sign in sign up
<<
Home , Angular frequency

International Journal of , 2018, Vol. 6, No. 2, 47-52 Available online at http://pubs.sciepub.com/ijp/6/2/4 ©Science and Education Publishing DOI:10.12691/ijp-6-2-4

On the Principle of Least

Vu B Ho*

Advanced Study, 9 Adela Court, Mulgrave, Victoria 3170, Australia *Corresponding author: [email protected] Abstract Investigations into the nature of the principle of least action have shown that there is an intrinsic relationship between geometrical and topological methods and the variational principle in classical . In this , we follow and extend this kind of mathematical analysis into the domain of quantum mechanics. First, we show that the identification of the of a quantum particle with the de Broglie in 2-dimensional would lead to an interesting feature; namely the action principle = 0 would be satisfied not only by the stationary path, corresponding to the classical , but also by any path. Thereupon the Bohr quantum condition possesses a topological character in the sense that the principal quantum훿푆 number is identified with the winding number, which is used to represent the fundamental group of paths. We extend our discussions into 3-dimensional space and show that the charge of a particle also possesses a topological character and푛 is quantised and classified by the homotopy group of closed surfaces. We then discuss the possibility to extend our discussions into with higher dimensions and show that there exist physical quantities that can be quantised by the higher homotopy groups. Finally we note that if Einstein’s field equations of general relativity are derived from Hilbert’s action through the principle of least action then for the case of = 2 the field equations are satisfied by any metric if the -momentum tensor is identified with the metric tensor, similar to the case when the momentum of a particle is identified with the curvature of the particle’s path. 푛 Keywords: principle of least action, Feynman path integral method, , quantum mechanics, general relativity Cite This Article: Vu B Ho, “On the Principle of Least Action.” International Journal of Physics, vol. 6, no. 2 (2018): 47-52. doi: 10.12691/ijp-6-2-4.

quantum condition can also be described in a classical way. In classical mechanics, the actual path of a particle is 1. Introduction found by extremising the action integral of the particle [8]. On the other hand, in quantum mechanics, the In , the principle of least action is a equation of a particle can be found by applying the variational principle that can be used to determine Feynman path integral formulation which assumes the uniquely the for various physical particle can take any trajectory [9]. The question then systems. However, recently the principle of least action arises as to whether it is possible for the action integral to that is associated with new concepts of Lagrangian be extremised by any path at the quantum level. In this symmetry has been proposed and studied by many authors. work we show that this problem may be investigated in It has also been shown that different formulations of the terms of geometry and topology, and it transpires that principle of least action can be constructed for a physical topology may play an important role in the determination system that is described by a particular system of of the nature of a quantum observable [10,11]. differential equations [1-5]. Furthermore, new approach to the theories of gravitation using the principle of least action has also been considered [6]. To extend these 2. Principle of Least Action in Classical investigations, in this work we will consider the case in Mechanics which the principle of least action can be applied to both classical physics and quantum physics. The investigation of the relationship between physics In the old quantum theory, the Bohr quantum condition and geometry had been carried out and culminated with = , where is the momentum of a particle, is the development of the principle of least action. In 1744, and is a positive integer, played a Euler developed and published his work on this crucial∮ 푝푑푠 role푛ℎ in the 푝quantum description of a physicalℎ variational principle for the of a particle moving system, although it had푛 been introduced into the quantum in a plane curve [12]. On the one hand, the dynamics of a theory in an ad hoc manner [7]. However, except for the particle can be studied by using the principle of least quantum condition imposed on the orbital angular action. On the other hand, the path of a particle can also be momentum, the Bohr model was based entirely on the determined by using geometrical methods, since a path classical dynamics of Newtonian physics. In this case, it can be constructed if the curvature is known at all points seems natural to raise the question as to whether the Bohr on the path. Euler showed the equivalence between these

International Journal of Physics 48 two methods by calculating the radius of curvature of the vector ( ), the unit tangent vector ( ), the unit principal path directly and by means of the variational principle. normal vector ( ) and the unit binormal vector ( ), Consider a particle moving in a plane under the influence defined퐫 by푠 the relation ( ) = (퐭)푠× ( ) , satisfy the of a . Let and be the per unit in the Frenet equations퐩 [14,15]푠 퐛 푠 x-direction and the y-direction, respectively. The normal 퐛 푠 퐭 푠 퐩 푠 푥 푦 ddtp d b 퐹 of the 퐹particle is given by =κκp,, =−+ tb =− p (4) ds ds ds 1 푛 − 푎 2 ( ) ( ) dy   dy 2 where and are the curvature and the aFnx= −+ F y 1  . (1) respectively, and = . is the linear element. If we dx   dx  consider휅 the푠 motion휚 푠 of a particle in a plane, as in the case 푑푠 √푑퐫 푑퐫 If is the radius of curvature, then = , of Bohr’s model of a hydrogen-like atom, the Frenet equations reduce to where is the of the particle. This result can also2 be 푛 obtained휌 by using the variational principle 푎 = 0−, 푣where⁄휌 ddtp 푣 =κκpt,. = − (5) is defined by ds ds 훿푆 2 푆 dy By differentiation, we obtain the following system of S= pds = p1. + dx (2) differential equations ∫∫dx dd2ttd(lnκ ) The condition = 0 leads to the stationary condition − +=κ 2t 0 (6) ds2 ds ds 훿푆 22 2 d ∂∂dy dy ddpp(lnκ ) d2 vv1+ −+=1  0. (3) − +=κ p 0. (7) dx dy dx∂ y  dx 2 ds ds ∂  ds dx If the curvature ( ) is assumed to vary slowly along Employing the relations = 2 and = the curve ( ), so that the condition (ln ) = 0 can 2 , the stationary condition2 given by Equation2 (3) be imposed, then (휅 )푠 and ( ) may be regarded as being 푥 = 휕푣 ⁄휕푥 퐹 휕푣 ⁄휕푦 oscillating퐫 with푠 a spatial period, or wavelength,푑 휅 ⁄푑푠 λ, whose then푦 leads to . This result reveals an intrinsic relationship퐹 between 2 geometrical methods and the relationship to the퐭 curvature푠 퐩 푠 is found as 푎푛 − 푣 ⁄휌 variational principle in classical mechanics. 2π κ = 휅 . (8) λ 3. Principle of Least Action in Quantum In the case of the Bohr’s planar model of a hydrogen- Mechanics like atom with circular orbits, the condition (ln ) = 0 is always satisfied, since the curvature remains constant 푑 휅 ⁄푑푠 We will now extend this geometrical analysis into for each of the orbits. In order to incorporate this the domain of quantum mechanics. We show that the elementary differential geometry into quantum mechanics, identification of the momentum of a quantum particle with we identify the wavelength defined in Equation (8) with the de Broglie wavelength leads to an interesting feature; the de Broglie’s wavelength of a particle. This seems to be namely the action principle = 0 is satisfied not only by a natural identification since the spatial period is the the stationary path corresponding to the classical motion, wavelength of the unit tangent vector ( ) . With this but also by any path. In 훿푆this case the Bohr quantum assumption, the momentum of the particle and휆 the condition possesses a topological character in the curvature are related through the relation퐭 푠 푝 sense that the principal quantum number is identified p = κ. (9) with the winding number, which is used to represent the 휅 fundamental group of paths [13]. Consider 푛a curve in the We now want to show how this result leads to Bohr’s three-dimensional spatial continuum R which is a postulate of the quantisation of . It topological image of an open segment of3 a straight line. should be mentioned here that this can only be discussed The curve can be represented by a real vector function in terms of Bohr’s quantum theory, or Feynman’s path ( ) . For the Bohr planar model, we utilise the integral methods, since these formulations do require fundamental homotopy group and it is convenient to concepts employed in classical physics, especially the 퐫consider푠 curves that may have points that correspond to concept of classical paths of a particle [9]. According to more than one value of the parameter . However, the canonical formulation of classical physics, the particle whenever the single-valuedness of the representation ( ) dynamics is governed by the action principle = is required for differential analysis of the푠 motion, the = 0 . Using the relationship = given in 훿푆 uncertainty principle in quantum mechanics may퐫 be푠 Equation (9) and the expression for the curvature of the invoked to shift the self-intersecting points into the third 훿path∫ 푝푑푠 ( ) of a particle in a plane, =푝 ′′ (ℏ1휅+ ′ ) , dimension to make the curve single-valued for the whole where = and = , the 휅 action2 3⁄2 domain of definition. This is a consequence of quantum integral푓 푥 ′ takes the form ′′ 2 휅 푓2 ⁄ 푓 푓 푑푓⁄푑푥 푓 푑 푓⁄푑푥 mechanics which does not allow both the momentum and '' 푆 f coordinate associated with the third dimension to vanish S=∫∫κ ds = dx. (10) simultaneously [10]. In differential geometry, the 1+ f '2

49 International Journal of Physics

It is shown in the calculus of variations that to 2 ddpp(lnκ ) d2 extremise the integral = ( , , , ) , the − +=κ p 0. (16) dτ 2 ddττ function ( ) must satisfy the differential′ equation′′ [16] 푆 ∫ 퐿 푓 푓 푓 푥 푑푥 If the temporal curvature ( ) is assumed to vary ∂∂LdL d2 ∂ L 푓 푥 −+ =0. (11) slowly along the curve ( ) , so that the condition ∂f dx ∂∂f' dx 2 f '' (ln ) = 0 can be imposed,휅 then휏 ( ) and ( ) may be regarded as being oscillating퐭 휏 with a temporal period , 푇 However, with the functional of the form given in 푑whose휅 ⁄relationship푑휏 with the temporal curvature퐭 휏 퐩 is휏 found = (1 + ) Equation (10), , it is straightforward from the differential equations given in Equation (15) 푇or to verify that the differential′′ equation′2 given in Equation (16) as 휅 (11) is satisfied퐿 by anyℏ 푓 function⁄ 푓 ( ). This result may be considered as a foundation for the Feynman’s path integral 2π κ = . (17) formulation of quantum mechanics,푓 푥 which uses all T classical trajectories of a particle in order to calculate the transition amplitude of a quantum mechanical system This result shows that the temporal curvature is [9,12]. Since any path can be taken by a particle moving actually the angular . In principle, the 휅 in a plane, if the orbits of the particle are closed, it is structure of the three-dimensional spatial manifold and the 휔 possible to represent each class of paths of the three-dimensional temporal manifold are identical, because, fundamental homotopy group of the particle by a circular without matter, both of them are just a three-dimensional path, since topologically, any path in the same equivalence Euclidean continuum. In order to incorporate this elementary class can be deformed continuously into a circular path differential geometry into quantum mechanics, we identify [13]. This validates Bohr’s assumption of the angular frequency defined in Equation (17) with the for the electron in a hydrogen-like atom. This assumption angular frequency in Planck’s quantum of energy = then leads immediately to the Bohr quantum condition of a particle. With this assumption, the energy of the particle and the curvature are related through퐸 ℏthe휔 ds relation ∮pds=∮κθ ds = ∮ =∮d = nh. (12) r 휅 E = κ. (18) The Bohr quantum condition possesses a topological As in the case of Bohr’s quantisation of angular character in the sense that the principal quantum number momentum, the quantisation of energy can be obtained is identified with the winding number which is used to from the relation given by Equation (18) using the principle represent the fundamental homotopy group of paths of the푛 of least action for the temporal manifold and it can be said electron in the hydrogen atom. that the quantisation of energy is a manifestation of It is interesting to note that our discussions for the , or , in the two-dimensional temporal dynamics of a particle in a three-dimensional spatial manifold. continuum can be extended to a three-dimensional In fact, the Feynman’s method of sum over random temporal manifold. Mathematically, a temporal manifold paths can be extended to higher-dimensional spaces to can be considered as a three-dimensional Euclidean formulate physical theories in which the transition amplitude continuum whose radial can be identified with the between states of a quantum mechanical system is the sum one-dimensional time in physics [17]. In this case we can over random hypersurfaces. This generalisation of the path ( ) also define the temporal position, or , vector , integral method in quantum mechanics has been developed ( ) the unit temporal tangent vector , the unit temporal and applied to other of physics, such as condensed ( ) 퐭 휏 principal normal vector and the unit temporal matter physics, quantum field theories and quantum ( ) 퐭푇 휏 ( ) = binormal vector , defined by the relation theories, mainly for the purpose of field quantisation. For 퐩 휏 ( ) × ( ) . These mathematical objects satisfy the example, although there are conceptual difficulties, the 퐛 휏 퐛 휏 temporal Frenet equations path integral approach to quantum gravity is carried 퐭푇 휏 퐩 휏 out by considering a sum over all field configurations, dtT ddpb =κκp,, =−+ tbT  =− p (13) and matter field , which are ddττ d τ determined by a metric consistent with the three-geometries at the boundaries of 휇휈 휇 where ( ) and ( ) are the temporal curvature and the the space-time [18]. On 푔the other hand, string휙 theories can temporal torsion, respectively, and = . is the be formulated in terms of a sum over random surfaces [19]. linear element휅 휏 of휚 a 휏temporal curve. If we only consider the In this case the surface integral method can be used in the motion of a particle in a plane, the 푑equations휏 √푑퐭 푑given퐭 in quantisation procedure, where the surface action of the form Equation (13) reduce to 2 µν α α S=−(1/2πσ)∫ d hh∂∂µν X X is used for bosonic strings, dtT dp µν α α µ α =κκpt, = − . (14) and S=−(1/2π) d2 σ ( hh∂∂− X X iψγ ∂ ψ) for ddττT ∫ µ ν αµ fermionic strings. In these action integrals, the spatial By differentiation we obtain the following system of coordinates ( = 1,2) describe a two-dimensional differential equations world sheet, the푖 quantities represent 2-dimensional 2 Dirac matrices,휎 the푖 quantities 휇 ( ) are mappings from ddttTTd(lnκ ) 2 − +=κ t 0 (15) the world manifold into the physical훾 휇 space-time, and 2 ddττ T dτ represents the geometry of the푋 2-휎dimensional manifold.휇휈 ℎ

International Journal of Physics 50

These surface actions are a generalisation of the familiar ∮PdA= nq. (22) action integral for a point particle, = , where the invariant interval is defined by the relation = From the result obtained in Equation (22), as in the case [20]. In the following, 푆however,−푚 ∫ 푑푠we focus2 of Bohr’s theory of quantum mechanics, we may consider attention휇 on휈 the general idea of a sum over random푑푠 a quantum process in which a physical entity transits from 휇휈 푔surfaces.푑푥 푑푥 This formulation is based on surface integral one surface to another with some radiation-like quantum methods by generalising the differential formulation as created in the process. Since this kind of physical process discussed for the Bohr’s model of a hydrogen-like atom. can be considered as a transition from one homotopy class Consider a surface in R defined by the relation = to another, the radiation-like quantum may be the result of ( , ) . The Gaussian3 curvature is given by 3the a change of the topological structure of the physical system, and so it can be regarded as a topological effect. relation1 2 = ( ( ) ) (1 + + ) , where푥 Furthermore, it is interesting to note that the action 푓 푥= 푥 and = 2 퐾 2[14]. 2Let2 be a 11 22 12 1 2 integral ( 4 ) is identical to Gauss’s law in three-dimensional퐾 휇 푓 푓physical− 푓 2quantity⁄ 휇 which휈푓 푓plays the role 휇 휇휈 푓of the휕푓 momentum⁄휕푥 푓 in the휕 two푓⁄휕푥-dimensional휕푥 space푃 action electrodynamics [21]. In this case the constant can be identified 푞with⁄ 휋 the∮ 퐾푑퐴charge of a particle, which represents integral. The quantity can be identified with the surface the topological structure of a physical system 푞and the density of a physical푝 quantity, such as charge. Since the charge of a physical system must exist in multiples of . momentum is proportional푃 to the curvature , which Hence, the charge of a physical system may depend on the determines the planar path of a particle, it is seen that in topological structure of the system and is classified by the푞 the three-dimensional푝 space the quantity should휅 be homotopy group of closed surfaces. This result may shed proportional to the Gaussian curvature , which is used to some light on why charge is quantised even in classical characterise a surface. If we consider a surface푃 action physics. As a further remark, we want to mention here that integral of the form = = ( 퐾2 ) , where in differential geometry, the Gaussian is related to the is a universal constant, which plays the role of Planck’s Ricci scalar curvature R by the relation R = 2 . And it constant, then we have푆 ∫ 푃푑퐴 ∫ 푞⁄ 휋 퐾푑퐴 푞 has been shown that the Ricci scalar 퐾curvature can be 2 identified with the potential of a physical퐾 system, q ff11 22−( f 12 ) 12 S = ∫ dx dx . (19) therefore our assumption of the existence of a relationship 2π 223/2 (1++ff12) between the Gaussian curvature and the surface density of a physical quantity can be justified [11]. Our discussions According to the calculus of variations, similar for random surfaces can also be generalised to a sum over to the case of path integral, to extremise the action random hypersurfaces of the form = ( , , … , ) integral = ( , , , ) , the functional in an (n+1)-dimensional Euclidean space. A generalised 푛+1 1 2 푛 ( , , , ) must satisfy the휇 differential1 2 equations [16] action integral is assumed to take the푥 form 푓 푥 푥 푥 푆 ∫ 퐿 푓 푓휇 푓휇휈 푥 푑푥 푑푥 휇 2 휇 휇휈 ∂LL ∂∂ ∂ ∂ L S= qnn K dA n, (23) 퐿 푓 푓 푓 푥 −+ =0. (20) ∫ ∂∂ffµ µν ∂ f ∂xµ ∂∂ xx µν where is a universal constant, which may be identified with some physical quantity, depending on the dimension Also as in the case of path integral, it is straightforward 푛 to verify that with the functional of the form of space,푞 and is the generalised Gaussian curvature defined as the product of the principal curvatures = 2 223/2 푛 L=( q/2π ) ( ff11 22 −( f 12 ) )/( 1 ++ f1 f 2 ) the differential … . In terms퐾 of the Riemannian curvature tensor, 퐾푛 equations given by Equation (20) are satisfied by any the generalised Gaussian curvature can be written as 푘1푘2 푘푛 surface. Hence, we can generalise Feynman’s postulate to µ…… µνν 11nn formulate a quantum theory in which the transition K= RR… (24) n n µµνν1 212 µn−− 1 µν nn 1 ν n amplitude between states of a quantum mechanical system 2 is a sum over random surfaces, provided the functional 2ng !det(µν ) in the action integral = is taken to be where , = 1,2, … and and are the proportional to the Gaussian curvature of a surface. 푃 푆 ∫ 푃푑퐴 metric tensor and the curvature tensor, respectively,푖 푖+1 푖 푖+1 of the Consider a closed surface and assume that we have many 휇푖 휈푖 푛 푔휇휈 푅휇 휇 휈 휈 such different surfaces which are described퐾 by the higher hyper-surface. From the theory of differential geometry in dimensional homotopy groups. As in the case of the higher dimensions, we have the generalised Gauss-Bonnet fundamental homotopy group of paths, we choose from theorem for the case when is even [22] among the homotopy class a representative spherical 푛 1 n surface, in which case we can write Knn dA= S χ, (25) ∫ 2 q PdA= dΩ, (21) where is the volume of an n-sphere and is the Euler ∮ π∮ 4 characteristic.푛 Hence, to be consistent with this result the where is an element of . Since variational푆 differential equation obtained from휒 the action depends on the homotopy class of the sphere that it integral given by Equation (23) must be satisfied by any represents,푑Ω we have = 4 , where is ∮ 푑theΩ hypersurface, since the Euler characteristic is a topological topological winding number of the higher dimensional invariant whose value depends only on the homotopy homotopy group. From this∮ 푑Ω result휋푛 we obtain a generalised푛 class, and not on the choice of an individual hypersurface. Bohr quantum condition In this case we have the generalised Bohr quantum condition

51 International Journal of Physics

∮Pnn dA= nq n (26) 0, and in this case the field equations given by1 Equation (27) reduces to where the universal constant connects the physical quantity 푅휇휈 − 2 푔휇휈푅 ≡ with the geometrical object . In higher-dimensional Λ 푛 = spaces, however, we do not푞 have a guiding relation, such Tgµν µν . (29) 푛 푛 κ as푃 de Broglie’s relation and Gauss’s퐾 law to identify the quantity . Nonetheless, this result also suggests that it It is indicated from Equation (29) that a physical entity may be possible to discuss a formalisation of the physics can be directly identified with a mathematical object. This 푛 that involves푞 manifolds of arbitrary dimension. interesting feature can be seen as an underlying principle for quantum physics. As long as the energy-momentum tensor is directly identified with the metric tensor through the relationship given by Equation (29) then for 4. Principle of Least Action in General 휇휈 휇휈 푇 1 푔 Relativity the case when = 2 we have Rµν−≡ gR µν 0, therefore 2 As a further examination, we will show in this section the resulting equations푛 from the principle of least action that Feynman’s postulate of path integral formulation of 1 Rµν− gR µν +=Λ g µνκ T µν are satisfied by any metric quantum mechanics, which can be verified by the 2 principle of least action as discussed above, may also be tensor . applied to the field equations of Einstein’s theory of general relativity. The field equations of general relativity 푔휇휈 are written in the form 5. Conclusion 1 Rµν− gR µν +=Λ g µνκ T µν (27) 2 In this work we have analysed the principle of least action that can be applied to various domains of physics, where is the energy-momentum tensor, is the namely the classical mechanics, the quantum mechanics metric tensor,휇휈 is the Ricci curvature tensor,휇휈 is the and the general relativity. We showed that the principle of 푇 푔 least action in the traditional formulation not only can be scalar curvature 휇휈and is the cosmological constant [23]. It is shown that 푅Equation (27) can be derived through푅 the applied to determine a unique path of a particle in classical principle of least actionΛ = 0, where the action is mechanics but also can be applied to determine the defined as quantum dynamics of a quantum particle with the 훿푆 푆 Feynman assumption that the quantum particle can take 1 4 any path to move in the spacetime continuum. These SR=∫( −+2Λ) M −gd x (28) 2κ results lead to the possibility to speculate that the mathematical formulation of physical laws is not only a where characterises matter fields [24]. In 1917, convenient mathematical method but there may be Einstein introduced the cosmological constant as an 푀 intrinsic relationships between physical entities and the additionℒ to his original field equations of general relativity mathematical objects. And the ultimate question that to retain the accepted view at the time that the universeΛ is arises from such speculation is whether it is possible to static. The reason for the addition is that if all matter formulate physics purely in terms mathematical objects by attracts each other than a static universe would not be able identifying them with physical entities. to remain static. The attractive gravity would cause the universe to collapse. Because the original field equations of general relativity contain only attractive forms of Acknowledgements gravity, a repulsive term is required. Einstein started his cosmological considerations by modifying Poisson’s I would like to thank the referees for their constructive = 4 equation with a repulsive term to form comments and suggestions for the improvement of the 2 = 4 the required equation , where denotes paper. I would also like to thank the administration of ∇ 휙 휋퐾휌 휆휙 a universal constant [23]2 . The modification of Poisson’s SciEP for their advices at various stages of the preparation ∇ 휙 − 휆휙 휋휅휌= (4 휆 ) equation results in the solution if for the publication of this work. matter was distributed uniformly through space with the 0 density . With these considerations,휙 Einstein− 휋휅 also⁄휆 noted휌 that in order to maintain the general covariance of the field 0 References equations,휌 the repulsive term that would be added to his field equations must be of the form . However, it is [1] S. Hojman, Problem of the identical vanishing of Euler-Lagrange observed that the requirements of repulsion and derivatives in field theory, Phys. Rev. D., 27, 451-453 (1983). 휇휈 covariance can be acquired by theΛ 푔energy-momentum [2] S. Hojman, Symmetries of Lagrangians and of their equations of tensor because if is considered to be some form of motion, J. Phys. A: Math. Gen., 17, 2399-2412 (1984). [3] I. Yu. Krivsky and V. M. Simulik, Noether analysis of zilch matter then it should be able to produce some form of conservation laws and their generalisation for the electromagnetic energy, in particular,Λ a . The fact that the field, Theor. Math. Phys. 80, 864-874 (1989). term can be replaced by an energy-momentum [4] I. Yu. Krivsky and V. M. Simulik, Noether analysis of zilch tensor can be demonstrated for the case when = 2 as conservation laws and their generalisation for the electromagnetic 휇휈 field, Theor. Math. Phys. 80, 912-921 (1989). follows.Λ푔 It is shown that when = 2 we have the identity 푛 푛 International Journal of Physics 52

[5] W. I. Fushchych, I. Yu. Krivsky and V. M. Simulik, On vector and [15] A. Prastaro, Geometry of PDEs and Mechanics (World Scientific, pseudo-vector Lagrangians for electromagnetic field, Nuovo Cim Singapore, 1996). B. 103, 423-429 (1989). [16] C. Lanczos, The Variational Principles of Mechanics (Dover [6] R. R. Lompay and A. N. Petrov, Covariant differential identities Publications, New York, 1970). and conservation laws in metric-torsion theories of gravitation, J. [17] Vu B Ho, A Temporal Dynamics: A Generalised Newtonian and Math. Phys. 54, 1-30 (2013). Wave Mechanics (Pre-print, ResearchGate, 2016), viXra [7] N. Bohr, Phil. Mag. 26, 1 (1913). 1708.0198v1. [8] H. Goldstein, Classical Mechanics (Addison-Wesley Inc., Sydney, [18] S. W. Hawking, General Relativity, An Einstein Centenary Survey, 1980). edited by S. W. Hawking and W. Israel (Cambridge University [9] R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948). Press, 1979). [10] Vu B Ho, Geometrical and Topological Methods in Classical and [19] A.M. Polyakov, Phys. Lett. B, 103, 207 (1981). Quantum Physics (PhD thesis, Monash University, Australia, [20] M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory 1996). (Cambridge University Press, Melbourne, 1988). [11] Vu B Ho, Spacetime Structures of Quantum Particles (Preprint, [21] J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, ResearchGate, 2017), viXra 1708.0192v1. New York, 1975). [12] W. Yourgrau and S. Mandelstam, Variational Principles in [22] M. Spivak, A Comprehensive Introduction to Differential Dynamics and Quantum Theory (Dover Publications, New York, Geometry (Publish or Perish, Berkerley, 1979). 1979). [23] A. Einstein, The Principle of Relativity (Dover Publications, New [13] I. M. Singer and J. A. Thorpe, Lectures Notes on Elementary York, 1952). Topology and Geometry (Scott,Foresman & Co., MIT, 1967). [24] R. M. Wald, General Relativity (The University of Chicago Press, [14] E. Kreyszig, Introduction to Differential Geometry and London, 1984). Riemannian Geometry (University of Toronto Press, 1975).