Energy momentum tensor

We shall be concerned with the doubled operators describing the energy-momentum tensor of the free Maxwell and Dirac fields according to the tilde conjugation rules, we have, respectively ... 

In fact, we will be interested in calculating only the (ll)-component, which corresponds to the physical energy-momentum tensor. 

Considering such choices for the parameters ay and using the explicit form of Gq1(x — x +y) in the 4-dimensional space-time (corresponding to N = 3), we obtain the renormalized o-dependenl, energy-momentum tensor in the general case, for both Maxwell and Dirac fields ... 

To show how powerful our method is, let us consider fermions confined in a tridimensional box at finite temperature. The energy-momentum tensor is a long expression for the general case of a parallelepiped box, but it follows from Eq. (29) that the Casimir energy for a cubic box of edge L is given by... 

We employ conformally coupled energy momentum tensor for the massless scalar field... 

The integration must go until the fluid-vacuum interface at rs, where an exterior vacuum solution continues the interior one. The details of matching conditions can be found in Ref. [14]. Applying them on the present problem we get the continuity of certain derivatives of the metric tensor, and those for the energy-momentum tensor result in the single equation,... 

Starting from the standard QED Lagrangian, the Hamiltonian characterizing a system of interacting electrons in a static external potential (x) is readily obtained as the 00-component of the energy-momentum tensor (see e.g. [35]),... 

In 0(3) electrodynamics, the stress energy momentum tensor is defined [11-20] as... 

For example, the energy momentum tensor T(z) = J2z n 2Ln = Y(lo,z) has the following operator product expansion ... 

For example, it can be shown that the energy momentum tensor due to A is [46]... 

The symmetric energy-momentum tensor (T, ) of electromagnetism in the vacuum can be defined from the Majorana equation using the matrices... 

Only eight of the nine matrices (850) are independent, and they form a basis for the SU(3) group, which is used for strong-field theory [46]. Therefore, the energy-momentum tensor is SU(3) invariant. 

In utilizing a complex three-vector (self-dual tensor) rather than a real antisymmetric tensor to describe the electromagnetic field, Hillion and Quinnez discussed the equivalence between the 2-spinor field and the complex electromagnetic field [63]. Using a Hertz potential [64] instead of the standard 4-vector potential in this model, they derived an energy momentum tensor out of which Beltrami-type field relations emerged. This development proceeded from the Maxwell equations in free homogeneous isotropic space... 

We will describe here the current status of the supernova research and outline ongoing projects to distinguish between a cosmological constant or a vacuum density contribution to the energy-momentum tensor in the Einstein equation. 

The energy-momentum tensor derived from Noether s theorem for electrodynamics... 

Energy-momentum conservation is expressed by dvT = 0 for a closed system. If Tfi were a symmetric tensor (when converted to 7 /x"), this would be assured because i f Tfi = 0 by construction. Since the gauge field part of the tensor deduced from Noether s theorem is not symmetric, this requires special consideration, as discussed below. A symmetric energy-momentum tensor is required for any eventual unification of quantum field theory and general relativity [422], The fermion field energy and momentum are... 

For the Maxwell field, the energy-momentum tensor Tfi(A) derived from Noether s theorem is unsymmetric, and not gauge invariant, in contrast to the symmetric stress tensor derived directly from Maxwell s equations [318], Consider the symmetric tensor 0 = T + AT, where... 

The Lagrangian density for a massless fermion field interacting with the SU 2) gauge field defines the Noether energy-momentum tensor... 

The S U(2) energy-momentum tensor can be symmetrized and made gauge invariant by adding an incremental tensor... 

Because 3MAdoes not reduce to terms that vanish even for a noninteracting field, this construction must be verified. The energy and 3-momentum of the gauge field derived from the resulting symmetric energy-momentum tensor are... 

Energy and momentum conservation can be directly deduced from the continuity equation for the energy momentum tensor [27, 28]. For the r" resulting from (2.1) one finds... 

---