Finitary Path Integrals, Causality & Curved Space-Time
Rock Brentwood
>> "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
FINITARY QUANTUM ACTION PRINCIPLE ("FEYNMAN CAUSALITY"):
exp(S(q0,q2)/(i h-bar)) = integral exp((S(q0,q1) + S(q1,q2))/(i h-
bar)) dq1.
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.
Rock Brentwood
> � � � � 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
>
> � � � � FINITARY QUANTUM ACTION PRINCIPLE ("FEYNMAN CAUSALITY"):
> � � � � exp(S(q0,q2)/(i h-bar)) = integral exp((S(q0,q1) + S(q1,q2))/(i h-
> bar)) dq1.
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.
Peter
...
> The action is obtained
> by summing over the "best" intermediate state
>
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,
G(r1,t1;r3,t3) = Integral G(r1,t1;r2,t2) G(r2,t2;r3,t3) dr2, t1 <
t2 < t3,
can be read also as 'no effect before cause', due to the time
ordering, t1<t2<t3.
I will add this to the next version of my paper, thank you for
bringing that formula to my attention!
Best wishes,
Peter