JHEP07(2021)057 Springer July 12, 2021 June 25, 2021 April 20, 2021 : : : Published Received Accepted symmetries of the boundary fields ) R Published for SISSA by , https://doi.org/10.1007/JHEP07(2021)057 SL(2 algebra on the light cone ) R , . SL(2 3 2104.05803 The Authors. c Classical Theories of Gravity, Models of Quantum Gravity

In a region with a boundary, the gravitational phase space consists of radiative , [email protected] variables on phase space. We compute the Dirac bracket and identify the Dirac ) R , E-mail: Institute for Quantum OpticsAustrian and Academy Quantum of Information Sciences, (IQOQI), Boltzmanngasse 3, 1090 Vienna, Austria Open Access Article funded by SCOAP ArXiv ePrint: observables on the light cone. Finally,in we an discuss various effective truncations way. to quantise the system Keywords: alters the boundary symmetries.we define To infer an the auxiliaryare phase Poisson brackets space, manifest. among where We Dirac the identifythe observables, the auxiliary gauge variables. generators andSL(2 All second-class gauge constraints generators that remove are at most quadratic in the fundamental edge modes and radiativeOur modes starting on point athe is null least surface the for restrictive definition the boundarythe of tetradic conditions radiative the Palatini-Holst modes possible. action. action aloneare and (two The dynamical. degrees its fixed of We boundary boundary freedom introduce data terms. per the consists covariant point). phase of We All space choose other and boundary explain fields how the Holst term Abstract: modes in the interiorto and explain edge how modes the at region the couples to boundary. its Such environment. edge In modes this are paper, necessary we characterise the Wolfgang Wieland Gravitational JHEP07(2021)057 , where , where 3 2 ]. The box 9 – 7 ], at a finite null symmetry of the 33 ) – ]. To compute the R , 18 35 , 34 SL(2 ] and [ 17 ] for the parity odd Palatini- 14 11 15 ]. This is to say that we describe – 2 6 2 – 10 25 3 ] deforms the 23 2 , 1 – 1 – holonomies ) 9 R , 15 5 18 SL(2 27 18 21 1 parametrisation of the boundary fields ) R boundary charges , ] action in causal domains. We identify the most minimal boundary conditions ) R 16 , , SL(2 15 The paper consists of two parts and a conclusion. The first part is section 3.6 Dirac observables from 3.1 3.2 Choice of3.3 time Kinematical phase3.4 space Constraints 3.5 Dirac analysis of the constraint algebra 2.1 Boundary conditions2.2 and quasi-local graviton Boundary field2.3 equations Radiative symplectic2.4 structure Gauge symmetries on the radiativeSL(2 phase space Immirzi parameter deforms the gaugefor symmetries, the radiative we data provide and a theboundary complete corner for characterisation data (edge the modes), Palatini-Holst seewe action. [ introduce The the second Diracboth part observables on of gravitational the the radiation null paper as boundary. is well section These as observables characterise gravitational memory [ we introduce theHolst covariant [ radiative phasealong space the null [ boundary. Twounconstrained. radiative modes We discuss are the kept resulting fixed,symplectic bulk all two-form plus and boundary other identify field boundary its theory, fields introduce gauge are the symmetries. pre- After explaining how the Barbero- boundary fields. Ourthe approach will gravitational be field quasi-localhas as [ a an boundary, open andoutgoing (dissipative) we radiation, which choose system escapes it in throughwe to the introduce a be null a finite boundary. phase null. At space, box the constraints In [ null and this boundary a way, itself, set it of is Dirac observables. easy to characterise the 1 Introduction The goal of thisexplain paper how is the to Barbero-Immirzi study parameter the [ gravitational phase space in a causal domain and 4 Summary and outlook 3 Contents 1 Introduction 2 Barbero-Immirzi parameter on the null cone JHEP07(2021)057 .  of ) TP and αβ z (2.3) (2.2) (2.1) F 0 ( ∈ frame 1 ∧ a M − β ξ e π ∧ U(1) onto a two- ., α symmetry of e C 1 ) − + cc R γ → ,  −  and a cosmological ]. The action con- ) P ν : γ e AB SL(2 38 , π ∧ Σ µ charge. . A vector field 37 e , we introduce a universal ∧ ) . The components of the . The pre-image 6 Λ , z 0 0 N ( C − B 1 AB 0 M − U(1) × Σ A µν vanishes, i.e. ∪ π ¯ 6 R F Σ Λ ( C = ∈ N ∧ − AB . T z ) β  ∪ γ e into an abstract and oriented three- 1 AB − → ∧ = 0 u, z − 1 α F ( a e N ξ AB M ∧ TP ∗ -valued curvature of the self-dual (complex- Σ : π 0 ) αβµν = is the self-dual component of the Plebański ∗  B AB C 0 2 1 π for all ⇔ , Σ A – 2 –  M ¯   z (2 AB ∂ M − sl M Σ R VP Z -violating Barbero-Immirzi term, which has no effect = ) = ∈ 1 πG 1 0 a , i.e. 16 CP u, z ξ + i) plays the role of a mere conversion factor between units is the BB (the base manifold) such that there are homeomorphisms ] and N = e )( γ B G ( φ S 36 C ∧ C , which is equipped with a surjection ◦ 0 ) , whose boundary consists of two spacelike manifolds A i π ( ∧ πγG AA , π M e 8 C To impose boundary conditions along C  A ( A P = , namely 0 i + a 0 B BB A on the base manifold is the null ray e AA A that satisfy ∧ C , e 0 P ∈ is the co-tetrad (soldering form) in spinor notation [ = d Ba . Newton’s constant AA 0 z A → e B Λ A A AA h C e F S × connected by a null surface In tetrad variables, the action is R 1 1 : any point is null (lightlike), if its push-forward under Universal structure. ruling, which allows usdimensional manifold to embed thedimensional oriented manifold manifold φ constant of length and unitsoriented of spacetime mass. region In theM following, we will studyboundary this inherit their action orientation in from a the compact bulk. and two-form where tains two coupling constants, namely the Barbero-Immirzi parameter where valued Ashtekar) connection [ on the field equations inmixes the the bulk, deforms null the dilation boundary charges field with theory an on otherwise2.1 a vanishing null surfaces and Boundary conditionsConsider and first quasi-local the graviton action in the bulk, Dirac bracket. Finally, we summarisethe the system paper at and the discuss quantum various level. strategies to approach 2 Barbero-Immirzi parameter onIn the this section, null we cone explain how the the boundary fields is manifest.tion. The There physical are phase space both is first-classthe obtained and gauge via second-class generators symplectic constraints. of reduc- The verticalrotations diffeomorphisms first-class of constraints along the are the co-frame nullconstraints. field. generators and To Besides compute the the gauge Poisson generators, brackets there among are Dirac a observables, few we second-class introduce the Poisson brackets, we introduce an auxiliary phase space, where the JHEP07(2021)057 , , ), ) N N a N ∗ ¯ 2.5 m ., (2.7) (2.8) (2.9) (2.4) (2.5) (2.6) T . The , (2.10) on metric a , which ab → 0 A +cc A k , m C ] from the a N 9  l M ) and ( ∪ ) ( ∗ contains also 1 ¯ T ) m − 2.4 is 1 and a : A (0 + +) C ` ¯ a m N ∗ N , ω + = T a ϕ , A . We call this vector a η N , m κ y such that N ∂ l a ( T k A is always future pointing. ( ∧ ` m. 0 a A l A ¯ ` . N , . A + 1 k = 1 ¯ C . m + ABab a + i ¯ ∧  = m. , Σ ¯ , then there is an associate co-basis b m = 0 ¯ 1 m m ∧ a ∗ N 0 ¯ AB m a ∧ ) , the co-basis ϕ l A a Σ = 1 a ∗ ¯ k ω ∂M m + m ∧ i ¯ 2 A m A A . We introduce these boundaries to cut , m ` ` = k ab a − C AB i l A ε = 2 and A a a + i − k Σ and ` 0 , π – 3 – ab k k AB m 6 Λ  a 0 1 ) = = ε ,m C  ) 1 A A a − − m 1 ` ¯ ω ` 2i A C = ,l + i ` ( A a , the corresponding dual basis of = are intrinsic to the boundary. From ( 0 and a (commuting) spinor determines the induced signature ` +i AB π = B admits the decomposition i = ( ab κ a ) = 0 = | k N κ q l F a a − B a ( A a a ∂M ` and additional auxiliary variables ∧ Aab ¯ η 1 2 k ) = m Aab m BA = η ,ω on m ) 0 , η ε 0 a − a a ) Aab AB ab C ]. First of all, we note that on a null surface, there always a η ( A m 9 Σ D AA ¯ π , m , m  m e  a a , ,N ∧ ∗ N a M , l A is the area two-form on the null surface and , k has two disconnected spacelike boundaries Z A ϕ A a ) ,` η  , i.e. l , m ( , ` such that a  N R P k : Aab . In addition, +i) ( N 0 η . We have, in fact, N Aab Z γ | and ∗ η ( C a N ( 0 N , which selects a specific dual null vector field in + ], the result is T ( a ∪ 2 k i AA = 1 1 of on Ω 40 πγG , − a 1 ,e ) 8 ∈ ab a ¯ is the area two-form. Besides C 39 m g Ba ¯ a , we proceed as in [ = m ab A ab and it satisfies = ∗ N , ε m ε N a A ϕ a h l N The boundary action is now constructed in the same way as in reference [ To introduce the boundary conditions and the corresponding boundary field theory The null surface = S ∂ , m a ab k boundary fields combined action for the bulk plus boundary field theory is given by Notice that all fields it is also possiblefound to in infer [ the pull-back of the tetrad to the null surface. Details can be field In the following, we restrictGiven ourselves the to co-basis configurations where where q where the one-form which is normalised such( that The complex-valued one-form exists a spinor-valued two-form where denotes the pull-back. If we introduce a second and linearly independent spinor are cross sections of off, before any caustics may form.by The induced orientation on the boundaries is determined along JHEP07(2021)057 ), . 2.10 (2.17) (2.18) (2.13) (2.14) (2.15) (2.16) (2.11) (2.12) + cc  ) are such A `  2.10 κ 1 2 , − .  ] a a . D ] ). . m ). First of all, consider ] , a  ν a null surface analogue of , m

m ∧ a +i C . ∗ 2.10 µ 2.12 , m , l A e a . The complex-valued one- a that vanish at the boundary η , ϕ l , b = 0 ¯ U(1) a a f are Lagrange multipliers that m = id N l ) is much simpler than ( e i X ζ , l . Z µ 1 a C a = 0 ab e

∼ f, − + Y A + κ / a a A π below. Before doing so, let us also 2.11 ] ν, ∗ a ξ ∂ a N a  is the ◦ ∗ ϕ ), the action simplifies Aab ∼ ∂ [ π + ) ¯ η ϕ ζm of the null surface and at the spacelike 1 γ , m AB a R ∼ , δ` ◦ a } Σ ⇔ + κ : = 2.10 ] − 1 [ , l π a admits the algebraic decomposition given a

∧ a a C A = 0 ) κ ∼ N , κ η [ κ ( VP , m 0 ]  y N 1 covariant derivative on the null surface and Aab AB a a Ba ∼ l C ∈ η = [ Ω Y Σ A ∗ ) – 4 – ∼ { ] ]. Although ( y a that only differ by such a vertical diffeomorphism a ] , m 6 Λ C g ∈ ξ ] a a , ϕ Diff( , δA X a 36 i a a , − , l , m ∈ κ , κ ∗ M a , a , m 2 , m a , ϕ κ ϕ , l a SL(2 AB 1 a  , l ):[ :[ = 0 , l a F κ a 1 : N κ R g ∧ , :[ ( κ ) = δ [ 0 :[ C → N AB = 0 C . ( 0 i Σ Diff → N ∀  → : = 0 ∈ N f Diff M 1 and the spinor-valued two-form ϕ N : ∀ Z C ∀ : )

 ζ is the exterior a C ∀ λ ξ : ∀ ∇ + i) = , where the boundary conditions are simply given by ∗ N N } 0 γ ( 1 ( ϕ vertical diffeomorphisms C 1

a Ω = ,M ξ i 0 ∈ πγG 8 D a M ). If we insert these constraints back into ( { denotes the interior product, i.e. ω , i.e. = Any two configurations of Let us now return to the definition of the equivalence class ( The boundary conditions for the extended bulk plus boundary action ( 2.5 A N S η y Finally, we have the complexified conformal transformations Next, there is a trivial shift symmetry Then, there are the dilations of the null normal of are to be identified. In other words, the group of which are generated by null vector fields fix the boundary conditions atdisks the corners that an equivalence class of boundary fields is kept fixed along the null boundary, We will identify the underlying equivalence relations We will see below thatthe the Ashtekar-Barbero connection one-form [ it is often more useful for us to work with the extended action ( l form impose constraints such thatin the ( one-form where JHEP07(2021)057 ) ]. ). 41 U(1) U(1) 2.12 2.10 (2.22) (2.23) (2.24) (2.19) (2.20) (2.21a) (2.21b) for ), which charge. In f 2.4 , λ along the null and ) U(1) A Ashtekar-Barbero ν , ` , on the other hand, Aab ab η A SU(2) ( N ¯ m. ∧ transforms as an abelian , and , , , A a A ). The theory in the bulk and ν. a CD κ N ω = 0 = 0 ), provided two distinct conditions N γ Σ -vacuum Einstein equations in spin- 2.5 0 y − l ) = Λ ) in the bulk and the additional gluing = AB λ A BA 2.13 + ( η N e Σ a ∂ A ABCD

∧ ∧ A ` ), ( Im

2.21a  γf ∇ ∧ ω ω + Ψ B 2.12 = A – 5 – + i + i Σ AB N µ, κ κ 3 ∂ Λ Σ

3 ν 1 2 1 2 -twisted boundary condition Λ − ) = γ − λ = B ( A = + = F Re AB A A  η F provided the following boundary field equations are satisfied, D` ∧ ) D A ) is stationary with respect to variations of the (anti-)self-dual . , ` N 2.10 Aab ]. We will see below that there is a corresponding ) implies that the one-form η satisfies the algebraic constraint ( ( 36 , denote the real and imaginary parts of the gauge parameter 2.18 Aab is the Weyl spinor. For our present purpose, we also need the evolution equa- 2 η , ν 1 ). The coupled bulk-boundary action is stationary with respect to variations of the and ABCD µ Ψ 2.13 Equation ( the field theoryare at derived the as an boundaryIn equation are of fact, connected motion the from viaconnection action the for the ( coupled given gluing bulk boundary plusare conditions conditions boundary satisfied: ( ( action namely the ( torsionless condition ( The boundary fieldboundary. equations propagate The the variationimplies boundary of spinors that the Lagrange multipliers and ( boundary spinors namely where tions for the boundary fields along the null boundary for given boundary conditions ( which imply that the self-dual curvature admits the algebraic decomposition, quantum theory, this charge, which measures the area2.2 of a cross section, Boundary is field quantisedThe equations [ equations of motion inconnection the variables, bulk are the familiar at the two corners of connection. This connectionconnection is [ the null surface analogue of the In addition, werotations will and also dilatations satisfy need the to impose that the gauge parameters where JHEP07(2021)057 . 2 ) a ), a and U(1) sub- , l a a (2.29) (2.25) (2.26) (2.27) (2.28) ) a 2.15 µl , ω . Going a ) 0 , m κ a A ] = ( ¯ ` a , l l ) below. A [ a is real. If we ` . Given the ). κ ( ( b a N 2.49 conf µ κ δ D 2.26 b on l a spin-connection on a a 0 ), ( Γ such that AA e N 2.24 . unique, see ( ) and a complexified confor- 0 SU(2) on ), ( A = i is ¯ are proportional to each other ` . If the torsionless equation is Γ 2.17 A a a connection . The set of bulk plus boundary N l `

. 2.23 ω
-twisted reality condition 1 a a ) . l ˆ is complex and γ = b a a . ), ( Ashtekar-Barbero connection, which U(1) a , ω Γ ), there is one additional constraint. ∇ N ( = Γ a a ω b

l l l ˆ ) a = 0 a 1 is defined as l 2.5 l ˆ Re spin connection is uniquely determined by the ( . ¯ Γ − is analogous to the extrinsic curvature. a 2.21b κ l connection γ cc + SU(2) ) l , = ( a − ), ( of − a κ κ ) ) SU(2) l a Γ N l ( – 6 –

) and ( i ( U(1) a ω to the null generators a κ l κ l ˆ − γ 2.21a b Γ 2.4 such that = ). Any such variation that satisfies the boundary γ + i) ∇ . This observation is important for us, because it al- ) = a b ), ( 0 l ˆ γ N γ κ = ( A a of the null generator from the boundary data 2.12 ¯ ` is not unique. However, this non-uniqueness is mild. In fact, if the l a 2.5 ) A l a ω ( ` Γ ( κ ), ( b , an infinitesimal dilatation ( . a γ D 2.4 b κ l is the null surface analogue of the ). Let us consider every such variation separately and demonstrate a Γ 2.18 ), we obtain, in fact is given, the pull-back of is therefore analogous of the , the connection of the boundary fields, we obtain the is a vector field in a ) is a combination of an infinitesimal and vertical diffeomorphism ( 2.23 N N a κ ) that couples the boundary degrees of freedom with the field theory in M ), we thus obtain on on µm 2.12 T ) ) 2.4 a a connection ) of the one-form ¯ ∈ ¯ 2.26 m m , ] = , a a a a l ˆ 2.16 m m m U(1) ( ( [ Finally, we also have to take into account the variations of the triple Besides the algebraic constraints ( Given a triad on a spatial hypersurface, the 2 conf µ matical histories other than thosefield already equations given is in given table by ( torsionless equation. Theredyad is no analogous statementdyad for the conditions ( shift ( mal transformation ( that the action is stationary withfield respect to equations such are variations provided satisfied. the bulk and Hence boundary there are no further constraints on the space of kine- The one-form is the sum ofBarbero-Immirzi the parameter intrinsic spin connection and the extrinsicject curvature to multiplied the by the boundary conditions ( back to ( The spatial hypersurface. On the other hand, We then also know that the non-affinity where satisfied, the two covariant derivatives agree, i.e. Therefore, the real and imaginaryvia parts the of Barbero-Immirzi the parameter one-form lows us to infer the non-affinity Going back to ( This condition implies that there exists a the bulk. At the kinematicalimpose level, that the the action boundary shouldδ one-form be stationary under conformal variations condition ( JHEP07(2021)057 ) 2.10 , e.g. ¯ (2.30) (2.31) (2.32) (2.33) a m κ ∧ A N A turns into the − N A can be extended A y ` will also preserve 0 η l . ) = 0 A ) a ∧ ¯ ` 0 + ¯ Aab m (recall the conditions
A η are not included in here, M A , m + cc η ∧ ω BA ` i a that are generated by a e i
C N ) ) − , l ] ω + i ∧ ∈ m a a Diff( =
A κ κ ¯ ζm + i ( , z ∈ k A , m ) ¯ ) is therefore stationary with ), which is shared between different 1 2 m B a + ξ z κ − ( ˆ N ϕ Σ − 1 , l ∧ ¯ = 0 1 2 k m a − 3 Λ . ) 0 2.10 ζ 0 , the action is trivially stationary = π A κ ( a [ ` − A AA ¯ = AA ∧ m = 0 = + η e e = 0 ζ A z -invariant scalar . = ( A A ) γ B f, a ∧ ∗ N ` a + ) under any such shifts is given by C A m of the triple F ϕ A ∂ fl , EOM a ∇ ∧ D` D η ( η )

∧ ] N 2.10 S ¯ N ζm ¯ ] = ] = [ m Z ( anyways. At a configuration where the bulk a a SL(2 0 l [ ∧ – 7 – κ ) [ . ] = shift ξ ), which generate the infinitesimal field variation + i) a δ m M Diff dilat , which consists of all solutions of the bulk plus boundary f κ γ , but preserve all other configuration variables, e.g. [ dilat f δ of all bulk and boundary variables. Since any such ( ) ∈ δ ] = 0 2.17 h kin a 0 ) ξ Diff( H shift ξ ϕ AA δ M , m of the boundary data i ⊂ 3 e ∈ a πγG [ . Since ] , l ), any such diffeomorphism ξ b ). On the other hand, it is also clear that the action ( 16 a ˆ ϕ ¯ Diff( phys m diffeo ξ κ − diffeo ξ a ( δ H = 0 [ δ ∈ 2.13 m ξ N ] = ˆ ∂ 2i ϕ | S [ a − , but ξ that do not preserve the light rays null flag soldering form self-dual connection dual (momentum) spinor Lagrange multiplier fields boundary data , = ϕ shift ξ 6= 0 δ ) ) ] ab a = 0 . The variation of the action ( a ε ab a A ξ , m ξ, l ∗ a a ] = 0 π 0 , l ,N a Ba , 0 a a A ] = [ . The action is a functional on the space of kinematical histories. Its saddle points define Aab AA A ω κ a N kinematical histories physical histories e ` A η ( ( AA l

[ e

T

[

Next, we consider the dilations ( The same statement also holds for the shift symmetry, which is generated by the field Any vertical diffeomorphism boundary bulk Diffeomorphisms ∈ 3 a shift diffeo ξ ξ because they would not preservepoints the universal on structure phase (the space. ruling of If the equations of motionarea are two-form satisfied, the with respect to any such variations, i.e. that annihilates all otherδ bulk and boundary configuration variables other than respect to those variations vertical diffeomorphism of δ variation, vertical diffeomorphisms reduces to the identityξ at the boundary of the boundary conditions ( is invariant under anyplus such boundary field equations are satisfied, the action ( field equations, which arefield equations. given by the Einstein equations in the bulk, and additional boundary into a diffeomorphism Table 1 the space of physical histories JHEP07(2021)057 ) ) ), = 2.23 2.10 2.19 . (2.37) (2.38) (2.39) (2.40) (2.34) (2.35) (2.36) ) for the = . +cc . The re- ] · [ = 2.26 . . wit respect to + cc EOM = dilat ., f  A δ f ε. +cc +cc ¯ Aab m N ν ε. EOM η N

y + cc ∧ ∂ l N I A  EOM that are generated by ∂ EOM

` ∧

I A )  A  η . a A EOM 1 ε A πG

y A η l η 8 ∧ η 1  , m y ν N + y πγG d ∧ i l a − A l 8 ν η A ∧ A , l − ∧ = 1 2 y a A l ¯ = . m ¯ κ A m` − i fN ( ∧ ∧ . ε ε N ∧ ν N a A + ∧ m 2 + cc ∧ A m Γ +i ∧ f )  − d N ε ) d into its real and imaginary parts ( ν, µ fN ν ν i a − ∧ = µω ∧ λ fε ∂ , + i − γ + i ( + a A = 1 γ A f γ − µω ` ν µ ∧ N d A ν − µ l d A + + ) is a total exterior derivative. If we compute y d d η l ∧ ν ν ∧ – 8 – D` ∧ 1 2 N ε d d ∧ ] = ] = ] = ( ε Z A i 2 a a a ∧ 2.34 − h ∧ ∧ ), we obtain a total exterior derivative, ` l A ) to go from the first to the second line. If ( [ ε ε κ   m η A A [ N [ γ γ η η i i Z + i) N N 2.4 2 2 + 2.34 conf γ λ Z Z 1 + conf λ γ δ 2 conf λ − − A ( δ δ    is real, we are left with a total derivative that turns into η A 1  ` y πγG a ) + i) + i) N N N l 8 Z Z Z Γ A γ γ ∧ ). In the last step, we used the reality cognition ( η ( ( 1 = πγG A ∧ +i) +i) +i) 2.5 16 i i N γ γ γ D ( ( ( πγG πγG − − EOM 8 8

. Since ] a = = ( = i i i S = = Γ [ πγG πγG πγG ε 8 8 8 ∧ conf λ EOM = = =

δ EOM ] i d

] S − [ , as given in ( S [ ) EOM , and obtain the field variation ). We decompose the gauge parameter a

] ) are satisfied, it follows that ( ν dilat ¯ f m S dilat f δ [ , δ 2.18 + i a 2.24 µ Let us summarise: at its saddle points, the coupled bulk plus boundary action ( Finally, we also need to consider the action of the complexified conformal transforma- conf λ , m δ a = k a surface integral, is invariant under those variations of the boundary data Going from the first( to the second line,abelian we connection used the decomposition of A short calculation yields tions ( λ Inserting this expression back into ( were we used theand gluing ( condition ( the expansion of the null surface, we obtain, in fact sulting derivative of the action is All other bulk and boundary configuration variables are annihilated by JHEP07(2021)057 . ∼ = N by / N ] a and a ξ N ∗ 0 (2.44) (2.42) (2.43) (2.41) = π . , m M : a . Given along , l N N a N + cc ., κ T and dilations Θ [ that generate  is the co-dyad ν ∈ ¯ ) m C +i a + cc . d C dilat µ ξ : ∈ δ  ∧ ¯ m A + cc N , z are ` d ( )  ) 1 A z ∧ A ( Ω is the integral of a sym- ` 1 N . A d − ) ∈ , ν, f N + δ A π a a ( ) to go from the first to the D` η Θ 0 A ξ A = ( m η M ` y z ∂M l 2.23 Θ ∧ γ I ∧ k − , ). The residual degrees of freedom, − a A ) − δ ¯ , where ), characterise the two physical degrees m ( N ε ] AB 1 2.20 a ) and ( h a A l ∧ M m d d + 2.12 [ d 2.4 a a d ∧ a k l a l m d − ) + Θ ) and ( a – 9 – is also bounded by the spacelike disks f AB − δ k ( κ Σ i 2 = = d N M 2.14 M a a ∧ + Z is the differential on the space of kinematical histories, m m  is unique only up to the addition of two-dimensional A ] κ = Θ is a one-form on field space and a two-form along d ` a d d l A that preserve the universal structure (the ruling of [ ), a possible choice for the pre-symplectic potential along has a boundary, there is a non-trivial cohomology and N , see ( α η ∧ + i) ] d . · phys N 1 2 [ ε ) γ , complexified conformal transformations ∂ H N i (

2 a 2.10 − C ] 0 = κ   S [ diffeo ξ π, γ f i ( δ δ N N of πγG , where Z Z = ] P and 8 · α is equipped with a universal ruling that is shared between different [ a ⊂ N l = ∂ + i) + i) N + d

shift , which is a one-form on field space and a three-form on ζ ∝ N M γ γ ν δ ] ] provide a more detailed discussions of such corner ambiguities in general ( ( N N Θ a for a summary. If j j are one-forms on field space. The pre-symplectic potential l [ 1 43 i i h = d , and πγG πγG 8 8 42 N ˜  and = 0 = = ), we obtain f . The boundary conditions for the gauge parameters N ] N . In other words, · ∂ [ phys , see table

Since the four-dimensional region denotes a linearised solution of the bulk plus boundary field theory (a tangent vector Θ a , we also have corresponding contributions to the pre-symplectic potential, N H δ 1 kin , ξ dilat f If to where we used thesecond boundary line. field equations ( M the null surface boundary is given by the one-form on field space plectic current an action, the pre-symplecticform, current i.e. is uniqueReference only [ up to therelativity. addition Given the of action an ( exact one- where is obtained from theof variation of physical the histories. actionthe and pre-symplectic Since the potential pull-back on on corner field terms. space to More the precisely, space the pre-symplectic potential the null surface are the same for allwe configurations thus on have the spaceon of (kinematical) histories makes no reference to thesurface interior boundary of spacetime. All fields are intrinsic2.3 to the abstract Radiative null symplecticThe structure null boundary spacetimes (points on phase space) such that the fibres light rays), shifts δ 0 which are given by the gaugeof equivalence freedom class at ( the null boundary. Notice that the definition of the equivalence vertical diffeomorphisms H JHEP07(2021)057 . ) A M N T D` T . On 2.26 ¯ A (2.50) (2.48) (2.49) (2.45) (2.46) (2.47) f stands ∈ . This ` admits ∈ ) ) l ), ( b ( a a b l ˆ l = σ ˆ ¯ N i m . The defi- connection a 2.23 ( − , on N ) m l . If we replace f , a ( ) ) 1 ), the one-form l l b σ U(1) = ( ( m − . ϑ − ϑ = 2 ˆ i 2 2 2.23 = cc ¯ = m. ab − a + b ˆ q − l ∧ l ˆ ) a l = b b a = to eliminate the second ( l ˆ k k b ) ] a ∇ l l . The symbol : ˆ b ( b a ˆ ` fϕ into vector fields ∇ a σ a b ˆ M , and [ b m k ∇ = 0 − , a ˆ denoting the T − m ] N ) . ∇ ab N c a ˆ γ b ˆ a m q C l ˆ R  m spin connection on b ∈ ˆ ¯ ˆ of the null surface solely in terms Γ T m : m − − ¯ ) a m ∇ ∧ l ) ∈ + b b ( a l N = = ˆ ¯ [ N m k (

l ˆ b ϑ ( a ) ε b b a σ l U(1) l l 2 l ˆ ( ˆ m a a m ∧ , with Ω ϑ + 2 a + a D D k 1 2 .  , which follows from ( ∇ ∈ Γ a a . a ) m b l a ) ( ε ¯ Γ and − l l m m cc ˆ ¯ ( m a ϑ b b = 0 is shifted into ` a ϑ m l ˆ − is useful to identify the components of − N m m 1 2 γ A ) ∧ – 10 – T ` , defines a b 0 − − = =  a N
A . = 2 i equals ε ∈ ¯ k − ¯ D m γm . ) cc A a ) = ) = l ∧ A γ l ( ` = ` cc + ¯ A A ( − d ϕ a ` ` + a ] l 0 0 A − b A A k D γm ¯ ¯ ) ˆ ` ` A m l a D` ( ( ` ( l a + [ a a A a A ϕ are the shear and expansion of the null congruence ` ¯ k ` ∇ D D D ) i ) b l l a a A a ( ( l − k ˆ ¯ ). Since ¯ ) between spinors and vectors on the null boundary to arrive at m ϑ m m ϕ A  a A A = l k ˆ ` ` 2.9 − 0 0 , the spin coefficient 2.42 m a and A A b b b = 2 = = 2 ¯ ¯ k k ) ∧ l ). Using the spinor formalism, the proof is immediate: ) ), we obtain, in fact ( l ¯ fm ( alone. An embedding of the boundary into the bulk is not necessary to d = = σ . It is important to note, however, that it is possible to compute shear ϕ , which are defined in a neighbourhood of 2.9 ) + A A 2.26 ` ` 2i a C N admits the decomposition M a a : C D D T fm N a a N ¯ ( + ∈ m m 1 on a A A a Ω ` ` k A ˆ m Another useful equation is to write the expansion Since we are primarily interested in the symplectic structure along the null boundary, ∈ equals on by D` a a A and the isomorphism ( the final result. of the exterior derivative of the area two-form where we used theterm Frobenius in the integrability second condition line. In addition, we used the boundary equations of motion ( the other hand, the spinintroduced coefficient in ( Notice that the one-form nition of this connection dependsk on a choice for the one-form for and expansion directlym from the exteriorinfer derivative shear of and expansion. the Inthe boundary fact, basis the intrinsic expansion exterior one-form derivative of the one-form where we extended inand the last steps extension of the vector fieldswith away shear from and expansion of the null congruence The components Going back to ( let us briefly discusssymplectic potential the ( geometric nature` of the various terms that appear in the pre- JHEP07(2021)057 ), .
] 2.52 0 (2.58) (2.59) (2.53) (2.54) (2.55) (2.56) (2.57) (2.51) (2.52) C [ f , and (iv) a K κ = − . ]  . 1 does not vanish  ¯ . , whose compo- m. C ] cc f [ · cc ). [ ∧ f − ) of the boundary and the connection − K ¯ m a m l ) dilat f 2.56 δ ∧ ∧ δ 2.33 ¯ m − k m ∧ ) l ∧ = ), ( ( k k , (iii) shifts of ( ) ) l d a ( f ϑ f δε 2.32 ∧ 1 2 . N , m f ϑ A A ∂ − a ν, I a κ D` = ( . +i) f ε. D` ∂ a A 1 γ A 1 C πG ¯ ( ` m − transformations, I . 8 f`  γ ∧ , ν m N − N − + i) k 1 Θ Z πG ] = 8 d γ U(1) ∧ ) = A ( ] = 0 ] = i ] =  A = a a a δε i l , we can then write the pre-symplectic poten- . We will see below that such gauge transforma- D` – 11 – πγG [ . ] = κ a N ∧ m [ ) N A [ D` N k Z a d C ` 16 [ Ω [ A , we then also know d Ω f ). Going back to the definition of shear and expan- U(1) ν f a a − U(1) ν l δ U(1) , m ν f` + i δ K f fl a δ dilat − f f πγG − d 2.33 δ , l 8 d ∧ a = ∧ ] = κ + ] = a δε ( l a δε ¯ [ l m κ defines an unphysical gauge direction, if the gauge parameters ( N d ), we obtain ] d Z · ) and ( ∧ a N [ next, there are the dilat k f ∧ Z k δ [ 1 ε 2.45 πG 2.32 dilat f 8 1 N πG δ dilat f Z 8 ), ( − δ , and = ∧ 1 1 ) = πG first of all, we consider the dilatations ( 2.47 − A 8 ,δ . has compact support. A large gauge transformation, where transformations of the boundary data − = ), ( D` R a A = l dilat f ` a δ 2.46 → ) in the following simplified form k N ( transformations: U(1) Θ N N Ω 2.42 : Taking into account U(1) The field variation f at the boundary, is generated by the dilatation charge given by ( Where we defined the corresponding charge, Therefore, nents are given by ( sion, ( Since Dilatations: data. An infinitesimal such variation defines the vector field , (ii) a tions are given by combinationsκ of (i) dilatations ofhorizontal the diffeomorphisms null of normal 2.4 Gauge symmetriesIn on this the section, radiative phase wewhich space identify are the the degenerate gauge directions symmetries of of the pre-symplectic two-form ( tial ( The corresponding pre-symplectic two-form is given by the exterior derivative on field space (i) (ii) JHEP07(2021)057 . a ). ), l R = =  → 2.30 2.41 . (2.63) (2.64) (2.65) (2.60) (2.61) (2.62)  ]. . C 47 : generator – + cc . We thus f  ) + cc 45 , ¯ , we obtain m shift ¯ ζ m U(1) δ ¯ m ∧ ∧ 41 , ∧ A m 2 , m ∧ 1 D` i k A ) − . l ` ( ) = ϑ . ∧  ε ¯ ζm k ( + ν δ U(1) ) = transforms according to ( ν δ . ¯ m a δε A ζ ( + i) κ ( + i) ∧ D` . γ , ν ε. ] ∧ ( γ A d (  C C [  ν δε I ν ` N f ) = 0 N N Z N Z Z Z ]. On the other hand, the γL , δ 1 πγG ] = i 44 8 1 1 A πG πγG 1 1 ] = shift ζ ), namely 8 πγG πγG δ is a vector under – 12 – are annihilated by the action of C 8 16 8 ( D` [ connection. On the other hand, the null vector ] = . a ) ), we then also know A f N
) are compact. Since the orbits are compact, the a + − − ` ] C 2.62 m [ [ Ω 0 K ¯ ν m ) = + δε δε δε C , 2.47 L [ U(1) 2.62 a , δ U(1) . Taking into account the variation of the co-dyad ( ν charge ν ∧ ∧ ∧ δ ¯ L m ν ν ν charge implies, therefore, the quantisation of area in units , m d d d shift ∧ ζ a − charge ( l δ denoting the Planck area. This observation provides a simple ] ( ( ) and ( U(1) N N N m 1 3 i Z Z Z N U(1) charge ( C − [ Ω U(1) 2.46 ν ), we obtain an important constraint between the dilatation G/c 1 1 1 = L ~ πγG πγG πγG ε ), ( π 8 8 8 U(1) δ 2.56 − − − − transforms as a = 8 2.45 a = = = 2 P under the shift symmetry, the one-form ) = ` ) and the γκ ), ( : , δ a κ 2.56 2.59 U(1) ν is the surface integral of the area density multiplied by the function , with δ ] ( 2 P C N [ γ` f Ω we obtain such that the shifttwo-form. symmetry defines a degenerate null direction of the pre-symplectic have On the other hand, The quantisation of the of explanation for the quantisation of area inShifts loop of quantum gravity [ All other boundary fields The orbits of the corresponding charge must beL quantised [ Going back tocharge ( ( Where we introduced the is charge neutral and the one-form Given ( Bringing everything together, and taking into account that Notice that (iii) JHEP07(2021)057 , =  . ) = 0 (2.73) (2.70) (2.71) (2.72) (2.66) (2.67) (2.68) (2.69) cc A −  D` ) . A ¯ m. ¯  ` m . ( ∧ ∧ ). We are thus y cc = ξ k k ( = − ξ ¯ ∧ . m 2.14 ]
L ¯ ξ B m ∧ ¯ = m ` ∧ C ∧ k [ ) A ∧ ¯ δ ` m A k is field independent, we ∧ ) , see ( k − ∧ a  ∧  D` N ∧ ) ξ . AB k  ∂ A ) A denotes the interior product F , ` cc B ∧ A y ) ( ` y ξ − δ  D` that vanish at the boundary of κ ξ A (  D` ` B y − ∧
` ξ ) A − ( ¯ ` A m AB ¯ D = 0 ( m ¯ and ` ¯ m, m ∧ ξ F ∧ A to the null surface satisfies the gluing ∧ a A ` y k 0 ∧ L ξ ` AB ( k . N ξ vanishes at ∗ ( ∧ y F , δ π ) m AA ξ a − y N ) = i ) + d e : ξ A = 0 ∧ ξ + Z ∧ A 0 ) κ m i A N − ) + k A D` N κ ¯ , ` ) ∧ ∧ ∂ l D` A 0 ξ

a A A ( T + i) ] a ` A D` – 13 – ). The second term can be simplified by taking ϑ L (d (d ( ` ξ γ AA ) D` ∈ A ) δ y y D` ( ∧ e ξ, l ` ) ω ξ ξ  a ω ( ∧ ∗ N y A ξ 2.50 ξ δε ξ A = = [ = ϕ + i + i L −  i a D`  κ )+ l κ m κ πγG D` κ -form. If such a vector field y N ξ ξ ( ξ δ ( ( 8 p N ∂ ξ δε y y y L Z Z L ∧ L ξ ξ 2( ξ ∧ − ε finally, we return to the vertical diffeomorphisms, which are 1 2    ξ κ κ ( +i) +i) L = = = = y γ γ h ∧ ξ ( ( ¯ ε   m N ξ N Z ∂ ∧ L Z i i k 1 N πG πγG πγG ∧ Z 8 8 8 1 πG ) , and obtain 8 − + − A 1 πG = is the Lie bracket of vector fields on 8 ) = D` ] ] = 0 · ), we obtain A a , ,δ = ` · ξ ξ [ ( [ ξ ξ δ L 2.23 C ( L , i.e. N Ω The first term on the rightence, hand which side follows is proportional directly tointo from the account ( expansion that of the the pull nullcondition back congru- of the tetrad This integral vanishes as asee constraint. ( In fact, taking into account that The two boundary terms disappear,left because with the last term, which is given by the integral between a vector fieldhave and a where generated by vertical vector fields N The resulting infinitesimal field variation is the Lie derivative Vertical diffeomorphisms: (iv) JHEP07(2021)057 . ) ) =0 onto 0 2.15 2.26 (2.77) (2.78) (2.74) (2.75) (2.76) deter- AA C a encodes v a 3 m d , we obtain κ → ∗ N describes two . ϕ 0 P = 0  ∼ . : / A AA ] . ` ξ cc a and it is equipped A π ` − = 0 , m C . k = a  . ×  , l ∧ and . a cc 0 = 0 0 R cc κ A A ξ [ ¯ − ` ¯ ` 0 − , such that A is nothing but the Raychaud- A 0 δC 0 ` ¯ m. 0 A AA − ¯ ` . k ` A ξ 0 ∧ AA of vertical diffeomorphisms ( 0 y A B = 0 ξ A ` ξ ξ + 0 A e i i ξ m B
) = A B C e ε ∧ − C ∝ e B ∧ , δ ) a e ) are satisfied. If we insert, in fact, the ∧ ∧ ξ is kept fixed. The one-form 0 ∧ k = 0 d ∧ 0 L . AB A ( AA N ¯ AB ` . The complex-valued one-form F ∧ AB e ∗ N C AA N A Σ y a 2.21b AB ) Σ ξ T ` N T ξ ξ on the null boundary. The inner product of F κ 3 ( 2 i T − N × ) = y ∈ b N N Z ξ 0 – 14 – ∈ a Z Z = = ( becomes ¯ N l m 0 0 a = 0 : Ω a ξ AA T l − ( e + i) C AA AA + i) + i) N × y m κ ) and taking into account the reality condition ( γ , which consists of a surjection v v ∂ ξ (

) γ γ 3 3 ∧ N  ( ( a d d ∗ )   = 2 , π 2.72 . The gauge equivalence class ε ∗ N , ξ T a C ϕ l ab ∧ ( ∈ i q πγG iΛ 1 = 0 ] P πγG πγG ∼ (d a a 8 24 8 l y a a ξ ξ − − + , m  ∀ m a . The boundary conditions are such that a gauge equivalence class = = that determines the ruling of the null boundary by its null rays , the generator N ), we thus see that the generator , l C a Z ξ a ω C C ∈ κ [ 2.75 1 ) back into to ( πG 8 P, z Before we go on and proceed to the second half of the paper, let us briefly = 2.73 ⊂ ξ is a vertical vector field, it satisfies ) vanishes, i.e. C z a ( a ξ l 1 − π and We will see in thehuri next equation section for that the the constraint null congruence Since Going back to ( vanishes as a constraint. Therefore, Its pull-back to the null surface is proportional to the null normal. In fact, The last line vanishes: the vector-valued volume three-form is given by The first two termsas on well the — right provided hand theEinstein side Einstein equations equations cancel in ( each addition other. to The third term vanishes Inserting ( for the one-form where the fields on the right hand side of this equation are all intrinsic to the null surface. = a z of boundary fields the non-affinity of the nullmines vector the field degenerate metric m physical degrees of freedom along the null boundary, which are the two degrees of freedom of summarise. The starting pointary of action this with sectionThe was the null the parity boundary definition of is oddwith the homeomorphic Holst a bulk to term plus universal thethe bound- in cartesian structure base product the manifold bulkγ and a boundary that is null. Summary. JHEP07(2021)057 ) ) ). R , 2.17 (3.1) (3.2) (3.3) (3.4) 2.78 SL(2 ), ( . The vertical . The next 2.65 ] , we introduce N a N ), ( , m a , on i ) remove two degrees of , l 2.61 ), there are two complex ) a f a gauge transformations gauge transformations κ 2.8 [ 2.18 ⊗ ¯ ), ( m and a compatible metric i , e a J U(1) U(1) 2.55 m = such that , see ( ( , 2 ) ) ) another two, the dilatations ( a = 0 f a , l ⊗ a , J, q 2.16 m ) , we introduce an internal and two- , a κ describes therefore two degrees of freedom V , which is a real-valued one-form intrinsic l ( ¯ ( a ij m ∼ N δ κ / Given the parity-odd action in the bulk, ] , , − a 2 1 = 4 f f m , m ij ( a q − , it is useful to work with auxiliary ] 2 , l i a , which will carry a natural representation of a ≡ √ R κ ]. ) ) = + ) = ). This is an important observation that pro- – 15 – , m j 1 2 − a = [ f f f lies parallel to the null rays, which are a universal structure, 41 1 ( ( , , l , g i 2.57 f over a J J N kinematical degrees of freedom per point on f 26 κ ( T . Finally, there is [ ⊗ V q ) ∈ N ∗ a m l T ) and ( + ¯ ∈ a 2 + 3 = 8 m ( 2.32 m × be an orthonormal basis of -symmetry among the boundary fields and the complexified conformal transformations ( 2 , and vertical diffeomorphisms, see ( ) 1 } a a R √ l 2 κ , , 1 + 2 symmetry is manifest. The final step is to impose the constraints = ) ) remove one of them, the shift symmetry ( e = 1 SL(2 R -valued differential form, parametrisation of the boundary fields , is an abelian connection with resect to both ] for the original derivations using the spin network representations and , i 2.14 boundary charges V i ) a 47 f ) R κ – { , R , shifts of SL(2 ) , is the Kronecker delta. Given the complex co-dyad a 45 , . We equipp this vector space with a complex structure ij ) . δ 2 , m SL(2 , R a N , 1 . Hence there are SL(2 κ The counting is immediate: since ( 4 N is characterised by a single number (a lapse function). Since . Let then a l components left to determine to diffeomorphisms ( remove the overall scale of freedom as well. Thealong gauge equivalence class the associated where variables on the null boundary.dimensional Over auxiliary every vector point space of SL(2 q and calculate the Diracphysical bracket observables. to identify the Poisson commutation relations among3.1 the To parametrise the boundary fields In this section,variables we on derive the the nulltify cone. Poisson an commutation The implicit relationsstep construction among is has to the three embed steps. configuration where the the The covariant phase first space step into is an to auxiliary iden- (kinematical) phase space, vides a simple geometric explanationSee for [ the quantisation ofmore area geometric in loop arguments given quantum in gravity. [ 3 we introduced the resulting pre-symplecticits two-form gauge along symmetries, the which nullof consist boundary of and dilatations studied of Notice that as well as dilations, see ( gravitational radiation crossing the null boundary. JHEP07(2021)057 (3.9) (3.5) (3.6) (3.7) (3.8) , but (3.16) (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) ) R , ). In the . We call a ) . A linear (2 a ) to raise and m sl ¯ R ( m , we now also i j , ) , δ ∈ a a a ¯ ¯ m J = m . m ( , SL(2 m a e mj is given by i m q . ) ( m matrix. We obtain the ) im R J 2 q , : × (2 generator ) = 2 S . m ij sl i ( q e i ( , m J ] , U(1) j  ¯ , , m k ) + ¯ i a , if it preserves the volume two-form. [ ¯ ) ¯ m . X. jm m i . In the following, we will also need m m ε R ij i 2 X, . = , n m ε , j J, 2 i m S. m 2 i ( − = m − S i + ¯ − k j ij = 1 SL(2 in and its inverse a m q = , , m ε m in terms of the co-dyad ¯ k k i m S ij with respect to the co-vector = ] = ] = +i = i ,PX a i – 16 – ¯ mi ¯ q = l i i, j m m m ¯ i j a q i i e X m i j  S e j i e with respect to the original complex-valued dyad m ] j = 0 m m i m J, X 1 = lm [ X, i S [ J ε − = i = = ¯ a e that annihilates the onto the complex-valued one-forms i is a complex and traceless S PJ e [ k k = k ) ): ) i i i X ¯ J R R R ij X X , , , ε . In other words, (2 ¯ sl X SL(2 SL(2 is equipped with a natural representation of , which is two-dimensional. Accordingly, ∈ → V and ) V S ∀ R X , . A basis in the corresponding Lie algebra generator and , and it is defined via (2 i j a δ sl defines, in fact, an element of : = ) are the components of V i U(1) P ¯ S . m m are the components of the complex structure: jm → ( ε i , i is a V m ) im : a m ε J J ¯ m S Going back to the decomposition of The vector space , a m have an induced actionthis of action where we suppressed thea projector vector indices otherwise preserves where commutation relations Such that the inverse matrix is given by where where map In components We then also have the internal volume two-form, ( where following, we will use thelower vector internal indices metric in Next, we decompose the one-form JHEP07(2021)057 , , u 1] µ , N and 1 a − (3.26) (3.22) (3.23) (3.24) (3.25) (3.17) (3.18) (3.19) (3.20) (3.21) [ . Since κ Ω = e 1 along − → 1 such that is given by C 1] N N , ∪ (0) a : 1 0 m u − C [ = → , but also on a N N ∂ m : . u is arbitrary and defines the and an overall scale, e.g. ρ is a one-form on in terms of a fiducial co-dyad ¯ ] = 0 a m, u e ) a ∂ l a ( (0) a k , σ m . 1 m , [ − , + a (0) ) is inferred from the Lie derivative , k ), but ¯ , = ) m and an overall conformal factor m 2 l a u a | (0) a ( 0 ) ∂ z ∧  . Γ z ϕ | C ) ) ρ 3.17 R a a

l l , δ l 1 ( γ ( , + ∂ (0) λ ϕ N = 1 + S . m − + m ) depends not only on . ) µ i l = µ ] = 0 + i λ SL(2 ( = – 17 – a a u − ) ρ , u l ) and ( ϕ ∂ ∂ is one-dimensional, we can parametrize any future ( e 2.51 = e [ → = Ω ( (0) a ϑ := a = a l 1 2 = 1 N Ω = e m o 3.21 N a v 1 are fixed background structures. Their field variations  m : κ 2 C

d = , δ S (0) a u . In addition, let us also choose the fiducial (unphysical) m m l 0 ] = 0 L u [ δ u > gauge potential a in terms of the fiducial null vector ∂ has the topology of a two sphere, a natural choice for a , which consists of two disjoint cross sections: ] a 0 l C l maps the unit sphere onto the complex plane. N [ U(1) is a gauge parameter for dilatations, ϕ transformation : i R e ) are the same as in ( N 2 ϑ R λ T → , , which can always be parametrised as follows: ∈ a and the ) above. Finally, we also need to choose a parametrisation for the null normal N κ a and ] : 1 SL(2 = cot 0 l − [ µ λ z 2.48 = , an As far as the covariant phase space is concerned both the time function The radiative symplectic potential ( a for which we now introduce a convenient parametrisation as well. Consider first the l a (0) . This can be done by introducing a fiducial time coordinate a a a as well as the fiducialvanish. co-dyad In other words, pointing null vector where lapse function coordinate such that it satisfies the boundary conditions at the boundary of the vector space of all null vectors on as in ( l which increases monotonically along thevector null generators: i.e. for every future pointing null where k where l one-form If the base manifold m Given the fiducial dyad, the corresponding fiducial volume element is The basic idea is to parametrise the physical co-dyad JHEP07(2021)057 is a into along (3.34) (3.30) (3.31) (3.32) (3.33) (3.27) (3.28) (3.29) π . The R 2 ◦ a , z e variables → ) = = 1 R N z , is completely : is given by its , i i u , a F a f U SL(2 U  2 Ω , along the null boundary. ) U ¯ z d . By introducing a (local) ) d ¯ z 1] 2 z, , ( − 1 L Ω , λ, − u, z, ) [ a  − ( ¯ z ∂ ) 1 2 a . → ) in terms of the l ¯ u F. a − l ) . = u, z, U ¯ λ z N 1 d ( , z, = d 0 − 2.51 , as given by the affine length :  1 ± ) z, µ l u a  . − ( ( u ( N a − Γ 1 l L e ) 1 l λ ] =: − ) = ) ( − = 0 u N ¯ ¯ − z z F γ i N d µ [ will be denoted as follows: if ). Otherwise, the coordinate 0 2 ). Hence, the non-affinity of a z, f 1 − , which is given by 1 l + ( – 18 – u , z, ∂ a e − a 1 a d l a L  L Z U ) takes values in a two-dimensional vector space, we 1 l 3.23 b 3.18 κ ± U ( u ( − =  a ∇ ). We may then introduce the new time coordinate κ R b e a U ) = a := l l 2 ¯ z = λ U ˙ F = − 3.25 a z, l µ ( ) b l ) = − ( L ¯ z e ∇ κ b 2 l ). Since on the base manifold, which naturally extends via 4 ). Therefore, L : u, z, C ) 3.5 l ( = ( 3.17 U κ → a U C b : , we have a fiducial coordinate system ∇ b z N ) and ( U is the conformal factor ( µ 2.49 is the lapse function ( ) l ), ( ( coordinate is unphysical. The only conditions are that it increase monotonically Ω = e N u 2.29 Let us also consider the corresponding null vector field introduced above. Consider firstthe the null Lie generators, derivative see of ( first the have vector-valued to co-dyad explainsimplest how possibility the is Lie derivative acts onto the basis vectors scalar function on the null surface, its time derivative is denoted by 3.3 Kinematical phaseIn space this section, we write the symplectic potential ( where expansion. Derivatives with respect to Its non-affinity is proportionalfollows. to Consider the first expansion of the null surface. This can be seensee as ( which satisfies the boundary conditions arbitrary. A morethe physical total time (affine) coordinate duration can of be a null introduced generator as of follows. Consider first where Above, we introduced a fiducialhomeomorphism time coordinate a coordinate on The and satisfy the boundary conditions ( 3.2 Choice of time JHEP07(2021)057 ). is a k 3.38 (3.47) (3.40) (3.41) (3.42) (3.43) (3.44) (3.45) (3.46) (3.35) (3.36) (3.37) (3.38) (3.39) of , ζ  )  j o e v . Going back 2 ∧ , 1 d + k  − ( o ∧ + U d v
= 2 d m ), back into ( k ∧ l d λ ) a )+ i l ), and the parametri- l m − j ( 1 a ∧ e µ e ϕ − k 3.21 . − ∧ m ) S i e 2.41 d k i d R 2 , ( 2 L ) , · j Ω dΩ 1 ). We now want to further d Ω  e l (2 S − ∧ j , see ( d ∧ h i i , see ( m a S sl l i e o · i ∧ 3.15 k ¯ 1 κ v 1 m ∈ i S j 2 − l h − ) e [Ω] d U, 1 S l S ~ X + L d · ∧ − ) ( d ) N l . The vectorial component L l i ( m S · Z a ( 1 S P e f l σ · k is inferred from σ − S − λ L h h S a Ω = + ¯ h − 1 k ij  = µ ¯ πγG m (d q l X , m e + ij 8 1
) P J l d q i Ω 2 u − L ( k, h 1 e 1 ~ γ ) − d γ σ l −   ρ ( – 19 – 2 , γ + + k + i γ − + Ω d ϕ Ω 2 , Ω U, l λ ˙ m 2 ij ) + ~
1 γ γ l − E, J ε − ( J L ) Ω ij L l µ  1 ϑ ( ε 1 γ d − U := Ω := d := := − d 1 2  1 + ϕ e of the one-form d e  + I − Ω e e ] component of Φ := Π := K E ) back into the symplectic structure, we obtain f K ¯ =  ζ d L p , N U m k u o 1 Z δ ∧ Ω = + d = is the projector defined in ( v − [d l − 2  j 3.37 i d ) λ ¯ S e d L e m 2 1 l πG · d 1 R ζ 2 ∧ ∧ 8 , − L o S o Ω i ) is a consequence of l + v v Ω + e (2 + Ω ij 2 2 u L ) and ( κ sl ε − d d d d  3.40 ρ = = N ∧ → − ∂ N 3.36 j ε λ k Z Z ) e − d N R µ ∧ Z , ∧ 1 1 − i πG πG e j e (2 8 8 e 1 − πG sl 8 − + ∧ : = i − ), we then finally obtain = e k P = N 3.38 N Θ Θ To make this expression more transparent, let us introduce the abbreviations The second identity ( sation arbitrary and the value ofto the ( In addition, we also note that where simply this expression. We insert the parametrisation of If we then insert ( where With this choice, we obtain JHEP07(2021)057 ) m . l e Π N 3.41 (3.51) (3.52) (3.53) (3.54) (3.55) (3.56) (3.48) (3.49) (3.50) -valued ). It is 3 = 10 ) ), where , ) R × ) , 3.50 3.50 . (2 x, y ( i x, y sl ). ( l N ˜ δ m N ˜ i δ ) 3.12 1 y ) ( 2 + 2 + 2 − y S ¯ e ¯ Π( d are given by XS · ) G ) and ( i x S πG π h denoting the )( 8 m 3.11 m l − l X . e Π ¯ Π m = = +8 ,S l ), ( + m

X l can be rearranged into the diagonal ) ) ¯ e ¯ e e Φ + y y Π ) X Π 3.10 . ( ( , z d equipped with its natural symplectic i e I ( + e · + Φ , 1 ,S E ) + Π ) h ) 1 − E m l R x
x u − π , d ¯ IJ d L X γ ¯ e Π( ¯ e Π( = d u e are now treated as functionally independent f K,E, Π . Notice that the first block of these equations,  Lie algebra-valued line density = 1 +  , z ) ) − SL(2 γ + S N – 20 – K ∗ f K R , L R p P , , ) , d , , T m ( ) ) l ] := d ) − · K [ ~ p e λ IJ x, y d x, y h x, y ( d , x, y SL(2 ( ( SL(2 ), simply says that the Hamiltonian vector field of ) ( = , , with ∧ N o N N ) ) ˜ δ N o ˜ v ˜ δ δ 1 → m ˜ δ ) 2 l v ) ) x, y − 3.54 y 2 ) d ( e y y Π x, y ( S d y ( ( ( N N N e d I ˜ ), ( δ : N ∂ N e Π( S on the null rays ˜ δ Z Z G i S S G e Π i π 3.53 P 1 1 πG XS π πG JS πG πγG πG πG 8 8 16 8 8 ), ( and and − − − + − = tr( e I . The commutation relations can be inferred immediately from the canon- m = = = = +4 = +8 = +8 ) l ) = 3.52 R e

, Π is the Dirac distribution on R ) ) ) ) ) ) , N ) , y y y y y y ), ( ( ( ( ( e Θ Φ SL(2 e I e ¯ e Φ( Π( f K , , x, y ,S ,S Θ , , SL(2 ) , ( ) ) 3.51 ) ) ) E x x x x x N x ( , ( ˜ δ → ( e e I Π( e e Π( Π( E f K K  The kinematical phase space is equipped with the symplectic potential ( Now, the basic idea to compute the resulting Poisson brackets is to introduce an     N p , :  K acts as a right-invariantS vector field on theical group-valued Poisson configuration commutation variable, which relationsstructure is of momentum in the matrix representation defined by ( where namely ( the constraints that bring us down to the twoimmediate true to physical calculate the degrees corresponding of Poissonbrackets brackets. freedom among The along only the non-vanishing canonical Poisson variables auxiliary (kinematical) phase space,p equipped with the symplecticcoordinates potential ( on phase space.dimensions. The Hence, physical phase this space auxiliary is obtained phase by space symplectic has reduction with respect to and off-diagonal components of an Given these definitions, some straight-forward algebra bringsinto the the symplectic potential following ( form, null rays. In addition, we have introduced in here the differential The line densities where the diacritic indicates that the corresponding variable defines a density along the JHEP07(2021)057 1 R − = → (3.64) (3.65) (3.57) (3.58) (3.59) (3.60) (3.61) (3.62) (3.63) ) back u ) via N 3.57 3.45 is a squeeze gauge trans- Λ: f K K , p U(1) , ) ) ¯ z ) acts on the second x, y , ( , , z, 1 N 3.48 ! ˜ δ connection instead ! − = 0 = 0 ) ( ) ) on the kinematical phase y   S ( R e E , E ϕ ! ~ ~
d d 3.46 G i γ 1 − is a null ray that starts at e ¯ hX, π 2 e ¯ hX SL(2 ) 0 f K + 16 + − , u -differential ( ) K ( ~ e ¯ I ¯ , − d X p z X )  ) e h  γ e h , x, y ) Λ λ ( + + on the base manifold. Inserting ( x, y x, y ( N ∧ ∧ ( ˜ δ x, y o o e C N e ϕJ ) ( N ϕJ v v ˜ δ ˜ δ y – 21 – 2 2 ∈ ) N ( ) y d d ˜ =: δ y ~ u ˜ h z d ( ( y 1 N N z ~ ˜ h d G γ − Z Z i ) to the physical phase space of the radiative modes Z S G π πG i ·

8 8 , πG π ] = 3.56 S − − λ indicates that the [ ~ [Λ] = d = = 0 = = 4 = 8 ) )–( C G :

) ) ) ) ) λ x, y y y y y y Λ: Next, we have a conformal constraint that imposes ( 3.51 ( ( ( ( ( ( ) = Pexp ∀ ∀ ¯ e e z e ¯ e e h h h ϕ ϕ N First of all, we have the Gauss constraint. Let , , , ˜ , , δ ) ) ) ) ) y is a test function of compact support. Notice that ), we introduce the constraint x x x ~ x x d ( ( u, z, e R e ( I I e e e Π( Π( Π(   S   3.56  ), we obtain → and projects down onto )–( 0 N 3.54 u is the path ordered exponential and : = 3.51 )–( λ u Pexp 3.51 For practical reasons, it is often more useful not to work with the group-valued con- where operator. we will see below thatformations this along constraint is the first-class null and boundary. generates local Conformal constraint. Gauss constraint. be a testspace function ( of compact support. To impose ( 3.4 Constraints In this section, we introduce theby constraints that the reduce the Poisson kinematical brackets phaseat space ( defined the null boundary. Thereconformal, are scalar five and different vectorial constraints, constraints. namely the We introduce Hamiltonian, Gauss, them below. where the notation argument. where ends at into ( figuration variables, but with a Lie algebra-valued flat such that (i) (ii) JHEP07(2021)057 ) ). 3.42 3.30 (3.72) (3.69) (3.70) (3.71) (3.66) (3.67) (3.68) ) and the ), the Ray- ), we obtain angle and projects ) and ( 0 3.58 dressing of the 3.32 u 2.75 3.57 , U(1) = , defined in ( a u ! U(1) = 0 U  ) and ( is the e h , ) only if we impose . ) , ) holonomy ( l Ω ¯ z ( ∆ 2.71 ¯ ) σ ! 3.37 ) = 0 + i l γ R ( , ends at ,  u, z, γ 1 σ ( ˜ ϕ 2 are defined in ( ) − l − ( , − − ˜ e ϕ σ SL(2 ϕ, = e Π ) ) ˜ ) l Φ 1 ¯ z 2 ( u u = 0  − ( ϑ , z a µ u u,z, Ω ¯ and 1 2 σ γ ( Z  σ λ to impose. All other constraints are real. ∧ − ) are essentially creation and annihilation ˜ Φ − N∂ o + e v = 2i∆ ) ) = 2 2 = ) − ¯ 3.67 ¯ z z l d Ω ( e a N is a complex constraint, hence we have two real – 22 – 2 ¯ 2 ζ ϑ z, ]) = 0 N ξ 2 ) ( ] ¯ l ζ Z U d ) is compatible with ( ¯ ∧ ( [ ζ ) as a constrained on phase space. This involves a u, z, L d [ o κ V 1 2 v V ( ] = ∆( 2 − 3.57 ) are functionally independent. The functional depen- 3.37 µ ) = d i [ ¯ z Im ) l N D ( Z 3.49 : ϑ h u, z, and µ l The Raychaudhuri equation determines the evolution of the ( ] = ∀ L σ ¯ ζ [ Besides the scalar constraint, we also need to impose the vectorial V is a test function and At the kinematical level, the : ). Equation ( on the base manifold. Strictly speaking, this ]) = 0 ¯ ), consider first the following vertical vector field ζ R is a test function of compact support and ζ [ ∀ C C V 3.37 → ( denoting a null ray that starts at 3.56 ∈ → Re z N )–( N N : : ⊂ -valued momentum ( 3.51 µ ζ z ) γ R , (2 is the rescaled shear.space To ( realise this equation as a constraint on the kinematical phase Taking into account that itschaudhuri expansion equation and turns non-affinity into are the related constraint by ( where conformal factor along the null generators. Going back to ( The expression simplifies once we evaluate it for the vector field constraints Notice also that theoperators vector with constraints the ( conformal factorthe playing oscillators. the role of the characteristic lengthHamiltonian scale of constraint. with down onto constraint is not necessary,straint analysis. but Notice we also will that see below that it greatly simplifies the con- where where respectively. Vectorial constraint. component of ( sl dence is obtained byscalar imposing constraint ( and ais complex-valued (momentum) given by constraint. The scalar constraint Scalar constraint. (v) (iv) (iii) JHEP07(2021)057 ≈   (3.75) (3.76) (3.77) (3.78) (3.79) (3.80) (3.73) (3.74) . ≈ subgroup is a lapse + cc ≈   . ≈ e ¯ h R  y l   u U(1) . m → + cc ∂ i ), we obtain the ) 1 l e Π ( − N  denotes the interior ¯ σ ) S : 3.64 l ¯ e y Π + cc − ( N u y ξ σ e ϕ N , ∂ u L y ∂  · u + i γ e is the generators of vertical Π N S γ e 2 I∂ ] u h d − ,  d ) and N 0 m 0 [ Ω λ l i + 2 2 e N ≈ H Π − 2 3.24 − γ e .  . Ω − ) + , , 2 , . ¯ γ σ 0 4 is the generator of an N   U L   σ d ξ f  K ˜ N d µ I ] ζ Λ λ = 0 L − + u u − u u u u 1 K ( d d d d d d d 2 2 d d p d d d C 2 of compact support this generator vanishes [ N

) are satisfied. Indeed, Ω Ω Ω N U N N N N N 2 2 N ξ N 2   d    2 2 2 U U d U L d d = 3.70 G C D V H − d d d – 23 – ] := Θ 0 − 2 ) − − − − − 1 2 C − ), (  Ω

N U e Φ ] [ = = = = =  ] ) imply that u u ) N d U ∂ H d

E [ ( ] ] ] ] [ 3.67 U 0 ¯ e N ζ Φ λ µ u N U 3.54 [ ξ [ [ [Λ] ξ ∂ N 1 γ d )–( [ ( L L ,V ,C ), ( ,D ,G ] − − ] on the kinematical phase space defined by the commutation ] ] . 1 γ U ,H   ] 3.64 d N N U N N UN − N [ [ [ [ u ξ 3.52 N N d  N ∧ [ ∂ H H H H o ∧ ∧ ∧ ∧     H v = o o o o 2  v v v v d 2 2 2 2 ). Going back to the definition of the constraints, we find, in fact, U = exp d d d d d N y Z N N N N N u 3.56 ϕ Z Z Z Z ∂ 1 πG )–( 4 1 1 1 1 is the fiducial vector field introduced in ( πG πG πG πG 8 8 8 8 stands for equality on the constraint hypersurface and . Going back to the definition of the Gauss constraint ( a u ) 3.51 ∂ ≈ ≈ ≈ − ≈ ≈ R ] = , N [ The Poisson brackets ( H where product, e.g. It is straightforward to check that forprovided every the constraints ( function that satisfies the boundary conditions Consider then the Lieto derivative with define respect the to generator this vector field such that we are able where SL(2 of It is immediate to checkdiffeomorphisms that the Hamiltonian constraint relations ( 3.5 Dirac analysis of the constraint algebra JHEP07(2021)057 , we U(1) (3.89) (3.90) (3.91) (3.92) (3.81) (3.82) (3.83) (3.84) (3.85) (3.86) (3.87) (3.88) E . and .  0  e Φ ζ ) 0 ~ . d ζ i + ¯ ζ e  Π ↔ 2  − e h − ζ + ), where we have to commute under the ( Ω  . First of all, we have Ω ~ d e ¯ h . If the shear vanishes, ¯ − V + Ω 1 e h, Ω 3.84 + i γ  − i 1 e h e γ h − Ω i) ] = 0 0 − and γ µ. Ω ¯ Ω ζ ~ − [ γ − d ¯ ζζ V e V i λ Π γ + i 2i∆ γ γ − ( 1 − h γ ∧ − e ¯ e + Π 1 ¯ o ζ 1 Ω e I − and v − −  2 µ , , , . 2 , , . . Ω e

Π − d Ω 0 ∧ 1  Ω o 0 N − = 0 = 0 = 0 = 0 = 0 = 0 = 0

2i∆ ≈ 2i∆ v Z Ω −

2 −

¯ ζζ ] = 0

] ] ] ] ] ] ] Poisson commutes with both e i ] e d  0 0 0 0 ¯ ¯ µ ζ ζ , λ µ ¯ [ [ [ ] ¯ ζ λ [ µ e [ +2i∆ ζ

ϕ 0 − [ [ N [ [Λ e πγG ζ µ D Z ,  [ 8 ,V ] ,V ,C ,D 2i∆ ] ,C – 24 – ,V ,G ,D ¯ ∧ ¯ ζ ] V ∧ ] ] − − [ λ and o ¯  [ λ ). ζ . A straight forward calculation yields e [Λ] µ [Λ] o [Λ] [ [ [ ¯ v V ζ [Λ] , } = πγG v C ] 2 G G  f G K 2 C 8 V ¯ ¯ 0 D G ζ  V d

  −  d [   ] ζ  K 3.68  ≈  µ p N V V, N [ Z  {

Z ∧ 0 ] equal on the constraint hypersurface ¯ ¯ ζ o ζ ,D [ ]  v πG πG 2 λ ) and ( ) is satisfied, the right hand side turns into [ ∧ d ,V ] o C = 4 N µ v , the constraints  [ + 16 2 3.67 Z 3.67

] d ) the only difficulty arises with equation ( D ¯ ζ = [  are conjugate, = 0 N stands for

Z ] 0 ,V 3.84 D ≈ U ] ζ = [ d µ [ )–( ¯

σ V are conjugate, but ] 1 , 0 and D ] ¯ − ζ  ¯ ζ [ E [ 3.81 C Ω V ,V ] { ≈ ¯ and ζ [ ˜ h e Φ V  We are now left to compute the Poisson brackets among All other constraints are second class. A few entries of the corresponding Dirac matrix Therefore, Finally, we also need to evaluate hence Poisson bracket. If the vector constraint ( where, once again, obtain that We then also find Since In proving ( take into account thatholonomy, see the equation constraint ( has been dressed byvanish identically, the introduction of an commutation relations JHEP07(2021)057 ) be α 3.58 } (3.98) (3.94) (3.95) (3.96) (3.97) (3.93) α Φ { holonomy ( ) be the Dirac matrix R , are therefore those that 6= 0 ,

SL(2 , ∗ . 0 } ζ 0 ,B ~ . d β ). The calculation is straight ¯ + cc ζ Φ : det ∆ S, S }  = 0 ∧ 0 { , , ) and the definition of the vectorial 3.58 o

αβ αβ v ,S ) 0 2 ¯ α ∆ = 0 = 0 ¯ z d ¯ 3.64

V ,

0 α  . The only entries of the inverse Dirac = ∆ ) ) N ¯ α ¯ ¯ z z Φ R Z , z } 0 α ), is ten-dimensional. The resulting physi- β u ∆ A, ( Φ → πG }  , n u, z, u, z, holonomy ( 3.56 j α ( ( holonomies α 16 − N ) – 25 – Φ m m ) ,S i i )–( :

{ R ) ). The physical phase space is obtained by im- S, V R , ¯ z ,  ,S ,S µ ≈ − ] ] − 3.51

λ µ A, B ] 3.56 [ [ 0 .  u, z, SL(2 = ζ C and D ( [ SL(2 )–(  ∗  = ¯ m V

R i 0 ∗ , ] S

¯ ζ 3.51  ) are second-class. The kinematical phase space, which is [ → α, β, . . . S, S V  { N A, B be its inverse. The resulting Dirac bracket is defined by removing 3.67  : α β δ λ ), ( = 3.66 µβ ), we get ∆ ), ( αµ 3.67 3.65 : ∆ denotes equality on the constraint hypersurface. is some unspecified multi-index. Summation (integration) is understood over all α ≈ αβ ∆ We now want to compute the Dirac bracket for a specific class of partial observables, Let us briefly summarise. The kinematical phase space is defined by the Pois- for all test functions matrix that we needcorrespond to to the evaluate vector in constraint. order Schematically, to calculate Furthermore, namely the matrix entriesforward: of first the ofcommute among all, themselves we with respect notespace. to In that the other Poisson the words, brackets matrix of the entries kinematical of phase the where repeated (De Witt) indices Finally, let usphase explain space. how The to physicala identify phase family space coordinates of is second-class (Dirac equippedand constraints with observables) and the on Dirac the bracket.the physical Let auxiliary directions from phase space, i.e. cal phase space is two-dimensionalthe and next spanned section, by we the compute Dirac the observables of Poisson the brackets between theory.3.6 such In observables. Dirac observables from son commutation relationsposing ( the constraints.Hamiltonian (Raychaudhuri equation) There and arestraints Gauss constraint ( both are first-class firstequipped class, and with all the second-class other Poisson brackets con- constraints. ( The constraint ( where Going back to the definition of the Gauss constraint ( JHEP07(2021)057 , is  . ) is the (3.99) on the x, y (3.104) (3.105) (3.102) (3.103) (3.100) (3.101) + cc )) }

y Θ( . ) and the holonomy holonomy , (  ) . ) ) 0 ¯ V,S z , π R R 3.56 ) , { , , 0 x +cc holonomy along ( holonomy is the conformal , z n )–( 0 π j , ) (  u ) Ω SL(2 SL(2 ) . ( R C x ) is immediate to cal- y , ) of the vector con- n 3.51 δ . ( translation generator ¯ U(1) z j ( } m )  j ¯ ≡ z, ! ) ,S ¯ R = 0 3.67 ¯ ) S XS w ) SL(2 , is the , u,  ) j ¯ V,S ) i w ( w, ¯ z ) { ¯ m z ( k i ] X x x, y  o m ( ( ), is the ) ) SL(2 , w, v j S i x o x 2 u, z, ) ( (2) ( u, z, ( d u S m 1 y ( on the null boundary, δ [ i ( ) back into the expression for the 3.57 ( k V − C i J 1 ¯ a V ) i XS Z ) Ω S −   ¯ e z , , ¯ z o ) ) u 0 0 )) x o d u 3.102 y )Ω d , and u d ) x 1 2 ≤ ) and ( 1 ∆( ¯ ( u, z, z 1 u, z, 2i∆( − ( 1 ± − − − ¯ ) ζ 1 , u > , u − Z Λ( x ) 1 2 1 2 3.68 u 1 ¯ ) e z u, z, u )Ω + − x ( z, Θ( ( ( d πG )) = – 26 – ]    ¯

ζ N 2i(∆( y 1 o 1 ) x,y ( v − ), the Poisson bracket − ( ¯ . Inserting ( u 2 w = = 4 e Z πG d  ) = is the complex-valued ), we obtain [ (2)

N 8 − u ) ) j δ × 3.67 C , w, ¯ ¯ ) z z − ) πG 1 Z 1 m x Θ( i 3.93 ( u u , if we know the ordinary Poisson bracket 16 = ( ∗ u d x,y m

} u, z, u, z, 1 0 = ) 1 ( ( ,V = Θ( − x k k ) ∗ i i Z ) above. ( ¯ j z

i ) S, S m 0 ,S ,S πG i ≡ { ] X ) = Θ( ¯ z 4 ] 3.11 , ¯ 0 ζ ,S N u, z, [Λ] − [ ] [ ( , defined via equations ( ¯ x, y ζ G , z [ = V 0 H m  i labels a point on ), we infer the commutation relations for the ) provides an important intermediate step, because it tells us that we i∆ ∗  u V Θ( e S ( )

 ) ¯  z n j y 3.99 is the integrand that appears in the definition ( 3.99 ( × n ) ,S j is the distribution ) ¯ z u, z, ), and ¯ z ( ) ,S u ) ≡ x u, z, 3.18 ( x Θ( ( u, z, ( m V i The physical phase space has two local degrees of freedom. The Equation ( m i S  S  has three functionally independent componentsIn and fact, it it is does not not quite commute a with Dirac the observable first-class yet. constraints. Instead, we have two-dimensional Dirac distribution on thefactor ( base manifold. Inthe addition, null generators, and that we introduced in ( Let us briefly summarisethat our determines notation the in conformalthe this class step equation: of function the co-dyad where Dirac bracket ( kinematical phase space.definition Given of the the fundamental vector Poissonculate. constraint brackets ( We ( obtain straint, i.e. know the Dirac bracket where and Inverting the right hand side of ( JHEP07(2021)057 , ) , β ij D e  2 R γ . can , ∧ × ij α γ D (3.109) (3.110) (3.106) (3.107) (3.108) e SL(2 . An ex- ) + cc , where a ∧ ¯ z ] A [ U(1) , z, 0 / ) αβ T ( F R dressed , , kl holonomy defined ) ¯ z )Ξ line dressing ¯ z SL(2 U(1) U(1) u, z, ] at the two boundaries ( T, z, ij 31 ) on the auxiliary internal ( γ – ij
3.1 ) 17 Ξ commutes with all second-class is the ¯ z  ) 0 Ω . i∆ j l e u, z, , z, z H ( j i Ω( n k (2) S − S, i δ H l ) m ) and ∆ 0 T are still only partial observables, because ]. J m S T k δ − γ , where the conformal factor takes the value ) is constructed using a ) ) 49 3.10 mn − m , ) entries, the commutation relations for ¯ z z q 1 – 27 – ( = e T D − 48 1 and := − H 3.108 × Θ( u, z, π ij = Ω H C γ ( = ). A short calculation yields πG Ω)( ij u z Ξ C 16 ∂ γ ( 3.99 − × u d = D ). Since the conformal factor ∗ 1 -invariant observable is given by the +1

− ) 0 , and since the inverse Dirac matrix has neither diagonal Z ] ¯ z 3.108 , µ U(1) 0 [ ) = is the two-dimensional internal metric ( D , z ¯ z 0 T ) are such that a gauge equivalence class of boundary fields is kept fixed. ( mn . Notice that both δ kl Let us briefly summarise. The first set of results were developed in section T, z, V ( , γ 2.12 = ij ) is an abbreviation for ¯ γ z denotes the complex structure ( ). Another natural example is the two-dimensional conformally rescaled metric, ij mn J q Ξ T, z, 3.68 ( Notice that the Dirac observable ( ij γ . Hence the area serves as a relational clock — Bondi’s luminosity distance — to define  in a finite domainand with introduced a the corresponding boundaryconditions boundary that ( term is along null.This the equivalence class We null is characterised specified surface. by two theand local The degrees boundary additional of boundary freedom conditions corner along the data null surface, — gravitational edge modes [ of perturbative gravity can be found in [ 4 Summary and outlook Summary. where we considered the Palatini action with the parity odd Holst term kinematical (gauge variant and partial)a observable gauge is invariant integrated observable. along A a recent light discussion ray of to such dressed obtain observables in the context be immediately inferred from ( where at those points ofT the null ray the Dirac observable ( constraints except entries nor any off-diagonal introduced following standard constructions. An example is given by which is the functional on phase space that evaluates the conformally rescaled metric where vector space neither of them commute with the Hamilton constraint. True Dirac observables can be ample for suchholonomy, an where via ( Imposing the Gauss constraint amounts to computing the quotient JHEP07(2021)057 ) , ) ¯ z (4.2) (4.1) , but → ∞ j z, 3.103 i ). The r ( j S k ] holonomy o ), which is 2.59 ) S [ currents, i.e. R ]. Finally, we , # ), ( 2.56 ) ) 2 35 R , − , 2.58 r SL(2 34 ( (2 holonomies Maurer-Cartan form in a vicinity of future O sl ), ( ), were we found the ) ) 0 + R R r> , , k ]. Given such a foliation, i } 2.57 ) between dilatations and ), where we saw that the 9 r 3.103 X N r 1 SL(2 { SL(2 2.63 ) 2.61 ¯ ) z , see [ ¯ z r z, N ( u, z, ) and quantise the asymptotic phase ( frame rotations, namely the Hamilton ) and ( (0) →∞ (0) Ω r ). The second class constraints impose ¯ σ 2.55 , we studied the Dirac observables on the Λ = 0 frame rotations ( U(1) + 3.64 2 k i = lim holonomy. At null infinity, its matrix elements ¯ X ± – 28 – , ) U(1) r 1 ) determines the fundamental area gap. R 0 I ) , r γ ¯ ) z ( ¯ z ) involves not only the O z, SL(2 ( u, z, ] to construct physical observables, which are invariant expansion ( ) + (0) ¯ z 3.103 (0) 49 ) and additional constraints on the Ω /r The first and perhaps simplest possibility is to restrict our- , such that z, σ 1 ). The entire construction is highly non-linear. Various trun- ( 48 ) is proportional to the generator of dilations ( ± + ). We computed the constraint algebra and defined the Dirac 3.65 (0) , which is itself obtained from the i k I ). The physical phase space is obtained by symplectic reduction 3.57 Ω σ δ 2.62 r " 3.67 3.53 ) = ) = ¯ ¯ z z )–( ) and ( ) and the Gauss constraint ( ) and ( charge ( 3.51 3.37 ). An important result was given in equation ( It seems an extremely daunting task to quantise the Dirac bracket ( gauge transformations and vertical diffeomorphisms. 3.75 u, r, z, u, r, z, 3.66 ( ) on the shear U(1) j 3.94 i Ω( , see ( H U(1) S (past) null infinity the radiative phase space atlimit null of infinity can the be quasi-local obtained radiativeconditions from for phase an the space various asymptotic introduced spin above. coefficientsaccount, of If we an obtain we adapted take the Newman-Penrose tetrad the into falloff selves to asymptotically flatspace spacetimes alone. ( Consider an auxiliary double null foliation 3.70 frame rotations is responsible for the quantisation of area in loop quantum gravity, . We introduced the pre-symplectic potential on the quasi-local radiative phase space, ~ d In the second part of the paper, section 1 N − cations seems necessary to quantisethree the possible algebra such in truncations an below. effective way.Asymptotic Let quantisation: us briefly consider Outlook. directly. The right handalso side the ( conformal factor.tion ( Yet the conformalS factor depends via the Raychaudhuri equa- bracket ( commutation relations among the describe both gravitational radiationintroduced as well a as line gravitational dressing memoryunder [ [ with respect to first-classthe and generators second-class of constraints. vertical diffeomorphisms Theconstraint and two ( first-class constraints are the conformal matchingequations ( ( U(1) where the Barbero-Immirzi parameter null surface. Tophase infer space the into a Poissonflux kinematical brackets, algebra phase we ( space found that contains it an useful auxiliary to embed the covariant and we identifiedimportant the observation gauge was symmetries madeBarbero-Immirzi of in parameter the deforms equation resulting the ( resulting pre-symplectic two-form.itself An a multiple of the cross-sectional area. The relation ( of JHEP07(2021)057 ) 1 − r (4.3) , the ( ). In O matrix is = 3.58 → ∞ ˜ ϕ S r spin network edge modes, see expansion of the ) R SU(2) ). For /r , 1 ) ], but it is difficult to ). Hence we can not ] on the tensor product ], we face the opposite 3.50 x, y SL(2 60 ( , is the (inverse) conformal 2.61 54 58 , , (2) gauge parameter ) into positive and negative Ω δ 59 53 ] that takes into account that , 57 ) ), 4.3 56 ) turn into the Lie algebra-valued 41 , , . Still, the question remains for x, y U(1) 55 30 Θ( 3.106 3.103 → ∞ πG 4 is usually ignored as well. A more appro- Ω . Once such coherent states are found, we − ) ˜ ϕ ¯ z . = z, ∗ ( is the Fock space for future (past) null infinity. – 29 –

holonomy ( o and ) + cc . A possible solution is to quantise the quasi-local S gauge parameter, see ( ), must be quantised as well. ± ) y ) b o ( Ω r is the initial condition that appears in ( R i H matrix in this framework. In fact, the ¯ ( m , (0) ) a O S ¯ σ ¯ R m U(1) , , ) edge modes see e.g. [ SL(2 x (0) Ω = ) ( limit, only the first two terms of the σ , where R zero-mode SL(2 † (0) , − = σ ) H ∈  R (0) ab , → ∞ ) ⊗ SL(2 σ holonomy enter the symplectic potential ( ¯ z In the spin network representation [ + Ω ) z, ]. H ( R SL(2 o , 52 S is the dressed – j i group-valued commutation relations ( SL(2 50 ) , H R 30 , problem. We have a straight-forwardmodes, and rigorous including quantisation of gravitationalcharacterise edge the two radiative modes.state on Consider, for a example partial an Cauchy hypersurface. Such a state consists of superpositions of Defining this tensor productsingle amounts Hilbert to space. join A thewell rigorous two be construction asymptotic possible of boundaries only such into viathe a a an gravitational joint entangling edge Hilbert product space modes, [ asymptotic may which boundaries very are (such as located at a joint cross section of the two peaked at appropriate valuescan of define asymptotichow states to by recover sending thea ordinary highly entangled boundaryHilbert state (linear space functional) [ violate the Heisenberg uncertaintycanonically relations conjugate that to tellimpose a us simultaneous gauge that conditions on theand both area the the two-form conformal factor is phase space at the full non-perturbative level and construct coherent states that are nature of the priate approach will alsoe.g. consider [ the Hilbert space of the In fact, if wethe ignore Hilbert the quantum space nature of of the the radiative conformal factor modes, and we only make consider a mistake, because we would is then simply theasymptotic shear state that is annihilatedFrom the by the perspective positivequantisation of frequency problematic. the part quasi-local The ofsent phase conformal the to space, factor infinity we is before find treated considering such as the an a quantum asymptotic classical theory. field In and addition, the quantum phase space defines the Poisson brackets of the radiativefrequency data modes, at we obtain null an infinity. infinite set of Separating harmonic ( oscillators. The Fock vacuum factor and the asymptotic dressed SL(2 commutation relations for the asymptotic shear. The result of this contraction on where Loop quantization: JHEP07(2021)057 (4.4) (QISS) , i C ~,~m, | ) ¯ z z, ; k , z k z (¯ (2) δ ], may provide a natural framework to + 1)) k 61 j ( k ) to generate the evolution of the boundary j q 3.70 – 30 – k X 3 Quantum Information Structure of Spacetime spin labels. Hence the conformal factor is quantised. ]. It would be very useful to quantise such a finite- γG/c 41 ~ SU(2) π ]. degrees of freedom [ 6 ) , = 8 5 R , i C It would be highly desirable to develop yet another truncation, where SL(2 ~,~m, ] that embed the finite-dimensional Hilbert spaces into larger and larger | are the usual ) ¯ z 63 , z, ( 62 ~,~m 2 b Ω and frequencies, seedimensional e.g. phase-space, [ and definemaps different [ UV completionstensor via product different spaces. refining we only consider aspace finite-dimensional that symplectic we subspace definedcutoffs. of If, in the for above. example, quasi-local the(round) phase cross metric, Such sections decompose a are all spherical, subspace configurationrespect we variables could to could into the choose spherical be an auxiliary harmonics found auxiliary (with metric), and by introduce various an UV effective UV cutoff for large spins study this problem in the spingravity, network the representation. problem of In how three-dimensional to euclidean explainhas such boundary been evolution studied for spin in network states [ where The question is then to construct acan representation use of the the radiative Hamiltonian modes, constraint suchstates that ( along we the null generators.of the Twisted relevant geometries, which admit a representations At the boundary ofquantum of the area partial is Cauchyarea excited. surface, operator, we the If would Wilson the have, e.g. resulting lines boundary terminate state and is a an eigenstate of the gravitational Wilson lines. Every such Wilson line represents a quantum of area. not necessarily reflect the views of the John Templeton Foundation. Acknowledgments Support from the Instituteacknowledged. for This Quantum research was OpticsTempleton supported Foundation, and as in part Quantum part of Information by the project the is (qiss.fr). ID gratefully 61466 The opinions grant expressed from in the this John publication are those of the author and do Other truncations: JHEP07(2021)057 33 ]. 214 (2016) , 14 Phys. ] 57 , SPIRE ]. IN Phys. Rev. D ][ , (1990) 725 ]. SPIRE ]. 31 IN ][ SPIRE Class. Quant. Grav. (2021) 095 IN , SPIRE J. Math. Phys. ][ IN 04 , Class. Quant. Grav. ][ arXiv:1710.04202 , Proc. Roy. Soc. Lond. A [ gr-qc/9911095 (2018) 13LT01 , [ JHEP 35 J. Math. Phys. , , arXiv:1005.3298 (2019) 807 [ gr-qc/9209012 Quasi-local holographic dualities in ]. [ gr-qc/9403028 Quasi-local holographic dualities in 938 [ (2000) 084027 ]. ]. SPIRE 61 ]. 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