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Continuous Wavelet Transform in Quantum Field Theory: An application to the Casimir effect
1 Mikhail V. Altaisky1,2
1 3

N at al i a E . K ap u t k i n a

3

2

Space Research Institute RAS, Profsoyuznaya 84/32, Moscow, 117997, Russia Joint Institute for Nuclear Research, Joliot Curie 6, Dubna, 141980, Russia National University of Science and Technology "MISiS", Leninsky prosp ect 4, Moscow, 119049, Russia altaisky@mx.iki.rssi.ru, nataly@misis.ru

Conference on Precesion Physics and Fundamental Physical Constants, Sep 6 ­ 11, 2013. St.Petersburg

1 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


Abstract
The Casimir force F = - 240lc , which attracts to each other two 4 perfectly conducting parallel plates separated by a distance a in vacuum, is one of the blueprints of the reality of vacuum fluctuations. The physical basis for calculation of the Casimir force is quantum electrodynamics defined on the space of point-dependent, possibly singular, functions (x ), the distributions. On this space the Green functions are defined. We have shown that it is possible to use region-dependent functions a (x ), where a is the size of the region centered around the space-time point x , constructed by means of the continuous wavelet transform, to get finite Green functions by construction. The singular limit of point-dependent fields is restored by integration over all scales a. We suggest that real measurement of the Casimir force between the parallel plates separated by the distance l is also dependent on the plate displacement l l - , i.e., F = F (l , /l ). The particular form of this dependence is derived.
2

2 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


R e f e r e n c e s:

T h i s t al k i s b as e 1. M.V.Altaisky. 2. M.V.Altaisky 3. A.M.Frassino 4. M.V.Altaisky 88(2013)025015

d on P h ys . R e v. D an d N . E . K ap u t and O.Panella. an d N . E . K ap u t

81(2010) 125003 kina. JETP Lett. 94(2011)341 Phys. Rev. D 85 (2012) 045030 k i n a. P h ys . R e v. D

3 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


Introduction
Casimir force results from the difference of the vacuum energy of the two different configurations: the rectangular volume bounded by two parallel conducting walls, and that not bounded by walls. Due to the Casimir effect two parallel conducting planes in vacuum attract each other with a force distance between the plates a [Cas48]: FC = - c 2 1.3 · 10-27 Nm =- 240a4 a4
2

In QED the Casimir effect is described by the loop diagrams

s(x) +

s(x) + s(x)

s(x)

s(x) + counterterms s(x)

where external field s(x) desribes the boundary conditions
4 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


QED as a field theory depending on scale
In quantum field theory the state of a quantum field | is described by the function phi (x ) = x | . The calculation of Feynman diagrams is performed in Fourier representation x | = x | p dp p |

It is also possible to use other locally compact groups to represent q u an t u m fi e l d s 1 U ( ) | g d µ( ) g | U ( ) | | = Cg G A particular case of the affine group G : x = ax + b is known as Wavelet transform a (x ) a (x ) = x , a ; g | (x )dx
0.8 g1(x) 0.6

0.4

0.2

0

-0.2

-0.4

1 g Ї a

x -b a

-0.6

-0.8 -4 -2 0 2 4

5 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


Wavelet-based regularization
Phys. Rev. D 81(2010) 125003, 88(2013)025015

(x ) a (x ) WW [Ja (x )] = S c al e - d e p e n d e n t fi e l d s

e

~ - S [ a ( x ) ] + J a ( x ) a ( x )

dadx a

D a (x )

(x ) (x ), where x is position, is resolution

~ each field (k ) will be substituted by the scale component ~ (k ) = g ( k )(k ). ~ ~ each integration in momentum variable will be accompanied by integration in corresponding scale variable: ddk ddk d . d (2 ) (2 )d each vertex is substituted by its wavelet transform.
6 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


Gr e e n f u n c t i o n s
n ln WW [Ja ] Ja1 (x1 ) . . . Jan (xn ) g ( a1 p ) g ( - a2 p ) ~ ~ . 2 + m2 p

a 1 ( x1 ) · · · a n ( xn )
(2)

c

=

.
J =0

G0 (a1 , a2 , p ) =

The integration over the internal scale variables daai in each loop is i performed from the minimal scale of all extermal lines a a 1 2

a

n

A = min(a1 , . . . , an ) up to infinity.
7 Mikhail V. Altaisky, Natalia E.Kaputkina

111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000

a

i

Continuous Wavelet Transform in Quantum Field Theory: An app


Effective cutoff
0.8 g1(x) 0.6

0.4

0.2

0

The aperture function g ( x ) = - xe
-x 2 /2

-0.2

-0.4

-0.6

,

-0.8 -4 -2 0 2 4

up to appropriate rescaling, after the integration in internal lines leads to the cutoff function FA ( p ) = e
-4 2 p
2

.
p

q+p/2 q-p/2

8 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


Regularization
Both integrals above, for the discrete and continuous spectra, are e vi d e n t l y i n fi n i t e , b u t t h e i r d i ff e r e n c e EQ - E0 E= , Lx Ly ­ the Casimir energy, can be regularized if the integrands are multiplied by some cutoff function f (k ), such that f (0) = 1, and f k 1 a0 0,

where 1/a0 is the inverse size of atom [IZ80]. This choice accounts for the fact, that the walls are made of real atoms.
The dimensional regularization is often used as well.

Regularized result gives the known values of the Casimir energy and the Casimir force: c 2 c 2 E (a ) = - , F (a ) = - 720a3 240a4
9 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


Scale-dependent Casimir energy
After the choice f (k ) = e E = c
-4 2 k
2

the regularized Casimir energy is:

F (n ) = =

2 1 F (n )dn , F (0) + F (1) + . . . - 4a3 2 0 2 du u + n2 exp -4 2 2 (u + n2 ) a 0 2 2 n + 4 e xp - 1 - erf 2 3 a a 2a

2

n

2

n. a

The difference between the sum and the integral above is evaluated by Euler-Maclaurin formula 1 F (0) + F (1) + . . . - 2
0

F (n )dn = -

1 1 B2 F (0) - B4 F (0) - . . . , 2! 4!

where Bn are the Bernoulli numbers.
10 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


Scale-dependent corrections to Casimir force
JETP Lett. 94(2011)341

This gives the Casimir energy 2 c 2 1+ E (a , ) = - 3 720a 7 + 3 28 2 a
4

2 a

2

+

Deviation of of unit area the "exact" line denotes with /a =
10 9 8 7 6 5 4

Casimir force b etween two plates in vacuum. The solid line denotes Casimir force ( = 0), the dashed the scale-dep endent Casimir force 0.1
"exact" 10% "accuracy"

+ ... ,
F, dyn/cm^2

and the Casimir force 10 c 2 1+ F (a , ) = - 4 240a 21 + 1 4 2 a
4

2 a

2

3

+

2 1 0 0.2 0.3 0.4 0.5 a, mkm 0.6 0.7 0.8

+ ... ,

11 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


Scale-dependence of Casimir force

z (t ) = z0 + cos wt What is the dependence of frequency shift on ? F (a , ) = - Standard theory c 2 F (a , ) = - 1-4 + 240a4 a 2 + 10 - ... a
12 Mikhail V. Altaisky, Natalia E.Kaputkina

E ( a2 ) - E ( a 1 ) , = a2 - a

1

Wavelet-regularized theory c 2 10 1+ 240a4 21 1 4 2 a 2 a
4 2

F (a , ) = - +

+

+ ... ,

Continuous Wavelet Transform in Quantum Field Theory: An app


THANK YOU FOR YOUR ATTENTION !!!

The authors is thankful to profs. N.V.Antonov, Yu.M.Pis'mak, O.V.Teryaev for useful discussions, and to prof. V.V.Nesterenko for critical comments. The research was supported in part by by RFBR Project 13-07-00409 and Programme of Creation and Development of the National University of Science and Technology "MISiS"

13 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app


H.B.G. Casimir, On the attraction between two perfectly conducting plates, Proc. Kon. Ned. Akad. Wet. 51 (1948), 793­795. C. Itzykson and J.-P. Zuber, Quantum field theory, McGraw-Hill, Inc., 1980.

13 Mikhail V. Altaisky, Natalia E.Kaputkina

Continuous Wavelet Transform in Quantum Field Theory: An app