Modification of quantum measure in area tensor Regge calculus and positivity

A comparative analysis of the versions of quantum measure in the area tensor Regge calculus is performed on the simplest configurations of the system. The quantum measure is constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. As we have found earlier, it is possible to implement also the correspondence principle (proportionality of the Lorentzian (Euclidean) measure to $e^{iS}$ ($e^{-S}$), $S$ being the action). For that a certain kind of the connection representation of the Regge action should be used, namely, as a sum of independent contributions of selfdual and antiselfdual sectors (that is, effectively 3-dimensional ones). There are two such representations, the (anti)selfdual connections being SU(2) or SO(3) rotation matrices according to the two ways of decomposing full SO(4) group, as SU(2) $\times$ SU(2) or SO(3) $\times$ SO(3). The measure from SU(2) rotations although positive on physical surface violates positivity outside this surface in the general configuration space of arbitrary independent area tensors. The measure based on SO(3) rotations is expected to be positive in this general configuration space on condition that the scale of area tensors considered as parameters is bounded from above by the value of the order of Plank unit.Comment: 10 pages, plain LaTe

Feynman path integral in area tensor Regge calculus and correspondence principle

The quantum measure in area tensor Regge calculus can be constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. This construction does not necessarily mean that Lorentzian (Euclidean) measure satisfies correspondence principle, that is, takes the form proportional to $e^{iS}$ ($e^{-S}$) where $S$ is the action. Requirement to fit this principle means some restriction on the action, or, in the context of representation of the Regge action in terms of independent rotation matrices (connections), restriction on such representation. We show that the representation based on separate treatment of the selfdual and antiselfdual rotations allows to modify the derivation and give sense to the conditionally convergent integrals to implement both the canonical quantisation and correspondence principles. If configurations are considered such that the measure is factorisable into the product of independent measures on the separate areas (thus far it was just the case in our analysis), the considered modification of the measure does not effect the vacuum expectation values.Comment: 9 pages, plain LaTe

The simplest Regge calculus model in the canonical form

Dynamics of a Regge three-dimensional (3D) manifold in a continuous time is considered. The manifold is closed consisting of the two tetrahedrons with identified corresponding vertices. The action of the model is that obtained via limiting procedure from the general relativity (GR) action for the completely discrete 4D Regge calculus. It closely resembles the continuous general relativity action in the Hilbert-Palatini (HP) form but possesses finite number of the degrees of freedom. The canonical structure of the theory is described. Central point is appearance of the new relations with time derivatives not following from the Lagrangian but serving to ensure completely discrete 4D Regge calculus origin of the system. In particular, taking these into account turns out to be necessary to obtain the true number of the degrees of freedom being the number of linklengths of the 3D Regge manifold at a given moment of time.Comment: LaTeX, 7 page

On the length expectation values in quantum Regge calculus

Regge calculus configuration superspace can be embedded into a more general superspace where the length of any edge is defined ambiguously depending on the 4-tetrahedron containing the edge. Moreover, the latter superspace can be extended further so that even edge lengths in each the 4-tetrahedron are not defined, only area tensors of the 2-faces in it are. We make use of our previous result concerning quantisation of the area tensor Regge calculus which gives finite expectation values for areas. Also our result is used showing that quantum measure in the Regge calculus can be uniquely fixed once we know quantum measure on (the space of the functionals on) the superspace of the theory with ambiguously defined edge lengths. We find that in this framework quantisation of the usual Regge calculus is defined up to a parameter. The theory may possess nonzero (of the order of Plank scale) or zero length expectation values depending on whether this parameter is larger or smaller than a certain value. Vanishing length expectation values means that the theory is becoming continuous, here {\it dynamically} in the originally discrete framework.Comment: 11 pages, plain LaTe

Area expectation values in quantum area Regge calculus

The Regge calculus generalised to independent area tensor variables is considered. The continuous time limit is found and formal Feynman path integral measure corresponding to the canonical quantisation is written out. The quantum measure in the completely discrete theory is found which possesses the property to lead to the Feynman path integral in the continuous time limit whatever coordinate is chosen as time. This measure can be well defined by passing to the integration over imaginary field variables (area tensors). Averaging with the help of this measure gives finite expectation values for areas.Comment: 9 pages, LaTeX, possible relation to quantisation of the usual length-based Regge calculus is discusse

On the area expectation values in area tensor Regge calculus in the Lorentzian domain

Wick rotation in area tensor Regge calculus is considered. The heuristical expectation is confirmed that the Lorentzian quantum measure on a spacelike area should coincide with the Euclidean measure at the same argument. The consequence is validity of probabilistic interpretation of the Lorentzian measure as well (on the real, i.e. spacelike areas).Comment: LaTeX, 7 pages, introduction and discussion given in more detail, references adde

On the Faddeev-Popov determinant in Regge calculus

The functional integral measure in the 4D Regge calculus normalised w.r.t. the DeWitt supermetric on the space of metrics is considered. The Faddeev-Popov factor in the measure is shown according to the previous author's work on the continuous fields in Regge calculus to be generally ill-defined due to the conical singularities. Possible resolution of this problem is discretisation of the gravity ghost (gauge) field by, e.g., confining ourselves to the affine transformations of the affine frames in the simplices. This results in the singularity of the functional measure in the vicinity of the flat background, where part of the physical degrees of freedom connected with linklengths become gauge ones.Comment: 5 pages, LaTe

Path integral measure in Regge calculus from the functional Fourier transform

The problem of fixing measure in the path integral for the Regge-discretised gravity is considered from the viewpoint of it's "best approximation" to the already known formal continuum general relativity (GR) measure. A rigorous formulation may consist in comparing functional Fourier transforms of the measures, i.e. characteristic or generating functionals, and requiring these to coincide on some dense set in the functional space. The possibility for such set to exist is due to the Regge manifold being a particular case of general Riemannian one (Regge calculus is a minisuperspace theory). The two versions of the measure are obtained depending on what metric tensor, covariant or contravariant one, is taken as fundamental field variable. The closed expressions for the measure are obtained in the two simple cases of Regge manifold. These turn out to be quite reasonable one of them indicating that appropriately defined continuum limit of the Regge measure would reproduce the original continuum GR measure.Comment: 9 pages, LaTeX, misprints in a formula (eq. (11)) remove

On the CP-odd Nucleon Potential

The CP-odd nucleon potential for different models of CP violation in the one meson exchange approximation is studied. It is shown that the main contribution is due to the $\pi$-meson exchange which leads to a simple one parameter CP-odd nucleon potential.Comment: 12 pages, RevTex, UM-P-92/114, OZ-92/3

Rotation and twist regular modes for trapped ghosts

A parameter-independent notion of stationary slow motion is formulated then applied to the case of stationary rotation of massless trapped ghosts. The excitations correspond to a rotation mode with angular momentum $J\neq 0$ and twist modes. It is found that the rotation mode, which has no parity, causes excess in the angular velocity of dragged distant coordinate frames in one sheet of the wormhole while in the other sheet the angular velocity of the ghosts is that of rotating stars: $2J/r^3$. As to the twist modes, which all have parity, they cause excess in the angular velocity of one of the throat's poles with respect to the other.Comment: 11 pages, 3 figures; General Relativity and Gravitation - 201