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Non-Lie and discrete symmetries of the Dirac equation

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Non-Lie and discrete symmetries of the Dirac equation
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  Nonlinear Mathematical Physics 1997, V.4, N 3–4, 436–444. Non-Lie and Discrete Symmetriesof the Dirac Equation Jiri NIEDERLE   ∗ and  Anatolii NIKITIN   ⋆ * Institute of Physics of the Academy of Sciences of the Czech Republik,Na Slovance 2, Prague 8, Czech Republic  ⋆  Institute of Mathematics of the National Academy of Sciences of Ukraina,3 Tereshchenkivs’ka Street, Kyiv-4, Ukraina  Abstract New algebras of symmetries of the Dirac equation are presented, which are formed bylinear and antilinear first–order differential operators. These symmetries are appliedto decouple the Dirac equation for a charged particle interacting with an external field. I. Introduction Symmetries of differential equations have important applications in construction of conser-vation laws [1], separation of variables [2], reduction of nonlinear problems to more simpleones [3], etc. All that causes the continuous interest of physicists and mathematicians inthe classical group-theoretical approach [4] and its modern generalizations.Early in the seventies, W.I. Fushchych proposed the fruitful concept of non-Lie symme-tries. It happens that even such well-studied subjects as the Maxwell and Dirac equationsadmit extended symmetry algebras which cannot be found using the classical Lie approach[5-7]. The distinguishing feature of these algebras is that they have usual Lie structuresin spite of the fact that their basis elements are not Lie derivatives and belong to classesof higher-order differential operators or even integro-differential operators.In recent paper [8] a new invariance algebra  D  of the Dirac equation was found. Beingthe algebra of the higher dimension than other known finite symmetry algebras of thisequation, the algebra  D  is formed by discrete symmetries like parity or charge conjugation.This algebra has useful applications in searching for hidden supersymmetries and reductionof the Dirac equation for a particle interacting with various external fields [8].In the present paper we continue the analysis of algebraic structures of discrete sym-metries and study their connections with non-Lie symmetries of the Dirac equation. Wefind a finite dimensional symmetry algebra of the Dirac equation, which unites both thenon-Lie [6, 7] and involutive discrete [8] symmetries. We also apply discrete symmetriesto decouple the Dirac equation for a particle interacting with an external field. Copyright   c  1997 by Mathematical Ukraina Publisher.All rights of reproduction in form reserved.  NON-LIE AND DISCRETE SYMMETRIES 437 II. Lie and non-Lie symmetries of the Dirac equation Let us start with the free Dirac equation Lψ  = 0 , L  =  γ  µ  p µ − m.  (2.1)Here  p µ  =  i ∂ ∂x µ ,  µ  = 0 , 1 , 2 , 3,  γ  µ are the Dirac matrices which we choose in the form γ  0  =   0  I I   0  , γ  a  =   0  − σ a σ a  0  , γ  4  =  iγ  0 γ  1 γ  2 γ  3  =   I  2  00  − I  2  ,  (2.2) σ a ( a  = 1 , 2 , 3) are the Pauli matrices,  I  2  is the 2 × 2 unit matrix.We say a linear operator  Q  is a  symmetry   of equation (2.1) if there exists such anoperator  α Q  that[ Q,L ] =  α Q L.  (2.3)In the classical Lie approach [4] symmetry operators are searched for in the form Q  =  a µ  p µ  +  b  (2.4)where  a µ are functions of   x  = ( x 0 ,x 1 ,... ) ,b  is a matrix dependent on  x . The maximalinvariance algebra of equation (2.1) in the class of operators (2.4) is the Poincar´e algebrawhose basis elements are P  µ  =  p µ , J  µν   =  x µ  p ν   − x ν   p µ  +  i 4 [ γ  µ ,γ  ν  ] .  (2.5)In other words, any symmetry of the Dirac equation, which has form (2.4), is a linearcombination of generators (2.5) (refer, e.g., to [9]). The related  α Q  in (2.3) are equal tozero.Supposing that coefficients  a µ in (2.4) are matrices, we find the simplest non-Lie sym-metry algebra for equation (2.1) which is generated by the following operators [7, 9]Σ µν   = 12 [ γ  µ ,γ  ν  ] + 1 m  (1 − iγ  4 )( γ  µ  p ν   − γ  ν   p µ ) , Σ 1  =  γ  4 −  im  (1 − iγ  4 ) γ  µ  p µ . (2.6)Operators (2.6) satisfy relations (2.3) for  α Σ µν = 1 m  ( γ  µ  p ν   − γ  ν   p µ ) and  α Σ 1  = − 1 mγ  4 γ  µ  p µ .Moreover, operators Σ µν   commute with Σ 1  and form the Lie algebra isomorphic to so(1,3).We notice that Lie symmetries (2.5) and non-Lie symmetries (2.6) can be united inframes of the 17-dimensional Lie algebra which includes (2.5) and (2.6) as subalgebras [9]. III. Algebras of discrete symmetries of the Dirac equation It is well known that the Dirac equation is invariant w.r.t. specific discrete transformationslike parity or charge conjugation. Let us analyze algebraic structures generated by thesesymmetries.  438 J. NIEDERLE, A. NIKITINConsider reflections of independent variables  x  = ( x 0 ,x 1 ,x 2 ,x 3 ): θ 0 x  = ( − x 0 ,x 1 ,x 2 ,x 3 ) , θ 1 x  = ( x 0 , − x 1 ,x 2 ,x 3 ) , θ 2 x  = ( x 0 ,x 1 , − x 2 ,x 3 ) ,θ 3 x  = ( x 0 ,x 1 ,x 2 , − x 3 ) , θx  = ( − x 0 , − x 1 , − x 2 , − x 3 ) . (3.1)The corresponding symmetry operators for equation (2.1) have the formΓ 0  =  γ  4 γ  0 ˆ θ 0 ,  Γ 1  =  γ  4 γ  1 ˆ θ 1 ,  Γ 2  =  γ  4 γ  2 ˆ θ 2 ,  Γ 3  =  γ  4 γ  3 ˆ θ 3 ,  Γ 4  =  iγ  4 ˆ θ  (3.2)where ˆ θ µ  and ˆ θ  are operators defined as follows:ˆ θ µ ψ ( x ) =  ψ ( θ µ x ) ,  ˆ θψ ( x ) =  ψ ( − x ) .  (3.3)Let us add the list of symmetries (3.2) by the following  antilinear   operatorΓ 5  =  C   =  iγ  2 c  (3.4)where  c  is the complex conjugation,  cψ ( x ) =  ψ ∗ ( x ).Operators (3.2), (3.4) generate very interesting algebraic structures. First, they satisfythe Clifford algebraΓ k Γ l  + Γ l Γ k  = 2 g kl  (3.5)where  g 00  = − g 11  = − g 22  = − g 33  =  g 44  =  g 55  = 1;  g kl  = 0,  k   =  l . Secondly, this Cliffordalgebra can be extended by adding the seventh basis elementΓ 6  = Γ 0 Γ 1 Γ 2 Γ 3 Γ 4 Γ 5  =  iC.  (3.6)Finally, the enveloping algebra of this seven-dimensional Clifford algebra is isomorphicto the algebra  gl (8 ,R ). In other words, there are 64 linearly independent products of theoperators Γ f  ( f   = 0 , 1 ,... 6):  Γ m ,  Γ m Γ n ,  Γ k Γ m Γ n ,  ˆ I   , k,l,m, = 0 , 1 ,... 6 (3.7)(ˆ I   is the unit operator) which form a basis of the Lie algebra isomorphic to  gl (8 ,R ). Thisisomorphism will be constructed explicitly in Section V.Thus the discrete symmetries of the Dirac equation generate a very extended Lie alge-bra. Restricting ourselves to linear symmetries we come to the 16-dimensional Lie algebraincluding the identity operator ˆ I   and the following 15 operators { Γ a ,  Γ b Γ c } , a,b,c  = 0 , 1 ,... 4 (3.8)with Γ a  defined in (3.2). Operators (3.8) form a basis of the algebra  so (2 , 4).We notice that the Dirac equation for a charged particle interacting with an externalfield( γ  µ π µ − m ) ψ  = 0 , π µ  =  p µ − eA µ ( x ) (3.9)still admits some of symmetries (3.7) provided functions  A µ ( x ) have definite parities w.r.t.the related reflections (3.1) or their combinations. Moreover, if the corresponding symme-try (3.7) is diagonalizable, then equation (3.9) can be reduced to two uncoupled subsystems[8]. We will demonstrate in Section VI that for some classes of vector-potentials  A µ  theDirac equation can be reduced to  eight   uncoupled equations.  NON-LIE AND DISCRETE SYMMETRIES 439 IV. The maximal present symmetry algebra for the Diracequation Thus there exist two symmetry algebras for the Dirac equation which are defined byrelations (2.6), (3.7) and which are of different srcin. Symmetries (2.6) are of the form of differential operators whereas (3.7) are functional operators of discrete transformations.Nevertheless, it is possible to find an algebraic structure which unify both of them.First let us note that it is impossible to include all symmetries (2.6) and (3.8) into afinite-dimensional Lie algebra. Indeed, commutators of operators (2.6) and (3.2) gener-ate second-order differential operators whose commutators give fourth-order differentialoperators, and so on. However the subset of symmetries (3.7) which commute with  γ  4 ,i.e.,Γ k Γ l ,  Γ 4 ,  Γ 4 Γ k Γ l , k,l  = 4 (4.1)can be united with operators (2.6) in framework of a 120-dimensional (!) Lie algebra withthe following basis Q 4 µ, 4 µ  =  C  Γ µ , Q 5 µ, 5 µ  =  C  Γ 4 Γ µ , Q 5 ν, 5 µ  =  g µµ C  Γ 4 Γ µ Σ µν  ,Q 4 ν, 4 µ  =  g µµ C  Γ µ Σ µν  , Q 4 ν, 5 µ  = − γ  4 Q 5 µ, 5 ν  , Q µν,µν   =  γ  µν  ˆ θ µ ˆ θ ν  ,Q 5 ν, 4 µ  = − γ  4 Q 4 µ, 4 ν  , Q 54 , 54  =  i Γ 4 , Q 54 ,µν   =  ε µνλσ γ  µ γ  ν  ˆ θ µ ˆ θ ν  Σ λσ ,Q µλ,µσ  =  g µµ Σ µλ γ  µ γ  ν  ˆ θ µ ˆ θ ν  , Q σλ,µν   =  ε µσνλ g µµ g νν  Σ 1 γ  µ γ  ν  ˆ θ µ ˆ θ ν  ,Q µν,mn  = Σ µν  , Q 54 ,mm  =  i Γ 4 . (4.2)Here  Q kl,mn  are tensors which are antisymmetric w.r.t. permutations of the first pairof indices and symmetric w.r.t. permutations of the second pair of indices and whosediagonal elements are equal, i.e.,  Q kn,ll  =  Q kn,mm  for any  l  and  m . The Greek indicesruns over the values 0 , 1 , 2 , 3 and no summation over repeated indices is assumed.Let us notice that algebra (4.2) can be extended by the following 136 elements Q 4 µ,λσ  =  C  Γ µ ˆ θ µλσ , Q 4 µ, 45  =  C  Γ µ ˆ θ νλσ , ν,λ,σ   =  µ,Q 5 µ,λσ  =  C  Γ 4 Γ µ ˆ θ µλσ , Q 5 µ, 45  =  C  Γ 4 Γ µ ˆ θ νλσ , ν,λ,σ   =  µ,Q µν, 4 λ  =  γ  µ γ  ν  ˆ θ µ ˆ θ ν  ˆ θ µνλ , Q µν, 5 λ  =  γ  µ γ  ν  ˆ θ µ ˆ θ ν  ˆ θ µνλ Γ 4 Σ 1 ,Q µν, 45  =  γ  µ γ  ν  ˆ θ λσ , Q 54 ,µν   =  i Γ 4 ˆ θ µν  (4.3)whereˆ θ µν   = 1 + 1 m  (1 − iγ  4 )( γ  µ  p µ  +  γ  ν   p ν  ) ˆ θ µ ˆ θ ν  , µ  =  ν, ˆ θ µνλ  = 1 + 1 m  (1 − iγ  4 )  γ  µ  p µ  +  γ  ν   p ν   +  γ  λ  p λ   ˆ θ µ ˆ θ ν  ˆ θ λ , µ  =  ν,µ  =  λ,ν    =  λ (no sum over repeated indices).These elements are new symmetries of equation (2.1), which cannot be expressed viacommutators of symmetries (2.6), (4.1).  440 J. NIEDERLE, A. NIKITINOperators (4.2), (4.3) form a basis of the 256-dimensional real invariance algebra of theDirac equation, defined over the field of real numbers. This algebra is characterized bythe following commutation relations  Q kl,mn ,Q k ′ l ′ ,m ′ n ′   = − 2[ δ  mm ′  g kk ′ Q ll ′ ,nn ′  +  g ll ′ Q kk ′ ,nn ′  − g kl ′ Q lk ′ ,nn ′  − g lk ′ Q kl ′ ,nn ′  + δ  nn ′  g kk ′ Q ll ′ ,mm ′  +  g ll ′ Q kk ′ ,mm ′  − g kl ′ Q lk ′ ,mm ′  − g lk ′ Q kl ′ ,mm ′  + δ  mn ′  g kk ′ Q ll ′ ,nm ′  +  g ll ′ Q kk ′ ,nm ′  − g kl ′ Q lk ′ ,nm ′  − g lk ′ Q kl ′ ,nm ′  + δ  nm ′  g kk ′ Q ll ′ ,mn ′  +  g ll ′ Q kk ′ ,mn ′  − g kl ′ Q lk ′ ,mn ′  − g lk ′ Q kl ′ ,mn ′  ]+ g mnm ′ n ′ gf   g kk ′ Q ll ′ ,gf   +  g ll ′ Q kk ′ ,gf   − g kl ′ Q lk ′ ,gf   − g lk ′ Q kl ′ ,gf   − 12  ( δ  mm ′ δ  nn ′  +  δ  mn ′ δ  nm ′ )  g kk ′ Q ll ′ ,ss  +  g ll ′ Q kk ′ ,ss − g kl ′ Q lk ′ ,ss − g lk ′ Q kl ′ ,ss  (4.4)where  m   =  m ′ ,  n   =  n ′ ,  g mnm ′ n ′ gf   is the totally symmetric unit tensor whose nonzerocomponents correspond to noncoinciding values of all indices, and no sum over  s,g,f  .For  m  =  n  ve have  Q kl,nn ,Q k ′ l ′ ,m ′ n ′   = − 2  g kk ′ Q ll ′ ,m ′ n ′  +  g ll ′ Q kk ′ ,m ′ n ′  − g kl ′ Q lk ′ ,m ′ n ′  − g lk ′ Q kl ′ ,m ′ n ′  . (4.5)Taking into account both the invariance algebra of the two-component Klein-Gordonequation and equivalence of this equation to the Dirac one, it is possible to shaw thatrelations (4.2), (4.4) present the maximally extended finite symmetry algebra for equation(2.1).Thus we found the most extended symmetry algebra for the Dirac equation, which in-cludes non-Lie symmetries (2.6), discrete symmetries (4.1) and their combinations. Sym-metries (4.2) form its 120-dimensional subalgebra and also satisfy relations (4.4), (4.5) forthe restricted set of values of indices defined in (4.3). Another its interesting subalgebrais 56-dimensional one formed by the linear (i.e., without complex conjugation) operatorswhich are presented as follows: { Q µν,λσ , Q µν, 54 , Q 54 ,µν  , Q 54 , 54 } , µ,ν,λ,σ  = 0 , 1 , 2 , 3 .  (4.6) V. Commutative discrete symmetries In order to describe all possible reductions of the Dirac for a charged particle interactingwith an external field we shall prove that operators (3.7). It is easy to see that thefour-component complex wave function of (3.9)Ψ =  column (Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 ) ,  Ψ k  = Ψ (1) k  +  i Ψ (2) k  , k  = 1 , 2 , 3 , 4 (5.1)is equivalent to the eight–component real function˜Ψ =  column  Ψ (1)1  , Ψ (2)1  , Ψ (1)2  , Ψ (2)2  , Ψ (1)3  , Ψ (2)3  , Ψ (1)4  , Ψ (2)4  .  (5.2)In representation (5.2) the Dirac matrices (2.2) are extended to the 8 × 8 real matrices γ  0  →  ˜ γ  0  =   0  I  4 I  4  0  , γ  a  →  ˜ γ  a  =   0  − ˜ σ a ˜ σ a  0  , γ  4  →  ˜ γ  4  =   I  4  00  I  4  ,
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