Chopin Soo

The Chern-Simons invariant as the natural time variable for classical and quantum cosmology

Abstract We propose that the Chern-Simons invariant of the Ashtekar-sen connection is the natural internal time coordinate for classical and quantum cosmology. The reasons for this are a number of interesting properties of this functional, which we describe here. (1) It is a function on the gauge and diffeomorphism invariant configuration space, whose gradient is orthogonal to the two physical degrees of freedom, in the metric defined by the Ashtekar formulation of general relativity. (2) The imaginary part of the Chern-Simons form reduces in the limit of small cosmological constant, Λ, and solutions close to DeSitter spacetime, to the York extrinsic time coordinate. (3) Small matter-field excitations of the Chern-Simons state satisfy, by virtue for the quantum constraints, a functional Schrodinger equation in which the matter fields evolve on a DeSitter background in the Chern-Simons time. We than propose this is the natural vacuum state of the theory for Λ ≠ 0. (4) This time coordinate is periodic on the configuration space of Euclideanized spacetimes, due to the large gauge transformations, which means that physical expectation values for all states in non-perturbative quantum gravity will satisfy the KMS condition, and may then be interpreted as thermal states. (5) Forms for the physical Hamiltonians and inner product which support the proposal are suggested, and a new action principle for general relativity, as a geodesic principle on the connection superspace, is found.

Wavefunction of the Universe and Chern–Simons perturbation theory

The Chern?Simons exact solution of four-dimensional quantum gravity with nonvanishing cosmological constant is presented in metric variables as the partition function of Chern?Simons theory with nontrivial source. The perturbative expansion is given, and the wavefunction is computed to the lowest order of approximation for the Cauchy surface which is topologically a 3-sphere. The state is well-defined even at degenerate and vanishing values of the dreibein. Reality conditions for the Ashtekar variables are also taken into account, and remarkable features of the Chern?Simons state and their relevance to cosmology are pointed out.

Superspace dynamics and perturbations around 'emptiness'

Superspace parametrized by gauge potentials instead of metric three-geometries is discussed in the context of the Ashtekar variables. Gauge-fixing conditions which lead to the natural geometrical separation of physical from gauge modes are derived with the use of the supermetric in connection-superspace. A perturbation scheme about an unusual background which is inaccessible to conventional variables is presented. The resultant expansion retains much of the simplicity of Ashtekar’s formulation of General Relativity. Quantum mechanically, the super-Hamiltonian constraint assumes the form of the Dirac equation with an “intrinsic time” tied to the signature of the supermetric.

New length operator for loop quantum gravity

An alternative expression for the length operator in loop quantum gravity is presented. The operator is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space. The expression for the regularized length operator can moreover be understood both from a simple geometrical perspective as the average of a formula relating the length to area, volume and flux operators, and also consistently as the result of direct substitution of the densitized triad operator with the functional derivative operator into the regularized expression of the length. Both these derivations are discussed, and the origin of an undetermined overall factor in each case is also elucidated.

Self-dual variables, positive-semidefinite action, and discrete transformations in four-dimensional quantum gravity.

A positive-semidefinite Euclidean action for arbitrary four-topologies can be constructed by adding appropriate Yang-Mills and topological terms to the Samuel-Jacobson-Smolin action of gravity with (anti-)self-dual variables. Moreover, on shell, the (anti-)self-dual sector of the new theory corresponds precisely to all Einstein manifolds in four dimensions. The Lorentzian signature action and its analytic continuations are also considered. A self-contained discussion is given on the effects of discrete transformations {ital C}, {ital P}, and {ital T} on the Samuel-Jacobson-Smolin action, and other proposed actions which utilize self- or anti-self-dual variables as fundamental variables in the description of four-dimensional gravity.

Intrinsic time gravity and the Lichnerowicz-York equation

We investigate the effect on the Hamiltonian structure of general relativity of choosing an intrinsic time to fix the time slicing. 3-covariance with momentum constraint is maintained, but the Hamiltonian constraint is replaced by a algebraic equation for the trace of the momentum. This reveals a very simple structure with a local reduced Hamiltonian. The theory is easily generalized; in particular, the square of the Cotton–York tensor density can be added as an extra part of the potential while at the same time maintaining the classic 2 + 2 degrees of freedom. Initial data construction is simple in the extended intrinsic time formulation; we get a generalized Lichnerowicz–York equation with nice existence and uniqueness properties. Adding standard matter fields is quite straightforward.

Generalized Painleve-Gullstrand metrics

An obstruction to the implementation of spatially flat Painleve–Gullstrand (PG) slicings is demonstrated, and explicitly discussed for Reissner–Nordstrom and Schwarzschild–anti-deSitter spacetimes. Generalizations of PG slicings which are not spatially flat but which remain regular at the horizons are introduced. These metrics can be obtained from standard spherically symmetric metrics by physical Lorentz boosts. With these generalized PG metrics, problematic contributions to the imaginary part of the action in the Parikh–Wilczek derivation of Hawking radiation due to the obstruction can be avoided.

Wigner rotations, Bell states, and Lorentz invariance of entanglement and von Neumann entropy

We compute, for massive particles, the explicit Wigner rotations of one-particle states for arbitrary Lorentz transformations; and the explicit Hermitian generators of the infinite-dimensional unitary representation. For a pair of spin 1/2 particles, Einstein–Podolsky–Rosen–ell entangled states and their behaviour under the Lorentz group are analyzed in the context of quantum field theory. Group theoretical considerations suggest a convenient definition of the Bell states which is slightly different from the conventional assignment. The behaviour of Bell states under arbitrary Lorentz transformations can then be described succinctly. Reduced density matrices applicable to systems of identical particles are defined through Yangs prescription. The von Neumann entropy of each of the reduced density matrix is Lorentz invariant; and its relevance as a measure of entanglement is discussed, and illustrated with an explicit example. A regularization of the entropy in terms of generalized zeta functions is also suggested.

Quantum field theory with and without conical singularities: black holes with a cosmological constant and the multi-horizon scenario

Boundary conditions and the corresponding states of a quantum field theory depend on how the horizons are taken into account. There is an ambiguity as to which method is appropriate because different ways of incorporating the horizons lead to different results. We propose that a natural way of including the horizons is to first consider the Kruskal extension and then define the quantum field theory on the Euclidean section. Boundary conditions emerge naturally as consistency conditions of the Kruskal extension. We carry out the proposal for the explicit case of the Schwarzschild-de Sitter manifold with two horizons. The required period is the interesting condition that it is the lowest common multiple of divided by the surface gravity of both horizons. Restricting the ratio of the surface gravity of the horizons to rational numbers yields finite . The example also highlights some of the difficulties of the off-shell approach with conical singularities in the multi-horizon scenario and serves to illustrate the much richer interplay that can occur among horizons, quantum field theory and topology when the cosmological constant is not neglected in black-hole processes.

BRST cohomology and invariants of four-dimensional gravity in Ashtekar variables

We discuss the BRST cohomologies of the invariants associated with the description of classical and quantum gravity in four dimensions, using the Ashtekar variables. These invariants are constructed from several BRST cohomology sequences. They provide a systematic and clear characterization of non-local observables in general relativity with unbroken diffeomorphism invariance, and could yield further differential invariants for four-manifolds. The theory includes fluctuations of the vierbein fields, but there exits a non-trivial phase which can be expressed in terms of Wittens topological quantum field theory. In this phase, the descent sequences are degenerate, and the corresponding classical solutions can be identified with the conformally self-dual sector of Einstein manifolds. The full theory includes fluctuations which bring the system out of this sector while preserving diffeomorphism invariance.