© Richard Sears, 1971, 2002
May 18, 1971
INTERNAL QUANTUM NUMBERS
AND
HOMOGENEOUS SPACES
1. INTRODUCTION
Several authors [1 - 17] have studied field theories on homogeneous spaces of the Poincaré group. In seeking explicit representations of this group one is naturally led to the idea of choosing different carrier spaces for the wave functions and, hence, to homogeneous spaces since Minkowski spacetime is a space of this sort. D. Finkelstein [1] has classified a large class of homogeneous spaces of the Poincaré group.
One of the primary motivations for extending spacetime in this manner is the fact that Minkowski space can carry only trivial spin representations using scalar valued wave functions; it has been shown explicitly [6, 13] that a non-trivial spin spectrum is carried by some of the larger homogeneous spaces of the Poincaré group.
The purpose of the present paper is to initiate the study of the possibility of finding explicit representations of the Poincaré group, which are labeled by additional discrete quantum numbers. It is immediately obvious, however, that to include quantum numbers such as those associated with isospin, a space is needed which is itself larger than the Poincaré group due to the circumstance that the maximal compact subgroup has already been pressed into service to carry the spin spectrum. We shall still be interested in finding explicit realizations of the Poincaré group; we shall, however, examine the action of that group on a space larger than itself.
To make this more precise, the following terminology is introduced. A free, elementary physical system is described by a unitary irreducible representation (UIR) of a kinematical group G in a Hilbert space H(C) defined over a manifold C called the carrier space. G is a (Lie) transformation group whose action πG on C is assumed to be effective but not necessarily transitive. A group extension G' of G whose action πG' on C is transitive is called a spectrum - generating group. Since G' does act transitively on C, then C is a homogeneous space of G', isomorphic to the coset space
C = G' / H'
where H' is some closed subgroup of G'. The free, or unperturbed, fields can then be constructed from the UIR of the kinematical group G [18] on the homogeneous space C of the spectrum-generating group G'.
It is intended here to examine an explicit example of a spectrum-generating group; the Poincaré group shall serve as the kinematical group. In this example, the Poincaré group has a diagonal action on a twelve-parameter carrier space C12. There are, in this case, two candidates for the spectrum - generating group, namely, ISO(1,3) x SO(3,3) and ISL(4,R). The corresponding twelve - parameter spaces are locally homeomorphic and are also globally homeomorphic if one considers suitable covering groups of these spectrum - generating groups. The former group was first suggested as a spectrum-generating group by A. Kihlberg [4, 19]. M. Flato and D. Sternheimer [20] have investigated the group SL(4, R).
In this paper our main objective is to examine the extent to which the formalism developed by Kihlberg may be generalized to the case of the larger spectrum - generating group.
In §2, the carrier space C12 is defined as a homogeneous space of ISO(1,3) x SO(3,3), and the subsequent action of the kinematical group on this space is examined. A Poincaré - invariant volume element is derived for C12. This invariant volume element is then used in §3 to construct unitary representations of the Poincaré group on C12. The irreducibility conditions are discussed and a set of operators, which is tentatively identified as the isospin operators, is introduced. Wave functions for massive particles and antiparticles are defined in §4 where the discrete transformations are also considered. Then in §5 the free fields are constructed and are shown to lead to the same problems with respect to microcausality as in the cases previously studied. In order to obtain some insight into the action of the kinematical group on the isospin parameters, which has been assumed, a naive model of elastic π - π scattering is examined in §6 for isospin invariance properties. It is shown, within the framework of conventional interaction theory, i.e., the (generalized) Dyson formula, that the straightforward generalization of Kihlberg's formalism presented here leads to unacceptably large violations of isospin invariance.
Some notational conventions are used in the following : trigonometric and hyperbolic functions are abbreviated according to the scheme
Greek indices take values from one to four. Lower case Latin indices take values from one to three. Capital Latin indices run from one to six.
The Minkowski metric is taken to be of signature ( - - - + ).
Semi-direct products of groups are denoted as follows:
2. THE CARRIER SPACE C12 AND THE ACTION OF THE KINEMATIC GROUP
The carrier space examined here is defined as the topological product space
C = M x Z
of Minkowski space-time M and an internal space Z which is a homogeneous space of the component (connected to the identity) SO0(3,3) of the pseudo - orthogonal group in six dimensions which leaves a quadratic form
- x12 - x22 - x32 + x42 + x52 + x62
invariant, or of a covering group of SO(3,3):
to be determined in such a manner that
i.) it carries a spin
spectrum;
ii.) it carries an isospin spectrum;
iii.) it possesses a volume element invariant under Poincare' transformations.
When we ask now for a group G' which contains the Poincaré group as a subgroup
and which has a transitive action on C, we are faced with a unification problem
[21]. In this example both a direct product and a semi-direct product
unification exist, namely
the latter unification being possible due to the local isomorphism of SO(3,3) and SL(4,R). In the following, we shall consider only the possibility of G1'. Some remarks concerning the action of G2' will be made, however, in this section.
A distinction ought to be made between the parameterizing ISO(1,3) which appears explicitly in the direct product which defines the spectrum generating group G1' and the physical ISO(1,3). The restrictive adjectives refer to the (possibly distinct) actions of different ISO(1,3) transformation groups. Thus
par ISO(1,3),
the parameterizing ISO(1,3), is that group whose parameters identify the external, Minkowski submanifold of the carrier space, i.e., it acts transitively on M and not at all on Z. The group of primary interest is
phys ISO(1,3),
the group of physical inhomogeneous Lorentz transformations; it is the UIR of phys ISO(1,3) which are to be identified as "free, elementary physical systems".
In G2', for example, one
would choose as phys ISO(1,3) the
which contained the rotations of the space-like dimensions x1, x2,
x3. However, as mentioned, we shall consider only the case G1'
in detail.
In this example, the physical action of ISO(1,3), which shall now be defined, is chosen for reasons of mathematical simplicity. It may be worthwhile to point out that other definitions are possible, and more exotic actions of the kinematical group on the carrier space could be of physical interest.
Let an element of the physical, restricted Poincaré group ISO(1,3) be denoted
by ,
where a is the translation and is the homogeneous Lorentz transformation.
The physical ISO(1,3) is defined as that ISO(1,3) which acts diagonally
on the carrier space. That is, if and , then
|
(1.1) |
A point is determined by a four-vector xμ; a point is parameterized in the G1' case by a 6x6 matrix FAB as will be shown later. (Greek indices run from one to four, capital Latin indices from one to six.) Then the element acts on C in the following fashion:
|
(1.2) |
where the somewhat clumsy notation in the second line is intended to refer to the fact that the Lorentz transformations have no effect on the indices 5, 6 that appear in FAB.
By making use of the Iwasawa decomposition of SO(3,3),
|
|
where the maximal compact subgroup is the six-parameter group [19]
|
, |
|
the abelian subgroup A is a three-parameter group, the nilpotent subgroup N is a six-parameter group. Then a general element of SO(3,3) is parameterized as
|
|
where the six generators NA of the nilpotent subgroup are defined as
|
|
By choosing the basis LAB of the Lie algebra so(3,3) in the following way as 6x6 matrices,
LAB
= eAB - eBA for A, B £ 3
LAB = -eAB + eBA for A, B > 3
LAB = eAB + eBA for A £ 3, B > 3
LAB = -eAB - eBA for A > 3, B £ 3
where the matrix eAB has all elements zero except for a 1 at the position (AB), and introducing the metric tensor
|
(1.4) |
the commutation relations for so(3,3) are then of the form
Then the expression (1.3) may be written in the form of a 6x6 matrix. Thus
(1.6)
(1.7)
|
(1.8) |
and where, by using the abbreviations
|
|
the matrix N6x6 is
|
(1.9) |
The action of an element of the kinematical group on the carrier space may now be calculated. Since this action is diagonal, we may consider separately the actions on the Minkowski space M and the internal space Z. Consider, for example, the effect on the internal space of the pure Lorentz transformation , or
|
(1.10) |
When applied to an element of Z, parameterized by the 6x6 matrix FAB obtained by multiplying together the matrices (6), (7), (8), (9), in that order, the result is a new 6x6 matrix F'AB, but now with values etc., of the internal parameters.
The matrix equality represents a set of 36 (dependent) equations for the transformed parameters. (These equations simplify upon pairwise addition of columns.) They have the following solutions:
where
The transformation formulas for the nilpotent parameters are not included since we will in any case not need them here.
Lorentz transformations may be written as a product of the above transformation and rotations; the rotation transforms the compact parameters according to the well-known SU(2) transformation laws. Since we shall need the Jacobian of the above transformation later on, we write it down here.
To begin with, observe that the Jacobian for the transformation (1.11) on the entire SO(3,3) group space has the form
where
;
|
|
|
|
and similarly for the other matrices. L, M, N are constant integers to be determined by the requirement of invariance. Thus the Jacobian (1.12) is of the form
|
|
From (1.11), where we have chosen
|
|
In addition,
|
|
If the constants L, M, N, are chosen as
|
|
then the Jacobian (1.12) is equal to one, i.e., the measure
|
|
is invariant under the homogeneous Lorentz transformations. In addition, since the translations do not affect the internal space, this measure is also invariant under the action of the physical Poincaré group.
It is now apparent that the homogeneous space of minimal dimension of SO(3,3) which fulfills the conditions i) - iii) is the eight-dimensional space Z8,
|
(1.13) |
where the stabilizer subgroup is the seven parameter subgroup generated by the following SO(3,3) generators:
|
(1.14) |
The Poincaré-invariant volume element dZ on this space is
We may point out that the nine-dimensional space Z9,
|
|
where the stabilizer subgroup is just the nilpotent subgroup appearing in the Iwasawa decomposition of SO(3,3), also fulfills conditions i) - iii), with Poincaré-invariant volume element (1.15).
3. REPRESENTATIONS OF THE KINEMATICAL GROUP
With the invariant volume element
|
(1.16) |
on the twelve-dimensional carrier space C12 = M x Z8, the representation space could be defined as the Hilbert space L2(C12) of equivalence classes of square-integrable functions on the carrier space C12. In this representation space the scalar product is
|
(1.17) |
We now define a transformation
where and g-1 y is the action of g by left multiplication. This transformation gives a representation of the kinematical group since
|
|
furthermore, the representation Tg is unitary :
|
|
due to the invariance of the measure dV.
It is not, however, irreducible. The number of degrees of freedom in an irreducible positive mass representation of phys ISO(1,3) is four [9], whereas the carrier space here is 12-dimensional. To find irreducible representations, then, eight restrictions on the function space must be found.
As a first step in this direction, we consider in more detail the realizations of the kinematical group. Using the transformation (1.18) and realizing the infinitesimal transformations as differential operators one finds the following expressions for the infinitesimal operators of
phys ISO(1,3) corresponding to its Lie algebra:
Here the Smn describe the effect of the physical Lorentz group on the internal space Z; due to the postulated diagonal action of the kinematical group on the carrier space, they form a subalgebra of the Lie algebra so(3,3). The commutation relations are
|
(1.20) |
where the gmr refer to the 4x4 matrix
|
|
the commutation relations for so(3,3) are given in Eq.(1.5).
Continuing in the same manner one obtains [19] the following realization of the LAB as differential operators on the space Z:
The remaining generators may be obtained by means of the commutation relations. Notice that the coefficients of the derivatives in a generator LAB do not depend on the parameters l, m; this has the effect that the operators
|
|
commute with the Lie algebra.
Turning now to the problem of finding suitable restrictions on the elements f of the representation space, the irreducibility requirement is to be met by requiring that the elements f be simultaneous eigenfunctions of a complete set of mutually commuting operators.
Now the set of operators
where
|
|
commute with phys ISO(1,3), since the first two are just the Casimir operators of the kinematical group itself, and the second two have already been seen to commute with the entire spectrum-generating group, and could therefore be used to reduce the representation space L2(C12). They are clearly not sufficient for a complete reduction, since they represent only four of the eight restrictions needed.
In addition, the set of operators (1.22) leads to difficulties due to the fact that the operators
|
|
have continuous spectra; their eigenfunctions will not be square integrable. To circumvent such difficulties one could attempt to choose a basis of elements, not of the Hilbert space L2(C12), but rather the component F* of the Gel'fand triplet
|
|
However, rather than following this line of investigation further, it is now proposed to take as representation space the mass irreducible space
|
|
i.e., the space of square integrable functions on the positive mass shell in momentum space which are also square integrable with respect to the Poincaré invariant measure dZ. In this case the Lie algebra (1.19) simplifies to
|
(1.23) |
Now
|
|
the other Casimir operator for phys ISO(1,3) gives the restriction
This equation may be satisfied by choosing a basis in which is diagonal and solving (1.24) for = 0. Wave functions with given momentum and spin may then be defined by means of a boost transformation.
In the rest frame ( where = 0 ) the spin operators may be identified as
|
(1.25) |
We shall further assume that the isospin operators in the rest frame Trest = (T1R, T2R, T3R), are to be identified as
|
(1.26) |
The rest state, denoted by
|
|
is labeled by the eigenvalues of the following set of commuting operators:
|
|
and by the eigenvalues a and b of two operators replacing in such a manner that the wave function is square integrable. By use of the expressions (1.21) one then finds for the following eigenvectors of the set (1.27) (and of the two operators in the abelian parameters
where
|
(1.29) |
The fact that we are able to interpret the eigenvalue label S as the spin quantum number is due to the fact that in the rest frame, where the orbital angular momentum vanishes, -S(S+1) is just the eigenvalue of
|
|
where the Lij are generators of the SO(3,3) group which parameterizes the internal space. In the boosted frame , S(S+1) is, of course, the eigenvalue of the Pauli - Lubanski operator. However, the spin operators are then no longer to be identified with iL12, iL13 and iL23 ; they are well known to act, in general, on both the M and Z spaces.
For states of positive mass m, the more general state with is now obtained by applying a boost operator to the rest state (28). Upon application of the helicity boost
where
|
|
and where in (1.28) we first make the substitutions of the parameter transformations (1.11) and then apply a rotation in the space-like parameters to arrive at an arbitrary momentum
|
|
the quantum number becomes the eigenvalue of the helicity operator
|
(1.31) |
The boost leaves the third component T3 of the isospin invariant,
|
|
|
Furthermore, under the action of the helicity boost (1.30) it is well known [22] that
|
(1.32) |
where
|
(1.33) |
Application of the helicity boost to the rest frame isospin generators i(L56, L64, L45) gives the following expressions for the relativistic isospin operators :
|
(1.34) |
where
|
|
The result of applying the boost to T2Rest shall be denoted by T2 , i.e.,
|
(1.35) |
In order to obtain half-integral spin and isospin numbers, one must actually find representations of the covering group
|
|
The eigenvalues then have the ranges
|
(1.36) |
In order to do this, the ranges of the parameters must be
|
(1.37) |
This choice enables us to get the representations of the covering group of instead of just phys ISO(1,3) itself [9].
4. THE DISCRETE TRANSFORMATIONS AND THE ANTIPARTICLE WAVE FUNCTIONS
The wave function for a massive particle with 4-momentum p, helicity σ and "third component" τ of isospin is
where
|
(1.39) |
and primed variables refer to variables transformed by the boost transformation (1.30), where N is a constant, and
The wave function of the antiparticle is defined as
This choice is dictated by the requirement that complex conjugation should give the antiparticle state with momentum -p in order to give crossing-symmetric S-matrix elements.
The (strong) parity, or CP reflection, is defined to act on the parameters precisely as in the Kihlberg space C8 [9]:
|
(1.41) |
we then put
|
(1.42) |
The time reversal T has the following action on in C8 :
|
(1.43) |
and
|
(1.44) |
Based on the consideration that here the spectrum generating group SO(3,3) leaves invariant a quadratic form of signature (---+++) and that the parameters are associated with the space-like signature, whereas the isospin parameters are associated with the time-like signature, we postulate that the action of CP on the isospin parameters should be entirely analogous to the action of T on the space-like parameters, and vice-versa :
|
(1.45) |
|
(1.46) |
In the rest frame, then
|
(1.47) |
and
|
(1.48) |
This is also consistent with the observation [23] that the total inversion " " in reference [9] should be identified with the ordinary CPT transformation.
5. THE FREE FIELDS
The one-particle states (1.38) and (1.40) have been defined with the helicity convention for the spin projections. Momentum space wave functions, introduced through
|
(1.49) |
give the momentum realization of the helicity states as
|
(1.50) |
|
(1.51) |
We introduce the annihilation operators for particles and antiparticles, respectively. The Hermitian conjugates are the corresponding creation operators. The nonzero commutation relations are
On the basis of the Newton-Wigner concept of elementarity [24], we may define free fields on C12, which transform according to UIR of the kinematical group as
Using the commutation relations (52) we can calculate the commutator
|
(1.55) |
It is found to be
In order that the fields defined in (1.53) and (1.54) should fulfill the requirement of microcausality, the commutator must vanish for space-like separations x1-x2. This is not, in general, true of the commutator (1.56) . It may be pointed out that (1.56) does vanish for certain relative orientations of the internal parameters z1 and z2 ; such "kinematic causality" has been discussed for field on homogeneous spaces of the Poincaré group by Fuchs. [16]
Kihlberg has investigated two methods for restoring the causal properties of the fields. First by relinquishing the requirement that the fields transform according to irreducible representations of the Poincaré group. The reducible fields in [10] satisfy microcausality and permit the construction of a relativistically invariant coupling, and hence to a set of rules for generalized Feynman diagrams associated with a perturbation expansion. However, the propagators obtained with this method include -functions in the internal variables, and all graphs containing closed loops are divergent. It is clear that the same remarks would apply to reducible fields on C12, defined in a manner similar to that of reference [10].
In [7] another remedy was proposed, wherein the complex "structure parameters" are chosen so that microcausality is satisfied, the fields transform according to irreducible representations of the Poincaré group. The condition for relativistic invariance of the free field propagators is then assured, and a modified definition of the coupling is used to retain invariance of the S-matrix.
A third approach [8] has been suggested: namely, to accept the breakdown of microcausality as a fact of life for field theories on the homogeneous spaces in question, and to leave the structure functions as undetermined smearing operators whose presence is responsible for the nonlocality of the resulting field theories.
Since both of the approaches to microcausal fields depend on some knowledge of the explicit form of the structure functions or on some reasonable and specific choice for the hitherto arbitrary L2 -functions, a meaningful investigation of the possibilities of extending these methods to the present case must be postponed until some further understanding of these functions is obtained. For the present, we shall make use of the expressions (53) and (54) as the definition of the free fields, and leave the study of the structure functions to a future paper.
We turn now to the definition of the free-field propagator. A straightforward generalization of the conventional definition is obtained by setting
|
(1.57) |
where T is the time-ordering operator. Then, from (1.53) and (1.54), together with the commutation relations (1.52) , one finds
|
(1.58) |
for and
|
(1.59) |
when This definition leads to a function, which is not, in general, relativistically invariant. (Cf. discussion on this point in ref [10].) Pending further investigation of the structure functions, however, the free-field propagator may be defined in a manner analogous to Weinberg's treatment [18] of the propagators in the higher spin case, setting ( for the case S = T = 0)
where is the usual Feynman propagator. One then proceeds in a purely formal manner from the representation
|
|
to write
in the case of zero spin and zero isospin. The discussion of this propagator is, of course, dependent upon possible singularities of the structure functions, which are p-dependent due to the primed variables.
6. INTERACTIONS
The existence of the Poincare'-invariant measure d4x dZ on the carrier space C12 implies that we may generalize the Dyson formula for the S-matrix as follows:
|
(1.61) |
where T is the modified time-ordering implicit in the definition (1.60) of the free-field propagator ; the Hamiltonian density H(x) is defined as
|
(1.62) |
and where
|
|
is the invariant measure on the internal space. This allows us to make use of Kihlberg's rules for the Feynman diagrams:
1) For each vertex insert a factor -ig ;
2) For each internal line, the propagator ;
3) For each external line, a wave function; and
4) Integrate over the internal variables z and the external variables x at each vertex
The graph
thus corresponds to the amplitude
where
|
(1.65) |
define Poincaré -invariant form factors. It has been assumed here that the exchanged particle has zero spin and isospin. Furthermore, we have introduced a new notation for the structure functions exhibiting explicitly the boost on the variables by the corresponding momentum vector.
We may make use of the graph (1.63) and the corresponding amplitude (1.64) to make an order-of-magnitude investigation of the charge independence properties of the theory. We shall examine the scattering matrix element for elastic scattering on the basis of the simple, one-particle exchange graph, the exchanged particle assumed to have both spin and isospin zero.
Now the Z-space part of the wave functions for spin 0 particles are, besides the structure functions F, just the Wigner D-functions in the isospin space, although these functions obtain distortions for nonzero momenta p. We begin by examining the effect of these distortions on the form factors V1 and V2. To do this we first assume that the structure functions F are independent of the isospin parameters as well as the spin space parameters and that the integrations over the parameters converge:
This can be achieved only approximately, since in order that the integrals converge the structure functions must depend on the variables and hence on the angles when they are boosted. However , by introducing convergence factors for large absolute values which are constant for smaller values of we can achieve the desired properties through a limiting procedure. Then
where here
|
|
and the Euler angles refer to the result of the boost transformation (1.30), the index in parenthesis referring to the different boosts to momenta p1, p2, p3, p4.
In the static limit ( p = 0 ) the wave functions on Z are just the conventional D - functions, i.e., the one-particle states are pure isospin states; the isospin space integrals in (1.67) are then of the form
|
(1.68) |
which vanishes if and fulfills the condition of charge independence since the attempt to couple states of different isospin will always result in a vanishing scattering matrix element.
However, in arriving at this result, two very restrictive assumptions have been made which must now be removed :
i) the static
approximation ( ) for both wave functions at each vertex;
ii) structure functions independent of isospin space parameters.
A further restriction was made, namely that the exchanged particle is an isoscalar. However, we shall not investigate possible generalizations of this point.
In the static limit, the isospin space parts of the pion wave functions are
|
(1.69) |
where we have chosen to set k = 0 for reasons of simplicity. For general values of the momentum , these functions become “distorted” by the action of the boost transformation (1.30). The exact relativistic expressions are
|
|
where
|
|
and, as before, are the polar angles of the 3-momentum .
To second-order terms in the variable
the charged pion wave function may be written
where
(1.71)
The presence of the factors
|
|
mixes the rest-frame wave functions with extraneous isospin states. In general, the correction terms do not depend on , and due to the choice k = 0, neither will they depend on for the pion wave functions.
In the case of the uncharged pion,
where the coefficients C are given by
We turn now to the form factors V1 and V2, given by (1.67), due to the assumption (1.67).
Substituting the corrected wave functions (1.70) and (1.72), gives
|
(1.74) |
With these form factors, we find
|
(1.76) |
|
(1.77) |
In the case of the elastic scattering of pions, isospin invariance implies [25] that, for example,
|
(1.80) |
Eq. (1.79) is obviously fulfilled, due to (1.75). However, in (1.78) where only the two "diagonal" elements are nonzero, we define the symmetry- breaking factor by
|
(1.81) |
This gives, to second order in x,
|
(1.82) |
Thus the percentage symmetry breaking is given by
|
(1.83) |
which for
gives 3%, in itself an acceptable amount [26]; however, that this result is somewhat misleading may be seen by examination of the coefficients C in (1.72) and (1.73) . They represent the amount of extraneous isospin states mixed in at successively higher momenta. In general, one should expect unacceptably large amounts of symmetry breaking already for momenta of the order of a pion rest mass.
We now allow the structure functions F to have some isospin space dependence, in the hope that this might improve the above result. Observe that the choice
where is an appropriate convergence factor, assumed to fulfill (1.66) (where, again, this may be done only approximately), restores the charged pion wave functions to pure isospin states as in the rest frame. If we make the same choice (1.84) of structure function for the neutral pion, we find the following form factors:
|
(1.85) |
where the wave function for the uncharged pion is now of the form
|
(1.86) |
These form factors give the amplitudes
|
(1.87) |
Comparison of (85) with (80), (81) gives the following expression for the symmetry-breaking factor :
|
(1.88) |
and
|
(1.89) |
which for gives 50% symmetry breaking, in accord with the above remarks.
These manipulations are to be seen as a tentative outline and as a demonstration of the fact that the present formalism may be extended to include the conventional attempts at the description of interactions. However, the most unsatisfactory aspect of the above description is the fact that it does not seem possible to maintain the isospin invariance of interactions to any acceptable degree.
This may be due to the postulated action of the kinematical group on the isospin parameters, or to the naive extension made here of conventional interaction formalism. These issues are postponed to the following paper, where we attempt to make a specific interpretation of the physical significance of the internal parameters.
Acknowledgements
I should like to express my gratitude to my advisor, Lektor Arne Kihlberg, for the guidance received in this work, as well as for his comments of more general nature which I have found significant. Many members of the Institute for Theoretical Physics have contributed their time to discuss this work with me, and I would like to thank all of them. Special gratitude is expressed to Professors Robert Anderson and Karl-Erik Eriksson, Drs. John Nagel and Ulf Ottoson.
I would also like to thank Professor David Finkelstein for several inspiring conversations.
Finally, my most profound gratitude to Professors Tullio Regge and Mario Rasetti for their very special inspiration and guidance.
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