Thermally correlated states in Loop Quantum Gravity Goffredo Chirco Paola Ruggiero and Carlo Rovelli arXiv:1408.0121 [gr-qc] September 3rd, 2014 Experimental Search for Quantum Gravity
framework probe the structure of space-time near the Planck scale look for observational signatures of the quantum gravity regime on matter fields neutrinos oscillations Lorentz invariance violations cosmology, primordial particles modified dispersion tensor modes fluctuations relations: XRays, GRBs CMB polarization, parity invariance breaking … gedanken experimental settings in quantum gravity ? LQG provides a nice framework … space-time is a manifestation of a physical field => geometry - area, volume, angles - is a manifestation of the gravitational field: use quantum geometry as laboratory to characterize/extract properties of quantum gravitational field at fundamental level.
spacetime thermodynamics … statistical mechanics of quantum GR QFT mechanical space-time system ? use SM thermal features of quantum a cluster of unsolved problems * field theory in curved spaces relying on localization algebraic QFT, Von Neumann algebras and KMS theory, quantum entanglement thermodynamics in the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a ρ ∼ e − h mathematical object called a Kubo-Martin-Schwinger (KMS) state
Kubo-Martin-Schwinger thermal state Bisognano – Wichmann theorem (BW): [Bisognano and Wichmann, 1976] 4d Minkowski [Haag, 1967] t fixed time localization z the restriction of the vacuum of a Lorentz-invariant quantum field theory to the algebra of field operators with support on the Rindler wedge z > |t| is a KMS (Kubo-Martin-Schwinger) state: a thermal equilibrium state, with inverse temperature 2 π , with respect to the flow generated by the boost operator K in the z direction
this process gives temperature to BHs Kubo-Martin-Schwinger thermal state the source of this effect are the quantum correlations in the field: Minkowski vacuum is not separable: highly nonlocal space like correlations, high entanglement . restricted Minkowski vacuum state the KMS state is described by reduced density matrix Gibbs state Rindler Hamiltonian Unruh effect T = Unruh temperature
do not focus on the meaning of temperature. focus on the relation local quantum correlations in the field btw thermality, local lorentz symmetry and structurere of correlation at fundamental level. long lasting project a state on a 3d spatial surface Σ has the Bisognano – Wichmann property if in any sufficient small patch of Σ can be locally written as a KMS state for any 2d surface S, after tracing over the degrees of freedom on one side of S. The source of this effect are the local quantum correlations in the field interest in quantum gravity ? a full background-independent non-perturbative theory must include states yielding conventional physics at low energy, including quantum field correlations we already know a lot in terms coherent states the ultraviolet structure 1 h 0 | φ ( x ) φ ( y ) | 0 i ⇠ | x � y | 2 , of these correlations on no sense! a background geometry in a quantum gravity theory (such as loop quantum gravity): no background metric defining the distance
motivations but KMS definition should make naturally sense in the LQG theory PROPOSAL we can use the KMS as a (partial) characterisation of “ good ” semiclassical states in quantum gravity (assuming this holds true for semiclassical states, for these to yield the expected low energy phenomenology) also … possible relation btw the structure of correlations of the quantum * gravitational state and the smooth structure of space-time geometry at the classical level coherent states: the expectation value of the the gravitational field * appropriately matches a given smooth geometry. little is known about states where also the fluctuations of the gravitational field, and especially the nonlocal correlations, match the ones of conventional field theory. [Thiemann, Livine and Speziale, Bianchi, Magliaro, and Perini, Ashtekar]
experimental setting space-time near the Planck scale
[Ponzano and Regge in 1968] quantum system: covariant formulation of LQG covariant formulation of LQG aims to provide a realization of the path-integral R R over geometries for 4d Lorentzian gravity → U e − → ω − e ∧ e ∧ F ? + 1 Z Z j e , i v => => S [ e , ω ] = e ∧ e ∧ F . γ e − → B f . quantum numbers M 2-complex compute Z on a foam with boundaries, as a function of the boundary spin network states Γ ∈ ∆ ∗ , X Y Y Z = (2 j f + 1) A v ( j e , i v ) , v j f ,i e f Space-time as a superposition of spin foams, which is a generalized Feynman diagram space-time as a process Z is a discretization of the path Conrady-Freidel PRD ’09 integral for quantum gravity: in EB-Magliaro-Perini PRD ’10 a suitable semiclassical limit Magliaro-Perini CQG ’11 Conrady-Freidel PRD ’09 Bahr-Dittrich-Hellmann-Kaminsky ’12
spin network states quantum states of the gravitational field on a 3-dimensional hypersurface. Γ Γ M link U[ ω ] node Σ the theory is invariant under local SL(2,C) transformations SN states on Γ defined to be (cylindrical) function Ψ [ ω ] = ψ (U[ ω ]) of the holonomy U[ ω ] ∈ SL(2,C) of the spin connection along the L links l of the graph The SL(2,C) generators BIJ = − BJI, I,J = 0, … ,3 associated to each oriented link play the role of the basic observables of the theory
spin network states look at spin networks by themselves It is convenient to pick the time gauge, which ties the normal to Σ to a direction t in the internal Minkowski space. Then BIJ split into ~ rotation generators and boost generators ~ L K At each node, the vector t determines a subgroup SU(2) ⊂ SL(2,C) that leaves it invariant The Hilbert space H(p,k) that carries the (p,k) representation decomposes into irreducible representations of SU(2) H ( p,k ) = � ∞ j = k H j , where Hj is the SU(2) representation of spin j
Despite the open puzzles in the dynamical sector, at the kinematical level LQG furnishes a nice picture of quantum space; it allows for computation of spectra of various geometric operators such as area, volume and length. spin network boundary states within H(p,k) the physical subspace of the theory is determined (in a given Lorentz frame) by the linear simplicity condition restriction on the set of the unitary representations K = �~ ~ L and picks a subspace within each representation l [ ! ] = ⌦ l D ( γ ( j l +1) ,j l ) ( g l [ ! ]) => Ψ j l m l m 0 j l m l ,j l m 0 l naturally isomorphic to the space L 2 [SU(2)] L : ( h l ) = ⌦ l D ( j ) l ( h l ) hl ∈ SU(2) and Dj(h) are Wigner matrices m l ,m 0 Y γ : H j ! H ( γ ( j +1) ,j ) | j, m i 7! | ( � ( j + 1) , j ); j, m i . L 2 [SU(2)] L is a Hilbert space and this isomorphism endows the physical state space with the scalar product needed to define a quantum theory.
spin network boundary states further restriction: gauge invariant states form the Hilbert space H Γ links nodes O O H Γ = L 2 [ SU (2) L /SU (2) N ] D j l ( h l ) · ( h l ) = ◆ n ∈ n l <=> very robust at the kinematical level LQG furnishes a nice picture of quantum geometry of the spatial section Σ where the states are represented by SU(2) spin networks. it allows for computation of spectra of geometric operators such as area, volume and length [Ashtekar, Lewandowski, Rovelli, Smolin, Bianchi]
spin network statistical states? statistical mechanics of quantum mechanical systems describing 3d space hyper surfaces now recall: a state on a 3d spatial surface Σ has the Bisognano – Wichmann property if in any sufficient small patch of Σ can be locally written as a KMS state for any 2d surface S, after tracing over the degrees of freedom on one side of S. question: can we reproduce the argument for SN states: namely build KMS SN states, hence characterize local quantum correlation of space(-time)?
quantum 3d geometry ~ QFT quantum many body system consider a 3d subregion in the manifold Σ : the cellular decomposition induces on the boundary of the region a tessellation in 2d cells. In the dual picture these are links l crossing the surface Σ fixed time localization Γ ( Σ ) S Σ B A S local approach: single link connecting two adjacent nodes, disregarding gauge invariance
single link state Each link l is oriented: we call n_S (source) and n_T (target) its initial and final nodes. The corresponding facet is equally oriented and separates a source cell S from a target cell T X | i = c mn | jmn i mn n S n T the single link Hilbert space factorize at fixed quantum number j l S T relate it to a local “ partition of space ” in two chunks, associated to the source and target nodes at the two ends of the link Given a state in this subspace, we can trace on one factor and define a density matrix over the other. Explicitly, tracing on the target factor, gives X c nm c nm 0 | j, m ih j, m 0 | on H S ⇢ = Tr T | ih | ⌘ j . on ecaus n
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