quantum vacuum energy calculations: from graphs to ...lkaplan/graphs_polyhedra.pdfquantum vacuum...

Quantum Vacuum Energy Calculations: From Graphs toPolyhedra and Beyond

Lev Kaplan

Tulane University

March 30, 2011

Mathematical Physics and Harmonic Analysis SeminarTexas A&M University

Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 1 / 49

Texas-Oklahoma-Louisiana Quantum VacuumCollaboration:

Ricardo Estrada (LSU)

Stephen A. Fulling (Texas A&M)

Lev Kaplan (Tulane)

Klaus Kirsten (Baylor)

Kimball A. Milton (U. Oklahoma)


Outline

1 IntroductionHistory and BackgroundMathematical PreliminariesBroad Objectives

2 ExamplesEx. 1: Quantum GraphsEx. 2: Rectangles, Pistons, and PistolsEx. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra

3 OutlookRegular PolygonsChaotic CavitiesOutside of a Box

4 Conclusions


Introduction History and Background

History and Background

Any quantum system has nonzero ground state energy, e.g. ~ω/2 forharmonic oscillator with frequency ω

In particular, vacuum state of a quantum field has nonzero energy,which is a function of boundary conditions

1948: Casimir-Polder force predicted between uncharged conductingplates

F/A = −π2~c/240a4

Measurement in 1958consistent with theory



History and Background

1997: More accurate experiment using plane and sphere

2002: Finally, original Casimir parallel plate experiment performedwith 15% precision



Vacuum energy beyond parallel plates:

Relation to van der Waals force in chemistry

1970s: QFT in curved spacetime

Chiral bag model of the nucleon

Dark energy in cosmology

Λ ∼ 10−26kg/m3 ∼ 10−120c5/~G 2

Why so small?Why not zero?Why comparable to present-day mass density of universe?



Stabilizing extra dimensions in brane world models

Nanotechnology applications: MEMS and NEMS

Static friction caused by vacuumenergy is major obstacle to furtherminiaturization of devices such asgears, etc.


Introduction Mathematical Preliminaries

Mathematical Preliminaries

Vacuum energy of scalar field in cavity

Work in units ~ = c = 1

Find all modes of field φ consistent with given boundary conditions,i.e. solve

−∇2φn = ω2nφn

Each mode n behaves as independent harmonic oscillator withfrequency ωn

Each mode has energy ωn(Nn + 12) Nn = 0, 1, 2, . . .

Total vacuum energy of the field is given formally by

Evac =1

2

∑n

ωn =1

2Tr√−∇2



Mathematical Preliminaries

Evac = 12

∑n ωn = 1

2 Tr√−∇2 divergent, must regularize

Cylinder (Poisson) kernel: Tt(x , y) = 〈x |e−t√−∇2 |y〉

Let Et = −1

2

∂

∂tTr Tt =

1

2

∑n

ωne−ωnt

Can separate finite and divergent parts as t → 0 and get t−independentfinite answer for physical forces if divergent terms can be shown to cancel[renormalization]

Similarly regularized local energy densityEt(x) = T00(x) = −1

2∂∂tTt(x , x) = 1

2

∑n ωn|φ(x)|2e−ωnt

Other components of stress-energy tensor can be obtained similarly



Real life is more complicated

In specific applications need to consider

vector fields (e.g. electromagnetic fields)

going beyond idealized boundary conditions

more physically realistic cutoffs (e.g. spatial dispersion)

Advantages of toy model

can address general conceptual issues independent of specificapplication

mathematical problem in spectral geometry: asymptotics of cylinderkernel, relation to zeta function, ...


Introduction Broad Objectives

Objectives/Questions

Role of periodic and closed classical orbits

quantum-classical correspondence

Sign of Casimir force in general situations

Relation of local and global quantities

nonuniform convergence

Systematic understanding of boundary, curvature, and corner effects

Proper renormalization

under what circumstances do divergent terms cancel?can all ∞’s be absorbed into boundary properties?

Combining “inside” + “outside” contributions


Examples Ex. 1: Quantum Graphs

Ex. 1: Quantum Graphs

S. A. Fulling, L.K., and J. H. Wilson, PRA (2007)G. Berkolaiko, J. M. Harrison, and J. H. Wilson, J Phys A (2009)J. H. Wilson, Texas A&M senior thesis



What is a quantum graph?

Set of line segments joined at vertices

Singular one-dimensional variety equipped with self-adjoint differentialoperator

Approximation for realistic physical wave systems

Chemistry: free electron theory of conjugated moleculesNanotechnology: quantum wire circuitsOptics: photonic crystals

Laboratory for investigating general questions about scattering,quantum chaos, spectral theory



What is a quantum graph?

H = −∇2 on each bond

Kirchhoff boundary conditions at each vertex

1 Continuity φj(0) = φα for all bonds j starting at vertex α2 Current conservation

∑j φ′j(0) = cαφα where sum is over all bonds j

starting at vertex α, and derivative is in outward direction

Neumann-like: cα = 0; Dirichlet: cα =∞



Direct calculation using spectrum

Two movable “pistons”

Focus on “a” region between pistons:Dirichlet (DD): ωn = nπ/a (n = 1 . . .∞)Neumann (NN): ωn = nπ/a (n = 0 . . .∞)

Tr Tt =∑∞

n=0,1 e−(nπ/a)t = a

πt ±12 + 1

12πta + O(t2)

⇒ Et = −1

2

∂ Tr Tt

∂t=

a

2πt2− π

24a+ O(t)



Vacuum Energy in simple QG

Et =a

2πt2− π

24a+ O(t)

First term divergent as t → 0

ETotalt = a+L1+L2

2πt2− π

24(a−1 + L1−1 + L2

−1) + O(t)

= const2πt2− π

24(a−1 + L1−1 + L2

−1) + O(t)

Divergent term is a-independent constant energy density⇒ force on piston is finite!



Vacuum Energy in simple QG

ETotal = const− π24(a−1 + L1

−1 + L2−1)

Can safely take L1, L2 →∞

ETotalt = const− π

24a

⇒ FDD = FNN = −∂E∂a

= − π

24a2(attractive)

If one piston is Dirichlet and the other Neumann,

⇒ FDN = +π

48a2(repulsive)



Periodic Orbit Perspective

Tr Tt =∫dx Tt(x , x)

Free cylinder kernel in 1D: T 0t (x , y) = t

π1

(x−y)2+t2

Then Tt(x , x) in problem with boundaries obtainable by method ofimages as sum over periodic and closed orbits:

Tt(x , x) = Re∑p

t

π

Ap

L2p + t2+ closed orbits

p = periodic orbit going through xLp = orbit lengthAp = product of scattering factors

Can obtain asymptotic t → 0 behavior term by term



Periodic Orbit Perspective

Taking trace and accounting for repetitions r

Tr Tt =∫dx Tt(x , x) = t

πat2

+ Re∑

p

∑∞r=1

tπ2Lp(Ap)r

(rLp)2+ O(t2)

Divergent (Weyl) term associated with zero-length orbit

For single line segment a, only one nonzero-length periodic orbitLp = 2a plus repetitions

FDD = FNN = − 14πa2

(+1 + 14 + 1

9 + · · · )FDN = − 1

4πa2(−1 + 1

4 −19 + · · · )

Periodic orbit sum converges

Sign of force can be read off from phase associated with shortest orbit(in general, boundaries + Maslov indices)



Vacuum Energy in Star Graphs

• N-like boundary at junction joining Bbonds

• N, D, or e iθj boundary at each piston

Each piston at distance aj from junction

Eigenvalues: solve∑B

j=1 tan(ωaj + θj) = 0

Exact solution for ωn when all aj equal

⇒ Then Evac = limt→0

[Et −

∑j aj

2πt2

]= lim

t→0

[1

2

∑n

ωne−ωnt −

∑j aj

2πt2

]

Convergence improved by Richardson extrapolation



Alternatively: use periodic orbit expansion

Contribution to vacuum energy from shortest orbit only (bouncing backand forth once in one bond):

Evac ≈ −1

2π

(2

B− 1

) B∑j=1

(±1)

aj

±1 for Neumann or Dirichlet pistons

Gives correct sign for Casimir forces at least for B > 3

repulsive for Neumann

attractive for Dirichlet



Comparison between shortest orbit approximation & exactanswer for equal bond case



Star Graphs: Periodic Orbits

Add up all repetitions of shortest orbits (Neumann):

E ≈ − 1

4π

∞∑r=1

1

r2

(2

B− 1

)r B∑j=1

1

aj≈ π

48

(1− 24 ln 2

π2B+ · · ·

) B∑j=1

1

aj

Compare with analytic result for B Neumann pistons with equal bondlengths:

E =π

48

(1− 3

B

)B

a

Shortest orbits give only leading contribution in 1/B expansion

⇒ need more orbits to obtain full answer for finite B



Star Graphs: Periodic Orbit Sum

B = 4 star graph with unequal bonds and all Neumann pistons

Sum over orbits Lp ≤ Lmax & compare with “exact” answer

0

0.1

0.2

5 10 15 20 25 30 35 40

Casim

ir En

ergy

Lmax

Star graph with B=4 and all Neumann pistons

Periodic orbit sumExact

0.01

0.1

1 10

Erro

r in

Casim

ir En

ergy

Lmax

Star graph with B=4 and all Neumann pistons

ErrorLmax

-1 power law

Error ∼ (Lmax)−1Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 24 / 49


Star Graphs: Periodic Orbit Sum

B = 4 star graph with unequal bonds and arbitrary e iθ pistons

Sum over orbits Lp ≤ Lmax & compare with “exact” answer

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

5 10 15 20 25 30 35 40

Casim

ir En

ergy

Lmax

Star graph with B=4 and random phase pistons

Periodic orbit sumExact

0.0001

0.001

0.01

1 10

Erro

r in

Casim

ir En

ergy

Lmax

Star graph with B=4 and random phase pistons

ErrorLmax

-3/2 power law

Error ∼ (Lmax)−3/2Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 25 / 49

Examples Ex. 2: Rectangles, Pistons, and Pistols

Ex. 2: Rectangles, Pistons, and Pistols

S. A. Fulling, L.K., K. Kirsten, Z. H. Liu, and K. A. Milton,J Phys A (2009); Z. H. Liu, Texas A&M Ph.D. thesis

Motivating Question: Naive renormalization (Lukosz 1971, ...) suggestsoutward force on sides of square or cubic box

Is this force real?



Ex. 2a: Rectangular cavity

No straightforward way to eval-uate Et = 1

2

∑n ωne

−ωnt di-rectly

Use classical path approach

Need all classical paths from x to x

Classify by number of bounces from a sides and number of bounces from bsides

Periodic paths: Even Even

Side paths: Even Odd or Odd Even

Corner paths: Odd Odd



Rectangle: periodic paths

Et,Periodic =ab

2πt3− ab

2π

∞∑k=1

(−1)η(2kb)2 − 2t2

[t2 + (2kb)2]5/2

− ab

2π

∞∑j=1

(−1)η(2ja)2 − 2t2

[t2 + (2ja)2]5/2

−ab

π

∞∑j=1

∞∑k=1

(−1)η(2ja)2 + (2kb)2 − 2t2

[t2 + (2ja)2 + (2kb)2]5/2

η = # of Dirichlet bouncesAssume all Neumann or all Dirichlet sides:

Et,Periodic =ab

2πt3− ζ(3)

16π

(a

b2+

b

a2

)− ab

8π

∞∑j=1

∞∑k=1

(a2j2 + b2k2

)−3/2+ O(t2)



Rectangle: add all paths

Et,Nonperiodic = ∓ 2a + 2b

8πt2± π

48

(1

a+

1

b

)+ O(t2)

Combining all terms:

Et =Area

2πt3∓ Perimeter

8πt2− ζ(3)

16π

(a

b2+

b

a2

)− ab

8π

∞∑j ,k=1

(a2j2 + b2k2

)−3/2 ± π

48

(1

a+

1

b

)+ O(t2)

Force on horizontal side:

F = −∂Et

∂a= divergent +

ζ(3)

16πb2− ζ(3)b

8πa3

+b

8π

∞∑j ,k=1

k2b2 − 2j2a2

(j2a2 + k2b2)5/2± π

48a2



Force on side of rectangle

F = divergent +ζ(3)

16πb2− ζ(3)b

8πa3+

b

8π

∞∑j ,k=1

k2b2 − 2j2a2

(j2a2 + k2b2)5/2± π

48a2

Naive “renormalization”: drop divergent terms and interprett−independent result as physical force on the side of box

a << b: attractive (like parallel plates)

Square box with Dirichlet sides: repulsive!

Problems:

Throwing away infinite terms

Ignoring outside of box



Ex. 2b: Rectangular piston

Add contributions from a× b rectangle and (L− a)× b rectangle

Divergent terms cancel (total area and perimeter are conserved)

One finite contribution from (L− a)× b rectangle survives as L→∞

Fpiston = 0 +ζ(3)

16πb2− ζ(3)b

8πa3

+b

8π

∞∑j ,k=1

k2b2 − 2j2a2

(j2a2 + k2b2)5/2± π

48a2− ζ(3)

16πb2



Ex. 2b: Rectangular piston

Finally (Cavalcanti 2004)

Fpiston = −ζ(3)b

8πa3+

b

8π

∞∑j ,k=1

k2b2 − 2j2a2

(j2a2 + k2b2)5/2± π

48a2

=π

b2

∞∑j ,k=1

k2K ′1

(2πjk

a

b

)[always attractive!]

= − ζ(3)b

8πa3+

π

48a2− ζ(3)

16πb2+πb

a3

∞∑j ,k=1

k2K0

(2πjk

b

a

)Decays exponentially for a� b; parallel plates for a� b



Ex. 2c: Casimir pistol

What would happen if exter-nal shaft is not present?

Here all dimensions� cutofft, except possibly c ∼ t

All Dirichlet boundaries

E =us

πt

∞∑k=1

1− 2k2u2

(1 + 4k2u2)5/2+

us

πt

∞∑j=1

1− 2j2s2

(1 + 4j2s2)5/2

+2us

πt

∞∑j=1

∞∑k=1

1− 2j2s2 − 2k2u2

(1 + 4j2s2 + 4k2u2)5/2

+s

2πt

∞∑j=1

−1 + 4j2s2

(1 + 4j2s2)2+

2r(l − s)

πt

∞∑k=1

1− 2k2r2

(1 + 4k2r2)5/2




Use scaled variablesc = rta = stb = utd = L− a = (l − s)t

E =us

πt

∞∑k=1

1− 2k2u2

(1 + 4k2u2)5/2+

us

πt

∞∑j=1

1− 2j2s2

(1 + 4j2s2)5/2

+2us

πt

∞∑j=1

∞∑k=1

1− 2j2s2 − 2k2u2

(1 + 4j2s2 + 4k2u2)5/2

+s

2πt

∞∑j=1

−1 + 4j2s2

(1 + 4j2s2)2+

2r(l − s)

πt

∞∑k=1

1− 2k2r2

(1 + 4k2r2)5/2




Horizontal force as func-tion of a:

narrow chamber a� b1/3c2/3: like parallel platesF ∼ −1/a2 (attractive)

longer chamber a� b1/3c2/3: gaps dominate:F is a−independent

c > 0.6t ⇒ attractivec < 0.6t ⇒ repulsive (believable???)


Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra

Example 3: Triangles, Triangular Cylinders, Prisms, andTetrahedra

[E. K. Abalo, K. A. Milton, and LK (PRD, 2010)]

Analytic expressions exist for spectrum of the following triangular cavitieswith Dirichlet or Neumann boundaries

Equilateral triangle (60◦ − 60◦ − 60◦)

Hemiequilateral triangle (30◦ − 60◦ − 90◦)

Isosceles right triangle (45◦ − 45◦ − 90◦)

For other triangles, Dirichlet spectrum may be evaluated numerically

Scaling method of Vergini et al. - code by Alex Barnett



Calculating vacuum energy of infinite cylinder

Given spectrum ωm,n for 2-dimensional cross-section,

Vacuum energy per unit length is

Evac =1

2limt→0

∑m,n

∫ ∞−∞

dk

2π

√k2 + ω2

m,ne−t√

k2+ω2m,n

=1

2limt→0

(− d

dt

)∫ ∞−∞

dk

2π

∑m,n

e−t√

k2+ω2m,n



E.g. Equilateral Triangular Cylinder

Spectrum:

ω2mnp =

2π2

3h2(m2 + n2 + p2) with m + n + p = 0

ω2m,n =

4π2

3h2(m2 + mn + n2)

Degeneracy factor of 1/6

Dirichlet: m 6= 0, n 6= 0, and m + n 6= 0

Neumann: m 6= 0 or n 6= 0




Vacuum energy:

Evac =1

2limt→0

(− d

dt

)∫ ∞−∞

dk

2π

∑m,n

e−t√

k2+ω2m,n

may be evaluated using Poisson’s summation formula

Isolate Divergences:

EDivvac =

(√3 h2

2π2t4±√

3 h

4πt3+

1

6πt2

)=

(3A

2π2t4± P

8πt3+

C

48πt2

)where A = Area, P = Perimeter, C =

∑i

(παi− αi

π

)




Finally, we have finite part:

E(D,N)vac =

1

144π2h2

±12πζ(3)− 10√

3

3ζ(4)− 16

√3

3

∞∑p,q=1

1 + (−1)p+q

(p2 + q2/3)2

Further simplifying:

E(D,N)vac =

1

96h2

[√3

9

[ψ′(2/3)− ψ′(1/3)

]± 8

πζ(3)

]

Evaluating:

E(D)vac =

0.017789

h2E(N)

vac = −0.045982

h2



Results for Dirichlet Triangular Cylinders

Plotting the scaled (dimensionless) vacuum energy εA vs. dimensionlessgeometric parameter A/P2, we observe “universal” behavior:

0.02 0.04 0.06 0.08

A

P2

-6

-4

-2

2

lnHE AL



Results for Dirichlet Right Triangles

0.02 0.04 0.06 0.08

A

P2

-3

-2

-1

1

2

logK A EO

Numerical vacuum energy for Dirichlet right triangles plotted alongwith exact results for the three special triangles, and the square

Numerical and exact results agree within 1% where exact results exist

Curve is proximity force approximation for acute angle of righttriangle → 0 (including contribution from two long sides only)



Analogous Calculations for three special Tetrahedra



Use General Poisson resummation formula:

∞∑m1,...,mn=−∞

e−τ√

(m+a)j Ajk (m+a)k

=2nπ(n−1)/2 Γ((n + 1)/2)

|det (A)|1/2∞∑

m1,...,mn=−∞

τ e−2πi mj aj

(τ2 + 4π2kj kj)(n+1)/2



E.g. for “Large Tetrahedron”

Spectrum:

ω2kmn =

π2

4a2[3(k2 + m2 + n2)− 2(km + kn + mn)

]Dirichlet boundaries: k > m > n > 0

Vacuum energy:

Divergent terms involving volume, perimeter, corners, appear as before

Finite part is

E (D)vac =

1

a

{− 1

96π2[Z3(2; 1, 1, 1) + Z3b(2; 1, 1, 1)

]+

1

8πζ(3/2)L−8(3/2)

+1

16πZ2b(3/2; 2, 1)− π

96− π√

3

72

}= −0.0468804266

aLev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 45 / 49

Outlook Regular Polygons

Outlook: Regular Polygons

Exact results for triangle and square only

Can we interpolate between these and the circle?What happens when number of sides � 1?

Work in progress!


Outlook Chaotic Cavities

Outlook: Chaotic cavities


Outlook Outside of a Box

Outlook: Outside of a Box

Consider concentric polygons of size a and b

Efinitet =

1

a

∞∑p,q=0

Cp,q

(ab

)p ( ta

)q=

(C0,0

a+

C1,0

b+

C2,0a

b2+ · · ·

)+ t

(C0,1

a2+

C1,1

ab+

C2,1

b2+ · · ·

)+ O(t2)

Take t � a� b and extract C0,0

Work in progress!


Conclusions

Conclusions

Careful regularization and renormalization (including inside and outsidecontributions) needed to obtain physically meaningful energies and forces

Classical orbit approach produces exact results in simple cases and mayallow for good approximations where exact solutions are nonexistent

Hope for intelligent combination of anlaytical and numerical tools for generalgeometries


quantum vacuum energy calculations: from graphs to ...lkaplan/graphs_polyhedra.pdfquantum vacuum...

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