- #1

- 73

- 1

G(x

_{1},x

_{2},...,x

_{n}) = ∫Φ(x

_{1})Φ(x

_{2})...Φ(x

_{n})e

^{i S[Φ]/ħ }D[Φ] / ∫Φ(x

_{n})e

^{i S[Φ]/ħ }D[Φ]

But I've been struggling to explain the existence of the 'i'. It seems like this is a postulate that can't be derived from anything else. It makes sense that the exponential is there since we are multiplying an infinite set of amplitudes.

Also, in classical physics the action is only defined up to a multiplicative factor - the Euler-Lagrange equations don't care about a multiplicative factor.

So if we replaced S[Φ] with 100 S[Φ] or 10000 S[Φ] or i S[Φ] we should get the same theory classically. But we should get different quantum field theories right? So how do we determine that there should be this 'i' here? It is often described as a 'phase' but this is not where phase enters into quantum mechanics. Phase comes from, for example, an electromagnetic waves emitted from oscillating source. So this is not really to do with phase.

Is there some kind of experiment which determines that this factor must be i/ħ or can it indeed be something else such as 1000? Or is it a factor like the electron mass or charge that actually does not have a fixed value but depends on your cut off?

One argument is that it comes from Unitarity and compares it to the unitary operator exp(iH) but if the action is, for example S[Φ]=∂Φ∂Φ, which doesn't come from a Hamiltonian then the argument is not valid. As far as I can tell Unitartity is only relevant to fermion wave functions (like Shrodingers equation) because it preserves particles number (and so probability).

For example if you replaced this 'i' with -1, what theoretical result would not then agree with experiment?

It seems if you replace the 'i' with a number α, then you add an extra factor of 1/α to all propagators and α to every interaction. How can this 'i' be derived from experiment?