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Symplectic reduction of p-form electrodynamics
Vol. 45 (2000)
REPORTS ON MATHEMATICAL
SYMPLECTIC
REDUCTION
No. 1
PHYSICS
OF p-FORM
ELECTRODYNAMICS
DARIUSZ CHRUSCI~~SKI Institute of Physics, Nicholas Copernicus University ul. Grudziadzka 5n, 87-100 Torud, Poland (e-mail: [email protected]) (Received
March
22,
1999
-
Revised
Seprember
14,
1999)
We propose a simple method to reduce a general p-form electrodynamics with respect to standard Gauss constraints. The canonical structure of the reduced theory displays a pdependent sign which makes the essential difference between theories with different parities of p. This feature was observed recently in the corresponding quantization condition for p-brane dyons. It suggests that these two structures are closely related.
1.
Introduction
The p-form electrodynamics is a simple generalization of ordinary electrodynamics in 4-dimensional Minkowski space-time M4 where the electromagnetic field potential l-form A, is replaced by a p-form in D-dimensional space-time [ 1, 21. The motivation to study such type of theories comes e.g. from string theory where one considers higher dimensional objects (the so called p-branes [3]) interacting with a gauge field. p-branes are natural objects which couple in a gauge invariant way to a p-form gauge potential and, therefore, they play a role of elementary extended sources. Recently the new input to study a p-form theory came from the electric-magnetic duality which started to play a prominent role in theoretical physics in the 90. (see e.g. [4]). Now, in order to have p-branes which carry both electric and magnetic charges (i.e. p-brane dyons) the dimension of space-time has to be equal to D = 2p + 2. It turns out [5, 61 that there is a crucial difference between p-form theories with different parity of p. Moreover, quantum mechanics implies the following quantization condition upon the electric and magnetic charges (ei , ~1) and (e2, g2) of any two dyons: e1g2
+ (-lFe2gl
= nh,
(1.1)
with an integer n. For odd p the above condition is a generalization of the famous Dirac condition [7], but for even p it was observed only recently [S-11]. Again, the parity of p plays a crucial role in (1 .l). The aim of the present paper is to show that this p-dependence is already present on the level of canonical structure of the underlying p-form theory. This feature 11211
122
D. CHRUklfiSKl
is however hidden in the standard formulation. ‘Therefore, we propose a different approach. Because the theory possesses a rich abelian gauge symmetry we propose to reduce it with respect to the corresponding Gauss constraints. Such a reduction was proposed in the case of ordinary electrodynamics (and linearized gravity) in [12] (it was applied to the variational formulation of electrodynamics of point-like charges in [13, 141). The method of this paper is a straightforward generalization of [12] to a p-form case. We show that the reduced canonical structure (see formula (4.28)) displays a p-dependent sign in a manner very similar to (1.1) which shows that the duality invariance, quantization condition and canonical structure are closely related. We postpone the study of this relation to the next paper. It would be also interesting to investigate relation between our approach and the standard two-potential formulation of a p-form theory (see e.g. [5, 151). The present paper is organized as follows: in Section 2 we introduce basic facts about p-form electrodynamics and present its (unreduced) canonical structure in Section 3. Then in Section 4 we describe the reduction procedure. As an example of a reduced p-form theory we present a Maxwell-type theory in Section 5. To make the presentation more transparent we discuss in the last section the p = 1 and p = 2 cases. They represent p odd and p even, respectively. All technical details and proofs are contained in appendixes. 2.
p-form
electrodynamics
Consider a p-form potential A defined in the D = (2p+2)-dimensional Minkowski space-time M2P+2 with the signature of the metric tensor (- , +, . . . , +). The corresponding field tensor is defined as a (p + I)-form by F = dA, F LLI...Pp+l
=
(2.1)
%4AW..Wp+ll
where the antisymmetrization is taken with a weight one, i.e. X[kl] := Xkl - Xlk. Having a Lagrangian L of the theory one defines another (p + 1)-form G as follows G@~-~LP+~
=
_(p
+
I)!
aF
a’
.
(2.2)
~l...llPfl
Now one may define the electric obvious way:
and magnetic
intensities
and inductions
in the
(2.3) 1 Bi,...ip
F.h-jpil
=
3
Dil...i, = Gi,...ipot
(2.5)
1 fi,...ip
= cp + 1j!
(2.4)
Eil...ipj14p+l
($-&+I
’
(2.6)
SYMPLECTIC
REDUCTION
OF p-FORM
ELECTRODYNAMICS
123
where the indices il, i2, . . . , j,, j2, . . run from 1 up to 2p + 1 and ti,i2.,_iZp+, is the Levi-Civita tensor in (2p + I)-dimensional Euclidean space, i.e. a space-like hyperplane C in the Minkowski space-time. The field equations are given by the Bianchi identities dF = 0, or in components (2.‘7) and the true dynamical
d + G = 0, or equivalently
equations
(2.8)
31~* G PI.-.Pp+ll - 0, in M2pf2
where the Hodge star operation .&WI-+,+1
1
=
is defined by ~l...ILp+l"I..."/~+l
(p+l)!
G
rl
(2.9)
w ...Up+I
and ~l-Ll~2+*~~+*is the covariantly constant volume form in the Minkowski spacetime Note that ~'l...i?p+l *.In terms of the electric and magnetic fields rloil-.,i2p+l. defined in (2.3)-(2.6) the field equations (2.7)-(2.8) have the following form: Eil...i&l...jJ~
vi, Bil...iP =
v~E~,,,,~
(2.10)
I’’
(2.11)
0,
(2.12) Dil...i,
v.
=
0
(2.13)
iI
where Vk denotes the covariant derivative on C compatible with the metric r/k/ induced from M2P+2. The Levi-Civita tensor density satisfies t12...2p+l = &, with g = det(rlkl). The field equations (2.10)-(2.13) may be rewritten using the language of differentials forms. Obviously D, B, E and H are p-forms on C. Let us define (p + I)forms: D := *=D,
t? := *=B,
(2.14)
where “*x” denotes the Hodge star on C: for any k-form, 1 (*CX)il...in_k
‘=
with IZ= 2p + 1. With this notation 8 = (-l)p ij=dH,
(n
_
@!
Eil...in_kjl...jk
xji
.-jk
9
(2.15)
one has: dE,
dt3 = 0,
(2.16)
d;l,=O.
(2.17)
124 3.
D. CHRU&X&KI
Canonical
structure
The phase space P of gauge invariant configurations (D, B), or equivalently (D, B), is endowed with a canonical structure. Using local coordinates (D’l+, Bjl...jp), the most general 2-form on P is given by J-+/d2P++2p+i
y [Xi,...i,j,...jP(x, y)6D’+p(x)
+ iYil,.,i,jl..,jp(X,
Y)6Di”.“p(X)
+ ~Zil,,,i,j,,,,j,(X,
Taking into account field XH~
A 6Dji”‘jp(Y)
Y)SB”‘...‘P(X) A SBj”“j’
the field equations
A GB”...‘p (y)
(Y)]a
(3.1)
(2.10) and (2.12), the hamiltonian
vector
XHpA 52, = -6H, is described
XH~ = Therefore,
(3.2)
by d2’+‘X
s
+ (_I)qkEjl-jP&
Vk Hjl “.jP &
Eil ,,,ipkj, ,,,j,
I,...lp>
I,...lp
. (3.3)
Eq. (3.2) is solved by HP and Qp defined as HP =
-!2p!
s
d2P+1x(~i’...i,Ei,~,,ip+ B'l-'PH. Il...lp) 7
(3.4)
and .n, = /d2P+1x/where G(x, y) denotes clidean space, i.e.
d2p+ly
~Dil...ip cx) A
the Coulomb
Green
SB.h...h
(Y)9,...ipkj,...jpVk
function
G(%
Y), (3.5)
in (2p + 1)-dimensional
A G(x, y) = -d(2p+1)(~ - y).
Eu-
(3.6)
The formula (3.5) uses only gauge-invariant quantities. It may be, however, considerably simplified by introducing a gauge potential. One solves (2.11) via @I-‘P
=
eil...ipjl...jp+l
v.
A, JI
J2...jpfl
9
(3.7)
which is the obvious generalization of B = V x A from the p = 1 electrodynamics. Now, modulo gauge-dependent boundary term, the symplectic form reads d2P+lx
Qp = (-1)P s
,j@l&
A
a~~,.,.~,
=
d2p+‘x
(_l)P+’
SGCL’+‘o A 6AW,+,,.
s
(3.8)
SYMPLECTIC
Having a symplectic brackets
REDUCTION
OF p-FORM
125
ELECTRODYNAMICS
structure in P one may easily derive the corresponding
Poisson
where F and 6 are functionals on P. Note that Dil...ip and Bil..,ip are not arbitrary but they have to satisfy the Gauss laws (2.11) and (2.13). The fundamental commutation relations implied by (3.9) read (@-ip(x),
Bjl-~jp(Y))p
= Eii...ipkjl...j~vks(2p+1)(x
(o”.%(x),
~‘l-jP(y))p
=
{B’~-~‘P(~),
Bjl...&?(y)],
_ =
y),
0,
(3.10) (3.11)
with x, y E C. 4.
Reduction
The idea of this section is to find a new set of electromagnetic variables using the philosophy of “2p + 1” decomposition (it is a straightforward generalization of [12]). These new variables solve partially the Gauss constraints (2.11) and (2.13). Moreover, they are canonical with respect to 52,. 4.1.
“2p + 1” decomposition
The “2p+ 1” decomposition is based on the observation sional Euclidean space C may be decomposed as
that the (2p+ l)-dimen-
IZ = S2P(l) x R, where S2P(l) denotes 2p-dimensional unit sphere direction. Let us introduce the spherical coordinates
(4.1) and IR represents on C:
A = 1,2, . . . ,2p,
the “radial”
(4.2) (4.3)
(~2~ denote spherical angles (to enumerate the angles we shall use capital letters A, B, C, . . .) (for more details see Appendix A). Let qkl denote the Minkowskian metric on C and let h, denote the corresponding volume element, denote the Levi-Civita tensor on i.e., h, = 4s (see (A.3)). Finally, let EA,...~~~ S2p(r) such that EA,...A~~:= E~A,...A~~ (we use the same letter to denote the Levi-
where ~Q,vPZ,...,
Civita tensors on C and S2P(r)). We shall denote by “I” the covariant derivative defined by qkl on C and by “I]” the covariant derivative on each S2p(r) defined by the induced metric I]AB.
126 4.2.
D.CHRU$CIfiSKI Symplectic reduction
Let us consider the evolution of the p-form electromagnetic field in the finite volume V E C. For simplicity we take V = K 2P+1(0, R) (a (2p + 1)-dimensional ball of radius R). Of course one may take any V topologically equivalent to K*P+‘(O, R), but then instead of the spherical coordinates one chooses coordinates adapted to V. To reduce np in V we use the “2p+l” decomposition (-l)‘+‘Qp
hp GGi’...i~o A GAi,,,,ip = -
= sV =--
s vhP
Using (3.7) one obtains
hp ~D”~..‘P A GAi,...i, s V
(PAD ri2...i’ A 6Ari2.,,ip + GDA1...ApA SAA,,,,A~) .
BAI...A, = @..&‘BI.JP
arAB,...B + p EA~~~~*,B~rBz~~JG ArB2,,,B ,,B P
P
I’
(4.4)
(4.5)
and hence ~A,.,_A~B,...B~B~‘~~~~~“~’ = p! a;‘,‘,::21 (-1)’ where we have used the following Eil...iNjl...j~E,
[&&,...c,
- P A~c~...c,IIc~]“~’ 7 (4.6)
identity =
rl...i~ll...l,+f
No
.
gjl...jM
._-I
[lI...l.&f] *-
.
gjl
[I, . . . al:]
(4.7)
with arbitrary N and M. Now, to simplify our consideration, let us choose the following gauge conditions for a p-form A A~...A~ and (p - 1)-form A rA2...A~ on each sphere S2p(r): AAI...A, ~rA2u.A~ [IAl = 0, llA2 = 0. (4.8) In such a gauge EA1...A,BI...B ’
=
BA,...A,IIB,
(_l)P+lp
p!
r-2
Ap_,
A’B2..,B~,
(4.9)
where Ap_l is defined by Ap_, := (p - l)! [r2VAVA - (p2 - l)]. This operator
has a clear geometrical
interpretation.
LEMMA 1. r-2Ap_1 is equal to the Luplace-Beltrami X = (XA2...Ap) on S2p(r) satisfying XA2...Ap~~~2= 0, i.e.
operator for (p - 1)-forms
r-*A p_1x = (d*d + dd*)X = d*dX. For proof see Appendix
(4.10)
(4.11)
B. The factor r 2 in (4.10) makes Ap_i r-independent,
i.e. [A,-,, LEMMA
2. The operator
%I = 0.
Ap_l is invertible.
(4.12)
SYMPLECTIC
REDUCTION
OF p-FORM
ELECTRODYNAMICS
For the proof see Appendix B. Denoting by A;:, obtain
Introducing
the inverse of A,_,
127
we finally
the variables: AI-A,
._
Q (1)
“‘1’ B2...Bp
*-
rD
**-
+
rA2...A,
Ai!,
(4.14)
, (tA1...~,~I...~,,
(4.15)
BA”““““B’)
we may rewrite the first integral in (4.4) as -
Jv
A., PJD ri2-.i~ ,,y&&+
=
1)P”
(-
L
s
hpUI;;~,~Ap A t3Q;i"'"".
(4.16)
V
To reduce the second term in (4.4) observe that in the special gauge (4.8), AA,, ,A,, may be expressed as AA~
=
...A.
EA ,... A,BI...B,,
p
BI..-B~,IJBI
(4.17)
Now, from (3.7) B rBz...B, Inserting obtain
(4.17)
=
cBz...BpC,...C P+’
&~...c,,+,
(4.18)
11~1 .
into (4.18) and using once more the gauge conditions B rBz...f$ = _P! r-2
Ap_l pBz...Bp,
(4.8) we (4.19)
and, therefore pB:...B”
=
__
r2
A;--,
Brh...B,,
(4.20)
P!
which, together
with (4.17), gives r2 AA ,,.. A,
=
--
(4.21)
CA ,... A,,B ,... B,
P!
Having
AA]
we may calculate
. ..A.,
-
s V
AP c~D~I-.-~PA SA
the second term in (4.4),
Al . ..A.
GA ,... APB ,... B,
SDAL-.-A~)~~~~
,,
y A;!,
&BrB2...BI,.
(4.22)
128
D. CHRU~CIfiSKl
Defining 4.-A,
._
Qc2)
.--rB
rA2...A,
1
(4.23)
(2)
I-I
(4.24)
B2...B,
we may rewrite
(4.22) as
-s s( h,
cYD~"'.~J' A SAAI.,.Ap
V
h,
s A,
Sl-l~;,,.Ap A 6 Q;",;--'"'.(4.25)
V
Now, taking into account
“F” =
=
(4.16) and (4.25) we have finally
Sl-l~;~~~Ap A SQ;ll;"'Ap
+
(-l)p+1SIl(,21,~~Ap A aQ;42;Ap)
(4.26)
V
Let us use a more compact
notation X .Y
Our result may be formulated
:=XA2...AP YA~.,.A~.
as
THEOREM 1. The reduced symplectic structure the reduced phase space pr” := (Qca,, l7@)),
a;* =
s A,
(4.27)
J2:” has the following form
(sn”’ . AJQ(I) + C-1)P+18ZZ(2). /GQC2,)
.
on
(4.28)
V
The above theorem THEOREM {F,
obviously
implies the following
2. The corresponding
E);* =
result.
reduced Poisson bracket (3.9) reads
s, [g) *--& +(-l,p+l-&*& - v = a].
This formula
A,
leads to the following
commutation
relations:
g(2~+‘)(~ _ y> (Q;",;-Ap(x), n;;,,,Bp(y)}red = 8A2-.Ap [Bz...&J {Q;',;'-'ApW,
and the remaining 4.3.
(4.29)
(4.30)
@;,,,Bp(y));* =(-1)p+16$;:-~ls(2p+1)(XP
y),
(4.3 1)
brackets vanish.
Reconstruction
Could we reconstruct D and B in terms of Q’s and II’s? The answer is positive by virtue of the following reconstruction.
129
SYMPLECTICREDUCTIONOF p-FORMELECTRODYNAMICS THEOREM 3. The variables (Qcml, II iu)., a = 1. 2) contain invariant information about the jields Dil...ip and BiI...‘p.
the entire gauge-
Proof: We show that Q(1) and n(2) encode the information about the D field. Due to (4.14) it is enough to show how to reconstruct the tangential part of 1). i.e. DAl...AIJ. Let us make the decomposition (see Appendix A): DAI...A,
=
U[A2...ApllA~]
LEMMA 3. The following
pVBZ...B,‘/15,~
(4.32)
r2DA,...~$j ,,A, = np_, UA”“Ap,
(4.33)
identities
For the proof see Appendix
hold:
~A1....4,llh
~2~AI...A,BI...B,
= a
(4.34)
VB2...B,.
P-l
B. Therefore,
VBp.B,,
Now, to reconstruct
+ cA~...A,,B~...B
=
-P!
r l-$..B,,
(4.35)
.
14A2...Aplet us use (2.13).
LEMMA 4. The Gauss constraints
DkA2...A~pkare equivalent
to
ar(&’ DrAz...Apj + r~~~-b&...~~,~,,A, = 0. The proof is given in Appendix
B. Finally, due to (4.36)
UA2...A, = _,2~,-’ p_,
a,. r2p-1
In the same way one shows that Q(z) and n(i) 4.4.
Counting
(4.36)
Q$i”.Ap)
reconstruct
(4.37) 3
the B field.
degrees of freedom
Let us count the total number Define a number nf by
N, of degrees
of freedom
for a p-form
nf :=
A p-form potential a gauge freedom
(4.38)
Ai,,,,ip on C has n,,’ independent
components.
However, due to
Ai,...i, + Ail...i,, + v,,,A~~,‘,.;,,I some of them may be gauged away e.g. to zero. A:,t)j,,._, carries n;_, but the equation (4.39) is still invariant under
A?’
I, . ..I.,-,
+ 4
theory.
(1) 1,.
lp__,
+
v,,,A!2’ Q...lr,-11‘
(4.39) components
(4.40)
130
D. CHRIkCItiSKl
Therefore, some of the components of A(‘) may be gauged away via AC2) which has r~,“_~components. The same argument applies for AC2) and one continues up to a O-form AP carrying obviously only one component. Therefore, the total number of degrees of freedom carried by Ai,,,.i, equals Np = npP- n,“_i + n;_2 - . . . + (-l)Pni. Using the standard mathematical
induction
(4.41)
it is easy to prove that
Np =
(4.42)
Now, note that Q(i) and Q(2) carry together 2P = -Np. P+l
(4.43)
Therefore, (4.44) Mp > Np, holds only for p = 1. It means that for p > 1, Q(a) are not
and the equality independent. LEMMA
5. For p > 1, the reduced
variables
Q(U) and II@) satisfy the con-
straints: VA2Q$Ap
= 0,
v%-I~;,,,Ap = 0.
(4.45)
The proof follows obviously from (2.11) and (2.13). Note that these constraints are trivially satisfied for p = 1. 5.
Maxwell
theory
The simplest p-form theory is of the Maxwell type, i.e. the constitutive between inductions (D, B) and intensities (E, H) are linear: Dil...i,
= ~‘1 ...i.
7
Bil...iP
= Hil...iP
In such a case it is easy to rewrite the field equations of Q’s and n’s. THEOREM
4. Equations A2
.A,
_
(5.1)
(2.10) and (2.12) in terms
(2.10) and (2.12) are equivalent
Q,,;'- -$I
relations
to:
A2...A,
n,,, )
(5.2) (5.3)
l-y’) A2._A,
fi (2) A2...A,
=
-A;:,
=
(-I)~A;!,
r-‘$(rQ~~,,,,,)
E
+ re2A,-,
[r-la~(rQ~~...A,)
-I Q~:...A PJ ’
+ rm2A,-l
QZ~...AJ l
(5.4) (5.5)
SYMPLECTIC
where
REDUCTION B2...Bp
QFJ..., := vA2B2 P
. . . qA,B,
Q(,)
For the proof see Appendix lowing Hamiltonian, H,, =
1
OF p-FORM
and
131
ELECTRODYNAMICS
17$j”Ap := qAzB2 . . . qApBp17f2!),B i’
C. These equations
may be derived
2(p - l)! s
kp rp2Qc,) . Q’“’ - r-2pal(r2P-1Q~a,)A~!~ [
- &&,_I
. @‘]
from the fol-
. &.(rQ@))
,
(5.6)
Q(a) = IQ(a), Hplrd.
(5.7)
n,,,
(5.8)
i.e.
= I&,,
HJfe”,
with the Poisson bracket defined in (4.29). One easily shows that the numerical value of HP is equal to the standard (i.e. obtained via the symmetric energy-momentum tensor) electromagnetic energy contained in V, i.e. HP
I
1
2P! s v
6.
p = 1 versus
A, (Di’-~ip Di ,,., i,, + Bi’...‘” Bi ,___ ;J
(5.9)
p = 2
The case p = 1 corresponds The symplectic form
to the ordinary
52, = (i = 1,2, 3) reduces
s
d3x8D’
electrodynamics
A 6Ai
in 4 dimensions.
(6.1)
to
.nl”” = where
.
s
h3(61-l(‘) A 6Qcl,
Q’s and l-I’s are functions
6l-l (2) A ~Q(2))t
+
(6.2)
(O-forms) on S2(r):
Q(l) = rD’, Q(2) = rB’, llcl) =rAO’(cABB
AIIB),
nc2) = -r&‘(EABD
and A, denotes the ordinary
Laplacian
on S2( l), i.e.,
A, := r2 VAVA Due to (4.41), Nt = 2 as it should.
“IIB),
(6.3)
132
D. CHRU.%2IdSKl
Now, p = 2 corresponds symplectic form
to 2-form electrodynamics
JZ2 = + (i, j = 1,2, . . . ,5) reduces
in D = 6. The corresponding
d5XJ D’j A 6Aij
s
(6.4)
to
fin;ed =
A5(81-l~’ A SQiq, - CUDS’A SQ;,),
(6.5)
s where
Q’s and D’s are now l-forms
on S4(r):
Q$,=rD
i-A
,
Q;,=rB
rA
,
l-I!) = r A;'(rABCDBAB”C), lib” = -rA,’ and A, denotes the Laplace-Beltrami l-forms XA satisfying XAllA = o),
(eABCDDAB’IC), operator
on S4(1) on co-closed
A, := r2 VAVA - 3. Due to (4.41), A72= 6. Now, note the differences
l-forms
(i.e.
(6.6)
in signs:
1. the difference between (6.1) and (6.4) is not essential because symplectic form may be multiplied by any (nonzero) number, 2. however, the difference between (6.2) and (6.5) is crucial. It implies that the p = 1 theory is duality invariant, whereas p = 2 is not [5]. Appendixes A.
Geometry
of S2p(r)
The relation between the Cartesian (y’, y2, . . . , y2J’+‘) and spherical (x’, x2, . . . , x2P+l) introduced in (4.2)-(4.3) reads as follows: y1 = r sin qzP sin q2P_i _. . sin
93
sin qp2sin (al,
y2 = r sin y3zPsin 4D2P_1. . . sin
(~3
sin 402cos
y3 = r sin ~2~ sin ‘p2P_1. . . sin q3 cos q2,
Y 2p = r sin ~2~ cos q7zP-1, Y 2p+1 = r cos ~2~.
rpl ,
coordinates
SYMPLECTIC
The Minkowskian
REDUCTION
OF p-FORM
133
ELECTRODYNAMICS
metric on C is diagonal: 2P qkk
=r2
sin2 q7l,
n
k=l,2
,...,
2P-1,
(A.1)
I=k+l ~2~.2~
Therefore,
=
(A-2)
rl r,=l.
r2,
the volume element hp =
,iG
= r2’ fi
sinA-’ PA.
(A.3)
A=1
One easily calculates symbols are:
the corresponding r,A,,
r;,
Christoffel
= r-‘SA,,
Now, one easily finds the Riemann RABCD
=
symbols. The only nonvanishing
r;s
= +-‘~A&
tensor on S2p(r), rp2(vACvDB
-
(A.51
~AD~CB)~
where qAB denotes the induced metric on each S2p(r). tensor RAB and the scalar curvature R read 2p - 1
RAB = ~
Finally, let
EAl . ..A*.,
(Ad)
The corresponding
Ricci
R = 2P(2P - 1) VABt
r2
r2
tensor density on S2p(r)
denote the Levi-Civita t12...2p
=
such that
hp.
field on S2p(r).
Let X = XA,.,,A, denote any p-form
(A.61
’
This field may be decomposed
X3 XA,...A,
Formula
=
V[A,~A,,,.A,,]
(A.7)
+ CA ,.., A,,B ,... Bp~B’~B2’~‘B”.
(A.7) follows from the Hodge theorem X = clcx + d*j3’ + h,
(A.8)
with cx being a (p - I)-form, /?’ a (p + 1)-form and h a harmonic p-form. Note that there is no harmonic term in (A.7) because the set of harmonic p-forms on S2p(r) is empty. Obviously, B’ = h * B for some constant h. To find h note that for even-dimensional Riemannian manifolds the adjoint d* of d reads as d*=--*d*
and, therefore,
(A.9)
d*b’ = -h * d * *,!I. Now, on a Riemannian
n-dimensional
* *wk = (-l)k’“-k’Wk, and hence for n = 2p, * * p = (- l)P+’ p. Therefore, implies A = (-l)p.
manifold (A.lO)
d*B’ = (-l)Ph
* clp which
134 B. B.1.
D. CHRU&.dSKI
Proofs Proof of Lemma
1
Consider any (p - 1)-form X = (X,,...A~) on S2P(r) such that VA2X~2...~p = 0. This condition may be rewritten as d’ X = 0. Therefore,
the Laplacian
on X reduces
03.1)
to (W
(d*d + dd*)X = d*d X. Due to (A.9), d*d X may be calculated 1 (*d * dX)&...A, =
(p + I)! 1
= (p + l)! p!(p x
+
t~2.,,~n~~,...~~V’A~~B’~~~Bp1c’~~~c~V~~~X~~...~~~
(p
A
l)!
I... BP (”
cA2...A,AB
BI...B,C~...CP
)
xc2,,,c
(~~~-.B,~I-~P~-+
EA2...ApAB,,,,Bp
_
vBI~AB~...B,CI...C,
x
t
vBP~BI...BP-I~CI...~PvIc,
-...-
1 =
~Az...A,A~,...~,V’A(*dX)E”--B,l
P!
=
as follows
I
P
_
p
+-JPG--C~VBI)
x
+ I)!
x V[C,xcz_.cpl
= -6
= To calculate
r”A:;:::z] VA% xc2...c,1
(p - l)!
(-vATA x A~...A,
+ (-1)PVAV,~2X~,...~,~~)
U3.3)
.
the second term note that = vA
VAV,A$A+Ap]A
-
Now, we use the following [VA,
VBlT
CI...Ck
(vA2XA3...A,A . . . -
well-known DAB
which holds for any tensor field Tcl-ck. VAVkX4...A,A
vA3XA2A4...ApA
VA,XA2.,.Ap-,A~A
property
TD&..Ck
= RC’
-
of the Riemann
+ . . . +
tensor
RCkDABTC2-Ck-lD,
03.5)
Using (B.5) one has RA3
= @B[VA, VB]X&...A,A = @B (
+ .. . +
(B.4)
) .
RAP CAB~A3...AP-~CA
+ RA
XCA4...ApA CAB CABX‘43-‘4PC
. >
03.6)
SYMPLECTIC
Using (AS)
REDUCTION OF p-FORM
135
ELECTRODYNAMICS
one obtains
,,AzBR~~ CABXCA4...A,A=r-2QW$W’(TIDAllcB
_ ljDBIJCA)XCA~...A,~A
=r -2XA2A4....$& = y-2(_l)~+lXA~....4,~, Exactly
(B.‘7)
the same result holds for the first (p - 2) terms in (B.6). The last term CABXA3...A,~C = rJAzB~CBXA3...A,C= +(_1)~(2~
+BRA
_ ~)XAC% .
(B.8)
due to (A.6). Finally, (- * d * dX)A2...A,
= =r
B.2.
(p
-
I)!
-2Ll
P-l
-
r-‘(p2
-
I)]
XA2...A,,
n
XAp..A,
(B.9)
Proof of Lemma 2
It is a well-known fact from the of Ak is nontrivial only for k = 0 electrodynamics in M4) is special which are simply constant functions
B.3.
[v”vA
theory of cohomology that on Y(r) the kernel and k = n. The p = 1 case (i.e. the ordinary because A, = r2VAVA possesses zero modes on S2(r). This problem was treated in [ 121.
Proof of Lemma 3
Let X = (XA,...A~) be a p-form may be decomposed as XA,...A,
=
on
V[A,~A2...Ap]
i.e. cz and p are (p - I)-forms
Due to the Hodge theorem,
S2p(r).
+
(A.8) it (B. 10)
CA ,... A,,B ,... ~?,,~~‘fi~“‘.~“~
on S’P(r), (B.l Y)
x =&Y+**dp. Let us choose the following
“gauge condition”
for 4 and B,
d*a = d*,3 = 0.
Now, taking into account
(B.11) and Lemma
(B.12)
1 one obtains
d*X = d*da = r-2A,_la
(B. 13)
and * dX = *d *d/Y = -d*d/3’ = -A,_,/% Rewriting
(B.13) and (B.14) in components
(B. 14)
one obtains
r2XA’A2”‘APIIA,= A,_,aA>...A,, ,
(B.15)
and
A,-,B
Bz...Bp
=
=E
_r2(.+dx)B2-.&
=
-1
(P + I>! AI-+'PB~-.~PV~, x~,,,,~,.
cB2...B,,BlA~...A “v[B,
XA,...Apl
0
(B.16)
136 B.4.
D. CHRUhXk3KI
Proof of Lemma gki2_..iP
4 ,k =
[email protected]
+
where the last term obviously DkA2...A,
,k =
aryAz...Ap +
+
1-E~~ji2-i~
+
cp
_
1)
+zgkii3--.ip,
(B.17)
vanishes. Therefore,
aBDBAz...Ap
r;rDrAz---Ap
+
r;BDBiz..-i,
+
riryA2...Ap
+
(B.18)
r;c~cA2..+
Now, taking into account (A.4), DkAp..A,
,k =
;
a,y-Aa...Ap
2P
DrAz...Ap
r = Y-2~ 8, ($J ~rh..A,
C.
Derivation
of Maxwell’s
+ (~B@&...A~
+ riCDCA2...Ap)
> ) + ~4 &...A, ,,A, .
0 (B.19)
equations
Using (2.12) one obtains . A2._.A, Q,,,
_
-rD
.
rA2...A,
=
’
_
rAZ...ApAI
-
Bl...B ‘BB1...BpllA, A2...A,
=
.
-A;!,$,
Cl)
P!
In the same way one proves (5.3) using (2.10). Now, fiT(‘) B2...B,
=
‘A;:, P!
= LA-’ P!
=
EA I... ,z,p B I...
P-1
B
p
vB1
(-I)+
p-1
VBI
JI...i,
)
VBi>DrB2...Bp_lB1
[arVB’DB1...Bp
+ p(p
+
-
I>!
(VBIDrBz...B,,
)I
&-D,,...,p
=rAi!l [
c”“cpVB1(t-l)pp!~rD~~...~~ [BL...BPI
[P!&-DB,...B,
P-1
=rAi!,VB1
k
Q.cz.,.cp
S
iA-1 -
D.
v
(~A1~~~AprC’~~~cf~~Dc,...c,
PC A~...A~c~rc2...c~Vcl
(-l)J’rA-’
p!
AI...A,kjl...j,
>
(p>2
=
I... BpBA’~‘.Ap”B’)
(-~)“~A~TI,FA,...A,B,...B~V~’
+ =
(tA ,.,. ApB
- (-1)PVi~2Dr -
rT2Ap-lDrBz...Bp]
(-1)y’1~~!Vc~D~~2...~p)
-
VB2DrBlB3...Bp
1
B3...Bp]B~
-
(C.2)
SYMPLECTIC
REDUCTION
OF p-FORM
13’7
ELECTRODYNAMICS
For any tensor field X,,,, “’ one has (C.3) To prove (C.3) note that and, therefore, (&XA,., “‘),,~‘lAB = & (VgXA... “‘) qAB = a, @I*~~B~A... ‘.‘) - a!. (vAB) ‘BxA... (C.4) But arVAB = and the formula fi”’
fqAB,
(C.3) follows. Using (C.3) one obtains
B2...B,
=A;!,
[re2ar
(‘2vB’DBI...Bp)
-
(C.5)
Y-*A~_&~:...~~].
Now,
=r
-2p+21]B2c2.. . ~Bpcpa,.(r2p~c1”~c~Jl~c,)
=-_r
-2p+2qB2c2 . . . qB,c,
a,2 (r2pDrC2.-Cp) ,
K.6)
due to Lemma 4. Finally, fp
B2...By = =
-)7B2C2
-A;:,
The same arguments
. . .
~B~c,A~!~
[+a,2
[r-2P+1$
(r2Pp1Q~~‘cp)
+
rp2Ap_,
Qz{“;‘]
(rQiij...B,> + r-2Ap-~Qi,!;...~,,].
(C.7) [7
apply to fic2).
Acknowledgements This work was partially
supported
by the KBN Grant no 2 P03A 047 15.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
C. Teitelboim: Phys. Left. B 167 (1986), 63, 69. M. Henneaux and C. Teitelboim: Foundations of Physics 16 (1986), 593. J. Polchinski: TAN Lectures on D-branes, hep-th/9611050. D. Olive: Nucl. Phys. (Proc. Suppl.) B 58 (1997). 43. S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim: Phys. Letr. B 400 (1997), G. W. Gibbons and D. A. Rasheed: Nucl. Phys. B 454 (1995), 18. P. A. M. Dirac: Proc. Roy. Sot. A 133 (1931), 60; Phys. Rev. 74 (1948). 757. E. Cremer, B. Julia, H. Lil and C. N. Pope: Nucl. Phys. B 523 (1998), 73. M. S. Bremer, H. Lil, C. N. Pope and K. S. Stelle: Nucl. Phys. B 529 (1998), 259.
80.
138 [lo] [I l] [12] [13] [14] [15]
D. CHRU.!XIfiSKI S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim: Nuct. Phys. B 520 (19981, 179. S. Deser, M. Henneaux and A. Schwimmer: Phys. Left. B 428 (1998), 284. J. Jezierski and J. Kijowski: Gen. Rd. Gruv. 22 (1990), 1284. J. Kijowski and D. ChruScifiski: Gen. Rel. Gruv. 27 (1995), 267. D. ChruSci6ski: Rep. M&h. Phys. 41 (1998), 13. J. H. Schwarz and A. Sen: Nucl. Phys. B 411 (1994), 35.
REPORTS ON MATHEMATICAL
SYMPLECTIC
REDUCTION
No. 1
PHYSICS
OF p-FORM
ELECTRODYNAMICS
DARIUSZ CHRUSCI~~SKI Institute of Physics, Nicholas Copernicus University ul. Grudziadzka 5n, 87-100 Torud, Poland (e-mail: [email protected]) (Received
March
22,
1999
-
Revised
Seprember
14,
1999)
We propose a simple method to reduce a general p-form electrodynamics with respect to standard Gauss constraints. The canonical structure of the reduced theory displays a pdependent sign which makes the essential difference between theories with different parities of p. This feature was observed recently in the corresponding quantization condition for p-brane dyons. It suggests that these two structures are closely related.
1.
Introduction
The p-form electrodynamics is a simple generalization of ordinary electrodynamics in 4-dimensional Minkowski space-time M4 where the electromagnetic field potential l-form A, is replaced by a p-form in D-dimensional space-time [ 1, 21. The motivation to study such type of theories comes e.g. from string theory where one considers higher dimensional objects (the so called p-branes [3]) interacting with a gauge field. p-branes are natural objects which couple in a gauge invariant way to a p-form gauge potential and, therefore, they play a role of elementary extended sources. Recently the new input to study a p-form theory came from the electric-magnetic duality which started to play a prominent role in theoretical physics in the 90. (see e.g. [4]). Now, in order to have p-branes which carry both electric and magnetic charges (i.e. p-brane dyons) the dimension of space-time has to be equal to D = 2p + 2. It turns out [5, 61 that there is a crucial difference between p-form theories with different parity of p. Moreover, quantum mechanics implies the following quantization condition upon the electric and magnetic charges (ei , ~1) and (e2, g2) of any two dyons: e1g2
+ (-lFe2gl
= nh,
(1.1)
with an integer n. For odd p the above condition is a generalization of the famous Dirac condition [7], but for even p it was observed only recently [S-11]. Again, the parity of p plays a crucial role in (1 .l). The aim of the present paper is to show that this p-dependence is already present on the level of canonical structure of the underlying p-form theory. This feature 11211
122
D. CHRUklfiSKl
is however hidden in the standard formulation. ‘Therefore, we propose a different approach. Because the theory possesses a rich abelian gauge symmetry we propose to reduce it with respect to the corresponding Gauss constraints. Such a reduction was proposed in the case of ordinary electrodynamics (and linearized gravity) in [12] (it was applied to the variational formulation of electrodynamics of point-like charges in [13, 141). The method of this paper is a straightforward generalization of [12] to a p-form case. We show that the reduced canonical structure (see formula (4.28)) displays a p-dependent sign in a manner very similar to (1.1) which shows that the duality invariance, quantization condition and canonical structure are closely related. We postpone the study of this relation to the next paper. It would be also interesting to investigate relation between our approach and the standard two-potential formulation of a p-form theory (see e.g. [5, 151). The present paper is organized as follows: in Section 2 we introduce basic facts about p-form electrodynamics and present its (unreduced) canonical structure in Section 3. Then in Section 4 we describe the reduction procedure. As an example of a reduced p-form theory we present a Maxwell-type theory in Section 5. To make the presentation more transparent we discuss in the last section the p = 1 and p = 2 cases. They represent p odd and p even, respectively. All technical details and proofs are contained in appendixes. 2.
p-form
electrodynamics
Consider a p-form potential A defined in the D = (2p+2)-dimensional Minkowski space-time M2P+2 with the signature of the metric tensor (- , +, . . . , +). The corresponding field tensor is defined as a (p + I)-form by F = dA, F LLI...Pp+l
=
(2.1)
%4AW..Wp+ll
where the antisymmetrization is taken with a weight one, i.e. X[kl] := Xkl - Xlk. Having a Lagrangian L of the theory one defines another (p + 1)-form G as follows G@~-~LP+~
=
_(p
+
I)!
aF
a’
.
(2.2)
~l...llPfl
Now one may define the electric obvious way:
and magnetic
intensities
and inductions
in the
(2.3) 1 Bi,...ip
F.h-jpil
=
3
Dil...i, = Gi,...ipot
(2.5)
1 fi,...ip
= cp + 1j!
(2.4)
Eil...ipj14p+l
($-&+I
’
(2.6)
SYMPLECTIC
REDUCTION
OF p-FORM
ELECTRODYNAMICS
123
where the indices il, i2, . . . , j,, j2, . . run from 1 up to 2p + 1 and ti,i2.,_iZp+, is the Levi-Civita tensor in (2p + I)-dimensional Euclidean space, i.e. a space-like hyperplane C in the Minkowski space-time. The field equations are given by the Bianchi identities dF = 0, or in components (2.‘7) and the true dynamical
d + G = 0, or equivalently
equations
(2.8)
31~* G PI.-.Pp+ll - 0, in M2pf2
where the Hodge star operation .&WI-+,+1
1
=
is defined by ~l...ILp+l"I..."/~+l
(p+l)!
G
rl
(2.9)
w ...Up+I
and ~l-Ll~2+*~~+*is the covariantly constant volume form in the Minkowski spacetime Note that ~'l...i?p+l *.In terms of the electric and magnetic fields rloil-.,i2p+l. defined in (2.3)-(2.6) the field equations (2.7)-(2.8) have the following form: Eil...i&l...jJ~
vi, Bil...iP =
v~E~,,,,~
(2.10)
I’’
(2.11)
0,
(2.12) Dil...i,
v.
=
0
(2.13)
iI
where Vk denotes the covariant derivative on C compatible with the metric r/k/ induced from M2P+2. The Levi-Civita tensor density satisfies t12...2p+l = &, with g = det(rlkl). The field equations (2.10)-(2.13) may be rewritten using the language of differentials forms. Obviously D, B, E and H are p-forms on C. Let us define (p + I)forms: D := *=D,
t? := *=B,
(2.14)
where “*x” denotes the Hodge star on C: for any k-form, 1 (*CX)il...in_k
‘=
with IZ= 2p + 1. With this notation 8 = (-l)p ij=dH,
(n
_
@!
Eil...in_kjl...jk
xji
.-jk
9
(2.15)
one has: dE,
dt3 = 0,
(2.16)
d;l,=O.
(2.17)
124 3.
D. CHRU&X&KI
Canonical
structure
The phase space P of gauge invariant configurations (D, B), or equivalently (D, B), is endowed with a canonical structure. Using local coordinates (D’l+, Bjl...jp), the most general 2-form on P is given by J-+/d2P++2p+i
y [Xi,...i,j,...jP(x, y)6D’+p(x)
+ iYil,.,i,jl..,jp(X,
Y)6Di”.“p(X)
+ ~Zil,,,i,j,,,,j,(X,
Taking into account field XH~
A 6Dji”‘jp(Y)
Y)SB”‘...‘P(X) A SBj”“j’
the field equations
A GB”...‘p (y)
(Y)]a
(3.1)
(2.10) and (2.12), the hamiltonian
vector
XHpA 52, = -6H, is described
XH~ = Therefore,
(3.2)
by d2’+‘X
s
+ (_I)qkEjl-jP&
Vk Hjl “.jP &
Eil ,,,ipkj, ,,,j,
I,...lp>
I,...lp
. (3.3)
Eq. (3.2) is solved by HP and Qp defined as HP =
-!2p!
s
d2P+1x(~i’...i,Ei,~,,ip+ B'l-'PH. Il...lp) 7
(3.4)
and .n, = /d2P+1x/where G(x, y) denotes clidean space, i.e.
d2p+ly
~Dil...ip cx) A
the Coulomb
Green
SB.h...h
(Y)9,...ipkj,...jpVk
function
G(%
Y), (3.5)
in (2p + 1)-dimensional
A G(x, y) = -d(2p+1)(~ - y).
Eu-
(3.6)
The formula (3.5) uses only gauge-invariant quantities. It may be, however, considerably simplified by introducing a gauge potential. One solves (2.11) via @I-‘P
=
eil...ipjl...jp+l
v.
A, JI
J2...jpfl
9
(3.7)
which is the obvious generalization of B = V x A from the p = 1 electrodynamics. Now, modulo gauge-dependent boundary term, the symplectic form reads d2P+lx
Qp = (-1)P s
,j@l&
A
a~~,.,.~,
=
d2p+‘x
(_l)P+’
SGCL’+‘o A 6AW,+,,.
s
(3.8)
SYMPLECTIC
Having a symplectic brackets
REDUCTION
OF p-FORM
125
ELECTRODYNAMICS
structure in P one may easily derive the corresponding
Poisson
where F and 6 are functionals on P. Note that Dil...ip and Bil..,ip are not arbitrary but they have to satisfy the Gauss laws (2.11) and (2.13). The fundamental commutation relations implied by (3.9) read (@-ip(x),
Bjl-~jp(Y))p
= Eii...ipkjl...j~vks(2p+1)(x
(o”.%(x),
~‘l-jP(y))p
=
{B’~-~‘P(~),
Bjl...&?(y)],
_ =
y),
0,
(3.10) (3.11)
with x, y E C. 4.
Reduction
The idea of this section is to find a new set of electromagnetic variables using the philosophy of “2p + 1” decomposition (it is a straightforward generalization of [12]). These new variables solve partially the Gauss constraints (2.11) and (2.13). Moreover, they are canonical with respect to 52,. 4.1.
“2p + 1” decomposition
The “2p+ 1” decomposition is based on the observation sional Euclidean space C may be decomposed as
that the (2p+ l)-dimen-
IZ = S2P(l) x R, where S2P(l) denotes 2p-dimensional unit sphere direction. Let us introduce the spherical coordinates
(4.1) and IR represents on C:
A = 1,2, . . . ,2p,
the “radial”
(4.2) (4.3)
(~2~ denote spherical angles (to enumerate the angles we shall use capital letters A, B, C, . . .) (for more details see Appendix A). Let qkl denote the Minkowskian metric on C and let h, denote the corresponding volume element, denote the Levi-Civita tensor on i.e., h, = 4s (see (A.3)). Finally, let EA,...~~~ S2p(r) such that EA,...A~~:= E~A,...A~~ (we use the same letter to denote the Levi-
where ~Q,vPZ,...,
Civita tensors on C and S2P(r)). We shall denote by “I” the covariant derivative defined by qkl on C and by “I]” the covariant derivative on each S2p(r) defined by the induced metric I]AB.
126 4.2.
D.CHRU$CIfiSKI Symplectic reduction
Let us consider the evolution of the p-form electromagnetic field in the finite volume V E C. For simplicity we take V = K 2P+1(0, R) (a (2p + 1)-dimensional ball of radius R). Of course one may take any V topologically equivalent to K*P+‘(O, R), but then instead of the spherical coordinates one chooses coordinates adapted to V. To reduce np in V we use the “2p+l” decomposition (-l)‘+‘Qp
hp GGi’...i~o A GAi,,,,ip = -
= sV =--
s vhP
Using (3.7) one obtains
hp ~D”~..‘P A GAi,...i, s V
(PAD ri2...i’ A 6Ari2.,,ip + GDA1...ApA SAA,,,,A~) .
BAI...A, = @..&‘BI.JP
arAB,...B + p EA~~~~*,B~rBz~~JG ArB2,,,B ,,B P
P
I’
(4.4)
(4.5)
and hence ~A,.,_A~B,...B~B~‘~~~~~“~’ = p! a;‘,‘,::21 (-1)’ where we have used the following Eil...iNjl...j~E,
[&&,...c,
- P A~c~...c,IIc~]“~’ 7 (4.6)
identity =
rl...i~ll...l,+f
No
.
gjl...jM
._-I
[lI...l.&f] *-
.
gjl
[I, . . . al:]
(4.7)
with arbitrary N and M. Now, to simplify our consideration, let us choose the following gauge conditions for a p-form A A~...A~ and (p - 1)-form A rA2...A~ on each sphere S2p(r): AAI...A, ~rA2u.A~ [IAl = 0, llA2 = 0. (4.8) In such a gauge EA1...A,BI...B ’
=
BA,...A,IIB,
(_l)P+lp
p!
r-2
Ap_,
A’B2..,B~,
(4.9)
where Ap_l is defined by Ap_, := (p - l)! [r2VAVA - (p2 - l)]. This operator
has a clear geometrical
interpretation.
LEMMA 1. r-2Ap_1 is equal to the Luplace-Beltrami X = (XA2...Ap) on S2p(r) satisfying XA2...Ap~~~2= 0, i.e.
operator for (p - 1)-forms
r-*A p_1x = (d*d + dd*)X = d*dX. For proof see Appendix
(4.10)
(4.11)
B. The factor r 2 in (4.10) makes Ap_i r-independent,
i.e. [A,-,, LEMMA
2. The operator
%I = 0.
Ap_l is invertible.
(4.12)
SYMPLECTIC
REDUCTION
OF p-FORM
ELECTRODYNAMICS
For the proof see Appendix B. Denoting by A;:, obtain
Introducing
the inverse of A,_,
127
we finally
the variables: AI-A,
._
Q (1)
“‘1’ B2...Bp
*-
rD
**-
+
rA2...A,
Ai!,
(4.14)
, (tA1...~,~I...~,,
(4.15)
BA”““““B’)
we may rewrite the first integral in (4.4) as -
Jv
A., PJD ri2-.i~ ,,y&&+
=
1)P”
(-
L
s
hpUI;;~,~Ap A t3Q;i"'"".
(4.16)
V
To reduce the second term in (4.4) observe that in the special gauge (4.8), AA,, ,A,, may be expressed as AA~
=
...A.
EA ,... A,BI...B,,
p
BI..-B~,IJBI
(4.17)
Now, from (3.7) B rBz...B, Inserting obtain
(4.17)
=
cBz...BpC,...C P+’
&~...c,,+,
(4.18)
11~1 .
into (4.18) and using once more the gauge conditions B rBz...f$ = _P! r-2
Ap_l pBz...Bp,
(4.8) we (4.19)
and, therefore pB:...B”
=
__
r2
A;--,
Brh...B,,
(4.20)
P!
which, together
with (4.17), gives r2 AA ,,.. A,
=
--
(4.21)
CA ,... A,,B ,... B,
P!
Having
AA]
we may calculate
. ..A.,
-
s V
AP c~D~I-.-~PA SA
the second term in (4.4),
Al . ..A.
GA ,... APB ,... B,
SDAL-.-A~)~~~~
,,
y A;!,
&BrB2...BI,.
(4.22)
128
D. CHRU~CIfiSKl
Defining 4.-A,
._
Qc2)
.--rB
rA2...A,
1
(4.23)
(2)
I-I
(4.24)
B2...B,
we may rewrite
(4.22) as
-s s( h,
cYD~"'.~J' A SAAI.,.Ap
V
h,
s A,
Sl-l~;,,.Ap A 6 Q;",;--'"'.(4.25)
V
Now, taking into account
“F” =
=
(4.16) and (4.25) we have finally
Sl-l~;~~~Ap A SQ;ll;"'Ap
+
(-l)p+1SIl(,21,~~Ap A aQ;42;Ap)
(4.26)
V
Let us use a more compact
notation X .Y
Our result may be formulated
:=XA2...AP YA~.,.A~.
as
THEOREM 1. The reduced symplectic structure the reduced phase space pr” := (Qca,, l7@)),
a;* =
s A,
(4.27)
J2:” has the following form
(sn”’ . AJQ(I) + C-1)P+18ZZ(2). /GQC2,)
.
on
(4.28)
V
The above theorem THEOREM {F,
obviously
implies the following
2. The corresponding
E);* =
result.
reduced Poisson bracket (3.9) reads
s, [g) *--& +(-l,p+l-&*& - v = a].
This formula
A,
leads to the following
commutation
relations:
g(2~+‘)(~ _ y> (Q;",;-Ap(x), n;;,,,Bp(y)}red = 8A2-.Ap [Bz...&J {Q;',;'-'ApW,
and the remaining 4.3.
(4.29)
(4.30)
@;,,,Bp(y));* =(-1)p+16$;:-~ls(2p+1)(XP
y),
(4.3 1)
brackets vanish.
Reconstruction
Could we reconstruct D and B in terms of Q’s and II’s? The answer is positive by virtue of the following reconstruction.
129
SYMPLECTICREDUCTIONOF p-FORMELECTRODYNAMICS THEOREM 3. The variables (Qcml, II iu)., a = 1. 2) contain invariant information about the jields Dil...ip and BiI...‘p.
the entire gauge-
Proof: We show that Q(1) and n(2) encode the information about the D field. Due to (4.14) it is enough to show how to reconstruct the tangential part of 1). i.e. DAl...AIJ. Let us make the decomposition (see Appendix A): DAI...A,
=
U[A2...ApllA~]
LEMMA 3. The following
pVBZ...B,‘/15,~
(4.32)
r2DA,...~$j ,,A, = np_, UA”“Ap,
(4.33)
identities
For the proof see Appendix
hold:
~A1....4,llh
~2~AI...A,BI...B,
= a
(4.34)
VB2...B,.
P-l
B. Therefore,
VBp.B,,
Now, to reconstruct
+ cA~...A,,B~...B
=
-P!
r l-$..B,,
(4.35)
.
14A2...Aplet us use (2.13).
LEMMA 4. The Gauss constraints
DkA2...A~pkare equivalent
to
ar(&’ DrAz...Apj + r~~~-b&...~~,~,,A, = 0. The proof is given in Appendix
B. Finally, due to (4.36)
UA2...A, = _,2~,-’ p_,
a,. r2p-1
In the same way one shows that Q(z) and n(i) 4.4.
Counting
(4.36)
Q$i”.Ap)
reconstruct
(4.37) 3
the B field.
degrees of freedom
Let us count the total number Define a number nf by
N, of degrees
of freedom
for a p-form
nf :=
A p-form potential a gauge freedom
(4.38)
Ai,,,,ip on C has n,,’ independent
components.
However, due to
Ai,...i, + Ail...i,, + v,,,A~~,‘,.;,,I some of them may be gauged away e.g. to zero. A:,t)j,,._, carries n;_, but the equation (4.39) is still invariant under
A?’
I, . ..I.,-,
+ 4
theory.
(1) 1,.
lp__,
+
v,,,A!2’ Q...lr,-11‘
(4.39) components
(4.40)
130
D. CHRIkCItiSKl
Therefore, some of the components of A(‘) may be gauged away via AC2) which has r~,“_~components. The same argument applies for AC2) and one continues up to a O-form AP carrying obviously only one component. Therefore, the total number of degrees of freedom carried by Ai,,,.i, equals Np = npP- n,“_i + n;_2 - . . . + (-l)Pni. Using the standard mathematical
induction
(4.41)
it is easy to prove that
Np =
(4.42)
Now, note that Q(i) and Q(2) carry together 2P = -Np. P+l
(4.43)
Therefore, (4.44) Mp > Np, holds only for p = 1. It means that for p > 1, Q(a) are not
and the equality independent. LEMMA
5. For p > 1, the reduced
variables
Q(U) and II@) satisfy the con-
straints: VA2Q$Ap
= 0,
v%-I~;,,,Ap = 0.
(4.45)
The proof follows obviously from (2.11) and (2.13). Note that these constraints are trivially satisfied for p = 1. 5.
Maxwell
theory
The simplest p-form theory is of the Maxwell type, i.e. the constitutive between inductions (D, B) and intensities (E, H) are linear: Dil...i,
= ~‘1 ...i.
7
Bil...iP
= Hil...iP
In such a case it is easy to rewrite the field equations of Q’s and n’s. THEOREM
4. Equations A2
.A,
_
(5.1)
(2.10) and (2.12) in terms
(2.10) and (2.12) are equivalent
Q,,;'- -$I
relations
to:
A2...A,
n,,, )
(5.2) (5.3)
l-y’) A2._A,
fi (2) A2...A,
=
-A;:,
=
(-I)~A;!,
r-‘$(rQ~~,,,,,)
E
+ re2A,-,
[r-la~(rQ~~...A,)
-I Q~:...A PJ ’
+ rm2A,-l
QZ~...AJ l
(5.4) (5.5)
SYMPLECTIC
where
REDUCTION B2...Bp
QFJ..., := vA2B2 P
. . . qA,B,
Q(,)
For the proof see Appendix lowing Hamiltonian, H,, =
1
OF p-FORM
and
131
ELECTRODYNAMICS
17$j”Ap := qAzB2 . . . qApBp17f2!),B i’
C. These equations
may be derived
2(p - l)! s
kp rp2Qc,) . Q’“’ - r-2pal(r2P-1Q~a,)A~!~ [
- &&,_I
. @‘]
from the fol-
. &.(rQ@))
,
(5.6)
Q(a) = IQ(a), Hplrd.
(5.7)
n,,,
(5.8)
i.e.
= I&,,
HJfe”,
with the Poisson bracket defined in (4.29). One easily shows that the numerical value of HP is equal to the standard (i.e. obtained via the symmetric energy-momentum tensor) electromagnetic energy contained in V, i.e. HP
I
1
2P! s v
6.
p = 1 versus
A, (Di’-~ip Di ,,., i,, + Bi’...‘” Bi ,___ ;J
(5.9)
p = 2
The case p = 1 corresponds The symplectic form
to the ordinary
52, = (i = 1,2, 3) reduces
s
d3x8D’
electrodynamics
A 6Ai
in 4 dimensions.
(6.1)
to
.nl”” = where
.
s
h3(61-l(‘) A 6Qcl,
Q’s and l-I’s are functions
6l-l (2) A ~Q(2))t
+
(6.2)
(O-forms) on S2(r):
Q(l) = rD’, Q(2) = rB’, llcl) =rAO’(cABB
AIIB),
nc2) = -r&‘(EABD
and A, denotes the ordinary
Laplacian
on S2( l), i.e.,
A, := r2 VAVA Due to (4.41), Nt = 2 as it should.
“IIB),
(6.3)
132
D. CHRU.%2IdSKl
Now, p = 2 corresponds symplectic form
to 2-form electrodynamics
JZ2 = + (i, j = 1,2, . . . ,5) reduces
in D = 6. The corresponding
d5XJ D’j A 6Aij
s
(6.4)
to
fin;ed =
A5(81-l~’ A SQiq, - CUDS’A SQ;,),
(6.5)
s where
Q’s and D’s are now l-forms
on S4(r):
Q$,=rD
i-A
,
Q;,=rB
rA
,
l-I!) = r A;'(rABCDBAB”C), lib” = -rA,’ and A, denotes the Laplace-Beltrami l-forms XA satisfying XAllA = o),
(eABCDDAB’IC), operator
on S4(1) on co-closed
A, := r2 VAVA - 3. Due to (4.41), A72= 6. Now, note the differences
l-forms
(i.e.
(6.6)
in signs:
1. the difference between (6.1) and (6.4) is not essential because symplectic form may be multiplied by any (nonzero) number, 2. however, the difference between (6.2) and (6.5) is crucial. It implies that the p = 1 theory is duality invariant, whereas p = 2 is not [5]. Appendixes A.
Geometry
of S2p(r)
The relation between the Cartesian (y’, y2, . . . , y2J’+‘) and spherical (x’, x2, . . . , x2P+l) introduced in (4.2)-(4.3) reads as follows: y1 = r sin qzP sin q2P_i _. . sin
93
sin qp2sin (al,
y2 = r sin y3zPsin 4D2P_1. . . sin
(~3
sin 402cos
y3 = r sin ~2~ sin ‘p2P_1. . . sin q3 cos q2,
Y 2p = r sin ~2~ cos q7zP-1, Y 2p+1 = r cos ~2~.
rpl ,
coordinates
SYMPLECTIC
The Minkowskian
REDUCTION
OF p-FORM
133
ELECTRODYNAMICS
metric on C is diagonal: 2P qkk
=r2
sin2 q7l,
n
k=l,2
,...,
2P-1,
(A.1)
I=k+l ~2~.2~
Therefore,
=
(A-2)
rl r,=l.
r2,
the volume element hp =
,iG
= r2’ fi
sinA-’ PA.
(A.3)
A=1
One easily calculates symbols are:
the corresponding r,A,,
r;,
Christoffel
= r-‘SA,,
Now, one easily finds the Riemann RABCD
=
symbols. The only nonvanishing
r;s
= +-‘~A&
tensor on S2p(r), rp2(vACvDB
-
(A.51
~AD~CB)~
where qAB denotes the induced metric on each S2p(r). tensor RAB and the scalar curvature R read 2p - 1
RAB = ~
Finally, let
EAl . ..A*.,
(Ad)
The corresponding
Ricci
R = 2P(2P - 1) VABt
r2
r2
tensor density on S2p(r)
denote the Levi-Civita t12...2p
=
such that
hp.
field on S2p(r).
Let X = XA,.,,A, denote any p-form
(A.61
’
This field may be decomposed
X3 XA,...A,
Formula
=
V[A,~A,,,.A,,]
(A.7)
+ CA ,.., A,,B ,... Bp~B’~B2’~‘B”.
(A.7) follows from the Hodge theorem X = clcx + d*j3’ + h,
(A.8)
with cx being a (p - I)-form, /?’ a (p + 1)-form and h a harmonic p-form. Note that there is no harmonic term in (A.7) because the set of harmonic p-forms on S2p(r) is empty. Obviously, B’ = h * B for some constant h. To find h note that for even-dimensional Riemannian manifolds the adjoint d* of d reads as d*=--*d*
and, therefore,
(A.9)
d*b’ = -h * d * *,!I. Now, on a Riemannian
n-dimensional
* *wk = (-l)k’“-k’Wk, and hence for n = 2p, * * p = (- l)P+’ p. Therefore, implies A = (-l)p.
manifold (A.lO)
d*B’ = (-l)Ph
* clp which
134 B. B.1.
D. CHRU&.dSKI
Proofs Proof of Lemma
1
Consider any (p - 1)-form X = (X,,...A~) on S2P(r) such that VA2X~2...~p = 0. This condition may be rewritten as d’ X = 0. Therefore,
the Laplacian
on X reduces
03.1)
to (W
(d*d + dd*)X = d*d X. Due to (A.9), d*d X may be calculated 1 (*d * dX)&...A, =
(p + I)! 1
= (p + l)! p!(p x
+
t~2.,,~n~~,...~~V’A~~B’~~~Bp1c’~~~c~V~~~X~~...~~~
(p
A
l)!
I... BP (”
cA2...A,AB
BI...B,C~...CP
)
xc2,,,c
(~~~-.B,~I-~P~-+
EA2...ApAB,,,,Bp
_
vBI~AB~...B,CI...C,
x
t
vBP~BI...BP-I~CI...~PvIc,
-...-
1 =
~Az...A,A~,...~,V’A(*dX)E”--B,l
P!
=
as follows
I
P
_
p
+-JPG--C~VBI)
x
+ I)!
x V[C,xcz_.cpl
= -6
= To calculate
r”A:;:::z] VA% xc2...c,1
(p - l)!
(-vATA x A~...A,
+ (-1)PVAV,~2X~,...~,~~)
U3.3)
.
the second term note that = vA
VAV,A$A+Ap]A
-
Now, we use the following [VA,
VBlT
CI...Ck
(vA2XA3...A,A . . . -
well-known DAB
which holds for any tensor field Tcl-ck. VAVkX4...A,A
vA3XA2A4...ApA
VA,XA2.,.Ap-,A~A
property
TD&..Ck
= RC’
-
of the Riemann
+ . . . +
tensor
RCkDABTC2-Ck-lD,
03.5)
Using (B.5) one has RA3
= @B[VA, VB]X&...A,A = @B (
+ .. . +
(B.4)
) .
RAP CAB~A3...AP-~CA
+ RA
XCA4...ApA CAB CABX‘43-‘4PC
. >
03.6)
SYMPLECTIC
Using (AS)
REDUCTION OF p-FORM
135
ELECTRODYNAMICS
one obtains
,,AzBR~~ CABXCA4...A,A=r-2QW$W’(TIDAllcB
_ ljDBIJCA)XCA~...A,~A
=r -2XA2A4....$& = y-2(_l)~+lXA~....4,~, Exactly
(B.‘7)
the same result holds for the first (p - 2) terms in (B.6). The last term CABXA3...A,~C = rJAzB~CBXA3...A,C= +(_1)~(2~
+BRA
_ ~)XAC% .
(B.8)
due to (A.6). Finally, (- * d * dX)A2...A,
= =r
B.2.
(p
-
I)!
-2Ll
P-l
-
r-‘(p2
-
I)]
XA2...A,,
n
XAp..A,
(B.9)
Proof of Lemma 2
It is a well-known fact from the of Ak is nontrivial only for k = 0 electrodynamics in M4) is special which are simply constant functions
B.3.
[v”vA
theory of cohomology that on Y(r) the kernel and k = n. The p = 1 case (i.e. the ordinary because A, = r2VAVA possesses zero modes on S2(r). This problem was treated in [ 121.
Proof of Lemma 3
Let X = (XA,...A~) be a p-form may be decomposed as XA,...A,
=
on
V[A,~A2...Ap]
i.e. cz and p are (p - I)-forms
Due to the Hodge theorem,
S2p(r).
+
(A.8) it (B. 10)
CA ,... A,,B ,... ~?,,~~‘fi~“‘.~“~
on S’P(r), (B.l Y)
x =&Y+**dp. Let us choose the following
“gauge condition”
for 4 and B,
d*a = d*,3 = 0.
Now, taking into account
(B.11) and Lemma
(B.12)
1 one obtains
d*X = d*da = r-2A,_la
(B. 13)
and * dX = *d *d/Y = -d*d/3’ = -A,_,/% Rewriting
(B.13) and (B.14) in components
(B. 14)
one obtains
r2XA’A2”‘APIIA,= A,_,aA>...A,, ,
(B.15)
and
A,-,B
Bz...Bp
=
=E
_r2(.+dx)B2-.&
=
-1
(P + I>! AI-+'PB~-.~PV~, x~,,,,~,.
cB2...B,,BlA~...A “v[B,
XA,...Apl
0
(B.16)
136 B.4.
D. CHRUhXk3KI
Proof of Lemma gki2_..iP
4 ,k =
[email protected]
+
where the last term obviously DkA2...A,
,k =
aryAz...Ap +
+
1-E~~ji2-i~
+
cp
_
1)
+zgkii3--.ip,
(B.17)
vanishes. Therefore,
aBDBAz...Ap
r;rDrAz---Ap
+
r;BDBiz..-i,
+
riryA2...Ap
+
(B.18)
r;c~cA2..+
Now, taking into account (A.4), DkAp..A,
,k =
;
a,y-Aa...Ap
2P
DrAz...Ap
r = Y-2~ 8, ($J ~rh..A,
C.
Derivation
of Maxwell’s
+ (~B@&...A~
+ riCDCA2...Ap)
> ) + ~4 &...A, ,,A, .
0 (B.19)
equations
Using (2.12) one obtains . A2._.A, Q,,,
_
-rD
.
rA2...A,
=
’
_
rAZ...ApAI
-
Bl...B ‘BB1...BpllA, A2...A,
=
.
-A;!,$,
Cl)
P!
In the same way one proves (5.3) using (2.10). Now, fiT(‘) B2...B,
=
‘A;:, P!
= LA-’ P!
=
EA I... ,z,p B I...
P-1
B
p
vB1
(-I)+
p-1
VBI
JI...i,
)
VBi>DrB2...Bp_lB1
[arVB’DB1...Bp
+ p(p
+
-
I>!
(VBIDrBz...B,,
)I
&-D,,...,p
=rAi!l [
c”“cpVB1(t-l)pp!~rD~~...~~ [BL...BPI
[P!&-DB,...B,
P-1
=rAi!,VB1
k
Q.cz.,.cp
S
iA-1 -
D.
v
(~A1~~~AprC’~~~cf~~Dc,...c,
PC A~...A~c~rc2...c~Vcl
(-l)J’rA-’
p!
AI...A,kjl...j,
>
(p>2
=
I... BpBA’~‘.Ap”B’)
(-~)“~A~TI,FA,...A,B,...B~V~’
+ =
(tA ,.,. ApB
- (-1)PVi~2Dr -
rT2Ap-lDrBz...Bp]
(-1)y’1~~!Vc~D~~2...~p)
-
VB2DrBlB3...Bp
1
B3...Bp]B~
-
(C.2)
SYMPLECTIC
REDUCTION
OF p-FORM
13’7
ELECTRODYNAMICS
For any tensor field X,,,, “’ one has (C.3) To prove (C.3) note that and, therefore, (&XA,., “‘),,~‘lAB = & (VgXA... “‘) qAB = a, @I*~~B~A... ‘.‘) - a!. (vAB) ‘BxA... (C.4) But arVAB = and the formula fi”’
fqAB,
(C.3) follows. Using (C.3) one obtains
B2...B,
=A;!,
[re2ar
(‘2vB’DBI...Bp)
-
(C.5)
Y-*A~_&~:...~~].
Now,
=r
-2p+21]B2c2.. . ~Bpcpa,.(r2p~c1”~c~Jl~c,)
=-_r
-2p+2qB2c2 . . . qB,c,
a,2 (r2pDrC2.-Cp) ,
K.6)
due to Lemma 4. Finally, fp
B2...By = =
-)7B2C2
-A;:,
The same arguments
. . .
~B~c,A~!~
[+a,2
[r-2P+1$
(r2Pp1Q~~‘cp)
+
rp2Ap_,
Qz{“;‘]
(rQiij...B,> + r-2Ap-~Qi,!;...~,,].
(C.7) [7
apply to fic2).
Acknowledgements This work was partially
supported
by the KBN Grant no 2 P03A 047 15.
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