self-coupled representations of the lorentz group and infinite ... - ICTP

REFERENUE

1

r

IC/67/58

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SELF-COUPLED REPRESENTATIONS OF THE LORENTZ GROUP AND INFINITE-COMPONENT FERMI FIELDS D. TZ. STOYANOV AND

I. T. TODOROV

1967 PIAZZA OBEROAN

TRIESTE

IC/67/58

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SELF-COUPLED REPRESENTATIONS OF THE LORENTZ GROUP AND INFINITE-COMPONENT FERMI FIELDS

D.Tz. STOYANOV* and I.T. TODOROV**

TRIESTE July 1967

* Joint List, for Nuclear koso^oh, Dubna, USSR. • On leave of absence from Joint Inst. for Nuclear Research, Dubna, USSR, and Physical Inst. of the Bulgarian Academy of Science, Sofia, Bulgaria.

ABSTRACT

A self-contained exposition is given of the theory of infinitecomponent Majorana fields.

The problem of spin and statistics for

such fields is specially analysed.

More general possibilities for in-

finite-component Fermi fields (than those transforming under an irreducible self-coupled representation of the Lorentz group) are indicated.

The formalism of ladder representations (involving Bose

creation and annihilation operators) is used throughout the paper. Most of the results are known (either from old or.-from recent publications) but are presented here in a unified way,their derivation simplified

sometimes being / . Among the few new points, we mention 1) the discussion of the quantization of Majorana fields: which specially concerns the Fourier components with space-like momenta; 2) the location of singularities of the matrix elements of some infinite-dimensional representations of SL,(2t C) for complex values of the group parameters and their relation to the failure of the axiomatic proof of TCP and spin and statistics for infinite-component fields.

-1-

TABLE OF CONTENTS Page 1.

2.

INTRODUCTION

4

1.1

Problems of infinite-component quantized fields

4

1.2

Historical remarks and references

6

1.3

Content of the present paper

8

SELF-COUPLED REPRESENTATIONS OF SL(2, C) 2. 1

Self-coupled representations of SL(2,, C) as representations of Sp(4, R)

2. 2 2. 3 3.

9

Description of Majorana representations in terms of creation and annihilation operators

12

P a i r s of coupled adjoint representations of SL(2, C)

17

QUANTIZED MAJORANA FIELD

21

3, 1

Statement of the results

21

3. 2

Complete set of solutions of the Majorana equation

22

3. 3

Quantization of the Majorana field

30

3. 4

Why the axioraatic proof of TCP and spin and statistics does not work for infinite-component fields

4.

9

CONCLUDING REMARKS

34 36

APPENDIX A Description of the irreducible representations of SL(2, C) in t e r m s of homogeneous functions

39

APPENDIX B "Schrodinger picture" for the self-coupled representations

47

Canonical basis

48

-2-

Complete set of solutions of the Majorana equation for c > 0

49

Solutions for the case

50

VO 0

APPENDIX C The ladder representation of U(2, 2) and reducible infinitecomponent fields REFERENCES

52 57

SELF-COUPLED REPRESENTATIONS OF THE LORENTZ GROUP AND INFINITE-COMPONENT FERMI FIELDS

1.

INTRODUCTION

1.1

Problems of infinite-component quantized fields In conventional relativistic quantum field theory, in both axio-

matic and Lagrangian approach,

it is assumed that a unitary re-

presentation U(a, A) of the covering of the Poincare group ISL(2, C) = SL(2,C).T 4 (A £ SL(2,C),i.e,det A = 1 , a = (a°,a) £ TA) is realized in the Hilbert space of states '"/v and that the field operators

f

(x)

transform covariantly under U(a,A)

U(ci,A)fcx> Here A

=

A(A) is the proper Lorentz transformation corresponding

to A

d.2)

{ 5^, is the 2 x 2 unit matrix,

(T. , j = 1, 2, 3 are the Pauli matrices)

and V(A) is a finite-dimensional representation of SL(2, C). It seems at first glance that this last assumption which asserts that the field has a finite number of components (and hence that V is a finite matrix), is purely technical and has no important physical idea behind it .

An actually used property of the representation

V which occurs in (1.1) appears to be the possibility of writing down an invariant non-degenerate hermitian form, in terms of the field operators

-4-

•^2,

(!• 3 )

1 / fl = / - i \ /

which means that the representation V is equivalent to its adjoint

V

=ft1/V1.

(1 4)

-

If we are dealing with strong and electromagnetic interactions only, then we have to require in addition the existence of a parity (spacereflection) operator in the representation space of V which implies that

jk - 1

i. e., that V(A) is equivalent to its parity-conjugate V(A mention that in general V(A

) j V

) (we

(A) so that (1. 5) is not a

consequence of (1.4)). Such a dropping of the requirement of finite dimensionality of V(A) looks quite attractive from the point of view of .Interpreting the present-day experimental data on elementary particle resonances. The number of resonances,which differ only with respect to spin and parity (having the same internal quantum numbers such as charge, hypercharge and isospin) is so large that it seems more advisable to consider them as a part of an infinite multiplet than to ascribe an independent field to each.* * There is of course a different and more common possibility, namely to describe all these particles and resonances as bound states of a few underlying fields (say of the quark fields) just as well as,in the case of the hydrogen atom,one is able to obtain theinfinity of spectral lines starring essentially with the two-component electron field.

It seems plausible however that both descriptions are possible (as

suggested recently for the case of the non-relativistic hydrogen atom) and then it is very desirable to develop this unconventional way of looking at the problem.

-5-

It turns out however that many important statements of conventional quantum, field theory such as TCP and spin and statistics theorems and usual crossing symmetry crucially depend on the assumption that we are using finite-dimensional representations ofSL(2, C) for the fields transformation law.

Furthermore, the only known

examples of infinite-component fields transforming under a unitary representation of SL(2, C) which can be quantized consistently in terms of anticommutators

are the fields satisfying first-order Majorana-type

equations which lead to a rather unreasonable (decreasing with spin) discrete mass spectrum, and^in addition;to a continuous spectrum of space-like momenta. On the other hand, the success of some rough first-order calculations for form factors and scattering in terms of infinite-component fields makes it desirable to get a better understanding of the difficulties of such type of theories in order to be able to circumvent them.

1, 2

Historical remarks and references The development of the idea of infinite multiplets seems to be

rather typical in that

a work of

the thirties, completely overlooked

in its time, is rediscovered step by step, first by mathematicians. and next by physicists {always independentlyl).

Hence, we find it instruct-

ive to make a brief digression into the history of the problem. Infinite component fields were first studied by MAJORANA' [1] (1932) who introduced the (only) two irreducible representations of SL(2, C) for which an invariant (linear) first-order differential equation can be written,

Majorana found the discrete spectrum of this equation

and realized that it also possesses a continuous set of solutions with space-like momenta.

This remarkable work of Majorana remained

practically unknown until 1966 when FRADKIN [2] revived it (on the suggestion of Amaldi) actually translating it into English and placing it in the context of later research.

In 1948 GEL'FAND and YAGLOM •

[3] (see also the exhaustive reviews in [4, 5]) rediscovered Majorana1 s -6-

results in the more general framework of the description of all irreducible representations of SL(2, C), but they apparently overlooked the existence of the continuous spectrum of solutions corresponding to space-like momenta of the infinite system of wave equations.* These space-like solutions (1949).

In [3] it

was

were

pointed out by BA.RGMANN [6]

first observed (though in a different termin-

ology) that the connection between spin and statistics is lost for infinite-component fields. The present-day interest in infinite multiplets arose mostly in connection with the search fora relativistic generalization of SU(6) (see [7-11] and further references quoted therein).

The breakdown

of the spin and statistics theorem has been re-emphasized by several authors (see B. ZUMINO 's contribution in [9] as well as [12,13]). It haa been argued in particular [12] that if a free local field transforms under a unitary representation of SL(2, C) then,at least for the case of an index-invariant theory, (which is incompatible with a Dirac-type equation) we are forced to use canonical commutation relations (i. e., Bose statistics) both for integer and half-integer spin.

It was noted in

[14, 15] that one can consider instead the "big" unitary field as a nonlocal collection of conventional finite-component free local fields for which, as we know, the spin and statisticstheorem is guaranteed. Examples of fields satisfying Majorana type equations which can be consistently quantized with anticommutators have been considered in [16] .

However, the discussion of this problem was not complete since

the rules of quantization of the space-like components of the field were not given. A physical understanding of infinite-component fields in terms of composite models has been attempted, particularly looking at the example * These solutions were originally overlooked by the present authors also - this invalidates the proof of the main assertion in our preprint E2-3289, Dubna, 1967. recent paper by FELDMAN & MAITHEWS ;(1967) (see [12])

Neither are they taken into account in the

of the non-relativistic hydrogen atom [17-20] .

As stressed by

P. BUDINIV, these examples correspond to a non-local (actually bi-local) infinite-component field except in the case of an infinitely heavy nucleon when all the poles of the Green function are going to infinity.

1. 3

Content of the present paper The main object of the present paper is a detailed study of the

quantized Majorana field.

The general features and problems of in-

finite-component fields are illustrated on this particular example. More complicated (and possibly more realistic) possibilities are only briefly indicated. Sec. 2 is devoted to some mathematical preliminaries about coupled and self-coupled representations of the Lorentz group.

It is

shown that the self-coupled representations of SL(2, C) are actually representations of the 10-parameter group of real symplectic transformations in four dimension, Sp(4, R).

The Lie algebra of the two self-

coupled representations [ 0 , i ] and [i,0] (the Majorana representations) as well as the canonical basis are described in terms of Bose creation and annihilation operators.

The well-known results about the class-

ification and description of the irreducible representations of SL(2, C) are summarized in Appendix A.

The only non-standard point in this

appendix is the expression of different four-vectors as differential operators in the space of function of two complex variables. in

2. 3

It is used

in the analysis of pairs of coupled mutually adjoint represent-

ations of SL(2, C).

The content of Appendix B is also related to Sec. 2.

The creation and annihilation operators are expressed in it as linear functions of z, z,

d/dz

and

bfbz, where z is a complex variable.

In this realization the scalar product in the representation space is defined in terms of an integral over the complex z plane.

* Private communication (see also [20]).

The main physical results are contained in Sec. 3.

The complete

set of solutions of the Majorana equation (both time-like and space-like) is described in a unified way and,afterwards, the possible ways of quantization are discussed, special attention being paid to the Fourier amplitudes for space-like momenta.

The reason why the axiomatic for infinite-component fields

proof of TCP and spin and statistics does not work I is explained in 3. 3. In Appendix C the ladder representations of the conformal group are described in terms of the same two complex variables which are used in Appendix A for the realization of an arbitrary representation of SL(2, C).

It is also shown that the analytic continuation of the matrix

elements of the ladder representation for complex values of the Lorentz parameters has singularities in the same points as the matrix elements of the irreducible representations of SL(2, C). corresponding to time-reflection.

These are the points

An analytic continuation to exactly

these points is used in the proof of TCP and spin and statistics in conventional theory of finite component fields.

2.

SELF-COUPLED REPRESENTATIONS OF SL(2, C)

2.1

Self-coupled representations of SL(2, C) as representations of Sp(4,R) We shall use the notation [ ^ , JL ] of ref. 4 for the irreducible

representations of SL{2, C) (see Appendix A).

A general (not neces-

sarily irreducible) representation of the Lorentz group will be denoted by

t. A representation V(A) of SL(2, C) (irreducible or not) is called

self-coupled if one can write a non-degenerate invariant hermitian form of the type if*~^ l / * ^ V

for a field ^ transforming according to (1.1)

-9-

t

S

:,L

L -U-'

(with the given V ) .

It is clear that a representation X of SL(2, C)

is self-coupled if and only if

it is contained in the direct product of f

with the four-vector representation [0, 2], There exist only two irreducible self-coupled representations [4] : [0,-t] and [£, 0] .

These are exactly the MAJORANA representations

(see [1, 2]). Both of them are infinite dimensional and unitary (see Subsection 2. 2 and Appendix A)-. In the representation space X of any of them a four-vector of operators LJ

can be defined

which, in addition to the

transformation law

(2-D

or in infinitesimal form, (writing V(A) = exp \ - i / _

S

!

-*>»y ( *)

foe commutation relations

Here S^ g^

are the generators of the homogeneous Lorentz group and

is the metric tensor in Minkowski space (g

= -g

K

=1,

K = 1,2,3). * One could be tempted to say that a field is transforming under a self-coupled represenution of SL(2, C) if an invariant first-order equation can be written down for it.

This definition, however, is not equi-

valent to the above because one can write invariant first-order equations for a lot of irreducible represent\

i f I It.

ations (for instance,for the four-vector representation [0, 2} one can write the equation Ou.Y' = 0 ) , while an invariant Lagrangian function of the above type exists only for two special irreducible representations. The difference is that we require the existence of a complete irreducible set of invariant first-order equations.

-10-

Eqs. (2. 2) and (2. 3) coincide with the commutation relations of the Lie algebra of the group of real symplectic transformations in four dimensions, Sp(4, R).

Indeed, the lowest faithful representation

of the commutation relations (2. 2) and (2. 3) is four-dimensional and may be written in terms of the Dirac matrices

((2. 2) and (2. 3) are simple consequences of the anticommutation rule .

which defines the Dirac matrices).

(2

.5)

It is known that each of the matrices

(2.4) satisfies the condition

^

a,b= 0,1,2,3,4,

(2.6)

where the superscript T stands for transposition and C is the antisymmetric charge-conjugation matrix defined by

For the group elements of Sp(4, R), (2. 6) gives CVC" = V*1 which implies the conservation of the antisymmetric bi-linear form ^ C Y) . Furthermore, in the Majorana basis (in which all Y^ and S ^

are

pure imaginary) all group elements V are real matrices, which completes the justification of our terminology. Sp(4, R) is the covering group of the de Sitter group SO(3, 2), the two groups having the same Lie algebra, defined by the commutation rules

-11-

.

>-?.fS±fA:'Mf.'M-

CX v

s~\ C 01 ~i

S ., S j = g

being the metric tensor in five dimensions:

Majorana representations are defined as such irreducible representations of Sp(4, R) which are irreducible also with respect to its subgroup SL(2, C). We mention that (2, 3) is just one of the many different ways to

close

the (Lie) algebra containing,in addition to the Lorentz

generators S^

, a four-vector

L/ .

Some other possibilities,

which can be relevant in the investigation of reducible self-coupled representations of SL(2, C), have been considered in [21] .

2. 2

Description of Majorana representations in terms of creation and annihilation operators We assume here the Pauli representation of y-matrices in

which V 0 is diagonal:

(2.9)

To describe the Majorana representations of Sp(4, R) we introduce the operators a

,

0( = 1, 2, satisfying the Bose type commut-12-

ation relations

,0^3 = 0,

[a,, Define a pair of four-component operator-valued spinors

''f and

by

''r-* \ Ol

(2.12)

I ,-v V-

Because of (2.11) they satisfy the commutation rules

(2.13)

The generators of the ladder type representation of Sp(4i R) are given by 0

•CLb

(2.14)

n

where s

ab

are the four-dimensional matrices (2,4) ,

checked (making use of (2.13) and (2. 6)) that Sa b commutation relations as s

It is easily

fulfil the same

;

* These operators should not be confused with particle creation and annihilation operators.

They do not

depend on co-ordinates (or momentum) and act only in the space of indices, which in our case is infinite dimensional (cf. Appendix C where the ladder representations of U(2, 2) are described in similar terms).

s :is

Here and further a, means either a , or a . .

'' 'Ihis realization of the Majorana representations has been recently used in [22].

It is immediately seen that the operators (2.15) are hermitian because of the identity

so that the corresponding representation of Sp(4, R) is unitary. The explicit expressions for the generators Sab in terms of the (*) creation and annihilation operators a read as follows:

(2.17)

:a)

The representation space X may be spanned by polynomials of a* acting on an SU(2)-invariant vector ] 0 )> defined by a^ [ 0 / = 0, M = 1,2. We shall use here the Fock variables

-14-

In these variables X is a space of entire analytic functions f( with scalar product

(Another realization of X (corresponding to the Schrodinger picture of quantum mechanics) is given in Appendix B. is defined by an integral.)

The scalar product in it

One can introduce in X a canonical ortho-

normal basis of monomials:

The generators (2.17), (2.18) act on this basis in the following way:

-15-

t lsy>=-{ + [ ^ D 1 - ^ ] ' ' ^ ! «>.},

> J •

(2,23)

We see from (2. 22) that the basis (2. 21) corresponds to the r e duction of the ladder representation with respect to SU(2). The vectors (2,21) have definite spin s and spin projection

The basis with these properties is called canonical basis. A consequence of (2. 22) and (2. 23) is that spin s may be transformed through multiple application of the generators into s + n, where n is an integer. Hence, starting with s = ^ = 0 we obtain an invariant subs pace X_ of X which contains vectors with integer spin only, and starting with s = £j = 1/2 we obtain another invariant subspace Xx t containing only half-integer spins. Moreover, it is easily checked that X is the direct sum of these two spaces

It can be verified that (2. 22) is a special case of the general formulae of GEL'FAND and NAIMARK [4, 5] corresponding to i i^ » 0 , A" + JL = 1/4 , so we see that our representation splits into the two

-16-

Majorana representations

([0,i:] acting in X_ and [i;, 0] acting in Xj. ),

This can also be seen by the calculation of the Casimir operators.

In

general, the Casimir operators of the Lorentz group are connected with the numbers

X

(see Appendix A).

and

/

by

On the other hand, a direct calculation using (2. 17)

gives for the ladder representation

**/i

implying

/ n*-n

_

*J

~

^

So we again find the solutions [ i A* - Q] .

' =0,

/f = + 1/2] and [ J? = + 1/2 ,

(We remark that in accordance with Appendix A the equi-

valence relation [ Ji , / ] ~ [- J^ ,-J/j holds for any two irreducible representations of SL(2, C), so that equivalent representations are contained in each of the above brackets.)

2. 3

Pairs of coupled adjoint representations of SL,(2, C) The Majorana representations studied in the previous section

are of a very special type.

We would like to consider here a wider

class of admissible representations (in accordance with the general principles stated in Subsection 1.1) which includes^among other things, the four-component Dirac field as well as some infinite-component fields. For this purpose we shall determine the set of all pairs of coupled irreducible representations of SL(2, C), [ JL # XJ and / _/ 0 1 [/Q L ], which satisfy conditions (1.4) and (1. 5) of the introduction.

-•;

7 -

Condition (1.4), i . e . , the condition of existence of a non-degenerate invariant hermitian form, leads to the requirement

(see [4]pt.II,§2. 9).

The condition of self-adjointness (1. 5) which is

necessary for any theory invariant with respect to space reflection, gives

(see [4] p t.II, §2. 6).

Conditions (2. 28) and (2. 29) can be fulfilled

simultaneously in two cases; j

= 1 (i.e.,

} -real)

/-arbitrary,

ix - -k1 (i. e., J^-pure imaginary),

")

f J)Q = 0 . J

(2.30)

If we impose in addition the requirement that the two represent ations are coupled, i . e . , that (A. 17) (see below, Appendix A) takes place, then we find the following sets of admissible pairs:

~i > 4 ] > (2.32)

It seems reasonable to consider (2. 31) and (2. 32) as the set of simplest representations which are appropriate to describe Fermi fields

or,

more generally, fields satisfying first-order equations in a theory of parity-conserving interactions. class (2. 31) for

JL = + 3/2 .

The Dirac field is contained in the The Majorana fields are also included -18-

in (2, 31) and (2. 32) being the only admissible pairs of equivalent representations and the only unitary representations of this class.

The

infinite-dimensional non-unitary representations used recently in [23] are included in (2. 32) for

>L half-integer.

Using the results of Appendix A, one can construct free field Lagrangians for fields transforming under (2. 31) and (2. 32) leading to first-order equations

X=

(2. 33)

l/L

where ft and P

have different meaning in the two cases.

For

transforming under (2. 31) we write

(2. 34)

% (X; H)

where ^ and ^ transform under [ i : , J[] and [-^, respectively, and the (continuous) spinor- variable z = (z , z ) substitutes the index of the field components (see Appendix A). For Y transforming under (2. 32) we have

0

(2.35)

•^7 where ^

and X A transform according to [ JlQ , j[ ] and [ / Q f - ^] ,

respectively.

-10-

In both cases the generators of the Lorentz group have the form

%

\

6 di

v c

6

f

\

\ I

]

I

in

OT'\

(see (A. 12)). We mention that, because the representations for and A. are adjoint, the corresponding matrix elements of M and N in the canonical basis are related by

lt,,)-fo ' O \ We introduce for each of the cases (2. 31) and (2. 32) a parity operator by

(2.38)

0 where the operators V

a r e given by (A. 19)-(A. 22). Using the results

of Appendix A we see that Pj" behave like vectors under the corresponding

reflection "}]" :

-20-

—I A '

whereas the quantities obtained from

/.

by changing the relative sign

of the non-vanishing (off-diagonal) elements are axial-vectors. There is only one common pair of representations of the two classes (2. 31) and (2.32), namely the pair



MMMHM-i]

-

considered in [23] .

3.

QUANTIZED MAJORANA FIELD

3. 1

Statement of'the results

*

In the next two subsections we shall consider an irreducible relativistic field transforming according to one of the Majorana re presentations and satisfying the Majorana equation

where L/

are infinite hermitian matrices with elements defined by

(2.23) (the infinite "spinor" index (s ^ ) is omitted). Eq. (3.1) can be obtained from variational principle with Lagrangian

As we shall see in Subsection 3. 2, the Majorana equation allows a continuous spectrum of space-like solutions in p-space.

If we quantize

the corresponding Fourier amplitudes in a canonical way, retaining the requirement of positive definiteness of the energy, we are forced to use Fermi statistics for both Majorana fields (independently of the value of -21-

j(

which determines the spin content).

In contradistinction

to the

scalar space-like field considered in [24], the free Majorana field is local.

If we drop the requirement of positive-definiteness of the energy

(bearingin mind that the spectral conditions are always violated, because of the space-like solutions) we obtain the possibility of defining a local field which is a superposition of annihilation operators only (Subsection 3.3). In Subsection 3.4 we discuss why the axiomatic proof of TCP and spin and statistics theorems does not work for infinite-component fields. We show that the well-known Bargmann-Hall-Wightman theorem is not valid when infinite-dimensional representations of the Lorentz group are involved because the matrix elements of such representations are shown explicitly to have singularities for some complex values of the group parameters.

3. 2

Complete set of solutions of the Majorana equation We first review the classical solutions of the Majorana equation

in momentum space*

(3.3)

)

We consider (3.3) as an eigenvalue problem for the energy p for fixed £ .

We mention that the operator

is hermitian with respect to the scalar product * These solutions have been recently described by W. RUHL [25] (who used a somewhat different realization of the representation space X ).

-22-

where (u, v) is the (positive definite) scalar product in the space X , where the unitary Majorana representation of SL(2, C) under consideration acts.

We recall that L

is a positive operator because of (2. 23)

and hence it has a positive inverse.

It follows that the eigenvectors

of H (or, which is the same thing, the solutions of (3.3)) corresponding to different eigenvalues of p

are orthogonal with respect to the product

(3.5). The simplest way to obtain the spectrum of eq. (3.3) is to multiply both sides by L p

+ & and to use

where w is the Pauli-Lubanski-Bargmann spin four-vector

To check (3. 6) one has to use the explicit expressions (2.17) and (2,18) of L ^

and S^

in terms of creation and annihilation operators.

Substituting (3.6) in the equation thus obtained we find

We have to consider essentially two different cases depending on the eigenvalue of p : the case of time-like and the case of space-like momentum p .

The intermediate case of light-like momentum can

be obtained by going to the limit from both sides.

-23-

In the case of time-like momenta, where p generated by w^

2

> 0 , the little group

is SU(2) and

W2- = -?*-s(S

+ L)

(3. 9)

with

\ °

1

lJ][^j

(3.10)

This immediately leads to the decreasing mass spectrum (for \C > 0)

Because of the positive-definiteness of L 0 of p appear in this case.

only positive eigenvalues

For p space-like the little group conserving p is SU(1, 1) ~ SL(2, R) 2 and we have the same formula for w , but with s in the range

These are exactly the values of s involved in the Sommerfeld-Watson integral in Regge theory of complex angular momentum. The limiting cases s ->

°G (CT->«SO

) correspond to solutions with

light-like momenta. Proceeding to the determination of the eigenvectors, we shall discuss here only the case , K> 0 in terms of the variables ^

(2.19).

Both this case and the case 1C = 0 are treated in terms of "Schrodinger variables" z and z in Appendix B. It is sufficient to consider the case when the momentum p is directed along the third axis, the general case being obtained from this through a three-dimensional rotation. -24-

We treat the discrete and the continuous spectrum simultaneously and put

Substituting (2.18), (2.19) and (3.13) in (3, 3) we obtain

(3.14) To get rid of the degeneracy we require, in addition, that u* be an eigenvector of the third spin component

(3.15)

Further, we make the change of variables {cf. (B. 6)}

and, putting

find the following equation for f

j- i :

(3.18) The solution of (3. 18),.regular for y = 0 , is

-25-

(3.19) where CD(a, c , z) is the confluent hypergeometric function defined by the Kummer series

(3.20) and C. p j r | is a normalization constant. The discrete time-like solutions are obtained from (3.19) for

i.e.

i$-±

= IjiT '

Then, in view of the Kummer identity z

we find

-26-

(3.21)

where L

(x) are the Laguerre polynomials (B. 9).

Requiring the normalization

;'