# Nonlinear non-signaling Schrödinger equation

Tamás Geszti  Department of the Physics of Complex Systems,
Eötvös University; H-1117 Budapest, Hungary
e-mail<EMAIL_ADDRESS>
(01 March 2024)

###### Abstract

In looking for new insight into the nature of the quantum-classical border
still unclear after a century of research, a new kind of nonlinear extension
of Schrödinger’s wave equation is proposed. It ensures non-signaling by
keeping linear the evolution of coordinate-diagonal elements of the density
matrix. The nonlinear part of the equation includes a negative kinetic energy
operator characterized by a universal critical mass $\mu$, estimated to be
about $2\cdot 10^{-23}~{}$kg. As some effective mass $M$ belonging to a
pointer variable $x$ grows beyond $\mu$, spreading of wave packets along $x$
turns into its reverse: collapsing, which marks the quantum-classical border.
Interference of large molecules is suggested for an experimental check of the
proposed framework. Phase differences among terms of a prepared superposition
are forced by the non-signaling condition to be preserved during collapse; a
possible reason for the observed remote correlations.

PACS numbers: 03.65.-w, 03.65.Ta, 04.40.-b

The nature of the quantum-classical border is one of the great unsolved
questions of 20-21th century physics, with various attempts competing, without
any broadly accepted outcome so far. Atoms, molecules and any kind of groups
of subatomic particles entangled by coming from a common source or just once
interacted, move like a single wave in their configuration space, accurately
described by Schrödinger’s linear wave equation. That explains many
properties, including spectra, chemical bonding, various kinds of
interferometry to an accuracy limited by our measuring and calculating
abilities only: wave-like motion is a fact. However, to observe those waves,
one needs some very special tool: detectors, which are metastable systems,
eventually flipped over by arriving waves of atoms, electrons, photons etc. to
give a signal of visible size. Then, as demonstrated by a huge amount of
experiments, nonlinear phenomena emerge: as soon as a superposition would grow
into macroscopically different terms - the famous ”Schrödinger’s cat” - it
collapses into an apparently randomly chosen term of it. More than that,
during measurement - in a so far unknown way - the incoming entanglement is
translated into strange correlations between remote events, often reminding of
some compact object - particle - going one way or another. Although everyday
philosophy might suggest that those correlations should be just consequences
of the common source acting as common cause, the necessarily nonlinear
dynamics describing the emergence of remote correlations could not be found
during many decades, giving ample space to fantasies dominated by the Einstein
bonmot ”spooky action-at-a-distance”. This Letter offers a way to see how the
common cause scenario can work, tracing back remote correlations to entangled
initial conditions, followed by fully local nonlinear dynamics Griffiths .

History of the topic has long been dominated by an extremely successful
phenomenology: the Copenhagen interpretation, connecting Schrödinger’s wave
function to measurement statistics through the shortcut of Born’s rule, at the
price of admitting to regard the wave function just a tool to calculate, not a
real physical quantity. That describes the observed statistics with high
accuracy, down to refined properties like all kinds of experiments violating
Bell-CHSH and GHZ inequalities Brunneretal ; ChshGhz . However, it opens up
the way towards mysteries like parallel worlds, or the idea of collapse being
a consequence of looking at the detector, or the cat; besides, attributing
physical reality to the seeming particles is a source of apparent backwards-
in-time causal influences Aharonov . From a more philosophical viewpoint,
restricting ourselves to statistics, and getting forced to give up any insight
to the enormous variety of individual events happening through wavelike
motion, is a legitimate source of dissatisfaction EprPbr . Admitting
nonlinearity would offer hope to find an escape from all that, clarifying the
dynamical origin of quantum randomness, and gaining insight into the whole
dynamical background of the quantum-classical transition, including Born’s
rule; last not least, offering an explanation to the mysterious-looking remote
correlations.

Beside a great variety of stochastic models that - by postulating some kind of
inherent randomness - offer a lot of insight beyond the Copenhagen
interpretation collapse , there have been various attempts to find valid
nonlinear extensions to Schrödinger-type wave equations Bialynicki ; Weinberg
; Polchinski ; Elze ; Vecsi , some exploring hydrodynamic analogies centered
around a kind of continuum equation Doebner ; Richardson ; Sbitnev  that might
offer some taste of locality. Although the proposed equations remain at the
nonrelativistic limit, a strong relativity-based requirement appeared soon
that has proved extremely useful in restricting the possibilities: one should
avoid the possibility of ”signaling”, i.e. influencing remote events by
locally imposed changes acting at a distance in no time; in the language of
relativity: at superluminal speed Gisin . General as that requirement sounds,
it appeared immediately in a technically useful form: noticing that signaling
scenarios typically build upon manipulating linear combinations of nonlinearly
evolving density matrices, one concludes that nonlinear dynamics of the wave
function can still be admitted, if a much looser necessary condition is
obeyed: one should preserve linear evolution of the density matrix

$\varrho\left(x,x^{\prime},t\right)=\Psi(x,t)\Psi^{*}(x^{\prime},t),$ (1)

where $\Psi(x,t)$ is a pure-state wave function, depending on time $t$ and
configuration space vector $x$, the multi-local components of which include
the full microscopic coordinates of all detectors involved in the measurement
process, along with all their environments Tsirelson . Although that approach
looks a highly promising way to follow, to take that escape from the catch of
signaling by any deterministic nonlinear wave equation has been demonstrated
to be impossible BassiHejazi . The present note is about opening a way round
that final obstacle, by taking one more step back, and admitting a relieved
version of the requirement of density matrix linearity that makes the task
solvable.

Our starting point is the observation that whenever using the density matrix
(1) to calculate the mean value of any local operator in the configuration
space, toward the end one always takes the limit $x^{\prime}\to x$.
Accordingly, to grant non-signaling one need not require linear evolution of
the whole density matrix; it suffices to assure linear evolution of its
coordinate-diagonal elements $\lim_{x^{\prime}\to
x}\varrho\left(x,x^{\prime},t\right)$.

To turn that into a practical requirement, we go back to definition (1) and
introduce the upper or lower index ”${nl}$” as a shorthand notation for
nonlinear contributions to a time derivative. Then the limit $x^{\prime}\to x$
can be taken directly, presenting the non-signaling condition in the final
form

$\lim_{x^{\prime}\to
x}\partial_{t}^{nl}\left(\Psi(x,t)\Psi^{*}(x^{\prime},t)\right)=\Psi^{*}\dot{\Psi}_{nl}+\Psi\dot{\Psi}_{nl}^{*}=0,$
(2)

a just-strong-enough necessary condition, which is mild enough to be
compatible with a deterministic nonlinear wave equation.

Looking for an appropriate nonlinear extension of Schrödinger’s equation that
satisfies the above condition, we observe that $\Psi^{*}\dot{\Psi}_{nl}$ and
$\Psi\dot{\Psi}_{nl}^{*}$ are complex conjugate quantities adding up to zero,
therefore - in accordance with Eq. (2) - they should satisfy a pair of
conjugate equations

$\Psi^{*}\dot{\Psi}_{nl}=-i~{}\mathbb{K}(\Psi,\Psi^{*});\qquad\Psi\dot{\Psi}_{nl}^{*}=i~{}\mathbb{K}(\Psi,\Psi^{*}),$
(3)

containing the same real-valued quantity $\mathbb{K}(\Psi,\Psi^{*})$ to be
specified by further requirements. Obviously, it should be nonlinear to admit
collapse; besides, to avoid any by-pass appearance of signaling, it should be
kept fully local, excluding any kind of integral or any scalar product of
operators.

To step further, some preliminary idea is needed about the way collapse may
happen. As a so far unexplored territory, we choose this: collapse is the
reverse of the well-known spreading of a wave packet, which is driven by the
kinetic energy, and can be reversed by adding some kind of negative kinetic
energy $(+\hbar^{2}/2\mu)\Delta$ with a characteristic mass $\mu$ negkin . A
simple real-valued, local expression, nonlinear in $\Psi(x,t)$ and
$\Psi^{*}(x,t)$, including that negative kinetic energy is this:

$\mathbb{K}:=\frac{\hbar}{4\mu}\left(\Psi\Delta\Psi^{*}+\Psi^{*}\Delta\Psi\right),$
(4)

where the strength of nonlinearity is characterized by the constant $\mu$,
postulated as a universal mass marking the quantum-classical border. Linear
quantum mechanics is valid for masses much smaller than $\mu$; nonlinearity is
taking the lead as an effective mass $M$ appearing in the linear kinetic
energy $(-\hbar^{2}/2M)\Delta$ is growing too large.

Accepting that choice, and including a linear Hamiltonian $H_{lin}$ carrying
all relevant properties of the system actually studied, we arrive at the
conjugate pair of nonlinear Schrödinger equations

$\dot{\Psi}=-\frac{i}{\hbar}\left(H_{lin}\Psi+\frac{\hbar^{2}}{4\mu}\left(\frac{\Psi}{\Psi^{*}}\Delta\Psi^{*}+\Delta\Psi\right)\right);\quad\dot{\Psi}^{*}=\frac{i}{\hbar}\left(H_{lin}\Psi^{*}+\frac{\hbar^{2}}{4\mu}\left(\frac{\Psi^{*}}{\Psi}\Delta\Psi+\Delta\Psi^{*}\right)\right)$
(5)

interconnecting the dynamics of the conjugate waves $\Psi(x,t)$ and
$\Psi^{*}(x,t)$, to be used in the rest of the present study normpreserving .

As a first technical check, we mention that for remote, uncorrelated objects
labeled by some index $\alpha$, the wave function is in a direct-product state

$\Psi(x,t)=\prod_{\alpha}\psi_{\alpha}\left(x_{\alpha},t\right).$ (6)

Since in that case the ratio $\Psi/\Psi^{*}$ is a direct product too, and

$\Delta=\sum_{\alpha}\Delta_{\alpha}$ (7)

is a sum over the configuration space dimensions belonging to the different
remote objects, Equations (5) are separated into local ones - a basic
requirement towards nonlinear extensions of Schrödinger’s equation.

The above equations apparently entail a continuity equation Doebner ;
Richardson ; Sbitnev  as required by matter conservation normpreserving .
There is an important bonus to that: although non-signaling would allow some
nonlinear current of zero divergence, here the conserved current is fully due
to linear quantum dynamics, with no contribution from nonlinearity, since
obviously
$(\Psi^{*}\nabla\Psi-\Psi\nabla\Psi^{*})+(\Psi\nabla\Psi^{*}-\Psi^{*}\nabla\Psi)$
is vanishing, not only its divergence. Luckily, this excludes violation of
mass additivity across the quantum-classical border, empirically known to be
valid from atomic to macroscopic scales, ever since the 19th-century uprise of
atom-based chemistry, linked to the discoveries of Dalton and Avogadro. To
violate that rule would be another possible harmful side effect of nonlinear
quantum dynamics; our approach avoids that catch too.

As mentioned above, the nonlinear terms in Eqs. (5) contain the Laplacian
divided by a positive $\mu$ acting as a kind of negative kinetic energy,
piloting wave packets towards collapse. As a basic example, take a particle
detector with an avalanche of growing effective mass $M$ that moves along some
”pointer variable” $x$ under driving force $-\nabla V(x)$; then a
$1\mathrm{D}$ toy model is described by an effective wave function
$\Psi(x,t)$, obeying the nonlinear Schrödinger equation

$\dot{\Psi}=-\frac{i}{\hbar}\left(-\frac{\hbar^{2}}{2M}\Delta\Psi+V(x)\Psi+\frac{\hbar^{2}}{4\mu}\left(\frac{\Psi}{\Psi^{*}}\Delta\Psi^{*}+\Delta\Psi\right)\right).$
(8)

As $M$ outgrows $\mu$, nothing happens to the nonlinear part of the
Hamiltonian; it is the linear kinetic energy
$-\left(\hbar^{2}/2M\right)\Delta$ which is going negligible. As a
consequence, the well-known spreading of wave packets, driven by quantum
kinetic energy under the control of Coulomb forces over the whole microscopic
quantum world, gradually disappears; instead, its reverse: shrinking takes the
lead.

Nonlinearity enters Eqs. (5) in the form of the ratio $\Psi/\Psi^{*}$. The
Madelung-Bohm representation Mad

$\Psi=A\exp(i\varphi)$ (9)

arranges conjugate pairs $\left(\Psi,\Psi^{*}\right)$ according to ”hyperbolic
coordinates” $A=\sqrt{\Psi\Psi^{*}}$ and
$\varphi=-i\ln{\sqrt{\Psi/\Psi^{*}}}$. The present approach is focusing on the
active dynamical role of phase, excluding any direct change of the amplitude
by nonlinearity, thereby satisfying the non-signaling requirement. In this
representation Eq. (8) is displayed in two coupled equations: a continuity
equation

$\partial_{t}A^{2}=-\vec{\nabla}\cdot\mathbf{j};\qquad\mathbf{j}~{}=~{}A^{2}~{}\frac{\hbar\vec{\nabla}\varphi}{M},$
(10)

which - as required by non-signaling - contains no nonlinear contribution to
the current, and an equation for the evolution of the phase $\varphi(x,t)$,

$\dot{\varphi}=\frac{\hbar}{2}\left(-\frac{1}{M}+\frac{1}{\mu}\right)\left(-\frac{\Delta
A}{A}+|\nabla\varphi|^{2}\right)-\frac{1}{\hbar}V.$ (11)

A striking thing to notice in the above couple of equations is that although
the conserved current $\mathbf{j}$ as it appears in Eq. (10) contains no
nonlinear part, its time derivative does have a nonlinear contribution through
phase dynamics, Eq. (11). That is a hint that the measurement process is an
interplay between nonlinear and linear dynamics, and visible collapse happens
in two steps: first changing phases with active contribution from
nonlinearity, then - as a consequence - amplitudes, changed by the linear
currents induced by phase changes. That seems to follow the Lagrangian two-
step way to Newtonian dynamics: coordinates determine forces that change
momenta, finally influencing coordinates.

In Eq. (11) the quantum-classical boundary appears as a kind of phase
transition, taking place as soon as its control variable: the growing
effective mass $M$ is crossing the critical value $\mu$. In particular, the
r.h.s. of Equation (11) suggests an inspiring analogy between quantum collapse
and the roughening of growing surfaces Rácz ; Vicsek , both being driven by a
term like $|\nabla\varphi|^{2}$ as soon as its coefficient turns positive,
making smoothness unstable in a noisy environment KPZ . As for collapse, that
noisy environment may turn out to be a back-action of the ”einselection”
process Zurek ; Paternostro .

Now we turn to check the chances of finding evidence for the nonlinear
scenario introduced here in non-measurement situations. This starts by an
order-of-magnitude estimate of the critical mass $\mu$; for that - following
Ref. Geszti2018  \- we consider the mass of the growing electron avalanche
desired to produce a clear signal by an avalanche photon detector. According
to review photondetector , that requires an avalanche of $2\cdot 10^{7}$
electrons, the mass of which is $\mu=2\cdot 10^{-23}~{}$kg; that is our first
estimate of the critical mass appearing in the above formalism to mark the
quantum-classical border. It is 15 orders of magnitude smaller than the Planck
mass, which seems to exclude any direct hint to gravitational origin grav ; it
is just a measure of nonlinearity, marking the borderline of the microscopic
quantum world. Looked from the other side: the present framework has a good
chance to build up peaceful coexistence with general relativity, like any
other approach in which randomness is emergent from deterministic-chaotic
dynamics.

Experiments approach the borderline drawn by the above estimate of mass $\mu$
from both sides: nanomechanical resonators from above, molecular interference
from below. As for nanomechanics NANO , doubly clamped carbon nanotube
resonators now reach the lowest mass of $10^{-21}$ kg, not too promising to
look for quantum nonlinearity, especially when you consider the side-effects
of mechanical nonlinearities. The lower side looks more interesting: molecules
of $10^{-25}$ kg have been demonstrated to interfere Arndt , mentioning that
to detect interference of molecules heavier by a factor of 100 looks
technically feasible, however, so far not yet confirmed. That seems to be a
hot territory to look for effects of quantum nonlinearity of the kind
suggested here; the actual effect may be an uprising tendency to turn
spreading into shrinking, that may block the beam widening necessary to
observe few-slit interference phenomena.

Although the above suggestion to a large-molecule interference test is the
main result of the present note, the way the non-signaling condition (2)
influences the dynamics described by Eqs. (5) is worth more discussion. The
main philosophy is this: while incoming entanglement is converted by nonlinear
dynamics into outcome statistics, non-signaling acts as a bunch of hard
restrictions, forbidding to forget incoming data. That is an extended variant
of Pearle’s Martingale principle Pearle , serving as a way to explain remote
correlations through preserved input, with no action-at-a-distance. For a
first insight into the way all that can be hosted by our mathematical
framework, consider an arrangement of incoming waves from the same source,
undergoing some kind of Stern-Gerlach-type separation (also called ”Von
Neumann measurement”). Subsequently each of the respectively placed detectors
meets its partial beam, thereby the detectors become part of the entangled
state

$\Psi=\sum_{k}u_{k}=\sum_{k}\prod_{\alpha}u_{k,\alpha},$ (12)

where $u_{k,\alpha}$ is the local state of the $\alpha$’th detector locally
entangled with the arriving partial beam, in the $k$’th term of the fully
entangled state. Substituting that structure as an Ansatz into the sum and
difference of Eqs. (3) and using the definition (4), one obtains two
equations, one for the non-signaling condition, the other for the dynamics
permitted by the first:

$\begin{split}\sum_{k}\sum_{l}\sum_{\alpha}\prod_{\beta\not=\alpha}&u^{*}_{k\beta}u_{l\beta}\bigl{(}u^{*}_{k\alpha}\partial_{t}^{nl}u_{l\alpha}+u_{l\alpha}\partial_{t}^{nl}u^{*}_{k\alpha}\bigr{)}~{}=~{}0;\\\
\sum_{k}\sum_{l}\sum_{\alpha}\prod_{\beta\not=\alpha}u^{*}_{k\beta}u_{l\beta}\bigl{(}u^{*}_{k\alpha}\partial_{t}^{nl}&u_{l\alpha}-u_{l\alpha}\partial_{t}^{nl}u^{*}_{k\alpha}+i\frac{\hbar}{2\mu}(u^{*}_{k\alpha}\Delta
u_{l\alpha}+u_{l\alpha}\Delta u^{*}_{k\alpha})\bigr{)}~{}=~{}0.\end{split}$
(13)

The above equations are apparently nonlocal in view of the nonlocal weights
$u^{*}_{k\beta}u_{l\beta}$ with $\beta\not=\alpha$. However, the structure
offers an escape from the catch: if at any place $\alpha$, any pair of local
components $(u^{*}_{k\alpha},u_{l\alpha})$ satisfies a pair of local
equations:

$\partial_{t}^{nl}\ln(u^{*}_{k\alpha}u_{l\alpha})~{}=~{}0$ (14)

for the non-signaling condition, and

$\partial_{t}^{nl}\ln\Bigl{(}\frac{u^{*}_{k\alpha}}{u_{l\alpha}}\Bigr{)}=i\frac{\hbar}{2\mu}\Bigl{(}\frac{\Delta
u^{*}_{k\alpha}}{u^{*}_{k\alpha}}+\frac{\Delta
u_{l\alpha}}{u_{l\alpha}}\Bigr{)}$ (15)

for the dynamics, then what happens at location $\alpha$ is going on
independently of whatever happens at different locations: ”spooky action-at-a-
distance” has been excluded. Eq. (14) immediately carries one more important
point: by the local Madelung-Born representation
$u_{k\alpha}=a_{k\alpha}\exp_{k\alpha}(i\varphi_{k\alpha})$ it is turned into
two equations:

$\partial_{t}^{nl}\ln(a_{k\alpha}a_{l\alpha})~{}=~{}0;\qquad\dot{\varphi}_{k}^{nl}-\dot{\varphi}_{l}^{nl}~{}=~{}0,$
(16)

the second of which is a particularly strong clue for the origin of remote
correlations: non-signaling forces nonlinear evolution to preserve initially
prepared phase differences among terms of the incoming superposition, much in
the Martingale way Pearle .

Further details of the two-step dynamics appearing in Eqs. (10,11) will be the
subject of forthcoming research. In the meantime we mention that besides the
routinely measurement- and non-measurement-based tests of various approaches
to the problem of the quantum-classical border, any proposed solution may
leave its footprint on several other phenomena. An example that seems worth
the closer investigation is local quantum scattering events leaving traces
spreading in their environment towards the macroscopic scale, as happens e.g.
in electric conduction; the slow dynamics of that can be a source of
$1/\mathrm{f}$ noise (flicker noise), a phenomenon explained in some special
cases but not understood in its observed generality.

A less studied aspect of noise dynamics is this: detector bias thresholds are
usually set empirically to eliminate noise that would violate Born’s rule.
Nonlinear quantum mechanics may have a chance to get insight into details of
that.

Finally, the answer to ”how does a detector know what the other is doing”, is
this: she does not know; they do correlated actions because they receive the
same command from a common source, through the channel of entanglement. To
analyze in details the way that happens, using the nonlinear extension of
Schrödinger’s equation proposed here, remains for forthcoming research. In the
meantime, a useful analogy is this: spreading the same belief towards remote
places may cause correlated actions of remote people who never had any
personal contact among themselves.

##  I Acknowledgments

I am deeply indebted to Lajos Diósi for unlimited discussions about quantum
foundations over many years. Many thanks to him and to Tamás Vicsek for
carefully reading a preliminary version of this paper, with important
suggestions to change. The present research had no financial support from any
source.

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