Difference between revisions of "Dispersion relations"
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where $\textbf{P}$ is the total momentum. The discretization of the lattice breaks Lorentz symmetry and as a result the relativistic dispersion relation is violated at finite lattice spacing. There a lattice dispersion relation expected describe the dispersion more accurately. The lattice dispersion relation for a single particle is | where $\textbf{P}$ is the total momentum. The discretization of the lattice breaks Lorentz symmetry and as a result the relativistic dispersion relation is violated at finite lattice spacing. There a lattice dispersion relation expected describe the dispersion more accurately. The lattice dispersion relation for a single particle is | ||
| − | $$ cosh (E(p)) = 1-cos(p) + cosh(m) $$ | + | $$ \cosh (E(p)) = 1-\cos(p) + \cosh(m) $$ |
where $m$ and $p$ are the mass and momentum respectively. There for the case of two degenerate particles, the lattice dispersion is | where $m$ and $p$ are the mass and momentum respectively. There for the case of two degenerate particles, the lattice dispersion is | ||
Latest revision as of 22:13, 13 December 2011
One method to compute the phaseshifts is to first compute the energy in a frame with non-zero total momentum and then compute the COM energy using the relativistic dispersion relation
$$ W_{CM}^2 = W_{LAB}^2-\textbf{P}^2 $$
where $\textbf{P}$ is the total momentum. The discretization of the lattice breaks Lorentz symmetry and as a result the relativistic dispersion relation is violated at finite lattice spacing. There a lattice dispersion relation expected describe the dispersion more accurately. The lattice dispersion relation for a single particle is
$$ \cosh (E(p)) = 1-\cos(p) + \cosh(m) $$
where $m$ and $p$ are the mass and momentum respectively. There for the case of two degenerate particles, the lattice dispersion is
$$ \cosh (E(p)/2) = 1-\cos(p) + \cosh(m) $$
where $m$ and $p$ are the mass and relative-momentum respectively. To test these relations, we compute the pion mass using twisted boundary conditions. The results are shown below. It is clear that the lattice dispersion relation more accurately describes the dispersion seen on the lattice.
