Exact Green’s Function for 2D Dirac Oscillator in Constant Magnetic Field Salah Haouat and Lyazid Chetouani D´epartement de Physique, Facult´e des Sciences Exactes, Universit´e Mentouri, Route Ain El-Bey, Constantine 25000, Alg´erie Reprint requests to L. C.; E-mail:
[email protected] Z. Naturforsch. 62a, 34 – 40 (2007); received October 16, 2006 The propagator of two-dimensional Dirac oscillator in the presence of a constant magnetic field is presented by means of path integrals, where the spin degree-of-freedom is described by odd Grassmannian variables and the gauge invariant part of the effective action has the form of the standard pseudoclassical action given by Berezin and Marinov. Then the path integration is carried out and the problem is solved exactly. The energy spectrum of the electron and the wave functions are extracted. – PACS numbers: 03.65.Ca, 03.65.Db, 03.65.Pm, 03.65.Ge. Key words: Path Integral; Dirac Oscillator; Exact Solutions.
1. Introduction For a long time, the harmonic oscillator has been considered as one of the most useful systems in quantum physics. So there are many relativistic generalizations such as the Dirac oscillator (DO) introduced by Itˆo et al. [1] and developed by Moshinsky and Szczepaniak [2]. By adding the vector potential (−imωβr), which is linear in coordinate and carries a matrix β , to Dirac equation, the authors have found a relativistic model, where the nonrelativistic limit reproduces the usual harmonic oscillator. This model was, during the last 15 years, the subject of many papers and has attracted the attention of many authors. Villalba studied the pure two-dimensional Dirac oscillator (2DDO) [3]. Later, Villalba and Maggiolo computed the energy spectrum of the 2DDO in the presence of a magnetic field [4]. Also, the problem of the 2DDO is solved in the presence of AharonovBohm potential [5]. The Dirac oscillator also has some applications, especially after its second time introduction by Moshinsky and Szczepaniak. For example, Gashimzade and Babaev used this model to express Kane-type semiconductor quantum dots [6]. However, in spite of the 2DDO has been much discussed, there is no well established path integral treatment. The purpose of this work is to set up a path integration for the problem of the 2DDO in the presence of a constant magnetic field. In Section 2 we give a path integral formulation for the problem in question and we
describe the spin degree-of-freedom by the odd Grassmannian variables. In Section 3, after integrating over odd trajectories, we calculate, exactly, Green’s function in Cartesian coordinates. Finally, we pass to the polar coordinates in order to extract the related spectrum and wave functions. 2. Path Integral Formulation of 2DDO The most useful path integral formulation for a relativistic spinning particle interacting with an external field is already proposed by Fradkin and Gitman [7]. They have presented the relative propagator according to the Feynman standard form
D(path) exp iS(path),
(1)
where S is a supersymmetric action, which describes at the same time the external motion and internal one related to the spin of the particle. However, for the problem of the 2DDO, that is governed by the Hamiltonian HDO = α (P − imωβ r) + β m,
(2)
it is convenient to rederive a path integral representation. Starting from the wave equation that can be written in the form µ γ Pµ + imω γ 0γ .r − m ψ (t,r) = 0, (3)
c 2007 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen · http://znaturforsch.com 0932–0784 / 07 / 0100–0034 $ 06.00
S. Haouat and L. Chetouani · Exact Green’s Function for 2D Dirac Oscillator
where Pµ = i∂ µ and the γ -matrices are given, in (2+1)dimension, in terms of Pauli matrices
γ 0 = σz ,
γ 1 = iσx ,
γ 2 = iσy ,
35
we express the propagator Sc (tb ,rb ,ta ,ra ) in the socalled global projection [8]
(4)
Sc (tb ,rb ,ta ,ra ) = K+ (tb ,rb ) Gc (tb ,rb ,ta ,ra ) , (14)
we define the propagator of the 2DDO in the presence of a constant magnetic field as a causal Green’s function Sc (tb ,rb ,ta ,ra ) solution of the equation γ 0 P0 −γ P−eA +imωγ 0γ .r−m Sc (tb ,rb ,ta ,ra ) (5) 2 = −δ (tb − ta) δ (rb −ra ) ,
where the new Green’s function Gc (tb ,rb ,ta ,ra ), that we suggest to calculate via path integration, is defined by
where the magnetic field is described by the vector potential A = Br u˜ ϑ , 2
(6)
Gc (tb ,rb ,ta ,ra ) = tb ,rb |
Ay =
Gc (tb ,rb ,ta ,ra ) = i
+∞ 0
dλ tb ,rb | (16) · exp(−iH(λ ))|ta ,ra ,
B x. 2
(7)
where the Hamiltonian H (λ ) is given by H(λ ) = λ (−P2 + m2 + m2 ω¯ 2 (x2 + y2)
Then, we present Sc (tb ,rb ,ta ,ra ) as a matrix element of an operator Sc Sc (tb ,rb ,ta ,ra ) = tb ,rb | Sc |ta ,ra ,
(8)
1 1 = −K+ , K− K− K+
(9)
where Sc = −
and the operators K+ and K− are given by eB eB K± = γ 0 P0 − γ 1 P1 + y − γ 2 P2 − x 2 2 (10) 0 1 0 2 + imω (γ γ x + γ γ y) ± m. Using properties of Pauli matrices, we get, after some calculations, K− K+ = P − m − m ω (x + y ) 2
2
2
¯2
2
2
+ 2mω¯ (Px y − Pyx) + 2imω¯ γ 1 γ 2 ,
(11)
where
ω¯ = ω +
eB . 2m
(12)
Introducing now the relation
dtdr |t,r t,r| = 1,
(15)
and has to be represented by the Schwinger proper time method as
that has the two components B Ax = − y, 2
−1 |ta ,ra K− K+
(13)
¯ 1 γ 2 ). − 2mω¯ (Px y − Pyx) − 2imωγ
(17)
The operator K+ (tb ,rb ) will eliminate the superfluous states caused by the product K− K+ . To present Gc (tb ,rb ,ta ,ra ) by means of a path integral, we write, in the beginning, exp(−iH(λ )) = [exp(−iH(λ )ε )]N , with ε = 1/N, and we insert (N − 1) resolutions of identity |x x| dx = 1 between all the operators exp (−iε H
(λ )) . Next, we introduce N additional integrations dλk δ (λk − λk−1 ) = 1. We get Gc (tb ,rb ,ta ,ra ) = i lim ·
+∞
N→∞ 0 ε →0 N
dλ0
dx1 dx2 . . . dxN−1
dλ1 dλ2 . . . dλN ∏ xk | exp (−iε H(λk )) |xk−1 k=1
(18)
· δ (λk − λk−1).
As ε is small, we can write xk | exp (−iε H (λk )) |xk−1 ≈ xk | 1 − iε H(λk ) |xk−1 , (19) and since the operator H (λ ) has a symmetric form with respect to operators X and P, the matrix element (19) will be expressed in terms of the Weyl symbols in the middle point x˜k = (xk + xk−1 )/2. In effect, using
S. Haouat and L. Chetouani · Exact Green’s Function for 2D Dirac Oscillator
36
identities that
|pk pk | dpk = 1 and taking into account
xk | pk =
1 (2π )
3/2
eipk xk ,
(20)
The multipliers in (18) are noncommutative due to the γ -matrix structure, so that we attribute formally the index k to γ -matrices, and we introduce the T-product which acts on γ -matrices. Then, using the integral representation for the δ -functions
the matrix element in (19) can be written in the form
dpk xk − xk−1 exp i p − H ( λ , x ˜ , p ) ε . k k k k ε (2π )3 (21)
δ (λk − λk−1) =
i 2π
eiπk (λk −λk−1 ) dπk ,
(22)
it is possible to gather all the multipliers, entering in (18), in one exponent and the Green’s function Gc can be expressed as follows: 1 ∞ Gc (tb ,rb ,ta ,ra ) = T dλ0 Dx Dp Dλ Dπ exp i dτ λ p2 − m2 − m2ω¯ 2 x2 + y2 0 0
(23) + λ 2mω¯ (px y − pyx) + λ 2mω¯ γ 1 γ 2 + px˙ + π λ˙ . To insert the γ -matrices by means of paths we introduce in the beginning an odd sources ρ µ (Grassmannian variable): 1 ∞ c dλ0 Dx Dp Dλ Dπ exp i dτ λ (p2 − m2 − m2 ω¯ 2 (x2 + y2 )) G (tb ,rb ,ta ,ra ) = 0 0
(24) 1 δ δ + λ 2mω¯ (px y − pyx) + px˙ + π λ˙ + λ 2mω¯ ρ ( τ ) γ d τ , T exp δ ρ1 δ ρ2 0 ρ =0
and we present the quantity T exp 01 ρ (τ ) γ dτ via a path integral over Grassmannian odd trajectories [7, 8]: 1 1 ∂ T exp ρ (τ )γ dτ = exp iγ µ lµ Dψ exp dτ ψµ ψ˙ µ − 2i ρµ ψ µ + ψµ (1)ψ µ (0) , (25) ∂θ 0 0 ψ (0)+ψ (1)=θ where the measure Dψ is given by 1 −1 Dψ = Dψ Dψ exp ψµ ψ˙ µ dτ , ψ (0)+ψ (1)=0
(26)
0
and θ µ and ψ µ are odd variables, anticommuting with γ -matrices. Finally, Green’s function Gc is presented in the Hamiltonian path integral representation 1 ∞ c µ ∂l G (tb ,rb ,ta ,ra ) = exp iγ d λ0 Dx Dp Dλ Dπ Dψ exp i d τ λ p 2 − λ m2 µ ∂θ ψ (0)+ψ (1)=θ 0 0 (27) 2 ¯2 2 2 µ 1 2 µ ˙ − λ m ω (x + y ) + 2λ mω¯ (px y − pyx) + px˙ + π λ − iψµ ψ˙ + iλ 8mω¯ ψ ψ + ψµ (1)ψ (0) . θ =0
We notice that, integrating over momenta and separating the gauge-fixing term π λ˙ and the boundary term ψµ (1) ψ µ (0), we obtain the supergauge invariant action [9, 10]
1 x˙2 A= + mω¯ (yx˙ − xy) ˙ − iψµ ψ˙ µ + iλ 8mω¯ ψ 1 ψ 2 dτ , (28) − 4λ 0 where the supersymmetric transformation is given by 1 α x˙ ε . 4λ In the next section, we give an exact calculation of Green’s function Gc (tb ,rb ,ta ,ra ).
δ xµ = iψ α ε , δ ψ α =
(29)
S. Haouat and L. Chetouani · Exact Green’s Function for 2D Dirac Oscillator
37
3. The Green’s Function Having shown how to formulate the problem of the 2DDO plus a magnetic field in the framework of Feynman path integrals, let us now go to the calculation of Green’s function Gc . First, we integrate over π , λ , x0 and p0 . We obtain Gc (tb ,rb ,ta ,ra ) =
∞ 0
dλ
dE −iE(tb −ta ) e × F(λ ) GE (xb , yb ; xa , ya ; λ ) , 2π
(30)
where the function GE is given only in terms of bosonic trajectories 1 dτ λ (E 2 − m2) − λ p2x − λ p2y − λ m2 ω¯ 2 (x2 + y2 ) GE (xb , yb ; xa , ya ; λ ) = DxDy Dpx Dpy exp i 0 (31) + 2λ mω¯ (px y − py x) + px x˙ + py y˙ , and the factor F (λ ) is given by
∂l F(λ ) = exp iγ ∂θ µ µ
ψ (0)+ψ (1)=θ
Dψ exp
1 0
˙µ
ψµ ψ − 4λ Fµν ψµ ψ
ν
dτ + ψµ (1)ψ (0) µ
. (32) θ =0
Here, the tensor F is defined by F12 = −F21 = mω¯ (all other elements are 0) and has to be understood as a matrix with lines marked by the first contravariant indices and with columns marked by the second covariant indices. In order to calculate F (λ ) we change, in the first stage, the integration variables from ψ to ξ , where
θ 1 ψ= ξ+ , 2 2
(33)
and the new variables ξ obey the following boundary conditions:
ξ (0) + ξ (1) = 0.
(34)
The factor F (λ ) will be then given through the Grassmann-Gaussian integral 1 1 ˙µ ∂l ξµ ξ − λ Fµν ξ µ ξ ν −2λ Fµν θ µ ξ ν dτ , (35) F(λ ) = exp iγ µ µ exp(−λ Fµν θ µ θ ν ) Dξ exp ∂θ 4 0 θ =0 that can be computed to be
1 µ ∂l 2 F(λ ) = det (cosh 2λ F) exp iγ ∂θ µ · [1 − λ Bµν θ µ θ ν ] ,
that
(36)
F mω¯
2 (1 − cos2λ mω¯ ) (38)
and F tan 2λ mω¯ . 2λ mω¯ We then get B=
θ =0
where the tensor B, that is understood as a matrix, is given by (see [11]) 1 B= tanh 2λ F. 2λ
cosh 2λ F = 1 +
(37)
From the definition of the tensor Fµν , it is easy to show
(39)
F (λ ) = cos (2mωλ ) + iγ 0 sin (2mωλ ) =
1 + sγ 0 exp (is2mωλ ) . 2 s=±1
∑
(40)
S. Haouat and L. Chetouani · Exact Green’s Function for 2D Dirac Oscillator
38
Thus, Green’s function Gc (tb ,rb ,ta ,ra ) can be expressed only through a bosonic path integral over the space coordinate and their corresponding momenta: 1 dE −iE(tb −ta ) 1 + sγ 0 ∞ Gc (tb ,rb ,ta ,ra ) = e d λ DxDy Dp Dp exp i dτ λ (E 2 − m2 ) x y ∑ 2 0 2π 0 s=±1 (41) 2 2 − λ py − λ px − λ m2 ω¯ 2 (x2 + y2 ) + 2λ mω¯ (px y − py x) + px x˙ + py y˙ + λ 2smω . Having succeeded to do integration over Grassmannian variables, let us now integrate over even (bosonic) trajectories. In the first stage, we integrate over momenta. The equation (31) becomes 1 2
x˙ y˙2 DxDy exp i dτ + + mω¯ (yx˙ − xy) ˙ . (42) GE (xb , yb ; xa , ya ; λ ) = expiλ E 2 − m2 4λ 4λ 0 The last path integral has a quadratic action and, consequently, is integrable. The result is mω¯ GE (xb , yb ; xa , ya ; λ ) = expiλ E 2 − m2 exp [−imω¯ (xa yb − ya xb )] ¯ ) 2π sin (2mωλ mω¯ 2 2 · exp i (xb − xa ) + (yb − ya ) . ¯ ) 2 tan (2mωλ
(43)
4. Energy Spectrum and Wave Functions In order to obtain the wave functions and their corresponding energies, we must express Green’s function Gc (tb ,rb ,ta ,ra ) in polar coordinates (x = r cos ϑ , y = r sin ϑ ). This passage from Cartesian coordinates to polar ones may be done with the help of the formula +∞ mω¯ rb ra mω¯ rb ra exp −i cos(∆ ϑ ) = ∑ I|k| (44) eik∆ ϑ . ¯ ) ¯ 2 tan(2mωλ i sin(2m ωλ ) k=−∞ We get Gc (tb ,rb ,ta ,ra ) =
1 + sγ 0 ∑ 2 s=±1 ·
∞ 0
dλ
eiλ (E
dE 2π
∑ exp i[E(tb − ta) + k(ϑb − ϑa )] k
2 −m2 +2mω ¯ k+2smω )
I|k|
¯ ) sin(2mωλ
(45) mω¯ rb ra mω¯ 2 2 ¯ ) . (r + ra ) cot(2mωλ exp i ¯ ) i sin(2mωλ 2 b
Using now the Hille-Hardy formula [12]
√ 1 ∞ 1 1+t t −α /2 2 xyt t n n!e− 2 (x+y) exp − (x + y) (xy)α /2 Lαn (x) Lαn (y) Iα =∑ 1−t 2 1−t 1−t Γ (n + α + 1) n=0
(46)
and taking t = e−i4λ mω¯ , x = mω¯ ra2 , y = mω¯ rb2 and α = |k|, we obtain a spectral decomposition of Green’s function Gc (tb ,rb ,ta ,ra ): Gc (tb ,rb ,ta ,ra ) = ∑ s
∞ 1 + sγ 0 ∑ 2 ∑ k n=0
∞ 0
dE exp iλ [E 2 − E2 ] exp i[k(ϑb − ϑa ) − E(tb − ta )] 2π (47) n! − m2ω¯ (rb2 +ra2 ) |k| |k| 2 |k| 2 e · (mω¯ ra rb ) Ln (mω¯ ra )Ln (mω¯ rb ), Γ (n + |k| + 1) dλ
S. Haouat and L. Chetouani · Exact Green’s Function for 2D Dirac Oscillator
where E ≡ En,k,s is given by En,k,s = m2 + 2mω¯ [2n + 1 − s + |k| − k]. (48) The quantity 12 1 + sγ 0 can be written in a product form of a spinor u and it’s conjugate u: ¯ 1 + sγ 0 = us u¯s , 2 where
0 , us=−1 = 1
(49) 1 us=+1 = . 0
(50)
Gc (tb ,rb ,ta ,ra ) = ∞
s
k n=0
dE ϕn,l,s (tb ,rb ) ϕ¯ n,l,s (ta ,ra ) , 2π E 2 − E2 + iε
(51)
where
ϕn,k,s (t,r) = us e−iEt eikϑ φn,k (r)
(52)
and the functions φn,k (r) are the radial part of wave functions relative to the nonrelativistic twodimensional radial harmonic oscillator |k| mω¯ 2 √ |k| φn,k (r) = Cn,k e− 2 r mω¯ r Ln mω¯ ra2 (53) with
Cn,k =
n! . Γ (n + |k| + 1)
(54)
Integrating over E, we obtain G (tb ,rb ,ta ,ra ) = c
∞
∞
ε ε (tb ,rb )ψ¯ n,k,s (ta ,ra ), ∑ ∑ Θ [ε (tb − ta)]ψn,k,s
(55)
where Θ (x) is the Heaviside step function and (56)
Now we must determine the Dirac oscillator states by ε acting the operator K+ on the functions ψn,k,s (t,r): ε Ψn,k,s
ε = NK+ ϕn,k,s (t,r) ,
where N is a normalization constant.
ε Ψn,k,s=−1 = i(k+1)ϑ ∂ k (60) ¯ − − m ω r −e r −iε Et ∂ r iNe φn,k (r). eikϑ (−ε E + m)
Using the differential formulas of the functions φn,k (r) (see Appendix), we obtain, finally, our solutions that can be classified as follows: (i) If k is a positive integer, we have ε Ψn,k,s=+1 = (61) (ε E + m)eikϑ φn,k (r) −iε Et iNe , i(k−1) ϑ mω¯ (n + k)φn,k−1 (r) −2 e ε Ψn,k,s=−1 = (62) 2 mω¯ (n + k + 1)ei(k+1)ϑ φn,k+1 (r) iNe−iε Et . ik ϑ (−ε E + m)e φn,k (r) ε ψs=−1
k=−∞ n=0
ε ψn,k,s (t,r) = us e−iε Et eikϑ φn,k (r).
ε Ψn,k,s=+1 = (ε E + m)eikϑ (59) −iε Et φn,k (r), iNe ∂ k i(k−1) ϑ −e + + mω¯ r ∂r r
ε , (ii) If k has a negative value, the two spinors ψs=+1 will be given by
∑ ∑
ε =±1 s=±1
Writing K+ in the form ∂ K+ = γ 0 i +m ∂t γ 1 + iγ 2 −iϑ ∂ 1 ∂ e (58) + imω¯ r i + + 2 ∂r r ∂ϑ γ 1 − iγ 2 iϑ ∂ 1 ∂ + e − imω¯ r , i − 2 ∂r r ∂ϑ ε ε the spinors ψs=+1 and ψs=−1 will be then written as
Then, by integration over λ we get the spectral decomposition of Green’s function:
∑∑ ∑
39
(57)
ε Ψn,k,s=+1 = (63) (ε E + m)eikϑ φn,k (r) iNe−iε Et √ , 2 mω¯ nei(k−1)ϑ φn−1,k+1 (r) ε Ψn,k,s=−1 = (64) ¯ (n + 1)ei(k+1)ϑ φn+1,k−1 (r) −iε Et −2 mω iNe . (−ε E + m)eikϑ φn,k (r)
Let us remark that our spinors are expressed in a fixed Cartesian representation and the passage to the diagonal (rotating) representation, that is used in [3] and [4], can be done by means of the following trans-
S. Haouat and L. Chetouani · Exact Green’s Function for 2D Dirac Oscillator
40
formation:
Ψdiag = where
Appendix: Important Differential Relations √ ε rS(ϑ )−1Ψn,k,s ,
ϑ 1 2 S(ϑ ) = exp − γ γ . 2
(65)
With the use of [12] x
(66)
[1] D. Itˆo, K. Mori, and E. Carreri, Nuovo Cim. A 51, 1119 (1967). [2] M. Moshinsky and A. Szczepaniak, J. Phys. A 22, 817 (1989). [3] V. M. Villalba, Phys. Rev. A 49, 586 (1994). [4] V. M. Villalba and A. R. Maggiolo, Eur. Phys. J. B 22, 31 (2001). [5] N. Ferkous and A. Bounames, Phys. Lett. A 325, 21 (2001). [6] F. M. Gashimzade and A. M. Babaev, Phys. Solid State 44, 162 (2002). [7] E. S. Fradkin and D. M. Gitman, Phys. Rev. D 44, 3220 (1991).
(67)
and k+1 Lkn (x) = Lk+1 n (x) − Ln−1 (x)
5. Conclusion We have solved the problem of the 2D Dirac oscillator in the presence of a constant magnetic field by using the Feynman-Beresin path integrals. In the first stage we have given a pseudoclassical action path integral, where we have described the spin degrees-offreedom by fermionic variables (Grassmannian variables). Then, the integration over odd trajectories has been done easily and the exact Green’s function is calculated in Cartesian coordinates. The passage to the polar coordinates permits to extract the energy spectrum of the electron and the corresponding wave functions. It is obvious that this path integral treatment has two advantages: the first one is to determine the relative Green’s function which leads to the good comprehension of the quantum behavior of this system, and the second one is that the spinors that we extract from the spectral decomposition of Green’s function have simple forms which make them suitable for variational calculations of physical quantities. In conclusion, the list of solvable relativistic problems in (2+1)-dimension by using path integrals has been extended to the Dirac oscillator in the presence of a magnetic field. Let us remark that the problem of the three-dimensional DO is under consideration.
∂ k L (x) = nLkn (x) − (n + k)Lkn−1 (x) ∂x n
we get
∂ |k| + + mω¯ r φn,k (r) ∂r r = 2 mω¯ (n + |k|)φn,k−1 (r),
∂ |k| + − mω¯ r φn,k (r) ∂r r = 2 mω¯ (n + 1)φn+1,k−1 (r), ∂ |k| − + mω¯ r φn,k (r) ∂r r √ = −2 mω¯ nφn−1,k+1 (r),
(68)
(69)
and
∂ |k| − − mω¯ r φn,k (r) ∂r r = −2 mω¯ (n + |k| + 1)φn,k+1 (r).
(70)
(71)
(72)
[8] C. Alexandrou, R. Rosenfelder, and A. W. Schreiber, Phys. Rev. A 59, 3 (1998). [9] F. A. Berezin and M. S. Marinov, JETP Lett. 21, 320 (1975); Ann. Phys. 104, 336 (1977). [10] L. Brink, S. Deser, B. Zumino, P. Di Vecchia, and P. Howe, Phys. Lett. B 64, 435 (1976); L. Brink, P. Di Vecchia, and P. Howe, Nucl. Phys. B 118, 76 (1977). [11] D. M. Gitman and S. I. Zlatev, Phys. Rev. D 55, 7701 (1997); D. M. Gitman, S. I. Zlatev, and W. D. Cruz, Brazilian J. Phys. 26, 419 (1996). [12] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York 1979.
