Local and gauge invariant observables in gravity arXiv:1503.03754 Igor Khavkine Department of Mathematics University of Trento 30 May 2015 LQP36 Leipzig
The need for local observables Consider a Classical or a Quantum Field Theory on an n -dim. spacetime M . ◮ In QFT, � ˆ φ ( x )ˆ φ ( y ) � is singular for some pairs of ( x , y ) . ◮ In classical FT, { φ ( x ) , φ ( y ) } is singular for some pairs of ( x , y ) . ◮ Instead, use smearing � φ ( x ) α ( x ) d ˜ φ (˜ α ) = x M so that � ˆ α )ˆ φ (˜ α ) , φ (˜ φ (˜ β ) � and { φ (˜ β ) } are always finite, provided α , ˜ ◮ ˜ β are smooth n -forms on M , α , ˜ ◮ ˜ β have compact supports. ◮ Smoothness diffuses singularities. Compactness ensures convergence of all integrals. ◮ Support of a functional: supp φ (˜ α ) = supp ˜ α ⊂ M . Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 1 / 11
Local observables ◮ Field φ is a section of some bundle π : F → M ( π k : J k F → M ). ◮ Local observables may be non-linear and depend on α = α ( x , φ ( x ) , ∂φ ( x ) , · · · ) d ˜ x on J k F derivatives (jets). An n -form ˜ � ( j k φ ) ∗ ˜ defines a local observable A φ = α, M provided supp A φ = π k supp ˜ α is compact! Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 2 / 11
Local observables in gauge theory (no gravity) ◮ Let G be the group of gauge transformations . ◮ Gauge transformations g ∈ G act on J k F (hence j k φ �→ g · j k φ ). ◮ No gravity: G fixes the fibers of π k : J k F → M . � ( j k φ ) ∗ ˜ A φ = α M is G -invariant provided g ∗ ˜ α = ˜ α + d ( · · · ) and supp A φ is compact! Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 3 / 11
No (such) local observables in gravity ◮ Gravity is General Relativity (GR), F = S 2 T ∗ M , G = Diff ( M ) . ◮ Diffeomorphisms do not fix the fibers of π k : J k F → M . In fact, diffeomorphisms act transitively on these fibers. ◮ M is never compact, as needed by global hyperbolicity . supp A φ = supp ˜ α compact ⇓ g ∗ ˜ α � = ˜ α + d ( · · · ) ⇓ � ( j k φ ) ∗ ˜ A φ = α M is not G -invariant ! Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 4 / 11
Relaxing locality: an explicit example ◮ Take dim M = 4. Write the dual Weyl tensor as ∗ cd = W abc ′ d ′ ε c ′ d ′ cd = ε aba ′ b ′ W a ′ b ′ cd . W ab ◮ Make use of curvature scalars (Komar-Bergmann 1960-61) b 1 = W ab b 3 = W ab cd W cd ab , cd W cd ef W ef ab , cd ∗ ef ∗ b 2 = W ab b 4 = W ab ab , cd W cd ab . W cd W ef ◮ Let ϕ be a generic metric (det | ∂ b i /∂ x j | � = 0) and let β = ( b 1 [ ϕ ]( x ) , b 2 [ ϕ ]( x ) , b 3 [ ϕ ]( x ) , b 4 [ ϕ ]( x )) for some x ∈ M . ◮ Take a : R 4 → R , with sufficiently small compact support containing β , let α = a ( b ) d b 1 ∧ d b 2 ∧ d b 3 ∧ d b 4 on J k ≥ 2 F ˜ � ( j k φ ) ∗ ˜ A φ = α. and M ◮ A φ is well-defined on a Diff -invariant neighborhood U ∋ ϕ among all metrics φ such that R [ φ ] ab = 0. A φ is Diff -invariant . δ A ′ δ A φ ◮ Peierls bracket well defined: { A , A ′ } φ = � δφ ( x ) · E φ ( x , y ) · δφ ( x ) . φ M × M Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 5 / 11
Some history of the idea ◮ Komar, Bergmann: PRL 4 432 (1960), RMP 33 510 (1961) Curvature scalars as coordinates, example with ( b 1 , b 2 , b 3 , b 4 ) . ◮ DeWitt: Ch.8 in Gravitation: Intro. Cur. Ris. (1963), L. Witten (ed.) Applied K-B idea to GR+Elasticity (matter as coordinates), computed Poisson brackets by Peierls method. ◮ Brown, Kuchaˇ r: PRD 51 5600 (1995) More matter (dust) as coordinates. ◮ Rovelli, Dittrich: PRD 65 124013 (2002), CQG 23 6155 (2006) Conceptual interpretation in terms of ‘partial’ observables, fields as coordinates in Hamiltonian formalism. ◮ Giddings, Marolf, Hartle: PRD 74 064018 (2006) Explicit perturbative computation on de Sitter, pointed out IR problems. ◮ Brunetti, Rejzner, Fredenhagen: [arXiv:1306.1058v4] (Apr 2015) Recalled K-B, B-K, R-D ideas in the context of the BV method. Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 6 / 11
New notion of local and gauge invariant observables ◮ Moduli space M k = J k F / G R − J k F , quotient by gauge sym-s. ← α = R ∗ ˜ β for some n -form ˜ ◮ Differential invariant ˜ β on M k . M ( j k φ ) ∗ ˜ α , with j k φ ( M ) ∩ supp ˜ ◮ A φ = � α compact for every φ ∈ U . ◮ A φ may be defined only on an open subset U ⊂ S of (covariant) phase space. Local charts! ◮ NB: Two metrics φ and ψ are Diff -equivalent iff R φ = R ψ in M k . Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 7 / 11
Differential invariants of fields (algebra) ◮ In any gauge theory, the group G of gauge trans. acts on J k F . ◮ Differential invariants : scalar G -invariant functions on J k F . ◮ Theorem (Lie-Tresse 1890s, Kruglikov-Lychagin 2011): ◮ (generically) all differential invariants (all k < ∞ ) are generated by ◮ a finite number of invariants and ◮ a finite number of differential operators satisfying ◮ a finitely generated set of differential identities . ◮ Examples ◮ Non-gauge theory: every function on J k F . ◮ Yang-Mills theory: invariant polynomials of curvature d A A . ◮ Gravity: curvature scalars, built from Riemann R , ∇ R , ∇∇ R , . . . α = a ( b 1 , . . . , b m ) d b 1 ∧ · · · ∧ d b n , ◮ Gauge invariant observables: let ˜ for some a : R m → R and differential invariants b i , i = 1 , . . . , m ≥ n , � ( j k φ ) ∗ ˜ then A φ = α is well-defined and gauge invariant , M provided supp [( j k φ ) ∗ ˜ α ] is compact. Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 8 / 11
Moduli spaces of fields (geometry) ◮ In any gauge theory, the group G of gauge trans. acts on J k F . ◮ Moduli space : quotient space M k = ( J k F \ Σ k ) / G ( Σ k is singular). ◮ Differential invariants are coordinates, separating points, on M k . ◮ Denote by R : J k F → M k the quotient map. Two (generic) field configurations φ and ϕ are gauge equivalent iff the images of R φ ( M ) and R ϕ ( M ) coincide as submanifolds of M k (for high k ). ◮ Differential identities among differential invariants define a PDE E k on n -dimensional submanifolds of M k , identifying submanifolds like R φ ( M ) . ◮ Finite generation means that there exists a k ′ such that all M k and E k ( k > k ′ ) can be recovered from M k ′ and E k ′ . α on M k and U such that φ ∈ U ◮ Choose compactly supported n -form ˜ implies R φ ( M ) ∩ supp ˜ α is compact. Then U is G -invariant, � ( j k φ ) ∗ R ∗ ˜ A φ = α is well-defined and gauge invariant , M and the A φ separate G -orbits in U . Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 9 / 11
Precise results, main limitations ◮ Goal: Subset of C ∞ ( S ) of gauge invariant fun-s, separating the G -orbits. ◮ The idea of generalized local observables has been around for a while. Can they give a complete solution? Not quiet. ◮ (1) Problem with highly symmetric configurations. Invariants do not separate all G -orbits. ◮ YM: non-trivial local holonomy. ◮ GR: Killing isometries. ◮ (2) Problem with infinitely repeating, nearly equivalent configurations. The integrals diverge. ◮ Good news! (IK [arXiv:1503.03754]) ◮ (1) and (2) are the only obstacles, generic configurations avoid them. ◮ Orbits of generic configurations are separated. ◮ A generic configuration has a neighborhood of generic configurations. ◮ Challenge: Precisely characterize generic configurations. ◮ (1) is easy: jet transversality theorem. ◮ (2) is harder: requires a variation on the density of embeddings. Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 10 / 11
Conclusion ◮ Local gauge invariant observables are important in both Classical (non-perturbative construction) and Quantum (perturbatively renormalized) Field Theory. ◮ Usual restriction on “compact support” excludes gravitational gauge theories. ◮ Relaxing the support conditions opens the door to a large class of gauge invariant observables (even for gravitational theories), defined using differential invariants or moduli spaces of fields. They separate gauge orbits on open subsets of the phase space. ◮ The Peierls formalism computes their Poisson brackets. ◮ Limitations: ◮ Observables may not be globally defined on all of phase space. ◮ Naive approach separates only generic phase space points (e.g., metrics without isometries and without near periodicity). ◮ Need to connect with operational description of observables. Igor Khavkine (Trento) GR Observables LQP36 30/05/2015 11 / 11
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