2,507 research outputs found
General methods and properties for evaluation of continuum limits of discrete time quantum walks in one and two dimensions
Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and non relativistic regimes (Di Molfetta and Arrighi, 2020). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in (Di Molfetta and Arrighi, 2020) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions ("coins") admit nontrivial continuum limits in 1D+1. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. We also demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk (CTQW) when the coin is allowed to transition through the continuum limit process. Finally, we introduce the Plastic Quantum Walk, or a quantum walk which admits both continuous time and continuous spacetime limits and, as a novel result, we use our 1D+1 results to obtain necessary and sufficient conditions concerning which DTQWs admit plasticity in 2D+1, showing the resulting Hamiltonians. We consider coin operators as general 4 parameter unitary matrices, with parameters which are functions of the lattice step size . This dependence on encapsulates all functions of for which a Taylor series expansion in is well defined, making our results very general
Continuous-time quantum walks on one-dimension regular networks
In this paper, we consider continuous-time quantum walks (CTQWs) on
one-dimension ring lattice of N nodes in which every node is connected to its
2m nearest neighbors (m on either side). In the framework of the Bloch function
ansatz, we calculate the spacetime transition probabilities between two nodes
of the lattice. We find that the transport of CTQWs between two different nodes
is faster than that of the classical continuous-time random walk (CTRWs). The
transport speed, which is defined by the ratio of the shortest path length and
propagating time, increases with the connectivity parameter m for both the
CTQWs and CTRWs. For fixed parameter m, the transport of CTRWs gets slow with
the increase of the shortest distance while the transport (speed) of CTQWs
turns out to be a constant value. In the long time limit, depending on the
network size N and connectivity parameter m, the limiting probability
distributions of CTQWs show various paterns. When the network size N is an even
number, the probability of being at the original node differs from that of
being at the opposite node, which also depends on the precise value of
parameter m.Comment: Typos corrected and Phys. ReV. E comments considered in this versio
Four-dimensional understanding of quantum mechanics and Bell violation
While our natural intuition suggests us that we live in 3D space evolving in
time, modern physics presents fundamentally different picture: 4D spacetime,
Einstein's block universe, in which we travel in thermodynamically emphasized
direction: arrow of time. Suggestions for such nonintuitive and nonlocal living
in kind of "4D jello" come among others from: Lagrangian mechanics we use from
QFT to GR saying that history between fixed past and future situation is the
one optimizing action, special relativity saying that different velocity
observers have different present 3D hypersurface and time direction, general
relativity deforming shape of the entire spacetime up to switching time and
space below the black hole event horizon, or the CPT theorem concluding
fundamental symmetry between past and future.
Accepting this nonintuitive living in 4D spacetime: with present moment being
in equilibrium between past and future - minimizing tension as action of
Lagrangian, leads to crucial surprising differences from intuitive "evolving
3D" picture, in which we for example conclude satisfaction of Bell inequalities
- violated by the real physics. Specifically, particle in spacetime becomes own
trajectory: 1D submanifold of 4D, making that statistical physics should
consider ensembles like Boltzmann distribution among entire paths, what leads
to quantum behavior as we know from Feynman's Euclidean path integrals or
similar Maximal Entropy Random Walk (MERW). It results for example in Anderson
localization, or the Born rule with squares - allowing for violation of Bell
inequalities. Specifically, quantum amplitude turns out to describe probability
at the end of half-spacetime from a given moment toward past or future, to
randomly get some value of measurement we need to "draw it" from both time
directions, getting the squares of Born rules.Comment: 13 pages, 18 figure
Quantum walks and non-Abelian discrete gauge theory
A new family of discrete-time quantum walks (DTQWs) on the line with an exact
discrete gauge invariance is introduced. It is shown that the continuous
limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac
fermion coupled to usual gauge fields in spacetime. A discrete
generalization of the usual curvature is also constructed. An alternate
interpretation of these results in terms of superimposed Maxwell fields
and gauge fields is discussed in the Appendix. Numerical simulations
are also presented, which explore the convergence of the DTQWs towards their
continuous limit and which also compare the DTQWs with classical (i.e.
non-quantum) motions in classical fields. The results presented in this
article constitute a first step towards quantum simulations of generic
Yang-Mills gauge theories through DTQWs.Comment: 7 pages, 2 figure
Quantum Walks and discrete Gauge Theories
A particular example is produced to prove that quantum walks can be used to
simulate full-fledged discrete gauge theories. A new family of walks is
introduced and its continuous limit is shown to coincide with the dynamics of a
Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic
interpretation is extended beyond the continuous limit by proving that these
DTQWs exhibit an exact discrete local gauge invariance and possess a
discrete gauge-invariant conserved current. A discrete gauge-invariant
electromagnetic field is also constructed and that field is coupled to the
conserved current by a discrete generalization of Maxwell equations. The
dynamics of the DTQWs under crossed electric and magnetic fields is finally
explored outside the continuous limit by numerical simulations. Bloch
oscillations and the so-called drift are recovered in
the weak-field limit. Localization is observed for some values of the gauge
fields.Comment: 7 pages, 7 figure
Electromagnetic lattice gauge invariance in two-dimensional discrete-time quantum walks
Gauge invariance is one of the more important concepts in physics. We discuss
this concept in connection with the unitary evolution of discrete-time quantum
walks in one and two spatial dimensions, when they include the interaction with
synthetic, external electromagnetic fields. One introduces this interaction as
additional phases that play the role of gauge fields. Here, we present a way to
incorporate those phases, which differs from previous works. Our proposal
allows the discrete derivatives, that appear under a gauge transformation, to
treat time and space on the same footing, in a way which is similar to standard
lattice gauge theories. By considering two steps of the evolution, we define a
density current which is gauge invariant and conserved. In the continuum limit,
the dynamics of the particle, under a suitable choice of the parameters,
becomes the Dirac equation, and the conserved current satisfies the
corresponding conservation equation
ABC of multi-fractal spacetimes and fractional sea turtles
We clarify what it means to have a spacetime fractal geometry in quantum
gravity and show that its properties differ from those of usual fractals. A
weak and a strong definition of multi-scale and multi-fractal spacetimes are
given together with a sketch of the landscape of multi-scale theories of
gravitation. Then, in the context of the fractional theory with
-derivatives, we explore the consequences of living in a multi-fractal
spacetime. To illustrate the behavior of a non-relativistic body, we take the
entertaining example of a sea turtle. We show that, when only the time
direction is fractal, sea turtles swim at a faster speed than in an ordinary
world, while they swim at a slower speed if only the spatial directions are
fractal. The latter type of geometry is the one most commonly found in quantum
gravity. For time-like fractals, relativistic objects can exceed the speed of
light, but strongly so only if their size is smaller than the range of
particle-physics interactions. We also find new results about log-oscillating
measures, the measure presentation and their role in physical observations and
in future extensions to nowhere-differentiable stochastic spacetimes.Comment: 20 pages, 1 figure. v2: typos corrected, minor improvements of the
tex
The Thermal Scalar and Random Walks in AdS3 and BTZ
We analyze near-Hagedorn thermodynamics of strings in the WZW model.
We compute the thermal spectrum of all primaries and find the thermal scalar
explicitly in the string spectrum using CFT twist techniques. Then we use the
link to the Euclidean WZW BTZ black hole and write down the Euclidean BTZ
spectrum. We give a Hamiltonian interpretation of the thermal partition
function of angular orbifolds where we find a reappearance of discrete states
that dominate the partition function. Using these results, we discuss the
nature of the thermal scalar in the WZW BTZ model. As a slight generalization
of the angular orbifolds, we discuss the string gas with a non-zero
chemical potential corresponding to angular momentum around the spatial cigar.
For this model as well, we determine the thermal spectrum and the Hagedorn
temperature as a function of chemical potential. Finally the nature of
corrections to the thermal scalar action is analyzed and we
find the random walk behavior of highly excited strings in this particular
background.Comment: 74 pages, v2: version accepted for publication in JHE
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