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(s/negationslash=r)/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
rs/bardbl=−d2/summationdisplay
s=1
(s/negationslash=r)Qr/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardblQT
r
=−Qr
d2/summationdisplay
s=1/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardbl
QT
r
=−QrQT
r
= 0 (399)
Taking the complex conjugate on both sides we find
1
d+1d2/summationdisplay
s=1
(s/negationslash=r)/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl= 0 (400)
Consequently
2
d+1d2/summationdisplay
s=1
(s/negationslash=r)/bardblgrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgrs/bardbl=1
d+1d2/summationdisplay
s=1
(s/negationslash=r)/parenleftig
/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl+/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
rs/bardbl
+/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
rs/bardbl+/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl/parenrightig
=¯Rr (401)48
Eq. (393) is proved similarly.
To prove the second group of identities we have to work a little harde r. Using
Eqs. (116) and (120) we find
1
d−1d2/summationdisplay
s=1
(s/negationslash=r)/an}bracketle{t/an}bracketle{ta/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardblb/an}bracketri}ht/an}bracketri}ht=d+1
d−1d2/summationdisplay
s=1/an}bracketle{t/an}bracketle{ta/bardblQs/bardblr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tr/bardblQs/bardblb/an}bracketri}ht/an}bracketri}ht
=(d+1)3
d2(d−1)d2/summationdisplay
s=1/parenleftig
TsarTsrb−K2
saK2
srTsrb
−K2
srK2
sbTsar+K2
saK4
srK2
sb/parenrightig
(402)
(where we used the fact that Qs/bardbls/an}bracketri}ht/an}bracketri}ht= 0 in the first step). After some algebra we
find
d2/summationdisplay
s=1TsarTsrb=d
d+1/parenleftigg/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
d/parenrightigg/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
d/parenrightigg
+Trba/parenrightigg
(403)
d2/summationdisplay
s=1K2
saK2
srTsrb=d
d+1/parenleftigg/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+2d+1
d(d+1)/parenrightigg/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
d/parenrightigg
+1
d+1Trba/parenrightigg
(404)
d2/summationdisplay
s=1K2
srK2
sbTsar=d
d+1/parenleftigg/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
d/parenrightigg/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+2d+1
d(d+1)/parenrightigg
+1
d+1Trba/parenrightigg
(405)
d2/summationdisplay
s=1K2
saK4