In the light of modern quantum field theoretical investigations it has become clear that all physical observables should be associated with extended but finite spacetime domains [169, 168]. Thus observables are always associated with open subsets of spacetime whose closure is compact, i.e. they are quasi-local. Quantities associated with spacetime points or with the whole spacetime are not observable in this sense. In particular, global quantities, such as the total energy or electric charge, should be considered as the limit of quasi-locally defined quantities. Thus the idea of quasi-locality is not new in physics. Although apparently in classical non-gravitational physics this is not obligatory, we adopt this view in talking about energy-momentum and angular momentum even of classical matter fields in Minkowski spacetime. Originally the introduction of these quasi-local quantities was motivated by the analogous gravitational quasi-local quantities [354, 358]. Since, however, many of the basic concepts and ideas behind the various gravitational quasi-local energy-momentum and angular momentum definitions can be understood from the analogous non-gravitational quantities in Minkowski spacetime, we devote the present section to the discussion of them and their properties.

To define the quasi-local conserved quantities in Minkowski spacetime, first observe that for any Killing vector the 3-form is closed, and hence, by the triviality of the third de Rham cohomology class, , it is exact: For some 2-form we have . may be called a ‘superpotential’ for the conserved current 3-form . (However, note that while the superpotential for the gravitational energy-momentum expressions of the next Section 3 is a local function of the general field variables, the existence of this ‘superpotential’ is a consequence of the field equations and the Killing nature of the vector field . The existence of globally defined superpotentials that are local functions of the field variables can be proven even without using the Poincaré lemma [388].) If is (the dual of) another superpotential for the same current , then by and the dual superpotential is unique up to the addition of an exact 2-form. If therefore is any closed orientable spacelike 2-surface in the Minkowski spacetime then the integral of on is free from this ambiguity. Thus if is any smooth compact spacelike hypersurface with smooth 2-boundary , then

depends only on . Hence it is independent of the actual Cauchy surface of the domain of dependence because all the spacelike Cauchy surfaces for have the same common boundary . Thus can equivalently be interpreted as being associated with the whole domain of dependence , and hence quasi-local in the sense of [169, 168] above. It defines the linear maps and by , i.e. they are elements of the corresponding dual spaces. Under Lorentz rotations of the Cartesian coordinates, and transform as a Lorentz vector and anti-symmetric tensor, respectively, whilst under the translation of the origin, is unchanged while . Thus and may be interpreted as the quasi-local energy-momentum and angular momentum of the matter fields associated with the spacelike 2-surface , or, equivalently, to . Then the quasi-local mass and Pauli-Lubanski spin are defined, respectively, by the usual formulae and . (If , then the dimensionally correct definition of the Pauli-Lubanski spin is .) As a consequence of the definitions holds, i.e. if is timelike then is spacelike or zero, but if is null (i.e. ) then is spacelike or proportional to .Obviously, we can form the flux integral of the current on the hypersurface even if is not a Killing vector, even in general curved spacetime:

Then, however, the integral does depend on the hypersurface, because this is not connected with the spacetime symmetries. In particular, the vector field can be chosen to be the unit timelike normal of . Since the component of the energy-momentum tensor is interpreted as the energy-density of the matter fields seen by the local observer , it would be legitimate to interpret the corresponding integral as ‘the quasi-local energy of the matter fields seen by the fleet of observers being at rest with respect to ’. Thus defines a different concept of the quasi-local energy: While that based on is linked to some absolute element, namely to the translational Killing symmetries of the spacetime and the constant timelike vector fields can be interpreted as the observers ‘measuring’ this energy, is completely independent of any absolute element of the spacetime and is based exclusively on the arbitrarily chosen fleet of observers. Thus, while is independent of the actual normal of , (for non-Killing ) depends on intrinsically and is a genuine 3-hypersurface rather than a 2-surface integral.If , the orthogonal projection to , then the part of the energy-momentum tensor is interpreted as the momentum density seen by the observer . Hence

is the square of the mass density of the matter fields, where is the spatial metric in the plane orthogonal to . If satisfies the dominant energy condition (i.e. is a future directed non-spacelike vector for any future directed non-spacelike vector , see for example [175]), then this is non-negative, and hence

can also be interpreted as the quasi-local mass of the matter fields seen by the fleet of observers being at rest with respect to , even in general curved spacetime. However, although in Minkowski spacetime for the four translational Killing vectors gives the four components of the energy-momentum , the mass is different from . In fact, while is defined as the Lorentzian norm of with respect to the metric on the space of the translations, in the definition of first the norm of the current is taken with respect to the pointwise physical metric of the spacetime, and then its integral is taken. Nevertheless, because of the more advantageous properties (see Section 2.2.3 below), we prefer to represent the quasi-local energy(-momentum and angular momentum) of the matter fields in the form instead of .Thus even if there is a gauge invariant and unambiguously defined energy-momentum density of the matter fields, it is not a priori clear how the various quasi-local quantities should be introduced. We will see in the second part of the present review that there are specific suggestions for the gravitational quasi-local energy that are analogous to , others to and some to .

In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not necessarily flat) spacetime (see for example [212, 396] and references therein) the configuration and momentum variables, and , respectively, are fields on a connected 3-manifold , which is interpreted as the typical leaf of a foliation of the spacetime. The foliation can be characterized on by a function , called the lapse. The evolution of the states in the spacetime is described with respect to a vector field (‘evolution vector field’ or ‘general time axis’), where is the future directed unit normal to the leaves of the foliation and is some vector field, called the shift, being tangent to the leaves. If the matter fields have gauge freedom, then the dynamics of the system is constrained: Physical states can be only those that are on the constraint surface, specified by the vanishing of certain functions , , of the canonical variables and their derivatives up to some finite order, where is the covariant derivative operator in . Then the time evolution of the states in the phase space is governed by the Hamiltonian, which has the form

Here is the induced volume element, the coefficients and are local functions of the canonical variables and their derivatives up to some finite order, the ’s are functions on , and is a local function of the canonical variables, the lapse, the shift, the functions , and their derivatives up to some finite order. The part of the Hamiltonian generates gauge motions in the phase space, and the functions are interpreted as the freely specifiable ‘gauge generators’. However, if we want to recover the field equations for (which are partial differential
equations on the spacetime with smooth coefficients for the smooth field ) on the phase
space as the Hamilton equations and not some of their distributional generalizations,
then the functional differentiability of must be required in the strong sense
of [387]^{1}.
Nevertheless, the functional differentiability (and, in the asymptotically flat case, also the existence) of
requires some boundary conditions on the field variables, and may yield restrictions on the form of
. It may happen that for a given only too restrictive boundary conditions would be able to ensure
the functional differentiability of the Hamiltonian, and hence the ‘quasi-local phase space’ defined with
these boundary conditions would contain only very few (or no) solutions of the field equations. In this case
should be modified. In fact, the boundary conditions are connected to the nature of the physical
situations considered. For example, in electrodynamics different boundary conditions must be imposed if the
boundary is to represent a conducting or an insulating surface. Unfortunately, no universal
principle or ‘canonical’ way of finding the ‘correct’ boundary term and the boundary conditions is
known.
In the asymptotically flat case the value of the Hamiltonian on the constraint surface defines the total
energy-momentum and angular momentum, depending on the nature of , in which the total divergence
corresponds to the ambiguity of the superpotential 2-form : An identically conserved
quantity can always be added to the Hamiltonian (provided its functional differentiability is preserved).
The energy density and the momentum density of the matter fields can be recovered as the
functional derivative of with respect to the lapse and the shift , respectively. In
principle, the whole analysis can be repeated quasi-locally too. However, apart from the promising
achievements of [7, 8, 327] for the Klein-Gordon, Maxwell, and the Yang-Mills-Higgs fields, as
far as we know, such a systematic quasi-local Hamiltonian analysis of the matter fields is still
lacking.

Suppose that the matter fields satisfy the dominant energy condition. Then is also non-negative for any non-spacelike , and, obviously, is zero precisely when on , and hence, by the conservation laws (see for example Page 94 of [175]), on the whole domain of dependence . Obviously, if and only if is null on . Then by the dominant energy condition it is a future pointing vector field on , and holds. Therefore, on has a null eigenvector with zero eigenvalue, i.e. its algebraic type on is pure radiation.

The properties of the quasi-local quantities based on in Minkowski spacetime are, however, more interesting. Namely, assuming that the dominant energy condition is satisfied, one can prove [354, 358] that

- is a future directed nonspacelike vector, ;
- if and only if on ;
- if and only if the algebraic type of the matter on is pure radiation, i.e. holds for some constant null vector . Then for some non-negative function , whenever , where and ;
- For the angular momentum has the form , where . Thus, in particular, the Pauli-Lubanski spin is zero.

Therefore, the vanishing of the quasi-local energy-momentum characterizes the ‘vacuum state’ of the classical matter fields completely, and the vanishing of the quasi-local mass is equivalent to special configurations representing pure radiation.

Since and are integrals of functions on a hypersurface, they are obviously additive, i.e. for example for any two hypersurfaces and (having common points at most on their boundaries and ) one has . On the other hand, the additivity of is a slightly more delicate problem. Namely, and are elements of the dual space of the translations, and hence we can add them and, as in the previous case, we obtain additivity. However, this additivity comes from the absolute parallelism of the Minkowski spacetime: The quasi-local energy-momenta of the different 2-surfaces belong to one and the same vector space. If there were no natural connection between the Killing vectors on different 2-surfaces, then the energy-momenta would belong to different vector spaces, and they could not be added. We will see that the quasi-local quantities discussed in Sections 7, 8, and 9 belong to vector spaces dual to their own ‘quasi-Killing vectors’, and there is no natural way of adding the energy-momenta of different surfaces.

If extends either to spatial or future null infinity, then, as is well known, the existence of the limit of the quasi-local energy-momentum can be ensured by slightly faster than (for example by ) fall-off of the energy-momentum tensor, where is any spatial radial distance. However, the finiteness of the angular momentum and centre-of-mass is not ensured by the fall-off. Since the typical fall-off of - for example for the electromagnetic field - is , we may not impose faster than this, because otherwise we would exclude the electromagnetic field from our investigations. Thus, in addition to the fall-off, six global integral conditions for the leading terms of must be imposed. At the spatial infinity these integral conditions can be ensured by explicit parity conditions, and one can show that the ‘conservation equations’ (as evolution equations for the energy density and momentum density) preserve these fall-off and parity conditions [364].

Although quasi-locally the vanishing of the mass does not imply the vanishing of the matter fields themselves (the matter fields must be pure radiative field configurations with plane wave fronts), the vanishing of the total mass alone does imply the vanishing of the fields. In fact, by the vanishing of the mass the fields must be plane waves, furthermore by they must be asymptotically vanishing at the same time. However, a plane wave configuration can be asymptotically vanishing only if it is vanishing.

For the (real or complex) linear massless scalar field and the Yang-Mills fields, represented by the symmetric spinor fields , , where is the dimension of the gauge group, the vanishing of the quasi-local mass is equivalent [365] to plane waves and the pp-wave solutions of Coleman [118], respectively. Then the condition implies that these fields are completely determined on the whole by their value on (whenever the spinor fields are necessarily null: , where are complex functions and is a constant spinor field such that ). Similarly, the null linear zero-rest-mass fields on with any spin and constant spinor are completely determined by their value on . Technically, these results are based on the unique complex analytic structure of the 2-surfaces foliating , where , and by the field equations the complex functions and turn out to be anti-holomorphic [358]. Assuming, for the sake of simplicity, that is future and past convex in the sense of Section 4.1.3 below, the independent boundary data for such a pure radiative solution consist of a constant spinor field on and a real function with one and another with two variables. Therefore, the pure radiative modes on can be characterized completely by appropriate data (the so-called holographic data) on the ‘screen’ .

These ‘quasi-local radiative modes’ can be used to map any continuous spinor field on to a collection of holographic data. Indeed, the special radiative solutions of the form (with fixed constant spinor field ) together with their complex conjugate define a dense subspace in the space of all continuous spinor fields on . Thus every such spinor field can be expanded by the special radiative solutions, and hence can also be represented by the corresponding family of holographic data. Therefore, if we fix a foliation of by spacelike Cauchy surfaces , then every spinor field on can also be represented on by a time dependent family of holographic data, too [365]. This fact may be a specific manifestation in the classical non-gravitational physics of the holographic principle (see Section 13.4.2).

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