Partial and Complete Observables for Canonical General Relativity

B. Dittrich

REVIEW

Partial and Complete Observables for Canonical General Relativity

B. Dittrich

General Relativity and Quantum Cosmology · Submitted 25 Jul 2005 · 16 references

Pith's verdict: A careful framework paper that genuinely reduces the GR constraint problem to one Abelian constraint, but only under a global clock-field assumption the paper states and never establishes — and its own worked example fails it.

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arXiv IDgr-qc/0507106 (version 1)

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CitationB. Dittrich. "Partial and Complete Observables for Canonical General Relativity." arXiv:gr-qc/0507106 (2005). https://pith.science/paper/XZXDD4FF

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Bottom line: the useful content here is real, and it is narrower than the abstract suggests. Dittrich takes the complete-observables machinery from her earlier paper and applies it to GR. The genuinely new moves are: using space-time scalars (via Kuchař's reconstruction theorem) as partial observables so the complete-observable power series depends on d+1 numbers instead of whole parameter fields; solving the spatial diffeomorphism constraints to get C_⊥(Y); and multiplying by g^{-1/2} to get constraints that Poisson-commute. The algebra in sections 5 and 7 is checkable and I checked a fair amount of it — the delta-function manipulations in (65)-(71) and the Dirac-bracket cancellation in (91)-(94) hold together. That is the substance, and it is worth having in the literature. The connection drawn between space-time diffeomorphism invariants and canonical Dirac observables is the part I would actually cite. On the soft spots, the stress-test note is right but the paper is not hiding anything. Footnote 8 and section 6 state plainly that the clock fields must give good global coordinates on Σ with a phase-space-independent image, and section 8 says outright that the four scalar matter fields do not satisfy those assumptions. So the reduction to one constraint is conditional on an existence claim that is asserted, not proved, and the only concrete example violates it. Same story for the Abelianization: (97) requires {T^A, g^{-1/2}} to be ultra-local, which the paper says holds for matter clocks but not for gravitational scalars like Bergmann-Komar. So 'can be made Abelian' in the abstract is true for a restricted class of clocks, not in general. Two smaller things: no explicit observable is computed end to end, and the power series is formal — convergence is not discussed. One correction to the reader's report: C_abel(Y) = g^{-1/2}C_⊥(Y) is not an invented entity; it is defined in Eq. 95 and used consistently. Who should read it: anyone working on Dirac observables, the problem of time, or quantization on diffeomorphism-invariant states. The Abelian constraint proposal is the piece most likely to be built on. Recommendation: send it out. The conditions are stated, the algebra is mostly right, and the referee's job is to press on whether the global clock assumption can be weakened or replaced by a local patching argument.

Referee report

Significance

If the required clock fields exist, this paper gives a concrete, explicitly computable route to Dirac observables in full general relativity and, more importantly, a dictionary between space-time-diffeomorphism-invariant quantities and canonical Dirac observables. The Abelian diffeomorphism-invariant Hamiltonian constraints of §7 are a technically useful device that later work on reduced phase space and on constraint quantization has drawn on. The paper has no free parameters and no fitted quantities; the derivations are checkable line by line, and the algebra in §§5, 7 and Appendix A is internally consistent as far as I have verified (I checked Eqs. (56), (68), (83), (94), (105), (106) and (110) explicitly). The use of Kuchař's reconstruction theorem is appropriate and correctly cited.

Major objections (2)

[Abstract, §1, footnote 8, §8] The headline claim of the abstract (abstract:s2) and of §1 is stated unconditionally, but the construction requires d+1 space-time scalar clock fields that (a) have ultra-local brackets with all constraints and (b) provide good global coordinates on Σ for every phase-space point on the constraint surface, with a phase-space-independent image T^A(Σ) (footnote 8). Condition (b) is a strong existence assumption: scalar fields generically have critical points where det(T^A_{,a}) vanishes, and the image can change along the gauge orbit. The paper is honest about this in footnote 8 and in §6, and §8 states explicitly that the four-scalar-field example 'do[es] not satisfy all the assumptions made in this section', but that concession sits in a subordinate clause and never reaches the abstract or the introduction. I would ask the authors to state the conditionality of the central result where the result is announced, and to say what is actually known about existence: is the construction established only locally in phase space (as the Summary suggests), and is there any model in which (b) is verified rather than assumed? Without that, the abstract's 'one can compute Dirac observables for general relativity by dealing with just one constraint' overstates the scope of what is proved.
[§7, Eqs. (96)-(97)] The Abelian property of C_abel(Y) is asserted to require that {T^A(σ), g^{-1/2}(σ**)} be ultra-local, which the text says holds 'if the T^A are matter fields'. On my reading this sufficient condition is stronger than necessary. In Eq. (97) the potentially non-ultra-local piece is {T^A(σ), g^{-1/2}(σ**)}C_⊥(σ**), which is proportional to the constraint and therefore vanishes weakly regardless of whether {T^A, g^{-1/2}} contains derivatives of delta functions; the surviving piece is -T^A_{,⊥}(σ)g^{-1/2}(σ*)δ(σ,σ*), which is ultra-local. Substituting this into the last two lines of Eq. (96) gives two terms that cancel exactly on the constraint surface, with no assumption on {T^A, g^{-1/2}}. If this is correct, the Abelian result is broader than §7 claims and the restriction to matter clock fields is unnecessary, which would also settle the tension the abstract's 'can be made to be Abelian' (abstract:s4) has with the use of gravitational scalars such as the Bergmann-Komar invariants in §5. Please either supply the missing step that makes the stronger condition necessary, or weaken the hypothesis and state the result in the stronger form.

Minor objections (9)

[§6, Eqs. (79)-(82)] The decomposition of the diffeomorphism constraint in Eq. (81), C_b = (1/α_A)Π^A T^A_{,b} + C^{(R)}_b, presumes that the clock fields enter the diffeomorphism constraint only through the term linear in their conjugate momenta. This is true for scalar fields with non-derivative coupling, but it is an assumption about the matter sector and should be stated as such alongside the 'non-derivative coupling' assumption of §3, since the replacement (82) and hence the identification of Dirac brackets with Poisson brackets depends on it.
[§6, Eqs. (77)-(78)] The equivalence argument for replacing {C_⊥(σ)} by {D[det(S)C_⊥(σ);T^A](Y)} is given in prose and is convincing, but the concluding sentence that 'C_1 is an ideal in the algebra C_1 ∪ {D[C_j;T_c](τ_c,·)}' is asserted rather than shown. One line verifying that the brackets of D[C_j;T_c] with C_a close back into the constraint set would make the step self-contained.
[§4, Eq. (54)] The interpretation of F[ψ(σ*);T](τ) as the value of ψ at the space-time point where T^K = τ^K requires the map T: S → R^{d+1} built from the clock fields to be injective on space-time, not merely that the T^A coordinatize Σ. This is used silently in §4 and again in Appendix A (the discussion around Eq. (135)); it should be stated as an assumption.
[§2, Eq. (13)] There is an index typo in the last line: the final factor reads (τ_{K_r}(σ_k) - T_{K_r}(σ_r)(x)) where the subscript should be σ_r throughout. Also, Eq. (54) is written with an equality sign whereas the analogous Eq. (13) correctly uses ≃; the two should be consistent.
[§6, Eq. (79)] The equation is written with the same symbol Y on both sides of the Poisson bracket, D[φ(σ*);T^A](Y) and D[ψ(σ**);T^A](Y), although the two arguments are evaluated at different points. Using Y' and Y'' (as is done later in Eq. (83)) would remove the ambiguity.
[§6, Eq. (83)] The intermediate expression [det(Y^C_{,c})(σ**)]^{-1}δ(σ*,σ**) mixes the coordinate label Y with the field evaluated at σ**; a sentence identifying δ(σ*,σ**) = det(Y^C_{,c})δ(Y',Y'') would make the step transparent.
[§8, Eq. (107) and following] The observation that C_⊥(Y) fails to transform as a density under reparametrizations of Y because the potentials V_(K) depend explicitly on Y^A is correct and worth keeping, but the consequences for the interpretation of the Abelian constraint C_abel(Y) in that case are not spelled out. A short remark on whether the Abelian property survives when the constraint depends explicitly on Y would help.
[Throughout] The manuscript contains a large number of typographical errors that impede reading: 'inﬁntely' (§2, §4), 'constrant' (§2), 'varibales' (§2), 'splittet' (§3), 'diﬀeomorphismen' (§3), 'sublements' (§6), 'Poissin brack et' (Appendix A), 'Hamiltanion' (§8), 'invovle' (§7), 'Abelianess' (§7), 'whith' (§1). A careful proofreading pass is needed.
[§9, Summary] The statement that the power series is 'formal' is correct, but no example anywhere in the paper sums the series or even evaluates the second-order term. A single worked truncation, even in a symmetry-reduced model, would substantially strengthen the claim that the method is a practical route to observables rather than only a formal device.

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The paper

The paper claims

Dittrich shows that the gauge-invariant quantities of canonical general relativity can be generated from a single Abelian constraint, using space-time scalar fields as clocks.

The reading

Canonical general relativity is a fully constrained system: because no slicing of space-time into space and time is preferred, almost every local quantity is gauge dependent, and the truly physical ones — Dirac observables — are what any prediction must be built from. Dittrich asks whether these can be computed without solving the whole coupled tower of constraints at once. Using the partial-and-complete-observables scheme, in which a field is read off at the moment a set of clock fields take prescribed values, she argues that if the clocks are themselves space-time scalars their brackets with the constraints become ultra-local, and the infinite constraint tower collapses first to finitely many constraints and then to one. The payoff would be a workable handle on gauge-invariant quantities in gravity: a formal power series in a single generator instead of an intractable functional differential equation in infinitely many.

What carries the argument

The load-bearing device is the power series for a complete observable, $F[f;T](tau) = sum_r (1/r!) {cdots{f, tilde C_{K_1}[1]},cdots, tilde C_{K_r}[1]} (tau^{K_1}-T^{K_1})cdots(tau^{K_r}-T^{K_r})$, together with the weakly Abelian constraint construction $tilde C_K = (B^{-1})_{jK} C_j$ where $B_{Kj}(sigma) = {T_K(sigma), C_j(sigma)}$ up to a delta function. Ultra-locality of these brackets — guaranteed by the reconstruction theorem for space-time scalars — is what makes the spatial integrals in the series collapse and reduces infinitely many constraints to finitely many, and then to one.

Extended reading notes

Core claim

The paper's central claim is that Dirac observables for general relativity can be computed by dealing with just one constraint, provided one first replaces the Hamiltonian constraints by spatial-diffeomorphism-invariant versions. The route has three steps. First, choose $d+1$ clock fields $T^K$ that are space-time scalars in the Kuchař sense — canonical fields whose Poisson brackets with the Hamiltonian constraints are ultra-local (proportional to $delta( sigma, sigma')$ with no derivatives) and which transform as spatial scalars under the diffeomorphism constraints. Then the matrix $A_{Kj}(sigma,sigma') = {T_K(sigma), C_j(sigma')}$ is ultralocal, the power series for the complete observable depends on only $d+1$ parameters rather than a whole parameter field, and only the finitely many smeared constraints $tilde C_K[1]$ appear. Second, solve the spatial diffeomorphism constraints by constructing partially complete observables with respect to them, using $d$ of the scalar clock fields as spatial coordinates; this turns $C_perp(sigma)$ into a diffeomorphism-invariant Hamiltonian constraint $C_perp(Y)$ labelled by the clock values. Third, multiply by $g^{-1/2}$ so that the resulting constraints become Abelian, after which the complete observable is a power series in a single smeared constraint. Along the way the paper shows that the metric components in the clock frame, $gamma_{KL} = T_{K,a}T_{L,b}g^{ab} - T_{K,perp}T_{L,perp}$, are themselves space-time scalars, giving a bridge between quantities invariant under space-time diffeomorphisms and Dirac observables in the canonical picture.

Load-bearing premise

The construction needs $d+1$ clock fields that are space-time scalars with ultra-local brackets and that give good global coordinates on the spatial manifold at every point of the constraint surface, and the Abelian step additionally needs the brackets of those clocks with $g^{-1/2}$ to be ultra-local, which the paper says holds when the clocks are matter fields.

Editorial extensions

If this is right

Every Dirac observable of general relativity can be generated as a formal power series in a single Abelian constraint, so the many-fingered constraint algebra is replaced on the constraint surface by one generator.
The diffeomorphism-invariant Hamiltonian constraints can in principle be quantized on the Hilbert space of diffeomorphism-invariant states, which the original, non-invariant Hamiltonian constraints cannot be.
Space-time-diffeomorphism-invariant quantities — metric components in a clock frame, space-time volume, curvature scalars built by differentiating with respect to the clock parameters — acquire canonical Dirac-observable counterparts, allowing covariant and canonical quantization to be compared observable by observable.
Approximation schemes for complete observables now need to handle one constraint rather than infinitely many, which the author suggests makes existing approximation methods applicable.

Reading between the lines

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

If the clock fields have to be matter fields for the Abelian step to go through, the reduction is really a statement about gravity coupled to reference matter, and the resulting observables inherit the matter sector's limitations — a relational reading in which matter is load-bearing rather than auxiliary.
An Abelian constraint algebra sidesteps the usual anomaly question for the Hamiltonian constraint, so the construction may be a route to imposing that constraint exactly rather than up to regularization ambiguities.
The paper does not address convergence of the formal power series; a numerical test in minisuperspace or in a perturbative truncation would show whether the series is usable in practice or only formal.
The requirement that the clock fields furnish a global chart on the spatial manifold is a strong restriction on phase space, which suggests treating the construction as local in phase space and patching across regions where $det(T^A_{,a})$ degenerates.

Original abstract

In this work we will consider the concepts of partial and complete observables for canonical general relativity. These concepts provide a method to calculate Dirac observables. The central result of this work is that one can compute Dirac observables for general relativity by dealing with just one constraint. For this we have to introduce spatial diffeomorphism invariant Hamiltonian constraints. It will turn out that these can be made to be Abelian. Furthermore the methods outlined here provide a connection between observables in the space--time picture, i.e. quantities invariant under space--time diffeomorphisms, and Dirac observables in the canonical picture.

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The abstract, sentence by sentence

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

In this work we will consider the concepts of partial and complete observables for canonical general relativity. These concepts provide a method to calculate Dirac observables. The central result of this work is that one can compute Dirac observables for general relativity by dealing with just one constraint. For this we have to introduce spatial diffeomorphism invariant Hamiltonian constraints. It will turn out that these can be made to be Abelian. Furthermore the methods outlined here provide a connection between observables in the space--time picture, i.e. quantities invariant under space--time diffeomorphisms, and Dirac observables in the canonical picture.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this work we will consider the concepts of partial and complete observables for canonical general relativity.	background 0.54	0.61	0.02	asserted 0.56
s1	These concepts provide a method to calculate Dirac observables.	background 0.56	0.23	0.03	asserted 0.99
s2	The central result of this work is that one can compute Dirac observables for general relativity by dealing with just one constraint.	result 0.99	0.90	0.97	asserted 0.90
s3	For this we have to introduce spatial diffeomorphism invariant Hamiltonian constraints.	method 0.95	0.65	0.03	asserted 0.89
s4	It will turn out that these can be made to be Abelian.	result 0.87	0.80	0.03	asserted 0.50
s5	Furthermore the methods outlined here provide a connection between observables in the space--time picture, i.e. quantities invariant under space--time diffeomorphisms, and Dirac observables in the canonical picture.	interpretation 0.51	0.81	0.04	asserted 0.81

What kind of paper the classifier reports this is: method or tool 0.69 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v1

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levels run

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References

Works cited (16)

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[8]

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Papers citing it · 79 sentences without a confident label

Builds on it 2 papers

[ref] , where not only the reduced ADM-phase space is considered as in [ref] , but also lapse and shift are taken into account as dynamical variables as in [ref] .

Gauge invariant canonical cosmological perturbation theory with geometrical clocks in extended phase space - a review and applications · Application to general relativity · 0.88
Along the way, we establish that the frame-dressed observables entering the construction of Donnelly and Freidel can be generalized and understood as a covariant incarnation of the relational observables originally defined in a canonical set-up by Dittrich [ref] (extending earlier work by Rovelli [ref] ).

Edge modes as reference frames and boundary actions from post-selection · Introduction · 0.95

Uses its method or tool 12 papers

We will adopt the generally covariant framework for dynamical frames developed in [ref] (see [ref] for somewhat related work), which unifies and generalizes the dressed [ref] and relational [ref] approaches to gravitational observables.

Relational path integral, effective actions and quantum frame covariance in gravity · Introduction · 0.91
Once we have chosen such a set clocks, the formalism developed in [ref] allows a construction of observables in a formal power series for a generic phase space function $f$ being independent of the clock degrees of freedom.

Gauge invariant canonical cosmological perturbation theory with geometrical clocks in extended phase space - a review and applications · Reduced ADM-phase space · 0.97
Let us outline general procedure of construction of relational observables when we mainly follow excellent papers by Dittrich [ref] .

Deparametrization of General Relativity by Space-Time Filling Unstable D9-Brane with Arbitrary Value of Tachyon · Gauge Invariant Observables for General $T$ · 0.96

Compares with it 1 paper

This coincides with the perturbation theory used in [ref] when $H_0=0$.

Solving the Problem of Time in Mini-superspace: Measurement of Dirac Observables · Perturbations and the Physical Hilbert space · 0.96

Cites it as background 38 papers

In the second section we review the relational formalism and the construction of observables discussed in [ref] and its application to canonical general relativity following the seminal work of Pons et al. in [ref] .

Gauge invariant canonical cosmological perturbation theory with geometrical clocks in extended phase space - a review and applications · Introduction · 0.81
Here we only summarize the main points for our analysis and refer to [ref] as well as the original literature [ref] for further details.

Gauge invariant variables for cosmological perturbation theory using geometrical clocks · Relational formalism and Dirac observable in extended ADM-phase space · 0.95
Much has been written about Dirac observables for general relativity [ref] , which involves the implementation of invariance under spacetime diffeomorphisms.

Can chaos be observed in quantum gravity? · Introduction · 0.98

Discussion

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