Entropy in Causal Fermion Systems
Magdalena Lottner
Fakultät für Mathematik
Universität Regensburg

June 19, 2020, LQP Workshop

Magdalena Lottner (Universität Regensburg)

Entropy in Causal Fermion Systems

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Preface

Joint work with Felix Finster and Robert Jonsson
Theory of causal fermion systems [3, p. 1]:
Novel approach to fundamental physics
Unification of interactions of the standard model with gravity
I Close connections to quantum field theoryII

Why consider entropy?
Plays important role in many physical effects
Many different definitions
I Correlations not always well understood (e.g. area law)
I Part of problems like black hole information paradox (see also [5]and [7])II

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Outline
1Preface

2

Preliminaries
Causal Fermion Systems
Causal Action Principle
Surface Layer Integrals

3

Definition of the Entropy
A Conserved Surface Layer Integral
Definition of the Entropy

4First Estimations
A more specific Form of σU
Minkowski Space Example
Generalization to the Logarithm

5Outlook

Magdalena Lottner (Universität Regensburg)

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3 / 18Causal Fermion Systems [3, p. 1, 3]
Definition (Causal Fermion System)
A causal fermion system (short CFS) of spin dimension n ∈ N ia atriple (H, F, ρ) consisting of a complex separable Hilbert space(H, h.|.iH ), the following subset of L(H):F := {x ∈ L(H) | x selfajoint with rank(x) ≤ 2n and at most n positive
and at most n negative eigenvalues} ,and a measure ρ on F, the so called universal measure.

Definition (Spacetime)
For a CFS (H, F, ρ) we define spacetime as:
M := supp(ρ) ⊆ F ⊆ L(H) .
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Causal Action Principle [3, p. 1-4]

Pä2xyxy
12nLagrangian: L : F × F → R , (x, y ) 7→ 4ni,j=1 |λi | − |λj |with λxy
i , i = 1, . . . , 2n (non-zero) eigenvalues of xy
non-negative
symmetricÄ,Definition (Causal Action Principle)
Minimize the action
S(ρ) :=Z ZFF

L(x, y )dρ(x)dρ(y ) ,by varying ρ under the constraints• Volume constraint: ρ(F) constant,• Trace constraint:

RF tr(x)dρ(x) constant,äR R ÄP2nxy 2• Boundedness constraint: F Fdρ(x)dρ(y ) ≤ C,
j=1 |λi |From now on we always assume that ρ is such a minimizer

Magdalena Lottner (Universität Regensburg)

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Surface Layer Integrals [3, p. 42-43]
1. CAUSAL FERMION SYSTEMS – AN OVERVIEWνNyδxΩ
ˆN· · · dµNbbΩˆΩdρ(x)ˆM \Ωdρ(y) · · · L(x, y)

Figure: Comparison of an ordinary surface integral and a surface layer

Figuresurface integral and a corresponding surface layer integral
integral 1.2.
[3, p. A42].

tain a corresponding
variationof the on
physicalwave functions. Following (1In CFS no canonicalmeasurehypersurface
1.4.5) gives a corresponding variation of the universal measure,−1i.e.
L decays fast for |x − y | > δ (for Dirac particles:∗δ ∼ m )ρτ = (Fτ )∗ ρwithFτ (x) := −Ψτ (x) Ψτ (x) .
→ gives integral over hypersurface smeared out on the scale of δρτ is the push-forward of the measure ρ, the volume constraint is clearly sat
der to verify the trace constraint, we note that formally,
ˆˆMagdalena LottnerRegensburg)in Causal
6 / 18tr(x)(Universitätdρ (x)=tr EntropyF (x)dρFermion(x) Systems

A conserved surface layer integral [3, S. 42-45], [4]For u ∈ H arbitrary set Au := |ui hu| ∈ L(H)symmetric
finite rank

II

→ One-parameter family of unitary transformations:(Uτ )τ ∈R := (exp(iτ Au ))τ ∈R→ Symmetry φ : R × F → F , (τ, x) 7→ Uτ xUτ−1 of L, i.e.:L(φ(τ, x), y ) = L(x, φ(−τ, y )) ,∀τ ∈ R, x, y ∈ M .

LemmaFor any Ω ⊆ M compact:
d0=dτ
=Z

Ω

ÇZ

dρ(x)

Ωdρ(x)Z

M\ΩZM\Ωådρ(y ) L(φ(τ, x), y ) − L(x, φ(τ, y ))

τ =0dρ(y ) D1,vu (x) − D2,vu (y ) L(x, y ) ,dwith vu (x) := dτφ(τ, x)|τ =0 for all x ∈ F.
Magdalena Lottner (Universität Regensburg)

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7 / 18Ω1/2 : past of Cauchy
surface ∂Ω1/2

∂Ω1
Ω12

Ω1

Ω1 in the future of Ω2
Meantime Ω12 := Ω1 \ Ω2
can be exhausted bycompact sets (Ω̃n )n∈N∂Ω2
Ω2

⇒ For the integrand decaying fast enough at spatial infinity:
0==Z

Ω12

Zdρ(x)Ω1ZZM\Ω1M\Ω12−Zdρ(y ) D1,vu (x) − D2,vu (y ) L(x, y )Ω2

RZM\Ω2R



D1,vu (x) − D2,vu (y ) L(x, y )dρ(y )dρ(x)→ hu, uiρ := Ω dρ(x) M\Ω dρ(y ) D1,vu (x) − D2,vu (y ) L(x, y )conserved for Ω past of Cauchy surface ∂ΩMagdalena Lottner (Universität Regensburg)

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Definition of the Entropy
In Minkowski space limiting case (see [4, Chapter 5]):hu, uiρ = chu|uiH∀u ∈ HF ⊆ H closed subspace .
U→ Assume this holds in general for HF suitable→ h., .iρ inner product on HF∂Ω

Spatial localization for U ⊆ M:
Z12 Ω∩U|ZM\Ω+Z ZΩ (M\Ω)∩U

D1,vu (x) − D2,vu (y ) L(x, y )dρ(x)dρ(y ){z}=:hu|uiU,ρ

Define localized state operator

hu|uiU,ρ = hu, σU uiρ ,∀u ∈ HF

Von-Neumann-like entropy (for reduced one-particle density) [6,(34)], [8, p. 400-401]Ää

S := trHF σU log(σU ) + (1 − σU ) log(1 − σU )Magdalena Lottner (Universität Regensburg)

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A more specific Form of σU [3, p. 39-40], [4, chapter 5]
Using a kernel Q the product h., .iρ can be rewritten as:hu|uiρ = 4Im

Zdρ(x)ZM\ΩΩdρ(y ) ≺ ψ u (x), Q(x, y )ψ u (y ) yAnd correspondingly:
Zhu|uiU,ρ = 2Imdρ(x)ΩZM\Ωdρ(y ) ≺ ψ u (x), Q(x, y )χU (y )ψ u (y ) y+ ≺ χU (x)ψ u (x), Q(x, y )ψ u (y ) y

→ More specific form for σU :σU =1((χU )∗ + χU ) ,2

with (χU )∗ adjoint of χU wrt h.|.iρ
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In vakuum Mionkowski space: σU = σσ̃U σ withIσ : L2 (R3 , C4 ) → HF projection operator with kernel σ(~x , ~y ) s.t.:Zσ(~x , ~z ) =d 3~y σ(~x , ~y )σ(~y , ~z ) .
R3Iσ̃U : L2 (R3 , C4 ) → L2 (R3 , C4 ) integral operator with kernelσ̃U (~x , ~y ) =1
χU (~x ) + χU (~y ) σ(~x , ~y ) .
2⇒ Assume in the follwoing that σU can always be represented in thisform

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Consider σU − (σU )2log(σU ) difficult to calculate
Why log-formula at all?
x − x 2 similar behaviour:y0.5−x log(x) − (1 − x) log(1 − x)x − x2

EVs 0, 1 don’t contribute→ Measues mixing
Iplus easier to calculate(similar as in [2, Theorem3.1]):

0x
00.5

1

tr(σU − (σU )2 ) = tr(σU − σU (σ − σR3 \U )) = tr(σU σR3 \U ) = tr(σ̃U σσ̃R3 \U σ)
1=4=ZZUR3ZR3ZR3ZR3χU (~x1 ) + χU (~x2 )

χR3 \U (~x3 ) + χR3 \U (~x4 ) ·

trC4 σ(~x1 , ~x2 )σ(~x2 , ~x3 )σ(~x3 , ~x4 )σ(~x4 , ~x1 ) d 3~x1 d 3~x2 d 3~x3 d 3~x43~d x

ZR3 \U

d 3~y trC4 σ(~x , ~y )σ(~y , ~x ) ,Magdalena Lottner (Universität Regensburg)

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Minkowski Space Example (see also [3, p. 25-34])
Vacuum Minkowski space: σ = Πε− regularized projector tonegative frequency solutions of Dirac equation−2m|~
x −~y|∼ e |~x −~y |3Important properties oftr(Πε− (~x , ~y )Πε− (~y , ~x )):|~x − ~y |tr Πε− (~x , ~y )Πε− (~y , ~x )only depends only on |~x − ~y |I diverges for |~x − ~y | → 0 and ε → 0
I decays fast for large |~x − ~y |I∼ ε−6→ Split surface layer integral with cutoff constant 0 < ε  δ  l:ZZBl (0) R3 \Bl (0)··· =ZZ

Bl (0) Bl (0)c ∩Bδ (~x )··· +ZZBl (0) Bl (0)c ∩Bδ (~x )c...

In the limit ε → 0 (physical case) fist term diverges, second one
bounded → focus on first term

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Simplifying integrals, the first term becomes:
Z3~d xBl (0)h=C · l 2Z δ0ZBl (0)c ∩Bδ (~x )d 3~y trC4 (Πε− (~x , ~y )Πε− (~y , ~x ))dr r 3 Fε (r 2 ) −112Z δ0idr r 5 Fε (r 2 ) ,with Fε : R → C such that Fε (r 2 ) = trC4 (Πε− (0, r ê1 )Πε− (0, r ê1 )) forall r ≥ 0First integral dominates for l  δ→ obtain area-law-like form:tr(σBl (0) − σB2 l (0) )= C(ε) · C0 (δ) · l 2 +| {z }−→ ∞ε→0Magdalena Lottner (Universität Regensburg)

C1 (δ)| {z }

C0 (δ)l 2 for lδ

+Entropy in Causal Fermion Systems

D(ε)| {z }·Corrections ,O(1) for ε→014 / 18

Generalization to the logarithm
Taylor expansion of logarithm:
tr (1 − σU ) ln(1 − σU ) = −∞X(−1)nn=1ntr(σUn ) +∞X(−1)nn=1ntr(σUn+1 ) ,→ Additivity:tr(σUn+1 ) = tr((σU )n (σ − σR3 \U )) = tr((σU )n σ) − tr((σU )n σR3 \U ) ,→ By definition of σU : tr((σU )n σ) = tr((σU )n )→ And similar as before: tr((σU )n σR3 \U ) = tr((χU σ)n (χR3 \U σ))→ Finially iteratively use:






tr (χU σ)n (χR3 \U σ) = tr (χU σ)n−2 (1 − χR3 \U )σ(χU σ)(χR3 \U σ)








=tr (χU σ)n−1 (χR3 \U σ) − tr (χU σ)n−2 (χR3 \U σ)(χU σ)(χR3 \U σ) ,Magdalena Lottner (Universität Regensburg)

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Ää

Proceed similarly with tr(σU log(σU )) = tr(σU log 1 − σR3 \U )→ After simplification this yields expansion:tr σU log(σU ) + (1 − σU ) ln(1 − σU ) =∞X

n=1ÄÄän ä

αn tr (χU σ)(χR3 \U σ)

Resulting traces correspond to higher order surface layer integrals(alternating integrals over inside and outside)
Conjecture:
Short range of integrand → for each pair of integrals over U and U conly small strips of space around ∂U relevant
I Surface layer integrals decay with growing order
I Only lower orders in n relevant
I Lowest order: Ordinary surface layer integral, which likely has
area-law-like form

I

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Outlook
Prove more general area-law-like form for area AU of U:ÇåS = C(ε) · C0 (δ) · AU +| {z }−→ ∞C1 (δ)| {z }

C0 (δ)AU for AU δ 2ε→0Ä+D(ε)| {z }·Corrections

O(1) for ε→0äShow that S := tr σU log(σU ) + (1 − σU ) log(1 − σU ) reallycorresponds to entanglement entropy→ Consequences for information paradox: General problem [5], [7]:Independent of the initial state a black hole evaporates by emitting
Hawking radiation
I Shrinking horizon → Bekenstein-Hawking entropy shrinks
I Thermal radiation → Entanglement-entropy grows
I Problem if Bekenstein-Hawking entopy and entanglement-entropy
are always equalI

Hope: Answer question whether they really are equal all the time
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[1]: CFS-website: https://causal-fermion-system.com/[2]: Chris Brislawn: Traceable Integral Kernels on Countably Generated Measure Spaces,
Pacific Journal of Mathematics, 150 (1991), no. 2.
[3]: Felix Finster: The Continuum Limit of Causal Fermion Systems,
arXiv:1605.04742 [math-ph], Fundamental Theories of Physics, vol. 186, Springer, 2016,(version 3, 2018).
[4]: Felix Finster, Johannes Kleiner: Noether-like theorems for causal variational principles,
arXiv:1506.09076 [math-ph], Calculus of Variations and Partial Differential Equations, 55(2016), no. 2, (version 2, 2016).
[5]: Daniel Harlow: Jerusalem Lectures on Black Holes and Quantum Information,
arXiv:1409.1231 [hep-th], Rev. Mod. Phys., 88 (2016), no. 1 (version 4, 2015).
[6]: Robert Helling, Hajo Leschke, Wolfgang Spitzer: A Special Case of a Conjecture byWidom with Implications to Fermionic Entanglement Entropy, arXiv:0906.4946 [math-ph],International Mathematics Research Notices, 2010, (version 2, 2010).
[7]: Leonard Susskind, James Lindesay, An Introduction To Black Holes, Information And
The String Theory Revolution: The Holographic Universe, World Scientific Publishing
Company, Singapur, 2005.
[8]: Walter Thirring: Quantum Mathematical Physics: Atoms, Molecules and Large
Systems, Springer, Berlin, Heidelberg 2002 (second edition, second printing 2003).

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