# A geometrical derivation of the Dirac equation

###### Abstract

We give a geometrical derivation of the Dirac equation by considering a spin- particle travelling with the speed of light in a cubic spacetime lattice. The mass of the particle acts to flip the multi-component wavefunction at the lattice sites. Starting with a difference equation for the case of one spatial and one time dimensions, we generalize the approach to higher dimensions. Interactions with external electromagnetic and gravitational fields are also considered. One logical interpretation is that only at the lattice sites is the spin- particle aware of its mass and the presence of external fields.

PACS: 03.65.Pm; 11.10.-z

## I Introduction

There are different ways to derive the Dirac equation. But probably there
is no derivation more elegant than the one Dirac gave in his
book.Dirac The derivation based on Wigner’s analysis of the
irreducible unitary representation of the “Poincare group” (the covering
group of the inhomogeneous proper othochronous Lorentz group) certainly is
also important.Wigner There is another intriguing derivation which
Feynman gave (for the case with one spatial dimension and one time
dimension) in his class^{2}^{2}2L. Brown, private communication (2001).
and which was given as a problem in his
book with HibbsHibbs .

In this paper we give another derivation of the Dirac equation. Our
approach bears some resemblance to Feynman’s and is based on Dirac’s
observation that the instantaneous velocity operators of the
spin-
particle (hereafter called by the generic name “the electron”) have
eigenvalues and that they anticommute.^{3}^{3}3The Hamiltonian
for a free electron is given by (in Dirac’s notation) with anticommuting and
. The -th component of the velocity is . Thus
has as eigenvalues
, corresponding to the eigenvalues of . This
result is actually implied by the uncertainty principle.
DiracDirac also shows that consists of two parts, a
constant part , connected with the momentum by the
classical relativistic formula, and an oscillatory part whose frequency is
high, being at least .
(Hereafter, unless clarity demands otherwise, we set the speed of light
, as well as Planck’s constant , equal to unity.)
We assume spacetime to be
“filled” with a four-dimensional cubic lattice with lattice length
. While it is natural to
take to be the Planck length ( cm), we will simply
take it to be a length very small compared to the electron’s Compton
wavelength, the only intrinsic length available in the problem for a free
electron. For the resulting difference equation, the zeroth order
term in gives a trivial identity while the first order term
yields the Dirac equation.

It is interesting that the Dirac equation is invariant under rotations and
Lorentz transformations, while the underlying spacetime lattice is not.
This situation is one which is not unfamiliar in the spatial dimensions
in condensed matter physics. (But a related and
more intriguing result was that found by SnyderSnyder more than half
a century ago, who showed that spacetime
being a continuum is not required by Lorentz invariance.) In our approach,
an electron’s propagation through spacetime can be visualized as
consisting of two steps: the electron transfers from one spatial lattice site
to a neighboring site in one unit of time (thus travelling with the speed of
light) and at a lattice site the electron (its multi-component
wavefunction, to be more precise) is ‘‘flipped’’ by a mass
operator^{4}^{4}4This is in consonance with the mass term
being equivalent to a helicity flip. and
interacts with external fields.

In the next section, we consider the case of one spatial dimension (and one time dimension). We treat the case of higher spatial dimensions in Section III. Interactions with external electromagnetic and gravitational fields are considered in Section IV. Further discussions are given in the last Section.

## Ii : (1 + 1)-dimensional case

We assume that the electron of mass moves with the speed of light from one lattice site to a neighboring (spatially left or right) site with time always increasing on the “cubic” spacetime (z,t) lattice. The wavefunction has two components

(1) |

where denotes the component arriving from the event while means arriving from .

Next we assume that, at the lattice site , the arriving components are partially turned around by a unitary matrix:

(2) |

with the “flip operator” defined by

(3) |

Here is a hermitian matrix which we call the “flip matrix” and give the most obvious form

(4) |

with being the first Pauli matrix. We will approximate Eq. (2) by a differential equation by first writing

(5) |

with the “transfer” operator given by

(6) |

where is the third Pauli matrix. Then the difference equation Eq. (2) takes the form

(7) |

The difference equation becomes a differential equation if we limit ourselves to the zeroth order (given by the identity ) and the first order term in . The first order equation is

(8) |

the Dirac equation^{5}^{5}5In our representation of the Dirac matrices,
the positive-energy spinor for
the plane-wave solution takes the form, aside from a normalization
constant, ,
with being the two-component spinor , and the
negative-energy spinor , with . in dimensions! There is no spin
in the
case, the little group of being trivial.
In passing we mention that,
for the Dirac equation to hold to all orders in , due to the
fact that and do not commute with each other, it is
necessary to replace in Eq. (7) by
their “symmetrical” product is
sufficiently small that higher order terms are negligible, i.e.,
the Dirac equation is a good approximation to the original difference
equation. . We will
not pursue this issue any further and will assume that

## Iii Higher spatial dimensions

We start by reminding ourselves that, for the Dirac equation, the velocity
operators not only
have eigenvalues , but they also anticommute with
each other.^{6}^{6}6Here an analogy with spin can be made. Just as one
cannot specify all three
components of the spin simultaneously in quantum mechanics without running
into inconsistencies (of predicting a spin times bigger in
some directions),
one cannot
specify all three components of the instantaneous velocity simultaneously
without running into inconsistencies of predicting a tachyonic speed of
times the speed of light. The
latter fact makes the generalization of our approach to more than one
spatial dimension non-trivial. Before we proceed to the -dimensional case, let us first discuss the two spatial dimensional
case.

As in the preceeding section (for the case), we use to give the dependence on the coordinate and for the flip operator. Of the three Pauli matrices, we have only left; so let us call the second spatial coordinate the coordinate. Now the problem is to express the dependence of on in the sense that gives the dependence on .

To solve this problem we appeal to rotational invariance and make explicit use of the spin- property of . Let be the rotation over from the axis to the axis. Then , which represents this rotation, is given by . Thus, to the term in the “transfer” operator we add

(9) |

and obtain

(10) |

Note that the flip operator, which is used in the preceeding section to invert the -motion, also inverts the -motion. In Dirac’s notationDirac , we identify , , .

Now we recall the general rulewein1 that spinors in dimensions and in dimensions have components. Thus for the case of three spatial dimensions and one temporal dimension (), we need -spinors. And we need, besides , , , an extra , all four of which anticommute with one another and have eigenvalues . Following DiracDirac , we introduce two independent sets of Pauli matrices and . The ’s and the ’s anticommute among each set, whereas the ’s and the ’s commute. As we want to make look like a flip matrix, we pick

(11) |

We complete the set (by the same argument we have used above for the case) with

(12) |

Here 1 is the unit matrix. In passing, we mention that it is easy to treat the case of -dimensional spacetime, as we can now identify with .

For the case of dimensions, the equation we obtain is

(13) |

It is of relevance to remark that the last three terms on the right hand side of Eq. (13) approximate the small finite steps of motion with the speed of light before the event is reached. The term represents the unitary transformation which takes place at that event (see Eqs. (2) and (3)). Thus the electron is not aware of the fact that it has a mass until it hits a lattice site. If it has no mass, then it is not flipped and it moves at the constant speed of light. If it is massive and is at “rest”, then it must be that the electron zigzags around with the speed of light and returns to its original spatial lattice site and wanders around again and returns again etc.

## Iv Interactions with external fields

To put the Dirac equation in a covariant form, we follow the usual procedure of writing (with ) and multiplying Eq. (13) by to yield

(14) |

where ( running over )

(15) |

with , the identity matrix and

The introduction of an electromagnetic field is straightforward by using the prescription of minimal substitution in Eq. (14)

(16) |

Although the term goes together with the
term in the minimal subsitution rule, it
is tempting to keep the term identified with the transfer
between lattice sites and put the term together
with
the flip term as taking place at the lattice sites. (But we should
keep in mind that, since the Dirac matrices do not commute among
themselves, beyond the first order term, there is a difference between
associating the interaction term with the “transfer” operator
or the “flip” operator .^{7}^{7}7For the
case, the term can be incorporated into
the flip operator in Eq. (2).)

To incorporate gravitational interactions one needs the tetrad (or vierbein) formalismUandK . One introduces at every event a set of local inertial coordinates with a tetrad of axes labelled by the Minkowski index running over .Schwinger Then the metric in any general noninertial coordinate system is given by in terms of the flat Minkowski metric . Gravitational interactions are introduced via the substitution rule

(17) |

where in terms of the affine connection and . At every spacetime lattice site labelled by , we have a tetrad . In our interpretation, the electron travels with the speed of light between lattice sites; this is represented by . Then at the lattice site there is a unitary transformation which, in addition to the mass “flip”, now contains the interaction term .

## V Discussions

We have presented a novel derivation of the Dirac equation, hoping to shed new light on the physics of the electron. Motivated by the distinct possibility that the underlying spacetime is discrete at small scales, we have started with a discrete “cubic” lattice. The resulting Dirac equation emerges as the lowest nontrivial order of approximation. Thus the observed Lorentz invariance does not preclude the existence of a discrete spacetime at small scales.

Is our approach useful? We think so. (1) The very fact that the underlying spacetime is discrete means that there is automatically an ultraviolet cutoff which may be used to ameliorate divergence problems in nonrenormalizable theories like (perhaps) quantum gravity. (2) Our starting point is a difference equation rather than a differential equation. While difference equations are more tedious to deal with analytically, they may hold some advantages in numerical calculations.

We conclude with some speculations and a couple of open questions.
In the scenario we have proposed, the electron travels between lattice sites
with the speed of light. Only at the lattice sites does the electron
‘‘feel’’ its mass and perhaps also the presence of all external
fields.^{8}^{8}8It is natural to visualize interactions taking place at
lattice sites
(in conjunction with the mass flip). After all, interactions occur at
spacetime points and spacetime points are the lattice sites if the
underlying spacetime is discrete.
(Since it is a Yukawa-type interaction which, via the Higgs
mechanism, generates mass for the electron, it seems reasonable to
assume that at least Yukawa-type interactions take place only at the
lattice sites where the mass operator makes its presence felt.) But if
gravitational
interactions also take place mainly at the lattice sites, does that mean
spacetime vertices somehow play an important role in
concentrating curvature? And if so, how is this description of geometry and
topology related to the
Regge calculusRegge , for example? These problems deserve further
investigations.

## Acknowledgements

We thank Giovanni Amelino-Camelia and E. Merzbacher for useful discussions, Laurie Brown for a useful correspondence, and C.N. Yang for his encouragement to understand the Dirac equation from a new perspective. We also thank Ted Jacobson and A. Rivero for calling our attention to some useful references. This work was supported in part by DOE under #DE-FG05-85ER-40219 and by the Bahnson Fund of University of North Carolina at Chapel Hill.

## References

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- (3) R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). Also see T. Jacobson, J. Phys. A17 (1984) 2433, ibid. A17 (1984) 375; I. Bialynicki-Birula, Phys. Rev. D49 (1994) 6920; L.H. Kauffman and H.P. Noyes, Phys. Lett. A218 (1996) 139.
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- (7) See, e.g., J. Schwinger, Particles, Sources, and Fields, Vol. I (Addison-Wesley, Reading, MA, 1970).
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