35 research outputs found
Time Gauge Fixing and Hilbert Space in Quantum String Cosmology
Recently the low-energy effective string theory has been used by Gasperini
and Veneziano to elaborate a very interesting scenario for the early history of
the universe (``birth of the universe as quantum scattering''). Here we
investigate the gauge fixing and the problem of the definition of a global time
parameter for this model, and we obtain the positive norm Hilbert space of
states.Comment: 13 pages, Plain TEX, no figure
Partial and Complete Observables for Hamiltonian Constrained Systems
We will pick up the concepts of partial and complete observables introduced
by Rovelli in order to construct Dirac observables in gauge systems. We will
generalize these ideas to an arbitrary number of gauge degrees of freedom.
Different methods to calculate such Dirac observables are developed. For
background independent field theories we will show that partial and complete
observables can be related to Kucha\v{r}'s Bubble Time Formalism. Moreover one
can define a non-trivial gauge action on the space of complete observables and
also state the Poisson brackets of these functions.
Additionally we will investigate, whether it is possible to calculate Dirac
observables starting with partially invariant partial observables, for instance
functions, which are invariant under the spatial diffeomorphism group.Comment: 38 page
Geometrodynamics of Schwarzschild Black Holes
The curvature coordinates of a Schwarz\-schild spacetime are turned
into canonical coordinates on the phase space of spherically
symmetric black holes. The entire dynamical content of the Hamiltonian theory
is reduced to the constraints requiring that the momenta vanish. What remains is a conjugate pair of canonical variables and
whose values are the same on every embedding. The coordinate is the
Schwarzschild mass, and the momentum the difference of parametrization
times at right and left infinities. The Dirac constraint quantization in the
new representation leads to the state functional which describes an unchanging superposition of black holes with different
masses. The new canonical variables may be employed in the study of collapsing
matter systems.Comment: 44 pages, Latex file, UU-REL-94/3/
Unconstrained Hamiltonian Formulation of SU(2) Gluodynamics
SU(2) Yang-Mills field theory is considered in the framework of the
generalized Hamiltonian approach and the equivalent unconstrained system is
obtained using the method of Hamiltonian reduction. A canonical transformation
to a set of adapted coordinates is performed in terms of which the
Abelianization of the Gauss law constraints reduces to an algebraic operation
and the pure gauge degrees of freedom drop out from the Hamiltonian after
projection onto the constraint shell. For the remaining gauge invariant fields
two representations are introduced where the three fields which transform as
scalars under spatial rotations are separated from the three rotational fields.
An effective low energy nonlinear sigma model type Lagrangian is derived which
out of the six physical fields involves only one of the three scalar fields and
two rotational fields summarized in a unit vector. Its possible relation to the
effective Lagrangian proposed recently by Faddeev and Niemi is discussed.
Finally the unconstrained analog of the well-known nonnormalizable groundstate
wave functional which solves the Schr\"odinger equation with zero energy is
given and analysed in the strong coupling limit.Comment: 20 pages REVTEX, no figures; final version to appear in Phys. Rev. D;
minor changes, notations simplifie
Hamiltonian reduction of SU(2) Dirac-Yang-Mills mechanics
The SU(2) gauge invariant Dirac-Yang-Mills mechanics of spatially homogeneous
isospinor and gauge fields is considered in the framework of the generalized
Hamiltonian approach. The unconstrained Hamiltonian system equivalent to the
model is obtained using the gaugeless method of Hamiltonian reduction. The
latter includes the Abelianization of the first class constraints, putting the
second class constraints into the canonical form and performing a canonical
transformation to a set of adapted coordinates such that a subset of the new
canonical pairs coincides with the second class constraints and part of the new
momenta is equal to the Abelian constraints. In the adapted basis the pure
gauge degrees of freedom automatically drop out from the consideration after
projection of the model onto the constraint shell. Apart from the elimination
of these ignorable degrees of freedom a further Hamiltonian reduction is
achieved due to the three dimensional group of rigid symmetry possessed by the
system.Comment: 25 pages Revtex, no figure
The York map as a Shanmugadhasan canonical transformation in tetrad gravity and the role of non-inertial frames in the geometrical view of the gravitational field
A new parametrization of the 3-metric allows to find explicitly a York map in
canonical ADM tetrad gravity, the two pairs of physical tidal degrees of
freedom and 14 gauge variables. These gauge quantities (generalized inertial
effects) are all configurational except the trace of
the extrinsic curvature of the instantaneous 3-spaces (clock
synchronization convention) of a non-inertial frame. The Dirac hamiltonian is
the sum of the weak ADM energy (whose density is coordinate-dependent due to the inertial
potentials) and of the first-class constraints. Then: i) The explicit form of
the Hamilton equations for the two tidal degrees of freedom in an arbitrary
gauge: a deterministic evolution can be defined only in a completely fixed
gauge, i.e. in a non-inertial frame with its pattern of inertial forces. ii) A
general solution of the super-momentum constraints, which shows the existence
of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge
group. It influences: a) the explicit form of the weak ADM energy and of the
super-momentum constraint; b) the determination of the shift functions and then
of the lapse one. iii) The dependence of the Hamilton equations for the two
pairs of dynamical gravitational degrees of freedom (the generalized tidal
effects) and for the matter, written in a completely fixed 3-orthogonal
Schwinger time gauge, upon the gauge variable ,
determining the convention of clock synchronization. Therefore it should be
possible (for instance in the weak field limit but with relativistic motion) to
try to check whether in Einstein's theory the {\it dark matter} is a gauge
relativistic inertial effect induced by .Comment: 90 page
Inflationary and Deflationary Branches in Extended Pre--Big Bang Cosmology
The pre--big bang cosmological scenario is studied within the context of the
Brans--Dicke theory of gravity. An epoch of superinflationary expansion may
occur in the pre--big bang phase of the Universe's history in a certain region
of parameter space. Two models are considered that contain a cosmological
constant in the gravitational and matter sectors of the theory, respectively.
Classical pre-- and post--big bang solutions are found for both models. The
existence of a curvature singularity forbids a classical transition between the
two branches. On the other hand, a quantum cosmological approach based on the
tunneling boundary condition results in a non--zero transition probability. The
transition may be interpreted as a spatial reflection of the wavefunction in
minisuperspace.Comment: 20 pages, latex, 3 figures available on reques
Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge
We define the {\it rest-frame instant form} of tetrad gravity restricted to
Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of
gauge transformations generated by the 14 first class constraints of the
theory, we define and solve the multitemporal equations associated with the
rotation and space diffeomorphism constraints, finding how the cotriads and
their momenta depend on the corresponding gauge variables. This allows to find
quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal
gauges and to find the Dirac observables for superspace in these gauges.
The construction of the explicit form of the transformation and of the
solution of the rotation and supermomentum constraints is reduced to solve a
system of elliptic linear and quasi-linear partial differential equations. We
then show that the superhamiltonian constraint becomes the Lichnerowicz
equation for the conformal factor of the 3-metric and that the last gauge
variable is the momentum conjugated to the conformal factor. The gauge
transformations generated by the superhamiltonian constraint perform the
transitions among the allowed foliations of spacetime, so that the theory is
independent from its 3+1 splittings. In the special 3-orthogonal gauge defined
by the vanishing of the conformal factor momentum we determine the final Dirac
observables for the gravitational field even if we are not able to solve the
Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted
to this completely fixed gauge.Comment: RevTeX file, 141 page
Hamilton's Formalism for Systems with Constraints
The main goal of these lectures is to introduce and review the Hamiltonian
formalism for classical constrained systems and in particular gauge theories.
Emphasis is put on the relation between local symmetries and constraints and on
the relation between Lagrangean and Hamiltonian symmetries.Comment: 52 pages, revised LATEX version, ETH-TH/93-48, Lectures given at the
Seminar "The Canonical Formalism in Classical and Quantum General
Relativity", Bad Honnef, September 9
Canonical and quantum FRW cosmological solutions in M-theory
We present the canonical and quantum cosmological investigation of a
spatially flat, four-dimensional Friedmann-Robertson-Walker (FRW) model that is
derived from the M-theory effective action obtained originally by Billyard,
Coley, Lidsey and Nilsson (BCLN). The analysis makes use of two sets of
canonical variables, the Shanmugadhasan gauge invariant canonical variables and
the ``hybrid'' variables which diagonalise the Hamiltonian. We find the
observables and discuss in detail the phase space of the classical theory. In
particular, a region of the phase space exists that describes a
four-dimensional FRW spacetime first contracting from a strong coupling regime
and then expanding to a weak coupling regime, while the internal space ever
contracts. We find the quantum solutions of the model and obtain the positive
norm Hilbert space of states. Finally, the correspondence between wave
functions and classical solutions is outlined.Comment: 32 pages, 11 figure