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ON SPINOR FIELDS ON THE SURFACES OF REVOLUTION

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At present time methods of the theory of surfaces penetrate into many areas of theoretical and mathematical physics and become an important and inherent part of the modern physics. The deep relation between the theory of surfaces and soliton theory
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  Proc. Int. Conf. ”Geometrization of Physics IV”Kazan State University, Kazan, October 4–8, 1999 ON SPINOR FIELDS ON THE SURFACES OF REVOLUTION V.V. Varlamov At present time methods of the theory of surfaces penetrate into many areas of theoretical andmathematical physics and become an important and inherent part of the modern physics.The deep relation between the theory of surfaces and soliton theory is well known, thereexists a numerous literature devoted to this question (see, for example, [Sym85, Bob94]).Recently, Konopelchenko and Taimanov introduced a new approach relating the methods of integrable systems with a theory of surfaces conformally immersed into 3- and 4-dimensionalspaces [Kon96, KT95, KT96, Tai97a, Tai97b]. The main tool of this approach is a gener-alized Weierstrass representation for a conformal immersion of surfaces into  R 3 or  R 4 . Aconsideration of the linear problem along with the Weierstrass representation allows oneto express integrable deformations of surfaces via such hierarchies of nonlinear differentialequations as a modified Veselov-Novikov hierarchy, Davy-Stewartson hierarchy and so on.This approach is also used for the study of the basic quantities related with 2D gravity suchas Polyakov extrinsic action, Nambu-Goto action, geometric action and Euler characteristic[CK96, KL98c, KL99]. There is a broad application of this approach in the study of constantmean curvature surfaces, Willmore surfaces, surfaces of revolution and in many other prob-lems related with differential geometry and physics [Kon96, Kon98, KL98a, KL98b, KT95,KT96, KL97, Mat98, Tai97a, Tai97b, Tai97c, Tai98, Var98, Yam99].The relation between generalized Weierstrass coordinates  X  ( z,  ¯ z  ) for a conformal immer-sion of the surface into a complex space C 4 and Dirac-Hestenes spinors  φ i  has been proposedin the recent author’s paper [Var98] and defined by the following expressions: φ 1  = Re X  4 − i Im X  3 ,φ 2  = Im X  2 − i Im X  1 ,φ 3  = Re X  3 − i Im X  4 ,φ 4  = Re X  1 +  i Re X  2 , (1)where X  1 =  i 2   Γ ( ψ 1 ψ 2 d ¯ z  − ϕ 1 ϕ 2 dz  ) ,X  2 = 12   Γ ( ψ 1 ψ 2 d ¯ z   +  ϕ 1 ϕ 2 dz  ) ,X  3 = 12   Γ ( ψ 1 ϕ 2 d ¯ z  − ϕ 1 ψ 2 dz  ) ,X  4 =  i 2   Γ ( ψ 1 ϕ 2 d ¯ z   +  ϕ 1 ψ 2 dz  ) ,  (2)and ψ αz  =  pϕ α ,ϕ α ¯ z  =  −  pψ α , α  = 1 , 2 (3)1  where Γ is a contour in a complex plane  C ,  ψ α ,ϕ α  and  p ( z,  ¯ z  ) are complex-valued functions.Further, the Dirac spinor field Φ = ( φ 1 ,φ 2 ,φ 3 ,φ 4 ) T  is understood as a minimal left ideal of the Dirac algebra  C 4 , 1   C 4   M 4 ( C ):  I  4 , 1  =  C 4 , 1 e 41  =  C 4 , 112 (1+Γ 0 ) 12 (1+ i Γ 12 ) ,  where  e 41 is a primitive idempotent of   C 4 , 1   C 4  [Lou81], Γ 12  = Γ 1 Γ 2  and Γ i  ( i  = 0 , 1 , 2 , 3) are matrixrepresentations of the units of   C 4 :Γ 0  =  I   00  − I   ,  Γ 1  =   0  σ 1 − σ 1  0  ,  Γ 2  =   0  σ 2 − σ 2  0  ,  Γ 3  =   0  σ 3 − σ 3  0  ,σ i  are the Pauli matrices,  I   is the unit matrix. In so doing, an explicit form of the Diracfield Φ ∈ I  4 , 1   M 4 ( C ) e 41  defined as followsΦ =  φ 12(1 + Γ 0 )12(1 +  i Γ 12 ) =  φ 1  0 0 0 φ 2  0 0 0 φ 3  0 0 0 φ 4  0 0 0  , where  φ  ∈  C +1 , 3    C 3 , 0    C 2  is a Dirac-Hestenes spinor field, the matrix representation of which has a form φ  =  φ 1  − φ ∗ 2  φ 3  φ ∗ 4 φ 2  φ ∗ 1  φ 4  − φ ∗ 3 φ 3  φ ∗ 4  φ 1  − φ ∗ 2 φ 4  − φ ∗ 3  φ 2  φ ∗ 1  , φ i  ∈ C . The formulae (2) define a conformal immersion of the 2d surface into a complex space  C 4 .The components of an induced metric have a form g zz  =  g ¯ z ¯ z  = 4  i =1 ( X  iz ) 2 = 0 , g z ¯ z  = 4  i =1 ( X  iz X  i ¯ z ) =  ψ 1 ψ 2 ϕ 1 ϕ 2 . Let us assume now that the surface immersed into  C 4 is a surface of revolution. Inthis case a potential  p  depends only one variable  x , and the functions  ψ α ,ϕ α  according to[KT95, KT96, Tai97a, Tai97b] take a form ψ α  =  r α ( x )exp( λy ) , ϕ α  =  s α ( x )exp( λy ) ,  (4)where  α  = 1 , 2;  λ  ∈  IR. Further, in the case of the surface of revolution the systems (3)(two-dimensional Dirac equations) ∂ ∂z ψ α  =  pϕ α ,∂ ∂  ¯ z ϕ α  =  −  pψ α ,α  = 1 , 2 , where ∂ ∂z   = 12   ∂ ∂x  +  i ∂ ∂y  , ∂ ∂  ¯ z   = 12   ∂ ∂x  − i ∂ ∂y  , 2  reduce to the systems ∂ ∂xr α  +  iλr α  =  us α ,∂ ∂xs α − iλs α  =  − ur α ,α  = 1 , 2 ,  (5)here  u  = 2  p . The system of the type (5) is nothing but a well-known  Zakharov-Shabat system  [ZS71]. One-soliton solutions of the systems (5) can be obtained via the linear Bargmannpotentials [Bar49] (see also [Lam80]): r α  =  e − iλx [4 iλ  +  a α ( x )] ,s α  =  e − iλx b α ( x ) , , α  = 1 , 2 .  (6)The fundamental solutions (Jost functions) of the systems (5) are defined by the followingexpressions: J  +1 α ( x,λ ) =  e − iλx 2 iλ  +  µ tanh( µx − φ )2 iλ − µ , J  +2 α ( x,λ ) = ± µe − iλx sech( µx − φ )2 iλ − µ .  (7)Analogously, for the Bargmann potentials of the form r α  =  e iλx c α ( x ) ,s α  =  e iλx [4 iλ  +  d α ( x )] , , α  = 1 , 2we have the fundamental solutions J  − 1 α ( x,λ ) = ± µe iλx sech( µx − φ )2 iλ − µ , J  − 2 α ( x,λ ) =  e iλx 2 iλ − µ tanh( µx − φ )2 iλ − µ .  (8)The following quantities T  ( λ ) = 1 c 12 ( λ ) , R ( λ ) =  c 11 ( λ ) c 12 ( λ )are called  the transmission coefficient and the reflection coefficient  , respectively. It is easyto verify that for the Jost functions (7)- (8) we have c 12  =  W  ( J  + α  ,J  − α  ) =  J  +1 α J  − 2 α − J  +2 α J  − 1 α  =  λ − iµ/ 2 λ  +  iµ/ 2 , c 11 ( λ ) =  c 22 ( λ ) = 0 . Therefore,  R ( λ ) = 0 and  u  is  a reflectionless potential  . The transmission coefficient  T  ( λ ) hasa pole at  λ  =  iµ/ 2 and Res[1 /c 12 ( iµ/ 2)] =  iµ . In so doing, localized Jost functions (7)-(8)(at the pole  λ  =  iµ/ 2) have a form J  +1 α ( x,iµ/ 2) =  12 e − µx/ 2 sech µx,J  +2 α ( x,iµ/ 2) =  ∓ 12 e µx/ 2 sech µx,J  − 1 α ( x,iµ/ 2) =  ∓ 12 e − µx/ 2 sech µx,J  − 2 α ( x,iµ/ 2) =  12 e µx/ 2 sech µx.  (9)Expressing the functions (4) via the localized Jost functions (9)  ψ α  =  J  +1 α ( x,iµ/ 2) e λy , ϕ α  = J  +2 α ( x,iµ/ 2) e λy and plugging into (2), we obtain after integration in (1) an explicit form of 3  the components of the Dirac spinor field Φ = ( φ 1 ,φ 2 ,φ 3 ,φ 4 ) T  on the surface of revolution: φ 1 ( x,y,iµ/ 2) = Im C   4 4 µ  cos µy  + 14 µ ( A ( x ) − i sech 2 µx )sin µy  + Re C  4 − i Im C  3 ,φ 2 ( x,y,iµ/ 2) = 12  B ( x ) − i Im C   1 4  sin µy  + 14(Im C   2 − i Re C   1 )cos µy  + Im C  2 − i Im C  1 ,φ 3 ( x,y,iµ/ 2) = 14 µ (sech 2 µx − iA ( x ))cos µy − i Im C   4 4 µ  sin µy  + Re C  3 − i Im C  4 ,φ 4 ( x,y,iµ/ 2) =  Im C   1 4 +  iB ( x )  cos µy − 14(Re C   1  +  i Im C   2 )sin µy  + Re C  1  +  i Re C  2 , where A ( x ) = tanh µx  + Re C   4 ,B ( x ) = 1 µ (arctan e µy − 12 sinh µx sech 2 µx ) + 12Re C   2 . Further, in soliton theory the system of the form (3) is known as a linear problem of amodified Veselov-Novikov (mVN) hierarchy. The first equation of the mVN-hierarchy has aform  p t  =  p zzz  + 3  p z ω  + 32  pω z  +  p ¯ z ¯ z ¯ z  + 3  p ¯ z ¯ ω  + 32  p ¯ ω ¯ z ,  (10)where  ω ¯ z  = (  p 2 ) z . In the case of the surface of revolution (  p  =  p ( x ) ,p  =  u/ 2) the equation(10) is reduced to a modified Korteweg-de Vries equation  u t  =  u xxx +  32 u 2 u x . Thus, a depen-dence of the potential  u  on a deformation parameter  t  has the form  u  = ± sech( µx − µ 3 t ). Itallows us to express a dependence of the Jost functions on  t . Thus, integrable deformationsof the spinor field Φ with components  φ i ( x,y,t,iµ/ 2) on the surface of revolution are definedby the modified Korteweg-de Vries hierarchy.Unfortunately, in within of fifteen-minute talk it is impossible to consider all furthergeneralizations of the proposed construction. We only briefly indicate the most interestingof them, such as multi-soliton solutions, in particular, two-soliton solutions defined by thequadratic Bargmann potentials, a consideration of integrable deformations of the field Φ withallowance for higher members of the mKdV-hierarchy, a definition of Willmore functionalsand so on. On the other hand, with the aim of identification of the field Φ with a ’physical’spinor field which depends on four variables  x,y,z,t , it pays to consider the components of aform  φ i ( x,y,z,t,iµ/ 2), where the variable  z   plays by analogy with  t  the role of a deformationparameter. The substitution of a so-defined ’wave function’ into the Dirac equation gives riseto a system of algebraic equations relating the constant  µ  with electron mass. In some senseit allows us to consider the electron as  a soliton surface of revolution  . This representationcompletely conforms with current trend of leaving from pointlike description of particles tothe side of extended objects such as strings, membranes and so on. References [Bar49] V. Bargmann,  On the connection between phase shifts and scattering potentials  ,Rev. Mod. Phys.  21 , 488-493 (1949).4  [Bob94] A.I. Bobenko,  Surfaces in terms of   2 × 2  matrices. Old and new integrable cases  ,in  Harmonic maps and integrable systems  (A. Fordy, J. Wood eds.), 83-127,Vieweg (1994).[CK96] R. Carroll, B.G. Konopelchenko,  Generalized Weierstrass-Enneper inducing, con- formal immersion and gravity  , Int. J. Modern Physics A  11 , 1183-1216 (1996).[Kon96] B.G. Konopelchenko,  Induced surfaces and their integrable dynamics  , Stud. Appl.Math.  96 , 9-51, (1996).[Kon98] B.G. Konopelchenko,  Weierstrass representation for surfaces in 4D spaces and their integrable deformations via the DS hierarchy  , preprint math.DG/9807129,(1998).[KL97] B.G. Konopelchenko, G. Landolfi,  On classical string configuration  , Mod. Phys.Lett.  A12 , 3161-3168 (1997).[KL98a] B.G. Konopelchenko, G. Landolfi,  Generalized Weierstrass representation for sur- faces in multidimensional Riemann spaces  , preprint math.DG/9804144, (1998); J.Geometry and Physics (to appear).[KL98b] B.G. Konopelchenko, G. Landolfi,  Induced surfaces and their integrable dynamics.II. Generalized Weierstrass representation in 4D spaces and deformations via DS hierarchy  , preprint math.DG/9810138, (1998).[KL98c] B.G. Konopelchenko, G. Landolfi,  Quantum effects for extrinsic geometry of strings via the generalized Weierstrass representation  , preprint hep-th/9810209(1998).[KL99] B.G. Konopelchenko, G. Landolfi,  On rigid string instantons in four dimensions  ,preprint hep-th/9901113 (1999).[KT95] B.G. Konopelchenko, I.A. Taimanov,  Generalized Weierstrass formulae, soliton equations and Willmore surfaces  , preprint N 187, Univ. Bochum (1995).[KT96] B.G. Konopelchenko, I.A. Taimanov,  Constant mean curvature surfaces via an integrable dynamical system  , J. Phys. A: Math. Gen.  29 , 1261-1265 (1996).[Lam80] G.L. Lamb, Jr.,  Elements of soliton theory  (Jhon Wiley & Sons, New York1980).[Lou81] P. Lounesto,  Scalar Products of Spinors and an Extension of Brauer-Wall Groups  ,Found. Phys.  11 , 721-740 (1981).[Mat98] S. Matsutani,  Dirac operators of a conformal surface immersed in   R 4 : further generalized Weierstrass relation  , preprint solv-int/9801006 (1998).[Sym85] A. Sym,  Soliton surfaces and their application   in: Soliton geometry from spectralproblems, Lecture Notes in Physics 239, Springer, Berlin 1985, 154-231.5
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