Lagrangian formulation for electric charge in a magnetic monopole distribution
G. Marmo, Emanuela Scardapane, A. Stern, Franco Ventriglia, Patrizia Vitale
aa r X i v : . [ h e p - t h ] J u l Lagrangian formulation for electric chargein a magnetic monopole distribution
G. Marmo ∗ , Emanuela Scardapane † , A. Stern ‡ , Franco Ventriglia , § and Patrizia Vitale , ¶ Dipartimento di Fisica E. Pancini Universit`a di Napoli Federico II,Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy Department of Physics, University of Alabama,Tuscaloosa, Alabama 35487, USA INFN Sez. di NapoliComplesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy
ABSTRACT
We give a Lagrangian description of an electric charge in a field sourced by a continuousmagnetic monopole distribution. The description is made possible thanks to a doubling of theconfiguration space. The Legendre transform of the nonrelativistic Lagrangian agrees with theHamiltonian description given recently by Kupriyanov and Szabo[1]. The covariant relativisticversion of the Lagrangian is shown to introduce a new gauge symmetry, in addition to standardreparametrizations. The generalization of the system to open strings coupled to a magneticmonopole distribution is also given, as well as the generalization to particles in a non-Abeliangauge field which does not satisfy Bianchi identities in some region of the space-time. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] Introduction
It is well known that a local Lagrangian description for an electric charge in the presence offields sourced by an electric charge distribution requires the introduction of potentials on theconfiguration space, introducing unphysical, or gauge, degrees of freedom in the field theory.If the field is sourced by a magnetic monopole, the description can be modified by changingthe topology of the underlying configuration space, see e.g.,[2],[3]. On the other hand, thisprocedure has no obvious extension when the fields are sourced by a continuous distribution ofmagnetic charge. In that case, auxiliary degrees of freedom can be added, possibly introducingadditional local symmetries. One possibility is to introduce another set of potentials followingwork of Zwanziger[4]. Another approach is to enlarge the phase space for the electric charge,and this was done recently by Kupriyanov and Szabo [1]. The result has implications for certainnongeometric string theories and their quantization, which leads to nonassociative algebras,see e.g.,[5]-[13].The analysis of [1] for the electric charge in a field sourced by magnetic monopole distri-bution is performed in the Hamiltonian setting. The formulation is made possible thanks tothe doubling of the number of phase space variables. In this letter we give the correspondingLagrangian description. It naturally requires doubling the number of configuration space vari-ables. So here if Q denotes the original configuration space, one introduces another copy, ˜ Q and writes down dynamics on Q × ˜ Q . While the motion on the two spaces, in general, can-not be separated, the Lorentz force equations are recovered when projecting down to Q . Theprocedure of doubling the configuration space has a wide range of applications, and actuallywas used long ago in the description of quantum dissipative systems [14]-[18]. The descriptionin [1] is nonrelativistic. Here, in addition to giving the associated nonrelativistic Lagrangian,we extend the procedure to the case of a covariant relativistic particle, as well as to particlescoupled to non-Abelian gauge fields that do not necessarily satisfy the Bianchi identity in aregion of space-time. As a further generalization we consider the case of an open string coupledto a smooth distribution of magnetic monopoles.The outline of this article is as follows. In section 2 we write down the Lagrangian fora nonrelativistic charged particle in the presence of a magnetic field whose divergence fieldis continuous and nonvanishing in a finite volume of space, and show that the correspondingHamiltonian description is that of [1]. The relativistic generalization is given in section 3.Starting with a fully covariant treatment we obtain a new time dependent symmetry, in ad-dition to standard reparametrization invariance. The new gauge symmetry mixes ˜ Q with Q .Gauge fixing constraints can be imposed on the phase space in order to recover the Poissonstructure of the nonrelativistic treatment on the resulting constrained submanifold. Furtherextensions of the system are considered in section 4. In subsection 4.1 we write down the ac-tion for a particle coupled to a non-Abelian gauge field which does not satisfy Bianchi identityin some region of space-time, whereas in 4.2 we generalize to field theory, by considering anopen string coupled to a magnetic monopole distribution, again violating Bianchi identity. In2oth cases we get a doubling of the configuration space variables (which in the case of theparticle in a non-Abelian gauge field includes variables living in an internal space), as well asa doubling of the number of gauge symmetries. We note that the doubling of the number ofworld-sheet degrees of freedom of the string is also the starting point of Double Field Theory,introduced by Hull and Zwiebach[19], and further investigated by many authors[20]-[25], inorder to deal with the T-duality invariance of the strings dynamics. This has its geometriccounterpart in Generalized and Double Geometry (see e.g. [26, 27] and [28]-[32]). Moreover,the doubling of configuration space has also been related to Drinfel’d doubles in the context ofLie groups dynamics [33]-[37] with interesting implications for the mathematical and physicalinterpretation of the auxiliary variables. We begin with a nonrelativistic charged particle on R in the presence of a continuous mag-netic monopole distribution. Say that the particle has mass m and charge e with coordinatesand velocities ( x i , ˙ x i ) spanning T R . It interacts with a magnetic field ~B ( x ) of nonvanishingdivergence ~ ∇ · ~B ( x ) = ρ M ( x ) . In such a case it is possible to show that the dynamics of theparticle, described by the equations of motion m ¨ x i = eǫ ijk ˙ x j B k ( x ) (2.1)cannot be given by a Lagrangian formulation on the tangent space T R because a vectorpotential for the magnetic field generated by the smooth monopoles distribution cannot defined,even locally. (A detailed discussion of this issue will appear in [38].) On the other hand, aLagrangian description is possible if one enlarges the configuration space to R × e R , and thisdescription leads to Kupriyanov and Szabo’s Hamiltonian formulation [1]. For this one extendsthe tangent space to T ( R × f R ) ≃ T R × g T R . We parametrize g T R by (˜ x i , ˙˜ x i ) , i = 1 , , L = m ˙ x i ˙˜ x i + eǫ ijk B k ( x )˜ x i ˙ x j , (2.2)correctly reproduces Eq. (2.1), together with an equation of motion for the auxiliary degreesof freedom ˜ x i m ¨˜ x i = eǫ ijk ˙˜ x j B k ( x ) + e (cid:16) ǫ jkℓ ∂∂x i B k − ǫ ikℓ ∂∂x j B k (cid:17) ˙ x j ˜ x ℓ (2.3)which are not decoupled from the motion of the physical degrees of freedom. Here we donot ascribe any physical significance to the auxiliary dynamics. There are analogous degreesof freedom for dissipative systems, and they are associated with the environment. Since oursystem does not dissipate energy, the same interpretation does not obviously follow. The La-grangian (2.2) can easily be extended to include electric fields. This, along with the relativisticgeneralization, is done in the following section.3n passing to the Hamiltonian formalism, we denote the momenta conjugate to x i and ˜ x i by p i = m ˙˜ x i − eǫ ijk ˜ x j B k ( x )˜ p i = m ˙ x i , (2.4)respectively. Along with x i and ˜ x i , they span the 12-dimensional phase space T ∗ ( R × g R ).The nonvanishing Poisson brackets are { x i , p j } = { ˜ x i , ˜ p j } = δ ij (2.5)Instead of the canonical momenta (2.4) one can define π i = p i + eǫ ijk ˜ x j B k ( x ) ˜ π i = ˜ p i , (2.6)which have the nonvanishing Poisson brackets: { x i , π j } = { ˜ x i , ˜ π j } = δ ij { π i , ˜ π j } = eǫ ijk B k { π i , π j } = e (cid:16) ǫ jkℓ ∂∂x i B k − ǫ ikℓ ∂∂x j B k (cid:17) ˜ x ℓ (2.7)The Hamiltonian when expressed in these variables is H = 1 m ˜ π i π i (2.8)Eqs. (2.7) and (2.8) are in agreement with the Hamiltonian formulation in [1].Concerning the issue of the lack of a lower bound for H , one can follow the perspective in[39], where a very similar Hamiltonian dynamics is derived. Namely, while it is true that H generates temporal evolution, it cannot be regarded as a classical observable of the particle.Rather, such observables should be functions of only the particle’s coordinates x i and itsvelocities ˜ π i /m , whose dynamics is obtained from their Poisson brackets with H ˙ x i = { x i , H } = 1 m ˜ π i ˙˜ π i = { ˜ π i , H } = em ǫ ijk ˜ π j B k (2.9)The usual expression for the energy, m ˜ π i ˜ π i , is, of course, an observable, which is positive-definite and a constant of motion. 4 Relativistic covariant treatment
The extension of the Lagrangian dynamics of the previous section can straightforwardly bemade to a covariant relativistic system. In the usual treatment of a covariant relativisticparticle, written on T R , one obtains a first class constraint in the Hamiltonian formulationwhich generates reparametrizations. Here we find that the relativistic action for a chargedparticle in a continuous magnetic monopole distribution, which is now written on T R × g T R ,yields an additional first class constraint, generating a new gauge symmetry. When projectingthe Hamiltonian dynamics onto the constrained submanifold of the phase space, and takingthe nonrelativistic limit, we recover the Hamiltonian description of [1].As stated above, our action for the charged particle in a continuous magnetic monopoledistribution is written on T R × g T R . Let us parametrize T R by space-time coordinates andvelocity four-vectors ( x µ , ˙ x µ ), and g T R by (˜ x µ , ˙˜ x µ ) , µ = 0 , , ,
3. So here we have includedtwo ‘time’ coordinates, x and ˜ x . Now the dot denotes the derivative with respect to somevariable τ which parametrizes the particle world line in R × f R . The action for a chargedparticle in an electromagnetic field F µν ( x ), which does not in general satisfy the Bianchi identity ∂∂x µ F νρ + ∂∂x ν F ρµ + ∂∂x ρ F µν = 0 is S = Z dτ n m ˙ x µ ˙˜ x µ √− ˙ x ν ˙ x ν + eF µν ( x )˜ x µ ˙ x ν + L ′ ( x, ˙ x ) o , (3.1) L ′ ( x, ˙ x ) is an arbitrary function of x µ and ˙ x µ . Indices are raised and lowered with the Lorentzmetric η =diag( − , , , τ , τ → τ ′ = f ( τ ), provided we choose L ′ appropriately. The actionis also invariant under a local transformation that mixes ˜ R with R , x µ → x µ ˜ x µ → ˜ x µ + ǫ ( τ ) ˙ x µ √− ˙ x ν ˙ x ν , (3.2)for an arbitrary real function ǫ ( τ ). The first term in the integrand of (3.1) changes by a τ − derivative under (3.2), while the remaining terms in the integrand are invariant.Upon extremizing the action with respect to arbitrary variations δ ˜ x µ of ˜ x µ , we recover thestandard Lorentz force equation on T R ˙˜ p µ = eF µν ( x ) ˙ x ν , (3.3)while arbitrary variations δx µ of x µ lead to˙ p µ = e ∂F ρσ ∂x µ ˜ x ρ ˙ x σ + ∂L ′ ∂x µ (3.4) p µ and ˜ p µ are the momenta canonically conjugate to x µ and ˜ x µ , respectively, p µ = m ( − ˙ x ρ ˙ x ρ ) / ( ˙ x µ ˙˜ x ν − ˙ x ν ˙˜ x µ ) ˙ x ν − eF µν ˜ x ν + ∂L ′ ∂ ˙ x µ p µ = m ˙ x µ √− ˙ x ν ˙ x ν (3.5)The momenta p µ and ˜ p µ , along with coordinates x µ and ˜ x µ , parametrize a 16 − dimensionalphase space, which we denote simply by T ∗ Q . x µ , ˜ x µ , p µ and ˜ p µ satisfy canonical Poissonbrackets relations, the nonvanishing ones being { x µ , p ν } = { ˜ x µ , ˜ p ν } = δ µν (3.6)˜ p µ satisfies the usual mass shell constraintΦ = ˜ p µ ˜ p µ + m ≈ , (3.7)where ≈ means ‘weakly’ zero in the sense of Dirac. Another constraint isΦ = p µ ˜ p µ + eF µν ( x )˜ p µ ˜ x ν ≈ , (3.8)where from now on we set L ′ = 0.The three-momenta π i and ˜ π i of the previous section can easily be generalized to four-vectors according to π µ = p µ + eF µν ( x )˜ x ν ˜ π µ = ˜ p µ (3.9)Their nonvanishing Poisson brackets are { x µ , π ν } = { ˜ x µ , ˜ π ν } = δ µν { π µ , ˜ π ν } = eF µν { π µ , π ν } = − e (cid:16) ∂∂x µ F νρ + ∂∂x ν F ρµ (cid:17) ˜ x ρ (3.10)Then the constraints (3.7) and (3.8) take the simple formΦ = ˜ π µ ˜ π µ + m ≈ = π µ ˜ π µ ≈ { Φ , Φ } = 0, and therefore Φ and Φ form a first class set of constraints.They generate the two gauge (i.e., τ − dependent) transformations on T ∗ Q . Unlike in thestandard covariant treatment of a relativistic particle, the mass shell constraint Φ does notgenerate reparametrizations. Φ instead generates the transformations (3.2), while a linearcombination of Φ and Φ generate reparametrizations. After imposing (3.7) and (3.8) on T ∗ Q ,one ends up with a gauge invariant subspace that is 12-dimensional, which is in agreement withthe dimensionality of the nonrelativistic phase space.Alternatively, one can introduce two additional constraints on T ∗ Q which fix the two timecoordinates x and ˜ x , and thus break the gauge symmetries. The set of all four constraintswould then form a second class set, again yielding a 12-dimensional reduced phase space, which6e denote by T ∗ Q . The dynamics on the reduced phase space is then determined from Diracbrackets and some Hamiltonian H . We choose H to be H = p = π − eF i ( x )˜ x i (3.12) p differs from π in the presence of an electric field. The latter can be expressed as a functionof the spatial momenta π i and ˜ π i , i = 1 , ,
3, after solving the constraints (3.11). The result is π = π i ˜ π i q ˜ π j + m , (3.13) π correctly reduces to the non-relativistic Hamiltonian (2.8) in the limit ˜ π j << m .In addition to recovering the non-relativistic Hamiltonian of the previous section, the gaugefixing constraints, which we denote by Φ ≈ ≈
0, can be chosen such that the Diracbrackets on T ∗ Q agree with the Poisson brackets (2.7) of the nonrelativistic treatment. Forthis take Φ = x − g ( τ ) Φ = ˜ x − h ( τ ) , (3.14)where g and h are unspecified functions of the proper time. By definition, the Dirac bracketsbetween two functions A and B of the phase space coordinates are given by { A, B } DB = { A, B } − X a,b =1 { A, Φ a } M − ab { Φ b , B } , (3.15)where M − is the inverse of the matrix M with elements M ab = { Φ a , Φ b } , a, b = 1 , ...,
4. Fromthe constraints (3.11) and (3.14) we get M − = 12(˜ π ) − π ˜ π π π − π − ˜ π (3.16)Substituting into (3.15) gives { A, B } DB = { A, B } − π ) π (cid:16) { A, x }{ ˜ π µ ˜ π µ , B } − { B, x }{ ˜ π µ ˜ π µ , A } (cid:17) − ˜ π (cid:16) { A, ˜ x }{ ˜ π µ ˜ π µ , B } − { B, ˜ x }{ ˜ π µ ˜ π µ , A } (cid:17) − π (cid:16) { A, x }{ ˜ π µ π µ , B } − { B, x }{ ˜ π µ π µ , A } (cid:17) ! (3.17)It shows that the Dirac brackets { A, B } DB and their corresponding Poisson brackets { A, B } are equal if both functions A and B are independent of π and ˜ π . We need to evaluate theDirac brackets on the constrained subsurface, which we take to be T R × g T R , parametrized7y x i , ˜ x i , π i and ˜ π i , i = 1 , ,
3. It is then sufficient to compute their Poisson brackets. Thenonvanishing Poisson brackets of the coordinates of T R × g T R are: { x i , π j } = { ˜ x i , ˜ π j } = δ ij { π i , ˜ π j } = eǫ ijk B k { π i , π j } = e (cid:16) ǫ jkℓ ∂∂x i B k − ǫ ikℓ ∂∂x j B k (cid:17) ˜ x ℓ + e (cid:16) ∂∂x i E j − ∂∂x j E i (cid:17) h ( τ ) , (3.18)where F ij = ǫ ijk B k , F i = E i and we have imposed the constraint Φ = 0. These Poissonbrackets agree with those of the nonrelativistic treatment, (2.7), in the absence of the electricfield. Here we extend the dynamics of the previous sections to 1) the case of a particle coupled to anon-Abelian gauge field violating Bianchi identities and 2) the case of an open string coupledto a smooth distribution of magnetic monopoles. Of course, another extension would be thecombination of both of these two cases, i.e., where an open string interacts with a non-Abeliangauge field that does not satisfy the Bianchi identities in some region of the space-time. Weshall not consider that here.
Here we replace the underlying Abelian gauge group of the previous sections, with an N dimensional non-Abelian Lie group G . We take it to be compact and connected with a simpleLie algebra. Given a unitary representation Γ of G , let t A , A = 1 , , ...N span the correspondingrepresentation ¯Γ of the Lie algebra, satisfying t † A = t A , Tr t A t B = δ AB and [ t A , t B ] = ic ABC t C , c ABC being totally antisymmetric structure constants. In Yang-Mills field theory, the fieldstrengths now take values in ¯Γ, F µν ( x ) = f Aµν ( x ) t A . A particle interacting with a Yang-Mills field carries degrees of freedom I ( τ ) associated with the non-Abelian charge, in additionto space-time coordinates x µ ( τ ). These new degrees of freedom live in the internal space¯Γ, I ( τ ) = I A ( τ ) t A . Under gauge transformations, I ( τ ) transforms as a vector in the adjointrepresentation of G , just as do the field strengths F µν ( x ), i.e., I ( τ ) → h ( τ ) I ( τ ) h ( τ ) † , h ( τ ) ∈ Γ.The standard equations of motion for a particle in a non-Abelian gauge field were given longago by Wong.[40] They consist of two sets of coupled equations. One set is a straightforwardgeneralization of the Lorentz force law˙˜ p µ = Tr (cid:16) F µν ( x ) I ( τ ) (cid:17) ˙ x ν , (4.1)8here ˜ p µ is again given in (3.5). The other set consists of first order equations describing theprecession of I ( τ ) in the internal space ¯Γ. Yang-Mills potentials are required in order to writethese equations in a gauge-covariant way.The Wong equations were derived from action principles using a number of different ap-proaches. The Yang-Mills potentials again play a vital role in all of the Lagrangian descriptions.In the approach of co-adjoint orbits, one takes the configuration space to be Q = R × Γ, andwrites[41],[3] I ( τ ) = g ( τ ) Kg ( τ ) † , (4.2)where g ( τ ) takes values in Γ, and K is a fixed direction in ¯Γ. Under gauge transformations, g ( τ ) transforms with the left action of the group, g ( τ ) → h ( τ ) g ( τ ), h ( τ ) ∈ Γ. The two sets ofWong equations result from variations of the action with respect to g ( τ ) and x µ ( τ ).Now in the spirit of [1] we imagine that there is a region of space-time where the Bianchiidentity does not hold, and so the usual expression for the field strengths in terms of theYang-Mills potentials is not valid. So we cannot utilize the known actions which yield Wong’sequations, as they require existence of the potentials. We can instead try a generalizationof (3.1), which doubles the number of space-time coordinates. This appears, however, to beinsufficient. In order to have a gauge invariant description for the particle, we claim that itis necessary to double the number of internal variables as well. Thus we double the entireconfiguration space, Q → Q × ˜ Q . Proceeding along the lines of the coadjoint orbits approach,we take ˜ Q to be another copy of R × Γ. Let us denote all the dynamical variables in thiscase to be x µ ( τ ), ˜ x µ ( τ ), g ( τ ) and ˜ g ( τ ), where both g ( τ ) and ˜ g ( τ ) take values in Γ and gaugetransformation with the left action of the group, g ( τ ) → h ( τ ) g ( τ ), ˜ g ( τ ) → h ( τ )˜ g ( τ ), h ( τ ) ∈ Γ.We now propose the following gauge invariant action for the particle S = Z dτ (cid:26) Tr Kg ( τ ) † ˙ g ( τ ) − Tr I ( τ ) ˙˜ g ( τ )˜ g ( τ ) † + m ˙ x µ ˙˜ x µ √− ˙ x ν ˙ x ν + Tr (cid:16) F µν ( x ) I ( τ ) (cid:17) ˜ x µ ˙ x ν (cid:27) , (4.3)where I ( τ ) is defined in (4.2). To see that the action is gauge invariant we note that the firsttwo terms in the integrand can be combined to: Tr Kg ( τ ) † ˜ g ( τ ) ddτ (cid:16) ˜ g ( τ ) † g ( τ ) (cid:17) , ˜ g ( τ ) † g ( τ ) beinggauge invariant. Variations of ˜ x µ in the action yields the Wong equation (4.1). Variations of x µ in the action gives a new set of equations defining motion on the enlarged configurationspace ˙ p µ = Tr (cid:16) ∂F ρσ ∂x µ I ( τ ) (cid:17) ˜ x ρ ˙ x σ , where p µ = m ( − ˙ x ρ ˙ x ρ ) / ( ˙ x µ ˙˜ x ν − ˙ x ν ˙˜ x µ ) ˙ x ν − Tr (cid:16) F µν I ( τ ) (cid:17) ˜ x ν (4.4)These equations are the non-Abelian analogues of (3.4). The remaining equations of motionresult from variations of the g ( τ ) and ˜ g ( τ ) and describe motion in Γ × Γ. Infinitesimal variationsof g ( τ ) and ˜ g ( τ ) may be performed as follows: For ˜ g ( τ ), it is simpler to consider variations9esulting from the right action on the group, δ ˜ g ( τ ) = i ˜ g ( τ )˜ ǫ ( τ ), ˜ ǫ ( τ ) ∈ ¯Γ. The action (4.3) isstationary with respect to these variations when ddτ (cid:16) ˜ g ( τ ) I ( τ )˜ g ( τ ) † (cid:17) = 0 , (4.5)thus stating that ˜ g ( τ ) I ( τ )˜ g ( τ ) † is a constant of the motion. For g ( τ ), consider variationsresulting from the left action on the group, δg ( τ ) = iǫ ( τ ) g ( τ ), ǫ ( τ ) ∈ ¯Γ. These variations leadto the equations of motion ˙ I ( τ ) = h I ( τ ) , ˙˜ g ( τ )˜ g ( τ ) † − F µν ( x )˜ x µ ˙ x ν i (4.6)The consistency of both (4.5) and (4.6) leads to the following constraint on the motion h I ( τ ) , F µν ( x ) i ˜ x µ ˙ x ν = 0 (4.7)This condition on T Q × T ˜ Q is a feature of the non-Abelian gauge theory, and is absent fromthe Abelian gauge theory. Finally we generalize the case of a particle interacting with a smooth magnetic monopoledistribution, to that of a string interacting with the same monopole distribution. Just aswe doubled the number of particle coordinates in the previous sections, we now double thenumber of string coordinates. We note that a doubling of the world-sheet coordinates of thestring, originally limited to the compactified coordinates, also occurs in the context of DoubleField Theory,[20] with the original purpose of making the invariance of the dynamics underT-duality a manifest symmetry of the action. The approach has been further extended tostrings propagating in so called non-geometric backgrounds [42],[43],[11],[12], which leads toquasi-Posson brackets, violating the Jacobi identity. The resolution involves a doubling of theworld-sheet coordinates, similar to what happens in the case under study.Whereas the configuration space for a Nambu-Goto string moving in d dimensions is R d ,which can have indefinite signature, here we take it to be R d × f R d . Denote the string coordi-nates for R d and f R d by x µ ( σ ) and ˜ x µ ( σ ), µ = 0 , , ..., d −
1, respectively, where σ = ( σ , σ )parametrizes the string world sheet, M . σ is assumed to be a time-like parameter, and σ aspatial parameter. In addition to writing down the induced metric g on T R d , g ab = ∂ a x µ ∂ b x µ , (4.8)where ∂ a = ∂∂σ a , a , b , ... = 0 , g on T R d × g T R d ,˜ g ab = ∂ a x µ ∂ b ˜ x µ (4.9)For the free string action we propose to replace the usual Nambu-Goto action by S = 12 πα ′ Z M d σ p − det g g ab ˜ g ab , (4.10)10here g ab denote matrix elements of g − and α ′ is the string constant.The action (4.10), together with the interacting term given below, is a natural generalizationof the point-particle action Eq. (3.1) because: • Just as with the case of the relativistic point particle action in section 3, it is relativisti-cally covariant. • Just as with the case of the relativistic point particle action in section 3, there is a newgauge symmetry, in addition to reparametrizations, σ a → σ ′ a = f a ( σ ), leading to newfirst class constraints in the Hamiltonian formalism. This new gauge symmetry mixes˜ R d with R d . Infinitesimal variations are given by δx µ = 0 δ ˜ x µ = ǫ a ( σ ) ∂ a x µ √− det g , (4.11)where ǫ a ( σ ) are arbitrary functions of σ , which we assume vanish at the string boundaries.This is the natural generalization of the τ − dependent symmetry transformation (3.2) forthe relativistic point particle. Invariance of S under variations (4.11) follows from: δS = 12 πα ′ Z M d σ p − det g g ab ∂ a x µ ∂ b (cid:16) ǫ c ∂ c x µ √− det g (cid:17) = 12 πα ′ Z M d σg ab (cid:18) g ac ∂ b ǫ c + ∂ a x µ ∂ b ∂ c x µ ǫ c − ∂ b det g g g ac ǫ c (cid:19) = 12 πα ′ Z M d σ (cid:18) ∂ c ǫ c + g ab (cid:16) ∂ a x µ ∂ b ∂ c x µ − ∂ c g ab (cid:17) ǫ c (cid:19) = 12 πα ′ Z ∂ M dσ a ǫ a , (4.12)which vanishes upon requiring ǫ a | ∂ M = 0 . • The action (4.10) leads to the standard string dynamics when projecting the equationsof motion to R d . Excluding for the moment interactions, variations of the action S withrespect to ˜ x µ ( σ ) away from the boundary ∂ M give the equations of motion ∂ a ˜ p a µ = 0 , ˜ p a µ = 12 πα ′ p − det g g ab ∂ b x µ (4.13)These are the equations of motion for a Nambu string. In addition to recovering theusual string equations on R d , variations of S with respect to x µ ( σ ) lead to another setof the equation of motion on R d × ˜ R d ∂ a p a µ = 0 , p a µ = 12 πα ′ p − det g n ( g ab g cd − g ad g bc − g ac g bd ) ˜ g cd ∂ b x µ + g ab ∂ b ˜ x µ o (4.14)Of course, (4.10) can be used for both a closed string and an open string. We now includeinteractions to the electromagnetic field. They occur at the boundaries of an open string, and11re standardly expressed in terms of the electromagnetic potential, which again is not possiblein the presence of a continuous magnetic monopole charge distribution. So here we take instead S I = e Z ∂ M dσ a F µν ( x )˜ x µ ∂ a x ν , (4.15)where F µν ( x ), is not required to satisfy the Bianchi identity in a finite volume of R d . We take −∞ < σ < ∞ , 0 < σ < π , with σ = 0 , π denoting the spatial boundaries of the string.Then the boundary equations of motion resulting from variations of ˜ x µ ( σ ) in the total action S = S + S I are (cid:16) ˜ p µ + eF µν ( x ) ∂ x ν (cid:17)(cid:12)(cid:12)(cid:12) σ =0 ,π = 0 , (4.16)which are the usual conditions in R d . The boundary equations of motion resulting fromvariations of x µ ( σ ) in the total action S = S + S I give some new conditions in R d × ˜ R d (cid:18) p µ + e (cid:16) ∂∂x µ F ρσ + ∂∂x σ F µρ (cid:17) ˜ x ρ ∂ x σ + eF µν ∂ ˜ x ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) σ =0 ,π = 0 (4.17)In the Hamiltonian formulation of the system π µ = p µ and ˜ π µ = ˜ p µ are canonically conju-gate to x µ and ˜ x µ , respectively, having equal time Poisson brackets n x µ ( σ , σ ) , π ν ( σ , σ ′ ) o = n ˜ x µ ( σ , σ ) , ˜ π ν ( σ , σ ′ ) o = δ µν δ ( σ − σ ′ ) , (4.18)for 0 < σ , σ ′ < π , with all other equal time Poisson brackets equal to zero. The canonicalmomenta are subject to the four constraints:Φ = ˜ π µ ˜ π µ + 1(2 πα ′ ) ∂ x µ ∂ x µ ≈ = ˜ π µ ∂ x µ ≈ = π µ ˜ π µ + 1(2 πα ′ ) ∂ x µ ∂ ˜ x µ ≈ = π µ ∂ x µ + ˜ π µ ∂ ˜ x µ ≈ and Φ generate the local symmetry trans-formations (4.11), while linear combinations of the four constraints generate reparametriza-tions. We have considered the problem of the existence of a Lagrangian description for the motionof a charged particle in the presence of a smooth distribution of magnetic monopoles. Themagnetic field does not admit a potential on the physical configuration space. Auxiliary vari-ables are employed in order to solve the problem, following a procedure commonly used to deal12ith dissipative dynamics. This is the Lagrangian counterpart of the Hamiltonian problem,addressed in [1], where the Bianchi identity violating magnetic field entails a quasi-Poissonalgebra on the physical phase space which does not satisfy Jacobi identity unless one doublesthe number of degrees of freedom. The problem was further extended to the relativistic case,as well as non-Abelian case. In the last section, we performed the generalization of the rel-ativistic point-particle action (3.1) to that of an open string interacting, once again, with aBianchi identity violating magnetic field. In order to circumvent the problem of the lack of apotential vector, the world-sheet degrees of freedom have been doubled analogous to the casein double field theory. Many interesting issues can be addressed, such as a possible relationshipwith double field theory, or the quantization problem, which relates Jacobi violation to non-associativity of the quantum algebra. We plan to investigate these aspects in a forthcomingpublication.
Acknowledgements . G.M. is a member of the Gruppo Nazionale di Fisica Matematica(INDAM),Italy. He would like to thank the support provided by the Santander/UC3M Excellence ChairProgramme 2019/2020; he also acknowledges financial support from the Spanish Ministry ofEconomy and Competitiveness, through the Severo Ochoa Programme for Centres of Excel-lencein RD (SEV-2015/0554).
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