arXiv:1606.03296v2 [gr-qc] 10 Mar 2017
Relativistic quantum mechanical spin-1 wave equation in 2+1 dimensional spacetime Mustafa Dernek, Semra Gurtas Dogan, Yusuf Sucu, Nuri Unal Abstract. In this study, we introduce a relativistic quantum mechanical wave equation of the spin-1 particle as an excited state of the zitterbewegung and show that it is consistent with the 2+1 dimensional Proca theory. At the same time, we see that in the rest frame this equation has two eigenstates, particle and antiparticle states or negative and positive energy eigenstates, respectively, and satisfy SO(2, 1) spin algebra. As practical applications, we derive the exact solutions of the equation in the presence of a constant magnetic field and a curved spacetime. From these solutions, we find Noether charge by integrating the constructed spin-1 particle current on hyper surface and discuss pair production from the charge. And, we see that the discussion on the Noether charge is useful tool for undersdantding the pair production phenomenon because the charge is derived from a probabilistic particle current. Keywords: Spin-1 particle wave equation; Proca theory; gravity; QED2+1 , pair production, Noether charge. Department of Physics, Faculty of Science, Akdeniz University, 07058 Antalya, Turkey. E-mail:
[email protected]
2 1. Introduction The physics in 2+1 dimensional spacetime presents many interesting and surprising results, both experimentally and theoretically. Therefore, in recent years, 2 + 1 dimensional theories have been widely studied in physical areas, such as gravity, high energy particle theory, condensed matter physics (e.g. monolayer structures), topological field theory, and string theory [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In the view of general relativity, 2+1 dimensional gravity has a number of solutions such as black hole [12], wormhole [13], and propagating gravitational wave [14, 15] . On the other hand, in the view of the 2+1 quantum electrodynamics (QED2+1 ), the properties of an electron in graphene can be described by the 2+1 dimensional Dirac equation [16, 17, 18, 19]. Also, the inter-quark potential in quantum chromodynamics is studied in these dimensions and it has a rich structure, as in the 3+1 dimensional spacetime [20, 21]. Additionally, in the QED2+1 context, massive or massless Dirac equation has provided important physical results in flat [22, 23, 24, 25, 26] and curved backgrounds [27, 28, 29, 30, 31]. Although the physical properties of the Dirac particle has been widely investigated in the context, the physical behavior of the massive or massless spin-1 particle has not almost been considered, quantum mechanically, in this framework, so far. Duffin-Kemmer-Petiau equation as a relativistic quantum mechanical wave equation for the spin-1 and spin-0 particles has a long history [32, 33, 34, 35, 36, 37, 38]. The equation has been discussed in the 3+1 spacetime dimensions whether there exists or not a classical correspondence in the context of Maxwell theory [39]. Also, the relativistic quantum mechanical wave equation for the spin-1 particles is derived as an excited state of zitterbewegung in the same dimensions, but it does not include spin-0 part [40, 41, 42]. The massless case of the equation was derived as a simple model of the zitterbewegung [43] and its equivalence with the Maxwell equations in a flat and curved background were discussed [43, 44]. The spin-1 equation was also solved in the exponentially expanding universe and showed that its solutions are identical with complexified Maxwell equations in the limit m2 → 0 [45]. The Maxwell theory is not valid in 2+1 dimensional spacetime because the magnetic field becomes pseudoscalar. Therefore, in 2+1 dimensional spacetime, the electrodynamical events are, classically, described by the 2+1 dimensional massive Proca gauge theory, which is called Maxwell-Chern-Simons theory, and it is completely different from the Maxwell theory. Also, contrary to even dimensional cases, in the topologically massive gauge theories, the gauge fields, F νρ , and potentials, Aµ , are related to each others as follows: 1 εµνρ F νρ , Aµ = 2m [46, 47, 48, 49, 50, 51] and, at the same time, the spin-1 particle wave equation was discussed in this context. Also, with the pseudoclassical approach, the spin-1 particle wave equation in the 2+1 dimensions was studied by canonical quantization [52, 53, 54]. Nevertheless, it has still been continued some discussions on such an
3 equation in the 2+1 dimensional spacetime structure, physically and mathematically [55, 56, 57, 58]. On all these points of view, a relativistic quantum mechanical wave equation for a spin-1 particle in the 2+1 dimensional spacetime should be expected to satisfy the Proca equation and the relation between vector potentials, Aµ , and tensor fields, F µν , in classical limit. With these motivations, we discuss, in detail, a relativistic quantum mechanical wave equation for a spin-1 particle directly derived from the 2+1 dimensional Barut-Zanghi model [40] and find a relation between this equation and the 2+1 dimensional Proca theory. As a practical application, we also discuss the exact solutions of the wave equation in a constant magnetic field and a curved background. Moreover, we write a current expression of the spin-1 particle in this context and calculate it from these solutions. And, finally, we find the Noether charge derived from the current and discuss pair production in terms of the charge. And, we see that such a discussion is useful tool for understanding the pair production phenomenon since this charge is derived from a probabilistic particle current. The outline of this study is as follows: In Section 2, in the 2+1 dimensional spacetime, we introduce a relativistic quantum mechanical wave equation for a spin1 particle as an excited state of the classical zitterbewegung model and construct its free particle solutions. We also discuss the particle and antiparticle solutions of the equation in the rest frame and show that the equation is equivalent to the Proca equation by defining vector potentials and electromagnetic fields in terms of the components of the spin-1 particle spinor. In Section 3, we obtain the exact solutions of the spin1 particle equation in a constant magnetic field. From these solutions we derive the energy eigenvalues of the particle and write the current densities and Noether charge. In Section 4, we find the exact solutions of the equation in the contracting and expanding 2+1 dimensional curved background. And, we derive the current and Noether charge from these solutions. In Conclusion, we evaluate the results of the study. 2. The spin-1 particle in the 2+1 dimensional flat spacetime A relativistic quantum mechanical wave equation for the spin-1 particle introduced in the 3+1 dimensions was discussed as an excited state of the classical zitterbewegung model [40, 41, 42]. As the classical model of the zitterbewegung [40] in the 2+1 dimensional spacetime, for which its symmetry and integrability properties are investigated [59], is quantized, it directly gives us the following relativistic quantum mechanical wave equation for the evolution of the free spin -1 particle [41]: {(σ µ ⊗ 1 + 1 ⊗ σ µ ) Pµ − (1 ⊗ 1) 2M}α′ α Ψαβ (x) = 0,
(1)
where M and Pµ are the mass and the energy-momentum of the particle, respectively, and σ µ is defined in terms of the Pauli matrices as σ µ = (σ 3 , iσ 1 , iσ 2 ) , and 1 is unit matrix. The matrices σ µ satisfy anti-commutation relations in 2+1 dimensional spacetime: {σ µ , σ υ } = 2ηµν ,
4 where ηµν =diag(1, −1, −1), is Minkowski metric in 2+1 dimensional spacetime. The spinor of the spin-1 particle, Ψαβ (x) , is defined as Ψαβ (r, t) = (ψ+ , ψ0 , ψ0 , ψ− )T .
(2)
where the T means transpose of the row matrix. In this model, the wave function, Ψαβ (x) , is the symmetric spinor with rank 2 and it is represented as a direct product of two Dirac spinors in the 2+1 dimensions and the quantization of the classical system requires that Ψαβ (x) should be symmetric with respect to the indices α, β, where the first (second) indices correspond to the first (second) set of the Dirac matrices. For this reason, Ψαβ (x) has three independent components, ψ+ , ψ0 , ψ− . If we choose a plane wave solutions as follows;
Ψ (r, t) =
φ+ φ0 φ0 φ−
−ip xµ e µ ,
(3)
then, the explicit form of the spin-1 particle equation is written in terms of the spinor components, φ+ , φ0 , φ− : (P0 − M) φ+ + (iP1 + P2 ) φ0 = 0,
(iP1 − P2 ) φ+ + (iP1 + P2 ) φ− − 2Mφ0 = 0,
(4)
(P0 + M) φ− − (iP1 − P2 ) φ0 = 0.
If we add and subtract the first and third rows and organize the second row of Eq (4), the following equations are obtained, respectively, P0 (φ+ − φ− ) + iP1 (2φ0 ) = M (φ+ + φ− ) ,
P0 (φ+ + φ− ) + P2 (2φ0 ) = M (φ+ − φ− ) ,
(5)
iP1 (φ+ + φ− ) − P2 (φ+ − φ− ) = M (2φ0 ) .
And, we reorganize Eq (5) as a first order partial differential equations system: ! ! √ 2φ0 φ+ − φ− 1 0 −∂ √ = M (φ+ + φ− ) , ∂ i √ M M ! ! √ φ+ + φ− 2φ0 2 0 −∂ √ = i M (φ+ − φ− ) , ∂ − √ M M ! ! √ φ+ + φ− φ+ − φ− 1 2 (6) ∂ − √ −∂ i √ = M (2φ0) . M M From this equation system, if we define the complex gauge potentials and fields in terms of the spinor components as 2φ0 φ+ − φ− φ+ + φ− A0 = √ , A1 = i √ , A2 = − √ (7) M M M and √ √ √ F 01 = M (φ+ + φ− ) , F 02 = i M (φ+ − φ− ) , F 12 = M (2φ0) , (8)
5 respectively, then, using the relations given Eqs (7) and (8), we see that Eq (6) satisfies the well known relations between the gauge fields and potentials as follows; ∂ µ Aν − ∂ ν Aµ = F µν .
(9)
According to the definitions in Eqs (7) and (8), then, Eq (6) becomes; ! √ √ 2 φ+ + φ− √ , =M i M ∂0 (φ+ − φ− ) + M ∂1 (2φ0 ) M ! √ √ 2 φ+ − φ− √ =M i M ∂0 (φ+ + φ− ) − i M ∂2 (2φ0 ) , (10) M ! √ √ 2φ 0 , M ∂1 (φ+ + φ− ) + i M ∂2 (φ+ − φ− ) = M 2 √ M and these equations are in the form of the massive Proca equation in the 2+1 dimensional spacetimes: ∂µ F µν + M 2 Aν = 0.
(11)
In particular, from the definitions in Eqs (7) and (8), we notice that the expressions of the vector potentials and fields in Eqs(7) and (8), respectively, are related to each other by the following relations; 1 µνρ Aµ = ǫ Fνρ , (12) 2M where ǫµνρ is the Levi-Civita symbol. These results are consistent with Ref. [51], and also if we eliminate the potentials by using Eq (12) in the Proca equation Eq (11), then, we derive the following equation; M (13) ∂µ F µν + ǫναβ Fαβ = 0. 2 Therefore, we say that these equations are compatible with the results of the topologically massive gauge theories [48, 49, 50, 51]. To discuss the free particle solutions of the spin-1 particle from Eq (4), at first, the components φ+ and φ− are written as π P φ± = (14) e∓i(ϕ+ 2 ) φ0 , P0 ∓ M where tan ϕ = P2 /P1 and P is the magnitude of the momentum vector, P = (P1 , P2 ). Then, substituting φ+ and φ− into the second row of Eq (4), we obtain the following free particle relativistic wave equation for φ0 ;
P02 − P12 − P22 − M 2 φ0 = 0.
(15)
This is the Proca equation for φ0 in the 2+1 dimensions and it has the correct energymomentum condition. Then, the normalized wave function for Eq (3) is obtained as (M + P0 ) e−i(ϕ+ 2 ) 1 P Ψ (r, t) = q P 2 |P0 | M π (M − P ) ei(ϕ+ 2 ) π
0
−ipµ xµ , e
(16)
6 where the normalization condition is given by |P0 | . M In the rest frame, the particle has only two states which are Ψ (r, t)∗T Ψ (r, t) =
Ψ (t) =
and
1 0 0 0
0 0 0 1
Ψ (t) =
−iM t e ,
iM t e ,
f or
f or
(17)
P0 = M
(18)
P0 = −M.
(19)
They are the particle and antiparticle solutions and, at same time, they are eigenstates of spin operator, S12 , with eigenvalues +1 and −1, respectively. On the other hand, Eq (4) can also be rewritten as matrix equation in the following form: (β µ Pµ − M) Ψ = 0, where the complex spinor Ψ has the three components in the representation: T √ Ψ (x1, x2, t) = ψ+ , 2 ψ0 , ψ− ,
and β matrices are Hermitian spin-1 matrices in the 2+1 dimensional spacetime and they can be defined as β0 =
1 0 0 0 0 0 , 0 0 −1
0 1 0 1 1 2 β 1 = √ i 1 0 1 , β = √ 2 0 1 0 2
0 1 0 −1 0 1 . 0 −1 0
(20)
And, it is clear that they satisfy the following SO(2, 1) spin-1 algebra: [β µ , β ν ] = −iǫµνρ βρ . These matrices also satisfy the following relation: ηµν β µ β ν = 2I. To discuss the conserved current for the spin-1 particle, we point out that the Hermitian conjugate of the β µ matrices obeys the following transformation: (β µ ) = γ (β µ )T ∗ γ, where γ is
1 0 0 γ = 0 −1 0 , 0 0 1
(21)
7 and the Hermitian conjugate of the wave function is Ψ = (Ψ)T ∗ γ.
(22)
Then, in this context, the conserved current for the spin-1 particle becomes J µ = Ψβ µ Ψ,
(23)
and the current components in terms of the spin-1 particle spinor components are explicitly given as follows: J 0 = |φ+ |2 − |φ− |2 , √ √ J 1 = i (φ+ + φ− )∗ 2φ0 − i 2φ∗0 (φ+ + φ− ) , √ √ J 2 = (φ+ − φ− )∗ 2φ0 + 2φ∗0 (φ+ − φ− ) .
(24)
By means of the definitions in Eqs (7) and (8), the components of the current may also be written in terms of the Proca fields as 2 2 1 01 (25) F − iF 02 − F 01 + iF 02 , J0 = 4M i ∗ ∗ i h 01 J 1 + iJ 2 = (26) F 01 + iF 02 , F − iF 02 F 12 − F 12 4M where J 0 corresponds to the difference between the right handed and left handed electric fields, J 1 and J 2 are the generalization of the Poynting vectors into the 2+1 dimensional complex fields. Furthermore, the current components can be expressed in terms of the gauge potentials and fields as follows; ∗ i ∗ i h ∗ 10 , (27) + A∗2 F 20 − A2 F 20 A1 F − A1 F 10 J0 = 4 ∗ i ∗ −i h ∗ 01 J1 = , (28) + A∗2 F 21 − A2 F 21 A0 F − A0 F 01 4 ∗ i ∗ −i h ∗ 02 J2 = . (29) + A∗1 F 12 − A1 F 12 A0 F − A0 F 02 4 In particular, it is also important that these results are consistent with the following expression for the conserved current of the Proca Fields in 3+1 dimensions [60, 61, 62, 63]: Jµ = ϕν G∗µν − Gνµ ϕ∗ν . 3. The spin-1 particle in a constant magnetic field The motion of charged particles in a magnetic field is an important problem in classical and quantum electrodynamics. Moreover, in 3+1 dimensions the problem of a charged particle in a constant magnetic field is a 2+1 dimensional problem because of the axial symmetry [64, 65]. Therefore, to discuss the relativistic quantum mechanical behavior of the charged spin-1 particle in a constant magnetic field, we write the relativistic wave equation for the spin-1 particle in the presence of electromagnetic fields as [(σ µ ⊗ I + I ⊗ σ µ ) πµ − 2(I ⊗ I)M] Ψ = 0,
(30)
8 where πµ is the generalized momentum of the particle and the explicit form of its in terms of an electromagnetic potential, Aµ , is πµ = pµ − eAµ , where e is the charge of the particle. Then, Eq (30) becomes
2(π0 − M) i(π1 − iπ2 ) i(π1 − iπ2 ) 0 i(π1 + iπ2 ) −2M 0 i(π1 − iπ2 ) i(π1 + iπ2 ) 0 −2M i(π1 − iπ2 ) 0 i(π1 + iπ2 ) i(π1 + iπ2 ) −2(π0 + M)
ψ+ ψ0 ψ0 ψ−
= 0. (31)
Letting π± = π1 ± iπ2 , and writing π0 , and π± in polar coordinates, (r, θ) , for a constant magnetic field,i.e, A0 = 0, A1 = 0, A2 =Br/2 as follows; π0 = iE, 1 ∂ ∂ √ ∓ i ρ), π± = Mζe−iθ (−i √ ± √ ∂ ρ ρ ∂θ q
2
E eBr eB . where B is magnitude of the constant magnetic field ζ = 2M 2, ǫ = M, ρ = 2 Using separation of variable method, the components of the general wave function can be written as follows;
ψ+ (ρ, θ) = exp(−iEt) exp(i(k + 1)θ)ϕ1 (ρ), ψ− (ρ, θ) = exp(−iEt) exp(i(k − 1)θ)ϕ−1 (ρ),
(32)
ψ0 (ρ, θ) = exp(−iEt) exp(ikθ)ϕ0 (ρ).
And substituting the expressions of π0 , and π± and the components of the general wave function into Eq (31), it becomes !
k 1 (ǫ − 1) d √ ϕ0 (ρ) = 0, ϕ1 (ρ) + 2 ρ + − ζ dρ 2ρ 2 !
k 1 d (ǫ + 1) √ ϕ0 (ρ) = 0, ϕ−1 (ρ) − 2 ρ − + ζ dρ 2ρ 2 √ d ζ ρ[( dρ −
Letting ϕ0 (ρ) =
√1 χ(ρ), ρ
1 (k 2ρ
d − 1) + 21 )ϕ1 (ρ) + ( dρ +
1 (k 2ρ
+ 1) − 21 )ϕ−1 (ρ)] = ϕ0 (ρ). (33)
we get the Whittaker differential equation:
d2 χ(ρ) 1 λ + − + + ρ2 4 ρ
1 4
2
− k4 χ(ρ) = 0. ρ2
(34)
Then, we can construct the general solution in terms of the Whittaker functions, Mλ, k (ρ), as follows; 2
ψ+ ψ0 = Nei(kθ−Et) ψ−
−2ζeiθ ǫ−1
k ρ
−λ) ( k+1 2 − 1 Mλ, k (ρ) + k+1 Mλ, k +1 (ρ) 2 2 √ Mλ, k (ρ)/ ρ 2 k+1 2ζe−iθ ( 2 −λ) Mλ, k +1 (ρ) (ǫ+1) k+1 2
,
(35)
9 2
ǫ 1 where λ− k2 = ǫ4ζ−1 2 − 2 = n+ 2 , and N is the normalization constant. From this solution, the energy eigenvalues are obtained as
E± = M(ζ 2 ±
q
ζ 4 + 1 + 2ζ 2 (2n + 1)),
(36)
Letting the asymptotic form of the solutions in Eq (32) [66], we get the components of the current in Eq (23) as follows; 0
J = |N|
2
Γ(k+1)2 4 2 2n+k−1 ζ ρ Γ(n+k+1)2 M
J 1 + iJ 2 = |N|2
1 Γ(k+1)2 4ζMρ2n+k− 2 Γ(n+k+1)2
n+1+ρ 2 ǫ−1
−
n ǫ+1
2
e−ρ ,
(37)
[2ǫn + (ǫ + 1)(1 + ρ)]e−iθ e−ρ ,
(38)
M (ǫ −1) × ((ǫ+1)2 [(k+1)2 −4+(2n+k+1)(18n+3k+7)]−(ǫ−1) 2 4(n+k+1)2 )1/2 .
(39)
where
|N| =
Γ(n + k + 1) Γ(2n + k)1/2 Γ(k + 1) 2
Under the strong magnetic field, i.e ζ 4 ≫ 1, the energy eigenvalues Eq (36) and the zeroth component of the current Eq (37) become, respectively, E+ ≈ 2Mζ 2 + M(2n + 1) + O(ζ −2),
(40)
E− ≈ −M(2n + 1) + O(ζ −2),
(41)
and J+0 =
2M ζ 2 πΓ(2n + k)
e−ρ ρ2n+k−1 [(n+1+ρ)2 −n2 ] × [(k+1)2 −4+(2n+k+1)(18n+3k+7)−4(n+k+1)2 ] ,
J−0 =
(42)
2M ζ 2 πΓ(2n + k) −ρ 2n+k 2
e ρ n (ρ+2(n+1)) × n2 [(k+1)2 −4+(2n+k+1)(18n+3k+7)−4(n+1) 2 (n+k+1)2 ] .
(43)
On the other hand, according to the weak magnetic field condition, i.e, ζ 4 ≪ 1, the positive and negative energy eigenvalues are E˜+ ≈ M(1 + 2(n + 1)ζ 2) + O(ζ 4),
(44)
E˜− ≈ −M(1 + 2nζ 2 ) + O(ζ 4),
(45)
and, then, the zeroth components of the current for the ± energy eigenvalues become J˜+0 =
2ζ 2 M e−ρ ρ2n+k−1 (n + ρ + 1)2 , πΓ(2n + k)[(k + 1)2 − 4 + (2n + k + 1)(18n + 3k + 7)]
(46)
J˜−0 =
2ζ 2 M e−ρ ρ2n+k−1 n2 , πΓ(2n + k)4(n + k + 1)2
(47)
10 respectively. Letting the currents, J±0 and J˜±0 , the Noether charges of these currents are computed by the following relations; Q± =
Z
(J 0 )± dS0 ,
(48)
where the Q± are Noether charges and the dS0 is a hypersurface [72, 73, 74]. If the currents, J±0 and J˜±0 , are integrated on the hypersurface, dS0 = ρdρdθ, we get the Noether charges, respectively, as follows; 2ζ 2 Q+ = M π (2n+1)+(2n+k)(4n+k+3) × [(k+1)2 −4+(2n+k+1)(18n+3k+7)−4(n+k+1) 2] ,
(49)
2ζ 2 Q− = M π 2
n (2n+k)(4n+k+3) × n2 [(k+1)2 −4+(2n+k+1)(18n+3k+7)]−4(n+1) 2 (n+k+1)2 ,
(50)
Q˜+ 2ζ 2((n + 1)2 + (2n + k)(4n + k + 3)) = , M π((k + 1)2 − 4 + (2n + k + 1)(18n + 3k + 7))
(51)
Q˜− ζ 2n2 = . M 2π(n + k + 1)2
(52)
And, we plot these charges as follows; 4x10 3x10 2x10 1x10
5
QM
5
5
n=2 n=3 n=10 n=50
5
0 -1x10 -2x10 -3x10
Q M 2.5x10 2.0x10
1.5x10
1.0x10
5.0x10
5
5
a)
5
5
n=1 n=1 n=2 n=3 n=10 n=50
Q+ M
5
5
5
4
b)
0 0
200
400
600
800
1000
Figure 1. The graph presents QM± as a function of ζ (ζ 4 ≫ 1, k = 23 ). a) The curve for QM− presents a negative increase for n = 1, and it has a positive maximum value for n = 2. It also decreases with increasing values of n. b) The QM+ curve reaches to a maximum value for n = 1. Increasing n shows a similar behaviour as QM− .
11 1.8x10
1.5x10
-3
~
Q+
n=700
-3
M
~
1.2x10
Q
-3
-
M 9.0x10
6.0x10
3.0x10
-4
-4
a) -4
0
~
Q M
1.8x10
1.5x10
-3
~
n=1
-3
Q
+
M 1.2x10
9.0x10
6.0x10
-3
-4
-4
b) 3.0x10
~
-4
Q-
M
0 0
Figure 2. The
0.02
Q˜± M
0.04
0.06
0.10
is plotted as a function of ζ ( for ζ 4 ≪ 1, k = 23 ).T he
for all of n values get the same values. Then curves approach to Q˜+ M
0.08
Q˜+ M
Q˜− M
Q˜+ M
curves
curves change with increasing n values.
curves. The curves merge for n ≥ 750.
The is plotted as a function of ζ ( for ζ 4 ≪ 1, k = 32 ) for n = 1 and n = 700 in respective panels. Q according to the strong and weak We consider the behaviour of the proportion M magnetic fields, see Figs: 1 and 2. From these figures, we see that the QM+ values coincide with the QM− values as from n ≥ 10 under the strong magnetic field conditions ˜
˜
while the QM+ values coincide with the QM− values as from n ≥ 750 under the weak magnetic field conditions, but, under both conditions, the difference between the QM+ values and the QM− values increase as the n values get smaller. Also, given the Noether charge derived from the probabilistic particle current, we can say that the magnetic field on the particle production is more effective in the small n values accoording to the antiparticle production, but for the large n values it is same. 4. The spin-1 particle in the (2+1) dimensional curved spacetime To discuss the physical and mathematical features of the spin-1 particle in a 2+1 dimensional spacetime curved background, the relativistic quantum mechanical wave equation of the spin-1 particle is easily generalized to a curved spacetime as follows; [(σ µ (x) × 1 + 1 ⊗ σ µ (x)) (Pµ − iΩµ ) − (1 ⊗ 1) 2M] Ψ (x) = 0,
(53)
where σ µ (x) are the spacetime dependent Dirac matrices, Ωµ is the spin connection of the spin-1 particle in 2+1 dimensions and its expression in terms of the spin connection of the spin − 21 particles, Γµ (x), is written as Ωµ = Γµ (x) ⊗ 1 + 1 ⊗ Γµ (x) .
(54)
To derive solutions of the spin-1 particle wave equation in a curved spacetime, as an
12 example, we consider the following 2+1 dimensional gravitational background [27, 67]: h
ds2 = l2 dτ 2 − cosh2 τ dθ2 + sin2 θ dφ2
i
,
(55)
where τ = t/l and τ ∈ (−∞, ∞), θ ∈ [0, π), φ ∈ [0, 2π) and l is the radius of the 1 . Also, the universe is a universe and related to the cosmological constant. Λ, as l = |Λ| 2 two spheres. It contracts to its minimum area of 4πl at τ = 0 and expands again [67]. Then, the metric tensor of the gravitational background and its inverse are written in the following way;
gµν = diag l2 , −l2 cosh2 τ, −l2 cosh2 τ sin2 θ ,
g µν = diag 1/l2 , −1/l2 cosh2 τ, −1/l2 cosh2 τ sin2 θ ,
(56)
g µυ = eµi (x) eνj (x) η ij .
(57)
respectively, and the g µυ is defined in terms of the triad fields, eµi (x) , as follows;
The triads of the curved background are explicitly written as eµi (x) = diag (1/l, 1/l cosh τ, 1/l cosh τ sin θ) .
(58)
The spacetime dependent Dirac matrices, σ µ (x) , are presented in terms of the triads and the constant Dirac matrices, σ i , as follows; σ µ (x) = eµi (x) σ i .
(59)
And, the explicit form of the Γµ (x) is i h 1 Γµ (x) = − gαν Γνβµ σ α (x) , σ β (x) , (60) 8 where Γνβ µ are the Christoffel symbols, and also its components in the curved background [27] are written as
Γ0 = 0, Γ1 = − 21 σ 0 σ 1 sinh τ, Γ2 = − 21 (σ 0 σ 2 sinh τ sin θ + σ 1 σ 2 cos θ).
(61)
Then, Eq (53) in the curved background is as follows; i cosh τ
(∂τ + tanh τ + iMl) ψ+ + i cosh τ
∂
∂θ + i sinφθ + cot θ ψ+ + i cosh τ
∂θ +
∂ i sinφθ
i cosh τ
∂
∂θ − i sinφθ ψ0 = 0,
∂
∂θ − i sinφθ + cot θ ψ− = −2iMlψ0 ,
(62)
ψ0 − (∂τ + tanh τ − iMl) ψ− = 0.
To find the general solution, we use the separation of variable method. In this connection, the rising and lowering operators of the spin-1 particle, ∂± , are defined as !
1 0 ∂φ + ∂± = ∓∂θ + i σ ⊗ 1 + 1 ⊗ σ 0 cot θ . sin θ 2
(63)
and their eigenfunctions in terms of the rotation group djλ,m (θ) are expressed as j Dλ,m (θ, φ) = hλ|R (θ, φ) |jmi = eiλφ djλ,m (θ) .
(64)
13 And, the irreducible representations of the rotation group djλ,m (θ) are given by djλ,m (θ) =
v
j−m u u
(−1) t (j + λ)! (j + m)! sin (θ/2)2j cot (θ/2)λ+m (λ + m)! (j − λ)! (j − m)!
×2 F1 λ − j, m − j; λ + m + 1; − cot2 (θ/2)
[71]. When the separation of variable method is employed, Ψ (τ, θ, φ) can be expanded in terms of the angular momentum eigenfunctions as j F (τ ) D+1,m (θ, φ) X (2j + 1) + j Ψ (τ, θ, φ) = 4π F0 (τ ) D0,m (θ, φ) j jm cosh τ F− (τ ) D−1,m (θ, φ)
(65)
j and the ∂± operators act on Dλ,m (θ, φ) as follows;
q
j ∂± Dλ,m (θ, φ) =
j (j ± λ + 1) (j ∓ λ)Dλ±1,m (θ, φ) .
Then, Eq (62) is rewritten as
(iMl + ∂τ ) F+ (τ ) − (iMl − ∂τ ) F− (τ ) +
MlF0 (τ ) −
1 2 cosh τ
q
q
i j cosh τ q i j cosh τ
(66)
(j + 1)F0 (τ ) = 0, (67)
(j + 1)F0 (τ ) = 0,
j (j + 1) (F− (τ ) − F+ (τ )) = 0.
From these equations, the F0 (τ ) can be defined as q 1 F0 (τ ) = j (j + 1) (F− (τ ) − F+ (τ )) . (68) 2Ml cosh τ and substituting it into the first and second rows of Eq (67), we get the following equations for the F± (τ ) : "
∂τ2
#
j (j + 1) F± (τ ) = 0. + 2 (tanh τ ) ∂τ ± 2iMl tanh τ + M l + cosh2 τ 2 2
(69)
These differential equations are satisfied the following solutions [68]: F+ (τ ) =
1−tanh τ 2
F− (τ ) = −
j+1
C+ ξ (τ )
1−tanh τ 2
j+1
l − iM 2
D+ ξ (τ )−
F1 (τ ) + D− ξ (τ )
iM l +1 2
iM l +1 2
F3 (τ ) + C− ξ (τ )
iM l 2
(70)
(71)
F2 (τ ) , F4 (τ ) ,
where the Fi (τ ) functions are an abbreviations of the Hypergeometric functions as follows; F1 (τ ) = 2 F1 (−j − iMl, −j − 1; −iMl; −ξ (τ )) , F2 (τ ) = 2 F1 (−j + iMl, −j + 1 ; 2 + iMl; −ξ (τ )) , F3 (τ ) = 2 F1 (−j − iMl, −j + 1 ; 2 − iMl; −ξ (τ )) , F4 (τ ) = 2 F1 (−j + iMl, −j − 1; iMl; −ξ (τ ))
1+tanh τ . Also, the coefficients C± and D± are related each other by the and ξ (τ ) = 1−tanh τ following relations:
Ml ±
C± = −
j+
1 2
i 2
2
2
+
−
1 4
1 4
D±
(72)
14 To construct the wave function when τ goes −∞, we consider the asymptotic forms of F+ (τ ) , F0 (τ ) and F− (τ ) in Eq (68), Eq (70) and Eq (71), then, the wave function is written as
Ψ (τ, θ, φ) =
X q
τ
2 2j + 1e
jm
j C+ e−iM lτ + e2τ D− eiM lτ D+1,m (θ, φ)
√
j(j+1)
− M l eτ {F+ e−iM lτ j +F− eiM lτ }D0,m (θ, φ)
j − e2τ D+ e−iM lτ + C−eiM lτ D−1,m (θ, φ)
,
(73)
where F+ = C+ + e2τ D+ and F− = C− + e2τ D− . The asymptotic expressions of the particle Ψ+ and antiparticle Ψ− solutions can be written separately as the following
q
Ψ+ = 2 2j + 1eτ −
q
Ψ− = 2 2j + 1eτ −
√
√
j C+ D+1,m (θ, φ) j(j+1) τ j e (C+ + e2τ D+ )D0,m Ml j −D+ e2τ D−1,m (θ, φ) .
(74)
(75)
−iM lτ , (θ, φ) e
j D− e2τ D+1,m (θ, φ)
iM lτ j , + e2τ D− )D0,m (θ, φ) e j −C− D−1,m (θ, φ)
j(j+1) τ e (C− Ml
where C+ and C− are normalization coefficients, and they are calculated as |C+ | = |C− | =
1 l
h
1−
2
2
j(j+1) 2τ e M 2 l2
e4τ − + 3 M 2jl2 (j+1) (1+M 2 l2 )
j 3 (j+1)3 e6τ M 4 l4 (1+M 2 l2 )
i−1/2
. (76)
To discuss the particle creation in this background, we construct the particle and the antiparticle solutions as τ → ∞. As τ goes to ∞, the wave function is ˜ (τ, θ, φ) = Ψ
X q
−τ
2 2j + 1e
jm
j ˜ − eiM lτ D+1,m C˜+ e−iM lτ + e−2τ D (θ, φ)
q
− M1 l j (j + 1)e−τ {F˜+ e−iM lτ j +F˜− eiM lτ }D0,m (θ, φ) j − e−2τ D˜ + e−iM lτ + C˜− eiM lτ D−1,m (θ, φ)
,
(77)
˜ + and F˜− = C˜− + e−2τ D ˜ − . The asymptotic expressions of the where F˜+ = C˜+ + e−2τ D particle + Ψ and anti-particle − Ψ solution to be written separately as the following j ˜+ D+1,m C (θ, φ) q √j(j+1) −iM lτ −τ j ˜ −τ −2τ ˜ , (78) + Ψ = 2 2j + 1e + )D0,m (θ, φ) e − M l e (C˜+ + e D j −2τ ˜ −e D+ D−1,m (θ, φ) . q
˜ = 2 2j + 1e−τ −
−Ψ
√
j ˜ − e−2τ D+1,m D (θ, φ)
j(j+1) −τ e (C˜− Ml
iM lτ j ˜ − )D0,m , + e−2τ D (θ, φ) e
j −C˜− D−1,m
(79)
(θ, φ)
˜ + = b1 D+ and D ˜ − = a2 D− . Here C˜+ and C˜− are where C˜+ = a1 C+ , C˜− = b2 C− , D normalization coefficients and the a1 , b1 , a2 and b2 are a1 =
Γ(−iM l)Γ(−iM l+1) , Γ(−iM l−j)Γ(−iM l+j+1)
a2 =
Γ(2+iM l)Γ(iM l−1) , Γ(−j+iM l)Γ(j+iM l+1)
b1 =
Γ(2−iM l)Γ(−iM l−1) , Γ(−iM l−j)Γ(−iM l+j+1)
b2 =
Γ(iM l)Γ(iM l+1) . Γ(−j+iM l)Γ(j+iM l+1)
15 Letting Eq (78) and Eq (79), we obtain the normalization coefficients as follows ˜ C+
= C˜− =
1 l
h
1−
j(j+1) −2τ e M 2 l2
2
2
e−4τ − + 3 M 2jl2 (j+1) (1+M 2 l2 )
j 3 (j+1)3 M 4 l4 (1+M 2 l2 )
e−6τ
i− 12
.(80)
The current for the spin-1 particle in 2+1 dimensional curved spacetime can be computed by using Eq (20) and Eq (23). Therefore, using Eq (74), Eq (75), Eq (76) and Eq (23), the zeroth component of the particle current, (J+0 ), and the antiparticle current, (J−0 ), are obtained by the following away; 1 j j (J 0 )± = ± 4e2τ (2j + 1) {|C± |2 D±1,m (θ, φ)∗ D±1,m (θ, φ) l j j − |D± |2 e4τ D∓1,m (θ, φ)∗ D∓1,m (θ, φ)},
(81)
and, using the orthogonality relation for the rotation group [71], then, the Noether charges from these currents are calculated as Q± =
Z
(J 0 )± dS0
j 2 (j + 1)2 1 e4τ =± 1− 2 2 2 2 l M l (1 + M l ) h j(j+1) j 2 (j+1)2 × 1 − M 2 l2 e2τ + 3 M 2 l2 (1+M 2 l2 ) e4τ − "
#
j 3 (j+1)3 M 4 l4 (1+M 2 l2 )
e6τ
!
j (j + 1) 2τ 1 ∼ 1+ e + O e4τ , =± l 4M 2 l2
i−1
(82)
where dS0 = l2 cosh2 τ sin θdθdφ. In the same way, from Eq (78), Eq (79) and Eq (23), the zeroth components of currents of particle (J˜+0 ) and antiparticle (J˜−0 ) as τ goes to ∞ are obtained as the following way: 2 n 1 j j (J˜0 )± = ± 4e−2τ (2j + 1) |C˜± D±1,m (θ, φ)∗ D±1,m (θ, φ) l j j ˜ ± 2 e−4τ D∓1,m (θ, φ)∗ D∓1,m (θ, φ)}, − D
(83)
and, from these currents, the Noether charges for the particle and antiparticle are calculated as ˜± = Q
Z
(J˜0 )± dS0
j 2 (j + 1)2 1 e−4τ =± 1− 2 2 l M l (1 + M 2 l2 ) h
"
× 1−
j(j+1) −2τ e M 2 l2
2
#
2
+ 3 M 2jl2 (j+1) e−4τ − (1+M 2 l2 ) !
1 j (j + 1) −2τ −4τ ∼ 1+ . e + O e =± l 4M 2 l2
j 3 (j+1)3 M 4 l4 (1+M 2 l2 )
e−6τ
i−1
(84)
16 ˜
These results show that as τ goes to ±∞, Q|Λ|± and Q|Λ|± , respectively, converge to ±1. This means that the universe only is composed of the ± |Λ| vacuum energies in its beginning and ending time. 5. Concluding remarks In this study, we have introduced a relativistic quantum mechanical wave equation of the spin-1 particle as an excited state of the zitterbewegung. This wave function has three independent components. And letting a complex vector potential, Aµ , and fields, F µν , in terms of three components, we show that the equation is consistent with the 2+1 dimensional Proca theory. At the same time, we see that this equation has two eigenstates, particle and antiparticle states or negative and positive energy eigenstates, respectively, in the rest frame and satisfy SO(2, 1) spin algebra. Apart from the free particle solution of the equation, we have derived the exact solutions of it in presence of the constant magnetic field and the curved background. From these solutions, we have constructed the current components and observed a spin-1 particle current in presence of a constant magnetic field that decreases by e−ρ , while the spin-1 particle current in the curved spacetime oscillates in time, which means that there are temporary particle creations. On the other hand, from the currents, we evaluate the Noether charges. And, we see that the QM+ values coincide with the QM− values as from n ≥ 10 under the ˜
˜
strong magnetic field conditions, while the QM+ values coincide with the QM− values as from n ≥ 750 under the weak magnetic field conditions, in this situation, the presence of magnetic field triggers the particle production more than antiparticle production in the small n values, but in the large values, the particle and antiparticle production are same. Also, in the curved background, as τ goes to ±∞, the Q|Λ|± and Q|Λ|± , respectively, converge to ±1. So, it can be said that, in the beginning and ending of the time, the universe may have, completely, been composed of the particle and antiparticle with the positive and negative |Λ| vacuum energies, respectively. Finally, besides to the results obtained in this study, we think that the consistent spin-1 particle wave equation in the 2+1 dimensional spacetime gains much interest in physical area [75]. Acknowledgments This work was supported by the Scientific Research Projects Unit of Akdeniz University. References [1] [2] [3] [4] [5]
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