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Canonical Quantization of Two-Dimensional Gravity S Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 1–16. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 5–21. Original Russian Text Copyright © 2000 by Vergeles. GRAVITATION, ASTROPHYSICS Canonical Quantization of Two-Dimensional Gravity S. N. Vergeles Landau Institute of Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia; e-mail: [email protected] Received March 23, 1999 Abstract—A canonical quantization of two-dimensional gravity minimally coupled to real scalar and spinor Majorana fields is presented. The physical state space of the theory is completely described and calculations are also made of the average values of the metric tensor relative to states close to the ground state. © 2000 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION quantization) for observables in the highest orders, will The quantum theory of gravity in four-dimensional serve as a criterion for the correctness of the selected space–time encounters fundamental difficulties which quantization route. have not yet been surmounted. These difficulties can be The progress achieved in the construction of a two- arbitrarily divided into conceptual and computational. dimensional quantum theory of gravity is associated The main conceptual problem is that the Hamiltonian is with two ideas. These ideas will be formulated below a linear combination of first-class constraints. This fact after the necessary notation has been introduced. makes the role of time in gravity unclear. The main computational problem is the nonrenormalizability of We shall postulate that space–time is topologically gravity theory. These difficulties are closely inter- equivalent to a two-dimensional cylinder. The time twined. For instance, depending on the computational coordinate t varies between minus infinity and plus procedure, the constraint algebra may or may not con- infinity while the spatial coordinate σ varies between 0 tain an anomalous contribution (central charge). The and 2π. All these functions are periodic with respect to the coordinate σ. The set of coordinates (t, σ) is presence or absence of an anomaly in the first-class µ constraint algebra has a decisive influence on the quan- denoted as {x }. The metric tensor in space–time is tization procedure and the ensuing physical picture. denoted by gµν so that the square of the interval is writ- These fundamental problems can be successfully ten as resolved using relatively simple models of generally 2 2 σ2 σ covariant theories in two-dimensional space–time. ds =g00dt ++g11d 2g01dtd . (1.1) These models particularly include two-dimensional gravity models, both pure and interacting with matter, Most of the formulas and notation in the Introduction and also two-dimensional string models (see, for exam- are taken from [2]. The metric tensor is then parame- ple [1–6] and the literature cited therein). trized as follows: In the present paper we present a canonical quanti- zation of two-dimensional gravity minimally coupled 22 2ρu–vv to real scalar and spinor Majorana fields. All the con- gµν =e, structions and calculations are given before the final v–1 (1.2) result is obtained. The physical states of the theory are 24ρ fully described. The complete state space has similar g≡detgµν =–ue. properties to the multidimensional Fock space in which η boson and fermion operators are acting. The calcula- Let i, j = 0, 1 and ij = diag(1, –1). We introduce the µ tions begin with the average values of the metric tensor orthonormalized basis {}e so that relative to states close to the ground state. i However, the present study should merely be per- µ ν η ceived as one of the first steps along the way to the gµνei ej = ij. (1.3) anomaly-free quantization (if this is at all possible) of some of the theories which, in more traditional quanti- To be specific we take zations, are anomalous. Only further studies, including a systematic study of the discrepancies in the predic- µ 1 – ρ 1 µ 0 e0 ==--- e , e1 –ρ. (1.4) tions (which are encountered in different approaches to u v e 1063-7761/00/9001-0001 $20.00 © 2000 MAIK “Nauka/Interperiodica” 2 VERGELES i The dyad {eµ} is uniquely determined by the equa- Cartan structure equation, we can easily establish the i ν ν i µ i relationship δ δ tions eµei = µ eµe j = j . Taking into account (1.4), we have –gRdt ∧ dσ = 2dω01. (1.11) ρ 0 ue 1 ρ–v Since the fields φ and χ in (1.8) are real and belong eµ = , eµ = e . (1.5) 0 1 to Grassmann algebra, we have We analyze the action φ(x)φ(x) = 0, χ(x)χ(x) = 0. (1.12) 2π Using (1.12), we can make the substitution 1 $ ψ ∂ ∂ µ ψ S = ∫dt ∫ dσ –g----------(ηR – 2λ) µ ( / x ) in (1.6). Thus, the fermion compo- 4πG nent of the action is proportional to the expression 0 (1.6) ρ e {φφ˙ + (u + v )φφ' + χχ˙ – (u – v )χχ'}. 1 µν∂ ∂ i µψγ j$ ψ + --g µ f ν f + --e j µ . ρ 2 2 In this last expression the factor e may be eliminated by substituting λ Here G is the gravitational constant, is the cosmo- φ e–ρ/2φ, χ e–ρ/2χ. logical constant, R is the scalar curvature of space– time, η and f are the real scalar fields, ψ is the two-com- On account of (1.12), no additional derivatives of the ponent spinor Majorana field, and {γi} are two-dimen- field ρ appear in the action when this substitution is sional Dirac matrices. We then assume that made. Consequently, the Lagrangian of this system has the form γ 0 = 0 1 , γ 1 = 0 –1 , 1 –1 1 0 1 0 + = ∫dσ---------- u η˙ (ρ˙ + v ρ' + v') (1.7) 2πG γ 5 ≡ γ 0γ 1 = 1 0 . v 2ρ 0 –1 + η' ---(ρ˙ + v ρ' + v') – u' – uρ' – λue u (1.13) The Majorana nature of the spinor field implies that 1 2 2 2 2 t + ----- [ f˙ + 2v f˙f ' – (u – v ) f ' ] ψ = γ0 ψ (the superscript t indicates transposition). In 2u our case, we have φ i ψ = , φ = φ†, χ = χ†. (1.8) + --[φφ˙ + (u + v )φφ' + χχ˙ – (u – v )χχ'] . χ 2 The covariant differentiation operation of the spinor π π π field is determined in accordance with the formula We denote by η, ρ, and the fields canonically conjugate to the fields π, ρ, and f, respectively. The ∂ $ ψ 1ω σij ψ fields u and v in (1.13) are Lagrangian factors. We µ = µ + -- ijµ , ∂x 2 obtain the Hamiltonian of the system (1.13) by a stan- (1.9) dard procedure: ij 1 i j σ = --[γ , γ ]. 4 * = ∫dσ(u% + v 3), ω The form of connectedness ijµ is obtained uniquely 1 2ρ from % = 2πGπηπρ + ----------[– (η'' – η'ρ') + λe ] 2πG ωi ωi ∧ ω j (1.14) d + j = 0, 1 2 2 i + --(π + f ' ) + --(– φφ' + χχ'), µ i i 2 2 where ω ≡ eµdx . Hence, we find i i v 3 = – πηη' + πρρ' + π f ' + --φφ' + --χχ' . ω = u' + uρ' – ---(ρ˙ + ρ'v + v') dt 2 2 01 u (1.10) 1 We make the following canonical transposition of vari- + --(ρ˙ + ρ'v + v')dσ. ables: u 0 1 λ –ρ Here and subsequently the dot and prime denote the λr = –-- ----------e (2η'coshΣ – 4πGπρ sinhΣ), partial derivatives ∂/∂t and ∂/∂σ, respectively. Using the 2 4πG JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 CANONICAL QUANTIZATION OF TWO-DIMENSIONAL GRAVITY 3 charge is present in the quantum algebra of the quanti- 1 1 λ –ρ λr = –-- ----------e (4πGπρ coshΣ – 2η'sinhΣ), ties (1.16). This implies that all the operators (1.16) 2 4πG (1.15) may be treated as first-class constraints in the Dirac 1 λ ρ sense. This new approach is used in the present study. π – r ' = -------e sinhΣ, 0 πG The idea of a new approach to quantization arose when studying a model describing pure gravity. This 0 λ ρ model is obtained from the model (1.16) by deleting the π + r ' = – -------e coshΣ. 1 πG second terms on the right-hand sides of the system (1.16). It was shown in [1–4, 6] that in pure gravity the- Here we have ory the central charge is zero if the scalar product is σ positive definite in the entire scalar state space. The rea- Σ σ π σ˜ π σ˜ son for this phenomenon is that in the new approach the ( ) = 2 G∫d η( ). operator ordering procedure in the quantities % and 3 0 differs radically from the ordering in traditional quanti- The variables describing the matter remain unchanged. zation. In the new variables the Hamiltonian (1.14) has the We now formulate the assumptions used as the basis form to develop the new quantization method. 1) The entire state space HC in which the fundamen- 1 2 0 2 2 1 2 a π π ψ % = -- – π + (r ') + π + (r ') tal operator fields r , a, f, , and act has a positive 2 0 1 definite scalar product. No indefinite metric is present in the HC space. 1 2 2 + --[π + f ' + i(– φφ' + χχ')], (1.16) In order to formulate the next assumption, we 2 denote the set of operators (1.16) by L and the set of all the material fields f, ψ by ψ. From the operators L we 3 (π 0' π 1') π i φφ i χχ = – 0r + 1r – f ' + -- ' + -- ' . eliminate the degrees of freedom describing the mate- 2 2 rial fields and we denote the set of operators thus So far the analysis has been classical.
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