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 may affect the dynamics of quantum systems.
[1][2] Later, following a different line of reasoning, Roger Penrose arrived at an estimation for the collapse time of a superposition due to gravitational effects, which is the same (within an unimportant numerical factor) as that found by Diósi, hence the name Diósi–Penrose model.
[3] In the Diósi model, the wave-function collapse is induced by the interaction of the system with a classical noise field, where the spatial correlation function of this noise is related to the Newtonian potential.
is a parameter introduced to smear the mass density function, required since taking a point-like mass distribution would lead to divergences in the predictions of the model, e.g. an infinite collapse rate[4][5] or increase of energy.
have been considered in the literature: a spherical or a Gaussian mass density profile, given respectively by and Choosing one or another distribution
Similarly to other collapse models, the Diósi–Penrose model shares the following two features: In order to show these features, it is convenient to write the master equation for the statistical operator
(1): It is interesting to point out that this master equation has more recently been re-derived by L. Diósi using a hybrid approach where quantized massive particles interact with classical gravitational fields.
, i.e. the time of decay goes to infinity, implying that states with well-localized position are not affected by the collapse.
To get an idea of the scale at which the gravitationally induced collapse becomes relevant, one can compute the time of decay in Eq.
s. Therefore, contrary to what might be expected considering the weaknesses of gravitational force, the effects of the gravity-related collapse become relevant already at the mesoscopic scale.
Recently, the model have been generalized by including dissipative[7][10] and non-Markovian[11] effects.
It is well known that general relativity and quantum mechanics, our most fundamental theories for describing the universe, are not compatible, and the unification of the two is still missing.
The standard approach to overcome this situation is to try to modify general relativity by quantizing gravity.
[3][4][9][12][13][14] The reasoning underlying this approach is the following one: take a massive system of well-localized states in space.
The key idea is that since space–time metric should be well defined, nature “dislikes” these space–time superpositions and suppresses them by collapsing the wave function to one of the two localized states.
To set these ideas on a more quantitative ground, Penrose suggested that a way for measuring the difference between two space–times, in the Newtonian limit, is where
, and its solution, one arrives at The corresponding decay time can be obtained by the Heisenberg time–energy uncertainty: which, apart for a factor
More recently, Penrose suggested a new and quite elegant way to justify the need for a gravity-induced collapse, based on avoiding tensions between the superposition principle and the equivalence principle, the cornerstones of quantum mechanics and general relativity.
In order to explain it, let us start by comparing the evolution of a generic state in the presence of uniform gravitational acceleration
One way to perform the calculation, what Penrose calls “Newtonian perspective”,[4][9] consists in working in an inertial frame, with space–time coordinates
Alternatively, because of the equivalence principle, one can choose to go in the free-fall reference frame, with coordinates
, solve the free Schrödinger equation in that reference frame, and then write the results in terms of the inertial coordinates
obtained in the Newtonian perspective are related to each other by Since the two wave functions are equivalent apart from an overall phase, they lead to the same physical predictions, which implies that there are no problems in this situation where the gravitational field always has a well-defined value.
However, if the space–time metric is not well defined, then we will be in a situation where there is a superposition of a gravitational field corresponding to the acceleration
However, when using the Einstenian perspective, it will imply a phase difference between the two branches of the superposition given by
, is problematic, since it is a non-relativistic residue of the so-called Unruh effect: in other words, the two terms in the superposition belong to different Hilbert spaces and, strictly speaking, cannot be superposed.
The only free parameter of the model is the size of the mass density distribution, given by
This Brownian-like diffusion is a common feature of all objective-collapse theories and, typically, allows to set the strongest bounds on the parameters of these models.
m to avoid unrealistic heating due to this Brownian-like induced diffusion.