Welcome SourceCodeOf_HumanGenome
www.GrammaticalPhysics.ac
Account
Access Detector
2 user(s) are online (1 user(s) are browsing Forum)

Members: 1
Guests: 1

SourceCodeOf_HumanGenome more...

# Correspondence between Solution and Action

SourceCodeOf_HumanGenome > Correspondence between Solution and Action @ 2013/4/15 14:42
A solution of the new grammar version of the Shrödinger equation:
Φ[χ]＝ΠkG(χ(k/α),k/α;x0,t0)
can be cartoonized as a picture of a starfish.

The integrand of the Feynman path integral:
exp{(i/h)S[χ]}＝ΠkG(χ(k/α + 1/α), k/α + 1/α; χ(k/α), k/α)
can be cartoonized as a picture of a centipede.

Using these two pictures as a hint, I made up the following functional as a solution nearly corresponding to the action.
Φ[χ]＝[Πk∫dxk∫dtkG(χ(k/α),k/α;xk,tk)]ΠjG(xj+1, tj+1; xj, tj)

This solution can be called a starfish having a centipede in place of a central disc.

k∫dξ(0,k/α) G(ξ(ε,k/α),ε;ξ(0,k/α),0)]Φ[ξ(0,□)]
= [Πk∫dξ(0,k/α) G(ξ(ε,k/α),ε;ξ(0,k/α),0)][Πs∫dxs∫dtsG(ξ(0,s/α),s/α;xs,ts)]ΠjG(xj+1, tj+1; xj, tj)
= [Πk∫dxk∫dtk∫dξ(0,k/α) G(ξ(ε,k/α),ε;ξ(0,k/α),0)G(ξ(0,k/α),k/α;xk,tk)]ΠjG(xj+1, tj+1; xj, tj)
= [Πk∫dxk∫dtk G(ξ(ε,k/α),k/α+ε;xk,tk)]ΠjG(xj+1, tj+1; xj, tj)
= [Πk'∫dx'k'∫dt'k' G(ξ(ε,k'/α-ε),k'/α;x'k',t'k')]ΠjG(x'j+1, t'j+1; x'j, t'j)　where k'=k+αε, x'k'=xk, t'k'=tk
= [Πk∫dx'k∫dt'k G(ξ(ε,k/α-ε),k/α;x'k,t'k)]ΠjG(x'j+1, t'j+1; x'j, t'j)
= [Πk∫dxk∫dtk G(ξ(ε,k/α-ε),k/α;xk,tk)]ΠjG(xj+1, tj+1; xj, tj)
= Φ[ξ(ε,□-ε)]
∵ Πj G(x'j+1, t'j+1; x'j, t'j)
= Πj' G(x'j'+1, t'j'+1; x'j', t'j')
= Πj' G(xj+1, tj+1; xj, tj)
= Πj G(xj+1, tj+1; xj, tj)
where j'=j+αε

Therefore,
Φ[χ]＝[Πk∫dxk∫dtkG(χ(k/α),k/α;xk,tk)]ΠjG(xj+1, tj+1; xj, tj)
is an approximate solution.

I hope that a path integral of this solution reproduce the same result as a path integral of exp{(i/h)S[χ]}.

---
Last edited at 2013/04/21/14:09

Category
Forum
Global

Notification method is private message [Change]

Site Search
Site Info
Webmasters wooder Recommend Us
{"ft_lang_label":"__label__en","ft_lang_prob":0.5831415,"math_prob":0.99497896,"size":2132,"snap":"2019-13-2019-22","text_gpt3_token_len":1083,"char_repetition_ratio":0.13486843,"word_repetition_ratio":0.07116105,"special_character_ratio":0.38039398,"punctuation_ratio":0.19122808,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9922556,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,2,null,2,null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-22T05:55:10Z\",\"WARC-Record-ID\":\"<urn:uuid:826fd9a6-f45a-4c2f-b6bc-c5f36e544e7e>\",\"Content-Length\":\"27763\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fc608425-c95d-4541-a19a-0ce2108ea24b>\",\"WARC-Concurrent-To\":\"<urn:uuid:ade3e7c8-3254-4050-94c8-f4fa7972b7e2>\",\"WARC-IP-Address\":\"133.130.64.24\",\"WARC-Target-URI\":\"http://grammaticalphysics.ac/vintage/xoops/qftt2.htm\",\"WARC-Payload-Digest\":\"sha1:XMZQG3EOGKTGQ57Q6HMKP5AXLVLADN4G\",\"WARC-Block-Digest\":\"sha1:RUB2OL46JYJ5VEJSTCYGY3ZLC4OHUM2G\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202635.43_warc_CC-MAIN-20190322054710-20190322080710-00138.warc.gz\"}"}