We study the problem of constructing a general hybrid quantum-classical bracket from a partial classical limit of a full quantum bracket. Introducing a hybrid composition product, we show that such a bracket is the commutator of that product. From this we see that the hybrid bracket will obey the Jacobi identity and the Leibniz rule provided the composition product is associative. This suggests that the set of hybrid variables belonging to an associative subalgebra with the composition product will have consistent quantum-classical dynamics. This restricts the class of allowed quantum-classical interaction Hamiltonians. Furthermore, we show that pure quantum or classical variables can interact in a consistent framework, unaffected by no-go theorems in the literature or the restrictions for hybrid variables. In the proposed scheme, quantum backreaction appears as quantum-dependent terms in the classical equations of motion.

中文翻译：

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量子经典动力学支架

我们研究了从完整量子支架的部分经典极限构造通用混合量子经典支架的问题。引入混合组合产品，我们证明这样的支架是该产品的换向器。由此我们看到，如果组合积是结合的，则混合括号将遵守雅可比恒等式和莱布尼茨规则。这表明属于具有组合乘积的关联子代数的混合变量集将具有一致的量子经典动力学。这限制了允许的量子-经典相互作用哈密顿量的类别。此外，我们表明纯量子或经典变量可以在一致的框架中相互作用，不受文献中的不通定理或混合变量的限制的影响。在提议的方案中，