An oscillator-representation of elementary particles

Two dynamical systems with same symmetry should have features in common, and as far as their shared symmetry is concerned, one may represent the other. The three light quark constituent of the hadrons, (a) have an approximate flavor SU(3)f symmetry, (b) have an exact color SU(3)c symmetry, and (c) as spin 1 2 particles, have a Lorentz SO(3, 1) symmetry. So does a 3D harmonic oscillator. (a) Its Hamiltonian has the SU(3) symmetry, breakable if the 3 oscillators are not identical. (b) The 3 directions of oscillation have the permutation symmetry. This enables one to create three copies of unbreakable SU(3) symmetry for each oscillator, and mimic the color of the elementary particles. (c) The Lagrangian of the 3D oscillator has the SO(3,1) symmetry. This can be employed to accommodate the spin of the particles. In this paper we propose a one-to-one correspondence (a) between the eigen modes of the Poisson bracket operator of the 3D oscillator and the flavor multiplets of the particles, and (b) between the permuted modes of the oscillator and the color and anticolor multiplets of the particles. The bi-colored gluons are represented by the generators of the color SU(3)c symmetry of the oscillator. Harmonic oscillators are common place objects and, wherever encountered, are analytically solvable. Elementary particles, on the other hand, are abstract entities far from one’s reach. Understanding of one may help a better appreciation of the other.


Introduction
, is the initiator of the map from matrices to quantum harmonic oscillators to expedite computation with Lie algebra representations [1]. Schwinger, 1952, evidently unaware of Jordan's work, represents the SU(2) algebra of the angular momentum by two uncoupled quantum oscillators [2]. Since then an extensive literature is created on the subject. The technique often bears the name of 'Jordan-Schwinger map'. In the majority of the existing literature, the oscillator is a quantum one. In their stellar system studies, however, Sobouti et al [3] and [4] associate the symmetries of their system of interest with those of the classical oscillators. They use Poisson brackets instead of the quantum commutation brackets, and work with complex functions in the phase space of the oscillator. Man'ko et al do the same, and give a realization of the Lie product in terms of Poisson brackets [5].
In this paper we follow the classic oscillator approach and explore the two-way association of SU(n)AEnD oscillators. The case n=2, of course, gives the oscillator representation of the angular momentum, albeit in a different space and different notation than that of Schwinger. The cases n=3, 4, ..., should be of relevance to particle physics. The flavor and color triplets of the three light quarks, (u, d, s), and their higher multiplets have the SU(3) symmetry. They can be given a 3D oscillator representation. By inviting in the heavier quarks there might also be room for higher SU(n) and higher nD oscillators.
In his seminal paper Schwinger writes: '... harmonic oscillator ... provides a powerful method for constructing and developing the properties of angular momentum eigen vectors. ... many known theorems are derived in this way, and some new results obtained. ' Schwinger can only be right in saying so, for wherever encountered, harmonic oscillators are exactly and easily solvable. It is in this spirit that we hope a harmonic representation of elementary particles might offer a simpler and easier understanding of at least the rudiments of the particle physics, if not lead to a different insight. Classical harmonic oscillators are common place objects and can be set up on table tops. Elementary particles, on the other hand, are highly abstract notions and far from one's intuition.

nD harmonic oscillators in phase space
Let (q i , p i ; i=1, 2, ..., n) be the canonically conjugate pairs of coordinates and momenta of an n dimensional (nD) harmonic oscillator, or equivalently of n uncoupled oscillators. The Hamiltonian and the Lagrangian are The time evolution of an attribute of the oscillator, a function f p t q t t i n , , ; 1, , say, on the phase space trajectory of the oscillator is governed by Liouville's equation, The last equality is the definition of the Poisson bracket operator, . Hereafter, it will be referred to as Liouville's operator. The reason for multiplication by i is to render  hermitian and talk of its eigen solutions. As we will see shortly, the eigen solutions are in general complex and  operate on functions of complex revariables p i ±iq i . It should also be noted that  is the sum of n linear first order differential operators ...
The transformation is linear, and a=[a ij ] and b=[b ij ] are two n×n matrices. The proof for n=3 is given in [3] and [4]. Its generalization to higher dimensions is a matter of letting the subscripts i and j in equations (2.1) and (2.2) span the range 1 to n. There are 2n 2 ways to choose the a-and b-matrices, showing that the symmetry group of  and thereof that of equation (2.1) is GL(n, c), the group of general n×n complex matrices. This statement is based on the fact that, as we will see shortly,  is defined on the complex plains (p i +iq i ; i=1, 2, ..., n) At this stage let us introduce  as the function space of all complex valued and square integrable functions, f (p i , q i ), in which the inner product is defined as ) is the energy scalar of the Hamiltonian operator of the oscillator. Associated with the transformation of equation (2.2), are the following generators on the function space , (insertion of −i in front of b jk is for later convenience). Again there are 2n 2 generators. All χʼs commute with  but not necessarily among themselves. Two notable subgroups of GL(n, c) are generated by (1) a ij antisymmetric, b ij antisymmetric, (2) a ij antisymmetric, b ij symmetric, Case 1 is the symmetry group of the Lagrangian. It is of Lorentz type, and in the 3D case reduces to SO(3, 1), the symmetry group of Minkowsky's spacetime and of Dirac's equation for spin 1 2 particles. We will come back to it briefly in the conclusion of this paper.
Case 2, antisymmetric a ij and symmetric b ij , is the symmetry of the Hamiltonian, the SU(n) group. Before proceeding further, however, let us give the proof of the last two statements. Under the infinitesimal transformation of equation There follows , and , 0if , and .
Coming back, an SU(n) is spanned by n 2 −1 linearly independent basis matrices. A convenient and commonly used basis for SU(n) is the generalized Gell-Mann's λ matrices. See e.g. [8] for their construction and see table 1 for a refresher. The generalized λ matrices consist of n n 1 This choice produces a set of (n 2 −1) linear differential operators, that are the oscillator representations of the SU(n) algebra in . The first 8 of them are as follows: All χʼs are hermitian. Their commutation brackets are the same as those of λʼs, The followings can be easily verified Let us use the shorthand notation z p iq Considering the fact that equation (2.1) is a linear differential equation, and ... So much for generalities. In section 3 we examine the 2D oscillator representation of SU(2) and reproduce Schwinger's model, albeit in a different notation. Next, we treat the 3D case and suggest a scheme, that we think, represents the flavor and color symmetries of the light quarks and higher particle multiplets.

2D oscillator and SU(2) -angular momentum
There are two complex planes to deal with, z 1 =p 1 +iq 1 and z 2 =p 2 +iq 2 . The first three operators of equations (2.5) are the ones to work with, and have the SU(2) algebra, The common eigen-states of J J , One may, of course, begin with any j m , ñ | and reach the other j 2 1 + ( ) members of the j-multiplet by operating on j m , ñ | with the raising and lowering ladders J ± . There is an unclassical feature to the eigenstates presented here; j can be integer or half integer, in sharp contrast to the classical angular momentum eigenvalues which have to be integers (see e.g. [9]). Quantum mechanics provides half integer eigenvalues by enlarging the angular momentum states via multiplication of the orbital angular states by an abstract spin one-half state. In our case, provision for half integer j comes from embedding of the angular momenta states in two complex spaces (p 1 , q 1 ) and (p 2 , q 2 ).

3D oscillator and Quark Flavor
There are three complex planes to deal with, ,a n d .
The collection of all eigen-states belonging to a given n=n 1 +n 2 +n 3 and m=m 1 +m 2 +m 3 will be called a multiplet and will be denoted by D(n, m). Below we examine some of these multiplets, compare their eigen characteristics with those of the known particle multiplets, and point out the one-to-one correspondence between the two. Table 3 displays the two triplets D(1, 0) and D(0, 1). The first of which consists of (z 1 , z 2 , z 3 ) and the second of z z z , , .

+
, an index to identify submultiplets within a multiplet. Table 4 and its geometrical representation, figure 1 display D(1, 1). It has nine members, (z z i j ; , 1, 2, 3 i j * = , or their linear combinations). D(1, 1) is identified as the pseudoscalar meson nonet. There are two submultiplets to it, characterized by the two Casimir numbers 4 and 0. Again the 9 members of the multiplet constitute a complete orthogonal set in the subspace of the pseudoscalar mesons. Table 5 and its geometrical representation, figure 2, is D(3, 0). It is identified with the Baryon decuplet of spin 3/2. It has ten members, z z z i j k ; , , 0, 1, 2, 3 i j k 1 2 3 = , constrained to i j k 3. + + = There are two submultiplet to the baryon decuplet, characterized by the two Casimir numbers 4 and 12. The ten members constitute a complete orthogonal set. The antibaryon decuplet is D(0, 3). It can be read from table 5 by simply interchanging the subscripts and superscripts in the first column of the table and changing the signs of the eigenvalues accordingly. This also means interchanging z z j j *  in table 5.

Color and color multiplets
By mid 1960 particle physicists had felt the need for an extra quantum number for quarks in order to comply with Pauli's exclusion principle and to justify coexistence of the like spin 1 2 quark flavors in baryons. The notion of color and color charge was introduced, [10][11][12], and [13]. The consensus of opinion nowadays is that each quark flavor comes in three colors, red, green and blue; and antiquarks in three anticolors, antired, antigreen and antiblue. Strong interactions are mediated by 8 bi-colored gluons, each carrying one color charge and one anticolor charge. Color is believed to be conserved in the course of strong interactions. Does the 3D harmonic oscillator has an attribute analogous to the color of the quarks, is that attribute conserved, and if so, what is the symmetry responsible for its conservation? What are the counterparts of gluons in the 3D oscillator? In section 2.1 we talked about the continuous symmetries of the Hamiltonian, the Lagrangian and of the relevant Liouville equation, and came up with a one-to-one correspondence between the fundamental eigen modes of the oscillator and the quantum numbers of the quark flavors. There are discrete symmetries to consider: A. Liouville's operator is antisymmetric under complex conjugation, a discrete transformation. This leads to The fact that z i and z i * are the eigen states of  with eigenvalues ±1 is due to the antisymmetry of  under the transformation of equation (5.1). In particle physics language, this is akin to the statement that if a particle is a reality, so is its antiparticle.
B. The total Hamiltonian and the total are symmetric under the permutation of the three dimension subscripts (1, 2, 3). To elucidate the point let us, for the moment, instead of talking of 3D oscillators, talk of three uncoupled oscillators (1, 2, 3) and the 3 coordinate directions in the (q, p) spaces. Let us the three directions as red(r), green(g), and blue(b). (The language we adopt is in the anticipation of finding a correspondence between the permutation symmetry of the 3D oscillator and the color symmetry of the quark triplets). One has the option to place any of the oscillators (1, 2, 3) in any of the colored directions, r, g, or b. The choices are: There are three copies of the oscillator 1: r(1), g(1), and b(1), and similarly of the oscillators 2 and 3. Each oscillator can be in a triplet color state. Defined as such, the color symmetry is exact and unbreakable. This is in contrast to the flavor symmetry in which one assumes the 3 oscillators are identical, while it may be broken by allowing the masses and spring constants of the oscillators to be different. As noted earlier the transformation of equation (5.1), amounts to going from the complex (p, q) plane to its complex conjugate, (p, q) * , plane. One may now commute the coordinates in this complex conjugated space and design an anticolor scheme, antired r ( ), antigreen g ( ), and antiblue b ( ), say. Thus,

Gluons
With the definition of the preceding section the color is now a direction in the (p, q) phase space. Each oscillator, while sharing the approximate flavor SU(3) f symmetry with the others, has its own exact fundamental triplet color SU(3) c symmetry. The adjoint color representation of SU(3) c is the same as the χ octet of equations (2.5) in  In equations (6.4) and (6.5), χ 3 and χ 8 are colorless and members of a color octet. In equations (6.6),  is colorless and a singlet.