30 research outputs found
Particle Creation at a Point Source by Means of Interior-Boundary Conditions
We consider a way of defining quantum Hamiltonians involving particle
creation and annihilation based on an interior-boundary condition (IBC) on the
wave function, where the wave function is the particle-position representation
of a vector in Fock space, and the IBC relates (essentially) the values of the
wave function at any two configurations that differ only by the creation of a
particle. Here we prove, for a model of particle creation at one or more point
sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian
can indeed be rigorously defined in this way without the need for any
ultraviolet regularization, and that it is self-adjoint. We prove further that
introducing an ultraviolet cut-off (thus smearing out particles over a positive
radius) and applying a certain known renormalization procedure (taking the
limit of removing the cut-off while subtracting a constant that tends to
infinity) yields, up to addition of a finite constant, the Hamiltonian defined
by the IBC.Comment: 41 page
Unconventional magnets in external magnetic fields
This short review surveys phenomena observed when a magnetic field is applied
to a system of localised spins on a lattice. Its focus is on frustrated magnets
in dimension . The interplay of field and entropy is illustrated in
the context of their unusual magnetocaloric properties, where field-tuned
degeneracies assert themselves. Magnetisation plateaux can reveal the physics
of fluctuations, with unusual excitations (such as local modes, extended string
defects or monopoles) involved in plateau termination. Field-tuning lattice
geometry is the final topic, where mechanisms for dimensional reduction and
conversion between different lattice types are discussed.Comment: Plenary Talk at HFM 2008 Conferenc
Linear independence of localized magnon states
At the magnetic saturation field, certain frustrated lattices have a class of
states known as "localized multi-magnon states" as exact ground states. The
number of these states scales exponentially with the number of spins and
hence they have a finite entropy also in the thermodynamic limit
provided they are sufficiently linearly independent. In this article we present
rigorous results concerning the linear dependence or independence of localized
magnon states and investigate special examples. For large classes of spin
lattices including what we called the orthogonal type and the isolated type as
well as the kagom\'{e}, the checkerboard and the star lattice we have proven
linear independence of all localized multi-magnon states. On the other hand the
pyrochlore lattice provides an example of a spin lattice having localized
multi-magnon states with considerable linear dependence.Comment: 23 pages, 6 figure
Complex charges, time reversal asymmetry, and interior-boundary conditions in quantum field theory
While fundamental physically realistic Hamiltonians should be invariant under
time reversal, time asymmetric Hamiltonians can occur as mathematical
possibilities or effective Hamiltonians. Here, we study conditions under which
non-relativistic Hamiltonians involving particle creation and annihilation, as
come up in quantum field theory (QFT), are time asymmetric. It turns out that
the time reversal operator T can be more complicated than just complex
conjugation, which leads to the question which criteria determine the correct
action of time reversal. We use Bohmian trajectories for this purpose and show
that time reversal symmetry can be broken when charges are permitted to be
complex numbers, where `charge' means the coupling constant in a QFT that
governs the strength with which a fermion emits and absorbs bosons. We pay
particular attention to the technique for defining Hamiltonians with particle
creation based on interior-boundary conditions, and we find them to generically
be time asymmetric. Specifically, we show that time asymmetry for complex
charges occurs whenever not all charges have equal or opposite phase. We
further show that, in this case, the corresponding ground states can have
non-zero probability currents, and we determine the effective potential between
fermions of complex charge.Comment: 22 pages LaTeX, 2 figure
Interior-boundary conditions for many-body Dirac operators and codimension-1 boundaries
We are dealing with boundary conditions for Dirac-type operators, i.e., first
order differential operators with matrix-valued coefficients, including in
particular physical many-body Dirac operators. We characterize (what we
conjecture is) the general form of reflecting boundary conditions (which
includes known boundary conditions such as the one of the MIT bag model) and,
as our main goal, of interior-boundary conditions (IBCs). IBCs are a new
approach to defining UV-regular Hamiltonians for quantum field theories without
smearing particles out or discretizing space. For obtaining such Hamiltonians,
the method of IBCs provides an alternative to renormalization and has been
successfully used so far in non-relativistic models, where it could be applied
also in cases in which no renormalization method was known. A natural next
question about IBCs is how to set them up for the Dirac equation, and here we
take first steps towards the answer. For quantum field theories, the relevant
boundary consists of the surfaces in -particle configuration space
on which two particles have the same location in
. While this boundary has codimension 3, we focus here on the
more basic situation in which the boundary has codimension 1 in configuration
space. We describe specific examples of IBCs for the Dirac equation, we prove
for some of these examples that they rigorously define self-adjoint
Hamiltonians, and we develop the general form of IBCs for Dirac-type operators.Comment: 33 pages LaTeX, no figures; v2: Section 2.4 and Appendix A adde