Elementary Notions of Relativistic Field Theory
A short reminder of the basics of classical relativistic field theory, mainly focused on the variational formalism and on the relation between symmetries and conserved currents. Includes definition (and properties) of the canonical energy-momentum and ang
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Elementary Notions of Relativistic Field Theory
The aim of this chapter is to introduce a few basic aspects of classic field theory, concerning in particular the covariant variational formalism and its main relativistic applications. The notions we will present here will be used for the subsequent formulation of a relativistic theory of the gravitational field. Our discussion will be focused on the symmetries of the Minkowski space–time, with the main purpose of showing how the definitions of the canonical energymomentum and angular momentum tensors arise, respectively, from the invariance of the action under global space–time translations and global Lorentz rotations. We will also provide explicit examples of energy-momentum tensors for simple systems of physical interest, such as scalar and vector fields, point particles and perfect fluids. We will always implicitly assume, throughout this chapter, that the gravitational interaction is absent (or negligible), and that the physical systems we are considering can be correctly described in a special-relativistic context using the appropriate representations of the Lorentz group in four-dimensional Minkowski space–time. For a useful reference to special relativity and to the related formalism we refer the readers to the excellent books [2, 4, 32, 45, 46] listed in the bibliography.
1.1 Symmetries and Conservation Laws We start considering a generic physical system represented by a field ψ(x), whose a classical dynamics is controlled by the action functional d 4 x L(ψ, ∂ψ, x), (1.1) S= Ω
where L is the Lagrangian density (depending ψ and on its gradients), and Ω is an appropriate four-dimensional integration domain on the Minkowski space–time. Here and in what follows we will collectively denote with x a generic dependence on all the space–time coordinates. Note that L has dimensions of energy density, so that M. Gasperini, Theory of Gravitational Interactions, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-88-470-2691-9_1, © Springer-Verlag Italia 2013
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Elementary Notions of Relativistic Field Theory
the above action has dimensions [S] = [energy] × [length], because of the factor c contained inside d 4 x = dx 0 d 3 x = cdt d 3 x. The canonical dimensions of the action ([energy] × [time]) can be easily restored by multiplying the integral (1.1) by the factor 1/c. Such a factor, however, is irrelevant for all the topics discussed in this chapter, and will be omitted. Let us first recall that the classical evolution of our physical system is described by the so-called Euler–Lagrange equations of motion. They are obtained by imposing on the action to be stationary with respect to local variations of the field ψ, with the constraint that such variations are vanishing on the border ∂Ω of the integration domain. We may consider, in particular, an infinitesimal transformation of the field ψ , at a fixed space–time position x, such that ψ(x) → ψ (x) = ψ(x) + δψ(x). The corresponding infinitesimal variation of the action is given by ∂L ∂L δS =
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