Re: Path integrals and statistical mechanics

From: Joao Leao <>
Date: Mon, 23 Jun 2003 13:44:06 -0400

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  :::
Harvard-Smithsonian Center for Astrophysics
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"All generalizations are abusive (specially this one!)"
Received on Mon Jun 23 2003 - 13:45:57 PDT

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