Why Quantum Mechanics Is An Inconsistent Theory

wikipedia  | 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, R. 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. However, it should be pointed out that while Diósi gave a precise dynamical equation for the collapse,^[2] Penrose took a more conservative approach, estimating only the collapse time of a superposition.^[3]

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. Penrose suggests an opposite approach, what he calls “gravitization of quantum mechanics”, where quantum mechanics gets modified when gravitational effects become relevant.^{[3][4][9][11][12][13]} The reasoning underlying this approach is the following one: take a massive system well-localized states in space. In this case, being the state well-localized, the induced space–time curvature is well defined. According to quantum mechanics, because of the superposition principle, the system can be placed (at least in principle) in a superposition of two well-localized states, which would lead to a superposition of two different space–times. 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

${\displaystyle \mathrm{\Delta }{E}_G=\frac{1}{G}\int d\mathit{r}(g_a(\mathit{r})-g_b(\mathit{r}){)}^2,}$

(9)

where ${\displaystyle g_i(\mathit{r})}$ is the Newtoninan gravitational acceleration at the point where the system is localized around ${\displaystyle i}$. The acceleration ${\displaystyle g_i(\mathit{r})}$ can be written in terms of the corresponding gravitational potentials ${\displaystyle {\mathrm{\Phi }}_i(\mathit{r})}$, i.e. ${\displaystyle g_i(\mathit{r})=-\mathrm{\nabla }{\mathrm{\Phi }}_i(\mathit{r})}$. Using this relation in Eq. (9), together with the Poisson equation ${\displaystyle {\mathrm{\nabla }}^2{\mathrm{\Phi }}_i(\mathit{r})=4\pi G{\mu }_i(\mathit{r})}$, with ${\displaystyle {\mu }_i(\mathit{r})}$ giving the mass density when the state is localized around ${\displaystyle i}$, and its solution, one arrives at

${\displaystyle \mathrm{\Delta }{E}_G=4\pi G\int d\mathit{r}\int d{\mathit{r}}^{\prime }\frac{[{\mu }_a(\mathit{r})-{\mu }_b(\mathit{r})][{\mu }_a({\mathit{r}}^{\prime })-{\mu }_b({\mathit{r}}^{\prime })]}{|\mathit{r}-{\mathit{r}}^{\prime }|}.}$

(10)

The corresponding decay time can be obtained by the Heisenberg time–energy uncertainty:

${\displaystyle {\tau }_G=\frac{\hslash }{\mathrm{\Delta }{E}_G},}$

(11)

which, apart for a factor ${\displaystyle 8\pi }$ simply due to the use of different conventions, is exactly the same as the time decay ${\displaystyle {\tau }_{\text{DP}}}$ derived by Diósi's model. This is the reason why the two proposals are named together as the Diósi–Penrose model.

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 ${\displaystyle \mathit{g}}$. One way to perform the calculation, what Penrose calls “Newtonian perspective”,^[4][9] consists in working in an inertial frame, with space–time coordinates ${\displaystyle (\mathit{r},t)}$ and solve the Schrödinger equation in presence of the potential ${\displaystyle V(\mathit{x})=m\mathit{g}\cdot \mathit{x}}$ (typically, one chooses the coordinates in such a way that the acceleration ${\displaystyle \mathit{g}}$ is directed along the ${\displaystyle z}$ axis, in which case ${\displaystyle V(z)=mgz}$). Alternatively, because of the equivalence principle, one can choose to go in the free-fall reference frame, with coordinates ${\displaystyle (\mathit{R},T)}$ related to ${\displaystyle (\mathit{r},t)}$ by ${\displaystyle \mathit{R}=\mathit{r}+\frac{1}{2}\mathit{g}{t}^2}$ and ${\displaystyle T=t}$, solve the free Schrödinger equation in that reference frame, and then write the results in terms of the inertial coordinates ${\displaystyle (\mathit{r},t)}$. This is what Penrose calls “Einsteinian perspective”. The solution ${\displaystyle \mathrm{\Psi }(\mathit{r},t)}$ obtained in the Einsteinian perspective and the one ${\displaystyle \psi (\mathit{r},t)}$ obtained in the Newtonian perspective are related to each other by

${\displaystyle \mathrm{\Psi }(\mathit{r},t)=e^{i\frac{m}{\hslash }\left(\frac{1}{6}{g}^2{t}^3+\mathit{g}\cdot \mathit{r}t\right)}\psi (\mathit{r},t).}$

(12)

Being the two wave functions equivalent apart for an overall phase, they lead to the same physical predictions, which implies that there are no problems in this situation, when the gravitational field has always 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 ${\displaystyle {\mathit{g}}_a}$ and one corresponding to the acceleration ${\displaystyle {\mathit{g}}_b}$. This does not create problems as far as one sticks to the Newtonian perspective. However, when using the Einstenian perspective, it will imply a phase difference between the two branches of the superposition given by ${\displaystyle e^{i\frac{m}{\hslash }\left(\frac{1}{6}({\mathit{g}}_a-{\mathit{g}}_b{)}^2{t}^3+({\mathit{g}}_a-{\mathit{g}}_b)\cdot \mathit{r}t\right)}}$. While the term in the exponent linear in the time ${\displaystyle t}$ does not lead to any conceptual difficulty, the first term, proportional to ${\displaystyle t^3}$, 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. Here is where the gravity-induced collapse plays a role, collapsing the superposition when the first term of the phase ${\displaystyle \frac{1}{6}(g_a-g_b{)}^2{t}^3}$ becomes too large.