We assume the space-time foam picture in which the vacuum is filled with a gas of virtual wormholes. It is shown that virtual wormholes form a finite (of the Planckian order) value of the energy density of zero-point fluctuations. However such a huge value is compensated by the contribution of virtual wormholes to the mean curvature and the observed value of the cosmological constant is close to zero. A nonvanishing value appears due to the polarization of vacuum in external classical fields. In the early Universe some virtual wormholes may form actual ones. We show that in the case of actual wormholes vacuum polarization effects are negligible while their contribution to the mean curvature is apt to form the observed dark energy phenomenon. Using the contribution of wormholes to dark matter and dark energy we find estimates for characteristic parameters of the gas of wormholes.

As is well known modern astrophysics (and, even more generally, theoretical physics) faces two key problems. Those are the nature of dark matter and dark energy. Recall that more than 90% of matter of the Universe has a nonbaryonic dark (to say, mysterious) form, while lab experiments still show no evidence for the existence of such matter. Both dark components are intrinsically incorporated in the most successful

As it was demonstrated recently [

Saving the dark matter component

It is necessary to point here out to the principle difference between actual and virtual wormholes. The principle difference is that a virtual wormhole exists only for a very small period of time and at very small scales and does not necessarily obey to the Einstein equations. It represents tunnelling event and therefore, the averaged null energy condition (ANEC) cannot forbid the origin of such an object. For the future we also note that a set of virtual wormholes may work as an actual wormhole opening thus the way for an artificial construction of wormhole-type objects in lab experiments.

In the present paper we describe a virtual wormhole as follows. From the very beginning we use the Euclidean approach (e.g., see [

In the region

In the present paper we assume the space-time foam picture in which the vacuum is filled with a gas of virtual wormholes. We show that virtual wormholes form a finite (of the Planckian order) value of the energy density of zero-point fluctuations. However such a huge value is compensated by the contribution of virtual wormholes to the mean curvature and the observed value of the cosmological constant should be close to zero.

To achieve our aim we, in Section

The basic aim of this section is to construct the generating functional which can be used to get all possible correlation functions. Consider the partition function which includes the sum over topologies and the sum over field configurations

Consider now the sum over topologies

The basic difficulty of the standard field theory is that the perturbation scheme based on (

And indeed, the above measure (

From (

Consider now the bias for a particular set of wormholes. For the sake of simplicity we consider the case when

In the rarefied gas approximation the total bias is additive; that is,

In the short-wave limit (

In other words, in the long-wave limit (

The action (

Let us consider the Fourier representation

At the present stage we still cannot evaluate the exact form for the cutoff function

By the construction the topological bias

Indeed, consider a single wormhole with parameters

Let

The above construction can be easily generalized to the presence of a set of wormholes. In the approximation of a dilute gas of wormholes we may neglect the influence of wormholes on each other (at least there always exists a sufficiently smooth map which transforms the family of lines of force for “independent” wormholes onto the actual lines). Then the total bias (projection) may be considered as the product

The projective nature of the bias operator

Thus taken into account that

The simplest choice gives merely

One may expect that the true cutoff function has a much more complex behavior. Indeed, some theoretical models in particle physics (e.g., string theory) have the property to be lower-dimensional at very small scales. The mean cutoff

Let us consider the total Euclidean action [

Consider a single wormhole whose metric is given by (

In the case of a set of wormholes (

In this section we consider the contribution from matter fields. In the case of a scalar field the stress energy tensor has the form

For the sake of simplicity we consider the massless case. Then by the use of the cutoff

To understand how wormholes remove divergencies, it will be convenient to split the bias function into two parts

Consider now the particular distribution of virtual wormholes (

From the above expression we see that to get the finite value of the cosmological constant

The value of

Consider now topology fluctuations in the presence of an external current. In the presence of an external current

The only unknown parameter in (

As we already pointed out the additional distribution of virtual wormholes (

Consider now the contribution to the dark energy from the gas of actual wormholes. Unlike the virtual wormholes, actual wormholes do exist at all times and, therefore, a single wormhole can be viewed as a couple of conjugated cylinders

Actual wormholes also produce two kinds of contribution to the dark energy. One comes from their contribution to the mean curvature which corresponds to an exotic stress energy momentum tensor. Such a stress energy momentum tensor reflects the dark energy reserved by additional virtual wormholes discussed in the previous section. Such energy is necessary to support actual wormholes as a solution to the Einstein equations. The second part comes from vacuum polarization effects by actual wormholes. The consideration in the previous section shows that for macroscopic wormholes the second part has the order

For rigorous evaluation of dark energy of the second type we, first, have to find the bias

Indeed, instead of the cylinders

From (

In this subsection we consider the Minkowsky space. Then the simplest actual wormhole can be described by the metric analogous to (

Now consider the simplest estimates. Actual wormholes seem to be responsible for the dark matter [

Thus, we see that virtual wormholes should indeed lead to the regularization of all divergencies in QFT which agrees with recent results [