Casimir effect and trace formula

Document Sample
Casimir effect and trace formula Powered By Docstoc
					                                                                                                                1
                    The Casimir effect as scattering problem

                                              Andreas Wirzba
                                           Institut fur Kernphysik
                                                     ¨
                                        Forschungszentrum Julich
                                                               ¨




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                  1
                      The Casimir effect as scattering problem

                                                Andreas Wirzba
                                             Institut fur Kernphysik
                                                       ¨
                                          Forschungszentrum Julich
                                                                 ¨

            1.   Introduction




· ◦   < ∧    >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                 1
                     The Casimir effect as scattering problem

                                               Andreas Wirzba
                                            Institut fur Kernphysik
                                                      ¨
                                         Forschungszentrum Julich
                                                                ¨

            1.   Introduction
            2.   Geometry dependence of the Casimir effect




· ◦   < ∧    >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                  1
                      The Casimir effect as scattering problem

                                                Andreas Wirzba
                                             Institut fur Kernphysik
                                                       ¨
                                          Forschungszentrum Julich
                                                                 ¨

            1.   Introduction
            2.   Geometry dependence of the Casimir effect
            3.   Map to a multi-scattering problem of a point particle




· ◦   < ∧    >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                  1
                      The Casimir effect as scattering problem

                                                Andreas Wirzba
                                             Institut fur Kernphysik
                                                       ¨
                                          Forschungszentrum Julich
                                                                 ¨

            1.   Introduction
            2.   Geometry dependence of the Casimir effect
            3.   Map to a multi-scattering problem of a point particle
            4.   Scalar Casimir effect for Dirichlet spheres and plates




· ◦   < ∧    >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                  1
                      The Casimir effect as scattering problem

                                                Andreas Wirzba
                                             Institut fur Kernphysik
                                                       ¨
                                          Forschungszentrum Julich
                                                                 ¨

            1.   Introduction
            2.   Geometry dependence of the Casimir effect
            3.   Map to a multi-scattering problem of a point particle
            4.   Scalar Casimir effect for Dirichlet spheres and plates
            5.   Conclusions




· ◦   < ∧    >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                    1
                        The Casimir effect as scattering problem

                                                  Andreas Wirzba
                                               Institut fur Kernphysik
                                                         ¨
                                            Forschungszentrum Julich
                                                                   ¨

            1.    Introduction
            2.    Geometry dependence of the Casimir effect
            3.    Map to a multi-scattering problem of a point particle
            4.    Scalar Casimir effect for Dirichlet spheres and plates
            5.    Conclusions



                                             Publications:
                        Aurel Bulgac & A.W., Phys. Rev. Lett. 87 (2001) 120404;
                  Aurel Bulgac, Piotr Magierski & A.W., Europhys. Lett. 72 (2005) 327;
                 Aurel Bulgac, Piotr Magierski & A.W., Phys. Rev. D 73 (2006) 025007;
                        A.W., A. Bulgac & P. Magierski, J. Phys. A (2006) 6815.



· ◦   < ∧    >     •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                         2
      • H.B.G. Casimir (1948): two conducting, but uncharged parallel plates attract
        each other in vacuum
                         L
                                        F || (L)                   c      π2             −7 1   µm4
                                                        =        − 4          ≈ −1.3 × 10     N
                                           A                      L       240              L4 cm2
                     A          A
                                                                   c      π2
                                         E || (L)       =        − 3          A
                                                                  L       720




· ◦    < ∧   >   •           Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                         2
      • H.B.G. Casimir (1948): two conducting, but uncharged parallel plates attract
        each other in vacuum
                         L
                                        F || (L)                   c      π2             −7 1   µm4
                                                        =        − 4          ≈ −1.3 × 10     N
                                           A                      L       240              L4 cm2
                     A          A
                                                                   c      π2
                                         E || (L)       =        − 3          A
                                                                  L       720


      • Reason: zero-point fluctuations of the e.m. field modified
        by the presence of the plates relative to the free case




· ◦    < ∧   >   •           Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                           2
      • H.B.G. Casimir (1948): two conducting, but uncharged parallel plates attract
        each other in vacuum
                         L
                                        F || (L)                   c      π2             −7 1   µm4
                                                        =        − 4          ≈ −1.3 × 10     N
                                           A                      L       240              L4 cm2
                     A          A
                                                                   c      π2
                                         E || (L)       =        − 3          A
                                                                  L       720


      • Reason: zero-point fluctuations of the e.m. field modified
        by the presence of the plates relative to the free case
        or rather relative to the infinitely separated case
                                                                                          1                              1
          ⇒ change in the energy of the vacuum:                                           2   ωk |plates(L) −            2   ωk |free(L→∞ !)




· ◦    < ∧   >   •           Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                           2
      • H.B.G. Casimir (1948): two conducting, but uncharged parallel plates attract
        each other in vacuum
                         L
                                        F || (L)                   c      π2             −7 1   µm4
                                                        =        − 4          ≈ −1.3 × 10     N
                                           A                      L       240              L4 cm2
                     A          A
                                                                   c      π2
                                         E || (L)       =        − 3          A
                                                                  L       720


      • Reason: zero-point fluctuations of the e.m. field modified
        by the presence of the plates relative to the free case
        or rather relative to the infinitely separated case
                                                                                          1                              1
          ⇒ change in the energy of the vacuum:                                           2   ωk |plates(L) −            2   ωk |free(L→∞ !)

      • Manifestation: Mesoscopic or even macroscopic realization of
                  quantum fluctuations of the vacuum




· ◦    < ∧   >   •           Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                           2
      • H.B.G. Casimir (1948): two conducting, but uncharged parallel plates attract
        each other in vacuum
                         L
                                        F || (L)                   c      π2             −7 1   µm4
                                                        =        − 4          ≈ −1.3 × 10     N
                                           A                      L       240              L4 cm2
                     A          A
                                                                   c      π2
                                         E || (L)       =        − 3          A
                                                                  L       720


      • Reason: zero-point fluctuations of the e.m. field modified
        by the presence of the plates relative to the free case
        or rather relative to the infinitely separated case
                                                                                          1                              1
          ⇒ change in the energy of the vacuum:                                           2   ωk |plates(L) −            2   ωk |free(L→∞ !)

      • Manifestation: Mesoscopic or even macroscopic realization of
                  quantum fluctuations of the vacuum

      • Distinctive Feature: Casimir calculations depend on the geometry in a
                   non-intuitive way.




· ◦    < ∧   >   •           Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                3

 Let us look at the other side of the “coin”:
    Take the geometry dependence as the guiding principle for the Casimir effect




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                3

 Let us look at the other side of the “coin”:
    Take the geometry dependence as the guiding principle for the Casimir effect


       Casimir energy ≡ vacuum energy of the geometry-dependent and
                        changeable part of the level density (d.o.s)




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                3

 Let us look at the other side of the “coin”:
    Take the geometry dependence as the guiding principle for the Casimir effect


       Casimir energy ≡ vacuum energy of the geometry-dependent and
                        changeable part of the level density (d.o.s)
                                            (↔ shell-correction energy in nuclear physics)




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                            3

 Let us look at the other side of the “coin”:
    Take the geometry dependence as the guiding principle for the Casimir effect


       Casimir energy ≡ vacuum energy of the geometry-dependent and
                        changeable part of the level density (d.o.s)
                                              (↔ shell-correction energy in nuclear physics)


            d.o.s.:    ρ(E) ≡                     δ(E − Ek ) = ρ0 (E) + ρbulk (E) + δρC (E, geom.-dep.)
                                           Ek
                                                                            free case          L→∞                L finite




· ◦   < ∧    >    •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                            3

 Let us look at the other side of the “coin”:
    Take the geometry dependence as the guiding principle for the Casimir effect


       Casimir energy ≡ vacuum energy of the geometry-dependent and
                        changeable part of the level density (d.o.s)
                                              (↔ shell-correction energy in nuclear physics)


            d.o.s.:    ρ(E) ≡                     δ(E − Ek ) = ρ0 (E) + ρbulk (E) + δρC (E, geom.-dep.)
                                           Ek
                                                                            free case          L→∞                L finite


→ Casimir Energy: EC               ≡            dE 1 E δρC (E, geom.-dep.)
                                                   2




· ◦   < ∧    >    •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                        3

 Let us look at the other side of the “coin”:
    Take the geometry dependence as the guiding principle for the Casimir effect


       Casimir energy ≡ vacuum energy of the geometry-dependent and
                        changeable part of the level density (d.o.s)
                                                 (↔ shell-correction energy in nuclear physics)


            d.o.s.:       ρ(E) ≡                      δ(E − Ek ) = ρ0 (E) + ρbulk (E) + δρC (E, geom.-dep.)
                                              Ek
                                                                               free case              L→∞                  L finite


→ Casimir Energy: EC                  ≡            dE 1 E δρC (E, geom.-dep.) = − 1
                                                      2                           2                                  dE NC (E, geom.-dep.)

                                                  E
       where            NC (E)        =                dE δρC (E ) (geom-dep. part of the integrated d.o.s)
                                                −∞

                                                                                         E
             with        N (E) ≡                      Θ(E − Ek ) =                           dE ρ(E )
                                              Ek                                     0



· ◦   < ∧    >      •    Andreas Wirzba   The Casimir effect as scattering problem       QFEXT07, Leipzig, 19-Sep-2007
 Calculation – Map onto a scattering problem                                                                       4


      • Casimir-energy calculation for two parallel plates of distance L is simple




· ◦    < ∧   >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Calculation – Map onto a scattering problem                                                                      4


      • Casimir-energy calculation for two parallel plates of distance L is simple
        since the problem is separable (i.e. quasi 1-dimensional).
      • More complicated geometries → in general, only proximity force approximation left.




· ◦    < ∧   >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Calculation – Map onto a scattering problem                                                                                                4


      • Casimir-energy calculation for two parallel plates of distance L is simple
        since the problem is separable (i.e. quasi 1-dimensional).
      • More complicated geometries → in general, only proximity force approximation left.
      • However, vacuum energy of a fluctating massless scalar field between non-overlapping
        spheres or cylinders with Dirichlet (or similar) boundary conditions still doable:
                                                                                                 d                        1
        Krein (1953) trace formula: variation of the d.o.s. δρ(E) ↔                              dE
                                                                                                    phase         shift   2i
                                                                                                                             ln detSn (E)




· ◦    < ∧   >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Calculation – Map onto a scattering problem                                                                                                4


      • Casimir-energy calculation for two parallel plates of distance L is simple
        since the problem is separable (i.e. quasi 1-dimensional).
      • More complicated geometries → in general, only proximity force approximation left.
      • However, vacuum energy of a fluctating massless scalar field between non-overlapping
        spheres or cylinders with Dirichlet (or similar) boundary conditions still doable:
                                                                                                 d                        1
        Krein (1953) trace formula: variation of the d.o.s. δρ(E) ↔                              dE
                                                                                                    phase         shift   2i
                                                                                                                             ln detSn (E)

                                               1 d
             δρ(E) = ρ(E) − ρ0 (E) =
                     ¯      ¯                        tr ln Sn (E)                   of the n-sphere S-matrix
                                              2πi dE




· ◦    < ∧    >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Calculation – Map onto a scattering problem                                                                                                4


      • Casimir-energy calculation for two parallel plates of distance L is simple
        since the problem is separable (i.e. quasi 1-dimensional).
      • More complicated geometries → in general, only proximity force approximation left.
      • However, vacuum energy of a fluctating massless scalar field between non-overlapping
        spheres or cylinders with Dirichlet (or similar) boundary conditions still doable:
                                                                                                 d                        1
        Krein (1953) trace formula: variation of the d.o.s. δρ(E) ↔                              dE
                                                                                                    phase         shift   2i
                                                                                                                             ln detSn (E)

                                               1 d
             δρ(E) = ρ(E) − ρ0 (E) =
                     ¯      ¯                        tr ln Sn (E)                   of the n-sphere S-matrix
                                              2πi dE
        Extract the geometry-dep. Casimir fluctuations out of the multi-scattering part




· ◦    < ∧    >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Calculation – Map onto a scattering problem                                                                                                    4


      • Casimir-energy calculation for two parallel plates of distance L is simple
        since the problem is separable (i.e. quasi 1-dimensional).
      • More complicated geometries → in general, only proximity force approximation left.
      • However, vacuum energy of a fluctating massless scalar field between non-overlapping
        spheres or cylinders with Dirichlet (or similar) boundary conditions still doable:
                                                                                                     d                        1
        Krein (1953) trace formula: variation of the d.o.s. δρ(E) ↔                                  dE
                                                                                                        phase         shift   2i
                                                                                                                                 ln detSn (E)

                                                     1 d
             δρ(E) = ρ(E) − ρ0 (E) =
                     ¯      ¯                              tr ln Sn (E)                 of the n-sphere S-matrix
                                                    2πi dE
        Extract the geometry-dep. Casimir fluctuations out of the multi-scattering part
        ⇒ Calculation mapped onto a quantum mechanical billiard problem:
          hyperbolical scattering of a point particle off n spheres or n discs
                            23132321

                                                2                    References:
                                                                     B. Eckhardt, J. Phys. A20 (1987);
                                                                     P. Gaspard & S. Rice, J. Chem. Phys. 90 (1989);
                                                                     P. Cvitanovi´ & B. Eckhardt, Phys. Rev. Lett. 63 (1989);
                                                                                 c
                                                        3            M. Henseler, A. Wirzba & T. Guhr, Ann. Phys. 258 (1997).
                                       1


                                 2313




· ◦    < ∧    >   •   Andreas Wirzba       The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                         5
 Digression: E.Beth & G.E. Uhlenbeck (1937)

predecessor of Krein (1962) formula for spherical potential:
                                                                                                              R      R

Idea: spherical scattering box minus empty reference box:                                  Lim
                                                                                            R−>


                                                           1
                                               `                        ´
                      uk (r)      ∼       sin kr −         2
                                                               π + η (k)
Asymptotically:
                       (0)                   `             1
                                                                 ´
                      uk (r)      ∼       sin kr −         2
                                                               π




· ◦   < ∧   >     •      Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                          5
 Digression: E.Beth & G.E. Uhlenbeck (1937)

predecessor of Krein (1962) formula for spherical potential:
                                                                                                              R       R

Idea: spherical scattering box minus empty reference box:                                  Lim
                                                                                            R−>


                                                            1
                                                `                        ´
                      uk (r)        ∼     sin kr −          2
                                                                π + η (k)                   and Dirichlet b.c.’s:
Asymptotically:                                                                                        (0)
                       (0)                   `              1
                                                                  ´                         uk (R) = uk (R) = 0.
                      uk (r)        ∼     sin kr −          2
                                                                π

                EV conditions ( (2 +1)-fold degenerate ) :
                                1
            ⇒      k ,n R −     2
                                    π + η (k         ,n )    =     πn,        n=0,1,2,···       (with potential)
                    (0)         1
                   k ,n R −     2
                                    π                        =     πn,        n=0,1,2,···       (without potential)

A change of the radial quantum number n by one unit, for fixed angular momentum , implies
                                                „                      «
                                                        ∂                                 (0)
                                          ∆k         R+    η (k)           = π = ∆k             R,
                                                        ∂k

such that the conditions ρ (k)∆k = ρ(0) (k)∆k(0) = 2 + 1 (note the averaging !)
                         ¯         ¯

                                               (0)              2 +1 ∂
leads to the formula          ρ (k) − ρ
                              ¯       ¯              (k) =        π ∂k
                                                                       η     (k)          (independently of R)


· ◦   < ∧   >     •      Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                6
 Digression 2: Semiclassical interpretation of Krein formula




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                6
 Digression 2: Semiclassical interpretation of Krein formula
Determinant of scattering matrix is semiclassically a sum over periodic orbits (+Weyl terms)


                                                R                                    R

                                                              −




                                                                         ˛
                                                     1    d          (n)
                                                                         ˛
                                                       Im    ln det S (k)˛
                                                    2π    dk             ˛
                                                                          k real




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                6
 Digression 2: Semiclassical interpretation of Krein formula
Determinant of scattering matrix is semiclassically a sum over periodic orbits (+Weyl terms)
Consider the difference of the densities of states of two bounded reference systems :
                                                R                                    R

                                                              −



                                       n                                  o
                                            (n)                           (0)
                                           ρ (k     ; R) − ρ (k     ; R)
                                                                     ˛
                                               1      d         (n)
                                                                     ˛
                                            =    Im     ln det S (k)˛
                                              2π     dk              ˛
                                                                      k real




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                6
 Digression 2: Semiclassical interpretation of Krein formula
Determinant of scattering matrix is semiclassically a sum over periodic orbits (+Weyl terms)
Consider the difference of the densities of states of two bounded reference systems :
                                                R                                    R

                                                              −

       • The container-induced periodic orbits cancel!




                                       n                                  o
                                            (n)                           (0)
                                           ρ (k     ; R) − ρ (k     ; R)
                                                                     ˛
                                               1      d         (n)
                                                                     ˛
                                            =    Im     ln det S (k)˛
                                              2π     dk              ˛
                                                                      k real




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                6
 Digression 2: Semiclassical interpretation of Krein formula
Determinant of scattering matrix is semiclassically a sum over periodic orbits (+Weyl terms)
Consider the difference of the densities of states of two bounded reference systems :
                                                R                                    R

                                                              −

       • The container-induced periodic orbits cancel!

       • However, ∃ further spurious periodic orbits whose lengths grow with increasing R.


                                       n                                  o
                                            (n)                           (0)
                                lim        ρ (k     ; R) − ρ (k     ; R)
                              R→∞
                                                                     ˛
                                               1      d         (n)
                                                                     ˛
                                            =    Im     ln det S (k)˛
                                              2π     dk              ˛
                                                                      k real




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                6
 Digression 2: Semiclassical interpretation of Krein formula
Determinant of scattering matrix is semiclassically a sum over periodic orbits (+Weyl terms)
Consider the difference of the densities of states of two bounded reference systems :
                                                R                                    R

                                                              −

       • The container-induced periodic orbits cancel!

       • However, ∃ further spurious periodic orbits whose lengths grow with increasing R.
       • Removal of long orbits by exponential damping or averaging:

                                       n                                  o
                                            (n)                           (0)
                        lim lim            ρ (k + i ; R) − ρ (k + i ; R)
                        →0+ R→∞
                                                                     ˛
                                               1      d         (n)
                                                                     ˛
                                            =    Im     ln det S (k)˛
                                              2π     dk              ˛
                                                                      k real




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                              6
 Digression 2: Semiclassical interpretation of Krein formula
Determinant of scattering matrix is semiclassically a sum over periodic orbits (+Weyl terms)
Consider the difference of the densities of states of two bounded reference systems :
                                                R                                    R

                                                              −

       • The container-induced periodic orbits cancel!

       • However, ∃ further spurious periodic orbits whose lengths grow with increasing R.
       • Removal of long orbits by exponential damping or averaging:

                                       n                                  o
                                            (n)                           (0)
                        lim lim            ρ (k + i ; R) − ρ (k + i ; R)
                        →0+ R→∞
                                                                     ˛
                                               1      d         (n)
                                                                     ˛
                                            =    Im     ln det S (k)˛
                                              2π     dk              ˛
                                                                      k real

                                                                                                  Note the order of the limits!


· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                    7

      1. “infinite” container:




· ◦    < ∧    >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                   7

      1. “infinite” container: ρ(E)=ρ0 (E) (background field)




· ◦    < ∧   >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                     7

      1. “infinite” container: ρ(E)=ρ0 (E) (background field)


      2. n bubbles (of radii ai ) “punched out” at “infinite” separation:
                         X n
         ρ(E) = ρ0 (E)+        ρW (E, ai ) (note the excluded volume !)
                          i=1
                               | {z }
                                  Weyl-Term




· ◦    < ∧    >    •     Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                     7

      1. “infinite” container: ρ(E)=ρ0 (E) (background field)


      2. n bubbles (of radii ai ) “punched out” at “infinite” separation:
                         X n
         ρ(E) = ρ0 (E)+        ρW (E, ai ) (note the excluded volume !)
                          i=1
                               | {z }
                                  Weyl-Term

      3. geometry-dependent arrangement of n bubbles:
                                   n
                                   X
             ρ(E) = ρ0 (E) +              ρW (E, ai ) + δρC (E, {ai }, {rij })
                                   i=1




· ◦    < ∧    >    •     Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                     7

      1. “infinite” container: ρ(E)=ρ0 (E) (background field)


      2. n bubbles (of radii ai ) “punched out” at “infinite” separation:
                         X n
         ρ(E) = ρ0 (E)+        ρW (E, ai ) (note the excluded volume !)
                          i=1
                               | {z }
                                  Weyl-Term

      3. geometry-dependent arrangement of n bubbles:
                                   n
                                   X
             ρ(E) = ρ0 (E) +              ρW (E, ai ) + δρC (E, {ai }, {rij })
                                   i=1


      4. Krein trace formula (note the averaging):                                        2iηn (E)
                                                           1 d z       }|    {
                       δρ(E)       =      ρ(E) − ρ0 (E) =
                                          ¯      ¯               ln det Sn (E)
                                                          2πi dE




· ◦    < ∧    >    •     Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                         7

      1. “infinite” container: ρ(E)=ρ0 (E) (background field)


      2. n bubbles (of radii ai ) “punched out” at “infinite” separation:
                         X n
         ρ(E) = ρ0 (E)+        ρW (E, ai ) (note the excluded volume !)
                          i=1
                               | {z }
                                  Weyl-Term

      3. geometry-dependent arrangement of n bubbles:
                                   n
                                   X
             ρ(E) = ρ0 (E) +              ρW (E, ai ) + δρC (E, {ai }, {rij })
                                   i=1


      4. Krein trace formula (note the averaging):                                        2iηn (E)
                                                           1 d z       }|    {
                       δρ(E)       =      ρ(E) − ρ0 (E) =
                                          ¯      ¯               ln det Sn (E)
                                                          2πi dE
                                                                                                                         det M † (k∗ )
                                                                                             z }| {
      5. Multiple scattering matrix
                                                                                                        Q
                                                                                             det Sn(E) = i det S1(E, ai ) det M (k)

                                                                                                            see A.W., Phys. Rep. 309 (1999)




· ◦    < ∧    >    •     Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                         7

      1. “infinite” container: ρ(E)=ρ0 (E) (background field)


      2. n bubbles (of radii ai ) “punched out” at “infinite” separation:
                         X n
         ρ(E) = ρ0 (E)+        ρW (E, ai ) (note the excluded volume !)
                          i=1
                               | {z }
                                  Weyl-Term

      3. geometry-dependent arrangement of n bubbles:
                                   n
                                   X
             ρ(E) = ρ0 (E) +              ρW (E, ai ) + δρC (E, {ai }, {rij })
                                   i=1


      4. Krein trace formula (note the averaging):                                        2iηn (E)
                                                           1 d z       }|    {
                       δρ(E)       =      ρ(E) − ρ0 (E) =
                                          ¯      ¯               ln det Sn (E)
                                                          2πi dE
                                                                                                                         det M † (k∗ )
                                                                                             z }| {
      5. Multiple scattering matrix
                                                                                                        Q
                                                                                             det Sn(E) = i det S1(E, ai ) det M (k)
                                                         „                                   «
                                      1                       d         `    ´
         → δ ρC (E, {ai }, {rij }) = − Im
             ¯                                                  ln det M k(E)                               see A.W., Phys. Rep. 309 (1999)
                                      π                      dE




· ◦    < ∧    >    •     Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                         7

      1. “infinite” container: ρ(E)=ρ0 (E) (background field)


      2. n bubbles (of radii ai ) “punched out” at “infinite” separation:
                         X n
         ρ(E) = ρ0 (E)+        ρW (E, ai ) (note the excluded volume !)
                          i=1
                               | {z }
                                  Weyl-Term

      3. geometry-dependent arrangement of n bubbles:
                                   n
                                   X
             ρ(E) = ρ0 (E) +              ρW (E, ai ) + δρC (E, {ai }, {rij })
                                   i=1


      4. Krein trace formula (note the averaging):                                        2iηn (E)
                                                           1 d z       }|    {
                       δρ(E)       =      ρ(E) − ρ0 (E) =
                                          ¯      ¯               ln det Sn (E)
                                                          2πi dE
                                                                                                                         det M † (k∗ )
                                                                                             z }| {
      5. Multiple scattering matrix
                                                                                                        Q
                                                                                             det Sn(E) = i det S1(E, ai ) det M (k)
                                                         „                                   «
                                      1                       d         `    ´
         → δ ρC (E, {ai }, {rij }) = − Im
             ¯                                                  ln det M k(E)                               see A.W., Phys. Rep. 309 (1999)
                                      π                      dE
         All determinants exists (although the relevant scattering matrices are infinite dimensional)
         since the associated T -matrices are trace-class.


· ◦    < ∧    >    •     Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                     7

      1. “infinite” container: ρ(E)=ρ0 (E) (background field)


      2. n bubbles (of radii ai ) “punched out” at “infinite” separation:
                         X n
         ρ(E) = ρ0 (E)+        ρW (E, ai ) (note the excluded volume !)
                          i=1
                               | {z }
                                  Weyl-Term

      3. geometry-dependent arrangement of n bubbles:
                                   n
                                   X
             ρ(E) = ρ0 (E) +              ρW (E, ai ) + δρC (E, {ai }, {rij })
                                   i=1


      4. Krein trace formula (note the averaging):                                        2iηn (E)
                                                           1 d z       }|    {
                       δρ(E)       =      ρ(E) − ρ0 (E) =
                                          ¯      ¯               ln det Sn (E)
                                                          2πi dE
                                                                                                                         det M † (k∗ )
                                                                                             z }| {
      5. Multiple scattering matrix
                                                                                                        Q
                                                                                             det Sn(E) = i det S1(E, ai ) det M (k)
                                           „                     «
                                      1       d          `     ´
         → δ ρC (E, {ai }, {rij }) = − Im
              ¯                                  ln det M k(E)          see A.W., Phys. Rep. 309 (1999)
                                      π      dE
      6. Casimir energy: Z ∞                   Z ∞               Z ∞
                                  1          1                1                     `       ´
                ∴ EC =       dE 2 E δ ρC = − 2
                                      ¯            dE N C =         dE Im ln det M k(E)
                           0                    0            2π 0


· ◦    < ∧    >    •     Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                             8


                       ∞                                                              ∞                                   ∞
                                1                              c                                     c
      EC   =            dE      2      ck δ ρC
                                            ¯        k(E) = −                          dk N C (k) =                       dk Im ln det M k
                   0                                          2                   0                 2π                0
                                ∞(1+i0+ )                                                 ∞(1−i0+ )
                    c                                                                                                           †
           =                                    dk ln det M k −                                       dk ln det M k
                   4π           0                                                     0
                                ∞
                    c
           =                        dk4 ln det M (ik4 ) for relativistic disp. E = ck and after Wick-rotation
                   2π       0




· ◦    < ∧     >        •           Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                             8


                       ∞                                                              ∞                                   ∞
                                1                              c                                     c
      EC   =            dE      2       ck δ ρC
                                             ¯       k(E) = −                          dk N C (k) =                       dk Im ln det M k
                   0                                          2                   0                 2π                0
                                ∞(1+i0+ )                                                 ∞(1−i0+ )
                    c                                                                                                           †
           =                                    dk ln det M k −                                       dk ln det M k
                   4π           0                                                     0
                                ∞
                    c
           =                        dk4 ln det M (ik4 ) for relativistic disp. E = ck and after Wick-rotation
                   2π       0


                                    †
                                                                   ´†                                  `       ∗
Note that det M (ik4 ) = det M (ik4 ) since det M (k) = det M (−k ) ;




· ◦    < ∧     >        •           Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                             8


                       ∞                                                              ∞                                   ∞
                                1                              c                                     c
      EC   =            dE      2       ck δ ρC
                                             ¯       k(E) = −                          dk N C (k) =                       dk Im ln det M k
                   0                                          2                   0                 2π                0
                                ∞(1+i0+ )                                                 ∞(1−i0+ )
                    c                                                                                                           †
           =                                    dk ln det M k −                                       dk ln det M k
                   4π           0                                                     0
                                ∞
                    c
           =                        dk4 ln det M (ik4 ) for relativistic disp. E = ck and after Wick-rotation
                   2π       0


                                    †
                                                                       ´†                              `       ∗
Note that det M (ik4 ) = det M (ik4 ) since det M (k) = det M (−k ) ; therefore corollary:
   Z ∞                                        Z ∞
 c          2n+1                         n c           2n+1
                                                            h
                                                                                               †
                                                                                                 i
       dk k      Im ln det M (k) = i(−1)         dk4 k4       ln det M (ik4 ) − ln det M (ik4 ) = 0
2π 0                                      4π 0

e.g. Casimir energy over modes with non-relativistic dispersion E = 2 k2 /2m integrates to
zero, unless there is a finite upper cutoff, as e.g. the Fermi momentum kF in the case of the
so-called fermionic Casimir effect
                                                                                      – see A. Bulgac & AW., Phys. Rev. Lett. 87 (2001).




· ◦    < ∧     >        •           Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                       9
 Multiple-scattering matrix for n spheres of radii aj & distances rjj (j, j                                          =1, 2, . . . n)
                                                                                                „        «2
  jj                                                         −l
                                                                p                                   aj         jl (kaj )
 Mlm,l   m
             = δ jj δll δmm + (1 − δ jj ) i2m+l                  4π(2l+1)(2l +1)                    aj         (1)
                                                                                                              hl (kaj )
               ∞ l
               X Xq        „                               «„                           «
                         ˜ ˜ l                  l      l       ˜
                                                               l           l        l                      (1)         m m ` (j) ´
             ×     2˜
                    l+1 il                                                                  Dm ,m (j, j ) h˜ (krjj ) Y˜ −˜ rjj
                                                                                             l
                                                                                                ˜                           ˆ
                                          0     0      0      m− m
                                                                 ˜         m
                                                                           ˜       −m                         l              l
                 ˜   ˜
                 l=0 m=−l
                                                                   M. Henseler, A. Wirzba & T. Guhr, Ann. Phys. 258 (1997) 286.




· ◦   < ∧    >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                9
 Multiple-scattering matrix for n spheres of radii aj & distances rjj (j, j                                               =1, 2, . . . n)
                                                                                                  „        «2
  jj                                                         −l
                                                                p                                     aj         jl (kaj )
 Mlm,l   m
             = δ jj δll δmm + (1 − δ jj ) i2m+l                  4π(2l+1)(2l +1)                      aj         (1)
                                                                                                                hl (kaj )
               ∞ l
               X Xq        „                               «„                           «
                         ˜ ˜ l                  l      l       ˜
                                                               l           l        l                      (1)         m m ` (j) ´
             ×     2˜
                    l+1 il                                                                  Dm ,m (j, j ) h˜ (krjj ) Y˜ −˜ rjj
                                                                                             l
                                                                                                ˜                           ˆ
                                          0     0      0      m− m
                                                                 ˜         m
                                                                           ˜       −m                           l                 l
                 ˜   ˜
                 l=0 m=−l
                                                                   M. Henseler, A. Wirzba & T. Guhr, Ann. Phys. 258 (1997) 286.

                                                      jj                 jj
                                                                                   `         jj
                                                                                                  ´             exp(ikrjj )
 In the limit of small scatterers:                M        (E) ≈ δ            − 1−δ                   fj (E)        rjj
                                                                                                                                   (+ p-wave)
                                                                                                      | {z }
                                                                                                      s-wave


 Two spheres of radius a and distance r




· ◦   < ∧    >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                        9
 Multiple-scattering matrix for n spheres of radii aj & distances rjj (j, j                                                       =1, 2, . . . n)
                                                                                                      „        «2
  jj                                                         −l
                                                                p                                         aj         jl (kaj )
 Mlm,l   m
             = δ jj δll δmm + (1 − δ jj ) i2m+l                  4π(2l+1)(2l +1)                          aj         (1)
                                                                                                                    hl (kaj )
               ∞ l
               X Xq        „                               «„                           «
                         ˜ ˜ l                  l      l       ˜
                                                               l           l        l                      (1)         m m ` (j) ´
             ×     2˜
                    l+1 il                                                                  Dm ,m (j, j ) h˜ (krjj ) Y˜ −˜ rjj
                                                                                             l
                                                                                                ˜                           ˆ
                                          0     0      0      m− m
                                                                 ˜         m
                                                                           ˜       −m                               l                     l
                 ˜   ˜
                 l=0 m=−l
                                                                   M. Henseler, A. Wirzba & T. Guhr, Ann. Phys. 258 (1997) 286.

                                                      jj                 jj
                                                                                   `         jj
                                                                                                      ´             exp(ikrjj )
 In the limit of small scatterers:                M        (E) ≈ δ            − 1−δ                       fj (E)            rjj
                                                                                                                                           (+ p-wave)
                                                                                                          | {z }
                                                                                                          s-wave
                                                                              a                                         a
 Two spheres of radius a and distance r
                                                                                                  r




· ◦   < ∧    >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                        9
 Multiple-scattering matrix for n spheres of radii aj & distances rjj (j, j                                                       =1, 2, . . . n)
                                                                                                      „        «2
  jj                                                         −l
                                                                p                                         aj         jl (kaj )
 Mlm,l   m
             = δ jj δll δmm + (1 − δ jj ) i2m+l                  4π(2l+1)(2l +1)                          aj         (1)
                                                                                                                    hl (kaj )
               ∞ l
               X Xq        „                               «„                           «
                         ˜ ˜ l                  l      l       ˜
                                                               l           l        l                      (1)         m m ` (j) ´
             ×     2˜
                    l+1 il                                                                  Dm ,m (j, j ) h˜ (krjj ) Y˜ −˜ rjj
                                                                                             l
                                                                                                ˜                           ˆ
                                          0     0      0      m− m
                                                                 ˜         m
                                                                           ˜       −m                               l                     l
                 ˜   ˜
                 l=0 m=−l
                                                                   M. Henseler, A. Wirzba & T. Guhr, Ann. Phys. 258 (1997) 286.

                                                      jj                 jj
                                                                                   `         jj
                                                                                                      ´             exp(ikrjj )
 In the limit of small scatterers:                M        (E) ≈ δ            − 1−δ                       fj (E)            rjj
                                                                                                                                           (+ p-wave)
                                                                                                          | {z }
                                                                                                          s-wave
                                                                              a                                         a
 Two spheres of radius a and distance r
                                                                                                  r
in the limit of small scatterers :
                  oo          1        oo     a2                    `    3´
                 NC (E) = − Im ln det M (E) ≈     sin[2(r − a)k] + O (ka) .
                              π               πr2




· ◦   < ∧    >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                        9
 Multiple-scattering matrix for n spheres of radii aj & distances rjj (j, j                                                       =1, 2, . . . n)
                                                                                                      „        «2
  jj                                                         −l
                                                                p                                         aj         jl (kaj )
 Mlm,l   m
             = δ jj δll δmm + (1 − δ jj ) i2m+l                  4π(2l+1)(2l +1)                          aj         (1)
                                                                                                                    hl (kaj )
               ∞ l
               X Xq        „                               «„                           «
                         ˜ ˜ l                  l      l       ˜
                                                               l           l        l                      (1)         m m ` (j) ´
             ×     2˜
                    l+1 il                                                                  Dm ,m (j, j ) h˜ (krjj ) Y˜ −˜ rjj
                                                                                             l
                                                                                                ˜                           ˆ
                                          0     0      0      m− m
                                                                 ˜         m
                                                                           ˜       −m                               l                     l
                 ˜   ˜
                 l=0 m=−l
                                                                   M. Henseler, A. Wirzba & T. Guhr, Ann. Phys. 258 (1997) 286.

                                                      jj                 jj
                                                                                   `         jj
                                                                                                      ´             exp(ikrjj )
 In the limit of small scatterers:                M        (E) ≈ δ            − 1−δ                       fj (E)            rjj
                                                                                                                                           (+ p-wave)
                                                                                                          | {z }
                                                                                                          s-wave
                                                                              a                                         a
 Two spheres of radius a and distance r
                                                                                                  r
in the limit of small scatterers :
                  oo          1        oo     a2                    `    3´
                 NC (E) = − Im ln det M (E) ≈     sin[2(r − a)k] + O (ka) .
                              π               πr2
Compare with the semiclassical result of the simplest periodic orbit (without repeats):
                oo              a2
              NC,scl (E) =               sin[2(r − 2a)k]    (Gutzwiller trace formula)
                           4πr(r − 2a)       | {z }
                                                                 Spo (k)/
                                                                                       A. Bulgac & AW, Phys. Rev. Lett 87 (2001)


· ◦   < ∧    >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                a                               a   10
  Two spheres:

                                                                                                  r




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                  a                                     a                     10
  Two spheres:

                                                                                                                    r
       oo
N.o.S NC (E) = − π Im ln det M oo (E)
                 1


                                    k = 10 / a                                                                      r = 2.5 a
               1.2                                                                         0.1
                                                 Semiclassical calculation                                                  Semiclassical approximation
                                                 Exact calculation                                                          Exact calculation
                                                                                          0.08
                1
                                                                                          0.06
               0.8
                                                                                          0.04
               0.6                                                                        0.02
       N(ε)




                                                                                  N(ε)
               0.4                                                                          0

                                                                                         −0.02
               0.2
                                                                                         −0.04
                0
                                                                                         −0.06

              −0.2                                                                       −0.08
                  0             5                     10                     15               0       0.5   1     1.5     2       2.5    3     3.5        4
                                        (r−2a) k                                                                          k a / 10




· ◦   < ∧             >   •   Andreas Wirzba       The Casimir effect as scattering problem       QFEXT07, Leipzig, 19-Sep-2007
                                                                                                  a                                     a                     10
  Two spheres:

                                                                                                                    r
                                                                          a2
       oo
N.o.S NC (E) = − π Im ln det M oo (E) ≈
                 1
                                                                      4πr(r−2a)
                                                                                           sin[2(r − 2a)k]
                                    k = 10 / a                                                                      r = 2.5 a
               1.2                                                                         0.1
                                                 Semiclassical calculation                                                  Semiclassical approximation
                                                 Exact calculation                                                          Exact calculation
                                                                                          0.08
                1
                                                                                          0.06
               0.8
                                                                                          0.04
               0.6                                                                        0.02
       N(ε)




                                                                                  N(ε)
               0.4                                                                          0

                                                                                         −0.02
               0.2
                                                                                         −0.04
                0
                                                                                         −0.06

              −0.2                                                                       −0.08
                  0             5                     10                     15               0       0.5   1     1.5     2       2.5    3     3.5        4
                                        (r−2a) k                                                                          k a / 10




· ◦   < ∧             >   •   Andreas Wirzba       The Casimir effect as scattering problem       QFEXT07, Leipzig, 19-Sep-2007
                                                                                                        a                                     a                     10
  Two spheres:

                                                                                                                          r
                                                                                a2
       oo
N.o.S NC (E) = − π Im ln det M oo (E) ≈
                 1
                                                                            4πr(r−2a)
                                                                                                 sin[2(r − 2a)k]
                                          k = 10 / a                                                                      r = 2.5 a
               1.2                                                                               0.1
                                                       Semiclassical calculation                                                  Semiclassical approximation
                                                       Exact calculation                                                          Exact calculation
                                                                                                0.08
                1
                                                                                                0.06
               0.8
                                                                                                0.04
               0.6                                                                              0.02
       N(ε)




                                                                                        N(ε)
               0.4                                                                                0

                                                                                               −0.02
               0.2
                                                                                               −0.04
                0
                                                                                               −0.06

              −0.2                                                                             −0.08
                  0                   5                     10                     15               0       0.5   1     1.5     2       2.5    3     3.5        4
                                              (r−2a) k                                                                          k a / 10

                                       µ(kF )
                                                                                 a2                         a2
                                  Z
                      oo                                  oo
                     EC     = −                 dE       NC (E)         ≈ −µ             j1 [2(r − 2a)kF ] ∼ 3
                                   0                                         2πr(r − 2a)                    L
                          Fermionic Casimir effect: oscillating because of presence of a second scale (chem. potential)
                                                                              A. Bulgac & AW, Phys. Rev. Lett 87 (2001)


· ◦   < ∧             >      •    Andreas Wirzba         The Casimir effect as scattering problem       QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                  11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions




· ◦    < ∧   >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                     11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions
      • However, 2-bounce orbit dominates in equilateral three- and four sphere systems



                                                                                          3
                                                                          3 x 12 :

                                                                                     1         2




· ◦    < ∧   >   •    Andreas Wirzba   The Casimir effect as scattering problem      QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                                       11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions
      • However, 2-bounce orbit dominates in equilateral three- and four sphere systems
                                                               (max. correction due to 3-bounce orbit is ∼ 10 % at r≈2.5a)


                                                  0                                                                                    3                       3
                                     10
                                                                                                                       3 x 12 :                    2 x 123 :
                                                                                                     123
                                                  −1                                                1212
                                     10                                                                                           1                      1
        Relative weight of various po in NC(ε )




                                                                                                                                            2                      2
                                                  −2
                                     10

                                                  −3
                                     10

                                                  −4
                                     10

                                                  −5
                                     10

                                                  −6
                                     10

                                                  −7
                                     10                0                        1                                2
                                                  10                      10                                   10
                                                                          r/a




· ◦    < ∧                                             >   •   Andreas Wirzba       The Casimir effect as scattering problem      QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                                       11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions
      • However, 2-bounce orbit dominates in equilateral three- and four sphere systems
                                                               (max. correction due to 3-bounce orbit is ∼ 10 % at r≈2.5a)


                                                  0                                                                                    3                       3
                                     10
                                                                                                                       3 x 12 :                    2 x 123 :
                                                                                                    123
                                                  −1                                                1212
                                     10                                                                                           1                      1
        Relative weight of various po in NC(ε )




                                                                                                                                            2                      2
                                                  −2
                                     10                                                                                                3
                                                                                                                       3 x 1212 :
                                                  −3
                                     10
                                                                                                                                  1         2
                                                                                                                                      x2
                                                  −4
                                     10

                                                  −5
                                     10

                                                  −6
                                     10

                                                  −7
                                     10                0                        1                                2
                                                  10                      10                                   10
                                                                          r/a




· ◦    < ∧                                             >   •   Andreas Wirzba       The Casimir effect as scattering problem      QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                                          11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions
      • However, 2-bounce orbit dominates in equilateral three- and four sphere systems
                                                               (max. correction due to 3-bounce orbit is ∼ 10 % at r≈2.5a)


                                                  0                                                                                    3                          3
                                     10
                                                                                                                       3 x 12 :                    2 x 123 :
                                                                                                    123
                                                  −1                                                1212
                                     10                                                                                           1                       1
        Relative weight of various po in NC(ε )




                                                                                                    1213                                    2                         2
                                                  −2
                                     10                                                                                                3                          3
                                                                                                                       3 x 1212 :                  3 x 1213 :
                                                  −3
                                     10
                                                                                                                                  1         2                 1       2
                                                                                                                                      x2
                                                  −4
                                     10

                                                  −5
                                     10

                                                  −6
                                     10

                                                  −7
                                     10                0                        1                                2
                                                  10                      10                                   10
                                                                          r/a




· ◦    < ∧                                             >   •   Andreas Wirzba       The Casimir effect as scattering problem      QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                                          11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions
      • However, 2-bounce orbit dominates in equilateral three- and four sphere systems
                                                               (max. correction due to 3-bounce orbit is ∼ 10 % at r≈2.5a)


                                                  0                                                                                    3                          3
                                     10
                                                                                                                       3 x 12 :                    2 x 123 :
                                                                                                    123
                                                  −1                                                1212
                                     10                                                                                           1                       1
        Relative weight of various po in NC(ε )




                                                                                                    1213                                    2                         2
                                                                                                    12123
                                                  −2
                                     10                                                                                                3                          3
                                                                                                                       3 x 1212 :                  3 x 1213 :
                                                  −3
                                     10
                                                                                                                                  1         2                 1       2
                                                                                                                                      x2
                                                  −4
                                     10
                                                                                                                                       3
                                     10
                                                  −5                                                                   6 x 12123 :

                                                                                                                                  1         2
                                                  −6
                                     10

                                                  −7
                                     10                0                        1                                2
                                                  10                      10                                   10
                                                                          r/a




· ◦    < ∧                                             >   •   Andreas Wirzba       The Casimir effect as scattering problem      QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                                           11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions
      • However, 2-bounce orbit dominates in equilateral three- and four sphere systems
                                                               (max. correction due to 3-bounce orbit is ∼ 10 % at r≈2.5a)


                                                  0                                                                                    3                          3
                                     10
                                                                                                                       3 x 12 :                    2 x 123 :
                                                                                                    123
                                                  −1                                                1212
                                     10                                                                                           1                       1
        Relative weight of various po in NC(ε )




                                                                                                    1213                                    2                          2
                                                                                                    12123
                                                  −2
                                     10                                                             121212                             3                          3
                                                                                                                       3 x 1212 :                  3 x 1213 :
                                                  −3
                                     10
                                                                                                                                  1         2                 1        2
                                                                                                                                      x2
                                                  −4
                                     10
                                                                                                                                       3                          3

                                     10
                                                  −5                                                                   6 x 12123 :                 3 x 121212 :

                                                                                                                                  1         2                 1        2
                                                  −6                                                                                                              x3
                                     10

                                                  −7
                                     10                0                        1                                2
                                                  10                      10                                   10
                                                                          r/a




· ◦    < ∧                                             >   •   Andreas Wirzba       The Casimir effect as scattering problem      QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                                           11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions
      • However, 2-bounce orbit dominates in equilateral three- and four sphere systems
                                                               (max. correction due to 3-bounce orbit is ∼ 10 % at r≈2.5a)


                                                  0                                                                                    3                          3
                                     10
                                                                                                                       3 x 12 :                    2 x 123 :
                                                                                                    123
                                                  −1                                                1212
                                     10                                                                                           1                       1
        Relative weight of various po in NC(ε )




                                                                                                    1213                                    2                          2
                                                                                                    12123
                                                  −2
                                     10                                                             121212                             3
                                                                                                    123123                                                        3
                                                                                                                       3 x 1212 :                  3 x 1213 :
                                                  −3
                                     10
                                                                                                                                  1         2                 1        2
                                                                                                                                      x2
                                                  −4
                                     10
                                                                                                                                       3                          3

                                     10
                                                  −5                                                                   6 x 12123 :                 3 x 121212 :

                                                                                                                                  1         2                 1        2
                                                  −6                                                                                                              x3
                                     10
                                                                                                                                       3
                                                  −7
                                     10                0                        1                                2
                                                                                                                       2 x 123123 :
                                                  10                      10                                   10
                                                                          r/a                                                     1        2
                                                                                                                                      x2




· ◦    < ∧                                             >   •   Andreas Wirzba       The Casimir effect as scattering problem      QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                                                     11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions
      • However, 2-bounce orbit dominates in equilateral three- and four sphere systems
                                                               (max. correction due to 3-bounce orbit is ∼ 10 % at r≈2.5a)


                                                  0                                                                                    3                                 3
                                     10
                                                                                                                       3 x 12 :                     2 x 123 :
                                                                                                    123
                                                  −1                                                1212
                                     10                                                                                           1                          1
        Relative weight of various po in NC(ε )




                                                                                                    1213                                    2                                    2
                                                                                                    12123
                                                  −2
                                     10                                                             121212                             3
                                                                                                    123123                                                               3
                                                                                                    121213             3 x 1212 :                   3 x 1213 :
                                                  −3
                                     10
                                                                                                                                  1         2                    1               2
                                                                                                                                      x2
                                                  −4
                                     10
                                                                                                                                       3                                 3

                                     10
                                                  −5                                                                   6 x 12123 :                  3 x 121212 :

                                                                                                                                  1         2                    1               2
                                                  −6                                                                                                                     x3
                                     10
                                                                                                                                       3                             3
                                                  −7
                                     10                0                        1                                2
                                                                                                                       2 x 123123 :             6 x 121213 :
                                                  10                      10                                   10
                                                                          r/a                                                     1        2             1                   2
                                                                                                                                      x2




· ◦    < ∧                                             >   •   Andreas Wirzba       The Casimir effect as scattering problem      QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                                                                                                            11
 Three and four spheres:
      • periodic orbit summation: ∃ of genuine three and more-body interactions
      • However, 2-bounce orbit dominates in equilateral three- and four sphere systems
                                                               (max. correction due to 3-bounce orbit is ∼ 10 % at r≈2.5a)


                                                  0                                                                                    3                                 3
                                     10
                                                                                                                       3 x 12 :                     2 x 123 :
                                                                                                    123
                                                  −1                                                1212
                                     10                                                                                           1                          1
        Relative weight of various po in NC(ε )




                                                                                                    1213                                    2                                    2
                                                                                                    12123
                                                  −2
                                     10                                                             121212                             3
                                                                                                    123123                                                               3
                                                                                                    121213             3 x 1212 :                   3 x 1213 :
                                                  −3
                                     10                                                             121323
                                                                                                                                  1         2                    1               2
                                                                                                                                      x2
                                                  −4
                                     10
                                                                                                                                       3                                 3

                                     10
                                                  −5                                                                   6 x 12123 :                  3 x 121212 :

                                                                                                                                  1         2                    1               2
                                                  −6                                                                                                                     x3
                                     10
                                                                                                                                       3                             3                              3
                                                  −7
                                     10                0                        1                                2
                                                                                                                       2 x 123123 :             6 x 121213 :                         3 x 121323 :
                                                  10                      10                                   10
                                                                          r/a                                                     1        2             1                   2                1         2
                                                                                                                                      x2

      • Billiard analogy : difficult to make long shots, especially with many bounces
                           – the slightest error ruins the shot.


· ◦    < ∧                                             >   •   Andreas Wirzba       The Casimir effect as scattering problem      QFEXT07, Leipzig, 19-Sep-2007
 The sphere-plate case for the scalar Casimir effect:                                                           12


The two-sphere case for identical spheres at a distance r contains the case of a sphere and a
plate with distance r/2:
                                                  vertical symmetry plane
                                         half−domain I           half−domain II
                                                                    oo
                                                                  L
                                               a                                a


                                                          Lo|
                                                           r




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 The sphere-plate case for the scalar Casimir effect:                                                           12


The two-sphere case for identical spheres at a distance r contains the case of a sphere and a
plate with distance r/2:
                                                  vertical symmetry plane
                                         half−domain I           half−domain II
                                                                    oo
                                                                  L
                                               a                                a


                                                          Lo|
                                                           r

Since C∞h → D∞v , there exist two classes of multi-scattering matrices in the half-domain:
                          ˛                                ˛
                   oo (m) ˛               (m)       oo (m) ˛            (m)
               Mll        ˛   = δll + All , Mll            ˛ = δll − All
                               N                                                      D




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 The sphere-plate case for the scalar Casimir effect:                                                             12


The two-sphere case for identical spheres at a distance r contains the case of a sphere and a
plate with distance r/2:
                                                    vertical symmetry plane
                                           half−domain I           half−domain II
                                                                      oo
                                                                    L
                                                 a                                a


                                                            Lo|
                                                             r

Since C∞h → D∞v , there exist two classes of multi-scattering matrices in the half-domain:
                          ˛                                ˛
                   oo (m) ˛               (m)       oo (m) ˛            (m)
               Mll        ˛   = δll + All , Mll            ˛ = δll − All ,
                                 N                                                      D

and the determinants factorize as
                          ∞
                          Y
     det M oo (k, a, r) =     det M oo (m) (k, a, r) = det M oo (k, a, r)|N det M oo (k, a, r)|D ,
                          m=−∞
                               “         ”             “                 ”˛
                                o|    o|            oo               o|
such that              det M k, a, L       = det M      k, a, r = 2(L +a) ˛
                                                                          ˛
                                                                           D
                                    Z ∞
and                   o|          c                   oo `                ´˛
                     EC (a, L) =         dk4 ln det M ik4 , a, r = 2(L+a) ˛D .
                                 2π 0


· ◦   < ∧   >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Scalar Casmir effect of Dirichlet-spheres and -plates – or How good is the PFA?                                13




· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Scalar Casmir effect of Dirichlet-spheres and -plates – or How good is the PFA?                                  13



M. Schaden, L. Spruch, PRA 58 (1998): short-distance PFA confirmed by semiclassics (periodic orbits)




· ◦   < ∧   >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Scalar Casmir effect of Dirichlet-spheres and -plates – or How good is the PFA?                                          13



H. Gies, K. Langfeld, L. Moyaerts, JHEP 0306 (2003): world-line approach (Feynman integral in x-space)
A. Scardicchio, R.L. Jaffe, NPB 704 (2005): optical ansatz (summation over closed orbits)

      −1440L ε
            2

       π3 h c a                                                           S1


                                                                         S2
                                                                                    (a)            (b)              (c)


                      L/a




· ◦   < ∧   >     •     Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Scalar Casmir effect of Dirichlet-spheres and -plates – or How good is the PFA?                                     13



H. Gies, K. Langfeld, L. Moyaerts, JHEP 0306 (2003): world-line approach (Feynman integral in x-space)
A. Scardicchio, R.L. Jaffe, NPB 704 (2005): optical ansatz (summation over closed orbits)

       −1440L ε
             2

        π3 h c a




                       L/a
Note: asymptotically ( L/a           1 ) s-wave scattering dominants:

                     π 3 c a 90          2             π 3 c a 90      π3 c a
      E(L)   ∼     −                                →−            ×2=−        × 1.847 · · ·
                     1440L2 π 4 (1 + a/L)(1 + a/2L)    1440L2 π 4      1440L2




· ◦   < ∧    >     •     Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Scalar Casmir effect of Dirichlet-spheres and -plates – or How good is the PFA?                                                                         13



H. Gies, K. Langfeld, L. Moyaerts, JHEP 0306 (2003): world-line approach (Feynman integral in x-space)
A. Scardicchio, R.L. Jaffe, NPB 704 (2005): optical ansatz (summation over closed orbits)

       −1440L ε
             2
                                                                                           2
        π3 h c a                                                                                                          asymptotical




                                                              -E/hbar c pi**3/1440 L**2
                                                                                          1.5
                                                                                                                                  s-wave
                                                                                                                          exact
                                                                                           1                                             semiclassical
                                                                                                                  PFA(plate)
                                                                                                          PFA(sphere)
                                                                                          0.5



                                                                                           0
                                                                                                -8   -6    -4    -2       0      2       4     6     8
                                                                                                                      log_2(L/a)
                       L/a
Note: asymptotically ( L/a           1 ) s-wave scattering dominants:

                     π 3 c a 90          2             π 3 c a 90      π3 c a
      E(L)   ∼     −                                →−            ×2=−        × 1.847 · · ·
                     1440L2 π 4 (1 + a/L)(1 + a/2L)    1440L2 π 4      1440L2




· ◦   < ∧    >     •     Andreas Wirzba   The Casimir effect as scattering problem                          QFEXT07, Leipzig, 19-Sep-2007
               Scalar Casmir effect of Dirichlet-spheres and -plates – or How good is the PFA?                                                                                                         13




                         2




                                                                                                                                         2
                        1.5
                                                                                                                                                                        asymptotical
ECasimir/EPFA(L/a<<1)




                                                                                                            -E/hbar c pi**3/1440 L**2
                                                                                                                                        1.5
                         1
                                                                                                                                                                                s-wave
                                                                                                                                                                        exact
                                   PFA plate-based                                                                                       1                                             semiclassical
                                   PFA sphere-based                                                                                                             PFA(plate)
                        0.5
                                   opt. approx. (Jaffe,Scardicchio’04)
                                                                                                                                                        PFA(sphere)
                                   worldline numerics
                                   "KKR" scattering approach (Wirzba’05)
                                                                                                                                        0.5

                         0
                         0.001         0.01              0.1               1       10
                                                                L/a
                                                                                                                                         0
                                 H. Gies & K. Klingmuller, 2006
                                                    ¨                                                                                         -8   -6    -4    -2       0      2       4     6     8
                                                                                                                                                                    log_2(L/a)



Note: asymptotically ( L/a                                                        1 ) s-wave scattering dominants:

                                                           π 3 c a 90          2             π 3 c a 90      π3 c a
                                 E(L)         ∼          −                                →−            ×2=−        × 1.847 · · ·
                                                           1440L2 π 4 (1 + a/L)(1 + a/2L)    1440L2 π 4      1440L2
                              while
                                                           5π 3 c a3 90
   Ep-wave (L)                                ∼          −                                   (i.e. no Casimir-Polder behavior in the scalar case)
                                                            1440L4 π 4


· ◦                              < ∧          >          •            Andreas Wirzba    The Casimir effect as scattering problem                          QFEXT07, Leipzig, 19-Sep-2007
 Asymptotics of Dirichlet sphere-plate problem:                                                                                                       with M. Bordag   14



                                  
                               a      5a    421 “ a ”2    535 “ a ”3 3083041 “ a ”4
        ED,l≥0       =     −       1+     +            +            +
                             8πR2     8R    144 R         1152 R      518400 R
                                                                                ff
                                       2741117 “ a ”5 557222415727 “ a ”6
                                     −               +                    + ···
                                       1382400 R         36578304000 R
                                                                  2




                                                                 1.8
                            E_C(a,L)/(-hbar c pi^3 a/1440 L^2)




                                                                 1.6




                                                                 1.4




                                                                 1.2




                                                                  1




                                                                 0.8
                                                                       1                       10                     100                   1000
                                                                                                         L/a



                     red: exact, green: up to O (a/R)4 , blue: up to O (a/R)6
                                               `      ´               `       ´



· ◦   < ∧   >    •       Andreas Wirzba                                    The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                  15
            Large distance expansion (a/R) of the scalar sphere-plate problem:
                         
                     a       5a     421 “ a ”2    535 “ a ”3 3083041 “ a ”4
  ED,l≥0    =    −        1+     +             +               +
                   8πR2      8R     144 R         1152 R           518400 R
                                                                                ff
                              2741117 “ a ”5 557222415727 “ a ”6
                            −                +                           + ···
                              1382400 R          36578304000 R
                       3
                          
                    5a        56 “ a ”2 597 “ a ”3 10453 “ a ”4 16557 “ a ”5
  ED,l>0    =    −         1+           −            +                   −
                   16πR4      25 R        640 R           1750 R             1600 R
                                                          ff
                              394844679647   “ a ”6
                            +                       + ···     (in coll. with Michael Bordag)
                               9144576000      R




· ◦   < ∧    >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                  15
            Large distance expansion (a/R) of the scalar sphere-plate problem:
                         
                     a         5a        421 “ a ”2     535 “ a ”3 3083041 “ a ”4
  ED,l≥0    =    −         1+       +               +               +
                   8πR2        8R        144 R         1152 R           518400 R
                                                                                     ff
                                2741117 “ a ”5 557222415727 “ a ”6
                              −                    +                          + ···
                                1382400 R             36578304000 R
                       3
                           
                    5a          56 “ a ”2 597 “ a ”3 10453 “ a ”4 16557 “ a ”5
  ED,l>0    =    −          1+               −             +                  −
                   16πR4        25 R           640 R           1750 R             1600 R
                                                               ff
                                394844679647      “ a ”6
                              +                          + ···     (in coll. with Michael Bordag)
                                  9144576000        R

                    10a3
                           
                                 63 “ a ”2 597 “ a ”3           4159 “ a ”4 271437 “ a ”5
  EN,l>0    =    −       4
                            1+                +             −                   −
                   16πR         100 R            320 R         14000 R             25600 R
                                                               ff
                                148355331834 “ a ”6
                              +                          + ···     (in coll. with Michael Bordag)
                                  2286144000        R
                                                   “ a ”ff
                     1
  EN,l=0    =    −       × 0.46066 . . . × 1 + O             (k4 → 0 and R/a → ∞ do not commute!)
                   4πR                                R




· ◦   < ∧    >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                  15
            Large distance expansion (a/R) of the scalar sphere-plate problem:
                          
                     a          5a        421 “ a ”2      535 “ a ”3 3083041 “ a ”4
  ED,l≥0    =    −          1+       +               +                 +
                   8πR2         8R        144 R          1152 R            518400 R
                                                                                        ff
                                 2741117 “ a ”5 557222415727 “ a ”6
                               −                    +                            + ···
                                 1382400 R              36578304000 R
                        3
                            
                     5a          56 “ a ”2 597 “ a ”3 10453 “ a ”4 16557 “ a ”5
  ED,l>0    =    −           1+               −              +                   −
                   16πR4         25 R           640 R            1750 R              1600 R
                                                                  ff
                                 394844679647      “ a ”6
                               +                           + ···      (in coll. with Michael Bordag)
                                   9144576000        R

                    10a3
                            
                                  63 “ a ”2 597 “ a ”3             4159 “ a ”4 271437 “ a ”5
  EN,l>0    =    −        4
                             1+                +               −                   −
                   16πR          100 R            320 R           14000 R             25600 R
                                                                  ff
                                 148355331834 “ a ”6
                               +                           + ···      (in coll. with Michael Bordag)
                                   2286144000        R
                                                    “ a ”ff
                     1
  EN,l=0    =    −        × 0.46066 . . . × 1 + O               (k4 → 0 and R/a → ∞ do not commute!)
                   4πR                                 R
                             3
                                          “ a ”ff
                   (3 + 6)a                            H.B.G. Casimir & D. Polder, Phys. Rev. 73 (‘48) 360;
EEM,l>0     =    −             × 1+O
                    16πR4                    R         T. Datta & L.H. Ford, Phys. Lett. 83A (’81) 314.


· ◦   < ∧    >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                  15
            Large distance expansion (a/R) of the scalar sphere-plate problem:
                          
                     a          5a        421 “ a ”2      535 “ a ”3 3083041 “ a ”4
  ED,l≥0    =    −          1+       +               +                 +
                   8πR2         8R        144 R          1152 R            518400 R
                                                                                        ff
                                 2741117 “ a ”5 557222415727 “ a ”6
                               −                    +                            + ···
                                 1382400 R              36578304000 R
                        3
                            
                     5a          56 “ a ”2 597 “ a ”3 10453 “ a ”4 16557 “ a ”5
  ED,l>0    =    −           1+               −              +                   −
                   16πR4         25 R           640 R            1750 R              1600 R
                                                                  ff
                                 394844679647      “ a ”6
                               +                           + ···      (in coll. with Michael Bordag)
                                   9144576000        R

                    10a3
                            
                                  63 “ a ”2 597 “ a ”3             4159 “ a ”4 271437 “ a ”5
  EN,l>0    =    −        4
                             1+                +               −                   −
                   16πR          100 R            320 R           14000 R             25600 R
                                                                  ff
                                 148355331834 “ a ”6
                               +                           + ···      (in coll. with Michael Bordag)
                                   2286144000        R
                                                    “ a ”ff
                     1
  EN,l=0    =    −        × 0.46066 . . . × 1 + O               (k4 → 0 and R/a → ∞ do not commute!)
                   4πR                                 R
                             3
                                          “ a ”ff
                   (3 + 6)a                            H.B.G. Casimir & D. Polder, Phys. Rev. 73 (‘48) 360;
EEM,l>0     =    −             × 1+O
                    16πR4                    R         T. Datta & L.H. Ford, Phys. Lett. 83A (’81) 314.


· ◦   < ∧    >   •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
 Further exact Casimir results for non-separable systems                                                           16



      • Casimir Interaction between a Plate and a Cylinder
        T. Emig, R.L. Jaffe, M. Kardar & A. Scardicchio, Phys. Rev. Lett. 96 (2006)
      • Exact zero-point interaction energy between cylinders
        F.D. Mazzitelli, D.A.R. Dalvit & F.C. Lombardo, New J. of Phys. 8 (2006)
      • Casimir effect for a sphere and a cylinder in front of a plane and corrections to the
        proximity force theorem
        M. Bordag, Phys. Rev. D73 (2006)
      • Casimir forces between arbitrary compact objects
        T. Emig, N. Graham, R.L. Jaffe & M. Kardar, arXiv:0707.1862
      • Casimir forces in a T operator approach
        O. Kenneth & I. Klich, preprint - 2007 · · · .
Casimir energy for N cylinders:
                                    ∞    dk         ∞
                                                                2 d −1 Im ln det M (k )
                              Z                 Z      q
                                                            1
                EC = LZ                               c k 2 + k⊥
                                                        dk⊥ 2                             ⊥
                               −∞ 2π      0                       dk⊥ π           |   {z    }
                                                                                  N-disk det.
                                   Z ∞      Z ∞
                                c                          k4
                      =    Lz 2        dk          dk4 q          ln det M (ik4 )
                              2π −∞           |k |       k42 − k2


                               c ∞
                                  Z
                      =    Lz         dk4 k4 ln det M (ik4 )
                              4π 0



· ◦    < ∧   >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                   17
 Conclusions:

      • Casimir energy re-defined as vacuum energy of the geometry-dependent part of the level
        density (connected to the multi-scattering phase shift by a modified Krein trace formula).
      • The non-overlapping (i.g. non-separable) N-sphere, sphere-plate, N-disk (and N-cylinder)
        Casimir problems can be solved exactly in the scalar (and also in the fermionic) case.
      • Calculation not plagued by subtraction of single-sphere contributions or by the removal of
        diverging ultraviolet contributions; all involved determinants exist and are finite since the
        pertinent T-matrices are trace-class.
      • Large-distance behavior dominated by s-wave scattering in the case of the scalar Casimir
        effect and by p-wave scattering for the EM Casimir effect.
      • The presented method can easily be applied to any number of spheres or cylinders with
        or without planes (in 2D disks with or without lines).
      • Moreover, the Dirichlet boundary conditions can be replaced by Neumann or mixed
        boundary conditons.
      • The spheres (or disks) can be replaced by other objects or even smooth potentials or
        non-ideal reflectors. The finite surface thickness can be booked as Weyl terms, as long
        as the objects do not overlap.
      • The Casimir energy is dominated by momenta k ∼ 1/L where L is the separation scale.
        For the scalar sphere-plate case the integration can be truncated at kmax ∼ 10/L
        corresponding to a truncation in the angular momentum lmax ≥ (e/2)kmax a ≈ 14a/L.


· ◦    < ∧   >    •    Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007
                                                                                                                18




Publications:
Aurel Bulgac & A.W., Phys. Rev. Lett. 87 (2001) 120404
              Physics News Update, No. 556, #1, Sep. 13, 2001.
Aurel Bulgac, Piotr Magierski & A.W., Europhys. Lett. 72 (2005) 372.
Aurel Bulgac, Piotr Magierski & A.W., Phys. Rev. D 73 (2006) 025007.
A.W., A. Bulgac & P. Magierski, J. Phys. A (2006) 6815.


· ◦   < ∧   >   •   Andreas Wirzba   The Casimir effect as scattering problem   QFEXT07, Leipzig, 19-Sep-2007