Uncertain about Heisenburg

Factual Questions
22

Yes, there is a relation of a sort, in that both spring from canonically conjugate pairs of variables—that is, pairs in which one variable generates translations of the other: thus, for instance, momentum generates translations in position, energy (or rather, the Hamiltonian) generates translations in time, and so on. If there is a symmetry in these translations, then Noether’s theorem implies a conservation law of the generator of the translation—so time-symmetry implies energy conservation, symmetry regarding spatial translation implies conservation of momentum, and so on.

Now, for a pair of conjugate variables, the Poisson bracket* is equal to one; and in quantum mechanics, the Poisson bracket is replaced by the commutator**. Thus, for a pair of conjugate variables, we always have a nonvanishing commutator; but for noncommuting variables, there is always an uncertainty relation.

  • The Poisson bracket can essentially be understood as giving the rate of change of one variable due to the translations induced by the other; for instance, the Poisson bracket of some quantity with the Hamiltonian, which generates time translations, is just its change over time.
    ** The commutator of two quantities A, B, written [A, B], is the difference between the two orderings of their product: [A, B] = AB - BA. This vanishes for all commuting quantities, but is nonzero if AB =/= BA, hence the name.
23

The Energy-Time uncertainty principle is somewhat problematic, but frequently good for hand-waving, though sometimes yielding incorrect results. The reason is that in non-relativistic QM, time is not an operator, it is a scalar parameter of the theory. In relativistic QM, where time and space have to be on an equal footing, the normal way to quantize is not to make time an operator, but rather to demote space coordinates to parameters as well by introducing quantum fields as time and space dependent operators.