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Is the imaginary time just a different convention to express the time evolution to make the calculations easier? Hawking also said that

"It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons." (From "The Universe In a Nutshell", UK, Bantam, 2001 edition, page 59, last paragraph.)

What are the predictions that we haven't been able to measure yet?

Thanks!!

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The main things that imaginary times gives us are "instantons". These are solutions of the classical equations that exist only in imaginary time. They describe tunneling processes that are classically forbidden, but quantum mechanically allowed.

For simple systems we can derive these tunneling processes by other means such as the WKB approximation, and the instanton predictions agree with both experiment and these alternative derivations. In field theory the alternative methods are harder to do, and for many effects have not been done.

There is at least one imaginary time prediction that we do not want to verify, and this is the decay of a "false vacuum state". Such a decay in our own universe would be fatal to all existing matter, and the life that depends on it.

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  • $\begingroup$ Thanks for the explanation! Are we able to verify the 'false vacuum state'? (I haven't learned that before but I'm just curious.) $\endgroup$ – ZR- Sep 21 '20 at 14:36
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    $\begingroup$ Not and survive! $\endgroup$ – mike stone Sep 21 '20 at 16:07

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