Towards a theory of wavefunction collapse Part 1: How the Diosi-Penrose criterion and Born's rule can be derived from semiclassical gravity, and how the criterion can be relativistically generalised with help of the Einstein-Hilbert action

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by  Garrelt Quandt-Wiese

Towards a theory of wavefunction collapse Part 1:
How the Dio´si-Penrose criterion and Born’s rule can be
derived from semiclassical gravity
How the Dio´si-Penrose criterion can be relativistically
generalised with the help of the Einstein-Hilbert action
Garrelt Quandt-Wiese 1
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1 Schlesierstr.16, 64297 Darmstadt, Germany
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[email protected]
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http://www.quandt-wiese.de
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e
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A new approach to wavefunction collapse is prepared by an analysis of semiclas-
1
sical gravity. The fact that, in semiclassical gravity, superposed states must share

]
h a common classical spacetime geometry, even if they prefer (according to general
p
relativity) differently curved spacetimes, leads to energy increases of the states,
-
t whentheirmassdistributionsaredifferent. Ifoneinterpretstheseenergyincreases
n
a divided by Planck’s constant as decay rates of the states, one obtains the lifetimes
u
of superpositions according to the Dio´si-Penrose criterion and reduction probabili-
q
[ ties according to Born’s rule. The derivation of Born’s rule for two-state superposi-

3 tionscanbeadaptedtothetypicalquantummechanicalexperimentswiththehelp
v ofacommonpropertyoftheseexperiments. Itisthattheyleadtonevermorethan
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two different mass distributions at one location referring e.g. to the cases that a
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3 particle ”is”, or ”is not”, detected at the location. From the characteristic energy
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of the Dio´si-Penrose criterion, an action is constructed whose relativistic general-
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. isation becomes obvious by a decomposition of the Einstein-Hilbert action to the
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0 superposedstates. InPart2,semiclassicalgravityisenhancedtotheso-calledDy-
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namical Spacetime approach to wavefunction collapse, which leads to a physical
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: mechanism for collapse.
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Keywords: Wavefunctioncollapse,semiclassicalgravity,quantummechanicsandrelativity,Born’srule.
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1 Introduction
Gravityisthemostoftendiscussedcandidateforaphysicalexplanationofwavefunction
collapse. The Dynamical Spacetime approach to wavefunction collapse, which is pre-
pared in this publication and developed in Part 2 [1], also assumes gravity as the driver
ofcollapse. ThatgravitycouldberesponsibleforcollapsewasfirstmentionedbyFeyn-
man [4] in the 1960s, and led to a first vague model formulated by Ka´rolyha´zi [5,15].
1 MyofficiallastnameisWiese.Fornon-officialconcerns,mywifeandIuseourcommonfamilyname:Quandt-Wiese.
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Concrete gravity-based models were developed by Dio´si [6] and Penrose [7] in the
1980s and 1990s. In Dio´si’s approach, fluctuations of the gravitational field are the
driver of collapse. In Penrose’s approach, the uncertainty of location in spacetime,
which occurs when superposed states prefer differently curved spacetimes due to dif-
ferent mass distributions, plays the central role, which leads to a fuzziness of energy
being responsible for the superposition’s decay. Interestingly, the approaches of Dio´si
and Penrose predict the same lifetimes of superpositions, which can be determined
with a characteristic gravitational energy depending on the mass distributions of the
superposed states. This rule of thumb, sometimes referred to as the Dio´si-Penrose
criterion, is often used for quantitative assessments of experimental proposals investi-
gating certain properties of wavefunction collapse [8–15].
Another starting point for a gravity-based collapse model is semiclassical gravity, in
which the gravitational field is not quantised and spacetime geometry is treated clas-
sically [16,17]. As a consequence, superposed states must share the same clas-
sical spacetime geometry, even if they prefer (according to general relativity) differ-
ently curved spacetimes, which is the case when their mass distributions are different.
This provokes a competition between the states for the curvature of spacetime. How-
ever, this mechanism alone cannot explain collapse, which is known from studies of
the Schro¨dinger-Newton equation displaying semiclassical gravity in the Newtonian
limit [18,19].
The purpose of this paper is to prepare the derivation of the Dynamical Spacetime
approach to wavefunction collapse in Part 2 [1]. This is carried out by an analysis
of semiclassical gravity. The Dynamical Spacetime approach enhances semiclassical
gravity by the so-called Dynamical Spacetime postulate, which enables it for an expla-
nation of wavefunction collapse. In [2], an overview on the derivation and proposed
experimental verification of the Dynamical Spacetime approach is given.
The question of whether the gravitational field must not to be quantised and spacetime
geometry can be treated classically, as assumed by semiclassical gravity and the Dy-
namical Spacetime approach, is still the subject of scientific debate [20,21] and has
not been decided by experiments so far [22].
The analysis of semiclassical gravity in this paper will show that there is a relation
betweensemiclassicalgravityandtheDio´si-Penrosecriterion. Thisrelationwillgiveus
an idea of how the Dio´si-Penrose criterion can be relativistically generalised with help
of the Einstein-Hilbert action. The analysis of semiclassical gravity will also show that
there is possibly a relationship between semiclassical gravity and Born’s rule, and that
the fact that all the experiments performed so far behave in accordance with Born’s
rule is related to a property that these experiments have in common.
—
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The remainder of this paper is structured as follows. In Section 2, we recapitulate the
Dio´si-Penrose criterion and the approaches of Penrose and Dio´si. In Section 3, we
start with the analysis of semiclassical gravity and show how the Dio´si-Penrose cri-
terion and Born’s rule can be derived for two-state superpositions. In Section 4, we
show how the Dio´si-Penrose criterion can be relativistically generalised with the help
of the Einstein-Hilbert action. In Section 5, we show how our derivation of Born’s rule
fortwo-statesuperpositionscanbegeneralisedforthetypicalquantummechanicalex-
periments, and give a preliminary explanation as to why all experiments so far behave
in accordance with Born’s rule.
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2 Approaches of Penrose and Dio´si
In this section, we recapitulate the approaches of Penrose and Dio´si. We begin with
the Dio´si-Penrose criterion for estimating the lifetimes of superpositions.
2.1 Dio´si-Penrose criterion
The Dio´si-Penrose criterion is an easy-to-use rule of thumb for estimating the lifetimes
of quantum superpositions. The lifetime of a superposition depends on how much the
mass distributions of its states differ from each other. A superposition of two states
with mass distributions ρ (x) and ρ (x) can be generated with the single-photon exper-
1 2
iment in the left-hand side of Figure 1, in which the detector displaces a rigid body for
photon detection. The mean lifetime of the superposition depends on a characteristic
gravitational energy E , which we call the Dio´si-Penrose energy. This energy divided
G12
by Planck’s constant can be thought of as a decay rate leading to the following lifetime
T of the superposition [7,23]:
G
h¯
T ≈ . Dio´si-Penrosecriterion (1)
G E
G12
The Dio´si-Penrose energy depends on the mass distributions of the superposition’s
states ρ (x) and ρ (x) as [7,13,23]
1 2
(cid:90)
(ρ (x) ρ (x))(ρ (y) ρ (y))
− −
E = ξG d3xd3y 1 2 1 2 , Dio´si-Penroseenergy (2)
G12 |x−y|
where G is the gravitational constant and ξ a dimensionless parameter in the order
of one. In their original publications, Dio´si and Penrose derived this dimensionless
parameter as ξ=1 [7,23]. In an overview article, Bassi however showed that Dio´sis ap-
proach leads to a Dio´si-Penrose energy with ξ=1 [15]. Here we will show that ξ can be
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Fig.1: Left: Experimenttogenerateasuperpositionofstateswithmassdistributionsρ (x)andρ (x).
1 2
Thedetectordisplacestherigidbodyforphotondetection.
Right: Illustrationoftheexperiment’sstatevector’sevolutioninconfigurationspace,whichsplits
intotwowavepacketsatt whenthephotonentersthebeamsplitter.
s
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consistently derived from Dio´si’s and Penrose’s approach and also from semiclassical
gravity to be ξ=1.
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The Dio´si-Penrose energy has different physical illustrations. One, which directly fol-
lows from Equation (2), is that it describes the gravitational self-energy resulting from
the difference of the states’ mass distributions ρ (x)−ρ (x). This illustration is not very
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intuitive, since this difference can be negative. Despite this fact, the Dio´si-Penrose
energy is always positive, as we will see later.
AmorehelpfulillustrationoftheDio´si-Penroseenergyisasfollows,whichonlyholdsfor
superposed rigid bodies, whose states are displaced against each other by a distance
∆s (i.e. ρ (x)=ρ (x−∆s)). Assuming hypothetically that the masses of the superpo-
2 1
sition’s states attract each other by the gravitational force, the Dio´si-Penrose energy
describes the mechanical work to pull the masses apart from each other over the dis-
tance of ∆s against their gravitational attraction. From this illustration, it follows that
for small displacements ∆s, where the gravitational force can be linearised, the Dio´si-
Penroseenergyincreasesquadraticallywiththedisplacement∆s(E ∝∆s2). Atlarge
G12
displacements ∆s, where the gravitational attraction vanishes, the Dio´si-Penrose en-
ergy converges to a constant value. It is important to note that this illustration of the
Dio´si-Penrose energy leads also to ξ=1 in Equation (2). The derivation of this illustra-
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tion of the Dio´si-Penrose energy is given in the appendix of [3].
Sometimes,itishelpfultoconverttheDio´si-Penroseenergy(Equation2)intoadifferent
form. WiththegravitationalpotentialsΦ (x)resultingfromthestates’massdistributions
i
ρ (x), which are given by
i
(cid:90)
ρ (y)
Φ (x) = −G d3y i , (3)
i |x−y|
Equation (2) can be converted for ξ=1 to
2
(cid:90)
1
E = d3x(ρ (x)−ρ (x))(Φ (x)−Φ (x)) , Dio´si-Penroseenergy (4)
G12 2 1 2 2 1
i.e. theDio´si-Penroseenergyisdescribedbytheintegralofthedifferenceofthestates’
mass distributions multiplied by the difference of their gravitational potentials.
2.2 Penrose’s approach
Penrose’s approach [7] is based on the argument that superposed states prefer (ac-
cording to general relativity) differently curved spacetimes when their mass distribu-
tions are different, which leads to an uncertainty in the location in spacetime. From this
uncertainty follows a fuzziness of the states’ energies, which Penrose accounts for the
superposition’s decay.
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Before coming to Penrose’s original derivation, an alternative derivation will be pro-
posed, which is not as precise as Penrose’s one, but expresses the idea quite well. In
the Newtonian limit, only the g -component of the metric field g (x) is of relevance,
00 µν
which describes the derivation of the physical time according to the time coordinate x0
√
as ds = g . The g -component can be expressed by the gravitational potential Φ(x)
dx0 00 00
as follows [24]:
ds √ Φ(x)
= g ≈ 1+ . (5)
dx0 00 c2
Since superposed states with different mass distributions have different gravitational
potentials, a clock runs with slightly different speeds depending on to whose state’s
spacetime geometry the clock is assigned. This uncertainty of the clock’s speed leads
to a fuzziness of the states’ energies. Multiplying the difference of the clock’s speed
(ds1−ds2) in State 1’s and 2’s spacetime geometries by the energy density of State 1
dx0 dx0
ρ (x)c2, we obtain an estimate for the fuzziness of State 1’s energy as follows:
1
(cid:90) (cid:90)
ds ds
∆E = d3x( 2 − 1)ρ (x)c2 = d3xρ (x)(Φ (x)−Φ (x)) . (6)
1 dx0 dx0 1 1 2 1
In the same way, the fuzziness of state 2’s energy yields ∆E =(cid:82) d3xρ (x)(φ (x)−φ (x)).
2 2 1 2
If we assume that in 50% of the cases State 1 decays to State 2, and in the other 50%
of the cases vice versa, the relevant fuzziness of energy is ∆E=(∆E +∆E )/2, which
1 2
is the Dio´si-Penrose energy in the form of Equation (4).
Now we turn to Penrose’s original derivation [7]. Penrose compares the gravitational
fields g (x) that arise in the differently curved spacetimes of States 1 and 2 as follows:
i
(cid:90)
1
E = d3x|g (x)−g (x)|2 . (7)
G12 8πG 1 2
Thefuzzinessofenergyfollowingfromthisapproachcanbephysicallyjustifiedwiththe
energy density that one can assign the gravitational field, which is given by 1 g2(x).
8πG
With g (x)=−∇Φ (x) and converting Equation (7) with Green’s first identity and Pois-
i i
son’sequation(∆Φ(x)=4πGρ(x)), weagainobtaintheDio´si-Penroseenergyintheform
of Equation (4). Penrose does not, in his derivation, refer to the energy density that
one can assign the gravitational field ( 1 g2(x)), and uses in Equation (7) the factor
8πG
1 instead of 1 , which leads to a Dio´si-Penrose energy with ξ=1 instead of ξ=1 in
4π 8π 2
Equation (2).
From Equation (7), it follows that the Dio´si-Penrose energy is always positive. This is
important, since otherwise the Dio´si-Penrose criterion (Equation 1) leads to negative
lifetimes.
A problematic point in Penrose’s approach, which was addressed by Penrose himself,
isasfollows. TocalculatetheuncertaintyofthegravitationalfieldaccordingtoEquation
(7), the points of the spacetime geometry of State 1 have to be identified with points
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of the spacetime geometry of State 2 for comparing the gravitational fields. Penrose
sees in this procedure a fundamental problem, which he expresses in his own words
as follows [25]:
The principle of general covariance tells us not only that there are to be no pre-
ferred coordinates, but also that, if we have two different spacetimes, representing
two physically distinct gravitational fields, then there is to be no naturally preferred
pointwiseidentificationbetweenthetwo-sowecannotsaywhichparticularspace-
timepointofoneistoberegardedasthesamepointassomeparticularspacetime
point of the other!
2.3 Dio´si’s approach
Dio´si adapts in his approach the work of Bohr and Rosenfeld, who investigated the
uncertainty of measuring an electromagnetic field by an apparatus obeying quantum
mechanics, to the measurement of a gravitational field. He found that a gravitational
field measured over a time ∆t on a volume V exhibits the following uncertainty δg [26]:
h¯G
(δg)2 ≥ . (8)
V∆t
Dio´si postulates that the gravitational field exhibits universal fluctuations according to
this uncertainty, i.e. a universal gravitational white noise. For the mathematical for-
mulation of his approach, he uses the framework of the dynamical reduction models,
which provide evolution equations for the density matrix, when stochastically fluctuat-
ing operators are introduced [27]. This leads to the following equation of motion for the
density matrix ρ [6,28]:
(cid:90) (cid:90)
dρ i G 1
= − [H,ρ]− d3xd3y [ρˆ(x),[ρˆ(y),ρ]] , (9)
dt h¯ 2h¯ |x−y|
in which H is the Hamiltonian, and ρˆ(x) the operator of mass density. From this equa-
tion, one can derive a characteristic decay time τ describing of how fast the inter-
d
ference between states of different mass distributions destroys. This decay time is
given by the DisiPenrose criterion, i.e. by τ =h¯/E , with a Dio´si-Penrose energy E
d G12 G12
(Equation 2) with ξ=1 [15] 2.
2
A problematic point in Dio´si’s approach is that the mass density operator ρˆ(x) in Equa-
tion (9) has to be modified to avoid divergences. The delta-shaped mass density oper-
ator has to be smeared, e.g. as in [6]
(cid:88) (cid:88) m
ρˆ(x) = m δ(x−xˆ ) ⇒ ρˆ(x) = i Θ(r −|x−xˆ |) , (10)
i i 4πr3 0 i
i i 3 0
2 Dio´siobtainsinhisoriginalworkaDio´si-Penroseenergywithξ=1[23]. Inanoverviewarticle[15],
Bassi however showed that Dio´si’s approach leads to a Dio´si-Penrose energy with ξ=1, which is also
2
intuitivelyexpectedbycomparingEquations(9)and(2).
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where r is the characteristic radius for the smearing and m , xˆ the mass and position
0 i i
operator of the i’s particle. The characteristic radius r was originally chosen by Dio´si
0
to be on the order of the nucleon’s radius, i.e. r ≈10−13cm [6], and was later revised by
0
Ghirardi to a much larger value of r ≈10−15cm [28]. This correction was necessary to
0
avoid a too-strong permanent increase of total energy [28]. A permanent increase of
total energy is characteristic of dynamical reduction models, and results from stochas-
tically fluctuating operators [27].
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3 Semiclassical gravity
We begin our analysis of semiclassical gravity by regarding first two-state superpo-
sitions in the Newtonian limit. In Section 3.1, we calculate the total energy of such
superpositions. In Section 3.2, we show how the states’ energies increase due to
the sharing of spacetime in semiclassical gravity. In Section 3.3, we derive the Dio´si-
Penrose criterion and Born’s rule from this result.
3.1 Total energy of a two-state superposition
The state vector |ψ> of the single-photon experiment in Figure 1 can be thought of
as a localised wavepacket in configuration space, which splits into two well separated
wavepackets|ψ >and|ψ >whenthephotonentersthebeamsplitteratt ,asshownin
1 2 s
the right-hand side of Figure 1. The state vector |ψ> will hereafter describe the entire
system, consisting for the experiment in Figure 1 of the photon, the beam splitter, the
detector and the rigid body the detector is displacing. After the photon was split by the
beam splitter, the state vector can be written as a superposition of the wavepackets
|ψ > and |ψ > as
1 2
|ψ >= c |ψ > +c |ψ > , (11)
1 1 2 2
where |ψ > and |ψ > correspond to the cases that the photon was reflected and re-
1 2
spectively transmitted at the beam splitter. The amplitudes c and c of this superposi-
1 2
tion fulfil the following normalisation:
|c |2 +|c |2 = 1 , (12)
1 2
since |ψ > and |ψ > shall be normalised (<ψ |ψ >=1). The mass distributions ρ (x) of
1 2 i i i
the states |ψ > and |ψ > can be calculated with the operator of mass density ρˆ(x) as
1 2
follows:
ρ (x) =< ψ |ρˆ(x)|ψ > . (13)
i i i
When the wavepackets corresponding to |ψ > and |ψ > are well separated in configu-
1 2
ration space, as in the right-hand side of Figure 1 for t>t , the mass distribution of the
s
superposition is given by the mean of the states’ mass distributions as follows.
ρ(x) = |c |2ρ (x)+|c |2ρ (x) . (14)
1 1 2 2
Since spacetime geometry is treated classically in semiclassical gravity, and the metric
fieldisintheNewtonianlimitdirectlylinkedtothegravitationalpotential(cf. Equation5),
the gravitational potential must also be treated classically. The gravitational potential
resulting from the mass distribution of Equation (14) is:
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Φ(x) = |c |2Φ (x)+|c |2Φ (x) , (15)
1 1 2 2
where Φ (x) and Φ (x) are the gravitational potentials resulting from the states’ mass
1 2
distributions according to Equation (3). This means that the gravitational potential of
the superposition is given by the mean of the states’ gravitational potentials.
The total energy of our two-state superposition can be calculated with
(cid:90)
E = d3xρ(x)(c2 + 1Φ(x)) , (16)
2
where ρ(x) and Φ(x) are the mass distribution and gravitational potential, respectively,
according to Equations (14) and (15). The first term in Equation (16) calculates the
masses’ rest energies (E=mc2). The masses’ kinetic energies are neglected, since
they are small compared to the rest energies. The second term calculates the grav-
itational energies between the masses (E =−Gm m /r), where the factor 1 avoids
G 1 2 2
gravitational energy between the two masses being counted twice during integration.
By inserting Equations (14) and (15) into Equation (16), we obtain after a short calcu-
lation 3 the following total energy of our two-state superposition:
E = |c |2E +|c |2E +|c |2|c |2E . (17)
1 1 2 1 1 2 G12
HereE andE arethetotalenergiesofStates1and2alone,which,similartoEquation
1 2
(16), are given by
(cid:90)
E = d3xρ (x)(c2 + 1Φ (x)) . (18)
i i 2 i
The term |c |2|c |2E , in which E is the Dio´si-Penrose energy according to Equa-
1 2 G12 G12
tion (4), expresses how much the superposition’s total energy increases due to the
sharing of the mean gravitational potential by the states. Since the sharing of gravita-
tional potential expresses the sharing of spacetime geometry in semiclassical gravity,
the Dio´si-Penrose energy can be regarded as a measure of how much the preferred
spacetime geometries of States 1 and 2 differ from each other, or of how strong the
states compete for spacetime geometry.
SincetheDio´si-PenroseenergyinEquation(4)correspondstoafactorofξ=1 inEqua-
2
tion (2), semiclassical gravity leads as the approaches of Penrose and Dio´si to a Dio´si-
Penrose energy with ξ=1.
2
3 The second term in Equation (16) can be transformed with the normalisation |c |2 +|c |2 = 1 as
1 2
follows:
(|c |2ρ +|c |2ρ )(˙|c |2Φ +|c |2Φ ) =
1 1 2 2 1 1 2 2
|c |2ρ (|c |2Φ +|c |2Φ )+|c |2ρ (|c |2Φ +|c |2Φ ) =
1 1 1 1 2 2 2 2 1 1 2 2
|c |2ρ (Φ −|c |2Φ +|c |2Φ )+|c |2ρ (|c |2Φ +Φ −|c |2Φ ) =
1 1 1 2 1 2 2 2 2 1 1 2 1 2
|c |2ρ Φ +|c |2ρ Φ +|c |2|c |2(ρ −ρ )(Φ −Φ ).
1 1 1 2 2 2 1 2 1 1 2 1
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