Recursive construction for a class of radial functions. II

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JOURNAL OF MATHEMATICAL PHYSICS VOLUME 43, NUMBER 5 MAY 2002

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Recursive construction for a class of radial functions.II. Superspace

Thomas Guhra)

Matematisk Fysik, LTH, Lunds Universitet, Box 118, 22100 Lund, Swedenand Max Planck Institut fu¨r Kernphysik, Postfach 103980, 69029 Heidelberg, Germany

Heiner Kohlerb)

Departamento de Fı´sica, Universidad Auto´noma de Madrid, Madrid, Spainand Max Planck Institut fu¨r Kernphysik, Postfach 103980, 69029 Heidelberg, Germany

~Received 19 October 2001; accepted for publication 1 February 2002!

We extend the recursion formula for matrix Bessel functions, which we obtainedpreviously, to superspace. It is sufficient to do this for the unitary orthosymplecticsupergroup. By direct computations, we show that fairly explicit results can beobtained, at least up to dimension 838 for the supermatrices. Since we introducea new technique, we discuss various of its aspects in some detail. ©2002 Ameri-can Institute of Physics.@DOI: 10.1063/1.1463218#

I. INTRODUCTION

In a previous work, we studied properties of matrix Bessel functions in ordinary space.1 Here,we generalize these investigations to superspace. For the introductory remarks and the macal and physical background relevant for the ordinary space, and also relevant as the basispresent study, we refer the reader to Ref. 1.

In mathematics, supersymmetry was pioneered by Berezin2 and, in particular group theoreticaaspects, by Kac.3,4 The theory of nonlinears models in spaces of supermatrix fields was devoped in physics of disordered systems by Efetov.5,6 Verbaarschot, Weidenmu¨ller, and Zirnbauer7,8

used his approach to study models in random matrix theory. In Ref. 9, the first supersymgeneralization of the Itzykson–Zuber integral10 was given. In Ref. 11, Gelfand–Tzetlicoordinates12 were constructed for the unitary supergroup. Extending Shatashvili’s13 method, thesupersymmetric Itzykson–Zuber integral was also rederived in Ref. 11 in its most generalUsing the techniques of Ref. 9, such a calculation was also performed in Ref. 14.

From a mathematical viewpoint, Efetov’s work5 is the basis for a harmonic analysis in certasupersymmetric coset spaces, the Efetov spaces, which are relevant for the nonlinears models. Inthe full superspaces, a technique involving convolution integrals and ingredients of thesponding harmonic analysis was introduced in Ref. 15. In the Efetov spaces, the theory omonic analysis, in both its mathematical and physical aspects, was developed by Zirnbaue16 andwas applied to disordered systems in Refs. 17 and 18. In the present contribution, we do noon the Efetov spaces, rather we address the full supergroup spaces. The supersymmetric ItzZuber integral9 and its application in Ref. 15 is the simplest example of a supermatrix Befunction appearing in this kind of harmonic analysis.

The matrix Bessel functions in superspace find direct application in random matrix theorgeneral reviews, see Refs. 19–21. In Ref. 22 it was shown that they are the kernels fsupersymmetric analog of Dyson’s Brownian motion.

The paper is organized as follows: In Sec. II, we introduce the supermatrix Bessel funand collect basic definitions and notations. In Sec. III, we extend the recursion formula of Rto superspaces. Since it is one of our goals to demonstrate that explicit results for supe

a!Electronic mail: [email protected]!Electronic mail: [email protected]

27410022-2488/2002/43(5)/2741/29/$19.00 © 2002 American Institute of Physics

4 Apr 2005 to 147.142.186.54. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp

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2742 J. Math. Phys., Vol. 43, No. 5, May 2002 T. Guhr and H. Kohler

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Bessel functions can indeed be obtained, we present, in some detail, such calculations forsupermatrix Bessel functions in Secs. IV and V, respectively. The asymptotics and the normtion are discussed in Sec. VI. We briefly comment on applications in Sec. VII and we summand conclude in Sec. VIII. Various calculations are shifted to the appendices.

II. SUPERMATRIX BESSEL FUNCTIONS

Similar to ordinary spaces,1 the superunitary case, i.e., integration over the supergrU(k1 /k2), is the simplest one. Since this was already discussed in detail in Refs. 9, 11, andrefrain from reconsidering it here. Thus, it turns out that we may restrict ourselves to the smatrix Bessel function of the unitary orthosymplectic group UOSp(k1/2k2). As discussed byKac,3,4 the supergroups U(k1 /k2) and UOSp(k1/2k2) exhaust almost all classical compact supgroups, apart from some exotic exceptions which are of little relevance for applications. Hthe integral we have to deal with is given by

Fk12k2~s,r !5E

uPUOSp(k1/2k2)exp~ i trgu21sur!dm~u!, ~2.1!

where dm(u) is the invariant measure. The arguments of the function~2.1! are the diagonalmatricess5diag(Acs1 ,A2cs2! and r 5diag(Acr1 ,A2cr2!. Here, we use Wegner’s notation23

and introduce the labelc561 to distinguish the two possible forms. We will return to this issThe matricess1 , s2 and r 1 , r 2 are given by

s15diag~s11,s21,...,sk11!, s25diag~s1212 ,...,sk2212!,

~2.2!r 15diag~r 11,r 21,...,r k11!, r 25diag~r 1212 ,...,r k2212!.

There is a twofold degeneracy ins2 andr 2 , because the matrixu21su or, equivalently,uru21 hasto be areal Hermitiansupermatrix23 of the form

s5FAcs (R) s (A)†

s (A) A2cs (HSd)G , c561. ~2.3!

The matricess (R) and s (HSd) have ordinary commuting entries, i.e., bosons, they are real smetric and Hermitian self-dual, respectively. The matrixs (A) has anticommuting or Grassmanentries, i.e., fermions, and is of the form

s (A)5@s1(A) , ...,sk1

(A)#, s i(A)5F s1i

(A)

s1i(A)*]

sk2i(A)

sk2i(A)*

G . ~2.4!

We can now appreciate the meaning of the parameterc which enters the definition~2.3! of the realHermitian matrices. Forc51, it yields the real symmetric and forc521 the Hermitian self-dualmatrix as boson–boson block, and vice versa for the fermion–fermion block. In the frameworandom matrix theory, we find the supermatrices corresponding to the Gaussian orthogosemble~GOE! for c511 and those for the Gaussian symplectic ensemble~GSE! for c521.

The infinitesimal volume element is given by

d@s#5)i 51

k1

)j 51

k2

ds i j(A)* ds i j

(A))i , j

ds i j(R))

i 51

k1

ds i i(R))

i , jd@s i j

(HSd)#)i 51

k2

ds i i(HSd) , ~2.5!


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2743J. Math. Phys., Vol. 43, No. 5, May 2002 Recursive construction. II

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where d@s i j(HSd)# is the product of the differentials of all independent elements of the quater

s i j(HSd) .

The supermatrix Bessel functions~2.1! are eigenfunctions of a wave equation in the curvspace of the eigenvaluess or r . As in the ordinary case, a supermatrix gradient]/]s is introducedand the Laplace operator is defined by

D5trgS ]

]s D 2

. ~2.6!

The plane waves exp(i trgsr) are the eigenfunctions, i.e., we have

D exp~ i trgsr!52trgr2 exp~ i trgsr!. ~2.7!

Here, boths andr are real Hermitian. As in ordinary space, the supermatrix Bessel functionobtained by averaging over the angular coordinates, i.e., over the diagonalizing group. Tplacian commutes with the average and we arrive at the differential equation

DsFk12k2~s,r !52trg r 2Fk12k2

~s,r !, ~2.8!

where the radial part of of the Laplacian~2.6! reads

Ds51

Bk1k2

(c) ~s!S (

p51

k1 ]

]sp1

Bk1k2

(c) ~s!]

]sp1

11

2(p51

k2 ]

]sp2

Bk1k2

(c) ~s!]

]sp2D . ~2.9!

The Jacobian or Berezinian is given by22

Bk1k2

(1) ~s!5uDk1

~s1!uDk2

4 ~ is2!

)p51k1 )q51

k2 ~sp12 isq2!2 , Bk1k2

(21)~s!5uDk1

~ is1!uDk2

4 ~s2!

)p51k1 )q51

k2 ~ isp12sq2!2 . ~2.10!

One easily convinces oneself thatDs depends onc only through a factorAc. Thus, without lossof generality, we setc51 and omit the indexc.

At this point, an important comment is in order. The normalization in ordinary space acing to Eq.~3.17! in Ref. 1,FN

(b)(x,0)51 andFN(b)(0,k)51, does not carry over to the supersym

metric case. This is due to the fact that the volume of some supergroups is zero2 resulting in thevanishing ofFk12k2

(0,s) for certain values ofk1 andk2 . This collides with the normalization othe plane waves~2.7! to unity at the origin. The reason for this contradiction is a well-knophenomenon in superanalysis. In going from Cartesian to angle eigenvalue coordinates, onadd additional terms to the measure to preserve the symmetries of the original integral. Thcalled Efetov–Wegner–Parisi–Sourlas terms in the physics literature. A full-fledged mathemtheory of these boundary terms was given by Rothstein.24

To solve this normalization problem, we use the following strategy. First, we evaluatsupermatrix Bessel functions without taking care of the normalization. We just multiply thegrals with a normalization constantGk12k2

. Having done the integrals, we determine the normization by comparing the asymptotics of the supermatrix Bessel function for large argumentthe Gaussian integral.

III. SUPERSYMMETRIC RECURSION FORMULA

We extend the recursion formula in ordinary space1 to superspace. After stating the resultSec. III A, we present the derivation and the calculation of the invariant measure in Secs. IIIIII C, respectively.


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A. Statement of the result

Let Fk12k2(s,r ) be defined through the group integral in Eq.~2.1!. It has two diagonal

matrices defined as in Eq.~2.2! as arguments. It can be calculated iteratively by therecursionformula

Fk12k2~s,r !5Gk12k2

E dm~s8,s! exp~ i ~ trgs2trgs8!r 11!F (k121)2k2~s8, r !, ~3.1!

where F (k121)2k2(s8, r ) is the group integral~2.1! over UOSp((k121)/2k2)) and Gk12k2

is anormalization constant, see Secs. II and VI. As in the ordinary case,1 the coordinatess8 areradialGelfand–Tzetlin coordinates. Again, they are different from theangular Gelfand–Tzetlin coordi-nates, which will be discussed elsewhere.25 We also introduced the diagonal matrix

r 5diag~r 21,...,r k11 ,ir 2!5diag~ r 1 ,i r 2! ~3.2!

such thatr 5diag(r11, r ) and the diagonal matrix

s85diag~s118 ,...,s(k121)18 ,is28!5diag~s18 ,is28!. ~3.3!

The invariant measure reads

dm~s8,s!52k211mB~s18 ,s1!mF~s28 ,s2!mBF~s8,s!d@j8#d@s18#,

mB~s18 ,s1!5Dk1

~s18!

A2)p51k1 )q51

k121~sp12sq18 !

,

~3.4!

mF~s28 ,s2!5Dk2

4 ~ is28!

)p51k2 )q51

k2 ~ isp22 isq28 !2 ,

mBF~s8,s!5)p51

k1 ) l 51k2 )q51

k121~ isl28 2sp1!~ isl22sq18 !

)p51k121

) l 51k2 ~ isl28 2sp18 !2 .

Here, we have introduced

d@j8#5 )p51

k2

djp8* djp8 , d@s18#5 )p51

k121

dsp18 . ~3.5!

The domain of integration for the bosonic variables is compact and given by

sp1<sp18 <s(p11)1, p51,...,~k121!. ~3.6!

The fermionic eigenvaluesisp28 are related to Grassmann variablesjp8 andjp8* through

ujp8u25 isp28 2 isp2 . ~3.7!

The Jacobian or Berezinian consists of three parts. One of them,mB(s18 ,s1), depends only onbosonic eigenvalues and one,mF(s18 ,s1), only on fermionic eigenvalues, i.e., only on Grassmavariables. The third part mixes commuting and anticommuting integration variables. To undonce more the difference between radial and angular Gelfand–Tzetlin coordinates whichpresent in superspace, we mention that the radial measure~3.4! is quite different from the angulaone.25


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As in ordinary space, the recursion formula is an exact map of the group integration oniteration exclusively in the radial space. Having done the iteration on the firstk1 levels, we havetreated all Grassmann variables. Thus, in the integrand, we are left with the matrix Bessel fuFk2

(4)(2 i2s2(k121) ,r 2) for USp(2k2) in ordinary space.1 We notice the occurrence of the facto

2 i2 in the argument ofFk2

(4)(2 i2s2(k121) ,r 2). Collecting everything, we arrive at

Fk12k2~s,r !5E )

n51

k121

dm~s(n),s(n21)!exp~ i ~ trgs(n21)2trgs(n)!r n1!

3exp~ is11(k121)r k11!Fk2

(4)~2 i2s2(k121) ,r 2!. ~3.8!

We have sets5s(0) and s85s(1). It is worthwhile to notice that the radial Gelfand–Tzetlcoordinates have a highly appreciated and valuable property: The Grassmann variables opear as moduli squared in the integrand. Thus, the number of integrals over anticommvariables is onlyhalf the number of the independent Grassmann variables. Moreover, advgeously, the exponential is a simple function of the integration variables. Thus, we may conthat the radial Gelfand–Tzetlin coordinates are the natural coordinates of the matrix ansupermatrix Bessel functions, because their intrinsic features are reflected.

B. Derivation

All crucial steps needed for the derivation of the supersymmetric recursion formula~3.1! carryover from the ordinary recursion formula in Ref. 1. We order the columns of the matruPUOSp(k1/2k2) in the form u5@u1 u2 ¯uk1

uk111¯uk11k2#. We also introduce a rectangula

matrix b5@u2¯uk1uk111¯uk11k2

# such thatu5@u1b#. Analogous to the ordinary case, we ha

b†b51(k121)2k2,

~3.9!

bb†5 (p52

k1

upup†1 (

p5k111

k11k2

upup†51k12k2

2u1u1† .

We define the square matrixs5b†sb and rewrite the trace in the exponent as

trgu†sur5trg s r 1s11r 11, ~3.10!

with s115u1†su1 . Similar to the ordinary case, the first term on the right-hand side of Eq.~3.10!

depends on the lastk1211k2 columnsup collected inb and the second term depends only onu1 .Thus, it is useful to decompose the invariant measure,

dm~u!5dm~b!dm~u1!, ~3.11!

and to write Eq.~2.1! in the form

Fk12k2~s,r !5E dm~u1!exp~ is11r 11!E dm~b!exp~ i trg s r !. ~3.12!

Since the coordinatesb are locally orthogonal tou1 , the measure dm(b) also depends onu1 .We now generalize the radial Gelfand–Tzetlin coordinates introduced in Ref. 1 for the

nary spaces to the superspace. Naturally, the projector reads (1k12k22u1u1

†) and we have thedefining equation

~1k12k22u1u1

†!s~1k12k22u1u1

†!ep85sp8ep8 , p51,...,k121,k111,...,k11k2 ~3.13!


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for the (k1211k2) radial Gelfand–Tzetlin coordinatessp8 and the corresponding vectorsep8 aseigenvalues and eigenvectors of the matrix (1k12k2

2u1u1†)s(1k12k2

2u1u1†) which has the gener

alized rankk1211k2 . Due tou1†ep850, we find

~1k12k22u1u1

†!sep85sp8ep8 , p51,...,k121,k111,...,k2 . ~3.14!

As in Ref. 11, the eigenvaluessp8 are calculated from the characteristic function

z~sp8!5detg~~1k12k22u1u1

†!s2sp8!52sp8 detg~s2sp8!u1†

1k12k2

s2sp8u1 , ~3.15!

which has to be discussed in the limits

z~sp8!→H 0 for p51,...,k121

` for p5k111,...,k11k2 .~3.16!

Thus, together with the normalizationu1†u151, these arek11k2 equations for the elements ofu1 .

The two parts of the integral~3.12! have to be expressed in terms of the radial GelfanTzetlin coordinatessp8 . In a calculation fully analogous to the ordinary case, we find

s115trgs2trgs8. ~3.17!

The eigenvaluestp , p51,...,k121,k111,...,k11k2 of s obtain from the characteristic functio

w~ tp!5detg~ s2tp!521

tpdetg~~1k12k2

2u1u1†!s2tp!. ~3.18!

Comparison with Eq.~3.15! shows that the characteristic functionsw(tp) and z(sp8) are, apartfrom the nonzero factor2tp , identical. This impliestp[sp8 , p51,...,k121,k111,...,k11k2 .Thus, by introducing the square matrixu which diagonalizess, we may write

s5b†sb5u†s8u. ~3.19!

By construction,u must be in the group UOSp(k121/2k2), becauses and s share the samesymmetries.

These intermediate results allow us to transform Eq.~3.12! into

Fk12k2~s,r !5E dm~s8,s!exp~ i ~ trgs2trgs8!r 11!E dm~b!exp~ i trg u†s8ur !, ~3.20!

where dm(s8,s) is, apart from phase angles, the invariant measure dm(u1), expressed in the radiaGelfand–Tzetlin coordinatess8. To do the integration overb, we view, for the moment, the vectou1 as fixed and observe that the measure dm(b) is the invariant measure of the group UOSp(k1

21/2k2) under the constraint thatb is locally orthogonal tou1 . The matrix uPUOSp(k1

21/2k2) is constructed fromb under the same constraint. Thus, sinceb and u cover the samemanifold, the integral overb in Eq. ~3.20! must yield the supermatrix Bessel functioF (k121)2k2

(s8, r ) and we arrive at the supersymmetric recursion formula~3.1!. In the last step, weused a line of arguing slightly different from the derivation in ordinary space. In this wayavoided a discussion related to the ill-defined supergroup volume. The invariance of the mis the crucial property we need for the proof and this holds both in superspace and in orspace.


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C. Invariant measure

In order to evaluate the invariant measure, we have to solve the system of equations~3.15! foruvp

(1)u25uup1u2, p51,...,k1 anduap(1)u25uu(k112p)1u21uu(k112p21)1u2, p51,...,k2 in terms of the

bosonic eigenvaluessp85sp18 , p51,...,k121 and the fermionic eigenvaluessk112p8 5sk112p218

5 isp28 , p51,...,k2 ,

15 (p51

k1

uvp(1)u21 (

p51

k2

uap(1)u2, ~3.21!

05 (q51

k1 uvq(1)u2

sq12sp181 (

q51

k2 uaq(1)u2

isq22sp18, p51,...,k121, ~3.22!

zp5 isp28)q51

k1 ~sq12 isp28 !

)q51k2 ~ isq22 isp28 !2 S (

q51

k1 uvq(1)u2

sq12 isp281 (

q51

k2 uaq(1)u2

isq22 isp28D , zp→`, p51,...,k2 .

~3.23!

In Appendix A, we sketch the solution of this system for small dimensions. Inspired by tsolutions one can conjecture the general solutions and verify them by plugging them directEqs.~3.21!–~3.23!; one finds

uvp(1)u25

)q51k121

~sp12sq18 !)q51k2 ~sp12 isq2!2

)q51k2 ~sp12 isq28 !2)q51,qÞp

k1 ~sp12sq1!, p51,...,k1 ,

~3.24!

uap(1)u252~ isp28 2 isp2!

)q51k121

~ isp22sq18 !)q51,qÞpk2 ~ isp22 isq2!2

)q51,qÞpk2 ~ isp22 isq28 !2)q51

k1 ~ isp22sq1!, p51,...,k2 .

These expressions are reminiscent of the ones derived in Ref. 11 for unitary matrices. Hoimportantly, all products in~3.24! involving fermionic eigenvalues are squared. This reflectsdegeneracy ofs in the fermion–fermion block. We have introduced new anticommuting variajp8 ,jp8* with ujp8u

25 isp28 2 isp2 according to definition~3.7!.From this point on, the invariant measure can be calculated in the same way as for the a

Gelfand–Tzetlin coordinates, see Ref. 11 for details. The result is summarized in Eq.~3.4!.

IV. THE FUNCTION F22„s ,r …

We use the recursion formula~3.1! to calculate the supermatrix Bessel function fUOSp(2/2). To avoid the imaginary unit in the exponent, we studyF22(2 is,r ). The recursionformula reads

F22~2 is,r !5G22E dm~s8,s!exp~~ trgs2trgs8!r 11!F12~2 is8, r !. ~4.1!

The functionF12(2 is8, r ) is easily found to be

F12~2 is8, r !5G12~122~r 212 ir 12!~s118 2 is128 !!exp~2r 12s12!. ~4.2!

The measure of the coset UOSp(2/2)/UOSp(1/2) is according to formula~3.4! given by

dm~s8,s!5~ is122s118 !)n51

2 ~ is128 2sn1!

A2)n512 ~s118 2sn1!~ is128 2s118 !2

ds118 dj18* dj18 . ~4.3!


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We do the Grassmann integration and find

F22~2 is,r !5G22exp~r 11~s111s21!22is12ir 12!

3Es11

s21mB~s8,s!)

q51

2

~ is122sq1!S 4)j 51

2

~ ir 122r j 1!22~ ir 122r 11!

3 (q51

21

is122sq112M11~s18 ,s1!D exp~s118 ~r 212r 11!!ds118 , ~4.4!

where we have introduced the operator

Mm j~s18 ,s1!51

~ ism22sj 18 ! S 1

2 (n51

k1 1

ism22sn12

1

ism22sj 182 (

n51nÞ j

k1 1

sj 18 2sn182

]

]sj 18 D . ~4.5!

For later purposes, we introduced general indicesm and j . Obviously, the Grassmann integratioyielded eigenvalues in the denominator. This is somewhat surprising because of the follobservation: we can always parametrize the group elementuPUOSp(2/2) in a noncanonical coseparametrization in the spirit of an Euler parametrization in ordinary space. Inserting this petrization into the defining equation of the supermatrix Bessel function~2.1! one can expand thetrace in all Grassmann variables. The expansion coefficients are polynomials in the commintegration variables and—more important—in the matrix elements ofs and r . The invariantmeasure can be expanded in the Grassmann variables as well. It does not depend onr and s.Although this procedure gets rapidly out of hand even for small groups, it is clear that the ouof this expansion will be polynomial in the eigenvalues ofs and r . In other words: eigenvaluecan only appear in the denominator by an integration over commuting variables and neveGrassmann integration. Therefore, before performing any integral over commuting variablesmust exist a form ofF22(2 is,r ), which is polynomial in the eigenvalues ofs and r .

To remove the denominators and to obtain such a polynomial expression, we use the folresult. Let f (s18) be an analytic, symmetric function ins1i8 , i 51,...,k1 . Furthermore, define theoperator

Lm~s!5(j 51

k1 1

ism22sj 1

]

]sj 1. ~4.6!

Then the action of the operator on the integral over the bosonic part of the measure is giv

Lm~s!Es11

s21¯E

s(k121)1

sk11mB~s8,s! f ~s18!d@s18#

52Es11

s21¯E

s(k121)1

sk11mB~s8,s! (

j 51

k121

Mm j~s18 ,s1! f ~s18!d@s18#. ~4.7!

This formula is derived in Appendix B.We now setf (s18)5exp(2s118 (r 212r 11)) and insert Eq.~4.7! into Eq. ~4.4!, we arrive at

F22~2 is,r !5G22exp~22is12ir 12!S 4)j 51

2

~ ir 122r j 1!~ is122sj 1!

22(q51

2

~ is122sq1!S ir 122r 212r 112]

]sq1D DF2

(1)~2 is1 ,r 1!, ~4.8!


metry

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whereF2(1)(s1 ,r 1) is the matrix Bessel function of the orthogonal group O~2! in ordinary space as

defined in Ref. 1. Although this can already be taken as the result, we underline the symbetweens and r by using the explicit form~3.21! of Ref. 1 forF2(s1 ,r 1),

F22~2 is,r !5G22expS trg rs2z

2D3S 4)

j 51

2

~ ir 122r j 1!~ is122sj 1!2 (q51

2

~ is122sq1! (p51

2

~ir122rp1!2zd

dzD 2pI 0~z/2!,

~4.9!

where we have introducedz5(s112s21)(r 112r 21) and the modified Bessel functionI 0 as definedin Ref. 26.

The result~4.7! was crucial in the derivation ofF22(2 is,r ). By means of this formula, thedenominator problem was overcome in one step. Because of its importance, we want to gaiinsight into this problem: In Appendix C, we rederiveF22(2 is,r ) in two other ways. It is clearthat the methods of Appendix C cannot be used for higher dimensionsk1 and 2k2 , but it will helpto understand the mechanisms needed when working with radial Gelfand–Tzetlin coordina

V. THE SERIES OF FUNCTIONS Fk 14„s ,r …

We calculate iteratively the four supermatrix Bessel functionsFk14(s,r ) for k151,2,3,4. Wedo this in Secs. V A–V D, respectively.

A. First level k 1Ä1

According to the recursion formulas~3.1! and ~3.8!, the starting point is the matrix Bessefunction for the unitary symplectic groupF2

(4)(s2 ,r 2), which was already calculated in Ref. 1. Uto a normalization, we have

F2(4)~ i2s2 ,r 2!5 (

vPS2S 1

D22~ is2!D2

2~v~ ir 2!!2

1

D23~ is2!D2

3~v~ ir 2!! Dexp~2tris2v~ ir 2!!.

~5.1!

Since the subgroup O~1! of UOSp(1/4) is trivial, no commuting integral has to be performedderive F14(2 is,r ). Inserting the measure~3.4! into the recursion formula and performing thGrassmann integrations yields straightforwardly

F14~2 is,r !5G14exp~ trg rs!S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D ~2~ is212s11!~ ir 212r 11!21!

3~2~ is222s11!~ ir 222r 11!21!2G14

exp~ trg~rs!!

D23~ ir 2!D2

3~ is2!1~ ir 12↔ ir 22!. ~5.2!

The exchange term (ir 12↔ ir 22) accounts for the permutation groupS2 in Eq. ~5.1!. Anticipatingthat the structure ofF14(2 is,r ) will, remarkably, survive on all levels up toF44(2 is,r ), we statethat F14(2 is,r ) essentially consists of two parts. A comparison with Eqs.~4.2!, and~5.1! showsthat the first part ofF14(2 is,r ) is a product of an exponential with three other terms. The fione,

S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D , ~5.3!

stems from the integral over the USp~4! subgroup. The other two terms can be identified withsupermatrix Bessel functions


cture oftheethe

tegra-y, we

t. With

duct

rs


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F12~2 is,r ! with s5diag~s11,is12,is12!, r 5diag~r 11,ir 12,ir 12! ~5.4!

and

F12~2 is,r ! with s5diag~s11,is22,is22!, r 5diag~r 11,ir 22,ir 22!. ~5.5!

The second part can be considered as a correction term, which destroys the product struF14(2 is,r ). We may identify the different parts of the product with the integrations overcorresponding subsets of the group. Thus,F2

(4)(2 is2 ,r 2) arises from the integration over thUSp~4! subgroup, the O~1! integration yields unity, and the other two factors come fromintegration over the coset UOSp(1/4)/(USp(4) O(1)).

B. Second level k 1Ä2

We now have to do one integration over a commuting variable. After the Grassmann intion, we are left with a considerable amount of terms. To arrange them in a convenient waintroduce the following notation for the product of two operatorsD1(s)D2(s) acting on a functionf (s), we define

@D1→~s!D2~s!# f ~s!5D1~s!D2~s! f ~s!2~D1~s!D2~s!! f ~s!. ~5.6!

This means, an operator with an arrow only acts on the terms outside the squared brackethis notation we can write

F24~2 is,r !5G24exp~ tr~r 2s2!1r 21~s111s21!!Es11

s21dmB~s18 ,s1!F)

i 51

2

)j 51

2

~ isi22sj 1!

3S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D3S 4)

i 51

2

~ ir 122r i1!12(k51

2r 212 ir 12

is122sk112M

11~s18 ,s1!D

3S 4)i 51

2

~ ir 222r i1!12(k51

2r 212 ir 22

is222sk112M21~s18 ,s1!D

1S 1

D22~ ir 2!D3

2~ is2!1

1

D23~ ir 2!D2

4~ is2! D3S 4

is122s118M21~s18 ,s1!2

2

is222s118M11~s18 ,s1! D

12

D23~ ir 2!D2

4~ is2!S 2 tr r 12tr ir 21(

i 51

21

is222si1D M11~s18 ,s1!

22

D23~ ir 2!D2

4~ is2!S 2 tr r 12tr ir 21(

i 51

21

is122si1D M21~s18 ,s1!

24

D23~ ir 2!D2

3~ is2! (k51

2

)j 51

2r 212 ir j 2

sk12 isj 2Gexp~s118 ~r 112r 21!!1~ ir 12↔ ir 22!.

~5.7!

As in Sec. IV, a denominator problem occurs. It becomes obvious in the proM 11

→ (s18 ,s1)M21(s18 ,s1). Thus, we expect an identity similar to formula~4.7!. This identity shouldmap a product of operatorsL1(s)L2(s) acting on the integral onto a product of operato


the

me

d to these of

vity of


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M11(s18 ,s1)M21(s18 ,s1) acting under the integral. Neither the outer operators,Lm(s), nor the innerones,Mm j(s), commute. Hence, the desired identity must be a nontrivial one. It is given byfollowing result.

We have the same conditions as in formula~4.7!, furthermore we define

@Lm→~s!Ll~s!#5 (

n51

k1

(q51

k1 1

~ ism22sn1!~ isl22sq1!

]2

]sn1]sq1. ~5.8!

Then the following formula holds

@Lm→~s!Ll~s!#E

s11

s21¯E

s(k121)1

sk11mB~s8,s! d@s18# f ~s18!

5Es11

s21¯E

s(k121)1

sk11mB~s8,s!F (

j 51

k121

(k51

k121

Mm j

~s18 ,s1!Mlk~s18 ,s1!

21

isl22 ism2(j 51

k121 S 1

ism22sj 18Ml j ~s18 ,s1!2

1

isl22sj 18Mm j~s18 ,s1! D

21

2 (kÞ j

k1211

~ ism22sk18 !~ ism22sj 18 !~ isl22sk18 !~ isl22sj 18 !G f ~s18! d@s18#. ~5.9!

The derivation is along the same lines as the one for formula~4.7!, it also involves formula~4.7!.With the identities~5.9! and~4.7! the denominator problem is again solved in one step. After sofurther manipulations we arrive at

F24~2 is,r !52pG24expS trg rs2z

2D S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D3F S 4)

i 51

2

~r i12 ir 12!~si12 is12!2(i 51j 51

2

~sj 12 is12!~r i12 ir 12!2z]→

]z D3S 4)

i 51

2

~r i12 ir 22!~si12 is22!2(i 51j 51

2

~sj 12 is22!~r i12 ir 22!2z]

]zD G I 0~z/2!

22pG24expS trg rs2z

2D 2

D23~ ir 2!D2

3~ is2!

3 (i 51k51

2

)j 51

2

~si12 isj 2!~r k12 ir j 2!I 0~z/2!22pG24expS trg rs2z

2D3

1

D23~ ir 2!D2

3~ is2!~~ trgs!~ trg r !21!z

]

]zI 0~z/2!1~ ir 12↔ ir 22!. ~5.10!

As in Sec. IV, we used the composite variablez5(s112s12)(r 112r 12). A comparison with Eqs.~4.8! and~5.2! shows the similarity in the structures ofF24(2 is,r ) andF14(2 is,r ). The formeralso decomposes into two parts. The first part is a product, whose factors can be assigneintegrations over the different submanifolds of the group in the same way as in the caF14(2 is,r ). The other one can be interpreted as a correction term due to the noncommutatithe operatorsLm in formula ~5.9!.


bease.essel, it isor

tionof the

cursion

.rmula


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C. Third level k 1Ä3

This structure ofFk14(2 is,r ) emerging in the previous calculations is likely to alsopresent for arbitraryk1 . However, fork1.2, we have so far not been able to treat the general cFortunately, in important physics applications, one matrix argument of the supermatrix Bfunction has an additional twofold degeneracy in the boson–boson block. In this casepossible to carry on the recursion up toF44(2 is,r ) by extending the techniques developed fk151 andk252. Thus, from now on, we restrict ourselves to this case.

At first sight, one might hope to achieve some simplification by applying the projecprocedure onto the degenerate matrix, because this results in a considerable simplificationinvariant measure. However, it turned out that the integrations are easier if one does the rewith the nondegenerate coordinates. Hence, we use the measure as it stands in Eq.~3.4!. WeconsiderF34(2 is,r ) in the case that

r 15diag~r 11,r 21,r 21!. ~5.11!

Having performed the Grassmann integral, one can arrange the terms in a way similar to Eq~5.7!.The complete expression and further details are given in Appendix D. We then can use fo~5.9! and find after some further algebra

F34~2 is,r !54 G34exp~ tr r 2s2!~r 212 ir 12!~r 212 ir 22!F S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D3S 4)

k51

2

~r k12 ir 12!)k51

3

~sk12 is12!12(k51

3

)j Þk

3

~sj 12 is12!

3S r 111r 212 ir 122]→

]sk1D D

3S 4)k51

2

~r k12 ir 22!)k51

3

~sk12 is22!12(k51

3

)j Þk

3

~sj 12 is22!

3S r 111r 212 ir 222]

]sk1D D 2

4

D23~ ir 2!D2

3~ is2! (i 51

3

)j 51j Þ i

3

~sj 12 is12!~sj 12 is22!

3S r 112 1r 21

2 1r 11r 212~r 111r 21!~ ir 121 ir 22!1 ir 12ir 22

2~r 111r 212 ir 122 ir 22!]

]si1D2

2

D23~ ir 2!D2

4~ is2!

3(i ,k

3

)j 51j Þ i

3

)l 51lÞk

3

~sj 12 is12!~sl12 is22!S ]

]si12

]

]sk1D G

3F3(1)~2 is1 ,r 1!, 1~ ir 12↔ ir 22!, ~5.12!

whereF3(1)(s1 ,r 1) is the matrix Bessel function of the orthogonal group O~3!. We notice that the

structure ofF14(2 is,r ) andF24(2 is,r ) reappears inF34(2 is,r ).

D. Fourth level k 1Ä4

In the calculation ofF44(2 is,r ), we again consider the case that the matrixr is degenerate,

r 15diag~r 11,r 11,r 21,r 21!. ~5.13!


in

lained

way as

mationhe, we


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The main problem is to find a convenient representation for the matrix Bessel functionF3(1)

(2 is18 , r 1) appearing on the third level in Eq.~5.12!. It turns out that the representation derivedAppendix B of Ref. 1 is very well suited to our purpose. Due to the degeneracy inr 1 , the originalthreefold integral can be reduced to an integral over just one single variable

F3(1)~2 is18 , r 1!5

exp~r 21 tr s18!

Aur 112r 21uE

2`

1`

dtexp~ i ~r 112r 21!t !

) i 513 Asi18 2 i t

. ~5.14!

Here, we again neglected the normalization because we want to fix it afterwards as exppreviously. Similarly,F4

(1)(2 is1 ,r 1) can be written as a double integral,

F4(1)~2 is1 ,r 1!5

exp~r 21 tr s1!

ur 112r 21uE

2`

1`

dt1E2`

1`

dt2ut12t2uexp~ i ~r 112r 21!~ t11t2!!

) i 514 )n51

2 Asi12 i t n

. ~5.15!

Singularities have to be taken care of appropriately. After inserting Eq.~5.10! into the recursionformula and performing the Grassmann integration, one can arrange the terms in a similarin the case ofF34(2 is,r ). At this point, we notice that formulas~4.7! and ~5.9! need to besupplemented by further identities. We state the most important one in the following.

The same conditions as for formula~4.7! apply. Moreover, we define the operator

Lm~s!5 (q51

k1 1

ism22sq1

]2

]sq12 1

1

2 (qÞn

1

~ ism22sq1!~sq12sn1! S ]

]sq12

]

]sn1D . ~5.16!

Then we have

Lm~s!Es11

s11¯E

s(k121)1

sk11mB~s8,s!d@s18# f ~s18!

52Es11

s21¯E

s(k121)1

sk11mB~s8,s!F(

j 51Mm j

→ ~s18 ,s1!]

]sj 18G f ~s18! d@s18#. ~5.17!

Again, the proof is along the same lines as the proof of formula~4.7! and the proof of formula~5.9! in Ref. 1.

Thus, there is a family of rules to transform operators symmetric insi1 Lm(s),Lm(s) actingonto an integral into an operator acting under the integral. We need one more such transforrule which tells us how the product@Lm(s) L l(s)# transforms into operators acting under tintegral. This formula and further details are given in Appendix E. Collecting everythingfinally arrive at

F44~2 is,r !54G44exp~2tr ~r 2s2!!)i , j

~r i12 ir j 2!F S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D3S 8)

i 51

2

~r i12 ir 12!)j 51

4

~sj 12 is12!14(i 51

4

)j Þ i

4

~sj 12 is12!S r 111r 212 ir 122]→

]si1D D

3S 8)i 51

2

~r i12 ir 22!)j 51

4

~sj 12 is22!14(i 51

4

)j Þ i

4

~sj 12 is22!S r 111r 212 ir 222]

]si1D D

216

D23~ ir 2!D2

3~ is2! (i 51

4

)j Þ i

4

~sj 12 is12!~sj 12 is22!S r 112 1r 21

2 1r 11r 212~ ir 121 ir 22!


esseldi-

s sec-the

ity. The

wnian


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3~r 111r 21!1 ir 12ir 2221

2trg r

]

]si1D2

8

D23~ ir 2!D2

4~ is2! (i 51j 51

4

)lÞ i

4

~sl12 is12!

3)lÞ j

4

~sl12 is22!S ]

]si12

]

]sj 1D GF4

(1)~2 is1 ,r 1!1~ ir 12↔ ir 22!. ~5.18!

We mention that in the derivation of this result we frequently used properties of the matrix BfunctionsF3

(1)(s1 ,r 1) and F4(1)(s1 ,r 1) that only hold for the case that one matrix has an ad

tional degeneracy.

VI. ASYMPTOTICS AND NORMALIZATION

The asymptotic behavior of the supermatrix Bessel functions calculated in the previoutions is a useful check which also allows us to fix the normalization constants. We find fromexpressions in Eqs.~4.1!, ~4.8! and in Eqs.~5.2!, ~5.10!, ~5.12!, and~5.18!,

lims→`r→`

Fk12k2~2 is,r !52k1k2Gk12k2

) l 51k1 )m51

k2 ~sl12 ism2!~r l12 ir m2!

Dk2

2 ~ is2!Dk2

2 ~ ir 2!

3det@exp~2si2r j 2!# i , j 51¯k2lim

s1→`r 1→`

Fk1

(1)~2 is1 ,r 1!. ~6.1!

In the degenerate case, each degenerate eigenvalue contributes according to its multiplicasymptotics of the matrix Bessel functions of the orthogonal group is given by27,28

limt→0

Fk1

(1)~2 is1 /t,r 1!5C(k1)t (k121)k1/4det@exp~sn1r m1 /t !#n,m51,...,k1

uDk1~s1!Dk1

~r 1!u1/2 , ~6.2!

where the constant can be found in Muirhead’s book,28

C(k1)5G~k1/2!

k1!pk1

2/22k1/4. ~6.3!

Thus we find

limt→0

Fk12k2~2 is/t,r !52k1k2t ((k122k2)21(k122k2))/4C(k1)Gk12k2

3det@exp~sn1r m1 /t !#n,m51,...,k1

det@exp~2si2r j 2 /t !# i , j 51¯k2

ABk1k2~s!Bk1k2

~r !~6.4!

for the asymptotic behavior.On the other hand, the supermatrix Bessel function relates to the kernel of Dyson’s Bro

motion in superspace.22 Due to the normalization of the Gaussian integral,

S p

2t D2((k122k2)21(k122k2))/4

2k222k22k1/2E d@s#expS 2

1

t~s2r! D51, ~6.5!

the kernel


by

ction

mentre, inere. Wectionsmetric

sked toussedrselves

ens in

on’sreen

tofinite

roach,er ofeviouss. It hasegrals,r for-


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Gk1k2~s,r ,t !5S p

2t D2((k122k2)21(k122k2))/4

2k222k22k1/2E

uPUOSp(k1/2k2)dm~u!expS 2

1

t~s2r! D

~6.6!

is also normalized. Since it is obviously connected with to the supermatrix Bessel function

Gk1k2~s,r ,t !5S p

2t D2((k122k2)21(k122k2))/4

2k222k22k1/2expS 2

1

t~ trgs21trg r 2! DFk12k2

~2 is/t,r !

~6.7!

we can fix the normalization by using the asymptotic behavior

limt→0

Gk1k2~s,r ,t !5S p

2D 2((k122k2)21(k122k2))/42k2

22k22k1/2

k1! k2!

3det@d~si12r j 1!# i , j 51¯k1

det@d~si22r j 2!# i , j 51¯k2

ABk1k2~s!Bk1k2

~r !~6.8!

of the kernel. Comparing Eq.~6.4! with Eq. ~6.8!, we find

Gk12k25

23k2(k22k1)1k12/425k2/22k1/2

p ((k122k2)212k1222k2)/4k2!G~k1/2!

. ~6.9!

We mention that this calculation also shows that the diffusion kernels of the one-point funand of the two-point function of Dyson’s Brownian motion,22 i.e., the functionG (2k)k(s,r ,t) whichwas denoted byGk(s,r ,t) in Ref. 22, indeed satisfy the proper initial condition.

VII. APPLICATIONS

Although we focus in this contribution on the mathematical aspects, we now briefly comon a particular kind of application. As the reader will realize, our results derived in Sec. VI asome sense, more general than what we need in those applications on which we focus htake this as an indication that explicit results for even more complex supermatrix Bessel funcan also be obtained. The results of the previous sections yield the kernels of the supersymanalog of Dyson’s Brownian Motion for the GOE and the GSE in the casesk51 andk52. We donot present the physics background here. The reader interested in these applications is aconsult Refs. 19–21 for generalities and Ref. 22, in particular Sec. 4.2, for the issue dischere. In the present contribution, we use the same notations and conventions. We restrict outo the transition toward the GOE and suppress the indexc. The corresponding formulas for thtransition toward the GSE are derived accordingly. We treat the one- and two-point functioSecs. VII A and VII B, respectively.

Forrester and Nagao29 derived expressions for generalized one-point functions of DysBrownian motion model with Poissonian initial conditions. They used an expansion of the Gfunction in terms of Jack polynomials. Datta and Kunz30 employed a supersymmetric techniqueaddress the two-level correlation function of the Poisson GOE transition. They arrive at anumber of ordinary and Grassmannian integrals which are still to be performed. In our appwe also arrive at a representation of the correlation function in terms of a finite numbintegrals. However, since we managed to integrate over almost all angular integrals in the prsections, our result contains considerably less integrals, in particular, no Grassmannian onea clear structure due to the fact that, apart from two integrals, all others are eigenvalue inti.e., live in the curved eigenvalue space of Dyson’s Brownian motion. Moreover, since oumulas for the kernel are valid on all scales, our result is also exact for finite level number.


y is

tyGOE,

l

onian

nalysis


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A. Level density

We use the result~4.9!, derived in Sec. IV, for the supermatrix Bessel functionF22

(2 is,r ). Using the replacementr→(x1J) and s→s/t and the relation~6.7!, we obtain thediffusion kernel for the level density

G1~s,x1J,t !5~2p!21/2J1

2texpS 2

1

t~s112x12J1!22

1

t~s212x12J1!21

2

t~ is122x11J1!2D

3S 22J1

t )j 51

2

~ is122sj 1!1 (q51

2

~ is122sq1!D . ~7.1!

We take the derivative with respect to the source variableJ1 and arrive at the level density

R1~x1 ,t !51

~2p!3/2t E expS 21

t~s112x1!22

1

t~s212x1!21

2

t~ is122x1!2D

3~~ is122s11!1~ is122s21!!B21~s! Z1(0)~s! d@s#, ~7.2!

where the Berezinian is given by Eq.~2.10! for k152 and k251. This result is exact for anarbitrary initial condition and for arbitraryN. In the case of a diagonal matrixH (0) as the initialcondition, we have for arbitraryk,

Zk(0)~s!5E d@H (0)#P~H (0)!)

n51

N) j 51

k ~ isj 22Hnn(0)!

) j 512k ~sj 11 i«2Hnn

(0)!1/2 ~7.3!

and analogously for the GSE. This has to be used in Eq.~7.2! for k51.In the limit t→` the stationary distribution of classical Gaussian random matrix theor

recovered. This can be seen by rewriting Eq.~7.1! for the rescaled energyx15x1 /t and therescaled source variableJ15J1 /t, see also Ref. 22. In this limit the initial condition yields uniand we arrive at an integral representation of the one-point correlation function of the pure

R1~x1!51

~2p!3/2IE exp~2~s112x1!22~s212x1!21~ is122x1!2!

3us112s21u

~ is122s11!~ is122s21!S 1

is122s111

1

is122s21D ~ is12!

N

~s111 i«!N/2~s211 i«!N/2 d@s#

~7.4!

where the symbolI denotes the imaginary part. Equation.~7.4! is equivalent to the classicaexpressions for the one-point functions as in Mehta’s book.19

Finally, we state an integral expression for the one-point function for the case of Poissinitial conditions, see Eq.~5.1! of Ref. 22. We have

Zk(0)~s!5S E dzp~z!

) j 51k ~ isj 22z!

) j 512k ~sj 11 i«2z!1/2D N

. ~7.5!

Inserting this initial condition fork51 into Eq. ~7.2! yields the level density of a transitioensemble between Poisson regularity and GOE in terms of a fourfold integral. A further anwill be published elsewhere.


ction

ther

very

that

om the

.plicit

thenergyn is,nt foren in

res inin thein the

ctions.


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B. Two-point function

The result~5.18!, derived in Sec. V, for the supermatrix Bessel functionF44(2 is,r ) gives,with the replacementr→(x1J) ands→s/t and according to Eq.~6.7!, the diffusion kernel for thetwo-point function

Gk~s,x1J,t !5expS 21

t~ trgs21trg ~x1J!2! DF44~22is/t,x1J!. ~7.6!

The derivative with respect to the source terms leaves us with the two-point correlation fun

R2~x1 ,x2 ,t !526G44

t2p2 IE SAB42~s!AuD4~s1!uZ2(0)~s!expS 2

1

t~ trs1

212x1212x2

222~ is122x1!2

22~ is222x2!2! D(k, j

F 1

~ is122sk1!~ is222sj 1! S 2x1

t2

]

]sj 1D S 2x2

t2

]

]sk1D

1t

2~x12x2!~ is122 is22!~ is122sk1!~ is222sj 1! S 2x1

t2

]

]sj 1D S 2x2

t2

]

]sk1D

321

4

t

~x12x2!~ is122sk1!~ is222sj 1!~ is122 is22!2 S ]

]sj 12

]

]sk1D G

3F4(1)~22is1 /t,x! D d@s#1~x1↔x2!. ~7.7!

The last line indicates that the integral withx1 and x2 interchanged has to be added. Sinceterms antisymmetric inx1 andx2 are antisymmetric insi2 ands22 as well this yields just a facto2 in Eq. ~7.7!. The symbolI denotes a certain linear combination ofR2(x1 ,x2 ,t) as explained inRef. 22. The normalization constant obtains from Eq.~6.9! and is given byG4452(2p)24. Thisresult is an exact expression for the two-point function of Dyson’s Brownian motion for einitial condition. Plugging in the initial condition of Eq.~7.5! for k52, we find an integralrepresentation of the two-point function for the transition toward the GOE. We noticeF4

(1)(22is1 /t,x) is, according to Eq.~5.15!, given as a double integral.In the previous discussion, we referred to properties of the kernels which can be seen fr

explicit formulas. In Ref. 22, only the explicit form of the kernel forb52 was available. How-ever, some of the general properties of the kernels forb51 andb54 could be anticipated in Ref22 from scaling relations of the supersymmetric version of Dyson’s Brownian motion. The exformulas derived in the present contribution allow one to derive the integral representations~7.2!and ~7.7! for the one-point and for the two-point function. Moreover, we emphasize thatkernels for the supersymmetric version of Dyson’s Brownian motion are the same on all escales.22 Thus, the integral representation derived here for the two-level correlation functioapart from the initial condition, the same on the so-called unfolded scale which is relevaphysics applications. The initial condition on the unfolded scale is found along the lines givRef. 22.

VIII. SUMMARY AND CONCLUSION

We extended the recursion formula of Ref. 1 to superspace. Due to the group structusuperspace, we could restrict ourselves to the unitary orthosymplectic supergroup. Asordinary case, the recursion formula is an exact map of a group integration onto an iterationradial space. We used it to calculate explicit expression for certain supermatrix Bessel fun


radialkely

at thegrals islicitof thesaw

urviveimilargroup

lenge

rsion.e the

kely tories ofursion

nse ofk in atHow-

son’s

atrixe recur-ion are.

ncialH. K.

t withor, yettrivial


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In ordinary space, we saw that the matrix Bessel functions are only special cases of thefunctions.1 We have not yet studied this further, but in our opinion a similar generalization is lito also exist in superspace.

It is a major advantage of the radial Gelfand–Tzetlin coordinates in superspace thGrassmann variables appear only as moduli squared. Thus, the number of Grassmann intea priori reduced by half. As we showed in detail, this is a highly welcome feature for expcalculations. As a remarkable consequence of this recursive way to proceed, the structuresupermatrix Bessel functions is only very little influenced by the matrix dimension. We alsothat the basic structures of the supermatrix Bessel function for smaller matrix dimensions sduring the iteration to higher ones. The matrix Bessel functions in ordinary space show sfeatures. There, the structure of the matrix Bessel functions is much more influenced by theparameterb than by the matrix dimension. However, as in ordinary space, it remains a chalto find the structure of these functions for arbitrary matrix dimension.

An interesting feature occurred which sheds light on the general properties of the recuTotal derivatives showed up in the integral over the commuting variables after having donGrassmann integration. Since similar terms already occurred in ordinary space, they are libe an intrinsic property of the recursion formula. Here, we succeeded in constructing a seoperator identities to remove them. This was a crucial step for the application of the recformula. A deeper understanding of these identities is highly desired.

It should be emphasized that the total derivatives are no boundary terms in the seRothstein. We showed in detail that such terms cannot occur because we always worcompact space. Thus, according to a theorem due to Berezin,2 the transformation of the invarianmeasure to our radial Gelfand–Tzetlin coordinates cannot yield Rothstein boundary terms.ever, if further integration over the eigenvalues is required, such terms can emerge.

As an application, we worked out some kernels for the supersymmetric analog of DyBrownian motion.

The radial Gelfand–Tzetlin coordinates are the natural coordinate system for the mBessel functions in superspace. This parametrization represents the appropriate tool for thsive integration of Grassmann variables. Once the particular features of this parametrizatbetter understood, they may allow for the evaluation of higher dimensional group integrals

ACKNOWLEDGMENTS

We thank B. Balantekin and Z. Pluhar for useful discussions. We acknowledge finasupport from the Deutsche Forschungsgemeinschaft—T. G. for a Heisenberg fellowship andfor a doctoral grant. H. K. also thanks the Max-Planck-Institute for financial support.

APPENDIX A: RADIAL GELFAND–TZETLIN COORDINATES FOR THE UNITARYORTHOSYMPLECTIC GROUP UOSp „k 1Õ2k 2…

We wish to express the moduli squared of the elements of an orthogonal (k1/2k2) dimensionalunit supervector in radial Gelfand–Tzetlin coordinates. To illustrate the mechanism, we starthe smallest nontrivial case, the group UOSp(2/4). We notice that there are at first sight mincrucial, differences to the calculation in Ref. 11 where we also started with the smallest noncase. The set of solutions of the Gelfand–Tzetlin equations~3.23! involves one bosonic and twofermionic eigenvalues. The eigenvalue equations read

15 (p51

2

~ uvp(1)u21uap

(1)u2!, ~A1!

05 (q51

2 S uvq(1)u2

sq12s1(1) 1

uaq(1)u2

isq22s1(1)D , ~A2!



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z15 is2(1))

q51

2~sq12 is2

(1)!

~ isq22 is2(1)!2 (

q51

2 S uvq(1)u2

sq12 is2(1) 1

uaq(1)u2

isq22 is2(1)D , ~A3!

where the last equation has to be solved in the limitz1→`. The bosonic equation~A2! has aunique solutions1

(1)5s118 . Takings118 as new parameter, Eqs.~A1! and ~A2! can be solved,

uvp(1)u25

sp12s118

sp12sq1S 12 (

k51

2isk22sq1

isk22s118uak

(1)u2D , p51,2. ~A4!

We insert these relations in Eq.~A3! and obtain

z15 is2(1)~s118 2 is2

(1)!)q51

2~sq12 is2

(1)!

~ isq22 is2(1)!2 S 11 (

k51

2ck

isk22 is2(1) uak

(1)u2D ~A5!

with z1→`. Here, we have defined the commuting variables

ck5)q51

2 ~ isk22sq1!

isk22s118, k51,2. ~A6!

It remains to determine the set of solutions of the fermionic eigenvalue equation~A5!. To this end,both sides are inverted

05 )q51

2

~ isq22 is2(1)!2S 12 (

k51

2ck

isk22 is2(1) uak

(1)u212)k51

2ck

isk22 is2(1) uak

(1)u2D . ~A7!

We can now take the square root on both sides

05 )q51

2

~ isq22 is2(1)!S 12

1

2 (k51

2ck

isk22 is2(1) uak

(1)u213

4 )k51

2ck

isk22 is2(1) uak

(1)u2D . ~A8!

The most general form of the fermionic eigenvalue is

is2(1)5a01 (

k51

2

akuak(1)u21a12)

k51

2

uak(1)u2. ~A9!

After inserting this ansatz in Eq.~A8!, we obtain two sets of solutions for the coefficientsai0 ,ai12

andai j with i 51,2,j 51,2

is128 5 is121S c11c1c2

is122 is22ua2

(1)u2D ua1(1)u2

2,

~A10!

is228 5 is221S c21c1c2

is222 is12ua1

(1)u2D ua2(1)u2

2.

Remarkably, we havea125a2150. This allows us to write the nilpotent part ofisk28 as themodulus squared of a new anticommuting coordinate,

isk28 5 isk21ujk8u2. ~A11!

We solve Eq.~A10! for uap(1)u2, insert the results in Eq.~A4! and arrive at


heytlin

e one

st, we

. This


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uvp(1)u25

~sp12s118 !)n512 ~sp12 isn2!2

~sp12sq1!)n512 ~sp12 isn28 !2,

~A12!

uap(1)u252~ isp28 2 isp2!

~ isp22s118 !~ isp22 isq2!2

~ isp22 isq28 !2)n512 ~ isp22sn1!

, p,q51,2 , qÞp.

The structure of Eq.~A12! indicates the form of the solutions for groups of higher order as twere stated in Eq.~3.24!. They are checked by inserting them directly into the Gelfand–Tzeequations~3.23!. The algebra needed is, although tedious, straightforward and similar to thhere.

APPENDIX B: DERIVATION OF FORMULA „4.7…

The technique we use is an extension of the one developed in Appendix D of Ref. 1. Firrewrite the integral in terms ofQ functions. The left hand side reads

Lm~s!E mB~s8,s! f ~s18!d@s18#)k< l

Q~sk12sl18 !)l ,n

Q~sl18 2sn1!. ~B1!

The integration domain is now the real axis for all variables. The action ofLm(s) on the integralyields:

E SmB~s8,s! (i 51

k1

(j 51

k1211

2

21

~ is122si1!~si12sj 18 !f ~s18!)

k< lQ~sk12sl18 !)

l ,nQ~sl18 2sn1!D d@s18#

1E mB~s8,s! (i 51

k1 1

ism22si1

]

]si1)k< l

Q~sk12sl18 !)l ,n

Q~sl18 2sn1!d@s18#. ~B2!

We decompose the first term in partial fractions and find

E SmB~s8,s! (i 51

k1

(j 51

k121 1

2

21

~ is122si1!~ is122sj 18 !f ~s18!2Dk1

~s18! f ~s18!

3 (j 51

k121 1

is122sj 18

]

]sj 18

1

A2) i 51k1 ~si12sj 18 !

)k< l

Q~sk12sl18 !)l ,n

Q~sl18 2sn1!d@s18#

1E mB~s8,s! f ~s8!(i 51

k1 1

ism22si1

]

]si1)k< l

Q~sk12sl18 !)l ,n

Q~sl18 2sn1!d@s18#. ~B3!

An integration by parts yields

E mB~s8,s! (j 51

k121

~2Mm j~s18 ,s1!!)k< l

Q~sk12sl18 !)l ,n

Q~sl18 2sn1!d@s18#1E mB~s8,s! f ~s8!

3S (i 51

k1 1

ism22si1

]

]si11 (

j 51

k1211

ism22sj 18

]

]sj 18D)

k< lQ~sk12sl18 !)

l ,nQ~sl18 2sn1!d@s18#.

~B4!

The derivatives of theQ functions yieldd distributions. Upon integration of thed distribution thetwo terms in the last integral cancel each other. Hence the last term vanishes identicallycompletes the proof.


ausen

g the

t

era-

rsiona-

expres-


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APPENDIX C: ALTERNATIVE DERIVATIONS OF F22„s ,r …

We present two different alternative derivations. We do this in some detail, becthe calculations give helpful informations on the roˆle played by the radial Gelfand–Tzetlicoordinates.

First, we use an angular parametrization of the coset UOSp(2/2)/UOSp(1/2) by writinfirst column ofuPUOSp(2/2) as

u153A12uau2 cosq

A12uau2 sinq

1

&a

1

&a*

4 . ~C1!

This is a canonical way to parametrize the supersphereS1u2 that is isomorphic to the coseUOSp(2/2)/UOSp(1/2), see Ref. 31. It coincides with a special choice of theangular Gelfand–Tzetlin coordinates. The invariant measure is in these coordinates simply dm(u1)5da* dadq.Thus, one directly obtains the the volumeV(S1u2)50, see Ref. 31. In the parametrization of thmeasure byradial Gelfand–Tzetlin coordinates~4.3!, one has to perform the Grassmann integtion and to apply formula~4.7! to achieve this result.

Although we use a different coordinate system, we still take advantage of the recuformula ~3.1!. To use it in the parametrization~C1!, one has to solve the Gelfand–Tzetlin equtions ~3.21!–~3.23! for the eigenvalues. The unique solution of the bosonic equation~3.22! is

s118 5a01) i 51

2 ~si12a0!

is122a0uau2, a05

s111s21

22

s112s12

2cos 2q. ~C2!

The fermionic equation yields

is128 5 is121) i 51

2 ~si12 is12!

is122a0uau2. ~C3!

After inserting Eqs.~C2! and ~C3! and the measure dm(u1) into the recursion formula~3.1!, theGrassmann integration can be performed. Remarkably, we arrive at the denominator–freesion

F22~s,r !5G22E0

2p

dqexpS trg rs2z

21

z

2cos 2q D

3F S )i 51

2

~r i12 ir 12!~si12 is12!11

2 (i 51j 51

2

~sj 12 is12!~r i12 ir 12!D2

1

2 S ir 1221

2~r 111r 21! D ~s112s21!cos 2q2

z

8~ ir 122r 21!~s112s21!sin2 2qG .

~C4!

To make contact with Eq.~4.9! one has to realize that in Eq.~C4! an additional total derivativeappears in the integrand. This becomes obvious if one adds and subtractsz/4 cos 2q in the squarebracket of Eq.~C4!,


, in

ion oneven

and ofordi-re,ad onlyhesemula.

yto be

holds

etlin


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F22~s,r !5G22E0

2p

dq expS trg rs2z

21

z

2cos 2q D

3S )i 51

2

~r i12 ir 12!~si12 is12!11

2 (i 51j 51

2

~sj 12 is12!~r i12 ir 12!2z

4cos 2q D

1ir 122r 21

r 212r 11E

0

2p

dq]2

]~2q!2 expS trg rs2z

2~12cos 2q! D . ~C5!

While the first integral reproduces Eq.~4.9!, the second one vanishes identically. In generalperforming Grassmann integrations, one has to take care of boundary contributions.2,24 Thesecontributions can appear whenever even coordinates are shifted by nilpotents and the functintegrates does not have compact support.2 However, in our case the basis space is always giby an n dimensional sphere, i.e., by a compact manifoldwithout boundary. Thus in a properlychosen coordinate system, no boundary terms should appear. With regard to Eq.~C5! this means:the fact that the last term vanishes is a direct consequence of the compactness of the circlethe analyticity of the function, that we integrate. However, in the radial Gelfand–Tzetlin conates, only the moduli squared of the vectoru1 are determined. Therefore, not the whole sphebut only a (2n11)th segment of it is covered by Eq.~3.24!. In our case, not the circle but onlyquarter of it is parametrized. This is allowed since the supermatrix Bessel functions depenon the moduli squareduui1u2. Nevertheless, one has to ensure that the introduction of tartificial boundaries does not alter the result. To this end we use the following integration for

Let s11,s118 ,s21 be real and letj8,j8* be anticommuting. Furthermore, define

f ~s118 ,j,j* !5 f 0~s118 !1 f 1~s118 !uju2, ~C6!

with two analytic functionsf 0(s118 ), f 1(s118 ). Then the integral

I 5Es11

s21ds118 dj* dj f ~s118 ,j,j* ! ~C7!

transforms under a shift ofs118 by nilpotents

s118 5y1g~y!uju2 ~C8!

in the following way:

I 5Es11

s21dy dj* dj

]s118

]yf ~y~s118 !,j,j* !2@ f 0~s21!g~s21!2 f 0~s11!g~s11!#. ~C9!

The proof is by direct calculation. The second term in Eq.~C9! is often referred to as boundarterm. It can be viewed as the integral of a total derivative, i.e., an exact one-form, that hasadded to the integration measure for functions with noncompact support.24 For functions of anarbitrary number of commuting and anticommuting arguments, a similar integral formulawith additional boundary terms.2 In going from the canonical coordinates (q,a,a* ) to the radialones (s118 ,j18 ,j18* ), in principle boundary terms can arise, since the bosonic Gelfand–Tzeigenvalue~C2! contains nilpotents. However, the crucial quantity isg(y) in formula ~C9! which,in our case, is given by

g~s118 !5) i 51

2 ~si12s118 !

is122s118. ~C10!


hen

e Eq.

s.

ed in

at theametri-. Cer-

ed thisotal de-atrix


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Thus,g(s118 ) causes the boundary term to vanish ats11 ands21. It is the product structure of theleft-hand side of Eq.~3.24! which always guarantees the vanishing of the boundary terms, wone goes from the Cartesian set of coordinates to the radial Gelfand–Tzetlin coordinates.

Therefore, one may think of the denominators, arising in Eqs.~4.4! and~4.5!, as belonging tototal derivatives of functions, which vanish at the boundaries. Keeping this in mind we deriv~4.9! in yet another way. We expand the product

)q51

k1

~ ism22sq1!5 (n50

k1 1

n!~ ism22sj 18 !n

]n

]~sj 18 !n )q51

k1

~sj 18 2sq1!, ~C11!

and insert it into the integral

Es11

s21¯E

s(k121)1

sk11mB~s18 ,s1!Km j~s18 ,s1! f ~s18!d@s18#

5Es11

s21¯E

s(k121)1

sk11mB~s18 ,s1!)

n51

k1

~ ism22sn1!Mm j~s18 ,s1! f ~s18! d@s18#. ~C12!

We can remove the term proportional to (ism22sj 18 )22 in the integrand by an integration by partThrough the expansion~C11!, the vanishing of the boundary terms is assured. We arrive at

Km j~s18 ,s1!52 (n52

k1 1

n!~ ism22sj 18 !n22

]n

]~sj 18 !n )q51

k1

~sj 18 2sq1!

1)q51

k1 ~ ism22sq1!2)q51k1 ~sj 18 2sq1!

ism22sj 18

3S 1

2 (q51

k1 1

ism22sq12

1

2 (q51

k1 1

sj 18 2sq12 (

q51qÞ j

k1 1

sj 18 2sq182

]

]sj 18 D . ~C13!

We notice that in the new operatorKm j(s18 ,s1) all denominators of the type (ism22sj 18 )21 havedisappeared. Fork152, we calculate

K1152~ is121s118 2s112s21!]

]sq18, ~C14!

which can be inserted into Eq.~4.4! by using the definition~C12!. Finally, the result~4.9! isreproduced by the substitution

s118 5s111s21

22

s112s12

2cos 2q, ~C15!

see Eq.~C2!. In other words, we have seen that the result of this procedure is summarizformula ~4.7!.

Finally, some remarks are in order: First, from this discussion, one might conclude thradial Gelfand–Tzetlin coordinates are less adapted to the problem than the canonical parzation ~C1!, because, in the latter, no denominators appear. We stress that this is not truetainly, the denominators appear due to the shift of the bosonic variable by nilpotents in Eq.~C2!.However, the difficulty in deriving Eq.~4.9! is the identification of the different parts of thintegrand after the Grassmann integration. Some of them belong to total derivatives anproblem exists in both parametrizations. Second, we emphasize that the appearance of trivatives in the integrand is not a peculiarity of supersymmetry. Already in Ref. 1 where the m


arancerpre-

to Eqs.

sult

to Eq.

t

we


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Bessel functions in ordinary space were treated we had to solve a similar problem. The appeof these total derivatives is an intrinsic property of the recursion formula. A geometrical intetation of this phenomenon is highly desired.

APPENDIX D: DETAILS FOR THE DERIVATION OF F34„À is ,r …

We always consider the case that one matrix has an additional degeneracy according~5.11! and ~5.13!. We introduce the notation

Si j 5~si12 isj 2!, Ri j 5~r i12 ir j 2!. ~D1!

Due to the degeneracy,F24(2 is, r ) simplifies enormously as compared to the general re~5.10!. We insert it into the recursion formula~3.1! and do the trivial integral over the O~2!subgroup. After performing the Grassmann integrals we arrive at an expression similar~5.7!,

F34~2 is,r !54G34exp~ tr r 2s21r 11~s111s21!!E dmB~s18 ,s1!)i 51

2

R1i )j 51

3

Sji

3F S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D S 4)i 51

2

Ri122(k51

3

R21Sk121

12(j 51

2

M1 j

~s18 ,s1!D S 4)i 51

2

Ri222(k51

3

R22Sk22112(

j 51

2

M2 j~s18 ,s1!D1S 1

D22~ ir 2!D2

3~ is2!1

1

D23~ ir 2!D2

4~ is2! D3(

j 51

2 S 4

is122sj 18M2 j~s18 ,s1!2

4

is222sj 18M1 j~s18 ,s1! D

2S 1

D22~ ir 2!D2

3~ is2!1

1

D23~ ir 2!D2

3~ is2! D )k51j 51

22

isk22sj 181

2

D23~ ir 2!D2

4~ is2!

3S trg r 1r 112(i 51

2

Si221D (

j 51

2

M1 j~s18 ,s1!22

D23~ ir 2!D2

4~ is2!

3S trg r 1r 112(i 51

2

Si121D (

j 51

2

M2 j~s18 ,s1!

24

D23~ ir 2!D2

3~ is2! (k51

3

)j 51

2

R2 jSk j21Gexp~~s218 1s118 !~r 212r 11!!1~ ir 12↔ ir 22!.

~D2!

Formulas~5.9! and~4.7! are needed to remove the denominators, in a way similar as forF24(s,r ).A single sum( j 51

2 M1 j (s18 ,s1) transforms according to formula~4.7!. Moreover, we observe thaparts of Eq.~D2! together with the product( j 51

2 M1 j

(s18 ,s1)(k512 M2k(s18 ,s1) yield exactly the

integrand of formula~5.9!. Thus, it can be transformed accordingly. After rearranging terms,arrive at the result~5.12!.


egen-

ng

uared

ntegral


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APPENDIX E: DETAILS FOR THE DERIVATION OF F44„À is ,r …

For the recursion, we needF34(s,r ) with degenerater 5diag(r11,r 21,r 21) according to Eq.~5.11!. Using the integral representation~5.14! for F3

(1)(2 is18 , r 1) we find the helpful identity

]

]si18

]

]sj 18exp~2r 21 tr s18!F3

(1)~2 is18 , r 1!51

2

1

si18 2sj 18S ]

]si182

]

]sj 18Dexp~2r 21trs18!F3

(1)~2 is18 , r 1!.

~E1!

We stress that this relation, which is crucial in the derivation, only holds, because of the deracy in the matrixr 1 . Employing Eq.~5.14! and another identity,

(i 51

3]

]si18exp~2r 21tr s18!F3

(1)~2 is18 , r 1!5~r 112r 21!exp~2r 21 tr s18!F3(1)~2 is18 , r 1!. ~E2!

We insertF34(s8, r ) into the recursion formula~3.1!. We then can arrange the terms emergifrom the Grassmann integration in a way similar to the former cases. We obtain

F44~2 is,r !54G44exp~ tr r 2s21r 11 tr s1!E dmB~s18 ,s1!)i 51

2

R2i)j 51

4

Sji

3F S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D3S 8R21R11

2 24R11R21(k51

4

Sk12114(

j 51

3 S R212]→

]sj 18D M1 j

→~s18 ,s1!

3S 8R22R122 24R12R22(

k51

4

Sk22114(

j 51

3 S R222]→

]sj 18D M2 j~s18 ,s1!D

116

D23~ ir 2!D2

4~ is2! (j 51

3

M1 j→~s18 ,s1!S 1

2R11R12S trg r 2(

i 51

4

Si121D

1S r 212r 112]

]sj 18D S R22R121R11R121R11R211

1

2~R121R22!(

i 51

4

Si121D D

216

D23~ ir 2!D2

4~ is2! (j 51

3

M2 j→~s18 ,s1!S 1

2R11R12S trg r 2(

i 51

4

Si221D

1S r 212r 112]

]sj 18D S R11R211R11R121R12R221

1

2~R121R22!(

i 51

4

Si221D D

216

D23~ ir 2!D2

3~ is2! (k51

4

)i , j

2

Ri j Sk j21exp~2r 11 tr s18!F3

(1)~2 is18 , r 1!1C~s,r !

1~ ir 12↔ ir 22!. ~E3!

Again, all operators with an arrow are understood to act only onto the term outside the sqbracket, i.e., onto exp(2r11 tr s18)F3

(1)(2 is18 , r 1). In the functionC(s,r ), we summarized theterms that are expected to arise due to noncommutativity of some operators acting on the iand some operators acting under the integral. The last two lines in formula~5.9! are examples ofsuch terms


sel


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C~s,r !54G44exp~ tr r 2s21r 11trs1!E dmB~s18 ,s1!)i 51

2

R2i)j 51

4

Sji F S 1

D22~ ir 2!D2

3~ is2!

11

D23~ ir 2!D2

4~ is2!D (

j 51

3 S R11R121~R111R12!~r 212r 11!2~R111R12!]→

]sj 18D

3S 16

is122sj 18M2 j~s18 ,s1!2

16

is222sj 18M1 j~s18 ,s1!D

2S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2!D

3)k51

2

)j 51

3 S R11R121~R111R12!~r 212r 11!2~R111R12!]→

]sj 18D

38

isk22sj 182S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2!D

3(j ,k

3 S ~r 212r 11!2]→

]sj 18D S ~r 212r 11!2

]→

]sk18D M1 j

→M2k28

D22~ ir 2!D2

3~ is2!

3(j 51

3 S ~r 212r 11!2]→

]sj 18D S (

k51

4

Sk221M j 1

→2 (k51

4

Sk121M j 2D

18

D22~ ir 2!D2

3~ is2!S (

iÞ jM j 1M j 2

→S ]→

]si182

]→

]sj 18D D Gexp~2r 11trs18!F3

(1)~2 is18 , r 1!.

~E4!

In order to evaluate Eqs.~E3! and ~E4! we need some more properties of the matrix Besfunction F4

(1)(2 is1 ,r 1). We investigate the action ofLk on F4(1)(2 is1 ,r 1) using the integral

representation~5.15!.After a straightforward calculation involving an integration by parts we find

Lk exp~2r 11tr s1!F4(1)~2 is1 ,r 1!

5(i 51

41

isk22si1S ~r 212r 11!

21~r 212r 11!]

]si1Dexp~2r 11 tr s1!F4

(1)~2 is1 ,r 1!. ~E5!

Now Eqs.~E3! and ~E4! can be enormously simplified by the observation that

~~r 212r 11!Lk2Lk!exp~2r 11trs1!F4(1)~2 is1 ,r 1!50, ~E6!

which follows directly from Eq.~E5!. We find for Eq.~E3!



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F44~2 is,r !54G44exp~ tr r 2s21r 11trs1!E dmB~s18 ,s1!)i 51

2

R1i )k51

4

Ski

3F S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D S R12R11S 8R12R2224R22(k51

4

Sk221D

3S 8R11R2124R21(k51

4

Sk12114(

j 51

3

M1 j→~s18 ,s1!D

1R12R11S 8R11R2124R21(k51

4

Sk121D

3S 8R22R1224R22(k51

4

Sk22114(

j 51

3

M2 j~s18 ,s1!D1(

j ,iR11S r 212r 112

]→

]sj 18D M1i

→~s18 ,s1!M2 j~s18 ,s1!

1(j ,i

R12S r 212r 112]→

]si18D M1i

→~s18 ,s1!M2 j~s18 ,s1!

1(j ,i

R21R22M1 j→~s18 ,s1!M2i~s18 ,s1!1

8

D23~ ir 2!D2

4~ is2!R11R12

3S trg r 2(i 51

4

Si121D (

j 51

3

~M1 j~s18 ,s1!2M2 j~s18 ,s1!!

216

D23~ ir 2!D2

3~ is2! (k51

4

)i , j

2

Ri j Sk j21G

3exp~2r 11tr s18!F3(1)~2 is18 , r 1!1C~s,r !1~ ir 12↔ ir 22!. ~E7!

The terms contained in Eq.~E4! simplify, too. We arrive at

C~s,r !54G44exp~ tr r 2s21r 11 tr s1!E dmB~s18 ,s1!)i 51

2

R1i)j 51

4

Sji

3F S 1

D22~ ir 2!D2

3~ is2!1

1

D23~ ir 2!D2

4~ is2! D3(

j 51

3 S R11R121~R111R12!~r 212r 11!2~R111R12!]→

]sj 18D

3S 16

is122sj 18M2 j~s18 ,s1!2

16

is222sj 18M1 j~s18 ,s1! D

2S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D3)

k51j 51

2 S R11R121~R111R12!~r 212r 11!2~R111R12!]→

]sj 18D

38

isk22sj 181

8

D22~ ir 2!D2

3~ is2! S (iÞ jM j 1

→~s18 ,s1!M j 2~s18 ,s1!S ]→

]si182

]→

]sj 18D D G

3exp~2r 11tr s18!F3(1)~2 is18 , r 1!. ~E8!


eigen-e

are

o this

tills


Downloaded 0

To further evaluate the expressions, we can now invoke a symmetry argument between thevaluesr 11 and r 21, respectively. Since the productR11R12 appears as a prefactor in front of thintegral~E7!, R21R22 must also appear as a prefactor in the final result. Thus, all terms in Eqs.~E7!and~E8! which do not containR21R22 as a factor must yield zero. The remaining terms whichproportional toR21R22 can again be treated using formulas~4.7! and ~5.9!. However, we want toshow explicitly that this line of arguing is correct and that the other terms indeed vanish. Tend, we need an additional identity to treat the operator product

(j 51

2]→

]sj 18M

1 j~s18 ,s1!(

k51

2

M2k~s18 ,s1!. ~E9!

The required identity is given by the following formula: The same conditions as for formula~4.7!apply, furthermore we define

Lm→~s!Ln~s!5(

i , j

1

~ ism22si1!~ isn22sj 1!

]3

]si1]sj 12 1

1

2 (i , j

1

~ ism22si1!~ isn22sj 1!

3]

]si1(kÞ j

k1 1

sj 12sk1S ]

]sj 12

]

]sk1D . ~E10!

Then we have

Lm→~s!Ln~s!E

s11

s21¯E

s(k121)1

sk11mB~s8,s!d@s18# f ~s18!

5Es11

s21¯E

s(k121)1

sk11 F (j 51

k121

(i 51

k121

Mmi→ ~s18 ,s1!

]→

]sj 18Mn j~s18 ,s1! f ~s18!2

1

isn22 ism2

3 (i 51

k121 S 1

ism22si18

]→

]si18Mni~s18 ,s1!2

1

isn22si18

]→

]si18Mmi~s18 ,s1! D

21

2 (kÞ l

1

~ ism22sk18 !~ isn22sk18 !~sk18 2sl18 !2

]→

]sk1

11

2 (kÞ l

1

~ ism22sk18 !~ isn22sl18 !~sk18 2sl18 !2

]→

]sk1G f ~s18!mB~s8,s!d@s18#. ~E11!

The proof is similar to the one of formula~4.7!. We notice that the arrow in Eq.~E10! is usedslightly differently than previously. The operatorLm

→(s) acts also on a part ofLn(s). This is notconsistent with the definition in Eq.~5.6!. However, since this is obvious where it occurs, we suse the same arrow. We can now translate the left-hand side of Eq.~E8! into an expression in termof F4

(1)(2 is,r ). After some further manipulations involving the identities in Eqs.~E6!, ~E1!, and~E2! we arrive at



Downloaded 0

F44~2 is,r !5G44exp~ tr r 2s21r 11 tr s1!)i , j

2

Rji )k51

4

SkiF S 1

D22~ ir 2!D2

2~ is2!1

1

D23~ ir 2!D2

3~ is2! D3S 8R12R2224R22(

k51

4

Sk22124L2~s!D S 8R11R2124R21(

k51

4

Sk12124L1~s!D

28

D23~ ir 2!D2

4~ is2!S trg r 2(

i 51

4

Si121D ~L1~s!2L2~s!!

216

D23~ ir 2!D2

3~ is2! (k51

4

)j 51

2

R2 jSk j21Gexp~2r 11tr s18!F4

(1)~s18 , r 1!~ ir 12↔ ir 22!.

~E12!

After rearranging terms this yields the result~5.18! for F44(2 is,r ).

1T. Guhr and H. Kohler, J. Math. Phys.43, 2707~2002!, preceding paper; math-ph/0011007.2F. A. Berezin,Introduction to Superanalysis, ~Reidel, Dordrecht, 1987!, Vol. 9.3V. C. Kac, Commun. Math. Phys.53, 31 ~1977!.4V. C. Kac, Adv. Math.26, 8 ~1977!.5K. B. Efetov, Adv. Phys.32, 53 ~1983!.6K. B. Efetov,Supersymmetry in Disorder and Chaos, ~Cambridge University Press, Cambridge, 1997!.7J. J. M. Verbaarschot and M. R. Zirnbauer, Ann. Phys.~N.Y.! 158, 78 ~1984!.8J. J. M. Verbaarschot, H. A. Weidenmu¨ller, and M. R. Zirnbauer, Phys. Rep.129, 367 ~1985!.9T. Guhr, J. Math. Phys.32, 336 ~1991!.

10C. Itzykson and J. B. Zuber, J. Math. Phys.21, 411 ~1980!.11T. Guhr, Commun. Math. Phys.176, 555 ~1996!.12I. M. Gelfand and M. L. Tzetlin, Dokl. Akad. Nauk SSSR71, 825 ~1950!.13S. L. Shatashvili, Commun. Math. Phys.154, 421 ~1993!.14J. Alfaro, R. Medina, and L. Urrutia, J. Math. Phys.36, 3085~1995!.15T. Guhr and H. A. Weidenmu¨ller, Ann. Phys.~N.Y.! 199, 412 ~1990!.16M. R. Zirnbauer, Commun. Math. Phys.141, 503 ~1991!.17M. R. Zirnbauer, Phys. Rev. Lett.69, 1584~1992!.18A. D. Mirlin, A. Mu ller-Groeling, and M. R. Zirnbauer, Ann. Phys.~N.Y.! 236, 325 ~1994!.19M. L. Mehta,Random Matrices, 2nd ed.~Academic, San Diego, 1991!.20F. Haake,Quantum Signatures of Chaos, 2nd ed.~Springer, Berlin, 2001!.21T. Guhr, A. Muller-Groeling, and H. A. Weidenmu¨ller, Phys. Rep.299, 189 ~1998!.22T. Guhr, Ann. Phys.~N.Y.! 250, 145 ~1996!.23F. J. Wegner, Z. Phys. B: Condens. Matter49, 297 ~1983!.24M. J. Rothstein, Trans. Am. Math. Soc.299, 387 ~1987!.25T. Guhr and H. Kohler~unpublished!.26M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions, 9th ed.~Dover, New York, 1970!.27 Harish-Chandra, Am. J. Math.80, 241 ~1958!.28R. J. Muirhead,Aspects of Multivariate Statistical Theory, ~Wiley, New York, 1982!.29P. J. Forrester and T. Nagao, Nucl. Phys. B532, 733 ~1998!.30N. Datta and H. Kunz, cond-mat/0006488.31M. R. Zirnbauer, J. Math. Phys.37, 4986~1996!.


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Recursive construction for a class of radial functions. II