I've come across a hole in my understanding.
Heisenberg's Uncertainty Principle can be expressed in terms of energy and time as
$$ \Delta E \, \Delta t \geq \frac{\hbar}{2} $$
where $\Delta E$ is the uncertainty in a particle's energy and $\Delta t$ is the time period over which this measurement is taken.
However, I know that the existence of virtual particles violating the law of conservation of energy is explained by this relation but on the surface, it does not make sense.
To explain, it is said that virtual particles (pairs) can "pop" into existence in the vacuum as long as they disappear (annihilate) within a time $t$. This is put down to the uncertainty relation given above. However, the uncertainty relation given above states that the uncertainty in a virtual particle's energy can be arbitrarily high for arbitrarily high times as long as their product exceeds $\frac{\hbar}{2}$.
I have seen a different relation given below which makes sense of this by turning around the inequality sign and getting rid of the $\Delta$ symbols but I do not see how to derive this properly. Any help would be appreciated as it's annoying me!
$$ E \, \tau \leq \frac{\hbar}{2} $$ where $\tau$ is the particle's lifetime and $E$ is the particle's actual energy now rather than the uncertainty in the energy.
