PHYSICAL
PRL 114, 031104 (2015)
REVIEW
week ending 23 JANUARY 2015
LETTERS
Emergent Gravity Requires Kinematic Nonlocality Donald M arolf Department o f Physics, University o f California, Santa Barbara, California 93106, USA (Received 26 September 2014; published 22 January 2015) This Letter refines arguments forbidding nonlinear dynamical gravity from appearing in the low energy effective description of field theories with local kinematics, even for those with instantaneous long-range interactions. Specifically, we note that gravitational theories with universal coupling to energy—an intrinsically nonlinear phenomenon—are characterized by Hamiltonians that are pure boundary terms on shell. In order for this to be the low energy effective description of a field theory with local kinematics, all bulk dynamics must be frozen and, thus, irrelevant to the construction. The result applies to theories defined either on a lattice or in the continuum, and requires neither Lorentz invariance nor translation invariance. DOI: 10.1103/PhysRevLett.ll4.031104
PACS numbers: 04.20.Cv, 04.60.-m
Introduction.—A ttem pts to directly quantize the gravi tational field encounter well-known difficulties associated with a lack o f perturbative renormalizability, the black hole inform ation problem , and the lack o f local observables due to invariance under diffeom orphism gauge symmetry (which, from the active point o f view, m oves spacetime points from one location to another). W hile it remains possible that any or all o f these issues m ay one day be surmounted, it is, nevertheless, interesting to ask whether diffeom orphism -invariant gravity— som etim es called background-independent gravity— m ight emerge an effective approxim ate description o f a system that is inherently better behaved at the m icroscopic level. M ost leading approaches to quantum gravity em body ideas along these lines. String theory, loop quantum gravity, causal sets [1], and causal dynam ical triangulations [2] (see, e.g., [3] for a recent overview) all propose that smooth classical geom etries arise only in appropriate semiclassical limits. (Asym ptotic safety is the m ost prom inent exception; see, e.g., [4,5] for recent reviews.) B ut the structures underlying these theories again involve novel physics that is difficult to control. So, it is natural to ask if gravity can arise from m ore fam iliar systems such as field theories with local kinem atics. Exam ples o f such proposals include [6-22]. Below, w e argue that such scenarios can succeed only if the m ap to gravitational degrees o f freedom involves long-range nonlocality, i.e., only if the notions o f locality are very different in the two descriptions. We em phasize that we focus here on w hether gravitational theories with an appropriate form o f diffeom orphism -invariance (or w hat is often called independence from background structures) can emerge as effective descriptions o f theories built on familiar background structures such as fixed (nondynam ical) spacetim e lattices or sm ooth spacetim es w ith a metric; we m ake no com m ent on the possible em ergence o f general relativity from discrete theories o f quantum gravity or on scenarios, such as in [23,24], where the entire notion o f tim e evolution is, itself, emergent. 0031 -90 0 7 /1 5 /1 1 4 (3 )/0 3 1104(5)
A s has been well know n for some time, the (spin-2) W einberg-W itten theorem [25] already excludes the em er gence o f gravity from local Poincare-invariant field theo ries. In particular, it forbids such theories from containing an interacting massless spin-2 degrees o f freedom in its spectrum o f asymptotic states. W hile clear and concise, the technical assum ption o f Poincare invariance appears to leave open many doors for exploration. For example, one m ight attem pt to evade the theorem by w orking on a lattice as in, e.g., [9,15,16,19,20], or by using other structures that break this symmetry. However, as noted in, e.g., [26,27], the lack o f local observables in quantum gravity suggests a m ore general result forbidding diffeom orphism -invariant gravity arising as the effective description o f any theory with sufficiently interesting local observables. O ur purpose, here, is to make this precise. Since any theory can be made diffeomorphism invariant via a process known as param etrization (see, e.g., [28,29,29-37]), we follow [38] in using the gravitational Gauss law to distinguish theories with sufficiently “inter esting” diffeom orphism invariance. The desired theories roughly correspond to what are often called “backgroundindependent” theories of gravity. Before stating our technical result in the next section, let us, therefore, take a m om ent to explain this idea in broadly accessible terms. First, we recall that (nonrelativistic) Newtonian gravity can be form ulated in terms o f a gravitational potential ([> that satisfies a Poisson equation V 2 = \n G p M sourced by the m ass density p M. As a result, the total m ass M — f v P u d V inside a volum e V (with volum e elem ent d V ) can be expressed as the boundary term M = (1 /4 uG ) f 9v d S n ‘d,(p where
PRL 114, 031104 (2015)
REVIEW
week ending 23 JANUARY 2015
LETTERS
Emergent Gravity Requires Kinematic Nonlocality Donald M arolf Department o f Physics, University o f California, Santa Barbara, California 93106, USA (Received 26 September 2014; published 22 January 2015) This Letter refines arguments forbidding nonlinear dynamical gravity from appearing in the low energy effective description of field theories with local kinematics, even for those with instantaneous long-range interactions. Specifically, we note that gravitational theories with universal coupling to energy—an intrinsically nonlinear phenomenon—are characterized by Hamiltonians that are pure boundary terms on shell. In order for this to be the low energy effective description of a field theory with local kinematics, all bulk dynamics must be frozen and, thus, irrelevant to the construction. The result applies to theories defined either on a lattice or in the continuum, and requires neither Lorentz invariance nor translation invariance. DOI: 10.1103/PhysRevLett.ll4.031104
PACS numbers: 04.20.Cv, 04.60.-m
Introduction.—A ttem pts to directly quantize the gravi tational field encounter well-known difficulties associated with a lack o f perturbative renormalizability, the black hole inform ation problem , and the lack o f local observables due to invariance under diffeom orphism gauge symmetry (which, from the active point o f view, m oves spacetime points from one location to another). W hile it remains possible that any or all o f these issues m ay one day be surmounted, it is, nevertheless, interesting to ask whether diffeom orphism -invariant gravity— som etim es called background-independent gravity— m ight emerge an effective approxim ate description o f a system that is inherently better behaved at the m icroscopic level. M ost leading approaches to quantum gravity em body ideas along these lines. String theory, loop quantum gravity, causal sets [1], and causal dynam ical triangulations [2] (see, e.g., [3] for a recent overview) all propose that smooth classical geom etries arise only in appropriate semiclassical limits. (Asym ptotic safety is the m ost prom inent exception; see, e.g., [4,5] for recent reviews.) B ut the structures underlying these theories again involve novel physics that is difficult to control. So, it is natural to ask if gravity can arise from m ore fam iliar systems such as field theories with local kinem atics. Exam ples o f such proposals include [6-22]. Below, w e argue that such scenarios can succeed only if the m ap to gravitational degrees o f freedom involves long-range nonlocality, i.e., only if the notions o f locality are very different in the two descriptions. We em phasize that we focus here on w hether gravitational theories with an appropriate form o f diffeom orphism -invariance (or w hat is often called independence from background structures) can emerge as effective descriptions o f theories built on familiar background structures such as fixed (nondynam ical) spacetim e lattices or sm ooth spacetim es w ith a metric; we m ake no com m ent on the possible em ergence o f general relativity from discrete theories o f quantum gravity or on scenarios, such as in [23,24], where the entire notion o f tim e evolution is, itself, emergent. 0031 -90 0 7 /1 5 /1 1 4 (3 )/0 3 1104(5)
A s has been well know n for some time, the (spin-2) W einberg-W itten theorem [25] already excludes the em er gence o f gravity from local Poincare-invariant field theo ries. In particular, it forbids such theories from containing an interacting massless spin-2 degrees o f freedom in its spectrum o f asymptotic states. W hile clear and concise, the technical assum ption o f Poincare invariance appears to leave open many doors for exploration. For example, one m ight attem pt to evade the theorem by w orking on a lattice as in, e.g., [9,15,16,19,20], or by using other structures that break this symmetry. However, as noted in, e.g., [26,27], the lack o f local observables in quantum gravity suggests a m ore general result forbidding diffeom orphism -invariant gravity arising as the effective description o f any theory with sufficiently interesting local observables. O ur purpose, here, is to make this precise. Since any theory can be made diffeomorphism invariant via a process known as param etrization (see, e.g., [28,29,29-37]), we follow [38] in using the gravitational Gauss law to distinguish theories with sufficiently “inter esting” diffeom orphism invariance. The desired theories roughly correspond to what are often called “backgroundindependent” theories of gravity. Before stating our technical result in the next section, let us, therefore, take a m om ent to explain this idea in broadly accessible terms. First, we recall that (nonrelativistic) Newtonian gravity can be form ulated in terms o f a gravitational potential ([> that satisfies a Poisson equation V 2 = \n G p M sourced by the m ass density p M. As a result, the total m ass M — f v P u d V inside a volum e V (with volum e elem ent d V ) can be expressed as the boundary term M = (1 /4 uG ) f 9v d S n ‘d,(p where
