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Gravity-induced decay

Sergei Viznyuk

Abstract

I suggest an object acted on by classical (i.e. non-coherent) force, in

gravitational field, unavoidably decays. I estimate decay rates in some scenarios.

Quantum phase shift between superposed states with different space geometries in

gravitational field inspired interest (Penrose, 1996) as well as controversy (Gao, 2013; Bonder,

Okon, & Sudarsky, 2016; Pikovski, Zych, Costa, & Brukner, 2015) since at least early 1960-ies

when (Feynman, 1962) first talked about role of gravity in breakdown of quantum mechanical

behavior. Despite the controversy, the nature of the phenomenon appears mundane: the states with

different space geometries in gravitational field experience phase shift with respect to each other.

If the shift is larger than the de-Broglie wave packet length the interference between packets with

different space geometries vanishes. It is the same effect as disappearance of interference pattern

in two-slit experiment when difference in optical lengths between two paths exceeds the coherence

length of the light source (Figger, Meschede, & Zimmermann, 2002). In such case, wave packets

(e.g. laser pulses) appear as if they travel down only one of the paths, as do classical particles.

However, the interference amplitude can be easily restored by introducing phase delay into one of

the optical paths. Perhaps one should not equate phase shift between wave packets or decrease in

interference amplitude with true de-coherence, which is an irreversible loss of phase relationships

between constituent states.

The observable effect of true de-coherence is disintegration of the object into wave packets

with no phase relationship between the packets. Such packets behave as separate objects.

True de-coherence arises if phases of constituent pure states no longer predictably relate to

each other, e.g. as a result of random dispersion. Random phase dispersion can be associated with:

1. Scattering (Uys, et al., 2010; Schlosshauer, 2007)

2. Brownian motion (Paz, 1994; Hornberger, 2009)

3. Dispersive media (Antonelli & al, 2011; Salemian & Mohammadnejad, 2011)

In this article I look into phase dispersion introduced by non-coherent coupling between external

non-gravitational fields and object’s constituent pure states, in the presence of gravitational field.

According to (Einstein, 1907) equivalence principle, the effect of the external non-gravitational

force on the object at rest in a [uniform] gravitational field is indistinguishable from the effect of

the same force on the object away from gravitating bodies, in object’s [accelerating] reference

frame. Thus, the influence of non-gravitational fields does not depend on presence of [uniform]

gravity. The inevitable conclusion is that any change to the state of the object, such as phase

dispersion between constituent pure states, can only be caused by non-gravitational fields:

It is clear, …, that, while some kind of decoherence might happen in specific

experimental situations, gravity, … is never the agent that causes the effect.

(Bonder, Okon, & Sudarsky, 2016)

Nevertheless, I suggest, gravity plays an important role, by establishing preferred basis in system’s

state vector space. Phase dispersion by coherent coupling between external field and pure states

retains coherence of the wave packet. If coupling is non-coherent, as e.g. with environment heat

bath, the phase dispersion includes random phase noise which, over sufficiently long time, results

in true de-coherence of the wave packet, and decay of the object is represents.

To demonstrate the principal feature, consider the object is in a stable state if no external

fields. If object is free-falling in a gravitational field, with respect to lab frame the state is a

superposition of eigenstates of Hamiltonian with field, i.e. in the preferred basis:

(1)

Thus in a lab, is no longer a static state, as can be seen from solution to Schrodinger’s equation:

(2)

An observer performing measurement of object’s vertical coordinate will find the expectation

value change according to:

, where

(3)

A two-level system is sufficient to demonstrate the main feature (Viznyuk, 2014):

(4)

From (4) the relation with acceleration of gravity is:

(5)

, where ought to be taken as a distance (radius) to the center of gravity at ,

and

.

Now consider the object is acted on by an external non-gravitational forces keeping the

object around fixed position in a lab frame. If external fields couple coherently, the evolution of

object’s state is still described by (2), where are eigenstates of Hamiltonian with all fields

included. With non-coherent coupling, the phases of -states no longer relate to each other as

(6)

Instead, non-coherent coupling can be modelled as a random phase walk, with phase

difference changing by during single act of interaction with external

field. Here has a meaning of mean free time between interactions, with

being a

scattering rate. During time there is scattering events, each with probability

to increment phase difference by in positive or negative direction. After

scattering events phase difference will be within standard deviation of its coherent

value, where

is the variance of binomial distribution

. So, instead of

(6), for non-coherent coupling I have:

(7)

Once random dispersion (7) grows to the , -states can be considered de-coherent

with respect to each other. From here I estimate time for the object to decay, considering it a

simple two-level system as in (4), (5):

, and decay rate

(8)

Eqs. (7-8) has been derived in approximation of large number of scattering events per rotation

period

, i.e. in approximation . In the opposite case I expect: as .

The inverse proportionality of decay rate on de-coherent scattering rate in (8) can be viewed

as manifestation of quantum Zeno effect (Misra B., 1977). Each scattering event constitutes the

measurement by the environment. The more frequent are the measurements performed on the

object, the more likely the object will be found in its initial state after fixed elapsed time.

Furthermore, the effect may not even be so “quantum” in nature. I can simulate it using the

following experiment: take a [fragile] wine glass and try to keep it vertically around the same

height with respect to lab floor by applying periodic kicks from below using some hard-surface

object, e.g. a wood board. In the mean free time between the kicks the wine glass will be in free

fall. The more frequent are the kicks, the lesser distance the wine glass will fall in between the

kicks, the lesser is the force of each kick required to return the wine glass to its initial position. In

the limit of infinite frequency of kicks, the wine glass will be simply standing steady on hard

surface. It can stay in such position forever. However, if the frequency of kicks is decreased, in

the mean free time the wine glass would fall greater distance. Each kick would have to be stronger

to return wine glass to its initial position; and with stronger kick it is more likely the wine glass

would eventually break into smaller pieces. As I expect, there is also a second limit, when ;

; corresponding to the case when wine glass is allowed to fall indefinitely without

experiencing kicks, and that would preserve wine glass too.

It is conceivable (8) could be applicable to any object, including elementary particles which

are otherwise considered stable, such as electrons and protons. I shall estimate from (8) in some

scenarios, using input parameters from the cited sources:

Scenario

Distance to the

center of gravity ()

Acceleration of

gravity ()

Scattering rate

(

Decay time

electron in Sun’s

photosphere

[21]

[10]

electron in Sun’s

corona

[21]

[10]

24 years

proton in Sun’s core

at the maximum

gravity location

[18]

2342

[9]

s

With solar mass distribution (Stix, 2004), the gravity reaches maximum of at about

of solar radius from the center of the Sun. The proton collision rate in Sun’s core is estimated

from (Knapp, 2011) to (Mullan, 2009). Thus from (8) the decay rate of

protons in Sun’s core could range from to . The claimed hydrogen

fusion rate of (Knapp, 2011; Mullan, 2009) in Sun’s core lays right in that

range. If the presented reasoning is valid, the thermonuclear fusion might not be the primary factor

behind Sun’s radiative output. Indeed, as radiative modes have higher entropy than heavy nuclei

the Second Law of Thermodynamics favors Sun evaporating into thermal radiation rather than

forming iron core and turning into a white dwarf or neutron star, a relatively low entropy states.

Also, the presented estimates show the proposed particle decay rates in Sun’s corona are many

orders of magnitude higher than in Sun’s photosphere, which may explain why the temperature of

Sun’s corona is , and up to in the hottest regions, in comparison with

photosphere which is about .

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https://www.academia.edu/6707979/New_QM_framework

21. Wikipedia (2016) Sun. https://en.wikipedia.org/wiki/Sun