Properties of some (3+1) dimensional vortex solutions of the CP^N model
PProperties of some (3+1) dimensional vortex solutions of the CP N model L. A. Ferreira (cid:63) , P. Klimas (cid:63) and W. J. Zakrzewski † ( (cid:63) ) Instituto de Física de São Carlos; IFSC/USP;Universidade de São PauloCaixa Postal 369, CEP 13560-970, São Carlos-SP, Brazil ( † ) Department of Mathematical Sciences,University of Durham, Durham DH1 3LE, U.K.
Abstract
We construct new classes of vortex-like solutions of the CP N modelin (3+1) dimensions and discuss some of their properties. These solutionsare obtained by generalizing to (3+1) dimensions the techniques well es-tablished for the two dimensional CP N models. We show that as the totalenergy of these solutions is infinite, they describe evolving vortices and anti-vortices with the energy density of some configurations varying in time. Wealso make some further observations about the dynamics of these vortices. a r X i v : . [ h e p - t h ] A ug Introduction
In this paper we present new classes of vortex-like solutions of the CP N model[1, 2] in (3+1) dimensions. Our results generalize those obtained in our previouspaper [3] where we presented a quite large class of exact solutions of CP N modelsin ( ) dimensions. These solutions were described by arbitrary functions oftwo variables, namely of the combinations x + i x and x + x , where x µ , µ =0 , , , are the Cartesian coordinates of four dimensional Minkowski space-time.Then we considered field configurations, which for fixed values of x + x wereholomorphic solutions of the CP N model in (2+0) dimensions. The dependenceon x + x was assumed to be in terms of phase factors ( e ik ( x + x ) ). These solutionsthen described straight vortices with waves traveling along them with the speed oflight. Solutions of that type were also constructed for an extended version of theSkyrme-Faddeev model [4, 5]. Our previous paper [3] contained other solutionsfor which the vortices and the waves were in more complicated interactions witheach other.In this paper we generalize the procedure of [3] and generate many morevortex-like solutions and also discuss solutions which correspond to configura-tions of parallel vortices and anti-vortices. Such structures interact with each otherand our solutions describe this interaction and the resultant dynamics. A noveltyof the paper is that we generalize to (3 + 1) dimensions a method for constructingsolutions which was originally proposed [2] in the context of the two dimensional CP N model. Given an holomorphic solution, i.e. a configuration depending onlyon x + i x and x + x , we are able to generate, using a projection operator,solutions depending on x + i x , x + x and also x − i x .As our solutions describe vortices their total energy is infinite so to comparevarious configurations of vortices it is convenient to talk of energy density orenergy per unit length. Then, as we discuss in this paper interesting phenomenathat can take place - the energy per unit length can stay constant, be periodic intime or even 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 infiniteand so is “constant” ( i.e. does not change). This observation complements theobservation of our previous paper [3] in which we pointed out that although theenergy per unit length of various parallel vortex configurations can depend on thedistance between them the vortices would still remain at rest.The paper is organized as follows. In the next section, for completeness, weintroduce our notation and recall some basic properties of the CP N models andof their classical solutions in (2+0) dimensions.1he next section presents our solutions and the following one discusses someproperties of these solutions. We finish the paper with a short section presentingour conclusions and further remarks. C P N model The CP N model in ( ) dimensional Minkowski space-time is defined in termsof its Lagrangian density L = M ( D µ Z ) † D µ Z , Z † · Z = 1 , (2.1)where M is a constant with the dimension of mass, Z = ( Z , . . . , Z N +1 ) ∈ C N +1 and it satisfies the constraint Z † · Z = 1 . . The covariant derivative 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 index µ runs here over the set µ = { , , , } and the Minkowski metric is(+,-,-,-). The Lagrangian (2.1) is invariant under the global transformation Z → U Z , with U being a ( N +1) × ( N +1) unitary matrix. One of the advantagesof the Z parametrization is that it makes this U ( N + 1) symmetry explicit [1, 2].It is also convenient to use the ‘un-normalized’ vectors ˆ Z with components ˆ Z i .Then Z = ˆ Z (cid:113) ˆ Z † · ˆ Z , (2.2)where the dot product involves the summation over all ( N + 1 ) components of ˆ Z .Sometimes, exploiting the full projective space symmetry of the model, we set u = ˆ Z ˆ Z N +1 and so use the parametrization Z = (1 , u , . . . , u N ) (cid:113) | u | + . . . + | u N | . (2.3)The u -field parametrization does not make the U ( N + 1) symmetry explicit but ithas 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 (1 + u † · u ) (cid:104) (1 + u † · u ) ∂ µ u † · ∂ µ u − ( ∂ µ u † · u )( u † · ∂ µ u ) (cid:105) . (2.4)2he classical solutions of the model are given by the N Euler-Lagrange equationswhich take the form: (1 + u † · u ) ∂ µ ∂ µ u k − u † · ∂ µ u ) ∂ µ u k = 0 . (2.5)The simplest CP case is given by one function u : Z = (1 , u ) √ | u | . In this paper we shall use the notation of [3] i.e. we define z ≡ x + i ε x , ¯ z ≡ x − i ε x , y ± ≡ x ± (cid:15) x (3.6)with ε a = ± , a = 1 , .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 Minkowski metric in the coor-dinates (3.6) becomes ds = − dz d ¯ z − dy + dy − . It then follows that (3.7) satisfiessimultaneously ∂ µ ∂ µ u i = 0 and ∂ µ u i ∂ µ u j = 0 for all i, j = 1 , . . . , N . Hence thisclass of solutions is quite large.However, these are not the only solutions we can construct very easily. Infact, we can exploit the construction [2] of the solutions of the CP N model in(2+0) dimensions (for N > ) to obtain further solutions. To do this we recall theconstruction in (2+0) dimensions:First we define a Gramm-Schmidt orthogonalising operator P z by its action onany vector f ∈ C N +1 , namely P z f = ∂ z f − f f † · ∂ z f | f | . (3.8)Then, if we take f = f ( z ) and consider ˆ Z = f ( z ) the corresponding u solvesthe equations (2.5). Note that as f ( z ) does not depend on y ± we have a solutionof the CP N model in (2+0) and in (3+1) dimensions. However, as is well known,(see e.g. [2] and the references therein) ˆ Z = P z f ( z ) (3.9)3efines further u ’s which also solve (2.5) in (2+0) dimensions. But, as the expres-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 kz f ( z ) , (3.10)where P kz f = P z ( P k − z f ) .To have more general solutions we observe that, like in [3], we can make thecoefficients of z in the original f ( z ) to be functions of one of y ± , say, y + . As y + isreal 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 give us further solutions of the equations (2.5)in (3+1) dimensions. This way for N > we can have holomorphic solutions andalso ‘mixed’ solutions.They are given, respectively, by u k ( z, y + ) = f k ( z, y + ) f N +1 ( z, y + ) (3.11)and u k ( z, ¯ z, y + ) ≡ P lz f k P z f N +1 . (3.12)Note that like in the (2+0) case the last (as we take larger l ) nonvanishingsolution would be antiholomorphic. Then the corresponding u k will be functionsof only ¯ z and y + . Let us first discuss briefly some quantities which we will use in the discussion ofvarious properties of our solutions.
The Hamiltonian density of the CP N model, when written in coordinates ( z , ¯ z , y + , y − ), takes the form H = H (1) + H (2) , (3.13)4here H (1) = 8 M (1 + u † · u ) (cid:104) ∂ ¯ z u † · ∆ · ∂ z u + ∂ z u † · ∆ · ∂ ¯ z u (cid:105) (3.14) H (2) = 8 M (1 + u † · u ) (cid:104) ∂ + u † · ∆ · ∂ + u + ∂ − u † · ∆ · ∂ − u (cid:105) (3.15)and ∆ 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 theHamiltonian density (3.15) that contains ∂ − drops out. For the holomorphic solu-tions the second part of (3.14) also drops out. For the ‘mixed’ solutions describedby (3.12) both parts of (3.14) are nonzero.Note that as our solutions depend on variables x and x only through thecombination y + it is useful to define the concept of energy per unit length whichinvolves the integration over x and x ( i.e. over the plane perpendicular to the x axis). This gives us E = (cid:90) R dx dx H = 8 πM (cid:104) I (1) + I (2) (cid:105) , where I ( a ) ≡ πM (cid:90) R dx dx H ( a ) , a = 1 , . As we are working with vortex configurations it is important to introduce the two-dimensional topological charge defined by the integral Q top = (cid:90) R dx dx ρ top (3.16)whose density is given by ρ top = 1 π ε ij ( D i Z ) † · ( D j Z ) = 1 π ε ij ∂ i u † · ∆ · ∂ j u (1 + u † · u ) == 1 π ∂ ¯ z u † · ∆ · ∂ z u − ∂ z u † · ∆ · ∂ ¯ z u (1 + u † · u ) . (3.17)The indices i and j here only take two values { , } . It is easy to see that for theholomorphic solution Q top = I (1) . 5 Vortex solutions of the
C P N model and some oftheir properties In [3] we studied some general classes of solutions of the CP model. Here, firstof all, we concentrate our attention on two classes of holomorphic solutions ofthe CP model and then look in some detail at the CP model concentrating ourattention this time on ‘mixed’ solutions (3.12). C P solutions In the CP model we have two functions f and f and in our discussion we cantake their ratio u = f f .Let us first consider the case when all the dependence on y + is in the form ofphase factors e ik i y + where k i are constant. Many interesting features are observedfor the configurations given by f ( z, y + ) = z + a z e ik y + , f ( z, y + ) = a z + a e ik y + , (4.18)where we have assumed, for simplicity, that all three parameters a , a and a arereal. The generalization to their complex values does not bring anything new tothe problem.The holomorphic solution u is then of the form u ( z, y + ) = z z + a e ik y + a z + a e ik y + . (4.19)The zeros of denominator do not lead to the singularities in the energy densityas both integrals I (1) and I (2) are invariant with respect to the inversion u → u .Next we look in detail at various special cases of this solution (4.19). First we consider the case of a = a = 0 . In this case the field configurationbecomes u = z a e − ik y + . (4.20)It is easy to convince oneself that this field configuration describes a vortexwith waves traveling along it with the speed of light. The profile of the energy6ensity is independent of y + . It has a maximum at a ring of radius r whichsatisfies r < r < r , where r = (cid:114) | a |√ is the radius of the circle at which theHamiltonian density H (1) has a maximum, and r = (cid:113) | a | corresponds to theradius of the circle at which H (2) has a maximum. The radius r depends on a and k . For k → it tends to r and for k → ±∞ it tends to r . The integral I (1) describes the topological charge of the vortex which for the solution consideredhere is I (1) = 1 π (cid:90) R dx dx a | z | ( a + | z | ) = 2 . The contribution to the energy per unit length that comes from the travelingwaves can be also calculated explicitly. We find I (2) = 1 π (cid:90) R dx dx k a | z | ( a + | z | ) = π k | a | . A modification of a solution of this type had been already studied in [3]. Anexample of such a solution is shown in Fig 1, where we plot the components ofthe isovector (cid:126)n = 11 + | u | (cid:16) u + u ∗ , − i ( u − u ∗ ) , | u | − (cid:17) (4.21)which depend on y + . As y + changes the images in Fig.1 rotate. In Fig. 2, we plotthe two contributions, topological and wave, of the energy density on the solution(4.20). A less trivial but still a very simple solution is obtained from (4.19) by putting a = 0 , and so u is given by u = 1 a ( z + a e ik y + ) . (4.22)In this case the integrals I (1) and I (2) can be calculated explicitly. They takethe values I (1) = 1 π (cid:90) R dx dx a | u | ) = 1 , (4.23) I (2) = 1 π (cid:90) R dx dx a a k (1 + | u | ) = a k . (4.24)7n 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 = re iϕ . Then | u | = 1 a (cid:104) r + a − a r cos( ϕ − k y + − π ) (cid:105) We note that the energy per unit length ( H integrated over the x x plane) doesnot depend on a or y + , whereas the energy density H does. The maximum ofthe energy density ( | u | = 0 ) is located at r = | a | and ϕ = k y + + π . Thecurve ( a cos ( k y + + π ) , a sin ( k y + + π ) , y + ) that joins the points at which theenergy density has a local maximum is a spiral. On this spiral not only H has amaximum but so do also both its contributions H (1) and H (2) . As y + = x + x ,we note that the spiral rotates around the x axis with the speed of light. Theonly effect of the dependence on y + is the rotation of the energy density. Thus theenergy per unit length calculated for e.g. y + = 0 is also valid for other values ofthe variable y + . For general values of a , a and a the expressions for the contributions to theenergy become rather complicated. We can write them as I (1) = 1 π (cid:90) R dx dx AC , I (2) = 1 π (cid:90) R dx dx BC , (4.25)where the expressions for A , B and C take the form (written in cylindrical coor-dinates ( r, ϕ, y + ) with z = re iϕ ) A = a a + 4 a r + a r + 2 a a a r cos [2 ϕ − ( k + k ) y + ]+ 4 a a r cos ( ϕ − k y + ) + 4 a a r cos ( ϕ − k y + ) (4.26) B = r a a ( k − k ) + r ( a a k + a k ) − a a ( k − k ) k r cos [ ϕ − k y + ]+ 2 a a a ( k − k ) k r cos [ ϕ − k y + ] − a a a k k r cos [( k − k ) y + ] (4.27) C = r (cid:104) r + 2 a r cos [ ϕ − k y + ] + a (cid:105) + a (cid:34) r + 2 a a r cos [ ϕ − k y + ] + (cid:18) a a (cid:19) (cid:35) . (4.28)To fully analyse these expressions requires numerical work. In Figs. 5 and 6we present the plots of the isovector (4.21) as well as of the energy densities for a8articular example of the above solution. However, even for a general configura-tion, it is possible to make a few analytical observations: • Rotations:Note that the energy per unit length depends on y + through periodic func-tions, involving four frequencies, namely k , k and k ± k . However, onecan isolate four situations where only one frequency is relevant and the timeevolution reduces to a rotation around the x -axis. In such cases, A , B and C depend on ϕ and y + only through the combination ϕ − ω y + , and the fourpossibilities when this happens are:1. k = k ≡ k and ω = k a = 0 and ω = k a = 0 and ω = k a = 0 and ω = k 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. • SingularityThe solution (4.19) exhibits an interesting property when a = a a . Indeed,in this case it reduces to u = z/a whenever ( k − k ) y + = 2 πn , with n integer. The case k = k is not interesting since it leads to a solutionindependent of y + . However, for k (cid:54) = k the solutions change their prop-erties, including the two dimensional topological charge (3.16), whenever y + = ξ n ≡ πnk − k . For those special values of y + the quantities (4.26)-(4.27)become A = a | (cid:126)r − (cid:126)r n | , B = a ( k − k ) r | (cid:126)r − (cid:126)r n | , C = ( r + a ) | (cid:126)r − (cid:126)r n | where (cid:126)r and (cid:126)r n are two-component vectors: (cid:126)r → ( x, y ) , and (cid:126)r n → ( x n , y n ) ,with x n = a a cos ( k ξ n + π ) , y n = a a sin ( k ξ n + π ) . (4.29)The expression | (cid:126)r − (cid:126)r n | then becomes | (cid:126)r − (cid:126)r n | = ( x − x n ) + ( y − y n ) = r − a a r cos ( ϕ − k ξ n − π ) + (cid:18) a a (cid:19) . (4.30)9he cancelation changes the degree of polynomials of variable z whichcauses the topological charge to jump from Q top = 2 down to Q top = 1 .The new topological charge is then given by the integral I (1) Q top ≡ I (1) = 1 π (cid:90) R dx dx a ( r + a ) = 1 . (4.31)Of course, such behaviour is well known 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 deltafunctions, which are not seen in (4.31). The interesting property here is thatthis process of the vortex shrinking to the delta function and then expandingagain is a function of time; i.e. is part of the dynamics of the system and isdescribed by our solution.The second and related important fact comes from the study of the integral I (2) . One can check that when the vortex shrinks to the delta function ( i.e. the cancellation takes place) the integral I (2) = 1 π (cid:90) R dx dx a ( k − k ) r ( r + a ) | (cid:126)r − (cid:126)r n | (4.32)diverges. This divergence comes from the singularity at the point (cid:126)r = (cid:126)r n which is responsible for the energy of the solution becoming infinite.Clearly, from a physical point of view such field configurations should beexcluded. • Anti-holomorphic solutionsWe can now also apply the transformation (3.8) to (4.19) and this would giveus an anti-holomorphic solution. Its properties are not very different fromwhat we had for the holomorphic one (except that the choice and meaningof parameters is different) so we do not discuss it here.
In our discussion so far we have assumed that all y + dependence of the 2-dimensional u ( z ) is of the form of phase factors exp( iky + ) ’s. There is, of course, no need tobe so restrictive. We could make the parameters of the 2-dimensional u ( z ) dependon y + in a more general way. Thus we could consider, for instance, also u ( z, y + ) = λ z − a ( y + ) , (4.33)10here a ( y + ) is an arbitrary function.Then, taking e.g. a ( y + ) = a y + would result in a vortex located at x = 0 , x = ax moving in the x direction with the velocity of light. Taking a morecomplicated function, e.g. a ( y + ) = a y would result in a curved vortex x = a ( x ) etc. One can also combine this dependence, for systems of more vortices,with the other dependences discussed above. This complicates the discussion butdoes not change its main features, hence in the remainder of this paper we returnto the discussion of the dependence on y + through the phase factors.One could naively think that infiniteness of the total energy of our solution isrelated to some “improper” choice of the dependence on y + . This is not true sincethe origin of the divergence comes from the topological nature of H (1) . The factthat H (1) is a total derivative prevents the dependence of H (1) on any parameters(including any depending on y + ). One can note that for some special cases like u = z exp ( − ay ) the contribution to the total energy coming from H (2) is finitebut the total energy remains infinite since H (1) contribution is always present. C P model Next we consider solutions of the CP model. First we look at the holomorphicones. The simplest CP model solution can be obtained by adding to the system (4.18)a constant third function, i.e. define f ( z, y + ) = z + a z e ik y + f ( z, y + ) = a z + a e ik y + f ( z, y + ) = a . (4.34)Then we can define holomorphic configurations as u i = f i f , i = 1 , , i.e. u ( z, y + ) = z + a z e ik y + a , u ( z, y + ) = a z + a e ik y + a . (4.35)Alternatively, we can interchange f ↔ f and consider the holomorphic config-urations ˜ u ( z, y + ) = z + a z e ik y + a z + a e ik y + , ˜ u ( z, y + ) = a a z + a e ik y + . (4.36)11ote from (2.3) that such an interchange corresponds to a phase transformationin Z , so both configurations describe the same solution of the CP model. Notealso (easier from (4.36)) that when a → this CP solution reduces to theholomorphic CP solution discussed before. In fact, ˜ u vanishes, and ˜ u becomesthe CP u -field.The integrals I (1) , I (2) for this CP holomorphic solution (using definition(4.35) or (4.36)) now take the form I (1) = 1 π (cid:90) R dx dx AC , I (2) = 1 π (cid:90) R dx dx BC (4.37)where A , B , C differ from A , B , C given by (4.26), (4.27) and (4.28) by termsproportional to a , i.e. A = A + a [ a + a + 4 r + 4 a r cos ( ϕ − k y + )] (4.38) B = B + a [ a k r + a k ] (4.39) C = C + a . (4.40)The Hamiltonian density H (2) , which is proportional to BC , is now regular at (cid:126)r = (cid:126)r n and y + = ξ n for a = a a (where previously we had a singularity) as now ittakes the value BC (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)r = (cid:126)r n , y + = ξ n = a a (cid:34) a a k + k (cid:35) . Hence we note that going to the CP manifold (by taking a (cid:54) = 0 ) has ‘filled inthe hole’ in the space of parameters ( i.e. as the system evolves none of its vorticesshrinks to the delta function).Note also that the energy density is independent of y + in four cases: k = k , a = 0 , a = 0 and a = 0 . Next we look at the ‘new’ mixed solutions. First we use (3.8) to calculate P z f .We find that for the system (4.34) they take the form P z f = a e ik y + (cid:104) ze − ik y + + a (cid:105) + e ik y + (cid:104) a + a ¯ ze ik y + (cid:105) (cid:104) a a + 2 a ze − ik y + + a z e − i ( k + k ) y + (cid:105) P z f = a a e ik y + ¯ z (cid:104) a + ¯ ze ik y + (cid:105) (cid:104) a a + 2 a ze − ik y + + a z e − i ( k + k ) y + (cid:105) P z f = − a e − ik y + (cid:104) a a + a ¯ ze ik y + + ¯ ze ik y + (¯ ze ik y + + a )(2 ze − ik y + + a ) (cid:105) When written in terms of u i this mixed solution is given by u ( z, ¯ z, y + ) = P z f P z f , u ( z, ¯ z, y + ) = P z f P z f . (4.41)Note that in the limit a → the mixed solution (4.41) becomes the anti-holomorphic solution of the CP model mentioned before. However for a (cid:54) = 0 the solution is different. This time the expressions for the energy density are quitecomplicated - so we do not present them here. However, we note that to guaranteethe convergence of the integral I (2) we have to require that a (cid:54) = 0 .To demonstrate that the energy per unit length does not depend on y + can bechecked without much effort. First, we observe that the overall factors e ik j y + in P z f k do not matter as they cancel in the expressions for | u j | and for | ∆ · u j | .Hence, the only relevant expressions are of the form ze − ik j y + = re i ( ϕ − k j y + ) and ¯ ze ik j y + = re − i ( ϕ − k j y + ) . When k = k ≡ k the energy density depends only onthe combination ( ϕ − ky + ) and r showing that the only effect of the dependenceon time is a rotation and, in consequence, the independence of the energy per unitlength on y + (or x for given x ). The other cases guaranteeing this are a = 0 and a = 0 . Finally we look at the corresponding anti-holomorphic solution. Such a solutionderived from the system (4.34) takes the form u (¯ z, y + ) = P z f P z f = a a e i ( k + k ) y + a a + ¯ ze ik y + (2 a + a ¯ ze ik y + ) (4.42) u (¯ z, y + ) = P z f P z f = − a e ik y + ( a + 2¯ ze ik y + ) a a + ¯ ze ik y + (2 a + a ¯ ze ik y + ) . (4.43)Note that, like for the ‘mixed case’, we have to require that a (cid:54) = 0 as otherwise H (1) = 0 , H (2) = 8 πM k a a ( a + a ) . In the next subsection we will produce an explicit example of these field con-figurations and discuss some of their properties. To avoid the problems mentioned13bove our example will have a (cid:54) = 0 . Note that in such a case the conditions of theindependence of the energy per unit length on y + are the same as for the mixedsolution. In our example we start with the set of functions (4.34) for which we have chosenthe following values of parameters: a = 2 . , a = 0 . , a = 1 . , a = 0 . k = 1 . and k = 2 . . The topological charge of the holomorphic solution is then Q top = 2 . The topological charge density at x = 0 (and for x = 0 ) has two peaks- one of them is localized at z = 0 , the other a bit further out - see Fig. 7. For theholomorphic solution the topological charge density is proportional to the energydensity and this leads to the energy per unit length being given by πM I (1) .The integrand H (1) / M is sketched in Fig. 8. The contribution coming fromthe waves H (2) / M is plotted in Fig. 9. The mixed solution generated by theapplication of the P z operator according to (3.8) leads to a solution which has Q top = 2 − and I (1) = 2 + 2 = 4 . As is easy to see from Fig. 7 theapplication of P z has changed two holomorphic peaks into two anti-peaks and inaddition it has generated two new peaks. The energy density H (1) thus has fourpeaks 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 two peaks of the topologicalcharge density into two anti-peaks and annihilates the previous anti-peaks. Thusthe anti-holomorphic solution is characterized by Q top = − and I (1) = 2 . Thecontribution to the energy per unit length M I (1) is the same as for the initial(holomorphic) case. Nevertheless, the total energies per unit length for these twosolutions differ since for solutions of the CP model the integrals I (2) are different( I (2)hol (cid:54) = I (2)anti − hol ). Let us note that our case has a time-dependent energy perunit length (calculated by the integration over the x x plane). It implies that thedependence of the energy density on y + is highly nontrivial. However, the energyper unit length is a periodic function of y + . Only for some special cases, like k = k etc. the energy per unit length is constant and so does not depend on y + .The time dependence of the energy density for all three solutions is shown in Fig.10. The energy density for the mixed solution for x = 0 and x = π/ , x = π , x = 7 π/ is plotted in Fig. 11. For the case x = π the peaks are maximallyseparated (this is not very clear without a detailed study of some other values x ).In this case the energy takes its maximal value, see Fig. 10.14igure 1: The tube solution. The part ( n , n ) (left) and the component n (right)of the isovector (cid:126)n for a = 0 , a = 0 , a = 2 , x = 0 , x = 0 and k = 2 . Theminimal value n = − occurs at the point x = 0 and x = 0 .Figure 2: The energy density of the tube solution - the topological part (left) andthe wave part (right). Here a = 0 , a = 0 , a = 2 , x = 0 , x = 0 and k = 2 .15igure 3: The spiral solution. The part ( n , n ) (left) and the component n (right)of the isovector (cid:126)n for a = 2 , a = 1 , a = 0 , x = 0 , x = 0 and k = 1 .The minimal value n = − occurs at the point x = − a cos ( k y + ) and x = − a sin ( k y + ) ; here x = − , x = 0 .Figure 4: The energy density of the spiral solution - the topological part (left) andthe wave part (right). Here a = 2 , a = 1 , a = 0 , and x = 0 , x = 0 and k = 1 . The maxima of the energy density for both contributions are located atthe same point on the plane x x corresponding with the minimum of n (see Fig3); here x = − , x = 0 . 16igure 5: The CP solution with all a k (cid:54) = 0 . The part ( n , n ) (left) and thecomponent n (right) of isovector (cid:126)n for a = 2 , a = 1 , a = 3 , x = 3 π/ , x = 0 , k = 1 and k = 2 .Figure 6: The energy density of the CP solution with all a k (cid:54) = 0 - the topologicalpart (left) and the wave part (right). Here a = 2 , a = 1 , a = 3 , x = 3 π/ , x = 0 , k = 1 and k = 2 . 17igure 7: The functions πρ top (where ρ top is a topological charge density) for x =0 , x = 0 , a = 2 . , a = 0 . , a = 1 . , a = 0 . , k = 1 and k = 2 . The leftpicture corresponds to the holomorphic solution, the central picture correspondsto the mixed solution and the right picture to the anti-holomorphic solution.Figure 8: The functions H (1) / M (proportional to topological contribution to theenergy density) for x = 0 , x = 0 , a = 2 . , a = 0 . , a = 1 . , a = 0 . , k = 1 and k = 2 . The left picture corresponds to the holomorphic solution,the central picture corresponds to the mixed solution and the right picture to theanti-holomorphic solution. 18igure 9: The functions H (2) / M (proportional to wave contribution to the en-ergy density) for x = 0 , x = 0 , a = 2 . , a = 0 . , a = 1 . , a = 0 . , k = 1 and k = 2 . The left picture corresponds to the holomorphic solution,the central picture corresponds to the mixed solution and the right picture to theanti-holomorphic solution. Figure 10: The integral I (2) as the function of x ∈ [0 , π ] . The other parametersread: x = 0 , a = 2 . , a = 0 . , a = 1 . , a = 0 . , k = 1 and k =2 . The left picture corresponds to the holomorphic solution, the central picturecorresponds to the mixed solution and the right picture to the anti-holomorphicsolution. 19igure 11: Time evolution of the mixed solution for a = 2 . , a = 0 . , a =1 . , a = 0 . , k = 1 and k = 2 . The functions H (1) / M (left column)and H (2) / M (right column) have been considered for x = 0 at the moments x = π/ (first row), x = π (second row), x = 7 π/ (third row). Their valuesfor x = 0 are sketched at the central pictures of Fig 8 and Fig 9.20 Conclusions and Further Comments
In this paper we have demonstrated that the CP N model in (3+1) dimensions hasmany classical solutions. Our construction has been based on the observation thatone can generalise ideas used in the construction of solutions of the CP N model in(2+0) dimensions and generate vortex and vortex-antivortex like solutions of thismodel in (3+1) dimensions. Like for the model in (2+0) dimensions we can gen-erate these solutions from field configurations described by polynomial functionsof x + i(cid:15) x . This time the coefficients of these functions could be also functionsof x + ε x . The energy of such configurations is infinite (as the energy densityis independent of x ) and so we interpret these solutions as describing systems ofvortices and antivortices.Of course our expressions solve equations in (3+1) dimensions and they alsodetermine the dynamics of these vortices.In this paper we have only looked at the simplest solutions (correspondingto very few vortices) with the time dependence being described by simple phasefactors. Even in this case the observed dynamics is quite complicated and hasexhibited various interesting properties. In particular, we have shown that thevortices can rotate in space (physical and internal) and their energy per unit lengthof the vortex can vary in time. During this time evolution some vortices canshrink to delta functions and then expand again often being characterised by avery periodical behaviour.One other unusual property is their dependence on the distance between thevortices: the energy density of two vortices can depend on the distance betweenthem and can possess a minimum at a specific value of this distance. This sug-gests that vortices which are located at non-minimal distances may be unstableand so could try to reduce their energy per unit length by moving towards this op-timal configurations. However, their configurations are solutions for any distanceas their infinite ‘inertia’ stops them from moving towards each other without anexternal push.We are now looking at other properties of these and other solutions. Acknowlegment:
L.A. Ferreira and W.J. Zakrzewski would like to thank theRoyal Society (UK) for a grant that helped them in carrying out this work. L.A.Ferreira is partially supported by CNPq (Brazil) and P. Klimas is supported byFAPESP (Brazil). 21 eferences [1] A. D’Adda, P. Di Vecchia and M. Luscher,
A 1/N expandable series of non-linear σ models with instantons , Nucl. Phys. B 146, (1978), 63[2] W.J. Zakrzewski, Low Dimensional Sigma Models (Hilger, Bristol, 1989).[3] L. A. Ferreira, P. Klimas and W.J. Zakrzewski,
Some (3+1)-dimensional vor-tex solutions of the CP N model , Phys. Rev. D 83 (2011) 105018[4] L. A. Ferreira,