Black holes, information, and the universal coefficient theorem

arXiv:1410.5291v2 [physics.gen-ph] 8 Mar 2017 Black Holes, Information and the Universal Coefficient Theorem Andrei T. Patrascu1 1University College London, Department of Physics and Astronomy, London, WC1E 6BT, UK General relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is based on the principle of unitary evolution. In this article I provide a possible answer to the black hole information paradox by means of homological algebra and pairings generated by the universal coefficient theorem. The unitarity of processes involving black holes is restored by demanding invariance of the laws of physics to the change of coefficient structures in cohomology. 1. INTRODUCTION The prescriptions of general relativity and quantum mechanics are taking away most of the absoluteness associated to choices of coordinates, trajectories followed by particles and states of physical systems in the absence of any accessible information about them. The mathematical language of differential form is perfectly suited for dealing with such prescriptions. It is my observation that there still remains an epistemological defect associated to these ideas. Not to all arbitrary conventions has been taken their absolute status away. In fact the connectivity of space is probably the last convention that still is considered absolute by many physicists. Mathematically however, one cannot assign an absolute topology to spacetime in the absence of a method for detecting such a topology. This obstruction is at the origin of several paradoxes and inconsistencies, notably the ”information paradox” for black holes. Because of this, in order to construct a consistent formulation of physics in a general context, it appears to be necessary for the laws of nature to be specified in a topology-covariant way. In the same way in which the language of differential forms allows us to transfer geometry-independent properties from one manifold to another, the topology-covariant language will allow us to transfer topology-independent properties from one topological space to another. In the same way in which differential forms allow us to construct objects that are coordinate independent, the universal coefficient theorems (and other theorems of homological algebra) will allow us to construct objects that are by construction, independent of choices of coefficients in (co)homology. This will provide us with a tool capable of identifying dualities in a more systematic way. In a more practical tone, one of the problems arising in the discussion of black holes in a quantum field theoretical context is the fact that the quantum prescription of unitarity may be lost in processes involving the thermal radiation of black holes [1]. In fact it can be shown that in a semi-classical approximation, each process involving the presence of a horizon may lead to outgoing thermal radiation [2]. An in-falling pure quantum state is then mapped into the external radiation which presents a thermal spectrum thus violating unitarity. I analyze here the origin of this problem and find that the semi-classical approximation is insufficient for a correct quantum description of phenomena involving space-time horizons. In fact, the solution appears to be related to topological properties of the transformation groups considered as acting on the given space. These properties are being analyzed in terms of group (co)homology within the group approach to quantization. The discussion continues for a group approach to the second quantization giving rise to a group theoretical analogue for quantum field theory. In order to realize a coefficient independent construction, I use the group extensions in the universal coefficient theorem to derive the connection between homology and cohomology. The resulting pairing between group homology and cohomology will provide us with a coefficient covariant construction as well as additional multipliers that will ensure this at the level of computations. The departure from a strict thermal spectrum in the context of a group approach to quantization for conformally invariant quantum field theories has been discussed in [33]. As the group approach to quantization implies the embedding of the (potentially curved) space Qwithin a larger differentiable structure containing the phase space of the theory, namely a group Gwhich will from now on drive the quantization procedure, the geometry of the physical space and its physical symmetries as well as possible gauge symmetries will be naturally taken care of in terms of group theoretical effects [34]. Also, a given phase space may possibly be embedded in different groups, each of these groups giving rise to different quantum theories having, in particular, non-equivalent vacua. A central extension of this group, namely G→˜ Gwill give rise to the quantizing group, the fundamental structure in the group approach to quantization. The centrally extended group ˜ Gplays a key role in characterizing the vacuum states in the curved space Q. In general, standard quantum field theories in curved spacetime do not have a preferred definition of particles. The origins of this ambiguity lie in the infinite, arbitrary, non-equivalent directions of polarization for the infinite dimensional 2 Heisenberg-Weyl subgroup. This means that arbitrary choices of annihilation operators by means of the canonical Bogolyubov transformations [7] lead to unitary non-equivalent vacua. However, the embedding of the curved space Q in a (centrally extended) group ˜ Gstrongly restricts the possible Bogolyubov transformations [34]. It is also possible to choose particular states which behave as vacua with respect to a given subgroup ˜ GK⊂˜ Gi.e. the states which are invariant under ˜ GKonly. The study of conformally invariant quantum field theories where Weyl (Poincare + dilatation) invariant pseudo-vacua (zero-mode coherent states) have been chosen is relevant for the Fulling-Unruh effect and vacuum radiation in relativistic accelerated frames [35]. Such a choice of particular pseudo-vacua corresponds to a symmetry breaking. References [33] and [34] also deal with the problem of second quantization from the perspective of the group approach. Indeed, as the choice of particular pseudo-vacua corresponds to a breakdown of the symmetry, it may result in a constrained version of our previous theory. At the second quantized level, we may, according to [34], introduce a corresponding constrained quantum field as well. The selection of a given Hilbert subspace Hǫ(˜ G)⊂H(˜ G) made of wave functions obeying a higher-order constraint Kψ =ǫψ where Kis a Casimir operator of ˜ GK⊂˜ Gmanifests itself at the level of second quantization in the form of a new quantum field theory. The vacuum for the new observables of this new (broken-symmetry) theory does not have to be identical to the vacuum of the original theory. The action of the other operators may produce vacuum radiation in this new theory. Such constructions have already shown how the Bogolioubov transformations can be restricted by demanding physical consistency and covariance to the arbitrary embeddings of spacetime in various groups. In this article I show how the choices of coefficient structures affect the sensitivity of the cohomology to more subtle cocycles. I showed in a previous article [9] how apparently trivial cohomology classes may actually be non-trivial when analyzed from the perspective of a different coefficient structure. An example of where this happens is the pseudo-cohomology, briefly presented in this article. I also study the topology of the groups involved by means of cohomology theories. The (co)homology with coefficients for the topology of manifolds with various dimensions has been studied for a long time. Relevant articles about the results and limitations of the associated methods are [26], [27], [28]. The analysis of groups by means of (co)homology is a somewhat newer idea, still presenting some interpretational gray areas. We know however that the second group cohomology of the group Gcan be shown to be isomorphic with the set of associated extensions [36]. As the group extensions may characterize the vacua and the coefficient structures in cohomology control the extension by means of the universal coefficient theorem, it is possible to employ different coefficients in order to characterize different vacua of the same theory or different quantum field theories altogether, maybe obtained one from the other by breaking certain symmetries. The pairing between homology and cohomology induced by the universal coefficient theorem leads to topology-covariant constructions that may be transfered from one topological space to another. This may lead to an algebraic method for describing dualities in quantum field theories and string theory. The covariant formulation with respect to some transformations and the related ideas leading to equivalence principles (Galilei, Lorentz, Poincare) are important in this context. In particular, it is possible to relate the existence of a simple manifest covariant formulation and, in a more extended way, of an ”equivalence principle” [4], to some topological properties of the transformation groups employed in the theory. The Unruh effect and the curved spacetime in the group approach to quantization have been described in [33], [34]. The novelty of this article is the extension of these ideas for the situation when a topological covariant description would make the computations clearer. 2. COVARIANCE PRINCIPLES IN PHYSICS The main developments of the past century (special relativity, general relativity and quantum mechanics) have brought to our attention the fact that abstract mathematical conventions should not stand at the fundaments of a description of reality. In general, the role of conventions is to facilitate the comprehension of physical reality and not to assign physical reality to conventional constructions [3]. This statement can be translated in modern terminology by using (co)homological algebraic notations. In order to do this let me follow reference [4] and define P=T r4◦L(1) to be the Poincare group where T r4is the four dimensional translation group and Lthe Lorentz group and G=T r4◦LG(2) to be the Galilei group where again T r4is the four dimensional translation group and LGis the group of galilean boosts and rotations. In contrast to the Poincare group, due to the absoluteness of time, the Galilei group admits 3 several semi-direct structures. One can use for example the decomposition G= (((T r3⊗B3)◦T)) ◦ R =H◦ R (3) where T r3is the 3 dimensional translation group, B3is the 3 dimensional boost group, Trepresents time translations and Rrepresents rotations. This allows one to define the mechanical evolution space as the homogeneous space parametrized by (t, x, ˙x). This evolution space is however not a homogeneous space for the Poincare group, because of the different cohomological properties of the Galilei and Poincare groups: while H2 0(G, U(1)) = Rfor the Galilei group, for the Poincare groups H2 0(P, U(1)) = 0. This difference in the cohomological structures of the Galilei and Poincare groups has as consequence the absence of any simple covariant formulation of Newtonian mechanics, as opposed to the Poincare case [4]. In this way, the existence of a special topological structure of the symmetry group of a theory is related to the existence of a simple enough covariant formulation. This is not to say that a covariant formulation for the Newtonian mechanics is impossible. In fact, it is possible, after certain choices regarding the probing of topological properties are made. With this example I show that the topological features of the kinematical symmetry group are of utmost importance and hence the cohomology must be capable of distinguishing them even at an incipient level. This is translated in the requirement that the cohomology detects not only standard non-trivial cocycles but also pseudo-cocycles i.e. cocycles that would become non-trivial only when certain operations on the groups are being performed. It is important to notice how this argument can be extended when one deals not only with covariance with respect to a symmetry group but with covariance to a change in the measurement technique for the topology of a group. Before entering this discussion it is important to put the observations above on firmer ground. Following reference [30] and [37] the Galilei group can in fact be obtained from the Poincare group by means of a group contraction when c→ ∞. Such a contraction gives rise to the notions of pseudo-cohomology and pseudo-extensions in the following way. Consider the trivial cohomology classes denoted by [[ξ]]. Inside these classes it is however sometimes possible to distinguish cohomology subclasses [ξ]∈[[ξ]]. These can be selected from the coboundaries according to some additional structure associated with the original group. Let me call now that original group G. The physical origin of pseudo-cohomology is as follows. We start from a given group Gfor which we know a central extension ˜ Gassociated with a two-cocycle ξcob generated by a function λon G. Now, consider there exists a well defined contraction limit of the group ˜ Gproducing ˜ Gcin the sense of Inonu and Wigner. Therefore the two-cocycle ξcob is well defined under the limit. It could however happen that the generating function λis ill-defined (divergent) in this limit. In that case the contracted two-cocycle is no longer a coboundary since there is no λcto generate it. This procedure therefore was among the first to generate non-trivial group cohomology. It also shows that the triviality of cohomology depends on the limit behavior of the generators of the coboundary. The Lie algebra structure constant associated with a pseudo-extension, i.e. a central extension characterized by a pseudo-cocycle, therefore differs from that of the trivial product. This fact requires the non-triviality of the gradient of λat the identity of the group G. Pseudo-cocycles are generated by functions which are, locally, linear functions. For finite-dimensional semisimple groups for which the Whitehead lemma applies, pseudo-cohomology is important. In the case of infinite dimensional semisimple Lie groups for which the Whitehead lemma does not apply, the group law for ˜ Gwill contain two-cocycles as well as pseudo-cocycles. The first physical example, connected to the discussion above appears in the contraction Poincare →Galileo where a special kind of trivial two-cocycles in the Poincare group become true two-cocycles for the Galilei group in the c→ ∞ limit. While the two-cocycle is well behaved in the limit, its generating function is not, initiating cohomology generation. Another simple physical example appears for the free non-relativistic particle with spin, where the Galilei group must be extended by a true two-cocycle to describe the canonical commutation relations between qand pas well as by a pseudo-cocycle associated with the Cartan subgroup of SU(2) to account for the spin degree of freedom. Such global properties identifiable by means of pseudo-cohomology and pseudo-extensions are also identifiable by means of obstructions to the naive change of coefficient groups in cohomology. 3. INDEPENDENCE OF TOPOLOGY AND THE UNIVERSAL COEFFICIENT THEOREM As argued in the previous chapters, the laws of physics should not depend on arbitrary choices. Specifically the choice of a particular coordinate system or a particular coefficient group in cohomology should not be relevant for the formulation of the laws of physics. I showed in a previous article [9] that specific choices of coefficient groups in cohomology may affect the observable connectedness of space-time (or generally of an abstract space or group) as measured by topological techniques. An interesting example for the role of the coefficient group in cohomology for the detection of topological properties is given in [31]. Here I focus on a different aspect, namely what changes should be made in a theory in order for it to describe the physical reality independent on the way one choses the coefficient structure in cohomology? As has been shown in [4] and as I argued in the previous sections, the existence of a 4 trivial second group cohomology associated to a symmetry group implies the existence of a straightforward covariant formulation of the associated theory. The sensitivity of cohomology in a given dimension however, is controlled by the choice of a coefficient structure in the cohomology. The effect of this choice is on its turn, encoded in the universal coefficient theorem by means of the extensions. The quantization prescription in general and the form taken by the unitarity constraints depend on the topology of the space where quantization is performed. The visibility of the respective topological structures also depends on the coefficients in cohomology. Therefore, formulating the quantization with respect to a cohomology that is not sensitive to certain topological features (e.g. black hole horizons) may restore unitarity and a straightforward quantization. However, such a construction will have to be corrected by topology dependent terms. These will become manifest when universal coefficient theorems are being employed. These together with their extensions and torsions will play the topology analogue of differential forms in geometry. It has been brought as an argument for the information paradox that a relatively ordered initial situation (dust or a star) leading to a black hole has as an inescapable final state the thermal radiation. Unless some ”emission of negative entropy” [1] by the black hole occurs, information should be lost. However, I showed in ref. [9] that the definition of entropy in a situation where different coefficient groups are required, must change. In fact, the entropy will have to include topological information as well. It will not be defined uniquely. Instead it will have different forms when regarded via different coefficient groups. This allows the changes in entropy required to restore unitarity in a global (topological) way. As stated in the previous chapters, geometric quantization is tightly bound to the existence of a classical limit. In the absence of such a limit the associated methods are often insufficient. In order to circumvent this problem a new quantization method has been developed [32] based on a group theoretical approach. The references [29] and [30] are also relevant for a better understanding of this. The main ingredient of such a method is a Lie group structure on the manifold replacing the quantum manifold of geometric quantization. This Lie group, which I call ˜ Galso appears to be a principal bundle with structure group U(1). However, in this more general approach ˜ G/U(1) is not forced to have a symplectic structure. Therefore non-symplectic symmetry parameters are naturally allowed giving rise to the corresponding operators (Hamiltonian, angular momentum, null charges, etc.). Another advantage is that on any Lie group there are always two sets of mutually commuting vector fields. The group approach to quantization is not meant to quantize a classical system (a phase space) but instead, the quantizing group is the primary quantity. The group approach to quantization is a more general approach based on the quantizing group ˜ Gwith a principal bundle structure ˜ G(M, T ) having Tas a structure group and Mits base. The group Tgeneralizes the phase invariance of quantum mechanics. It will encode constraints that may introduce topological properties that have to be taken into account. Of course the simplest yet general case remains T=U(1). The group law for ˜ G={˜g= (g, ζ)/g ∈ G, ζ ∈U(1)}is ˜g′∗˜g= (g′∗g, ζ′ζeiξ(g′,g)) (4) the group operation in Gbeing g” = g′∗gand ζ(g′, g) is a two co-cycle of Gwith the property that ξ(g2, g1) + ξ(g2∗g1, g3) = ξ(g2, g1∗g3) + ξ(g1, g3), gi∈G(5) We say the central extensions are trivial if the two co-cycles are coboundaries which can be written in the form of ξ(g′, g) = δ(g′∗g)−δ(g′)−δ(g) (6) where δ(g) is the generating function of the co-boundary. The group ˜ Gacting on itself on the right and on the left provide two sets of mutually commuting invariant vector fields ˜ XL ˜gi=∂˜g”j ∂˜gi˜g=e ∂ ∂˜gj(7) ˜ XR ˜gi=∂˜g”j ∂˜g′i˜ g′=e ∂ ∂˜gj(8) [˜ XL ˜gi,˜ XR ˜gj] = 0 (9) where {˜gj}is a parametrization of ˜ G. Next, the left invariant quantization 1-form Θ associated with the central generator ˜ XL ζ=˜ XR ζ, ζ ∈Tnamely the T-component of ˜ θL(ζ)of the canonical left invariant 1-form ˜ θLon ˜ G. 5 The differential dΘ is a presymplectic form and its characteristic module Ker(Θ) ∩Ker(dΘ) is generated by a left subalgebra GΘ. The quotient group ( ˜ G, Θ)/GΘis a quantum manifold. The trajectories generated by the vector fields in GΘare the generalized equations of motion of the theory. The Noether invariants under those equations are F˜gj=i˜ XR ˜gjΘ. One may consider the set of complex-valued T-functions on ˜ Gin the sense of principal bundle theory: ψ(ζ∗˜g) = DT(ζ)ψ(˜g), ζ ∈T(10) where DTis the natural representation of Ton the complex numbers. The representation of ˜ Gon the set of complex valued T-functions generated by GR={˜ XR}is called Bohr quantization. However, this quantization is as it stands, reducible. To obtain an irreducible representation one may impose a full polarization P ˜ XLψp= 0,∀˜ XL∈ P (11) which is a maximal, horizontal left subalgebra of ˜ GLwhich contains GΘ. The existence of a full polarization for the whole subalgebra GΘis not guaranteed. In this case a higher order polarization will be required which is a polarization for the enveloping algebra UGLwhich contains GΘ. This higher order polarization will be made of the extended vector fields corresponding to the momentum variables, spacetime symmetries and internal symmetries together with a deformation of the vector field ˜ XL tassociated with the temporal evolution which, usually, can be chosen to be the Casimir operator of G PHO =<˜ XHO t,˜ XL hi>(12) The group ˜ Gcan be irreducibly represented on the space H(˜ G) = {|ψi} of polarized wavefunctions and on its dual H∗(˜ G) = {hψ|}. The coordinates of the ket and the bra in the representation defined through the polarization Pare ψp(˜g) =<˜gP|ψ > ψ′∗ p(˜g) =< ψ′|˜gP>(13) This leads to the inner product of the form < ψ′|ψ >=Z˜ G µ(˜g)ψ′∗ P(˜g)ψP(˜g) (14) with µ(˜g) = θL ˜gi∧θL ˜gj∧... (15) The closure relation in ˜ Gis 1 = Z˜ G |˜gPiµ(˜g)h˜gP|(16) The group ˜ Ghas a unitary representation ρsuch that h˜gP|ρ(˜g′)|ψi=ψP(˜g′−1∗˜g) (17) Enlarging the the structure group Tallows us to introduce constraints in the theory. These constraints should always include U(1). The constraints are being introduced by means of the T-function conditions ρ(˜ t)|ψi=D(ǫ) T(˜ t)|ψi,˜ t∈T(18) For example quantum mechanics on a non-simply connected manifold Qmay be recovered from quantum mechanics on its universal covering ¯ Qby choosing T=π1(Q)⊗U(1) as the structure group. This is well known and leads to the so called topological quantum effects known as the θ-structure. If the structure group Tis non-central, not all operators ˜ XR ˜gpreserve the constraints imposed within T. For this to happen for a subgroup ˜ GT⊂˜ Gwe need that [˜ GT, T ]⊂Ker(Dǫ T) (19) The classes of inequivalent two-cocycles for the topologically trivial quantization define the second cohomology group H2(G;U(1)). However, as showed above (and also following from [32]), in order to introduce constraints which may 6 imply non-trivial topological structure the group U(1) may be replaced with a more general structure. This would lead to a construction of the type H2(G;T) which expands on the standard cohomology and replaces the usual coefficient structure with another one, capable to detect additional topological structure. Modified coefficient groups therefore allow the precise choice of the level of refinement demanded from a particular cohomology theory. However, there exist physical properties which are basically independent of a particular topology. Dualities relating open and closed strings are relevant examples. In order to construct such topology-independent theories, it appears to be necessary to employ the universal coefficient theorems. These theorems state that a specific framework, constructed by the choice of a coefficient group in (co)homology is (up to (extension) torsion in (co)homology) equivalent with the choice of an integer coefficient group. One result of this theorem is that distinct classes in (co)homology under one coefficient group may appear as identified under another coefficient group. There are several ways in which we can generalize the usual pairings relating vectors and 1-forms to pairings relating homology and cohomology. Such more general pairings will involve the universal coefficient theorems and the required covariance will be translated into rules relating the possible extensions arising in the associated exact sequence. One such possible pairing is defined as <, >:Hq(G;M)×Hq(G)→M(20) which relates homology with cohomology. This pairing is bilinear and its adjoint is a homomorphism Hq(G;M)→Hom(Hq(G); M) (21) Universal coefficient theorems, among other things, provide a measure of how this adjoint fails to be an isomorphism in terms of Ext and T or [10]. Here qrepresents the dimension of the space for which the (co)homology is calculated. In the context of the group approach to quantization the main topological tool to be used is group (co)homology. This cohomology measures the extent to which the invariants of the groups in an exact sequence do not respect the original exact sequence. Consider for example an abelian group Mtogether with a group action of the group Gon M. The elements of Gare acting as an automorphism of M. One may consider the submodule of G-invariant elements of M MG={x∈M|∀g∈G:gx =x}(22) Consider Nas a G-submodule of M. The invariants in M/N are in general not the quotient of the invariants in Mby the invariants in N. The invariance modulo Nis a broader concept, measured precisely by the first group cohomology H1(G, N). In order to describe the homology groups for trivial group action of Gon Min terms of the homology groups for trivial group action of Gon a reference group, say Zone may use the universal coefficient theorem for group homology 0→Hp(G;Z)⊗M→Hp(G;M)→T or(Hp−1(G;Z), M)→0 (23) The sequence splits, although not naturally giving Hp(G;M)∼ =(Hp(G;Z)⊗M)⊕T or(Hp−1(G;Z), M) (24) However, to connect homology with cohomology in a way useful to the study of how the scalar products change at a change of coefficients a more useful result is that of the dual universal coefficient theorem, linking the homology groups for trivial group action of Gon Zand the cohomology group for trivial action of Gon M 0→Ext(Hp−1(G;Z), M)→Hp(G;M)→Hom(Hp(G;Z), M)→0 (25) Here as well, the sequence splits although not naturally Hp(G;M)∼ =Hom(Hp(G;Z)⊕Ext(Hp−1(G;Z), M) (26) In the case of second order cohomology, with Gagain a group and Aan abelian group we have 0→Ext1(Gab, A)→H2(G;A)→Hom(H2(G;Z), A)→0 (27) where Hom(H2(G;Z), A) called the second cohomology group up to isoclinism is the group of group homomorphisms from H2(G;Z) to A. The group H2(G;Z) is the second homology group for the trivial group action. The extension 7 group Ext1(Gab, A) describes abelian group extensions with normal subgroup Aand quotient group Gab. The map Ext1(Gab, A)→H2(G;A) leads us from the abelian group extension with normal subgroup Aand quotient group Gab to the extension of Gby A. A short exact sequence of groups given as 0→A→E→G→1 (28) with Ethe central extension, meaning that the image of Ain Eis a central subgroup of E, defines a natural homomorphism β:M(G)→Awith M(G) = H2(G;Z). Suppose then that we fix the abelian group Aand the group Gand let the central extension group Evary. The structures that Ecan take up to congruence correspond to H2(G;A). For each extension we obtain an element of Hom(M(G), A). Congruent group extensions define the same homomorphism. Therefore we obtain our homomorphism H2(G;A)→Hom(M(G), A) which is surjective. Therefore, the universal coefficient theorem appears to be a tool for identifying the changes in the extensions. Moreover, it can be interpreted as a pairing, therefore demanding particular multipliers originating in the various Ext groups appearing on the left, which practically encode non-trivial two-cocycles. These will be the multiplicative elements arising in the scalar product formulas and leading to non-thermal corrections to the Hawking radiation. 4. BLACK HOLES AND THE UNITARITY PROBLEM The previous sections showed that when using equivalence principles and covariant formulations of theories, one usually relies on specific topological properties of the symmetry groups. Especially the second group-cohomology, when trivial, allows for a simple covariant formulation as the one used in the bra-ket formalism or in the tensorial construction of general relativity. However, not in all situations is the second group-cohomology trivial. The sensitivity of the second cohomology group to various topological features (non-trivial cocycles, pseudo-cocycles, etc.) depends on the coefficient structure chosen in order to describe the cohomology itself. When the second cohomology of the required group is non-trivial one can still formulate a covariant theory provided one uses a proper coefficient structure, giving the desired sensitivity to the cohomology groups. The universal coefficient theorem applied to group cohomology proves to be useful in analyzing what happens when the coefficient structure is changed. In this section, I present some physical arguments for the necessity of a coefficient independent construction and, implicitly, of theories that do not depend on how precisely cohomology can probe certain topological features. If in the case of general relativity and quantum mechanics the covariance had to be implemented with respect to a symmetry group, in order to implement the topological covariance one has to consider the coefficient structures in (co)homology and the associated extensions. Probably the most important object for which the current discussion is relevant is a black hole. The problem of information conservation was discussed in the context of quantized fields over a given background in [1]. I partially follow the discussion presented therein, pinpointing the aspects where an extension of that treatment is necessary due to some ignored topological aspects. Considering, in agreement with [1] a massless Hermitian scalar field and an uncharged non-rotating black hole, after quantization one obtains a scalar field operator φwhich satisfies the wave equation φ= 0 (29) Given the background metric associated to the Schwarzschild spacetime [5] where the considered black hole is present one can rewrite this as (−g)1 2∂µ[(−g)1 2gµν∂νφ] = 0 (30) One can also define a conserved scalar product of the form (φ1, φ2) = iZdn−1x|g|1/2g0νφ∗ 1(x, t)←→ ∂νφ2(x, t) (31) the integral being over a constant thypersurface. When φ1and φ2are solutions of the field equation above and vanish at spatial infinity, then (φ1, φ2) is conserved. The existence of a flow of particles originating at a small affine distance from the event horizon has been derived in [2]. One particularity of this derivation is that the average number of outgoing particles in each mode is distributed in accordance with a thermal spectrum. Moreover, the full probability distribution, not just the average, of the emitted particles is that of thermal radiation. This observation creates a conflict with standard quantum mechanics when one considers the process of an in-falling object together with the radiation emitted on the external part of the horizon. The main issue is that this process does not preserve unitarity. 8 If the in-falling system is in a pure quantum state, the out-coming radiation is in a naturally mixed state. The full information related to the in-falling object is forever hidden behind the horizon. This result, however, appears only when one does not consider the process as described in a topologically covariant way. Using some of the observations in [9] I show here that there exists a special choice of coefficients in the quantization group cohomology for which the quantization prescription (particularly the group approach to quantization) allows a unitary connection between the outgoing radiation and the in-falling system. This suggests that the quantum information is in fact conserved, albeit not in the obvious way, but instead in a way visible only by means of cohomology with carefully chosen coefficients. In order to show this I continue the derivation of the spectrum of the Hawking radiation underlining the modifications in the way of thinking that must be considered in order to obtain the correct result. This method is in agreement with the AdS/CFT solution but its construction allows for a higher degree of generality. Let me now take the quantum fields used in the field equation above and decompose them as φ=Zdω(aωfω+a+ ωf∗ ω) (32) where fωand f∗ ωform a complete set of solutions of the field equation and are normalized according to (fω1, fω2) = δ(ω1−ω2) (33) The aωoperators are time independent. The standard method of quantization (second quantization) would be [aω1, a+ ω2] = δ(ω1−ω2) 0 = [a+ ω1, a+ ω2] = [aω1, aω2] (34) Let me chose the fωsuch that at early times and large distances they form a complete set for the incoming positive frequency solutions of energy ω. It is possible to compute the spectrum of the created particles by making an expansion of the field in terms of the late time positive frequency solutions. Let pωbe the solutions of the field equation that have zero Cauchy data on the event horizon and are asymptotically out-coming with positive frequency. Again, consider that in this domain pωand p∗ ωform a complete set of solutions. The normalization condition is (pω1, pω2) = δ(ω1−ω2) (35) There must also be an in-coming component of the solution at the event horizon at late times. Let me call this set of solutions qω. The superposition of these components at late times is localized on the horizon and has zero Cauchy data on the distant region. The components qωand q∗ ωform a complete set on the horizon and are normalized as (qω1, qω2) = δ(ω1−ω2) (36) The two components, being defined in disjoint regions are assumed to have null scalar product (qω1, pω2) = 0 (37) The expansion of the fields in terms of the above components is then φ=Zdω{bωpω+cωqω+b+ ωp∗ ω+c+ ωq∗ ω}(38) where bωand cωare the associated annihilation operators. The commutation relations are now [bω1, b+ ω2] = δ(ω1−ω2) [cω1, c+ ω2] = δ(ω1−ω2) (39) all other commutators are vanishing. The spectrum of the outgoing particles is determined by the coefficients of the Bogolubov transformation relating bωto aω′and a+ ω′. One may define the operators cωand c+ ωas the annihilation and creation operators for particles falling into the black hole. However, this definition is ambiguous due to the fact that the positive frequency components for the in-falling matter are not well defined. The physical meaning of these operators should therefore be taken as symbolic. Using the complete set given by fωand f∗ ωone can write pω=Zdω′(αωω′fω′+βωω′f∗ ω′) (40) 9 where αand βare complex numbers, independent of the coordinates. We can therefore calculate bω= (pω, φ) (41) and expressing φand pωin terms of fω′and f∗ ω′one can obtain bω=Zdω(α∗ ωω′aω′−β∗ ωω′a+ ω′) (42) and the invariant becomes (pω1, pω2) = Zdω′(α∗ ω1ω′αω2ω′−β∗ ω1ω′βω2ω′) (43) It is worthwhile noticing that the coefficients can be expressed as βωω′=−(f∗ ω′, pω) αωω′= (fω′, pω) (44) The discussion up to this point is unsurprising. The calculation of the coefficients above can be used in order to derive the average number of created particles observed at later times. However the exact form in which the previous calculations are being performed does not take the fact into account that the topology as encoded by cohomology groups changes when a black hole forms. Moreover, it is not clear that the coefficient groups in the associated cohomology are good enough to take into account the new topology. While the curvature of spacetime is correctly taken into account in the previous discussion, there are certain modifications required for the pairings to be isomorphically translated from the language of flat or curved spacetime to the language of spacetime with a horizon. It is important to notice that there are several possible choices of topologies over a space. One possible choice would be to consider any two points joined together in a subset for a specific topology if they can be connected by light in both directions. The space filled with low density dust before the formation of a black hole has every point connected in such a topology. Once a horizon forms the topology defined in the above way changes. Moreover, after the horizon is formed, any topology that, prior to the formation of the horizon, connected two points on different sides of what is now the horizon, must change in order to consider the new situation. Such a choice of topology would naturally incorporate causality. Various choices of topology have been discussed in [11-17]. For a discussion of homology with non-trivial coefficients and suggestive examples of the applications of the universal coefficient theorem ref. [18-21] could be relevant for the reader. For discussions on topology changes references [22-25] are recommended. Because of this change of topology, each of the constructions defined above has to be carefully analyzed. For this I will employ a group approach to quantization and adapt this language to the second quantization prescription. In doing this I mainly follow reference [34]. However, already at the level of the Poincare group, we face a problem when dealing with relativistic quantum mechanics. There appears to be no position operator ˆxsatisfying the commutation relation [ˆx, ˆp] = i~ˆ 1 with the ordinary momentum operator ˆp. The group approach to quantization is formulated in terms of the extended group ˜ Gwhich is a principal bundle with fiber U(1). In the case of geometric quantization one requires a symplectic form ωand the existence of a polarization i.e. a maximal (half the dimension of the manifold) isotropic distribution of vector fields with respect to the symplectic form. When such a polarization does not exist, geometric quantization cannot be performed. In the group approach, the symplectic form is replaced by dΘ where Θ is the left 1-form dual to the vertical generator (the generator tangent to the fiber). The non-polarizability of the geometric quantization is translated here into the absence of a first order full polarization i.e. a maximal left subalgebra containing Ker(Θ) and excluding the U(1)-generator. However, here, this anomaly can be avoided by adding additional operators in the left enveloping algebra to a non-full first order polarization and so defining a higher order polarization. Moreover, higher-order polarizations have been shown to be useful for representing a physical system in an equivalent but different realization than that given by the first order full polarizations. In the group approach to quantization, as the whole spacetime is embedded into a larger group structure containing the phase space of the problem, the group cohomology becomes a key element in identifying possible obstructions to quantization. Particularly, a situation not considered up to now is that in the process of black hole formation, the cohomology must be sensitive enough to detect the formation of a horizon. Therefore, whatever coefficient group was employed before the formation of the black hole, thereafter we need a coefficient group that gives sufficient sensitivity to the cohomology and allows a proper quantization by means of the group approach. Therefore, a set of requirements arise. First, a universal coefficient theorem for group cohomology must exist, which provides us with the properties 16 not defined to belong in the same open set then modifications must be implemented. Let Gand Kbe two abstract groups. A group ˜ Gis said to be an extension of Gby Kif Kis an invariant subgroup of ˜ Gand ˜ G/K =G. In terms of exact sequences this means that 1→K→˜ G→G→1 (94) is exact i.e. Kis injected into ˜ Gand ˜ Gis projected onto Gby the canonical homomorphism so that G=˜ G/K. However, the mere knowledge of Kand Gdoes not define ˜ Guniquely. In order to be able to discern extensions one has to define two exact sequences 1→Ki1 −→ ˜ G1 π1 −→ G→1 (95) 1→Ki2 −→ ˜ G2 π2 −→ G→1 (96) If the two group extensions are related via an isomorphism ˜ f: ˜ f:˜ G1→˜ G2(97) and the injective maps i1,2and the projections π1,2satisfy i2=˜ f◦i1 π1=π2◦˜ f(98) then the extensions are equivalent. Consider now the two group extensions, defined by two different two-cocycles ξ1 and ξ2with their group laws defined separately with simple brackets (...) for the first group and square brackets [...] for the second group: (g′, θ′)(g, θ) = (g′g, θ′+θ+ξ1(g′, g)),[g′, θ′][g, θ] = [g′g, θ′+θ+ξ2(g′, g)] (99) If there exists an isomorphism ˜ fas defined above and if we can rewrite (g, θ) = (e, θ)(g, 0) (100) (e, 0) being the identity of this law, ˜ fis completely determined when the images of (e, θ) and (g, 0) are given. From the conditions on the injection and projection above one obtains ˜ f◦i1=i2⇒˜ f(e, θ) = [e, θ] π2◦˜ f=π1⇒˜ f(g, 0) = [g, η(g)] (101) This implies a general form for ˜ fnamely ˜ f(g, θ) = [g, θ +η(g)] (102) The knowledge of ηdetermines the knowledge of ˜ f. However, ˜ fis also a homomorphism hence ˜ f(g′g, θ′+θ+ξ1(g′, g)) = [g′g, θ′+θ+ξ1(g′, g) + η(g′g)] (103) must be equal to ˜ f(g′, θ′)˜ f(g, θ) = [g′, θ′+η(g′)][g, θ +η(g)] = = [g′g, θ′+θ+ξ2(g′, g) + η(g′) + η(g)] (104) 17 and hence ξ1(g′, g) = ξ2(g′, g) + η(g′) + η(g)−η(g′g) = =ξ2(g′, g) + ξcob(g′, g)(105) where the notation ξcob(g′, g) is used for the two-coboundary generated by η(g). The calculation above gives a condition for the equivalence of extensions. One can see that proportional two-cocycles ξ2=λξ1may define equivalent groups but inequivalent extensions. In the case of the black hole formation the inequivalent extensions are those considered when constructing the pairings resulting in the definition of the out-going radiation. Therefore what we require is to have on one side the trivial situation far away from a horizon, and, on the other side of the pairing, the non-trivial situation, close to the horizon. We do not demand triviality or non-triviality of the cohomology in any of these cases. What is required however is that no matter what the sensitivity of cohomology, the pairing between homology and cohomology must remain covariant and objectively take into account such a transition from a region characterized by one topology to another region characterized by another. In this case we obtain a nontrivial factor that will correct the thermal nature of the out-coming radiation. In order to make the connection with the bracket construction and to classify the extensions one has to rely on a fiber bundle definition of the extension. Let therefore Gand Kbe abstract general groups and ˜ Gbe the extension of Gby K. One can relate the cosets of Kin ˜ G, each defining an element g∈Gwith the fibers over gof a fiber bundle that defines the extension. The fiber through ˜g0∈˜ Gis given by π−1(π(˜g0)) = {˜g|˜g=k˜g0, k ∈K}(106) A section of ˜ G(K, ˜ G/K =G) s:G→˜ G, s : (g)→s(g)(107) selects an element in ˜ Gin each fiber. Now, given a fiber π(s(g′′)) = π(s(g′)s(g)) (108) thus there exists a factor ω(g′, g)∈Ksuch that s(g′)s(g) = ω(g′, g)s(g′, g) (109) and this relation defines the factor ω(g′, g). One can define ω(g′, e) = ω(e, g) = s(e) and take s(e) = ˜e∈˜ G. Thus, one obtains the normalized section. Similarly one can obtain, for a normalized section, also a normalized factor: ω(g, e) = ω(e, g) = ω(e, e) = e∈K(110) As a general statement, relative to any normalized trivializing section s:G→˜ Gone can associate a factor system ω:G×G→Ksatisfying ω(g′′, g)ω(g′′g′, g) = ([s(g′′)]ω(g′, g))ω(g′′, g′g) (111) where [s(g)]k=s(g)ks(g)−1∀k∈K. According to this fiber bundle representation of the extensions, the group law of the group extension can be defined in terms of the factor system as (g′′, k′′) = (g′, k′)∗s(g, k) = (g′g, k′[s(g′)]kω(g′, g)) (112) Returning to the physical problem, the invariant bracket defined above, (φ1, φ2) = iZdn−1x|g|1/2g0νφ∗ 1(x, t)←→ ∂νφ2(x, t) (113) must be extended in order to obtain a topologically covariant description. The definition of the adjoint of the topological bracket can be identified as the right hand side of the universal coefficient theorem. When a choice of coefficients is considered such that the horizon of the black hole becomes visible one obtains a correction to the bracket as given by the factor that characterizes the extension of the homology group in a dimension smaller by one unit. It 18 will be this extension that will generate the quantization prescription to be used in the physical situation. The bracket is defined now with a correction in the group operation associated to its defining symmetry. Hence a topological factor is missing in the construction used in [1]. I underline that this factor is purely topological. Hence one has to extend the scalar bracket when a topological covariance is required: (φ1, φ2)′=< φ1, φ2> ω(φ1, φ2)(φ1, φ2) (114) where the < ... > notation refers to the topological invariant and ω(φ1, φ2) refers to the factor system that characterizes the extension and depends on the choice of the coefficient structure. This factor will appear also in the coefficients defining the probability of particle detection far from the black hole horizon. I must add that this is the first derivation of the fact that such a transformation is required, using almost only topological arguments. This, by itself is a very important conclusion. However, I am aware that a detailed calculation of fluxes might also be beneficial. This will be the subject of a future work. To make these considerations more accurate I will follow [1]. These results can also be interpreted in terms of the group approach to the second quantization. Consider the vacuum state at the infinite past as |0−>=XXλAB|AI>|BH>(115) where |AI>is the outgoing state with nja particles in the jth outgoing mode and |BH>is the horizon state with nkb particles in the kth mode going into the hole. Otherwise stated |AI>=Qj(nja!)−1/2(b+ j)nja |0I> |BH>=Qk(nkb!)−1/2(c+ k)nkb |0I>(116) One can chose an observable at the far future, composed only of {bj}and {b+ j}and operating only on the vectors |AI>. The expectation value of this observable can be written as <0−|Q|0−>=XXρAC QCA (117) where QCA =< CI|Q|AI>is the matrix element of the observable in the Hilbert space of the outgoing states. The density matrix is ρAC =XλAB ¯ λCB (118) and is associated to measurements in the far future but not to measurements of systems falling into the black hole. But, as has been shown above, the propagators for a transition from incoming matter to out-going radiation ∆(+) P(˜g′,˜g) =<˜g′|˜gP>=Pn∈IψP,n(˜g′)ψ∗ P,n(˜g) ∆(−) P(˜g′,˜g) =<˜g′∗|˜g∗ P>= ∆(+) P(˜g, ˜g′)(119) are matrices associated to non-trivial two-cocycles detected by cohomology with certain coefficients. Writing therefore the associated matrix elements in a topology covariant way implies adding additional factors correcting precisely the distribution of the outgoing radiation. But the visibility of the cocycles is controlled by the coefficients in cohomology. It is at this point where several extensions of the standard prescription are necessary. The above density matrix does not encode the full information that can be obtained in the far future. It does encode however everything that can be obtained from non-topological considerations. A particular form of the universal coefficient theorem is 0→Ext(Hi−1(G;R), M)→Hi(G;M)h −→ Hom(Hi(G;R), M)→0 (120) This can be interpreted in a form that resembles the interpretation of the non-commutativity of some physical observables: the third arrow Hi(G;M)h −→ Hom(Hi(G;R), M) (121) 19 maps the cohomology with coefficients in the group Minto the homomorphisms between the homology with coefficients in Rand the group M. The sequence is exact, hence this map is a surjection. This means there are no elements in the set of homomorphisms from the homology with coefficients in Rto the group Mnot represented in the cohomology with coefficients in M. However, there are elements in the cohomology that can be mapped into the same element of Hom. The second arrow Ext(Hi−1(G;R), M)→Hi(G;M) (122) is an injection. Hence the extension encodes the way in which the use of a coefficient structure instead of another changes the classes of the cohomology. This implies a change in the factors of the inner products used. One may ask if locality is preserved in this situation. Indeed, the problem of locality when unitarity is restored appears to be fundamental to the AdS/CFT solution of the information paradox [6], [8]. The information, in the approach of this work, is encoded in the global topological structure of the field in such a way that it is not accessible by any local measurements. One has to remember that the quantum field is not a measurable quantity. There is no physically observable ”quantum field” in the same way in which there is no physically observable wavefunction. Nevertheless, the global, topological properties of the fields (and wavefunctions) are important and encode relevant information. Any local measurement can be seen as a ”small” (weak) measurement. Can such a measurement reveal the global information? The correct answer to this question is no. Any weak measurement will reveal a weak information that will not provide any access to the information encoded globally and retrievable only via a statistical topological measurement. If one choses a coefficient structure for which the global non-triviality is invisible, locality is regained. Information is conserved but only in the factors appearing due to the use of the extension group. Hence unitarity is still preserved but in a ”hidden” form (in the extension). If one choses a suitable coefficient structure the global information becomes accessible due to the manifest visibility of the global non-triviality. However, one cannot recover the information unless one performs a probing of the topology. This may look non-local in a sense but the information obtained in this way concerns topologically non-trivial field (wavefunction) structures hence this ”non-locality” is not a physical one but rather one related to a choice of performing certain measurements. 5. CONCLUSION As a conclusion, I have shown that topological corrections to the thermal radiation of a black hole as given by the requirement of topological covariance of the laws of physics can account for a factor in the coefficients defining the thermal radiation. This factor imposes non-trivial changes in the form of the distribution function that amount to non-thermal corrections. 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