Struct
NumCosmoMathCSQ1DState
Description [src]
struct NcmCSQ1DState {
/* No available fields */
}
A point $(\alpha, \gamma)$ of the hyperbolic plane at a time $t$ in a given
NcmCSQ1DFrame. The point is a complex structure $J_{ab}$, which fixes a mode of the
oscillator up to a time-dependent phase, see ncm_csq1d_state_get_J().
Instance methods
ncm_csq1d_state_compute_distance
The hyperbolic distance $d$ between the points of state and state1, which must have
the same frame and time:
$$\cosh d = \cosh\alpha\cosh\alpha_1\cosh(\gamma - \gamma_1) - \sinh\alpha\sinh\alpha_1.$$.
ncm_csq1d_state_get_J
Gets the components of the complex structure $J_{ab}$ of state, the symmetric,
positive definite matrix of unit determinant the point $(\alpha, \gamma)$
represents. For the mode $(\phi, P_\phi)$ of ncm_csq1d_state_get_phi_Pphi(),
$J_{11} = 2\vert\phi\vert^2$, $J_{22} = 2\vert P_\phi\vert^2$ and
$J_{12} = \phi P_\phi^ + \phi^ P_\phi$.
ncm_csq1d_state_get_circle
Sets cstate to the point at hyperbolic distance r from state in the direction
theta; $\theta = 0$ increases $\alpha$ by r at fixed $\gamma$. cstate keeps the
frame and time of state.
ncm_csq1d_state_get_minkowski
Gets the spatial coordinates of state on the hyperboloid $x_0^2 - x_1^2 - x_2^2 = 1$,
with $x_0 = \cosh\alpha\cosh\gamma$.
ncm_csq1d_state_get_phi_Pphi
Gets the eigenvector $(\phi, P_\phi)$ of the complex structure of state in the phase
where $\phi$ is real and positive,
\begin{align}
\phi &= \sqrt{\frac{e^{-\gamma}\cosh\alpha}{2}}, \
P_\phi &= -\tanh\alpha\,\sqrt{\frac{e^{\gamma}\cosh\alpha}{2}} - i\sqrt{\frac{e^{\gamma}}{2\cosh\alpha}},
\end{align}
normalized by $\phi P_\phi^ - \phi^ P_\phi = i$. In this phase
$e^{-i\theta(t)}(\phi, P_\phi)$ solves the equations of motion, with
$\theta = \int\nu\,\mathrm{d}t + \delta\theta$ from ncm_csq1d_eval_int_nu() and
ncm_csq1d_eval_delta_theta_at(). This phase differs from the published one by a
factor that depends on time through $\alpha$; see the
CSQ1D Formalism page.
ncm_csq1d_state_get_poincare_disc
Gets state as a point of the Poincaré disc, the projection of the hyperboloid point of
ncm_csq1d_state_get_minkowski() from $(-1, 0, 0)$.
ncm_csq1d_state_get_poincare_half_plane
Gets state as the point $x + iy$ of the Poincaré upper half-plane, where
$J_{11} = (x^2 + y^2)/y$, $J_{12} = -x/y$ and $J_{22} = 1/y$.
ncm_csq1d_state_set_um
Sets state from $(\chi, U_-)$; $e^{U_-}$ is the component $J_{11}$ of the complex
structure, see ncm_csq1d_state_get_J().
ncm_csq1d_state_set_up
Sets state from $(\chi, U_+)$; $e^{U_+}$ is the component $J_{22}$ of the complex
structure, see ncm_csq1d_state_get_J().