We give a geometrical derivation of the Dirac equation by considering a
spin-1/2 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-1/2 particle aware of its mass and the presence of external
fields.Comment: 9 pages, LaTeX, version accepted for publication in Phys. Lett. 

Bialynicki-Birula

Dirac

Feynman

H. van Dam

Jacobson

Kauffman

Kibble

Regge

Schwinger

Snyder

Utiyama

Weinberg

Wightman

Y.Jack Ng

English

arXiv

Crossref

A geometrical derivation of the Dirac equation

We give a geometrical derivation of the Dirac equation by considering a spin- 1/2 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- 1/2 particle aware of its mass and the presence of external fields

Ng, Y. Jack

van Dam, H.

Carolina Digital Repository

arXiv:hep-th/0211002v2  4 Feb 2003A geometrical derivation of the Dirac equationY. Jack Ng∗ and H. van DamInstitute of Field Physics,Department of Physics and Astronomy,University of North Carolina,Chapel Hill, NC 27599-3255AbstractWe give a geometrical derivation of the Dirac equation by considering a spin-12 particle travellingwith the speed of light in a cubic spacetime lattice. The mass of the particle acts to flip themulti-component wavefunction at the lattice sites. Starting with a difference equation for thecase 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 logicalinterpretation is that only at the lattice sites is the spin-12 particle aware of its mass and thepresence of external fields.PACS: 03.65.Pm; 11.10.-z∗ E-mail: yjng@physics.unc.edu (Y.J. Ng)1I. INTRODUCTIONThere are different ways to derive the Dirac equation. But probably there is no derivationmore elegant than the one Dirac gave in his book.[1] The derivation based on Wigner’sanalysis of the irreducible unitary representation of the “Poincare group” (the covering groupof the inhomogeneous proper othochronous Lorentz group) certainly is also important.[2]There is another intriguing derivation which Feynman gave (for the SO(1, 1) case with onespatial dimension and one time dimension) in his class1 and which was given as a problemin his book with Hibbs[3].In this paper we give another derivation of the Dirac equation. Our approach bears someresemblance to Feynman’s and is based on Dirac’s observation that the instantaneous ve-locity operators of the spin-12particle (hereafter called by the generic name “the electron”)have eigenvalues ±c and that they anticommute.2 (Hereafter, unless clarity demands oth-erwise, we set the speed of light c, as well as Planck’s constant h̄, equal to unity.) Weassume spacetime to be “filled” with a four-dimensional cubic lattice with lattice length∆x = ∆y = ∆z = ∆t = l. While it is natural to take l to be the Planck length (∼ 10−33cm), we will simply take it to be a length very small compared to the electron’s Comptonwavelength, the only intrinsic length available in the problem for a free electron. For theresulting difference equation, the zeroth order term in ∆t gives a trivial identity while thefirst order term yields the Dirac equation.It is interesting that the Dirac equation is invariant under rotations and Lorentz trans-formations, while the underlying spacetime lattice is not. This situation is one which is notunfamiliar in the spatial dimensions in condensed matter physics. (But a related and moreintriguing result was that found by Snyder[4] more than half a century ago, who showed thatspacetime being a continuum is not required by Lorentz invariance.) In our approach, anelectron’s propagation through spacetime can be visualized as consisting of two steps: the1 L. Brown, private communication (2001).2 The Hamiltonian for a free electron is given by H = cαipi + ρ3mc2 (in Dirac’s notation) with anticom-muting αi and ρ3. The i-th component of the velocity isdxidt= (ih̄)−1[xi, H ] = cαi. Thusdxidthas aseigenvalues ±c , corresponding to the eigenvalues ±1 of αi. This result is actually implied by the uncer-tainty principle. Dirac[1] also shows that cαi consists of two parts, a constant part c2piH−1, connectedwith the momentum by the classical relativistic formula, and an oscillatory part whose frequency is high,being at least mc2/πh̄.2electron transfers from one spatial lattice site to a neighboring site in one unit of time (thustravelling with the speed of light) and at a lattice site the electron (its multi-componentwavefunction, to be more precise) is “flipped” by a mass operator3 and interacts with exter-nal fields.In the next section, we consider the case of one spatial dimension (and one time dimen-sion). We treat the case of higher spatial dimensions in Section III. Interactions with externalelectromagnetic and gravitational fields are considered in Section IV. Further discussions aregiven in the last Section.II. SO(1, 1): (1 + 1)-DIMENSIONAL CASEWe assume that the electron of mass m moves with the speed of light from one lattice siteto a neighboring (spatially left or right) site with time t always increasing on the “cubic”spacetime (z,t) lattice. The wavefunction has two componentsψ(z, t) =ψ+(z, t)ψ−(z, t) , (1)where ψ+ denotes the component arriving from the event (z −∆t, t−∆t) while ψ− meansarriving from (z +∆t, t−∆t).Next we assume that, at the lattice site (z, t), the arriving components are partiallyturned around by a unitary matrix:ψ+(z, t)ψ−(z, t) = Fψ+(z −∆t, t−∆t)ψ−(z +∆t, t−∆t) , (2)with the “flip operator” F defined byF ≡ e−iFm∆t. (3)Here F is a hermitian 2×2 matrix which we call the “flip matrix” and give the most obviousformF = σx ≡ σ1 =0 11 0 , (4)3 This is in consonance with the mass term being equivalent to a helicity flip.3with σ1 being the first Pauli matrix. We will approximate Eq. (2) by a differential equationby first writingψ+(z −∆t, t−∆t)ψ−(z +∆t, t−∆t) = T ψ(z, t), (5)with the “transfer” operator T given byT = e−∆t( ∂∂t+σ3 ∂∂z ), (6)where σ3 ≡ diag(1,−1) is the third Pauli matrix. Then the difference equation Eq. (2)takes the formψ(z, t) = FT ψ(z, t). (7)The difference equation becomes a differential equation if we limit ourselves to the zerothorder (given by the identity ψ(z, t) = ψ(z, t)) and the first order term in ∆t. The first orderequation isi∂∂tψ(z, t) = mσ1ψ(z, t)− iσ3∂∂zψ(z, t), (8)the Dirac equation4 in (1 + 1) dimensions! There is no spin in the SO(1, 1) case, the littlegroup of pµ being trivial. In passing we mention that, for the Dirac equation to hold toall orders in ∆t, due to the fact that σ1 and σ3 do not commute with each other, it isnecessary to replace FT in Eq. (7) by their “symmetrical” product e−∆t(imσ1+ ∂∂t+σ3 ∂∂z ). Wewill not pursue this issue any further and will assume that ∆t is sufficiently small that higherorder terms are negligible, i.e., the Dirac equation is a good approximation to the originaldifference equation.III. HIGHER SPATIAL DIMENSIONSWe start by reminding ourselves that, for the Dirac equation, the velocity operators1ih̄[~x,H ] = c~α not only have eigenvalues ±c, but they also anticommute with each other.5The latter fact makes the generalization of our approach to more than one spatial dimension4 In our representation of the Dirac matrices, the positive-energy spinor for the plane-wave solution takesthe form, aside from a normalization constant, u(p) ∼ (−σ1√p2 +m2 − iσ2p −m)u(0), with u(0) beingthe two-component spinor (1,−1), and the negative-energy spinor v(p) ∼ (σ1√p2 +m2 + iσ2p−m)v(0),with v(0) = (1, 1).5 Here an analogy with spin can be made. Just as one cannot specify all three components of the spinsimultaneously in quantum mechanics without running into inconsistencies (of predicting a spin√3 times4non-trivial. Before we proceed to the (3 + 1)-dimensional case, let us first discuss the twospatial dimensional SO(2, 1) case.As in the preceeding section (for the SO(1, 1) case), we use σ3 to give the dependenceon the z coordinate and σ1 for the flip operator. Of the three Pauli matrices, we have onlyσ2 left; so let us call the second spatial coordinate the y coordinate. Now the problem is toexpress the dependence of ψ on y in the sense that σ3 gives the dependence on z.To solve this problem we appeal to rotational invariance and make explicit use of thespin-12property of ψ. Let R be the rotation over π/2 from the y axis to the z axis. ThenU(R), which represents this rotation, is given by U(R) = 1√2(1 − iσ1). Thus, to the σ3 ∂∂zterm in the “transfer” operator T we addU−1(R)σ3∂∂yU(R) = σ2∂∂y, (9)and obtaini∂∂tψ(y, z, t) =(mσ1 − iσ2∂∂y− iσ3∂∂z)ψ(y, z, t). (10)Note that the flip operator, which is used in the preceeding section to invert the z-motion,also inverts the y-motion. In Dirac’s notation[1], we identify σ1 = αm, σ2 = α2, σ3 = α3.Now we recall the general rule[5] that spinors in 2n dimensions and in 2n+1 dimensionshave 2n components. Thus for the case of three spatial dimensions and one temporal dimen-sion (SO(3, 1)), we need 4-spinors. And we need, besides αm, α2, α3, an extra α, all fourof which anticommute with one another and have eigenvalues ±1. Following Dirac[1], weintroduce two independent sets of Pauli matrices ~ρ and ~σ. The ρ’s and the σ’s anticommuteamong each set, whereas the ρ’s and the σ’s commute. As we want to make αm look like aflip matrix, we pickαm = ρ1σ1 =0 0 0 10 0 1 00 1 0 01 0 0 0. (11)We complete the set (by the same argument we have used above for the SO(2, 1) case) withα1 = ρ2σ1, α2 = ρ3σ1, α3 = σ31. (12)bigger in some directions), one cannot specify all three components of the instantaneous velocity simul-taneously without running into inconsistencies of predicting a tachyonic speed of√3 times the speed oflight.5Here 1 is the 2 × 2 unit matrix. In passing, we mention that it is easy to treat the case of(4 + 1)-dimensional spacetime, as we can now identify α4 with σ21.For the case of (3 + 1) dimensions, the equation we obtain isi∂∂tψ =(αmm− α1i∂∂x− α2i∂∂y− α3i∂∂z)ψ. (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 (x, y, z, t)is reached. The term αmm represents the unitary transformation e−iFmdt which takes placeat that event (see Eqs. (2) and (3)). Thus the electron is not aware of the fact that it has amass until it hits a lattice site. If it has no mass, then it is not flipped and it moves at theconstant speed of light. If it is massive and is at “rest”, then it must be that the electronzigzags around with the speed of light and returns to its original spatial lattice site andwanders around again and returns again etc.IV. INTERACTIONS WITH EXTERNAL FIELDSTo put the Dirac equation in a covariant form, we follow the usual procedure of writingαm = γ0 (with (γ0)2 = 1) and multiplying Eq. (13) by γ0 to yield(iγµ∂µ −m)ψ = 0, (14)where (µ running over 0, 1, 2, 3)γµ ≡ γ0αµ, (15)with α0 ≡ I, the identity matrix and ∂µ ≡(∂∂t, ∂∂x, ∂∂y, ∂∂z). From the way we have derivedthe Dirac equation, we can trace the iγµ∂µ term to the “transfer” of the electron at thespeed of light between the lattice sites while the m term comes from the “flip” unitarytransformation at the lattice sites.The introduction of an electromagnetic field Aµ is straightforward by using the prescrip-tion of minimal substitution in Eq. (14)i∂µ ⇒ i∂µ + eAµ. (16)Although the eαµAµ term goes together with the αµ∂µ term in the minimal subsitution rule,it is tempting to keep the ∂µ term identified with the transfer between lattice sites and put6the eαµAµ term together with the Fm flip term as taking place at the lattice sites. (Butwe 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 termwith the “transfer” operator T or the “flip” operator F .6)To incorporate gravitational interactions one needs the tetrad (or vierbein) formalism[6].One introduces at every event x a set of local inertial coordinates with a tetrad eµa ofaxes labelled by the Minkowski index a, b, c running over 0, 1, 2, 3.[7] Then the metric inany general noninertial coordinate system is given by gµν = eaµηabebν in terms of the flatMinkowski metric ηab. Gravitational interactions are introduced via the substitution ruleγa∂a ⇒ γaeνa(∂ν −14iωbcνσbc), (17)where ωbaν = (∂νeµa + Γµνλeλa)ebµ in terms of the affine connection Γµνλ and σbc = 12i[γb, γc]. Atevery spacetime lattice site labelled by x, we have a tetrad eµa . In our interpretation, theelectron travels with the speed of light between lattice sites; this is represented by γaeνa∂ν .Then at the lattice site there is a unitary transformation which, in addition to the mass“flip”, now contains the interaction term γaeνaωbcνσbc.V. DISCUSSIONSWe have presented a novel derivation of the Dirac equation, hoping to shed new lighton the physics of the electron. Motivated by the distinct possibility that the underlyingspacetime is discrete at small scales, we have started with a discrete “cubic” lattice. Theresulting Dirac equation emerges as the lowest nontrivial order of approximation. Thus theobserved Lorentz invariance does not preclude the existence of a discrete spacetime at smallscales.Is our approach useful? We think so. (1) The very fact that the underlying spacetime isdiscrete means that there is automatically an ultraviolet cutoff which may be used to ame-liorate divergence problems in nonrenormalizable theories like (perhaps) quantum gravity.(2) Our starting point is a difference equation rather than a differential equation. While dif-ference equations are more tedious to deal with analytically, they may hold some advantagesin numerical calculations.6 For the SO(1, 1) case, the Aµ term can be incorporated into the flip operator in Eq. (2).7We conclude with some speculations and a couple of open questions. In the sce-nario we have proposed, the electron travels between lattice sites with the speed oflight. Only at the lattice sites does the electron “feel” its mass and perhaps also thepresence of all external fields.7 (Since it is a Yukawa-type interaction which, via theHiggs mechanism, generates mass for the electron, it seems reasonable to assume thatat least Yukawa-type interactions take place only at the lattice sites where the massoperator makes its presence felt.) But if gravitational interactions also take place mainlyat the lattice sites, does that mean spacetime vertices somehow play an important rolein concentrating curvature? And if so, how is this description of geometry and topologyrelated to the Regge calculus[8], for example? These problems deserve further investigations.AcknowledgementsWe thank Giovanni Amelino-Camelia and E. Merzbacher for useful discussions, LaurieBrown for a useful correspondence, and C.N. Yang for his encouragement to understandthe Dirac equation from a new perspective. We also thank Ted Jacobson and A. Rivero forcalling our attention to some useful references. This work was supported in part by DOEunder #DE-FG05-85ER-40219 and by the Bahnson Fund of University of North Carolinaat Chapel Hill.[1] P.A.M. Dirac, Quantum Mechanics, 4th ed. (Oxford University Press, London, 1958).[2] See, e.g., A.S. Wightman, in Dispersion Relations and Elementary Particles, edited by C.DeWitt and R. Omnes (Wiley and Sons, New York, N.Y., 1960).[3] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, NewYork, 1965). Also see T. Jacobson, J. Phys. A17 (1984) 2433, ibid. A17 (1984) 375; I. Bialynicki-7 It is natural to visualize interactions taking place at lattice sites (in conjunction with the mass flip). Afterall, interactions occur at spacetime points and spacetime points are the lattice sites if the underlyingspacetime is discrete.8Birula, Phys. Rev. D49 (1994) 6920; L.H. Kauffman and H.P. Noyes, Phys. Lett. A218 (1996)139.[4] H.S. Snyder, Phys. Rev. 71 (1947) 38.[5] See, e.g., S. Weinberg, The Quantum Theory of Fields, Vol. I (Cambridge University Press,Cambridge, 1995).[6] R. Utiyama, Phys. Rev. 101 (1956) 1597; T.W.B. Kibble, J. Math. Phys. 2 (1961) 212.[7] See, e.g., J. Schwinger, Particles, Sources, and Fields, Vol. I (Addison-Wesley, Reading, MA,1970).[8] T. Regge, Nuovo Cimento 19 (1961) 558.9

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A geometrical derivation of the Dirac equation

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