
aiss-matricks

bhautiki mein, "s"-matricks yaa scattering matricks ack scattering prakriya seey gujarane vaali bhautik pranali kii praarambhik sthiti aur antim sthiti seey sambandhit haye. iska upayog quantum yantriki, prakirnan siddhant aur quantum kshetra siddhant (qft) mein kiya jaataa haye.
adhik aupchaarik roop seey, qft key sandarbh mein, "s"-matricks koo bhautik avasthaon key hilbert sthaan mein asymptotic roop seey mukt kann avasthaon ("inn-states" aur "out-states") key seton koo jodane vaale ekaatmak matricks key roop mein paribhaashit kiya gayaa haye. ack bahu-kann avastha koo "mukt" (gair-antahkriya) kahaa jaataa haye yadi yeh ack tensor utpaad key roop mein lorentz parivartan key tahat parivartit hoe jaataa haye, yaa bhautiki kii bhasha mein "pratyaksh utpaad", jaisa kii neeche diye gaye sameekaran dwara nirdhaarit kiya gayaa haye. "asymptotically free" kaa matlab haye kii raajya kaa yeh roop yaa tou door key ateet mein yaa door key bhavishya mein haye.
jabaki "aiss"-matricks koo kissy bhee prishthabhoomi (spacetime) key liye paribhaashit kiya jaa sakata haye joe asymptotic roop seey hull karane yogya haye aur jisamen koee ghatana kshitij naheen haye, iska minkovsky sthaan key mamle mein ack saral roop haye. iss vishesh mamle mein, hilbert sthaan asamaroop lorentz samooh (poinkere samooh) key aparivartaniya ekaatmak pratinidhitva kaa ack sthaan haye; "s"-matricks sootra _ 1 (door key ateet), aur sootra _ 2 (door key bhavishya) key beach vikas prachaalak haye. ise kewal shoonya oorjaa ghanatva (yaa anant kann prithakkaran doori) kii seema mein paribhaashit kiya gayaa haye.
yeh dikhaaya jaa sakata haye kii yadi minkovsky sthaan mein ack quantum kshetra siddhant mein dravyamaan antar haye, tou asymptotic ateet mein aur asymptotic bhavishya mein sthiti donon koo fock space dwara varnit kiya jaataa haye.
itihas.
aiss"-matricks koo pahali baar 1937 key pepper mein johnn archibald wheeler dwara "anunaad samooh sanrachana kii vidhi dwara prakaash naabhik key ganitiya vivaran parr" pesh kiya gayaa thaa. iss pepper wheeler mein ack "scattering matricks" pesh kiya gayaa-gunank kaa ack ekaatmak matricks joe "ack maanak roop key samadhanon key saath ack manmaane vishesh samadhan [abhinn samikaranon key] key asymptotic vyavhaar koo jodata haye", lekin ise poori tarah seey vikasit naheen kiya.
1940 key dashak mein, werner heisenberg ney swatantra roop seey "aiss"-matricks key vichaar koo vikasit aur pramaanit kiya. usa samay quantum kshetra siddhant mein maujood samasyagrast vichalanon key kaaran, heisenberg koo "siddhant kii aavashyak visheshtaon" koo alag karane key liye prerit kiya gayaa thaa joe bhavishya key parivartanon seey prabhaavit naheen honge kyonki siddhant vikasit huaa thaa. aisaa karate hue, unhein ack ekaatmak "visheshta" "s"-matricks pesh karane key liye prerit kiya gayaa.
haalaanki, aaj, sateek "aiss"-matricks parinam anuroop kshetra siddhant, samakalan pranali, aur quantum kshetra siddhant aur string siddhant key kaee aur kshetron kii ack pramukh upalabdhi hain. "aiss"-matrick ack kshetra-saiddhaantik upchaar key liye vikalp naheen hain, balki, aise antim parinamon key poorak hain.
prerana.
uchch-oorjaa kann bhautiki mein ack vyakti prakirnan prayogon mein vibhinn parinamon kii sambhavana kii ganana karane mein ruchi rakhta haye. inn prayogon koo teen charanon mein vibhajit kiya jaa sakata haih
jiss prakriya seey aaney vaale kanon koo (unaki antahkriya key madhyam seey) baahar jaane vaale kanon mein parivartit kiya jaataa haye, usey prakirnan kahaa jaataa haye. kann bhautiki key liye, inn prakriyaon kaa ack bhautik siddhant vibhinn aaney vaale kanon key vibhinn urjaaon key saath takaraane parr vibhinn baahar jaane vaale kanon kii sambhavana kii ganana karane mein saksham honah chaahiye.
quantum kshetra siddhant mein "s"-matricks thheek isi koo praapt kartaa haye. yeh maanaa jaataa haye kii inn mamalon mein laghu-oorjaa-ghanatva kaa anuman maanya haye.
upayog karein.
s"-matricks quantum yantriki mein sankraman sambhavana ayam aur vibhinn antahkriyaon key cross section seey nikatata seey sambandhit haye; "s"-matricks mein tatvon (vyaktigat sankhyaatmak pravishtiyon) koo prakirnan ayam key roop mein jaanaa jaataa haye. jatil-oorjaa tala mein "aiss"-matricks key dhruvon kii pehchaan baadhya avasthaon, aabhaasi avasthaon yaa anunaad key saath kii jaati haye. jatil-oorjaa tala mein "aiss"-matricks kii shakha katauti ack prakirnan channel key udghaatan seey judee hoti haye.
quantum kshetra siddhant key liye hamiltonian drishtikon mein, "s"-matricks kii ganana paraspar kriya chitra mein ekikrit hamiltonian key samay-crum ghatank key roop mein kii jaa sakati haye; ise fenman key path samakalan kaa upayog karke bhee vyakt kiya jaa sakata haye. donon hii mamalon mein, "s"-matricks kii vichalit karane vaali ganana seey fenman aarekh banate hain.
scattering theory mein, "aiss"-matricks ack operator haye joe heisenberg chitra mein mukt kann "out-states" (scattering channel) key liye mukt kann "inn-states" kaa maanchitran kartaa haye. yeh bahut upayogi haye kyonki aksar humm baatchit kaa varnan naheen curr sakate hain (cumm seey cumm, sabse dilchasp naheen).
ack-ayami quantum yantriki mein.
ack saral prototype jisamen "aiss"-matricks 2-ayami haye, pahale chitran key uddeshyon key liye maanaa jaataa haye. isamen, 1-ayami quantum yantriki key niyamon key anusaar ack sthaaniya kshamata seey tej oorjaa vaale kann bikhare hue hain. pahale seey hii yeh saral model adhik samanya mamalon kii kuchh visheshtaon koo pradarshit kartaa haye, lekin ise sambhalana aasaan haye.
pratyek oorjaa ack matricks pradaan karati haye joe iss parr nirbhar karati haye. iss prakaar, kull "s"-matricks koo, alankarik roop seey, ack upayukt aadhar parr, pratyek tatva shoonya key saath ack "nirantar matricks" key roop mein dekhaa jaa sakata haye, sivaay kissy diye gaye key liye vikarn key saath-block key.
paribhasha.
ack sthaaniya ack ayami sambhaavit baadha parr vichaar karein, joe oorjaa key saath quantum kanon kii kiran key adheen haye. yeye kann baaen seey daaen sambhaavit baadha parr ghatana hain.
sambhaavit baadha key baahar schroedinger key sameekaran key samadhan samtal tarangen hain joe
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sambhaavit baadha key baain or key kshetra key liye, aur
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sambhaavit baadha key adhikaar key kshetra key liye, jahan
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tarang sadish haye. hamaare avalokan mein samay nirbharta kii aavashyakta naheen haye aur isaliye ise chhod diyaa gayaa haye. gunank vaalaa shabd aaney vaali tarang kaa pratinidhitva kartaa haye, jabaki gunank vaalaa shabd baahar jaane vaali tarang kaa pratinidhitva kartaa haye. yeh paraavartit tarang key liye khadaa haye. choonki humm aaney vaali tarang koo sakaratmak disha mein (baain or seey aaney vaali) nirdhaarit karate hain, isaliye yeh shoonya haye aur ise chhoda jaa sakata haye.
prakirnan ayam", i.ee.aaney vaali tarangon key saath baahar jaane vaali tarangon kaa sankraman ativyaapi ack raikhik sambandh haye joe "s"-matricks koo paribhaashit kartaa haye,
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uparokt sambandh koo iss prakaar likhaa jaa sakata haye -
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kahaan
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sambhaavit baadha key prakirnan gunon koo poori tarah seey visheshta dete hain.
ekaatmak sampatti.
s"-matricks kii ekaatmak sampatti seedhe quantum yantriki mein sambhaavyata dhaara key sanrakshan seey sambandhit haye.
tarang falan kii sambhaavyata vartamaan ghanatva koo iss prakaar paribhaashit kiya gayaa haye
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baadha key baain or sootra 11 kaa sambhaavyata vartamaan ghanatva sootra _ 10 haye
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jabaki baadha key daaeen or sootra 14 kaa sambhaavyata vartamaan ghanatva sootra _ 13 haye
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sambhaavyata dhaara key sanrakshan key liye,. iska tatparya haye kii "s"-matricks ack ekaatmak matricks haye.
samay-vipareet samroopata.
yadi kshamata vaastavik haye, tou pranali mein samay-vipareet samroopata hoti haye. iss sthiti mein, yadi schroedinger key sameekaran kaa samadhan haye, tou yeh bhee ack samadhan haye.
samay-ulat samadhan dwara diyaa jaataa haye
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sambhaavit baadha key liye baain or key kshetra key liye, aur
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sambhaavit baadha key adhikaar key kshetra key liye,
jahan gunank key saath shabd, aaney vaali tarang kaa pratinidhitva karate hain, aur gunank key saath shabd, baahar jaane vaali tarang kaa pratinidhitva karate hain.
way phir seey "s"-matricks seey sambandhit hain,
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vaha haye,
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abb, sambandh
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ack saath ack sthiti utpann karein
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iss sthiti kaa, ekatmakta sambandh key sanyojan seey, tatparya haye kii "s"-matricks samamit haye, samay ulat samroopata key parinamswarup,
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samroopata aur ekatmakta key sanyojan seey, aiss-matricks koo iss roop mein vyakt kiya jaa sakata haih
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sootra _ 24 aur sootra _ 25 key saath. isaliye aiss-matricks teen vaastavik maapdandon dwara nirdhaarit kiya jaataa haye.
sthanantaran matricks.
sthanantaran matricks" sootra _ 26 samtal tarang sootra _ 27 aur sootra _ 28 koo prakirnan kshamata key "daaeen" taraf samtal tarang sootra _ 29 aur sootra _ 30 koo "baain" taraf seey sambandhit kartaa haih 
sootra _ 31 aur isake ghatakon koo aiss-matricks key ghatakon seey sootra _ 32 aur sootra _ 33 key madhyam seey praapt kiya jaa sakata haye, jisake madhyam seey samay-vipareet samroopata maani jaati haye.
samay-prativarti samroopata key mamle mein, sthanantaran matricks sootra _ 26 koo teen vaastavik maapdandon dwara vyakt kiya jaa sakata haih
sootra _ 35 
sootra _ 36 aur sootra _ 37 key saath (yadi r = 1 haye tou baain aur daaeen or key beach koee sambandh naheen hoga)
parimit varg achchi tarah seey.
dravyamaan m key saath ack kann kii samay-vipareet samroopata key saath ack-ayami, gair-saapeksh samasya joe ack (sthir) parimit varg "achchi tarah seey" taka pahunchati haye, kaa sambhaavit kaary v haye. 
mukt kann key tarang paiket koo samtal tarang sootra mein vighatit karke prakirnan koo hull kiya jaa sakata haye, jisamen tarang sankhya sootra 42 key saath baain or seey aaney vaali samtal tarang key liye yaa isi tarah daaeen or seey sootra 43 (door) key liye.
tarang sankhya k key saath samtal tarang key liye aiss-matricks kaa samadhan haih
jisake dwara sootra _ 49 varg kuein key ander samtal tarang kii (badhee hui) tarang sankhya haye, kyonki samtal tarang seey judey oorjaa igenvalue sootra _ 50 koo sthir rehna haih sootra _ 51
sancharan sootra haye _ 52
sootra 53 key mamle mein phir sootra 54 aur isaliye sootra 55 aur sootra 56 i.ee. tarang sankhya k key saath ack samtal tarang binaa paraavartan key kuein seey gujarati haye yadi sootra _ 58 key liye sootra _ 57 haye.
seemit varg baadha.
varg "baadha" varg key samaan haye jisamen sootra _ 60 key liye sootra _ 59 kaa antar haye.
samtal tarangon key oorjaa igenvalue sootra _ 61 key aadhar parr teen alag-alag mamle hain (tarang sankhya k resp key saath. -key) baadha seey doorah
sancharan gunank aur paraavartan gunank.
sambhaavit baadha key baain or seey sancharan gunank haye, jaba,
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isaliye inn aur out avasthaon kii maang poorna hamiltonian key eigenstate hain joe mishrit kann oorjaa shabdon kii anupasthiti key kaaran aavashyak roop seey gair-paraspar kriya curr rahey hain. uparokt khand mein charcha seey pataa chalata haye kii raajyon mein aur raajyon key baahar aisaa honah chaahiye kii
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badey sakaratmak aur nakaratmak key liye, mukt-kann avasthaon dwara darshaae gaye sambandhit package kii upasthiti hoti haye, joe gati mein chikani aur upayukt roop seey sthaaniyakrit maani jaati haye. tarang package aavashyak hain, anyatha samay vikas seey kewal ack charan kaarak praapt hoga joe mukt kanon kaa sanket detaa haye, joe aisaa naheen hoe sakata haye. daahine haath kii or seey nimnalikhit haye kii ander aur baahar kii avasthaaen oopar prati hamiltonian key eigenstate hain. iss aavashyakta koo aupchaarik banaane key liye, yeh maan lein kii poorna hamiltonian koo dou shabdon mein vibhajit kiya jaa sakata haye, ack mukt-kann hamiltonian aur ack antahkriya, jaise kii saamaanyikaran aur lorentz parivartan gunon key sambandh mein eigenstate kii upasthiti inn-aur out-states key samaan hoe,
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inn aur out raajyon koo poorna hamiltonian key eigenstate key roop mein paribhaashit kiya gayaa haye,
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santoshjanak
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key liye yaa kramshah. paribhaashit karein
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taba
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yeh antim abhivyakti kewal tarang packages kaa upayog karke kaam karegi.inn paribhashaon seey pataa chalata haye kii ander aur baahar kii avasthaaen usee tarah samanya hoe jaati hain jaise mukt-kann avasthaaen,
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aur teen sett ekaatmak roop seey samatulya hain. abb igenvalue sameekaran koo phir seey likhein,
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jahan lhs parr operator koo vyutkramaniya banaane key liye shartein jodee gaee hain. kyonki inn aur out raajyon key liye mukt-kann raajyon koo cumm curr diyaa,
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rhs parr praapt karane key liye
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phir mukt kann avasthaon kii poornata kaa upayog karein,
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anta mein praapt karane key liye
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yahaan mukt kann avasthaon parr isake igenvalue dwara pratisthaapit kiya gayaa haye. yeh lipman-schwinger sameekaran haye.
raajyon mein baahar key raajyon key roop mein vyakt kiya gayaa.
praarambhik avasthaon kaa vistaar antim avasthaon key aadhar parr kiya jaa sakata haye (yaa isake vipareet). poornata sambandh kaa upayog karate hue,
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jahan sambhavana haye kii antahkriya badal jaati haye
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mein
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quantum yantriki key samanya niyamon key anusaar,
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aur koee likh sakata haye
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vistaar gunank sateek roop seey neeche paribhaashit kiye jaane vaale "s"-matricks tatva hain.
aiss"-matricks.
s"-matricks abb dwara paribhaashit kiya gayaa haye
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yahaan aur sankshipt naam hain joe kann samagri kaa pratinidhitva karate hain lekin vyaktigat label koo dabaa dete hain. s"-matricks seey judaa huaa vahaan s-operator dwara paribhaashit haye
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jahan mukt kann avasthaaen hain. yeh paribhasha paraspar kriya chitra mein upayog kiye gaye pratyaksh drishtikon key anuroop haye. ekaatmak samatulyata key kaaran bhee,
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bhautik aavashyakta key roop mein, ack ekaatmak prachaalak honah chaahiye. yeh quantum kshetra siddhant mein sambhavana key sanrakshan kaa ack kathan haye. lekin
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taba poornata seey,
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tou "s" inn-states seey out states mein ekaatmak parivartan haye.
lorentz invariance "s"-matricks parr ack aur mahatvapurna aavashyakta haye. aiss-operator praarambhik "inn" avasthaon key antim "out" avasthaon key quantum canonical parivartan kaa pratinidhitva kartaa haye. isake alawa, vacuum avastha koo aparivartaniya chhod detaa haye aur "-space feeled mein" "-space feeled mein" badal jaataa haye,
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nirmaan aur vinash prachaalak key sandarbh mein, yeh bunn jaataa haye
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isaliye
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ack samaan abhivyakti taba hoti haye jaba ack baahari raajya parr baain or sanchaalit hoti haye. iska matlab haye kii "s"-matricks koo iss prakaar vyakt kiya jaa sakata haye
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yadi ack antahkriya kaa sahee varnan kiya jaataa haye, tou yeye guna bhee sahee honey chaahieh
vikas prachaalak "yuu".
ack samay-nirbhar nirmaan aur vinash prachaalak koo nimnanusar paribhaashit karein,
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tou, kheton key liye,
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kahaan
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humm ack charan antar kii anumati dete hain, joe dwara diyaa gayaa haye
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kyonki,
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spasht abhivyakti koo pratisthaapit karate hue, ack haye
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jahan sootra _ 121 hamiltonian kaa antahkriya bhaag haye aur sootra _ 122 samay crum haye.
nireekshan seey, yeh dekhaa jaa sakata haye kii yeh sootra spasht roop seey sahparivartan naheen haye.
dyson shrinkhala.
aiss"-matricks key liye sabse vyaapak roop seey upayog kii jaane vaali abhivyakti disson shrinkhala haye. yeh "s"-matricks prachaalak koo shrinkhala key roop mein vyakt kartaa haih
sootra _ 123
kahanh
nott-"aiss"-matricks.
choonki black holl seey hocking vikiran mein kanon key parivartan koo "aiss"-matricks key saath varnit naheen kiya jaa sakata thaa, isaliye stephen hocking ney ack "nott-" aiss "-matricks kaa prastaav rakhaa, jisake liye unhone dollar chihn ($) kaa upayog kiya, aur isaliye jise" dollar matricks "bhee kahaa jaataa thaa.