Amplitude systems for spin-1/2 particles

HAL Id: jpa-00210988

https://hal.archives-ouvertes.fr/jpa-00210988

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Amplitude systems for spin-1/2 particles

Michael J. Moravcsik, Jutta Pauschenwein, Gary R. Goldstein

To cite this version:

Michael J. Moravcsik, Jutta Pauschenwein, Gary R. Goldstein. Amplitude systems for spin-1/2 par-

ticles. Journal de Physique, 1989, 50 (10), pp.1167-1194. �10.1051/jphys:0198900500100116700�. �jpa-

00210988�

Amplitude systems ^for spin-1/2 particles

Michael J. Moravcsik (1,*), Jutta Pauschenwein (1) ^and Gary ^{R. Goldstein} (2,**)

(1) Institut für theoretische Physik, Universität Graz, A-8010 Graz, Austria

(2) Department of Theoretical Physics, University ^of Oxford, Oxford 0X1 3NP, G.B.

(Reçu ^{le 25 avril 1988,} révisé le 6 décembre 1988, accepté ^{le 13} janvier 1988)

Resume.

On discute les propriétés de différents systèmes d’amplitudes qui ^sont utilisés pour la

description des réactions entre particules ^de spin 1/2. ^On compare les différentes représentations ^et

on présente les transformations entre les différents ensembles sous forme de matrices.

Abstract. 2014 The properties of various amplitude systems ^{used to} describe reactions involving spin-1/2 particles ^are described, ^the _systems compared, ^{and the} transformation matrices among them tabulated.

Classification

Physics ^Abstracts

9nnn

1. Introduction.

The analysis ^of particle ^reactions (in ^{nuclear- and} high energy physics) ^is facilitated by describing ^{these reactions in a} way which explicitely displays ^the dependence ^{of the} dynamics

on all the quantum numbers and parameters ^{which are} firmly established. In particular, ^the

evidence is compelling ^{that the structure} of such reactions which pertains ^to spin ^and angular

momentum is rich and complex, and that its understanding ^{is a} necessary part ^of eventually discovering ^the fundamental dynamical ^{laws of} particles, ^and, specifically, ^{that of} strong interactions.

The characterization of this spin- ^and angular momentum structure can be done in at least two ways. The time-honored, traditional way has been in ^terms of a partial ^wave expansion ^of

the reaction amplitudes, ^which yields ^the phase shifts. This approach ^has brought ^{us much} insight for many reactions, both in nuclear and in particle physics. ^The approach, ^however,

also has some disadvantages ^{which are} gaining increasing importance nowadays. ^{Some of}

these can be summarized ^as follows :

a) ^at all but the lowest energies ^the partial ^wave description requires ^{a very} large ^{number of} parameters ;

b) ^{in most,} though ^not all, of the reactions the truncation of the partial ^wave expansion

cannot be done cleanly ^on physical grounds, ^{and the} mathematical criteria for doing ^the

truncation are not unambiguous ^{either ;}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500100116700

c) ^{while in some} reactions the decomposition into definite angular momentum states appears ^to illuminate some significant features of the underlying dynamics, ^{in most reactions}

this does not appear ^to be the case ;

d) when there are large ^{and numerous inelastic channels} competing with the elastic channel

being described, unitarity (which is the basis for a phase ^shift description) ^{becomes too} complicated ^{to be a} simplifying feature in the partial ^wave description ;

e) ^{in order to} perform ^{a reliable} partial ^wave analysis, especially ^at higher energies, ^one

needs an extensive set of data in ^a wide range of the ^momentum transfer (or scattering angle).

The second way ^to describe particle reactions is in terms of spin amplitudes ^{which are}

functions of the kinematic parameters (energy ^and angle) of the reaction. While this approach (contrary ^{to the} partial ^wave expansion) ^does not connect data at different angles ^{and thus}

loses some unifying power, it shows many advantages ^{in other} respects. Some of these can be summarized as follows :

a) the number of parameters needed for the description of the reaction is independent ^of

the energy ^at which the analysis is carried out ;

b) ^{data at} single angles ^{can thus be} analyzed, something ^{that is not} possible ^{in the} partial

wave approach ;

c) ^{there is a} great variety ^of amplitude systems which allows one to choose the one most suited for the particular purpose ^at hand ;

d) dynamic ^{models can be} compared ^with phenomenological descriptions easier and in ^a

more definitive way ;

e) ^the analysis ^is independent of any inelastic channels that may be présent ;

f) specific ^kinematic configurations (e.g. ^{forward- or backward} reactions, ^{or reactions at}

90°) ^can be studied much easier in terms of amplitudes ;

g) ^the description requires ^no approximations ^{like the} angular ^{momentum truncation}

needed in the partial ^wave approach.

For these and other reasons, amplitude analysis ^has gained importance ^and popularity ⁱⁿ

recent years.

A frequently occurring ^and important class of reactions is the one containing only spin-1/2 particles. Nucleon-nucleon and nucleon-antinucleon reactions are perhaps ^{the most} promi-

nent, ^{as well as} boson-nucleon and boson-hyperon reactions. In other instances, ^{like in} electron-proton, electron-electron, ^and electron-positron scattering, the reaction type ^{is also} the same but is further constrained by ^{the known} electromagnetic ^{nature of the} dynamics. ^In

any ^case, descriptions of reactions involving only spin-1/2 particles pervade the literature.

In this practice, ^a variety of different amplitude systems have been used. As remarked

earlier, they ^{are not so much} competitors ^but they ^rather complement ^each other, ^{since for} different purposes different amplitude systems turn out to be the best to use. This fact, together ^{with the} large ^{number of} specific analyses which have been carried out in the past ⁱⁿ the various amplitude systems, suggests that it is likely ^{that most of these} systems ^will continue to be used in the future. As ^a result, it is useful to offer a comparative discussion of

them, and it is perhaps ^{even more useful to} display explicitely ^{how one can make a transition}

from ^one of these systems ^to another. That is the aim of the present paper.

2. Général properties ^{of the} amplitude systems.

The number of amplitudes needed for the description ^{of a reaction} is, ^of course, independent

of what amplitude system ^{we use, and} depends only ^on the values of the spins ^{of the} particles

in the process, and ^on what conservation laws constrain the reaction matrix. In this paper ^our

interest, ^for practical ^reasons, will be in reactions with both parity conservation and time

reversal invariance constraining the reaction matrix, ^that is, in the reactions of the type A + B --> A + B with parity conservation. The conceptual conclusions of ^our discussion can,

however, be carried over also to reactions which, ^for example, ^{are not} constrained by ^time

reversal invariance or are not even parity conserving.

With such constraints from parity conservation and time reversal invariance, ^{for A with} spin ^{1/2 and B with} spin 0, the number of amplitudes ^is 2, while if both A and B have spins ^of 1/2, the number of amplitudes is 6. In what follows, ^{we will} specifically ^discuss only ^the

reaction with four spin-1/2 particles. ^The simpler reaction with only ^two spin-1/2 ^{and two} spin-

0 particles ^has obviously analogous properties ^{on a} reduced scale which can easily ^{be deduced}

from the present discussion.

The parity-conserving and time-reversal-constrained reactions with four spin-1/2 particles

fall in two additional subsets : one in which A

B (and the number of amplitudes ^{reduces to} 5), ^{and one} in which this is not the case. The classic example ^{for the} first is p-p scattering (or

p-p scattering ^if charge conjugation constraints are taken into account), ^{while an} example ^for

the second is n-p scattering (if charge independence ^{is not} taken into account). ^{Because of the}

difference in the number of amplitudes, and because of the somewhat different amplitude conventions, ^{the two cases} require ^somewhat separate ^treatment.

In both cases, however, ^{one can} divide the various amplitude systems used in the past ^into several classes and by several criteria.

One way ^to make the classification is according ^{to the} simplicity ^{of the} relationship ^between

the bilinear combination of amplitudes ^{and the} experimental observables. There are

amplitude systems ^{which are «} optimal » [1] ^{in this} respect in that for them the matrix that represents ^this relationship ^{is as close to} diagonal ^as possible. ^Other amplitude systems ^are non-optimal ^{in this} respect, in that for them the matrix is either not in the form of a string ^of

submatrices along ^{the main} diagonal, ^or it is but these submatrices are larger than what is

minimally required.

For the reactions under consideration in the present discussion, ^the difference between

optimal ^and non-optimal representations is small. To understand why, ^{let us} remember that in the optimal representations the M-matrix is spanned by basis matrices in each of which only

one element is non-zero, and the density ^{matrices are} spanned by basis matrices with only ^one

non-zero element along ^the diagonal (the other elements being zero) ^and by matrices with two

symmetrically placed off-diagonal ^elements being ^non-zero (and the other elements being zéro).

Now for spin-1/2 particles these matrices are so small (two-by-two) that the above optimal

basis matrices and the more usual non-optimal basis matrices (e.g. ^{the Pauli} matrices) ^differ only ^{to a} slight ^{extent. As} long ^as the three coordinate directions utilized in these basis matrices (e.g. ^{the x,} y, and z in the Pauli matrices) ^{are chosen to be} orthogonal ^{to each} other, the resulting non-optimal representation will differ from the optimal ^ones (as ^{far as} simplicity

is concerned) only slightly ^{or not at} all, especially ^when parity conservation and time reversal invariance are imposed ^also beside Lorentz invariance. Other advantages ^{of the} optimal

formalism (e.g. ^the flexibility in the choice of bases matrices, ^see the earlier discussion) ^will

however remain.

The other way of making ^a distinction among the various amplitude systems ^{in use is} according ^{to whether} they ^use space-fixed ^or particle-fixed coordinate systems. ^{In the} former,

the three coordinate directions ^are fixed ^once and for all with respect ^to the space in which the reaction takes place. They ^{can, for} example, be defined in terms of the two directions of, a) ^the incoming particle ^momentum, b) the normal to the reaction plane. Thereafter all

particles (that is, ^{both the} incoming ^and outgoing particles) ^use this fixed coordinate system

for the three directions appearing ^{in the tensors} describing ^them.

In the particle-fixed representations ^each particle ^{has its own} coordinate directions, adjusted, ^for example, ^{to its own momentum} direction. There may be constraints ^on the relative position ^of these directions between ^two different particles, resulting ^from constraints from conservation laws, but even so, all particles in the reaction are on an equal footing ^with respect ^{to the} definition of the coordinate directions. This is quite different from the situation described earlier for the space-fixed amplitude systems ^{where the} particle ^{which is} initially

used to define the coordinate system for all four particles is treated preferentially.

One obvious consequence of the space-fixed coordinate systems is that in it the relationship

of the final state particle ^momenta (if ^the system ^{was fixed} through ^{an initial} particle) ^{to the}

coordinate directions of this final state particle depends ^{on the reaction} angle, while this is not the case for the particle-fixed systems. Consequently the transformation matrices between

space-fixed amplitude systems ^and particle-fixed amplitude systems will contain the reaction

angle, while the transformation matrices between two space-fixed ^{or two} particle-fixed amplitude systems ^{will not} contain that angle. ^{As we will} see, ^one system (the singlet-triplet system) is neither pure space-fixed ^nor pure particle-fixed, and hence its relationship ^{to all}

other systems contains the reaction angle.

For many purposes particle-fixed amplitude systems ^are simpler ^{and « more natural », and}

hence many of the ^more modern systems ^{are of such} type. Among ^{these are} the whole class of

optimal formalisms ^as defined in reference [1], and hence also the helicity formalism which is

a special ^{case of the} optimal formalism. The so-called « transversity ^{formalism »} in the strict

sense of the word is also particle-fixed, namely ^{if it is meant to be an} optimal formalism with the quantization ^axes in the direction normal to the scattering plane. ^{There are, however, also} non-optimal formalisms, space-fixed, ^{where the} quantization ^{axes are} also in the direction normal to the reaction plane. ^An example ^{would be a} formalism which uses the Pauli spinors

as bases with the coordinate (usually ^called z) which is connected with the diagonal ^Pauli

matrix being identified with the direction normal to the reaction plane. ^One needs, therefore,

be careful with the terminology.

The various formalisms also differ from each other according ^{to the} properties they ^focus

on in their definitions. For example, ^the helicity ^{formalism was} defined because of its simple properties ^under coordinate transformations, and because ^a partial ^wave decomposition ^{in it}

appears simple if written in terms of the tabulated d-functions, whereas the corresponding decomposition, ^for example ^{in some of the} space-fixed amplitude systems, ^contains explicitely

many 9- j symbols ^{and other} purely geometrical ^{factors which are} hidden from the eye in the d- . functions occurring ^{in the} helicity formalism. On the other hand, ^the helicity formalism is not

particularly natural with regard ^{to the} parity constraints, and it does not appear ^to be

particularly natural either in describing ^the dynamical mechanisms occurring ⁱⁿ strong interaction processes. Some of the space-fixed systems ^were introduced at a time when relativistic effects were small in the then available energy range in the reactions which ^were of interest at that time, and hence such systems, written in terms of quantities well known from non-relativistic quantum mechanics, ^were preferred.

Some other systems ^were specifically constructed to be simple ^{in a} particular dynamical

model. For example, ^the optimal ^« magic » amplitude system ^{was defined} specifically ^{for the}

purpose of ascertaining ^the extent to which one-particle-exchange (OPE) mçchanisms

dominate the reaction. Another example ^{is the} exchange amplitude system, originally

introduced during ^the Regge-era ^of particle physics, ^which appeared natural in that dynamical

framework. The planar-transverse system ^{was, on the other} hand, introduced not ^{on a} priori grounds, ^but only after ^an analysis ^of data, ^{and its} justification ^{is in its} exhibiting ^some striking

and simple ^features qf ^{the data} actually ^obtained by experiment.

Some types ^of systems ^show advantages ^{even. in} a purely phenomenological analysis ^{of the}

data, ^without regard ^{to the} dynamics. ^For example, ^the optimal transversity system ^{is a} natural one to use in such phenomenological analysis of any parity conserving reaction, ^since

its quantization ^direction (normal ^{to the reaction} plane) ^{coincides with the} preferential

direction in the symmetry operation ^of reflection which is the basis for parity conservation.

Thus we see indeed that there is a good ^{reason to have such a multitude of different} amplitude systems. ^{In the next} section, therefore, ^an overview of a list of these systems ^is given and their connections are specified.

3. Définitions and transformation matrices.

°

We will now define the various amplitude systems ^{and then} display their transformation matrices into each other. For some previous contributions to this task, ^see references [2, 3].

The systems ^{we will consider are} tabulated in table I. There are ten systems ^{for the case} with Lorentz invariance, parity conservation, time reversal invariance, and identical particles,

and seven systems ^{for the case} when the last (identical particle) constraint is not imposed.

NORMALIZATION. 2013 The normalization of the various ^sets of amplitudes is done in many different ways in the literature. In ^{some cases} the amplitudes themselves are dimensionless,

Table I.

The amplitude systems considered in this _paper. The de finitions of ^{the various} amplitudes ^are given ^{in the text.}

Identical Particles

Non-identical particles

JOURNAL DE PHYSIQUE. - T. 50, N 10, 15 MAI 1989

and are normalized to some dimensionless number like 1 (but ⁱⁿ some cases 1/2 ^or 2). ^{In other}

usage the amplitudes ^{have a} dimension of the square ^root of cross sections, ^that is, ^a

dimension of length. It would be impossible ^{to cover all the} possibilities which appear in the literature. Hence we concentrated in this paper ^on the relative normalizations of the various

amplitude systems. In table 1 ^we give, ^{for each} set, ^{that sum} of the bilinear combination of

amplitudes which should be taken to have the same value for all the different sets. This combination is the one that enters in the totally unpolarized differential cross section for that

amplitude system. This way the different amplitude systems ^{can be} compared, and it is this normalization which served ^{as a} basis for the transformation matrices given ⁱⁿ this paper.

When then some amplitude system ^{from a} paper in the literature is used, the first task should be ascertain from that paper what the expression there is for the totally unpolarized

differential cross section, and then the amplitudes from there should be renormalized to agree with the normalization used in this paper. Then such amplitudes ^can be used in the transformation matrices given in this paper.

THE PROPERTIES OF THE AMPLITUDES SYSTEMS. - We will ^now briefly ^{comment on each of}

the nine amplitude systems ^and give their definitions.

1 ) ^The Saclay system [4]. ^{- Named so} because it appears in reference [2] with authors mainly

from Saclay. The M-matrix here is given by

where

This is for A different from B. For the identical particle ^{case we have}

This system ^uses the non-relativistic notation (which, ^{however, as} it has been shown [5], ^can

also handle the relativistic situation if interpreted the proper way). If the ^{z direction} (the ^one

for which the Pauli matrix is the diagonal one) ^{is taken to} be the normal to the reaction plane,

then this system ^{is of the} transversity type ^{but not with a} particle-fixed coordinate system (and

hence not a member of the set of optimal systems). ^The system ^is important because the

Saclay group, which defined it, ^has performed many experiments ^{of the} type ^under consideration which are then published within the framework of this formalism.

2) ^The Wolfenstein system [6].

^Defined only for the identical particle case, in it the M-

matrix is given by

where S is the spin-singlet projection operator, ^{T the} spin-triplet projection operator, ^{with the} definitions

This is basically ^{of the same} type ^{as the} Saclay system, and is very simply ^{related to} it, ^with several ^{of the} amplitudes ^{in the two} systems being ^{the same} except ^{for a} numerical factor. Its

significance ^is historical, in that the first complete ^reaction analyses in the 1950’s were carried out using ^this amplitude system. ^It has also been used since.

3) The Hoshizaki system [7].

Also defined only for the identical particle ^case, in it the matrix is given by

This is even more similar to the Saclay system than the Wolfenstein system ^is. Indeed, ^the

difference is only ^{in two of the} amplitudes which, ⁱⁿ the Hoshizaki system, ^{are sums and} differences of two amplitudes ^{in the} Saclay systems, the others being ^{the same in the two} systems except for factors of 2. The importance ^{of this} system is in the fact that Hoshizaki has

published ^{a number of} analyses of nucleon-nucleon scattering ^at energies higher ^than others,

and these analyses ^{use this} amplitude system.

The three above sets differ little from each other.

4) ^The Singlet-triplet system [2, 3].

^This system ^is significantly different from the previous

ones (as ^{well as from the} upcoming ones) in that it focuses on a simple description ^{of the two}

particle systems occurring in the initial and final states. The M-matrix here has the following

form in the « singlet-triplet space » :

This is for A different from B ; for identical particle MST

^0.

One then needs angular ^{momentum addition} geometry ^to relate these (larger ^number of)

matrix elements to the (smaller ^number of) amplitudes ^{in the} previous systems. ^{As will be}

seen, this ST system ^{is not} quite space-fixed (as ^the previous systems were), ^{but not} quite particle-fixed ^either (as ^the upcoming systems ^will be), and hence the transformation matrices between ST and any of the others will contain dependence ^on the reaction angle.

5) ^The Helicity system [8].

This is the first particle-fixed system ^that gained ^{wide use. It}

also featured easy Lorentz transformation properties ^{which are} perhaps nowadays ^{not as} important ^as they ^{were at} that time. It is the ^« natural ^» system for massless particles. ^There

are several conventions in use about some aspects of the definition of the helicity amplitudes,

as can be seen from references [8, 9]. In reference [2] ^{the five} amplitudes Ml, ^...,

M5 ^are used. In this paper ^{we use} (a, b, ^c, d, e).

Table II.

The relative normalization of the various amplitude systems used in this paper.

Identical Particles

Non-identical particles

In reference [3] ^six helicity amplitudes ^{are used, which are related to the ones used in this}

paper by

The amplitudes used in this paper, in ^terms of the helicity ^{values of} particles C, A, ^{D and} B, respectively, ^{in the reaction A + Bu C + D, are defined as follows :}

The helicity system ^{is a} special ^{case of the} planar optimal systems, ^a family ^of systems ^which also includes the planar-transverse ^{and the} « magic » systems (see below).

6) ^The transversity system [10, 11].

^{This is a} special ^{case of the} optimal systems ^{which is} particularly ^{useful for the} phenomenological analysis ^of parity-conserving reactions, ^{since in it}

the parity constraints on the amplitudes ^are simple, ^{and since} in it the simple polarization (with only ^{one of the} particles ^{in the reaction} being polarized) ^is given ^{in terms of} magnitude-

squares of reaction amplitudes, ^thus making ^the determination of these magnitudes easy.

In references [2] ^and [3] ^the transversity amplitudes Tl, ^..., T6 ^{are used.} They ^{are related to}

the transversity amplitudes ^used in this paper by ^the correspondence

In particular, ^{the former set} is defined (see ^Ref. [15]), ^{in terms of the} spin projections ^{of the} particles C, A, D, B, in the reaction A + B --. C + D, ^{as follows :}

7) The Planar-transverse system [12]. ^{- This} system ^{is a} planar optimal system in which the

quantization ^{axis is} perpendicular ^{to the} helicity direction. There is growing ^evidence that, ^for

a variety ^of strong interaction processes, the reaction amplitudes ^{have a} strikingly simple

structure in this system, the relative phases among amplitudes being multiples ^{of 90°. The} amplitudes ^{are defined} analogously ^{to the} helicity amplitudes (see above).

8) ^{The «} Magic » system [13]. ^{- This} planar optimal system ^{is used for} one-particle-exchange

tests, since in it these tests have a particularly simple structure. It can be obtained from the

helicity system by ^a rotation of the quantization axis in the reaction plane by ^an angle which, however, depends ^{on the} reaction energy and reaction angle. ^{For this reason the}

transformation from the « magic » system ^to any other system is somewhat more involved, ^as

the values of the elements of the transformation matrices depend ^on reaction energy and reaction angle. Therefore, in this paper, ^we give only ^the transformation matrix between the

helicity system ^{and the «} magic system ^». The transformation matrix between the ^« magic » system and any other system ^can be obtained straightforwardly by ^the multiplication ^{of the}

transformation matrix between ^« magic » ^and helicity ^{with the} transformation matrix between

helicity and that other system.

The ^« magic » ^frame amplitudes ^{are defined in terms of the} planar amplitudes ^{for which}

rotation angles ^{in the} plane ^{are related to} crossing angles (between ^{s and t} channels). ^{For the}

case of A + B the six magic amplitudes (a m, b i , c m, d m, e m, b2) ^{are related to} helicity

amplitudes ^via

where

When A and B are identical spin-1/2 particles, ^{the six} amplitudes ^{reduce to five with}

so that the transformation matrix becomes

The angles by ^{which the} quantization ^{axes of the «} magic » system ^are rotated with respect

to the helicity ^or particle ^{momentum direction are} given by

where s

(pA ⁺ PB)2 is the square of the center-of-mass energy, t

(PA’ - P A)2 is ^{the square}

of the four-momentum transfer, ^the primes ^{refer to the} outgoing particles, ^and mA and MB ^are the masses of particles A and B. When we have A

B, then the single angle ^of

rotation becomes

9) ^The exchange amplitude system.

This has its origin ^{in the once} popular Regge ^model,

since it explicitely emphasizes ^the quantum numbers of certain exchange processes.

10) ^{The Arndt} system [14].

This has been used by Arndt in his extensive analysis ^of

nucleon-nucleon elastic scattering. His program (SAID) provides spin amplitudes ^{in this}

form. The definitions of his amplitudes Hl, ^..., HS ^{are related to} helicity amplitudes ^as

follows :

k... center-of-mass momentum.

It is useful to note that in reference [14] ^the fi amplitudes ^{are related} directly ^{not to the} helicity amplitudes ^{but to the} Saclay amplitudes.

11 ) ^{The use} of ^the transformation ^matrices.

^{In what now} follows, ^we give the matrices which transform the amplitudes ^{from one set to} another. The notation is as follows : in order to obtain the amplitudes ⁱⁿ system ^W from those in system V, ^{one needs the two column}

vectors (called ^{Zv and} Zw) of the five (or six) amplitudes ⁱⁿ systems ^{V and} W, respectively,

and then the transformation matrix (five-by-five ^or six-by-six) Q y -+ W from the collection of such matrices to be given below. The amplitude ^{vectors are} given ^{in table} I, ^{in row form for} the sake of a more economical display ^{in a} printed table. The order of the amplitudes ^{in these}

vectors is, ^{of course, the} important piece of information. We thus have

The transformation matrices now follow.

JOURNAL DE PHYSIQUE. - T. 50, ^N’ 10, 15 MAI 1989

References

[1] GOLDSTEIN G. R. and MORAVCSIK M. J., ^Ann. Phys. ^{N.Y. 98} (1976) ^128.

[2] ^{BYSTRICKY J.,} LEHAR F. and WINTERNITZ P., ^J. Phys. ^{France 39} (1978) ^1.

[3] ^{LA FRANCE P. and} WINTERNITZ P., ^J. Phys. ^{France 41} (1980) ^1391.

[4] ^{OEHME R.,} Phys. ^{Rev. 98} (1954) ^{147, 216.}

PUZIKOV L. D., RYNDIN R. M. and SMORODINSKII Ya. A., Zh. Eksp. Teor. Fiz 32 (1957) ^592.

LEHAR F. and WINTERNITZ P., Fortschr. Phys. ¹⁵ (1967) 495 ; ^reference [2], equation (2.1).

[5] ^{STAPP H. P.,} Phys. ^{Rev. 103} (1965) ^425.

[6] WOLFENSTEIN L. and ASHKIN J., Phys. ^{Rev. 85} (1952) 947 ; ^reference [2], equation (2.9).

[7] ^HOSHIZAKI N., Suppl. Prog. ^Theor. Phys. ⁴² (1968) 1, 107 ; ^reference [2], equation (2.6).

[8] JACOB M. and WICK G. C., ^Ann. Phys. ^{N. Y. 7} (1959) 404 ; ^reference [2], equation (2.23).

[9] COHEN-TANNOUDJI G., MOREL A. and NAVELET H., ^Ann. Phys. ^{N. Y. 46} (1968) ^239.

[10] ^{KOTANSKI A., Acta} Phys. ^{Pol. 29} (1966) ^{699 ; reference} [2], equation (2.30).

[11] GOLDSTEIN G. R. and MORAVCSIK M. J., ^Ann. Phys. N. Y. 142 (1982) ^219.

[12] GOLDSTEIN G. R. and MORAVCSIK M. J., Phys. Lett. B 102 (1981) ^189.

[13] GOLDSTEIN G. R. and MORAVCSIK M. J., Phys. Rev. D 30 (1984) ^55.

[14] ^{ARNDT R.,} Phys. Rev. D 28 (1983) ^97.

[15] GOLDSTEIN G. R. and MORAVCSIK M. J., J. Phys. Colloq. ^{France 46} (1985) ^C2-251.