Classical Field Theory


 Erik Richardson
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1 Classcal Feld Theory Before we embark on quantzng an nteractng theory, we wll take a dverson nto classcal feld theory and classcal perturbaton theory and see how far we can get. The reader s expected to be famlar wth the Hamltonan and Lagrangan formalsms of classcal mechancs for a fnte collecton of degrees of freedom q, and for the Hamltonan formalsm, ther conjugate momenta p. In the Lagrangan formalsm we defne a functonal L({ q }, {q }) of the coordnates and ther tme dervatves. From ths functonal we defne an acton S that defnes the equatons of moton va a mnmzaton over paths {q (t)} wth a mnmzaton condton: S = dtl wth 0 = ds = dtâ dq q d (82) dt q Snce the varatons dq are arbtrary (up to boundary condtons on the trajectory), the mnmzaton condton provdes a set of dfferental equatons whose solutons are the classcal equatons of moton: d dt q q = 0. (83) From the Lagrangan formulaton, one can derve a Hamltonan va a Legendre transform wth respect to the veloctes q. 23,24 The Hamltonan s defned n terms of generalzed coordnates and ther conjugate momenta: H = Â p q (p, q ) L[ q (p, q ), q ] (84) where the dependence of q on the p and q are determned by nvertng the relatonshp q = p, whch arses from maxmzng H over q. 25 The conjugate momenta are related to q and ther dervatves by: Hamlton s equatons of moton are: dp dt p = q (85) = H q and dq dt = H p (86) 23 One can also derve the Hamltonan formulaton from frst prncples, usng the mathematcal language of dfferental forms and symplectc geometry. I hghly recommend the text by Arnold, Mathematcal Methods of Classcal Mechancs" f you are nterested n a formal treatment of the topc. 24 Recall the defnton of the Legendre transform of a convex functon f (x): f (x)! f (x )=sup x [x x f (x)]. 25 For the Legendre transform to be well defned, L must be a convex functon of the q.
2 36 quantum feld theory notes (hubsz  sprng 2016) Generalzaton to the contnuum These approaches to classcal mechancs can be generalzed to systems wth an nfnte number of degrees of freedom  as s needed for classcal feld theory  wthout great dffculty. 26 We consder a sngle classcal feld f(x). 27 In ths case we can derve the equatons of moton from a Lagrangan densty, L( µ f, f), that gves an acton: S = dt d 3 xl( µ f, f) (87) 26 Examples nclude densty of a flud r(x), dsplacements of a sold x(x), or classcal electromagnetc waves. 27 It s not hard to generalze to a collecton of felds. whose varaton s zero when evaluated on the classcal soluton: 0 = ds classcal = d 4 xdl( µ f, f) = d 4 xdf µ. f ( µ f) (88) The second term s obtaned after ntegraton by parts, assumng that df boundary = 0. Snce ths must apply ndependent of the form of df, we have the EulerLagrange equatons of moton: µ ( µ f) f = 0. (89) Wth defne the Hamltonan densty as the Legendre transform of the Lagrangan densty wth respect to ḟ: 28 H(p, f) =pḟ[p, f] L(ḟ[p, f], f) (90) wth the enforced condton = p used to elmnate ḟ from H and ḟ relate f and ts dervatves to p. 29 and wth classcal Hamlton equatons gven by ṗ(x) = H f and ḟ(x) = H p (91) 28 Ths requres that the Lagrangan densty be a convex functon of ḟ, whch wll n turn be necessary to have a spectrum of states wth postve norm n the quantum theory. 29 Takng the Legendre transform wth respect to p then gves back the Lagrangan densty, as the Legendre transform s a nvolutve transformaton: f (x) = f (x). Applcaton to scalar feld theory Let us consder a Lagrangan 30 gven by L = 1 2 gµn µ f n f V(f) = 1 2 ḟ2 1 2 ( ~rf) 2 V(f). (92) We refer to V as the potental for f, and we refer to the full g µn µ f n f term as the knetc term. 31 The potental can nclude a mass term for f: V m2 f 2, or t can contan terms that are hgher polynomals n f, e.g. V 3 4! l f4. Terms that are nonquadratc n the felds are typcally referred to as nteracton terms. 32 In ths case, gong to the Hamltonan formalsm, we have ḟ[p] =p, and H = 1 2 p ( ~rf) 2 + V(f). (93) 30 We wll start neglectng the densty,", and just refer to L as the Lagrangan. 31 We could n prncple have a Lagrangan wth more dervatves on f, however these requre more boundary condtons to solve the equatons of moton. In quantum mechancal systems, terms wth hgher dervatves can lead states wth negatve norm that eventually have negatve consequences for the stablty of the theory. 32 Ths could nclude a mass term for f: m 2 f 2, or also terms that are beyond quadratc n f, e.g. lf 4.
3 classcal feld theory 37 Note that n the Hamltonan formalsm, Lorentz nvarance s not as manfest n the descrpton. We pcked out a specal drecton n space tme when we pcked out ḟ as the varable to be replaced by p n the Legendre transform. The EulerLagrange equatons for the scalar feld are: µ (g µn n f)+ V f = 0 (94) In the case that V(f) = 1 2 m2 f 2, we obtan the KlenGordan scalar feld equaton: f + m 2 f = 0 (95) Noether s Theorem It s often the case that a Lagrangan has some symmetres assocated wth t. Perhaps these mght represent symmetres under rotatons, or for the case of elementary partcle physcs, nvarance under Lorentz transformatons. A theory mght have approxmate global symmetres (.e. rotatons whch mx collectons of felds amongst each other). In classcal feld theory, there s a correspondence between contnuous symmetres of a theory and conservaton laws. Noether s theorem establshes ths correspondence between symmetry transformatons and conserved charges." Let us consder some nfntesmal change of a feld f under some transformaton: f 0 = f + adf (96) Ths transformaton s a symmetry f t leaves the equatons of moton nvarant, whch s ensured f the acton s nvarant. Ths s the case so long as under transformatons of the form (96) leave the Lagrangan nvarant up to a total 4dvergence: L! L + a µ J µ (x). (97) Pluggng (96) nto a Lagrangan that s a functon of f and ts dervatves, we have: adl = f adf + µ f µ(adf) = µ f µ f adf + a µ df (98) µ f Snce the frst term n brackets vanshes when evaluated on solutons to the EulerLagrange equatons, we can equate 33 µ µ f df = µj µ (99) 33 Note that t can be the case that the Lagrangan s nvarant on ts own, and J µ s vanshng, n whch case the RHS of ths equaton s zero.
4 38 quantum feld theory notes (hubsz  sprng 2016) Usng the equalty above, we see that there s a conserved current j µ assocated wth the transformaton adf: µ j µ = 0 for j µ = µ f df J µ (100) For a collecton of felds f n all transformng as f n! f n + adf n, the current s the sum over contrbutons from each of the df n : j µ = Â df n J µ. (101) n µ f n When there are symmetres that depend on several transformaton parameters a, there s a separate conserved current assocated wth each transformaton. 34 We can reexpress the conservaton law n terms of a total charge whose tme dervatve s vanshng. Takng Q = d 3 xj 0, (102) 34 Examples nclude 3D rotatons, 4D Lorentz nvarance, global symmetres SO(N) wth N > 2, etc, etc... We fnd that t Q = d 3 x 0 j 0 = d 3 x~r ~ j = 0 (103) where the last equalty s vanshng as t s a total 3dvergence. Example: Complex scalar feld Let us begn by consderng a Lagrangan that s a functon of a complex scalar feld f. The Lagrangan we study s L = µ f 2 m 2 f 2 (104) Ths Lagrangan s nvarant under a phase rotaton f! e a f (and f! e a f ), where a s a constant phase. Ths s a symmetry of the Lagrangan (and thus the acton). For an nfntesmal transformaton (small a), we have f! f af, so we dentfy df = f, and df = f. 35 Snce dl = 0 under ths transformaton, we have J µ = 0, and thus j µ = apple apple µ f df + Pluggng n our expressons for df, we have µ f df (106) j µ = f µ f + µ f f (107) Checkng the dvergence of ths current, we fnd: 35 We could also consder two real scalar felds f 1 and f 2 wth the Lagrangan L = 1 2 ( µf ) m2 f 2 (105) where the ndex s summed over. The physcs of ths classcal system are dentcal, and the assocated symmetry transformaton s under SO(2) rotatons f1 of the vector ~f =. f 2 µ j µ = f f + f f = 0 (108) where the equatons of moton for f, ( + m 2 )f ( ) = 0, (109) have been enforced n the last equalty.
5 classcal feld theory 39 Noether s Theorem: For a classcal feld theory wth an acton that s nvarant under some contnuous transformaton, there s an assocated current (and thus total charge) that s conserved when the equatons of moton are satsfed. Noether s Theorem apples when: The symmetres are contnuous, and the transformaton parameters a can be taken nfntesmally small The equatons of moton are satsfed Note that t apples when the parameters are local transformatons as well  when the a are functons of x. 36 Example: Translaton nvarance and T µn 36 Ths s of great mportance when we go on to dscuss gauge theores, where vector felds can couple only to conserved currents. If you try to quantze the theory otherwse, there are ssues wth untarty. An mportant example of contnuous symmetres are those assocated wth the regon n whch our felds lve." Translaton nvarance n space and tme are part of the symmetres of full Poncaré nvarance. Let us consder the change n a Lagrangan due to a translaton n spacetme coordnates x µ! x µ x µ. 37 Under such a translaton, the 37 If x s a functon of coordnates, then shft n a scalar feld f s f(x)! f(x + x) f(x)+x µ µ f(x), where the last term s smply the Taylor expanson. Note that there are 4 transformaton parameters, one for each Mnkowsk coordnate. We can wrte the transformaton x µ = a e n, where = 0,,3and the en span a bass for Mnkowsk space. Thus d f = e µ µf. Also note that the Lagrangan s not nvarant, snce L s also a scalar feld: dl = x µ µ L, so we have a nonzero µ J µ = µ (e µ L). The remanng part of the current s apple ( apple ) j µ = Â e r n µ f rf n e µ L = er g rn n Â n f n g µn L n µ f n (110) The term n brackets on the rght we dentfy as the stress energy tensor: 38 T µn = Â n f n g µn L (111) n µ f n By Noether s theorem, the stress energy tensor s conserved: µ T µn = 0. Note that ths corresponds to 4 Noether currents  one each for the four possble translatons. The four conserved charges" are Q µ = d 3 xt 0µ (112) and they correspond to total energy (Q 0 ) and 3momentum (Q ). ths s a dffeomorphsm transformaton. Here we take x to be a global transformaton, or a constant shft. 38 Ths defnton of the stressenergy tensor s not necessarly symmetrc, but can be made so through addng terms as s done n the defnton of the BelnfanteRosenfeld stress energy tensor (whch s the one that couples to gravty).
6 40 quantum feld theory notes (hubsz  sprng 2016) We now have an explanaton for the rules of conservaton of energy and momentum. If physcs s the same at any place n Mnkowsk space, then energy and momentum are conserved. Currents and Sources Let us make a couple comments about the dfferent defntons of currents" that we typcally employ. Frst, currents can be functons of felds assocated wth a contnuous symmetry va Noether s Theorem Second, currents can be confguratons of (possbly movng) external charges. By external, we mean that they are fxed functons of tme and space, and that they act then as source terms n the equatons of moton. I.e., n Maxwell s equatons we have µ F µn = J n, where J n s a conserved current. If J n s nonzero, t s a source term for electromagnetc dynamcs. Fnally, t s often convenent n nteractng theores to employ a shorthand for the set of nteractons of a vector feld. For example, under electromagnetsm there are contrbutons to the Noether currents that couple to A µ for every sngle partcle type that carres electrc charge, and t s convenent to wrte all these nteractons as L 3 A µ J µ. Coulomb s Law Before embarkng on the quantum mechancs of feld theory, let us try a calculaton n classcal feld theory that wll serve as a warmup to ts quantzed counterpart. We can derve Coulomb s Law usng technques n classcal feld theory (many of whch we contnue to employ n quantum feld theory later). We start wth a charge e at the orgn, whch corresponds to an external source/current: j 0 = r(x) =ed (3) (~x) ~ j(x) =0 (113) The Lagrangan assocated wth Maxwell s equatons s L = 1 4 F µnf µn A µ j µ, (114) F Notng that µn r A s = d µr d ns d nr d µs, and F2 µn r A s = 2F µn F µn r A s, we have for the Euler Lagrange equaton: r F rs = A s s ( r A r )=j s (115) Makng the gauge choce r A r = 0 (Lorenz gauge), we have A µ = j µ or A µ = 1 j µ (116)
7 classcal feld theory 41 To obtan the second equaton, we have nverted the operator. Whle t s not mmedately clear what ths means (unless you are wellversed n Fourer analyss), we mght ntut that somethng very nonlocal s nvolved, as the nverse of dervatve operators s an ntegral operaton. We say that A µ s determned by the propagaton of the feld from the localzed sources, and we call 1 the propagator for A µ. To understand better what ths means, let us solve the problem n momentum space. Fourer transformng the LHS of the orgnal equaton A µ = j µ, we have or d 4 xe kx A µ = k 2 d 4 xe kx A µ = k 2 d 4 xe kx A µ =(2p)ed(k 0 ) (117) Ã µ (k) =(2p)e d(k 0) k 2 (118) Now we perform the nverse Fourer transform to fnd A µ (x) =e = e 4p 2 = e 8p 2 1 r d 3 k e ~k ~x (2p) 3 ~ k 2 d k d cos qe ~ k r cos q = e 8p 2 1 r lm d!0 dk ekr " e kr k dk ekr k + d e kr # (119) The last ntegral can be performed usng contours that close ether above or below the real axs wth a contour at. For postve r, we must close the frst exponental ntegral above to get a vanshng contrbuton from the contour at, whch msses the pole, and thus gves no contrbuton. For the other term, we close below, whch pcks up the pole at k = d, gvng a factor ( 2p)( e dr ) where the factor of 2p comes from the resdue theorem assocated wth the negatvely orented contour that encrcles the pole. The result of the ntegral s then (after takng the d! 0 lmt): A 0 (x) = e 1 4p r (120) whch s, of course, the value of the electrc potental assocated wth a postve pont charge at the orgn, and ts gradent gves the expresson for the electrc feld of a pont charge.
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