Conventional descriptions of transverse waves in an elastic solid are limited
by an assumption of infinitesimally small gradients of rotation. By assuming a
linear response to variations in orientation, we derive an exact description of
a restricted class of rotational waves in an ideal isotropic elastic solid. The
result is a nonlinear equation expressed in terms of Dirac bispinors. This
result provides a simple classical interpretation of relativistic quantum
mechanical dynamics. We construct a Lagrangian of the form L=-E+U+K=0, where E
is the total energy, U is the potential energy, and K is the kinetic energy.Comment: 9 pages; Added references in revisio

Close, R. A.

English

arXiv

R. A. Close

Crossref

Adv. Appl. Cliﬀ ord Algebras 21 (2011), 273–281© 2010 The Author(s) This article is published with open access at Springerlink.com 0188-7009/020273-9published online November 13, 2010DOI 10.1007/s00006-010-0249-1Exact Description of Rotational Waves in anElastic SolidR. A. CloseAbstract. Conventional descriptions of transverse waves in an elasticsolid are limited by an assumption of inﬁnitesimally small gradients ofrotation. By assuming a linear response to variations in orientation,we derive an exact description of a restricted class of rotational wavesin an ideal isotropic elastic solid. The result is a nonlinear equationexpressed in terms of Dirac bispinors. This result provides a simpleclassical interpretation of relativistic quantum mechanical dynamics. Weconstruct a Lagrangian of the form L = −E + U +K = 0, where E isthe total energy, U is the potential energy, and K is the kinetic energy.Mathematics Subject Classiﬁcation (2010). 74B05, 81P10.Keywords. Elastic solid, rotational waves, transverse waves, Dirac equa-tion.1. IntroductionThe ideal elastic solid has been an important model in the history of physics.It is a good approximation for many processes in condensed matter. It wasalso the basis of early theories of light because of its ability to support trans-verse waves [14]. Aether models fell into disuse at the end of the 19th centurywhen experiments failed to detect variations in light speed relative to thedirection of the earth’s motion through space [8]. However, de Broglie’s wavehypothesis for matter [2] predicts a null result for such aether-drift experi-ments whether or not the waves propagate through a material medium.It was recently proposed that a lattice of elastic cells would yield fermi-ons and gauge ﬁelds equivalent to the Standard Model of particle physics [12].This mechanical model is similar to the one James Maxwell used to developthe equations of electromagnetism [14]. While a homogeneous elastic solidmay be regarded as a continuum limit of such a lattice, it is fundamentallysimpler in structure.It is current practice to describe physical events in terms of discreteelementary particles rather than continuous waves. However, the Dirac andAdvances inApplied Cliﬀ ord Algebras274 R. A. Close Adv. Appl. Cliﬀ ord AlgebrasKlein-Gordon equations which are used to compute the correlations betweenparticle events also determine the wave-like evolution of continuously dis-tributed physical quantities such as momentum, energy, and angular mo-mentum densities [13, 5]. Furthermore, the ﬁrst-order Dirac equation maybe regarded as a representation of a second-order wave equation with thebispinor wave function encoding the ﬁrst derivatives [7, 9, 10, 11, 3]. Half-integer spin is attributable to the fact that waves propagating in oppositedirections are independent states separated by 180-degree rotation. Thus itis quite possible that a single wave equation may determine the evolution ofdynamical quantities everywhere in space.An ideal homogeneous, isotropic, elastic solid (with no dissipation andconstant elasticity) is the simplest possible model for wave-like processes inthree spatial dimensions. Hence the physics of an ideal elastic solid shouldbe the foundation for any analysis of Lorentz-covariant processes in three-dimensional space. Yet this simple model has never been properly understood.Prior analyses of the elastic solid rely on approximations of inﬁnitesi-mally small rotations or small rates of variation of rotations [6]. Analysis ofﬁrst-order stress and strain yields an elastic wave equation for displacementa (r, t):ρ∂2t a = (2μ + λ)∇ [∇ · a] − μ∇× [∇× a] . (1)This equation has two obvious ﬂaws. First, since the displacement ﬁeldis deﬁned with respect to positions in space rather than particular elementsof the solid, the value of the ﬁeld is transported with the solid as it moves.Inclusion of convection by velocity u and rotation by vorticity w yields:ρ∂2t a + ρu · ∇∂ta− ρw × ∂ta = (2μ + λ)∇ [∇ · a] − μ∇× [∇× a] . (2)The second ﬂaw is more problematic. A rigid rotation by angle Θ yieldsa non-zero divergence of the displacement ﬁeld (∇ · a = 2 (cosΘ − 1)). If Θvaries with position either radially or along the rotation axis, then the motionpreserves volume and the elastic response should be proportional to the shearmodulus μ. Instead, the term (2μ + λ)∇ [∇ · a] fails to distinguish betweencompressional and rotational origins of divergence of the displacement ﬁeld.This is why the equation is only valid for inﬁnitesimally small gradients ofrotation.The goal of this paper is to formulate an exact description of rotational(or transverse) waves in an ideal elastic solid. We will accomplish this taskby describing incompressible motion in terms of rotations rather than dis-placements.2. Basic AssumptionsWe make the following basic assumptions:1. The elastic solid is characterized by an inertial density ρ and coeﬃcient ofelasticity μ, with characteristic wave speed c =√μ/ρ.2. There is a linear response to variations of orientation angle Θ relativeVol. 21 (2011) Description of Rotational Waves in an Elastic Solid 275to equilibrium. This means that an initial static perturbation (with velocityu (r) = 0) would yield the response:∂2tΘ = c2∇2Θ. (3)3. The velocity ﬁeld u has no divergence (∇ · u =0). Therefore the velocitymay be written as the curl of a vector ﬁeld:u =12ρ[∇× J¯] . (4)The vector ﬁeld J¯ is called the conjugate angular momentum density. It diﬀersfrom the usual deﬁnition of angular momentum density J = r × ρu in thatit is independent of the choice of origin and can have arbitrary direction. Ifu falls to zero suﬃciently rapidly toward inﬁnity, then integration by partsyields the alternative expressions for kinetic energy:K =12∫d3r ρu2 =12∫d3rw · J¯ (5)where w = ∇ × u/2 is the angular velocity, or vorticity. Hence J¯ is thevariable conjugate to angular velocity for a Lagrangian which depends on uonly through the (positive) kinetic energy.4. We assume that there are no velocity-dependent forces (such as frictionaldamping). Hence velocity only enters the equations of evolution through theconvection and rotation of ﬁelds. We neglect any eﬀect of the symmetrictensor (∂iuj +∂jui) on the orientation angle or angular momentum direction.Additional assumptions will be introduced in order to simplify the math-ematics, and these may limit the generality of the results.3. Equation of EvolutionStarting from (3), we deﬁne an angular potential Q such that: ∇2Q = −4ρΘ.The static condition for Q is:∇2 {∂2tQ− c2∇2Q} = 0 ( if u (r) = 0 everywhere) . (6)Deﬁne the spin angular momentum as:S ≡ ∂tQ. (7)When motion is present, it contributes to the time derivative ∂tS only throughconvection (−u · ∇S) and rotation (w × S):∇2 {∂2tQ− c2∇2Q + u · ∇S−w × S} = 0. (8)From here on, we will consider only wave-like solutions satisfying:∂2tQ− c2∇2Q + u · ∇S−w × S = 0. (9)For oscillatory solutions to this equation, the ﬁrst two terms are always inphase (∂2tQ − c2∇2Q), whereas the velocity-dependent terms may have dif-ferent phase. However, if the velocity-dependent terms do not add to zero276 R. A. Close Adv. Appl. Cliﬀ ord Algebrasthen they must have the same temporal phase as the linear terms:u · ∇S−w × S = Ω2 (r)Q (10)where Ω2 (r) is some function of position (more generally, Ω2 (r) could havediﬀerent values for each component of Q). Substitution yields:∂2tQ− c2∇2Q + Ω2 (r)Q = 0. (11)If Ω2 (r) is constant and positive, then this is the Klein-Gordon equation.Now our only remaining task is to solve for the velocity in terms ofother wave variables. To do this we note that the above equation (9) can bewritten in terms of a four-component complex Dirac bispinor (ψ) using thefollowing identiﬁcations:Sj ≡ ∂tQj ≡ 12[ψ†σjψ]c [∇ ·Q] ≡ −12[ψ†γ5ψ]c2 {∇ ×∇×Q}j ≡ −i2cijk{[∂iψ†] γ5σkψ − ψ†γ5σk∂iψ} . (12)This identiﬁcation between bispinors and classical variables was made previ-ously [4], where it was also assumed that the velocity could be derived fromthe spin. In this paper we will derive an expression for velocity from theconjugate momentum.The matrices σj are the Dirac spin matrices:σ1 =⎛⎜⎜⎝0 1 0 01 0 0 00 0 0 10 0 1 0⎞⎟⎟⎠ , σ2 =⎛⎜⎜⎝0 −i 0 0i 0 0 00 0 0 −i0 0 i 0⎞⎟⎟⎠ ,σ3 =⎛⎜⎜⎝1 0 0 00 −1 0 00 0 1 00 0 0 −1⎞⎟⎟⎠ . (13)The matrices cγ5σj are the Dirac velocity matrices. One representation forγ5 is:γ5 =⎛⎜⎜⎝0 0 1 00 0 0 11 0 0 00 1 0 0⎞⎟⎟⎠ . (14)The above identiﬁcations provide seven constraints on the eight free parame-ters of the Dirac bispinor. In terms of bispinors, the rotational wave equation(9) is:∂∂t[ψ†σjψ]+ c∂j[ψ†γ5ψ]− icijk {∂iψ†γ5σkψ − ψ†γ5σk∂iψ}+ u · ∇ [ψ†σjψ]− kijwk [ψ†σiψ] = 0. (15)Vol. 21 (2011) Description of Rotational Waves in an Elastic Solid 277Expanding the derivatives yields:ψ†σj[∂tψ + cγ5σ · ∇ψ + u · ∇ψ + w · iσ2 ψ]+ c.c. = 0 (16)where (c.c.) represents the complex conjugate. The Hermitian conjugate wavefunction may be regarded as an independent variable (the independent realand imaginary parts of the wave function are linear combinations of elementsof ψ and ψ†). Validity for arbitrary ψ† requires the terms in brackets to sumto zero. This yields the Dirac equation:∂tψ + cγ5σ · ∇ψ + u · ∇ψ + iw · σ2 ψ + iχψ = 0 (17)where χ may be any operator with the property:Re(ψ†σj iχψ)= 0. (18)Since χ has no eﬀect on the original equation for Q, we assume it to be zero.Now we construct a Lagrange density L . Lagrange’s equations of mo-tion for a ﬁeld variable ψ are:∂t∂L∂ [∂tψ]+∑j∂j∂L∂ [∂jψ]− ∂L∂ψ= 0. (19)A similar equation holds with ψ† replacing ψ. A possible Lagrange densityfor rotational waves is therefore:L = −iψ†∂tψ + ψ†[−icγ5σ · ∇]ψ + ψ† [−iu · ∇ + w · σ2]ψ. (20)Using the Hermitian conjugate of (19) yields simply ∂L /∂ψ† = 0. Since thecomplex conjuate of the equation of evolution is also valid, we can requirethe Lagrange density to be real:L = Re{−iψ†∂tψ + ψ†[−icγ5σ · ∇]ψ + ψ† [−iu · ∇ + w · σ2]ψ}. (21)The conjugate momentum to the ﬁeld ψ is pψ:pψ =∂L∂ [∂tψ]= −iψ†. (22)We assume that we can neglect boundary terms in the integration by partsof: ∫d3rw · S = 12∫d3r [∇× u] · S = 12∫d3r u · [∇× S] . (23)The conjugate momentum for r is:pr =∂L∂u= Re(−ψ†i∇ψ)+ 12∇× ψ†σ2ψ = ρu. (24)This conjugate momentum was derived under the assumption that u is anindependent variable. Identiﬁcation of pr with ρu is justiﬁed by the lack ofexternal forces, implying that ρu enters the Lagrangian only through thekinetic energy.278 R. A. Close Adv. Appl. Cliﬀ ord AlgebrasMaking u a function of ψ introduces a factor of 1/2:L = Re{−iψ†∂tψ + ψ† [−icγ5σ · ∇] ψ}+12ρ[Re(−ψ†i∇ψ)+ 12∇× ψ†σ2ψ]2. (25)Variation of this Lagrange density determines the evolution of rotationalwaves:∂tψ + cγ5σ · ∇ψ + u · ∇ψ + iw · σ2 ψ = 0 (26)where velocity u and vorticity w = ∇×u/2 are functions of ψ as determinedfrom (24).4. Dynamical Variables4.1. Angular MomentumRecall that the velocity is the curl of an origin-independent angular momen-tum:u =12ρ∇× J¯. (27)We can separate the conjugate angular momentum (J¯) into orbital (L¯) andspin (S) components by identifying the orbital (uL) and spin (uS) contribu-tions to velocity:uL =1ρRe(−ψ†i∇ψ) = 12ρ∇× L¯uS =12ρ∇× ψ†σ2ψ =12ρ∇× S. (28)The incompressibility condition ∇ · u = 0 places an additional restriction onthe wave function:∇·ρuL = 12∇·[−ψ†i∇ψ + i [∇ψ] ψ] = i2{−ψ†∇2ψ + [∇2ψ†] ψ} = 0. (29)The usual deﬁnition of angular momentum J = L+S depends on the choiceof origin. If we deﬁne a rotational velocity as uR = w × r and note thatthe vorticity is the instantaneous angular velocity (w = dΘ/dt), then from(19) the conjugate angular momentum would have the same form as foundin relativistic quantum mechanics:pΘ =∂L∂ [∂tΘ]= ψ†{−r× i∇ + σ2}ψ = L + S. (30)4.2. Energy and momentumIn the Lagrange density deﬁned in (25), the velocity-dependent term is clearlythe kinetic energy density. This observation suggests that the Lagrangian hasthe form:L = Re{−iψ†∂tψ + ψ†[−icγ5σ · ∇] ψ + 12ρu2}= −E + U + K (31)Vol. 21 (2011) Description of Rotational Waves in an Elastic Solid 279where E ≡ Re (iψ†∂tψ) is the total energy, U ≡ Re (−ψ† [icγ5σ · ∇] ψ) ispotential energy, and K is kinetic energy density. We will simply adopt thesedeﬁnitions without further justiﬁcation.The Hamiltonian is the negative of the energy:H = pψ∂tψ −L = Re{ψ†[icγ5σ · ∇ + 12(iu · ∇ −w · σ2)]ψ}= −{U +12ρu2}. (32)Hamilton’s equation for the wave function is:∂tψ =∂H∂pψ=∂H∂ [−iψ†] ={−cγ5σ · ∇ − u · ∇ − iw · σ2}ψ. (33)We can also deﬁne a Hamiltonian operator with ∂tψ = iHψ (note oppositesign convention from quantum mechanics):H = icγ5σ · ∇ + iu · ∇ −w · σ2. (34)The Hamiltonian is a special case (T 00 ) of the energy-momentum tensor:Tμν =∂L∂ [∂μψ]∂νψ −L δμν . (35)The dynamical momentum density P is derived from the orbital part of theangular momentum:T 0i = Pi = −iψ†∂iψ. (36)This is identical to the dynamical momentum of relativistic quantum me-chanics. Unlike the conjugate momentum, it does not include the rotationalmotion associated with the spin angular momentum.The sign of the Hamiltonian (and Lagrangian) is simply a convention.We chose signs to make the conjugate momentum parallel (rather than anti-parallel) to velocity, thus requiring the Hamiltonian to be the negative of theenergy. In analogy with plane waves, the function cos (ωt− kx) with ω > 0is equivalent to cos (ωt + kx) with ω < 0. We could change the sign of theLagrangian and Hamiltonian and still preserve the sign of the momentumby using covariant derivatives (∂μ = (∂t,−∂i)). However, such a relativis-tic construction is unnecessary and perhaps misleading since we are in factdealing with Galilean space-time (the laws of relativity apply to the spaceof measurements, not to the absolute space-time). The energy-momentumtensor components T 0μ = (−E , Pi) may still be regarded as a covariant vectorwhose magnitude is the scalar E 2−P 2. The equation of evolution is of courseunaﬀected by the choice of sign of the Lagrangian.5. DiscussionThe description of rotational waves derived here provides a simple classicalinterpretation of Dirac bispinors and quantum mechanical operators. Due tothe nonlinear nature of the equation of evolution, we expect that an elastic280 R. A. Close Adv. Appl. Cliﬀ ord Algebrassolid has a discrete spectrum of fermionic soliton solutions. If each soliton isassigned its own independent wave function, then cancellation of interferencebetween these independent ‘particles’ would yield the Pauli exclusion princi-ple and require the introduction of interaction potentials [4]. Characterizationof these interaction potentials is beyond the scope of this paper.Correlations between rotational soliton waves would be computed in thesame manner as in relativistic quantum mechanics [4]. The reason is that thebispinor wave functions, and not the measurement values, satisfy a ﬁrst orderwave equation which may be used to directly compute variations in space andtime. Therefore Bell’s Theorem [1] is not applicable.The quantum mechanical Dirac equation is also equivalent to a deter-ministic equation for the evolution of spin angular momentum density. Com-parison of the classical and quantum equations for spin density may yieldnew understanding of the physical properties of elementary particles.6. ConclusionsA nonlinear Dirac equation describes evolution of rotational waves in an idealisotropic elastic solid. The equation is derived by assuming a linear responseto variations in orientation and incorporating the inﬂuence of motion throughconvection and rotation. The Hamiltonian is represented as a sum of potentialand kinetic energy. This result provides a simple classical interpretation ofDirac bispinors and the associated dynamical operators of relativistic quan-tum mechanics.AcknowledgmentThe author is grateful to Damon Merari for his interest and encouragement.References[1] J. S. Bell, On the Einstein Podolsky Rosen Paradox. Phys. 1 (1964), 195-200.[2] L. de Broglie, Recherches sur la The´orie des Quanta. PhD Thesis, Universityof Sorbonne, Paris, 1924.[3] R. A. Close, Torsion waves in three dimensions: quantum mechanics with atwist. Found. Phys. Lett. 15 (2002), 71-83.[4] R. A. Close, A Classical Dirac Equation. In: Ether Space-time & Cosmology,Vol 3. (Physical vacuum, relativity, and quantum physics) eds. M.C. Duﬀy andJ. Levy, Apeiron, Montreal 2009, 49-73.[5] D. Hestenes, Local observables in the Dirac theory. J. Math. Phys. 14 (7) (1973),893-905.[6] H. Kleinert, Gauge Fields in Condensed Matter. vol II. World Scientiﬁc, Sin-gapore, 1989, 1240-64.[7] S. Matsutani and H. Tsuru, Physical relation between quantum mechanics andsolitons on a thin elastic rod. Phys. Rev. A 46 (1992), 1144-7.[8] A. A. Michelson and E. W. Morley, On the relative motion of the earth and theluminiferous ether. Am. J. Sci. (3rd series) 34 (1887), 333-45.Vol. 21 (2011) Description of Rotational Waves in an Elastic Solid 281[9] P. Rowlands, The physical consequences of a new version of the Dirac equation.In: Causality and Locality in Modern Physics and Astronomy: Open Ques-tions and Possible Solutions (Fundamental Theories of Physics, vol 97) eds G.Hunter, S. Jeﬀers, and J-P. Vigier, Kluwer Academic Publishers, Dordrecht,1998, 397-402.[10] P. Rowlands, J. P. Cullerne, The connection between the Han-Nambu quarktheory, the Dirac equation and fundamental symmetries. Nuclear Phys. A 684(2001), 713-715.[11] P. Rowlands, Removing redundancy in relativistic quantum mechanics.Preprint: arXiv:physics/0507188 (2005).[12] I. Schmelzer, A condensed matter interpretation of SM fermions and gaugeﬁelds. Found. Phys. 39 (1) (2009), 73-107.[13] Y. Takabayashi, Relativistic hydrodynamics of the Dirac matter. Suppl. Prog.Theor. Phys. 4 (1) (1957), 1-80.[14] E. Whittaker, A History of the Theories of Aether and Electricity. Vol 1.Thomas Nelson and Sons Ltd., Edinburgh, 1951, 128-69.R. A. Close4110 SE Hawthorne Blvd. #232Portland, OR 97214USAe-mail: robert.close@classicalmatter.orgReceived: July 16, 2009.Accepted: October 8, 2009.Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Advances in Applied Clifford Algebras

Exact Description of Rotational Waves in an Elastic Solid

Springer - Publisher Connector

Adv. Appl. Cliﬀ ord Algebras 21 (2011), 273–281
© 2010 The Author(s) This article is published with 
open access at Springerlink.com 
0188-7009/020273-9
published online November 13, 2010
DOI 10.1007/s00006-010-0249-1
Exact Description of Rotational Waves in an
Elastic Solid
R. A. Close
Abstract. Conventional descriptions of transverse waves in an elastic
solid are limited by an assumption of inﬁnitesimally small gradients of
rotation. By assuming a linear response to variations in orientation,
we derive an exact description of a restricted class of rotational waves
in an ideal isotropic elastic solid. The result is a nonlinear equation
expressed in terms of Dirac bispinors. This result provides a simple
classical interpretation of relativistic quantum mechanical dynamics. We
construct a Lagrangian of the form L = −E + U +K = 0, where E is
the total energy, U is the potential energy, and K is the kinetic energy.
Mathematics Subject Classiﬁcation (2010). 74B05, 81P10.
Keywords. Elastic solid, rotational waves, transverse waves, Dirac equa-
tion.
1. Introduction
The ideal elastic solid has been an important model in the history of physics.
It is a good approximation for many processes in condensed matter. It was
also the basis of early theories of light because of its ability to support trans-
verse waves [14]. Aether models fell into disuse at the end of the 19th century
when experiments failed to detect variations in light speed relative to the
direction of the earth’s motion through space [8]. However, de Broglie’s wave
hypothesis for matter [2] predicts a null result for such aether-drift experi-
ments whether or not the waves propagate through a material medium.
It was recently proposed that a lattice of elastic cells would yield fermi-
ons and gauge ﬁelds equivalent to the Standard Model of particle physics [12].
This mechanical model is similar to the one James Maxwell used to develop
the equations of electromagnetism [14]. While a homogeneous elastic solid
may be regarded as a continuum limit of such a lattice, it is fundamentally
simpler in structure.
It is current practice to describe physical events in terms of discrete
elementary particles rather than continuous waves. However, the Dirac and
Advances in
Applied Cliﬀ ord Algebras
274 R. A. Close Adv. Appl. Cliﬀ ord Algebras
Klein-Gordon equations which are used to compute the correlations between
particle events also determine the wave-like evolution of continuously dis-
tributed physical quantities such as momentum, energy, and angular mo-
mentum densities [13, 5]. Furthermore, the ﬁrst-order Dirac equation may
be regarded as a representation of a second-order wave equation with the
bispinor wave function encoding the ﬁrst derivatives [7, 9, 10, 11, 3]. Half-
integer spin is attributable to the fact that waves propagating in opposite
directions are independent states separated by 180-degree rotation. Thus it
is quite possible that a single wave equation may determine the evolution of
dynamical quantities everywhere in space.
An ideal homogeneous, isotropic, elastic solid (with no dissipation and
constant elasticity) is the simplest possible model for wave-like processes in
three spatial dimensions. Hence the physics of an ideal elastic solid should
be the foundation for any analysis of Lorentz-covariant processes in three-
dimensional space. Yet this simple model has never been properly understood.
Prior analyses of the elastic solid rely on approximations of inﬁnitesi-
mally small rotations or small rates of variation of rotations [6]. Analysis of
ﬁrst-order stress and strain yields an elastic wave equation for displacement
a (r, t):
ρ∂2t a = (2μ + λ)∇ [∇ · a] − μ∇× [∇× a] . (1)
This equation has two obvious ﬂaws. First, since the displacement ﬁeld
is deﬁned with respect to positions in space rather than particular elements
of the solid, the value of the ﬁeld is transported with the solid as it moves.
Inclusion of convection by velocity u and rotation by vorticity w yields:
ρ∂2t a + ρu · ∇∂ta− ρw × ∂ta = (2μ + λ)∇ [∇ · a] − μ∇× [∇× a] . (2)
The second ﬂaw is more problematic. A rigid rotation by angle Θ yields
a non-zero divergence of the displacement ﬁeld (∇ · a = 2 (cosΘ − 1)). If Θ
varies with position either radially or along the rotation axis, then the motion
preserves volume and the elastic response should be proportional to the shear
modulus μ. Instead, the term (2μ + λ)∇ [∇ · a] fails to distinguish between
compressional and rotational origins of divergence of the displacement ﬁeld.
This is why the equation is only valid for inﬁnitesimally small gradients of
rotation.
The goal of this paper is to formulate an exact description of rotational
(or transverse) waves in an ideal elastic solid. We will accomplish this task
by describing incompressible motion in terms of rotations rather than dis-
placements.
2. Basic Assumptions
We make the following basic assumptions:
1. The elastic solid is characterized by an inertial density ρ and coeﬃcient of
elasticity μ, with characteristic wave speed c =
√
μ/ρ.
2. There is a linear response to variations of orientation angle Θ relative
Vol. 21 (2011) Description of Rotational Waves in an Elastic Solid 275
to equilibrium. This means that an initial static perturbation (with velocity
u (r) = 0) would yield the response:
∂2tΘ = c
2∇2Θ. (3)
3. The velocity ﬁeld u has no divergence (∇ · u =0). Therefore the velocity
may be written as the curl of a vector ﬁeld:
u =
1
2ρ
[∇× J¯] . (4)
The vector ﬁeld J¯ is called the conjugate angular momentum density. It diﬀers
from the usual deﬁnition of angular momentum density J = r × ρu in that
it is independent of the choice of origin and can have arbitrary direction. If
u falls to zero suﬃciently rapidly toward inﬁnity, then integration by parts
yields the alternative expressions for kinetic energy:
K =
1
2
∫
d3r ρu2 =
1
2
∫
d3rw · J¯ (5)
where w = ∇ × u/2 is the angular velocity, or vorticity. Hence J¯ is the
variable conjugate to angular velocity for a Lagrangian which depends on u
only through the (positive) kinetic energy.
4. We assume that there are no velocity-dependent forces (such as frictional
damping). Hence velocity only enters the equations of evolution through the
convection and rotation of ﬁelds. We neglect any eﬀect of the symmetric
tensor (∂iuj +∂jui) on the orientation angle or angular momentum direction.
Additional assumptions will be introduced in order to simplify the math-
ematics, and these may limit the generality of the results.
3. Equation of Evolution
Starting from (3), we deﬁne an angular potential Q such that: ∇2Q = −4ρΘ.
The static condition for Q is:
∇2 {∂2tQ− c2∇2Q} = 0 ( if u (r) = 0 everywhere) . (6)
Deﬁne the spin angular momentum as:
S ≡ ∂tQ. (7)
When motion is present, it contributes to the time derivative ∂tS only through
convection (−u · ∇S) and rotation (w × S):
∇2 {∂2tQ− c2∇2Q + u · ∇S−w × S} = 0. (8)
From here on, we will consider only wave-like solutions satisfying:
∂2tQ− c2∇2Q + u · ∇S−w × S = 0. (9)
For oscillatory solutions to this equation, the ﬁrst two terms are always in
phase (∂2tQ − c2∇2Q), whereas the velocity-dependent terms may have dif-
ferent phase. However, if the velocity-dependent terms do not add to zero
276 R. A. Close Adv. Appl. Cliﬀ ord Algebras
then they must have the same temporal phase as the linear terms:
u · ∇S−w × S = Ω2 (r)Q (10)
where Ω2 (r) is some function of position (more generally, Ω2 (r) could have
diﬀerent values for each component of Q). Substitution yields:
∂2tQ− c2∇2Q + Ω2 (r)Q = 0. (11)
If Ω2 (r) is constant and positive, then this is the Klein-Gordon equation.
Now our only remaining task is to solve for the velocity in terms of
other wave variables. To do this we note that the above equation (9) can be
written in terms of a four-component complex Dirac bispinor (ψ) using the
following identiﬁcations:
Sj ≡ ∂tQj ≡ 12
[
ψ†σjψ
]
c [∇ ·Q] ≡ −1
2
[
ψ†γ5ψ
]
c2 {∇ ×∇×Q}j ≡ −
i
2
cijk
{[
∂iψ
†] γ5σkψ − ψ†γ5σk∂iψ} . (12)
This identiﬁcation between bispinors and classical variables was made previ-
ously [4], where it was also assumed that the velocity could be derived from
the spin. In this paper we will derive an expression for velocity from the
conjugate momentum.
The matrices σj are the Dirac spin matrices:
σ1 =
⎛
⎜⎜⎝
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
⎞
⎟⎟⎠ , σ2 =
⎛
⎜⎜⎝
0 −i 0 0
i 0 0 0
0 0 0 −i
0 0 i 0
⎞
⎟⎟⎠ ,
σ3 =
⎛
⎜⎜⎝
1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 −1
⎞
⎟⎟⎠ . (13)
The matrices cγ5σj are the Dirac velocity matrices. One representation for
γ5 is:
γ5 =
⎛
⎜⎜⎝
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
⎞
⎟⎟⎠ . (14)
The above identiﬁcations provide seven constraints on the eight free parame-
ters of the Dirac bispinor. In terms of bispinors, the rotational wave equation
(9) is:
∂
∂t
[
ψ†σjψ
]
+ c∂j
[
ψ†γ5ψ
]− icijk {∂iψ†γ5σkψ − ψ†γ5σk∂iψ}
+ u · ∇ [ψ†σjψ]− kijwk [ψ†σiψ] = 0. (15)
Vol. 21 (2011) Description of Rotational Waves in an Elastic Solid 277
Expanding the derivatives yields:
ψ†σj
[
∂tψ + cγ5σ · ∇ψ + u · ∇ψ + w · iσ2 ψ
]
+ c.c. = 0 (16)
where (c.c.) represents the complex conjugate. The Hermitian conjugate wave
function may be regarded as an independent variable (the independent real
and imaginary parts of the wave function are linear combinations of elements
of ψ and ψ†). Validity for arbitrary ψ† requires the terms in brackets to sum
to zero. This yields the Dirac equation:
∂tψ + cγ5σ · ∇ψ + u · ∇ψ + iw · σ2 ψ + iχψ = 0 (17)
where χ may be any operator with the property:
Re
(
ψ†σj iχψ
)
= 0. (18)
Since χ has no eﬀect on the original equation for Q, we assume it to be zero.
Now we construct a Lagrange density L . Lagrange’s equations of mo-
tion for a ﬁeld variable ψ are:
∂t
∂L
∂ [∂tψ]
+
∑
j
∂j
∂L
∂ [∂jψ]
− ∂L
∂ψ
= 0. (19)
A similar equation holds with ψ† replacing ψ. A possible Lagrange density
for rotational waves is therefore:
L = −iψ†∂tψ + ψ†
[−icγ5σ · ∇]ψ + ψ† [−iu · ∇ + w · σ
2
]
ψ. (20)
Using the Hermitian conjugate of (19) yields simply ∂L /∂ψ† = 0. Since the
complex conjuate of the equation of evolution is also valid, we can require
the Lagrange density to be real:
L = Re
{
−iψ†∂tψ + ψ†
[−icγ5σ · ∇]ψ + ψ† [−iu · ∇ + w · σ
2
]
ψ
}
. (21)
The conjugate momentum to the ﬁeld ψ is pψ:
pψ =
∂L
∂ [∂tψ]
= −iψ†. (22)
We assume that we can neglect boundary terms in the integration by parts
of: ∫
d3rw · S = 1
2
∫
d3r [∇× u] · S = 1
2
∫
d3r u · [∇× S] . (23)
The conjugate momentum for r is:
pr =
∂L
∂u
= Re
(−ψ†i∇ψ)+ 1
2
∇× ψ†σ
2
ψ = ρu. (24)
This conjugate momentum was derived under the assumption that u is an
independent variable. Identiﬁcation of pr with ρu is justiﬁed by the lack of
external forces, implying that ρu enters the Lagrangian only through the
kinetic energy.
278 R. A. Close Adv. Appl. Cliﬀ ord Algebras
Making u a function of ψ introduces a factor of 1/2:
L = Re
{−iψ†∂tψ + ψ† [−icγ5σ · ∇] ψ}
+
1
2ρ
[
Re
(−ψ†i∇ψ)+ 1
2
∇× ψ†σ
2
ψ
]2
. (25)
Variation of this Lagrange density determines the evolution of rotational
waves:
∂tψ + cγ5σ · ∇ψ + u · ∇ψ + iw · σ2 ψ = 0 (26)
where velocity u and vorticity w = ∇×u/2 are functions of ψ as determined
from (24).
4. Dynamical Variables
4.1. Angular Momentum
Recall that the velocity is the curl of an origin-independent angular momen-
tum:
u =
1
2ρ
∇× J¯. (27)
We can separate the conjugate angular momentum (J¯) into orbital (L¯) and
spin (S) components by identifying the orbital (uL) and spin (uS) contribu-
tions to velocity:
uL =
1
ρ
Re
(−ψ†i∇ψ) = 1
2ρ
∇× L¯
uS =
1
2ρ
∇× ψ†σ
2
ψ =
1
2ρ
∇× S. (28)
The incompressibility condition ∇ · u = 0 places an additional restriction on
the wave function:
∇·ρuL = 12∇·
[−ψ†i∇ψ + i [∇ψ] ψ] = i
2
{−ψ†∇2ψ + [∇2ψ†] ψ} = 0. (29)
The usual deﬁnition of angular momentum J = L+S depends on the choice
of origin. If we deﬁne a rotational velocity as uR = w × r and note that
the vorticity is the instantaneous angular velocity (w = dΘ/dt), then from
(19) the conjugate angular momentum would have the same form as found
in relativistic quantum mechanics:
pΘ =
∂L
∂ [∂tΘ]
= ψ†
{
−r× i∇ + σ
2
}
ψ = L + S. (30)
4.2. Energy and momentum
In the Lagrange density deﬁned in (25), the velocity-dependent term is clearly
the kinetic energy density. This observation suggests that the Lagrangian has
the form:
L = Re
{
−iψ†∂tψ + ψ†
[−icγ5σ · ∇] ψ + 1
2
ρu2
}
= −E + U + K (31)
Vol. 21 (2011) Description of Rotational Waves in an Elastic Solid 279
where E ≡ Re (iψ†∂tψ) is the total energy, U ≡ Re (−ψ† [icγ5σ · ∇] ψ) is
potential energy, and K is kinetic energy density. We will simply adopt these
deﬁnitions without further justiﬁcation.
The Hamiltonian is the negative of the energy:
H = pψ∂tψ −L = Re
{
ψ†
[
icγ5σ · ∇ + 1
2
(
iu · ∇ −w · σ
2
)]
ψ
}
= −
{
U +
1
2
ρu2
}
. (32)
Hamilton’s equation for the wave function is:
∂tψ =
∂H
∂pψ
=
∂H
∂ [−iψ†] =
{
−cγ5σ · ∇ − u · ∇ − iw · σ
2
}
ψ. (33)
We can also deﬁne a Hamiltonian operator with ∂tψ = iHψ (note opposite
sign convention from quantum mechanics):
H = icγ5σ · ∇ + iu · ∇ −w · σ
2
. (34)
The Hamiltonian is a special case (T 00 ) of the energy-momentum tensor:
Tμν =
∂L
∂ [∂μψ]
∂νψ −L δμν . (35)
The dynamical momentum density P is derived from the orbital part of the
angular momentum:
T 0i = Pi = −iψ†∂iψ. (36)
This is identical to the dynamical momentum of relativistic quantum me-
chanics. Unlike the conjugate momentum, it does not include the rotational
motion associated with the spin angular momentum.
The sign of the Hamiltonian (and Lagrangian) is simply a convention.
We chose signs to make the conjugate momentum parallel (rather than anti-
parallel) to velocity, thus requiring the Hamiltonian to be the negative of the
energy. In analogy with plane waves, the function cos (ωt− kx) with ω > 0
is equivalent to cos (ωt + kx) with ω < 0. We could change the sign of the
Lagrangian and Hamiltonian and still preserve the sign of the momentum
by using covariant derivatives (∂μ = (∂t,−∂i)). However, such a relativis-
tic construction is unnecessary and perhaps misleading since we are in fact
dealing with Galilean space-time (the laws of relativity apply to the space
of measurements, not to the absolute space-time). The energy-momentum
tensor components T 0μ = (−E , Pi) may still be regarded as a covariant vector
whose magnitude is the scalar E 2−P 2. The equation of evolution is of course
unaﬀected by the choice of sign of the Lagrangian.
5. Discussion
The description of rotational waves derived here provides a simple classical
interpretation of Dirac bispinors and quantum mechanical operators. Due to
the nonlinear nature of the equation of evolution, we expect that an elastic
280 R. A. Close Adv. Appl. Cliﬀ ord Algebras
solid has a discrete spectrum of fermionic soliton solutions. If each soliton is
assigned its own independent wave function, then cancellation of interference
between these independent ‘particles’ would yield the Pauli exclusion princi-
ple and require the introduction of interaction potentials [4]. Characterization
of these interaction potentials is beyond the scope of this paper.
Correlations between rotational soliton waves would be computed in the
same manner as in relativistic quantum mechanics [4]. The reason is that the
bispinor wave functions, and not the measurement values, satisfy a ﬁrst order
wave equation which may be used to directly compute variations in space and
time. Therefore Bell’s Theorem [1] is not applicable.
The quantum mechanical Dirac equation is also equivalent to a deter-
ministic equation for the evolution of spin angular momentum density. Com-
parison of the classical and quantum equations for spin density may yield
new understanding of the physical properties of elementary particles.
6. Conclusions
A nonlinear Dirac equation describes evolution of rotational waves in an ideal
isotropic elastic solid. The equation is derived by assuming a linear response
to variations in orientation and incorporating the inﬂuence of motion through
convection and rotation. The Hamiltonian is represented as a sum of potential
and kinetic energy. This result provides a simple classical interpretation of
Dirac bispinors and the associated dynamical operators of relativistic quan-
tum mechanics.
Acknowledgment
The author is grateful to Damon Merari for his interest and encouragement.
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R. A. Close
4110 SE Hawthorne Blvd. #232
Portland, OR 97214
USA
e-mail: robert.close@classicalmatter.org
Received: July 16, 2009.
Accepted: October 8, 2009.
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial 
License which permits any noncommercial use, distribution, and reproduction in any medium, provided the 
original author(s) and source are credited.


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