# A New Approach to Quantum Gravity from a Model of an Elastic Solid

###### Abstract

We show that the dynamics of an elastic solid embedded in a Minkowski space consist of a set of coupled equations describing a spin- field, , obeying Dirac’s equation, a vector potential, , obeying Maxwell’s equations and a metric, , which satisfies the Einstein field equations. The combined set of Dirac’s, Maxwell’s and the Einstein field equations all emerge from a simple elastic model in which the field variables , and are each identified as derived quantities from the 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 appear in the Lagrangian. We demonstrate the approach in a three dimensional setting where explicit solutions of the Dirac field in terms of fractional derivatives are obtained. A higher dimensional version of the theory would provide an alternate approach to theories of quantum gravity.

###### pacs:

PACS numbers:## I Introduction

In constructing a quantum field theory, the usual prescription is to start with a known equation of motion, such as Dirac’s equation, and ”invent” a suitable Lagrangian that reproduces the equation of motion when Lagrange’s equations are applied. While this prescription has been successful it is not unique. In other words it is possible for two different Lagrangian’s to lead to the same equation of motion. For example, section of Reference[] provides a good example of two different Lagrangian’s that lead to the same equation of motion for the density variations in an acoustic field.

In a theory of quantum gravity, this traditional approach would involve using a Lagrangian with an appropriate set of terms such that when Lagrange’s equations are applied, the Einstein field equations are reproduced. The Lagrangian obtained in this manner explicitly contains the gravitational metric, as well as any other field variables that are coupled to it. For instance an attempt at merging gravity with QED would produce a Lagrangian that explicitly includes the Dirac Spinor field , the electromagnetic vector potential and the gravitational metric .

In this paper we demonstrate an alternate approach to quantum gravity based on a model of an elastic solid. In the this model, the only field variables that appear in the Lagrangian are the field displacements, , that occur in elasticity theory. Using the methods of fractional calculus, we will show that the equations of motion of the system describe excitations that can be identified as massless, non-interacting, spin- particles obeying Dirac’s equation. We then assume that one of our coordinates is periodic and use a dimensional reduction technique to reduce the dimensionality from three-dimensions to two-dimensions.

When terms beyond the linear approximation are included, this dimensional reduction produces a new set of equations in which the spin field, , is shown to interact via a vector potential and a metric with field variables . The compatibility equations of St. Venant are shown to reproduce Maxwell’s equation for and the Einstein field equations for . We quantize the field displacements using standard approaches and thereby produce a quantization of , and even though none of these quantities appears explicitly in the Lagrangian. We demonstrate the basic methods in a three-dimensional setting where exact expressions for the Dirac field can be obtained.

When quantized, this theory provides a low dimensional version of a quantum description electrodynamics coupled to gravity. If this procedure could be extended to higher dimensions it 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 an elastic bodyref:Love ; ref:Sokolnikoff ; ref:Landau_Lifshitz ; ref:Green_Zerna ; ref:Novozhilov . We assume that the material points of a body are continuous and can be assigned a unique label . For a three-dimensional solid each point of the body may be labeled with three coordinate numbers with .

If this three dimensional elastic body is placed in a large ambient three dimensional space then the material coordinates can be described by their positions in the 3-D fixed space coordinates with . We imagine that the solid is free to distort within the fixed ambient space described with coordinates . In this description the material points are functions of . A deformation of the elastic body results in infinitesimal displacements of these material points. If before deformation, a material point is located at fixed space coordinates then after deformation it will be located at some other coordinate . The deformation of the medium is characterized at each point by the displacement vector

which measures the displacement of each point in the body after deformation. We will assume that our elastic solid is periodic in the coordinate and at various points in this paper we will Fourier transform the coordinate.

It is one of the aims of this paper to take this model of an elastic medium and derive from it equations of motion that have the same form as Dirac’s equation. In doing so we have to 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 equations of motion. In the undeformed state we may take the external coordinates to coincide with the material coordinates . The approach that we will use in this paper is to derive equations of motion using the fixed space coordinates and then translate this to the internal coordinates of our space.

### ii.1 Strain Tensor

Let us assume that we have an elastic solid embedded in a three-dimensional Minkowski space with metric

(1) |

We first consider the effect of a deformation on the measurement of distance. After the elastic body is deformed, the distances between its points changes as measured with the fixed space coordinates. If two points which are very close together are separated by a radius vector before deformation, these same two points are separated by a vector afterwards. The squared distance between the points before deformation is then . Since these coincide with the material points in the undeformed state, this can be written . The squared distance after deformation can be writtenref:Landau_Lifshitz

where is

(2) |

and the presence of the matrix simply reflects the fact that we are assuming our solid is embedded in a Minkowski space with a pseudo-Euclidean metric.

The quantity 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 the fixed space coordinates. As is well known in elasticity theory, we could just as well treat the fixed space coordinates as functions of the material coordinates. In this case, the strain tensor has the formref:Sokolnikoff

(3) |

These two different approaches to the strain tensor are known in elasticity theory as the Lagrangian and Eulerian perspectives. In this work we will derive the equations of motion using the fixed space coordinates which simplifies the derivation and we will translate the result, when necessary, to the internal coordinates.

In most treatments of elasticity it is assumed that the displacements as well as their derivatives are infinitesimal so the last term in Equation (2) is dropped. In this work, we will treat the strain components as small but finite. We will then examine the structure of the equations of motion when the higher order terms are treated as small perturbations on the infinitesimal strain results.

### ii.2 Metric Tensor

The quantity

is the metric for our system and determines the distance between any two points. One interesting aspect of the elasticity theory approach is that it provides a natural metric on the system in terms of the strain components expressed entirely in terms of the internal coordinates of the elastic body. This means that at any point in space the distance measurement can be made without reference to the fixed space coordinates. In other words if you were an ant living in this elastic medium, Equation (II.2) would be the metric that you would use.

Even though the metric in Equation (II.2) does not have the Euclidean form, the space in which we are working is still intrinsically flat. The metric that we derived is due simply to a coordinate transformation and so cannot describe the curved space of general relativity. That this metric is simply the result of a coordinate transformation from the Minkowski metric can be seen by writing the metric in the formref:Millman_Parker

where

and is the Jacobian of the transformation. Later in section VI, however we will use a dimensional 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 for the Fourier modes of this two dimensional system is not a simple coordinate transformation.

The inverse metric, which is written with upper indices as , can be obtained by explicitly inverting Equation (II.2) or we can write where

(5) |

This yields for the inverse metric

(6) |

Equation (6) shows that we can write the inverse matrix directly in terms of derivatives of with respect to the fixed space coordinates. This form of the inverse metric will be useful in later sections.

#### ii.2.1 Internal vs. External Coordinates and Summation Convention

The change in form of the metric between that given in Equation (1) and Equation (II.2) is due simply to a change in coordinates between the fixed space coordinates and the material coordinates. In this regard the transformation is similar to changing from Cartesian to spherical coordinates. This change is useful because it allows us to derive equations in the fixed space coordinates where the calculations are simplified, and then when necessary we can switch to the internal coordinates using .

We would like to be able to use the notation that a raised index on a variable indicates a contraction with the metric tensor and that a raised index and a lower index with the same label 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 we will be explicit in each section as to which coordinate system the raised indices refer to. For instance we can write Equation (2) more compactly as

(7) |

where so that upper/lower indices indicate contraction with the fixed space metric, Equation (1).

### ii.3 St Venant’s Equations of Compatibility

In Section (II.4) we will derive the equations of motion of the elastic solid using the Lagrangian formalism. There are, however, additional constraints that an elastic solid must also satisfy. These constraints are called the St Venant equations of compatibility in classical elasticity theoryref:Sokolnikoff . The usual description of these compatibility equations is that they are integrability conditions or, a restriction on the strain components such that they can be considered partial derivatives of a function as displayed in Equation (2). In other words if is a function that is composed of the partial derivatives of then it has to satisfy certain conditions and these are the compatibility equations.

However, from a geometric standpoint these equations are simply a re-statement of the fact that space is flatref:Novozhilov . In other words the compatibility equations are equivalent toref:Novozhilov

(8) |

where is the Riemann Curvature tensor and are the Christoffel symbols given byref:Schutz

In the above equations an upper/lower index implies a contraction with the metric in the internal 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 (II.2), and a statement about the curvature of space, Equation (8). We get these equations even though, as we will see shortly, the metric is not a dynamical variable appearing in the Lagrangian.

### ii.4 Equation of Motion

In the following we will use the notation

and therefore the strain tensor is

(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 the lagrangian density of our system. This approach leads to the usual equations of equilibrium in elasticity theoryref:Love ; ref:Novozhilov . The strain energy is quadratic in the strain tensor and therefore the Lagrangian can be written

The quantities are known as the elastic stiffness constants of the materialref:Sokolnikoff . For an isotropic space most of the coefficients are zero and in fact there are only two independent elastic constants in a three-dimensional isotropic space. The lagrangian density then reduces to

(10) |

where and are known as Lamé constantsref:Sokolnikoff .

We first derive the equations of motion of the system in the approximation where the strain components are infinitesimal. In the infinitesimal strain approximation, the quadratic terms in Equation (9) are dropped giving

The usual Lagrange equations,

apply with each component of the displacement vector, , treated as an independent field variable.

Using the above form of the Lagrangian one can write

where the divergence of the displacement field is . In classical elasticity theory is known as the dilatation and physically represents the fractional change in density of a medium due to a deformation.

We now have three field equations (one for each value of ),

(11) |

where

(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) to Dirac’s equation and compare this method to the traditional Dirac reduction. But first we turn our attention to quantizing the field displacements in this elastic model.

### ii.5 Quantization

The Lagrangian density of the system is given by Equation (10). The coordinate plays the role of time in this three-dimensional space so the canonical momenta associated with the field variable are given by

which gives

(13) | |||||

The Hamiltonian density is defined as

and the total Hamiltonian is the integral over all space of the Hamiltonian density

Inverting Equation (13) allows us to replace the variables with in the result. This gives

(14) | |||||

We now Fourier transform the field variables making the assumption that one of our coordinates, is compact with the topology of a circle. Therefore, when we Fourier transform the field variables, the component is associated with a discrete spectrum while the other two coordinates are continuous. Writing out the coordinate dependencies explicitly we have,

The Fourier transform results in terms in the Hamiltonian that mix field variables associated with and . For instance, the contribution to the total Hamiltonian from the term in Equation (14) becomes

The total Hamiltonian then becomes

where is written symmetrically in and as

(17) | |||||

Since terms in the Hamiltonian with different values of are not mixed, the function in Equation (17) can be solved independently for each . is a bilinear function in the variables and and can be diagonalized exactly, using the methods of Biougliobovref:Kittel ; ref:Tsallis ; ref:Tikochinsky .

#### ii.5.1 Exact Diagonalization

The methods used in diagonalizing the Hamiltonian in Equation (17) are summarized in the referencesref:Tsallis ; ref:Tikochinsky . The idea is to rewrite the Hamiltonian in terms of a set of creation and annihilation operators, and such that the Hamiltonian has the form

and the operators satisfy the commutation relations

The details of this procedure are included in Appendix (A). Of particular interest are the energy eigenvalues of the modes. There are three distinct positive energies for the states . They are

(18) | |||||

(19) | |||||

(20) |

With the operators calculated, the field variables can be written as a linear combination of creation and annihilation operators as

(21) |

where the are coefficients given in the appendix and is the energy of a given mode. These eigenstates are the linear approximation obtained from keeping the lowest terms in the strain components in Equation (10). The higher order terms that were left out of the Lagrangian can be incorporated by treating them as perturbations. In other words we can use standard perturbation theory to find new strain components that are nonlinear in the creation and annihilation operators. These field components can then serve as the basis for a theory of finite strain as we do in the next section.

Strictly speaking the field displacements as expressed in Equation (21) are not energy eigenstates since the have different energies. We will mainly be concerned however, with a low energy approximation of the spectrum of this elastic solid. For positive values of and and , the energies can be arbitrarily small compared to and . For instance with and , the energies and are more than times greater than . This suggests that in a low energy theory only the excitation corresponding to energies will be present for suitably defined Lame constants. We will not investigate the mechanical properties of such a solid but merely point out that in such a theory at low energies, both the field displacements and the dilatation, , are energy eigenstates. We also note that each of the energies is proportional to with taking on discrete values. So one would expect in the lowest energy approximation that only the modes with will be present and at slightly higher energies the mode with will be 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 any quantity, that depends on the field displacements, quantum mechanically. For instance we can now calculate the metric given in Equation (6) using the form of the field decomposition given in Equation (21) even though the metric itself is not a dynamical variable appearing in the Lagrangian.

In the finite strain theory treated in Section (V) we will need to take Fourier transforms of field variables in the internal coordinates rather than the fixed space coordinates. Since we will be keeping the nonlinear terms in all of our equations, then for consistency we assume that the field variables in Equation (21) have been properly treated to the same order in perturbation theory. We will not explicitly calculate the field variables in perturbation theory rather we will focus on the form of the field equations when terms beyond the linear approximation are kept.

We will now give a new derivation of Dirac’s equation as the equation of motion of the elastic solid.

## Iii Derivation of Dirac’s Equation of Motion

### iii.1 Cartan’s Spinors

The concept of Spinors was introduced by Eli Cartan in 1913ref:Cartan . In Cartan’s original formulation spinors were motivated by studying isotropic vectors which are vectors of zero length. In three dimensional Minkowski space the equation of an isotropic vector is

(22) |

for generally complex quantities . A closed form solution to this equation is realized as

(23) |

where the two quantities are then

The two component object has the rotational properties of a spinorref:Cartan and any equation of the form (22) has a spinor solution.

In the following we use the notation and the wave equation is written

This equation can be viewed as an isotropic vector in the following way. The components of the vector are the partial derivative operators acting on the quantity . As long as the partial derivatives are restricted to acting on the scalar field it has a spinor solution given by

(24) |

and

(25) |

where the ”hat” notation indicates that the quantities are operators. Let us now introduce the variables

Equations (24) and (25) are now

and

These are equations of fractional derivatives of order (also called semiderivatives) denoted and . Fractional derivatives have the property thatref:Miller_Ross

and various methods exist for writing closed form solutions for these operatorsref:Miller_Ross ; ref:Oldham_Spanier . The exact form for these fractional derivatives however, is not important here. The important thing to note is that a solution to the wave equation can be written in terms of spinors which are fractional derivatives.

One of the interesting properties of suitably defined fractional derivatives (for instance the Weyl fractional derivative) that will be exploited in later sections is their action on the exponential function. While the derivative of an exponential is given by

the semiderivative of the exponential function is given by

(26) |

This will prove useful later when we Fourier transform the equations of motion.

### iii.2 Matrix Form

It can be readily verified that our spinors satisfy the following equations

and in matrix form

(27) |

The matrix

is equal to the dot product of the vector with the pauli spin matrices

where

are proportional to the Pauli matrices and satisfy the anticommutation relations

(28) |

where is the identity matrix.

Equation (27) can be written

(29) |

This equation has the form of Dirac’s equation in three-dimensions for a noninteracting, massless, spin- field, .

### iii.3 Relation to the Dirac Decomposition

The fact that the wave equation and Dirac’s equation are related is not new. However the decomposition used here is not the same as that used by Dirac. The usual method of connecting the second order wave equation to the first order Dirac equation is to operate on Equation (29) from the left with giving

(30) | |||||

where is a two component spinor and Equation (28) has been used in the last step.

This shows that Dirac’s equation does in fact imply the wave equation. The important thing to note about Equation (30) however, is that the three-dimensional Dirac’s equation implies not one wave equation but two in the sense that each component of the spinor satisfies this equation. Explicitly stated, Equation (30) reads

for the independent scalars .

Conversely, if one starts with the wave equation and tries to recover Dirac’s equation, it is necessary to start with two independent scalars each independently satisfying the wave equation. In other words, using the usual methods, it is not possible to take a single scalar field that satisfies the wave equation and recover Dirac’s equation for a two component spinor.

What has been demonstrated in the preceding sections is that starting with only one scalar quantity satisfying the wave equation, Dirac’s equation for a two component spinor may be derived. Furthermore any medium (such as an elastic solid) that has a single scalar that satisfies the wave equation must have a spinor that satisfies Dirac’s equation and such a 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 the spinor (represented as a fractional derivative) when the field displacements are Fourier transformed. In the infinitesimal theory of elasticity, all terms in beyond the linear term are dropped. In the infinitesimal theory therefore, no distinction is made between transforming the coordinates and . Later when we assume small but finite strain components, we will need to distinguish these coordinates.

The Dirac field in Equation (29) is given explicitly by

(31) |

with and . Because the semiderivatives are independent of we can bring through the operator when we Fourier transform .

Transforming the dilatation first in the periodic coordinate , Equation (29) becomes

(32) | |||||

where we used , and .

Equation (32) is equal to zero only if the coefficients of are zero for each value of giving

(33) |

where and satisfies the conditions for a two dimensional metric with

(34) |

Equation (33) shows that the fourier modes of the elastic solid obey a two dimensional version of Dirac’s equation for spin- particles with a mass . The two continuous variables left in the problem are and with playing the role of time.

Let us examine the form of the spinor further by Fourier transforming the two continuous coordinates and . Using Equation (26) we further transform the spinor as

(35) | |||||