Quantum Oppenheimer-Snyder model

Włodzimierz Piechocki1, ∗ and Tim Schmitz2, †1Department of Fundamental Research, National Centre for Nuclear Research, Pasteura 7, 02-093 Warszawa, Poland

2Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany(Dated: August 10, 2020)

We construct two reduced quantum theories for the Oppenheimer-Snyder model, respectivelytaking the point of view of the comoving and the exterior stationary observer, using affine coherentstates quantization. Investigations of the quantum corrected dynamics reveal that both observerscan see a bounce, although for the exterior observer certain quantization ambiguities have to bechosen correctly. The minimal radius for this bounce as seen from the stationary observer is thenshown to always be outside of the photon sphere. Possible avenues to lower this minimal radiusand reclaim black holes as an intermediate state in the collapse are discussed. We demonstratefurther that switching between the observers at the level of the quantum theories can be achievedby modifying the commutation relations.

I. INTRODUCTION

Singularities are an unavoidable feature of general rel-ativity, but it is widely believed that they cannot occur ina full theory of quantum gravity. In the absence of a com-plete such theory we have to rely on specialized modelsto investigate how singularities could be avoided, whatthat means conceptually for the full theory, and wherepotential observational windows to it could be. In thispaper we present such a model by quantizing the pro-totypical example for gravitational collapse to a blackhole: the Oppenheimer-Snyder (OS) model for spheri-cally symmetric, homogeneous dust collapse.

Quantization of the OS model has been discussed be-fore with regard to its spectrum and possible singularityavoidance [1–5]. Here we want to focus in particular onquantum corrected dynamics.

In Ref. [6] we have presented a canonical formulation ofthe OS model restricted to a flat interior, and in particu-lar implemented the comoving and the exterior stationaryobservers explicitly. In this paper we want to quantizethe OS model in this form, leading to two different de-parametrized quantum theories, one for each observer.As we have already discussed in Ref. [6], our investiga-tion of a different collapse model in Ref. [7] shows thatquantum corrections as seen by the comoving observerwill cause the dust cloud to bounce instead of collaps-ing to a singularity. Further we have found indicationsthat from the point of view of the exterior observer thedust cloud can also bounce. In the following we want toinvestigate this in more detail.

These kind of quantum bounces avoiding classical sin-gularities have in fact emerged in various approaches toquantum gravity. In addition to the results mentionedabove, in the Wheeler-DeWitt approach null shells havebeen shown to bounce as well [8, 9]. Similar results fornull shells can also be obtained by using an effective one-

∗ wlodzimierz.piechocki@ncbj.gov.pl† tschmitz@thp.uni-koeln.de

loop action for quantum corrected gravity [10]. Loopquantum cosmology can yield bounces [11] which havebeen discussed in the context of collapse models [12], al-though the robustness of these bounces has been calledinto question recently [13]. Further there have been inves-tigations of effective models for bouncing collapse basedon the results above [14–18].

There are many open questions regarding the consis-tency of bouncing collapse. The most important one isthe lifetime of the temporary black hole produced by thecollapse; in various approaches to bouncing collapse ithas been shown that the transition from collapse to ex-pansion has a lifetime proportional to the mass of thedust cloud [19–22]. In general this would lead to ex-tremely short lived black holes. There are proposals formechanisms that could increase this lifetime [20, 23] butno consensus has been reached so far. Related questionsconcern the behavior of the horizon during the bounce[8, 14, 24–27] and by what mechanism quantum gravita-tional effects even propagate to the horizon [26–28]. Fora review of bouncing collapse and all it entails see Ref.[29].

For most of these open questions, particularly for thelifetime, the point of view of the stationary observer isespecially relevant. This is why we explicitly implementthat observer here in the quantum theory; we hope tocomplement and build on our discussion of the comovingobserver [6, 7] and shed a bit more light on the aforemen-tioned questions in bouncing collapse.

For some of the quantum theories producing bouncesmentioned above the exterior observer is the preferredone, but this seems to be limited to the cases where nullcollapse was investigated. To our knowledge a quanti-zation of a model for massive collapse from the point ofview of the stationary observer has not been done before.

In addition, having access to two observers with theirquantum theories will allow us to discuss the relationshipbetween them. With this we can at the very least decidewhether our method of implementing these observers byswitching between them classically and quantizing thereduced theories is consistent. Further we can investigatehow this switch could be realized within the quantum

arX

iv:2

004.

0293

9v2

[gr

-qc]

7 A

ug 2

020

2

theories.Due to the unusual form of the Hamiltonian for the

stationary observer we apply a quantization methodthat might be unfamiliar to the reader: affine coherentstates quantization (ACSQ). Coherent states quantiza-tion schemes rely on the identification of phase space witha Lie group. With the help of this Lie group one canconstruct a family of coherent states by letting a uni-tary irreducible representation of it on a Hilbert spaceact on some fixed state. Phase space functions are thenmapped to operators on this Hilbert space by insertingthem into the resolution of the identity that the coherentstates bring with them. This allows also more compli-cated phase space functions to be paired up with opera-tors, at least formally. Because the phase space here is ahalf plane we choose the affine group acting on the realline as the Lie group. For a more detailed explanationwith some simple examples we refer the reader to [30–32],or our introduction to the method below.

This quantization scheme has also been used in quan-tum cosmology, for example in Refs. [33–36], and quitereliably seems to replace the singularity by a bounce.

Finally we want to mention that for simplicity we willrestrict ourselves to the flat OS model, as we did in Ref.[6]. A generalization to curved Friedmann interiors wouldbe interesting, but we do not believe that it will changeour results significantly.

We proceed here as follows. In Sec. II we summarizethe classical considerations from Ref. [6] that lead to theHamiltonians we will discuss for the rest of the paper. InSec. III we will quickly introduce the uninitiated reader toaffine coherent states quantization, and apply this quan-tization scheme to the Hamiltonians relevant for the co-moving observer in Sec. IV and the stationary observerin Sec. V. We will primarily focus on the quantum cor-rected dynamics. How the two quantum theories can berelated we discuss in Sec. VI, before we finally concludein Sec. VII.

In the following we will use units where G = c = 1.

II. CANONICAL FORMULATION OF THE OSMODEL

Here we will briefly recapitulate the canonical formu-lation of the OS model developed in Ref. [6]. We startedfrom a general spherically symmetric spacetime. Fol-lowing the usual Arnowitt-Deser-Misner procedure, theEinstein-Hilbert action for this class of spacetimes canbe brought into canonical form, see Ref. [37]. This in-volves arbitrarily foliating the spacetime by spatial hy-persurfaces characterized by a fiducial coordinate framewith label time t and radial coordinate r.

We included then into this canonical formulationBrown-Kuchař dust [38], characterized by the canonicalpair of dust proper time τ and dust density. To imple-ment a discontinuity in the matter content we followedRef. [39] and partially gauge fixed the fiducial coordinate

frame such that the surface of the dust cloud we want todescribe is always at a fixed r = rS , splitting the space-time into interior and exterior.

Let us first focus on the exterior, r > rS , where thedust density vanishes. One can bring the constraintsthere into a fully deparametrizable form by promotingthe Schwarzschild coordinates to phase space variables.This was done with the help of a series of canonical trans-formations developed in Ref. [37]. The constraints arethen the momenta conjugate to the Schwarzschild coor-dinates. Both classically and in the quantum theory thissimply tells us that the exterior is exactly Schwarzschild.

Through the fall-off behavior of the exterior canonicalvariables as they approach rS one can make sure thatinterior and exterior are smoothly matched. In particularwe chose those conditions in such a way that the usualmatching between a FLRW interior and a Schwarzschildexterior across r = rS , see e.g. Ref. [40], are appropriatelyreproduced in the canonical formalism.

Since we have fully deparametrized the exterior at theclassical level, it is clear that the matching in this formwill not hold up through quantization. In fact, it can noteven be guaranteed that at the level of quantum correc-tions the matching can be made exact again by allowinga non-vanishing energy-momentum tensor on the collaps-ing body’s surface, as was done e.g. in Ref. [16]. We willbriefly discuss at the end of this section how this restric-tion could be lifted in future work.

In the interior, r < rS , where the dust density doesnot vanish, we imposed a further symmetry-reduction tohomogeneity. To this end we restricted the dust densityto be homogeneous, and the spacetime metric to be ofFLRW form. This allowed us to integrate out the ra-dial degree of freedom, giving us essentially a canonicalformulation of Friedmann models with dust as matter,described by the scale factor a and its momentum pa,and the dust proper time τ and its momentum Pτ . Wefurther identified Pτ as the total mass of the collapsingbody M > 0.

The Hamiltonian constraint in the interior can then beexpressed in the form H+Pτ , and is hence deparametriz-able with regard to dust proper time. H therein is ourfirst physical Hamiltonian. Restricting to flat interiors itis given by

H = −P2

2R, (1)

where R = arS is the curvature radius of the dust cloud,and P = pa/rS its canonical momentum. This Hamilto-nian then describes the dust cloud from the point of viewof the comoving observer.

The Hamiltonian is always negative, but this does notpose a problem. Classically it can be identified as −M ,hence our Hamiltonian is negative but our notion of en-ergy is not.

Note that the same Hamiltonian describes theLemaître-Tolman-Bondi model for inhomogeneous,spherically symmetric dust collapse, when one separates

3

the dust cloud into a continuum of dust shells. This wehave discussed in Ref. [7], where the above Hamiltonianwas quantized using Dirac’s canonical quantization.Large parts of this discussion carry over to the OSmodel, as already mentioned in Ref. [6]. For complete-ness we will also discuss this Hamiltonian using ACSQand compare the results to Ref. [7].

Additionally we promoted the Painlevé-Gullstrand co-ordinate transformation, which connects comoving timeto Schwarzschild Killing time T , to a canonical transfor-mation. Thereby we obtain T as a canonical variable inthe interior, given by

T = τ ± 2√

2Pτ

[√R−

√Pτ2

ln

∣∣∣∣∣√R+√

2Pτ√R−√

2Pτ

∣∣∣∣∣].

Further we imposed PT = Pτ to keep the convenientphysical interpretation of this quantity as M . To makethe canonical transformation complete one also needs tofind a new momentum canonically conjugate to R thathas a vanishing Poisson bracket with T . This can be doneby finding an appropriate generating function. The exactform of this new momentum is of no further importancehere, details can be found in Ref. [6]. For simplicity wewill also call this new momentum P .

The interior Hamiltonian constraint can then bebrought into the form

HT = PT −R

2

{tanh2 P

R , R > 2PTcoth2 P

R , R < 2PT. (2)

How the above can be deparametrized with regard toKilling time will be discussed in Sec. V. The resultingHamiltonian will then describe the evolution of the dustcloud as seen from a stationary exterior observer.

It is straightforward to see that the constraint (2)yields the expected behavior for the dust cloud. In par-ticular we want to point out that as the momentum Pgrows to infinity, the constraint goes to R = 2PT = 2M :the dust cloud asymptotically approaches the horizon.

Note that the above cannot be easily adapted to theLemaître-Tolman-Bondi model. Switching the observeras above would apply to its outermost dust shell, but itis unclear how one would extend this to the other shells,since the Painlevé-Gullstrand coordinate transformationonly applies up to the dust cloud’s surface.

In summary we want to say that this setup to describedust collapse canonically is convenient due its simplicity,but this simplicity comes with restrictions on the model.For the remainder of this section we want examine howone could go beyond some of these restrictions.

Firstly we want to discuss how our efforts could beadapted to non-flat interiors. Up to the Hamiltonian forthe comoving observer such a generalization is straight-forward, it simply adds a potential term linear in R to Eq.(1), with its sign determined by the interior’s curvature.

It is when switching observers where the restriction toflat interiors simplifies matters considerably. The comov-

ing time from the Painlevé-Gullstrand transformation de-scribes comoving observers that start with zero velocityat infinity. Hence it can only be used for the flat case,where this initial condition is applicable. Dust cloudsdescribed by positively or negatively curved FLRW lineelements fulfill instead other such conditions: the pos-itive curvature case starts collapsing with zero velocityfrom a finite radius, and the negative curvature case hasa non-zero velocity everywhere.

Coordinates that are applicable to these initial condi-tions are available, see Ref. [41] for the negative curvaturecase and Ref. [42] for the positive curvature case. Espe-cially the latter requires some additional care since it isonly valid up to the initial radius of the collapsing bodyand would need to be extended beyond this region forour purposes. For simplicity we thus restricted ourselvesto flat interiors.

Further restrictions are that the exterior is not al-lowed to differ from Schwarzschild, and that the inte-rior is always homogeneous. In Ref. [6] we have alreadydiscussed indications that homogeneity could necessarilybreak near the classical singularity. It seems to us thatthe most fruitful avenue for discussing these questions isto emulate the quantum corrected dynamics of the in-terior that we will derive here with an effective mattercontribution. On the one hand, one can then investigatewhat exterior can be matched to it, as e.g. done in Refs.[14, 16–18]. On the other hand, one can implement inho-mogeneity either perturbatively or by letting every dustshell in the cloud evolve separately, as discussed in Ref.[7]. These considerations are outside of the scope of thepresent efforts.

III. AFFINE COHERENT STATESQUANTIZATION

Coherent states can be defined with the help of a Liegroup by letting a unitary irreducible representation of iton some Hilbert space act on an arbitrary state from thatspace [43]. In coherent states quantization one can thenuse these coherent states to quantize a classical canonicaltheory by identifying the classical phase space with thisLie group.

ACSQ is a coherent states quantization scheme for theLie group being the affine group, and below we give aquick introduction to it. More details can be found e.g.in Refs. [30–32] and references therein.

We start from a canonical theory whose phase spaceis the half plane: One canonical variable R is restrictedto the positive half line, and its conjugate momentum Ptakes values from all the reals. This phase space can beidentified with the affine group of the real line, acting ony ∈ R as

(P,R) · y =y

R+ P.

Note that this identification of the phase space with theaffine group is highly ambiguous, and different choices

4

lead to unitarily inequivalent quantum theories [44]. Wewill make use of this ambiguity later, but for now westick to the one given above. It has the advantage ofbehaving similarly to Dirac’s prescription for canonicalquantization, as we will see below.

One can let this group act on the Hilbert space H =L2(R+, dx) by using the representation

U(P,R) · ψ(x) =ei~Px

√Rψ(xR

).

Affine coherent states (ACS) can then be constructed as|P,R〉 = U(P,R) |Φ〉, where initially the so-called fiducialvector |Φ〉 is a normalized element of the Hilbert space,fixed but arbitrary. Later more conditions on |Φ〉 willbe implemented as needed, such that certain numericalfactors emerging from the quantum theory are finite.

Choosing the fiducial vector can be seen as a quan-tization ambiguity in ACSQ, roughly corresponding tothe factor ordering problem in Dirac’s canonical quan-tization, as we will see later. In practice one usuallyconsiders a whole family of fiducial vectors to see howdifferent choices impact the quantum theory.

With the help of the ACS one can now construct aresolution of the identity on the Hilbert space. To thisend we need in addition to the ACS a measure on phasespace invariant under the action of the affine group,

dR′dP ′ = dRdP, (P ′, R′) = (π, ρ)(P,R).

This measure is left invariant and allows us to write

1 =1

2π~cΦ−1

∫ ∞0

dR

∫ ∞−∞

dP |P,R〉 〈P,R| , (3)

where we define

cΦα =

∫ ∞0

dx

x2+α|Φ(x)|2.

Readers can convince themselves that this is indeed aresolution of the identity by confirming that it commuteswith the irreducible representation U(P,R). By Schur’slemma it then has to be at least a multiple of the identity.For a complete proof of (3) see Ref. [32].

In order for cΦ−1 to be finite the first additional condi-tion on Φ presents itself: Φ has to be square-integrablewith regard to the measure x−1dx.

The quantization procedure now builds on (3). We as-sign functions f(P,R) on phase space to operators actingon the Hilbert space by

f 7→ f =1

2π~cΦ−1

∫ ∞0

dR

∫ ∞−∞

dP f(P,R) |P,R〉 〈P,R| .

The operators constructed in this way are automati-cally symmetric, and if the function f is at least semi-bounded then f even has a self-adjoint extension inthe Friedrich extension of the quadratic form defined asf(φ, ψ) = 〈φ|f |ψ〉. For details see Ref. [30] and referencestherein.

As an example, consider how R, P and the dilation D,where D = RP , act:

Rψ(x) =cΦ0cΦ−1

xψ(x),

Pψ(x) = −i~ψ(x)′ − i~γ ψ(x)

x,

Dψ(x) = −i~ cΦ0

cΦ−1

xψ(x)′ − i~λψ(x),

where γ and λ are constants that depend on the fiducialvector. For details on the derivation see App. A. For realfiducial vectors γ vanishes and λ takes the value cΦ0 /2cΦ−1,and the operators associated with position, momentumand dilation match those from Dirac quantization, asidefrom numerical factors. This direct correspondence be-tween the two quantization schemes will not hold formore complicated phase space functions, as we will seelater.

Because the momentum operator matches its coun-terpart from Dirac quantization it cannot be made self-adjoint on the half line, and is thus not strictly speakingan observable. This is why we also explicitly consider thedilation here, which does not have that problem.

The canonical and affine commutation relations arealso intact, regardless of Φ(x) being real,

[R, P ] = i~cΦ0cΦ−1

,

[R, D] = i~cΦ0cΦ−1

R,

aside from the same numerical factor depending on |Φ〉as above. This factor can be made to disappear bya convenient choice of |Φ〉, or simply absorbed by areparametrization of R in the affine group.

The last concept that we will need in the following isthe notion of the lower symbol of a phase-space function.It is the expectation value of the operator associated withthis function with regard to ACS,

f = 〈P,R| f |P,R〉

=1

2π~cΦ−1

∫ ∞0

dR′∫ ∞−∞

dP ′ f(P ′, R′) | 〈P ′, R′|P,R〉 |2.

Consider once again as an example the lower symbols ofposition and momentum,

R =cΦ−3c

Φ0

cΦ−1

R,

P = P,

D = D − i~α,

where details can once again be found in App. A. Asidefrom a prefactor and a constant imaginary shift i~α in

5

D, which vanishes for real fiducial vectors, these lowersymbols then match the original phase space functions(apart from the rescaling of R we already know fromthe commutation relations). As for the operators, thiswill generally not be the case for other functions, andthe lower symbols will e.g. acquire additional terms ascompared to their classical counterpart.

In particular the lower symbol of the Hamiltonian hasa special significance as it is often taken to generate quan-tum corrected time evolution in the classical phase space,see e.g. Refs. [32, 33]. The viewpoint most commonlyadopted seems to be that this quantum corrected sectoris not to be identified within the full quantum theory,but is rather a theory of its own. This theory can beconstructed by taking the action for full quantum me-chanics and restricting the states that are to be variedto coherent states. Details can be found in Ref. [45] andreferences therein.

As we discuss in App. B, one can also show thatevolution of the coherent states according to the fullSchrödinger equation implies evolution of their parame-ters R and P according to the lower symbol of the Hamil-tonian as an approximation. The accuracy of the approx-imation thereby depends on the stability of the coherentstates. In this way one can also obtain the quantum cor-rected phase space picture without leaving the full quan-tum theory. The difference between the two viewpointsis purely conceptual and does not have any operationalconsequences for the following.

Note that the same phase space picture emerges whenone investigates coherent state propagators using a pathintegral approach, see e.g. Refs. [46, 47] for a discussionof this for canonical coherent states. Conceptually thiscould be assigned to either of the viewpoints above.

Lastly we want to note that it is often useful to workwith a centered fiducial vector, which means that it ful-fills

1 = 〈Φ|x|Φ〉 =

∫ ∞0

dx x |Φ(x)|2 = cΦ−3

0 = 〈Φ|dx|Φ〉 = −i~∫ ∞

0

dx Φ∗(x)

(x∂

∂x+

1

2

)Φ(x)

The second condition can be realized by choosing a realfiducial vector. For more details see Ref. [45].

IV. QUANTUM THEORY FOR THECOMOVING OBSERVER

As recapitulated in Sec. II, the dynamics of the OSmodel from the point of view of the comoving observercan be described by the Hamiltonian

H = −P2

2R.

The Hamilton operator in the corresponding quantumtheory accordingly takes the form

H =1

2π~cΦ−1

∫ ∞0

dR

∫ ∞−∞

dP H(P,R) |P,R〉 〈P,R| ,

and acts on wave functions as

Hψ(x) = − 1

2π~cΦ−1

∫ ∞0

dR

∫ ∞−∞

dP

∫ ∞0

dyP 2

2R2

ei~ (x−y)PΦ

(xR

)Φ∗(yR

)ψ(y)

=~2

2cΦ−1

∫ ∞0

dR

R2Φ(xR

) ∂2

∂x2Φ∗(xR

)ψ(x),

where we have used the identity∫∞−∞ dP P 2ei(x−y)P =

−2π δ′′(x− y).For simplicity we will only consider real fiducial vec-

tors. The Hamiltonian can then be simplified by par-tial integration. We understand our assumption that noboundary terms emerge from these partial integrationsas additional boundary conditions on the fiducial vector.Finally we arrive at

Hψ(x) =~2

2cΦ−1

(1− cΦ′−4

x3ψ(x)− 1

x2ψ′(x) +

1

xψ′′(x)

)

=~2

2cΦ−1

x−1∓

√cΦ′−4

∂

∂xx

1±2√cΦ′−4

∂

∂xx−1∓

√cΦ′−4 ψ(x).

It is obvious that the Hamiltonian is the same as theone obtained from Dirac’s quantization in Ref. [7], apartfrom an irrelevant prefactor, restricted to a specific classof factor orderings (a + 2b = 1 in the notation of Ref.[7]). There a Hilbert space with measure x1−a−2bdx wasinvestigated, which then matches under the aforemen-tioned constraint our Hilbert space here. Therefore wesuspect that one can achieve a perfect match between thetwo approaches by allowing measures in our Hilbert spacedifferent from the standard one, but this is not necessaryfor our purpose.

Self-adjoint extensions of the Hamiltonian were dis-cussed in Ref. [7], as well as solutions to the correspond-ing Schrödinger equation. It was further shown that thenorm squared of those solutions behaves towards x = 0

to leading order as x2+2√cΦ′−4 , as long as 4 cΦ

′

−4 ≥ 9 or onechooses a specific self-adjoint extension of the Hamilto-nian (θ = π in the notation of Ref. [7]). Should theseconditions not be fulfilled, the solutions behave instead

like x2−2√cΦ′−4 . Note that the cases where 2

3

√cΦ′−4 ∈ Z

had to be excluded from the discussion. Interpreting thenorm squared of wave functions as the probability dis-tribution for R in analogy to Dirac’s canonical quanti-zation, the above shows that the classical singularity isavoided in the quantum theory when one either choosesthe θ = π self-adjoint extension, or the fiducial vectorsuch that 4 cΦ

′

−4 ≥ 9 or cΦ′

−4 < 1.

6

We have further shown in Ref. [7] that the spectrumof the Hamiltonian above is the negative real line, poten-tially with an additional positive eigenvalue. This pos-itive eigenvalue only occurs for some self-adjoint exten-sions, and its value strongly depends on this extensionand the chosen factor ordering. We can hence excludeit as unphysical. Since the Hamiltonian is interpreted asthe negative mass of the dust cloud, as discussed in Sec.II, the energy of the system is therefore positive definite.

In Ref. [7] we further discussed quantum corrected dy-namics of the system by considering expectation valuesof a wave packet. Here we want to compare these resultsto the quantum corrected dynamics obtained from thelower symbol of the Hamiltonian.

Let us find this lower symbol, keeping in mind we haverestricted ourselves to real fiducial vectors:

H = 〈P,R| H |P,R〉

=~2

2RcΦ−1

∫ ∞0

dx e−i~PxΦ

(xR

)(

1− cΦ′−4

x3− 1

x2

∂

∂x+

1

x

∂2

∂x2

)ei~PxΦ

(xR

)= −P

2

2R−cΦ′

−1 + cΦ1 (cΦ′

−4 − 1)

2cΦ−1

~2

R3. (4)

H consists of the classical Hamiltonian and an additionalpotential, vanishing in the classical limit ~ → 0. Notethat for more complicated phase space functions the clas-sical limit is obtained by also taking a limit in the chosenfamily of fiducial vectors, as we will see in Sec. V.

The potential in (4) will always be repulsive. cΦ′

−4 ispositive and we can rewrite

cΦ′

−1 − cΦ1 =

∫ ∞0

dx

x3(Φ(x)− xΦ(x)′)

2 ≥ 0.

In general, this identity holds up to a boundary term dueto partial integration. However, since we have alreadyimplicitly restricted the behavior of the fiducial vectorfor x → 0 and x → ∞ by assuming the finiteness of thecΦα ’s appearing above, this boundary term vanishes. Itfollows that the numerical factor in front of the effectivepotential in (4) is always positive.

In summary we can say that ACS evolve as determinedby an effective Hamiltonian

H = −P2

2R− ~2δ

R3, (5)

where the factor δ > 0 depends on the choice of fidu-cial vector. Note that this Hamiltonian matches the onefor Friedmann models found in Ref. [45] using Klauder’senhanced quantization. Further, in Ref. [48] a quantumcorrected Hamilton-Jacobi equation equivalent to (5) wasfound from a Born-Oppenheimer approximation on min-isuperspace, albeit with a slightly different repulsive po-tential.

To solve the equations of motion given by (5), we takeinto account that the Hamiltonian itself is a constant ofmotion, −M with M > 0, and find(

dR

dτ

)2

=2M

R− 2~2δ

R4,

giving the solution

R(τ) =

(~2δ

M+

9M

2(τ − τ0)2

) 13

.

Remarkably, this exactly matches the quantum correcteddust trajectories in Ref. [7]: For large R, the trajectoryreproduces the classical expanding/collapsing trajecto-ries, which are connected by a bounce replacing the col-lapse to a singularity. The minimal radius depends onquantization ambiguities, here the fiducial vector, andscales with the energy as M−

13 .

Depending on the factor δ, the dust cloud can tem-porarily fall behind its horizon R = 2M and a stage ofthe collapse similar to a black hole is reached. As alreadymentioned in the introduction, it is important to knowhow long this stage lasts. We can estimate this lifetimeby

∆τ = τ+ − τ−,

where R(τ±) = 2M . This leads to the expression

∆τ =8M

3

√1−

(R0

2M

)3

,

where R0 = R(τ0). When the minimal radius R0 is sig-nificantly smaller than the horizon we recover the resultfrom Ref. [7]: from the point of view of the comoving ob-server, the lifetime of the black hole stage is proportionalto the mass of the dust cloud.

Also concerning the quantum corrected trajectory inmomentum space P (τ) we have the same agreement be-tween the two approaches. As follows from the Hamilto-nian equations of motion for (5),

P = −R dR

dτ= −3M

R(τ − τ0).

In conclusion, we can say that the ACSQ of this specificHamiltonian quite successfully reproduces the results ofthe usual Dirac quantization discussed in Ref. [7]. Find-ing the quantum corrected dynamics of the system is eveneasier in ACSQ, since one only has to compute the lowersymbol of the Hamiltonian.

V. QUANTUM THEORY FOR THESTATIONARY OBSERVER

From the point of view of the stationary observer wehave the Hamiltonian constraint

HT = PT −R

2

{tanh2 P

R , R > 2PTcoth2 P

R , R < 2PT. (6)

7

We first want to discuss how we can arrive at a de-parametrized quantum theory with regard to T . In thecurrent form (6), the effective Hamiltonian still dependson PT through the position of the split between insideand outside of the horizon. As already noted in Ref.[6], having the explicit split in the constraint is unnec-essary, since the conditions R < 2PT and R > 2PT areimplemented automatically by the constraint itself: Onthe constraint surface we have 2PT = R tanh2 P

R < Rand respectively on the inside. We can then identify aneffective Hamiltonian, albeit a multivalued one,

H = −R2

{tanh2 P

R

coth2 PR

. (7)

It is furthermore convenient to bring the Hamiltonianinto a slightly different form with a simple canonicaltransformation: we introduce A = 1

2R2, proportional to

the surface area of the dust cloud, and its canonical mo-mentum PA = P/R. The Hamiltonian then takes theform

H = −√A

2

{tanh2 PAcoth2 PA

.

If we want to directly quantize this Hamiltonian, wealso have to allow multivalued quantum states, bothbranches evolving with respect to a branch of the Hamil-tonian operator. Such a construction has been discussedbefore in Ref. [49]. There it has been shown that in such amultivalued quantum theory unitary time evolution canstill be implemented with boundary conditions on thewave function at the Hamiltonian’s branching points. Be-cause we are mainly interested in the quantum correcteddynamics of the theory, we will not explicitly implementthe construction from Ref. [49], but it would certainlybe possible. The branching points for our Hamiltonianwould be at PA → ±∞. Alternatively one can view thefollowing as a quantization of two completely differentsystems. As it turns out, what specific viewpoint is takendoes not make much of a difference.

There are other approaches to quantizing multivaluedHamiltonians: Adding constraints implementing the ve-locities as phase space coordinates, eventually leading tothe use of Dirac brackets, has been used e.g. in Refs.[50, 51]. Alternatively, in Ref. [52] an effective Hamilto-nian was found as a sum of the different branches of theoriginal multivalued one by considering the path integral.Unfortunately all these methods rely on the multivaluedHamiltonian emerging from a single-valued Lagrangian,as e.g. in modified gravity where the velocities occur inpowers higher than two. This is not the case for ourHamiltonian, leaving us only with the approach from Ref.[49].

As is apparent, even with the problem of de-parametrization out of the way this Hamiltonian stillpresents some challenges if one wants to follow Dirac’sprescription for canonical quantization. We will insteademploy ACSQ to tackle this problem. Earlier we have

seen that there at least formally every phase space func-tion has an associated operator, so the fact that the hy-perbolic tangent and cotangent both do not have a Taylorseries that is defined everywhere is not a problem. Fur-thermore, since both branches of the Hamiltonian aresemi-bounded the resulting operators will have a self-adjoint extension. The fact that the Hamiltonian hasa complicated dependency on the momentum is hencenot a problem, at least on the formal level.

Unfortunately the Hamilton operators as acting in po-sition space are the sum of a multiplicative and an un-bounded integral operator. It is thus quite challenging tofind eigenfunctions or make statements about the spec-trum, but we can still discuss the quantum corrected dy-namics of the system. To this end we will consider thelower symbols of both branches of the Hamiltonian sep-arately from each other.

We will call the outside branch of the HamiltonianH+(PA, A) and the inside branch H−(PA, A). Their re-spective lower symbols are

H+ = − 1

2π~cΦ−1

∫ ∞0

dA

∫ ∞−∞

dPA∣∣〈PA, A|PA, A〉∣∣2[√

A

2

(tanh2 PA − 1

)+

√A

2

]

and

H− = − 1

2π~cΦ−1

∫ ∞0

dA

∫ ∞−∞

dPA∣∣〈PA, A|PA, A〉∣∣2[√

A

2

(coth2 PA − 1

)+

√A

2

],

where

∣∣〈PA, A|PA, A〉∣∣2 =

∫ ∞0

dx

∫ ∞0

dyei~ (PA−PA)(x−y)

AA

Φ(xA

)Φ∗(xA

)Φ∗(yA

)Φ(yA

).

We have rewritten the Hamiltonian slightly to make useof the following identities:∫ ∞−∞

dPA(tanh2 PA − 1

)e−

i~ PA(x−y)

= −∫ ∞−∞

dPAe−

i~ PA(x−y)

cosh2 PA= −

π~ (x− y)

sinh(π2~ (x− y)

) ,∫ ∞−∞

dPA(coth2 PA − 1

)e−

i~ PA(x−y)

=

∫ ∞−∞

dPAe−

i~ PA(x−y)

sinh2 PA= −

π~ (x− y)

tanh(π2~ (x− y)

) .The first Fourier transformation can straightforwardly beobtained by contour integration. In computing the sec-ond integral one has to be a bit more careful, because the

8

integrand diverges for PA → 0. We have regularized it byperforming two contour integrations, one over a contourincluding this divergence and one excluding it, and aver-aging over the results. The full derivation can be foundin App. C.

The lower symbols then take the form

H± = −cΦ− 5

2

cΦ− 12

cΦ−1

√A

2+

1

2√

2~2cΦ−1A∫ ∞0

dA√A

∫ ∞0

dx

∫ ∞0

dyx− y

F±(π2~ (x− y)

)ei~PA(x−y)Φ

(xA

)Φ∗(yA

)Φ∗(xA

)Φ(yA

), (8)

where

F+(x) = sinh(x) and F−(x) = tanh(x)

To progress any further we need to specify a fiducialvector. A convenient choice, borrowed from Ref. [45], is

Φ(x) =(2β)β√Γ(2β)

xβ−12 e−βx,

where β is a positive real parameter. The relevant con-stants are then

cΦ−1 =2β

2β − 1,

cΦ− 12

= 2√

2β32

Γ(2β − 3

2

)Γ(2β)

,

cΦ− 52

=Γ(2β + 1

2

)√

2βΓ(2β),

cΦ−3 = 1.

For these constants to be finite we have to impose β > 34 .

Note that this fiducial vector is centered.We can then perform the A integration,∫ ∞

0

dA√A

Φ∗(xA

)Φ(yA

)=

(2β)2β

Γ(2β)(xy)

β− 12

∫ ∞0

dA

A2β− 12

e−βA

(x+y)

=22ββ

32

Γ(2β)

(xy)β− 1

2

(x+ y)2β− 3

2

Γ(2β − 3

2

).

This gives as the lower symbols

Γ(2β − 1) Γ(2β)

Γ(2β + 1

2

)Γ(2β − 3

2

) H±(PA, A)

= −√A

2+

24β− 52 β2β+ 1

2

~2Γ(2β + 1

2

)A2β

∫ ∞0

dx

∫ ∞0

dy

x− yF±(π2~ (x− y)

) (xy)2β−1

(x+ y)2β− 3

2

ei~PA(x−y)− β

A (x+y). (9)

These quantum corrected Hamiltonians can be investi-gated numerically. Before solving the equations of motionwe can first consider phase space portraits. To this endwe identify as the mass of the quantum corrected dustcloud

M = − Γ(2β − 1) Γ(2β)

Γ(2β + 1

2

)Γ(2β − 3

2

) H±(PA, A). (10)

After all, for |PA| → ∞ the second term in the Hamil-tonians vanishes, since Fourier transforms of integrablefunctions vanish at infinity, leading to A = 2M2; theclassical horizon is still present at least kinematically, sowe can find the mass of the dust cloud from the area ofthis horizon. See Fig. 1 and Fig. 2 for the phase spaceportraits.

Firstly we want to note that the figures we show hereare restricted to masses close to the Planck scale, andsimilarly for the parameter β, since this makes the nu-merical computations considerably more stable. We willshow analytically at the end of this section that the re-sults discussed in the following also apply to astrophysicalscales, at least when one chooses β accordingly.

For lower β as compared toM the outside branch givenby H+ in Fig. 1 is qualitatively very close to the classi-cal portrait; kinematically the system is split into twoparts, asymptotically collapsing toward the horizon andexpanding away from it. Solving the equations of mo-tion shows that also dynamically the quantum correctedtrajectory behaves similarly to the classical one, see Fig.3d. When one chooses a higher β, this picture changesdramatically: The branches collapsing from and expand-ing to infinity connect, as well as those near the horizon.This suggests a bounce of the dust cloud when collaps-ing from infinity, and a recollapse when expanding awayfrom the horizon. This prediction is confirmed by thetrajectories in Fig. 3a and Fig. 3b. By comparison to theclassical trajectories one can also see that the recollapseis a slower process than the bounce.

The inside branch given by H− behaves very similarlyto the outside branch and is in sharp contrast to theclassical case not confined to the inside of the horizon,see Fig. 2. The only qualitative difference to the out-side branch is that the asymptotic approach toward thehorizon can also happen from the inside, after a bounceclose to the classical singularity. For fixed β this seemsto take place for higherM than the approach to the hori-zon from the outside. All of this can also be seen in thetrajectories in Fig. 3.

We also want to note that these quantum correctedtrajectories demonstrate that we do not need to considerexplicitly constructing multivalued states at this stage.It seems that the branching points of the HamiltonianP → ±∞ can never be reached in finite time.

Next we want to take a closer look at how this tran-sition in behavior depends on M and β. To this end weplot phase space portraits at PA = 0 as a function of Mfor different β, see Fig. 4. When for givenM and β thereare two values of A on the phase space contour we have

9

(a) M = 0.4 and β = 1.

(b) M = 0.4 and β = 5.

FIG. 1: The quantum corrected phase space portraitsfor the outside branch of the Hamiltonian as given by(10) (full green line) as compared to their classical

counterpart (dotted red line), given by

M =√

A2 tanh2 PA, and the horizon M =

√A2 (dashed

blue line), in Planck units.

a bounce and a recollapse, and when there are none wehave an asymptotic approach to the horizon.

With this in mind we see that for every β all massesup to some critical mass bounce and recollapse, and allabove it do not. This critical mass grows with increasing

β. Aside from the fact that the minimal area of thebounce grows slower with decreasing M for the insidebranch than for the outside branch, there is no qualitativedifference between the two branches for this aspect of thedynamics either.

In addition we can read off from Fig. 4. that the mini-mal area of bouncing dust clouds lies outside of the pho-ton sphere, and the maximal area of recollapsing onesbetween photon sphere and horizon. This is somewhatdiscouraging with regard to bouncing collapse: roughlyspeaking, everything outside of the photon sphere is vis-ible to an outside observer, so these quantum correcteddynamics suggest that the dust cloud will never resem-ble anything even close to a black hole. Furthermore theminimal area grows with increasing β, while the maximalarea decreases, suggesting that the details of the dynam-ics strongly depend on the fiducial vector.

The above suggests that if we want all dust clouds tobounce regardless of mass, or at least those with massesrelevant for astrophysical considerations, we need to con-sider the case of very high β relative to unity. For thesevalues of β and M a numerical treatment seems to bequite challenging. Luckily we can estimate the integralsin H± for the case β → ∞ by using a saddle point ap-proximation, allowing us to check the behavior of thesystem analytically.

First we note that the factor in front of the Hamil-tonians in (9) approaches 1 when β → ∞. Further weslightly rewrite the Hamiltonians,

H±(PA, A) = −√A

2+

β2β+ 12

252 ~2Γ

(2β + 1

2

)∫ ∞

0

dx

∫ ∞0

dyx− y

F±(π2~ (x− y)

) (x+ y)32

xy

ei~PA(x−y)+β[− x+y

A +2 ln( 4A

xyx+y )].

The function in the exponential multiplied by β has asingle critical point at x = y = A, where it takes thevalue 2 ln 2−2. The eigenvalues of its Hessian are −1/A2

and −2/A2, meaning the critical point is a maximum.The Hamiltonians for β →∞ can then be approximatedas

H±(PA, A) ∼ −√A

2+

√A3

πβ~2,

where we have used that

Γ(az + b) ∼√

2πe−az(az)az+b−12

for a > 0 and β →∞ [53], and

x

F± (ax)

∣∣∣∣x=0

=1

a.

The end result is the same for both the inside and theoutside branch. This matches our earlier observation that

10

(a) M = 0.4 and β = 1. (b) M = 1 and β = 1.

(c) M = 0.4 and β = 5.

FIG. 2: The quantum corrected phase space portraits for the inside branch of the Hamiltonian as given by (10) (full

green line) as compared to their classical counterpart (dotted red line), given by M =√

A2 coth2 PA, and the horizon

M =√

A2 (dashed blue line), in Planck units.

11

(a) Bouncing trajectories compared to classicalexpansion away from and collapse toward the horizon,

for M = 0.4 and β = 5.

(b) Recollapsing trajectories compared to classicalexpansion away from and collapse toward the horizon,

for M = 0.4 and β = 5.

(c) A trajectory collapsing from infinity andapproaching the horizon from inside, compared to the

classical approaches to the horizon both from inside andoutside, for M = 1 and β = 1.

(d) Trajectories asymptotically approaching the horizonfrom outside, compared to the classical counterpart, for

M = 0.4 and β = 1.

FIG. 3: The quantum corrected trajectories for the outside branch (full green lines) and inside branch (dotteddashed yellow lines) of the Hamiltonian (9) for different initial conditions as compared to their classical counterparts

(dotted red lines) and the horizon M =√

A2 (dashed blue lines), in Planck units.

12

(a) outside branch (b) inside branch

FIG. 4: The quantum corrected phase space portraits at PA = 0 for the outside and inside branch of theHamiltonian as given by (10) for different β (green lines) as compared to the horizon M =

√A2 (dashed blue line)

and the photon sphere M =√

2A3 (full red line), in Planck units.

the inside and outside branches show similar behavior forhigher β. The Hamiltonian furthermore does not dependon the momentum in this limit. Since we are mainlyinterested in the PA = 0 region of phase space this is noobstacle; based on the series of the complex exponentialin the Hamiltonians for x → y we can deduce that thesubleading order in the β → ∞ expansion will dependlinearly on PA and hence vanish for PA = 0, leaving uswith the above.

We are interested in real, positive solutions of

0 = R− R3√2πβ~2

− 2M,

where we have replaced A by the radius of the dustcloud, A = 1

2R2. The polynomial above has a mini-

mum at R− = − 1√3(2πβ~2)

14 and a maximum at R+ =

1√3(2πβ~2)

14 . Depending on the value of the polynomial

at these extrema we have either one, two or three solu-tions.

ForM > 0 the minimum can never be positive, but themaximum can be negative for

√2πβ~2 < 27M2. In this

case we have a single root with R < R− < 0, and henceno positive solution. This reflects what we have seen inthe numerical investigation: for small β as compared toM , the dust cloud does not bounce.

When√

2πβ~2 > 27M2 the maximum is positive, andwe have in addition to the negative root a positive rootat R > R+ > 3M , and one at R− < R < R+ < 3M .

Since for R = 0 the polynomial is negative, the last roothas to be positive. It is easy to see that any positive roothas to fulfill R > 2M . This also agrees with our pre-vious results; for higher β the dust cloud bounces witha minimal radius outside of the photon sphere or recol-lapses with maximal radius between horizon and photonsphere, depending on the initial conditions. Furthermore,the maximal radius grows with increasing β while theminimal radius shrinks towards the horizon.

At√

2πβ~2 = 27M2 both positive roots then join atR = R+ = 3M , the minimal radius of the bounce and themaximal radius of the recollapse meeting at the photonsphere.

If one finally takes the limit β → ∞, the minimal ra-dius gets pushed out to infinity, and the maximal radiusto the horizon. The only valid solutions are then dustclouds stuck at the horizon, implying that one can neverchoose a single β such that dust clouds with arbitrarymasses bounce.

Note that the same results can be obtained for a differ-ent fiducial vector, see App. D, demonstrating a certainrobustness of the behavior described above with regardto the quantization ambiguities.

Lastly we want to mention that the classical limit inACSQ is not simply ~ → 0, but also requires taking alimit in the family of fiducial vectors such that |Φ(x)|2 →δ(x−1). For our choice of fiducial vector this translates toβ →∞, but obviously taking ~→ 0 in the approximatedHamiltonian above does not yield the classical limit. This

13

is because the two limits ~ → 0 and β → ∞ cannotsimply be taken one after the other to get the classicallimit, one rather has to let β → ∞ as a function of ~as ~ → 0, as also discussed in Ref. [33]. Details can befound in App. E.

VI. CONNECTING THE TWO OBSERVERS

In the last sections we have developed two differentquantum theories for OS collapse, one for the comov-ing observer and one for the stationary exterior observer.When we assume that those two theories are differentsectors in a full theory for a quantum OS model, fully co-variant and thus incorporating every possible observer, itbecomes necessary to find a way to connect the two spe-cialized theories. Here we want to explore a possible suchconnection; we want to find a way to switch observers inthe quantum theory.

To this end we first note that

{R,Π±} =

(1−

Π2±R2

),

where we define Π+ = R tanh PR and Π− = R coth P

R .Hence we can express the Hamiltonian for the stationaryobserver (7) as

Heff = −Π2

2R, (11)

with the modified Poisson bracket

{R,Π} =

(1− Π2

R2

). (12)

It is apparent that the Hamiltonian is identical to theone for the comoving observer. The difference betweenthe two classical deparametrized theories is then only inthe Poisson bracket; for the comoving observer we havethe usual {R,P} = 1, and for the stationary observerwe have the above. It seems reasonable to assume thatthis observation could be generalized also to other timecoordinates, corresponding to other observers.

As a side note we want to mention that in this formboth branches of the Hamiltonian are unified. The dy-namics of the outside are recovered if we restrict to|Π| < R, and the ones of the inside for |Π| > R. Itwill become apparent later that difficulties with findinga similarly unified quantum representation hinders theconstruction of a quantum theory with both branches,once again only leaving the prescription with multival-ued states to accomplish this.

In the following we want to show that the above ob-servation carries over from the classical theories to thequantum theories: we will demonstrate that the quan-tum theory corresponding to the stationary observer asdiscussed in Sec. V can also be considered as a quan-tization of (11) with the modified Poisson bracket (12),reproduced in the commutator [R, Π].

Note that (12) also emerges, in a slightly modifiedform, in the context of noncommutative spaces for the socalled Snyder models. In fact we can exactly reproducethe modified bracket corresponding to the anti-Snydermodel by using A = 1

2R2. For a discussion of these mod-

els, including how they can be quantized, see Ref. [54]and references therein. Here we will employ ACSQ tofulfill these commutation relations.

It turns out that it is useful to follow what we did clas-sically; there we promoted the functions Π±(P,R) to thephase space coordinate Π, so let us proceed analogouslyfor their operators.

Let us first consider the operator associated with Π+

using the quantization map from previous sections,

Π+ψ(x) =1

2π~cΦ−1

∫ ∞0

dR

∫ ∞−∞

dP

∫ ∞0

dy tanhP

R

ei~ (x−y)PΦ

(xR

)Φ∗(yR

)ψ(y)

Π=R tanh PR=

1

2π~cΦ−1

∫ ∞0

dR

∫ R

−R

dΠ

1− Π2

R2

∫ ∞0

dyΠ

R

ei~ (x−y)R2 ln

(1+ Π

R

1−ΠR

)Φ(xR

)Φ∗(yR

)ψ(y),

where we have expressed the inverse hyperbolic tan-gent in terms of the logarithm. The last expressionabove looks suspiciously like an ACSQ of Π in a newparametrization of the affine group by the phase spacevariables as discussed in [44]. Along the same line wecan proceed for Π−,

Π−ψ(x) =1

2π~cΦ−1

∫ ∞0

dR

∫ ∞−∞

dP

∫ ∞0

dy cothP

R

ei~ (x−y)PΦ

(xR

)Φ∗(yR

)ψ(y)

Π=R coth PR=

1

2π~cΦ−1

∫ ∞0

dR

∫R\[−R,R]

dΠΠ2

R2 − 1

∫ ∞0

dy

Π

Rei~ (x−y)R2 ln

(ΠR

+1

ΠR−1

)Φ(xR

)Φ∗(yR

)ψ(y).

The above suggests using the quantization maps

fψ(x) =1

2π~cΦ−1

∫ ∞0

dR

R

∫I

dΠ∣∣1− Π2

R2

∣∣∫ ∞

0

dy f(R,Π)

ei~ (x−y)R2 ln

∣∣∣∣ 1+ ΠR

1−ΠR

∣∣∣∣Φ(xR

)Φ∗(yR

)ψ(y), (13)

assigning f to the phase space function f(Π, R), where

I = [−R,R] or I = R\[−R,R].

The above is indeed equivalent to ACSQ with an al-ternative parametrization of the affine group by

R ∈ R+ ,R

2ln

∣∣∣∣∣1 + ΠR

1− ΠR

∣∣∣∣∣ ∈ R,

14

leading to coherent states

〈x|Π, R〉 =1√Rei~x

R2 ln

∣∣∣∣ 1+ ΠR

1−ΠR

∣∣∣∣ψ(xR

).

That these states indeed lead to a resolution of theidentity with the measure in (13) directly follows fromthe fact that we constructed them by substitution fromthe coherent states used in the previous sections. Thequantization map (13) is therefore well defined. For de-tails see Ref. [44].

We necessarily need to restrict Π either to |Π| > R or|Π| < R in order for the parametrization above to coverthe affine group only once. This is what we alluded toearlier: at this point the inside and outside branch onceagain split up.

As one can see from how we constructed the quanti-zation maps (13), the operator associated with a phasespace function f(R,Π) will be equivalent to one of twooperators quantized according to the ACSQ prescriptionwe have used in the previous chapters, one associatedwith f(Π+(P,R), R) and one with f(Π−(P,R), R), de-pending on which of the maps we are using. It is theneasy to see that the operator corresponding to the Hamil-tonian (11) in these new quantization maps is equivalentto one of the two branches of (7) quantized with the orig-inal quantization map.

Let us now demonstrate that these quantization mapsindeed reproduce (12). R still acts as a multiplicationoperator,

Rψ(x) =cΦ0cΦ−1

xψ(x).

We can then write

[R,Π] =cΦ0

2π~cΦ−12

∫ ∞0

dR

R

∫I

dΠ∣∣1− Π2

R2

∣∣∫ ∞

0

dy (x− y)

Π ei~ (x−y)R2 ln

∣∣∣∣ 1+ ΠR

1−ΠR

∣∣∣∣Φ(xR

)Φ∗(yR

)ψ(y)

= − icΦ0

2πcΦ−12

∫ ∞0

dR

R

∫I

dΠ

∫ ∞0

dy sgn(

1− Π2

R2

)

Π∂

∂Πei~ (x−y)R2 ln

∣∣∣∣ 1+ ΠR

1−ΠR

∣∣∣∣Φ(xR

)Φ∗(yR

)ψ(y)

=icΦ0

2πcΦ−12

∫ ∞0

dR

R

∫I

dΠ∣∣1− Π2

R2

∣∣∫ ∞

0

dy

(1− Π2

R2

)

ei~ (x−y)R2 ln

∣∣∣∣ 1+ ΠR

1−ΠR

∣∣∣∣Φ(xR

)Φ∗(yR

)ψ(y)

= i~cΦ0cΦ−1

{R,Π}.

Note that to suppress any boundary terms that arise fromthe partial integration we need make use of the usualregularization of Fourier transformations for Π → ±R,corresponding to the momentum in the Fourier transfor-mation going to ±∞. If we choose the inside branch of

the quantization map we also need to suppress boundaryterms at Π → ±∞, corresponding to the regularizationof coth(x) at x = 0 we made use of in the last section.

The calculation above further suggests that one caneasily implement other modified Poisson brackets inACSQ by replacing the momentum in our original quan-tization by the integral of the right hand side of the mod-ified bracket over the new momentum, as long as the re-sulting function covers the real line at least once. Asidefrom leading further support to the feasibility of gener-alizing this procedure to different observers, this couldbe of use when studying e.g. the aforementioned Snyderspaces, or theories with Dirac brackets.

At this point we can conclude that the quantizationmap works as designed: It reproduces the modified Pois-son brackets (12) and maps (11) to the Hamilton opera-tors used in Sec. V. Our quantization in Sec. V also madeuse of a simple canonical transformation, but as alreadyargued in Ref. [55] this does not affect the underlyingquantum theory. In App. F we explicitly demonstratethat also the quantum corrected considerations emergeunscathed from the new quantum theory: the modifi-cation of the Poisson bracket carries over as is to thequantum corrected equations.

We have thus demonstrated that the reduced quantumtheories from Secs. IV and V can be related by modify-ing the commutation relations. In ACSQ this is furtherequivalent to switching between parametrizations of theaffine group in the quantization map.

By referencing the quantization map we still involvethe classical theory, but conceptually it would be moresatisfactory to discuss the relation between the quantumtheories purely at the quantum level. This is possible byconsidering what we discussed above as a map T on theoperator algebra. Lastly we want to share a few obser-vations about this map.

We know how it acts at least on operators correspond-ing to phase space functions f(P,R),

T (f) = f±,

where f±(P,R) = f(Π±(P,R), R), and hats exclusivelydenote quantization via the original map from now on.T has to be linear because the quantization map is,

and it has to be bijective, at least when restricted tooperators corresponding to phase space functions. Fur-thermore we know that it cannot be an isomorphism: itchanges the commutation relations by construction butleaves the identity invariant:

T ([R, P ]) = i~ T (1) = i~1 6= [T (R), T (P )] = [R, Π].

This map is furthermore not equivalent to any transfor-mation, including non-linear ones, acting on the Hilbertspace H. This can be seen as follows. Such an equivalenttransformation T would be defined by

〈ψ, T (O)χ〉 = 〈T (ψ), OT (χ)〉

15

for all ψ, χ ∈ H and all operators O. For clarity we switchfrom Dirac notation to explicitly writing out the scalarproduct on H. The above has to hold in particular forthe identity operator 1 = T (1), such that

〈ψ, χ〉 = 〈T (ψ), T (χ)〉

It is then easy to see that T has to be linear. Sincelinear operators have an adjoint, one can then find fromthe above that T further has to be unitary. As followsdirectly from Ref. [44], this is not possible. Even withoutreference to Ref. [44] it leads to a contradiction, becauseit would imply that T is an isomorphism of the operatoralgebra,

〈ψ, T ([O1, O2])χ〉 = 〈T ψ, [O1, O2] T χ〉= 〈ψ, T †[O1, O2]T χ〉= 〈ψ, [T †O1T , T †O2T ] T χ〉= 〈ψ, [T (O1), T (O2)]χ〉 .

We can thus say that the switch between the two ob-servers happens not at the level of the Hilbert space, butof the operator algebra and in particular its interpreta-tion in terms of physical quantities. We want to stressthat the above rests on the fact that the modification ofthe commutation relation is built into the quantizationmap. Dropping this feature and sticking to the two dif-ferent Hamiltonians we used originally, there might exista transformation on H that maps the two Hamiltoniansonto each other. For the explicit construction of such atransformation for a different system see Ref. [56].

In conclusion we can say that our two quantum theo-ries, each corresponding to one of the observers, can berelated by changing the quantization map in such a waythat the canonical commutation relations are modified.This relation can not be represented as a transformationbetween Hilbert spaces, and is then in particular not uni-tary.

This is unfortunate, since one generally expects statesrepresenting different observer in a full quantum grav-ity theory to be connected by unitary transformations.We can offer in defense of our construction that at thelevel of such deparametrized theories as we discuss here aunitary relation between observers might not be strictlynecessary, as long as predictions made still show someform of consistency between observers.

Here we have only demonstrated a very basic suchconsistency: we have shown that both observers see abounce when the quantization ambiguities are chosen ac-cordingly. Before we can really say with confidence thatwhat we discussed here could be developed into a candi-date for a notion of quantum covariance, the predictionsmade for each observer need to be compared more care-fully.

The most obvious way forward in this regard seemsto be an application of this idea to more straightfor-ward coordinate transformation than we have considered

here. Beyond its elaborate functional form, the Painlevé-Gullstrand transformation also brings with it the furtherconceptual complication of splitting up the system intoinside and outside of the horizon. It might thus be in-teresting to consider models collapsing to a naked sin-gularity, such as the Lemaître-Tolman-Bondi model, tocircumvent the latter problem.

Furthermore, leaving the realm of black holes andfocusing on time-reparametrization invariant systemsmight simplify matters further. Applying our construc-tion to quantum cosmology proper is not straightforward,since there Hamiltonian constraints are generally not de-parametrizable. However, one could as a first step discussonly cosmological models generated by Brown-Kuchařdust. We leave this for future work.

VII. CONCLUSIONS

In this paper we have discussed a quantum OS modelusing ACSQ based on our previous classical considera-tions in Ref. [6]. More specifically we have quantized twoHamiltonians describing the flat OS model from the pointof view of two observers, one comoving with the dust andone stationary outside of it.

Especially quantizing the latter Hamiltonian has nicelyillustrated the benefits of ACSQ. In Dirac quantizationthis Hamiltonian does not even have a well defined op-erator due to the occurrence of hyperbolic functions andsquare roots. In ACSQ every phase space function can atleast formally be identified with an operator, and if thefunction is at least semi-bounded we can even be surethat it has a self-adjoint extension. This is particularlyuseful here, because the Hamiltonian operator turns outto be a complicated integral operator for which it is dif-ficult to find eigenfunctions. One can further find quan-tum corrected dynamics of the system without explicitlyconsidering the Hamilton operator by working with thelower symbol of the Hamiltonian.

We want to emphasize again that while usually thesequantum corrected dynamics are interpreted as exact intheir own semiclassical theory [45], we have demonstratedhere that they can also be understood within the fullquantum theory; they follow from the Schrödinger equa-tion as approximate dynamics of coherent states.

Of course if one wants to go beyond quantum cor-rections, directly working with the operator cannot behelped. In particular determining how accurate the semi-classical approximation is depends on the stability of thecoherent states. Determining this stability seems to bea problem close in complexity to actually solving theSchrödinger equation. Nevertheless, using the lower sym-bol of the Hamiltonian is a convenient way to find quan-tum corrected dynamics of a system.

The first Hamiltonian, describing the model as seenby a comoving observer, had already been quantized us-ing Dirac’s prescription for canonical quantization in thecontext of the Lemaître-Tolman-Bondi model in Ref. [7].

16

Here we have shown that for this particular HamiltonianACSQ and Dirac quantization lead to the same Hamil-ton operator. Further the quantum corrected dynamicsfound in ACSQ exactly match those found by investi-gating a wave packet in Dirac quantization in Ref. [7].These show that the dust cloud as seen by the comovingobserver with quantum corrections bounces at a minimalradius scaling inversely with M

13 , where M is the total

mass of the dust cloud.For the second Hamiltonian, corresponding to the exte-

rior stationary observer, the quantum corrected dynam-ics predict that the dust cloud as seen by the exteriorobserver can also bounce, where the minimal radius alsogrows larger with decreasing M . This is encouragingsince it establishes a rudimentary consistency for our ap-proach of switching observers in the classical canonicaltheory as described in Ref. [6]. Complementary to thebounce there emerges also a recollapse of dust cloudsstarting close to the horizon and expanding away fromit. These dust clouds reach a maximal radius and thenstart approaching the horizon again.

The other details of the dynamics are unfortunatelyless satisfactory. The bounce is far less robust underquantization ambiguities than for the comoving observer.For any given mass the parameter β in the family of fidu-cial vectors we used had to be chosen higher than a crit-ical value ∝M4 for a bounce to occur. For smaller β thequantum corrected dynamics are qualitatively identicalto the classical asymptotic approach to the horizon. Asa consequence one can never choose a β such that thedust cloud bounces for all M .

A feature of the bounce that we have shown to bequite robust with regard to quantization ambiguities isthe fact that when the bounce occurs, the minimal ra-dius is outside of the photon sphere R = 3M . As seenby the exterior observer, the horizon never forms and thecollapse never reaches a stage that even remotely resem-bles a black hole. The notion of a lifetime can then noteven be defined. In a way, this bounce evades most ofthe conceptual problems of bouncing collapse somewhatelegantly by avoiding the horizon altogether. Thereby itunfortunately directly contradicts observations.

As a consequence we also have to concede that thead hoc construction to find the lifetime from the pointof view of the exterior observer in [7], leading to a life-time ∝M3, does not agree with the more rigorous imple-mentation of this observer here. It is possible that thislifetime could be recovered when one goes beyond quan-tum corrections since it relies to some extent on ‘discrete’states; maybe transitions between our semiclassical statesand states within the horizon are allowed in the theorybut do not emerge in the quantum corrected dynamics.However, as mentioned above the unwieldy form of theHamilton operator makes such investigations quite de-manding.

We could now start speculating that the inclusion ofpre-Hawking radiation could save the model, after all ithas been shown to also affect classical collapse quite dras-

tically [57]. However, in hindsight our results are notvery surprising. Our canonical formulation fixes the ex-terior of the dust cloud to be exactly Schwarzschild, andimposes the Painlevé-Gullstrand coordinate transforma-tion between our two observers. Under these conditionsavoiding the horizon altogether from the point of viewof the exterior observer seems to be the only straightfor-ward way to implement a bounce consistently betweenthe two observers. While the matching conditions to theSchwarzschild exterior are not implemented explicitly,they influence our results implicitly in this way throughthe Painlevé-Gullstrand transformation. However, thedeeper meaning of the photon sphere as the minimal ra-dius remains unclear to us.

Note that the same line of reasoning does not applyto the comoving observer, for which why we did notfind a lower bound for the minimal radius. To find theHamiltonian generating the corresponding dynamics theSchwarzschild exterior is not relevant at all, it is identicalto that of a Friedmann model.

A similar conclusion was drawn in Refs. [16–18], wherethe matching conditions between an effective bouncinginterior and an arbitrary exterior was investigated. Itemerged that the surface of the collapsing body must bein an untrapped region of the exterior at the time of thebounce. As already mentioned at the end of Sec. II, itseems to us that this procedure of finding an exteriorto an effective bouncing interior through matching con-ditions is a very promising approach to constructing abouncing collapse model that is both conceptually andphenomenologically consistent, as was demonstrated inthe aforementioned references. Coincidentally, it poten-tially makes it possible to also incorporate Hawking ra-diation according to Ref. [57].

Summarizing the parts of our paper relevant for bounc-ing collapse we can say that crucial ingredients seem tobe missing from the program, and unfortunately this pa-per does not do much to fill in the blanks. These negativeresults however suggest where to look next, and we stillbelieve that explicitly investigating the two distinguishedobservers in the OS model is a promising approach.

Lastly we want to comment on some aspects of thispaper going beyond bouncing collapse. The Hamiltonianrelevant for the stationary observer is multivalued andconsists of two branches, one describing the behavior ofthe dust cloud outside of the horizon and one inside. Inprinciple one would have to implement this Hamiltonianin the quantum theory by also considering multivaluedstates, but for our purposes this has turned out to be ir-relevant: the quantum corrected trajectories never makeuse of the freedom to switch between the two branches,which allowed us to ignore this aspect for the discussionabove. The results are the same as when we would haveignored the inside branch from the beginning.

The quantum corrected dynamics of the inside branchare interesting insofar as they behave very similarly totheir outside counterpart. For M and β chosen accord-ingly there is a bounce and a recollapse outside of the

17

horizon also for the inside branch. Also the asymptoticapproach to the horizon can be obtained. The only qual-itative difference between the two branches is that onecan choose M and β in such a way that the collapse con-tinues through the horizon and a bounce occurs close tothe singularity followed by an asymptotic approach tothe horizon from the inside.

Why the two branches of the quantum correctedHamiltonian would behave so similarly when their clas-sical counterparts occupy completely disjunct regions ofphase space is puzzling to us. Because the most drasticdepartures of the quantum corrected inside branch fromthe classical trajectories occur close to PA = 0, wherethe classical Hamiltonian diverges, we cannot exclude thepossibility that our regularization of this branch in Sec.V has led to this unintuitive behavior.

However, since it is not necessary to consider bothbranches to find at least an internally consistent picturefor the bounce, contradicting indications we found in Ref.[6], this is not of primary importance. The disagreementwith Ref. [6] in this respect is in our view not problem-atic, but rather a cautionary tale that excessively refor-mulating the classical constraints without any physicalmotivation can lead to wildly different quantum theories.

We have also observed that our two reduced quantumtheories can be related by modifying the canonical com-mutation relation. The switch between observers at thelevel of these quantum theories then takes place on theoperator algebra and its intepretation as physical quanti-ties. It is impossible to implement it as a transformationon the Hilbert space.

It would be interesting to see whether this construc-tion can be generalized beyond the present specializedcase. If it can, another feature of ACSQ could becomerelevant; it seems to be relatively simple to implementat least some modified commutation relations by chang-ing the parametrization of the affine group by the phasespace. This is also of some interest for the quantizationof other theories with modified Poisson brackets or Diracbrackets, see e.g. Refs. [50, 51, 54].

ACKNOWLEDGMENTS

The authors would like to thank Claus Kiefer andRoberto Casadio for valuable discussions and feedbackon the manuscript, and are grateful to anonymous ref-erees for constructive criticism. TS is grateful to NickKwidzinski, Yi-Fan Wang and Nick Nussbaum for fur-ther helpful discussions. This work was supported by theGerman-Polish bilateral project DAAD and MNiSW, No57391638.

Appendix A: Operators to elementary phase spacefunctions

Here we want to investigate the elementary phase spacefunctions R, P and D = RP in the context of ACSQ.

Consider first how R, P and D act on wave functions,

Rψ(x) =1

2π~cΦ−1

∫ ∞0

dR

∫ ∞−∞

dP

∫ ∞0

dy ei~ (x−y)P

Φ(xR

)Φ∗(yR

)ψ(y)

=1

cΦ−1

∫ ∞0

dR∣∣Φ( xR)∣∣2 ψ(x) =

cΦ0cΦ−1

xψ(x),

Pψ(x) =1

2π~cΦ−1

∫ ∞0

dR

∫ ∞−∞

dP

∫ ∞0

dyP

Rei~ (x−y)P

Φ(xR

)Φ∗(yR

)ψ(y)

= − i~cΦ−1

∫ ∞0

dR

RΦ(xR

) ∂

∂xΦ∗(xR

)ψ(x)

u= xR= −i~ψ(x)′ − i~ψ(x)

cΦ−1x

∫ ∞0

du Φ(u)Φ∗(u)′

≡ −i~ψ(x)′ − i~γ ψ(x)

x,

Dψ(x) =1

2π~cΦ−1

∫ ∞0

dR

∫ ∞−∞

dP

∫ ∞0

dy P ei~ (x−y)P

Φ(xR

)Φ∗(yR

)ψ(y)

= − i~cΦ−1

∫ ∞0

dR Φ(xR

) ∂

∂xΦ∗(xR

)ψ(x)

u= xR= −i~ c

Φ0

cΦ−1

xψ(x)′ − i~ψ(x)

cΦ−1

∫ ∞0

du

uΦ(u)Φ∗(u)′

≡ −i~ cΦ0

cΦ−1

xψ(x)′ − i~λψ(x),

where we have used the identities∫∞−∞ dP Pei(x−y)P =

−2πi δ′(x− y) and∫∞−∞ dP ei(x−y)P = 2π δ(x− y). Fur-

thermore we have assumed that the fiducial vector is alsosquare integrable with regard to the measure x−2dx, suchthat cΦ0 is finite. Note that for real fiducial vectors γvanishes and λ takes the value cΦ0 /2cΦ−1. The partial in-tegrations that have to be performed above do not leadto boundary terms, since the fiducial vector vanishes atzero and at infinity due to cΦ−1 being finite.

Further we want to compute the lower symbols for

18

these operators,

R =cΦ0cΦ−1

∫ ∞0

dx

Rx |Φ

(xR

)|2 =

cΦ−3cΦ0

cΦ−1

R,

P = −i~∫ ∞

0

dx

Re−

i~PxΦ∗

(xR

)( ∂

∂x+γ

x

)ei~PxΦ

(xR

)v= x

R= P − i~R

∫ ∞0

dv(|Φ(v)|2

)′= P,

D =−i~∫ ∞

0

dx

Re−

i~PxΦ∗

(xR

)( cΦ0cΦ−1

x∂

∂x+ λ

)ei~PxΦ

(xR

)v= x

R=cΦ−3c

Φ0

cΦ−1

RP − i~λ− i~ cΦ0

cΦ−1

∫ ∞0

dv v Φ∗(v)Φ(v)′

≡cΦ−3c

Φ0

cΦ−1

D − i~α,

where we used the same behavior for Φ(x) at the bound-aries as above to find P . The factor α is always real, andfor real fiducial vectors it vanishes.

Appendix B: Quantum corrected dynamics

Suppose our to be quantized classical theory has aHamiltonian H(P,R). The time evolution in the quan-tum theory is then given by the Schrödinger equation

i~d

dt|ψ(t)〉 = H |ψ(t)〉 .

Let us further assume that one can choose |Φ〉 such thatthe ACS |P,R〉 are approximately stable under the timeevolution above, |P,R, t〉 ≈ |P (t), R(t)〉 with |P,R, 0〉 =|P,R〉. We want to compare the trajectories of theseapproximately stable ACS to the dynamics generated byH.

We have seen above that the parameters R(t) andP (t) can be extracted from |P (t), R(t)〉 by computing thelower symbols of the phase space functions R, P and D.Because the momentum operator on the half line cannotbe made self-adjoint, we will use here R and D. Sincethe lower symbols are simply the expectation values ofoperators with regard to the ACS, their time evolutionfollows a Heisenberg equation,

R(t) =cΦ−1

cΦ−3cΦ0

˙R(t)

=i

~cΦ−1

cΦ−3cΦ0

〈P (t), R(t)| [H, R] |P (t), R(t)〉 , (B1)

D(t) =cΦ−1

cΦ−3cΦ0

˙D(t)

=i

~cΦ−1

cΦ−3cΦ0

〈P (t), R(t)| [H, D] |P (t), R(t)〉 .(B2)

These equations of motion can indeed be rewritten interms of the lower symbol of the Hamiltonian H as fol-lows. First, let us write out the lower symbol of thecommutator between H and R, suppressing the time de-pendency of R(t) and P (t),

〈P,R| [H, R] |P,R〉 = − cΦ0

2π~cΦ−12∫ ∞

0

dR

∫ ∞−∞

dP

∫ ∞0

dx

∫ ∞0

dyH(P , R)

RR

(x− y) ei~ (x−y)(P−P )Φ

(xR

)Φ∗(xR

)Φ∗(yR

)Φ(yR

).

Replacing

(x− y) ei~ (x−y)(P−P ) = i~

∂

∂Pei~ (x−y)(P−P )

we recognize

〈P,R| [H, R] |P,R〉 = −i~ cΦ0

cΦ−1

∂

∂PH(P,R).

Analogously we proceed for D,

〈P,R| [H, D] |P,R〉 = − icΦ0

2πcΦ−12∫ ∞

0

dR

∫ ∞−∞

dP

∫ ∞0

dx

∫ ∞0

dyH(P , R)

RR

e−i~ (xP+yP )Φ∗

(xR

)Φ∗(yR

)(−x ∂

∂x+ y

∂

∂y

)ei~ (xP+yP )Φ

(xR

)Φ(yR

). (B3)

We first perform a partial integration, flipping the x-derivative. No boundary terms are produced since thefiducial vector vanishes at infinity and the origin. Thisis not an additional restriction on the fiducial vector, itfollows from cΦ−1 being finite. We then focus on the termsin the integral that have derivatives acting on them,(

∂

∂xx+ y

∂

∂y

)e−

i~ (x−y)PΦ∗

(xR

)Φ(yR

)=

(1 + P

∂

∂P−R ∂

∂R

)e−

i~ (x−y)PΦ∗

(xR

)Φ(yR

).

The only step left is now to commute the derivative in Rwith the remaining factor 1/R in the integral by using

1

R

(1−R ∂

∂R

)f(R) = −R ∂

∂R

f(R)

R,

It follows for (B3),

〈P,R| [H, D] |P,R〉

= −i~ cΦ0

cΦ−1

(P∂

∂P−R ∂

∂R

)H(P,R).

19

Inserting these results into the Heisenberg equations ofmotion (B1) and (B2) gives

R(t) =1

cΦ−3

∂

∂PH(P,R).

D(t) =1

cΦ−3

(P∂

∂P−R ∂

∂R

)H(P,R),

and replacing D = RP we find

P (t) = − 1

cΦ−3

∂

∂RH(P,R).

Indeed we see that the dynamics of stable ACS are gov-erned by the effective quantum corrected HamiltonianH. For centered fiducial vectors as discussed in Sec. IIIone does not even need to rescale the lower symbol, sincecΦ−3 = 1.

It should be noted that the stability of ACS is far froma trivial problem. Since the accuracy of the semiclassicalapproximation as discussed here depends on this approx-imate stability, this topic certainly deserves further inves-tigation. Here we do not explicitly consider this problemsince we will not go beyond the quantum corrected phasespace picture.

Appendix C: Fourier transform of the squares of thehyperbolic cosecant and cotangent

Let us start with the integral

I+(x− y) = −∫ ∞−∞

dPAe−

i~ PA(x−y)

cosh2 PA,

and solve it using contour integration. We can close thecontour in the lower half plane for x − y > 0 and in theupper half plane for x− y < 0. The hyperbolic cosine iszero at P 0

A = iπ2 (2k+1), k ∈ Z, and behaves close to these

roots as cosh2 PA ∼ −(PA−P 0A)2 such that the integrand

above has second order poles there. The residue at thesepoles is then − i

~ (x− y)eπ2~ (2k+1)(x−y). This gives us

I+(x− y) = −2π

~|x− y|

∞∑k=0

e−π2~ (2k+1)|x−y|

= −2π~ |x− y|

eπ2~ |x−y| − e− π

2~ |x−y|

= −π~ (x− y)

sinh(π2~ (x− y)

) ,where we recognized the geometric series.

Analogously we proceed with the second integral

I−(x− y) =

∫ ∞−∞

dPAe−

i~ PA(x−y)

sinh2 PA.

An important difference to the first integral is thatstrictly speaking I− does not converge due to a singu-larity of the integrand at PA = 0. We will regularize

this integral by taking the average over two contour inte-grals, one where the contour on the real line is deformedto include PA = 0 and one where we exclude it. Theintegrand has second order poles at P 0

A = iπk, wheresinh2 PA ∼ (PA−P 0

A)2, the residue is − i~ (x−y)e

π~ k(x−y),

and we close the contours as above. The integral is then

I−(x− y) = −2π

~|x− y|

(1

2+

∞∑k=1

e−π~ k|x−y|

)

= −π~|x− y|1 + e−

π~ |x−y|

1− e−π~ |x−y|

= −π~ (x− y)

tanh(π2~ (x− y)

) ,where the first term in the sum had to be weighted with12 due to the regularization described above.

Appendix D: Bouncing behavior for a differentfiducial vector

To make sure our characterization of the bounce inSec. V is robust we want to also check a different fiducialvector. We choose

Φ(x) =e−

ln2 x4σ

(2πσx2)14

,

where σ is a positive real parameter. Note that the clas-sical limit for this fiducial vector involves taking the limitσ → 0. The relevant constants are given by

cΦα = eσ2 (2+α)2

.

This fiducial vector is not centered, so we have to rescaleH± accordingly.

For the lower symbol of the Hamiltonians (8) we alsoneed the following integral:∫ ∞

0

dA√A

Φ∗(xA

)Φ(yA

)=

1√2πσxy

∫ ∞0

dA√A e−

14σ (ln2 x

A+ln2 yA )

= (xy)14 e

9σ8 −

18σ (ln x−ln y)2

.

This gives us

e−3σ4 H±(PA, A) = −

√A

2+

e−σ8

4~2√πσ∫ ∞

0

dx

∫ ∞0

dyx− y

F±(π2~ (x− y)

) e i~PA(x−y)

(xy)14

e− 1

4σ

[12 (ln x

A−ln yA )

2−ln2 xA−ln2 y

A

].

The function in the exponential multiplied by 1/σ hasa single critical point at x = y = A, where it vanishes.

20

The eigenvalues of its Hessian are −1/A2 and −1/2A2,meaning the critical point is a maximum. Using the sad-dle point approximation the Hamiltonians for σ → 0 canthen be estimated as

H±(PA, A) ∼ −√A

2+

√2σA3

π~2.

Under the identification β = 1/2σ we recover all resultsfrom the fiducial vector used in Sec. V.

Appendix E: Classical limit

The classical limit in ACSQ involves not only ~ → 0,but also taking a limit in your family of fiducial vectorssuch that |Φ(x)|2 → δ(x − 1). For our choice of fiducialvector in Sec. V this can be achieved by β →∞. Takingthese two limits one after the other does not produce thecorrect classical limit anymore, in contrast to Sec. IV.The easiest way to obtain the correct classical limit, asalso discussed in [33], is to let β → ∞ as a function of~ when taking ~ → 0. Let us take a closer look at thedefinition of the lower symbol to find this relation:

f =1

2π~cΦ−1

∫ ∞0

dA′∫ ∞−∞

dP ′A f(P ′A, A′)

| 〈P ′A, A′|PA, A〉 |2.

To obtain he original phase space function from the lowersymbol we then need to take the limits mentioned abovein such a way that

| 〈P ′A, A′|PA, A〉 |2

2π~cΦ−1

→ δ (A−A′) δ (PA − P ′A)

For the fiducial vector chosen here we have

| 〈P ′A, A′|PA, A〉 |2

2π~cΦ−1

=2β − 1

4πβ~

1

4

(√A′

A+

√A

A′

)2

+AA′

4β2~2(PA − P ′A)

2

]−2β

(E1)

Since the lower symbol of 1 is again 1, the above is nor-malized with regard to integration over the half plane.For A = A′ and PA = P ′A it reads

1

2π~cΦ−1

| 〈PA, A|PA, A〉 |2 =2β − 1

4πβ~

and thus diverges regardless of how exactly β depends on~ when ~ → 0. For the correct classical limit to emergewe now only need (E1) to vanish when A 6= A′ or PA 6=P ′A when the limit is taken. Note to this end that

1

4

(√A′

A+

√A

A′

)2

> 1

for all A,A′ > 0 where A 6= A′. Since the second termin the bracket in (E1) is non negative, (E1) then alwaysvanishes exponentially for β → ∞, as long as we do notlet β depend on ~ logarithmically. For A = A′ and PA 6=P ′A we only require that the product β~ does not diverge.Then the term in the bracket is bigger than 1 and thewhole expression vanishes again in the limit. The aboverequirements can be fulfilled by demanding that β ∝ 1/~for ~→ 0, but other choices are also possible.

Appendix F: Quantum corrected dynamics withmodified commutation relations

Here we want to check whether our investigations of thelower symbol of the Hamiltonian from Sec. V carries overto the alternate quantization map leading to modifiedcommutation relations constructed in Sec. VI. We havealready noted in Sec. VI that one can identify operatorsto phase space functions f(Π, R) in the new parametriza-tion with operators to f±(P,R) = f(Π±(P,R), R) in theold parametrization by using

|P,R〉old = |Π±(P,R), R〉new .

We write this identification as

“f = f±,

where the rounded hat denotes the quantization map us-ing the new parametrization, and the usual hat the oldone. From the above straightforwardly follows that onecan also identify lower symbols,

f(Π±(P,R), R) = 〈Π±(P,R), R| “f |Π±(P,R), R〉new= 〈P,R|f±|P,R〉old= f±(P,R).

At the level of the lower symbols the reparametriza-tion emerges as the original coordinate transformationon phase space.

This allows us to write,

R(t) =1

cΦ−3

∂

∂PH±(P,R) =

1

cΦ−3

∂

∂PH(Π±(P,R), R),

P (t) = − 1

cΦ−3

∂

∂RH±(P,R) = − 1

cΦ−3

∂

∂RH(Π±(P,R), R).

We can see that we can easily arrive at the old equationsof motion from the new lower symbol of the Hamiltonianby identifying Π = Π±, completely analogous to the clas-sical case. Furthermore we can now see how the equationsof motion look in the new phase space variables. To thisend we identify

P = P± = P,

21

where P is given by

P =R

2ln

∣∣∣∣∣1 + ΠR

1− ΠR

∣∣∣∣∣ .We then get the equations of motion

R(t) =1

cΦ−3

(1− Π2

R2

)∂

∂ΠH(Π, R),

Π(t) = − 1

cΦ−3

(1− Π2

R2

)∂

∂RH(Π, R),

from a coordinate transformation (R,P ) to (Π, R). Theform of these equations is completely equivalent to theclassical equations of motion with the modified Poissonbrackets (12).

[1] F. Lund, Phys. Rev. D 8, 3253 (1973).[2] C. Corda and F. Feleppa, arXiv:1912.06478 [gr-qc].[3] Y. Peleg, Phys. Lett. B 356, 462 (1995).[4] Y. Peleg, arXiv:9303169 [hep-th].[5] Y. Peleg, arXiv:9402036 [hep-th].[6] T. Schmitz, Phys. Rev. D 101, 026016 (2020).[7] C. Kiefer and T. Schmitz, Phys. Rev. D 99, 126010

(2019).[8] P. Hájíček and C. Kiefer, Int. J. Mod. Phys. D 10, 775

(2001).[9] P. Hájíček, “Quantum Theory of Gravitational Collapse

(Lecture Notes on Quantum Conchology),” in QuantumGravity: From Theory to Experimental Search, editedby D. J. W. Giulini, C. Kiefer, and C. Lämmerzahl(Springer, Berlin, Heidelberg, 2003) pp. 255–299.

[10] V. Frolov and G. Vilkovisky, Phys. Lett. B 106, 307(1981).

[11] A. Ashtekar, J. Phys.: Conf. Ser. 189, 012003 (2009).[12] C. Rovelli and F. Vidotto, Int. J. Mod. Phys. D 23,

1442026 (2014).[13] M. Bojowald, Universe 6, 36 (2020).[14] C. Bambi, D. Malafarina, and L. Modesto, Phys. Rev.

D 88, 044009 (2013).[15] Y. Liu, D. Malafarina, L. Modesto, and C. Bambi, Phys.

Rev. D 90, 044040 (2014).[16] J. Ben Achour, S. Brahma, and J.-P. Uzan, J. Cosmol.

Astropart. Phys. 03, 041 (2020).[17] J. Ben Achour and J.-P. Uzan, arXiv:2001.06153 [gr-qc].[18] J. Ben Achour, S. Brahma, S. Mukohyama, and J.-P.

Uzan, arXiv:2004.12977 [gr-qc].[19] M. Ambrus and P. Hájíček, Phys. Rev. D 72, 064025

(2005).[20] M. Christodoulou and F. D’Ambrosio, arXiv:1801.03027

[gr-qc].[21] M. Christodoulou, C. Rovelli, S. Speziale, and I. Vilen-

sky, Phys. Rev. D 94, 084035 (2016).[22] C. Barceló, R. Carballo-Rubio, and L. J. Garay, Class.

Quantum Grav. 34, 105007 (2017).[23] C. Barceló, R. Carballo-Rubio, and L. J. Garay, J. High

Energy Phys. 01, 157 (2016).[24] C. Bambi, D. Malafarina, and L. Modesto, Eur. Phys.

J. C 74, 2767 (2014).[25] P. Hájíček, Nucl. Phys. B 603, 555 (2001).[26] C. Barceló, R. Carballo-Rubio, L. J. Garay, and

G. Jannes, Class. Quantum Grav. 32, 035012 (2015).[27] C. Barceló, R. Carballo-Rubio, and L. J. Garay, Int. J.

Mod. Phys. D 23, 1442022 (2014).[28] H. M. Haggard and C. Rovelli, Phys. Rev. D 92, 104020

(2015).[29] D. Malafarina, Universe 3, 48 (2017).

[30] H. Bergeron and J. Gazeau, Ann. Phys. (NY) 344, 43(2014).

[31] J. P. Gazeau and R. Murenzi, J. Math. Phys. 57, 052102(2016).

[32] C. R. Almeida, H. Bergeron, J. P. Gazeau, and A. C.Scardua, Ann. Phys. (NY) 392, 206 (2018).

[33] H. Bergeron, A. Dapor, J. P. Gazeau, and P. Małkiewicz,Phys. Rev. D 89, 083522 (2014).

[34] H. Bergeron, E. Czuchry, J. P. Gazeau, andP. Małkiewicz, Universe 6, 7 (2019).

[35] A. Góźdź, W. Piechocki, and G. Plewa, Eur. Phys. J. C79, 45 (2019).

[36] A. Góźdź and W. Piechocki, Eur. Phys. J. C 80, 142(2020).

[37] K. V. Kuchař, Phys. Rev. D 50, 3961 (1994).[38] J. D. Brown and K. V. Kuchař, Phys. Rev. D 51, 5600

(1995).[39] P. Hájíček and J. Kijowski, Phys. Rev. D 57, 914 (1998).[40] N. Kwidzinski, D. Malafarina, J. J. Ostrowski,

W. Piechocki, and T. Schmitz, Phys. Rev. D 101, 104017(2020).

[41] K. Martel and E. Poisson, Am. J. Phys. 69, 476 (2001).[42] R. Gautreau and B. Hoffmann, Phys. Rev. D 17, 2552

(1978).[43] A. M. Perelomov, Commun. Math. Phys. 26, 222 (1972).[44] A. Góźdź, W. Piechocki, and T. Schmitz,

arXiv:1908.10039 [math-ph].[45] J. R. Klauder, Enhanced Quantization (World Scientific,

Singapore, 2015).[46] Y. Weissman, J. Phys. A: Math. Gen. 16, 2693 (1983).[47] J. R. Klauder, Phys. Rev. D 19, 2349 (1979).[48] R. Casadio, Int. J. Mod. Phys. D 09, 511 (2000).[49] A. Shapere and F. Wilczek, Phys. Rev. Lett. 109, 200402

(2012).[50] L. Zhao, P. Yu, and W. Xu, Mod. Phys. Lett. A 28,

1350002 (2013).[51] S. Ruz, R. Mandal, S. Debnath, and A. K. Sanyal, Gen.

Relativ. Gravit. 48, 86 (2016).[52] M. Henneaux, C. Teitelboim, and J. Zanelli, Phys. Rev.

A 36, 4417 (1987).[53] DLMF, “NIST Digital Library of Mathematical Func-

tions,” http://dlmf.nist.gov/, Release 1.0.25 of 2019-12-15, f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier,B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller,B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

[54] S. Mignemi, J. Phys.: Conf. Ser. 343, 012074 (2012).[55] J. R. Klauder, Int. J. Theor. Phys. 33, 509 (1994).[56] A. Vanrietvelde, P. A. Hoehn, F. Giacomini, and

E. Castro-Ruiz, Quantum 4, 225 (2020).[57] L. Mersini-Houghton, Phys. Lett. B 738, 61 (2014).