8,912 research outputs found
The Fermionic Signature Operator in the Exterior Schwarzschild Geometry
The structure of the solution space of the Dirac equation in the exterior
Schwarzschild geometry is analyzed. Representing the space-time inner product
for families of solutions with variable mass parameter in terms of the
respective scalar products, a so-called mass decomposition is derived. This
mass decomposition consists of a single mass integral involving the fermionic
signature operator as well as a double integral which takes into account the
flux of Dirac currents across the event horizon. The spectrum of the fermionic
signature operator is computed. The corresponding generalized fermionic
projector states are analyzed.Comment: 26 pages, LaTeX, 1 figure, minor improvements, references added
(published version
Dynamical Gravitational Coupling as a Modified Theory of General Relativity
A modified theory of general relativity is proposed, where the gravitational
constant is replaced by a dynamical variable in space-time. The dynamics of the
gravitational coupling is described by a family of parametrized null geodesics,
implying that the gravitational coupling at a space-time point is determined by
solving transport equations along all null geodesics through this point.
General relativity with dynamical gravitational coupling (DGC) is introduced.
We motivate DGC from general considerations and explain how it arises in the
context of causal fermion systems. The underlying physical idea is that the
gravitational coupling is determined by microscopic structures on the Planck
scale which propagate with the speed of light.
In order to clarify the mathematical structure, we analyze the conformal
behavior and prove local existence and uniqueness of the time evolution. The
differences to Einstein's theory are worked out in the examples of the
Friedmann-Robertson-Walker model and the spherically symmetric collapse of a
shell of matter. Potential implications for the problem of dark matter and for
inflation are discussed. It is shown that the effects in the solar system are
too small for being observable in present-day experiments.Comment: 43 pages, LaTeX, 9 figures, 6 ancillary file
An Integral Spectral Representation of the Massive Dirac Propagator in the Kerr Geometry in Eddington-Finkelstein-type Coordinates
We consider the massive Dirac equation in the non-extreme Kerr geometry in
horizon-penetrating advanced Eddington-Finkelstein-type coordinates and derive
a functional analytic integral representation of the associated propagator
using the spectral theorem for unbounded self-adjoint operators, Stone's
formula, and quantities arising in the analysis of Chandrasekhar's separation
of variables. This integral representation describes the dynamics of Dirac
particles outside and across the event horizon, up to the Cauchy horizon. In
the derivation, we first write the Dirac equation in Hamiltonian form and show
the essential self-adjointness of the Hamiltonian. For the latter purpose, as
the Dirac Hamiltonian fails to be elliptic at the event and the Cauchy horizon,
we cannot use standard elliptic methods of proof. Instead, we employ a new,
general method for mixed initial-boundary value problems that combines results
from the theory of symmetric hyperbolic systems with near-boundary elliptic
methods. In this regard and since the time evolution may not be unitary because
of Dirac particles impinging on the ring singularity, we also impose a suitable
Dirichlet-type boundary condition on a time-like inner hypersurface placed
inside the Cauchy horizon, which has no effect on the dynamics outside the
Cauchy horizon. We then compute the resolvent of the Dirac Hamiltonian via the
projector onto a finite-dimensional, invariant spectral eigenspace of the
angular operator and the radial Green's matrix stemming from Chandrasekhar's
separation of variables. Applying Stone's formula to the spectral measure of
the Hamiltonian in the spectral decomposition of the Dirac propagator, that is,
by expressing the spectral measure in terms of this resolvent, we obtain an
explicit integral representation of the propagator.Comment: 31 pages, 1 figure, details added, references added, minor
correction
Universal upper bounds on the Bose-Einstein condensate and the Hubbard star
For hard-core bosons on an arbitrary lattice with sites and
independent of additional interaction terms we prove that the hard-core
constraint itself already enforces a universal upper bound on the Bose-Einstein
condensate given by . This bound can only be attained for
one-particle states with equal amplitudes with respect to the
hard-core basis (sites) and when the corresponding -particle state
is maximally delocalized. This result is generalized to the
maximum condensate possible within a given sublattice. We observe that such
maximal local condensation is only possible if the mode entanglement between
the sublattice and its complement is minimal. We also show that the maximizing
state is related to the ground state of a bosonic `Hubbard star'
showing Bose-Einstein condensation.Comment: to appear in Phys. Rev.
Influence of the Fermionic Exchange Symmetry beyond Pauli's Exclusion Principle
Pauli's exclusion principle has a strong impact on the properties of most
fermionic quantum systems. Remarkably, the fermionic exchange symmetry implies
further constraints on the one-particle picture. By exploiting those
generalized Pauli constraints we derive a measure which quantifies the
influence of the exchange symmetry beyond Pauli's exclusion principle. It is
based on a geometric hierarchy induced by the exclusion principle constraints.
We provide a proof of principle by applying our measure to a simple model. In
that way, we conclusively confirm the physical relevance of the generalized
Pauli constraints and show that the fermionic exchange symmetry can have an
influence on the one-particle picture beyond Pauli's exclusion principle. Our
findings provide a new perspective on fermionic multipartite correlation since
our measure allows one to distinguish between static and dynamic correlations.Comment: title has been changed; very close to published versio
The Fermionic Signature Operator and Quantum States in Rindler Space-Time
The fermionic signature operator is constructed in Rindler space-time. It is
shown to be an unbounded self-adjoint operator on the Hilbert space of
solutions of the massive Dirac equation. In two-dimensional Rindler space-time,
we prove that the resulting fermionic projector state coincides with the
Fulling-Rindler vacuum. Moreover, the fermionic signature operator gives a
covariant construction of general thermal states, in particular of the Unruh
state. The fermionic signature operator is shown to be well-defined in
asymptotically Rindler space-times. In four-dimensional Rindler space-time, our
construction gives rise to new quantum states.Comment: 27 pages, LaTeX, more details on self-adjoint extension (published
version
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