Relational Evolution of Observables for Hamiltonian-Constrained Systems
Andrea Dapor, Wojciech Kamiński, Jerzy Lewandowski, Jedrzej Świeżewski
aa r X i v : . [ g r- q c ] O c t Relational evolution of observables for Hamiltonian-constrained systems
Andrea Dapor, ∗ Wojciech Kamiński,
1, †
Jerzy Lewandowski,
1, 2, ‡ and Jędrzej Świeżewski
1, § Faculty of Physics, University of Warsaw, Hoża 69, 00-681 Warszawa, Poland Institute for Quantum Gravity (IQG), FAU Erlangen – Nurnberg, Staudtstr. 7, 91058 Erlangen, Germany (Dated: October 17, 2018)Evolution of systems in which Hamiltonians are generators of gauge transformations is a notionthat requires more structure than the canonical theory provides. We identify and study this ad-ditional structure in the framework of relational observables (“partial observables”). We formulatenecessary and sufficient conditions for the resulting evolution in the physical phase space to bea symplectomorphism. We give examples which satisfy those conditions and examples which donot. We point out that several classic positions in the literature on relational observables containan incomplete approach to the issue of evolution and false statements. Our work provides usefulclarification and opens the door to studying correctly formulated definitions.
I. INTRODUCTION: RELATIONAL DIRACOBSERVABLES
This paper is about physical systems in which theHamiltonian is a generator of a gauge transformation.The most important example is general relativity. Thephysical evolution of those systems is a notion that re-quires more care than in a regular canonical theory andan additional structure. On one hand, theoretical physi-cists dealing with such systems usually know some meth-ods which work in the examples they are interested in.On the other hand, there is a diversity of formulationsof frameworks for the Hamiltonian-constrained systems[1–5], each one aiming to capture the peculiarity of thisgauge-time evolution in a general way. The formulationwe consider in this work originates from Rovelli’s idea of“partial observables” [6]. We prefer to call it “relationalobservables.” A systematic approach to the relationalobservables framework was developed in a series of pa-pers and books [7–10]. Unfortunately, we found an errorin those attempts [7]. The error has passed to the lit-erature and still requires a correction. We would like toemphasise that, in specific examples, the idea of the re-lational observables is usually applied in a correct way[11]. It is the general theory that needs our correction.We present it in our current paper.
1. Kinematical phase space
In the (classical) canonical framework, the states forma phase space Γ . The phase space is a manifold endowedwith a differential 2-form Ω – a symplectic form – whichby definition is closed, d Ω = , (I.1) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] and not degenerate, X ⌟ Ω = ⇒ X = , (I.2)at every point of Γ .
2. Constrained systems and gauge transformations
In the case of a constrained system, the physical statesare subject to constraints, and they form a constraintsurface Γ C contained in the phase space Γ C ⊂ Γ . (I.3)On the constraint surface, there is a naturally induced2-form Ω C , the pullback of Ω . Provided Ω C is nondegen-erate, the pair Γ C and Ω C becomes the physical phasespace. Often, however, the 2-form Ω C has null directions,i.e., there is a nonvanishing vector ℓ tangent to Γ C , suchthat ℓ ⌟ Ω C = . (I.4)In this case Ω C is degenerate; it cannot be a symplecticform, and Γ C cannot play the role of a physical phasespace. The maps Γ C → Γ C which define the flow of anull vector field ℓ are considered gauge transformations.The flow of each null vector field Lie-drags the 2-form Ω C , L ℓ Ω C = , (I.5)so Ω C is gauge invariant.
3. Physical phase space If ℓ and ℓ are two null vector fields, then so is theircommutator, [ ℓ , ℓ ] ⌟ Ω C = L ℓ ( ℓ ⌟ Ω C ) − ℓ ⌟ L ℓ Ω C = . (I.6)Therefore, the null directions define a foliation of Γ C .The set of the leaves of that foliation is the physical phasespace ¯Γ . We have the natural projection Π ∶ Γ C → ¯Γ . (I.7)We will be assuming that ¯Γ is a manifold and that thenatural projection is smooth. Then, there is a 2-form ¯Ω on ¯Γ such that Π ∗ ¯Ω = Ω C . (I.8)We will be also assuming that there is a global section ofthe projection Π , that is an embedding σ ∶ ¯Γ → Γ C , (I.9)such that combined with the natural projection Π , it isthe identity Π ○ σ = id . (I.10)In other words, our considerations will be local and willconcern generic points. We will not address the issues ofpossible singular, or even non-Hausdorf points of ¯Γ , orpossible topological nontriviality of the projection Π .The pair ( ¯Γ , ¯Ω ) is the physical phase space. Its pointscorrespond to the physical states, and ¯Ω − defines thephysical Poisson bracket: { ¯ f , ¯ g } phys = ( ¯Ω − ) IJ ∂ I ¯ f ∂ J ¯ g. (I.11)FIG. 1: The pictures illustrate the constructionpresented in the introduction. The picture (a) showsthe phase space, the constraint surface, orbits of gaugetransformations, the physical phase space, and theprojection onto it. The picture (b) shows the physicalphase space, the constraint surface, and the( τ -dependent) sections which embed one into the other.
4. Hamiltonians
The dynamics of a canonical theory is defined by aHamiltonian, a parameter t -dependent function H ( t ) de-fined on the phase space Γ . At every value of t , theHamiltonian H ( t ) defines a vector field X H ( t ) tangent to Γ , such that X H ( t ) ⌟ Ω = dH ( t ) . (I.12) For every function F defined on Γ , the Hamiltonian de-fines its evolution t ↦ F ( t ) via ddt F ( t ) = X H ( t ) ( F ( t )) , F ( t ) = F. (I.13)In the case of constraints, the vector field X H ( t ) is tan-gent to Γ C at every instant of t , and its flow preserves the Ω C including its null directions. The Hamiltonian H ( t ) is constant along the leaves of the null directions, ℓ ( H ( t )) = ℓ ⌟ dH ( t ) = ℓ ⌟ ( X H ( t ) ⌟ Ω ) = . (I.14)Therefore, the projection Π ∗ X H ( t ) defines a unique vec-tor field X ¯ H ( t ) on ¯Γ , H ( t ) defines a unique function ¯ H ( t ) on ¯Γ , and X ¯ H ( t ) ⌟ ¯Ω = d ¯ H ( t ) . (I.15)In fact, in a constrained system, the Hamiltonian vec-tor field X H ( t ) is defined on Γ C modulo a null vectorfield by a nonunique Hamiltonian (that explains the plu-ral “Hamiltonians”), but its projection onto the physicalphase space ¯Γ is unique. As a consequence, upon thosesmoothness assumptions, we end up with the physicalphase space, symplectic form, and, respectively, Hamil-tonian: ¯Γ , ¯Ω , ¯ H ( t ) .
5. Hamiltonian-constrained systems
There is a catch, however. In some constrained sys-tems, the Hamiltonian vector field X H ( t ) satisfies at ev-ery point of Γ C , X H ( t ) ⌟ Ω = . (I.16)Then, the projected Hamiltonian vector field is identi-cally zero: X ¯ H ( t ) = . (I.17)A system with this property is called Hamiltonian-constrained. An example is canonical general relativity,in which Hamiltonians (labelled by free functions – lapseand shift – representing the choice of space-time coordi-nates) are projected to in ¯Γ .
6. Relational Dirac observables
In order to introduce dynamics in the case of aHamiltonian-constrained system, one needs some extrastructure. Sometimes this extra structure is incorrectlycharacterized.We explain now this misunderstanding andidentify the correct structure. We start by introducingthe original construction used in the literature. Considera global section of the natural projection Π , σ ∶ ¯Γ → Γ ⊂ Γ C (I.18)where we denoted Γ ∶= σ ( ¯Γ ) . Every function f definedon Γ C determines a function ¯ f defined on ¯Γ by the re-striction of f to the slice Γ , ¯ f ∶= σ ∗ f. (I.19)Suppose T is a family of functions T m , m ∈ M , on Γ C such that the slice Γ is defined as their common zero set Γ = { γ ∈ Γ C ∶ T m ( γ ) = , m ∈ M } . (I.20)We can then consider any other slice defined by T m = τ m ∈ R ; (I.21)that is, Γ τ ∶= { γ ∈ Γ C ∶ T m ( γ ) = τ m , m ∈ M } . (I.22)In this way, for every family τ of numbers τ m ∈ T m ∶= ( τ m , τ m ) ⊂ R , the clock system T defines a global section σ τ . Functions T m are called in the literature clock func-tions. Then, for every τ we obtain from a function f afunction on ¯Γ ; generalizing Eq. (I.19), it is ¯ f τ ∶= σ ∗ τ f. (I.23)On the other hand, every function ¯ f on ¯Γ can be equiv-alently expressed by the function Π ∗ ¯ f on Γ C extendedarbitrarily to Γ whenever it is useful. Π ∗ ¯ f is gauge in-variant (i.e., constant along each null direction), and wecall it a (weak) Dirac observable. In the relational ob-servables literature, the Dirac observable correspondingto such a function ¯ f τ is denoted by the symbol F [ f,T ] ( τ ) ∶= Π ∗ ¯ f τ . (I.24)In this way, given a function f on Γ and a system T of clock functions T m , one defines for every family τ ofnumbers τ m the Dirac observable (I.24).
7. Gauge transforming a relational Dirac observable
Given a function f ∶ Γ C → R , the dependence on τ ofthe Dirac observable F [ f,T ] ( τ ) can be interpreted as anevolution, τ ↦ F [ f,T ] ( τ ) , (I.25)or in terms of the physical phase space, as an evolutionof the physical observable, τ ↦ ¯ f τ . (I.26)More generally, for every gauge transformation, that is,a map α ∶ Γ C → Γ C which preserves each null leaf, wecan define a transformed gauge invariant function: α ↦ F [ α ∗ f,T ] ( τ ) . (I.27) In this framework [7] one restricts to the gauge trans-formations preserving the foliation of Γ C defined by theclock system T , which is such that α ∗ T m = T m + τ ′ m , τ ′ m ∈ R , m ∈ M . (I.28)Then, the gauge transformation amounts to τ ′ ↦ F [ f,T ] ( τ + τ ′ ) . (I.29)Therefore, this definition does not really add to Eq. (I.25)anything more; however, we keep it for the consistencywith [7].
8. Dirac bracket
One more element bridging the kinematical phasespace Γ with the physical phase space ¯Γ with the help ofthe system T of the clock functions is the Dirac bracket, C ( Γ ) ∋ f, g ↦ { f, g } ∗ ∈ C ( Γ ) , which may be defined using Eq. (I.23) by the followingequality: { f, g } ∗ τ = { ¯ f τ , ¯ g τ } phys which is required to be true for every τ . The relationalconstruction of observables F [ ⋅ , ⋅ ] ( ⋅ ) , the Dirac bracket { ⋅ , ⋅ } ∗ , the Poisson bracket in Γ , and the “gauge trans-formations” (I.29) are consistent in the following way [7]: { F [ f,T ] ( τ + τ ′ ) , F [ g,T ] ( τ + τ ′ )} = F [{ f,g } ∗ ,T ] ( τ + τ ′ ) (I.30)[of course τ ′ could be erased without any loss of informa-tion, but we keep it for the consistency with Ref. [7] andEq. (8.20) therein].
9. Incorrect statements
It is often stated in the relational observables literatureabout this action of the gauge transformations (I.27, I.29)and the evolution (I.25, I.26) that:1. They naturally pass to a map defined in the set
D ⊂ C ( Γ C ) of the gauge invariant functions on Γ C , ˆ α ∶ D ∋ F [ f,T ] ( τ ) ↦ F [ α ∗ f,T ] ( τ ) ∈ D (I.31)by a fixed gauge transformation α , or, respectively,to a map ˆ α ∶ D ∋ F [ f,T ] ( τ ) ↦ F [ f,T ] ( τ + τ ′ ) ∈ D , (I.32)where τ ′ is defined by Eq. (I.28) (see Eq. (8.15) inRef. [7]).2. For every α or τ ′ , the corresponding maps (I.31,I.32) preserve the Poisson bracket { ⋅ , ⋅ } restrictedto D (see the interpretation of Eq. (8.20) in Ref.[7]).Let us explain why item 1 above is not true. Givena function F [ f,T ] ( τ ) on Γ C , the function f is neitherknown nor unique. The would-be evolution (I.31) wouldbe well-defined if the right-hand side of Eq. (I.31) wereindependent of that ambiguity. Unfortunately, the right-hand side does depend on that choice; given the function F [ f,T ] ( τ ) , we can choose functions f and f such that F [ f ,T ] ( τ ) = F [ f ,T ] ( τ ) = F [ f,T ] ( τ ) ; nonetheless, F [ f ,T ] ( τ + τ ′ ) ≠ F [ f ,T ] ( τ + τ ′ ) . The maps (I.31, I.32) are therefore ill-defined.Since item 1 above is not true, item 2 is pointless. Butit is even worse than that. Itself, the consistency Eq.(I.30) is true, and one could think that it will eventu-ally imply 2 as soon as the definition 1 is fixed. In thispaper we correct the idea of item 1 of the action of thegauge transformations in D . In a quite clear and naturalway, we fix item 1. We identify the additional structuresbehind the resulting gauge transformations induced in D , and we define and characterize all the possibilities.Ironically however, the resulting gauge transformationsin general do not satisfy item 2. We find the conditionsupon which 2 is satisfied and alternative conditions uponwhich 2 is not satisfied. Obviously Eq. (I.30) continuesto be true regardless of the case we consider. What iswrong is inferring II. CORRECTED APPROACH
Our aim is to use the idea of (I.25, I.26) to define anevolution in the physical phase space ¯Γ correctly. We justneed to modify the idea suitably, so that it leads to a con-sistent definition. Therefore, we use here the structuresand notation introduced in the previous section.
10. Reference functions
The weakness of the attempt to apply Eq. (I.26)was an overcompleteness of the set of all the functions f on Γ C [more precisely, of their restrictions to a slice σ τ ( ¯Γ ) ⊂ Γ C ]. To cure it, we choose a functionally com-plete, but not overcomplete , set of functions on Γ C . Thatis, we fix a system θ of functions θ I , I ∈ I , defined in Γ C ,such that their restrictions to the slices σ τ ( ¯Γ ) form a co-ordinate system for each value of τ . In other words, theassumption is that the pullbacks σ ∗ τ θ I onto the physicalphase space ¯Γ provide a coordinate system. We will call θ a system of reference functions.
11. Map ¯ f ↦ f With this additional structure θ , for every function ¯ f on ¯Γ , we can define a unique function f on Γ C , which will be used in Eq. (I.25). To do this, we write ¯ f in terms ofthe coordinates, ¯ f ( ¯ γ ) = ˜ f (( σ ∗ θ I )( ¯ γ )) , ¯ γ ∈ ¯Γ , (II.1)where ˜ f is thus defined uniquely. This leads to a uniqueextension f of the function ¯ f to Γ C , f ( γ ) = ˜ f ( θ I ( γ )) , γ ∈ Γ C . (II.2)This is an “extension” in the sense that the function f coincides with Π ∗ ¯ f on the slice σ ( ¯Γ ) , f ∣ σ ( ¯Γ ) = Π ∗ ¯ f ∣ σ ( ¯Γ ) . (II.3)
12. Relational evolution defined in C ( ¯Γ ) Finally, with C ( ¯Γ ) → C ( Γ C ) , (II.4) ¯ f ↦ f, (II.5)every τ defines a map C ( ¯Γ ) → C ( ¯Γ ) (II.6) ¯ f ↦ ¯ f τ ∶ = σ ∗ τ f. (II.7)This last map is an automorphism of the associative (notPoisson) algebra of functions as long as σ ∗ τ θ I are coor-dinates on ¯Γ . In particular, for τ = τ given by τ m = , m ∈ M , the map is the identity: ¯ f ↦ ¯ f τ = ¯ f . (II.8)In this way, we have completed our definition of the an-ticipated evolution (I.31) ( ¯ f , τ ) ↦ ¯ f τ (II.9)in the Poisson algebra C ( ¯Γ ) of functions on the physicalphase space ¯Γ .
13. Dependence of the evolution on the choice of θ The question we should ask is what the map (II.9) de-pends on. Our construction uses the system T of clockfunctions T m and the system θ of reference functions θ I (coordinates on the slices defined via T as T m = τ m ).Now, recall that on Γ C , we have gauge transformations,the maps α ∶ Γ C → Γ C preserving the leaves generatedby the null directions. Clearly every two pairs ( T, θ ) and ( T ′ , θ ′ ) which are gauge-equivalent (i.e., such that T ′ = α ∗ ( T ) and θ ′ = α ∗ ( θ ) for a gauge transformation α )define the same evolution (II.9). Ignoring possible topo-logical nontrivialities, every two systems T and T ′ are To convince the reader of this, consider ¯ f in C ( ¯Γ ) and twoassociated functions in C ( Γ C ) , f and f ′ , such that f = ˜ f ○ θ and gauge-equivalent. Therefore, without loss of generality,we can fix a clock function system T , and the freedomof choosing the reference function system θ is enough toconstruct all maps (II.9).
14. Relation between θ and ∂∂T We can do even better. Indeed, not all the informationinvolved in θ is relevant for the evolution (II.9). Given θ ,consider θ ′ such that θ ′ I ′ = ˜ θ ′ I ′ ( θ I ) . (II.10)The evolution (II.9) defined by θ ′ is the same as that de-fined by θ . The relevant part of the choice of θ is encodedin the vector fields ∂∂T m corresponding to the coordinatesystem set by the functions θ I , T m on the constraint sur-face Γ C . Conversely, for a given system T of clock func-tions, let us consider a set of vector fields ∂ m on Γ C satisfying ∂ m ( T n ) = δ nm , [ ∂ m , ∂ n ] = . (II.11)Due to the Frobenius theorem, locally, there is a localsolution θ I to the equations ∂ m θ I = , I ∈ I . That is enough if we keep ignoring the global nontriviali-ties. In this case, however, owing to Eq. (II.10), our def-inition of the evolution (II.7) extends consistently fromone local chart ¯ θ τ to another local chart θ ′ I ′ τ . III. SYMPLECTOMORPHICITY CONDITIONS
15. (Non)preserving of {⋅ , ⋅} phys As shown, item 1 of Sec. I 9 has been cured by theintroduction of a system θ of reference functions. Whatabout item 2? Does the map (II.7) preserve the physicalPoisson bracket (I.11) defined on C ( ¯Γ ) ? For every fixedfamily of numbers τ , we have defined coordinates on ¯Γ , ¯ θ Iτ ∶ = σ ∗ τ θ I . (III.1)In terms of them, the evolution (II.7) reads ¯ θ I ↦ ¯ θ Iτ . (III.2) f = ˜ f ′ ○ θ ′ , where the tilded functions are defined by Eq. (II.1)as ¯ f = ˜ f ○ θ ○ σ and ¯ f = ˜ f ′ ○ θ ′ ○ σ ′ . Then, saying that theevolution is the same is equivalent to saying that σ ∗ τ f = σ ′∗ τ f ′ ,or f ○ σ τ = f ′ ○ σ ′ τ . Now, recalling that σ τ = α ○ σ ′ τ and θ ′ I = θ I ○ α , we can write the following sequence of identities: f ′ ○ σ ′ τ = ˜ f ′ ○ θ ′ ○ α − ○ σ τ = ˜ f ′ ○ θ ○ σ τ = ˜ f ○ θ ○ σ τ = f ○ σ τ . In thesecond-to-last step, we used the fact that ˜ f ′ = ˜ f coming from ˜ f ○ θ ○ σ = ¯ f = ˜ f ′ ○ θ ′ ○ σ ′ = ˜ f ′ ○ θ ○ σ . The 2-form Ω C used in the previous section to define thephysical Poisson bracket can be written as Ω C = Ω IJ ( θ K , T n ) dθ I ∧ dθ J + dT m ∧ ( ω mI dθ I + ω mn dT n ) . (III.3)On the physical phase space ¯Γ , the formula for the phys-ical symplectic form ¯Ω reads ¯Ω = Ω IJ ( ¯ θ Kτ , τ n ) d ¯ θ Iτ ∧ d ¯ θ Jτ . (III.4)The physical Poisson bracket between two functions ¯ θ Iτ and ¯ θ Jτ reads { ¯ θ Iτ , ¯ θ Jτ } phys = ( Ω − ) IJ ( ¯ θ Kτ , τ n ) . (III.5)Therefore, the Poisson bracket is preserved if and onlyif the 2-form Ω C decomposed according to Eq. (III.3)satisfies ∂∂T n Ω IJ ( θ K , T m ) = . (III.6)Generically, this condition is not satisfied, and the map(II.7) is not a symplectomorphism; hence, item 2 is nottrue. In fact, in example 2 we will see that every arbi-trarily chosen family of maps C ( ¯Γ ) → C ( ¯Γ ) labelled by τ , ¯ f ↦ ¯ f τ can be obtained as the evolution (II.9).
16. Identities
The condition d Ω C = implies conditions on the termsof the decomposition (III.3) Ω C . In particular we have d ( θ ) ( Ω IJ dθ I ∧ dθ J ) = (III.7)where by d ( θ ) we denoted the part of the exterior deriva-tive involving only the derivatives ∂∂θ I . This condition isequivalent to simply d ¯Ω = . (III.8)Another condition is ( ∂∂T m Ω IJ ) dθ I ∧ dθ J = d ( θ ) ( ω mI dθ I ) (III.9)
17. Symplectomorphicity and Hamiltonians
Therefore, a reference system θ and a clock system T do satisfy the symplectomorphism condition (III.6) if andonly if d ( θ ) ( ω mI dθ I ) = . (III.10)In this case, there are (locally) defined Hamiltonians H m , ω mI dθ I = d ( θ ) H m ( θ, T ) , (III.11)which project to τ -dependent Hamiltonians on ¯Γ : ¯ H m ( τ ) = σ ∗ τ H m , m ∈ M . (III.12)We will go back to this and specifically to how the corre-sponding Hamiltonian vector fields fit into this approachafter the following three examples. IV. EXAMPLES
18. Example 1: Trivial evolution
Let ¯ θ I be coordinates on ¯Γ . Define the system θ ofreference functions to consist of the functions θ I = Π ∗ ¯ θ I . (IV.1)In terms of those functions Ω C = Ω IJ ( θ K ) dθ I ∧ dθ J . (IV.2)The resulting evolution is the identity ¯ θ Iτ = ¯ θ I . (IV.3)In this case we even did not need to fix on Γ C any system T of clock functions. The result is independent of thatchoice.
19. Example 2: Arbitrary evolution
Suppose there is given an arbitrary family of maps C ( ¯Γ ) → C ( ¯Γ ) , ¯ f ↦ ¯ f τ (IV.4)labelled by the family τ of numbers τ m such that:• the family τ is consistent with a system T of clockfunctions T m on Γ C ;• for τ = , the map (IV.4) is the identity;• ¯ f τ ( ¯ γ ) is differentiable in each τ m for every ¯ γ andevery differentiable function ¯ f .We construct now a system θ of reference functions θ I on Γ C , such that the corresponding evolution (II.9) willcoincide with ( ¯ f , τ ) ↦ ¯ f τ given by Eq. (IV.4).Let ¯ θ I be coordinates in ¯Γ . The map (IV.4) maps eachof them appropriately: ¯ θ I ↦ ¯ θ Iτ . (IV.5)The suitable reference functions are defined at each point γ ∈ Γ C as follows: θ I ( γ ) ∶ = ¯ θ Iτ ( Π ( γ ))∣ τ = T ( γ ) . (IV.6)
20. Example 3: Standard choice
In practice, we have at our disposal the auxiliary kine-matical phase space Γ endowed with coordinate system ( p χ , q χ ) , where χ runs over a labelling set X , such thatthe symplectic form takes the canonical form Ω = ∑ χ dp χ ∧ dq χ . (IV.7)Suppose that on the constraint surface Γ C the coordi-nates p m , m ∈ M ⊂ X (IV.8)are determined by the remaining coordinates. Split theset of coordinates accordingly: p χ = p i , p m q χ = q i , q m . (IV.9)Hence, p m ∣ Γ C = H m , m ∈ M (IV.10)where H m ∶ Γ → R is for every m ∈ M a function suchthat ∂ p n H m = , n ∈ M . Suppose that the index m ranging the subset M labelsalso the null directions tangent to Γ C . This is what hap-pens when Γ C is defined by first class constraints. Choosefor a system of clock functions T m = q m . (IV.11)Let us choose for reference functions θ I = q i , p j . (IV.12)The 2-form Ω C expressed by those functions is Ω C = dp i ∧ dq i + dT m ∧ dH m . (IV.13)Remarkably, somewhat for free, it satisfies the condition(III.6) for the corresponding evolution in ¯Γ to be Hamil-tonian. The corresponding Hamiltonians are the func-tions p m restricted to slices of Γ C such that T m ′ = const , m ′ ∈ M . This method is often used in practice (e.g., Refs. [11,12]) due to its simplicity and efficiency.
21. Example 4: Nontrivial clock function
The starting point is similar to the previous example,namely Ω = dp ∧ dq + dP ∧ dQ (IV.14)and Γ C is defined by the equation P − h ( q, p ) = . (IV.15)Let the clock function T = ˜ T ( q, p, Q, P ) , be defined in the whole Γ , and, conversely, Q = ˜ Q ( q, p, T, P ) . Finally, let θ I = q, p. Then, in the new coordinates q, p, T, P , Ω C = ( + h ,p ˜ Q ,q − h ,q ˜ Q ,p ) dp ∧ dq + ˜ Q ,T dh ∧ dT. The symplectomorphicity condition ( h ,p ˜ Q ,q − h ,q ˜ Q ,p ) ,T = , generically is not satisfied.
22. Message
We learn from the examples that our relational evo-lution can be trivial (example 1) or arbitrary (example2). If the gauge fixing functions are just some of thecoordinates and for the physical degrees of freedom wechoose another subset of the coordinate system in whichthe symplectic 2-form had the canonical form, then, ina case of first class constraints, the corresponding evolu-tion is symplectomorphic (example 3). Finally, if in ex-ample 3 we make a nontrivial choice of clock function butleave the coordinates parametrizing the physical degreesof freedom, then, generically, the symplectomorphicitycondition will be violated (example 4).
V. FROM ( T, θ ) TO THE GENERATORS OFEVOLUTION
23. Generators
Given a reference system θ and a clock system T on theconstraint surface Γ C , the corresponding evolution de-fined by the relational framework is generated just by thevector fields ∂∂T m , m ∈ M . What is nontrivial about thosevector fields are their ( τ -depending) projections onto thephysical phase space ¯Γ . They can be calculated by usingthe decomposition of ∂ T m into the null part and the parttangent to the slices T m ′ = τ m ′ , m ′ ∈ M , (V.1)that is, by ∂ T m = ∂ T m − X m + X m , (V.2) such that ( ∂ T m − X m ) ⌟ Ω C = , X m = X Im ∂ θ I . (V.3)The evolution defined by ( θ, T ) in C ( ¯Γ ) is generated bythe τ -dependent vector fields on ¯Γ labelled by m ∈ M , ¯ X m ( τ ) = Π ∗ ∂ T m ∣ T = τ = Π ∗ X m ∣ T = τ = X Im ( τ ) ∂ ¯ θ Iτ . (V.4)
24. Calculation of ¯ X m The vector field X m , can be calculated from Eqs.(III.3, V.3) by inverting the equality X Im Ω IJ = ω mJ (V.5)(notice that Ω IJ is necessarily invertible).
25. Additional conditions
There are additional consistency conditions on Ω IJ , ω mI , and ω mn , namely Eq. (V.3) implies ω nn = , ω mn − ω nm = X n ⌟ ω m (V.6)and d Ω C = in addition to Eqs. (III.7, III.9) imply ∂∂T m ω nI − ∂∂T n ω mI − ∂∂θ I ω [ mn ] = = ω [ mm ′ ,m ′′ ] . (V.7)
26. Symplectomorphisms once again
Now, in the case (III.11) considered before when thecorresponding evolution C ( ¯Γ ) → C ( ¯Γ ) preserved thePoisson bracket, we have ¯ X m ( τ ) ⌟ ¯Ω = ω mI ( T = τ ) d ¯ θ Iτ = d ¯ H m ( τ ) , (V.8)where the functions ¯ H m were defined on ¯Γ in Eq. (III.12).Hence, indeed, in this case ¯ X m ( τ ) are Hamiltonian vectorfields, and the Hamiltonians are the functions we hadalready defined before. VI. SUMMARY
We have fixed the inconsistent definition of a rela-tional evolution of the Dirac observables of Hamiltonian-constrained systems [Ref. [7], Eq. (8.15)]. To this end,in addition to a system, say, T of clock functions usedin the relational observables framework, we introduced asystem of reference functions, say, θ . With this structure,the gauge transformations preserving the constant-valuesurfaces of the clock functions do induce a family of move-ments of the physical phase space ¯Γ . In general and infact even generically, the movements are not symplecto-morphisms of ¯Γ (unlike what could be concluded fromRef. [7], Eq. (8.20), assuming that Ref. [7], Eq. (8.15) istrue). While the induced movements are defined by pairs ( T, θ ) , all the clock function systems T are gauge equiv-alent to each other. Therefore, one can fix any system ofclock functions on Γ C and vary only the reference func-tion systems. The reference function systems which doinduce symplectomorphisms in the physical phase space ¯Γ are characterized by some special form taken by the2-form Ω C . One can read off the corresponding Hamil-tonians in that case. On the other hand, every family ofmaps ¯Γ → ¯Γ (not necessarily symplectomorphic) labelledby the labelling set of the clock function system can beobtained as the relational evolution corresponding to asuitable reference function system. One of the exampleswe illustrate our construction with shows in what waya naive choice of the clock and, respectively, referencesystems provides a symplectomorphic relational evolu- tion. Another example shows that a nontrivial choiceof a clock function system induces in the physical phasespace a nonsymplectic evolution.Our work provides the correctly defined evolution ofHamiltonian-constrained systems. The issue of the physi-cal evolution of those systems is still an outstanding prob-lem of general relativity. We hope that our correction ofthe relational approach will help to solve this problem. VII. ACKNOWLEDGEMENTS
We benefitted from discussions with Alex Stottmeis-ter, Kristina Giesel, Hanno Sahlmann, and Thomas Thie-mann. This work was partially supported by the grant ofPolish Ministerstwo Nauki i Szkolnictwa Wyższego nr NN202 104838 and by the grant of Polish Narodowe Cen-trum Nauki nr 2011/02/A/ST2/00300. [1] P A M Dirac -
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