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I read from this website that the uncertainty principle implies that seemingly empty space is filled with energy, called vacuum energy . The relevant equation I can think of is $$\Delta E\Delta t \geq \frac{\hbar}{2}.$$ Here, $\Delta E$ is standard deviation of energy for a system in quantum state $|\psi \rangle$ and $\Delta t$ is the approximate amount of time it takes for the expectation value of energy to change by a standard deviation as the system evolves.

How is this related to the statement that empty space is filled with energy?

Qmechanic
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TaeNyFan
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    "that empty space is filled with energy" is not correct. The HUP applies to results of measurements, to measure you need interactions ,i.e. supply energy. – anna v Nov 15 '19 at 14:45
  • @AnnaV Correct, but QFT the same issue pops up as zero point energy. In my opinion absent in a sufficiently careful formulation. – my2cts Nov 15 '19 at 15:12
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    While some pop-sci sources appeal to the HUP to explain the QFT vacuum, that is layering hand-waving on imprecision on vagueness. – dmckee --- ex-moderator kitten Nov 15 '19 at 17:07
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    Essentially a duplicate of https://physics.stackexchange.com/q/474545/50583 – ACuriousMind Nov 15 '19 at 18:27
  • @AnnaV Fair enough, but as far as I know the mainstream physics opinion is that there is a cosmological constant problem. – my2cts Nov 15 '19 at 18:38
  • I might have misunderstood, but as I recall in A Brief History of Time it is implied that a part of space having zero magnetic and electric field and zero time derivatives for those fields implies a violation of the uncertainty principle. Been a long time. – R. Romero Nov 15 '19 at 21:45

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