It is an annoying fact of field theorists' life that one can solve exactly only the most trivial, toy models. For more interesting models, even for the simplest such as $\phi^4$ or QED, it is not known how to exactly evaluate the generating functional ("path") integral without resorting to a perturbative expansion or some kind of heuristic, semiclassical evaluation.

Based on methods initiated and developed by Schwinger and Fradkin, school of physicists has developed that is trying to perform exactly as many steps in the evaluation of the generating functional (and related quantities) as possible, before resorting to approximations. By insisting on such an approach, it is possible to a get set of approximate non-perturbative results, as in the case of eikonal (UV) like approximations.

For class of simple models (which includes $\phi^4$ and gauge theories) it is possible to express the generating functional as formal, exact series of several, consecutive functional operations. These involve performing Gaussian linkage operations (equivalent to path integrals with Gaussian measures), a useful representation of exact Green's functions $G_c[A]$ in fixed (but arbitrary) external fields, and a knowledge of the corresponding loop generating functionals $L[A] \sim \det[G_c[A]]$ . The exact representations of such a Green's functions are known (Fradkin, Fried \& Gabellini) but their evaluation is still mainly an unsolved task.

My current research is centered about the methods to evaluate and approximate these exact representations of Green's functions $G_c(x,y|A)$ in arbitrary external fields. Based on the recent non-perturbative, eikonal-inspired hierarchy of approximations to $G_c[A]$ of Fried \& Gabellini, I have devised an independent counterpart to their hierarchy of approximations. Most basic of those new approximations, the $<0>$-approximation (where the propagating field is behaving as extremely UV), is still giving exact results for cases of constant external field $A$, as well as for monochromatic plane waves (lasers).

Applied to the $\phi^4$ model, the $<0>$-approximation gives an intriguing result in $1+3$ dimensions: the quantized model is free (noninteracting) and massless. In other numbers of dimensions, the conclusion is different, as in the case of $1+1$ where both mass and interaction survive quantization. Higher approximations (that are gradual improvements of $<0>$-approximation), preserve the free\&massless result in $1+3$ dimensions, and it is tempting to consider this as the simplest analytical proof of triviality of the $\phi^4$ model.

Another closely-related direction of my research is devising a hierarchy of approximations of the exact Green's functions, when the external field is gauge (as in QCD) or gravitational. Both cases involve additional technical difficulties due to non-commutativity, conveniently focused in (path) Ordered Exponentials (OEs). It is possible to map such matrix OEs into vertex operators, and to extract some additional information from them, due to better knowledge of how to approximately evaluate such operators.

One natural check of the above approximations is found in the Schwinger-DeWitt asymptotic expansion, which was successfully applied twenty-five years ago to the case of gravity (propagation in a curved background). For the case of a scalar background, both methods (functional approximation and Schwinger-DeWitt asymptotic expansion) are giving same results, reaffirming each other.

One of the goals of my present research is a comparison of the effective quantum behavior of particles and fields in presence of certain backgrounds with their classical motion in those fields. In particular, I have concentrated on laser-like external fields (in cases of scalar, electromagnetic and gravitational nature), finding new classical solutions. One line of research is considering the effects of gravitational waves on an electromagnetic laser: I have found the exact classical solution for that problem.

Although the development of the abovementioned hierarchy of approximations to $G_c[A]$ was just a one small logical step in exploring possibilities of functional approach, I have found it more valuable than expected. Presently, I am exploring properties and possibilities for application of the first few members of that hierarchy. One important application is the evaluation of functional determinants in external fields, and among them the approximate (valid in the long-wave domain) bosonization in $1+d$ ($d \ge 1$) dimensions (in the spirit of recent research by Schaposnik on bosonization in $d=2$).

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