Local densities are formed by acting several times on the scalar
and vector nonlocal densities with the relative momentum operator
and taking the limit of
.
Using the spherical representation, the possible
coupled
-tensors (10) (up to sixth order in derivatives)
are those given in Table 1
(replacing
with
).
Acting with on the scalar nonlocal density
gives 16 different local
densities up to N
LO (one for every term in
Table 1). They are listed in Table 3.
When acting with
on the vector nonlocal densities
, one has to construct all
possible ways of coupling the
-tensors with the vector density.
Obviously, each of the 4 scalar (
) derivative operators gives
one local density, while each of the 12 non-scalar (
) derivative
operators gives three local densities. Altogether, from the vector
density one obtains 40 local densities up to N
LO. They are listed
in Table 4.
No. |
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density | ![]() |
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1 | ![]() |
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0 | 0 | 0 | 0 | 1 | 1 | |
2 | ![]() |
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1 | 1 | 0 | 1 | ![]() |
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|
3 | ![]() |
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2 | 0 | 0 | 0 | 1 | 1 | |
4 | ![]() |
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2 | 2 | 0 | 2 | 1 | 1 | |
5 | ![]() |
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3 | 1 | 0 | 1 | ![]() |
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|
6 | ![]() |
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3 | 3 | 0 | 3 | ![]() |
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|
7 | ![]() |
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4 | 0 | 0 | 0 | 1 | 1 | |
8 | ![]() |
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4 | 2 | 0 | 2 | 1 | 1 | |
9 | ![]() |
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4 | 4 | 0 | 4 | 1 | 1 | ||
10 | ![]() |
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5 | 1 | 0 | 1 | ![]() |
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|
11 | ![]() |
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5 | 3 | 0 | 3 | ![]() |
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||
12 | ![]() |
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5 | 5 | 0 | 5 | ![]() |
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||
13 | ![]() |
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6 | 0 | 0 | 0 | 1 | 1 | |
14 | ![]() |
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6 | 2 | 0 | 2 | 1 | 1 | ||
15 | ![]() |
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6 | 4 | 0 | 4 | 1 | 1 | ||
16 | ![]() |
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6 | 6 | 0 | 6 | 1 | 1 |
No. |
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density | ![]() |
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|||
17 | ![]() |
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0 | 0 | 1 | 1 | ![]() |
1 | |
18 | ![]() |
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1 | 1 | 1 | 0 | 1 | ![]() |
|
19 | ![]() |
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1 | 1 | 1 | 1 | 1 | ![]() |
|
20 | ![]() |
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1 | 1 | 1 | 2 | 1 | ![]() |
|
21 | ![]() |
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2 | 0 | 1 | 1 | ![]() |
1 | |
22 | ![]() |
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2 | 2 | 1 | 1 | ![]() |
1 | |
23 | ![]() |
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2 | 2 | 1 | 2 | ![]() |
1 | |
24 | ![]() |
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2 | 2 | 1 | 3 | ![]() |
1 | |
25 | ![]() |
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3 | 1 | 1 | 0 | 1 | ![]() |
|
26 | ![]() |
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3 | 1 | 1 | 1 | 1 | ![]() |
|
27 | ![]() |
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3 | 1 | 1 | 2 | 1 | ![]() |
|
28 | ![]() |
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3 | 3 | 1 | 2 | 1 | ![]() |
|
29 | ![]() |
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3 | 3 | 1 | 3 | 1 | ![]() |
|
30 | ![]() |
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3 | 3 | 1 | 4 | 1 | ![]() |
|
31 | ![]() |
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4 | 0 | 1 | 1 | ![]() |
1 | |
32 | ![]() |
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4 | 2 | 1 | 1 | ![]() |
1 | |
33 | ![]() |
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4 | 2 | 1 | 2 | ![]() |
1 | |
34 | ![]() |
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4 | 2 | 1 | 3 | ![]() |
1 | |
35 | ![]() |
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4 | 4 | 1 | 3 | ![]() |
1 | |
36 | ![]() |
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4 | 4 | 1 | 4 | ![]() |
1 | ||
37 | ![]() |
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4 | 4 | 1 | 5 | ![]() |
1 |
No. |
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density | ![]() |
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|||
38 | ![]() |
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5 | 1 | 1 | 0 | 1 | ![]() |
|
39 | ![]() |
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5 | 1 | 1 | 1 | 1 | ![]() |
|
40 | ![]() |
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5 | 1 | 1 | 2 | 1 | ![]() |
|
41 | ![]() |
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5 | 3 | 1 | 2 | 1 | ![]() |
|
42 | ![]() |
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5 | 3 | 1 | 3 | 1 | ![]() |
||
43 | ![]() |
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5 | 3 | 1 | 4 | 1 | ![]() |
||
44 | ![]() |
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5 | 5 | 1 | 4 | 1 | ![]() |
||
45 | ![]() |
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5 | 5 | 1 | 5 | 1 | ![]() |
||
46 | ![]() |
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5 | 5 | 1 | 6 | 1 | ![]() |
||
47 | ![]() |
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6 | 0 | 1 | 1 | ![]() |
1 | |
48 | ![]() |
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6 | 2 | 1 | 1 | ![]() |
1 | |
49 | ![]() |
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6 | 2 | 1 | 2 | ![]() |
1 | ||
50 | ![]() |
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6 | 2 | 1 | 3 | ![]() |
1 | ||
51 | ![]() |
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6 | 4 | 1 | 3 | ![]() |
1 | ||
52 | ![]() |
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6 | 4 | 1 | 4 | ![]() |
1 | ||
53 | ![]() |
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6 | 4 | 1 | 5 | ![]() |
1 | ||
54 | ![]() |
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6 | 6 | 1 | 5 | ![]() |
1 | ||
55 | ![]() |
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6 | 6 | 1 | 6 | ![]() |
1 | ||
56 | ![]() |
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6 | 6 | 1 | 7 | ![]() |
1 |
One can also act on each of the local densities with derivative operators
of Table 1, and then couple ranks
and
to the
total rank
, i.e.,
In Tables 3 and 4, for completeness we also
show the time-reversal () and space-inversion (
) parities defined as,
Local densities constructed above are complex. Taking the complex conjugations gives relations derived in Appendix B:
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(27) |