Covariant singularities in Quantum Field Theory and Quantum Gravity
aa r X i v : . [ h e p - t h ] F e b Covariant singularities in Quantum FieldTheory and Quantum Gravity
Roberto Casadio ab ∗ , Alexander Kamenshchik abc † , and Iberê Kuntz ab ‡ a Dipartimento di Fisica e Astronomia, Università di Bolognavia Irnerio 46, 40126 Bologna, Italy b I.N.F.N., Sezione di Bologna, I.S. FLAGviale B. Pichat 6/2, 40127 Bologna, Italy c L.D. Landau Institute for Theoretical Physicsof the Russian Academy of Sciences119334 Moscow, Russia
February 23, 2021
Abstract
It is rather well-known that spacetime singularities are not covariant under fieldredefinitions. A manifestly covariant approach to singularities in classical gravity wasproposed in [1]. In this paper, we start to extend this analysis to the quantum realm.We identify two types of covariant singularities in field space corresponding to geodesicincompleteness and ill-defined path integrals (hereby dubbed functional singularities).We argue that the former might not be harmful after all, whilst the latter makes allobservables undefined. We show that the path-integral measure is regular in any four-dimensional theory of gravity without matter or in any theory in which gravity iseither absent or treated semi-classically. This might suggest the absence of functionalsingularities in these cases, however it can only be confirmed with a thorough analysis,case by case, of the path integral. We provide a topological and model-independentclassification of functional singularities using homotopy groups and we discuss examplesof theories with and without such singularities. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] Introduction
The complexity of Nature and its infinitude of details give rise to an uncountable number ofpossibilities to describe it using physical models. From a practical point of view, narrowingdown these possibilities amounts to the almost artistic job of identifying relevant hypothesesand then testing them against observations. Along the history of physics, this processhas resulted in very important fundamental principles, such as coordinate-independenceand gauge invariance. With the advent of modern geometry, the former has practicallybecome a no-brainer and has achieved such a level of confidence that it is usually takenfor granted. Gauge invariance under Lie symmetries has similarly gained a lot of trust afterculminating in the Standard Model of particle physics [2–5]. Geometrically, gauge invariancedoes not seem much different from coordinate-independence, as it reflects the ambiguityin choosing coordinates (or more precisely local trivialization) of the vector bundle. Bothnotions definitely share the same idea that physics should not depend on the artificial choiceswe make to describe it.There is a third instance of this seemingly general idea of coordinate-independence,namely the ambiguity in the choice of fields. Albeit not as popular as the other two, fieldredefinitions play prominent roles in both classical and quantum field theories. Their ap-plication in gravity has led to the discovery of both Starobinsky’s model of inflation [6, 7],whose inflaton is hidden in the square of the Ricci curvature, and Higgs inflation [8, 9],which results from a field redefinition of the non-minimal coupling of the Higgs boson togravity. Field redefinitions have also been employed to show the equivalence between parti-cle dark matter and modified gravity [10]. In quantum field theory, field redefinitions havebeen used for simplifying the renormalization procedure by taming UV divergences [11–15]and for simplifying the action of effective field theories [16–19]. This is in fact justified byBorchers’ theorem [20, 21], which states that S -matrix elements are invariant under fieldredefinitions interpolating between fixed asymptotic states. Nonetheless, there is no guaran-tee that more general field redefinitions would leave the S -matrix invariant, let alone otherquantities in the theory, such as the effective action. This problem is actually intimatelyrelated to the gauge-dependence of the standard effective action, which motivated the in-troduction of a covariant effective action under general field redefinitions by Vilkovisky andDeWitt [22, 23, 25]. More recently, field-covariant formulations of quantum field theory haveregained some attention [26–29].In spite of the usefulness of field redefinitions, their interpretation in gravity has notreached a consensus yet. There is in fact a long-standing debate concerning whether theJordan frame or the Einstein frame should be regarded as physical [30–36]. While someargue in favour of the former and some in favour of the latter, many others hold the positionthat they are actually equivalent. Ideally, this type of discussion should be framed in termsof physical observables. Nevertheless, what comprises the list of observables in gravity isa rather subtle and non-trivial subject per se . The best way to phrase the equivalence (ornon-equivalence) between different frames is then through the action. It turns out that theclassical action S = S [ φ ] , which describes the dynamics of a set of classical fields φ , is a2calar under field redefinitions. In fact, for any one-to-one transformation φ → ˜ φ = f ( φ ) , (1.1)the classical action transforms as ˜ S [ ˜ φ ] = S [ φ ] . (1.2)Furthermore, the equations of motion transform covariantly, δ ˜ S [ ˜ φ ] δ ˜ φ = δφδ ˜ φ δS [ φ ] δφ , (1.3)thus solutions in the frame φ are taken into solutions in the frame ˜ φ and vice-versa. Inthis sense, φ and ˜ φ are just coordinates in the field space. We have no means to tell themapart, regardless of whether the set of fields φ includes the spacetime metric. At the formallevel, one could simply require that classical observables be scalars under field redefinitions.This demands, of course, great care when interpreting objects in a theory. Some equationsand quantities might very well look completely different in different field-space coordinates.Fundamental concepts might change or even lose their meaning. Must we not forget that,in the end of the day, the fate of a theory is dictated by its experimentally observablepredictions, everything else is just instrumental to obtaining them.At the quantum level, on the other hand, the above considerations might change quiteconsiderably. Notwithstanding Borchers’ theorem, scattering amplitudes are not invariantunder general field redefinitions, but only those that keep the asymptotic states fixed. Thiswould suggest a preferred set of coordinate systems in field space. Furthermore, S -matrixelements are not the only observables in a quantum field theory. Correlation functions, thatis in-in amplitudes, calculated in the Schwinger-Keldysh formalism are the actual observablesin cosmology and astrophysics. The question of coordinate-independence in field space thusbecomes of fundamental importance for studying quantum effects in these contexts. Contraryto the classical action, the effective action, which encodes all the information of a quantumsystem, explicitly turns out not to be a scalar under field redefinitions. This clearly opensup two possibilities as either i) physics depends on field-space coordinates at the quantumlevel or ii) the effective action formalism is not complete.In this paper, we shall take the position that physics at the fundamental level shouldnot depend on the way fields are parameterised. There are at least two important reasonsfor this. For one, as explained above, the classical action does not depend on the fieldparameterization, thus it is reasonable to keep this property in the quantum theory as well.Secondly, and more importantly, a dependence on the parameterisation is suggestive of ananthropic viewpoint, in which one attaches a special meaning to one’s choice (in this case,the choice of coordinates in field space), whereas no choice appears physically favoured apriori . In fact, the results of any experiment are numerical readings that do not imply any We remark that this does not prevent the existence of extremals of S [ φ ] that cannot be mapped into ex-tremals of ˜ S [ ˜ φ ] when the mapping (1.1) is not regular. This also calls for attention to boundary contributionsin the definition of the action, like the famous Gibbons-Hawking-Brown-York term [37]. µ, ν, ρ, . . . ) shall denotespacetime indices; lowercase mid-alphabet Latin indices (e.g. i, j, k, . . . ) collectively repre-sent both discrete indices (denoted by the corresponding capital Latin letters I, J, K, . . . ),and the continuum spacetime coordinates x ≡ x µ . This association can be schematicallyrepresented as i = ( I, x ) , so that φ i = φ I ( x ) are the coordinates of a field configuration.Repeated mid-alphabet lowercase indices results in summations over all the discrete indicesand integration over the spacetime Ω of dimension dim(Ω) = n . We shall usually denotepoints P ∈ Ω with their coordinates. For example, we will write x ∈ Ω as a shorthand nota-tion for x ( P ) ∈ U (Ω) , where U ∈ R n is the domain of the chart of Ω including the point P ofcoordinates x . Lowercase Latin indices of the beginning of the alphabet (e.g. a, b, c, . . . ) shallbe reserved to internal indices in gauge theories and indices corresponding to the beginning Indices will be omitted overall when no confusion can arise.
4f the Greek alphabet (e.g. α, β, γ, . . . ) will denote spinor indices.
Functional methods [38, 39], like the Schwinger action principle and the Feynman path in-tegral, concentrate on the transition amplitude between two quantum states. More specifi-cally, the spacetime Ω is assumed to be globally hyperbolic and admits foliations in (spatial)Cauchy hypersurfaces Σ t , which we label by a time coordinate t for simplicity. We shallalso assume that all the fields φ are spacetime tensors and their action S is a scalar func-tional of them. One then considers the transition amplitude between a certain quantumstate | ζ in , t in i defined on the initial hypersurface Σ t in and a given final state | ζ out , t out i on adifferent hypersurface Σ t out , h ζ out , t out | ζ in , t in i = h ζ in , t in | U − ( t out , t in ) | ζ in , t in i , (2.1)where the unitary operator evolving states along the foliation is given by U ( t , t ) = exp (cid:18) − i ~ S [ φ ; t , t ] (cid:19) , (2.2)with the action S computed on the spacetime volume ¯Ω ⊆ Ω between the hypersurfaces Σ t and Σ t . Since the transition amplitude (2.1) only depends on the initial and final states, anyvariation δφ of the fields which vanishes on the initial and final hypersurfaces must leave itunchanged. This yields the operator equations of motion for the fields φ from the Schwingervariational principle δS = 0 . Of course, the field equations for the operators φ do notdepend on the states | ζ in , t in i and | ζ out , t out i , but one is eventually interested in computingthe transition amplitudes themselves or the expectation values of quantum observables forthe chosen quantum states.In Feynman approach, one computes how field excitations evolve between the initial andfinal quantum states by first coupling each of the fields φ i to an external current J i , that is S J [ φ ] = S [ φ ] + J i φ i . (2.3)The insertion of these currents results in the path integral h ζ out , t out | ζ in , t in i [ J ] = Z "Y i d φ i exp (cid:26) i ~ S J [ φ ] (cid:27) := exp (cid:26) i ~ W [ J ] (cid:27) , (2.4)where all field configurations compatible with the boundary conditions at Σ t in and Σ t out aresummed over. Functional derivatives of W [ J ] with respect to the currents J i (evaluated at One might be interested in the evolution forward in time, so that t out > t in , or backward in time, with t out < t in . i = 0 ) then yield correlation functions and quantum corrected equations of motion for the(expectation values of) the fields φ i . In the functional integral (2.4), one can change howthe fields φ are represented as long as the redefinition is not singular and can be inverted.Hence, the very starting point of functional quantisation shows how deeply entangled arethe geometry of spacetime (represented by the foliation Σ t and the expectation values of thefields φ ) and the geometry of field space in which the φ i act as coordinates.In this construction, the role of the initial and final quantum states is crucial andshould not be overlooked. For instance, high energy physicists are usually interested inscattering processes occurring in a (supposedly) well defined vacuum | ζ in , t in → −∞i = | ζ out , t out → + ∞i ≡ | i . Moreover, the question one is usually trying to answer is theprobability that incoming particles of given momenta result in certain final states that canbe experimentally detected. Each scattering process is therefore characterised by certaintypes and numbers of incoming and outgoing field excitations (particles) which are formallycreated and absorbed in the asymptotic vacuum | i by the external currents J i . Transitionamplitudes, as well as the S -matrix which maps initial into final asymptotic states, do notdepend on the explicit parameterisation of the fields φ , but the correlations functions dodepend on the choice of the currents J i , which in turn must be such that J i φ i is a scalarunder both field redefinitions and changes of spacetime coordinates.In cosmology, or in the study of the gravitational collapse, the question one would like toanswer is instead what final states | ζ out , t out i are more likely to develop from an initial state | ζ in , t in i of interest if the field dynamics is driven by a specific action S = S [ φ ] (and then,of course, which states and dynamics best fit the experimental data). For example, | ζ in , t in i could be the quantum state representing the asymptotically flat regular space of a star(including all information about the matter source), and one is then interested in computingthe probability that this system evolves into a singular configuration, for which some of theobservables develop diverging expectation values at some t out > t in . The choice of relevantobservables to compute depends on the physical system, and one could consider the energy-momentum tensor of matter for a collapsing star or for a cosmological model. In any case,observables will be given by functions of the fields and one should therefore compute thesequantities for all possible final states | ζ out , t out i in order to weigh their relative probability ofoccurring. We remark that this problem is further complicated by the fact that the foliation Σ t is also part of what one is investigating, particularly so if one is interested in the possibledevelopment of spacetime singularities. Strictly speaking, the emergence of the latter wouldin fact determine the topology of spacetime by removing (sets of) points from the spacetimemanifold.It should be clear from this brief review of the formalism, that the background fieldmethod, in which one restricts the calculation to transition amplitudes between quantumstates corresponding to the evolution of the background field ϕ i := δW [ J ] δJ i , (2.5) In cosmology, one is usually interested in the development of singularities going backward in time, thatis for t out < t in .
6s possibly the only workable assumption. The definition of the background field (2.5) allowsfor the introduction of the Legendre transform Γ[ ϕ ] = W [ J ] − J i ϕ i , (2.6)which satisfies δ Γ[ ϕ ] δϕ i = − J i . (2.7)When the external current vanishes J i = 0 , Eq. (2.7) plays the role of the quantum generali-sation of the Euler-Lagrange equations, which justifies calling Γ[ ϕ ] the effective (or quantum)action and Eq. (2.7) its corresponding effective equations of motion. From Eqs. (2.4) and(2.6), one finds the integro-differential equation for Γ[ ϕ ] , exp { i Γ[ ϕ ] } = Z "Y i d φ i exp (cid:26) i ~ (cid:20) S [ ϕ ] − (cid:0) φ i − ϕ i (cid:1) δ Γ[ ϕ ] δϕ i (cid:21)(cid:27) , (2.8)whose exact solution is known only in very simple cases. In practical terms, one assumes thatboth the initial state | ζ in , t in i and the final state | ζ out , t out i are such that the correspondingexpectation values of the fields are well approximated by small perturbations φ about abackground configuration ϕ , that is φ i → ϕ i + φ i . (2.9)The functional machinery can then be used to estimate the expectation values of the ob-servables by taking functional derivatives of Γ[ ϕ ] with respect to the ϕ . The effective action Γ[ ϕ ] turns out to play a central role in quantum field theory. For one, it is the generator ofone-particle irreducible diagrams, which makes the study of renormalisation much easier andallows one to readily calculate scattering amplitudes. Secondly, as we have seen, it gener-alises the classical action by generating effective equations of motion for the evolution of thebackground fields which account for the backreaction of quantum fluctuations to arbitraryloop order.For our subsequent analysis, it is important to remark that the dependence on the quan-tum states | ζ in , t in i and | ζ out , t out i of the effective action in Eq. (2.8) is now hidden in thevery definition of the background field, namely ϕ i = h ζ out , t out | φ i | ζ in , t in i . (2.10)Like the full quantum generator W [ J ] would be obtained by functionally integrating overall field configurations φ respecting the boundary values implied by | ζ in , t in i and | ζ out , t out i ,the effective quantum action Γ[ ϕ ] only requires integrating over the quantum fields (ideally The background configuration need not be a solution of the classical equations of motion. In fact,the background field solves the effective equations of motion in the Schwinder-Keldysh formalism, in whichquantum corrections are taken into account. The relevance of the background field method therefore goesbeyond the study of scattering amplitudes. Σ t in and Σ t out ). In either case, the result should not depend on the explicitvariables φ or ϕ we choose for representing the fields, as long as they also remain spacetimetensors. The S -matrices are indeed invariant under a change of fields. Nonetheless, the term ( φ i − ϕ i ) in Eq. (2.8) is not, thus making the effective action a non-covariant object underfield redefinitions. In order to study objects invariant under field redefinitions, we need to set out the geometryof field space M [40–45], which follows in analogy with Riemannian geometry, where fieldsplay the role of mere coordinates in the geometrical space M (see Ref. [21] for a detailedreview on the geometrical aspects of the field space). There is, however, a crucial differencewith the usual theory of manifolds regarding the dimensionality of M . Because coordinates inthis scenario consist of a set of fields which are themselves functions of spacetime, φ i = φ I ( x ) ,the field space M is infinite-dimensional. For every fixed spacetime point x ∈ Ω , the spacecomprised by all φ I ( x ) forms nonetheless a finite-dimensional manifold N . As suggestedby the notation φ i = φ I ( x ) , one can thus imagine that the topology of M is given by infinitecopies of N , M = Y x ∈ Ω N ( x ) . (3.1)The above construction concerns only the topological structure of the field space. Nothing sofar has been said about geometrical structures, such as the metric, nor has it been required.There is, however, one reason to introduce a metric in field space. Loop corrections invariablyrequire the calculation of functional determinants of the Hessian det H ij , (3.2)where H ij = δ S [ ϕ ] δϕ i δϕ j , (3.3)denotes the Hessian matrix, which is a bilinear form, i.e. it carries two covariant field-space indices. The determinant of bilinear forms transforms as a tensor density, leading toa dependence on the basis of the tangent space of M in Eq. (3.2). Because the discreteindices in H ij generally contain spacetime indices, not only does the effective action failto be invariant under field redefinitions, but it also fails to be invariant under spacetimecoordinate transformations. Determinants of linear operators, i.e. objects containing mixedindices, on the other hand, are invariant under basis transformations. Thus, to make thedeterminant of the Hessian invariant under coordinate transformations, one must transformone of its covariant indices into a contravariant index. This requires the introduction of ametric in field space. An example is the space of non-linear σ -models. M , hereby denoted G ij , must be seen as part of the definition of the theory,along with the classical action. The line element is defined as usual as d s = G ij d φ i d φ j = Z Ω d n x Z Ω d n x ′ G IJ ( x, x ′ ) d φ I ( x ) d φ J ( x ′ ) . (3.4)We shall require that G ij be invariant under the same gauge symmetries used to define theclassical action. This is particularly important to enforce these symmetries at the quantumlevel via the path integral measure, which takes a factor p det G ij to cancel out the Jacobiandeterminant from the field redefinition, thus preventing gauge anomalies. Apart from sym-metry, there is no other guiding principle to help us choose among all infinite possibilitiesfor a field-space metric. Nonetheless, should we extend the topological construction (3.1) togeometrical structures, it is natural to assume ultralocality G ij = G IJ δ ( x, x ′ ) , (3.5)where G IJ depends only on the fields φ I and none of their derivatives. This is an enormoussimplification as we reduced the determination of the metric in the infinite-dimensionalfield space M to the determination of the metric defined on the finite-dimensional space N .Unfortunately, with this level of generality, there are still infinitely many choices one canmake, thus we shall restrict to the simplest cases where one need not include additionaldimensionful parameters.For a general field space, there is no unique way of defining a connection over M and thereis a priori no reason for adopting the Levi-Civita connection. General connections require,however, additional mathematical structures separate from the metric. Thus, for simplicity,we shall make again the minimal choice which entails the torsionless and metric-preservingLevi-Civita connection Γ i jk = 12 G il ( ∂ j G kl + ∂ k G jl − ∂ l G jk ) . (3.6)Our choices imply that the entire geometry of M is determined by the field-space metric G ij .In particular, the ultralocality of the field-space metric extends to the connection Γ i jk = Γ IJK δ ( x I , x J ) δ ( x I , x K ) , (3.7)where x I denotes the argument x of the field φ I = φ I ( x ) and Γ IJK = 12 G IL (cid:18) ∂G LK ∂φ J + ∂G JL ∂φ K − ∂G JK ∂φ L (cid:19) . (3.8)The functional Riemannian tensor is defined in the usual way R ijkl = ∂ k Γ i lj − ∂ l Γ ikj + Γ i km Γ mlj + Γ i lm Γ mkj , (3.9) We are making a slight abuse of notation as G ij = G IJ ( x, x ′ ) = G IJ ( φ ( x )) δ ( x, x ′ ) . Thus, obviously, G IJ ( x, x ′ ) and G IJ ( φ ) = G IJ ( φ ( x )) are different objects. We shall nevertheless use the same tensorialnotation but with different arguments to distinguish them and avoid heavy notations. R jl = R i jil and R = R ii being the the functional Ricci tensor and functional Ricci scalar,respectively. Let us note that, because of the assumption of ultralocality, many contractionswill diverge as δ ( x, x ) . This is rather expected and only reflects the infinite dimension offield space dim( M ) ≡ G ij G ij = N V (Ω) δ ( x, x ) , (3.10)where N = dim( N ) is the dimension of the finite-dimensional space N and V (Ω) denotes theinfinite volume of the spacetime Ω . To make sense of contracted quantities, one must thendefine densities, such as R ijkl R ijkl dim( M ) = 1 N V (Ω) Z Ω d n x R IJKL R IJKL , (3.11)in which divergences due to the infinite dimension of field space, stemming from both the δ ( x, x ) and the infinite spacetime volume, are canceled out. This clearly has nothing to dowith singularities in field space as it is a property of every theory and not of a specific fieldconfiguration. Since the aforementioned densities can always be defined, we shall implicitlyemploy the above procedure and focus on contracted quantities of the finite-dimensionalspace N . In the following, we shall exemplify the above formalism with non-abelian Yang-Mills theories and gravitational theories.For SU ( N g ) gauge theories in flat spacetime, the fields are represented by φ A ( x ) = A aµ ( x ) and our minimal choice for the metric leads to G ij = η µν δ ab δ ( x, x ′ ) , (3.12)where a, b = 1 , , . . . , N g − and η µν is the Minkowski metric. The minimal choice hasled to a field-independent metric G ij with a trivial geometry, i.e. vanishing connection andcurvature. The metric (3.12) is incidentally also obtained from the kinetic term of theclassical action. Vilkovisky has indeed suggested that G IJ , and ultimately G ij , be identifiedfrom the highest-order minimal operator present in the classical action [22].For metric theories of gravity, one identifies φ I ( x ) = g µν ( x ) . The assumption ofsimplicity then leads to the one-parameter family of field-space metrics G ij = 12 ( g µρ g σν + g µσ g ρν + c g µν g ρσ ) δ ( x, x ′ ) , (3.13)which involves only a dimensionless parameter c . Such a metric first appeared in the litera-ture in the DeWitt’s seminal paper [46] on the canonical quantisation of gravity. Its inverseis found by solving G ij G jk = δ ki , which gives G ij = 12 (cid:18) g µρ g σν + g µσ g ρν − c n c g µν g ρσ (cid:19) δ ( x, x ′ ) , (3.14) One could define the effective action on domains ¯Ω ⊆ Ω of finite volume but we shall not go into thesedetails here. One could also formally deal with undefined products of the Dirac delta by regarding it as the limit of asequence of functions. One can then perform all calculations before taking the limit to the Dirac distribution. A common convention in the literature is to take φ I ( x ) = g µν ( x ) with spacetime covariant ratherthan contravariant indices. This generally leads to confusion as field-space covariant indices correspond tospacetime contravariant indices and vice-versa. n is the spacetime dimension. Note that the DeWitt metric is only invertible for c = − /n . The parameter c cannot be determined without some additional assumption.The aforementioned procedure proposed by Vilkovisky [47] gives c = − for the Einstein-Hilbert action, but it would be different for higher-derivative gravity. We stress, once again,that Vilkovisky’s procedure is not a necessary requirement, thus we shall leave c unspecified.For the DeWitt metric, the Ricci tensor is given by R IJ = 14 (cid:0) g µν g ρσ − n g µ ( ρ g σ ) ν (cid:1) (3.15)and the Ricci scalar reads G IJ R IJ = n − n − n . (3.16)It is standard practice in General Relativity to analyse curvature invariants, like theKretschmann scalar, in order to decide whether a singularity is physical or just a coordinatesingularity. Since diffeomorphism invariants are the same in all coordinate systems, only“true” singularities would affect them. Analogously, we can seek a scalar functional inorder to investigate the appearance of singularities in the field space. We could, for example,consider the functional Kretschmann scalar, K = R IJKL R IJKL , (3.17)to assess whether a singularity is real or only a consequence of a bad choice of field variables.In Ref. [1], we have indeed calculated K for the DeWitt metric and found K = n (cid:18) n n − (cid:19) . (3.18)The fact that K is finite everywhere suggests the absence of covariant singularities in theoriesof gravity without matter. At this point, we should stress again that the metric (3.13) is notunique. In fact, any metric of the form G ij = 12 ( − g ) ǫ ( g µρ g σν + g µσ g ρν + c g µν g ρσ ) δ ( x, x ′ ) , (3.19)with g = det g µν , would satisfy ultralocality and simplicity for any value of ǫ . The choice ǫ = 0 leads to the functional measure originally proposed by Misner [48], whereas the casewith ǫ = 1 / was put forward by DeWitt [38–40]. Nevertheless, any value of ǫ is a priori allowed at the classical level [49]. The functional Kretschmann scalar for an arbitrary ǫ turnsout to be given by K = n − g ) − ǫ (cid:18) n n − (cid:19) . (3.20)Notice that a covariant singularity in K would be present where g = 0 , for ǫ > , or where | g | → ∞ , for ǫ < . Therefore, the case ǫ = 0 is the only possibility which excludes a Singularities in the scalars derived from the Riemann tensor also signal the possible divergence of tidalforces, which makes them particularly relevant for physics. K , regardless of the spacetime metric. Since the field-space metric is part ofthe definition of the theory, one could in principle take the case ǫ = 0 as the definition ofa gravitational theory at the classical level. Nonetheless, at the quantum level, the pathintegral measure p det G ij is expected to preserve the diffeomorphism symmetry of grav-ity [50–56], thus requiring ǫ = 1 / . In any case, it is not clear whether a singularity in K at, say, g µν = g s µν would be a real issue. Removing g s µν from the field-space manifold M anddefining covariant singularities in terms of geodesic incompleteness could, in fact, only meanthat g s µν cannot be realised in Nature. Contrary to the usual picture of spacetime singular-ities, which face the philosophical impasse of the sudden termination of physics, covariantsingularities would only signify the absence of a certain field configuration.A more useful definition of singularities should be given in terms of physical observables.Since all observables of a quantum field theory can be computed directly from the quantumaction, Γ[ ϕ ] is a natural candidate to seek a proper and useful definition for singularity. Onethen needs a covariant formulation of the effective action, which will be reviewed in the nextsection. Although, in principle, there can be solutions with non-singular observables but witha singular functional Kretschmann scalar K , they would not have any practical consequencefor physics and their dangerousness would rather be an epistemological question. We shalldiscuss the differences between these types of singularities in depth in Section 5. High-energy physicists are very rarely interested in quantities other than S -matrix elementsand cross-sections. The S -matrix contains in fact all accessible information of scatteringprocesses performed by colliders. These objects, being defined on-shell, are invariant underfield redefinitions that interpolate between a fixed choice of asymptotic states [57]. Theimportance of scattering amplitudes in cosmology is however secondary. Unfortunately, onedoes not have the same level of control to throw cosmological particles against each otherand observe the output. Observational cosmology is largely based on the measurement ofstatistical correlation functions and one is thus rather interested in the evolution of off-shellquantities resulting from the backreaction of quantum fluctuations. Quantum effects for bothon-shell and off-shell quantities are however contained in the same single object: the effectiveaction. Although S -matrices are guaranteed to be invariant under field redefinitions, off-shellcorrelation functions are not, which would imply the possible existence of a preferred fieldparameterisation.As we remarked in the Introduction, this is not a problem at the tree level becausethe classical action is manifestly a scalar under the field redefinitions (1.1). Nonetheless,the one-loop effective action acquires a new term proportional to the classical equations ofmotion [22], Γ[ ϕ ] → Γ ′ [ ϕ ] = S [ ϕ ] + i ~ (cid:20) δ Sδϕ m δϕ n + δf i δϕ m δf k δϕ n δ ϕ l δf i δf k δSδϕ l (cid:21) + O ( ~ ) , (4.1)in which we employed the notation of Eq. (2.9) and denoted the background fields with ϕ .The additional term of order ~ is not important for scattering amplitudes, since δS/δϕ = 0 a priori to prefer one field parameterisation over the others. This will be particularlyimportant in the study of singularities under field redefinitions, as we shall see in the nextsection.One way to overcome this issue is to use the geometrical apparatus of Section 3 in orderto enforce the invariance of the effective action under field redefinitions. This was indeedthe approach adopted by Vilkovisky [22], where a connection in the field space is introducedto compensate for the second term between brackets in Eq. (4.1). With a connection in fieldspace, one should then replace the functional derivative with the corresponding covariantfunctional derivative δ Sδϕ m δϕ n → ∇ m ∇ n S := δ Sδϕ m δϕ n − Γ i mn δSδϕ i , (4.2)and modify the definition of the effective action accordingly. Since Vilkovisky’s effectiveaction only works for a flat field space, DeWitt later on proposed the improved effectiveaction [23] exp (cid:26) i ~ Γ[ ϕ ] (cid:27) = Z d µ [ φ ] exp (cid:26) i ~ (cid:0) S [ φ ] − σ i ( ϕ, φ ) ( C − ) ji [ ϕ ] ∇ j Γ[ ϕ ] (cid:1)(cid:27) , (4.3)where σ i ( ϕ, φ ) = 12 (cid:18)Z γ ( ϕ,φ ) d s (cid:19) (4.4)is the geodetic interval (which is analogous to Synge’s world function [24]), calculated alongthe geodesic γ with end-points ϕ and φ , and C ij = h∇ j σ i ( ϕ, φ ) i T . The angular bracketshere denote the functional average, which, for any functional F [ ϕ, φ ] , is given by h F [ ϕ, φ ] i T = exp (cid:26) − i ~ Γ[ ϕ ] (cid:27) Z d µ [ φ ] F [ ϕ, φ ] exp (cid:26) i ~ (cid:0) S [ φ ] + T i [ ϕ, φ ] ∇ i Γ[ ϕ ] (cid:1)(cid:27) , (4.5)where T i [ ϕ, φ ] = σ i ( ϕ, φ )( C − ) ji [ ϕ ] . Note that the definition of C ij is recursive because C ij also appears in the definition of the functional average. Finding an explicit expressionfor C ij is thus utterly difficult and generally requires expansions in series ( e.g. the loopexpansion). The geodetic interval σ i ( ϕ, φ ) transforms as a vector at ϕ and as a scalar at φ ,thus making the effective action invariant under both redefinitions of the background ϕ andof the quantum field φ . The functional measure for a gauge theory reads [21] d µ [ φ ] = [d φ ] M [ φ ; χ ] (4.6)with [d φ ] := Y k d φ k | det G ij ( φ ) | / (4.7)13nd M [ φ ; χ ] := (cid:0) det Q αβ (cid:1) ˜ δ [ χ α ] , (4.8)where det Q αβ is the Faddeev-Popov determinant with Q αβ = χ α,i [ φ ] K iβ [ φ ] defined in termsof the gauge fixing χ α and the generator of gauge transformations K iβ . The functional Diracdistribution ˜ δ [ χ α ] is defined analogously to the standard case as Z Y α d χ α ! ˜ δ [ χ α ] = 1 . (4.9)As usual, the gauge fixing ensures that the domain of integration in the functional integralbe restricted to the orbit space, namely the space of distinct equivalence classes unrelatedby a gauge transformation. The gauge fixing is thus chosen so to pick up only one memberof each equivalence class, which is tantamount to demanding that χ α [ φ ǫ ] = χ α [ φ ] , (4.10)where φ iǫ = K iα δǫ α , has only the solution δǫ α = 0 for a given φ i . From this requirement, byexpanding the left-hand side of Eq. (4.10) to first order in δǫ α , it is straightforward to showthat Q αβ must satisfy det Q αβ = 0 . (4.11)The presence of the Faddeev-Popov determinant det Q αβ and the gauge condition χ γ = 0 imposed by the functional Dirac distribution guarantee the gauge invariance of the measure.The determinant of the field space metric in the measure is crucial for obtaining a pathintegral measure invariant under field reparameterization. The covariant measure togetherwith σ i ( ϕ, φ ) and the covariant functional derivative then make the quantum action Γ[ ϕ ] ascalar functional under redefinitions of both the background ϕ i and the quantum field φ i . Inthe one-loop approximation, Eq. (4.3) leads to Γ[ ϕ ] = S [ ϕ ] + i ~ ∇ i ∇ j S [ ϕ ] + O ( ~ ) (4.12)as expected. We see that the replacement (4.2) has been automatically accounted for.The Vilkovisky-DeWitt effective action, as defined in Eq. (4.3), is not automatically thegenerator of one-particle irreducible (1PI) diagrams. An additional improvement was madein Ref. [25], where it was shown that in order for the Vilkovisky-DeWitt effective action togenerate 1PI diagrams, it requires the generalised definition of the background field v i [ ϕ ] = δW [ J ] δJ i , (4.13)thus implying the modified Legendre transform Γ[ ϕ ] = W [ J ] − J i v i [ ϕ ] . (4.14)14uch a modification is indeed expected as the current J i in Eq. (2.6) couples to the factor ( φ i − ϕ i ) in Eq. (2.8). The effective equations of motion then becomes ∇ i Γ[ ϕ ] = − J k ∇ i v k [ ϕ ] . (4.15)This modification will play a crucial role in the topological study of Section 6.The quest for singularities now amounts to calculating the on-shell quantum action Γ[ ϕ ] at any desirable solution of the full effective equations of motion. Whilst this approach isvery general, finding the exact effective equations of motion (or even the effective actionitself) is rather non-trivial. In practice, one employs approximative methods to obtainthe effective action and its corresponding equations of motion and solutions. In the semi-classical approximation, for example, one can calculate the solutions order by order in aloop expansion and singularities can then be studied at each order. Note that the meaningof going on-shell in this case depends on the loop order under consideration. Scatteringamplitudes are typically worked out on the mass shell, i.e. when the classical equations ofmotion are valid. On the other hand, the on-shell quantum action refers to the quantumaction evaluated on the solution of the quantum equations of motion. This means thatthe on-shell quantum action at n -loop order is evaluated on the solution of the quantumequations of motion at ( n − -loop order. With a proper definition for the effective action, we can investigate singularities in field spacewhose covariance is now manifest. We shall define a covariant singularity ϕ = ϕ as a solutionof the effective equations of motion in which the Vilkovisky-DeWitt effective action Γ[ ϕ ] evaluated at that point is undefined, that is Γ[ ϕ ] does not attain any value or is divergent.As we have already anticipated, this might happen for two different reasons. On one hand,the field space M might be geodesically incomplete, in which case ϕ would correspond toa point at the boundary of M . Because ϕ does not belong to M , the effective action Γ[ ϕ ] ,which takes values from M to the real numbers, is obviously undefined at ϕ . This typeof covariant singularity thus reflects the absence of certain field configuration ϕ , which, aswe shall see, is not particularly worrying. On the other hand, covariant singularities mightalso be manifested as a result of the path integral in Eq. (4.3). In this case, the covariantsingularity ϕ does belong to M but corresponds to an existing configuration with undefinedobservables. In this case, ϕ shall be called a functional singularity.From the definition (4.3), one can see that a functional singularity occurs whenever thepath integral measure (4.6) either vanishes identically or diverges at some configuration φ = φ . From Eq. (4.11), these conditions take place whenever [d φ ]( φ ) = Y i d φ i q | det G ij [ φ ] | = ( ∞ . (5.1) Since Γ is a scalar under field redefinitions, ∇ i Γ = ∂ i Γ ≡ δ Γ /δϕ i . [d φ ] is the functional measure of a non-gauge theory, the same conditions (5.1) applyregardless of the presence of gauge symmetry. Under the assumption that the Jacobian ofany field redefinition is regular ( i.e. finite and non-zero), the former possibility translatesinto q | det G ij [ φ ] | = 0 , ∀ φ ∈ M . (5.2)Because this condition must be valid for all field configurations, the singularity would be aproperty of the entire field-space geometry rather than of some pathological configuration.This case is easily remedied by a proper choice of G ij which satisfies p | det G ij [ ¯ φ ] | 6 = 0 forat least one field configuration φ = ¯ φ . Therefore, the condition (5.2) is not a useful proxyfor investigating singularities.On the other hand, the divergence of the measure at a single field configuration φ = φ (again for regular Jacobians) lim φ → φ q | det G ij [ φ ] | = ∞ (5.3)is a sufficient condition for the singular behaviour of observables. From Eq. (4.6), thepresence of a functional singularity implies that path integrals and their corresponding ob-servables, such as correlation functions and S -matrix elements, are undefined. As opposedto the standard singularities in spacetime, whose existence can depend on the chosen fieldvariables and might not affect the observables after all, functional singularities not onlymake observables undefined, but also make the entire path integral formalism meaningless.Needless to say, this type of singularities is far more dangerous than the typical ones inspacetime.Note that our definition of functional singularity is not based on the geodesic completenessof field space. It is rather a direct way of formalising under what conditions observables arewell-defined. The geodesic completeness has nonetheless important consequences for theformalism. In fact, if the field space M is geodesically incomplete, then the geodetic interval σ ( ϕ, φ ) used in the definition (4.3) of the Vilkovisky-DeWitt effective action does not existfor all points in M and consequently the Vilkovisky-DeWitt effective action is not definedeverywhere in the field space. The question of whether a finite-dimensional manifold isgeodesically complete depends on the signature of the metric defined upon it. For Euclideanmetrics, geodesic completeness is guaranteed by the Hopf-Rinow theorem which, however,does not hold in infinite dimensions [58]. On the other hand, for pseudo-Riemannian metricswe have the Hawking-Penrose theorem [59], but whether this result can be extended to theinfinite-dimensional field space is an open question (see Appendix A for a derivation of thefocusing theorem in field space). Overall, geodesic completeness is not guaranteed in fieldspace and the best we can do is to search for singularities in the curvature invariants caseby case, as we did for the DeWitt metric (see Eqs. (3.18) and (3.20)), in addition to lookingfor boundaries of the field-space metric G ij . Let us recall that G ij is part of the definitionof the theory, thus one can play this game until a geodesic-complete metric is found.Despite the technical (as opposed to physical) issue in the definition of the Vilkovisky-DeWitt effective action, geodesic incompleteness leads to no serious issues for physics. For Note that the complex exponential in the integrand of the path integral is bounded. φ would have measurableeffects upon φ . Moreover, a straightforward remedy for the effective action on geodesicallyincomplete field spaces can be achieved by replacing the geodesic in the definition of σ ( ϕ, φ ) by some other curve such that its tangent vector at ϕ i is proportional to ( ϕ i − φ i ) . Finally, we should stress that all observables do remain well-defined and finite everywherein geodesically incomplete field spaces.One natural question is whether functional singularities are at all related to spacetimesingularities. The explicit calculation of the determinant of G ij indeed reveals the possiblerelation between functional and spacetime singularities. In practice, this will largely dependon the choice for field space metric G ij . In the following, we shall give examples of thesimplest ( i.e. with no new dimensionful parameter) choices for each type of field, namelyscalar fields, abelian and non-abelian gauge fields, spinors and the metric field of gravity. Weshall impose the symmetries present at the classical level on the field-space metric as well.Because of gravity, this will demand the presence of the spacetime volume element √− g forevery field. In particular, this will lead to the choice ǫ = 1 / in Eq. (3.19).The simplest choice for a scalar field theory, with φ i = φ ( x ) , would then be G s ij = √− g δ ( x, x ′ ) , (5.4)where g = det g µν is the determinant of the spacetime metric. Any other choice for the fieldspace metric would invariably introduce additional dimensionful parameters. Calculatingthe determinant of G s ij explicitly from Eq. (5.4) points at the relation between singularitiesin the spacetime Ω and functional singularities det G s ij = Y x ∈ Ω √− g . (5.5)Functional singularities thus correspond to divergences of g , which in turn implies that atleast one of the eigenvalues of the spacetime metric is singular. It would thus appear thatspacetime singularities with g → ∞ would not be removable by field redefinitions. Nonethe-less, because the spacetime metric in this case is not dynamical (we are just quantising ascalar field in curved spacetime), the path integral measure is constant in field space. Quan-tities of interest are ratios of path integrals, thus the divergences g → ∞ are canceled out bythe normalisation factor. Such a cancelation will always take place when gravity is treatedas external, as we shall now see for the other types of matter fields.For a Dirac spinor φ i = ( ψ α ( x ) , ¯ ψ α ( x )) , the most obvious choice for a metric would be G f ij = √− g (cid:18) ε αβ ε αβ (cid:19) δ ( x, x ′ ) , (5.6) This is required in order to recover the standard effective action in the limit of flat field space. Here Greek letters denote spinor indices. ε αβ is the inverse of the two-dimensional Levi-Civita tensor. Its determinant thenreads det G f ij = Y x ∈ Ω ( − g ) D , (5.7)where D is the spinor dimension.For an abelian Yang-Mills theory, the field is φ i = A µ ( x ) and the simplest field-spacemetric reads G YM ij = √− g g µν δ ( x, x ′ ) , (5.8)whose determinant is det G YM ij = Y x ∈ Ω ( − n g n − . (5.9)For a Yang-Mills theory with gauge group SU ( N g ) , one identifies the field-space coordinatesas φ A ( x ) = A aµ ( x ) . We adopt the field space metric G nYM ij = √− g g µν δ ab δ ( x, x ′ ) , (5.10)which is the simplest choice in this case. The determinant can then be readily calculated tobe det G nYM ij = Y x ∈ Ω ( − n ( N g − g ( n − N g − . (5.11)Note that none of the above field-space metrics depend on the dynamical fields ( i.e. ∂ k G ij =0 ), thus the Levi-Civita connection and consequently the Riemann tensor vanish. Moreover,their determinants are constant in field space, resulting in a factor that gets trivially canceledout by the normalisation factor in the path integral. As we have anticipated, spacetime sin-gularities do not propagate to the observables which are themselves finite even for singularspacetime metrics. It is astonishing that this happens even at the classical level. This picturemight however change when gravity is quantised, which is the subject of the following.In the case of gravitational theories, the determinant of the DeWitt metric (3.19) (with ǫ = 1 / ) reads det G DW ij = Y x ∈ Ω ( − n − (cid:16) c n (cid:17) g ( n − n +1) , (5.12)and again we can see the relation between functional singularities and spacetime singularitiesfor any spacetime dimension n . Amusingly, in four dimensions det G ij becomes constant forany metric configuration, suggesting the absence of singularities in this case. Apart fromthe four-dimensional case, functional singularities do appear for singular spacetime metrics.Notice that the spacetime metric is no longer external, thus one can no longer cancel it outby the normalisation factor of path integrals. We should also stress that the DeWitt metricaccounts only for the gravitational sector. In real systems, matter is nonetheless alwayspresent, which is expected to lead to different conclusions. A complete field-space metric,corresponding to a theory with N s scalars, N f fermions, N YM abelian gauge fields, N nYM G ij = diag (cid:0) N s z }| { G s IJ , . . . , G s IJ , N f z }| { G f IJ , . . . , G f IJ , N YM z }| { G YM IJ , . . . , G YM IJ , N nYM z }| { G nYM IJ , . . . , G nYM IJ , G DW IJ (cid:1) δ ( x, x ′ ) . (5.13)The resulting determinant then reads det G ij = Y x ∈ Ω ( − α (cid:16) cn (cid:17) ( − g ) β , (5.14)where α = 12 N s + D N f + n N YM + n (cid:0) N g − (cid:1) N nYM + n − (5.15) β = 12 N s + D N f + ( n − N YM + ( n − (cid:0) N g − (cid:1) N nYM + 14 ( n − n + 1) . (5.16)Therefore, the presence of functional singularities in an arbitrary theory is parameterised by β , which depends solely on the particle content of the model. Note, in particular, that β isstrictly positive for n ≥ , thus functional singularities could be present in a theory of quan-tum gravity coupled to matter with an arbitrary action and whose field-space metric is givenby Eq. (5.13). This conclusion is nonetheless dependent on the actual choice of field-spacemetric. Avoiding singularities thus require deviating from the simplest diagonal choice madein Eq. (5.13). We should stress that the condition (5.3) imposed on the geometry of field-space is a sufficient but not necessary condition for the presence of functional singularities.In fact, the final outcome of the path integral depends on global contributions and boundaryconditions in addition to the local value of the functional measure. This makes Eq. (5.3) agood proxy to reveal functional singularities, but not a good one to infer their absence. Forthat, we need topological methods that we study in the next section. In the last section, we have laid out the connection between functional singularities and thegeometry of field space. We shall now relate functional singularities to the topology of mapsbetween the field space and the real circle S , as suggested by the definition of the effectiveaction. This will lead to the classification of functional singularities in terms of a windingnumber defined in the field space M , which allows for the elaboration of some strategies toobtain regular quantum field theories. For the purposes of this section, we set ~ = 1 forsimplicity.From Eq. (4.3), it is natural to define the functional ψ [ ϕ ] := e i Γ[ ϕ ] , (6.1)in order to investigate functional singularities. Indeed, the presence of a functional singularityat ϕ = ϕ translates into ψ [ ϕ ] = 0 or ψ [ ϕ ] = ∞ , at which points the effective action19ecomes undefined. In fact, Eq. (6.1) has no solutions for Γ[ ϕ ] for these values of ψ [ ϕ ] , thusthe effective action at ϕ does not exist. We shall call ψ [ ϕ ] the functional order parameterbecause ψ plays the analogous role of an order parameter in the theory of phase transitionsin ordered media or cosmology [60–62]. The field space M can be thought of as the orderedmedium itself, whereas functional singularities correspond to topological defects. Let usrecall, however, that M is an infinite-dimensional space. Since Γ is real, the functional orderparameter ψ defines the map ψ : M → S , (6.2)from the field space to the unit circle, the latter playing the role of the order parameterspace. If we encircle an exact solution ϕ with a d -dimensional hypersurface γ d ( ϕ ) ⊂ M whose topology is S d , the functional order parameter restricted to γ d ( ϕ ) induces the map ψ | γ d : S d → S (6.3)between higher-dimensional spheres centred at ϕ and the circle. Covariant singularities canthus be classified using the homotopy groups π d ( S ) .Apart from the fundamental group π ( S ) = Z , which is isomorphic to the integers, allhigher homotopy groups of the circle are trivial, namely π d ( S ) = 0 for d > . This meansthat all hyperspheres in M can be continuously contracted to a point with the exceptionof the circle γ . The latter may find obstructions that prevent it from being continuouslycontracted to a point. Such obstructions are precisely the functional singularities definedbefore. They could be given by strings (or higher-dimensional submanifolds) in field space, i.e. extended higher-dimensional objects, along which the effective action is undefined. Since π ( S ) = Z , the homotopy classes are labeled by the winding number W = 12 π i I ψ [ γ ] δψψ = 12 π Z π d θ Z Ω d n x ∂ L ( x ) ∂ϕ I ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ I ( x )= γ I ( x ; θ ) d γ I ( x ; θ )d θ , (6.4)where ψ [ γ ] denotes the image of γ under the map ψ [ ϕ ] . The field configurations γ i = γ I ( x ; θ ) are an explicit parameterisation of γ in terms of the angle ≤ θ ≤ π such that γ I ( x ; 0) = γ I ( x ; 2 π ) and, of course, Γ[ ϕ ] = Z Ω d n x L ( ϕ I , ∂ µ ϕ I , . . . ) , (6.5)with L the effective Lagrangian density. Note that, because δ Γ = δψ/ψ is an exact form,the winding number does not depend on the curve γ .A functional singularity exists whenever W = 0 . Note that, in general, the closed curve γ runs over points ϕ ∈ M that are not solutions of the exact equations of motion. On the This definition of the winding number holds for a functional singularity with ψ = 0 . Nonetheless, thewinding number around ψ → ∞ can be analogously obtained by shifting the order parameter ψ [ ϕ ] + ψ = e i Γ[ ϕ ] with ψ → ∞ . The final result turns out to be the same Eq. (6.4). γ about the point ϕ that is restricted only tosolutions of the full equations of motion (4.15) with J i = 0 , then Eq. (6.4) implies that ϕ isnot singular. Similarly, if there is a curve γ whose tangent vector is everywhere normal to δ Γ /δϕ i , then ϕ is non-singular. In principle, Eq. (6.4) establishes a well-defined procedureto determine whether a certain field configuration is singular.Computing the effective action in general, however, is far from trivial and one usuallyneeds to rely on approximate methods. We shall now see a simple example where we cancompute W and then move to the case of gravity. In these examples, we do not calculatethe effective action from first principles, we rather assume that it has been given. This is,however, enough to show the power of our formalism. In this sense, both examples shouldbe considered as toy models. Whether these toy models are realised in real systems does notconcern us here. Let us consider a field theory with two real scalars ϕ i = ( ϕ ( x ) , ϕ ( x )) and the effectiveaction Γ[ ϕ , ϕ ] = Z Ω d x (cid:20) ∂ µ ϕ ∂ µ ϕ + 12 ∂ µ ϕ ∂ µ ϕ − V ( ϕ , ϕ ) (cid:21) , (6.6)where the potential V ( ϕ , ϕ ) = Λ arctan (cid:18) ϕ ϕ (cid:19) , (6.7)and Λ is a mass parameter accounting for the correct dimensions. Note that the potential,thus the effective action, is undefined at the trivial solution ϕ I ( x ) = (0 , , which makessuch a point a field singularity by our definition. It is then convenient to consider γ givenby homogeneous and static configurations encircling the origin ϕ i = (0 , , that is γ I ( x ; θ ) = ( A cos θ, A sin θ ) , (6.8)where A is an arbitrary positive constant. The variation of the effective action yields δ Γ[ ϕ , ϕ ] δϕ I ( x ) = − (cid:3) ϕ I ( x ) − ∂V∂ϕ I ( x ) , (6.9)which, when evaluated along γ , gives δ Γd θ = Z Ω d x ∂ L ∂ϕ I (cid:12)(cid:12)(cid:12)(cid:12) ϕ I = γ I d γ I d θ = Z Ω d x ∂V ( ϕ ) ∂ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ I ( x )= γ I sin θ − ∂V ( ϕ ) ∂ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ I ( x )= γ I cos θ ! A = V (4) Λ , (6.10) Of course, this explicitly depends on the field-space metric G ij . δ Γ / d θ denotes the functional derivative along the curve γ and V (4) is the 4-volumeof the whole spacetime Ω . Note that the kinetic terms vanish along γ , because we areconsidering static and homogeneous configurations (which, furthermore, are not solutions ofthe effective equations of motion). Note also that, for this simple case, the same result canbe obtained from the direct calculation of the effective action on the encircling configura-tions (6.8), that is Γ( θ ) = − V (4) Λ arctan(cot θ ) = V (4) Λ θ , (6.11)and then use W = 12 π Z π δ Γd θ d θ . (6.12)Finally, the winding number (6.4) is simply given by W = 1 , (6.13)provided we set V (4) = Λ − . Let us recall that the winding number is an integer by definition,thus we must choose V (4) and Λ − to comply with this fact. This is formally reflected in thenormalisation of the parameter θ along γ as an angle. For our purpose, what matters is thatthe formalism for functional singularities can distinguish between field spaces with W = 0 everywhere and those with W = 0 for paths encircling specific configurations, non-zerowinding numbers with different magnitudes being fundamentally equivalent.The above calculation shows that the field configuration ϕ i = (0 , is indeed a functionalsingularity of non-zero topological charge. Note that in obtaining this result we assumed thatthe effective action had already been calculated and handed to us in the closed form (6.6).The calculation of the effective action, however, depends on the geometry of field space via thepath integral measure. It is clear from Eq. (5.5) that the functional singularity (6.13) couldnot have resulted from the divergent measure for the simplest field-space metric (5.4), because det G s ij is regular in flat spacetime. It is also important to stress that a non-zero windingnumber is a necessary and sufficient condition for the existence of functional singularities,whereas a divergent det G ij is only sufficient. Therefore, the result (6.13) could either reflecta more complicated field-space metric than (5.4), whose determinant diverges, or a resultthat is not captured solely by the path integral measure. A typical example of a spacetime singularity in which the determinant of the spacetimemetric vanishes, namely g = 0 , is the system of a homogeneous massless scalar field φ = φ ( t ) minimally coupled to the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. Forsimplicity, we shall assume the effective action ˜Γ = Z Ω d x √− g (cid:18) R π G N − ∂ µ φ ∂ µ φ (cid:19) , (6.14)22here G N denotes Newton’s constant and R is the Ricci scalar. In the ADM decomposition,the spatially flat FLRW metric is given by d s = − N d t + a (cid:2) (d x ) + (d x ) + (d x ) (cid:3) (6.15)where we have set the shift functions N i to zero, N = N ( t ) denotes the lapse function and a = a ( t ) the scale factor. The Ricci scalar for the metric (6.15) is given by R = 6 N ¨ aa − ˙ a ˙ Na N + ˙ a a ! (6.16)where we adopted the standard notation ˙ a = d a/ d t for time derivatives. Plugging the FLRWmetric (6.15), along with (6.16), in the effective action (6.14) leads to Γ ≡ ˜Γ V (3) = 12 Z d t " κ a ¨ aN − a ˙ a ˙ NN + a ˙ a N ! + a ˙ φ N − d F d t = − Z d t a ˙ a κ N − a ˙ φ N ! ≡ Z d t L , (6.17)where we have defined κ = 8 π G N / and V (3) is the spatial volume corresponding to thespatial isometries of the chosen field configurations, in analogy with the previous example.Note that, in obtaining Eq. (6.17), we have included the total derivative of F = a ˙ aκ N (6.18)in order to remove second derivatives of a , which further eliminates ˙ N . The variation ofthe Lagrangian then reads δL = a κ N − ¨ aa − ˙ a a + ˙ a ˙ Na N − κ ˙ φ ! δa − dd t a ˙ φN ! δφ + 12 a ˙ a κ N − a ˙ φ N ! δN , (6.19)which gives the equations of motions of each degree of freedom when δL = 0 . For N = 1 ,this system has solutions of the form a ± ( t ) = ± √ κ p φ t (6.20) φ ± ( t ) = ± √ κ log (cid:18) ± tt (cid:19) , (6.21) In fact, N is not a dynamical variable but a Lagrange multiplier corresponding to the time-reparameterisation invariance of the model. t is an integration constant and p φ = a ˙ φ is a constant of motion that follows from theequation for φ . Note that we have taken p φ positive for simplicity, since its sign does not affectour analysis. The positive (negative) sign of the scale factor a in Eq. (6.20) corresponds to anexpanding (contracting) universe with < t < ∞ (respectively, −∞ < t < ). For simplicity,we have also chosen the integration constants for a such that a ± (0) = 0 . This allows us tojoin the contracting and expanding solutions at t = 0 to form the “bouncing” configurationwhich we denote as ϕ i s = ( a s ( t ) , φ s ( t )) for brevity. Note that the Ricci curvature (6.16) forthe solutions (6.20) diverges for t → , indicating the presence of a spacetime singularityat the bounce in t = 0 , where furthermore the scalar field | φ s | → ∞ . Notably, the effectiveaction (6.17) is however finite for the solutions (6.20)-(6.21) (indeed Γ[ ϕ s ] = Γ[ a ± , φ ± ] = 0 ),suggesting that the spacetime singularity at t = 0 has no corresponding functional singularity.In the following, we shall confirm this by showing that the winding number for the bouncingconfiguration ϕ s is indeed equal to zero.Let us recall that to calculate the winding number for a certain field configuration, wemust encircle that specific configuration with a curve γ . Similarly to the previous example,we shall parameterise γ around ϕ s as γ I ( t ; θ ) = ( a s ( t ) + A cos θ, φ s ( t ) + A sin θ, , (6.22)for all values of t for which a = a s ( t ) and φ = φ s ( t ) are defined, and A is a positive constant.Note that the Lagrangian L in Eq. (6.17) diverges (like t − ) for t → when computed onthe above configurations (for θ fixed). This makes the calculation of the effective action Γ along the encircling configurations (6.22) quite tricky because the time integral in Eq. (6.17)diverges. In order to avoid such complications, we can exploit the freedom to add totalderivatives to L and, in particular, we replace F in Eq. (6.18) by F = α ˙ φ + β ˙ a (6.23)and take α = A θ (6.24) β = A θ . (6.25)This choice for the total derivative cancels out the divergence in the time integral on theconfigurations (6.22), while keeping Γ[ ϕ s ] = 0 . Moreover, we remark that F ( γ I ( t ; θ )) → for t → ±∞ . The resulting effective action evaluated along (6.22) then vanishes identically,namely Γ( θ ) = 0 (6.26) Let us recall that total derivatives affect neither the (effective) equations of motion nor scatteringamplitudes. One could also regularise the time integral in Eq. (6.17) with a cut-off | t | > T > and take T → atthe end. It can be easily verified that this procedure also leads to W = 0 . W = 0 follows. As we expected, the spacetime singularity at t = 0 does not correspondto a functional singularity. The most striking consequence of this result is that all physicalobservables remain finite and well-defined for the bouncing solution, even at t = 0 . Thisindeed reflects the existence of field-space coordinates in which the spacetime singularitycompletely disappears [63–65]. Note that this conclusion cannot be reached from the func-tional measure associated to (5.5), as the vanishing of the measure at a single configurationis not sufficient to yield a vanishing path integral. The fact that W = 0 indeed results fromnon-trivial global contributions to the path integral. The winding number W can be expressed in terms of the external current, which brings thepossibility of evaluating W without the need of Γ[ ϕ ] . In fact, using the effective equationsof motion (4.15) in the expression (6.4) for the winding number, we find W = − π I γ J k v k ; i d γ i . (6.27)From the Stokes theorem, one then obtains W = − π Z A ( J k v k ; i ) ,j d ϕ j ∧ d ϕ i = − π Z A ξ ij d ϕ j ∧ d ϕ i , (6.28)where A is a surface with boundary ∂ A = γ and we used the antisymmetry of the wedgeproduct, with ξ ij [ ϕ ] := ( J k v k ; i ) ,j − ( J k v k ; j ) ,i . (6.29)We can now proceed as before and parameterise the surface A by ζ I ( x ; r, θ ) in terms of theradius r > and the angle ≤ θ < π . In this parameterization, Eq. (6.28) becomes W = − π Z A d r ∧ d θ Z d n x Z d n x ′ ξ IJ [ r, θ ] ∂ζ I ( x ; r, θ ) ∂r ∂ζ J ( x ′ ; r, θ ) ∂θ , (6.30)with ξ IJ [ r, θ ] = Z d n x K ( δδϕ J ( x J ) (cid:20) J K ( x K ) ∇ ϕ I ( x I ) v K ( x K ) (cid:21) ϕ I ( x I )= ζ I ( x I ; r, θ ) (6.31) − δδϕ I ( x I ) (cid:20) J K ( x K ) ∇ ϕ J ( x J ) v K ( x K ) (cid:21) ϕ I ( x I )= ζ I ( x I ; r, θ ) ) , (6.32) A rigorous definition of functional integration is a well-known open problem in mathematics, thus theformal application of Stokes theorem to infinite-dimensional spaces must be carried out with great care. Inour case, we recall that γ is finite-dimensional (and so is A ), thus one can apply the Stokes theorem asusual. i = ( I, x I ) to keep track of the different indices. Recall that both J I ( x ) = J I [ ϕ ( x )] and v I ( x ) = v I [ ϕ ( x )] are functionals of ϕ i . This expression can be further simplifiedif we assume the standard effective equations, namely v k ; i ≡ ∇ i v k = δ ki , in which case weobtain ξ IJ [ r, θ ] = δJ I ( x I ) δϕ J ( x J ) (cid:12)(cid:12)(cid:12)(cid:12) ζ I ( x I ; r, θ ) − δJ J ( x J ) δϕ I ( x I ) (cid:12)(cid:12)(cid:12)(cid:12) ζ I ( x I ; r, θ ) . (6.33)This is however only valid when the field-space geometry satisfies R ijkl v l = 0 , which includesthe case of a flat field space. From Eqs. (6.30) and (6.33), we see that W can be interpretedas the flux of the curl of the external current J i across the area determined by the circuit γ . Note that the vanishing curl of the external current, ξ IJ [ r, θ ] = 0 , is a sufficient (but notnecessary) condition for the absence of singularities.The above topological consideration shows that there is an infinite number of possiblefunctional singularities, each one labeled by the winding number W . These are potentialsingularities, but their actual presence depends on the specifics of the path integral andfield-space geometry, as outlined in Section 5. Given the effective action, one is then ableto determine explicitly whether its corresponding theory contains functional singularities aswe did in the examples of Sections 6.1 and 6.2. The crucial point in such a topologicalclassification is that one can now assess whether spacetime singularities, predicted by theHawking-Penrose theorem, lead to any catastrophic consequence for the theory itself. Aswe showed in Section 6.2, it is indeed possible that known spacetime singularities turn outto be removable when W = 0 . Because the Vilkovisky-DeWitt effective action is invariantunder field redefinitions, so is the winding number. The absence of functional singularitiesin combination with the invariance of the winding number under field redefinitions suggeststhe existence of some coordinates in field space in which the spacetime singularity is insteadregular. The formalism developed in this section can thus be used to enforce the vanishing ofthe winding number, W = 0 , to mitigate functional singularities and, ultimately, spacetimesingularities. This might indeed be helpful in the construction of a regular field-space metric,in relation to the description of functional singularities in Section 5. Spacetime singularities have since long been pointed as one of the reasons General Relativityneeds replacement. The generality of the Hawking-Penrose theorem makes it difficult toovercome spacetime singularities even in modified gravity theories, with the exception ofsome very special models. With the principle of covariance in field space, it becomes crucialto put singularities under the microscope and analyse them under this new perspective. Asit is somewhat expected, the Hawking-Penrose theorem does not survive field redefinitions.The calculation of the field-space Kretschmann scalar in pure gravity has also suggested thatclassical observables, defined as scalar functionals in field space, are finite for the special case ǫ = 1 / of the DeWitt metric [1].In this paper, we have taken another step in understanding the meaning of singularitiesin physics. We considered the effective action as the onset for the investigation of singular-26ties since it encodes the information about all physical observables. A closer look into itsdefinition has revealed two potential types of singular behaviour. The first, and less dan-gerous one, takes the form of points where geodesics become incomplete for a finite value ofthe affine parameter. They are thus analogous to spacetime singularities and are the onesthat would have been revealed by the functional Kretschmann scalar. At the classical level,geodesic completeness appears to be sufficient for the proper definition of classical observ-ables. On the other hand, at the quantum regime there is another source of singularity as thepath integral can become divergent or vanish identically, in which case the effective actioncannot be defined. It is important to stress that such divergences are not UV divergencesof perturbative quantum field theory. Functional singularities are rather defined at the non-perturbative level and they are reminiscent of divergent path-integral measures which inherittheir properties from the corresponding geometry of field space. It is somewhat surprisingthat both the functional Kretschmann scalar and the path-integral measure remain regularin four spacetime dimensions for the DeWitt metric. This suggests that n = 4 stands at aspecial place from the perspective of the geometry of field space. The presence of matter,however, changes this situation considerably as a functional singularity always exists whenthe determinant of the spacetime metric g → ∞ , unless gravity is treated classically orsemi-classically. We must again emphasise that the effective action stems from the interplaybetween the field-space geometry, the classical action and the boundary conditions in thepath integral. For this reason, a regular field space alone does not guarantee the absence offunctional singularities.The fundamental group of the functional order parameter space, on the other hand, takesinto account all the three ingredients above. It provides a topological classification of func-tional singularities, which are then labeled by the winding number. Whether such functionalsingularities are really present in a given theory, however, depends on the resulting effectiveaction, which ultimately hinges on the particular geometry of the field space, on the classi-cal action as well as on boundary conditions. Generally, an effective action with vanishingwinding number is free of functional singularities. We showed that the winding numberindeed vanishes for the class of field theories with functionally irrotational external sourcesand whose functional Riemann tensor satisfies R ijkl v l = 0 . We thus conclude by remarkingthat the topological classification of functional singularities, along with the geometry of fieldspace, serves as an important tool in the construction of a consistent theory of quantumgravity. Acknowledgments
I.K., A.K. and R.C. are partially supported by the INFN grant FLAG. The work of R.C. hasalso been carried out in the framework of activities of the National Group of MathematicalPhysics (GNFM, INdAM) and COST action Cantata. A.K. is partially supported by theRussian Foundation for Basic Research grant No 20-02-00411.27
Focusing theorem in field space
The existence of caustics in M can be understood in terms of the convergence of a familyof geodesics with the aid of the Raychaudhuri equation in field space, which ultimatelytranslates into a condition on the functional Ricci tensor. The proof follows the same logic ofthe standard result in spacetime, which we review in the following for the case of Riemannianmetrics. Let us assume that the field space M can be foliated by hypersurfaces orthogonalto geodesics so that d s = (d φ ) + G ¯ ı ¯ d φ ¯ ı d φ ¯ , (A.1)where φ is a fiducial but otherwise arbitrary direction taken to be orthogonal to theothers (denoted by barred indices). The condition for the existence of a focal point, namelya point where the geodesic congruence converges, is given by det G ¯ ı ¯ = 0 . (A.2)Due to the ultralocality of the field-space metric, the focusing condition (A.2) can be trans-lated into a condition on the metric of the finite-dimensional manifold N , Y x ∈ Ω det G ¯ I ¯ J ( x ) = 0 ⇐⇒ det G ¯ I ¯ J = 0 , (A.3)so that a focal point in N is also a focal point in M . We can thus study the convergence of ageodesic congruence in terms of G ¯ I ¯ J . Since G ¯ I ¯ J is the metric of the finite-dimensional space N , one can repeat the reasoning used to demonstrate the focusing theorem in spacetime aswe shall now review.Let X ¯ I be the normalized vector field tangent to the geodesic congruence, parameterisedby an affine parameter λ along geodesics, and orthogonal to the hypersurface φ = φ for agiven λ . The functional covariant derivative of X ¯ I can be split into the irreducible represen-tations of the group SO ( N ) as ∇ ¯ I X ¯ J = ς ¯ I ¯ J + Ω ¯ I ¯ J + ϑN − h ¯ I ¯ J , (A.4)where N is the dimension of N , h ¯ I ¯ J = G ¯ I ¯ J − X ¯ I X ¯ J and ϑ = G ¯ I ¯ J ∇ ¯ I X ¯ J (A.5) ς ¯ I ¯ J = ∇ (¯ I X ¯ J ) − ϑN − h ¯ I ¯ J (A.6) Ω ¯ I ¯ J = ∇ [ ¯ I X ¯ J ] (A.7)denote the functional expansion parameter, the functional shear tensor and the functionaltwist tensor of the congruence, respectively. The functional Raychaudhuri equation thenreads δϑ d λ = − ϑ N − − ς ¯ I ¯ J ς ¯ I ¯ J + Ω ¯ I ¯ J Ω ¯ I ¯ J − R ¯ I ¯ J X ¯ I X ¯ J , (A.8) Analogous results can be found for time-like and null-like geodesics in Lorentzian field spaces. One can obviously choose any of the fields φ I , different choices correspond to different foliations of M . δ/ d λ is the functional derivative along the congruence. 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