Generalized Kronecker and permanent deltas, their spinor and tensor equivalents and
applications
R. L. Agacy
Citation: Journal of Mathematical Physics 40, 2055 (1999); doi: 10.1063/1.532851
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JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 40, NUMBER 4
APRIL 1999
Generalized Kronecker and permanent deltas, their spinor
and tensor equivalents and applications
R. L. Agacya)
42 Brighton Street, Gulliver, Townsville, Qld 4812, Australia
~Received 14 September 1998; accepted for publication 8 December 1998!
The aim of this paper is fourfold: ~i! to introduce a generalized permanent delta on
an equal footing with the generalized Kronecker delta, to use for the symmetries of
any tensor or spinor, ~ii! to cite an ancillary reference source of comprehensive
tensorial and spinorial combinatorial formulas for both, ~iii! to table spinor equivalents of these individual tensors and give examples of their usage, and ~iv! to
tabulate the tensor equivalents of various useful combinations of their spinor forms.
© 1999 American Institute of Physics. @S0022-2488~99!03303-4#
I. INTRODUCTION
The generalized Kronecker delta ~gKd! is well known as an alternating function or antisymmetrizer, e.g., d abc
de f X abc 53!X @ de f # . In contrast, although symmetric tensors are seen constantly in
the mathematical language for general relativity ~GR!, there does not appear to be employment of
any symmetrizer, for e.g., X (abc) , in analogy with the antisymmetrizer. Historically, nomenclature
for both date back to Cauchy 1812 in terms of ‘‘fonctions symétriques alternées’’ and ‘‘fonctions
symétriques permanentes.’’ 1 A permanent symmetrized tensor was probably first introduced by
Cramlet,2,3 but seems to have been neglected, never acquiring the same prominence as the gKd.
However, it is exactly the combination of both types of symmetrizers, treated equally, that give us
the flexibility to describe any type of tensor symmetry, being exactly the symmetrizers needed to
describe Young tableaux tensors.4 We define such a ‘‘permanent’’ symmetrizer as a generalized
permanent delta ~gpd! below. Our purpose is to restore an imbalance between the gKd and gpd
and cite a reference of combinatorial formulas for them, most of which is not in the literature. The
contents of this Ref. 5, referred to as PAPS, which contains a comprehensive tabulation of gKd
and gpd formulas is amplified at the end of this section. In Sec. II we define the gpd and gKd and
state some very basic relations. In Sec. III we illustrate the usage of the generalized deltas in the
symmetries of tensors. In Sec. IV examples are given of finding spinor equivalents of simple
tensors almost instantly, using gKd and gpd spinors. A table of spinor equivalents of individual
tensor generalized deltas are tabulated in Appendix A. Converse to finding spinor equivalents, we
next tabulate the tensor equivalents of spinor generalized deltas in Appendix B. They can be
applied to obtain a variety of spinor formulas.
General indices range from 1,...,n. All index sets are permutations of each other. Twocomponent spinor indices are in capital lower case roman.
The PAPS reference5 contains the following parts and sections: Part I; formulas for
n-dimensions, ~1! Introduction, ~2! Definitions and illustrations of the gKd and gpd, ~3! Generalized Kronecker d, ~4! Generalized permanent d, ~5! Combined gKd and gpd, Part II; formulas for
tensors and spinors in the mathematical language of GR, ~6! Spinor equivalents of gKd and gpd
tensors, ~7! Tensor equivalents of gKd and gpd spinors. Appendixes A–F provide tabulations of
all formulas. The derivation of each and every formula is provided in the above sections.
II. DEFINITIONS AND BASICS OF THE gKd AND gpd
Complete symmetry of ~a tensor’s! indices, as opposed to total antisymmetry, is manifested by
all positive signs in any p-linear expression. Whereas the gKd’s antisymmetry comes about
a!
Electronic mail: ragacy@ultra.net.au
0022-2488/99/40(4)/2055/9/$15.00
2055
© 1999 American Institute of Physics
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J. Math. Phys., Vol. 40, No. 4, April 1999
R. L. Agacy
through a determinant ~interchanges of rows/columns or indices changes the sign!, total positive or
pure or permanent symmetry, as we term it, comes about through the use of a permanent. Then in
complete analogy to the gKd we introduce the generalized permanent delta or gpd. This is
defined, like the gKd determinant of a matrix, except that we take all positive signs. We use the
kernel letter p to denote a gpd and double vertical lines for the permanent of the defining matrix.
The gKd and gpd are completely complementary to each other and are defined, for p(<n) distinct
indices, respectively by,
U U I I
i
i ¯i
1
p
i
i
•
•
d j1
d j1
•
•
d j1
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
d jp
•
•
d jp
1
d j1 ¯ jp 5
i
d j1
i
d jp
1
p
1
,
i ¯i
p j1 ¯ jp 5
1
p
i
i
d jp
p
1
p
,
i
p
where the first is a determinant and the second a permanent. The gKd has value 11 ~21! depending on whether ( j 1 ,..., j p ) is an even ~odd! permutation of (i 1 ,...,i p ). The gpd has the permanent
value 11 for any permutation of the index sets. Note that p ab 5 d ab . The gpd is a complete
symmetrizer ~permanent symmetrizer! and now one easily sees that p abc
de f X abc 53!X (de f ) .
A simple but important interaction between the gKd and gpd is that in any expression containing them as a product, where there is a summation on a pair of indices between them, such
expression vanishes. For
............
...a...b... ............
...b...a... ............
...a...b... ............
d ...a...b...
............ p ...a...b... 5 d ............ p ...b...a... 52 d ............ p ...b...a... 52 d ............ p ...a...b... 50.
From this it also obviously follows that the product of any gKd and any gpd with two or more
common contracted indices vanishes.
If A5 @ a ij # is an n3n matrix its determinant and permanent are
det A5
1 i1
i
j ¯j
a ¯a jn d i 1¯i n ,
n 1
n
n! j 1
per A5
1 i1
i
j ¯j
a ¯a jn p i 1¯i n .
n
1
n
n! j 1
III. gKd’s AND gpd’s IN SYMMETRIES OF TENSORS
The use of symmetrizing parentheses and antisymmetrizing brackets for specification of tensor symmetries can become both convoluted and ambiguous. For example 4T @ (i u j u # k) can be confusing at first sight; taken to mean performing symmetry in i, k and skew symmetry in i, j, it is still
ambiguous, depending on which operation is performed first. It can be appreciated that symmetry
specifications in this way for tensors with a large number of indices and intertwining brackets can
be quite horrendous. Specification of tensor symmetries with generalized deltas ~gd’s—
collectively gKd’s and gpd’s! gives clean, unambiguous expressions. Performing symmetry then
antisymmetry and vice versa on 4T @ (i u j u # k) gives two quite different expressions:
pn
pn
2 ~ T @ i j # k 1T @ k j # i ! 5 ~ T i jk 1T k ji 2T jik 2T jki ! 5 p ik
~ T p jn 2T j pn ! 5 d lm
p j p ik T lmn ,
ln pm
2 ~ T ~ i u j u k ! 2T ~ j u i u k ! ! 5 ~ T i jk 2T jik 1T k ji 2T ki j ! 5 d ipm
j ~ T pmk 1T km p ! 5 p pk d i j T lmn .
Specification of what one means by the gd’s is unequivocal. They are also exactly what is
needed for tensors obeying the symmetry of Young tableaux illustrated next.
The Riemann tensor R abcd is a Young tableau ~YT! tensor in its algebraic symmetries,
expressible as4,6 R abcd 5 121 R $ ac,bd % , obeying the partial symmetries and antisymmetries as determined by its tableau $ac,bd%. Fully expanded, it is4
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R. L. Agacy
2057
R abcd 5 121 R $ ac,bd %
5 121 @ R abcd 1R adcb 1R cbad 1R cdab 2R abdc 2R acdb 2R dbac 2R dcab
2R bacd 2R bdca 2R cabd 2R cdba 1R badc 1R bcda 1R dabc 1R dcba # .
Its symmetries are inbuilt. It is easy to see the skewsymmetry in ~a,b! and in ~c,d!. Then too
the interchange (a,b)⇔(c,d) can also be seen, while the cyclic symmetry R abcd 1R acdb
1R adbc 50, though tedious, is easy enough to verify. Conversely, using these symmetries in the
rhs does indeed produce the lhs.
The above expression can be put into a convenient form involving gKd’s and gpd’s. Let E abcd
be the expression on the rhs within brackets:
E abcd 5R abcd 1R cbad 1R adcb 1R cdab 2R abdc 2R dbac 2R acdb 2R dcab
2R bacd 2R cabd 2R bdca 2R cdba 1R badc 1R dabc 1R bcda 1R dcba
pq
5 d ab
@ R pqcd 1R cqpd 1R pdcq 1R cd pq 2R pqdc 2R dq pc 2R pcdq 2R dc pq #
pq rs
5 d ab
d cd @ R pqrs 1R rq ps 1R psrq 1R rspq #
pq rs eg
5 d ab
d cd p pr @ R eqgs 1R esgq #
pq rs eg f h
5 d ab
d cd p pr p qs R e f gh ,
so that
pq rs eg f h
d cd p pr p qs R e f gh
R abcd 5 121 d ab
and there are no intertwined, or indeed any parentheses or brackets.
As for recognition of symmetries from this, it is evident that because of the gKd’s the
expression is skew in ~a,b! and in ~c,d!. With (a,b)⇔(c,d) and slight index manipulation it is
also visible that the expression is symmetric. The cyclic identity is not obvious from either the 16
term expression or its compacted gd equivalent. But it is here that the PAPS reference tables come
into play.
pq rs eg f h
d cd p pr p gs above and formula m3 from the table in PAPS ApObserving the gd factor d ab
bcd pq rs ..
pendix C 7.3.2, i.e., d ... d .b d cd p pr p ..qs 50 ~dots allow any free indices!, we can immediately write
down the symmetry/condition as d bcd
i jk R abcd 50. But let us show it in reverse ~in fact deriving the
m3 identity!.
Multiply the above relation for R abcd by d bcd
i jk to get
1 bcd pq rs
eg f h
d bcd
i jk R abcd 5 12 d i jk d ab d cd p pr p qs R e f gh
pq eg f h
5 61 d brs
i jk d ab p pr p qs R e f gh
~ or see PAPS-formula k8 or, better, Eq. ~ 4 !!
p
prs q
eg f h
5 61 @ d qrs
i jk d a 2 d i jk d a # p pr p qs R e f gh
50
~ two repeated indices for a gKd and a gpd! .
Expanding the lhs ~six terms!, but using the antisymmetry in the last two indices, produces the
cyclic identity. This reverse derivation relied on ‘‘already knowing’’ the cyclic identity of the
Riemann tensor ~by applying d bcd
i jk ).
If instead we have the tensor
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J. Math. Phys., Vol. 40, No. 4, April 1999
R. L. Agacy
U abcd 5T abcd 2T cbad 1T dbac 2T dbca 1T cbda 2T abdc
1T bacd 2T bcad 1T bdac 2T bdca 1T bcda 2T badc
pgh
5 d acd
~ T pbgh 1T b pgh !
pgh e f
5 d acd
p pb T e f gh
one may, with some scrutiny, discern the antisymmetry in indices ~a,c! and in ~a,d! ~and hence
total antisymmetry in all three of these indices! from the expanded expression. The antisymmetry
in ~a,c,d! is immediately obvious from the compacted gd expression. However, there is another
‘‘hidden’’ symmetry, unobvious from either expression for U abcd . It is
U abcd 2U bcda 1U cdab 2U dabc 50.
But how can this be discovered? Here, again, inspection of the identity o9 in the same table in
PAPS, i.e., d e....f gh d ep..f g p ..hp 50, provides the answer. Adapted to our indices by (e, f ,g,h)
→(a,c,d,b) we can write ~sign changes do not matter here!
pgh e f
d abcd
i jkl d acd p pb T e f gh 50.
Thus the ‘‘hidden’’ identity is d abcd
i jkl U abcd 50. One may wish to leave it like this; however,
expansion of it, and using antisymmetry in first, third, and fourth indices produces the 4-term
alternating cyclic identity above ~with subscripts i,j,k,l!. Further, we also remark that while the
antisymmetries can be encompassed within the bracketed notation, U abcd 5U @ a u b u cd # , what can be
suggested to accommodate the hidden symmetry? Clearly the gd notation to express and ‘‘discover’’ symmetries of tensors seems superior.
The simpler Lanczos tensor7,8 satisfies algebraic ~nondifferential, or not involving covariant
differentiation! relations L i jk 52L jik , L i jk 1L jki 1L ki j 50, and L i j j 50 ~optional algebraic gauge
condition, often accepted!. In any case, it is the symmetries of the ~free! 3-index object that
interests us. These first two symmetries qualify it as a YT tensor, expressible as4 L i jk 5 31 L $ ik, j % .
This again is a one-line expression for the tensor, encompassing the first two symmetries. Fully
expanded, we have
L i jk 5 31 L $ ik, j % 5 31 @ L i jk 2L jik 1L k ji 2L ki j #
5 13 d ipm
j @ L pmk 1L km p #
ln
5 13 d ipm
j p k p L lmn .
jk
)
It is then easy enough to verify the skewsymmetry and the cyclic symmetry ~multiply by d iabc
from this equation—and in reverse; that with these symmetries, employed on the rhs, we do get
the lhs. The cyclic relation for the Lanczos tensor gives rise to the identity d i...jk d ip.j p ..k p 50, recorded
as identity o2 in the PAPS reference document.
Other than determining symmetries of tensors, the use of gd’s may possibly help in ‘‘seeing’’
the number of independent components of a tensor with various symmetries.
Some computer algebra packages allow for symmetrization and antisymmetrization of indices.
It is suggested that perhaps specific gpd and gKd objects ~over and above a single Kronecker delta!
be constructed in these packages allowing easy expansions of products of them over summed
indices. This would be most useful for checking identities and formulas of various sorts.
IV. SPINOR AND TENSOR EQUIVALENTS OF gKd’s AND gpd’s AND EXAMPLES
A. Spinor equivalents of tensors
It is quite easy to construct spinor equivalents of the gKd’s and gpd’s from the basic spinor
equivalent of the Kronecker d, i.e., d ab ⇔ d AB d B 88 , and the use of determinants and permanents. This
A
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R. L. Agacy
2059
leads to a tabulation of spinor equivalents of individual gKd’s and gpd’s in Appendix A. The
derivation of such formulas can be done by sight and by simple manipulations.
Apart from this table it is also our purpose to demonstrate in a couple of examples that use of
the gd’s can facilitate extremely quick derivation of spinor equivalents.
In general, if one expresses the symmetries of a tensor using gd’s then ~using the table of
Appendix A! we can write down their spinor equivalents, replacing all tensor indices by corresponding spinor equivalent ones. Once this is done the spinor equivalent is technically found.
However, it may involve several relations and consequent algebraic manipulations in order to get
decompositions, such as the Weyl and Ricci parts for the Riemann tensor equivalent.
The two illustrations of obtaining the spinor equivalents of 2-index permanent symmetric and
skewsymmetric tensors follow.
First, we determine the spinor equivalent of the skewsymmetric tensor F ab 52F ba 5F @ ab # .
This can be written as F ab 5 21 F cd d cd
ab in terms of a gKd. Then taking the spinor equivalent ~see D 2
in Appendix A! gives
CD
8 8
8 8
F ABA 8 B 8 5 41 F CDC 8 D 8 @ d CD
AB p A 8 B 8 1 p AB d A 8 B 8 #
C D
C D
52F BAB 8 A 8 5F @ AB #~ A 8 B 8 ! 1F ~ AB !@ A 8 B 8 # 5 e AB c A 8 B 8 1 e A 8 B 8 f AB ,
where c A 8 B 8 5 21 F X X (A 8 B 8 ) and f AB 5 21 F (AB)X 8 X 8 . If F ab is real, F ab 5F̄ ab , then c A 8 B 8 5 f̄ A 8 B 8
5 f AB , so that the spinor equivalent of the real skew-tensor ~Maxwell tensor! F ab is
F ab 5F @ ab # ⇔ e AB f̄ A 8 B 8 1 e A 8 B 8 f AB .
Second, we determine the spinor equivalent of a permanent symmetric ~or just symmetric by
common usage if the context is clear! tensor T ab 5T ba 5T (ab) . This can be written T ab
5 21 T cd p cd
ab in terms of a gpd. Then taking the spinor equivalent ~see P 2 in Appendix A! gives
CD
8 8
8 8
T ABA 8 B 8 5 41 T CDC 8 D 8 @ d CD
AB d A 8 B 8 1 p AB p A 8 B 8 #
C D
C D
5T BAB 8 A 8
5T @ AB #@ A 8 B 8 # 1T ~ AB !~ A 8 B 8 !
5 41 e AB e A 8 B 8 T1T ~ AB !~ A 8 B 8 ! ,
where T5T X X X 8 X 8 5T XX 8 XX 8 , which is also the trace of the tensor T a a . Thus
T ab 5T ~ ab ! ⇔ 41 e AB e A 8 B 8 T1T ~ AB !~ A 8 B 8 ! .
If one writes R in place of T, then the spinor equivalent of the trace-free Ricci tensor is
R ab 2 14 g ab R⇔R ~ AB !~ A 8 B 8 ! .
B. Tensor equivalents of spinors
The tensor equivalents of gd spinors in combination are elegantly derived in PAPS. The
results, along with related formulas are accumulated in the table in Appendix B.
We feel it worth mentioning at least one formula, one that enables us to easily remember the
usual rather complicated formula for the spinor equivalent of the alternating tensor—by using
gKd/gpd spinors. Abbreviate by putting
AB
D5 d CD
AB
P5 p CD
D 8 5 d C88 D88
P 8 5 p C88 D88 .
A B
A B
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J. Math. Phys., Vol. 40, No. 4, April 1999
R. L. Agacy
The formula, with the understanding that the first relation is a symbolic equivalence, is ~see
full derivation in PAPS!
i
i AB A 8 B 8
i DP
AB A 8 B 8
d D 8 P 8 [ ~ DP 8 2PD 8 ! 5 ~ d CD
p C 8 D 8 2 p CD
d C 8 D 8 ! ⇔ e ab cd .
2
2
2
To give an example from the table, consider the tensor equivalent of the spinor ~last but one
table entry in Appendix B!
ab
ab
d AC d BD d DA 88 d CB 88 ⇔ 21 ~ p ab
cd 2g g cd 2i e cd ! .
Multiply the lhs by H ABA 8 B 8 and the rhs by its tensor equivalent H ab and one immediately
obtains ~see Ref. 6, p. 153!
1
ab
ab
ab
a
H CDD 8 C 8 ⇔ 21 ~ p ab
cd 2g g cd 2i e cd ! H ab 5 2 ~ H cd 1H dc 2g cd H a 2i e abcd H ! .
Now take the conjugate of the spinor/tensor equivalent relation, to get d AD d BC d C88 d D88 ⇔ 21 ( p ab
cd
2g ab g cd 1i e ab cd , multiply as before by H ABA 8 B 8 and its equivalent H ab , and arrive at the different tensor equivalent ~with unprimed indices interchanged!
A
B
H DCC 8 D 8 ⇔ 21 ~ H cd 1H dc 2g cd H a a 1i e abcd H ab ! .
APPENDIX A: SPINOR EQUIVALENTS FOR THE gKd AND gpd
In the reference formulas below it should be mentioned that there is a good deal of interplay
between the gKd and gpd in such specifications, there being a variety of ways to express expansions of some gKd’s, gpd’s ~and also their spinor equivalents!.
1. Spinor equivalents of the gKd’s
D 1 5 d ab ⇔ d AB d B 88 ,
A
D 2 5 d ab
cd 5
U
d ad
d bd
D 3 5 d abc
de f 5
d cd
U
d ac
d bc
d ae
d be
d ce
U
U
d AC d CA 88
d ad
⇔ B B8
d bd
d Cd C8
U
d AD d DA 88
d BD d DB 88
AB
AB
5 21 @ d CD
p C88 D88 1 p CD
d C88 D88 # ,
A B
A B
U
d af
d bf
d cf
1
a bc
a bc
A
BC
8 BC 8 8
8 8
5 d ad d bc
e f 2 d e d d f 1 d f d de ⇔ 2 @ d D d D 8 ~ d EF p E 8 F 8 1 p EF d E 8 F 8 !
A
B C
B C
BC
BC
BC
BC
2 d AE d E 88 ~ d DF
p D88 F88 1 p DF
d D88 F88 ! 1 d AF d F 88 ~ d DE
p D88 E88 1 p DE
d D88 E88 !# .
A
B C
B C
A
B C
B C
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2061
We can expand the gKd with 4 ~upper/lower! indices by a Laplace expansion of its first two rows
and complementary minors. The result is
U
d ae
d be
D 4 5 d abcd
5
e f gh
d ce
d de
d af
d bf
d cf
d df
U
d ag
d bg
d cg
d dg
d ah
d bh
d ch
d dh
cd
ab cd
ab cd
ab cd
ab cd
ab cd
5 d ab
e f d gh 2 d eg d f h 1 d eh d f g 1 d f g d eh 2 d f h d eg 1gd gh d e f .
The rhs can be written as the sum/difference of permanents of gKd’s, if desired
I
II
d ab
ef
d cd
ef
II
d ab
d ab
gh
eg
cd 2
d gh
d cd
eg
I
d ab
d ab
fh
eh
cd 1
dfh
d cd
eh
d ab
fg
.
d cd
fg
The spinor equivalent of d abcd
e f gh is
AB
CD
CD
AB
AB
8 8
8 8
8 8
8 8
8 8
8 8
D 4 ⇔ 41 @~ d AB
EF p E 8 F 8 1 p EF d E 8 F 8 !~ d GH p G 8 H 8 1 p GH d G 8 H 8 ! 2 ~ d EG p E 8 G 8 1 p EG d E 8 G 8 !
A B
A B
C D
C D
A B
A B
CD
CD
AB
AB
CD
CD
3 ~ d FH
p F 88H 88 1 p FH
d F 88H 88 ! 1 ~ d EH
p E 88 H88 1 p EH
d E 88 H88 !~ d FG
p F 88G 88 1 p FG
d F 88G 88 !
C D
C D
A B
A B
C D
C D
AB
AB
CD
CD
AB
AB
1 ~ d FG
p F 88 G88 1 p FG
d F 88 G88 !~ d EH
p E 88H 88 1 p EH
d E 88H 88 ! 2 ~ d FH
p F 88 H88 1 p FH
d F 88 H88 !
A B
A B
C D
C D
A B
A B
CD
CD
AB
AB
CD
8 8
8 8
3 ~ d EG
p E 88G 88 1 p EG
d E 88G 88 ! 1 ~ d GH
p G88 H88 1 p GH
d G88 H88 !~ d CD
EF p E 8 F 8 1 p EF d E 8 F 8 !# .
C D
C D
A B
A B
C D
C D
2. Spinor equivalents of the gpd’s
P 1 5 p ab 5 d ab ⇔ d AB d B 88 ,
A
P 2 5 p ab
cd 5
I
I
d ad
d bd
P 3 5 p abc
de f 5
d cd
I
I
d AC d CA 88
d ad
⇔ B B8
d bd
d Cd C8
d ac
d bc
d ae
d be
d ce
d af
d bf
d cf
I
d AD d DA 88
d BD d DB 88
I
AB
AB
5 21 @ d CD
d C88 D88 1 p CD
p C88 D88 # ,
A B
A B
1
a bc
a bc
A
BC
8 BC 8 8
8 8
5 d ad p bc
e f 1 d e p d f 1 d f p de ⇔ 2 @ d D d D 8 ~ d EF d E 8 F 8 1 p EF p E 8 F 8 !
A
B C
B C
BC
BC
BC
BC
1 d AE d E 88 ~ d DF
d D88 F88 1 p DF
p D88 F88 ! 1 d AF d F 88 ~ d DE
d D88 E88 1 p DE
p D88 E88 !# .
A
B C
B C
A
B C
B C
Since permanents only involve positive signs, the following Laplace expansion is also clear:
P 4 5 p abcd
e f gh 5
I
d ae
d be
d ce
d de
d af
d bf
d cf
d df
d ag
d bg
d cg
d dg
d ah
d bh
d ch
d dh
I
cd
ab cd
ab cd
ab cd
ab cd
ab cd
5 p ab
e f p gh 1 p eg p f h 1 p eh p f g 1 p f g p eh 1 p f h p eg 1 p gh p e f .
The rhs can be written as the sum of permanents of permanents, if desired
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2062
J. Math. Phys., Vol. 40, No. 4, April 1999
I
R. L. Agacy
II
p ab
ef
p cd
ef
p ab
p ab
gh
eg
cd 1
p gh
p cd
eg
II
I
p ab
p ab
fh
eh
cd 1
pfh
p cd
eh
p ab
fg
.
p cd
fg
The spinor equivalent of p abcd
e f gh is
AB
CD
CD
AB
AB
8 8
8 8
8 8
8 8
8 8
8 8
P 4 ⇔ 41 @~ d AB
EF d E 8 F 8 1 p EF p E 8 F 8 !~ d GH d G 8 H 8 1 p GH p G 8 H 8 ! 1 ~ d EG d E 8 G 8 1 p EG p E 8 G 8 !
A B
A B
C D
C D
A B
A B
AB
AB
CD
CD
CD
CD
3 ~ d FH
d F 88H 88 1 p FH
p F 88H 88 ! 1 ~ d EH
d E 88 H88 1 p EH
p E 88 H88 !~ d FG
d F 88G 88 1 p FG
p F 88G 88 !
C D
C D
A B
A B
C D
C D
AB
AB
CD
CD
AB
AB
p F 88 H88 !
1 ~ d FG
d F 88 G88 1 p FG
p F 88 G88 !~ d EH
d E 88H 88 1 p EH
p E 88H 88 ! 1 ~ d FH
d F 88 H88 1 p FH
A B
A B
C D
C D
A B
A B
CD
CD
AB
AB
CD
8 8
8 8
3 ~ d EG
d E 88G 88 1 p EG
p E 88G 88 ! 1 ~ d GH
d G88 H88 1 p GH
p G88 H88 !~ d CD
EF d E 8 F 8 1 p EF p E 8 F 8 !# .
C D
C D
A B
A B
C D
C D
APPENDIX B: SPINOR⇔TENSOR EQUIVALENTS
The mixed mode spinor appears with its covariant form below it.
Spinor
Tensor
d AB
e AB
d ab
d AB d BA 88
e AB e A 8 B 8
g ab
A8B8
d AB
CD 5D, d C 8 D 8
e AB e CD 5 e AC e BD 2 e AD e BC
A8B8
p AB
CD 5P, p C 8 D 8 5P 8
e AC e BD 1 e AD e BC
8 1PD 8 )
e AC e BD e A 8 C 8 e B 8 D 8 2 e AD e BC e A 8 D 8 e B 8 C 8
d ab
cd
1
1
AB A 8 B 8
AB A 8 B 8
2 ( CD C 8 D 8 1 p CD p C 8 D 8 )5 2 (DD 1PP
e AC e BD e A 8 C 8 e B 8 D 8 1 e AD e BC e A 8 D 8 e B 8 C 8
p ab
cd
g ac g bd 1g ad g bc
1
1
AB A 8 B 8
AB A 8 B 8
2 ( CD p C 8 D 8 1 p CD C 8 D 8 )5 2 (DP
d
d
d
d
8
8)
g ac g bd 2g ad g bc
i DP
A B
A B
i~dACdBDdD88dC882dADdBCdC88dD88!5 dD8P8
2
i( e AC e BD e A 8 D 8 e B 8 C 8 2 e AD e BC e A 8 C 8 e B 8 D 8 )
e cd ab 5 e ab cd
A8B8
d AB
CD d C 8 D 8 5DD 8
e AB e CD e A 8 B 8 e C 8 D 8
g ab g cd
g ab g cd
A8B8
d AB
CD p C 8 D 8 5DP 8
e AC e BD e A 8 C 8 e B 8 D 8 2 e AD e BC e A 8 D 8 e B 8 C 8
1 e AC e BD e A 8 D 8 e B 8 C 8 2 e AD e BC e A 8 C 8 e B 8 D 8
5 e AB e CD ( e A 8 C 8 e B 8 D 8 1 e A 8 D 8 e B 8 C 8 )
ab
d ab
cd 2i e cd
A8B8
p AB
CD d C 8 D 8 5PD 8
e AC e BD e A 8 C 8 e B 8 D 8 2 e AD e BC e A 8 D 8 e B 8 C 8
2 e AC e BD e A 8 D 8 e B 8 C 8 1 e AD e BC e A 8 C 8 e B 8 D 8
5( e AC e BD 1 e AD e BC ) e A 8 B 8 e C 8 D 8
ab
d ab
cd 1i e cd
8 8
p AB
CD p C 8 D 8 5PP 8
( e AC e BD 1 e AD e BC )3( e A 8 C 8 e B 8 D 8 1 e A 8 D 8 e B 8 C 8 )
ab
2 p ab
cd 2g g cd
2g ac g bd 12g ad g bc 2g ab g cd
d AC d BD d DA 88 d CB 88
e AC e BD e A 8 D 8 e B 8 C 8
1
ab
ab
ab
2 ( p cd 2g g cd 2i e cd )
1
2 (g ac g bd 1g ad g bc 2g ab g cd 2i e abcd )
A B
e abcd
g ac g bd 2g ad g bc 2i e abcd
g ac g bd 2g ad g bc 1i e abcd
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J. Math. Phys., Vol. 40, No. 4, April 1999
R. L. Agacy
2063
T. Muir, The Theory of Determinants in the Historical Order of Development ~Macmillan and Co., Limited, London,
1906!, Vol. 1, p. 94.
2
C. M. Cramlet, Invariant tensors and their application to the study of determinants and allied tensor functions, Ph.D.
thesis, University of Washington ~1926!.
3
C. M. Cramlet, ‘‘Applications of the determinant and permanent tensors to determinants of general class and allied tensor
functions,’’ Am. J. Math. 49, 87–96 ~1927!.
4
R. L. Agacy, Generalized Kronecker, permanent delta and Young tableaux applications to tensors and spinors;
Lanczos–Zund spinor classification and general spinor factorizations, Ph.D. thesis, London University ~1997!. The
1
expression L i jk 5 3 L $ ik, j % is correctly stated on p. 28 of this reference, but incorrectly stated on p. 26 with L $ i j,k % on the
rhs.
5
See AIP document No. PAPS JMAPAQ-Vol. 40-033903 for 33 pages of the document entitled ‘‘Generalized Kronecker
and Permanent deltas, their spinor and tensor equivalents—Reference Formulae.’’ Order by PAPS number and journal
reference from the American Institute of Physics, Physics Auxiliary Publications Service, 500 Sunnyside Boulevard,
Woodbury, NY 11797-2999. Fax: 516-576-2223, email: paps@aip.org. The price is $1.50 for each microfiche or $5.00
for photocopies of up to 30 pages, and $0.15 for each additional page over 30 pages. Airmail additional. Make checks
payable to the American Institute of Physics.
6
In this connection we remark that our definition for the algebraic symmetries of the Riemann tensor R abcd
1
5 12 R $ ac,bd % , agrees with the result in R. Penrose and W. Rindler, Spinors and Space-time ~Cambridge University Press,
3
England, 1984!, Vol. 1, p. 144, 4 R abcd 5R( āc̄ ) . This is because the symmetrization on two letters introduces a factor of
1
( bI dI )
1/2, which together with 2 rows gives a factor of 1/4. Antisymmetrization of columns produces another factor of 1/4.
3
1
Hence the rhs of the latter is 1/16 of our result, i.e., 4 R abcd 5 16 R $ ac,bd % , agreeing precisely with our definition.
7
S. B. Edgar and A. Höglund, ‘‘The Lanczos potential for the Weyl curvature tensor: existence, wave equation, and
algorithms,’’ Proc. R. Soc. London, Ser. A 453, 835–851 ~1997!.
8
P. Dolan and C. Kim, ‘‘The wave equation for the Lanczos potential,’’ Proc. R. Soc. London, Ser. A 447, 557–575
~1994!.
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