Girardi Theory - Rimini - Weber

The Girardi-Rimini-Weber theory or GDV theory ( English Ghirardi-Rimini-Weber theory, GRW ) is one of the theories of the objective collapse of the wave function in quantum mechanics. The theory tries to solve the measurement problem and fill the gap in the Copenhagen interpretation by answering the question of how the wave function collapses.

The GDV theory differs from other theories of objective collapse in that the collapse of the wave function occurs spontaneously, without the intervention of an external measurement. Such an approach makes it possible to solve the measurement problem, in particular, to answer the question of where and when the quantum system, which was initially in a superposition state, goes over to the unambiguous results observed at the macroscopic level using a measuring device.

GDV theory was proposed in 1985 by Italian physicists Giancarlo Girardi , Alberto Rimini and Tullio Weber ^[1] ^[2] .

Content

Theory Formulation

In the theory of GDV, it is believed that a particle described by a wave function can undergo spontaneous, random localization (collapse). This localization is a process, as a result of which the superposition of the quantum state in which the particle is located is destroyed, and the wave function becomes a certain eigenstate of the coordinate operator. Due to spontaneity, such localization does not depend on whether coordinates were previously measured. On the contrary, the Copenhagen interpretation postulates that the collapse of the wave function occurs as a result of the measurement on the system, therefore, performing measurements of the same observable several times, the same result will be obtained.

GDV theory claims that the spatial wave function $N$ ${\ displaystyle N}$ particles $\Psi (t,\mathbf {r} _{1},\mathbf {r} _{2},...,\mathbf {r} _{N})$ ${\ displaystyle \ Psi (t, \ mathbf {r} _ {1}, \ mathbf {r} _ {2}, ..., \ mathbf {r} _ {N})}$ evolves in time according to the Schrödinger equation , but sometimes it can experience a “jump” and go to another wave function $\Psi '$ ${\ displaystyle \ Psi '}$ with probability $N/\tau$ ${\ displaystyle N / \ tau}$ per unit of time. Value $\tau$ ${\ displaystyle \ tau}$ represents a new fundamental constant having the dimension of time. Since spontaneous collapse was never observed in microscopic systems, Girardi, Rimini, and Weber postulated that $\tau$ ${\ displaystyle \ tau}$ must have a very large value, of the order of 10 ¹⁵ seconds (that is, the frequency of spontaneous collapse for an individual particle will be of the order of one case in one hundred million years) ^[3] . With increasing $N$ ${\ displaystyle N}$ (transition to macroscopic systems) the likelihood of spontaneous localization also increases. The wave function is localized in an extremely short period of time, therefore, a superposition of the states of the macroscopic system will also exist only for a very short time, which practically excludes the observation of such states. New “reduced”, or “collapsed”, wave function $\Psi '$ ${\ displaystyle \ Psi '}$ in theory, GDV has the form:

\Psi '_{n}(t,\mathbf {r} _{1},...,\mathbf {r} _{N})={\frac {j(\mathbf {x} -\mathbf {r} _{n})\Psi (t,\mathbf {r} _{1},...,\mathbf {r} _{N})}{R_{n}(\mathbf {x} )}}

{\ displaystyle \ Psi '_ {n} (t, \ mathbf {r} _ {1}, ..., \ mathbf {r} _ {N}) = {\ frac {j (\ mathbf {x} - \ mathbf {r} _ {n}) \ Psi (t, \ mathbf {r} _ {1}, ..., \ mathbf {r} _ {N})} {R_ {n} (\ mathbf {x })}}}

,

Where $\mathbf {r} _{n}$ ${\ displaystyle \ mathbf {r} _ {n}}$ randomly selected from a set $\{\mathbf {r} _{1},...,\mathbf {r} _{N}\}$ ${\ displaystyle \ {\ mathbf {r} _ {1}, ..., \ mathbf {r} _ {N} \}}$ , $j(\mathbf {x} )$ ${\ displaystyle j (\ mathbf {x})}$ Is the function normalized to unity from space $L^{2}$ ${\ displaystyle L ^ {2}}$ , but $R_{n}$ ${\ displaystyle R_ {n}}$ - normalizing factor equal to

|R_{n}(\mathbf {x} )|^{2}=\int \mathrm {d} \mathbf {r} _{1}...\mathrm {d} \mathbf {r} _{N}|j\psi |^{2}

{\ displaystyle | R_ {n} (\ mathbf {x}) | ^ {2} = \ int \ mathrm {d} \ mathbf {r} _ {1} ... \ mathrm {d} \ mathbf {r} _ {N} | j \ psi | ^ {2}}

.

Collapse Center $\mathbf {x}$ ${\ displaystyle \ mathbf {x}}$ randomly selected by probability density $|R_{n}(\mathbf {x} )|^{2}$ ${\ displaystyle | R_ {n} (\ mathbf {x}) | ^ {2}}$ . As a function $j(\mathbf {x} )$ ${\ displaystyle j (\ mathbf {x})}$ Girardi, Rimini and Weber suggested using a Gaussian :

j(\mathbf {x} )=Ke^{-{\frac {\mathbf {x} ^{2}}{2a^{2}}}},

{\ displaystyle j (\ mathbf {x}) = Ke ^ {- {\ frac {\ mathbf {x} ^ {2}} {2a ^ {2}}}},}

Where $a$ ${\ displaystyle a}$ - the second fundamental constant arising in the theory of GDV and a component of the order of 10 ⁻⁷ meters.

Using the assumptions of the GDV theory formulated here, it can be proved that its predictions do not contradict the predictions of quantum mechanics obtained in the framework of the Copenhagen interpretation. The difference is that the GDV theory mathematically describes the collapse of the wave function, while the Copenhagen interpretation considers it only empirically ^[3] .

Theory Problems

The main problem of the initial model of spontaneous localization of the Girardi, Rimini, and Weber wave function is its inability to describe symmetric or antisymmetric permutations of identical particles ^[3] . In 1990, the theory of GDV was generalized to the case of such systems of Girardi, Perlom, and Rimini, which proposed a model of continuous spontaneous localization (NSL, English continuous spontaneous localization, CSL ). Another problem remains the construction of the relativistic theory of collapse: similar models were independently proposed by Roderich Tumulka and Giancarlo Girardi, however, there are still active discussions in the scientific community around these models.

Notes

↑ Ghirardi GC, Rimini A., Weber T. A model for a unified quantum description of macroscopic and microscopic systems // Quantum Probability and Applications. - Springer, 1985. - P. 223–232. - DOI : 10.1007 / BF02817189 .
↑ Ghirardi GC, Rimini A., Weber T. Unified dynamics for microscopic and macroscopic systems // Phys. Rev. D. - 1986. - Vol. 34. - P. 470-491. - DOI : 10.1103 / PhysRevD.34.470 .
↑ ¹ ² ³ Bell JS Speakable and Unspeakable in Quantum Mechanics. - Cambridge University Press, 2004. - P. 201–212.

Literature

Bassi A., Ghirardi GC Dynamical Reduction Models (Eng.) // Phys. Rep. - 2003. - Vol. 379. - P. 257-426. - DOI : 10.1016 / S0370-1573 (03) 00103-0 . - arXiv : quant-ph / 0302164 .
Bassi A., Lochan K., Satin S., Singh TP, Ulbricht H. Models of Wave-function Collapse, Underlying Theories, and Experimental Tests (Eng.) // Rev. Mod. Phys. - 2013 .-- Vol. 85. - P. 471-527. - DOI : 10.1103 / RevModPhys.85.471 . - arXiv : 1204.4325 .
Ghirardi GC Un'occhiata alle carte di Dio: Gli interrogativi che la scienza moderna pone all'uomo. - Il Saggiatore, 1997.
Penrose R. Shadows of the Mind = Shadows of Mind. - Izhevsk: IKI, 2005 .-- S. 688.

[1] Ghirardi GC, Rimini A., Weber T. A model for a unified quantum description of macroscopic and microscopic systems // Quantum Probability and Applications. - Springer, 1985. - P. 223–232. - DOI : 10.1007 / BF02817189 .

[2] Ghirardi GC, Rimini A., Weber T. Unified dynamics for microscopic and macroscopic systems // Phys. Rev. D. - 1986. - Vol. 34. - P. 470-491. - DOI : 10.1103 / PhysRevD.34.470 .

[bell-3] ¹ ² ³ Bell JS Speakable and Unspeakable in Quantum Mechanics. - Cambridge University Press, 2004. - P. 201–212.