Stochastic Quantization and AdS/CFT
aa r X i v : . [ h e p - t h ] D ec Stochastic Quantization and AdS/CFT
Diego S. Mansi and Andrea Mauri Dipartimento di Fisica Teorica, Universita degli Studi di MilanoVia Celoria, Milano 20133, Italy
Anastasios C. Petkou Department of Physics, University of Crete, Heraklion 71003, Greece
Abstract
We argue that there is a relationship between stochastic quantization and AdS/CFT,and we present an explicit calculation to support our claim. In particular, we showthat a conformally coupled scalar with φ interaction on AdS is related, via stochasticquantization as well as via AdS/CFT, to a massless scalar with φ interaction in 3d.We show that our results have an underlying geometric origin, which might help toelucidate further the proposed relationship between stochastic quantization and holog-raphy. The possibility that stochastic quantization is related to AdS/CFT has been discussed be-fore (e.g. [1, 2]), however the discussion has not picked up momentum mainly due to theabsence of an explicit example. It is not hard to anticipate such a relationship. In thestochastic quantization scheme of Parisi & Wu ([3]) the correlation functions of an Eu-clidean d -dimensional field theory arise as equilibrium configurations, for large “fictitious”times, of the corresponding correlation functions of a d +1-dimensional field theory describedby a Fokker-Planck action. On the other hand, in AdS/CFT the generating functional forconnected correlation functions of a d -dimensional field theory arises as the appropriatelyrenormalized on-shell action of a d + 1-dimensional gravitational theory. Therefore, a con-nection could be established if the stochastically quantized action is somehow related to theboundary action of AdS/CFT and if the Fokker-Planck and the holographic bulk actionsare also related. Clearly, we also need to relate the stochastic ”time” to the holographicdirection. Then, such a relationship would imply a profound connection between stochasticprocesses and gravitation. [email protected] [email protected] [email protected] Related ideas have recently appeared in the lattice approach to quantum gravity [4].
1n this work we revisit the idea that stochastic quantization is related to AdS/CFT and wepresent an explicit example to support it. We start by sketching a simple formal correspon-dence between stochastic quantization and AdS/CFT. Namely, we show that the partitionfunction of stochastic quantization corresponds to an average over holographic partitionfunctions, if we identify the Fokker-Planck and the bulk actions, and also the initial clas-sical action with the holographic boundary effective action. Our explicit example involvesa conformally coupled scalar with φ interaction in fixed AdS . We show that the leadingterms in a large coupling expansion of the holographic effective action of the model, give the3d action of a massless scalar with φ interaction written, curiously, in an unconventionalmanner. Next, starting from the latter 3d action we use stochastic quantization to arrive atits corresponding 4d Fokker-Planck action. The leading terms in a large coupling expansionof that Fokker-Planck action give precisely (i.e. including the numerical coefficients), theinitial 4d action of a massless scalar with φ interaction. The latter action is actually equiva-lent to that of a conformally coupled scalar with φ interaction on AdS . Hence, our explicitexample demonstrates that the above 3d and 4d field theories are related both via AdS/CFTas well as via stochastic quantization. We consider our results as a strong indication thatstochastic quantization and AdS/CFT are intimately related. We then discuss the generalconditions under which such a relationship might arise focusing on the role of boundaryconditions. Finally, we point out that our results above have a geometric origin. Indeed,both the 4d and 3d actions that are involved in our example are merely disguised 4d and3d gravitational actions for conformally flat metrics. Then, the boundary condition that en-ables the calculation of the boundary effective action in AdS/CFT arises as the stationaritycondition for a system involving bulk and boundary gravity. This observation might helpto elucidate further the relationship between stochastic quantization and AdS/CFT in ourspecific example. It is not hard to sketch a formal connection between stochastic quantization (see Appendix Afor a condensed review) and AdS/CFT. The Boltzmann weight for a d -dimensional Euclideantheory of the scalar field φ ( ~x ) is (we set henceforth ~ = 1) P [ φ ] ≡ Z d e − S cl [ φ ] , with Z d = Z [ D φ ] e − S cl [ φ ] , (1)or equivalently R [ D φ ] P [ φ ] = 1. The extended scalar field φ ( ~x ) φ ( t, ~x ) satisfies theLangevin equation ∂φ ( t, ~x ) ∂t + κ δS cl [ φ ] δφ ( t, ~x ) = η ( t, ~x ) , (2)2here κ is a generic kernel. The source η ( t, ~x ) is a “white noise” defined by the followingpartition function and correlation functions Z = Z [ D η ] exp (cid:20) − κ Z − T d t Z d d x η ( t, ~x ) (cid:21) , (3) h η ( t, ~x ) i = 0 , (4) h η ( t , ~x ) η ( t , ~x ) i = 2 κδ d ( ~x − ~x ) δ ( t − t ) . (5)To make the connection with AdS/CFT we need to depart slightly from the standard stochas-tic quantization procedure where the fictitious ”time” interval is taken to be t ∈ [0 , T ]. Inthat case, one fixes the initial field configurations at t = 0 and lets the system evolve in t . The crucial point is then [3] that at T → ∞ the fields and their equal ”time” correla-tions functions relax to their equilibrium values - the latter being identified with properlyquantized configurations.Here instead we take the fictitious ”time” interval to be t ∈ [ − T,
0] such that starting fromany finite initial t = − T , the fields evolve via the Langevin towards their values at t = 0which we denote as φ − T (0 , ~x ). Sending then the initial ”time” − T → −∞ , the systemreaches thermal equilibrium at t = 0 i.e. the field configurations at t = 0 are properlyquantized. Accordingly, we have to impose an initial distribution for the field φ ( t, ~x ) at t = − T by means e.g. of a delta function as P t = − T [ φ ] = Π x (cid:8) δ d [ φ ( − T, ~x )] (cid:9) . (6)Here we have chosen a vanishing initial configuration having in mind to take a large T limit.The t=0 correlation functions, which are evaluated as stochastic averages over the whitenoise with Boltzmann weight that of (3), relax into those of the d -dimensional theory (1),namely lim T →∞ h φ − T (0 , ~x ) φ − T (0 , ~x ) ...φ − T (0 , ~x n ) i η = h φ ( ~x ) φ ( ~x ) ...φ ( ~x n ) i S cl . (7)One can turn the stochastic averages over the white noise into ”path integrals” over thescalar fields changing variables η φ . After a straightforward calculation (see e.g. [5]) andwith our choice of initial data we get Z = Z [ D φ (0)] e − S cl [ φ (0)] / Z [ D φ ] e −S F P , (8)where the Fokker-Planck action S F P and the ”path integral” measure are S F P = Z − T dt Z d d ~x " κ ˙ φ + κ (cid:18) δS cl δφ (cid:19) − κ δ S cl δφ , (9)[ D φ ] = Y − T
1. Hence, a path integral over φ , corresponding to the quantizationof φ , will generically produce inconsistencies such as negative probabilities or negative normstates. Nevertheless, there are known cases where φ is a normalizable mode as well andhence it can correspond to an operator ˜ O with dimension ˜∆ = d − ∆ > d/ −
1. A well-known example is the conformally coupled scalar field in 4-dimensions. In such cases theEuclidean functional W d [ φ ] itself can be used to construct a well defined Boltzmann weightfor a d -dimensional theory. In other words, W d [ φ ] can be interpreted as an effective action i.e. we can write W d [ φ ] ≡ Γ d [ φ ]. Now, the leading term of the effective action Γ d [ φ ] is aclassical action that we denote as I d [ φ ]. Then, it is natural to take a further step and define(see also [8]) Z ′ = Z [ D φ ] e − I d [ φ ] Z hol [ φ ] = Z [ D φ ] e − I d [ φ ] Z [ D φ ] φ e − S d +1 [ φ ] , (13)as an average, with weight I d [ φ ], of the holographic partition functions.We notice now a strong formal similarity between (8) and (13) provided we make the followingcorrespondences: S . Q . : AdS / CFT (14) S F P [ φ ] ↔ S d +1 [ φ ] S cl [ φ ] ↔ I d [ φ ]stochastic ”time” ↔ holographic direction . The determinant gives rise generically to infinities of the form δ d (0) that - when properly regularized -act as counterterms to some of the divergences that arise in the perturbative expansion [6].
4t a first glance, the above formal similarity may appear too optimistic. As we have previ-ously commented, any conventional d -dimensional action S cl would lead to a non-relativisticFokker-Planck action S F P . This seems irreconcilable with the standard AdS/CFT dictio-nary where both the bulk S d +1 and the boundary I d actions are relativistic. Furthermore,the boundary action is always conformal. This means that the presumed relationship be-tween stochastic quantization and AdS/CFT is non-generic. Indeed, the explicit examplewe present below is special and has a geometric origin. Nevertheless, we do believe that therelationship between stochastic quantization and AdS/CFT even for such special cases couldshed light into certain quantum properties of spacetime. To give precise meaning to the formal correspondence sketched above we consider the modelstudied in [9, 10] of a conformally coupled scalar with φ interaction on fixed Euclidean AdS I = Z d x √ g (cid:18) g µν ∂ µ φ ∂ ν φ + 12 m ℓ φ + λ φ (cid:19) , x µ = ( r, ~x ) , (15)where ℓ is the radius of AdS, determining the mass scale m ℓ ℓ = −
2. The dimensionlesscoupling λ is kept general. Upon introducing Poincar´e coordinates and rescaling the field as ds = ℓ r ( dr + d~x ) , g µν = Ω − ( x ) η µν , Ω( x ) = rℓ , φ = Ω( x ) f , (16)the action becomes I = I f + I div = Z ∞ dr Z d ~x (cid:18) η µν ∂ µ f ∂ ν f + λ f (cid:19) + Z d ~x f r (cid:12)(cid:12)(cid:12)(cid:12) ∞ . (17)The last term is divergent and needs to be renormalized by the addition of appropriatecounterterms (see e.g. [11]). Hence, this simple model essentially reduces to a masslesstheory on ”half” 4-dimensional flat space. The asymptotic boundary resides at r = 0 , ∞ andis isomorphic to S . As usual in the AdS/CFT [12], we remove the point at r = ∞ and weare left with a theory living on R (the space at r = 0). This is consistent with conformalinvariance. Physically sensible boundary conditions must imply this regularity condition ,namely the vanishing of the fields at the ”horizon” point r = ∞ of AdS. The reverse doesnot hold in general.For λ >
0, the equations of motion for the action (17) − ∂ µ ∂ µ f + λf = 0 , (18) Under the conformal inversion x µ ˆ I x ≡ x µ /x , scalar fields with dimension ∆ > φ ( x ) x − φ ( ˆ I x ). Hence, the finiteness of fields in the origin is preserved under conformal transformationsif the fields vanish at infinity. f ( r, ~x ) = k b − b + ( r + r ) + ( ~x − ~x ) , k = r λ . (19)The istantonic nature of this type of solutions requires λ to be finite. The parameter b determines the instanton size, while ( − r , ~x ) may be viewed as the coordinates of theinstanton center. The solution is regular for all r > r > b > r = 0 as f ( r, ~x ) = φ ( ~x ) + rφ ( ~x ) + O ( r ) , (20)where φ ( ~x ) and φ ( ~x ) are the two arbitrary data necessary to determine the general solutionof the 2nd order differential equation (18). It was shown in [9, 10] that evaluating (17) on-shell as a functional of φ ( ~x ) yields the 3-dimensional effective action for a composite scalaroperator with dimension ˜∆ = 1 as I on − shellf [ φ ] = − Γ d [ φ ] . (21)To achieve that we need to impose boundary conditions relating φ ( ~x ) to φ ( ~x ). Suchconditions can be generically expressed as F ( φ , φ ) = 0 for some functions F [11]. There isa general method to evaluate the boundary on-shell action with given boundary conditions,which is essentially the Hamilton-Jacobi method for field theory [13]. However, in our simplemodel we can take a more direct approach making use of the exact solution (19). Considerthe Hamiltonial formulation of our model which arises in a standard and simple way fromthe finite part in the rhs of (17). In this case we have I f = Z ∞ dr Z d ~x [ π ∂ r f − H ] , H = 12 (cid:18) π − ∂ i f ∂ i f − λ f (cid:19) . (22)Suppose now that on-shell the following condition holds H on − shell ( π, f ) = ∂ i V i ( f ) , (23)for some functional V i ( f ). This would greatly simplify the calculation of the on-shell value of(22) since, from the one hand it implies that the contribution of the Hamiltonian in the on-shell action is a total spatial derivative and hence vanishes, and from the other hand it givesan on-shell relationship between π and f such that the kinetic term in (22) can be written(at least term-by-term in some expansion) as a total r -derivative. Generically, the functional V i ( f ) in (23) can be calculated in a spatial derivative expansion and the coefficients are fixedrequiring consistency with the e.o.m. [13]. Here, we use input from the exact solution (19)on which the Hamiltonian density isˆ H = 12 (cid:18) ˆ π − ∂ i ˆ f ∂ i ˆ f − λ f (cid:19) = − ∂ i ∂ i ˆ f , (24) Notice that for a solution of of (18) to exist for λ > R . π = ∂ r f . Hence, if we are interested incalculating the on-shell action up to two spatial derivatives we could use (24) for genericsolutions of the e.o.m.As mentioned above, (24) is at the same time a boundary condition for the generic solution(20), since it relates φ ( ~x ) to φ ( ~x ). Explicitly the latter relation yields (we use now unhattedvariables denoting a general solution of the e.o.m.) π ( r, ~x ) = ± r λ f + ∂ i f ∂ i f − ∂ i ∂ i f = π ( ~x ) + r π ( ~x ) + O ( r ) , (25) π ( ~x ) = φ ( ~x ) = ± r λ φ + ∂ i φ ∂ i φ − ∂ i ∂ i φ . (26)An alternative but revealing way to express the boundary condition (26) is π ( ~x ) = ± r λ φ ( ~x ) " λ R / = ± r λ φ ( ~x ) (cid:18) λ R + O ( λ − ) (cid:19) , (27)where R ≡ R (cid:2) g = φ η (cid:3) = − φ − h ∂ i ln φ ( ~x ) ∂ i ln φ ( ~x ) + 4 ∂ i ∂ i ln φ ( ~x ) i , (28)is the scalar curvature of a conformally flat 3-dimensional metric g ij ( ~x ) = φ ( ~x ) η ij . We term(24) Hamiltonian boundary condition since it implies that the Hamiltonian density retainsthe form it has on an exact solution i.e. on a solution where both φ and φ are completelyfixed. Finally, we can substitute these results in (22) and consider a large λ expansion toobtain I on − shellf [ φ ] = − Γ[ φ ] = − r λ Z d ~x (cid:18) λφ + 1 φ ∂ i φ ∂ i φ + . . . (cid:19) , (29)where the dots denote terms O (1 /λ ). This effective action has the intriguing property thatit is a disguised form of a well-known action in 3d. Indeed, defining Φ = √ λ φ we findfrom (29) Γ[ φ ] = Γ[Φ] = Z d ~x (cid:18) ∂ i Φ ∂ i Φ + g Φ + . . . (cid:19) , (30)and the coupling constant g = (cid:0) λ (cid:1) . The terms shown in (29) and (30) are then the classicalaction of a 3d conformal theory. In the AdS/CFT example above the bulk theory is holographically related to a boundary ac-tion written in terms of elementary fields. The description passes through the non-canonical7d action (29). We will show now that this property can be understood, in the oppositedirection, in terms of stochastic quantization. Namely, we will associate the Fokker-Plankaction to the action of the bulk theory and the boundary effective action to S cl . The Hamil-tonian boundary conditions (24) will play a crucial part in this correspondence. This waywe will provide an explicit realization of the formal correspondence (14) between stochasticquantization and AdS/CFT.To implement our idea we apply the stochastic quantization procedure to the following 3dclassically conformal action S cl [ φ ] = 2 √ λ Z d ~x (cid:18) φ ∂ i φ∂ i φ + λφ (cid:19) . (31)Notice that our starting action is two times the holographic effective action we found in (29),in accordance with our identification in (14). As explained at the end of the previous Section,this unconventional action is related to a canonical φ model by a simple field redefinition.It is then straightforward to compute the 4d Fokker-Plank action inserting (31) in (9): (cid:18) δS cl δφ (cid:19) = 29 (cid:2) λφ − φ (cid:3) φ + 6 ∂ i φ∂ i φ + . . . (cid:3) , (32) S F P [ φ ] = Z dt Z d ~x (cid:20) ∂ µ φ∂ µ φ + λ φ + . . . (cid:21) . (33)Here we have chosen for definiteness κ = 1 / λ limit. Hence we see that the leading terms for large- λ in the stochastic quantizationof S cl [ φ ] reproduce precisely (i.e. including numerical coefficients!) the bulk action (17) forthe conformally coupled scalar field, if we identify the stochastic ”time” with the holographicdirection r .Now we would like to better understand why this connection takes place trying to givea unified picture. In order to do so we consider the general Fokker-Plank system (9) inHamiltonian formalism I = Z −∞ dt Z d ~x h ˙ φπ − H F P i H F P = 12 π − (cid:18) δS cl δφ (cid:19) + 12 δ S cl δφ ! , (34)where π = ˙ φ is the standard canonical momentum and again we chose κ = 1 /
2. We noticeat first that the theory lives in “half” 4-dimensional flat space. In section 3 we have shownthat the conformally coupled scalar model on
AdS can be reduced to a theory on this samespace. Then the 4-dimensional space generated by the addition of the fictitious time directionin stochastic quantization exactly reproduces the space-time associated to our initial bulksystem. More generally it is possible to think of adding suitable time direction(s) and allowfor more general fictitious time evolutions to reproduce different space-time structures to beassociated to other gravitational systems. Having this in mind, we would like to perform As mentioned before, the term (11) gives rise to δ d (0) infinities in field theory and is irrelevant here.
8n holographic analysis of the simple FP model in (34), now considered as our initial bulktheory. Therefore we evaluate the variation of the action on-shell: δI o.s. = − Z d ~x δφ π . (35)Now we consider a solution φ ( r, ~x ) of the Fokker-Plank equations of motion for which π (0 , ~x ) = − δS cl δφ . (36)It is clear from (35) that this selects a particular class of solutions with specific boundaryconditions such that we simply have I o.s. = S cl . This is the identification we were looking forin (14), with the correct overall coefficient. In this case the holographic generating functionalof connected diagrams for the boundary theory is directly related to the stochastichallyquantized action. In our scalar model, the generating functional can also be interpreted asan effective action. In any case, it is clear that the specific boundary condition (36) is thekey element to obtain an exact correspondence between the action S cl and the boundaryeffective action of holography.It is easy to see that the Hamiltonian boundary condition (24) we had to introduce in ourscalar example exactly coincides with (36). In fact we have14 (cid:18) δS cl δφ (cid:19) = 118 λ (cid:0) λ∂ i φ∂ i φ − λφ (cid:3) φ + 9 λ φ + . . . ) ≡ π (0 , ~x ) = ˙ φ , (37)and again we had to consider a large λ limit.One way to understand the boundary condition (36) from the point of view of stochasticquantization is the following. The field configuration is constrained to satisfy the Langevinevolution (2). After infinite time the field will eventually relax to the equilibrium configura-tion at the boundary t = 0. At the equilibrium, the role of quantum oscillation is played bythe noise average. Taking the average of the Langevin equation we directly read the bound-ary conditions in (36), which are now valid for the ”quantum” configuration of the field. Thekey role played by the choice of boundary conditions deserves some more analysis and inthe next section we will provide a geometrical interpretation for them from the holographicpoint of view. There is a simple, but possible far reaching geometrical origin behind the relationship be-tween the 4-dimensional φ theory and the 3-dimensional φ theory. Consider the Euclidean In the context of AdS/CFT it can be shown that an action such as (34) can arise considering a non-conformally coupled scalar on fixed AdS and taking the non-relativistic limit. R I (4) EH = − πG Z ∞ dr Z d ~x √ g ( R − ) . (38)It is well-known that in order to setup a proper Dirichlet problem for the metric at theboundary r = 0 , ∞ , i.e. in order that the variation of bulk on-shell action vanishes when wefix the metric at the boundary, we have to add to (38) boundary the Gibbons-Hawking term[14] I (4) GH = 18 πG Z ∂ M d ~x √ gg ij K ij , (39)where g ij ( ~x ) is the restriction of the bulk metric to the boundary and K ij ( ~x ) is the extrinsiccurvature. From now on we take all fields to vanish at r = ∞ , hence the boundary ∂ M = R is at r = 0. Consider conformally flat metrics g µν ( x ) = ϕ ( x ) η µν . An explicit calculationgives I = I (4) EH + I (4) GH = − π Z d x (cid:18) η µν ∂ µ φ∂ ν φ − λ φ (cid:19) , (40)where we have defined ϕ ( x ) = p G φ ( x ) , λ = G Λ = − λ . (41)The GH term cancelled exactly the boundary term arising from I (4) EH . As we have mentionedbefore, (40) is equivalent to the 1st-order action (22). Hence, its on-shell variation yields δI on − shell = 34 π Z R d ~x δφ ( ~x ) π ( ~x ) , (42)where the boundary values of the canonical variables have been defined in Section 3.However, if we do not wish to fix the boundary metric, the only way to make (40) stationaryis to impose (Neumann) boundary conditions on π ( ~x ) e.g. the above case we should require π ( ~x ) = 0. This gives us intriguing possibilities, for example by adding the appropriateboundary functionals we can impose boundary conditions satisfied by exact non-perturbativesolutions of the bulk equations of motion.With this in mind we consider extending (40) by just the 3d gravity in the boundary withaction I (3) EH = − πG Z R d ~x p ˆ g ( R − ) . (43)Since there are no boundary terms now, the variation of (43) gives δI (3) EH = − πG Z d ~x p ˆ gδ ˆ g ij (cid:18) R ij −
12 ˆ g ij R + Λ ˆ g ij (cid:19) . (44)Take now the boundary metric to be conformally flat and related to the bulk one asˆ g ( ~x ) ij = ϕ ( ~x ) η ij , (45)10nd using (41) we find after some algebra δI (3) EH = 3Λ G π s G G Z R d ~x δφ ( ~x ) (cid:18) − G R (cid:19) φ ( ~x ) , (46)where R is given by (28). Quite remarkably, the Hamiltonian boundary condition (24)arises as the stationarity condition for the total action I = I + I (3) EH , (47)which is nothing but the sum of bulk and boundary gravity in 4d and 3d. Matching thecoefficients gives the following relationships between 4d and 3d quantities λ = − Λ G , λ G = G , Λ = 23 Λ , (48)hence we need a negative cosmological constant in the boundary as well.To summarize, we presented a rather simple example of an explicit correspondence betweena 4d bulk and a 3d boundary theory. We have argued that it gives support to our claimthat there is a relationship between stochastic quantization and AdS/CFT, at least undercertain conditions. We pointed out that our results have a geometric origin since they canbe obtained by the coupling of 4d and 3d conformal gravities.We need not stress again that our results must be taken only as an incentive to study furtherthe relationship between stochastic quantization and holography . In particular, one wouldneed to understand further the subleading term in λ and how they match between 3d and 4d,where we suspect we could find the evidence for the presence of Gaussian noise. Moreover,one could study correlation functions and extend our analysis to gauge fields and gravity.Such issues are currently under study. Acknowledgments:
The work of ACP was partially supported by the research grant with KA 2745 from the Universityof Crete. A. C. P. would like to thank G. Kofinas and G. Semenoff, for useful discussions, and J.Ambjorn and P. Damgaard for interesting correspondence.
AppendixA Stochastic Quantization
In many ways stochastic quantization may be viewed as an application of Stoke’s theoremin field theory [15]. Consider a ( d + 1)-dimensional manifold X with a non empty boundary For a related but slightly different approach see [17]. X . Then, for any d -form Ω [ d ] the following equality holds Z X dΩ [ d ] = Z ∂ X Ω [ d ] . (49)Now suppose that Ω [ d ] is the lagrangian of a d -dimensional Euclidean theory with action S cl [ φ ] involving a single scalar field φ , S cl [ φ ] = Z ∂ X Ω [ d ] [ φ ; ~x ] , (50)where ~x are the coordinates of the boundary. The above imply that (49) provides an al-ternative definition of the d -dimensional theory S cl [ φ ] using d + 1-forms that depend on the d + 1 coordinates { t, ~x } . However, despite the fact that we have added a new dimension -the stochastic ”time” coordinate t - the physical content of the theory still resides on theboundary. The latter property implies the presence of a topological invariance in the ( d + 1)-dimensional description of the theory: the action cannot depend on the specific extension ofthe field in the bulk. Explicitly, the d +1-dimensional action in terms of the field Φ ∈ C ∞ ( X )which is the extension on X of φ ∈ C ∞ ( ∂ X ) constrained by Φ (cid:12)(cid:12)(cid:12) ∂ X = φ , is defined as S d +1 [Φ] = Z X dΩ [ d ] . (51)The topological invariance can thus be phrased as S d +1 [Φ + δ Φ] = S d +1 [Φ] for any variationvanishing on the boundary δ Φ (cid:12)(cid:12)(cid:12) ∂ X = 0. The idea is to gauge-fix such a symmetry a-la BRST.To be more concrete consider a manifold X with a cylindrical structure, X = B × [0 , T ] wherethe base manifold B is parameterized by the coordinates ~x and the additional coordinate t runs from 0 to T . The boundary is given by the d -chain consisting in the two copies of B at t = 0 and at t = T with the correct orientations, ∂ X = B T − B . Explicitly S d +1 [Φ] = (cid:18)Z B T − Z B (cid:19) Ω [ d ] = S cl [ φ T ] − S cl [ φ ] = ˜ S cl [ φ ] , (52)where φ = Φ (cid:12)(cid:12)(cid:12) B and φ T = Φ (cid:12)(cid:12)(cid:12) B T . For Ω [ d ] a pure d -form on the base manifold we haveΩ [ d ] = 1 n ! Ω i ...i d ( t, ~x )d x i ∧ · · · ∧ d x i d , dΩ [ d ] = d t ∧ ˙Ω [ d ] . (53)But since Ω [ d ] depends on t only through the extension Φ of the scalar field we simply havethat S d +1 [Φ] = Z X d t ∧ ˙Ω [ d ] = Z X d t ∧ ˙Φ δS cl δ Φ . (54)Next we choose a convenient gauge-fixing condition E [Φ; t, ~x ] corresponding to a particularchoice of the extension Φ of the scalar field. For example, we may consider an instantonicgauge-fixing condition given by the Langevin equation E [Φ; t, ~x ] = ˙Φ( t, ~x ) ε [ d ] ( ~x ) + κ δS cl δ Φ( t, ~x ) , (55)12here α is a constant kernel and ε [ d ] is the volume form on the boundary. Classically wewould like to take E [Φ; t, ~x ] = 0 but is not a good quantum condition. Hence the idea is tolet it hold only as an average, h E [Φ; t, ~x ] i = 0 and hence we are led to introduce a white noise η as a source for the Langevin equation, E [Φ; t, ~x ] = η ( t, ~x ) ε [ d ] ( ~x ). We thus introduce theghost ψ which is an anti-commuting scalar, the corresponding anti-ghost ¯ ψ , a source η forthe gauge fixing condition which is typically given by a white noise, and a BRST nilpotentoperator Q such that Q Φ = ψ , Q ¯ ψ = η, Qψ = 0 , Qη = 0 . (56)It is easy to show that the action S d +1 [Φ] is invariant under the fermionic transformation δ ǫ ≡ ¯ ǫQ provided that ψ satisfies periodic boundary conditions . We thus add a Q -exactterm in the action which does not modify the δ ǫ invariance of the theory S d +1 [Φ] → S d +1 [Φ] + ( QS cl ) [Φ , η, ψ, ¯ ψ ] , with S cl [Φ , η, ¯ ψ ] = 12 κ Z X ¯ ψ (cid:0) ηε [ d ] − E (cid:1) . (57)Doing so we arrive at( QS cl ) [Φ , η, ψ, ¯ ψ ] = 12 κ Z X d t ∧ (cid:20) η (cid:0) ηε [ d ] − E (cid:1) + 2 ¯ ψ δEδ Φ ψ (cid:21) = 1 κ Z X d t ∧ (cid:20)
12 ( η − ∗ d E ) + 12 ( ∗ d E ) (cid:21) ε [ d ] ++ 1 κ Z Z X d t ∧ ¯ ψ ( t, ~x ) (cid:20) δ d ( ~x − ~y ) ε [ d ] ( ~y ) ∂ t + κ δ S cl δ Φ( t, ~x ) δ Φ( t, ~y ) (cid:21) ψ ( t, ~y ) . (58)Consider then the partition function for the theory Z [ φ , φ T ] = Z D Φ D η D ψ D ¯ ψ e − { S d +1 [Φ]+( QS cl )[Φ ,η,ψ, ¯ ψ ] } . (59)If κ > S eff [Φ , ψ, ¯ ψ ] = Z X d t ∧ (cid:20) ˙Φ δS cl δ Φ + 12 κ ( ∗ d E ) ε [ d ] (cid:21) + F [Φ , ψ, ¯ ψ ] , (60)where F [Φ , ψ, ¯ ψ ] is given by the last line of (58). Then, using( ∗ d E ) = ˙Φ + 2 κ ˙Φ ∗ d δS cl δ Φ + κ (cid:18) ∗ d δS cl δ Φ (cid:19) , (61)we obtain S eff [Φ , ψ, ¯ ψ ] = Z X d t ∧ " κ ˙Φ ε [ d ] + 2 ˙Φ δS cl δ Φ + κ (cid:18) ∗ d δ S cl δ Φ (cid:19) ε [ d ] + F [Φ , ψ, ¯ ψ ] . (62) For simplicity we will assume the fields ψ and ¯ ψ to vanish on the boundary. S eff [Φ , ψ, ¯ ψ ] = 1 κ Z X d t ∧ "
12 ˙Φ ε [ d ] + κ (cid:18) ∗ d δS cl δ Φ (cid:19) ε [ d ] + F [Φ , ψ, ¯ ψ ] . (63)The fermonic part of the action can be formally integrated out to give Z D ψ D ¯ ψ e − F [Φ ,ψ, ¯ ψ ] = det (cid:18) δEδ Φ (cid:19) ∼ exp (cid:20) κ Z X d t ∧ δ S cl δ Φ( t, ~x ) (cid:21) − exp (cid:20) − κ Z X d t ∧ δ S cl δ Φ( t, ~x ) (cid:21) . (64)In this step one must be consistent with the choice of periodicity for the fields on theboundary. Collapsing the forms we can then write the end result as Z = Z D Φ h e − S + F P − e − S − F P i , S ± F P [Φ] = Z d d +1 x " κ ˙Φ + κ (cid:18) δS cl δ Φ (cid:19) ± κ δ S cl δ Φ . (65)It’s important to stress that this derivation of the partition function for the FP system leadsto the supersymmetric realization of stochastic quantization [16]. Supersymmetry followsfrom the choice of periodic boundary conditions for the fields ψ and Φ. It’s straightforwardto reintroduce in the partition function the dependence on general choices of boundary valuesfor the fields and consider only forward propagation in time. In this more general settingone is free to fix a given initial configuration for the fields and let them evolve according tothe Langevin equation. In Section 2 we explicitly chose an initial condition for the field φ generally breaking supersymmetry. References [1] D. Polyakov, Class. Quant. Grav. , 1979 (2001) [arXiv:hep-th/0005094].[2] G. Lifschytz and V. Periwal, JHEP (2000) 026 [arXiv:hep-th/0003179].[3] G. Parisi and Y. s. Wu, Sci. Sin. , 483 (1981).[4] J. Ambjorn, R. Loll, W. Westra and S. Zohren, Phys. Lett. B , 359 (2009)[arXiv:0908.4224 [hep-th]].[5] P. H. Damgaard and H. Huffel, Phys. Rept. , 227 (1987).[6] J. Zinn-Justin, Nucl. Phys. B (1986) 135.[7] P. Horava, Phys. Rev. D (2009) 084008 [arXiv:0901.3775 [hep-th]].[8] G. Compere and D. Marolf, Class. Quant. Grav. , 195014 (2008) [arXiv:0805.1902[hep-th]]. 149] S. de Haro and A. C. Petkou, JHEP (2006) 076 [arXiv:hep-th/0606276].[10] S. de Haro, I. Papadimitriou and A. C. Petkou, Phys. Rev. Lett. (2007) 231601[arXiv:hep-th/0611315].[11] I. Papadimitriou, JHEP (2007) 075 [arXiv:hep-th/0703152].[12] E. Witten, Adv. Theor. Math. Phys. (1998) 253 [arXiv:hep-th/9802150].[13] J. Parry, D. S. Salopek and J. M. Stewart, Phys. Rev. D (1994) 2872[arXiv:gr-qc/9310020].[14] G. W. Gibbons and S. W. Hawking, Phys. Rev. D (1977) 2752.[15] L. Baulieu and B. Grossman, Phys. Lett. B (1988) 351.[16] E. Gozzi, Phys. Rev. D (1983) 1922.[17] R. Dijkgraaf, D. Orlando and S. Reffert, Nucl. Phys. B824