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Self-gravitating interferometry and intrinsic decoherence Gooding, David William Francisco 2015

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Self-gravitating Interferometry andIntrinsic DecoherencebyDavid William Francisco GoodingB.Sc., Simon Fraser University, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2015c© David William Francisco Gooding 2015AbstractTo investigate the possibility that an intrinsic form of gravitational decoher-ence can be theoretically demonstrated within canonical quantum gravity,we develop a model of a self-gravitating interferometer, and analyze theWKB regime of its reduced phase space quantization. We search for evi-dence in the resulting interference pattern that general relativity necessarilyplaces limits on coherence, due to the inherent ambiguity associated withforming superpositions of geometries. We construct the “beam” of the in-terferometer out of WKB states for an infinitesimally thin shell of matter,and work in spherical symmetry to eliminate the occurrence of gravitationalwaves. For internal consistency, we encode information about the beam op-tics within the dynamics of the shell itself, by arranging an ideal fluid on thesurface of the shell with an equation of state that enforces beam-splittingand reflections.The interferometric analysis is performed for single-mode inputs, and co-herence is shown to be fully present regardless of gravitational self-interaction.Next we explore the role coordinate choices play in our description of the in-terferometer, by considering a family of generalized coordinate systems andtheir corresponding quantizations. Included in this family are the Painleve´-Gullstrand coordinates, which are related to a network of infalling observersthat are asymptotically at rest, and the Eddington-Finkelstein coordinates,which are related to a network of infalling observers that travel at the speedof light. We then introduce another model, obtained by adding to the shell aharmonic oscillator as an internal degree of freedom. The internal oscillatorevolves with respect to the local proper time of the shell, and therefore servesas a clock that ticks differently depending on the shell’s position and mo-mentum. If we focus only on the external dynamics, we must trace out theclock degree of freedom, and this results in a form of intrinsic decoherencethat shares some features with a recently-proposed “universal” decoherencemechanism attributed to gravitational time dilation. We discuss some vari-ations of this proposal, and point out a way to bootstrap the gravitationalcontribution to the time dilation decoherence with self-gravitation. We in-terpret this as a fundamentally gravitational intrinsic decoherence effect.iiPrefaceAll of the research presented in this thesis was conducted under the super-vision of Bill Unruh while the author attended UBC.Much of the material in Chapters 1 and 2 has been published in Phys-ical Review D under the title “Self-gravitating interferometry and intrinsicdecoherence” [1]. This publication was co-authored by Bill Unruh. WithBill’s direction, the author of this dissertation performed the calculations,generated the figures, and did the majority of the writing.Some of the material in Chapter 3 was done in collaboration with Friede-mann Queisser, a postdoctoral fellow at UBC. Friedemann first brought tothe author’s attention the generalized family of coordinates presented in [68],and contributed notes to Sections 3.4-3.5.The content of Chapter 4 went into a paper entitled “Bootstrapping TimeDilation Decoherence,” co-authored by Bill Unruh, that was submitted onJan. 25th, 2015 as an invited contribution to a special issue of Foundationsof Physics (“Probing the limits of quantum mechanics: theory and experi-ment”). The material was written entirely by the author of this dissertation,with guidance from Bill.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Preliminary Considerations . . . . . . . . . . . . . . . . . . . 11.2 The Universe of this Nutshell . . . . . . . . . . . . . . . . . . 21.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Self-gravitating Interferometry . . . . . . . . . . . . . . . . . 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Theory of Self-gravitating Spherical Shells . . . . . . . . . . 112.2.1 Action Principle . . . . . . . . . . . . . . . . . . . . . 112.2.2 Hamiltonianization . . . . . . . . . . . . . . . . . . . 142.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . 152.2.4 Phase Space Reduction . . . . . . . . . . . . . . . . . 172.2.5 Boundary Terms . . . . . . . . . . . . . . . . . . . . . 202.2.6 Constructing Classical Spacetime . . . . . . . . . . . 212.3 Single-mode Interferometry . . . . . . . . . . . . . . . . . . . 222.3.1 Equation of State Determination . . . . . . . . . . . . 222.3.2 The WKB Approximation . . . . . . . . . . . . . . . 262.3.3 Flat Spacetime Limit . . . . . . . . . . . . . . . . . . 292.3.4 General Relativistic Picture . . . . . . . . . . . . . . 352.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41ivTable of Contents3 The Meaning of Time in Reduced Phase Space . . . . . . . 433.1 Coordinate Generalizations and Time Transformations . . . 433.2 Phase Space Reduction in a Family of Coordinate Systems . 463.3 Boundary Terms . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Reduced Equations of Motion . . . . . . . . . . . . . . . . . 503.5 Transcendental Hamiltonian Approximations . . . . . . . . . 513.6 Interferometry in the Painleve´-Gullstrand Family . . . . . . 533.6.1 Flat Spacetime Limit . . . . . . . . . . . . . . . . . . 533.6.2 General Relativistic Corrections . . . . . . . . . . . . 623.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 Superpositions of Clocks and Intrinsic Decoherence . . . . 714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Classical Action . . . . . . . . . . . . . . . . . . . . . . . . . 734.3 Hamiltonianization . . . . . . . . . . . . . . . . . . . . . . . 754.4 Reduced Phase Space Quantization . . . . . . . . . . . . . . 764.5 Time Dilation Decoherence . . . . . . . . . . . . . . . . . . . 804.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 Denouement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.1 Reflections and Resolutions . . . . . . . . . . . . . . . . . . . 84Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89AppendicesA Probability Current Conservation . . . . . . . . . . . . . . . 96A.1 Standard Nonrelativistic Quantum Mechanics . . . . . . . . 96A.2 General Systems Quadratic in Momenta . . . . . . . . . . . . 99B Multi-mode Interferometry . . . . . . . . . . . . . . . . . . . 101B.1 General Relativistic Wave-packets . . . . . . . . . . . . . . . 101B.2 Localize, Normalize, Propagate . . . . . . . . . . . . . . . . . 104B.3 Fringe Visibility and Path Retrodiction . . . . . . . . . . . . 108C The Exact WKB Phase . . . . . . . . . . . . . . . . . . . . . . 111vList of Figures2.1 Schematic representation of the splitting, reflecting, and re-combining that occur in our shell interferometer. The innerand outer mirrors are located at X− and X+, respectively,and the beam-splitter is located at Xδ. The initial split, de-picted near the bottom of the diagram, is characterized bya transmission coefficient T← and a reflection coefficient R←.These coefficients determine the amplitudes of the transmit-ted and reflected components of the initial state, which prop-agate along the different interferometer arms. The two arrowsat the top of the diagram represent the final interferometeroutputs, after recombination. . . . . . . . . . . . . . . . . . . 242.2 A sample equation of state represented by (2.60). There is anegative pressure peak p1 for some low density σ2 that causesthe shell to reflect inwards whenever X gets sufficiently large,there is a positive pressure peak p2 for some high density σ5 toreflect the shell outwards whenever X gets sufficiently small,and there is an intermediate pressure peak p2 that serves asa beam-splitter for our interferometer. . . . . . . . . . . . . . 252.3 A sample mass function Mˆ is plotted with respect to theshell radius X. The mass function Mˆ serves to parametrizean equation of state of the form depicted in Figure 2.2. Theapproximate step function near X = 2 serves as a beam-splitter, and the steep quadratic sides correspond to the innerand outer reflectors of the interferometer. . . . . . . . . . . . 292.4 Sample interference pattern, for M+ = 15, v+ = 0.003, X− =5000, L− = 20, and M− chosen to satisfy (2.108), plot-ted against the outer mirror position, X+. The alternat-ing pattern from 0 to 1 of both the final reflection and fi-nal transmission coefficients indicates that complete construc-tive/destructive interference occurs in this parameter region,from which we can conclude that coherence is fully present. . 38viList of Figures2.5 Sample interference pattern, for M+ = 15, v+ = 0.01, X− =200, L− = 20, and M− chosen to satisfy (2.108), plottedagainst the outer mirror position, X+. As in the previous fig-ure, there is an alternating pattern of the final reflection/transmissioncoefficients from 0 to 1 as the outer arm length changes, in-dicating that coherence is still fully present as we bring theinterferometric range closer to the Schwarzschild radius of theshell. In this region of the parameter space, we begin to seethe sum of the final reflection and transmission coefficientsfail to add up to exactly unity, due to the gradual breakdownof our approximation to the probability current. . . . . . . . . 382.6 Sample interference pattern, for M+ = 0.05, v+ = 0.0001,X− ≈ 1, L− ≈ 0.997, and M− chosen to satisfy (2.108), plot-ted against the outer mirror position, X+. The node spacingdecreases as the interferometric range is pushed closer to theSchwarzschild radius of the shell, and asymptotes to a fixedvalue ∆Ln = pi/4M+v+ as we approach spatial infinity. This“gravitational node squeezing” was present, though not visu-ally detectable, in the previous interference patterns. . . . . . 403.1 Sample reflection and transmission coefficients for the ini-tial beam-splitting using the exact flat spacetime momentumgiven by (3.59), for M+ = 1 and different combinations ofM− and E. The coefficients are plotted against the asymp-totic observer velocity, v∞. . . . . . . . . . . . . . . . . . . . 573.2 Sample reflection and transmission coefficients for the initialbeam-splitting, using the flat spacetime WKB momentumgiven by (3.61), for M+ = 1 and different combinations ofE and M−. The coefficients are plotted against the asymp-totic observer velocity, v∞. . . . . . . . . . . . . . . . . . . . 583.3 Various WKB momenta in flat spacetime, for M+ = 1 and dif-ferent combinations of E and M−, plotted against the asymp-totic observer velocity, v∞. The curved lines are WKB mo-menta which converge at the point where they become imag-inary; at this point, they become significantly different fromthe exact momenta (straight lines), which are real and distinctfor all regions of the parameter space. To avoid cluttering they-axis with multiple labels, the y-values of the plots, whichare various types of momenta, are specified in the legends. . . 60viiAcknowledgementsThe author would like to thank his supervisor, Bill Unruh, whose guidanceand patience have been sincerely appreciated. Also, thanks to the Natu-ral Sciences and Engineering Research Council of Canada (NSERC) andthe Templeton Foundation (Grant No. JTF 36838) for financial support.Finally, the author would like to thank Friedemann Queisser for collaborat-ing, Dan Carney, Philip Stamp, and the Aspelmeyer and Brukner groupsfor stimulating discussions, Bob Wald for clarifying some subtleties of phasespace reduction, Roger Penrose for inspiring this entire thesis, and my fam-ily, for sporadic help along the way.viiiDedicationTo Ashley, for her encouragement and support.ixChapter 1Introduction1.1 Preliminary ConsiderationsEver since the discovery by Hawking in the early 1970s that black holesemit thermal radiation due to quantum effects [2], it has been a majorgoal in theoretical physics to understand the connections between generalrelativity, quantum theory, and thermodynamics. The four laws of black holemechanics [3] themselves closely resemble thermodynamic laws, with horizonarea playing the role of an entropy (as pointed out by Bekenstein [4]), andHawking’s discovery of black hole evaporation indicated that this was morethan just a resemblance: a black hole was shown to have a temperatureequal to 1/2pi times its surface gravity, and an entropy of 1/4 times the areaof its event horizon (in natural units, c = G = ~ = 1).There still remain many open questions about the role of entropy ingravitational systems, as well as possible quantum-mechanical origins. Onemight hope that a complete theory of quantum gravity would answer suchquestions, but until such a theory is formulated we have to settle for partialpictures coming from approximation schemes. Black hole radiation, for in-stance, was demonstrated within the framework of quantum field theory incurved spacetime, which approximates the spacetime as being fixed and thequantum field as being a negligible perturbation that does not significantlyaffect the curvature of the spacetime.One of the major goals of this thesis is to determine the consequencesof eliminating the assumption of a fixed background spacetime on the ap-plication of quantum theory to gravitational systems. It is not possible toaccomplish this in general, so we will find an appropriate (idealized) arenafor our exploration, and work within a particular approximation scheme inorder to formulate tractable problems. Though it may be incorrect to simply“quantize” general relativity, as the Einstein equation could emerge from anunderlying theory as an equation of state [5], for the purposes of this thesiswe will consider the spacetime metric a fundamental degree of freedom, andwork within canonical quantum gravity.Apart from the specific issues raised by quantum field theory in curved11.2. The Universe of this Nutshellspacetimes, it is in general becoming more and more important to under-stand systems for which both quantum and general relativistic effects areimportant (see [6]-[8] for some recent experimental examples, though an ex-haustive discussion of such systems is beyond the scope of this thesis). It hasalso become increasingly clear that our current conceptual understanding ofquantum gravitational systems is severely lacking. This lack of understand-ing inspires us to study such systems, and the clash between quantum theory(QT) and general relativity (GR), in hope that we may find guidance to-wards a resolution of the many technical and conceptual problems one faceswhen attempting to unify these two pillars of physics.1.2 The Universe of this NutshellThe specific purpose of this work is to explore an ambiguity that resultsfrom taking both QT and GR seriously: quantum superpositions of differentmatter states are associated with different spacetime geometries, and hencedifferent definitions of time evolution; how, then, can we use a single time-evolution operator to evolve superpositions of distinct spacetimes? Sincethe 1980s, Roger Penrose has been arguing that this ambiguity results in aninstability, and that in turn this instability leads to a type of “decay” thatreduces the system to a state with a single well-defined geometry [9].Let us sketch the line of reasoning Penrose uses to reach this conclusion[10]. Suppose we have two quantum states, |ψ〉 and |χ〉, which representtwo static mass distributions, each with the same energy E but localized indifferent places. Neglecting gravity, both of these states will be “stationary,”meaning that each will be an eigenfunction of the appropriate Hamiltonianoperator H. Accordingly, we will have H|ψ〉 = E|ψ〉 and H|χ〉 = E|χ〉, andby the superposition principle we will also haveH (α|ψ〉+ β|χ〉) = E (α|ψ〉+ β|χ〉) , (1.1)for any (complex) α, β. This tells us that the arbitrary superposition α|ψ〉+β|χ〉 is also stationary, as it is an eigenstate of the Hamiltonian H withenergy E.In GR, gravity is a consequence of the manner in which a distribution ofmass/energy curves space and time. Thus, to incorporate gravity into ourdescription, we associate each mass distribution with a spacetime geometry,and therefore each of the states |ψ〉 and |χ〉 a geometry state (denoted by|Gψ〉 and |Gχ〉, respectively). The superposed state then takes the formα|ψ〉|Gψ〉+ β|χ〉|Gχ〉. (1.2)21.2. The Universe of this NutshellIs this superposition “stationary”? One should keep in mind that in GR,a spacetime is considered stationary if there exists a timelike Killing vectorT , which generates time-translations. Penrose then regards this T as thedifferential operator ∂/∂t for the spacetime, and interprets quantum statesthat are eigenstates of T to be stationary states, since they have well-definedenergies. Explicitly, |Ψ〉 is considered stationary ifT |Ψ〉 = −iEΨ|Ψ〉, (1.3)for some energy EΨ.One then faces a problem, when trying to understand the superposedstate (1.2): there is no longer a specific timelike Killing vector, since thereis no longer a specific spacetime. GR tells us that we cannot in generalcompare two spacetimes unambiguously through point-wise identification,so we therefore do not have an unambiguous differential operator with whichto define time-translation. This also prevents us from defining “stationarystates” as the eigenvectors of such an operator.If the states |ψ〉 and |χ〉 are very similar, then we can identify pointsin the corresponding spacetimes in an approximate way. In the Newtonianlimit, for instance, we can identify the time coordinate t of the two space-times, but even then we can not identify individual points of the spatialgeometries. In other words, there is a correspondence between the spatialslices of one spacetime with the spatial slices of the other, but not betweenthe points of a spatial slice of one spacetime with the points of a spatial sliceof the other. This ambiguity in point-wise identification of spatial slices di-rectly implies an essential ill-definedness in the notion of a time-translation,since although the time coordinates of the two spacetimes can be identified,it is the structure of the remaining (spatial) coordinates that defines thetime-translation operator.Penrose then suggests that we can quantify the difference between two(similar) spacetimes with the scalar quantity(fψ − fχ)2 = (fψ − fχ) · (fψ − fχ) , (1.4)with fψ and fχ being the acceleration 3-vectors of geodesic (free-fall) mo-tions of the associated spacetimes. From a Newtonian perspective, fψ and fχrepresent the Newtonian gravitational force-per-unit-test-mass (also calledsimply the gravitational field), at a particular point, in each space-time [10].The total measure of incompatibility (or “uncertainty,” as Penrose callsit) can be obtained by integrating (1.4) over a spatial slice. We will call the31.2. The Universe of this Nutshellresulting quantity ∆G. For concreteness, Penrose uses a flat spatial metricfor the integration, in which case the spatial integral of (1.4) yields∆G = −4pi∫ ∫d3x d3y(ρψ(x)− ρχ(x)) (ρψ(y)− ρχ(y))|x− y|, (1.5)with the mass density ρ connected to the gravitational field f via the New-tonian gravitational equation (Gauss’s law) ∇·f = −4piρ. The quantity ∆Gis effectively the gravitational self-energy of the difference between the twomass distributions. Penrose thinks of ∆G as an energy uncertainty associ-ated with the superposition of geometries, and hypothesizes that this energyuncertainty makes the superposition unstable. In analogy with a radioactivenucleus, or any other unstable particle, we can use the time-energy uncer-tainty principle to estimate the lifetime of the superposition, ∆t ∼ 1/∆G,after which there is a significant probability that the superposition will “de-cay” into one geometry or the other. A consequence of this lifetime estimateis that as the geometries associated with each element of the superpositionbecome arbitrarily different, the superposition becomes arbitrarily unstable,and decays after an arbitrarily short time.It is not clear from Penrose’s work, however, whether some sort of “col-lapse” occurs, or whether there is simply a form of “intrinsic” decoher-ence that removes phase correlations between states associated with suf-ficiently different geometries. In this thesis, we consider the latter, anddiscuss whether or not a direct application of both QT and GR is enoughto demonstrate the existence of this new type of “intrinsic” decoherence.By “intrinsic” decoherence, we mean a decoherence effect that arisessolely out of the internal behaviour of an isolated system, and not due toits interaction with the external world. For example, if we use a buckyballin a double-slit experiment, and prepare one of the slits to excite internaldegrees of freedom of the buckyball, then the internal degrees of freedomcarry “which-way” information and decohere the center-of-mass degree offreedom [11]. More generally, if a system carries an internal clock and isin a superposition of states corresponding to two paths that have differentproper times associated with them, then again the internal clock read at theinterference screen could provide which-way information, and decohere thecenter-of-mass [12]-[14].Whereas the decoherence produced by entangling internal degrees offreedom to a center-of-mass coordinate could be considered “third-party”decoherence [15], what we are primarily concerned with here is whether ornot there is something about the quantum effect of a system’s gravity onthe system itself that could lead to such intrinsic decoherence. Penrose’s41.2. The Universe of this Nutshellintuition says yes: the path a mass takes alters the associated spacetimeand especially the flow of time. Since the quantum phase is determined bythe flow of time, the phase evolution is also altered by which path the masstakes. When one tries to interfere the two paths, these “random” phases(because it is impossible to uniquely map one spacetime onto another) causedecoherence. It is this gravitational intrinsic decoherence that we explorehere.There are several proposals in the literature for a mechanism to describeintrinsic forms of gravitational decoherence [16]-[18], and other proposalsfor intrinsic decoherence mechanisms that are merely inspired by tensionbetween QT and GR [19]-[21]. Many of these approaches incorporate alter-ations to known physics, such as adding stochastic [21] or nonlinear [22]-[24]terms to the Schro¨dinger equation, in order to achieve the desired decoher-ence effect. Such alterations are often ad hoc, and have historically faceddifficulties maintaining consistency with experimental constraints; nonlinearadditions to the Schro¨dinger equation, for instance, have been shown undera wide range of conditions to lead to either superluminal signal propagationor to communication between different branches of the wavefunction [25].While it may be possible to obtain a sensible theory that allows communi-cation between different wavefunction branches [26], it remains to be seenwhether a consistent interpretation results from this alteration. Instead, wetake Penrose’s initial arguments at face value, and entertain the possibilitythat a consistent combination of QT and GR can explain (gravitational)intrinsic decoherence without any assumptions about new physics.Now, it is well-known that gravitational waves can carry away informa-tion from a system in a manner analogous to standard decoherence [27],[28]. Penrose’s suggestion is independent of such standard gravitational-wave-induced decoherence, so to distinguish between the two we will workin spherical symmetry. The restriction to spherical symmetry is not onlya technical simplification, but avoids the occurrence of gravitational wavesaltogether.Because this exploration requires both QT and GR, we will naturally befaced with some serious difficulties, which we will have to either overcome,sidestep, or ignore [29]. For instance, we will avoid issues with the factor-ordering ambiguity by working in the WKB1 regime (as in [30]), and wewill avoid issues with perturbative non-renormalizability by working in min-isuperspace (i.e. enforcing spherical symmetry) and employing a reduced1By “WKB,” we mean “Wentzel-Kramers-Brillouin.”51.2. The Universe of this Nutshellphase space approximation (as in [31])2. It is still unclear what the exactconnection is between the reduced phase space approximation, obtained bysolving the GR constraints classically and then quantizing the reduced the-ory, and the standard Dirac quantization, obtained by quantizing the theoryin the full kinematical Hilbert space and then enforcing the constraints atthe quantum level. Following Hawking’s path integral approach to quantumgravity [32], Halliwell has made some progress elucidating the connectionbetween reduced phase space minisuperspace quantization and the standardDirac approach in special cases [33], but in general the connection is not wellunderstood. Nonetheless, the limit we will work in has a rich structure, andin this thesis we will explore whether or not it has a rich enough structureto contain evidence of intrinsic decoherence caused by gravity.Since we aim to test whether or not gravity places a fundamental limit onthe coherence of quantum systems, we develop a model of a self-gravitatinginterferometer. Interferometers are ideal for studying coherence, becauseinterference is a key feature of coherent systems. We describe how the sameinterferometer would behave in the absence of gravity, and then we inves-tigate the consequences of general relativistic corrections to this behaviour.In an interferometric setting, the intrinsic decoherence we seek to under-stand manifests itself as a phase-scrambling along different interferometerarms (for a general discussion see [34]), which in this case is attributed togravity. According to Penrose, we should expect that the (interferometric)coherence should decay as the arm-length increases indefinitely, since thiswould correspond to a superposition of arbitrarily different spacetimes. Wefocus on the possibility that no collapse occurs, so we will simply analyzethe interference pattern and search for departures from non-gravitationalbehaviour that indicate coherence loss. Conceptually, we are testing theidea that when one forms superpositions of geometries in the interferome-ter, the nature of time in GR leads at the quantum level to an imprint ofwhich-way information, which is accompanied by a loss of fringe visibility[35].Still, an objection may be raised that if one describes the interferome-ter as a closed quantum system without tracing out over any physical de-grees of freedom, then QT implies that coherence must prevail, regardlessof whether the system is general relativistic. This objection was raised byBanks, Susskind, and Peskin [36] in the context of black hole evaporation,2As we will see in Chapter 2, the combination of the minisuperspace restriction and thereduced phase space approximation reduce the number of (continuous) degrees of freedomfrom infinity to one, thus sidestepping any renormalization issues.61.2. The Universe of this Nutshellbut it was later pointed out not only that the arguments in [36] were incon-clusive, and that we have reason to support the possibility that pure statescan effectively evolve into mixed states in black hole systems [37].The more radical idea entertained here is that one might find pure statesevolving to mixed states in gravitational systems without horizons. In gen-eral, this “dissipationless” type of decoherence has been explored to somedegree [38]-[42], but even the fact that it is possible has not been widelyappreciated. Nonetheless, one can observe that the thermal character of ac-celeration radiation is approximately present even without the involvementof Rindler horizons (for recent analyses see [43]-[46]), and by the equivalenceprinciple one might expect to find a gravitational analog of this thermal be-haviour. This means, then, that one might expect that gravity generatesan intrinsic form of entropy, even in systems without the horizon structurethat one usually associates with entropy in black hole thermodynamics.With this in mind, we will construct our interferometer, theoretically,out of a self-gravitating, spherically symmetric, infinitesimally thin shell ofmatter. The interferometer “optics” are encoded internally, by adding tan-gential pressure to the fluid that lives on the surface of the shell. The result-ing model is reminiscent of an idea Einstein first proposed in 1939 [47], butin our case, the tangential pressure satisfies an equation of state that pro-duces a beam-splitter and perfect reflectors. The fluid is ideal, in the sensethat one obtains a perfect-fluid stress-energy tensor, if one projects the fullfour-dimensional spacetime stress-energy tensor onto the three-dimensionalhistory of the shell. This approach ensures that at the classical level, theinterferometric setup is manifestly invariant under coordinate transforma-tions.The configuration we construct resembles that of a Michelson interfer-ometer in optics. Thus, we will send initial states at a beam-splitter, atwhich point the transmitted and reflected components travel in opposite di-rections until they encounter “mirrors.” The components will then reflect,travel back towards each other, and encounter the splitter once more. Therewill be two possible outputs, corresponding to final transmission and finalreflection, which are comprised of different combinations of the initially splitwave components.What we mean by (interferometric) coherence, in this system, is thesustained phase relationships between different wave components that canallow us, for instance, to completely cancel either of the final outputs. Inother words, if we are unable to obtain complete constructive or destructiveinterference in our interferometer (as predicted by Penrose), we can concludethat coherence is being limited in the system. The goal of our current71.3. Outlineinvestigation is to determine whether or not general relativistic effects coulddemonstrably produce such a limitation.1.3 OutlineOur investigation begins in the following chapter (Chapter 2) with the con-struction of our self-gravitating interferometer model, followed by an anal-ysis of the resulting interference for single-mode input states, in the WKBregime. The stationary nature of the probability distributions for the single-mode states makes the notion of a probability current very useful, and wewill exploit this usefulness to study the interference patterns in our system.Chapter 2 makes use of a specific coordinate choice during the processof phase space reduction. The reduced phase space is classically equiva-lent to the full phase space, modulo coordinate transformations. Does thisproperty hold at the quantum level? Chapter 3 generalizes the original co-ordinate choice to a family of similar coordinate choices, at which point wecan determine which aspects of our system (if any) depend on the coordi-nate choice. Along the way, we will encounter several serious obstacles thatobscure the answers to these questions, and discuss what we can learn fromthe difficulties that arise.We then turn our attention to a more complicated model, presented inChapter 4, obtained by adding an internal degree of freedom to the shell.The internal degree of freedom is chosen to be a harmonic oscillator thatevolves with respect to the proper time of the shell. As such, the oscillatorbehaves as a local clock, and evolves differently depending on the shell’s(external) motion due to time dilation. If one is only interested in makinga measurement of the external properties of the shell, it is necessary to takethe partial trace over the clock degree of freedom. We show that even in the(gravity-free) limit of flat spacetime, this leads to the type of intrinsic deco-herence previously referred to as “third-party” decoherence [15], due to theacceleration produced by the tangential pressure on the surface of our shell.Though in this limit the system exhibits “third-party” decoherence, it doesnot result from a fundamental limit to coherence brought about by generalrelativity, and is thus not the type of intrinsic decoherence studied in the ear-lier chapters. However, by incorporating gravitational self-interaction, wecan demonstrate that the system exhibits “third-party” decoherence evenwhen the tangential pressure vanishes, because of the gravitational influ-ence of the system on itself. We interpret this “third-party decoherencewithout the third party” as a fundamentally gravitational intrinsic decoher-81.3. Outlineence effect.The thesis then finishes in Chapter 5 with a reflection on the results ofthe previous chapters: the implications, remaining questions, and overalllessons that can be learned from this work.9Chapter 2Self-gravitatingInterferometry2.1 IntroductionThis chapter introduces the model we use to construct our self-gravitatinginterferometer. To remind the reader, this model has spherical symmetry,to avoid gravitational wave decoherence, and involves an infinitesimally thinshell of matter, so that we only have a single (continuous) physical degreeof freedom to work with. The physical degree of freedom in this case is the“areal” radial coordinate of the shell, defined as the square root of the shellarea divided by 4pi. This “areal” radius is closely connected with the shell’senergy density.In our construction of the interferometer, we will embed “mirrors” anda beam-splitter into the dynamics of the shell itself, to avoid any artificialexternal influences on the system. This can be accomplished by adding tan-gential pressure to the surface of the shell, with an equation of state thathas the necessary features to describe reflections and beam-splitter. Thus,the “mirrors” and beam-splitter are designed to be triggered by internal fea-tures of the shell, like energy density, rather than being located at particularcoordinate points; this removes all possible coordinate dependence from themotion of the shell and the locations of the interferometer’s optical compo-nents. Since we are using an “areal” radial coordinate to describe the shellposition, the correspondence between the shell area and the energy densityfixes the optical components at specific radial coordinate values.The reduced Hamiltonian treatment of a spherically symmetric systemwas originally done by Berger, Chitre, Moncrief and Nutku [48] and wascorrected by Unruh [49], who also applied the technique to a spherical shell[private communication]. This is similar to Kraus and Wilczek [50]-[52] whoused it to quantize a pressureless spherical shell. The coordinate choices theymade (the Painleve´-Gullstrand form) simplified the Hamiltonian equationssignificantly, and left the Hamiltonian that of a single degree of freedom (the102.2. Theory of Self-gravitating Spherical Shellsposition of the shell). This approach also works for a shell with tangentialpressure, which is what we will use in this thesis.Much of this chapter will focus on the construction of a general relativis-tic action principle for our model, followed by the Hamiltonian formulationof the system, and subsequent phase space reduction. Once in its reducedform, the system is simple enough to be analyzed as an interferometer. Wewill first describe this interferometer in the flat spacetime limit, and pointout how coherence manifests itself in the interference pattern. Finally, wewill calculate general relativistic corrections to the flat spacetime behaviour,to search for evidence of a fundamental limitation on the system caused bygravity.2.2 Theory of Self-gravitating Spherical Shells2.2.1 Action PrincipleThe perfect-fluid shell model we now develop is a generalization of the dustshell model used by Kraus and Wilczek in their attempt to calculate self-interaction corrections to standard Hawking radiation [50]-[52]. Generalizingthe Kraus and Wilczek approach to include the required pressure effectsis not without complications, even in the classical theory. In contrast tothe approach to thin-shells pioneered by Israel that involves stitching twospacetimes together along the shell’s history [53], the starting point for ourtheoretical considerations is an action that is composed of a gravitationalpart given by the Einstein-Hilbert action, plus some action for the shell thatwe can initially leave unspecified, written (in natural units) asI =116pi∫d4x√−g(4) R(4) + Ishell. (2.1)The superscripts on the metric determinant g and the Ricci scalar R indi-cate that these quantities are constructed from the full (3 + 1)-dimensionalspacetime metric components {gµν}, with µ, ν ∈ {0, 1, 2, 3}.We will express the metric in Arnowitt-Deser-Misner (ADM) form [54],which in spherical symmetry is given bygµνdxµdxν = −N2dt2 + L2 (dr +N rdt)2 +R2dΩ2, (2.2)where N is the lapse function, N r is the radial component of the shift vector,and L2 and R2 are the only nontrivial components of the spatial metric.The angular variables are taken to be the polar angle θ and the azimuthalangle φ, such that the angular metric takes the form dΩ2 = dθ2 + sin2 θdφ2.112.2. Theory of Self-gravitating Spherical ShellsThe shell action studied by Kraus and Wilczek takes the formIdust = −m∫dλ√−gµνdxµdλdxνdλ, (2.3)with m being the rest-mass of the shell and all metric quantities evaluated onthe shell history. The arbitrary parameter λ can be chosen to coincide withthe coordinate time t, which simplifies the integrand for the shell action.To describe a more general fluid than dust, we need a more generalaction. There are well-established variational principles for regular perfectfluids in GR [55], but the authors are unaware of any satisfactory variationalprinciples for the perfect fluid shells we wish to describe. The stress-energytensor for a perfect fluid with density σ and pressure p is given bySab = σuaub + p(γab + uaub), (2.4)where ua are the components of the fluid proper velocity in coordinates thatcover the fluid history. For our purposes, the geometry along the fluid historyof our shell is described by an induced metric γabdyadyb = −dτ2 + Rˆ2dΩ2,with τ being the shell proper time. This induced metric obeys the relationγab = eµaeνbgµν , (2.5)with the introduction of projectors onto the shell history given byeµa =∂xµ∂ya= uµδτa + δµΩδΩa . (2.6)Here and elsewhere, the repeated Ω denotes a sum over angular coordinates.These projectors allow us to express the full spacetime stress-energy tensorof our perfect fluid shell asTµν = Sabeµaeνb δ(χ), (2.7)where we have introduced a Gaussian normal coordinate χ in the directionof the outward-pointing space-like unit normal ξ, with the shell locationdefined by χ = 0.We want to obtain an action, expressed in terms of the full spacetimequantities, that yields the tensor (2.4) in the intrinsic coordinates of the shellhistory. To convert derivative expressions from the intrinsic coordinates tothe ADM coordinates given in equation (4.3), we can write infinitesimalchanges in r and t asdt = utdτ + ξtdχ, dr = urdτ + ξrdχ. (2.8)122.2. Theory of Self-gravitating Spherical ShellsTaking advantage of the fact that ξ satisfies uµξµ = 0 and ξµξµ = 1, andsuppressing the (vanishing) angular components for brevity, the outwardnormal can be written asξα =√g2tr − gttgrr(−ur, ut)= N2L2(−ur, ut). (2.9)For radial integration within an ADM slice, one has dt = 0, and in thiscase we can solve for drdχ in equation (2.8) to obtain [56]drdχ= ξr −urutξt. (2.10)Also, since uµ = (dt/dτ)(1, X˙, 0, 0), the 4-velocity normalization uµuµ = −1(evaluated on the shell) implies(dtdτ)2= (ut)2 =(N2 − L2(N r + X˙)2)−1. (2.11)This allows conversion of the delta function appearing in our expression(2.7) for the full spacetime stress-energy tensor:δ(χ) =drdχδ(r −X) =√N2 − L2(N r + X˙)2NLδ(r −X). (2.12)Using equation (2.7), we find that our stress-energy tensor takes the formTµν =(σuµuν + pgΩΩδµΩδνΩ)δ(χ), (2.13)where the repeated Ω indices denote a single sum over angular coordinates.In expression (2.13), the “tangential” nature of the pressure is manifest, sincethe projection of this tensor onto the space-like normal ξ clearly vanishes.The action we seek, then, yields (2.13) upon taking variations with re-spect to the metric, in accordance with the definitionδI =12∫d4x√−4gTµνδgµν . (2.14)We are especially interested in the contribution from the tangential pressure,which takes the formδIp = 8pi∫dt dr NLδ (χ) pRδR. (2.15)132.2. Theory of Self-gravitating Spherical ShellsBy inspection, we find that the actionIshell = −∫dλ√−gµνdxµdλdxνdλM(R), (2.16)with all quantities evaluated on the shell history, yields the appropriatestress-energy tensor: the relevant variational derivative of (4.1) with respectto the metric isδIshell,p = −∫dt dr NLδ (χ)M ′(R)δR, (2.17)from which it follows that one has the pressure identification p = −M ′(R)/8piR,along with the usual density identification σ = M(R)/4piR2. We will usethe freedom in choosing the function M(R) to parametrize an equation ofstate that relates the density and pressure of our fluid. It should be notedthat R is not a coordinate, but a metric component that serves as a measureof the shell’s internal energy.The action (4.1) is reparametrization-invariant, as well as invariant undergeneral (spherically symmetric) coordinate transformations, even with theinclusion of an R-dependent ‘mass’. As mentioned above, this is because R,when evaluated on the shell, is nothing more than the reduced area of theshell, and this area is independent of coordinate choices.2.2.2 HamiltonianizationFollowing the canonical formalism [54], one can perform a Legendre transfor-mationH = PX˙−L, for the shell variables. Here L is the Lagrangian definedby (4.1), subject to the condition that the shell history is parametrized byt. One then findsL = −∫dr√N2 − L2(N r + X˙)2M(R)δ(r −X), (2.18)and it follows that the momentum conjugate to the shell position X for theunreduced problem is given byP =∂L∂X˙=∫drL2(N r + X˙)M(R)√N2 − L2(N r + X˙)2δ(r −X). (2.19)Explicitly, we can determine the Hamiltonian H to beH = PX˙ − L =∫dr (NHs0 +NrHsr ), (2.20)142.2. Theory of Self-gravitating Spherical Shellswith the definitionsHs0 =√L−2P 2 +M(R)2δ(r −X),Hsr = −Pδ(r −X). (2.21)Similarly, we can Hamiltonianize the gravitational action, and expressthe total action asI =∫dt PX˙ +∫dt dr(piRR˙+ piLL˙−NH0 −NrHr), (2.22)for H0 = Hs0 +HG0 and Hr = Hsr +HGr , such thatHG0 =Lpi2L2R2−piLpiRR+(RR′L)′−(R′)22L−L2,HGr = R′piR − Lpi′L. (2.23)2.2.3 Equations of MotionOnce in Hamiltonian form, the equations of motion for the system are ob-tained by varying the action with respect to the variables N , N r, piL, piR,L, and R. Explicitly, these variations (respectively) lead toH0 = 0,Hr = 0,L˙ =NR(LpiLR− piR)+ (N rL)′ ,R˙ = −NpiLR+N rR′, (2.24)L˙ =N2(1−pi2LR2−(R′)2L2)−N ′RR′L2+N rpi′L +NP 2δ(r −X)L2√P 2 + L2M2,R˙ =NpiLR2(LpiLR− piR)−N(R′L)′−(N ′RL)′+ (N rpiR)′ −NM dMdR δ(r −X)√L−2P 2 +M2.The first two equations are the Hamiltonian and momentum constraints,whereas the next four are the dynamical equations of motion for the gravi-tational variables.152.2. Theory of Self-gravitating Spherical ShellsFor the shell variables, the equation of motion for X can be easily ob-tained by varying the action with respect to P , or simply by solving equation(2.19) for X˙. The result isX˙ =∫dr(NPL√P 2 + L2M2−N r)δ(r −X)=NˆPLˆ√P 2 + Lˆ2Mˆ2− Nˆ r, (2.25)with hats indicating that one evaluates the quantities at r = X.The equation of motion for P is more subtle, since a standard variationof the action with respect to X is formally ambiguous, as noted in [57].The ambiguity arises because one must evaluate quantities on the shell (L′,(N r)′, N ′ and R′) that are (possibly) discontinuous at r = X:P˙ =(N rP −N√L−2P 2 +M2)′shell. (2.26)However, it has been demonstrated in [58] that this ambiguity can beremoved by requiring consistency with the constraints and the gravitationalequations of motion (2.24), at least for the case of a dust shell. The ar-gument described in [58] shows that one must average the discontinuousquantities when interpreting the equation of motion for the shell momen-tum, and similar reasoning leads to the same conclusion for the arbitraryperfect fluid shell described here. One then has the equation of motionP˙ = ¯(N r)′P −N¯ ′Lˆ√P 2 + Lˆ2Mˆ2 +Nˆ(P 2L¯′ − Lˆ3MˆM¯ ′)Lˆ2√P 2 + Lˆ2Mˆ2, (2.27)with the average taken over (N r)′ in the first term of the right-hand-side,and the last term containing the factor M¯ ′ defined as M¯ ′ = ˆdMdR R¯′.Let us briefly sketch the argument that leads to this result. To start,we take the time derivative of the (integrated and rearranged) momentumconstraint:P˙ = −∆piLddt(Lˆ)− Lˆddt(∆piL) . (2.28)Then, by continuity of L˙, we haveddtL(X) = L′(X ± )X˙ + L˙(X ± ) = L¯′X˙ + ¯˙L. (2.29)162.2. Theory of Self-gravitating Spherical ShellsAveraging the equation of motion for L, noting that ddt(∆piL) = ∆(pi′L)X˙ +∆( L˙), and calculating ∆( L˙) from the equation of motion for piL, we obtainP˙ = P˙ + Φ, (2.30)with P˙ representing the right side of equation (2.27), and Φ defined suchthatΦ = − PNˆRˆLˆR¯ +Nˆ∆R′R¯′Lˆ+ ∆N ′R¯′RˆLˆ− Lˆ∆pi′L(Nˆ r + X˙) +NˆMˆ ˆdMdR R¯′√Lˆ−2P 2 + Mˆ2. (2.31)To then demonstrate that Φ vanishes, one needs to take the jump of themomentum constraint across the shell to obtain Lˆ∆pi′L = R¯′∆piR + R¯∆R′,then integrate the equation of motion for piR across the shell, and use theresult, combined with the fact that the delta contribution to R˙ is given by−X˙(∆piR)δ(r −X) [57].2.2.4 Phase Space ReductionWe now seek a description of the system in terms of only the shell coordinateX and a conjugate momentum Pc. Note that it is not necessarily true thatPc will coincide with the conjugate momentum P for the unreduced problem,as will become clear in what follows.To proceed with the Hamiltonian reduction, we will make use of theLiouville form F and the symplectic form Ω, which on the full phase space(denoted by Γ) can be written asF = PδX +∫dr (piLδL+ piRδR) (2.32)andΩ = δP ∧ δX +∫dr (δpiL ∧ δL+ δpiR ∧ δR), (2.33)respectively, with δ denoting an exterior derivative in the associated func-tional space (see [57] for more details). The reduced phase space Γ¯ is definedas the set of equivalence classes in Γ under changes of coordinates, and each(permissible) choice of coordinates defines a hypersurface H¯ ⊆ Γ that istransversal to the orbits generated by coordinate transformations; this en-sures that there exists an isomorphism between Γ¯ and the representativehypersurface H¯.172.2. Theory of Self-gravitating Spherical ShellsAt this point we can determine the symplectic form Ω¯ induced on H¯ asfollows: first, consider the pullback of F to H¯; this yields a quantity whichwe denote by FH¯ . Then, the symplectic form ΩH¯ on the representativehypersurface H¯ (corresponding to Ω¯) takes the formΩH¯ = δFH¯ . (2.34)This quantity defines the canonical structure of the reduced phase space.To explicitly determine the (nonlocal) contribution of the gravitationalvariables to the dynamics on the reduced phase space, we can solve theGR constraints for the gravitational momenta, insert the solutions into theLiouville form on the full phase space, and perform the integration to expressthe gravitational contribution solely in terms of the (local) shell variables.Away from the shell, take the following linear combination of the constraints:−R′LH0 −piLRLHr =M′, (2.35)forM(r) =pi2L2R+R2−R(R′)22L2. (2.36)The quantityM(r) corresponds to the ADM mass H when evaluated outsideof the shell, and vanishes inside the shell. This enables us to solve for thegravitational momenta piL, piR away from the shell. The result ispiL = ±R√(R′L)2− 1 +2MR, piR =LR′pi′L. (2.37)One then makes a coordinate choice, to pick out a representative hyper-surface H¯. The coordinates we will use resemble the flat-slice coordinates{L = 1, R = r} described in [51] (also known as Painleve´-Gullstrand coor-dinates), though we will have a deformation region X−  < r < X explicitlyincluded, in order to both satisfy the constraints and yield a continuousspatial metric. The deformation region is related to a jump in R′ across theshell. This can be seen by first integrating the Hamiltonian and momentumconstraints across the shell. Doing so yields, respectively,∆R′ = −VˆRˆ, ∆piL = −PLˆ, (2.38)where V =√P 2 +M2L2 and ∆ indicates the jump of a quantity across theshell. We therefore takeL = 1, R(r, t) = r −XVˆ G(X − r), (2.39)182.2. Theory of Self-gravitating Spherical Shellsfor a function G having support in the interval (0, 1) with the propertydG(z)/dz → 1 as z ↘ 0. Outside of the deformation region, these coincidewith flat-slice coordinates. For concreteness, let us suppose G(z) takes theformG(z) = ze−z2/(1−z2) (2.40)for all z ∈ (0, 1).In what follows, it will be useful to note that Mˆ = M(Rˆ) = M(X), andthat now P is considered to be a function of X and H, as a consequence ofthe gravitational constraints. We can implicitly determine this function byinserting the gravitational momentum solutions away from the shell givenby equation (4.22) into the jump equations (2.38) and squaring. We arethen left withH = Vˆ +Mˆ22X− P√2HX. (2.41)With this coordinate choice, the only gravitational contribution to theLiouville form comes from the piR term, and only from within the deforma-tion region. Keeping in mind that we only care about terms that remainnonzero in the → 0 limit, we have, in the deformation region,piR =XR′′√(R′)2 − 1+O(1), (2.42)since R = X + O() and R′′ = O(−1). One can also note that δR =(1−R′) δX +O(), and express the gravitational contribution to the Liou-ville form as∫ XX−dr piRδR = XδX∫ XX−drR′′ (1−R′)√(R′)2 − 1+O(). (2.43)To evaluate this integral, one can change the integration variable from r tov = R′: ∫ XX−dr piRδR = XδX∫ R′−1dv(1− v)√v2 − 1+O(), (2.44)with R′− being R′ evaluated just inside the shell. The integration is thenstraightforward, and after applying (2.41) and making some rearrangementsone arrives atXδX[−PX−√2HX+ ln(1 +√2HX+Vˆ + PX)], (2.45)192.2. Theory of Self-gravitating Spherical Shellsplus terms that vanish as → 0. This completes the calculation of FH¯ , thepullback of the full Liouville form F to H¯:FH¯ = PcδX, (2.46)with the reduced canonical momentum evidently given byPc = −√2HX +X ln(1 +√2HX+Vˆ + PX). (2.47)This result agrees with [57] in the limit of a dust shell (Mˆ ′ = 0).To connect this with the expression derived by Kraus and Wilczek, weneed only apply the expression (2.41) to the argument of the logarithm,which leads toPc = −√2HX −X ln(X + Vˆ − P −√2HXX). (2.48)This form of the reduced momentum coincides with [50] in the dust-shelllimit.2.2.5 Boundary TermsTo obtain a well-defined variational principle for the reduced problem, wemust be careful with boundary terms, as first noted in [59] and [49]. In[50], it is observed that for asymptotically-flat spacetimes, we simply needto subtract the ADM mass (denoted suggestively by H) from our reducedLagrangian. Specifically, as mentioned in [57], a nonzero boundary variationresults from integrating the term∫dtdrN rL(δpiL)′ (which is part of themomentum constraint) by parts. The only contribution comes from infinity,and in this case we have N r → N√2H/r, N → 1, andδ(piL)→ δ(√2Hr) =√r2HδH, (2.49)so the variation of the boundary term is cancelled if we add to the actionthe termIbdry = −∫dtH. (2.50)Including the boundary term in the action defined by FH¯ , one obtainsthe reduced actionIreduced =∫dt(PcX˙ −H), (2.51)202.2. Theory of Self-gravitating Spherical Shellswith the reduced momentum given by (2.48). From the form of the reducedaction (2.51), we can see that the ADM mass is the reduced Hamiltonian,defined implicitly by (2.48) and (2.41).Since (2.41) has more than one solution P = P (X,H), our conjugatemomentum Pc in turn becomes a multi-valued function of X and H, as oneexpects from a theory that allows the degree of freedom to either increaseor decrease. Explicitly, P is given byP =11− 2HX(√2HX(H −Mˆ22X))±11− 2HX√√√√(H −Mˆ22X)2− Mˆ2(1−2HX) , (2.52)while the combination Vˆ −P that appears in the reduced momentum (2.48)isVˆ − P =H − Mˆ22X ∓√(H − Mˆ22X)2− Mˆ2(1− 2HX)1 +√2HX. (2.53)2.2.6 Constructing Classical SpacetimeSuppose one can find a solution X(t) to the classical equations of motion forthe reduced system (2.51). Then, the gravitational constraints and equationsof motion (2.24) can be solved to determine all the metric components gµν .Therefore, from the reduced system solution X(t) one can construct theclassical spacetime structure, as we will now demonstrate.By inserting the gravitational momenta solutions (4.22) into the grav-itational equations of motion (2.24), one can obtain the lapse function Nand the radial shift component N r that correspond to our coordinate choice(2.39).Outside of the shell, one finds the familiar Schwarzschild structure, inflat-slice coordinates. The lapse function is constant, and unity if we wanta time coordinate that increases towards the future, while the radial shift isgiven byN r(r ≥ X) = ±√2Hr. (2.54)The ± here indicates two possible time-slicings, though we will often takethe upper sign (this means that the gravitational momenta solution (4.22)should take the upper sign as well, to ensure N → 1 as r →∞).212.3. Single-mode InterferometryAlong with the expression (2.41) for P in terms of X and H, we now haveenough information to determine the classical path X(t), since H is constantalong such paths. Specifically, the equation of motion for X becomesX˙ =P√P 2 + Mˆ2−√2HX, (2.55)which leads to the expressiondtdX=√2HXX − 2H(2.56)±H − Mˆ22X(1− 2HX)√(H − Mˆ22X)2− Mˆ2(1− 2HX).Therefore, finding the classical path X(t) has been reduced to quadratureand inversion.With the classical path known, one can also calculate the classical action,as done for the case of dust in [50]:S(t,X(t)) = S(0, X(0)) +∫ t0dt˜[Pc(t˜)X˙(t˜)−H], (2.57)withPc(0) =∂S∂X(0, X(0)). (2.58)Unlike the (massless) dust case, however, our classical path X(t) is not anull geodesic of the flat-slice metricds2 = −dt2 +(dr +√2Hrdt)2, (2.59)and so we cannot so easily determine explicit expressions for our shell tra-jectories.2.3 Single-mode Interferometry2.3.1 Equation of State DeterminationUp until this point, the function M(X) has been left unspecified, thoughwe have established the identifications σ = M(X)/4piX2 for the density and222.3. Single-mode Interferometryp = −M ′(X)/8piX for the pressure. We would like to exploit this freedomfor the purposes of interferometry. To maintain internal consistency, thereshould be a relationship p = p(σ), which represents an equation of state forour fluid shell. The function M(X) parametrizes this relationship, thoughnot every choice of M(X) yields a consistent (let alone physical) equationof state.The interferometric setup resembles that of Michelson, except we onlyhave one spatial dimension to work with, since our system is sphericallysymmetric. Still, we would like the equation of state to produce two ‘reflec-tors’ - one to reflect the shell outward if it gets too small, and one to reflectthe shell inward if it gets too large. Also, we would like the equivalent of a‘half-silvered mirror’ to be in between the two reflectors, to act as a beam-splitter. This is depicted schematically in Figure 2.1, with X± being theshell radii that correspond to the reflectors, and Xδ the radius correspond-ing to the splitter. Accordingly, our equation of state p = p(σ) must havea large positive peak for some large density, a large negative peak for somesmall density, and an intermediate peak (serving as the beam-splitter) forsome intermediate density.It would be convenient to use delta functions for these purposes, butdue to the conversion between δ(σ − σ0) and δ(X − X0) and the resultingappearance of products of delta functions, this possibility seems problem-atic. Therefore, we have been considering the simplest alternative one couldthink of: rectangular barriers. These can be described with the use of stepfunctions, which we will define such that Θ(x < 0) = 0 and Θ(x > 0) = 1.The equation of state, then, takes the formp = p1 (Θ(σ − σ1)−Θ(σ − σ2))+p2 (Θ(σ − σ3)−Θ(σ − σ4))+p3 (Θ(σ − σ5)−Θ(σ − σ6)) , (2.60)with σi+1 > σi, p1 < 0, and p2, p3 > 0. We may as well take p1 = −p3, sinceboth of these peaks serve the same purpose of reflecting, but we will not yetimpose this condition.One would now like to find the function M(X) that parametrizes theequation of state (2.60). If we could express (2.60) as p =∑i p˜iΘ(X −Xi),then the identification p = −M ′(X)/8piX would implyM(X) = M0 + 4pi∑ip˜i(X2i −X2)Θ(X −Xi), (2.61)232.3. Single-mode InterferometryFigure 2.1: Schematic representation of the splitting, reflecting, and recom-bining that occur in our shell interferometer. The inner and outer mirrorsare located at X− and X+, respectively, and the beam-splitter is locatedat Xδ. The initial split, depicted near the bottom of the diagram, is char-acterized by a transmission coefficient T← and a reflection coefficient R←.These coefficients determine the amplitudes of the transmitted and reflectedcomponents of the initial state, which propagate along the different interfer-ometer arms. The two arrows at the top of the diagram represent the finalinterferometer outputs, after recombination.242.3. Single-mode InterferometryFigure 2.2: A sample equation of state represented by (2.60). There is anegative pressure peak p1 for some low density σ2 that causes the shell toreflect inwards whenever X gets sufficiently large, there is a positive pressurepeak p2 for some high density σ5 to reflect the shell outwards whenever Xgets sufficiently small, and there is an intermediate pressure peak p2 thatserves as a beam-splitter for our interferometer.which would yield a density given byσ =M04piX2+∑ip˜i(X2iX2− 1)Θ(X −Xi). (2.62)The problem with this possibility is that, in general, it isn’t necessarily truethat Θ(X −Xi) produces the same (reversed) ordering as Θ(σ − σi), giventhat σi = M(Xi)/4piX2i . This problem can be avoided by making sure thatthe density σ is a monotonically decreasing function of X. This leads to theconditionM04pi≥ −∑ip˜iX2i Θ(X −Xi). (2.63)Figure 2.2 illustrates the desired step function peaks, to enable our systemto operate as an interferometer.To understand what this means in terms of the pressure peaks in ourequation of state (2.60), we first note that if σ monotonically decreases inX, then step functions can be converted by Θ(σ − σi) = 1 − Θ(X − Xi).This allows us to conclude that p˜2 = −p˜1 = p1, p˜4 = −p˜3 = p2, and252.3. Single-mode Interferometryp˜6 = −p˜5 = p3. Then, one finds that monotonicity is maintained as long asM04pi> max{p3X25,6, p3X25,6 + p2X23,4,p3X25,6 + p2X23,4 − p1X22}, (2.64)where the notation X2i,j = X2i −X2j was introduced, for brevity.Since an equation of state (2.60) is described by the pressure as a functionof density, one should translate the conditions for monotonicity in terms ofthe step locations {σi} and the step amplitudes {pi}. To convert betweenthe {Xi} and the {σi}, one can use the relationsX26 =M04piσ6,X25 =M04piσ6(σ6 + p3)(σ5 + p3),X24 =M04piσ6(σ6 + p3)(σ5 + p3)σ5σ4,X23 =M04piσ6(σ6 + p3)(σ5 + p3)σ5σ4(σ4 + p2)(σ3 + p2),X22 =M04piσ6(σ6 + p3)(σ5 + p3)σ5σ4(σ4 + p2)(σ3 + p2)σ3σ2,X21 =M04piσ6(σ6 + p3)(σ5 + p3)σ5σ4(σ4 + p2)(σ3 + p2)σ3σ2(σ2 + p1)(σ1 + p1). (2.65)With these expressions, one can write the monotonicity conditions in themuch simpler form{σ5 > 0, σ3 > 0, σ1 + p1 > 0}. (2.66)Thus, as long as we keep the density σ positive, it will be monotonic in Xprovided σ1 + p1 > The WKB ApproximationNow that we have fully defined the reduced action (2.51) of our interfer-ometer, we can seek a quantum description of the system. As mentionedabove, there are immediate complications that one is faced with: the Hamil-tonian is only implicitly defined, which makes even writing down an explicitSchro¨dinger equation impossible, and even if we could write down an explicitSchro¨dinger equation we would have to deal with factor-ordering ambigu-ities. Fortunately, both of these issues are for the most part avoided by262.3. Single-mode Interferometryfocusing one’s attention on the parameter space region of validity of theWKB approximation (the so-called “WKB regime”). Let us now clarifywhat we mean by this.First, we will consider the time-independent Schro¨dinger equation,HΨ = EΨ, (2.67)for a Hamiltonian that is quadratic in momentum:H(X,P ) = H0(X) +H1(X)P +H2(X)P2. (2.68)For our purposes, we will approximate a general Hamiltonian H(X,P )by the first three terms in a Taylor expansion in P , given byHw = H(X, 0) +(∂H∂P)P +12(∂2H∂P 2)P 2, (2.69)with the P -derivatives evaluated at P = 0.To make sure this Hamiltonian becomes Hermitian in the quantum the-ory, we must order the operators appropriately. We can symmetrize theterm linear in P , such that(∂H∂P)P →12(ˆ(∂H∂P)Pˆ + Pˆˆ(∂H∂P)), (2.70)as well as ordering the quadratic term as(∂2H∂P 2)P 2 → Pˆˆ(∂2H∂P 2)Pˆ . (2.71)In the weak field limit, i.e. X large compared to the Schwarzschild radius2E, the WKB Hamiltonian for our shell system is given byHw ∼(Mˆ −Mˆ218X)−23√2MˆXP +(12Mˆ+13X)P 2. (2.72)We will postpone the derivation of the WKB Hamiltonian (2.72) until Sec-tion 3.5, as it will be instructive to consider the details in a more generalsetting. For brevity, we have dropped the subscript c on the reduced mo-mentum here and for the rest of the chapter.The WKB approximation we seek is obtained by considering the ~→ 0limit of (2.67). We will rewrite the wavefunction Ψ asΨ = ei~∞∑n=0~nSn(X), (2.73)272.3. Single-mode Interferometrywhere for the purposes of keeping track of asymptotic orders we are explicitlywriting the “~”s. If we take the usual coordinate representation of themomentum operator Pˆ = −i~ ddX , it follows thatPˆΨ = −i~ddXΨ =(∞∑n=0~nS′n)Ψ (2.74)andPˆ 2Ψ =(−i~ddX)2Ψ =(−i~∞∑n=0~nS′′n)Ψ +(∞∑n=0~nS′n)2Ψ. (2.75)Equation (2.67) then tells us thatE = H0 +H1∞∑n=0~nS′n −i~H ′22− i~H ′2∞∑n=0~nS′n+H2(∞∑n=0~nS′n)2− i~H2(∞∑n=0~nS′′n). (2.76)At order ~0, equation (2.76) yieldsE = H0 +H1S′0 +H2(S′0)2. (2.77)Upon comparison with the classical Hamiltonian form (2.68), we can deducethat S′0 = P±(E,X), with P± being the two momentum solutions to equa-tion (2.68) with the value H = E. We have thus arrived at the standardWKB phase,S0 =∫dX P±(E,X). (2.78)By inspection, one can also see that this WKB phase holds for an arbitraryHamiltonian H(X,P ), as long as it is possible to expand H(X,P ) in powersof P .At order ~, we haveH1S′1 −12iH ′1 − iH′2S′0 − iS′′0H2 + 2H2S′0S′1 = 0, (2.79)from which we can obtainS′1 =i2ddX (H1 +H2S′0)H1 +H2S′0=i2ddXln∣∣H1 +H2S′0∣∣. (2.80)282.3. Single-mode InterferometryFigure 2.3: A sample mass function Mˆ is plotted with respect to the shellradius X. The mass function Mˆ serves to parametrize an equation of stateof the form depicted in Figure 2.2. The approximate step function nearX = 2 serves as a beam-splitter, and the steep quadratic sides correspondto the inner and outer reflectors of the interferometer.Comparing again with the classical Hamiltonian form (2.68), we can writeH1+H2S′0 = ∂H/∂P (evaluated at H = E), and up to an irrelevant constantthe next term in the WKB expansion for Ψ can be written asS1 =i2ln∣∣∣∣∂H∂P∣∣∣∣. (2.81)If we only include contributions from S0 and S1, we arrive at the formof the WKB approximation used to describe modes in our interferometer:ΨE =ei~∫dX P±(E,X)√∣∣∂E∂P∣∣. (2.82)2.3.3 Flat Spacetime LimitTo determine whether or not gravity produces some form of decoherencein our interferometer, let us first clarify the manner in which coherencemanifests itself in the absence of gravity. In this case spacetime is flat, andalong the arms of the interferometer defined by (2.61) the shell behaves asa free particle.As evident from Figure 2.3, the mass of the “free” shell is different oneach interferometer arm. Let us call the inner mass M−, and the outer mass292.3. Single-mode InterferometryM+, such that M− > M+. For simplicity, suppose the reflectors are perfect,which for this system means that the quadratic walls of the mass functionare large and steep. Similarly, let the quadratic beam-splitter interval beapproximated by a step function, to ensure that only constant mass functionbasis states need to be used in the quantum analysis.Further, let us treat each element of the interferometer separately, in asimilar manner to that which is done in optical systems. The initial state willfirst encounter the splitter, at which point each incoming mode will trans-form into a reflected mode with a factor R← and a transmitted mode with afactor T← (subscripts are used here because the reflection/transmission co-efficients depend on the direction the incoming state encounters the splitterfrom).The split initial state components will then perfectly reflect off of theouter/inner reflectors, and travel back towards one another to the beam-splitter. Upon recombination there will be further splitting of the compo-nents coming from each direction of the splitter, which produces two outputs(one going in each direction from the splitter) that are themselves composedof two parts; it is the interference between these two parts of each outputthat we are interested in.Let us now describe the process in detail. For the purposes of this section,we will restrict our attention to a single-mode input, since the multi-modeanalysis is more involved and will be discussed in Appendix B. As mentionedin Section 2.3.2, we will approximate the single-mode input by an ingoingWKB state:Ψ0 =ei∫dXP−+√|∂E/∂P−+|≡ ψ−+, (2.83)where the first set of plus/minuses of the reduced momentum P indicatingoutgoing/ingoing, and the second set indicating evaluations of P as X ap-proaches Xδ from above/below. We will define the integration such that thelower bound in X is Xδ.Treating the first splitting on its own, let us consider the wavefunctionΨ ={ψ−+ +R←ψ++ : X > XδT←ψ−− : X < XδThe (classical) flat spacetime Hamiltonian satisfies H =√P 2 + Mˆ2, whichin the nonrelativistic limit yields H ≈ Mˆ +P 2/2Mˆ . Applying wavefunctioncontinuity at Xδ, and integrating the nonrelativistic Schro¨dinger equationi∂ψ∂t= Mˆψ −12∂∂X(1Mˆ∂ψ∂X)(2.84)302.3. Single-mode InterferometryacrossXδ, one can obtain the reflection and transmission amplitudesR← andT←. The equations take a simpler form after transforming to the variablesR¯← and T¯←, which are defined byR¯← ≡√√√√∣∣∣∣∣∂E∂P−+∂E∂P++∣∣∣∣∣R←, T¯← ≡√√√√∣∣∣∣∣∂E∂P−+∂E∂P−−∣∣∣∣∣T←. (2.85)One can then easily solve for the new variables:R¯← =M−P−+ −M+P−−M+P−− −M−P++, T¯← =M−(P++ − P−+)M−P++ −M+P−−. (2.86)For convenience, we can also derive the reflection and transmission ampli-tudes from the left, which are found to beR¯→ =M−P++ −M+P+−M+P−− −M−P++, T¯→ =M+(P+− − P−−)M−P++ −M+P−−, (2.87)using the similar definitionsR¯→ ≡√√√√∣∣∣∣∣∂E∂P+−∂E∂P−−∣∣∣∣∣R→, T¯→ ≡√√√√∣∣∣∣∣∂E∂P+−∂E∂P++∣∣∣∣∣T→. (2.88)Let us call the outgoing state after the split Ψ(i)+ and the ingoing stateΨ(i)− . We then can consider the first splitting a transformation of the wave-function such thatΨ0 = ψ−+ →(Ψ(i)+Ψ(i)−)=(R←ψ++T←ψ−−). (2.89)This splitting should preserve the probability current, for consistency. Inthe nonrelativistic, flat spacetime limit, the probability current J satisfiesthe continuity equation∂∂t(|ψ|2)+∂∂XJ = 0 (2.90)and is given by the usual quantum mechanics expression12im(ψ∗ψ′ − ψψ∗′). (2.91)312.3. Single-mode InterferometryTherefore, in this limit we have an input probability current ofJ0 = |Ψ0|2P−+M+. (2.92)After first encountering the beam-splitter, the probability current (2.92)splits into reflected and transmitted components|J (i)+ | =12iM+(Ψ(i)∗+(Ψ(i)+)′−Ψ(i)+(Ψ(i)+)∗′)= |J0|R¯2← (2.93)and|J (i)− | =12iM−(Ψ(i)∗−(Ψ(i)−)′−Ψ(i)− (Ψ−(i))∗′)= |J0|(M+P−−M−P−+)T¯ 2←. (2.94)The splitting preserves probability current, as can be confirmed by observingthat∣∣∣∣J(i)+J0∣∣∣∣+∣∣∣∣J(i)−J0∣∣∣∣ is unity. The terms∣∣∣∣J(i)+J0∣∣∣∣ and∣∣∣∣J(i)−J0∣∣∣∣ are usually called thereflection and transmission coefficients, respectively.The second transformation propagates the modes along the interferom-eter arms, such that(Ψ(i)+Ψ(i)−)→(Ψ(ii)+Ψ(ii)−)=((E,++ )−1/2R←eiΦ++(E,−− )−1/2T←eiΦ−−). (2.95)For brevity, the notation E,±± was used to denote ∂E/∂P±±, and it isunderstood that we are evaluating these quantities at the outer walls of theinterferometer. We have also introduced the quantities Φ±± = φ±±−Et±±,for φ±± =∫ X±XδdXP±±, where t++ and t−− denote the travel times fromthe splitter to X+ and X−, respectively.The modes then reflect off of the outer walls, as(Ψ(ii)+Ψ(ii)−)→(Ψ(iii)+Ψ(iii)−)=((E,−+ )−1/2R←eiΦ++R→(E,+− )−1/2T←eiΦ−−R←). (2.96)The outer wall reflection amplitudes (R→, R←) only depend on continuityof the wavefunction. To obtain the reflection amplitude from the left, forinstance, consider the wavefunctionΨ = ψ++ →{0 : X > X+(ψ++ +R→ψ−−) : X < X+322.3. Single-mode InterferometryBy applying wavefunction continuity at X+, one immediately obtains R→.R← can be similarly determined, and the results areR¯→ = −ei(φ+++φ−+), R¯← = −ei(φ+−+φ−−), (2.97)with help of the simplifying definitionsR¯→ ≡√√√√∣∣∣∣∣∂E∂P++∂E∂P−+∣∣∣∣∣R→, R¯← ≡√√√√∣∣∣∣∣∂E∂P−−∂E∂P+−∣∣∣∣∣R←. (2.98)The phases are defined such that φ±∓ =∫ XδX∓dXP±∓ (signs chosen together).We will refer to the modes after reflection from the outer walls as Ψ(iii)± .Propagation along the arms back to the splitter then proceeds as(Ψ(iii)+Ψ(iii)−)→(Ψ(iv)+Ψ(iv)−), (2.99)for (Ψ(iv)+Ψ(iv)−)=((E,−+ )−1/2R←eiΦ++R→eiΦ−+(E,+− )−1/2T←eiΦ−−R←eiΦ+−). (2.100)In this expression, the quantities E,−+ and E,+− are evaluated at the split-ter, and we have used the definitions Φ±∓ = φ±∓−Et±∓ (signs again chosentogether). Here t−+ and t+− denote the travel times from X+ to the splitterand from X− to the splitter, respectively.The second encounter with the splitter occurs as it did before, as(Ψ(iv)+Ψ(iv)−)→(Ψ(v)+Ψ(v)−)=(R¯← T¯→T¯← R¯→)(Ψ(iv)+Ψ(iv)−). (2.101)At the order we are working at in ~, the derivatives of our final outputssatisfyddXΨ(v)± = iP±±Ψ(v)± , (2.102)and so the currents for our final outputs are given byJ (v)± =12iM±(Ψ(v)∗±(Ψ(v)±)′−Ψ(v)±(Ψ(v)±)∗′)=P±±M±∣∣∣Ψ(v)±∣∣∣2. (2.103)332.3. Single-mode InterferometryWe then have enough information to calculate the final reflected andtransmitted probability currents, which can be written|J (v)+ | = |J0|[1− 4R¯2←(1− R¯2←)sin2 ϕ](2.104)and|J (v)− | = |J0|4R¯2←(1− R¯2←)sin2 ϕ, (2.105)where we have defined ϕ = φ+++φ−+−φ+−−φ−− and made use of the iden-tity R¯2←+M+P−−M−P−+T¯ 2← = 1. The flat-spacetime interferometer thus manifestlyconserves probability current in all regions of the parameter space.One can now search for a nice region in the parameter space that cancelsone of the outputs. First, we would like to avoid regions of the parameterspace that don’t describe splitting, i.e. complete initial reflection or trans-mission by the beam-splitter. We can accomplish this in a simple way byenforcing an equal splitting condition, R¯2← = 1/2. This leads to compactexpressions for the final reflection and transmission coefficients, given byRf ≡|J (v)+ ||J0|= cos2 ϕ (2.106)andTf ≡|J (v)− ||J0|= sin2 ϕ, (2.107)respectively.We should also make sure that our shell velocity doesn’t approach thespeed of light, since we are working in the nonrelativistic limit. For smallshell speeds, given an outer mass M+ and an initial speed v+, the initialsplitting will be equal provided the inner mass satisfiesM− ≈M+[1 +(6√2− 8)v2+ −(99√2− 140)v4+]. (2.108)In the quantum context, the “speed” v+ is defined such that E = M+ +12M+v2+, for a WKB state with energy E.If we denote the interferometer arm lengths by L± ≡ ±(X± − Xδ), wecan see from the form of the reflection and transmission coefficients that oneof the outputs will be completely cancelled ifϕ = 2L+√2M+ (E −M+)− 2L−√2M− (E −M−)= 2L+M+v+ − 2L−M−v−=npi2, (2.109)342.3. Single-mode Interferometryfor n ∈ Z. Thus, as the outer arm length is increased or decreased, theoutputs are alternately cancelled out for each value of n (odd values cancelthe transmitted output, and even values cancel the reflected output), withpartial interference for intermediate arm lengths that don’t correspond tosolutions of (2.109). This behaviour is a direct reflection of coherence in theflat spacetime interferometer.2.3.4 General Relativistic PictureThe current framework was designed to facilitate the inclusion of general rel-ativistic corrections. Several expressions become messier once one includesgravity, and some expressions fundamentally change in structure. For in-stance, the standard probability current Js = 12im (ψ∗ψ′ − ψψ∗′) given by(2.91) is no longer conserved in systems with more general Hamiltonians.In fact, a probability current for an arbitrary Hamiltonian system has neverbeen constructed; only special cases are known, such as the standard currentJs for the nonrelativistic Hamiltonian H = P 2/2m + V (X). For any otherHamiltonian, the current Js does not satisfy the continuity equation (2.90),and therefore does not conserve probability.A special case that is less well-known is for the first relativistic correctionto the nonrelativistic Hamiltonian H = P 2/2m+V (X), which is of the formαP 4 (with α = −1/8m3, to match the next term in the Taylor expansion of√P 2 +m2 in P ). The Schro¨dinger equation i∂ψ/∂t = Hψ, along with itsconjugated counterpart −i∂ψ∗/∂t = H†ψ∗, imply that∂ρ∂t= i[ψ(H†ψ∗)− ψ∗ (Hψ)], (2.110)and so linearity in H allows us to determine the correction to the currentJs coming from the extra term αP 4 independently of the first terms in theHamiltonian. If one observes thatψ∂4ψ∗∂X4− ψ∗∂4ψ∂X4= −∂∂X(ψ∗∂3ψ∂X3−∂ψ∗∂X∂2ψ∂X2+∂2ψ∗∂X2∂ψ∂X−∂3ψ∗∂X3ψ)(2.111)then one can deduce that the current correction takes the formiα(ψ∗∂3ψ∂X3−∂ψ∗∂X∂2ψ∂X2+∂2ψ∗∂X2∂ψ∂X−∂3ψ∗∂X3ψ). (2.112)Similar (somewhat more complex) constructions exist for Hamiltoni-ans with higher powers of the momentum, but our system of interest has352.3. Single-mode Interferometryposition-dependent coefficients when expanded in powers of the momentum.If we take the operator ordering of the approximate form (2.69) as an exactHamiltonian, we end up with a more general quadratic Hamiltonian thanH = P 2/2m + V (X), since now we have a term linear in momentum witha function of X as a coefficient, as well as another function of X as a coef-ficient of the term quadratic in momentum. As demonstrated in AppendixA, we can find a probability current J for this more general Hamiltonianthat satisfies the continuity equation (2.90). We can express this probabilitycurrent asJ =(∂H∂P)|Ψ|2 +12i(∂2H∂P 2)(Ψ∗Ψ′ −ΨΨ∗′). (2.113)The P -derivatives in this expression are again evaluated at P = 0, andfor the special case of {(∂H∂P)= 0,(∂2H∂P 2)= 1/m}, we are left with thenonrelativistic, flat spacetime limit described by (2.91).If we use the WKB Hamiltonian (2.72), we find that the generalizedprobability current is given byJ ∼ −23√2MˆX|Ψ|2 +(1 +2Mˆ3X)Js, (2.114)with Js being the standard (nonrelativistic) expression (2.91) for the proba-bility current. Note that although the functional form of Js with respect toΨ is the same as the nonrelativistic current (2.91), in the above expressionwe are inserting the general relativistic WKB wavefunction Ψ.Since our Schro¨dinger equation now takes the asymptotic formHwΨ = i∂∂tΨ, (2.115)taking the operator ordering mentioned above, we no longer have the simplereflection and transmission amplitudes obtained in the previous section. Forinstance, integrating (2.115) across Xδ yields[(32Mˆ+1X)Ψ′]δ= i√2Xδ[√Mˆ]δΨ(Xδ). (2.116)Here, [·]δ represents the jump of a quantity across Xδ.To the order we are working at in ~, the new reflection and transmissionamplitudes for scattering from the right are given byR¯← =√2Xδ[√Mˆ]δ+(32M−+ 1Xδ)P−− −(32M++ 1Xδ)P−+−√2Xδ[√Mˆ]δ−(32M−+ 1Xδ)P−− +(32M++ 1Xδ)P++(2.117)362.3. Single-mode InterferometryandT¯← =(32M++ 1Xδ)(P++ − P−+)−√2Xδ[√Mˆ]δ−(32M−+ 1Xδ)P−− +(32M++ 1Xδ)P++. (2.118)Similarly, for scattering from the left we haveR¯→ =√2Xδ[√Mˆ]δ+(32M−+ 1Xδ)P+− −(32M++ 1Xδ)P++−√2Xδ[√Mˆ]δ−(32M−+ 1Xδ)P−− +(32M++ 1Xδ)P++(2.119)andT¯→ =(32M−+ 1Xδ)(P+− − P−−)−√2Xδ[√Mˆ]δ−(32M−+ 1Xδ)P−− +(32M++ 1Xδ)P++. (2.120)Reflection from the outer walls is again described by (2.98), though itshould be noted that since the partial derivatives in (2.98) now have X-dependence, they are to be evaluated at the outer walls (X+ for R→ andX− for R←).Given the definition of probability current in this (more general) setting,we haveJ (v)± =((1 +2M±3Xδ)P±±M±−23√2M±Xδ)∣∣∣Ψ(v)±∣∣∣2. (2.121)Just as in the flat spacetime limit, the final output states are given by(2.101), except that now the reflection/transmission amplitudes and theWKB phases are more complicated.The initial current can be expressed asJi =(1−2M+3P−+√2M+Xδ+2M+3Xδ)J0, (2.122)with J0 being the nonrelativistic initial current (2.92), and so the final reflec-tion and transmission coefficients Rf ≡|J(v)+ ||Ji|and Tf ≡|J(v)− ||Ji|(respectively)are fully determined.Another similarity to the flat spacetime limit is that the oscillatory partof the final reflection and transmission coefficients is defined by ϕ = φ++ +372.3. Single-mode InterferometryFigure 2.4: Sample interference pattern, for M+ = 15, v+ = 0.003,X− = 5000, L− = 20, and M− chosen to satisfy (2.108), plotted againstthe outer mirror position, X+. The alternating pattern from 0 to 1 of boththe final reflection and final transmission coefficients indicates that completeconstructive/destructive interference occurs in this parameter region, fromwhich we can conclude that coherence is fully present.Figure 2.5: Sample interference pattern, for M+ = 15, v+ = 0.01, X− = 200,L− = 20, and M− chosen to satisfy (2.108), plotted against the outer mirrorposition, X+. As in the previous figure, there is an alternating pattern ofthe final reflection/transmission coefficients from 0 to 1 as the outer armlength changes, indicating that coherence is still fully present as we bringthe interferometric range closer to the Schwarzschild radius of the shell. Inthis region of the parameter space, we begin to see the sum of the finalreflection and transmission coefficients fail to add up to exactly unity, dueto the gradual breakdown of our approximation to the probability current.382.3. Single-mode Interferometryφ−+−φ+−−φ−−, with the various φ terms involving integrals of the generalrelativistic momentum (2.48). In the weak-field limit, the initial ingoingmomentum is given byP−+ ∼ −√H2 −M2+ +23√2HXH−(H2 −M2+/2)H√H2 −M2+X, (2.123)to second order in 1/√X. Care should be taken with these approximations,however, because our probability current (2.113) is exactly conserved onlyin the quadratic momentum limit, which for the shell system is defined by(2.72). Also, the WKB solutions only approximately satisfy the Schro¨dingerequation. Because of this, in order to control the errors involved in theapproximations we find it useful to consider the “WKB momentum,” whichwe define by solving (2.72) for P , and expanding to second order in 1/√X.For the initial ingoing momentum, the WKB momentum takes the formPw−+ ∼ −√2M+ (H −M+) +23√2M+XM+−√2M+(H −M+)(7M+ − 4H)12X. (2.124)To understand the interference pattern described by Rf and Tf , let usconsider what ϕ looks like in the weak-field limit, for slow speeds (v± → 0):ϕ ∼ 2L+M+v+ − 2L−M−v−+M2+v+ln(X+Xδ)−M2−v−ln(XδX−). (2.125)Let us further imagine that we vary the outer arm length L+, while keepingall other parameters constant. If the phase condition (2.109) from flat space-time still approximately holds, then the corresponding expression (2.125) inthe weak-field limit tells us that successive values of n (say, n to n + 1)are associated with outer arm length values L+n and L+(n+1). Subtractingϕn = npi/2 from ϕn+1 = (n+ 1)pi/2 yieldspi2= 2M+v+∆Ln +M2+v+ln(X+n + ∆LnX+n), (2.126)392.3. Single-mode InterferometryFigure 2.6: Sample interference pattern, for M+ = 0.05, v+ = 0.0001,X− ≈ 1, L− ≈ 0.997, and M− chosen to satisfy (2.108), plotted againstthe outer mirror position, X+. The node spacing decreases as the interfer-ometric range is pushed closer to the Schwarzschild radius of the shell, andasymptotes to a fixed value ∆Ln = pi/4M+v+ as we approach spatial infin-ity. This “gravitational node squeezing” was present, though not visuallydetectable, in the previous interference patterns.with the definitions X+n = Xδ + L+n and ∆Ln = L+(n+1) − L+n. The dis-tance between nodes of the interference pattern, denoted by ∆Ln, is some-what less than the outer mirror radius X+, for the cases we are interestedin; thus, we can expand the logarithm and solve for ∆Ln, which gives us∆Ln ≈pi4M+(v+ +M+2v+X+n) . (2.127)This result shows that gravity causes the node spacing in the interferencepattern to increase with increasing outer arm length. In the flat space limit(i.e. as X± →∞), we obtain the equal node spacing ∆Ln = pi/4M+v+, forall n ∈ Z.One can see from Figures 2.4 and 2.5 that as we go from the essentiallyflat limit (X± → ∞) to less than 10 Schwarzschild radii, we can still alter-nately cancel the reflection and transmission coefficients, even though theapproximations lead to a probability current that is not fully conserved (notethat the sum of the final probability currents differs from the initial currentby about 1%). We take this as a direct indication that coherence is fullypresent in the single-mode system even with general relativistic correctionstaken into account.402.4. DiscussionIt is not clear from Figures 2.4 and 2.5, but the node spacing is indeedchanging as (2.127) suggests. The reason it is not visible from these plots isthat the node spacing changes noticeably only over a range of many wave-lengths. Under more extreme circumstances, as depicted in Figure 2.6, thereare visible changes in node spacing, though this represents a situation thatis of less physical interest, since the de Broglie wavelength of the shell islarger than the interferometer arms.2.4 DiscussionThere are two problems with taking this result of no loss of coherence asthe definitive answer to whether or not gravity, by itself, could decohere asystem. The first is that these single-mode states correspond in some senseto energy eigenstates; one might expect it is only a superposition of energiesthat leads to decoherence, since from the above analysis one can see that thetime-dependence cancels out of the final expressions for output probabilitiesin the interferometer. As mentioned above, we discuss how wave-packetsbehave in this model in Appendix B.The second problem is that Penrose’s intuition ties the loss of coherenceto the inability to map one spacetime in any unique way onto a differentspacetime. By our coordinate choice we have, in effect, chosen a uniqueway: two spacetime points are the same if they have the same coordinates.However, this is of course arbitrary and depends on the coordinate choicemade. While the Painleve´-Gullstrand coordinates have many advantageousfeatures, they are not the only possible choice. Do all coordinate choicesproduce the same maxima and minima in the interference pattern? Thefull classical action that describes our interferometer is independent of thecoordinate choice, but it is not obvious whether this is enough to ensurecoordinate independence in the reduced phase space quantization. Theseissues will be examined in Chapter 3.It could also be argued that the reduced phase space approximation leadsto an artificial form of time-evolution that is not entirely consistent withthe “timeless” structure of canonical quantum gravity. For instance, thelack of a satisfactory interpretation of reduced phase space minisuperspacequantum cosmology was discussed in [60]. One might then be drawn to theconclusion that in the limited setting of our approximations, the evolutionwill necessarily be unitary (by construction), and we will escape Hawking’soriginal arguments about pure states evolving into mixed states [61] by virtueof our approximation scheme.412.4. DiscussionCertainly, our simple model does not have the features often associatedwith nonunitary modifications to standard Hamiltonian evolution (such asthe inclusion of microscopic wormhole interactions [62]), but there is stillreason to believe the evolution defined by (2.48) could in principle exhibitdecoherence. For one thing, we have in some places used an approximateHamiltonian (2.72) that is quadratic in momenta and strictly Hermitian,but it may not be possible to define a Hermitian Hamiltonian operator thatexactly corresponds to the solution of (2.48) (which is transcendental). Foranother, even if one could solve (2.48), the resulting Hamiltonian would benon-polynomial in both the momenta and the coordinate X. This meansthat the time evolution of the wavefunction at X is not described by a finitenumber of derivatives at X, and is thus nonlocal, in the sense that the evo-lution equation is equivalent to an integro-differential equation with finitely-many derivatives [63]-[66]. While some systems can be nonlocal in this wayand yet maintain coherence (such as in the case of relativistic particles inflat spacetime [63], [64]), in other such systems there can be unexpected be-haviour such as “nonlocally-induced randomness” [65], [66], which would inour case be attributable to gravity. These studies are still in their infancy, soit remains an open question whether or not this type of nonlocal behaviourcan be connected with gravitational intrinsic decoherence.42Chapter 3The Meaning of Time inReduced Phase Space3.1 Coordinate Generalizations and TimeTransformationsIn Section 2.2.4, we chose a specific coordinate system when we carriedout the Hamiltonian reduction. One naturally wonders whether there iscoordinate dependence in any of the results we have presented. The canon-ical momentum in the reduced system certainly depends on the coordinatechoice, but one can show that a broad set of choices lead to the same reducedclassical action [58].What about the quantization associated with the specific coordinatechoice we made? It is often claimed that different choices of a time variablelead to unitarily inequivalent quantizations (see [67], for instance). Doesthis imply that different coordinate choices lead to fundamentally differentinterference patterns? For each choice of coordinates, we can uniquely definea (classical) network of observers whose worldlines span the spacetime, soa different coordinate choice produces a reduced phase space quantizationdefined with respect to a different observer network. Which features of eachquantization reflect properties of the shell system itself, and which featuresare due entirely to the properties of the observer network?We will investigate differences that may arise due to different coordinatechoices by considering a family of coordinate systems that generalize thePainleve´-Gullstrand coordinates used in the previous chapter. We will re-peat many of the previous calculations, this time for an arbitrary member ofthe family of coordinate systems. In doing so, we will face some serious dif-ficulties interpreting the results and connecting the generalized calculationswith the previous ones.Let us now describe the infinite family of coordinate systems that gen-eralized our previous coordinates: in terms of the metric variables definedabove, we can define the coordinate family (as demonstrated by Martel and433.1. Coordinate Generalizations and Time TransformationsPoisson [68]) byL = λ, R = r, N =1λ, N r = ±√1− λ2fλ2, f = 1−2Mr, (3.1)withM being the (enclosed) ADM mass and 0 < λ ≤ 1 corresponding to theset of Painleve´ family members. In the λ→ 0 limit, the coordinates definedby (3.1) become the familiar (null) Eddington-Finkelstein coordinates 3.In the upper limit, λ → 1, the coordinates (3.1) reduce to the Painleve´-Gullstrand coordinates used in the previous chapters. We will choose thepositive sign for the shift vector N r, for simplicity.The line element in these coordinates then takes the formds2 =1λ2dt2λ + λ2(dr +1λ2√1− λ2fdtλ)2+ r2dΩ2. (3.2)This generalized Painleve´-Gullstrand coordinate system still possessesthe connection between coordinate lines and infalling observers, as with thestandard λ = 1 coordinates. The quantity λ is related to the “initial velocityat infinity”4 of an infalling observer:λ =√1− v2∞. (3.3)This initial velocity is in turn related to the geodesic observer’s energy perunit rest mass E˜ through the standard expression E˜ = 1/√1− v2∞ [68].Any particular generalized coordinate system in the family can therefore beinterpreted as a network of geodesic observers, each with the same energyper unit rest mass. Any such network has the property that its observersall have proper 4-velocities equal to a constant times the gradient of a timefunction tλ, with the time function given bytλ = T +∫dr√1− λ2ff. (3.4)Here T is the Schwarzschild time, which is related to the standard (λ = 1)Painleve´-Gullstrand time t byt = T + 4M(√r2M+12ln∣∣∣∣∣√ r2M − 1√ r2M + 1∣∣∣∣∣). (3.5)3Also known as Penrose-Eddington-Finkelstein coordinates, due to Penrose’s explicitinitial use of them [69].4Following the convention from [68], we define positive observer velocity to be radiallyinward, such that v∞ takes on values from 0 to 1 as λ varies from 1 to 0. This is theopposite convention as the one used for the shell velocity.443.1. Coordinate Generalizations and Time TransformationsThe time transformation (3.4) also implies that the relationship betweenthe time coordinates (at the shell) specified by two different choices of λ istλ − tλ′ = Fλ − Fλ′ , (3.6)such thatFλ =∫dX√1− λ2ff= 4H[X4H√1− λ2f + ln( √f1 +√1− λ2f)+(1− λ22)√1− λ2ln(√X2H(√1− λ2 +√1− λ2f)) (3.7)and f = 1 − 2H/X. This expression differs from the corresponding (incor-rect) expression presented by Martel and Poisson in [68], though one needonly perform some simple differentiation to see that (3.7) is the correct one.In the context of investigating the connections between quantizations fordifferent time choices, the transformation defined by (3.6) and (3.7) has somepeculiar features. Most notably, the relation between two time coordinatesinvolves both the shell position X and the Hamiltonian H. This makes itunclear under what conditions this transformation can be implemented; atthe quantum level, both X and H are operators, which implies that eitherthe relation between different time choices is operator-valued, or that wemay only be able to perform the transformation at the classical level. Thenit is no simple matter connecting the evolution operators for different timechoices, since each operator takes the formUλ(tλ) = e−iHtλ . (3.8)Will the quantization for one choice of λ be interpreted from a differentchoice λ′ as having an operator-valued time “parameter”?To explore this issue, among others, we will quantize the shell system foran arbitrary time choice in the reduced phase space approximation. Beforewe can do this, we need to determine the reduced Hamiltonian structure inthe generalized coordinates, which we will now focus our attention on.453.2. Phase Space Reduction in a Family of Coordinate Systems3.2 Phase Space Reduction in a Family ofCoordinate SystemsThe starting point in the phase space reduction associated with the general-ized Painleve´-Gullstrand coordinates (3.1) is again to consider the Liouvilleform F on the full phase space Γ; just as in Chapter 2, this is given byF = PδX +∫dr (piLδL+ piRδR). (3.9)We can then pull this back to the representative hypersurface H¯λ ⊆ Γ associ-ated with the particular coordinate choice for a given λ. The pulled-back Li-ouville form Fλ induces a Liouville form on the reduced phase space throughthe isomorphism between the reduced phase space Γ¯ and the representativehypersurface H¯λ ⊆ Γ mentioned in Chapter 2. The induced Liouville formon Γ¯ yields the canonical structure of our reduced system. As usual, thereduced phase space Γ¯ is defined as the set of equivalence classes in Γ underchanges of coordinates.To avoid repetition with Section 2.2.4, we will just sketch the steps in-volved in the calculation of Fλ.As with our previous coordinate choice, we must take care to includea deformation region (in R) near the shell (X −  < r < X), in order tosatisfy the gravitational constraints. Away from the shell, the gravitationalmomenta solutions (4.22) evaluated in our new coordinates (3.1) are givenbypiL = ±R√(R′λ)2− 1 +2M(r)R, piR =λRpi′L, (3.10)with M(r) being a function that equals the ADM mass outside of the shelland 0 inside.We can then determine what conditions the gravitational constraintsdefined in Section 2.2.3 impose on the metric function R by integrating theseconstraints across the shell, and assuming both continuity of the (spatial)metric and finiteness of the gravitational momenta. One then finds theconditions∆R′ = −V¯Rˆ, ∆piL = −Pλ, (3.11)where V¯ =√P 2 + M¯2, M¯ = Mˆλ, and ∆ indicates the jump of a quantityacross the shell. By inspection, the metric function R defined in equation463.2. Phase Space Reduction in a Family of Coordinate Systems(3.12) can be generalized asR(r, t) = r −XV¯ G(X − r), (3.12)for a function G having the same properties as in Section 2.2.4, i.e.limz→0+dG(z)dz= 1 (3.13)limz→0−dG(z)dz= 0 , (3.14)from which followslim→0R′(X − ) = 1 +V¯X(3.15)lim→0R′(X + ) = 1 . (3.16)As can be expected from the form of the coordinate choice (3.1), thepiR term integrated over the deformation region will again give the onlycontribution to the pullback of the Liouville form:Fλ = PδX +∫ XX−dr piRδR. (3.17)In the → 0 limit, we have, in the deformation region,piR =XR′′λ√(R′λ)2− 1+O(1), (3.18)which allows us to express the gravitational contribution to the Liouvilleform as∫ XX−dr piRδR = XδX∫ XX−drR′′ (1−R′)λ√(R′λ)2− 1+O(). (3.19)We can then change the integration variable from r to v = R′, which yields∫ XX−dr piRδR =XδXλ∫ R′−1dv(1− v)√(vλ)2− 1+O(), (3.20)473.3. Boundary Termswith R′− being R′ evaluated just inside the shell. Integrating and rearrangingthen leads toδX[−P −Xλ√w +X(√1− λ2 − ln(1 +√1− λ2))+ X ln(1 + λ√w +V¯ + PX)](3.21)(plus terms that vanish as → 0), where we have used the definitionw ≡1λ2− 1 +2HX. (3.22)This completes the calculation of Fλ, the pullback of the full Liouville formF to H¯λ:Fλ = PλδX, (3.23)with the reduced canonical momentum evidently given byPλ = − Xλ√w +X[√1− λ2 − ln(1 +√1− λ2)]+ X ln(1 +V¯ + PX+ λ√w). (3.24)In the reduced phase space, the unreduced momentum P becomes a con-strained function of H and X. One can obtain this function by inserting thegravitational momentum solutions away from the shell given by equation(3.10) into the jump equations (3.11) and squaring. We then find that P isconstrained to solveλ2H = V¯ +M¯22X− Pλ√w. (3.25)It is straightforward to check that this reduces to the previous case intro-duced in Chapter 2 when λ = 1.3.3 Boundary TermsIn the previous Hamiltonian reduction, the boundary term we added to theinitial reduced action was (very conveniently) equal to −∫dtH, with Hbeing the ADM mass. Thus, the ADM mass was identified with the reducedHamiltonian of the system. We will now demonstrate that this propertyholds for an arbitrary Painleve´ family member.As mentioned in Section 2.2.5, a nonzero boundary variation resultsfrom integrating by parts the term∫dtλ dr N rL(δpiL)′, which is part of483.3. Boundary Termsthe momentum constraint. The boundary in this case is spatial infinity,and as r → ∞ we now have N r →√1− λ2f/λ2, N → 1/λ, and piL →√1/λ2 − 1 + 2M/r. It then readily follows thatδ(piL)→δH√1λ2 − 1 +2Mr, (3.26)and so δ(piL)N rL → δH. From this we can conclude that the variation ofthe boundary term is cancelled if we add to the action the termIbdry = −∫dtλH, (3.27)which has the same form as before, with the previous time t replaced by thePainleve´ family member time tλ.We can then determine the reduced action associated with each familymember by adding the boundary term to the action defined by Fλ. Theresult isIλreduced =∫dtλ(PλdXdtλ−H), (3.28)with the reduced momentum now given by (3.24). As before, we can seefrom the form of the reduced action (3.28) that the ADM mass is the reducedHamiltonian for an arbitrary Painleve´ family member.Let us now explicitly write the unreduced momentum P that satisfiesthe constraint (3.25). The result is a function of X, H, and λ:P =√1− λ2fh±√h2 −M2ff, (3.29)with f = 1− 2H/X and h = H −M2/2X. One can easily confirm that thisreduces to the previous case given by equation (2.52) when λ = 1.Though the equations seem somewhat more complicated, they are still ofthe same form as before: P still obeys a quadratic equation, and the reducedcanonical momentum can still be explicitly determined. This remarkable factreadily allows the previous calculations to be performed for arbitrary λ (for0 < λ ≤ 1), so that one can straightforwardly investigate which propertiesof the system depend on the particular choice of coordinates.493.4. Reduced Equations of Motion3.4 Reduced Equations of MotionIn order to derive Hamilton’s equations of motion for the reduced system,we will make use of the expressionP = X√(V¯X+ 1)2− λ2 −X√1− λ2f, (3.30)which results from the gravitational momentum solution piL (3.10) and thesecond jump condition (3.11). With (3.30), along with the implicit relation(3.24), we can obtainX˙ =∂H∂Pλ=1λ2[PV¯−√1− λ2f]. (3.31)Solving (3.31) for P and substituting into (3.30) yieldsHλM(X)−M(X)λ2X=1−(X˙λ2 +√1− λ2f)√1− λ2f√1−(X˙λ2 +√1− λ2f)2(3.32)The line element (3.2) gives the relation between the proper time of theshell, τ , and the coordinate time, tλ:dτ2 =1λ2dt2λ − λ2(dr +1λ2√1− λ2fdtλ)2, (3.33)from which it follows thatdtλdτ=1fdXdτ√1− λ2f +√(dXdτ)2+ f . (3.34)Using (3.34) to express X˙ as X˙ = dXdτdτdtλand rearranging relation (3.30), wefindHM(X)−M(X)2X=√(dXdτ)2+ f (3.35)which is the correct classical equation of motion for the shell.503.5. Transcendental Hamiltonian Approximations3.5 Transcendental Hamiltonian ApproximationsThough it is not possible to solve the relation (3.24) for the Hamiltonian Hwhen X is finite, in Chapter 2 we presented an approximate Hamiltonian(2.72) associated with the weak-field limit (large X) of the system defined bythe canonical momentum (2.48). We will now derive this result as a specialcase of the results for arbitrary λ.In the X → ∞ limit (flat spacetime), the Hamiltonian is given exactlybyH =11− v2∞(−v∞Pλ +√M(X)2(1− v2∞) + P2λ). (3.36)For finite but large X, it is possible to obtain the Hamiltonian for smallmomentum Pλ. To this end, we choose the ansatzH = H0 +H1Pλ +H2P2λ (3.37)with some functions H0(X), H1(X), and H2(X). We substitute this ansatzinto the relation (3.24) and expand for large X and small Pλ. Then wecompare the coefficients for the various powers of Pλ and deduce for λ = 1H0 = M −M218X+2M3405X2+O(1/X3) (3.38)H1 = −23√2MX−M3/2135√21X3/2+161M5/248600√2X5/2+O(1/X7/2)(3.39)H2 =12M+13X+M270X2+O(1/X3) . (3.40)When the coordinate system corresponds to an observer with finite velocityat infinity, we findH0 ∼M√1− v2∞−√1− v2∞M324v2∞X2(3.41)H1 ∼ −v∞1− v2∞−M2Xv∞√1− v2∞+M26v3∞X2(3.42)H2 ∼12M(1− v2∞)3/2+12X(1− v2∞)−3M16X2v2∞√1− v2∞. (3.43)Note that these asymptotic expansions do not have the correct limit forv∞ → 0. The reason for this is that for large X and finite v∞, we have513.5. Transcendental Hamiltonian Approximations√1− λ2f ∼ 1/X (modulo a constant) whereas for vanishing v∞, we find√1− f ∼ 1/√X. However, it is possible to find to the order to which weexpanded the Hamiltonian an interpolating function. Specifically, a possibleinterpolating Hamiltonian isH int0 =M√1− v2∞+A0√v4∞ +B0X2+C0X3−A0v2∞ (3.44)H int1 = −A1 −B1√v2∞ +C1X(1 + v2∞E1X)+3D1v3∞2X−D1(v6∞ +1X)3/2−√v2∞ +E1X(3.45)H int2 = −A22−B2v2∞2 (1 + v2∞X)−C22√v4∞ +D2X2−12√v4∞ +E2X4(3.46)with the term zeroth-order in Pλ given explicitly by{A0 = −M27, B0 =9M24√1− v2∞, C0 = −2M35}, (3.47)the first-order coefficient given byA1 = v∞(− 1−M3/2v8∞135√2+11− v2∞+40(1− v2∞)3/2(2√2M − 3√M(1−v2∞)1/4)2M5/2(√2(1− v2∞) + 30√M(1− 4v2∞)))(3.48)B1 =40(1− v2∞)7/2(2√2M − 3√m(1−v2∞)1/4)2 (90√M − (√2 + 120√M)(1− v2∞))M5/2(1− v2∞)2(√2(1− v2∞) + 30√M(1− 4v2∞))2(3.49)C1 =M5(√2(1− v2∞) + 30√M(1− 4v2∞))214400(1− v2∞)3(2√2M − 3√M(1−v2∞)1/4)2 (3.50)D1 =M3/2135√2(3.51)E1 =M√1− v2∞, (3.52)523.6. Interferometry in the Painleve´-Gullstrand Familyand the quadratic coefficient given byA2 = −1M (1− v2∞)3/2+ v2∞(−1 +16√1− v2∞−27M + 48√1− v2∞)(3.53)B2 = −23(3.54)C2 =16√1− v2∞3(9M − 16√1− v2∞)(3.55)D2 =M218225(3.56)E2 =(9M − 16√1− v2∞)264(1− v2∞). (3.57)We now have all of the generalized asymptotics necessary to re-performthe interferometric calculations of Chapter 2, this time for an arbitrarymember of the Painleve´-Gullstrand family of coordinate systems.3.6 Interferometry in the Painleve´-GullstrandFamily3.6.1 Flat Spacetime LimitBefore exploring the implications of our generalized coordinates for the self-gravitating system, let us see how the flat spacetime interferometer is af-fected. We will consider single-mode inputs, as in Chapter 2, for simplicity.In the flat spacetime limit our Hamiltonian takes the form (3.36), which wecan then take the nonrelativistic limit of (Pλ → 0) to obtainH ∼Mˆ√1− v2∞−v∞1− v2∞Pλ +12Mˆ (1− v2∞)3/2P 2λ . (3.58)Unlike the situation for finite X described in Section 3.5, this Hamiltonianhas the correct behaviour as v∞ → 0 (or, equivalently, λ → 1), and re-produces the nonrelativistic, flat spacetime Hamiltonian given in Section2.3.3.The (exact) momentum Pλ that corresponds to equation (3.36) is givenbyPλ = v∞H ±√H2 − Mˆ2. (3.59)533.6. Interferometry in the Painleve´-Gullstrand FamilyThis expression has the usual flat spacetime momentum in the v∞ → 0 limit,and as v∞ → 1 (λ→ 0) leads to the “null observer” HamiltonianH =P 20 + Mˆ22P0. (3.60)It should be noted that we still use the same symbol H here because theactual value of the (classical) Hamiltonian is the same, despite being ex-pressed in terms of a different momentum. Perhaps unsurprisingly, in thenull observer limit the exact momentum (3.59) is always positive for mas-sive shells, regardless of whether the shell is ingoing or outgoing. In thecoordinate representation, the operator Pˆ0 is represented as −i∂/∂X, soone might expect the inverse operator “1/Pˆ0” to be nonlocal. Whether ornot a nonlocal Hamiltonian can produce an evolution that is equivalent (orperhaps just approximately equivalent) to a local Hamiltonian evolution isan open question.We will also make use of the “WKB momentum,” as in Section 2.3.4,which in this context is defined as the solution of (3.58) (treated as anequality) for Pλ:Pwλ =Mˆv∞√1− v2∞±(1− v2∞)3/4√√√√2Mˆ[H − Mˆ(1− 12v2∞√1− v2∞)].(3.61)For a small observer velocity v∞, the special relativistic boost factors in theWKB momentum expression (3.61) are approximately unity, and we are leftwithPwλ ≈ Mˆv∞ ±√2Mˆ(H − Mˆ). (3.62)This is just as one might expect: the shell momentum defined with respect toour generalized Painleve´-Gullstrand coordinate system is given by the usualnonrelativistic expression, offset by a momentum Mˆv∞ that is attributed tothe shell due to the infalling nature of the coordinates.The null observer limit of the WKB momentum (3.61), however, is ill-defined. One can see from (3.58) that the quadratically truncated Hamilto-nian has divergent terms as v∞ → 1. Thus, as we approach this limit wemust use the exact momentum (3.59) in our WKB modes, to avoid patho-logical behaviour. As for the Hamiltonian, as we approach the null observerlimit we should use the exact Hamiltonian, though for simplicity we will543.6. Interferometry in the Painleve´-Gullstrand Familyexpand in λ up to second order, such thatH ≈Pλ2(1 +λ24)+Mˆ22Pλ(1−Mˆ2λ24P 2λ). (3.63)The Hamiltonian (3.63) has a perfectly good limit as λ → 0, but looksslightly unwieldy. Even with the inverse powers of the momentum, we cansolve the corresponding Schro¨dinger equation in each section where Mˆ isconstant (i.e. each side of the beam-splitter) by making use of the momen-tum representation. When gravitational corrections are added, however,inverse powers of X prevent us from avoiding a confrontation with nonlocaloperators. We will keep this in mind when considering whether or not thereare fundamental differences to the quantum evolution introduced by generalrelativity.For much slower asymptotic observers, with v∞ sufficiently less thanunity, the Hamiltonian (3.58) is well-behaved, and leads to the (simpler)Schro¨dinger equationi∂∂tλΨ =Mˆ√1− v2∞Ψ +iv∞1− v2∞Ψ′ −12 (1− v2∞)3/2(1MˆΨ′)′, (3.64)with Hermitian factor-ordering on the last term. As well as being continuous,one can immediately see by integrating (3.64) across the splitter at Xδ thatthe wavefunction Ψ obeys the same jump condition as in Section 2.3.3;namely, [1MˆΨ′]δ= 0, (3.65)with [·]δ indicating the jump of a quantity across Xδ. If we consider thesimple scattering problem from the right,Ψ ={ψλ−+ +Rλ←ψλ++ : X > XδTλ←ψλ−− : X < Xδwe realize that determination of the reflection and transmission amplitudesis trivial, because they obey the same relations as in Section 2.3.3:R¯λ← =M−Pλ−+ −M+Pλ−−M+Pλ−− −M−Pλ++, T¯λ← =M−(Pλ++ − Pλ−+)M−Pλ++ −M+Pλ−−, (3.66)where now the momentum terms are given by (3.59), and the barred ampli-tudes are related to the unbarred amplitudes viaR¯λ← ≡√√√√∣∣∣∣∣∂E∂Pλ−+∂E∂Pλ++∣∣∣∣∣Rλ←, T¯λ← ≡√√√√∣∣∣∣∣∂E∂Pλ−+∂E∂Pλ−−∣∣∣∣∣Tλ←. (3.67)553.6. Interferometry in the Painleve´-Gullstrand FamilyAs with our previous definitions, the first set of plus/minuses indicate out-going/ingoing, and the second set indicate M±.The same pattern occurs for scattering from the left, and the correspond-ing amplitudes areR¯λ→ =M−Pλ++ −M+Pλ+−M+Pλ−− −M−Pλ++, T¯λ→ =M+(Pλ+− − Pλ−−)M−Pλ++ −M+Pλ−−, (3.68)using the similar definitionsR¯λ→ ≡√√√√∣∣∣∣∣∂E∂Pλ+−∂E∂Pλ−−∣∣∣∣∣Rλ→, T¯λ→ ≡√√√√∣∣∣∣∣∂E∂Pλ+−∂E∂Pλ++∣∣∣∣∣Tλ→. (3.69)The details of sending an initial state through the interferometer thusexactly parallel the nonrelativistic, flat spacetime case presented in Section2.3.3. The reflection off of the inner and outer walls (X− and X+, respec-tively) is described by the same reflection coefficientsR¯λ→ = −ei(φλ+++φλ−+), R¯λ← = −ei(φλ+−+φλ−−), (3.70)with the definitions φλ±± =∫ X±XδdXPλ±±, φλ±∓ =∫ XδX∓dXPλ±∓ (signschosen together), andR¯λ→ ≡√√√√∣∣∣∣∣∂E∂Pλ++∂E∂Pλ−+∣∣∣∣∣Rλ→, R¯λ← ≡√√√√∣∣∣∣∣∂E∂Pλ−−∂E∂Pλ+−∣∣∣∣∣Rλ←. (3.71)The probability current Jλ still satisfies the continuity equation∂∂tλ(|Ψλ|2)+∂∂XJλ = 0, (3.72)but this time the current is described by the generalized form (2.113), whichin this case isJλ = −v∞1− v2∞|Ψλ|2 +12iMˆ (1− v2∞)3/2(Ψ∗λΨ′λ −ΨλΨ∗′λ)= −v∞1− v2∞|Ψλ|2 +1(1− v2∞)3/2Jλs. (3.73)In the last line, we used Jλs to denote the standard (nonrelativistic) ex-pression (2.91), evaluated with the λ-dependent WKB state Ψλ. Expression(3.73) implies that for real momenta, the input probability current isJλ0 =|Ψλ0|21− v2∞(Pλ−+M+√1− v2∞− v∞), (3.74)563.6. Interferometry in the Painleve´-Gullstrand Family(a) E = 1.01, M− = 1.0097 (b) E = 1.1, M− = 1.0997Figure 3.1: Sample reflection and transmission coefficients for the initialbeam-splitting using the exact flat spacetime momentum given by (3.59),for M+ = 1 and different combinations of M− and E. The coefficients areplotted against the asymptotic observer velocity, v∞.which certainly differs in form from the previous flat spacetime result (2.92).In this case the difference is superficial, because our initial WKB state Ψλ0 =ψλ−+ has a modulus-squared given by|Ψλ0|2 =∣∣∣∣∂H∂Pλ−+∣∣∣∣−1=∣∣∣∣∣Pλ−+M+ (1− v2∞)3/2−v∞1− v2∞∣∣∣∣∣−1, (3.75)from which we can immediately deduce Jλ0 = −1.5Let us take a moment to consider the reflection and transmission co-efficients associated with the initial split. Since |Jλ0| = 1, the reflectionand transmission coefficients are given by |J (i)λ+| and |J(i)λ−| (respectively). Asimple calculation then confirms that |J (i)λ+| = |Rλ←|2 and |J (i)λ−| = |Tλ←|2.Figures 3.1 and 3.2 highlight the drastic differences in the reflection andtransmission coefficients for the initial splitting, depending on whether oneuses the exact momentum (3.59) or the WKB momentum (3.61). What’s5The previous initial current (2.92) is also equal to −1, although this was not explicitlymentioned.573.6. Interferometry in the Painleve´-Gullstrand Family(a) E = 1.01, M− = 1.0097 (b) E = 1.1, M− = 1.0997Figure 3.2: Sample reflection and transmission coefficients for the initialbeam-splitting, using the flat spacetime WKB momentum given by (3.61),for M+ = 1 and different combinations of E and M−. The coefficients areplotted against the asymptotic observer velocity, v∞.happening here is that the discriminant from (3.61) is changing signs, and sothe WKB momenta are taking on imaginary components. This is illustratedin Figure 3.3. In the process, for the first set of parameters Figure 3.2ashows that as v∞ increases from 0, the splitter continuously goes from beinga 50−50 splitter, to totally reflecting, and then to totally transmitting. Thisis in stark contrast with the behaviour shown in Figure 3.1a for the sameparameters, which shows the splitter smoothly transitioning from being a50−50 splitter for v∞ = 0 to being somewhat more reflecting (Rλ← → 1) asthe observer velocity reaches about half of the speed of light (v∞ = 1). Afterthis point, the shell momenta are too relativistic to be approximated by ourprobability current (which is conserved for Hamiltonians that are quadraticin momenta), resulting in a breakdown of the probabilistic interpretation ofour reflection and transmission coefficients. Figure 3.3 depicts the ingoingand outgoing WKB momenta merging as the discriminant approaches zero,after which there cease to be real WKB momenta. The straight lines arethe exact momenta, which continue (linearly) up to v∞ = 1.583.6. Interferometry in the Painleve´-Gullstrand FamilyIn the entire parameter range of v∞, the sum of the reflection and trans-mission coefficients is identically 1 if we use only the WKB momenta inthe expressions for Rλ← and Tλ← (we will demonstrate this in a more gen-eral context in the next section). If we use the exact momenta, we noticethat the probability current is not quite conserved, as the current expres-sion was derived for systems with Hamiltonians that are exactly quadraticin momentum. On the other hand, the phase in a WKB state is moreaccurately the momentum obtained from inverting a Hamiltonian H(x, p)with respect to p than it is the momentum obtained from solving H(x, p) ≈H0(x) +H1(x)p+H2(x)p2 for p (= pw). Since phase information is crucialfor interferometry, and at the same time probability conservation is crucialfor a sensible interpretation of a quantum system, we will use the variousmomenta carefully.It should be mentioned that in Figures 3.1 and 3.2, the initial probabilitycurrent is equal to −1 for all v∞ ∈ [0, 1], though the initial state is definedto be an ingoing WKB mode (with the same energy for each individualplot). Strictly speaking, since each choice of v∞ defines a different coordi-nate system (and therefore a different notion of time), this means that theinitial state is different for each value of v∞ being plotted. We shouldn’t besurprised, then, that the reflection and transmission probabilities vary withv∞, because this is not an indication that the coordinates used to describethe initial scatter change these probabilities for a given initial state. We willaddress this issue in the following sections.Let us focus now on the final interferometer outputs. In the gener-alized coordinates, we are no longer able to exploit the “identity” R¯2← +M+P−−M−P−+T¯ 2← = 1, which led to the simple form of (2.104) and (2.105), sinceit is no longer valid in the new coordinates. Instead, we must calculate thefinal probability currents from the final output states defined by (2.101) and(2.100). Using the fact that for single-mode WKB states in flat spacetimethe derivatives of the final states obeyddXΨ(v)λ± = iPλ±±Ψ(v)λ±, (3.76)we can write the term Jλs asJλs =Pλ±±M±∣∣∣Ψ(v)λ±∣∣∣2. (3.77)This means we can write the output currents asJ (v)λ± =∣∣∣Ψ(v)λ±∣∣∣2(−v∞1− v2∞+1M± (1− v2∞)3/2Pλ±±)(3.78)593.6. Interferometry in the Painleve´-Gullstrand Family(a) WKB momenta and exact momenta for M− = 1.0097, E =1.01(b) WKB momenta and exact momenta for M− = 1.0997, E = 1.1Figure 3.3: Various WKB momenta in flat spacetime, for M+ = 1 and dif-ferent combinations of E and M−, plotted against the asymptotic observervelocity, v∞. The curved lines are WKB momenta which converge at thepoint where they become imaginary; at this point, they become significantlydifferent from the exact momenta (straight lines), which are real and dis-tinct for all regions of the parameter space. To avoid cluttering the y-axiswith multiple labels, the y-values of the plots, which are various types ofmomenta, are specified in the legends.603.6. Interferometry in the Painleve´-Gullstrand Familywhich we can readily interpret to beJ (v)λ± =∣∣∣Ψ(v)λ±∣∣∣2(∂H∂Pλ)±±, (3.79)with the partial derivative evaluated at Pλ±±. The expression (3.79) makesintuitive sense: Hamilton’s equation for X˙ and the identification of theprobability density ρ with∣∣∣Ψ(v)λ±∣∣∣2yield the familiar current relationJ = ρv, (3.80)where we have used v = X˙ to denote the shell velocity.Taking the modulus-squared of the output states (2.101) gives us∣∣∣Ψ(v)λ+∣∣∣2=∣∣∣∣∂H∂Pλ++∣∣∣∣−1 (R¯4λ← + T¯2λ←T¯2λ→ + 2R¯2λ←T¯λ←T¯λ→ cos 2ϕλ)(3.81)and∣∣∣Ψ(v)λ−∣∣∣2=∣∣∣∣∂H∂Pλ−+∣∣∣∣−1T¯ 2λ←(R¯2λ← + R¯2λ→ + 2R¯λ←R¯λ→ cos 2ϕλ), (3.82)with the definition ϕλ = (φλ++ + φλ−+ − φλ+− − φλ−−). Using (3.79), wecan express the final reflected and transmitted probability currents asJ (v)λ+ =(∂H∂Pλ++)∣∣∣∂H∂Pλ−+∣∣∣(R¯4λ← + T¯2λ←T¯2λ→ + 2R¯2λ←T¯λ←T¯λ→ cos 2ϕλ)(3.83)andJ (v)λ− =(∂H∂Pλ−−)∣∣∣∂H∂Pλ−+∣∣∣T¯ 2λ←(R¯2λ← + R¯2λ→ + 2R¯λ←R¯λ→ cos 2ϕλ). (3.84)These final output currents immediately determine the final reflection andtransmission coefficients, since Rλf = |J(v)λ+ | and Tλf = |J(v)λ− |. Implicit inthese expressions, as well as several other expressions in this section, is theassumption that the momenta are real. We will enforce this reality conditionfor the rest of this thesis, because apart from the above comments regardingthe discriminant of the WKB momenta switching signs in some regions of theparameter space, we regard these imaginary components as an unnecessary(and in this case, unphysical) addition to the interferometric analysis.613.6. Interferometry in the Painleve´-Gullstrand FamilyHow does the interference pattern in this case compare with the flatspacetime case presented in Chapter 2? If we use the WKB momenta, thefinal reflection and transmission coefficients take the simpler formsRλf =[1− 4R¯2λ←(1− R¯2λ←)sin2 ϕλ](3.85)andTλf = 4R¯2λ←(1− R¯2λ←)sin2 ϕλ, (3.86)which bear a striking resemblance to the previous flat spacetime expressions(2.104) and (2.105). Continuing the same steps as in Section 2.3.3 wouldlead us to the phase conditionϕλ = 2L+Pλ++ + 2L−Pλ−− =npi2(3.87)(for n ∈ Z), which is the generalized form of (2.109). We can then deducethe distance between nodes in the interference pattern to be∆Lλn =pi4Pλ++, (3.88)and again observe that coherence is fully present in the flat spacetime inter-ferometer.The node spacing (3.88) varies with the asymptotic observer velocity, butwe should be careful what we conclude from this, because there is still theissue of a changing initial state to deal with. We will will see that the initialstate problem in question presents us with a serious obstacle in our investiga-tion of the coordinate dependence of our description of the interferometer:different coordinate choices define different notions of time, and differentdefinitions of time evolution. We expected this to be the case because of(3.4)-(3.8), but we will see in Section 3.7 that the time transformation (3.6)has some very peculiar consequences indeed.3.6.2 General Relativistic CorrectionsWe will worry about the initial state issue in the next section; first, let us usethe generalized Hamiltonian asymptotics from Section 3.5 to determine thefinal output reflection and transmission coefficients for our shell interferome-ter by transforming the (tentative) input state accordingly. The Schro¨dingerequation for the arbitrary quadratic Hamiltonian (3.37) takes the formi∂∂tλΨ = H0Ψ−i2(2H1Ψ′ +H ′1Ψ)−(H2Ψ′)′ , (3.89)623.6. Interferometry in the Painleve´-Gullstrand Familyand upon integration across Xδ one obtains the jump condition[H2Ψ′]δ = −i2[H1]δ Ψ(Xδ). (3.90)To calculate the reflection and transmission amplitudes for the initialscatter off of the beam-splitter, we consider the wavefunctionΨ ={ψλ−+ +Rλ←ψλ++ : X > XδTλ←ψλ−− : X < XδIf we apply continuity at Xδ and the jump condition (3.108) we can deter-mine the reflection amplitude Rλ← and transmission amplitude Tλ←. Mak-ing use of the notation∆(∂H∂Pλ)=12((∂H∂Pλ)++−(∂H∂Pλ)−−), (3.91)∆ (H2Pλ)± =12H2± (Pλ+± − Pλ−±) , (3.92)andH2± = lim→0H2∣∣∣X=Xδ±, (3.93)we can express the barred amplitudes compactly asR¯λ← =2∆ (H2Pλ)+ −∆(∂H∂Pλ)∆(∂H∂Pλ) (3.94)andT¯λ← =2∆ (H2Pλ)+∆(∂H∂Pλ) , (3.95)for scattering from the right. Similarly, we can express the barred amplitudesasR¯λ→ =2∆ (H2Pλ)− −∆(∂H∂Pλ)∆(∂H∂Pλ) (3.96)andT¯λ→ =2∆ (H2Pλ)−∆(∂H∂Pλ) , (3.97)for scattering from the left.633.6. Interferometry in the Painleve´-Gullstrand FamilyBefore working out the specific interferometric properties of our systemin the Painleve´-Gullstrand family of coordinates, let us explore the generalfeatures of scattering with the Hamiltonian (3.37). We will assume, however,that the arbitrary quadratic Hamiltonian still has a discontinuity at Xδ, toserve as a beam-splitter. Dropping the λ subscript temporarily and solving(3.37) for P gives us the WKB momentum,P = −H12H2±√(H −H0)H2+(H12H2)2. (3.98)For brevity, we will denote this by P = P0± ± P±, with the ± subscriptsreferring to which side of the splitter the quantity is evaluated at, and theother ± indicating outgoing/ingoing. We can then express the WKB P -derivatives as(∂H∂P)±±= H1± + 2H2±P±± = ±2H2±P±. (3.99)We can tell from (3.99) that the WKB P -derivatives satisfy∣∣∣∣∂H∂P∣∣∣∣+±=∣∣∣∣∂H∂P∣∣∣∣−±, (3.100)which implies that R¯← = R←, as well as R¯→ = R→, R¯→ = R→, andR¯← = R←.In the previous section it was pointed out that the initial probabilitycurrent is −1, and that the reflection and transmission coefficients for theinitial scatter off the beam-splitter are R2λ← and T2λ←, respectively. We cannotice that this remains true for arbitrary quadratic Hamiltonians, since tothe order we are working at in ~, the derivative relationddXΨλ± = iPλ±±Ψλ± (3.101)holds, which in turn implies that the current relationJ (v)λ± =∣∣∣Ψ(v)λ±∣∣∣2(∂H∂Pλ)±±(3.102)also holds. The aforementioned expressions for the reflection and transmis-sion coefficients then directly follow.643.6. Interferometry in the Painleve´-Gullstrand FamilyProbability is conserved in this description of the initial split providedthe reflection and transmission coefficients add up to 1, since by definitionthese coefficients give the splitting probabilities. Specifically, we findR2← + T2← =∣∣∣∣∂H∂P∣∣∣∣−1−+(R¯2←∣∣∣∣∂H∂P∣∣∣∣+++ T¯ 2←∣∣∣∣∂H∂P∣∣∣∣−−)=∣∣∣∣∂H∂P∣∣∣∣−1−+(2∆ (H2P )+ −∆(∂H∂P)∆(∂H∂P))2 ∣∣∣∣∂H∂P∣∣∣∣+++(2∆ (H2P )+∆(∂H∂P))2 ∣∣∣∣∂H∂P∣∣∣∣−−=8∆ (H2P )2+ − 4∆ (H2P )+(∂H∂P)++ +(∂H∂P)++ ∆(∂H∂P)∣∣∂H∂P∣∣−+ ∆(∂H∂P)=1(2H2+P+) 12 (2H2+P+ + 2H2−P−)[8P 2+H22+−2 (2H2+P+) (2P+)H2+ +H2+P+ (2H2+P+ + 2H2−P−)]=4P 2+H2+ − 4H2+P2+ + P+ (H2+P+ +H2−P−)P+ (H2+P+ +H2−P−)= 1, (3.103)thus confirming that probability is conserved for the initial split, if we usethe WKB momenta in our scattering expressions.The final output states are determined by equations (2.101) and (2.100),which yieldΨ(v)λ+ = ψλ++(Rλ←eiΦλ++Rλ→eiΦλ−+Rλ←+ Tλ←eiΦλ−−Rλ←eiΦλ+−Tλ→), (3.104)andΨ(v)λ− = ψλ−−(Rλ←eiΦλ++Rλ→eiΦλ−+Tλ←+ Tλ←eiΦλ−−Rλ←eiΦλ+−Rλ→), (3.105)with the mode functions ψλ±± evaluated at X = Xδ + 0±.Taking the modulus-squared of the final states (3.104) and (3.105) gives653.6. Interferometry in the Painleve´-Gullstrand Familyus∣∣∣Ψ(v)λ+∣∣∣2= |ψλ++|2[∣∣∣R2λ←R˜λ→∣∣∣2+∣∣∣Tλ←Tλ→R˜λ←∣∣∣2+ 2 <{R2λ←R˜λ→(Tλ←Tλ→R˜λ←)∗}], (3.106)and∣∣∣Ψ(v)λ−∣∣∣2= |Tλ←ψλ−−|2[∣∣∣Rλ←R˜λ→∣∣∣2+∣∣∣Rλ→R˜λ←∣∣∣2+ 2 <{Rλ←R˜λ→(Rλ→R˜λ←)∗}], (3.107)where <{·} denotes the real part of a quantity and (·)∗ denotes complexconjugation. We have also used the abbreviations R˜λ→ = Rλ→ei(Φλ+++Φλ−+)and R˜λ← = Rλ←ei(Φλ−−+Φλ+−), for brevity.The same reasoning we used to establish that the probability is conservedfor the initial split can be easily extended to show that the final interferom-eter output probabilities also sum to unity if the WKB momenta are used,but we will not belabour that point here. We will instead return to the taskof determining the final output probabilities in the Painleve´-Gullstrand fam-ily of coordinates. The case λ = 1 was presented in Chapter 2, so we willrestrict our attention to 0 < λ < 1. Using the asymptotic Hamiltonianexpressions (3.41)-(3.43), the jump condition (3.90) becomes[(1λMˆ+1X)Ψ′]δ=iλ∆Mv∞XδΨ(Xδ), (3.108)with ∆M = (M+ −M−) /2. This parallels the previous gravitational jumpcondition (2.116), though we can see from the right hand side of (3.108) thatthe v∞ → 0 limit is ill-defined.The reflection and transmission amplitudes for scattering from the rightcan be written asR¯λ← =λ∆Mv∞Xδ+(1λM−+ 1Xδ)Pλ−− −(1λM++ 1Xδ)Pλ−+− λ∆Mv∞Xδ −(1λM−+ 1Xδ)Pλ−− +(1λM++ 1Xδ)Pλ++(3.109)andT¯λ← =(1λM++ 1Xδ)(Pλ++ − Pλ−+)− λ∆Mv∞Xδ −(1λM−+ 1Xδ)Pλ−− +(1λM++ 1Xδ)Pλ++. (3.110)663.6. Interferometry in the Painleve´-Gullstrand Familywhich correspond to the previous gravitational reflection and transmissionamplitudes (2.117) and (2.118). Similarly, for scattering from the left wehaveR¯λ→ =λ∆Mv∞Xδ+(1λM−+ 1Xδ)Pλ+− −(1λM++ 1Xδ)Pλ++− λ∆Mv∞Xδ −(1λM−+ 1Xδ)Pλ−− +(1λM++ 1Xδ)Pλ++(3.111)andT¯λ→ =(1λM−+ 1Xδ)(Pλ+− − Pλ−−)− λ∆Mv∞Xδ −(1λM−+ 1Xδ)Pλ−− +(1λM++ 1Xδ)Pλ++, (3.112)which are the analogs of (2.119) and (2.120). Remarkably, as v∞ → 0,the transmission amplitudes from both the left and right approach zero, forany finite values of the remaining parameters. This does not reproduce thePainleve´-Gullstrand limit presented in the previous chapter, and so is clearlynot the correct behaviour as v∞ → 0; the singular nature of the Hamiltonianasymptotics (3.41)-(3.43) is causing our WKB scattering approximation tobreak down.To resolve this issue, we can use the interpolating Hamiltonian we derivedin Section 3.5. Though the interpolating Hamiltonian has the correct limitas v∞ → 0, its use leads to some ugly expressions, which we will not presenthere. Nonetheless, even with the interpolating Hamiltonian, one still endsup with reflection and transmission probabilities that are λ-dependent, andit is unclear exactly how to interpret this, given the λ-dependence of theinitial state.A similar issue arises when using the reflection and transmission am-plitudes to determine the node spacing in the interference pattern. Theoscillatory part of the final reflection and transmission coefficients involvesthe phaseϕ¯λ = Φλ++ + Φλ−+ − Φλ+− − Φλ−−, (3.113)with Φλ±±˜ = φλ±±˜ − Htλ±±˜ and φ±±˜ = ±±˜∫ X±˜XδdX Pλ±±˜. Modulo apossible factor of 2, this means that if we fix all other system parametersand vary the outer interferometer arm length L+ = X+ − Xδ, the nodespacing in phase space for the nth node is given by ϕ¯λ(n+1) − ϕ¯λn = pi, forϕ¯λn =∫ X+nXδdX (Pλ++ − Pλ−+)−∫ XδX−dX (Pλ+− − Pλ−−)−H (tλ++ + tλ−+ − tλ+− − tλ−−) . (3.114)673.7. DiscussionIf we make the assumption of equal travel times, i.e. we recombine thequantum states after a definite (shared) coordinate time has elapsed, thenthe node spacing condition becomespi = ϕ¯λ(n+1) − ϕ¯λn =∫ X+(n+1)X+ndX (Pλ++ − Pλ−+) . (3.115)We can further simplify the analysis by working in a regime where the nodespacing is much smaller than the outer interferometer arm length; in thiscase we can use the definitions ∆Ln = L+(n+1) − L+n, L+n = X+n − Xδ,∆(Pλ)± = (Pλ+± − Pλ−±)/2, and ∆(Pλ)+n = ∆(Pλ)+|X=X+n to approxi-mate the integral in (3.115) aspi = 2∫ X+n+∆LnX+ndX ∆ (Pλ)+≈ 2∆ (Pλ)+n ∆Ln +(∂∆ (Pλ)+n∂∆Ln)(∆Ln)2 . (3.116)When solved for the node spacing ∆Ln, the approximation (3.116) yields∆Ln =−∆ (Pλ)+n ±√(∆ (Pλ)+n)2+ pi(∂∆(Pλ)+n∂∆Ln)(∂∆(Pλ)+n∂∆Ln) . (3.117)As the outer interferometer arm length increases, the node spacing changesas∂∆Ln∂L+n=∂∆Ln∂X+n, (3.118)which clearly has a rather complicated dependence on λ.3.7 DiscussionWhat can we conclude from the λ-dependence of this interference pattern?Since each λ specifies a coordinate choice, and this coordinate choice can beassociated with a network of infalling observers, one might wonder if coher-ence itself could be observer-dependent. At first sight, this seems like quitean unintuitive possibility, but can we rule it out on consistency grounds?If a beam-splitter has a transmission probability of 50% according to oneobserver network, then that observer network could be used to make a series683.7. Discussionof output measurements, and there would be roughly 50% recorded trans-missions. Is this not an objective property of the beam-splitter, agreed uponby any other observer network?To address these questions, let us return to the issue about the initialstate mentioned earlier in the Chapter. In Chapter 2, we chose an initial(ingoing) WKB state of the formΨ0 =ei∫dX P−+√|∂H/∂P−+|≡ ψ−+, (3.119)evaluated at H = E, which we generalized in the previous section toΨλ0 = ψλ−+. (3.120)The problem with this generalization is that although the states ψ−+ andψλ−+ have the same energy, for λ 6= 1 they are, strictly speaking, differentstates. The time dependence of the generalized initial state, for instance, isdefined byΨλ−+ = ψλ−+e−iEtλ , (3.121)with tλ being related to the previous Painleve´-Gullstrand time byt− tλ =∫dX√2E/Xf−∫dX√1− λ2ff(3.122)as a result of the transformations presented in Section 3.1. Here f = 1 −2E/X, and the integration can be performed to obtaint− tλ = 4E(√X2E+12ln∣∣∣∣∣√X/2E − 1√X/2E + 1∣∣∣∣∣)−4E[X4E√1− λ2f + ln( √f1 +√1− λ2f)+(1− λ22)√1− λ2ln(√X2E(√1− λ2 +√1− λ2f)) .(3.123)One can then see that t = 0 corresponds to a generalized time tλ thatdepends on the values of X and E, which implies that the (classical) trans-formation between the two definitions of time depends on the location inphase space. At the quantum level, the state of a system with respect to aparticular coordinate choice at a specific value of the associated coordinate693.7. Discussiontime does not in general correspond to a system state with respect to a dif-ferent coordinate choice at any individual value of the time associated withthis other coordinate choice.6 The very notion of “initial state” thereforeseems to be coordinate-dependent, which would potentially render the ques-tion of how the coordinate choice affects the propagation of a given initialstate through the interferometer ill-defined.To avoid this paradoxical conclusion, one could use an augmented familyof coordinate systems that had equivalent constant-time hypersurfaces inthe region where the initial state is defined, but were different everywhereelse. For instance, one could have a family that were equal to Painleve´-Gullstrand coordinates (λ = 1) for r > r0, but for r < r0 the coordinatesystems in this new family range from the λ = 1 to the λ = 0 coordinatesof the original family. The different coordinate systems for r < r0 thencontinuously connect to the same (λ = 1) coordinates for r > r0, whichenables us to define the same initial state for each member of the new family(in the r > r0 region). This idea was suggested very recently (by Bill Unruh),and until it has been fully implemented we will have to accept that theanalysis of coordinate dependence in our description of the self-gravitatinginterferometer is inconclusive.6Given the connection between coordinate choices and observer networks, we mightsummarize this behaviour with the (somewhat Rovellian) maxim, “Everyone else’s timeis an operator except one’s own.”70Chapter 4Superpositions of Clocks andIntrinsic Decoherence4.1 IntroductionTime dilation is one of the most profound consequences of relativity theory.In classical systems, time dilation effects are fairly well understood, but inquantum systems there are still many questions that remain unanswered.One such question is how to properly incorporate the effects of time dilationinto the quantum evolution of composite systems, either in the limit of flatspacetime or in situations where gravitational effects are significant.In a series of papers by Pikovski et al. [12]-[14], for instance, a “univer-sal” decoherence mechanism was proposed for composite general relativisticsystems, due to gravitational time dilation. In [12], Pikovski et al. presentan approximate quantum description of such a composite system that asa whole behaves as a point particle (located at the center-of-mass of thesystem) with a well-defined proper time, with internal degrees of freedomthat are defined in the rest frame of the system. The system is placedin the gravitational field of the earth, and the authors postulate that thequantum evolution of the system with respect to a laboratory frame on thesurface of the earth should be given by a Schro¨dinger equation of the formi DDτΨ = HrestΨ, with τ being the proper time of the system (treated as apoint particle), and Hrest being the rest-frame Hamiltonian. Expressing theproper time derivative in terms of the lab-frame time t induces a couplingbetween the internal degrees of freedom and the center-of-mass coordinate,and if one only keeps track of the center-of-mass dynamics, tracing out overthe internal degrees of freedom leads to a novel form of the “third-partydecoherence” described in Chapter 1. It is this effect, and variations on thetheme, that we focus on in this chapter.It is unclear whether any inconsistencies result from simply replacingthe usual time derivative in the Schro¨dinger equation with a proper timederivative for the system’s center-of-mass, since this is not the standard714.1. Introductionprocedure for quantizing general relativistic systems. Rather than makeuse of the same model used by Pikovski et al., we will explore similar ideaswith a model that generalizes the self-gravitating spherical perfect fluid shellintroduced in Chapter 2. Whereas the original model was introduced tostudy the consequences of general relativity on massive interferometers, herewe extend the model to include an “internal” harmonic oscillator, to analyzethe quantum structure of composite relativistic systems. By “internal,” wemean that the harmonic oscillator is described by an internal coordinate qthat oscillates in an abstract space that is not part of the spacetime; such aninternal degree of freedom could represent the values of a single sphericallysymmetric mode of an oscillating field confined to the surface of the shell,for instance. The internal coordinate oscillates harmonically with respect tothe proper time of the (external) shell position, and therefore the oscillatorserves as a clock, evolving based on the local flow of time determined bythe external motion. Unlike the postulated evolution in [12], however, ourinternal coordinates evolve classically according to the proper time of theshell, and then a Hamiltonian H is defined with respect to a coordinate timet, which then allows us to quantize the system with a standard Schro¨dingerequation i ∂∂tΨ = HΨ. This avoids any extra postulates about how suchsystems evolve quantum-mechanically.We should keep in mind that the internal oscillator contributes to theexternal shell dynamics as well, which in turn affects the spacetime; in otherwords, the very ticking of our clock influences the manner in which it ticks.This is especially relevant in the quantized system, because uncertainties inclock readings become intimately connected with uncertainties in spacetimegeometry.We will explore some of the ambiguities associated with the quantumtheory of this generalized shell system in reduced phase space, and then re-late an approximate form of our reduced Hamiltonian with the Hamiltonianpresented in [12]. We exploit this parallel to demonstrate time dilation deco-herence in our system, and observe that when the fluid pressure is nonzero,the (external) shell position decoheres even in the (gravity-free) limit of flatspacetime, because of the acceleration caused by the pressure. This indicatesthat the proposed effect results from proper time differences alone, and assuch is not necessarily related to gravity.Further, we can use our generalized shell model to include self-interactioncorrections to the time dilation decoherence, such that the decoherence isaltered by the manner in which our shell and its clock influence the state oftheir own geometry. We find that even without pressure, the self-gravitationof the shell leads to the nonzero acceleration required to produce the time di-724.2. Classical Actionlation decoherence. We interpret this “self-decoherence” as a fundamentallygravitational effect.4.2 Classical ActionFor context, before adding an internal oscillator, the self-gravitating spher-ical perfect fluid shell model introduced in Chapter 2 is described by theaction Ix + IG, where Ix is the shell actionIx = −∫dλ√−gµνdxµdλdxνdλM(R), (4.1)with all quantities evaluated on the shell history, and IG is the Einstein-Hilbert actionI =116pi∫d4x√−g(4) R(4). (4.2)Here superscripts on the metric determinant g and the Ricci scalar R indi-cate that these are constructed from the full (3 + 1)-dimensional spacetimemetric components {gµν}. We make use of the ADM form of the metric inspherical symmetry,gµνdxµdxν = −N2dt2 + L2 (dr +N rdt)2 +R2dΩ2, (4.3)where N is the lapse function, N r is the radial component of the shiftvector, and L2 and R2 are the only nontrivial components of the spatialmetric [54]. It is then clear that R is the “radius” of the shell, obtainedfrom the area 4piR2 of symmetry two-spheres. The shell contribution Ix isanalogous to a free relativistic particle action, except with a mass M thatdepends on the position-dependent metric function R; the function M(R)serves to parametrize the relationship between the density σ = M(R)/4piR2and pressure Pσ = −M ′(R)/8piR of the fluid.We add an internal oscillator to our shell with the actionIq =12∫dτ[m(dqdτ)2− kq2], (4.4)with τ being the proper time evaluated on the shell history, and q being aninternal coordinate that does not take values in the (external) spacetime.The quantity k is related to ω0, the natural frequency of the oscillator,via k = mω20. The action (4.4) is manifestly invariant under coordinatetransformations, as it only makes use of the proper time of the shell.734.2. Classical ActionIt simplifies the description to parametrize the shell history with the co-ordinate time t, such that the classical shell motion is defined by a trajectoryr = X(t). We can use the shell 4-velocity uµ = (dt/dτ)(1, X˙, 0, 0) to expressthe proper time differentials asdτ = −uµdxµ = τ˙ dt (4.5)anddqdτ= −uµ∂µq = τ˙−1dqdt. (4.6)Now q is being treated as a function solely of coordinate time t, to reflectour choice of parametrization. The 4-velocity normalization uµuµ = −1 thenimplies that one can express the derivative of the proper time with respectto the coordinate time asτ˙ =∫dr√N2 − L2(N r + X˙)2δ(r −X)=√Nˆ2 − Lˆ2(Nˆ r + X˙)2. (4.7)An overhat denotes that a quantity is to be evaluated on the shell history;likewise, it is understood that the overdots denote coordinate-time deriva-tives along the shell trajectory.If we define the original shell Lagrangian asLx = −∫dr√N2 − L2(N r + X˙)2M(R)δ(r −X)= −τ˙ Mˆ (4.8)and the oscillator Lagrangian asLq =12∫dr τ˙[m(q˙τ˙)2− kq2]δ(r −X)=12(mq˙2τ˙− kτ˙q2), (4.9)then the shell-oscillator action is given byIshell =∫dtL =∫dt (Lx + Lq). (4.10)744.3. Hamiltonianization4.3 HamiltonianizationWe can Hamiltonianize the shell-oscillator system with the Legendre trans-formation H = PX˙ + pq˙ − L. The momentum p conjugate to the internalcoordinate q is given byp ≡∂L∂q˙= m∫drq˙τ˙δ(r −X) = mq˙τ˙, (4.11)and that the momentum conjugate to the shell position X is given byP ≡∂L∂X˙= −Mˆ∂τ˙∂X˙−12m(q˙τ˙)2 ∂τ˙∂X˙−12kq2∂τ˙∂X˙= −∂τ˙∂X˙(Mˆ +p22m+12kq2)= −∂τ˙∂X˙(Mˆ +Hq), (4.12)where the internal “clock” Hamiltonian Hq = p2/2m+kq2/2 takes the formof a (free) harmonic oscillator. Introducing the notation M˜ = Mˆ +Hq, onecan observe that our shell-oscillator system becomes very similar to the shellsystem without the oscillator, subject to the transformation Mˆ → M˜ .More explicitly, the shell momentum P can be expressed asP =∫drL2(N r + X˙)M˜√N2 − L2(N r + X˙)2δ (r −X), (4.13)from which we can solve for X˙ to obtainX˙ =∫dr(NPL√P 2 + L2M˜2−N r)δ (r −X)=NˆPLˆ√P 2 + Lˆ2M˜2− Nˆ r. (4.14)The Legendre transformation then gives us the HamiltonianH = PX˙ + pq˙ − L =∫dr (NHst +NrHsr ), (4.15)with the definitionsHst =√L−2P 2 + M˜2δ(r −X),Hsr = −Pδ(r −X). (4.16)754.4. Reduced Phase Space QuantizationWe remind the reader that M˜ = Mˆ +Hq, so our clock Hamiltonian adds tothe (position-dependent) shell mass Mˆ to alter the Hamiltonian constraintfrom the form it took in Chapters 2 and 3.Hamiltonianizing the gravitational sector as well, as in Chapter 2, leadsto the total actionI =∫dt(PX˙ + pq˙)+∫dt dr(piRR˙+ piLL˙−NHt −NrHr), (4.17)for Ht = Hst +HGt and Hr = Hsr +HGr , such thatHGt =Lpi2L2R2−piLpiRR+(RR′L)′−(R′)22L−L2,HGr = R′piR − Lpi′L. (4.18)4.4 Reduced Phase Space QuantizationSince we are working in spherical symmetry, the metric itself has no actualdegrees of freedom, because although there are only two gravitational con-straints (Ht = 0 and Hr = 0), there are also only two independent metricfunctions (L and R). Accordingly, one can obtain an unconstrained descrip-tion of the system by making a coordinate choice, solving the constraints forthe corresponding gravitational momenta, and inserting the solutions intothe Liouville form F on the full phase space,F = PδX + pδq +∫dr (piLδL+ piRδR). (4.19)This amounts to a pullback of the full Liouville form to the representativehypersurface defined by the coordinate choice. From the Liouville pullback,denoted by F˜ , we can deduce the canonical structure of the reduced phasespace, which only depends on the shell-oscillator variables X and q (andtheir momenta).To solve the gravitational constraints, first consider the following linearcombination of the constraints, away from the shell:−R′LHt −piLRLHr =M′, (4.20)forM(r) =pi2L2R+R2−R(R′)22L2. (4.21)764.4. Reduced Phase Space QuantizationThe quantity M(r) corresponds to the ADM mass H when evaluated out-side of the shell, and vanishes inside the shell. We can now solve for thegravitational momenta piL, piR away from the shell. The result ispiL = ±R√(R′L)2− 1 +2MR, piR =LR′pi′L. (4.22)Assuming a continuous metric and singularity-free gravitational mo-menta, we can integrate the gravitational constraints (Ht = 0 and Hr = 0)across the shell, from which we obtain the jump conditions∆R′ = −V˜Rˆ, ∆piL = −PLˆ, (4.23)where V˜ =√P 2 + M˜2. We use ∆ to denote the jump of a quantity acrossthe shell (at r = X(t)).The coordinates we will use resemble the Painleve´-Gullstrand coordi-nates {L = 1, R = r}, though the jump conditions force us to include adeformation region (X −  < r < X) near the shell. By inspection, therequired metric function R can be generalized asR(r, t) = r −XV˜ G(X − r), (4.24)for a function G having the propertieslimz→0+dG(z)dz= 1 (4.25)limz→0−dG(z)dz= 0 , (4.26)from which it follows thatlim→0R′(X − ) = 1 +V˜X(4.27)lim→0R′(X + ) = 1 . (4.28)By inserting the gravitational momentum solutions (4.22) associatedwith the coodinate choice (4.24) into the jump equations (4.23) and squar-ing, one findsH =√P 2 + M˜2 +M˜22X− P√2HX. (4.29)774.4. Reduced Phase Space QuantizationThis implies that the unreduced momentum P is implicitly defined as afunction of the reduced phase space quantities X, q, p, and H: one caneasily solve (4.29) to obtainP =11− 2HX(√2HX(H −M˜22X))(4.30)±11− 2HX√√√√(H −M˜22X)2− M˜2(1−2HX) .The ± in (4.30) indicates whether the shell is outgoing (+) or ingoing (-),with respect to our choice of coordinates.Let us now calculate the pullback of the full Liouville form to the rep-resentative hypersurface defined by our coordinate choice. The conditionL = 1 and the fact that R = r outside of the deformation region impliesthatF˜ = PδX + pδq +∫ XX−dr piRδR. (4.31)We can then simplify the remaining integral by changing the integrationvariable from r to v = R′, which yields∫ XX−dr piRδR = XδX∫ R′−1dv(1− v)√v2 − 1+O(), (4.32)with R′− being R′ evaluated just inside the shell. Now the integration istrivial, and we can easily obtain the desired Liouville form pullback,F = PcδX + pδq, (4.33)with the reduced canonical momentum for the shell position satisfyingPc = −√2HX +X ln(1 +V˜ + PX+√2HX). (4.34)This expression gives an implicit definition of the Hamiltonian H on thereduced phase space, as a function of the shell-oscillator variables (X andq), along with the momenta that are conjugate to them in the reduced phasespace (Pc and p, respectively).Just as the expressions (2.47) and (2.48) from Chapter 2 were shown to beequivalent, the expression (4.34) for the reduced canonical shell momentum784.4. Reduced Phase Space QuantizationPc is equivalent toPc = −√2HX −X ln(X + V˜ − P −√2HXX), (4.35)despite the different minus sign placement.To gain some intuition for how the presence of the oscillator alters thereduced dynamics, let us consider the flat spacetime limit of the systemdefined by (4.35). Keeping in mind the similarities with the relativistic-particle-like structure of our shell system, it should be unsurprising that inthis limit (4.35) becomesPc = ±√H2 − M˜2 = ±√H2 −(Mˆ +Hq)2, (4.36)and therefore the Hamiltonian is given byH =√P 2c + M˜2 =√P 2c +(Mˆ +Hq)2. (4.37)In the nonrelativistic regime (i.e. small Pc), the Hamiltonian can then beexpressed asH ≈ Mˆ +Hq +P 2c2(Mˆ +Hq)= Mˆ +12kq2 +p22m+P 2c2(Mˆ + 12kq2 + p22m) . (4.38)The last term in this approximate Hamiltonian is an effective coupling be-tween the internal oscillator variables (q and p) and the external shell vari-ables (X and Pc). The coupling is of course produced by the fact that theinternal “clock” oscillates harmonically with the shell’s proper time, the flowof which is influenced by the external variables.Even in the flat spacetime nonrelativistic limit, one can tell from theappearance of the clock Hamiltonian Hq in the denominator of the lastterm in (4.38) that exact quantization will require nonstandard techniques.The Hamiltonian (4.38) leads to the following Schro¨dinger equation, in thecoordinate basis:i∂∂tΨ =(Mˆ +12kq2 −12m∂2∂q2)Ψ−12∂∂X[1Mˆ + 12kq2 − 12m∂2∂q2]∂∂XΨ. (4.39)794.5. Time Dilation DecoherenceThe factor-ordering in the last term of (4.39) was chosen to make the differ-ential operator Hermitian, but there is still some ambiguity in the meaning ofthe bracketed factor between the X-derivatives, since the formal expressionhas q-derivatives in the denominator.To formulate a more tractable problem, let us consider the followingapproximation to the Hamiltonian (4.38):H = Mˆ +Hq +P 2c2Mˆ(1 + HqMˆ)≈ Mˆ +Hq +P 2c2Mˆ(1−HqMˆ), (4.40)which should be valid as long as the shell mass Mˆ is sufficiently largerthan the clock energy Hq. In this case the system decomposes into a morestandard formH = H0 +Hxq = Hx +Hq +Hxq, (4.41)with the approximate shell Hamiltonian Hx = Mˆ + P 2c /2Mˆ , the internal“clock” Hamiltonian Hq = p2/2m+ kq2/2, and the interactionHxq = −P 2c2Mˆ2Hq, (4.42)which is induced by the clock oscillation being defined with respect to theproper time of the shell.4.5 Time Dilation DecoherenceThe decomposition (4.41) is of the same form as the one used recently byPikovski et al. [12] to demonstrate decoherence due to gravitational timedilation for composite systems, though the interpretation of the system vari-ables is different. In the (gravity-free) limit of flat spacetime, the main dif-ference is simply that the “external” coordinate of our system is the shellradius instead of the (somewhat ill-defined) center-of-mass coordinate. Inthe next section, however, the self-gravitation of the shell-plus-clock systemis taken into account, so both the shell and the clock influence the spacetimegeometry, which is therefore no longer fixed. Nonetheless, we can exploitthe similarity enough to demonstrate a decoherence effect in our systemanalogous to that described by Pikovski et al., as will become clear in whatfollows.804.5. Time Dilation DecoherenceRepresented by a density operator, the full state ρ obeys the von Neu-mann equation,ρ˙ = −i [H, ρ] . (4.43)We can then change the frame to primed coordinates, as in [12], which aredefined by ρ′(t) = eit(H0+h)ρ(t)e−it(H0+h), for h(X,Pc) = Trq [Hxqρq(0)].We are assuming that the initial state of the system is of the product formρ(0) = ρx(0)ρq(0), i.e. initially uncorrelated. Denoting the average clockenergy Trq [Hqρq(0)] by E¯q and the shell part of the interaction by Γ =−P 2c /2Mˆ2, we obtain the expression h = Γ(X,Pc)E¯q. The transformed vonNeumann equation isρ˙′(t) = i[H ′0(t) + h′(t), ρ′(t)]− i[H ′0(t) +H′xq(t), ρ′(t)]= −i[H ′xq(t)− h′(t), ρ′(t)], (4.44)with h′(t) = h(X ′(t), P ′c(t)). If we integrate and iterate equation (4.44), weare led to the integro-differential equationρ˙′(t) = −i[H ′xq(t)− h′(t), ρ′(0)](4.45)−∫ t0ds[h˜(t),[h˜(s), ρ′(s)]],using the definition h˜(t) = H ′xq(t)−h′(t). At this point Pikovski et al. traceover the internal variables, which for us describe the clock, and make useof the Born part of the Born-Markov approximation, keeping only terms upto second order in the interaction Hamiltonian Hxq, and replacing the ρ′(s)in the integral by ρ′x(s)ρ′q(0). This application of the Born approximationassumes weak coupling, but in contrast to the full Born-Markov it does notignore memory effects. For a detailed discussion of this approximation, see[71]. The reduced equation for the external shell evolution is then given byρ˙′x(t) = Trq[ρ˙′(t)]≈ −∫ t0ds Trq{[h˜(t),[h˜(s), ρ′(s)]]}= −∫ t0ds Trq{(Hq − E¯q)2 [Γ′(t),[Γ′(s), ρ′(s)]]}= − (∆Eq)2∫ t0ds[Γ′(t),[Γ′(s), ρ′x(s)]], (4.46)with Γ′(s) = Γ(X ′(s), P ′c(s)) and(∆Eq)2 = Trq[(Hq − E¯q)2ρq(0)]. (4.47)814.6. DiscussionOne can then transform back to the unprimed frame, whereby the substitu-tion s→ t− s leads to the expressionρ˙x(t) = −i[Hx + ΓE¯q, ρx(t)](4.48)− (∆Eq)2∫ t0ds[Γ, e−isHx [Γ, ρx(t− s)] eisHx].In general, the reduced evolution equation (4.48) exhibits decoherencedue to the nonunitary contribution of the last term on the right. Undersome special circumstances this term vanishes, leaving the reduced systemto evolve unitarily; for example, such a circumstance occurs for initial statesthat are eigenstates of the internal (clock) Hamiltonian, of course then theoscillator is not much of a clock, as it never changes with time (modulo aphase).4.6 DiscussionIn the last section, we demonstrated intrinsic decoherence due to time dila-tion, in the (gravity-free) limit of flat spacetime. In the system described byPikovski et al. [12], which includes spacetime curvature caused by the ex-ternal gravitational field of the earth, the time dilation decoherence shouldvanish in the absence of the earth’s gravitational influence: without theearth, the center-of-mass coordinate they use defines the origin of an iner-tial frame, and in that frame the proper time associated with the center-of-mass coordinate is equal to the coordinate time.7 However, in our system,decoherence (in the position basis) is present even without an external grav-itational field, because of the nonzero acceleration of the shell due to theposition dependence of the mass (Mˆ = M(X)). We are then led to concludethat the time dilation decoherence proposed in [12] is not necessarily relatedto gravity, but produced by proper time differences in composite systemswith nonzero accelerations.The preceding analysis can be extended to include self-interaction effects,by making use of the Hamiltonian asymptotics derived in Chapter 3, alongwith the transformation Mˆ → M˜ . We then find that the components Hx7Of course, quantum fluctuations of the center-of-mass motion will still produce deco-herence in the momentum basis for the reduced system, but coherence will remain for thecenter-of-mass position itself; similarly, for our shell system in the absence of both gravityand pressure, the effective interaction 4.42 will lead to decoherence in the momentum basisof the reduced system, but coherence will remain for the (external) shell position.824.6. Discussionand Γ of the decomposition (4.41) generalize toHx →(Mˆ +P 2c2Mˆ)+P 2c3X−23√2MˆXPc −Mˆ218X (4.49)andΓ→(−P 2c2Mˆ2)+−Mˆ9X−13Mˆ√2MˆXPc , (4.50)to second order in 1/√X. The bracketed term in each of these expressionsoriginates from self-gravitation. From this we can observe that even in theconstant Mˆ limit, where the fluid pressure vanishes, (4.48) (with the gener-alized components Hx and Γ) indicates that the time dilation decoherenceremains present, in this case because the self-gravitation produces a nonzeroacceleration of the shell position. Conceptually, such an effect should occurfor any composite general relativistic system that has internal motion that(classically) evolves according to the proper time associated with the sys-tem’s external motion, since the alteration of the local flow of time causedby the system’s influence on its own spacetime geometry induces an effec-tive coupling between the internal and external degrees of freedom. It is thiseffect that is fundamentally gravitational in nature, as it is present even inthe absence of any other interactions.We have therefore arrived at a type of intrinsic decoherence similar to the“third-party” decoherence described by Stamp [15], though in contrast tothe use of the earth as the third party as proposed by Pikovski et al. [12], wehave bootstrapped the idea by incorporating gravitational self-interaction,effectively producing third-party decoherence without the third party.83Chapter 5Denouement5.1 Reflections and ResolutionsIn the preceding chapters, we have explored some of the consequences offorming superpositions of quantum states that correspond to different space-time geometries, in the context of a general relativistic model of a self-gravitating spherical perfect fluid shell. It has long been argued by Penrose(and others) that there is an inherent ambiguity associated with formingsuch superpositions [9], [10]. However, if the corresponding spacetime ge-ometries are nearly identical, we know from standard quantum mechanicsthat there is effectively a well-defined notion of time evolution. When suchan effective structure exists (as it does in almost every conceivable quantumsystem within technological reach), it has been extensively confirmed ex-perimentally that quantum coherence is possible, which enables interferencebetween distinct elements of a superposition.When the spacetime geometries are sufficiently different, Penrose arguesthat the effective structure of quantum-mechanical time evolution ceases tobe well-defined, at least with the current interpretation given to how weusually apply our quantization techniques. Penrose speculates that this suf-ficient difference in spacetime geometries leads to an instability that causesthe interference between the corresponding quantum states to decay, thoughit is unclear from his work whether the system undergoes a “collapse” orjust an intrinsic form of gravitational decoherence. Since the former possibil-ity seems to necessarily require altering the foundations of quantum theory,this thesis focused on the possibility that a suitably-interpreted quantumdescription of time evolution in an idealized general relativistic system canexhibit this intrinsic form of gravitational decoherence, without the additionof any exotic or untested physics. Accordingly, we worked within canonicalquantum gravity; for simplicity, as well as practicality, we restricted ourattention to minisuperspace, and used the reduced phase space approach toquantization.We used our self-gravitating fluid shell model to perform an interfer-ometric analysis in Chapter 2 for single-mode input states, but ended up845.1. Reflections and Resolutionsconcluding that even though the classical model was general-relativisticallycorrect, we did not observe any limitations on the coherence of the systemattributable to gravity. This outcome does not necessarily imply that ourmodel is incapable of producing such a limitation, as discussed in Section 2.4,though it does mean that our approximation scheme did not capture the es-sential features necessary to confirm Penrose’s proposed decoherence effect.The fact that we did not observe any limitation on the coherence of ourself-gravitating interferometer could also be explained if Penrose’s argumentabout the ill-definedness of time evolution for superpositions of geometriesis avoided by the very structure of canonical quantum gravity itself. Pen-rose recognizes this possibility, as well as the fact that superpositions of3-geometries are ubiquitous in canonical quantum gravity, but considers thepicture one ends up with insufficient to describe the physics within such asuperposition of spaces [10]. Rather than being this dismissive, the perspec-tive we have entertained in this thesis is that canonical quantum gravity isprecisely the arena to obtain a resolution of the issues Penrose raises, andperhaps the resolution entails no fundamental decoherence whatsoever, byproviding a well-defined description not only of how superpositions of 3-geometries behave, but also of what this implies for the physics within sucha superposition.To understand how this could be the case, let’s recall the details of thesingle-mode analysis. We followed the standard quantum mechanical pre-scription, and assumed that the system evolves according to a time evolutionoperator ∂/∂t that is the same for both components of our superposition.Penrose’s argument suggests that it is exactly this identification that is in-herently ambiguous, since by “∂/∂t” we do not mean the timelike Killingvector associated with any specific spacetime; how, then, can we justify theuse of a single operator for both components?A natural possibility is that once a coordinate choice is made that de-fines a physical time evolution, one can construct a quantum theory thatreflects the geometric structure in the (physical) Hamiltonian operator, whiletreating the associated time as a parameter that is no longer endowed withnontrivial geometric meaning. In our shell system, then, the position depen-dence and non-polynomial form of our Hamiltonian encodes the geometriccontent about spacetime, while using the coordinate time t as a parameterthat flows forward independent of the quantum state of the geometry, thusdefining an operator ∂/∂t that acts in the same way on any particular state.As long as we take into account how the Hamiltonian operates differentlyon states corresponding to different geometries, we could potentially escapePenrose’s argument by construction, since the time evolution has a unique855.1. Reflections and Resolutionsdefinition once a particular coordinate choice is made that fixes the meaningof the Hamiltonian.Regardless of what the true theoretical explanation is, there are exper-imental investigations already underway to test for signatures of gravity-induced intrinsic decoherence in various optical systems [72]. These exper-iments generalize the infamous Colella-Overhauser-Werner (COW) experi-ment [73]-[75], which measures the gravitational phase shift experienced byneutrons in an interferometer that has one path at a different gravitationalpotential than the other. The original COW experiment was the first ex-periment to include both quantum and gravitational effects, though it onlyrequired Newtonian gravity to describe it. The more recent experiments,such as the efforts to observe macroscopic superposition effects in optome-chanical systems [76], [77], intend to test the role of general relativity inquantum systems. There are many technical obstacles to overcome to mini-mize the effects of standard environmental decoherence, which obscures thedesired behaviour, but there is hope that these types of experiments willbear fruit within the next decade [78].One of the remaining questions from the analysis presented in Chapter 2is about the role of our coordinate choice on the properties of the inter-ference pattern: does our approximation scheme describe coherence in acoordinate-invariant manner, or are our conclusions about the coherence ofthe system dependent on the coordinates used? In Chapter 3, we attemptedto answer this question by re-performing the previous analysis, this timeusing an infinite family of similar coordinate systems. We interpreted thesefamily members as different networks of infalling observers, each with a dif-ferent asymptotic velocity. Upon doing so, we encountered some perplexingobstacles due to the nebulous connection between quantizations associatedwith different family members. The main problem was that the very con-cept of an “initial state” seemed to be coordinate-dependent, which directlyimpeded our ability to compare the interference patterns we obtained usingdifferent coordinate choices.Another issue worth mentioning is the identification of travel times forthe two interferometer paths (tλ++ + tλ−+ = tλ−− + tλ+−). We enforcedthis identification for our interferometric analysis in both Chapter 2 andChapter 3, but a difficulty arises when we consider the time transformationdefined by equation (3.123): if the travel times are equal in one coordinatesystem, the dependence of the transformation on the Hamiltonian H andthe shell position X makes it unlikely that the travel times will remain equalin another coordinate system. This difficulty therefore must also be over-come before any conclusive statements can be made about the coordinate865.1. Reflections and Resolutionsdependence of our interferometer.So, though we have found indications that the interferometric coherencemight be “observer-dependent,” and speculated about the implications ofthe difficulties we encountered trying to obtain a concrete determination ofthe possible observer-dependence, we are ultimately forced to accept that theinvestigation of coordinate dependence in our interferometer is inconclusive(at least tentatively).It is also possible that the only way to describe the coherence of a gen-eral relativistic interferometer in a coordinate-independent way is to in someway define the quantum evolution with respect to a physical clock that isembedded within the system. An initial realization of this idea was exploredby Page and Wooters [79], and later extended by Gambini, Porto and Pullin[80], [18] by incorporating a generalization of the relational interpretation ofquantum mechanics put forward by Rovelli [81], [82]. There have even beenexperimental proposals to test the validity of the relational interpretationof quantum evolution [83], though nothing concrete has yet been producedto this end. The main contention of these approaches is that one shouldformulate quantum time evolution using conditional probabilities, such thatinstead of determining the probability that an observable takes on a partic-ular value at a coordinate time “t”, one should determine the probabilitythat an observable takes on a particular value, given the condition that theclock observable takes on a particular value. It is this conditional probabil-ity, in these proposals, that yields a physically meaningful description of theevolution of quantum predictions.In Chapter 4, we explored the effects of adding such a “clock” degreeof freedom to our shell system, in the form of an internal oscillator. Uponanalysis of the combined system, we obtained a clear picture of some di-rect consequences of forming superpositions of geometries, though ratherthan attempting to formulate the evolution using conditional probabilities(which it is not clear we should necessarily do), we applied the same quanti-zation techniques used in the previous chapters. We first showed that timedilation induces an effective coupling between the external variables (thatdescribe the shell’s motion) and the internal (clock) variables, and identifieda regime where the reduced Hamiltonian of our system has the same formas the Hamiltonian used in a recent proposal of a “universal” decoherencemechanism due to gravitational time dilation [12]. For the situation whereonly the external variables are observed, one can follow the calculation pre-sented in [12], and trace out the clock variables. This leads to an interestingform of intrinsic decoherence called “third-party” decoherence [15], whichwe showed in Section 4.5 in the (gravity-free) limit of flat spacetime.875.1. Reflections and ResolutionsSince the proposed time dilation decoherence effect was present even inthe flat spacetime limit of our system, we concluded that the effect wasnot necessarily related to gravity. Though the specific application of theeffect mentioned in [12] was gravitational, with the role of the “third-party”played by the earth, the effect relies only on proper time differences. Ourshell system generally has tangential pressure that produces acceleration,and this alone leads to time dilation decoherence, even without gravity.As a variation on this theme of time dilation decoherence, we then con-sidered what happens when one takes into account the gravitational self-interaction of the shell. We found that the same calculation, with suitablygeneralized components, provided us with a demonstration that the timedilation decoherence still remains present when there is no tangential pres-sure whatsoever (or any other interactions), by virtue of the quantum con-sequences of self-gravitation. We interpreted this example of “third-partydecoherence without the third party” to be a manifestly gravitational effect.One can then connect this gravitational “self-decoherence” to the ideasbrought up by Penrose: conceptually, tracing out the clock variables effec-tively averages the contribution of the internal oscillation to the ADM massof the spacetime (as indicated by the appearance of E¯q in equation 4.48),while producing a nonunitary contribution to the (external) evolution of thereduced density matrix that is proportional to the square of the clock en-ergy uncertainty. Thus, the reduced dynamics associated with the externaldegree of freedom experiences an averaged geometry due to our ignorance ofthe internal evolution, and also decoheres due to the resulting uncertainty inthe spacetime geometry. 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Wehner, “Equivalence of wave-particleduality to entropic uncertainty,” Nature Comm. 5 5814 (2014).[89] These words were written by Fermat in his copy of the ancient textArithmetica by Diophantus, in reference to what is now known as “Fer-mat’s Last Theorem.”95Appendix AProbability CurrentConservationA.1 Standard Nonrelativistic QuantumMechanicsThroughout this thesis, we make use of the concept of a probability current.In nonrelativistic quantum mechanics, the probability current takes the formJ(x, t) =12im[ψ∗ψ′ − ψ (ψ∗)′], (A.1)where the prime indicates differentiation with respect to x (we’ll restrict ourattention to one dimension, for simplicity). In Dirac notation, this can alsobe expressed asJ(x) =12im[〈ψ|x〉 〈x |pˆ|ψ〉+ 〈x|ψ〉(〈x|pˆ†|ψ〉)∗], (A.2)from which we can deduce, provided the momentum operator pˆ is Hermi-tian 8, that the current J(x) is the expectation value of a simple operator:J(x, t) = 〈ψ|Jˆ(x)|ψ〉, (A.3)for the operatorJˆ(x) =12m(|x〉〈x|pˆ+ pˆ†|x〉〈x|). (A.4)Note that in these expressions, a hat denotes an operator, in contrast withthe majority of this thesis, in which a hat indicates that a quantity is eval-uated along the shell trajectory.8Note that in this appendix, we are denoting the momentum operator by pˆ, whichcorresponds to the classical momentum p. This should not be confused with our use of pin other parts of the thesis to denote pressure.96A.1. Standard Nonrelativistic Quantum MechanicsThe utility of the probability current (A.1) comes from the fact that itsatisfies the continuity equation,∂ρ(x, t)∂t+∂J(x, t)∂x= 0, (A.5)with ρ(x, t) = |ψ|2 being the probability density. This can be establishedstraightforwardly using the standard nonrelativistic Schro¨dinger equation,i∂∂t|ψ〉 = Hˆ|ψ〉 =(pˆ22m+ Vˆ)|ψ〉, (A.6)and its dual-space counterpart− i∂∂t〈ψ| = 〈ψ|Hˆ†, (A.7)where Hˆ is the Hamiltonian operator and Vˆ is the potential energy operator.Taking the time derivative of the probability density yields∂∂t|ψ|2 =∂ψ∗∂tψ + ψ∗∂ψ∂t= i〈ψ|Hˆ†|x〉〈x|ψ〉 − i〈ψ|x〉〈x|Hˆ|ψ〉= i(12m〈ψ|(pˆ†)2|x〉+ 〈ψ|Vˆ †|x〉)〈x|ψ〉−i〈ψ|x〉(12m〈x|pˆ2|ψ〉+ 〈x|Vˆ |ψ〉)= i(−12m(ψ∗)′′ + ψ∗V (x))ψ − iψ∗(−12mψ′′ + V (x)ψ)= i12m[ψ∗ψ′′ − (ψ∗)′′ψ]= −∂∂x(12im[ψ∗ψ′ − ψ(ψ∗)′]), (A.8)which demonstrates that the probability current defined by (A.1) indeedsolves the continuity equation (A.5).It will also be useful to consider an analogy between an approximategeneral relativistic probability current (derived in the next section) and theprobability current for a (nonrelativistic) charged scalar particle in (3 + 1)dimensional flat spacetime. Suppose we have this charged scalar particle inan electromagnetic field defined by a vector potential A(r, t) and a scalarpotential φ(r, t). In this case the effect of the potentials on our particle97A.1. Standard Nonrelativistic Quantum MechanicsHamiltonian is to replace the momentum p by the kinetic momentum P =p−qA(r, t), with q being the particle charge. The total Hamiltonian is thengiven byHˆ =12m(pˆ− qAˆ)2+ qφˆ =Pˆ22m+ qφˆ, (A.9)and a calculation analogous to (A.8) proceeds as∂∂t|ψ|2 =∂ψ∗∂tψ + ψ∗∂ψ∂t= i〈ψ|Hˆ†|r〉〈r|ψ〉 − i〈ψ|r〉〈r|Hˆ|ψ〉= i(12m〈ψ|(Pˆ2)†|r〉+ q〈ψ|φˆ†|r〉)〈r|ψ〉−i〈ψ|r〉(12m〈r|Pˆ2|ψ〉+ q〈r|φˆ|ψ〉)=12im(ψ∗〈r|Pˆ2|ψ〉 − 〈ψ|(Pˆ2)†|r〉ψ). (A.10)In the coordinate basis, we can represent the operator Pˆ by −i(∇ − iqA),and the calculation continues as∂∂t|ψ|2 = −12im(ψ∗[(∇− iqA)2 ψ]−[(∇+ iqA∗)2 ψ∗]ψ)= −∇ ·(12im[ψ∗∇ψ − ψ∇ (ψ∗)])+q2mψ∗ [∇ · (Aψ) + A · ∇ψ]+q2m[∇ · (A∗ψ∗) + A∗ · ∇ψ∗]ψ. (A.11)At this point it is common to simplify the expressions by working in aparticular gauge. We will be most interested in the Coulomb gauge, ∇·A =0, which for real A leads to∂∂t|ψ|2 = −∇ ·(12im[ψ∗∇ψ − ψ∇ (ψ∗)])+qm(ψ∗A · ∇ψ + ψA · ∇ψ∗)= −∇ ·[Js −qmA|ψ|2], (A.12)where we have denoted the standard (3+1)-dimensional nonrelativistic prob-ability current byJs =12im[ψ∗∇ψ − ψ∇ (ψ∗)] . (A.13)98A.2. General Systems Quadratic in MomentaThe total probability current operator is therefore given byJˆ(r) =12m(|r〉〈r|pˆ + pˆ†|r〉〈r|)−qm|r〉Aˆ〈r|=12m(|r〉〈r|Pˆ + Pˆ†|r〉〈r|), (A.14)and the associated continuity equation takes the form∂ρ(r, t)∂t+∇ · J(r, t) = 0. (A.15)A.2 General Systems Quadratic in MomentaNow suppose our system is effectively (1+1)-dimensional, and has a Hamil-tonian given byH = H0(x) +H1(x)p+H2(x)p2, (A.16)where {Hi(x)} are (for now) arbitrary functions of x. We will order theoperators in the quantum Hamiltonian as in Section 2.3.2, in an attempt toenforce Hermiticity. For this Hamiltonian, calculation of the time derivativeof the probability density becomes∂∂t|ψ|2 = i〈ψ|Hˆ†|x〉〈x|ψ〉 − i〈ψ|x〉〈x|Hˆ|ψ〉= i〈ψ|[(1/2)(Hˆ1pˆ+ pˆHˆ1) + pˆHˆ2pˆ]†|x〉ψ−iψ∗〈x|[(1/2)(Hˆ1pˆ+ pˆHˆ1) + pˆHˆ2pˆ]|ψ〉= iψ[i2∂∂x(H1ψ∗) +i2H1∂ψ∗∂x−∂∂x(H2∂ψ∗∂x)]−iψ∗[−i2H1∂ψ∂x−i2∂∂x(H1ψ)−∂∂x(H2∂ψ∂x)]= −∂∂x(H1|ψ|2)−∂∂x(1iH2[ψ∗ψ′ − ψ (ψ∗)′]). (A.17)Thus, our system possesses a probability current of the same form as (A.12),given byJ(x, t) = H1|ψ|2 +1iH2[ψ∗ψ′ − ψ (ψ∗)′], (A.18)from which we can deduce the probability current operatorJˆ(x) = |x〉Hˆ1〈x|+ Hˆ2|x〉〈x|pˆ+ pˆ†|x〉〈x|Hˆ2. (A.19)99A.2. General Systems Quadratic in MomentaWe can gain a different perspective on the result (A.19) by consideringthe Hamiltonian (A.16) to be the second-order truncation of a Taylor seriesexpansion of a general Hamiltonian in powers of p (provided such a thingexists for a given Hamiltonian). Indeed, this is the context in which (A.16)is used in this thesis. The coefficient functions that multiply powers of pcan then be interpreted asH0(x) = H|p=0 , H1(x) =(∂H∂p)∣∣∣∣p=0, H2(x) =12(∂2H∂p2)∣∣∣∣p=0. (A.20)This enables us to express the probability current asJ(x, t) = ψ∗(∂H∂p)0ψ+12ψ∗(∂2H∂p2)0(pˆψ) +12(pˆψ)∗(∂2H∂p2)0ψ, (A.21)where the subscript 0 indicates an evaluation at p = 0, and it is understoodthat the action of the operator pˆ on the coordinate-space wavefunction ψis given by pˆψ = −i∂ψ/∂x. Since the general p-derivative of the truncatedHamiltonian (A.16) is(∂H∂p)=(∂H∂p)∣∣∣∣p=0+(∂2H∂p2)∣∣∣∣p=0p, (A.22)we can re-express the probability current operator (A.19) asJˆ(x) = |x〉(∂̂H∂p)〈x|, (A.23)as long as we make sure to symmetrize the term linear in momentum, tomaintain consistency with (A.21). Keeping in mind Hamilton’s equationfor the time evolution of x (i.e. dx/dt = ∂H/∂p), we arrive at an intuitiveexpression for the probability current operator:Jˆ(x) = |x〉vˆ〈x|, (A.24)with vˆ being an operator version of the velocity v = dx/dt. Taking theexpectation value yields the corresponding probability current,J(x, t) = ψ∗vˆψ, (A.25)with the action of the operator vˆ understood to be in the coordinate basis.This has the same form as the current for a charged scalar particle (A.14),since the kinetic momentum is connected to the particle velocity via P =mv.100Appendix BMulti-mode InterferometryB.1 General Relativistic Wave-packetsHere we consider the properties of our self-gravitating interferometer withmulti-mode input states, to explore whether or not having a superpositionof ADM masses could affect the conclusions of the single-mode analysis. Wewill form localized wave-packets as initial states by taking a Gaussian distri-bution of energies as weighting factors of the previously introduced ingoingWKB modes. We emphasize that since we are superposing WKB modeswith different energies, we are actually superposing states that correspondto spacetime geometries with different ADM masses (recall that the classicalvalue of our Hamiltonian was found to be equal to the ADM mass of theassociated geometry). We are not, then, merely doing quantum mechan-ics on a fixed curved spacetime; even in the single-mode analysis we wereworking with a superposition of geometries, because each path through theinterferometer (classically) defined a different geometry. For single-mode in-put states, however, the corresponding spacetimes had the same ADM mass,and therefore beyond the outer reflector of the interferometer the spacetimeswe superposed were equivalent. With multi-mode input states, this will nolonger be the case. The purpose of this appendix is to identify possible waysthat evidence of intrinsic gravitational decoherence could manifest itself inthe behaviour of the multi-mode system.Suppose that our wave-packets are much narrower than the interferom-eter arm lengths - this allows us to treat each element of the interferometerseparately, mode by mode, as we did in the single-mode analysis. The initialwave-packet will first encounter the splitter, at which point each mode inthe wave-packet will transform into a reflected mode with a factor R← anda transmitted mode with a factor T←. We remind the reader that subscriptsare used here because the reflection/transmission coefficients depend on thedirection each mode of the wave-packet encounters the splitter from.The split wave-packets will then perfectly reflect off of the outer/innerreflectors, and travel back towards one another at the beam-splitter. Uponrecombination there will be splitting of each set of modes coming from each101B.1. General Relativistic Wave-packetsdirection of the splitter, which produces two outputs (one going in eachdirection from the splitter) that are themselves composed of two parts.It is clear that we will not observe interference in the multi-mode inter-ferometer unless the wave-packets travelling on each of the interferometerarms reach the splitter for recombination at approximately the same time;there must be sufficient overlap of the parts of the output states associatedwith the two paths through the interferometer in order for cancellation ofeither of the final outputs to be possible. There is some parameter freedomleft in this system, so let us attempt to make use of this freedom to makethe (classical) travel times along the interferometer arms be the same. Fromequation (2.56), we find that the time can be found as a function of H andX:t =∫dX(dtdX)=∫dX(X˙−1)=∫dX√2HXf±hf√h2 − Mˆ2f , (B.1)with h = H − Mˆ2/2X and f = 1− 2H/X. The first term is independent ofwhether the shell is outgoing or ingoing, and can be integrated exactly as∫dX√2HXf= 4H(√X2H− arctanh√X2H)= 4H√X2H−12ln∣∣∣∣∣∣1 +√X2H1−√X2H∣∣∣∣∣∣ . (B.2)Let us denote the term (B.2) by t0. The second term in (B.1), which gives acontribution that we will denote by t±, depends both on whether the shellis outgoing or ingoing as well as on the mass function Mˆ . For weak-fieldregions of constant Mˆ , we can expand the integrand in inverse powers of√X and integrate term-by-term to obtain the asymptotic contributiont± ∼ ±1√H2 − Mˆ2HX +(4H4 + Mˆ4 − 6Mˆ2H2)2(H2 − Mˆ2) ln(X2H)+H(9Mˆ6 − 60H2Mˆ4 + 80H2Mˆ4 − 32H6)8X(H2 − Mˆ2)2 . (B.3)102B.1. General Relativistic Wave-packetsThe total travel time for the path through the interferometer that ini-tially reflects from the beam-splitter is given by t++ + t−+, and the time forthe path that initially transmits is t−− + t+−. We then havet++ + t−+ = t+(X+)− t+(Xδ+) + t−(Xδ+)− t−(X+)= 2t+(X+)− 2t+(Xδ+), (B.4)as well ast−− + t+− = t−(X−)− t−(Xδ−) + t+(Xδ−)− t+(X−)= 2t+(X−)− 2t+(Xδ−), (B.5)with Xδ± = lim→0 (Xδ ± ). The difference in travel times can then beexpressed as∆t =12((t++ + t−+)− (t−− + t+−))= t+(X+)− t+(Xδ+)− t+(X−) + t+(Xδ−). (B.6)In the flat spacetime, nonrelativistic limit, we haveX˙±±˜ =P±±˜M±˜= ±√2M±˜(E −M±˜)M±˜, (B.7)and so equating the travel times (∆t = 0) gives us the following conditionfor the lengths of the interferometer arms:L+√M+E −M+= L−√M−E −M−. (B.8)Here, the interferometer arm lengths are defined such that L± ≡ ±(X± −Xδ), and the use of ± and ±˜ above is to remind us that the signs can bechosen independently.The arm length condition (B.8) (or its relativistic generalization) can beapplied for a single mode, but for a wave-packet all we can do is apply thecondition for the peak energy (or expectation value, perhaps), and try tokeep the energy variance small enough that the deviations from equal traveltimes for different energy modes will be negligible. We also want to keep theenergy variance small so that the travel time will be less than the coherencetime for the wave-packet, to make sure that dispersion will not significantlyaffect the interference pattern.103B.2. Localize, Normalize, PropagateB.2 Localize, Normalize, PropagateWe imagine an initial state for the shell to be a superposition of ingoingWKB modes, with a Gaussian distribution in energy:Ψ0 = N∫dE eiX˜(E−E0)e−(E−E0)24σ2E ψ−+. (B.9)Here the parameter X˜ has been used to control the peak location in X-space.Before propagating the initial state through the interferometer, we shoulddetermine the relationship between the peak location parameter X˜ and thetrue peak location in X-space. Since the relevant mode function is given byψ−+ = (E,−+ )−1/2 ei∫dX P−+ , (B.10)the integral in (B.9) is no longer just a Gaussian. However, if we suppose thatthe Gaussian prefactor to the mode function varies on a broader energy scalethan the mode function itself, we can approximate the integral by evaluatingthe square root term at the peak energy E0 and expanding the WKB phaseto linear order in (E−E0). The resulting Gaussian can be easily integrated,and in flat spacetime one obtainsΨ0 = 2σE√piM+P+N˜e− (X−X0)24σ2X , (B.11)with X-variance σ2X and peak location X0 given byσ2X =√2(E0 −M+)M+14σ2E, (B.12)X0 =√2(E0 −M+)M+X˜ −i(E0 −M+)σ2E. (B.13)Note that the “physical” peak location in X-space is given by the real partof X0, and we have absorbed irrelevant factors into N˜ coming from constantsof integration. If we normalize the state such that 〈Ψ0 |Ψ0〉 = 1, the factorN˜ is found to be given by∣∣∣N˜∣∣∣−2=√2piσXe√2M+(E0−M+)3/2σ2E . (B.14)The general relativistic initial state can be similarly analyzed. It simpli-fies matters to further approximate the WKB phase by expanding in inverse104B.2. Localize, Normalize, Propagatepowers of√X. Performing the X-integration term by term then leads toterms that diverge as X →∞, so at this point we can safely neglect inversepowers of√X when we evaluate the Gaussian energy integral. The fully in-tegrated initial state then has an exponential with a bi-quadratic structurein√X,9 so we can complete the square to obtain the peak locationX˜0 ∼2(E20 −M2+)E01 +√√√√1 +Xˆ2√E20 −M2+2, (B.15)withXˆ ≡ X˜ − lnX∂∂E0(E20 −M2+/2)E0√E20 −M2+ . (B.16)In the flat spacetime limit, this reduces to the real part of the previousresult:X0 =√E20 −M2+X˜E0. (B.17)Let us use our initial wave-packet to calculate the initial probabilitycurrent. For simplicity, we will work in the flat spacetime, nonrelativis-tic limit. Using the standard expression (2.91), as well as the definitionX˜0 ≡ X˜√2(E0 −M+)/M+, one can express the current associated with theapproximate X-Gaussian (B.11) asJ0 = −σE√2pi(2(E0 −M+)M+)1/4e−(X−X˜0)22σ2X . (B.18)Actually, we can simplify things more than that for arbitrary peak widthsby taking advantage of the fact that we only care about what happens asX → Xδ: if we take the lower integration boundary for X-integrals to beX = Xδ, then as X ↘ Xδ, we haveΨ0 = N∫dEeiX˜(E−E0)e−(E−E0)24σ2E√∣∣∣∂E∂P−+∣∣∣. (B.19)9Actually, there is a term logarithmic in X, but for the purpose of finding an approx-imate peak location we ignore the variation of this logarithm compared to√X and X,since we work in the weak-field (X →∞) limit.105B.2. Localize, Normalize, PropagateLet us change variables to z = (1/2σE)(E − E0 − 2iσ2EX˜). We can thenexpress the initial state asΨ0 = 2NσE√pie−σ2EX˜2e14d2dz2[∣∣∣∣∂E∂P−+∣∣∣∣−1/2], (B.20)evaluated at z = 0. We can similarly express the initial derivative (withrespect to X) asΨ′0 = 2iNσEe−σ2EX˜2∫dzP−+∣∣∣∣∂E∂P−+∣∣∣∣−1/2e−z2= iN0e14d2dz2[P−+∣∣∣∣∂E∂P−+∣∣∣∣−1/2], (B.21)for N0 ≡ 2NσE√pie−σ2EX˜2.We are then led to the probability current J0 as X ↘ Xδ given byJ0 =|N0|2M+<(N∗+N˜+), (B.22)where < indicates the real part and we have made the definitionsN+ ≡ e14d2dz2∣∣∣∣∂E∂P−+∣∣∣∣−1/2(B.23)andN˜+ ≡ e14d2dz2(P−+∣∣∣∣∂E∂P−+∣∣∣∣−1/2). (B.24)The initial current expression (B.22) is valid for arbitrary wave-packet widths,and can be generalized easily to include gravity. If our wave-packet is highly-peaked about a particular energy E0, then the probability current takes thesimpler formJ00 = −|N0|2. (B.25)Now that we have parametric control over the localization of the initialstate, we can propagate this state mode by mode through the interferometer.Each mode transforms in the same way as described in Chapter 2, and soone finds the final output states(Ψ(v)+Ψ(v)−)=∫dE NE(ψ(v)+ψ(v)−), (B.26)106B.2. Localize, Normalize, Propagatewith the definitionsNE = Nei(E−E0)X˜e−(E−E0)24σ2E , (B.27)ψ(v)+ = ψ++(R←eiΦ++R→eiΦ−+R←+ T←eiΦ−−R←eiΦ+−T→), (B.28)ψ(v)− = ψ−−(R←eiΦ++R→eiΦ−+T←+ T←eiΦ−−R←eiΦ+−R→). (B.29)Here, the mode functions are evaluated at Xδ.How do the probability currents transform at the beam-splitter? Thereflected and transmitted probability currents for the initial split can becalculated in the same way as the initial current, and the reflection andtransmission coefficients can be written as∣∣∣∣∣J (i)+J0∣∣∣∣∣=<(N (i)∗+ N˜+(i))<(N∗+N˜+) (B.30)and∣∣∣∣∣J (i)−J0∣∣∣∣∣=M+M−<(N (i)∗− N˜−(i))<(N∗+N˜+) , (B.31)using the definitionsN (i)+ ≡ e14d2dz2(R←∣∣∣∣∂E∂P++∣∣∣∣−1/2)(B.32)N˜+(i)≡ e14d2dz2(R←P++∣∣∣∣∂E∂P++∣∣∣∣−1/2)(B.33)N (i)− ≡ e14d2dz2(T←∣∣∣∣∂E∂P−−∣∣∣∣−1/2)(B.34)N˜−(i)≡ e14d2dz2(T←P−−∣∣∣∣∂E∂P−−∣∣∣∣−1/2). (B.35)In the limit that our initial wave-packet is highly-peaked about an energyE0, the reflection and transmission coefficients take the simpler forms∣∣∣∣∣J (i)+J0∣∣∣∣∣0= R2←,∣∣∣∣∣J (i)−J0∣∣∣∣∣0= T 2←. (B.36)107B.3. Fringe Visibility and Path RetrodictionIt is then easy to demonstrate that R2← + T2← = 1.For arbitrary peak widths, things are not so simple. The time depen-dence of the probability currents implies that the output currents do notbalance at each point in time, as they did in the stationary (single-mode)case. Thus, the continuity equation does not imply probability current con-servation, and this makes our single-mode analysis inappropriate for themulti-mode system.B.3 Fringe Visibility and Path RetrodictionUnlike the probability current, the total probability is still conserved (inprinciple), even if our approximations break down in some regions of theparameter space and indicate otherwise. What will prove useful, then, in thistime-dependent setting, are the “inner” and “outer” output probabilities,which are defined in the obvious way: the inner output probability (denotedby P−) is the probability of finding the shell on the inner side of the splitter(X− < X < Xδ) after recombination, and the outer output probability(denoted by P+) is the probability of finding the shell on the outer side ofthe splitter (Xδ < X < X+). To calculate P− and P+, we can use theoutput states (B.26), except rather than evaluating the mode functions inthe outputs at Xδ, we will need to integrate the magnitude-squared outputsin their respective regions:P± = ±∫ X±XδdX∣∣∣Ψ(v)±∣∣∣2. (B.37)One must be careful, of course, to make sure enough time has elapsed af-ter recombination so that all of the wave that will eventually transmit hastransmitted, and similarly for the final reflected part.How can we extract information about the coherence of the multi-modesystem from these output probabilities? A simple approach is to use ananalogy with a Mach-Zehnder interferometer: as we vary a system parameter(outer arm length, for instance), there is a phase shift induced betweenthe different interferometric paths, and this affects how the waves interfereupon recombination [14]. Let us call this (generalized) phase shift δΦ. Wecan then quantify the coherence in the system with a quantity V, which isdefined as the amplitude of the oscillation of P+ (or, equivalently, P−) asδΦ is varied. The quantity V is referred to as the “fringe visibility” (or just“visibility”) of the interference pattern. Since the inner and outer outputprobabilities P± necessarily take values between 0 and 1, the visibility V is108B.3. Fringe Visibility and Path Retrodictionbounded from below by 0 (no visibility of the interference fringes) and fromabove by 1 (full visibility of the interference fringes).In the version of the Mach-Zehnder described in [14], an internal degreeof freedom is added to the particle used in the interferometer. Just as inChapter 4, the evolution of the internal degree of freedom is defined withrespect to the proper time of the particle, and therefore serves as a localclock. Without the internal degree of freedom, the standard Mach-Zehnderhas maximal visibility (V = 1), but after the internal degree of freedom isadded the visibility is reduced due to the proper time evolution accumulatinginformation about which path was taken through the interferometer. As theproper time evolutions of different interferometer paths become more andmore dissimilar, the visibility goes down; however, another quantity, calledthe “distinguishability” and denoted by D, goes up. The distinguishabilityquantifies the “which-way” information known about a system, in the senseof providing a measure of the probability of correctly predicting which pathwas taken through the interferometer (i.e. path retrodiction). Hence, thereis a natural trade-off between path retrodictability and visibility of the in-terference pattern, and this is generally considered to be a manifestation ofwave-particle duality.For the Mach-Zehnder with an added internal degree of freedom, forinstance, one can demonstrate that a duality relation holds, of the formV2 + D2 = 1. Duality relations like this one have been around at least asfar back as the 1970s [84], and there have been various extensions, such asbroadening the applicability to include asymmetric beam-splitting [85]. Still,even though some work has gone into quantifying coherence in a unified way(for a recent analysis, see [86]), there has been much confusion on the truemeaning of the duality relations, and their connections to different formsof uncertainty principles. In particular, it is often claimed that the dualitybetween visibility and distinguishability is conceptually distinct from theHeisenberg uncertainty principle (HUP), since several derivations of suchduality relations existed in the literature that did not assume that the HUPholds.Very recently, however, it was demonstrated that there was indeed acommon conceptual origin of both the duality relations and the HUP [87],[88]. It was discovered that one can derive the various duality relations, aswell the HUP, from a generalized uncertainty relation defined in terms ofentropies. This so-called “entropic uncertainty” relation provides the con-ceptual link between the duality relations and the HUP that was previouslyabsent in the literature.A possible implication of this link is that it could potentially fill in the109B.3. Fringe Visibility and Path Retrodictiongap in Penrose’s argument between the time uncertainty induced by super-posing geometries and the resulting decoherence, since it is currently unclearexactly how they are connected. An in-depth analysis of this possibility isbeyond the scope of this appendix, and is the subject of future work.110Appendix CThe Exact WKB PhaseTo facilitate the approximation schemes used in this thesis, it was neces-sary to use several asymptotic forms of the reduced phase space canonicalmomentum. The exact expression,P±±˜ = −√2HX −X ln(1−√2HX+β±±˜X), (C.1)with the definitionsβ±±˜ =h±˜ ∓√h2±˜−M2±˜f1 +√2HX, (C.2)h±˜ = H −M2±˜2X , and f = 1 − 2H/X, is often difficult to work with. Forinstance, the WKB state we make use of,ψ±±˜ =eiI±±˜√∣∣∣∂H∂P±±˜∣∣∣, (C.3)has a phase (the “WKB phase”) that involves the integral of the reducedcanonical momentum:I±±˜ =∫dX P±±˜. (C.4)We remind the reader that the first set ± indicates outgoing (for +) oringoing (for −), whereas the second set ±˜ indicates which side of the splitterlocation Xδ one is considering.Rather than finding asymptotic expressions for the WKB phase, let ussee how far we can get towards a more explicit exact expression. Afterintegrating the square root term in (C.1) and applying some simple algebraic111Appendix C. The Exact WKB Phasemanipulations we can write the WKB phase (C.4) asI±±˜ = −23X√2HX +∫dX X ln(1 +√2HX)−∫dX X ln1−2HX+(h±˜ ∓√h2±˜−M2±˜f)X . (C.5)One can notice that the first two terms on the right hand side of this equationare independent of the shell mass, and represent non-perturbative contribu-tions to the WKB phase. The second of these two terms can be straightfor-wardly integrated to obtain∫dX X ln(1 +√2HX)=(12X2 − 2H2)ln(√X +√2H)−12XH+(H +16X)√2HX −14X2 lnX, (C.6)but the last term in expression (C.5) requires a bit more work. First wewill separate out some factors of X in the argument of the logarithm, toget rid of inverse powers of X; then we can separate the logarithms andintegrate the part that came from the factors of X. Suppressing indices forthe moment, the remaining part involves the integral∫dX X ln(X2f +X(h∓√h2 −M2f)). (C.7)If we integrate by parts, this becomes12X2 lnF −12∫dXX2 dFdXF, (C.8)for a function F defined byF = X2f +X(h∓√h2 −M2f). (C.9)Putting all the pieces together, we are led to the following expression for theWKB phase:I =√2HX(H −X2)−12XH −14X2 +(12X2 − 2H2)ln(√X +√2H)+12X2 ln(X3/2F)+12∫dXX2 dFdXF. (C.10)112Appendix C. The Exact WKB PhaseThe terms are arranged such that the first line of (C.10) contributes to thephase in the same way as a massless dust shell, whereas the second linecompletely encodes the information about the special equation of state wemake use of for the interferometry presented in several chapters of this thesis.In the flat spacetime limit, the function F simplifies toF → X2 +X (H − P ) , (C.11)with P = ±√H2 −M2, and the last unevaluated integral in the WKB phase(C.10) is found to be∫dXX2 dFdXF→ X2 + (H − P ) [(H − P ) ln (X +H − P )−X] . (C.12)Of course, if we only care about the flat spacetime limit, we can startfrom the momentum P = ±√H2 −M2, which immediately yields the phaseI = ±√H2 −M2X, or, for nonrelativistic speeds, I ≈ ±√2M(H −M)X.The utility of (C.12) presents itself when we search for approximations tothe WKB phase that are more accurate than the usual perturbative ap-proximations we were forced to use in this thesis for practical reasons; inother words, using our previous perturbation approach to correct (C.12) andinserting it into (C.10) provides us with a better approximation to the ex-act behaviour than would be obtained by exclusively using the perturbationapproach.If we really wish to be accurate, we can go one step further and performthe (somewhat challenging) integral (C.12) exactly. This is possible, andhas been done (by the author), but will not be presented here due to thelengthy and unenlightening form of the result.1010“I have discovered a truly remarkable proof of this proposition that this margin is toosmall to contain.” [89]113


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