Relativistic Quantum Mechanics

Chapte Relativistic Quantum Mechanics In this Chapte we will addess the issue that the laws of physics must be fomulated in a fom which is Loentz invaiant, i.e., the desciption should not allow one to diffeentiate between fames of efeence which ae moving elative to each othe with a constant unifom velocity v. The tansfomations beween such fames accoding to the Theoy of Special Relativity ae descibed by Loentz tansfomations. In case that v is oiented along the x axis, i.e., v = v ˆx, these tansfomations ae x = x v t, t = t v c v c x v c, x = x ; x 3 = x 3. which connect space time coodinates x, x, x 3, t in one fame with space time coodinates x, x, x 3, t in anothe fame. Hee c denotes the velocity of light. We will intoduce below Loentz-invaiant diffeential equations which take the place of the Schödinge equation of a paticle of mass m and chage q in an electomagnetic field [c.f. efeq:ham, 8.45] descibed by an electical potential V, t and a vecto potential A, t i [ t ψ, t = m i q ] A, c t + qv, t ψ, t. The eplacement of. by Loentz invaiant equations will have two supising and extemely impotant consequences: some of the equations need to be fomulated in a epesentation fo which the wave functions ψ, t ae vectos of dimension lage one, the components epesenting the spin attibute of paticles and also epesenting togethe with a paticle its anti-paticle. We will find that actually seveal Loentz invaiant equations which eplace. will esult, any of these equations being specific fo cetain classes of paticles, e.g., spin paticles, spin paticles, etc. As mentioned, some of the equations descibe a paticle togethe with its anti-paticle. It is not possible to uncouple the equations to descibe only a single type paticle without affecting negatively the Loentz invaiance of the equations. Futhemoe, the equations need to be intepeted as actually descibing many paticle-systems: the equivalence of mass and enegy in elativistic fomulations of physics allows that enegy convets into paticles such that any paticle descibed will have companions which assume at least a vitual existence. Obviously, it will be necessay to begin this Chapte with an investigation of the goup of Loentz tansfomations and thei epesentation in the space of position and time t. The epesentation 87

88 Relativistic Quantum Mechanics in Sect.. will be extended in Sect..4 to cove fields, i.e., wave functions ψ, t and vectos with functions ψ, t as components. This will povide us with a geneal set of Loentz invaiant equations which fo vaious paticles take the place of the Schödinge equation. Befoe intoducing these geneal Loentz invaiant field equations we will povide in Sects..5,.7 a heuistic deivation of the two most widely used and best known Loentz invaiant field equations, namely the Klein Godon Sect..5 and the Diac Sect..7 equation.. Natual Repesentation of the Loentz Goup In this Section we conside the natual epesentation of the Loentz goup L, i.e. the goup of Loentz tansfomations.. Rathe than stating fom., howeve, we will povide a moe basic definition of the tansfomations. We will find that this definition will lead us back to the tansfomation law., but in a setting of epesentation theoy methods as applied in Secti. 5 to the goups SO3 and SU of otation tansfomations of space coodinates and of spin. The elements L L act on 4 dimensional vectos of position and time coodinates. We will denote these vectos as follows x µ def = x, x, x, x 3.3 whee x = ct descibes the time coodinate and x, x, x 3 = descibes the space coodinates. Note that the components of x µ all have the same dimension, namely that of length. We will, hencefoth, assume new units fo time such that the velocity of light c becomes c =. This choice implies dimtime = dimlength. Minkowski Space Histoically, the Loentz tansfomations wee fomulated in a space in which the time component of x µ was chosen as a puely imaginay numbe and the space components eal. This space is called the Minkowski space. The eason fo this choice is that the tansfomations. leave the quantity s = x x x x 3.4 invaiant, i.e., fo the tansfomed space-time codinates x µ = x, x, x, x 3 holds x x x x 3 = x x x x 3..5 One can intepete the quantity s as a distance in a 4 dimensional Euclidean space if one chooses the time component puely imaginay. In such a space Loentz tansfomations coespond to 4-dimensional otations. Rathe than following this avenue we will intoduce Loentz tansfomations within a setting which does not equie eal and imaginay coodinates. The Goup of Loentz Tansfomations L = O3, The Loentz tansfomations L descibe the elationship between space-time coodinates x µ of two efeence fames which move elative to each othe with unifom fixed velocity v and which might be eoiented elative to each othe by a otation aound a common oigin. Denoting by x µ the

.: Natual Repesentation of the Loentz Goup 89 coodinates in one efeence fame and by x µ the coodinates in the othe efeence fame, the Loentz tansfomations constitute a linea tansfomation which we denote by 3 x µ = L µ νx ν..6 ν= Hee L µ ν ae the elements of a 4 4 matix epesenting the Loentz tansfomation. The uppe index close to L denotes the fist index of the matix and the lowe index ν futhe away fom L denotes the second index. [ A moe conventional notation would be L µν, howeve, the latte notation will be used fo diffeent quantities futhe below.] The following possibilities exist fo the positioning of the indices µ, ν =,,, 3: 4-vecto: x µ, x µ ; 4 4 tenso: A µ ν, A µ ν, A µν, A µν..7 The eason fo the notation is two-fold. Fist, the notation in.6 allows us to intoduce the so-called summation conventon: any time the same index appeas in an uppe and a lowe position, summation ove that index is assumed without explicitly noting it, i.e., y µ x µ } {{ } new = 3 y µ x µ µ= } {{ } old ; A µ νx ν } {{ } new = 3 A µ νx ν ν= } {{ } old ; A µ νb ν ρ } {{ } new = 3 A µ νb ν ρ ν= } {{ } old..8 The summation convention allows us to wite.6 x µ = L µ νx ν. The second eason is that uppe and lowe positions allow us to accomodate the expession.5 into scala poducts. This will be explained futhe below. The Loentz tansfomations ae non-singula 4 4 matices with eal coefficients, i.e., L GL4, R, the latte set constituting a goup. The Loentz tansfomations fom the subgoup of all matices which leave the expession.5 invaiant. This condition can be witten x µ g µν x ν = x µ gµν x ν.9 whee g µν = Combining condition.9 and.6 yields Since this holds fo any x µ it must be tue = g.. L µ ρg µν L ν σ x ρ x σ = g ρσ x ρ x σ.. L µ ρg µν L ν σ = g ρσ.. This condition specifies the key popety of Loentz tansfomations. We will exploit this popety below to detemine the geneal fom of the Loentz tansfomations. The subset of GL4, R, the

9 Relativistic Quantum Mechanics elements of which satisfy this condition, is called O3,. This set is identical with the set of all Loentz tansfomations L. We want to show now L = O3, GL4, R is a goup. To simplify the following poof of the key goup popeties we like to adopt the conventional matix notation fo L µ ν L = L µ ν = L L L L 3 L L L L 3 L L L L 3 3 L 3 L 3 L 3 L 3 3 Using the definition. of g one can ewite the invaiance popety. Fom this one can obtain using gl T gl = and, hence, the invese of L L = g L T g = The coesponding expession fo L T is obviously To demonstate the goup popety of O3,, i.e., of..3 L T gl = g..4 g =.5 L L L L 3 L L L L 3 L L L L 3 L 3 L 3 L 3 L 3 3..6 L T = L T = g L g..7 O3, = { L, L GL4, R, L T gl = g },.8 we note fist that the identity matix is an element of O3, since it satisfies.4. We conside then L, L O3,. Fo L 3 = L L holds L T 3 g L 3 = L T L T g L L = L T L T gl L = L T g L = g,.9 i.e., L 3 O3,. One can also show that if L O3,, the associated invese obeys.4, i.e., L O3,. In fact, employing expessions.6,.7 one obtains L T g L = glgggl T g = glgl T g.. Multiplying.4 fom the ight by gl T and using.5 one can deive L T glgl T multiplying this fom the left by by gl T yields = L T and L g L T = g. Using this esult to simplify the.h.s. of. esults in the desied popety L T g L = g,. i.e., popety.4 holds fo the invese of L. This stipulates that O3, is, in fact, a goup.

.: Natual Repesentation of the Loentz Goup 9 Classification of Loentz Tansfomations We like to classify now the elements of L = O3,. Fo this pupose we conside fist the value of det L. A statement on this value can be made on account of popety.4. Using det AB = det A det B and det A T = det A yields det L = o det L = ±..3 One can classify Loentz tansfomations accoding to the value of the deteminant into two distinct classes. A second class popety follows fom.4 which we employ in the fomulation.. Consideing in. the case ρ =, σ = yields L L L L 3 =..4 o since L + L + L 3 it holds L. Fom this we can conclude L o L,.5 i.e., thee exist two othe distinct classes. Popeties.3 and.5 can be stated as follows: The set of all Loentz tansfomations L is given as the union L = L + L + L L.6 whee L +, L +, L, L ae disjunct sets defined as follows L + = { L, L O3,, det L =, L } ;.7 L + = { L, L O3,, det L =, L } ;.8 L = { L, L O3,, det L =, L } ;.9 L = { L, L O3,, det L =, L }..3 It holds g L and L as one can eadily veify testing fo popety.4. One can also veify that one can wite L = gl + = L + g ;.3 L + = L + ;.3 L = gl + = L + g.33 whee we used the definition am = {M, M, M M, M = a M }. The above shows that the set of pope Loentz tansfomations L + allows one to geneate all Loentz tansfomations, except fo the tivial factos g and. It is, hence, entiely suitable to investigate fist only Loentz tansfomations in L +. We stat ou investigation by demonstating that L + foms a goup. Obviously, L + contains. We can also demonstate that fo A, B L + holds C = AB L +. Fo this pupose we conside the value of C = A µb µ = 3 j= A jb j + A B. Schwatz s inequality yields 3 3 A jb j A 3 j B j..34 j= j= j=

9 Relativistic Quantum Mechanics Fom. follows B 3 j= Bj = o 3 j= Bj = B. Similaly, one can conclude fom. 3 j= A j = A..34 povides then the estimate 3 A jb j j= [ A ] [ B ] < A B..35 One can conclude, theefoe, 3 j= A jb j < A B. Since A and B, obviously A B. Using the above expession fo C one can state C >. In fact, since the goup popety of O3, ascetains C T gc = g it must hold C. The next goup popety of L + to be demonstated is the existence of the invese. Fo the invese of any L L + holds.6. This elationship shows L = L, fom which one can conclude L L +. We also note that the identity opeato has elements µ ν = δ µ ν.36 whee we defined δ µ ν = { fo µ = ν fo µ ν.37 It holds, = and, hence, L +. Since the associative popety holds fo matix multiplication we have veified that L + is indeed a subgoup of SO3,. L + is called the subgoup of pope, othochonous Loentz tansfomations. In the following we will conside solely this subgoup of SO3,. Infinitesimal Loentz tansfomations The tansfomations in L + have the popety that they ae continously connected to the identity, i.e., these tansfomations can be paametized such that a continuous vaiation of the paametes connects any element of L + with. This popety will be exploited now in that we conside fist tansfomations in a small neighbohood of which we paametize by infinitesimal paametes. We will then employ the Lie goup popeties to geneate all tansfomations in L +. Accodingly, we conside tansfomations L µ ν = δ µ ν + ɛ µ ν ; ɛ µ ν small..38 Fo these tansfomations, obviously, holds L > and the value of the deteminant is close to unity, i.e., if we enfoce.4 actually L and det L = must hold. Popety.4 implies + ɛ T g + ɛ = g.39 whee we have employed the matix fom ɛ defined as in.3. To ode Oɛ holds ɛ T g + g ɛ =..4 It should be noted that accoding to ou pesent definition holds δ µν = g µρδ ρ ν and, accodingly, δ = and δ = δ = δ 33 =.

.: Natual Repesentation of the Loentz Goup 93 Using.5 one can conclude which eads explicitly This elationship implies ɛ ɛ ɛ ɛ 3 ɛ ɛ ɛ ɛ 3 ɛ ɛ ɛ ɛ 3 ɛ 3 ɛ 3 ɛ 3 ɛ 3 3 ɛ T = g ɛ g.4 = ɛ µ µ = ɛ ɛ ɛ ɛ 3 ɛ ɛ ɛ ɛ 3 ɛ ɛ ɛ ɛ 3 ɛ 3 ɛ 3 ɛ 3 ɛ 3 3 ɛ j = ɛ j, j =,, 3..4 ɛ j k = ɛ k j, j, k =,, 3.43 Inspection shows that the matix ɛ has 6 independent elements and can be witten w w w 3 ɛϑ, ϑ, ϑ 3, w, w, w 3 = w ϑ 3 ϑ w ϑ 3 ϑ..44 w 3 ϑ ϑ This esult allows us now to define six geneatos fo the Loentz tansfomationsk =,, 3 The geneatos ae explicitly J = K = J k = ɛϑ k =, othe five paametes zeo.45 K k = ɛw k =, othe five paametes zeo..46 ; J = ; K = ; J 3 = ; K 3 = These commutatos obey the following commutation elationships.47.48 [ J k, J l ] = ɛ klm J m.49 [ K k, K l ] = ɛ klm J m [ J k, K l ] = ɛ klm K m. The opeatos also obey J K = J J + J J + J 3 J 3 =.5

94 Relativistic Quantum Mechanics as can be eadily veified. Execise 7.: Demonstate the commutation elationships.49,.5. The commutation elationships.49 define the Lie algeba associated with the Lie goup L +. The commutation elationships imply that the algeba of the geneatos J k, K k, k =,, 3 is closed. Following the teatment of the otation goup SO3 one can expess the elements of L + though the exponential opeatos L ϑ, w = exp ϑ J + w K ; ϑ, w R 3.5 whee we have defined ϑ J = 3 k= ϑ kj k and w K = 3 k= w kk k. One can eadily show, following the algeba in Chapte 5, and using the elationship J k =.5 L k whee the 3 3 matices L k ae the geneatos of SO3 defined in Chapte 5, that the tansfomations.5 fo w = coespond to otations of the spatial coodinates, i.e., L ϑ, w = = R ϑ..53 Hee R ϑ ae the 3 3 otation matices constucted in Chapte 5. Fo the paametes ϑ k of the Loentz tansfomations holds obviously ϑ k [, π[, k =,, 3.54 which, howeve, constitutes an ovecomplete paametization of the otations see Chapte 5. We conside now the Loentz tansfomations fo ϑ = which ae efeed to as boosts. A boost in the x diection is L = expw K. To detemine the explicit fom of this tansfomation we evaluate the exponential opeato by Taylo expansion. In analogy to equation 5.35 it issufficient to conside in the pesent case the matix since L Using the idempotence popety = exp w exp w K = exp = = n= w n n! L n.55..56 =.57

.: Natual Repesentation of the Loentz Goup 95 one can cay out the Taylo expansion above: L = n= w n n! + n= w n+ n +! = cosh w + sinh w = cosh w sinh w sinh w cosh w..58 The conventional fom. of the Loentz tansfomations is obtained though the paamete change v = sinh w cosh w = tanh w.59 Using cosh w sinh w = one can identify sinhw = cosh w and coshw = sinh w +. Coespondingly, one obtains fom.59 v = cosh w cosh w = sinh w sinh w +..6 These two equations yield cosh w = / v ; sinh w = v / v,.6 and.56,.59 can be witten exp w K = v v v v v v.6 Accoding to.3,.6,.5 the explicit tansfomation fo space time coodinates is then x = x v t v, t = t v x v, x = x, x 3 = x 3.63 which agees with.. The ange of the paametes w k can now be specified. v k defined in.59 fo the case k = coesponds to the elative velocity of two fames of efeence. We expect that v k can only assume values less than the velocity of light c which in the pesent units is c =. Accodingly, we can state v k ], [. This popety is, in fact, consistent with.59. Fom.59 follows, howeve, fo w k w k ], [..64 We note that the ange of w k -values is not a compact set even though the ange of v k -values is compact. This popety of the w k -values contasts with the popety of the paametes ϑ k specifying otational angles which assume only values in a compact ange.

96 Relativistic Quantum Mechanics. Scalas, 4 Vectos and Tensos In this Section we define quantities accoding to thei behaviou unde Loentz tansfomations. Such quantities appea in the desciption of physical systems and statements about tansfomation popeties ae often extemely helpful and usually povide impotant physical insight. We have encounteed examples in connection with otational tansfomations, namely, scalas like = x + x + x 3, vectos like = x, x, x 3 T, spheical hamonics Y lm ˆ, total angula momentum states of composite systems like Y lm l, l ˆ, ˆ and, finally, tenso opeatos T km. Some of these quantities wee actually defined with espect to epesentations of the otation goup in function spaces, not in the so-called natual epesentation associated with the 3 dimensional Euclidean space E 3. Pesently, we have not yet defined epesentations of Loentz tansfomations beyond the natual epesentation acting in the 4 dimensional space of position and time coodinates. Hence, ou definition of quantities with special popeties unde Loentz tansfomations pesently is confined to the natual epesentation. Nevetheless, we will encounte an impessive example of physical popeties. Scalas The quantities with the simplest tansfomation behaviou ae so-called scalas f R which ae invaiant unde tansfomations, i.e., f = f..65 An example is s defined in.4, anothe example is the est mass m of a paticle. Howeve, not any physical popety f R is a scala. Counteexamples ae the enegy, the chage density, the z component x 3 of a paticle, the squae of the electic field E, t o the scala poduct of two paticle positions. We will see below how tue scalas unde Loentz tansfomations can be constucted. 4-Vectos The quantities with the tansfomation behaviou like that of the position time vecto x µ defined in.3 ae the so-called 4 vectos a µ. These quantites always come as fou components a, a, a, a 3 T and tansfom accoding to Examples of 4-vectos beside x µ ae the momentum 4-vecto p µ = E, p, E = a µ = L µ ν a ν..66 m v, p = m v v.67 the tansfomation behaviou of which we will demonstate futhe below. A thid 4-vecto is the so-called cuent vecto J µ = ρ, J.68 whee ρ, t and J, t ae the chage density and the cuent density, espectively, of a system of chages. Anothe example is the potential 4-vecto A µ = V, A.69 whee V, t and A, t ae the electical and the vecto potential of an electomagnetic field. The 4-vecto chaacte of J µ and of A µ will be demonstated futhe below.

.: Scalas, 4 Vectos and Tensos 97 Scala Poduct then 4-vectos allow one to constuct scala quantities. If a µ and b µ ae 4-vectos is a scala. This popety follows fom.66 togethe with. a µ g µν b ν.7 a µ gµν b ν = L µ ρ g µν L ν σa ρ b σ = a ρ g ρσ b σ.7 Contavaiant and Covaiant 4-Vectos It is convenient to define a second class of 4-vectos. The espective vectos a µ ae associated with the 4-vectos a µ, the elationship being a µ = g µν a ν = a, a, a, a 3.7 whee a ν is a vecto with tansfomation behaviou as stated in.66. One calls 4-vectos a µ covaiant and 4-vectos a µ contavaiant. Covaiant 4-vectos tansfom like whee we defined a µ = g µν L ν ρg ρσ a σ.73 g µν = g µν..74 We like to point out that fom definition.7 of the covaiant 4-vecto follows a µ = g µν a ν. In fact, one can employ the tensos g µν and g µν to aise and lowe indices of L µ ν as well. We do not establish hee the consistency of the ensuing notation. In any case one can expess.73 a µ = L µ σ a σ..75 Note that accoding to.7 L σ µ is the tansfomation invese to L σ µ. In fact, one can expess [L T ] µ ν = L ν µ and, accodingly,.7 can be witten L ν µ = L µ ν..76 The 4-Vecto µ An impotant example of a covaiant 4-vecto is the diffeential opeato µ = x µ = t,.77 The tansfomed diffeential opeato will be denoted by µ def =..78 x µ To pove the 4-vecto popety of µ we will show that g µν ν tansfoms like a contavaiant 4- vecto, i.e., g µν ν = L µ ρg ρσ σ. We stat fom x µ = L µ νx ν. Multiplication and summation of x µ = L µ νx ν by L ρ σg ρµ yields, using., g σν x ν = L ρ σg ρµ x µ and g µσ g σν = δ µ ν, x ν = g νσ L ρ σg ρµ x µ..79 This is the invese Loentz tansfomation consistent with.6. We have duplicated the expession fo the invese of L µ ν to obtain the coect notation in tems of covaiant, i.e., lowe, and

98 Relativistic Quantum Mechanics contavaiant, i.e., uppe, indices..79 allows us to detemine the connection between µ and µ. Using the chain ule of diffeential calculus we obtain µ = 3 ν= x ν x µ x ν = gνσ L ρ σg ρµ ν = L ν µ ν.8 Multiplication by g λµ and summation ove µ togethe with g λµ g ρµ = δ λ ρ yields i.e., µ does indeed tansfom like a covaiant vecto. g λµ µ = L λ σg σν ν,.8 d Alembet Opeato We want to constuct now a scala diffeential opeato. Fo this pupose we define fist the contavaiant diffeential opeato µ = g µν ν = t,..8 Then the opeato µ µ = t.83 is a scala unde Loentz tansfomations. In fact, this opeato is equal to the d Alembet opeato which is known to be Loentz-invaiant. Poof that p µ is a 4-vecto We will demonstate now that the momentum 4-vecto p µ defined in.67 tansfoms like.66. Fo this pupose we conside the scala diffeential It holds fom which follows One can wite dτ = dx µ dx µ = dt d.84 p = E = The emaining components of p µ can be witten, e.g., dτ = v.85 dt d dτ = d v dt..86 m = m dt v v dt..87 p = m v = m dx v v dt..88 One can expess then the momentum vecto p µ = m dx µ v dt = m d dτ xµ..89

.3: Relativistic Electodynamics 99 The opeato m d dτ tansfoms like a scala. Since xµ tansfoms like a contavaiant 4-vecto, the.h.s. of.89 alltogethe tansfoms like a contavaiant 4-vecto, and, hence, p µ on the l.h.s. of.89 must be a 4-vecto. The momentum 4-vecto allows us to constuct a scala quantity, namely Evaluation of the.h.s. yields accoding to.67 o p µ p µ = p µ g µν p ν = E p.9 E p = which, in fact, is a scala. We like to ewite the last esult m v m v v = m.9 p µ p µ = m.9 E = p + m.93 o E = ± p + m..94 In the non-elativistic limit the est enegy m is the dominant contibution to E. Expansion in m should then be apidly convegent. One obtains E = ±m ± p m p p 3 4m 3 + O 4m 5..95 This obviously descibes the enegy of a fee paticle with est enegy ±m, kinetic enegy ± p m and elativistic coections..3 Relativistic Electodynamics In the following we summaize the Loentz-invaiant fomulation of electodynamics and demonstate its connection to the conventional fomulation as povided in Sect. 8. Loentz Gauge In ou pevious desciption of the electodynamic field we had intoduced the scala and vecto potential V, t and A, t, espectively, and had chosen the so-called Coulomb gauge 8., i.e., A =, fo these potentials. This gauge is not Loentz-invaiant and we will adopt hee anothe gauge, namely, t V, t + A, t =..96 The Loentz-invaiance of this gauge, the so-called Loentz gauge, can be demonstated eadily using the 4-vecto notation.69 fo the electodynamic potential and the 4-vecto deivative.77 which allow one to expess.96 in the fom µ A µ =..97 We have poven aleady that µ is a contavaiant 4-vecto. If we can show that A µ defined in.69 is, in fact, a contavaiant 4-vecto then the l.h.s. in.97 and, equivalently, in.96 is a scala and, hence, Loentz-invaiant. We will demonstate now the 4-vecto popety of A µ.

3 Relativistic Quantum Mechanics Tansfomation Popeties of J µ and A µ The chage density ρ, t and cuent density J, t ae known to obey the continuity popety t ρ, t + J, t =.98 which eflects the pinciple of chage consevation. This pinciple should hold in any fame of efeence. Equation.98 can be witten, using.77 and.68, µ J µ x µ =..99 Since this equation must be tue in any fame of efeence the ight hand side must vanish in all fames, i.e., must be a scala. Consequently, also the l.h.s. of.99 must be a scala. Since µ tansfoms like a covaiant 4-vecto, it follows that J µ, in fact, has to tansfom like a contavaiant 4-vecto. We want to deive now the diffeential equations which detemine the 4-potential A µ in the Loentz gauge.97 and, theeby, pove that A µ is, in fact, a 4-vecto. The espective equation fo A = V can be obtained fom Eq. 8.3. Using t A, t = t A, t togethe with.96, i.e., A, t = t V, t, one obtains t V, t V, t = 4πρ, t.. Similaly, one obtains fo A, t fom 8.7 using the identity 8.8 and, accoding to.96, A, t = t V, t t A, t A, t = 4 π J, t.. Combining equations.,., using.83 and.69, yields µ µ A ν x σ = 4 π J ν x σ.. In this equation the.h.s. tansfoms like a 4-vecto. The l.h.s. must tansfom likewise. Since µ µ tansfoms like a scala one can conclude that A ν x σ must tansfom like a 4-vecto. The Field Tenso The electic and magnetic fields can be collected into an anti-symmetic 4 4 tenso E x E y E z F µν = E x B z B y E y B z B x..3 E z B y B x Altenatively, this can be stated F k = F k = E k, F mn = ɛ mnl B l, k, l, m, n =,, 3.4 whee ɛ mnl = ɛ mnl is the totally anti-symmetic thee-dimensional tenso defined in 5.3. One can eadily veify, using 8.6 and 8.9, that F µν can be expessed though the potential A µ in.69 and µ in.8 as follows F µν = µ A ν ν A µ..5

.3: Relativistic Electodynamics 3 The elationships.3,.4 establishe the tansfomation behaviou of E, t and B, t. In a new fame of efeence holds F µν = L µ α L ν β F αβ.6 In case that the Loentz tansfomation L µ ν is given by.6 o, equivalently, by.63, one obtains E x Ey v B z Ez+v B y v v F µν = E x B z v E y v E y v B z v E z +v B y v B y +v E z v B z v E y v B x B y+v E z B v x.7 Compaision with F µν = E x E y E z E x B z B y E y B z B x E z B y B x.8 yields then the expessions fo the tansfomed fields E and B. The esults can be put into the moe geneal fom E = E, E = E + v B v B = B, B = B v E v.9. whee E, B and E, B ae, espectively, the components of the fields paallel and pependicula to the velocity v which detemines the Loentz tansfomation. These equations show that unde Loentz tansfomations electic and magnetic fields convet into one anothe. Maxwell Equations in Loentz-Invaiant Fom One can expess the Maxwell equations in tems of the tenso F µν in Loentz-invaiant fom. Noting µ F µν = µ µ A ν µ ν A µ = µ µ A ν ν µ A µ = µ µ A ν,. whee we used.5 and.97, one can conclude fom. µ F µν = 4π J ν.. One can eadily pove that this equation is equivalent to the two inhomogeneous Maxwell equations 8., 8.. Fom the definition.5 of the tenso F µν one can conclude the popety σ F µν + µ F νσ + ν F σµ =.3 which can be shown to be equivalent to the two homogeneous Maxwell equations 8.3, 8.4.

3 Relativistic Quantum Mechanics Loentz Foce One impotant popety of the electomagnetic field is the Loentz foce acting on chaged paticles moving though the field. We want to expess this foce though the tenso F µν. It holds fo a paticle with 4-momentum p µ as given by.67 and chage q dp µ dτ = q m p ν F µν.4 whee d/dτ is given by.86. We want to demonstate now that this equation is equivalent to the equation of motion 8.5 whee p = m v/ v. To avoid confusion we will employ in the following fo the enegy of the paticle the notation E = m/ v [see.87] and etain the definition E fo the electic field. The µ = component of.4 eads then, using.4, o with.86 de dτ = q m p E.5 de dt Fom this one can conclude, employing.93, = q E p E..6 de dt = d p dt = q p E.7 This equation follows, howeve, also fom the equation of motion 8.5 taking the scala poduct with p p d p = q p dt E.8 whee we exploited the fact that accoding to p = m v/ v holds p v. Fo the spatial components, e.g., fo µ =,.4 eads using.3 dp x dτ = q m EE x + p y B z p z B y..9 Employing again.86 and.67, i.e., E = m/ v, yields dp x dt = q [ E x + v B x ]. which is the x-component of the equation of motion 8.5. We have, hence, demonstated that.4 is, in fact, equivalent to 8.5. The tem on the.h.s. of. is efeed to as the Loentz foce. Equation.4, hence, povides an altenative desciption of the action of the Loentz foce.

.4: Function Space Repesentation of Loentz Goup 33.4 Function Space Repesentation of Loentz Goup In the following it will be equied to decibe the tansfomation of wave functions unde Loentz tansfomations. In this section we will investigate the tansfomation popeties of scala functions ψx µ, ψ C 4. Fo such functions holds in the tansfomed fame ψ L µ νx ν = ψx µ. which states that the function values ψ x µ at each point x µ in the new fame ae identical to the function values ψx µ in the old fame taken at the same space time point x µ, i.e., taken at the pais of points x µ = L µ νx ν, x µ. We need to emphasize that. coves solely the tansfomation behaviou of scala functions. Functions which epesent 4-vectoso othe non-scala entities, e.g., the chage-cuent density in case of Sect..3 o the bi-spino wave function of electon-positon pais in Sect..7, obey a diffeent tansfomation law. We like to expess now ψ x µ in tems of the old coodinates x µ. Fo this pupose one eplaces x µ in. by L µ ν xν and obtains ψ x µ = ψl µ ν xν.. This esult gives ise to the definition of the function space epesentation ρl µ ν of the Loentz goup ρl µ νψx µ def = ψl µ ν xν..3 This definition coesponds closely to the function space epesentation 5.4 of SO3. In analogy to the situation fo SO3 we seek an expession fo ρl µ ν in tems of an exponential opeato and tansfomation paametes ϑ, w, i.e., we seek an expession which coesponds to.5 fo the natual epesentation of the Loentz goup. The esulting expession should be a genealization of the function space epesentation 5.48 of SO3, in as fa as SO3, is a genealization otation + boosts of the goup SO3. We will denote the intended epesentation by L ϑ, w def = ρl µ ν ϑ, ϑ w = ρ e J + w K.4 which we pesent in the fom L ϑ, w = exp ϑ J + w K..5 In this expession J = J, J, J 3 and K = K, K, K 3 ae the geneatos of L ϑ, w which coespond to the geneatos J k and K k in.47, and which can be constucted following the pocedue adopted fo the function space epesentation of SO3. Howeve, in the pesent case we exclude the facto i [cf. 5.48 and.5]. Accodingly, one can evaluate J k as follows J k = lim ϑ k [ ] ρ e ϑ kj k ϑ.6 and K k K k = lim w k w [ ρ e w k K k ]..7

34 Relativistic Quantum Mechanics One obtains J = x 3 x 3 ; K = x + x which we like to demonstate fo J and K. In ode to evaluate.6 fo J we conside fist J = x 3 x 3 ; K = x + x J 3 = x x ; K 3 = x 3 + x 3.8 e ϑ J = e ϑ J = cosϑ sinϑ sinϑ cosϑ.9 which yields fo small ϑ ρ e ϑ J ψx µ = ψx, x, cosϑ x + sinϑ x 3, sinϑ x + cosϑ x 3 = ψx µ + ϑ x 3 x 3 ψx µ + Oϑ..3 This esult, obviously, epoduces the expession fo J in.8. One can detemine similaly K stating fom e w K = e w K = coshw sinhw sinhw coshw..3 This yields fo small w ρ e w K ψx µ = ψcoshw x + sinhw x, sinhw x + coshw x, x, x 3 = ψx µ + w x + x ψx µ + Ow.3 and, obviously, the expession fo K in.6. The geneatos J, K obey the same Lie algeba.49 as the geneatos of the natual epesentation, i.e. [ J k, J l ] = ɛ klm J m [ K k, K l ] = ɛ klm J m [ J k, K l ] = ɛ klm K m..33 We demonstate this fo thee cases, namely [J, J ] = J 3, [K, K ] = J 3, and [J, K ] = K 3 : [ J, J ] = [x 3 x 3, x 3 x 3 ] = [x 3, x 3 ] [x 3, x 3 ] = x + x = J 3,.34

.4: Function Space Repesentation of Loentz Goup 35 [ K, K ] = [x + x, x + x ] = [x, x ] [x, x ] = x + x = J 3,.35 [ J, K ] = [x 3 x 3, x + x ] = [x 3, x ] [x 3, x ] One-Dimensional Function Space Repesentation = x 3 + x 3 = K 3..36 The exponential opeato.5 in the case of a one-dimensional tansfomation of the type Lw 3 = exp w 3 K 3,.37 whee K 3 is given in.8, can be simplified consideably. Fo this pupose one expesses K 3 in tems of hypebolic coodinates R, Ω which ae connected with x, x 3 as follows a elationship which can also be stated x = R coshω, x 3 = R sinhω.38 R = { + x x 3 if x x x 3 if x <.39 and tanhω = x3 x, x cothω = x 3..4 The tansfomation to hypebolic coodinates closely esembles the tansfomation to adial coodinates fo the geneatos of SO3 in the function space epesentation [cf. Eqs. 5.85-5.87]. In both cases the adial coodinate is the quantity conseved unde the tansfomations, i.e., x + x + x 3 in the case of SO3 and x x 3 in case of tansfomation.37. In the following we conside solely the case x. The elationships.39,.4 allow one to expess the deivatives, 3 in tems of R, Ω. We note and The chain ule yields then = R x 3 = R x 3 R x Ω x 3 = Ω x = R + R + = x R, R x 3 = x R Ω tanhω tanhω x 3 = cosh Ω x Ω cothω.4 cothω x = sinh Ω x 3..4 Ω x Ω = x R R sinh Ω x 3 Ω Ω x 3 Ω = x3 R R + cosh Ω x Ω..43

36 Relativistic Quantum Mechanics Inseting these esults into the definition of K 3 in.8 yields K 3 = x 3 + x 3 = Ω..44 The action of the exponential opeato.37 on a function fω C is then that of a shift opeato Lw 3 fω = exp w 3 fω = fω + w 3..45 Ω.5 Klein Godon Equation In the following Sections we will povide a heuistic deivation of the two most widely used quantum mechanical desciptions in the elativistic egime, namely the Klein Godon and the Diac equations. We will povide a deivation of these two equations which stem fom the histoical development of elativistic quantum mechanics. The histoic oute to these two equations, howeve, is not vey insightful, but cetainly is shot and, theefoe, extemely useful. Futhe below we will povide a moe systematic, epesentation theoetic teatment. Fee Paticle Case A quantum mechanical desciption of a elativistic fee paticle esults fom applying the coespondence pinciple, which allows one to eplace classical obsevables by quantum mechanical opeatos acting on wave functions. In the position epesentation the coespondence pinciple states E = Ê = i t p = ˆ p = i.46 which, in 4-vecto notation eads p µ = ˆp µ = i t, = i µ ; p µ = ˆp µ = i t, = i µ..47 Applying the coespondence pinciple to.9 one obtains the wave equation µ µ ψx ν = m ψx ν.48 o µ µ + m ψx ν =..49 whee ψx µ is a scala, complex-valued function. The latte popety implies that upon change of efeence fame ψx µ tansfoms accoding to.,.. The patial diffeential equation.5 is called the Klein-Godon equation. In the following we will employ so-called natual units = c =. In these units the quantities enegy, momentum, mass, length, and time all have the same dimension. In natual units the Klein Godon equation.5 eads µ µ + m ψx µ =.5

.5: Klein Godon Equation 37 o t + m ψx µ =..5 One can notice immeadiately that.5 is invaiant unde Loentz tansfomations. This follows fom the fact that µ µ and m ae scalas, and that as postulated ψx µ is a scala. Unde Loentz tansfomations the fee paticle Klein Godon equation.5 becomes µ µ + m ψ x µ =.5 which has the same fom as the Klein Godon equation in the oiginal fame. Cuent 4-Vecto Associated with the Klein-Godon Equation As is well-known the Schödinge equation of a fee paticle is associated with a consevation law fo paticle pobability whee i t ψ, t = m ψ, t.53 t ρ S, t + j S, t =.54 ρ S, t = ψ, t ψ, t.55 descibes the positive definite pobability to detect a paticle at position at time t and whee j S, t = mi [ ψ, t ψ, t ψ, t ψ, t ].56 descibes the cuent density connected with motion of the paticle pobability distibution. To deive this consevation law one ewites the Schödinge equation in the fom i t m ψ = and consides Im [ψ i t ] m ψ =.57 which is equivalent to.54. In ode to obtain the consevation law connected with the Klein Godon equation.5 one consides Im [ ψ µ µ + m ψ ] =.58 which yields which coesponds to whee ψ t ψ ψ t ψ ψ ψ + ψ ψ = t ψ t ψ ψ t ψ + ψ ψ ψ ψ =.59 t ρ KG, t + j KG, t =.6 ρ KG, t = i m ψ, t t ψ, t ψ, t t ψ, t j KG, t = mi ψ, t ψ, t ψ, t ψ, t..6

38 Relativistic Quantum Mechanics This consevation law diffes in one impotant aspect fom that of the Schödinge equation.54, namely, in that the expession fo ρ KG is not positive definite. When the Klein-Godon equation had been initially suggested this lack of positive definiteness woied physicists to a degee that the Klein Godon equation was ejected and the seach fo a Loentz invaiant quantum mechanical wave equation continued. Today, the Klein-Godon equation is consideed as a suitable equation to descibe spin paticles, fo example pions. The pope intepetation of ρ KG, t, it had been ealized late, is actually that of a chage density, not of paticle pobability. Solution of the Fee Paticle Klein Godon Equation Solutions of the fee paticle Klein Godon equation ae ψx µ = N e ip µx µ = N e i p E o t..6 Inseting this into the Klein Godon equation.5 yields E o p m ψ, t =.63 which esults in the expected [see.93] dispesion elationship connecting E, p, m The coesponding enegy is E = m + p o..64 E o p o, ± = ± m + p o.65 This esult togethe with.6 shows that the solutions of the fee paticle Klein-Godon e- quation.5 ae actually detemined by p o and by the choice of sign ±. We denote this by summaizing the solutions as follows µ µ + m ψ o p, λ x µ =.66 ψ o p, λ x µ = N λ,p e i p λe o pt E o p = m + p o, λ = ± The spectum of the Klein Godon equation.5 is a continuum of positive enegies E m, coesponding to λ = +, and of negative enegies E m, coesponding to λ =. The density ρ KG p, λ associated with the coesponding wave functions ψ o p, λ x µ accoding to.6 and.66 is ρ KG p, λ = λ E o p m ψ o p, λ x µ ψ o p, λ x µ.67 which is positive fo λ = + and negative fo λ =. The pope intepetation of the two cases is that the Klein Godon equation descibes paticles as well as anti-paticles; the anti-paticles cay a chage opposite to that of the associated paticles, and the density ρ KG p, λ actually descibes chage density athe than pobability. Geneating a Solution Though Loentz Tansfomation A paticle at est, i.e., with p =, accoding to?? is decibed by the independent wave function ψ o p =, λ x µ = N e iλmt, λ = ±..68

.6: Klein Godon Equation with Electomagnetic Field 39 We want to demonstate now that the wave functions fo p in?? can be obtained though appopiate Loentz tansfomation of.68. Fo this pupose we conside the wave function fo a paticle moving with momentum velocity v in the diection of the x 3 axis. Such wave function should be geneated by applying the Loentz tansfomation in the function space epesentation.45 choosing p m = sinhw3. This yields, in fact, fo the wave function.68, using.38 to eplace t = x by hypebolic coodinates R, Ω, Lw 3 ψ o p =, λ x µ = exp w 3 N e iλmrcoshω Ω = N e iλmrcoshω+w3..69 The addition theoem of hypebolic functions coshω+w 3 = coshω coshw 3 + sinhω sinhw 3 allows us to ewite the exponent on the.h.s. of.69 iλ m coshw 3 R coshω iλ m sinhw 3 R sinhω..7 The coodinate tansfomation.38 and the elationships.6 yield fo this expession m m v iλ v x iλ v x3..7 One can intepet then fo λ = +, i.e., fo positive enegy solutions, p = mv/ v.7 as the momentum of the paticle elative to the moving fame and m m = v v = m + m v v = m + p = E o p.73 as the enegy [c.f..66] of the paticle. In case of λ = + one obtains finally Lw 3 ψ o p =, λ = + x µ = N e ipx3 E o px.74 which agees with the expession given in.66. In case of λ =, i.e., fo negative enegy solutions, one has to intepete p = mv/ v.75 as the momentum of the paticle and one obtains Lw 3 ψ o p =, λ = x µ = N e ipx3 + E o px..76.6 Klein Godon Equation fo Paticles in an Electomagnetic Field We conside now the quantum mechanical wave equation fo a spin paticle moving in an electomagnetic field descibed by the 4-vecto potential A µ x µ = V, t, A, t ; A µ x µ = V, t, A, t.77

3 Relativistic Quantum Mechanics fee classical fee quantum classical paticle in quantum paticle in paticle field V, A paticle field V, A enegy E E qv i t i t qv momentum p p q A ˆ p = i ˆ p q A = ˆ π 4-vecto p µ p µ qa µ i µ i µ qa µ = π µ Table.: Coupling of a paticle of chage q to an electomagnetic field descibed by the 4-vecto potential A µ = V, A o A µ = V, A. Accoding to the so-called minimum coupling pinciple the pesence of the field is accounted fo by alteing enegy, momenta fo classical paticles and the espective opeatos fo quantum mechanical paticles in the manne shown. See also Eq..47. To obtain the appopiate wave equation we follow the deivation of the fee paticle Klein Godon equation above and apply again the coespondence pinciple to.93, albeit in a fom, which couples a paticle of chage q to an electomagnetic field descibed though the potential A µ x ν. Accoding to the pinciple of minimal coupling [see.69] one eplaces the quantum mechanical opeatos, i.e., i t and i in.5, accoding to the ules shown in Table.. Fo this pupose one wites the Klein-Godon equation.5 g µν i µ i ν + m ψx µ =..78 Accoding to the eplacements in Table. this becomes which can also be witten g µν i µ qa µ i ν A ν ψx µ = m ψx µ.79 g µν π µ π ν ; m ψx µ =..8 In tems of space-time deivatives this eads [ i t qv, t ψ, t = i qa, ] t + m ψ, t..8 Non-Relativistic Limit of Fee Paticle Klein Godon Equation In ode to conside futhe the intepetation of the positive and negative enegy solutions of the Klein Godon equation one can conside the non-elativistic limit. Fo this pupose we split-off a facto exp imt which descibes the oscillations of the wave function due to the est enegy, and focus on the emaining pat of the wave function, i.e., we define ψ, t = e imt Ψ, t,.8 and seek an equation fo Ψ, t. We will also assume, in keeping withnthe non-elativistic limit, that the mass m of the paticle, i.e., it s est enegy, is much lage than all othe enegy tems, in

.6: Klein Godon Equation with Electomagnetic Field 3 paticula, lage than i t Ψ/Ψ and alge than qv, i.e., i tψ << m, q V << m..83 Ψ The tem on the l.h.s. of.8 can then be appoximated as follows: i t qv e imt Ψ = i t qv me imt Ψ + e imt i t Ψ qv e imt Ψ = m e imt Ψ + me imt i t Ψ qv e imt Ψ +me imt i t Ψ e imt Ψ qv e imt i t Ψ me imt qv Ψ e imt i t qv Ψ + q V e imt Ψ m e imt Ψ mqv e imt Ψ me imt i t Ψ.84 whee we neglected all tems which did not contain factos m. The appoximation is justified on the gound of the inequalities.83. The Klein-Godon equation.8 eads then [ [ˆ p qa, i t Ψ, t = ] t] + qv, t Ψ, t.85 m This is, howeve, identical to the Schödinge equation. of a non-elativistic spin- paticle moving in an electomagnetic field. Pionic Atoms To apply the Klein Godon equation.8 to a physical system we conside pionic atoms, i.e., atoms in which one o moe electons ae eplaced by π mesons. This application demonstates that the Klein Godon equation descibes spin zeo paticles, e.g., spin- mesons. To manufactue pionic atoms, π mesons ae geneated though inelastic poton poton scatteing p + p p + p + π + π +,.86 then ae slowed down, filteed out of the beam and finally fall as slow pions onto elements fo which a pionic vaiant is to be studied. The pocess of π meson captue involves the so-called Auge effect, the binding of a negative chage typically an electon while at the same time a lowe shell electon is being emitted π + atom atom e + π + e..87 We want to investigate in the following a desciption of a stationay state of a pionic atom involving a nucleus with chage +Ze and a π meson. A stationay state of the Klein Godon equation is descibed by a wave function ψx µ = ϕ e iɛt..88 Inseting this into.8 yields we assume now that the Klein Godon equation descibes a paticle with mass m π and chage e fo qv, t = Ze and A, t [ ] ɛ + Ze + m π ϕ =..89

3 Relativistic Quantum Mechanics Because of the adial symmety of the Coulomb potential we expess this equation in tems of spheical coodinates, θ, φ. The Laplacian is = + sin θ θsinθ θ + sin θ φ = ˆL..9 With this expession and afte expanding ɛ + Ze one obtains d d ˆL Z e 4 + ɛze + ɛ m π The opeato ˆL in this equation suggests to choose a solution of the type ϕ = R l φ =..9 Y lm θ, φ.9 whee the functions Y lm θ, φ ae spheical hamonics, i.e., the eigenfunctions of the opeato ˆL in.9 ˆL Y lm θ, φ = l l + Y lm θ, φ..93.9 leads then to the odinay diffeential equation d d ll + Z e 4 + ɛze + ɛ m π R l =..94 Bound state solutions can be obtained eadily noticing that this equation is essentially identical to that posed by the Coulomb poblem potential Ze fo the Schödinge equation d ll + d + m πze + m π E R l =.95 The latte poblem leads to the well-known spectum E n = m π Ze n ; n =,,... ; l =,,... n..96 In this expession the numbe n defined though n = n l.97 counts the numbe of nodes of the wave function, i.e., this quantity definitely must be an intege. The similaity of.94 and.95 can be made complete if one detemines λ such that λl λl + = l l + Z e 4..98 The suitable choice is λl = + l + Z e 4.99

.6: Klein Godon Equation with Electomagnetic Field 33 and one can wite.94 d λl λl + d + ɛze + ɛ m π R l =.. The bound state solutions of this equation should coespond to ɛ values which can be obtained fom.96 if one makes the eplacement One obtains E ɛ m π m π, l λl, e e ɛ m π.. ɛ m π m π m π Z e 4 ɛ m = π n + λl +.. Solving this fo ɛ choosing the oot which endes ɛ m π, i.e., which coesponds to a bound state yields ɛ = m π + Z e 4 n + λl+ ; n =,,... ; l =,,.....3 E KG n, l, m = Using.97,.99 and defining E KG = ɛ esults in the spectum m π + Z e 4 n l + l+ Z e 4 n =,,... l =,,..., n m = l, l +,..., +l.4 In ode to compae this esult with the spectum of the non-elativistic hydogen-like atom we expand in tems of the fine stuctue constant e to ode Oɛ 8. Intoducing α = Z e 4 and β = l +.4 eads + and one obtains the seies of appoximations + α n β + β α + +.5 α n β + β α α n α β + Oα α n α β n + Oα

34 Relativistic Quantum Mechanics + α + α + Oα n βn 3 3 + α n + α βn 3 α n α 8n 4 + Oα 3 α βn 3 + α 8n 4 + α 4n 4 + Oα3..6 Fom this esults fo.4 [ E KG n, l, m m mz e 4 n mz4 e 8 n 3 l + ] 3 + OZ 6 e..7 4n Hee the fist tem epesents the est enegy, the second tem the non-elativistic enegy, and the thid tem gives the leading elativistic coection. The latte tem agees with obsevations of pionic atoms, howeve, it does not agee with obsevations of the hydogen spectum. The latte spectum shows, fo example, a splitting of the six n =, l = states into goups of two and fou degeneate states. In ode to descibe electon specta one must employ the Loentz-invaiant wave equation fo spin- paticles, i.e., the Diac equation intoduced below. It must be pointed out hee that ɛ does not denote enegy, but in the pesent case athe the negative of the enegy. Also, the π meson is a pseudoscala paticle, i.e., the wave function changes sign unde eflection..7 The Diac Equation Histoically, the Klein Godon equation had been ejected since it did not yield a positive-definite pobability density, a featue which is connected with the nd ode time deivative in this equation. This deivative, in tun, aises because the Klein Godon equation, though the coespondence pinciple, is elated to the equation E = m + p of the classical theoy which involves a tem E. A moe satisfactoy Loentz invaiant wave equation, i.e., one with a positive-definite density, would have only a fist ode time deivative. Howeve, because of the equivalence of space and time coodinates in the Minkowski space such equation necessaily can only have then fist ode deivatives with espect to spatial coodinates. It should featue then a diffeential opeato of the type D = iγ µ µ. Heuistic Deivation Stating fom the Klein-Godon Equation An obvious stating point fo a Loentz-invaiant wave equation with only a fist ode time deivative is E = ± m + p. Application of the coespondence pinciple.46 leads to the wave equation i t Ψ, t = ± m Ψ, t..8 These two equation can be combined i t + m i t m Ψ, t.9

.7: Diac Equation 35 which, in fact, is identical to the two equations.8. Equations.8,.9, howeve, ae unsatisfactoy since expansion of the squae oot opeato involves all powes of the Laplace opeato, but not an opeato i γ as suggested by the pinciple of elativity equivalence of space and time. Many attempts wee made by theoetical physicists to lineaize the squae oot opeato in.8,.9, but fo a long time to no avail. Finally, Diac succeeded. His solution to the poblem involved an ingenious step, namely, the ealization that the lineaization can be caied out only if one assumes a 4-dimensional epesentation of the coefficients γ µ. Initially, it was assumed that the 4-dimensional space intoduced by Diac could be linked to 4- vectos, i.e., quantities with the tansfomation law.66. Howeve, this was not so. Instead, the 4-dimensionsional epesentation discoveed by Diac involved new physical popeties, spin- and anti-paticles. The discovey by Diac, achieved though a beautiful mathematical theoy, stengthens the believe of many theoetical physicists today that the popeties of physical matte ultimately deive fom a, yet to be discoveed, beautiful mathematical theoy and that, theefoe, one oute to impotant discoveies in physics is the ceation of new mathematical desciptions of natue, these desciptions ultimately meging with the tue theoy of matte. Popeties of the Diac Matices Let us now tace Diac s steps in achieving the lineaization of the squae oot opeato in.8. Stating point is to boldly factoize, accoding to.9, the opeato of the Klein Godon equation µ µ + m = P + m P m. whee P = iγ µ µ.. Obviously, this would lead to the two wave equations P mψ = and P + mψ = which have a fist ode time deivative and, theefoe, ae associated with a positive-definite paticle density. We seek to identify the coefficients γ µ. Inseting. into. yields g µν µ ν m = iγ µ µ + miγ µ µ m = γ µ γ ν µ ν m = γµ γ ν µ ν + γ ν γ µ ν µ m = γµ γ ν + γ ν γ µ µ ν m. whee we have changed dummy summation indices, exploited µ ν = ν µ, but did not commute the, so fa, unspecified algebaic objects γ µ and γ ν. Compaing the left-most and the ight-most side of the equations above one can conclude the following popety of γ µ γ µ γ ν + γ ν γ µ = [ γ µ, γ ν ] + = g µν.3 We want to detemine now the simplest algebaic ealization of γ µ. It tuns out that no 4-vecto of eal o complex coefficients can satisfy these conditions. In fact, the quantities γ, γ, γ, γ 3 can only be ealized by d d matices equiing that the wave function Ψx µ is actually a d dimensional vecto of functions ψ x µ, ψ x µ,... ψ d x µ. Fo µ = ν condition.3 eads { γ µ µ = =..4 µ =,, 3

36 Relativistic Quantum Mechanics Fom this follows that γ has eal eigenvalues ± and γ j, j =,, 3 has imaginay eigenvalues ±i. Accodingly, one can impose the condition Fo µ ν.3 eads γ is hemitian ; γ j, j =,, 3 ae anti-hemitian..5 γ µ γ ν = γ ν γ µ,.6 i.e., the γ µ ae anti-commuting. Fom this one can conclude fo the deteminants of γ µ detγ µ γ ν = det γ ν γ µ = d detγ ν γ µ = d detγ µ γ ν..7 Obviously, as long as detγ µ the dimension d of the squae matices γ µ must be even so that d =. Fo d = thee exist only thee anti-commuting matices, namely the Pauli matices σ, σ, σ 3 fo which, in fact, holds σ j = ; σ j σ k = σ k σ j fo j k..8 The Pauli matices allow one, howeve, to constuct fou matices γ µ fo the next possible dimension d = 4. A pope choice is γ = ; γ j σ j = σ j,.9 Using popety.8 of the Pauli matices one can eadily pove that condition.3 is satisfied. We will ague futhe below that the choice f γ µ, except fo similaity tasnfomations, is unique. The Diac Equation Altogethe we have shown that the Klein Godon equation can be factoized fomally iγ µ µ + m iγ µ µ m Ψx µ =. whee Ψx µ epesents a 4-dimensional wave function, athe than a scala wave function. Fom this equation one can conclude that also the following should hold which is the celebated Diac equation. The Adjoint Diac Equation iγ µ µ m Ψx µ =. The adjoint equation is Ψ x µ iγ µ µ + m =. whee we have defined Ψ = ψ, ψ, ψ 3, ψ 4 and whee µ denotes the diffeential opeato µ opeating to the left side, athe than to the ight side. One can eadily show using the hemitian

.7: Diac Equation 37 popety of the Pauli matices γ = γ and γ j = γ j fo j =,, 3 which, in fact, is implied by.5. This popety can also be witten γ µ = γ γ µ γ..3 Inseting this into. and multiplication fom the ight by γ yields the adjoint Diac equation Ψ x µ γ iγ µ µ + m =..4 Similaity Tansfomations of the Diac Equation - Chial Repesentation The Diac equation can be subject to any similaity tansfomation defined though a non-singula 4 4 matix S. Defining a new epesentation of the wave function Ψx µ leads to the new Diac equation Ψx µ = S Ψx µ.5 i γ µ µ m Ψx µ =.6 whee γ µ = S γ µ S.7 A epesentation often adopted beside the one given by.,.9 is the socalled chial epesentation defined though and γ = Ψx µ = S Ψx µ ; S = ; γ j = σ j σ j.8, j =,, 3..9 The similaity tansfomation.7 leaves the algeba of the Diac matices unaffected and commutation popety.3 still holds, i.e., [ γ µ, γ ν ] + = g µν..3 Execise.7.: Deive efeq:diac-intoa fom.3,.7. Schödinge Fom of the Diac Equation Anothe fom in which the Diac equation is used often esults fom multiplying. fom the left by γ i t + i ˆ α ˆβ m Ψ, t =.3

38 Relativistic Quantum Mechanics wheeˆ α has the thee components α j, j =,, 3 and σ j ˆβ = ; ˆα j = σ j, j =,, 3..3 This fom of the Diac equation is called the Schödinge fom since it can be witten in analogy to the time-dependent Schödinge equation i t Ψx µ = H o Ψx µ ; H o = ˆ α ˆ p + ˆβ m..33 The eigenstates and eigenvalues of H coespond to the stationay states and enegies of the paticles descibed by the Diac equation. Cliffod Algeba and Diac Matices The matices defined though satisfy the anti-commutation popety d j = iγ j, j =,, 3 ; d 4 = γ.34 d j d k + d k d j = { fo j = k fo j k.35 as can be eadily veified fom.3. The associative algeba geneated by d... d 4 is called a Cliffod algeba C 4. The thee Pauli matices also obey the popety.35 and, hence, fom a Cliffod algeba C 3. The epesentations of Cliffod algebas C m ae well established. Fo example, in case of C 4, a epesentation of the d j s is d = ; d = d 3 = i i ; d 4 = i i.36 whee denotes the Konecke poduct between matices, i.e., the matix elements of C = A B ae C jk,lm = A jl B km. The Cliffod algeba C 4 entails a subgoup G 4 of elements ± d j d j d js, j < j < < j s s 4.37 which ae the odeed poducts of the opeatos ± and d, d, d 3, d 4. Obviously, any poduct of the d j s can be bought to the fom.37 by means of the popety.35. Thee ae including the diffeent signs 3 elements in G 4 which we define as follows Γ ± = ± Γ ± = ±d, Γ ±3 = ±d, Γ ±4 = ±d 3, Γ ±5 = ±d 4 Γ ±6 = ±d d, Γ ±7 = ±d d 3, Γ ±8 = ±d d 4, Γ ±9 = ±d d 3 Γ ± = ±d d 4, Γ ± = ±d 3 d 4 Γ ± = ±d d d 3, Γ ±3 = ±d d d 4, Γ ±4 = ±d d 3 d 4, Γ ±5 = ±d d 3 d 4 Γ ±6 = ±d d d 3 d 4.38

.8: Loentz Invaiance of the Diac Equation 39 These elements fom a goup since obviously any poduct of two Γ s can be expessed in tems of a thid Γ. The epesentations of this goup ae given by a set of 3 4 4 matices which ae equivalent with espect to similaity tansfomations. Since the Γ j ae hemitian the similaity tansfomations ae actually given in tems of unitay tansfomations. One can conclude then that also any set of 4 4 matices obeying.35 can diffe only with espect to unitay similaity tansfomations. This popety extends then to 4 4 matices which obey.3, i.e., to Diac matices. To complete the poof in this section the eade may consult Mille Symmety Goups and thei Application Chapte 9.6 and R.H.Good, Rev.Mod.Phys. 7, 955, page 87. The eade may also want to establish the unitay tansfomation which connects the Diac matices in the fom.36 with the Diac epesentation.9. Execise 7.: Demonstate the anti-commutation elationships.8 of the Pauli matices σ j. Execise 7.3: Demonstate the anti-commutation elationships.8 of the Diac matices γ µ. Execise 7.4: Show that fom.4 follows that γ has eal eigenvalues ± and can be epesented by a hemitian matix, and γ j, j =,, 3 has imaginay eigenvalues ±i and can be epesented by an anti-hemitian matix..8 Loentz Invaiance of the Diac Equation We want to show now that the Diac equation is invaiant unde Loentz tansfomations, i.e., the fom of the Diac equation is identical in equivalent fames of efeence, i.e., in fames connected by Loentz tansfomations. The latte tansfomations imply that coodinates tansfom accoding to.6, i.e., x = L µ νx ν, and deivatives accoding to.8. Multiplication and summation of.8 by L µ ρ and using.76 yields ρ = L ν ρ ν, a esult one could have also obtained by applying the chain ule to.6. We can, theefoe, tansfom coodinates and deivatives of the Diac equation. Howeve, we do not know yet how to tansfom the 4-dimensional wave function Ψ and the Diac matices γ µ. Loentz Tansfomation of the Bispino State Actually, we will appoach the poof of the Loentz invaiance of the Diac equation by testing if thee exists a tansfomation of the bispino wave function Ψ and of the Diac matices γ µ which togethe with the tansfomations of coodinates and deivatives leaves the fom of the Diac equation invaiant, i.e., in a moving fame should hold iγ µ µ m Ψ x µ =..39 Fom invaiance implies that the matices γ µ should have the same popeties as γ µ, namely,.3,.5. Except fo a similaity tansfomation, these popeties detemine the matices γ µ uniquely, i.e., it must hold γ µ = γ µ. Hence, in a moving fame holds iγ µ µ m Ψ x µ =..4

3 Relativistic Quantum Mechanics Infinitesimal Bispino State Tansfomation We want to show now that a suitable tansfomation of Ψx µ does, in fact, exist. The tansfomation is assumed to be linea and of the fom Ψ x µ = SL µ ν Ψx µ ; x µ = L µ ν x ν.4 whee SL µ ν is a non-singula 4 4 matix, the coefficients of which depend on the matix L µ ν defining the Loentz tansfomation in such a way that SL µ ν = fo L µ ν = δ µ ν holds. Obviously, the tansfomation.4 implies a similaity tansfomation Sγ µ S. One can, hence, state that the Diac equation. upon Loentz tansfomation yields isl η ξ γ µ S L η ξ L ν µ ν m Ψ x µ =..4 The fom invaiance of the Diac equation unde this tansfomation implies then the condition SL η ξγ µ S L η ξ L ν µ = γ ν..43 We want to detemine now the 4 4 matix SL η ξ which satisfies this condition. The pope stating point fo a constuctiuon of SL η ξ is actually.43 in a fom in which the Loentz tansfomation in the fom L µν is on the.h.s. of the equation. Fo this pupose we exploit. in the fom L ν µg νσ L σ ρ = g µρ = g ρµ. Multiplication of.43 fom the left by L σ ρg σν yields SL η ξγ µ S L η ξ g ρµ = L σ ρg σν γ σ..44 fom which, using g ρµ γ µ = γ ρ, follows One can finally conclude multiplying both sides by g ρµ SL η ξγ ρ S L η ξ = L σ ργ σ..45 SL η ξγ µ S L η ξ = L ν µ γ ν..46 The constuction of SL η ξ will poceed using the avenue of infinitesimal tansfomations. We had intoduced in.38 the infinitesimal Loentz tansfomations in the fom L µ ν = δ µ ν + ɛ µ ν whee the infinitesimal opeato ɛ µ ν obeyed ɛ T = gɛg. Multiplication of this popety by g fom the ight yields ɛg T = ɛg, i.e., ɛg is an anti-symmetic matix. The elements of ɛg ae, howeve, ɛ µ ρg ρν = ɛ µν and, hence, in the expession of the infinitesimal tansfomation L µν = g µν + ɛ µν.47 the infinitesimal matix ɛ µν is anti-symmetic. The infinitesimal tansfomation SL ρ σ which coesponds to.47 can be expanded Sɛ µν = i 4 σ µνɛ µν.48 Hee σ µν denote 4 4 matices opeating in the 4-dimensional space of the wave functions Ψ. Sɛ µν should not change its value if one eplaces in its agument ɛ µν by ɛ νµ. It holds then Sɛ µν = i 4 σ µνɛ µν = S ɛ νµ = + i 4 σ µνɛ νµ.49

.8: Loentz Invaiance of the Diac Equation 3 fom which we can conclude σ µν ɛ µν = σ µν ɛ νµ = σ νµ ɛ µν, i.e., it must hold σ µν = σ νµ..5 One can eadily show expanding SS = to fist ode in ɛ µν that fo the invese infinitesimal tansfomation holds S ɛ µν = + i 4 σ µνɛ µν.5 Inseting.48,.5 into.46 esults then in a condition fo the geneatos σ µν i 4 σ αβγ µ γ µ σ αβ ɛ αβ = ɛ νµ γ ν..5 Since six of the coefficients ɛ αβ can be chosen independently, this condition can actually be expessed though six independent conditions. Fo this pupose one needs to expess fomally the.h.s. of.5 also as a sum ove both indices of ɛ αβ. Futhemoe, the expession on the.h.s., like the expession on the l.h.s., must be symmetic with espect to intechange of the indices α and β. Fo this pupose we expess ɛ νµ γ ν = ɛαµ γ α + ɛβµ γ β = ɛαβ δ µ βγ α + ɛαβ δ µ αγ β = ɛαβ δ µ βγ α δ µ αγ β..53 Compaing this with the l.h.s. of.5 esults in the condition fo each α, β [ σ αβ, γ µ ] = i δ µ βγ α δ µ αγ β..54 The pope σ αβ must be anti-symmetic in the indices α, β and opeate in the same space as the Diac matices. In fact, a solution of condition.54 is σ αβ = i [ γ α, γ β ].55 which can be demonstated using the popeties.3,.6 of the Diac matices. Execise 7.5: Show that the σ αβ defined though.55 satisfy condition.54. Algeba of Geneatos of Bispino Tansfomation We want to constuct the bispino Loentz tansfomation by exponentiating the geneatos σ µν. Fo this pupose we need to veify that the algeba of the geneatos involving addition and multiplication is closed. Fo this pupose we inspect the popeties of the geneatos in a paticula epesentation, namely, the chial epesentation intoduced above in Eqs..8,.9. In this epesentation the Diac matices γ µ = γ, γ ae σ j γ = ; γ j = σ j, j =,, 3..56

3 Relativistic Quantum Mechanics One can eadily veify that the non-vanishing geneatos σ µν ae given by note σ µν = σ νµ, i.e. only six geneatos need to be detemined σ j = i iσ [ γ j σ l, γ j ] = iσ j ; σ jk = [ γ j, γ k ] = ɛ jkl σ l..57 Obviously, the algeba of these geneatos is closed unde addition and multiplication, since both opeations convet block-diagonal opeatos A.58 B again into block-diagonal opeatos, and since the algeba of the Pauli matices is closed. We can finally note that the closedness of the algeba of the geneatos σ µν is not affected by similaity tansfomations and that, theefoe, any epesentation of the geneatos, in paticula, the epesentation.55 yields a closed algeba. Finite Bispino Tansfomation The closedness of the algeba of the geneatos σ µν defined though.48 allows us to wite the tansfomation S fo any, i.e., not necessaily infinitesimal, ɛ µν in the exponential fom S = exp i4 σ µνɛ µν..59 We had stated befoe that the tansfomation S is actually detemined though the Loentz tansfomation L µ ν. One should, theefoe, be able to state S in tems of the same paametes w and ϑ as the Loentz tansfomation in.5. In fact, one can expess the tenso ɛ µν though w and ϑ using ɛ µν = ɛ µ ρg ρν and the expession.44 ɛ µν = w w w 3 w ϑ 3 ϑ w ϑ 3 ϑ w 3 ϑ ϑ.6 Inseting this into.59 yields the desied connection between the Loentz tansfomation.5 and S. In ode to constuct an explicit expession of S in tems of w and ϑ we employ again the chial epesentation. In this epesentation holds i 4 σ µνɛ µν = i σ ɛ + σ ɛ + σ 3 ɛ 3 + σ ɛ + σ 3 ɛ 3 + σ 3 ɛ 3 = w i ϑ σ w + iϑ..6 σ We note that this opeato is block-diagonal. Since such opeato does not change its block-diagonal fom upon exponentiation the bispino tansfomation.59 becomes in the chial epesentation S w, ϑ e w i ϑ σ =.6 e w + i ϑ σ This expession allows one to tansfom accoding to.4 bispino wave functions fom one fame of efeence into anothe fame of efeence.

.8: Loentz Invaiance of the Diac Equation 33 Cuent 4-Vecto Associated with Diac Equation We like to deive now an expession fo the cuent 4-vecto j µ associated with the Diac equation which satisfies the consevation law µ j µ =..63 Stating point ae the Diac equation in the fom. and the adjoint Diac equation.4. Multiplying. fom the left by Ψ x µ γ,.4 fom the ight by Ψx µ, and addition yields Ψ x µ γ iγ µ µ + iγ µ µ Ψx µ =..64 The last esult can be witten i.e., the consevation law.63 does hold, in fact, fo The time-like component ρ of j µ µ Ψ x ν γ γ µ Ψx ν =,.65 j µ x µ = ρ, j = Ψ x µ γ γ µ Ψx µ..66 ρx µ = Ψ x µ Ψx µ = 4 ψ s x µ.67 has the desied popety of being positive definite. The consevation law.63 allows one to conclude that j µ must tansfom like a contavaiant 4-vecto as the notation implies. The eason is that the.h.s. of.63 obviously is a scala unde Loentz tansfomations and that the left hand side must then also tansfom like a scala. Since µ tansfoms like a covaiant 4-vecto, j µ must tansfom like a contavaiant 4-vecto. This tansfomation behaviou can also be deduced fom the tansfomation popeties of the bispino wave function Ψx µ. Fo this pupose we pove fist the elationship s= S = γ S γ..68 We will pove this popety in the chial epesentation. Obviously, the popety applies then in any epesentation of S. Fo ou poof we note fist S w, ϑ = S w, ϑ = e w i ϑ σ e w + i ϑ σ.69 One can eadily show that the same opeato is obtained evaluating γ S w, ϑ γ = e w + i ϑ σ e w i ϑ σ We conclude that.68 holds fo the bispino Loentz tansfomation. We will now detemine the elationship between the flux..7 j µ = Ψ x µ γ γ µ Ψ x.7

34 Relativistic Quantum Mechanics in a moving fame of efeence and the flux j µ in a fame at est. Note that we have assumed in.7 that the Diac matices ae independent of the fame of efeence. One obtains using.68 j µ = Ψ x µ S γ γ µ SΨx µ = Ψ x µ γ S γ µ SΨx µ..7 With S L η ξ = SL η ξ one can estate.46 S L η ξγ µ S L η ξ = L νµ γ ν = L ν µ γ ν = L µ νγ ν..73 whee we have employed.76. Combining this with.7 esults in the expected tansfomation behaviou j µ = L µ ν j ν..74.9 Solutions of the Fee Paticle Diac Equation We want to detemine now the wave functions of fee paticles descibed by the Diac equation. Like in non-elativistic quantum mechanics the fee paticle wave function plays a cental ole, not only as the most simple demonstation of the theoy, but also as poviding a basis in which the wave functions of inteacting paticle systems can be expanded and chaacteized. The solutions povide also a complete, othonomal basis and allows one to quantize the Diac field Ψx µ much like the classical electomagnetic field is quantized though ceation and annihilation opeatos epesenting fee electomagnetic waves of fixed momentum and fequency. In case of non-elativistic quantum mechanics the fee paticle wave function has a single component ψ, t and is detemined though the momentum p R 3. In elativistic quantum mechanics a Diac paticle can also be chaacteized though a momentum, howeve, the wave function has fou components which invite futhe chaacteization of the fee paticle state. In the following we want to povide this chaacteization, specific fo the Diac fee paticle. We will stat fom the Diac equation in the Schödinge fom.3,.3,.33 i t Ψx µ = H o Ψx µ..75 The fee paticle wave function is an eigenfunction of H o, a popety which leads to the enegy momentum dispesion elationship of the Diac paticle. The additional degees of feedom descibed by the fou components of the bispino wave function equie, as just mentioned, additional chaacteizations, i.e., the identification of obsevables and thei quantum mechanical opeatos, of which the wave functions ae eigenfunctions as well. As it tuns out, only two degees of feedom of the bispino fou degees of feedom ae independent [c.f..8,.83]. The independent degees of feedom allow one to choose the states of the fee Diac paticle as eigenstates of the 4-momentum opeato ˆp µ and of the helicity opeato Γ σ ˆ p/ ˆ p intoduced below. These opeatos, as is equied fo the mentioned popety, commute with each othe. The opeatos commute also with H o in.33. Like fo the fee paticle wave functions of the non-elativistic Schödinge and the Klein Godon equations one expects that the space time dependence is govened by a facto exp[i p ɛt]. As pointed out, the Diac paticles ae descibed by 4-dimensional, bispino wave functions and we need to detemine coesponding components of the wave function. Fo this pupose we conside

.9: Solutions of the Fee Paticle Diac Equation 35 the following fom of the fee Diac paticle wave function Ψx µ φx = µ φo χx µ = χ o e i p ɛt.76 whee p and ɛ togethe epesent fou eal constants, late to be identified with momentum and enegy, and φ o, χ o each epesent a constant, two-dimensional spino state. Inseting.76 into.3,.3 leads to the 4-dimensional eigenvalue poblem m σ p φo φo = ɛ..77 σ p m To solve this poblem we wite.77 explicitly χ o χ o ɛ m φ o σ p χ o = σ p φ o + ɛ + m χ o =..78 Multiplication of the st equation by ɛ +m and of the second equation by σ p and subtaction of the esults yields the -dimensional equation [ ɛ m σ p ] φ o =..79 Accoding to the popety 5.34 of Pauli matices holds σ p = p. One can, hence, conclude fom.79 the well-known elativistic dispesion elationship which has a positive and a negative solution ɛ = m + p.8 ɛ = ± E p, E p = m + p..8 Obviously, the Diac equation, like the Klein Godon equation, epoduce the classical elativistic enegy momentum elationships.93,.94 Equation.78 povides us with infomation about the components of the bispino wave function.76, namely φ o and χ o ae elated as follows φ = χ o = σ p ɛ m χ.8 σ p ɛ + m φ,.83 whee ɛ is defined in.8. These two elationships ae consistent with each othe. In fact, one finds using 5.34 and.8 σ p ɛ + m φ = σ p ɛ + m ɛ + m χ = p ɛ m χ = χ..84 The elationships.8,.83 imply that the bispino pat of the wave function allows only two degees of feedom to be chosen independently. We want to show now that these degees of feedom coespond to a spin-like popety, the socalled helicity of the paticle.

36 Relativistic Quantum Mechanics Fo ou futhe chaacteization we will deal with the positive and negative enegy solutions [cf..8] sepaately. Fo the positive enegy solution, i.e., the solution fo ɛ = +E p, we pesent φ o though the nomalized vecto u φ o = = u C, u u = u + u =..85 u The coesponding fee Diac paticle is then descibed though the wave function Ψ p, + x µ u = N + p σ p E p + m u e i p ɛt, ɛ = +E p..86 Hee N + p is a constant which will be chosen to satisfy the nomalization condition Ψ p, + γ Ψ p, + =,.87 the fom of which will be justified futhe below. Similaly, we pesent the negative enegy solution, i.e., the solution fo ɛ = E p, though χ o given by u χ o = = u C, u u = u + u =..88 u coesponding to the wave function Ψ p, x µ = N p σ p E p + m u u e i p ɛt, ɛ = E p..89 Hee N p is a constant which will be chosen to satisfy the nomalization condition Ψ p, + γ Ψ p, + =,.9 which diffes fom the nomalization condition.87 in the minus sign on the.h.s. The fom of this condition and of.87 will be justified now. Fist, we demonstate that the poduct Ψ p, ±γ Ψ p, ±, i.e., the l.h.s. of.87,.9, is invaiant unde Loentz tansfomations. One can see this as follows: Let Ψ p, ± denote the solution of a fee paticle moving with momentum p in the laboatoy fame, and let Ψ, ± denote the coesponding solution of a paticle in its est fame. The connection between the solutions, accoding to.4, is Ψ p, ± = S Ψ, ±, whee S is given by.59. Hence, Ψ p, ± γ Ψ p, ± = Ψ, ± S γ S Ψ, ± = Ψ, ± γ S γ γ S Ψ, ± = Ψ, ± γ Ψ, ±..9 Note that we have used that, accoding to.68, S = γ S γ and, hence, S = γ S γ. We want to demonstate now that the signs on the.h.s. of.87,.9 should diffe. Fo this pupose we conside fist the positive enegy solution. Employing.86 fo p = yields, using γ as given in.9 and u u = [c.f..85], Ψ, + γ Ψ, + = N + u, γ u = N +..9

.9: Solutions of the Fee Paticle Diac Equation 37 The same calculation fo the negative enegy wave function as given in.89 yields Ψ, γ Ψ, = N, u γ = N u..93 Obviously, this equies the choice of a negative side on the.h.s. of.9 to assign a positive value to N. We can also conclude fom ou deivation N ± =..94 We want to detemine now N ± p fo abitay p. We conside fist the positive enegy solution. Condition.87 witten explicitly using.86 is [ ] N+ p u T σ p T, E p + m u γ o u σ p E p + m u =.95 Evaluating the l.h.s. using γ as given in.9 yields [ N+ p u u u σ p ] E p + m u =..96 Replacing σ p by p [c.f. 5.34] and using the nomalization of u in.85 esults in [ N+ p p ] E p + m =.97 fom which follows Noting N + p = m + E p m + E p p..98 m + E p p = m p + me p + E p = m + E p m.99 the nomalization coefficient.98 becomes N + p = m + E p m..3 This esult completes the expession fo the wave function.86. Execise 7.6: Show that the nomalization condition [ ] N + p u T σ p T, E p + m u u σ p E p + m u =.3 yields the nomalization coefficient N + p = m + E p E p..3

38 Relativistic Quantum Mechanics We conside now the negative enegy solution. Condition.9 witten explicitly using.89 is [ ] N p σ p T σ p E p + m u, u T γ o E p + m u =.33 u Evaluating the l.h.s. yields [ N p u σ p ] E p + m u u u =..34 This condition is, howeve, identical to the condition.96 fo the nomalization constant N + p of the positive enegy solution. We can, hence, conclude m + E p N p =.35 m and, theeby, have completed the detemination fo the wave function.89. The wave functions.86,.89,.3 have been constucted to satisfy the fee paticle Diac equation.75. Inseting.86 into.75 yields H o Ψ p, λ x µ = λ E p Ψ p, λ x µ,.36 i.e., the wave functions constucted epesent eigenstates of H o. eigenstates of the momentum opeato i µ, i.e., The wave functions ae also i µ Ψ p, λ x µ = p µ Ψ p, λ x µ.37 whee p µ = ɛ, p. This can be veified expessing the space time facto of Ψ p, λ x µ in 4-vecto notation, i.e., exp[i p ɛt] = expip µ x µ. Helicity The fee Diac paticle wave functions.86,.89 ae not completely specified, the two components of u indicate anothe degee of feedom which needs to be defined. This degee of feedom descibes a spin attibute. This attibute is the so-called helicity, defined as the component of the paticle spin along the diection of motion. The coesponding opeato which measues this obsevable is Λ = σ ˆp..38 ˆp Note that ˆp epesents hee an opeato, not a constant vecto. Rathe than consideing the obsevable.37 we investigate fist the obsevable due to the simple opeato σ ˆp. We want to show that this opeato commutes with H o and ˆp to ascetain that the fee paticle wave function can be simultaneously an eigenvecto of all thee opeatos. The commutation popety [ σ ˆp, ˆp j ] =, j =,, 3 is faily obvious. The popety [ σ ˆp, H o ] = follows fom.33 and fom the two identities σ σ ˆp σ σ ˆp =.39

.9: Solutions of the Fee Paticle Diac Equation 39 and σ σ = ˆp σ σ σ ˆp σ ˆp ˆp σ σ ˆp σ σ σ ˆp σ ˆp ˆp =..3 We have shown altogethe that the opeatos ˆp, H o and σ ˆp commute with each othe and, hence, can be simultaneously diagonal. States which ae simultaneously eigenvectos of these thee opeatos ae also simulteneously eigenvectos of the thee opeatos ˆp, H o and Λ defined in.38 above. The condition that the wave functions.86 ae eigenfunctions of Λ as well will specify now the vectos u. Since helicity is defined elative to the diection of motion of a paticle the chaacteization of u as an eigenvecto of the helicity opeato, in pinciple, is independent of the diection of motion of the paticle. We conside fist the simplest case that paticles move along the x 3 diection, i.e., p =,, p 3. In this case Λ = σ3. We assume fist paticles with positive enegy, i.e., ɛ = +E p. Accoding to the definition 5.4 of σ 3 the two u vectos, T and, T ae eigenstates of σ3 with eigenvalues ±. Theefoe, the wave functions which ae eigenstates of the helicity opeato, ae Ψpê 3, +, +, t = Ψpê 3, +,, t = N p N p p m + E p p m + E p eipx3 E p t eipx3 E pt.3 whee ê 3 denotes the unit vecto in the x 3 -diection and whee E p = m + Ep m + p ; N p = m..3 We assume now paticles with negative enegy, i.e., ɛ = E p. The wave functions which ae eigenfunctions of the helicity opeato ae in this case Ψpê 3,, +, t = Ψpê 3,,, t = whee E p and N p ae defined in.3. N p N p p m + E p p m + E p eipx3 + E p t eipx3 + E p t.33

33 Relativistic Quantum Mechanics To obtain fee paticle wave functions fo abitay diections of p one can employ the wave functions.3,.33 except that the states, T and, T have to be eplaced by eigenstates u ± p of the spin opeato along the diection of p. These eigenstates ae obtained though a otational tansfomation 5., 5.3 as follows u p, + = exp i ϑ p σ,.34 u p, = exp i ϑ p σ.35 whee ϑ p = ê3 p p ê 3, p.36 descibes a otation which aligns the x 3 axis with the diection of p. [One can also expess the otation though Eule angles α, β, γ, in which case the tansfomation is given by 5..] The coesponding fee paticle wave functions ae then Ψ p, +, +, t = Ψ p, +,, t = Ψ p,, +, t = Ψ p,,, t = N p N p N N p whee E p and N ae again given by.3. u p, + p m + E p u p, + e i p E pt.37 u p, p m + E p u p, e i p Ept.38 p m + E p u p, + u p, + e i p + E pt.39 p m + E p u p, u p, Geneating Solutions Though Loentz Tansfomation e i p + Ept.3 The solutions.3,.3 can be obtained also by means of the Loentz tansfomation.6 fo the bispino wave function and the tansfomation.3. Fo this pupose one stats fom the solutions of the Diac equation in the chial epesentation.6,.9, denoted by, fo an independent wave function, i.e., a wave function which epesents fee paticles at est. The coesponding wave functions ae detemined though i γ t m Ψt =..3 and ae Ψp =, +, t = e imt,

.9: Solutions of the Fee Paticle Diac Equation 33 Ψp =, +, t = Ψp =,, t = Ψp =,, t = e imt, e+imt, e+imt..3 The eade can eadily veify that tansfomation of these solutions to the Diac epesentationsas defined in.8 yields the coesponding solutions.3,.33 in the p limit. This coespondence justifies the chaacteization ±, ± of the wave functions stated in.3. The solutions.3 can be witten in spino fom φo χ o e imt, φ o, χ o {, }.33 Tansfomation.6 fo a boost in the x 3 diection, i.e., fo w =,, w 3, yields fo the exponential space time dependence accoding to.74,.76 imt i p 3 x 3 Et.34 and fo the bispino pat accoding to.6 φo χ o e w 3σ 3 e w 3σ 3 φo χ o = e w 3σ 3 φ o e w 3σ 3 χ o..35 One should note that φ o, χ o ae eigenstates of σ 3 with eigenvalues ±. Applying.34,.35 to.33 should yield the solutions fo non-vanishing momentum p in the x 3 diection. Fo the esulting wave functions in the chial epesentation one can use then a notation coesponding to that adopted in.3 Ψpw 3 ê 3, +, +, t = Ψpw 3 ê 3, +,, t = e w 3 e w 3 e w 3 e w 3 eipx3 E p t eipx3 E p t

33 Relativistic Quantum Mechanics Ψpw 3 ê 3,, +, t = e w 3 e w 3 Ψpw 3 ê 3,,, t = e w 3 e w 3 whee accoding to.6 pw 3 = m sinhw 3. means of.8 yields Ψpw 3 ê 3, +, +, t = Ψpw 3 ê 3, +,, t = Ψpw 3 ê 3,, +, t = Ψpw 3 ê 3,,, t = Employing the hypebolic function popeties eipx3 + E pt eipx3 + E p t.36 Tansfomation to the Diac epesentation by cosh w 3 sinh w 3 cosh w 3 sinh w 3 sinh w 3 cosh w 3 sinh w 3 cosh w 3 eipx3 E pt eipx3 E p t eipx3 + E p t eipx3 + E pt.37 cosh x = coshx +, sinh x = coshx,.38 the elationship.6 between the paamete w 3 and boost velocity v 3, and the expession.3 fo E p one obtains cosh w 3 = = v3 + = m + m v 3 v 3 m + m = Ep + m + v 3 v 3 m +.39

.9: Invaiance of Diac Equation Revisited 333 and similaly sinh w 3 = Ep m m = p m Ep + m.33 Inseting expessions.39,.33 into.37, indeed, epoduces the positive enegy wave functions.3 as well as the negative enegy solutions.33 fo p. The change of sign fo the latte solutions had to be expected as it was aleady noted fo the negative enegy solutions of the Klein Godon equation.68.76. Invaiance of Diac Equation Revisited At this point we like to povide a vaiation of the deivation of.43, the essential popety stating the Loentz invaiance of the Diac equation. Actually, we will deive this equation only fo infinitesimal tansfomations, which howeve, is sufficient since it must hold then fo any finite tansfomation, and since the calculations following.43 consideed solely the limit of infinitesimal tansfomations anyway. The eason why we povide anothe deivation of.43 is to familiaize ouselves with a fomulation of Loentz tansfomations of the bispino wave finction Ψx µ which teats the spino and the space-time pat of the wave function on the same footing. Such desciption will be essential fo the fomal desciption of Loentz invaiant wave equations fo abitay spin futhe below. In the new deivation we conside the paticle descibed by the wave function tansfomed, but not the obseve. This tansfomation, efeed to as the active tansfomation, expesses the system in the old coodinates. The tansfomation is Ψ x µ = SL η ξ ρl η ξ Ψx µ.33 whee SL η ξ denotes again the tansfomation acting on the bispino chaacte of the wave function Ψx µ and whee ρl η ξ denotes the tansfomation acting on the space-time chaacte of the wave function Ψx µ. ρl η ξ has been defined in.3 above and chaacteized thee. Such tansfomation had been applied by us, of couse, when we geneated the solutions Ψ p, λ, Λ x µ fom the solutions descibing paticles at est Ψ p =, λ, Λ t. We expect, in geneal, that if Ψx µ is a solution of the Diac equation that Ψ x µ as given in.33 is a solution as well. Making this expectation a postulate allows one to deive the condition.43 and, theeby, the pope tansfomation SL η ξ. To show this we ewite the Diac equation. using.33 i SL η ξ γ µ S L η ξ ρl η ξ µ ρ L η ξ m Ψ x µ =.33 Hee we have made use of the fact that SL η ξ commutes with µ and ρl η ξ commutes with γ µ. The fact that any such Ψ x µ is a solution of the Diac equation allows us to conclude SL η ξγ µ S L η ξ ρl η ξ µ ρ L η ξ = γ ν ν.333 which is satisfied in case that the following conditions ae met ρl η ξ µ ρ L η ξ = L ν µ ν ; SL η ξγ µ S L η ξ L ν µ = γ ν..334

334 Relativistic Quantum Mechanics We will demonstate now that the fist condition is satisfied by ρl η ξ. The second condition is identical to.43 and, of couse, it is met by SL η ξ as given in the chial epesentation by.6. As mentioned aleady we will show condition.334 fo infinitesimal Loentz tansfomations L η ξ. We will poceed by employing the geneatos.8 to expess ρl η ξ in its infinitesimal fom and evaluate the expession + ɛϑ J + ɛ w K µ ɛϑ J ɛ w K = µ + ɛm ν µ ν + Oɛ.335 The esult will show that the matix M ν µ is identical to the geneatos of L ν µ fo the six choices ϑ =,,, w =,,, ϑ =,,, w =,,,..., ϑ =,,, w =,,. Inspection of.335 shows that we need to demonstate [J l, µ ] = J l ν µ ν ; [K l, µ ] = K l ν µ ν..336 We will poceed with this task consideing all six cases: [J, µ ] = [x 3 x 3, µ ] = [J, µ ] = [x 3 x 3, µ ] = [J 3, µ ] = [x x, µ ] = [K, µ ] = [x + x, µ ] = [K, µ ] = [x + x, µ ] = [K 3, µ ] = [x 3 + x 3, µ ] = µ = µ = 3 µ = µ = 3 µ = 3 µ = µ = µ = 3 µ = µ = µ = µ = 3 µ = µ = µ = µ = 3 µ = µ = µ = µ = 3 3 µ = µ = µ = µ = 3.337.338.339.34.34.34 One can eadily convince oneself that these esults ae consistent with.336. We have demonstated, theefoe, that any solution Ψx µ tansfomed accoding to.33 is again a solution of the Diac equation, i.e., the Diac equation is invaiant unde active Loentz tansfomations.

.: Diac Paticles in Electomagnetic Field 335. Diac Paticles in Electomagnetic Field We like to povide now a desciption fo paticles govened by the Diac equation which includes the coupling to an electomagnetic field in the minimum coupling desciption. Following the espective pocedue developed fo the Klein-Godon equation in Sect..6 we assume that the field is descibed though the 4-vecto potential A µ and, accodingly, we eplace in the Diac equation the momentum opeato ˆp µ = i µ by i µ qa µ whee q is the chage of the espective paticles see Table. in Sect.6 above. Equivalently, we eplace the opeato µ by µ + iqa µ. The Diac equation. eads then [ iγ µ µ + iqa µ m ] Ψx ν =.343 One may also include the electomagnetic field in the Diac equation given in the Schödinge fom.33 by eplacing i t by see Table. i t qv and ˆ p by ˆ π = ˆ p q A..344 The Diac equation in the Schödinge fom eads then i t Ψx µ = ˆ α ˆ π + qv + ˆβ m Ψx µ.345 whee ˆ α and ˆβ ae defined in.3. Non-Relativistic Limit We want to conside now the Diac equation.345 in the so-called non-elativistic limit in which all enegies ae much smalle than m, e.g., fo the scala field V in.345 holds qv << m..346 Fo this pupose we choose the decomposition Ψx µ = φx µ χx µ..347 Using the notation σ = σ, σ, σ 3 T one obtains then i t φ = σ ˆ π χ + qv φ + mφ.348 i t χ = σ ˆ π φ + qv χ mχ..349 We want to focus on the stationay positive enegy solution. This solution exhibits a timedependence exp[ im + ɛt] whee fo ɛ holds in the non-elativistic limit ɛ << m. Accodingly, we define φx µ = e imt Φx µ.35 χx µ = e imt X x µ.35

336 Relativistic Quantum Mechanics and assume that fo the time-deivative of Φ and X holds t Φ Φ << m, t X X << m..35 Using.35,.35 in.348,.349 yields i t Φ = σ ˆ π X + qv Φ.353 i t X = σ ˆ π Φ + qv X mx..354 The popeties.346,.35 allow one to appoximate.354 and, accodingly, one can eplace X in.353 by σ ˆ π Φ m X..355 to obtain a closed equation fo Φ i t Φ X σ ˆ π m σ ˆ π m Φ.356 Φ + qv Φ..357 Equation.356, due to the m facto, identifies X as the small component of the bi-spino wave function which, hencefoth, does not need to be consideed anymoe. Equation.357 fo Φ can be efomulated by expansion of σ ˆ π. Fo this pupose we employ the identity 5.3, deived in Sect. 5.7, which in the pesent case states Fo the components of ˆ π ˆ π holds ˆ π ˆ π We want to evaluate the latte commutato. One obtains σ ˆ π = ˆ π + i σ ˆ π ˆ π..358 l = ɛ jkl π j π k π k π j = ɛ jkl [π j, π l ]..359 [π j, π k ] = [ i j + qa j, i k + qa k ] = [ i j, i k] } {{ } = Fo an abitay function f holds + q [A j, i k] + q [ i j, A k ] + q [A j, A k ] } {{ } = = q i [A j, k ] + q i [ j, A k ]..36 [A j, k ] + [ j, A k ] f = j A k A k j + A j k k A j f..36

.: Diac Paticles in Electomagnetic Field 337 Using j A k f = j A k f + A k j f k A j f = k A j f + A j k f whee j denotes confinement of the diffeential opeato to within the backets, one obtains [A j, k ] + [ j, A k ] f = [ j A k k A j ] f.36 o, using.36 and A µ = V, A, [π j, π k ] = q i ja k k A j = q i A ɛ jkl = q l i B l ɛ jkl.363 whee we employed B, t = A, t [see 8.6]. Equations.344,.358,.359,.363 allow us to wite.357 in the final fom [ [ˆ p qa, i t Φ, t ] t] q m m σ B, t + q V, t Φ, t.364 which is efeed to as the Pauli equation. Compaision of.364 govening a two-dimensional wave function Φ C with the coesponding non-elativistic Schödinge equation. govening a one-dimensional wave function ψ C, eveals a stunning featue: the Pauli equation does justice to its two-dimensional chaacte; while ageeing in all othe espects with the non-elativistic Schödinge equation. it intoduces the exta tem q σ B Φ which descibes the well-known inteaction of a spin- paticle with a magnetic field B. In othe wods, the spin- which emeged in the Loentz-invaiant theoy as an algebaic necessity, does not leave the theoy again when one takes the non-elativistic limit, but athe emains as a steady guest of non-elativistic physics with the pope inteaction tem. Let us conside biefly the consequences of the inteaction of a spin- with the magnetic field. Fo this pupose we disegad the spatial degees of feedom and assume the Schödinge equation The fomal solution of this equation is i t Φt = q σ B Φt..365 Φt = e iqt B σ Φ..366 Compaision of this expession with 5., 5.3 shows that the popagato in.366 can be intepeted as a otation aound the field B by an angle qtb, i.e., the inteaction q σ B induces a pecession of the spin- aound the magnetic field. Diac Paticle in Coulomb Field - Spectum We want to descibe now the spectum of a elativistic electon q = e in the Coulomb field of a nucleus with chage Ze. The espective bispino wave function Ψx µ C 4 is descibed as the stationay solution of the Diac equation.343 fo the vecto potential A µ = Ze,,,..367

338 Relativistic Quantum Mechanics Fo the pupose of the solution we assume the chial epesentation, i.e, we solve [ i γ µ µ + iqa µ m ] Ψx µ =.368 whee Ψx µ and γ µ ae defined in.8 and in.9, espectively. Employing π µ as defined in Table. one can wite.368 γ µ π µ m Ψx µ =..369 Fo ou solution we will adopt pesently a stategy which follows closely that fo the spectum of pionic atoms in Sect..6. Fo this pupose we squae the Diac equation, multiplying.369 fom the left by γ ν π ν + m. This yields [ i γ µ µ + iqa µ + m ] [ i γ µ µ + iqa µ m ] Ψx µ = γ µˆπ µ γ ν ˆπ ν m Ψx µ =..37 Any solution of.368 is also a solution of.37, but the convese is not necessaily tue. Howeve, once a solution Ψx µ of.37 is obtained then is a solution of.369. This follows fom accoding to which follows fom.37 [ i γ µ µ + iqa µ + m ] Ψx µ.37 [ i γ µ µ + iqa µ + m ] [ i γ µ µ + iqa µ m ] = [ i γ µ µ + iqa µ m ] [ i γ µ µ + iqa µ + m ].37 [ i γ µ µ + iqa µ m ] [ i γ µ µ + iqa µ + m ] Ψx µ =.373 such that we can conclude that.37, indeed, is a solution of.369. Equation.37 esembles closely the Klein-Godon equation.8, but diffes fom it in an essential way. The diffeence aises fom the tem γ µˆπ µ γ ν ˆπ ν in.37 fo which holds γ µˆπ µ γ ν ˆπ ν = 3 γ µ ˆπ µ + µ= µ,ν= µ ν γ µ γ ν ˆπ µˆπ ν..374 The fist tem on the.h.s. can be ewitten using, accoding to.3, γ = and γ j =, j =,, 3, 3 γ µ ˆπ µ = ˆπ ˆ π..375 µ= Following the algeba that connected Eqs. 5.3, 5.3 in Sect. 5.7 one can wite the second tem in.374, noting fom.3 γ µ γ ν = γ ν γ µ, µ ν and alteing dummy summation

.: Diac Paticles in Electomagnetic Field 339 indices, µ,ν= µ ν = 4 = 4 γ µ γ ν ˆπ µˆπ ν = µ,ν= µ ν µ,ν= µ ν µ,ν= µ ν γ µ γ ν ˆπ µˆπ ν + γ ν γ µˆπ ν ˆπ µ γ µ γ ν ˆπ µˆπ ν γ ν γ µˆπ µˆπ ν + γ ν γ µˆπ ν ˆπ µ γ µ γ ν ˆπ ν ˆπ µ [ γ µ, γ ν ] [ˆπ µ, ˆπ ν ].376 This expession can be simplified due to the special fom.367 of A µ, i.e., due to A =. Since which follows eadily fom the definition.344, it holds 4 µ,ν= µ ν = 4 = [ˆπ µ, ˆπ ν ] = fo µ, ν =,, 3.377 [ γ µ, γ ν ] [ˆπ µ, ˆπ ν ] 3 [ γ, γ j ] [ˆπ, ˆπ j ] + 4 j= 3 [ γ j, γ ] [ˆπ j, ˆπ ] j= 3 [ γ, γ j ] [ˆπ, ˆπ j ]..378 j= Accoding to the definition.9, the commutatos [ γ, γ j ] ae [ γ, γ j σ j σ j ] = σ j σ j σ j = σ j.379 The commutatos [ˆπ, ˆπ j ] in.378 can be evaluated using.367 and the definition.344 [ˆπ, ˆπ j ] = i t + qa, i j ] = [ t + iqa j j t + iqa ] f = i j qa f.38 whee f = f, t is a suitable test function and whee denotes the ange to which the deivative is limited. Altogethe, one can summaize.376.38 γ µ γ ν σ ˆπ µˆπ ν = i qa σ.38 µ,ν= µ ν Accoding to.367 holds qa = ˆ Ze..38

34 Relativistic Quantum Mechanics whee ˆ = / is a unit vecto. Combining this esult with.38,.374,.375 the squaed Diac equation.368 eads [ ] t i Ze + σ ˆ Ze + i σ ˆ m Ψx µ =.383 We seek stationay solutions of this equation. Such solutions ae of the fom Ψx µ = Φ e iɛt..384 ɛ can be intepeted as the enegy of the stationay state and, hence, it is this quantity that we want to detemine. Insetion of.384 into.383 yields the puely spatial fou-dimensional diffeential equation [ ] ɛ + Ze + σ ˆ Ze + i σ ˆ m Φ =..385 We split the wave function into two spin- components Ψ = φ+ φ.386 and obtain fo the sepaate components φ ± [ ] ɛ + Ze + ± i σ ˆ Ze m φ ± =..387 The expession.89 fo the Laplacian and expansion of the tem esult in the twodimensional equation [ ˆL ] Z e 4 i σ ˆ Ze + Ze ɛ + ɛ m φ ± =..388 Except fo the tem i σ ˆ this equation is identical to that posed by the one-dimensional Klein- Godon equation fo pionic atoms.9 solved in Sect..6. In the latte case, a solution of the fom Y lm ˆ can be obtained. The tem i σ ˆ, howeve, is genuinely two-dimensional and, in fact, couples the obital angula momentum of the electon to its spin-. Accodingly, we expess the solution of.388 in tems of states intoduced in Sect. 6.5 which descibe the coupling of obital angula momentum and spin { Y jm j, ˆ, Y jmj +, ˆ, j =, 3... ; m = j, j +,... + j }.389 Accoding to the esults in Sect. 6.5 the opeato i σ ˆ is block-diagonal in this basis such that only the states fo identical j, m values ae coupled, i.e., only the two states {Y jm j, ˆ, Y jmj + ˆ} as given in 6.47, 6.48. We note that these states ae also eigenstates of the angula,

.: Diac Paticles in Electomagnetic Field 34 momentum opeato ˆL [cf. 6.5]. We select, theefoe, a specific pai of total spin-obital angula momentum quantum numbes j, m and expand Using φ ± = h ± Y jm j, ˆ + g ± Y jm j +, ˆ.39 σ ˆ Y jm j ±, ˆ = Y jmj, ˆ.39 deived in Sect. 6.5 [c.f. 6.86], popety 6.5, which states that the states Y jm j ±, ˆ ae eigenfunctions of ˆL, togethe with the othonomality of these two states leads to the coupled diffeential equation [ + Ze ɛ + ɛ m j j + Z e 4 ±ize ±i Ze j + j + 3 Z e 4 ] h± g ± =..39 We seek to bing.39 into diagonal fom. Any similaity tansfomation leaves the fist tem in.39, involving the unit matix, unalteed. Howeve, such tansfomation can be chosen as to diagonalize the second tem. Since, in the pesent teatment, we want to detemine solely the spectum, not the wave functions, we equie only the eigenvalues of the matices j B ± = j + Z e 4 ±ize ±i Ze j + j + 3,.393 Z e 4 but do not explicitly conside futhe the wavefunctions. Obviously, the eigenvalues ae independent of m. The two eigenvalues of both matices ae identical and can be witten in the fom λ j [λ j + ] and λ j [λ j + ].394 whee λ j = λ j = j + Z e 4.395 j + Z e 4.396 Equation.39 eads then in the diagonal epesentation λ,j[λ, j + + ɛze + ɛ m f, =.397 This equation is identical to the Klein-Godon equation fo pionic atoms witten in the fom., except fo the slight diffeence in the expession of λ, j as given by.395,.396 and.99, namely, the missing additive tem, the values of the agument of λ,j being j =, 3,... athe than l =,,... as in the case of pionic atoms, and except fo the fact that we have two sets of values fo λ, j, namely, λ j and λ j.. We can, hence, conclude that the

34 Relativistic Quantum Mechanics spectum of.397 is again given by eq..3, albeit with some modifications. Using.395,.396 we obtain, accodingly, ɛ = ɛ = m ;.398 Z + e 4 n + + j+ Z e 4 m ;.399 Z + e 4 n + j+ Z e 4 n =,,,..., j =, 3,..., m = j, j +,..., j whee ɛ coesponds to λ j as given in.395 and ɛ coesponds to λ j as given in.396. Fo a given value of n the enegies ɛ and ɛ fo identical j-values coespond to mixtues of states with obital angula momentum l = j and l = j +. The magnitude of the elativistic effect is detemined by Z e 4. Expanding the enegies in tems of this paamete allows one to identify the elationship between the enegies ɛ and ɛ and the non-elativistic spectum. One obtains in case of.398,.399 ɛ m ɛ m mz e 4 n + j + 3 + OZ4 e 8.4 mz e 4 n + j + + OZ4 e 8.4 n =,,,..., j =, 3,..., m = j, j +,..., j. These expessions can be equated with the non-elativistic spectum. Obviously, the second tem on the.h.s. of these equations descibe the binding enegy. In case of non-elativistic hydogen-type atoms, including spin-, the stationay states have binding enegies E = mz e 4 n, n =,,... l =,,..., n m = l, l +,..., l m s = ±..4 In this expession n is the so-called main quantum numbe. It is given by n = n + l + whee l is the obital angula momentum quantum numbe and n =,,... counts the nodes of the wave function. One can equate.4 with.4 and.4 if one attibutes to the espective states the angula momentum quantum numbes l = j + and l = j. One may also state this in the following way:.4 coesponds to a non-elativistic state with quantum numbes n, l and spin-obital angula momentum j = l ;.4 coesponds to a non-elativistic state with quantum numbes n, l and spin-obital angula momentum j = l +. These consideations ae summaized in the following equations E D n, l, j = l, m = m ;.43 Z + e 4 n l + l+ Z e 4 E D n, l, j = l +, m = m ;.44 Z + e 4 n l + l Z e 4 n =,,... ; l =,,..., n ; m = j, j +,..., j

.: Diac Paticles in Electomagnetic Field 343 main obital spin- non-el. el. spect. quantum angula obital binding binding notation numbe mom. ang. mom. enegy / ev enegy / ev n l j Eq..4 Eq..45 s s p p 3 3s 3p 3p 3 3d 3 3d 5-3.6583-3.66-3.446-3.45 3-3.447 3 -.576 -.5578 3 3 3 -.5577 3 3 5 3 -.5576 Table.: Binding enegies fo the hydogen Z = atom. Degeneacies ae denoted by. The enegies wee evaluated with m = 5.4 kev and e = /37.36 by means of Eqs..4,.45. One can combine the expessions.43,.44 finally into the single fomula E D n, l, j, m = m Z + e 4 n j j+ + n =,,... l =, {,..., n fo l = j = l ± othewise m = j, j +,..., j.45 In ode to demonstate elativistic effects in the spectum of the hydogen atom we compae in Table. the non-elativistic [cf..4] and the elativistic [cf..45] spectum of the hydogen atom. The table enties demonstate that the enegies as given by the expession.45 in tems of the non-elativistic quantum numbes n, l elate closely to the coesponding nonelativistic states, in fact, the non-elativistic and elativistic enegies ae hadly discenible. The eason is that the mean kinetic enegy of the electon in the hydogen atom, is in the ange of ev, i.e., much less than the est mass of the electon 5 kev. Howeve, in case of heavie nuclei the kinetic enegy of bound electons in the gound state scales with the nuclea chage Z like Z such that in case Z = the kinetic enegy is of the ode of the est mass and elativistic effects become impotant. This is clealy demonstated by the compaision of non-elativistic and elativistic specta of a hydogen-type atom with Z = in Table.3.

344 Relativistic Quantum Mechanics main obital spin- non-el. el. spect. quantum angula obital binding binding notation numbe mom. ang. mom. enegy / kev enegy / kev n l j Eq..4 Eq..45 s s p p 3 3s 3p 3p 3 3d 3 3d 5-36. -6.6-34. -4. 3-35. 3-5. -7.9 3 3 3-5.8 3 3 5 3-5.3 Table.3: Binding enegies fo the hydogen-type Z = atom. Degeneacies ae denoted by. The enegies wee evaluated with m = 5.4 kev and e = /37.36 by means of Eqs..4,.45. Of paticula inteest is the effect of spin-obit coupling which emoves, fo example, the nonelativistic degeneacy fo the six p states of the hydogen atom: in the pesent, i.e., elativistic, case these six states ae split into enegetically diffeent p and p 3 states. The p states with j = involve two degeneate states coesponding to Y m, ˆ fo m = ±, the p 3 states with j = 3 involve fou degeneate states coesponding to Y 3 m, ˆ fo m = ±, ± 3. In ode to investigate futhe the deviation between elativistic and non-elativistic specta of hydogen-type atoms we expand the expession.45 to ode OZ 4 e 8. Intoducing α = Z e 4 and β = j +.45 eads +.46 α n β + β α The expansion.6 povides in the pesent case E D n, l, j, m m mz e 4 n mz4 e 8 n 3 [ j + ] 3 + OZ 6 e..47 4n This expession allows one, fo example, to estimate the diffeence between the enegies of the states p 3 and p cf. Tables.,.3. It holds fo n = and j = 3, E p 3 E p mz4 e 8 3 [ ] = mz4 e 8 3..48

.: Diac Paticles in Electomagnetic Field 345 Radial Diac Equation We want to detemine now the wave functions fo the stationay states of a Diac paticle in a 4-vecto potential A µ = V,,,.49 whee V is spheically symmetic. An example fo such potential is the Coulomb potential V = Ze / consideed futhe below. We assume fo the wave function the stationay state fom Ψx µ = e iɛt Φ,.4 X whee Φ, X C descibe the spatial and spin- degees of feedom, but ae time-independent. The Diac equation eads then, accoding to.3,.345, σ ˆ p X + m Φ + V Φ = ɛ Φ.4 σ ˆ p Φ mx + V X = ɛ X.4 In this equation a coupling between the wave functions Φ and X aises due to the tem σ ˆ p. This tem has been discussed in detail in Sect. 6.5 [see, in paticula, pp. 68]: the tem is a scala ank zeo tenso in the space of the spin-angula momentum states Y jm j ±, ˆ intoduced in Sect. 6.5, i.e., the tem is block-diagonal in the space spanned by the states Y jm j ±, ˆ and does not couple states with diffeent j, m-values; σ ˆ p has odd paity and it holds [c.f. 6.97, 6.98] σ ˆ p f Y jm j +, ˆ = i [ + j + 3 σ ˆ p g Y jm j, ˆ = i [ + These equations can be bought into a moe symmetic fom using ] f Y jm j, ˆ j ] g Y jm j +, ˆ..43.44 + = which allows one to wite.43,.44 σ ˆ p f Y jm j +, ˆ = i [ + j + σ ˆ p g Y jm j, ˆ = i [ j + ] f Y jm j, ˆ ] g Y jm j +, ˆ..45.46

346 Relativistic Quantum Mechanics The diffeential equations.4,.4 ae fou-dimensional with -dependent wave functions. The aguments above allow one to eliminate the angula dependence by expanding Φ and X in tems of Y jm j +, ˆ and Y jmj, ˆ, i.e., Φ a Y jm j + =, b ˆ + Y jm j, ˆ X c Y jm j +, d..47 ˆ + Y jm j, ˆ In geneal, such expansion must include states with all possible j, m values. Pesently, we conside the case that only states fo one specific j, m pai contibute. Inseting.47 into.4,.4, using.45,.46, the othonomality popety 6.57, and multiplying by esults in the following two independent pais of coupled diffeential equations [ ] i i j + [ + j + ] d + [ m + V ɛ ] a = a + [ m + V ɛ ] d =.48 and i i [ + j + [ j + ] ] c + [ m + V ɛ ] b = b + [ m + V ɛ ] c =..49 Obviously, only a, d ae coupled and b, c ae coupled. Accodingly, thee exist two independent solutions.47 of the fom Φ i f Y jm j + = X g Y jm j.4 Φ i f Y jm j = X g Y jm j +.4 whee the factos i and have been intoduced fo convenience. Accoding to.48 holds fo f, g [ ] [ j + + j + ] g + [ ɛ m V ] f = f [ ɛ + m V ] g =.4

.: Diac Paticles in Electomagnetic Field 347 and fo f, g [ [ + j + j + ] g + [ ɛ m V ] f = ] f [ ɛ + m V ] g =.43 Equations.4 and.43 ae identical, except fo the opposite sign of the tem j + ; the equations detemine, togethe with the appopiate bounday conditions at = and, the adial wave functions fo Diac paticles in the potential.49. Diac Paticle in Coulomb Field - Wave Functions We want to detemine now the wave functions of the stationay states of hydogen-type atoms which coespond to the enegy levels.45. We assume the 4-vecto potential of pue Coulomb type.367 which is spheically symmetic such that equations.4,.43 apply fo V = Ze /. Equation.4 detemines solutions of the fom.4. In the non-elativistic limit, Φ in.4 is the lage component and X is the small component. Hence,.4 coesponds to states Ψx µ i f Y jm j +, ˆ,.44 i.e., to states with angula momentum l = j +. Accoding to the discussion of the spectum.45 of the elativistic hydogen atom the coesponding states have quantum numbes n =,,..., l =,,..., n. Hence,.4 descibes the states p, 3p, 3d 3, etc. Similaly,.43, detemining wave functions of the type.4, i.e., in the non-elativistic limit wave functions Ψx µ i f Y jm j, ˆ,.45 coves states with angula momentum l = j and, coespodingly the states s etc., s, p 3, 3s We conside fist the solution of.4. The solution of.4 follows in this case fom the same pocedue as that adopted fo the adial wave function of the non-elativistic hydogen-type atom. Accoding to this pocedue, one demonstates fist that the wave function at behaves as γ fo some suitable γ, one demonstates then that the wave functions fo behaves as exp µ fo some suitable µ, and obtains finally a polynomial function p such that γ exp µp solves.4; enfocing the polynomial to be of finite ode leads to discete eigenvalues ɛ, namely, the ones given in.45., 3p 3, 3d 5,

348 Relativistic Quantum Mechanics Behaviou at We conside fist the behaviou of the solutions f and g of.4 nea =. We note that.4, fo small, can be witten [ ] [ j + + j + ] g + Ze f = f Ze g =..46 Setting f a γ, g b γ.47 yields γ b γ j + b γ + Ze a γ = γ a γ + j + a γ Ze b γ =..48 o γ + j + Ze Ze γ j + a b =..49 This equation poses an eigenvalue poblem eigenvalue γ fo pope γ values. One obtains γ = ± j + Z e 4. The assumed -dependence in.47 makes only the positive solution possible. We have, hence, detemined that the solutions f and g, fo small, assume the -dependence in.47 with γ = j + Z e 4..43 Note that the exponent in.47, in case j + < Ze, becomes imaginay. Such -dependence would make the expectation value of the potential d ρ.43 infinite since, accoding to.66,.67,.4, fo the paticle density holds then ρ γ =..43 Behaviou at Fo vey lage values.4 becomes g = ɛ m, f f = ɛ + m g.433

.: Diac Paticles in Electomagnetic Field 349 Iteating this equation once yields g = m ɛ g f = m ɛ f.434 The solutions of these equations ae f, g exp± m ɛ. Only the exponentially decaying solution is admissable and, hence, we conclude f e µ, g e µ, µ = m ɛ.435 Fo bound states holds ɛ < m and, hence, µ is eal. Let us conside then fo the solution of.434 Insetion into.434 esults in which is obviously coect. f = m + ɛ a e µ, g = m ɛ a e µ..436 m ɛ m + ɛ a m ɛ m + ɛ a = m + ɛ m ɛ a m + ɛ m ɛ a =.437 Solution of the Radial Diac Equation fo a Coulomb Potential To solve.4 fo the Coulomb potential V = Ze / We assume a fom fo the solution which is adopted to the asymptotic solution.436. Accodingly, we set f = m + ɛ e µ f.438 whee µ is given in.435. Equation.4 leads to g = m ɛ e µ g.439 m ɛ [ j + ] g + m + ɛ Ze f + m ɛ m + ɛ g m ɛ m + ɛ f =.44 m + ɛ [ + j + ] f + m ɛ Ze g m + ɛ m ɛ f + m + ɛ m ɛ g =.44 The last two tems on the l.h.s. of both.44 and.44 coespond to.437 whee they cancelled in case f = g = a. In the pesent case the functions f and g cannot be chosen identical due to the tems in the diffeential equations contibuting fo finite. Howeve, without loss of geneality we can choose f = φ + φ, g = φ φ.44 which leads to a patial cancellation of the asymptotically dominant tems. We also intoduce the new vaiable ρ = µ..443

35 Relativistic Quantum Mechanics Fom this esults afte a little algeba [ ρ j + m + ɛ ρ ] φ Ze φ m ɛ ρ φ + φ + φ =.444 [ ρ + j + m ɛ ρ ] φ Ze + φ + m + ɛ ρ φ φ φ =..445 Addition and subtaction of these equations leads finally to the following two coupled diffeential equations fo φ and φ ρ φ + j + ρ ρ φ + j + ρ φ + φ ɛze m ɛ ρ φ mze m ɛ ρ φ + We seek solutions of.446,.447 of the fom s n φ ρ = ρ γ mze m ɛ ρ φ =.446 ɛze m ɛ ρ φ φ =.447 s= n φ ρ = ρ γ s= α s ρ.448 β s ρ.449 fo γ given in.43 which confom to the pope behaviou detemined above [c.f..46.43]. Inseting.448,.449 into.446,.447 leads to [ s + γ α s + j + β ɛze s m ɛ α s s ] mze m ɛ β s ρ s+γ =.45 [s + γ β + j + α s + mze m ɛ α s + ɛze m ɛ β s β s ] =.45 Fom.45 follows Fom.45 follows β s = α s β s = s + γ + mze m ɛ j + s + γ ɛze β s + m ɛ..45 ɛze m ɛ m Z e 4 j + m ɛ s + γ ɛze m ɛ β s = s + γ + Z e 4 j + β s..453 s + γ ɛze m ɛ

.: Diac Paticles in Electomagnetic Field 35 Using.43 one can wite this β s = s + γ ɛze m ɛ s s + γ β s..454 Defining one obtains s o = ɛze m ɛ γ.455 β s = =. = s s o ss + γ β s s s o s s o s s s + γs + γ β s s o s o... s s o s! γ + γ + γ + s β.456 Fom.45 follows α s = j + mze m ɛ s o s o s o... s s o s o s! γ + γ + γ + s β.457 One can elate the polynomials φ ρ and φ ρ defined though.448,.449 and.456,.457 with the confluent hypegeometic functions F a, c; x = + a c x + aa + cc + x! +....458 o, equivalently, with the associated Laguee polynomials It holds L α n = F n, α +, x..459 j + φ ρ = β mze m ɛ ρ γ F s o, γ + ; ρ.46 s o φ ρ = β ρ γ F s o, γ + ; ρ..46 In ode that the wave functions emain nomalizable the powe seies.448,.449 must be of finite ode. This equies that all coefficients α s and β s must vanish fo s n fo some n N. The expessions.456 and.457 fo β s and α s imply that s o must then be an intege, i.e., s o = n. Accoding to the definitions.43,.455 this confinement of s o implies discete values fo ɛ, namely, ɛn m =, n =,,,....46 Z + e 4 n + j+ Z e 4

35 Relativistic Quantum Mechanics This expession agees with the spectum of elativistic hydogen-type atoms deived above and given by.45. Compaision with.45 allows one to identify n = n j + which, in fact, is an intege. Fo example, fo the states p, 3p, 3d 3 holds n =,,. We can, hence, conclude that the polynomials in.46 fo ɛ values given by.45 and the ensuing s o values.455 ae finite. Altogethe we have detemined the stationay states of the type.4 with adial wave functions f, g detemined by.438,.439,.44, and.46,.46. The coefficients β in.46,.46 ae to be chosen to satisfy a nomalization condition and to assign an oveall phase. Due to the fom.4 of the stationay state wave function the density ρx µ of the states unde consideation, given by expession.67, is time-independent. The nomalization integal is then d π π sin θdθ dφ Φ + X =.463 whee Φ and X, as in.4, ae two-dimensional vectos detemined though the explicit fom of the spin-obital angula momentum states Y jm j ±, ˆ in 6.47, 6.48. The othonomality popeties 6.57, 6.58 of the latte states absob the angula integal in.463 and yield [note the / facto in.4] d f + g =.464 The evaluation of the integals, which involve the confluent hypegeometic functions in.46,.46, can follow the pocedue adopted fo the wave functions of the non-elativistic hydogen atom and will not be caied out hee. The wave functions.4 coespond to non-elativistic states with obital angula momentum l = j +. They ae descibed though quantum numbes n, j, l = j +, m. The complete wave function is given by the following set of fomulas Ψn, j, l = j +, m xµ = e iɛt if Y j,m j +, ˆ G Y j,m j, ˆ.465 F = F κ, G = F + κ, κ = j +.466 whee { [ n F ± κ = N µ γ e µ ] + γm κ F n, γ + ; µ ɛ } ± n F n, γ + ; µ N = µ 3 Γγ + m ɛγγ + n + κ n! 4m n +γm ɛ n +γm ɛ.467.468 This fomula has been adapted fom Relativistic Quantum Mechanics by W. Geine, Spinge, Belin, 99, Sect. 9.6.

.: Diac Paticles in Electomagnetic Field 353 and µ = m ɛm + ɛ γ = j + Z e 4 n = n j m ɛ =..469 + Z e 4 n +γ We want to conside now the stationay states of the type.4 which, in the non-elativistic limit, become Ψx µ e ɛt i f Y jm j, ˆ..47 Obviously, this wavefunction has an obital angula momentum quantum numbe l = j and, accodingly, descibes the complementay set of states s, s, p 3, 3s, 3p 3, 3d 5, etc. not not coveed by the wave functions given by.465.469. The adial wave functions f and g in.4 ae govened by the adial Diac equation.43 which diffes fom the adial Diac equation fo f and g solely by the sign of the tems j + /. One can veify, tacing all steps which lead fom.4 to.469 that the following wave functions esult Ψn, j, l = j, m xµ = e iɛt if Y j,m j, ˆ G Y j,m j +, ˆ.47 F = F κ, G = F + κ, κ = j.47 whee F ± κ ae as given in.467.469. We have, hence, obtained closed expessions fo the wave functions of all the stationay bound states of elativistic hydogen-type atoms.

354 Relativistic Quantum Mechanics