What is the Dirac equation?
M. Burak Erdo˘gan ∗ William R. Green † Ebru Toprak ‡ July 15, 2021
at all times, and hence the model needs to be first or- der in time, [Tha92]. In addition, it should conserve In the early part of the 20th century huge advances the L2 norm of solutions. Dirac combined the quan- were made in theoretical physics that have led to tum mechanical notions of energy and momentum vast mathematical developments and exciting open operators E = i~∂t, p = −i~∇x with the relativis- 2 2 2 problems. Einstein’s development of relativistic the- tic Pythagorean energy relation E = (cp) + (E0) 2 ory in the first decade was followed by Schr¨odinger’s where E0 = mc is the rest energy. quantum mechanical theory in 1925. Einstein’s the- Inserting the energy and momentum operators into ory could be used to describe bodies moving at great the energy relation leads to a Klein–Gordon equation speeds, while Schr¨odinger’stheory described the evo- 2 2 2 4 lution of very small particles. Both models break −~ ψtt = (−~ ∆x + m c )ψ. down when attempting to describe the evolution of The Klein–Gordon equation is second order, and does small particles moving at great speeds. In 1927, Paul not have an L2-conservation law. To remedy these Dirac sought to reconcile these theories and intro- shortcomings, Dirac sought to develop an operator1 duced the Dirac equation to describe relativitistic quantum mechanics. 2 Dm = −ic~α1∂x1 − ic~α2∂x2 − ic~α3∂x3 + mc β Dirac’s formulation of a hyperbolic system of par- tial differential equations has provided fundamental which could formally act as a square root of the Klein- 2 2 2 models and insights in a variety of fields from parti- Gordon operator, that is, satisfy Dm = −c ~ ∆ + 2 4 cle physics and quantum field theory to more recent m c . This is possible if the coefficients αj, β are 2 2 applications to nanotechnology, specifically the study 4 × 4 matrices satisfying αj = β = I and the anti- of graphene. commutation relationship (for j 6= k)
αjαk = −αkαj, αjβ = −βαj. Dirac Equation Typically the Pauli matrices, To formulate the Dirac equation consistent with a quantum mechanical interpretation, the wave func- 0 −i 0 1 1 0 σ1 = , σ2 = , σ3 = , tion at time t = 0 should determine the wave function i 0 1 0 0 −1
∗The first author is a professor of mathematics at the Uni- are used to formulate the Dirac system versity of Illinois at Urbana-Champaign. His email address is 3 [email protected]. X †The second author is an associate professor of mathematics 2 i~ψt(x, t) = − ic~ αk∂xk + mc β ψ(x, t), (1) at Rose-Hulman Institute of Technology. His email address is k=1 [email protected]. ‡The third author is a Hill Assistant Professor at Rutgers 1For concreteness, we only consider the case of three spatial 3 University. Her email address is [email protected]. dimensions, that is when x ∈ R .
1 where A model dispersive PDE is the free Schr¨odinger equation: 2 IC 0 0 σi β = , αi = . 0 −I 2 σ 0 C i iψt = −∆ψ. (3)
For notational convenience we choose physical A plane wave of the form ψ(x, t) = exp(ik · x − iωt) units c = = 1, and denote P3 α ∂ = α · ∇, ~ k=1 k xk formally solves the equation provided that the “dis- to write the equation as persion relation”, ω = |k|2, holds. Writing exp(ik · x − iωt) = exp(ik · (x − kt)), this relationship may be iψ (x, t) = (−iα · ∇ + mβ)ψ(x, t). (2) t interpreted as saying the speed of the wave is equal By the representation (2), it is clear that the Dirac to the frequency of vibration. operator does not act on a complex-valued wave func- Using Fourier transform techniques, we can repre- tion as the Schr¨odingeroperator does, but rather acts sent the initial data ψ(x, 0) = f(x) as a superposition 4 of plane waves exp(ik · x) for a broad class of ini- on spinor fields. That is the solution to (2) is a C - valued function. tial data. Now, the dispersion relation dictates that In contrast to the Schr¨odingermodel, in which the higher frequency (|k| large) portions of the solution move more quickly than lower frequency portions of free operator H0 = −∆ is non-negative, the free Dirac the solution. A by-product is that a concentrated operator Dm = −iα·∇+mβ is unbounded both above initial profile f(x) will spread out, becoming smaller and below. In particular, the spectrum is σ(Dm) = (−∞, −m] ∪ [m, ∞). and smoother in some sense. The smoothing effect is These peculiarities lead to an ambiguity in the more subtle compared to parabolic equations such as physical interpretation of the solution ψ. One inter- the heat equation since the linear evolution preserves 2 pretation is that the solution couples the evolution the L -based Sobolev norms. Various quantifications of a quantum particle with its anti-particle, namely of this decay and smoothing for linear equations are coupling the evolution of electrons and positrons, useful in the study of nonlinear counterparts. Heuris- [Tha92]. This interpretation predicted the existence tically, for equations with power nonlinearities the of the positron before its discovery in 1932. nonlinear terms become smaller compared to the lin- The unboundedness in both directions creates chal- ear terms due to the decay. lenges in the analysis of perturbations. For non- The best way to measure this spread is to use the linear perturbations, the concentration-compactness Fourier transform techniques to represent the solu- n method fails for Dirac as it relies on the operator be- tion of (3) as a convolution integral. For x ∈ R , ing bounded below. Moreover, one can not define the Friedrichs extension for symmetric perturbations Z − n i|x−y|2/4t of Dirac operators since they are not bounded below, ψ(x, t) = (−4πit) 2 e f(y) dy. (4) n [RS79]. R Using (4), one can see the global dispersive estimate The Dirac equation as a dispersive PDE it∆ − n ke fkL∞ ≤ Cn|t| 2 kfkL1 . The Dirac equation is an example of a dispersive par- x x tial differential equation. By dispersive, we mean This L1 → L∞ bound reflects the dispersion present equations for which different frequency components in the Schr¨odingerequation, since the sup norm de- of the initial data (wave) propagate with different ve- cays in time as the waves of varying frequencies locities. We can envision the solution as being made spread out in space. These estimates can be used up of wave packets, where the frequency of vibration to obtain Strichartz estimates of the form is directly related to the speed at which the wave it∆ packet moves. ke fk q r ≤ Ckfk 2 , Lt Lx Lx
2 for admissible exponents q, r ≥ 2, (q, r, n) 6= [EGT19]. See [GM01] for a more thorough discus- 2 1 1 (2, ∞, 2), with q = n( 2 − r ). sion of further spectral issues. Strichartz estimates were first obtained by Dispersive estimates discussed in the previous sec- Strichartz via Fourier restriction theory of Stein. tion for (5) do not hold in general if the opera- Later, it was established that they can be obtained tor H has eigenvalues. One can project onto the using L1 → L∞ estimates. These mixed space-time continuous spectrum to recover dispersive estimates, estimates provide another way to quantify the dis- though threshold eigenvalues or resonances (at ener- persive nature of the equation, and are ubiquitous in gies λ = ±m) are known to affect the time decay, the study of nonlinear equations. [EGT19]. One can also measure the dispersion via smooth- One can study the perturbed evolution through ing estimates. Since a wave packet with frequency a generalized eigenfunction expansion, or through k moves with speed ∼ |k| it spends at most ∼ |k|−1 the resolvent operators and functional calculus tech- units of time on a ball of radius one. Averaging in niques. The free resolvent operators are defined by −1 time implies the Kato smoothing estimate R0(z) = (Dm − z) for z ∈ C \ σ(Dm). Recall- − 1 − 1 it∆ ing that Dm = −iα · ∇ + mβ, and the the anti- khxi 2 (−∆) 4 e fk 2 ≤ CkfkL2 . Lt,x x commutation relations of the Pauli matrices, we see that D2 = −∆ + m, and This estimate can be interpreted to say that on av- m erage the solution is half a derivative smoother than (D − λ)(D + λ) = −∆ + m2 − λ2. the initial data. m m The dispersion in the Dirac equation is weaker than Formally, this leads us to the relationship that in the Schr¨odingerdue to the hyperbolic nature of the equation. Since the square of the free Dirac R (λ) = (D + λ)R (λ2 − m2), (6) 2 2 0 m 0 operator Dm = −∆ + m is a diagonal system of Klein-Gordon operators, there is finite speed of prop- −1 where R0(z) = (−∆ − z) is the Schr¨odingerresol- agation and the large frequency behavior of solutions vent. By Agmon’s limiting absorption principle for does not travel arbitrarily fast as in the Schr¨odinger Schr¨odingeroperators, we can define the limiting op- evolution. This leads to weaker time decay and loss erators as z approaches the essential spectrum from of derivatives in dispersive estimates. upper and lower half plane. In the massive case, when m > 0, for |λ| close to Linear perturbations m, the dominant contribution of the Dirac resolvent behaves like the Schr¨odingerresolvent. This leads to To account for particle interactions or external elec- − 3 a generic time-decay rate of size |t| 2 in three spa- tromagnetic fields in three dimensions, one perturbs tial dimensions. When λ is away from the thresh- (1) with a 4 × 4 matrix-valued potential i.e., old energies ±m, solutions behave like solutions to a Klein–Gordon equation; hence smoothness of the iψt = (Dm + V )ψ. (5) initial data is required to obtain a time-decay. In the For simplicity, we will assume that V is Hermi- massless case, m = 0, the equation behaves like the tian. Under mild decay and local singularity as- wave equation which has decay rate |t|−1 in R3. sumptions on V the perturbed Dirac operator H := Although the relationship between the resolvents, Dm + V is essentially self-adjoint, and σess(H) = (6), leads one to expect that there should be analo- σ(Dm) by Weyl’s Theorem. In contrast to other gous results for the dynamics of solutions to the Dirac dispersive equations, such perturbations can pro- equation to those known about the Schr¨odingerequa- duce infinitely many eigenvalues in the spectral gap tion, at least at the linear level, the analysis of the (−m, m), [Tha92]. Faster decay of the potential Dirac equation presents many non-trivial technical ensures there are only finitely many eigenvalues, challenges.
3 Perturbations of linear differential operators with f(τ) = τ) nonlinearities are Lorentz covariant and the Coulomb potential appear naturally in many ar- exhibit a null-structure, comparable to other geomet- eas of physics. For the Dirac operator in R3 the ric nonlinear wave equations. γ Coulomb potential takes the form V (x) = |x| I4 mod- Scattering results, that is the series of conclusions els electric and/or gravitational interaction between that large time behavior of solutions approaches that the electron and proton in hydrogenic atoms. The of a free wave, are known for (7) with f(τ) = τ in 1 low dimensions provided the initial data is small in gradient scales like |x| , so the Coloumb potential is critical for the Dirac operator. The coupling constant an appropriate Sobolev norm, [BH15]. γ determines whether the corresponding operator is A solitary wave is a solution of (7) of the form γ well-defined or not. In fact, Dm + I4 is essentially −iωt √ |x| ψ(x, t) = φ(x)e , self-adjoint if and only if |γ| ≤ 3 , [Tha92]. 2 where ω is a real-valued parameter and φ solves a A similar situation occurs for the Schr¨odingerop- stationary nonlinear Dirac equation. Such solitary erator with inverse square potential, −∆ + β . This |x|2 waves are known to exist for broad classes of nonlin- 3 3 operator is essentially self-adjoint in R if β ≥ 4 , earities, but the stability results are less developed. 1 3 [RS79]. For − 4 ≤ β < 4 , one can define Friedrichs Linearizing about such stationary solutions leads to β extension since −∆ + |x|2 remains a positive opera- linear equations of the form considered in (5). Spec- tor. Since the Dirac operator is unbounded below, tral stability is known for classes of nonlinearity f the Friedrichs extension can not be defined. There√ that are polynomial-like, [BC19]. Stronger stabil- are infinitely many self-adjoint extensions if |γ| > 3 . ity results, such as orbital and asymptotic stability √ 2 3 of these solutions for general f is thus far incom- However, when 2 < |γ| < 1, there is a unique distinguished self-adjoint extension analogous to the plete and less developed than other dispersive equa- Friedrichs extension, see e.g. [Gal17]. tions, [Dod19]. One cannot use the concentration- compactness arguments that are standard for other dispersive equations, [NS11], among other challenges. Non-linear models Though, somehow unexpectedly, there seems to be a Often a nonlinear Schr¨odingerequation is formed by lack of blow-up phenomenon in nonlinear Dirac equa- adding a nonlinear term: tions which commonly occurs in other dispersive non- linear models, [BC19].
iut = −∆u ± f(u)u, Honeycomb potentials typically of the form f(τ) = |τ|γ . There are more choices for how to create the nonlinearity out of the Finally, we briefly note the appearance of a two- C4-valued spinor. The Soler model from quantum dimensional Dirac equation in the study of waves in field theory is periodic structures inspired by the study of graphene, a material with layers of hexagonal lattices of car- ∗ iψt = Dmψ − f(ψ βψ)ψ, (7) bon atoms. The evolution of waves in graphene is modeled by a two-dimensional Schr¨odinger equation with f real-valued satisfying f(0) = 0, ψ∗ = ψT , −∆ + V with a periodic “honeycomb” potential, i.e. and β is the matrix used in defining the Dirac opera- V is periodic, even, and invariant under 2π/3 rota- tor. One dimensional analogues of this model include tions. In contrast to the previous discussion, V does the Gross-Neveu and massive Thirring equations for not decay as |x| → ∞. scalar and vector self-interaction respectively. The When decomposing the solution of the Schr¨odinger Soler model is well-posed provided the initial data equation into Floquet-Bloch states there are Dirac has enough Sobolev regularity, ψ(x, 0) ∈ Hs(Rn) points of the honeycomb structure, where the disper- n for s > 2 , [BC19]. The Thirring and Soler (with sion surfaces intersect. This leads to an intersection
4 of successive spectral energy bands at some energy µ [FW14] Charles L. Fefferman and Michael I. Weinstein, and the time-independent operator −∆+V −µ has a Wave packets in honeycomb structures and two- two-dimensional nullspace spanned by Φ (x), Φ (x). dimensional Dirac equations, Comm. Math. Phys. 1 2 326 (2014), no. 1, 251–286. MR3162492 A wave-packet spectrally localized at this energy is [Gal17] Matteo Gallone, Self-adjoint extensions of Dirac op- of the form (with δ a small parameter) erator with Coulomb potential, Advances in quantum mechanics, 2017, pp. 169–185. MR3588049 2 iµt X [GM01] V. Georgescu and M. M˘antoiu, On the spectral the- ψ(x, t) ≈ e δαj(δx, δt)Φj(x), ory of singular Dirac type Hamiltonians, J. Operator j=1 Theory 46 (2001), no. 2, 289–321. MR1870409 [NS11] Kenji Nakanishi and Wilhelm Schlag, Invariant where αj are the amplitudes. Writing α = (α1, α2) ∈ manifolds and dispersive Hamiltonian evolution C2, the evolution of α is governed by a two- equations, Zurich Lectures in Advanced Mathemat- dimensional Dirac equation with zero mass. The ics, European Mathematical Society (EMS), Z¨urich, dynamics of such wave packets is believed to be re- 2011. MR2847755 sponsible for the remarkable properties of graphene, [RS79] Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt [FW14]. Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR529429 headshots, images and permissions [Tha92] Bernd Thaller, The Dirac equation, Texts and Mono- graphs in Physics, Springer-Verlag, Berlin, 1992. The AMS requires documentation that the MR1219537 owner/copyright holder of each image gives No- tices and the AMS permission to use it. It is the responsibility of authors to secure this permission. For each image, please provide (1) a credit line and (2) forwarded correspondence from the owner/copyright holder indicating that Notices is granted reprint permission. (See the Permissions Tips document.)
References
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