Modular theory and Bekenstein’s boundRoberto Longo

Okinawa, OIST, Strings 2018, June 2018
Partly based on a joint work with Feng Xu

Thermal equilibrium states
A primary role in thermodynamics is played by the equilibriumdistribution.
Gibbs states
Finite quantum system: A matrix algebra with Hamiltonian H andevolution τt = Ade itH . Equilibrium state ϕ at inverse temperature
β is given by the Gibbs propertyϕ(X ) =Tr(e −βH X )Tr(e −βH )What are the equilibrium states at infinite volume where there isno trace, no inner Hamiltonian?von Neumann algebras
H a Hilbert space. B(H) algebra of all bdd linear operators on H.
M ⊂ B(H) is a von Neumann algebra if it is a ∗-algebra and is
weakly closed. Equivalently (von Neumann density theorem)
M = M00
with M0 = {T ∈ B(H) : TX = XT

∀X ∈ M} the commutant.

A C ∗ -algebra is only closed in norm.
Observables are selfadjoint elements X of M, states arenormalised positive linear functionals ϕ,ϕ(X ) = expected value of the observable X in the state ϕM abelian ⇔ M = L∞ (X , µ).
A factor M is in general not of type I, i.e. not isomorphic to B(H)KMS states (HHW, Baton Rouge conference 1967)
Infinite volume. A a C ∗ -algebra, τ a one-par. automorphism groupof A. A state ϕ of A is KMS at inverse temperature β > 0 if forX , Y ∈ A ∃ function FXY s.t.
(a) FXY (t) = ϕ X τt (Y )(b) FXY (t + iβ) = ϕ τt (Y )XFXY bounded analytic on Sβ = {0 < =z < β}ϕ τt (Y )Xβϕ X τt (Y )KMS states generalises Gibbs states, equilibrium condition forinfinite systems

Tomita-Takesaki modular theory
M be a von Neumann algebra on H, ϕ = (Ω, ·Ω) normal faithfulstate on M. Embed M into HM
X →X ΩyX 7→X ∗−−−−−→isometricM
yX →X ΩS :X Ω7→X ∗ Ω

H −−0−−−−−−→ H
non isometric

S = S̄0 , ∆ = S ∗ S > 0 positive selfadjointt ∈ R 7→ σtϕ ∈ Aut(M)σtϕ (X ) = ∆it X ∆−itintrinisic dynamics associated with ϕ (modular automorphisms).

Tomita-Takesaki modular theory
By a remarkable historical accident, Tomita announced thetheorem at the 1967 Baton Rouge conference. Soon later Takesaki
completed the theory and charcterised the modular group by theKMS condition.
• σ ϕ is a purely noncommutative object• If ϕ(X ) = Tr(ρX ) (type I case) then σtϕ (X ) = ρit X ρ−it andlog ∆ = log ρ − log ρ0• σ ϕ is characterised by the KMS condition at inverse temperature
β = −1 with respect to the state ϕ.
• σ ϕ is intrinsic modulo scaling, the inverse temperature given byϕβ the rescaled group t 7→ σ−t/β
is physicalBekenstein’s boundFor decades, modular theory has played a central role in theoperator algebraic approach to QFT, very recently several physicalpapers in other QFT settings are dealing with the modular group,
although often in a heuristic (yet powerful) way!
I will discuss the Bekenstein bound, a universal limit on theentropy that can be contained in a physical system with given sizeand given total energy
If R is the radius of a sphere that can enclose our system, while E
is its total energy including any rest masses, then its entropy S isbounded byS ≤ λREThe constant λ is often proposed λ = 2π (natural units).

Casini’s argument
Subtract to the bare entropy of the local state the entropy
corresponding to the vacuum fluctuations. V bounded region.
The restriction ρV of a global state ρ to von Neumann algebraA(V ) has formally entropy given by
S(ρV ) = − Tr(ρV log ρV ) ,known to be infinite. So subtract the vacuum state entropy
SV = S(ρV ) − S(ρ0V )with ρ0V the density matrix of the restriction of the vacuum state.
Similarly, K Hamiltonian for V , consider
KV = Tr(ρV K ) − Tr(ρ0V K )Bekenstein bound is now SV ≤ KV which is equivalent to thepositivity of the relative entropy
S(ρV |ρ0V ) ≡ Tr ρV (log ρV − log ρ0V ) ≥ 0 ,Araki’s relative entropy
An infinite quantum system is described by a von Neumann
algebra M typically not of type I so Tr does not exists; however
Araki’s relative entropy between two faithful normal states ϕ andψ on M is defined in general byS(ϕ|ψ) ≡ −(η, log ∆ξ,η η)
where ξ, η are cyclic vector representatives of ϕ, ψ and ∆ξ,η is therelative modular operator associated with ξ, η.
S(ϕ|ψ) ≥ 0
positivity of the relative entropy
Relative entropy is one of the key concepts. We take the view that
relative entropy is a primary concept and all entropy notions arederived concepts

Analog of the Kac-Wakimoto formula (L. ‘97)
The root of our work relies in this formula for the incremental freeenergy of a black hole (cf. the Kac-Wakimoto formula,Kawahigashi, Xu, L.)Hρ be the Hamiltonian for a uniformly accelerated observer in theMinkowski spacetime with acceleration a > 0 in representation ρ(localised in the wedge for Hρ )(Ω, e −tHρ Ω) t=β = d(ρ)
with Ω the vacuum vector and β = 2π
a the inverse Hawking-Unruh
temperature. d(ρ)2 is Jones’ index.
The left hand side is a generalised partition formula, so log d(ρ)
has an entropy meaning in accordance with Pimsner-Popa work.
Here we generalise this formulaCP maps, quantum channels and entropy
N , M vN algebras. A linear map α : N → M is completelypositive ifα ⊗ idn : N ⊗ Matn (C) → M ⊗ Matn (C)is positive ∀n (quantum operation)
ω faithful normal state of M and α : N → M CP map as above.
SetXHω (α) ≡ supS(ω|ωi ) − S(ω · α|ωi · α)
(ωi )

iP
supremum over all ωi with i ωi = ω.
The conditional entropy H(α) of α is defined byH(α) = inf Hω (α)ωinfimum over all “full” states ω for α. Clearly H(α) ≥ 0 becauseHω (α) ≥ 0 by the monotonicity of the relative entropy .
α is a quantum channel if its conditional entropy H(α) is finite.

Bimodules and CP maps
Let α : N → M be a completely positive, normal, unital map andω a faithful normal state of M∃! N − M bimodule Hα , with a cyclic vector ξα ∈ H and left andright actions `α and rα , such that(ξα , `α (n)ξα ) = ωout (n) ,(ξα , rα (m)ξα ) = ωin (m) ,with ωin ≡ ω, ωout ≡ ωin · α. Converse is true.
CP map α ←→ cyclic bimodule HαWe haveH(α) = log Ind(Hα )(Jones’ index)

Promoting modular theory to the bimodule setting

H an N − M-bimodule with finite Jones’ index Ind(H)Given faithful, normal, states ϕ, ψ on N and M, I define themodular operator ∆H (ϕ|ψ) of H with respect to ϕ, ψ as
∆H (ϕ|ψ) ≡ d(ϕ · `−1 ) d(ψ · r −1 · ε) ,Connes’ spatial derivative, ε : `(N )0 → r (M) is the minimal
conditional expectationlog ∆H (ϕ|ψ) is called the modular Hamiltonian of the bimodule H,
or of the quantum channel α if H is associated with α.

Properties of the modular Hamiltonian
If N , M factors
ϕ∆itH (ϕ|ψ)`(n)∆−it
H (ϕ|ψ) = ` σt (n)ψ∆itH (ϕ|ψ)r (m)∆−itH (ϕ|ψ) = r σt (m)(implements the dynamics)∆itH (ϕ1 |ϕ2 ) ⊗ ∆itK (ϕ2 |ϕ3 ) = ∆itH⊗K (ϕ1 |ϕ3 )(additivity of the energy)−i2t∆itH (ϕ1 |ϕ2 )∆itH̄ (ϕ2 |ϕ1 ) = dH

If T : H → H0 is a bimodule intertwiner, thenT ∆itH (ϕ1 |ϕ2 ) = (dH0 /dH )it ∆itH0 (ϕ1 |ϕ2 )TpConnes’s bimodule tensor product w.r.t. ϕ2 ; dH = Ind(H)Physical Hamiltonian
We may modify the modular Hamiltonian in order to fulfil the rightphysical requirements (additivity of energy, invariance under charge
conjugation,...)K (ϕ1 |ϕ2 ) = − log ∆H (ϕ1 |ϕ2 ) − log dis the physical Hamiltonian (at inverse temperature 1).
The physical Hamiltonian at inverse temperature β > 0 is given by−β −1 log ∆ − β −1 log d
From the modular Hamiltonian to the physical Hamiltonian:shifting

scaling− log ∆ −−−−→ − log ∆ − log d −−−−→ β −1 − log ∆ − log dThe shifting is intrinsic, the scaling is to be determined by thecontext!

Modular and Physical Hamiltonians for a quantum channel
We now are going to compare two states of a physical system, ωinis a suitable reference state, e.g. the vacuum in QFT, and ωout is astate that can be reached from ωin by some physically realisableprocess (quantum channel).
α : N → M be a quantum channel (normal, unital CP map withfinite entropy) and ωin a faithful normal state of M. ωout = ωin · αlog ∆α ≡ log ∆HαKα = β −1 KHα = β −1 − log ∆Hα − log dHα(physical Hamiltonian at inverse temperature β)Kα may be considered as a local Hamiltonian associated with αand the state transfer with input state ωin .

Thermodynamical quantities
The entropy S ≡ Sα,ωin of α is
ˆ log ∆α ξ)ˆS = −(ξ,where ξˆ is a vector representative of the state ωin · r −1 · ε in Hα .
The quantityˆ K ξ)
ˆE = (ξ,is the relative energy w.r.t. the states ωin and ωout .
The free energy F is now defined by the relative partition function
ˆ e −βK ξ)ˆF = −β −1 log(ξ,F satisfies the thermodynamical relation
F = E − TS

A form of Bekenstein bound
As F = 12 β −1 H(α), we haveF ≥0(positivity of the free energy)

becauseH(α) ≥ 0(monotonicity of the entropy)

So the above thermodynamical relation
F = E − β −1 Sentails the following general, rigorous version of the BekensteinboundS ≤ βE
To determine β we have to plug this general formula in a physicalcontextFixing the temperature in QFT
O a spacetime region s.t. the modular group σtω of the local von
Neumann algebra A(O) associated with vacuum ω has a geometricmeaning. So there is a geometric flow θs : O → O and are-parametrisation of σtω that acts covariantly w.r.t θ.
Well known illustration concerns a Rindler wedge region O of theMinkowski spacetime. The vacuum modular group ∆−it of A(O)w.r.t. the vacuum state is here equal to U(βt), with U the boost
unitary one-parameter group acceleration a and β the Unruhinverse temperature. Re-parametrisation of the geometric flow is
the rescaling by inverse temperature β = 2π/a.
Connes and Rovelli suggested to define locally the inversetemperature bydθs
βs =dsthe Minkowskian length of the tangent vector to the modularorbit. Namely dτ = βs ds with τ proper time

Schwarzschild black hole
Schwarzschild-Kruskal spacetime of mass M > 0, namely theregion inside the event horizon, and N ≡ A(O) the local von
Neumann algebra associated with O on the underlying Hilbert
space H, O Schwarzschild black hole region, ω vacuum state
H is a N − N bimodule, indeed the identity N − N bimodule
L2 (N ) associated with ω.
The modular group of A(O) associated with ω is geometric andcorresponds to the geodesic flow. KMS Hawking temperature is
T = 1/8πM = 1/4πR
with R = 2M the Schwarzschild radius, thenS ≤ 4πREwith S the entropy associated with the state transfer of ω by aquantum channel, and E the corresponding relative energy.

Conformal QFT
Conformal Quantum Field Theory on the Minkowski spacetime,
any spacetime dimension. OR double cone with basis a radius
R > 0 sphere centered at the origin and A(OR ) associated localvN algebra.
The modular group of A(OR ) w.r.t. the vacuum state ω has ageometrical meaning (Hislop, L. 1982):∆−isOR = U ΛOR (2πs)
with U is the representation of the conformal group and ΛOR is a
one-parameter group of conformal transformation leaving OR
globally invariant and conjugate to the boost one-parameter groupof pure Lorentz transformations.

dThe inverse temperature βR = ds
ΛOR (s)x s=0 in OR is maximal
on the time-zero basis of OR , in fact at the origin x = 0 with value

βR = πR
SoS ≤ πREwith S and E the entropy and energy associated with any
quantum channel by the vacuum state.

Boundary CFTThe analysis is less complete. Yet it shows up new aspects as thetemperature depends on the distance from the boundary.
1+1 dimensional Boundary CFT on the right Minkowski half-planex > 0. The net A+ of von Neumann algebras on the half-plane isassociated with a local conformal net A of von Neumann algebras
on the real line (time axes) byA+ (O) = A(I ) ∨ A(J) ;Here I , J are intervals of the real line at positive distance withI > J. We fix O = I × J.

IOJA double−cone
O=I x J

Figure: BCFT

(More generally a finite-index extension of A is needed).
There is a natural state with geometric modular action (Martinetti,Rehren, L.), that corresponds to the chiral “2-interval state” andgeometric action of the double covering of the Möbius group.

With R > 0, let OR be the dilated double cone associated with theintervals RI , RJ
The maximal inverse temperatures are related byβ OR = R β O .
By choosing the KMS inverse temperatures equal to the maximaltemperature, with S and E the entropy and energy in OR withrespect to the geometric state and a quantum channel, we haveS ≤ λO RE
where the constant λO is equal to βO .

Summary

von Neumann algebra ←→ quantum system
CP map with finite entropy between q. systems ←→ quantum channel
quantum channel ←→ finite index bimodule
finite index bimodule and state −→ modular Hamiltonian
modular Hamiltonian & physical functoriality −→ phys. Hamiltonian
modular and physical Hamiltonians −→ F = E − TS
F = E − TS & autom. positivity of the free energy F −→ S ≤ βES ≤ βE & geometrical modular flow −→ Bekenstein’s bound