- Email: [email protected]
Gravitons in loop quantum gravity
N UCLEAR
P H Y S I CS B
Nuclear Physics B 378 (1992) 288—308 North-Holland
_________________
Gravitons in ioop quantum gravity Joost Zegwaard
*
Institute for Theoretical Physics, Princetonplein 5, P.O. Box 80006, 3508 TA Utrecht, The Netherlands Received 18 July 1991 (Revised 5 March 1992) Accepted for publication 9 March 1992
Recently, Ashtekar, Rovelli and Smolin developed a loop representation for linearized quantum general relativity. They succeeded in deriving the Fock space structure for gravitons. based on 1oop functionals. In this article we study the relation of this theory to the full loop representation for quantum general relativity of Rovelli and Smolin. We show, after having reformulated the full loop representation by defining it on loop triplets, how the graviton loop states can he embedded in it. Also we investigate which operators on the full theory and operators on the graviton states correspond to each other. An explicit linear embedding is constructed, mapping states and operators of full- and linearized theory into another. This embedding helps us in interpreting states and operators of loop quantum general relativity.
1. Introduction Over the past few years an entirely new, non-perturbative approach to the quantization of general relativity has been developed. In 1986 Ashtekar introduced new phase-space variables, simplifying the canonical formulation of general relativity [1]. Using these variables, Jacobson and Smolin succeeded in finding solutions of the Hamilton constraint of the canonical formulation of quantum general relativity, based on holonomies of Ashtekar’s connection over certain loops [21. Shortly afterwards, a new representation for quantum general relativity based on this solution set was discovered, the so-called loop representation, which led to the discovery of a set of solutions to all constraints simultaneously [31. Although the loop representation seems to be an important breakthrough in the development of a non-perturbative theory for quantum gravity, it tends to obscure the relation of the model to physical reality: Rovelli and Smolin did not find any meaningful physical observable, nor the correct inner product. Similar techniques have been applied to other contexts in order to test the validity of the method in situations where the physics is well known from more established theories. For instance, a loop quantization procedure for Maxwell theory was developed [4] and loop methods were applied to lattice gauge theories [5]. And, only recently, a loop *
E-mail: zegwaard~fys.ruu.nl
0550-3213/92/$05.00 © 1992
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Elsevier Science Publishers B.V. All rights reserved
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representation was found for the linearized theory of gravity [6]; in this case the quantization procedure led to a Hilbert space of graviton states on a flat background, entirely equivalent to the usual Hilbert space for massless helicity ±2 particles, including the inner product and observables. A surprising and probably important aspect of this program is that it never introduces Poincaré invariance explicitly. Nevertheless, the correct graviton Fock space could be deduced. A quite natural question at this point is, if the Fock (graviton) representation for linearized gravity is indeed the correct linearization for the full theory of quantum gravity. As remarked in ref. [6], there is no principle guaranteeing that states which look semiclassical at large scales continue to do so at the Planck scale. As a consequence, it is not a priori clear whether the quantization of the theory of linear perturbations around a flat background is sensible at this scale. Therefore it should be investigated if, and in what sense, the representation in ref. [6] can be seen as the linearization of the full loop gravity theory in ref. [3]. Rovelli suggested how to incorporate the linearized theory into the full one [7]; using his ideas we will in this article perform explicit calculations showing that the linearized loop theory is indeed an approximation to a subtheory of the full loop quantum gravity theory. We will show that operators have linearized equivalents; also we will see which exact loop states correspond to gravitons moving around on a flat background. Before we actually show how the linearized loop representation can be emdedded in the full representation, we will first review shortly in sect. 2 the linearized representation of ref. [6], in order to introduce these relatively new ideas once more and to fix the notation and conventions. Then we will introduce a map E, which should actually describe the embedding of the linearized theory, and derive a number of properties it has to fulfill; we will propose an explicit form for it. Using the embedding we will show that a number of operators in the full theory (e.g. the small T-algebra of ref. [3]) has a linearized equivalent. Furthermore, we will investigate which states in the full theory correspond to the graviton states in the graviton loop theory. Finally, we will briefly consider the inner product and we will draw a number of conclusions.
2. Gravitons and loops In this section we will give a brief review of the recent work of Ashtekar et al. [6]. We will pay extra attention to aspects which we think to be of special importance, and we also already add some extra definitions we will need in subsequent sections. For reasons of compatibility with the literature we will follow the notational conventions of ref. [6] closely. The procedure followed in ref. [6] essentially consists of two main steps. First, starting from Ashtekar’s variables a linearization is chosen and new canonically
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conjugate, linearized variables are introduced. Second, for the linearized variables another loop representation is defined in such a way that again the commutation relations of the loop representation essentially copy the classical Poisson bracket relations. The new loop representation closely resembles the original in ref. [3]; however, an important technical difference will be introduced, namely that in the linearized loop representation loop functionals will be functionals of triplets of multiloops, i.e. triplets ~ (m, 172’ 113) wherein the separate components ~j, itself are multiloops and independent of each other In the next section it will be shown that it is convenient to describe the full loop representation in terms of functionals on loop triplets too, especially when one is looking for relations between both loop representations. The canonical variables for general relativity [1] consist of a pair of fields (A~, E~)on a three-manifold M, satisfying the Poisson bracket relation =
~.
(1) a, b,... are space indices on M, i, j,... internal S0(3) algebra indices. To linearize the classical theory we choose flat background variables A and E such 5aI) (q~~,) is the three metric, which can be derived from the that F 0 and ~ ~ basic fields using (det q)q” E’E”’). The simplest choice possible is =
=
=
(2)
(3) Now we set EJ”
=
+ e~’,
(4)
2e~f~~3.
(5)
and furthermore we define =
Using this relation it can be checked that the field h” is related to the usual graviton field y~,by the equation =
_~~~ul’ + ~ah~c
(6)
For the linearized connection we will use the same symbol At,. The fundamental Poisson bracket relation in the linearized theory is {A~(x), e~(y)}~=i~j~3(x-y). *
The word “loop” will be used both for single and for multiple 1oops, depending on the context.
(7)
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The constraints can be written down in the linearized variables. It turns out that the original SO(3) symmetry, expressed in the Gauss law constraint, decomposes into three independent U(1) symmetries; in other words, the linearized connection A essentially consists of three U(1) connections A’, which moreover do not change under transformations generated by the linearized scalar and vector constraints. Using a suitable gauge it can be shown that the classical physical degrees of freedom of the linearized variables are the symmetric, tracefree and transverse parts of e~’and A~.Therefore it is convenient to choose the standard complex unit basis vectors k”, rn”(k) and ifi”(k) in momentum three space, satisfying the conditions ma(k)ka=0, ma(k)~”(k)
=
1,
ma(k)m~’(k)=0,
(8)
-~—~—j-.
(9)
k”
=
The decomposition of connection and linearized dreibein into the physical degrees of freedom goes, in this gauge, as follows: A~(k) =A~(k)ma(k)mi,(k) +A(k)~a(k)~ñh(k), e~(k)
=
e~(k)ma(k)mh(k)
+
e(k)~ia(k)~h(k),
(10) (11)
with A,,h =A~/h, cab e~~/h (we will often use li-symbols implicitly for raising, lowering and interchanging spatial and group indices). The fields with superscript + correspond to positive, those with to negative helicity. We will construct the linearized equivalent of the classical loop variables of ref. [3], the so-called Tt~variables. It is very natural to choose =
—
t’(y)
exP[G ~ ds A~(Y(s))~(s)].
(12)
This is an abelian holonomy, since the linearized connections A’ are U(1)-valued. Moreover, t’(y) is invariant under transformations generated by scalar and vector constraints, so in contrast to the original Ttt variable, t’ is a physical observable. We do not have to choose a variable corresponding to the T’ variables of ref. [3], since h~1~ itself is already invariant under transformations generated by the Gauss law constraint. The principal Poisson bracket relation in terms of these fields reads {hah(X),
t’( y)}
=
—
2iG(f’ds li3( ~
—
y(s))~lib)1)t1(y),
(13)
where no summation over i is implied in the right-hand side. For notational simplicity we define Fa(x, y)
=
ftds li3(x
—
y(s))~(s),
(14)
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and its Fourier transform, Fa(k,
~)
1 =
=
(2w)
3/2
(2~)3/2
fdx etkxFa(x, y)
f’ds elk
Y(s)~a(S)
(15)
The field Fa(k, y) is called the form factor of the loop ‘y. Now we must define the quantum loop representation for the linearized theory. As mentioned before, the loop space for this representation consists of loop triplets i’~E ~ where the Z means that loops in this space must be identified if they define the same U(1) holonomy. The space of functionals on this loop space will be called .9°.Note the distinction with the original loop representation of ref. [3], where loops belong to the holonomic loop space of SU(2) (or SO(3)). Before we can define the quantum operators corresponding to the classical variables, we must first introduce a new notation. We define a union of a loop triplet i~ and a loop singlet y as follows: 1U1y=(nUy,172,173)
(16)
and correspondingly we define U 2 and U Also we define the composition ~ o~y in the same way. The difference between union and composition is that for two single loops 11 and y the union 17 U y consists of two loops, while the composition 17 o y is again a single loop (and only defined unambiguously if 17 and y intersect once). In the U(1) holonomic space they are identified, in the SO(3) holonomic space, however, they are not. The correct definitions for the quantum operators I’ and fr’~appear to be ~.
(I’(y)~)(i~) (~ah(k)~,)(~)
U
1y),
=
~i(i~
=
2hGF(a(k, 171)li”~’~(1).
(17) (18)
A straightforward calculation shows that these operators satisfy the right commutation relation: [~ah(k)
I’(y)]
=
2hGF(a(k, y)li’~1’(y).
(19)
We will define an extra operator in the quantum representation, which was not defined in ref. [61. It is the operator ê~’,corresponding to the classical linearized dreibein e’. Note that this field is not invariant under transformations generated by the Gauss-law constraint. This certainly violates the spirit of the original papers on the loop representation, where it was emphasized that the new basic phase-space
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variables (specifically, the T-algebra) were invariant with respect to transformations generated by this constraint on the classical level. However, we will only use the definition of ê~’to show equivalence of certain operators in the linearized and the full loop representation, not to define physical observables; for the latter its gauge invariant part I”~is sufficient. The quantum action of ê,” is defined as (ê~(k)~)(~) hGF”(k, =
(20)
17~)~1(T~)
from which the classical eq. (5) is directly reproduced as a quantum operator equation: /ah
=
2e~~~8h)I.
(21)
As a following step the space is defined as the space of vector fields F”(k) in momentum space which satisfy the reality condition (for the Fourier transform) Fa(k)=Fa(~~k)and which are transversal: Fa(k)ka 0. Since all form factors belong to functionals x on determine a unique functional ~/i on via .~
=
~
.~
~
(22) where F”(k, ~ is considered to be an element of by defining (F”(k, ii), Fa(k, 17~). ~i is called a regular loop state if it is derived from a functional x admitting a convergent Taylor expansion on It is proved in ref. [6] that the map from these functionals x to regular loop states is bijective, which means that it is allowed to switch back and forth unambiguously between the subset of regular states in the loop representation and the representation of convergent functional series on The physical states are precisely those regular loop states ~1(i which ~
=
.~.
~
can be written in the form
~),
~/J(T~) =~(F~(k,
F~ and F
F(k,
n)).
are the tracefree transverse components of F”(k, ~): 1(k, 17’), F(k, ii) ~a~iF” F(k, ii) =mam ~F~(k, 17’).
(23)
=
(24)
The set of all physical states is denoted by ~ On this physical subspace an inner product can be defined; it is essentially determined by the physical reality conditions. A graviton Fock space based on this inner product can be constructed with creation operators aL(k), a11(k) and annihilation operators a ~(k), a _(k), which are defined in terms of h”~and I’(y) using an appropriate limiting procedure. The normal ordered hamiltonian operator reads i~=fd3k h 1k [a~(k)a+(k)
+
a~(k)a(k)].
(25)
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~s
0
/
Gravitons in loop quantum gravity
is defined, and a general one-graviton state can be
written in the Fock space as t If)
=
(k)) 0)
(26)
ds)~)(~),
(27)
(fd3kf±(k)a~(k)+f(k)a
or as a loop functional:
=
~
where L 1(x) is the Fourier transform of the “wave function” (in momentum space) fai(k)
=
~
-f~(k)~a(k)rnj(k)).
(28)
For an n-graviton state, f will depend on n variables k1,. .., k,, and carry 2n indices a1 a,,, i1 i,,. As described in ref. [6], this representation contains divergences which can be solved by introducing a regularized form factor F”(k, y). The dependence on r cancels in any calculation of a physical quantity, essentially because r does not appear in the graviton Fock space, where the inner product is defined. . . .
. . .
3. Embedding in the full quantum theory In the previous section we have seen that it is possible to write down a brand-new loop representation for the linearized theory of quantum general relativity, using the linearized Ashtekar variables. But, a priori one does not know if this loop representation has anything to do with the original loop representation of Rovelli and Smolin, though there are many similarities, both in the spirit and in the technical details of the papers. The results of the linearized loop representation could be compared to the usual theory for gravitons on a flat background and led to the expected graviton Fock space. If we could now show that both loop representations coincide in the relevant region, i.e. if we could find a procedure to “linearize” the full loop representation and uncover the linearized loop representation again, this would tell us more about the validity of the loop representation for quantum gravity. It would imply that at least a part of this loop representation has a physically correct and well-understood meaning, and moreover, it might help us in interpreting the loop representation in other regimes, for instance on other backgrounds or in incorporating non-perturbative effects. This linearization of the full loop theory is the subject of this and following sections, and we will show that indeed the linearized loop representation can be seen as a very special case of the
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full loop theory. This not only gives us confidence in the validity of the loop theory, it also gives us some clues for the interpretation of the physical states and for the identification of the physical operators and the inner product, at least on the linearized sector. Denote the solution set of the loop representation in ref. [31by ~. Let us investigate which conditions the (at this point fictitious) map E from the set of “exact” loop functionals 4 ~ into the set of linearized loop functionals 1/5 EY’~’ has to fulfill. In the first place, E has to be a linear map: if 1/~and 1/2E~’are ~‘
mapped by E on
i4s~ and
1/’2’
respectively, then we expect 1/
+
‘/~2
tobe mapped on
+ 1/’2~ Furthermore, we must show that E maps an operator Q on ~ to the corresponding operator P on S~~0h;“corresponding” means that the classical variable P is exactly the linearization of the classical variable Q We thus want to have ~.
(29)
EQ=PE
on the elements of describing gravitons on a flat background. For instance, the classical Ashtekar dreibein is decomposed in the linearized theory as follows: ~‘
(30) so the inverse densitized three metric can be written as (det q)q”
Qab
=
E,”E”
=
li” + 2e~’li~’ + e,aeL~~.
(31)
After quantization and using the notation 2e~’~ h~~h the following equation must hold: =
EQab
=
(li” +
+
e~ehl)E.
(32)
Two other examples involve the loop variable of the linearized theory, t’(y). This variable is, on the classical level, the linearization of the loop variable T°(y); the latter can be expanded: T°(y)
Here
Tr P exp Gç6A
=
3
+ Tr(Gri~A1+ ~G2T1TJ(~A1
)
3). (~A1)) +
(33)
0(
T 1
*
=
and
stand for the generators of the S0(3) algebra, which are traceless
In fact, the word linearization is here somewhat misleading since we will do the calculations up to second order.
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and normalized according to Tr(r1r~) li,1, and E denotes the order of magnitude of the linear perturbations. We can express this result in abelian holonomies: =
3)
T°(y)= ~~(ex~ =
t’
~(t’(y)
~
G~A’+exp G~(_A’)) + t’(y~)) +
+ 0(E
(34)
0(~).
is also proportional to the linearization of the T’ variable of ref. [3], since T~(y)(s)
=
Tr[(P exp Gf’~’A~(y(t))~(t)dt)Ea(y(s))] +
=
e~(y(s)))(t’(y)
-
‘))
t’(y
+ 0(E~).
(35)
So, in the quantum theory we will have to prove the equations E~°(y)
=
ET~(y)(s)
=
(~
~(I’(y) +
+
I’(y ‘)))E,
e~(y(s))][I’(y)
-
(36) Il(y1)]
E.
(37)
In constructing the embedding E we encounter the technical obstruction that loop functionals in the linearized representation are functionals of loop triplets ~q, while the original loop representation of the full theory acts on singkts of (multiple) loops only. To make the construction of E possible we will propose a new representation of the full theory, which is suggested by the form of eq. (36). The loop functionals 1/ will become functionals of three independent multiple loops, and operators P on this representation will be defined as follows: (/~1~)(Tl)
=
E(I~)(ii),
(38)
where J~,. is the original operator acting on the functional defined by considering only the dependence on the ith loop, not on the other two. In calculating this action three terms appear, in each of which two loops are taken to be fixed and the third one to be variable. To be more specific, take for instance the operator T°(li). Its definition on the “usual” loop functionals is as follows: 7~°(li)ç~(y) =q~(yUli).
(39)
The corresponding definition on loop triplet-functionals is (t0(li)4)(a)
=
E~(~U 1li),
(40)
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where / E In appendix A we show that this “extended” loop representation essentially leads to the same results as the original one, which is based on functionals of singlets of multiple loops. We have to make an assumption about the way the operator E acts on We will introduce some fixed “background loop” A and propose that E acts as follows: .~‘.
.~‘.
(E~)(’q) =cb(A
(41)
i~).
°
The operator a connects the corresponding components to another, so ~ is connected to ~ etc. Eventually this equation will have to be defined by using a certain regularization to spread ~ over a distance big enough to ensure that always “hits” A,. At this point however we will assume that the right-hand side is defined by introducing line segments p1, which are straight (unknotted) lines 7L• If we want eq. (32) to be fulfilled we must have the equality connecting A,. and ~,
(E~”~)(~) which for ~
=
+ ê~~êh~) E~}(n),
+
=
(42)
0, the “trivial” loop triplet, leads to the equation Qab~(
A)
=
li*~~4( A).
(43)
We have used the fact that h”t/i(O) ê~’t/i(0) 0. Using the definition of Q” (see ref. [8] and appendix A) extended to functionals of loop triplets we get the formal expression =
~ah(x)~(
A)
=
=
(hG)2[fds
~ [fdt
li3(x
li3(x
-
-
A 1(t))A~(t)]~(A)~
(44)
where summation over i is implicit. Since the right-hand side, being a product of delta functions, is not well defined, we introduce a regularization for each term alone instead of for their product. Therefore we seek a “loop” A satisfying 3(x
—
At(s) )A~(s)”
=
8~,
(45)
“hGfds li where quotation marks denote the necessity of smearing with a some function over a sufficiently large scale. The freedom in choosing a dreibein (in this case 8,”) introduces a freedom in the choice of the precise loop components; this freedom corresponds to local rotations of A. We now see that is it convenient to choose A such that A~a 8,”, constraining A, to be a set of straight lines in the x, direction in
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M. Of course, this A, does not consist of closed loops for an infinite manifold M, but this can be solved easily by compactifying M, for instance from ~ to S3. Suppose further that these lines pass the plane x, 0 through a number of points, arranged in an infinite square lattice with distance d between neighboring lattice points (this was suggested by Rovelli [7]). We will take M ~ because this is the usual configuration space for gravitons. The set of loops A 1 can be parametrized by a continuous parameter s El ~, +~[ and two discrete parameters n1, n2 E as follows: =
=
—
A’~1”2(s)
=
(s, n1d, n,d).
(46)
Integrating the formal expression (45) over a test function f leads to hGfdsf(A~(s))A’(s) Choose now, for instance, i 1 (i use the parametrization to find =
=
(47)
8~fd3xf(x).
2, 3 would lead to the same final result) and
=
3xf(x).
~
fdsf(s, n1d, n2d)
fl,,fl
=8~fd
(48)
lvi
2
Assume that the test function f varies slowly on scales smaller than Planck’s length, so that the summation on the left-hand side can be approximated by the corresponding integration. The equation then becomes hGfds dt duf(s, td, ud)
=
hGfds dt du
f(s, 2t, u) d
=
fMd~~),(49)
which immediately gives the equality
=~
d=v~~=1~,
(50)
where ~ denotes Planck’s length (c 1). So if we define E as explained above, with the loop A defined as three perpendicular sets of straight lines, intersecting the transversal planes in square lattices with lattice distance Ii,, then the flat background is reproduced in the full theory in the right way, through the operator ~ah and using the appropriate regularization. However, it remains to be shown that this is also true for the fluctuations on the background; in fact, we will perform the calculation shortly. It should be noted that we could have chosen, instead of 8,”, another “root” of =
in eq. (45). This would have led to the same A, but rotated over a certain angle in loop space. So the orientation of A cannot have any influence on the physical 8ah
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results of the calculations. Moreover, after deforming A by means of a diffeomorphism the calculation gives a transformed, but still flat geometry, as was explained for a general loop in ref. [8] (the calculation in this article survives the regularization). In particular, the physical lattice distance d is invariant under diffeomorphisms acting on A.
4. Linearization of operators Since we have a candidate for a mapping E relating the full- and linearized loop representation, it becomes interesting to find out which operators and which states in those two representations are related to each other. We will investigate corresponding states in sect. 5; here we will check if the variables T°,T” and Qab can indeed be linearized correctly in the quantum theory, using this mapping E. As was explained in sect. 3, “correctly” means that the quantization of a variable is related to the quantization of its linearized version. Starting with the operator T1~, we can write (E~0(y)~)(ii)
=
(f°(y)~)(A
=
L~((A
=
E ~[~(A
=
°
i~)
o
°
U~y)
(~a
y)) + ~(A
a
(~ o
[(E~)(~a 1y)+(E~)(~o1y1)I
=
+ I’(y~))Ecb](ii).
~[(I’(y)
(51)
We implicitly introduced line segments to be able to define ~ o~ y; they cancel in 11 the linearizedtoloop representation. corresponds the right operator onThe derivation shows straightforwardly that T Considering the second variable, T”, we can calculate (using the notation of ref.
[3]) (Eta(y)(S)~)(~)
=
fa(y)(5)~(A
=
~hG~”(y,
a
A~
°
17~)(s)[~(Aa
ii
o~y)—~(Aa
~
a~y~)]
(52)
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where hG~i”(y,A~a
hG~”(y, A~)(s)+hG~”(y, ?7~)(S)
?7~)(S)
83(y(s)
~hGfdt
=
-
A~(t))A~(t)
+~hGfdt 83(y(s)
17~(t))~(t)
hGF”(y(s),
+
=
(53)
17~)),
again using the regularization we described earlier. With this result we can write (E(y)(s)~)(~)
=
+
hGFa(y(s), 171))[(E~)(~U ~y)
=
+
hGFa(y(s), 17~))[(t’(y)
+
=
e~(y(s))}[i’(y)
-
-
-
(E~)(~U
i’(y’))E~J(1)
1’(y’)] E~)(i),
(54)
showing the correct transformation of 7” using the map E. Note that the correct factor ordering is important; the equality would not be correct if the order of e,~’(y(s))and I’(y) or I’(y~)had been changed. As a third example, we will also show the transferability of the operator Q’°~’ between S~’°” and To accomplish that we have to check eq. (42) on loops i~ which are not necessarily zero. Expanding this equation we get ~.
(E~ah(x)~)(~) ~ab(~)~(A
a
=
(8~+ hGF”(x, ?7’))(8~+ hGFb(x, 17~))~(Ao 2Fa(x, + 2hGF(”(x, 17’)8~~ + (hG) 17~)Fb(x,17’))~(A =
=
(8ah
*
~).
(55)
This result must be compared with the right-hand side of eq. (42), which yields (liab
+ h~’(X) +
ê~êb1(x))E4j(~) =
(liab
+
2hGF(”(x, 17’)87)
2F”(x, +(hG)
17~)Fb(x,17’))~(A a ~),
(56)
the same result as for the full theory. As a consequence the operator for the three metric, Qab is correctly transferable between the representations ~ and ,9”r’~’• The loop A can be seen as a background loop for flat minkowskian space; its importance in the loop theory is, that it establishes the transformation from the topological knot relations of the full theory of quantum gravity to some kind of
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301
metrical relations in the linearized theory. For instance, using this background one can make a distinction between two loops y1 and Y2 in loop space if their distance in the direction perpendicular to A, is larger than Planck’s length li,, by using the fact that A ° ‘ and A ° ~ belong to two different knot classes. This means that a loop state 1/’ in the linearized loop theory will in general discriminate between them.
5. Graviton states in the loop representation Before we can interpret E as a map of a subset of in ,/.~‘, and thus its inverse of an embedding of 5”°~’in .4~,we must first check if the image of E consists of loop functionals satisfying the diffeomorphism constraint, i.e. loop functionals that only depend on know classes (see ref. [31).More concretely, we have to take a general graviton state and show that the corresponding full loop state, which one gets by applying E I to it, can be seen as a functional on smooth knot classes. To ensure that E ‘ is well defined, we will restrict E to states f E supported only on loops a which are of the form a A o ~ or which can be transformed in this form by a diffeomorphism on M. On this subset E is indeed an injective mapping. Only these states will correspond to the graviton states of the linearized theory. Considering the equation .~
—
~‘
=
(E4)(i1) =~(A
°
ii),
(57)
and taking into account that we would like 4 to be a functional on knot classes, we conclude that only those states 1/’ E.~9”~’can be embedded in the full theory that satisfy the equation I/f(Dr’ll)=l/J(ll),
(58)
where Dr is a diffeomorphism on loop space; it leaves the background A invariant. So Dr keeps the intersection points of A, and the 1oops ~, fixed, and it shifts ‘17, in such a way that it does not cross the fixed background A,; Dr will be called a “restricted diffeomorphism”. One might expect any graviton state to satisfy this equation, because these states are solutions to the linearized diffeomorphism constraint. However, since the linearized constraint neglects term of order ~2, the full loop states corresponding to the graviton states will appear to satisfy the full diffeomorphism constraint only up to linear order; the graviton functionals are thus mapped into functionals closely approximating diffeomorphism-invariant states in the full representation. A physical (regular) loop state in the linearized theory is a loop functional derived from a graviton state. For simplicity we take a one-graviton state; n-gravi-
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ton states will essentially give the same results (with more variables):
=
(~ ff~
1(ms))~(s
ds)~o(i)~
(59)
which was already mentioned in sect. 2. This graviton state corresponds to the loop state satisfying the equation E4~ 1/it, or more explicitly, ~.,
=
4f(A
°
‘q) _llif(i~).
(60)
Because the vacuum functional 1/’~~ which appears in every graviton state, gives zero for all non-zero loops, we will simply renormalized it to one. One might see it as the contribution of the infinite background loop A. We deliberately do not introduce a regularization parameter r here, as in ref. [6], because it would spoil our graviton calculations, in which the delta functions in the Fourier-transformed form factors are already smeared out by the graviton wave functions, and therefore do not need further regularization. So, from now on we put 1/i~(ii) I for all E~ in the regular loop states. In fact, this also restores the symmetry between the two helicities, since the remaining part essentially consists of Hermite polynomials in both the F~and the F fields. We will show that -ff(~q)is in lowest order invariant under the action of restricted diffeomorphisms Dr on the loop triplet i~. This can be computed directly, using the fact that f is transverse, trace-free and symmetric. However, we choose an alternative approach, by showing that the full diffeomorphism constraint is mapped by E to the linearized one, so that is a solution to the first one if the corresponding I/If solves the latter. The full vector constraints reads, in terms of the T-variables: =
C~(x) ~ =
G~li2Ea~Tx,~O)~
(61)
where y~ is a loop in the plane perpendicular to the unit vector ê~starting in x with radius 8 (positively oriented with respect to e’). The derivation of this expression is similar to the derivation of the Hamilton constraint by Rovelli and Smolin [3]. It can be quantized directly, and then the corresponding linearized constraint can be derived using eq. (54):
~(
x)
=
lim
G~82 Eahe~8~[t’(y,’~)-
We only considered the linear order in
,
I’(( y,’~)~‘)].
(62)
since the linearized constraint was
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derived in ref. [6] by neglecting all higher orders. Rewriting this equation using loop derivatives we get ~(x)
=
(63)
~E,,iC8C~
here we also used that li/8y —li/liy, which follows from ~‘(y (I’(y)Y1 We reproduced the known expression for the linearized vector constraint in ref. [6], where is was denoted by ~(x). This result suffices to show diffeomorphism invariance of the full state corresponding to the graviton state ~ To see this, extend the definition of thf in eq. (60) to diffeomorphism classes, by defining cbf(a) t/If() if a E [A a ii] ~, for some finite loop triplet and zero otherwise. Now consider the diffeomorphism constraint: =
‘)=
=
i~,
0
=
=
~
=
(~E~f)()
+ O(E2)
(E~,,~f)()
=
(~a~f)(A
a
)+
0(E2),
(64)
showing that solves the vector constraint to linear order. This means that can be considered as an approximation to a full diffeomorphism-invariant state I f) E .~4’,which actually corresponds to the original one-graviton state. To be more concrete: to calculate /I(a), where a E [A a ], one should find a diffeomorphism D explicitly satisfying Da A a This D is defined up to restricted diffeomorphisms, resulting in different possible loops but the graviton functionals t/Fj are precisely such that the choice of this -q will only change terms of at least quadratic order in E in the final result. The graviton Fock space is therefore a valid description for a subset of the full loop states if the perturbations on the flat background are small; of course this is exactly what one expects. Besides, by assuming the test function in the regularization of F”(x, A) to be smooth on Planck scales we already restricted ourselves to the situation of small perturbations. It is possible to decompose I f) into a basis of knot classes: i~.
=
i~,
If)=
~
c[~]I[A
o
1]),
(65)
[A
where the state [A
a
~~])represents the loop functional if~e[Ao] ~O otherwise.
*
[A
i~]
is a 1oop triplet knot class, see appendix A.
‘1”[A.
~
satisfying (66)
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/ Gra,itons in
loop quantum gravity
(see e.g. ref. [8]). The value of the constants c1,~1is =
E
f L~(171(s))~9(s)ds,
(67)
which is in lowest order independent of the choice of an element from the restricted know class [11],, consisting of loop triplets related to by a restricted diffeomorphism. The state I f) is now, for some precise choice of all coefficients an exact state of quantum gravity, approximating the one-graviton functional The vacuum functional I/’~~ corresponds to the state 4~in defined by taking all coefficients c[1] equal to one. In other words, 411(a) is one if a E [A a i]~for some (finite) loop triplet and zero otherwise. A general n-graviton state is governed by a real function f~ a,,i, It satisfies the same conditions as the one-graviton function, hut in more indices, and it defines in exactly the same way a linear approximation to a full diffeomorphism-invariant loop state in quantum gravity. The inner product is well defined in the graviton Fock space: if I f~)) and I g~) are two one-graviton states, then their inner product is (in linear order in e) i~
.~
i~,
I gW)
= fd3k
fai(
—k)g”(k)
=
fd3x~,t(x)~”’(x).
(68)
We expect that in the exact (to all orders) inner product the integration will become a summation, since the exact solution set of the full loop representation has a countable basis. For the loop functionals t/i~ and t/Ig corresponding to these kets we have
K~fI~g)fd3x~~(x)ga~(x).
(69)
=
For two n-graviton states this becomes (f(fl)
I g(~I))
=
3x,,
~
“‘“
fd3x1 ...d
(70) Furthermore, different graviton sectors are orthogonal, which means that we have in general Kf~”~g~~~)) a 8mm. On the linear sector of quantum loop gravity the inner product is thus well defined. However, we are interested in finding an inner product for the full theory. This inner product should be defined directly on diffeomorphism invariant loop functionals, and should correspond to the abovementioned inner product on the graviton sector of the theory. Despite serious efforts we have still not succeeded in finding such an inner product.
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6. Conclusions The main goal of this paper was to find the relation between the loop representation for linearized quantum gravity introduced in ref. [6], and the representation for the full theory in ref. [31.The most important result of this paper is that we have achieved this aim by finding an embedding from the linearized solutions into a subset of the full solutions. Furthermore, we showed that by using this embedding we could transfer a number of operators from one representation into the other. This could help us in finding and interpreting the physical observables in the full theory, because we know how to interpret a number of physical observables in the graviton theory. For the same reason we can identify a subset of full loop solutions as gravitons moving on a flat background. Therefore we have gained some physical intuition for the meaning of the physical states in the loop representation. We have also found a background loop which is closely connected to the flat background we chose. There are still some open questions. Probably the most important one is that we still do not have an inner product on the full loop representation. The inner product we proposed in ref. [8] seems not to correspond to the inner product on the graviton sector. We have not been able to find an inner product that does. The correct inner product will have to deal with the fact that the space of full loop states has a countable basis. Also, in this paper we did not consider nonperturbative effects in loop quantum gravity, although these must be an essential part of the full theory. It might be interesting to consider other background metrics and to perturb canonical quantum gravity around these new backgrounds. One would then expect that a different background leads to a different loop A and thus to a different embedding E. However, every background should lead to a theory which can be seen as a subtheory of the full theory of quantum gravity. And this full theory should not only consist of the union of these subtheories, but should also contain essentially non-perturbative information, allowing for instance background changing. The linearized theories might help in developing physical intuition in interpreting operators and states in quantum gravity, and finally they should give us information about physics on the Planck scale. I would like to thank G.
‘t
Hooft for stimulating discussions.
Note added After completion of this work a review article by Rovelli appeared [9], in which research on the linearization of the loop representation analogous to ours is described. Besides the extra attention Rovelli pays to the regularization there are some differences in the precise calculations, leading to a definition of the mapping corresponding to our E which is slightly different from the one in eq. (41).
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Gravitons in loop quantum gravity
Appendix A
LOOP TRIPLET REPRESENTATION
A peculiar feature of the linearized loop representation is that it involves functionals in triplets of multiple 1oops. To construct the embedding of the linearized representation into the full one it appears to be convenient to define the full representation by functionals on loop triplets too. In this appendix we will find out what the physical operators and states of this extended representation are and show that we can essentially copy the results of the loop singlet representation; furthermore we will show that functionals of loop triplets has the important advantage as compared to the original functionals that the operator Q”” involving the (densitized) three metric is defined more consistently. Let us denote the space of loop singlets by 2”, of loop triplets by 2’s. We do not specify the holonomy, which would force us to make certain identifications in these spaces. We must first check if operators on 2’~ satisfy the same commutation relations as the corresponding operators on 2~I. First remember that an operator P on 2’~ is defined by P= ~
(A.1)
where P, is the loop operator which acts precisely the same as P but only on the ith loop, with the other two loops fixed. For instance, if P acts like (P~)(a) =F(a)I/i(a’), (F is some functional of loops) then
P, is defined by
(P,~)(a)=F(a,)I/i(a~, a2, as), and similarly
(A.2)
P2
and P3. Because the three ments, we will certainly have [i~, It follows that for three operators P, [i~,
Q1]
P, operators act on different argu-
a~
Q and
(A.3)
(A.4)
R on 2’s, which satisfy
~j =k,
(A.5)
one has the following relation as well: (A.6)
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Gravitons in loop quantum gravity
307
with R = E,R,. The commutation relations will therefore not change in the new representation. Which functionals on 2’~ can be identified as the physical states? Take a constraint operator C, which must now also be decomposed into C,. A physical state has to satisfy, for every loop triplet a
E (~~1/)(a)= 0.
(A.7)
Applying this to vector and scalar constraint implies that physical states are functionals 4(ot) which have support on smooth loop triplets a and are invariant on loop triplet knot classes [a]: /I(a) =q~(Da).
(A.8)
Here D y in general stands for the loop satisfying (D ~)(s) = D(y(s)). This means for instance that ~(a,, a2, a3) and 1/(D a~,D a2, D a3) should yield the same value, but /I(D a~,a2, a3) should in general not. The original knot classes of Rovelli and Smolin have now become diffeomorphism classes of loop triplets, which, however, are still knot classes since D does not change the way the loop components a, are knotted into another; they transform simultaneously. As compared to the original ones the new knot classes contain extra information in their splitting into three loop components. This extension of loop functionals has an important advantage in the calculation of expectation values of the operator Qab = E,”E~’’. In ref. [8] the unregulated action was calculated (h and G are now included): 2Fa(x, y)Fb(x, y)~(y). (A.9) = (hG) where the notation of this article has been used. On the right-hand side we have the tensor product of a vector with itself; it is clear that this severely restricts the possible values. For instance, no vector v exists such that ~ = 8ab since the determinant of the product is always zero. On functionals of loop triplets the formula becomes =
(E(hG)2Fa(x,
aj)Fh(x,
ai))~(a).
(A.10)
The set of possible outcomes is now not a priori restricted, since to any tensor field we can connect a dreibein such that L’1”
tab = ~
(A.11)
So in this representation any reasonable value for the expectation value of
Q””
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Gravitons in loop quantum gravity
(modulo the regularization of F) can be obtained: for instance, in sect. 3 we found a loop A such that Q””ç6(A) = 8”~/I(A),using an appropriate regularization.
References 111
A. Ashtekar, Phys. Rev. D36 (1987) 1587 [2] T. Jacobson and L. Smolin, NucI. Phys. B299 (1988) 295 [3] C. Rovelli and L. Smolin, NucI. Phys. B331 (1990) 80 [4] A. Ashtekar and C. Rovelli, Loop representation for the quantum Maxwell field, Syracuse preprint, 1991 [5] C. Rovelli and L. Smolin, Loop representation for lattice gauge theory, Pittsburgh and Syracuse preprint, 1990 161 A. Ashtekar, C. Rovelli and L. Smolin, Phys. Rev. D44 (1991) 1740 171 C. Rovelli, Quantum gravity in the loop representadon, talk given at Les Journées Relativistes, Cargèse, May 1991 [8] J. Zegwaard, Class. Quant. Gray. 8 (1991) 1327 [9] C. Rovelli, Class. Quant. Gray. 8 (1991) 1613
P H Y S I CS B
Nuclear Physics B 378 (1992) 288—308 North-Holland
_________________
Gravitons in ioop quantum gravity Joost Zegwaard
*
Institute for Theoretical Physics, Princetonplein 5, P.O. Box 80006, 3508 TA Utrecht, The Netherlands Received 18 July 1991 (Revised 5 March 1992) Accepted for publication 9 March 1992
Recently, Ashtekar, Rovelli and Smolin developed a loop representation for linearized quantum general relativity. They succeeded in deriving the Fock space structure for gravitons. based on 1oop functionals. In this article we study the relation of this theory to the full loop representation for quantum general relativity of Rovelli and Smolin. We show, after having reformulated the full loop representation by defining it on loop triplets, how the graviton loop states can he embedded in it. Also we investigate which operators on the full theory and operators on the graviton states correspond to each other. An explicit linear embedding is constructed, mapping states and operators of full- and linearized theory into another. This embedding helps us in interpreting states and operators of loop quantum general relativity.
1. Introduction Over the past few years an entirely new, non-perturbative approach to the quantization of general relativity has been developed. In 1986 Ashtekar introduced new phase-space variables, simplifying the canonical formulation of general relativity [1]. Using these variables, Jacobson and Smolin succeeded in finding solutions of the Hamilton constraint of the canonical formulation of quantum general relativity, based on holonomies of Ashtekar’s connection over certain loops [21. Shortly afterwards, a new representation for quantum general relativity based on this solution set was discovered, the so-called loop representation, which led to the discovery of a set of solutions to all constraints simultaneously [31. Although the loop representation seems to be an important breakthrough in the development of a non-perturbative theory for quantum gravity, it tends to obscure the relation of the model to physical reality: Rovelli and Smolin did not find any meaningful physical observable, nor the correct inner product. Similar techniques have been applied to other contexts in order to test the validity of the method in situations where the physics is well known from more established theories. For instance, a loop quantization procedure for Maxwell theory was developed [4] and loop methods were applied to lattice gauge theories [5]. And, only recently, a loop *
E-mail: zegwaard~fys.ruu.nl
0550-3213/92/$05.00 © 1992
—
Elsevier Science Publishers B.V. All rights reserved
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Gracitons in loop quantum grauty
289
representation was found for the linearized theory of gravity [6]; in this case the quantization procedure led to a Hilbert space of graviton states on a flat background, entirely equivalent to the usual Hilbert space for massless helicity ±2 particles, including the inner product and observables. A surprising and probably important aspect of this program is that it never introduces Poincaré invariance explicitly. Nevertheless, the correct graviton Fock space could be deduced. A quite natural question at this point is, if the Fock (graviton) representation for linearized gravity is indeed the correct linearization for the full theory of quantum gravity. As remarked in ref. [6], there is no principle guaranteeing that states which look semiclassical at large scales continue to do so at the Planck scale. As a consequence, it is not a priori clear whether the quantization of the theory of linear perturbations around a flat background is sensible at this scale. Therefore it should be investigated if, and in what sense, the representation in ref. [6] can be seen as the linearization of the full loop gravity theory in ref. [3]. Rovelli suggested how to incorporate the linearized theory into the full one [7]; using his ideas we will in this article perform explicit calculations showing that the linearized loop theory is indeed an approximation to a subtheory of the full loop quantum gravity theory. We will show that operators have linearized equivalents; also we will see which exact loop states correspond to gravitons moving around on a flat background. Before we actually show how the linearized loop representation can be emdedded in the full representation, we will first review shortly in sect. 2 the linearized representation of ref. [6], in order to introduce these relatively new ideas once more and to fix the notation and conventions. Then we will introduce a map E, which should actually describe the embedding of the linearized theory, and derive a number of properties it has to fulfill; we will propose an explicit form for it. Using the embedding we will show that a number of operators in the full theory (e.g. the small T-algebra of ref. [3]) has a linearized equivalent. Furthermore, we will investigate which states in the full theory correspond to the graviton states in the graviton loop theory. Finally, we will briefly consider the inner product and we will draw a number of conclusions.
2. Gravitons and loops In this section we will give a brief review of the recent work of Ashtekar et al. [6]. We will pay extra attention to aspects which we think to be of special importance, and we also already add some extra definitions we will need in subsequent sections. For reasons of compatibility with the literature we will follow the notational conventions of ref. [6] closely. The procedure followed in ref. [6] essentially consists of two main steps. First, starting from Ashtekar’s variables a linearization is chosen and new canonically
290
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Gratitons in loop quantum grat’ity
conjugate, linearized variables are introduced. Second, for the linearized variables another loop representation is defined in such a way that again the commutation relations of the loop representation essentially copy the classical Poisson bracket relations. The new loop representation closely resembles the original in ref. [3]; however, an important technical difference will be introduced, namely that in the linearized loop representation loop functionals will be functionals of triplets of multiloops, i.e. triplets ~ (m, 172’ 113) wherein the separate components ~j, itself are multiloops and independent of each other In the next section it will be shown that it is convenient to describe the full loop representation in terms of functionals on loop triplets too, especially when one is looking for relations between both loop representations. The canonical variables for general relativity [1] consist of a pair of fields (A~, E~)on a three-manifold M, satisfying the Poisson bracket relation =
~.
(1) a, b,... are space indices on M, i, j,... internal S0(3) algebra indices. To linearize the classical theory we choose flat background variables A and E such 5aI) (q~~,) is the three metric, which can be derived from the that F 0 and ~ ~ basic fields using (det q)q” E’E”’). The simplest choice possible is =
=
=
(2)
(3) Now we set EJ”
=
+ e~’,
(4)
2e~f~~3.
(5)
and furthermore we define =
Using this relation it can be checked that the field h” is related to the usual graviton field y~,by the equation =
_~~~ul’ + ~ah~c
(6)
For the linearized connection we will use the same symbol At,. The fundamental Poisson bracket relation in the linearized theory is {A~(x), e~(y)}~=i~j~3(x-y). *
The word “loop” will be used both for single and for multiple 1oops, depending on the context.
(7)
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Gratitons in loop quantum grwity
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The constraints can be written down in the linearized variables. It turns out that the original SO(3) symmetry, expressed in the Gauss law constraint, decomposes into three independent U(1) symmetries; in other words, the linearized connection A essentially consists of three U(1) connections A’, which moreover do not change under transformations generated by the linearized scalar and vector constraints. Using a suitable gauge it can be shown that the classical physical degrees of freedom of the linearized variables are the symmetric, tracefree and transverse parts of e~’and A~.Therefore it is convenient to choose the standard complex unit basis vectors k”, rn”(k) and ifi”(k) in momentum three space, satisfying the conditions ma(k)ka=0, ma(k)~”(k)
=
1,
ma(k)m~’(k)=0,
(8)
-~—~—j-.
(9)
k”
=
The decomposition of connection and linearized dreibein into the physical degrees of freedom goes, in this gauge, as follows: A~(k) =A~(k)ma(k)mi,(k) +A(k)~a(k)~ñh(k), e~(k)
=
e~(k)ma(k)mh(k)
+
e(k)~ia(k)~h(k),
(10) (11)
with A,,h =A~/h, cab e~~/h (we will often use li-symbols implicitly for raising, lowering and interchanging spatial and group indices). The fields with superscript + correspond to positive, those with to negative helicity. We will construct the linearized equivalent of the classical loop variables of ref. [3], the so-called Tt~variables. It is very natural to choose =
—
t’(y)
exP[G ~ ds A~(Y(s))~(s)].
(12)
This is an abelian holonomy, since the linearized connections A’ are U(1)-valued. Moreover, t’(y) is invariant under transformations generated by scalar and vector constraints, so in contrast to the original Ttt variable, t’ is a physical observable. We do not have to choose a variable corresponding to the T’ variables of ref. [3], since h~1~ itself is already invariant under transformations generated by the Gauss law constraint. The principal Poisson bracket relation in terms of these fields reads {hah(X),
t’( y)}
=
—
2iG(f’ds li3( ~
—
y(s))~lib)1)t1(y),
(13)
where no summation over i is implied in the right-hand side. For notational simplicity we define Fa(x, y)
=
ftds li3(x
—
y(s))~(s),
(14)
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Gratitons in loop quantum grar’ity
and its Fourier transform, Fa(k,
~)
1 =
=
(2w)
3/2
(2~)3/2
fdx etkxFa(x, y)
f’ds elk
Y(s)~a(S)
(15)
The field Fa(k, y) is called the form factor of the loop ‘y. Now we must define the quantum loop representation for the linearized theory. As mentioned before, the loop space for this representation consists of loop triplets i’~E ~ where the Z means that loops in this space must be identified if they define the same U(1) holonomy. The space of functionals on this loop space will be called .9°.Note the distinction with the original loop representation of ref. [3], where loops belong to the holonomic loop space of SU(2) (or SO(3)). Before we can define the quantum operators corresponding to the classical variables, we must first introduce a new notation. We define a union of a loop triplet i~ and a loop singlet y as follows: 1U1y=(nUy,172,173)
(16)
and correspondingly we define U 2 and U Also we define the composition ~ o~y in the same way. The difference between union and composition is that for two single loops 11 and y the union 17 U y consists of two loops, while the composition 17 o y is again a single loop (and only defined unambiguously if 17 and y intersect once). In the U(1) holonomic space they are identified, in the SO(3) holonomic space, however, they are not. The correct definitions for the quantum operators I’ and fr’~appear to be ~.
(I’(y)~)(i~) (~ah(k)~,)(~)
U
1y),
=
~i(i~
=
2hGF(a(k, 171)li”~’~(1).
(17) (18)
A straightforward calculation shows that these operators satisfy the right commutation relation: [~ah(k)
I’(y)]
=
2hGF(a(k, y)li’~1’(y).
(19)
We will define an extra operator in the quantum representation, which was not defined in ref. [61. It is the operator ê~’,corresponding to the classical linearized dreibein e’. Note that this field is not invariant under transformations generated by the Gauss-law constraint. This certainly violates the spirit of the original papers on the loop representation, where it was emphasized that the new basic phase-space
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293
variables (specifically, the T-algebra) were invariant with respect to transformations generated by this constraint on the classical level. However, we will only use the definition of ê~’to show equivalence of certain operators in the linearized and the full loop representation, not to define physical observables; for the latter its gauge invariant part I”~is sufficient. The quantum action of ê,” is defined as (ê~(k)~)(~) hGF”(k, =
(20)
17~)~1(T~)
from which the classical eq. (5) is directly reproduced as a quantum operator equation: /ah
=
2e~~~8h)I.
(21)
As a following step the space is defined as the space of vector fields F”(k) in momentum space which satisfy the reality condition (for the Fourier transform) Fa(k)=Fa(~~k)and which are transversal: Fa(k)ka 0. Since all form factors belong to functionals x on determine a unique functional ~/i on via .~
=
~
.~
~
(22) where F”(k, ~ is considered to be an element of by defining (F”(k, ii), Fa(k, 17~). ~i is called a regular loop state if it is derived from a functional x admitting a convergent Taylor expansion on It is proved in ref. [6] that the map from these functionals x to regular loop states is bijective, which means that it is allowed to switch back and forth unambiguously between the subset of regular states in the loop representation and the representation of convergent functional series on The physical states are precisely those regular loop states ~1(i which ~
=
.~.
~
can be written in the form
~),
~/J(T~) =~(F~(k,
F~ and F
F(k,
n)).
are the tracefree transverse components of F”(k, ~): 1(k, 17’), F(k, ii) ~a~iF” F(k, ii) =mam ~F~(k, 17’).
(23)
=
(24)
The set of all physical states is denoted by ~ On this physical subspace an inner product can be defined; it is essentially determined by the physical reality conditions. A graviton Fock space based on this inner product can be constructed with creation operators aL(k), a11(k) and annihilation operators a ~(k), a _(k), which are defined in terms of h”~and I’(y) using an appropriate limiting procedure. The normal ordered hamiltonian operator reads i~=fd3k h 1k [a~(k)a+(k)
+
a~(k)a(k)].
(25)
294
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~s
0
/
Gravitons in loop quantum gravity
is defined, and a general one-graviton state can be
written in the Fock space as t If)
=
(k)) 0)
(26)
ds)~)(~),
(27)
(fd3kf±(k)a~(k)+f(k)a
or as a loop functional:
=
~
where L 1(x) is the Fourier transform of the “wave function” (in momentum space) fai(k)
=
~
-f~(k)~a(k)rnj(k)).
(28)
For an n-graviton state, f will depend on n variables k1,. .., k,, and carry 2n indices a1 a,,, i1 i,,. As described in ref. [6], this representation contains divergences which can be solved by introducing a regularized form factor F”(k, y). The dependence on r cancels in any calculation of a physical quantity, essentially because r does not appear in the graviton Fock space, where the inner product is defined. . . .
. . .
3. Embedding in the full quantum theory In the previous section we have seen that it is possible to write down a brand-new loop representation for the linearized theory of quantum general relativity, using the linearized Ashtekar variables. But, a priori one does not know if this loop representation has anything to do with the original loop representation of Rovelli and Smolin, though there are many similarities, both in the spirit and in the technical details of the papers. The results of the linearized loop representation could be compared to the usual theory for gravitons on a flat background and led to the expected graviton Fock space. If we could now show that both loop representations coincide in the relevant region, i.e. if we could find a procedure to “linearize” the full loop representation and uncover the linearized loop representation again, this would tell us more about the validity of the loop representation for quantum gravity. It would imply that at least a part of this loop representation has a physically correct and well-understood meaning, and moreover, it might help us in interpreting the loop representation in other regimes, for instance on other backgrounds or in incorporating non-perturbative effects. This linearization of the full loop theory is the subject of this and following sections, and we will show that indeed the linearized loop representation can be seen as a very special case of the
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full loop theory. This not only gives us confidence in the validity of the loop theory, it also gives us some clues for the interpretation of the physical states and for the identification of the physical operators and the inner product, at least on the linearized sector. Denote the solution set of the loop representation in ref. [31by ~. Let us investigate which conditions the (at this point fictitious) map E from the set of “exact” loop functionals 4 ~ into the set of linearized loop functionals 1/5 EY’~’ has to fulfill. In the first place, E has to be a linear map: if 1/~and 1/2E~’are ~‘
mapped by E on
i4s~ and
1/’2’
respectively, then we expect 1/
+
‘/~2
tobe mapped on
+ 1/’2~ Furthermore, we must show that E maps an operator Q on ~ to the corresponding operator P on S~~0h;“corresponding” means that the classical variable P is exactly the linearization of the classical variable Q We thus want to have ~.
(29)
EQ=PE
on the elements of describing gravitons on a flat background. For instance, the classical Ashtekar dreibein is decomposed in the linearized theory as follows: ~‘
(30) so the inverse densitized three metric can be written as (det q)q”
Qab
=
E,”E”
=
li” + 2e~’li~’ + e,aeL~~.
(31)
After quantization and using the notation 2e~’~ h~~h the following equation must hold: =
EQab
=
(li” +
+
e~ehl)E.
(32)
Two other examples involve the loop variable of the linearized theory, t’(y). This variable is, on the classical level, the linearization of the loop variable T°(y); the latter can be expanded: T°(y)
Here
Tr P exp Gç6A
=
3
+ Tr(Gri~A1+ ~G2T1TJ(~A1
)
3). (~A1)) +
(33)
0(
T 1
*
=
and
stand for the generators of the S0(3) algebra, which are traceless
In fact, the word linearization is here somewhat misleading since we will do the calculations up to second order.
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and normalized according to Tr(r1r~) li,1, and E denotes the order of magnitude of the linear perturbations. We can express this result in abelian holonomies: =
3)
T°(y)= ~~(ex~ =
t’
~(t’(y)
~
G~A’+exp G~(_A’)) + t’(y~)) +
+ 0(E
(34)
0(~).
is also proportional to the linearization of the T’ variable of ref. [3], since T~(y)(s)
=
Tr[(P exp Gf’~’A~(y(t))~(t)dt)Ea(y(s))] +
=
e~(y(s)))(t’(y)
-
‘))
t’(y
+ 0(E~).
(35)
So, in the quantum theory we will have to prove the equations E~°(y)
=
ET~(y)(s)
=
(~
~(I’(y) +
+
I’(y ‘)))E,
e~(y(s))][I’(y)
-
(36) Il(y1)]
E.
(37)
In constructing the embedding E we encounter the technical obstruction that loop functionals in the linearized representation are functionals of loop triplets ~q, while the original loop representation of the full theory acts on singkts of (multiple) loops only. To make the construction of E possible we will propose a new representation of the full theory, which is suggested by the form of eq. (36). The loop functionals 1/ will become functionals of three independent multiple loops, and operators P on this representation will be defined as follows: (/~1~)(Tl)
=
E(I~)(ii),
(38)
where J~,. is the original operator acting on the functional defined by considering only the dependence on the ith loop, not on the other two. In calculating this action three terms appear, in each of which two loops are taken to be fixed and the third one to be variable. To be more specific, take for instance the operator T°(li). Its definition on the “usual” loop functionals is as follows: 7~°(li)ç~(y) =q~(yUli).
(39)
The corresponding definition on loop triplet-functionals is (t0(li)4)(a)
=
E~(~U 1li),
(40)
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where / E In appendix A we show that this “extended” loop representation essentially leads to the same results as the original one, which is based on functionals of singlets of multiple loops. We have to make an assumption about the way the operator E acts on We will introduce some fixed “background loop” A and propose that E acts as follows: .~‘.
.~‘.
(E~)(’q) =cb(A
(41)
i~).
°
The operator a connects the corresponding components to another, so ~ is connected to ~ etc. Eventually this equation will have to be defined by using a certain regularization to spread ~ over a distance big enough to ensure that always “hits” A,. At this point however we will assume that the right-hand side is defined by introducing line segments p1, which are straight (unknotted) lines 7L• If we want eq. (32) to be fulfilled we must have the equality connecting A,. and ~,
(E~”~)(~) which for ~
=
+ ê~~êh~) E~}(n),
+
=
(42)
0, the “trivial” loop triplet, leads to the equation Qab~(
A)
=
li*~~4( A).
(43)
We have used the fact that h”t/i(O) ê~’t/i(0) 0. Using the definition of Q” (see ref. [8] and appendix A) extended to functionals of loop triplets we get the formal expression =
~ah(x)~(
A)
=
=
(hG)2[fds
~ [fdt
li3(x
li3(x
-
-
A 1(t))A~(t)]~(A)~
(44)
where summation over i is implicit. Since the right-hand side, being a product of delta functions, is not well defined, we introduce a regularization for each term alone instead of for their product. Therefore we seek a “loop” A satisfying 3(x
—
At(s) )A~(s)”
=
8~,
(45)
“hGfds li where quotation marks denote the necessity of smearing with a some function over a sufficiently large scale. The freedom in choosing a dreibein (in this case 8,”) introduces a freedom in the choice of the precise loop components; this freedom corresponds to local rotations of A. We now see that is it convenient to choose A such that A~a 8,”, constraining A, to be a set of straight lines in the x, direction in
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Gravitons in ioop quantum grality
M. Of course, this A, does not consist of closed loops for an infinite manifold M, but this can be solved easily by compactifying M, for instance from ~ to S3. Suppose further that these lines pass the plane x, 0 through a number of points, arranged in an infinite square lattice with distance d between neighboring lattice points (this was suggested by Rovelli [7]). We will take M ~ because this is the usual configuration space for gravitons. The set of loops A 1 can be parametrized by a continuous parameter s El ~, +~[ and two discrete parameters n1, n2 E as follows: =
=
—
A’~1”2(s)
=
(s, n1d, n,d).
(46)
Integrating the formal expression (45) over a test function f leads to hGfdsf(A~(s))A’(s) Choose now, for instance, i 1 (i use the parametrization to find =
=
(47)
8~fd3xf(x).
2, 3 would lead to the same final result) and
=
3xf(x).
~
fdsf(s, n1d, n2d)
fl,,fl
=8~fd
(48)
lvi
2
Assume that the test function f varies slowly on scales smaller than Planck’s length, so that the summation on the left-hand side can be approximated by the corresponding integration. The equation then becomes hGfds dt duf(s, td, ud)
=
hGfds dt du
f(s, 2t, u) d
=
fMd~~),(49)
which immediately gives the equality
=~
d=v~~=1~,
(50)
where ~ denotes Planck’s length (c 1). So if we define E as explained above, with the loop A defined as three perpendicular sets of straight lines, intersecting the transversal planes in square lattices with lattice distance Ii,, then the flat background is reproduced in the full theory in the right way, through the operator ~ah and using the appropriate regularization. However, it remains to be shown that this is also true for the fluctuations on the background; in fact, we will perform the calculation shortly. It should be noted that we could have chosen, instead of 8,”, another “root” of =
in eq. (45). This would have led to the same A, but rotated over a certain angle in loop space. So the orientation of A cannot have any influence on the physical 8ah
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results of the calculations. Moreover, after deforming A by means of a diffeomorphism the calculation gives a transformed, but still flat geometry, as was explained for a general loop in ref. [8] (the calculation in this article survives the regularization). In particular, the physical lattice distance d is invariant under diffeomorphisms acting on A.
4. Linearization of operators Since we have a candidate for a mapping E relating the full- and linearized loop representation, it becomes interesting to find out which operators and which states in those two representations are related to each other. We will investigate corresponding states in sect. 5; here we will check if the variables T°,T” and Qab can indeed be linearized correctly in the quantum theory, using this mapping E. As was explained in sect. 3, “correctly” means that the quantization of a variable is related to the quantization of its linearized version. Starting with the operator T1~, we can write (E~0(y)~)(ii)
=
(f°(y)~)(A
=
L~((A
=
E ~[~(A
=
°
i~)
o
°
U~y)
(~a
y)) + ~(A
a
(~ o
[(E~)(~a 1y)+(E~)(~o1y1)I
=
+ I’(y~))Ecb](ii).
~[(I’(y)
(51)
We implicitly introduced line segments to be able to define ~ o~ y; they cancel in 11 the linearizedtoloop representation. corresponds the right operator onThe derivation shows straightforwardly that T Considering the second variable, T”, we can calculate (using the notation of ref.
[3]) (Eta(y)(S)~)(~)
=
fa(y)(5)~(A
=
~hG~”(y,
a
A~
°
17~)(s)[~(Aa
ii
o~y)—~(Aa
~
a~y~)]
(52)
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where hG~i”(y,A~a
hG~”(y, A~)(s)+hG~”(y, ?7~)(S)
?7~)(S)
83(y(s)
~hGfdt
=
-
A~(t))A~(t)
+~hGfdt 83(y(s)
17~(t))~(t)
hGF”(y(s),
+
=
(53)
17~)),
again using the regularization we described earlier. With this result we can write (E(y)(s)~)(~)
=
+
hGFa(y(s), 171))[(E~)(~U ~y)
=
+
hGFa(y(s), 17~))[(t’(y)
+
=
e~(y(s))}[i’(y)
-
-
-
(E~)(~U
i’(y’))E~J(1)
1’(y’)] E~)(i),
(54)
showing the correct transformation of 7” using the map E. Note that the correct factor ordering is important; the equality would not be correct if the order of e,~’(y(s))and I’(y) or I’(y~)had been changed. As a third example, we will also show the transferability of the operator Q’°~’ between S~’°” and To accomplish that we have to check eq. (42) on loops i~ which are not necessarily zero. Expanding this equation we get ~.
(E~ah(x)~)(~) ~ab(~)~(A
a
=
(8~+ hGF”(x, ?7’))(8~+ hGFb(x, 17~))~(Ao 2Fa(x, + 2hGF(”(x, 17’)8~~ + (hG) 17~)Fb(x,17’))~(A =
=
(8ah
*
~).
(55)
This result must be compared with the right-hand side of eq. (42), which yields (liab
+ h~’(X) +
ê~êb1(x))E4j(~) =
(liab
+
2hGF(”(x, 17’)87)
2F”(x, +(hG)
17~)Fb(x,17’))~(A a ~),
(56)
the same result as for the full theory. As a consequence the operator for the three metric, Qab is correctly transferable between the representations ~ and ,9”r’~’• The loop A can be seen as a background loop for flat minkowskian space; its importance in the loop theory is, that it establishes the transformation from the topological knot relations of the full theory of quantum gravity to some kind of
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301
metrical relations in the linearized theory. For instance, using this background one can make a distinction between two loops y1 and Y2 in loop space if their distance in the direction perpendicular to A, is larger than Planck’s length li,, by using the fact that A ° ‘ and A ° ~ belong to two different knot classes. This means that a loop state 1/’ in the linearized loop theory will in general discriminate between them.
5. Graviton states in the loop representation Before we can interpret E as a map of a subset of in ,/.~‘, and thus its inverse of an embedding of 5”°~’in .4~,we must first check if the image of E consists of loop functionals satisfying the diffeomorphism constraint, i.e. loop functionals that only depend on know classes (see ref. [31).More concretely, we have to take a general graviton state and show that the corresponding full loop state, which one gets by applying E I to it, can be seen as a functional on smooth knot classes. To ensure that E ‘ is well defined, we will restrict E to states f E supported only on loops a which are of the form a A o ~ or which can be transformed in this form by a diffeomorphism on M. On this subset E is indeed an injective mapping. Only these states will correspond to the graviton states of the linearized theory. Considering the equation .~
—
~‘
=
(E4)(i1) =~(A
°
ii),
(57)
and taking into account that we would like 4 to be a functional on knot classes, we conclude that only those states 1/’ E.~9”~’can be embedded in the full theory that satisfy the equation I/f(Dr’ll)=l/J(ll),
(58)
where Dr is a diffeomorphism on loop space; it leaves the background A invariant. So Dr keeps the intersection points of A, and the 1oops ~, fixed, and it shifts ‘17, in such a way that it does not cross the fixed background A,; Dr will be called a “restricted diffeomorphism”. One might expect any graviton state to satisfy this equation, because these states are solutions to the linearized diffeomorphism constraint. However, since the linearized constraint neglects term of order ~2, the full loop states corresponding to the graviton states will appear to satisfy the full diffeomorphism constraint only up to linear order; the graviton functionals are thus mapped into functionals closely approximating diffeomorphism-invariant states in the full representation. A physical (regular) loop state in the linearized theory is a loop functional derived from a graviton state. For simplicity we take a one-graviton state; n-gravi-
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ton states will essentially give the same results (with more variables):
=
(~ ff~
1(ms))~(s
ds)~o(i)~
(59)
which was already mentioned in sect. 2. This graviton state corresponds to the loop state satisfying the equation E4~ 1/it, or more explicitly, ~.,
=
4f(A
°
‘q) _llif(i~).
(60)
Because the vacuum functional 1/’~~ which appears in every graviton state, gives zero for all non-zero loops, we will simply renormalized it to one. One might see it as the contribution of the infinite background loop A. We deliberately do not introduce a regularization parameter r here, as in ref. [6], because it would spoil our graviton calculations, in which the delta functions in the Fourier-transformed form factors are already smeared out by the graviton wave functions, and therefore do not need further regularization. So, from now on we put 1/i~(ii) I for all E~ in the regular loop states. In fact, this also restores the symmetry between the two helicities, since the remaining part essentially consists of Hermite polynomials in both the F~and the F fields. We will show that -ff(~q)is in lowest order invariant under the action of restricted diffeomorphisms Dr on the loop triplet i~. This can be computed directly, using the fact that f is transverse, trace-free and symmetric. However, we choose an alternative approach, by showing that the full diffeomorphism constraint is mapped by E to the linearized one, so that is a solution to the first one if the corresponding I/If solves the latter. The full vector constraints reads, in terms of the T-variables: =
C~(x) ~ =
G~li2Ea~Tx,~O)~
(61)
where y~ is a loop in the plane perpendicular to the unit vector ê~starting in x with radius 8 (positively oriented with respect to e’). The derivation of this expression is similar to the derivation of the Hamilton constraint by Rovelli and Smolin [3]. It can be quantized directly, and then the corresponding linearized constraint can be derived using eq. (54):
~(
x)
=
lim
G~82 Eahe~8~[t’(y,’~)-
We only considered the linear order in
,
I’(( y,’~)~‘)].
(62)
since the linearized constraint was
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303
derived in ref. [6] by neglecting all higher orders. Rewriting this equation using loop derivatives we get ~(x)
=
(63)
~E,,iC8C~
here we also used that li/8y —li/liy, which follows from ~‘(y (I’(y)Y1 We reproduced the known expression for the linearized vector constraint in ref. [6], where is was denoted by ~(x). This result suffices to show diffeomorphism invariance of the full state corresponding to the graviton state ~ To see this, extend the definition of thf in eq. (60) to diffeomorphism classes, by defining cbf(a) t/If() if a E [A a ii] ~, for some finite loop triplet and zero otherwise. Now consider the diffeomorphism constraint: =
‘)=
=
i~,
0
=
=
~
=
(~E~f)()
+ O(E2)
(E~,,~f)()
=
(~a~f)(A
a
)+
0(E2),
(64)
showing that solves the vector constraint to linear order. This means that can be considered as an approximation to a full diffeomorphism-invariant state I f) E .~4’,which actually corresponds to the original one-graviton state. To be more concrete: to calculate /I(a), where a E [A a ], one should find a diffeomorphism D explicitly satisfying Da A a This D is defined up to restricted diffeomorphisms, resulting in different possible loops but the graviton functionals t/Fj are precisely such that the choice of this -q will only change terms of at least quadratic order in E in the final result. The graviton Fock space is therefore a valid description for a subset of the full loop states if the perturbations on the flat background are small; of course this is exactly what one expects. Besides, by assuming the test function in the regularization of F”(x, A) to be smooth on Planck scales we already restricted ourselves to the situation of small perturbations. It is possible to decompose I f) into a basis of knot classes: i~.
=
i~,
If)=
~
c[~]I[A
o
1]),
(65)
[A
where the state [A
a
~~])represents the loop functional if~e[Ao] ~O otherwise.
*
[A
i~]
is a 1oop triplet knot class, see appendix A.
‘1”[A.
~
satisfying (66)
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/ Gra,itons in
loop quantum gravity
(see e.g. ref. [8]). The value of the constants c1,~1is =
E
f L~(171(s))~9(s)ds,
(67)
which is in lowest order independent of the choice of an element from the restricted know class [11],, consisting of loop triplets related to by a restricted diffeomorphism. The state I f) is now, for some precise choice of all coefficients an exact state of quantum gravity, approximating the one-graviton functional The vacuum functional I/’~~ corresponds to the state 4~in defined by taking all coefficients c[1] equal to one. In other words, 411(a) is one if a E [A a i]~for some (finite) loop triplet and zero otherwise. A general n-graviton state is governed by a real function f~ a,,i, It satisfies the same conditions as the one-graviton function, hut in more indices, and it defines in exactly the same way a linear approximation to a full diffeomorphism-invariant loop state in quantum gravity. The inner product is well defined in the graviton Fock space: if I f~)) and I g~) are two one-graviton states, then their inner product is (in linear order in e) i~
.~
i~,
I gW)
= fd3k
fai(
—k)g”(k)
=
fd3x~,t(x)~”’(x).
(68)
We expect that in the exact (to all orders) inner product the integration will become a summation, since the exact solution set of the full loop representation has a countable basis. For the loop functionals t/i~ and t/Ig corresponding to these kets we have
K~fI~g)fd3x~~(x)ga~(x).
(69)
=
For two n-graviton states this becomes (f(fl)
I g(~I))
=
3x,,
~
“‘“
fd3x1 ...d
(70) Furthermore, different graviton sectors are orthogonal, which means that we have in general Kf~”~g~~~)) a 8mm. On the linear sector of quantum loop gravity the inner product is thus well defined. However, we are interested in finding an inner product for the full theory. This inner product should be defined directly on diffeomorphism invariant loop functionals, and should correspond to the abovementioned inner product on the graviton sector of the theory. Despite serious efforts we have still not succeeded in finding such an inner product.
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6. Conclusions The main goal of this paper was to find the relation between the loop representation for linearized quantum gravity introduced in ref. [6], and the representation for the full theory in ref. [31.The most important result of this paper is that we have achieved this aim by finding an embedding from the linearized solutions into a subset of the full solutions. Furthermore, we showed that by using this embedding we could transfer a number of operators from one representation into the other. This could help us in finding and interpreting the physical observables in the full theory, because we know how to interpret a number of physical observables in the graviton theory. For the same reason we can identify a subset of full loop solutions as gravitons moving on a flat background. Therefore we have gained some physical intuition for the meaning of the physical states in the loop representation. We have also found a background loop which is closely connected to the flat background we chose. There are still some open questions. Probably the most important one is that we still do not have an inner product on the full loop representation. The inner product we proposed in ref. [8] seems not to correspond to the inner product on the graviton sector. We have not been able to find an inner product that does. The correct inner product will have to deal with the fact that the space of full loop states has a countable basis. Also, in this paper we did not consider nonperturbative effects in loop quantum gravity, although these must be an essential part of the full theory. It might be interesting to consider other background metrics and to perturb canonical quantum gravity around these new backgrounds. One would then expect that a different background leads to a different loop A and thus to a different embedding E. However, every background should lead to a theory which can be seen as a subtheory of the full theory of quantum gravity. And this full theory should not only consist of the union of these subtheories, but should also contain essentially non-perturbative information, allowing for instance background changing. The linearized theories might help in developing physical intuition in interpreting operators and states in quantum gravity, and finally they should give us information about physics on the Planck scale. I would like to thank G.
‘t
Hooft for stimulating discussions.
Note added After completion of this work a review article by Rovelli appeared [9], in which research on the linearization of the loop representation analogous to ours is described. Besides the extra attention Rovelli pays to the regularization there are some differences in the precise calculations, leading to a definition of the mapping corresponding to our E which is slightly different from the one in eq. (41).
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Appendix A
LOOP TRIPLET REPRESENTATION
A peculiar feature of the linearized loop representation is that it involves functionals in triplets of multiple 1oops. To construct the embedding of the linearized representation into the full one it appears to be convenient to define the full representation by functionals on loop triplets too. In this appendix we will find out what the physical operators and states of this extended representation are and show that we can essentially copy the results of the loop singlet representation; furthermore we will show that functionals of loop triplets has the important advantage as compared to the original functionals that the operator Q”” involving the (densitized) three metric is defined more consistently. Let us denote the space of loop singlets by 2”, of loop triplets by 2’s. We do not specify the holonomy, which would force us to make certain identifications in these spaces. We must first check if operators on 2’~ satisfy the same commutation relations as the corresponding operators on 2~I. First remember that an operator P on 2’~ is defined by P= ~
(A.1)
where P, is the loop operator which acts precisely the same as P but only on the ith loop, with the other two loops fixed. For instance, if P acts like (P~)(a) =F(a)I/i(a’), (F is some functional of loops) then
P, is defined by
(P,~)(a)=F(a,)I/i(a~, a2, as), and similarly
(A.2)
P2
and P3. Because the three ments, we will certainly have [i~, It follows that for three operators P, [i~,
Q1]
P, operators act on different argu-
a~
Q and
(A.3)
(A.4)
R on 2’s, which satisfy
~j =k,
(A.5)
one has the following relation as well: (A.6)
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with R = E,R,. The commutation relations will therefore not change in the new representation. Which functionals on 2’~ can be identified as the physical states? Take a constraint operator C, which must now also be decomposed into C,. A physical state has to satisfy, for every loop triplet a
E (~~1/)(a)= 0.
(A.7)
Applying this to vector and scalar constraint implies that physical states are functionals 4(ot) which have support on smooth loop triplets a and are invariant on loop triplet knot classes [a]: /I(a) =q~(Da).
(A.8)
Here D y in general stands for the loop satisfying (D ~)(s) = D(y(s)). This means for instance that ~(a,, a2, a3) and 1/(D a~,D a2, D a3) should yield the same value, but /I(D a~,a2, a3) should in general not. The original knot classes of Rovelli and Smolin have now become diffeomorphism classes of loop triplets, which, however, are still knot classes since D does not change the way the loop components a, are knotted into another; they transform simultaneously. As compared to the original ones the new knot classes contain extra information in their splitting into three loop components. This extension of loop functionals has an important advantage in the calculation of expectation values of the operator Qab = E,”E~’’. In ref. [8] the unregulated action was calculated (h and G are now included): 2Fa(x, y)Fb(x, y)~(y). (A.9) = (hG) where the notation of this article has been used. On the right-hand side we have the tensor product of a vector with itself; it is clear that this severely restricts the possible values. For instance, no vector v exists such that ~ = 8ab since the determinant of the product is always zero. On functionals of loop triplets the formula becomes =
(E(hG)2Fa(x,
aj)Fh(x,
ai))~(a).
(A.10)
The set of possible outcomes is now not a priori restricted, since to any tensor field we can connect a dreibein such that L’1”
tab = ~
(A.11)
So in this representation any reasonable value for the expectation value of
Q””
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Gravitons in loop quantum gravity
(modulo the regularization of F) can be obtained: for instance, in sect. 3 we found a loop A such that Q””ç6(A) = 8”~/I(A),using an appropriate regularization.
References 111
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