Many Lagrangians of physical theories can be expressed as eigenvalues of certain, relatively simple, matrices involving Dirac gamma matrices. We give concrete examples for Lagrangian corresponding to a point particle coupled to electromagnetic field, electrodynamics, nonabelian gauge theories, extended objects and gravity. We also discuss (in case of a point particle) what are the implications of the least action principle applied to matrix Lagrangians

Trzetrzelewski, M.

MPG.PuRe

arXiv:1209.0332v1  [math-ph]  3 Sep 2012AEI-2012-051A note on the principle of least action andDirac matricesMaciej Trzetrzelewski ∗Max-Planck-Institut fu¨r Gravitationsphysik,Albert-Einstein-Institut,Mu¨hlenberg 1, D-14476 Potsdam,GermanyAbstractMany Lagrangians of physical theories can be expressed as eigen-values of certain, relatively simple, matrices involving Dirac gammamatrices. We give concrete examples for Lagrangian correspondingto a point particle coupled to electromagnetic field, electrodynamics,nonabelian gauge theories, extended objects and gravity. We also dis-cuss (in case of a point particle) what are the implications of the leastaction principle applied to matrix Lagrangians.1 MotivationA nice property of Dirac matrices in 4 dimensions is that a large classof Lagrangians appearing in classical physics can be understood aseigenvalues of some matrices involving them. As for the Lagrangianfor a point particle this observation is almost trivial considering thecontext in which Dirac introduced gamma matrices - it also worksin arbitrary spacetime dimension. In the following sections we wouldlike to note some less trivial observations, in particular that the La-grangians of 4D gauge theories and general relativity as well as ex-tended objects can also be represented as eigenvalues of certain ma-trices. Lastly we consider the variation procedure for actions that are∗e-mail: maciej.trzetrzelewski@gmail.com1matrices (not eigenvalues of matrices) and discuss conditions underwhich one recovers classical equations of motion.The fact that Dirac matrices appear so naturally at the level ofclassical theories (although they originate from Dirac’s attempt toformulate relativistically invariant quantum theory) is somewhat in-triguing. If not purely accidental and formal it must be a consequenceof some underlying physical reason. For the letter a good explanationwould a claim that the spin (and hence the Dirac/spin matrices) is theproperty of space-time itself which is then inherited by particles. Un-der these circumstances it would be natural to expect that relativisticfield theories involve (in a certain way) Dirac matrices already at theclassical level.2 Lagrangians as eigenvalues2.1 Point particlesFor a point particle with massm, charge q and coordinates xµ, coupledto the electromagnetic field Aµ(x), in curved spacetime given by themetric gµν(x), one simply takesP := mγµ(x)x˙µ + 1qAµ(x)x˙µ, {γµ(x), γν(x)} = 2gµν(x). (1)Here γµ(x) are x dependent Dirac matrices. One possible choice isto use the vielbein eaµ(x), gµν(x) = eaµ(x)ebν(x)ηab where ηab is theMinkowski metric (we use the many minus convention when needed)and to take γµ(x) = eaµ(x)γa where γa are usual (x independent) Diracmatrices {γa, γb} = 2ηab.Because the square of the mass term of P is just m2gµν(x)x˙µx˙νand since the potential term commutes with the mass term we con-clude that there are two doubly degenerated eigenvalues of P and thespectrum isspec(P ) = {P+, P+, P−, P−}, P± = ±m√gµν(x)x˙µx˙ν + qAµ(x)x˙µ.Hence the Lagrangian for a point-particle can be obtained by takingthe smallest eigenvalue, P−, of P so that the least action principlegives the correct equations of motion.22.2 Gauge theoriesA similar result may be obtained for a Lagrangian of electrodynamicsin flat space in 4 dimensions. One takesE :=18(γabFab)2 = −14FabFab1−14iγ5Fab∗F ab (2)where γab := [γa, γb]/2, Fab := ∂[aAb] is the field strength and the dualtensor is ∗F ab := 12ǫabcdFcd where ǫabcd is completely antisymmetricwith ǫ0123 = 1. The second equality in (2) is unique for 4-dimensionalspacetimes and relies on the following identity for gamma matrices{γab, γcd} = −2(ηacηbd − ηadηbc)1− 2iγ5ǫabcd. (3)The eigenvalues of E are again double degenerated and its spectrumisspec(E) = {E+, E+, E−, E−}, E± = −14FabFab ±14iFab∗F ab. (4)(where we use the fact that there are two doubly degenerated eigen-values of a matrix a + γ5b: a± b). The second term in E± is a totalderivative and hence does not affect the equations of motions there-fore to obtain the formulation of electrodynamics one can take E+ orE−1. Before going further let us note that by equation (2) it is notpossible to couple a pseudo-scalar field (the axion) to the F ∗F term,leaving the FF term unchanged. Therefore from this perspective thecoupling to the axion field does not appear here quite naturally.For non-abelian gauge theories we may takeEYM := −14(γabGAab)2 = −14GAabGab A1−14iγ5GAab∗Gab Awhere A is the adjoint index of the gauge group and GAab is the A’thcomponent of the field strength (reducing to Fab if the group is U(1)).Here the conclusions are similar to (4) with one difference that theG∗G term must be kept due to the presence of instantons. Thereforethe θ angle is fixed and equal to θ = 8π2 (using the conventions wherethe θ term is given by− θi32pi2GAab∗Gab A).1One can also take ∗Fab instead of Fab in the definition of E obtaining equivalent result.32.3 General RelativityFor the Lagrangian of General Relativity we find that a good choiceisG :=14Rµνρσγµν(x)γρσ(x) = R1− 2iγ5 ∗R = R1. (5)where γµν(x) = eµaeνbγab and ∗R is a dual to the Riemann tensor. Toprove that (5) holds one again uses (3). Moreover, in 4 dimensionsthat dual ∗R is zero due to the Bianchi identities∗R := ǫµνρσRµνρσ = 8(R0123 +R0231 +R0312) = 0therefore matrix G is actually a unit matrix proportional to the Ricciscalar.2.4 Extended objectsFor extended objects (with p spatial dimensions) the task is moretricky since the corresponding Lagrangians are more complicated (com-pared to a point-like particle). Their Lagrangians are given by [1, 2, 3]L = −Λ√(−1)p detGαβ , Gαβ = ∂αXµ∂βXνgµν(X)and the action is obtained by integrating L over the world-volumeparametrized by σα, α = 0, . . . , p (for strings, p = 1, for membranes,p = 2). Here Xµ are the coordinates of the object and gµν(X) is themetric of spacetime. Dimensional constant Λ is the tension. Usingobservations made in [4] we find that a good choice of the matrixLagrangian isPstring =Λ2!γµν(X){Xµ,Xν}, (6){Xµ,Xν} = ǫαβ∂αXµ∂βXνfor a string andPmembrane =Λ3!γµνρ(X){Xµ,Xν ,Xρ}, (7){Xµ,Xν ,Xρ} = ǫαβγ∂αXµ∂βXν∂γXρfor a membrane, where γµνρ(X) = eµaeνb eρcγabc, γabc =13(γµνγρ+cycl.).We note a natural appearance of the Poisson and Nambu brackets in(6) and (7) respectively. For higher dimensional objects p > 2 (which4would require the embedding spacetime to have more dimensions then4) one simply takes Λ(p+1)!γµ1...µp+1{Xµ, . . . ,Xµp+1}.Let us now argue why the eigenvalues of matrices (6) and (7) givethe corresponding Lagrangians. Using identity (3) one immediatelyverifies that the square of Pstring is equal to − detGαβ , α, β = 0, 1.To prove the same result for a membrane it is useful to use anotheridentity{γabc, γa′b′c′} = 2ǫabcdǫa′b′c′d′ηdd′1 = −2∣∣∣∣∣∣ηaa′ηab′ηac′ηba′ηbb′ηbc′ηca′ηcb′ηcc′∣∣∣∣∣∣ 1which follows form the fact that in 4 dimensions γabc = iǫabcdγdγ5.Now it is easy to see that the square of Pmembrane is equal to detGαβ ,α, β = 0, 1, 2.3 Action principle for matrices?Let us apply the least action principle directly to a matrix Lagrangian(not its eigenvalue) of a neutral particle in curved spaceS = m∫γµ(x)x˙µdτ. (8)In the case of a flat space the matrix Lagrangian in S is a total deriva-tive hence δS = 0 for all trajectories - which is of course inconsistentwith the geodesic equation.In curved space the conclusion is different. Varying S with respectto xµ and integrating by parts we obtainδS =∫(x˙µ∂νγµ(x)− γ˙ν)δxνdτ (9)hence the equations of motion are x˙µ∂νγµ(x) = γ˙ν which can be alsowritten asx˙µfµν = 0, fµν := ∂[µγν](x). (10)However the only solution of this equation is x˙µ = 0 which can beseen by expanding the matrix fµν in terms of γa’s asfµν = faµνγa, faµν := ∂[µeaν]5therefore equation (10) implies x˙µfaµν = 0 for every a hence x˙µ = 0.This shows that the direct application of the least action principle tomatrices via (8) is inconsistent with the geodesic equation.In the remaining part of this section we argue that the geodesicequation can nevertheless be obtained for certain geometries if theequations of motion (10) are modified as followsx˙µfµν = Cν . (11)Here Cν is some constant, nonzero matrix and can be obtained byevaluating the l.h.s. of (11) at some time τ0. Therefore matrix Cν iscompletely determined by initial conditions of the trajectory xµ(τ) viaCν = x˙µfµν |τ=τ0 . This modification could be argued from equation (9)if we generalize the variation procedure by saying that the integrand in(9) is at most a traceless (but nor necessarily 0) matrix. The matrix Cνcan be expanded in terms of γa’s and therefore satisfies this condition.Let us now find solutions of (11). Ideally, we would like to find theinverse of fµν so that equation (11) could be inverted. This can bedone by noting that in 4 dimensions there exist a unique identity forantisymmetric rank-2 tensors and their duals, namely [5]faµν∗f νρ a = −(faνσ∗f νσ a)δρµ, a fixed,where ∗fµν a is a dual tensor ∗fµν a := ǫµνρσfaρσ/2. This identityimplies that{fµν ,∗f νρ} = −2δρµ(f∗f), (f∗f) := faνσ∗f νσ bηab, (12)where ∗fµν is a dual matrix ∗fµν :=∗ fµν aγa. Applying (12) to (11)we find thatx˙ρ = −12(f∗f){Cν ,∗ f νρ}, (13)where we assumed that the geometry under consideration is such that(f∗f) 6= 0. That assumption implies that the metric gµν(x) is nonstandard (i.e. not diagonal). To see this explicitly let us assume thatthe metric is given by gµν(x) = ηµνf2µ(x) (µ fixed) hence eaµ = δaµfµ(x).In this case it follows that (f∗f) is(f∗f) = [−(∂0−1fµ)(∂2−3fν)+(∂0−2fµ)(∂1−3fν)−(∂0−3fµ)(∂1−2fν)]ηµν(where ∂a−b := ∂a−∂b) which is 0. Therefore equation (13) applies tovery special geometries.6Equation (13) is an autonomic one i.e. it gives the formula forthe 4-velocities x˙µ in terms of xµ’s. To obtain the solutions for xµone would have to integrate (13) once, which however can be doneexplicitly only when the metric is specified.Let us now show that xµ satisfying (11) satisfies also the geodesicequation. To see this it is useful to differentiate (modified) equationsof motion (11) w.r.t. τ . We find thatx¨µfµν + x˙µx˙σ∂σfµν = 0.Applying (12) to the above equation we obtainx¨ρ + Γ˜ρµσx˙µx˙σ = 0, Γ˜ρµσ :=12(f∗f)∗fρν a∂(σfbµ)νηab.Therefore we arrive at the geodesic equation provided Γ˜ρµσ coincidewith Christoffel symbols Γρµσ .4 AcknowledgementsThis work was supported by DFG (German Science Foundation) viathe SFB grant.References[1] P. A. M. Dirac, An Extensible Model of the Electron, Proc. Roy.Soc. A268 (1962) 57-67.[2] Y. Nambu, Duality and hydrodynamics, Copenhagen SummerSymposium, 1970.[3] T. Goto, Relativistic quantum mechanics of one-dimensional me-chanical continuum and subsidiary condition of dual resonance,Prog. Theor. Phys. 46 (1971) 1560.[4] M. Trzetrzelewski, Dirac equation for membranes, Phys. Rev. D84, 087701 (2011), arXiv:1103.1964v2.[5] J. D. Bjorken, S. Drell, Relativistic Quantum fields, McGraw-Hill(1965), ISBN 0-07-005494-0.7

A note on the principle of least action and Dirac matrices

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