S3Action.hpp File Reference

Calculate S3 bulk actions on 3D Delaunay Triangulations. More...

#include <cmath>
#include <stdexcept>
#include <utility>
#include "Mpfr_value.hpp"

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Classes
class	cdt::s3_action::PhysicalParameters
	Finite physical couplings used to evaluate the Euclidean action. More...

Namespaces
namespace	cdt
	clang-15 does not support std::format

Functions
auto	cdt::s3_action::detail::make_finite_couplings (long double const k, long double const lambda) -> std::pair< long double, long double >
auto	cdt::s3_action::make_physical_parameters (long double const alpha, long double const k, long double const lambda) -> PhysicalParameters
	Validate physical couplings for three-dimensional action APIs.
auto	cdt::s3_action::s3_bulk_action (Int_precision const n1_tl_count, Int_precision const n3_31_13_count, Int_precision const n3_22_count, PhysicalParameters const &parameters) -> mpfr_values::Value
	Calculates the generalized S3 bulk action in terms of \(\alpha\), \(k\), \(\lambda\), \(N_1^{TL}\), \(N_3^{(3,1)}\), and \(N_3^{(2,2)}\).
auto	cdt::s3_action::s3_bulk_action_alpha_minus_one_imaginary_coefficient (Int_precision const n1_tl_count, Int_precision const n3_31_13_count, Int_precision const n3_22_count, long double const k_value, long double const lambda_value) -> mpfr_values::Value
	Calculates the coefficient of \(i\) in the \(\alpha=-1\) S3 bulk action.
auto	cdt::s3_action::s3_bulk_action_alpha_one (Int_precision const n1_tl_count, Int_precision const n3_31_13_count, Int_precision const n3_22_count, long double const k_value, long double const lambda_value) -> mpfr_values::Value
	Calculates S3 bulk action for \(\alpha\)=1.

Detailed Description

Calculate S3 bulk actions on 3D Delaunay Triangulations.

Author

Adam Getchell

Calculates the S3 Bulk (and later, boundary) actions. Uses the GNU MPFR library for arbitrary precision arithmetic on floating point numbers. See http://www.mpfr.org for more details. Note: for performance reasons, variables should not hold successively increasing values. We avoid this by setting each variable only once. See https://gmplib.org/manual/Efficiency.html#Efficiency for details.

See also

Regge calculus

Three-dimensional CDT

Definition in file S3Action.hpp.

Function Documentation

make_finite_couplings()

auto cdt::s3_action::detail::make_finite_couplings ( long double const k, long double const lambda ) -> std::pair< long double, long double >

inlinenodiscard

Definition at line 86 of file S3Action.hpp.

make_physical_parameters()

auto cdt::s3_action::make_physical_parameters ( long double const alpha, long double const k, long double const lambda ) -> PhysicalParameters

inlinenodiscard

Validate physical couplings for three-dimensional action APIs.

Parameters

alpha	Wick-rotation parameter; must be finite and greater than 1/2.
k	Inverse Newton coupling; must be finite.
lambda	Cosmological coupling; must be finite.

Returns

Validated coupling values.

Exceptions

std::invalid_argument	if any coupling is non-finite.
std::domain_error	if alpha is not greater than 1/2.

Definition at line 68 of file S3Action.hpp.

s3_bulk_action()

auto cdt::s3_action::s3_bulk_action ( Int_precision const n1_tl_count, Int_precision const n3_31_13_count, Int_precision const n3_22_count, PhysicalParameters const & parameters ) -> mpfr_values::Value

inlinenodiscard

Calculates the generalized S3 bulk action in terms of \(\alpha\), \(k\), \(\lambda\), \(N_1^{TL}\), \(N_3^{(3,1)}\), and \(N_3^{(2,2)}\).

The formula is:

\begin{eqnarray*}S^{(3)} &=& 2\pi k\sqrt{\alpha}N_1^{TL} \\ &+& N_3^{(3,1)}\left[-3k\text{arcsinh}\left(\frac{1}{\sqrt{3} \sqrt{4\alpha +1}}\right)-3k\sqrt{\alpha}\text{arccos}\left(\frac{2\alpha+1} {4\alpha+1}\right)-\frac{\lambda}{12}\sqrt{3\alpha+1}\right] \\ &+& N_3^{(2,2)}\left[2k\text{arcsinh}\left(\frac{2\sqrt{2}\sqrt{2\alpha+1}} {4\alpha +1}\right)-4k\sqrt{\alpha}\text{arccos}\left(\frac{-1}{4\alpha+1} \right)-\frac{\lambda}{12}\sqrt{4\alpha +2}\right]\end{eqnarray*}

Parameters

n1_tl_count	\(N_1^{TL}\) is the number of timelike links
n3_31_13_count	\(N_3^{(3,1)}\) is the number of (3,1) and (1,3) simplices
n3_22_count	\(N_3^{(2,2)}\) is the number of (2,2) simplices
parameters	Validated physical parameters for the action

Returns

\(S^{(3)}(\alpha)\) as a 256-bit MPFR value

See also

Regge calculus

Three-dimensional CDT action

Definition at line 253 of file S3Action.hpp.

s3_bulk_action_alpha_minus_one_imaginary_coefficient()

auto cdt::s3_action::s3_bulk_action_alpha_minus_one_imaginary_coefficient ( Int_precision const n1_tl_count, Int_precision const n3_31_13_count, Int_precision const n3_22_count, long double const k_value, long double const lambda_value ) -> mpfr_values::Value

inlinenodiscard

Calculates the coefficient of \(i\) in the \(\alpha=-1\) S3 bulk action.

The Lorentzian expression at \(\alpha=-1\) is purely imaginary. This real-valued API returns its coefficient of \(i\); it does not return a complex number or a Wick-rotated Euclidean action. The formula is:

\[S^{(3)}(\alpha=-1)=-2\pi ik N_1^{TL}+N_3^{(3,1)}\left(2.673ik+0.118i \lambda\right)+N_3^{(2,2)}\left(7.386ik+0.118i\lambda\right) \equiv iS^3_{EDT}\]

Parameters

n1_tl_count	\(N_1^{TL}\) is the number of timelike links
n3_31_13_count	\(N_3^{(3,1)}\) is the number of (3,1) and (1,3) simplices
n3_22_count	\(N_3^{(2,2)}\) is the number of (2,2) simplices
k_value	\(k=\frac{1}{8\pi G_{Newton}}\)
lambda_value	\(\lambda=k*\Lambda\) where \(\Lambda\) is the Cosmological constant

Returns

The real coefficient \(S^{(3)}(\alpha=-1)/i\) as a 256-bit MPFR value

Definition at line 119 of file S3Action.hpp.

s3_bulk_action_alpha_one()

auto cdt::s3_action::s3_bulk_action_alpha_one ( Int_precision const n1_tl_count, Int_precision const n3_31_13_count, Int_precision const n3_22_count, long double const k_value, long double const lambda_value ) -> mpfr_values::Value

inlinenodiscard

Calculates S3 bulk action for \(\alpha\)=1.

The formula is:

\[S^{(3)}(\alpha=1)=2\pi k N_1^{TL}+N_3^{(3,1)}\left(-3.548k-0.167\lambda \right)+N_3^{(2,2)}\left(-5.355k-0.204\lambda\right)\]

Parameters

n1_tl_count	\(N_1^{TL}\) is the number of timelike links
n3_31_13_count	\(N_3^{(3,1)}\) is the number of (3,1) and (1,3) simplices
n3_22_count	\(N_3^{(2,2)}\) is the number of (2,2) simplices
k_value	\(k=\frac{1}{8\pi G_{Newton}}\)
lambda_value	\(\lambda=k*\Lambda\) where \(\Lambda\) is the Cosmological constant

Returns

\(S^{(3)}(\alpha=1)\) as a 256-bit MPFR value

Definition at line 180 of file S3Action.hpp.