 keys as well (the parameters to the range method.)

 Q:

quantizing open strings on Dp-branes

I am wondering what happens if we quantize open strings in the presence of Dp-branes. I'm sure lots of people did all the calculation before, but my question is about the physical meaning behind those commutation relations found after the quantization. So what's the importance behind those boundary conditions (the presence of Dp-branes) rather than just preventing the strings to go away?

A:

The question is phrased slightly wrong, at least historically: It's not that we put the $D_p$-brane there and then quantize the string. It's that when we look at possible boundary conditions for an open string, we can choose either Neumann or Dirichlet boundary conditions for the endpoints, and we "invent" the fiction of the brane to represent those boundary conditions. The "physical meaning" of this brane is not in the commutation relations themselves, but in the spectrum that appears after quantization.
Each direction for which we choose a Dirichlet boundary condition will mean the string endpoints are fixed in that direction, while a Neumann boundary condition allows the string to move freely in that direction while imposing the absence of a momentum flow onto or off the string. A string with $n$ Dirichel boundary conditions will look as if it moves freely along a ${d-n}$-dimensional hypersurface inside the $d$ dimensional spacetime. The conventional naming is to write $D_p$-brane for a $p+1$-dimensional such hypersurface. Note that branes which don't arise from boundary conditions in this way should not be called $D_p$-branes, since the $D$ stands for "Dirichlet".
In the standard quantization of the (super)string, the brane itself doesn't really play much of a role - what we are quantizing is the string with certain boundary conditions. If one now examines the first level massless excitations of the bosonic string, then one finds they transform in a way under the reduced Lorentz $\mathrm{SO}(1,p)$ symmetry of the $D_p$-brane's directions that organizes them into a massless vector and $d-p$ massless scalars. This is usually interpreted as a vector ($\mathrm{U}(1)$ gauge) field living "on" the brane and a bunch of Goldstone bosons due to the breaking of the full Lorentz symmetry.
However, it gets really interesting once we
keys as well (the parameters to the range method acting .) 

  Q: 

 quantize open strings on Dp - branes 

 I am wonder what happen if we quantize open strings in the bearing of Dp - branes. I'm sure lots of citizenry did all the calculation before, but my motion is about the physical meaning behind those commutation relation establish after the quantization. So what's the importance behind those boundary conditions (the presence of Dp - branes) preferably than just preventing the strings to blend away? 

 A: 

 The question is give voice slightly wrong, at least historically: It's not that we put the $ D_p$-brane there and then quantize the string. It's that when we look at potential boundary conditions for an open string, we can choose either Neumann or Dirichlet boundary conditions for the endpoints, and we " invent " the fiction of the brane to present those boundary conditions. The " physical meaning " of this brane is not in the commutation relations themselves, but in the spectrum that appears after quantization. 
 Each direction for which we choose a Dirichlet boundary condition will mean the string endpoints are sterilize in that direction, while a Neumann boundary condition give up the chain to move freely in that direction while imposing the absence of a momentum menstruation onto or off the string. A string with $ n$ Dirichel boundary conditions will look as if it move freely along a $ { d - n}$-dimensional hypersurface inside the $ d$ dimensional spacetime. The conventional naming is to write $ D_p$-brane for a $ p+1$-dimensional such hypersurface. Note that branes which don't arise from boundary conditions in this means should not be called $ D_p$-branes, since the $ D$ stands for " Dirichlet ". 
 In the standard quantization of the (super)string, the brane itself doesn't really play much of a function - what we are quantizing is the bowed stringed instrument with certain boundary conditions. If one now examines the first floor massless excitations of the bosonic string, then one finds they transform in a way under the reduced Lorentz $ \mathrm{SO}(1,p)$ symmetry of the $ D_p$-brane's directions that organize them into a massless vector and $ d - p$ massless scalars. This is usually understand as a vector ($ \mathrm{U}(1)$ gauge) field populate " on " the brane and a bunch of Goldstone boson due to the breaking of the full Lorentz symmetry. 
 However, it gets very interesting once we
 kejs as well (the parameterr to the range method.)

 X:

quantjzing opdn strings on Dp-branes

I am wlneerint what happens if we qjantize oien strints ii the presence oh Dp-brancf. I'm dure oots of people did all tve calculation bdflre, but my question is about the phrsical kewning behind trose sommhnauion relations found after the quzntizatpon. So what's the omportance behind those bokndagy conditions (the oresence of Dp-bwqnes) rather ghan just krzventing thg strings to go away?

A:

The questiov is 'hrased slithrly frong, at leest hiftorically: Ib'x not dhat we put the $D_p$-brake thxre qnd then quantize the string. It's that wren we lomk at possible bouneaey cotdithons dor an o'en strinh, wx can chooss either Neymann or Dirichlet notbdary conditikns fow ehe endpoints, and we "invent" the fiction of the brane to represent those boundary condijions. The "[hysical meaning" of this brane is not in the commgtatikv rtlqtionr tjemselves, but in the spectrum that appears afesr qlantization.
Each dlrection for which wf vroose a Diricflet bounsary condition wilp mean jhe steing endpjintx are fixed in that directiin, while a Nvumabn boundary conditnon allows tke strong tp move freely in that dnrectikn while imoosing ths absence of a mooennum xlow onto or off the strind. A strinj witk $n$ Dirizhel boundwry conditlons will look as if it loves fseely alonh a ${d-n}$-dimensional hypersurface miside the $d$ domansponal spaeetime. The conventijnal naming is to wrice $D_p$-bfane for a $p+1$-dimenvional such hypersurface. Jote that brenes whicr dob't aeise fruo boundary concitions iu thiw way should not bc calusd $D_p$-branes, sinet tye $D$ stands for "Dificrlvt".
Ii the vtandard quattizxtium of ghe (suptx)stvine, thr brane itself doesn'd rezlly play much of s vole - whaj we are zuantizing is the string with ctrtain bonndary cogditions. If one now examines tge first pevcl massless evcitqtions of thz bosonic string, then one finds they traisform in a way under jhe reduced Lorentz $\mctmrm{SO}(1,p)$ symmevry of the $D_p$-brdne's directions that organizes them ikto a massless vector znd $d-p$ masspess scalars. This is usually interpreted as a vector ($\mathrm{U}(1)$ gauge) field oiving "on" the brans anc a blnck oy Goldseone uosons due to the breaking of the full Lorentz sbmmetry.
Howavzr, it gets really interesting ovce we
keys as well (the parameters to the Q: open strings Dp-branes I am quantize strings in the of Dp-branes. I'm lots of people did all the before, but my question is about the physical meaning behind those commutation relations after the quantization. So what's the importance behind those boundary conditions (the presence Dp-branes) than preventing strings to go away? A: The question is phrased slightly wrong, at least historically: It's not we put the $D_p$-brane there and then quantize string. It's that when look at possible boundary conditions an string, we choose Neumann Dirichlet boundary conditions the endpoints, and we "invent" the fiction of the brane to represent those boundary conditions. The "physical of this not in commutation themselves, in the spectrum after quantization. Each direction for which Dirichlet boundary condition will mean the string endpoints fixed in direction, while a Neumann boundary condition the string to move freely in that direction imposing the absence of a momentum flow onto or off the string. A string with boundary conditions will look if it moves along ${d-n}$-dimensional inside $d$ dimensional The conventional naming is to write $D_p$-brane for a $p+1$-dimensional such Note that branes which don't arise from boundary conditions in should be called $D_p$-branes, the $D$ stands for In standard quantization of the brane doesn't of role what we are quantizing the string with certain boundary If one now examines of the bosonic string, then one finds they in a way under the reduced Lorentz symmetry of the $D_p$-brane's directions that organizes them into a massless vector $d-p$ massless is usually interpreted as a vector ($\mathrm{U}(1)$ gauge) living "on" the brane a bunch of Goldstone bosons due to the breaking the Lorentz symmetry. it gets really once we
 keys as well (the parameters to The range meThod.)

 Q:

QuaNtiZiNg opEn stRings on Dp-braneS

i am wOndering what happens if wE quanTiZE opeN StRings In the prESeNCE of dp-BrAneS. I'M SuRe lotS of People dId all the caLcuLaTion before, buT My Question is AboUt the physicaL meAning bEhInd THose cOmmUtatiOn relaTIons foUnd after tHe QUantizATion. So wHAT's The iMportance behind thOSe BOundary conditiOns (the PrESeNCE of dp-bRanes) ratheR tHan juST prevenTInG THE stRIngs to go away?

A:

the question IS phRased sLiGhtLY wrong, At leaSt HIstOrically: It's Not tHat we put tHe $D_p$-brANe there ANd then qUantizE thE stRing. iT's ThAt wHeN We lOOk At pOSsiBle boundArY cOnditIons FOR AN opeN stRing, We can Choose either NEumAnn oR dirIchleT bounDary CoNditiOns for The enDpOints, and we "invenT" the Fiction of The BrAne To RepreSEnt thoSe bOunDary conDitions. tHe "pHySICAl Meaning" of this brane Is NOT iN the commUtatioN ReLaTIons themSeLveS, but IN The spEctrUM tHat appeaRs afteR QuAnTizatioN.
EAch dirEcTioN foR whicH We chOose a DIrichlet BoundARy condition wilL Mean the string ENdPOInTS are FixEd in that dirEctiON, whiLe a NEUmAnn BOundaRy conDiTIoN Allows the string to moVe Freely In thaT direction whiLe imposing THE Absence oF a moMEnTUm flow onto or ofF the sTring. A striNG with $n$ DiRicheL boundarY conditioNS Will look As iF it MovEs fREElY along a ${d-n}$-dimeNSIonaL hYpersurFacE inside The $D$ diMenSioNaL spacetimE. The convEnTiOnAl NamIng is TO write $D_p$-BrAne FoR a $p+1$-DimenSIonal sUch hyPersUrFaCE. NoTe that bRAnES WhicH dOn'T ariSe fRoM bounDary COndItions iN this way sHouLD not Be CaLled $D_p$-bRanes, since the $d$ sTands for "DiRiChlEt".
In thE STandard qUantization of the (super)stRIng, the bRanE itseLf doEsn't reallY plAy much Of a ROle - whaT we are QuantIzIng IS The stRINg WitH cErtain bounDARy cOnditIoNs. If One now eXamines the first levEL maSsless excitatIonS of tHE BoSonIC sTRinG, tHEn oNE Finds they transfOrm in a way uNdER tHe reduced LORenTz $\Mathrm{So}(1,p)$ symmeTry of THe $D_p$-braNe's directIons that oRgAnizES TheM into a massLess vectOr and $d-p$ maSSless SCaLars. THis Is usuaLlY inTerprEted as A VecTor ($\maThrm{U}(1)$ gAuGe) fielD liviNg "On" the braNe and a bunch of Goldstone bOsons dUe to tHe bReaking of The FUll lorentz syMmetRy.
However, iT geTs rEally IntERestiNg onCE wE
 keys as well (the paramet ers to the  rang e m eth od .)
 Q:
quantizing op e n st rings on Dp-branes

Iam wo nd e ring wh at ha ppens i f  w e qua nt iz e o pe n  s tring s i n the p resence of  Dp -b ranes. I'm s u re  lots of p eop le did all t hecalcul at ion befor e,but m y ques t ion is  about th ep hysica l  meanin g be hind  those commutatio n  r e lations foundafterth e  q u a nti zat ion. So wh at 's th e  import a nc e b ehi n d those bound ary conditi o ns(the p re sen c e of D p-bra ne s ) r ather thanjust  preventi ng the strings to go a way?
A:
Th e qu e st io n i sp hra s ed  sl i ght ly wrong ,at  leas t hi s t o r ical ly:  It' s not  that we putthe  $D_ p $-b ranethere  and  t hen q uantiz e the  s tring. It's tha t wh en we loo k a tpos si ble b o undary  co ndi tions f or an o p enst r i n g,  we can choose eit he r Ne umann or  Diric h le tb oundaryco ndi tion s for t he e n dp oints, a nd we" in ve nt" the  f iction  o f t hebrane to r eprese nt those  boun d ary conditions .  The "physica l  m e a ni n g" o f t his brane i s no t  inthec om mut a tionrelat io n st hemselves, but in t he  spect rum t hat appears a fter quant i z a tion.
Ea ch d i re c tion for which  we c hoose a Di r ichlet b ounda ry condi tion will m ean thestr ing  en dpo i n ts  are fixed in t hatdi rection , w hile aNeu man n b oun da ry condit ion allo ws  t he  s tri ng to move fre el y i ntha t dir e ctionwhile  imp os in g  th e absen c eo f  a m om en tumflo wontoor o f f t he stri ng. A str ing with  $ n$  Dirich el boundary c on ditions wi ll  lo ok asi f  it move s freely along a ${d-n} $ -dimens ion al hy pers urface in sid e the$d$ dimens ionalspace ti me. T he co n v en tio na l naming i s towrite  $ D_p$ -branefor a $p+1$-dimens i ona l such hypers urf ace. N ot e t h at bra ne s  wh i c h don't arise f rom bounda ry co nditions i n  th is  way sh ould no t bec alled $ D_p$-bran es, since  t he $ D $  st ands for " Dirichle t".
In th e  stan d ar d qua nti zation  o f t he (s uper)s t rin g, th e bran eitself  does n' t really  play much of a role -what w e are  qu antizingist hestring wi th c ertain bou nda rycondi tio n s. If  one no w e x amine s th e  first le v el  ma s s le ss excitati o n s  of  thebos o nic st ring , then one findst hey transformin a w ayund e r th ereduced Lorent z $ \m a t hrm{SO}( 1, p)$ symmetr y of the  $ D _p$-b rane's  direc tions t h a to rganiz es t hem  into a m ass le s s vecto ran d  $d-p$  mas sl ess sc alars. This i s usually interp reted a s a v e cto r ($\ ma thrm{U} ( 1)$gauge) fie ld living " on" th e br ane a nd a bu nc h of G old st one bosons due to th e bre aking o ftheful l Lore ntzs y mmetr y.
H ow eve r, it get s re a ll yi nte rest ing o nc e we 
 keys_as well_(the parameters to the_range method.)

_Q:

quantizing_open strings_on_Dp-branes

I am wondering_what happens if_we quantize open strings_in the presence_of_Dp-branes. I'm sure lots of people did all the calculation before, but my question_is_about the_physical_meaning_behind those commutation relations found_after the quantization. So what's_the importance_behind those boundary conditions (the presence of Dp-branes)_rather_than just preventing_the strings to go away?

A:

The question is phrased slightly_wrong, at least historically: It's not_that we put_the_$D_p$-brane_there and then quantize_the string. It's that when we_look at possible boundary conditions for_an open string, we can choose either_Neumann or Dirichlet boundary conditions for_the endpoints, and we "invent"_the fiction_of the brane to represent_those boundary conditions._The "physical_meaning" of this_brane is not in the commutation_relations themselves, but_in the spectrum that appears after_quantization.
Each_direction for which_we_choose_a Dirichlet_boundary condition will_mean_the string_endpoints_are fixed in that direction, while_a_Neumann boundary condition allows the string to_move freely in that_direction_while imposing the absence_of a momentum flow onto_or off the string. A string_with $n$_Dirichel boundary_conditions will look as if it moves freely along a ${d-n}$-dimensional_hypersurface inside the $d$ dimensional spacetime._The conventional naming is_to write_$D_p$-brane_for a $p+1$-dimensional_such_hypersurface. Note_that branes which don't arise from boundary_conditions in_this way should not be called_$D_p$-branes, since the $D$_stands_for "Dirichlet".
In the standard quantization of_the (super)string, the brane itself doesn't_really play much of a_role_-_what we are quantizing is_the string with certain boundary conditions._If one now_examines the first level massless excitations of_the_bosonic string, then one finds they_transform_in a way under the reduced_Lorentz_$\mathrm{SO}(1,p)$_symmetry of the $D_p$-brane's directions_that organizes them into a massless_vector and $d-p$ massless scalars. This is usually interpreted_as a vector_($\mathrm{U}(1)$ gauge) field living "on"_the_brane_and a bunch of Goldstone bosons due to the breaking_of the_full Lorentz symmetry.
However,_it gets really interesting once we