The factorization problem in Jackiw-Teitelboim gravity
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Citation Harlow, Daniel and Daniel Jafferis. "The factorization problem in Jackiw-Teitelboim gravity." Journal of High Energy Physics 2020, 2 (February 2020): 177 © 2020 The Authors
As Published https://doi.org/10.1007/JHEP02(2020)177
Publisher Springer Science and Business Media LLC
Version Final published version
Citable link https://hdl.handle.net/1721.1/128911
Terms of Use Creative Commons Attribution
Detailed Terms https://creativecommons.org/licenses/by/4.0/ JHEP02(2020)177 Springer February 12, 2020 February 27, 2020 November 14, 2019 : : : Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP02(2020)177 b [email protected] , . 3 1804.01081 and Daniel Jafferis The Authors. a c 2D Gravity, AdS-CFT Correspondence, Black Holes
In this note we study the 1 + 1 dimensional Jackiw-Teitelboim gravity in , [email protected] Center for the FundamentalCambridge, Laws of MA Nature, 02138, Harvard U.S.A. University, E-mail: Center for Theoretical Physics, MassachusettsCambridge, Institute MA of 02138, Technology, U.S.A. b a Open Access Article funded by SCOAP ArXiv ePrint: works using the SYKpure model. Einstein gravity Finally in weCFT 2 suggest dual, + consistent that 1 with similar dimensions, the comments which results should we’d of apply then Maloney to and concludeKeywords: Witten. also cannot have a appears to be in tensionthe with tension the is existence acute: ofwell-defined gauge Lorentzian we constraints bulk argue in the path thathas bulk. integral, JT which wormholes In gravity this has but is model no a ithint CFT sensible as does dual. to quantum not what theory, In have could based bulk be black on language, added hole a to it microstates. rectify these issues, It and does we give however an give example some of how this Lorentzian signature, explicitly constructing thethe gauge-invariant quantum classical Hilbert phase space space andHawking and Hamiltonian. wave We function also in semiclassicallyresults compute two to the different illustrate Hartle- bases thethe of gravitational Hilbert this version space Hilbert of of space. the the factorization two-boundary We system then problem tensor-factorizes use of on these AdS/CFT: the CFT side, which Abstract: Daniel Harlow The factorization problem ingravity Jackiw-Teitelboim JHEP02(2020)177 19 12 12 5 – 1 – 9 13 ]. But in fact as we move from the left to the central . It is sometimes said that in the context of the two-sided 14 22 – 1 18 21 ]. 17 – , we see that we can evolve the boundary times as far to the future as ]. Gravitational backreaction also provides the mechanism by which the 15 , 1 7 1 9 ]. This has led to much progress in understanding the correspondence, 14 25 8 – – 1 10 ] for a recent review, but sooner or later we will need to confront the fact that the 9 One especially confusing aspect of gravitational physics is that time translations are 3.3 Hartle-Hawking state 2.2 Phase space and symplectic form 3.1 Hilbert3.2 Space and energy eigenstates Euclidean path integral 2.1 Solutions we like without ever having the bulkthe time interior slice part go of behind the the horizon. slicethis up is It that precisely is we not the start until part we toof move of directly general the see relativity. evolution physics which How behind is are thewhen generated horizon, we from by but to the the distinguish CFT Hamiltonian the point constraint slices of in view the they center describe and precisely right the diagram, same quantum state? gauge transformations: much of the interestingFor dynamics example is tied consider up in figure theAdS-Schwarzschild gauge geometry, constraints. we canboundary see times the forward interiordiagram [ of in the figure wormhole by evolving both physics from the point ofeffects there view [ of the outsideholographic observer encoding unusually of sensitive the to higher-dimensionaltheory breaks gravitational bulk down into if we the try lower-dimensional toentropy preserve boundary bounds bulk locality [ beyond what is allowed by holographic Most discussions of bulkbackground physics [ in AdS/CFTsee focus [ on perturbative fieldsbulk about theory is a gravitational; fixed inas generic an states afterthought. gravitational In backreaction particular cannot be the strong treated redshift effects near black hole horizons make 6 Conclusion 1 Introduction 4 Factorization and the range of5 the time Embedding shift in SYK 3 Quantum Jackiw-Teitelboim gravity Contents 1 Introduction 2 Classical Jackiw-Teitelboim gravity JHEP02(2020)177 ]. This is the observation 28 , 27 ] also are too non-unique to do [ 26 – 2 – factorization problem 1 ]. Where precisely does it form? A naive answer would be at the 24 , 23 . Different kinds of time evolution in the bulk and boundary: the endpoints of a bulk ]. The recently studied “holographic screens” [ 25 In this paper we will be primarily interested in a third issue raised by considering grav- Another interesting question related to the black hole interior and gauge constraints is A related point is that the entire formation and evaporation of a small black hole in There is clearly at least some approximate sense of “where” the edge of a black hole currently is, for 1 example the event horizon telescopesimulates will black soon hole image merger theinteresting disc events to of using understand Sagittarius code A* the whichor and generality excises not the of LIGO some a the team formal kind already the underlying definition of firewall assumptions could black arguments. in be hole such given region. calculations, which It and applies would whether in be sufficiently generic situations to be relevant for the job. Ifwhere indeed (and there also are how) they firewalls, form. there should beitational a physics gauge-invariant in prescription AdS/CFT:that for the the presence of gauge constraints in the bulk poses a potential obstacle to the exis- of a huge shell ofevaporates. collapsing It matter, seems which doubtful, willa to not trick say collapse as the until this. least, longactually that One after apparent we might our horizons could also black are remove suggest hole highly aslice that non-unique firewall [ firewalls since by form they so depend at silly on “the” a apparent choice horizon, of but Cauchy the following. Say thatdevelops we a believe firewall that [ aevent black horizon, hole but which this evolves is unlikely fornotion, to a which actually long be for enough correct. example time The can event be horizon is modified a by teleological putting our evaporating black hole inside intersects the Hawking radiation after theboth complete of evaporation of which the asymptote resultingthe black to Hawking hole, process the would same beral time complementary diffeomorphisms slice to are of the imagined moreboundary the to standard time boundary. be one evolutions. gauge in Such fixed, which a tempo- directly description tying of together the bulk and AdS is spacelike tothe some Hamiltonian boundary constraints. timecollapsing In slice, matter prior other and to words, thus the there must formation of is be an a describable event spatial horizon purely and slice via another that spatial intersects slice that the Figure 1 timeslice are moved upevolved and using down the using Hamiltonianstates, the the constraint. ADM second two Hamiltonian, Since slices while the describe the the Hamiltonian interior same constraint CFT of is state. the zero slice on is physical JHEP02(2020)177 (1.4) (1.1) (1.2) (1.3) has a M , it is always possible is physical: in fact it is I a × R . 2. , µν ) approaches a constant on each ν x F L/ = 0. After quantization, this system δA µν = t µν x a, . F dxF at µ µ r + L L 0 γr ∝ Z Et = √ − dt x – 3 – ∂M | For our example on = ∂M µ ∞ Z −∞ x 2 A Z A 1 4 = 0 to ⊃ − − x = δS = 0, which leads to an empty theory. ) for the gauge transformations which we actually quotient S E π . In fact the most natural choice is to require that these con- ), and in particular it has no tensor product decomposition into R ∂M = 0 gauge by a gauge transformation that vanishes at the endpoints of the 0 is the (outward pointing) normal form. To formulate a good variational problem, A is a constant which could be removed by an “illegal” gauge transformation Λ = µ a r . Since we are not allowed to do such gauge transformations, Of course pure Maxwell theory is not expected to have a gravity dual anyways, so the In AdS/CFT this is motivated by wanting to preserve boundary locality. Quotienting by gauge transfor- 2 ax mations which approach nonzero constantsin amounts the to boundary imposing CFT, aserious singlet which constraint, condition violates but for locality. we aenting still global When by want symmetry gauge the to transformations boundary avoid whichcomponents projections is approach requires that 0 independent us + mix nonzero 1 to constants the just dimensional on two set locality the asymptotic different is boundaries. boundary a Quoti- less gauge group is U(1), not degrees of freedom to the left and right ofnon-factorization the of line this systemreaching may consequences: at first the appear Einstein-Maxwell uninteresting. theory on But the in two-sided fact AdS-Schwarzschild it has far- where − nothing but the Wilson linejust from becomes the quantum mechanics of a particle on a circle (here we are assuming the stants are all zeroby: (modulo 2 transformations wherewhich they act are nontrivially nonzero onto are phase go then space. to viewed asinterval. The asymptotic equation symmetries of motion then requires that is to take These boundary conditionsmust are at not least preserved requireconnected under that component any general of gauge gauge transformation transformations: Λ( we where we need to impose boundarythe space conditions of such configurations that obeying thisthese the term boundary boundary conditions, vanishes conditions. for the There natural variations are choice within various for options AdS/CFT for (the “standard quantization”) spacetime, but its value cannoteven-dimensional. be To the find only the dynamical otherthe variable since dynamical boundary phase variable conditions: space we is need theboundary always to variation term be of more the careful Maxwell about action on any spacetime structure of the boundarysimple field example, consider theory 1 when + 1 studied dimensional on Maxwell a theory disconnected on a spacetime. line interval As times a time: The equations of motion for this theory tell us that the electric field is constant throughout tence of a CFT dual, since such constraints might not be consistent with the tensor product JHEP02(2020)177 ], ]. ]. 45 35 , 29 – (1.5) , 34 27 41 , 39 , 38 ) is particularly ]. See also [ R , 33 ] for a nice review 36 SL(2 / ) 1 S ), which acts on the JT theory R , . + 2) R + Φ( – 4 – R 0 = Φ L ], with bulk Lagrangian density ], while the Schwarzian action on elements of diff( 32 42 – – 30 This lack of factorization implies that the theory cannot have a CFT dual, nev- 37 3 We will make almost noneither mention as of a the global group symmetry nor SL(2 group. as a natural subgroup of the gauged diffeomorphism We work primarily in Lorentzianshell signature, degrees focusing of freedom. on identifying the physicalThe on- “Schwarzian” Lagrangian willSchwarzian make theory no is appearance not in sensibleview our by it itself analysis. is in an Indeed Lorentzian artifact the signature, of from a our particular point way of of evaluating the Euclidean path integral. Finally we discuss some of the possible implications of our work for higher-dimensional There has been considerable recent interest in this model, see [ The theory we will study is the 1 + 1 dimensional Jackiw-Teitelboim theory of dilaton A similar analysis of the asymptotically-Minkowski CGHS model was done in [ • • • 3 techniques will be useful even to SYK-oriented readers. gravity. In particular, we will argue that there is awho quite discussed close degrees analogy of between freedom JT related to gravity those we find here in a one-sided context. model [ well-suited for understanding how the JTWe nonetheless theory include is a embedded section in where thatversion we model of explain [ the how our SYK analysis model,Schwarzian fits language. and into We the we hope Lorentzian will that our there analysis explain of how the JT to theory understand with our “more conventional” results in the These differences arise because our analysis is not particularly motivated by the SYK it could have a CFTand the dual; canonical then gravity the analysis JT would Lagrangian eventually would exit be itsand a regime further low of references. energy validity. approximation Ourfrom approach this however literature: is rather different in method and emphasis Maxwell theory in this model, whichspace. we will see similarly doesertheless not it have a is factorized a Hilbert spectrum. self-consistent There quantum is mechanical nothing system,JT in albeit description. the one However, gravitational we with analysis will a that comment requires continuous on a what breakdown might of be the added to the theory so that gravity [ The first two sections of our paper will simply repeat the analysis sketched above for ory, and which tells us,one-sided among other states things, must that exist in withNonetheless gravitational all it theories gauge with would charges CFT be allowed duals, tional by nice charge model, to quantization which concretely [ at realizedual. least the Our somewhat factorization main plausibly problem goal in might in this have a paper been gravita- is hoped to to do have precisely a this. CFT geometry in any spacetime dimension has a zero mode sector which is equivalent to this the- JHEP02(2020)177 . is 1) (2.3) (2.2) (2.4) (2.1) − M K Φ. Φ( | | is just the γ | γ µν | 0 p δg p dx dt R Φ 2 2 − ∂M ∇ Z , Φ i µν δ g αβ − δγ 5 + 2) µ Φ + 2)+2 V ν αβ R µ γ r R ∇ | µ γ + ( | Φ) Φ( the outward-pointing normal form ∇ p g µν − x µ 1 r − δg , Φ − Φ + ν ν d √ r d x µν ∇ µ 2 µν ν R d ∇ ∂M r is a holographic renormalization, which en- Z − Rg M µν = K 1 2 Z γ – 5 – µν µ + Φ + ( g − ≡ V δ µ is outward-pointing if it is spacelike but inward-pointing if it µν K 1) appear only on the asymptotically-AdS parts of the boundary, K |∇ µ | R g r − | γ + 2)Φ | αβ is the trace of the extrinsic curvature of the boundary, ]. From a two-dimensional point of view Φ p 0 K R x p δγ ( d Φ K 2( d 49 1 2 αβ , h − | γ ∂M M γ 32 Z Z | g p is spacelike/timelike. − Φ + Φ + 2 δ dx µ √ 2 r x gR | ]. 2 ∂M γ − | d Z 48 √ – p Z + x , where dx 2 46 ν = d r µ ∂M r M R δS the induced metric on the boundary and Z ∓ is a large positive constant, which in a situation where we obtained this theory by − The boundary term not involving 0 µν µν 0 4 g γ Previous attempts to quantize Jackiw-Teitelboim gravity with other boundary condi- For spacetimes with additional boundaries which are not asymptotically-AdS, such as the time slice Σ In our conventions the normal vector ≡ 5 4 = Φ µν holds regardless of theγ signature of the boundary. The inducedwe will metric use in is computing related thethe Hartle-Hawking to terms wave function the below, ordinary thissince equation one remains it correct by is except only that there that we include the holographic renormalization counterterm is timelike. This ensures that Stokes theorem with there. sures that the actionconditions and we Hamiltonian will are soon finite discuss. on The configurations variation of obeying this the action boundary is dimensional reduction would correspond toof the the volume in compact higher-dimensional directionscoefficient Planck [ units of the topologicalfield Einstein-Hilbert we part will of call thedefined the as action. dilaton. Φ is a dynamical scalar given by S Here Φ tions include [ 2 Classical Jackiw-Teitelboim gravity The Jackiw-Teitelboim action on a 1 + 1 dimensional asymptotically-AdS spacetime precise path integral descriptionsa in two-sided the Hilbert bulk, spacecounted both by which have the does wormhole Bekenstein-Hawking not formula,the solutions, answers factorize, and both to neither neither these have have questions have CFTleave become black for duals. more future hole standard In work. microstates once both matter cases is added, something we in 1 + 1 dimensions and pure Einstein gravity in 2 + 1 dimensions: both seem to have JHEP02(2020)177 µ L ξ H (2.8) (2.9) (2.5) (2.6) (2.7) (2.10) and dilaton µν γ ) at each asymptotic 2.6 must be a time translation. . ] ∂M 50 ∂M is analogous to the AdS radius | . b , φ . Φ CFT µν δS − = 0 c , δγ Φ r | b 2 c λ 2 c ∂M µν r | ) Φ = 0 r φ + 2 = 0 γ ) ∇ β λ µν CFT = = R 2 ξ r ≡ g γ α | ( – 6 – − ∂M ∂M p ∇ | | µν ν CFT µν β tt γ Φ is the sum of left and right ADM Hamiltonians γ ν ∇ 3 ≡ c γ γ µ r to each component of α H ∇ fixed and positive. µ µ ( µν CFT γ = 2 ξ b T φ µν CFT T with ) vanish for any variation in the space of configurations obeying 1], on which the metric and dilaton obey ( , 2.4 → ∞ [0 c r × R ) we then apparently have . component of this (which is the only component) at each boundary is the AdS ana- 2.4 R tt H To understand these Hamiltonians more concretely, we can then define a “CFT metric” sum of these Hamiltonians. Fromare now on precisely we specialize two to asymptotically-AdSwith boundary boundaries: conditions topology where there weboundary, are and then the full restricting Hamiltonian toand spacetimes From ( The logue of the ADM Hamiltonian for that boundary, and the full canonical Hamiltonian is the in terms of which we can define a boundary stress tensor [ energy). We thus expectact boundary nontrivially time on translations phase space: toon the be indeed respective asymptotic they boundaries. symmetries will be which generated by theat ADM each Hamiltonians boundary, which means that the pullbackAs of in electromagnetism, wetime translations will are only trivial, with actually(note the also quotient motivation that again by otherwise being we diffeomorphisms to would preserve be where boundary left with these locality a boundary theory with no states of nonzero and then taking in Planck units inboundary conditions higher are dimensions, onlywhich it preserved approach by will an the be isometry subset of large of the infinitesimal in boundary diffeomorphisms the metric, semiclassical limit. These these boundary conditions. The obviousΦ choice at is the to AdS fix boundary, the which induced we metric can do by imposing As in theboundary electromagnetic terms case, in we ( need to choose boundary conditions such that the so after some simplification the equations of motion are JHEP02(2020)177 ). 2.5 (2.15) (2.16) (2.11) (2.13) (2.14) can be ), since . When 2 2.6 → ∞ . AdS 2 X . As is normal with . −∞ ) to the hypersurfaces. The → ∞ , to become singular, collapsing 2 h X . x 2 B,C 2 Φ − − dx τ dX 1 + τ CX, A, τ. 2) rotation in the embedding space, so + = 1 (2.12) + − , cos sin + , 2 2 2 2 1 cos 2 2 2 T x x 2 X = ( dT dτ h x ) BT 2 − µ − x 1 + 1 + n 2 2 + 2 – 7 – 1 and some with Φ = 1 + T 1 p p x, dT Φ = Φ p + . Its maximal extension involves infinitely many ∞ (1 + = = = h AT − 2 1 1 2 2 − T = X T T for a reason which will be apparent momentarily. = 2 = 0 by an SO(1 Φ = 2 h Φ = Φ ds C ds = to Φ ), the slices of constant Φ are given by the intersections of the B A 2.5 ) with a family of hyperplanes 2.12 is spacelike or null will never obey our boundary conditions ( µ n are three fixed real parameters which label the solution: we can think of A, B, C is the universal cover of the induced geometry on the surface We can present these solutions more concretely by choosing coordinates We may then ask what the set of possible solutions for Φ look like: the answer is 2 is timelike, we can set µ We illustrate thisboundary solution in regions, figure someReissner-Nordstrom-type with solutions, Φ we expect = thatis small included) + matter will cause fluctuations the (once “inner matter horizons”, where Φ = in terms of which we have n we can restrict to solutions of the form where we have relabelled where them as parametrizing thesolutions where normal vector Φ will be negative almost everywhere on one of the AdS boundaries at in this Minkowski space. The two AdS boundaries arethat at for any solution ofembedding ( surface ( AdS There are various waysOne to nice describe way the is setnegative to of curvature, observe solutions which that of meansobtained the the that via equations first an it embedding of equation is into motion requires described 1 ( + the by 2 metric dimensional a Minkowski to piece space, have of with constant metric AdS 2.1 Solutions JHEP02(2020)177 are h (2.18) (2.17) . In pure Jackiw- 2 6 are the “inner horizon”. The h 2 s , Φ ) r ) − 2 t t s s − r r dr 2 r , and we indicate the value of the dilaton 2 + ) is still only well-defined in the green 2 1 sinh( 1 cosh( 2.6 dt − − ) ) is well-defined only in the shaded green region. 2 s 2 2 r ) ) 2.6 s s – 8 – − 2 s r/r r/r r r. ( ( ( b − φ r/r p p = = = = 2 1 2 Φ = X T T ds . The dilaton profile in the wormhole solution of Jackiw-Teitelboim gravity. Between In addition to these “global” coordinates, we can also go to “Schwarzschild” coordi- Were we not to do this, then we would need to include additional degrees of freedom to keep track of 6 what happens on the other pieces of the boundary. in terms of which we have it simplest to just truncate the spacetime at thenates, inner via horizon. the geometry down to justTeitelboim gravity the there wormhole is region no shadedtwo matter green asymptotically-AdS which in boundaries can figure do obeyingregion, this, since ( but additional the boundary dynamical dataregion. problem would Since with be we needed are tofor primarily extend gravity in the interested higher solution in dimensions, out constructing where of a the this inner theory horizon which is is indeed a always singular, good we model find the “outer horizon”,dynamical while problem the with dashed boundary conditionsIf red ( we lines assume where thatasymptotically-AdS the Φ boundaries. inner = horizon is singular, this solution describes a wormhole connecting two Figure 2 the two vertical blackon lines the various geometry surfaces. is In global AdS Reissner-Nordstrom language, the dashed black lines where Φ = Φ JHEP02(2020)177 , a R c in r H a = − (2.23) (2.19) (2.21) (2.22) (2.20) r L . More L H H + R ), we in fact did respectively has H . This parameter h R ) on each boundary t ) just lies at 2.16 ) by 2.6 2.10 and are also slices of constant L 2.16 t r , ) R t s r . 1 2 h b , φ 2Φ 2 h b . cosh( φ Φ and the “left” and “right” Schwarzschild on all solutions, so the operator R = . t = h b τ = R R 2 φ ) + Φ H R H L L t t = H s cannot be the only parameter on the space of = + – 9 – r s 1 L L = r h = H H δ L H cosh( = = H τ bifurcate outer horizon. The parameters of these solutions cos h . , these coordinates cover the “right exterior” piece of the green 3 ∞ ) tell us that , which as usual lies between the right asymptotic boundary and 2 2.20 < t < becomes the boundary time. ) it is easy to evaluate the boundary stress tensor ( −∞ t at the AdS boundaries is 2.18 and R , which tells us how long we evolved the solution ( t s δ , L t From ( r > r so a slice which is attached to the left and right boundaries at We illustrate this in figure will call explicitly, the relationship between globaltimes time the electromagnetic example.are In time the translations present on the discussion,that left the we and have only right discovered asymptotic asymptotic two boundaries. newBut symmetries parameters: in So fact at our equation first phase ( it spacegenerates might still no seem seems evolution odd-dimensional! on phase space. Thus we have only one new parameter, which we solutions: phase spacein must the be Maxwell even-dimensional. example,an arises The illegal because other gauge in parameter, going transformation.removed analogous to this to the The way coordinates easiest is ( approaches way to an to act asymptotic restore with symmetry any at anothertion infinity: solution illegal (modulo the parameters gauge legal parameters gauge we transformation, of transformations) of this will the gauge become transforma- class the which gravitational analogue of In the previoustheory, subsection labeled we by described theis a value analogous of one-parameter to the family dilaton theAs of on electric in solutions the field the bifurcate of in Maxwell horizon, our the example Φ 1 JT + however, 1 Φ Maxwell example: it is locally measurable. so the full canonical Hamiltonian evaluates to 2.2 Phase space and symplectic form Φ, so in particularand the moreover cutoff surface in the boundary conditions ( for these solutions, one finds shaded region in figure the right part ofare the related Φ via = Φ These coordinates have the nice feature that slices of constant For JHEP02(2020)177 . 4 (2.26) (2.24) (2.25) ]. . For any Hamilto- 35 . , shown in figure , δ δ ∞ 34 < δ < . L H −∞ − 0, R > H H, h b δ, . ∂ 2 h b ba φ = 2 ) 2Φ 1 R − ˆ t = 1 = 0 = ω – 10 – ˙ δ − h H ˙ Φ R = ( t a is defined by ˙ x ab ω . Using a geodesic, shown in red, to measure Figure 4 this geodesic arrives at the right boundary. We then have R ˆ t is the time where our time slice intersects the right boundary. Thus we can think . Different time slices of the wormhole solution can correspond to different initial data for R t as measuring the “relative time shift” between the two boundaries: from now on we We thus have arrived at the following two-dimensional Hamiltonian system: There is another somewhat more operational way of describing δ The ranges of these phasenian space system coordinates the symplectic are form Φ where of will refer to it asquite the similar “time to shift the operator” one-sided From “hydrodynamic this point modes” view, discussed the in time [ shift operator is The idea is tofire start a geodesic at into the thewhat point bulk time which on is the orthogonal left to surfaces boundary of where constant Φ, our and time then slice see at is attached, Figure 3 the JT gravity. Hereand the third first are and the second same slices since are they different differ points by evolution in by phase space, while the first JHEP02(2020)177 (2.31) (2.32) (2.33) (2.34) (2.35) (2.30) (2.27) (2.28) (2.29) , so then R t , . The Hamil- E = ∞ 2 b φ L to t , giving us , log H ! −∞ − δ to b h φ E 2 !! δ b s φ E
2 . ) s τ , 2 .
tanh , ∂M x | L h cos ). We then find E − 7 Φ 2 c b : e dx d x φ cosh 1 + b P dP, 2 2 we can always choose dH. 2.23 φ ∧
√ . ∧ ∧ p c 1 + L c + dδ x 2 log(2Φ H x and H b h − p 2 dδ dL b − φ h – 11 – Z ) = L φ P b 2 − ) and ( 4Φ = = = 2 log = and reflect off of the potential, and indeed that is R ω ω bare = = Φ , via = δ/φ H L 2.22 ) c L ∞ h ). ω b H r bare b limit we need to be a bit more careful. We will defined by solving ≡ L = φ c L δ/φ via ( r 2.16 L h h δ Φ tanh (Φ → ∞ h c r cosh (Φ = 2Φ is indeed a well-defined function on our two-dimensional phase space. P L = 2 log ) we apparently have L ) and led to a much more unpleasant formula for the Hamiltonian. are canonically conjugate variables, both ranging from ), we have P 2.25 2.17 determined in terms of is simply the canonical conjugate of c determined in terms of 2.16 δ x and τ Before moving on to the quantum theory, it is convenient to here introduce another We thank Henry Lin for pointing out an error in the first version of this paper, which originated in L 7 non-relativistic particle moving inwith an exponential waves that potential! come Thiswhat in is happens from a in scattering the problem, solutions ( equation ( so tonian takes a very nice form in terms of In terms of the renormalized geodesic length, JT gravity becomes just the mechanics of a the symplectic form becomes which shows that A calculation then shows that if we define with and a “renormalized geodesic distance”, Using the symmetryfrom generated ( by Thus pair of coordinates onbetween this phase the space. two Roughlydistance endpoints speaking is these of infinite are a in the the time geodesic distance slice and its canonical conjugate, but since that which is more elegantly written by changing coordinates from Φ so from ( JHEP02(2020)177 (3.6) (3.3) (3.4) (3.5) (3.1) (3.2) 0, such that = 0. E > , while at large positive . L E . ) 2 L ( L/ , , with − E i L . e b Ψ E ) 4 | b ) E Eφ 0 are constructed using modified b 2 b √ Eφ ] for a more rigorous version of this i Eφ Eφ 2 ) = 2 2 . 51 L √ E > Re i √ ( i √ i i 2 2 E E + | ) which vanish at − K Ψ ∂ is hermitian on this Hilbert space then tells L E ) ∂E Γ(2 b E L b δ Γ( i ( b − = b Eφ e ψ 2 ≡ i b – 12 – Eφ Eφ Eφ 2 √ 2 2 δ φ E 2 i | √ − √ √ i i i e 2 H 4 2 ) + − − 1 ≈ L − 2 ( ) Γ( 00 E = 2 L ( Ψ . The energy eigenstates have wave functions which can be were self-adjoint, then we could exponentiate it to obtain R E b = 0 in finite time. L , though hermitian, must not be self-adjoint. In fact this ) = φ δ 1 Ψ δ 2 L H ( , which we could use to lower the energy as much as we like, − E Ψ iaδ e . Classically this generates a good flow on phase space, so it should correspond L hits the boundary at These subtleties may be avoided if we instead use the renormalized geodesic distance The reader may (rightly) be uncomfortable with this however: there is an old argument δ -normalizable functions of we have 2 L with reflection coefficient The normalizable solutions ofBessel functions, this in equation the with usual scattering normalization we have These wave functions decay doubly exponentially at large negative L determined from the Schrodinger equation: by operator to a self-adjoint operator andAs thus usual have in a single-particle basis quantum of mechanics, (delta-function we normalized) can eigenstates. construct the Hilbert space out of to the Hamiltonian in aThe quantum mechanical argument system is whose energy trivial:the is set bounded if from of below. contradicting operators the lower boundargument). on the Therefore energy (see our problem [ is visible already in the classical system: the vector flow on phase space generated in the energy representation. Requiringus that that we must restrict to wave functions due to Pauli that there can be no self-adjoint “time operator” which is canonically conjugate We then can define the time shift operator as We now discuss the quantization of JT Gravity, starting3.1 with the Hilbert space Hilbert formalism. SpaceThe and most energy straightforward eigenstates proposal forit the is Hilbert spanned space by of a the set quantum of JT delta-function theory normalized is that states 3 Quantum Jackiw-Teitelboim gravity JHEP02(2020)177 one (3.7) (3.8) (3.9) (3.14) (3.10) (3.11) (3.13) (3.12) . 1)] . Namely we sum − β K + Φ( ] h 2 b . . φ Φ K ) 45 b 0 h φ 2 β = π [Φ 4 2 s 2 s ..., γ r + Φ r ), no more needs to be said about + 2 b 0 has periodicity 0 √ + − φ Φ dr 3.4 2 2 s c π E 2 (Φ dx t r r 4 = r e π , 1 2 b c + ∂M 2 φ c r ≈ 2 . 1 Z 2 b r 2 E = 4 β π E φ π β 2 S dt 4 = Z ) − = = 1 + 2 s e E = ) tells us that this reflection coefficient is a pure = t r 2 s ∗ – 13 – s E Φ r z t ∂M r Z − | + log γ D r, − + 2)] + 2 2 c Φ g b i r r 2 c log L φ R D r = Γ( β H ∗ ∂ = ( h p Z ) 2 − β z Φ = + Φ( ≡ = ds = = ] R β i K 0 requires [ S L s Z [Φ r H g h = √ r x 2 d to get asymptotically-AdS boundary conditions. The Euclidean action is M Z = → ∞ E c S r − If we interpret thisnamic as formulas a to thermal find partition the function, energy then and we entropy: can use standard thermody- so evaluating the Euclidean action on this solution one finds [ The extrinsic curvature at the boundary is where smoothness at and take The saddle point for this path integral is the Euclidean Schwarzschild solution at the boundary. We then again take the dilaton to obey quantum JT gravity. To compareto to consider what how we standard do Euclidean inbegin gravity higher methods this dimensions are however, discussion related it toasymptotic by is our boundary, useful reviewing exact on the solution. which the We Euclideanover time geometries path coordinate with integral the for topology of JT the gravity disk, with and with induced metric phase, as is necessary sinceas providing there an is exact no solution transmission. of These quantum expressions JT3.2 can gravity be with thought two Euclidean asymptotic of path boundaries. It integral may seem that given the scattering wave functions ( The Gamma function identity Γ( JHEP02(2020)177 . 3.1 . β (3.17) (3.16) (3.15) ]. They can be . We can compute 53 , 5 52 . ) L E b φ , L √ . So we need to pick boundary + , h βE 0 2 − (Φ x e π ) , which we discussed in subsection 4 L e L 1 + E ( ≈ p L ) that it passes through the bifurcate horizon h and ρ L above: these states are labelled by a real E L ( E (i) S = 0 = Φ – 14 – ) we know that the energy is a simple function e dE 3.1 ) Σ Σ there. To achieve these we will require that | | L ∞ 0 Φ Φ h 2.21 E Z ( µ 0 ∂ S ≈ µ ] n ≡ β [ ) Z L E ( 2 and a “bulk” boundary Σ, which we interpret as a time-slice of the L β/ ρ c below to what extent these can actually be interpreted as thermal r 4 , and are collectively called the Hartle-Hawking state [ basis calculation. From ( basis calculation is conceptually simpler but technically harder, so we begin β E L . The Hartle-Hawking state: we sum over geometries and dilaton configurations with an that the dilaton is equal to Φ (ii) The The basic idea of the Hartle-Hawking state is illustrated in figure conditions on the bulk sliceand Σ which ensure semiclassically compute the wave functionthe of two-boundary the Hilbert Hartle-Hawking space, state labelled by in the two bases of with the of the value of the dilaton at the bifurcate horizon, Φ from either side an observer sees a black holethe in wave equilibrium at function inverseis of temperature to the fix Hartle-Hawking thecomputes state the induced in wave metric function various in on the bases, the Wheeler-de Witt the bulk representation. slice traditional In Σ, this choice section, together we with will any matter fields, which Whether or not thealways Euclidean use path it integral to define definessystem a a which natural thermal family we partition ofparameter constructed function, states we in in can the section Hilbertinterpreted space as of the describing two-boundary a wormhole connecting the two asymptotic boundaries, where with 3.3 Hartle-Hawking state We discuss in section energy and entropy, but wesuggestive manner note now that one can rewrite the semiclassical result in the Figure 5 AdS boundary of length two-boundary system. The boundarythe conditions wave on function in. Σ depend on which basis we wish to compute JHEP02(2020)177 β (3.20) (3.18) (3.19) and are the h i θ . ). ! ) 3.9 i θ ), since both bound- − π 2.4 ( , ) i i is the argument in its wave θ X . The kink in the boundary h − h b φ Φ + 2 , it is a “sliver” of the Euclidean π 6 = )) ( γK i s Φ terms are not present since this is . We emphasize that here Φ x r δ √ E − dx + Φ( /β, b and 0 ∂M is the number of boundaries, and 2) and Z b πφ means no corner). These corner contributions (Φ αβ -basis Hartle-Hawking wave function. π i , β/ δγ + 2 – 15 – E X = 2 (0 = αβ ) should hold at the peak of the wave function. h i gR γ ∈ θ Φ + 2 √ E E 3.18 x t direction there. To proceed further we need to evaluate 2 S d 2 and a piece of bulk boundary Σ with vanishing normal is the genus, M any g at its minimum on Σ. In the end we may then substitute β/ Z → − c ) with
r h E S π 1 -function contribution. This suggests that we should upgrade our ) with corner terms 4 3.10 − δ = 3.9 . Saddle point for the b − g 2 2 to get the wave function in terms of − picks up a Figure 6 2 E/ K is the normal vector to Σ. In the second equation we have chosen “global” coor- does not violate the boundary conditions since it happens at the minimum of Φ: b ≡ φ related via µ s is the Euler character, χ r n p χ = We begin our discussion of corner terms by recalling the Gauss-Bonnet theorem in the The saddle point for this calculation is shown in figure not = r labels which Hartle-Hawking state we are considering and Φ h can be derived bycurvature smoothing out theEuclidean corner JT and action then ( taking a limit where the extrinsic Here interior opening angles of any corners ( the Euclidean action of thishas solution, corners, but which require this additional is terms complicated in by the the fact actionpresence that not of the present corners: in solution ( β function. We expect however that ( Schwarzschild solution ( at the derivative of Φ vanishes in derivative of Φ andΦ Φ = Φ are dinates on the slice,These which boundary is conditions not are consistent really withary necessary, the terms but action vanish variation it (remember ( that is thenot perhaps Φ an useful AdS to boundary). beof More concrete. AdS concisely, we boundary want of to length integrate over geometries with a piece where JHEP02(2020)177 i θ 6 so = 0 ) in − 2 π 2 (3.22) (3.23) (3.24) (3.25) (3.21) K = . Φ 3.25 π/ ). γ ] not as a θ β √ h b [ 3.19 is integrated φ Φ Z dx 2 β E B t . Z − ) which arise from since it would not 2 π j θ 1 5 ); in fact we recover − both have ) C − h 2 K . 3.13 2 π C i + Φ 4 0 + Φ) )) j 0 x , βE/ (Φ 2 h b ] from ( γ φ β − Φ [ √ + Φ( 2 β E Z is 1)] + 2(Φ b 0 ) holds. Moreover if we square it and dx θ φ − ) we at last arrive at the semiclassical − 2 (Φ ) 2 π h +Σ 3.18 the set of corners of the second type, and K C 2 ), and remembering that h b B 2.21 ∈ 2 X φ j Z Φ p 2 + Φ C β 3.12 0 + + Φ( , then the full Euclidean action for computing 0 = – 16 – ] for some more discussion of this). If we denote (Φ ) + 2 K ). Here however we are interpreting B Φ i θ 0 π 54 θ , but in fact π 2 π [Φ − + 2)] + 2 h 3.15 = 2 γ π R E √ S were )) ( dx i θ − x + Φ( B 2 as in equation ( ) = exp Z R E 0 ( E/ terms can be simplified by using the Gauss-Bonnet theorem ( b + Φ( β + 2 [Φ 0 φ 0 Ψ g 8 p gR (Φ √ 1 √ x = C 2 x ∈ X d i 2 h , since we need the corners to cancel when we glue two states together to , we recover the “partition function” d i M θ E Z M + 2 2, we find Z − = 0 π β/ E below. S = Φ 4 the set of corners of the first type and − E 1 Returning now to our saddles for the energy-basis wave function, the saddle in figure ] precisely in the representation ( S The calculation of the Φ C 8 β [ − Z thermal trace, but instead asproduced the norm by of the the Hartle-Hawking Euclideansection state with path the integral. normalization We further discuss the meaning of ( As expected, this wave functionintegrate is over peaked when ( from 0 to so substituting Φ Hartle-Hawking wave function there. We thus have Evaluating this on our saddle point using ( contribute if its internal angle so we do have a contribution. Away from this kink the bulk slice Σ is a geodesic, so has corners of boththey don’t types, contribute. but The fortunately kink the at corners the of horizon type is a corner of type by also the AdS piecewave functions of is the boundary by which is describing an overlapfunction of however, two there states. are In additionalcutting using corners an the (such “overlap” path as type integral those pathinstead to integral: in compute of figure for a these wave thecompute corner an prescription overlap involves (see section 3.2 of [ and indeed this is the correct prescription for corners in the action for a path integral JHEP02(2020)177 (3.27) (3.28) (3.26) (3.29) tells us 0 E, t such that our β and = 0 by convention, so L 0 E, t = 0 (remember again that , 0 µν E, t δγ + is obtained by solving the equation λ . There is a two-parameter family of s , We now want Σ to be a geodesic of r 1 as functions of 7 ). These boundary conditions are again − , λ 9 , c = 0 and s 2 β tanh r r x 2 b s ds K J x φ r L/ cosh e sin = 0 = ) whose boundary has a piece which is asymp- 2 s b and β -basis Hartle-Hawking wave function. r 4 s and the AdS component of the boundary indeed φ Σ Σ – 17 – = L | r 4 J γ + L a 3.10 K 2 ≡ ≡ arctan J + 2 log(2 a x s 1 r p L ), since now 2, and a piece which is a geodesic through the bulk, of -basis calculation. ) = ) = 2.4 λ λ β/ L ( ( c r r E t . We illustrate this in figure , so we now define Σ by requiring Φ terms are not present since Σ is not an AdS boundary). . Saddle point for the δ L L this closest approach happens. We can set being equal to − s E t 2. After some algebra, we find that and β/ Figure 7 c αβ r δγ tells us how close our geodesic approaches the center of the disk and αβ J γ The saddle points for this calculation are a bit more involved, we want a piece of We now proceed to the This calculation is something of an aside to the main points of our paper, so casual readers may wish 9 to skim the remainder of this section. has length with where at what value of to construct a solutiongeodesic we indeed has need renormalized to length give renormalized length geodesics in this geometry, parametrizing by proper length we have with the range of consistent with the variation ( the Φ the Euclidean Schwarzschild geometry ( totically AdS, with length renormalized length JHEP02(2020)177 . θ c s , as r πr and 2 π β 2 (3.33) (3.34) (3.35) (3.31) (3.32) (3.36) (3.37) (3.30) E we have = ) in two. 1. When s r 2 of the AdS ). This wave 3.10 a > β/ c → −∞ 3.29 r L 2 at each corner for π/ , → θ ) happens when ..., x ) and using ( + x ]. Near this peak we have ... 2 2 3.29 β ) tan [ + 3.28 Z ) since the length ..., 2 + . we have peak ). There are now no corners of type now make a nontrivial contribution. 2 s 2 2) + , π L r x 2 , , ) of (0 , π/ 2 b − C − 3.21 s 2 φ 4 ∈ b 2 L/ − L β βr 2 → ∞ L/ φ e ( β is the semiclassical limit. − b 3.13 x e π x b s c 8 φ b b ( b r r by solving ( b L φ φ φ β s πφ 1, or in other words that + 2 4 φ − 2 2, which from ( β tan β 0 – 18 – e 8 π = c ≤ ≥ Φ π/ r = 2 increases monotonically. q − − a 2 in the bulk, so we need only compute the corner π β s δL β = 2 and − − r = we are potentially sensitive to a subleading term in x , 2 π L c J = r L b = R φ θ ) = exp ) = = constant = constant j L ( x S β − ψ , which corresponds to cutting the Euclidean solution ( and decrease π β 2 β is = 0 on Σ and = L ). Indeed a short calculation tells us that we have K c terms, and the terms at the AdS boundary. The result is s r /r 0 , but since Φ( is positive, and that we must have L (1 a O = 0 when this inequality is saturated, and no solution exists for . If we fix , or in other words determined as a function of π s and β 4 r /π x β Thus we see that the Hartle-Hawking states fit nicely into Hilbert space of the Jackiw- Finally to evaluate the action, we again use ( , but we will see that the two corners of type → = 2 1 s which is consistent with the idea that large Teitelboim gravity, with (reasonably) simple semiclassical wave functions in the so the width in with function has a unique maximumexpected. at This peak will dominateis the consistent integrated with square the of saddle the point wave function, evaluation which ( again so there will be awe nontrivial again corner have contribution. Theterms, rest the of Φ the action is easy to evaluate, any which is This should ber what we find is the peak ofC the wave function.This is As not obvious, since we expect that as with a we find that Note that component of the boundaryA must unique be solution less exists than provided that or equal to the full boundary length and JHEP02(2020)177 ], ), β [ (4.2) (4.3) (4.1) Z 3.25 ). So far the The one-sided 10 3.13 ], so the wave function ). Instead we have equa- . i ) by labeling states by the 55 L 4.1 E 4.1 | 2 / L are their conjugates under a two-sided R L βE i i i − | ∗ ) i i L | L i E ∗ there can be no boundary theory dual to βE b i | φ − i e √ 2 βE + ] for more explanation of this. i 0 − X 57 e (Φ π – 19 – 2 i ] = e . Were one to have existed, there would have been X β L [ = ) is half that of the two-sided energy used in ( Z ], we can interpret the unnormalized Hartle-Hawking dE i β 56 ∞ ψ 2.20 0 | Z i ∝ β ψ | , which by ( 11 L E , since there is no matter in the pure JT theory all energy is sourced by R E are energy eigenstates of the “right” CFT and 6= R ), which we can write in a manner more similar to ( i i L | E 3.25 The answer to this last question is no. The reason is that the Hilbert space of two- Here Readers who have casually followed the recent SYK developments may be puzzled, since naively one bases. It would be interesting to extend these calculations to one-loop, in fact the 10 11 version of CPT which exchanges the two sides, seemight [ have gotten theSchwarzian impression theory”. that the This two-boundary isdoes JT not wrong, theory make basically sense. should because be We a give dual single the to precise Schwarzian “two theory statements copies in in of the Lorentzian the following signature section. This is not awhere state in athe bifurcate tensor-product horizon. Hilbert We space: thereforepure conclude quantum indeed that Jackiw-Teitelboim gravity theresuch are a no factorization. states at all tion ( one-sided energy to get but this is nothing but thein one-sided the thermal Jackiw-Teitelboim partition gravity? function. Is this interpretation valid boundary Jackiw-Teitelboim gravity, whichdoes is not just tensor-factorize a intoHartle-Hawking single-particle state a quantum exists, product there mechanics, is of no one-boundary analogue of Hilbert equation spaces. ( Although the in the tensor productpath Hilbert integral space is of then two the copies norm of of this the state, boundary CFT. state as corresponding to the unnormalized thermofield double state We now return to thewhose interpretation semiclassical of value the in single-boundaryonly the Euclidean Hilbert Jackiw-Teitelboim path gravity space integral isHartle-Hawking interpretation given state we by in ( have the givenpath two-boundary integral it without Hilbert is any space, rescaling.for as as In this the produced AdS/CFT path norm by however integral: there of the is following Euclidean the another [ interpretation unnormalized normalization of theitself Hartle-Hawking might state be is as well. one-loop exact [ 4 Factorization and the range of the time shift L JHEP02(2020)177 ) 4.6 (4.7) (4.8) (4.9) (4.4) (4.5) (4.6) moving on a ] we computed m . ] β [ β [ Z R Z ] β [ L Z . = β πm ? (4.10) 2 R 2 ] , r β R βH [ R − H Z e . = ⊗ R = R m E L Tr 2 I βE is similarly noncompact, leading to a con- , βE − ⊗ H r + − L e n R ) e R L L R ) E I = H ( E βH – 20 – ( p ⊗ ) = − = e tot L E L ( dE ρ H H ρ is not well-defined. Let’s illustrate this in a simpler Tr ∞ would both be given by the function . In our Jackiw-Teitelboim quantum mechanics with dEρ = 0 R Z ∞ tot = H Z 0 Z Z ] = → ∞ βH β [ and − R e Z L Z Tr ) would indeed hold. In other words, what would the two-sided ) need to be such that we indeed had ≡ E 4.6 ( ] β [ tot ), the dynamical coordinate . Momenta is quantized as ρ ] we can then attempt to compute by computing the thermal trace in our tot R β Z [ 2.35 ) is divergent while the right hand side is finite. This is another illustration tot 4.6 Z ). It may seem that we deserved the nonsense we got in attempting to test equation ( There is another interesting illustration of the non-factorization of the JT gravity 3.13 is continuous, the trace in equation to do somethingmodified more interesting: such that we can ( density ask of how states the theory would need to be equation ( of the non-factorization of the two-boundary Jackiw-Teitelboim Hilbert space. in JT gravity, sincealready after know that all the that JT equation Hilbert was space derived doesn’t assuming factorize. factorization and But we in fact we can use this The key point isvergent that in the the density limitHamiltonian of ( that states, and thereforetinuous spectrum the with partition a function, divergent are density di- of states, so the left hand side of our putative The thermal partition function is therefore example: the quantum mechanics ofcircle a of free radius non-relativistic particle of mass so we have a density of states Both sides ofAssuming this factorization, identity arein computable ( in JTtwo-sided gravity, Hilbert so space. we ThereH can is however test an if immediate problem: it is since the true. spectrum of we have the partition function identity Hilbert space. In any tensor product Hilbert space for which the Hamiltonian is a sum of the form JHEP02(2020)177 ] β [ ) we Z (4.11) 2.32 ] for more 29 ), this equation . From ( 4.8 ) ] is able to correctly L without the Jackiw- β E [ ( ) S L Z 2 E e ( S 2 ), our renormalized geodesic e , 4.6 ) L ). In light of ( a trace of the Hilbert space of the E ( S must be in a factorized theory using 2 3.14 not δ e ]: it is the time it takes for the time evolution ≈ 58 , 2 21 – 2) 19 The idea that exponentially long time evolu- E/ ( – 21 – L 12 ρ is the entropy ( ≈ ) For gravity we might therefore expect that achieving S E ( 13 ]. tot ρ ) and 27 3.16 is quite natural from the point of view of the proposal that exponentially complex ) for times which are longer than of order L E ( R S 2 H ]. e to reach maximal circuit complexity, and is also the time it takes the thermofield double + cannot really be larger than some length of order 59 t L L L H iH was defined in ( − = e L ] is as the norm of the unnormalized Hartle-Hawking state: what we have learned ρ β H [ In this last argument it may seem that we have gotten “something for nothing”, since Strictly speaking we also need to introduce a UV cutoff as well since no continuum quantum field theory The timescale Z 13 12 operations should disrupt the structureoperator of spacetime [ state to reach maximal state complexity. on a connected space hasis a best factorized Hilbert understood space.discussion in In of terms quantum this. field of theory whether the question or of not factorizability the theory obeys the “split property”, see [ and the Wilson line whichof stretches these from one dynamical boundary chargesfactorizability to [ is the as other simple can as be introducing split matter by fields, a pair and in some sense this is true. How might we attain a factorizableis version indeed of a the partition Jackiw-Teitelboim function? gravity,the in answer For which is the simple: 1 + we 1 needunit Maxwell to of theory introduce U(1) discussed new gauge in matter charge. fields theboundary which introduction, This is possess modifies the no fundamental Gauss’s longer law required such to that be the equivalent electric to flux the on electric the flux left on the right boundary, here is that factorizationthermal is partition the function. key assumption which allows us to re-interpret5 this as a Embedding in SYK count black hole microstatespossible using only the because only that Euclidean thebulk path low effective integral theory is energy with bulk onespace effective asymptotic exists. action. boundary: Given This in only the factto is bulk in theory, JT the gravity only no Hilbert such space Hilbert interpretation we can give discussed in [ we learned what the rangeonly of the the time-shift Jackiw-Teitelboim operator pathsame integral. old miracle This by is which the indeed Euclidean miraculous, path integral but evaluation of in fact it is the then also learn thatnian we cannot evolve theTeitelboim description Hartle-Hawking breaking state down. withphysics” our At in total least any Hamilto- then, theorytion if which is not factorizes. in sooner, tension there with must the be semiclassical “new description of the Hartle-Hawking state was also where has a natural interpretation: in aobservable factorized theory obeying ( In the semiclassical approximation, this can happen only if JHEP02(2020)177 (5.2) (5.3) (5.1) In Eu- 14 runs as before from 1 to . d . a , interacting with Hamilto- χ a c # χ 2 quantum mechanics living on χ ) , b d χ N χ a abcd c χ , where J χ ( a i b χ χ abcd a J χ X a 0 J ij ij X ) ij 0 Σ)] ): Z t, t Σ G ) and ( 0 4 4 R t, t / / G, dt ij ( 3 ij 3 1 ( . , t, t X G ij − − G 1 ( 0 Z # iS diff( 2 Σ 2 a