Conservation Laws in Relativistic Field Theory
In sect. 5.9 we derived the conservation laws for energy and momentum of the electromagnetic field, with only a hint at angular momentum. In the present chapter we are going to show quite generally that conservation of energy, momentum, and angular moment
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Conservation Laws in Relativistic Field Theory
In sect. 5.9 we derived the conservation laws for energy and momentum of the electromagnetic field, with only a hint at angular momentum. In the present chapter we are going to show quite generally that conservation of energy, momentum, and angular momentum, as well as the law of motion for the center of mass( -energy) is intimately related to Poincare covariance of the Lagrangian formulation of the dynamics. More precisely, we shall be able to associate a divergence-free symmetric energy-momentum tensor with any physical system whose dynamics derives from a 'principle of stationary action' that is Poincare-covariant: translational covariance produces a divergence-free tensor, and rotational covariance allows to symmetrize it. There is a general connection between symmetries and conservation laws. This connection is most natural and direct in the formalism of quantum mechanics: since any semilinear operator commuting with the Hamiltonian of a system is conserved in time, this is the case, in particular, for any semi unitary operator commuting with the Hamiltonian. For each one-parameter group of such symmetries, the corresponding Hermitian generator is likewise conserved. This latter version, the conservation of the 'infinitesimal' generator, also holds classically in the Hamiltonian formalism. However, the Hamiltonian formalism is less suitable for making the relativistic symmetry manifest; for that purpose, the Lagrangian formulation in terms of an action principle is optimal. Here we have again a relation between symmetries and conservation laws, known as E. Noether's theorem: If the dynamical equations can be written as the Euler equations of an action principle, then to each one-parameter invariance group of the action integral there is a conservation law. In recent years, it has been (re)discovered that it is possible to set up a 'covariant Hamiltonian formalism', avoiding the usual transition via the Legendre transformation which breaks manifest covariance. Roughly, this is achieved by taking as the phase space the space of solutions of the dynamical equations, rather than the space of canonical initial data: it is possible to describe the important structures of phase space directly in terms of the space of solutions. See, e.g., J. Lee, R. M. Wald, J. Math. Phys. 31, 725 (1990).
Since the proof of Noether's theorem yields an explicit construction of the conserved quantities whose quantum analogs can, in many cases, be simply guessed, we shall present it here. The conserved quantities so obtained behave additively for composite but noninteracting systems. (Those quantum mechanically conserved quantities stemming from symmetries which cannot be imbedded into connected symmetry groups of the Hamiltonian behave multiplicatively; e.g., parity.) In this chapter weI shall mainly proceed deductively, treating applications in the exercises. 1 'We'
includes the reader.
R. U. Sexl et al., Relativity, Groups, Particles © Springer-Verlag Wien 2001
10 Conservation Laws
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