Dimensional analysis, regularization and extra dimensions in electrostatics

Transcription

1 Dimensional analysis, regularization and extra dimensions in electrostatics Daniel Erni (BA 34, Allgemeine und Theoretische Elektrotechnik (ATE) Abteilung für Elektrotechnik und Informationstechnik Fakultät für Ingenieurwissenschaften Universität Duisburg-Essen -/3- -/3- «Curled-up higher spatial dimensions made visible within a simple electrostatic Gedanken experiment» Agenda A simplistic prelude to string theory and quantum field theory. Cutoff regularization. Dimensional regularization. A charged mini-sphere within a dielectric slab is capable to reveal some mysteries of advanced particle physics! Concluding remarks.

2 A popular prelude to string theory Space and its extra dimensions String theory (or M-theory) intends to reconcile general relativity with quantum theory along a highly-sophisticated quantum-field theory. Space consists of a foam-like structure with the 4 macroscopic space-time dimensions and 7 microscopic extra dimensions. The extra dimensions are thought to be «curled up» into a complicated manifold (e.g. Calabi-Yau manifolds, cf. figure). P The microscopic manifold is manifest only for ultra-small length scales around the Planck length ( P =.6 35 m). Mimicking the curling-up in electrostatics! A prelude to quantum field theory Some tools used in QFT -3/3- -4/3- QFT is mostly dealing with divergent (path) integrals. Integrals are used in scattering processes (being formalized with Feynman diagrams). QFT has developed a powerful methodological framework to get rid of emerging infinities, such as e.g.: Dimensional analysis Cutoff regularization Dimensional regularization Using such techniques in electrostatics to tackle well known infinities!

3 Potential field of a line charge I A diverging integral in electrostatics (A) Coulomb integral: (constant line charge) = 4 + z This integral is logarithmically divergent! (B) Scale invariance of the integral: ( k) = 4 = k 4 + z k z zk d z k = + zk : transversal scale invariance! : longitudinal scale invariance! zk:=u Potential field of a line charge II A diverging integral in electrostatics -5/3- -6/3- (C) Strange flavor of the «infinities»: From transversal scale invariance: ( )= ( ) (D) Dealing with «infinities»: ( ) ( )= = C But what is the meaning of the following? u = ( ) E = Regularization is a formal measure for dealing with the ambiguous infinities. Neither difference nor differentials are directly tractable from infinities. 3

4 Cutoff regularization I (B) Differences and differentials: (A) Introducing a regulator : = 4 u = lim { ( ) ( )}= = ln 4 = + z = ln = + + ( ) (well known relation) Cutoff regularization II Intermezzo: «doing the math» = lim ln + ( ) + ( ) + ( ) + ( ) ( ) ( ) ln ln ++ + ln = ln = + ( ) ( ) ( ) + ( ) ( ) -7/3- -8/3-4

5 Cutoff regularization III (B) Differences and differentials: E = 4 + z e Intermezzo: = + z u d z = + z du = f( u)du +u Cutoff regularization IV (B) Differences and differentials: (intermezzo) f( u) du = u f( u) du = d +u 4 { f( u)du} du = f u du d d d ( z )= + z + z = 4 = d = z z + z + = z + z = -9/3- -/3-5

6 Cutoff regularization V (B) Differences and differentials: E = lim 4 e + z = lim e + = e (well known relation) Cutoff regularization VI -/3- -/3- (C) Observations and conclusions: u = ln 4 E = e The regulator allows to extract the difference between infinities. The expressions are independent of. 4 = ln + + z + + z + z 4 + z + + z +z +z + + (cf. pp. 7) ( ) Cutoff regularization breaks the translation symmetry, namely the expressions are not invariant to the translation z z + z. This means that the potential () erroneously depends on z. 6

7 Dimensional regularization I Using dimension as variable (A) Motivation and underlying ideas: ooking for a regularization technique that preserves the translation symmetry. Carrying out the integration for higher dimensions hoping for tractable infinities. Dimensional regularization means to calculate the Coulomb integral for () in n dimensions, where n is not necessarily an integer! -dimension: n-dimensions: dv = d dv n = d n z n dv = d R R z = = R dv 3 = d 3 R z = 4 R3 3 Dimensional regularization II Using dimension as variable -3/3- -4/3- (B) The n-dimensional integration: dv = d z = = dv n = d n z n = n z n n = (C) The n-dimensional Coulomb integral: = 4 = + z 4 d n = n ( n +) n z n n μ μ is an auxiliary scale factor for maintaining the dimension of + z 7

8 Dimensional regularization III Using dimension as variable (D) Solving the n-dimensional Coulomb integration: = 4 d n z n = = n μ + z (E) Dimensional regularization around dimension: = 4 lim{ ( z) }= z ( n ) ( ) n μ n= (-function is singular for vanishing z = representing a formalized kind of infinity). (complicated derivation and, hence, not shown here) 4 = 4 n ( ) n μ μ μ : dimensional scale factor : dimensional regulator Dimensional regularization IV Using dimension as variable -5/3- -6/3- (F) Differences and differentials: u = lim { ( )}= μ lim 4 μ = ln (well known relation) 4 E = lim e = lim + μ 4 e = 4 e Both expressions are independent of μ and! (well known relation) 8

9 Dimensional regularization V Intermezzo: «translation invariance» = 4 = 4 = 4 d n d n d n z n n μ + z = n z + z = μ n + ( z + z ) u n du n μ + u = u := z + z du = Note: The proof is actually more complicated. One has to show that the transformation of the integration limits can be absorbed into the scale factor. Dimensional regularization preserves the translation symmetry, namely () is invariant to the translation z z + z with () being thus independent of z. Extra dimensions I Radial decay of the electric field Spatial, object and field dimensions -7/3- -8/3- Example D obj E r D eff point charge line charge r r 3 (A) Effective dimension: The dimension that is «felt» by the electric field around the object. sheet charge r D eff = D space D obj Objects are capable to compactify the 3 dimensional space according to their boundaries to yield a subspace of reduced effective dimensions D eff, which becomes effective as field domain in electrostatics. (B) Radial decay of the field: E( r) r D eff 9

10 Extra dimensions II Intuitive example: «dielectric slab» COMSO Simulation D. Schäfer Extra dimensions III Intuitive example: «dielectric slab» Radial decay of the electric field -9/3- -/3- Electric field strength r r r

11 Extra dimensions IV Intuitive example: «dielectric slab» Radial decay of the electric field -/3- Deviation of the real field decay compared to r n r r r cm cm Extra dimensions V Probing the extra dimensions -/3- The compactification of the infinite dimension to a finite dimension, yields an effective dimensional space. The field «feels» the effective dimension. Probe the field with a characteristic length scale *: «Curled-up» extra dimensions are effective only below a characteristic length scale.

12 Conclusion Dimensional analysis, regularization and extra dimensions -3/3- We have got an idea how to deal with infinities. We can handle now n-dimensional integration! We have seen how fields are processing both the dimension of the space and of the objects. We got a taste of extra dimensions. And, hopefully we had some fun Further reading: F. Olness, R. Scalise, «Regularization, renormalization, and dimensional analysis: Dimensional regularization meets freshman E&M», Am. J. Phys., vol. 79, no. 3, pp. 36-3, March.