PHYSICAL REVIEW D 97, 126006 (2018) Antipodal correlation on the meron wormhole and a bang-crunch universe Panagiotis Betzios* Crete Center for Theoretical Physics, Institute for Theoretical and Computational Physics, Department of Physics, University of Crete, 71003 Heraklion, Greece † Nava Gaddam Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, 3508 TD Utrecht, Netherlands ‡ Olga Papadoulaki Mathematical Sciences and STAG Research Centre, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom (Received 13 January 2018; published 11 June 2018) We present a covariant Euclidean wormhole solution to Einstein Yang-Mills system and study scalar perturbations analytically. The fluctuation operator has a positive definite spectrum. We compute the Euclidean Green’s function, which displays maximal antipodal correlation on the smallest three sphere at the center of the throat. Upon analytic continuation, it corresponds to the Feynman propagator on a compact bang-crunch universe. We present the connection matrix that relates past and future modes. We thoroughly discuss the physical implications of the antipodal map in both the Euclidean and Lorentzian geometries and give arguments on how to assign a physical probability to such solutions. DOI: 10.1103/PhysRevD.97.126006 I. INTRODUCTION whose few proposed resolutions suffer from closed time- like curves and other intricacies [10]. Euclidean wormholes [1–3] are extrema of the Euclidean Inspired by these questions, we study a Euclidean meron action whose interpretation is still partially shrouded in wormhole [11] in light of the antipodal Z2 mapping mystery. Originally they were proposed as a resolution to proposed by ’t Hooft. We find a positive definite spectrum the cosmological constant problem [4], and as objects that for scalar perturbations, whose Euclidean Green’s function lead to a loss of quantum coherence causing an inherent exhibits large antipodal correlation localized near the uncertainty in the fundamental constants of nature [2,5,6]. smallest sphere at the center of the throat after performing An alternative interpretation was given in [7] within the the antipodal map. The analytic continuation of this context of the Wheeler-de Witt equation. Besides several solution results in a finite bang-crunch geometry [12], problems specific to these proposals, a basic general with temporal and spatial boundaries, opening up a question that remains unanswered is whether such solutions possible handle on the problem of observables. are stable (and thus minima of the Euclidean action), or if they should alternatively be thought of as bounces (or maxima) that might nevertheless contribute to the path II. THE MERON WORMHOLE integral in some nonperturbative fashion akin to [8]. Consider the Euclidean Einstein-Yang Mills system: On a parallel note, defining quantum gravitational Z observables in closed universes is an acute problem [9] pffiffiffi 1 1 4 a 2 S ¼ d x g − R þ ðFμνÞ 16πGN 4 * [email protected] a † in units ℏ ¼ c ¼ 1, where R is the Ricci scalar and Fμν, the [email protected] ‡ field strength for the SUð2Þ (possibly embedded in SUðNÞ) [email protected] a a gauge field Aμ. The field strength is defined as Fμν ≔ ∂ a − ∂ a þ ϵabc b c Published by the American Physical Society under the terms of μAν νAμ gYM AμAν. In addition to the Yang- the Creative Commons Attribution 4.0 International license. μ a Mills equations of motion D Fμν ¼ 0 and the correspond- Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, ing Bianchi identities, the Einstein equations of motion are and DOI. Funded by SCOAP3. given by 2470-0010=2018=97(12)=126006(6) 126006-1 Published by the American Physical Society BETZIOS, GADDAM, and PAPADOULAKI PHYS. REV. D 97, 126006 (2018) 1 throat. The IR divergence can be evaded by considering Rμν − gμνR ¼ 8πG Tμν 2 N similar solutions in the presence of a small cosmological constant that provides a natural regulator for such IR ¼ λ ¼ a aρ − 1 ð a Þ2 where Rμν Rμλν and Tμν FμρFν 4 gμν Fμν . The divergences. This is one possible and known way of a ¼ η −1 x−2 meron configuration Aμ aμνxνgYM yields the field making sense of similar configurations; however, a strength satisfactory solution for asymptotically flat space-times is not known. Inspired by the work of Kosterlitz and 1 f1 f2 Thouless [21] (whose relevance to merons is sketched in a ¼ η þð η − η Þ Fμν aμν 2 xμ aργxγ xρ aμγxγ 2 ; [19]), we propose that the seemingly infinite classical gYM x x action must be compared with the entropy of such configurations. A known issue with the pure meron back- with f1 ¼ f2 ¼ −1. Here, ηaμν are the ’t Hooft Eta symbols [13]—whose conventions we follow—that mix the gen- ground [19] is that one needs to regulate the meron core erators of space-time SOð4Þ with those of the SUð2Þ gauge with an ad-hoc procedure which results in a violation of the symmetry. The summation over all lower indices is carried equations of motion for the regulated configurations. This UV problem is evaded by the meron wormhole owing to its out by the flat metric δμν; however, owing to the conformal nature of the Yang-Mills equations, this is a solution in any finite-sized throat, a fact that will become even more transparent in our novel interpretation—using an antipodal conformally flat background. The meron configuration was — one of the first proposed mechanisms for confinement identification of the wormholes as holes of nothing. [14,15]. This meron-pair solution has a unit topological instanton charge split equally between the two points at A. Symmetries of the background x ¼ 0 and x ¼ ∞; the pair may be brought closer together Flat space has ten killing vectors (KVs) and five with a conformal transformation [16]. Unlike the Belavin- conformal KVs (CKVs). Since the wormhole is confor- Polyakov-Schwartz-Typkin (BPST) instanton [13,17,18], mally flat, its symmetries may be found by studying if the this solution is not self-dual. The meron’s magnetic field former remain KVs and if any of the latter get promoted to lines provide the required locally-negative energy density KVs. It may be checked that while rotations remain killing to support a wormhole throat and source the right hand side vectors of the wormhole background, translations become of the Einstein equations which are solved by the metric CKVs. Additionally, none of the five CKVs (dilatation and special conformal transformations) of flat space are pro- 2 2 2 r1 μ ν 2 −2 moted to KVs. Finally, the wormhole background pos- ds ¼ 1 þ δμνdx dx r1 ¼ πG g ; μ 2 μ −2 x N YM sesses a discrete inversion symmetry under x → r1x x . In the r coordinates, it reads r → −r. This is an antipodal 2 ¼ 2 þð 2 þ 4 2Þ Ω2 ð Þ ds dr r r1 d 3 1 mapping when appended with appropriate maps on the pffiffiffiffiffiffiffiffiffi three-sphere, similar to the one considered in [22,23] in the x ¼ μ ≥ 0 ¼ x − 2x−1 with xμx and r r1 . This transforma- context of ’t Hooft’s black hole S-Matrix. tion maps the regions x ≶ r1 to the two sides of the ≶ 0 x ¼ wormhole r and the smallest sphere to r1. III. SCALAR PERTURBATIONS The geometry is that of a two-sided Euclidean space (with the two asymptotic regions at r ¼∞) connected by a The action governing the fluctuations of aR chargedpffiffiffi scalar ð2Þ ¼ 4 Lð2Þ ≡ 2r1 Rfield in this background, is given by Sϕ d x g ϕ wormhole throat of smallest size ; it has the curious 4 pffiffiffi ⋆ property that the regime of its semiclassical validity d x gϕ Mϕϕ with coincides with the perturbative regime of the YM theory. ð2Þ 2 2 2 The metric admits a natural analytic continuation r ¼ it Lϕ ¼½jDμϕj þðm þ ζRÞjϕj ; into a big bang-big crunch universe, with singularities at ¼2 ϕ ≕ ð∇ þ Þϕ r r1 [12]. It is an open problem to find general where Dμ μ igYMAμ;a . In order to explicitly 2 multimeron solutions specified by functions f1ðx Þ evaluate the spectrum of the operator Mϕ, it is first useful 2 and f2ðx Þ, constrained by both Einstein and Yang-Mills to define the following space-time operators, which re- equations. present rotations in the two invariant SU(2) subgroups of Let us mention here that the Euclidean action is the rotation group SO(4): logarithmically divergent. This is easy to see since the 1 1 Ricci scalar vanishes (R ¼ 0) and, from the Yang-Mills a a L ≔− iη μνxμ∂ν and L ≔− iη¯ μνxμ∂ν Lagrangian, one finds the usual log-divergence of the 1 2 a 2 2 a meron configuration ∼ logðL=aÞ where L is the system ½ a b¼ δ ϵabc c 2 ¼ 2 ¼ 2 ¼ −ð1 8Þ size (IR-cutoff) and a is a UV-cutoff [19]. This problem has with Lp;Lq i pq Lp and L1 L2 L = × 2 been discussed in the past, for example in [11] and later in ðxμ∂ν −xν∂μÞ . Owing to the SUð2Þ projection of these [20]. The UV problem is evaded by the finite size of the operators, the eigenvalues of L2 are given in terms of 126006-2 ANTIPODAL CORRELATION ON THE MERON WORMHOLE … PHYS. REV. D 97, 126006 (2018) half-integers l via L2 ¼ lðl þ 1Þ. Furthermore, for scalars, breaks all the noncompact symmetries of flat space, there isospin rotations are generated by the operators Ta with are no zero modes either. The vacuum spectrum is eigenvalues T2 ¼ tðt þ 1Þ, with an arbitrary total isospin t.