I have some few questions to ask in imaginary time and quantum theory :

- What is imaginary time which is very difficult to imagine at all ?
- How Stephen Hawking says that we can calculate the history of universe in imaginary time ?
- What does Quantum theory and Quantum principle mean ?
- What is the relation and difference between the real time and imaginary time ?

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TopNewestAnswering the 1s:

Imaginary time enters as a calculating trick to put time on the same footing as spatial dimensions. Depending on your convention, the spacetime interval \(ds^2\) is written as \(\left(dx^2+dy^2+dz^2-dt^2\right)\). This looks similar to the rule for combining perpendicular distances in a 4d Euclidean space. If we instead used imaginary time so that \(\tau/i=t\), we could simply write \(\left(dx^2+dy^2+dz^2+d\tau^2\right)\), which has time and space appearing in equivalent ways. In general, it is profitable to think of ways to bring new things into agreement with old things, so that the methods for old things can be used to solve new things.

You run into an analogous situation in quantum mechanics compared to statistical physics.

In both, you can calculate the probability of an event happening, for example, the state \(\mid A\rangle\) becoming the state \(\mid B\rangle\).

In QM, the procedure is to calculate the expectation value of a factor that describes transition probabilities. It has the form \(e^{-iHt/\hbar}\), where \(t\) is time, \(H\) is the total energy (Hamiltonian), \(\hbar\) is Planck's constant, and \(i\) is \(\sqrt{-1}\). In statistical physics, the probability of an event occurring corresponds to the expected value of \(e^{-\beta H}\) where, again, \(H\) is the Hamiltonian, and \(\beta\) is the inverse temperature of the system.

As you can see, these two approaches can be built in the same way if we opt for imaginary time \(\tau = it\) in the QM case. I.e. problems that exist in the space where time and space are on opposite footing in the metric, can be transformed into problems in a Euclidean space where time and space are on the same footing.

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