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Path integral for star exponential functions of quadratic forms. (English) Zbl 1039.53105
Mladenov, Ivaïlo 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 (ISBN 954-90618-4-1/pbk). 330-340 (2003).
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 \(\mathbb C^2\), 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 \(D=b^2-ac\neq 0\).
For the entire collection see [Zbl 1008.00022].
53D55 Deformation quantization, star products
81S40 Path integrals in quantum mechanics