Scalars: Spin 0 Fields

Transcription

1 40 Chapter 3. Scalars: Spn 0 Felds Chapter 3 vers 5/17/18 copyrght of Robert D. Klauber Scalars: Spn 0 Felds..f I look back at my lfe as a scentst and a teacher, I thnk the most mportant and beautful moments were when I say, ah-hah, now I see a lttle better ths s the joy of nsght whch pays for all the trouble one has had n ths career. ctor F. Wesskopf Quarks, Quasars, and Quandares 3.0 Prelmnares Ths chapter presents the most fundamental concepts n the theory of quantum felds, and contans the very essence of the theory. Master ths chapter, and you are well on your way to masterng that theory Background Early efforts to ncorporate specal relatvty nto quantum mechancs started wth the nonrelatvstc Schrödnger equaton, p ħ ħ H where H, (3-1) t m m and attempted to fnd a relatvstc, rather than non-relatvstc, form for the Hamltonan H. 1 One mght guess that approach would lead to a vald relatvstc Schrödnger equaton. Ths s, n essence, true but there s one problem, as we wll see below. In specal relatvty, the 4-momentum vector s Lorentz covarant, meanng ts length n 4D space s nvarant. For a free partcle (.e., = 0), E/c 1 p 1 3 E m c p p m c g p p E/c p p p p p. (3-) c 3 p Changng dynamcal varables over to operators (as happens n quantzaton),.e., one fnds, from the RHS of (3-), E H and p ħ, (3-3) Seekng a relatvstc quantum theory? Try relatvstc Hamltonan n Schrödnger equaton Relatvstc energy E Relatvstc E relatvstc operator H 1 Actually, Schrödnger frst attempted to fnd a wave equaton that was relatvstc and came up wth what later came to be known as the Klen-Gordon equaton, whch we wll study n ths chapter. He dscarded t because of problems dscussed later on heren, and because t gave wrong answers for the hydrogen atom. Shortly thereafter, he deduced the non-relatvstc Schrödnger equaton we are famlar wth. Some tme afterwards, other researchers then tred to relatvze that equaton, as dscussed heren.

2 Secton 3.1 Relatvstc Quantum Mechancs: A Hstory Lesson 41 4 H ħ c m c, (3-4) seemngly the only form a relatvstc Hamltonan could take. Unfortunately, takng the square root of terms contanng a dervatve s problematc, and dffcult to correlate wth the physcal world. The soluton to the problem of fndng a relatvstc Schrödnger equaton has been found, however, and as we wll see n the next three chapters, turns out to be dfferent for dfferent spn types. Ths was qute unexpected at frst, but has snce become a cornerstone of relatvstc quantum theory. (See frst row of Wholeness$Chart 1- n Chap. 1, pg. 7.) Partcles wth zero spn, such as -mesons (pons) and the famous Hggs boson, are known as scalars, and are governed by one partcular relatvstc Schrödnger equaton, deduced by (after Schrödnger, actually), and named after, Oscar Klen and Walter Gordon. Partcles wth ½ spn, such as electrons, neutrnos, and quarks, and known as spnors, by a dfferent relatvstc Schrödnger equaton, dscovered by Paul Drac. And partcles wth spn 1, such as photons and the W s and Z s that carry the weak charge, and known as vectors, by yet another relatvstc Schrödnger equaton, dscovered by Alexandru Proça. The Proça equaton reduces, n the massless (photon) case, to Maxwell s equatons. We wll devote a separate chapter to each of these three spn types and the wave equaton assocated wth each. We begn n ths chapter wth scalars Chapter Overvew RQM frst, where we wll look at deducng the Klen-Gordon equaton, the frst relatvstc Schrödnger equaton, usng the relatvstc H, solutons (whch are states = wave functons) to the Klen-Gordon equaton, probablty densty and ts connecton to the funny normalzaton constant n the solutons, and the problem wth negatve energes n the relatvstc solutons. Then QFT, usng the classcal relatvstc L (Lagrangan densty) for scalar felds, and the Legendre transformaton to get H (Hamltonan densty), from L and the Euler-Lagrange equaton, fndng the same Klen-Gordon equaton, wth the same mathematcal form for the solutons, but ths tme the solutons are felds, not states, from nd quantzaton, fndng the commutaton relatons for QFT, determnng relevant operators n QFT: H = H d 3 x, number, creaton/destructon, etc., showng ths approach avods negatve energy states, seeng how the vacuum s flled wth quanta of energy ½ħ, dervng other operators (probablty densty, 3-momentum, charge) and pckng up relevant loose ends (scalars = bosons, Fock (multpartcle) space). And then, seeng quantum felds n a dfferent lght, as harmonc oscllators. Wth fnally, and mportantly, fndng the Feynman propagator, the mathematcal expresson for vrtual partcles. Free (no force) Felds In ths chapter, as well as Chaps. 4 (spn ½) and 5 (spn 1), we wll deal only wth felds/partcles that are not nteractng,.e., feel no force = free. Thus, we wll take potental energy = 0. In Chap.$7, whch begns Part of the book, we wll begn to nvestgate nteractons. Bad news: Relatvstc H has square root of a dfferental operator But answer has been found, as we wll see Each spn type has ts own relatvstc wave equaton RQM overvew (scalars) QFT overvew (scalars) We study free (no nteractons) case frst 3.1 Relatvstc Quantum Mechancs: A Hstory Lesson Two Possble Routes to RQM Recall from Chaps. 1 and, that 1 st quantzaton, for both non-relatvstc and relatvstc partcle theores, entals ) usng the classcal form of the Hamltonan as the quantum form of the

3 4 Chapter 3. Scalars: Spn 0 Felds Hamltonan, and ) changng Posson brackets to commutators. We recall also from Prob. 6 of Chap. 1 that non-commutaton of dynamcal varables means those varables are operators (because ordnary numbers commute.) For example, j j equvalent p,x ħ p ħ (3-5) as the RHS above s the only form that satsfes the LHS, and t s an operator. One mght expect that ths s the route we would follow to obtan RQM,.e., 1 st quantzaton of relatvstc classcal partcle theory. However, hstorcally, t was done dfferently. That s, RQM was frst extrapolated from NRQM, not from classcal theory. As llustrated n Fg. 3-1, t can be done ether way. In ths book, to save space and tme, we wll only show one of these paths, the hstorcal one represented by the lowest arrow n Fg Non-commutatng varables must be operators Classcal Non-Relatvstc Partcle Theory Change H to relatvstc No commutators Classcal Relatvstc Partcle Theory 1st Quantzaton: H stays the same + nvoke commutators 1st Quantzaton: H stays the same + nvoke commutators Route we could take, but wll not NRQM Change H to relatvstc Same commutators RQM Route we wll take n ths chapter Fgure 3-1. Dfferent Routes to Relatvstc Quantum Mechancs 3.1. Deducng the Klen-Gordon Equaton As we saw n Sect , when we try to use a relatvstc Hamltonan n the Schrödnger equaton, we have the problem of the partal dervatve operator (see (3-4)) beng under a square root sgn. So, rather than use H, Klen and Gordon, n 197, dd the next best thng. They used H nstead. That s, they squared the operators (operate on each sde twce rather than once) n the orgnal Schrödnger equaton (3-1) and thus from (3-), obtaned ħ H 4 operc m c t ħ t p, (3-6) whch becomes from the square of (3-4) ħ m c m c ħ 0. (3-7) c t X X x x0 x x ħ Re-arrangng, we have the Klen-Gordon equaton (expressed n two equvalent ways wth slghtly dfferent notaton) m c 0 or 0, m n nat. unts. (3-8) x x ħ As noted n Chap., Prob.$4, the operaton s called the d Alembertan operator, and s the 4D Mnkowsk coordnates analogue of the 3D Laplacan operator Cartesan coordnates. of Let s square operators on both sdes of Schrödnger eq Then use operator form for H To get the Klen-Gordon equaton

4 Secton 3.1 Relatvstc Quantum Mechancs: A Hstory Lesson 43 In 1934, Paul and Wesskopf 1 showed that the Klen-Gordon equaton specfcally descrbes a spn-0 (scalar) partcle. Ths should become evdent to us as we study the Drac and Proça equatons, for spn ½ and spn 1, later on, and compare them to the Klen-Gordon equaton The Solutons to the Klen-Gordon Equaton A soluton set to (3-8), readly checked by substtuton nto (3-8) (whch s good practce when usng contravarant/covarant notaton), s (where E n n m p ) 1 Ent pnx Ent pnx x Ae ħ n Bneħ, (3-9) n1 E n / ħ absent n NRQM where we wll dscuss the funny lookng normalzaton factor n front, contanng the volume and the energy of the nth soluton, later. The coeffcents A n and B n are constants, and a complex conjugate form for the coeffcent of the last term above,.e., B, s used because t wll prove advantageous later. Ths s a dscrete set of solutons, typcal for cases wth waves constraned nsde a volume, though can be taken as large as one wshes. Each dscrete wavelength n the summaton of (3-9) fts an nteger number of tmes nsde the volume. Contnuous (ntegral rather than sum) solutons, for waves not constraned nsde a specfc volume, exst for (3-8) as well, but we are not concerned wth them at ths pont. Ths soluton set s also specfcally for plane waves. We wll not consder alternatve soluton forms for other wave shapes that would exst n problems wth cylndrcal or sphercal geometres. The soluton (3-9), because we are workng n RQM, s a state,.e., (x) above = (x), for a sngle partcle. Each ndvdual term n the summaton s an egenstate. (x) s a general state superposton of egenstates. Note that n NRQM, we only had terms n the counterpart to (3-9) that had the exponental form of (Ent pn x)/ħ, because that was the only form that satsfed the non-relatvstc Schrödnger equaton. Because we are usng the square of the relatvstc Hamltonan n RQM, we get addtonal solutons of exponental form +(Ent pn x)/ħ that also solve the relatvstc Klen-Gordon equaton. You should do Prob. 1, at the end of the chapter, to justfy the statements n ths paragraph to yourself. Wth an am towards usng natural unts, we note the followng relatons, where wave number k = / and we use the debrogle relaton p. = ħk., E/c E/c ħ/c nat. unts E p k p k p ħ, (3-10) p ħk p k and recall the notaton ntroduced n Chap., px p x Et px Et p x p x Et px p kx kx t k x x k x ħ ħ ħ n nat. unts E, p k, p k, px kx. n (3-11) It s then common to re-wrte (3-9) n natural unts wth the above notaton. In dong so, we also swtch the dummy summaton varable n, whch represents each ndvdual wave n the summaton, to the 3D vector quantty k, representng the wave number and drecton of each possble wave. For free felds, a gven wave wth wave number vector k has a partcular energy (see (3-) wth p = k n natural unts), and we can desgnate that energy va ether Ek or k. It s common practce for scalars to use k (rather than p) and (rather than Ep or Ek.) Klen-Gordon equaton s specfcally for scalars Solutons to Klen-Gordon equaton (dscrete) Contnuous solutons also exst Only plane wave solutons here Solutons n RQM are states (partcles) Relatvstc form has extra set of solutons Relatons for p and k Notaton revew 1 Paul, W. and Wesskopf,., Helv. Phys. Acta 7, 709 (1934). Translaton n Mller, A. I., Early Quantum Electrodynamcs: A Source Book, Cambrdge U. Press, New York (1994)

5 44 Chapter 3. Scalars: Spn 0 Felds The Klen-Gordon equaton solutons (3-9) then become, n natural unts 1 kx kx x Ake Bke. (3-1) k k Except for Box 3-1, whch revews NRQM, we wll henceforth, n ths chapter, use natural unts. Defnton of Egensolutons As noted prevously, n RQM, the soluton of (3-1) s that of a general (sum of egenstates) sngle partcle state. Each egenstate has mathematcal form (where we are gong to omt the k part here, because of what s comng) kx e or k,a k,b kx e. (3-13) Each of these forms has what s called unt norm. That s, for k,a (and smlarly, for k ),,B 3 1 kx kx 3 k,a k,a d x e e d x 1, (3-14) or more generally, all such egenstates are orthonormal,.e., ther nner products are 1 k, k d x e e d x kk. (3-15) 3 kx kx 3 A,A Smlar relatons to (3-15) exst for k, and every,b k,a s orthogonal to every k. Work ths,b out by dong Prob.. Relatons (3-13) to (3-15) should look famlar from NRQM. There, (3-14) was the ntegral of the probablty densty for a partcle n an egenstate. In RQM, however, thngs are a lttle dfferent, as we wll see, and we use the term unt norm for the property dsplayed n (3-14). Unt norm egenstates were advantageous n NRQM, and they wll be n QFT as well. That s the reason we omtted the k part of our solutons (3-1) n formng our defntons (3-13). By so dong, the egenstates then have unt norm, and thngs just turn out easer later on Probablty Densty n RQM We are gong to nvestgate probablty densty n RQM, but frst look over Box 3-1, and be sure you understand how probablty densty s derved n NRQM. Probablty Densty Usng the Klen-Gordon Equaton For RQM, we start wth the Klen-Gordon equaton rather than Schrödnger equaton. Frst postmultply t by, then subtract the complex conjugate equaton post-multpled by,.e., t (3-16), t and note that = 0. The LHS of the result can be replaced wth the new LHS n (3-17) below, and the RHS wth (3-18). t t t t t t t t t LHS of result above 0 new LHS RHS of result above 0 new RHS (3-17) (3-18) Natural unts form of Klen- Gordon solutons Egenstates of Klen-Gordon equaton Egenstates have unt norm and are orthogonal We defned egenstates to have unt norm because t wll be advantageous Deduce RQM probablty densty usng relatvstc wave equaton

6 Secton 3.1 Relatvstc Quantum Mechancs: A Hstory Lesson 45 Box 3-1. Revew of Non-Relatvstc QM Probablty Densty In non-relatvstc quantum mechancs (NRQM), we encountered 1) the wave functon soluton to the Schrödnger equaton, and ) the partcle probablty densty (or equvalently when s a scalar quantty, *) We revew here the dervaton of that relaton for probablty densty. Conserved quanttes n feld theory: Recall the contnuty equaton of contnuum mechancs and electromagnetsm, t j 0 d x constant n tme, (B3-1.1) mples 3 where s densty (mass or charge densty), j s the 3D current densty (mass/area-sec or charge/area-sec), and s all space, or at least large enough so that everywhere outsde t, for all tme, = 0. s fxed n space and tme, whereas can change n space and tme nsde. Any conserved quantty (such as total mass M or total charge Q) obeys (B3-1.1). The general procedure: Use the governng quantum wave equaton to deduce another equaton havng the form of the contnuty equaton (B3-1.1), and we wll then know that, whatever t turns out to be n that case, must represent a conserved quantty. Its ntegral over all space s constant n tme. If we normalze such that when ntegrated over all space, the result equals one, we can conjecture that s the partcle probablty densty (whch when ntegrated over all space equals the probablty that we wll fnd the partcle somewhere n all space,.e., one.) Then throughout tme, as our partcle evolves, moves, and rearranges ts probablty densty dstrbuton, the total probablty of fndng t somewhere n space s always one. It turns out, from experment, that the conjecture that ths quantty n NRQM equals probablty densty s true. Probablty Densty Usng the Schrödnger Equaton: Frst, pre-multply the Schrödnger equaton by the complex conjugate of the wave functon,.e., 1 ħ t ħ M Then, post-multply the complex conjugate of the Schrödnger equaton by the wave functon 1 ħ t ħ M where the potental s real so =. Addng (B3-1.) to (B3-1.3), we get or 1 ħ 1 ħ t t ħ M M ħ ħ t M ħ ħ 0 snce Ths s the same as the contnuty equaton (B3-1.1) f we take as our probablty densty, and as our probablty current densty (sometmes just probablty current) ħ j M Ths s how the commonly used relaton (B3-1.6) s found. (B3-1.) (B3-1.3) (B3-1.4) (B3-1.5) (B3-1.6). (B3-1.7)

7 46 Chapter 3. Scalars: Spn 0 Felds Equatng the new LHS of (3-17) to the new RHS of (3-18), and to make future work easer, multplyng both sdes by the constant, gves the form of the contnuty equaton j 0, (3-19) t t t t where probablty densty and the probablty current for a Klen-Gordon partcle are 0 j, and (3-0) t t,, j j,,. (3-1) Importantly, and perhaps surprsngly, the relatvstc form of the probablty densty (3-0) s not the same as (B3-1.6), the NRQM probablty densty. 4 Currents We ntroduce 4D notaton for the scalar and 3D vector of (3-19) and defne the scalar 4-current 0 j,, j j j j The 4D contnuty equaton form of (3-19) s then j x, j j 0. (3-), (3-3) where we have shown three common notatonal ways to desgnate partal dervatve. (3-3) tells us the mportant fact that the 4-dvergence of the 4-current of any conserved quantty (total probablty n ths case) s zero. Probablty for Klen-Gordon Dscrete Solutons For a sngle partcle state n RQM, we are gong to assume at frst, for smplcty, that the soluton (3-1), has only terms wth coeffcents Ak,.e., the general state contans no egenstates shown wth coeffcents Bk. Probablty densty (3-0) s then (where prmes do not denote dervatves wth respect to spatal coordnates, merely dfferent summaton dummy varables) kx k x k x kx A e A e A e A e k k k k k k, (3-4) k k k k k k k k where the k and k came from the tme dervatves. If we ntegrate over the volume (whch s large enough to encompass the entre state), the result must equal 1. When we do so, all terms wth k k go to zero, so the k k and cancel out. The term n the denomnator cancels n the ntegraton over the volume, and the two terms result n a factor of that cancels wth the n the denomnator. The result s 3 k k d x A 1. (3-5) Thus A k s the probablty of measurng the kth egenstate, smlar to what the coeffcents of egenstates represented n NRQM. Dfference from NRQM Note that n RQM 3 Ak 3 d x 1 but d x A 1 (RQM) t t k, (3-6) k k k whereas n NRQM, we had Manpulatons of the wave equaton lead to an equaton lke the contnuty equaton From that, we deduce form of RQM probablty densty 4-current and 4D form of contnuty equaton 4-dvergence of 4-current of conserved quantty always = 0 Scalar probablty densty n terms of frst Klen-Gordon soluton set Square of absolute value of coeffcent Ak = probablty of fndng kth egenstate Comparng probablty n NRQM and RQM

8 Secton 3. The Klen-Gordon Equaton n Quantum Feld Theory 47 Normalzaton Factors d 3 x A k 1 (NRQM). (3-7) k Obtanng the RHS of (3-6) s the reason for the normalzaton factors 1/ k used n the soluton of (3-1) and (3-9). Those factors result n a total probablty of one for a sngle partcle and A k as the probablty for measurng the respectve egenstate. That s, the form of the relatvstc feld equaton gave us the form of the probablty densty n (3-0) (and (3-6)), and the need to have total probablty of unty gave us the normalzaton factors n the solutons. Relatvstc Invarance of Probablty Ths total probablty value of unty n (3-5) (and (3-6)) s a relatvstc nvarant (.e., a world scalar.) If we change our frame, the energy spectrum (.e., the k values) wll change (knetc energy for each energy-momentum egenstate looks dfferent). But these changes cancel out n the probablty calculaton, snce the k cancel, and always result n a total probablty of one for any frame. Further, the A k here are constants that do not vary wth frame, so the probablty of fndng any partcular state s also ndependent of what frame the measurements are taken n. Note that ths means the normalzaton factors chosen provde relatvstc nvarance of total probablty, whch we would not have had wth any other choce Negatve Energes n RQM If we take our tradtonal operator form for H as / t and operate on one of our Klen-Gordon soluton egenstates of (3-1) and (3-13), we should get the energy egenvalue k. When we do ths for the egenstates wth exponents n kx, all looks as expected. kx kx k,a e e,a,a,a k k,a,a,a Hk Ek k k Ek k. (3-8) t t However, when we do t for the egenstates wth exponents n +kx, we have an uh-oh,.e., H E t t E kx kx k,b e e k,b k,b k,b k k,b k,b k,b k. (3-9) Snce k s always a postve number, we have states wth negatve energes n RQM. We mght have expected ths, snce we used the square of the Hamltonan as the bass of RQM, and square roots typcally have both postve and negatve sgns. The bottom lne: Ths s not an attrbute of what a good theory has been expected to have,.e., solely postve energes as we see n our world. As we wll shortly see, QFT solved ths dlemma (as well as others delneated n Chap. 1.) Negatve Probabltes n RQM Do Prob. 3 to prove to yourself that a partcle contanng only egenstates of the exponental form +(Ent pn x)/ħ = kx (.e., those wth coeffcents Bk n (3-1)) has total probablty of beng measured of 1. The extra states n RQM have physcally untenable negatve probabltes! Tme to move on to QFT. 3. The Klen-Gordon Equaton n Quantum Feld Theory 3..1 States vs Felds It should come as no surprse, to those who have read Chap. 1, that the fundamental scalar wave equaton of RQM, the Klen-Gordon equaton (3-8), s also the fundamental scalar wave equaton of QFT, except that theren s consdered a feld, nstead of a state. The word feld n classcal theory means an entty that, unlke a partcle, s spread out,.e., s a functon of space (t has dfferent values at dfferent spatal locatons) and typcally also a functon of tme. The state of NRQM and RQM certanly flls that bll, but n quantum theory we don t use the word feld for ths, we use the word state (or wave functon or ket or partcle.) RQM normalzaton factors arse from need to have total probablty = 1 and A k = probablty of kth state Total probablty and A k are frame ndependent (relatvstcally nvarant) Half of our RQM egenstates have negatve energy Half of our RQM egenstates have negatve probablty densty States & felds both spread out n space. But n quantum theores, feld also means operator

9 48 Chapter 3. Scalars: Spn 0 Felds The word feld n quantum theory refers to a quantty that s spread out n space, but also, mportantly, as we wll soon see, s an operator n QFT. More properly, t s called a quantum feld or an operator feld, though the short term feld s far more common. Confusngly, we use the same symbol n QFT for a feld as we used for a state n NRQM and RQM. Notaton In QFT, symbols such as, whch are not part of a ket symbol, do not represent states, but felds. Unless otherwse explctly noted, n QFT notaton, symbolzes a state (partcle) and symbolzes a feld (operator), On the other hand, n NRQM and RQM, both symbols above represented the same thng, a state. We wll understand these dstnctons a lttle better later, but for now understand that formally, the Klen-Gordon equaton n QFT s called a feld equaton, because ts soluton s a (quantum or operator) feld. See the second and thrd rows of Wholeness$Chart 1- n Chap. 1, pg. 7. There are two common ways to derve ths equaton, whch we present n the followng two sectons, plus a thrd, whch s a good check on the theory and can be found n the Appendx A. 3.. From RQM to QFT Fg. 3- llustrates, schematcally, the two basc routes to QFT. The quckest s at the bottom of the fgure, for whch we smply postulate that the soluton of the Klen-Gordon equaton (3-8) descrbes a feld (nstead of a partcle). Ths s reasonable, snce s a functon of spatal locaton (and often tme),.e., t s a feld n the formal mathematcal sense. Notatonal dfference between states and felds. In QFT, s not a state, but a feld Classcal Relatvstc Partcle Theory Partcles to felds Classcal Relatvstc Feld Theory Two dfferent routes to QFT 1st Quantzaton: H stays the same + nvoke commutators nd Quantzaton: H stays the same + nvoke commutators Quantzaton way to QFT RQM States to felds Same wave equaton [p,x j] [ r, s] QFT Short way to QFT Fgure 3-. Dfferent Routes to Quantum Feld Theory We then must apply the commutaton relatons for felds (see Chap., pg. 31, Wholeness$Chart -5, 6 th column = 3 rd column on rght hand page), nstead of the commutaton relatons for partcle propertes (same chart, 3 rd column on left hand page). When we do ths, and smply crank the mathematcs, we obtan QFT. Because the QFT we then obtan descrbes the real world so well, t justfes the orgnal postulate. The formal mathematcs are much the same as for the alternatve route, llustrated on the RHS of Fg. 3-, and treated n the next secton From Classcal Relatvstc Felds to QFT Classcal Scalar Felds The classcal Lagrangan densty for a free (no forces), real, relatvstc scalar feld has form L ɺɺ ɺɺ, (3-30) 0 0 K K K Short route: RQM QFT. Smlar math as nd quantzaton below nd quantzaton route: Classcal felds QFT Start wth classcal Lagrangan densty for free scalar feld