FROZEN GAUSSIAN APPROXIMATION FOR THE

FROZEN GAUSSIAN APPROXIMATION FOR THE DIRAC EQUATION IN SEMI-CLASSICAL REGIME LIHUI CHAI∗ , EMMANUEL LORIN† ‡ , AND XU YANG§ Abstract. This paper is devoted to the derivation and analysis of the Frozen Gaussian Approximation (FGA) for the Dirac equation in the semi-classical regime. Unlike the strictly hyperbolic system studied in [J. Lu and X. Yang, Comm. Pure Appl. Math., 65, 759–789, 2012], the Dirac equation possesses eigenfunction spaces of multiplicity two, which demands more delicate expansions for deriving the amplitude equations in FGA. Moreover, we prove that the nonrelativistic limit of the FGA for the Dirac equation is the FGA of the Schr¨ odinger equation, which shows that the nonrelativistic limit is asymptotically preserved after one applies FGA as the semiclassical approximation. Numerical experiments including Klein-Paradox are presented to illustrate the method, and confirm part of the analytical results. AMS subject classifications.

Notations • i: imaginary unit • ε: semiclassical parameter • α, ˆ α ˆ j , βˆ0 : Dirac matrices • V (or A): scalar (or vector) potential • boldface Greek alphabets (lower and upper cases) such as ψ ε and Υ : vectors in C4 • boldface English alphabets (lower and upper cases) such as Q and x: vectors in R3 • ∂u B = {∂uj Bk }j,k 1. Introduction. This paper is devoted to the derivation and analysis of the Frozen Gaussian Approximation (FGA) for the linear Dirac equation modeling quantum particles subject to a classical electromagnetic field. The Dirac equation is a linear with constant coefficient, Hermitian 4-equation hyperbolic system, with two pairs of double eigenvalues ±c, where c denotes the speed of light. Physically, this equation models a relativistic quantum wave equation for half-spin particles, such as fermions (e.g. electrons) [33]. There is growing interest in Physics for the study of this equation in particular for modeling heavy-ion collisions [30, 18, 2], for pair production using strong laser fields [11, 10, 16], or for modeling Graphene [21]. These active research fields in Physics, have motived recent computational works [2, 34]. Let us mention [22, 14, 15], where an embarassingly parallel quantum lattice Boltzmann methods were developed and applied to the simulation of the interaction of relativistic electron with an intense external electric field. For similar purposes, efficient spectral and finite element methods were developed such as the ones presented in [35, 1, 28, 31, 27, 29, 7, 3]. Galerkin methods with balanced basis functions is proposed in [17] for solving the time dependent Dirac equation (TDDE). Graphene-related simulations using similar techniques are also proposed in [13, 12]. Regarding to the numerical approximation of the TDDE in the non-relativistic regime, there exists an extended literature; see for instance [6, 4, 5]. The approximation of the Dirac equation in the semi-classical regime has also recently brought some attention, in particular in [36], where a Gaussian beam-based method is proposed based on adiabatic approximation. In this work, the Dirac equation will be approximated in the semi-classical regime, using the celebrated Frozen Gaussian Approximation (FGA). The latter was originally developped by Herman-Kluk (HK) [20] for the Schr¨ odinger equation in the semi-classical regime. The mathematical analysis was then proposed in [32] showing the accuracy and efficiency of this anzatz in particular when the initial data is localized in phase space. HK formalism was later developed of several types of partial differential equations, such as wave equations [24], linear hyperbolic systems of conservation laws [26], elastic wave equations and seismic tomography [8, 9, 19]. Some applications and analysis on the Schr¨odinger equations were also proposed in [23, 37]. Unlike the aforementioned equations studied using FGA, the Dirac equation possesses two eigenfunction spaces of multiplicity two, which demands more delicate expansions for deriving the amplitude equations in the FGA formulation. As a deeper investigation, we also prove that the nonrelativistic limit of 1 School

of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China ([email protected]) of Mathematics and Statistics, Carleton University, Ottawa, Canada ([email protected]) 3 Centre de Recherches Math´ ematiques, Universit´ e de Montr´ eal, Montr´ eal, Canada 4 Department of Mathematics, University of California, Santa Barbara, CA 93106, USA ([email protected]) 2 School

1

the FGA for the Dirac equation is the FGA for the Schr¨odinger equation, which shows that the nonrelativistic limit is asymptotically preserved after one applies FGA as the semiclasscial approximation. In the end, we present several numerical experiments including Klein Paradox to illustrate the method, and confirm part of the analytical results. Recall first that, in the quantum regime, the Dirac equation reads ˆ i∂t ψ(t, x) = Hψ(t, x),

(1) where

 
ˆ = α · cp − eA(t, x) + βmc2 + I4 Vc (t, x) + V (x) , H

(2) and



0 αγ = σγ

(3)

σγ 0



 I , β= 2 0

 0 . −I2

The σγ ’s are the usual 2 × 2 Pauli matrices defined as      0 1 0 −i 1 (4) σx = , σy = and σz = 1 0 i 0 0

 0 , −1

while I2 is the 2 × 2 unit matrix. The momemtum operator is denoted p = −i∇. The speed of light c and fermion mass m are kept explicit in Eq. (2), allowing to easily adapt the method to natural or atomic units (a.u.). In (1), e is the electric charge (with e = −|e| for an electron), I4 is the 4 × 4 unit matrix and α = (αγ )γ=1,··· ,4 , β are the Dirac matrices. This equation models a relativistic electron of mass m subject to an interaction potential Vc and electromagnetic field V (x), A(t, x), and where for fixed time t, ψ(t, x) ∈ L2 (R3 ) ⊗ C4 is the coordinate (x = (x, y, z)) dependent four-spinor. We denote by h , i the L2 (R3 , C4 )-inner product. In principle, the Dirac equation must be coupled to Maxwell’s equations, modeling the evolution of the electromagnetic field [22]. In the following, the coupling is neglected, and we then assume that the EM propagates as in vacuum. 2. Frozen Gaussian Approximation for the Dirac equation. This section is devoted to the construction of the FGA for the Dirac equation in the semi-classical regime. We first recall some basic facts about Frozen Gaussian Approximations (FGA) for evolution equations in the semi-classical regime. We then derive the FGA for the field-particle Dirac equation in the relativistic regime. For ε > 0, we reformulate the Dirac equation as follows:   (5) iε∂t ψ ε (t, x) = −icεα ˆ·∇−α ˆ · A(x) + mβˆ0 c2 + V (x) ψ ε (t, x),   i (6) SI (x) . ψ ε (0, x) = ϕεI (x) = ω I (x) exp ε Here ψ ε = (ψ1ε , ψ2ε , ψ3ε , ψ4ε )T ∈ C4 is the spinor. In the following, we will assume that m = 1, corresponding to the mass of the electron in atomic unit. 2.1. Characteristic fields. The symbol of the Dirac operator which is studied in this paper reads (7)

D(q, p) := α ˆ · (pc − A(q)) + mβˆ0 c2 + V (q) = α ˆ · pc + B(q),

where we have denoted (8)

B(q) := −α ˆ · A(q) + mβˆ0 c2 + V (q)I4 .

We easily show that D is a Hermitian matrix, and it has two double (real) eigenvalues q 2 (9) h± (q, p) = ±λ(q, p) + V (q), where λ(q, p) = |pc − A(q)| + c4 , 2

Denote the corresponding eigenvectors as Υ m , m = ±1, ±2, and let u = pc − A(q), then  Υ +1 =

1 r+

Υ −1 =

1 r−

(10)



 u3   q u1 + iu2   , 2 2 4  |u| + c − c  0   −u3  −u1 − iu2  q   ,  |u|2 + c4 + c2  0

Υ +2 =

1 r+

Υ −2 =

1 r−

 u1 − iu2   −u3   ,  0  q 2 |u| + c4 − c2   −u1 + iu2   u3    , 0 q  2

|u| + c4 + c2

s   q 2 2 4 4 where r± = 2 |u| + c ∓ |u| + c . Other details can be found in Appendix. 2.2. Derivation of the FGA. Using the eigenvector of the Dirac operator symbol, we can now define the FGA for the Dirac equation, which reads as follows ψ ε (t, x) = (11)

XZ 1 (am (t, q, p) + εβ m (t, q, p)) (2πε)9/2 m R9   i × exp Φm (t, x, y, q, p) vm (y, q, p) dy dq dp, ε

where Φm (t, x, y, q, p) = S (t, q, p) + (12)

i 2 |x − Qm (t, q, p)| + P m · (x − Qm (t, q, p)) 2

i 2 |y − q| − p · (y − q), 2 am (t, q, p) = am (t, q, p) Υ m (Qm , P m ), +

(13)

vm (y, q, p) = Υ m (q, p) · ψ εI (y).

(14)

In this subsection, we respectively derive the evolution equations respectivelly 1) for the Gaussian profiles Q and momentum functions P , then 2) for the action function S, and finally 3) for the FGA amplitude a. For the sake of simplicity, we consider one branch (e.g. , “+” branch) and drop the subscription without causing any confusion. Let h be the eigenvalue and Υ 1 and Υ 2 be the corresponding eigenvectors. 1. Gaussian profile Q and momentum function P , evolution equation. As for any FGA, the bi-center Q and P simply satisfy the Hamiltonian flow:

(15)

 dQ   = ∂P h(Q, P ), Q(0, q, p) = q, dt   dP = −∂Q h(Q, P ), P (0, q, p) = p. dt

We then determine the evolution equation for the action function. 2. Action function S, evolution equation. In order to derive the evolution equation for the action function S, preliminary computations are needed. By definition of Φ, we have (16) (17)

∂t Φ = ∂t S − P · ∂t Q + (x − Q) · ∂t (P − iQ),

∇x Φ = i(x − Q) + P , 3

Then taking derivatives to (11) gives  Z  i ∂t ψ ε = ∂t a + ε∂t β + ∂t Φ(a + εβ) eiΦ/ε v dy dq dp ε  Z  i (18) = ∂t a + (∂t S − P · ∂t Q + (x − Q) · ∂t (P − iQ)) a eiΦ/ε v dy dq dp ε Z + (ε∂t β + i (∂t S − P · ∂t Q + (x − Q) · ∂t (P − iQ)) β) eiΦ/ε v dy dq dp Z i ε (19) (i(x − Q) + P ) (a + εβ) eiΦ/ε v dy dq dp ∇x ψ = ε We now expand B(x) about Q 1 2 B(x) = B(Q) + (x − Q) · ∂Q B(Q) + (x − Q)2 : ∂Q B(Q) + O(x − Q)3 . 2

(20)

Substitute these into (5), to the order of ε0 , we get Z Z iΦ/ε − (∂t S − P · ∂t Q)a e (21) v dy dq dp = (cα ˆ · P + B(Q))a eiΦ/ε v dy dq dp. As a (t, q, p) = a1 (t, q, p) Υ 1 (Q, P ) + a2 (t, q, p) Υ 2 (Q, P ), then the above equation holds if the action function S satisfies the following evolution equation ∂t S = P · ∂t Q − h(Q, P ).

(22)

3. Amplitude evolution equation. We now construct the evolution equation for a. This is a much more technical step, and it requires an additional definition. Definition 2.1. Two functions f , and g are said equivalent if Z Z iΦ/ε f ∼g ⇔ fe dy dq dp = geiΦ/ε dy dq dp . Based on this definition, we easily show: Lemma 1. −1 b · (x − Q) ∼ −ε∂zk (bj Zjk ),

(23)

−1 (x − Q) · G(x − Q) ∼ ε∂zk Ql Glj Zjk + ε2 ...,

(24)

where Einstein’s summation convention has been used, and where ∂z = ∂q − i∂p ,

(25)

Z = ∂z (Q + iP ).

Moveover, (x − Q)a ∼ O(ε|a|−1 ) for |a| > 2.

To the order of ε1 ,   Z i iε ∂t a + (x − Q) · ∂t (P − iQ)a + i(∂t S − P · ∂t Q)β eiΦ/ε v dy dq dp ε  Z  1 2 = icα ˆ · (x − Q)a + (x − Q) · ∂Q Ba + (x − Q)2 : ∂Q Ba + εDβ eiΦ/ε v dy dq dp, 2

which implies, Z  (26)

 1 ∂t a + (x − Q) · ∂t (Q + iP )a + i(D(Q, P ) − h(Q, P ))β eiΦ/ε v dy dq dp ε  Z  c 1 1 2 2 = α ˆ · (x − Q) + (x − Q) · ∂Q B + (x − Q) : ∂Q B a eiΦ/ε v dy dq dp . ε iε 2iε 4

Thus Z h

  i −1 ∂t av − ∂zk ∂t (Qj + iPj )Zjk av + i(D(Q, P ) − h(Q, P ))βv eiΦ/ε dy dq dp  Z      i −1 −1 −1 = −∂zk cˆ αj Zjk av + i∂zk ∂Qj BZjk av − ∂zk Ql ∂Ql ∂Qj B Zjk av eiΦ/ε dy dq dp, 2 that is (27)


−1 i i −1 ∂t (Qj + iPj ) − cˆ αj + i∂Qj B Zjk av av + ∂zk Ql ∂Ql ∂Qj B Zjk 2 ∼ i(h(Q, P ) − D(Q, P )) β v.

∂t av − ∂zk

h

We define for n = 1, 2,
F nj = ∂t (Qj + iPj ) − cˆ αj + i∂Qj B Υ n


= ∂Pj h(Q, P ) − i∂Qj h(Q, P ) − cˆ αj + i∂Qj B Υ n
= ∂Pj h(Q, P ) − cˆ αj − i∂Qj h(Q, P ) + i∂Qj B Υ n ,

(28)

then (27) can be written as (29)


−1 i i −1 av v + ∂zk Ql ∂Ql ∂Qj B Zjk a1 F 1j + a2 F 2j Zjk 2 ∼ i(h(Q, P ) − D(Q, P )) β v .

∂t av − ∂zk

h

Solvability for β gives, for m = 1, 2   h
−1 i i 2 † 1 −1 Υ m ∂t av − ∂zk a1 F j + a2 F j Zjk v + ∂zk Ql ∂Ql ∂Qj B Zjk av = 0. (30) 2 Let Υ ∈ Span{Υ 1 , Υ 2 }. Since (cα ˆ · P + B(Q))Υ (Q, P ) = h(Q, P )Υ (Q, P ), then cˆ αj Υ + (cα ˆ · P + B) ∂Pj Υ = ∂Pj h Υ + h ∂Pj Υ ,

∂Qj B Υ + (cα ˆ · P + B) ∂Qj Υ = ∂Qj h Υ + h ∂Qj Υ , and thus for m = 1, 2,

Υ †m


Υ †m cˆ αj − ∂Pj h Υ = 0,
∂Qj B(Q) − ∂Qj h Υ = 0 .

Thus for m, n = 1, 2, Υ †m F nj = 0, and h i −1 −1 −1 Υ †m ∂zk an F nj Zjk v = Υ †m ∂zk F nj Zjk an v = −∂zk Υ †m F nj Zjk (31) an v . Notice that (32)


Υ †m ∂t (an Υ n ) = δmn ∂t an + Υ †m ∂Pj h∂Qj Υ n − ∂Qj h∂Pj Υ n an .

Then (30) gives the equation for am (33)


δmn ∂t an − Υ †m ∂Qj h∂Pj Υ n − ∂Pj h∂Qj Υ n an i −1 −1 Υ n an . = − ∂zk Υ †m F nj Zjk an − Υ †m ∂zk Ql ∂Ql ∂Qj B Zjk 2 5

In the vector form d dt

(34)

    a1 a = (L + M + N ) 1 , a2 a2

where L, M, N are 2 × 2 matrices i † −1 Υ ∂z Ql ∂Ql ∂Qj B Zjk Υ n, 2 m k = Υ †m (∂Q h · ∂P Υ n − ∂P h · ∂Q Υ n ) ,

Lmn =

(35)

Nmn

−1 Mmn = −∂zk Υ †m F nj Zjk .

This concludes the construction of the FGA. Remark 2.1. It is easy to see that L is diagonal by the orthogonality of the eigenvectors.

We now study the structure of the matrix M when A is null. This will be a fundamental piece of information in order to determine the nonrelativistic limit of the FGA for the Dirac equation. We first show that M is diagonal when V are null, and then extend the result for non-null potentials. Proposition 2.1. Assume that V and A are null, then M is diagonal.

Proof. When there is no potential, it is easy to show that P (t) = p,

∂z P = −iI, Z = 2I − it Let us set T =

it 2λ

 I−

p⊗p λ2



Q(t) = q + q

p

∂z Q = I − it ! I p⊗p − 3/2 . λ1/2 λ

, then Z −1 =

t,

2

|p| + 1

! p⊗p − 3/2 , λ1/2 λ I

1 (I + T + T 2 + T 3 + ...). Notice now that 2

2 p⊗p p⊗p 2 p⊗p , = I − cp ⊗ p = I − 2 2 + |p| 2 λ λ λ4  n p⊗p I− = I − dp ⊗ p. λ2 

I−

Thus, we can see that there exists a c such that Z −1 =

1 (I + c1 (I − d1 p ⊗ p) + c2 (I − d2 p ⊗ p) + ...). 2

Since Υ does not depend on Q, ∂z Υ n = −i∂P Υ n . For m 6= n, −1 −1 Mmn = −∂zk Υ †m F nj Zjk = −∂z Υ †m (∂Pj h − a ˆj )Υ n Zjk .

Consider the positive branch and take m = 1, n = 2, M12 = Tr(−∂z Υ †1 (∂P h − a ˆ)Υ 2 Z −1 ), and  0 1 ∂P Υ †1 = 0 r 1

 1 P1 /λ 0 1 1 1 −i P2 /λ 0 − ∂P r ⊗ Υ †1 , = A − ∂P r ⊗ Υ †1 . r r r 0 P3 /λ 0 6

This leads to 

  1 1 A − ∂P r ⊗ Υ †1 (∂P h − a ˆ)Υ 2 Z −1 r r     1 1 = −Tr ˆ)Υ 2 Z −1 A(∂P h − a ˆ)Υ 2 Z −1 + Tr ∂P r ⊗ Υ †1 (∂P h − a r r
1 ˆ)Υ 2 Z −1 + 0, = − Tr A(∂P h − a r q  

M12 = −Tr

(A(∂P h − a ˆ))

q

 |P | + 1 =   q 

−P2 P3 0

0 P2 P3

2

2

|P | + 1 − P22 − 1

−P1 P2

P1 P2

−P1 P3

0

 P1 P3 + i 

P22 −

0 P12 −

q

2

|P | + 1 + 1

2

|P | + 1 + 1   −P1 P2  0



P 1 P2 q

 2 |P | + 1 − P12 − 1  .  0


Thus Tr (A(∂P h − a ˆ)Υ 2 p ⊗ p) = 0 . Finally, we have M12 = Tr A(∂P h − a ˆ)Υ 2 Z −1 = 0, which concludes the proof. 2 We now consider the case when A is null. Proposition 2.2. Assume that A is null, then M is diagonal. In addition, M11 = M22 = −Tr(Z −1 ∂z P Re(A11 )), † aj )Υ n }jk . where {Amn jk } = {∂Pk Υ m (∂Pj h − cˆ Proof. When the vector potential A is null, since Υ does not depend on Q, ∂z Υ n = ∂z P · ∂P Υ n . For −1 −1 m 6= n, Mmn = −∂zk Υ †m F nj Zjk = −∂zk Υ †m (∂Pj h − cˆ aj )Υ n Zjk = −Tr(∂z P · ∂P Υ †m (∂P h − cˆ a)Υ n Z −1 ) . q p 2 Let λ = c2 |P | + c4 and r = 2(λ2 − c2 λ), then ∂P Υ †1 (∂P h − cˆ a)Υ 1

r2 λ r2 λ † =∂ Υ (∂ h − cˆ a )Υ P P 2 2 c2 c2  2    2 2 P 2 + P3 + c −P1 P2 −P1 P3 |c|  c2  q −P1 P2 P12 + P32 + c2 −P2 P3 = − 1 2 2 2 2 2 −P1 P3 −P2 P3 P1 + P2 + c |P | + c q   2 0 c2 − |cP | + c4 + P32 −P2 P3   q  2 + ic2   |cP | + c4 − c2 − P 2 0 P P , 1

3

−P1 P3

P2 P3

∂P Υ †1 (∂P h − cˆ a)Υ 2

3

0

r2 λ r2 λ † = − ∂ Υ (∂ h − cˆ a )Υ P P 1 2 c2 c2  q 2 |cP | + c4 − c2 − P22 0 P2 P3    0 P1 P2 =c2    q −P2 P3 2 c2 − |cP | + c4 + P22 −P1 P2 0  0 P1 P3 q −P1 P2  2 2 2  −P P 0 c − |cP | + c4 + P12 1 3 + ic  q 2 P1 P2 |cP | + c4 − c2 − P12 0 

7

  . 

† Let {Amn aj )Υ n }jk , and then jk } be the matrix {∂Pk Υ m (∂Pj h − cˆ −1 −1 mn −1 Mmn = −(∂zk Pl Amn ∂z P Amn ). lj Zjk ) = −(Zjk ∂zk Pl Alj ) = −Tr(Z

Lemma 2. Z −1 ∂z P is symmetric. As A is anti-symmetric and B is symmetric, then Tr(AB) = 0. Thus, we can obtain the following Mmn = − Tr(Z −1 ∂z P Amn ) = 0,

(36)

M11 = − Tr(Z

(37)

−1

11

for m 6= n

∂z P Re(A )) = M22 .

This concludes the proof. 2 Notice that when c → ∞, Re(A11 ) → − 12 I, which implies M11 → 21 Tr∂z P Z −1 . We now consider the general situation. Proposition 2.3. Assume A is not null. Then M is in general not diagonal, except when A is only time-dependent. In addition, M11 = M22 = −Tr(Z −1 ∂z P Re(A11 )), † where {Amn aj )Υ n }jk . jk } = {∂Pk Υ m (∂Pj h − cˆ q 2 Proof. Let U = cP − A(Q) and λ = c2 |U | + c4 , then from (28),


F nj = c(∂Uj λ − α ˆ j ) + i∂Qj A · (∂U λ − α) ˆ Υn
= c(∂Uj λ − α ˆ j ) + i∂Qj Al (∂Ul λ − α ˆl ) Υ n, which brings −1 Mmn = −∂zk Υ †m F nj Zjk


−1 ˆ j ) + i∂QjAl (∂Ul λ − α ˆ l ) Υ n Zjk = −∂zk Υ †m c(∂Uj λ − α
−1 = −∂zk Uν ∂UνΥ †m c(∂Uj λ − α ˆ j ) + i∂QjAl (∂Ul λ − α ˆ l ) Υ n Zjk


−1 = −(∂zk Pν − ∂zk Qµ ∂QµAν ) ∂UνΥ †m c(∂Uj λ − α ˆ j ) + i∂QjAl (∂Ulλ − α ˆ l ) Υ n Zjk i h ˆ + i∂Q A · (∂Uλ − α)) ˆ Υ n Z −1 = −Tr (∂z P − ∂z Q ∂Q A) ∂UΥ †m (c(∂U λ − α) h i = −Tr Z −1 (∂z P − ∂z Q ∂Q A) ∂UΥ †m (c(∂U λ − α) ˆ + i∂Q A · (∂Uλ − α)) ˆ Υn . Since Z = ∂z (Q + iP ), i
icZ −1 (Z − ∂z Q) + Z −1 ∂z Q ∂Q A ∂UΥ †m (c(∂U λ − α) ˆ + i∂Q A · (∂Uλ − α)) ˆ Υn h i
= Tr ic(I3 − Z −1 ∂z Q) + Z −1 ∂z Q ∂Q A ∂UΥ †m (c(∂U λ − α) ˆ + i∂Q A · (∂Uλ − α)) ˆ Υn h i = Tr ic ∂UΥ †m (c(∂U λ − α) ˆ + i∂Q A · (∂Uλ − α)) ˆ Υn h i
− Tr icZ −1 ∂z Q − Z −1 ∂z Q ∂Q A ) ∂UΥ †m (c(∂U λ − α) ˆ + i∂Q A · (∂Uλ − α)) ˆ Υn ,

Mmn = Tr

h

which concludes the proof. 2 8

3. Nonrelativistic limit of the FGA for Dirac. In this section, we are interested in the nonrelativistic limit of the Dirac equation in the semi-classical regime. The purpose is to prove that due to the linearity of this equation, we can derive the FGA for the Schr¨odinger equation in the semi-classical and nonrelativistic regime from the FGA for the Dirac equation. In other words, we show the commutativity of the formal diagram Fig. 1. We will also provide some mathematical properties of the FGA in the nonrelativistic regime. We consider the case of a free particle (no interaction potential) with no external magnetic field A, but subject to an external space-dependent potential. The initial data is ψ ε (0, x) = ω I (x) exp

 i SI (x) , ε

and the FGA reads ψ ε (t, x)

=

1 P R m R9 9/2 (2πε)


am (t, q, p) + εβ m (t, q, p) i  × exp Φm (t, x, y, q, p) vm (y, q, p)dydqdp, ε

with Φm , am and vm defined in (2.6)-(2.8). For the Dirac equation, the center of Gaussian profiles and momemtum functions satisfy the Hamiltonian flow:  dQ   dt   dP dt

=

∂P h(Q, P ),

=

−∂Q h(Q, P ),

Q(0, q, p) = q, P (0, q, p) = p.

where one branch (h = h+ ) has been selected, and where i) h± (q, p) = ±λ(p) + V (q) and ii) λ(p) = p p2 c2 + c4 with p := kpk2 .

Relativistic Dirac Equation

c → +∞

ε→0

ε→0

FGA semi−classical Dirac Equation

Schroedinger Equation

c → +∞

FGA semi−classical Schroedinger Equation

Fig. 1. Commutation scheme: FGA for Schr¨ odinger from i) semi-classical limit of the nonrelativistic Dirac equation, and from ii) relativistic limit of the semi-classical limit of the Dirac equation.

3.1. Nonrelativistic limit of the FGA for Dirac equation. We plan to derive a FGA for the Schr¨ odinger equation, by taking the nonrelativistic limit, i.e. with c → +∞. We will use the ∼-notation for the FGA in the nonrelativistic limit. We now state an important result of this paper. Theorem 3.1. The nonrelativistic limit of the FGA for the field-free linear Dirac equation is the FGA for the Schr¨ odinger equation. 9

Proof. First, for c large  1  ∂P h(Q, P ) = P p (38) 1 + P 2 /c2  ∂Q h(Q, P ) = ∂Q V (Q),

=


 P2 P 1 − 2 + O P 4 c−4 , 2c

where P := kP k2 . Then by keeping the leading order terms in (38), we define the Gaussian profiles and momentum centers as solutions to the following Hamiltonian flow, in the semiclassical and nonrelativistic limits:  e  dQ  e, e q, p) = q,  = P Q(0, dt (39) e  dP  e e (0, q, p) = p.  = −∂Q V (Q), P dt Next, the evolution of the action function ∂t S(t, Q, P )

= P · ∂t Q − h(Q, P )

can be rewritten in the nonrelativistic limit, as follows ∂t S(t, Q, P )

=


P2 − V (Q) − c2 + O P 4 c−2 , 2

where we have used (38), and that s h± (Q, P ) = ±c2

1+


P2 P2 + V (Q) ± c2 + O c−2 P 4 . + V (Q) = ± 2 c 2

An evolution equation of the action function in the nonrelativistic limit can also be defined as: (40)

e P e) e Q, ∂t S(t,

= c2 +

Pe2 e . − V (Q) 2

We now focus on the evolution equation for the amplitude which requires a bit more careful analysis. As A is assumed null, we can rewrite B as: 1 2 B(x) = βb + V (Q)I4 + (x − Q) · ∂Q V (Q) + (x − Q)2 : ∂Q V (Q) + O(x − Q)3 . 2 Next, by taking m = n in (3.18), we get for n = 1, 2 ∂t an (41)


= Υ†n ∂Qj h∂Pj Υn − ∂Pj h∂Qj Υn an i −1 −1 Υn an , +Υ†n ∂zk F nj Zjk Υn an − Υ†n ∂zk Ql ∂Ql ∂Qj V Zjk 2

where
F nj = ∂Pj h(Q, P ) − cb αj − i∂Qj h(Q, P ) + i∂Qj V (Q) Υn . As Υ is Q-independent, ∂Qj Υn = 0 , so that the second term in the RHS of (41) is also null, and the latter can be rewritten as i −1 −1 Υn an . ∂t an = ∂Qj V Υ†n ∂Pj Υn an + Υ†n ∂zk F nj Zjk Υn an − Υ†n ∂zk Ql ∂Ql ∂Qj V Zjk 2 Now we make use of the relations (A.8), recalling that u = cP , then ∂P = c∂u , we then get ∂t an

=

i −1 −1 2 ∂Qj V Υ†n ∂Pj Υn an + Υ†n ∂zk F nj Zjk Υn an − Υ†n ∂zk Ql ∂Q V Zjk Υn a n . l Qj 2 10

As we look at the positive branch, we have r 

2 P 2 c2 + c4 − c2

r+ (P ) =

p

 √
P 2 c2 + c4 = c2 2 + O P 2 c−2 ,

and asymptotically get Υ†n ∂P Υn =



i i P2 , −P1 , 0 + 2 P1 , P2 , 0 . 2 P 4c

Case 1. V is identically null. In this case ∂t an

i −1 −1 2 V Zjk Υn a n . Υ†n ∂zk F nj Zjk an − Υ†n ∂zk Ql ∂Q l Qj 2

=

Let us focus on the term ∂zk F nj . We notice that the expression of F nj can be simplified into
F nj = ∂Pj h(Q, P ) − cb αj Υn ,
so that ∂zk F nj = ∂Pj h(Q, P ) − cb αj ∂zk Υn + ∂zk ∂Pj h(Q, P )Υn . Now, (42)

−1 Υ†n ∂zk F nj Zjk

= =


−1 −1 Υ†n ∂Pj h(Q, P ) − α bj ∂zk Υn Zjk + Υ†n ∂zk Pj Υn Zjk 
† −1 −1 . ∂zk Υn ∂Pj h(Q, P ) − α bj Υn Zjk + Υ†n ∂zk Pj Υn Zjk

We make use of the relation
−1 −1 −1 Υ†n ∂zk F nj Zjk = −∂zk Υ†n F nj Zjk = −∂zk Υ†n ∂Pj h(Q, P ) − α bj Υn Zjk . Thus 

† −1 −1 −1 . bj Υn Zjk − ∂zk Υn ∂Pj h(Q, P ) − α bj Υn Zjk = Υ†n ∂zk Pj Υn Zjk −∂zk Υ†n ∂Pj h(Q, P ) − α Recall that the α ˆ j are hermitian, and that ∂zk = ∂qk = i∂pk , so that 
 −1 −1 −2Re ∂zk Υ†n ∂Pj h(Q, P ) − α bj Υn Zjk = Υ†n ∂zk Pj Υn Zjk . As we have assumed, that V is null, then P (t) = p, Q(t) = pt + q, and as a consequence Z = I(2 − it), and −1 Zjk = δjk /(2 − it). We then deduce that 
   bj Υn Z −1 = tr Z −1 ∂z P . tr Re∂z Υ†n ∂P h(Q, P ) − α 
 In order to conclude the proof, we need to show that tr Im∂z Υ†n ∂P h(Q, P ) − α bj Υn Z −1 is at most a O(c−1 ). After some symbolic computations, we acutally show that 
 p2 + O(c−2 ). tr Im∂z Υ†n ∂P h(Q, P ) − α bj Υn Z −1 = 2c
In the nonrelativistic limit using (38), we get ∂zk F nj = Pj − cb αj ∂zk Υn + ∂zk Pj Υn + O(P 2 c−2 ), then 1 Υ†n ∂zk F nj = Υ†n ∂zk Pj Υn + O(P −1 c−1 ) . 2 We then define the amplitude e an and its corresponding evolution equation by (43)

de an dt

=

1 e j Z −1 e ∂z P jk an . 2 k 11

We deduce from (39), (40), and (43) a nonrelativistic limit of the FGA for Dirac equation consistent with the one for the Schr¨ odinger equation [37], where in particular, the evolution equation for the amplitude reads
 de a 1  −1 e . = e atr Z (∂z P dt 2

(44)

This concludes the proof, when V is null. Case 2. V is not identically null. The proof is essentially identical, but now relies on Proposition 2.2. We start from
d a1 Υ1 + a2 Υ2 dt

= M a1 Υ1 + a2 Υ2 ).

According to Proposition 2.2, M is diagonal and for c large, we have established that M11 = M22 that when c → +∞ goes to Tr∂z P Z −1 /2 so that the evolution equation for the amplitude is yet (44). According to Proposition 2.3, the result remains valid for t-dependent only electric potentials A. 2 In a bounded domain Ωε , non-reflecting conditions are trivially established for the Dirac equation in both relativistic or nonrelativistic regimes. Whenever Q(t) ∈ / Ωε , the contribution of this Gaussian function is simply removed from the reconstruction of ψ ε , see [37] for details in the case of the Schr¨odinger equation. 3.2. FGA for Klein-Gordon and nonrelativistic. A balance-type approach is usually used to derive the nonrelativistic limit of the Klein-Gordon equation. For the sake of simplicity, we assume below
T that A(Q) = 0 and V (Q) = 0. We set ψ ε = φε , χε which satisfies 

iε∂t φε iε∂t χε

= =

−icεσ · pχε + βmc2 φε , −icεσ · pφε + βmc2 χε .

As V is null, φε (as well as χε ) naturally satisfies the Klein-Gordon equation.
ε2 ∂t2 φε = c2 ε2 4 + m2 c4 I2 φε . Using similar elimation process, the FGA for Klein-Gordon can be directly constructed from the FGA for Dirac. In this goal, we could construct “from scratch” the FGA for Klein-Gordon, and will to that independently, it can be deduced from the one for the potential-free Dirac equation.
(1,2) (2)
T (1) (1,2) (2) T = am Υm , and Υm = Υ(1) . Under the above We now set am = am , am , with am m , Υm assumptions, in the evolution equation for a only involves the matrix M which simply reads for (m, n) ∈ {1, 2}2 Mmn

=

−∂zk Υ†m ∂Pj h(Q, P ) .

The amplitude equation, as for the Dirac equation then reads dan dt

=

f n, Ma


 f = M11 = M22 = tr ∂z Υ† ∂P h − α where we have set M ˆ Υn Z −1 . Considering again the positive branch n
d a1 Υ1 + a2 Υ2 dt

f a1 Υ1 + a2 Υ2 ), = M

direct calculations show that the FGA for Klein-Gordon coincides with the one for Dirac assuming V null. It was proven in Section 2, that when c goes to infinity, M tends to tr[∂z P Z −1 ]/2 which is the amplitude equation for the potential-free Schr¨ odinger equation. As a consequence, one deduces that the nonrelativistic limit of the FGA for Klein-Gordon is the FGA for Schr¨odinger. 12

3.3. Mathematical properties in nonrelativistic regime. We present some mathematical properties of the FGA in the nonrelativistic regime. We recall [33] that for smooth external and interaction potential, the Dirac operator is self-adjoint on C0∞ (R3
; C4 ), while for Kato’s potentials singular at x = 0, the Dirac operator is self-adjoint on C0∞ R3 − {0}; C4 . It is also well known that the Dirac operator is an unitary operator. We first discuss the preservation of the `2 -norm in time. Start from ψ ε (0, ·) in L2 (R3 ) ⊗ C4 , and assume that P4 R ε 2 kψ εI k22 := j (0, x)| dx j=1 R3 |ψ 2 R P4 ε ε (0, q, p)Υ (q, p) · F ψ (q, p) = dqdp a m m 6 I m=1 P4 R R 2 = j=1 R3 |ωI,j (x)| dx + O(ε) = 1. The O(ε)-term is coming from the fist order term in (2.5). This can expressed as well as ε kD0,1,δ ψ εI kL2 (R3 ;C4 ) = kF ε ∗ (χδ F ε ψ εI )kL2 (R3 ;C4 ) ,

where F ε is the FBI transform (which is a L2 (R3 )-unitary operator) for some smooth cutoff function χδ ε defined in [25] and D0,1 is the FGA at order 1 and time t = 0. Next, kψ ε (t, ·)k22

=

2 1 P R R iΦm (t,x,y,p,q)/ε a (t, p, q)v (y)e dydqdp dx . m m 3 9 m R R 9 (2πε)

Recall that vm (y, q, p) = Υm (q, p) · ψ εI (y). Now [25] as kF ε f kL2 (R6 ;C4 ) = kf kL2 (R3 ;C4 ) , we get kψ ε (t, ·)k2L2 (R3 ;C4 )

= kψ εI k2L2 (R3 ;C4 ) + O(c−2 ) .

ε Notice that it is proven (in a more general form) in [25], that denoting Ht (resp. Ht,1 ) a strictly hyperbolic (resp. corresponding FGA at order 1) propagator, the following estimate holds for any T > 0 and m = 0, ε max kHt ψ εI − Ht,1,δ ψ εI kL2 (R3 ;C4 ) ≤ CT ε,

0≤t≤T

where kψ εI kL2 (R3 ;C4 ) = 1. We next discuss the numerical dispersion issues, refered in the physics literature as Fermion doubling, which is responsible for the generation of spurious states [14]. Recall that the dispersion relation for the Dirac equation reads Ek = ±c2 p2k + c4 (m is taken equal to 1). Although, it is a problem usually treated at the discrete level, it must also be explored at the continuous level for FGA. Proposition 3.1. Assume that A and V are null. The nonrelativistic FGA for Dirac is non-dispersive up to a O(c−2 ). Proof. As A and V are assumed to be null, P = p is constant, and from time 0 to t Q(t, q, p)

= q ± tp

p 1 + p2 /c2

.

The action function reads S(t, Q, P )

c2 = S(0, Q, P ) ∓ t p , 1 + p2 /c2

and
P m · x − Qm (t, q, p)


pm = pm · x − q m ∓ t p 2 2 1 + p /c
p2m . = P m · x − Qm (0, q, p) ∓ t p 1 + p2 /c2 13

That is, for c large Φm (t, x, y, q, p)

Φm (0, x, y, q, p) ∓ p

p2m + c2

t 1 + p2m /c2 Φm (0, x, y, q, p) ∓ (p2m + c2 )t + O(p2m c−2 ) .

= =

Recall that (3.19) degenerates into da/dt = Ma, with −1 Mnn = −∂zk Υ†n F nj Zjk ,

and where M is diagonal, according to Proposition 2.1. In addition, for V , and c large Z is diagonal and reads

Zjk = ∂qj − i∂pj Qk + iPk  c2 δjk  pj pk c4 t = 2δjk ± i
3/2 − p 2 2 p c + c4 p2 c2 + c4 c2 δjk = 2δjk ∓ i p t + O(c−2 ) . p2 c2 + c4 e = (e As a consequence, denoting a a1 , e a2 )T , de an dt

=

1 e j Z −1 e ∂z P jk an 2 k

=

3 e an , 2(t + 2i)

as an (t, q, p)) = (2i + t)3/2 an (0, q, p). We deduce that the dispersion relation is satisfied by the FGA for c and ε−1 large enough, that is  i  Arg a(t, q, p) exp Φm (t, x, y, q, p) ε

=

 i  Arg (2i + t)3/2 exp Φm (0, x, y, q, p) ε ∓p2m c2 t + O(p2m c−2 t) .

Notice that the (2i + t)3/2 is a low frequency term, so that at the continuous level and ε small enough, the dispersion relation is satisfied up to a O(p2 c−2 ) term. 2 4. Relativistic regime. We propose in this section some analytical results related to FGA in the relativistic regime. We are in particular interested in the relevance of the FGA in multiscale regimes. 4.1. Error analysis. For fixed c, the order-K FGA error is estimated in [25] for strictly hyperbolic systems, while since the Dirac equation is not strictly hyperbolic for d > 1 and is not homogeneous, the same proof in [25] can not directly apply to the Dirac equation. However, following the same type of arguments in [25], one may conjecture that the error estimate remains valid. That is, Conjecture 4.1. Assume that ψ I ∈ L2 (Rd ), we get the following error estimate for the order-K FGA: ε max kHt ψ εI − Ht,1,δ ψ εI kL2 (R3 ;C4 ) ≤ CT,K kψ I kεK ,

0≤t≤T

for some for some CT,K > 0. The rigorous proof to the conjecture is lengthy and technically involved, and thus we shall leave it as a future work. According to [25], if m = 0, the constant CT,K is linear in c, while CT,K is quadratic in c for m 6= 0. In other words, K should be taken large enough or/and ε small enough, such that c2 εK is small. This error estimates tells us that in the relativistic limit, either K should be taken very large or ε must be small enough to get an accurate FGA. This important problem is addressed in the following subsection dedicated to the relevance of the FGA depending on the relative values of c and ε. 14

b 2 ) gives 4.2. Multiscale FGA in relativistic regime. In the relativistic regime, the mass-term (βmc rise to the so-called zitterbewegung effect which corresponds to a fluctuation of the electron position with a frequency of 2mc2 . In order to accurately capture this effect, the time step of a numerical solver must be taken smaller than 1/mc2 corresponding physically to the zepto-second regime (10−21 second). More specifically, as numerically c ≈ 137 a.u., then 1/c2 ≈ 5.3 × 10−5 . We discuss this question for FGAs in the relativistic of an external electromagnetic field. The eigenvalues are of the form q regime in presence 2 2 λ(q, p) = ±c p − A(q)/c + c . We consider 3 cases cε  1, cε = O(1) and cε  1. Initially the FGA was derived in (t, x)-variable from the Dirac equation, by introducing the (t0 , x0 )-variables, by setting t0 = tε and x0 = εx. Basically, the FGA as derived above is accurate if cε  1. In particular, the scaling x0 = εx makes sense if the initial state, the external or interaction potentials possess high wavenumbers. We will assume below that ε = c−1/a with a > 0 and discuss the relevance of the FGA depending on the value of a. b · (x − Q)/ε behaves as ε−a , and is same order or dominant against the other ε−1 term, and The term cα βb0 c2 is of order ε−2a . Case 1. ε = c−1/a with 0 < a < 1/2. First, we notice that the computation of the Gaussian and momentum functions do not impose any c-related constraints on the time step, as this system explicitly reads: 
dQ P − A(Q)/c    , Q(0, q, p) = q,  dt = ± p 1 + |P − A(Q)/c|2
(45) dP Pl (Q) − Al (Q)/c ∂Q Al (Q)    p , P (0, q, p) = p.  dt = −∂Q V (Q) ∓ 1 + |P − A(Q)/c|2 Next, the action function reads ∂t S(t, Q, P )

=

p P · (P − A(Q)/c) p − c2 |cP + A(Q)|2 /c4 + 1 . 1 + |P − A(Q)/c|2

In order to solve this equation, we can proceed as follows. We first determine S solution to ∂t S(t, Q, P )

p
P · (P − A(Q)/c) p − c2 |cP + A(Q)|2 /c4 + 1 − 1 , 2 1 + |P − A(Q)/c|

=

from which we have removed the stiff-term c2 from the action function equation. Then, we get S(t, Q, P ) = c2 t + S(t, Q, P ). The overall FGA should be then constructed as in (2.6-2.8) except that Φm is rewritten as Ψm (t, x, y, q, p)

Φm (t, x, y, q, p)


i S(t, q, p) + |x − Qm (t, q, p)|2 + P m · x − Qm (t, q, p) 2 i 2 + |y − q| − p · (y − q), 2 = Ψm (t, x, y, q, p) + c2 t .

=

Let us now focus on the evolution equation for the amplitude. The latter involves L, M and N as defined in (3.19). From (45), we have deduced that the computation of Q and P is not restricted by c2 , so that the same conclusion holds for Z −1 . Recall that the eigenvectors are normalized. Let us start with L Lmn

=

i † −1 Υ ∂z Ql ∂Ql ∂Qj BZjk Υn , 2 m k

2 b · A + c2 βb0 + V I4 . As the mass term c2 βb0 is an order 0 operator, the computation of ∂Q B(Q) where B = −α is also free from c2 -constraints. The same conclusion holds for N . Regarding to M, the main term to study is F nj which is the sum of several terms including cb αj , while the other terms are O(1) in c. However,

∂zk Υm = ∂zk Qj ∂Qj Υm + ∂zk Pj ∂Pj Υm , 15

and   Pj − Aj (Q)/c − cb αj Υn . F nj = p 1 + |P − A(Q)/c|2 The conclusion is that the main multiscale constraint is in the reconstruction of the phase of the FGA: 2

ε

ψ (t, ·)

=

eic t/ε P R R iΨm (t,x,y,p,q)/ε dydqdpdx . 3 9 am (t, p, q)vm (y)e (2πε)9 m R R

This highly oscillating phase does not impose any restriction on the computation of the amplitude a, the shifted phase S, or on the Hamiltonian flow. In this first case, the FGA is kept unaltered. Case 2. ε = O(c−2 ). In this case, the βb0 c2 -term is the only term in O(ε−1 ) in B (8), then D, (7). This naturally modifies the construction of the FGA. As a consequence the double eigenvalues to D becomes h± (q, p) = ±|cp − A(q)| + V (q). The eigenvectors have to be recomputed accordingly, denoting again u = cp − A(q) and r =     ±u3 ±(u1 − iu2 )   1  ±(u1 + iu2 )  ∓u3 .  , Υ±2 = 1  Υ±1 =      0 |u| r r |u| 0

√

2|u|,

The rest of the algorithm is identical to the one described in Section 1. Case 3. ε = O(c−1 ). In this case, the FGA must in principle be re-derived from scratch. Indeed, the b · p term behaves as α b · p/ε. Moreover, βb0 c2 -term is now a O(ε−2 ) (8), and the term cα
c −1 b · (x − Q) = −∂zk α α bj Zjk av , ε which is then a ε−2 term and should be removed from the previous expression of F nj , that is,
F nj = ∂Pj h(Q, P ) − i∂Qj h(Q, P ) + i∂Qj B(Q) Υn , then from the evolution equation for a. Now α bj Υn = ∂Pj hΥn , so that

−1 −1 −∂zk α bj Zjk av = −∂zk ∂Pj hZjk av

−1 −1 = −∂zk ∂Pj hZjk a1 Υ1 v − ∂zk ∂Pj hZjk a2 Υ2 v , and from (3.16)
−1 −1 −∂zk ∂Pj hZjk a1 Υ1 v = −Υ2 ∂zk Υ†2 ∂Pj hZjk a1 Υ1 v,
−1 −1 −∂zk ∂Pj hZjk a2 Υ2 v = −Υ1 ∂zk Υ†1 ∂Pj hZjk a2 Υ2 v . Then the action function equation still reads, as Υ†2,1 Υ1,2 = 0, ∂t S = P · Q − h(Q, P ) . The term c2 βb0 now behaves βb0 ε−2 , and we can argue that the nonrelativistic limit should be considered in the first place. Case 4. ε < O(c−1 ). In this case all the c-terms would have to be “removed” from the FGA construction, as they would correspond to high order terms and would then have to be treated independently. In this case and as above, we should instead take the nonrelativistic limit. 16

5. Numerical experiments. This section is devoted to numerical experiments to illustrate the accuracy and relevance of the FGA for the Dirac equation. 5.1. A simple accuracy FGA test. We set A ≡ 0 and V ≡ 0 in the Dirac equation (5), and set the initial condition (6) as    T 2 (46) ω I (x) = exp −A |x − x0 | , 0, 0, 0 , and SI (x) = p0 · x, with A = 32, x0 = (1, 1, 1), and p0 = (1, 0, 0). We compute the solutions using FGA for different ε’s and compare them with the reference solutions computed by Fourier spectral method. In this first set, we set c = 1, which allows in particular, for a removal of the computational difficulty due to the very small timescale (1/c2 ) coming from the β0 c2 -term. The solutions with ε = 2−7 are shown in Fig 2, and the relative L2 errors are listed in Table 1 illustrating the accuracy of the FGA.

(a) FGA solution

(b) Reference solution Fig. 2.

Table 1 The L2 -errors at t = 0.2.

ε

2−4

2−5

2−6

2−7

kψ ε − ψ εF GA k2

2.63 × 10−2

7.62 × 10−3

4.21 × 10−3

3.19 × 10−3

Remark 5.1. Let us remark that an partially explicit solution can be constructed for specific initial data: 2

ψ(0, x, z) = N [1, 0, 0, 0]T × e−((x−0.5)

+(z−1)2 )/4∆2

.

The corresponding Fourier transform is given by b px , pz ) = 4π∆2 N [1, 0, 0, 0]T × e−∆2 p2 , ψ(0, where p2 = p2x + p2z . The solution to
b px , pz ) = cα b px , pz ), b · p + βc2 /ε ψ(t, i∂t ψ(t, is given by b px , pz ) ψ(t,

= =

  b 2 t/ε ψ(0, b px , pz ) b · pt − iβc exp − icα i h b2 b · pε + βc cα b px , pz ), sin(Eε t) ψ(0, I4 cos(Eε t) − i εEε 17

p p with p2 c2 + c4 /ε2 . We set p0x := εpx and p0z := εpz , t0 := t/ε, p0 := (p0x )2 + (p0z )2 , E 0 = p Eε = 0 (p0 )2 c2 + c4 and ∆ := ∆/ε, then h i b2 b · p0 + βc cα 0 0 2 0 0 4π∆2 I4 cos(E 0 t0 ) − i sin(E t ) N [1, 0, 0, 0]T × e−(∆ p ) , 0 E p so that, denoting x0 = xε, z 0 = zε and r0 = (x0 )2 + (y 0 )2 , 
R c2  0 0 0 0 2 ∞ 0 −(∆0 p0 )2 0 0 0  sin(E 0 t0 ) dp0 , ψ (t , x , z ) = 2N (∆ ) p e J (p r) cos(E t ) − i  1 0  0 0  E    ψ2 (t0 ,0 x0 , z 0 ) = 0,
0 R ∞ 0 −(∆0 p0 )2 cp0 0 0 0 0 2 0 0 0  ψ (t , x , z ) = −2N (∆ ) sin(θ) p e J (p r) sin(E t ) dp ,  3 1 0 0  E0      ψ4 (t0 , x0 , z 0 ) = −2N (∆0 )2 sin(θ) R ∞ p0 e−(∆0 p0 )2 J1 (p0 r) cp sin(E 0 t0 )
dp0 , 0 E0 (47)

b t, p0 /ε, p0 /ε ψ x z




=

where J0,1 are the zeroth and first order Bessel functions, θ = tan−1 (z 0 /x0 ). For instance taking r0 = 0, using the Erfi-function, we get ψ 1 (t0 , 0, 0)

=

h   εct0 2   εct0 i √ εct0 N 1− π exp − Erfi . 2∆ 2∆ 2∆

5.2. Barrier potential. This test is devoted to the propagation of a wavepacket toward a smooth barrier potential. We denote the energy of a wavepacket with wavenumber (kx , ky , kz ), by p Eε = k 2 c2 + c4 /ε, q where k := kx2 + ky2 + kz2 and the barrier potential is defined as (48)

V (x, y, z) =

V0  x
 1 + tanh . 2 LV

In the semi-classical regime, the wavepacket should be totally reflected if V0 /ε > Eε − c2 /ε, while it should be totally transmitted otherwise. Notice that we do not expect Klein paradox to occur in this regime, see next subsection. We consider both situation in a 3-d simulation on a spatial domain [0, 2]3 , of a wavepacket defined by

ψ(0, x, y, z) = N [1, 0, 0, 0]T × exp − (x − x0 )2 + (y − y0 )2 + (z − z0 )2 /4∆2 exp(ikx x), √ where x0 = 0.5, y0 = z0 = 1, kx = 0.5 and ∆ = 4 2. In this test we take c = 1, Lv = 0.2. We report −6 the modulus of the first component ψ√ , corresponding 1 of the FGA solution at time T = 1.5, with ε = 2 to wavepacket energy equal to Eε = 1.25/ε, when V0 = (resp. V0 = 0). The time step is taken equal to ∆t = 10−2 As expected, total transmission (resp. reflection) occurs Fig. 3 (Left) (resp. (Middle)), when V0 /ε < Eε − 1/ε (resp. V0 /ε > Eε − 1). We also report in Fig. 3 (Right) the FGA without potential, that is V0 = 0. 5.3. Nonrelavistic limit. This test is dedicated to the computation of the FGA making vary values of c. We compare the FGA solution, with ε = 2−7 , from c small to c large, and observe in particular that the FGA is convergent when c goes to infinity (to the FGA for Schr¨odinger, see Theo. 3.1). We represent the first component ψ1 of the FGA solution, with Cauchy data

ψ(0, x, y, z) = N [1, 0, 0, 0]T × exp − (x − x0 )2 + (y − y0 )2 + (z − z0 )2 /4∆2 exp(ikx x), where x0 = y0 = z0 = 1, kx = 0.5, and ∆ = 4. We report the solution for c = 0.1, 0.5, 1, 5, 10, 10000 in Fig. 4. As numerically observed on this example, for c large enough, the FGA solution becomes independent of c, and corresponds to the FGA solution for the Schr¨odinger equation, according to Theorem 3.1. 18

Fig. 3. Barrier potential (Left) V0 = 0.2 corresponding to an almost null reflection. (Middle) V0 = 1 corresponding to a total reflection. (Right) No-potential.

Fig. 4. Modulus of first component. For c respectively equal to 0.1, 0.5, 1, 5, 10, 10000.

5.4. Klein Paradox. The Klein paradox refers to the partial transmission of an electronic wavepacket through a potential energy barrier higher than the kinetic energy Ep of an incoming electron [7]. We will perform two series of tests in 1-d and 2-d. We show in particular, that the direct FGA does not allow to capture this quantum effect. 5.4.1. One-dimensional tests. The electronic kinetic energy is given here by (ε)

Ekx =

p

c2 kx2 + c4 /ε .

In order to apply the FGA, we regularize the barrier potential as follows V (x) =

x
 V0  1 + tanh , 2 LV

where LV is a small parameter characterizing the gradient of the potential. In theory, transmission occurs (ε) when V0 /ε > Ekx + 2c2 /ε. The transmission T and reflection R coefficients for a barrier potential located 19

at 0, are respecitively given by R R+

T = R

R

ψ† ψ

ψ† ψ

R R−

R= R

,

R

ψ† ψ

ψ† ψ

,

where (49)

T =−

sinh(πk (ε) LV ) sinh(πκ(ε) LV )

, sinh π(V0 /cε + k (ε) + κ(ε) )LV /2 sinh π(V0 /cε − k (ε) − κ(ε) )LV /2

k (ε) =

q 1 (ε) Ekx − V0 /ε)2 − c4 /ε2 , c

and κ(ε) = −

q
2 1 E (ε) − V0 /ε − c4 /ε2 . c

As a preliminary test, we perform a test in the quantum regime using a method described in [14]. We assume that ε = 1/8, which is close to a semi-classical regime. The spatial domain is [−20, 20], LV = 10−4 , and V0 = 5×104 . The time step is given by ∆t = 2.23×10−6 and space step ∆x = 1.53×10−4 , corresponding to N = 262144 grid points. The initial data is a Gaussian centered at x0 = −5 and is given by
ψ(0, x) = [1, 0, 0, C]T × exp − (x − x0 )2 /4∆2 exp(ikx x), with width ∆ = 1, wavenumber kx /ε = 500 and C=

c2

ckx p . + c4 + 2kx2

The theoretical transmission coefficient is given by T = 0.322, while the numerical one is given by Th = 0.327. At total of 8 × 104 time iterations is necessary to performed this computation, corresponding to ≈ 2160.9 seconds (on 16 processors), which is relatively very resource demanding for a such a simple test. 5.4.2. Two-dimensional tests. We here consider a quantum and semi-classical regime simulation. Quantum regime. We assume that ε = 1, c ≈ 137.0359895. The spatial domain is [−1, 1] × [−2, 2], LV = 10−4 , z
 V0  V (x, z) = 1 + tanh , 2 LV with V0 = 4.64 × 104 . The time step is given by ∆t = 2.85 × 10−5 and space step ∆x = ∆z = 2 × 10−3 . The initial data is a Gaussian centered at (x0 , z0 ) = (0, −1.0) and is given by
(50) ψ(0, x, z) = [1, 0, 0, C]T × exp − ((x − x0 )2 + (z − z0 ))2 /4∆2 exp(ikx x), with width ∆ = 0.2, wavenumber kx = 200 and C=

c2

ckx p . + c4 + 2kx2

(ε)

Then Ekx = 3.3224 × 103 and κ(ε) = 133.9374. The transmission coefficient which is computed numerically is given by Th ≈ 0.661 and the theoretical one is T = 0.65. The computation is performed on 8 processors with a total of 1230.4 seconds. Semi-classical regime. We consider the initial condition (50), with ∆ = 1/64, x0 = z0 = 1, and kx = 0.5. We assume that ε = 2−7 and c = 1. The barrier potential is defined by (48) with Lv = 0.2, and V0 = 4, and as a consequence V0 /ε > Eε + 2c2 /ε. In the quantum regime, the transmission coefficient gives τ ≈ 0.485, when Lv small enough. In the semi-classical limit, we numerically report the FGA wavepacket at time T = 1. The wavepacket is totally reflected illustrating the inability for the direct FGA, which is a semi-classical solution, to simulate the Klein paradox which is a quantum effect. A special treatment must be implemented in order to allow the transmission. 20

Fig. 5. Klein paradox in one-dimension in quantum regime (ε = 1/8). First and fourth spinor component: initial data and final time is T = 0.22.

Fig. 6. Klein paradox in two-dimension in quantum regime (ε = 1). Density is T = 0.0171 (right).

P4

i=1

|ψ i |2 : initial data (left) and final time

6. Direct spectral method. The computational domain is [−ax , ax , ] × [−ay , ay ] × [−az , az ], where ax > 0, ay > 0 and az > 0 . We denote the grid-point set by  DNx ,Ny ,Nz = xk1 ,k2 ,k3 = (xk1 , yk2 , zk3 ) (k ,k ,k )∈O , 1

2

3

Nx Ny Nz

where  ONx Ny Nz = (k1 , k2 , k3 ) ∈ N3 , : k1 = 0, · · · , Nx − 1; k2 = 0, · · · , Ny − 1; k3 = 0, · · · , Nz − 1 . Then, we define the space steps as hx = xk1 +1 − xk1 = 2ax /Nx , hy = yk2 +1 − yk2 = 2ay /Ny , hz = zk3 +1 − zk3 = 2az /Nz . 21

Fig. 7. Klein paradox in two-dimension in quantum regime (ε = 2−7 ). Density |ψ 1 |2 : initial data (left) and final time is T = 1 (right).

The corresponding discrete wavenumbers are defined by ξ := (ξp , ξq , ξr ), where ξp = pπ/ax with p ∈ {−Nx /2, · · · , Nx /2−1}, ξq = qπ/ay with q ∈ {−Ny /2, · · · , Ny /2−1} and ξr = rπ/az with r ∈ {−Nz /2, · · · , Nz /2− 1}. We denote next by αγ = Πγ Λγ ΠTγ , for ν = x, y, z and where   1 0 0 0  0 1 0 0   Λγ =   0 0 −1 0  . 0 0 0 −1 The matrices Πγ are defined as follows.    0 1 1 0 0 1  1 0 0 −1  1  1    Πx = √  ,Π = √  1 0 0 1  y −i 2 2 0 1 −1 0 0

−i 0 0 1

−i 0 0 −1

  0 1 1  1   , Πz = √  0 i  2 1 0 0

0 −1 0 1

0 −1 0 −1

 −1 0  . 1  0

A first or second order splitting method is used in combination with the Fast Fourier Transform. The first order splitting reads as follows. We denote by tn∗ := tn + ∆t. Naturally, in order to capture the high frequencies ∆t must be chosen smaller than c2 /ε, naturally motivating the use of computational method in the semi-classical limit. We denote by ψ nh , the approximate spinor at time tN . 1. First step: integration of the source from tn to tn1 = tn + ∆t. 
 ψ nh1 = exp − i∆t βc2 /ε + Vh /ε − α · Ah ψ nh . This step is performed exactly, up to the approximation of A (resp. V ) by their projection Ah (resp. Vh ) on a finite difference grid. 2. Second step: integration of the generalized transport equation in Fourier space. One sets φnh1 := ΠTx ψ nh1 , and solve ∂t φ + Λx ∂x φ = 0,   from time tn to tn2 = tn +∆t, basically by discretizing in space: φ(tn+1 , ·) = Fx−1 e−iξ∆tΛx Fx (φ(tn , ·)) , where Fx is the Fourier transform in the x-direction that is for ` = 1, · · · , 4, such that φ` denotes (`),n the `th component of φ. We denote φh 1 , the `th component φh at time n1 . (`),n2

φh

=

1 PNx /2−1  −iξp ∆tλ(x) PNx −1 (`),n1 −iξp (xk +ax )  iξp (x+ax ) j , ` 1 e e k1 =0 φh,k1 e Nx p=−Nx /2 22

(x)

where λ` is the `th eigenvalues of Λx . 3. Similarly one computes, from tn to tn3 = tn + ∆t (`),n3

φh

=

1 PNy /2−1  −iξq ∆tλ(y) PNy −1 (`),n∗2 −iξq (yk +ay )  iξq (y+ay ) j , ` 2 e e k2 =0 φh,k2 e Ny q=−Ny /2

n∗

where φh2 = ΠTy Πx φnh2 and finally (`),n+1

φh

=

1 PNz /2−1  −iξr ∆tλ(z) PNz −1 (`),n∗3 −iξr (zk +az )  iξr (z+az ) j , ` 3 e e k3 =0 φh,k3 e Nz r=−Nz /2

n∗

where φh3 = ΠTz Πy φnh3 . Finally ψ n+1 = Πz φn+1 . n h Alternatively, it is possible to directly apply 3d FFTs. By construction, the `2 -norm is naturally conserved. 7. Conclusion. This paper derives and analyzes the frozen Gaussian approximation (FGA) for the linear Dirac equation in the semi-classical limit. Strong connections of the FGA for Dirac and Klein-Gordon equations with the Schr¨ odinger equation are established in the non-relativistic limit. Important physical properties, such as numerical dispersion, are also mathematically analyzed and illustrated by numerical experiments. In future works, FGA for the Dirac equation will be used to solve realistic strong fields problems in quantum relativistic physics. Acknowledgment. XY was partially supported by the NSF grant DMS-1818592. Appendix A. Eigenvalues and eigenvectors. A.1. Eigenvectors computation. In order to determine M and N in (34), we need to explicitly construct ∂h and ∂Υ . First notice (51)

c(Pl c − Al )∂Q Al + ∂Q V, ∂Q h± = ∓ q 2 |P c − A(Q)| + c4

(P c − A) ∂P h± = ± q . 2 |P c − A(Q)| + c4

(Pj c − Aj )(P c − A) · ∂Qk A c∂Qk Aj ∂Pj ∂Qk h± = ±  , 3/2 ∓ q 2 2 |P c − A(Q)| + c4 |P c − A(Q)| + c4

(52)

cδjk c(Pj c − Aj )(Pk c − Ak ) ∂Pj ∂Pk h± = ∓  , 3/2 ± q 2 2 |P c − A(Q)| + c4 |P − A(Q)| + c4 ∂Qj ∂Qk h± = ∓

(P c − A) · ∂Qj A (P c − A) · ∂Qk A  3/2 2 |P c − A(Q)| + c4

±

∂Qj A · ∂Qk A − (P c − A) · ∂Qj ∂Qk A q + ∂Qj ∂Qk V, 2 |P c − A(Q)| + c4

Hence (53)

∂z Υ = ∂z Qj ∂Qj Υ + ∂z Pj ∂Pj Υ . q 2 Let λ = |u| + c4 , and u = P c − A(Q). We now determine the positive and negative branches. A.2. Positive branch. The corresponding eigenvectors read     u3 u1 − iu2   u1 + iu2 −u3 1 1     (54) , Υ = Υ 1 = q 2   , 2 0 r  |u| + c4 − c2  r q  2 2 4 |u| + c − c 0 23

s   q 2 2 where r = 2 |u| + c4 − c2 |u| + c4 , and u = P c − A(Q). We hve

(55)

(56)

 0 1 0 ∂u Υ 1 = r 1  1 1 ∂u Υ 2 = −i r 0

 0 1 0 − ∂u r ⊗ Υ 1 , r 0  0 u1 /λ 1 0 u2 /λ − ∂u r ⊗ Υ 2 , r 0 u3 /λ

1 u1 /λ i u2 /λ 0 u3 /λ 0 0 −1

and (57)

  1 1 q − 2 u. ∂u r = − r 2 |u| + c4

Thus
i −u2 , u1 , 0 , 2 r
i † 0 , Υ 2 ∂u Υ 2 = 2 u2 , −u1 , r
1 † u1 + iu2 , Υ 2 ∂u Υ 1 = 2 −u3 , −iu3 , r
1 † Υ 1 ∂u Υ 2 = 2 u3 , −iu3 , −u1 + iu2 . r Υ †1 ∂u Υ 1 =

(58)

We define the following Berry connection matrices:      1 −iu2 u3 iu1 −iu3 0 B= 2 , , (59) −u3 iu2 −iu3 −iu1 u1 + iu2 r

−u1 + iu2 0



A.3. Negative branch. The corresponding eigenvectors read     −u3 −u1 + iu2   u3 1 1   −u1 − iu2   (60) , Υ = Υ 1 = q 2 ,   2 0 r  |u| + c4 + c2  r q  2 4 + c2 |u| + c 0 s   q 2 2 where r = 2 |u| + c4 + c2 |u| + c4 .

(61)

(62)

(63)

 0 1 0 ∂u Υ 1 = r −1  −1 1 i ∂u Υ 2 = r 0

 u1 /λ 0 1 u2 /λ 0 − ∂u r ⊗ Υ 1 , r u3 /λ 0  0 0 u1 /λ 1 0 0 u2 /λ − ∂u r ⊗ Υ 2 , r 1 0 u3 /λ −1 −i 0

 1 q ∂u r = r

 1 2

|u| + 24

+ 2 u. c4

.

Thus
i −u2 , u1 , 0 , 2 r
i † 0 , Υ 2 ∂u Υ 2 = 2 u2 , −u1 , r
1 † u1 + iu2 , Υ 2 ∂u Υ 1 = 2 −u3 , −iu3 , r
1 † Υ 1 ∂u Υ 2 = 2 u3 , −iu3 , −u1 + iu2 . r Υ †1 ∂u Υ 1 =

(64)

We define the following Berry connection matrices:      1 −iu2 u3 iu1 −iu3 0 B= 2 , , (65) −u3 iu2 −iu3 −iu1 u1 + iu2 r

−u1 + iu2 0

 .

We conclude this appendix, by providing some working values, necessary to determine the non-relativistic limit of the FGA. q p
A.4. Intermediate values. Denoting again r = 2 p2 c2 + c4 − c2 p2 c2 + c4 , and for the positive branch     0 c     c 0  ∂ p1 r  ∂p1 r 1 1 −  − p1 c2 0 ∂p1 Υ1 =  Υ , ∂ Υ = Υ2 1 p1 2     2 2 r p 2 2 r r r2   p c 4 2 p c +c p 0 p2 c2 + c4   1 ∂p2 Υ1 =  r  p

  1 ∂p3 Υ1 =  r  p

0 ic p2 c2

  ∂ p2 r − Υ1 ,  r2  2 2 4 p c +c 0 c 0 p 3 c2

−ic 0 0 p2 c2





 1 ∂p2 Υ2 =  r  p

 1 ∂p3 Υ2 =  r 

  ∂p3 r − Υ1 ,  r2 p2 c2 + c4  0

p

  ∂p2 r − Υ2  r2 

p2 c2 + c4 0 −c 0 p3 c2







   ∂p3 r − Υ2 .  r2 

p2 c2 + c4

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