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N, s=1,2,"·,n. 23) öJlJr = 0, s = 1,2,···,n everywhere. These are the n conservation laws stated in Noether's theorem. 22), (Jn = (Ps,js), s = 1,2,,,', n, where Ps is viewed as acharge density. 23) implies that ddqs = t r 8 ps dx = J[{3r yr. js dx = O. 25) One of the most importance situations is that the system is invariant under translations, x'l1- = xl' +wl', u'(x') = u(x). 18) gives us U~ = 0 and X~ = 8~. : = 0, 81'TI'v = 0; 81'T; = 0, 81'TI'v = 0; /I = 0, 1,2,3. 5) the component function T8 is the Hamiltonian density, (E,P), (PI') E l l [{3 Pi = [{3 Tgdx, TPdx.

23) for a non-relativistic particle to give a quantum-mechanical description of a relativistic particle. However, there are two difficulties. 22), we need to study it in the context of quantum field theory. 25) describes the field distribution of certain spin-O particles so that cp is viewed as a quantum field and 1if>1 2 is proportional to the number of particles present. 3 Maxwell equations In this subsection, we present a simple mathematical derivation of the Maxwell equations. For this purpose, we consider a sourceless electromagnetic vector field F, V'.

21) more explicitly. It is readily seen that 81g1 _I 1JlII 8gJlII - 9 9 . 27) 36 1. Primer of Field Theory On the other hand, from goß goß = 4, we have ßgoß goß ßg/lV = _goß ßgoß = -goß8~8ß = _g/lv. 28), it is easy to verify that 2 0/l ßv ßC _ 9 9 ßC ßV ßgoß - ß(ß/lu)' U. 2 in Hat spacetime limit. 29) may weIl be recognized as the curved spacetime version of the energy-momentum tensor. 25) automatically. The basie principle whieh led Einstein to write down his fundamental equations for gravitation states that the geometry of aspacetime is determined by the matter it contains.

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Algebre des ensembles by Sierpinski W.

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