. One thus has (in one dimension)

$\backslash psi(x)=\backslash frac\{1\}\{(\backslash pi\backslash sigma)^\{\backslash frac\{1\}\{4\}\}\}\backslash ,e^\{-\backslash frac\{(x-a)^2\}\{2\backslash sigma^2\}\}\backslash quad\backslash longrightarrow\backslash quad\backslash psi\_a(x)=\backslash hat\{L\}\_\{x=a\}\backslash ,\backslash psi(x)=\{\backslash cal\; N\}e^\{-\backslash frac\{(x-a)^2\}\{2r\_C^2\}\}\backslash ,e^\{-\backslash frac\{(x-a)^2\}\{2\backslash sigma^2\}\}$ ,

where $\{\backslash cal\; N\}$ is a normalization factor. Let us further assume that the initial state is delocalised, i.e. that $\backslash sigma\backslash gg\; r\_C$. In this case one has

$\backslash psi\_a(x)\backslash simeq\{\backslash cal\; N\}\text{'}e^\{-\backslash frac\{(x-a)^2\}\{2r\_C^2\}\}$,

where $\{\backslash cal\; N\}\text{'}$ is another normalization factor. One thus finds that after the sudden jump has occurred, the initially delocalised wave function has become localized.

Another interesting case is when the initial state is the superposition of two Gaussian states, centered at $x=-a$ and $x=a$ respectively: $\backslash psi(x)=\backslash frac\{1\}\{2(\backslash pi\backslash sigma)^\{\backslash frac\{1\}\{4\}\}\}\backslash ,\backslash left[e^\{-\backslash frac\{(x+a)^2\}\{2\backslash sigma^2\}\}+e^\{-\backslash frac\{(x-a)^2\}\{2\backslash sigma^2\}\}\backslash right]$. If the localization occurs e.g. around $x=a$ one has

$\backslash psi\_a(x)=\{\backslash cal\; N\}e^\{-\backslash frac\{(x-a)^2\}\{2r\_C^2\}\}\backslash left[e^\{-\backslash frac\{(x+a)^2\}\{2\backslash sigma^2\}\}+e^\{-\backslash frac\{(x-a)^2\}\{2\backslash sigma^2\}\}\backslash right]=\{\backslash cal\; N\}\backslash left[e^\{-\backslash frac\{\backslash sigma^2+r\_C^2\}\{2\backslash sigma^2r\_C^2\}\backslash ,\backslash left(x+\backslash frac\{\backslash sigma^2-r\_C^2\}\{\backslash sigma^2+r\_C^2\}a\backslash right)^2-\backslash frac\{2a^2\}\{\backslash sigma^2+r\_C^2\}\}+e^\{-\backslash frac\{\backslash sigma^2+r\_C^2\}\{2\backslash sigma^2r\_C^2\}\backslash ,(x-a)^2\}\backslash right]$.

If one assumes that  each Gaussian is localized ($\backslash sigma\backslash ll\; r\_C$) and that the overall superposition is delocalised ($2a\backslash gg\; r\_C$), one finds

$\backslash psi\_a(x)\backslash simeq\{\backslash cal\; N\}\text{'}\backslash left[e^\{-\backslash frac\{\backslash left(x+a\backslash right)^2\}\{2\backslash sigma^2\}-\backslash frac\{2a^2\}\{r\_C^2\}\}+e^\{-\backslash frac\{(x-a)^2\}\{2\backslash sigma^2\}\}\backslash right]$.

We thus see that the Gaussian that is hit by the localization is left unchanged, while the other is exponentially suppressed.

This is one of the most important features of the GRW theory, because it allows us to recover classical mechanics for macroscopic objects. Let us consider a rigid body of $N$ particles whose statistical operator evolves according to the master equation described above. We introduce the center of mass ($\backslash hat\{Q\}$) and relative ($\backslash hat\{r\}\_i$) position operators, which allow us to rewrite each particle's position operator as follows: $\backslash hat\{q\}\_i=\backslash hat\{Q\}+\backslash hat\{r\}\_i$. One can show that, when the system Hamiltonian can be split into a center of mass Hamiltonian $H\_\{\backslash mathrm\{CM\}\}$ and a relative Hamiltonian $H\_r$, the center of mass statistical operator $\backslash rho\_\{\backslash mathrm\{CM\}\}$ evolves according to the following master equation:

$\backslash frac\{d\}\{dt\}\backslash rho\_\{\backslash mathrm\{CM\}\}(t)=-\backslash frac\{i\}\{\backslash hbar\}[\backslash hat\{H\}\_\{\backslash mathrm\{CM\}\},\backslash rho\_\{\backslash mathrm\{CM\}\}(t)]-\backslash sum\_\{i=1\}^N\backslash lambda\_i\backslash left(\backslash rho\_\{\backslash mathrm\{CM\}\}(t)-\backslash hat\{T\}\_\{\backslash mathrm\{CM\}\}[\backslash rho\_\{\backslash mathrm\{CM\}\}(t)]\backslash right)$,

where

$\backslash hat\{T\}\_\{\backslash mathrm\{CM\}\}[\backslash rho\_\{\backslash mathrm\{CM\}\}(t)]=\backslash left(\backslash frac\{1\}\{\backslash pi\; r\_C^2\}\backslash right)^\{\backslash frac\{3\}\{2\}\}\backslash int\_\backslash infty^\backslash infty\; d^3x\backslash ,e^\{-\backslash frac\{(\backslash hat\{Q\}-x)^2\}\{2r\_C^2\}\}\backslash ,\backslash rho\_\{\backslash mathrm\{CM\}\}(t)\backslash ,e^\{-\backslash frac\{(\backslash hat\{Q\}-x)^2\}\{2r\_C^2\}\}$.

One thus sees that the center of mass collapses with a rate $\backslash Lambda$ that is the sum of the rates of its constituents: this is the amplification mechanism. If for simplicity one assumes that all particles collapse with the same rate $\backslash lambda$, one simply gets $\backslash Lambda=N\backslash ,\backslash lambda$.

An object that consists of in the order of the Avogadro number of nucleons ($N\backslash simeq10^\{23\}$) collapses almost instantly: GRW's and Adler's values of $\backslash lambda$ give respectively $\backslash Lambda=10^7\backslash ,\backslash mathrm\{s\}^\{-1\}$ and $\backslash Lambda=10^\{15\}\backslash ,\backslash mathrm\{s\}^\{-1\}$. Fast reduction of macroscopic object superpositions is thus guaranteed, and the GRW theory effectively recovers classical mechanics for macroscopic objects.

• The GRW theory makes different predictions than standard quantum mechanics,{{refn|It has "as its practically unique effect, that of inducing a very fast suppression of coherence among macroscopically distinguishable states", see }} and as such can be tested against it (see CSL model).{{cite journal |title=Quantum Superposition of Massive Objects and Collapse Models |first=Oriol |last=Romero-Isart |journal=Phys. Rev. A |volume=84 |page=052121 |date=2011 |issue=5 |doi=10.1103/PhysRevA.84.052121 |arxiv=1110.4495|bibcode=2011PhRvA..84e2121R |s2cid=118401637 }}
• The collapse noise repeatedly kicks the particles, thus inducing a diffusion process (Brownian motion). This introduces a steady amount of energy in the system, thus leading to a violation of the energy conservation principle. For the GRW model, one can show that energy grows linearly in time with rate $\backslash lambda\backslash hbar^2/4mr\_C^2$ , which for a macroscopic object amounts to $\backslash simeq\; 10^\{-14\}\; \backslash mathrm\{erg\backslash ,\backslash ,s\}^\{-1\}$. Although such an energy increase is negligible, this feature of the model is not appealing. For this reason, a dissipative extension of the GRW theory has been investigated.{{Cite journal|last1=Smirne|first1=Andrea|last2=Vacchini|first2=Bassano|last3=Bassi|first3=Angelo|date=2014-12-31|title=Dissipative extension of the Ghirardi-Rimini-Weber model|journal=Physical Review A|volume=90|issue=6|pages=062135|doi=10.1103/PhysRevA.90.062135|arxiv=1408.6115 |bibcode=2014PhRvA..90f2135S |hdl=2434/314893|s2cid=52232273|hdl-access=free}}
• The GRW theory does not allow for identical particles. An extension of the theory with identical particles has been proposed by Tumulka.{{Cite journal|last=Tumulka|first=Roderich|date=2006-06-08|title=On spontaneous wave function collapse and quantum field theory|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=462|issue=2070|pages=1897–1908|doi=10.1098/rspa.2005.1636|arxiv=quant-ph/0508230|bibcode=2006RSPSA.462.1897T |s2cid=16123332}}
• GRW is a non relativistic theory, its relativistic extension for non-interacting particles has been investigated by Tumulka,{{Cite journal|last=Tumulka|first=Roderich|date=2006-11-01|title=A Relativistic Version of the Ghirardi–Rimini–Weber Model|journal=Journal of Statistical Physics|language=en|volume=125|issue=4|pages=821–840|doi=10.1007/s10955-006-9227-3|arxiv=quant-ph/0406094|bibcode=2006JSP...125..821T |s2cid=13923422|issn=1572-9613}} while interacting models are still under investigation.
• The master equation of the GRW theory describes a decoherence process according to which the off-diagonal elements of the statistical operator are suppressed exponentially. This is a feature that the GRW theory shares with other collapse theories: those involving white noises are associated to Lindblad master equations,{{Cite journal|last=Lindblad|first=G.|title=On the generators of quantum dynamical semigroups|journal=Communications in Mathematical Physics|year=1976|language=en|volume=48|issue=2|pages=119–130|doi=10.1007/BF01608499|bibcode=1976CMaPh..48..119L |s2cid=55220796|issn=0010-3616|url=http://projecteuclid.org/euclid.cmp/1103899849 }} while the coloured QMUPL model follows a non-Markovian Gaussian master equation.{{Cite journal|last1=Diósi|first1=L.|last2=Ferialdi|first2=L.|date=2014-11-12|title=General Non-Markovian Structure of Gaussian Master and Stochastic Schr\"odinger Equations|journal=Physical Review Letters|volume=113|issue=20|pages=200403|doi=10.1103/PhysRevLett.113.200403|pmid=25432028|arxiv=1408.1273|bibcode=2014PhRvL.113t0403D |s2cid=14535901}}{{Cite journal|last=Ferialdi|first=L.|date=2016-03-22|title=Exact Closed Master Equation for Gaussian Non-Markovian Dynamics|journal=Physical Review Letters|volume=116|issue=12|pages=120402|doi=10.1103/PhysRevLett.116.120402|pmid=27058061|arxiv=1512.07244|bibcode=2016PhRvL.116l0402F |s2cid=206271698}}

• Quantum decoherence
• Penrose interpretation
• Interpretations of quantum mechanics

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Category:Interpretations of quantum mechanics

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