Summary
The system of a particle constrained on a general hypersurface is quantized according to the path integral method. The two forms of equations deseribing the hypersurface,f(xi)=const andf(xi)=0, bring quite a different aspect to the formulation, but it is proven that both are exactly equivalent.
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References
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By time discretization of the delta function in eq. (8) we have\(\partial \left( {\dot f\left( x \right)} \right) \sim \int {\prod\limits_j {d\lambda _j \prod\limits_j {exp\left[ {i\varepsilon \lambda _i \tfrac{{f\left( {t_j } \right) - f\left( {t_{j - i} } \right)}}{\varepsilon }} \right]} } } \).
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Shimizu, A., Inamoto, T. & Miyazaki, T. Path integral quantization of a nonrelativistic particle constrained on a general hypersurface. Nuov Cim B 107, 973–976 (1992). https://doi.org/10.1007/BF02899298
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DOI: https://doi.org/10.1007/BF02899298