Properties of some (3+1) dimensional vortex solutions of the CP^N model

Properties of some (3+1) dimensional vortex solutions of the C P N model L. A. Ferreira ? , P . Klimas ? and W . J. Zakrzewski † ( ? ) Instituto de Física de São Carlos; IFSC/USP; Uni versidade de São P aulo Caixa Postal 369, CEP 13560-970, São Carlos-SP , Brazil ( † ) Department of Mathematical Sciences, Uni versity of Durham, Durham DH1 3LE, U.K. Abstract W e construct ne w classes of vorte x-like solutions of the C P N model in (3+1) dimensions and discuss some of their properties. These solutions are obtained by generalizing to (3+1) dimensions the techniques well es- tablished for the two dimensional C P N models. W e show that as the total energy of these solutions is infinite, the y describe e v olving vortices and anti- vortices with the energy density of some configurations v arying in time. W e also make some further observ ations about the dynamics of these v ortices. 1 Intr oduction In this paper we present ne w classes of vorte x-like solutions of the C P N model [1, 2] in (3+1) dimensions. Our results generalize those obtained in our previous paper [3] where we presented a quite large class of exact solutions of C P N models in ( 3 + 1 ) dimensions. These solutions were described by arbitrary functions of two variables, namely of the combinations x 1 + i x 2 and x 3 + x 0 , where x µ , µ = 0 , 1 , 2 , 3 are the Cartesian coordinates of four dimensional Minko wski space-time. Then we considered field configurations, which for fixed values of x 3 + x 0 were holomorphic solutions of the C P N model in (2+0) dimensions. The dependence on x 3 + x 0 was assumed to be in terms of phase factors ( e ik ( x 3 + x 0 ) ). These solutions then described straight v ortices with wav es traveling along them with the speed of light. Solutions of that type were also constructed for an extended version of the Skyrme-F addee v model [4, 5]. Our pre vious paper [3] contained other solutions for which the vortices and the wav es were in more complicated interactions with each other . In this paper we generalize the procedure of [3] and generate many more vorte x-lik e solutions and also discuss solutions which correspond to configura- tions of parallel vortices and anti-vortices. Such structures interact with each other and our solutions describe this interaction and the resultant dynamics. A nov elty of the paper is that we generalize to (3 + 1) dimensions a method for constructing solutions which was originally proposed [2] in the conte xt of the two dimensional C P N model. Giv en an holomorphic solution, i.e. a configuration depending only on x 1 + i x 2 and x 3 + x 0 , we are able to generate, using a projection operator , solutions depending on x 1 + i x 2 , x 3 + x 0 and also x 1 − i x 2 . As our solutions describe vortices their total energy is infinite so to compare v arious configurations of vortices it is conv enient to talk of energy density or energy per unit length. Then, as we discuss in this paper interesting phenomena that can take place - the ener gy per unit length can stay constant, be periodic in time or e ven grow with time. At first sight this may seem surprising but, in fact, this is not in contradiction of any principles, as the total energy remains infinite and so is “constant” ( i.e. does not change). This observation complements the observ ation of our previous paper [3] in which we pointed out that although the energy per unit length of v arious parallel vorte x configurations can depend on the distance between them the vortices w ould still remain at rest. The paper is or ganized as follo ws. In the next section, for completeness, we introduce our notation and recall some basic properties of the C P N models and of their classical solutions in (2+0) dimensions. 1 The next section presents our solutions and the following one discusses some properties of these solutions. W e finish the paper with a short section presenting our conclusions and further remarks. 2 General r emarks about the C P N model The C P N model in ( 3 + 1 ) dimensional Minko wski space-time is defined in terms of its Lagrangian density L = M 2 ( D µ Z ) † D µ Z , Z † · Z = 1 , (2.1) where M 2 is a constant with the dimension of mass, Z = ( Z 1 , . . . , Z N +1 ) ∈ C N +1 and it satisfies the constraint Z † · Z = 1 . . The co v ariant deri v ati ve D µ acts on any N component vector Ψ and so also on Z , according to D µ Ψ = ∂ µ Ψ − ( Z † · ∂ µ Z )Ψ . The inde x µ runs here ov er the set µ = { 0 , 1 , 2 , 3 } and the Minkowski metric is (+,-,-,-). The Lagrangian (2.1) is in variant under the global transformation Z → U Z , with U being a ( N + 1) × ( N +1) unitary matrix. One of the advantages of the Z parametrization is that it makes this U ( N + 1) symmetry e xplicit [1, 2]. It is also con venient to use the ‘un-normalized’ vectors ˆ Z with components ˆ Z i . Then Z = ˆ Z q ˆ Z † · ˆ Z , (2.2) where the dot product in v olves the summation o ver all ( N + 1 ) components of ˆ Z . Sometimes, e xploiting the full projectiv e space symmetry of the model, we set u = ˆ Z ˆ Z N +1 and so use the parametrization Z = (1 , u 1 , . . . , u N ) q 1 + | u 1 | 2 + . . . + | u N | 2 . (2.3) The u -field parametrization does not make the U ( N + 1) symmetry explicit b ut it has the advantage that it brings out the real degrees of the freedom of the model. In terms of u i ’ s the Lagrangian density (2.1) takes the form L = 4 M 2 (1 + u † · u ) 2 h (1 + u † · u ) ∂ µ u † · ∂ µ u − ( ∂ µ u † · u )( u † · ∂ µ u ) i . (2.4) 2 The classical solutions of the model are gi ven by the N Euler-Lagrange equations which take the form: (1 + u † · u ) ∂ µ ∂ µ u k − 2( u † · ∂ µ u ) ∂ µ u k = 0 . (2.5) The simplest C P 1 case is gi ven by one function u : Z = (1 , u ) √ 1+ | u | 2 . 3 Some solutions In this paper we shall use the notation of [3] i.e. we define z ≡ x 1 + i ε 1 x 2 , ¯ z ≡ x 1 − i ε 1 x 2 , y ± ≡ x 3 ±  2 x 0 (3.6) with ε a = ± 1 , a = 1 , 2 . It is easy to check that any set of functions ˆ Z k and so u k that depend on coor- dinates x µ in a special way , namely u k = u k ( z , y + ) (3.7) is a solution of the system of equations (2.5). The Minko wski metric in the coor- dinates (3.6) becomes ds 2 = − dz d ¯ z − dy + dy − . It then follo ws that (3.7) satisfies simultaneously ∂ µ ∂ µ u i = 0 and ∂ µ u i ∂ µ u j = 0 for all i, j = 1 , . . . , N . Hence this class of solutions is quite large. Ho we ver , these are not the only solutions we can construct v ery easily . In fact, we can exploit the construction [2] of the solutions of the C P N model in (2+0) dimensions (for N > 1 ) to obtain further solutions. T o do this we recall the construction in (2+0) dimensions: First we define a Gramm-Schmidt orthogonalising operator P z by its action on any v ector f ∈ C N +1 , namely P z f = ∂ z f − f f † · ∂ z f | f | 2 . (3.8) Then, if we take f = f ( z ) and consider ˆ Z = f ( z ) the corresponding u solv es the equations (2.5). Note that as f ( z ) does not depend on y ± we have a solution of the C P N model in (2+0) and in (3+1) dimensions. Ho wev er , as is well kno wn, (see e.g . [2] and the references therein) ˆ Z = P z f ( z ) (3.9) 3 defines further u ’ s which also solv e (2.5) in (2+0) dimensions. But, as the e xpres- sion for u does not depend on y ± these functions also solve the equations (2.5) in (3+1) dimensions. This procedure can then be repeated, namely we can take ˆ Z = P k z f ( z ) , (3.10) where P k z f = P z ( P k − 1 z f ) . T o hav e more general solutions we observ e that, like in [3], we can make the coef ficients of z in the original f ( z ) to be functions of one of y ± , say , y + . As y + is real the operation of applying P z operator does not introduce the other y ± , i.e . y − , and so the corresponding ˆ Z and so u giv e us further solutions of the equations (2.5) in (3+1) dimensions. This way for N > 1 we can ha ve holomorphic solutions and also ‘mixed’ solutions. They are gi ven, respecti vely , by u k ( z , y + ) = f k ( z , y + ) f N +1 ( z , y + ) (3.11) and u k ( z , ¯ z , y + ) ≡ P l z f k P z f N +1 . (3.12) Note that like in the (2+0) case the last (as we tak e larger l ) non vanishing solution would be antiholomorphic. Then the corresponding u k will be functions of only ¯ z and y + . 3.1 Some pr operties of our solutions Let us first discuss briefly some quantities which we will use in the discussion of v arious properties of our solutions. 3.1.1 The energy of the solutions The Hamiltonian density of the C P N model, when written in coordinates ( z , ¯ z , y + , y − ), takes the form H = H (1) + H (2) , (3.13) 4 where H (1) = 8 M 2 (1 + u † · u ) 2 h ∂ ¯ z u † · ∆ 2 · ∂ z u + ∂ z u † · ∆ 2 · ∂ ¯ z u i (3.14) H (2) = 8 M 2 (1 + u † · u ) 2 h ∂ + u † · ∆ 2 · ∂ + u + ∂ − u † · ∆ 2 · ∂ − u i (3.15) and ∆ 2 ij ≡ (1 + u † · u ) δ ij − u i u ∗ j . For solutions depending on y + i.e. described by u k ( z , ¯ z , y + ) the part of the Hamiltonian density (3.15) that contains ∂ − drops out. For the holomorphic solu- tions the second part of (3.14) also drops out. F or the ‘mixed’ solutions described by (3.12) both parts of (3.14) are nonzero. Note that as our solutions depend on v ariables x 0 and x 3 only through the combination y + it is useful to define the concept of ener gy per unit length which in volv es the integration o ver x 1 and x 2 ( i.e. o ver the plane perpendicular to the x 3 axis). This giv es us E = Z R 2 dx 1 dx 2 H = 8 π M 2 h I (1) + I (2) i , where I ( a ) ≡ 1 8 π M 2 Z R 2 dx 1 dx 2 H ( a ) , a = 1 , 2 . 3.1.2 The topological charge As we are working with v ortex configurations it is important to introduce the two- dimensional topological charge defined by the inte gral Q top = Z R 2 dx 1 dx 2 ρ top (3.16) whose density is gi ven by ρ top = 1 π ε ij ( D i Z ) † · ( D j Z ) = 1 π ε ij ∂ i u † · ∆ 2 · ∂ j u (1 + u † · u ) 2 = = 1 π ∂ ¯ z u † · ∆ 2 · ∂ z u − ∂ z u † · ∆ 2 · ∂ ¯ z u (1 + u † · u ) 2 . (3.17) The indices i and j here only tak e two values { 1 , 2 } . It is easy to see that for the holomorphic solution Q top = I (1) . 5 4 V ortex solutions of the C P N model and some of their pr operties In [3] we studied some general classes of solutions of the C P 1 model. Here, first of all, we concentrate our attention on tw o classes of holomorphic solutions of the C P 1 model and then look in some detail at the C P 2 model concentrating our attention this time on ‘mixed’ solutions (3.12). 4.1 C P 1 solutions In the C P 1 model we have two functions f 1 and f 2 and in our discussion we can take their ratio u = f 1 f 2 . Let us first consider the case when all the dependence on y + is in the form of phase factors e ik i y + where k i are constant. Man y interesting features are observ ed for the configurations gi ven by f 1 ( z , y + ) = z 2 + a 1 z e ik 1 y + , f 2 ( z , y + ) = a 2 z + a 3 e ik 2 y + , (4.18) where we ha ve assumed, for simplicity , that all three parameters a 1 , a 2 and a 3 are real. The generalization to their complex values does not bring anything ne w to the problem. The holomorphic solution u is then of the form u ( z , y + ) = z z + a 1 e ik 1 y + a 2 z + a 3 e ik 2 y + . (4.19) The zeros of denominator do not lead to the singularities in the energy density as both integrals I (1) and I (2) are in variant with respect to the in version u → 1 u . Next we look in detail at v arious special cases of this solution (4.19). 4.1.1 The tube solution First we consider the case of a 1 = a 2 = 0 . In this case the field configuration becomes u = z 2 a 3 e − ik 2 y + . (4.20) It is easy to con vince oneself that this field configuration describes a vorte x with wa ves traveling along it with the speed of light. The profile of the energy 6 density is independent of y + . It has a maximum at a ring of radius r 0 which satisfies r 1 < r 0 < r 2 , where r 1 = r | a 3 | √ 3 is the radius of the circle at which the Hamiltonian density H (1) has a maximum, and r 2 = q | a 3 | corresponds to the radius of the circle at which H (2) has a maximum. The radius r 0 depends on a 3 and k 2 . F or k 2 → 0 it tends to r 1 and for k 2 → ±∞ it tends to r 2 . The integral I (1) describes the topological char ge of the vorte x which for the solution considered here is I (1) = 1 π Z R 2 dx 1 dx 2 4 a 2 3 | z | 2 ( a 2 3 + | z | 4 ) 2 = 2 . The contribution to the energy per unit length that comes from the tra veling wa ves can be also calculated explicitly . W e find I (2) = 1 π Z R 2 dx 1 dx 2 k 2 2 a 2 3 | z | 4 ( a 2 3 + | z | 4 ) 2 = π 4 k 2 2 | a 3 | . A modification of a solution of this type had been already studied in [3]. An example of such a solution is sho wn in Fig 1, where we plot the components of the isov ector ~ n = 1 1 + | u | 2  u + u ∗ , − i ( u − u ∗ ) , | u | 2 − 1  (4.21) which depend on y + . As y + changes the images in Fig.1 rotate. In Fig. 2, we plot the two contrib utions, topological and w ave, of the ener gy density on the solution (4.20). 4.1.2 The spiral solution A less tri vial but still a very simple solution is obtained from (4.19) by putting a 3 = 0 , and so u is gi ven by u = 1 a 2 ( z + a 1 e ik 1 y + ) . (4.22) In this case the integrals I (1) and I (2) can be calculated explicitly . They take the v alues I (1) = 1 π Z R 2 dx 1 dx 2 1 a 2 2 1 (1 + | u | 2 ) 2 = 1 , (4.23) I (2) = 1 π Z R 2 dx 1 dx 2 1 a 2 2 a 2 1 k 2 1 (1 + | u | 2 ) 2 = a 2 1 k 2 1 . (4.24) 7 In Fig. 3 we plot the components of the isovector (4.21) for the solution (4.22). In order to analyze the energy density let us introduce the parameterization z = r e iϕ . Then | u | 2 = 1 a 2 2 h r 2 + a 2 1 − 2 a 1 r cos( ϕ − k 1 y + − π ) i W e note that the energy per unit length ( H inte grated o ver the x 1 x 2 plane) does not depend on a 2 or y + , whereas the energy density H does. The maximum of the energy density ( | u | 2 = 0 ) is located at r = | a 1 | and ϕ = k 1 y + + π . The curve ( a 1 cos ( k 1 y + + π ) , a 1 sin ( k 1 y + + π ) , y + ) that joins the points at which the energy density has a local maximum is a spiral. On this spiral not only H has a maximum but so do also both its contributions H (1) and H (2) . As y + = x 3 + x 0 , we note that the spiral rotates around the x 3 axis with the speed of light. The only ef fect of the dependence on y + is the rotation of the energy density . Thus the energy per unit length calculated for e .g. y + = 0 is also valid for other v alues of the v ariable y + . 4.1.3 The general case (4.19) For general v alues of a 1 , a 2 and a 3 the expressions for the contributions to the energy become rather complicated. W e can write them as I (1) = 1 π Z R 2 dx 1 dx 2 A C 2 , I (2) = 1 π Z R 2 dx 1 dx 2 B C 2 , (4.25) where the e xpressions for A , B and C take the form (written in cylindrical coor- dinates ( r , ϕ, y + ) with z = r e iϕ ) A = a 2 1 a 2 3 + 4 a 2 3 r 2 + a 2 2 r 4 + 2 a 1 a 2 a 3 r 2 cos [2 ϕ − ( k 1 + k 2 ) y + ] + 4 a 1 a 2 3 r cos ( ϕ − k 1 y + ) + 4 a 2 a 3 r 3 cos ( ϕ − k 2 y + ) (4.26) B = r 2 a 2 1 a 2 3 ( k 1 − k 2 ) 2 + r 4 ( a 2 1 a 2 2 k 2 1 + a 2 3 k 2 2 ) − 2 a 1 a 2 3 ( k 1 − k 2 ) k 2 r 3 cos [ ϕ − k 1 y + ] + 2 a 2 1 a 2 a 3 ( k 1 − k 2 ) k 1 r 3 cos [ ϕ − k 2 y + ] − 2 a 1 a 2 a 3 k 1 k 2 r 4 cos [( k 1 − k 2 ) y + ] (4.27) C = r 2 h r 2 + 2 a 1 r cos [ ϕ − k 1 y + ] + a 2 1 i + a 2 2 " r 2 + 2 a 3 a 2 r cos [ ϕ − k 2 y + ] +  a 3 a 2  2 # . (4.28) T o fully analyse these expressions requires numerical work. In Figs. 5 and 6 we present the plots of the iso vector (4.21) as well as of the ener gy densities for a 8 particular example of the above solution. Ho we ver , e ven for a general configura- tion, it is possible to make a fe w analytical observations: • Rotations: Note that the energy per unit length depends on y + through periodic func- tions, in volving four frequencies, namely k 1 , k 2 and k 1 ± k 2 . Howe ver , one can isolate four situations where only one frequenc y is rele vant and the time e v olution reduces to a rotation around the x 3 -axis. In such cases, A , B and C depend on ϕ and y + only through the combination ϕ − ω y + , and the four possibilities when this happens are: 1. k 1 = k 2 ≡ k and ω = k 2. a 1 = 0 and ω = k 2 3. a 2 = 0 and ω = k 1 4. a 3 = 0 and ω = k 1 Note that the spiral solution (4.22) belongs to the last case and the tube so- lution (4.20) corresponds to the case when none of the frequencies matters. • Singularity The solution (4.19) exhibits an interesting property when a 1 = a 3 a 2 . Indeed, in this case it reduces to u = z /a 2 whene ver ( k 2 − k 1 ) y + = 2 π n , with n inte ger . The case k 1 = k 2 is not interesting since it leads to a solution independent of y + . Howe ver , for k 1 6 = k 2 the solutions change their prop- erties, including the two dimensional topological charge (3.16), whene ver y + = ξ n ≡ 2 π n k 2 − k 1 . F or those special values of y + the quantities (4.26)-(4.27) become A = a 2 2 | ~ r − ~ r n | 4 , B = a 2 3 ( k 1 − k 2 ) 2 r 2 | ~ r − ~ r n | 2 , C = ( r 2 + a 2 2 ) | ~ r − ~ r n | 2 where ~ r and ~ r n are two-component vectors: ~ r → ( x, y ) , and ~ r n → ( x n , y n ) , with x n = a 3 a 2 cos ( k 1 ξ n + π ) , y n = a 3 a 2 sin ( k 1 ξ n + π ) . (4.29) The expression | ~ r − ~ r n | 2 then becomes | ~ r − ~ r n | 2 = ( x − x n ) 2 + ( y − y n ) 2 = r 2 − 2 a 3 a 2 r cos ( ϕ − k 1 ξ n − π ) +  a 3 a 2  2 . (4.30) 9 The cancelation changes the degree of polynomials of v ariable z which causes the topological char ge to jump from Q top = 2 do wn to Q top = 1 . The ne w topological charge is then gi ven by the inte gral I (1) Q top ≡ I (1) = 1 π Z R 2 dx 1 dx 2 a 2 2 ( r 2 + a 2 2 ) 2 = 1 . (4.31) Of course, such behaviour is well kno wn from the study of topological soli- tons [6]. The space of parameters of the field configuration is not complete (has ‘holes’) and the integrand of the charge density has corresponding delta functions, which are not seen in (4.31). The interesting property here is that this process of the v ortex shrinking to the delta function and then e xpanding again is a function of time; i.e. is part of the dynamics of the system and is described by our solution. The second and related important fact comes from the study of the integral I (2) . One can check that when the v ortex shrinks to the delta function ( i.e . the cancellation takes place) the inte gral I (2) = 1 π Z R 2 dx 1 dx 2 a 2 3 ( k 1 − k 2 ) 2 r 2 ( r 2 + a 2 2 ) 2 | ~ r − ~ r n | 2 (4.32) di ver ges. This div ergence comes from the singularity at the point ~ r = ~ r n which is responsible for the energy of the solution becoming infinite. Clearly , from a physical point of vie w such field configurations should be excluded. • Anti-holomorphic solutions W e can no w also apply the transformation (3.8) to (4.19) and this would giv e us an anti-holomorphic solution. Its properties are not very dif ferent from what we had for the holomorphic one (except that the choice and meaning of parameters is dif ferent) so we do not discuss it here. 4.1.4 Further Comments In our discussion so far we hav e assumed that all y + dependence of the 2-dimensional u ( z ) is of the form of phase f actors exp( ik y + ) ’ s. There is, of course, no need to be so restricti ve. W e could make the parameters of the 2-dimensional u ( z ) depend on y + in a more general way . Thus we could consider , for instance, also u ( z , y + ) = λ 1 z − a ( y + ) , (4.33) 10 where a ( y + ) is an arbitrary function. Then, taking e .g. a ( y + ) = a y + would result in a v ortex located at x 2 = 0 , x 1 = ax 3 moving in the x 3 direction with the velocity of light. T aking a more complicated function, e.g. a ( y + ) = a y 2 + would result in a curved vortex x 1 = a ( x 3 ) 2 etc. One can also combine this dependence, for systems of more vortices, with the other dependences discussed above. This complicates the discussion but does not change its main features, hence in the remainder of this paper we return to the discussion of the dependence on y + through the phase factors. One could naiv ely think that infiniteness of the total ener gy of our solution is related to some “improper” choice of the dependence on y + . This is not true since the origin of the di ver gence comes from the topological nature of H (1) . The fact that H (1) is a total deri v ati ve prev ents the dependence of H (1) on any parameters (including any depending on y + ). One can note that for some special cases like u = z 2 exp ( − ay 2 + ) the contribution to the total ener gy coming from H (2) is finite but the total ener gy remains infinite since H (1) contribution is al ways present. 4.2 The C P 2 model Next we consider solutions of the C P 2 model. First we look at the holomorphic ones. 4.2.1 The holomorphic solutions The simplest C P 2 model solution can be obtained by adding to the system (4.18) a constant third function, i.e. define f 1 ( z , y + ) = z 2 + a 1 z e ik 1 y + f 2 ( z , y + ) = a 2 z + a 3 e ik 2 y + f 3 ( z , y + ) = a 4 . (4.34) Then we can define holomorphic configurations as u i = f i f 3 , i = 1 , 2 , i.e. u 1 ( z , y + ) = z 2 + a 1 z e ik 1 y + a 4 , u 2 ( z , y + ) = a 2 z + a 3 e ik 2 y + a 4 . (4.35) Alternati vely , we can interchange f 2 ↔ f 3 and consider the holomorphic config- urations ˜ u 1 ( z , y + ) = z 2 + a 1 z e ik 1 y + a 2 z + a 3 e ik 2 y + , ˜ u 2 ( z , y + ) = a 4 a 2 z + a 3 e ik 2 y + . (4.36) 11 Note from (2.3) that such an interchange corresponds to a phase transformation in Z , so both configurations describe the same solution of the C P 2 model. Note also (easier from (4.36)) that when a 4 → 0 this C P 2 solution reduces to the holomorphic C P 1 solution discussed before. In f act, ˜ u 2 v anishes, and ˜ u 1 becomes the C P 1 u -field. The integrals I (1) , I (2) for this C P 2 holomorphic solution (using definition (4.35) or (4.36)) no w take the form I (1) = 1 π Z R 2 dx 1 dx 2 A C 2 , I (2) = 1 π Z R 2 dx 1 dx 2 B C 2 (4.37) where A , B , C dif fer from A , B , C giv en by (4.26), (4.27) and (4.28) by terms proportional to a 2 4 , i.e. A = A + a 2 4 [ a 2 1 + a 2 2 + 4 r 2 + 4 a 1 r cos ( ϕ − k 1 y + )] (4.38) B = B + a 2 4 [ a 2 1 k 2 1 r 2 + a 2 3 k 2 2 ] (4.39) C = C + a 2 4 . (4.40) The Hamiltonian density H (2) , which is proportional to B C 2 , is now regular at ~ r = ~ r n and y + = ξ n for a 1 = a 3 a 2 (where pre viously we had a singularity) as now it takes the v alue B C 2     ~ r = ~ r n , y + = ξ n = a 2 3 a 2 4 " a 2 3 a 4 2 k 2 1 + k 2 2 # . Hence we note that going to the C P 2 manifold (by taking a 4 6 = 0 ) has ‘filled in the hole’ in the space of parameters ( i.e . as the system e volv es none of its v ortices shrinks to the delta function). Note also that the energy density is independent of y + in four cases: k 1 = k 2 , a 1 = 0 , a 2 = 0 and a 3 = 0 . 4.2.2 The mixed solution Next we look at the ‘ne w’ mix ed solutions. First we use (3.8) to calculate P z f . W e find that for the system (4.34) the y take the form P z f 1 = a 2 4 e ik 1 y + h 2 z e − ik 1 y + + a 1 i + e ik 1 y + h a 3 + a 2 ¯ z e ik 2 y + i h a 1 a 3 + 2 a 3 z e − ik 1 y + + a 2 z 2 e − i ( k 1 + k 2 ) y + i P z f 2 = a 2 a 2 4 12 − e ik 2 y + ¯ z h a 1 + ¯ z e ik 1 y + i h a 1 a 3 + 2 a 3 z e − ik 1 y + + a 2 z 2 e − i ( k 1 + k 2 ) y + i P z f 3 = − a 4 e − ik 2 y + h a 2 a 3 + a 2 2 ¯ z e ik 2 y + + ¯ z e ik 2 y + ( ¯ z e ik 1 y + + a 1 )(2 z e − ik 1 y + + a 1 ) i When written in terms of u i this mixed solution is gi ven by u 1 ( z , ¯ z , y + ) = P z f 1 P z f 3 , u 2 ( z , ¯ z , y + ) = P z f 2 P z f 3 . (4.41) Note that in the limit a 4 → 0 the mix ed solution (4.41) becomes the anti- holomorphic solution of the C P 1 model mentioned before. Ho we ver for a 4 6 = 0 the solution is dif ferent. This time the expressions for the ener gy density are quite complicated - so we do not present them here. Ho we ver , we note that to guarantee the con ver gence of the integral I (2) we hav e to require that a 2 6 = 0 . T o demonstrate that the ener gy per unit length does not depend on y + can be checked without much effort. First, we observe that the ov erall factors e ik j y + in P z f k do not matter as they cancel in the expressions for | u j | 2 and for | ∆ · u j | 2 . Hence, the only relev ant expressions are of the form z e − ik j y + = r e i ( ϕ − k j y + ) and ¯ z e ik j y + = re − i ( ϕ − k j y + ) . When k 1 = k 2 ≡ k the energy density depends only on the combination ( ϕ − ky + ) and r sho wing that the only effect of the dependence on time is a rotation and, in consequence, the independence of the energy per unit length on y + (or x 0 for gi ven x 3 ). The other cases guaranteeing this are a 1 = 0 and a 3 = 0 . 4.2.3 The anti-holomorphic solution Finally we look at the corresponding anti-holomorphic solution. Such a solution deri ved from the system (4.34) tak es the form u 1 ( ¯ z , y + ) = P 2 z f 1 P 2 z f 3 = a 2 a 4 e i ( k 1 + k 2 ) y + a 1 a 3 + ¯ z e ik 1 y + (2 a 3 + a 2 ¯ z e ik 2 y + ) (4.42) u 2 ( ¯ z , y + ) = P 2 z f 2 P 2 z f 3 = − a 4 e ik 2 y + ( a 1 + 2 ¯ z e ik 1 y + ) a 1 a 3 + ¯ z e ik 1 y + (2 a 3 + a 2 ¯ z e ik 2 y + ) . (4.43) Note that, like for the ‘mix ed case’, we ha ve to require that a 2 6 = 0 as otherwise H (1) = 0 , H (2) = 8 π M 2 k 2 2 a 2 3 a 2 4 ( a 2 3 + a 2 4 ) 2 . In the next subsection we will produce an explicit example of these field con- figurations and discuss some of their properties. T o av oid the problems mentioned 13 abov e our example will ha ve a 2 6 = 0 . Note that in such a case the conditions of the independence of the energy per unit length on y + are the same as for the mix ed solution. 4.2.4 An example In our example we start with the set of functions (4.34) for which we ha ve chosen the follo wing v alues of parameters: a 1 = 2 . 5 , a 2 = 0 . 6 , a 3 = 1 . 0 , a 4 = 0 . 01 k 1 = 1 . 0 and k 2 = 2 . 0 . The topological charge of the holomorphic solution is then Q top = 2 . The topological charge density at x 0 = 0 (and for x 3 = 0 ) has two peaks - one of them is localized at z = 0 , the other a bit further out - see Fig. 7. F or the holomorphic solution the topological charge density is proportional to the energy density and this leads to the energy per unit length being giv en by 8 π M 2 I (1) . The integrand H (1) / 8 M 2 is sketched in Fig. 8. The contribution coming from the w av es H (2) / 8 M 2 is plotted in Fig. 9. The mixed solution generated by the application of the P z operator according to (3.8) leads to a solution which has Q top = 2 − 2 = 0 and I (1) = 2 + 2 = 4 . As is easy to see from Fig. 7 the application of P z has changed two holomorphic peaks into two anti-peaks and in addition it has generated two ne w peaks. The energy density H (1) thus has four peaks and H (2) only three (with the zero in the place of the fourth H (1) one). The next application of the P z operator changes tw o peaks of the topological charge density into two anti-peaks and annihilates the pre vious anti-peaks. Thus the anti-holomorphic solution is characterized by Q top = − 2 and I (1) = 2 . The contribution to the energy per unit length 8 M 2 I (1) is the same as for the initial (holomorphic) case. Ne vertheless, the total energies per unit length for these two solutions differ since for solutions of the C P 2 model the integrals I (2) are different ( I (2) hol 6 = I (2) anti − hol ). Let us note that our case has a time-dependent energy per unit length (calculated by the integration over the x 1 x 2 plane). It implies that the dependence of the ener gy density on y + is highly nontri vial. Howe ver , the ener gy per unit length is a periodic function of y + . Only for some special cases, like k 1 = k 2 etc. the energy per unit length is constant and so does not depend on y + . The time dependence of the ener gy density for all three solutions is sho wn in Fig. 10. The energy density for the mixed solution for x 3 = 0 and x 0 = π / 4 , x 0 = π , x 0 = 7 π / 4 is plotted in Fig. 11. For the case x 0 = π the peaks are maximally separated (this is not v ery clear without a detailed study of some other v alues x 0 ). In this case the energy tak es its maximal value, see Fig. 10. 14 Figure 1: The tube solution. The part ( n 1 , n 2 ) (left) and the component n 3 (right) of the isov ector ~ n for a 1 = 0 , a 2 = 0 , a 3 = 2 , x 0 = 0 , x 3 = 0 and k 2 = 2 . The minimal v alue n 3 = − 1 occurs at the point x 1 = 0 and x 2 = 0 . Figure 2: The energy density of the tube solution - the topological part (left) and the wa ve part (right). Here a 1 = 0 , a 2 = 0 , a 3 = 2 , x 0 = 0 , x 3 = 0 and k 2 = 2 . 15 Figure 3: The spiral solution. The part ( n 1 , n 2 ) (left) and the component n 3 (right) of the isovector ~ n for a 1 = 2 , a 2 = 1 , a 3 = 0 , x 0 = 0 , x 3 = 0 and k 1 = 1 . The minimal v alue n 3 = − 1 occurs at the point x 1 = − a 1 cos ( k 1 y + ) and x 2 = − a 1 sin ( k 1 y + ) ; here x 1 = − 2 , x 2 = 0 . Figure 4: The ener gy density of the spiral solution - the topological part (left) and the wa ve part (right). Here a 1 = 2 , a 2 = 1 , a 3 = 0 , and x 0 = 0 , x 3 = 0 and k 1 = 1 . The maxima of the energy density for both contributions are located at the same point on the plane x 1 x 2 corresponding with the minimum of n 3 (see Fig 3); here x 1 = − 2 , x 2 = 0 . 16 Figure 5: The C P 1 solution with all a k 6 = 0 . The part ( n 1 , n 2 ) (left) and the component n 3 (right) of isov ector ~ n for a 1 = 2 , a 2 = 1 , a 3 = 3 , x 0 = 3 π / 4 , x 3 = 0 , k 1 = 1 and k 2 = 2 . Figure 6: The ener gy density of the C P 1 solution with all a k 6 = 0 - the topological part (left) and the wa ve part (right). Here a 1 = 2 , a 2 = 1 , a 3 = 3 , x 0 = 3 π / 4 , x 3 = 0 , k 1 = 1 and k 2 = 2 . 17 Figure 7: The functions π ρ top (where ρ top is a topological charge density) for x 0 = 0 , x 3 = 0 , a 1 = 2 . 5 , a 2 = 0 . 6 , a 3 = 1 . 0 , a 4 = 0 . 01 , k 1 = 1 and k 2 = 2 . The left picture corresponds to the holomorphic solution, the central picture corresponds to the mixed solution and the right picture to the anti-holomorphic solution. Figure 8: The functions H (1) / 8 M 2 (proportional to topological contrib ution to the energy density) for x 0 = 0 , x 3 = 0 , a 1 = 2 . 5 , a 2 = 0 . 6 , a 3 = 1 . 0 , a 4 = 0 . 01 , k 1 = 1 and k 2 = 2 . The left picture corresponds to the holomorphic solution, the central picture corresponds to the mixed solution and the right picture to the anti-holomorphic solution. 18 Figure 9: The functions H (2) / 8 M 2 (proportional to wav e contribution to the en- ergy density) for x 0 = 0 , x 3 = 0 , a 1 = 2 . 5 , a 2 = 0 . 6 , a 3 = 1 . 0 , a 4 = 0 . 01 , k 1 = 1 and k 2 = 2 . The left picture corresponds to the holomorphic solution, the central picture corresponds to the mixed solution and the right picture to the anti-holomorphic solution. 0 1 2 3 4 5 6 x0 11.0 11.5 12.0 12.5 13.0 0 1 2 3 4 5 6 x0 100 120 140 160 180 200 0 1 2 3 4 5 6 x0 100 120 140 160 180 200 Figure 10: The integral I (2) as the function of x 0 ∈ [0 , 2 π ] . The other parameters read: x 3 = 0 , a 1 = 2 . 5 , a 2 = 0 . 6 , a 3 = 1 . 0 , a 4 = 0 . 01 , k 1 = 1 and k 2 = 2 . The left picture corresponds to the holomorphic solution, the central picture corresponds to the mix ed solution and the right picture to the anti-holomorphic solution. 19 Figure 11: T ime e volution of the mix ed solution for a 1 = 2 . 5 , a 2 = 0 . 6 , a 3 = 1 . 0 , a 4 = 0 . 01 , k 1 = 1 and k 2 = 2 . The functions H (1) / 8 M 2 (left column) and H (2) / 8 M 2 (right column) hav e been considered for x 3 = 0 at the moments x 0 = π / 4 (first row), x 0 = π (second row), x 0 = 7 π / 4 (third row). Their v alues for x 0 = 0 are sk etched at the central pictures of Fig 8 and Fig 9. 20 5 Conclusions and Further Comments In this paper we hav e demonstrated that the C P N model in (3+1) dimensions has many classical solutions. Our construction has been based on the observ ation that one can generalise ideas used in the construction of solutions of the C P N model in (2+0) dimensions and generate vorte x and vortex-anti vorte x like solutions of this model in (3+1) dimensions. Like for the model in (2+0) dimensions we can gen- erate these solutions from field configurations described by polynomial functions of x 1 + i 1 x 2 . This time the coef ficients of these functions could be also functions of x 3 + ε 2 x 0 . The energy of such configurations is infinite (as the energy density is independent of x 3 ) and so we interpret these solutions as describing systems of vortices and anti vortices. Of course our expressions solve equations in (3+1) dimensions and the y also determine the dynamics of these vortices. In this paper we have only looked at the simplest solutions (corresponding to v ery few vortices) with the time dependence being described by simple phase factors. Even in this case the observed dynamics is quite complicated and has exhibited v arious interesting properties. In particular , we have sho wn that the vortices can rotate in space (ph ysical and internal) and their ener gy per unit length of the vorte x can vary in time. During this time ev olution some v ortices can shrink to delta functions and then expand again often being characterised by a very periodical beha viour . One other unusual property is their dependence on the distance between the vortices: the energy density of two v ortices can depend on the distance between them and can possess a minimum at a specific v alue of this distance. This sug- gests that vortices which are located at non-minimal distances may be unstable and so could try to reduce their ener gy per unit length by moving to wards this op- timal configurations. Howe ver , their configurations are solutions for any distance as their infinite ‘inertia’ stops them from moving to wards each other without an external push. W e are no w looking at other properties of these and other solutions. Acknowlegment: L.A. Ferreira and W .J. Zakrzewski would like to thank the Royal Society (UK) for a grant that helped them in carrying out this work. L.A. 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