Volume 110A, number 4 PHYSICS LETTERS 22 July 1985 TRANSFORMATION OF THE FREE PROPAGATOR TO THE QUADRATIC PROPAGATOR G. JUNKER 1 and A. INOMATA Department of Physics, State University of New York at Albany, Albany, NY 12222, USA Received 18 April 1985; accepted for publication 14 May 1985 A space and time transformation is found, which changes the classical action for a quadratic lagrangian into that for a free particle. It is shown that the propagator for a time-dependent damped oscillator can be obtained from the free propagator. As is well known, the propagator of a quantum system is in principle obtainable from its classical lagrangian by path integration [ 1]. In particular, if the lagrangian is quadratic, the propagator can be evaluated directly from the classical action S c via the van Vleck-Pauli formula [ 1,2], ( i a2Sc) 1/2 K(x",t";x',t')= 21rh Ox'ax" exp[(i/~)Sc(X"'t";x"t')] " (1) Although this procedure for a quadratic system is exact and unambiguous, the actual calculation of the classical action is not always simple. Therefore, calculations of the propagators for various quadratic systems have been tirelessly appearing in the literature [3-8]. Recently, the technique of changing "space" and "time" variables in path integration has been proven useful for non-quadratic systems such as the hydrogen atom [9,10], the Morse oscillator [ 11], and the Dirac-Coulomb problem [ 12], to which formula (1) is not immediately applicable. It is certainly interesting to explore a similar transformation technique for quadratic systems. For non-quadratic systems used are local transformations of short time intervals which are usually non-integrable. In contrast, a transformation that relates the classical equa- tion of motion for one quadratic system to that for another quadratic system, if available, is globally meaningful in quantization, since the propagator given by (1) depends only on the classical solutions. In this paper, we pro- pose to utilize such a global transformation of "space" and "time" variables, say, y = y(x, t) and s = s(t), for finding the propagator of the second quadratic system K2(x", t";x', t') from that of the first Kl(X", t"; x', t') as l~ tl I tl ft I K2(x , t ;x , t') = [(ay'/ax')(ay"/ax")] 1/2 Kl(y , s ; y , s'). (2) First, we present a space and time transformation which changes a quadratic action into a free particle action. Then we show that the propagator for a quadratic lagrangian can be obtained from the propagator for a free particle by the transformation. The most general quadratic lagrangian is [1] L(~, q, t) = a(t)~ 2 + b(t)?lq + c(t)q 2 + d(t)~ + e(t)q +f(t), (3) where a, b, c, d, e and f are all well-behaved functions of time and a :/= 0. As a physical lagrangian, (3) is a little more general than necessary. The equation of motion for a system described by (3) is given by Z/" + ~(t)t~ + 6o2(t)q = g(t), (4) 1 Present address: Physikalische Institut der Universitat Wiarzburg, D-8700 Wiarzburg, West Germany. 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 195