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The Functions in QMGxX

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Laurin J. Felder edited this page Apr 20, 2023 · 10 revisions

Matrix Configurations

qmgXsu2

Generates the matrix configuration of the fuzzy sphere.

Arguments

Argument Type Description
Nim Int the dimension of the representation
Normalized=True Bool True: the matrices are normalized such that the sum over all squares is the identity; False: the matrices are not normalized

Output

The output is X.

Output Type Description
X MathConf(3,Nim) the matrix configuration of the fuzzy sphere

Description

qmgXsu2 generates the matrix configuration of the fuzzy sphere. Compare to [1] section 2.3.2.

Example(s)

An example for Nim=4:

X=qmgXsu2[4]

qmgXsu3

Generates the matrix configuration of the fuzzy CP2.

The following function has been derived from https://arxiv.org/abs/0908.3864 by Richard Shurtleff, licensed under CC-BY-4.0.

Arguments

Argument Type Description
Pin Int Pin as in the (Pin,Qin) irreducible representation of SU(3)
Qin Int Qin as in the (Pin,Qin) irreducible representation of SU(3)
Normalized=True Bool True: the matrices are normalized such that the sum over all squares is the identity; False: the matrices are not normalized

Output

The output is X.

Output Type Description
X MathConf(8,(Pin+1)(Qin+1)(Pin+Qin+2)/2) the matrix configuration of the fuzzy CP2

Description

qmgXsu3 generates the matrix configuration of the fuzzy $\mathbb{C}P^2$. Compare to [1] section 2.3.5.

Example(s)

An example for (Pin,Qin)=(3,0):

X=qmgXsu5[3,0]

qmgXcs

Generates the matrix configuration of the fuzzy torus.

Arguments

Argument Type Description
Nim Int the dimension of the representation

Output

The output is X.

Output Type Description
X MathConf(4,Nim) the matrix configuration of the fuzzy torus

Description

qmgXcs generates the matrix configuration of the fuzzy torus. Compare to [1] section 4.5.

Example(s)

An example for Nim=4:

X=qmgXcs[4]

qmgXrand

SeedRandom in advance recommended; generates a random matrix configuration.

Arguments

Argument Type Description
Dim Int the dimension of the target space
Nim Int the dimension of the Hilbert space
componentMax Real the maximal magnitude of the components

Output

The output is X.

Output Type Description
X MathConf(Dim,Nim) a random mantrix configuration

Description

qmgXrand generates a random matrix configuration. Since random numbers are involved, it is recomended to seed a random state in advance to maintain reproducability. Compare to [1] section 4.2.

Example(s)

An example for Dim=5 and Nim=4:

SeedRandom[1];
X=qmgXrand[5,4,0.5]

Sets in Target Space

qmgxCartesianCoordinates

Generates Cartesian coordinate lines in target space.

Arguments

Argument Type Description
Dim Int the dimension of target space
directionP Int an integer between 1 and Dim. The direction in which the coordinate lines point
directionsT Int(k) a list of length k consisting of distinct integers between 1 and Dim, directionP excluded. The directions in which the coordinate lines are stacked
lengthP Real the step length between two points in the coordinate line
lengthT Real the step length between two different coordinate lines in each direction in directionsT
nP Int the number of points in the coordinate lines
nT Int the number of coordinate lines in each direction in directionsT
center={} Either {} or Real(Dim) the center of the coordinate lines, either {} (then {0,...,0} is used as center) or a point in target space

Output

The output is xs.

Output Type Description
xs Real((2*nP+1)*(2*nT)^k,Dim) a list of points in target space

Description

qmgxCartesianCoordinates generates Cartesian coordinate lines in target space.

Example(s)

An example for Dim=4 and main direction 2:

xs=qmgxCartesianCoordinates[4,2,{1,3},5,5,100,3,{0,0,1}]

qmgx3DSphericalCoordinates

Generates three dimensional spherical coordinate lines in target space.

Arguments

Argument Type Description
rMin Int the minimal radius
rMax Int the maximal radius
nr Int the number of points in the radial direction
ntheta Int the number of poins in the polar direction
nphi Int the number of points in the azimuthal direction

Output

The output is xs.

Output Type Description
xs Real((nr+1)*(ntheta+1)*phi,3) a list of points in target space

Description

qmgx3DSphericalCoordinates generates spherical coordinate lines in $\mathbb{R}^3$.

Example(s)

An example for radial coordinate lines:

xs=qmgx3DSphericalCoordinates[1,3,100,4,4]

An example for polar coordinate lines:

xs=qmgx3DSphericalCoordinates[1,3,4,100,4]

An example for azimuthal coordinate lines:

xs=qmgx3DSphericalCoordinates[1,3,4,4,100]

qmgx3DSphericalCoordinatesSector

Generates a sector of three dimensional spherical coordinate lines in target space.

Arguments

Argument Type Description
rBord={0.5,1.5} Real(2) a list consisting of the minimal radius and the maximal radius
thetaBord={0.2Pi,0.8Pi} Real(2) a list consisting of the minimal polar angle and the maximal polar angle
phiBord={-0.015Pi,0.015Pi} Real(2) a list consisting of the minimal azimuthal angle and the maximal azimuthal angle
nr Int the number of poins in the radial direction
ntheta Int the number of points in the polar direction
nphi Int the number of points in the azimuthal direction

Output

The output is xs.

Output Type Description
xs Real((nr+1)*(ntheta+1)*(phi+1),3) a list of points in target space

Description

qmgx3DSphericalCoordinatesSector generates a sector of spherical coordinate lines in $\mathbb{R}^3$.

Example(s)

An example for radial coordinate lines:

xs=qmgx3DSphericalCoordinatesSector[100,4,4]

An example for azimuthal coordinate lines and custom bounds:

xs=qmgx3DSphericalCoordinatesSector[{0.1,0.5},{0.3,1},{0.2,0.4},100,4,4]

qmgxSplitHemispheres

Splits given points in target space into the upper and lower hemisphere.

Arguments

Argument Type Description
xs Real(k,Dim) a list of length k consisting of points in target space

Output

The output is {xsUpper,xsLower}.

Output Type Description
xsUpper Real(r,Dim) the points x in xs with x(Dim)>0
xsLower Real(k-r,Dim) the points x in xs with x(Dim)≤0

Description

qmgxSplitHemispheres splits the points in the upper hemisphere of $\mathbb{R}^{Dim}$ from the points in the lower hemisphere.

Example(s)

An example for Dim=3:

xs=qmgx3DSphericalCoordinates[1,3,100,4,4];
qmgxSplitHemispheres[xs]

qmgxRandomBall

SeedRandom in advance recommend; generates random points in a ball in target space.

Arguments

Argument Type Description
Dim Int the dimension of target space
radius Real the radius of the ball
n Int the number of random points
center={} Either {} or Real(Dim) the center of the ball, either {} (then {0,...,0} is used as center) or a point in target space

Output

The output is xs.

Output Type Description
xs Real(n,Dim) a list of points in target space

Description

qmgxRandomBall generates random points in a ball in target space. Since random numbers are involved, it is recomended to seed a random state in advance to maintain reproducability.

Example(s)

An example for Dim=4:

SeedRandom[1]
xs=qmgxRandomBall[4,3.5,1000,{0,0,0,1}]

qmgxRandomCube

SeedRandom in advance recommend; generates random points in a cube in target space.

Arguments

Argument Type Description
Dim Int the dimension of target space
length Real the side length of the cube
n Int the number of random points
center={} Either {} or Real(Dim) the center of the cube, either {} (then {0,...,0} is used as center) or a point in target space

Output

The output is xs.

Output Type Description
xs Real(n,Dim) a list of points in target space

Description

qmgxRandomCube generates random points in a cube in target space. Since random numbers are involved, it is recomended to seed a random state in advance to maintain reproducability.

Example(s)

An example for Dim=4:

SeedRandom[1]
xs=qmgxRandomCube[4,2.5,1000,{0,1,0,1}]
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