Introducing SU(3) color charge in undergraduate quantum mechanics
IIntroducing SU(3) color charge in undergraduate quantum mechanics
Brandon L. Inscoe ∗ and Jarrett L. Lancaster † Department of Physics, High Point University, One University Parkway, High Point, NC 27262 USA (Dated: July 30, 2019)We present a framework for investigating effective dynamics of SU(3) color charge. Two- and three-bodyeffective interaction terms inspired by the Heisenberg spin model are considered. In particular, a toy model for athree-source “baryon” is constructed and investigated analytically and numerically for various choices of inter-actions. VPython is used to visualize the nontrivial color charge dynamics. The treatment should be accessibleto undergraduate students who have taken a first course in quantum mechanics, and suggestions for independentstudent projects are proposed.
I. INTRODUCTION
It is quite notable that one can earn a college degreein physics without significant exposure to two of the fourknown fundamental interactions. Gravity and electromag-netism are covered in any introductory sequence, and at leastone semester is typically devoted to an in-depth study of elec-tromagnetism. While an investigation of gravity as a field the-ory is typically reserved for a course on general relativity, it isstriking that the strong and weak nuclear interactions are rarelymentioned in any meaningful way until one gets to a graduate-level course in quantum field theory.However, it is possible to introduce the relevant structureof nontrivial gauge theories by leaning heavily on the frame-work of spin, which typically constitutes a significant portionof the standard undergraduate experience in quantum mechan-ics. Operationally, the quantum mechanics of the spin de-gree of freedom is built around the group SU (2) . In the cur-rently accepted theory of the strong nuclear interaction, quan-tum chromodynamics , the relevant gauge group is SU (3) . Theunderlying quantum field theory bares a formal similarity tothat of electromagnetism (i.e., Maxwell’s equations) but witha much richer structure due to the nonabelian gauge group.Rather than attempt to introduce quantum chromodynamics asa fully formed field theory, the present aim is to explore thestructure imposed by SU (3) in the context of quantum mechan-ics. To this end, we present a quantum mechanical system ofthree sources of SU (3) “color” charge and explore the dynam-ics generated by simple, effective interactions. In addition topedagogical value, this model may be relevant for simulationsinvolving cold atoms in optical lattices.This paper is organized as follows. Some basic aspects ofquantum mechanical spin are summarized in Sec. II. Whilemuch of this material is fairly standard, our treatment of SU (3) dynamics follows this setup presented very closely. Section IIIintroduces the notion of “color charge” as a generalization ofelectric charge which shares similarities with the structure ofspin. By analogy with the interaction of a spin with mag-netic fields, effective interactions for color charges with ex-ternal fields and other color charges are proposed. In Sec. Vwe explore the color charge dynamics in systems of interact-ing color sources with particular attention paid to a three-bodysystem which serves as a toy model for a baryon. Lastly, con-clusions and suggestions for student projects are contained inSec. VI II. SPIN
Before defining of color charge, it is useful to review somebasic aspects of the theory of quantum spin. Here we presentsome fairly standard material on the quantum mechanics ofspin from which our treatment of SU (3) sources and interac-tions will follow completely by analogy. Specifically, we re-quire the basic properties of several interacting, spin- degreesof freedom. A. One spin
A single spin- degree of freedom | 𝜒 ⟩ belongs to the fun-damental representation of SU (2) and can be represented by atwo-component column vector | 𝜒 ⟩ = 𝑎 | ↑ ⟩ + 𝑏 | ↓ ⟩ ̇ = ( 𝑎𝑏 ) (1)for some complex numbers 𝑎 and 𝑏 satisfying | 𝑎 | + | 𝑏 | = 1 .Here the symbol ̇ = stands for “represented by” in the sensethat the abstract two-dimensional quantum state may be repre-sented by a familiar two-component column vector. Operatorsacting on these states will be represented by matrices, andthe action of operators on states translates to ordinary matrixmultiplication in this representation.Given a state | 𝜒 ⟩ , the expectation value of observables suchas the three components of spin can be computed as the innerproduct ⟨ ̂𝑆 𝑥 ⟩ = ⟨ 𝜒 | ̂𝑆 𝑥 | 𝜒 ⟩ . The spin operators are ̂𝑆 𝜈 ̇ = ℏ 𝜎 𝜈 ,where 𝜈 = 𝑥, 𝑦, 𝑧 and the Pauli matrices 𝜎 𝜈 given by 𝜎 𝑥 = ( ) , 𝜎 𝑦 = ( 𝑖𝑖 ) , 𝜎 𝑧 = ( ) . (2)To generate nontrivial dynamics, the spin must interact withother spins or external magnetic fields. Postponing discussionof multiple sources to the next subsection, the interaction of asingle spin with an external magnetic field is encoded in theHamiltonian ̂𝐻 ̂𝐻 = − 𝜆 𝐁 ⋅ ̂ 𝐒 , (3)where 𝜆 is a constant related to the magnetic moment of thesource and 𝐁 = 𝐵 𝑥 ̂𝐱 + 𝐵 𝑦 ̂𝐲 + 𝐵 𝑧 ̂𝐳 is an applied magnetic field. a r X i v : . [ phy s i c s . e d - ph ] J u l We employ the shorthand ̂ 𝐒 = ̂𝑆 𝑥 ̂𝐱 + ̂𝑆 𝑦 ̂𝐲 + ̂𝑆 𝑧 ̂𝐳 where each co-efficient ̂𝑆 𝜈 is actually a matrix given by ℏ times Eq. (2).Thus, the Hamiltonian Eq. (3) takes the form , ̂𝐻 ̂ = − 𝜆ℏ ( 𝐵 𝑧 ( 𝐵 𝑥 − 𝑖𝐵 𝑦 )( 𝐵 𝑥 + 𝑖𝐵 𝑦 ) − 𝐵 𝑧 ) . (4)As a simple example, let us take 𝐁 to be a constant. Thefundamental problem in quantum mechanics is to obtain thetime-dependent expectation value of observables 𝑂 ( 𝑡 ) = ⟨ 𝜒 ( 𝑡 ) | ̂𝑂 | 𝜒 ( 𝑡 ) ⟩ for some initial state | 𝜒 (0) ⟩ ≡ || 𝜒 ⟩ . For a gen-eral state given by Eq. (1), the spin expectation values can becomputed by using the Pauli matrices, ⟨ 𝜒 ( 𝑡 ) | ̂𝑆 𝑥 | 𝜒 ( 𝑡 ) ⟩ = 12 ( 𝑎 ∗ ( 𝑡 ) 𝑏 ( 𝑡 ) + 𝑏 ∗ ( 𝑡 ) 𝑎 ( 𝑡 ) ) , (5) ⟨ 𝜒 ( 𝑡 ) | ̂𝑆 𝑦 | 𝜒 ( 𝑡 ) ⟩ = 12 𝑖 ( 𝑎 ∗ ( 𝑡 ) 𝑏 ( 𝑡 ) − 𝑏 ∗ ( 𝑡 ) 𝑎 ( 𝑡 ) ) , (6) ⟨ 𝜒 ( 𝑡 ) | ̂𝑆 𝑧 | 𝜒 ( 𝑡 ) ⟩ = 12 (| 𝑎 ( 𝑡 ) | − | 𝑏 ( 𝑡 ) | ) . (7)Explicit forms for the complex amplitudes 𝑎 ( 𝑡 ) and 𝑏 ( 𝑡 ) areobtained at arbitrary times by solving the time-dependentSchrödinger equation 𝑖ℏ 𝜕 | 𝜒 ( 𝑡 ) ⟩ 𝜕𝑡 = ̂𝐻 | 𝜒 ( 𝑡 ) ⟩ . (8)In the present situation, Eq. (8) reduces to a system of twocoupled (linear) differential equations for 𝑎 ( 𝑡 ) and 𝑏 ( 𝑡 ) , 𝑖 ̇𝑎 ( 𝑡 ) = − 𝜆 [ 𝐵 𝑧 𝑎 ( 𝑡 ) + ( 𝐵 𝑥 − 𝑖𝐵 𝑦 ) 𝑏 ( 𝑡 ) ] ,𝑖 ̇𝑏 ( 𝑡 ) = − 𝜆 [ ( 𝐵 𝑥 + 𝑖𝐵 𝑦 ) 𝑎 ( 𝑡 ) − 𝐵 𝑧 𝑏 ( 𝑡 ) ] . (9)Let us take 𝐵 𝑦 = 𝐵 , 𝐵 𝑥 = 𝐵 𝑧 = 0 and take an initial stategiven by Eq. (1) with 𝑎 (0) = 1 and 𝑏 (0) = 0 . Then Eq. (9)gives | 𝜒 ( 𝑡 ) ⟩ ̇ = ( cos( 𝜔𝑡 )− sin( 𝜔𝑡 ) ) , (10)with 𝜔 = 𝜆𝐵 . One may then show ⟨ ̂𝑆 𝑥 ( 𝑡 ) ⟩ = − ℏ 𝜔𝑡 ] , ⟨ ̂𝑆 𝑦 ( 𝑡 ) ⟩ = 0 , ⟨ ̂𝑆 𝑧 ( 𝑡 ) ⟩ = ℏ 𝜔𝑡 ] , (11)so that the spin precesses about the external magnetic field.While this is not a particularly complicated example, the stepsinvolved in obtaining Eqs. (11) are virtually identical to thosewe will take with SU (3) sources in later sections. B. Multiple spins
A single spin can only interact with external magnetic fieldswhich couple to its magnetic moment. When brought into the vicinity of another spin, the two spins may also interact witheach other. Before addressing interactions, we summarize theformalism for representing a system of two spins. Given twospins in quantum states || 𝜒 ⟩ and || 𝜒 ⟩ , the total quantum stateis constructed by forming a tensor product of the individualspin states, | 𝜒 ⟩ = || 𝜒 ⟩ ⊗ || 𝜒 ⟩ . (12)It is convenient to represent the tensor product as a Kroneckerproduct so that || 𝜒 ⟩ ⊗ || 𝜒 ⟩ ̇ = ( 𝑎 𝑏 ) ⊗ ( 𝑎 𝑏 ) ≡ ⎛⎜⎜⎜⎝ 𝑎 𝑎 𝑎 𝑏 𝑏 𝑎 𝑏 𝑏 ⎞⎟⎟⎟⎠ . (13)Operators acting on the full state are built from single-spin op-erators and must have the same overall dimension as the totalHilbert space. For example, the expectation value of the 𝑥 -component of the first particle’s spin can be obtained from theinner product 𝑆 𝑥 = ⟨ 𝜒 | ̂𝑆 𝑥 ⊗ ̂𝐼 | 𝜒 ⟩ , (14)where ̂𝐼 is the identity operator, represented in this context bythe identity matrix. The total 𝑥 component of spin forthe two-spin system is represented by ̂𝑆 𝑥 = ̂𝑆 𝑥 + ̂𝑆 𝑥 = ̂𝑆 𝑥 ⊗ ̂𝐼 + ̂𝐼 ⊗ ̂𝑆 𝑥 . (15)The simplest phenomenological interaction for two spins isknown as the Heisenberg Hamiltonian, which takes the form ̂𝐻 = − 𝐽 ̂ 𝐒 ⋅ ̂ 𝐒 , where the dot product is shorthand for ̂𝐻 = − 𝐽 [ ̂𝑆 𝑥 ⊗ ̂𝑆 𝑥 + ̂𝑆 𝑦 ⊗ ̂𝑆 𝑦 + ̂𝑆 𝑧 ⊗ ̂𝑆 𝑧 ] , (16)Here 𝐽 is an effective interaction energy, and the explicit repre-sentation of ̂𝐻 takes the form of a ×2 = 4×4 matrix. Pro-ceeding in this manner, one can construct spin systems of anysize with arbitrary interactions. However, the resulting Hamil-tonian grows explosively in size with the dimension being 𝑁 ,representing a practical upper limit on the sizes of quantumspin systems which can be simulated effectively on classicalcomputers. In the present work, we consider small systemswith 𝑁 ≤ . III. COLOR CHARGE
Up to this point, we have not made any significant referenceto the group SU (2) or its properties. Formally, the Pauli matri-ces in Eqs. (2) are the generators of SU (2) , which is the groupof unitary matrices with unit determinant. In other words,every such matrix can be represented by a linear combinationof the Pauli matrices and the identity matrix. The groupSU (3) is simply the group of unitary matrices with de-terminant equal to one. Some familiarity with the Pauli ma-trices, perhaps gained through a study of spin, provides somevaluable intuition for the mechanics of working with objectsin SU (3) .In place of the Pauli matrices, the eight generators of SU (3) are known as the Gell-Mann matrices 𝜆 ( 𝛼 ) . The actual genera-tors are of more immediate use for our purposes and differ fromthe Gell-Mann matrices by a simple factor of via 𝑡 ( 𝛼 ) = 𝜆 ( 𝛼 ) , ̂𝑡 (1) = 12 ⎛⎜⎜⎝ ⎞⎟⎟⎠ ̂𝑡 (2) = 12 ⎛⎜⎜⎝ 𝑖 𝑖 ⎞⎟⎟⎠ , (17) ̂𝑡 (3) = 12 ⎛⎜⎜⎝ ⎞⎟⎟⎠ ̂𝑡 (4) = 12 ⎛⎜⎜⎝ ⎞⎟⎟⎠ , (18) ̂𝑡 (5) = 12 ⎛⎜⎜⎝ 𝑖 𝑖 ⎞⎟⎟⎠ ̂𝑡 (6) = 12 ⎛⎜⎜⎝ ⎞⎟⎟⎠ , (19) ̂𝑡 (7) = 12 ⎛⎜⎜⎝ 𝑖 𝑖 ⎞⎟⎟⎠ ̂𝑡 (8) = 12 √ ⎛⎜⎜⎝ ⎞⎟⎟⎠ . (20)For brevity, we shall refer to the ̂𝑡 ( 𝛼 ) as the Gell-Mann ma-trices. By analogy with spin, a single source of SU (3) “color”charge belongs to the fundamental representation of SU (3) andis represented by a three-component vector | 𝜓 ⟩ = 𝛼 | 𝑟 ⟩ + 𝛽 | 𝑏 ⟩ + 𝛾 | 𝑔 ⟩ ̇ = ⎛⎜⎜⎝ 𝛼𝛽𝛾 ⎞⎟⎟⎠ . (21)Here the basis states corresponding to “up” and “down” inSU (2) are known as “red,” “blue” and “green.” The relation-ship of these basis states to color will be explored below. It isuseful to label explicitly the three basis vectors which span thecolor space | 𝑟 ⟩ ̇ = ⎛⎜⎜⎝ ⎞⎟⎟⎠ , | 𝑏 ⟩ ̇ = ⎛⎜⎜⎝ ⎞⎟⎟⎠ , | 𝑔 ⟩ ̇ = ⎛⎜⎜⎝ ⎞⎟⎟⎠ . (22)Just as each Pauli matrix corresponds to a component ofspin, each Gell-Mann matrix corresponds to a component ofcolor charge, 𝑞 ( 𝛼 ) with 𝑞 ( 𝛼 ) → ⟨ 𝜓 | ̂𝑡 ( 𝛼 ) | 𝜓 ⟩ . (23)Color charge is a type of generalization of electric charge pos-sessed by quarks. In addition to color charge, quarks also haveelectric charge, flavor and spin. We will use the term “quark”to refer to point sources of color charge, but we are only consid-ering the color charge degree of freedom. Note that the colorcharge is actually an eight-dimensional vector which lives inthe abstract “gauge space” rather than in real space. UsingEq. (23), one may show that at most only two components arenonzero for each of the three basis states in Eq. (22). Denot-ing 𝑄 ( 𝛼 ) r = ⟨ 𝑟 | ̂𝑡 ( 𝛼 ) | 𝑟 ⟩ , and similarly for | 𝑔 ⟩ and | 𝑏 ⟩ , the matrix q q Q g Q r Q b Q g_ Q b_ Q r_ FIG. 1: Graphic depiction of the vectors 𝐐 r , 𝐐 g , 𝐐 b and correspond-ing “anticolors,” 𝐐 r , 𝐐 g and 𝐐 b . Only the 𝑞 (3) and 𝑞 (8) componentsare nonzero. multiplications using Eq. (20) yield 𝐐 r = 12 ̂ 𝐞 (3) + 12 √ ̂ 𝐞 (8) , (24) 𝐐 g = − 12 ̂ 𝐞 (3) + 12 √ ̂ 𝐞 (8) , (25) 𝐐 b = − 1 √ ̂ 𝐞 (8) , (26)where ̂ 𝐞 ( 𝛼 ) is a unit vector in the 𝛼 direction in gauge space.We note that the vanishing of all but the third and eighth com-ponents of charge in Eqs. (24)–(26) is analogous to the 𝑥 - and 𝑦 -components of spin expectation values being zero for the ba-sis states | ↑ ⟩ and | ↓ ⟩ . While 𝜎 𝑧 is the only diagonal Pauli ma-trix (corresponding to the only nonzero component of spin inthe 𝑧 -basis), there are two diagonal Gell-Mann matrices, ̂𝑡 (3) and ̂𝑡 (8) . As diagonal matrices, the corresponding color chargeoperators will commute and thus can be observed simultane-ously. That is, 𝑞 (3) and 𝑞 (8) correspond to well-defined quan-tum numbers. A plot of the color charge vectors 𝐐 r,g,b is shownin Figure 1.The relationship of color charge to visual color can be un-derstood from the vector nature of these charges. Just as red,blue and green light combine to form white (colorless) light,these three vectors add to zero. Additionally, to each quarkthere is also an antiquark containing the “opposite” charge.While the electron has electric charge − 𝑒 , the positron car-ries charge −(− 𝑒 ) = + 𝑒 . The same basic reasoning applies tocolor charge so that an “anti-red” quark carries color charge 𝐐 r = − 𝐐 r . Our VP YTHON visualization routine colors thesources by mapping the angle from the positive 𝑞 (3) axis to aparticular hue angle on the color wheel. In the Hue-Saturation-Value (HSV) color model, a hue is defined by an angle 𝜃 withred, green and blue corresponding to 𝜃 r = 0 ◦ , 𝜃 g = 120 ◦ and 𝜃 b = 120 ◦ , respectively. The vectors 𝐐 r,g,b in Figure 1 corre-spond to angles 𝜙 r = 30 ◦ , 𝜙 g = 150 ◦ and 𝜙 b = 270 ◦ , respec-tively, as measured from the 𝑞 (3) axis in the ( 𝑞 (3) , 𝑞 (8) ) plane.The colors depicted in Figure 1 are obtained by calculating thecorresponding hue angle 𝜃 = 𝜙 − 30 ◦ , so that tan( 𝜃 + 30 ◦ ) = 𝑞 (8) 𝑞 (3) . (27)We take as our initial state a combination of all three basisquarks, | 𝜓 ⟩ = | 𝑟 ⟩ ⊗ | 𝑔 ⟩ ⊗ | 𝑏 ⟩ , (28)which has no net color charge. This state is meant to mimicthat of a baryon, which is also colorless. It should be notedthat actual baryons are in a color singlet configuration, whichis a totally antisymmetric superposition state analogous the thespin singlet state | , ⟩ = √ ( | ↑ ⟩ | ↓ ⟩ − | ↓ ⟩ | ↑ ⟩ ) . Explicitly, ||| 𝜓 singlet ⟩ = 1 √ | 𝑟𝑔𝑏 ⟩ − | 𝑟𝑏𝑔 ⟩ + | 𝑔𝑏𝑟 ⟩ − | 𝑔𝑟𝑏 ⟩ + | 𝑏𝑟𝑔 ⟩ − | 𝑏𝑔𝑟 ⟩ ) , (29)where we have employed shorthand notation | 𝑟𝑔𝑏 ⟩ ≡ | 𝑟 ⟩ ⊗ | 𝑔 ⟩ ⊗ | 𝑏 ⟩ , and similarly for other terms, for brevity. The sig-nificance of naturally-occurring bound states employing thesinglet color configuration cannot be understated, and this isintimately related to isolated color charge being essentially un-observable to realistic experiments. Indeed the singlet config-uration is invariant with respect to gauge transformations inSU (3) , whereas the simple product state | 𝜓 ⟩ is not. By usingsuperposition to take full advantage of the underlying symme-try, we will see that the singlet configuration actually possessesa lower ground state energy for our choice of effective inter-actions. Because gauge transformations amount to a redefini-tion of color charge components, gauge invariant dynamicsrequire net color charge itself to be effectively unobservable,much like the scalar and vector potentials in electromagnetism.Unless otherwise noted, we will consider the product state | 𝜓 ⟩ as the initial state for simplicity. This theoretical choice is forpedagogical simplicity, but the resulting calculations are notentirely decoupled from experimental reality. While actualquarks are found in the singlet state, nothing forbids simulatingnon-singlet states using cold atoms and artificial gauge fields. IV. INTERACTIONSA. Two-body terms
In this section we explore effective interactions between thethree quarks in the product state | 𝜓 ⟩ , given by Eq. (28). It should be noted that the strong nuclear interaction is highlymodified by quantum processes in a process known as renor-malization . A naïve exploration of the corresponding clas-sical Lagrangian leads to the prediction of a Coulomb-likepotential. The interactions between sources and the me-diating fields result in a highly-modified confining potentialwhich makes is effectively impossible to separate two quarks inexperiments. In what follows, we consider a reduced modelin which the sources interact directly through phenomenolog-ical interactions which are motivated below.Looking to the Heisenberg Hamiltonian in Eq. (16) for in-spiration, we can postulate the existence of an effective two-body interaction between two color sources of the form ̂𝐻 = 𝐽 ∑ 𝛼 =1 ̂𝑡 ( 𝛼 ) ⊗ ̂𝑡 ( 𝛼 ) , (30)for some constant 𝐽 . A possible motivation for such an ex-pression can be seen by considering what how the interactionenergy behaves for several simple situations. First, considertwo arbitrary sources in the state, | 𝜓 ⟩ = || 𝜓 ⟩ ⊗ || 𝜓 ⟩ . The in-teraction energy is given by the expectation value of Eq. (30) 𝐸 int [ 𝜓 ] = ⟨ 𝜓 | 𝐽 ∑ 𝛼 =1 ̂𝑡 ( 𝛼 ) ⊗ ̂𝑡 ( 𝛼 ) | 𝜓 ⟩ . (31)One property of the tensor product is that [ ̂𝐴 ⊗ ̂𝐵 ] [| 𝜒 ⟩ ⊗ | 𝜙 ⟩] = ( ̂𝐴 | 𝜒 ⟩) ⊗ ( ̂𝐵 | 𝜙 ⟩) , so the in-teraction can be written 𝐸 int [ 𝜓 , 𝜓 ] = 𝐽 𝐐 ⋅ 𝐐 , (32)where the components 𝑄 ( 𝛼 )1 , ≡ ⟨ 𝜓 , || ̂𝑡 ( 𝛼 ) || 𝜓 , ⟩ . Eq. (32) isformally a generalization of the Coulomb potential in whichthe notion of “like” and “opposite” charges has been gener-alized to an inner product of color charge vectors in gaugespace. By treating only the color degree of freedom, wedo not have any information about the variation of the inter-action energy with spatial separation of sources. However,on general grounds one may expect an inverse-linear depen-dence on source separation ( 𝐽 ∝ 𝑅 −1 ) in three spatial dimen-sions before including the effects of dynamical gluons. For 𝐐 ⋅ 𝐐 > , this interaction energy is lowered by increasingthe separation, so the resulting force is repulsive. Conversely,if 𝐐 ⋅ 𝐐 < , the interaction energy is lowered by decreas-ing the separation between sources, and the resulting force isattractive.The above reasoning applies to any color states || 𝜓 , ⟩ , butlet us specialize to the pure red, green and blue states inEq. (22) where the only nonzero color charge components ex-ist in the two-dimensional ( 𝑞 (3) , 𝑞 (8) ) plane. From Figure 1, itis clear that the inner product of a quark’s color charge vectorwith that of its antiquark will be maximally negative. As withelectrons and protons, Eq. (32) predicts that particles and theirantiparticles should attract via the strong interaction.Without loss in generality, consider 𝐐 r . The angular dis-placement between 𝐐 r and either 𝐐 g or 𝐐 b is 120 ◦ , and since cos 120 ◦ < , there will be an attractive force between the redand blue and between the red and green color source. Identicalreasoning can be used for the other quarks to show that thereis mutual attraction between any two of these quarks. Thistype of calculation is based on a simplified, effective interac-tion. Consequently, the predictions should not be trusted foraccurate results given the qualitative changes to the interac-tions when dynamical gluons are included in the theory. How-ever, it is worth noting the qualitative features of quarks beingmutually attractive and a three-particle bound state seemingplausible at this level are consistent with reality. Furthermore,these types of calculations are essentially equivalent to thetree-level calculations one can perform in quantum chromo-dynamics that do not take into account quantum fluctuations. The Hamiltonian in Eq. (30) acts on states of two colorsources. In a system of three sources, this pairwise interac-tion would contribute to each of the three possible pairings, sothe appropriate operator for a system of three sources is ̂𝐻 pairs = 𝐽 ∑ 𝛼 =1 [ ̂𝑡 ( 𝛼 )1 ⋅ ̂𝑡 ( 𝛼 )2 + ̂𝑡 ( 𝛼 )1 ⋅ ̂𝑡 ( 𝛼 )3 + ̂𝑡 ( 𝛼 )2 ⋅ ̂𝑡 ( 𝛼 )3 ] . (33)Here the notation is shorthand with appropriate identity oper-ators not explicitly written. For example, ̂𝑡 ( 𝛼 )1 ⋅ ̂𝑡 ( 𝛼 )3 ≡ ̂𝑡 ( 𝛼 ) ⊗ 𝐼 ⊗ ̂𝑡 ( 𝛼 ) . (34)The corresponding interaction energy takes the form 𝐸 int [ 𝜓 , 𝜓 , 𝜓 ] = 𝐽 ( 𝐐 ⋅ 𝐐 + 𝐐 ⋅ 𝐐 + 𝐐 ⋅ 𝐐 ) . (35)Let us calculate explicitly the interaction energy for the state | 𝜓 ⟩ = | 𝑟 ⟩ ⊗ | 𝑔 ⟩ ⊗ | 𝑏 ⟩ . Eqs. (35) and (24)–(26) give 𝐸 int [ 𝑟, 𝑔, 𝑏 ] = 𝐽 ( 𝐐 r ⋅ 𝐐 g + 𝐐 r ⋅ 𝐐 b + 𝐐 g ⋅ 𝐐 b ) = − 𝐽 . (36)Now consider the singlet state in Eq. (29). It was claimedabove that this is the actual color state realized by three quarkswithin a baryon and that it minimizes the interaction energy.Though we cannot prove this statement for the full theory ofquantum chromodynamics, we can show that the singlet doescorrespond to a lower energy than the simple product state forinteractions given by Eq. (33). For a superposition state suchas ||| 𝜓 singlet ⟩ , we must use Eq. (33) rather than the “classical”limit given by Eq. (35). To see this, let us explore a single termthat arises as ̂𝐻 pairs acts on the first term of the singlet state, ̂𝑡 (1) ⊗ ̂𝑡 (1) ⊗ ̂𝐼 ( | 𝑟 ⟩ ⊗ | 𝑔 ⟩ ⊗ | 𝑏 ⟩ ) = 14 | 𝑔 ⟩ ⊗ | 𝑟 ⟩ ⊗ | 𝑏 ⟩ . (37)If we were considering the product state | 𝑟𝑔𝑏 ⟩ , such a termwould vanish upon taking the inner product with ⟨ 𝑟𝑔𝑏 | . How-ever, the singlet state contains a term √ | 𝑔𝑟𝑏 ⟩ , so this termdoes contribute to the energy. The Gell-Mann matrices arefairly sparse, so most such cross terms still vanish. However,there are a number to keep track of, and after either a fairlycareful accounting or a few minutes with a computer algebrapackage, one finds 𝐸 int [ 𝜓 singlet ] = −2 𝐽 , (38) which is, as advertised, a lower energy than the bare prod-uct state. Using terminology loosely, one can think of a statesuch as | 𝑟𝑔𝑏 ⟩ being “classical” in the sense that a measure-ment of 𝑞 (3) or 𝑞 (8) on any of the three sources would alwaysreturn the same answer, since each quark is in a simultane-ous eigenstate of both ̂𝑞 (3) and ̂𝑞 (8) . The singlet state givesan expectation value of zero for each charge component foreach source. However, as with any quantum mechanical ob-servable, an individual measurement of ̂𝑞 (3) and ̂𝑞 (8) would re-turn one of the operator’s eigenvalues as a result. The mea-surement forces one source to “pick” one of the three positivecolor charge eigenstates. Because the singlet state is entan-gled, the measurement of source one also affects the state ofsources two and three. Entanglement is sometimes viewed asone of the fundamentally “quantum” features systems can ex-hibit, and the singlet state is in some sense more inherently“quantum” than the product state. The singlet is also pos-sesses a higher degree of symmetry than the product state,being antisymmetric with respect to the exchange of sources, ||| 𝜓 singlet ( 𝑟, 𝑏, 𝑔 ) ⟩ = − ||| 𝜓 singlet ( 𝑟, 𝑔, 𝑏 ) ⟩ . Though beyond thescope of this work, one may show that it is invariant with re-spect to SU (3) gauge transformations which redefine the colorcharge components. A similar situation arises with the ground state of the one-dimensional Heisenberg antiferromagnet, which is composedof 𝑁 spin- degrees of freedom with interactions betweennearest neighbors described by the Hamiltonian ̂𝐻 Heisenberg = 𝐽 𝑁 −1 ∑ 𝑗 =1 [ ̂𝑆 𝑥𝑗 ̂𝑆 𝑥𝑗 +1 + ̂𝑆 𝑦𝑗 ̂𝑆 𝑦𝑗 +1 + ̂𝑆 𝑧𝑗 ̂𝑆 𝑧𝑗 +1 ] . (39)A plausibly simple candidate for a ground state is the Néelstate, | ↑↓↑ ⋯ ↓↑↓ ⟩ . One reason this cannot be the ground stateis shown by observing the inversion symmetry of ̂𝐻 Heisenberg isnot respected by this state. That is, another equally good can-didate is | ↓↑↓ ⋯ ↑↓↑ ⟩ . A more problematic feature of the Néelstate is that while it minimizes the energetic contribution fromthe ̂𝑆 𝑧𝑗 ̂𝑆 𝑧𝑗 +1 terms, it is not even an eigenstate of the full Hamil-tonian. To see this, note that the first two terms in Eq. (16) maybe recast as ̂𝑆 𝑥𝑗 ̂𝑆 𝑥𝑗 +1 + ̂𝑆 𝑦𝑗 ̂𝑆 𝑦𝑗 +1 = ̂𝑆 + 𝑗 ̂𝑆 − 𝑗 +1 + ̂𝑆 − 𝑗 ̂𝑆 + 𝑗 +1 , (40)where ̂𝑆 ± 𝑗 ≡ ̂𝑆 𝑥𝑗 ± 𝑖 ̂𝑆 𝑦𝑗 are the spin raising/lowering opera-tors. Each term raises (lowers) a particular spin and lowers(raises) its right neighbor. This process is analogous to the ex-ample for the three-source SU (3) color charge system consid-ered above in the singlet state. However, the equivalent “sin-glet” state for the case of 𝑁 spins is substantially more com-plicated than that of three SU (3) color charges, and a fairlyelaborate technique known as the Bethe ansatz must be usedto obtain the ground state. B. Three-body interactions
In addition to the two-body interactions which arise at theclassical level, higher-order interactions are also possible for-mally. Indeed, the full theory of quantum chromodynam-ics leads to many-body interactions between sources of colorcharge. The effective interactions considered here amount toneglecting the dynamics of the mediating fields, known as gluons . Including dynamical quantum mechanical degrees offreedom for these mediating fields amounts to a legitimatequantum field theory and is necessarily much more complexthan the idealized model presented here.In a system of three sources, one would expect such termsto have the general structure ̂𝐻 = ∑ 𝛼,𝛽,𝛾 𝑔 𝛼𝛽𝛾 ̂𝑡 ( 𝛼 ) ⊗ ̂𝑡 ( 𝛽 ) ⊗ ̂𝑡 ( 𝛾 ) , (41)where 𝑔 𝛼𝛽𝛾 is some tensor of coefficients. The interactionsin quantum chromodynamics, or any gauge theory, are highlyconstrained by symmetry considerations. As a group, SU (3) possesses sets of numbers known as structure constants de-fined by [ ̂𝑡 ( 𝛼 ) , ̂𝑡 ( 𝛽 ) ] = 𝑖 ∑ 𝛾 𝑓 𝛼𝛽𝛾 ̂𝑡 ( 𝛾 ) , (42) { ̂𝑡 ( 𝛼 ) , ̂𝑡 ( 𝛽 ) } = 13 𝛿 𝛼𝛽 + ∑ 𝛾 𝑑 𝛼𝛽𝛾 ̂𝑡 ( 𝛾 ) , (43)where [ 𝐴, 𝐵 ] ≡ 𝐴𝐵 − 𝐵𝐴 is the commutator and { 𝐴, 𝐵 } = 𝐴𝐵 + 𝐵𝐴 is the anticommutator. The Kronecker delta is de-fined by 𝛿 𝑎𝑏 = 1 for 𝑎 = 𝑏 and 𝛿 𝑎𝑏 = 0 otherwise. By virtueof the antisymmetry (symmetry) of the commutator (anticom-mutator) with respect to indices, one may verify that the 𝑑 𝛼𝛽𝛾 are totally symmetric while the 𝑓 𝛼𝛽𝛾 are totally antisymmetricwith respect to the indices 𝛼 , 𝛽 , 𝛾 . There are a total of = 512 possible index combinations for each, but most turn out to bezero. The following values are obtained 𝑑 = 𝑑 = 𝑑 = − 𝑑 = 1 √ , (44) 𝑑 = 𝑑 = 𝑑 = 𝑑 = 𝑑 = 12 , (45) 𝑑 = 𝑑 = 𝑑 = − 12 , (46) 𝑑 = 𝑑 = 𝑑 = 𝑑 = − 12 √ , (47) 𝑓 = 1 , (48) 𝑓 = 𝑓 = 𝑓 = 𝑓 = 12 , (49) 𝑓 = 𝑓 = − 12 , (50) 𝑓 = 𝑓 = √ . (51)Aside from entries which may be obtained from symmetry viacyclic permutations, 𝑑 𝛼𝛾𝛽 = 𝑑 𝛽𝛼𝛾 = 𝑑 𝛼𝛽𝛾 , 𝑓 𝛾𝛽𝛼 = 𝑓 𝛼𝛾𝛽 = 𝑓 𝛽𝛼𝛾 = − 𝑓 𝛼𝛽𝛾 , and 𝑓 𝛾𝛼𝛽 = 𝑓 𝛽𝛾𝛼 = 𝑓 𝛼𝛽𝛾 , all other componentsare zero. One may compute gauge scalars (or pseudoscalars) fromthese structure constants of the forms ∑ 𝛼,𝛽,𝛾 𝑑 𝛼𝛽𝛾 ̂𝑡 ( 𝛼 ) ̂𝑡 ( 𝛽 ) ̂𝑡 ( 𝛾 ) , ∑ 𝛼,𝛽,𝛾 𝑓 𝛼𝛽𝛾 ̂𝑡 ( 𝛼 ) ̂𝑡 ( 𝛽 ) ̂𝑡 ( 𝛾 ) . (52)The symmetric scalar resemble the component form of the or-dinary scalar products between two three-dimensional vectors, 𝐀 ⋅ 𝐁 = ∑ 𝑖,𝑗 𝛿 𝑖𝑗 𝐴 𝑖 𝐵 𝑗 , (53)(54)which is invariant with respect to three-dimensional rotations.Accordingly the expressions in (52) provide suitable candi-dates for interaction terms involving three ̂𝑡 ( 𝛼 ) operators. Suchinteractions involve all three states and are fundamentally dif-ferent from familiar two-body interaction. In quantum chro-modynamics, three-body interaction terms between the medi-ating gluon fields appear in the definition of the Lagrangian, which should result in effective three-body interactions be-tween sources at the level of the effective model presentedhere. It should be noted that the combination involving theasymmetric structure constants should also contain factorswhich depend on the spin on the sources, as it is not gaugeinvariant by itself. Indeed, the gauge scalar obtained from 𝑓 𝛼𝛽𝛾 is analogous to a pseudoscalar obtained from an ordinaryvector cross product. Our focus is on exploring dynamics withminimum complications, so we will adopt both terms as pos-sible forms for three-body interactions. V. DYNAMICS
Our goal in this section is to repeat the basic steps fromSec. II for a three-quark product state has the form || 𝑞 , 𝑞 , 𝑞 ⟩ = || 𝑞 ⟩ ⊗ || 𝑞 ⟩ ⊗ || 𝑞 ⟩ , (55)with time evolution generated by the Hamiltonian ̂𝐻 = ̂𝐻 pairs + Δ ̂𝐻 , (56) = 𝐽 ∑ 𝛼 [ ̂𝑡 ( 𝛼 )1 ⋅ ̂𝑡 ( 𝛼 )2 + ̂𝑡 ( 𝛼 )1 ⋅ ̂𝑡 ( 𝛼 )3 + ̂𝑡 ( 𝛼 )2 ⋅ ̂𝑡 ( 𝛼 )3 ] + Δ 𝐽 ∑ 𝛼,𝛽,𝛾 𝑔 𝛼𝛽𝛾 ̂𝑡 ( 𝛼 ) ⊗ ̂𝑡 ( 𝛽 ) ⊗ ̂𝑡 ( 𝛾 ) , (57)where Δ is a dimensionless parameter to control the relativestrength of the three-body interaction terms. The task at handis to solve the time-dependent Schrödinger equation for a giveninitial state and compute the expectation values of observables(i.e., color charge components). Before proceeding, we mustobtain explicit representations of the initial state and relevantoperators which are suitable for numerical or analytic analysis. A. Two-body interactions (
Δ = 0 ) Let us first consider the case
Δ = 0 so that only two-bodyinteractions are at play. This case admits an analytic solutionfor the color charge components through straightforward, if te-dious steps. The general state of Eq. (55) is a tensor product ofthree color states and is represented by a = 27 -dimensionalcolumn vector || 𝜓 ⟩ ̇ = ⎛⎜⎜⎜⎝ 𝑐 𝑐 ⋮ 𝑐 ⎞⎟⎟⎟⎠ , (58)where the 𝑐 𝑖 are complex numbers. For a simple product stateof the three basis vectors, || 𝜓 ⟩ = | 𝑟 ⟩ ⊗ | 𝑔 ⟩ ⊗ | 𝑏 ⟩ , one finds 𝑐 = 1 , with 𝑐 𝑖 = 0 for 𝑖 ≠ . In principle, it is possibleto work out the action of the two-body Hamiltonian operatorin Eq. (33) on the general state | 𝜒 ⟩ , obtaining a set of coupleddifferential equations for the coefficients 𝑐 𝑖 via the Schrödingerequation 𝑖ℏ 𝜕𝜕𝑡 | 𝜓 ⟩ = ̂𝐻 pairs | 𝜓 ⟩ . (59)Upon substituting Eq. (58) into Eq. (59), one may use the ex-plicit forms of the Gell-Mann matrices in Eq. (20) to obtain theaction of ̂𝐻 pairs on | 𝜓 ⟩ . The resulting 27 equations split intodecoupled sets, so that only those equations involving 𝑐 arenontrivial. All others involve only quantities which have beeninitialized to zero and will never evolve into nonzero values. Inwhat follows, we work in units where ℏ → for brevity. Thedynamics of the system are then contained in the equations ofmotion 𝑖 ̇𝑐 = 𝐽 ( − 𝑐 + 𝑐 + 𝑐 + 𝑐 ) , (60) 𝑖 ̇𝑐 = 𝐽 ( 𝑐 − 𝑐 + 𝑐 + 𝑐 ) , (61) 𝑖 ̇𝑐 = 𝐽 ( 𝑐 − 𝑐 + 𝑐 + 𝑐 ) ,𝑖 ̇𝑐 = 𝐽 ( 𝑐 + 𝑐 − 𝑐 + 𝑐 ) , (62) 𝑖 ̇𝑐 = 𝐽 ( 𝑐 + 𝑐 − 𝑐 + 𝑐 ) , (63) 𝑖 ̇𝑐 = 𝐽 ( 𝑐 + 𝑐 + 𝑐 − 𝑐 ) . (64)We note that the six indices appearing above correspond tothe six ways of ordering the red, green and blue sources in theproduct state || 𝑞 𝑞 𝑞 ⟩ . For the state | 𝑔𝑟𝑏 ⟩ , we would have 𝑐 → with all others zero, resulting in the same equationsof motion. The singlet state corresponds to 𝑐 = − 𝑐 = − 𝑐 = 𝑐 = 𝑐 = − 𝑐 = 1 √ . (65)This linear system can be solved explicitly for the nontrivialcoefficients using the initial conditions 𝑐 (0) = 1 with all oth- ers zero, giving 𝑐 ( 𝑡 ) = 16 𝑒 − 𝑖𝐽𝑡 − 16 𝑒 𝑖𝐽𝑡 , (66) 𝑐 ( 𝑡 ) = 16 𝑒 − 𝑖𝐽𝑡 + 16 𝑒 𝑖𝐽𝑡 − 13 𝑒 𝑖 𝐽𝑡 , (67) 𝑐 ( 𝑡 ) = 16 𝑒 − 𝑖𝐽𝑡 + 16 𝑒 𝑖𝐽𝑡 − 13 𝑒 𝑖 𝐽𝑡 , (68) 𝑐 ( 𝑡 ) = 16 𝑒 − 𝑖𝐽𝑡 − 16 𝑒 𝑖𝐽𝑡 , (69) 𝑐 ( 𝑡 ) = 16 𝑒 − 𝑖𝐽𝑡 − 16 𝑒 𝑖𝐽𝑡 , (70) 𝑐 ( 𝑡 ) = 16 𝑒 𝑖𝐽𝑡 + 16 𝑒 𝑖𝐽𝑡 + 23 𝑒 𝑖 𝐽𝑡 . (71)Expectation values of charge components follow from gener-alizing Eq. (23) to the case of three sources. For example, thecharge components of the first source are given by 𝑄 ( 𝛼 )1 ( 𝑡 ) = ⟨ 𝜓 ( 𝑡 ) | ̂𝑡 ( 𝛼 ) ⊗ ̂𝐼 ⊗ ̂𝐼 | 𝜓 ( 𝑡 ) ⟩ , (72)and will depend on the nonzero 𝑐 𝑖 ( 𝑡 ) . Explicitly, setting allother coefficients to zero, we obtain 𝑄 (3)1 ( 𝑡 ) = 12 [| 𝑐 ( 𝑡 ) | − | 𝑐 ( 𝑡 ) | − | 𝑐 ( 𝑡 ) | + | 𝑐 ( 𝑡 ) | ] , (73) 𝑄 (3)2 ( 𝑡 ) = 12 [ − | 𝑐 ( 𝑡 ) | + | 𝑐 ( 𝑡 ) | + | 𝑐 ( 𝑡 ) | − | 𝑐 ( 𝑡 ) | ] , (74) 𝑄 (3)3 ( 𝑡 ) = 12 [| 𝑐 ( 𝑡 ) | − | 𝑐 ( 𝑡 ) | − | 𝑐 ( 𝑡 ) | + | 𝑐 ( 𝑡 ) | ] , (75) 𝑄 (8)1 ( 𝑡 ) = √ [ − 13 | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | − 13 | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | ] (76) 𝑄 (8)2 ( 𝑡 ) = √ [ | 𝑐 ( 𝑡 ) | − 13 | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | − 13 | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | ] (77) 𝑄 (8)3 ( 𝑡 ) = √ [ | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | + 16 | 𝑐 ( 𝑡 ) | − 13 | 𝑐 ( 𝑡 ) | − 13 | 𝑐 ( 𝑡 ) | ] (78)Here 𝐐 , , ( 𝑡 ) is the color charge vector of the source whichwas initially red, green, blue. Using the explicit solutions for 𝑐 𝑖 ( 𝑡 ) in Eqs. (66)–(71), these expectation values reduce to 𝑄 (3)1 ( 𝑡 ) = − 𝑄 (3)2 ( 𝑡 ) = 12 𝑓 ( 𝑡 ) , (79) 𝑄 (3)3 ( 𝑡 ) = 0 , (80) 𝑄 (8)1 ( 𝑡 ) = 𝑄 (8)2 ( 𝑡 ) = 12 √ 𝑓 ( 𝑡 ) , (81) 𝑄 (8)3 ( 𝑡 ) = − 1 √ 𝑓 ( 𝑡 ) , (82)where 𝑓 ( 𝑡 ) ≡ ( [ 𝐽 𝑡 ]) . Equations (79)–(82)predict oscillations in the charge components. Already thisis a qualitatively different type of behavior than occurs inelectrostatics in which the charge is a scalar which does notpossess any dynamics. These oscillations describe a sort offlip-flop behavior with the color sliding along the directionsdefined by Figure 1 and switching between “positive” and“negative,” where “positive” corresponds to parallel to thecolor’s initial direction and “negative” being antiparallel.Such nontrivial charge dynamics also persists in the corre-sponding classical field theory, as has been demonstratedexplicitly for the case of SU (2) dynamics. B. Three-body interactions ( Δ ≠ ) The case in which three-body interactions are included issignificantly more complex, so we employ a numerical ap-proach to make the treatment as accessible as possible. AJ
UPYTER notebook which makes heavy use of several conve-nient N
UMPY functions is included in the supplemental mate-rial and allows the reader to recreate all cases studied hereby simply changing parameters. To reduce the number of freeparameters in what follows, we restrict attention to the anti-symmetric three-body interactions, 𝑔 𝛼𝛽𝛾 → 𝑓 𝛼𝛽𝛾 , in what fol-lows.The basic steps required to obtain 𝑄 ( 𝛼 )1 , , ( 𝑡 ) are to (1) buildthe Hamiltonian generating time evolution, (2) construct anexplicit vector representation initial state || 𝜓 ⟩ , (3) solve theSchrödinger equation to obtain | 𝜓 ( 𝑡 ) ⟩ , and (4) compute ap-propriate inner products of the form ⟨ 𝜓 ( 𝑡 ) | ̂𝑂 | 𝜓 ( 𝑡 ) ⟩ where ̂𝑂 is some 3-body operator corresponding to a component ofone source’s color charge. Particularly helpful with steps (1)and (4) is the N UMPY function kron(A,B) which computesthe Kronecker product of two matrices, A and B . The Kro-necker product provides an explicit representation of the ab-stract tensor product in Eqs. (55) and (57), so much of the te-dious work can be performed behind the scenes, leading to afairly compact program. The interested reader can find an ex-plicit scheme for constructing appropriate single-site matricesin multi-site systems in Ref. 20 which also generalizes to oper-ators which cannot be decomposed into a Kronecker product.For an 𝑁 -dimensional Hamiltonian ̂𝐻 with energy eigen-values 𝜖 𝑛 and corresponding eigenstates || 𝜙 𝑛 ⟩ , Hermiticityguarantees that any state | 𝜓 ⟩ may be written as a linear com-bination of the eigenstates || 𝜓 ⟩ = 𝑁 ∑ 𝑛 =1 𝑐 𝑛 || 𝜙 𝑛 ⟩ , (83)where the coefficients 𝑐 𝑛 = ⟨ 𝜙 𝑛 || 𝜓 ⟩ represent the overlap be-tween the state || 𝜓 ⟩ and the 𝑛 th eigenstate. Since the eigen-states have trivial time evolution || 𝜙 𝑛 ( 𝑡 ) ⟩ = 𝑒 − 𝑖𝜖 𝑛 𝑡 || 𝜙 𝑛 ⟩ as sta-tionary states, the full time-dependence of an arbitrary state (a)(b)(c) FIG. 2: Color charge components as a function of time for
Δ = 0 . for (a) 𝑄 ( 𝛼 )1 ( 𝑡 ) ; (b) 𝑄 ( 𝛼 )2 ( 𝑡 ) ; (c) 𝑄 ( 𝛼 )3 ( 𝑡 ) ; Only the 𝛼 = 3 , componentsbecome nonzero, so other components are not shown. FIG. 3: Using random couplings in the three-body term ̂𝐻 𝑚𝑏𝑜𝑥 𝑏𝑜𝑑𝑦 with 𝜉 ∈ [0 , . allows the charge components with 𝛼 ≠ , togradually become nonzero. The time required for these other compo-nents to become comparable in size to 𝑄 (3 , ) decreases with increas-ing 𝜉 . || 𝜓 ⟩ can be written as | 𝜓 ( 𝑡 ) ⟩ = ∑ 𝑛 𝑐 𝑛 𝑒 − 𝑖𝜖 𝑛 𝑡 || 𝜙 𝑛 ⟩ . (84)The built-in N UMPY routine w,v = eigh(H) provides a listof eigenvalues w and matrix v whose columns are the corre-sponding eigenvectors of a (Hermitian) matrix H . Accordingly,we use this diagonalization procedure to solve the Schrödingerequation numerically.We employ three-dimensional arrays A[i,j,k] to store thecoefficients or 𝑓 𝛼𝛽𝛾 and 𝑑 𝛼𝛽𝛾 as well as the Gell-Mann matri-ces. For the latter, we define an array ts such that ts[:,:,n] is the 𝑛 th Gell-Mann matrix. By indexing these matrices, thesums in Eq. (57) can be easily written as an unrestricted sumand coded as a series of nested loops.Operators corresponding to one-body observables are con-structed by applying nested instances of kron() to a Gell-Mann matrix and two factors of the identity matrix (stored as t0 in the program), ̂𝑡 (2) ⊗ ̂𝐼 ⊗ ̂𝐼 → kron(ts[:,:,2],kron(t0,t0)) . (85)Figure 2 depicts typical charge component dynamics for Δ ≠ . We still observe periodic behavior, though it is some-what more complex than that for Δ = 0 (c.f, Eqs. (79)–(82)). The patient reader may verify that applying the sametedious steps as for the two-body interactions also leads to aclosed-form analytic solution. Defining three-body interaction strength 𝑉 = Δ 𝐽 for brevity, we find 𝑄 (3)1 ( 𝑡 ) = 2 sin ( √ 𝑉 𝑡 ) √ [ cos ( 𝐽 𝑡 ) − cos ( √ 𝑉 𝑡 )] , (86) 𝑄 (8)1 ( 𝑡 ) = − 2 cos ( √ 𝑉 𝑡 ) √ [ cos ( √ 𝑉 𝑡 ) + cos ( 𝐽 𝑡 )] + 13 √ ] . (87)Other source components can be obtained by applying a rota-tion of 120 ◦ to the vector with components ( 𝑄 (3)1 ( 𝑡 ) , 𝑄 (8)1 ( 𝑡 )) .Computer algebra software such as M ATHEMATICA is quiteuseful in dealing with the significant algebra involved in ob-taining Eqs. (86)–(87).The reader could be forgiven for being mystified by the par-ticular choices of three-body interaction coefficients 𝑓 𝛼𝛽𝛾 and 𝑑 𝛼𝛽𝛾 . With the numerical approach, one is not bound to studyhighly symmetric and analytically tractable situations. Fig-ure 3 depicts the numerical solution for the case in which thecoefficients of the three-body interaction in Eq. (57) are re-placed by random numbers 𝑓 𝛼𝛽𝛾 → 𝜉 . In this case we drawfrom a uniform distribution 𝜉 ∈ [0 , . . For short times,the system essentially follows the two-body solution. Grad-ually, however, the random three-body couplings cause theother charge components to become nonzero. The reader mayverify using the provided program that using only two-bodycouplings with random values is not sufficient to turn on thecomponents with 𝛼 ≠ , . C. Visualization
The plots in Figures 2 and 3 are not the most lucid ways ofdepicting the dynamics. From Eqs. (86)–(87) we see that thedynamics should be in general quasiperiodic . The linear com-bination of trigonometric functions 𝑎 cos( 𝜔𝑡 ) + 𝑏 sin( 𝜔 ′ 𝑡 ) willonly itself be periodic if the ratio 𝜔𝜔 ′ is a rational number. Thisperiodicity is satisfied for 𝑉𝐽 = Δ 𝑛 = 2 √ 𝑛 for any integer 𝑛 . Figure 4 shows several examples of periodic (closed) orbits.Even in the quasiperiodic case, the angular separation betweentwo color charge vectors is always ◦ . It is interesting to notethat for 𝑛 divisible by 3, the three trajectories coincide result-ing in a single closed curve along which all three color chargesmove in the ( 𝑞 (3) , 𝑞 (8) ) plane. Additionally, the point 𝑛 = 1 re-sults in the same dynamics as Δ = 0 . For general values of Δ not corresponding to Δ 𝑛 , the quasiperiodic dynamics leads tospace-filling curves that never close.The provided J UPYTER notebook also makes use ofV
PYTHON routines to animate the dynamics with the colorcharge trajectories traced in real time. While Figure 4 depictsthe individual trajectories, the VP YTHON visualization showsthe individual color charges with the color assigned accordingto ( 𝑞 (3) , 𝑞 (8) ) coordinates as described in Sec. III with Eq. (27).0 FIG. 4: Periodic orbits in the ( 𝑞 (3) , 𝑞 (8) ) plane for various values of 𝑛 and Δ 𝑛 = 2 √ 𝑛 . The case where 𝑛 is a multiple of 3 is particu-larly degenerate, with all three source vectors moving along the sameclosed curve. The initial state of each source (red, blue, green) fixesthe color of the trajectory. That is, whenever a color charge arrives at an angle 𝜙 = 30 ◦ ,it is colored in red with continuous changes in color through-out the evolution according to the value of 𝜙 = tan −1 𝑞 (8) 𝑞 (3) . Itshould be emphasized that the actual dynamics being investi-gated are of each source’s components of color charge. Thatis, the sources themselves are not moving in real space. Tomake this more transparent, we fix the three sources a points inspace and attach a vector to each source representing its colorcharge vector. This vector will change magnitude, directionand color in the visualization while the sources remain fixedin place. Several screenshots of the resulting animation for aperiodic orbit with 𝑛 = 6 are shown in Figure 5. In the secondpanel, one observes a color change in process (for example,what began as the green charge is turning into a red charge).The anti-colors form Figure 1 emerge in the third panel. FIG. 5: Snapshots of the VP
YTHON animation for periodic orbit. Thecolor charges evolve under a periodic orbit with 𝑛 = 19 . Each arrowdirection follows the orientation of that source’s color charge vectorin ( 𝑞 (3) , 𝑞 (8) ) space. VI. DISCUSSION
We have presented a reduced, (0 + 1) -dimensional effectivemodel for the strong interaction in which the components ofcolor charge can be computed as a function of time for a giveninitial state. Both two- and three-body interactions have beenconsidered, and several choices resulted in compact, analyticsolutions. More general types of interactions and initial states have been treated numerically.Though the details of the analytic calculations are some-times fairly tedious, the basic steps involved are no more so-phisticated than those used to study spin dynamics in under-graduate quantum mechanics. Accordingly, extensions of thework presented could provide motivation for interesting inde-pendent projects. In particular, the spin degree of freedomcould also be included to provide a more faithful represen-1tation of actual quarks. The attached J
UPYTER notebook al-lows one to investigate symmetric three-body interactions, aswell as the antisymmetric three-body terms considered here.A much larger phase space is accessible to the curious studentthan what has been presented.Another possible line of inquiry is examining how the timerequired for the off-diagonal charge components to becomecomparable in magnitude to 𝑞 (3 , depends on the relativestrength of asymmetric interactions, such as the random cou- plings considered here. Lastly, investigation of dynamics re-sulting from the singlet initial configuration is another line ofinquiry not developed in this work. The color charge expecta-tion values vanish for all components of all sources at arbitrarytime when the initial state is the singlet. That does not implythat nothing interesting happens. One might look to two- orthree-point correlation functions of the charge components fornontrivial dynamics. ∗ Present address: Joint School of Nanoscience & Nanoengineering,2907 E Gate City Boulevard, Greensboro, NC 27401 † Electronic mail: [email protected] For a fairly recent survey of relevant literature, see A. S. Kronfeldand C. Quigg, “Resource Letter QCD-1: Quantum chromodynam-ics,”
Am. J. Phys. (11), 1081–1116 (2010). Here “nonabelian” refers to the nonabelian (or, noncommutative)nature of the gauge group, SU (3) . For two elements 𝑎, 𝑏 ∈ SU (3) ,in general one has 𝑎 ◦ 𝑏 ≠ 𝑏 ◦ 𝑎 where ◦ denotes the binary operationof the group. Electrodynamics also possesses a gauge symmetry,but the relevant gauge group is the abelian group U (1) in whichelements 𝑎 may be parameterized by a phase 𝜙 as 𝑎 → 𝑒 𝑖𝜙 and thegroup operation ◦ takes the form of ordinary multiplication, whichis commutative, or “abelian.” Classically, a spin can be thought of as a small loop of currentwhich creates a magnetic field. The interaction of two spins witheach other can be pictured as the interaction of one spin with themagnetic field created by the other (or vice versa). For a particularly readable introduction, see K. Joel, D. Kollmarand L. F. Santos, “An introduction to the spectrum, symmetries,and dynamics of spin-1/2 Heisenberg chains,”
Am. J. Phys. (6),450–457 (2013). The group SU ( 𝑁 ) contains 𝑁 − 1 generators, which gives 3 gen-erators (Pauli matrices) for SU (2) and eight (Gell-Mann matrices)for SU (3) . M. K. Agoston,
Computer Graphics and Geometric Modeling:Implementation and Algorithms , (Springer, London, UK, 2005). D. J. Griffiths,
Introduction to Elementary Particles , 2nd ed.(Wiley-VHC, Weinheim, Germany, 2008). A. D. Boozer, “Classical Yang-Mills theory,”
Am. J. Phys. (9),925–931 (2011). V. Galitski, I. Spielman and G. Juzeliunas, “Artificial gauge fieldswith ultracold atoms,”
Physics Today (1), 38–44 (2019). V. Rubakov,
Classical Theory of Gauge Fields , (Princeton Univer-sity Press, Princeton, NJ, 2002). F. Halzen and A. D. Martin,
Quarks and Leptons: An IntroductoryCourse in Modern Particle Physics , (Wiley, Hoboken, NJ, 1984). This is a big omission. Unlike electromagnetism, chromodynam-ics is qualitatively altered by virtual processes related to gluon ex-change. In electromagnetism, such effects are weak and only alterserve to make the electromagnetic interaction weaker at low ener-gies. Quite the contrary, the strong force becomes much strongerat low energies and we are essentially working in the high-energylimit by using such expressions. D. J. Griffiths and D. Schroeter,
Introduction to Quantum Mechan-ics , 3rd ed. (Cambridge University Press, Cambridge, UK, 2018). T. Giamarchi,
Quantum Physics in One Dimension , (ClarendonPress, Oxford, UK, 2004). A. Zee,
Quantum Field Theory in a Nutshell , 2nd ed., (PrincetonUniversity Press, Princeton, NJ, 2010). D. Peskin and D. Schroeder,
Introduction to Quantum Field The-ory , (Westview Press, Boulder, CO, 1995). V. Dmitra˘sinovic, “Cubic Casimir operator of SU C (3) and con-finement in the nonrelativistic quark model,” Phys. Lett. B ,135–140 (2001). Note that the equivalent problem in electrostatics is the trivial sce-nario of two electric charges separated by a fixed distance in whichno dynamics occur in any measurable variable. http://physics.highpoint.edu/ ∼ jlancaster/research/colorcharge/ D. Candela, “Undergraduate computational physics projects onquantum computing,”
Am. J. Phys. , 688–702 (2015). Python array indices start with 0 instead of 1, and the actual pro-gram uses shifted indices so that ts[:,:,0] corresponds to ̂𝑡 (1) ,and so on in this way. An alternative is to package the identitymatrix as ts[:,:,0] so that the Gell-Mann matrix labels line upwith the array indices. Since we have to perform a sum over termslike 𝑓 𝛼𝛽𝛾 ̂𝑡 ( 𝛼 ) ⊗ ̂𝑡 ( 𝛽 ) ⊗ ̂𝑡 ( 𝛾 ) , we chose to live with the offset to keepthe array indices consistent among the coefficients and matrices.22