Heredegen’s algebraic approach to Casimir
effectWhat two plates and two delta like systems tells as about?

Kamil Ziemiankziemianfvt@gmail.com
Virtual LQP Workshop, 17 VI 2020
Jagiellonian University in Cracow

Table of contents

1. Casimir effect in algebraic settings

2. Herdegen’s approach

3. Two plates system

4. Two delta like systems

2

General remarks
The main aim of this talk is to present foundations of Herdegen’s
framework and interesting results obtained in it. On the mathematicalside only main steps are presented, often in simplified form. Also some
important results are only mention or even omitted.
All results presented here are derived in as rigorous way as of today’smathematics requires. Precise definitions, results and they proofs canbe found in the works listed at the end of presentation.
I assume general knowledge of algebraic approach to quantum physics,
but for the sake of clarity I will repeat well know points, stressing thisthat are important here.

3

Casimir effect in algebraic
settings

Algebraic formalism
In this talk quantum system will be described by three elements.
1) C∗ algebra A.
2) Representation π of algebra A as operators acting on Hilbert spaceH.
π : A → π(A)(1)3) Time evolution implemented by family of automorphism of A andunitary transformations of H.
αt : A → A, A 7→ αt A,A∈A∗π(αt A) = U(t)π(A)U (t), U(t) = exp(itH),
where H is hamiltonian of the system. We assume that it isnonnegative.
5

Algebraic formalism
Two systems Q0 and Q1 are physically comparable if and only if they
have common algebra A and equivalent representations π0 and π1 . If
not, comparison of such systems can leads to “unreasonably”
phenomena of physical origin, like physically justified infinities.
Consider for Minkowski spacetime at thermal equilibrium with
T = 0 K and T = 1 K. Transition from first state to the second require
infinite amount of energy delivered to the system, so we shouldn’t be
surprised if compering this two states leads to infinite results.
We call interaction V singular if it introduction to the system change
representation to nonequivalent to π0 . In other cases we call it
nonsingular.

6

Algebraic problems of two “plates” system [Her01]
Consider massless scalar field φ in 1 + 1 dimensions (1 + 3 isanalogous) with two metal “plates” at distance x = 0 and x = a. In
standard analysis of Casimir effect we require that field obey Dirchletboundary conditions.
∂t2 φ(x, t) = −∆φ(x, t)(2)

φ(0) = φ(a) = 0(3)

How algebra C∗ for such case looks like?

7

Algebraic problems of two plate system [Her01]
In current context algebra for two “plates” with Dirichlet boundary
conditions should be build on the top of symplectic space of functions
from L2 (R, dx), that are regular enough and vanish at the “plates”
positions 0 and a (more precisely there are sum of functions from H01 ).
But this can give us different algebraic model for every value of a!
Situation is even worse when we want to find common algebra forplate position vary in range (−l, l) and (a − ε, a + ε). The functions
must now vanish on both these intervals, which give us trivial theory.
Maybe I will have time go back to this problem.

8

Moral of the story
Dirichlet boundary conditions belongs to singular class of interactions.
I will argue that for these reason we shouldn’t be surprised beappearing of infinities in such linear system, since they are probably
physical.
Also, renormalization of physical inifinites can lead to obscuring oreven losing of important information’s about system.
In the story part I will sketch Herdegen’s formalism that allow us toinvestigate systems free from all this problems, which are “closeenough” to Dirichlet case to recover canonical Casimir term. I will also
explain what mean “close enough”.

9Herdegen’s approachHerdegen framework. Assumptions [Her05]
Let Q be quantum system with Hamiltonian H0 and M be macroscopic
system described by classical variables collectively denoted a. For
example a can by just separation of the plates. We make following
assumptions about they interaction.
1) Introduction of system M is nonsingular for the Q.
2) When M is in fixed state described by a1 time evolution is given by
αa1 , t A 7→ A,
π(αa1 ,t A) = Ua1 (t)π(A)Ua∗1 (t),(4)
Ua1 (t) = exp(itHa1 ).

(5)

Since we can always add C id to Hamiltonian there is big ambiguity indefinition of Ha :Ha → Ha + C(a) id,(6)with arbitrary function C(a). Fortunately, this not introduce ambiguity
to physical predictions.
11Herdegen framework. Assumptions [Her05]
3) We will consider only adiabatic change of parameters a(t).
4) When joint system Q-M evolves, energy of Q is as previously given
by H0 , but due to interactions with M quantum states evolution is
given by family of operators HM (t).
5) Due to adiabatic changes of a(t) we can use approximation
HM (t) = Ha(t) ,(7)where Ha(t) was defined previously as Hamiltonian for fixed state of M
with parameters a(t).

12Herdegen framework. Assumptions [Her05]
6) For fixed t consider eigenstate of Ha(t)Ha(t) ψa(t) = Ea(t) ψa(t) , for fixed t.

(8)

We assume that this eigenstates are not degenerated and depend
continuously on a(t). Since we are interested in ground state this is notvery restrictive assumption. Adiabatic evolution ψ(t) of state ψa(0) is
given by
ψ(t) = exp iϕ(Ea(t) , ψa(t) , t) ψa(t) .
(9)7) Expectation value of any observable O in the state ψa(t) is given by
hOit = (ψa(t) , Oψa(t) ).

(10)

This expression is independent of any ambiguity in choosing of Ha(t) .
13

Herdegen framework. Assumptions [Her05]
8) Casimir energy is given by expectation value of H0 (Hamiltonian ofisolated system Q) in its current ground state Ωa(t) .
Ea(t) = (Ωa(t) , H0 Ωa(t) )(11)9) Casimir force is minus derivative of energy with respect to a(t).
Fa(t) = −∂Ea(t)∂a(t)(12)

Since we are interested in how energy and force depend on a, not on t,
from this moment I will omitting time dependence in a.

14

Constructions of algebra and representation [Her05]
This was list of our wishes, but if they can’t come true, it is void.
Fortunately there exist quit general construction.
We need real Hilbert space R and operator selfadjoint operator h withdomain D(h). Operator h need to be strictly positive or nonnegative(more about this latter): h > 0 or h ≥ 0.
Consider L = D(h) ⊕ R ⊂ R ⊕ R. We denote Vi = vi ⊕ ui , vi ∈ D(h),ui ∈ R. L is symplectic space with symplectic product:σ(V1 , V2 ) = (v2 , u1 ) − (v1 , u2 ).

(13)

Dynamics on L is given by Hamiltonian1[(hv, hv) + (u, u)].
2
We can solve equation of motions for this system and find time
evolution family of transformations Tt .
H(V) =(14)

15

Constructions of algebra and representation [Her05]
Let K = R ⊕ iR be complexification of R. We define operator
j : L → K,
j(V) = j(v ⊕ u) = h1/2 v − ih−1/2 u.
Operator j requires that h > 0. In special cases we can allow it to have 0
eigenvalue.
After making this more precise and putting some work we can use
standard construction of Weyl C∗ algebra corresponding to symplectic
space L with representation in Fock space with K as “one-particle
space” (see the second volume of Bratteli, Robinson book [BR87]).
When you introduce macroscopic bodies M we only need to change
operator h to ha and repeat all above construction. This is easy to say,
but hard to follow.
16

Energy and number of particles [Her05]
Ground state representations for algebraic systems constructed thisway are equivalent if and only if
−1/2
Na = (Ωa , NΩa ) = Tr h−1/2 (ha − h)h−1< ∞.
a (ha − h)h(15)

In the context of quantum field this formula have clear meaning:
introduction of macroscopic bodies create only finite number of freeparticles.
Physics requires that Casimir energy is also finite.
Ea = (Ωa , H0 Ωa ) = Tr (ha − h)h−1
a (ha − h) < ∞(16)

Proving of r.h.s. of this formulas takes some time.
17Local energy density for quantum field [Her05]
In this case ground state Ωa defines distribution on pairs of test
functions f , g with L2 scalar product.
Ta ( f , g) =

 1

1
−1
f , (ha − h)g + ∇f , (h−1
)∇g
a −h
4
4

(17)

By definition, outside singular support of distribution this give us
kernel function Ta (~x, ~y). In such region, and only in it, we can defined
local energy density:εa (~x) = Ta (~x, ~x).
(18)

18

What this framework give us?
• We can recover in rigorous manner textbook results for two plates.
• Since energy is equal (Ωa , H0 Ωa ) it must be positive. There is no
dubious “negative energy”.
• All physical quantities are under control.
• We can quite well understand relation between global and local
energy.
• It is hard to find nonsingular interactions that looks simple and“natural”.
• To understand how to use this formalism you need to put some
work.

19

What you need to do in practice?
1) Find promising system described by operators h and ha . Show that
Na < ∞ and Ea < ∞.
2) Introduce rescaled family of systems hλ, a , λ ∈ [1, 0) such thath1, a = ha .
3) Show that for all λ ∈ [1, 0) you have Nλ, a < ∞, Eλ, a < ∞.
4) Show that in limit λ & 0 your family tends in the sens of resolvent
limit (or other well defined way) to interesting system: hλ, a → hI . This
limit case is most probably singular.
5) Derive asymptotic expansion of Casimir energy Eλ, a around λ = 0.
6) Compute energy density ελ, a (~x). Distribution Tλ, a ( f , g) mostprobably will be regular on R3 .
7) Compute energy density limit for λ & 0. If limit case is singular,
singular support of limit distribution will be probably nonempty.
20

Solved systems• Multidimensional harmonic oscillator, Andrzej Herdegen, 2005,[Her05].
• Two plate system for scalar and electromagnetic field, Herdegen,2006, [Her06], Andrzej Herdegen and Mariusz Stopa 2010,[HS10].
• Two point like objects for scalar field, Ziemian, 2020, to be
published.

21

Two plates system

Herdegen and Stopa approach to two plates [HS10]
Plates are parallel to xy plane, so interesting dynamics is only in z
directions. After “integreting out xy directions over unit area”,
consistency conditions take form
Tr (hz, a − hz )2 < ∞,Tr (hz, a − hz )hz (hz, z − hz ) < ∞.

(19)
(20)

Energy per unit area
Ea =



1
Tr (hz, a − hz )(2hz, a + hz, a )(hz, a − hz )
24π

(21)

In this special case, since we don’t have take h−1
a , it can have 0
eigenvalue.
23

Herdegen and Stopa approach to two plates [HS10]
We will consider simpler case of scalar field. We have R = L2R (R3 , d3 x),√
K = L2 (R3 , d3 x), h = −∆.
ha2 = −∆ + V(22)

V is projection operator of finite rank. In position space it is integral
operator with kernelhiV(z, y) = f (z − b)f (y − b) + f (z + b)f (y + b) .
(23)
f (z) is smooth complex function with compact support.
Rescaled system is defined byfλ (z) = λ−1 f

z
λ

(24)

For λ & 0 system converge in resolvent sens to field with Dirichletboundary conditions in planes z = ±b.
24

Herdegen and Stopa approach to two plates [HS10]
Asymptotic expansion of Casimir energy
Eλ, a =

E∞
c
π2
+
−
+ O(λ),λ3
λa2
1440a3(25)

where E∞ , c are constant.
E∞ /λ3 is two times energy of single plate in vacuum, per unit area,−π 2 /1440a3 this term in this expansion recovers standard formula for
Casimir force.
Constant c depends on choice of function f and typically c > 0, so force
become repulsive for large a.
Since λ is roughly thickens of the plate, this model is only valid when
λ < a and this guaranties that energy is always positive.

25

Herdegen and Stopa approach to two plates [HS10]For λ 6= 0 local energy density ελ, a (~x) is defined on whole R3 . In thelimit λ & 0 singular support of distribution is equal to {−b, +b} (onthe x axis).
For this reason integration over whole space is mathematicallyunjustified. But, if we do it anyway we arrive at resultZ
ε0, a (x) dx = −π2.
1440a3(26)

This expression give us canonical Casimir term for force, but notrecover total formula. It is also negative, while global energy is always
positive.

26

Two delta like systems

Two delta like systems
In this case we know only results for scalar field.
This problem is in many ways like two plates problem.
√
R = L2R (R3 , d3 x), K = L2 (R3 , d3 x), h = −∆.
h~a2 = −∆ + V
V is again projection operator of finite rank.
hiV(~x, ~y) = σ(g) g(~x − ~b)g(~x − ~b) + g(~x + ~b)g(~x + ~b)

(27)

(28)

g(~x) is smooth, positive, spherical symmetric function with compact
support.
Rescaled version of the modelgλ (~x) = λ−3 g

 
~x
λ

(29)

In limit λ & 0 we have two δ system, well know in literature [Alb88].
28

Two delta like systems
Asymptotic expansion of Casimir energy
+∞
 Z
2α χ
e−2l dlE(a, λ) = Eself (λ) + 3+
π λ(γ + l)[(γ + l)2 − e−2l ]
0

b1 χ
+
γ

+∞
Z

l [3(γ + l) e − e−4l ]
2
dl −
2
2
−2l
2
γ
(γ + l) [(γ + l) − e ]
2

2 −2l

0

+

1
γ

+∞
Z

+∞
Z

le−2l dl+(γ + l)[(γ + l)2 − e−2l ]0(1 − l)e−2l
dl + O(λ)(γ + l)2 − e−2l


0

(30)
Eself (λ) is two times energy of single δ in vacuum,a is distance between centers of two bodies,
α is parameter of δ interaction [Alb88],
γ = αa/2π 2 > 1 is constrain on distance a [Alb88],
χ > 0 and b1 > 0 are constants describing properties of function g.
29

Two delta like systems
Presented expression for global energy is dominated by model
dependent terms which can’t be removed. Numerical analysis showthat force is repulsive. This contradict previous result found inliterature that predicts universal attractive force [Sca05].
Also we can show that local energy density have universal limit, which
exclude possibility of recovering global energy by integrating local[FP18], [Fer19].

30

Open problems• How to compute electromagnetic Casimir effect for two delta like
objects?
• How this framework relates to alternative algebraic formalism ofClaudio Dappiaggi, Gabriele Nosari and Nicola Pinamonti[DNP16]?
• Investigate limit λ & 0 for single sphere.

31Thank you. Any question?

Papers on Herdegen’s approach
Herdegen, A., “No-nonsens Casimir force”, Acta Phys. Polon., B32,
55–65, 2001, hep-th/0008207.
Herdegen, A., “Quantum backreaction (Casimir) effect I. What areadmissible idealizations?”, Ann. Henri Poincare, 6, 657, 2005, DOI:
10.1007/s00023-005-0219-7, arXiv: hep-th/0412132.
Herdegen, A., “Quantum backreaction (Casimir) effect II. Scalarand electromagnetic fields”, Ann. H. Poincare, 7, 253–301, DOI:
https://doi.org/10.1007/s00023-005-0249-1, arXiv: hep-th/0507023.
Herdegen, A., and Stopa, M., “Global vs local casimir effect”, Ann.
Henri Poincare, 11, 1171, 2010, DOI: 10.1007/s00023-010-0053-4,
arXiv: 1007.2139.
Ziemian, K., “Algebraic Approach to Casimir Force Between Two
δ-like Potentials”, to be published in Ann. Henri Poincare.

Bibliography
Albeverio, S., Gesztesy, F., Høegh-Krohn, R., and Holden, H.,
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Tokyo, 1988.
Bordag., M., Mohideen, U., and Mostepanenko, V. M., “Newdelepoments in the casimir effect”, Physics Reports, 353, 1, 2001,
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Bibliography
Fermi, D., “The Casimir energy anomaly for a point interaction”,
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Fermi, D., Pizzocchero, L., Local zeta regularization and the scalar
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Bibliography
Scardicchio, A., “Casimir dynamics: Interactions of surfaces with
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If you have more questions, please write to me
at kziemianfvt@gmail.com.