7 research outputs found
Discrete Spacetime and Relativistic Quantum Particles
We study a single quantum particle in discrete spacetime evolving in a causal
way. We see that in the continuum limit any massless particle with a two
dimensional internal degree of freedom obeys the Weyl equation, provided that
we perform a simple relabeling of the coordinate axes or demand rotational
symmetry in the continuum limit. It is surprising that this occurs regardless
of the specific details of the evolution: it would be natural to assume that
discrete evolutions giving rise to relativistic dynamics in the continuum limit
would be very special cases. We also see that the same is not true for
particles with larger internal degrees of freedom, by looking at an example
with a three dimensional internal degree of freedom that is not relativistic in
the continuum limit. In the process we give a formula for the Hamiltonian
arising from the continuum limit of massless and massive particles in discrete
spacetime.Comment: 6 page
Causal Fermions in Discrete Spacetime
In this paper, we consider fermionic systems in discrete spacetime evolving
with a strict notion of causality, meaning they evolve unitarily and with a
bounded propagation speed. First, we show that the evolution of these systems
has a natural decomposition into a product of local unitaries, which also holds
if we include bosons. Next, we show that causal evolution of fermions in
discrete spacetime can also be viewed as the causal evolution of a lattice of
qubits, meaning these systems can be viewed as quantum cellular automata.
Following this, we discuss some examples of causal fermionic models in discrete
spacetime that become interesting physical systems in the continuum limit:
Dirac fermions in one and three spatial dimensions, Dirac fields and briefly
the Thirring model. Finally, we show that the dynamics of causal fermions in
discrete spacetime can be efficiently simulated on a quantum computer.Comment: 16 pages, 1 figur
Quantum equilibration in finite time
It has recently been shown that small quantum subsystems generically
equilibrate, in the sense that they spend most of the time close to a fixed
equilibrium state. This relies on just two assumptions: that the state is
spread over many different energies, and that the Hamiltonian has
non-degenerate energy gaps. Given the same assumptions, it has also been shown
that closed systems equilibrate with respect to realistic measurements. We
extend these results in two important ways. First, we prove equilibration over
a finite (rather than infinite) time-interval, allowing us to bound the
equilibration time. Second, we weaken the non degenerate energy gaps condition,
showing that equilibration occurs provided that no energy gap is hugely
degenerate.Comment: 7 page
Quantum Systems Equilibrate Rapidly for Most Observables
Considering any Hamiltonian, any initial state, and measurements with a small
number of possible outcomes compared to the dimension, we show that most
measurements are already equilibrated. To investigate non-trivial equilibration
we therefore consider a restricted set of measurements. When the initial state
is spread over many energy levels, and we consider the set of observables for
which this state is an eigenstate, most observables are initially out of
equilibrium yet equilibrate rapidly. Moreover, all two-outcome measurements,
where one of the projectors is of low rank, equilibrate rapidly.Comment: Main Text: 5 pages, 1 figure. Appendices: 7 pages, 1 figur