Foundations of Physics Letters, Vol. 15, No. 1, February 2002
Torsion Waves in Three Dimensions:
Quantum Mechanics With a Twist
R. A. Close

This paper presents a mathematical description of torsion waves in three dimensions.
Torsion waves are described by a four-component wave function which satisfies a Dirac
equation. The term normally associated with electron mass introduces rotation or
oscillation of the propagation direction. The relation to quantum mechanics is discussed.

1. INTRODUCTION
Much of our understanding of physical processes is by analogy with simple models.
For example, the harmonic oscillator can be used to illustrate a wide variety of physical
phenomena. Progress in quantum theory, however, has come primarily from mathematical
inference rather than physical modeling. No suitable model has been found to illustrate the
behavior of the quantum wave functions, leading some to remark that “quantum theory
needs no ‘interpretation’”.(1) The motivation for this paper is that the art of physical
modeling seems to have been sorely neglected.
Suppose we attempt to develop a physical model for quantum mechanics in general and
particle physics in particular. Where should we start? We know that matter consists of
waves, so let’s start with the simplest medium which can support waves: a homogeneous
elastic solid. The neglect of physical modeling in the atomic age is so severe that even this

1
simple model has never been completely analyzed. Hence it is unknown to what extent, if
any, this is an adequate model for particle physics and quantum mechanics. There are well
known descriptions of compression waves and vector shear waves in three dimensions, and
of torsion waves in one dimension, but evidently no one has ever constructed a complete
description of torsion waves in three dimensions. Instead, the “tangential approximation”
has been used to approximate the rotation field by the curl of a displacement field. (2) The
limitation of this representation is that as one varies the magnitude of the displacement
field a given point moves along a straight line rather than a circle. Hence this is a valid
representation of rotations only for infinitesimal displacements. The purpose of this paper
is to find an exact description of torsion waves in three dimensions.
We will start by considering one-dimensional torsion waves on a torsion wave machine
(or a stretched-out rubber band). A torsion wave machine has at least one intriguing
parallel with particle physics. If one rotates a single rod near the center of the wire, a righthanded twist propagates in one direction and a left-handed twist propagates in the other
direction, analogous to the production of particles and anti-particles. Indeed, the notion that
torsion should be associated with matter is widely accepted.(3) Therefore there is reason to
believe that a mathematical analysis of torsion waves might provide some clues to the
interpretation of quantum mechanics.
2
2. ANALYSIS OF TORSION WAVES
2.1. Wave Equation
If the moment of inertia per unit length is I, and the torsion spring constant of the wire
(or rubber band ) is K, then the wave equation is given by:
I
 2 z, t 
t 2
K
 2 z, t 
(1)
z 2
where z,t is the orientation at axial position z and time t. The wave speed is given by
c= K I .
Like linear waves, a unique frequency and wavelength cannot be defined for torsion waves
unless many cycles are produced in succession. If one end of the wave machine is rotated
at a constant rate , the torsion waves propagate along the machine with uniform
wavelength =c/. Each rod along the machine rotates with the constant driving frequency
. The angular momentum per unit length  is therefore   I  Ick  Ic  z . The
angular momentum is therefore proportional to the spatial derivative of the angle. The
angular momentum of a twist from 0 to  0 can be obtained by integrating over angle:
L0 
z   0 

z   0 
I
z   0 
0
d
d
dz 
dz   Icd  Ic0
 Ic
dt
z   0  dz
0
(2)
Thus we see that the total angular momentum of a twist of given angle is independent
of frequency.
3
A twist propagating with constant wavelength has no torque, so the kinetic and
potential energies remain constant as the wave propagates. The kinetic energy per unit
length
I2/2
is
and
the
potential
energy
per
unit
length
is
1
K  / z 2  2  2 K / 2  I2 / 2 . Integration from 0 to  0 yields for the total energy:
2

z    0 

z   0 
I
z    0 
0
d
d
dz 
dz   Icd  Ic0  L0
 Ic
dt
z   0  dz
0
(3)
The wave energy is equal to the wave angular momentum times the angular frequency.
This is in agreement with the energy quantum of . At this point we make the
identifications:
I
L0
c 0
(4)
Lc
K c I  0
0
2
so that the wave equation is simply:
 2 z, t 
t 2
 c2
 2 z, t 
(5)
z 2
Incidentally, although we have been describing torsion waves along a wire, this is the
same equation as for torsion waves in a cylindrical rod (see e.g. Feynman et al. (4)). The
equation for torsion waves is independent of the diameter of the rod! However, at any
given point the interactions must be local. Therefore Eq. 5 is valid not only for a torsion
wave machine but also for local rotations in an isotropic elastic medium.
4
2.2. Complex Wave Functions
Now suppose we try to write the waves in harmonic form   0ei kz  t  . This
representation is appropriate for oscillations but not for sustained twists. For example, if a
rod is rotated from =0 to =, the system has not returned to its original state but rather
has added a complete twist. Therefore we must make a substitution of variables if we want
to use a harmonic representation.
One possible transformation is z, t   ei . Solving for d/dz yields:

i

 * 
   *

z
2
z
z 
(6)
Recall that the general solution to the one-dimensional wave equation is a combination
of arbitrary functions of the form f(z±ct). Each of these solutions has the property that:
f z  ct 
f z  ct 
 c
t
z
(7)
Substituting -d/dt for cd/dz on the right side of Eq. 6 yields:

i 

 * 

*


z 2c 
t
t 
(8)
Interestingly, the right hand side of this equation looks just like the ‘probability
density’ of the Klein-Gordon equation. The lack of positive definiteness for this quantity
has made a probabilistic interpretation of the Klein-Gordon equation problematic, but here
5
we have the simple explanation that positive and negative values indicate different
directions of twist (i.e. polarizations).
If the sign of d/dz is always positive, we can try a different substitution
z
z, t    dz z, t  or equivalently    z . For an arbitrary rotation angle the
2
2
normalization is:

 dz z, t   0
2
(9)

Separating the phase  from the amplitude yields:
z, t  
 i
e
z
(10)
With this construction the wave function  looks remarkably like a one-dimensional
Schrödinger wave function with a ‘probability density’ of |d/dz|.
This scalar definition of  only accounts for a single polarization (d/dz > 0). Let’s
call this right-handed (R). We could also have a left-handed wave (L) with d/dz < 0. Note
that the handedness of the wave is independent of orientation. If we also allow for
propagation in both the positive (+) and negative (–) z-directions, then there are a total of
four possible ‘polarizations’: R+, R–, L+, and L–. Since solutions to the wave equation in
one dimension are of the form f(z±ct), we have (d/dz)R+ ~ fR+(z-ct), etc. By identifying the
6
polarization with the direction the wave is traveling, the sign in the argument (z±ct) is the
opposite of the polarization label.
2.3. Conservation Laws
Let hd=| hd|2=|d/dz|hd for each handedness (h) and direction (d). Suppose that L+=
L=0 so that only right handed polarizations are present. If we superpose equal amplitude
waves propagating in opposite directions, the twist pattern will remain stationary. More
generally, the phase velocity is the difference between the (+) and (–) amplitudes divided
by the total rate of twist:
vz  c R   R   R   R  
(11)
We can define a one-dimensional “twist” density  and twist current density Jz as:
  R  R
(12)
J z  c R    R    v z
Because each component satisfies a wave equation, the continuity equation holds,
indicating conservation of total twist:
 
 R     R 
 R  
 
 J z   R  c


c
0
t z
z   t
z 
 t
(13)
Combining the R and L polarizations, there are at least two conserved quantities of interest,
one which is positive definite (like an energy or probability density) and one which is
signed (like electric charge or spin):
7
 p  R  R  L  L
(14)
s  R   R   L  L
The current associated with each of these densities is obtained by multiplying by c and
changing the sign of the waves propagating in the minus z-direction:
J p  c R    R    L    L  
(15)
J s  c R    R    L    L  
2.4. Derivation of the Dirac Equation
The wave function for all four polarizations can be written as the vector:
2
 R     R  


2
 



   R    R 2 
 L  
L 

 
  L     L  2 
(16)
The order is of course arbitrary. Note that the (+) and (–) states for each handedness
differ by a 180 degree rotation. If the states were orthogonal then these would describe a
spin 1/2 system. However, spin 1/2 systems require complex numbers, so ||2 cannot
transform with spin 1/2 but  (or some linear combination of i’s) may. In order to make
the connection with quantum mechanics, we need to show that the wave functions 
satisfy a Dirac equation.
First, we define two special functions. Using  † to denote the Hermitian conjugate
of  , the rate of angular rotation of the wave is clearly given by:
8
 z
 c zR    zR    zL 
t


†
  zL    c 


0

0 1 0 0 
  c †  z 

0 0 1 0

0 0 0  1
1
0
0
(17)
In quantum mechanics this quantity is proportional to the spin angular momentum.
The positive-density current Jp of Eq. 15 (with added z subscript) is:
J pz  c  zR    zR    zL 
1

† 0
  zL    c 
0

 0
0
1
0
0
0

0 0
  c † j pz 

1 0

0  1
0
(18)
Now we are ready to look at the first-order wave equation. Keeping track of the signs in
the arguments of (+) and (–) waves yields:
 R    1

 
1   R    0

c t  L    0

 

 L   0
0 0 0   R  
 R  
 

 
 1 0 0  R

   j pz   R  
0  1 0  z  L  
z  L  
 



0 0 1   L  
 L  
(19)
Substituting |hd|2=hd yields:
   2 
 2
R
  R  2 


2

  R 
  
 cj pz   R  2 

2
t  L 
z   L  




   2 
 2
 L 
  L  
(20)
It is convenient to regard the wave vectors as diagonal matrices  in order to expand
the derivatives. The vector form can be recovered by contracting each side of the equation
9
with a vector consisting of all ones. Since any two diagonal matrices commute the
derivatives can be expanded to yield:
†



 †
    †  c † j pz
  cj pz

t
t
z
z
(21)
Rearranging yields:
 




 †    cj pz        †  cj pz  † 
z 
z
 t
 t

(22)
Since the quantities in brackets are the complex conjugates of each other, they must be
pure imaginary in order to cancel the minus sign. Therefore:


  cj pz   i
t
z
(23)
where  is a real matrix (not necessarily constant). The vector form of the equation is
obtained simply by replacing the matrix  with the vector . This result is similar to the
Dirac equation but involves the matrix jpz instead of the Dirac matrix z. In order to use
the familiar Dirac matrices of quantum mechanics,(5) we seek a unitary transformation
which will preserve the form of the rate of twist in Eq. 17 but change the equation for Jpz to
one involving z:
10
U †  zU   z



†
U j pzU   z  


0 0
0 0
1 0
0 1
1 0

0  1
0 0

0 0 
(24)
U † U  
 new  U † old
The matrix  will be computed below. It is straightforward to show that the transformation
must have the form:



U 


a
0
c
0
0 a 0 

b 0  b
0 c 0 

d 0 d 
For simplicity we take a  b  c  d  1
(25)
2 . This transformation yields the desired
equations for the new wave function:
 z
 c †  z 
t
J pz  c †  z 
(26)


   c z   i
t
z
11
The physical interpretation of the new wave functions is:
2
 ZR  
 ZR  
 ZL  
 ZL  
1   3
2
2
2  4
2
2
1   3
2
2
2  4
2
(27)
It is clear that the choice of z-axis was arbitrary. Therefore we are justified to generalize
the last of Equations 26 to an arbitrary axis to obtain a three-dimensional Dirac equation:

  c    i
t
(28)
where  represents the vector of Dirac matrices:
0

0
x  
0

1
0 0 1
0 0


0 1 0
0 0
, y  

1 0 0
0 i


0 0 0
i 0
0  i
0 0


i 0
0 0
, z  

0 0
1 0


0 0
0 1
1 0

0  1
0 0

0 0 
(29)
Equation 28 is the general equation for the torsion wave function and is the main result of
this paper.
12
2.5. Physical Interpretations
We can obtain the second-order wave equation for each component by applying the time
derivative twice to obtain:
2
t
2
  c2   2  ic    
(30)
If we require each component to obey the same second order equation, then we have:
2  M 2 I
    0
(31)
where I is the identity matrix and M is a scalar. A solution is:
1

0
  M
0

0
0

1 0 0
 M
0 1 0 

0 0  1
0 0
(32)
For electrons M  me c 2  . Hence the Dirac equation for a free electron represents a
special case of the torsion wave equation:

  c    iM
t
(33)
13
To determine the physical interpretation of the wave components, let’s look at the socalled ‘probability current’ of the Dirac equation:

 
J pz  c †  z   c 1* 3  *31  *2  4  *4  2

1
1
1
2
2
2
2
1
 c  1   3  1   3   2   4   2   4 
2
2
2
2


 
J py  c †  y   ic 1* 4  *4 1  *2  3  *3 2

1
1
1
2
2
2
2
1
 c  1  i 4  1  i 4   2  i 3   2  i 3 
2
2
2
2


 
J px  c †  x   c 1* 4  *4 1  *3 2  *2  3
(34)

1
1
1
2
2
2
2
1
 c  1   4  1   4   2   3   2   3 
2
2
2
2

Each of these has precisely the same form as Jp in Eq. 15 above. Note that if all the
signs are changed to positive then we obtain c times the density p of Eq. 14 for any
propagation direction.
This decomposition into positive and negative currents is not
unique, however.
The z-components of twist have already been determined in Eq. 27. The components
for other axes then follow from rotations of –90 degrees about the x-axis and +90 degrees
about the y-axis. The rotation matrices are (see e.g. Feynman et al.(6)):
U z y
 1 i 0 0 


 i 1 0 0 

0 0 1  i


 0 0 i 1 


U zx
1 1

1 1

0 0

0 0

0 0

0 0
1  1

1 1 
Applying these to the z-eigenfunctions in Eq. 27 yields:
14
(35)
2
 ZR  
 ZR  
 ZL  
 ZL  
1   3
2
2
2  4
2
2
1   3
2
2
2  4
2
  i 2   3  i 4
 1
2
2
i1   2  i 3   4
2
2
YL  
1  i 2   3  i 4
2
2
i1   2  i 3   4
2
2
YL  
YR 
YR  
 XR 
  2  3  4
 1
2
 XR  
2
 1   2   3   4
2
2
(36)
2
 XL  
1   2   3   4
2
 XL  
 1   2   3   4
2
2
Note that each of these terms corresponds to an eigenvector of one of the velocity
operators:
c z  ZR   
c z  ZR   
c z  ZL   
c z  ZL   
1
1
0
 
c  
c i 
c y YR   
2   1
2   1
 
 
0
 i 
0
i 
1
 
c  
c1
c y YR   
2  i 
20
 
 
  1
  1
1 
1
 0

c  
c i
c y YL     
2 1
2 1 
 

 0
i 
 0
i 
1 

c  
c 1
c y YL   
2 i 
2  0
 

1 
1
1
 
c1
c x  XR   
2   1
 
  1
  1
 
c1
c x  XR   
21 
 
  1
1

c 1
c x  XL     
2 1

1
  1
 
c1
c x  XL   
2   1
 
1
(37)
Each of the above velocity eigenvectors corresponds to propagation of a torsion wave
with definite handedness parallel to the axis. Utilizing the fact that c can be interpreted
as a velocity operator, the Dirac equation can be written as:
15
d
  iM 
dt
where
(38)
d

  c   is the convective derivative.
dt t
2.6. Interpretation of Electron Mass
Since the mass factor M does not appear in the wave equation in real space, it is part of the
solution rather than part of the problem. Consider the effect of mass on the time derivative
of the p-current:


d †
 α   2iM †α   c † α   α   α α   
dt
(39)
If we neglect the spatial gradients then we have (assuming one component i is zero):






 
 
 
d
J pz  2iMc  1* 3   3* 1   2* 4   4* 2
dt
d
J py  2 Mc  1* 4   4* 1   2* 3   3* 2  2MJ px
dt
d
J px  2iMc  1* 4   4* 1   2* 3   3* 2  2 MJ py
dt
(40)
Thus we see that the mass term introduces a rotation of Jp in the x-y plane with
frequency 2M. The behavior of the z-component is less obvious. If we suppose that
1 ~ eiMt and  3 ~ e iMt then J pz ~ cos 2 Mt and dJ pz dt ~  sin 2 Mt . Therefore Jpz
oscillates at the same frequency as the circulation of Jpx and Jpy. Hence the Dirac equation
for a free electron is equivalent to a description of a torsion wave in which the current
components oscillate or rotate with frequency 2M.
16
3. DISCUSSION
We have developed a mathematical description of torsion waves in an elastic medium
and shown that the torsion wave function satisfies a Dirac-type equation. The Dirac
equation for a free electron is a special case of the general torsion wave equation. The mass
term is proportional to the rate of rotation or oscillation of the wave direction.
While torsion wave analysis has obvious application to solid-state physics, the most
intriguing application of the formulation presented here is in modeling quantum
mechanics. The correspondence between the torsion angular velocity and quantum angular
momentum operators indicates that the similarity between torsion and electron wave
functions is not mere coincidence. If elementary particles are assumed to be soliton
solutions of torsion waves, then it is plausible that the likelihood of detecting a ‘particle’ at
a given point would be proportional to the rate of twist at that point. Further study is
needed of the forms and interactions of torsion solitons.
Incidentally, the interpretation of matter as twists in an elastic medium also suggests a
rudimentary explanation of gravity. Ordinarily if one twists a wire (or rubber band) under
constant tension the length will contract slightly, thereby increasing the density and
reducing the wave speed. If torsion waves in an elastic medium are accompanied by such
compression, then refraction due to the gradient in wave speed would cause a mutual
attraction between waves.
It is already known that a space with infinitesimal torsion and curvature can be
modeled as a crystal with plastic deformations and defects.(7) The torsion wave model of
particles proposed here also treats the vacuum as a solid but allows it to be continuous and
17
isotropic. While torsion waves may prove insufficient as an interpretation of quantum
mechanics, it will certainly be interesting to learn how the vacuum differs from a simple
three-dimensional elastic solid.
4. CONCLUSIONS
Torsion waves in three dimensions are described by a four-component wave function
which satisfies a Dirac-type equation. Along any axis the four wave polarizations represent
right- and left-handed waves propagating up and down the axis. The Dirac equation for a
free electron can be interpreted as describing a torsion wave whose direction rotates or
oscillates with frequency proportional to the mass.
ACKNOWLEDGEMENTS
I would like to thank Damon Merari for his encouragement and insightful discussions
during the development of the ideas presented in this paper. I am also grateful to Hagen
Kleinert for previewing this manuscript.
18
REFERENCES
1. C. A. Fuchs and A. Peres, “Opinion: quantum theory needs no ‘interpretation’,” Physics
Today 53(3), 70-71 (2000).
2. H. Kleinert, Gauge Fields in Condensed Matter (World Scientific, Singapore, 1989),
Vol. II, Part IV, p. 1270.
3. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics
(World Scientific, Singapore, 1995), Second edition, p. 420.
4. R. P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on
Physics, Vol. II (Addison-Wesley, Reading, 1963), p. 38-8.
5. P. A. M. Dirac, Proc. Roy. Soc. (London), Vol. A117, 610 (1928).
6. R. P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on
Physics, Vol. III (Addison-Wesley, Reading, 1963), p. 6-10.
7. H. Kleinert, Gauge Fields in Condensed Matter (World Scientific, Singapore, 1989),
Vol. II, Part IV, p. 1362.
19