Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part, we wll use the same notaton as Mandl and Shaw. The components of a contravarant four-vector x are denoted by x µ for µ = 0, 1,, 3 where x 0 = ct s the tme component and x j for j = 1,, 3 are the three spatal components. Unless otherwse specfed, any tme a Greek letter ndex s used, t wll range over 0, 1,, 3 and any tme a Latn letter ndex s used, t wll range over 1,, 3. We wll also use the bold face x to denote the three spatal coordnates of the four-vector x. The covarant metrc tensor we use s gven by 1 0 0 0 g = 0 1 0 0 0 0 1 0. (1.1) 0 0 0 1 The notaton g µν refers to the entry n the µth row and νth column. A covarant vector s defned from the contravarant vector by the usual ndex-lowerng x µ = g µν x ν (1.) where the usual Ensten summaton conventons are used. The contravarant metrc tensor s defned by the equaton { g λµ g µν = δν λ 1 ν = λ = (1.3) 0 ν λ so t follows that g µν = g µν for every µ, ν,.e. the contravarant and covarant metrc tensors are the same. The metrc tensor s used to defne the generalzed scalar product of two four-vectors. For two four-vectors a and b, ther scalar product s defned as ab := a µ b µ = a µ g µν b ν = g µν a µ b ν = a 0 b 0 (a 1 b 1 + a b + a 3 b 3 ). (1.4) A Lorentz transformaton s a matrx Λ that preserves the scalar product xx for any four-vector x. Ths means that g µν (Λx) µ (Λx) ν = g µν Λ µ αx α Λ ν βx β = g µν x µ x ν. (1.5) We also nsst that each entry of the Lorentz transformaton s real. Lorentz transformatons also preserve the scalar product ab for any four-vectors a, b. We now defne some operators that we wll need later. µ := x µ (1.6) µ := x µ (1.7) := µ µ = 1 c t. (1.8) We also adopt the notaton that f an ndex s preceded by a comma, t means we are consderng the dervatve wth respect to that ndex. So for example, F,µ means F x µ. 1
Classcal pcture We start wth N felds φ r for 1 r N. We consder each feld to be a scalar feld on four-dmensonal spacetme. We assume that the system can be descrbed by a Lagrangan densty L (φ r, φ r,α ), (.1).e. the Lagrangan densty s only a functon of the felds and ther frst dervatves wth respect to tme and space. For an arbtrary regon Ω n spacetme, the acton ntegral s defned by S(Ω) := d 4 xl (φ r, φ r,α ). (.) Ω By mposng the usual prncple of least acton and nsstng that δs(ω) = 0, we can derve the usual Euler-Lagrange equatons L ( ) L φ r x α = 0 (.3) φ r,α for 1 r N and 0 α 3. We want to ntroduce the noton of a conjugate feld (analogous to the conjugate momentum to a generalzed coordnate n classc Lagrangan mechancs), but the problem s that system we are workng wth has uncountably many degrees of freedom. To get around ths, we approxmate t by a system wth only countably many degrees of freedom n the followng way. For a fxed tme t, decompose the three-dmensonal space nto small cells ndexed by, each of volume δx. If x denotes the center pont of the th cell, we approxmate each feld φ r by lettng φ r take the value φ r (x ) everywhere n the th cell. What ths accomplshes s that now we have a countable set of generalzed coordnates q r (t) := φ r (t, x ) =: φ r (t, ) (.4) that descrbe the system. We can also approxmate the spatal dervatves of φ r (t, ) n terms of the values of φ r n the adjacent cells. Thus the Lagrangan densty for the th cell takes the form L (φ r (t, ), φ r (t, ), φ r (t, )) (.5) where the dot represents the tme dervatve and denotes the ndex of any cell adjacent to the th. Then the total Lagrangan for the system s gven by L(t) = δx L (φ r (t, ), φ r (t, ), φ r (t, )) (.6) Now that we have dscrete varables descrbng the system, t s possble to defne the conjugate momenta n the usual way. We defne p r := L = L q r φ r (t, ) = δx π r (t, ) (.7) where π r (t, ) s defned to be π r (t, ) := We now can defne a Hamltonan densty and the usual Hamltonan L φ r (t, ). (.8) H := π r (t, ) φ r (t, ) L (.9) H = p r q r L = ( δx π r (t, ) φ ) r (t, ) L = δx H. (.10)
At ths pont, we want to brng our approxmaton closer to the actual system by takng a lmt as δx 0. Ths gves us the followng defntons and relatons. π r (x) := L φ (feld conjugate to φ r ) (.11) r L = d 3 xl (φ r (x), φ r,α (x)) (Lagrangan) (.1) H (x) := π r (x) φ r (x) L (φ r (x), φ r,α (x)) (Hamltonan densty) (.13) H = d 3 xh (x) (Hamltonan). (.14) Note that n analogy to classcal mechancs, f the Lagrangan densty does not depend explctly on tme, then the Hamltonan s constant n tme. 3 Quantum pcture Recall the dscrete approxmatons to the system that we started wth. We now want to quantze the model by nterpretng the generalzed coordnates and conjugate momenta as operators and mposng commutaton relatons on them. The commutaton relatons are chosen n analogy to the usual quantum-mechancal commutaton relatons. Now agan we take a lmt as δx 0 and we get 4 Example [φ r (t, ), π s (t, j)] := δ rsδ j δx (3.1) [φ r (t, ), φ s (t, j)] := [π r (t, ), π s (t, j)] := 0. (3.) [φ r (t, x), π s (t, x )] := δ rs δ(x x ) (3.3) [φ r (t, x), φ s (t, x )] := [π r (t, x), π s (t, x )] := 0. (3.4) Ths secton wll be dedcated to workng out an example of the above theory for a specfc system. Consder a system wth one real-valued feld φ and Lagrangan densty L = 1 ( φ,α φ α, µ φ ) (4.1) where µ s a constant. It turns out that ths Lagrangan densty corresponds to a spnless neutral boson wth mass µ/c. Usng the equaton of moton (.3), we have L φ = ( ) L x α (4.) µ φ = ( ) 1 x α φ, α (4.3) = 1 α α φ (4.4) = 1 φ (4.5) so the equaton of moton s ( ) 1 + µ φ = 0. (4.6) 3
Ths s the Klen-Gordon equaton. The feld conjugate to φ defned by (.11) s The Hamltonan densty s The commutaton relatons become 5 Conservaton laws π(x) = L φ = 1 c φ(x). (4.7) H = π(x) φ(x) L (4.8) = 1 c φ(x) 1 ( φ,α φ α, µ φ ) (4.9) = 1 ( c π(x) + ( φ) + µ φ ). (4.10) [φ(t, x), φ(t, x )] = δ(x x ) (4.11) [φ(t, x), φ(t, x )] = 0 (4.1) [π(t, x), π(t, x )] = 1 c 4 [ φ(t, x), φ(t, x )] = 0. (4.13) It s a general prncple of physcs that any mathematcal symmetres n the Lagrangan of the system correspond to some conserved quantty n the physcal system. For example, n classcal mechancs, a translatonnvarant Lagrangan corresponds to the conservaton of energy and a rotaton-nvarant Lagrangan corresponds to the conservaton of angular momentum. We can also apply ths dea to the quantum case. Consder a transformaton of a feld φ of the form Ths wll cause the Lagrangan densty to change lke δl φ(x) φ(x) + δφ(x). (5.1) = L L δφ + δφ,α φ = α If the orgnal transformaton s a symmetry, then we wll have δl = 0, so where f α s defned as ( ) L δφ. (5.) α f α = 0 (5.3) f α := L δφ. (5.4) Now we want to nvestgate whch quantty wll be conserved as a result of ths symmetry. Defne F α (t) := d 3 xf α (t, x). (5.5) From equaton (5.3), we have It follows that the quantty s conserved. 1 df 0 (t) c dt = d 3 x j f j (t, x) = 0. (5.6) F 0 = d 3 xf 0 (t, x) (5.7) R 3 = d 3 x L δφ (5.8) φ = d 3 xπ(t, x)δφ (5.9) 4
6 Example In ths secton we wll consder an example of the theory developed n the prevous secton. If φ s a complexvalued feld, then we treat φ and φ as ndependent felds, where φ denotes the complex conjugate of φ. We wll suppose that the Lagrangan densty L s nvarant under nfntesmal rotatons of the form so that n the notaton of the above secton, we have φ exp(ɛ)φ (1 + ɛ)φ (6.1) φ exp( ɛ)φ (1 ɛ)φ (6.) δφ = ɛφ (6.3) δφ = ɛφ. (6.4) Now the conserved quantty from equaton (5.9) becomes F 0 = ɛc d 3 x ( π(x)φ(x) π(x)φ(x) ). (6.5) We can scale by any constant factor we want, so we rename Q := q d 3 x ( π(x)φ(x) π(x)φ(x) ) (6.6) where q s an undetermned constant at the moment. We want to see how the operator Q acts on our orgnal feld φ, so we compute the commutator [Q, φ(x)] = q d 3 x [π(x ), φ(x)]φ(x) (6.7) = qφ(x). (6.8) Ths result ndcates that when the operator Q acts on the feld φ, t scales t by a factor of q. Smlarly, f t were to act of φ, t would scale t by a factor of q. Because of ths, the operators φ and φ can be nterpreted as creaton and absorpton operators for electrc charge. References [1] F. Mandl and G. Shaw. Quantum Feld Theory. Wley, 1984. 5