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THE LOBACHEVSKY SPACES IN THE NON-COMMUTATIVE MINKOVSKY SPACES
M. A. Olshanetsky, V.-B. K. Rogov
Institute Theor. Exper. Physics, Moscow, Russia Moscow State University of Communications (MIIT), Russia
The final goal of this work is the solutions of the Klein-Gordon equations on NCMS in terms of the horospheric coordinates [1], [2]. By analogy with the classical case, the solutions are products of g-cylindric functions. The reduction of these solutions to NCLS, NCILS and the non-commutative cone is straightforward.
Notations.
Classical variables are denoted by small letters, while their non-commutative deformation (quantization) by capital letters. We do not introduce a special notation for the non-commutative multiplication.
The coordinates (xi,x2,xs,x4) or (yo, yiiUiiU'i.) of the Minkowski space M4 we identify with the matrix elements of the matrix x
The choice 1 or i in front of ya defines the signature of the Minkowski space. The generators of the non-commutative Minkowski space we also arrange in the matrix form
x-(££)- (2) The deformation parameter is q = exp9 € (0,1], or q — expi9, (|g| = 1).
1 Horospheric coordinates on the classical Minkowski spaces
There are two types of Minkowski spaces with the signature (+, —, —, —) and (+, +, —, —). The first one allows us to describe the Lobachevsky space L and the Imaginary Lobachevsky space IL. We will consider first of them.
1.1 Minkowski space in the horospheric description
The Minkowski space M1,3 can be identified with the space of Hermitian matrices M1’3 = {x € Mate | = x} j (®i — xi, X2 = X3 , X4 = X4).
x =
X\ X2 X3 X4
= y0Id + ^ eayaaa , ea = 1, or i,
The metric is ds2 — det(dx) = dx\dx$ — dx2dx$. Another set of coordinates is ya, (a = 0,... 3) corresponds to the choice ea = 1 in (1)
3
x = £^> oq = Id.
a=0
It leads to the metric
ds2 = dy2-YJ dv] ■ j=i
The group SL(2, C) is the double covering of the proper Lorentz group SOo(l, 3) and acts on the Minkowski space M1,3 = {yo,..., y3} as
x-»9tx9, 9 € SL(2, C), (3)
where is the Hermitian conjugated matrix. The action preserves
det x. = yl-y\-yl-yl = xix± - X3X2
and thereby the metric on M1,3.
The time-like part M1,3+ of M1,3 corresponds to the matrices with detx > 0, while
detx < 0 corresponds to the space-like part M1,3~. The equation detx = 0 selects the light
C1’3 = {x : det x = x\x± — X2X$ = 0} . (4)
We introduce the horospheric coordinates x ~ (r, h,z,z). If x\ ^ 0 then
x\ = rh , X2 = rhz , £3 = rhz , (5)
X4 = r(h\z\2 + eh-1).
Here
z E C, h £ R\0, e = ±1,0, r2e = detx.
and
z = X2X^X , z = X3X11, r = i/| detx|, for det x^0,
, _ J XI (I det xl)-1/2 for £ = ±1,
\ x\ for e = 0.
The case e — 1 corresponds to the time-like part of M1,3, £ = — 1 corresponds to the space-like
part and e = 0 to the light-cone C1,3.
The horospheric coordinates on the light-cone C1,3 are (h, z, z)
x\ = h, X2 = hz, £3 = £2 , X4 = h\z\2 . (6)
To describe the case x\ — 0 we put e = —1, h —> 0, r < 00 and z —>■ 00 such that lim hz =
exp (it), a: 2 = r exp(ii) and X4 takes an arbitrary real value. Thus, the horospheric description
has the form
(r, exp(ii), £4), x2 = rexp(it) £3 = r exp(—it).
Consider the commutative algebra <S(M1,3) of the Schwartz functions on M1,3. The invariant integral with respect to the SL(2,C) action on <S(M1,3)
1(f) = I f(xi,X2, x3, x4)dxidx2dxsdxi takes the form in the horospheric coordinates
1(g) = J g(z, z, h,r)r3hdrdhdzdz , g 6 5(M1,3).
1.2 Homogeneous spaces, embedded in M1’3
The action of SL(2, C) (3) leads to the foliation of M1,3. The orbits are defined by fixing detx. The quadric
L = {det x = r% > 0, x\ > 0}
is the upper sheet of the two-sheeted hyperboloid. It is a model of the Lobachevsky space. The metric on L is the restriction of the invariant metric dx\dx± — dx2dx^ on r = const. In what follows we assume ro = 1. The horospheric coordinates on L have the restrictions h > 0. Since SU(2) leaves the point yo = 1, ya — 0 the Lobachevsky space is the coset L ~ SL(2, C)/SU(2) ~ SO0(l, 3)/SO(3).
Consider the commutative algebra <S(L) of Schwartz functions on L. Functions from <S(L) are infinitely differentiable with all derivatives tending to zero when \z\ —»■ oo, h —>• oo, h —>• 0 faster than any power. Let Ir2 be the ideal in <S(M1,3) generated by /(detx — rfi) = 0. The algebra <S(L) can be described as the factor-algebra <S(M1,3)//x with the additional condition x\ > 0. In the similar way we describe the upper sheet of the light-cone C1,3 as ¿>(M1,3)//o. The horospheric coordinates (5) being restricted on C1,3+ satisfy the condition (r = 1, h > 0, e = 0). C1,3+ is the quotient SL(2, C)/Bc, where Be is the subgroup of the form
The space
IL = {det x = —1}
is called the Imaginary Lobachevsky space. The corresponding quadric is y\ — Va = "1-It is the de Sitter space:
IL~SL(2,C)/SU(1,1) ~SO0(l,3)/SO0(l,2),
since
g^a3g = cr3 , for g € SU(1,1).
As before, >S(IL) ~ <S(M1,3)//_i, but in contrast with the L and C+ the horospheric radius h of IL can take an arbitrary value h € R\0. We partially compactify IL with respect to the coordinate h. Two ’’limiting” spaces = {/i -> ±00} are called absolutes. It follows from (4) and (6) that can be considered as the projectivization of the cone C1,3. The both absolutes are homeomorphic to C and therefore can be compactify to E ~ CP1. Note, that while H are two components of the boundary of the IL, H+ is the boundary of C1,3+ and the L.
2 Laplace operator and its eigenfunctions
In this work we generalize to the non-commutative case the following facts concerning the eigenfunctions of the Laplace operator.
The solutions of the Klein-Gordon equation on M4
Afu(xi,X2,X3,X4) = V2fv(x 1,12, £3,0:4),
_ d2 d2
dx\dx4 8x38x2
are the exponents
fl/{xl,X2,X3,XA) = exp(^£) , (£c) = ^2(iXi ,
1/2 = £l£l - 66 ■
We will consider A and its eigenfunctions in the horospheric coordinates. The metric on M1,3 in the horospheric coordinates takes the form
ds2 = gjkdxjdxk = edr2 — er2h~2dh2 — r2h2dzdz.
Then one can rewrite A = ^det^1/2o>jgJ'fc(det g)l^2dk and we come the eigenvalue problem
r~2
'2 d2 d 2 2 <92 о 9
dh?+ dh+ £ dzdz ~r dr
fv{z,h,z\r) = v2fv(z,h,z;r). (7)
Let Zv(x) be a cylindric function. We will prove the non-commutative analog of the following statement
Proposition 1 The basic harmonics of the eigen-value problem (7) are
fu(z,h,z\r) = r~lh~l exp(i/j,z + iflz) x Za(rv)Za(2iell2\n\h~l),e = ±1, (8)
ft,(z, h, z\ r) = h01^1 exp(i/j,z + ip,z), e = 0, a2 = u2 + 1,
where fj,, a G C.
It follows from (7) that the restrictions of the Klein-Gordon equation to the homogeneous spaces assume the form
.2 a2 , 9 , . . a2 \ ,=, _,,
hav+3ah+ich' mai)Mh'z'~z) = (" -VMh,z,z),
L —^ £ — 1, IL —^ £ = —1 .
Thus, we come to the following statement
Corollary 1 The basic harmonics on L, IL and the light-cone C1,3 are
f„(z, h, z) = h~l exp(i/j,z + ifiz)Zv2_x (2ie 2 |/z|/i_1), e — ±1, (9)
and
f„(z,h,z) = /ia_1 exp(ifiz + ifiz), £ = 0, a2 = i/2 + 1. (10)
• • • 13
3 Non-commutative 4d Minkowski space ML’
UjLl
3.1 Definition
We define an algebra generated by matrix elements of (2).
• • • 13
Definition 1 The non-commutative 4d Minkowski space ML’ , 0 < q < 1, S £ N is the unital
0,1/
associative algebra with the anti-involution * and four generators Xj, j = with the
quadratic relations
XlX3 = q~SX3Xi , XxX2 = q*X2X 1,
[X2,X3] = qS-2(l-q2)X1X4, (11)
*2*4 = qS~2XAX2 , X3*4 = q~5+2X4*3 ,
[X\,*4] = 0,x* = Xi, x% = x3, x; = x4.
This space was described in [3]. Following this approach we cast the relations in in the
form of the reflection equation. Consider the basis in Mat(2)
E' = {1 ¡0 -*-(!! £)■*"(! o)’Ei=(o !)■
Define two ii-rnatrices
R(q) = q ^(Ei E\ + E4 <8> E4) + (Ei ® E4 + E4 <8> Ei) + q 1 (1 — q2)E3 <2> E2 ,
i?(2)(g) = {Ei ®Ei+E4® Ei) + q5~l(Ei ®E4 + E4® Ei).
It can be checked straightforwardly that the relations (11) are equivalent to the reflection equation
R(q)X^R^(q)X^ = ¿2(2>(g)XWR\q),
where X*1) = X®Id and X<2) = Id®X and
(v) = q~1(Ei ® Ei + e4 ® e4)
+ (E\ ® E4 + E4 <8> Ei) + q *(1 — q2)E2 <8> E3 .
13 * •
The algebra M5’? has two independent Casimir elements
Kx = Xf-2X54 , K2 = XiX4 - q~sX3X2 . (12)
The Casimir operator K2, see (12), is the quantum determinant K2 = det(/ X. In an irreducible module over Ml’ this operator is a scalar: K2 = sr2 £ R. It allows us to define the time-like part (e = 1), the space-like part M$~, (e = — 1), and the light cone
(e = 0).
• 13
3.2 Quantum Lorentz group action on M^’
We start with a pair of the standard Uq(SL2) Hopf algebra [4]. The first one is generated by A, B, C, D and the unit 1 with relations
AD — DA = 1, AB = qBA, BD = qDB,
AC = q~lCA, CD = q~lDC, (13)
There is a copy of this algebra £Y*(SL2) generated by A*,B*,C*,D* with the relations coming from (13) U*V* = (VU)*. They commute with A,B,C,D. The pair Uq(SL2), U*(SL2) forms a Hopf algebra ZYy^(SL2), where the coproduct and the antipode are twisted in the consistent way
A{A) = A® A,
A {B) = A®B + B®D(A*)S, (14)
A(C) =A®C+C® D(A*)~S.
A B \ _ ( D —q~1(A*)~sB
C D )~\ —q(A*)sC A
The counit on f7q^(SL(2,C)) assumes the form
e(A) = 1, e(B,C) = 0. (15)
There is the Casimir element in (7<7^(SL(2, C)) which commutes with any u £ Uq(S'L2 (C)):
o [q~l + q)(A2 + A~2) - A 1 , „ . , ,
üq - ------W1 - W---------- 2'{BC + CB) ■ (16)
it is a right module over the Hopf algebra UqS\SL(2, C)).
We define the action of the quantum group W^(SL(2,C)) on mJ’3:
Xi X2\ A=( q*X 1 q~hx2 \ x3 X4 J- [ q\xz q-hx4 ) ’ *1 *2 \ B = f 0 X\
X3 XA J V 0 *3
Xl X2 ) r = ( X2 0
x3 x4 ; v ^4 0
*1 X2 \ .* ( q~X1 q~X2
A* = . (17)
X3 X.4 / y q s X3 q * X4 J
1 3
The direct calculations show that the commutation relations in ML’ are compatible with the
coproduct in ^s^(SL(2, C)).
Similarly, one can define the left action of WgS^(SL(2,C)) on Let
w(m, k, I, n) = X?X$X[X$
1 3
be the ordered monomial. Define the Schwartz space <S(M5’9) as the series with the rapidly decreasing coefficients
= {\f(XX\, X4, X2)X = ^ Min)i am,k,l,n G C} , (18)
with
\am,k,l,n\ < (1 + W -\- k + I +71 Y,
for any j 6 N, when \m\, |fc|, |/|, |n| —> oo .
Proposition 2 The Jackson integral
(/) = J dq2X3dq2Xidq2X4dq2X2tf{Xi,X2,X3,X4)t (19)
is an invariant functional on ¿’(Ml’3) with respect to the following action ofUqS\SL(2, C)); (f.u) = e(u)(f) , where e(u) is the counit (15).
4 Horospheric description
4.1 Horospheric generators
We introduce another set of generators - the non-commutative analog of the horospheric coordinates (Z*,H,Z,R), {H* = H, R* = R, (Z*)* = Z)
Xi = RH, X2 = RHZ , X3 = RZ*H, (20)
XA = R{Z*HZ + eH~x), £ = ±1,0. (21)
The defining relations
ZH — q~sHZ, Z*H = qsHZ* ,
[R,H] = [R,Z] = [R,Z*}=:Q,
ZZ* = q25~2Z*Z - eqs-2(1 - q2)H~2 . (22)
yield the relations (11). The Casimir elements are
K2 = eR2 , Ki = R25-2H5~2{Z*HZ + eH~1)i . (23)
The inverse relations assume the form
H = R~xX\, Z = X^X2 , Z* == X3X^, *
R = eK\.
In terms of the horospheric generators the action of ZY^(SL(2, C)) takes the form
Z*.A = z*, H.A = qïH, Z.A = q~lz,
Z*.A* = qMf1Z\ H.A* = q6-^H, Z.A* = Z,
Z*.B = 0, H.B = 0, Z.B = q- 2,
Z*.C = ql~ôH-2 , H.C = HZ, Z.C = -qïZ2,
R.A = R, R.A* = R, R.B = 0, R.C = 0. (24)
It follows from these relations that R is invariant with respect to the ^s^(SL(2. C)) action R.u = e(u)R.
Define the analog of the Schwartz space <S(Mj’3), see (18), in terms of the ordered monomial w(m,k,n) — Z*mHkZn. Since R is a center element its position is irrelevant. Let
tf {Z -, H, Z, -R)t — ^ k, n)R , cim,k,n,i SC. (25)
For <S(M^’3) the coefficients satisfy the condition
\am,k,n,i\ < (1 +m2 + k2 + l2 + n2)j,
for any j € N, when |m|, |fc|, \l\, |n| -> oo.
The invariant integral (19) is well defined functional on (25). It assumes the form
Iq2{f) = J dq2Z*dq2Hdq2Zdq2Rtf(Z*,H,Z,R)Ht.
4.2 Homogeneous spaces
Consider an irreducible representation of algebra (22). Then one can fix the Casimir operator (23) K2 = eR2, R2 = r2 G R+. It allows us to define the non-commutative analog of Lobachevsky spaces and the cone. Let us fix the ideal Ie = {K2 — er2 = 0}. Then
S(M^q)/h ~ L (e = 1),
~ IL (e = -l),
~ CS;93 (e = o).
As we observed above the action of the quantum Lorentz group preserves these spaces. It justifies the notion of homogeneous spaces in the noncommutative situation.
1 3
We can directly define their generators using the horospheric description of M^’.
Definition 2 The non-commutative Lobachevsky space L(Hz), the non-commutative Imaginary Lobachevsky space IL^q (dS3) and the non-commutative cone C*’5 are the associative unital algebras with an anti-involution and the defining relations
ZH = q-SHZ, Z*H = qSHZ* ,
ZZ* = q25-2Z*Z - eq5~2( 1 - q2)H~2 .
(Z)* = Z*, H* = H, ll
'6,q
H3 ~ e = 1, dS3 ~ e — — 1, Cl ’¡I ~ e = 0.
In addition we define the non-commutative absolute.
Definition 3 The non-commutative absolute is the associative algebra with two gener-
ators and the commutation relation
ZZ* = q~2+2SZ*Z. (26)
5 The Laplace operator and its eigen-functions
5.1 The Laplace operator on
Consider the action of W^(SL(2, C)) on the ordered monomials w(m, k, l,n) = X^X^X^X^. It follows from (17) that
m+k — l — n
w{m, k) Z, n).A — q 2 tu(ra, ,
(<5— l)(m — fc-H — n)
w(m,k,l,n).A = q * w(m,kj,n),
w(m,k,l,n).B = (27)
m+k-t- + ¿(„_1)1---+ ^ _|_
1 — q
rn+fc-5l+3n+5 sn /|n II') 1 — q21 , i i 1 \
q2 l+ri+i) —l-w(m + l^k,l - l,n),
l — ql
r
i 2k
w(m,k,l,n).C = q ^ ±-+<5n_^—^r-w(m,k — l,l,n + 1)
1 — qz
-3ro-3fc+3i-n+3 I rn I I „-I 1 — q m . ill -, \
+ q 2 +"'.fe l+n>—--------—w(m — l,k,l + l,n).
I — qz
1 3
Introduce the group-like operator M that acts on Mtf’ as
? i» W («**«}* V (28)
XZ XA ) ?2l3 92l4 j V '
It has the following properties
M* = M, A (M) = M0AT, e(M) = 1,
S(M) =M~\
Evidently, M commutes with A,B,C and A*. Define the Hopf algebra £/g(GL(2,C)) generated by A,B,C and M. It is the quantum deformation of the classical algebra GL(2, C). Let i i
o . — q^M)2
'= ---(9-1 _ q)2------• (29)
Consider the following Casimir element of W9(GL(2,C))
Aq = - ilqtM]R~2. (30)
This operator is the quantum analog of the Laplace operator A, see (7). Define the partial
differentiation acting on w(m,k,l,n) in such a way that it does not break the ordering. It means, in particular, that the differentiation of the ordered monomial with respect, for example, X\ takes the form
1 — q2k
Dxxwim^k^l^n) = -------2-w(m,k — 1 ,l,n).
Let Txf(X) = f(qX).
ij~t /3 T rtrrxlnrtsi ■#/»/»> s\nrt
• • • 13
Proposition 3 The action of the Laplace operator on Mr’ assumes the form
1 (rp— lrp— lrp rp— 1 rp—lrp rp rjl 1
9 \1Xi1X2lx31Xi J-Xi1X21X31X4:
f(X3,XuXA,X2).Aq= (31)
1
(.q-q-1)2
I rp—lrp rp—lrp—1 rp—1 /71 — 1 /7-1 — 1 /7-1 — 1 \ ^( rp— 1 rp nr fP I rTT rp rp
+1X1 1X2-lX31X4 - 1X11X21X31X4) - Xi iXiJ-XiJ-Xi + J-x1ix2J-x3lxi +Tx1Tx2Tx:iTx\ -~T~lTX2TXiTx])\ +ql+%^-S+1^T^Tlxi~S)Dx2DX3
+q5-sT^{S+1)Tx2Tx3T-l}3~5)Dx1Dxi}/№, XU X4, X2).
Remark 1 In the classical limit lim^i Aq = A, see (7).
5.2 The Laplace operator on in terms of the horospheric generators
Define the ordered monomial
w(m, k, n) = (Z*)mHkZn ,
and let
F{Z*,H,Z) = £ amfi,nw(m,k,n).
m,k,n
Consider the action of the operator Aq on the Schwartz space (18).
Proposition 4 The action of the Casimir operator Aq in terms of horospheric generators takes the form
F(Z*,H,Z,R). Aq = (32)
(1 ,;,Vj2 [q~lTH -q2+ qTHl} F(Z*,H, Z, R)
+eql-sDz*Dz \H-2T&h-1F{Z\ H, Z, i?)t.
Remark 2 In the classical limit (32) takes the form of the Laplace operator in horospheric coordinates lim9_>.i Aq = A, see (7).
Our main goal is to find the eigen-functions of Aq
FU{Z*,H, Z,R).Aq = [Jl2 FU(Z*,H,Z,R). (33)
Z q2
These functions are expressed through the ^-exponents and the three types of g-cylindric functions. For |r/| ^ 1 they can be defined by the expansion
00 n(2—6)m(m+a) „a+2m
q
Z«){Z) (1 - q2)«rg2(a + 1) 2 (q2,q2)m(q2a+2,q2)m2«+2m ’ (34)
i = ~^2 + ^ + 2’ where T92(a + 1) is the ç2-r-function. We assume that
2(TJr?)<1 f°r i = 2'
The non-commutative analog of the horospheric elementary harmonics (8) has the following form
Proposition 5 The basic solutions of (33) are defined as
FV{Z\ H, Z, R) = e(ilZ*)Va(H)e(ixZ)EUta(R), (e ^ 0),
where ¡i,a G C,
V„(H) = H-1zS,)(2(-e)1/2|^|?-i/2if“1),
S uAR) = •
Represent the solutions in the form
Fu(Z*,H,Z,R) = Va(Z*,H,Z)E„>a(R).
Substituting it in (33) and using the comultiplication relations (14), we find
(Va(Z*,H, Z).Qq) (E„,a(R).Çlq) R-2
- (Va(Z*,H, Z).nq<M) (3v,a(R)Slq,M) R-2
12 Va{Z\H,Z)EUtQ{R) = 0.
It follows from (24) that it can be rewritten as
CVa(Z*,H,Z).nq)EI/!a(R)R-2 -Va(Z*,H,Z) {~v,a{R).nq,MR-2)
Va(Z*,H,Z)E„iQ(R)= 0.
S^(Д).п9>м + E*AR) ( ) =0 (36)
L2J
In this way we come to the equations
„—a+2 _ 9/j2 i .jCt+2
Va(Z*,H, Z).nq - ---------- ^y1---------Va(Z*,H, Z)= 0, (35)
'q-v+2 _ 2q2 + qu+2 2 ^-Q+2 - 2?2 + ?Q+2'
(1 — <72)2 * (1^
From (28) and (29) one rewrites the equation (36) as
qSVia{q-lR) - (qa+2 + q-°+2)SVfil(R) + qZEv,a{qR) = ^“"(l - <f?Ev,a{R).
Put
z = 2q~^2(l-qu)(l-q‘2y1R.
/o\
Then we come to the difference equation for g-Bessel functions with 8 = 1 and Zq (z) = EUia(R)R.
Consider now (35) and put
Va(Z*, H, Z) = e(j2Z*)Va(H)efaZ). (37)
Assume that e = ±1 and
Substituting this expression in (37), we express Va in terms of monomials w(m,k,n). Using the action of Qq on monomials we obtain
00 /i 2\2k—2 e(/xZ*)jrCfc -2 k7-MT2--2Tg~a~2fc+2(i - q2)(i - <?°‘+2k)H-a-2k-lefaz)
“ (q2,q2)k(q2a+2,q2)k
oo 2\2fc
- E|„|2e(AZ-) Y, = 0 .
fc=o (q2^2h(q2a+2^2)k Then the coefficients c/.. satisfy the recurrence relation
<*+! = -w92“+4‘+2-i(“+2i+2).
Ck = (-e)*|/i|sV2-w*+“>-<*-
Then
jc °° ^(2—<J)Ar(A;-fa) (1 n2\2k x
«.(*) = **w- £<-*>*:f**)
A:=0
These series coincide with (34) up to a constant multiplier after replacing 2z by (—e)1/2q~5/2H
-l
Remark 3 In the classical limit we come to Proposition 2.1
lim e(p,Z*)Va(H)e{iJ,Z)E^a(R) = exp(ifiz + ip,z)va{h)xv,a(r) ■
q-> 1
As in the classical situation one can restrict the operator Aq on the non-commutative homogeneous spaces.
Corollary 2 The restrictions of Aq assume the form
L^> : A<? = (1 -q2)2 - 292 + &H ] + 1l~SDz*DzH~2T^fl,
ILg,s, : Aq = _* [q3TH - 2q2 + qT~1} - ql~6D^DzH^T^1.
Then we obtain the non-commutative deformations of the classical formulas (9), (10).
Corollary 3 The basic harmonics on the non-commutative L, IL and the light-cone Cj’^ are
F„(z, h, z) = e{fiZ*)H-1Z^'>{2i£2\n\H-1)e(fiZ), e = ±1,
and
F„(z,h,z) =e{nZ*)Ha~le(nZ), e = 0.
Here v2 = o? — 1.
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