Hideki Kamo

Necessary and sufficient conditions for the existence of a lagrangian. The case of quasi-linear field equations

Abstract Necessary and sufficient conditions for the existence of the Lagrangian associated with given field equations of motion are investigated. For the quasi-linear equations Aabμν(xλ, φc, φϱc)φμνb + Ba(xμ, φb, φνb) = 0, the complete necessary and sufficient conditions are obtained, resorting to the formalism of an exterior derivative. It is emphasized that, to find expressions of these conditions, the anti-symmetric parts of the second derivatives of a Lagrangian, R μν ab = ( ∂ 2 L ∂φ μ a ∂φ ν b − ∂ 2 L ∂φ ν a ∂φ μ b )/2 , which disappear in the field equations, take an important role. The procedure to construct the Lagrandian associated with the field equations is also presented.

Schwinger's variational principle in the theory of quantized quasi-linear fields: The case of Hermitian scalar fields

Abstract As an ectension of the previous work, Schwingers variational principle is formulated for the quantized fields which correspond to the classical real scalar fields described by the Lagrangian density L c (∂ τ φ, φ) = − 1 2 g ab (φ)(∂ μ φ a )(∂ μ φ b ) − v(φ) . The c-number variations of field operations φ a (x) and space-time coordinatei x are sufficient to give the laws of the quantized fields. The Euler-Lagrange equations, the canonical equations of motion and the canonical commutation relations are derived from this principle. The energy-momentum tensor Tμν, energy-momentum vector Pμ and angular-momentum tensor Mμν are introduced in a natural way. The differential conservation law ∂μTμν = 0 is also derived and it is inferred that Pμ and Mνλ should obey the commutation relations which characterize the inhomogeneous Lorentz group. A quantized version of Noethers theorem, which relates the conservation law with the invariance of the action, is established. All resulting relations are consistent with one another, although there arises a pathological situation in the explicit calculation of ∂μTμν. Further, it is shown that our formulation is covariant under an arbitrary change of field variables. The appropriate choice of the quantal Lagrangian density is essential to ensure internal consistency and covariance. The ambiguities in the quantal form of the Lagrangian density are discussed. A comment on the use of Schwingers criterion for Lorentz invariance is also given.