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Finite Lorentz transformations in quantum field theory
Vol.
11 (1977)
REPORTS
ON MATHEMATICAL
FINITE LORENTZ QUANTUM
No. 1
PNYSICS
TRANSFORMATIONS FIELD THEORY*
IN
A. A. BOGUSH and F. I. FEDOROV Academy of Sciences of Byeolorussian S.S.R., Institute of Physics, Minsk, U.S.S.R. (Received March 13, 1976)
The basic relations defining the vector parametrization of the complex Lorentz group SO(4.C) and its subgroups are presented. The main attractive properties of this parametrization are demonstrated. The simple solutions of some basic problems in elementary particle theory based on direct use of the finite Lorentz transformations (covariant spin state description, direct coviariant calculation and exclusion of the kinematic singularities in scattering amplitudes for polarized particles, proof of the Bargmann-Hall-Wightman theorem in axiomatic QFIJ are given. Introduction There exist many complicated problems in the elementary particle theory which can be solved in simple and natural way by making direct use of the finite symmetry group transformations, but not the infinite small ones. Unfortunately, at this time, contrary to the known and very well worked out infinitesimal approach, the general method of the finite transformations in the continuous group theory is far from satisfying all current needs and requirements of theoretical physics. The effectivity of such a method depends on the convenient choice of the group parameters. The vector parametrization forming
the foundation of the present treatment gives an example of such a parametrization finite transformations in the theory of the space-time symmetry groups. 1. Complex Lore&
The homogeneous
group and its vector parametrization
of
[l]-[13]
proper Lorentz group can be considered as a group SO(3.Z) of
the orthogonal transformation matrices L (L,f = LL = I, detL = IL1= +I) in the four-dimensional space V(3.Z) with one imaginary coordinate (x4 = it, c = 1). Thereby the SO(3.Z) group can be identified with one of the possible subgroups of the SO(4.C). In framework of the vector parametrization under consideration (see [6], [7]) the SO(4.C) group is parametrized by means of two independent complex three-dimensional vectors q = a+ib and g = c+id. -\
* Extended version of this paper will appear in the Proceedings held in Karpacz, February 1976. L371
of XIII Winter School of Physics
A. A. BOGUSH and F. I. FEDOROV
38
First of all it should be noticed that by simple redefinition of the group vector parameters and by taking into account the corresponding limitations for their allowed values, one can extend the results obtained for one orthogonal group in four-dimensional space and its finite-dimensional representations to all other groups of the same dimensionality and their subgroups. This is illustrated schematically by the following table (see [S], [7j): L = Z(% $9, (a) SO(4.C):
(b) SO(3.Z):
q = afib, g = c+id, 1+(-q*, s>” # 0, 1+(-g”, gy # 0; L = L(q, y”), g =* q* = a-ib, q = ai-ib, l-9 = l+(-q*,qy # 0; L = L(a,c),
(c) SO(4.R):
(d) SO(3.C):
q=q*=
a,
g=eg*=c,
L -_
L I
0.3)
O(d 0
O(q) E SO(3.C),
1’
0
q = g = a+ib,
(1.4)
I +<-q”, q)’ # 0; L = L(n, n), O(n) 0
I
L= [ 0 (e) SO(3.R);
1’
n’+ (n”>” ON) = 1f2--lfn2 E s0(3.~),
q = q* = g = g* z n,
WI
0 < InI < co.
The usefullness of the vector parametrization main properties :
can be demonstrated
by listing their
1. The vector parametrization of the SO(4.C) group gives the simple relations establishing the one-to-one correspondence between the finite Lorentz transformation matrices and their parameters :
(1+q+) (1-f-g-) L
q+
=
[",
=
Ml*
;I,
g) =
g-
(1 +q2)1/2
=
[:'
-3,
(1 +gZ)l/Z
'
Ccr").L' =
FObe%,
FINITE LORENTZ
39
TRANSFORMATIONS
where (see [5])
(1.7) and L-L SpL
=
&l++gJ =+ _
(q+g)” q-g *
_(q_g)
[
where Lat = JCL= I,
I
(1.8) .
,
detL = jL1 = +I.
Thus, due to the known local isomorphic correspondence SO(4.C) @SO(3.C), any finite Lorentz transformation matrix L is represented as two matrix polynomials of the first order relative to q+ and g_ mutually That corresponds to the decomposition of the SO(4.C) Lie algebra into isomorphic to the SO(3.C) Lie algebra.
(1.9) N SO(3.C)@ a product of commuting. two algebras
2. The group operation LL’ = L”
(1.10) connecting the three transformation matrices L = L(q, g), L’ = L(q’, g’) and L” = L(q”, g”) belonging to the SO(4.C) group is here in one-to-one correspondence with the simple operation over the vector parameters defining the above group elements. The first obtained (see [2]-[4] for the case of the SO(3.Z) group) very simple composition law of the parameters of the Lorentz group allows us to operate in many cases solely with the vector parameters, not dealing with explicit expressions for the finite transformation matrices. It should be noticed that in the case of the SO(4.C) its two vector parameters q and g are composed independently lq
q’>
=
9
q+q’ + hn’l = 1-w’
q”*
cg
g’>
=
g
+
9
g’+ kg’1 = l-f%’
g”
by means of the same rule which holds for the SO(3.R) vector parameter example, [11).
(1.11) n (see, for
3. Any Lorentz transformation matrix L = L(q, q*) E SO(3.1) (see [2]-[4]) can be always written down as a product of two simple transformations of evident meaning L = L(q, q*) = L(n, n)L(iu, -iu),
q =, L(iu, -iu)
Here the transformation Ql
= -9:
(1.12)
q* = (n, - iu>.
defined by a pure imaginary vector (101< 1)
=
<-q*,
l+(l+(_q*,q}Z)p-
Q>
=
iu
(1.13)
40
A. A. BOGUSH
describes the hyperbolic
rotation
and F. I. FEDOROV
with imaginary rotation
angle 0 around
the axis ui
= n/In1 u = lulu1, corresponding
1111 = tanhi
(1.14)
to the motion of two inertial frames with the relative velocity v (Iv] < 1) v=
-i<-q*,
q),
u=
V
l+(l-VZ)“’
(1.15)
’
The matrix Z(n, n) in (1.12) is defined by the pure real vector q. = q$ = (q, -io)
=
q(l +q*2)1’2+q*(1
and describes the three-dimensional space rotation tion n, = n/In] with real rotation angle q: In] = tang,
n = Inlnl,
+qy
= n
(1 +q*)l’* + (1 +q*)l’*
(1.16)
(see [I] and (1.5)) around the direc-
0 Q InI G co.
(1.17)
Thus, the physical meaning of the introduced vector parameters of the SO(3.Z) group is demonstrated. 4. Within the framework of the vector parametrization SO(3.Z) the simple condition ([5]) l-v2
=
1+(-q*,
q>* =
11+q*l*
(l+Jq)*)*
for the real Lorentz group
# O
(1.18)
I’Sintroduced which excludes ail the values of the complex vector q’ which correspond to the physically impossible velocity values v (see (1.15)) equal to the light velocity z, c = 1. Condition (1.18) is responsible for the non-compactness of the SO(3.Z) group, icludes all the values of q ([5]) 1.
1 +q2 = 0,
2.
]q/* = co,
q* = a2+b2 c 00;
lq*]* = (a2-b2)2+4(ab)2
< M
(1.19)
under which the matrix L = L(q, q*) may have the infinite large elements and lose, thereby, its meaning as the transformation matrix belonging to the Lorentz group. 5. The general problem of constructing all the finite-dimensional (irreducible and reducible) representation operators taking the explicit polynomial form has been solved for the Lorentz group SO(3.Z) (see [9], [ll]) and for the SO(4.C) group (see [6], [7]) in very simple way by making the direct use of the results obtained previously for the SO(3.R) group (see [IO]). In accordance with the local isomorphic correspondence SO(4.C) N SO(3.c)@ @SO(3.c), the finite-dimensional representation operators T = T,~J, (q, g) for the SO(4.C) group have the form 7’u.r,(q, g) = z(q)@ TV(g)
(1.20)
FINITE
LORENTZ
TRANSFORMATIONS
41
where
PI(QJ T&l)
are
the
obtained
SO(3.C)
=
(1
representation
for SO(3.R)
PI&>
+q2)l
’
T&d
operators
with
in [IO] and PI = PI(~) 21
h(G) = are the 21 (21’) polynomials equations of the form
=
the
(1
(1.21)
+g2)l
weights
I, I’ = 0, l/2,
1, 3/2, . . .
(P&))
21 @(q2)(“-k’/2
cc k=O
(4)”
(1.22)
n=k
to 4 = 4.3. (i = g,,J,) satisfying
with respect
the minimal
21+1 af’(q2)‘2’+1-k)/2(il)k
=
0,
(1.23)
J. (a = 1, 2, 3) being the infinitesimal representation operators for the SO(3). The coefficients b$, in (1.22) are expressed in terms of the coefficients a$” in (1.23) by means of the recurrence relations obtained in [IO], [Ill. 6. From the general formula (1.20) the convenient expressions have obtained for the unimodular 2 x2-matrices (see (1.21) for I, I’ = l/2)
been
directly
(1.24)
T,
defining
in the convenient WI,
12 (g)
4d
vector-parametrical
g) - A(qEMg)
the known correspondence between covering group SL(2.C)@SL(2.C). example, [ 141, [ 151) 2 x 2-matrices A = apa,, with the subsidiary
=
(Uq,
=
(1.25)
(11+;q,2
form
([6], [A)
g)z = z’ -
A(q)z&g)
= Z’)
(1.26)
the complex Lorentz group SO(4.C) and its universal Contrary to the usually used in this case (see, for
B = b,a,,
(a,) = (ch -if,),
(1.27)
conditions ai-a2
= 1>
b$-b2
= 1,
(1.28)
the matrices A = A(q) (1.24) and B = A(g) (1.25) are defined by means of the independent parameters of the group SO(4.C). Thereby, it is not necessary to establish the correspondence between the complex Lorentz group transformation matrices L and its universal covering group ones as is usually done (see, for example, [14], [15]). It is also evident that the general formulas (1.20), (1.21) give the possible finitedimensional representation operators of the universal covering group SL(Z.C)@SL(2.C).
A. A. BOGUSH
42
and F. 1. FEDOROV
7. There exists the direct and natural correspondence between the introduced parameters q and g of the complex Lorentz group and the complex quatemions By considering the known relation t’ = Atias the definition of the transformation 2 = z,e,+ze
(1.29)
law for the complex quaternions
= z,+z,
eieo = eoei = ei,
vector ([13]).
ez=
Z = z,e,-ze 1,
= z,-z, (1.30)
ei ek 1= c&lei - 6ike, ,
and by taking into account the normalization conditions AA= 1 and Bg= 1 for the quaternions in (1.29), one can introduce the explicit normalized quatemions in the form (see (1.24), (1.25)) A = Ao+A = (l+q2)I,2
= A(q),
1+g
B = Bo+B = (l+g2)‘,2
Using these definitions and the known multiplication immediately obtain z’ = z;+z’
= A(q)&&
or after going over to the box-matrix
= A(g).
(1.31)
rules for the quaternions,
one can
= (n+dz,)+(~z+z,u),
(1.32)
form
[:;I=[:;][:]-
(1.33)
Here the matrix l-%+(q+gY+q~g+g~q
G
:I =+[
-(Q-g)--[%?I y = (1 +qzy,
p = (1 +gy
q-g-h.4
I+%%!
13
(1.34)
coincides with the expanded expression (see [2] for the case of the SO(3.Z) group) for the transformation matrix L = L(q, g) (1.6) of the complex Lorentz group SO(4.c) written in the vector-parametrical form. 8. Due to the relations (1.7) completely defining all the allowed values of the SO(4.c) group complex vector parameters q and g, the vector parametrization ensures essentially the explicit regular parametrization of the finite transformation matrices L = L(q, g) (1.6), (1.34) in the theory of the complex Lorentz group SO(4.C), its universal covering group SL(2.C)@SL(2.C) (1.24)-(1.26) and their arbitrary (reducible and irreducible) finite-dimensional representations (1.20)-( 1.22). This means that the vector parametrization ensures automatically the validity of all the basic requirements which are usually imposed by introducing specially the regular parametrization of the Lorentz group in physics (see, for example, [151). These requirements are fulfilled here not only locally for infinite small transformations but also for all the allowed finite transformations of the SO(4.C) group,
FINITE
LORENTZ
2. Simple (plane) Lorentz transformations
43
TRANSFORMATIONS
( [ 161~[241)
In order to apply the finite Lorentz transformations in elementary particle theory it is necessary, first of all, to express Lorentz matrices and corresponding group representation operators in terms of the quantities used by description of the particle states, that is, in terms of the energy-momentum and spin vectors. The problem is solved very simply by use of the vector parametrization under consideration: we need to find the corresponding values of the vector parameters. Making use of the decomposition (1.12), one can find (see [16]) that the proper Lorentz transformations L = L(q, q*) connecting two arbitrary fixed four-vectors p = (p, ipo) and p’ = (p’, i&J (p’ = p”) uq,
Q*)P = P’
are defined, in general, by the vector parameters q = (Po+Kdn+3p-p’+
(2.1)
([16]) in, P+P’I)
po+p;,-i(np-up’)
’
(2.2)
where n is an arbitrary real vector parameter of the SO(3.R) group. The significance of this result consists in its generality. Thus, substituting the value p = (0,ipo= im) (p = 0, m is the“rest mass of the particle) into formula (2.2), we obtain the general definition (up to an arbitrary space rotation) of the vector parameters giving so-called boost transformations. The simple boost transformation is obtained when (n = 0) q=iu=-ip’.
(2.3)
m-+-p0
Another important
consequence follows when taking
p =p’, uq, This leads to the general formula ([16]) Q
=n+ik!!i , PO
q*b
= P-
P=p,=v PO
(2.4)
(2.5)
for the vector parameters of the little Lorentz group transformations which is now well known and widely used in relativistic kinematics. With the help of the composition law (see (1.11)) the vector parameter of the Wigner rotation can also be defined in a simple form ([17]) q=q*=
[UPI Po+m-up
.
Gw
It is easy to verify that the vector parameters of the space rotation q. = qb = n (1.16), pure Lorentz transformations (boost) q1 = -q: = iu (1.13), little Lorentz groups (2.5) and Wigner rotation (2.6) satisfy the condition (see 171, [18]) q2 = q*2,
(2.7)
44
A. A. BOGUSH and F. I. FEDOROV
defining so-called simple Lorentz transformations. Thus all the finite Lorentz transformations, in fact, used in physics, may be considered in a unique way as the simple transformations of the Lorentz group. This is a base for developing the general theory of simple transformations, forming the foundation of the finite transformation approach in the high orthogonal group theory. It should be noticed that the simple transformations of the Lorentz group (such as for arbitrary SO(n.C) group) have some attractive properties. The simple transformations of an arbitrary SO(n.C) group are parametrized in terms of so-called simple anti-symmetrical n x n matrices by means of the universal relations. Any simple transformation can always be written down in the universal (but not single-valued) form of the so-called pldne transformation matrix. For example, in the case of the simple Lorentz transformations we obtain ([19])
where
* (P$-P’) + P' . P L = uq, q*) = L[P’, PI = I-2 -tP+P’) (P+p,)2 ---7-’ P
(2.8) (2.9)
The plane transformations L = L[p’,p] (2.8) have a natural geometrical interpretation. They describe the rotations in the plane defined by two vectors p andp’ (by matrix p' *p-p *p’) with rotation angle 8 which is defined by the relations ([7], [18]) (2.10) where (2.11) is the parameter 4 x 4 matrix corresponding to the vector parameter (2.9). An arbitrary finite-dimensional representation operator T = Tcr.r,(q, q*) of the Lorentz group SO(3.1) corresponding to the simple (plane) transformation L = L(q, q*) = L($ (q2 = q*2) has the same universal polynomial structure as the corresponding reducible representation operator of the SO(3.C) group (see (1.21), (1.22)) ([20], [241) (2.12) Here Jj“ are the representation (I, irreducible SO(3) (2.12), we obtain
known infinitesimal operators of the SO(3.Z) group finite-dimensional Z’) under consideration, lo is the greatest value of the weights of the representations which form the reducible one. Substituting (2.11) into ([20], [24])
T,,(q) = Zo[p’, p] = -$p2
+pp’) 2
2
“=II k=n
bE,, ( ;I ---I
)I0 (&
(2.13)
FINITE LORENTZ
45
TRANSFORMATIONS
It is noteworthy that formulas (2.12), (2.13) may be easily generalized to the case where arbitrary SO(n.C) groups are considered. Due to that, one can utihze them by constructing the finite-dimensional representation operators of general type for the SO(n.C) group in the exphcit polynomial form (see [203_[24]). 3. Some applications of the finite Lore& transformations in the elementary particle theory 1. The covariant spin state description
([25], [29])
The first problem solved with the help of the finite Lorentz transformations was connected with the description of the spin states of the moving particles. The description of free particle states with the rest mass m and spin s is conveniently accomplished by use of the relativistic wave equations of the first order in universal matrix form {fi = c = 1). (a~~~+~)~(x) = 0 (3.1) Then the particle state corresponding to the definite values of the energy-momentum four-vector p = (p, ip,) (p’ = pz -pi = -m2) and spin projection Sk can be obtained in the compact covariant form of the projective dyadic-matrix [2s] zI&= um,(- $) &(o)
=
plk
(p = Y+q, 7 being a matrix of the bilinear Lorentz-invariant ously the two algebraic matrix equations -ijhlk
=
arlk = with the compatibility
(3.2)
. plk
form) satisfying simuitane-
ml tlk,
(3.3)
Sktlk,
(3.4)
condition [i$, uJ_ = i$-a$
= 0.
(3.5)
Here - 3 = - ipPaP is the “mass operator” with the eigenvalues ml = km corresponding to two possible energy values p. = +(P~+Bz~)~‘~ and c defines the spin projection operator with 2s+ I values Sk = -s, - (s- l), . . . , s- 1, s. The projection operators a,,(- $) and /Is,(u) appearing in (3.2) are defined from the corresponding minimal equations for the operators- ip^and cr (see 1251 and also [26& When the ordinary spin projection operator (T=cm=
i - --Q&J 2
bc
(n’ = I)
(a,b,c=
1,2,3),
(3.6)
Jab being the representation inkitesimal operators of the Lorentz group SO(3.1) corresponding to the space rotation, is used, then due to, the compatibility condition (3.5) there exist only two possibilities: 1. p= 0, spin projection 2. p # 0: n = p//pi.
axis vector n may be arbitrary,
(3.7) (3.8)
46
A. A. BOGUSH and F. I. FEDOROV
In order to describe the spin states of the moving particle with arbitrarily directed projection axis, one needs to introduce more general expressions for the spin projection operator. One can obtain such a covariant operator by performing the transformation on the outgoing operator (3.6) with the help of the representation operator T = T[r’, r] corresponding to the boost transformation L[r’, r]r = r’,
r’ = (r’, jr;)
r = (0, i),
(rz = rlz = -1).
Hence for the explicitly covariant spin projection operator 0’ = - t sPYGo r; n: JQ”, after making the use of boost transformation and n’ = (n’, in;) (nZ = n” = l), we obtain 0’~
fin=
(
r;n--
(3, =
nr’ . r’ l+rA
,I -l_iEabc
1
JbC9
(3.9
obtained in this way ([27]) (3.10)
matrix L[r’, r]n = n’, where n = (n, 0) t-l=-, P’
a-i[nr’]T, z,
=
m
(3.11)
-iJa4.
This operator permits to describe the spin states of a moving particle with spin projection values in arbitrary direction. The other approach in the spin theory based on direct use of the little Lorentz group vector parameters q = c+i[cp,], p1 = p/p0 (2.5) is developed in [28], [29]. In this case the arbitrary three-dimensional real vector c, up to which the vector parameter of the little Lorentz group is defined, may be identified with the spin projection axis of arbitrary direction. 2. Direct covariant computation of the scattering amplitudes ([27], [30]-[40]) Any matrix element can be always written in the trace form (3.12) M 1-2 = %Qff’ul = Sp(Qq,. %). One of the most attractive forms of expressing the dyadic matrix Yy, . F2 from (3.12) is based on making use of the Lorentz group representation operators T(q, q*) = T(q) realizing the transition [27] T(@yl
= Fz,
!&T(-q)
= !&.
(3.13)
Then the matrix element (3.12) takes the form (11271,[36]) Ml.+1 = Sp(Qu’, . %) = Sp(Q4 W-q))
= Sp(QT(-WL).
(3.14)
Here rl, = ilSp = Y1 * PI and 11, = ilc, = u/, - p2 are the projective operators (see (3.5) and [25]) defining the free particle states before (!Pi = P(p)) and after (u/, = y’(p’)) the interaction, Q is the interaction top operator. All the operators appearing in (3.14) are expressed in terms of matrices ap of the wave equatior (3.1) for the particle under consideration. The general problem of computation of the scattering amplitudes in quantum electrodynamics has been recently solved in [33]-[36]. The following most general and simple
FINITE LORENTZ
formula is obtained for arbitrary h!i142
--
Y(p’,
47
TRANSFORMATIONS
interaction
matrix element in QED ([36])
c’)QY’(p, c) = Sp(Q&,
c)Z-(-q))
= F(M,+MC),
(3.15)
where all the spin variables c, c’ are contained only in the scalar factor F = (I/m) (2(1 +cc’) (p;+m)
and in the three-dimensional
(3.16)
(p;+m))“’
vector C _
[cc’]+i(c+c’) I+cc’
(3.17)
*
It is clear that this general result removes all the computational difficulties appearing usually in general study of the polarized particles. The generality of the approach under consideration allows to extend it to the particles with the spin value 1 (see [32], [37]) and 3/2 (see [38], [39]$. On these grounds, the general computation of the interaction matrix elements for particles with spin 0, l/2 and 1 can be performed (see [31], [29], [36], [32]). By using the above results, a wide set of concrete processes of electromagnetic interaction involving leptons and hadrons has been considered. As a consequence, some subsidiary possibilities of obtaining new information in experimental study of polarized elementary particle interaction has been established. It should be emphasized that the direct computation of the interaction matrix elements may also be applied when studying other interaction processes (not only electromagnetic) where a conventional perturbation method is not applicable. Thus, for example, it very often happened that by considering the binary reaction 1+2 + 3+4,
(3.18)
(a+a’ -+ a+a’)
where u and u’ are the spin values of the 1,3 and 2,4 particles one could write its scattering amplitude in the form of the following decomposition ([40]) (3.19) Here !P, = !F(p,)
are the free particle states with four-momenta p, = (p,, ipo,) (p; and spin projections s, (r = 1,2, 3,4), i$f is the known M-function, r?’ and Bf”” define some “top operators” expressed in terms of the corresponding wave equation matrices a$@ and tl(“‘). Fi(s, r) are some functions of the invariant variables &r= -(PI +p#, t = -(PI -is); responsible for the dynamics of the process under = pr’-p&
= -m:)
consideration. (3.19) can It is evident that any “kinematic factor” appearing in the decomposition be always written down according to the above approach of direct calculation of the matrix elements in the foh3wing form ([40]):
(p’, ri(“‘ujr,) (p, B;““!P2) = Sp(&@‘Y, - !&) x Sp(BI”“Y2 . !&) = Sp(bf”“r:““F/,
. i&)
= Sp(#“@““Y2
. y/_,) = @by’),
(3.20)
48
A.A.
where the obvious notations
BOGUSH and F.I.
FEDOROV
are introduced
a?) = Sp(r:“‘!Py, * !&) = (!F3T,c”?Fl),
6!“” = Sp(BI”“!P~ * FJ
(3.21)
= (FJ3pP2).
(3.22)
Thereby, the calculation of the scattering amplitudes (3.19) for binary reaction with polarized particles, for example, with spin c, u’ = 0, l/2, 1, may be reduced to utilizing the previously obtained results for the expressions of the type (3.21) and (3.22) (see [31], [29], [36], [321). 3. On the kinematic singularities of the scattering amplitudes for binary reactions 1241 As it is known, one of the possible approaches to exclude the kinematic singularities is based on establishing the relations between helicity and spinor amplitudes (see, for example, [ 141). The desired connection for binary reactions may be expressed by the following relation [24] (~4,24;
PJ,
&MP,
7 1, ; ~2 7 A2)
= ~qP4)~yp&kP;
9 Pi 3 Pi 9 Pk) WP,)
WP2)
x
x T14(-9)D~rl,(Qk>T13(-9)D~1,(9;)T12(-Q)0~1*(q~)X
Here DJfx’,(q;) are the representation function spaces under consideration d =
(3.23)
~‘+a?s:l‘w
operators of the little Lorentz group in the state defined by the corresponding vector parameters
c,+ik,Prlhr
(r = 1,2,3,4),
where the representation weight values I, coincide with the spin values s, of particles, which perform the transition of spin states IpI, s,) to the helicity states Jpr, A,) : lpr, sr)D$‘, = Ipr, A,). The appearance in (3.23) of the Lorentz group representation operators F’*)(q) corresponds to the Lorentz transformations L(q) performing the transition of L(q)p, = pi. Introducing the two sets of basic orts n, = k,/(k:)‘J’,
where a, b = 1,2,3
n: = k:/(k:‘)‘/‘,
(3.24)
n,&, = 8&r
and
k, = p&ri,
kl = (1-n,.n,)p,,
kS = (l-n,*n,-n2*n2)p3,
k’, = pijim;,
k; = (l-n;
k; = (1 -n;
* ni)pi,
. n’, -n;.
(3.25) n;)p;,
one can perform the desired transition by means of transformations 2%,) = WIJ)
Wq2)
(3.26)
mll)
with the parameter 4 x 4-matrices defining the plane Lorentz transformations 0
0
il
=
n1an,-n,.n, 1 +n, Fz,
0
,
42 = n;*n2-n2. 1 +n,n;-
n’2 ’
63 =
n;
0
. n, -n,
1+&z;
f n’l
(see (2.11)) (3.27)
49
FINITE LORENTZ TRANSFORMATIONS
It is evident
that
when the
known
decomposition
of the spinor
M-function (3.28)
(ii being the kinematic covariants) is realized in such a way that the coefficients F&, 2, U) do not contain kinematic singularities, then the singularities are to be in remaining parts of the amplitude. Due to ambiguity of the operators T(s) in (3.23) one can define them in such a manner that the kinematic singularities appearing in
p2, AL) by extraction
of the it?(pl, p2, p3, po) with the helicity state functions y’“‘(p,) will enter only the above transformation operators T(q). Taking into account that the representation operators T”*‘(q,) appearing in (3.26) are defined with help of the formula (2.13), we shall use the relations (see [5], [24]) (1 +q*)*_ + 0 (I+ IQI*)*
1-v* =
1+q2
or
#O
(3.29)
1+ 141*
defining the allowed values of the complex vector parameters q in the case where the simple (plane) Lorentz transformations (q* = q**) are considered. Analysing the limitations which follow from the general condition (3.29) for the values of the square of the parameters (3.27) 0 qf=
_
l-y,
f-
1
q- -
1 +tl,n,
0
-n*n;
l-f-n,4
qg = _ ’
0
l-y;
)
(3.30)
li-n,n\
using the definitions (3.24), (3.25), one can easily see that the threshold (pseudothreshold) values for the Mandelstam variables appear at such values of the vectors q,, when the corresponding Lorentz transformations and their representation operators are defined at points belonging to the light cone [24]. 4. On the proof of the Burgmann-Hull-
Wightman
THEOREM.Suppose that under the following . . . . 2” = {z>, L(q’, q’*)zj
= z;,
theorem ([123, 171)
transformation
L(q’, q’*) E SO(3.1)
of n complex
vectors
0’ = 1) 2, .‘.) n),
zl
,
...
(3.31)
the function f of these vectors is transformed by means of some irreducibleJinite-dimensional representation (I, I’) of the proper redI Lorentz group SO(3.1) f@(q’,
P’*)z* 3 *.*f Uq’, q’*)zj,
..-, Jqq’, q’*)z,) = Ttl,I*,(q’, q’*)f(z,,
which is analytic
in the future
..-) Zj, ...) i.)
(3,32)
tube
T.’ = (zj = .Uj+iyj, Xj E 1/(3.1), yj E V+(3.1)),
(3.33)
50
A. A. BOGUSH
where V(3.1) is the Minkowski 01” = Y2--Y~(0,_03. Then the function f(z) tube domain
and F. I. FEDOROV
space and V+(3.1)
is the domain inside future
allows the single-valued y-n =
u
analytic
L(q,g)(z,,
. . ..zj.
continuation
light
cone
into the extended (3.34)
. . ..z.)
LESO(4.C)
which transforms f(L(q9g)z19
according
to the formula
..-*L(q*$)zj3
...9L(q9dzn)=
T~f,l*)(q,glf(zl,
.*a9 97
..-rZJ,
where L = L(q, g) E SO(4.C) is an arbitrary complex Lorentz transformation is the corresponding transformation in the finite-dimensional representation this complex group.
(3.35)
and T,&q,g) space (I, I’) of
Unlike in the usual approach in the proof of this theorem (see, for example, [15]) taking into account the regularity of all the jinite transformations L(q, g) E SO(4.C) and their finite-dimensional representation operators T,,,,., (q, g), one can conclude that the relation (see (3.35)) 3g&,,,)
f(Uq,,
= T,t.dq,,
will be valid not only for the smaZ1 but also for all the finite mations L(q,, , gL2) E SO(4.C) satisfying the conditions Qq,,,
g,,)-(1,
=
~(217
(3.36)
9gulf&,,)
=(I),
~(2)
complex
Lorentz
transfor-
(3.37)
E X.
In virtue of the above regularity of the finite representation operators T~~,~,,(q, g), the analyticity of the right-hand side expression in (3.35), both inside and outside of the outgoing domain T,,+ (3.33) including the points of the extended tube F,, (3.34) for ah L(q, g) E SO(4.C) follows immediately. Hence, we can consider the expression To, &q, g) x x f(z) in (3.35) as a definition of the analytic continuation f(L(q, g)z) of the function f(z) by means of the holomorphic function for all points in F” (3.34). Let z(r) and zt2) be the two points in T.’ (3.33) chosen in such a way that there exist two matrices L(q, , g,) and L(q,, g2) satisfying the conditions L(q1, g&o,
= z,
Lcq2,g,)z(2,
=
z(l),
z,
~(2)
E Tn+
9
Due to the above-mentioned property we can now define the analytical the function f of point z (3.34) by means of the relations f(U%
3fz,)qu)
= T,l,P,(ql9 gJf(z,1J,
f(%
7g,)+,)
= T,,.dq,
(3.38)
ZEF,.
continuation
of
(3.39)
7 gJf(zm)
with analytic functions on the right-hand side. Then from the existence of the two transformations (3.37) the existence of the finite complex Lorentz transformation WI,,, $2
.a,) =
(-n2,s,>t
=
L(--q,,-g,)L(q1, &I2
fzl)? =
(--g2tgJ
(3.40)
FINITE LORENTZ TRANSFORMATIONS
and its representation
51
operator
(3.41) To,&l12, g,,) = Tc,19(--q,, --g*)~r,1&l1, 8,) always follow, for which the relation (3.36) will be valid. Hence, using the group properties of the transformations in question, we immediately obtain the equality f(Mi* 3g&,2,)
= ~,LP,olZ 3gd_f@,2,) = ~w,(Qz,
&l_fbwll2
9g&o,)
= ~~r.I.hh~ &)G1*1&I12~la,l.&,,! = ~,I.d%
3 &lf(r,J
= .f(J%h 9gl)z(, ,)
(3.42)
proving the single-valuedness of the obtained analytic continuation. Hence, by using the vector parametrization of the complex Lorentz group SO(4.C) and by taking into account the essential regularity of the finite transformations of this group and their finite-dimensional representation operators for all the allowed values of the group vector parameters q and g (see (1.7)), we obtain an evident simplification of the proof of the Bargmann-Hall-Wightman theorem in comparison with the usual one (see [15], [41], [42]). There is no need to perform the subsidiary constructions, connected with the proof of the relation (3.36) and requiring to take into account the topological properties of the complex Lorentz group, to introduce the equivalence classes and to use the normal form for the transformation matrices of this group. Finally, it should be emphasized that the examples given above are far from exhausting all the possible applications of the finite Lorentz transformations in the elementary particle theory by making use of the vector parametrization under consideration. REFERENCES [1] F. I. Fedorov: Doklady Akademii Nauk BSSR 2 (1958) 208. [2] -: DAN BSSR 5 (1961) 101. [3] -: ibid. 5 (1961) 194. [4] -: Doklady Akademii Nauk SSSR 143 (1962) 56. [5] -: DAN BSSR 17 (1973) 208. [6] A. A. Bogush, F. I. Fedorov: Teoreticheskajy i matematicheskaja physica 13 (1972) 67. I 171 -, -: Complex Lorent? group and its applications in, axiomatic quantum field theory, Lect. at High Energy Phys. School’(BSSR, Home1 1973)iDubna JINR 1,2-7642jbubna 1973, p. 5. [8] $. I. Fedorov: Some methods of relativistic kinematics, Lst. at Hight Energy Phys. School (BSSR / Homel, 1971); Dubna JINR 2-6371, Dubna 1972, p. 3. 191 -: ‘Vesti Akademii Nauk BSSR. ser. phys.-mathem. nauk N 2 (1967) 85. [lOI F. I. Fedorov, E. E. Tcharev: VAN BSSR, s.ph.-m.n. 1 (1967) 101. [ll] E. E. Tcharev, F. I. Fedorov: Yud. Phys. 5 (1967) 1112. 1121 A.A. Bogush, A.A. Mironenko: K&V BSSR, s. ph.-m.n. 6 (1973) 56. [131 A. A. Bogush, Yu. A. Kurochkin, A. K. Lapkovski, F. I. Fedorov: VAN BSSR, s.ph.-m.n. 1 (1976) 69. 1141 Yu. V. Novoshilov: Introduction to the elementary particle theory, Nauka, Moscow 1972. [I51 N. N. Bogolubov, A. A. Logunov, I. T. Todorov: Foundation of the axiomatic approach in quantum field theory, Nauka, Moscow 1969.
52
A. A. BOGUSH
and F. I. FEDOROV
[16] F. I. Fedorov, A. A. Bogush: DAN BSSR 6 (1962) 690. [17] F. I. Fedorov: TMPH 2 (1970) 343. [18] A.A. Bogush, L. G. Moroz, F. I. Fedorov: VAN BSSR, s.ph.-m.n 4 (1970) 85. [19] A. A. Bogush, F. I. Fedorov: DAN BSSR 5 (1961) 241. [20] A. A. Bogush, F. I. Fedorov, A. M. Fedorowych: DAN SSSR 214 (1974) 85. [21] A. A. Bogush, F. I. Fedorov: ibid. 206 (1972) 1033. [22] A.A. Bogush: DAN ESSR 17 (1974) 996. [23] A. A. Bogush, V. S. Otchik, F. I. Fedorov: DAN SSSR 227 (1976) 265. [24] A. A. Bogush, Yu. A. Kurochkin: VAN BSSR. s.ph.-m.n. 5 (1974) 75. [25] F. I. Fedorov: JETP 35 (1958) 483. [26] A. A. Bogush, L. G. Moroz: Introduction to the classicalfield theory, Nauka i technika, Minsk 1968. [27] A. A. Bogush, F. I. Fedorov: VAN BSSR. s.ph.-t.n. ‘2 (1962) 26. [28] F. I. Fedorov, E. E. Tcharev: Yad. Phys. 7 (1968) 189. [29] F. I. Fedorov: VAN BSSR, s.ph.-m.n. 1 (1974) 86. [30] A.A. Bogush: ibid. 2 (1964) 29. [31] -: DAN BSSR 7 (1963) 520. 1321 A.A. Bogush, A.I. Bolsun: DAN SSSR. 155 (1964) 1046. [33] F. I. Fedorov: Yadphys. 17 (1973). 882. 1341 -: TMPH 18 (1974) 329. [35] -: VAN BSSR, s.ph.-m.n. 2 (1974) 58. 1361 -: ibid. 3 (197.5) 51. [37] F.I. Fedorov, S. Engelman: ibid. 3 (1974) 66. [38] V. K. Gronski: Autorefirat of rhe dissertation. Minsk 1965. [39] N. I. Gurin, F. I. Fedorov: VAN BSSR, s. ph.-m.+. 4 (1976) 72. [40] A. A. Bogush, Yu.A. Kurochkin, F. I. Fedorov: DAN SSSR 231 (1976) 312. [41] R. F. Streater, A. S. Wightman: CPT, spin and statistics and all that, New York 1964. [42] R . Jost : The general theory of quantizied fields. Prov., Isl., 1965.
11 (1977)
REPORTS
ON MATHEMATICAL
FINITE LORENTZ QUANTUM
No. 1
PNYSICS
TRANSFORMATIONS FIELD THEORY*
IN
A. A. BOGUSH and F. I. FEDOROV Academy of Sciences of Byeolorussian S.S.R., Institute of Physics, Minsk, U.S.S.R. (Received March 13, 1976)
The basic relations defining the vector parametrization of the complex Lorentz group SO(4.C) and its subgroups are presented. The main attractive properties of this parametrization are demonstrated. The simple solutions of some basic problems in elementary particle theory based on direct use of the finite Lorentz transformations (covariant spin state description, direct coviariant calculation and exclusion of the kinematic singularities in scattering amplitudes for polarized particles, proof of the Bargmann-Hall-Wightman theorem in axiomatic QFIJ are given. Introduction There exist many complicated problems in the elementary particle theory which can be solved in simple and natural way by making direct use of the finite symmetry group transformations, but not the infinite small ones. Unfortunately, at this time, contrary to the known and very well worked out infinitesimal approach, the general method of the finite transformations in the continuous group theory is far from satisfying all current needs and requirements of theoretical physics. The effectivity of such a method depends on the convenient choice of the group parameters. The vector parametrization forming
the foundation of the present treatment gives an example of such a parametrization finite transformations in the theory of the space-time symmetry groups. 1. Complex Lore&
The homogeneous
group and its vector parametrization
of
[l]-[13]
proper Lorentz group can be considered as a group SO(3.Z) of
the orthogonal transformation matrices L (L,f = LL = I, detL = IL1= +I) in the four-dimensional space V(3.Z) with one imaginary coordinate (x4 = it, c = 1). Thereby the SO(3.Z) group can be identified with one of the possible subgroups of the SO(4.C). In framework of the vector parametrization under consideration (see [6], [7]) the SO(4.C) group is parametrized by means of two independent complex three-dimensional vectors q = a+ib and g = c+id. -\
* Extended version of this paper will appear in the Proceedings held in Karpacz, February 1976. L371
of XIII Winter School of Physics
A. A. BOGUSH and F. I. FEDOROV
38
First of all it should be noticed that by simple redefinition of the group vector parameters and by taking into account the corresponding limitations for their allowed values, one can extend the results obtained for one orthogonal group in four-dimensional space and its finite-dimensional representations to all other groups of the same dimensionality and their subgroups. This is illustrated schematically by the following table (see [S], [7j): L = Z(% $9, (a) SO(4.C):
(b) SO(3.Z):
q = afib, g = c+id, 1+(-q*, s>” # 0, 1+(-g”, gy # 0; L = L(q, y”), g =* q* = a-ib, q = ai-ib, l-9 = l+(-q*,qy # 0; L = L(a,c),
(c) SO(4.R):
(d) SO(3.C):
q=q*=
a,
g=eg*=c,
L -_
L I
0.3)
O(d 0
O(q) E SO(3.C),
1’
0
q = g = a+ib,
(1.4)
I +<-q”, q)’ # 0; L = L(n, n), O(n) 0
I
L= [ 0 (e) SO(3.R);
1’
n’+ (n”>” ON) = 1f2--lfn2 E s0(3.~),
q = q* = g = g* z n,
WI
0 < InI < co.
The usefullness of the vector parametrization main properties :
can be demonstrated
by listing their
1. The vector parametrization of the SO(4.C) group gives the simple relations establishing the one-to-one correspondence between the finite Lorentz transformation matrices and their parameters :
(1+q+) (1-f-g-) L
q+
=
[",
=
Ml*
;I,
g) =
g-
(1 +q2)1/2
=
[:'
-3,
(1 +gZ)l/Z
'
Ccr").L' =
FObe%,
FINITE LORENTZ
39
TRANSFORMATIONS
where (see [5])
(1.7) and L-L SpL
=
&l++gJ =+ _
(q+g)” q-g *
_(q_g)
[
where Lat = JCL= I,
I
(1.8) .
,
detL = jL1 = +I.
Thus, due to the known local isomorphic correspondence SO(4.C) @SO(3.C), any finite Lorentz transformation matrix L is represented as two matrix polynomials of the first order relative to q+ and g_ mutually That corresponds to the decomposition of the SO(4.C) Lie algebra into isomorphic to the SO(3.C) Lie algebra.
(1.9) N SO(3.C)@ a product of commuting. two algebras
2. The group operation LL’ = L”
(1.10) connecting the three transformation matrices L = L(q, g), L’ = L(q’, g’) and L” = L(q”, g”) belonging to the SO(4.C) group is here in one-to-one correspondence with the simple operation over the vector parameters defining the above group elements. The first obtained (see [2]-[4] for the case of the SO(3.Z) group) very simple composition law of the parameters of the Lorentz group allows us to operate in many cases solely with the vector parameters, not dealing with explicit expressions for the finite transformation matrices. It should be noticed that in the case of the SO(4.C) its two vector parameters q and g are composed independently lq
q’>
=
9
q+q’ + hn’l = 1-w’
q”*
cg
g’>
=
g
+
9
g’+ kg’1 = l-f%’
g”
by means of the same rule which holds for the SO(3.R) vector parameter example, [11).
(1.11) n (see, for
3. Any Lorentz transformation matrix L = L(q, q*) E SO(3.1) (see [2]-[4]) can be always written down as a product of two simple transformations of evident meaning L = L(q, q*) = L(n, n)L(iu, -iu),
q =
Here the transformation Ql
= -9:
(1.12)
q* = (n, - iu>.
defined by a pure imaginary vector (101< 1)
=
<-q*,
l+(l+(_q*,q}Z)p-
Q>
=
iu
(1.13)
40
A. A. BOGUSH
describes the hyperbolic
rotation
and F. I. FEDOROV
with imaginary rotation
angle 0 around
the axis ui
= n/In1 u = lulu1, corresponding
1111 = tanhi
(1.14)
to the motion of two inertial frames with the relative velocity v (Iv] < 1) v=
-i<-q*,
q),
u=
V
l+(l-VZ)“’
(1.15)
’
The matrix Z(n, n) in (1.12) is defined by the pure real vector q. = q$ = (q, -io)
=
q(l +q*2)1’2+q*(1
and describes the three-dimensional space rotation tion n, = n/In] with real rotation angle q: In] = tang,
n = Inlnl,
+qy
= n
(1 +q*)l’* + (1 +q*)l’*
(1.16)
(see [I] and (1.5)) around the direc-
0 Q InI G co.
(1.17)
Thus, the physical meaning of the introduced vector parameters of the SO(3.Z) group is demonstrated. 4. Within the framework of the vector parametrization SO(3.Z) the simple condition ([5]) l-v2
=
1+(-q*,
q>* =
11+q*l*
(l+Jq)*)*
for the real Lorentz group
# O
(1.18)
I’Sintroduced which excludes ail the values of the complex vector q’ which correspond to the physically impossible velocity values v (see (1.15)) equal to the light velocity z, c = 1. Condition (1.18) is responsible for the non-compactness of the SO(3.Z) group, icludes all the values of q ([5]) 1.
1 +q2 = 0,
2.
]q/* = co,
q* = a2+b2 c 00;
lq*]* = (a2-b2)2+4(ab)2
< M
(1.19)
under which the matrix L = L(q, q*) may have the infinite large elements and lose, thereby, its meaning as the transformation matrix belonging to the Lorentz group. 5. The general problem of constructing all the finite-dimensional (irreducible and reducible) representation operators taking the explicit polynomial form has been solved for the Lorentz group SO(3.Z) (see [9], [ll]) and for the SO(4.C) group (see [6], [7]) in very simple way by making the direct use of the results obtained previously for the SO(3.R) group (see [IO]). In accordance with the local isomorphic correspondence SO(4.C) N SO(3.c)@ @SO(3.c), the finite-dimensional representation operators T = T,~J, (q, g) for the SO(4.C) group have the form 7’u.r,(q, g) = z(q)@ TV(g)
(1.20)
FINITE
LORENTZ
TRANSFORMATIONS
41
where
PI(QJ T&l)
are
the
obtained
SO(3.C)
=
(1
representation
for SO(3.R)
PI&>
+q2)l
’
T&d
operators
with
in [IO] and PI = PI(~) 21
h(G) = are the 21 (21’) polynomials equations of the form
=
the
(1
(1.21)
+g2)l
weights
I, I’ = 0, l/2,
1, 3/2, . . .
(P&))
21 @(q2)(“-k’/2
cc k=O
(4)”
(1.22)
n=k
to 4 = 4.3. (i = g,,J,) satisfying
with respect
the minimal
21+1 af’(q2)‘2’+1-k)/2(il)k
=
0,
(1.23)
J. (a = 1, 2, 3) being the infinitesimal representation operators for the SO(3). The coefficients b$, in (1.22) are expressed in terms of the coefficients a$” in (1.23) by means of the recurrence relations obtained in [IO], [Ill. 6. From the general formula (1.20) the convenient expressions have obtained for the unimodular 2 x2-matrices (see (1.21) for I, I’ = l/2)
been
directly
(1.24)
T,
defining
in the convenient WI,
12 (g)
4d
vector-parametrical
g) - A(qEMg)
the known correspondence between covering group SL(2.C)@SL(2.C). example, [ 141, [ 151) 2 x 2-matrices A = apa,, with the subsidiary
=
(Uq,
=
(1.25)
(11+;q,2
form
([6], [A)
g)z = z’ -
A(q)z&g)
= Z’)
(1.26)
the complex Lorentz group SO(4.C) and its universal Contrary to the usually used in this case (see, for
B = b,a,,
(a,) = (ch -if,),
(1.27)
conditions ai-a2
= 1>
b$-b2
= 1,
(1.28)
the matrices A = A(q) (1.24) and B = A(g) (1.25) are defined by means of the independent parameters of the group SO(4.C). Thereby, it is not necessary to establish the correspondence between the complex Lorentz group transformation matrices L and its universal covering group ones as is usually done (see, for example, [14], [15]). It is also evident that the general formulas (1.20), (1.21) give the possible finitedimensional representation operators of the universal covering group SL(Z.C)@SL(2.C).
A. A. BOGUSH
42
and F. 1. FEDOROV
7. There exists the direct and natural correspondence between the introduced parameters q and g of the complex Lorentz group and the complex quatemions By considering the known relation t’ = Atias the definition of the transformation 2 = z,e,+ze
(1.29)
law for the complex quaternions
= z,+z,
eieo = eoei = ei,
vector ([13]).
ez=
Z = z,e,-ze 1,
= z,-z, (1.30)
ei ek 1= c&lei - 6ike, ,
and by taking into account the normalization conditions AA= 1 and Bg= 1 for the quaternions in (1.29), one can introduce the explicit normalized quatemions in the form (see (1.24), (1.25)) A = Ao+A = (l+q2)I,2
= A(q),
1+g
B = Bo+B = (l+g2)‘,2
Using these definitions and the known multiplication immediately obtain z’ = z;+z’
= A(q)&&
or after going over to the box-matrix
= A(g).
(1.31)
rules for the quaternions,
one can
= (n+dz,)+(~z+z,u),
(1.32)
form
[:;I=[:;][:]-
(1.33)
Here the matrix l-%+(q+gY+q~g+g~q
G
:I =+[
-(Q-g)--[%?I y = (1 +qzy,
p = (1 +gy
q-g-h.4
I+%%!
13
(1.34)
coincides with the expanded expression (see [2] for the case of the SO(3.Z) group) for the transformation matrix L = L(q, g) (1.6) of the complex Lorentz group SO(4.c) written in the vector-parametrical form. 8. Due to the relations (1.7) completely defining all the allowed values of the SO(4.c) group complex vector parameters q and g, the vector parametrization ensures essentially the explicit regular parametrization of the finite transformation matrices L = L(q, g) (1.6), (1.34) in the theory of the complex Lorentz group SO(4.C), its universal covering group SL(2.C)@SL(2.C) (1.24)-(1.26) and their arbitrary (reducible and irreducible) finite-dimensional representations (1.20)-( 1.22). This means that the vector parametrization ensures automatically the validity of all the basic requirements which are usually imposed by introducing specially the regular parametrization of the Lorentz group in physics (see, for example, [151). These requirements are fulfilled here not only locally for infinite small transformations but also for all the allowed finite transformations of the SO(4.C) group,
FINITE
LORENTZ
2. Simple (plane) Lorentz transformations
43
TRANSFORMATIONS
( [ 161~[241)
In order to apply the finite Lorentz transformations in elementary particle theory it is necessary, first of all, to express Lorentz matrices and corresponding group representation operators in terms of the quantities used by description of the particle states, that is, in terms of the energy-momentum and spin vectors. The problem is solved very simply by use of the vector parametrization under consideration: we need to find the corresponding values of the vector parameters. Making use of the decomposition (1.12), one can find (see [16]) that the proper Lorentz transformations L = L(q, q*) connecting two arbitrary fixed four-vectors p = (p, ipo) and p’ = (p’, i&J (p’ = p”) uq,
Q*)P = P’
are defined, in general, by the vector parameters q = (Po+Kdn+3p-p’+
(2.1)
([16]) in, P+P’I)
po+p;,-i(np-up’)
’
(2.2)
where n is an arbitrary real vector parameter of the SO(3.R) group. The significance of this result consists in its generality. Thus, substituting the value p = (0,ipo= im) (p = 0, m is the“rest mass of the particle) into formula (2.2), we obtain the general definition (up to an arbitrary space rotation) of the vector parameters giving so-called boost transformations. The simple boost transformation is obtained when (n = 0) q=iu=-ip’.
(2.3)
m-+-p0
Another important
consequence follows when taking
p =p’, uq, This leads to the general formula ([16]) Q
=n+ik!!i , PO
q*b
= P-
P=p,=v PO
(2.4)
(2.5)
for the vector parameters of the little Lorentz group transformations which is now well known and widely used in relativistic kinematics. With the help of the composition law (see (1.11)) the vector parameter of the Wigner rotation can also be defined in a simple form ([17]) q=q*=
[UPI Po+m-up
.
Gw
It is easy to verify that the vector parameters of the space rotation q. = qb = n (1.16), pure Lorentz transformations (boost) q1 = -q: = iu (1.13), little Lorentz groups (2.5) and Wigner rotation (2.6) satisfy the condition (see 171, [18]) q2 = q*2,
(2.7)
44
A. A. BOGUSH and F. I. FEDOROV
defining so-called simple Lorentz transformations. Thus all the finite Lorentz transformations, in fact, used in physics, may be considered in a unique way as the simple transformations of the Lorentz group. This is a base for developing the general theory of simple transformations, forming the foundation of the finite transformation approach in the high orthogonal group theory. It should be noticed that the simple transformations of the Lorentz group (such as for arbitrary SO(n.C) group) have some attractive properties. The simple transformations of an arbitrary SO(n.C) group are parametrized in terms of so-called simple anti-symmetrical n x n matrices by means of the universal relations. Any simple transformation can always be written down in the universal (but not single-valued) form of the so-called pldne transformation matrix. For example, in the case of the simple Lorentz transformations we obtain ([19])
where
* (P$-P’) + P' . P L = uq, q*) = L[P’, PI = I-2 -tP+P’) (P+p,)2 ---7-’ P
(2.8) (2.9)
The plane transformations L = L[p’,p] (2.8) have a natural geometrical interpretation. They describe the rotations in the plane defined by two vectors p andp’ (by matrix p' *p-p *p’) with rotation angle 8 which is defined by the relations ([7], [18]) (2.10) where (2.11) is the parameter 4 x 4 matrix corresponding to the vector parameter (2.9). An arbitrary finite-dimensional representation operator T = Tcr.r,(q, q*) of the Lorentz group SO(3.1) corresponding to the simple (plane) transformation L = L(q, q*) = L($ (q2 = q*2) has the same universal polynomial structure as the corresponding reducible representation operator of the SO(3.C) group (see (1.21), (1.22)) ([20], [241) (2.12) Here Jj“ are the representation (I, irreducible SO(3) (2.12), we obtain
known infinitesimal operators of the SO(3.Z) group finite-dimensional Z’) under consideration, lo is the greatest value of the weights of the representations which form the reducible one. Substituting (2.11) into ([20], [24])
T,,(q) = Zo[p’, p] = -$p2
+pp’) 2
2
“=II k=n
bE,, ( ;I ---I
)I0 (&
(2.13)
FINITE LORENTZ
45
TRANSFORMATIONS
It is noteworthy that formulas (2.12), (2.13) may be easily generalized to the case where arbitrary SO(n.C) groups are considered. Due to that, one can utihze them by constructing the finite-dimensional representation operators of general type for the SO(n.C) group in the exphcit polynomial form (see [203_[24]). 3. Some applications of the finite Lore& transformations in the elementary particle theory 1. The covariant spin state description
([25], [29])
The first problem solved with the help of the finite Lorentz transformations was connected with the description of the spin states of the moving particles. The description of free particle states with the rest mass m and spin s is conveniently accomplished by use of the relativistic wave equations of the first order in universal matrix form {fi = c = 1). (a~~~+~)~(x) = 0 (3.1) Then the particle state corresponding to the definite values of the energy-momentum four-vector p = (p, ip,) (p’ = pz -pi = -m2) and spin projection Sk can be obtained in the compact covariant form of the projective dyadic-matrix [2s] zI&= um,(- $) &(o)
=
plk
(p = Y+q, 7 being a matrix of the bilinear Lorentz-invariant ously the two algebraic matrix equations -ijhlk
=
arlk = with the compatibility
(3.2)
. plk
form) satisfying simuitane-
ml tlk,
(3.3)
Sktlk,
(3.4)
condition [i$, uJ_ = i$-a$
= 0.
(3.5)
Here - 3 = - ipPaP is the “mass operator” with the eigenvalues ml = km corresponding to two possible energy values p. = +(P~+Bz~)~‘~ and c defines the spin projection operator with 2s+ I values Sk = -s, - (s- l), . . . , s- 1, s. The projection operators a,,(- $) and /Is,(u) appearing in (3.2) are defined from the corresponding minimal equations for the operators- ip^and cr (see 1251 and also [26& When the ordinary spin projection operator (T=cm=
i - --Q&J 2
bc
(n’ = I)
(a,b,c=
1,2,3),
(3.6)
Jab being the representation inkitesimal operators of the Lorentz group SO(3.1) corresponding to the space rotation, is used, then due to, the compatibility condition (3.5) there exist only two possibilities: 1. p= 0, spin projection 2. p # 0: n = p//pi.
axis vector n may be arbitrary,
(3.7) (3.8)
46
A. A. BOGUSH and F. I. FEDOROV
In order to describe the spin states of the moving particle with arbitrarily directed projection axis, one needs to introduce more general expressions for the spin projection operator. One can obtain such a covariant operator by performing the transformation on the outgoing operator (3.6) with the help of the representation operator T = T[r’, r] corresponding to the boost transformation L[r’, r]r = r’,
r’ = (r’, jr;)
r = (0, i),
(rz = rlz = -1).
Hence for the explicitly covariant spin projection operator 0’ = - t sPYGo r; n: JQ”, after making the use of boost transformation and n’ = (n’, in;) (nZ = n” = l), we obtain 0’~
fin=
(
r;n--
(3, =
nr’ . r’ l+rA
,I -l_iEabc
1
JbC9
(3.9
obtained in this way ([27]) (3.10)
matrix L[r’, r]n = n’, where n = (n, 0) t-l=-, P’
a-i[nr’]T, z,
=
m
(3.11)
-iJa4.
This operator permits to describe the spin states of a moving particle with spin projection values in arbitrary direction. The other approach in the spin theory based on direct use of the little Lorentz group vector parameters q = c+i[cp,], p1 = p/p0 (2.5) is developed in [28], [29]. In this case the arbitrary three-dimensional real vector c, up to which the vector parameter of the little Lorentz group is defined, may be identified with the spin projection axis of arbitrary direction. 2. Direct covariant computation of the scattering amplitudes ([27], [30]-[40]) Any matrix element can be always written in the trace form (3.12) M 1-2 = %Qff’ul = Sp(Qq,. %). One of the most attractive forms of expressing the dyadic matrix Yy, . F2 from (3.12) is based on making use of the Lorentz group representation operators T(q, q*) = T(q) realizing the transition [27] T(@yl
= Fz,
!&T(-q)
= !&.
(3.13)
Then the matrix element (3.12) takes the form (11271,[36]) Ml.+1 = Sp(Qu’, . %) = Sp(Q4 W-q))
= Sp(QT(-WL).
(3.14)
Here rl, = ilSp = Y1 * PI and 11, = ilc, = u/, - p2 are the projective operators (see (3.5) and [25]) defining the free particle states before (!Pi = P(p)) and after (u/, = y’(p’)) the interaction, Q is the interaction top operator. All the operators appearing in (3.14) are expressed in terms of matrices ap of the wave equatior (3.1) for the particle under consideration. The general problem of computation of the scattering amplitudes in quantum electrodynamics has been recently solved in [33]-[36]. The following most general and simple
FINITE LORENTZ
formula is obtained for arbitrary h!i142
--
Y(p’,
47
TRANSFORMATIONS
interaction
matrix element in QED ([36])
c’)QY’(p, c) = Sp(Q&,
c)Z-(-q))
= F(M,+MC),
(3.15)
where all the spin variables c, c’ are contained only in the scalar factor F = (I/m) (2(1 +cc’) (p;+m)
and in the three-dimensional
(3.16)
(p;+m))“’
vector C _
[cc’]+i(c+c’) I+cc’
(3.17)
*
It is clear that this general result removes all the computational difficulties appearing usually in general study of the polarized particles. The generality of the approach under consideration allows to extend it to the particles with the spin value 1 (see [32], [37]) and 3/2 (see [38], [39]$. On these grounds, the general computation of the interaction matrix elements for particles with spin 0, l/2 and 1 can be performed (see [31], [29], [36], [32]). By using the above results, a wide set of concrete processes of electromagnetic interaction involving leptons and hadrons has been considered. As a consequence, some subsidiary possibilities of obtaining new information in experimental study of polarized elementary particle interaction has been established. It should be emphasized that the direct computation of the interaction matrix elements may also be applied when studying other interaction processes (not only electromagnetic) where a conventional perturbation method is not applicable. Thus, for example, it very often happened that by considering the binary reaction 1+2 + 3+4,
(3.18)
(a+a’ -+ a+a’)
where u and u’ are the spin values of the 1,3 and 2,4 particles one could write its scattering amplitude in the form of the following decomposition ([40]) (3.19) Here !P, = !F(p,)
are the free particle states with four-momenta p, = (p,, ipo,) (p; and spin projections s, (r = 1,2, 3,4), i$f is the known M-function, r?’ and Bf”” define some “top operators” expressed in terms of the corresponding wave equation matrices a$@ and tl(“‘). Fi(s, r) are some functions of the invariant variables &r= -(PI +p#, t = -(PI -is); responsible for the dynamics of the process under = pr’-p&
= -m:)
consideration. (3.19) can It is evident that any “kinematic factor” appearing in the decomposition be always written down according to the above approach of direct calculation of the matrix elements in the foh3wing form ([40]):
(p’, ri(“‘ujr,) (p, B;““!P2) = Sp(&@‘Y, - !&) x Sp(BI”“Y2 . !&) = Sp(bf”“r:““F/,
. i&)
= Sp(#“@““Y2
. y/_,) = @by’),
(3.20)
48
A.A.
where the obvious notations
BOGUSH and F.I.
FEDOROV
are introduced
a?) = Sp(r:“‘!Py, * !&) = (!F3T,c”?Fl),
6!“” = Sp(BI”“!P~ * FJ
(3.21)
= (FJ3pP2).
(3.22)
Thereby, the calculation of the scattering amplitudes (3.19) for binary reaction with polarized particles, for example, with spin c, u’ = 0, l/2, 1, may be reduced to utilizing the previously obtained results for the expressions of the type (3.21) and (3.22) (see [31], [29], [36], [321). 3. On the kinematic singularities of the scattering amplitudes for binary reactions 1241 As it is known, one of the possible approaches to exclude the kinematic singularities is based on establishing the relations between helicity and spinor amplitudes (see, for example, [ 141). The desired connection for binary reactions may be expressed by the following relation [24] (~4,24;
PJ,
&MP,
7 1, ; ~2 7 A2)
= ~qP4)~yp&kP;
9 Pi 3 Pi 9 Pk) WP,)
WP2)
x
x T14(-9)D~rl,(Qk>T13(-9)D~1,(9;)T12(-Q)0~1*(q~)X
Here DJfx’,(q;) are the representation function spaces under consideration d =
(3.23)
~‘+a?s:l‘w
operators of the little Lorentz group in the state defined by the corresponding vector parameters
c,+ik,Prlhr
(r = 1,2,3,4),
where the representation weight values I, coincide with the spin values s, of particles, which perform the transition of spin states IpI, s,) to the helicity states Jpr, A,) : lpr, sr)D$‘, = Ipr, A,). The appearance in (3.23) of the Lorentz group representation operators F’*)(q) corresponds to the Lorentz transformations L(q) performing the transition of L(q)p, = pi. Introducing the two sets of basic orts n, = k,/(k:)‘J’,
where a, b = 1,2,3
n: = k:/(k:‘)‘/‘,
(3.24)
n,&, = 8&r
and
k, = p&ri,
kl = (1-n,.n,)p,,
kS = (l-n,*n,-n2*n2)p3,
k’, = pijim;,
k; = (l-n;
k; = (1 -n;
* ni)pi,
. n’, -n;.
(3.25) n;)p;,
one can perform the desired transition by means of transformations 2%,) = WIJ)
Wq2)
(3.26)
mll)
with the parameter 4 x 4-matrices defining the plane Lorentz transformations 0
0
il
=
n1an,-n,.n, 1 +n, Fz,
0
,
42 = n;*n2-n2. 1 +n,n;-
n’2 ’
63 =
n;
0
. n, -n,
1+&z;
f n’l
(see (2.11)) (3.27)
49
FINITE LORENTZ TRANSFORMATIONS
It is evident
that
when the
known
decomposition
of the spinor
M-function (3.28)
(ii being the kinematic covariants) is realized in such a way that the coefficients F&, 2, U) do not contain kinematic singularities, then the singularities are to be in remaining parts of the amplitude. Due to ambiguity of the operators T(s) in (3.23) one can define them in such a manner that the kinematic singularities appearing in
p2, AL) by extraction
of the it?(pl, p2, p3, po) with the helicity state functions y’“‘(p,) will enter only the above transformation operators T(q). Taking into account that the representation operators T”*‘(q,) appearing in (3.26) are defined with help of the formula (2.13), we shall use the relations (see [5], [24]) (1 +q*)*_ + 0 (I+ IQI*)*
1-v* =
1+q2
or
#O
(3.29)
1+ 141*
defining the allowed values of the complex vector parameters q in the case where the simple (plane) Lorentz transformations (q* = q**) are considered. Analysing the limitations which follow from the general condition (3.29) for the values of the square of the parameters (3.27) 0 qf=
_
l-y,
f-
1
q- -
1 +tl,n,
0
-n*n;
l-f-n,4
qg = _ ’
0
l-y;
)
(3.30)
li-n,n\
using the definitions (3.24), (3.25), one can easily see that the threshold (pseudothreshold) values for the Mandelstam variables appear at such values of the vectors q,, when the corresponding Lorentz transformations and their representation operators are defined at points belonging to the light cone [24]. 4. On the proof of the Burgmann-Hull-
Wightman
THEOREM.Suppose that under the following . . . . 2” = {z>, L(q’, q’*)zj
= z;,
theorem ([123, 171)
transformation
L(q’, q’*) E SO(3.1)
of n complex
vectors
0’ = 1) 2, .‘.) n),
zl
,
...
(3.31)
the function f of these vectors is transformed by means of some irreducibleJinite-dimensional representation (I, I’) of the proper redI Lorentz group SO(3.1) f@(q’,
P’*)z* 3 *.*f Uq’, q’*)zj,
..-, Jqq’, q’*)z,) = Ttl,I*,(q’, q’*)f(z,,
which is analytic
in the future
..-) Zj, ...) i.)
(3,32)
tube
T.’ = (zj = .Uj+iyj, Xj E 1/(3.1), yj E V+(3.1)),
(3.33)
50
A. A. BOGUSH
where V(3.1) is the Minkowski 01” = Y2--Y~(0,_03. Then the function f(z) tube domain
and F. I. FEDOROV
space and V+(3.1)
is the domain inside future
allows the single-valued y-n =
u
analytic
L(q,g)(z,,
. . ..zj.
continuation
light
cone
into the extended (3.34)
. . ..z.)
LESO(4.C)
which transforms f(L(q9g)z19
according
to the formula
..-*L(q*$)zj3
...9L(q9dzn)=
T~f,l*)(q,glf(zl,
.*a9 97
..-rZJ,
where L = L(q, g) E SO(4.C) is an arbitrary complex Lorentz transformation is the corresponding transformation in the finite-dimensional representation this complex group.
(3.35)
and T,&q,g) space (I, I’) of
Unlike in the usual approach in the proof of this theorem (see, for example, [15]) taking into account the regularity of all the jinite transformations L(q, g) E SO(4.C) and their finite-dimensional representation operators T,,,,., (q, g), one can conclude that the relation (see (3.35)) 3g&,,,)
f(Uq,,
= T,t.dq,,
will be valid not only for the smaZ1 but also for all the finite mations L(q,, , gL2) E SO(4.C) satisfying the conditions Qq,,,
g,,)-(1,
=
~(217
(3.36)
9gulf&,,)
=(I),
~(2)
complex
Lorentz
transfor-
(3.37)
E X.
In virtue of the above regularity of the finite representation operators T~~,~,,(q, g), the analyticity of the right-hand side expression in (3.35), both inside and outside of the outgoing domain T,,+ (3.33) including the points of the extended tube F,, (3.34) for ah L(q, g) E SO(4.C) follows immediately. Hence, we can consider the expression To, &q, g) x x f(z) in (3.35) as a definition of the analytic continuation f(L(q, g)z) of the function f(z) by means of the holomorphic function for all points in F” (3.34). Let z(r) and zt2) be the two points in T.’ (3.33) chosen in such a way that there exist two matrices L(q, , g,) and L(q,, g2) satisfying the conditions L(q1, g&o,
= z,
Lcq2,g,)z(2,
=
z(l),
z,
~(2)
E Tn+
9
Due to the above-mentioned property we can now define the analytical the function f of point z (3.34) by means of the relations f(U%
3fz,)qu)
= T,l,P,(ql9 gJf(z,1J,
f(%
7g,)+,)
= T,,.dq,
(3.38)
ZEF,.
continuation
of
(3.39)
7 gJf(zm)
with analytic functions on the right-hand side. Then from the existence of the two transformations (3.37) the existence of the finite complex Lorentz transformation WI,,, $2
.a,) =
(-n2,s,>t
=
L(--q,,-g,)L(q1, &I2
fzl)? =
(--g2tgJ
(3.40)
FINITE LORENTZ TRANSFORMATIONS
and its representation
51
operator
(3.41) To,&l12, g,,) = Tc,19(--q,, --g*)~r,1&l1, 8,) always follow, for which the relation (3.36) will be valid. Hence, using the group properties of the transformations in question, we immediately obtain the equality f(Mi* 3g&,2,)
= ~,LP,olZ 3gd_f@,2,) = ~w,(Qz,
&l_fbwll2
9g&o,)
= ~~r.I.hh~ &)G1*1&I12~la,l.&,,! = ~,I.d%
3 &lf(r,J
= .f(J%h 9gl)z(, ,)
(3.42)
proving the single-valuedness of the obtained analytic continuation. Hence, by using the vector parametrization of the complex Lorentz group SO(4.C) and by taking into account the essential regularity of the finite transformations of this group and their finite-dimensional representation operators for all the allowed values of the group vector parameters q and g (see (1.7)), we obtain an evident simplification of the proof of the Bargmann-Hall-Wightman theorem in comparison with the usual one (see [15], [41], [42]). There is no need to perform the subsidiary constructions, connected with the proof of the relation (3.36) and requiring to take into account the topological properties of the complex Lorentz group, to introduce the equivalence classes and to use the normal form for the transformation matrices of this group. Finally, it should be emphasized that the examples given above are far from exhausting all the possible applications of the finite Lorentz transformations in the elementary particle theory by making use of the vector parametrization under consideration. REFERENCES [1] F. I. Fedorov: Doklady Akademii Nauk BSSR 2 (1958) 208. [2] -: DAN BSSR 5 (1961) 101. [3] -: ibid. 5 (1961) 194. [4] -: Doklady Akademii Nauk SSSR 143 (1962) 56. [5] -: DAN BSSR 17 (1973) 208. [6] A. A. Bogush, F. I. Fedorov: Teoreticheskajy i matematicheskaja physica 13 (1972) 67. I 171 -, -: Complex Lorent? group and its applications in, axiomatic quantum field theory, Lect. at High Energy Phys. School’(BSSR, Home1 1973)iDubna JINR 1,2-7642jbubna 1973, p. 5. [8] $. I. Fedorov: Some methods of relativistic kinematics, Lst. at Hight Energy Phys. School (BSSR / Homel, 1971); Dubna JINR 2-6371, Dubna 1972, p. 3. 191 -: ‘Vesti Akademii Nauk BSSR. ser. phys.-mathem. nauk N 2 (1967) 85. [lOI F. I. Fedorov, E. E. Tcharev: VAN BSSR, s.ph.-m.n. 1 (1967) 101. [ll] E. E. Tcharev, F. I. Fedorov: Yud. Phys. 5 (1967) 1112. 1121 A.A. Bogush, A.A. Mironenko: K&V BSSR, s. ph.-m.n. 6 (1973) 56. [131 A. A. Bogush, Yu. A. Kurochkin, A. K. Lapkovski, F. I. Fedorov: VAN BSSR, s.ph.-m.n. 1 (1976) 69. 1141 Yu. V. Novoshilov: Introduction to the elementary particle theory, Nauka, Moscow 1972. [I51 N. N. Bogolubov, A. A. Logunov, I. T. Todorov: Foundation of the axiomatic approach in quantum field theory, Nauka, Moscow 1969.
52
A. A. BOGUSH
and F. I. FEDOROV
[16] F. I. Fedorov, A. A. Bogush: DAN BSSR 6 (1962) 690. [17] F. I. Fedorov: TMPH 2 (1970) 343. [18] A.A. Bogush, L. G. Moroz, F. I. Fedorov: VAN BSSR, s.ph.-m.n 4 (1970) 85. [19] A. A. Bogush, F. I. Fedorov: DAN BSSR 5 (1961) 241. [20] A. A. Bogush, F. I. Fedorov, A. M. Fedorowych: DAN SSSR 214 (1974) 85. [21] A. A. Bogush, F. I. Fedorov: ibid. 206 (1972) 1033. [22] A.A. Bogush: DAN ESSR 17 (1974) 996. [23] A. A. Bogush, V. S. Otchik, F. I. Fedorov: DAN SSSR 227 (1976) 265. [24] A. A. Bogush, Yu. A. Kurochkin: VAN BSSR. s.ph.-m.n. 5 (1974) 75. [25] F. I. Fedorov: JETP 35 (1958) 483. [26] A. A. Bogush, L. G. Moroz: Introduction to the classicalfield theory, Nauka i technika, Minsk 1968. [27] A. A. Bogush, F. I. Fedorov: VAN BSSR. s.ph.-t.n. ‘2 (1962) 26. [28] F. I. Fedorov, E. E. Tcharev: Yad. Phys. 7 (1968) 189. [29] F. I. Fedorov: VAN BSSR, s.ph.-m.n. 1 (1974) 86. [30] A.A. Bogush: ibid. 2 (1964) 29. [31] -: DAN BSSR 7 (1963) 520. 1321 A.A. Bogush, A.I. Bolsun: DAN SSSR. 155 (1964) 1046. [33] F. I. Fedorov: Yadphys. 17 (1973). 882. 1341 -: TMPH 18 (1974) 329. [35] -: VAN BSSR, s.ph.-m.n. 2 (1974) 58. 1361 -: ibid. 3 (197.5) 51. [37] F.I. Fedorov, S. Engelman: ibid. 3 (1974) 66. [38] V. K. Gronski: Autorefirat of rhe dissertation. Minsk 1965. [39] N. I. Gurin, F. I. Fedorov: VAN BSSR, s. ph.-m.+. 4 (1976) 72. [40] A. A. Bogush, Yu.A. Kurochkin, F. I. Fedorov: DAN SSSR 231 (1976) 312. [41] R. F. Streater, A. S. Wightman: CPT, spin and statistics and all that, New York 1964. [42] R . Jost : The general theory of quantizied fields. Prov., Isl., 1965.