Various authors have considered a conformal extension CGo of the Galilei group which in some sense is the nonrelativistic limit of the conformal extension of the Poincare group. and have also established an invariance group for the free-particle SchrOdinger equation. the "Schrodinger group." Here we establish the most general conformal extension C G of the Galilei group, which is found to be identical to the group of the most general coordinate transformations that permit the use of noninertial frames of reference and of curvilinear coordinates in Galilei-invariant theories, which was considered by one of us some time ago, and is a gauge group containing a number of arbitrary functions. Both CGo and the Schrodinger group are subgroups of C G containing the Galilei group, but otherwise they do not overlap. The Hamilton-Jacobi and Schrodinger equations for particles which are free or interact via inverse-square potentials are shown to be invariant under the Schrodinger group, and a further invariance of the Hamilton-Jacobi equation is established.482
938REVIEWS OP 1lttt. 013ERN PHYSICS~OCTOBER 1964 equations. In the quantized version of this formulation of the theory one would look for an operator representation for the I" s which would reproduce the classical commutator algebra between the various F's obtained from their Pb's. Since now one has many more observables than degrees of freedom the observables are not all independent of one another and so one has certain consistency conditions to satisfy that are not present when one eliminates degrees of freedom from the theory directly. It is not clear at present whether or not one can satisfy these consistency requirements.If they can be satisfied then the BK procedure would have the advantage over the other schemes of quantization that it does not require a weak-field approximation procedure to obtain its results.
ACKNOWLEDGMENTSIIe Discussion o~~~~~~~~~~~~~~~~~~~~~~e~~~~~~~~~~~~9 38 940 941 941 942 943 944 945 945 946 946 948 949 951 952 955 955 957 958 960 963 quirement of invariance under all coordinate transformations we are following the original terminology of A. Einstein, Ann. Phys. 49, 769 (1916).Some authors prefer a narrower interpretation of this principle, as discussed in Sec. VII, but the physical considerations involved should be kept apart from questions of terminology, 'We use, as customary, the terms "three-dimensional" and "four-dimensional" for brevity to distinguish between notations which treat space and time coordinates on a diferent or on the same footing, respectively. No physical distinction between the formulations is i!nplied by these terms, however; it is one of the main purposes of this paper to clarify the relation between different formulations and the physical content of a theory.
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