Inquiring electromagnetic quantum fluctuations about the orientability of space
aa r X i v : . [ h e p - t h ] S e p Inquiring electromagnetic quantum fluctuations about theorientability of space
C.H.G. Bessa, N.A. Lemos, and M.J. Rebou¸cas Departamento de F´ısica, Universidade Federal de Campina Grande, Caixa Postal 500858429-900 Campina Grande – PB, Brazil Instituto de F´ısica, Universidade Federal Fluminense, Av. Litorˆanea, S/N24210-340 Niter´oi – RJ, Brazil Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 15022290-180 Rio de Janeiro – RJ, Brazil (Dated: September 10, 2020)
Abstract
Orientability is an important global topological property of spacetime manifolds. It is oftenassumed that a test for spatial orientability requires a global journey across the whole 3 − spaceto check for orientation-reversing paths. Since such a global expedition is not feasible, theoreticalarguments that combine universality of physical experiments with local arrow of time, CP viola-tion and CPT invariance are usually offered to support the choosing of time- and space-orientablespacetime manifolds. In this paper, we show that it is possible to access spatial orientability ofMinkowski empty spacetime through local physical effects involving quantum vacuum electromag-netic fluctuations. To this end, we study the motions of a charged particle and a point electricdipole subjected to these electromagnetic fluctuations in Minkowski spacetime with orientable andnon-orientable spatial topologies. We derive analytic expressions for the velocity dispersion forboth of these point-like particles in the two inequivalent spatially flat topologies. For the chargedparticle, we show that it is possible to distinguish the orientable from the non-orientable topologyby contrasting the time evolution of the respective velocity dispersions. This is a significant resultthat makes apparent that it is possible to access orientability through electromagnetic quantumvacuum fluctuations. However, the answer to the central question of the paper, namely how tolocally probe the orientability of Minkowski 3 − space intrinsically comes about only in the studyof the motions of an electric dipole. For this point-like particle, we find that a characteristicinversion pattern exhibited by the velocity dispersion curves is a signature of non-orientability.This important result makes it clear that it is possible to locally unveil spatial non-orientabilitythrough the inversion pattern of velocity dispersion curves of a point electric dipole under quan-tum vacuum electromagnetic fluctuations. Our findings open the way to a conceivable experimentinvolving quantum vacuum electromagnetic fluctuations to locally probe the spatial orientabilityon the microscopic scale of Minkowski empty spacetime. PACS numbers: 03.70.+k, 05.40.Jc, 42.50.Lc, 04.20.Gz, 98.80.Jk, 98.80.Cq . INTRODUCTION The Universe is modeled as a four-dimensional differentiable manifold, which is a topolog-ical space with an additional differential structure that permits to define locally connections,metric and curvature with which the gravitation theories are formulated. Geometry is a lo-cal attribute that brings about curvature, whereas topology is a global feature of a manifoldrelated, for example, to compactness and orientability. Geometry constrains but does notspecify the topology. So, topologically different manifolds can have a given geometry. Since topology antecedes geometry, it is important to determine whether, how and towhat extent physical phenomena depend upon or are somehow affected, induced, triggered,or even driven by a nontrivial topology. The net role played by the spatial topology ismore clearly ascertained in the static spatially flat Friedmann-Lemaˆıtre-Robertson-Walkerspacetime, whose dynamical degrees of freedom are frozen. Thus, in this work we focus onMinkowski spacetime, whose spatial geometry is Euclidean.Although the topology of the spatial section, M , of Minkowski spacetime, M = R × M ,is usually taken to be the simply-connected Euclidean space E , it is a mathematical factthat it can also be any one of the possible 17 topologically distinct quotient (multiply-connected) manifolds M = E / Γ, where Γ is a discrete group of isometries or holonomiesacting freely on the covering space E [8, 9]. The action of Γ tessellates or tiles the coveringmanifold into identical domains or cells which are copies of what is known as fundamentalpolyhedron (FP) or fundamental cell or domain (FC or FD). On the covering manifold E , themultiple connectedness of M is taken into account by imposing periodic boundary conditions(repeated cells) that are determined by the action of the group of discrete isometries Γ onthe covering space E .In a manifold with periodic boundary conditions only certain modes of fields can exist.Thus, a nontrivial topology may leave its mark on the expectation values of local physicalquantities. A case in point is the Casimir effect of topological origin [10–15].Quantum vacuum fluctuations of the electromagnetic field in Minkowski spacetime seem Despite our present-day inability to predict the spatial topology of the Universe from a fundamentaltheory, one should be able to probe it through cosmic microwave background radiation (CMBR) or (and)stochastic primordial gravitational waves [1, 2], which should follow some basic detectability conditions [3].For recent topological constraints from CMBR data we refer the readers to Refs. [4–6]. For some limitson the circles-in-the-sky method designed for the searches of cosmic topology through CMBR see Ref. [7]. changes of topological nature in the background 3 − space are made to allow forstochastic motions, as for example the insertion of perfectly reflecting planes into the three-dimensional spatial section, the resulting mean squared velocity of a charged test particledoes not vanish [17–23].In a recent paper, the question as to whether a nontrivial topology of the spatial sectionof Minkowski spacetime allows for stochastic motion of test charged particles under vacuumfluctuations of the electromagnetic field has been addressed [24]. It was shown that thesevacuum fluctuations do indeed give rise to stochastic motions of charged particles as longas the spatial topology of Minkowski spacetime is nontrivial. In this way, either by in-serting perfectly reflecting boundaries or by having any of the classified nontrivial 3 − spacetopologies, the crucial background attribute to enable stochastic motions of charged parti-cles under quantum vacuum electromagnetic fluctuations is a nontrivial 3 − space topologyof Minkowski spacetime. Orientability is an important global topological property of a spacetime manifold. It iswidely assumed, implicitly or explicitly, that a manifold modeling the physical spacetime isglobally orientable in all respects. Namely, that it is spacetime orientable and, additionally,that it is separately time and space orientable. Besides, it is also generally assumed that,being a global property, the 3 − space orientability cannot be tested locally. Thus, to disclosethe spatial orientability one would have to make a trip along some specific closed pathsaround the whole 3 − space to check, for example, whether one returns with left- and right-hand sides exchanged. This reasoning is at first sight open to an intriguing criticism: sincesuch a global journey across the whole 3 − space is not feasible one might think that spatialorientability cannot be probed. In the face of this hurdle, one would have either to derive itfrom a fundamental theory of physics or answer the orientability question through cosmo-logical observations or local experiments. Thus, it is conceivable that spatial orientabilitymight be subjected to local experimental tests. In Refs. [17–23] one has a nontrivial inhomogeneous quotient orbifold space topology while in Ref. [24]we have nontrivial flat smooth manifolds. We also note in this regard that the classification of three-dimensional Euclidean spaces was first taken up in the field of crystallography [26–28] and completed in1934 [29]. For a recent exposition the reader is referred to Refs. [30–33]. One can certainly take advantage of gedanken experiments to reach theoretical conclusions, but not as areplacement to actual experimental evidence in physics [34]. − space topology of Minkowski spacetime through the velocity disper-sion of the motions of test charged particles [24], and given that 8 out of the possible 17quotient flat 3 − manifolds are non-orientable [8], a question that naturally arises is whetherthese quantum vacuum fluctuations could be also used to reveal locally specific topologicalproperties such as orientability of 3 − space. Our chief goal in this article is to address this question by inquiring the electromagneticquantum fluctuations about the spatial orientability of Minkowski spacetime. To this end, weinvestigate stochastic motions of a charged particle and of an electric dipole under quantumfluctuations of the electromagnetic field in Minkowski spacetime with two inequivalent spatialtopologies, namely the orientable slab ( E ) and the non-orientable slab with flip ( E ). These topologies turn out to be suitable to show that one can unveil orientability and non-orientability signatures through the motion of point-like particles in Minkowski spacetime.In Section II we introduce the notation and present some key concepts and results re-garding topologies of three-dimensional manifolds, which will be needed in the rest of thepaper. In Section III we present the physical systems along with the background geom-etry and topology, and derive the mean square velocity dispersion for motion of both acharged particle and an electric dipole under quantum vacuum electromagnetic fluctuationsin Minkowski spacetime with E and E flat 3 − space topologies. For the charged particle,we show that by comparing the time evolution of the velocity dispersions for a particle in E and E one can discriminate the orientable from the non-orientable topology. This resultis significant in that it makes apparent the strength of our approach to access orientabilitythrough electromagnetic quantum vacuum fluctuations. However, the answer to the centralquestion of the paper, namely how to locally probe the orientability of Minkowski 3 − spaceper se, comes about only in the study of the stochastic motions of an electric dipole. Inthis regard, the most important finding is that the spatial non-orientability can be locallyunveiled through the inversion pattern of velocity dispersion curves for a point electric dipoleunder quantum vacuum electromagnetic fluctuations. Section IV is dedicated to conclusions In mathematical terms, this amounts to identifying signatures of non-translational covering holonomiesthrough the motion of point-like particles under electromagnetic vacuum fluctuations. In the next section we present a summary of flat three-dimensional topologies and some of their maintopological properties. For a more detailed account on these topologies we recommend Refs. [30–33].
II. TOPOLOGICAL PREREQUISITES
Our primary aim in this section is to introduce the notation and give some basic def-initions and results concerning the topology of flat three-dimensional manifolds that areused throughout this paper. The spatial section M of the Minkowski spacetime manifold M = R × M is usually assumed to be the simply connected Euclidean space E . But itcan also be a multiply-connected quotient 3 − manifold of the form M = E / Γ, where E isthe covering space and Γ is a discrete and fixed-point-free group of discrete isometries (alsoreferred to as the holonomy group [8, 9]) acting freely on the covering space E [8].Possibly the best known example of three-dimensional quotient Euclidean manifold withnontrivial topology is the 3 − torus T = S × S × S = E / Γ, whose fundamental polyhedron(FP) is a parallelepiped with sides a, b, c (say), the opposite faces of which are identifiedthrough translations. In any multiply-connected quotient flat 3 − manifold the fundamentalpolyhedron tiles (tessellates) the whole infinite simply-connected covering space E . Thegroup Γ = Z × Z × Z consists of discrete translations associated with the face identification.The periodicities in the three independent directions are given by the circles S .In forming the quotient manifolds M an essential point is that they are obtained fromthe covering manifold E by identifying points that are equivalent under the action of thediscrete isometry group Γ. Hence, each point on the quotient manifold M represents allthe equivalent points on the covering space. The multiple connectedness leads to periodicboundary conditions on the covering manifold E (repeated cells) that are determined by theaction of the group Γ on the covering manifold. Clearly, different isometry groups Γ definedifferent topologies for M , which in turn give rise to different periodicity on the coveringmanifold (different mosaic of the covering space E ).Another important point in forming the flat quotient manifolds M is that every coveringisometry γ ∈ Γ can be expressed (in the covering space E ) through translation, rotation,reflection (flip) and combinations thereof. A screw motion, for example, is a combinationof a rotation R ( α, b u ) by an angle α around an axis b u , followed by a translation along avector L = L b w , say. A general glide reflection is combination of a reflection followed by a R is a topological space while E is a geometrical space, i.e. R endowed with the Euclidean metric. P = ( x, y, z ) γP = ( − x, y, z ) + (0 , , c ), where c is a constant.If c = 0 we have a simple reflection or flip.In dealing with metric manifolds in mathematical physics two concepts of homogeneityarise. Local homogeneity is a geometrical characteristic of metric manifolds. It is formulatedin terms of the action of the group of local isometries. In dealing with topological spaces, wehave the global homogeneity of topological nature. A way to characterize global homogeneityof the quotient manifolds is through distance functions. Indeed, for any x ∈ M the distancefunction ℓ γ ( x ) for a given discrete isometry γ ∈ Γ is defined by ℓ γ ( x ) = d ( x , γ x ) , (1)where d is the Euclidean metric defined on M . The distance function provides the length ofthe closed geodesic that passes through x and is associated with a holonomy γ . In globallyhomogeneous manifolds the distance function for any covering isometry γ is constant. Inglobally inhomogeneous manifolds, in contrast, the length of the closed geodesic associatedwith at least one γ is non-translational (screw motion or flip, for example) and depends onthe point x ∈ M , and then is not constant.When the distance between a point x and its image γ x (in the covering space) is aconstant for all points x then the holonomy γ is a translation, that is, all elements of thecovering group Γ in globally homogeneous spaces are translations. This means that in thesemanifolds the faces of the fundamental cells are identified through independent translations.In this paper, we shall consider the topologically nontrivial spaces E and E . The slabspace E is constructed by tessellating E by equidistant parallel planes, so it has only onecompact dimension associated with a direction perpendicular to those planes. Taking the x -direction as compact, one has that, with n x ∈ Z and a >
0, points ( x, y, z ) and ( x + n x a, y, z )are identified in the case of the slab space E . The slab space with flip E involves anadditional inversion of a direction orthogonal to the compact direction, that is, one directionin the tessellating planes is flipped as one moves from one plane to the next. Letting the flipbe in the y -direction, the identification of points ( x, y, z ) and ( x + n x a, ( − n x y, z ) definesthe E topology. In this way, the slab space E is globally homogeneous, whereas the slabspace with flip, E , is globally inhomogeneous since the covering group Γ contains a flip,which clearly is a non-translational discrete isometry.Orientability is another very important global (topological) property of a manifold that6easures whether one can choose consistently a clockwise orientation for loops in the mani-fold. A closed curve in a manifold M that brings a traveler back to the starting point mirror-reversed is called an orientation-reversing path. Manifolds that do not have an orientation-reversing path are called orientable , whereas manifolds that contain an orientation-reversingpath are non-orientable [35]. Most surfaces that we encounter, such as cylinders, planesand tori are orientable, whereas the M¨obius strip and Klein bottle are non-orientable sur-faces. For three-dimensional quotient manifolds, when the covering group Γ contains atleast an holonomy γ that is a reflection (flip) the corresponding quotient manifold is non-orientable. Thus, for example, the slab space is orientable while the slab space with flip isnon-orientable. Clearly non-orientable manifolds are necessarily globally inhomogeneous asthe covering group Γ contains a reflection, which obviously is a non-translational coveringholonomy.In Table I we collect the symbols and names used to refer to the manifolds together withthe number of compact independent dimensions and information concerning their globalhomogeneity and orientability. These are the three-dimensional manifolds with nontrivialtopologies that we shall be concerned with in this paper. Symbol Name Compact Dim. Orientable Homogeneous E Slab space 1 yes yes E Slab space with flip 1 no noTABLE I: Symbols and names of two multiply-connected flat orientable and non-orientable Eu-clidean quotient manifolds M = E / Γ along with the number of compact dimensions, orientabilityand global (topological) homogeneity.
Having set the stage for our investigation, in the next section we proceed to show that thetopological (global) non-orientability property of the spatial section of Minkowski spacetimemanifold can be locally probed through the study of the motions of a charged test particleor a point electric dipole under quantum vacuum fluctuations of the electromagnetic field.
III. NON-ORIENTABILITY FROM ELECTROMAGNETIC FLUCTUATIONS
Quantum vacuum fluctuations of the electromagnetic field in Minkowski spacetime withnontrivial spatial topologies give rise to stochastic motions of charged particles. In thissection, we address the main underlying question of this paper, which is whether thesefluctuations offer a suitable way of discovering a putative non-orientability of Minkowski7 − space. We take up this question through the study of stochastic motions of a chargedparticle and an electric dipole under electromagnetic quantum fluctuations in Minkowskispacetime with two inequivalent spatial topologies, namely the orientable slab space ( E )and the non-orientable slab space with flip ( E ). In the following we present the details ofour investigation and main results. A. NON-ORIENTABILITY WITH POINT CHARGED PARTICLE
We first consider a nonrelativistic test particle with charge q and mass m locally subjectedto vacuum fluctuations of the electric field E ( x , t ) in the topologically nontrivial spacetimemanifold equipped with the Minkowski metric η µν = diag(+1 , − , − , − E , but here we take for M each of the two manifolds inTable I.Locally, the motion of the charged test particle is determined by the Lorentz force. Inthe nonrelativistic limit the equation of motion for the point charge is d v dt = qm E ( x , t ) , (2)where v is the particle’s velocity and x its position at time t . We assume that on the timescales of interest the particle practically does not move (has a negligible displacement), sowe can ignore the time dependence of x . Thus, the particle’s position x is taken as constantin what follows [17, 23]. Assuming that the particle is initially at rest ( t = t = 0) the integration of Eq. (2) gives v ( x , t ) = qm Z t E ( x , t ′ ) dt ′ , (3)and the mean squared velocity, velocity dispersion or simply dispersion in each of the threeindependent directions i = x, y, z is given by h ∆ v i i = q m Z t Z t h E i ( x , t ′ ) E i ( x , t ′′ ) i dt ′ dt ′′ . (4)Following Yu and Ford [17], we assume that the electric field is a sum of classical E c andquantum E q parts. Because E c is not subject to quantum fluctuations and h E q i = 0, the The corrections arising from the inexactness of this assumption are negligible in the low velocity regime. By definition, h ∆ v ( x , t ) i ≡ h v ( x , t ) · v ( x , t ) i − h v ( x , t ) i · h v ( x , t ) i . h E i ( x , t ) E i ( x ′ , t ′ ) i in equation (4) involves only the quantum part of theelectric field [17].It can be shown [36] that locally h E i ( x , t ) E i ( x ′ , t ′ ) i = ∂∂x i ∂∂x ′ i D ( x , t ; x ′ , t ′ ) − ∂∂t ∂∂t ′ D ( x , t ; x ′ , t ′ ) (5)where, in Minkowski spacetime with M = E , the Hadamard function D ( x , t ; x ′ , t ′ ) is givenby D ( x , t ; x ′ , t ′ ) = 14 π (∆ t − | ∆ x | ) . (6)The subscript 0 indicates standard Minkowski spacetime, ∆ t = t − t ′ and | ∆ x | ≡ r is thespatial separation for topologically trivial Minkowski spacetime: r = ( x − x ′ ) + ( y − y ′ ) + ( z − z ′ ) . (7)In Minkowski spacetime with a topologically nontrivial spatial section, the spatial separa-tion r takes a different form that captures the periodic boundary conditions imposed on thecovering space E by the covering group Γ, which characterize the spatial topology. In con-sonance with Ref. [12], in Table II we collect the spatial separations for the two topologicallyinequivalent Euclidean spaces we shall address in this paper. Spatial topology Spatial separation r for Hadamard function E - Slab space ( x − x ′ − n x a ) + ( y − y ′ ) + ( z − z ′ ) E - Slab space with flip ( x − x ′ − n x a ) + ( y − ( − n x y ′ ) + ( z − z ′ ) TABLE II: Spatial separation in Hadamard function for the multiply-connected flat orientable( E ) and its non-orientable counterpart ( E ) quotient Euclidean manifolds. The topologicalcompact length is denoted by a . The numbers n x are integers and run from −∞ to ∞ . VELOCITY DISPERSION – SLAB SPACE WITH FLIP E For the sake of brevity, we give here only the detailed calculations of the components ofthe velocity dispersion (4) for a charged particle in Minkowski spacetime with E spatialtopology. The corresponding expressions for E spatial topology can then be easily obtainedfrom those for E as we show below. The reader is referred to Refs. [31–33] for pictures of the fundamental cells and further properties of allpossible three-dimensional Euclidean topologies.
9o obtain the correlation function for the electric field that is required to compute thevelocity dispersion (4) for slab space with flip E , we replace in Eq. (5) the Hadamardfunction D ( x , t ; x ′ , t ′ ) by its renormalized version given by [24] D ren ( x , t ; x ′ , t ′ ) = ∞ ′ X n x = −∞ π (∆ t − r ) (8)in which the prime indicates that the term of the sum with n x = 0 is omitted, ∆ t = t − t ′ ,and, from Table II, the spatial separation is r = ( x − x ′ − n x a ) + ( y − ( − n x y ′ ) + ( z − z ′ ) . (9)The term with n x = 0 in the sum (8) that defines the renormalized Hadamard function D ren ( x , t ; x ′ , t ′ ) has been subtracted out from the sum because it would give rise to aninfinite contribution to the velocity dispersion.Thus, from equation (5) the correlation functions h E i ( x , t ) E i ( x ′ , t ′ ) i = ∂∂x i ∂∂x ′ i D ren ( x , t ; x ′ , t ′ ) − ∂∂t ∂∂t ′ D ren ( x , t ; x ′ , t ′ ) (10)are then given by h E x ( x , t ) E x ( x ′ , t ′ ) i = ∞ ′ X n x = −∞ ∆ t + r − r x π [∆ t − r ] , (11) h E y ( x , t ) E y ( x ′ , t ′ ) i = ∞ ′ X n x = −∞ (3 − ( − n x ) ∆ t + (1 + ( − n x ) r − − n x r y π [∆ t − r ] , (12) h E z ( x , t ) E z ( x ′ , t ′ ) i = ∞ ′ X n x = −∞ ∆ t + r − r z π [∆ t − r ] , (13)where ∆ t = t − t ′ and r x = x − x ′ − n x a , r y = y − ( − n x y ′ , r z = z − z ′ , r = q r x + r y + r z . (14)The components of the velocity dispersion, given by Eq. (4), can then be computed withthe help of the integrals [24] I = Z t Z t dt ′ dt ′′ t − r ) = t r ( t − r ) (cid:26) rt − r − t ) ln ( r − t ) ( r + t ) (cid:27) (15)and J = Z t Z t dt ′ dt ′′ ∆ t (∆ t − r ) = t r ( t − r ) (cid:26) rt + ( r − t ) ln ( r − t ) ( r + t ) (cid:27) , (16)10n which ∆ t = t ′ − t ′′ .Inserting equations (11) to (16) into Eq. (4) and taking the coincidence limit x ′ → x wefind h ∆ v x i E = ∞ ′ X n x = −∞ q t π m r ( t − r ) (cid:26) rt (¯ r x + r ) + ( t − r )(3¯ r x − r ) ln ( r − t ) ( r + t ) (cid:27) , (17) h ∆ v y i E = ∞ ′ X n x = −∞ q t π m r ( t − r ) (cid:26) rt (¯ r y + (3 − ( − n x ) r )+( t − r )[3¯ r y − (3 − ( − n x ) r ] ln ( r − t ) ( r + t ) (cid:27) , (18) h ∆ v z i E = ∞ ′ X n x = −∞ q t π m r ( t − r ) (cid:26) rt (¯ r z + r ) + ( t − r )(3¯ r z − r ) ln ( r − t ) ( r + t ) (cid:27) , (19)where r = p n x a + 2(1 − ( − n x ) y , (20)¯ r x = r − r x = − n x a + 2(1 − ( − n x ) y , (21)¯ r y = (1 + ( − n x ) r − − n x (1 − ( − n x ) y = (1 + ( − n x ) n x a + 8(1 − ( − n x ) y , (22)¯ r z = r − r z = r , (23)with the use of equations (14) in the coincidence limit.As can be seen from Eqs. (20) – (23), the summands in Eqs (17) – (19) are even functionsof the sum index n x , therefore each sum equals twice the corresponding sum over positive n x only. Thus we can write the components of the velocity dispersion in the form h ∆ v x i E = ∞ X n =1 q t π m r n ( t − r n ) (cid:26) r n t ( ξ n + r n ) + ( t − r n )(3 ξ n − r n ) ln ( r n − t ) ( r n + t ) (cid:27) , (24) h ∆ v y i E = ∞ X n =1 q t π m r n ( t − r n ) (cid:26) r n t [ η n + (3 − ( − n ) r n ]+( t − r n )[3 η n − (3 − ( − n ) r n ] ln ( r n − t ) ( r n + t ) (cid:27) , (25) h ∆ v z i E = ∞ X n =1 q t π m r n ( t − r n ) (cid:26) r n t + 2( t − r n ) r n ln ( r n − t ) ( r n + t ) (cid:27) , (26)11here r n = p n a + 2(1 − ( − n ) y , (27) ξ n = − n a + 2(1 − ( − n ) y , (28) η n = (1 + ( − n ) n a + 8(1 − ( − n ) y . (29)Since r n , ξ n and η n depend on y , it follows that all components of the velocity dispersiondepend on the flipped coordinate which gives rise to non-orientability.Before proceeding to E topology, we discuss the topological Minkowskian limit for thevelocity dispersion in E . We begin by recalling that compact lengths associated withEuclidean quotient manifolds are not fixed. Different values of a correspond to different3 − manifolds with the same topology. By letting a → ∞ the topological Minkowskian limitfor the velocity dispersion is attained. From Eq. (27) it follows that letting a → ∞ amountsto letting r n → ∞ . For very large r n each term of the sum (24) consists of a fraction whosenumerator is dominated by a power of r n not bigger than the fourth (the logarithmic termtends to zero as r n → ∞ ) whereas the denominator becomes proportional to r n . Thereforeeach term of the sum vanishes in the limit a → ∞ and the dispersion h ∆ v x i is zero, whichis the topological Minkowskian limit for the velocity dispersion of a charged particle inMinkowski spacetime with spatial topology E . The same argument shows that the othercomponents of the velocity dispersion also vanish in the limit a → ∞ . This result makesit clear that the vanishing of the velocity dispersion in the topological Minkowskian limitalso holds for the non-orientable and globally inhomogeneous E topology. This extendsthe results obtained in Ref. [24] for globally homogeneous and orientable topologies, such as E for instance. VELOCITY DISPERSION – SLAB SPACE E The factors of ( − n x that appear in equations (12) and (18) arise from derivatives withrespect to y ′ in Eq. (10) contributed by the separation r given by Eq. (9). Hence, theresults for E are immediately obtained from those for E by simply replacing ( − n by 112verywhere in Eqs. (24) to (29). This leads to h ∆ v x i E = − q t π m ∞ X n =1 n a ln ( na − t ) ( na + t ) , (30) h ∆ v y i E = h ∆ v z i E = q t π m ∞ X n =1 (cid:26) tn a ( t − n a ) + 1 n a ln ( na − t ) ( na + t ) (cid:27) , (31)in agreement with the results obtained in [24, 25]. ANALYSIS OF THE RESULTS
The velocity dispersions are singular at t = r n , where r n = na for E whereas it isgiven by Eq. (27) for E . These singularities correspond to the time light takes to traveleach of the infinitely many distances r n that arise from the periodic boundary conditionsimposed to take account of the identifications ( x, y, z ) ↔ ( x + n x a, y, z ) or ( x, y, z ) ↔ ( x + n x a, ( − n x y, z ) in the covering space E . In the case in which E is split into twoidentical domains by the presence of a reflecting plane (nontrivial quotient orbifold), thevelocity dispersion exhibits only one singularity [17], which has been ascribed to the localproperties of the physical system. Accordingly, the introduction of a function of time toswitch on and off the interaction between the particle and the electromagnetic field has beensuggested to regularize the singularity [21, 23, 37]. This does not seem appropriate to copewith singularities that arise from global topological features of spacetime. Nevertheless,as remarked in [24], it may be possible to recast the switching function in terms of thetopological parameter a (and presumably n x ) in order to smooth out the divergences withdue regard to their topological origin. It is also conceivable that more realistic boundaryconditions than those brought about by the method of images may be necessary to smearout the singularities. It is further to be noted that for both E and E the dispersionsare negative for certain values of t . Therefore, an adequate regularization should render thevelocity dispersions free of singularities and also positive. These are thorny issues that lieoutside the purview of the present work, though.A very important question that arises at this point is what then we ultimately learnfrom the above calculations regarding the local test of spatial orientability by studying themotions of a charged particle under electromagnetic quantum fluctuations. In other words,what these fluctuations are teaching us about orientability through velocity dispersions,equations (24) – (29) for the non-orientable E topology and equations (30) – (31) for13he orientable E space topology. Let us now discuss this capital issue. Clearly the chiefconclusion can only be extracted through comparisons between the stochastic motions ofthe charged test particles lying in space manifolds with each of the two topologies. Inthis regard, a first difficulty one encounters is how to make a proper comparison because E is globally homogenous whereas E is not (cf. Table I). This means that the velocitydispersion does not depend on the particle’s position for E , but it does when the particlelies in a space with the globally inhomogeneous topology E . The functional dependenceof the dispersion on the particle’s position coordinates in these manifolds makes apparentthe first difficulty. Indeed, the components of the velocity dispersions (24) – (29) for E depend on the y -coordinate, while the components (30) – (31) for E do not. Thus, onehas to suitably choose the point P = ( x, y, z ) in E for the particle’s position in order tomake a proper comparison between the dispersions curves for the topologically homogeneous E and the topologically inhomogeneous E manifolds. From the identification of ( x, y, z )and ( x + n x a, ( − n x y, z ) that defines the E topology, clearly a suitable way to freeze outthe global inhomogeneity degree of freedom, and thus isolate the non-orientability effect,is by choosing as the particle’s position the point P = ( x, , z ). Since our chief concern isorientability, in all figures in this paper but one (Fig. 1) we choose this point as the particle’sposition when dealing with E topology.Having circumvented this particle position difficulty related to the topological inhomo-geneity of E , we illustrate in Fig. 1 how it affects the x -component curves of the normalizedvelocity dispersion h ∆ v ( x , t ) i n ≡ m q h ∆ v ( x , t ) i . (32)For two particle’s positions, one with y = 0 and another with y = 1 / a = 1, in E one has different velocity dispersions curves. This shows that the velocitydispersion is able to capture the non-homogeneity of topological origin as has been indicatedin [24]. By way of clarification, Figures 1 and 2 arise from Eqs. (24)–(29) as well as (30)and (31) with compact length a = 1 and n ranging from 1 to 50.Figure 1 also shows that for the particle’s position P in E the dispersion curves coincidewith those for a generic point in globally homogenous E . This shows that for P = ( x, , z ),where the global inhomogeneity degree of freedom is frozen, the x -component of the disper-sion cannot be used to distinguish between the two orientable and non-orientable 3 − spaces.With different pattern of curves, the same difficulty holds true for the z -components of the14 IG. 1: Time evolution of the x -component of the normalized velocity dispersion h ∆ v ( x , t ) i n fora test particle with mass m and charge q in Minkowski spacetime with spatial section endowedwith the non-orientable and globally inhomogeneous E and orientable E topologies, both withcompact length a = 1. We show one curve for the globally homogeneous E (dashed line) andtwo curves for E : dotted and solid lines, for the particle at the positions P = ( x, , z ) and P = ( x, / , z ), respectively. The figure illustrates the topological inhomogeneity of E , and showsthat when the degree of inhomogeneity is frozen the dispersion curves for E [for the particle at P = ( x, , z )] and E [for the particle at generic P = ( x, y, z ) ] coincide. dispersion for E and E , but we do not show this figure for the sake of brevity. We notethat this result on these two components of the dispersion are contained in our equations.Indeed, for the particle’s position at P = ( x, , z ) for E , taking account of (27) – (29) itis straightforward to show that Eqs. (24) and (26) reduce, respectively, to (30) and (31).However, it should be noticed that even for P = ( x, , z ) the y -component of the dispersion h ∆ v y i E does not reduce to h ∆ v y i E . This means that in order to extract informationregarding orientability from the motion of a charged particle under electromagnetic fluctu-ations the proper comparison should be between the y -components of the dispersion, as wedo in Fig. 2. This was to be expected from the outset since the reflection holonomy for E is in the y -direction (cf. Table II).Figure 2 displays the y -components of the velocity dispersion of a charged test particle inMinkowski space whose spatial section has E (orientable) and E [non-orientable, P =( x, , z )] topologies. Clearly the chief conclusion it that the component along the direction ofthe flip (cf. Table II) can be used to find out whether the particle lies in Minkowski spacetimewith orientable or non-orientable 3 − space. Figure 2 also shows different dispersion curvesfor E and E which, in both cases, repeat themselves periodically. For the two topologies15 IG. 2: Time evolution of the y -component of the normalized velocity dispersion for a chargedtest particle in Minkowski spacetime with spatial section endowed with the orientable E andnon-orientable E topologies, both with compact length a = 1. We show curves for E (dashedline) and for E with the particle at P = ( x, , z ) (dotted line). The dispersion curves exhibitsimilar periodic patterns for time intervals of the order of one, but it is possible to distinguish thetwo topologies by contrasting the y -component of their velocity dispersions. the overall patterns of the velocity dispersion curves are similar.Although the above result is valuable to the extent that it makes clear the strength of ourapproach to access orientability through electromagnetic vacuum fluctuations, it demands acomparison between the dispersion curves for the two given spatial manifolds for a decisionabout orientability. Thus, it does not provide a conclusive answer to the central question ofthis paper, namely how to locally probe the orientability of the spatial section of Minkowskispacetime in itself through a physical experiment with electromagnetic vacuum fluctuations.Given the directional properties of a point electric dipole, a pertinent question that emergeshere is whether it would be more effective to use it for testing the individual non-orientabilityof a generic 3 − space. If so, after a suitable renormalization is worked out, we could envisagea local experiment to probe the orientability of a 3 − space per se. In the next subsection weshall investigate this remarkable possibility. B. NON-ORIENTABILITY WITH POINT ELECTRIC DIPOLE
A noteworthy outcome of the previous section is that the time evolution of the velocitydispersion for a charged particle can be used to locally differentiate an orientable ( E )from a non-orientable ( E ) spatial section of Minkowski spacetime. However, it cannotbe used to decide whether a given 3 − space manifold is or not orientable. In this way, it16annot be taken as a definite answer to our central question about the spatial orientabilityof Minkowski spacetime. So, a question that naturally arises here is whether the velocitydispersion of a different type of point-like particle could provide a suitable local signatureof non-orientability. As a point electric dipole has directional properties, one would expectthat its velocity dispersion could potentially bring about unequivocal information regardingnon-orientability. To examine this issue we now turn our attention to topologically inducedmotions of an electric dipole under quantum vacuum electromagnetic fluctuations.Newton’s second law for a point electric dipole of mass m in an external electric fieldreads m d v dt = p · ∇ E ( x , t ) (33)where p is the electric dipole moment. With the same hypotheses as for the point chargeand assuming the dipole is initially at rest, integration of Eq. (33) yields v ( x , t ) = 1 m p j Z t ∂ j E ( x , t ′ ) dt ′ (34)with ∂ j = ∂/∂x j and summation over repeated indices implied.The mean squared speed in each of the three independent directions i = x, y, z is givenby h ∆ v i i = p j p k m Z t Z t h (cid:0) ∂ j E i ( x , t ′ ) (cid:1)(cid:0) ∂ k E i ( x , t ′′ ) (cid:1) i dt ′ dt ′′ , (35)which can be conveniently rewritten as h ∆ v i i = lim x ′ → x p j p k m Z t Z t ∂ j ∂ ′ k h E i ( x , t ′ ) E i ( x ′ , t ′′ ) i dt ′ dt ′′ (36)where ∂ ′ k = ∂/∂x ′ k .Now we proceed to the computation of the velocity dispersion for a point dipole in spaces E and E . The space E has two topologically conspicuous directions: the compact x -direction and the flip y -direction associated with the non-orientability of E . To probe thenon-orientability of E by means of stochastic motions, it seems most promising to choosea dipole oriented in the y -direction, since the orientation of the dipole would also be flippedupon every displacement by the topological length a along the compact direction. Indeed,it is for a dipole oriented in the flip direction that the effect of the non-orientability is mostnoticeable, as we show in the following.For a dipole oriented along the y -axis we have p = (0 , p,
0) and h ∆ v x i ( y ) = lim x ′ → x p m Z t Z t ∂ y ∂ y ′ h E x ( x , t ′ ) E x ( x ′ , t ′′ ) i dt ′ dt ′′ , (37)17here the superscript within parentheses indicates the dipole’s orientation. With the helpof Eq. (11) the x -component of the velocity dispersion for the slab space with flip E takesthe form h ∆ v x i ( y ) E = lim x ′ → x p π m ∞ ′ X n x = −∞ Z t Z t dt ′ dt ′′ ∂ y ∂ y ′ ∆ t + r − r x (∆ t − r ) . (38)with r defined by Eq. (9) and ∆ t = t ′ − t ′′ , while r x is given by Eq. (14). Making use of ∂ y ∂ y ′ ∆ t + r − r x (∆ t − r ) = − − n x (cid:20) t − r ) + 3 r − r x + 6 r y (∆ t − r ) + 24 ( r − r x ) r y (∆ t − r ) (cid:21) . (39)we find h ∆ v x i ( y ) E = − p π m ∞ ′ X n x = −∞ ( − n x (cid:26) I + 3( r − r x + 6 r y ) I + 24( r − r x ) r y I (cid:27) , (40)where, with ∆ t = t ′ − t ′′ , I = I = Z t Z t dt ′ dt ′′ (∆ t − r ) = t (cid:20) tr ( t − r ) + 3 r ln ( r − t ) ( r + t ) (cid:21) , (41) I = Z t Z t dt ′ dt ′′ (∆ t − r ) = 16 r ∂I ∂r = t (cid:20) t (9 r − t ) r ( t − r ) − r ln ( r − t ) ( r + t ) (cid:21) , (42) I = Z t Z t dt ′ dt ′′ (∆ t − r ) = 18 r ∂I ∂r = t (cid:20) t (57 t − r t + 87 r ) r ( t − r ) + 105 r ln ( r − t ) ( r + t ) (cid:21) . (43)Similar calculations lead to h ∆ v y i ( y ) E = − p π m ∞ ′ X n x = −∞ ( − n x (cid:26) (5 − − n x ) I + 6[ r + (7 − − n x ) r y ] I +48[ r − ( − n x r y ] r y I (cid:27) (44)and h ∆ v z i ( y ) E = − p π m ∞ ′ X n x = −∞ ( − n x (cid:26) I + 3( r + 6 r y ) I + 24 r r y I (cid:27) . (45)Since the coincidence limit x ′ → x has been taken, it follows from Eq. (14) that in Eqs. (40)to (45) one must put r = p n x a + 2(1 − ( − n x ) y , r x = n x a , r y = 2(1 − ( − n x ) y . (46)It can be immediately checked that, as for the point charge, in the Minkowskian limit( a → ∞ ) the velocity dispersion for a dipole is zero.18 IG. 3: Time evolution of the x -component of the normalized velocity dispersion h ∆ v ( x , t ) i n for apoint electric dipole oriented in the flip y -direction in Minkowski spacetime with orientable E andnon-orientable E spatial topologies, both with compact length a = 1. The solid and dashed linesstand, respectively, for the dispersion curves for a dipole in 3 − space with E and E topologies.For the globally inhomogeneous topology E the dipole is at P = ( x, , z ), thus freezing out thetopological inhomogeneity. Both dispersion curves show a periodicity, but the curve for E exhibitsa different kind of periodicity characterized by a distinctive inversion pattern. Non-orientability isresponsible for this pattern of successive inversions, which is absent in the dispersion curve for theorientable E . For the slab space E the components of the dipole velocity dispersion are obtained fromthose for E by setting r x = r , r y = 0, and replacing ( − n x by 1 everywhere. Therefore,we have h ∆ v x i ( y ) E = − p π m ∞ ′ X n x = −∞ I , (47) h ∆ v y i ( y ) E = − p π m ∞ ′ X n x = −∞ ( I + 3 r I ) , (48) h ∆ v z i ( y ) E = − p π m ∞ ′ X n x = −∞ (2 I + 3 r I ) , (49)in which r = | n x | a . ANALYSIS OF THE RESULTS
Now we ask ourselves what these fluctuations can reveal about spatial orientabilitythrough the velocity dispersion equations (40) – (45) for the dipole in the non-orientable3 − space with E topology, and equations (47) – (49) for the dipole in the orientable 3 − spacewith E topology. In the remainder of this section we shall focus on this fundamental ques-tion. 19 IG. 4: Time evolution of the y -component of the normalized velocity dispersion for a point electricdipole oriented in the y -direction under the same conditions as those of Fig. 3. The velocitydispersion curve for E also displays a characteristic inversion pattern but which is different fromthe one for the x -component shown in Fig. 3. For the y -component of the velocity dispersion thesignature of non-orientability can be recognized in the pattern of successive upward and downward“horns” formed by the dashed curve. We begin by noting that the expressions for the components of the dispersion for E and E topologies are too involved to lend themselves to a straightforward interpretation.Nevertheless, something significant can be said: for a dipole located at P = ( x, , z ) allcomponents of the velocity dispersion for E are different from those for E because eachsummand in Eqs. (40), (44) and (45) contains the prefactor ( − n x which is absent fromthe corresponding Eqs. (47) – (49) for E . Since not much further can be read from ourequations, in order to demonstrate our main result, which is ultimately stated in terms ofpatterns of curves for the dispersion, we begin by plotting figures for the components of thevelocity dispersion. Figures 3 to 5 come from Eqs. (40) – (45) as well as (47) – (49), withthe topological length a = 1 and n x = 0 ranging from −
50 to 50. The normalized velocitydispersion in these figures is defined by h ∆ v ( x , t ) i n ≡ m p h ∆ v ( x , t ) i . (50)In the three figures the solid lines stand for the dispersion curves for the dipole in Minkowskispacetime with E orientable spatial topology, whereas the dashed lines indicate dispersioncurves for the dipole located at P = ( x, , z ) in a 3 − space with E non-orientable topology.In the case of the x -component, the time evolution curves of the dispersion for E and E , shown in Fig. 3, present a common periodicity but clearly with distinguishable patterns.The dispersion curve for E displays a distinctive sort of periodicity characterized by an20 IG. 5: Time evolution of the z -component of the normalized velocity dispersion for a pointelectric dipole oriented in the y -direction under the same conditions as those of Fig. 3. For the z -component of the velocity dispersion the non-orientability of E manifests itself by an inversionpattern similar to the one for the y -component shown in Fig. 4, namely the pattern of alternatingupward and downward “horns” formed by the dashed curve. inversion pattern. Non-orientability gives rise to this pattern of consecutive inversions, whichis not present in the dispersion curve for the orientable E .The differences become more salient when one considers the other two components ofthe velocity dispersion, shown in Figs. 4 and 5. For both of these components, the non-orientability of E is disclosed by an inversion pattern whose structure is more strikingthan the one for the x -component. The dispersion curves for E form a pattern of alter-nating upward and downward “horns”, making the non-orientability of E unmistakablyidentifiable.From the above analysis of Figs. 3 to 5 as compared with the corresponding analysis ofFig. 2, we see that the point dipole is a much more efficient non-orientability probe thanthe point particle. Furthermore, the characteristic inversion pattern exhibited by the dipoledispersion curves makes it possible to identify the non-orientability of E by itself, withouthaving to make a comparison with the dispersion curves for its orientable counterpart. Wehave checked, although we do not show the calculations, that for a dipole located at P =( x, , z ) and oriented either in the x -direction or in the z -direction, only the y -component ofthe dispersion for E is different from the one for E . Thus, it is for a dipole oriented inthe flip direction that the non-orientability of E is most sharply exposed.This discussion leads us to the remarkable conclusion that it is possible to unveil apresumed spatial non-orientability by local means, namely by the stochastic motions of21oint particles caused by quantum electromagnetic vacuum fluctuations. If the motion ofa point electric dipole is taken as probe, non-orientability can be intrinsically discerned bythe inversion pattern of the dipole’s velocity dispersion curves. IV. CONCLUSIONS AND FINAL REMARKS
In general relativity and quantum field theory spacetime is modeled as a differentiablemanifold, which is a topological space equipped with an additional differential structure. Ori-entability is an important topological property of spacetime manifolds. It is often assumedthat the spacetime manifold is orientable and, additionally, that it is separately time andspace orientable. The theoretical arguments usually offered to assume orientability combinethe space-and-time universality of local physical experiments with physically well-defined(thermodynamically, for example) local arrow of time, violation of charge conjugation andparity (CP violation) and CPT invariance [38–40]. One can certainly use such reasonings insupport of the standard assumptions on the global structure of spacetime. Nevertheless,it is reasonable to expect that the ultimate answer to questions regarding the orientabilityof spacetime should rely on cosmological observations or local experiments, or should evencome from a fundamental theory of physics.In the physics at daily and even astrophysical length and time scales, we do not findany sign or even hint of non-orientability. This being true, the remaining open questionis whether the physically well-defined local orientations can be extended continuously tocosmological and/or microscopic scales.At the cosmological scale, one would think at first sight that to disclose spatial ori-entability one would have to make a trip around the whole 3 − space to check for orientation-reversing paths. Since such a global journey across the Universe is not feasible one mightthink that spatial orientability cannot be probed globally. However, a determination of thespatial topology through the so-called circles in the sky [43], for example, would bring outas a bonus an answer to the 3 − space orientability problem at cosmological scale. This universality can be looked upon as a topological assumption of global homogeneity, which in turnrules out spatial non-orientability of 3 − space. See Ref. [41] for a dissenting point of view, and also the related Ref. [42]. In the searches for these circles so far undertaken, including the ones carried out by the Planck Collabo-ration [5, 6], no statistically significant pairs of matching circles have been found (see Ref. [4] for the most
22n this paper we have addressed the question as to whether electromagnetic quantumvacuum fluctuations can be used to bring out on a microscopic scale the spatial orientabilityof Minkowski spacetime. To this end, we have studied the stochastic motions of point-like particles under quantum electromagnetic fluctuations in Minkowski spacetime with theorientable slab space ( E ) and the non-orientable slab space with flip ( E ) topologies (cf.Tables I and II).For a point charged particle, we have derived analytic expressions for the velocity dis-persion, namely Eqs. (24) – (29) for E space topology, and Eqs. (30) – (31) for E spacetopology. From these equations we have made Fig. 1 and Fig. 2. Using these equationsand figures we have shown that it is possible to distinguish the orientable from the non-orientable topology by contrasting the time evolution of the respective velocity dispersionsalong the flip direction of E . In spite of being a significant result in that it makes apparentthe power of our approach to access orientability through electromagnetic quantum vacuumfluctuations, it is desirable to be able to decide about the orientability of a given spatialmanifold in itself. In this way, the results concerning the motion of a charged particle donot afford a conclusive answer to the central question posed in this paper, which is how toprobe the orientability of Minkowski 3 − space per se through a local physical experimentwith electromagnetic vacuum fluctuations.To tackle the central question, motivated by a dipole’s directional properties, we havethen examined whether the study of stochastic motion of a point-like electric dipole wouldbe more effective for testing the non-orientability of a generic 3 − space individually, i.e.without having to make a comparison of the results for an orientable space with those forits non-orientable counterpart.To this end, we have derived the velocity dispersion equations (40) – (45) for the dipoleoriented in the flip direction in the non-orientable 3 − space with E topology, and equa-tions (47) – (49) for the dipole in the orientable 3 − space with E topology. From theseequations we have calculated and plotted Figures 3 to 5. As a result of the detailed analysisof these equations and figures we have found that there exists a characteristic inversion pat-tern exhibited by the velocity dispersion curves in the case of E , making apparent that the extensive search yet, and also references therein for the other searches). These negative observationalresults, however, are not sufficient to exclude the possibility that the Universe has a detectable (orientableor non-orientable) nontrivial topology (see Ref. [7] for some limits of these searches). E can be identified per se. The inversion pattern of the velocity dis-persion curves for the dipole is ultimately a signature of the reflection holonomy, and oughtto be present in the dispersion curves for the dipole in all remaining seven non-orientabletopologies with flip, namely the four Klein spaces ( E to E ) and those in the chimney-with-flip class ( E to E ). Clearly the inversion patterns for the electric dipole changewith the associated topology: different topologies give rise to velocity dispersion curves withdistinct inversion patterns.Observation of physical phenomena and experiments are fundamental to our understand-ing the physical world. Our results make it clear that it is possible to locally unveil spatialnon-orientability through the stochastic motions of point-like particles under electromagneticquantum vacuum fluctuations. The present paper is a step on the way to a conceivable ex-periment involving these fluctuations to locally probe the spatial orientability of Minkowskiempty spacetime.
Acknowledgments
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