By Bunge M.

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Letters 36, 1322 (1976); M. C. Temperature Physics, Ontaniemi, Finland 1975, edited by M. Kusius, M. Vuorio (North Holland, Amsterdam (1975), Vol. 1, p. 29. [30] W. F. Brinkman, J. W. Serene, P. W. Anderson, Phys. Rev. A 10, 2386 (1974); M. T. Bealmonod, D. R. Fredkin, S. K. Ma, Phys. Rev. 174, 227 (1968); Y. Kuroda, A. D. S. Nagi, J. Low. Temp. 23, 751 (1976). [31] D. Rainer, J. W. Serene, Phys. Rev. B13, 4745 (1976); Y. Kuroda, A. D. S. Nagi, J. Low Temp. Phys. 25, 569 (1976). 38 2 Relativistic Fields [32] For a review see: D.

63) this is no longer necessary. Thus we shall also study the real scalar field with an action A= d4 xL(x) = 1 2 d4 xφ(x)(−∂ 2 − M 2 )φ(x). 65) In this case, a prefactor 12 is the normalization convention for the field. Also, we have used the letter φ(x) to denote the real field, as is commonly done. 1 The fields have two polarization degrees of freedom (right and left polarized) and are described by the usual electromagnetic action. Historically, this was the very first example of a relativistic classical field theory.

80) An analogous transformation law holds for rotations so that we can write, in general, A µ (x ) = Λµ ν Aν (x). 32) in the way envisaged above for particles with non-zero intrinsic angular momentum. The field has several components. It points in the same spatial direction before and after the coordination change. This is ensured by its components changing in the same way as the coordination of the point xµ . 32). Indeed ∂ µ Aµ (x ) = (Λµ ν ∂ ν )Λµ λ Aλ (x) = ∂ ν Aν (x). 65). The invariance of the first term is shown similarly A ν (x )∂ 2 Aν (x ) = Λν λ Aν (x)∂ 2 Λν κ Aκ (x ) = Aν (x)∂ 2 Aν (x) = Aν (x)∂ 2 Aν (x).