1. Review: Relational Observables in Quantum Gravity Donald Marolf UCSB May 23, 2007

2. Classical gravity is non-local? 1. Diffeomorphism Invariance means no local observables: E.g.,    . 2. Constraints! E.g., can measure H at boundary. But “easy” to gauge fix. Use “physical” coordinates attached to reference structure; E.g. GPS frame near earth. Note: may use arbitrarily small ripple … or asymptotic boundary (Minkowski,AdS) Classical gauge fixing fails only when both i. spacetime has symmetries and ii. Cauchy surfaces are compact. Generically, classical gravity is effectively local. Physical information is relational. Equations remain local (hyperbolic).

3. Quantum Gravity How local is it? How local is it? Can one repeat the relational story? (perturbative gravity, toy models) Can one repeat the relational story? (perturbative gravity, toy models) In which quantum states can one gauge fix? In which quantum states can one gauge fix? Can one explore locality without gauge-fixing? Can one explore locality without gauge-fixing? Classical gauge fixing fails only when both i. spacetime has isometries and ii. Cauchy surfaces are compact.

4. Relational Observables Relational info vs. Relational Observables (Operators) Observable : A gauge-invariant self-adjoint operator on the Hilbert space. Relational Observable : An observable which captures relational information. Canonical Example : 1+1 free relativistic particle toy model 0 = H = -P T 2 + P X 2 + m 2 X( ) = X 0 + P X, T( ) = T 0 + P T X( ) = X 0 + P X, T( ) = T 0 + P T = proper time, residual gauge symmetry:   + const = proper time, residual gauge symmetry:   + const A Classical Relational Observable : X when T = t [X] T=t = X 0 + (t-T 0 )P X /P T = X + (t-T)P X /P T Any quantum observable with such a classical limit is a relational quantum observable. E.g., different factor orderings of T/P T give different quantum relational observables with the same classical limit.

5. E.g., for free particle More complicated examples? Solving dynamics is hard… Solving dynamics is hard… Generalize? E.g., in [X] T=t, why force T = t precisely? Generalize? E.g., in [X] T=t, why force T = t precisely? Single Integral Obsevables M O = A(x) for some density A(x). take A( ) = X( )  (T – t) dT/d take A( ) = X( )  (T – t) dT/d get O = [X] T=t or smear out… take A( ) = X( ) f(T – t) dT/d take A( ) = X( ) f(T – t) dT/d get O = [X] T ~ t f

6. Field Theory M O = A(x) for some density A(x). Single Integral Obsevables If |  > has 1  -particle and 1  -particle, expect to describe  where they meet!   f Reference Fields: (Z-model) Reference Particles: Given scalars Z a (a=1,..,d) and , take A(x) =  (x) f(Z a -z a ) |dZ b /dx c | get O = [  ] Z a ~z a most useful in states w/ nice Z’s, but defined more generally Given scalars  take A(x) = -g  (x)   (x)   (x) Protolocal observables

7. Results  x  (L p (d-2) R 2 ) 1/d, # of cells ~ (R/L p ) (d-2)(d-1)/d << (R/L p ) (d-2) R (Giddings, Marolf, Hartle) Without Gravity: In a limit, protolocal correlators  local correlators. However, requires large energy density! However, requires large energy density! With Gravity (and avoid grav collapse): ``Instrument’’ a region R of finite size R Best resolution is Note: Region R need not be entire universe U. However, vacuum noise can be large when U is exponentially large! = dx 1 dx 2 = dx 1 dx 2 ~ dx 1 dx 2 ~ dx 1 dx 2 ~ const x Vol( U ) Note: const = 0 in pert. theory about static spacetimes. AdS, Minkowski may be OK, but not dS. Boltzmann

8. now more from Mike Gary…..