Partial and Complete Observables for Hamiltonian Constrained Systems

B. Dittrich (Max Planck Institute; Perimeter Institute; Potsdam; Waterloo)

REVIEW

Partial and Complete Observables for Hamiltonian Constrained Systems

B. Dittrich (Max Planck Institute, Potsdam, Perimeter Institute, Waterloo)

General Relativity and Quantum Cosmology · Submitted 2 Nov 2004 · 19 references

Pith's verdict: The paper that turns Rovelli's one-clock construction into a workable n-constraint calculus; the core algebra holds up, and the soft spots are the ones the author already names.

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arXiv IDgr-qc/0411013 (version 1)

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CitationB. Dittrich (Max Planck Institute, Perimeter Institute, Potsdam, Waterloo). "Partial and Complete Observables for Hamiltonian Constrained Systems." arXiv:gr-qc/0411013 (2004). https://pith.science/paper/W7YN7NA7

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Bottom line: this is the paper that turns Rovelli's one-clock trick into something you can actually compute with when there are n constraints, and the pieces people cite downstream — the PDE system, the weak abelianization, the Dirac bracket identity — are all here and all correct as far as I can check them line by line. If you work on Dirac observables in canonical gravity, this is the reference you want on the shelf next to Kuchař's bubble-time paper, because section 7 is what connects the two literatures explicitly rather than by folklore. The generalization in section 4, the derivation of (5.10) done twice (displacement in parameter space, then differentiating the defining equation), the consistency proof (5.28) via Jacobi and the first-class property, and the Poisson-algebra homomorphism statement (8.13)–(8.14) are the real content. Section 9, on starting from partially invariant partial observables, is the part I did not expect and it is genuinely useful — it tells you when a degenerate set of clocks is good enough, which matters for the spatial-diffeomorphism-invariant route sketched in section 10. On the stress-test note: it lands, but only partly. The proof of Theorem 4.1 in (4.4) does need the exponential of the constraint span to sweep the whole orbit, which is stronger than invertibility of A_kj. The paper states exactly that as an assumption in the sentence after (4.2), so it is not hidden, and section 6's abelianized constraints are what make it usable. What the text does not say out loud is that the general-C_j proof is really established through the abelianized set; that is a presentational gap worth a paragraph, not a correctness failure. Related and more honest: Theorem 4.1's hypothesis — f constant on the intersection of the level set with the orbit — is close to its conclusion, so the theorem is nearly tautological and the substance is the construction, not the theorem. The author says as much in section 11 about global clocks and about the unproved convergence of (5.45). Example 3 (baker's map) is inconclusive and the author admits it. Section 10 is largely a re-derivation of Torre and Kuchař, so read it as interpretation, not new physics. Who should read it: anyone doing canonical quantization of constrained systems, and especially anyone who needs the observable algebra rather than just existence. Send it out.

Referee report

Major objections (2)

[§4, Eq. (4.2) and the sentence following it; §6] The load-bearing assumption is not merely invertibility of A_{kj}={C_k,T_j} but that the exponential parametrization β ↦ α_{Σ_j β_j C_j}(x) sweeps out the whole gauge orbit G_x. The paper states this explicitly after (4.2), which is good, but it deserves more prominence because it is what makes the definition (4.2) well posed at all parameter values. I want to record that the proof of Theorem 4.1 in (4.4) is actually robust to the non-abelian case: the step 'α_{ε_l C_l}∘α_{β_j C_j}(x) is still a point on the gauge orbit' uses only that α_ε is a bijection of G_x onto itself, so the image set {α_ε∘α_β(x)} equals G_x and the intersection with the level set {T_i=τ_i} is unchanged; it does not require α_ε∘α_β to be of the form α_{β'}. The genuine gap is in the definition, not the invariance proof, and §6's weak abelianization C̃_m=A^{-1}_{mj}C_j is the right resolution. The authors should say this explicitly so that a reader does not mistake the exponential-coverage assumption for a property of the proof.
[§8, Eqs. (8.8)–(8.13)] The Dirac-bracket identity (8.13) is one of the paper's most useful results, but its proof is sketched. The step from (8.10) to (8.11) rests on the assertion that f* and F[f,T_i](τ_i,·) 'coincide up to the first order in (T_j−τ_j)', justified only by the remark that 'one can also calculate (8.8) explicitly and compare it to the power series of F in (5.45)'. Since (8.13) is stated as an equality of complete observables and is used to conclude that F[T_i](τ_i) is a Poisson-algebra homomorphism (8.14), the explicit comparison should be carried out in the text, at least to the order needed to establish the identity on the gauge-fixed surface. As written, the chain (8.10)–(8.13) is plausible but not fully verified.

Minor objections (7)

[Title] The title contains a typo: 'Constrai ned' should read 'Constrained'.
[§5, Example numbering] The examples jump from 'Example 5' to 'Example 7'; there is no Example 6. Either renumber or supply the missing example.
[§5, Eq. (5.32)] In the first line of (5.32) the factor 'm_3 p_k^{3-1}_3 q^{n_3}_3' appears to be a typographical corruption of 'm_3 p^{m_3-1}_3 q^{n_3}_3'; please check the exponents and indices against (5.29).
[Throughout] There are numerous typographical errors that should be corrected in a final pass, including 'diﬀeormorphism', 'transcedental', 'alreday', 'neccessarily', 'repitition', 'intial', 'constaint', 'Poissonbracket', 'non-degenrate', 'inﬁnte', 'neccessary', and 'constraint hypersur face'.
[§5, Eq. (5.45) and §11] The formal series (5.45) is described as providing 'a local definition of a complete observable', but this is conditional on convergence, which the paper acknowledges only in §11. It would be clearer to state the convergence hypothesis inline at (5.45) so that the claim is not read as unconditional.
[§10, Eqs. (10.21)–(10.22)] The pullback argument that produces Π_ψ(r')[τ_0,τ_1] from Ψ(τ_0(r'),τ_1(r')) is compressed; a sentence spelling out which of the two functional derivatives is being inverted, and why the result is independent of the choice of embedding, would help the reader follow the derivation.
[References] Reference [13] is cited as 'to appear' for the statement that a special case of Example 5 was quantized; since this is not load-bearing for the central claim, no action is required, but the citation should be updated if a published version exists by the time of publication.

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The paper

The paper claims

Any gauge theory with enough usable clock variables can be turned into a family of genuine observables, computed as a power series in what those clocks read.

The reading

General relativity and other gauge theories are written in variables that carry no physical meaning until the gauge freedom is removed; the quantities that survive are Dirac observables, and for gravity almost none are known explicitly. This paper takes an idea of Rovelli's — treat some phase-space functions as clocks, then ask what another function reads at the moment those clocks show prescribed values — and turns it into a working construction for any finite number of first-class constraints. The author proves that the resulting object, the complete observable $F[f,T_1,...,T_n](t_1,...,t_n,x)$, is invariant under the flows generated by the constraints, derives a system of partial differential equations it obeys, and solves that system formally as a power series in the clock readings. Along the way the construction is identified with Kuchař's Bubble Time formalism for parametrized field theories, its Poisson brackets are shown to be governed by the Dirac bracket, and the method is applied to cylindrical gravitational waves, where the mode coefficients come out as Dirac observables.

What carries the argument

The load-bearing object is the bracket matrix $A_{kj} = {C_k,T_j}$ and its inverse. It does three jobs at once: it certifies that the clocks parametrize the gauge orbit (its determinant must not vanish on the constraint surface), it converts the constraint algebra into the weakly abelian set $A^{-1}_{mj}C_j$ whose flows commute on that surface and along which the clocks move linearly, and it supplies the coefficients of the partial differential equation $partial F/partial t_m = A^{-1}_{mk}{C_k,f}$ that the complete observable satisfies. Iterating that equation yields the formal series $F = sum_k (1/k!) g_{(k)}(y) prod_i (t_i - T_i(y))^{k_i}$, where $g_{(k)}$ is obtained by repeatedly applying the operators $h mapsto A^{-1}_{jl}{C_l,h}$. The same matrix reappears in the Poisson-bracket result ${F[f,T],F[g,T]} = F[{f,g}_*,T]$, tying the algebra of complete observables to the Dirac bracket on the gauge-fixed surface.

Extended reading notes

Core claim

The central claim is Theorem 4.1. Given $n$ independent first-class constraints $C_j$, $n$ clock functions $T_i$ whose bracket matrix $A_{kj} = {C_k,T_j}$ is invertible on the constraint surface, and a function $f$ that returns the same number at every point of a gauge orbit where the clocks read a prescribed tuple $(t_1,...,t_n)$, the complete observable $F[f,T_1,...,T_n](t_i,x)$ is well defined and commutes weakly with every constraint. The proof is a flow argument: composing the constraint flow with a further gauge transformation shifts the flow parameter, so the same intersection of the clock level set with the gauge orbit is selected, and the value of $f$ there is unchanged. The paper then shows that the derivative of a complete observable with respect to a clock reading is again a complete observable, giving the system $A_{kj},partial_j F = {C_k,f}$ evaluated at the point where the clocks read $t_i$, proves the system's integrability conditions hold at least on the constraint surface, and writes the solution as a formal power series whose coefficients are iterated brackets built from $A^{-1}$. A by-product is that the constraints can be replaced by the weakly abelian set $A^{-1}_{mj}C_j$, with respect to which the clocks advance linearly — the same device Kuchař used in the Bubble Time formalism.

Load-bearing premise

The construction needs $n$ clock functions whose bracket matrix with the constraints is invertible everywhere on the constraint surface, and the paper itself concedes in its discussion that such clocks may not exist globally — for general relativity they provably do not.

Editorial extensions

If this is right

Dirac observables for a constrained Hamiltonian system can be computed, at least locally and formally, without solving the constraints first: pick clocks, invert $A_{kj}$, and iterate the bracket operators.
The Poisson algebra of Dirac observables is the Dirac bracket algebra of the underlying phase-space functions, so a quantization of the reduced theory can be sought as a representation of that bracket.
Gauge transformations act nontrivially on complete observables by shifting the clock parameters, which gives a precise sense in which 'evolving constants of motion' generalize from one constraint to many.
For background-independent field theories the construction reproduces Kuchař's Bubble Time formalism, so that formalism can be read as a special case of partial and complete observables.
If the partial observable and some clocks are already invariant under a subalgebra of the constraints, the remaining equations decouple and the complete observable depends on fewer parameters — the case relevant to spatially diffeomorphism-invariant quantities in gravity.

Reading between the lines

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

The power series is the practical entry point: truncated at low order it gives approximate Dirac observables built from nested Poisson brackets, which is exactly the shape a perturbative treatment of canonical gravity would need.
The Baker-transformation example in the text shows the construction degrading gracefully — the computed quantity is invariant only over a finite stretch of the flow — which suggests treating the method as producing observables valid on a patch of the gauge orbit rather than globally.
Because the clocks can be chosen to be spatial-diffeomorphism-invariant fields, the framework offers a route to splitting the full constraint algebra of general relativity into a solved part and an unsolved part, reducing the number of functional equations one must handle.
The formal series and the PDE system give a concrete test bed: one could ask, for a fixed model, at what order the truncated series stops commuting with the constraints, turning the convergence question into a computable diagnostic.

Original abstract

We will pick up the concepts of partial and complete observables introduced by Rovelli in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kucha\v{r}'s Bubble Time Formalism. Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions. Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.

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We will pick up the concepts of partial and complete observables introduced by Rovelli in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kucha\v{r}'s Bubble Time Formalism. Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions. Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We will pick up the concepts of partial and complete observables introduced by Rovelli in order to construct Dirac observables in gauge systems.	contribution 0.48	0.54	0.09	asserted 0.85
s1	We will generalize these ideas to an arbitrary number of gauge degrees of freedom.	contribution 0.98	0.81	0.05	asserted 0.96
s2	Different methods to calculate such Dirac observables are developed.	contribution 0.77	0.90	0.10	asserted 0.99
s3	For background independent field theories we will show that partial and complete observables can be related to Kucha\v{r}'s Bubble Time Formalism.	result 0.64	0.83	0.15	asserted 0.98
s4	Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions.	contribution 0.57	0.88	0.13	hedged possibility 0.54
s5	Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.	contribution 0.92	0.71	0.09	hedged possibility 0.96

What kind of paper the classifier reports this is: method or tool 0.55 · evidence from proof or derivation 0.98.

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References

Works cited (19)

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[5]

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[6]

I. Kouletsis, K.V. Kuchaˇ r: “Symplectomorphisms in His tory Phase Space: Bosonic String Model”e-print: gr-qc/0108022

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[7]

M. Henneaux, C. Teitelboim, “Quantization of Gauge Syst ems”, Princeton University Press 1992

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K.V. Kuchaˇ r, “Time and Interpretations of Quantum Gra vity”, in Proceedings of the 4th Canadian Conference on General Relativity and Relativistic A strophysics, eds. G. Kunstat- ter, D. Vincent and J. Williams (World Scientiﬁc, Singapore , 1992)

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[11]

C.G. Torre, “A complete set of observables for cylindri cally symmetric gravita tional ﬁelds”, Class. Quantum Grav. 8, 1895, (1991)

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L. Smolin, “Time, Measurement and Information Loss in Q uantum Cosmology” in College Park 1993, Directions in general relativity, Vol. 2 , ed. by B.L. Hu et. al. (Cambridge Univ. Press, 1993), p. 237-292, e-print: gr-qc/9301016

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[13]

B. Dittrich, T. Thiemann: “Testing the Master Constrai nt Programme for Loop Quantum Gravity II: Finite Dimensional Examples”, to appear

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K. V. Kuchaˇ r, “Canonical Quantization of Cylindrical Gravitational Waves”, Phys. Rev. D4, 955, (1971)

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[16]

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[19]

C.G. Torre, “Gravitational Observables and Local Symm etries”, Phys. Rev. D48, (1993), 2373, e-print: gr-qc/9306030 38

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Builds on it 2 papers

In short, the relational Dirac observable (generalising [ref] and [ref]) defined as [ref] [equation] is invariant under the flows generated by the constraints [ref]. $F_f_φ,{χ_J}({χ_J^0})$ gives the value of $f_φ$ when the dynamical clocks $χ_J$ take the values $χ_J^0$ for all $J$'s (the curly brackets in [ref] are meant to emphasise that we deal with a set of clocks and not with the $J$-th clock only).

Relational dynamics and Page-Wootters formalism in group field theory · Many reparametrisation symmetries and multi-fingered time · 0.83
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 18 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
We are considering a general clock $T̂=T̂|_g=0+ħ h(Ĥ_C)$ where $h(ϵ)=d/dϵg(ϵ)$ whose states are [ref] [equation] where the coefficients of the Fourier transform, or the plane waves in this case, are [equation] The gauge-dependence of $f̂_S$ dictates [equation] To find out what the gauge-dependent system observable $f̂_S$ looks like when the clock $T$ reaches the target time $τ$, we must evolve the system from $t$ to $τ$ as [equation] where [equation] Then the gauge-invariant relational observable associated to $f̂_S$ is derived by projecting the above into the clock states [ref] [equation] or equivalently [equation] where [equation] and in our case [equation] Eqs. [ref] or [ref] yields the gauge-invariant value associated to $f̂_S$ when the clock operator $T̂$ has eigenvalue $τ$.

Relational quantum dynamics of the black hole interior: singularity resolution and quantum bounce · General treatment of gauge-invariant observables · 0.88
Employing the strategy of “evolving constants of motion” [ref] , we could define classical physical observables (which Poisson-commute with the Hamiltonian constraint) as relational Dirac observables [ref] .

Relational dynamics and Page-Wootters formalism in group field theory · Discussion · 0.97

Compares with it 3 papers

With different notation, this expression appeared in the literature in [ref] as his equation (2.8)and in [ref] as her equation (5.23).

Revisiting observables in generally covariant theories in the light of gauge fixing methods · The observables associated with fields other than the lapse and shift · 0.82
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
With different notation, this expression ([ref]) appeared in the literature in [ref] as his equation (2.8) and in [ref] as her equation (5.23).

Observables in classical canonical gravity: folklore demystified · Constructing observables · 0.87

Cites it as support 1 paper

As has been shown in [ref] such observables $𝒪_f,T[τ]$ are gauge invariant, as we will discuss later.

Gauge invariant canonical cosmological perturbation theory with geometrical clocks in extended phase space - a review and applications · Relational formalism and Dirac observables · 0.99

Cites it as background 35 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

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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