ЯДЕРНАЯ ФИЗИКА, 2012, том 75, № 10, с. 1370-1375
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
FIRST INTEGRALS OF MOTION IN A GAUGE COVARIANT FRAMEWORK, KILLING-MAXWELL SYSTEM AND QUANTUM ANOMALIES
© 2012 M. Visinescu*
Department of Theoretical Physics, National Institute for Physics and Nuclear Engineering, Magurele, Bucharest, Romania Received May 5, 2011
Hidden symmetries in a covariant Hamiltonian framework are investigated. The special role of the Stackel—Killing and Killing—Yano tensors is pointed out. The covariant phase—space is extended to include external gauge fields and scalar potentials. We investigate the possibility for a higher-order symmetry to survive when the electromagnetic interactions are taken into account. A concrete realization of this possibility is given by the Killing—Maxwell system. The classical conserved quantities do not generally transfer to the quantized systems producing quantum gravitational anomalies. As a rule the conformal extension of the Killing vectors and tensors does not produce symmetry operators for the Klein—Gordon operator.
1. INTRODUCTION
The evolution of a dynamical system is described in the phase—space and from this point of view it is natural to go in search of conserved quantities corresponding to symmetries of the complete phase—space, not just the configuration one. Such symmetries are associated with higher-rank symmetric Stackel—Killing (SK) tensors which generalize the Killing vectors. These higher-order symmetries are known as hidden symmetries and the corresponding conserved quantities are quadratic, or, more general, polynomial in momenta. Also Killing tensors play a very important role in the Hamilton—Jacobi theory of separation of variables and the integrability of finite-dimensional Hamiltonian systems [1]. Another natural generalization of the Killing vectors is represented by the antisymmetric Killing—Yano (KY) tensors which in many aspects are more fundamental than the SK tensors. The existence of higher-rank KY tensors indicates the presence of dynamical symmetries which are not isometries.
The conformal extension of the SKtenzor equation determines the conformal Stackel—Killing (CSK) tensors which define first integrals of motion of the null geodesics. Investigations of the hidden symmetries of the higher-dimensional space—times have pointed out the role of the conformal Killing—Yano (CKY) tensors to generate background
E-mail: mvisin@theory.nipne.ro
metrics with black hole solutions (see, e.g., [2] for a brief review).
In the study of the dynamics of particles in external gauge fields it has been proved that a gauge-covariant Hamiltonian approach of the symmetries [3] is more convenient and productive. The covariant framework permits us to find conditions of electromagnetic field to maintain the symmetries of the system. A concrete realization of these constraints is provided by the Killing-Maxwell (KM) system [4].
Passing from the classical motions to the hidden symmetries of a quantized system it is necessary to investigate the corresponding quantum conserved quantities and separability of the equations of motion. Especially in the case of hidden symmetries there can appear anomalies representing discrepancies between the conservation laws at the classical level and the corresponding ones at the quantum level.
The plan of the paper is as follows. In Section 2 we establish the generalized Killing equations in a covariant framework including external gauge fields and scalar potentials. In Section 3 we discuss the special role of the KY tensors and in the next section we describe the KM system. In Section 5 the relationship between conformal symmetries in a curved space background and the corresponding quantum operators is presented in connection with quantum gravitational anomalies. Finally, the last section is devoted to conclusions.
2. SYMMETRIES AND CONSERVED QUANTITIES
Let (M, g) be a n-dimensional manifold equipped with a (pseudo-)Riemmanian metric g and denote by
H = \gijPiPj, (1)
the Hamilton function describing the geodesic motions in M. In terms of the phase—space variables (xi,pi) the canonical symplectic structure u of T*M is u = dpi A cX and the Poisson bracket of two observables P, Q of T*M is
{P,Q} =
dP dQ
dxi dpi
dP dQ
dpi dxi '
(2)
Let us consider a conserved quantity of motion expanded as a power series in momenta:
1
K = K0 + —Kil"'ik(x)pi1
k=l
•Pik
(3)
It has vanishing Poisson bracket with the Hamilto-nian, {K, H} = 0, which implies
K(il-ik ;i) = 0, (4)
where a semicolon denotes the covariant differentiation corresponding to the Levi—Civita connection V and round brackets indicate full symmetrization over the indices enclosed. A symmetric tensor K%1"'%k satisfying (4) is called an SK tensor of rank k. The SK tensors represent a generalization of the Killing vectors and are responsible for the hidden symmetries of the motions, connected with conserved quantities of the form (3) polynomials in momenta. Indeed, using Eq. (4), for any geodesic 7 with tangent vector xi = p1
(5)
■■ik-
ik
qk = Kii
is constant along 7.
In the presence of an electromagnetic field Fij, Souriau [5] has proposed to replace the canonical symplectic structure, u, by the twisted symplectic structure uF = dpi A dxi + 2 ]%j dx i A dxj and the Poisson bracket of two observables is now
' dx1 dpi dpi dx1
+ Fi
ij
dP ^ dQ dpi dpj '
is to replace the Hamiltonian by
H = ^(pt-At)(pJ-AJ) + V(x), (8)
work with the original Poisson bracket (2) and consider the polynomials (3) in the variables (pi — A) for i = l,...,n [6]. For completeness, in (8) we included a scalar potential V(x).
The disadvantage of this approach is that the canonical momenta pi and implicitly the Hamilton equations of motion are not manifestly gauge co-variant. This inconvenience can be removed using van Holten's receipt [3] by introducing the gauge-invariant momenta:
n = Pi — Ai. (9)
The Hamiltonian (8) becomes
h = ^mlu3 + v(x), (io)
and the equations of motion are derived using the Poisson bracket
{P,Q} =
dP dQ
dP dQ dxi dni
dP dQ
(11)
dni dx
- + qFt
ij
dni dn
Now the fundamental Poisson brackets are
{xi,xj} = 0, {xi, n} = Sj, (12)
{ni, nj} = Fij,
showing that the momenta ni are not canonical.
Searching for conserved quantities (3) expanded rather into powers of the gauge-invariant momenta ni we get the following series of constraints
KiVi = 0, K, + FjiKj = Kij Vj
K(ii-ii;ii+i) + F (il+1 Kii"4l)j =
(13a) (13b) (13c)
(l + 1)
K h-ñ+d V j,
(6)
for l = 1,...,(p - 2),
K(ii-ip-i;ip) + f. (ipKii"'ip-i)j = 0, (13d)
K (ii---ip;ip+i) = 0.
(13e)
The traditional means to deal with the coupling to a gauge field Fij expressed (locally) in terms of the potential l-form Ai
F = dA, (7)
We remark that the last equation is satisfied by a SK tensor (4), while the rest of the equations mixes up the terms of K with the gauge field strength Fij and derivatives of the potential V(x). Also it is worth mentioning that Eq. (13) separate into two groups:
p
1
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one involves the terms of K of odd degree in the momenta n and the other involves only terms of K of even degree in the momenta [7].
Several applications using van Holten's covariant framework [3] are given in [8—11].
3. KILLING-YANO TENSORS
KY tensors are a different generalization of Killing vectors which can be defined on a manifold. They were introduced by Yano [12] from a purely mathematical perspective and later on it turned out they have many interesting properties relevant to physics [13]. Here, we shall point out the role of KY tensors in construction of conserved quantities paying a special attention to the KM system introduced by Carter [4].
A KY tensor is a p-form Y(p < n) which satisfies
VxY = (14)
p + 1
for any vector field X, where "hook" operator j is dual to the wedge product. This definition is equivalent with the property that Vj Yil...ip is totally antisymmetric or, in components,
(ip ;j) =0- (15)
1p-1 yip
The first connection with the symmetry properties of the geodesic motion is the observation that along every geodesic 7 in M, Yil...ip-1jxj is parallel.
These two generalizations SK and KY of the Killing vectors could be related. Let Yil...ip be a KY tensor, then the symmetric tensor field
K — Y Y
Kij ~ • •ip Yj
12-• • ip
(16)
Vj Y —
1
p + 1
-X_idY -
1
n — p + 1
Xb A d*Y, (17)
where Xb denotes the 1-form dual with respect to the metric to the vector field X and d* is the exterior co-derivative. Let us recall that the Hodge dual maps the space of p-forms into the space of (n — p)-forms. The square of * on a p-form Y is either +1 or —1 depending on n,p and the signature of the metric [13, 23]
**Y — epY,
*-1Y — er.
* Y,
with the number
ep — (—1)p *
! detg
|detg|
(18)
(19)
With this convention, the exterior co-derivative can be written in terms of d and the Hodge star:
(20)
d*Y — (—1)p *-1 d * Y.
is a SK tensor and it sometimes refers to this SK tensor as the associated tensor with the KY tensor Yil..,ip. That is the case of the Kerr metric [14, 15] or the Euclidean Taub-NUT space [16, 17]. However, the converse statement is not true in general: not all SK tensors of rank 2 are associated with a KY tensor. A counterexample is the generalized Taub-NUT space [18] which admits SK tensors but no KY tensors [19].
Having in mind the special role of null geodesic for the motion of massless particles, it is convenient to look for conformal generalization of KY tensor. Let us mention also that recently a lot of interest focuses on higher-dimensional black holes. It was demonstrated the remarkable role of the CKY tensors in the study of the properties of such black holes (see, e.g., [20-22] and the cites contained therein).
A CKY tensor of rank p is a p-form which satisfies
These operators can be expressed in terms of the connection V. If {X'} is any basis for vector fields, with {e'} the natural du
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