Abstract
We offer a clear physical explanation for the emergence of the quantum operator formalism, by revisiting the role of the vacuum field in quantum mechanics. The vacuum or random zero-point radiation field has been shown previously—using the tools of stochastic electrodynamics—to be central in allowing a particle subject to a conservative binding force to reach a stationary state of motion. Here we focus on the stationary states, and consider the role of the vacuum as a driving force. We observe that the particle responds resonantly to certain modes of the field. A proper Hamiltonian analysis of this response allows us to unequivocally trace the origin of the basic quantum commutator, \(\left[ x,p\right] =i\hbar \), by establishing a one-to-one correspondence between the response coefficients of x and p and the respective matrix elements. The (random) driving field variables disappear thus from the description, but their Hamiltonian properties become embodied in the operator formalism. The Heisenberg equations establish the dynamical relationship between the response functions.
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Notes
Quoting from Ref. [7]: “Typically, the external (field) force used in experiments is small with respect to the internal ones (in a crystal), so that the system is weakly perturbed. Thus, the dominant term is the linear response function. If we are able to disentangle it, the linear-response function returns us information on the ground state and the excitation spectrum, their symmetry properties, the strength of correlations.” It may of course happen that the intensity of the applied field is so high (as is the case with current high-intensity laser pulses) that the response of the system to it becomes nonlinear. This case falls beyond the scope of the present discussion.
In fact, materials are known to resonate in general to a series of frequencies, but the responses are usually analysed separately, for simplicity.
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Cetto, A.M., de la Peña, L. & Valdés-Hernández, A. On the physical origin of the quantum operator formalism. Quantum Stud.: Math. Found. 8, 229–236 (2021). https://doi.org/10.1007/s40509-020-00241-7
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DOI: https://doi.org/10.1007/s40509-020-00241-7