The Ghirardi–Rimini–Weber theory (GRW) is a spontaneous
collapse theory in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, proposed in 1986 by
Giancarlo Ghirardi
Giancarlo Ghirardi (28 October 1935 – 1 June 2018) was an Italian physicist and emeritus professor of theoretical physics at the University of Trieste.
He is well known for the Ghirardi–Rimini–Weber theory (GRW), which he proposed in 1985 t ...
, Alberto Rimini, and Tullio Weber.
Measurement problem and spontaneous collapses
Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
has two fundamentally different dynamical principles: the linear and deterministic
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
, and the nonlinear and stochastic
wave packet reduction postulate. The orthodox interpretation, or
Copenhagen interpretation
The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as featu ...
of quantum mechanics, posits a
wave function collapse
In quantum mechanics, wave function collapse occurs when a wave function—initially in a quantum superposition, superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is ...
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
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
. The aforementioned issues constitute the
measurement problem
In quantum mechanics, the measurement problem is the problem of how, or whether, wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key se ...
in quantum mechanics.
Collapse theories
Objective-collapse theories, also known as models of spontaneous wave function collapse or dynamical reduction models, are proposed solutions to the measurement problem in quantum mechanics. As with other theories called interpretations of quan ...
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 Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of findi ...
). The imprecise talk of “observer” and a “measurement” that plagues the orthodox interpretation is 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 the field developed and different models were proposed, among which the
CSL model,
which is formulated in terms of identical particles; the
Diósi–Penrose model The Diósi–Penrose model was introduced as a possible solution to the measurement problem, where the wave function collapse is related to gravity. The model was first suggested by Lajos Diósi when studying how possible gravitational fluctuations ...
,
which relates the spontaneous collapse to gravity; the QMUPL model,
that proves important mathematical results on collapse theories; the coloured QMUPL model, the only collapse model involving coloured stochastic processes for which the exact solution is known.
The theory
The first assumption of the GRW theory is that the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
(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
An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraord ...
, and distinguishes it from
hidden variable theories
In physics, hidden-variable theories are proposals to provide explanations of quantum mechanical phenomena through the introduction of (possibly unobservable) hypothetical entities. The existence of fundamental indeterminacy for some measurem ...
, like the
de Broglie–Bohm theory
The de Broglie–Bohm theory, also known as the ''pilot wave theory'', Bohmian mechanics, Bohm's interpretation, and the causal interpretation, is an interpretation of quantum mechanics. In addition to the wavefunction, it also postulates an act ...
, 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. For more philosophical issues related to the GRW theory and to
collapse theories
Objective-collapse theories, also known as models of spontaneous wave function collapse or dynamical reduction models, are proposed solutions to the measurement problem in quantum mechanics. As with other theories called interpretations of quan ...
in general one should refer to.
Working principles
* Each particle of a system described by the multi-particle wave function
independently undergoes a spontaneous localization process (or jump):
,
where
is the state after the operator
has localized the
-th particle around the position
.
* The localization process is random both in space and time. The jumps are
Poisson distributed in time, with mean rate
; the probability density for a jump to occur at position
is
.
* The localization operator has a
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
form:
,
where
is the position operator of the
-th particle, and
is the localization distance.
* In between two localization processes, the wave function evolves according to the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
.
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
there is a probability
that a collapse occurs, i.e. that the pure state
is transformed into the following statistical mixture:
.
In the same time interval, there is a probability
that the system keeps evolving according to the Schrödinger equation. Accordingly, the GRW master equation for
particles reads
,
where
is the Hamiltonian of the system, and the square brackets denote a
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
.
Two new parameters are introduced by the GRW theory, namely the collapse rate
and the localization distance
. 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
,
while more recently
Stephen L. Adler proposed that the value
(with an uncertainty of two orders of magnitude) is more adequate. There is a general consensus on the value
for the localization distance. This is a mesoscopic distance, such that microscopic superpositions are left unaltered, while macroscopic ones are collapsed.
Examples
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
, centered at
, and let us assume that this undergoes a localization process at the position
. One thus has (in one dimension)
,
where
is a normalization factor. Let us further assume that the initial state is delocalised, i.e. that
. In this case one has
,
where
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
and
respectively: