Posted: Oct 11, 2012 6:48 pm
twistor59 wrote:zaybu wrote:twistor59 wrote:
Yes, that's my impression too. As soon as you're dealing with a 4 dimensional spacetime, presumably you've got yourself a block universe?
Twistor, take a second look at that paper. It's full of nonsense. Here's one statement that got me jumped off the floor: "The received view has it that Schrödinger’s equation is Galilean invariant, so it is generally understood that NRQM resides in Galilean spacetime and therefore respects absolute simultaneity."
Well, I posted it because section 2 contains a description of the blockworld, which was the subject of this thread.
But anyway, when they write "NRQM resides in Galilean spacetime" I would interpret that to mean "the Hilbert space of NRQM carries a unitary representation of the Galilean group". However, Galilean invariance of the Schroedinger equation is a bit awkward, since if you perform Galilean transformations on the wavefunctions, you have to fuck about with their phases to retain covariance:
http://redshift.vif.com/JournalFiles/V13NO4PDF/V13N4OST.pdf
Maybe it means that quantum mechanics was invented on the shores of the sea of Galilee?
If the theory is invariant under a symmetry, it is with the Lagrangian that we must work it out. In the case of lorentzian symmetry, you perform the transformation:
(1) xμ → x'μ= Λμνxν , where Λμν is part of the Lorentz group.
For a scalar field, this means,
(2) ϕ(x) → ϕ'(x) = ϕ(Λ-1 x)
Substitute (1) and (2) into the Langrangian,
L = 1/2 Nμν ∂μϕ∂νϕ − 1/2 m2ϕ2, where Nμν is the Minkowski metric.
If you can show that L = L', then you have an invariance, which for the Lorentz transformation, this turns out to be true.
But the paper talks about the invariance of the Schroedinger equation. I know of no textbooks that ever did that.