In this pedagogical note, we discuss obstacles to the usual Palatini
formulations of gauge and gravity theories in presence of odd-derivative order,
Chern-Simons, terms.Comment: 4 pages. Dedicated to Rafael Sorkin on his 60th Birthda

Deser S

S Deser

English

arXiv

In this pedagogical note, we discuss obstacles to the usual Palatini formulations of gauge and gravity theories in the presence of odd-derivative order, Chern–Simons terms

Deser, S.

Caltech Authors - Main

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY
Class. Quantum Grav. 23 (2006) 5773–5776 doi:10.1088/0264-9381/23/18/N01
COMMENTS, REPLIES AND NOTES
A note on first-order formalism and odd-derivative
actions
S Deser
Department of Physics, Brandeis University, Waltham, MA 02454, USA
and
Lauritsen Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
Received 5 June 2006
Published 22 August 2006
Online at stacks.iop.org/CQG/23/5773
Abstract
In this pedagogical note, we discuss obstacles to the usual Palatini formulations
of gauge and gravity theories in the presence of odd-derivative order, Chern–
Simons terms.
PACS numbers: 03.50.−z, 04.20.−q, 04.20.Fy, 04.60.Kz
1. Introduction
Writing second-order actions in the first-order form is a particular case of the general
Ostrogradski procedure of lowering derivative order by adding new variables. This method has
proven particularly illuminating in gauge theories, as exemplified by the Palatini formulation
of general relativity, keeping metric and connection as independent field variables. Indeed, it
has become a major activity for generalized gravity models, having long been well-understood
for standard GR, where the primary distinction between the two formulations consists of
relative matter contact terms when spinors are present [1]. In this note, we consider a more
fundamental distinction, quite apart from matter couplings, in the presence of odd-derivative
order, Chern–Simons (CS) terms, particularly in D = 3, where they play a direct kinematical
role. We shall see that major changes take place, in part, because gauge invariance can be lost
in the Ostrogradski process. We will deal first with the simplest, Abelian vector, topologically
massive model (TME), then with the full nonlinear gravitational one (TMG); both have been
exhaustively analysed in their non-Palatini avatars in [2], whose notations we follow.
2. Vectors
The equivalent Maxwell Lagrangians are
M2(A) = − 14f 2µν(A), fµν(A) ≡ ∂µAν − ∂νAµ, (1a)
M1(A, F ) = − 12Fµνfµν(A) + 14F 2µν. (1b)
0264-9381/06/185773+04$30.00 © 2006 IOP Publishing Ltd Printed in the UK 5773
5774 Comments, Replies and Notes
Here Fµν is an independent gauge-invariant variable. The Chern–Simons (CS) contributions
can also be written in two ways,
C1(A) = −m2 ε
µναfµν(A)Aα (2a)
C0(A, F ) = −m2 ε
µναFµνAα. (2b)
The ‘zeroth-order’ form (2b) is no longer gauge invariant (unlike the integral of (2a)). (A
modification of (1b), (2b) that we will not consider here, replaces Fµν by the combination
fµν(B), where Bµ is a second potential; this retains gauge invariance, at the price of keeping
derivative order.) The combination (1a) + (2a) is of course the original TME with a single
massive excitation. This same model is reproduced by taking (1b) + 2(a); the CS term does not
affect the Fµν = fµν(A) field equation. However, (1a) + (2b) represents a drastic modification:
we learn that Aµ and consequently Fµν both vanish; (loss of gauge invariance of (2b) forces
vanishing of Aµ). Finally, the doubly Palatini variant (1b) + (2b) has the field equations
∂µF
µν + m∗Fν = 0, ∗Fα ≡ 12 εµναFµν. (3a)
Fµν = fµν(A) + m εµν αAα. (3b)
Upon eliminating Fµν , we find a combined TME/Proca equation,
∂µf
µα + m εµναfµν − m2Aα = 0. (4)
This is known [3] to describe a topological-ordinary mass mix, with two massive excitations,
of masses m(
√
2 ± 1), as against the value m for the single gauge invariant one of TME.
In summary, the differences in kinematic content of the four candidate ‘TMEs’ are
extreme, ranging from ordinary TME (in two cases) to no excitation to two massive ones,
depending on where fµν(A) → Fµν is inserted.
3. Gravity
Let us first list the candidate Lagrangians in the (fully nonlinear) gravitational 2+1 TMG
of [2]:
E2(g) =
√−g gµνRµν(g) (5a)
E1(g, ) =
√−g gµνRµν(). (5b)
Here Rµν(g) is the usual metric Ricci tensor, while Rµν() is the purely affine version
in terms of the (symmetric) connection αµν . Although Rµν() is not symmetric (because
∂µ 
α
να = ∂ν αµα), only its symmetric projection survives in (5b). The equivalence of (5a) and
(5b) rests on the Palatini identity
δRµν() ≡ Dα()δαµν − Dµ()δανα, (6)
where Dµ() is the (affine) covariant derivative and δ, being the difference of two affinities
at a point, transforms as a tensor. Varying the affinity in (5b) then implies that Dα()gµν = 0,
determining αµν to be the purely metric Christoffel symbol
{ α
µν
}
, at least if gµν is invertible
[4]. Now we adjoin to (5a), (5b) the two versions of the gravitational CS term. They also
differ from vector CS in that the purely affine version is both gauge invariant and depends only
Comments, Replies and Notes 5775
on the  and not at all on the metric:
C1() = − 12 µ−1ελµν ρλσ
(
∂µ
σ
ρν +
2
3 
σ
µβ
β
νρ
) (7a)
C3(g) ≡ C1
(
αµν =
{ α
µν
})
. (7b)
The C1 version is of first-derivative order (in ) while the metric version (7b) involves three
derivatives. The usual [2] TMG is the sum (5a) + (7b), with a single gauge excitation of
mass µ. The combination (5a) + (7a), metric Einstein plus ‘Palatini’ CS, is—surprisingly—
equivalent to pure Einstein gravity, Rµν(g) = 0, plus a totally decoupled and rather empty
connection sector, with
εµνα Rλσµν() + (αλ) = 0, (8)
where Rλσµν() is the Riemann tensor of . This states essentially that  is integrable (but still
not metric). The opposite version, Palatini Einstein and metric CS, (5b) + (7b), is in fact TMG
in disguise: varying the affinity, that only appears in (5b), tells us that  is (g), whereupon
the δg equation states that
√−g Gµν(g) + µ−1Cµν(g) = 0, Cµν(g) ≡ εµαβgβσDα
(
Rσν − 14gσνR
)
. (9)
So far, then, we have two separate versions of TMG and one of pure Einstein. The final, and
more difficult, combination is the fully Palatini model of (5b) + (7a). Since the metric only
appears in (5b), its variation says that R(µν)() = 0. Varying  gives two terms:
Dα()(g
µν
√−g)δµα Dβ(gβν
√−g) + µ−1{ελσνRµαλσ () + (νµ)
} = 0. (10)
This means that the affinity is no longer a metric one, but rather obeys a far more
complicated relation, one we have been unable to solve. (Of course, a consistent solution is to
assume each part of (10) vanishes separately, which would reduce the system to pure (metric)
Einstein.) Unfortunately, the special fact of D = 3 metric geometry, that metric Riemann and
Ricci tensors are equivalent,
1
4 ε
µαβ Rαβλσ (g) ε
λσν = det g Gµν(g), (11)
does not at all hold at purely affine level.
In summary, we have extended the TME/TMG vector/tensor D = 3 models to allow for
independence of affinity and potential. In each case, we have seen quite dramatic differences.
For vectors, making the higher derivative (Maxwell term) Palatini keeps the TME structure,
while the opposite choice forbids any excitations. Double Palatini corresponds to TME +
Proca, with two excitations of different masses.
The gravitational situation is different. If we let the ‘highest derivative’ CS term only
become Palatini, we find pure metric Einstein gravity. The other case in which CS remains
metric just reproduces standard TMG, in analogy to the vector effect. Finally, double Palatini
here becomes quite involved, although it does allow for pure Einstein as a consistent solution.
Acknowledgments
I thank R Jackiw for very useful correspondence, particularly regarding the details of Palatini
TMG. This work was supported in part by NSF grant PHY04-01667.
5776 Comments, Replies and Notes
References
[1] Weyl H 1950 Phys. Rev. 77 699
Deser S and Zumino B 1976 Phys. Lett. B 62 335
[2] Deser S, Jackiw R and Templeton S 1982 Ann. Phys. 140 372
[3] Deser S and Tekin B 2002 Class. Quantum Grav. 19 L97 (Preprint hep-th/0203273) and references therein
[4] Deser S 2006 Class. Quantum Grav. 23 4539 (Preprint gr-qc/0603125)


First-order Formalism and Odd-derivative Actions

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