^{1}

^{3}.

In a spacetime divided into two regions

Motivated by the development of black hole thermodynamics [

Despite its motivation, this article does not contribute by elongating the promising road towards quantum gravity emerging from the gauge/gravity correspondence, and our work is classical in substance.

A recurrent theme in this study is that a hypersurface locally splits spacetime into two regions and can be thought of as the communication channel between them.

Spacetime localized properties of the field will be of our interest. Since measuring devices live in spacetime as well, measurement will be understood as the interaction between the system of interest (a certain field) and the measuring device (a field or a detector modeled otherwise). Think for example of a beam interacting with a screen for a short period of time; a spacetime description of the situation takes place in a bounded spacetime domain

From this point of view, there could be field configurations in

We will also play with the idea of having the ability of separating a system into subsystems in arbitrary ways as indicated by splitting a spacetime region into subregions like

From the study of how perturbations propagate through communicating hypersurfaces a condition for gauge equivalence naturally arises. We complement it with requirements of locality and relativity of measurement to give rise to a notion of gauge vector fields which is suited to work on spacetime confined domains. This notion of gauge becomes a cornerstone for the version of Lagrangian field theory presented in this work and for the study of observables and holography given in the rest of the article.

A close relative to our proposal is the discovery of holographic behavior in gauge theories confined to bounded domains; more precisely, that the presence of boundaries in gauge theories leads to “would be gauge degrees of freedom” living at the boundary [

Our interest on communicating hypersurfaces

In bounded spacetime domains

It is natural to wonder about observables calculated from conserved currents which depend only on the field at the codimension two surface

Before we finish setting up the context of this study, we have to give a pair of remarks. The first one regards the causal structure. General relativity is one of the theories that we intend to cover in our study, and it sets spacetime geometry as a dynamical field interacting with matter fields. Since causal structure follows from spacetime geometry, at the initial stage of our setting we consider a spacetime

The second remark is about the term “local”. In this introductory section we already mentioned “spacetime localized” properties of the field; it refers to properties of the field inside a bounded spacetime region

The organization of this paper is as follows. In Section

In this section we give a brief review of the version of Lagrangian classical field theory that will be used in this work. It will also be useful to fix the notation for the rest of the article. Local functionals and the variational principle are elegantly treated in the jet bundle using the tools of the variational bicomplex. Vector fields in the space of solutions play an important role in our work, in particular because observables and observable currents have associated Hamiltonian vector fields. In order to correctly model those vector fields we have to step out of the geometrical formalism of the jet bundle and work at the level of sections using analytical methods. Here we will give a minimal description of the part of this formalism that is essential for presenting our results. For an excellent short introduction the variational bicomplex see [

We will aim to have a

There is a bundle over spacetime

Points in the

Different jets are related by projection maps. The projection

Smooth functions on

Given a section

In classical field theory the main objects are not points of

A vector field in

A convenient basis for differential forms in

Let us consider Hamilton’s principle of extremal action for a Lagrangian density of

The field equation is written as

Vector fields in

For example, if we are interested in using two vector fields

When dealing with nonlocal objects (like nonlocal vector fields or nonlocal currents) we will work at the level of sections and modify the notation as done with the functional written above. Thus, symbols with a tilde denote

According to the terminology that we declared in the introductory section, the functional

This example will take place considering that spacetime is a cylinder,

The fields that we will consider are a

The evaluation of the prolongation of the section in the jet is written as

The basis for tangent vectors in the jet induced by the chart can be written as

We live as an exercise to the reader to calculate the horizontal and vertical differentials of the coordinate functions that we chose for

We consider the first order action

We start calculating

The space of solutions in this example is formed by pairs consisting of a constant function (the

This example is completely different from general relativity in the sense that vector fields in the space of solutions can be modeled by local vector fields. The field equations and the linearized field equations are linear, and every class of solutions modulo gauge contains local representatives. Then if we work with only local fields and local vector fields the description continues being physically appropriate.

In the next section we will be able to comment on the space of solutions modulo gauge. Then it will be clear why is that Horowitz called the quantization of this class of theories quantum cohomology.

Consider a solution of the field equations

Let us study the cut and paste operation just described. Consider two domains intersecting at a hypersurface

Thus, the extremality of the action is equivalent to demanding first extremality with respect to variations in

Because of our interest in the type of relative measurements described in Section

It is interesting to characterize pairs of perturbations

Now let us consider a spacetime domain

Relativity and locality of measurement motivate the second condition needed in our definition of gauge perturbations. The simplest way to present it is recalling that we were searching for conditions for two variations of the field

Now we comment on locality of measurement. Consider a situation in which spacetime

A vector field

(i)

(ii)

Isolated gravitational systems may be modeled over a spacetime domain of the type

The standard definition for gauge perturbations used in Lagrangian field theory (appropriate in manifolds with no boundary) is in terms of families of symmetries of the Lagrangian depending on a parameter that may be locally varying, and through Noether’s second theorem this is related to ambiguities and over determination in the field equation (see [

It can be verified that this definition of gauge perturbations leads to a Lie subalgebra of the algebra of vector fields in

Now it may be illustrative for the reader to go back to the two-dimensional abelian BF theory example. We have already calculated

Recall that in the example spacetime was a cylinder

It is natural to consider functions of the field calculated integrating currents over hypersurfaces. If the currents depend locally on the field and its partial derivatives those functions may be written as

A physical observable

There are very few hypersurface local observables, but we will see in this section that there are plenty of (possibly nonlocal) observables calculated integrating currents. One reason to suspect it is that

Here we will consider this type of observables where the domain of definition of the current may be a proper open subset of the space of fields [

Since in this work we will refer multiple times to (possibly nonlocal) gauge invariant conserved currents defined in an open domain of the space of solutions, we will use the term

In order to make definitions a bit more concrete let us consider as an example two-dimensional abelian BF theory as defined in Section

Notice that if Condition (ii) for gauge vector fields were not present we would not have a gauge invariant observables measuring

The resulting family of observables is a large family capable of distinguishing solutions which are not gauge related. Thus, the family of observables includes Noether charges for systems with simple Lagrangian symmetries whose generators are defined everywhere in the domain of the Lagrangian density, and it includes many more observables. Two aspects of our treatment are essential for proving separability of points in the space of solutions modulo gauge: The first one is properly modeling the space of variations of a given solution and the space of vector fields in

The linearized field equation is a nonlinear partial differential equation, which is linear only when not studied in the whole jet, but only on

It turns out that, in field theories with local degrees of freedom, observable currents can distinguish gauge inequivalent solutions. A local version this statement is as follows: Consider any given solution

There are two nonexclusive possibilities for

Previous works on similar approaches to classical field theory argue that there are not many physical observables arising from integrating conserved currents besides Noether charges (see for example [

The case that was not covered in the proof given above is that relation (

Before closing this section we comment on the type of measurements of the bubble chamber, which are not properly modeled integrating currents. We consider that if the field of interest

We saw that a perturbation of the field modeled by an almost locally Hamiltonian vector field

The results described in the previous section show that the family of observable currents and perturbations satisfying (

Consider observables calculated integrating conserved currents of the type

Let us recall the terminology declared in the introductory section now that we have developed concepts and notation that let us be more explicit. A physical observable

Consider general relativity on a spacetime domain

Then the observables related to a gauge vector field

Anderson and Torre [

In the next section we will exhibit a family of holographic gravitational observables and a large family of nonholographic gravitational observables.

In contrast to the case of general relativity, as discussed in Section

Khavkine has proposed a generalized notion of spacetime local observables and exhibited a large family of gravitational observables which are spacetime local according to his definition [

For the sake of concreteness, in this section we will consider a spacetime domain

There are many gravitational observables in our setting. According to the conventions stated in the previous paragraph, the components of the induced metric on the boundary (the pull back of the spacetime metric to the boundary)

This fact just talks about our choice of reference system. Part of our motivation to impose (

In Section

In the previous section we asked if hypersurface nonlocal gravitational observables could be approximated by local ones implying that nonlocal gravitational observables are also holographic. With the aide of (

A large family of observables is given by the symplectic product of physical perturbations (see [

Below we give another family of examples of gravitational observables that can be defined thanks to the existence of a reference at the boundary. If the reader would like to have further motivation for considering bounded spacetime domains, it could be illuminating to read Appendix

When the location of individual points inside

On the other hand, if the points

Similar arguments could be used to define physical observables measuring areas of minimal surfaces determined by curves fixed at

Are observables in these families holographic? They are not. One way to see that this might be true is to observe that the requirement for the functional to be an observable is that

We started reviewing the notion of what a gauge perturbation is in the context of Lagrangian field theories defined on confined spacetime domains. The initial assumptions included that we are working in a covariant setting in which spacetime geometry may be one of the dynamical fields, implying that there is no causal structure fixed a priori. The initial consideration to determine which perturbations are considered gauge was a covariant form of a determinism principle thoroughly explained in Section

At first glance, it may seem that both aspects of holography mentioned in the previous paragraph are disjointed, and that the term holography refers to completely are unrelated phenomena in both instances. However, the work presented in this article shows the intimate relationship between them: first of all, from our definition of gauge vector fields and our considerations of observables which are only locally defined in the space of solutions, we could give a straight forward argument proving that observables calculated integrating observable currents

In Section

Observables in nonlinear theories with gauge redundancies defined in spacetime domains foliated by Cauchy surfaces with no boundary are expected to be nonlocal in the sense of depending on infinitely many derivatives of the field. A brief discussion of the reasons behind this expectation is given in Section

Can every gravitational observable be approximated by local observables, inheriting their holographic behavior? This issue is addressed in Section

Is the family of holographic observables capable of separating points in the space of solutions modulo gauge? The arguments given above tell us that this question is equivalent to asking if for any given solution the tangent space to the space of solutions based on it is generated by gauge vector fields together with would be gauge vector fields, which is clearly not the case.

In this appendix we motivate working with bounded spacetime domains from the point of view of a covariant initial value formulation on a given hypersurface with boundary which may be thought of as a laboratory where data is retrieved at a given time. We comment on the correspondence between this formalism and the one used in the main body of this article emphasizing interpretational issues which arise when setting up the correspondence and their relation with two key aspects of our formalism: considering locally defined observables and the notion of gauge.

For the sake of concreteness, consider the following scenario for gravitational thought measurements on earth: Earth’s southern hemisphere is covered by laboratories (covering a layer from a height of

In this appendix the submanifold of the jet in which the field equation

Here we consider the case in which the curve of solutions

(i)

for every solution

(ii)

for some solution

Now we choose an almost locally Hamiltonian vector field

The equation

For a more detailed presentation, see [

No data were used to support this study.

There are no conflicts of interest associated with the publication of this work.

I would like to thank Igor Khavkine for important clarifications on the subject of local observables. I acknowledge correspondence and discussions about the subject of the article with Jasel Berra, Homero Daz, Laurent Fridel, Alberto Molgado, Robert Oeckl, Michael Reisenberger, Charles Torre, José A. Vallejo, and Luca Vitagliano. This work was partially supported by Grants PAPIIT-UNAM IN 109415 and IN100218 and by a sabbatical grant by PASPA-UNAM.