Wigner D-Matrices

Wigner D-matrices are a complete set of functions that can be used to represent functions defined on the surface of a sphere, in terms of rotations, which specify the orientation of a rotation relative to some reference frame. Wigner D-matrices represent the transformation of a vector or tensor under a rotation as a matrix multiplication. However, Wigner D-matrices can not form a complete basis for the space of angular momentum states, there are functions in the space of angular momentum states that cannot be expressed as a linear combination of Wigner D matrices. This is because the space of angular momentum states includes not only the angular momentum states of a system, but also its spin states. The Wigner D matrices do not fully describe the spin states of a system, and additional degrees of freedom are needed to account for the transformation of spin states under rotations.

Why Wigner D-matrices Are Redundant？

There are multiple D-matrices that can represent the same rotation. For example, consider a rotation of 90 degrees around the z-axis followed by a rotation of 90 degrees around the x-axis. This can be represented by two different sets of Euler angles: (90 degrees, 0 degrees, 0 degrees) and (0 degrees, 90 degrees, 90 degrees). Both of these sets of Euler angles will give the same rotation, but they will result in different D-matrices.

Spherical Harmonics

Spherical harmonics are a complete set of functions that can be used to represent functions defined on the surface of a sphere, labeled/in terms of angular variables. Spherical harmonics is a generalization of form of the Fourier series, where the sines and cosines are replaced by more general functions that are defined in terms of spherical coordinates rather than Cartesian coordinates. Fourier series represents a periodic function as a sum of sines and cosines, which are periodic functions with a fixed frequency. Spherical Harmonics form a complete basis for the space of angular momentum states, any function in the space of angular momentum states can be expressed as a linear combination of spherical harmonics.

Why Spherical Harmonics Are Not Redundant？

There is only one spherical harmonic function for each degree and order, which is uniquely defined.

The transformation of angular momentum states under rotations can be represented using the Wigner D-matrices or the spherical harmonics. The Wigner D-matrices provide a matrix representation of the rotation group, while the spherical harmonics provide a basis for the space of functions on the sphere that transform in a particular way under rotations.

In the context of protein backbone structure, spherical harmonics may be more suitable for representing the rotational symmetries of the protein because they have simple mathematical properties and can be easily combined using the rules of angular momentum. On the other hand, Wigner D-matrices may be more suitable for representing the transformations of angular momentum states under rotations, although this may not be directly relevant to protein backbone structure.