In this article spherical functions are replaced by polynomials that have been well known in electrostatics since the time of Maxwell and associated with multipole moments.
Here, properties of tensors, including high-rank moments as well, are considered to repeat basically features of solid spherical functions but having their own specifics.
Using of invariant polynomial tensors in Cartesian coordinates, as shown in a number of recent studies, is preferable and simplifies the fundamental scheme of calculations [11] [12] [13] .
The rules for using harmonic symmetric tensors are demonstrated that directly follow from their properties.
These rules are naturally reflected in the theory of special functions, but are not always obvious, even though the group properties are general .
[15] At any rate, let us recall the main property of harmonic tensors: the trace over any pair of indices vanishes [9] .
[16] Here, those properties of tensors are selected that not only make analytic calculations more compact and reduce 'the number of factorials' but also allow correctly formulating some fundamental questions of the theoretical physics [9] .
There is an obvious property of contraction that give rise to a theorem simplifying essentially the calculation of moments in theoretical physics.
[15] The first term in the formula, as is easy to see from expansion of a point charge potential, is equal to The remaining terms can be obtained by repeatedly applying the Laplace operator and multiplying by an even power of the modulus
The coefficients are easy to determine by substituting expansion in the Laplace equation .
This form is useful for applying differential operators of quantum mechanics and electrostatics to it.
It is convenient to write the differentiation formula in terms of the symmetrization operation.
A symbol for it was proposed in,[12] with the help of sum taken over all independent permutations of indices: As a result, the following formula is obtained:
Following [11] one can find the relation between the tensor and solid spherical functions.
In perturbation theory, it is necessary to expand the source in terms of spherical functions.
If the source is a polynomial, for example, when calculating the Stark effect, then the integrals are standard, but cumbersome.
When calculating with the help of invariant tensors, the expansion coefficients are simplified, and there is then no need to integrals.
It suffices, as shown in,[14] to calculate contractions that lower the rank of the tensors under consideration.
The following rank reduction formula is useful: where symbol [m] denotes all left (l-2) indices.
If the brackets contain several factors with the Kronecker delta, the following relation formula holds: Calculating the trace reduces the number of the Kronecker symbols by one, and the rank of the harmonic tensor on the right-hand side of the equation decreases by two.
Repeating the calculation of the trace k times eliminates all the Kronecker symbols: The Laplace equation in four-dimensional 4D space has its own specifics.
Here, the decomposition of the tensor power up to the rank l=6 is presented: To derive the formulas, it is useful to calculate the contraction with respect two indices, i.e., the trace.
has the form Also useful is the frequently occurring contraction over all indices, which arises when normalizing the states.
The decomposition of tensor powers of a vector is also compact in four dimensions: When using the tensor notation with indices suppressed, the last equality becomes Decomposition of higher powers is not more difficult using contractions over two indices.
Ladder operators are useful for representing eigen functions in a compact form.
[11] It can be obtained from expansion of point-charge potential: Straightforward differentiation on the left-hand side of the equation yields a vector operator acting on a harmonic tensor:
- fold application to unity, the harmonic tensor arises: written here in different forms.
The commutator in the scalar product on the sphere is equal to unity: To calculate the divergence of a tensor, a useful formula is whence (
variable is convenient for physical problems: In particular, The scalar product of the ladder operator
[18] They perform Lie algebra as the angular momentum and the Laplace-Runge-Lenz operators.