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There is an old intriguing result in non-relativistic QM, stating (roughly) that there is an Heisenberg Time-Energy Uncertainty Relation.

Unfortunately, in QM time is not an operator like space, and an old result shows that if there was one, it would imply negative values of the Energy operator.

However, if I remember well, in the Dirac equation negative energy does pop up, in the infamous Dirac's sea. Moreover, time and space are indeed unified by relativistic constraints.

Thus I wonder: is there some setup of relativistic QM where the above T-E relation pops up in some form or another, as a genuine analogue of the other one X-P between space and momentum?

PS I am well aware that the standard path to relativistic QM is to demote space to the same rank of time, ie as parameters, not to promote time to an operator. But I know that in principle something like this could also be done, thought it would lead to a more complicated picture.

ADDENDUM The mathematical rationale behind my question is basically this: I would like to think of the E-T pair as some kind of rotation of the X-P operators pair

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    $\begingroup$ Probably going to get better answers on physics.stackexchange.com. $\endgroup$
    – Qiaochu Yuan
    Commented May 1, 2016 at 21:12
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    $\begingroup$ ok thanks Qiaochu. Let me keep it here a little longer, one never knows. But I think you are probably right, this is a physics question (although there is an operator algebra side of it in my addendum: even if the answer turns out to be physically nil, would be nice to know if there is a math toy model along these lines....) $\endgroup$
    – Mirco A. Mannucci
    Commented May 1, 2016 at 21:21
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    $\begingroup$ Thinking of the Dirac equation in relativistic QM as a complete analog of the Schroedinger equation in non-relativistic QM is overly simplistic. In any physically reasonable way, the Hamiltonian of a single Dirac particle or of the Dirac quantum field is positive definite. This means that you would encounter the same issues with the time operator as in non-relativistic QM. For what it's worth, the tachionic ($m^2 < 0$) Klein-Gordon field does have energy unbounded from both above and below. $\endgroup$
    – Igor Khavkine
    Commented May 2, 2016 at 2:40
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    $\begingroup$ A dynamical time operator in Dirac’s relativistic quantum mechanics arxiv.org/pdf/0908.2789v3.pdf and Time in Quantum Mechanics and Quantum Field Theory arxiv.org/pdf/quant-ph/0211047.pdf (both describe the operator for relativisitic QM). $\endgroup$
    – Chris Gerig
    Commented May 2, 2016 at 6:34
  • $\begingroup$ Thanks Chris! Very very valuable refs.....Yes Igor, good points. $\endgroup$
    – Mirco A. Mannucci
    Commented May 2, 2016 at 9:25

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You don't really need quantum mechanics to address this issue, in classical mechanics you would ask whether time and energy can be thought of as canonically conjugate variables (because quantization would then produce an uncertainty relation between these operators). You would then want to treat energy $E$ as a dynamical variable, like position $x$, which means that the Hamiltonian $H(x,E;p,t)=H_0(x,p;t)-E$ must depend explicitly on time $t$, like it does on momentum $p$. The Hamilton equations of motion, $$\dot{q}=\partial H/\partial p,\;\;\dot{p}=-\partial H/\partial q,$$ $$\dot{E}=\partial H/\partial t,\;\dot{t}=-\partial H/\partial E,$$ tell you that, indeed, $E$ and $t$ are canonically conjugate like $q$ and $p$.

There is a fun discussion along these lines at Physics SE, in response to the question "Is energy actually momentum in the direction of time?"

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  • $\begingroup$ Thanks Carlo. I find the insertion of E as a dynamical variable a bit awkward, but perhaps I just have to let it sink in. Great link to the amusing discussion on Physics SE! $\endgroup$
    – Mirco A. Mannucci
    Commented May 3, 2016 at 10:52

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