SECOND-ORDER LAGRANGIAN FORMULATION OF LINEAR FIRST-ORDER FIELD EQUATIONS

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1 SECOND-ORDER LAGRANGIAN FORMULATION OF LINEAR FIRST-ORDER FIELD EQUATIONS CONSTANTIN BIZDADEA, SOLANGE-ODILE SALIU Department of Physics, University of Craiova 13 Al. I. Cuza Str., Craiova 00585, Romania Received December 15, 014 A second-order Lagrangian formulation with respect to a set of linear first-order field equations either relativistic or not is proposed. The general formalism is illustrated in the case of first-order field equations containing a single spatial derivative. Key words: field equations, Lagrangian approach, second-class constraints. PACS: Ef. The relationship between the Lagrangian [1] and Hamiltonian [] formalisms stands for a subject of interest in theoretical physics in the context of both degenerate and non-degenerate dynamical systems [3 5]. It is well-known that the first-order Hamilton equations usually originate from a second-order Lagrangian formulation. For instance, the Klein Gordon [6, 7], Maxwell [8 1] written in terms of potentials, or Einstein [13] actions can be expressed in both second- and first-order form. Recently it has been shown that Schrödinger s equation [14] also allows for a secondorder Lagrangian formulation [15]. On the other hand, neither of Dirac [16], chiral bosons in two dimensions [17], or chiral self-dual p-forms [18] actions allows an equivalent local second-order form. Thus, there appears to be an antisymmetry between the first- and second-order formulations of dynamics. In this paper we investigate the possibility to derive a second-order Lagrangian formulation of independent, linear first-order field equations relativistic or not, not necessarily in a Hamiltonian form. Let Q A t,x 1,,x n be a set of fields that parameterize the time-evolution of a dynamical system. We denote by ε A the Grassmann parity of Q A. Assume that the considered system is described by the system of independent, linear first-order equations H A Q A k s=1 Γ A j 1 j s B j1 js Q B m A BQ B = 0, 1 where the objects Γ A j 1 j s B and m A B may be functions of x = x 1,,x n. RJP Rom. 61Nos. Journ. Phys., 1-, Vol , Nos , P. c 7 36, 016 Bucharest, - v.1.3a*

2 8 Constantin Bizdadea, Solange-Odile Saliu Let us try some solutions of the form of equations 1, where we used the notations Ô A B = Q A = Φ A + ÔA BΦ B, k s=1 Γ A j 1 j s B j1 js + m A B. 3 Under these considerations, the following result can be shown to hold: Q A given by are solutions to the first-order equations given in 1 if and only if Φ A are solutions to the second-order equations E A Φ A ÔA CÔC BΦ B = 0. 4 Moreover, it is easy to show that formulas express the general form of the solutions to equations 1. The above result emphasizes that the Φ A s play the role of potentials associated to the observables Q A. Now, we investigate the following problem: can the second-order equations 4 originate in some Euler Lagrange equations? In view of this, we consider a constant and invertible matrix ρ AB with the symmetry properties ρ AB = ε A+1ε B +1 ρ BA. 5 Then, we easily find the relations ρ AB E B = L 1 ρ AB Φ A ΦB t Φ ρ AB Ô B A CÔC DΦ D, 6 which suggest us to try a Lagrangian of the type L 0 = 1 ρ AB Φ A ΦB V Φ, i Φ, i j Φ,, 7 and to require that the second-order equations of the type 4 to be expressed in the form ρ AB E B = δl L0 δφ A. 8 From 6 and 8 we obtain the equations δ L V δφ A = ρ ABÔB CÔC DΦ D. 9 In consequence, we find the following answer to above mentioned problem: the second-order equations as in 4 originate in some Euler Lagrange equations of the form 8 if and only if there exists a constant and invertible matrix ρ AB that satisfies 5 such that equations 9 possess solutions. RJP 61Nos. 1-, c v.1.3a*

3 3 Second-order Lagrangian formulation oflinear first-order field equations 9 On the one hand, from 9 we get that Φ A δl V δφ A = ρ ABΦ A Ô B CÔC DΦ D. 10 Meanwhile, for any non-integrated density the standard formula holds for some K i, where ˆN denotes the counting operator ˆN = n 0 Then, from we deduce that Φ A δl V δφ A = ˆN V + i K i, 11 j1 jn Φ A L j1 jn Φ A. 1 ˆN V + i K i = ρ AB Φ A Ô B CÔC DΦ D. 13 Assume that equations 9 posses solutions. In consequence, we have that V has to be quadratic in the fields and their derivatives, such that ˆN V = V. 14 Substituting 14 in 13 and taking into account that V is defined up to a total derivative, we arrive at V = 1 ρ ABΦ A Ô B CÔC DΦ D. 15 The above considerations lead to the following conclusion: there exists a secondorder Lagrangian formulation of the first-order equations 1 in terms of the of the potentials Φ A if and only if there exists a constant and invertible matrix ρ AB that satisfies 5 such that V given by 15 is solution to equations 9. In the sequel we focus on the particular case of first-order equations with a single spatial derivative, i.e., on equations of the type H A Q A Γ A j B jq B m A BQ B = In this situation equations 9 take the concrete form δ L V δφ A = λij AB i j Φ B + ν j AB jφ B + µ AB Φ B, 17 where we used the notations λ ij AB = 1 ρ AC Γ C DΓ i D j B + ΓC j D ΓD B i ν j AB = ρ AC Γ C D i i Γ D j B + ΓC j µ AB = ρ AC Γ C j D jm D B + m C Dm D B, 18, 19 D md B + m C DΓ D j B RJP 61Nos. 1-, c v.1.3a* , 0

4 30 Constantin Bizdadea, Solange-Odile Saliu 4 while formula 15 reads as V = 1 ΦA λ ij AB i j Φ B + ν j AB jφ B + µ AB Φ B. 1 After some computations we find that 1 is solution to equations 17 if and only if the relations 1 λ ij AB = ε Aε B λ ij BA, ν i AB + ε Aε B νba i = ε A ε B j λ ij BA, 3 µ AB ε Aε B µ BA = 1 i ν i AB ε Aε B ν i BA. 4 are fulfilled. For instance, in the case of Dirac equations H 0 ψ + iγ 0 mψ iγ j j ψ = 0, 5 H 0 ψ i m ψ + j i j ψγ γ 0 = 0, 6 relations 4 are satisfied with the choice ρ AB = For this example the potentials Φ A take the form χ Φ A =, 8 χ with χ a Dirac spinor, such that V is given by V = j χ j χ m χχ. 9 Using 7 9 in 7 we derive the Klein Gordon Lagrangian from which we obtain the second-order equations L 0 = µ χ µ χ m χχ, 30 E + m χ = 0, 31 Ē + m χ = 0. 3 Based on formulas we infer that in the context of the example under study the following relations hold ψ = 0 χ iγ 0 mχ iγ j j χ, 33 ψ = 0 χ + i m χ + i j χγ j γ We remark that if the spinors χ, χ satisfy equations 31 3, then the spinors iγ 0 χ, i χγ 0 obey exactly the same equations. Consequently, if we perform the RJP 61Nos. 1-, c v.1.3a*

5 5 Second-order Lagrangian formulation oflinear first-order field equations 31 transformation χ iγ 0 χ, χ i χγ 0, 35 which leaves invariant also the Lagrangian 30, formulas lead to the expressions ψ = iγ µ µ χ + mχ, 36 ψ = i µ χγ µ + m χ, 37 which emphasize the next general relationship between Dirac and Klein Gordon equations: ψ, ψ given by are solutions to the Dirac equations 5 6 if and only if χ, χ are solutions to the Klein Gordon equations In the final part of this paper we investigate the first-order equations expressed like in 16 in the particular case Γ A j B = 0 Γ a j b Γ a j b 0, m A B = 0 m a b m a b 0 In this situation the fields Q A split into two subsets of the type such that equations 16 read as. 38 Q A = Q a,p a, 39 H a Q a Γ a j b jp b m a b P b = 0, 40 H a P a + Γ a j b jq b + m a b Qb = Similarly, the potentials Φ A split into two subsets Φ A = ϕ a,φ a, 4 while equations 4 are expressed by E a ϕ a + 1 Γ a c i Γ c j b + Γa c j Γ c b i i j ϕ b + Γ a c i i Γ c j b + +Γ a c j m c b + ma cγ c j b j ϕ b + + Γ a c j j m c b + ma cm c b ϕ b = 0, 43 Ē a φ a + 1 Γ a c i Γ c j b + Γa c j Γ c b i i j φ b + Γ a c i i Γ c j b + +Γ a c j m c b + ma cγ c j b j φ b + + Γ a j c j m c b + ma cm c b φ b = RJP 61Nos. 1-, c v.1.3a*

6 3 Constantin Bizdadea, Solange-Odile Saliu 6 In this setting formula yields the relations Q a = ϕ a + Γ a j b jφ b + m a b φb, 45 P a = φ a Γ a j b jϕ b m a b ϕb. 46 Due to the fact that equations have exactly the same form, we can set such that formulas yield ϕ a = φ a, 47 Q a = ϕ a + Γ a j b jϕ b + m a b ϕb, 48 P a = ϕ a Γ a j b jϕ b m a b ϕb. 49 In consequence, in this particular case we are able to describe the solutions to equations only in terms of the subset of potentials denoted by ϕ a. By means of the notations λ ij ab = 1 ρ ac Γ c d i Γd j b + Γc j d Γd b i, 50 ν j ab = ρ ac Γ c d i iγ d j b + Γc j d md b + mc d Γd j b, 51 µ ab = ρ ac Γ c j d jm d b + mc d md b, 5 and applying a line similar to that employed with respect to equations 16, we get that the second-order equations of the type 43 may be expressed as Euler Lagrange equations if and only if the following relations hold 1 λ ij ab = εaε b λ ij ba, 53 ν i ab + εaε b νba i = ε aε b j λ ij ba, 54 µ ab εaε b µ ba = 1 i ν i ab εaε b ν i ba. 55 Equations are automatically verified in the case where Γ a i b = ρac Γ i cb, ma b = ρac m cb, ρ ab ρ bc = δ c a, 56 with Γ i cb and m cb some constants subject to relations Γ i ac ρcd Γ j [db] + Γi [ac] ρcd Γ j db + Γj ac ρcd Γ i [db] + Γj [ac] ρcd Γ i db = 0, 57 Γ j ac ρcd m db + Γ j [ac] ρcd m [db] + m ac ρ cd Γ j db + m [ac]ρ cd Γ j [db] = 0, 58 RJP 61Nos. 1-, c v.1.3a* m [ac] ρ cd m db + m ac ρ cd m [db] = 0, 59

7 7 Second-order Lagrangian formulation oflinear first-order field equations 33 where we employed the notations Γ i ab = Γi ab + Γi ba, Γi [ab] = Γi ab Γi ba, 60 m ab = m ab + m ba, m [ab] = m ab m ba. 61 We emphasize that formulas are valid for both boson and fermionic systems. Under these considerations, equations become H a Q a ρ ac Γ j cb jp b + m cb P b = 0, 6 H a P a + ρ ac Γ j cb jq b + m cb Q b = It is easy to see that equations are fulfilled for Γ i ab = Γi ba, m ab = m ba. 64 For purely bosonic systems in the context of relations 56 and 64, the Lagrangian that generates equations 43 takes the expression L 0 = 1 ρ ab ϕ a ϕ b + 1 Γ i 4 acρ cd Γ j db + Γj acρ cd Γ i db i ϕ a j ϕ b 1 Γ j acρ cd m db + m ac ρ cd Γ j db ϕ a j ϕ b 1 m acρ cd m db ϕ a ϕ b. 65 In this manner Lagrangian 65 precisely provides a second-order description of the first-order equations as in 6 63 in terms of the potentials involved with relations Consider now a Lagrangian of the form L 0 = 1 ρ ab ϕ a ϕ b 1 ρ abq a Q b + Q a Γ j ab jϕ b + m ab ϕ b. 66 Defining the canonical momenta in the standard manner by π a = L 0 ϕ a, Π a = L 0 Q a, from the canonical analysis of Lagrangian 66 we infer the constraints a Π a 0, Θ a ρ ab Q b + Γ j ab jϕ b + m ab ϕ b 0, 67 as well as the canonical Hamiltonian 1 H 0 = d D 1 x ρab π a π b + 1 ρ abq a Q b Q a Γ j ab jϕ b + m ab ϕ b. 68 It is easy to see that constraints 67 are second-class. Using the Dirac procedure we RJP 61Nos. 1-, c v.1.3a*

8 34 Constantin Bizdadea, Solange-Odile Saliu 8 find that the only nonvanishing Dirac brackets are given by [ϕ a t,x,π b t,y] = δ a b δd 1 x y, 69 [Q a t,x,π b t,y] = ρ ac Γ j cb x j + m cb δ D 1 x y, 70 such that the equations of motion read as ϕ a ρ ab π b, 71 π a 1 Γ i acρ cd Γ j db + Γj acρ cd Γ i db i j ϕ b Γ j acρ cd m db + m ac ρ cd Γ j db j ϕ b m ac ρ cd m db ϕ b, 7 Q a ρ ac ρ bd Γ j cb jπ d + m cb π d, 73 Π a The concrete expressions of the Dirac brackets imply that we may take any of the pairs ϕ a,π a or respectively Q a,π a as independent variables. If we consider the pairs ϕ a,π a as independent variables, then the equations of motion are expressed by relations 71 7 viewed as strong equalities, while the canonical Hamiltonian from 68 becomes H 0 = d D 1 x ρab π a π b 1 4 Γ j acρ cd m db + m ac ρ cd Γ j db Γ i acρ cd Γ j db + Γj acρ cd Γ i db ϕ a j ϕ b + 1 m acρ cd m db ϕ a ϕ b i ϕ a j ϕ b It is rather obvious that this situation signifies nothing but the Hamiltonian version of the theory with the Lagrangian 65. If we take now the pairs Q a,π a as independent variables, then the equations of motion read as Q a = ρ ac ρ bd Γ j cb jπ d + m cb π d, 76 π a = Γ j ac j Q c m ac Q c, 77 while the canonical Hamiltonian is expressed by H 0 = d D 1 x 1 ρ ab π a π b ρ ab Q a Q b. 78 Equations are nothing but the first-order equations 6 63 modulo the identification P b = ρ bd π d. 79 In consequence, the second-order Lagrangian formulation 65 in terms of the potentials and equations 6 63 may be respectively identified with the parame- RJP 61Nos. 1-, c v.1.3a*

9 9 Second-order Lagrangian formulation oflinear first-order field equations 35 terisations ϕ a,π a and Q a,π a of the phase-space associated with the degenerate Lagrangian 66. To conclude with, in this paper we emphasized the conditions that grant that a set of independent, first-order field equations not necessarily in a Hamiltonian form allows for a second-order Lagrangian formulation. In this respect we implemented two main steps: i we expressed the general solution to the first-order equations in terms of the general solution to some second-order equations; ii we inferred the general conditions that must be fulfilled in order to establish that the second-order equations originate in some Euler Lagrange equations. Next, we applied this twostep procedure to first-order equations containing a single spatial derivative. In this context we determined the general relationship between Klein Gordon and Dirac equations. Moreover, we showed that for a particular form of the first-order equations with a single spatial derivative there exists a second-order Lagrangian formulation subject to purely second-class constraints. In this special case the first-order equations can be expressed in Hamiltonian form in terms of the Dirac bracket built with respect to this second-class constraint set. In a future work we hope to extend the previous results to first-order systems of the type recently investigated in [19 8]. REFERENCES 1. J. L. Lagrange, Mécanique analitique Cambridge University Press, Cambridge, W. R. Hamilton, Phil. Trans. R. Soc. London 15, P. A. M. Dirac, Can. J. Math., P. A. M. Dirac, Lectures on Quantum Mechanics Academic Press, New York, M. Henneaux, C. Teitelboim, Quantization of Gauge Systems Princeton University Press, Princeton, O. Klein, Z. Phys. 37, W. Gordon, Z. Phys. 40, J. C. Maxwell, Philos. Mag. 1, J. C. Maxwell, Philos. Mag. 1, J. C. Maxwell, Philos. Mag. 1, J. C. Maxwell, Philos. Mag. 3, J. C. Maxwell, Philos. Mag. 3, A. Einstein, Annalen Phys. 354, E. Schrödinger, Annalen Phys. 81, A. A. Deriglazov, Phys.Lett. A 373, P. A. M. Dirac, Proc. R. Soc. A 117, R. Floreanini, R. Jackiw, Phys. Rev. Lett. 59, M. Henneaux, C. Teitelboim, Phys. Lett. B 06, E. M. Cioroianu, S. C. Sararu, JHEP 0507, E. M. Cioroianu, S. C. Sararu, Rom. Rep. Phys. 57, E. M. Cioroianu, S. C. Sararu, Int. J. Mod. Phys. A 1, RJP 61Nos. 1-, c v.1.3a*

10 36 Constantin Bizdadea, Solange-Odile Saliu 10. E. M. Cioroianu, Mod. Phys. Lett. A 6, S. C. Sararu, AIP Conf. Proc. 147, E. M. Cioroianu, Int. J. Mod. Phys. A 7, S. C. Sararu, Int. J. Theor. Phys. 51, E. M. Cioroianu, AIP Conf. Proc. 1564, S. C. Sararu, Rom. J. Phys. 58, S. C. Sararu, Cent. Eur. J. Phys. 11, RJP 61Nos. 1-, c v.1.3a*