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Message from discussion Hamiltonian Dynamics = Adiabatic Processes Only?
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Kevin A. Scaldeferri  
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 More options Jun 28 2000, 3:00 am
Newsgroups: sci.physics.research
From: ke...@cco.caltech.edu (Kevin A. Scaldeferri)
Date: 2000/06/28
Subject: Re: Hamiltonian Dynamics = Adiabatic Processes Only?
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In article <4Q5ClMhU7...@cc.usu.edu>, Charles Torre <to...@cc.usu.edu> wrote:
>In article <8j0e95$...@gap.cco.caltech.edu>, ke...@cco.caltech.edu (Kevin A. Scaldeferri) writes:
>> In article <UFS$3V8YH...@cc.usu.edu>, Charles Torre <to...@cc.usu.edu> wrote:

>>>Nah. If there is a Lagrangian formulation then there is a
>>>Hamiltonian formulation and vice versa.

>> I'm not sure this is true.  I can write down a Lagrangian with an
>> infinite number of time derivatives.  It's rather unclear how to turn
>> this into a Hamiltonian.  (This is a particularly perverse case, but
>> not completely without interest.)

>Well, ahem, hmmm. Okay. But you are stretching the definition of
>"Lagrangian" a bit from its usual one ( I guess I thought
>that the more or less standard classical mechanics kind of
>Lagrangian was being discussed). As usual, most disagreements
>are more about definitions than consequences of definitions. Normally
>one defines a Lagrangian as a local function of the spacetime coordinates,
>the dynamical variables and a finite number of their derivatives, which
>is what I had in mind.

You are right, this is a non-local Lagrangian.  To let the cat out of
the bag, what I am thinking of is the sort of Lagrangian encountered
in non-commutative field theories.

>I would be interested in hearing some more about
>cases where it is useful to think in terms of Lagrangians with an
>infinite number of derivatives. Oh. Maybe you are thinking about
>effective actions in quantum theory? Fair enough. (Still,
>even though an infinite number of derivatives may arise in a
>derivative expansion of the "Lagrangian", one usually truncates
>to a finite number of terms in any perturbative computation.
>Then what I said still applies.)

These cases are different from an effective action.  There is not a
momentum expansion the way there usually is in an effective theory.

OTOH, as I said, this case is a little perverse as the theories with
infinite numbers of time derivatives are not unitary.  The theories
with infinite space derivatives are okay, though.

--
======================================================================
Kevin Scaldeferri                       Calif. Institute of Technology
                      The INTJ's Prayer:
     Lord keep me open to others' ideas, WRONG though they may be.


 
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