Quantization, holography, and the universal coefficient theorem

arXiv:1404.1800v3 [gr-qc] 31 Jul 2014 Quantization, Holography and the Universal Coefficient Theorem Andrei T. Patrascu1 1University College London, Department of Physics and Astronomy, London, WC1E 6BT, UK I present a method of quantization using cohomology groups extended via coefficient groups of different types. This is possible according to the Universal Coefficient Theorem (UCT). I also show that by using this method new features of quantum field theory not visible in the previous treatments emerge. The main argument is that several constructions considered as absolute until now may appear as relative, depending on individual choices of group structures needed to probe a topology. The universal coefficient theorem also gives information about how these structures as measured by different choices of groups, relate to each other. This may result in the formulation of new dualities and a deeper understanding of the relation between quantum field theories and gravity. PACS numbers: 03.70.+k, 04.60.-m, 11.15.-q, 11.25.Tq INTRODUCTION The quantization of gravity is a major unsolved problem [1]. The equivalence principle [2], the black hole information paradox [3], the holographic conjecture [4], emergence of space-time [5] or coarse graining of observables [6] are only a few concepts that followed from it. I present here a method that makes use of a theorem of algebraic topology and homological algebra (the universal coefficient theorem) in order to prove that some theoretical constructions used in previous descriptions of quantum gravity may not have an absolute meaning independent of some arbitrary choices of groups of coefficients. These choices of coefficients may induce different topological structures, therefore assuming independence of coefficient groups implies a form of independence of topology. The reason for considering this invariance as important in a quantum theory of gravity is the fact that there exist arbitrary choices that may make the connectivity of a space change. One can cite the formation of a black hole that makes matter in a region of spacetime collapse onto itself. After the collapse passes the horizon there is no method of avoiding the central region where quantum effects like spacetime topology change may appear. Another example is the choice of making extremely accurate length measurements in space. This implies adding energy in a given region. This may in the end generate horizons which imply the collapse of matter towards a region where quantum gravity and changes of topology are assumed to be possible. One may assume therefore that a full theory of quantum gravity may not depend on arbitrary choices of this kind in the same way in which the formal aspects of general relativity should not depend on a choice of a coordinate system. The applicability of the theorem is not restricted to space-time itself but can be used generally to field-spaces, groups, various manifolds or discrete spaces. Its use in these different situations will be made implicitly. The main idea of this paper is that the identification of relevant physical observables in the QFT context is strongly dependent on the choice of coefficient groups associated to (co)homology groups of the field space. The (co)homological structure of a field theory can be described with various coefficient groups, each inducing some indexation over the field space. It is well known that some choices are better than other. In general one uses a Z2-group when orientation is not relevant or a R-coefficient structure when continuum properties of the analyzed space appear to be relevant. However, there are more subtle applications of the coefficient groups. I show here that the choice of one coefficient group instead of another can hide a set of physically relevant observables in the quantization procedure. Also, the logical assignment of observables in an equivalence class dictated by the availability of a practical measurement of its spectrum by an observer may allow, by using the axiom of choice, the construction of predictors for the spectrum of other observables in the same equivalence class [7]. As a result, it appears to be impossible to assign an absolute topology to a space (be it “physical” spacetime or the space of field configurations) in the absence of an arbitrary choice of a coefficient group. I start with a field theoretical context. At this level already some aspects must be clarified. When quantizing a one particle theory one may use for example Feynman’s path integral formulation. This implies the existence of an “expectation catalogue” for positions in space-time indexed in some way. As no information about the intermediate steps is available one uses the principle of quantum mechanics that states that no actual state can be assigned to an object unless that state can be actually empirically confirmed to be realized. In this case the integration that gives rise to the quantum amplitude must be a sum over all possible configurations. An extension of this principle was necessary due to the Lorentz group. As one was not able to discuss in the context of special relativity about a predefined or fixed number of particles, quantum fields had to be introduced. These are simply extensions of the “expectation catalogues” of simple one-particle quantum mechanics. They are not “measurable” in any physical sense individually, but their interference and their topol- 2 ogy is probed statistically by the rules of quantum mechanics. It should be well known that the statistics of an experiment (say Bohm-Aharonov) depends on the topology of the field space (the regions where the wavefunction is defined). In the end, the statistics must probe all connected components of all possible configurations. In the case of quantum gravity there are different approaches on how a quantum field theoretical formulation should look like. It is however clear that such a formulation should exist. I refer here to the works on string field theory [21]. There the “quantum field” becomes a world-sheet-string-field “expectation catalogue” which is expanded even more with respect to the previous situations. While a string-field theoretical approach exists, it is not clear how the various configurations interrelate and what configurations can exist in various situations. Dualities are supposed to help in this aspect by identifying configurations and simplifying the overall problem. It appears to me that there exists a general method of constructing such dualities based on the ideas presented in this article. It also appears to me that the constructed dualities will have an applicability restricted to specific arbitrary choices of group-structures in topology. This is conjectured to be valid also for the holographic principle. It is the universal coefficient theorem that will in the end provide a description of what configurations can be simultaneously known and what configurations will interfere at the level of the “catalogue of expectations”. It also appears that the change of topology is of major importance in quantum gravity as one expects a change in the topology of spacetime during the formation of a black hole. However, the form of the laws of nature should not depend on a specific topology. I partially follow in this introduction reference [8]. First construct a functor Efrom the category of spacetimes (Loc) to the category of local convex vector spaces (V ec). This functor associates to each spacetime Ma configuration space E(M) of fields defined on it. The isometric embeddings χ:M→Nare mapped into pullbacks χ∗:E(N)→E(M). The space of the observables called Fwill be the space of the functionals F:E(M)→R. It is at this point that one also has to define the topological structure of the space (or space-time M). Physically this remains uncertain unless a choice of a coefficient group in (co)homology is made. This will define the topology and will allow a specific definition of the observables. Essentially the “experimental setup” (or a coefficient group choice) tells spacetime how to connect. This connection tells quantum mechanics how the correlations between “expectation catalogues” should be constructed (what observables make physical sense). What follows is standard quantum mechanics which (via the universal coefficient theorem) tells the experimentalist how to connect the results obtained with one group structure to possible results obtained by other observers using other group structures. This is important when one compares, for example, the observations made when falling towards a black hole to those of a far away observer. Finally, accurate measurements and probing of spacetime at small scales implies adding energy in a small region of space which in the end may alter the topology of spacetime itself. One can observe that in principle a topology induced by a choice of a coefficient group (via a particular experimental setup) results in a modified set of observables and a modified algebra for the resulting quantum (field) theory. Also, the geometry of the (field) space imposes restrictions on possible topologies (for example extreme curvature may imply restrictions over the allowed topologies). One can summarize this as T opology probed by quantum mechanics induced by a choice of a coefficient group ⇆Geometry well defined local quantum observables quantum operator algebras  In this context the main question for quantum gravity is “how do different geometries correlate?” To this question one can give an answer when one considers the topology of the field space and the fact that this topology is not given in an absolute sense. The acceptance of the non-universality of topology (as proved clearly by the universal coefficient theorem) leads to different “counting rules” for different contexts. In what follows one defines the class of functionals called “local functionals” as F(φ) = ZM dvolMf(jx(φ)) (1) where jx(φ) = (x, φ(x), ∂φ(x), ...) is the jet of φat the point x. Let Lbe a suitably defined Lagrangean. We can define an associated action functional S[L[φ]]. The field equation becomes in this context S′ M(φ) = 0 where the prime denotes the Euler-Lagrange derivative. The space of solutions of this equation forms a subspace of E(M) called ES(M). In the context of classical field theory 3 one is interested in the space of local functionals over ES(M) called FS(M). This space can be defined as the quotient FS(M)=F(M)/F0(M) where F0(M) is the space of functionals that vanish on-shell (on ES(M)). A (co)homological interpretation for the FS(M) space is required. For this one needs a vector field structure on the configuration space. The action of the vector fields X[.] on the space of smooth functionals C∞(E(M)) is ∂XF[φ] =< F[φ], X[φ]>(2) One can associate to the action functional a map from the set of test functions over the spacetime manifold to the space of “observable”-functionals δS:D(M)→F(M) such that φ7→< S′ M[φ], X[φ]>=δS(X)(φ) (3) where S′ Mis the Euler-Lagrange derivative of the action. Suppose there is an action Ssuch that F0(M) = δS(D(M)). Then FS(M) = F(M)/F0(M) = F(M)/Im(δS) (4) From this one can construct the chain complex 0→D(M)δS −→ F(M)→0 (5) This can be associated with the Batalin-Vilkovisky complex used in the geometric quantization. The 0-order homology of this complex is FS(M) = F(M)/F0(M). The set of critical points of the action functional {φ∈D(M)|δS[φ] = 0}(6) contains connected components that can be identified by the first homotopy group π0({φ∈D(M)|δS[φ] = 0}) (7) The functionals on the classes of this group are the gauge invariant observables. One can see that the correct identification of possible maps as well as homotopically equivalent structures is extremely important for the correct construction of the field space in the phase preceding actual quantization. Probably the best mathematical formalization of quantum mechanics is offered by what is known as “geometric quantization” [9]. In this formulation one starts with a classical theory and follows a set of steps that assure the consistency of the resulting quantum theory. One may start with a general classical action depending on a set of fields S[φ]. This implies the existence of a symplectic manifold. The main idea is to realize the symplectic form of this manifold as the curvature of a U(1) principal bundle with a connection. We obtain the pre-quantum Hilbert space as the Hilbert space of square integrable sections of the principal line bundle. One has to pick for each point in this space a certain subspace of the complexified tangent space at that point. One defines the quantum Hilbert space to be the space of all square integrable sections of the line bundle that give 0 when differentiated covariantly at that point in the direction of any vector of the tangent space. As basic quantum mechanics teaches us there exist two sets of variables that become non-commutative operators when quantizing. These may be called “positions” and “momenta” although their physical meaning may be rather different. The next step is the choice of a polarization i.e. the choice of “positions” and “momenta”. This choice is not unique. Once a polarization is available one can form a Hilbert space of states as the space of sections of the associated line bundle. The last step would be to associate to the classical variables actual quantum operators on the quantum Hilbert space. This amounts to the quantization of observables while mapping Poisson brackets to commutators. This procedure is in general not well defined for all operators. Strictly speaking the method of geometric quantization is not properly defined in the context of quantum gravity. The definition of a fieldspace or a space of configurations is extremely complicated and the integration over such a structure appears to be ill-defined. However, it is precisely the method presented in this article that may add some extra structure to this space (for example via the addition of new dualities) such that its rigorous definition might become possible. Several attempts of using geometric quantization in the context of string theory are known [23],[24] but the subject remains open for future research. Given a BV complex and some quantum observables in the context of a choice of a coefficient structure I now state the following Lemma Lemma 1 (The Universal Coefficient Theorem) If Cis a chain complex of free abelian groups, then there are natural short exact sequences 0→Hn(C)⊗G→Hn(C;G)→T or(Hn−1(C), G)→0 (8) ∀n,G, and these sequences split. Here T or(Hn−1(C), G) is the torsion group associated to the homology. In this way homology with arbitrary coefficients can be described in terms of homology with the “universal” coefficient group Z♭ This lemma is also valid for cohomology groups where it is formulated as 0→Ext(Hi−1(C), G)→Hi(C;Z)⊗Gh −→ Hi(C;G)r −→ Hom(Hi(C), G)→0 (9) 4 where now the T or group on the right is replaced by the Ext group on the left. Moreover, this theorem is a property of algebraic topology independent of the existence of an underlying manifold structure for the spaces or groups on which it may be applied. For a proof in both the homology and the cohomology cases see reference [10]. The following example shows how the choice of the coefficient group can affect the correct identification of the homotopy type of a function. Example 2 (Homotopy and coefficient group) Take a Moore space M(Zm, n) obtained from Snby attaching a cell en+1 by a map of degree m. The quotient map f:X→X/Sn=Sn+1 induces trivial homomorphisms on the reduced homology with Zcoefficients since the nonzero reduced homology groups of Xand Sn+1 occur in different dimensions. But with Zmcoefficients the situation changes, as we can see considering the long exact sequence of the pair (X, Sn), which contains the segment 0 = ˜ Hn+1(Sn;Zm)→˜ Hn+1(X;Zm)f∗ −→ ˜ Hn+1(X/Sn;Zm) (10) Exactness requires that f∗is injective, hence non-zero since ˜ Hn+1(X;Zm) is Zm, the cellular boundary map Hn+1(Xn+1, Xn;Zm)→Hn(Xn, Xn−1;Zm) (11) being exactly Zm m −→ Zm(12) One can see that a map f:X→Ycan have induced maps f∗that are trivial for homology with Zcoefficients but not so for homology with Zmcoefficients for suitably chosen m. This means that homology with Zmcoefficients can tell us that fis not homotopic to a constant map, information that would remain invisible if one used only Z-coefficients. ♭ As the final step of this introduction I state here the main theorems of this article as well as a conjecture. Theorem 1 (Relativity of Observables) There exist observables visible using some choices of coefficient groups and invisible using other choices. ♭ Theorem 2 (Relativity of distinguishability) There exists no unequivocal measure of distinguishability of quantum states that is independent of the choice of the coefficient group. Distinguishability is relative. ♭ Theorem 3 (Relativity of Symmetry) A particular choice of a coefficient group makes a specific symmetry structure in the field space manifest. There exists no absolute symmetry. ♭ Conjecture (Relativity of Holography) There is no general unequivocal mapping of any consistent geometric structure in a space-time volume to its surface. In the full context of quantum gravity the existence of a holographic principle is an undecidable statement depending on particular choices of the coefficient groups. “Strong-weak” dualities can however be constructed and generalized in a case-by-case way ♭ The proofs of the theorems as well as validity arguments for the conjecture are provided in the following chapters. The method of proof is as follows: I make a choice of a coefficient group in cohomology (i.e. a choice of topology). I try to construct standard quantum mechanics (eventually using geometric quantization). If geometric quantization is impossible I can always switch to a different topology where this method is possible and see how it relates to the topology where it was impossible via the universal coefficient theorem. This may bring new insights about the geometric quantization prescription. I construct a set of observables and physical states using a particular choice of the coefficient group. I obtain a set of physical states obeying some properties (distinguishability, etc.). I make another choice of the group structure where the above stated properties are not valid any more. By the Universal Coefficient Theorem it follows that the considered properties are relative i.e. cannot be associated to a full theory of quantum gravity. PATH INTEGRAL QUANTIZATION AND FIELD THEORIES One method of quantization is given by what is known under the name of “Feynman path integral” [22]. This has been generalized although not completely, for string field theory [21]. For an introduction I partly follow [22]. I assume that the standard prescription of computing quantum probabilities using quantum amplitudes is well known. If Pac is the quantum probability of measuring event cwhen it follows the measurement of event athen the probability must be calculated as Pac =|ϕac|2where ϕac =Pbϕabϕbc where the sum is over the possible intermediate states bwhich, I emphasize, following Feynman (ref. [22], page 3 in manuscript) have no meaningful independent value. In a 1-space and 1-time dimensional context a succession of measurements may represent a succession of the space-coordinate xat successive times t1, t2, ..., where ti+1 =ti+ǫ. Let the observed value at ti be xi. Classically the successive values of x1, x2, ... define a path x(t) when ǫ→0. If the intermediate positions are actually measured one may talk about such a path with a well defined set of observed positions x1, x2, ... and the probability that the specified path P(...xi, xi+1, ...) lies in a region Ris given by the classical formula P=ZR P(...xi, xi+1, ...)...dxidxi+1... (13) where the integral is taken over the ranges of the variables which lie within the region R. If the intermediate positions are not measured then one cannot assign a value to them. In this case the probability of finding the outcome of a measurement in Ris |ϕ(R)|2and ϕ(R), i.e. the 5 probability amplitude, is calculated as ϕ(R) = lim ǫ→0ZR Φ(...xi, xi+1, ...) (14) where Φ(...xi, xi+1, ...) defines the path. In the given limit this object becomes a path functional. There should be no mystery nowadays that the probability amplitude should be calculated as ϕ(R) = lim ǫ→0ZR exp[i ~X i S(xi+1, xi)]...dxi+1 A dxi A... (15) where Sis the action functional for the given path segment. In order to go a step further and define the wavefunction in this context I will continue to follow Feynman’s paper [22]. The region Rconsidered above can be divided into future and past with respect to a choice of a time position t. One can define the region R′as the past and the region R′′ as the future. The probability amplitude connecting these regions will be ϕ(R′, R′′) = Zχ∗(x, t)ψ(x, t)dx (16) where ψ(xk, t) = lim ǫ→0ZR′ exp[i ~ k−1 X i=−∞ S(xi+1, xi)]dxk−1 A dxk−2 A... (17) and χ∗(xk, t) = lim ǫ→0ZR′′ exp[i ~ ∞ X i=k S(xi+1, xi)] 1 A dxk+1 A dxk+2 A... (18) In this way one can separate the “past” and the “future” via the functions ψand χ. One may also construct a closer equivalence to the matrix representation of quantum mechanics by introducing matrix elements of the form < χt′′ |F|ψt′>S= lim ǫ→0Z... Zχ∗(x′′, t′′)F(x0, ...xj)exp[i ~ j−1 X i=0 S(xi+1, xi)]ψ(x′, t′)dx0 A...dxj−1 Adxj(19) In the limit ǫ→0, Fis a functional of the path x(t). At this moment one can define various equivalences between functionals. These are to be associated to operator equations in the matrix formulation. One can of course define ∂F ∂xkand one can calculate the associated matrix element using an action functional S. Using the fact that the action functional appears as exp(i ~S) one obtains matrix equations as, say < χt′′ |∂F ∂xk|ψt′>S=−i ~< χt′′ |F∂S ∂xk|ψt′>S(20) which can be stated as a functional relation defined for an action Sas ∂F ∂xk↔ −i ~F∂S ∂xk (21) Using the fact that S=Pj−1 i=0 S(xi+1, xi) one can rewrite ∂F ∂xk↔ −i ~F[∂S(xk+1, xk) ∂xk +∂S(xk, xk−1) ∂xk ] (22) In the case of a simple 1-dimensional problem one can write ∂S(xk+1, xk) ∂xk =−m(xk+1 −xk)/ǫ (23) and ∂S(xk, xk−1) ∂xk = +m(xk−xk−1)/ǫ −ǫV ′(xk) (24) Neglecting terms of order ǫone obtains m(xk+1 −xk) ǫxk−m(xk−xk−1) ǫxk↔~ i(25) The important aspect here is that the order of terms in a matrix operator product corresponds to the order in “time” of the corresponding factors in a functional. The order of the factors in the functional is of no importance as long as the indexation of these factors is reflected in the ordering of the operators in the matrix representation. This means the left-most term in the above equation must change order so that one obtains the well known commutation relation px −xp =~ i(26) One may observe that the choice of a specific indexation of the measurement outcomes, according to a time index (i.e. Z-group), leads to the well known commutation relations. The ideas behind path integral quantization are kept intact when going to the relativistic context. However, when we have to go to a gravitational context the sum over configurations (geometries) 6 becomes non-trivial. In this sense one has to construct the (co)homology structure of the space and one has to deal with the universal coefficient theorem. This theorem states that a specific framework, constructed through 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. However, some choices of coefficient groups may make some observables manifest while others may hide them. Moreover, simple order relations as the ones used in the proof above are no longer uniquely defined. What was identified by Feynman as a natural choice (time ordering) may in fact be just the result of a given coefficient group. Other ordering relations (like radial ordering in the case of CFT’s) are also known. It is visible in this context that the construction of a path integral prescription using another coefficient group will change the quantization prescription (as formulated via the algebra of operators). Quantization doesn’t mean only algebra of operators, as has been made obvious in the definition of geometric quantization. In an ideal situation one would expect a physical motivation that determines the operator algebra. This might appear in the context of the application of universal coefficient theorems. The group structure imposed over the configuration space can be chosen for example as R/Zcase in which one arrives at a continuous cyclic structure. This will present a somehow altered operator algebra. One may ask what is the physical meaning of the coefficient group? In fact, it is an extra layer of information that has to be dealt with when performing quantization. It appears that it is not sufficient to simply integrate over non-equivalent field configurations as done in nongravitational models. The coefficient structure adds new “degrees of freedom” to the problem. These must be considered when performing path integral quantization in order to obtain suitable unitary results. From this point of view, the extra-structure appears to be a step forward towards the unambiguous solution of the unitarity problem (also known as “information paradox”). In a less formal tone, the “information” describing the system is encoded not only in the actual system but also in the set of rules one chooses in order to “read” that information. I stressed in the above digression that the intermediate states in the path integral formulation must be added to the amplitude while keeping all possible outcomes, mainly because one cannot assign an outcome before a measurement is performed. The same considerations are valid when dealing with coefficient groups. While one can certainly prepare an experiment that involves a special choice of a coefficient group one will obtain a result dependent of this choice. When no practical choice is made one cannot assign any “physical” value to the choices of coefficients but one must consider them when calculating quantum amplitudes. From this perspective the question of the existence of a “Planck scale topology” is void of meaning. “Microscopic geometries” are to be associated to choices of coefficient groups and these choices are arbitrary. However, the universal coefficient theorem generates classes of topologies that can be identified in the sense of having the same Ext and T or groups. This may lead to an overall simplification of the path integral formulation as many configurations will appear as connected by dualities. One should notice that both string theory and quantum loop gravity assume special choices of topology as being absolute (Lie group topology for string theory as the “string worldsheet” and discrete topology for LQG). I consider these choices as an epistemological issue. In string theory one starts by postulating a fundamental string. This implies a continuous group structure and a well defined topology. By the universal coefficient theorem however, this is simply a convention. Using that convention one arrives at an algebra of operators (say, Virasoro algebra). It should be clear now that this choice has nothing fundamental to it. In quantum loop gravity one starts the other way around: one fixes the canonical quantization prescription involving the standard algebra and obtains in the end a particular topology (a discrete topology). Again, one arbitrary choice determines the other. There is nothing fundamental to it either. One cannot assign a precise topology to any space unless one makes a choice of a coefficient group in cohomology. In order to do this one must consider the universal coefficient theorem and its T or and Ext groups. Any fixation on an absolute topology would be equivalent with the postulation of the “ether” in special relativity i.e. void of meaning. One may notice that quantum gravity cannot be defined using a fixed (non-dynamical) spacetime manifold. In fact, analysis in terms of the universal coefficient theorem makes the spacetime highly dynamical allowing even changes of topology. These can be seen if one considers for example coefficient groups of finite torsion degrees. The larger (but finite) the torsion degree of the group the more “non-local” will the associated “observables” look. The ”non-local” behaviour in extreme conditions (black holes) is essentially the result of a specific choice of topology. This will persist until clearer information about the group structure imposed by a particular experiment is given. When this happens is for the experiment to decide. The situation is similar to the supposed ”objective collapse of the wavefunction” which is assumed (wrongly) to actually happen at some scale. This mistake vanishes when one understand that the wavefunction is to be interpreted as a ”expectation catalogue”. In the same way, when information about the connectivity of spacetime and of the ”field-space” becomes manifest one will have to adopt the local structure at hand. Of course, topologically disconnected macroscopic black holes may retain (from the perspective of an observer lying outside) some apparent non-local aspects as their internal structure is inaccessible. 7 One may ask if my method has as result the identification of different representations for the same algebra of operators. This is not the case. As can be seen from Feynman’s example the specific ordering of the events generates some commutation relations which define the algebra of operators. If one generalizes this to different choices of coefficient groups for probing the field space one can see that the algebra of operators will not be preserved. Indeed, one can use coefficients in a continuous group. In this case one can recover the string-theoretical case where a continuous line-like object appears as “fundamental” and in fact the algebra of its operators is rather different. The associated group is generally not easily connected to the local algebra as the Exp map is not always easily defined. Continuous group coefficients are useful. It is well known that one uses continuous coefficient groups when one wishes to avoid unnecessary complications due to the low-scale behaviour of the space to be studied. In fact, a claimed advantage of working with string-like objects is its so called “UV-completeness”. Of course, from the perspective of coefficient-group-extended quantization this property is just a trade-off between using continuous groups in order to have UV-completeness and the complications that appear in the BRST-cohomology treatment of string theory. RELATIVITY OF OBSERVABLES As shown in the introduction, the physical observables are to be identified with the functionals over the classes of the homotopy group associated to the critical points of the action functional. Example 2 already showed how this identification is relativized by the UCT. I give here a more detailed proof. Take a set of observables obtained after geometric quantization A={A1, A2, ..., An}(27) where A ⊂ FS. While in the classical case FSis to be associated with a space of local functionals, in the case of quantum gravity the locality condition may be relaxed (see ref. [11]). One can observe that the BV-complex 0→D(M)ι −→ F(M)γ −→ FS(M)→0 (28) with FS(M) = F(M)/F0(M) and δS=γ◦ιcan be represented as the complex of example 2 0→˜ Hn+1(X;Zm)f∗ −→ ˜ Hn+1(X/Sn;Zm)→... (29) In the last case f∗is the induced map over the homology groups of the map f:X7→ X/Snover the analyzed spaces. In the case of the BV-complex the original maps would be the functionals F:ES7→ ESwhich are to be associated to the physical observables of the quantum theory. In the same way as in example 2 one can define the map as a function of degree m. One may remark that observables that cannot be distinguished in Zwill be visible if the choice of coefficients is Zm. In order to have a correct representation of the actual set of observables one must redefine Aas ˜ A={[A1],[A2], ..., [An]}(30) where each term [Ai] may be a set of observables on its own, the elements of which may not be discernible given a specific choice of coefficients. It has been noted in reference [11] that for example classes of microscopical observables of black holes may be inaccessible to independent measurement due to large energies or long times required for accurate probing. While this is certainly possible I show here that the same can happen due to certain choices of coefficient groups. While it is certainly always possible to change the coefficient group with which one probes the field space this change may involve a change in the physical experimental setup. This would make a simultaneous use of two coefficient groups in the same experiment impossible. As indiscernability of observables (coarse graining) may imply emergent locality (as shown in [11]) it may look like the UCT assures some form of locality at all levels. However, I am cautious in calling this “locality” with its proper name. I am also cautious when speaking about “emergent locality” or even more drastically, “emergence of space-time” (see ref. [5]) The reasons for this caution are expressed in the following section. RELATIVITY OF DISTINGUISHABILITY Ongoing research in quantum information has led to various alternative definitions of distinguishability of quantum states. One recent paper [11] argues that physical criteria like extreme energy requirements or long waiting times would make some distinctions between quantum states impractical. I show here that in fact distinguishability of quantum states is mainly related to choices of the coefficient groups of (co)homology. There exist possible predictors that allow “guesses” concerning the presence of different physical states in the same equivalence classes associated to some observers [7]. Using quantum information tools one observes that given a set of observables Aone cannot distinguish a random pure micro-state in a microcanonical ensemble HEof dimension dEfrom the maximally entangled state ΩE=IE dE unless the number of different outcomes of the operator N(A) scales as √dE. Whenever N(A)∼√dEone would require a long time or very large energies to achieve the accuracy that would allow the distinction of these states. These statements presented also in [11] are partially correct. While one can follow the standard path of constructing normed or semi-normed spaces that would 8 predict how “far away” quantum states are in a given configuration I show here that these measures must be relative considering the fact that the arbitrary choice of a coefficient group may make the difference between distinguishability and indistinguishability of two quantum states relative. This statement is in full agreement with the uncertainty principle and in the spirit of quantum mechanics as it extends the concept of uncertainty to the arbitrary choice of a coefficient group. In this section I follow ref. [11] in order to introduce the concepts I require. Consider a finite dimensional subspace HE⊂Hof dimension dEconsisting of all pure states ψ=|ψ >< ψ|that live in a microcanonical ensemble of energy [E−δE, E +δE]. I may assume that the Hamiltonian describing the unitary time evolution of the system has non-degenerate energy gaps. Consider again the set of observables A={A1, A2, ..., An}. One may ask what are the necessary conditions for such a set to distinguish a random pure state ψ∈HEfrom a maximally mixed state in HE. One can follow two obvious paths and one less obvious path to quantify the difference between quantum states ψ∈HE. What one obviously could do is to measure the expectation value of some operator A∈ A. However, the measurement of expectation values of an observable is not sensitive enough to distinguish any different quantum states. A quantum measurement in general offers a set of eigenvalues aappearing with some probabilities pa. Most of the information about the quantum system is encoded in the probability spectrum {pa}. Hence in order to distinguish two quantum states ρand σ using a particular observable Aone can define a measure as DA(ρ, σ) = 1 2X a|tr(|a >< a|ρ)−tr(|a >< a|σ)|(31) |a > being the eigenvectors of A. This measure is defined so that it encodes the information of the entire spectrum {pa}. One can extremize the definition in order to define a measure over a whole set of observables DA(ρ, σ) = max A∈A DA(ρ, σ) (32) If Aincludes the entire set of observables in the Hilbert space one may define the distinguishability of two quantum states in general as D(ρ, σ) = 1 2tr|ρ−σ|A(33) where |ρ−σ|Ais the maximal difference in probability spectra over the entire set of available observables. If I continue to use this language it will be impossible to identify the restrictions due to the universal coefficient theorem. In fact one has to go a step back and to remember that quantisation implies summation over inequivalent field configurations and this implies the construction of (co)homology groups. Physical observables are identified with the functionals over the classes of these groups. Different choices of coefficient groups in the (co)homology may lead to identification of functionals (they may appear as homotopic to the identity) while using other groups may make them appear in different classes (i.e. being different observables). Considering that special features of the field space induced by mappings of finite degree cannot be ignored in the procedure of quantization one may have for a complex like 0→˜ Hn+1(X;Zm)f∗ −→ ˜ Hn+1(X/Sn;Zm)→... (34) a set of observables A={A1, A2, ..., An}while under 0→˜ Hn+1(X;Z)f∗ −→ ˜ Hn+1(X/Sn;Z)→... (35) another set ˜ A={[A1...Ai1],[Ai2...Ai3]..., [Aik...Ain]} where the observables in the square brackets represent the classes of observables that cannot be distinguished in the given coefficient setup. One may imagine that the choice of a coefficient group induces a forgetful functor between the category of observables Aand ˜ A. This functor also maps the discernability measure from DA(ρ, σ) = max A∈A DA(ρ, σ) (36) towards D˜ A(ρ, σ) = max A∈˜ A DA(ρ, σ) (37) One may observe that although the definition is still valid, the set of available observables changed significantly. One may look at this as a change of topological basis although this analysis may be beyond the scope of this article. In the last section I invited to caution in using terms like locality in relation to indiscernability of observables and entanglement. Indeed, the prescription of maximization used in the definition of the measure above is not trivial. Following the universal coefficient theorem, in order to establish the maximum over the set of observables, one will always have to pick one element from an equivalence class. One may not be aware of the existence of more than one element in the given class but the class exists and a choice has to be made in order to be able to compare in the end representatives from various classes. In order to be able to do this (as the elements of one class are supposed to be indiscernable so one cannot define a choice function) one has to invoke the axiom of choice. However, associating probability theory and the axiom of choice in the context of quantum mechanics is probably the most non-trivial task in mathematical logics. Examples of how the axiom of choice reflects on the mathematics of coordinated inference can be found in [7]. A suitable analysis of these problems in the realm of quantum information is the subject of a future paper. What I may add here is that the indexation of operators 9 in Aand ˜ Amay give an order relation in terms of, for example, energy. In this sense one may define the order over the operators in Aas A1≺A2≺... ≺An(38) This ordering implies the visibility at a given energy. However, the deformation of some observables such that they enter a single homotopy class after the application of a new coefficient group may alter this order. In fact, one will have to define an order relation between equivalence classes where the choice of representatives is not unambiguously defined in the absence of the axiom of choice [Ai1][Ai2]... [Ain] (39) Nothing stops this new ordering to invert the previous one in some instances such that observables invisible at some energy and choice of coefficients become visible under another choice of coefficients. It follows that new “strong-weak” dualities can be constructed using the method of coefficient groups. Their applicability goes beyond quantum gravity to subjects like condensed matter or many particle systems. Everything one has to do is to re-quantize the theory using a different coefficient setup and to take into account possible torsion groups in homology. While theoretically this is possible it remains to be seen if there are practical difficulties. This will be the subject of another article to be developed in the near future. Another aspect that might be important in this context is the similarity of these problems with the “hat problems” discussed in [7]. The main idea is that although it may look unlikely, there might exist predictors that after a finite set of trials are always capable of assigning the equivalence class of an operator and determine an order of occurrence. These predictors however, depend on the availability of the axiom of choice so they are outside of the scope of this paper. However, their existence may suggest that exact locality may be dependent of some very particular choices. One may also ask if the renormalization prescription is affected by the indiscernability of states induced by choices of (co)homology. Possible emergence of new “topological” Ward identities (i.e. having their origin in some remaining “invariance” under change of topology, prescribed by the UCT) may have important roles in a possible renormalization of gravity. RELATIVITY OF SYMMETRY Symmetries are of major importance in physics in general and in quantum field theories in particular. They manifest themselves in the quasi-invariance of an action under the transformations of a group. The fact that one has quasi-invariance (i.e. invariance up to a total derivative) of the action under a group may be irrelevant classically, however, it is important in quantum mechanics as it allows the construction of group-invariant quantum equations (Schrodinger-equations when the group is the non-relativistic Galilei group for example). One may notice that the existence of a quantum formulation of the laws of physics is related to the existence of non-trivial (phase) factors (i.e. additive terms in the composition rule of the group operation, see [16]) that cannot be reduced to zero for all group elements (i.e. they form nontrivial classes in the second cohomology of the transformation group). One also observes that the existence of basic quantum effects is a result of the global (topological) properties of the groups associated to the supposed “natural” symmetries (Galilei group, Lorentz group, conformal group, etc.). These properties are probed via group (co)homologies. Information about a group (or in general a space) is not only encoded in the group (space) itself but also in the way in which the group (space) acts (is mapped) into some reference module (space). This is why one can study group properties by analysing the actions of the group on an associated space. On that space one can construct a CW complex and analyse it via combinatorial techniques. Moreover, information about a group (space) may also be encoded in the way in which one probes that group (space). One can classify the various ways in which information about a group fails to be encoded geometrically (i.e. non-topologically)1by using cohomology groups of different orders. For example the classes of the second cohomology group H2(G, U(1)) i.e. the cohomology group of the maps between the analyzed group Gand the unitary 1-dimensional group U(1) encode the global character of the factors in the composition rule of the group-operation in Gi.e. the way in which they fail to vanish globally [16]. The non-trivial third cohomology group H3(G, U(1)) encodes the failure of the associativity property of the composition rule [16]. Also, the existence of not globally vanishing (phase) factors induces super-selection rules. They are induced in standard quantum mechanics by the presence of nontrivial operators that commute with all the observables and thus belong to any complete set of commuting observables. As a result, these operators decompose the Hilbert space of all possible states of a system into coherent subspaces characterized by their eigenvalues. The superposition principle holds only inside these superselection subspaces and no observable may have non-zero matrix elements between states of different superselection eigenvalues. As an example one may consider the mass of particles in a space acted upon by a Galilei group. Bargmann superselection rules arising due to the topology of the Galilei group forbid for example mass decay (i.e. physical subspaces corresponding to different mass 1I contrast here geometrical and topological results although they might be related, see for example Gauss-Bonnet theorem, etc. 16 with a classical action Scl =Zd4x(−1 4Fα µν Fαµν +Lmatter[A, ψ, ¯ ψ]) (64) where ψ,¯ ψare the matter fields, Ais the gauge field and Fµν is the field strength tensor (also for a non-abelian theory). Suppose there exists a gauge anomaly and suppose one adds a local counter-term of order 3 in the coupling constant gcalled ∆Γ such that Scl →Scl +1 6Zd4pd4q∆Γµνρ αβγ(−p−q, p, q)Aα µ(−p−q)Aβ ν(p)Aγ ρ(q) (65) At order g3such a term modifies the 3-point vertex function as Γµνρ αβγ →[Γµνρ αβγ]new = Γµνρ αβγ + ∆Γµνρ αβγ (66) If one can find a local ∆Γ such that (pµ+ qµ)[Γµνρ αβγ]new(−p−q, p, q) = 0 then one says the anomaly is irrelevant. Whenever such a local counter-term does not exist the anomaly is relevant. One may notice that the “relevance” of anomalies is due to their failure to be cancelled locally. As stated in the main paper, relevant anomalies can be associated to non-trivial BRST cohomology classes at ghost number one. Let now [Γµνρ αβγ]new →[c]. The arrow maps the transformed 3point vertex function to a (co)homology class of the group Hn(X) where Xis the associated space. The description here is formal; only the reasoning is of importance. Using the UCT one can see that the cohomology group is determined via the short exact sequence: 0→Ext(Hi−1(X), A)→Hi(X;Z)⊗Ah −→ Hi(X;A)r −→ Hom(Hi(X), A)→0 (67) One can now chose Asuch that the map X→X/([c]∼ id) becomes trivial. In this case one cannot distinguish the class of the previously “relevant” anomaly from the identity over X. This assures that there exists a coefficient structure over the cohomology that trivializes the anomaly. This comes at a cost. One must introduce the extension group on the left Ext(Hi−1(X), A). The extension group is generally defined in association with the Ext functor. Its definition is not particularly involved: let Rbe a ring and let ModRbe the category of modules over R. Consider B∈ModR, take a fixed A∈ModR and define T(B) = HomR(A, B) as the set of homomorphisms over Rfrom Ato B. The Ext functor is defined as Extn R(A, B) = (RnT)(B) (68) This can easily be calculated considering the injective resolution 0→B→I0→I1→... (69) and computing 0→HomR(A, I0)→HomR(A, I1)→... (70) where we excluded HomR(A, B) from the complex. Then the extension (RnT)(B) is the homology of this complex. So, in the particular case above, the existence of anomalies is “shifted” into the way in which one can nontrivially map a general group into an abelian group. The relevant information is in this case encoded not in one of the two groups but in the topology of the maps between them. This facilitates calculations for field theories quantized over cohomologies with particular coefficient groups while preserving the non-trivial information related to quantization in the Ext part of the sequence above. One should notice that the second arrow in the UCT formula above is an injection i.e. while all the elements of the Ext group must have a correspondence in Hi(X;Z)⊗A, the latter group might have different elements with no correspondence in Ext. This may suggest that Ext may be a better measure for the true (physical) anomalies. Indeed, in the standard model gauge anomalies introduced by chiral fermions cancel naturally when all the fermions are included. However, there appears to be a more general rule suggesting a more accurate method of predicting “true” particles while avoiding to fall in the trap of considering fictitious objects, “needed” in order to cancel anomalies, as “physical particles”. 17 Beyond the Holographic principle Finally one may ask what this idea brings new with respect to the interpretation of the holographic principle. In order to answer this I may turn again to the idea of performing a quantum-gravity experiment. Assume one has a topological measuring device using a particular choice of a group structure for the coefficients. It remains to be seen how such a device can be implemented practically. Assume also one performs the measurements at a scale where quantum gravity is irrelevant and in a region where there are no black holes to talk of. In this case the choice of the coefficient group is irrelevant. The extension and torsion is always trivial and one obtains the same results known from simple quantum mechanics. One can chose a complete set of commuting observables and start making predictions considering also the effects of possible non-commuting observables as it is the custom in standard quantum mechanics. Now consider a different region of space-time where either because one excites gravitational modes that can alter the topology of spacetime or because one has a black hole somewhere, the topology of the space-time stops being trivial. In this case one has to perform a topological measurement with an apparatus that will provide information about how the (co)homology or homotopy of the region looks like when seen through the specific choice of the coefficient group. According to this measurement one has to design restrictions on the observables allowed by classical quantum mechanics. The Ext and T or parts of the chain will not be trivial and will have to be considered when designing further lower-scale experiments using the space-time measured via the coefficient groups. Not all observables will exist in this situation (due to merging of equivalence classes). A somehow metaphorical way of looking at this is considering the group choice as a choice of coefficients in a polynomial. Classical quantum measurements after a choice is made are metaphorically equivalent to finding solutions of these equations. If one chooses for example rational coefficients, the number e(the basis of the natural logarithm) will be transcendental (i.e. no polynomial with rational coefficients can have eas a root). Experimental verification The idea of adding uncertainty to the topology of space-time itself has, as I showed before, many implications. Unfortunately most of these are not easily verifiable. While this article is fundamentally theoretical I try here to pinpoint some possible experiments where this subject may become useful. It is known that topology is not only associated to space-time itself. As I showed before and in the main article, one may probe via (co)homology or homotopy with coefficients (of course in an abstract sense) also field-spaces, groups and other abstract spaces. A more accessible experiment where topological features are important is the Bohm-Aharonov experiment. There, one may observe the effects of a nontrivial topology generated by a magnetic field, in a region where the given magnetic field vanishes. If one could manage to create a magnetic field in a state of quantum superposition between a situation with trivial topology and one with non-trivial topology one could check if the measurement of the shift of the observed interference pattern will fix the degrees of freedom of the system or if new quantum restrictions may appear due to the quantization of the topology itself. One should notice that the topological superposition should ideally be obtained without an entanglement with a local object (like the spin of an electron, etc). Also, possible verifications could be provided by the study of the topological phases of matter. I expect the procedure given by the UCT to be particularly important for the classifications of these phases and for the possible discovery of new ones. The fractional quantum hall effect may also have an interpretation in terms of rational Ext groups. One may ask what happens with the theoretical prediction of magnetic monopoles in the context of uncertain topology. Are they still possible? If future experiments will succeed in proving the fundamental limitations of topological measurements one can safely extend this principle towards space-time itself. CONCLUSION As a conclusion, in this paper I show an aspect of quantization that has been probably overlooked but that may have major implications not only in the description of quantum gravity but also in the theory of quantum information. On the quantum information side problems like the “hat problems” may have some interesting quantum representations. Also possible new “strong-weak” dualities may result to be important in fields like condensed matter or many particle physics. The discussion of other possible applications in quantum gravity or condensed matter physics will be the main subject of a future article. 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