Talk:Path integral formulation
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I believe Dyson was the one that showed the approaches to be equivalent JeffBobFrank 01:21, 18 Feb 2004 (UTC)
The last paragraph says some contentious things. The sum-over-histories method is hardly "unpopular". The "sum-over-histories interpretation", however - that is, the attempt to elevate the sum-over-histories formalism into a physical ontology - is indeed little-known; I don't think I've ever seen it outside that paper coauthored by Sorkin. Let me quote the paper's last paragraph:
"... the sum-over-histories formulation goes a long way toward taking the 'mystery' out of quantum mechanics, or at least reducing it to the mystery inherent in the notion of probability itself. No doubt that mystery is enhanced somewhat by the presence of non-positive amplitudes and references to two-way paths, but the fundamental idea... remains the same..."
In my opinion this indicates the sophistical character of this sum-over-histories "interpretation". I'm reminded of a cartoon: a physicist stands at a blackboard, in front of a crowd of skeptical colleagues. In the middle step of his derivation, he has written, THEN A MIRACLE OCCURS. "See? It's all just probabilities. Of course, some of them are negative probabilities, a concept which makes no sense under either the frequentist or the subjectivist interpretation of the concept of probability; but that just shows that further research is required..."
There is something to the claim that "[this is] the only form of the theory which can explain [the EPR] paradox without breaking locality". The individual paths appearing in the formalism are indeed built purely from ontologically local entities (point particles, local field values), something which is not true in any formalism which countenances, say, entangled quantum states. Nonetheless, the paper by Sinha and Sorkin (in its concluding analysis) in fact expresses some doubt as to whether sum-over-histories is local after all, given the "global character" of how the final probabilities are calculated.
Wikipedia is hardly the place in which theoretical debates of this sort should be adjudicated, but I hope it's clear why I find that last paragraph somewhat problematic. I also want to emphasize again, for absolute clarity, that the sum-over-histories method is not being criticised here, because it is only an algorithm. It's the attempt to turn it into an ontology (an "interpretation") which is deeply problematic. I leave it to more experienced Wikipedians to decide what the just solution here is. Mporter 21 Feb 2004, 5.55pm AEST
- As a sidelight, apropos your comments about negative probabilities, you may enjoy Feynman "Negative probability" in Quantum Implications, eds Hiley and Peat, where he makes a case for allowing them, as long as such an event is not measurable/verifiable. Like having negative dollars as you add up your bills, it may be calculationally allowed as long as certain restrictions on the state are true.GangofOne 07:04, 10 Jun 2005 (UTC)
Should this article actually be merged with Functional integral (QFT)? While it is in principle the same subject, that article is both very specific in its application to quantum field theory (as opposed to, say, nonrelativistic single-particle QM), and is also very technical. This seems to be more the place for an introduction to the path-integral formulation. (If we do want to merge the articles, I say the other one should come here, and not the reverse, since this article has the more general title.) And I'd rather do it sooner than later. --Matt McIrvin 04:06, 27 Sep 2004 (UTC)
Well, I went ahead and did it... --Matt McIrvin 06:13, 27 Sep 2004 (UTC)
The material formerly in Functional integral (QFT) is now incorporated into a section here, and I've tried to write some introductory matter to make the symbols a little clearer, though the heavily mathematical part further down still needs a lot more explanatory text. I've put in an introduction and reorganized the whole page into sections and subsections; my new section on single-particle mechanics needs more development but is a start. Diagrams would be nice. I've kept the controversial section on QM interpretation at the very end; I'll let other people argue over that for now. --Matt McIrvin 07:15, 27 Sep 2004 (UTC)
Attempted to NPOVify the interpretation section. --Matt McIrvin 05:35, 2 Oct 2004 (UTC)
Is <math> \lim_{\Delta t \rightarrow 0 } \int_{-\infty}^{+\infty} dx_1 \int_{-\infty}^{+\infty} dx_2 \int_{-\infty}^{+\infty} dx_3 \ldots \int_{-\infty}^{+\infty} dx_{n-1} \ e^{\frac{i}{\hbar}I(H(x_j, t))} <math> realy correct?
Wouldn't it rather be like <math> \lim_{\Delta t \rightarrow 0 } \int_{-\infty}^{+\infty} dx_1 \int_{-\infty}^{+\infty} dx_2 \int_{-\infty}^{+\infty} dx_3 \ldots \int_{-\infty}^{+\infty} dx_{n-1} \ \prod_{j=1}^{n-1}e^{\frac{i}{\hbar}I(H(x_j, t_j))} <math>
with <math> t_j = j \Delta t<math>
or is it
<math> \lim_{\Delta t \rightarrow 0 } \int_{-\infty}^{+\infty} dx_1 \int_{-\infty}^{+\infty} dx_2 \int_{-\infty}^{+\infty} dx_3 \ldots \int_{-\infty}^{+\infty} dx_{n-1} \ e^{\frac{i}{\hbar}I(H(x_1, \ldots, x_{n-1}, t_j))} <math>
with different H for each n ?
- The way I wrote it is perhaps not the best way of putting it; it needs to be more explicit. What I really wanted to get across is that in the integrand, <math>H<math> is the function of time represented by a set of straight segments connecting the <math>x_j<math> at times <math>t_j<math>, and <math>I<math> is actually the integral of the Lagrangian <math>L(x, \dot x, t)<math> over that path. I suppose in practice it would end up being the product of the exponential for each little segment, but that form is further from the spirit of the thing.
- I probably should have abandoned the generic use of <math>H<math> at that point... my mind's too fuzzy right now to make it better. --Matt McIrvin 00:27, 11 Oct 2004 (UTC)
- Also each little segment would depend on <math>x_j<math> and <math>x_{j+1}<math>... --Matt McIrvin 15:01, 11 Oct 2004 (UTC)
- This is not a necessity, the limit inherent to integration would take care of this as <math>x_{j+1} = x_j +{\rm d}x<math>, see Riemann sums).
- I have searched the net but didn't find anything better than stated here so I have tried some own thoughts.
- Starting from the <math>\sum_{\rm all\ paths}e^{{\rm i} S}<math> approach I came up with <math>\int_{\bar{\varphi} \in \{\bar\varphi | \bar\varphi(0) = \bar a; \bar\varphi(1) = \bar b\}}e^{{\rm i}\int_{\lambda = 0}^{1}\bar E \bar\varphi(\lambda){\rm d}\lambda}{\rm d}\mu(\bar\varphi)<math>
where <math>\bar\varphi<math> varies over all paths in spacetime starting from <math>\bar\varphi(0)=\bar a<math> and ending in <math>\bar\varphi(1)=\bar b<math>, <math>\bar E<math> denoting the energy four-vector and <math>\mu<math> is an aproptiate measure on the set of possible paths. With the paths approximated by segments of straight lines we are likely to end up with the official thing but with an additional benefit of a clearer understanding.
- Alas, I am stuck on <math>\mu<math> as well as on <math>\bar E<math>, especially in case where we have zero rest mass.
- Can anyone do better please? 217.94.149.179 20:05, 20 Oct 2004 (UTC)
Pavel V. Kurakin (Keldysh Institute of Applied Mathematics, Russian Academy of Sciences (http://www.keldysh.ru), me (http://www.keldysh.ru/departments/dpt_17/kurakin.html)).
My idea is that many-paths are physically real, but in sub-quantum (not observed by us) world. Many-paths, amplified by transactional interpretation of quantum mechanics (TIQM) (http://en.wikipedia.org/wiki/Transactional_interpretation) by John Cramer lead me to a 3rd new idea (after 1st: many-paths and 2nd: transactions). 3 together they constitute, I believe, an original theory, letting to explain quantum superposition of states, state vector reduction and non-local correlations like EPR (see quantum entenglement (http://en.wikipedia.org/wiki/Quantum_entanglement)).
Shortly speaking, signals move in vacuum in so-called 'hidden time', which is not equivalent to our physical time. They move between all sources, which are to emit particles, and all (possible) detectors. In the simplest case we have one source and a set of possible detectors. How will a particle chose one of many detectors?
It explores the space and counts how much it likes different detectors, in full accordance with Feynman many-paths. While it explores (many copies of that particle travel and explore), phisical time does not tick. Finally the source prefers some definite detector. Copies of the particle (more strictly - signals) are killed all but one. This one ultimately comes to a detector we physically see our particle at.
How long can signals explore the space? Infinite time! :) -- In 'hidden' time. Physiacl time does tick (at detecting point) only when 'ultimate decision signal' comes to that detector.
More accurate arguments were published this year by Keldysh Institute of Applied Mathematics, Russian Academy of Sciences in my preprint (http://www.geocities.com/bellstheorem/index.html).
I would be happy to know any criticism :)