Nijenhuis-Type Variants of Local Theory of Background Independence

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F ... , r := σ σ ( q ( pendence r p +¯ +¯ − p q ) multivector 1 ) ) ) t9,ec on each et[9], − , (3) (2) (1) Definition 2 The Nijenhuis–Lie derivative of P ∈ Xp with respect to V ∈ X1 is N £V P := [ V , P ]SN . (4) Structure 3 The algebras formed by equipping (graded) vector spaces V with SN brackets are a subcase of Gersten- haber algebras (defined in Sec 4), so we refer to them as SNG-algebras. Let us finally also extend consideration from algebras to algebroids [21], using the phrase ‘algebraic structures’ as a portmanteau of the two. Each of classical deformation theory [21, 29], kinematical quantization [19], and GR producing a constraint algebroid – the Dirac algebroid [6] – even before either of the previous are involved, justify this more generalized scope. SNG-algebroids and SNG-algebraic structures are thus in play.

3 Nijenhuis Local Background Independence

1) We here employ Nijenhuis–Lie derivatives (4) to encode Relationalism. A) In the canonical case, we work with changes of conﬁguration Q. (5) ˙ in place of velocities Q = Q. /dt so as to stay free from time variables for the reasons given in Article I of [28]. B) We correct by Nijenhuis–Lie derivative along physically irrelevant group g’s changes a. , Q −→ Q − NQ . . £a. . (6) C) We know what form these corrections take by solving the generalized Killing–Nijenhuis equation £Nσ =0 (7) for geometrical level of structure σ to obtain the corresponding physically irrelevant automorphism group in question, g. D) We complete this with a move using all of g to obtain g-invariant objects. [C.f. group averaging or Article II of [28] for a detailed review of all of B) to D)]. 2 In the spacetime counterpart, we are free to use plain auxiliary corrections A in place of change corrections a. on spacetime objects S: N S −→ S − £AS , (8) with steps B) and C) then applying unaltered.

2) Closure is assessed using a) the SN bracket (2). b) An ‘SN Algorithm’ analogue of the Dirac [6] and Lie [1, 30] Algorithms. This permits six types of equation to arise from an initial set of generators G or constraints C, as follows. i) Inconsistencies: equations reducing to 0 = 1 as envisaged by Dirac [6]. ii) Identities: equations reducing to 0 = 0. ′ ′ iii) New secondary generators G or secondary constraints C . iv) ‘SN speciﬁer equations’ are also possible if there is an appending process. I.e. a generalization of Dirac’s appending of constraints to Hamiltonians H using Lagrange multipliers Λ, i.e. H −→ H + Λ · C , (9) by which equations relating these a priori for Λ can also arise from one’s algorithm. v) Rebracketing using SN–Dirac brackets in the event of encountering SN-secondary objects, i.e. ones that do not close under SN brackets. vi) Topological obsruction terms: Nijenhuis Mathematics’ analogue of anomalies.

Termination occurs if the SN Algorithm (using [30]’s terminology) 0) hits an inconsistency, I) cascades to inconsistency, II) cascades to triviality, or III) arrives at an iteration producing no new objects while retaining some degrees of freedom. Successful candidates – terminating at iii) – produce an ‘SNG’ class of Gerstenhaber algebraic structures of generators gSNG , (10) or of SN-ﬁrst-class constraints, F (those that do close under SN brackets) in the canonical case.

3) We next consider observables O [24, 3, 15, 26] deﬁned in the present context by zero-commutant SN brackets with the generators, G (including with the F constraints as a subcase) G O [ , ]SN ‘=’ 0 . (11) 2We consider both canonical and spacetime settings for the structures discussed in the current Article; see e.g. [26, 28] for discussions of the merit of each of these positions, as well as for inter-relations between these two positions.

2 ‘=’ is here the portmanteau of Dirac’s notion of weak equality ≈ [6] – from equality up to a linear combination of constraints to [30] equality up to a linear combination of generators – and of strong vanishing: the standard notion of equality, =. In the constrained canonical subcontext, these are zero-commutants of the SN-ﬁrst-class constraints. In this context, these equations can moreover be recast as explicit PDEs to be solved using the Flow Method [12, 23]. This particular application gives an SN integral theory of invariants. I.e. the SN version of Lie’s Integral Approach to Invariants [1, 30], albeit in both cases elevated to restricting functional dependence on a function space over the phase space geometry in question. The spacetime version is, rather, over the space of spacetimes. Such functions are, in each case, observables. These functions moreover constitute SNG-algebras: observables SNG-algebras ObsSNG , (12) whether canonical or spacetime. Each theory’s lattice of SNG generators or constraints subalgebraic structures additionally induces a dual lattice of SNG observables subalgebras. (If the reader is unsure what this means, they should consult the third Article of [28] for the usual Lie subcase.)

4) Deformation of SNG-algebraic structures encountering Rigidity [8, 9, 21] gives a means of Constructing more structure from less. This generalizes, ﬁrstly, Spacetime Construction from space’s ‘passing families of theories through the Dirac Algorithm’ approach. Secondly, obtaining more structure from less for each of space and spacetime separately. (See e.g. Article IX in [28] for Dirac and Lie Mathematics examples of each of these). On the one hand, we can pose rigidity for SNG-algebraic structures – that under deformations of generators G −→ Gα = G + α φ . (13) for parameter α and functions φ, the cohomology group condition H2(gNSG, gNSG)=0 (14) diagnoses rigidity. On the other hand, we are currently rather lacking in theorems for this SNG-algebra case. The idea is moreover to take Rigidity to be [29] a selection principle in the Comparative Theory of Background Independence.

5) Reallocation of Intermediary-Object (RIO) Invariance gives the general SNG-algebraic structures’ commuting- pentagon analogue of posing Refoliation Invariance for GR. RIO refers to whether, in going from an initial object to a ﬁnal object, proceeding via intermediary object 1 or intermediary object 2 causes one to be out by at most just an automorphism of the ﬁnal object, fin − fin fin O21 O12 = Aut(O ) . (15) Refoliation Invariance is then the case in which our objects are spatial slices within a given spacetime. Consult Article III in [28] for further details about Refoliation Invariance, or [29] and Article XII in [28] for RIO Invariance more generally. This is again a selection principle whose outcome remains unexplored in the SN case.

4) and 5) are thus identiﬁcations of new research directions, whereas 1), 2) and 3) constitute new conﬁrmed structures and results.

4 Conclusion

We have considered Nijenhuis Mathematics, with concrete focus on Schouten–Nijenhuis brackets and subsequent algebraic structures and algorithms. This is a useful generalization and robustness check – a natural next port of call in developing the Comparative Theory of Background Independence – and also a stepping stone to further cases of interest that follow from introduction of each of the following two further structural elements.

A) The Vinogradov bracket [13, 20] [,]V uniﬁes the Schouten–Nijenhuis and Frölicher–Nijenhuis brackets. B) The general Gerstenhaber bracket [7, 25], is a bilinear product taking one between spaces of k-linear maps HCk(V ) of a graded vector space V: p¯ × q¯ −→ p¯+¯q [,]G : HC HC HC . (16) Like the SN bracket, this is of graded Lie bracket type, thus obeying graded-antisymmetry and graded-Jacobi identities similar to (2, 3). Such k-linear maps are moreover the basic objects entering Hochschild cohomology [25, 18], which then plays a major role in Deformation Quantization [11, 25, 17] and the theory of quantum-mechanically relevant operator algebras [19].

A) and B) are known to support, ﬁrstly, Vinogradov algebraic structures and Gerstenhaber algebraic structures. Secondly – as new results – a Vinogradov Algorithm and a Gerstenhaber Algorithm: analogues of the Dirac and Lie Algorithms. Thirdly – as new notions (as far as the author is aware) – Vinogradov and Gerstenhaber notions of observables. I.e. the zero commutants with the corresponding type of algebraic structure’s generators under the corresponding deﬁning type of bracket. Fourthly, one can moreover at least pose Vinogradov and Gerstenhaber

3 deformations (not to be confused with Gerstenhaber having already worked out the deformations of simpler algebras). And then ask, as a selection principle, which subcases of each exhibit Rigidity. Finally, one can likewise pose RIO Invariance in both the Vinogradov and general Gerstenhaber contexts.

Acknowledgments I thank various friends for support and proofreading.

References

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