Towards Quantum Field Theory in Curved Spacetime for an Arbitrary Observer
TTowards Quantum Field Theory in Curved Spacetimefor an Arbitrary Observer
Hui Yao [email protected]
DAMTP, University of Cambridge
Abstract
We propose a new framework of quantum field theory for an arbitrary observer incurved spacetime, defined in the spacetime region in which each point can both receivea signal from and send a signal to the observer. Multiple motivations for this pro-posal are discussed. We argue that radar time should be applied to slice the observer’sspacetime region into his simultaneity surfaces. In the case where each such surface is aCauchy surface, we construct a unitary dynamics which evolves a given quantum stateat a time for the observer to a quantum state at a later time. We speculate on possibleloss of information in the more general cases and point out future directions of our work.
Keywords : quantum field theory in curved spacetime, observer-dependence, simul-taneity, notion of particles, dynamics of quantum states, information loss.
Contents a r X i v : . [ g r- q c ] J u l Motivation
The Universe is ultimately observer-participatory, each part of which is communicating withthe others and never at rest. Existence are not simply “things just out there”, but realityfor an observer is his dynamical construction. Both quantum theory and relativity, the twopillars of our modern conceptual understanding of Nature, taught us so, although the greatunion of the two remains a mystery still. We shall hence aim at a further understanding ofthe observer’s quantum description of matter in a general relativistic background. We firstdiscuss the multiple motivations of this paper. • Quantum field theory in curved spacetime is the necessary first step towards a conceptualunderstanding of a quantum theory of gravity. Even though such a framework is ultimatelyincomplete, we expect some features of the theory to remain. We shall treat the quantumfield like a “test field” with the background spacetime completely classical and unaffectedby the matter field. • We view physics as a theory of an arbitrary observer’s dynamical description of his system.Therefore any physical theory should be formulated explicitly in terms of an observer’s ownphysical quantities at a fundamental level. There have been great conceptual advance-ments in understanding the observer-dependent aspects of quantum field theory [1-6].Rather than treating observer-dependence as an emergent feature, we now push these de-velopments further by taking observer-dependence to be a cornerstone of our formulationof quantum field theory in curved spacetime. • Working at a semi-classical level, we shall understand an observer as a history of spacetimeevents, in other words, a timelike curve in a classical curved spacetime. A quantum stateis a quantum description of the physical system for the observer at some proper time ofhis. We consider dynamics as the updating of his description of the system rather thanas the “changing of things in themselves”. Above all, we would like to construct a theoryspecifying a one-parameter family of quantum states along a timelike worldline. • Dynamics is the specification of a one-parameter family of physical states, whether theybe points in classical phase spaces or vectors in Hilbert spaces or some other mathemat-ical representation. Therefore intrinsic to dynamics is a notion of “time”. A notion oftime is usually only available in special cases. One may restrain oneself to the notion ofasymptotic past/future only, but this does not define a parameter of physical time. Thereis a “natural” choice of time if the spacetime manifold admits a special symmetry, like theKilling time for stationary spacetimes or the usual cosmic time of Friedmann-Robertson-Walker (FRW) cosmology. However, the last two choices are rather mathematical andonly apply to special cases. We would instead like a single physically-motivated definitionapplicable to all situations. The only possibility of such a definition is the proper timeon an observer’s clock. Such a choice in fact forces one to adopt an observer-dependentdescription of physics, which is precisely what we have been aiming for. • In this paper we shall take a Hilbert space approach in that we consider Hilbert spaces tobe the most fundamental objects mathematically representing our quantum description ofphysical systems. Quantum states are vectors or more generally density matrices in thesespaces. Such a mathematical framework is especially suitable for discussing quantum stateevolution, where a two-parameter mapping U ( t , t ) relates a quantum state in a Hilbert1pace at some time t for an observer to another state in a possibly different Hilbert spaceat time t . The Hilbert space approach is also particularly suitable for discussing anypossible loss of information in the form of a pure state evolving to a mixed density matrix. • There are two important conceptual lessons that we should draw from Hawking’s semi-classical analysis [7, 8] of black hole information loss. Firstly, Hawking has insisted thatany quantum state for the observer outside the black hole must be physically meaningfulfor him: to obtain the correct description of the quantum system according to the observer,we must trace out all degrees of freedom that he cannot causally access. As we shall see,our quantum theory is precisely defined over the spacetime region causally connected tothe given observer. Secondly, that there is a loss of information when black hole evaporatesindicates that the loss of information is rooted in the “evolution” of a horizon rather thansimply the presence of a horizon. To make precise sense of evolution one again needs anotion of time, which we have chosen to be the proper time of an observer. • We shall speculate that there will be a non-unitary evolution and hence information lossfor an observer, precisely when his surface of simultaneity evolves from a Cauchy surfaceto a non-Cauchy surface for the spacetime region to which the observer has causal access.Instead of resolving information loss as a paradox, we propose to forcefully carry Hawk-ing’s argument [7, 8] through and speculate that the possibility of information loss is afundamental feature of quantum field theory in curved spacetime rather than special toblack hole evaporation. • To obtain physical quantities from the point of view of an observer, one normally has tostudy the response of a model particle detector following the observer’s worldline. Wewould instead like to have a fundamental theory in which the particle content at anytime can be simply read off from the mathematical representation of the physical state.Furthermore a quantum state, which completely encodes all information about the possiblemeasurement outcomes and their probabilities of occurrence, tells much more than simplythe particle spectrum, in particular whether the quantum state is pure or mixed. • Wald [9] has forcefully argued that there is no natural choice of Fock space of particles inquantum field theory in a most general spacetime, and that this is in analogy to choosing acoordinate system on a manifold in general relativity which cannot be physical. But for agiven observer, as we shall show, there indeed exists a natural notion of “particles”. Therehave been serious difficulties in attempting to define a notion of particles in a most generalspacetime, but this does not imply that the idea of “particle” itself is not fundamental, aslong as one takes an observer-dependent viewpoint of quantum field theory. • A notion of particles is usually only available in special cases such as if the spacetime admitsa special symmetry or special asymptotic behaviours, or if the physical situation admitsan adiabatic approximation. However generalisation is essential, as elements particular tospecial cases can obscure what the fundamental features are. The formulation which wewill present can be applied rather generally to a wide class of observers without requiringany special symmetries or asymptotic behaviours of the spacetime. • Unlike in e.g. [10], here we shall make no fundamental distinction between particle cre-ation due to the motion of the observer and that due purely to gravitational fields. Weview spacetime as a geometric and causal background in which the observer’s worldline is2efined. All observers are regarded as completely equivalent at the fundamental level ofquantum field theory in curved spacetime. • Ashtekar and Magnon [1] have constructed a one-parameter family of Fock spaces intheir formulation of quantum field theory in curved spacetime. The authors found theirconstruction to depend on a choice of timelike vector field and hence of the correspondingintegral curves. They were therefore forced to conclude that their theory depends on afield of observers. Rather than be led to a conclusion of observer-dependence, we havetaken as our conceptual starting point the construction of a quantum field theory withobserver-dependence. • One might interpret Ashtekar and Magnon’s construction [1], which depends on a congru-ence of timelike curves, to be for a family of observers. However, we insist any physicaltheory should be formulated for an arbitrary single observer. Mathematically, as we shallshow, the construction of the family of Fock spaces depends only on a choice of scalarfunction t and a corresponding foliation. Physically, if one starts with a single observer, ina most general situation it is far from clear how to choose a family which he is a memberof. Furthermore, specifying an arbitrary family of observers has no direct physical inter-pretation as one can never set up an experiment with an uncountably many number ofdetectors which trace out a congruence of curves covering the entire spacetime. Finally,one expects that different observers within one family, even if there is such a preferredgrouping, would differ in their description of a physical system. One therefore would likethis difference to be naturally accounted for and built into the fundamental theory. • This work in some sense parallels Einstein’s construction of special relativity. Just asthe principle of relativity is the guiding principle of special relativity, it is our conceptualstarting point that the same laws of quantum field theory should apply to any arbitraryobserver, although the observers’ dynamical quantum descriptions may differ. Secondly,just as Einstein recognised the inseparable connection between time and the signal velocity,we shall apply radar time, which is operationally defined using light signal communication,to formulating quantum field theory for an arbitrary observer in a general spacetime.The plan of the paper is given as follows.In section 2, we shall construct a one-parameter family of Fock spaces based on theformalism of Ashtekar and Magnon [1]. We shall show that the formalism requires a choiceof scalar function t .In section 3, we shall apply radar time to operationally define this function t for eachpoint which a given observer can both send a signal to and receive a signal from. We regarddefining quantum field theory in the spacetime region causally connected to a given observeras an axiom of our framework. In section 3.1 we discuss the importance of Cauchy surfacesin the formulation of a unitary theory and the preservation of information.In section 4, we construct the dynamics of quantum states in our theory. We definequantum evolution in terms of a two-parameter mapping from one Fock space to another,each associated with a time for an observer, and we show this mapping satisfies certainnecessary physical conditions.Finally in section 5, we summarise our main results and consider possible directions inwhich our work might be developed.We shall use natural units throughout this paper. The sign convention in general rela-tivity is the same as that of [11], in particular, η ab = ( − , +1 , +1 , +1).3 The Hilbert Spaces of the Free Real Scalar Field
In this section, we shall construct a one-parameter family of Fock spaces of the free realscalar field based on the work of Ashtekar and Magnon [1]. Definitions and notations areintroduced which will be used throughout the subsequent sections. We shall summarise theformalism in a form most simple and ready for our purpose of the Hilbert space approach asmotivated in section 1. In particular, we shall not start from the *-algebra of abstract fieldoperators of [1].Let V be the vector space of all well-behaved real-valued solutions of Klein-Gordonequation ( ∇ a ∇ a − m ) φ = 0 (2.1)on a given globally hyperbolic spacetime. Let Σ be an arbitrary spacelike Cauchy surfaceof the given spacetime, with arbitrary coordinates { x i } and unit future-directed normal n a .The induced metric on Σ is h ab . A symplectic form ω on V is defined as ω ( φ, ψ ) = (cid:90) Σ ( ψ ∇ a φ − φ ∇ a ψ ) n a √ h d x. (2.2)Let J be a complex structure on the real vector space V , i.e. an automorphism on V whichsatisfies J = − . J endows V with the structure of a complex vector space, which we shalldenote as V J . We shall use | (cid:105) to distinguish an element | φ (cid:105) in V J from its counterpart φ in V . Hence in our notation we have, for example, i | φ (cid:105) = | J φ (cid:105) . An inner-product (cid:104) | (cid:105) can bedefined on V J as (cid:104) φ | ψ (cid:105) = ω ( φ, J ψ ) + i ω ( φ, ψ ) . (2.3)This indeed defines an inner-product if and only if the complex structure J is compatiblewith the symplectic form ω , i.e. ω ( φ, ψ ) = ω ( J φ, J ψ ) , ∀ φ, ψ ∈ V (2.4) ω ( φ, J φ ) > ∀ φ ∈ V \{ } . (2.5)The Cauchy completion of the complex inner-product space ( V J , (cid:104) | (cid:105) ) is a Hilbert spacewhich we shall denote as H J .We now define the n -particle space to be the Hilbert space ⊗ ns H J , i.e. the n th-ranksymmetric tensor over H J . The space of all quantum states is then the symmetric Fockspace F s ( H J ) based on the Hilbert space H J : F s ( H J ) = ⊕ ∞ n =0 ⊗ ns H J . (2.6)Creation and annihilation operators are defined as mappings on this F s ( H J ) in the usualway; see e.g. [9]. We shall denote the creation and annihilation operators associated with | φ (cid:105) as C µ ( φ ), where µ = +1 is for creation and µ = − The use of the index Ashtekar and Magnon [1] have assumed that all solutions in V are smooth and induce, on any spacelikeCauchy surface, initial data sets of compact support. The assumption about compact support is used toensure convergence of various integrals and to discard various surface terms when integrating by parts. For typographical convenience, we did not choose the notation of C µ ( | φ (cid:105) ). Although it should beunderstood that C µ ( φ ) depends on | φ (cid:105) ∈ H J rather than on φ ∈ V . µ will become clear in section 4. One can show from their definitions these opera-tors satisfy the following properties: (i) ( C µ ( φ ) ) † = C − µ ( φ ); (ii) each creation/annihilationoperator is complex-linear/anti-linear in its argument; and (iii)[ C + ( φ ) , C + ( ψ )] = [ C − ( φ ) , C − ( ψ )] = 0 , [ C − ( φ ) , C + ( ψ )] = (cid:104) φ | ψ (cid:105) . (2.7)To summarise, we have constructed a Fock space for each choice of a complex structure J on V which is compatible with the symplectic form ω in the sense of (2.4) and (2.5). Tofinish the construction, it remains to specify a one-parameter family of complex structures J t satisfying (2.4) and (2.5).We now summarise, in a slightly different form, the construction of J t due to Ashtekarand Magnon [1]. Let t be a scalar function on the spacetime such that each constant t hypersurface Σ t is a spacelike Cauchy surface and the set { Σ t } foliates the given spacetime.Let n a = N − (cid:0) ∂∂t (cid:1) a be the unit future-directed normal to Σ t , where N = (cid:113) − g ab (cid:0) ∂∂t (cid:1) a (cid:0) ∂∂t (cid:1) b . (2.8)To construct J t , we introduce a t -dependent Hamiltonian operator H t on H t defined by H t | φ (cid:105) t := − i | ˜ H t φ (cid:105) t = −| J t ˜ H t φ (cid:105) t (2.9)where ˜ H t is a real-linear operator on V which is defined as follows. If φ has on Σ t the Cauchydata φ | t = f , n a ∇ a φ | t = g , then ˜ H t φ ∈ V is the solution with Cauchy data ˜ H t φ | t = N g , n a ∇ a ( ˜ H t φ ) | t = − N − Θ f on the same Cauchy surface, whereΘ := − N h ab D a D b − N h ab ( D a N ) D b + m N ; (2.10) h ab is the induced metric on Σ t ; and D a is the covariant derivative on (Σ t , h ab ). It is easy tosee that H t is complex-linear on H t if and only if ˜ H t is a real-linear operator on V and ˜ H t commutes with J t , i.e. [ ˜ H t , J t ] = 0.It is proven [1] that there exists a unique complex structure J t satisfying (2.4) and (2.5)such that (cid:104) φ | H t | φ (cid:105) t is real for any | φ (cid:105) t ∈ H t . If φ is the solution with Cauchy data ( f, g ) onΣ t as defined before, then J t φ is the solution with Cauchy data (Θ − N g, − N − Θ f ) on thesame Cauchy surface. One can check that J t does indeed commute with ˜ H t : [ J t , ˜ H t ] = 0.Furthermore, the following relation holds automatically (cid:104) φ | H t | φ (cid:105) t = (cid:90) Σ t T ab N n a d Σ b , (2.11)where the energy-momentum tensor T ab is given by T ab = ∇ a φ ∇ b φ − g ab ( ∇ c φ ∇ c φ + m φ ) . (2.12)In applying the above formalism to obtain the quantum theory as motivated and outlinedin section 1, we need to solve two more problems.First of all, we would like to interpret the above mathematical formalism physically foran arbitrary single observer in a given spacetime. In the above construction of the one-parameter family of Fock spaces, there is an ambiguity in the choice of the scalar function t . The subscript t on H t is to remind us that it is constructed out of J t as described previously.
5e need to specify the scalar t and understand its operational meaning for a given observer.This we shall discuss in the next section.Secondly, we would like to understand the dynamics of the theory. A differential formof the dynamics has been discussed in [1], however we will not follow that approach. Thequestion we would like to answer is: given a quantum state for an observer at his propertime t , what is the evolved quantum state at another time t ? In other words, we need toconstruct a two-parameter mapping from F s ( H t ) to F s ( H t ) satisfying certain properties,which we shall discuss in section 4. In the previous section, we have constructed a one-parameter family of Fock spaces. Theconstruction relies on a choice of scalar function t and a corresponding slicing of the spacetime { Σ t } . In this section, we would like to understand this t for a spacetime event p as the“time” of p for a given arbitrary observer, and Σ t as the set of all events “simultaneous”to the observer at his proper time t . We would like to understand how the events of agiven spacetime are directly related to the local physical quantities of the observer in anoperational way.To understand the physical meaning of this “time” and “simultaneity”, we first look atan inertial observer C ( t ) in Minkowski spacetime. Let t p be the earliest time when C ( t )can receive a signal from an event p . Let t (cid:48) p be the latest time for C ( t ) to send a signal toreach p . We define t q and t (cid:48) q analogously for another event q . Then the events p and q aresimultaneous for the inertial observer C ( t ) if and only if the relation t p − t q = t (cid:48) q − t (cid:48) p holds.This captures the intuition that if p and q are simultaneous then the difference in inquiringtime and response time would equal, and this difference is purely due to any difference inthe distances from p and q to the observer. This construction, known as radar time , hasbeen advocated by Bondi [12]. Applications of radar time to an observer-dependent particleinterpretation in quantum field theory has recently been pioneered by Dolby and Gull [13]. (cid:45)(cid:54)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) (cid:64)(cid:64)(cid:64) (cid:0)(cid:0)(cid:0) (cid:64)(cid:64)(cid:64) (cid:0)(cid:0)(cid:0) p qq (cid:48) x C ( t ) t p t (cid:48) p t q t (cid:48) q Figure 1: The events p and q are simultaneous, but p and q (cid:48) are not.The above characterisation of simultaneity is readily generalisable to an arbitrary observerin a curved spacetime. Let an observer C ( t ) be a timelike curve parametrised by his proper I would like to thank David Wiltshire for letting me be aware of [13] after completion of a draft of thispaper. t and consider a spacetime event p . Let τ p be the earliest proper time of the observersuch that there is a future-directed null geodesic originating from p to C ( τ p ), and let τ (cid:48) p bethe latest time such that there exists a future-directed null geodesic originating from C ( τ (cid:48) p )to p . If there exist both τ p and τ (cid:48) p , then we say that p is in causal contact with the observer.Following [14], we shall call the set of all points in causal contact with the observer C ( t )the diamond of C ( t ). The diamond is thus defined to be the set of all points which can bothreceive light signals from and send light signals to the observer C ( t ), in other words, thosepoints which can exchange information with C ( t ). Equivalently, the diamond is defined tobe the intersection of the past of C ( t ) and the future of C ( t ). We may also talk about thehorizon for an observer C ( t ), most generally, as the boundary of the diamond of the observer.We define the scalar function t ( p ) for each point p in the diamond of an observer to be t ( p ) := ( τ p + τ (cid:48) p ) / , (3.1)and we say p is simultaneous to the observer at t ( p ). In the case when the point p is lying on C ( t ), i.e. p = C ( t ) for some t , we define t ( p ) = t . Given a time t , we define the surface ofsimultaneity Σ t to be the set of all points p in the diamond for which t ( p ) = t . Since eachpoint p of the diamond belongs to exactly one Σ t ( p ) , we obtain a foliation { Σ t } of the entirediamond. Furthermore, we have a timelike vector field ∂/∂t everywhere orthogonal to { Σ t } .The simultaneity of our definition depends on the observer’s entire history; that is, thesurface of simultaneity Σ t at time t depends on the observer’s trajectory C ( t ) for all t .This should not be taken as a flaw of our definition. Firstly, our definition using light signalcommunication has a clear operational meaning. Secondly, in Minkowksi spacetime with theusual coodinates ( t, x i ), the constant t surfaces are the usual planes of simultaneity for aninertial observer with constant x i , and the specification of the entire trajectory of constant x i is essential. Thirdly, if two observers locally coincide around a common point, their surfacesof simultaneity through that point will also be locally the same.One could instead naively define simultaneity surface Σ t to be the set of all points lyingin geodesics orthogonal to the observer’s trajectory C ( t ) at t . This definition of Σ t dependsonly on the point C ( t ) and its tangent vector, and for an arbitrary observer C ( t ) in Minkowskispacetime it gives the usual simultaneity plane for the inertial observer tangent to C ( t ) at C ( t ). However, the physical meaning of these surfaces for an arbitrary observer becomesunclear far from the worldline. Furthermore, these surfaces may intersect [11] within theobserver’s diamond, so that the “time” of the point of intersection becomes ambiguous underthis definition. Our definition of simultaneity does not suffer from the same problem, as eachpoint in the diamond of the observer belongs to exactly one simultaneity surface.We have defined a scalar function t and a corresponding foliation { Σ t } of the diamondregion of an arbitrary observer. Application of this idea to the formalism summarised insection 2 implies that we should treat the diamond as the physically relevant part of spacetimefor the observer, on which the vector space V of real Klein-Gordon solutions is defined.Operationally, only those events in the diamond of an observer are physically relevantto him. If there exists a future-directed null geodesic originating from a point in C ( t ) to apoint p , then one may naively say that the observer can send a signal to p . However this has It may happen that a future-directed null geodesic originating from a point p intersects an observer’sworldline multiple times. For example, in cylindrical spacetime with metric ds = − dt + dθ , a null geodesicfrom a point p intersects a constant θ observer infinitely many times. In this case, our definition of τ p is thefirst future point of intersection. Similar remarks hold for τ (cid:48) p .
7o causal effect on him if there is no future-directed null geodesic from p to C ( t ). The onlyway that the observer is able to operationally determine whether p has received his signalis if in principle he can receive a reply from p . Similarly, any event q to which there is nofuture-directed null geodesic from C ( t ) connecting is physically irrelevant.It is in fact a more or less usual practice in quantum field theory that one interprets onlythe degrees of freedom in the region causally connected to the observer as the physicallyrelevant ones. In the example of the Rindler observers, only the field in the Rindler wedge isconsidered relevant [2]. For observers staying outside of a black hole, only the field outsidethe black hole horizon is relevant [7, 8]. And for an inertial Minkowski observer of a finite-lengthed worldline, only the field in the observer’s diamond region is relevant [14]. Ratherthan interpreting the physically relevant spacetime region for a given observer on a case bycase basis, we here regard defining quantum field theory on only the diamond for an arbitraryobserver as an axiom of our framework. The formalism in section 2 requires that each surface Σ t of the foliation be a Cauchy surface.Our slicing of the diamond for an arbitrary observer defined earlier in this section, however,does not sastisfy this requirement in general. It may indeed be the case that all simultaneitysurfaces of the observer are Cauchy surfaces for his diamond, as in the case of the Rindlerobserver. But it may also be that at least one but not all simultaneity surfaces are Cauchysurfaces, as in the case of an inertial Minkowski observer with a finite worldline [14]; or thatnone of the simultaneity surfaces are Cauchy surfaces, such as a co-moving observer in theMilne universe, even though the spacetime is globally hyperbolic.In the next section, we shall construct a unitary quantum theory in the case that allsimultaneity surfaces are Cauchy surfaces. The construction for the more general cases hasto remain future work. However, we argue here that the limitation of the current formulationshould not be regarded as a fundamental flaw of our conceptual approach. We insteadinterpret this restriction as revealing the importance of Cauchy surfaces in the formulationof a unitary theory and the preservation of information. We shall now speculate on thequalitative features of the theory for the more general cases.Let t and t be an observer’s proper time with t > t . Generally, when surface Σ t is notcontained in the future Cauchy development D + (Σ t ) of surface Σ t , one cannot determinethe data on Σ t and hence the physical state at t from the data on Σ t , and therefore thedynamics of the theory is non-deterministic. Conversely, if all of Σ t lies in D + (Σ t ) butnot all of Σ t lies in D − (Σ t ), where D − (Σ t ) is the past Cauchy development of Σ t , thenevolution of the field from Σ t to Σ t is deterministic but non-invertible and hence thereis a loss of information. There will exist timelike curves crossing Σ t but not Σ t , whichphysically means that there will exist worldlines of objects leaving the observer’s horizonbefore he reaches t .In the case where only one of the simultaneity surfaces Σ t is a Cauchy surface, such asthe example in [14], one can determine the field throughout the entire diamond from Σ t .One can predict the future of Σ t , but information is lost as the field evolves from Σ t to afuture simultaneity surface. One can also “postdict” at t what happened before that time,although before t he cannot predict what happens at t : there is more for him to learn, butin retrospect nothing is surprising. 8e now consider the implications of the above discussion for the example of a gravita-tionally collapsing black hole without evaporation. As shown in Figure 2 (a), an arbitraryobserver that does not fall into the black hole has a diamond corresponding to spacetimeoutside the black hole horizon. If the causal structure and the simultaneity surfaces arecorrectly depicted in our figure where all simultaneity surfaces are Cauchy surfaces, then wesee that the observer’s quantum field theory is unitary and that no information is lost. Thisis consistent with the intuition that an observer remaining outside the black hole never inhis finite proper time sees an object falling behind the horizon and no information carriedby the in-falling object could ever disappear from the sight of the observer. Even thoughthe observer cannot access spacetime behind the horizon, that region is physically irrelevantto him: he does not know everything, but at least he knows what he knew.(a) (b)Figure 2: Diagrams of black hole spacetimes: (a) without evaporation; and (b) with evapo-ration. Two typical simultaneity surfaces Σ t , Σ t for the observer C ( t ) are shown. In the spacetime of black hole with evaporation, however, we speculate that not allsimultaneity surfaces will be Cauchy surfaces for the diamond of the observer staying outsideof the black hole, as depicted in Figure 2 (b). We expect that in the figure, Σ t will lie in D + (Σ t ), but Σ t will not be contained in D − (Σ t ). Hence it follows that evolution will bedeterministic, but information will be lost, a profound and long renowned proposal due toHawking [7, 8].Wald [9] has already pointed out that there should generally be a loss of information as-sociated with evolution of a Cauchy surface to a non-Cauchy surface. Here we have furtherdeveloped the idea. With our one-parameter family of hypersurfaces { Σ t } constructed previ-ously, we are able to provide Wald’s “evolution of surfaces” a precise mathematical meaning.Physically, we give these “surfaces” an operational meaning as the simultaneity surfaces of agiven observer. Moreover, the question of whether or not a surface is a Cauchy surface is nowaddressed with respect to the observer’s diamond. We have emphasised that the observershould play a fundamental role in the discussion of a possible non-unitary quantum theory.9e have proposed that Hawking’s original insight that the laws of physics may allowfor loss of information can be much generalised. Information can be lost not only in blackhole evaporation but precisely whenever one’s surface of simultaneity evolves from a Cauchysurface to a non-Cauchy surface, whether this be due to the background spacetime or to theobserver’s motion in that background. We emphasise that because our formulation reducesto the standard quantum field theory for inertial observers in Minkowski spacetime, allowingfor a non-unitary theory for general observers in curved spacetime cannot be in violationof the usual laws of physics in flat spacetime. And as we have argued that quantum fieldtheory should be formulated in an observer-dependent way, the notion that information lossis observer-dependent does not mean that it is not fundamental.Here we do not intend to claim a resolution to the paradox of “black hole informationloss”. Rather, we have merely speculated on the qualitative features that a general quantumfield theory for an arbitrary observer might possess. To consolidate our ideas about infor-mation loss, we would need a mathematical scheme for tracing out field degrees of freedomalong the worldline of the observer and a corresponding detailed analysis. Moreover, wehave adopted a semi-classical approximation, although we expect some features of the cur-rent theory, in particular the importance of causal structure on possible loss of information,to remain in a more complete theory.We have aimed at constructing a quantum theory for an arbitrary observer. To comparephysical descriptions between arbitrary observers it is therefore very important to studyquantum state transformation: specifically given a quantum state for an observer C ( t ) attime t , we would like to determine the corresponding state for another observer C ( s ) at time s . It is conceivable that when the simultaneity surface Σ s for observer C ( s ) is a subsetof Σ t for observer C ( t ), the state transformation from C ( t ) at t to C ( s ) at s shouldbe defineable. Furthermore, one would expect tracing out the extra degrees of freedom onΣ t to be necessary, so that a pure state for C ( t ) would in general be a mixed state for C ( s ). Finally, we require such a state transformation to be consistent with quantum stateevolution. We will not however develop these any further. We have constructed a one-parameter family of Fock spaces. The scalar function t on whichthis construction depends is the time of each event for the observer defined in section 3, andthe corresponding foliation of spacetime is the simultaneity surfaces of constant t . We havealso argued that the vector space V of real solutions of Klein-Gordon equation should bedefined on the diamond of the observer, which is the set of all points that can both receivelight signals from and send light signals to the observer.In this section, we turn to constructing the dynamics of quantum states for our theory:given a quantum state for an observer at his proper time t , what is the evolved quantumstate at another time t ? In other words, we need to construct a two-parameter mapping U ( t , t ) : F s ( H t ) → F s ( H t ) such that:(a) U ( t , t ) is an isomorphism, i.e. a complex-linear bijection;(b) the inner product of U ( t , t ) | φ (cid:105) t and U ( t , t ) | ψ (cid:105) t is equal to (cid:104) φ | ψ (cid:105) t , for all | φ (cid:105) t and | ψ (cid:105) t in F s ( H t ); and(c) U ( t , t ) = U ( t , t ) U ( t , t ). 10t then follows from these properties that U ( t, t ) = t on F s ( H t ) and that U ( t , t ) = U − ( t , t ).For notational simplicity, we will concentrate on some arbitrary choice of t = 0 and t = t . We use subscript t on various symbols to denote its dependence on t and/or J t , andsubscript 0 for the case of t = 0. In addition, we denote U ( t,
0) as U t , although it should beunderstood that U t implicitly depends on a choice of t = 0.Consider a state of the form C +0 ( φ ) . . . C +0 ( φ n ) | (cid:105) , where | (cid:105) is the vacuum in F s ( H ).We define our U t by U t (cid:16) C +0 ( φ ) . . . C +0 ( φ n ) | (cid:105) (cid:17) = (cid:16) U t C +0 ( φ ) (cid:17) . . . (cid:16) U t C +0 ( φ n ) (cid:17) (cid:16) U t | (cid:105) (cid:17) , (4.1)where we have used the same symbol U t to denote the mapping from F s ( H ) to F s ( H t ) aswell as the mapping from the complex vector space A := { α C +0 ( φ ) + β C − ( ψ ) : α, β ∈ C , φ, ψ ∈ V } (4.2)to A t defined similarly to (4.2). The nature of U t , i.e. whether it is on F s ( H ) or A , shouldbe clear from the argument on which it is acting.Once we have defined U t on both A and | (cid:105) , generalising the action of U t to the entireFock space F s ( H ) via complex-linearity and to density matrices should be straightforward.We shall then show that the mapping is well-defined and satisfies the three properties (a),(b), and (c).We now construct the map from A to A t . Our construction is guided by the followingintuition: there is a “natural” change of creation/annihilation operators under a change of J t ; and there is a change of creation/annihilation operators induced by the evolution of theirunderlying classical fields.We now capture the contribution of J t to U t . Define A µt := { C µt ( φ ) : φ ∈ V } , (4.3)where µ = ±
1. We have A t ∼ = A + t ⊕ A − t , where ∼ = denotes isomorphism. The subspace A + t is naturally isomorphic to V t via C + t ( φ ) ∼ = | φ (cid:105) t (4.4)where recall V t is the complex vector space constucted out of ( V, J t ) as in section 2. Similarly,the subspace A − t is isomorphic to ¯ V t , the conjugate of V t .Observe that there is also a natural isomorphism from V t to the (+ i ) J t -eigensubspace of V C ∼ = V ⊕ ( iV ). We shall denote this eigensubspace as V + t , and clearly V + t = { P + t φ : φ ∈ V } where P + t := (1 − iJ t ) / V C onto V + t . This isomorphism is given by | φ (cid:105) t ∼ = P + t φ. (4.5)Similarly, ¯ V t is isomorphic to V − t , the ( − i ) eigensubspace of V C , via P − J := (1 + iJ t ) / A t and V C . We denote this isomorphismas F t , that is: F t C µt ( φ ) = P µt φ, ∀ φ ∈ V, µ = ± . (4.6) We emphasise here that C µ ( φ ) and C νt ( φ ) are mappings on different Fock spaces and hence cannot bedirectly related by “=”. J t corresponds to different notions of “+” and “-”, embodied as different projections inour formalism. It is important to notice that V C is independent of J t . This disentanglesthe contribution of J t from the contribution of the classical evolution to U t , which we nowdescribe.Let u ( t , t ) : V → V be a two-parameter automorphism on V . We denote its complex-ification also by u ( t , t ). It is obvious that this complexification defines an automorphismon V C . Again, we denote u ( t,
0) as u t for some arbitrary t = 0, t = t .To capture the contribution of u t to U t , combined with that of J t , we now define the map U t : A → A t to be U t := F − t u t F : A U t −−−−−→ A tF (cid:121) (cid:120) F − t V C u t −−−−−→ V C (4.7)Having defined U t , we now determine the image of operators in A under this map. Since P + t + P − t is identity on V C : U t C µ ( φ ) = F − t ( P + t + P − t ) u t F C µ ( φ )= F − t ( P + t u t P µ φ ) + F − t ( P − t u t P µ φ ) , (4.8)where we used (4.6). It is then straightforward to show that the action of U t on A is U t C µ ( φ ) = (cid:88) α C αt ( φ µα | t ) , (4.9)where φ µα | t = ( u t − µα J t u t J ) φ, (4.10)with all Greek indices taking the value ±
1. We shall henceforth drop the summation sign inany equation with repeated α or β indices, with one index up and the other down.We have defined U t : A → A t and determined its action on A as given by (4.9). Tofinish the construction of U t on F s ( H ), it remains to specify how vacuum evolves. Denote | χ (cid:105) t to be the state evolved from vacuum: | χ (cid:105) t = U t | (cid:105) . We shall only impose that | χ (cid:105) t isa uniquely defined unit vector in F s ( H t ) and that it satisfies the condition (cid:16) U t C − ( φ ) (cid:17) | χ (cid:105) t = C αt ( φ − α | t ) | χ (cid:105) t = 0 ∀ φ ∈ V. (4.11)Before specifying | χ (cid:105) t and u t , and before showing that our construction satisfies (a), (b),and (c), we first say a few words about vacuum and particle creation. From (4.9), we seethat C +0 ( φ ) will evolve to a pure creation operator if and only if φ + −| t = 0 and that C − ( φ )will evolve to a pure annihilation operator if and only if φ − + | t = 0. But φ + −| t = φ − + | t , so thetwo conditions are equivalent. If no C µ ( φ ) gets mixed, we have ( u t + J t u t J ) φ = 0 for all φ ∈ V , i.e. J t = u t J u − t . (4.12)In this case, U t C µ ( φ ) = C µt ( u t φ ). Additionally, (4.11) reduces to C − t ( u t φ ) | χ (cid:105) t = 0 ∀ φ ∈ V. (4.13)12he vector | χ (cid:105) t is then fixed to be | (cid:105) t , the vacuum in F s ( H t ), since u t is an automorphismon V . We then have U t | φ . . . φ n (cid:105) = | ( u t φ ) ( u t φ ) . . . ( u t φ n ) (cid:105) t . (4.14)That is, a n -particle state in ⊗ ns H will evolve to a state in ⊗ ns H t for all n : particles are notcreated. The mapping U t in (4.14) is indeed complex-linear, as illustrated by U t i | φ φ . . . φ n (cid:105) = U t | ( J φ ) φ . . . φ n (cid:105) = | ( u t J φ ) ( u t φ ) . . . ( u t φ n ) (cid:105) t = | ( J t u t φ ) ( u t φ ) . . . ( u t φ n ) (cid:105) t = i | ( u t φ ) ( u t φ ) . . . ( u t φ n ) (cid:105) t = i U t | φ φ . . . φ n (cid:105) (4.15)Therefore, we see that particle creation is precisely due to J t failing to evolve in a way sat-isfying (4.12).There are two conditions that the automorphism u t must satisfy. We now discuss thefirst condition motivated by condition (b) on U t . To begin with, we have that (cid:104) φ | ψ (cid:105) = (cid:104) | [ C − ( φ ) , C +0 ( ψ )] | (cid:105) , for | φ (cid:105) , | ψ (cid:105) ∈ H , (4.16)and that the inner product between U t | φ (cid:105) and U t | ψ (cid:105) is given by (cid:104) χ | (cid:16) U t C − ( φ ) (cid:17)(cid:16) U t C +0 ( ψ ) (cid:17) | χ (cid:105) t , (4.17)where we have used the property (cid:16) U t C µ ( φ ) (cid:17) † = U t (cid:16) C µ ( φ ) (cid:17) † . (4.18)To see this property, notice that the right-hand side of (4.18) is U t C − µ ( φ ) = C − αt ( φ − µ − α | t )= C − αt ( φ µα | t )= (cid:16) C αt ( φ µα | t ) (cid:17) † , (4.19)where the final expression is simply the left-hand side of (4.18). Condition (b) then implies (cid:104) | [ C − ( φ ) , C +0 ( ψ )] | (cid:105) = (cid:104) χ | (cid:16) U t C − ( φ ) (cid:17)(cid:16) U t C +0 ( ψ ) (cid:17) | χ (cid:105) t . (4.20)Using (4.11), we can write the right-hand side of (4.20) as (cid:104) χ | (cid:104) U t C − ( φ ) , U t C +0 ( ψ ) (cid:105) | χ (cid:105) t . (4.21)Noticing that (cid:104) χ | χ (cid:105) t = 1 and that by (2.7) the commutator in (4.21) is a multiple of t on F s ( H t ), we see (4.20) is equivalent to[ U t C − ( φ ) , U t C +0 ( ψ )] = U t (cid:16) [ C − ( φ ) , C +0 ( ψ )] (cid:17) (4.22)13here we have defined on the right-hand side U t = t . (4.23)On the other hand, for (4.1) to be well-defined, it is necessary that[ U t C +0 ( φ ) , U t C +0 ( ψ )] = 0 (4.24)That U t preserves the commutation relationship on all A then follows from (4.18), (4.22),(4.24), and complex-linearity. We now derive the condition on U t for this commutation tobe preserved. A straightforward calculation using (2.3), (2.4), and (2.7) shows that[ C µt ( φ ) , C νt ( ψ )] = i ω ( P µt φ, P νt ψ ) t , (4.25)where we have extended the symplectic form ω given by (2.2) to V C via complex-linearity.Applying (4.9) and using the fact that both sides of (4.25) are bilinear, we have[ U t C µ ( φ ) , U t C ν ( ψ )] = (cid:104) C αt ( φ µα | t ) , C βt ( ψ νβ | t ) (cid:105) = i ω (cid:16) P αt φ µα | t , P βt ψ νβ | t (cid:17) t . (4.26)Since P αt φ µα | t = F t C αt ( φ µα | t )= F t U t C µ ( φ )= u t F C µ ( φ )= u t P µ φ, (4.27)where we have used (4.6), (4.7), and (4.9), we have[ U t C µ ( φ ) , U t C ν ( ψ )] = i ω ( u t P µ φ, u t P ν ψ ) t . (4.28)Hence [ U t C µ ( φ ) , U t C ν ( ψ )] = U t (cid:16) [ C µ ( φ ) , C ν ( ψ )] (cid:17) (4.29)will be satisfied if and only if ω ( u t P µ φ, u t P ν ψ ) = ω ( P µ φ, P ν ψ ) . (4.30)Summing (4.30) over µ, ν , one can show that this in turn is equivalent to ω ( u t φ, u t ψ ) = ω ( φ, ψ ) ∀ φ, ψ ∈ V. (4.31)That is, U t preserves the commutation relation on A if and only if u t preserves the symplecticform ω on V (or equivalently on V C ). With u t satisfying (4.31), it is easy to generalise theproof that U t satisfies condition (b) for states in H to the general case of all states in F s ( H ).Hence, in order for definition (4.1) of U t to be well-defined and for U t to satisfy condition(b), the first condition on u t which we impose is that it satisfies (4.31).The second condition, motivated by condition (c), which we shall impose on u t is: u ( t , t ) = u ( t , t ) u ( t , t ) . (4.32)14ith a choice of automorphism u t on V satisfying (4.31) and (4.32), and a choice of | χ (cid:105) t satisfying (4.11), we shall now show our U t on F s ( H ) satisfies (a), (b), and (c).First of all, we have shown (b) is satisfied. This in particular implies U t is injective.Complex-linearity is obvious. Hence to show condition (a) is satisfied, it remains to showthat U t is onto.We now show U t is onto. We first show there exists a | ˜ χ (cid:105) such that U t | ˜ χ (cid:105) = | (cid:105) t , thevacuum in F s ( H t ). Let | ˜ χ (cid:105) be | ˜ χ (cid:105) := U (0 , t ) | (cid:105) t where U (0 , t ) is defined precisely in thesame way as U ( t,
0) except for a change of “0” and “ t ”. Then by construction using (4.11), (cid:16) U (0 , t ) C − t ( φ ) (cid:17) | ˜ χ (cid:105) = 0 ∀ φ ∈ V. (4.33)Since by using (4.11) and (4.29) we have U t (cid:16) C α ( φ α ) | ψ (cid:105) (cid:17) = (cid:16) U t C α ( φ α ) (cid:17) U t | ψ (cid:105) ∀ φ α ∈ V, | ψ (cid:105) ∈ F s ( H ) , (4.34)we may apply U t on both sides of (4.33) to obtain (cid:16) U t U (0 , t ) C − t ( φ ) (cid:17)(cid:16) U t | ˜ χ (cid:105) (cid:17) = 0 . (4.35)Observe that U t U (0 , t ) on A t is F − t u ( t, F F − u (0 , t ) F t = t (4.36)where u ( t, u (0 , t ) = on V follows from (4.32) and the fact that u ( t , t ) is an automor-phism on V . Equation (4.35) now becomes C − t ( φ ) ( U t | ˜ χ (cid:105) ) = 0 ∀ φ ∈ V (4.37)with a unique solution U t | ˜ χ (cid:105) = | (cid:105) t . (4.38)We now conclude U t on F s ( H ) is onto, using (4.38), (4.34), and the fact that U t from A to A t is an isomorphism. Furthermore on F s ( H t ) we have U (0 , t ) = U − t . (4.39)Finally, we prove that condition (c) is satisfied. Applying U ( t (cid:48) , t ) for an arbitrary t (cid:48) to (cid:16) U t C − ( φ ) (cid:17) U t | (cid:105) = 0 ∀ φ ∈ V (4.40)which is simply (4.11), we have that (cid:16) U ( t (cid:48) , t ) U t C − ( φ ) (cid:17) U ( t (cid:48) , t ) U t | (cid:105) = 0 , (4.41)where we have used (4.34) again. Observe that, using (4.7), the condition (4.32) on u t isequivalent to condition (c) as a mapping on A . We have that (cid:16) U t (cid:48) C − ( φ ) (cid:17) U ( t (cid:48) , t ) U t | (cid:105) = 0 . (4.42) Note that we did not make any similar assumptions on U t for our proof. U (0 , t (cid:48) ) = U − t (cid:48) to (4.42) and use (4.34) to obtain C − ( φ ) (cid:16) U (0 , t (cid:48) ) U ( t (cid:48) , t ) U t | (cid:105) (cid:17) = 0 , ∀ φ ∈ V. (4.43)Hence, U (0 , t (cid:48) ) U ( t (cid:48) , t ) U t | (cid:105) = | (cid:105) . (4.44)Using (4.39) again, this is simply U ( t (cid:48) , t ) U t | (cid:105) = U t (cid:48) | (cid:105) . (4.45)We finally conclude that condition (c) is satisfied over all F s ( H ), using (4.45), (4.34), andthe fact that condition (c) is satisfied as mappings on A .We now specify the evolution of vacuum from | (cid:105) to | χ (cid:105) t satisfying (4.11). To do so, weshall first extend the definition of the evolution map U t : A → A t to sums of products ofoperators in A via U t (cid:16) (cid:89) i C µ i ( φ i ) (cid:17) := (cid:89) i U t C µ i ( φ i ) (4.46)and complex linearity. Clearly this mapping is well defined, following from definition (4.23)and that U t is linear on A and preserves commutation (4.29). Furthermore, this extended U t is invertible, satisfies the desired composition as in condition (c) at the beginning of thischapter, and commutes with taking the adjoint of its argument as in (4.18).The evolution of vacuum is then given by applying (4.46). Observe that the vacuumdensity matrix can formally be written as | (cid:105)(cid:104) | = ∞ (cid:89) n =1 ( − N /n ) (4.47)where the total number operator N on F s ( H ) is defined by N | φ n (cid:105) := n | φ n (cid:105) for all | φ n (cid:105) ∈ ⊗ ns H . This can be seen by noticing that (cid:104) | (cid:105)(cid:104) | (cid:105) = 1 and (cid:104) ψ m | (cid:105)(cid:104) | φ n (cid:105) = 0 aretrue for both sides of (4.47), for | φ n (cid:105) ∈ ⊗ ns H and | ψ m (cid:105) ∈ ⊗ ms H with at least one of m or n being non-zero. The evolved state is then given by ρ t := U t (cid:0) | (cid:105)(cid:104) | (cid:1) = ∞ (cid:89) n =1 (cid:16) t − ( U t N ) /n (cid:17) . (4.48)The operator U t N is also defined using (4.46). We write N as N = (cid:88) k C +0 ( φ k ) C − ( φ k ) (4.49)where summation is over all k such that {| φ k (cid:105) } is an orthonormal basis for H . Then wehave that U t N = (cid:88) k (cid:16) U t C +0 ( φ k ) (cid:17)(cid:16) U t C − ( φ k ) (cid:17) . (4.50)One can show that this definition of U t N is independent of a choice of an orthonormal basis {| φ k (cid:105) } for H . 16n general, (4.48) does not define a finite-trace operator on F s ( H t ), in which case strictlyspeaking our construction of unitary dynamics fails mathematically, although physicallymeaningful predictions can still be made. This, our only problem with infinity, resembles therenormalisation problem in the usual field theory of quantum operators. In a more completetheory, the dimension of the physical Hilbert space should probably be replaced by a finiteone, so we believe this difficulty does not indicate that our formulation is fundamentallytowards a wrong direction.In the case when (4.48) does converge to a finite-trace operator, however, we shall showthat ρ t defined in (4.48) is a pure density matrix. First of all, (4.46) maps hermitian operatorsto hermitian operators, using (4.18). It then follows ρ † t = ρ t . Secondly, we have ρ t = ρ t ,since U t (cid:0) | (cid:105)(cid:104) | (cid:1) U t (cid:0) | (cid:105)(cid:104) | (cid:1) = U t (cid:0) | (cid:105)(cid:104) | | (cid:105)(cid:104) | (cid:1) = U t (cid:0) | (cid:105)(cid:104) | (cid:1) (4.51)where we have used (4.46). It follows from these two properties that ρ t is a projector witheigenvalues 1 or 0, so that we can write ρ t in an orthonormal basis as ρ t := n (cid:88) k =1 | k (cid:105)(cid:104) k | (4.52)where n is the trace of ρ t if it is finite. We shall now show in this case n = 1 and that ρ t defined in (4.48) is a pure density matrix. Evolving the identity ρ t | i (cid:105)(cid:104) i | = | i (cid:105)(cid:104) i | for anarbitrary i between 1 and n by U (0 , t ) we obtain | (cid:105)(cid:104) | U (0 , t ) (cid:0) | i (cid:105)(cid:104) i | (cid:1) = U (0 , t ) (cid:0) | i (cid:105)(cid:104) i | (cid:1) (4.53)where we have used (4.46). Similarly, U (0 , t ) (cid:0) | i (cid:105)(cid:104) i | (cid:1) | (cid:105)(cid:104) | = U (0 , t ) (cid:0) | i (cid:105)(cid:104) i | (cid:1) (4.54)Hence U (0 , t ) (cid:0) | i (cid:105)(cid:104) i | (cid:1) is proportional to | (cid:105)(cid:104) | , that is, | i (cid:105)(cid:104) i | is proportional to U t | (cid:105)(cid:104) | forall i . This is only possible if n = 1, i.e. ρ t = | χ (cid:105)(cid:104) χ | t for some pure state | χ (cid:105) t as desired.We have specified the evolution of vacuum as in (4.48), and we have demonstrated thatit indeed defines a pure state | χ (cid:105) t . It remains to check that | χ (cid:105) t satisfies condition (4.11),i.e. C αt ( φ − α | t ) | χ (cid:105) t = 0. This is the case if and only if C αt ( φ − α | t ) | χ (cid:105)(cid:104) χ | t C − βt ( φ − β | t ) = 0 . (4.55)But by (4.46) the left hand side of this is U t C − ( φ ) U t (cid:0) | (cid:105)(cid:104) | (cid:1) U t C +0 ( φ ) = U t (cid:16) C − ( φ ) | (cid:105)(cid:104) | C +0 ( φ ) (cid:17) (4.56)which is simply 0 as desired.We say a final word about particle creation. If the quantum state was vacuum at timezero, then the expected particle number at time t is tr( | χ (cid:105)(cid:104) χ | t N t ), where N t is the totalnumber operator for F s ( H t ) defined in the same way as N for F s ( H ). For the | φ (cid:105) t particlenumber operator C + t ( φ ) C − t ( φ ), using that U t is trace-preserving, we havetr (cid:16) | χ (cid:105)(cid:104) χ | t C + t ( φ ) C − t ( φ ) (cid:17) = tr (cid:16) | (cid:105)(cid:104) | (cid:16) U (0 , t ) C + t ( φ ) C − t ( φ ) (cid:17)(cid:17) . (4.57)17t is then straightforward to calculate that the expected | φ (cid:105) t particle number seen by theobserver at his proper time t is given by the square of the norm of | φ − + | (cid:105) , where φ − + | := (cid:0) u − t + J u − t J t (cid:1) φ. (4.58)See (4.10) for a comparison of definitions.Finally it remains to specify the classical evolution u t , the automorphism on V , satisfying(4.31) and (4.32). One natural choice seems to be the following. For each φ ∈ V , we define u t φ ∈ V to be the unique solution with Cauchy data on Σ t u t φ | Σ t = φ | Σ √ h t n a ∇ a ( u t φ ) | Σ t = √ h n a ∇ a φ | Σ (4.59)where we have used a natural point identification map from Σ to Σ t given by the inte-gral curves of ∂/∂t . Clearly this mapping is defined independent of the labelling of the3-parameter family of the integral curves of ∂/∂t . It can be shown that it indeed is anautomorphism on V satisfying (4.31) and (4.32). However, it remains future work to checkwhether this choice of u t , although self-consistent, leads to the same physical conclusions asthose in the various well-known cases.[Aside. One may attempt to define the classical evolution by “pulling” Cauchy data backfrom Σ t to Σ along ∂/∂t , instead of “pushing” Cauchy data as above. More precisely, onemay define ˜ u t such that ˜ u t φ | Σ = φ | Σ t √ h n a ∇ a (˜ u t φ ) | Σ = √ h t n a ∇ a φ | Σ t . (4.60)It can be shown that ˜ u t is also an automorphism on V and satisfies (4.31). However, it doesnot satisfy (4.32). In fact ˜ u t = u − t , hence u ( t (cid:48) , t ) u t = u t (cid:48) implies that ˜ u t ˜ u ( t (cid:48) , t ) = ˜ u t (cid:48) , whichis not the physically desired composition law.] We summarise the main results of our discussion here.The goal of this paper is to propose a quantum field theory for an arbitrary observer incurved spacetime. To this end, we constructed a one-parameter family of Fock spaces basedon the formalism of Ashtekar and Magnon [1]. Each Fock space is based on a Hilbert spaceconstructed from the vector space V of real-valued Klein-Gordon solutions and a parameter-dependent complex structure J t . This construction requires a choice of scalar function t suchthat each constant t hypersurface Σ t is a spacelike Cauchy surface.We then applied this mathematical formalism for an arbitrary observer to the region ofspacetime which the observer can both send signals to and receive signals from. Following[14], we have used the terminology “diamond” to refer to this region. We argue that radartime should be applied for the above function t used in the construction of the Fock spaces. Here we have included a factor of √ h in the definition of Cauchy data which is slightly different fromthe convention in section 2. t , the set of points in the diamond with radar time t , is the set ofall events “simultaneous” to the observer at his proper time t . Our definition using lightsignal communication has a clear operational meaning and directly relates each point tothe observer’s local physical quantities. Our slicing applies to all observers and reflects thecausal structure of the underlying spacetime.Although the diamond of an observer in general may not cover the entire spacetime, itis operationally the only region physically relevant to the observer. We therefore define thevector space V of real Klein-Gordon solutions and our quantum field theory on the diamondof a given observer. We regard this as an axiom of our framework.In the case where all simultaneity surfaces of an observer are Cauchy surfaces of hisdiamond, we have constructed a unitary dynamics where no information is lost: given aquantum state for an observer at his proper time t , we constructed a two-parameter mapping U ( t , t ) from F s ( H t ) to F s ( H t ) that will give us the evolved state at time t . We require ourmapping to satisfy three conditions: that U t is an isomorphism; that the mapping preservesthe inner product; and that the mapping satisfies U ( t , t ) U ( t , t ) = U ( t , t ).The action of U t on F s ( H ) is specified by its action on the vector space A of creationand annihilation operators given by (4.2), as well as its action on the vacuum in F s ( H ).The construction is guided by the intuition that there is an evolution of operators in A according to a change in the complex structure J , as well as according to an evolution u t of the operators’ underlying classical fields, i.e. an automorphism on V C . The evolution ofvacuum state is given by (4.48) and satisfies (4.11). We have deduced that particle creationwill take place if and only if J t fails to evolve according to J t = u t J u − t .We have also speculated on features that the theory covering the more general cases mighthave. We have further developed Wald’s insight [9] that there should generally be a loss ofinformation when there is evolution of a Cauchy surface to a non-Cauchy surface. Our one-parameter foliation of the diamond gives this “evolution of surfaces” a precise mathematicalmeaning. Physically, we give these “surfaces” an operational meaning as the simultaneitysurfaces of a given observer. Moreover, the question of whether or not a surface is a Cauchysurface is now addressed with respect to the observer’s diamond.Above all, we speculate that information will be lost precisely whenever an observer’ssurface of simultaneity evolves from a Cauchy surface to a non-Cauchy surface, whetherthis be due to the background spacetime or to the observer’s motion in that background.This generalises Hawking’s original insight [7, 8] and takes the information loss of black holeevaporation as just a special case.There is much scope for future generalisations and applications of this work. • We would like to generalise our current formulation to the cases when not all surfaces ofsimultaneity of an observer are Cauchy surfaces for his diamond. As we evolve data froma Cauchy surface to a non-Cauchy surface, we need a general scheme to trace out the fielddegrees of freedom lost along the worldline of the observer. Such a scheme would alsoconsolidate our speculations about information loss. • It is important to apply our framework to various examples, especially to see how ourresults would compare with experiments and with the results obtained by other formula-tions, such as with canonical quantization or with model particle detectors.19n particular, it would be very important to explore quantum field theory in FRW cos-mology using our formalism, where the simultaneity surfaces for a co-moving observer arenot the usual surfaces of constant energy density. This offers the opportunity to test ourtheory with cosmological observations. • We would like to generalise our formalism to higher-spin fields and interacting fields. • One important extension of our work is to study quantum state transformation betweenarbitrary observers and its operational meaning. Given a quantum state for an observer C ( t ) at time t , we would like to know the corresponding state for another observer C ( s )at time s . We require such a transformation to be consistent with the state evolution wehave already constructed. • We constructed a unitary dynamics relating a quantum state at some time of an observerto a state at a later time. However we have not discussed, for an arbitrary observerin curved spacetime, how a quantum measurment projects quantum states nor how thenotion of “wavefunction collapse” should be understood. • One of the most intriguing aspects of quantum field theory in curved spacetime is the yetto be understood deep relationship between causal horizons and thermodynamics. Firstly,it has been argued [5] that the ultimate significance of the thermodynamics of black holehorizons hangs on the issue of its generalisation. In our framework, the diamond for anarbitrary observer provides a natural and most general notion of causal horizons, andby defining quantum theory over this region, our formalism naturally establishes a linkbetween quantum theory and causal horizons.Secondly, thermodynamics of causal horizons can be studied using a statistical mechanicsapproach. This requires a concept of particles, and our formulation indeed provides anotion of particles for each observer. On the other hand, it would also be very enlighteningfrom our formalism to obtain an expression directly relating thermodynamic quantities andproperties of causal horizons without needing to consider any particle spectrum.Finally, it has been argued that entropy is an observer-dependent quantity [6]. Ourobserver-dependent quantum field theory may be an ideal framework for a further in-vestigation of the observer-dependence of thermodynamic quantities. • We propose that observer-dependence as a fundamental feature of quantum field theoryshould be taken much further. Since the quantum states of particles in general depend onthe observer, therefore different observers will have different (cid:104) T µν (cid:105) , the expectation valuesof the energy-momentum associated with their respective quantum states. By Einstein’sfield equations, the quantum field’s back-reaction on the spacetime metric will also beobserver-dependent and hence so will the spacetime metric itself. Indeed, the possibilitythat spacetime itself may be observer-dependent has been suggested by Gibbons andHawking [3]. Such a profound suggestion merits further investigation which our frameworkmay be well-suited to pursue, and this investigation may indeed be the correct path leadingto a complete and consistent union of quantum theory and relativity. Acknowledgements
I would like to thank Rex Liu for discussions, David Wiltshire and Steffen Gielen for com-ments and Jonathan Oppenheim for criticisms.20 eferences [1] A. Ashtekar and A. Magnon, Proc. Roy. Soc. Lond. A , 375 (1975).[2] W. G. Unruh, Phys. Rev. D , 870 (1976).[3] G. W. Gibbons and S. W. Hawking, Phys. Rev. D , 2738 (1977).[4] L. Susskind, L. Thorlacius and J. Uglum, Phys. Rev. D , 3743 (1993) [arXiv:hep-th/9306069].[5] T. Jacobson and R. Parentani, Found. Phys. , 323 (2003) [arXiv:gr-qc/0302099].[6] D. Marolf, D. Minic and S. F. Ross, Phys. Rev. D , 064006 (2004) [arXiv:hep-th/0310022].[7] S. W. Hawking, Phys. Rev. D , 2460 (1976).[8] S. W. Hawking, Commun. Math. Phys. , 395 (1982).[9] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermody-namics , (Chicago, 1994).[10] P. H´aj´ıˇcek, Phys. Rev. D , 2757 (1977).[11] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation , (Freeman, 1973).[12] H. Bondi,
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