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Zbl 1039.53105
Yoshioka, Akira; Matsumoto, Toshio
Path integral for star exponential functions of quadratic forms.
(English)
[A] Mladenov, Iva\"ilo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6--15, 2002. Sofia: Coral Press Scientific Publishing. 330-340 (2003). ISBN 954-90618-4-1/pbk

The $*$-exponential function $$e_*^{t\frac{H}{i\hbar}}=\sum_{n=0}^\infty \frac{t^n}{n!}\left(\frac{H}{i\hbar}\right)^n_*$$ for the Moyal $*$-product $$f*g=f\exp{\left(\frac{i\hbar}{2}\overleftarrow{\partial_x}\wedge \overrightarrow{\partial_y}\right)}g$$ of functions on $\Bbb C^2$, where $(H)^n_*=H*\cdots *H$ -- $n$-times, is studied. Proving the Trotter type formula $$\lim_{N\to\infty}e^{\frac{t}{N}\widetilde{H}}*\cdots *e^{\frac{t}{N}\widetilde{H}}= \frac{1}{\cosh{\frac{t}{2}}}e^{2\widetilde{H}\tanh{\frac{t}{2}}},$$ for $\widetilde{H}=\frac{y^2-x^2}{2i\hbar}$, the authors conclude that $$e_*^{t\frac{H}{i\hbar}}= \frac{1}{\cosh{\sqrt{D}t}}\exp{\left(\frac{H}{\sqrt{D}}\tanh{\sqrt{D}t}\right)}$$ for any quadratic function $H=ax^2+2bxy+cy^2$ with $D=b^2-ac\ne 0$.
[Janusz Grabowski (Warszawa)]
MSC 2000:
*53D55 Deformation quantization, star products
81S40 Path integrals in quantum mechanics

Keywords: Moyal product; deformation quantization; star-exponential

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