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Massless particle with rigidity as a model for the description of bosons and fermions
Volume 243, number 4
PHYSICS LETTERS B
5 July 1990
Massless particle with rigidity as a model for the description of bosons and fermions M.S. Plyushchay Institutefor High Energy Physics, SU-142 284 Protvino (Moscowregion), USSR Received 15 March 1990
It is shown that the model of a massless particle with rigidity, whose action depends on the curvature of a particle world trajectory, describes on the quantum level massless states of either integer or half-integer spin depending on the value of the quantized parameter of the model.
In ref. [ 1 ] a m o d e l o f a massless particle with rigidity was proposed. The action o f this m o d e l d e p e n d s on the curvature o f a particle world trajectory, while its lagrangian contains a d e p e n d e n c e on the secondorder derivatives over the evolution parameter. The peculiarity o f this m o d e l is that its classical equations o f m o t i o n are consistent only for superrelativistic m o t i o n o f a particle. The particle in this case moves at the velocity o f light along the m o m e n t u m vector p and simultaneously performs a circular m o t i o n in a transverse plane. On the h a m i l t o n i a n level, the system is described by first-class constraints, one o f which is the mass-shell c o n d i t i o n : / 9 2 ~ 0. The set o f the constraints as a whole guarantees that besides " e x t e r n a l " observables, i.e. the vector o f particle mom e n t u m a n d N e w t o n - W i g n e r coordinate (which evolves at a velocity o f light), the only " i n t e r n a l " observable is the helicity 2. The helicity o f the particle is connected with the p a r a m e t e r o f the model, a , by means o f the equality 22 = a 2. In ref. [ 1 ] some definite scheme o f quantization was considered, in which the p a r a m e t e r ot was q u a n t i z e d and took an arbitrary fixed integer value n > 0. As a result, in that scheme the q u a n t u m states o f the system were massless states o f integer helicities 2 = + n. In the present p a p e r we shall show that there is a m o r e general scheme o f q u a n t i z a t i o n o f the m o d e l o f ref. [ 1 ], leading to a description o f m a s s l e s s states o f either integer or half-integer helicities. First we shall consider a canonical quantization on a reduced phase Elsevier Science Publishers B.V. (North-Holland)
space, a n d afterwards we shall realize an explicitly covariant quantization o f the model. The m o d e l o f a massless particle with rigidity is described by the action
S=-ot f kds,
(1)
where ot is a dimensionless p a r a m e t e r (in units h = c = l ) , d s 2 = - d x U d x P g u u , //, lp=0, 1, 2, 3, g~,~= diag( - l, l, l, 1 ); k is the curvature o f the world trajectory, k 2 = (d2x~/ds2) 2. In the following, for definiteness, we suppose ot > 0. In the p a r a m e t r i z a t i o n x ~ = x U ( z ) , action ( 1 ) is rewritten in the form
S=LdT,
L = - o t ~
2,
where ± U = d x ~ / d z , £~=.~-xU.(.~5c)/5c 2, a n d we suppose that ±z > O, 2 2 >i 0 (see ref. [ 1 ] ). The h a m iltonian constraints following from the lagrangian L, have the form [ 1 ] ~1 Ol = r t q ~ 0 ,
~2 = 1 (7~2q2- a 2 ) ~,~0 ,
~4 = p T t ~ 0 ,
~5 = p 2 - ~ 0 .
~3 = p q ~ 0 , (2)
Here the quantities q~=Sc ~ play the role o f independent variables in the phase space o f the system and ~ Action ( 1) was considered independently in ref. [2], where the hamiltonian constraints were found. However, neither the physical content nor the quantization of the system were investigated there. 383
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by pU and n u we denote the canonical momenta, conjugated to x u and qu, respectively. The nonzero Poisson brackets (PBs) for the phase space coordinates have the form {x u, p ~} = gU~, { qU, n ~} = gU~, and the role of the total hamiltonian [ 3 ], generating the evolution of the system over parameter z by means o f PBs, is played by the quantity [ 1 ] (3)
H = 0 3 "~-Vl ~1 -~-v2~2 ,
where Va= Va(Z), a = 1, 2, are arbitrary functions. The canonical m o m e n t u m p u is simultaneously the conserved e n e r g y - m o m e n t u m vector o f the system, while the conserved angular m o m e n t u m tensor is
MU~=xUpP- xPpU + qU~r~-q~ u .
(4)
Let us introduce the classical Pauli-Lubanski vector
wU= ½¢u~'~PM~,~pa, E0123= l .
(5)
Then, taking into account the constraints Ok~ 0, k = 3, 4, 5, we obtain the relation
wU=2p u ,
(6)
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terms of independent variables, let us introduce the following set o f gauge conditions:
ZI=X°--z~O,
z2=q°-l~0,
Zs=n°~0,
z4=q2-1~0.
(10)
The requirement of their conservation, d Z k / d ' t ' ~ 0 , k = 1, 2, 3, 4, leads to the fixing o f the multipliers vl and v2 in the hamiltonian (3): UI=0 ,
l)2=--a--2p 0 •
(11)
Let us pass on now from the PBs to the Dirac brackets (DBs) [3]. To do so, use the gauges (10) and the constraints q~k~ 0, k = 1, 3, 4, 5, forming a set o f second-class constraints. After that all these constraints and gauges will be considered to be equal to zero in the strong sense. The simplest way to calculate the DBs consists in the restriction of the symplectic twoform o f the system, 27= dPu ^ dx u + dn u ^ dq u, on the surface given by the above-mentioned set of the gauges and constraints. The restriction sought-for has the form a = dpi ^ dzi + d2 ^ d~0. Here
where
2
z i = x i + 2 m l O i m 2 - - ~ 6 ( m ~ c o s ~ 0 - m ~ sin~0) , (12) 2 = (qn~) (ttn2) - (qn2) (Trnl) .
(7)
Here we have introduced the normals n u, a = 1, 2, depending on pU and characterized by the properties nap=O, nanb=Oab. Together with pU and nU_, n 2 = n _ n a = 0 , n_ p = 1, they form a complete tetrad on the surface p2=O: gU~=pUn~ + p~nU__ "]-nana. u We can take the quantities n u_ and nau in the form
and we have used the following parametrization of the phase space coordinates on the surface under consideration:
xU=(z,x),
pU=(pO=+_~,p),
(13)
qU=(1,mlcos~o+m2sin~o+~oo),
1
nU_ = ~--7 ( - 1, 0, 0, 1 ) , ng=
,rJ~, -
p+=pOWp3,
j = 1, 2 .
ztu= (0, k( - m l sin ~0+m2 cos ~0) ) ,
, (8)
From relation (6) we conclude that the quantity (7) is the classical helicity o f the system. And since the constraint ¢2 ~ 0 on the surface o f the remaining constraints can be rewritten in the equivalent form q]'2= ~ ( 2 2 - s t 2 ) ~ 0 ,
(9)
where 0 ~<~0< 2n, - oo < k < oo; the three-dimensional vectors raa, a = 1, 2, depend on the direction of the m o m e n t u m p and have the properties: ma'P=O, m a ' m b : r a b , ml X m z = p / P °. Substituting qU and n u from (14 ) into (4) and using definition (5), we find that the independent variable 2 here is a helicity. From the two-form a we obtain the following nonzero DBs for the independent variables on the reduced phase space: { z ' , p J ) * = 6 o,
the values o f the classical helicity are determined by the parameter a: 2 = + st. To pass on to the quantization of the system in 384
(14)
{~,2}*=1.
(15)
With the help of the brackets (15) and equalities ( 1 2 ) - ( 1 4 ) one can find the DBs f o r x u, pU, qU and
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PHYSICS LETTERS B
n u, which, however, will not be necessary in the further consideration. The quantities 2, p and z, the latter being an analogue of the Newton-Wigner coordinate [4 ] (see ref. [ 1 ] ), have zero DBs with the remaining constraint 02~0, which now is rewritten in the form of (9) where 2 should be treated as an independent variable. In other words, these quantities form a complete set of gauge-invariant quantities (observables) of the system. Besides, note that the reduced phase space consists here of two parts, on which pO= + x/~5 and pO= _ x / ~ - In the following we shall write simply pO, implying either the first or the second equality. Taking into account equalities (11 ) and the explicit z-dependence of the gauge Z~, we arrive at the hamiltonian [ 1 ] /-t-----pO( 1 -- O/-2~2 )
generating the evolution on the reduced phase space by means of DBs, on which constraint (9) is treated as a condition on the initial data of the system. In the quantum case the remaining constraint (9) turns into an equation for the wave functions of the physical states [ 3 ]: (16)
( £ 2 - - 0 £ 2 ) ~rJ= 0 .
In the representation where the operator/~ is diagonal, the operator ~ can be realized in the form ~= i0/ 0p. Taking into account that ~0varies between 0 and 2 it, we shall work on the space of 27t-periodic functions of the form
5 July 1990
states (17), introduce the scalar product ~2 2n
<~I ]~z>----(2g)-| J d3P j d~o ~'(p, ~0)~2(p,~0) , 0 (19) with respect to which operator (18) is hermitian. The wave functions ~t"(p)ei"~ are eigenfunctions of ~( with eigenvalues 2 = m + c. With the help of ( 12 ) - ( 14 ) and ( 4 ), construct now the angular momentum operator. Its components have the form
)~Oi=,~i ½(~O~i.~i~O)_[_2m I ~i~i2
'
(20)
- t h , 8ilh2/~+ rh2 Calculating the commutators [)Qua, ~/,~p] and [~ru~, /~'~], we can convince ourselves that the operators/~u and ~ u ~ satisfy the commutation relations of the Poincar6 algebra. Constructing then a quantum analogue of the vector (5), we obtain the relation flu= £/~u. Hence, £ is the helicity operator of the system. And therefore, the condition of the relativistic invariance of the quantum theory leads to a restriction on the constant c in ( 18 ): c = ½n, where n is an arbitrary integer. Obviously, it is enough to restrict ourselves to the case n = 0 and n = 1 without any loss of generality. Then the equation for the wave functions of physical states (16) takes the form
[(m+½n)2-a 2] ~u"(p)
eim~=0.
(22)
m
~'t(ll~, ~O)-----~ ~ , m ( p ) e i m ~ ,
meZ.
(17)
m
Since the operator ~, acting on the wave function of the form (17), makes this function leave the space of 2n-periodic functions, we shall consider the operators sin (b and cos ~ instead. Then the commutators [sin ~b, ~] = i c o s (b and [cosO, £] = - i s i n ~bwill correspond on the quantum level to the DBs ( 15 ) for 2 and ~0. The operator ~, satisfying these commutation relations, can be realized in the general case as ,~=-i~
+c,
(18)
where c is an arbitrary real constant. On the space of
Thus, we arrive at the condition of quantization of the parameter or: c~=k+ ½n, where k is some arbitrary fixed integer, k > 0 when n = 0 , and k~> 0 in the case when n = I. As a result, we find that the wave functions ~(k+n/2) (p, ~) = ~k(p) eik~,
~-(k+n/z)(p, ~0)=~--(k+n)(p)e - i ( k + n ) ~ satisfy the condition (22), i.e. they describe the ~2 In accordance with the above, here we imply that the space of states is the direct sum of spaces, differing in the sign o f p °, and the scalar product (19) is introduced on each of these spaces, orthogonal to each other.
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physical states of the system, which, in their turn, are eigenfunctions of ,( with eigenvalues 2 = + ( k + ½n), respectively. So, choosing the constant ot = k> 0 and using representation (18) with c= 0, we obtain that the states of integer helicities 2 = + k are physical ones (in ref. [ 1 ] only this case was considered). But, putting a = k + ½ , k>~0, and c=½ in (18), we obtain physical states of half-integer helicities 2 = + (k + ½). Before passing on to the construction of an explicitly covariant quantum theory we note, that representation (18) for c ~ 0 and the representation ~o =-iO/O~o (c=O) can be connected through the relation Oo(~o),(oO~-l(~0)=,(,
O~((o)=e ~ .
(23)
In the case c = n, (23) is a unitary transformation. But if c # 0 , relation (23) should be treated as a formal one, since in this case the action of the operator O~(~) on the space of states ( 17 ) is not defined. In an explicitly covariant scheme of quantization, constraints (2) turn into the equations for the wave functions of physical states: /~2~V=0 , l / ~ v = 0, ½( ~ + ~ )
/~7~v=0 ,
~=0,
[ 21(q27~2-F~2qZ)-F
l - - a 2] ~r-/-----O.
~u= i _
op,
~u=
•
are the solutions of eq. (24), where 386
ei~=r-l(nlq+inzq) , r= [ (ntq)Z + (nzq) 2] 1/2,
(28)
and the quantities n~u, a = 1, 2, are given in (8). The action of the helicity operator, ~= (h, 8) (h2 z~) - (h2~) (h, ~ ) ,
(29)
on the wave functions (27) has the form .0
,(~(P, q ) = F (p, q ) ( - l ~-~)f(p, ei~') . And since on the wave functions (27) condition (25) is reduced to the equation
(£z_ot2) ~t(p, q)=0,
(30)
we conclude that the parameter at is quantized: a = k > 0 [otherwise eq. (30) would have no solutions different from zero on the class of single-valued wave functions ]. So, we find that in the representation (26) the physical states are described by the wave functions of the form
7t+k(p, q) =F(p, q) ~/+k(p) [r-i (nlq+_in2q) ]k, (31)
(25)
which have integer helicities ;t = + k. Thus, we reproduce the results obtained in the noncovariant approach for the case of integer values of the parameter o~and c = 0 in the representation (18). And to establish the complete correspondence between the covariant and noncovariant schemes of quantization, introduce on the space of physical states the scalar product (see ref. [ 5 ] )
(26)
Then the wave functions of the form
~'(p, q)=F(p, q)f(p, e~)
F(p, q) = 6 ( p 1) J(pq) (q2) -1/1,
(24)
Here the quantum analogous of the constraints ~ and ~2 are written in a hermitian form, and besides, using the uncertainty of ordering the noncommuting operators when constructing a quantum analogue of the constraint ~2, we add a definite constant (equal to 1 ) to achieve agreement with the above scheme ofquantization (see below). The quantum analogues of constraints (2) form an algebra of first-class constraints, and therefore the set of quantum conditions (24), (25) turns out to be consistent. Let us work in the representation where the operators ~0u and Ou are diagonal, and ~u and ~u are realized in a "standard form" 0
5 July 1990
(27)
(~1, 7-'2)phr~=(hu~ ILil 7-'2> , ( ~ I ~2 ) = ~ d4p d4q ~T(P, q) ~z(P, q ) ,
J=½(0+d+), 0 = ( 2 n ) - l 8 ( ~ 2 - 1) 6(h_O) t~(ri_~)/) °
x [ ~ O ( ~ t ° - z ) + 6 ( ~ ° - z ) ~1,
~=/~°/l#°l • (32)
Putting now ~+k(p)=O(p°)~u+-k~), or ~ ± k ( p ) = 0 ( _ p 0 ) ~±k(p) in the wave functions of the physical states (31), where 0(p°) = 1 at p ° > 0 and O(p °) = 0 at p°
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PHYSICS LETTERS B
posite signs o f p °, are orthogonal to each other, while for the states with the energy o f the same sign, the scalar product (32) coincides with the product (19) for the physical states in the noncovariant approach. Now, in order to obtain the states with half-integer helicities as physical states of the system, let us realize the operators ~u and ~u in the form ~u=i--
0p.
+cr-2q~[ (n2q) 0 u n l ~ - (n~q) 0Un2~] ,
(33)
+cr-2[ (nlq) n~-(n2q) n~]
(34)
~u=-i
-----
Oqu
instead o f (26). Here c is an arbitrary real constant and r is defined in (28). By a direct calculation one can get convinced that the operators (33), (34) and /~u, c~u satisfy the canonical permutation relations, i.e. [~u,/)~] = [~u, ~ ] =igU~ and all the other c o m m u tators are zeros. Representation ( 33 ), (34) with c ~ 0 and representation (26) (c = 0) can be connected by the relation 0c(~o) 0 o 0 ; -~ (~o) = 0~. Here 0~ is the operator (33) or (34), and 0o is the corresponding operator from (26); 0~(~o)=eic% and e i~' is defined in ( 28 ). As in ( 23 ), O~(q~) here is a unitary operator in the case when c = n, otherwise this operator is not single-valued, and therefore, at c ~ n this relation is to be considered as formal. In representation (33), (34) the solutions of eqs. (24) have the same form (27), and the helicity operator (29) acts on functions (27) after the law
,f~(p, q) = F ( p ,
(
q) \ - i
~0
+c)f(p, e i~)
are the solutions o f eq. (35), where d+ = 0 , d_ = 1. The wave functions (36) are the eigenfunctions o f the helicity operator with the eigenvalues 2= +(k+½). Thus, we have shown that on the q u a n t u m level the model of a massless particle with rigidity describes either states o f integer helicities, 2 = + k > 0, or those of half-integer helicities, 2 = + ( k + ½), k~> 0, depending on whether the parameter o f the model, or, is integer or half-integer. Besides, it has been demonstrated that quantization on a reduced phase space and quantization without fixing gauges can be realized in a co-ordinate way. In ref. [6] a nongrassmannian model o f the spinning particle was constructed by analogy with the pseudoclassical mechanics of spin-½ massless particle [ 7 ] a3. It has the same physical content as model [ 1 ] and is described by the same set ofhamiltonian constraints (2). The only difference is that model [ 6 ] permits zero value for the parameter a. Therefore, it is obvious that the schemes of quantization considered here are applicable also to the model of ref. [6 ], and in the case when ot = 0, the model o f ref. [ 6 ] describes spinless states on the q u a n t u m level. In conclusion we note that it seems interesting to perform secondary quantization o f the models ofrefs. [ 1,6], especially in the case o f half-integer values o f the parameter or. I am grateful to V.I. Borodulin, A.V. R a z u m o v and S.N. Storchak for valuable discussions. #3
Since we work on the class o f single-valued wave functions, the requirement o f relativistic invariance o f the quantum theory leads to the restriction c = ½n. Putting c=½ (i.e., n = l ) in (33), (34), we rewrite condition (25) in the equivalent form
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Model [6] is given by the lagrangian L=jc2/2e-otx/~, where e is the Lagrange multiplier, k 2 = ±2_ (.in) 2, n u= q u/ x/~ and the space-like vector qu is an independent vector in configuration space of the system, which serves to describe the spin degrees of freedom.
References
F(p,q)
-i~-~ +
- a 2 f ( p , e i~') = 0 .
(35)
This equation has solutions on the class o f single-valued functions, different from zero, only when or= k + ½, k>~ 0; and in this case the wave functions o f the form
~[,t_+(k+l/Z)(p, q)=F(p, q) ~t+(p) e -+i(k+a-+)~ (36)
[ 1] M.S. Plyushchay, Mod. Phys. Lett. A 4 (1989) 837. [2] C. Batlle, J. Gomis and N. Roman-Roy,J. Phys. A 21 (1988) 2693. [3] P.A.M. Dirac, Lectures on quantum mechanics, Belfer Graduate School of Science, Yeshiva University (New York, 1964). [4] T.D. Newton and E.P. Wigner, Rev. Mod. Phys. 21 (1949) 400. 387
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[ 5 ] V.G. Budanov, A.V. Razumov and A. Yu. Taranov, Proc. IV Seminar on Problems of high energy physics and quantum field theory, Vol. 2 (Protvino, 1981 ) p. 273. [6] M.S. Plyushchay, Phys. Lett. B 236 (1990) 291. [7] F.A. Berezin and M.S. Marinov, JETP Lett. 21 (1975) 321; Ann. Phys. (NY) 104 (1977) 336; R. Casalbuoni, Nuovo Cimento 33A ( 1976 ) 369;
388
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L. Brink, S. Deser, B. Zumino, P. Di Vecchia and P. Howe, Phys. Lett. B 64 (1976) 435; L. Brink, P. Di Vecchia and P. Howe, Phys. Lett. B 65 (1976) 471.
PHYSICS LETTERS B
5 July 1990
Massless particle with rigidity as a model for the description of bosons and fermions M.S. Plyushchay Institutefor High Energy Physics, SU-142 284 Protvino (Moscowregion), USSR Received 15 March 1990
It is shown that the model of a massless particle with rigidity, whose action depends on the curvature of a particle world trajectory, describes on the quantum level massless states of either integer or half-integer spin depending on the value of the quantized parameter of the model.
In ref. [ 1 ] a m o d e l o f a massless particle with rigidity was proposed. The action o f this m o d e l d e p e n d s on the curvature o f a particle world trajectory, while its lagrangian contains a d e p e n d e n c e on the secondorder derivatives over the evolution parameter. The peculiarity o f this m o d e l is that its classical equations o f m o t i o n are consistent only for superrelativistic m o t i o n o f a particle. The particle in this case moves at the velocity o f light along the m o m e n t u m vector p and simultaneously performs a circular m o t i o n in a transverse plane. On the h a m i l t o n i a n level, the system is described by first-class constraints, one o f which is the mass-shell c o n d i t i o n : / 9 2 ~ 0. The set o f the constraints as a whole guarantees that besides " e x t e r n a l " observables, i.e. the vector o f particle mom e n t u m a n d N e w t o n - W i g n e r coordinate (which evolves at a velocity o f light), the only " i n t e r n a l " observable is the helicity 2. The helicity o f the particle is connected with the p a r a m e t e r o f the model, a , by means o f the equality 22 = a 2. In ref. [ 1 ] some definite scheme o f quantization was considered, in which the p a r a m e t e r ot was q u a n t i z e d and took an arbitrary fixed integer value n > 0. As a result, in that scheme the q u a n t u m states o f the system were massless states o f integer helicities 2 = + n. In the present p a p e r we shall show that there is a m o r e general scheme o f q u a n t i z a t i o n o f the m o d e l o f ref. [ 1 ], leading to a description o f m a s s l e s s states o f either integer or half-integer helicities. First we shall consider a canonical quantization on a reduced phase Elsevier Science Publishers B.V. (North-Holland)
space, a n d afterwards we shall realize an explicitly covariant quantization o f the model. The m o d e l o f a massless particle with rigidity is described by the action
S=-ot f kds,
(1)
where ot is a dimensionless p a r a m e t e r (in units h = c = l ) , d s 2 = - d x U d x P g u u , //, lp=0, 1, 2, 3, g~,~= diag( - l, l, l, 1 ); k is the curvature o f the world trajectory, k 2 = (d2x~/ds2) 2. In the following, for definiteness, we suppose ot > 0. In the p a r a m e t r i z a t i o n x ~ = x U ( z ) , action ( 1 ) is rewritten in the form
S=LdT,
L = - o t ~
2,
where ± U = d x ~ / d z , £~=.~-xU.(.~5c)/5c 2, a n d we suppose that ±z > O, 2 2 >i 0 (see ref. [ 1 ] ). The h a m iltonian constraints following from the lagrangian L, have the form [ 1 ] ~1 Ol = r t q ~ 0 ,
~2 = 1 (7~2q2- a 2 ) ~,~0 ,
~4 = p T t ~ 0 ,
~5 = p 2 - ~ 0 .
~3 = p q ~ 0 , (2)
Here the quantities q~=Sc ~ play the role o f independent variables in the phase space o f the system and ~ Action ( 1) was considered independently in ref. [2], where the hamiltonian constraints were found. However, neither the physical content nor the quantization of the system were investigated there. 383
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by pU and n u we denote the canonical momenta, conjugated to x u and qu, respectively. The nonzero Poisson brackets (PBs) for the phase space coordinates have the form {x u, p ~} = gU~, { qU, n ~} = gU~, and the role of the total hamiltonian [ 3 ], generating the evolution of the system over parameter z by means o f PBs, is played by the quantity [ 1 ] (3)
H = 0 3 "~-Vl ~1 -~-v2~2 ,
where Va= Va(Z), a = 1, 2, are arbitrary functions. The canonical m o m e n t u m p u is simultaneously the conserved e n e r g y - m o m e n t u m vector o f the system, while the conserved angular m o m e n t u m tensor is
MU~=xUpP- xPpU + qU~r~-q~ u .
(4)
Let us introduce the classical Pauli-Lubanski vector
wU= ½¢u~'~PM~,~pa, E0123= l .
(5)
Then, taking into account the constraints Ok~ 0, k = 3, 4, 5, we obtain the relation
wU=2p u ,
(6)
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terms of independent variables, let us introduce the following set o f gauge conditions:
ZI=X°--z~O,
z2=q°-l~0,
Zs=n°~0,
z4=q2-1~0.
(10)
The requirement of their conservation, d Z k / d ' t ' ~ 0 , k = 1, 2, 3, 4, leads to the fixing o f the multipliers vl and v2 in the hamiltonian (3): UI=0 ,
l)2=--a--2p 0 •
(11)
Let us pass on now from the PBs to the Dirac brackets (DBs) [3]. To do so, use the gauges (10) and the constraints q~k~ 0, k = 1, 3, 4, 5, forming a set o f second-class constraints. After that all these constraints and gauges will be considered to be equal to zero in the strong sense. The simplest way to calculate the DBs consists in the restriction of the symplectic twoform o f the system, 27= dPu ^ dx u + dn u ^ dq u, on the surface given by the above-mentioned set of the gauges and constraints. The restriction sought-for has the form a = dpi ^ dzi + d2 ^ d~0. Here
where
2
z i = x i + 2 m l O i m 2 - - ~ 6 ( m ~ c o s ~ 0 - m ~ sin~0) , (12) 2 = (qn~) (ttn2) - (qn2) (Trnl) .
(7)
Here we have introduced the normals n u, a = 1, 2, depending on pU and characterized by the properties nap=O, nanb=Oab. Together with pU and nU_, n 2 = n _ n a = 0 , n_ p = 1, they form a complete tetrad on the surface p2=O: gU~=pUn~ + p~nU__ "]-nana. u We can take the quantities n u_ and nau in the form
and we have used the following parametrization of the phase space coordinates on the surface under consideration:
xU=(z,x),
pU=(pO=+_~,p),
(13)
qU=(1,mlcos~o+m2sin~o+~oo),
1
nU_ = ~--7 ( - 1, 0, 0, 1 ) , ng=
,rJ~, -
p+=pOWp3,
j = 1, 2 .
ztu= (0, k( - m l sin ~0+m2 cos ~0) ) ,
, (8)
From relation (6) we conclude that the quantity (7) is the classical helicity o f the system. And since the constraint ¢2 ~ 0 on the surface o f the remaining constraints can be rewritten in the equivalent form q]'2= ~ ( 2 2 - s t 2 ) ~ 0 ,
(9)
where 0 ~<~0< 2n, - oo < k < oo; the three-dimensional vectors raa, a = 1, 2, depend on the direction of the m o m e n t u m p and have the properties: ma'P=O, m a ' m b : r a b , ml X m z = p / P °. Substituting qU and n u from (14 ) into (4) and using definition (5), we find that the independent variable 2 here is a helicity. From the two-form a we obtain the following nonzero DBs for the independent variables on the reduced phase space: { z ' , p J ) * = 6 o,
the values o f the classical helicity are determined by the parameter a: 2 = + st. To pass on to the quantization of the system in 384
(14)
{~,2}*=1.
(15)
With the help of the brackets (15) and equalities ( 1 2 ) - ( 1 4 ) one can find the DBs f o r x u, pU, qU and
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n u, which, however, will not be necessary in the further consideration. The quantities 2, p and z, the latter being an analogue of the Newton-Wigner coordinate [4 ] (see ref. [ 1 ] ), have zero DBs with the remaining constraint 02~0, which now is rewritten in the form of (9) where 2 should be treated as an independent variable. In other words, these quantities form a complete set of gauge-invariant quantities (observables) of the system. Besides, note that the reduced phase space consists here of two parts, on which pO= + x/~5 and pO= _ x / ~ - In the following we shall write simply pO, implying either the first or the second equality. Taking into account equalities (11 ) and the explicit z-dependence of the gauge Z~, we arrive at the hamiltonian [ 1 ] /-t-----pO( 1 -- O/-2~2 )
generating the evolution on the reduced phase space by means of DBs, on which constraint (9) is treated as a condition on the initial data of the system. In the quantum case the remaining constraint (9) turns into an equation for the wave functions of the physical states [ 3 ]: (16)
( £ 2 - - 0 £ 2 ) ~rJ= 0 .
In the representation where the operator/~ is diagonal, the operator ~ can be realized in the form ~= i0/ 0p. Taking into account that ~0varies between 0 and 2 it, we shall work on the space of 27t-periodic functions of the form
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states (17), introduce the scalar product ~2 2n
<~I ]~z>----(2g)-| J d3P j d~o ~'(p, ~0)~2(p,~0) , 0 (19) with respect to which operator (18) is hermitian. The wave functions ~t"(p)ei"~ are eigenfunctions of ~( with eigenvalues 2 = m + c. With the help of ( 12 ) - ( 14 ) and ( 4 ), construct now the angular momentum operator. Its components have the form
)~Oi=,~i ½(~O~i.~i~O)_[_2m I ~i~i2
'
(20)
- t h , 8ilh2/~+ rh2 Calculating the commutators [)Qua, ~/,~p] and [~ru~, /~'~], we can convince ourselves that the operators/~u and ~ u ~ satisfy the commutation relations of the Poincar6 algebra. Constructing then a quantum analogue of the vector (5), we obtain the relation flu= £/~u. Hence, £ is the helicity operator of the system. And therefore, the condition of the relativistic invariance of the quantum theory leads to a restriction on the constant c in ( 18 ): c = ½n, where n is an arbitrary integer. Obviously, it is enough to restrict ourselves to the case n = 0 and n = 1 without any loss of generality. Then the equation for the wave functions of physical states (16) takes the form
[(m+½n)2-a 2] ~u"(p)
eim~=0.
(22)
m
~'t(ll~, ~O)-----~ ~ , m ( p ) e i m ~ ,
meZ.
(17)
m
Since the operator ~, acting on the wave function of the form (17), makes this function leave the space of 2n-periodic functions, we shall consider the operators sin (b and cos ~ instead. Then the commutators [sin ~b, ~] = i c o s (b and [cosO, £] = - i s i n ~bwill correspond on the quantum level to the DBs ( 15 ) for 2 and ~0. The operator ~, satisfying these commutation relations, can be realized in the general case as ,~=-i~
+c,
(18)
where c is an arbitrary real constant. On the space of
Thus, we arrive at the condition of quantization of the parameter or: c~=k+ ½n, where k is some arbitrary fixed integer, k > 0 when n = 0 , and k~> 0 in the case when n = I. As a result, we find that the wave functions ~(k+n/2) (p, ~) = ~k(p) eik~,
~-(k+n/z)(p, ~0)=~--(k+n)(p)e - i ( k + n ) ~ satisfy the condition (22), i.e. they describe the ~2 In accordance with the above, here we imply that the space of states is the direct sum of spaces, differing in the sign o f p °, and the scalar product (19) is introduced on each of these spaces, orthogonal to each other.
385
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physical states of the system, which, in their turn, are eigenfunctions of ,( with eigenvalues 2 = + ( k + ½n), respectively. So, choosing the constant ot = k> 0 and using representation (18) with c= 0, we obtain that the states of integer helicities 2 = + k are physical ones (in ref. [ 1 ] only this case was considered). But, putting a = k + ½ , k>~0, and c=½ in (18), we obtain physical states of half-integer helicities 2 = + (k + ½). Before passing on to the construction of an explicitly covariant quantum theory we note, that representation (18) for c ~ 0 and the representation ~o =-iO/O~o (c=O) can be connected through the relation Oo(~o),(oO~-l(~0)=,(,
O~((o)=e ~ .
(23)
In the case c = n, (23) is a unitary transformation. But if c # 0 , relation (23) should be treated as a formal one, since in this case the action of the operator O~(~) on the space of states ( 17 ) is not defined. In an explicitly covariant scheme of quantization, constraints (2) turn into the equations for the wave functions of physical states: /~2~V=0 , l / ~ v = 0, ½( ~ + ~ )
/~7~v=0 ,
~=0,
[ 21(q27~2-F~2qZ)-F
l - - a 2] ~r-/-----O.
~u= i _
op,
~u=
•
are the solutions of eq. (24), where 386
ei~=r-l(nlq+inzq) , r= [ (ntq)Z + (nzq) 2] 1/2,
(28)
and the quantities n~u, a = 1, 2, are given in (8). The action of the helicity operator, ~= (h, 8) (h2 z~) - (h2~) (h, ~ ) ,
(29)
on the wave functions (27) has the form .0
,(~(P, q ) = F (p, q ) ( - l ~-~)f(p, ei~') . And since on the wave functions (27) condition (25) is reduced to the equation
(£z_ot2) ~t(p, q)=0,
(30)
we conclude that the parameter at is quantized: a = k > 0 [otherwise eq. (30) would have no solutions different from zero on the class of single-valued wave functions ]. So, we find that in the representation (26) the physical states are described by the wave functions of the form
7t+k(p, q) =F(p, q) ~/+k(p) [r-i (nlq+_in2q) ]k, (31)
(25)
which have integer helicities ;t = + k. Thus, we reproduce the results obtained in the noncovariant approach for the case of integer values of the parameter o~and c = 0 in the representation (18). And to establish the complete correspondence between the covariant and noncovariant schemes of quantization, introduce on the space of physical states the scalar product (see ref. [ 5 ] )
(26)
Then the wave functions of the form
~'(p, q)=F(p, q)f(p, e~)
F(p, q) = 6 ( p 1) J(pq) (q2) -1/1,
(24)
Here the quantum analogous of the constraints ~ and ~2 are written in a hermitian form, and besides, using the uncertainty of ordering the noncommuting operators when constructing a quantum analogue of the constraint ~2, we add a definite constant (equal to 1 ) to achieve agreement with the above scheme ofquantization (see below). The quantum analogues of constraints (2) form an algebra of first-class constraints, and therefore the set of quantum conditions (24), (25) turns out to be consistent. Let us work in the representation where the operators ~0u and Ou are diagonal, and ~u and ~u are realized in a "standard form" 0
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(27)
(~1, 7-'2)phr~=(hu~ ILil 7-'2> , ( ~ I ~2 ) = ~ d4p d4q ~T(P, q) ~z(P, q ) ,
J=½(0+d+), 0 = ( 2 n ) - l 8 ( ~ 2 - 1) 6(h_O) t~(ri_~)/) °
x [ ~ O ( ~ t ° - z ) + 6 ( ~ ° - z ) ~1,
~=/~°/l#°l • (32)
Putting now ~+k(p)=O(p°)~u+-k~), or ~ ± k ( p ) = 0 ( _ p 0 ) ~±k(p) in the wave functions of the physical states (31), where 0(p°) = 1 at p ° > 0 and O(p °) = 0 at p°
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posite signs o f p °, are orthogonal to each other, while for the states with the energy o f the same sign, the scalar product (32) coincides with the product (19) for the physical states in the noncovariant approach. Now, in order to obtain the states with half-integer helicities as physical states of the system, let us realize the operators ~u and ~u in the form ~u=i--
0p.
+cr-2q~[ (n2q) 0 u n l ~ - (n~q) 0Un2~] ,
(33)
+cr-2[ (nlq) n~-(n2q) n~]
(34)
~u=-i
-----
Oqu
instead o f (26). Here c is an arbitrary real constant and r is defined in (28). By a direct calculation one can get convinced that the operators (33), (34) and /~u, c~u satisfy the canonical permutation relations, i.e. [~u,/)~] = [~u, ~ ] =igU~ and all the other c o m m u tators are zeros. Representation ( 33 ), (34) with c ~ 0 and representation (26) (c = 0) can be connected by the relation 0c(~o) 0 o 0 ; -~ (~o) = 0~. Here 0~ is the operator (33) or (34), and 0o is the corresponding operator from (26); 0~(~o)=eic% and e i~' is defined in ( 28 ). As in ( 23 ), O~(q~) here is a unitary operator in the case when c = n, otherwise this operator is not single-valued, and therefore, at c ~ n this relation is to be considered as formal. In representation (33), (34) the solutions of eqs. (24) have the same form (27), and the helicity operator (29) acts on functions (27) after the law
,f~(p, q) = F ( p ,
(
q) \ - i
~0
+c)f(p, e i~)
are the solutions o f eq. (35), where d+ = 0 , d_ = 1. The wave functions (36) are the eigenfunctions o f the helicity operator with the eigenvalues 2= +(k+½). Thus, we have shown that on the q u a n t u m level the model of a massless particle with rigidity describes either states o f integer helicities, 2 = + k > 0, or those of half-integer helicities, 2 = + ( k + ½), k~> 0, depending on whether the parameter o f the model, or, is integer or half-integer. Besides, it has been demonstrated that quantization on a reduced phase space and quantization without fixing gauges can be realized in a co-ordinate way. In ref. [6] a nongrassmannian model o f the spinning particle was constructed by analogy with the pseudoclassical mechanics of spin-½ massless particle [ 7 ] a3. It has the same physical content as model [ 1 ] and is described by the same set ofhamiltonian constraints (2). The only difference is that model [ 6 ] permits zero value for the parameter a. Therefore, it is obvious that the schemes of quantization considered here are applicable also to the model of ref. [6 ], and in the case when ot = 0, the model o f ref. [ 6 ] describes spinless states on the q u a n t u m level. In conclusion we note that it seems interesting to perform secondary quantization o f the models ofrefs. [ 1,6], especially in the case o f half-integer values o f the parameter or. I am grateful to V.I. Borodulin, A.V. R a z u m o v and S.N. Storchak for valuable discussions. #3
Since we work on the class o f single-valued wave functions, the requirement o f relativistic invariance o f the quantum theory leads to the restriction c = ½n. Putting c=½ (i.e., n = l ) in (33), (34), we rewrite condition (25) in the equivalent form
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Model [6] is given by the lagrangian L=jc2/2e-otx/~, where e is the Lagrange multiplier, k 2 = ±2_ (.in) 2, n u= q u/ x/~ and the space-like vector qu is an independent vector in configuration space of the system, which serves to describe the spin degrees of freedom.
References
F(p,q)
-i~-~ +
- a 2 f ( p , e i~') = 0 .
(35)
This equation has solutions on the class o f single-valued functions, different from zero, only when or= k + ½, k>~ 0; and in this case the wave functions o f the form
~[,t_+(k+l/Z)(p, q)=F(p, q) ~t+(p) e -+i(k+a-+)~ (36)
[ 1] M.S. Plyushchay, Mod. Phys. Lett. A 4 (1989) 837. [2] C. Batlle, J. Gomis and N. Roman-Roy,J. Phys. A 21 (1988) 2693. [3] P.A.M. Dirac, Lectures on quantum mechanics, Belfer Graduate School of Science, Yeshiva University (New York, 1964). [4] T.D. Newton and E.P. Wigner, Rev. Mod. Phys. 21 (1949) 400. 387
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[ 5 ] V.G. Budanov, A.V. Razumov and A. Yu. Taranov, Proc. IV Seminar on Problems of high energy physics and quantum field theory, Vol. 2 (Protvino, 1981 ) p. 273. [6] M.S. Plyushchay, Phys. Lett. B 236 (1990) 291. [7] F.A. Berezin and M.S. Marinov, JETP Lett. 21 (1975) 321; Ann. Phys. (NY) 104 (1977) 336; R. Casalbuoni, Nuovo Cimento 33A ( 1976 ) 369;
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L. Brink, S. Deser, B. Zumino, P. Di Vecchia and P. Howe, Phys. Lett. B 64 (1976) 435; L. Brink, P. Di Vecchia and P. Howe, Phys. Lett. B 65 (1976) 471.