Rotationally invariant integrals of arbitrary dimensions

Transcription

1 September 1, 14 Rotatioally ivariat itegrals of arbitrary dimesios James D. Wells Physics Departmet, Uiversity of Michiga, A Arbor Abstract: I this ote itegrals over spherical volumes with rotatioally ivariat desities are computed. Exploitig the rotatioal ivariace, ad usig idetities i the itegratio over Gaussia fuctios, the geeral -dimesioal itegral is solved up to a oe-dimesioal itegral over the radial coordiate. The volume of a -sphere with uit radius is computed aalytically i terms of the Γz) special fuctio, ad its scalig properties that deped o the umber of dimesios are discussed. The geometric properties of -cubes with volumes equal to that of their correspodig -spheres are also derived. I particular, oe fids that the legth of the side of such a -cube asymptotes to zero as icreases, whereas the logest straight lie that ca fit withi the cube asymptotes to the costat value eπ Fially, itegrals over power-law form factors are computed for fiite ad ifiite radial extet. 1 Prelimiaries Our iterest is to cosider itegrals over a volume i -dimesioal space that is rotatioally ivariat ad has a desity that is rotatioally ivariat. The first requiremet implies that we itegrate over the volume of S. S is defied 1 to be S { r R r 1}. 1) where r x 1, x,..., x ) ad r x 1 + x x. For itegrals over a fiite radius oe rescales the coordiates such that radius of the sphere itegrated over is r 1. The secod requiremet is that the mass desity ρ r) that is assumed throughout the volume, i.e., the itegrad of the itegral, must be rotatioally ivariat ad therefore ca oly deped o the distace from the origi: ρ r) ρr). Therefore, the types of itegrals uder cosideratio are I [ρr)] ρr)dx 1 dx dx. ) S 1 We use the otatio S rather tha S to distiguish from the commo defiitio by topologists that S is uit sphere i R +1 such that x 1 + x + + x The defiitio S used here is equivalet to the uit ball B defiitio by topologists Mukres ). 1

2 Volumes of S ad S 3 If oe is ot clever, solvig itegrals of the eq. ) type is very hard as the umber of dimesios icreases. Let s illustrate this with the simple example of computig the volume of S : V I [1] dx 1 dx 3) S where V i geeral deotes the volume of S. S is just a filled i circle with r 1 ad π is its volume. Solvig for V usig cartesia coordiate systems is paiful, as the derivatio below suggests: +1 ) + 1 x 1 V dx dx 1 4) x 1 ) 1 x 1 dx 1 [x 1 1 x 1 + si 1 x 1 ] +1 + π ) π ) π This approach is uecessarily difficult, ad ofte prohibitively difficult, as the umber of dimesios icreases or more complex fuctios ρr) are itroduced. A obviously easier approach is to take advatage of the symmetries ivolved ad use polar coordiates, where x r cos θ ad y r si θ. 5) where r is of course the radius ad θ is the coordiate agle that sweeps over the etire surface. Its rage is to π. The differetial volume elemet i polar coordiates must be computed. The aswer is dx 1 dx rdrdθ 6) Our itegral the becomes 1 V dx 1 dx S rdrdθ S 7) This itegral is ow separable, meaig the radial part ca be itegrated separately ad idepedetly from the agular sweep variable. The result is trivial to solve: π ) 1 ) V dθ rdr π 1 π. 8) Note I am usig volume V techically here as the itegral over S, whereas we would ormally ad more colloquially say that the filled i circle of S is a area. The volume of S 1 is, whereas otechically we would say it is a legth.. We shall defie the techical term area A later.

3 Let s ow compute the volume of S 3. To make the computatio simpler we agai exploit the symmetry ad defie a coordiate r which is fixed legth radius from the origi, ad agular coordiates φ ad θ that sweep over the etire surface of the sphere at fixed r. The stadard spherical coordiates used for this purpose are x 1 r cos φ si θ 9) x r si φ si θ x 3 r cos θ Agular coverage requires φ < π ad θ < π. The volume elemet is ad the volume itegral over S 3 becomes π V 3 dx 1 dx dx 3 S 3 dx 1 dx dx 3 r dr si θdθdφ 1) π ) 1 ) si θdθ dφ r dr 11) π) π, 1) 3 which is the well-kow result of the volume of a sphere of uit radius i three dimesios. 3 Isolatig Radial Itegral The key to the itegrals over S ad over S 3, ad ideed over S i geeral, is to utilize a coordiate system that has the radial distace from the origi as oe coordiate, which we call r. The fuctio ρr) depeds oly o that coordiate. The remaiig coordiates ad how they are defied is ot so importat. The oly requiremet is that they agularly sweep over the etire surface at fixed radius r. We foud θ variable i the S case ad φ ad θ i the S 3 case, but we could have chose differet variables ad it would ot have materially affected the difficulty of the problem, or chaged the symmetry compatibility of the coordiates to the problem. For this reaso it is helpful to defie a geeralized differetial solid agle dω 1, which is defied to be a differetial area o the surface of S. Whe itegrated it gives the total area of surface of S : A dω 1 13) S where S deotes the full area of the surface of S. I the case of S ad S 3 we foud S : dω 1 dθ A 1 π 14) S 3 : dω si θdθdφ A 4π 15) 3

4 I geeral, ay itegral of the kid give i eq. ca be rewritte as the area of S multiplied by radial itegral: I [ρr)] ρr)dx 1 dx dx 16) S ) 1 ρr)r 1 dr where, S dω 1 A R [ρr)] R [ρr)] 1 ρr)r 1 dr 17) If we fid a way to tabulate A values, the the difficult -dimesioal itegrals I [ρr)] become the much easier to solve oe-dimesioal itegrals R [ρr)]. However, determiig A ca be difficult for high dimesioal spaces or fractioal, as is sometimes eeded i quatum field theory, for example. 4 Solid Agle Itegrals over S Oe path to determie A for arbitrary real value of is through maipulatios of a multidimesioal gaussia itegral. Let us defie where G G e x 1 dx1 e x dx + e x 1 +x +...+x ) dx 1 dx dx + e x dx G 1 ). 18) e x dx π, therefore G π. 19) However, G ca be writte also i geeralize spherical coordiates sice the itegrad is spherically symmetric ad the itegral is over a sphere with radius at ifiity. ) G e r r 1 dr A e r r 1 dr. ) S dω 1 The radial itegral has a form similar to the Gamma fuctio, defied as Arfke 1) Γz) e t t z 1, Rez) >. 1) The two most importat properties of the Γ fuctio that we will have occasio to use later are zγz) Γz + 1) ad Γz + 1) z!, ) 4

5 where z! is the ormal factorial fuctio z! 1 3 z whe z is a iteger, yet it is still defied whe z is o-iteger by the Γz + 1) fuctio. Defiig t r eables us to recast the radial itegral result of eq. ) i terms of the Gamma fuctio e r r 1 dr 1 1 Γ Combiig eqs. 19), ) ad 3) oe ca solve for A : e t t 1 dt ) 3) A π Γ ). 4) This is the area of the surface of S, which is equivalet to the itegral over the solid agle factor dω 1. We are ow i positio to use this result to tur I [ρr)] from a -dimesioal itegral to a much easier oe dimesioal itegral over r. 5 Summary of Result Let us ow go back to the origial itegral of eq. ). I summary, we have leared we ca ow rewrite this itegral as I [ρr)] ρr)dx 1 dx dx A R [ρr)] S π Γ ) 1 ρr)r 1 dr 5) which is ow relatively easy to compute, sice it oly ivolves a oe-dimesioal itegral over the radius. 6 Example: Areas ad Volumes of S Recall that to compute the volume of a S sphere, we must compute I [1]: V I [1] ρr)dx 1 dx dx 6) S ) 1 r 1 dr S dω 1 A R [1] π Γ ) 1 π Γ + 1) 5

6 where the Gamma fuctio idetity zγz) Γz +1) has bee used. The volume V is defied o S, which has radius of uity. The relatio betwee V ad A is simply V 1 A. If we assume the radius is a the volume becomes V a) a V. Let us compute the area ad volume of the S for the first 1 itegers. We also compute l, which is the legth of a side of a -cube that has the same volume as the S, ad is defied to be l V ) 1/. The values ±l / are the places where the cube itersects each x i axis, ad it is these values we put i the table. Usig Stirlig s series expasio Bartle 1964) we ca compute l aalytically at large, l! l + l π + O 1 ) 7) l eπ.6637 large ) 8) where e.7188 is the ormal e 1 expoetial costat. The value of l / teds to as, which suggests that the faces of the cubes pass ifiitesimally closely to the origi as, which is somewhat ati-ituitive of a cube with a uit volume. Note, for > 1 the cube itersects the x i axis at a positio iside the radius of the sphere. This is expected sice these are the closest poits of the cube s surface to the origi, ad the cube has yet further poits away, such as at vertices of the cube, which are always at a distace further tha 1 to compesate ad reder the overall volume equal that of S. The distace from the origi to the cube s vertices is the -cube hypoteuse h. This value is h l ) + l ) + + At large the value of h asymptotes to l ) l ) h l 9) h eπ.6637 limit). 3) I other words, the uit volume -cube does exted past the S s uit radius i places, but it ever exteds beyod a distace of.7, where the vertices reside. I Fig. 1 the values of l / ad h are plotted as a fuctio of. Aother iterpretatio of h is that h is the legth of the logest straight lie that ca be fully cotaied withi a -cube of the same volume as S. This lie exteds from a vertex of the cube at positio x passig through the origi to the correspodig vertex o the opposite side of the cube at positio x. Whe 1,, 3 ad these logest possible straight-lie distaces are, π.51, 48π ) 1/6.79, ad eπ 4.13, respectively. These are always loger legths tha the logest legth of a straight lie that ca fit ito the S sphere, which is of course for the sphere of uit radius, correspodig to ay straight lie that passes through the origi. 6

7 Figure 1: The lower blue curve is a plot vs. umber of dimesios of half the legth of a -cube, l /, that has the same volume as a uit radius S sphere. This legth asymptotes to l as, which is idicative of the well-kow property of S that its volume asymptotes to zero for large dimesios. The upper gree curve is a plot of the distace to the furthest poit of the surface of the same -cube to the origi. This furthest poit is ay oe of the vertices of the cube. Its value asymptotes to h eπ/.66 as. Equivaletly, the logest straight lie that ca fit wholly iside the -cube has legth h ad exteds from oe vertex at positio x to the opposite vertex at poit x. 7 Example: Form Factor Itegrals Let us ed this discussio by cosiderig aother example beyod volume computatios: a k ρ k r) ρ 31) r + a ) k where a is a smoothig radius parameter ad ρ is a costat with uits of -volume desity. I will call itegrals over eq. 31) form factor itegrals sice ρ k r) is remiiscet of form factors i physics, where ρ ρ is early costat ear the ceter whe r a but the falls off rapidly by the power law ρ 1/r k whe r a. Let us assume that i this case the radius 7

8 A V l / h π π π π π 1 π π 8 15 π π π π π π4 1 4 π π π π5 1 1 π Table 1: Computatios of A, V ad l for iteger dimesioality from 1 through 1. V is the volume of uit radius S, ad A is the area of the surface of S. For spheres of radius a, these are rescaled to A a) a 1 A ad V a) a V. The quatity l is the legth of a side of a -dimesioal cube that has the same volume as the S sphere. It is defied to be l V ) 1/. The cube s itersectio poits o each x i axis are ±l /, ad it is this value that we put i the table. h is the furthest poit that the cube reaches from the origi, ad is the -dimesioal hypoteuse. For large the values of A, V ad l asymptote to, whereas h asymptotes to h eπ/.66. itegral is from to r. I other words, we wish to compute a k I [ρ k ; r ] ρ r + a ) dx 1dx k dx 3) S r ) A R [ρ k ; r ] where the otatio S r ) meas a -sphere of radius r, ad I [ρ; r ] ad R [ρ; r ] idicate itegrate of r from to r. I this otatio if r is left ustated, as was the case for the volume itegrals over uit radius S, its value is assumed to be r 1. We ca rewrite R [ρ k ; r ] as a itegral over ξ from the stadard rage of to 1 by simple trasformatio of variables ξ r/r, leadig to ) a k A R [ ρ k ] 33) I [ρ k ; r ] ρ r r where R [ ρ k ] F, k, ξ ) 1 ξ 1 dξ ξ + ξ ) k where ξ a /r. 34) 8

9 The geeral solutio to F, k, ξ ) with arbitrary argumets ivolves the evaluatio of hypergeometric fuctios. However, for ay particular values of the argumets the itegratio is ofte possible to carry out aalytically for small iteger values of ad k, ad usually easy to do umerically for ay values of ad k. I the case that a the itegral diverges uless > k. It is frequetly the case that oe wishes to itegrate r from to, ad i that case ad with a the itegratio ca be simplified by the substitutio β r/a, yieldig ρ a k r + a ) k r 1 dr ρ a G, k), where 35) G, k) β 1 dβ β + 1) k. 36) Agai, it is usually easy to solve this itegral umerically for arbitrary values of ad k, ad it is ofte easy to compute aalytically for low values of ad k, such as G3, ) π 4. The itegral diverges uless < k. I summary, for itegratios of ρ k r) of eq. 31) from to fiite r i dimesios oe fids ) a I [ρ k ; r ] ρ r k ) A F, k, a 37) r where the fuctio F is defied i eq. 34) ad A is defied i eq. 4). For itegratios of ρ k r) from to oe fids I [ρ k ; ] ρ a A G, k) 38) where G, k) is defied i eq. 36). Refereces Arfke, G.B., Webber, H.J. 1). Mathematical Methods for Physicists, 5th ed. New York: Harcourt. Bartle, R.G. 1964). The Elemets of Real Aalysis. New York: Joh Wiley & Sos. Deery, P., Krzywicki, A. 1996). Mathematics for Physicists. Mieola, NY: Dover. Mukres, J.R. ). Topology, d ed. Upper Saddle River, NJ: Pretice Hall. r 9