On Tuesday, April 14, 2020 at 11:07:42 AM UTC-5, Jason wrote:
>
> There has been controversy <https://arxiv.org/pdf/quant-ph/0105049.pdf> in
> the meaning/interpretation of the Time-Energy uncertainty relation in
> quantum mechanics, but relatively none regarding the meaning of the
> position-momentum uncertainty.
>
> However, can these not be viewed equivalently in terms of a 4-dimensional
> space time?
>
> For example, I have seen some describe mass/energy as momentum through
> time. Massless particles don't age, and have no momentum through time.
>
> Similarly, cannot a point-in-time measurement be viewed as a measurement
> of position in the time dimension?
>
> In my view, you can go from the position-momentum uncertainty to the
> time-energy uncertainty simply by flipping the time-space orientation. Is
> this valid? Is there something I am missing?
>
> Jason
>
The Fourier transform of time and frequency would naively mean there is
negative frequency, which by E = ħω, and if we restrict the angular
frequency away from negative then the energy is positive. That is one
departure. If we did have a time operator such as T = iħ∂/∂E it would mean
that energy is a generator of time. There would then be time eigenstates
|t> such that T|t> = t|t>. We can think then of the time eigenstate |t> =
e^{it(E - E_0}/ħ} such that energy is a continuous generator. This forbids
the existence of discrete bound states.
As a result, we do not normally think of a time operator. This operator
would then have some Schrödinger equation of the form
iħ∂ψ/∂E = Tψ
If we can’t have a continuous energy then we can’t have a continuous time
either. The existence of a time operator then requires that it have a
discrete measure and that time and energy be bounded away. Is there
something of this form? Yes, it is called the Taub-NUT spacetime, but it is
not the universe we observe.
The Taub-NUT spacetime is analogous to a black hole, but where the horizon
condition occurs with time rather than with radius. There is also only one
horizon. So this time version the black hole has only one black hole, at
least if we take the spacetime as a global condition. I attach an image of
this spacetime below. The green region is a region that has chronology
protected and no timelike curves. The yellow region has closed timelike
curves. The green region has this cyclicity condition on time, and I wrote
a short letter on how a limited sort of time operator exists for a discrete
time that cycles around. One of the oddities is that what plays the role of
mass is a dual to mass, called the NUT parameter μ. This is analogous to
the magnetic monopole. It shares with the gravitation mass m = GM/c^2 the
S-duality condition m
mμ = 2πħ
laid down by Montenen and Olive.
[image: Taub-NUT spacetime1.PNG]
This spacetime does not reflect our observable universe. However, as a
local region that “bolts” a de Sitter spacetime to an anti-de Sitter
spacetime it may have some applicability We exist in a spacetime that is at
least approximately de Sitter, which has no closed timelike curves etc. The
inflationary spacetime is dS as well. A dS and AdS may have some
correspondence with a “bolt” between them that is a Taub-NUT spacetime. It
this is so there may then be some topological charge corresponding to the
NUT parameter μ. This of course will probably be more mercurial to find
than the EM magnetic monopole, if it exists.
The nice thing is that in this setting there is ultimately an equivalency
between momentum-position and energy-time conjugate variables. However, in
our observable world, certainly on the vacuum of low energy or physical
vacuum, the physics we observe is constrained away from any such
equivalency.
LC
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