- Email: [email protected]
Four-valued representations of the three-dimensional rotation group in Euclidean space
I7.A.2Nuclear
Physics
Bl (1967)
594-596.
North-Holland
Publ.
FOUR-VALUED REPRESENTATIONS THREE-DIMENSIONAL ROTATION IN EUCLIDEAN SPACE
Comp..
Amsterdam
OF THE GROUP
S. S. SANNIKOV PhJjsico-Technical Ukrainian
Institute. Academy of Sciences. SSR. Khav’kov. USSR
Received
3 February
1967
Abstract: A new representation of the rotation group R3 corresponding to a weight -$ is constructed. It is an infinite-dimensional, irreducible, non-unitary,four-valued representation of the R3 group (or two-valued representation of SU(2) group). The approach considered here is a basis for the construction of rotation group representations with complex spin and complex angular momentum.
The new representations of the JZ4 Lorentz group was constructed in an infinitesimal form in ref. [ 11. Here we study in detail the global structure of such a representation in the case of the rotation group R3 (compact subgroup of the Lorentz group). The generators of considered representation have a form (L+ = L1 f iL2) L3 = h(2z$
L
+ l),
= 1,2 +2,
_
These operators L,, cy= 1,2,3 satisfy commutation It follows from (1) that
relations
[La,LPI =
ir,p,L,.
L2=-&k(,+l).
(2)
Hence, the representation considered here has a weight h = - a or X = -i . Let us consider a class of analytic functions f(z) of one complex variable z = x + iy. A global representation corresponding to generators (1) is defined by the formula T(r)f(z)
= f(z’)
K(e;“/) ,
(3)
where Y = (cu, 3, y) E R 3, Q, P, y are the Euler angles of the R3 group (0 < (Y, y < 2a, 06 p s a). In (3) z’ = az +b(d/dz), a = exp(-$(a-ty))cos$p, b= -i exp( -ii (Q - y)) sin $3, and K (2;
Obviously
T(Y)
Y)
=
(COS
satisfies
T(y1y2). * See
footnote
on next page.
+a,-+
exp[-$i(cr+y)
+ $iz2
eDZy tg *p] .
all group axiom * and in particular
T(rl)T(y2)
(3’) =
THREE-DIMENSIONAL
ROTATION
5%
We now may define more precisely the @’ class of functions in which the representation (3) is constructed. It is a class of entire analytic functions of an order p d 2 and the type of 0 G T < m (refs. [2,3]). The representation with X = -$ is realized by functions which are even with respect to z -3 -2, and the representation with h = -2 is realized by odd functions. A canonical basis of the representation with X = -) is formed by functions n = 0,1,2,3
,...
(4)
ffp&?f+l 7
(5)
These functions satisfy equations L3fn = (n + I)&,
L-&
L+f?z = %+lfn+l~
where fffl = -id-&z - $) (ref. [4]). An operator T(Y) acts on the functions
=
(4) as
Tb')fn(d = ~&2;Y)K(GY) ,
(6)
where Pa are polynomials of degree n in z 2 expressed polynomials H2, by a relation
-” tg $p)” HZn( ._&
through Hermitean
-in)) .
,-++
(7)
The matrix elements of representation (3) may be obtained from (6) by decomposition of the right part of (6) into the canonical basis functions (4)
T(Y)f&) = ifiq),&)f,W
(8)
9
m=O where Dyn,(y)
d,,(P) = (-1) ;
1:gy#_
I;
bl,.
=
,-i(m+B)r
d,,(P)
s+p /s!r(s+p+$)
)
e-i(n+a)Q ,
(T)“”
(q)iP
pt, K)(x),
(s+p)!r(s+$)
K are Jacobi polynomials
[5], and
p
=
Iwz-nl,
K
=
-(fiz+n+t),
follows from (4), (5) and (9) that the representation considered here is (i) an infinite-dimensional irreducible representation of the rotation group and (ii) a four-valued representation of the R3 group (or two-valued representation of the SU(2) group which is a covering group of the R3 group). The latter circumstance means that there exists a “covering group” on which the matrix elements (9) are one-valued functions. The Euler angles of this group vary in a region 0 G cy,/3 < 8n, 0 d /3 g 71. It is any Riemannian surface of the SU(2) group. The existence of such surfaces follows from the fact that 1 footnote from preceding
page. Strictly speaking we have here to do with some gcncralization of representations with so-called projective representation which is charwhere x(rI. ~2) is a number function acterized by T(rI)T(r2) = X(VI,~~)T(YIY~). satisfying a condition
S. S. SANNIKOV
596
Lie group is an analytical many-fold (fifth Hilbert problem [6]). Obviously this representation is non-unitary (compare with finite-dimensional representations [7] which are equivalent to unitary representations). In fact the matrix elements (9) are singular functions on a group and therefore non-measurable. An invariant scalar product has a form every
with a measure dp(z) = l/n exp(- ?z [ 2, dz, dz = dx dy, and Iv(z) = cp(iz). We have * [f,‘, cp]f = [c~,f]Z, n = 1. The scalar product (10) exists iff(z) E @ and q(z) E 4, where 4 is a class of entire analytical functions of the order p < 2 and the type of 0 C T < m. The form (10) is a strong-indefinite metric: The representation considered is unitary in IX?? -fmlz= g this metric *“,” = (-l)’g- 6?!?’ go+(r) - o(r-l) = o-l(r) (ref. [4]). The elements from +’ are functionals (generalized functions) on a space + (ref. [8]). The spaces $Jand @ arelinear topological spaces. Obviously the mapping Y - T(r) is continuous (functions T(r)f , f E $’ as functions on a group are analytic functions). Further, the linear operators T(r) are unbounded in the topology of the space $’ (which is induced by the topology of the Riemannian surface). In conclusion we note that the representation considered here is an analytic continuation in the group parameters of the representation of the J?2 Lorentz group which was considered in ref. [9] and is related to a onedimensional quantum oscillator, so that the problem of the oscillator may be considered from the compact group point of view. The author is indebted to M. A. Naimark and D. P. Zhelobenko discussions.
for useful
REFERENCES [l] [Z] [3] [4] [5] [6] [7] [8] [9]
S.S.Sannikov, Nucl.Phys. 1B (1967) 577. V. Bargmann, Comm. Pure Appl. Math. 14 (1961) 187. S.S.Sannikov. Ukr.Phys.J.10 (1965) 684. S.S.Sannikov, J.Nucl.Phys.(USSR) 2 (1965) 570. New York, H.Bateman, Higher transcendental functions, Vol. 2 (McGraw-Hill. 1953). L.S.Pontragin, Continued groups (Moscow, 1954). M.A. Naimark, Linear representations of the Lorentz group (Moscow, 1958). I.M.Gel’fand and G.E.Shilov, Distributions, Vol.1 (Moscow, 1959). S.S.Sannikov, ZhETF (USSR) 49 (1965) 1913.
* fdenotes complex conjugation of I. ** t’denotes Hermitean conjugation.
Physics
Bl (1967)
594-596.
North-Holland
Publ.
FOUR-VALUED REPRESENTATIONS THREE-DIMENSIONAL ROTATION IN EUCLIDEAN SPACE
Comp..
Amsterdam
OF THE GROUP
S. S. SANNIKOV PhJjsico-Technical Ukrainian
Institute. Academy of Sciences. SSR. Khav’kov. USSR
Received
3 February
1967
Abstract: A new representation of the rotation group R3 corresponding to a weight -$ is constructed. It is an infinite-dimensional, irreducible, non-unitary,four-valued representation of the R3 group (or two-valued representation of SU(2) group). The approach considered here is a basis for the construction of rotation group representations with complex spin and complex angular momentum.
The new representations of the JZ4 Lorentz group was constructed in an infinitesimal form in ref. [ 11. Here we study in detail the global structure of such a representation in the case of the rotation group R3 (compact subgroup of the Lorentz group). The generators of considered representation have a form (L+ = L1 f iL2) L3 = h(2z$
L
+ l),
= 1,2 +2,
_
These operators L,, cy= 1,2,3 satisfy commutation It follows from (1) that
relations
[La,LPI =
ir,p,L,.
L2=-&k(,+l).
(2)
Hence, the representation considered here has a weight h = - a or X = -i . Let us consider a class of analytic functions f(z) of one complex variable z = x + iy. A global representation corresponding to generators (1) is defined by the formula T(r)f(z)
= f(z’)
K(e;“/) ,
(3)
where Y = (cu, 3, y) E R 3, Q, P, y are the Euler angles of the R3 group (0 < (Y, y < 2a, 06 p s a). In (3) z’ = az +b(d/dz), a = exp(-$(a-ty))cos$p, b= -i exp( -ii (Q - y)) sin $3, and K (2;
Obviously
T(Y)
Y)
=
(COS
satisfies
T(y1y2). * See
footnote
on next page.
+a,-+
exp[-$i(cr+y)
+ $iz2
eDZy tg *p] .
all group axiom * and in particular
T(rl)T(y2)
(3’) =
THREE-DIMENSIONAL
ROTATION
5%
We now may define more precisely the @’ class of functions in which the representation (3) is constructed. It is a class of entire analytic functions of an order p d 2 and the type of 0 G T < m (refs. [2,3]). The representation with X = -$ is realized by functions which are even with respect to z -3 -2, and the representation with h = -2 is realized by odd functions. A canonical basis of the representation with X = -) is formed by functions n = 0,1,2,3
,...
(4)
ffp&?f+l 7
(5)
These functions satisfy equations L3fn = (n + I)&,
L-&
L+f?z = %+lfn+l~
where fffl = -id-&z - $) (ref. [4]). An operator T(Y) acts on the functions
=
(4) as
Tb')fn(d = ~&2;Y)K(GY) ,
(6)
where Pa are polynomials of degree n in z 2 expressed polynomials H2, by a relation
-” tg $p)” HZn( ._&
through Hermitean
-in)) .
,-++
(7)
The matrix elements of representation (3) may be obtained from (6) by decomposition of the right part of (6) into the canonical basis functions (4)
T(Y)f&) = ifiq),&)f,W
(8)
9
m=O where Dyn,(y)
d,,(P) = (-1) ;
1:gy#_
I;
bl,.
=
,-i(m+B)r
d,,(P)
s+p /s!r(s+p+$)
)
e-i(n+a)Q ,
(T)“”
(q)iP
pt, K)(x),
(s+p)!r(s+$)
K are Jacobi polynomials
[5], and
p
=
Iwz-nl,
K
=
-(fiz+n+t),
follows from (4), (5) and (9) that the representation considered here is (i) an infinite-dimensional irreducible representation of the rotation group and (ii) a four-valued representation of the R3 group (or two-valued representation of the SU(2) group which is a covering group of the R3 group). The latter circumstance means that there exists a “covering group” on which the matrix elements (9) are one-valued functions. The Euler angles of this group vary in a region 0 G cy,/3 < 8n, 0 d /3 g 71. It is any Riemannian surface of the SU(2) group. The existence of such surfaces follows from the fact that 1 footnote from preceding
page. Strictly speaking we have here to do with some gcncralization of representations with so-called projective representation which is charwhere x(rI. ~2) is a number function acterized by T(rI)T(r2) = X(VI,~~)T(YIY~). satisfying a condition
S. S. SANNIKOV
596
Lie group is an analytical many-fold (fifth Hilbert problem [6]). Obviously this representation is non-unitary (compare with finite-dimensional representations [7] which are equivalent to unitary representations). In fact the matrix elements (9) are singular functions on a group and therefore non-measurable. An invariant scalar product has a form every
with a measure dp(z) = l/n exp(- ?z [ 2, dz, dz = dx dy, and Iv(z) = cp(iz). We have * [f,‘, cp]f = [c~,f]Z, n = 1. The scalar product (10) exists iff(z) E @ and q(z) E 4, where 4 is a class of entire analytical functions of the order p < 2 and the type of 0 C T < m. The form (10) is a strong-indefinite metric: The representation considered is unitary in IX?? -fmlz= g this metric *“,” = (-l)’g- 6?!?’ go+(r) - o(r-l) = o-l(r) (ref. [4]). The elements from +’ are functionals (generalized functions) on a space + (ref. [8]). The spaces $Jand @ arelinear topological spaces. Obviously the mapping Y - T(r) is continuous (functions T(r)f , f E $’ as functions on a group are analytic functions). Further, the linear operators T(r) are unbounded in the topology of the space $’ (which is induced by the topology of the Riemannian surface). In conclusion we note that the representation considered here is an analytic continuation in the group parameters of the representation of the J?2 Lorentz group which was considered in ref. [9] and is related to a onedimensional quantum oscillator, so that the problem of the oscillator may be considered from the compact group point of view. The author is indebted to M. A. Naimark and D. P. Zhelobenko discussions.
for useful
REFERENCES [l] [Z] [3] [4] [5] [6] [7] [8] [9]
S.S.Sannikov, Nucl.Phys. 1B (1967) 577. V. Bargmann, Comm. Pure Appl. Math. 14 (1961) 187. S.S.Sannikov. Ukr.Phys.J.10 (1965) 684. S.S.Sannikov, J.Nucl.Phys.(USSR) 2 (1965) 570. New York, H.Bateman, Higher transcendental functions, Vol. 2 (McGraw-Hill. 1953). L.S.Pontragin, Continued groups (Moscow, 1954). M.A. Naimark, Linear representations of the Lorentz group (Moscow, 1958). I.M.Gel’fand and G.E.Shilov, Distributions, Vol.1 (Moscow, 1959). S.S.Sannikov, ZhETF (USSR) 49 (1965) 1913.
* fdenotes complex conjugation of I. ** t’denotes Hermitean conjugation.