Relativistic Quantum Mechanics


 Roderick Pearson
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1 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 Loentzinvaiant 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 antipaticle. 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 antipaticle. 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 paticlesystems: 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
2 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 spacetime 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 4dimensional 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 spacetime 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
3 .: 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: 4vecto: x µ, x µ ; 4 4 tenso: A µ ν, A µ ν, A µν, A µν..7 The eason fo the notation is twofold. Fist, the notation in.6 allows us to intoduce the socalled 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 nonsingula 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
4 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 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.
5 .: 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=
6 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 =.
7 .: 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 [ 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
8 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 =.57
9 .: 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.
10 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 socalled 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 socalled 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. 4Vectos The quantities with the tansfomation behaviou like that of the position time vecto x µ defined in.3 ae the socalled 4 vectos a µ. These quantites always come as fou components a, a, a, a 3 T and tansfom accoding to Examples of 4vectos beside x µ ae the momentum 4vecto 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 4vecto is the socalled 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 4vecto A µ = V, A.69 whee V, t and A, t ae the electical and the vecto potential of an electomagnetic field. The 4vecto chaacte of J µ and of A µ will be demonstated futhe below.
11 .: Scalas, 4 Vectos and Tensos 97 Scala Poduct then 4vectos allow one to constuct scala quantities. If a µ and b µ ae 4vectos 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 4Vectos It is convenient to define a second class of 4vectos. The espective vectos a µ ae associated with the 4vectos 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 4vectos a µ covaiant and 4vectos a µ contavaiant. Covaiant 4vectos 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 4vecto 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 4Vecto µ An impotant example of a covaiant 4vecto 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 4vecto 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
12 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 Loentzinvaiant. Poof that p µ is a 4vecto We will demonstate now that the momentum 4vecto 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
13 .3: Relativistic Electodynamics 99 The opeato m d dτ tansfoms like a scala. Since xµ tansfoms like a contavaiant 4vecto, the.h.s. of.89 alltogethe tansfoms like a contavaiant 4vecto, and, hence, p µ on the l.h.s. of.89 must be a 4vecto. The momentum 4vecto 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 nonelativistic 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 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 Loentzinvaiant 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 socalled Coulomb gauge 8., i.e., A =, fo these potentials. This gauge is not Loentzinvaiant and we will adopt hee anothe gauge, namely, t V, t + A, t =..96 The Loentzinvaiance of this gauge, the socalled Loentz gauge, can be demonstated eadily using the 4vecto notation.69 fo the electodynamic potential and the 4vecto deivative.77 which allow one to expess.96 in the fom µ A µ =..97 We have poven aleady that µ is a contavaiant 4vecto. If we can show that A µ defined in.69 is, in fact, a contavaiant 4vecto then the l.h.s. in.97 and, equivalently, in.96 is a scala and, hence, Loentzinvaiant. We will demonstate now the 4vecto popety of A µ.
14 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 4vecto, it follows that J µ, in fact, has to tansfom like a contavaiant 4vecto. We want to deive now the diffeential equations which detemine the 4potential A µ in the Loentz gauge.97 and, theeby, pove that A µ is, in fact, a 4vecto. The espective equation fo A = V can be obtained fom Eq 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 4vecto. 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 4vecto. The Field Tenso The electic and magnetic fields can be collected into an antisymmetic 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 antisymmetic theedimensional 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
15 .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 LoentzInvaiant Fom One can expess the Maxwell equations in tems of the tenso F µν in Loentzinvaiant 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.
16 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 4momentum 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 xcomponent 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.
17 .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 4vectoso othe nonscala entities, e.g., the chagecuent density in case of Sect..3 o the bispino wave function of electonpositon 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 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
18 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
19 .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 ] OneDimensional Function Space Repesentation = x 3 + x 3 = K The exponential opeato.5 in the case of a onedimensional 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 ]. 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
20 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 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 4vecto 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, complexvalued function. The latte popety implies that upon change of efeence fame ψx µ tansfoms accoding to.,.. The patial diffeential equation.5 is called the KleinGodon equation. In the following we will employ socalled 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
21 .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 4Vecto Associated with the KleinGodon Equation As is wellknown 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
22 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 KleinGodon 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 KleinGodon 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 KleinGodon 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 antipaticles; the antipaticles 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
23 .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 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 4vecto potential A µ x µ = V, t, A, t ; A µ x µ = V, t, A, t.77
24 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 4vecto 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 4vecto potential A µ = V, A o A µ = V, A. Accoding to the socalled 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 KleinGodon 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 spacetime deivatives this eads [ i t qv, t ψ, t = i qa, ] t + m ψ, t..8 NonRelativistic 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 nonelativistic limit. Fo this pupose we splitoff 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 nonelativistic limit, that the mass m of the paticle, i.e., it s est enegy, is much lage than all othe enegy tems, in
25 .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 KleinGodon 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 nonelativistic 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 socalled 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
26 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 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 wellknown 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 The suitable choice is λl = + l + Z e 4.99
27 .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 =,, 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 nonelativistic hydogenlike 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 α n β + β α α n α β + Oα α n α β n + Oα
28 34 Relativistic Quantum Mechanics + α + α + Oα n βn 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 nonelativistic 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 Loentzinvaiant 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 positivedefinite 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 positivedefinite 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 KleinGodon Equation An obvious stating point fo a Loentzinvaiant 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
29 .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 4dimensional epesentation of the coefficients γ µ. Initially, it was assumed that the 4dimensional 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 4dimensionsional epesentation discoveed by Diac involved new physical popeties, spin and antipaticles. 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 positivedefinite 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 leftmost and the ightmost 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 4vecto 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
30 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 antihemitian..5 γ µ γ ν = γ ν γ µ,.6 i.e., the γ µ ae anticommuting. 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 anticommuting 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 4dimensional 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
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