# Correction terms for propagators and d’Alembertians due to spacetime discreteness

###### Abstract

The causal set approach to quantum gravity models spacetime as a discrete structure — a causal set. Recent research has led to causal set models for the retarded propagator for the Klein-Gordon equation and the d’Alembertian operator. These models can be compared to their continuum counterparts via a sprinkling process. It has been shown that the models agree exactly with the continuum quantities in the limit of an infinite sprinkling density — the continuum limit. This paper obtains the correction terms for these models for sprinkled causal sets with a finite sprinkling density. These correction terms are an important step towards testable differences between the continuum and discrete models that could provide evidence of spacetime discreteness.

###### pacs:

04.60.-m,11.10.-z,03.65.Pm^{†}

^{†}: Class. Quantum Grav.

## 1 Introduction

Causal set theory is an approach to quantum gravity in which the continuum manifold description of spacetime is replaced by a discrete description based on a causal set. A causal set is a locally finite partially ordered set in which the set elements represent spacetime events and the partial order represents the causality relations between them. See [1, 2, 3, 4] for full details.

In recent years, causal set models have been developed for the retarded and Feynman propagators of the Klein-Gordon equation [11, 12] and the d’Alembertian operator [14, 15, 16, 17, 18]. These models are defined implicit to a causal set but their behaviour can be compared to their continuum counterparts when the causal sets are generated by a sprinkling process from a Lorentzian manifold. Sprinkling is a random process that depends on a single parameter, the sprinkling density, that measures the expected number of causal set elements that will be sprinkled into a spacetime volume.

The parameters in the models are chosen so that their expectation value, averaged over sprinkled causal sets, agrees with the corresponding continuum quantity for causal sets generated by sprinkling into Minkowski spacetime. In most instances, the agreement is exact only in the limit of infinite sprinkling density (i.e. the continuum limit). The results of this paper provide the correction terms for large, finite sprinkling densities.

These correction terms are an important step towards phenomenological results from causal set theory in which the effects of a fundamental spacetime discreteness could be detected. They also provide an analytical approach to analyse the behaviour of the models for very large sprinkling densities, something that is difficult to do through direct computer simulations.

An alternative view of causal sets is as Lorentz-invariant discretizations of Lorentzian manifolds. In this view, the correction terms derived here may offer a natural cut-off to tame the divergences in continuum quantum field theories.

### 1.1 Preliminaries

A *causal set* (or *causet*) is a locally finite partial order, i.e.
a pair where is a set and a relation on which is (i) reflexive (); (ii) antisymmetric (); (iii) transitive (); and (iv) locally finite () for all . Here is a causal interval and denotes the cardinality of a set . We write if and . A *link* between (written ) is a relation such that there exists no with . A finite *chain* of length is a sequence of distinct elements . A finite *path* of length is a sequence of distinct elements

The set represents the set of spacetime events and the partial order represents the causal order between pairs of events. If we say “ is to the causal past of ”. The causal relation of a Lorentzian manifold (without closed causal curves) satisfies conditions (i)-(iii). It is condition (iv) that enforces spacetime discreteness—each causal interval contains only a finite number of events.

*Sprinkling* is a way to generate a causal set from a -dimensional Lorentzian manifold . Points are placed at random within using a Poisson process (with sprinkling density ) so the expected number of points in a region of -volume is . This generates a causal set whose elements are the sprinkled points and whose partial order relation is “read off” from the manifold’s causal relation restricted to the sprinkled points.

Here we shall only consider causal sets generated by sprinkling into -dimensional Minkowski spacetime, . To fix notation, we let be a point in with coordinates and use the quadratic form . With this metric signature the d’Alembertian operator is

(1) |

The causal partial order for is defined as: . The solid forward light cone at a point is the set

Quantities calculated on a sprinkled causal set can be averaged over multiple sprinklings (with the same sprinkling density ) to give a function in . The behaviour of this expectation value allows the discrete and continuum theories to be compared. We shall focus on the expectation values for the retarded propagator and the d’Alembertian operators for sprinklings into . The behaviour of these expectation values for large, finite values of will be examined.

## 2 Chains and Paths

To begin it will be useful to look at the expected number of chains and paths in a causal set generated by sprinkling into with sprinkling density . To assist us, we define the following important functions:

(2) |

(3) |

(4) |

The function is the “causal step function” — the indicator function for the closed forward light cone . The function is the -dimensional volume of the causal interval and the function denotes the probability that the sprinkled causal set has a link between and .

For two functions on their convolution is defined as:

(5) |

The expected number of chains of length one from to is . The expected number of chains of length from to is given by

(6) |

where there are copies of and convolutions in the final expression. From the definition we have .

The expected number of paths of length one from to is given by . The expected number of paths of length from to is given:

(7) |

where there are copies of and convolutions.

We note that these functions can be used to define regular distributions through convolution. For example, defines a distribution for suitable test functions with action .

We recall the known results for , calculated explicitly in [5, Theorem III.2, p50]. For we have:

(8) |

where and

(9) |

for is a real dimensionless constant.

To the author’s knowledge, there is no known closed-form expression for . Power series based on recurrence relations have been obtained in [13, Sec A.2] and an approximation for the total number of paths between and in was derived in [6, eq 2.5.15].

Useful insight into the function can be obtained by expanding it in terms of the functions to give:

(10) |

where we have used (8).

We can now apply the d’Alembertian operator using the observation that for even dimensions we have [8, eq 3.4 with ]:

(11) |

where is the -dimensional delta-function and is a constant. From this and (6) we have

(12) |

Combining with (10) it immediately follows that

(13) |

The infinite sum can be evaluated and is equal to

(14) |

where is a differential operator.

## 3 Propagators and d’Alembertians

The function plays a central role in a number of areas of causal set research. We now briefly describe two relevant areas.

### 3.1 Propagators

Propagators for the Klein-Gordon equation are solutions to the equation:

(17) |

The *retarded propagator* is the solution that is supported only in the forward light cone . In this is:

(18) |

where is a Bessel function of the first kind.

In [11] a causal set model for the retarded propagator was presented for and . In this model involved a quantum mechanical path integral that summed amplitudes assigned to paths in the causal set. The expectation value of the propagator for a causal set generated by sprinkling into with density was equal to:

(19) |

It was established in [11] that

(20) |

In particular, the massless propagator is related to through:

(21) |

### 3.2 Discrete d’Alembertians

In recent years a number of researchers have worked on causal set models that reproduce the behavior of the continuum d’Alembertian operator.

In these models, a field is represented by values assigned to each causal set element. By computing an alternating sum of these field values over layers of elements in the causal set the model is able to approximate the d’Alembertian operator applied to the field: .

More specifically, the expectation value of the models for sprinklings into with density is an integral kernel that satisfies

(22) |

for suitable functions . The kernels are the sum of a delta-function and the function acted upon with appropriate differential operators.

This was extended, in [15, eq (4)], to :

(24) |

We note that equation (16) provides a surprisingly simple expression for in :

(25) |

This connection has been called the sweety-salty duality and will be used in section 5.2.

The models have been extended to higher dimensions [16, 17] as:

(26) |

where, for even we have:

(27) |

and for odd we have . The and are constants chosen to give the correct answers in dimensions. They are given in closed-form in [17].

Generalizations of these models have also been proposed recently [18]. The d’Alembertian work has also been applied to model the scalar curvature of a causal set, as discussed in [15].

The notation for this work sometimes use a sprinkling density as the variable, sometimes a discreteness length scale . We have therefore clarified our operators with a subscript. In [16, 17] they use a differential operator which is related to ours as . The sign of also depends on the metric signature used for . In addition, the work is often phrased using , a function supported on past light cone, rather than the future light cone. We hope the conventions used here are clear.

The results of section 5.2 give the correction terms to the limit in (22). Previous approaches to obtaining the correction terms involve Taylor-expanding the fields and calculating the integral term-by-term. Each of these integrals typically diverge so are evaluated with a large-scale cut-off. This cut-off breaks the Lorentz-invariance of the theory so is then removed with care. The methodology used in 5.2 is manifestly Lorentz-invariant and gives the correction terms without direct reference to the field .

## 4 Large expansion

The models just described both require taking the infinite limit to give agreement with the continuum quantities. The central premise of causal set theory is that spacetime *is* fundamentally discrete. As such, the most interesting regime is when is large but *finite*. Indeed, we would expect to be sufficiently large so that discreteness effects have gone undetected so far. Often the discreteness scale is assumed to be based on the Planck-length. To sprinkle into with a Planckian density we could take to be the inverse of the Planck 4-volume: which corresponds to causal set elements per cubic-metre-second of spacetime — certainly a large number.

We now look at the large behaviour of . We will then apply these results to look at the large behaviour of the propagator and d’Alembertian models. To get started, we summarize two tools we shall use extensively: the Riesz distributions and the Laplace transform.

### 4.1 Riesz Distributions

An important class of Lorentz invariant retarded distributions are the Riesz distributions (see [8] for full review). These are related to powers of the quadratic form and have an action on test functions defined by:

(28) |

where is a normalization constant:

(29) |

With this normalization, the map is a complex-analytic mapping with the following properties:

(30) |

(31) |

(32) |

We note that the expected number of chains, defined in (8) and treated as a distribution, is related to the through .

Another important distribution is the infinitesimal generator of given by:

(33) |

This has been analyzed thoroughly in the classification of Lorentz invariant distributions supported on the forward light cone [7, 8]. For our purposes, it will also be useful to define a related distribution , using an -independent constant , as:

(34) |

The distribution is defined up to the addition of multiples of the delta function [7, Sec 7.2] so explicit spacetime expressions can only be given for test functions that vanish at 0. It is related to the pullback of derivatives of the delta function on through [8, eq 6.3]:

(35) |

In , we have the following spacetime expression for [8, sec 1.0, Remark 2], [7, sec 7.2]:

(36) |

For even dimensions , we have (compare [8, p283]):

(37) |

From the properties of , we have which, in other notation, is

(38) |

which gives:

(39) |

Combining this with (34) and (37) gives:

(40) |

### 4.2 Laplace Transform

Following [9, I,1;1], the Laplace transform of a function defined on is:

(41) |

where and . The Fourier transform of can be obtained from the Laplace transform in the limit as the imaginary part of tends to zero.

The Laplace transform converts convolutions into products:

(42) |

and satisfies

(43) |

where .

The Laplace transform of retarded Lorentz invariant functions supported on the forward light cone has been obtained in [9]. We can apply this expression to by recognizing that, for , is equal to:

(47) |

(compare (3) and (4)). From [9, I,2;1] the Laplace transform of is therefore given by the one-dimensional integral:

(48) |

where is a modified Bessel function of the second kind.

This will be the central equation which will enable us to find the large behaviour of . We split the discussion depending on whether is even or odd.

### 4.3 Even Dimensions

For even, it is useful to evaluate (48) with the substitution to get:

(49) |

For (), the Bessel function in (49) has integer order with a series expansion of the form [10, 8.446]:

(50) | ||||

where is the digamma function [10, 8.330]. This satisfies and where is the Euler-Mascheroni constant.

Substituting the Bessel function expansion (50) into (49) results in an infinite series with terms of the form:

(51) |

and

(52) |

These integrals can be evaluated as:

(53) |

(54) |

Combining these gives an infinite series for of:

(55) | ||||

where and .

#### 4.3.1 Dimension

For , we have and:

(56) |

(57) |

(58) |

(59) |

Combining these gives:

(60) | ||||

(61) |

The first few terms are, in descending powers of (noting that ):

(62) |

Translating this into position space, using the results of section 4.2, gives

(63) |

We also note that can be evaluated in closed-form as:

(64) |

where is the Exponential Integral.

#### 4.3.2 Dimension

For , we have and:

(65) |

(66) |

(67) |

(68) |

(69) |

Combining these gives:

(70) |

The first few terms, in descending powers of , are:

(71) |

where is the Euler-Mascheroni constant.

Translating this into position space, using the results of section 4.2, gives

(72) |

where is the massless retarded propagator: .

### 4.4 Odd Dimensions

For odd, it is useful to evaluate (48) with the substitution to get:

(73) |

For (), the Bessel function in (73) is of half-integer order . These can be expressed in terms of exponentials and powers to give [10, 8.468]:

(74) |

Combining this with (73) gives as:

(75) |

with . By expanding the in the integrand, an infinite series for can be generated in powers of :

(76) | ||||

(77) |

where .

#### 4.4.1 Dimension

For , we have and:

(78) |

(79) |

which combine to give

(80) | ||||

we have

(81) |

therefore

(82) |

The first few terms, in descending powers of are:

(83) |

Translating to position space we have:

(84) |

## 5 Analysis

Using the large expansions for we can find the large behaviour of the propagators and discrete d’Alembertians discussed in sections 3.1 and 3.2.

### 5.1 Propagators

Recall that the expectation value of the causal set retarded propagator is: