Path integral for star exponential functions of quadratic forms. (English) Zbl 1039.53105

Yoshioka, Akira; Matsumoto, Toshio

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].

Reviewer: Janusz Grabowski (Warszawa)

MSC:

53D55	Deformation quantization, star products
81S40	Path integrals in quantum mechanics

Keywords:

Moyal product; deformation quantization; star-exponential