On Oct 22, 2:03 am, Koobee Wublee <koobee.wub...@gmail.com> wrote:
>> "There is no known formulation ofpathintegralsto >> curvedspacetimes."
In reply to Yablon's question -- the generalization of the Fourier Theorem was, in fact, the very impetus behind the recent work in generalizing the perturbation theory and renormalization formalisms to curved spacetime. It was largely heralded by Brunetti and Fredenhagen and works off of what I believe is called microlocal analysis.
> I have not read the article, but from the subject, I am certainly > entitled to make comments about that. It is indeed stupid to make > that statement. <shrug>
Then that negates the entitlement you're claiming.
In fact, it is a major open issue. The Osterwalder-Shrader theorem does not generalize to curved space-times. Some recent work on this is may be found in quant-ph/9904094. It's also discussed in passing in Week 146 of This Week's Finds.
One of the major open issues in the study of quantum field theory on curved spacetimes, particularly the question of how to carry out renormalization and perturbation theory in a curved setting. This has only been resolved in recent times -- and only then in the Epstein- Glaser-Bogoliubov causal framework; NOT in the setting of the path integral formalism.
One possible approach naturally suggests itself for generalizing the path integral formalism to a form suitable for application to curved spacetime.
It not only removes the need for the Osterwalder-Shrader theorem, but actually links it to the "causality" principle of Bogoliubuv-Epstein- Glaser -- providing an analogue to this principle.
The key is contained in theorem 1, section 7 of LNP 107, which states a result that amounts to being the classical version of the desired result.
The theorem stated there is the action principle in finitary form. It can be generalized to quantum settings, by replacing the right-hand side of the equation presented in the theorem by suitable finitary form of the path integral.
The result is a decomposition formula that recursively expresses the action over a region in terms of the actions over the components of a finite foliation of that region.
This is a summary of the process and results: One starts with a COMPACT region W (emphasis on compact) over base space M which possesses a timelike foliation t |-> W(t). (The compactness means that all the W(t)'s share a common spacelike 2- surface as a boundary, Bdy(W(t)) = H = horizon).
LNP 107 restricts attention to the case where the configuration space bundle Q has connected, simply connected fibres Q_t.
Each subregion W(t1,t2), comprising the foliation layers over the integral [t0,t1] has a symplectic structure given by omega(t0,t1) = dp1 ^ dq1 - dp0 ^ dq0. The dynamics are given by the condition that omega(t0,t1) = 0. Correspondingly, the canonical 1-form theta(t0,t1) = p1 dq1 - p0 dq0 is exact and the Poincare' lemma, one has generating functions S (q0,q1), dS(q0,q1) = p1 dq1 - p0 dq0.
Theorem 1 is the finitary action principle. Noting that one has a cancellation theta(t0,t2) = theta(t0,t1) + theta(t1,t2) omega(t0,t2) = omega(t0,t1) + omega(t1,t2) one has a situation similar to Stokes' Theorem. The action is obtained by summing over the "best" intermediate state S(q0,q2) = S(q0,q1) + S(q1,q2) exrtremized over q1.
This principle bears the analogue of the "causality principle" in Bogoliubov-Epstein-Glaser perturbation. It quantizes to the following recursive system
Thus, we arrive at an EXACT formulation of the Path integral approach suitable for curved space.
It gives you a recursive decomposition of the generating functions S over the different subregions in terms of those for its subregions.
An interesting exercise -- not yet carried out -- is to run this formulation on "Example 1" in Section 7 of LNP 107. Example 1 provided an illustration of the classical version, above, of the finitary action principle. So, running this through the quantized version, one should end up with the quantized simple harmonic oscillator.
On Oct 31, 12:50 pm, Rock Brentwood <markw...@yahoo.com> wrote:
> S(q0,q2) = S(q0,q1) + S(q1,q2) exrtremized over q1.
[Classical Theorem]
> This principle bears the analogue of the "causality principle" in > Bogoliubov-Epstein-Glaser perturbation. It quantizes to the following > recursive system
This is a bit hasty. The path integral is giving you a fancy kernel for the vacuum-expectation-of-time-ordered operator, <0| T[ ... ] |0> in the form integral { Dq exp((i/h-bar) S) [...] } = integral { Dq exp((i/h-bar) S) } <0| T[...] |0>.
I posted an article a while back on how one might do away with all pretense toward any kind of limit relation and just simply graft in the Epstein-Glaser formulation for T[...] on the right hand side.
Here, I'm trying to push this down one level further and directly incorporate the E-G decomposition formula by exploiting the classical decomposition formula.
For the simple harmonic oscillator, one can in fact directly see the decomposition. The total action (over short enough time intervals to avoid problems with periodicity) is S(t1, t2) = (mw)/(2S) ((q0^2 + q1^2) C - 2 q0 q1) where q0 = q(t0), q1 = q(t1) and (C, S) = (cos w(t1-t0), sin w(t1-t0))
The sum S(t2, t1) + S(t1, t0) can be directly verified to have an extremum at q1* = (q0 S12 + q2 S01)/S02, where S12 = sin w(t2-t1), S01 = sin w(t1-t0), S02 = sin w(t2-t0). And at the extremum, one indeed has S(t2, t1*) + S(t1*, t0) = S(t2, t0).
So, this is the classical version of the causality principle; and Theorem 1 in section 7 of LNP 107.
In the path integral expression on the left, the action S(t1,t0) is taken over an interval [t0,t1] and incorporates a sum over all histories. A kind of causality principle is implemented by the decomposition Dq_{t2 t0} = Dq_{t2 t1} Dq_{t1 t0} where Dq_{t' t''} is the (infinite) product Pi_{t = t' to t''} dq(t).
The integral on the left breaks down into a product of integrals associated with the intervals [t0,t1] and [t1,t2] and is stitched together with an integral over q1 -- something like this
integral Dq_{t2 t0} [...] ==> integral dq1 (integral Dq_{t2 t1} [...]) (integral Dq_{t1 t0} [...])
where the respective path integrals are taken with fixed endpoints (q2,q0) on the left, (q2,q1) and (q1,q0) on the right and the intermediate variable is integrated out.
The problem is that the equivalent expression on the right is under <0| ... |0>'s. So, there's no easy way to get a self-contained recursive decomposition formula that completely does away with any kind of limit definition or infinitary element.
On 31 Okt., 19:50, Rock Brentwood <markw...@yahoo.com> wrote: ...
> The action is obtained > by summing over the "best" intermediate state > S(q0,q2) = S(q0,q1) + S(q1,q2) exrtremized over q1.
> This principle bears the analogue of the "causality principle" in > Bogoliubov-Epstein-Glaser perturbation...
This formula looks like another formulation of Huygens' Principle (invented by Huygens for mechanical purposes; it makes the velocity of a body to be continuous), cf Feynman's paper pioneering the path integral method (Rev. Mod. Phys. 1948) and my papers on Huygens' Principle as a most general principle for transport and propagation processes (Eur. J. Phys. 1996; Lat. Am. J. Phys. Educ. Vol. 3, No. 1, Jan. 2009 19 / http://www.journal.lapen.org.mx).
Indeed, the Chapman-Kolmogorov equation for the propagator (Green's function), G,
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