Derivation of an Operator-Based SpatialNoncommutativity ParameterA thesis submitted in partial fulfillment of the requirementfor the degree of Bachelor of Science with Honors inPhysics from the College of William and Mary in Virginia,byKevin Carr Sapp.Accepted for: HonorsAdvisor: Dr. Carl E. CarlsonDr. Jeffrey NelsonDr. Nahum ZobinWilliamsburg, VirginiaApril 23, 2008AbstractThe idea that the operators defining spacetime could be noncommuting has gainedpopularity in recent years. The formulation in which the commutators themselves area set of commuting numbers has been applied to a number of quantum phenomenato determine what effects it might have. Most of the work overall has focused on thisset of commuting numbers, but some earlier theories of noncommutative coordinatesestablished the coordinate commutator as having an operator-based matrix value in-volving the angular momentum operator, and D¨oplicher et al. obtained a very similaralgebra without mention of the angular momentum. Beginning from the relativisticgeometry used to derive this result, we hope to explore an alternative foundation fora noncommutative geometry of space and determine its effects on some well-studiedquantities in atomic physics.iAcknowledgementsI would like to thank my advisor, Dr. Carl Carlson, for his help and support in undertakingthis project. I am also very grateful to Dr. Jeffrey Nelson, Dr. Nahum Zobin, and Dr.William J. Kossler for sitting on my honors committee. I extend thanks also to the Collegeof William and Mary, for providing me with this opportunity for learning both inside andoutside a class setting, to those who helped me on short notice and outside their field (AshwinRastogi, Brian DeSalvo), and to all my friends and family who have supported me throughthis process.ii1 IntroductionThe commutativity of space-time has been challenged in at least two different contexts in thehistory of modern physics, and each of these instances resulted in a different interpretationof this noncommutativity. First, the concept of a quantized space-time was already devel-oped and minimally laid out near the end of the 1940s [1], using a system of noncommutingcoordinate operators which produced a commutator proportional to the angular momen-tum operator (this is analogous to the noncommuting directional components of the angularmomentum operators themselves in standard quantum mechanics). This form of noncom-mutativity has only minimally developed beyond these roots [2], although it has largelyentered the realm of algebraic topology [3]. The second development of noncommutativequantum mechanics has been one stemming from recent developments in string theory [4],especially contingent on a non-Abelian version of Yang-Mills theory. This theory leads tocoordinate commutators given by a set of C -numb ers, or commuting numbers, described bythe definition[ˆxµ, ˆxν] = iθµν(1)The quantity on the right-hand side is a set of ordinary numbers, and so this equation is notLorentz covariant. Though this Lorentz-violating noncommutative geometry results, thisset of numbers θµνhas been widely used to derive possible theories of noncommutativity,partly justified by its derivation from recent string theory results [5]. However, an alternativeoperator-based noncommutative theory has also developed; thus, one might also be able tostudy phenomenology using such an operator-based commutator as described by Snyder,[ˆxµ, ˆxν] = ia2Lµν. (2)This commutator will significantly increase the complexity of manipulations within any me-chanics affected by it, but it will also produce a Lorentz-invariant space-time missing from1other theories. The C -number theory generates a space with some coordinate transforma-tion definable by the matrix elements θµν, as well as a momentum dependence [6] whichexemplifies Lorentz violation. The operator construction, on the other hand, produces ageometric form known as the “fuzzy sphere” [3], which at large scales reduces to the typicalcontinuous spatial representation, but at very small scales (out of necessity, smaller thanthat with which we can currently interact), the spatial coordinates become noncommutativeand our definition of position becomes weaker. The reliance of this system on the angularmomentum operator L is one possibility which does not reflect a lorentz violation, as Eqn. 2is symmetric under lorentz transformation.1.1 The Hamiltonian of a C -number θµνA significant amount of work has been done to formulate a modified basic quantum mechanicsfrom the C -number commutator mentioned previously, iθµν, and the method of developingsuch a system has been well-documented, especially given a simple1rpotential [6]. We wishhere to give an example of such a system. A typical set of commutators is given,[ˆxµ, ˆxν] = iθµν[ˆxµ, ˆpν] = iδµν[ˆpµ, ˆpν] = 0,(3)which are valid on the same Hilbert spaces as the usual commutative coordinates. To usethis to perform meaningful calculations, we can reformulate the Hamiltonian based on thesecoordinates. Since these coordinates can be mapped onto a Hilbert space which containstheir commutative counterparts, a new reference frame can be defined which allows thecoordinate operators to commute in the usual way ([ˆxµ, ˆxν] = 0):xµ= ˆxµ+12θµνˆpν, pµ= ˆpµ. (4)2Thus, we can state the Hamiltonian in the noncommutative space and transform it using theabove definition into a commuting coordinate system, converting the antisymmetric matrixθµνinto a vector θi≡ ²ijkθjk:Hµ=ˆpµˆpµ2m+ V (ˆxµ) ;V (ˆxµ) = −Ze2√ˆxµˆxµ;V (ˆxµ) = −Ze2q¡xµ−12θµνpν¢¡xµ−12θµλpλ¢= −Ze2r− Ze2~L ·~θ4r3+ O(θ2),(5)and dropping the terms of order θ2, we produce the noncommutative Hamiltonian. Theperturbative energy can be calculated from this Hamiltonian by assuming electron and protonspin applies here, and thus j = l ±12, and simplified by choosing a coordinate system so thez-axis is parallel to~θ:∆EHNC= −hnl0jj0z|Ze24~~L ·~θr3|nl jjzi=−(Zα2)4mec24θzλ2ejzµ1 ∓12l + 1¶·1n3l(l +12(l + 1)δll0δjzj0z,(6)which should then be able to provide perturbative energies to known transition values, de-pendent on the precision of the transition energy.It should be noted, however, that building the Hamiltonian on the noncommutative spacewill alter its implementation, as typical vector and operator multiplication on this space willbe replaced by the Moyal star product, defined byf ? g(x) =