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the and same space-time the the by ter of described invarianc in positioning diffeomorphism reformulated the Lattice be pos of can are derivatives. action points lattice se lattice space-time and discrete the the a When manifold, on uous fields. regularized lattice is bosonic gravity or quantum for integral epooealtiecutrato iemrhs symmetry diffeomorphism of counterpart lattice a propose We .INTRODUCTION I. atc iemrhs invariance diffeomorphism Lattice hlspewg1,D610Heidelberg D-69120 16, Philosophenweg ntttfu hoeicePhysik Theoretische f¨ur Institut nvri¨tHeidelbergUniversit¨at .Wetterich C. incnb vroei hsapoc,adi suitable the the not here a are follow if objects we geometrical and reason where ac- approach, approach this the alternative For not this of is exists. in It limit boundedness overcome continuum of be [10]. problem simplices can the of tion if edges a however, under of action lengths clear, the the of invariance of the change Regge- diffeomorphism on lattice In based of is 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“fundamental” degrees of freedom in the functional inte- a lattice equivalent of diffeomorphism symmetry for actions gral. Geometrical objects are then not available for a for- of the type S2 (In d dimensions those are constructed as mulation of diffeomorphism invariance and an alternative integrals over d-forms.) We will relate this to different pos- has to be followed. sibilities of positioning the abstract lattice pointsz ˜ on the In this paper we propose a general form of lattice quan- coordinate manifold. tum gravity where the functional integral is based on lattice In the continuum, the invariance of the action under gen- fields, i.e. variables (˜z) for every pointz ˜ of a space-time eral coordinate transformations states that it should not lattice. The basic fieldsH may be bosons or fermions and matter if fields are placed at a point x or some neighbor- they behave as “scalars” with respect to diffeomorphism ing point x + ξ, provided that all fields are transformed symmetry. This makes the formulation of the measure simultaneously according to suitable rules. In particular, for the functional integral straightforward. Lattice spinor scalar fields (x) are simply replaced by (x + ξ). After gravity [4] belongs to this class of models. What is needed an infinitesimalH transformation the new scalarH field ′(x) is a lattice equivalent of diffeomorphism symmetry of the at a given position x is related to the original scalarH field action. This “lattice diffeomorphism invariance” should be (x) by a special property of the lattice action that guarantees dif- H ′ µ feomorphism symmetry for the continuum limit and the (x)= (x ξ)= (x) ξ ∂µ (x). (3) quantum effective action. Its formulation is the key topic H H − H − H of this paper. Diffeomorphism symmetry states that the action is the The type of models that we consider are not based on same for (x) and ′(x). Implicitly the general coordi- a fluctuating geometry. The fluctuating degrees of free- nate transformationsH H assume that the same rule for forming dom are fields that “live” in some manifold. Since there derivatives is used before and after the transformation. is a priori no geometry and metric the manifold is a pure We want to implement a similar principle for a lattice for- coordinate manifold and we choose cartesian coordinates mulation. For this purpose we associate the abstract lattice xµ. This coordinate manifold should not be confounded pointsz ˜ = (˜z0, z˜1, z˜2, z˜3), with integerz ˜µ, with points on a with physical space-time - the latter emerges only once a manifold. As mentioned before, we consider here a piece of d µ 0 1 d−1 dynamical metric is found which determines geometry and Ê with cartesian coordinates x = (x , x ,...x ), but topology. In general, physical space-time will be curved, we do not specify any metric a priori, nor assume its exis- µ and physical distances do not coincide with the coordinate tence. A mapz ˜ xp (˜z) defines the positioning of lattice d → distances in Ê . As an example, the geometry of a sphere points in the manifold. We can now compare two different µ µ can be described by an appropriate metric for cartesian positionings, as a regular lattice xp (˜z)=˜z ∆, or some ir- ′µ coordinates, where the behavior for x is related to regular one with different coordinates xp (˜z). In general, the pole opposite to the one at x =| 0.|(A → ∞full description there are many different possible positionings for the same can be achieved by a second coordinate patch covering the abstract lattice pointsz ˜. In particular, we can compare missing pole. Knowing the metric everywhere except for two positionings related to each other by an arbitrary in- ′µ µ µ the missing pole the completion of topology is unique - we finitesimal shift xp = xp + ξp (x). We emphasize that the require that physics is not affected by removing or adding notion of an infinitesimal neighborhood requires a contin- a point of a manifold. Our setting resembles is this aspect uous manifold and cannot be formulated for the discrete the emergence of geometry from correlation functions dis- abstract lattice pointsz ˜. cussed in ref. [11].) Having at our disposal only fields on Positioning of the lattice points on a manifold is also re- a coordinate manifold our setting is very close to standard quired for the notion of a lattice derivative. One can define lattice field theories as lattice gauge theories or scalar field the meaning of two neighboring lattice pointsz ˜1 andz ˜2 in theories where the metric does not fluctuate. an abstract sense. (This involves some type of “incidence The issue of diffeomorphism symmetry in the absence of matrix” but no manifold.) A lattice derivative of a field a metric may be demonstrated for the continuum action of will involve the difference between field values at neigh- an abelian gauge theory. We may compare two (euclidean) boring sites, (˜z ) (˜z ). For the definition of a lattice H 1 − H 2 actions involving the gauge field strength Fµν , derivative ∂ˆµ we need, in addition, some quantity with dimension ofH length. This is only provided by the position- µρ νσ S1 = Fµν Fρσ δ δ (1) ing on the manifold. For our cartesian coordinates we use Zx a simple definition of the lattice derivative by requiring for suitable pairs (˜z1, z˜2) the relation and µ µ ˆ (˜z1) (˜z2)= xp (˜z1) xp (˜z2) ∂µ , (4) S = ϕ2F F ǫµνρσ . (2) H − H − H 2 Z µν ρσ
x with summation over repeated indices implied. We select The first one involves a fixed inverse metric δµν and is locally d pairs for a model in d dimensions, and solve the not diffeomorphism symmetric, while the second does not system of d independent equations (4) for the d different involve any metric and is diffeomorphism symmetric. (A derivatives ∂ˆµ . This yields the local derivatives ∂ˆµ (˜y) 2 H H squared scalar field ϕ is employed for preventing S2 to be in terms of suitable cartesian distances and differences of a total derivative.) The task of the present paper is to find lattice variables. 3

Finally, the positioning ofz ˜ on a manifold is a crucial mation. In a sense, we treat continuous functions similar to ingredient for the formulation of a continuum limit, where numerical simulations where they have to be specified by a one switches from (˜z) to (x) and derivatives thereof. finite amount of information. In our formulation diffeomor- H H If the lattice action is originally formulated in terms of phism transformations are nothing else than the possibility (˜z) only, its expression in terms of lattice derivatives will to move the lattice points, where the information about the inH general depend on the positioning, since the relation function is given by its value at these points, within a man- between (˜z) and lattice derivatives (4) depends on the ifold. Diffeomorphism invariance is realized if the action in positioning.H We can now state the principle of “lattice terms of fields and their derivatives does not notice this diffeomorphism invariance”. A lattice action is lattice dif- change in positions. feomorphism invariant if its expression in terms of lattice This paper is organized as follows. In sects. II-VIII derivatives does not depend on the positioning of the lat- we discuss lattice diffeomorphism invariance in two dimen- tice points. For infinitesimally close positionings the lattice sions. The particular features of two-dimensional gravity action is independent of ξp. (For a mathematically unam- do not play a role for the issue of diffeomorphism invari- biguous definition of lattice diffeomorphism invariance we ance. The discussion in these sections can be extended in a need a few further specifications. This is best done by in- straightforward way to four dimensions. Notation and ge- vestigating below particular examples.) ometric visualization are simplest in two dimensions, how- In short, lattice diffeomorphism invariance has the sim- ever. In sect. II we introduce our concept of lattice dif- ple intuitive meaning that it does not matter how the ab- feomorphism invariance for a scalar field theory, typically stract lattice points are placed on the manifold. A similar a non-linear σ-model. In the following sections III-VII we interpretation is possible for diffeomorphism symmetry in show that lattice diffeomorphism invariance induces diffeo- the continuum: it does not matter how (scalar) fields are morphism symmetry for the continuum limit of the “classi- placed on a given coordinate manifold. This makes the cor- cal action”, and, most important, for the quantum effective respondence between lattice diffeomorphism invariance and action. diffeomorphism symmetry in the continuum rather natu- In sect. III we introduce interpolating fields and a suit- ral. (Diffeomorphism symmetry in the continuum can be able definition of derivatives as a central tool for this argu- viewed from different aspects, for example as a symmetry ment. The interpolation fields are used in sect. IV in order transformation among field variables. Not all these aspects to establish the diffeomorphism symmetry of the contin- find a direct correspondence in lattice diffeomorphism in- uum limit of the classical action. Sect. V turns to the variance.) quantum effective action for the scalar fields and estab- The usual discussion of lattice theories considers implic- lishes its diffeomorphism symmetry in the continuum limit. itly a given fixed positioning, for example a regular lat- In sect. VI we introduce the metric as the expectation value tice. We investigate here a much wider class of position- of a collective field. The quantum effective action for the ings. Only the comparison of different positionings allows metric, its diffeomorphism symmetry and the covariance of the formulation of lattice diffeomorphism invariance in our the gravitational field equations are discussed in sect. VII. setting. In sect. VIII we extend our discussion to lattice spinor We will see that the continuum limit of a lattice diffeo- gravity in two dimensions. Four dimensional models, both morphism invariant action exhibits diffeomorphism sym- for fundamental scalars and fermions, are discussed in sect. metry. Also the quantum effective action is invariant un- IX. This section is kept short since besides a more involved der general coordinate transformations. This extends to algebra no new concepts are needed. We draw our conclu- the effective action for the metric which appears in our set- sions in sect. X. ting as the expectation value of a suitable collective field. We will show that lattice diffeomorphism invariance im- plies diffeomorphism symmetry for the quantum effective II. LATTICE DIFFEOMORPHISM INVARIANCE action of the metric in the continuum limit. The gravita- IN TWO DIMENSIONS tional field equations are therefore covariant with a similar general structure as in general relativity. Basic construction principles of a lattice diffeomorphism In order to show diffeomorphism symmetry of the con- invariant action can be understood in two dimensions. We 0 1 tinuum limit and the effective action we use the concept label abstract lattice points by two integersz ˜ = (˜z , z˜ ), 0 1 of interpolating functions. We define a version of partial withz ˜ +z ˜ odd. We first consider models with scalar derivatives of interpolating functions that takes into ac- fields. We will be mainly interested in continuous fields, count the lack of knowledge of details of the interpolation. but the basic issue of positioning independence can already At the positions of lattice cells these derivatives equal the be understood for discrete fields. lattice derivatives. For smooth interpolating fields they co- Our simplest example employs three real bosonic discrete lattice fields k(˜z) = 1 , k = 1, 2, 3. For the formula- incide with the standard definition of partial derivatives. H ± In this view, the lattice does not reflect a basic discrete- tion of the functional integral we define the measure as a ness of space and time. It rather expresses the fact that product of independent sums only a finite amount of information is available in practice, and that arbitrarily accurate continuous functions are an = . (5) Z DH idealization since they require an infinite amount of infor- Yz˜ Yk Hk(˜Xz)=±1 4

This measure does obviously not depend on the position- The volume corresponds to the surface inclosed by straight ing. For a finite number of lattice points the partition lines joining the positions of the four lattice pointsx ˜j (˜y) function is a finite sum of the cell in the orderx ˜1, x˜2, x˜4, x˜3. For simplicity we restrict the discussion to “deformations” of the regular lat- µ µ Z = exp( S), (6) tice, xp =z ˜ ∆, where V (˜y) remains always positive and Z DH − the path of one point during the deformation never touches another point or a straight line between two other points and therefore well defined for finite S. at the boundary of the surface. We use the volume V (˜y) We define the action as a sum over local cells located at for the definition of an integral over the relevant region of y˜ = (˜y0, y˜1), withy ˜µ integer andy ˜0 +˜y1 even, the manifold

S = (˜y). (7) d2x = V (˜y), (10) X L Z y˜ Xy˜ Each cell consists of four lattice points that are nearest where we define the region by the surface covered by the neighbors ofy ˜, denoted byx ˜j (˜y), j = 1 ... 4. Their lat- cells appearing in the sum. tice coordinates arez ˜ x˜ (˜y) = (˜y0 1, y˜1) , z˜ x˜ (˜y) = We next express the action (7), (8) in terms of average 1 − 2 (˜y0, y˜1 1) , z˜ x˜ (˜y) = (˜
y0, y˜1 + 1), andz ˜ x˜ (˜y)
= fields in the cell − 3 4 (˜y0 + 1, y˜1). The local
term (˜y) involves lattice fields
1 on the four sites of the cell thatL we denote by (˜x ). We k(˜y)= k x˜j (˜y) (11) k j H 4 X H choose H j

α klm and lattice derivatives (4) associated to the cell (˜y)= ǫ k(˜x )+ k(˜x )+ k(˜x )+ k(˜x ) L 48 H 1 H 2 H 3 H 4   ˆ 1 1 1 ∂0 k(˜y) = (x3 x2) k(˜x4) k(˜x1) l(˜x4) l(˜x1) [ m(˜x3) m(˜x2) + c.c., (8) H 2V (˜y)n − H − H ×H − H  H − H 
1 1 such that the action specifies a general Ising type model. (x x ) k(˜x3) k(˜x2) , − 4 − 1 H − H (The complex conjugate is omitted for real fields and is
o 1 0 0 needed only for the complex fields discussed later.) At ∂ˆ k(˜y) = (x x ) k(˜x ) k(˜x ) 1 2V (˜y) 4 1 3 2 this point no notion of a manifold is introduced. We spec- H n − H − H
0 0 ify only the connectivity of the lattice by grouping lattice (x3 x2) k(˜x4) k(˜x1) . (12) pointsz ˜ into cellsy ˜ such that each cell has four points − − H − H
o and each point belongs to four cells. This defines neigh- For the pairs (˜xj1 , x˜j2 ) = (˜x4, x˜1) and (˜x3, x˜2) the lattice boring cells as those that have two common lattice points. derivatives obey Neighboring lattice points belong to at least one common µ µ ˆ k(˜xj1 ) k(˜xj2 ) = (x x )∂µ k(˜y). (13) cell. H − H j1 − j2 H Our setting can be extended to continuous fields, for In terms of cell average and lattice derivatives all quantities example by replacing the constraint k(z) = 1 by in (˜y) depend on the cell variabley ˜ or the associated po- 2 H ± Σk (z) = 1. This is a generalized non-linear sigma- L µ Hk sition of the cell xp (˜y) that we take somewhere inside the model or Heisenberg model. The partition function is surface of the cell, the precise assignment being unimpor- symmetric with respect to global SO(3)-rotations between tant at this stage. In this form we denote (˜y) by ˆ(˜y; xp) the three components , provided we replace the func- L L k or ˆ(x; x ), where ˆ(x) only depends on quantities with tional measure (5) by theH standard SO(3)-invariant mea- p supportL on discreteL points in the manifold corresponding sure for everyz ˜. A generalization to unbounded fields, to the cell positions. We indicate explicitly the dependence e.g. < (˜z) < , is more problematic because of k on the choice of the positioning by the argument x . the unboundedness−∞ H of the∞ action (8) even for a finite num- p The action appears now in a form referring to the posi- ber of lattice sites. A further generalization uses complex ∗ tions on the manifold fields k with constraint (˜z) k(˜z) = 1. For a suitable H Hk H invariant measure the action and partition function are in- 2 2 S(xp)= d x ¯(˜y; xp)= d x ¯(x; xp), (14) variant under unitary transformations of the group SU(3). Z L Z L The action (7), (8) is real and we concentrate on real α. with We now proceed to an (almost) arbitrary positioning of 2 ˆ the lattice points on a piece of Ê by specifying positions ¯ ¯ (˜y; xp) µ (˜y; xp)= (x; xp)= L . (15) xp (˜z). This associates to each cell a “volume” V (˜y), L L V (˜y; xp) Lattice diffeomorphism invariance states that for fixed 1 µ µ ν ν V (˜y)= ǫµν (x4 x1 )(x3 x2 ), (9) (˜y) and ∂ˆµ (˜y) the ratio ¯(˜y; xp) in eq. (15) is inde- 2 − − H H L pendent of the positioning, or independent of ξp for in- µ ′ with ǫ = ǫ = 1 and x shorthands for the posi- finitesimal changes of positions x = xp + ξp, 01 − 10 j p µ µ tions of the sitesx ˜j of the celly ˜, i.e. xj = xp z˜ x˜j (˜y) . ¯(˜y; xp + ξp)= ¯(˜y; xp) , S(xp + ξp)= S(xp). (16) 
 L L 5

This definition provides the necessary specification for the states that the action remains invariant with respect to this precise meaning of lattice diffeomorphism invariance. map if the change of the cell volume is taken into account The ξp-independence of ¯(˜y; xp) means that the depen- properly. These aspects are discussed in more detail in the L dence of V (˜y; xp) and ˆ(˜y; xp) on the positioning xp must appendix. cancel. Inserting eqs. (11),L (13) in eq. (8) yields

α klm µν ˆ(˜y)= ǫ V (˜y) k(˜y)ǫ ∂ˆµ l(˜y)∂ˆν m(˜y)+c.c., (17) III. INTERPOLATING FIELDS L 12 H H H and we find indeed that the factor V (˜y) cancels in ¯(˜y)= As a key result of this note we state that the contin- L ˆ(˜y)/V (˜y). Thus the action (7), (8) is lattice diffeomor- uum limit of both the action and the quantum effective Lphism invariant. This property is specific for a certain action is diffeomorphism symmetric if the lattice action is class of actions - for example adding to ǫklm a quantity lattice diffeomorphism invariant. We will specify the pre- sklm which is symmetric in l m would destroy lattice cise meaning of this statement and detail our argument in diffeomorphism symmetry. ↔ the following. A central ingredient is the observation that For all typical lattice theories the formulation of (˜y) diffeomorphism transformations can be realized by reposi- only in terms of next neighbors and common cells (notL us- tionings of the lattice variables, without transforming the ing a distance) does not refer to any particular positioning. lattice variables themselves. However, once one proceeds to a positioning of the lattice We first want to show that the continuum limit of a points and introduces the concept of lattice derivatives, the lattice diffeomorphism invariant model exhibits diffeomor- independence on the positioning of ¯(˜y; xp) for fixed (˜y) phism symmetry. The continuum limit will be defined in L H and ∂ˆµ (˜y)) is not shared by many known lattice theories. terms of interpolating functions. Consider a system of func- For example,H standard lattice gauge theories are not lattice tions f(x) that are completely determined by their values diffeomorphism invariant. This is also the basic difference fn = f(xn) at points xn in Ê2, while they interpolate in some specifiedN way inbetween those points. We take to of the actions S1 and S2 (eqs. (1), (2)) mentioned in the N introduction: only for S2 it is possible to find a lattice dif- be equal to the number of lattice points. We further define feomorphism invariant action such that it is obtained in which points are neighbors in a way that allows a two- the continuum limit. dimensional ordering similar to the lattice pointsz ˜. Be- We may also introduce higher lattice derivatives and ex- yond the notion of neighborship the position of the points press xn in Ê2 is arbitrary. We identify the points xn with the arbitrary positions xp(˜zn) of the lattice pointsz ˜n in N ˜ k(˜x1) k(˜x2) k(˜x3)+ k(˜x4)= k(˜y) Ê . The information contained in some lattice field f(˜z) H − H − H H EH 2 2 ˆ2 ˆ ˆ ˜ is equivalent to the values fn , with fn = f˜(˜zn). = c1(x)(∂0 ∂1 ) k(˜y)+ c2(x)∂0∂1 k(˜y). (18) { } − H H The map between the lattice functions f˜(˜z) and the in- The functions c1 and c2 depend on the positions xj of terpolating fields f(x) depends both on the choice of inter- the cell pointsx ˜j . (For the regular lattice one has c1 = polation and on the selection of points xn = xp(˜zn). This 2 ∆ ,c2 = 0.) Using eqs. (11), (13) we can write the lat- reflects the fact that a complete specification of the inter- tice fields k(˜z) in terms of s,k(˜y) = k(˜y)), ∂ˆ k(˜y), polating function f(x) requires the specification of (i) the H K {H 0H ˆ ˆ2 ˆ2 ˆ ˆ points xn, (ii) the values f(xn), and (iii) an interpolation ∂1 k(˜y), (∂0 ∂1 ) k(˜y), ∂0∂1 k(˜y) ,s = 1 ... 5. One canH therefore− expressH an arbitraryH (˜}y) in terms of lattice description. Even for a given prescription of interpolation derivatives. Lattice diffeomorphismL invariance is realized the function f(x) depends on the locations xn where it ˜ if V (˜y) (˜y), once expressed in terms of s,k(˜y), does not takes the values fn. For a given lattice function f(˜z) two L K depend on the positions xp(˜z). We note that the lattice different positionings of the lattice points in Ê2 will lead variables s,k(˜y) are not independent. First, for a given to two different interpolating functions f(x). K positioning only one particular linear combination of In contrast to the values of f at the points xn the in- K4 and appears and we may set the orthogonal one to zero. formation about the behavior of f(x) for x = xn is not K5 6 Second, a given (˜z) can be expressed in terms of s(˜y) for contained in the values of the lattice field f˜(˜z ). It de- H K n all four cellsy ˜ to whichz ˜ belongs. This constraint relates pends on the specific interpolation which is chosen. This variables s(˜y) in different cells to each other. Imposing interpolation may be continuous and differentiable, or only K these constraints one may consider the change from (˜z) piecewise differentiable in cells or parts of cells. For the H to s(˜y) as a change of basis for the lattice variables. For lattice regularized theory nothing should depend on the K our particular action (8) the higher derivatives 4 and 5 particular choice of interpolation since the latter involves K K are absent. In this case we can restrict s =1 ... 3. information not available in the lattice model. In this sense From this perspective we consider a family of basis space becomes “fuzzy”. All interpolations should represent changes for the lattice variables (˜z) s(˜y). The spe- the same physics. H → K cific basis s(˜y) is determined by a specific positioning of For a given choice of interpolation we next define K the lattice points. A repositioning of the lattice points can the notion of an “interpolation derivative” for an in- be considered as a map between two members of this fam- terpolating function f(x). We select two vectors fields (1) (2) µ µ ily s (˜y) (˜y). Lattice diffeomorphism symmetry η (x), η (x) that are nowhere parallel or nonvanishing, K → K2 1 2 6

µ ν ǫµν η1 (x)η2 (x) = 0. Furthermore, we introduce two ad- the same for all cellsy ˜ within a region , up to small higher 6 µ µ R ditional fields ζ1 (x), ζ2 (x). Interpolation derivatives at x order corrections. (This can be realized for the generalized are defined by the two relations (a =1, 2) non-linear σ-model, not for the generalized Ising model.) In this case the interpolation derivatives ∂ˆ f(x) become µ µ f x + ζa(x)+ ηa(x) f(x + ζa(x) = η (x)∂ˆµf(x). − a independent of the choice of ηa, again up to small correc-

(19) tions. (This holds provided that x, x + ζa and x + ζa + ηa all belong to the region . We concentrate here on those R This definition becomes the usual definition of partial x and ηa for which ∂ˆµf can be expressed in terms of the derivatives for infinitesimal ηa and ζa. (In this limit ζ lattice fields f˜(˜z) as discussed above.) Indeed, we note for drops out since ∂µf(x) and ∂µf(x + ζ) are not different η′ in eq. (22) the relation for infinitesimal ζ.) In contrast to the usual differentiation 1 ′ we do not take here the vectors ηa, ζa to be infinitesimal η1(x)= η1(x)+ η1 x + η1(x) . (23) but keep finite values. This is justified by the “fuzziness”
′ of space, since for infinitesimal ηa, ζa the derivatives would Defining the derivatives with the set of vectors η1, η2 strongly depend on the chosen interpolation. Of course, amounts to use for eq. (19) with x + η1 = x + η1(x), x˜ = ˆ the interpolation derivatives ∂µf(x) will now depend on x + ζ1(x) the particular choice of ηa(x) and ζa(x).
Let us consider the particular case where for each posi- f x˜ + η (x)+ η (x + η f(˜x) 1 1 1 − tion of a lattice cell x = xp(˜y) the vectors ηa correspond µ µ
ˆ to the diagonals in the associated celly ˜. More specifically, = η1 (x)+ η1 (x + η1) ∂µf(x)
we consider = f x˜ + η1(x)+ η1(x + η1) f(˜x + η1)
− +f(˜x + η1) f(˜x) η1(x) = xp z˜ x˜4(˜y) xp z˜ x˜1(˜y) , − − µ ˆ ′ µ ˆ



= η1 (x + η1)∂µf(˜y )+ η1 (x)∂µf(˜y) η2(x) = xp z˜ x˜3(˜y) xp z˜ x˜2(˜y) , − ηµ(x)+ ηµ(x + η ) ∂ˆ f(˜y). (24) ζ (x) = x z˜x˜ (y)

x (˜y),

1 1 1 µ 1 p 1 p ≈


− ζ2(x) = xp z˜ x˜2(˜y) xp(˜y). (20) We conclude that the relation (21) also holds, up to small − ′

corrections, for the derivative formed with (η1, η2), which With this choice the interpolation derivatives at x = xp y˜) is therefore the same as the one formed with (η1, η2). This can be expressed in terms of two differences of values of can easily be generalized to other vectors ηa linking points ′ the lattice field in the celly ˜, namely between positionsx ˜4 xn within the region . The corrections ∂ˆµf˜(˜y ) ∂ˆµf˜(˜y) andx ˜ orx ˜ andx ˜ , respectively. By virtue of eq. (13) the R ∼ − 1 3 2 can be linked to higher derivatives as ∂ˆµ∂ˆν f(x), but we will interpolation derivatives are given by the lattice derivatives not discuss this issue here. Our definition of interpolation derivatives (19) adapts ∂ˆ f(x)= ∂ˆ f˜(˜y), (21) µ µ the notion of a derivative to a situation where only limited wherey ˜ denotes the cell associated to x. Eq. (20) fixes the information about a function is available - in our case the points xn and the values fn = f(xn). Our notion of smooth vectors ηa and ζa for all positions yn = xp(˜yn) of the cells N functions adapts the concept of differentiability to this sit- y˜n. For values of x inbetween the points yn we may again uation. On a large enough scale of coordinate distances in choose some interpolation for ηa(x), ζa(x). The values of 2 Ê the vectors η can be considered as infinitesimal, and ∂ˆ f(x) for x = y will depend on the interpolation. a µ n the independence on the choice of η reflects the indepen- The choice6 (20) is particular since only information con- a dence on the limiting procedure for differentiable functions. tained in a lattice field is required for the computation of On distance scales of the order η , however, this view is interpolation derivatives at the points y . On the other a n no longer possible and particular| vectors| as the ones in eq. hand, there are other choices for ηa(x) which allow the ˆ (20) have to be selected in order to retain the computability computation of ∂µf(x) in terms of the information con- of the derivatives in terms of the pairs (x ,f ). ˜ n n tained in a lattice function f(˜z). For example, we may With this perspective the lattice regularization may not replace η1(x) by be considered as a sign of a discrete nature of space and time. It rather reflects the fact that only a limited amount η′ (x) = x z˜ x˜ (˜y′) x z˜ x˜ (˜y) , 1 p 4 p 1 of information is available for the specification of contin- ′ − y˜ =y ˜ + (2 , 0).



(22) uous functions. In short, our definitions of interpolating fields and their interpolation derivatives are chosen such In fact, whenever both vectors ηa(x) join two pairs of points that they coincide with the lattice field and lattice deriva- ˆ xn = xp(˜zn) (and are not parallel), the derivatives ∂µf(x) tives for all cell locations. Away from the cell locations ˜ can be expressed in terms of the lattice field f(˜z) associated both f(x) and ∂ˆµf(x) are no longer computable in terms to to f(x). (We discuss here x = xp(˜y) and keep the same of the lattice function f˜(˜z) and the positions xn = xp(˜zn). ζa(x) as before.) They depend on the specific choice of an interpolation. We will now focus on smooth lattice fields f˜(˜z) character- The definition of interpolation derivatives (19) can also ized by the property that the lattice derivatives ∂ˆµf˜(˜y) are be used for discrete lattice variables, as fn = 1. In ± 7 this case, however, the notion of smooth lattice fields is displacements ξµ(x) in eq. (3) can be considered effectively no longer available for the microscopic degrees of freedom. as arbitrary infinitesimal functions of x. This will be different for the expectation values of the dis- In a more microscopic approach the notion of indepen- crete variables which are continuous real numbers. dent infinitesimal displacements ξµ(x) for every point x is In summary of this section, we have defined interpolat- no longer meaningful in view of the limited amount of in- ing functions and their interpolation derivatives with the formation available for the interpolating functions and the following properties: (i) The interpolating functions coin- associated fuzziness of space. Since the lattice information cide with the lattice functions for all positions xn of lattice concerns only a finite number of points xn, we only should points. (ii) The interpolation derivatives of the interpolat- consider the infinitesimal displacements ξ(xn) = ξn as ing functions coincide with the lattice derivatives for all cell independent parametersN of general coordinate transforma- positions yn. (iii) The interpolation derivatives coincide tions. The associated interpolating functions ξ(x) can be with the usual definition of partial derivatives for smooth constructed with the same interpolation prescription as for functions. (iv) The functional integral over interpolating (x). The infinitesimal displacements ξn can then be iden- H functions involves integrals over the values fn = f(xn). tified with a change of positioning of the lattice points The precise interpolating function f(x) that is specified by xp(˜z) xp(˜z)+ ξp(˜z), with ξn = ξp(˜zn). The displace- → the set fn depends on the positions xn and on the details ments specified in this way by the points xn and ξn = ξ(xn) of the interpolation{ } prescription. (v) For a given interpo- are associated to fuzzy diffeomorphism transformations. lation prescription a change of positions xn results for We will establish in this section the invariance of the con- { } given fn in a change of the function f(x). Such changes tinuum action under fuzzy diffeomorphisms. Before doing will be{ associated} in sect. IV with fuzzy diffeomorphism so, we will specify further our choice of interpolation. transformations. For smooth enough fields the continuum action (25) co- incides with the lattice action (7), (8) or (14), (17). The derivative ∂µ k(x) can be identified with ∂ˆµ k(˜y) for all IV. DIFFEOMORPHISM SYMMETRY OF H H x in a neighborhood of xp(˜y) and k(x) is well approxi- CONTINUUM ACTION H mated by the cell average k(˜y). Only for fields varying on a length scale comparableH to a typical coordinate distance Coming back to the lattice model (7), (8) we associate between lattice points the relation between the continuum interpolating fields k(x) to the lattice fields k(˜z). With action and the lattice action will depend on details of the H H the choice of interpolation derivatives (19), (20) and eq. interpolation prescription. For any specific interpolation (17) we can define the continuum action as a functional of one may express the lattice action as a functional of the in- the interpolating fields terpolating fields. This “interpolating action”, which may α be rather complicated, typically differs from the continuum S = d2xǫklmǫµν (x)∂ˆ (x)∂ˆ (x)+ c.c. (25) action (25) for strongly varying fields. Its precise form will Z k µ l ν m 12 H H H depend on the interpolation. It will not be relevant for smooth fields for which the difference to the continuum For smooth fields we replace ∂ˆµ ∂µ. In this limit we associate the continuum action (25)→ to the continuum limit action disappears. of the lattice action (7), (8). We will discuss next a specific class of interpolations for We consider the lattice action as the basic object since it which the interpolating action and the continuum action is computable in terms of the available information encoded coincide. For this choice the functional (25) for the inter- polating fields equals precisely the lattice action (7), (8). in k(˜z). The continuum action should mainly be consid- eredH as a very useful approximation for smooth enough For our purpose we impose the condition fields. In a certain sense the usual role of space as a man- 2 klm µν d xǫ ǫ k(x)∂ˆµ l(x)∂ˆν m(x) ifold is realized only on distance scales large compared to Z H H H the distance between lattice points. Interpolating fields cell and the value of the continuum action (25) for these fields klm µν = ǫ ǫ V (˜y) k(˜y)∂ˆµ l(˜y)∂ˆν m(˜y). (26) are formally defined for arbitrarily small coordinate dis- H H H tances x1 x2 in Ê2. However, the dependence on the Interpolations obeying the condition (26) are called “con- choice| of interpolation− | introduces physical ambiguity on gruent action interpolations”. It is straightforward to see very small distance scales if no information on a particular that for congruent action interpolations the continuum ac- “physical interpolation prescription” is available. tion (25) equals precisely the lattice action (7), (8) for arbi- We observe that the continuum limit of the action (25) trary configurations of the lattice fields k(˜z). In this case is diffeomorphism symmetric in the usual sense if the fields the continuum action is computable inH terms of the infor- k(x) transform as scalars under general coordinate trans- mation contained in the lattice fields without the need to H formations (3). Then ∂µ k(x) transforms as a vector, and know additional details of the interpolation prescription. the contraction with theHǫ-tensor implies an invariant ac- We notice that despite of the appearance the continuum tion. The usual notion of diffeomorphism transformations action is not invariant under continuous rotations. The in- can be realized for smooth enough “macroscopic fields” terpolation derivatives in eq. (25) are the ones that we have µ µ where the derivatives ∂µ (x) coincide with the standard defined for interpolating fields by specific vectors η , ζ . H a a definition of partial derivatives, while ηa(x) as well as the Continuous rotation symmetry arises effectively only for 8 smooth fields. Even though the use of congruent action action is not lattice diffeomorphism invariant the contin- interpolations is not crucial for the properties of smooth uum action is not diffeomorphism symmetric. Thus lattice fields it can be used as a convenient tool for later proofs diffeomorphism invariance is a necessary condition for ex- of diffeomorphism invariance of the quantum effective ac- act diffeomorphism symmetry of the continuum action. For tion. We will use this class of interpolation functions in the the continuum limit of smooth fields this condition may be following. weakened. It is sufficient that possible violations of exact It may be instructive to discuss briefly a particular ex- diffeomorphism symmetry of the continuum action vanish ample for a congruent action interpolation. We choose the in the continuum limit. value of f at the cell position y = xp(˜y) to be equal to the On the other hand, we will show that lattice diffeomor- cell average f˜(˜y)=Σj f˜(˜xj )/4, i.e. f(y) = f˜(˜y). Drawing phism invariance implies the invariance of the continuum action under “fuzzy diffeomorphisms”. Fuzzy diffeomor- straight lines between the cell position y and the positions µ xj of the four lattice points belonging to the cell we cut the phisms are characterized by parameters ξf (x) that are a cell into four triangles. For each triangle we have now three subset of the arbitrary displacements ξµ(x) that character- values of f at the corners, e.g. f(x1),f(x2) and f(y) for ize the general coordinate transformations. Every smooth µ the triangle (x1, x2,y). For all values of x within this trian- displacement ξ (x) can be well approximated by a suitable µ gle we interpolate f(x) by a plane in the three dimensional fuzzy diffeomorphism transformation ξf (x). For the con- space spanned by the coordinates (x0, x1,f). The orienta- tinuum limit of smooth fields and smooth displacements tion and position of this plane are fixed by the three values lattice diffeomorphism invariance therefore implies diffeo- of f(x) at the corners of the triangle. We apply this proce- morphism symmetry of the continuum action. dure to all four triangles of the cell. For this interpolation We recall in this context that “smooth” relates to weak the values of f(x) on each edge of a triangle are given by a variations on the scale of the lattice distance. The latter linear interpolation between the values of f(x) at the two may be associated to the Planck length or even a smaller corners bounding a given edge. Since on each boundary of length scale. Seen from a description at a larger physi- the triangle f(x) depends only on those two values, the in- cal arbitrarily varying fields and displacements are smooth. terpolating function f(x) is continuous on the edges of each We proof the statements of the preceding paragraph in the triangle, taking the same value for neighboring triangles. following. This includes the edge joining two neighboring cells. Thus Fuzzy diffeomorphisms are defined as interpolating func- µ f(x) is continuous, while ∂µf(x) is typically discontinuous tions for infinitesimal repositionings of lattice points ξp (˜z). for this type of interpolation. Similar to the interpolating fields the fuzzy diffeomor- Our procedure fixes the interpolation prescription up to phisms are specified at a discrete set of lattice points. At the choice of the cell position y = xp(˜y). Fixing the cell these points they equal the infinitesimal repositioning of position amounts to two conditions for the values of y0 and lattice points 1 y . As a first condition we impose the relation (26). Since µ µ µ µ ξ (xn)= ξ (xn)= x (˜zn) x (˜zn), (27) the integral over interpolating functions depends on the f p p,2 − p,1 cell position xp(˜y) the relation (26) can indeed be seen as where xp,2(˜zn) and xp,1(˜zn) are two infinitesimally close a condition for xp(˜y). Besides the condition (26) we may positions of a particular lattice pointz ˜n. Inbetween the impose one further condition for determining xp(˜y). For example,we may require that the sum of the surfaces of points xn the fuzzy displacements are interpolated accord- two triangles within the cell that do not share a common ing to some suitable interpolation description that we may choose to be the same one as for the interpolating fields. side equals half the surface of the cell or comes as close as µ possible to this value. Different interpolation prescriptions lead to different ξf (x) for x = x . This explains the name of “fuzzy diffeomor- There are many other possible prescriptions for congru- n phisms”.6 ent action interpolations. In the following we will assume A given positioning of the lattice points, together with a that the interpolation is continuous and differentiable ev- given interpolation prescription, results in a map from the erywhere in R2. This has the advantage that standard lattice field (˜z) to a continuous interpolating function partial derivatives ∂ (x) are defined for the interpolating µ (x). A generalH lattice action (not necessarily lattice dif- functions. (This contrastsH with our example of a piecewise feomorphismH invariant) can be expressed in terms of (x) continuous and differentiable interpolation for which ∂ µ ˆ H ˆ H and the interpolation derivatives ∂µ (x), is discontinuous at the boundaries of triangles, while ∂µ H H is always well defined.) The choice of differentiable lattice S[ (˜z)] = S[ (x); g(x)]. (28) congruent interpolations is not important for the contin- H H uum limit, but it will facilitate some of the formal proofs In general, the coefficients in front of the different terms in the following. will depend on the positions xp(˜z). These “position de- A condition of the type (26) can be imposed for inter- pendent couplings” are denoted in eq. (28) by g(x). Let polating functions for rather arbitrary lattice actions, not us now consider two infinitesimally close positionings lead- necessarily being lattice diffeomorphism invariant. How- ing to two sets of fields and couplings (x),g (x) and H1 1 ever, the diffeomorphism symmetry of the continuum ac- 2(x),g2(x) . From eq. (28) one infers
tion is a direct consequence of lattice diffeomorphism in- H
variance. More precisely, we will see that if the lattice S[ (x); g (x)] = S[ (x); g (x)]. (29) H1 1 H2 2 9

The two fields 2(x) and 1(x) are related by a fuzzy (fuzzy) diffeomorphism invariance of the effective action. diffeomorphism transformationH H This follows a simple basic idea: If ¯(˜y) or the associated continuum action does not notice whereL the lattice points 2(x) 1(x)= δf (x)= (x ξf ) (x). (30) are placed on the manifold, this property will also hold for H − H H H − − H the effective action. In other words, if no information on In other words, a fuzzy coordinate transformation maps the positioning is contained in ¯(˜y), the effective action an interpolating field (x) to another interpolating field L H1 will also not involve such information. Or, in an equivalent 2(x) which corresponds to the same lattice field (˜z). view, if the continuum action S[ (x)] does not involve po- TheH two interpolating fields (x) and (x) haveH the H H1 H2 sition dependent couplings there is no way how a position same values n at the points xn, but the position of the H dependence of couplings could arise on the level of the effec- points xn in the manifold differs. (They also obey the tive action. This simple result is crucial since it guarantees same interpolation prescription.) Fuzzy diffeomorphisms diffeomorphism symmetry of the quantum effective action are maps in the space of all interpolating functions for ar- in the continuum limit. We will show in sect. VII that it bitrary sets of points xn . For differentiable interpolating generalizes to the effective action for the metric, thereby functions (x) one has{ the} usual relation H providing the basic ingredient for general relativity. µ The generating functional for the connected Greens func- δf (x)= ξ (x)∂µ (x), (31) H − f H tions of lattice variables is defined in the usual way by in- troducing sources where ∂µ denotes here the usual partial derivative (not the interpolation derivative). Arbitrary smooth displacements ∗ µ W J(˜z) = ln exp S + k(˜z)J (˜z)+ c.c , ξµ(x) are approximated by a suitable ξ (x) and we recover Z DH − H k f    Xz˜
eq. (3). (35) A repositioning of the lattice points also changes the cou- with pling functions g(x), ∂W = k(˜z) = hk(˜z). (36) ∂J ∗(˜z) hH i g (x) g (x)= δ˜f g(x)= g(x ξf ) g(x). (32) k 2 − 1 − − (We omit the complex conjugate source term for real k.) According to eq. (29) the continuum action is invariant In the continuum limit the source term becomes H under the simultaneous variation of fields and couplings ∗ ∗ k(˜z)Jk (˜z)= k(x)jk (x). (37) δf S[ (x); g(x)] + δ˜f S[ (x); g(x)] =0, (33) H Z H H H Xz˜ x with One also may define ∗ Γ[h,J]= W [J]+ hk(˜z)J (˜z)+ c.c. , (38) δf S[ (x); g(x)] = S[ (x ξf ); g(x)] S[ (x); g(x)], − k H H − − H Xz˜
δ˜f S[ (x); g(x)] = S[ (x); g(x ξf )] S[ (x); g(x)].(34) H H − − H which becomes the usual quantum effective action Γ[h] Fuzzy diffeomorphisms act only on the fields, not on the (generating functional of 1PI-Greens functions) if we solve couplings, similar to general coordinate transformations. eq. (36) for J ∗(˜z) as a functional of h(˜z) and insert this We conclude that the continuum action is invariant un- solution into eq. (38), der fuzzy diffeomorphisms precisely if the couplings do not Γ[h]=Γ h,J[h] . (39) depend on the positioning, δ˜ S = 0. f   The possibility to express the continuum action in terms The (“functional”) derivative of the effective action with of (˜x) without coordinate dependent couplings reflects respect to h yields the exact quantum field equation theH property of lattice diffeomorphism invariance. If ¯(˜y) L ∂Γ ∗ would show an explicit dependence on coordinates, this = Jk (z). (40) would show up in the continuum action (with interpolation ∂hk(z) (26)). In analogy to the lattice action the coordinate de- We next use an interpolating field j(x) that coincides pendence in the definition (19) of interpolation derivatives with jn = j(xn) = j(˜z) for all positions of lattice points cancels against the coordinate dependence in the volume xn = xp(˜zn). Here the relation between j(˜z) and J(˜z) in- factor. This concludes the proof of our statements concern- volves an approximate volume factor according to eq. (10). ing the relation between lattice diffeomorphism invariance The interpolation prescription for j(x) is chosen such that and diffeomorphism symmetry of the continuum action. eq. (37) holds for arbitrary interpolating fields and sources (not necessarily smooth), similar to eq. (26). The expec- tation value h(x) is given by the same congruent action V. EFFECTIVE ACTION interpolation for the lattice field h(˜z) as used for relating (x) to (˜z), such that In this section we discuss the quantum effective action H H for scalar fields that correspond to the expectation val- h (˜z)J ∗(˜z)= h (x)j∗(x). (41) z k Z k k ues of the interpolating fields k(x) . We establish the X x hH i z˜ 10

We are interested in Γ[h(x)] as a functional of the inter- The same argument applies to W j(x) , polating fields h(x). The lattice field equation (40) is then   transmuted to an equation for continuous fields which in- W j(x) = ln (˜z)exp S (x) volves a functional derivative Z   DH  − H  δΓ ∗ ∗ + k(x)j (x)+ c.c , (48) = jk (x). (42) Z H k δhk(x) x
We next want to show that the quantum effective action which is diffeomorphism symmetric when only j(x) is trans- Γ[h(x)] is invariant under fuzzy diffeomorphisms, formed as a scalar density. This transformation property is compatible within δf Γ[h(x)] = Γ[h(x ξf )] Γ[h(x)]=0. (43) − − δW [j] For smooth differentiable fields h(x) and smooth displace- = hk(x), (49) δj∗(x) ments ξµ(x) this implies diffeomorphism symmetry of the k effective action under the transformation (3). For this pur- pose we write such that the solution of eq. (49), j(x) h(x) , transforms indeed as a scalar density if h(x) is transformed  as a scalar. Thus Γ h(x) is invariant when one transforms only the exp Γ h(x), j(x) = (˜z)exp S (x) − Z DH {− H field h(x) to h(x)+ δf h(x).      The last step involves the continuum limit of smooth ∗ + k(x) hk(x) jk (x)+ c.c , (44) functions. Since an arbitrary general coordinate transfor- Z H − x
mation ξµ(x) is well approximated by a suitable fuzzy coor- µ with S (x) the diffeomorphism symmetric continuum dinate transformation ξf (x) we conclude that the quantum action (25).H The integrand on the r.h.s. is diffeomorphism effective action is diffeomorphism symmetric in the contin- symmetric if h(x) and (x) transform as scalars and j(x) uum limit. as a scalar density. TheH functional measure (˜z) is in- In turn, the lattice version of the effective action DH variant under diffeomorphisms, such that ΓRh(x), j(x) is ¯ ˆ diffeomorphism symmetric, and this extends to Γ h(x). Γ[h(˜z)] = V (˜y)LΓ h(˜y), ∂µh(˜y) ... (50) More in detail, we consider fuzzy diffeomorphisms and Xy˜   define the transformation of the source functions δ j(x) f ¯ such that the r.h.s. of eq. (37) is invariant, is lattice diffeomorphism invariant. If not, LΓ would in- volve x-dependent couplings. The x-dependence of the cou- µ µ ¯ δf j(x)= ∂µξf (x)j(x) ξf (x)∂µj(x). (45) plings would remain if we express LΓ in terms of interpo- − − lating fields h(x). This is in contradiction with the diffeo- In the continuum limit j(x) transforms as a scalar density morphism symmetry which only transforms h(x), and the with respect to the usual general coordinate transforma- general invariance with respect to repositioning if both cou- tions. With plings and lattice derivatives are transformed. The two in-

µ variances are compatible only for x-independent couplings. δf h(x)= ξ (x)∂µh(x) (46) − f we compute the variation of the effective action VI. METRIC δf Γ[h(x), j(x)] = Γ[h(x)+ δf h(x), j(x)+ δf j(x)] Γ[h(x), j(x)]. (47) So far we have used the coordinates xµ only for − the parametrization of a region of a continuous two- ∗ In eq. (44) the term x hj is invariant under this dimensional manifold. We have not used the notion of a ∼ ∗ fuzzy coordinate transformation,R while the term x j metric and the associated “physical distance”. (The phys- also remains invariant if we shift simultaneously R(x)H ical distance differs from the coordinate distance x y , H → | − | (x)+ δf (x). The shift in affects the term involving except for the metric gµν = δµν .) The notion of a metric HS (x) .H However, for any givenH lattice field (˜z) we can and the associated physical distance, topology and geome- H H achieve  the shift (x) (x ξf ) by a repositioning of try can be inferred from the behavior of suitable correlation the lattice points,Hwithout→ Hchanging− the lattice field (˜z). functions [11]. Roughly speaking, for a euclidean setting µ H (This holds precisely for ξf , being realized as interpola- the distance between two points x and y gets larger if a tions of repositionings.) Since S is diffeomorphism invari- suitable properly normalized connected two-point function ant, δf S[ (x)] = 0, we conclude that the integrand, and G(x, y) gets smaller. This is how one world “measure” dis- thereforeH the integral, remains invariant under the trans- tances intuitively. formation of the fields h h + δf h, j j + δf j. This For the non-linear σ-model we may consider the two means that Γ is invariant→ if only the fields→ h and j are point function transformed. It is therefore a diffeomorphism invariant functional δf Γ[h(x), j(x)] = 0. G(x, y)= k(x) k(y) c, (51) hH H i 11 with k(x) suitable interpolating fields. Following ref. [11] interpolation derivatives in eq. (52) are replaced by partial the metricH is related to the behavior of G(x, y) for x y derivatives and the metric has the standard transformation and can be defined as → property under general coordinate transformations. As a particular positioning we can use the regular lat- 1 ∗ µ µ gµν (x) = Gµν (x)+ G (x) , tice x (˜z) = ∆˜z . This corresponds to a fixed choice of 2h µν i coordinates in general relativity. With this choice one has −2 Gµν (x) = µ ∂ˆµ k(x)∂ˆν k(x). (52) 2 0 H H V (˜y)=2∆ and Xk 1 The real normalization constant µ−1 has dimension of aµ˜(x)= δµ˜. (57) 0 µ 2∆ µ length such that Gµν and gµν are dimensionless. For real − k one has for the diagonal elements gµµ 0. We gener- 2 2 H ≥ Choosing µ0 = 4∆ /3, the collective field Gµν in eq. (53) alize here the setting of ref. [11] and also admit complex L coincides with the lattice metric G( ) in eq. (54). where g can be negative. The signature of the metric µν k µµ As an illustration, we may consider a particular config- isH not defined a priori. Points where det g (x) = 0 in- µν uration of lattice fields where for a given celly ˜ one has dicate singularities - either true singularities or coordinate
singularities. More generally, the geometry and topology (˜x )= (˜x ) = (˜x )= (˜x )= H (˜y), (e.g. singularities, identification of points etc.) of the space H1 1 H1 2 H1 3 H1 4 1 (˜x ) (˜x ) = H (˜y) , (˜x )= (˜x ), can be constructed from the metric [11]. We also adapt to H2 4 − H2 1 2 H2 3 H2 2 the lattice formulation and use in eq. (52) interpolation 3(˜x3) 3(˜x2) = H3(˜y) , 3(˜x4)= 3(˜x1). (58) derivatives. H − H H H L The metric is the central object in general relativity and For this configuration the lattice metric Gµν (˜y) is diagonal, appears in our setting as the expectation value of a suitable collective field. In practice, we do not need its relation to 1 2 1 2 G00 = H , G11 = H , G01 = G10 =0, (59) the behavior of correlation functions and we can simply 3 2 3 3 take eq. (52) as the definition of a metric. while the cell action (8) is given by For x coinciding with the position of one of the cells yn = xp(˜yn) the interpolation derivative ∂ˆµ k(x) is given by the α H (˜y)= H1H2H3 + c.c. (60) lattice derivative ∂ˆµ k(˜y). For these values of x the field L 12 H Gµν (x)= Gµν (˜y) can be expressed by lattice quantities For appropriate choices of H1,H2,H3 the combination −2 αH H H can be real and negative, thus giving a sub- Gµν (˜y) = µ ∂ˆµ k(˜y)∂ˆν k(˜y) 1 2 3 0 H H Xk stantial contribution to the functional integral. For ex- ample, this can be achieved for α > 0, real positive H3, = µ−2aµ˜(x)aν˜(x)G(L), (53) 0 µ ν µ˜ν˜ purely imaginary H1 and H2 with H1H2 < 0. In this with “lattice metric” case one finds a Minkowski signature of the lattice metric G00 < 0, G11 > 0. Euclidean signature obtains, for exam- (L) 1 ple, for real positive H2 and real negative H1. Gµ˜ν˜ = pk,µ˜pk,ν˜ (54) 3 For the non-linear σ-model an SO(3)-rotation of the L and components k yields the same Gµν and (˜y). There are, of course, manyH other configurations. ForL example, a cell pk,0 = k(˜x4) k(˜x1) , pk,1 = k(˜x3) k(˜x2). (55) with (˜x ) = (˜x ) = (˜x ) = (˜x ) for one H − H H − H k 4 k 1 k 3 k 2 valueH of k does notH contribute−H to GL (˜y)−H or (˜y). If con- Here the functions aµ˜(x) are defined implicitly by µν µ figurations of the type (58) dominate one mayL expect an µ˜ expectation value for the metric gµν of the type (59). ∂ˆµ k(˜y)= a (x)pk,µ(˜y). (56) H µ Properties of the metric can often be extracted from sym- They depend on the positions xj of the points of a cell and metries. If the expectation values preserve lattice transla- are described in more detail in the appendix. tion symmetry the metric gµν (x) will be independent of x. On the other hand, for x = yn the field Gµν and therefore Invariance under a parity reflection implies g01 = g10 = 0. 6 the metric gµν will depend on the specific interpolation Symmetry of the expectation values under lattice rota- prescription. This is again an expression of the fuzziness tions would imply a flat euclidean metric g00 = g11. A of space due to the lack of information beyond the lattice Minkowski metric gµν = ηµν requires that the expectation fields k(˜z). values violate the euclidean rotation symmetry. SimilarH to the lattice derivatives, the x-dependence of µ˜ the metric arises only through the functions aµ(x) which reflect the positioning of the lattice points. For interpo- VII. EFFECTIVE ACTION FOR GRAVITY AND GRAVITATIONAL FIELD EQUATIONS lating functions k(x) transforming as scalars under fuzzy general coordinateH transformations the metric (52) trans- forms as a covariant second rank symmetric tensor with re- The quantum effective action for the metric, Γ[gµν ], can spect to fuzzy diffeomorphisms. In the continuum limit the be constructed in the usual way by introducing sources for 12 the collective field, This concludes our discussion of lattice diffeomorphism invariance for two-dimensional non-linear σ-models. Two W [T˜] = ln exp S + GR (x)T˜µν (x) , dimensions are special for gravity, since the graviton does Z Z µν DH  − x not propagate. However, all our constructions generalize ˜ to four dimensions as we will discuss in sect. IX. Before R 1 ∗ δW [T ] Gµν = (Gµν + Gµν ) , = gµν (x). (61) doing so we will investigate lattice spinor gravity in two 2 δT˜µν (x) dimensions as a second type of model with a lattice diffeo- µν morphism invariant action. Solving formally for T˜ as a functional of gµν , the effective action obtains by a Legendre transform VIII. LATTICE SPINOR GRAVITY µν Γ[gµν ]= W + gµν (x)T˜ (x). (62) − Zx We next formulate quantum gravity in two dimensions The metric obeys the exact quantum field equation based on fundamental fermions instead of bosons. The problem of boundedness of the action is completely ab- δΓ sent for fermionic theories, where the partition function = T˜µν (x), (63) δgµν (x) becomes a Grassmann functional integral. We may use for every lattice point two species, a =1, 2, of two-component ˜µν a and we realize that T can be associated to the energy complex Grassmann variables ϕα(˜z), α = 1, 2, or equiv- µν ˜µν 1 µν a momentum tensor T by T = 2 √gT , g = det gµν . alently eight real Grassmann variables ψγ (˜z),γ = 1 ... 4, | | a a a a a a The effective action Γ[gµν (x)] is invariant under fuzzy with ϕ1 (˜z) = ψ1 (˜z)+ iψ3 (˜z) , ϕ2(˜z) = ψ2 (˜z)+ iψ4 (˜z). diffeomorphisms. This can be shown in analogy to the The functional measure (5) is replaced by effective action for scalar fields in sect. V. We only sketch the proof here for the continuum limit of smooth fields. ψ = dψ1(˜z)dψ2 (˜z) . (65) Z D γ γ Under a general coordinate transformation k(x) trans- Yz˜ Yγ
forms as a scalar H We introduce the bilinears (with Pauli matrices τk) ν δξ k(x)= ξ ∂ν k(x). (64) a ab b H − H k(˜z)= ϕα(˜z)(τ2)αβ (τ2τk) ϕβ(˜z), (66) R H This implies that ∂µ k and Gµν transform as covariant such that the action (8) contains now terms with six Grass- vectors and second rankH symmetric tensors, respectively. ˜µν mann variables. We keep the definitions (11), (12) and con- In consequence, T transforms as a contravariant ten- clude that this fermionic action is lattice diffeomorphism sor density, with T µν a symmetric second rank tensor. R µν µν invariant, leading to eq. (25) in terms of Grassmann fields Thus G T˜ and g T˜ are diffeomorphism invari- a a x µν x µν ϕ (x). In terms of ψ the action is an element of a real R R ˜ α γ ant, and Γ[gµν ] is diffeomorphism invariant if W [T ] is dif- Grassmann algebra. feomorphism invariant. This is indeed the case for T˜µν The lattice derivatives for the Grassmann variables are transforming as a tensor, since in eq. (61) the continuum defined similar to eq. (13) by the two relations action S[ (x)] is diffeomorphism invariant. For arbitrary H a a µ µ ˆ a interpolating fields (not necessarily smooth) the invariance ϕα(˜xj1 ) ϕ˜α(˜xj2 ) = (xj1 xj2 )∂µϕα(˜y) (67) − − of Γ[g (x)] under fuzzy diffeomorphisms follows if T˜µν is µν for (j , j )=(4, 1) and (3, 2). With R ˜µν 1 2 transformed such that x Gµν T remains invariant. a a For a given positioningR of the lattice points and a given k(˜xj1 ) k(˜xj2 )= ϕ (˜xj1 )+ ϕ (˜xj2 ) (τ )αβ H − H α α 2 prescription for the interpolation procedure the functional ab b b
(τ τ ) ϕ (˜x 1 ) ϕ (˜x 2 ) , (68) integral (61) is well defined and regularized for a finite 2 k β j β j × −
number of lattice points. (This holds for arbitrary func- and using reordering of the Grassmann variables, one ob- tions T˜µν (x) provided the exponent in eq. (61) remains tains from eq. (8) bounded.) Therefore also Γ gµν (x) is a well defined a a ab (y) = 8iαA(˜y) ϕ (˜x ) ϕ (˜x ) (τ )αβ(τ ) functional that is, in principle, unambiguously calculable. L − α 4 − α 1 2 2 b b
(More precisely, this holds for all metrics for which the sec- ϕβ(˜x3) ϕβ (˜x2) + ..., (69) ond equation (61) is invertible.) A key question concerns × −
the general form of the effective action Γ[gµν ]. If Γ is diffeo- with morphism invariant and sufficiently local in the sense that 1 1 2 2 A(˜y)=ϕ ¯1(˜y)¯ϕ2(˜y)¯ϕ1(˜y)¯ϕ2(˜y), (70) an expansion in derivatives of gµν yields a good approxima- a tion for slowly varying metrics, then only a limited number andϕ ¯α(˜y) the cell average. The dots indicate terms that do of invariants as a cosmological constant or Einstein’s cur- not contribute in the continuum limit. In terms of lattice vature scalar R contribute at long distances. The signature derivatives (67) one finds the action S = d2x ¯(˜y), of the metric is not fixed a priori. For g = 0 the inverse R L µν 6 µν µν a ab b metric g is well defined. The existence of g opens the ¯(˜y)= 8iαA(˜y)ǫ ∂ˆµϕ (˜y)(τ )αβ(τ ) ∂ˆν ϕ (˜y)+ ... L − α 2 2 β possibility that Γ[gµν ] also involves the inverse metric. (71) 13

For fixed spinor lattice derivatives (67) the leading term We define the lattice derivatives by the four relations (71) is again lattice diffeomorphism invariant. + − µ The continuum limit (25) can be expressed in terms of (˜y + vν ) (˜y vν ) = (x x ) ∂ˆµ (˜y), (77) H − H − ν − ν H spinors using ∂µ k(x)=2ϕ(x)τ τ τk∂µϕ(x), where the 2 2 ± + − H ⊗ where x = xp(˜y vν ). With ∆ν = (x x )/2 the cell first 2 2 matrix E in E F acts on spinor indices α, the ν ± ν − ν second×F on flavor indices⊗a. With volume amounts to µ ν ρ σ Fµν = A∂µϕτ τ ∂ν ϕ (72) V (˜y) = 2ǫµνρσ∆ ∆ ∆ ∆ − 2 ⊗ 2 0 1 2 3 ′ ′ ′ ′ 1 µ ν ρ σ µ ν ρ σ one obtains = ǫ ǫ ∆ ′ ∆ ′ ∆ ′ ∆ ′ . (78) 12 µνρσ µ ν ρ σ 2 µν S =4iα d xǫ Fµν + c.c., (73) 4 Z Using d x = y˜ V (˜y) one finds indeed that the action does notR dependP on the positioning of the lattice points, in accordance with eq. (71). Two comments are in or- der: (i) For obtaining a diffeomorphism invariant contin- α˜ − S = d4xǫµνρσ ˆ+ ˆ + c.c., (79) uum action it is sufficient that the lattice action is lattice 3 Z Fµν Fρσ diffeomorphism invariant up to terms that vanish in the continuum limit. (ii) The definition of lattice diffeomor- with phism invariance is not unique, differing, for example, if ˆ± 1 klm ¯± ˆ ± ˆ ± we take fixed lattice derivatives (12) for spinor bilinears or µν (˜y)= ǫ k (˜y)∂µ l (˜y)∂ν m(˜y). (80) F 12 H H H the ones (67) for spinors. It is sufficient that the action is lattice diffeomorphism invariant for one of the possible The continuum limit ¯ , ∂ˆµ ∂µ, is diffeomorphism definitions of lattice derivatives kept fixed. invariant due to the contractionH → H of→ the partial derivatives We finally note that A and Fµν are invariant under with the ǫ-tensor. ± SO(4, C) transformations. This symmetry group rotates For bosons one may employ real or complex fields a Hk among the four complex spinors ϕα, with complex infinites- and define a non-linear σ-model by the condition imal rotation coefficients. For real coefficients, one has ± ∗ ± SO(4), whereas other signatures as SO(1, 3) are realized ( k ) k =1. (81) if some coefficients are imaginary. The continuum action H H

(73) or (25) exhibits a local SO(4, C) gauge symmetry. A For complex fields the non-linear σ-model is invariant un- − subgroup of SO(4, C) is the two-dimensional Lorentz group der a global SU(3)+ SU(3) symmetry acting on the + −× SO(1, 1). The action (8) is therefore a realization of lattice index k of k and k , respectively. For complex bosons H H + ∗ − − ∗ spinor gravity [4] in two dimensions. it is possible to identify ( k ) with k or ( k ) , thus reducing the “flavor symmetry”H to SUH(3). ForH real bosons k the symmetry is reduced to SO(3) SO(3). IX. LATTICE DIFFEOMORPHISM INVARIANCE H For spinor gravity one uses eq. (74) and× defines IN FOUR DIMENSIONS ± ± ± Fµν = A Dµν , (82)

We generalize the lattice to four dimensions with in- ± 0 1 2 3 µ with A defined as in eq. (70) and tegers (˜z , z˜ , z˜ , z˜ ) and µ z˜ odd. Each cell located aty ˜ = (˜y0, y˜1, y˜2, y˜3), withP y˜µ even, contains eight ± µ Dµν = ∂µϕ±τ2 τ2∂ν ϕ±. (83) lattice points, located at theP nearest neighbors ofy ˜ at − ⊗ ν ν y˜ v0,... y˜ v3, with (vµ) = δµ. We double the number This yields for the continuum limit ± ± + − of degrees of freedom with bosons k and k , or fermions ± ± a a H H ˆ 4iF . (84) ϕ+α, ϕ−α, using Fµν →± µν ± The lattice action for spinor gravity proposed in ref. [4] k = ϕ±C± τ2τkϕ± , C± = τ2. (74) H ⊗ ± corresponds to eq. (76) if we replace in eq. (75) the cell We further introduce average ¯(˜y) by the plane average (˜y+vµ)+ (˜y vµ)+ H H H − ± 1 ± (˜y + vν )+ (˜y vν ) /4. The difference vanishes in the (˜y) = ǫklm ¯ (˜y) H H − Fµν 12 Hk continuum limit.  ± ± + (˜y + vµ) (˜y vµ) For spinor gravity Fµν is invariant under SO(4, C)+ × Hl − Hl − −  ± ±  transformations acting on ϕ+, while Fµν is invariant m(˜y + vν ) m(˜y vν ) , (75) under SO(4, C)− transformations acting on ϕ−. Since × H − H −  with ¯(˜y) the cell average. (Note that (y) in eq. (8) A+A− involves already the maximal number of eight H L a a obeys = α 01 + c.c.). A lattice diffeomorphism invariant different complex spinors ϕ+α, ϕ−α at a given point, actionL in fourF dimensions can be written as all inhomogeneous terms vanish for local transforma- tions [4]. Thus the action (79) is invariant under local

α˜ µνρσ + −

C C C S = ǫ + c.c. (76) SU(2, C)+ SU(2, )− SU(2, )L SU(2, )R trans- 24 Fµν Fρσ × × × Xy˜ formations, where ϕ+ transform as (2, 1, 2, 1) and ϕ− as

14

C C (1, 2, 1, 2). Here SO(4, C) = SU(2, ) SU(2, )L and rank tensors which are candidates for the metric. We first

+ + ×

C C C SO(4, C)− = SU(2, )− SU(2, )R, and SU(2, ) de- have different possible choices V . We could also use gµν = × −2 ˜m ˜n notes the SU(2)-transformations with three arbitrary com- µ0 Eµ Eν ηmn instead of eq. (87). Finally we could plex infinitesimal parameters, equivalent to six real param- employh aih latticei metric of the type (54), with a suitable eters. We can identify the generalized Lorentz group with choice of σ , σ = (+, ),

1 2 −

C C SO(4, C)= SU(2, ) SU(2, )−, such that ϕ and ϕ− + × + transform as two inequivalent Weyl spinors with opposite (L) 1 σ1 σ2 G = p p + (µ ν) , (88) handedness. The standard Lorentz group is the SO(1, 3) µ˜ν˜ k,µ˜ k,ν˜ 6 ↔
subgroup of SO(4, C). Choosing real transformation pa- with rameters, the subgroup SU(2)L SU(2)R can be identified × with a local gauge symmetry. σ σ σ p = (˜y + vµ) (˜y vµ). (89) The continuum limit of the action of lattice spinor grav- k,µ˜ Hk ˜ − Hk − ˜ ity, as defined by eqs. (79), (84), namely The metric obtains then by a generalization of eq. (53). This metric can be employed for bosons as well. (For 16˜α − S = d4xA+A D + c.c., fermions the different candidates are not all independent, Z 3 due to the possibility to reorder the fermions.) We could µνσρ + − D = ǫ Dµν Dρσ, (85) even discuss effective models with several distinct metrics. We expect, however, that in general only one particular differs from earlier versions of spinor gravity [12], [13], and linear combination will remain massless, whereas the oth- also from first attempts to formulate a diffeomorphism in- ers are massive. The massless mode can be associated with variant action in terms of spinors without employing a met- the physical metric and the graviton. ric [14–16]. It realizes local Lorentz symmetry and has the For a vanishing energy momentum tensor one expects property that the difference of signature between time and that the expectation values of fields are invariant under space only arises from the dynamics rather than being im- some of the symmetries of the action, but not under all posed in the action. In this respect it resembles the higher of them. (Certain symmetries will be spontaneously bro- dimensional model of ref. [17]. ken.) The conserved symmetries determine the form of The action (85) can be expressed in terms of suitable the metric to a large extent. This may be illustrated by vierbein bilinears a discussion of the consequences of discrete symmetries in ˜m m m case of spinor gravity or for complex bosons k. Let us Eµ = ϕC+γM V ∂µϕ ϕC−γM V ∂µϕ, (86) 0 H ⊗ ± ⊗ distinguish a “time coordinate” x from the three “space k m − coordinates” x . A reflection of one of the space coordi- with ϕ = (ϕ+, ϕ ), γM the usual 4 4 Dirac matrices k k and V suitable 2 2 flavor matrices. (This× reformulation nates, x x , has to be accompanied by a transfor- × mation → − acting on the “internal indices” of . will be described in a separate publication, see ref. [18] H → PH H for conventions.) The vierbein bilinears are vectors with This is needed in order to leave the action invariant. The required condition S[ ] = S[ ] can be achieved by respect to general coordinate transformations and vectors PH − H with respect to global Lorentz transformations. Diffeomor- changing the sign of an odd number of components k, for example − −, + +. The transformationH phism symmetry and local Lorentz symmetry of the action Hk → −Hk Hk → Hk leaves gµν invariant, such that invariance under guarantee that only invariants similar to Cartan’s formu- H→PH space reflections, ∂k ∂k, implies that the metric is di- lation of general relativity in terms of vierbeins [19] can →− appear in this expression of the continuum action. The agonal. If we further assume that the expectation values presence of several vierbein-type objects (e.g. for different are invariant under π/2-rotations among the space coordi- V ) induces, however, new interesting features of a mix- nates one infers gkl = aδkl. Lattice translation symmetry ing between geometrical and flavor aspects. For a suitable of expectation values implies that a is independent of x. choice of the fermion bilinear E˜m one may associate its Furthermore, the action is also invariant under “diagonal µ reflections” x0 xk which are accompanied by an internal expectation value with the vierbein of general relativity. ↔ From the vierbein bilinear we can construct a collective transformation , with S[ ] = S[ ]. An ex- ample is =Hi →, which DH changesDH the sign− ofH the metric. field for the metric as DH H Then invariance under the diagonal reflections, ∂0 ∂k − ↔ 2 ˜m ˜n implies, g = gkk. The combined symmetries therefore Gµν = µ0 Eµ Eν ηmn. (87) 00 − lead to flat Minkowski space, gµν = aηµν . It remains to be (The special role of the symmetry SO(1, 3) suggested by seen if the non-linear σ-model or spinor gravity with action the presence of ηmn = diag( 1, 1, 1, 1) is only apparent. (76) lead indeed to a “ground state” of this type. − ˜0 One could define a euclidean vierbein by multiplying Eµ by m a factor i, thus replacing γM by euclidean Dirac matrices. X. CONCLUSIONS In terms of those eq. (87) involves δmn instead of ηmn.) We may again define the metric by eq. (52) and define the diffeomorphism invariant effective action Γ[gµν ] similar We have formulated the property of lattice diffeomor- to eqs. (61), (62). We note in this respect that there are phism invariance for lattice models of quantum gravity that several collective fields transforming as symmetric second do not involve a fundamental metric or geometric degrees 15 of freedom. The functional integrals are defined in terms of the restriction that we consider only those infinitesimal µ variables at discrete lattice points which behave as scalars displacements ξf (x) that can be obtained as interpolat- under diffeomorphisms. We have constructed lattice diffeo- ing functions of a repositioning of the lattice points on the morphism invariant actions for discrete bosons, continuous manifold. In the continuum limit of smooth fields this re- bosons and fermions, both for two and four dimensions. We striction plays no role. The lattice functions (˜z) are not have shown that lattice diffeomorphism invariance implies affected by diffeomorphism transformations. ThisH guaran- diffeomorphism symmetry of the continuum action and the tees the diffeomorphism invariance of a functional measure quantum effective action. defined in terms of (˜z). General coordinate transforma- H The metric gµν arises as the expectation value of a suit- tions act only on the interpolating fields and correspond to 2 able collective field. The effective action for the metric is a change of positioning of lattice points xn in Ê . diffeomorphism invariant, which implies that the gravita- In this appendix we investigate lattice diffeomorphism tional field equations are covariant with respect to general transformations that act directly on the discrete lattice coordinate transformations. For a finite number of lattice variables, rather than on interpolating functions. For sim- points the functional integral is fully regularized. The effec- plicity of the presentation we concentrate on two dimen- tive action Γ[gµν ] (62) for the metric is then well defined. sions. Again, infinitesimal lattice diffeomorphism trans- The correlation functions of the metric are, in principle, formations correspond to an infinitesimal change xp(˜z) → computable - even though this may be hard in practice. xp(˜z)+ ξp(˜z) of the positioning of the lattice points. We This also holds for the quantum field equations which ob- study here the change of the discrete lattice derivatives tain from the variation of the effective action. Consistent ∂ˆµ (˜y) resulting from such a repositioning. For “restricted geometries correspond to solutions of these field equations latticeH diffeomorphisms” we keep the cell averages (˜y) H with an energy momentum tensor as a source. fixed. If we keep also the positions xp(˜y) of the cells fixed The bosonic Ising type or non-linear σ-models are suit- (for example on a regular lattice xp(˜y) = ∆˜y) the geo- able for numerical lattice simulations. Of key interest is a metric meaning of the restricted diffeomorphisms can be computation of expectation value and correlation functions visualized as moving the the positions of the lattice points of the metric. These can be compared to the ones that fol- z˜ at fixed position fory ˜. This changes the shape and vol- low from a diffeomorphism invariant action in a derivative ume of each cell, but not its position. (The cell position expansion. In particular, in four dimensions one expects discussed in this appendix may differ from the one used for the long range correlations appropriate for a massless gravi- some given interpolating description.) ton. For real fields k the signature of the metric is neces- The transformation of the lattice derivatives follows from sarily euclidean. ForH the non-linear σ model with complex µ the change of positions xj in eq. (12), while the cell average bosons k we may investigate the case of Minkowski sig- H (11) is kept fixed. Similar to the usual general coordinate nature as well. A numerical evaluation of Grassmann inte- transformations one can define the notion of “lattice vec- grals is more involved, in particular for the non-Gaussian tors” and “lattice tensors” with respect to these restricted action (76). Analytical approximations for the quantum transformations. For a scalar lattice function f˜(˜z) the lat- effective action could be based on the 2P I-formulation [20] tice derivatives transform as a lattice vector. Defining for the bosonic effective action [21], as sketched in ref. [22]. This requires the solution of gap equations in the presence p0(˜y)= f˜(˜x4) f˜(˜x1) , p1(˜y)= f˜(˜x3) f˜(˜x2) (A.1) of bosonic background fields as the metric. − − A priori, all solutions of the gravitational field equa- and tion (63) are viable if we include appropriate sources and boundary terms in the definition of the functional inte- dµ = xµ xµ , dµ = xµ xµ, (A.2) gral. This includes cosmological solutions. For an effec- 0 4 − 1 1 3 − 2 tive action Γ[g ] that is invariant under general coordinate µν we can write the lattice derivative as transformations and sufficiently local, one expects for long distances the validity of a derivative expansion with cos- ∂ˆ f(˜y)= aµ˜(x)p , (A.3) mological constant, Einstein’s curvature scalar and higher µ µ µ˜ order terms. If this turns out to be the case the lattice diffeomorphism invariant actions realize the construction with of regularized quantum gravity. 1 aµ˜(x)= ǫ ǫµ˜ν˜dν . (A.4) µ 2V (˜y) µν ν˜

APPENDIX: RESTRICTED LATTICE The lattice derivative depends on the positions of the lat- DIFFEOMORPHISMS µ˜ tice points through the xj -dependent coefficients aµ(x) which multiply the differences of lattice variables pµ˜. We In the main text we have discussed diffeomorphism trans- observe the relations formations on the level of interpolating functions. Differ- entiable interpolating fields obey the usual rules for trans- 1 µ˜ν˜ µ ν V (˜y)= ǫµν ǫ d d , formations of scalars, densities, covariant vectors and ten- 4 µ˜ ν˜ µ ν sors and so on. The lattice regularization imposes only ǫµν dµ˜dν˜ =2Vǫµ˜ν˜, (A.5) 16 and transforms in the same way as 1/V (˜y),

µ˜ µ ρσ ˆ ˆ ˆ µ ρσ ˆ ˆ aµdµ˜ =2, δp ǫ ∂ρf(˜y)∂σg(˜y) = ∂µξp ǫ ∂ρf(˜y)∂σg(˜y) , (A.13)
−
1 µ µν µ˜ ν˜ µ˜ρ˜ ν˜σ˜ ν such that V (˜y)Aµν (˜y) is invariant. ǫ aµaν = 2 ǫµν ǫ ǫ dρ˜dσ˜ , 4V As a consequence, the “cell action” ˆ(˜y) in eq. (17) is µν µ˜ ν˜ 1 L ǫ ǫµ˜ν˜aµaν = . (A.6) invariant under restricted lattice diffeomorphisms. This is V another facet of lattice diffeomorphism invariance. In fact, With respect to infinitesimal repositionings one has every lattice action can be expressed in terms of suitable cell averages and lattice derivatives. In general, however, δ dµ = ξµ(˜x ) ξµ(˜x ) , δ dµ = ξµ(˜x ) ξµ(˜x ), (A.7) the couplings multiplying different terms will depend on p 0 p 4 p 1 p 1 p 3 p 2 µ − − the coordinates xp (˜z) of the lattice points. Since the orig- and infers for the change of the cell volume inal lattice action does not “know” about the positioning, the expression of the action in terms of cell averages and ˆ µ derivatives, ˆ(˜y), is invariant under a repositioning if both δpV (˜y)= ∂µξp (˜y)V (˜y). (A.8) the cell averagesL and associated cell derivatives are trans- ν formed by lattice diffeomorphism transformations, and the Here ∂ˆµξ (˜y) is formed with the standard prescription p couplings are transformed according to their dependence (A.3) for lattice derivatives, using for pµ˜ differences of ˆ ξν (˜z). For the functions aµ˜(x) a change of positioning re- on xp. For a lattice diffeomorphism invariant model (˜y) p µ transforms as 1/V (˜y) if only the fields (˜y) and associ-L sults in H ated derivatives ∂ˆµ (˜y) are transformed. In this case Lˆ(˜y) µ˜ ν µ˜ H δpa (x)= ∂ˆµξ (˜y)a (x). (A.9) has to transform as V (˜y) if the couplings are transformed, µ − p ν while (˜y) and ∂ˆµ (˜y) are kept fixed. One can then write H H Since pµ˜ does not depend on the positioning one obtains ˆ(˜y)= V (˜y) ¯(˜y) and conclude that ¯(˜y) does not involve L L L from eq. (A.3) any couplings that depend on xp. This is precisely the char- acterization of lattice diffeomorphism invariance in sect. II. ˆ ˆ ν ˆ δp∂µf(˜y)= ∂µξp (˜y)∂ν f(˜y). (A.10) At this point a general principle for the construction of − lattice diffeomorphism invariant actions becomes visible. ˆ ν The transformation property δpvµ(˜y) = ∂µξp (˜y)vν (˜y) Since δp (˜y) = 0, all lattice actions with defines a lattice vector with respect to restricted− lattice H ¯ µν ˆ ˆ diffeomorphism transformations. For smooth enough lat- (˜y)= f (˜y) ǫ ∂µ l(˜y)∂ν m(˜y) (A.14) ν L H H H tice functions vµ(˜y) and ξ (˜y) the lattice derivatives be-
p transform V (˜y)−1 under lattice diffeomorphism transfor- come standard partial derivatives of interpolating func- ∼ tions. If we extend the discussion beyond the restricted mations for arbitrary functions f. The associated contin- uum limit replaces (˜y) by the interpolating field (x), transformations the repositioning also associates to the H H and ∂ˆµ ˜(˜y) ∂ˆµ (x). The expression of the associ- celly ˜ a new position xp(˜y) + ξp(˜y). From δξvµ(x) = H → H δpvµ(˜y)+ vµ(x ξ) vµ(x) one recovers for the associ- ated ˆ(˜y) = ¯(˜y)V (˜y) in terms of the lattice field (˜z) ated interpolating− functions− the standard transformation is straightforwardL L and does not involve the positions ofH the property of a covariant vector with respect to general co- lattice points. This construction is easily generalized to ordinate transformations four dimensions or any other number of dimensions. In d dimensions it formulates a diffeomorphism invariant ac- ν ν δξvµ = ∂µξ vν ξ ∂ν vµ. (A.11) tion as an integral over a d-form. It is not known if other − − forms of lattice diffeomorphism invariant actions exist that We conclude that the transformation of lattice vectors do not follow this construction. under restricted lattice diffeomorphism transformations We conclude that lattice diffeomorphism invariant ac- translates for the associated interpolating functions to the tions are those for which ¯(˜y) transforms as transformation law of covariant vectors under general coor- L ¯ ˆ µ ¯ dinate transformations. This holds in the continuum limit δp (˜y)= ∂µξp (˜y) (˜y) (A.15) ν L − L of smooth vµ and ξ . Tensors with respect to restricted lattice diffeomor- under restricted lattice diffeomorphism transformations. phisms (“lattice tensors”) can be obtained in a standard Here we recall that δp is taken for fixedy ˜, such that ˆ way from multiplication of lattice vectors. For example, δp (˜y) = 0 while ∂p(∂µ ) is given by eq. (A.10). H H ∂ˆµf(˜y)∂ˆν g(˜y)+ ∂ˆν f(˜y)∂ˆµg(˜y) transforms as a second rank Using the congruent action interpolation one obtains the symmetry lattice tensor. This is the transformation of the continuum action metric field G in eq. (53). The antisymmetric lattice µν ¯ tensor S = (x), (A.16) Zx L

Aµν (˜y) = ∂ˆµf(˜y)∂ˆν g(˜y) ∂ˆν f(˜y)∂µg(˜y) ˆ − ˆ where the lattice derivative ∂µf(˜y) is replaced by the “in- ρσ = ǫµν ǫ ∂ˆρf(˜y)∂ˆσ g(˜y) (A.12) terpolation derivative” ∂ˆµf(x) defined by eq. (19) and 17 obeying the relation (21), and (˜y) is replaced by (x). This is the transformation property of a contravariant vec- This continuum action coincidesH with the lattice action.H tor. Contraction with any invariant objectw ˜µ˜(˜y) yields a For the continuum action we have to take into account contravariant vector the volume factor in eq. (26) such that with eq. (A.8) the µ µ µ˜ variation under a (restricted) repositioning of lattice points w (˜y)= bµ˜(x)˜w (˜y), (A.24) becomes which transforms as δ S = ∂ ξµ(x) ¯(x)+ δ ¯(x) . (A.17) p Z µ p p µ ν ˆ µ x L L
δpw (˜y)= w (˜y)∂ν ξp (˜y). (A.25) For a lattice diffeomorphism invariant action obeying The product of a covariant and a contravariant vector is eq. (A.15) the continuum action is invariant, δpS = 0. invariant We emphasize that δp ¯(x) is a well defined operation for any ¯(x) that can be writtenL as a sum of terms for which µ L δp(vµw )=0. (A.26) each term is a polynomial in derivatives ∂ˆµf(x), multiplied by an arbitrary function of f(x). (Generalizations to sev- eral functions fa(x) are obvious.) It simply transforms all Withw ˜0 = J(˜x ) J(˜x ) , w˜1 = J(˜x ) J(˜x ), and derivatives of functions according to eq. (A.10), without 4 − 1 3 − 2 µ˜ any further change, vµ = ∂ˆµ (˜y)= a pµ, H µ ˜ ˆ ˆ ν µ ˆµ µ µ˜ δp ∂µf(x) = ∂µξp (x)∂ν f(x). (A.18) w = ∂ J(˜y)= bµ˜w˜ , (A.27)
− For congruent action interpolations the use of (continu- one finds ous) functions (x) that are defined for all points of a con- H µ µ˜ tinuous manifold is a simple rewriting of the lattice model ∂ˆµ ∂ˆ J =p ˜µw˜ = [ (˜x ) (˜x )][J(˜x ) J(˜x )] H ˜ H 4 − H 1 4 − 1 in a different picture. In this picture the action is given as +[ (˜x3) (˜x2)][J(˜x3) J(˜x2)]. (A.28) a functional of (x). In general, different positionings of H − H − H the lattice points result in different pictures, characterized Contravariant lattice vectors ∂µˆJ(˜y) related to sources in- by different continuum actions. Restricted lattice diffeo- troduce new possibilities for constructing lattice diffeomor- morphism transformations correspond to a repositioning phism invariants in Γ[h, j] and therefore also for the effec- of lattice points on a manifold and therefore to a switch tive action Γ[h]. between different pictures. For a lattice diffeomorphism invariant action all pictures for arbitrary positionings are the same. We next turn to the restricted lattice diffeomorphism transformations of sources. We start with sources for cell 1 averages, J(˜y) = 4 Σj J x˜j (˜y) . If J(˜y) transforms as a scalar, the ratio j(˜y) = J(˜y)/V
(˜y) transforms as a scalar density, with

ν δpj(˜y)= ∂ˆν ξ (˜y)j(˜y), (A.19) − p or, for smooth lattice fields,

ν δξj(x)= ∂ν ξ (x)j(x) . (A.20) −
Lattice diffeomorphism invariance is realized if ¯ trans- L forms as a scalar density when only ∂ˆµ (˜y) are transformed according to restricted lattice diffeomorphisms.H This ex- tends to a source term for cell averages, ¯ ∗ ∗ j (˜y)= k(˜y)jk(˜y)+ k(˜y)jk (˜y) , (A.21) L −H H
provided the sources transform according to eq. (A.19). µ˜ We further notice that we can consider aµ(x) as a regular 2 2 matrix, since the definition implies det a =1/2V > 0. × µ The inverse matrix bµ˜(x) obeys

µ˜ ν ν µ ν˜ ν˜ aµbµ˜ = δµ , bµ˜aµ = δµ˜, (A.22) and transforms as µ ν ˆ µ δpbµ˜(x)= bµ˜(x)∂ν ξp (˜y). (A.23) 18

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