arXiv:1602.05499v2 [gr-qc] 1 Jul 2016 sltdhrznbudr condition boundary horizon isolated ymti pta emty sopsdt aihn spati vanishing a to shoul variables. opposed it Ashtekar-Barbero peake as that is geometry, which out vacuum spatial a point symmetric constructing tr and of r way canonical state simple of a this of context provides of family the content 1-parameter from in physical meaning results additional the precise previous an a some by given recall modified be will can we state kno paper, been Kodama this has and In vacuum It the [7]. variables. on Ashtekar t acting complex action of of Chern-Simons context One the the welcome. in 6] highly [5, are state spacetimes o classical model certain a to from solids of dyna behaviour fundamental the the extracting of to graining i compared coarse challenge a main d from The very it emerges gravity. proven space quantum While has loop it theory full scales, spacetime. from candidate smallest of a the at provides structure geometry 4] quantum quantum 3, fundamental 2, the [1, on gravity quantum Loop Introduction 1 fmxmlsmer.Tesmnlwr 8 led commented already [8] constru work state seminal The Kodama horizo symmetry. locate the maximal not of in does condition point enforced boundary particular horizon symmetry imple isolated in maximal is We condition the this to which gravity. on quantum state vacuum loop (kinematical) in entropy hole enn unu oios oee o ieetreasons. different for however horizons, quantum defining ∗ [email protected] et ehglgttesmlrt ftecntuto leadi construction the of similarity the highlight we Next, ie h opeiyo hspolm vnfra proposals formal even problem, this of complexity the Given oentso h oaasae aia ymty n the and symmetry, maximal state, Kodama the on notes Some ihih h iiaiyo hscntuto ihteisol the maximal with sense) construction suitable gi this a of be (in similarity can a the variables on highlight connection peaked real vacuum we for a them, implements state on Kodama Based the variables. that canonical defining in ambiguity n testa ti,i gemn iherirwr,inadeq work, earlier with agreement in is, horizon. it that stress and aut fPyis nvriyo asw atua5 02-09 5, Pasteura Warsaw, of University Physics, of Faculty ercl oewl n oels nw eut bu h Koda the about results known less some and well some recall We sltdhrznbudr condition boundary horizon isolated F .Bodendorfer N. ∝ ac 9 2018 29, March Σ Abstract hc aosyetr aydrvtoso black of derivations many enters famously which 1 tdhrznbudr condition boundary horizon ated ∗ aet en h oino quantum a of notion the define to uate s u eeyefre ata notion partial a enforces merely but ns, sntiges hnteepnnilof exponential the than else nothing is etdo vr ufc.Comparing surface. every on mented e a entes aldKodama called so the been has hem oudrtn o lsia curved classical how understand to s aesm omnshighlighting comments some make lmti si h tnadcs of case standard the in as metric al e rcs enn n htit that and meaning precise a ven na nacransne maximally sense, certain a in a, on d h ieauewihso htthe that show which literature the be can that task formidable a mics, eapeitdmc oe sit as more, much appreciated be d ffiuts a oetatlwenergy low extract to far so ifficult atoms. f ysmercgoer.W also We geometry. symmetric ly nfrain.W ilhighlight will We ansformations. gt h oaasaet the to state Kodama the to ng o aefntoscorresponding functions wave for nbfr ntecneto QCD of context the in before wn nteiaeuc of inadequacy the on o unu rvt,focussing gravity, quantum for a stkrBreovariables Ashtekar-Barbero eal ffr nitiun itr of picture intriguing an offers u htoecncntuta construct can one that out asaeadterelated the and state ma to,w ihih htthe that highlight we ction, ,Wra,Poland Warsaw, 3, F F ∝ ∝ Σ θ Σ for This paper is organised as follows: We start by reviewing some background material on the Kodama state in section 2. Next, we point out the notion of maximal symmetry enforced by the real Kodama state (section 3.1), comment on possible solutions to the Hamiltonian constraint (section 3.2), and compare to the isolated horizon boundary condition (section 3.3). In an appendix, we generalise our comments to the higher dimensional connection variables introduced in [9].

2 Known results on the θ-ambiguity and the Kodama state

The Ashtekar-Barbero variables [10, 11] are a 1-parameter family of variables which coordinatise a the phase space of 3 + 1-dimensional . They are given by a densitised triad Ei i and an SU(2)-connection Aa, related to geometric variables as

ab 2 ai bj i i i qq = β E E δij, Aa =Γa + βKa, (2.1)

i i 1 ijk where qab is the spatial metric, Kab = K ebi the extrinsic curvature, and Γ = ǫ Γjk the spin a a − 2 connection constructed from the undensitized co-triad ei derived from Ea. β R 0 is a free a i ∈ \{ } parameter known as the Barbero- [11, 12]. For the purpose of quantisation using techniques, see e.g. [2], it is of paramount importance that the only non-vanishing Poisson bracket is given by

i b (3) b i Aa(x), Ej (y) = δ (x,y) δaδj. (2.2) n o Given this, one can show that the Ashtekar-Lewandowski measure [13, 14] induces a positive i a linear functional on the holonomy-flux algebra constructed from Aa and Ei . A Hilbert space representation then follows via the GNS construction. On the other hand, one is free to modify the canonical variables as long as (2.2) remains the only non-vanishing Poisson bracket. In particular, a second free parameter θ R can be 1 ∈ introduced as a a abc i Pi := Ei + θǫ Fbc, (2.3) i 1 ijk i where F = ǫ Fbcjk is the curvature of A . It can be checked by direct calculation that bc − 2 a i b (3) b i Aa(x), Pj (y) = δ (x,y) δaδj (2.4) n o is the only non-vanishing Poisson bracket2. One thus has a 2-parameter family of connection variables labelled by β and θ. The idea for (2.3) dates back to Yang-Mills theory and its θ-ambiguity [7]. Within the loop quantum gravity literature, it was first3 discussed in [16] and [5, 17], see also [18, 6, 19, 20, 21, 22] for more recent work. Mostly, it is approached from the point of view of the so called Kodama state 3 Λ := exp SCS(A) , (2.5) | i 2Λ Z  Σ which has originally been a proposal for a physical wave function within the context of self-dual Ashtekar variables. This state is nothing else than the exponential of the Chern-Simons functional

1The first reference known to the author where this was explicitly stated in the context of real connection variables is [15], although the idea is immediate given [16] and [11]. 2 i j The Poisson bracket of two P s integrated over the spatial slice Σ against arbitrary smearing functions µa, νb a i b j i ab 2 actually evaluates to {Pi [µa], Pj [νb ]} = θ R∂Σ µaνbiǫ d x. In order for it to vanish, we need to choose the smearing functions of the P s to vanish on ∂Σ, or simply work on a spatial slice without boundary. 3While [5] was printed before [16], the available preprint of [16] predates the submission date of [5].

2 2 SCS = Tr A dA + A A A integrated over the spatial slice Σ. Formally, the Hamiltonian ∧ 3 ∧ ∧ constraint with a  in self-dual variables annihilates this state, since

ˆ ade δ δ ilm 2Λ δ abc ˆi H Λ = ǫ l m ǫ i + ǫ Fbc Λ = 0. (2.6) | i δAd δAe  3 δAa  | i There are however several technical problems which have prohibited to make this precise, including that there is so far no Hilbert space representation of the self-dual variables which implements their reality conditions, see [2] for more discussion. Moreover, the direct analogue of the Kodama state for complex Ashtekar variables in Yang-Mills theory has unphysical properties [23]. On the other hand, the Kodama state has much better properties once it is considered in the context of th