Table Of ContentCorrigendum: The Plebanski sectors of the EPRL vertex
(2011 Class. Quant. Grav. 28 225003, arXiv:1107.0709)
∗
Jonathan Engle
Department of Physics, Florida Atlantic University, Boca Raton, Florida, USA
3
1 January 11, 2013
0
2
Abstract
n
a Wecorrectwhatamountstoasignerrorintheproofofpart(i.) oftheorem3. ThePlebanski
J sectorsisolatedbythelinearsimplicityconstraintsdonotchange—theyarestillthethreesectors
0 (deg), (II+), and (II-). What changes is the characterization of the continuum Plebanski two-
1 form corresponding to the first two terms in the asymptotics of the EPRL vertex amplitude for
Regge-like boundary data. These two terms do not correspond to Plebanski sectors (II+) and
] (II-),butrather to thetwo possible signs of theproduct of thesign of thesector — +1 for (II+)
c
q and −1 for (II-)— and thesign of the orientation ǫIJKLBIJ ∧BKL determined byBIJ. This is
- consistentwithwhatonewouldexpect,asthisisexactlythesignwhichclassically relatestheBF
r
g action, in Plebanski sectors (II+) and (II-), to the Einstein-Hilbert action, whose discretization
[ is theRegge action appearing in theasymptotics.
1
v The error and the corrected final result
4
1
2 Theerrorliesinpart(i.) oftheorem3ofthepaper. Inordertostatethiserror,letusdefineanumbered
2
4-simplex to be a geometrical 4-simplex with vertices numbered arbitrarily, and each tetrahedron
.
1 numbered by the vertex it does not contain. An ‘ordered4- simplex’ as defined in definition 3 is then
0 a numbered4-simplexthatadditionallysatisfiesaconditionrelatingthe numberingto orientation. In
3
order for the argument for part (i.) of theorem 3 to be valid, the numbered 4-simplex gauranteed by
1
thereconstructiontheoremmustbe‘ordered’,becauseitisthenusedtocalculatethePlebanskisector
:
v of the geometrical bivectors, whose well-definition requires this. But, in general, the reconstructed
i
X 4-simplex will not be ordered.
r This is the error in the paper. As we will see, it can be easily corrected, and upon correction,
a
the interpretation of the terms in the asymptotics of the vertex amplitude will no longer involve only
Plebanski sectors, but also the orientation ǫ BIJ ∧BKL determined by the continuum two-form
IJKL
BIJ reconstructed from the discrete data at the critical points. Specifically, let
µν
ω(B ):=sgn ˚ǫαβγδǫ BIJBKL
µν IJKL αβ γδ
(cid:0) (cid:1)
where˚ǫαβγδ isthefixedorientationonM ∼=R4,andletν(Bµν)=+1,−1ifBµIJν isinPlebanskisector
(II+) or (II-), respectively, and let ν(B ) = 0 otherwise. Then the first and second terms in the
µν
asymptotics of equation (3.10) correspond to critical points where ων =+1 or −1, respectively.
∗[email protected]
1
Note that this modified result is exactly what one would expect: The first and second terms in
equation (3.10) are respectively eiSRegge and e−iSRegge, where SRegge is the Regge action. The Regge
action is a discretization of the Einstein-Hilbert action S , and the relation of the BF action S
EH BF
to the Einstein-Hilbert action, in Plebanski sectors (II+) and (II-), is precisely S =ωνS .
BF EH
Details of the correction
In the following, {B } shall always denote a “discrete Plebanski field” in the sense of [1] — that
ab
is, a set of so(4) algebra elements BIJ = −BJI satisfying closure ( BIJ = 0) and orientation
ab ab b:b6=a ab
(BIJ =−BIJ). The algebraindices IJ willusually be suppressed. TPhe algebraelementsB arealso
ab ba ab
referredto as bivectors due to the antisymmetry ofthe algebraindices. Let B ({B },σ) denote the
µν ab
unique so(4)-valued two form, constant with respect to ∂ , such that its integral over each triangle
a
∆ (σ)ofthenumbered4-simplexσisequaltothealgebraelementB . Theexistenceanduniqueness
ab ab
of the two-form B satisfying these conditions is ensured by Lemma 1 of [1]. The proof of Lemma
µν
1 does not depend on σ being ordered; see also the related work in [2]. When defining the Plebanski
sector and orientation of a set of algebra elements {B }, however, we will see that it is necessary
ab
to restrict σ to be ordered, but for the mere reconstruction of B itself, we can and do omit this
µν
restriction.
We begin by noting that the proof of theorem 1 in [1] actually succeeds in proving the following
much stronger statement.
Theorem 1, stronger statement. For any numbered4-simplex σ, B (Bgeom(σ),σ) is in Plebanski
µν ab
sector (II+) and has orientation ω =+1.
Let us next prove two lemmas, which will make the corrected proof of part (i) of Theorem 3 a
single line. For these two lemmas, let P denote any orientation-reversing diffeomophim such that
P ◦P is the identity.
Lemma 3 Given any discrete Plebanski field {B } and any numbered 4-simplex σ,
ab
B ({B },Pσ)=−P∗B ({B },σ). (1)
µν ab µν ab
Proof. Asmentionedin[1],theonlybackgroundstructuresusedintheconstructionofthecontinuum
two-formB ({B },σ)aretheflatconnection∂ andthefixedorientation˚ǫαβγδ. Webeginbymaking
µν ab a
the fixed orientation an explicit argument in the construction B (B ,σ,˚ǫ), so that, given {BIJ},
µν ab ab
(σ,˚ǫ)7→BIJ is covariantunder the symmetry group of ∂ , that is, under all of GL(4). In particular,
µν a
for P ∈GL(4), it follows that
B ({B },Pσ,P˚ǫ)=P∗B ({B },σ,˚ǫ). (2)
µν ab µν ab
Furthermore, by definition of the reconstructed two-forms (and introducing the orientation˚ǫ as an
explicit argument also of each oriented triangle ∆ (σ,˚ǫ)), one has
ab
Z B({Ba′b′},Pσ,˚ǫ) := Bab =:Z B({Ba′b′},Pσ,P˚ǫ)
∆ab((Pσ),˚ǫ) ∆ab((Pσ),P˚ǫ)
= −Z B({Ba′b′},Pσ,P˚ǫ)=−Z P∗B({Ba′b′},σ,˚ǫ)
∆ab((Pσ),˚ǫ) ∆ab((Pσ),˚ǫ)
where the second to last equality holds because the sole effect of replacing P˚ǫ with ˚ǫ in the argu-
ment for triangle ∆ is to reverse the orientation of the triangle and hence negate the value of
ab
the integral, and the last equality holds because of (2). Because the continuum two-forms are con-
stant with respect to ∂ and are completely determined by the values of the above integrals for all
a
a,b [1],it followsthat the integrandsofthe firstand lastexpressionsareequal, which,combinedwith
2
B ({B },σ):=B ({B },σ,˚ǫ), implies the claimed result (1). (cid:4)
µν ab µν ab
Inordertounderstandthesignificanceoftheabovelemma,wefirstnotethat,forB inPlebanski
µν
sector(II+)or(II-),theactionofP onB flipstheorientationofB whileleavingitsPlebanksisec-
µν µν
tor unchanged, andnegationofB flips its Plebanskisectorwhile leavingits orientationunchanged.
µν
These facts, together with the above lemma imply
ω(B ({B }),Pσ)=−ω(B ({B }),σ) and ν(B ({B }),Pσ)=−ν(B ({B }),σ). (3)
µν ab µν ab µν ab µν ab
Because of the above equations, if we wish to use B ({B },σ) to define a Plebanski sector and
µν ab
orientation for a given set of algebra elements {B }, a restriction must be placed on the numbered
ab
4-simplex σ suchthatit notpossible to use botha 4-simplex σ′ andits parity reversalPσ′; otherwise
the Plebanskisectorandorientationof{B }will be ill-defined. The restrictionusedispreciselythat
ab
σ beordered inthesenseof[1]. Oncethisrestrictionismade,ν(B ({B }),σ)andω(B ({B }),σ)
µν ab µν ab
are independent of the remaining freedom in σ. This was proven for ν(B ({B }),σ) in Lemma
µν ab
2 of [1]. For ω(B ({B }),σ), the proof follows from the same argument, together with the fact
µν ab
that, for any orientation preserving diffeomorphism ϕ, ω(ϕ∗B ) = ω(B ). Thus, one may define
µν µν
ν({B }):=ν(B ({B }),σ) and ω({B }):=ω(B ({B }),σ) where any ordered σ may be used.
ab µν ab ab µν ab
(The significance of the ordering condition on σ in this context is essentially that, by imposing a
certain compatibility between the numbering and the orientation, the ordering condition endows the
numbering of vertices with orientation information which turns out to be essential in extracting the
Plebanski sector and dynamical orientation from the algebra elements {B }.)
ab
Lemma 5 For any numbered 4-simplex σ, ω({Bgeom(σ)})ν({Bgeom(σ)})=1.
ab ab
Proof.
Case 1, σ is ordered: Thenν({Bgeom(σ)}):=ν(B ({Bgeom(σ)},σ))=+1andω({Bgeom(σ)}):=
ab µν ab ab
ω(B ({Bgeom(σ)},σ))=+1 where,in eachofthese equations,the firstequality followsby definition
µν ab
and the final equality is implied by the above stronger version of theorem 1.
Case 2, σ is not ordered: Then Pσ is an ordered 4-simplex, so that
ν({Bgeom(σ)}):=ν(B ({Bgeom(σ)},Pσ))=−ν(B ({Bgeom(σ)},σ))=−1
ab µν ab µν ab
and
ω({Bgeom(σ)}):=ω(B ({Bgeom(σ)},Pσ))=−ω(B ({Bgeom(σ)},σ))=−1
ab µν ab µν ab
where, in each of the above equations, the first equality follows by definition, the second equality
follows from equation (3), and the final equality is implied by the above stronger version of theorem
1.
In both cases, one has ω(Bgeom(σ))ν(Bgeom(σ))=1, as claimed. (cid:4)
ab ab
The corrected statement and proof of part (i.) of theorem 3 are then as follows.
Theorem 3, part (i), corrected. Suppose {A ,n } is a set of non-degenerate reduced boundary
ab ab
data satisfying closure and {X±} are such that orientation is satisfied. If {X−} 6∼ {X+}, then
a a a
{Bphys(A ,n ,X±)} is either in Plebanski sector (II+) or (II-). Furthermore, the sign µ in the
ab ab ab a
reconstruction theorem equals ων.
Proof. Letσ denotethenumbered4-simplexgauranteedbythe reconstructiontheorem,unique upto
translation, rotation, and inversion. Using the relation Bphys =µBgeom(σ) between the physical and
ab ab
geometrical bivectors in the reconstruction theorem, and using Lemma 5, one has
ω Bphys ν Bphys =ω(Bgeom(σ))(µ·ν(Bgeom(σ)))=µ.
ab ab ab ab
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:4)
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Acknowledgements
The author would like to thank the Institute for Quantum Gravity at the University of Erlangen-
Nuernberg for an invitation to give a seminar on this topic, the preparation for which led to the
realization of the error corrected here, and would like to thank Carlo Rovelli and John Barrett for
emphasizing the importance of this result and the importance of publishing its correction quickly.
This work was supported in part by NSF grant PHY-1237510 and by NASA through the University
of Central Florida’s NASA-Florida Space Grant Consortium.
References
[1] J. Engle, “The Plebanski sectors of the EPRL vertex,” Class. Quant. Grav., vol. 28, p. 225003,
2011.
[2] J. Barrett, W. Fairbairn, and F. Hellmann, “Quantum gravity asymptotics from the SU(2) 15j
symbol,” Int. J. Mod. Phys. A, vol. 25, pp. 2897–2916,2010.
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