I have no idea what the motivation for this would be or what such a theory would look like, but it seems like the kind of crazy idea that someone would have played around with at some point.
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$\begingroup$ I played with it. if you are interested, send me an email. $\endgroup$– Arnold NeumaierAug 10, 2017 at 9:28
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What would come the most close to that would be Julian Schwinger's Quantum Action Principle, see for example this pdf: https://arxiv.org/abs/1503.08091. Or see his paper "the Theory of quantized Fields I".
What he essentially does is, he promotes the Action to be an Operator, and states that the variation of this Operators matrix elements equals the variation of the transition amplitudes between the two states that the matrix elements are calculated in between.
AccidentalFourierTransform
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answered Jul 6, 2017 at 10:06