In this paper we address the relation between the star exponentials emerging within the Deformation Quantization formalism and Feynman's path integrals associated with propagators in quantum dynamics. In order to obtain such a relation, we start by visualizing the quantum propagator as an integral transform of the star exponential by means of the symbol corresponding to the time evolution operator and, thus, we introduce Feynman's path integral representation of the propagator as a sum over all the classical histories. The star exponential thus constructed has the advantage that it does not depend on the convergence of formal series, as commonly understood within the context of Deformation Quantization. We include some basic examples to illustrate our findings, recovering standard results reported in the literature. Further, for an arbitrary finite dimensional system, we use the star exponential introduced here in order to find a particular representation of the star product which resembles the one encountered in the context of the quantum field theory for a Poisson sigma model.
· i ~ En(tt0)n(x)n(x0) , Xn=0 where the functions, {n(x)} n=0 L2(R), comprise a set of eigenfunctions of H with eigenvalues En and the series converges in a distributional sense, providing c a representation of the propagator in terms of the spectral decomposition of the Hamiltonian operator . Our next task is to express the quantum propagator K(xf , tf , x0, t0), as an integral transform of the star exponential via the symbol associated to the evolution operator through the inverse Weyls quantization map. Since the star product obtained in (11) denes a homomorphism between the classical observables, C (R2), and linear operators 9 (27) (28) (29) (30) (31) (32) (33) Star exponentials from propagators and path integrals acting on the Hilbert space L2(R), this implies that the symbol corresponding to the formal evolution operator 1 2! (cid:18) b is given by the star exponential , , it ~ · i ~ t H = 1 H + c it ~ (cid:19) 2 H 2 + ,
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(cid:18) f,0 = |x0i hxf | be a nondiagonal density operator, with |x0i and |xf i being Let X |x0i = x0 |x0i and eigenvectors associated to the position operator X |xf i = xf |xf i, respectively, where x0, xf R. Using the inverse Weyls inversion formula (6), the corresponding nondiagonal Wigner function related to the density c operator it ~ it ~ H + i ~ tH H H + . = 1 Exp (cid:19) (cid:18) (cid:19) X, such that, b c c f,0 reads b f,0(x, p) := Q1 W (|x0i hxf |) = 1 2~e i ~ (xf x0)p (x (xf + x0)/2) . Using the integral property of the Wigner function (9) in the sense of distributions, the propagator may be identied as follows i ~ (tf t0)H(x, p) K(xf , tf , x0, t0) = f,0(x, p) Exp dxdp R2 1 2~ (cid:19) i ~ (tf t0)H ((xf + x0)/2, p) (cid:19) (cid:18) Z i ~ (xf x0)p Exp e dp . =
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R (cid:18) The former expression shows that the propagator can be formulated as the Fourier transform of a star exponential of the Hamiltonian function dened on the classical phase space . To proceed further, we introduce the new variables q = (xf + x0)/2, q = (xf x0)/2 and we set, for the sake of simplicity, tf = t and t0 = 0. By computing the inverse Fourier transform of the formula (37), we have Z i ~ tH(q, p) ~ 2qpK(q + q, t, q q, 0) dq . e i Exp = 2 R (cid:18) (cid:19)
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Z Equation (38) provides us with an advantageous approach to calculate star exponentials without relying on convergence of formal series, since as mentioned in the previous section, this can become rather dicult. Instead, it is possible to make use of the Feynmans representation of the propagator in terms of path integrals, i.e., express the propagator, K(q + q, t, q q, 0), as a weighted sum over all possible histories of the classical motion in the phase space since, as we will observe in the following subsection, certain aspects of quantum mechanics become more transparent within this formulation such as for instance, the fact that probability amplitudes of quadratic Lagrangians involve the classical action, as well as the convenience of using this framework in order to generalize to the case of eld theories [15, 27].
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