A New Approach to Quantum Gravity from a Model of an Elastic Solid
aa r X i v : . [ m a t h - ph ] O c t A New Approach to Quantum Gravity from a Model of an ElasticSolid
John M. Baker
Abstract
We show that the dynamics of an elastic solid embedded in a Minkowski space consist of a set ofcoupled equations describing a spin-1 / A µ ,obeying Maxwell’s equations and a metric, g µν , which satisfies the Einstein field equations. Thecombined set of Dirac’s, Maxwell’s and the Einstein field equations all emerge from a simple elasticmodel in which the field variables Ψ, A µ and g µν are each identified as derived quantities fromthe field displacements of ordinary elasticity theory. By quantizing the elastic field displacements,a quantization of all of the derived fields are obtained even though they do not explicitly appearin the Lagrangian. We demonstrate the approach in a three dimensional setting where explicitsolutions of the Dirac field in terms of fractional derivatives are obtained. A higher dimensionalversion of the theory would provide an alternate approach to theories of quantum gravity. . INTRODUCTION In constructing a quantum field theory, the usual prescription is to start with a knownequation of motion, such as Dirac’s equation, and ”invent” a suitable Lagrangian that repro-duces the equation of motion when Lagrange’s equations are applied. While this prescriptionhas been successful it is not unique. In other words it is possible for two different Lagrangian’sto lead to the same equation of motion. For example, section 12 − A µ and thegravitational metric g µν .In this paper we demonstrate an alternate approach to quantum gravity based on a modelof an elastic solid. In the this model, the only field variables that appear in the Lagrangianare the field displacements, u i , that occur in elasticity theory. Using the methods of fractionalcalculus, we will show that the equations of motion of the system describe excitations thatcan be identified as massless, non-interacting, spin-1 / A µ and a metric with field variables g µν . The compatibility equations of St.Venant are shown to reproduce Maxwell’s equation for A µ and the Einstein field equationsfor g µν . We quantize the field displacements using standard approaches and thereby producea quantization of Ψ, A µ and g µν even though none of these quantities appears explicitly inthe Lagrangian. We demonstrate the basic methods in a three-dimensional setting whereexact expressions for the Dirac field can be obtained.When quantized, this theory provides a low dimensional version of a quantum description2lectrodynamics coupled to gravity. If this procedure could be extended to higher dimensionsit would provide an alternate approach to theories of quantum gravity. II. ELASTICITY THEORY
The theory of elasticity is usually concerned with the infinitesimal deformations of anelastic body . We assume that the material points of a body are continuous and canbe assigned a unique label ~a . For a three-dimensional solid each point of the body may belabeled with three coordinate numbers a i with i = 1 , , a i can be described by their positions in the 3-D fixed spacecoordinates x i with i = 1 , ,
3. We imagine that the solid is free to distort within thefixed ambient space described with coordinates x i . In this description the material points a i ( x , x , x ) are functions of ~x . A deformation of the elastic body results in infinitesimaldisplacements of these material points. If before deformation, a material point a is locatedat fixed space coordinates x , x , x then after deformation it will be located at some othercoordinate x , x , x . The deformation of the medium is characterized at each point by thedisplacement vector u i = x i − x i which measures the displacement of each point in the body after deformation. We willassume that our elastic solid is periodic in the coordinate a and at various points in thispaper we will Fourier transform the a coordinate.It is one of the aims of this paper to take this model of an elastic medium and derive fromit equations of motion that have the same form as Dirac’s equation. In doing so we haveto distinguish between the intrinsic coordinates of the medium which we will call ”internal”coordinates and the fixed space coordinates which facilitates our derivation of the equationsof motion. In the undeformed state we may take the external coordinates to coincide withthe material coordinates a i = x i . The approach that we will use in this paper is to deriveequations of motion using the fixed space coordinates and then translate this to the internalcoordinates of our space. 3 . Strain Tensor Let us assume that we have an elastic solid embedded in a three-dimensional Minkowskispace with metric η ij = − (1)We first consider the effect of a deformation on the measurement of distance. Afterthe elastic body is deformed, the distances between its points changes as measured withthe fixed space coordinates. If two points which are very close together are separated bya radius vector dx i before deformation, these same two points are separated by a vector dx i = dx i + du i afterwards. The squared distance between the points before deformationis then ds = − ( dx ) + ( dx ) + ( dx ) . Since these coincide with the material points inthe undeformed state, this can be written ds = − ( da ) + ( da ) + ( da ) = P i,j da i η ij da j .The squared distance after deformation can be written ds ′ = X ij (cid:16) dx i (cid:17) η ij (cid:16) dx j (cid:17) = X ij (cid:16) da i + du i (cid:17) η ij (cid:16) da j + du j (cid:17) = X ij da i + X k ∂u i ∂a k da k ! η ij da j + X l ∂u j ∂a l da l ! = X ij η ij da i da j + X ijk η ij ∂u i ∂a k da k da j + X ijl η ij ∂u j ∂a l da i da l + X ijkl η ij ∂u i ∂a k ∂u j ∂a l da k da l = X ij η ij + X k η ik ∂u k ∂a j + η jk ∂u k ∂a i ! + X kl η kl ∂u k ∂a i ∂u l ∂a j ! da i da j = X ik ( η ij + 2 ǫ ij ) da i da j where ǫ ij is ǫ ij = 12 X k η ik ∂u k ∂a j + η jk ∂u k ∂a i + X l η kl ∂u k ∂a i ∂u l ∂a j ! . (2)and the presence of the matrix η ij simply reflects the fact that we are assuming our solid isembedded in a Minkowski space with a pseudo-Euclidean metric.4he quantity ǫ ik is known as the strain tensor. It is fundamental in the theory of elasticity.In the above derivation, the material or internal coordinates were treated as functions of thefixed space coordinates. As is well known in elasticity theory, we could just as well treatthe fixed space coordinates as functions of the material coordinates. In this case, the straintensor has the form ǫ ij = 12 X k η ik ∂u k ∂x j + η jk ∂u k ∂x i − X l η kl ∂u k ∂x i ∂u l ∂x j ! (3)These two different approaches to the strain tensor are known in elasticity theory as theLagrangian and Eulerian perspectives. In this work we will derive the equations of motionusing the fixed space coordinates which simplifies the derivation and we will translate theresult, when necessary, to the internal coordinates.In most treatments of elasticity it is assumed that the displacements u i as well as theirderivatives are infinitesimal so the last term in Equation (2) is dropped. In this work, wewill treat the strain components as small but finite. We will then examine the structure ofthe equations of motion when the higher order terms are treated as small perturbations onthe infinitesimal strain results. B. Metric Tensor
The quantity g ij = η ij + X k η ik ∂u k ∂a j + η jk ∂u k ∂a i + X l η kl ∂u k ∂a i ∂u l ∂a j ! (4)= η ij + 2 ǫ ij is the metric for our system and determines the distance between any two points. One in-teresting aspect of the elasticity theory approach is that it provides a natural metric on thesystem in terms of the strain components expressed entirely in terms of the internal coordi-nates of the elastic body. This means that at any point in space the distance measurementcan be made without reference to the fixed space coordinates. In other words if you were anant living in this elastic medium, Equation (4) would be the metric that you would use.Even though the metric in Equation (4) does not have the Euclidean form, the space inwhich we are working is still intrinsically flat. The metric that we derived is due simply toa coordinate transformation and so cannot describe the curved space of general relativity.5hat this metric is simply the result of a coordinate transformation from the Minkowskimetric can be seen by writing the metric in the form g µν = ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a − ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a = J T ηJ where ∂x µ ∂a ν = δ µν + ∂u µ ∂a ν . and J is the Jacobian of the transformation. Later in section VI, however we will use adimensional reduction technique borrowed from Kaluza-Klein theories to reduce the three-dimensional flat space to a two-dimensional curved space. We will show that the metric forthe Fourier modes of this two dimensional system is not a simple coordinate transformation.The inverse metric, which is written with upper indices as g ik , can be obtained by explicitlyinverting Equation (4) or we can write ( g ik ) = ( J − ) η − ( J − ) T where J − = ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x ∂a ∂x (5)This yields for the inverse metric g µν = η µν + X α − η να ∂u µ ∂x α − η µα ∂u ν ∂x α + X αβ η αβ ∂u µ ∂x α ∂u ν ∂x β (6)Equation (6) shows that we can write the inverse matrix directly in terms of derivatives of u i with respect to the fixed space coordinates. This form of the inverse metric will be usefulin later sections.
1. Internal vs. External Coordinates and Summation Convention
The change in form of the metric between that given in Equation (1) and Equation (4) isdue simply to a change in coordinates between the fixed space coordinates and the material6oordinates. In this regard the transformation is similar to changing from Cartesian tospherical coordinates. This change is useful because it allows us to derive equations in thefixed space coordinates where the calculations are simplified, and then when necessary wecan switch to the internal coordinates using u i = x i − a i .We would like to be able to use the notation that a raised index on a variable indicates acontraction with the metric tensor and that a raised index and a lower index with the samelabel implies a summation (ie the Einstein summation convention). We have to be careful,however, to point out which set of coordinates, and hence which metric we are using, so wewill be explicit in each section as to which coordinate system the raised indices refer to. Forinstance we can write Equation (2) more compactly as ǫ ij = 12 ∂u i ∂a j + ∂u j ∂a i + ∂u l ∂a i ∂u l ∂a j ! . (7)where u l u l = P k η lk u l u k so that upper/lower indices indicate contraction with the fixed spacemetric, Equation (1). C. St Venant’s Equations of Compatibility
In Section (II D) we will derive the equations of motion of the elastic solid using theLagrangian formalism. There are, however, additional constraints that an elastic solid mustalso satisfy. These constraints are called the St Venant equations of compatibility in classicalelasticity theory . The usual description of these compatibility equations is that they areintegrability conditions or, a restriction on the strain components ǫ ij such that they can beconsidered partial derivatives of a function u as displayed in Equation (2). In other words if ǫ ij is a function that is composed of the partial derivatives of u then it has to satisfy certainconditions and these are the compatibility equations.However, from a geometric standpoint these equations are simply a re-statement of thefact that space is flat . In other words the compatibility equations are equivalent to R αβµν ≡ ∂ g αν ∂a β a µ − ∂ g αµ ∂a β a ν + ∂ g βµ ∂a α a ν − ∂ g βν ∂a α a µ ! + Γ ρβµ Γ ρ,αν − Γ ραµ Γ ρ,βν = 0 , (8)where R αβµν is the Riemann Curvature tensor and Γ ραβ are the Christoffel symbols givenby Γ ραβ = g λρ ∂g λα ∂x β + ∂g λβ ∂x α − ∂g αβ ∂x λ ! .
7n the above equations an upper/lower index implies a contraction with the metric in theinternal coordinates.One of the interesting aspects of an elastic solid is that this setting gives you for ”free”an explicit expression for the metric, Equation (4), and a statement about the curvature ofspace, Equation (8). We get these equations even though, as we will see shortly, the metricis not a dynamical variable appearing in the Lagrangian.
D. Equation of Motion
In the following we will use the notation u µν = ∂u µ ∂x ν and therefore the strain tensor is ǫ µν = 12 (cid:16) u µν + u νµ + u lµ u lν (cid:17) . (9)and all contractions are with the fixed space metric, Equation (1).We work in the fixed space coordinates and take the negative of the strain energy as thelagrangian density of our system. This approach leads to the usual equations of equilibriumin elasticity theory . The strain energy is quadratic in the strain tensor ǫ µν and thereforethe Lagrangian can be written L = − X µναρ C µναρ ǫ µν ǫ αρ The quantities C µναρ are known as the elastic stiffness constants of the material . For anisotropic space most of the coefficients are zero and in fact there are only two independentelastic constants in a three-dimensional isotropic space. The lagrangian density then reducesto L = ( λ + 2 µ ) h ǫ + ǫ + ǫ i + 2 λ [ ǫ ǫ − ǫ ǫ − ǫ ǫ ] + 4 µ h ǫ − ǫ − ǫ i (10)where λ and µ are known as Lam´e constants .We first derive the equations of motion of the system in the approximation where the straincomponents u ij are infinitesimal. In the infinitesimal strain approximation, the quadraticterms in Equation (9) are dropped giving ǫ µν = 12 ( u µν + u νµ ) . X ν ddx ν ∂L∂u ρν ! − ∂L∂u ρ = 0 , apply with each component of the displacement vector, u ρ , treated as an independent fieldvariable.Using the above form of the Lagrangian one can write ∂L∂u ρν = 2 λ ( ση ρν ) + 4 µǫ ρν where the divergence of the displacement field is σ ≡ ( − u + u + u ). In classical elasticitytheory σ is known as the dilatation and physically represents the fractional change in densityof a medium due to a deformation.We now have three field equations (one for each value of ρ ), ∂∂x ν ∂L∂u ρν = (2 λ + 2 µ ) ∂∂x ρ σ + 2 µ ∇ ( u ρ ) = 0 (11)where ∇ = − ∂ ∂x + ∂ ∂x + ∂ ∂x . Applying the operator ∂/∂x ρ to Equation (11) yields thewave equation − ∂ ∂x + ∂ ∂x + ∂ ∂x ! σ = 0 (12)Equation (12) shows that the classical dilatation in the medium obeys the wave equation.In Section (III) we will demonstrate a new method for reducing the wave equation (12) toDirac’s equation and compare this method to the traditional Dirac reduction. But first weturn our attention to quantizing the field displacements u i in this elastic model. E. Quantization
The Lagrangian density of the system is given by Equation (10). The coordinate x playsthe role of time in this three-dimensional space so the canonical momenta associated withthe field variable u ρ are given by P ρ = ∂L∂u ρ which gives P = − λ ( ǫ − ǫ − ǫ ) − µǫ (13)9 = 4 µǫ P = 4 µǫ . The Hamiltonian density is defined as H = X ρ P ρ u ρ − L = − P u + P u + P u − L and the total Hamiltonian is the integral over all space of the Hamiltonian density H = Z d x H Inverting Equation (13) allows us to replace the variables u ρ with P ρ in the result. Thisgives H = 3 P λ + 2 µ ) + P + P µ − λ ( u + u ) P λ + 2 µ − P u − P u + µ ( u + u ) − λ ( u + u ) λ + 2 µ + 2 λu u +( λ + 2 µ ) (cid:16) u + u (cid:17) (14)We now Fourier transform the field variables making the assumption that one of our coor-dinates, a is compact with the topology of a circle. Therefore, when we Fourier transformthe field variables, the q component is associated with a discrete spectrum while the othertwo coordinates are continuous. Writing out the coordinate dependencies explicitly we have, P ρ ( x , x , x ) = X q Z dq Z dq P ρ,~q e ı~q · ~x u ρ ( x , x , x ) = X q Z dq Z dq u ρ,~q e ı~q · ~x . The Fourier transform results in terms in the Hamiltonian that mix field variables associatedwith q and − q . For instance, the contribution to the total Hamiltonian from the P term inEquation (14) becomes Z d xP = Z d x X q Z dq Z dq X q ′ Z dq ′ Z dq ′ P ,~q P ,~q ′ e ı ( ~q + ~q ′ ) · ~x = X q Z dq Z dq P ,~q P , − ~q H = X q H q where H q is written symmetrically in q and − q as H q = − q u , − q u ,q λ λ + 2 µ − q q u ,q u , − q λ λ + 2 µ − q q u , − q u ,q λ λ + 2 µ − q u , − q u ,q λ λ + 2 µ (15)+ iq P ,q u , − q λ λ + 2 µ ) − iq P , − q u ,q λ λ + 2 µ ) + q u , − q u ,q λ + iq P ,q u , − q λ λ + 2 µ )+ q q u ,q u , − q λ − iq P , − q u ,q λ λ + 2 µ ) + q q u , − q u ,q λ + q u , − q u ,q λ + 3 P , − q P ,q λ + 2 µ ) + P , − q P ,q µ + P , − q P ,q µ + 12 iq P ,q u , − q (16)+ 12 iq P ,q u , − q − iq P , − q u ,q − iq P , − q u ,q + 2 µq u , − q u ,q + µq u , − q u ,q + µq q u ,q u , − q + µq q u , − q u ,q + µq u , − q u ,q + 2 µq u , − q u ,q (17)Since terms in the Hamiltonian with different values of q are not mixed, the function H q inEquation (15) can be solved independently for each q . H q is a bilinear function in the vari-ables u i, ± q and P i, ± q and can be diagonalized exactly, using the methods of Biougliobov .
1. Exact Diagonalization
The methods used in diagonalizing the Hamiltonian in Equation (15) are summarized inthe references . The idea is to rewrite the Hamiltonian in terms of a set of creation andannihilation operators, b i,q and b † i,q such that the Hamiltonian has the form H q = X i ω i,q b † i,q b i,q and the operators satisfy the commutation relations[ b i,q , b † j,q ′ ] = iδ i,j δ q,q ′ . The details of this procedure are included in Appendix (A). Of particular interest are theenergy eigenvalues of the modes. There are three distinct positive energies for the states b i .They are E ,q = 14 q q + q (18)11 ,q = 14 vuuut − µ ( λ + µ ) q µ ( λ + µ )( λ + 2 µ ) q q + q (19) E ,q = 14 vuuut µ ( λ + µ ) q µ ( λ + µ )( λ + 2 µ ) q q + q (20)With the operators b i calculated, the field variables u i can be written as a linear combi-nation of creation and annihilation operators as u i = X q =1 6 X i =1 (cid:16) c ij b i e i~q · ~x + c ′ ij b † i e − i~q · ~x (cid:17) (21)where the c ij are coefficients given in the appendix and q is the energy of a given mode.These eigenstates are the linear approximation obtained from keeping the lowest terms inthe strain components u ij in Equation (10). The higher order terms that were left out of theLagrangian can be incorporated by treating them as perturbations. In other words we canuse standard perturbation theory to find new strain components that are nonlinear in thecreation and annihilation operators. These field components can then serve as the basis fora theory of finite strain as we do in the next section.Strictly speaking the field displacements as expressed in Equation (21) are not energyeigenstates since the b i have different energies. We will mainly be concerned however, witha low energy approximation of the spectrum of this elastic solid. For positive values of λ and µ and λ > (cid:16) √ (cid:17) µ , the energies E ,q can be arbitrarily small compared to E ,q and E ,q . For instance with λ = 13 µ and µ = .
1, the energies E ,q and E ,q are more than20 times greater than E ,q . This suggests that in a low energy theory only the excitationcorresponding to energies E ,q will be present for suitably defined Lame constants. We willnot investigate the mechanical properties of such a solid but merely point out that in such atheory at low energies, both the field displacements u i and the dilatation, ∇ · ~u , are energyeigenstates. We also note that each of the energies is proportional to q q + q with q takingon discrete values. So one would expect in the lowest energy approximation that only themodes with q = 0 will be present and at slightly higher energies the mode with q = 1 willbe present. This low energy approximation will be exploited in later sections.One of the things that we have gained from this formalism is the ability to calculate anyquantity, that depends on the field displacements, quantum mechanically. For instance wecan now calculate the metric given in Equation (6) using the form of the field decomposition12iven in Equation (21) even though the metric itself is not a dynamical variable appearingin the Lagrangian.In the finite strain theory treated in Section (V) we will need to take Fourier transforms offield variables in the internal coordinates rather than the fixed space coordinates. Since wewill be keeping the nonlinear terms in all of our equations, then for consistency we assumethat the field variables in Equation (21) have been properly treated to the same order inperturbation theory. We will not explicitly calculate the field variables in perturbationtheory rather we will focus on the form of the field equations when terms beyond the linearapproximation are kept.We will now give a new derivation of Dirac’s equation as the equation of motion of theelastic solid. III. DERIVATION OF DIRAC’S EQUATION OF MOTIONA. Cartan’s Spinors
The concept of Spinors was introduced by Eli Cartan in 1913 . In Cartan’s originalformulation spinors were motivated by studying isotropic vectors which are vectors of zerolength. In three dimensional Minkowski space the equation of an isotropic vector is − x + x + x = 0 (22)for generally complex quantities x i . A closed form solution to this equation is realized as x = ξ + ξ , x = ξ − ξ , and x = − ξ ξ (23)where the two quantities ξ i are then ξ = ± s x + x ξ = ± s x − x . The two component object ξ = ( ξ , ξ ) has the rotational properties of a spinor and anyequation of the form (22) has a spinor solution.In the following we use the notation ∂ µ ≡ ∂/∂x µ and the wave equation is written (cid:16) − ∂ + ∂ + ∂ (cid:17) φ = 0 . ∂/∂x µ acting on the quantity φ . As long asthe partial derivatives are restricted to acting on the scalar field φ it has a spinor solutiongiven by ˆ ξ = 12 ∂∂x + ∂∂x ! (24)and ˆ ξ = 12 ∂∂x − ∂∂x ! (25)where the ”hat” notation indicates that the quantities ˆ ξ are operators. Let us now introducethe variables z = x + x and z = x − x Equations (24) and (25) are now ˆ ξ = ∂∂z and ˆ ξ = ∂∂z . These are equations of fractional derivatives of order 1 / ξ = D / z and ˆ ξ = D / z . Fractional derivatives have the property that D / z D / z = ∂∂z and various methods exist for writing closed form solutions for these operators . Theexact form for these fractional derivatives however, is not important here. The importantthing to note is that a solution to the wave equation can be written in terms of spinors whichare fractional derivatives.One of the interesting properties of suitably defined fractional derivatives (for instancethe Weyl fractional derivative) that will be exploited in later sections is their action on theexponential function. While the derivative of an exponential is given by ∂∂x e αx = αe αx the semiderivative of the exponential function is given by D / x e αx = √ αe αx (26)This will prove useful later when we Fourier transform the equations of motion.14 . Matrix Form It can be readily verified that our spinors satisfy the following equations " ˆ ξ ∂∂x + ˆ ξ ∂∂x + ∂∂x ! φ = 0 " ˆ ξ ∂∂x − ∂∂x ! − ˆ ξ ∂∂x φ = 0and in matrix form ∂∂x ∂∂x + ∂∂x ∂∂x − ∂∂x − ∂∂x ˆ ξ ˆ ξ φ = 0 (27)The matrix X = ∂∂x ∂∂x + ∂∂x ∂∂x − ∂∂x − ∂∂x is equal to the dot product of the vector ∂ µ ≡ ∂/∂x µ with the pauli spin matrices X = ∂∂x γ + ∂∂x γ + ∂∂x γ where γ = − , γ = , γ = − are proportional to the Pauli matrices and satisfy the anticommutation relations { γ µ , γ ν } = 2 Iη µν . (28)where I is the identity matrix.Equation (27) can be written X µ =1 ∂ µ γ µ ˆ ξφ = 0 . (29)This equation has the form of Dirac’s equation in three-dimensions for a noninteracting,massless, spin-1 / ξφ . C. Relation to the Dirac Decomposition
The fact that the wave equation and Dirac’s equation are related is not new. Howeverthe decomposition used here is not the same as that used by Dirac. The usual method of15onnecting the second order wave equation to the first order Dirac equation is to operate onEquation (29) from the left with P ν =1 γ ν ∂ ν giving0 = X µ,ν =1 γ ν γ µ ∂ ν ∂ µ Ψ( x )= X µ,ν =1
12 ( γ ν γ µ + γ µ γ ν ) ∂ ν ∂ µ Ψ( x )= (cid:16) − ∂ + ∂ + ∂ (cid:17) Ψ( x ) (30)where Ψ = ( α , α ) is a two component spinor and Equation (28) has been used in the laststep.This shows that Dirac’s equation does in fact imply the wave equation. The importantthing to note about Equation (30) however, is that the three-dimensional Dirac’s equationimplies not one wave equation but two in the sense that each component of the spinor Ψsatisfies this equation. Explicitly stated, Equation (30) reads − ∂ + ∂ + ∂ − ∂ + ∂ + ∂ α α = 0for the independent scalars α , α .Conversely, if one starts with the wave equation and tries to recover Dirac’s equation, itis necessary to start with two independent scalars each independently satisfying the waveequation. In other words, using the usual methods, it is not possible to take a single scalarfield that satisfies the wave equation and recover Dirac’s equation for a two componentspinor.What has been demonstrated in the preceding sections is that starting with only onescalar quantity satisfying the wave equation, Dirac’s equation for a two component spinormay be derived. Furthermore any medium (such as an elastic solid) that has a single scalarthat satisfies the wave equation must have a spinor that satisfies Dirac’s equation and sucha derivation necessitates the use of fractional derivatives. IV. DIMENSIONAL REDUCTION IN INFINITESIMAL STRAIN
In this section we take a closer look at the equation of motion, Equation (29), and thespinor (represented as a fractional derivative) when the field displacements are Fourier trans-16ormed. In the infinitesimal theory of elasticity, all terms in u ij beyond the linear term aredropped. In the infinitesimal theory therefore, no distinction is made between transformingthe coordinates a i and x i . Later when we assume small but finite strain components, we willneed to distinguish these coordinates.The Dirac field in Equation (29) is given explicitly byˆ ξφ ( x , x , x ) = D / z D / z φ ( x , x , x ) (31)with z = x + x and z = x − x . Because the semiderivatives D / z ,z are independent of x we can bring e ıx through the operator ˆ ξ when we Fourier transform φ .Transforming the dilatation first in the periodic coordinate q , Equation (29) becomes0 = X µ =1 γ µ ∂ µ ˆ ξφ = X µ =1 γ µ ∂ µ + γ ∂ ˆ ξφ = X q e ıq x X µ =1 γ µ ∂ µ + γ ıq ˆ ξφ q = X q e ıq x X µ =1 γ γ µ ∂ µ + ıq ˆ ξφ q (32)where we used φ = P q φ q e ıq x , and γ γ = 1.Equation (32) is equal to zero only if the coefficients of e ıq x are zero for each value of q giving X µ =1 ıγ ′ µ ∂ µ − q ˆ ξφ q = 0 (33)where γ ′ µ = γ γ µ and satisfies the conditions for a two dimensional metric { γ ′ µ , γ ′ ν } = 2 Iη ′ µν with η ′ µν = − (34)Equation (33) shows that the fourier modes of the elastic solid obey a two dimensional versionof Dirac’s equation for spin-1 / q . The two continuous variables leftin the problem are x and x with x playing the role of time.17et us examine the form of the spinor further by Fourier transforming the two continuouscoordinates x and x . Using Equation (26) we further transform the spinor asˆ ξφ q ( x , x ) = D / z D / z φ q ( x , x )= Z dq dq D / z e ı ( q x + q x ) D / z e ı ( q x + q x ) φ q ,q ,q = Z dq dq D / z e ız ( q + q )+ ız ( q − q ) D / z e ız ( q + q )+ ız ( q − q ) φ q ,q ,q = Z dq dq e ı ( q x + q x ) √ q + q √ q − q √ ıφ q ,q ,q (35)When the fields b i are viewed as a time dependent quantity (ie the Heisenberg picturein quantum mechanics), the wavevector q is equal to the energy of the q th mode . Thisallows us to write the column vector in Equation (35) as u ( q ) = √ E + q √ E − q Compare this formula to the expression for the four dimensional Dirac spinor in a definitestate of helicty u ( p ) = √ E + P √ E − P The dimensional reduction in the infinitesimal theory of elasticity has produced a twodimensional version of Dirac’s equation for a particle with a bare mass q and a spinor thathas a consistent form to the known 4 dimensional version.Incidentally, with the identification q as a mass term, the energy eigenstates given inEquation (18) are seen to have the relativistic form E q ∼ q q + m .The only problem with the interpretation as relativistic particles is the question of quan-tum statistics. Relativistic spin-1 / b i that we have defined obey commutation relations. This situation could be”rescued” by defining new operators c k = θ k b k θ k are complex Grassman numbers. Grassman numbers satisfy θ k θ k ′ = − θ k ′ θ k , ( θ k ) = 0this would imply { c k , c k ′ } = { θ k b k , θ k ′ b k ′ } = θ k θ k ′ b k b k ′ + θ k ′ θ k b k ′ b k = θ k θ k ′ b k b k ′ − θ k θ k ′ b k ′ b k = θ k θ k ′ [ b k , b k ′ ]= 0Similarly we have, n c † k , c † k ′ o = 0 , and n c, c † k ′ o = ıθ k θ ⋆k ′ δ k,k ′ With this definition, the fields c k satisfy appropriate quantum statistics and still obeysDirac’s equation. We do not wish to dwell on this admittedly ad-hoc procedure for gettingfermion statistics into this theory. Rather we wish to focus on the form of the equations ofmotion that are derived and demonstrate that the all ingredients for a theory of quantumgravity are present in this model.In this section, we have demonstrated that the equation of motion of this model of anelastic solid, in the infinitesimal strain approximation has as its equation of motion a twodimensional version of Dirac’s equation for spin-1 / / V. FINITE STRAINA. Internal Coordinates
In this section we will need to Fourier transform our field variables in a and thereforeneed to translate the equations of motion from the fixed space coordinates to the internal19oordinates. For clarity and to adopt a more consistent convention, in the remainder of thistext we change notation and write the internal coordinates not as a i but as x ′ i and the fixedspace coordinates will continue to be unprimed and denoted x i . Now using u i = x i − x ′ i wecan write ∂∂x i = X j ∂x ′ j ∂x i ∂∂x ′ j = X j ∂x j ∂x i − ∂u j ∂x i ! ∂∂x ′ j = X j δ ij − ∂u j ∂x i ! ∂∂x ′ j = ∂∂x ′ i − X j ∂u j ∂x i ∂∂x ′ j (36)Equation (36) relates derivatives in the fixed space coordinates x i to derivatives in thematerial coordinates x ′ i . We can now re-write the three-dimensional Dirac’s equation as X µ =1 γ µ ∂ µ Ψ = X µ =1 γ µ ∂ ′ µ − X ν ∂u ν ∂x µ ∂ ′ ν ! Ψ= X µ =1 γ ′ µ ∂ ′ µ Ψ = 0 (37)where Ψ ≡ ˆ ξφ , ∂ ′ µ = ∂/∂x ′ µ and γ ′ µ is given by γ ′ µ = γ µ − X α =1 u µα γ α . (38)The γ ′ µ are simply the gamma matrices expressed in the primed coordinate system. Theanticommutator of these matrices is { γ ′ µ , γ ′ ν } = { γ µ − X α u µα γ α , γ ν − X β u νβ γ β } = { γ µ , γ ν } − X β u νβ { γ µ , γ β } − X α u µα { γ α , γ ν } + X αβ u µα u νβ { γ α , γ β } = 2 I η µν − X β u νβ η µβ − X α u µα η αν + X α X β u µα u ν β η αβ Comparison with Equation (6) shows that { γ ′ µ , γ ′ ν } = 2 Ig µν (39)20hese gamma matrices have the form of the usual dirac’s matrices in a curved space .To further develop the form of Equation (37) we have to transform the spinor propertiesof ξ . Much like a normal vector, the components of a spinor are altered under a change ofcoordinates and as currently written ξ is a spinor with respect to the x i coordinates not the x ′ i coordinates. To transform its spinor properties we assume (similar to Brill and Wheler )a real similarity transformation and write Ψ = S Ψ ′ where S is a transformation that takesthe spinor in x µ to a spinor in x ′ µ .We then have ∂ ′ µ Ψ = ( ∂ ′ µ S )Ψ ′ + S∂ ′ µ Ψ ′ . Equation (37) then becomes 0 = γ ′ µ [ S∂ ′ µ Ψ ′ + ( ∂ ′ µ S )Ψ ′ ]= γ ′ µ S [ ∂ ′ µ Ψ ′ + S − ( ∂ ′ µ S )Ψ ′ ]= S − γ ′ µ S [ ∂ ′ µ Ψ ′ + S − ( ∂ ′ µ S )Ψ ′ ]Using ( ∂ ′ µ S − ) S = − S − ( ∂ ′ µ S ). This can finally be written˜ γ µ [ ∂ ′ µ − Γ µ ]Ψ ′ = 0 (40)where Γ µ = ( ∂ ′ µ S − ) S and ˜ γ µ = S − γ ′ µ S . The new gamma matrices ˜ γ µ still satisfy theappropriate anticommutation condition { ˜ γ µ , ˜ γ ν } = 2 Ig µν (41)What we have done in the above manipulations is re-write Equation (29) completely interms of the internal coordinates of the elastic solid. Equation (40) has the same physicalcontent as Equation (29), it is just expressed in different coordinates.Notice however that Equation (40) now superficially has the form of the Einstein-Diracequation in three-dimensions for a massless, noninteracting, spin-1 / . The quantity ∂ ′ µ − Γ µ is the covariant derivative for an object with spin. In order to make this identification,the field Γ µ must satisfy the additional equation ∂ ˜ γ µ ∂x ν + ˜ γ β Γ ′ µβν − Γ ν ˜ γ µ + ˜ γ µ Γ ν = 0where Γ ′ µβν is the usual Christoffel symbol. That this equation holds is shown in Appendix (B).Just as was done in the infinitesimal strain approach we will use a dimensional reductionto transform Equation (40). In this case however, the gamma matrices couple the metricinto the problem and this metric will also need to be reduced from three to two dimensions.21 I. DIMENSIONAL REDUCTION IN FINITE STRAIN
The dimensional reduction method that we use is borrowed from Kaluza Klein theory. Itconsists of two parts. First we need to remove the dependence of the field variables on thecoordinate x ′ . This is accomplished by Fourier transforming the field variables as before.The second part consists of reducing the 3 × g µν , in three dimensions to a 2 × ×
3, three dimensional metric as ˜ g µν and the 2 ×
2, two dimensionalmetric as g µν . We now write the Kaluza Klein ansatz ˜ g αβ = g αβ − Φ A α − Φ A α − Φ A β − Φ ˜ g αβ = g αβ − A α − A β (cid:16) − Φ − + A µ A µ (cid:17) (42)where the vector A µ is the electromagnetic vector potential, Φ is a scalar and the upper/lowerindices indicate a contraction with g µν .With this ansatz for the metric, it can be shown that the flat space condition R αβ = 0 inthree dimensions, reduces to the Einstein Field equations in curved two-dimensional spaceand Maxwell’s equations for the field A µ . The details of this derivation are given by Liu andWesson where the reduction is performed in a five dimensional setting. This is a standarddimensional reduction used in Kaluza Klein theories and has been shown to produce aconsistent treatment of four-dimensional gravity and electromagnetism from five-dimensionalflat space. Therefore, we will not repeat this derivation here only noting that there is nothingin the derivation that is particular to five dimensions. The exact same treatment works in adimensional reduction from three to two dimensions so that we will freely quote the results ofthat work which gives the following equations for the metric and the electromagnetic vectorpotential F λα ; λ = − − Φ λ F λα R αβ = −
12 Φ F λβ F λβ + Φ − Φ α ; β where F αβ ≡ A β ; α − A α ; β and we have used the notation where a comma indicates an ordinaryderivative and a semicolon indicates covariant differentiation.22he second of these equations, may be written in the more traditional form G αβ ≡ R αβ − g αβ R = 8 π (cid:16) T αβem + T αβs (cid:17) (43)where the quantities T αβem and T αβs are effective energy momentum tensors given by T αβem = −
12 Φ (cid:18) F αλ F βλ − g αβ F µν F µν (cid:19) T αβs = Φ − (cid:16) Φ α ; β − g αβ Φ µ ; µ (cid:17) So what we see is that by using the dimensional reduction technique of Kaluza Klein theorywe automatically obtained the Einstein field equations and Maxwell’s equations. What wenow show is that the same dimensional reduction technique not only produces these equationsbut also changes the non-interacting Einstein-Dirac equation in three dimensions, into aninteracting theory for a massive spin-1 / A. Dimensional Reduction of the Dirac Equation
We begin by rewriting Equation (40) as X µ =1 ˜ γ µ [ ∂ ′ µ − Γ µ ]Ψ ′ + ˜ γ [ ∂ ′ − Γ ]Ψ ′ = 0 (44)We now rewrite the matrix ˜ γ . From Equations (42) and (41) we can write˜ γ ˜ γ = g = A µ A µ − Φ − This allows us to write ˜ γ = X α =1 ˜ γ α A α + γ ⊥ Φ − (45)where the matrix γ ⊥ satisfies { ˜ γ , γ ⊥ } = 0 , { ˜ γ , γ ⊥ } = 0 , γ ⊥ = 1An explicit expression for γ ⊥ is given in Appendix (C). It can be seen that this form for ˜ γ correctly gives ˜ γ ˜ γ = g . We can now write Equation (44) as0 = X µ =1 ˜ γ µ [ ∂ ′ µ − Γ µ ]Ψ ′ + ( X α =1 ˜ γ α A α + γ ⊥ Φ − )[ ∂ ′ − Γ ]Ψ ′ = X µ =1 ˜ γ µ [ ∂ ′ µ + A µ ( ∂ ′ − Γ ) − Γ µ ]Ψ ′ + γ ⊥ Φ − [ ∂ ′ − Γ ]Ψ ′ = X µ =1 γ ′′ µ [ ∂ ′ µ + A µ ( ∂ ′ − Γ ) − Γ µ ]Ψ ′ + Φ − [ ∂ ′ − Γ ]Ψ ′ (46)23here γ ′′ µ = γ ⊥ ˜ γ µ . This equation has the same form as the classical Einstein-Dirac-Maxwellequation . The metric enters in the equation through the γ ′′ µ matrices which satisfy { γ ′′ µ , γ ′′ ν } = 2 Ig µν and the electromagnetic vector potential enters in the same way as theminimal coupling prescription ∂ µ → ∂ µ + ıe/cA µ . Additionally, a mass term has been createdin Equation (46) with m = Φ − [ ∂ ′ − Γ ].The field variables in Equation (46) still are dependent on x ′ . To complete the derivationwe need to remove this dependence by Fourier Transforming our field variables. We firstwrite the spinor field Ψ = X q Ψ q e ıqx ′ this gives X q e ıqx ′ X µ =1 γ ′′ µ [ ∂ ′ µ + A µ ( ıq − Γ ) − Γ µ ]Ψ ′ + Φ − [ ıq − Γ ]Ψ ′ = 0Now we write A µ = P k A µ,k e ıkx ′ , Γ µ = P k Γ µ,k e ıkx ′ , γ ′′ µ = P k ′ γ ′′ µk ′ e ık ′ x ′ , Φ − = P k ′ Φ − k ′ e ık ′ x ′ Which gives X q,k,k ′ ,k ′′ e ıx ′ ( q + k + k ′ + k ′′ ) X µ =1 γ ′′ µk ′ [ ∂ ′ µ δ k, δ k ′′ , + A µ,k ( ıqδ k ′′ , − Γ ,k ′′ ) − Γ µ,k δ k ′′ , ]Ψ ′ q + δ k ′′ , Φ − k ′ [ ıqδ k, − Γ ,k ]Ψ ′ q o = 0This equation is true only if the coefficients of the exponential are independently true. Let m = q + k + k ′ + k ′′ then X k,k ′ ,k ′′ X µ =1 γ ′′ µk ′ [ ∂ ′ µ δ k, δ k ′′ , + A µ,k ( ı ( m − k − k ′ − k ′′ ) δ k ′′ , − Γ ,k ′′ ) − Γ µ,k δ k ′′ , ]Ψ ′ m − k − k ′ − k ′′ + δ k ′′ , Φ − k ′ [ ı ( m − k − k ′ − k ′′ ) δ k, − Γ ,k ]Ψ ′ m − k − k ′ − k ′′ o = 0 . (47)This is a series of equations, one for each distinct value of m . The dynamics containedin Equation (47) describes an infinite series of spin particles Ψ m − k − k ′ − k ′′ interacting via ainfinite series of vector potentials A µ,k . Up until now we have kept all terms in this series ofequations. We will now examine Equation (47) in the low energy approximation.24 . Spectrum of Lowest modes Each of the quantities in Equation (47) is a function of the field displacements u i . Weshowed in previous sections that the energy of the Fourier modes of field displacementsincreases with q . We therefore expect that in a system where the energy is arbitrarily low,not all of the modes in Equation (47) will be excited. At the lowest energies only the mode ~u q =0 will be excited, as the energy of the system increases modes ~u q = ± becomes excitedand so on. As this energy is increased Equation (47) rapidly becomes more complex but atsufficiently low energies its form is quite simple.We illustrate this by considering a theory in which only the lowest modes are present.Strictly speaking, our Hamiltonian formalism in Section (II E) produced energy eigenstatesfor modes ~u q in a basis e ıqx while we want to make a statement about the energy of themodes, ~u q ′ in the basis e ıq ′ x ′ , with x ′ i = x i − u i . In Appendix (D) we show that the two arerelated and that if the only mode present in the system is q = 0 in the first basis then, theonly mode present is q ′ = 0 in the second.Let us now consider the case where there is insufficient energy to excite the mode ~u q = ± and only ~u q =0 is excited. This would represent the lowest energy possible in our system.In this case the wavevectors in Equation (47) must equal to m = k = k ′ = k ′′ = 0 andEquation (47) reduces to X µ =1 γ ′′ µ [ ∂ ′ µ − A µ, Γ , − Γ µ, ]Ψ ′ − Φ − Γ , Ψ ′ = 0 . (48)From Equation (6) we also have˜ g µν = η µν + X α =1 − η να ∂u µ ∂x α − η µα ∂u ν ∂x α + X β =1 η αβ ∂u µ ∂x α ∂u ν ∂x β , ( µ, ν = 1 , , g µν = η µν + X α =1 − η να ∂u µ ∂x α − η µα ∂u ν ∂x α + X β =1 η αβ ∂u µ ∂x α ∂u ν ∂x β , ( µ, ν = 1 ,
2) (49)and A µ = X α =1 − η µα ∂u ∂x α + X β =1 η αβ ∂u µ ∂x α ∂u ∂x β , ( µ = 1 ,
2) (50)25nd where η µν ≡ − . (51)Note that the 2D metric in Equation (49) is no longer due to a simple coordinate change.There is no coordinate transformation (or any other transformation) that involves only thecoordinates x ′ and x ′ that will remove the Fourier transforms in this equation and globallycreate the form given in Equation (1).Let us summarize what we have done in this section. We began with an equation forDirac’s equation in three-dimensions, Equation (40), which described a spin-1 / / g µν and the electromagnetic potential A µ .We also began with a trivial flat space metric in three-dimensions, Equation (4) andusing dimensional reduction, derived a two-dimensional curved metric, Equation (49), andan electromagnetic vector potential, Equation (50).Equations (48), (49) and (50) represent the main result of this paper. They provide aquantum mechanical treatment of a combined system of the Dirac Equation, electromag-netism and gravity, albeit in a low dimensional setting. We note however that this wholeprocedure can be carried out in higher dimensions. The only part of the derivation that islacking in higher dimensions is an explicit solution of the Dirac spinor in terms of fractionalderivatives like those given in Equations (III A). If a higher dimensional version of this the-ory is constructed it would provide an alternate approach to current theories of quantumgravity. VII. CONCLUSIONS
We have taken a model of an elastic medium and derived an equation of motion thathas the same form as Dirac’s equation in the presence of electromagnetism and gravity. Wederived this equation by using the formalism of Cartan to reduce the quadratic form ofthe wave equation to the linear form of Dirac’s equation. We showed that the dimensionalreduction technique from Kaluza Klein theory produces not only the Einstein Field and26axwell’s equations but also induces both mass and interaction terms into Dirac’s Equation.The formalism demonstrates that a quantum mechanical treatment of the Einstein-Dirac-Maxwell equations can be derived from the equations of motion of the Fourier modes of anelastic solid and provides a new approach to theories of quantum gravity.
APPENDIX A: QUANTIZATION COEFFICIENTS
In diagonalizing the Hamiltonian given in Equation (15) we first write the field operatorsin terms of an intermediate set of ladder operators P n,q = ı √ (cid:16) a † n,q − a n, − q (cid:17) u n,q = 1 √ (cid:16) a n,q + a † n, − q (cid:17) Now define the vectors Q q = (cid:16) a ,q , a ,q , a ,q , a , − q , a , − q , a , − q , a † ,q , a † ,q , a † ,q , a † , − q , a † , − q , a † , − q (cid:17) Q † q = (cid:16) a † ,q , a † ,q , a † ,q , a † , − q , a † , − q , a † , − q , a ,q , a ,q , a ,q , a , − q , a , − q , a , − q (cid:17) This allows us to write H q = Q † q AQ q where the matrix A is given by A = T SS T and the nonzero elements of the matrices T and S are T , = 316( λ + 2 µ ) T , = µq λ + 2 µ ) T , = µq λ + 2 µ ) T , = 8 (2 q + q ) µ + 2 µ + λ (4 (4 q + q ) µ + 1)16 µ ( λ + 2 µ ) T , = µ (3 λ + 2 µ ) q q λ + 2 µ ) T , = − µq λ + 2 µ ) T , = µ (3 λ + 2 µ ) q q λ + 2 µ ) 27 , = 8 ( q + 2 q ) µ + 2 µ + λ (4 ( q + 4 q ) µ + 1)16 µ ( λ + 2 µ ) T , = − µq λ + 2 µ ) T , = − µq λ + 2 µ ) T , = − µq λ + 2 µ ) T , = 316( λ + 2 µ ) T , = µq λ + 2 µ ) T , = 8 (2 q + q ) µ + 2 µ + λ (4 (4 q + q ) µ + 1)16 µ ( λ + 2 µ ) T , = µ (3 λ + 2 µ ) q q λ + 2 µ ) T , = µq λ + 2 µ ) T , = µ (3 λ + 2 µ ) q q λ + 2 µ ) T , = 8 ( q + 2 q ) µ + 2 µ + λ (4 ( q + 4 q ) µ + 1)16 µ ( λ + 2 µ ) S , = − (2 λ + 2 µ ) q λ + 2 µ ) S , = − (2 λ + 2 µ ) q λ + 2 µ ) S , = − λ + 2 µ ) S , = − (2 λ + 2 µ ) q λ + 2 µ ) S , = 2 µ (4 µ (2 q + q ) −
1) + λ (4 µ (4 q + q ) − µ ( λ + 2 µ ) S , = µ (3 λ + 2 µ ) q q λ + 2 µ ) S , = − (2 λ + 2 µ ) q λ + 2 µ ) S , = µ (3 λ + 2 µ ) q q λ + 2 µ ) S , = 2 µ (4 µ ( q + 2 q ) −
1) + λ (4 µ ( q + 4 q ) − µ ( λ + 2 µ ) S , = − λ + 2 µ ) 28 , = (2 λ + 2 µ ) q λ + 2 µ ) S , = (2 λ + 2 µ ) q λ + 2 µ ) S , = 2 µ (4 µ (2 q + q ) −
1) + λ (4 µ (4 q + q ) − µ ( λ + 2 µ ) S , = µ (3 λ + 2 µ ) q q λ + 2 µ ) S , = (2 λ + 2 µ ) q λ + 2 µ ) S , = µ (3 λ + 2 µ ) q q λ + 2 µ ) S , = 2 µ (4 µ ( q + 2 q ) −
1) + λ (4 µ ( q + 4 q ) − µ ( λ + 2 µ ) S , = (2 λ + 2 µ ) q λ + 2 µ )The methods outlined by Tikochinsky and Tsallis can now be applied to the matrix A .The final results allow us to define six creation operators b † i and six annihilation operators b i that satisfy [ b i,q , b † j,q ′ ] = iδ i,j δ q,q ′ [ b i,q , b j,q ′ ] = 0 [ b † i,q , b † j,q ′ ] = 0These operators are Eigenstates of the Hamiltonian with eigenvalues E ,q = 14 q q + q E ,q = 14 q q + q E ,q = 14 vuuut − √ q µ ( λ + 2 µ ) (2 λ + 2 µ )( λ + 2 µ ) q q + q E ,q = 14 vuuut − √ q µ ( λ + 2 µ ) (2 λ + 2 µ )( λ + 2 µ ) q q + q E ,q = 14 vuuut √ q µ ( λ + 2 µ ) (2 λ + 2 µ )( λ + 2 µ ) q q + q E ,q = 14 vuuut √ q µ ( λ + 2 µ ) (2 λ + 2 µ )( λ + 2 µ ) q q + q H q = X i E i,q b † i,q b i,q The field displacement operators, u ij can be written in terms of these ladder operators.Denote the vector of field and ladder operators as X q = ( P ,q , P ,q , P ,q , u ,q , u ,q , u ,q , P , − q , P , − q , P , − q , u , − q , u , − q , u , − q ) B q = (cid:16) b ,q , b ,q , b ,q , b ,q , b ,q , b ,q , b † ,q , b † ,q , b † ,q , b † ,q , b † ,q , b † ,q (cid:17) . This allows us to write X i,q = X j c i,j B i,q where the coefficients c i,j are listed below. In writing the c i,j coefficients we have defined thefollowing quantities a = √ q µ ( λ + 2 µ ) (2 λ + 2 µ ) λ = vuuut − √ q µ ( λ + 2 µ ) (2 λ + 2 µ )( λ + 2 µ ) λ = vuuut √ q µ ( λ + 2 µ ) (2 λ + 2 µ )( λ + 2 µ ) + 1 d = 8 µ (2 λ + 2 µ ) (cid:16) µ + 16Eq λµ − µ − λ (cid:17) + a (cid:16) µ + 32Eq λµ − µ − λ (cid:17) d = 8 µ (2 λ + 2 µ ) (cid:16) µ + 16Eq λµ − µ − λ (cid:17) + a (cid:16) − µ + 10 µ + λ (cid:16) − µ (cid:17)(cid:17) d = a (cid:16) µ + 32Eq λµ − µ − λ (cid:17) − µ (2 λ + 2 µ ) (cid:16) µ + 16Eq λµ − µ − λ (cid:17) d = 12 aµ + ( λ + 2 µ ) (cid:16) µ + 96Eq λµ + 4 (cid:16) λ + 3 (cid:17) µ + 18 λµ − λ (cid:17) d = ( λ + 2 µ ) (cid:16) µ + 96Eq λµ + 4 (cid:16) λ + 3 (cid:17) µ + 18 λµ − λ (cid:17) − aµn = 2 √ vuut µ ( q + q ) / q (2 µω q − n = 2 √ vuut µ ( q + q ) / q (2 µω q − = 8 √ (cid:18) µ (2 λ + 2 µ ) ω q (2(2 µ ( λ + 2 µ ) λ ω q + 2 µ (3 λ + 4 µ ) ω q + λ (16 ω q µ + 2 µ + λ (16 µ ω q + 1)) ω q ) a − ( λ + 2 µ )(4 µ ( λ + 2 µ )(2 λ + 2 µ ) λ ω q +4 µ (2 λ + 2 µ )( λ + 10 µ ) ω q + ( λ + 2 µ ) λ (2 µ (16 µ ω q + 5) + λ (32 µ ω q + 2)) ω q ) a +4 µ ( λ + 2 µ ) (2 λ + 2 µ )(2 µ ( λ + 2 µ ) λ ω q + 2 µ (3 λ + 4 µ ) ω q + λ (16 ω q µ + 2 µ + λ (16 µ ω q + 1)) ω q )) (cid:19) / / (cid:18) ( λ + 2 µ ) q ( a (32 ω q µ − µ + λ (32 µ ω q − − µ (2 λ + 2 µ )(16 ω q µ − µ + λ (16 µ ω q − (cid:19) / n = 8 (cid:18) − µ ( λ + 2 µ ) (8( λ + 2 µ )(2 λ + 2 µ )(4 λω q − λ ω q ) µ + a ( λ ω q ((32 µ ω q + 2) λ + 6 µ (16 µ ω q − λ + 4 µ (16 µ ω q − − λ − µ ) µ (2 λ + 2 µ ) ω q )) (cid:19) / / (cid:18) (12 aµ + ( λ + 2 µ )((32 µ ω q − λ + 6 µ (16 µ ω q + 3) λ + 4 µ (16 µ ω q + 3))) (cid:19) / n = 8 √ (cid:18) µ (2 λ + 2 µ ) ω q (2(2 µ ( λ + 2 µ ) λ ω q + 2 µ (3 λ + 4 µ ) ω q + λ (16 ω q µ + 2 µ + λ (16 µ ω q + 1)) ω q ) a + ( λ + 2 µ )(4 µ ( λ + 2 µ )(2 λ + 2 µ ) λ ω q + 4 µ (2 λ + 2 µ )( λ + 10 µ ) ω q + ( λ + 2 µ ) λ (2 µ (16 µ ω q + 5) + λ (32 µ ω q + 2)) ω q ) a + 4 µ ( λ + 2 µ ) (2 λ + 2 µ )(2 µ ( λ + 2 µ ) λ ω q + 2 µ (3 λ + 4 µ ) ω q + λ (16 ω q µ + 2 µ + λ (16 µ ω q + 1)) ω q )) (cid:19) / / (cid:18) ( λ + 2 µ ) q (8 µ (2 λ + 2 µ )(16 ω q µ − µ + λ (16 µ ω q − a (32 ω q µ − µ + λ (32 µ ω q − (cid:19) / n = 8 (cid:18) µ ( λ + 2 µ ) (8( λ + 2 µ )(2 λ + 2 µ )(3 λ ω q − λω q ) µ + a ( λ ω q ((32 µ ω q + 2) λ + 6 µ (16 µ ω q − λ + 4 µ (16 µ ω q − − λ − µ ) µ (2 λ + 2 µ ) ω q )) (cid:19) / / (cid:18) (( λ + 2 µ )((32 µ ω q − λ + 6 µ (16 µ ω q + 3) λ + 4 µ (16 µ ω q + 3)) − aµ ) (cid:19) / Finally, the nonzero components of the matrix c ij are c , = i √ ω q ( a (2 λ − µ )( λ +2 µ ) λ − µ (2 λ +2 µ )(4 λµ ( λ +2 µ ) − a ( λ − µ )) ω q )( λ +2 µ ) d n q c , = − i √ ω q ( a (2 λ − µ )( λ +2 µ ) λ +4 µ (2 λ +2 µ )( a ( λ − µ )+4 λµ ( λ +2 µ )) ω q )( λ +2 µ ) d n q c , = i √ ( aµ +( λ +2 µ ) ( − λ +18 µλ +12 µ +4 µ (2 a +( λ +2 µ )(2 λ +2 µ )) λ ω q )) d n c , = i √ ( ( λ +2 µ ) ( − λ +18 µλ +12 µ +4 µ (( λ +2 µ )(2 λ +2 µ ) − a ) λ ω q ) − aµ ) d n , = − i √ q n q − µn q ω q c , = − i √ λ +2 µ ) q (( λ +2 µ )( − λλ +6 µλ +4 µ (2 λ +2 µ ) ω q ) − aµω q ) d n ω q c , = − i √ λ +2 µ ) q (8 aµω q +( λ +2 µ )( − λλ +6 µλ +4 µ (2 λ +2 µ ) ω q )) d n ω q c , = − i √ q n q − µn q ω q c , = i √ q ( a (2 λ +10 µ +4 µ (2 λ +2 µ ) λ ω q ) − µ ( λ +2 µ )(2 λ +2 µ )(2 µλ ω q +1)) d n q c , = − i √ q (8 µ ( λ +2 µ )(2 λ +2 µ )(2 µλ ω q +1)+ a (2 λ +10 µ +4 µ (2 λ +2 µ ) λ ω q )) d n q c , = i √ n − µn ω q c , = − i √ λ +2 µ ) q (( λ +2 µ )( − λλ +6 µλ +4 µ (2 λ +2 µ ) ω q ) − aµω q ) d n ω q c , = − i √ λ +2 µ ) q (8 aµω q +( λ +2 µ )( − λλ +6 µλ +4 µ (2 λ +2 µ ) ω q )) d n ω q c , = i √ n − µn ω q c , = i √ a (2 λ +10 µ +4 µ (2 λ +2 µ ) λ ω q ) − µ ( λ +2 µ )(2 λ +2 µ )(2 µλ ω q +1)) d n c , = − i √ µ ( λ +2 µ )(2 λ +2 µ )(2 µλ ω q +1)+ a (2 λ +10 µ +4 µ (2 λ +2 µ ) λ ω q )) d n c , = √ µ ( λ +2 µ ) ω q (( λ +2 µ )(2 λ +2 µ )(4 µω q − λ ) − aλ ) d n c , = √ µ ( λ +2 µ ) ω q (2 aλ +( λ +2 µ )(2 λ +2 µ )(4 µω q − λ )) d n c , = − √ µ (2 λ +2 µ ) ω q (4 µλ ω q a + a − µ ( λ +2 µ )) d n q c , = √ µ (2 λ +2 µ ) ω q (4 µλ ω q a + a +4 µ ( λ +2 µ )) d n q c , = √ µq ω q n q − µn q ω q c , = √ µ (2 λ +2 µ ) q ω q ( a ( λ +4 µω q ) − µ (( λ +2 µ ) λ +4 µ (2 λ +2 µ ) ω q )) d n q c , = √ µ (2 λ +2 µ ) q ω q ( a ( λ +4 µω q )+4 µ (( λ +2 µ ) λ +4 µ (2 λ +2 µ ) ω q )) d n q c , = − √ µq ω q n q − µn q ω q c , = − √ µq (2 aλ +( λ +2 µ )(2 λ +2 µ )( − λ +6 µ +4 µ ( λ +2 µ ) λ ω q )) d n c , = − √ µq (( λ +2 µ )(2 λ +2 µ )( − λ +6 µ +4 µ ( λ +2 µ ) λ ω q ) − aλ ) d n c , = − √ µω q n − µn ω q c , = √ µ (2 λ +2 µ ) ω q ( a ( λ +4 µω q ) − µ (( λ +2 µ ) λ +4 µ (2 λ +2 µ ) ω q )) d n c , = √ µ (2 λ +2 µ ) ω q ( a ( λ +4 µω q )+4 µ (( λ +2 µ ) λ +4 µ (2 λ +2 µ ) ω q )) d n c , = √ µω q n − µn ω q c , = − √ µq (2 aλ +( λ +2 µ )(2 λ +2 µ )( − λ +6 µ +4 µ ( λ +2 µ ) λ ω q )) d n c , = − √ µq (( λ +2 µ )(2 λ +2 µ )( − λ +6 µ +4 µ ( λ +2 µ ) λ ω q ) − aλ ) d n , = i √ ( aµ +( λ +2 µ ) ( − λ +18 µλ +12 µ +4 µ (2 a +( λ +2 µ )(2 λ +2 µ )) λ ω q )) d n c , = i √ ( ( λ +2 µ ) ( − λ +18 µλ +12 µ +4 µ (( λ +2 µ )(2 λ +2 µ ) − a ) λ ω q ) − aµ ) d n c , = i √ ω q ( a (2 λ − µ )( λ +2 µ ) λ − µ (2 λ +2 µ )(4 λµ ( λ +2 µ ) − a ( λ − µ )) ω q )( λ +2 µ ) d n q c , = − i √ ω q ( a (2 λ − µ )( λ +2 µ ) λ +4 µ (2 λ +2 µ )( a ( λ − µ )+4 λµ ( λ +2 µ )) ω q )( λ +2 µ ) d n q c , = − i √ q n q − µn q ω q c , = i √ q ( a (2 λ +10 µ +4 µ (2 λ +2 µ ) λ ω q ) − µ ( λ +2 µ )(2 λ +2 µ )(2 µλ ω q +1)) d n q c , = − i √ q (8 µ ( λ +2 µ )(2 λ +2 µ )(2 µλ ω q +1)+ a (2 λ +10 µ +4 µ (2 λ +2 µ ) λ ω q )) d n q c , = − i √ q n q − µn q ω q c , = − i √ λ +2 µ ) q (( λ +2 µ )( − λλ +6 µλ +4 µ (2 λ +2 µ ) ω q ) − aµω q ) d n ω q c , = − i √ λ +2 µ ) q (8 aµω q +( λ +2 µ )( − λλ +6 µλ +4 µ (2 λ +2 µ ) ω q )) d n ω q c , = i √ n − µn ω q c , = i √ a (2 λ +10 µ +4 µ (2 λ +2 µ ) λ ω q ) − µ ( λ +2 µ )(2 λ +2 µ )(2 µλ ω q +1)) d n c , = − i √ µ ( λ +2 µ )(2 λ +2 µ )(2 µλ ω q +1)+ a (2 λ +10 µ +4 µ (2 λ +2 µ ) λ ω q )) d n c , = i √ n − µn ω q c , = − i √ λ +2 µ ) q (( λ +2 µ )( − λλ +6 µλ +4 µ (2 λ +2 µ ) ω q ) − aµω q ) d n ω q c , = − i √ λ +2 µ ) q (8 aµω q +( λ +2 µ )( − λλ +6 µλ +4 µ (2 λ +2 µ ) ω q )) d n ω q c , = √ µ (2 λ +2 µ ) ω q (4 µλ ω q a + a − µ ( λ +2 µ )) d n q c , = − √ µ (2 λ +2 µ ) ω q (4 µλ ω q a + a +4 µ ( λ +2 µ )) d n q c , = − √ µ ( λ +2 µ ) ω q (( λ +2 µ )(2 λ +2 µ )(4 µω q − λ ) − aλ ) d n c , = − √ µ ( λ +2 µ ) ω q (2 aλ +( λ +2 µ )(2 λ +2 µ )(4 µω q − λ )) d n c , = √ µq ω q n q − µn q ω q c , = √ µq (2 aλ +( λ +2 µ )(2 λ +2 µ )( − λ +6 µ +4 µ ( λ +2 µ ) λ ω q )) d n c , = √ µq (( λ +2 µ )(2 λ +2 µ )( − λ +6 µ +4 µ ( λ +2 µ ) λ ω q ) − aλ ) d n c , = − √ µq ω q n q − µn q ω q c , = − √ µ (2 λ +2 µ ) q ω q ( a ( λ +4 µω q ) − µ (( λ +2 µ ) λ +4 µ (2 λ +2 µ ) ω q )) d n q c , = − √ µ (2 λ +2 µ ) q ω q ( a ( λ +4 µω q )+4 µ (( λ +2 µ ) λ +4 µ (2 λ +2 µ ) ω q )) d n q c , = − √ µω q n − µn ω q c , = √ µq (2 aλ +( λ +2 µ )(2 λ +2 µ )( − λ +6 µ +4 µ ( λ +2 µ ) λ ω q )) d n , = √ µq (( λ +2 µ )(2 λ +2 µ )( − λ +6 µ +4 µ ( λ +2 µ ) λ ω q ) − aλ ) d n c , = √ µω q n − µn ω q c , = − √ µ (2 λ +2 µ ) ω q ( a ( λ +4 µω q ) − µ (( λ +2 µ ) λ +4 µ (2 λ +2 µ ) ω q )) d n c , = − √ µ (2 λ +2 µ ) ω q ( a ( λ +4 µω q )+4 µ (( λ +2 µ ) λ +4 µ (2 λ +2 µ ) ω q )) d n APPENDIX B: AUXILIARY EQUATION
We wish to show that ∂ ˜ γ µ ∂x ν + ˜ γ β Γ ′ µβν − Γ ν ˜ γ µ + ˜ γ µ Γ ν = 0We first consider the equation ∂γ µ ∂x ν = 0true in the unprimed coordinate system. But since the unprimed coordinate system isEuclidean space, the Christoffel symbols are identically zero. This allows us to write ∂γ µ ∂x ν + γ β Γ µβν = 0Since this is a tensor equation true in all frames, in the primed coordinate system we canimmediately write ∂ ′ ν γ ′ µ + γ ′ β Γ ′ µβν = 0Using γ ′ µ = S ˜ γ µ S − , we have ∂ ′ ν ( S ˜ γ µ S − ) + ( S ˜ γ β S − )Γ ′ µβν = 0or ( ∂ ′ ν S )˜ γ µ S − + S ( ∂ ′ ν ˜ γ µ ) S − + S ˜ γ µ ( ∂ ′ ν S − ) + ( S ˜ γ β S − )Γ ′ µβν = 0 . Multiplying by S − on the left and S on the right yields S − ( ∂ ′ ν S )˜ γ µ + ( ∂ ′ ν ˜ γ µ ) + ˜ γ µ ( ∂ ′ ν S − ) S + ˜ γ β Γ ′ µβν = 0Finally, using Γ ν = ( ∂ ′ ν S − ) S and again noting that ∂ ′ ν S − S = − S − ∂ ′ ν S we have,˜ γ µ Γ ν − Γ ν ˜ γ µ + (cid:16) ∂ ′ ν ˜ γ µ + ˜ γ β Γ ′ µβν (cid:17) = 0 (B1)34 PPENDIX C: DECOMPOSITION OF ˜ γ ′ In decomposing the matrix ˜ γ ′ we seek a matrix γ ⊥ that satisfies { ˜ γ , γ ⊥ } = 0 , { ˜ γ , γ ⊥ } = 0 , γ ⊥ = 1 (C1)where from Equations (41) and (38) we can write˜ γ = S − (cid:18) (1 − u ) γ + u γ + u γ (cid:19) S ˜ γ = S − (cid:18) u γ + (1 − u ) γ + u γ (cid:19) S We can treat the matrices γ µ as vectors and use the cross product formula to compute anorthogonal vector. In other words we can write γ ⊥ = ˜ γ × ˜ γ using the rules γ × γ = γ γ × γ = γ γ × γ = − γ . This gives v ⊥ = S − (cid:18) [ u u − u (1 − u )] γ + [ u u − (1 − u ) u ] γ + [(1 − u )(1 − u ) − u u ] γ (cid:19) S with γ ⊥ ≡ v ⊥ | v ⊥ | and | v ⊥ | = q v ⊥ . It can be verified directly that γ ⊥ satisfies Equation (C1). APPENDIX D: FOURIER COMPONENTS
We wish to relate the Fourier components of the field displacements u i , when expandedin the basis e ıq ′ x ′ to those expanded in the basis e ıqx . The two coordinate systems are relatedby u µ = x µ − x ′ µ where the field displacements u i are assumed to be small. We write u µ = X q u µq e ıqx = X q u µq e ıq · ( x ′ + u ) X q u µq e ıqx ′ (1 + ıq · u )= X q u µq e ıqx ′ (1 + ı X k q · u k e ık · x ) ≈ X q u µq e ıqx ′ (1 + ı X k q · u k e ık · x ′ )= X qk u µq ( δ k, + ıq · u k ) e ı ( q + k ) · x ′ correct to second order in u µ,q .Setting q + k = q ′ we have u µ = X q ′ k u µ,q ′ − k ( δ k, + ı ( q ′ − k ) · u k ) e ıq ′ · x ′ so that the components of the field displacements in the primed frame are u m ′ = X k u m ′ − k ( δ k, + ı ( m ′ − k ) u k )We can now relate the energy eigenstates which are expressed in the unprimed frame to theFourier components in the primed frame. In a low energy theory in which only u is present,to second order in u i , the only Fourier component in the primed frame that is nonzero is u m ′ =0 . If there is enough energy to excite the fields u and u ±