Fock representations from U(1) holonomy algebras

30 Pages·0.26 MB·

by  Madhavan Varadarajan

Fock representations from holonomy algebras
U(1)
Madhavan Varadarajan
Raman Research Institute, Bangalore 560 080, India.
[email protected]
November 1999
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ABSTRACT

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We revisit the quantization of U(1) holonomy algebras using the abelian C∗ algebra
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J based techniques which form the mathematical underpinnings of current efforts to

9 construct loop quantum gravity. In particular, we clarify the role of “smeared
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loops” and of Poincare invariance in the construction of Fock representations of

1 these algebras. This enables us to critically re-examine early pioneering efforts to
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construct Fock space representations of linearised gravity and free Maxwell theory
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5 from holonomy algebras through an application of the (then current) techniques of
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loop quantum gravity.
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1. Introduction
In the early nineties [1, 2, 3] linearised gravity in terms of connection variables and
free Maxwell theory on flat spacetime, were treated as useful toy models on which
to test techniques being developed for loop quantum gravity[4] . Significant progress
has been made in the field of loop quantum gravity since then[5]. Hence, it is useful
to revisit these systems using current techniques to clarify certain questions which
arise in the context of those pioneering but necessarily non-rigorous efforts.
Two important (and related) questions are:
(I) How did similar techniques for the quantization of general relativity and for its
linearization about flat space, result in a non Fock representation for the (kinematic
sector) of the former and a Fock representation for the latter? In particular, what is
the role of Poincare invariance in obtaining the Fock representation? (This last point
was a puzzle to the authors themselves [1]).
(II)What is the role of “smeared” loops in [1] in obtaining a Fock representation?
In this work, we use the abelian C∗ algebra techniques [6, 8] which constitute the
mathematically rigorous framework of the loop quantum gravity program today, to
investigate (I) and (II) above. It is also our aim to clarify the role of the different
mathematical structures in the quantization procedure which determine whether a
Fock or non Fock representation results. Although we restrict attention to U(1)
theory on a flat spacetime, we believe that our results should be of some relevance to
the case of linearised gravity.
This work is motivated by the following question in loop quantum gravity: how do
Fockspacegravitonsonflatspacetimearisefromthenon-FockstructureoftheHilbert
space which serves as the kinematical arena for loop quantum gravity? Admittedely,
the answer to this question must await the construction of the full physical state
space (i.e. the kernel of all the constraints) of quantum gravity. Nevertheless, this
work may illuminate some facets of the issues involved.
The starting point for our analysis is the abelian Poisson bracket algebra of U(1)
holonomies around loops on a spatial slice. This algebra is completed to the abelian
1
C∗ algebra, of [6, 8]. Hilbert space representations of are in determined by
HA HA
continuous positive linear functions (PLFs) on . We review the construction of
HA
and of the PLF introduced in [6, 8] (which we shall call the Haar PLF) in section
HA
2. The resulting representation is a non Fock representation in which the Electric
flux is quantized [7].
Insection 3weconstruct anabelianC∗ algebra , basedonthePoissonbracket
r
HA
algebra of holonomies around the “Gaussian smeared ” loops of [1].1 Next, we derive
the key result of this work, namely that there exists a natural C∗ algebraic isomor-
phism, I : with the property that I ( ) = .
r r r r
HA → HA HA HA
The standard flat spacetime Fock vacuum expectation value restricts to a positive
linear function on . We are unable to show the continuity or lack thereof, of this
r
HA
Fock PLF on . Nevertheless, since the GNS construction needs only a * algebra
r
HA
(as opposed to a C∗ algebra), we can use the Fock PLF to construct a representation
of the * algebra . In section 4 we show that this representation is indeed the
r
HA
standardFockrepresentation even though isapropersubalgebra ofthestandard
r
HA
Weyl algebra for U(1) theory.
Using the map, I , we can define a Haar PLF on . We construct the resulting
r r
HA
representation in section 5a. Finally, we use I to define a Fock PLF on . The
r
HA
resulting representation is, in a precise sense, an approximation to the standard Fock
representation. We study it in section 5b.
Section 6 is devoted to a discussion of our results in the context of the questions
(I) and (II). Some useful lemmas are proved in Appendices A1 and A2.
In this work the spacetime of interest is flat R4 and we use global cartesian co-
ordinates (t,xi), i = 1,2,3. The spatial slice of interest is the initial t = 0 slice
and all calculations are done in the spatial cartesian coordinate chart (xi). We use
units in which both the velocity of light and Plancks constant, h¯, are equal to 1.
We freely raise and lower indices with the flat spatial metric. The Poisson bracket
between the U(1) connection A (~x),a = 1,2,3 and its conjugate electric field Eb(~y)
a
is A (~x),Eb(~y) = eδbδ(~x,~y) where e is a constant with units of electric charge.
{ a } a
1 r is a small length which characterises the width of the Gaussian smearing function in [1].
2
2. Review of the construction and representation
theory of .
HA
We quickly review the relevant contents of [6, 8]. We refer the reader to [6, 8],
especially appendix A2 of [8] for details.
The mathematical structures of interest are as follows.
is the space of smooth U(1) connections on the trivial U(1) bundle on R3.2 We
A
restrict attention to connections A (x) whose cartesian components are functions of
a
rapid decrease at infinity.
is the space of unparametrized, oriented, piecwise analytic loops 3 on R3 with
Lx0
basepoint x~ . Composition of a loop α with a loop β is denoted by α β. Given a
0
◦
loop α , the holonomy of A (x) around α is H (A) := exp(i A dxa).
∈ Lx0 a α α a
H
α˜ is the holonomy equivalence class (hoop class) of α i.e. α,β define the same
hoop iff H (A) = H (A) for every A (x) .
α β a
∈ A
is the group generated by all hoops α˜, where group multiplication is hoop
HG
˜
composition i.e. α˜ β := α β.
◦ ◦
g
is the abelian Poisson bracket algebra of U(1) holonomies.
HA
is the free algebra generated by elements of , with product law αβ :=
FLx0 Lx0
α β. With this product, all elements of are expressible as complex linear
◦ FLx0
combinations of elements of .
Lx0
K is a 2 sided ideal of , such that
FLx0
N N
a α K iff a H (A) = 0 for every A (x) , (1)
X i i ∈ X i αi a ∈ A
i=1 i=1
where a are complex numbers.
i
is quotiented by K to give the algebra /K. The K equivalence class
FLx0 FLx0
of α is denoted by [α]. As abstract algebras, and /K are isomorphic.
HA FLx0
N N
( a [α ])∗ := a∗[α−1] (2)
i i i i
X X
i=1 i=1
2Thus a minor changeof notation fromA2 of [8] is that we denote ofthat reference by by .
0
A A
3This is in contrast to the C1 loops of A2 of [8].
3
defines a relation on .
∗ HA
N N
a [α ] := sup a H (A) (3)
||Xi=1 i i || A∈A|Xi=1 i αi |
defines a norm on . is the abelian C∗ algebra obtained by defining on
HA HA ∗ HA
and completing the resulting algebra with respect to .
∗ || ||
∆ is the spectrum of . ∆ is also denoted by / where denotes the U(1)
HA A G G
gauge group and is a suitable completion of the space of connections modulo gauge,
/ . From Gel’fand theory, ∆ is the space of continuous, linear, multiplicative
A G ∗
homeomorphisms, h, from to the (C∗ algebra of) complex numbers C. From [8]
HA
the elements of ∆ are also in 1-1 correspondence with homeomorphisms from to
HG
U(1).
Given X , h(X) is a complex function on ∆. ∆ is endowed with the weakest
∈ HA
topology in which h(X) for all X are continuous functions on ∆. In this
∈ HA
topology,∆isacompact, Hausdorff spaceandthefunctions h([α]), α aredense
∈ Lx0
in the C∗ algebra, C(∆), of continuous functions on ∆. Further, C(∆) is isomorphic
to . Every continuous cyclic representation of is in 1-1 correspondence with
HA HA
a continuous positive linear functional (PLF) on . Since = C(∆), every
∼
HA HA
continuous PLF so defined on C(∆) is in correspondence, by the Riesz lemma, with
some regular measure dµ on ∆ and Hˆ is represented on ψ L2(∆,dµ) as unitary
α
∈
operator through (Hˆ ψ)(h) = h([α])ψ(h).
α
In particular, the continuous ‘Haar’ PLF [8]
Γ(α) = 1 if α˜ = o˜
= 0 otherwise (4)
(where o is the trivial loop), corresponds to the Haar measure on ∆.
∆ = / can also be constructed as the projective limit space [9] of certain
A G
finite dimensional spaces. Each of these spaces is isomorphic to n copies of U(1) and
is labelled by n strongly independent hoops. Recall from [8] that α˜ i = 1..n are
i
strongly independent hoops iff α are strongly independent loops; α , i = 1..n
i ∈ Lx0 i
4
are strongly independent loops iff each α has at least one segment which intersects
i
α at most at a finite number of points. The Haar measure on ∆ is the projective
j6=i
limit measure of the Haar measures on each of the finite dimensional spaces. 4 Then
the considerations of [10] show that the electric flux Eads through a surface S can
S a
R
be realised as an essentially self adjoint operator on the dense domain of cylindrical
functions 5 as
∂ψ
Eˆads ψ = e N(S,α )h([α ]) {[αi]} (5)
ZS a {[αi]} Xi i i ∂h([αi])
where N(S,α ) is the number of intersections between α and S.
i i
3. and the isomorphism I
r r
HA
In section 3a we recall the definition of ‘smeared’ loops and their holonomies from
[1] and construct the ‘smeared’ loop related structures α˜ , K , , and ∆ .
r r r r r
HA HA
In section 3b, using Appendix A2, we show that an isomorphism exists between the
structures α˜, K, , ,∆ and their ‘smeared’ versions.
HA HA
3a. The construction of
r
HA
In the notation of [1],
H (A) = expi Xa(~x)A (~x)d3x, (6)
α Z γ a
R3
Xa(~x) := dsδ3(~γ(s),~x)γ˙a, (7)
γ I
γ
where s is a parametrization of the loop γ, s [0,2π]. Xa(~x) is called the form factor
∈ γ
of γ. Its Fourier transform is
1
Xa(~k) := d3xXa(~x)e−i~k·~x
γ 2π23 ZR3 γ
4 Note that the proof of continuity of the Haar PLF in [8] is incomplete in that it applies only if
the loopsα ofA.7 of[8]areholonomicallyindependent. Nevertheless,ifasinthis work,werestrict
j
attention to piecewise analytic loops, continuity of the Haar PLF can immediately be inferred from
its definition through the Haar measure.
5Cylindrical functions on ∆ are of the form ψ :=ψ(h([α ])..h([α ])), where α ,i=1..n, are
{[αi]} 1 n i
a finite number of strongly indendent loops and ψ is a complex function on U(1)n.
5
1
= dsγ˙a(s)e−i~k·~γ(s). (8)
2π23 Iγ
The Gaussian smeared form factor [1] is defined as
Xa (~x) := d3yf (~y ~x)Xa(~y) = dsf (~γ(s) ~x)γ˙a(s) (9)
γ(r) ZR3 r − γ Iγ r −
where
1 −x2
fr(~x) = 2π3r3e2r2 x := |~x| (10)
2
approximates the Dirac delta function for small r. The Fourier transform of the
smeared form factor is
Xa (~k) = e−k22r2Xa(~k) (11)
γ γ
(r)
and the smeared holonomy is defined as
H (A) = expi Xa (~x)A (~x)d3x
γ(r) ZR3 γ(r) a
= expi Xa ( ~k)A (~k)d3k. (12)
ZR3 γ(r) − a
~
where A (k) is the Fourier transform of A (~x).
a a
We define α˜ , K , , ,∆ as follows.
r r r r r
HA HA
α˜ is the r-hoop class of α i.e. α,β define the same r-hoop iff H (A) = H (A)
r α β
(r) (r)
for every A (x) . is the group generated by all r-hoops α˜ where group
a r r
∈ A HG
multiplication is r-hoop composition i.e.
˜
α˜ β := (α β) . (13)
r r r
◦ ◦
g
Note that the above definition is consistent because, from (12) and the definition of
r-hoop equivalence, it follows that
H (A)H (A) = H (A) (14)
α(r) β(r) (α·β)(r)
Notethatfrom(13),itfollowsthattheidentityelement of iso˜ andthat(α˜ )−1 =
r r r
HG
α−1 .
r
g
is theabelian Poisson bracket algebra of ther-holonomies, H (A), A ,
r α a
HA (r) ∈ A
α .
∈ Lx0
6
Recall that with the product law defined in section 2, all elements of are
FLx0
expressible as complex linear combinations of elements of . We define the 2 sided
Lx0
ideal of K , through
r ∈ FLx0
N N
a α K iff a H (A) = 0 for every A (x) , (15)
X i i ∈ r X i αi(r) a ∈ A
i=1 i=1
where a are complex numbers. The K equivalence class of α is denoted by [α] . It
i r r
can be seen that, as abstract algebras, and /K are isomorphic.
HAr FLx0 r
It can be checked that the relation defined on by
r r
∗ HA
N N
( a [α ] )∗r := a∗[α−1] (16)
i i r i i r
X X
i=1 i=1
is a relation. Note that from (12), the complex conjugate of H (A) is H (A)
α α−1
∗ (r) (r)
and hence the abstract relation just encodes the operation of complex conjugation
r
∗
on the algebra .
r
HA
Next we define the norm as
r
|| ||
N N
a [α ] := sup a H (A) . (17)
||Xi=1 i i r||r A∈A|Xi=1 i αi(r) |
It is easily verified that is indeed a norm on the algebra with relation
r r
|| || ∗ HA ∗
defined by (16). Completion of with respect to gives theabelianC∗ algebra
r r
HA || ||
.
r
HA
Next, we characterize the spectrum ∆ of as the space of all homomorphisms
r r
HA
from to U(1).
r
HG
Let h ∆ . Thus h is a linear, multiplicative, continuous * homorphism from
r
∈
to C.
r
HA
h([α] )h([α−1] ) = h([o] ). (18)
r r r
⇒
Choosing α = o, h([o] )2 = h([o] ) h([o] ) = 1. (19)
r r r
⇒ ⇒
1
h([α−1] ) = = h∗([α] ). (20)
r r
⇒ h([α] )
r
7
(20) implies that h([α] ) = 1 and this, coupled with the fact that is commuta-
r r
| | HG
tive, shows that every h ∆ defines a homomorphism from to U(1).
r r
∈ HG
Conversely, let h be a homomorphism from to U(1). Its action can be ex-
r
HG
tended by linearity to elements of so that h( N a [α ] ) := N a h([α ] ). 6
HAr i=1 i i r i=1 i i r
P P
It is also easy to see that h([α−1] ) = h∗([α] ). These properties and the fact that h
r r
is a homomorphism from to U(1) C, imply that h is a linear, multiplicative,
r
HG ⊂
* homomorphism from to C.
r
HA
Finally we show that h extends to a continuous homomorphism on . From [6]
r
HA
it follows that for α , i = 1..n, there exist strongly independent β , j = 1..m
i ∈ Lx0 j
such that each α is the composition of some of the β . From this fact and Lemma
i j
{ }
2 of Appendix A1, it can be shown that, for a given N a [α ] and any
i=1 i i r ∈ HAr
P
δ > 0, there exists A(ai,δ,r) such that
a ∈ A
N
a (h([α ] ) H (A(ai,δ,r))) < δ. (21)
i i r α
|X − i(r) |
i=1
From (21), it is straightforward to show that
N N N
a (h([α ] ) sup a H (A) . = a [α ] . (22)
|Xi=1 i i r | ≤ A∈A|Xi=1 i αi(r) | ||Xi=1 i i r||r
Since is dense in , (22) implies that h can be extended to a continuous
r r
HA HA
(linear, multiplicative) homorphism from to C.
r
HA
Thus ∆ can be identified with the set of all homomorphisms from to U(1).
r r
HG
3b. The isomorphism I
r
We show that
(i) K = K : Let
r
N
a H (A) = 0 for every A (x) . (23)
X i αi a ∈ A
i=1
6h can be defined on [α] because K equivalence subsumes r-hoop equivalence.
r r
8
From Lemma 3 of Appendix A1, given A , there exists A such that
a a(r)
∈ A ∈ A
N N
a H (A) = a H (A ). (24)
i αi(r) i αi (r)
X X
i=1 i=1
(23) and (24) imply that K K .
r
⊂
Let
N
a H (A) = 0 for every A (x) . (25)
X i αi(r) a ∈ A
i=1
Given A ,B ,
a a
⇒ ∈ A
N N N
a H (A) = a H (A) a H (B) . (26)
|X i αi | |X i αi −X i αi(r) |
i=1 i=1 i=1
Choose, B = Aǫ where Aǫ is defined in Lemma 1, A1.
a a a
Then
N a H (A) N a ǫ for every ǫ > 0. N a H (A) = 0 and hence,
| i=1 i αi | ≤ i=1| i| ⇒ i=1 i αi
P P P
K K.
r
⊂
Thus K = K , [α] = [α] and α˜ = α˜
r r r
(ii) N a [α ] = N a [α ] :
|| i=1 i i || || i=1 i i r||r
P P
Let N a [α ] = c . Then c N a H (A) for every A . Further,
|| i=1 i i r||r r r ≥ | i=1 i αi(r) | a ∈ A
P P
for every τ > 0 there exists (τ)A such that c N a H ((τ)A) τ.Then,
a ∈ A r −| i=1 i αi(r) | ≤
P
from Lemma 3, A1, there exists (τ)A such that
a(r)
∈ A
N
0 c a H ((τ)A ) τ. (27)
≤ r −X i αi (r) ≤
i=1
N N N
sup a H (A) c a [α ] a [α ] (28)
⇒ A∈A|Xi=1 i αi | ≥ r ⇒ ||Xi=1 i i || ≥ ||Xi=1 i i r||r
Let N a [α ] = c. Then for every τ > 0 there exists (τ)A such that
|| i=1 i i || 2 a ∈ A
P
N τ
c−|XaiHαi((τ2)A)| ≤ 2. (29)
i=1
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