Position-Momentum vs. Time-Energy Uncertainty

[Jason Resch]

There has been controversy 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

[Bruce Kellett]

You are missing the fact that energy is bounded below, whereas momentum can take on any value between plus and minus infinity. Time is not an operator in quantum mechanics.

Bruce

Jason

 

[Alan Grayson]

Isn't there a valid interpretation/ application of the time-energy uncertainty relation in the context of emission of radiation? If so, what is it? TIA, AG 

Jason

 

[Lawrence Crowell]

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.
 

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

[smitra]

On 15-04-2020 04:20, Alan Grayson wrote:
> On Tuesday, April 14, 2020 at 4:28:23 PM UTC-6, Bruce wrote:
>
>> On Wed, Apr 15, 2020 at 2:07 AM Jason Resch <jason...@gmail.com>
>> wrote:
>>

>>> There has been controversy [1] 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?
>>
>> You are missing the fact that energy is bounded below, whereas
>> momentum can take on any value between plus and minus infinity. Time
>> is not an operator in quantum mechanics.
>>
>> Bruce
>
> Isn't there a valid interpretation/ application of the time-energy
> uncertainty relation in the context of emission of radiation? If so,
> what is it? TIA, AG
>

The rigorous versions of these interpretations involve having some
physical object included in the system that serves as a clock. So, if
you actually perform a measurement involving time, then the measured
time is represented by a physical clock. So, by including a quantum
mechanical description of a simplified model clock, you then do get an
observable for the measured time, despite the fact that there is no
observable that allows you to measure the parameter t in the Schrodinger
equation.

Saibal



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[Alan Grayson]

Can you give a concrete example where the time-energy form of the UP can be applied to? I once had an example, but can't recall what it was. TIA, AG

[Brent Meeker]

It's used all the time in interpreting collision spectra in particle physics.  A sharp resonant line in the energy spectrum implies the generation of a long live particle, while a broad line implies a short lifetime.

Brent

[Bruce Kellett]

But that is applying a generalised UP to an ensemble of similar short-lived particles. There is no significant uncertainty in the energy of each particle, although it may be uncertain when it will decay.

Bruce

[Alan Grayson]

If time isn't an operator, why does this work? And I'm not sure how to interpret it physically. If one waits some time t, and measures in some interval t, t + delta t, do we get a spread of energies? And of what? TIA, AG