# Re: Path integrals and statistical mechanics

```There is another interesting piece of the puzzle. Laurent Nottale has
developed an interesting unification of Relativity and Quantum Physics
using the notions of Fractal Spacetime. IIRC, particles move in a 2D
time manifold (which is fractal, not Euclidean ie think "plane filling
curves") and this seems to tend to a QM limit ine one direction, and a
classical relativistic limit in the other.```
```
Being 2D, this does imply that "complex time" might have some legs in
it.

Cheers

Joao Leao wrote:
>
> The so-called "Wick rotation" which is often employed to turn the
> unruly measure of path integrals into a regular summable measure
> and is represented as
>
> t ---> -it
>
> has no direct relation with the so-called "Weyl unitarity trick"
> which is used to turn the bilinear anti-symmetric non-positive definite
> Lorentz metric dt^2 - dx^2- dy^2 - dz^2  to a unitary one:
> -dt^2-dx^2 - etc...
>
> though they have the same flavor as mathematical expedients
> without physical (empirical) meaning. The formal analogy
> between Quantum Field Theory and Stat Mech depends indeed on
> the first of these tricks.  Unless, of course, if you believe
> in "imaginary time" it will be hard for you to know what
> you are talking about when you speak of a rotation "in the
> complex plane of t". We would most likely need 4-dimensional
> wrist watches to display the "current iTime" ( though Apple is
> probably at work on an iClock as we speak !).
>
> -Joao Leao
>
>
>
> George Levy wrote:
>
> > Hi Doriano,
> >
> > Welcome to the list.
> >
> > You raise an interesting problem and. I don't know the answer to your
> > question. However, I just want to point out that an observer in relative
> > motion observes the rotation in the complex plane of space-time
> > geodesics. Could there be a connection between quantum and relativistic
> > rotations?
> >
> > George
> >
> > Doriano Brogioli wrote:
> >
> > > Hi to everybody. I subscribed to this mailing list yesterday, but I'd
> > > like to pose a question since I think it _must_ be the right place.
> > >
> > > Quantum mechanics can be formulated in terms of path integrals
> > > (Feinmann integrals). By substituting the time t with an (Euclidean)
> > > immaginary time i s, that is, a real value s times the imaginary root
> > > mean square of -1, the path integral changes to the Boltzmann
> > > distribution, where the energy is the (classical) energy of a
> > > continuum (classical) mechanical system, at temperature 1/h.
> > >
> > > From this fact, someone claims that quantum world is simply a
> > > classical world, but rotated by pi/2 in the complex plane of t: the
> > > real world is classical, but we see it at the wrong angle. In
> > > particular, something similar happens near the event horizon of a
> > > black hole, and it should be the ultimate origin of Hawking radiation.
> > >
> > > I tried to derive this relation, or some kind of this, and I concluded
> > > that it holds only at a formal level. Has anyone any idea about this
> > > topic?
> > >
> > > Doriano Brogioli
> > >
> > >
>
> --
>
> Joao Pedro Leao  :::  [EMAIL PROTECTED]
> Harvard-Smithsonian Center for Astrophysics
> 1815 Massachussetts Av. , Cambridge MA 02140
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>
>
>

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