S. P. Gavrilov, D. M. Gitman, and A. A. Smirnov
Citation: Journal of Mathematical Physics 45, 1873 (2004); doi: 10.1063/1.1699483 View online: http://dx.doi.org/10.1063/1.1699483
View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/45/5?ver=pdfcov Published by the AIP Publishing
Green functions of the Dirac equation with
magnetic-solenoid field
S. P. Gavrilov,a)D. M. Gitman,b)and A. A. Smirnovc)
Institute of Physics, University of Sao Paulo, P. O. Box 66318, 05315-970 Sao Paulo, Brazil
共Received 10 October 2003; accepted 19 February 2004; published 2 April 2004兲
Various Green functions of the Dirac equation with a magnetic-solenoid field共the superposition of the Aharonov–Bohm field and a collinear uniform magnetic field兲 are constructed and studied. The problem is considered in 2⫹1 and 3⫹1 dimen-sions for the natural extension of the Dirac operator共the extension obtained from the solenoid regularization兲. Representations of the Green functions as proper time integrals are derived. The nonrelativistic limit is considered. For the sake of com-pleteness the Green functions of the Klein–Gordon particles are constructed as well. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1699483兴
I. INTRODUCTION
In the present paper we continue our previous study1–3of the Dirac equation with a magnetic-solenoid field, constructing and studying various Green functions of this equation. We recall that the magnetic-solenoid field is the collinear superposition of the constant uniform magnetic field and the Aharonov–Bohm 共AB兲 field. The AB field is a field of an infinitely long and infinitesi-mally thin solenoid. Recently the interest in such a field configuration has been renewed in connection with planar physics problems, quantum Hall effect, and the Aharonov–Bohm effect in cyclotron and synchrotron radiations.4 –9
In principle, the Green functions can be constructed whenever complete sets of solutions of the Dirac equations are available. In this connection, one should recall that solutions of the Dirac equation with the magnetic-solenoid field in 2⫹1 and 3⫹1 dimensions were obtained in Ref. 1. The singularity of the AB field demands a special attention to the correct definition of the Dirac operator. The need for self-adjoint extensions in the case of the Dirac Hamiltonian with the pure AB field in 2⫹1 dimensions was recognized in Refs. 10 and 11 where certain boundary condi-tions at the origin were established. The regularized case and peculiarities of the behavior of a spinning particle in the presence of the magnetic string were considered in Refs. 12 and 13. The problem of the self-adjoint extension of the Dirac operator with the magnetic-solenoid field was studied in Refs. 2, 3, and 14. In 2⫹1 dimensions, a one-parametric family of self-adjoint Dirac Hamiltonians specified by the corresponding boundary conditions at the AB solenoid was con-structed, and the spectrum and eigenfunctions for each value of the extension parameter were found. In 3⫹1 dimensions, a two-parametric family of the self-adjoint Dirac Hamiltonians was constructed on the condition that the spin polarization is conserved. The corresponding spectrum and eigenfunctions for each value of the extension parameters were found as well. In Refs. 2 and 3 the procedure of solenoid regularization was also considered. The procedure implies considering the finite solenoid and then making its radius go to zero. This procedure specifies some particular boundary conditions. The values of the extension parameters corresponding to the solenoid regu-larization case were determined in 2⫹1 and 3⫹1 dimensions. Further, we call the corresponding extension the natural extension. Nonrelativistic propagators for the spinless and spin-1/2 particle a兲Present address: Departamento de Fisica e Quimica, UNESP, Campus de Guaratingueta, Brazil. On leave from Tomsk
State University, 634041, Russia. Electronic mail: [email protected] b兲Electronic mail: [email protected]
c兲Electronic mail: [email protected]
1873
moving in the pure AB field were considered mainly in the relation to the AB effect. The propa-gator of the spinless particle was found in Refs. 15, 16, and 17 as a sum of partial propapropa-gators corresponding to homotopically different paths in the covering space of the physical background. The nonrelativistic propagator of the spin-1/2 particle in the AB field for a particular value of the self-adjoint extension parameter was discussed in Ref. 18. The relativistic scalar case for the AB field was studied in Ref. 19. The propagators and the AB effect in general gauge theories were considered in Refs. 20 and 21. Recently, vacuum polarization effects in the AB field have aroused great interest, see, for example, Refs. 22 and 23 and references therein.
In the present paper, we construct and study the Green functions of the Dirac particle in the magnetic-solenoid field in 2⫹1 and 3⫹1 dimensions. The physical importance of the problem is stressed by the fact that the knowledge of the Green functions in such a configuration allows one to study quantum共and quantum field兲 effects in the magnetic-solenoid field on a regular base. A technical specificity of the problem is related to the necessity to take into account all the pecu-liarities related to the self-adjoint extension problem of the Dirac operator in the background under consideration. In Sec. II we consider the (2⫹1)-dimensional case in detail. Here, constructing the Green functions, we use the exact solutions of the Dirac equation that are related to the specific values of the extension parameter. These values correspond to the natural extension, see above. The representations of the Green functions as proper time integrals are derived. In addition, we calculate the nonrelativistic Green functions as well. In Sec. III we extend the results to the (3
⫹1)-dimensional case. In the Appendix, for the sake of completeness, we present the Green
functions of the relativistic scalar particle.
We note that the magnetic-solenoid field belongs to such type of fields that do not violate the vacuum stability. For such fields a unique stable vacuum exists, and quantum field definitions of the Green functions below hold true.24 In particular, the causal propagator Sc(x,x
⬘
) and the anticausal propagator Sc¯(x,x⬘
) are defined by the expressionsSc共x,x
⬘
兲⫽i具
0兩Tˆ共x兲C 共x⬘
兲兩0典
, 共1兲 S¯c共x,x⬘
兲⫽i具
0兩ˆ共x兲C 共x⬘
兲T兩0典
, 共2兲 whereˆ (x) is the quantum spinor field in the Furry representation, satisfying the Dirac equation with the magnetic-solenoid field, 兩0典 is the vacuum in this representation. The symbol of the T-product acts on both sides: it orders the field operators to its right-hand side and antiorders them to its left-hand side. The functions Sc(x,x⬘
), S¯c(x,x
⬘
) can be expressed via the functions S⫿(x,x⬘
),Sc共x,x
⬘
兲⫽共⌬x0兲S⫺共x,x⬘
兲⫺共⫺⌬x0兲S⫹共x,x⬘
兲, ⌬x0⫽x0⫺x0⬘
, 共3兲 S¯c共x,x⬘
兲⫽共⫺⌬x0兲S⫺共x,x⬘
兲⫺共⌬x0兲S⫹共x,x⬘
兲, 共4兲 and the latter can be calculated via a complete set⫾a(x) of solutions of the Dirac equation withthe magnetic-solenoid field as
S⫿共x,x
⬘
兲⫽i兺
a ⫾a共x兲⫾
¯a共x
⬘
兲. 共5兲The solutions with the subscript共⫹兲 belong to the positive energy spectrum, whereas the solutions with the subscript共⫺兲 belong to the negative energy spectrum. Via a all possible quantum num-bers are denoted.
The Dirac equation with the magnetic-solenoid field has the form
共␥P
Here P⫽i⫺qA(x), x⫽(x), q is an algebraic charge, for electrons q⫽⫺e⬍0, M is the electron mass, and A(x) are potentials of the magnetic-solenoid field. In the (3⫹1)-dimensional case ⫽0,1,2,3 and ␥ are the corresponding gamma-matrices. In the (2⫹1)-dimensional case ⫽0,1,2 and in what follows, we employ the letter ⌫ to denote the gamma-matrices. We use for
these matrices the following representation:
⌫0⫽3, ⌫1⫽i2, ⌫2⫽⫺i1,
where i are the Pauli matrices. In cylindric coordinates (,r), x1⫽r cos, x2⫽r sin, the potentials of the magnetic-solenoid field have the form
A0⫽0, eA1⫽关l0⫹⫹A共r兲兴 sin r , eA2⫽⫺关l0⫹⫹A共r兲兴 cos r , 共A3⫽0 in 3⫹1兲, A共r兲⫽eBr2/2. 共7兲
Here B is the magnitude of the uniform magnetic field, and the magnitude BABof the AB field is given by the expression BAB⫽⌽␦(x1)␦(x2), where ⌽ is the AB-solenoid flux, (l0⫹)
⫽⌽/⌽0, ⌽0⫽2/e. It is supposed that l0 is integer and 0⭐⬍1.
The functions S⫿(x,x
⬘
) obey the Dirac equation共6兲, whereas the causal and anticausal propa-gators obey the nonhomogeneous Dirac equations:共␥P
⫺M 兲Sc共x,x
⬘
兲⫽⫺␦共x⫺x⬘
兲 , 共␥P⫺M 兲S¯c共x,x⬘
兲⫽␦共x⫺x⬘
兲.We note that the commutation function S(x,x
⬘
), the advanced Sadv(x,x⬘
) and the retarded Sret(x,x⬘
) Green functions can be expressed in terms of Sc(x,x⬘
), S¯c(x,x⬘
) as follows:S共x,x
⬘
兲⫽S⫺共x,x⬘
兲⫹S⫹共x,x⬘
兲⫽sgn共⌬x0兲关Sc共x,x⬘
兲⫺S¯c共x,x⬘
兲兴, 共8兲 Sadv共x,x⬘
兲⫽⫺共⫺⌬x0兲S共x,x⬘
兲, Sret共x,x⬘
兲⫽共⌬x0兲S共x,x⬘
兲. 共9兲II. 2¿1 DIMENSIONAL CASE A. Sets of exact solutions
First we study the (2⫹1)-dimensional case, for which, as known,2,3the Dirac operator with the magnetic-solenoid field in 2⫹1 dimensions possesses a one-parameter family of self-adjoint extensions. That provides a one-parameter family of boundary conditions at the origin. Following Refs. 2 and 3, we denote the extension parameter as⌰. Generally speaking, the AB symmetry is violated for the spinning particle, which is therefore sensible to the solenoid flux sign. As was demonstrated in Refs. 2 and 3, the values ⌰⫽⫾/2 correspond to the natural extension, ⌰
⫽⫺/2 if the flux is positive and ⌰⫽/2 if the flux is negative. Below we present a set of solutions ⫾a(x) of 共6兲 which we will use for Green function construction according to the
formulas共5兲. We consider the problem separately for two values of the extension parameter. We start with the case ⌰⫽⫺/2. The positive energy spectrum is given by ⫹ and the negative energy spectrum is given by⫺ ,
⫹⫽⫺⫺⫽
冑
M2⫹. 共10兲Both branches are determined by the spectrum of the quantity which is defined below. The solutions⫾a(x) can be expressed via the solutions u(x) of the squared Dirac equation. The latter solutions have the form
⫾um,l,共x兲⫽e⫺i⫾x
0
um,l,共x⬜兲,
共11兲
where um,l,共x⬜兲⫽
冑
␥gl共兲m,l,共r兲, l⫽0, um,0,⫹1共x⬜兲⫽冑
␥g0共兲m,0,⫹1共r兲⫹1, um,0,⫺1共x⬜兲⫽冑
␥g0共兲m,ir⫺1共r兲⫺1, ␥⫽e兩B兩, and gl共兲⫽ 1冑
2 exp再
i冋
l⫺l0⫺ 1 2共1⫹ 3兲册冎
, ⫹1⫽冉
1 0冊
, ⫺1⫽冉
0 1冊
.The functionsm,l,(r), m,ir⫺1(r) are expressed via the Laguerre functions Im⫹␣,m() as m,l,共r兲⫽Im⫹兩兩,m共兲 , m,⫺1
ir 共r兲⫽I
m⫺,m共兲,
共12兲
⫽␥r2/2 , ⫽⫹l⫺共1⫹兲/2.
We recall that the Laguerre functions Im⫹␣,m() are related to the Laguerre polynomials Lm␣(x)
关8.970, 8.972.1 共Ref. 25兲兴 as
Im⫹␣,m共x兲⫽
冑
m!⌫共m⫹␣⫹1兲e⫺x/2x␣/2Lm␣共x兲.
For the magnetic field B⬎0, the spectrum ofcorresponding to the functions um,l,(x⬜) is
⫽
再
2␥共m⫹l⫹兲, l⫺共1⫹兲/2⭓0,2␥共m⫹共1⫹兲/2兲, l⫺共1⫹兲/2⬍0, 共13兲
except the functions um,0,⫺1(x⬜) for which the spectrum ofis
⫽2␥m. 共14兲
Then the complete set⫾a with a⫽(m,l) has the form
⫾m,l共x兲⫽N共⌫P⫹M 兲⫾um,l,⫺1共x兲. 共15兲
The latter form provides correct expressions both for ⫽0 and ⫽0, since the states with
⫽0 can only be expressed in terms of the spinors with ⫽⫺1 关we note that⫹⬅0 for ⫽0,
nevertheless it is convenient to remain in共11兲 u⫹ with⫹⫽M]. The normalization factor with respect to the usual inner product (,
⬘
)⫽兰†(x)⬘
(x)dx readsN⫽
再
关2兩⫾兩共兩⫾兩⫺M 兲兴⫺1/2, ⫽0,
关2M兴⫺1, ⫽0.
The quantum number l characterizes the angular momentum of the particle, m is the radial quantum number, see Ref. 1.
For B⬍0 the spectrum of states differs nontrivially from the expressions given by Eqs. 共13兲 and共14兲. Herecorresponding to um,l,(r) is
⫽
再
2␥共m⫺l⫹1⫺兲, l⫺共1⫹兲/2⬍0,2␥共m⫹共1⫺兲/2兲, l⫺共1⫹兲/2⭓0, 共16兲
except the functions um,0,⫺1(x⬜) for which the spectrum ofis
⫽2␥共m⫹1⫺兲. 共17兲
Now we go to the case with the extension parameter⌰⫽/2. We recall that one needs for self-adjoint extensions of the radial Dirac Hamiltonian only in the subspace l⫽0 to which we refer to as the critical subspace. Thus, the only solutions in the l⫽0 subspace must be subjected to the one of asymptotic condition from a one-parametric family of boundary conditions as r→0. By this reason for ⌰⫽/2, the solutions only differ from共11兲 in the subspace l⫽0,
um,0,⫹1共x⬜兲⫽
冑
␥g0共兲irm,⫹1共r兲⫹1, m,ir⫹1共r兲⫽Im⫹⫺1,m共兲,共18兲
um,0,⫺1共x⬜兲⫽
冑
␥g0共兲m,0,⫺1共r兲⫺1,where the spectrum for um,0,⫹1(x⬜) is given as
⫽2␥共m⫹兲, B⬎0, 共19兲
⫽2␥m, B⬍0. 共20兲
B. Construction of Green functions
The main point in constructing the Green functions is the summations in the representation
共5兲. In the case under consideration, this summation can be done with the help of special relations
which can be established for the solutions of the Dirac equation.
Let us start with the calculation of the Green functions for the extension parameter ⌰
⫽⫺/2 and B⬎0. In this case, taking into account that the eigenfunctions u of the equation
关(⌫P⬜)2⫹兴u⫽0 corresponding to any⫽0 obey the equations
⌫P⬜ ⫾um⫹,l,⫺共x兲⫽⫺i
冑
⫾um⫺,l,共x兲, l⭐0 , P⬜⫽共0,P1, P2兲,共21兲 ⌫P⬜ ⫾um,l,⫺共x兲⫽i
冑
⫾um,l,共x兲, l⭓1, m⫾⫽m⫹共1⫾兲/2,and the explicit form of the solutions ⫾m,l, one can verify that for 兩兩⫽M the following
relations hold true:
⫾m,l共x兲⫾¯m,l共x
⬘
兲⫽共⌫P⫹M 兲 1 2⫾e ⫺i⫾⌬x0兺
⫽⫾1m⫺,l,共x⬜,x⬜⬘
兲⌶, l⭐0, 共22兲 ⫾m,l共x兲⫾¯m,l共x⬘
兲⫽共⌫P⫹M 兲 1 2⫾e ⫺i⫾⌬x0兺
⫽⫾1m,l,共x⬜,x⬜⬘
兲⌶, l⭓1, where m,l,共x⬜,x⬜⬘
兲⫽ ␥ 2e i[l⫺l0⫺(1⫹)/2]⌬I m⫹␣,m共兲Im⫹␣,m共⬘
兲, 共23兲 ⌬⫽⫺⬘
, ␣⫽再
⫹l⫺共1⫹兲/2, l⭓1, ⫺关⫹l⫺共1⫹兲/2兴, l⭐0, ⌶⫾1⫽共1⫾3兲/2.The above relations and Eqs. 共3兲 and 共5兲 allow us to represent the causal Green function in the following form: Sc共x,x
⬘
兲⫽共⌫P⫹M 兲⌬c共x,x⬘
兲, 共24兲 ⌬c共x,x⬘
兲⫽i兺
m,l,冋
共⌬x 0兲e ⫺i⫹⌬x0 2⫹ ⫺共⫺⌬x 0兲e ⫺i⫺⌬x0 2⫺册
m,l,共x⬜,x⬜⬘
兲⌶. Then we can use the representations共⌬x0兲e ⫺i⫹⌬x0 2⫹ ⫺共⫺⌬x 0兲e ⫺i⫺⌬x0 2⫺ ⫽ 1 2i
冕
⫺⬁ ⬁ e⫺ip0⌬x0 2⫺p 0 2⫺i⑀dp0, 共25兲 1 2⫺p 0 2⫺i⑀⫽i冕
0 ⬁ e⫺i(2⫺p0 2)s ds, 共26兲in Eq.共24兲. Integrating over p0, we obtain finally
⌬c共x,x
⬘
兲⫽冕
0 ⬁ f共x,x⬘
,s兲ds, 共27兲 f共x,x⬘
,s兲⫽ 1 2共s兲1/2e 共⫺i⌬x02/4s兲 ei/4e⫺iM2si兺
m,l, e⫺ism,l,共x⬜,x⬜⬘
兲⌶.The path of the integration over s is deformed so that it goes slightly below the singular points sk⫽k/␥, k⫽1,2, . . . .
Using共5兲, 共22兲, and the representation
⫺共⫺⌬x0兲e ⫺i⫹⌬x0 2⫹ ⫹共⌬x 0兲e ⫺i⫺⌬x0 2⫺ ⫽ 1 2i
冕
⫺⬁ ⬁ e⫺ip0⌬x0 2⫺p 0 2⫹i⑀dp0, 共28兲 1 2⫺p 0 2⫹i⑀⫽i冕
⫺0 ⫺⬁ e⫺i(2⫺p0 2)s ds,instead of共25兲 and 共26兲 we obtain from 共4兲,
S¯c共x,x
⬘
兲⫽共⌫P⫹M 兲⌬¯c共x,x⬘
兲, ⌬¯c共x,x⬘
兲⫽冕
⫺0 ⫺⬁
f共x,x
⬘
,s兲ds, 共29兲 where f (x,x⬘
,s) is given by Eq.共27兲. The negative values for s are defined as s⫽兩s兩e⫺i, and the path of integration over s is deformed so that it goes slightly below the singular points⫺sk.We now consider the summations in共27兲. Applying the formula 关8.976共1兲 共Ref. 25兲兴 we can sum over m to get
兺
m⫽0
⬁
e⫺i2m␥sIm⫹␣,m共兲Im⫹␣,m共
⬘
兲⫽exp再
i 2共⫹
⬘
兲cot共␥s兲冎
ei␣␥sei␥s 2i sin共␥s兲e ⫺ i␣/2J ␣共z兲, z⫽冑
⬘
/sin共␥s兲, 共30兲where J␣(z) are the Bessel functions 关8.402 共Ref. 25兲兴, and for negative s we take arg s⫽⫺
Similar results can be obtained for the case B⬍0. Here one should use the solutions corre-sponding to the spectrum of共16兲 and 共17兲. Then these results can be united to obtain expressions which hold true for any sign of B,
f共x,x
⬘
,s兲⫽兺
l⫽⫺⬁ ⬁ fl共x,x⬘
,s兲, fl共x,x⬘
,s兲⫽A共s兲兺
⫽⫾1⌽l,共s兲e ⫺ieBs⌶ , 共31兲 A共s兲⫽ eB83/2s1/2sin共eBs兲exp
再
i 4 ⫺iM 2s⫺il 0⌬冎
⫻exp再
⫺i共⌬x0兲2 4s ⫹ ieB 4 共r 2⫹r⬘
2兲cot共eBs兲冎
,⌽l,共s兲⫽eil⌬e⫺i(l⫹)eBse⫺ i兩l⫹兩/2J兩l⫹兩共z兲, l⫽l⫺共1⫹兲/2, l⫽0,
共32兲 ⌽0,⫹1共s兲⫽e⫺i⌬ei(1⫺)eBse⫺ i(1⫺)/2J1⫺共z兲, ⌽0,⫺1共s兲⫽e⫺ieBsei/2J⫺共z兲.
Now we consider the summation over l. One can see that the following relations hold true:
兺
l⫽1 ⬁ ⌽l,⫺1共s兲⫽兺
l⫽1 ⬁⌽l⫹1,⫹1共s兲⫽e⫺ieBsY共z,⌬⫺eBs,兲,
兺
l⫽⫺1 ⫺⬁ ⌽l,⫺1共s兲⫽兺
l⫽⫺1 ⫺⬁⌽l⫹1,⫹1共s兲⫽e⫺ieBsY共z,⫺⌬⫹eBs,⫺兲,
where
Y共z,,兲⫽a1共z兲⫹Y˜共z,,兲, Y˜共z,,兲⫽
兺
l⫽2
⬁
al共z兲, al共z兲⫽eil共⫺i兲l⫹Jl⫹共z兲.
共33兲
The evaluation of the sum in共33兲 can be done in a similar way to what was done in Ref. 26. There exist all zal(z) on the half-line, 0⬍z⬍⬁, and the relation 关8.471 共2兲 共Ref. 25兲兴, zJ(z)
⫽关J⫺1(z)⫺J⫹1(z)兴/2, can be used. The series Y˜(z,,) converges and the series of
deriva-tives 兺l⬁⫽2zal(z) converges uniformly in 共0,⬁兲. It is a sufficient condition to write down
zY˜ (z,,)⫽兺⬁l⫽2zal(z). Thus, one arrives to a differential equation with respect to Y (z,,),
d
dzY共z,,兲⫽⫺Y共z,,兲i cos⫹ 1 2共⫺i兲
关⫺ieiJ
共z兲⫹J1⫹共z兲兴. 共34兲
that is true on the half-line, 0⬍z⬍⬁. The solution of 共34兲 reads
Y共z,,兲⫽1 2共⫺i兲
冕
0
z
ei(y⫺z)cos关⫺ieiJ共y兲⫹J1⫹共y兲兴dy. 共35兲
This is also valid for Y (z,⫺,⫺).
It is useful to introduce the following function:
fnc共x,x
⬘
,s兲⫽兺
l⫽0fl共x,x
⬘
,s兲.It defines the part of the Green functions that is the same for all extensions. With the help of the function Y (z,,) 共33兲, 共35兲 one can write
fnc共x,x
⬘
,s兲⫽A共s兲e⫺ieBse⫺ieBs3
兵Y共z,⌬⫺eBs,兲⫹Y共z,⫺⌬⫹eBs,⫺兲
⫹关e⫺ i/2J
共z兲⫺e⫺i(⌬⫺eBs)e⫺ i(1⫺)/2J1⫺共z兲兴⌶⫹1其. 共36兲
The function f0(x,x
⬘
,s) is specific for each extension. It is reasonable to mark it with a superscriptthat assumes the values of the extension parameter. Thus, for ⌰⫽⫺/2, f0(⫺/2)共x,x
⬘
,s兲⫽A共s兲e⫺ieBs关e⫺i⌬e⫺ i(1⫺)/2J1⫺共z兲⌶⫹1⫹e⫺ieBs3
ei/2J⫺共z兲⌶⫺1兴.
共37兲
Accordingly, the function f (x,x
⬘
,s) acquires the same superscript, f(⫺/2)共x,x⬘
,s兲⫽ fnc共x,x⬘
,s兲⫹ f0(⫺/2)
共x,x
⬘
,s兲. 共38兲For the extension parameter⌰⫽/2, one obtains
f0(/2)共x,x
⬘
,s兲⫽A共s兲e⫺ieBs关e⫺i⌬e⫺ i(⫺1)/2J⫺1共z兲⌶⫹1⫹e⫺ieBs3e⫺ i/2J共z兲⌶⫺1兴,共39兲
f(/2)共x,x
⬘
,s兲⫽ fnc共x,x⬘
,s兲⫹ f0(/2)共x,x
⬘
,s兲.
Besides, one can consider particles with ‘‘spin-down’’ polarization in 2⫹1 dimensions. The corresponding wave functions(⫺1)(x) can be presented as
(⫺1)共x兲⫽1共⌫P⫺M 兲u共x兲 ,
where u(x) are solutions 共11兲 of the squared Dirac equation. The propagator related to such particles can be expressed in terms of the function⌬c(x,x
⬘
) 共24兲,S(c⫺1)共x,x
⬘
兲⫽⫺1共⌫P⫺M 兲⌬c共x,x⬘
兲1. At this point we should make some remarks.One can see that there exists a simple relation between scalar Green functions and Green functions of the squared Dirac equation 共for the above considered extensions兲. Consider this relation in the example of causal Green functions. First of all, we note that the Klein–Gordon equation differs from the squared Dirac equation by the Zeeman interaction term. Then we can see
共remembering the origin of the quantum number l for both spinning and spinless particles兲 that the
scalar propagator can be derived from⌬c(x,x
⬘
) by only retaining the terms with⫽⫺1 only. The term eB3, which is responsible for the Zeeman interaction with the uniform magnetic field, has to be removed. The Zeeman interaction with the solenoid flux, influencing the terms with l⫽0, depends on the flux sign and can be repulsive or attractive. The repulsive contact interaction case is physically equivalent to the spinless case, since in both cases the corresponding wave functions vanish at the origin. The necessary boundary condition is realized for the extension parameter⌰⫽/2. Thus, one can obtain the scalar Green functions using the coefficients of ⌶⫺1 in fl(x,x
⬘
,s) 共31兲, 共32兲 and f0(/2)
(x,x
⬘
,s) 共39兲. By following such prescriptions, one arrives at the expression共A1兲 obtained by direct calculation.In the spinless case there is no physically preferred orientation of the plane x1x2. Therefore, the solenoid flux direction does not matter, i.e., the AB symmetry, l0→l0⫹1, is conserved. The
direction of the uniform magnetic field does not matter as well. This can be observed from the explicit form of the Green functions共A1兲 where the change B→⫺B is equivalent to the choice of the opposite orientation of the plane, l→⫺l, ⌬→⫺⌬, ⌽→⫺⌽. In the spinning case the given spin direction breaks the symmetry related to the plane orientation. The Zeeman interaction of the spin with the background violates the AB symmetry as well as the symmetry with respect to the change B→⫺B.
As is known, influence of the solenoid flux on the particle is observed only when the flux is not equal to an integral number of quanta (⫽0). In this connection it is instructive to consider
the Green functions for the particular case ⫽0. We note that the part fnc(x,x
⬘
,s) 共36兲 of thefunction f (x,x
⬘
,s) is regular everywhere, while the part f0(x,x⬘
,s) is singular at the origin. Thus,taking the limit→0 in 共36兲 and using the relation J1( y )⫽⫺J0
⬘
(y ) we get fnc共x,x⬘
,s兲⫽A共s兲e⫺ieBs3
兵e⫺iz cos(⌬⫺eBs)⫺J0共z兲⫹关J0共z兲⫹ie⫺i(⌬⫺eBs)J1共z兲兴⌶⫹1其.
The corresponding expression for f0(x,x
⬘
,s) can be obtained in the following way. We restrict therange of z to 0⬍␦⬍z⬍⬁, where␦Ⰶ1. Then we take the limit→0 and use the continuity of the Bessel functions with respect to its index. At the end we construct the analytic continuation of the obtained expressions over the interval共0,␦兲. Thus, starting from either 共37兲 or 共39兲 we get
f0共x,x
⬘
,s兲⫽A共s兲关⫺ie⫺i⌬J1共z兲⌶⫹1⫹eieBsJ0共z兲⌶⫺1兴,
where the superscript is no longer necessary. Thus, the explicit form of f (x,x
⬘
,s) isf共x,x
⬘
,s兲⫽ eB83/2s1/2sin共eBs兲exp
再
i 4 ⫺ i共⌬x0兲2 4s ⫺iM 2s⫺ieBs3冎
⫻exp再
⫺il0⌬⫹ ieB 4 共r2⫹r
⬘
2兲cot共eBs兲⫺ieBrr⬘
cos共⌬⫺eBs兲2 sin共eBs兲
冎
. 共40兲 Making a transformation to Cartesian coordinates in 共40兲 and setting l0⫽0, one can obtain theknown result of the uniform magnetic field, see for example, Ref. 27.
C. Nonrelativistic case
Consideration of the Green functions in the background under question in the nonrelativistic case is important for various physical applications. Below we study this case in detail. The solutions of the Schro¨dinger equation for ‘‘spin-up’’ particles共⫹兲 and antiparticles 共⫺兲 in the case
⌰⫽⫺/2 read ⫹m,l共x兲⫽e⫺iEx 0
冑
␥ 2e i(l⫺l0⫺1) m,l,⫹1共r兲, E⫽ m,l, 2 M , 共41兲 ⫺m,l共x兲⫽e⫺iEx 0冑
␥ 2e ⫺i(l⫺l0) m,l,⫺1共r兲, l⫽0, ⫺m,0共x兲⫽e⫺iEx 0冑
␥ 2e il0 m,⫺1 ir 共r兲, 共42兲where the valuesm,l, are defined by m,l, with the help of formulas共13兲, 共14兲 for B⬎0, and
共16兲, 共17兲 for B⬍0. The solutions⫹m,l(x) (⫺m,l(x)) for the ‘‘spin-down’’ case can be obtained
from the solutions⫺m,l(x) (⫹m,l(x)) for the ‘‘spin-up’’ case with the change→⫺ in共41兲,
共42兲.
The retarded Green functions for particles and antiparticles are defined as
Sret,(⫾)共x,x
⬘
兲⫽共⌬x0兲兺
l Sl (⫾)共x,x⬘
兲, S l (⫾)共x,x⬘
兲⫽i兺
m ⫾m,l共x兲⫾m,l * 共x⬘
兲, 共43兲 Snc(⫾)共x,x⬘
兲⫽兺
l⫽0 Sl(⫾)共x,x⬘
兲,where the part Snc(⫾)(x,x
⬘
) is the same for all extensions, whereas S0(⫾)(x,x⬘
) is specific for each extension. Carrying out the summations in共43兲 one obtainsSl(⫾)共x,x
⬘
兲⫽Anr共x,x⬘
兲e⫿i␥e⫾i(l⫾⫺l0)⌬e⫺i兩l⫾⫹兩␥e⫺ i兩l⫾⫹兩/2J兩l ⫾⫹兩共znr兲, Anr共x,x⬘
兲⫽ ␥ 4sin共␥兲 exp冋
i 2共⫹⬘
兲cot共␥兲册
, 共44兲Snc(⫹)共x,x
⬘
兲⫽Anr共x,x⬘
兲e⫺il0⌬e⫺i(1⫹)eB兵e⫺ i/2J共znr兲⫺e⫺i⌬eieBe⫺ i(1⫺)/2J1⫺共znr兲⫹Y共znr,⌬⫺eB,兲⫹Y共znr,⫺⌬⫹eB,⫺兲其, 共45兲
Snc(⫺)共x,x
⬘
兲⫽Anr共x,x⬘
兲eil0⌬ei(1⫺)eB兵Y共znr,⫺⌬⫺eB,兲⫹Y共znr,⌬⫹eB,⫺兲其,共46兲
znr⫽
冑
⬘
/sin共␥兲, ⫽⌬x0/2M , l⫾⫽l⫺共1⫾1兲/2, l⫽0,whereas for l⫽0,
S0(⫹)(⫿/2)共x,x
⬘
兲⫽Anr共x,x⬘
兲e⫺i(l0⫹1)⌬e⫺ieBe⫿i(1⫺)/2J⫾(1⫺)共znr兲, 共47兲S0(⫺)(⫿/2)共x,x
⬘
兲⫽Anr共x,x⬘
兲eil0⌬ei(1⫺)eBe⫾i/2J⫿共znr兲. 共48兲The Green function in the ‘‘spin-down’’ case can be obtained with the change ⌬→⫺⌬ in
共44兲–共48兲 and with the change S(⫾)by S(⫿) in all the functions S(x,x
⬘
) in共44兲–共48兲. Thus, onecan see that the Green functions for the nonrelativistic particle is irregular at r⫽0 when the contact interaction is attractive.
We note that for the limiting case B⫽0 共the uniform magnetic field is absent兲, Sl(⫹)(⫺/2)(x,x
⬘
) coincide with the known expression for the spinless particle,15–17which is natu-ral in the case of a repulsive contact interaction. Sl(⫹)(/2)(x,x⬘
) for B⫽0 coincide with the corresponding expressions obtained in Ref. 18.III. 3¿1 DIMENSIONAL CASE
To obtain the Green functions in 3⫹1 dimensions we use the orthonormalized solutions
⫾⌿p3,m,l,(x) of the Dirac equation found in Refs. 2 and 3. The quantum numbers m, l have the
same meaning as in the (2⫹1)-dimensional case, p3 is the x3 component of the momentum, and
is the spin quantum number. The positive energy spectrum is given by ⫹ and the negative energy spectrum is given by ⫺ . They both are expressed via the quantityas
⫹⫽⫺⫺⫽
冑
M2⫹p3 2⫹. 共49兲
The spectra ofare given in共13兲, 共14兲 for B⬎0, and in 共16兲, 共17兲 for B⬍0. For⫽0, one can present the solutions⫾⌿p3,m,l, in the following form:
⫾⌿p3,m,l,共x兲⫽N共␥P⫹M 兲⫾Up3,m,l,共x兲, ⫾Up3,m,l,共x兲⫽ 1
冑
2 e ⫺i⫾x0⫺ip3x3U m,l,共x⬜兲, Um,l,共x⬜兲⫽冉
um,l,共x⬜兲 3u m,l,共x⬜兲冊
, N⫽关2兩⫾兩共兩⫾兩⫹p3兲兴⫺1/2, 共50兲 whereas for ⫽0, ⫾⌿p3,0,l,⫺共x兲⫽N共␥P⫹M 兲⫾Up3,0,l,⫺共x兲, ⫽sgn共B兲,where um,l,(x⬜) are the two-spinors defined in共11兲.
We are going to construct the Green functions using the solutions that correspond to the natural extensions of the Dirac operator, i.e., for the extension parameters chosen as⌰⫹1⫽⌰⫺1
⫽⌰, and ⌰⫽⫾/2. First we consider the case⌰⫽⫺/2, and B⬎0. We note that for⫽0, ␥P
⬜Um,l,⫺⫽i
冑
Um,l,, l⭓1,共51兲
␥P
⬜Um⫹,l,⫺⫽⫺i
冑
Um⫺,l,, l⭐0,where P⬜⫽(0,P1, P2,0). The summations in共5兲 can be done similarly to the (2⫹1)-dimensional
case by the help of some important relations derived by us for the solutions共50兲. Namely, for the states with a given⫽0, the following relations hold true:
兺
⫽⫾1 ⫾⌿p3,m,l,共x兲⫾⌿ ¯p 3,m,l,共x⬘
兲 ⫽兺
⫽⫾1 1 2⫾ 共␥ P ⫹M 兲 1 2共1⫹⌺ 3兲 ⫾p3,m,l,共x,x⬘
兲, l⭓1, 共52兲兺
⫽⫾1 ⫾⌿p3,m⫹,l,⫺共x兲⫾⌿¯p3,m⫹,l,⫺共x⬘
兲 ⫽兺
⫽⫾1 1 2⫾ 共␥ P ⫹M 兲 1 2共1⫹⌺ 3兲 ⫾p3,m⫹,l,⫺共x,x⬘
兲, l⭐0,and for⫽0, we have
⫾⌿p3,0,l,⫺1共x兲⫾⌿¯p3,0,l,⫺1共x
⬘
兲⫽ 1 2⫾ 共␥ P ⫹M 兲 1 2共1⫺⌺ 3兲 ⫾p3,0,l,⫺1共x,x⬘
兲, where ⫾p3,m,l,共x,x⬘
兲⫽ 1 2e ⫺i⫾⌬x0⫺ip3⌬x3 m,l,共x⬜,x⬜⬘
兲, ⌬x3⫽x3⫺x⬘
3. 共53兲The functionsm,l,(x⬜,x⬜
⬘
) are defined in共23兲. Therefore,Sc共x,x
⬘
兲⫽共␥P⫹M 兲⌬c共x,x⬘
兲, ⌬c共x,x⬘
兲⫽i兺
m,l,冕
⫺⬁ ⬁ dp31 2共1⫹⌺ 3兲冋
共⌬x0兲 1 2⫹⫹p3,m,l,共x,x⬘
兲 ⫺共⫺⌬x0兲 1 2⫺⫺p3,m,l,共x,x⬘
兲册
. 共54兲Applying the relations共25兲,共26兲, one obtains the proper time integral representation for ⌬c,
⌬c共x,x
⬘
兲⫽冕
0 ⬁ f共x,x⬘
,s兲ds, f 共x,x⬘
,s兲⫽兺
l⫽⫺⬁ ⬁ fl共x,x⬘
,s兲, fl共x,x⬘
,s兲⫽D共s兲兺
⫽⫾1⌽l,共s兲e ⫺ieBs1 2共1⫹⌺ 3兲,D共s兲⫽ eB
162s sin共eBs兲 exp
再
i 4s关共⌬x3兲 2⫺共⌬x 0兲2兴⫺iM2s冎
⫻exp再
⫺il0⌬⫹ ieB 4 共r 2⫹r⬘
2兲cot共eBs兲冎
, 共55兲where⌽l,(s) are defined in共32兲.
Carrying out similar calculations for B⬍0 one can verify that 共55兲 is valid for both signs of B. Therefore, for any sign of B, we get
fnc共x,x
⬘
,s兲⫽兺
l⫽0fl共x,x
⬘
,s兲⫽D共s兲e⫺ieBs(⫹⌺3)
再
Y共z,⌬⫺eBs,兲⫹Y共z,⫺⌬⫹eBs,⫺兲
⫹关e⫺ i/2J
共z兲⫺e⫺i(⌬⫺eBs)e⫺ i(1⫺)/2J1⫺共z兲兴
1 2共1⫹⌺
3兲
冎
,f0(⫺/2)共x,x
⬘
,s兲⫽12D共s兲e⫺ieBs关e⫺i⌬e⫺ i(1⫺)/2J1⫺共z兲共1⫹⌺3兲⫹eieBsei/2J
⫺共z兲共1⫺⌺3兲兴,
f(⫺/2)共x,x
⬘
,s兲⫽ fnc共x,x⬘
,s兲⫹ f0(⫺/2)共x,x⬘
,s兲, 共56兲 Using the corresponding solutions for the case⌰⫽/2, we obtainf0(/2)共x,x
⬘
,s兲⫽ 12D共s兲e⫺ieBs关e⫺i⌬e⫺ i(⫺1)/2J⫺1共z兲共1⫹⌺3兲⫹eieBse⫺ i/2J
共z兲共1⫺⌺3兲兴,
f(/2)共x,x
⬘
,s兲⫽ fnc共x,x⬘
,s兲⫹ f0(/2)共x,x
⬘
,s兲. 共57兲
IV. SUMMARY
Various Green functions of the Dirac equation with the magnetic-solenoid field are con-structed as sums over exact solutions of this equation. We stress that doing that we had to take into account all the peculiarities related to the self-adjoint extension problem of the Dirac operator in the background under consideration. Both 2⫹1 and 3⫹1 dimensional cases are considered. Compact form for the Green functions was obtained thanks to the important relations共22兲 and 共52兲 derived by us for the exact solutions under consideration. The representations of the Green func-tions as proper time integrals are constructed. The kernels of the proper time integrals are repre-sented both as infinite sums over the orbital quantum number l and as simple integrals. The Green functions are obtained for two natural self-adjoint extensions, one for the positive solenoid flux and the other one for the negative solenoid flux. The physical motivation for the choice of these extensions is their correspondence to the presence of the point-like magnetic field at the origin and their close relation to the MIT boundary conditions.23,28,29Thus, the considered cases are of most interest for applications. Other values of the extension parameter correspond to additional contact interactions,30 and some of the values are of physical interest as well. To find a closed form of Green functions for the arbitrary value of the extension parameter is a more complicated task. The spectra of the corresponding extensions in the critical subspace are no longer periodic for such a situation that requires to apply more exquisite calculation methods. We suppose to consider this issue in the future.
In addition, the nonrelativistic Green functions are constructed. The latter Green functions are represented for all possible types of 2⫹1 dimensional nonrelativistic particles.
ACKNOWLEDGMENTS
D.M.G. is grateful to the Erwin Schro¨dinger Institute for hospitality and to the foundations CNPq and FAPESP for permanent support. A.A.S. and S.P.G thank FAPESP for support.
APPENDIX
For the sake of completeness we consider here the Green functions for the scalar particle. They are defined by Eqs.共3兲, 共4兲, 共8兲, and 共9兲, where S⫿(x,x
⬘
) readS⫿共x,x
⬘
兲⫽⫾i兺
⫾n共x兲⫾n*共x⬘
兲,and ⫾n(x) form a complete set of orthonormalized solutions of the Klein–Gordon equation. Here we consider the natural extension of the Klein–Gordon operator for which solutions in 2
⫹1 dimensions and the related spectrum read3
⫾m,l共x兲⫽ 1
冑
2 e ⫺i⫾x0冑
␥ 2 e i(l⫺l0)I m⫹兩l⫹兩,m共兲, ⫾⫽⫾冑
M2⫹, ⫽␥关1⫹2m⫹兩l⫹兩⫹共l⫹兲兴, l⫽0,⫾1,⫾2, . . . , m⫽0,1,2, . . . .Using Eqs. 共25兲, 共26兲, 共28兲, and 共30兲, we calculate the causal and anticausal propagators. They have the form
Sc共x,x
⬘
兲⫽冕
0 ⬁ fsc共x,x⬘
,s兲ds, S¯c共x,x⬘
兲⫽冕
⫺0 ⫺⬁ fsc共x,x⬘
,s兲ds, fsc共x,x⬘
,s兲⫽兺
l flsc共x,x⬘
,s兲 ,flsc共x,x
⬘
,s兲⫽A共s兲eil⌬e⫺i(l⫹)eBse⫺ i兩l⫹兩/2J兩l⫹兩, 共A1兲 fsc共x,x⬘
,s兲⫽A共s兲e⫺ieBs关e⫺ i/2J共z兲⫹Y共z,⌬⫺eBs,兲⫹Y共z,⫺⌬⫹eBs,⫺兲兴,
where A(s) is given in共31兲, and Y(z,,) in共33兲, 共35兲. The expression 共A1兲 can be generalized for the (D⫹1)-dimensional case, where D is the number of spacial dimensions, with the substi-tution A(s) in共A1兲 by A(D)(s),
A(D)共s兲⫽A共s兲exp
再
i 4sk兺
⫽3 D 共⌬xk兲2冎
冉
e⫺ i/2 4s冊
(D⫺2)/2 , D⭓3. 1V. G. Bagrov, D. M. Gitman, and V. B. Tlyachev, J. Math. Phys. 42, 1933共2001兲.2
S. P. Gavrilov, D. M. Gitman, and A. A. Smirnov, Phys. Rev. A 67, 024103共2003兲; Eur. Phys. J. direct C 30, 009 共2003兲. 3
S. P. Gavrilov, D. M. Gitman, and A. A. Smirnov, Focus on Mathematical Physics共Nova Science, New York, in press兲; hep-th/0308093.
4Y. Nambu, Nucl. Phys. B 579, 590共2000兲. 5H.-P. Thienel, Ann. Phys.共N.Y.兲 280, 140 共2000兲. 6
R. M. Cavalcanti, Comment on ‘‘Quantum mechanics of an electron in a homogeneous magnetic field and asingular magnetic flux tube,’’ quant-ph/0003148.
7M. Hirokawa and O. Ogurisu, J. Math. Phys. 42, 3334共2001兲.
8V. G. Bagrov, D. M. Gitman, A. Levin, and V. B. Tlyachev, Nucl. Phys. B 605, 425共2001兲. 9P. Exner, P. Sˇt’ovicˇek, and P. Vytrˇas, J. Math. Phys. 43, 2151共2002兲.
10
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