In the present study, we use four elementary building blocks
to construct the EDF, namely, the scalar and vector nonlocal
densities,
and
,
along with the total derivative
and relative
momentum
(6).
Spherical representations of the building blocks can be defined by
using standard convention of spherical tensors [26] as
In order to motivate the best suitable choice of the phase convention,
in Tables 21 and 22 we present relations between
the spherical and Cartesian representations of densities and terms in
the EDF, respectively. All NLO densities in the Cartesian representation,
which are listed in Table 21, are real. It is then clear
that the phase convention, which would render all NLO densities in
the spherical representation real does not exist. However, for the
phase factors ,
,
, and
equal to
or
, in the spherical representation all NLO densities and
terms in the EDF are either real or imaginary.
No. |
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Cartesian representation [25,30,24] | Phase | ||||
1 |
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= |
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= |
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= |
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2 |
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= |
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= |
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= |
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= |
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3 |
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= |
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= |
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17 |
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= |
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= |
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= |
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18 |
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= |
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= |
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= |
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19 |
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= |
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= |
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21 |
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22 |
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= |
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No. |
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Phase | |||
1 | ![]() |
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= |
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+1 | |
2 | ![]() |
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= |
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+1 | |
3 | ![]() |
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= |
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+1 | |
4 | ![]() |
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= |
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+1 | |
5 | ![]() |
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= |
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+1 | |
6 | ![]() |
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= |
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+1 | |
7 | ![]() |
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= |
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+1 | |
8 | ![]() |
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= |
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+1 | |
9 | ![]() |
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= |
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+1 | |
10 | ![]() |
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= |
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+1 | |
11 | ![]() |
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= |
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+1 | |
12 | ![]() |
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= |
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+1 | |
13 | ![]() |
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= |
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+1 | |||||||
14 | ![]() |
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= |
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+1 |
Among many options of choosing the phase convention, in the present study we set
At the same time, all vector densities in Table 21 and
vector operators in Eqs. (89)-(91) are
consistently characterized by phase factors connecting the spherical
and Cartesian representations.
Phase conventions (92) also lead to very simple phase properties, which our spherical tensors have with respect to complex conjugation. Indeed, spherical tensors (88)-(91) obey standard transformation rules under complex conjugation [26],
Standard rule (107) propagates through the angular
momentum coupling, i.e., if signs and
characterize tensors
and
, respectively,
then the coupled tensor,
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(110) |