Spin effects and compactification
aa r X i v : . [ g r- q c ] F e b Spin effects and compactification
Alexander J. Silenko ∗ Research Institute for Nuclear Problems,Belarusian State University, Minsk 220030, Belarusand Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna 141980, Russia
Oleg V. Teryaev † Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna 141980, Russia
Abstract
We consider the dynamics of Dirac particles moving in the curved spaces with one coordinatesubjected to compactification and thus interpolating smoothly between three- and two-dimensionalspaces. We use the model of compactification, which allows us to perform the exact Foldy-Wouthuysen transformation of the Dirac equation and then to obtain the exact solutions of theequations of motion for momentum and spin in the classical limit. The spin precesses with thevariable angular velocity, and a “flick” may appear in the remnant two-dimensional space once ortwice during the period. We note an irreversibility in the particle dynamics because the particlecan always penetrate from the lower-dimensional region to the higher-dimensional region, but notinversely.
PACS numbers: 04.20.Jb, 03.65.Pm, 11.10.Ef, 11.25.Mj ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Low-dimensional structures are now under scrutiny in nonperturbative QCD, cosmology,high-energy physics, and condensed matter physics. Properties of particles placed into suchstructures are usually described by considering quantum theory in two dimensions. However,there is no doubt that real space remains three dimensional, which may lead to qualitativedifferences in some observables.This especially concerns the particle spin properties, which are crucially different at twoand three spatial dimensions (see, e.g., Refs. [1–3]). Thus, transition to (2+1)-dimensionalspacetimes leads to losses of a significant part of such properties. At the same time, in thetwo-dimensional space, anyons [4] may appear.In the present work, we investigate the problem of transformation of the spin propertiesunder the compactification of some spatial dimension. This problem is generally very difficultbecause the spin dynamics depends on many factors. To extract some common properties, weconsider the toy model [5] of the curved space of variable dimensionality smoothly changingfrom three to two. A great preference of the model used is a possibility to obtain exact quantum-mechanical solutions.We use the conventional Dirac equation for a consistent description of spin-1/2 particlemotion in the curved space and take into account relativistic effects. While such effects arenot too important in condensed matter physics (except for graphene), we keep in mind theirfurther applications to the processes at Large Hadron Collider in the case [6, 7] of variable(momentum) space dimension. We use the relativistic method [8] of the Foldy-Wouthuysen(FW) transformation [9] to derive exact quantum-mechanical equations of motion and obtaintheir classical limit.In this work, we focus our attention on the spin properties. We show that, in contrastto a “naive” estimation, the spin in an effectively two-dimensional space may precess aboutthe noncompactified dimensions and therefore a “flick” may appear in the remnant spaceonce or twice during the period. 2
I. HERMITIAN HAMILTONIANS FOR THE METRIC ADMITTING THE EF-FECTIVE DIMENSIONAL REDUCTION
Let us start with the following metric proposed by Fiziev [5]: ds = c dt − ρ ( z ) d Φ − ρ ( z ) d Φ − ρ ( z ) dz , (2.1)where ρ ( z ) = 1 + ρ ′ ( z ) + ρ ′ ( z ) , the primes define derivatives with respect to z , and ρ i arethe functions of z . The spatial coordinates vary in the limits −∞ < z < ∞ , < Φ , < π .We suppose ρ i ( z ) to be positive. The (3+1)-dimensional manifold defining this metricis a hypersurface in a flat pseudo-Euclidean (5+1)-dimensional space. The tetrad e b =1 , e b ji = δ ij √ g ii allows us to define the local Lorentz (tetrad) frame. This considerablysimplifies an analysis of results from possibly using the rescaled Cartesian coordinates dX = ρ ( z ) d Φ , dY = ρ ( z ) d Φ , dZ = ρ ( z ) dz in the neighborhood of any point.Taking the limit ρ ( z ) → ρ ( z ) → d = 3 to d = 2. We consider the case when the compactificationof the e ( e ) direction results in the confinement of the particle in a narrow interval ofΦ (Φ ) angles.The transverse part of the metric (if z is assumed to be a longitudinal coordinate) has thestructure of the Clifford torus, which is the product of two unit circles in the fourdimensionalEuclidean space: y + y = y + y = 1 . (2.2)The Clifford tori are used for analyzing twisted materials [10] and vesicles [11–13]. Thereis also some qualitative similarity to projection of a tube in a six-dimensional space ontoa three-dimensional space, which was used for the construction of the quasicrystals theory[14].We consider Clifford tori as a toy model of dimensional reduction. We are not necessarilyassigning the physical sense to all of the intermediate values of z except the asymptoticsfor z → ±∞ corresponding to the three- and two-dimensional spaces. Here, varying thedimension plays the same role as varying the coupling constant for the case of an adiabaticswitch on the interaction.To describe the spin-1/2 particles, we use the conventional covariant Dirac equation (seeRef. [15] and references therein). To find the Hamiltonian form of this equation, one can3ubstitute the given metric into the general equation for the Hermitian Dirac Hamiltonian(Eq. (2.21) in Ref. [16]). For the metric (2.1), the Hermitian Dirac Hamiltonian was firstderived in Ref. [17]. It can be presented in the form H D = βmc − i ~ cρ α ∂∂ Φ − i ~ cρ α ∂∂ Φ − i ~ c α (cid:26) ρ , ∂∂z (cid:27) , (2.3)where { . . . , . . . } denotes an anticommutator.We transform this Hamiltonian to the FW representation by the method elaborated inRef. [8] which was earlier applied in our previous works [15, 16, 18]. After the exact FWtransformation, we get the result H F W = β √ a + ~ Σ · b , (2.4)where a = m c + c p ρ + c p ρ + c (cid:26) ρ , p (cid:27) , b = b e + b e = c ρ ′ ρ ρ p e − c ρ ′ ρ ρ p e , (2.5)and ( p , p , p ) = (cid:18) − i ~ ∂∂ Φ , − i ~ ∂∂ Φ , − i ~ ∂∂z (cid:19) is the generalized momentum operator.Primes denote derivatives with respect to z . The e , e , e vectors form the spatial partof the orthonormal basis defining the local Lorentz (tetrad) frame. For the given time-independent metric, the operators H F W , p , and p are integrals of motion.Neglecting a noncommutativity of the a and b operators allows us to omit anticommuta-tors and results in H F W = β (cid:16) √ a + ~ b + √ a − ~ b (cid:17) + Π · b b (cid:16) √ a + ~ b − √ a − ~ b (cid:17) , (2.6)where Π = β Σ is the spin polarization operator. It can be proven that extra terms ap-pearing from the above noncommutativity are of order of | ~ / ( p z l ) | , where p z is the particlemomentum and l is the characteristic size of the nonuniformity region of the external field(in the z direction). With this accuracy, H F W = β (cid:18) √ a − ~ b a / (cid:19) + ~ Π · b √ a . (2.7)The second term proportional to ~ is important even when it is relatively small. This termcontributes to the difference between gravitational interactions of spinning and spinlessparticles and therefore violates the weak equivalence principle. Its importance relative to4he main term is defined by the ratio ( ~ b/a ) . The weak equivalence principle is also violatedby the spin-dependent Mathisson force (see Refs. [15, 19] and references therein) defined bythe third term in Eq. (2.7). While the third term is usually much bigger than the secondone, it vanishes for unpolarized spinning particles. The second term proportional to ( Π · b ) is always nonzero. An analysis of Eqs. (2.5) and (2.7) leads to the conclusion that this termcan be comparable with the main one (proportional to √ a ) when l ∼ λ B , where λ B is the deBroglie wavelength. The existence of the term proportional to ~ is not a specific propertyof the toy model used. The appearance of such terms in the FW Hamiltonians describing aDirac particle in Riemannian spacetimes was noticed in several works [18, 20, 21], whereasits relation to the spin-originated effect leading to the violation of the weak equivalenceprinciple was never mentioned.The equation of spin motion is given by d Π dt = Ω × Π , Ω = β b √ a . (2.8)As a result, the spin rotates relative to e i vectors ( i = 1 , ,
3) with the angular velocity Ω . Its motion relative to the Cartesian axes is much more complicated.It has been proven in Ref. [22] that finding a classical limit of relativistic quantum me-chanical equations reduces to the replacement of operators by respective classical quantitieswhen the condition of the Wentzel-Kramers-Brillouin approximation, ~ / | pl | ≪
1, is satisfied.It has also been shown that the classical limit of the FW Hamiltonians for Dirac [15, 16, 18]and scalar [23] particles in Riemannian spacetimes coincides with the corresponding purelyclassical Hamiltonians.
III. MOTION OF PARTICLE AT VARIABLE DIMENSIONS
Let us first study the motion of the particle by neglecting the influence of the spin onto itstrajectory. Since p and p are integrals of motion, they can be replaced with the eigenvalues P and P , respectively. Let us choose the e axis as the compactified dimension and supposethat ρ ( z ) is a decreasing function ( ρ ( z ) → z → ∞ ). We can neglect a dependenceof ρ on z , assuming that this function changes much more slowly. We denote initial valuesof all parameters by additional zero indices and consider the general case when the initialvalue of the metric component, ρ ≡ ρ ( z ), is not small.5he classical limit of the Hamiltonian is given by H = s m c + c P ρ + c P ρ + c p ρ . (3.1)The possibility of making general conclusions with the special model used is based on thefact that the Hamiltonian of a particle in an arbitrary static spacetime is given by H = s c ( m c + g ij p i p j ) g , i, j = 1 , , . (3.2)Equation (3.2) covers spinless [24] and spinning [15, 16] particles in classical gravity as wellas the classical limit of the corresponding quantum-mechanical Hamiltonians for scalar [23]and Dirac [15] particles. For spinning particles, the term s · Ω should be added to thisHamiltonian [15, 16]. When the metric is diagonal, g ii = 1 /g ii and Eq. (3.2) takes the sameform as Eq. (3.1).To describe the compactification, we can introduce the compactification radius δ so thatthe “compactification point” z c can be defined by ρ ( z c ) = δ . Due to the energy E conser-vation, the particle can reach this point if E ≥ s m c + c P δ + c P ρ ( z c ) . (3.3)Note that the decrease of compatification radius δ while E remains finite implies the corre-sponding decrease of P .The particle velocity is equal to v z ≡ dzdt = ∂ H ∂p = c p Eρ = c sgn ( p ) Eρ ( z ) p E − m c − c R ( z ) , R ( z ) = P ρ ( z ) + P ρ ( z ) . (3.4)Different signs correspond to the two different directions of the longitudinal particle motion.Note that the arrival to the compactification point with zero velocity ( z c = z f being thefinal point of particle trajectory) corresponds to the equality sign in Eq. (3.3).A tedious but simple calculation allows us to obtain the longitudinal component of theparticle acceleration: a z ≡ d zdt = − c E ρ (cid:18) R ′ p ρ ′ ρ (cid:19) . (3.5)It is obvious that p ( z f ) = 0 , R ′ ( z f ) ≥ R ( z )),so that a z ( z f ) ≤
0. Therefore, z f is the turning (if R ′ ( z f ) >
0) or attracting (if R ′ ( z f ) = 0)6oint. For nonmonotonic R ( z ) there is a possibility of passage to the region z > z f due topossible growth of ρ ( z ). The particle motion is then limited by the point ˜ z f correspondingto the neglect of the motion in the e direction E = s m c + c P ρ (˜ z f ) . (3.6)The important particular case of Eq. (3.1) corresponds to P = 0. The particle penetratesinto the region of the effective dimensional reduction ( z → ∞ ) and does not reverse thedirection of its motion.In this study, as was mentioned above, we consider that the smooth adiabatic transitionfrom the three-dimensional space to the effectively two-dimensional one does not necessarilyattribute the physical sense to all intermediate points in particle motion. At the same time,the true change of the dimensionality was discussed in cosmology (see Refs. [7, 25–27]) andin connection with experiments at the LHC (see Refs. [6, 7, 28, 29]). Our analysis can alsobe applicable at the LHC.Note also that the motion in the opposite direction of increasing dimension does notimpose any conditions for the initial state of the particle. One may say that the regionof lower dimension is “repulsive” whereas the region of higher dimension is “attractive”,implying a sort of irreversibility in the particle dynamics. This property emerges becauseof the appearance of ρ in the expression for the Hamiltonian in the denominator. Such asituation is a general one that can be seen from Eq. (3.2) in the case of diagonal metric.This may give additional support to the hypothesis [25, 26] that such a transition from thelower dimensionality to the higher one leaded to the evolution of the Universe. IV. SPIN EVOLUTION AT VARIABLE DIMENSIONS
In the classical limit, the angular velocity of spin precession is given by Ω = b E = c Eρ (cid:18) P ρ ′ ρ e − P ρ ′ ρ e (cid:19) . (4.1)Because d s /dt = v z ( z )( d s /dz ), Eqs. (3.4) and (4.1) define an easily solvable system offirst-order homogeneous linear differential equations.Equation (4.1) is rather informative about details of the compactification. Only the Ω component contains parameters of the compactified dimension. Although |P | / |P | ≪
1, the7resence of additional factors does not allow for neglecting Ω as compared with Ω (underthe condition that P = 0).When ρ ( z ) = const , Ω = 0 and the spin rotates about the e axis, the spin projectiononto the e e surface, which is the spatial part of the (2+1)-dimensional spacetime, oscil-lates. The spin appears in this surface only once (in the special case when the cone of spinprecession is tangent to this surface) or twice per rotation period. Evidently, the origin ofthis spin “flickering”, as well as the appearance of pseudovector, is completely unexplainablein terms of the two-dimensional space.The model used allows to obtain an exact analytical description of the spin evolution. Itis characterized by a change of the angle ϕ defining the direction of the spin in the planeorthogonal to Ω : ∆ ϕ ( z ) = Z Ω( t ) dt = Z zz Ω( y ) v z ( y ) dy. (4.2)The problem of spin evolution at the effective dimensional reduction can be solved in ageneral form. To simplify the analysis, let us consider the case of ρ ( z ) = ρ = const . Inthis case, the exact value of the integral is∆ ϕ ( z ) = arcsin c P Aρ ( z ) − arcsin c P Aρ , (4.3)where A = s E − m c − c P ρ = c s p ρ + P ρ . (4.4)Since ρ ( z f ) = c |P | (cid:18) E − m c − c P ρ (cid:19) − / , (4.5)the total spin turn ( z = z f ) is given by∆ ϕ = sgn ( P ) · π − arctan P ρ ρ p . (4.6)The passage of the particle to the region of compactification implies, as was discussed above,the relative smallness of the second term so that the spin rotates by about 90 ◦ .If P = 0, the spin projection onto the e direction is always conserved. The spin can,however, rotate about the e direction if ρ depends on z . In this case, the angle of the spin8urn is equal to∆ φ ( z ) = − arcsin c P Bρ ( z ) + arcsin c P Bρ , B = √ E − m c = s c p ρ + c P ρ . (4.7)The total spin turn ( z = z f ) is given by∆ φ = arctan P ρ ρ p − sgn ( P ) · π . (4.8) V. CONCLUSIONS AND OUTLOOK
We considered the Dirac fermion dynamics in the curved space model of variable dimen-sion. The advantage of the toy model used is the possibility of performing the exact FWtransformation of the Dirac equation and then obtaining the exact solutions of the equationsof motion for momentum and spin in the classical limit. At the same time, the obtainedHamiltonian (3.1) is similar to the generic one (3.2) so that one can expect that qualita-tive features of spin and momentum dynamics will persist for other compactification-relatedmetrics as well.The analysis of particle momentum evolution allows us to describe the motion at theboundary between the regions of space having different dimensions. The passage to theregion of lower dimension is more natural in the special case when the generalized momentumin the compactified direction P = 0. At the same time, the transition to the region of higherdimension (considered in Refs. [25, 26] as a possible way of the evolution of the Universe)does not impose the constraints for its initial state, manifesting a sort of irreversibility.The particle motion (especially near the turning point) is characterized by the three mainproperties which cannot be naturally explained from the point of view of observer residing inthe compactified spacetime: i) a reversion of the direction of motion; ii) a rather quick mo-tion along the compactified direction, which may be seen as a sort of “zitterbewegung”; iii) the appearance of a pseudovector of spin in the compactified (2+1)-dimensional space andits rotation or flickering [when the spin pseudovector crosses the remnant (2+1)-dimensionallayer].The experimental tests of the emerging spin effects may be performed by studies of spinpolarizations of Λ (and, probably, also Λ c ) hyperons produced in the high-energy collisionswhere the compactification [6, 7] takes place. This may bear a resemblance to the recently9roposed [30] tests of the vorticity in heavy-ion collisions, although a detailed analysis isrequired.We can finally conclude that the transition to (2+1)-dimensional spacetime leads to thenontrivial behavior of spin which, generally speaking, cannot be adequately described fromthe point of view of an observer residing at (2+1) dimensions. Acknowledgments
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