Gravity-Matter Entanglement in Regge Quantum Gravity 6 1 Nikola Paunkovi´c1 and Marko Vojinovi´c2 0 1 2 SQIG— Security and Quantum Information Group, Instituto deTelecomunica˜coes, and Departamento deMatema´tica, Instituto SuperiorT´ecnico, Universidade deLisboa, Avenida n Rovisco Pais 1049-001, Lisboa, Portugal a 2 J Grupo deF´ısica Matema´tica, Faculdade deCiˆencias daUniversidade deLisboa, Campo Grande, Edif´ıcio C6, 1749-016 Lisboa, Portugal 2 2 E-mail: [email protected], [email protected] ] c Abstract. We argue that Hartle–Hawking states in the Regge quantum gravity model q generically contain non-trivial entanglement between gravity and matter fields. Generic - impossibility to talk about “matter in a point of space” is in line with the idea of an emergent r g spacetime, and as such could be taken as a possible candidate for a criterion for a plausible [ theory of quantum gravity. Finally, this new entanglement could be seen as an additional “effectiveinteraction”,whichcouldpossiblybringcorrectionstotheweakequivalenceprinciple. 1 v 1 3 Introduction. The unsolved problems of interpreting quantum mechanics (QM) and 8 6 formulating quantum theory of gravity (QG) are arguably the two most prominent ones of 0 the twentieth century theoretical physics. Up to date, most of the efforts to solve the two . 1 were taken independently. Indeed, the majority of the interpretations of QM do not involve 0 explicit dynamical effects (with notable exceptions of the spontaneous collapse and the de 6 Broglie–Bohm theories), while the researchers from the QG community often adopt the many- 1 : world interpretation of QM. Nevertheless, the two problems share a number of similar unsolved v questions and counter-intuitive features, such as nonlocality: entanglement-based quantum i X nonlocality, as well as the anticipated explicit dynamical nonlocality in QG (a consequence r of quantum superpositions of different gravitational fields, i.e., different spacetimes and their a respective causal orders). We analyse the generic entanglement between gravitational and matter fieldsin theRegge modelof quantum gravity, andits possibleimpact to thefundamental questions regarding QM and QG. Regge quantumgravity model. Asimpletoymodelofquantumgravitywithmatterfields is the Regge quantum gravity with one real scalar field, whose construction can be motivated by the Loop Quantum Gravity research program [1, 2]. The path integral of the model is ZT = DL DΦ exp iSRegge(L)+iSmatter(L,Φ) , (1) Z Z h i where L are the lengths of the edges of the triangulation T of a 4-manifold M , and Φ are 4 the values of the scalar field in 4-simplices of T. The measure terms DL and DΦ are defined via discretization induced by T. The actions S and S represent lattice discretizations Regge matter of the Einstein–Hilbert action for gravity and an action of the scalar field coupled to gravity, respectively. See [3, 4] for details. A generic state of the system, belonging to the kinematical Hilbert space HG⊗HM, is |Ψi= Dl Dφ Ψ(l,φ) |li|φi. (2) Z Z However, since gravity is a theory with constraints, not every kinematical state is allowed, so we must choose the coefficients Ψ(l,φ) such that |Ψi is an element of the physical Hilbert space Hphys ⊂ HG⊗HM. One such class of states are the Hartle–Hawking (HH) states [5], defined by the following choice of the coefficients, for a given triangulation T: Ψ(l,φ) = Ψ (l,φ) ≡ DL DΦ exp iS (L,l)+iS (L,Φ,l,φ) . (3) HH Regge matter Z Z h i This expression differs from (1) in that the triangulation T is now assumed to have a nontrivial 3-dimensionalboundary∂T,andthatthevariablesl,φlivingontheboundaryarenotintegrated over, in contrast to the bulk variables L and Φ. Using (2) and (3), one can calculate the reduced density matrix of the Hartle–Hawking state, ′ ∗ ′ ′ ρˆM = TrG|ΨihΨ| = Dφ Dφ Dl ΨHH(l,φ)ΨHH(l,φ) |φihφ|, Z Z (cid:20)Z (cid:21) ′ where the integral in the brackets can be denoted as ZT∪T¯(φ,φ). The resulting density matrix can then be tested for entanglement by checking if the trace of its square equals one [6]. In the Regge quantum gravity model the path integrals reduce to a finite number of ordinary integrals, which can then in principle be evaluated. For a generic triangulation, we obtain TrM ρˆ2M = Dφ Dφ′ ZT∪T¯(φ,φ′) 2 6= 1, Z Z (cid:12) (cid:12) (cid:12) (cid:12) i.e., the gravitational and scalar degrees of freedom in the generic HH state are entangled. Discussion. In [7] Penrose argues that gravity-matter entanglement is at odds with (class- ical) spacetime, seen as a (four-dimensional) differentiable manifold. In light of this, our result could be seen as a quantitative indicator that in quantum gravity one cannot talk of “matter in a point of space”, i.e., this result could beseen as a confirmation of a “spacetime as an emergent phenomenon”. Thus, generic gravity-matter entanglement could be seen as a possible candidate for a criterion for a plausible theory of quantum gravity. Entanglement is in standard quantum mechanics a generic consequence of the interaction. This new entanglement can be regarded as a consequence of an effective interaction (such as the “exchange interactions”, which are a consequence of quantum statistics). This additional “effective interaction” can potentially lead to corrections to the weak equivalence principle. Acknowledgments NP was partially supported, under the CV-Quantum internal project at IT, by FCT PEst- OE/EEI/LA0008/2013 and UID/EEA/50008/2013. MV was supported by the FCT grant SFRH/BPD/46376/2008, the FCT project PEst-OE/MAT/UI0208/2013, and partially by the projectON171031 oftheMinistryofEducation, ScienceandTechnological Development, Serbia. References [1] RovelliC 2004 Quantum Gravity (Cambridge: Cambridge UniversityPress) [2] RovelliC and Vidotto F2014 Covariant Loop Quantum Gravity (Cambridge: Cambridge UniversityPress) [3] Mikovi´cA and Vojinovi´c M 2012 Class. Quant. Grav. 29 165003 [4] Mikovi´cA 2013 Rev. Math. Phys. 25 1343008 [5] HartleJ B and Hawking S W 1983 Phys. Rev. D 28 2960 [6] NielsenMAandChuangIL2000QuantumComputationandQuantumInformation(Cambridge: Cambridge UniversityPress) [7] Penrose R 1996 Gen. Relativ. Gravit. 28 581