$\Delta E \Delta t \geq \hbar /2$ and energy conservation

Question

As I understand it, time is not an observable in quantum mechanics, therefore $\Delta E \Delta t \geq \hbar/2$ is just a way to say that, given any operator $\hat{A}$, $$\sigma_H\sigma_A\geq\frac{\hbar}{2}\left|\frac{d}{dt}\langle A\rangle \right|$$ $$\Rightarrow \sigma_H\left(\frac{\sigma_A}{\left| \frac{d}{dt}\langle A\rangle\right|}\right)\geq\frac{\hbar}{2},$$ and since $\sigma_A / \left| d/dt\langle A\rangle\right|$ has the same unit as time we say that $\Delta E \Delta t \geq \hbar/2$, but $\Delta t$ is just a way to say how quick the expected value of the observable $A$ is changing and the expression for the uncertainty principle is some kind of statement about the stability of the system.

At the same time, I've seen some physicists say that this expression tell us that, for a small time interval, we can "violate" the conservation of energy by a small amount. But how exactly, in this context, should we think about the "small time interval"? Is it just an observable whose expected value varies quickly in respect to time? If so, can we say that for such an observable the conservation of energy is violated?

Answer 1

Since you have already presented the correct interpretation for $\Delta t$, there is little more technical to say about that part - claims that the energy-time uncertainty principle somehow allows us to "violate conservation of energy" are simply wrong. What follows is the most charitable interpretation I have to offer for why people nevertheless like to say this:

There is a different energy-time relation for unstable states, so-called resonances, where the lifetime of the state $\delta t$ is inversely proportional to its energy width $\delta E$, and we can also write that as $$ \delta t \delta E \geq \hbar,$$ where we should note that I could just as well have chosen the more usual $\Gamma$ and $\tau$ for the energy width and lifetime and then this wouldn't look so much like the uncertainty principle. But especially in pop-sci contexts, this looks enough like the uncertainty principle that waving our hands and claiming that it is is easier than spending an hour explaining to people what the heck the energy width of a scattering resonance is.

Therefore, I suspect this is what people are actually thinking about when they say this strange claim about violation of energy conservation: A stable particle with $\delta t = \infty$ has a definite mass-energy, i.e. $\delta E = 0$. The unstable ones don't have definite mass-energies, and if you squint and forget that this is an energy width in a differential cross-section and not a direct detection of an actual particle state, then this looks like "violation of energy conservation" since the unstable particle can have mass-energies different from the "definite" one it "should" have. Hence "short-lived particles can violate energy conservation".

Answer 2

Αt the same time, I've seen some physicists say that this expression tell us that, for a small time interval, we can "violate" the conservation of energy by a small amount

The $ΔEΔt>h/4π$ is the Heisenberg uncertainty principle for the time and energy variables , and there are problems in deriving it from the formal mathematical operator theory.

Let us take the simpler space momentum HUP

$ΔpΔx>h/4π$

Does demanding that it holds for a given measurement lead to violations of momentum conservation? Clearly not, as the HUP is for limiting the possibility of measuring the momentum with accuracy if the space is measured very accurately. For each measurement building the distribution of momentum with the same boundary conditions, momentum is conserved within the $Δp$ , and the accumulated distribution will show what the $Δp$ is.

Objectively the same should be true with the energy-time HUP.

Now when going to the mathematics from which the HUP can be derived, and the statements ""violate" the conservation of energy" one has to keep in mind for the thought experiment whether one is talking of virtual exchanges or on mass shell exchanges. To get real four vectors of energy-momentum an interaction is required. The models of pair creation out of the vacuum cannot hold if energy is not supplied by real incoming four vectors.