Dislocations in the Spacetime Continuum: Framework for Quantum Physics

Transcription

1 Issue 4 October PROGRESS IN PHYSICS Voume Disocations in the Spacetime Continuum: Framework for Quantum Physics Pierre A. Miette PierreAMiette@aumni.uottawa.ca, Ottawa, Canada This paper provides a framework for the physica description of physica processes at the quantum eve based on disocations in the spacetime continuum within STCED Spacetime Continuum Eastodynamics. In this framework, photon and partice sefenergies and interactions are mediated by the strain energy density of the disocations, repacing the roe payed by virtua partices in QED. We postuate that the spacetime continuum has a granuarity characterized by a ength b corresponding to the smaest STC eementary Burgers disocation-dispacement vector. Screw disocations corresponding to transverse dispacements are identified with photons, and edge disocations corresponding to ongitudina dispacements are identified with partices. Mixed disocations give rise to wave-partice duaity. The strain energy density of the disocations are cacuated and proposed to expain the QED probem of mass renormaization. 1 Introduction In a previous paper [1], the deformabe medium properties of the spacetime continuum STC ed us to expect disocations, discinations and other defects to be present in the STC. The effects of such defects woud be expected to manifest themseves mosty at the microscopic eve. In this paper, we present a framework to show that disocations in the spacetime continuum are the basis of quantum physics. This paper ays the framework to deveop a theory of the physica processes that underie Quantum Eectrodynamics QED. The theory does not resut in the same formaism as QED, but rather resuts in an aternative formuation that provides a physica description of physica processes at the quantum eve. This framework aows the theory to be feshed out in subsequent investigations. 1.1 Eastodynamics of the Spacetime Continuum As shown in a previous paper [1], Genera Reativity eads us to consider the spacetime continuum as a deformabe continuum, which aows for the appication of continuum mechanica methods and resuts to the anaysis of its deformations. The Eastodynamics of the Spacetime Continuum STCED [1 7] is based on anayzing the spacetime continuum within a continuum mechanica and genera reativistic framework. The combination of a spacetime continuum deformations resuts in the geometry of the STC. The geometry of the spacetime continuum of Genera Reativity resuting from the energy-momentum stress tensor can thus be seen to be a representation of the deformation of the spacetime continuum resuting from the strains generated by the energy-momentum stress tensor. As shown in [1], for an isotropic and homogeneous spacetime continuum, the STC is characterized by the stress-strain reation µ ε µν λ g µν ε = T µν 1 where T µν is the energy-momentum stress tensor, ε µν is the resuting strain tensor, and ε = ε α α is the trace of the strain tensor obtained by contraction. The voume diatation ε is defined as the change in voume per origina voume [8, see pp ] and is an invariant of the strain tensor. λ and µ are the Lamé eastic constants of the spacetime continuum: µ is the shear moduus and λ is expressed in terms of κ, the buk moduus: λ = κ µ / 3 in a four-dimensiona continuum. As shown in [1], energy propagates in the spacetime continuum as wave-ike deformations which can be decomposed into diatations and distortions. Diatations invove an invariant change in voume of the spacetime continuum which is the source of the associated rest-mass energy density of the deformation. On the other hand, distortions correspond to a change of shape of the spacetime continuum without a change in voume and are thus massess. Thus deformations propagate in the spacetime continuum by ongitudina diatation and transverse distortion wave dispacements. This provides a natura expanation for wave-partice duaity, with the transverse mode corresponding to the wave aspects of the deformation and the ongitudina mode corresponding to the partice aspects of the deformation [7]. The rest-mass energy density of the ongitudina mode is given by [1, see Eq.3] ρc = 4 κ ε 4 where ρ is the rest-mass density, c is the speed of ight, κ is the buk moduus of the STC the resistance of the spacetime continuum to diatations, and ε is the voume diatation. This equation demonstrates that rest-mass energy density arises from the voume diatation of the spacetime continuum. Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics 87

2 Voume PROGRESS IN PHYSICS Issue 4 October The rest-mass energy is equivaent to the energy required to diate the voume of the spacetime continuum. It is a measure of the energy stored in the spacetime continuum as mass. The voume diatation is an invariant, as is the rest-mass energy density. This is an important resut as it demonstrates that mass is not independent of the spacetime continuum, but rather mass is part of the spacetime continuum fabric itsef. Mass resuts from the diatation of the STC in the ongitudina propagation of energy-momentum in the spacetime continuum. Matter does not warp spacetime, but rather, matter is warped spacetime i.e. diated spacetime. The universe consists of the spacetime continuum and energy-momentum that propagates in it by deformation of its STC structure. Note that in this paper, we denote the STCED spacetime continuum constants κ, λ, µ, ρ with a diacritica mark over the symbos to differentiate them from simiar symbos used in other fieds of Physics. This aows us to retain existing symbos such as µ for the eectromagnetic permeabiity of free space, compared to the Lamé eastic constant µ used to denote the spacetime continuum shear moduus. 1. Defects in the Spacetime Continuum As discussed in [1], given that the spacetime continuum behaves as a deformabe medium, there is no reason not to expect disocations, discinations and other defects to be present in the STC. Disocations in the spacetime continuum represent the fundamenta dispacement processes that occur in its structure. These fundamenta dispacement processes shoud thus correspond to basic quantum phenomena and provide a framework for the description of quantum physics in STCED. Defect theory has been the subject of investigation since the first haf of the XX th century and is a we-deveoped discipine in continuum mechanics [9 14]. The recent formuation of defects in soids in based on gauge theory [15, 16]. The ast quarter of the XX th century has seen the investigation of spacetime defects in the context of string theory, particuary cosmic strings [17, 18], and cosmic expansion [, 1]. Teeparae spacetime with defects [18,, 3] has resuted in a differentia geometry of defects, which can be foded into the Einstein-Cartan Theory ECT of gravitation, an extension of Einstein s theory of gravitation that incudes torsion [19, ]. Recenty, the phenomenoogy of spacetime defects has been considered in the context of quantum gravity [4 6]. In this paper, we investigate disocations in the spacetime continuum in the context of STCED. The approach foowed ti now by investigators has been to use Einstein-Cartan differentia geometry, with disocations transationa deformations impacting curvature and discinations rotationa deformations impacting torsion. The disocation itsef is modeed via the ine eement ds [17]. In this paper, we investigate spacetime continuum disocations using the underying dispacements u ν and the energy-momentum stress tensor. We thus work from the RHS of the genera reativistic equation the stress tensor side rather than the LHS the geometric tensor side. It shoud be noted that the genera reativistic equation used can be the standard Einstein equation or a suitaby modified version, as in Einstein-Cartan or Teeparae formuations. In Section of this paper, we review the basic physica characteristics and dynamics of disocations in the spacetime continuum. The energy-momentum stress tensor is considered in Section.. This is foowed by a detaied review of stationary and moving screw and edge disocations in Sections 3, 4 and 5, aong with their strain energy density as cacuated from STCED. The framework of quantum physics, based on disocations in the spacetime continuum is covered in Section 6. Screw disocations in quantum physics are considered in Section 6. and edge disocations are covered in Section 6.3. Section 7 covers disocation interactions in quantum physics, and Section 8 provides physica expanations of QED phenomena provided by disocations in the STC. Section 9 summarizes the framework presented in this paper for the deveopment of a physica description of physica processes at the quantum eve, based on disocations in the spacetime continuum within the theory of the Eastodynamics of the Spacetime Continuum STCED. Disocations in the Spacetime Continuum A disocation is characterized by its disocation-dispacement vector, known as the Burgers vector, b µ in a four-dimensiona continuum, defined positive in the direction of a vector ξ µ tangent to the disocation ine in the spacetime continuum [14, see pp.17 4]. A Burgers circuit encoses the disocation. A simiar reference circuit can be drawn to encose a region free of disocation see Fig. 1. The Burgers vector is the vector required to make the Burgers circuit equivaent to the reference circuit see Fig.. It is a measure of the dispacement between the initia and fina points of the circuit due to the disocation. It is important to note that there are two conventions used to define the Burgers vector. In this paper, we use the convention used by Hirth [14] referred to as the oca Burgers vector. The oca Burgers vector is equivaenty given by the ine integra b µ u µ = ds 5 s C taken in a right-handed sense reative to ξ µ, where u µ is the dispacement vector. A disocation is thus characterized by a ine direction ξ µ and a Burgers vector b µ. There are two types of disocations: an edge disocation for which b µ ξ µ = and a screw disocation which can be right-handed for which b µ ξ µ = b, or efthanded for which b µ ξ µ = b, where b is the magnitude of the Burgers vector. Arbitrary mixed disocations can be decom- 88 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics

3 Issue 4 October PROGRESS IN PHYSICS Voume Fig. 1: A reference circuit in a region free of disocation, S: start, F: finish Fig. : A disocation showing the Burgers vector b µ, direction vector ξ µ which points into the paper and the Burgers circuit, S: start, F: finish S F F S ξ µ b µ posed into a screw component, aong vector ξ µ, and an edge component, perpendicuar to vector ξ µ. The edge disocation was first proposed by Orowan [7], Poanyi [8] and Tayor [9] in 1934, whie the screw disocation was proposed by Burgers [3] in In this paper, we extend the concept of disocations to the eastodynamics of the spacetime continuum. Edge disocations correspond to diatations ongitudina dispacements and hence have an associated rest-mass energy, whie screw disocations correspond to distortions transverse dispacements and are massess [1]..1 Disocation dynamics In three-dimensiona space, the dynamic equation is written as [31, see pp ], T i j, j = X i ρ ü i 6 where ρ is the spacetime continuum density, X i is the voume or body force, the comma, represents differentiation and u denotes the derivative with respect to time. Substituting for ε µν = 1 uµ;ν u ν;µ in 1, using and u µ ;µ = ε µ µ = ε in this equation, we obtain µ u i µ λ ε ;i = X i ρ ü i 7 which, upon converting the time derivative to indicia notation and rearranging, is written as µ u i ρ c u i, µ λ ε ;i = X i. 8 We use the arrow above the naba symbo to indicate the 3- dimensiona gradient whereas the 4-dimensiona gradient is written with no arrow. Using the reation [1] µ c = 9 ρ in the above, 8 becomes µ u i u i, µ λ ε ;i = X i 1 and, combining the space and time derivatives, we obtain µ u i µ λ ε ;i = X i. 11 This equation is the space portion of the STCED dispacement wave equation 51 of [1] µ u ν µ λ ε ;ν = X ν. 1 Hence the dynamics of the spacetime continuum is described by the dynamic equation 1, which incudes the acceerations from the appied forces. In this anaysis, we consider the simper probem of disocations moving in an isotropic continuum with no voume force. Then 1 becomes µ u ν µ λ ε ;ν =, 13 where is the four-dimensiona operator and the semi-coon ; represents covariant differentiation. Separating u ν into its ongitudina irrotationa component u ν and its transverse soenoida component uν using the Hemhotz theorem in four dimensions [3] according to u ν = u ν uν, 14 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics 89

4 Voume PROGRESS IN PHYSICS Issue 4 October 1 can be separated into a screw disocation dispacement transverse equation by In terms of the stress tensor, the inverse of is given µ u ν = 15 and an edge disocation dispacement ongitudina equation u ν = µ λ µ ε ;ν. 16. The energy-momentum stress tensor The components of the energy-momentum stress tensor are given by [33]: T = H T j = s j T i = g i T i j = σ i j 17 where H is the tota energy density, s j is the energy fux vector, g i is the momentum density vector, and σ i j is the Cauchy stress tensor which is the i th component of force per unit area at x j. From the stress tensor T µν, we can cacuate the strain tensor ε µν and then cacuate the strain energy density of the disocations. As shown in [3], for a genera anisotropic continuum in four dimensions, the spacetime continuum is approximated by a deformabe inear eastic medium that obeys Hooke s aw [31, see pp. 5 53] E µναβ ε αβ = T µν 18 where E µναβ is the eastic modui tensor. For an isotropic and homogeneous medium, the eastic modui tensor simpifies to [31]: E µναβ = λ g µν g αβ µ g µα g νβ g µβ g να. 19 For the metric tensor g µν, we use the fat spacetime diagona metric η µν with signature as the STC is ocay fat at the microscopic eve. Substituting for 19 into 18 and expanding, we obtain T = λ µ ε λ ε 11 λ ε λ ε 33 T 11 = λ ε λ µ ε 11 λ ε λ ε 33 T = λ ε λ ε 11 λ µ ε λ ε 33 T 33 = λ ε λ ε 11 λ ε λ µ ε 33 T µν = µ ε µν, µ ν. ε 1 [ = 3 λ µ T 4 µ λ µ λ T 11 T T 33 ] ε 11 1 [ = 3 λ µ T 11 4 µ λ µ λ T T T 33 ] ε 1 [ = 3 λ µ T 4 µ λ µ λ T T 11 T 33 ] ε 33 1 [ = 3 λ µ T 33 4 µ λ µ λ T T 11 T ] ε µν = 1 µ T µν, µ ν. 1 where T i j = σ i j. We cacuate ε = ε α α from the vaues of 1. Using η µν, 3 and T α α = ρc from [], we obtain 4 as required. This confirms the vaidity of the strain tensor in terms of the energy-momentum stress tensor as given by 1. Esheby [34 36] introduced an eastic fied energy-momentum tensor for continuous media to dea with cases where defects such as disocations ead to changes in configuration. The dispacements u ν are considered to correspond to a fied defined at points x µ of the spacetime continuum. This tensor was first derived by Morse and Feshback [37] for an isotropic eastic medium, using dyadics. The energy fux vector s j and the fied momentum density vector g i are then given by [34, 37]: s j = u k σ k j g i = ρ u k,i u k b i j = L δ i j u k,i σ k j where ρ is the density of the medium, in this case the spacetime continuum, L is the Lagrangian equa to K W where W is the strain energy density and K is the kinetic energy density H = K W, and b i j is known as the Esheby stress tensor [38, see p. 7]. If the energy-momentum stress tensor is symmetric, then g i = s i. In this paper, we consider the case where there are no changes in configuration, and use the energy-momentum stress tensor given by 17 and. 9 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics

5 Issue 4 October PROGRESS IN PHYSICS Voume Fig. 3: A stationary screw disocation in cartesian x, y, z and cyindrica poar r, θ, z coordinates [14, see p. 6]. We now consider the previous screw disocation, moving a- ong the x-axis, parae to the disocation, at a constant speed v x = v. Equation 13 then simpifies to the wave equation for massess transverse shear waves for the dispacements u z aong the z-axis, with speed c t = c given by 9, where c t is the speed of the transverse waves corresponding to c the speed of ight. If coordinate system x, y, z, t is attached to the uniformy moving screw disocation, then the transformation between the stationary and the moving screw disocation is given by [14] x = y = y z = z x vt 1 v /c 1/ 5 t = t vx/c 1 v /c 1/. which is the specia reativistic transformation. The ony non-zero component of the deformation in cartesian coordinates is given by [14, see pp ] 3 Screw disocation 3.1 Stationary screw disocation We consider a stationary screw disocation in the spacetime continuum, with cyindrica poar coordinates r, θ, z, with the disocation ine aong the z-axis see Fig. 3. Then the Burgers vector is aong the z-axis and is given by b r = b θ =, b z = b, the magnitude of the Burgers vector. The ony non-zero component of the deformations is given by [14, see pp. 6 61] [13, see p. 51] u z = b π θ = b π tan1 y x. 3 This soution satisfies the screw disocation dispacement equation 15. Simiary, the ony non-zero components of the stress and strain tensors are given by respectivey. σ θz = b π µ r ε θz = b 1 4π r 3. Moving screw disocation 4 where u z = b π tan1 x vt, 6 γ = 1 v c. 7 This soution aso satisfies the screw disocation dispacement equation 15. It simpifies to the case of the stationary screw disocation when the speed v =. Simiary, the ony non-zero components of the stress tensor in cartesian coordinates are given by [14] σ xz = b µ π σ yz = b µ π x vt γ y γx vt x vt γ y. 8 The ony non-zero components of the strain tensor in cartesian coordinates are derived from ε µν = 1 uµ;ν u ν;µ [1, see Eq.41]: ε xz = b 4π ε yz = b 4π x vt γ y γx vt x vt γ y, in an isotropic continuum. Non-zero components invoving time are given by ε tz = ε zt = 1 uz ct u t z ε tz = b v 4π c x vt γ y 9 3 where u t = has been used. This assumes that the screw disocation is fuy formed and moving with veocity v as described. Using, the non-zero stress components invoving time are given by σ tz = σ zt = b µ π v c x vt γ y. 31 Screw disocations are thus found to be Lorentz invariant. Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics 91

6 Voume PROGRESS IN PHYSICS Issue 4 October 3.3 Screw disocation strain energy density We consider the stationary screw disocation in the spacetime continuum of Section 3.1, with cyindrica poar coordinates r, θ, z, with the disocation ine aong the z-axis and the Burgers vector aong the z-axis b z = b. Then the strain energy density of the screw disocation is given by the transverse distortion energy density [1, see Eq. 74] E = µ e αβ e αβ 3 Fig. 4: A stationary edge disocation in cartesian x, y, z and cyindrica poar r, θ, z coordinates [14, see p. 74]. where from [1, see Eq. 33], e αβ = ε αβ e s g αβ 33 where e s = 1 4 εα α is the diatation which for a screw disocation is equa to. The screw disocation is thus massess E =. The non-zero components of the strain tensor are as defined in 4. Hence Substituting from 4, E = µ εθz ε zθ. 34 E = µ b 1 = E. 35 8π r We now consider the more genera case of the moving screw disocation in the spacetime continuum of Section 3., with cartesian coordinates x, y, z. The non-zero components of the strain tensor are as defined in 9 and 3. Substituting in 3, the equation becomes [1, see Eqs ] E = µ εtz ε xz ε yz. 36 Substituting from 9 and 3 into 36, the screw disocation strain energy density becomes E = µ b γ = E. 37 8π x vt γ y This equation simpifies to 35 in the case where v =, as expected. In addition, the energy density which is quadratic in energy as per [1, see Eq.76] is mutipied by the specia reativistic γ factor. 4 Edge disocation 4.1 Stationary edge disocation We consider a stationary edge disocation in the spacetime continuum in cartesian coordinates x, y, z, with the disocation ine aong the z-axis and the Burgers vector b x = b, b y = b z = see Fig. 4. Then the non-zero components of the deformations are given in cartesian coordinates by [14, see p. 78] u x = b π u y = b π 1 tan 1 y x µ λ 1 µ λ x y µ λ x y xy µ λ x y µ ogx µ λ y. 38 This soution resuts in a non-zero R.H.S. of the edge disocation dispacement equation 16 as required. Equation 16 can be evauated to give a vaue of ε in agreement with the resuts of Section 4.3 as shown in that section. The cyindrica poar coordinate description of the edge disocation is more compex than the cartesian coordinate description. We thus use cartesian coordinates in the foowing sections, transforming to poar coordinate expressions as warranted. The non-zero components of the stress tensor in cartesian coordinates are given by [14, see p. 76] σ xx = b µ π σ yy = b µ π σ zz = 1 = b µ π σ xy = b µ π µ λ y3x y µ λ x y µ λ yx y µ λ x y λ σxx σ yy µ λ λ µ λ y x y µ λ µ λ xx y x y. 39 The non-zero components of the strain tensor in cartesian coordinates are derived from ε µν = 1 uµ;ν u ν;µ [1, see 9 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics

7 Issue 4 October PROGRESS IN PHYSICS Voume Eq.41]: ε xx = b π y x y 1 µ λ x y µ λ x y = by 3 µ λ x µ y π µ λ x y ε yy = b π = by π ε xy = b π µ y µ λ x y µ λ x µ y µ λ x y µ λ xx y µ λ x y in an isotropic continuum. 4. Moving edge disocation 1 µ λ µ x x y 4 We now consider the previous edge disocation, moving a- ong the x-axis, parae to the z-axis, aong the sip pane xz, at a constant speed v x = v. The soutions of 13 for the moving edge disocation then incude both ongitudina and transverse components. The ony non-zero components of the deformations in cartesian coordinates are given by [11, see pp. 39 4] [39, see pp ] where u x = bc πv tan 1 γ y x vt α tan 1 u y = bc πv γ og [ x vt γ y] α γ og [ x vt γ y ], x vt 41 α = 1 v c, 4 γ = 1 v c 43 and c is the speed of ongitudina deformations given by c = µ λ ρ. 44 This soution again resuts in a non-zero R.H.S. of the edge disocation dispacement equation 16 as required, and 16 can be evauated to give a vaue of ε as in Section 4.3. This soution simpifies to the case of the stationary edge disocation when the speed v =. The non-zero components of the stress tensor in cartesian coordinates are given by [14, see pp ] [11, see pp. 39 4] σ xx = bc y λ γ 3 µ λ γ πv x vt γ y µ α γ x vt γ y σ yy = bc y µ λ γ 3 λ γ πv x vt γ y µ α γ x vt γ y σ zz = 1 = λ b π = b π λ σxx σ yy µ λ c c γ y x vt γ y λ µ µ λ γ y σ xy = µ bc x vt πv α γ 1/γ x vt γ y x vt γ y γ x vt γ y. 45 It is important to note that for a screw disocation, the stress on the pane x vt = becomes infinite at v = c. This sets an upper imit on the speed of screw disocations in the spacetime continuum, and provides an expanation for the speed of ight imit. This upper imit aso appies to edge disocations, as the shear stress becomes infinite everywhere at v = c, even though the speed of ongitudina deformations c is greater than that of transverse deformations c [14, see p. 191] [11, see p. 4]. The non-zero components of the strain tensor in cartesian coordinates are derived from ε µν = 1 uµ;ν u ν;µ [1, see Eq.41]: ε xx = bc y πv ε yy = bc y πv ε xy = bc x vt πv in an isotropic continuum. γ x vt γ y α γ x vt γ 3 x vt γ y α γ x vt γ x vt γ y α γ 1/γ x vt 46 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics 93

8 Voume PROGRESS IN PHYSICS Issue 4 October Non-zero components invoving time are given by ε tx = ε xt = 1 ux ct u t x ε ty = ε yt = 1 uy ct u t y ε tx = b c γ y π v x vt γ y α x vt γ y ε ty = b c γ x vt π v x vt γ y α γx vt γ x vt γ y 47 where from [1, see Eq. 36] the energy-momentum stress tensor T αβ is decomposed into a stress deviation tensor t αβ and a scaar t s, according to t αβ = T αβ t s g αβ 5 where t s = 1 4 T α α. Then the diatation strain energy density of the edge disocation is given by the massive ongitudina diatation energy density 5 and the distortion massess strain energy density of the edge disocation is given by the transverse distortion energy density Stationary edge disocation energy density We first consider the case of the stationary edge disocation of Section 4.1. The voume diatation ε for the stationary edge disocation is given by where u t = has been used. This assumes that the edge disocation is fuy formed and moving with veocity v as described. Using, the non-zero stress components invoving time are given by σ tx = b µ π σ ty = b µ π α γ c γ y v x vt γ y α x vt γ y c γ x vt v x vt γ y. γx vt x vt γ y 4.3 Edge disocation strain energy density 48 As we have seen in Section 3.3, the screw disocation is massess as ε = and hence E = for the screw disocation: it is a pure distortion, with no diatation. In this section, we evauate the strain energy density of the edge disocation. As seen in [1, see Section 8.1], the strain energy density of the spacetime continuum is separated into two terms: the first one expresses the diatation energy density the mass ongitudina term whie the second one expresses the distortion energy density the massess transverse term: where E = E E 49 E = 1 κ ε 1 3 κ ρc 1 κ t s 5 where ε is the voume diatation and ρ is the mass energy density of the edge disocation, and E = µ e αβ e αβ 1 4 µ t αβ t αβ. 51 ε = ε α α = ε xx ε yy 53 where the non-zero diagona eements of the strain tensor are obtained from 4. Substituting for ε xx and ε yy from 4, we obtain ε = b µ y π µ λ x y. 54 In cyindrica poar coordinates, 54 is expressed as ε = b µ sin θ. 55 π µ λ r We can disregard the negative sign in 54 and 55 as it can be eiminated by using the FS/RH convention instead of the SF/RH convention for the Burgers vector [14, see p. ]. As mentioned in Section 4.1, the voume diatation ε can be cacuated from the edge disocation dispacement ongitudina equation 16, viz. u ν = µ λ µ ε ;ν. For the x-component, this equation gives u x = u x x Substituting for u x from 38, we obtain Hence and u x = b π u x y = µ λ µ ε,x. 56 µ λ xy µ λ x y = µ λ ε,x. 57 µ ε,x = b π ε = b π µ xy 58 µ λ x y µ µ λ xy dx. 59 x y 94 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics

9 Issue 4 October PROGRESS IN PHYSICS Voume Evauating the integra [4], we obtain ε = b π µ y 6 µ λ x y in agreement with 54. Simiary for the y-component, substituting for u y from 38, the equation gives u y = u y x ε,y = b π Evauating the integra [4] ε = b µ π µ λ we obtain ε = b π u y y = µ λ µ ε,y 61 µ x y µ λ x y. 6 x y dy, 63 x y µ y 64 µ λ x y again in agreement with 54. The mass energy density is cacuated from 4 ρc = 4 κ ε = λ µ ε 65 where 3 has been used. Substituting for ε from 54, the mass energy density of the stationary edge disocation is given by ρc = 4b κ µ y π µ λ x y. 66 In cyindrica poar coordinates, 66 is expressed as ρc = 4b π κ µ sin θ. 67 µ λ r Using 54 in 5, the stationary edge disocation ongitudina diatation strain energy density is then given by E = b π κ µ µ λ y x y. 68 In cyindrica poar coordinates, 68 is expressed as where e s = 1 4ε is the voume diatation cacuated in 54 and e αβ e αβ = ε αβ 14 εgαβ ε αβ 1 4 εg αβ. 7 For g αβ = η αβ, the off-diagona eements of the metric tensor are, the diagona eements are 1 and 7 becomes E = µ ε xx 1 4 ε ε yy 1 4 ε ε xy. 73 Expanding the quadratic terms and making use of 53, 73 becomes E = µ ε xx ε yy 3 8 ε ε xy 74 and finay 5 E = µ 8 ε ε xx ε yy ε xy. 75 Substituting from 4 and 54 in the above, E = 5 8 b µ π µ µ λ y x y b µ π y [ 3 µ λ µ λ x 4 µ x y µ y4] µ λ x y 4 b µ µ λ x x y π µ λ x y. 4 which becomes E = b µ 1 π µ λ x y 4 { 5 4 µ y x y y [ 3 µ λ µ λ x 4 µ x y µ y4] µ λ x x y } E = b π κ µ µ λ sin θ r. 69 The distortion strain energy density is cacuated from 51. The expression is expanded using the non-zero eements of the strain tensor 4 to give E = µ exx e yy e xy e yx. 7 As seen previousy in 33, e αβ = ε αβ e s g αβ 71 In cyindrica poar coordinates, 77 is expressed as { E = b µ 5 sin θ π µ λ 4 µ sin θ r r [ 3 µ λ µ λ cos 4 θ µ cos θ sin θ µ sin4 θ ] µ λ cos θ cos θ sin θ } r 78 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics 95

10 Voume PROGRESS IN PHYSICS Issue 4 October or { E = b µ 5 π µ λ 4 µ sin θ r [ 3 µ λ µ λ cos 4 θ sin θ r µ cos θ sin4 θ µ sin 6 θ ] r r µ λ cos θ cos θ r } Moving edge disocation energy density 79 We next consider the genera case of the moving edge disocation in the spacetime continuum of Section 4., with cartesian coordinates x, y, z. We first evauate the voume diatation ε for the moving edge disocation. The voume diatation is given by ε = ε α α = ε xx ε yy 8 where the non-zero diagona eements of the strain tensor are obtained from 46. Substituting for ε xx and ε yy from 46 in 8, we notice that the transverse terms cance out, and we are eft with the foowing ongitudina term: ε = bc y πv This equation can be further reduced to ε = bc πv v c γ 3 γ x vt γ y 81 γ y x vt γ y 8 and finay, using c /c = µ / µ λ see 9 and 44, The distortion strain energy density is cacuated from 51. The expression is expanded using the non-zero eements of the strain tensor 46 and 47 and, from 71 and 7, we obtain [1, see Eqs ] [ E = µ ε xx 1 4 ε ε yy 14 ε ] ε tx ε ty ε xy. 87 Expanding the quadratic terms and making use of 53 as in 74, 87 becomes E = µ ε xx ε yy 3 8 ε ε tx ε ty ε xy. 88 Substituting from 46, 47 and 8, b c { E = µ 3 v γ y π v 8 c x vt γ y γ y 4 x vt γ α y x vt γ y γ 3 4 y x vt γ α y x vt γ y v γ y c x vt γ α y x vt γ y v γ x vt α γx vt c x vt γ y γ x vt γ y γ x vt x vt γ α } γ 1/γx vt y x vt γ y which simpifies to 89 εx i, t = b π µ γ y. 83 µ λ x vt γ y As seen previousy, the mass energy density is cacuated from 65: ρc = 4 κ ε = λ µ ε. 84 Substituting for ε from 83, the mass energy density of an edge disocation is given by b c 4 { α 4 3 γ E = µ π v 4 x vt γ y 3 1 γ γ x vt γ α γ v γ γ y c x vt γ y x vt γ y 3 γ γ x vt α γ x vt γ y v 4 c 4 γ y }. 9 ρx i, t c = b π 8 κ µ γ y. 85 µ λ x vt γ y Using 83 in 5, the edge disocation ongitudina diatation strain energy density is then given by E = 1 κ b µ γ y. 86 π µ λ x vt γ y We consider the above equations for the moving edge disocation in the imit as v. Then the terms and x vt γ y sin θ r x vt x vt γ y cos θ r Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics

11 Issue 4 October PROGRESS IN PHYSICS Voume in cyindrica poar coordinates. Simiary for the same terms with γ instead of γ. The voume diatation obtained from 83 is then given in cyindrica poar coordinates r, θ, z by ε b π µ sin θ. 93 µ λ r The mass energy density is obtained from 85 to give ρc b π 8 κ µ sin θ. 94 µ λ r From 86, the edge disocation diatation strain energy density is then given by E b π κ µ µ λ sin θ r. 95 These equations are in agreement with 55, 67 and 69 respectivey. The edge disocation distortion strain energy density in the imit as v is obtained from 89 by making use of 91 and 9 as foows: b c 4 E µ 4π v 4 4 sin θ r v c v c Simpifying, sin θ { 3 α sin θ r α sin θ r r cos θ γ α r γ v 4 c 4 sin θ r 4 γ sin θ r cos θ γ α γ 1 r γ b c 4 E µ 4π v 4 cos θ r cos θ { α sin θ r r v 4 c 4 }. sin θ r α sin θ r 4 γ α sin θ r v 1 α sin θ c r γ v c α cos θ γ r γ α γ 1 cos } θ. γ r Using the definitions of γ, γ and α from 7, 4 and 43 respectivey, using the first term of the Tayor expansion for γ and γ as v, and negecting the terms mutipied by v /c in 97 as they are of order v 6 /c 6, 97 becomes b c 4 { [ E µ 3 v 4 4π v 4 c v4 4 c v 1 v ] sin θ c c r v 1 v cos } θ. c r Squaring and simpifying, we obtain { 5 v 4 b c 4 E µ 4π v 4 c v4 c 4 v4 c 4 v4 c v4 4 c 4 v4 sin θ 4 c c r } 99 cos θ r c c and further b { E µ 1 c 5 π c c c4 c c 4 cos θ r c 4 c 4 }. sin θ r 1 Using c /c = µ / µ λ see 9 and 44, 1 becomes b { µ E µ 1 π µ λ 5 µ sin θ µ λ r µ µ cos } θ. µ λ µ λ r This equation represents the impact of the time terms incuded in the cacuation of 87 and the imit operation v used in Curved disocations In this section, we consider the equations for generay curved disocations generated by infinitesima eements of a disocation. These aow us to hande compex disocations that are encountered in the spacetime continuum. 5.1 The Burgers dispacement equation The Burgers dispacement equation for an infinitesima eement of a disocation d = ξd in vector notation is given by [14, see p. 1] ur = b 4π 1 4π A ˆR da 1 R 4π [ µ λ µ λ C C b d R b R d ] 1 R Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics 97

12 Voume PROGRESS IN PHYSICS Issue 4 October where u is the dispacement vector, r is the vector to the dispaced point, r is the vector to the disocation infinitesima eement d, R = r r, b is the Burgers vector, and cosed oop C bounds the area A. In tensor notation, 1 is given by u µ r ν = 1 b µ 8π A 1 b β ɛ µβγ R dx γ 8π C 1 µ λ b β ɛ 4π µ λ βαγ C x α R da α R x µ x α dx γ 13 where ɛ αβγ is the permutation symbo, equa to 1 for cycic permutations, 1 for anti-cycic permutations, and for permutations invoving repeated indices. As noted by Hirth [14, see p. 13], the first term of this equation gives a discontinuity u = b over the surface A, whie the two other terms are continuous except at the disocation ine. This equation is used to cacuate the dispacement produced at a point r by an arbitrary curved disocation by integration over the disocation ine. 5. The Peach and Koeher stress equation The Peach and Koeher stress equation for an infinitesima eement of a disocation is derived by differentiation of 13 and substitution of the resut in [14, see p ]. In this equation, the disocation is defined continuous except at the disocation core, removing the discontinuity over the surface A and aowing to express the stresses in terms of ine integras aone. σ µν = µ b α ɛ βαµ 8π C x β R dx ν µ b α ɛ βαν 8π C x β R dx µ µ 14 µ λ b α ɛ 4π µ λ βαγ C 3 R x β x µ x ν δ µν x β R dx γ. This equation is used to cacuate the stress fied of an arbitrary curved disocation by ine integration. 6 Framework for quantum physics In a soid, disocations represent the fundamenta dispacement processes that occur in its atomic structure. A soid viewed in eectron microscopy or other microscopic imaging techniques is a tange of screw and edge disocations [1, see p. 35 and accompanying pages]. Simiary, disocations in the spacetime continuum are taken to represent the fundamenta dispacement processes that occur in its structure. These fundamenta dispacement processes shoud thus correspond to basic quantum phenomena and provide a framework for the description of quantum physics in STCED. We find that disocations have fundamenta properties that refect those of partices at the quantum eve. These incude sef-energy and interactions mediated by the strain energy density of the disocations. The roe payed by virtua partices in Quantum Eectrodynamics is repaced by the interaction of the energy density of the disocations. This theory is not perturbative as in QED, but rather cacuated from anaytica expressions. The anaytica equations can become very compicated, and in some cases, perturbative techniques are used to simpify the cacuations, but the avaiabiity of anaytica expressions permit a better understanding of the fundamenta processes invoved. Athough the existence of virtua partices in QED is generay accepted, there are physicists who sti question this interpretation of QED perturbation expansions. Weingard [41] argues that if certain eements of the orthodox interpretation of states in QM are appicabe to QED, then it must be concuded that virtua partices cannot exist. This foows from the fact that the transition ampitudes correspond to superpositions in which virtua partice type and number are not sharp. Weingard argues further that anaysis of the roe of measurement in resoving the superposition strengthens this concusion. He then demonstrates in detai how in the path integra formuation of fied theory no creation and annihiation operators need appear, yet virtua partices are sti present. This anaysis shows that the question of the existence of virtua partices is reay the question of how to interpret the propagators which appear in the perturbation expansion of vacuum expectation vaues scattering ampitudes. [4] The basic Feynman diagrams can be seen to represent screw disocations as photons, edge disocations as partices, and their interactions. The exchange of virtua partices in interactions can be taken as the forces resuting from the overap of the disocations strain energy density, with suitaby modified diagrams. The perturbative expansions are aso repaced by finite anaytica expressions. 6.1 Quantization The Burgers vector as defined by expression 5 has simiarities to the Bohr-Sommerfed quantization rue C p dq = nh 15 where q is the position canonica coordinate, p is the momentum canonica coordinate and h is Panck s constant. This eads us to consider the foowing quantization rue for the STC: at the quantum eve, we assume that the spacetime continuum has a granuarity characterized by a ength b corresponding to the smaest eementary Burgers disocationdispacement vector possibe in the STC. The idea that the existence of a shortest ength in nature woud ead to a natura cut-off to generate finite integras in QED has been raised 98 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics

13 Issue 4 October PROGRESS IN PHYSICS Voume before [43]. The smaest eementary Burgers disocationdispacement vector introduced here provides a ower bound as shown in the next section. Then the magnitude of a Burgers vector can be expressed as a mutipe of the eementary Burgers vector: b = nb. 16 We find that b is usuay divided by π in disocation equations, and hence we define b = b π, 17 and simiary for the eementary Burgers disocation-dispacement vector b, b = b π Screw disocations in quantum physics Screw disocations in the spacetime continuum are identified with massess, transverse deformations, specificay photons. Consider the dispacement of a stationary screw disocation as derived in Section 3.1: u z = b θ = b θ. 19 π Taking the derivative with respect to time, we obtain u z = v z = b π θ = b ω. 11 π The speed of the transverse dispacement is c, the speed of ight. Substituting for ω = πν, 11 becomes Hence c = b ν. 111 b = λ, 11 the waveength of the screw disocation. This resut is iustrated in Fig. 5. It is important to note that this reation appies ony to screw disocations. The strain energy density of the screw disocation is given by the transverse distortion energy density derived in Section 3.3. For a stationary screw disocation, substituting 17 into 35, E = µ b 1 r. 113 The tota strain energy of the screw disocation is then given by W = E dv 114 V where the voume eement dv in cyindrica poar coordinates is given by rdr dθ dz. Substituting for E from 113, 114 becomes µ b W = rdr dθ dz. 115 r V Fig. 5: A waveength of a screw disocation. From 16, b can be taken out of the integra to give W = µ b Λ b 1 r dr dθ dz 116 θ z where Λ is a cut-off parameter corresponding to the radia extent of the disocation, imited by the average distance to its nearest neighbours. The strain energy per waveength is then given by and finay W λ = µ b og Λ b π dθ 117 W λ = µ b 4π og Λ b. 118 The impications of the tota strain energy of the screw disocation are discussed further in comparison to Quantum Eectrodynamics QED in Section Edge disocations in quantum physics The strain energy density of the edge disocation is derived in Section 4.3. The diatation massive strain energy density of the edge disocation is given by the ongitudina strain energy density 5 and the distortion massess strain energy density of the edge disocation is given by the transverse strain energy density 51. For the stationary edge disocation of 79, using 17 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics 99

14 Voume PROGRESS IN PHYSICS Issue 4 October into 79, we have E = b µ µ λ { 5 4 µ sin θ r [ 3 µ λ µ λ cos 4 θ sin θ r µ cos θ sin4 θ µ sin 6 θ ] r r µ λ cos θ cos θ r }. 119 The distortion strain energy of the edge disocation is then given by W = E dv 1 V where the voume eement dv in cyindrica poar coordinates is given by rdr dθ dz. Substituting for E from 119 and taking b out of the integra, 1 becomes W = b µ µ λ z θ Λ b { 5 4 µ sin θ r [ 3 µ λ µ λ cos 4 θ sin θ r µ cos θ sin4 θ µ sin 6 θ ] r r µ λ cos θ cos θ r } rdr dθ dz 11 where again Λ is a cut-off parameter corresponding to the radia extent of the disocation, imited by the average distance to its nearest neighbours. Evauating the integra over r, b µ W = µ λ og Λ π { 5 b 4 µ sin θ [ 3 µ λ µ λ cos 4 θ sin θ µ cos θ sin 4 θ µ sin6 θ ] } µ λ cos θ cos θ dθ dz. z 1 Evauating the integra over θ [4], we obtain 13 at the top of the next page. Appying the imits of the integration, both the coefficients of λ and µ λ are equa to and ony the coefficient of µ is non-zero. Equation 13 then becomes b µ W = µ λ og Λ b where is the ength of the edge disocation. 9π 4 µ dz. 14 Evauating the integra over z, we obtain the stationary edge disocation transverse strain energy per unit ength W = 9π b µ µ µ λ og Λ b. 15 We find that the stationary edge disocation transverse strain energy per unit ength where we have added the abe E W E = 9 8π b µ µ µ λ og Λ b 16 is simiar to the stationary screw disocation transverse strain energy per unit ength W S = 1 4π b µ og Λ b 17 except for the proportionaity constant. Simiary, the ongitudina strain energy of the stationary edge disocation is given by W E = E dv. 18 Substituting for E from 69, this equation becomes W E = V V b π κ µ µ λ sin θ r dv. 19 Simiary to the previous derivation, this integra gives W E = 1 π b κ µ µ λ og Λ b. 13 The tota strain energy of the stationary screw and edge disocations have simiar functiona forms, with the difference residing in the proportionaity constants. This is due to the simper nature of the stationary disocations and their cyindrica poar symmetry. This simiarity is not present for the genera case of moving disocations as evidenced in equations 37, 86 and 9. For the moving edge disocation in the imit as v, subsituting for 11 in 1 and using 17, we have W E b µ { z θ Λ b µ µ λ 5 4 µ µ λ rdr dθ dz µ sin θ µ λ r µ } cos θ µ λ r 131 where again Λ is a cut-off parameter corresponding to the radia extent of the disocation, imited by the average distance to its nearest neighbours. 3 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics

15 Issue 4 October PROGRESS IN PHYSICS Voume b µ W = µ λ og Λ b 3 µ λ µ λ θ µ θ µ µ λ θ sin θ 1 64 sin θ 3 64 [ 5 θ z 4 µ 1 4 θ sin θ 1 64 sin 4θ 1 19 sin 4θ 1 19 sin θ 1 sin 4θ 19 sin 6θ sin 6θ sin θ 1 16 sin 4θ 1 48 ] π sin 6θ dz sin 6θ 13 Evauating the integra over r, W E b µ og Λ b { µ µ λ 5 4 z µ µ λ π dθ dz µ sin µ λ θ µ } cos µ λ θ. 13 Evauating the integra over θ [4] and appying the imits of the integration, we obtain W E b µ og Λ dz b { µ µ λ 5 4 µ µ λ µ π µ λ µ } π µ λ 133 and evauating the integra over z, we obtain the moving edge disocation transverse strain energy per unit ength in the imit as v W E 3 4π b µ µ µ λ µ µ λ where is the ength of the edge disocation. 6.4 Strain energy of moving disocations og Λ b 134 In the genera case of moving disocations, the derivation of the screw disocation transverse strain energy and the edge disocation transverse and ongitudina strain energies is more difficut. In this section, we provide an overview discussion of the topic Screw disocation transverse strain energy The transverse strain energy of a moving screw disocation, which aso corresponds to its tota strain energy, is given by W S = V E S dv 135 where the strain energy density E S is given by 113, viz. E S = 1 b µ γ x vt γ y 136 and V is the 4-dimensiona voume of the screw disocation. The voume eement dv in cartesian coordinates is given by dx dy dz dct. Substituting for E S, 135 becomes W S = V 1 b γ µ dx dy dz dct. 137 x vt γ y As before, b is taken out of the integra from 16, and the integra over z is handed by considering the strain energy per unit ength of the disocation: W S = b µ ct y x b x y Λ γ dx dy dct 138 x vt γ y where is the ength of the disocation and as before, Λ is a cut-off parameter corresponding to the radia extent of the disocation, imited by the average distance to its nearest neighbours. Evauating the integra over x [4], W S = b µ γ ct y dy dct [ 1 ] Λ x vt y arctan y b 139 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics 31

16 Voume PROGRESS IN PHYSICS Issue 4 October where the imits corresponding to the maximum cut-off parameter Λ and minimum cut-off parameter b are stated expicity. Appying the imits of the integration, we obtain W S b µ = γ dy dct ct y { 1 arctan Λ y vt 1 } arctan y b vt. 14 This integration over y is not eementary and ikey does not ead to a cosed anaytica form. If we consider the foowing simper integra, the soution is given by 1 x vt y arctan dy = i [Li i x vt Li i x vt ] 141 where Li n x is the poyogarithm function. As pointed out in [44], [t]he poyogarithm arises in Feynman diagram integras and, in particuar, in the computation of quantum eectrodynamics corrections to the eectrons gyromagnetic ratio, and the specia cases n = and n = 3 are caed the diogarithm and the triogarithm, respectivey. This is a further indication that the interaction of strain energies are the physica source of quantum interaction phenomena described by Feynman diagrams as wi be seen in Section Edge disocation ongitudina strain energy The ongitudina strain energy of a moving edge disocation is given by W E = dv 14 E E V where the strain energy density E E is given by 86, viz. E E = 1 κ µ γ y b µ λ x vt γ y 143 and V is the 4-dimensiona voume of the edge disocation. The voume eement dv in cartesian coordinates is given by dx dy dz dct. Substituting for E E, 14 becomes W E = V 1 κ µ γ y b µ λ x vt γ y dx dy dz dct. 144 As before, b is taken out of the integra from 16, and the integra over z is handed by considering the strain energy per unit ength of the disocation: W E µ = κ b µ λ γ y 145 dx dy dct x vt γ y ct y x b x y Λ where is the ength of the disocation and as before, Λ is a cut-off parameter corresponding to the radia extent of the disocation, imited by the average distance to its nearest neighbours. The integrand has a functiona form simiar to that of 138, and a simiar soution behaviour is expected. Evauating the integra over x [4], W E µ = κ b µ λ [ 1 x vt x vt γ y ct 1 ] Λ x vt y γ y arctan γ y y b y dy dct 146 where the imits corresponding to the maximum cut-off parameter Λ and minimum cut-off parameter b are stated expicity. Appying the imits of the integration, we obtain W E µ = κ b dy dct µ λ ct y { 1 Λ y vt Λ y vt γ y 1 y b vt y b vt γ y 1 γ y arctan Λ y vt γ y 1 } γ y arctan y b vt γ y. 147 This integration over y is again found to be intractabe, incuding that of 14, and ikey does not ead to a cosed anaytica form. In the arctan Λ integra of 14 and 147, we can make the approximation Λ y Λ and evauate this term as seen in 141: 1 Λ vt y γ y arctan dy = γ y i [Li i Λ vt Li i Λ vt ] 148 γ y γ y where Li n x is the poyogarithm function as seen previousy. 3 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics

17 Issue 4 October PROGRESS IN PHYSICS Voume Edge disocation transverse strain energy The transverse strain energy of a moving edge disocation is given by W E = E E dv 149 where the strain energy density E E is given by 9, viz. { E E = µ b c4 α 4 3 γ v 4 x vt γ y 3 1 γ γ x vt γ α γ v γ γ y c x vt γ y x vt γ y 3 γ γ x vt α γ x vt γ y V v 4 c 4 γ y } 15 and V is the 4-dimensiona voume of the edge disocation. The voume eement dv in cartesian coordinates is given by dx dy dz dct. Substituting for E E as before, taking b out of the integra from 16, and handing the integra over z by considering the strain energy per unit ength of the disocation, 149 becomes W E = µ b c4 v 4 { α 4 3 γ x vt γ y dx dy dct ct y x b x y Λ 3 1 γ γ x vt γ α γ v γ γ y c x vt γ y x vt γ y 3 γ γ x vt α γ x vt γ y v 4 c 4 γ y } 151 where is the ength of the disocation and as before, Λ is a cut-off parameter corresponding to the radia extent of the disocation, imited by the average distance to its nearest neighbours. Again, the integrand has functiona forms simiar to that of 138 and 145. A simiar, but more compex, soution behaviour is expected, due to the additiona compexity of Disocation interactions in quantum physics As mentioned is Section 6, the basic Feynman diagrams can be seen to represent screw disocations as photons, edge disocations as partices, and their interactions. More specificay, the externa egs of Feynman diagrams that are on mass-she representing rea partices correspond to disocations, whie the virtua off mass-she partices are repaced by the interaction of the strain energy densities. The exchange of virtua partices in QED interactions can be taken as the perturbation expansion representation of the forces resuting from the overap of the strain energy density of the disocations. The Feynman diagram propagators are repaced by the disocation strain energy density interaction expressions. The properties of Burgers vectors and disocations [14, see pp. 5-6] have rues simiar to those of Feynman diagrams, but not equivaent as virtua partices are repaced by disocation strain energy density interactions. A Burgers vector is invariant aong a disocation ine. Two Burgers circuits are equivaent if one can be deformed into the other without crossing disocation ines. The resutant Burgers vector within equivaent Burgers circuits is the same. Disocation nodes are points where mutipe disocations meet. If a the disocation vectors ξ i are taken to be positive away from a node, then N ξ i = 15 i=1 for the N disocations meeting at the node. Burgers vectors are conserved at disocation nodes. In this section, we consider the interactions of disocations which are seen to resut from the force resuting from the overap of their strain energy density in the STC [14, see p. 11]. 7.1 Parae disocation interactions From Hirth [14, see pp ], the energy of interaction per unit ength between parae disocations incuding screw and edge disocation components is given by W 1 µ π µ π = µ π b 1 ξ b ξ og R R Λ µ λ µ λ b 1 ξ b ξ og R R Λ µ λ [b 1 ξ R] [b ξ R] µ λ R 153 where ξ is parae to the z axis, b i ξ are the screw components, b i ξ are the edge components, R is the separation between the disocations, and R Λ is the distance from which the disocations are brought together, resuting in the decrease in energy of the system. The components of the interaction force per unit ength between the parae disocations are obtained by differentiation: F R = W 1/ R F θ = 1 R W 1 /. θ 154 Pierre A. Miette. Disocations in the Spacetime Continuum: Framework for Quantum Physics 33