Oscillatory modes of quarks in baryons for 3 quark flavors u, d, s

Transcription

1 Oscillatory modes of quarks in baryons for quark flavors u, d, s Peter Minkowski Albert Einstein Center for Fundamental Physics - ITP, University of Bern Abstract The present notes are meant ( to illustrate the oscillatory modes of N fl = light quarks, u, d, s, using the SU N fl = 6 ) ( ) SO L broken symmetry classification. L = N fl n = 1 L n stands for the space rotation group generated by the sum of the individual angular momenta of quarks in their c.m. system. The motivation arises from modeling the yields of hadrons in heavy ion collisions at RHIC and LHC, necessitating at the respective highest c.m. energies per nucleon pairs an increase of heavy hadron resonances relative to e.g. SPS energies, whence the included hadrons are treated as a noninteracting gas. Notes p. 1

2 cont-1 List of contents 1 Introduction 8 Modes of quarks in baryons according to the SU ( N fl = 6 ) SO ( L ) broken symmetry classification -1 The reduction of an SUN group R-fold product representation through the symmetric group S R Young tableaux Y N R - The main N P = 56 + baryon valence quark configuration multiplet the entry point - The first oribitally excited, negative parity, N P = 70 based, N = 10 baryon valence quark configuration multiplet the confirmation from experimental spectroscopy? -4 The first oribitally excited, negative parity, N P = 0 based, N = 0 baryon valence quark configuration multiplet about SU fl singlet baryons -4-1 The unitary scalar product adapted to rotational, combinatorial and c.m. related conditions of valence quark u,d,s modes in baryons From oscillatory modes to counting of states -1 Counting the number of points with nonnegative coordinates on a unit grid in R = 6 dimensions Incorrect procedure according to subsectin -1(-1) superseded by subsection -1-d p.

3 cont- List of contents continued (I) -1-1 Exact recursive combinatorics 5-1-1a The power of the set (N, R) is summable, i.e. calculable in terms of the sum of integer powers p over the integers 0, 1,,, K denoted I (K, p) below -1-1b The binomial reduction formula I (K, p+1) I (K, p) 9-1- Preparing the summation variables and tools to perform the nested N 0 = N N 1 N N R 1 N R = 0 summation of (N, R) in eq Continuum inegral approximations 45-1-a The barycentric 6 spatial oscillatory variables and the set (N, R) 46-1-b -1-c The barycentric 6 spatial oscillatory variables and their symmetries with respect to the quark positions The barycentric 6 spatial oscillatory variables and their symmetries with respect to the quark positions in dimensionless universal variables -1-d The barycentric coordinates dimension by dimension : cartesian and skew hexagonal coordinates as appropriate for quark position variables p.

4 cont- -rec -res -res- List of contents continued (III) Reconstruction of the two-dimensional irreducible unitary representation of S from 1 spacelike dimension Results on the reconstruction of the two-dimensional irreducible unitary representation of S from 1 spacelike dimension Choosing a complex basis for transforming the basis derived in eq. 14 for the two-dimensional irreducible unitary representation of S from 1 spacelike dimension -res- Extending 1 spatial dimension to 81 4 Partial countings of oscillatory modes of quarks in baryons Characteristic transformation properties of the basis functions ( ) ψ n 1, n ζ, ζ under the permutation group S ( ) Transformation properties of the basis functions ψ n 1, n ζ, ζ under the transpositions T 1, T, T C Consolidation of transformation properties of S as elaborated in subsection -rec 4-C-1 Schur s lemma p. 4

5 cont-4 List of contents continued (IV) 5 Counting oscillatory modes using the circular oscillatory wave function basis 105 enforcing overall Bose symmetry under the combined permutations of SU6 (fl spin) barycentric coordinates 5-1 Traces of irreducible representations of S Aligning statistics between the u, d, s SU6 (fl spin) group and oscillator modes in 6 barycentric configuration space variables 5--1 Associating symmetric and antisymmetric representations 108 of SU6 (fl spin) to oscillator wave functions in the circular pair-mode basis 5-ins Reducible traces from direct product representations of S Back to subsection Associating the mixed 70-representation of SU 6 ( f l spin) to oscillator mode wave functions in the circular pair-mode basis p. 5

6 cont-5 A1 A A-f A List of contents continued (V) Appendix 1 : Some checks on results in subsection -1-1b The binomial reduction formula I (K, p+1) I (K, p) Appendix : A Fortran program to calculate (N, R),the power of the set of states of the oscillatory modes of u, d, s - valence quarks in baryons, displayed in eqs. 45, 59 Incorrect procedure according to subsetion -1(-1) superseded by subsection -1-d The Fortran code of the file sumbar01.f for N = with extended details in comment statements Appendix : The x representation of S in the basis denoted by the collection d π in eq. 11 Associating the collections D π in eq. 14 and d π in eq. 11 for elements beyond unity of the x representation of S 140 From d π= to d π= 141 R1 References p. 6

7 Figures 1 List of figures Fig 1 The symmetric Young tableau for SU6 and rank R = 10 Fig The mixed Young tableau for SU6 and rank R = 11 Fig The antisymmetric Young tableau for SU6 and rank R = 1 Fig 4 The hexagonal logic in the (ξ, ξ 1 ) plane 56 p. 7

8 Introduction The notes developed here have their root in my discussion of the oscillatory modes of quarks in baryons in ref. [1-1980], as well as the presentation of two hadron resonance collections used in ref. [-010], worked out subsequently in a notefile similar to this one in ref. [-011]. In comparing the two collections used in ref. [-010], denoted Ntype=65 Ntype=6, turn out both to be too small to account for the hadron abund ances, as measured at RHIC and LHC, comparing with the Hadron Resonance Gas ( HRG ) approach ( see e.g. ref. [4-010] ), with noninteracting hadrons. The detailed description of the collection of hadron resonances used in thermal fits to all, including RHIC and LHC hadron abundances is explicitely presented in ref. [5-009]. The selection consists of includeing all hadron resonances in the PDG tables of 008 [6-008]. Here we propose to consider the three light flavors of quark u, d, s, extending the modes discussed in ref. [1-1980] as a first step, in the next section. - Modes of quarks in baryons according to the SU ( N fl = 6 ) SO ( L ) broken symmetry classification The strategy to select hadron resonance followed in 009 in ref. [5-009], as described in section 1, and strategies grouping togther, e.g. baryon states, between in refs. [ ], [8-1977], [9-1979] for flavors of quarks, using the SU ( N fl = 6 ) SO classification, forming the title of section, reveal a break of tradition. ( L ) broken symmetry p. 8

9 -1 Here we propose to illustrate with new data with respect to 1980 the reliability of the PDG with respect to missing as well as incorrectly included hadron resonances, as compared with spectroscopic theoretical valence quark models with extended broken symmetries The reduction of an SUN group R-fold product representation through the symmetric group S R Young tableaux Y N R Alfred Young + 16 April 187 in Widnes, Lancashire, England 15 December 1940 in Birdbrook, Essex, England, cited from ref. [1-01]. The three irreducible Young tableaux in figures 1 - below, arise as tensors of rank within a symmetry group of SU ( 6 = SU spin SUN fl = ). Its broken character is discussed later. Rank three corresponds to the wave function of a baryon formed from three valence quarks, confined with respect to color, exclusively. In the construction of this wave function the width of the clearly involved resonance shall be set approximately to zero. We turn towards the functions asociated with these Young tableaux, depending on 6 integer arguments (1) D 6 (m 6, m 5,, m 1 ) = D (m 6, m 5,, m 1 ) m 6 > m 5 > m 1 > 0 : integers D (m 6, m 5,, m 1 ) = Π 6 j= Π k = j 1 k=1 ( m j m j k ) The D - functions for symmetric, mixed and antisymmetric representations of SU6 will be discussed after the figures below. p. 9

10 - 7 Young tableau Y^{6}_{} for SU6 and R = m_{6} = 6 + r_{6} = 9 m_{5} = 5 + r_{5} = 5 m_{4} = 4 + r_{4} = 4 m_{} = + r_{} = m_{} = + r_{} = m_{1} = 1 + r_{1} = 1 D^{6}_{} ( 9, 5, 4,,, 1 ) Fig 1 : The symmetric Young tableau for SU6 and rank R = p. 10

11 - 7 Young tableau mix Y^{6}_{} for SU6 and R = 6 m_{6} = 6 + r_{6} = 8 5 m_{5} = 5 + r_{5} = 6 4 m_{4} = 4 + r_{4} = 4 m_{} = + r_{} = D^{6}_{} ( 8, 6, 4,,, 1 ) m_{} = + r_{} = 1 m_{1} = 1 + r_{1} = Fig : The mixed Young tableau for SU6 and rank R = p. 11

12 -4 7 Young tableau mix Y^{6}_{} for SU6 and R = m_{6} = 6 + r_{6} = 7 m_{5} = 5 + r_{5} = 6 m_{4} = 4 + r_{4} = 5 m_{} = + r_{} = m_{} = + r_{} = m_{1} = 1 + r_{1} = 1 D^{6}_{} ( 7, 6, 5,,, 1 ) Fig : The antisymmetric Young tableau for SU6 and rank R = p. 1

13 -5 The significance of Young tableaux with respect to an SUN transformation group lies in the one to one correspondence of irreducible representations of this group formed by the R-fold tensor product of the defining one, symmetrized first along the rows and antisymmetrized thereafter along the columns associated with any given Young tableau, forming a representaion of the symmetric group S R, The three Young tableaux in figures 1 - each determine such an irreducible representation of SU6. The D functions, defined in eq. 1, determine the respective dimensions of these representations. We list the three D functions in eq. below : D (9, 5, 4,,, 1) = d 5 : D (8, 6, 4,,, 1) = d 4 : D (7, 6, 5,,, 1) = d () d j = D (j, j 1,, 1) = (j 1)!d j 1 = Π j 1 k=1 k! d 1 = d = 1, d =,d 4 = 1, d 5 = 4 1, d 6 = This concludes the discussion of the main premises contained in Young tableaux. p. 1

14 -6 The dimensions of irreducible SUN representations belonging to a specific Young tableau are given by () dim ( Y-t) = D (m R, m R 1,, m 1 ) D (R, R 1,, 1) Substituting eq. in eq. it follows, always for SU6 dim ( ) = = 56 5! (4) dim ( ) = !5! = 70 dim ( ) = !4!5! = 0 p. 14

15 The main N P = 56 + baryon valence quark configuration multiplet the entry point The most stable baryon multiplet, restricted to the light three flavors of u, d, s quarks, shall be labelled by the total number of stetes including spin and flavor, with multiplicity and parity denoted N and by the superfix P respectively. The landmark pertaining to the J P = states within this multiplet is the discovery picture of the Omega in 1964, shown in figure 4 below, as administered electronically today by Wikipedia, the free encyclopedia, ref. [1-196] for the authors of the discovery of ref. [ ]. J denotes the total angular momentum formed from spins and oribital angular momenta. The strangeness baryon was predicted in 196 together with its mass region for a decuplet with J = The configuration space wave functions of the 56 + combined SU6 SU in refs. [14-196]. ( L ) - ground state multiplet are ( assumed ) independent of angular relative momenta of the valence quarks and all equal as a consequence, and thus form an octet of J = 1 and a decuplet of J = overall 56 states of equal parity, positive by convention. i.e. a multiplet of This concludes the entry point presentation of Young tableaux. The nontrivial higher configurations shall be discussed in subsequent sections. p. 15

16 -8 Fig 4 : The bubble chamber picture of the Omega in 1964, see ref. [ ] p. 16

17 The first oribitally excited, negative parity, N P = 70 based, N = 10 baryon valence quark configuration multiplet the confirmation from experimental spectroscopy? The N = 10 negative parity u, d, s multiplet is well described in the current PDG review Quark Model by C. Amsler, T. De Grand and B. Krusche in ref. [17-011]. The mixed SU6 spin fl Young talbleau yielding the 70 representation is combined with an SU ( L ) ; L = 1 orbital angular momentum wave function in a way compatible with icluding color overall fermion permutation antisymmetry of three valence quarks u, d, s. (5) L = L 1 + L + L ; L = L 1 = 1 ; L + 1 = N = (L + 1) N = 10 The N = 70 representation of SU6 spin fl decomposes with respect to total quark spin multiplicity and ( times ) SU fl irreducible representations as (6) 70 = This in turn combines the respective total spins 1 and to total angular momenta J P (7) spin = 1 : J P = 1 spin = : J P = 5 1 p. 17

18 -10 Finally also combining total orbital angular momentum and total spin the N = 10 multiplet decomposes according to (8) N = 10 = 10, ( ) ( 1 ) + # 60 8, ( ) ( 1 ) + # 48 8, ( 5 ) ( ) ( 1 ) + # 96 1, ( ) ( 1 ) # 6 10 p. 18

19 -11 S = 0 I = 1 S = 0 I = S = 1 I = 0 S = 1 I = 1 S = I = 1 S = I = 0 J P = ( M = 150 J P = ( 1 M = 155 J P = ( 1 M = 1650 J P = ( M = 1700 ) ) ) ) J P = ( M = 1700 J P = ( 1 M = 160 ) ) J P = ( M = 1690 J P = ( 1 M = 1670 J P = ( 1 M = 1405 J P = ( M = 150 ) ) ) ) J P = ( M = 1670 ) J P = ( ) 1 M = 1750 ) J P = ( M = 1940? J P = ( ) M = 180 J P = ( 5 M = 1675 ) J P = ( 5 M = 180 ) J P = ( 5 M = 1775 ) Table 1 : Candidate states belonging to the mixed N = 70 negative parity Young tableau M in MeV (9) p. 19

20 -1 The lowest in mass negative parity states, without heavy flavor content, according to the current PDG tables [16-01], are listed in Table 1, eq. 9. We list the numbers for SU fl octets, decuplets and singlets in the N = 70 ; N = 10 multiplet of states (10) octets J P = ( 1 ) : # octets J P = ( ) : # 64 octets J P = ( 5 ) : 1 # 48 decuplets J P = ( 1 ) : 1 # 0 decuplets J P = ( ) : 1 # 40 singlets J P = ( 1 ) : 1 # singlets J P = ( ) : 1 # 4 # 10 p. 0

21 -1 Accepting all states listed in Table 1 and assigning Ξ () ; M = 180 to an octet, Σ () ; M = 1949 to a decuplet and Λ () ; M = 150 and Λ (1) ; M = 1405 to a singlet, the following states are either missing or corresponding quantum numbers cannot be assigned (11) Ω () decuplet # 4 Ω (1) decuplet # Ξ () decuplet # 8 Ξ (1) decuplet # 4 Ξ (5) octet # 1 Ξ () octet # 8 Ξ (1) # missing states :76 out of10 octet # 8 Σ (1) decuplet # 6 Σ () octet # 1 Σ (1) octet # 6 Λ () octet # 4 Λ (1) octet # p. 1

22 -14 Only states with nonvanishing strangeness are among the missing or non-assignable. In order to put this number ( 76 ) in perspective we list among the N = 10 states those with vanishing strangeness : a quartet per decuplet and a doublet per octet. Thus we have for the nonstrange number of states (1) ( ) decuplet (1 ) decuplet N (5 ) octlet N ( ) octlet N (1 ) octlet # 16 # 8 # 1 # 16 # 8 # strange states :150 # nonstrange states : 60 Let me associate a quality factor relative to the spectroscopic recognition of the states with nonvanishishing strangeness, pertaining to the negative parity N = 70 ; N = 10 multiplet of states, abbreviated by { 70 } p.

23 -15 restricted also to three u, d, s flavored valence quark configurations, lowest in mass (1) Qf (S < 0) = 1 74 = 150 # correctly assigned strange states # all strange states { 70 } nonstrange states = % The result in eq. 1 prompts two remarks 1) The quality factor Qf (S < 0) 49.%,defined in eqs. 11-1, is clearly insufficient. ) This calls for explanations in the first instance by the experimenters having performed the pertinent experiments. p.

24 The first oribitally excited, negative parity, N P = 0 based, N = 0 baryon valence quark configuration multiplet about SU fl singlet baryons First we study the following symmetric (pseudo-) tensor structure obtained from a triplet of configuration space vectors x 1,, subject to the c.m. restriction (14) T mn L = = + (x 1 ) m (x x ) n + (x 1 ) n (x x ) m + (x ) m (x 1 x ) n + (x ) n (x 1 x ) m + (x ) m (x x 1 ) n + (x ) n (x x 1 ) m δ mn Det(x 1, x, x ) i=1 x i = 0 = Det(x 1, x, x ) = 0 Under rotations of the configuration vectors x i R x i ; i = 1,, the functions T mn L= (x 1, x, x ) form the 5 orbital angular momentum wave functions with L =. The full wave functions, not restricted to oscillatory modes, as described in ref. [1-1980], but with quantum numbers specified according to the SU ( N fl = 6 ) SO classification, and associated Young tableau symmetry relations ( L) broken symmetry p. 4

25 -17 are then constructed as follows, where the orbital part here T mn L = defined in eq. 14 is a factor Ψ µν (x 1,,x, x ) = T mn L= (x 1, x, x )ψ (x 1, x, x ; L = ) ψ (x 1, x, x ; L = ) ψ (x 1, x, x ) : to simplify notation ψ (x 1, x, x ) = ψ (Rx 1, Rx, Rx ) orbital rotation symmetry ψ (x 1, x, x ) = ψ (x j 1, x j, x j ) permutations 1 j 1 j j i=1 x i = 0 overall center of mass condition (15) Whence the rotational, combinatorial and c.m. related conditions, displayed in the last lines of eq. 15, are satisfied they are broken by the asymmetries of the quark masses m u, m d, m s and independently by the breaking of SU6 spin fl the remaining dynamical structure of QCD resides in the specific form of the residual wave function ψ (x 1, x, x ). It is a Gaussian function for oscillatory modes [1-1980]. p. 5

26 The unitary scalar product adapted to rotational, combinatorial and c.m. related conditions of valence quark u,d,s modes in baryons The present subsection fits nicely in the context of this chapter, even if it is a digression, preparing discussion of positive parity SU ( N fl = 6 ) SO ( L ) broken symmetry baryon multiplets. To this end we replace the specific wave function Ψ µν (x 1,,x, x ) defined in eq. 15 by a collection of generic ones (16) Ψ µν (x 1,,x, x ) (α) Ψ (α) (x 1,,x, x ) The unique unitary scalar product, compatible with all rotational, combinatorial and c.m. related conditions is then of the form (17) Ψ () Ψ (1) = () / Π i=1 d x i δ (x 1 + x + x ) Ψ () (x 1,,x, x )Ψ (1) (x 1,,x, x ) We calculate the the square norm of a Gaussian, using the barycentric coordinates for three valence quarks in the c.m. system (18) z 1 = 1 (x 1 x ), z = 1 6 (x 1 + x x ) z = 1 (x 1 + x + x ) = X c.m. 0 p. 6

27 -19 It is good to remember from ref. [1-1980] that only out of the three barycentric three-vector variables as well as their conjugate momenta, i.e. (19) z 1, z π 1, π exhibit oscillatory ( confined ) motion, wheres the c.m. related position and conjugate momentum are left in free motion. Thus the universal, i.e. N c independent oscillatory frequency appears in the form of the induced M operator in the N c (= here) dependent combination M = [ ] N c 1 ν=1 K N c π ν + (K N c ) 1 Λ z ν ( (0) K N c = N c / (N c 1) = here) π µ = 1 i z µ ; µ = 1, Finally in the identificationi of oscillatory variables we reduce to the actual oscillator ones through a canonical transformation (1) z µ = λz µ π µ = λ 1 π µ ; µ = 1, under which p. 7

28 -0 M in eq. 0 becomes () M = N c 1 ν=1 K N c λ π ν + λ Λ K N c z ν The quantity λ of dimension length, introduced in eqs. 1 and, is to be chosen such that () K N c λ = λ Λ K N c λ 4 = K N c whereupon M in eqs. 0, assumes the reduced universal form M = Λ N c 1 ν=1 [ π ν + z ν ] Λ λ = K N c Λ (4) z µ = λz µ π µ = λ 1 π µ ; µ = 1, ; λ = iπ µ = z µ z µ ; µ = 1, K N c Λ p. 8

29 -1 The associated ( universal ) oscillator vector-variables derive from the relation on the last line in eq. 4 (5) a µ k = 1 ( iπ µ k + z µ k ) ; a µ k = 1 ( iπ µ k + z µ k ) µ = 1, ; k = 1,, which in turn yields the relativistic structure of the oscillatory M operator, substituting in eqs. 0, and 4 (6) M = (Λ) ν=,k= ν=1, k=1 [ ] a ν k a ν k + 1 ; Λ = 1/α 1 The contribution of the zero mode oscillations, for each dimension of configuration space ( = ) is inherent to the classical limiting form of oscillatory motion, and is accompanied in the sense of a long range approximation ( especially given finite quark masses ) by a constant correction.the latter remains non zero also in the limit of vanishing quark masses, as discussed in ref. [1-1980]. In eq. 6 1/α denotes the inverse of the Regge slope. p. 9

30 - If we determine it from the positive parity Λ trajectory from the present PDG tables [16-01] (7) Λ, J P : M j : M j : M : and average the two half mass square difference entries in the last line of eq. 7 with weights two to one we obtain (8) 1/α = 1 ( M M 1 ) ( M M ) 1.06 GeV I remark that in ref. [16-01] Λ 9 + has only three stars, and furthermore the trajectory contains only three entries, whereas I think to remember that it contained four sometimes back a. Λ 1 + would extrapolate to.755 GeV using eq. 8. a Tempora mutantur nos et mutamur in illis. p. 0

31 - We will come back to eigenvalues, number partitions and counting of flavor-, spin- and 6 orbital oscillator-states, deriving from the mass square operator as characterized in eq. 6 in a subsequent subsection and return to the scalar product of the ground state oscillator wave function with itself. The latter is constructed from the orbital oscillator operators specified in eq. 5, neglecting here flavor and spin quantum numbers within the approximations discussed in this subsection. p. 1

32 -1 - From oscillatory modes to counting of states The form of the mass square operator, as displayed in eqs. 0-6 is as a long distance approximation not specified, in particular with respect to the oscillatory zero modes, as well as other constant contributions in the configuration space distances at large. The next step can be inferred from the way the universal inverse Regge slope is determined in eqs. 7-8 and amounts to the parametrization starting with eq. 6 repeated below (9) M = (Λ) ν=,k= ν=1, k=1 From eqs. 6, 9 we introducuce the decomposition [ ] a ν k a ν k + 1 ; Λ = 1/α (0) M = M + M (0) ; M (0) = (Λ) C (0) M = (Λ) ν=,k= ν=1, k=1 extended to spin and flavor [a ν k a ν k ] The extension to include spin and flavor degrees of freedom in the universal part M as diplayed in eq. 0 corresponds to the definition of a direct product label A (1) A = µ dim spin flavor = {1,,, 6} p.

33 - The universal part M defined in eq. 0 for valence quarks u, d, s in baryons thus becomes () M = (Λ) R A=1 a A a A ; R = 6 This prompts the reparametrization of M and subsequent approximate universality extension of the operator C (0) and its eigenvalues, both defined in eq. 0 () M = (Λ) 1 M = R A=1 a A a A + C (0) C + C (0) C = R A=1 a A a A ; R = 6 The approximate universality extension of th eigenvalues of the operator C (0), mentioned above, implies for the eigenvalues the Ansatz E (M ) = N + C (0) (4) N = R A=1 n A ; n B = 0, 1 ; B = 1.., R C (0) approximately independent of {n1, n,, n R } The approximation defined in eqs. 9-4 reduces the counting of baryon states to the counting of points on a unit grid in R = 6 dimensions. p.

34 Counting the number of points with nonnegative coordinates on a unit grid in R = 6 dimensions Incorrect procedure according to subsetion -1(-1) superseded by subsection -1-d The sought approximate counting we are envisaging is the number of points (N, R) with coordinates (5) (n 1, n,, n R ) ; n B = 0, 1 ; B = 1.., R and the condition for a given posiive integer N (6) S = A n A N By reflection symmetry on any one of the R coordinates ( quadrant symmetry for R = ) we can extend the coordinates in eq. 5 to negative integer values so that the condition in eq. 6 becomes, but only approximatively. eflection symmetry is reduced for all points with some n B 0. (7) It follows (8) S ± = A n A N ; n B = 0, ±1 (N, R) R ± (N, R) Since here N may be smaller than R = 6 care must be taken especially when trying to circumscube a R-sphere to the R-cube of length N to majorize ± (N, R). The R-volume of the R-cube is (9) V R cube = (N ) R p. 4

35 Exact recursive combinatorics The power of the set ± (N, R) defined in eq. 8 as the number of signed integers {n 1, n,,n R } ; n B = 0, ±1 ; B = 1,,, R which satisfy the condition given in eq. 7 repeated below (40) S ± = A n A N ; n B = 0, ±1 ; B = 1.., R satisfies succesive recursion relations on and N and R ± (N, R)= N N 1 =0 (N N 1 + 1) ± (N 1, R 1) = N N 1 =0 (N N 1 + 1) N 1 N =0 (N 1 N + 1) ± (N, R ) (41) = = N N 1 =0 (N N 1 + 1) N 1 N =0 (N 1 N + 1) N R N R 1 =0 (N R N R 1 + 1) ± (N R 1, 1) p. 5

36 -5 From eq. 41 we can anchor the recursion at R = 1 (4) ± (N R 1, 1) = N R Introducing the fixed endpoints N = N 0, N R = 0 we obtain the base form of ± (N, R) ± (N, R) = (N N 1 + 1) (N 1 N + 1) (N N N 1 N R 1 0 R N R 1 + 1) (N R 1 N R + 1) N 0 = N, N R = 0 fixed end points ; summation variables :N 1,, N R 1 (4) Since eq. 4 is exact and exact recursion relations for both ± (N, R) ( eq. 41 ) and (N, R) exist, contrary to eq. 8, which is only approximate is I proceed to derive the corresponding relations for (N, R) R below. p. 6

37 -6 The recursion relations for (N, R) is of the form (N, R)= N N 1 =0 (N N 1 + 1) ± (N 1, R 1) = N N 1 =0 (N N 1 + 1) N 1 N =0 (N 1 N + 1) ± (N, R ) (44) = = N N 1 =0 (N N 1 + 1) N 1 N =0 (N 1 N + 1) and arises from eq. 41 by the substitution N R N R 1 =0 (N R N R 1 + 1) (N R 1, 1) (N κ 1 N κ ) N κ 1 N κ ; forκ = 0,1,,R. The algebraic expression analogous to eq. 4 but for (N, R) arises through the same substitution and takes the form p. 7

38 -7 (45) (N, R) = (N N 1 + 1) (N 1 N + 1) (N N N 1 N R 1 0 R N R 1 + 1) (N R 1 N R + 1) N 0 = N, N R = 0 fixed end points ; summation variables :N 1,, N R 1-1-1a - The power of the set (N, R) is summable, i.e. calculable in terms of the sum of intieger The quantities powers p over the integers 0, 1,,, K denoted I (K, p) below (46) I (K, p) = K r=0 r p ; K, p, r : integers, with K, p 1 to which we proceed to reduce the counting of baryon states with masses up to the limit (47) ( ) 1/α M limit C (0) = N α, C (0) in eq. 47 beeing defined in eqs. 8 and respectively, and which is given by the integer quantity (N, R) defined in eq. 45, are classical ones in number theory. For a modern presentation see e. g. ref. [19-01]. p. 8

39 -8-1-1b - The binomial reduction formula I (K, p+1) I (K, p) We postpone to the next subsection the calculational reduction of (N, R) in eq. 45, iin order to focus on the systematic reduction with respect to the power p of the quantities I (K, p),the sum of powers of integers as given in eq. 46. I (K +1, p+1) = I (K, p+1) + (K +1) p+1 (48) = K+1 r=r +1 = 1 (r + 1) p+1 = K r =0 (r + 1) p+1 Next we express the summands in the last line of eq. 48 (r + 1) p+1 through their binomial expansion (49) (r + 1) p+1 = (r ) p+1 + p q=0 p + 1 q (r ) q Subtracting I (K, p+1) from both sides of eq. 48 we obtain p + 1 I (K, q ) = (K +1) p+1 (50) p q=0 q p. 9

40 -9 add In eq. 50 repeated below for clarity the value I (K, q = 0) needs special care, relative to all positive values q > 0. p (51) q=0 p + 1 I (K, q ) = (K +1) p+1 q This can be seen decomposing eq. 49 (5) (r + 1) p+1 = (r ) p+1 + p q=1 p + 1 q (r ) q + 1 Inserting the expression on the right hand side of eq. 5 into the last line on the right hand side of eq. 48 repeated below I (K +1, p+1) = I (K, p+1) + (K +1) p+1 (5) it follows = K+1 r=r +1 = 1 (r + 1) p+1 = K r =0 (r + 1) p+1 p. 40

41 -9 add1 (54) I (K +1, p+1) = I (K, p+1) + + p p + 1 q=1 q I (K, q ) + K r =0 (1) The last term on the right hand side of eq. 54 serves to define the limiting values for q 0 K r =0 (1) = K + 1 (55) I (K, 0) = K + 1 ; p + 1 = 1 0 It appears surprising that singling out the last term in the sum on the left hand side of eq. 50 the recursion index has dropped from p + 1 to p (56) (p + 1) I (K, p) = (K +1) p+1 p 1 q=0 p + 1 q I (K, q ) p. 41

42 -9 We substitute the summation variable q = p q and rewrite eq. 56 (p + 1) I (K, p) = (K +1) p+1 p p + 1 (57) q =1 p q I (K, p q ) Finally we substitute p ( p + 1 in eq. ) 57 to promote the recursion I (K, p + 1) I K, p p q q (58) I (K, p + 1) = = (p + ) 1 p+1 q=1, dropping the prime on the dummy summation variable (p + ) 1 (K +1) p+ p + I (K, p + 1 q ) p + 1 q I (K, 0) = K + 1 ; p = 1 for p 0 This ends the subsection on recursions of power sums on the power. p. 4

43 Preparing the summation variables and tools to perform the nested N 0 = N N 1 N N R 1 N R = 0 summation of (N, R) in eq. 45 First I repeat eq. 45, the defining equation for the nested summation to be carried out below (59) (N, R) = (N N 1 + 1) (N 1 N + 1) (N N N 1 N R 1 0 R N R 1 + 1) (N R 1 N R + 1) N 0 = N, N R = 0 fixed end points ; summation variables :N 1,, N R 1 Next we want to distinguish fixed nonnegative integers : N = N 0 ; N R = 0 from variables. also nonnegative integers : N 1,, N R 1 as defined and restricted in eqs. 45 = 59. To this end we make the notational identifications (60) (N 1,, N R 1 ) (V 1,, V R 1 ) nonnegative summation variables (V 1,, V R 1 ) and rewrite the quantity (N, R) in eq. 59 using the variables substituted in eq. 60 p. 4

44 -10-1 p. 44

45 -11 First we repeat eq. 45 ( = 59 ) below Continuum integral approximations (61) (N, R) = (N N 1 + 1) (N 1 N + 1) (N N N 1 N R 1 0 R N R 1 + 1) (N R 1 N R + 1) N 0 = N, N R = 0 fixed end points ; summation variables :N 1,, N R 1 In order to check the exact result for the quantity (N, R) the power of the set of states as oscillatory modes with main quantum number (6) N = 5 κ=1 N κ N as worked out in detail in Appendix ( eq. 05 ) we consider an interpolation of the integer valued summands in eq. 61 and replace the integer logic, nested step-function sum, by a 5-fold nested integral, an approximation to the exact result. p. 45

46 a - The barycentric 6 spatial oscillatory variables and the set (N, R) It is preferable to work out first the barycentroc oscillatory variables as conditioned by the quark flavor Young tableu reduced flavor spin multiplets as derived in eq. 4 dim ( ) = = 56 5! (6) dim ( ) = !5! = 70 dim ( ) = !4!5! = 0 We recall the definition of the barycentric variables in eqs and 8 p. 46

47 -11- M = Λ N c 1 ν=1 [ π ν + z ν ] (64) z µ = λz µ π µ = λ 1 π µ ; µ = 1, ; λ = iπ µ = z µ z µ ; µ = 1, K N c Λ K N c = N c / (N c 1) ( = here) ; Λ = 1/α 1.06 GeV It is worthwhile to perform the calculation of the universal scale factors ( λ, λ 1 ) step by step, using the GeV fm conversion factor (65) c = (44) GeV fm p. 47

48 -11- We add at this point the Regge slope determination from the mesonic trajectory, analogous to the Λ baryonic one in eq. 7, J P : 1 5 M j : ± ± ± 0.05 (66) M j : ± : M : ± : For the inverse error-square weighted average of the root mean square of the two 1 M determinations in eq. 66 we obtain (67) 1 M = ( ± ) GeV 1/α (1.16 ± 0.007) GeV The error in eq, 67 represents the statistical error only, it does not account for the systematic error caused by the widths of the resonances involved. The value of 1/α can be compared wit the one obtained for the Λ trajectory in eq. 8. p. 48

49 -11-4 If we use the value for 1 called π(4) at a mass (68) / α also for the pion trajectory, we expect what in todays nomenclature is m π (4) m π =.1 GeV yet no entry exists in the present PDG tables in ref. [16-01], while a J PC = 4 +, I = 1 resonane was listed near this mass in earlier PDG tables. Here I must admit that I have never checked the quality of this π(4) resonance in the mass range derived in eq. 68. After this digression we return to eqs. 64 and 65, in order to complete the scale relation of configuration space variables to the dimensionless barycentric variables, defined in eq. 18 repeated below (69) z 1 = 1 (x 1 x ), z = 1 6 (x 1 + x x ) z = 1 (x 1 + x + x ) = X c.m. 0 Using rational units for which = c = 1 and choosing as energy and momentum units (70) [E ] = [p] = 1 GeV the unit of length follows from eq. 65 (71) [L] = 1 GeV 1 = (44) fm 1 5 fm p. 49

50 -11-5 From eq. 64 we obtain z µ = λz µ π µ = λ 1 π µ ; µ = 1, λ = K N c α = GeV 1 = fm (7) λ 1 = = GeV for : ( ) α 1 = 1.06 GeV -1-b - The barycentric 6 spatial oscillatory variables and their symmetries with respect to the quark positions We extend the barycentric dimensionloss coordinates to include a general c.m. position (7) x 1 k = λx 1 k x k = λx k x k = λx k X k = λx k = 1 (x 1 k + x k + x k ) k = 1,, : configuration space coordinate labels p. 50

51 -11-6 In the following we will suppress the configuration space coordinate labels k = 1,, displayed in eq. 7 and restrict configuration space three vectors to their dimensionless representatives x 1,, and functions thereof. The first thing to do is to transform to universal dimensionless configuration space variables the scalar product defined in eqs. 17, 18 repeated below Ψ () Ψ (1) () / Π i=1 d x i δ (x 1 + x + x ) = Ψ () (x 1,,x, x )Ψ (1) (x 1,,x, x ) (74) z 1 = 1 (x 1 x ), z = 1 6 (x 1 + x x ) z = 1 (x 1 + x + x ) = X c.m. 0 z j = λz j, x j = λx j ; j = 1,, The normalizing factor () / in the integral in eq. 17 and 74, which were missing in earlier versions, are now included in order to generate a conventionally normalized 6 dimensional L space, which we do next step by step. p. 51

52 -11-7 The scalar product in eq. 74 becomes ( ) Ψ () Ψ (1) λ 6 Π i=1 d ξ i δ ξ = Ψ () (x 1,,x, x )Ψ (1) (x 1, x, x ) (75) ξ 1 = 1 (x 1 x ), ξ = 1 6 (x 1 + x x ) ξ = 1 (x 1 + x + x ) = X c.m. 0 z j = λz j, x j = λx j ; j = 1,, ; X c.m. = λx c.m. = 1 i=1 x i In order to eliminate the scale factor λ we redefine the wave functions Ψ ( 1 ), () in eq. 75 (76) ψ ( 1 ), ( ) = λ Ψ ( 1 ), () Using the dimensionless wave functions (77) ψ ( 1 ), ( ) (x 1, x, x ) defined in eq. 76 the scalar product ( eq. 75 ) becomes p. 5

53 -11-8 (78) ψ () ψ (1) = Π i=1 d ξ i δ ( ξ ) ψ () (x 1,,x, x )ψ (1) (x 1, x, x ) -1-c - The barycentric 6 spatial oscillatory variables and their symmetries with respect to the quark positions in dimensionless universal variables The central properies under the quark position permutation group S can perfectly be discussed according to the dimensionless variables x i ; i = 1,, π 1 (79) x 1 x x i 1 i i x i 1 x i x i To this end we invert the linear relations in eq. 75 x 1 = 1 ξ ξ (80) x = 1 ξ ξ x = 6 ξ = 1 (x x 1 x ) ( ) The relation on the last line of eq. 80 takes into account the vanishing of i=1 x i. p. 5

54 -11-9 For completeness we give both barycentric variable transformations alongside ( eqs. 75 and 80 ) ξ 1 = 1 (x 1 x ), ξ = 1 6 (x 1 + x x ) ξ = 1 (x 1 + x + x ) = X c.m. 0 (81) z j = λz j, x j = λx j ; j = 1,, ; X c.m. = λx c.m. = 1 i=1 x i x 1 = 1 ξ ξ 6 x = 1 ξ ξ x = 6 ξ p. 54

55 d - The barycentric coordinates dimension by dimension : cartesian and skew hexagonal coordinates as appropriate for quark position variables I shall sketch the hexagonal versus cartesion coordinate association simplifying first to one space dimesnion and two oscillator variables (ξ, ξ 1 ) in their two dimensional representation in figure 4 below p. 55

56 Fig 4 : The hexagonal logic in the (ξ, ξ 1 ) plane p. 56

57 -pmodes-1 Modes of a pair of onedimensional oscillators pairmodes and the complex plane The even dimension already for just 1 space dimension :, of the barycentric relative coordinates with vanishing values for the c.m. coordinate(s) derive from the base positions of the quarks bound in baryons, as discussed here in subsections -4, and all subsections of section. The pairing mode allows to reveal explicity the hidden SU symmetry (8) ζ = 1 (x + iy ) ; x = ξ, y = ξ 1 ) a = ( 1 ζ + ζ ; b = ( 1 ζ + ζ ) [a, b] = 0 In eq. 8 a and b are two independent ( commuting ), bosonic, absorption oscillator operators acting from the left on a wave function ψ as defined in eq. 8 (8) ( ) ζ, ζ x = 1 (ζ + ζ ). They can also be expressed in the real variables x and y ; y = i 1 (ζ ζ ζ = 1 ( x i y ) ; ζ = 1 ( x + i y ) ) p. 57

58 -pmodes- Thus we arrive at the (x, y ) representation of the paired oscillators (a, b) (a 1, a ). We do this assembling the parts of a 1, as defined in eq. 8 in the table-equations below 1 ζ = 1 (x + iy ) 1 ζ = 1 (x iy ) 1 ζ = 1 ( x + i y ) 1 ζ = 1 ( x i y ) ( ) 1 ζ + ζ = 1 x + x + +i(y + y ) ( ) 1 ζ + ζ = 1 x + x i(y + y ) (84) Further it follows for the adjoint operators from eq. 84 (85) ( ) a = 1 ζ + ζ ( ) b = 1 ζ + ζ ; a = 1 ; b = 1 x x i (y y ) x x + +i (y y ) ( ) = 1 ζ ζ ( ) = 1 ζ ζ p. 58

59 -pmodes- The polynomial basis of normalized wave function associated with the two paired absorption oscillators (a 1, a ) follows from the associated construction of the creation oscillators a 1, in eq. 85 (86) ) ψ n 1, n (ζ ( ) n, ζ = N 1 (n 1 + n ) 1 ( ) n ζ ζ ζ ζ exp ( ζ ζ ζ ζ = 1 ( x + y ) In eq. 86 N denotes the normalization constant of the ground state with N = 0 ( ) N = (n 1!) (n!) dζ dζ exp ζ ζ 1 ) (87) 1 dζ dζ == 1 4 dζ dζ 1 (dx + idy ) (dx idy ) = 1 i = dζ dζ = dxdy N = (n 1!) (n!) dxdy exp ( x y ) = (dx dy ) = (n 1!) (n!) π 0 d e = (n 1!) (n!) π p. 59

60 -pmodes-4 Finally we come to the paired oscillator mode orthogonal polynomials, beeing not Hermite polynomials, which prevail for unpaired modes, but simple monomials. This structure is derived from substituting the two expressions for the creation operators a 1, : ( ζ ζ ) n 1 and (ζ ζ ) n in eq. 86, as combined operators inside the powers, acting on the left on the given paired mode ground state (88) ( ) a ( ζ 1 = ζ ) ( ) ζ = exp ζ ζ exp ζ ζ ) ) ( ) a = ζ ζ (ζ = exp ζ ( ζ exp ζ ζ The expression for the paired wave function ψ n 1, n (ζ, ζ ) in eq. 86 then takes the form (89) ) ψ n 1, n (ζ, ζ = ( ) ( ζ = N 1 (n 1 + n ) ) ) n n exp ζ ζ 1 ( ( ζ exp ζ ζ = N 1 (n 1 + n ) ζ n ( ) 1 ζ n exp ζ ζ ) We use polar coordinates, as they are representing finite rotations of the complex ζ - plane p. 60

61 -pmodes-5 leading to the wave function representation (90) ψ n 1, n (ζ, ζ = ) = (n 1 + n ) π(n 1!)(n!) 1 exp (i (n n 1 ) ϕ) [ (n 1 + n ) exp ( )] = ζ ; ϕ = arg (ζ ) The functions ψ n 1, n in eq. 90 form a complete basis in the space L (ζ 1, ζ ). They are combined with the restrictions from overall Fermi statistics including an overall color antisymmetric selection rule in conjunction with the three Young tableaux as displayed in eq. 6. Thus we study the action of the symmetric group S on these base functions, as defined in eq. 79. (91) U π 1 = ψ n 1, n i 1 i i π ψ n 1, n ( ) ζ, ζ 1 i 1 i i = 1 ( ) ζ, ζ p. 61

62 -1 Eq. 91 needs to be elaborated, as follows (9) π 1 i 1 i i 1 = π i 1 i i 1 Next we identify the subgroup of even permutations A = Z of S Z = π 1 or π (9) Z = π 1 or π 1 ; Z = 1 1 Z 1 = Z ; Z = Z Here is the place to emphasize that the discussion within subsection -1-d is for the time beeing restricted to oscillatory modes in one space dimension, to be generalized to three subsequently, but after finishing the selection rules staying with 1 space dimension for the time beeing. We proceed to identify the permutation Z as defined in eq. 9 with a rotation of the ζ plane by 10 degrees, completing the action of S displayed in eq. 91 p. 6

63 -1 (94) U π 1 i 1 i i ψ n 1, n ( ζ, ζ = D m 1 m n 1 n (π (.)) ψ m 1, m (ζ, ζ ) ) = and for the abelian cyclic subgroup A of even permutations eq. 94 becomes (95) U π 1 1 ψ n 1, n = D n 1 n (Z ) ψ n 1, n (ζ, ζ ( ζ, ζ ) ) = = ψ n 1, n (Z 1 ζ, Z ζ ) Eqs. 90 and 95 imply (96) It follows from eq. 96 D n 1 n (Z ) ψ n 1, n (ζ, ζ ) = = exp (i (n 1 n ) (π /)) ψ n 1, n (ζ, ζ ) = ψ n 1, n (Z 1 ζ, Z ζ ) = (97) D n 1 n (Z ) = Z n 1 n, Z = exp (i (π /)) p. 6

64 -14 Next we decompose the action of Z on ζ into real and imaginary parts (98) ζ Z ζ = ζ : ζ = ξ ξ 1, ζ = 1 1 ξ ξ 1 ξ = 1 ξ ξ 1 ξ 1 = ξ 1 ξ 1 We recall eq. 81, repeating it below (99) ξ 1 = 1 (x 1 x ), ξ = 1 6 (x 1 + x x ) ξ = 1 (x 1 + x + x ) = X c.m. 0 Substituting the expressions on the first line in eq. 99 in eq. 98 we obtain (100) ξ = 1 6 (x 1 + x x ) (x 1 x ) ξ 1 = 6 (x 1 + x x ) 1 (x 1 x ) p. 64

65 -15 and arranging the factors yielding the result (101) ξ = 1 6 [(x 1 + x x ) + (x 1 x )] ξ 1 = 1 [(x 1 + x x ) (x 1 x )] The final result is compared with the initial choice of barycentric variables in eq. 99 (10) ξ = 1 6 (x + x x 1 ) ξ = 1 6 (x 1 + x x ) ξ 1 = 1 (x x ) ξ 1 = 1 (x 1 x ) The choice, marked by or in eq. 9, is revealed inspecting the substitution of the ξ j indices from the right hand - to the left hand side of eq. 10, corresponding to the cyclic permutation associated with the actions of Z and Z 1 Z Z : ζ Z ζ π 1 1 (10) Z : ζ Z ζ π p. 65

66 -16 -rec - Reconstruction of the two-dimensional irreducible unitary representation of S from 1 spacelike dimension The actions of Z and Z 1 Z, defined in eq. 10 allows to assoiate two representation matrices with the corresponding permutations, using eq. 98 ) π ) π 1 1 (104) 1 1 4) π Z + Z Z 1 T 1 In eq. 104 the last column denotes a shorthand name for the constructed elements of S, while the signs in the second last column are + for even and for odd permutations respectively, equal to the determinant of the representation matrices. The assignment of the barycentric variables to the numbering of the associated 1-dimensional coordinates x 1,, as given in eq. 10 singles out the representation of the transposition T 1 1 as beeing associated with the diagonal Pauli matrix σ. p. 66

67 -17 The remaining elements including the identity permutation, denoted, also for its representing unit matrix, can be found from multiplications of the elements defined in eq. 104 π 1 = Z = ( Z 1 ) = (T1 ) 1 (105) 1) π The numbering ) - 4) in eq. 104 and 1) in eq. 105 serves to segregate the subgroup of even permutations A Z corresponding to the entries numbered 1), ), ) from the odd ones, to be completed next. This we do step by step. First we ddetermine the product, using the symbol : ) 4) ) 4) = π 1 π 1 = π 1 T (106) p. 67

68 -18 Thus we multiply the matrices associated with the elements ) and 4) in eq. 104 ) ; 4) (107) ) 4) 1 1 and check compatibility of eqs. 106 and 107 implied by the -dimensional unitary representation of S. This is done in an analogous way to the substitutions relative to A applied in eqs.10 and 10 (108) ζ ξ + iξ 1 ξ ξ 1 = 1 with the identifications (eqs. 98, 101) ξ ξ 1 1 ; ) 4) : ζ ξ ξ 1 ξ ξ 1 p. 68

69 -19 ξ 1 = 1 (x 1 x ), ξ = 1 6 (x 1 + x x ) (109) ξ = 1 (x 1 + x + x ) = X c.m. 0 Then eq. 108 becomes and ξ j ξ j ; x j x j ; j = 1,, ξ = 1 ξ + ξ 1 ; ξ 1 = ξ + 1 ξ 1 (110) ξ = 1 6 (x 1 + x x ) + (x 1 x ) ξ 1 = + 6 (x 1 + x x ) + 1 (x 1 x ) Rendering the factors commensurable in the last lines of eq. 110 we obtain ξ = 1 6 (x (x 1 + x ) + (x 1 x )) (111) = ξ 1 = 1 6 (x + x 1 x ) 1 ((x 1 + x x ) + (x 1 x )) = 1 (x 1 x ) p. 69

70 -0 It remains to substitute the variables ξ permutation (11) ξ = ξ 1 = 1, ( 1 x x x ) 1 (x 1 x on the left hand side of eq. 111 to identify the associated ) = = 1 6 (x 1 + x x ) 1 (x 1 x ) which follows from the ordering of the indices of the barycentric variables x j ; j = 1,, appering on the rightmost side of eq. 11 as (11) ) 4) π 1 1 T ( ) Hence we can consistently identify the second odd permutaion, which becomes the fifth constructed permutation according to the definition ) 4) = 5) in accordance with eqs. 107 and 11 5) π 1 1 (114) T 1 1 p. 70

71 -1 It remains to construct the third odd permutation and its asociated unitary representation matrix. We choose among several paths to consider the multiplication ) 4) closely follow the previous multiplication ) 4) in eq. 106 ) 4) = π 1 1 (115) π 1 = π 1 1 T 1 1 The resulting remaining odd permutation T 1 as result of the ) 4) multiplication is not surprising. Thus we multiply the matrices associated with the elements ) and 4) analogous to eq. 107 for ) 4) (116) ) 1 ) 4) 1 1 ; 4) p. 71

72 - The analog for ) 4) to eq. 108 for ) 4) becomes (117) ζ ξ + iξ 1 ξ ξ 1 = 1 ξ ξ 1 ; ) 4) : ζ 1 ξ ξ 1 ξ ξ 1 Next we adapt eqs relative to ) 4) to ) 4), which yields ξ 1 = 1 (x 1 x ), ξ = 1 6 (x 1 + x x ) (118) as well as (119) ξ = 1 (x 1 + x + x ) = X c.m. 0 and ξ j ξ j ; x j x j ; j = 1,, ξ = 1 ξ ξ 1 ; ξ 1 = ξ = 1 6 (x 1 + x x ) ξ 1 = ξ + 1 ξ 1 (x 1 x ) 6 (x 1 + x x ) + 1 (x 1 x ) p. 7

73 - The third equation (eq. 111) relative to ) 4) is replaced for ) 4) by ξ = 1 6 (x (x 1 + x ) (x 1 x )) (10) = ξ 1 = = 1 6 (x + x x 1 ) 1 (( x 1 x + x ) + (x 1 x )) 1 (x x ) Substituting the variables ξ 1, with ) 4), in analogy to eq. 11 for ) 4) ξ ( 1 = x x x (11) ) ξ 1 = on the left hand side of eq. 10 we identify the permutation associated 1 (x 1 x ) = = 1 6 (x + x x 1 ) 1 (x x ) which follows from the ordering of the indices of the barycentric variables x j ; j = 1,, appering on the rightmost side of eq. 11 in continuing the analogy tp eqs for ) 4) = 5) p. 7

74 -4 (1) ) 4) π 1 1 T 1 ( ) Hence we can consistently identify the third odd permutaion, which becomes the six th and last constructed permutation according to the definition ) 4) = 6) in accordance with eqs. 116 and 1 analogous with eqs. 107 and 11 for the product ) 4) = 5) 6) π 1 (1) T 1 -res - Results on the reconstruction of the two-dimensional irreducible unitary representation of S from 1 spacelike dimension In this subsection we collect the results first in the representation of the permutation group elements as in eqs. 104 (), ), 4)), 105 (1)), 114 (5)), 1 (6)), in the order 1) - 6). p. 74

75 -5 (14) 1) π ) π ) π 4) π 5) π 6) π Z + Z Z T T 1 T 1 p. 75

76 -6 The main ingredients to the construction of irreducible representations of finite groups, here S, leading in summary to eq. 14, are to my knowledge due to Issai Schur, as displayed also in ref. [-006]. -res- - Choosing a complex basis for transforming the basis derived in eq. 14 for the two-dimensional irreducible unitary representation of S from 1 spacelike dimension Here we invert the decomposition of the complex numbers ζ, ζ into real and imaginary parts, as displayed ( e.g. ) in eqs. 98 and 108 (15) ζ Z ζ = ζ : ζ = ξ ξ 1, ζ = 1 1 ξ ξ 1 ξ = 1 ξ ξ 1 ξ 1 = ξ 1 ξ 1 repeated above and on next page below p. 76

77 -7 (16) ζ ξ + iξ 1 ξ ξ 1 = 1 ξ ξ 1 1 ; ) 4) : ζ ξ ξ 1 ξ ξ 1 back to the original complex- and complex conjugate variables ζ, ζ, as defined on the left hand side of the first relation in eq 16 (17) ξ ζ = ξ + iξ 1 = 1 i ξ ξ 1 ζ = ξ iξ 1 1 i ξ 1 M = 1 1 MM = M M = ( ) i i Thus the unitary x matrix (18) u = 1 M = M ξ ξ 1 is a unitary x matrix which generates the similarity transformation p. 77

78 -8 through the following steps, denoting by D π ; π S the six x unitary matrices in the basis given in eq. 14 and likewise by d π ; π S the six transformed x unitary matrices, associated with the basis as described in eq. 17 (19) π d π : ζ d π ζ ζ ζ ζ = M ζ ξ ξ 1 = d π M ξ ξ 1 π D π : ξ D π ξ ξ 1 ξ 1 Next we multiply the last relation in eq. 19 by M from the left (10) π D π : ζ ζ MD π ξ ξ 1 = MD π M 1 ζ ζ p. 78

79 -9 Comparing eq. 10 with the first relation in eq. 19 we obtain the sought similarity transformation d π = MD π M 1 = ud π u 1 (11) u = 1 1 i 1 i, u 1 = i i The detailed calculations of the matrix product associated with the sixth permutation representation matrices d π from the basis formed by D π given in eq. 14 are performed in Appendix. The collection of x representation matrices d π ; π = 1,,6 is displayed in eq. 1 below p. 79

80 -0 1)d π=1 π )d π= π )d π= π 4)d π=4 π 5)d π=5 π 6)d π=6 π e +i(π/) 0 + Z 0 e i(π/) e i(π/) 0 + Z Z 1 0 e +i(π/) 0 1 T e +i(π/) T e i(π/) 0 0 e i(π/) T 1 e +i(π/) 0 (1) p. 80

81 -1- We go back to the subsection on page -pmodes-1 : -res- - Extending 1 spatial dimension to Modes of a pair of onedimensional oscillators pairmodes and the complex plane which for spatial dimension becomes ց Modes of pairs of onedimensional oscillator-pairmodes and the complex -plane The 1 complex variable ζ associated with the 1 pairmode as appropriate for 1 space dimension and defined in eq. 8, for space-time dimensions with axes (X), (Y),(Z), thus becomes a complex three vector (1) ζ = ( ζ (X), ζ (Y ), ζ (Z) ) The notation (X), (Y),(Z) for the three orthogonal axes of the -dimensional configuration space in the c.m. system is chosen in order to prevent confusing these with the 1-dimensional quantities denoted x, y, z, as introduced for 1 spatial dimension and defined in eqs. 8 and 81, which become -vectors for space dimensions. The extension of the various space variables from 1 to dimensions we shall do in segmented steps : p. 81

82 -- 1-1) The center of mass position variables in 1 spatial dimension These variables appear (last) in eq. 75 repeated below ( ) Ψ () Ψ (1) λ 6 Π i=1 d ξ i δ ξ = Ψ () (x 1,,x, x )Ψ (1) (x 1, x, x ) ξ 1 = 1 (x 1 x ), ξ = 1 6 (x 1 + x x ) ξ = 1 (x 1 + x + x ) = X c.m. 0 z j = λz j, x j = λx j ; j = 1,, ; X c.m. = λx c.m. = 1 i=1 x i (14) 1-) Extension of the last three relations in eq. 14 to spatial dimensions The extension takes the form x j x j with x j = X c.m. X c.m. with X c.m. = (15) ( ( x (X) j, x (Y ) j, x (Z) j ) X (X) c.m., X (Y ) c.m., X (Z) c.m. Again note that X c.m. and the axis superfix (X) denote very different objects. ; j = 1,, ) = 0 p. 8

83 -- 1-) (continued) The configuration space -vectors x 1,,, X c.m. in eq. 15 have dimension [ mass 1 ] in rational units. They can be reduced to dimensionless configuration space variables, as given in eqs. 0-6 for 1 spatial dimesnsion and in the case of spatial dimensions follows straightforwardly from eqs. 0, 4 and 8 (16) x j = λ 1 x j ; j = 1,, ; λ 1 = Λ K N c 1/ K N c = N c / (N c 1) ( = here) ; Λ = 1/α 1.06 GeV -1) Dimensionless barycentric coordinates in 1 space dimension We recall the definition of the dimensionless barycentric coordinates asociated with the dimensionless quantities x 1,,, X c.m. = 0 for 1 spatial dimension in eqs. 81 and 15 in point 1-1) p. 8

84 -4- -1) (continued) ξ 1 = 1 (x 1 x ), ξ = 1 6 (x 1 + x x ) ξ = 1 (x 1 + x + x ) = X c.m. 0 (17) x j = λx j ; j = 1,, ; X c.m. = λx c.m. = 1 i=1 x i = 0 x 1 = 1 ξ ξ 6 x = 1 ξ ξ x = 6 ξ -) Extension of the dimensionless variables in eq. 17 to spatial coordinates The extension of the configuration variables x 1,, x 1,, from d = 1 to d= dimensions is defined in point 1-). Similarly the barycentric coordinates for d = 1 become three vectors for d =. p. 84