Quantum Spin Dynamics (QSD) II

T. Thiemann

REVIEW

Quantum Spin Dynamics (QSD) II

T. Thiemann

General Relativity and Quantum Cosmology · Submitted 29 Jun 1996 · 24 references

Pith's verdict: A careful, honest continuation paper: the kernel classification and the symmetry lemma are real work, the Euclidean self-adjointness proof is nearly complete, and the Lorentzian case is openly left as a program.

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arXiv IDgr-qc/9606090 (version 1)

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CitationT. Thiemann. "Quantum Spin Dynamics (QSD) II." arXiv:gr-qc/9606090 (1996). https://pith.science/paper/VISZGD2B

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Bottom line: this is the second QSD paper and it earns its place. The classification of the constraint kernel (Theorem 1.1, the Type I/II split with the derived spin-net construction) is careful combinatorics, and Lemma 3.1 is the good idea in the paper: vertices added by the constraint have co-linear incident tangents, so the volume operator kills them, and that is exactly what makes the modified loop assignment symmetric. Theorem 3.3's conjugation k = Pc is explicit and checkable, and the paper is candid about what it has not done. The remark that 0 belongs to the residual rather than the point spectrum, with the creation-operator analogy, is the right way to frame the empty-kernel problem and I think it is correct. On the stress-test: the specific worry that the segments s_i(Delta) might themselves be exceptional edges does not land. They are segments of analytic edges incident at a non-exceptional vertex, and Definition 3.2 requires both endpoints to be exceptional vertices, so P carries no factor for them. But there is a related bookkeeping question the paper does not address. The output f_pm is cylindrical on gamma' = gamma union a_ij(Delta), and a_ij is itself a smooth exceptional edge with spin 1/2, so the parity operator for gamma' carries an extra factor e^{2 pi i /2} = -1 that the displayed phase product in Theorem 3.3 omits. That flips the sign in P H^E = -H^E P and would break the commutator with k as written. It looks fixable by a phase convention or a redefinition of k, but it needs to be written out, since this proof is the template for the Lorentzian route. The larger soft spots are acknowledged in the text: Lorentzian self-adjointness is a Kato-Rellich program with the relative bound unchecked; the kernel of the symmetric operator is deferred to [21]; the claim that the non-symmetric operator has no nontrivial distributional solutions rests on the worked trivalent example, and the paper itself calls the co-planar kernel a conjecture. The smooth-exceptional-edge device leaves the analytic category, which the paper says it would rather avoid. And the volume operator matrix elements in [24] are load-bearing and still a preprint. Who should read it: anyone in the loop quantum gravity program, and anyone who wants to see what a rigorously defined constraint operator looks like. Send it out.

Referee report

Significance

If the construction holds, this is a substantial contribution: it is one of the few places where a Wheeler-DeWitt constraint operator for full Lorentzian gravity is defined on the whole Hilbert space and where spectral questions are posed in a mathematically disciplined way. The genuine strengths deserve naming. Theorem 3.3 is a real proof, not a sketch: the conjugation k = P c is written down explicitly in Eq. (3.4), its domain and boundedness are checked, cylindrical consistency of the family (P̂_γ) is verified on spin-network states, and the sign flip under j → j ± 1/2 is tracked term by term. Lemma 3.1 and Theorem 3.1 are clean and short. The paper is also unusually candid about what it has not done: it labels the co-planar kernel a conjecture, states that uniqueness of the self-adjoint extension is unknown, says the Kato-Rellich relative-bound analysis is 'beyond the scope of the paper,' and confesses that the smooth-exceptional-edge device is a cost it would prefer to avoid. That candour is a virtue and should be preserved in revision. The classification of Theorem 1.1 and the computational recipe of §2 give later work a concrete target, and the QED-Fock analogy of §4 is a he

Major objections (6)

[Abstract; §3.3 opening; §3.4] The abstract states that 'for the Euclidean Wheeler-DeWitt operator as well as for the generator of the Wick transform from the Euclidean to the Lorentzian regime we prove existence of self-adjoint extensions.' The Euclidean half is delivered by Theorem 3.3. The Wick-generator half is not. §3.3 opens by asserting that 'without changing formula [1],(4.1), K̂ is now automatically a symmetric operator and it has self-adjoint extensions,' with no conjugation constructed, no deficiency indices computed, and no proof in this paper; §3.4 then uses this as a premise ('Although Ĉ, Ĥ^E were shown to possess self-adjoint extensions'). Either supply the argument here — the conjugation k = P c of Theorem 3.3 is the natural candidate, since K̂ ∝ [V̂, Ĥ^E(1)] and V̂ preserves representations — or soften the abstract so that it claims a proof only for the Euclidean operator and a strategy for the Wick generator.
[§3.3.3, Eqs. (3.8)-(3.11)] The group-averaging construction of the physical inner product rests on the statement that 'the operators corresponding to different vertices commute (in the diffeomorphism invariant context).' This is asserted, not proved, and it is not obvious: Ĥ_v and Ĥ_{v'} each enlarge the graph by exceptional edges attached at their own base points, and the two orders of action produce graphs that are diffeomorphic but not identical, so the commutativity is a statement about the induced action on diffeomorphism-invariant states that needs an argument. Since the n-dimensional Abelian group in (3.8) and the Haar integral in (3.9) presuppose exactly this commutativity, the construction needs either a proof or an explicit citation. Relatedly, (3.9) integrates over N_v ∈ ℝ with Haar measure; convergence of that integral, and the sense in which [f] is a well-defined distribution, are not discussed.
[Theorem 1.1; §2] The abstract's 'complete kernel' is, in the body, the solution space of the linear system (1.3), whose matrices m^(n) are never computed. §2 gives a recipe ('straightforward but rather tedious') and defers explicit coefficients to [21]. Two things would strengthen the claim. First, the proof of Theorem 1.1 establishes that (1.3) is the condition for [f] to be a solution, but does not establish that the solution space is nontrivial at any level n > 0; that requires the matrices m^(n) to have a kernel, which is nowhere shown. Second, the theorem's exhaustiveness rests on the assertion in Definition 1.3 that 'for each w ∈ W^0 there is precisely one n > 0 such that w ∈ W^(n)(w0(w))', which is stated without proof. Either exhibit at least one nonzero Type II solution, or state explicitly that the kernel is characterized but its dimension is left to [21].
[§3.2.1, Definitions 3.1-3.2] The symmetry proof depends on enlarging the graph category to include 'smooth exceptional edges,' which the authors themselves flag as a cost ('we confess ... a technique that prevents us from introducing the notion of a smooth exceptional edge ... would be strongly preferred. Unfortunately, at the moment we do not have such a technique'). Two consequences need more discussion than they receive. First, cylindrical functions on extended graphs do not form an algebra, as the authors note, so the standard holonomy-algebra machinery of [5-7] does not directly apply to the domain on which the symmetric operator is defined; the paper should say what, if anything, is lost. Second, the physical inner product of §3.3.3 is built by group averaging, which in [9] is set up on the analytic category; the paper should state explicitly why the averaging and the resulting Hilbert space are unaffected by the non-analytic edges.
[Theorem 3.3, proof] The commutator check [k, Ĥ^E] = 0 turns on the parenthetical that 'the spins of all other exceptional edges are unchanged (this is because the other edges s_i(Δ) involved in Ĥ^E_{ijk,Δ} are non-exceptional edges).' This is the step that makes P̂_{γ'} f_± = -p f_± hold, and it is asserted rather than argued. The ingredients to close it are present in the paper — Definition 3.2 together with the remark that a smooth exceptional edge cannot be analytic, while the s_i(Δ) are segments of analytic edges of the skeleton — but the reader has to supply the inference. Please state it: no s_i(Δ) can be, or contain, a smooth exceptional edge of γ', hence the only exceptional edge whose spin changes is e = a_ij(Δ).
[§3.1] The argument that the non-symmetric operator admits no nontrivial distributional solutions beyond the co-planar-vertex states is a plausibility argument from the trivalent case: the text shows that an infinite sum over spins on a fixed graph fails because each of the twelve terms in (3.1) must vanish separately, and asserts that adding diffeomorphic images or extra graphs does not help because the resulting graphs are topologically distinct. The paper is candid that the co-planar kernel is a conjecture, and the distinction between the point-spectrum kernel and the distributional kernel of Theorem 1.1 is easy to lose. Since the motivation for the whole symmetric construction in §3 rests on this argument, a short explicit statement of what is proved and what is conjectured at the head of §3.1 would help the reader considerably.

Minor objections (6)

[§1, first paragraph] The introduction promises that 'we will define the notion of an observable and give explicit, non-trivial examples of those,' but the only example given after Definition 1.5 is the projector onto Type I solutions, which the text itself calls trivial. Either supply a non-trivial example or drop the promise.
[Definition 1.2 and throughout] The term 'vertex contractor' is used where the spin-network literature says 'intertwiner'. This is a terminology slip that will confuse readers and should be corrected throughout.
[§3.4] The sentence 'Therefore it follows from Nelson's theorem that there exists a dense set of analytic vectors for Ĉ' runs the theorem in the wrong direction: Nelson's analytic-vector theorem gives self-adjointness from a dense analytic set for a closed symmetric operator. The inference should be routed explicitly through the self-adjoint extension whose existence was established, and the phrase 'for some t > 0' should be tied to the analyticity radius of that extension.
[Eq. (3.5)] The overall normalization -64/(3(iℓ_p²)³) in the definition of t̂_Δ is not motivated. A one-line check that the expression reduces to the classical Lorentzian constraint density in the appropriate limit would help the reader trust the factor.
[References] The paper depends heavily on [1] for the operator definition, the anomaly-freeness proof, and the cylindrical-consistency arguments, and on [21] for the explicit coefficients and the spectral analysis. A short paragraph summarizing exactly which definitions are imported from [1] would make the paper more self-contained.
[Throughout] There are numerous typographical errors that should be corrected: 'Whe eler-DeWitt' and 'no n-perturbative' in the abstract, 'picturized' in Definition 1.1, 'viewn' after Definition 1.5, 'explictly' in §3.1, 'Avcademic Press' in [18], 'to to' in [22], and the malformed subscript 'γıΓe' in the proof of Theorem 3.3.

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The paper

The paper claims

A modified loop assignment makes the Wheeler–DeWitt constraint operator of four-dimensional Lorentzian quantum gravity symmetric on a dense domain, gives it self-adjoint extensions, and pins down its exact kernel

The reading

The paper tries to show that the Wheeler–DeWitt constraint of four-dimensional Lorentzian canonical gravity, regulated on a graph with a loop assignment, can be turned into an operator that is symmetric on a dense domain of the connection Hilbert space and that admits self-adjoint extensions. The first half classifies the exact kernel of a non-symmetric version of that operator: the solutions are group-averaged spin networks whose graphs either have co-planar incident tangents at every vertex or are built from such states by repeated action of the constraint, and the paper writes down the inner product these solutions inherit. The second half diagnoses why the original loop assignment fails symmetry, namely that the constraint adds edges to a graph, so the assignment made for a graph can never coincide with the assignment made for its enlarged image, and repairs it by declaring the added arcs smooth exceptional edges: marked non-analytic edges that the constraint recognises and does not act on again. With that repair the operator no longer grows graphs without limit, symmetry follows from the volume operator annihilating the newly created vertices, and self-adjoint extensions follow from von Neumann's criterion with an explicit conjugation. The payoff, if it holds, is that the Lorentzian constraint becomes an operator that can be diagonalised on a finite graph and exponentiated for group averaging.

What carries the argument

The load-bearing device is the notion of a smooth exceptional edge together with Lemma 3.1. A vertex is called exceptional when it has at least valence three, all incident edges have co-linear tangents, and exactly two of them join analytically; an edge joining two such vertices is exceptional, and it is smooth-exceptional when it meets the analytic edges in a $C^∞$ fashion. Because such an edge cannot itself be analytic, it acts as a permanent marker: the constraint's loop assignment for any graph in the family $Γ_e(γ)$ restricted to the analytic skeleton reduces to the assignment for $γ$. Lemma 3.1 supplies the vanishing: for $v$ in $V(γ') - V(γ)$ the volume operator term $V_v$ kills the contribution, since all edges at $v$ have co-linear tangents, so the constraint simply ignores the extra vertices. The second device is the conjugation $k = P c$ built from the exceptional-edge spin operator $J_e = √{(1/4) - Δ_e} - 1/2$, which makes $H^E$ commute with an anti-linear norm-preserving involution and therefore forces equal deficiency indices.

Extended reading notes

Core claim

The central claim is that the Wheeler–DeWitt constraint admits a version that is a genuine symmetric operator on the whole Hilbert space rather than merely a formal expression, and that this is achieved by changing the triangulation rather than the factor ordering. Thiemann shows that the obstruction to symmetry is structural: because the regulated constraint appends arcs to the graph a cylindrical function lives on, the loop assignment for $f_γ$ and for $γ' = γ ∪ a_{ij}(Δ)$ can never agree, so the two matrix elements $<g_{γ'}, H^E_γ(N) f_γ>$ and $<H^E_{γ'}(N) g_{γ'}, f_γ>$ fail to match. The fix is to require each arc $a_{ij}(Δ)$ to join its two segments in at least a $C^1$ fashion, making it a kink and, once it is smooth but not analytic, a smooth exceptional edge; Lemma 3.1 then shows that the vertices such edges introduce all have co-linear incident tangents, hence are annihilated by the volume operator, hence $H^E_{γ'} f = H^E_γ f$ and symmetry follows. Existence of self-adjoint extensions is proved with von Neumann's theorem using the conjugation $k = P c$, where $c$ is complex conjugation and $P$ multiplies each exceptional edge by $e^{2π i J_e}$, a parity that flips sign exactly when the constraint shifts that edge's spin by $1/2$.

Load-bearing premise

The claim that the non-symmetric operator has no non-trivial distributional solutions beyond the co-planar-vertex states is argued from a single worked trivalent example and from the observation that the resulting graphs are topologically distinct, and the paper itself calls the completeness of that kernel a conjecture.

Editorial extensions

If this is right

The kernel of the non-symmetric Lorentzian constraint is exactly the space spanned by group-averaged spin networks in the sets $W_0$ and $W^{(n)}(w_0)$, with a physical inner product $<[f],[f']>_{phys} = [f](f')$ induced by the group-averaging measure.
Every Lorentzian solution also solves the Euclidean constraint, because the two parts $H^E$ and $T$ of $H$ must vanish separately.
With the symmetric operator, repeated action of the constraint keeps a graph inside a finite family $Γ_e(γ)$, so the spectral problem on diffeomorphism-invariant states reduces to a separable Hilbert space with finitely many degrees of freedom.
Because the constraint acts only on spin labels and the diffeomorphism constraint only on the graph, the constraint algebra of the symmetric Hamiltonian constraint is effectively Abelian, which is what makes the unitary operators $U[\vec N] = \prod_v e^{iN_v H_v}$ and group averaging available.
The generator of the Wick rotation from the Euclidean to the Lorentzian regime also has self-adjoint extensions, so the formal relation $H = p W^{-1} H^E W$ with $W = e^{-C/\hbar}$ can at least be posed, subject to domain questions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the symmetric construction survives scrutiny, the practical bottleneck moves entirely to the volume operator: the spectrum of the constraint at a vertex is largely fixed by the spectrum of $V$, so the programme's near-term testability rests on the matrix elements computed in the companion volume-operator work.
The author's own worry about leaving the analytic category is worth taking seriously as a research direction: cylindrical functions on extended graphs do not form an algebra, so one could ask whether an alternative repair keeps analyticity by enlarging the graph category once and for all rather than by marking individual edges.
The conjecture that the co-planar-vertex states are the complete zero-eigenvalue kernel is stated as a conjecture; a direct computation on a four-valent vertex with three linearly independent tangents would settle whether the kernel is really that small.
The comparison with the earlier zero-eigenvalue solutions suggests a concrete check: those states are annihilated by the volume operator, so if the present operator annihilates them too, the two approaches agree on a common subspace and disagree only on what else is in the kernel.

Original abstract

We continue here the analysis of the previous paper of the Wheeler-DeWitt constraint operator for four-dimensional, Lorentzian, non-perturbative, canonical vacuum quantum gravity in the continuum. In this paper we derive the complete kernel, as well as a physical inner product on it, for a non-symmetric version of the Wheeler-DeWitt operator. We then define a symmetric version of the Wheeler-DeWitt operator. For the Euclidean Wheeler-DeWitt operator as well as for the generator of the Wick transform from the Euclidean to the Lorentzian regime we prove existence of self-adjoint extensions and based on these we present a method of proof of self-adjoint extensions for the Lorentzian operator. Finally we comment on the status of the Wick rotation transform in the light of the present results.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We continue here the analysis of the previous paper of the Wheeler-DeWitt constraint operator for four-dimensional, Lorentzian, non-perturbative, canonical vacuum quantum gravity in the continuum. In this paper we derive the complete kernel, as well as a physical inner product on it, for a non-symmetric version of the Wheeler-DeWitt operator. We then define a symmetric version of the Wheeler-DeWitt operator. For the Euclidean Wheeler-DeWitt operator as well as for the generator of the Wick transform from the Euclidean to the Lorentzian regime we prove existence of self-adjoint extensions and based on these we present a method of proof of self-adjoint extensions for the Lorentzian operator. Finally we comment on the status of the Wick rotation transform in the light of the present results.

background contribution method result interpretation main claim

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s0	We continue here the analysis of the previous paper of the Wheeler-DeWitt constraint operator for four-dimensional, Lorentzian, non-perturbative, canonical vacuum quantum gravity in the continuum.	background 0.99	0.41	0.03	asserted 0.81
s1	In this paper we derive the complete kernel, as well as a physical inner product on it, for a non-symmetric version of the Wheeler-DeWitt operator.	result 0.53	0.97	0.55	asserted 1.00
s2	We then define a symmetric version of the Wheeler-DeWitt operator.	contribution 0.90	0.94	0.06	asserted 0.97
s3	For the Euclidean Wheeler-DeWitt operator as well as for the generator of the Wick transform from the Euclidean to the Lorentzian regime we prove existence of self-adjoint extensions and based on these we present a method of proof of self-adjoint extensions for the Lorentzian operator.	result 0.64	0.93	0.25	asserted 1.00
s4	Finally we comment on the status of the Wick rotation transform in the light of the present results.	interpretation 0.97	0.68	0.08	unclear 0.86

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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full text
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References

Works cited (24)

[21]

T. Thiemann, “Spectral Analysis of the Wheeler-DeWitt constraint operator”, Harvard University preprint

View on Pith
[24]

T. Thiemann, “Complete formula for the matrix elements of the volume oper- ator in canonical quantum gravity”, Harvard Preprint HUTMP -96/B-393 27

View on Pith
[1]

T. Thiemann, “Quantum Spin Dynamics (QSD)”, Harvard Pre print HUTMP- 96/B-351, The previous paper in this volume or in the gr-qc ar chive

View on Pith
[9]

A. Ashtekar, J. Lewandowski, D. Marolf, J. Mour˜ ao, T. Th iemann, “Quantiza- tion for diﬀeomorphism invariant theories of connections w ith local degrees of freedom”, Journ. Math. Phys. 36 (1995) 519-551

View on Pith 1995
[2]

A. Ashtekar, Phys. Rev. Lett. 57 2244 (1986), Phys. Rev. D36, 1587 (1987)

View on Pith 1986
[3]

T. Thiemann, Class. Quantum Grav. 13 (1996) 1383-1403

View on Pith 1996
[4]

A. Ashtekar, “A generalized Wick transform for gravity” , preprint CGPG- 95/12-1, gr-qc/9511083

View on Pith Pith review arXiv
[5]

A. Ashtekar and C.J. Isham, Class. & Quan. Grav. 9, 1433 (1992)

View on Pith 1992
[6]

A. Ashtekar and J. Lewandowski, “Representation theory of analytic holonomy C ⋆ algebras”, in Knots and quantum gravity , J. Baez (ed), (Oxford University Press, Oxford 1994) 26

View on Pith 1994
[7]

J. Baez, Lett. Math. Phys. 31, 213 (1994); “Diﬀeomorphism invariant gener- alized measures on the space of connections modulo gauge tra nsformations”, hep-th/9305045, in the Proceedings of the conference on qua ntum topology, D. Yetter (ed) (World Scientiﬁc, Singapore, 1994)

View on Pith Pith review arXiv 1994
[8]

A. Ashtekar, J. Lewandowski, Journ. Geo. Physics 17 (1995) 191, J. Math. Phys. 36, 2170 (1995)

View on Pith 1995
[10]

C. Rovelli, L. Smolin, Phys. Rev. Lett. 72 (1994) 446

View on Pith 1994
[11]

A. Ashtekar, J. Lewandowski, “Regularization of the Ha miltonian constraint”, in preparation

View on Pith
[12]

T. Thiemann, “The length operator in canonical quantum gravity”, Harvard Preprint HUTMP-96/B-394

View on Pith
[13]

T. Thiemann, “A Hamiltonian operator for canonical qua ntum gravity”, Har- vard University Preprint

View on Pith
[14]

T. Thiemann, “A regularization of canonical Yang-Mill s quantum theory”, Har- vard University Preprint

View on Pith
[15]

A. Higuchi Class. Quant. Grav. 8, 1983 (1991), Class. Quant. Grav. 8, 2023 (1991)

View on Pith 1991
[16]

D. Marolf, “The spectral analysis inner product for qua ntum gravity,” preprint gr-qc/9409036, to appear in the Proceedings of the VIIth Mar cel-Grossman Conference, R. Ruﬃni and M. Keiser (eds) (World Scientiﬁc, S ingapore, 1995), Class. Quant. Grav. (1995) “Almost Ideal Clocks in Quantum Cosmology: A Brief Derivati on of Time,” preprint gr-qc/9412016

View on Pith Pith review arXiv 1995
[17]

J. Baez, S. Sawin, “Functional Integration on Spaces of Connections”, q- alg/9507023

View on Pith
[18]

M. Reed, B. Simon, “Functional Analysis”, Mod. Meth. Ma th. Phys. Vol II, Avcademic Press, New York, 1970

View on Pith 1970
[19]

B. Br¨ ugmann, J. Pullin, Nucl. Phys. B 363 221

View on Pith
[20]

B. Br¨ ugmann, J. Pullin, R. Gambini, Phys. Rev. Lett. 68 (1992) 431, Nucl. Phys. B 385 (1992) 1199

View on Pith 1992
[22]

R. Loll, “A real alternative to to quantum gravity in loo p space”, Preprint DFF 244/02/96, gr-qc/9602041

View on Pith Pith review arXiv
[23]

T. Thiemann, “The ADM Hamiltonian operator for canonic al quantum grav- ity”, Harvard University Preprint

View on Pith

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How the literature uses it

34 arXiv papers cite this in 60 sentences

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Papers citing it · 40 sentences without a confident label

Builds on it 1 paper

The construction of a set of rigorously defined solutions to the diffeomorphism and the Hamiltonian constraint described in [ref] thus immediately generalises to our case.

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Uses its method or tool 2 papers

In order to avoid this problem, Thiemann proposes [ref] to use a set of bounded variables for quantizing Higgs scalars.

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The quantization strategy we will describe in the following is due to Thiemann [ref] , but draws on important earlier work and ideas by Rovelli, Smolin, Lewandowski and others.

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Compares with it 1 paper

The approximation used in [ref] is $Im(A(β)) ≈ϵ^2F_ab^jτ_j+𝒪(ϵ^4)$, while the other one - which is in case of a flat background metric equivalent - is $Re(A(β)) ≈d_n +ϵ^4 F_ab^iF_i^ab + 𝒪(ϵ^6)$.

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Cites it as background 12 papers

In this section we review the quantization of Hamiltonian constraint, as proposed by Thiemann in [ref] .

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In the formulation of general relativity in terms of real SU(2) connection, Thiemann discussed loop quantization of standard model fields [ref] .

Matter in Loop Quantum Gravity · Fermions · 0.88
These multipole operators provide a further tool to probe the geometry of spin networks, on top of the standard area operators, volume operators playing a crucial role in the quantization of the Hamiltonian constraints à la Thiemann [ref] and $(N)$-operators which define basic exchange of quanta of areas between spin network edges [ref] .

Probing the Shape of Quantum Surfaces: the Quadrupole Moment Operator · Introduction · 0.94

Discussion

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[1] [21]

T. Thiemann, “Spectral Analysis of the Wheeler-DeWitt constraint operator”, Harvard University preprint

View on Pith

[2] [24]

T. Thiemann, “Complete formula for the matrix elements of the volume oper- ator in canonical quantum gravity”, Harvard Preprint HUTMP -96/B-393 27

View on Pith

[3] [1]

T. Thiemann, “Quantum Spin Dynamics (QSD)”, Harvard Pre print HUTMP- 96/B-351, The previous paper in this volume or in the gr-qc ar chive

View on Pith

[4] [9]

A. Ashtekar, J. Lewandowski, D. Marolf, J. Mour˜ ao, T. Th iemann, “Quantiza- tion for diﬀeomorphism invariant theories of connections w ith local degrees of freedom”, Journ. Math. Phys. 36 (1995) 519-551

View on Pith 1995

[5] [2]

A. Ashtekar, Phys. Rev. Lett. 57 2244 (1986), Phys. Rev. D36, 1587 (1987)

View on Pith 1986

[6] [3]

T. Thiemann, Class. Quantum Grav. 13 (1996) 1383-1403

View on Pith 1996

[7] [4]

A. Ashtekar, “A generalized Wick transform for gravity” , preprint CGPG- 95/12-1, gr-qc/9511083

View on Pith Pith review arXiv

[8] [5]

A. Ashtekar and C.J. Isham, Class. & Quan. Grav. 9, 1433 (1992)

View on Pith 1992

[9] [6]

A. Ashtekar and J. Lewandowski, “Representation theory of analytic holonomy C ⋆ algebras”, in Knots and quantum gravity , J. Baez (ed), (Oxford University Press, Oxford 1994) 26

View on Pith 1994

[10] [7]

J. Baez, Lett. Math. Phys. 31, 213 (1994); “Diﬀeomorphism invariant gener- alized measures on the space of connections modulo gauge tra nsformations”, hep-th/9305045, in the Proceedings of the conference on qua ntum topology, D. Yetter (ed) (World Scientiﬁc, Singapore, 1994)

View on Pith Pith review arXiv 1994

[11] [8]

A. Ashtekar, J. Lewandowski, Journ. Geo. Physics 17 (1995) 191, J. Math. Phys. 36, 2170 (1995)

View on Pith 1995

[12] [10]

C. Rovelli, L. Smolin, Phys. Rev. Lett. 72 (1994) 446

View on Pith 1994

[13] [11]

A. Ashtekar, J. Lewandowski, “Regularization of the Ha miltonian constraint”, in preparation

View on Pith

[14] [12]

T. Thiemann, “The length operator in canonical quantum gravity”, Harvard Preprint HUTMP-96/B-394

View on Pith

[15] [13]

T. Thiemann, “A Hamiltonian operator for canonical qua ntum gravity”, Har- vard University Preprint

View on Pith

[16] [14]

T. Thiemann, “A regularization of canonical Yang-Mill s quantum theory”, Har- vard University Preprint

View on Pith

[17] [15]

A. Higuchi Class. Quant. Grav. 8, 1983 (1991), Class. Quant. Grav. 8, 2023 (1991)

View on Pith 1991

[18] [16]

D. Marolf, “The spectral analysis inner product for qua ntum gravity,” preprint gr-qc/9409036, to appear in the Proceedings of the VIIth Mar cel-Grossman Conference, R. Ruﬃni and M. Keiser (eds) (World Scientiﬁc, S ingapore, 1995), Class. Quant. Grav. (1995) “Almost Ideal Clocks in Quantum Cosmology: A Brief Derivati on of Time,” preprint gr-qc/9412016

View on Pith Pith review arXiv 1995

[19] [17]

J. Baez, S. Sawin, “Functional Integration on Spaces of Connections”, q- alg/9507023

View on Pith

[20] [18]

M. Reed, B. Simon, “Functional Analysis”, Mod. Meth. Ma th. Phys. Vol II, Avcademic Press, New York, 1970

View on Pith 1970

[21] [19]

B. Br¨ ugmann, J. Pullin, Nucl. Phys. B 363 221

View on Pith

[22] [20]

B. Br¨ ugmann, J. Pullin, R. Gambini, Phys. Rev. Lett. 68 (1992) 431, Nucl. Phys. B 385 (1992) 1199

View on Pith 1992

[23] [22]

R. Loll, “A real alternative to to quantum gravity in loo p space”, Preprint DFF 244/02/96, gr-qc/9602041

View on Pith Pith review arXiv

[24] [23]

T. Thiemann, “The ADM Hamiltonian operator for canonic al quantum grav- ity”, Harvard University Preprint

View on Pith