Thermally correlated states in Loop Quantum Gravity Goffredo Chirco - - PowerPoint PPT Presentation

Goffredo Chirco

Thermally correlated states in Loop Quantum Gravity

Paola Ruggiero and Carlo Rovelli September 3rd, 2014

Experimental Search for Quantum Gravity

arXiv:1408.0121 [gr-qc]

framework

look for observational signatures of the quantum gravity regime on matter fields probe the structure of space-time near the Planck scale Lorentz invariance violations

particles modified dispersion relations: XRays, GRBs

cosmology, primordial tensor modes fluctuations

CMB polarization, parity invariance breaking…

neutrinos oscillations 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. gedanken experimental settings in quantum gravity ? LQG provides a nice framework…

QFT GR SM

thermal features of quantum field theory in curved spaces

spacetime thermodynamics…

*

?

a cluster of unsolved problems

ρ ∼ e−h

localization algebraic QFT, Von Neumann algebras and KMS theory, quantum entanglement thermodynamics statistical mechanics of quantum mechanical space-time system use relying on 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 mathematical object called a Kubo-Martin-Schwinger (KMS) state

Kubo-Martin-Schwinger thermal state

Bisognano–Wichmann theorem (BW): z t 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 2π, with respect to the flow generated by the boost operator K in the z direction fixed time localization 4d Minkowski

[Bisognano and Wichmann, 1976] [Haag, 1967]

restricted Minkowski vacuum state

.

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

Rindler Hamiltonian T = Unruh temperature

Gibbs state

Unruh effect the KMS state is described by reduced density matrix

this process gives temperature to BHs

local quantum correlations in the field

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 the ultraviolet structure

• f these correlations on

a background geometry

h0|φ(x)φ(y)|0i ⇠ 1 |x y|2 ,

no sense! in a quantum gravity theory (such as loop quantum gravity): no background metric defining the distance

do not focus on the meaning of temperature. focus on the relation btw thermality, local lorentz symmetry and structurere of correlation at fundamental level. long lasting project we already know a lot in terms coherent states

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)

motivations

KMS definition should make naturally sense in the LQG theory PROPOSAL 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]

but

space-time near the Planck scale

experimental setting

M

[Ponzano and Regge in 1968]

covariant formulation of LQG aims to provide a realization of the path-integral

• ver geometries for 4d Lorentzian gravity

compute Z on a foam with boundaries, as a function of the boundary spin network states

Z = X

jf ,ie

Y

f

(2jf + 1) Y

v

Av(je, iv),

Z is a discretization of the path integral for quantum gravity: in a suitable semiclassical limit

quantum system: covariant formulation of LQG

Γ

∈ ∆∗,

space-time as a process

Space-time as a superposition of spin foams, which is a generalized Feynman diagram

Conrady-Freidel PRD’09 EB-Magliaro-Perini PRD’10 Magliaro-Perini CQG’11 Bahr-Dittrich-Hellmann-Kaminsky ’12

Conrady-Freidel PRD’09

2-complex

ω −

→ Ue

− →

e −

→ Bf .

R R S[e, ω] =

Z

e ∧ e ∧ F? + 1 γ

Z

e ∧ e ∧ F .

je, iv

=> =>

quantum numbers

quantum states of the gravitational field on a 3-dimensional hypersurface. SN states on Γ defined to be (cylindrical) function Ψ[ω] = ψ(U[ω])

• f the holonomy U[ω] ∈ SL(2,C) of the spin connection along the L

links l of the graph

spin network states

Γ

M Σ link node Γ 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 U[ω]

the theory is invariant under local SL(2,C) transformations

The Hilbert space H(p,k) that carries the (p,k) representation decomposes into irreducible representations of SU(2)

spin network states

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 ~

K ~ L

At each node, the vector t determines a subgroup SU(2) ⊂ SL(2,C) that leaves it invariant

look at spin networks by themselves

H(p,k) = ∞

j=kHj,

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

~ K = ~ L

within H(p,k) the physical subspace of the theory is determined (in a given Lorentz frame) by the linear simplicity condition

Ψjlmlm0

l[!] = ⌦l D(γ(jl+1),jl)

jlml,jlm0

l

(gl[!])

naturally isomorphic to the space L2 [SU(2)]L :

(hl) = ⌦l D(j)

ml,m0

l(hl)

=>

restriction on the set of the unitary representations and picks a subspace within each representation hl ∈ SU(2) and Dj(h) are Wigner matrices

Yγ : Hj ! H(γ(j+1),j) |j, mi 7! |((j + 1), j); j, mi.

L2 [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.

further restriction: gauge invariant states form the Hilbert space HΓ

HΓ = L2[SU(2)L/SU(2)N]

spin network boundary states

(hl) = O

l

Djl(hl) · O

n

◆n

∈

nodes links

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] very robust at the kinematical level

<=>

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. now recall: question: can we reproduce the argument for SN states: namely build KMS SN states, hence characterize local quantum correlation of space(-time)?

spin network statistical states?

statistical mechanics of quantum mechanical systems describing 3d space hyper surfaces

Γ (Σ) S A B

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 Σ S local approach: single link connecting two adjacent nodes, disregarding gauge invariance

l

S T

nT nS

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

single link state

the single link Hilbert space factorize at fixed quantum number j

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

| i = X

mn

cmn|jmni

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

⇢ = TrT | ih | ⌘ X

n

cnmcnm0|j, mihj, m0|

• n HS

j .

ecaus

• n

Since the restriction of to Hj is given by , because of the simplicity conditions, we can define the density matrix

Bisognano - Wichmann property on a single link

e2⇡ ~

K·~ z =

X

m

e2⇡m|j, mihj, m|.

where here |j,m⟩ is a basis of eigenstates of

trT [| ih |] = e2⇡ ~

KS·~ n.

trS[| ih |] = e2⇡ ~

KT ·~ n0.

We say that a link state ψ with spin j has the Bisognano-Wichmann property if there are and such that

~ n ~ n0

definition

~ L · ~ z ~ K ~ L

let’s find a class of states {|ψ⟩} satisfying the conditions above

thermal when traced

• n either side

we would like this state to be KMS with respect to local Lorentz boost flow…

Bisognano-Wichmann states on a single link

for a given , set . Sandwiching between eigenstates of gives the equation

· hjm | trT [| ih |] |jm0i = e2⇡mmm0

( )

X

n

cnmcnm0 = e2⇡m mm0. cc† = ΛΛ†

solved for any unitary matrix U by c = ΛU

cmn = X

k

D(~ n)mke⇡kUkn cmn = X

k

Vmke⇡kD†(~ n0)kn | i = X

mnl

D(~ n)lme⇡mD†(~ n0)mn|j, l, ni

| j~

n~ n0i =

X

m

e⇡m|j, mi~

n ⌦ |j, mi~ n0

=> wide class of (spin j) states labelled by two arbitrary vectors and the SU(2) representation j, that satisfy the Bisognano-Wichmann property

~ z ~ n = ~ n0 ≡ ~ z ~ KS · ~ z

eigenbasis transform

=>

how coherent?

estimate the mean value and the dispersion of the geometrical operators on the states: The mean value, properly normalized, reads

~ LS ⌘ h~ LSi Nj = Pj

m=j e2⇡mm

Pj

m=j e2⇡m

~ z.

to understand if the state become sharp for large j, we look at the relative dispersion of in the plane orthogonal to the direction identified by the mean value.

hLxi = hLyi = 0, (Lx) h~ L2i = (Ly) h~ L2i = 1 2 h(~ L2 L2

z)i

h~ L2i .

) i = 1 2 j(j + 1)

Pj

m=j m2e2γπm

Pj

m=j e2γπm

j(j + 1) ,

which goes to zero in the limit j → ∞

~ LS

with

the vector operator points in the direction identified by the state!

, ~ LS ⇠ j + O(j). given by:

it scales correctly with j in the large j limit

good semiclassical behaviour - overcomplete basis

BW link states form an overcomplete basis for each j, in the Hilbert space Hj ⊗ Hj∗: the resolution of the identity is given by

Ij = d2

j

(4⇡)2Nj Z

S2 d2~

n Z

S2 d2~

n0 | j~

n~ n0ih j~ n~ n0|

it indicates that every state can be expressed as a superposition of states with semiclassical labels.

~ n ~

n0

integration is over the two-sphere of the normalized vectors , , with the standard R3 measure restricted to the unit sphere

now we want to extend the definition from a single link state to a full spin network state: combine Bisognano-Wichmann states associated to the links that join on a single node n

single node: take the tensor product of a Bisognano-Wichmann link state per each of the links meeting at n and project on the SU(2) gauge invariant subspace

thermally correlated spin network states

The projection is performed by integrating over the local gauge group SU(2),

|Ψ(n)

jl,~ nl,~ n0

li =

Z dh O

l2n

D(h)| jl~

nl~ n0

li.

The BW graph state is then determined by a spin associated to each link and two vectors and ′ associated to the source and the target of each link

|Ψjl,~

nl,~ n0

li =

Z Y

n

dhn O

l⌘hnl,n0

li

Djl(hnl)Djl†(hn0

l)| jl~

nl~ n0

li

full spin network gauge invariant state

~ n ~ n0

statistical coherent states

crucial difference with the common intrinsic Livine-Speziale states on the graph

Hjl = O

n

Hn

Hn = InvSU(2)[ O

l

Hjl].

|jl,~ nl,~ n0

li =

O

n

|◆n

jl,~ nl,~ n0

li

Livine-Speziale states are tensor states with respect to this decomposition

the space of the states with fixed spin is the tensor product of one intertwined space per node with

|jl,~ nl,~ n0

li

does not act as a rotation of the vectors , since the adjoint representation acts with the same group element, whereas here we have two different SU(2) elements ( ). the density matrix in general is not pure!

[D(h)e2⇡~

Ll·~ nlD†(˜

h)]

h, ˜ h ~ nl

BW states do not factorize: reduce the density matrix of the full state to the intertwined space Hn, by tracing over the external representation spaces of the links

⇢ = tr[|Ψ(n)

jl,~ nl,~ n0

lihΨ(n)

jl,~ nl,~ n0

l|] =

= O

l2n

Z dh Z d˜ h [D(h)e2⇡~

Ll·~ nlD†(˜

h)]

the combination

statistical coherent states

=> BW spin network states carry nontrivial quantum correlations

what about not adjacent nodes? computed numerically the correlations on a chain of N = 7 nodes, fixing all jl = 1/2

probe correlations for adjacent nodes

correlations are non-vanishing. This is the main property we were seeking.

hP (0)

i

P (0)

j

i hP (0)

i

ihP (0)

j

i

P(0) = |ι0⟩⟨ι0| is the projector on the first element of the recoupling basis on the i-th node

~ L(a) · ~ L(b)

nA nB ~ L(1)

S

~ L(2)

S

~ L(3)

T

~ L(4)

T

l1 l2 l3 l4

probe the correlation looking at the dihedral angle observable on a simple graph (dipole graph)

h(~ L(1)

S

· ~ L(2)

S )A(~

L(3)

T

· ~ L(4)

T )Bi

6= h(~ L(1)

S

· ~ L(2)

S )Aih(~

L(3)

T

· ~ L(4)

T )Bi.

1.5 2.0 2.5 3.0 3.5 4.0 N N1 N2 107 105 0.001 PN1PN2c

• FIG. 2. Fit of the correlation function (hP(N1)P(N2)ic =

hP(N1)P(N2)i hP(N1)ihP(N2)i ) as a function of the dis- tance between nodes (∆N = N1 N2). The scale is linear

• logarithmic. Fit model: f(∆N) = a exp(b∆N). Fit results:

a = 0, 73; b = 5, 20.

correlations interpreted as the result the interplay between the thermal correlations

• n the single links and the effect of gauge-invariance at nodes, which ties the links

in quadruples and allows for the propagation of the those thermal correlations among far nodes coupling via gauge!

correlations behaviour for not adjacent nodes

SUMMARY and DISCUSSION We have defined a family of states in loop quantum gravity which are peaked on an (intrinsic) geometry and have non trivial correlations between distinct nodes

*

These correlations are such that tracing on one node yields, on a neighbouring node, a thermal state with respect to a flow related, via the simplicity conditions, to the boost generator Correlations extends to non-neighbouring nodes and get mixed up by gauge invariance

* * *

we have defined states at fixed spin j. conjugate momentum, namely the extrinsic curvature at the facet, has to be fuzzy. extend to extrinsic coherent states?

*

internal gauge/Minkowsky Lorentz symmetry? gauge fixing: the boost generator can also be interpreted as the generator

• f physical boosts: it evolves a state on a surface to the state on a boosted

surface, which is to say to the surfaces of a boosted observer SUMMARY and DISCUSSION The 2π in the Bisognano–Wichmann temperature is related to the Minkowski geometry and its complex extension, as well as the corner terms of the action

• n the splitting surface

* *

what about near-equilibrium thermodynamics: open quantum spin network system?

*

“spin network thermal field theory”in 2nd quantisation formalism: GFT?

~ K

*

correlations propagates via gauge coupling: study thermalization

• n spin networks (thermal time problem)

Thank you!