Derivation of the most general relativistic transformation law of quantum fields

Abstract

In this note we derive the most general relativistic transformation law of quantum fields from the Relativity Principle. In the special case of frame-independent fields it reduces to the familiar forms [compare equation (8)] of the current QFT (-quantum field theory), due originally to Wigner. This is a modified, and perhaps simpler, version of that reported in a previous paper (Ingraham, 1962). Just a word of motivation. This more general relativistic formalism is not just a mathematical curiosity, but should be of vital interest because it allows the incorporation of a cut-off]- into quantum fields and thus into the S-matrix, while the conventional formalism does not (the latter is not proven, but forty years of vain effort in field theory teaches us this). In other words, we suggest that the unnecessarily restrictive form of relativistic invariance currently used is mainly responsible for QFT's well-known pathology. Start with the set of equivalent 'observers', or frames, 5r s162 5r .... ~o stands for a space-time frame and comprises also the 'coordinate frame', or basis, in state vector Hilbert Space J4 ~ when we are talking about QFT. Equivalent means that the theory should not prefer one to another, that they are intrinsically indistinguishable. Consider the orthonormal basis~ ]0), ]k), ]ki,k2), etc. where ]0) is the no-meson state (vacuum), ]k) is a free incoming one-meson state with 3-momentum k relative to ~'s axes, ]ki,k2) is an incoming two-meson state with 3-momenta kl, k2 relative to 5r etc. Similarly let [0)' (= ]0)), Ik')', ]kx', k2')', �9 �9 denote an orthonormal basis for any other frame ~', where k', etc. are the free meson 3-momenta relative to frame ~'. Consider the state ]k)', where k are the same three numbers that appear in ]k): these states are called subjectively identical for frames ~' and ~,o, with the corresponding definition for two and highermeson states. [k)' and [k~ are different (-objectively different) states of course, since if ~ sees a meson moving in the direction k, then these same three numbers for 5r define a (in general) different direction. But the name is justified because the state [k)' 'looks the same' to ~,o, as the state [k) does to s ~