Perturbative Quantum Gravity on Complex Spacetime
aa r X i v : . [ g r- q c ] S e p Perturbative Quantum Gravity on ComplexSpacetime
Mir FaizalDepartment of Mathematics, Durham University,Durham, DH1 3LE, United Kingdom,[email protected] 4, 2018
Abstract
In this paper we will study non-anticommutative perturbative quan-tum gravity on spacetime with a complex metric. After analysing theBRST symmetry of this non-anticommutative perturbative quantum grav-ity, we will also analyse the effect of shifting all the fields. We will con-struct a Lagrangian density which is invariant under the original BRSTtransformations and the shift transformations in the Batalin-Vilkovisky(BV) formalism. Finally, we will show that the sum of the gauge-fixingterm and the ghost term for this shift symmetry invariant Lagrangian den-sity can be elegantly written down in superspace with a single Grassmannparameter.
Key words: BV-Formalism, Non-anticommutative Quantum GravityPACS number: 04.60.-m
Noncommutative field theory arises in string theory due to presence of the
N S antisymmetric tensor background [1]-[3]. However, other backgrounds like the RR background can generate non-anticommutative field theory [4, 5]. Quan-tum gravity on noncommutative spacetime has been thoroughly studied [6]-[9].In fact, it is hoped that noncommutativity predicts the existence of the cos-mological constant which is of the same order as the square of the Hubble’sconstant [10]. Perturbative quantum gravity on noncommutative spacetime hasbeen already analysed [11]. It was found that the graviton propagator was thesame as that in the commutative case. However, the noncommutative nature ofspacetime was experienced at the level of interactions.The idea of noncommutativity of spacetime has been generalized to non-anticommutativity. In addition to this, quantum field theory has been studiedon non-anticommutative spacetime [12]. Non-anticommutativity of spacetimeoccurs if the metric is complex. Spacetime with complex metric has been stud-ied as an interesting example of nonsymmetric gravity [13]-[15]. Even thoughnonsymmetric quantum gravity was initially studied in an attempt to unify1lectromagnetism and gravity [16, 17], it is now mainly studied due to its rele-vance to string theory [18]-[20]. Quantum gravity on this non-anticommutativecomplex spacetime has also been discussed before [21].In this paper we will discuss the perturbative quantum gravity on this non-anticommutative complex spacetime. We will first analyse the BRST symmetryof this theory and then study its the invariance under the original BRST andshift transformations in the BV-formalism. Consequently, we will express ourresults in superspace formalism.The BRST symmetry for Yang-Mills theories [22]-[25] and spontaneouslybroken gauge theories [26] has already been analysed in noncommutative space-time. The invariance of a theory, under the original BRST transformations andshift transformations, that occurs naturally in background field method can beanalysed in the BV-formalism [27]-[30]. Both the BRST formalism [31, 32] andthe BV-formalism can be given geometric meaning by the use of superspace[33]-[36].BV-formalism has been used for quantizing W gravity [37]-[39]. It has alsobeen used in quantizing metric-affine gravity in two dimentions [40]. The BRSTsymmetry for perturbative quantum gravity in four-dimensional flat spacetimehave been studied by a number of authors [41]-[43] and their work has beensummarized by N. Nakanishi and I. Ojima [44]. The BRST symmetry in two-dimensional curved spacetime has been studied thoroughly [45]-[47]. Similarly,the BRST symmetry for topological quantum gravity in curved spacetime [48,49] and the BRST symmetry for perturbative quantum gravity in both linearand non-linear gauge’s has also been analysed [50]. However, so far no work hasbeen done in analysing the non-anticommutative perturbative quantum gravityin superspace BV-formalism. This is what we aim to do in this paper. We shall analyse perturbative quantum gravity with the following hyperboliccomplex metric [21], g ( f ) bc = b ( f ) bc + ωa ( f ) bc , (1)where ω is the pure imaginary element of a hyperbolic complex Clifford algebrawith ω = +1. Here ω forms a ring of numbers and not a field which theusual system of complex numbers do. The advantage of using ω is that thenegative energy states coming from the purely imaginary part of the metric willbe avoided.Now we can introduce non-anticommutativity as follows,[ˆ x a , ˆ x b ] = 2ˆ x a ˆ x b + iωτ ab , (2)where τ ab is a symmetric tensor. We use Weyl ordering and express the Fouriertransformation of this metric as,ˆ g ( f ) ab (ˆ x ) = Z d kπe − ik ˆ x g ( f ) ab ( k ) . (3)Now we have a one to one map between a function of ˆ x to a function of ordinarycoordinates y via g ( f ) ab ( x ) = Z d kπe − ikx g ( f ) ab ( k ) . (4)2o, the product of ordinary functions is given by g ( f ) ab ( x ) ♦ g ( f ) ab ( x ) = exp ω τ ab ∂ a ∂ b ) g ( f ) ab ( x ) g ( f ) ab ( x ) | x = x = x . (5)Now R ( f ) abcd given as, R ( f ) abcd = − ∂ d Γ abc + ∂ c Γ abd + Γ aec ♦ Γ ebd − Γ aed ♦ Γ ebc , (6)and we also get R bc = R dbcd . Thus finally R ( f ) is given by R ( f ) = g ( f ) ab ♦ R ( f ) ab . (7)The Lagrangian density for pure gravity with cosmological constant λ can nowbe written as, L c = √ g ( f ) ♦ ( R ( f ) − λ ) , (8)where we have adopted units, such that 16 πG = 1. In perturbative gravity onflat spacetime one splits the full metric g ( f ) ab into η ab which is the metric for thebackground flat spacetime and h ab which is a small perturbation around thebackground spacetime, g ( f ) ab = η ab + h ab . (9)Here both η ab and h ab are complex. The covariant derivatives along with thelowering and raising of indices are compatible with the metric for the backgroundspacetime. The small perturbation h ab is viewed as the field that is to bequantized. All the degrees of freedom in h ab are not physical as the Lagrangian density for h ab is invariant under the following gauge transformations, δ Λ h ab = D eab ♦ Λ e = [ δ eb ∂ a + δ ea ∂ b + g ce ♦ ( ∂ c h ab ) + g ec ♦ h ac ∂ b + η ec ♦ h cb ∂ a ] ♦ Λ e . (10)In order to remove these unphysical degrees of freedom, we need to fix a gaugeby adding a gauge-fixing term along with a ghost term. In the most generalcovariant gauge the sum of the gauge-fixing term and the ghost term can beexpressed as, L g = s Ψ , (11)where Ψ = c a ♦ (cid:18) ∂ b h ab − k∂ a h + 12 b a (cid:19) , (12)with k = 1. Now the sum of the ghost term, the gauge-fixing term and the orig-inal classical Lagrangian density is invariant under the following BRST trans-formations s h ab = D eab ♦ c e , s c a − c b ♦ ∂ b c a ,s c a = − b a , s b a = 0 . (13)3V-formalism is used to analyse the extended BRST symmetry. This ex-tended BRST symmetry for perturbative quantum gravity can be obtained byfirst shifting all the original fields as, h ab → h ab − ˜ h ab , c a → c a − ˜ c a ,c a → c a − ˜ c a , b a → b a − ˜ b a , (14)and then requiring the resultant theory to be invariant under both the originalBRST transformations and these shift transformations. This can be achievedby letting the original fields transform as, s h ab = ψ ab , s c a = φ a ,s c a = φ a , s b a = ρ a , (15)and the shifted fields transform as s ˜ h ab = ψ ab − D ′ eab ♦ ( c e − ˜ c e ) , s ˜ c a = φ a + ( c b − ˜ c b ) ♦ ∂ b ( c a − ˜ c a ) ,s ˜ c a = φ a + ( b a − ˜ b a ) , s ˜ b a = ρ a . (16)Here ψ ab , φ a , φ a , and ρ a are ghosts associated with the shift symmetry and theirBRST transformations vanish, s ψ ab = s φ a = s φ a = s ρ a = 0 . (17)We define antifields with opposite parity corresponding to all the original fields.These antifields have the following BRST transformations, s h ∗ ab = b ab , s c ∗ a = B a ,s c ∗ a = B a , s b ∗ a = b a . (18)Here b ab , B a , B a , and b a are Nakanishi-Lautrup fields and their BRST transfor-mations vanish too, s b ab = s B a = s B a = s b a = 0 . (19)It is useful to define h ′ ab = h ab − ˜ h ab , c ′ a = c a − ˜ c a ,c ′ a = c a − ˜ c a , b ′ a = b a − ˜ b a . (20)The physical requirement for the sum of the gauge-fixing term and the ghostterm is that all the fields associated with shift symmetry vanish. This can beachieved by choosing the following Lagrangian density,˜ L g = − b ab ♦ ˜ h ab − h ∗ ab ♦ ( ψ ab − D ′ eab ♦ ( c ′ e )) − B a ♦ ˜ c a + c ∗ a ♦ (cid:0) φ a + ( c ′ b ) ♦ ∂ b ( c a ′ ) (cid:1) + B a ♦ ˜ c a − c ∗ a ♦ (cid:16) φ a + ( b a ′ ) (cid:17) + B a ♦ ˜ b a + b ∗ a ♦ ρ a . (21)The integrating out the Nakanishi-Lautrup fields in this Lagrangian density willmake all the shifted fields vanish. 4f we choose a gauge-fixing fermion Ψ, such that it depends only on theoriginal fields and furthermore define L g = s Ψ, then we have L g = − δ Ψ δh ab ♦ ψ ab + δ Ψ δc a ♦ φ a + δ Ψ δc a ♦ ψ a − δ Ψ δb a ♦ ρ a . (22)The total Lagrangian density is given by L = L c ( h − ˜ h ) + ˜ L g + L g . (23)After integrating out the Nakanishi-Lautrup fields, this total Lagrangian densitycan be written as, L = L c ( h − ˜ h ) + h ∗ ab ♦ D abe c e + c ∗ a ♦ c b ♦ ∂ b c a − c ∗ a ♦ b a − (cid:18) h ∗ ab + δ Ψ δh ab (cid:19) ♦ ψ ab + (cid:18) c ∗ a + δ Ψ δc a (cid:19) ♦ φ a − (cid:18) c ∗ a − δ Ψ δc a (cid:19) ♦ φ a + (cid:18) b ∗ a − δ Ψ δb a (cid:19) ♦ ρ a . (24)Now integrating out the ghosts associated with the shift symmetry, we get thefollowing expression for the antifields, h ∗ ab = − δ Ψ δh ab , c ∗ a = δ Ψ δc a ,c ∗ a = − δ Ψ δc a , b ∗ a = δ Ψ δb a . (25)These equations along with Eq. (12) fix the exact expressions for the antifieldsin terms of the original fields. In this section we will express the results of the previous section in superspaceformalism with one anti-commutating variable. Let θ be an anti-commutatingvariable, then we can define the following superfields, ω ab = h ab + θψ ab , ˜ ω ab = ˜ h ab + θ ( ψ ab − D ′ eab ♦ ( c ′ e )) ,η a = c a + θφ a , ˜ η a = ˜ c a + θ ( φ a + ( c ′ b ) ♦ ∂ b ( c ′ a )) ,η a = c a + θφ a , ˜ η a = ˜ c a + θ ( φ a + ( b ′ a )) ,f a = b a + θρ a , ˜ f a = ˜ b a + θρ a , (26)and the following anti-superfields,˜ ω ∗ ab = h ∗ ab − θb ab , ˜ η ∗ a = c ∗ a − θB a , ˜ η ∗ a = c ∗ a − θB a , ˜ f ∗ a = b ∗ a − θb a . (27)From these two equations, we have ∂∂θ ˜ ω ∗ ab ♦ ˜ ω ab = − b ab ♦ ˜ h ab − h ∗ ab ♦ ( ψ ab − D ′ eab ♦ ( c ′ e )) ,∂∂θ ˜ η ∗ a ♦ ˜ η a = − B a ♦ ˜ c a + c ∗ a ♦ ( ψ a + ( c ′ b ) ♦ ∂ b ( c ′ a )) , − ∂∂θ ˜ η a ♦ ˜ η ∗ a = B a ♦ ˜ c a − c ∗ a ♦ ( φ a + ( b a ′ )) , − ∂∂θ ˜ f ∗ a ♦ ˜ f a = b a ♦ ˜ b a + b ∗ a ♦ ρ a . (28)5ow we can express ˜ L g given by Eq. (21) as,˜ L g = ∂∂θ (˜ ω ∗ ab ♦ ˜ ω ab + ˜ η ∗ a ♦ ˜ η a − ˜ η a ♦ ˜ η ∗ a − ˜ f ∗ a ♦ ˜ f a ) . (29)Furthermore, if we define Ψ as,Φ = Ψ + θs Ψ= Ψ + θ (cid:18) − δ Ψ δh ab ♦ ψ ab + δ Ψ δc a ♦ φ a + δ Ψ δc a ♦ ψ a − δ Ψ δb a ♦ ρ a (cid:19) , (30)then we can express L g given by Eq. (22) as, L g = ∂∂θ Φ . (31)Now the complete Lagrangian density in the superspace formalism is given by, L = ∂∂θ Φ + ∂∂θ (˜ ω ∗ ab ♦ ˜ ω ab + ˜ η ∗ a ♦ ˜ η a − ˜ η a ♦ ˜ η ∗ a − ˜ η a ♦ ˜ η ∗ a − ˜ f ∗ a ♦ ˜ f a )+ L c ( h ab − ˜ h ab ) . (32)Upon elimination of the Nakanishi-Lautrup fields and the ghosts associated withshift symmetry, this Lagrangian density is manifestly invariant under the BRSTsymmetry as well as the shift symmetry. In this paper we analysed non-anticommutative perturbative gravity with acomplex metric. As this theory contained unphysical degrees of freedom, weadded a gauge-fixing term and a ghost term to it. We found that the sum ofthe original classical Lagrangian density, the gauge-fixing term and the ghostterm was invariant under the BRST transformations. As the shifting of fieldsoccurs naturally in the background field method, we analysed the effect of theshift symmetry in the BV-formalism. Finally, we expressed our results in thesuperspace formalism using a single Grassmann parameter.It is well known that the sum of the original classical Lagrangian density, thegauge-fixing term and the ghost term for most theories posing BRST symmetryis also invariant under another symmetry called the anti-BRST symmetry [44].It will be interesting to investigate the anti-BRST version of this theory. Fur-thermore, the invariance of a gauge theory under the BRST and the anti-BRSTtransformations along with the shift transformations has already been analysedin the superspace BV-formalism [33]. Thus, after analysing the anti-BRST sym-metry for this theory, the invariance of this theory under the original BRST andthe original anti-BRST transformations along with shift transformations can bestudied in the superspace BV-formalism.It will also be interesting to generalise the results of this paper to generalcurved spacetime. The generalisation to arbitrary spacetime might not be sosimple as it is still not completely clear how the BRST symmetry works forgeneral curved spacetime. There will also be ambiguities due to the defini-tion of vacuum state. We know it is possible to define a vacuum state called6he Euclidean vacuum in maximally symmetric spacetime [51]. We also knowthe ghosts in anti-de Sitter spacetime do not contain any infrared divergence.Therefore, the generalisation of this work to anti-de Sitter spacetime can bedone easily [52]. However, as the ghosts in de Sitter spacetime contain infrareddivergence, this work can not be directly extended to de Sitter spacetime [52].In order to generalize this work to de Sitter spacetime we will have to modifythe BRST transformations accordingly.
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