.-; G.

THE HAMILTONIAN OF QUANTUM ELECTRODYNAMICS

Iwo Bialynicki-Birula

Institute for Th~oretical PhysicsPolish Academy of SciencesLotnikow 32/46 02-668 Warsaw. Poland

1. INTRODUCTION

The purpose of these lectures is to establish a bridge betweenthe full. relativistic QED and its nonrelativistic approximation, "used in quantum optics and atomic physics. Usually, one introducesthe nonrelativistic QED by minimally coupling the quantized electro-magnetic field to the quantum-mechanical particle (or particles).Such a procedure, certainly, leads most directly to the final result,but it has two shortcomings. First, it does not directly generalizeto an improved theory, in which the lowest order relativistic correc-ions are included. Second, it does not explain what is the connec-tion between the fundamental theory of relativistic QED and its lesssophisticated counterpart--the nonrelativistic QED.

These two shortcomings will be overcome here with the help ofa clear mathematical procedure, based on a series of unitary trans-formations. I will introduce a systematic perturbation theory inthe parameter v/c. The lowest order approximation will yield thenonrelativistic theory. Higher-order approximations give the rela-tivistic corrections.

Various pieces of my approach, probably even all of them, maybe found scattered in the literature. In these notes they are alljoined into one logical reasoning for the benefit of those whoprefer to learn it from one s0urce.

2. THE ORIGIN OF THE QED HAMILTONIAN

Undoubtedly, the most important and universal physical quantityis the energy. The total energy of the system expressed in terms

41

Proceedings of the NATO Advanced Study Institute onQuantum Electrodynamics and Quantum Optics in Boulder COEdited by A. O. Barut, Plenum Press New York 1984

of canonical variables--the Hamiltonian--generates the time evolutionin classical and in quantum theories. In quantum theories, the solu-tion of the eigenvalue problem for the energy operator gives theenergy spectrum and a very important family of quantum states--thestationary states.

The role of the Hamiltonian as the generator of time transla-tions is embodied in the equation:

. 1F=IVf[F,H] (1)

where F is any operator representing a physical variable, whichdoes not depend explicitly on time. This fundamental equation isobtained directly from its classical counterpart by the standardprescription of canonical quantization--the replacement of thePoisson bracket by the commutator divided by ~~.

The Hamiltonian of a given physical system is not unique. Itsform depends on the choice of canonical variables. In classicaltheories these variables may always be subject to a canonical trans-formation. In quantum theories the role of canonical transformationsis played by unitary transformations. Under a general unitary trans-formation, which may also depend explicitly on time, Eq.(2) trans-forms as follows ( ~ = 1 ):

F = 1[ F, H ] (2)

where

and

(4)

(3)

Time-dependent unitary transformations result in a change ofpicture; the Schrodinger, Heisenberg and Dirac pictures being themost important special cases. Time-independent unitary transforma-tions result merely in a change of representation within the samepicture. The unitary transformations fo the Hamiltonian of the form:

H -+ H = U-1 H U (5)

do not change the energy spectrum, but do change the form of theHamiltonian. The eigenvalue problem with the new Hamiltonian maybe easier to handle. Of course, the form of H, as a function of thenew canonical variables, is the same as the form of H expressed in

42

Iell

e'I

'olutionie solu-.he-the

terms of the old variables. Time-independent unitary transformationswill serve as an important and convenient tool in our study of theHamiltonian in QED.

is La-

The starting point of my derivation of the QED Hamiltonian isthe full relativistic field theory, in which the electrons and thephotons are described by the electron field 0/ and the electromagneticfield, respectively. These fields obey the set of coupled Diracand Maxwell equations ( f!. = 1 = e)

(1) ~ yP a - m ) W = e y~ A w~ ~

a fPV = jVu

(6a)

:h)U isarde

(6b)

where

Its f = a Apv ~ v

a Av ~

(7)

caltrans-

rmationsy trans-rans-

and

j~ =~e[~, y~wJ (8)

(2)

The commutator in the definition of j~ is necessary to ensure theinvariance under charge conjugation. More about charge conjugationwill be said later. My notation is summarized in the Appendix A.

(3)

Since we will be studying the Hamiltonian, not the S matrix,an explicitly relativistic formalism will be of lesser importance.The following form of the field equations, in which the time var-iable is singled out, will be more suited for our considerations,

(5)

-+

~ at w = [ m S - ~ ~.( ~ - ~ e A )] w + e Ao W , (9a)

-+ -+ -+at E = ~ x B - j , (9b)

-+~·E = p . (9c)-+ -+

at B = - ~ x E , (9d)

-+~·B = 0 (ge)

(4)

~e of~ thesforma-samethe form:

then mayn of thesed in

I have replaced here the formula (7) by its equivalent--the homogen-eous Maxwell equations (9d) and (ge)--in order to stress the time-evolutionary character of the field equations.

43

When writing the field equations, I have disregarded all intri-cate problems connected with infinities and renormalization. Theywill be discussed later.

In theories with gauge fields and quantum electrodynamics is,of course, the most distinguished example of such a theory, thenotion of time evolution requires an elaboration. Since the basicmathematical objects in QED, namely the electron field ~, its her-mitian conjugate ~t and the electromagnetic potentials Av are allaffected by gauge transformations, the field equations (9) do notgive a unique solution of the initial value problem. This compli-cation is already present at the classical level of the theory.Given the set of variables ~, ~t, ~, AO and E at time ~, we mayevaluate with the help of Eqs. (9a), (9b) and (9d) the values of~, Band E at ~+ V~, but we can not determine uniquely the valuesof the potentials A and AO. The potentials are defined by BandE only up to gauge transformations:

(10)

Therefore, we must fix the potentials, by imposing a gauge condition,in order to obtain the solution of the initial value problem forEqs. (9).

Following Heisenberg and Pauli, I shall choose the following,very simple gauge condition

A = 0a (11)

which is best suited for the canonical formulation. The potentialssatisfying this condition are nowadays referred to as the potentialsin the temporal gauge. The condition (11) does not fix the vectorpotential uniquely. We are still free to perform gauge transforma-tions with an arbitrary, time-independent function A. However, thefreedom of such time-independent gauge transformations does notaffect the solution of the initial value problem, which now has aunique solution. In order to see explicitly how it works, let mewrite down in the temporal gauge Eqs.(9a), (9b) and the relationbetween the vector potential and the electric field,

1 -+-+d~ ~ = -e- ( m s ,l a·O ) ~ (12a),{.

-+(vxA)-1d~ E = V x (12b)

-+ -+d~ A = - E (12c)

44

intri-rbey

where+ +D=Il-,[e.A (13)

is,easictler-allnotpli-

These are two basic time-evolution equations of our system.What remains to be done is to secure the validity of Gauss's law(9c). This will be done in the next Section.

The evolution equations (12) may be written in the canonicalform (1). To this end ( w,wt) and ( ~,t)are chosen as the pairsof canonical variables obeying the following anti commutation andcommutation relations:

Yofuesnd

t+ + +W (~I) } = 0 o(~ - ~I)S as (14a){ wa(it) ,

[ E. (it) , A, pt ') ] = ,[ 0" 0 (h: - Jt I ).{. j .{.j

(14b)

(10) All the remaining commutators (anticommutators) vanish. The Hamil-tonian is chosen as the energy operator in the temporal gauge, ex-pressed in terms of canonical variables. It is simply equal to thesum of the energy of the electron field and the energy of the electro-magnetic field,

ldition,'or

ring, + * + +H = ~!d3~ [ Wt, ( m S - ,[ «-u ) 1/J ] + ~!d3~ ( E2 + 82) • (15)

(11) Again, the commutator of the electron operators is needed for thecharge-conjugation invariance.

Itialstentials~ctorEorma-r, theItasat meion

It is now a matter of straightforward calculations to check thatthe commutators of W, t and A with the Hamiltonian (15) are equal tothe r.h.s. of Eqs.(12) multiplied by,[.

3. GAUSS'S LAW AND THE COULOMB INTERACTION

The time-evolution equations (12) do not guarantee the validityof Gauss's law, but it does follow from these equations that

(12a) 'at; G(t) = 0

(12b) where

G(it) = Il·E(h:) - p(it)(12c)

(16)

(17)

45

Thus, in classical physics the equation G(~) = 0 will be preservedin time, once it is assumed to hold for the initial conditions. Inquantum physics such a simple implementation of Gauss's law is notpossible; it would be inconsistent with the commutation relations.We can not just set ~'E equal to p, even for one value of Z. Whatcan be done instead is to select a subspace of states, which isannihilated by the operators G(t),

Get) IjI = 0 (18)

There is no clash between this condition and the eigenvalue problemfor the Hamiltonian, because Eq.(16) tells us that all the operatorsG(t) commute with the Hamiltonian, so that we may diagonalize H inthe subspace of "good" states, on which Gauss's law is valid.

+The commutativity of G(~) and H has an interesting interpreta-tion in the context of gauge transformations. The operator G,

(19)

is the generator of gauge transformations of the operators wand A,

(20a)

(20b)

Therefore, the fact that G(t) are constants of motion means also thatthe Hamiltonian+is invariant under the simultaneous gauge transforma-tions of wand A. In turn, the subsidiary condition (18) may now beinterpreted as the gauge-invariance condition of the state vectors.

The subspace of gauge invariant state vectors can be explicitlyfound. To this end, let us introduce the field analog of the Schrod-inger representation for the electromagnetic oper~tors. In thisrepresentation A(t) acts as a multiplication and E(t) acts as a func-tional differentiation,

(21)

++Next,+let us decompose the vector potential A(~) into its longitud-inal AL and +

transverse Ay parts,

A AL +~+ +, ~ x A = 0 ~'Ay = 0 (22)L

46

idInIt

This decomposition carries over to E,-+ -+Er = i%Ar-+ -+

EL = io/oAL(23)

It-+

Gauss's law involves only the longitudinal part of E. In the Schrod-inger representation the subsidiary condition (18) reads:

[18)-+ -+ -+

( V· i%AL - p ) ~[AL'Ar] = 0 (24)

lemtorsIn

The general solution of this equation is

~[AL'~]= e-i¢ ~[~]

ta- where

(19)

ft.,

-+ -1¢ =-fd3~ v·AL~ p

-1In my shorthand notation ~ stands for the standard Green's functionof the Laplacian.

(20a)-+

Having established the dependence of the state vectors on AL'we may easily find the effective Hamiltonian He acting in the sub-space spanned by the vectors ~[AI]. The subscript e stands here forthe Coulomb, because, as we shal see below, in He the Coulomb inter-action between the charges appears explicitly. In order to obtainHe, we must pull through H the operator exp(-i¢). This is accomp-lished with the help of the following formulas:

(20b)

thatorma-rw berrs , ei¢ ~ e-i¢= exp( ie~-1v.A) ~ (27a)

ei¢ ~te-i¢= ~t exprie~-1v.A) . (27b)

'-+' -+ 1e~¢ E e-~¢= E + ~- Vp (27c)

:itly:hrod-

func-

(21) which follow directly from the canonical commutation relations (14).Eqs.(27) lead to the following expression for the Coulomb Hamilton-ian:

tud-

(22)

He = ei¢ H e-i¢ = ~fd3~[~t,( m 8 - i ~.(v - i e ~ )) ~]1 -+ 1 -+ -+

+8iTfd3~ fd3~' p(~) p(Jt.) + ~fd3~ ( q. + B2 ) . (28)1T pt - ft· I

(25)

(26)

47

-+In this formula I have-+dropped the operator E1. in the electromagneticfield energy, because EL gives zero when acting on the state vectorsIJI[IY]'

The operator He is often referred to as the Hamiltonian in theCoulomb gauge. However, it is clear from our derivation, that theoperator Hc has a gauge invariant meaning. It is the Hamiltonianreduced to the subspace of gauge invariant state vectors.

The electromagnetic interactions, as described by the Hamilton-ian He ' consists of the interaction of charges with the electromag-metic field (interaction via the exchange of photons) and of theinstantaneous interaction of charges via the Coulomb forces.

4. PARTICLES, ANTIPARTICLES, AND CHARGE CONJUGATION

Having established the form of the Hamiltoniart, we may, inprinciple, proceed with the solution of the eigenvalue problem. How-ever, this problem is so difficult that no progress can be madewithout a thorough understanding of the physical content of thetheory. The difficulties are greatly enhanced by the fact that thetheory is ill-defined and only a judicious use of the physical in-sight enables one to arrive at meaningful conclusions. Therefore,in addition to the definition of the Hamiltonian in terms of thecanonical field operators, we also need an interpretation of thebasic objects appearing in the Hamiltonian--the field operators ~and ~t describing the charged particles.

I will approach this problem in cautious manner, making one stepat a time, because a complete solution of QED is not known and aimingat ultimate answers may lead us astray.

The use of the field operators (or creation and annihilationoperators) to describe the electron degrees of freedom is necessaryin a relativistic theory. Otherwise, we immediately run into theproblems caused by the negative-energy solutions of the Dirac equa-tion. After all, this equation does not describe a single particle,but a many-particle system. Interactions are accompanied by theappearance of pairs of particles and antiparticles from the vacuumand their disappearance into the vacuum. Only in the nonrelativisticapproximation we can separate the motion of particles from that ofantiparticles. This nonrelativistic dynamics is deeply buried inthe formalism of relativistic QED and unearthing it will requiresome effort. These complications arise, because the nonrelativisticparticles are rather complex, collective excitations of the vacuum.Their electromagnetic interactions are much more involved than thoseof the "original particles", whose creation and annihilation oper-ators enter the initial Hamiltonian.

•I••••

I11

C

I:I

1ill

J1:c

]

t

]

1

tE

19netic,ctors

These remarks bring us to the first important question: Whatare the main characteristics of those "original", or "bare", part-icles? In order to answer this question, I will tkae a closer lookat the various terms in the Hamiltonian He. Let me start with thefirst term--the mass term.

l thethe.an

In the Dirac representation (see the Appendix A), which I amusing here, the S matrix is diagonal,

s = diag (1,1,-1,-1 ) (29)

Llton-romag+the

We expect both particles and antiparticles to give positive contrib-utions to the mass term. This can be achieved by interchanging thecreation and the annihilation operators for the lower components ofthe Dirac bispinors ~ and ~t, which leads to the following decompos-ition of these operators,

inHow- ~ = r ~t 1

L.

~t = ( <Pt , X

deet thein-

are,hehes ~

The conjugation, denoted by t, means here both the hermitian conjug-ation of each component and the transition between the column- androw-arrangement of the components. The sign - denotes the transitionbetween columns and rows (transposition) and an additional mUltipli-cation by the antisymmetric matrix io2,

x = ( Xl Xz )io2 = ( -X2 ' Xl )

me stepl aiming xt• -~cr, l:! 1 · [-:~1.onss sarytheequa-~ticle ,:heacuumtivisticrat ofIi inireivisticacuum.n thoseoper-

The multiplication by i02 is necessary in order to have the sametransformation properties for ~ and X under rotations.

In this new notation the mass term has the form

Hma6~ = m Jd3~ ( ~t~ + xtx ) + const .

The (infinite) constant appearing in this formula results from therearrangement of the creation and annihilation operators, in orderto obtain their normal ordering. It is an analog of the zero-pointenergy of the electromagnetic field.

Now, let me show that our interpretation of ~ in terms of theannihilation operators of particles and the creation operators of

(30)

(3Ia)

(3Ib)

(32)

49

antiparticles is also consistent with our understanding of the lastterm in HC --the energy of the electrostatic interaction betweenthe charge densities. With the use of the formulas (30), we obtainfor the charge density,

5

t

I

of

p = ~ e [ Wt, w] = e ( ~t~ - xtx ) (33) Ass1.:t~fem

Notice that there is no additional constant in this expression, be-cause the two infinite contributions from the rearrangements of thefield operators cancel out. This cancellation is a direct result ofthe charge-conjugation invariance of the theory. I have made surefrom the very beginning that the theory be invariant under chargeconjugation, consisting in the interchange of particles and anti-particles and the sign-reversal of the electromagnetic field: TI

~t ~ x+ (34)0:

b:i,01u:t

The charge density must be, clearly, odd under charge conjugationand, therefore, it can not contain a constant term.

The remaining term of the electron Hamiltonian has no simpleinterpretation. It contains the pair-creation and the pair-annihil-ation terms, which shows that the particles created by ~t and xt arequite different from the electrons and positrons observed in experi-ments. However, since we have gotten the quantum numbers correctly,our tentative interpretation may serve as a sound and consistentstarting point. The rest will, anyway, be determined by the detailedstudy of the Hamiltonian.

H

uPtP

The full Hamiltonian expressed in terms of the operators ~ andX reads

-+ -+ -+-+H = m J ( <PH + x+x ) + ~ J ( E2 + B2 -,{. J ( ~t cr'O xt

+ - -+ -+ ~ ) (35)X 0'0where I have dropped the additive constant. Unmarked integrals will,from now on, denote the d3~ integration over the whole space.

The unitary transformations, which will be introduced in thenext Section, are easier to apply when the Hamiltonian is notrestricted to the subspace of gauge-invariant states, but at theend I will always impose Gauss's law and obtain the physical Hamil-tonian HC'

5. THE MAKING OF THE NONRELATIVISTIC HAMILTONIAN

Let me begin with making a somewhat unorthodox decomposition

50

;t of the Hamiltonian (35) into Ho and HI'In HO = Hma.6-6+ H frteR.d HI = Hpa,Ur. (36)

33) As I have already said before, the full Hamiltonian of QED will besubject to certain unitary transformations. The role of these uni-tary transformations is to bring the Hamiltonian to the diagonalform, which will contain only the terms conserving separately thenumber of particles and the number of antiparticles,

e-heof

'e[ Hdiag , ! ~t~ ] = [ Hdiag , ! xtx ] = 0 (37)

:34)

The terms diagonal and ooo-diagonal refer here to the decompositionof the full Hilbert space into the direct sum of subspaces labelledby the total number of particles and the total number of antipart-icles. Each subspace is an invariant subspace of every diagonaloperator. The HO part of H is already diagonal. The role of theunitary transformations will be to eliminate all the off-diagonalterms in the Hamiltonian.

1

Lei111-are

eri-tly,

Aiming at the description of low-energy phenomena, we may treatHI when compared to the mass term, as a small perturbation, because-iD/m is of the order of vie. We are unable to find the desiredunitary transformation in one step, but the existence of this smallparameter will enable us to effectively perform the elimination ofthe off-diagonal terms of the Hamiltonian in successive orders ofperturbation theory in 11m.

ailed

and The best method to explain this elimination procedure is tosimply show how it works in practice. To this end, I will expandthe unitary transformation of the QED Hamiltonian (35) into thepower series in the (antihermitian) generator F of U,

(35) eF ( HO + HI )e-F = HO + HI + [ F, HO ] + [ F, HI ]will,

+J.z[ F, [ F, HO]]+'" (38)

Ie In perturbation theory this infinite series will be reduced to afinite sum, if F is a polynomial in the expansion parameter.e

1Ili1- The off-diagonal small term HI will be eliminated in the lowestorder of perturbation theory, if we choose F in such a way that

[ F, HO ] + HI = 0 (39)

on

51

:;••.j,Ii

I';..

* .i

The effective method of solving this equation will be based onthe lemma: Lemma. The solution of the commutator equation:

[ X , A ] = B (40)

can be written in the following form:In

X = ~ Jdt sgn(t) e-s!tl iAt B -iAt (41)e

(43a)an

(43b)

will wh

0, fosu1118

(44) fi

where the integral is extended over the whole real axis and the limits + 0 is tacitly assumed.

The (formal) proof of this lemma consists in taking the commu-tator of the operator appearing under the integral with the operatorA, noting that the result may be written as a derivative with res-pect to t, and finally integrating by parts.

The application of the lemma to Eq.(39) gives

(42)

where the dependence of H1(t) on t is generated by the HamiltonianHO. This time evolution may be viewed as a version of the Dirac(interaction) picture. In this picture the electron and positronoperators depend on time as follows:

4>(t) = e-hnt 4>

X (-r) = e-imt: X

4>t (z) = ehnt 4>t

hntXt(t) = e x+

The explicit form of the time evolution of the vector potentialnot be needed. It suffices to know its first derivative at t =

+ + + + +A(t) = A + t d~ + ••. = A - t E +.•. ,

since the higher derivatives in the Taylor expansion contributehigher order terms in 11m. This is seen from the following form-ula:

(45)

The first two terms in the expansion of F in inverse powers of m,calculated from the formula (42), are

52

we

ed on

(40)

(41)

~ limit

JIIIIIIU-

er ato rres-

(42)

lianlC

ron

(43a) •f(43b)

Iwill :

I• 0,

(44)i

rrm- I(45) I

m, JI•,

F - -~ ( + + - - + +1 - 2m I ¢t X-O xt - X a-O ¢ )

F ~e ( + + - - + +2 = 4m21 ¢t a-E xt + X a-E ¢

In order to obtain the transformed Hamiltonian,

- 1H = HO + ~ [ Fl ' Hl ] + O(~)

we need only to calculate the commutator:

[ ] 1 [ + + -Fl, Hl = - m If ¢t a-O xt X d-n ¢ ]

1 + + + +- - I ( cpt (a-OF q, - X (a_O)2 xi")m

1 ++ . ++ 1 ++= - - I (¢t (a-SF ¢ + xt (a-O*)2 x) + - Tr(a-DFm m

where the trace operation refers to both the space variables and thespin indices_ The positron term has been reordered with the help ofthe identity:

a d a = - crT2 2

and by integration by parts_I will write the Hamiltonian (47) in the final, Coulomb form,

which is obtained by performing the same transformation as in theformula (28)_ Note that this procedure is justified, because thesubsidiary condition (18) is not modified by the unitary transfor-mation (38) of the Hamiltonian (F is gauge invariant)_ In thefinal form of the Hamiltonian I will also reinstate e and ~-

He = me2 f k2¢t¢ + xtx ) - 2m I + +1

¢t O~ ¢ + xtOT X )

- 2: I ( <j>t&q,- xtdx ) -B + 8l'Tf !! P (~) pt - Jt I 1-1 p (Jt I )

+ ++ ~ f ( E2 + 82 ) + vac

T

where

(46a)

(46b)

(47)

(48)

(49)

(50)

53

and vac stands for the divergent vacuum energy (the trace term in(48)), which does not depend on the electron and positron operators.

The interpretation of this Hamiltonian will be given in thenext Section.

6. THE MEANING OF THE NONRELATIVISTIC HAMILTONIAN

I will begin this Section with the discussion of two distinctinterpretations of the connection between the original Hamiltonian(25) and the transformed Hamiltonian (50).

According to the first interpretation the Hamiltonian is act-ually changed by the unitary transformation, only its spectrumremains the same. This point of view may be called the active inter-pretation of the unitary transformation of the Hamiltonian. Thosewho would prefer to deal with just one Hamiltonian, may choose thepassive interpretation of the unitary transformation.

The distinction between the active and the passive interpre-tations is well known from geometry. There, we may view the trans-formation of the components, say, of a vector, as being due eitherto the rotation of the vector or to the (inverse) rotation of thecoordinate system.

According to the passive interpretation of the unitary trans-formation of the Hamiltonian, the operator H does not change, butits form changes when one expresses H in terms of a different setof canonical variables. In other words, the canonical variables,not the Hamiltonian, are subject to the unitary transformation.

In order to see this more explicitly, let us go back to thetransformation formula (5) and rewrite it in the form

H = U H U-1

As I have noted before, the new Hamiltonian H and the old Hamil-tonian H have the same form when they are expressed in terms of newand old canonical variables, respectively. Let us regard the unit-ary transformation U in Eq.(52) as a function of the new canonicalvariables. Then, the right hand side of Eq.(52) looks very muchthe same as the unitary change of the Hamiltonian, which we havestudied in the last Section, except that U is replaced by U-1.Therefore, by just assuming that all the field operators are thenew canonical variables, connected by a unitary transformation tothe old ones, and by changing F and -F , we arrive at the same form

54

(51)

(52)

III

I•

••:II

III

•1

'.I•

•

••••••

tl

••e•a1

(51)of the Hamiltonian as in Eq.(50). However, according to this new,passive interpretation, this Hamiltonian is viewed as a new expres-sion (in terms of new canonical variables) of the same old operator.

inat ors . The two interpretations are not only completely equivalent, but

even all the specific calculations are exactly the same in bothapproaches. I have chosen the active interpretation mostly forits typographical simplicity; one does not have to distinguish inprint between the old and new canonical variables.

Ile

inctnf.an

Our starting point required the use of field operators todescribe the charged particles, but the final form of the Hamiltonianmay easily be translated into the quantum-mechanical language. Sincethe new Hamiltonian (50) is diagonal, we may consider its actionseparately in each subspace of a given number of particles and anti-particles. In this subspace we may replace each one-particle oper-ator by its quantum-mechanical counterpart according to the rule:act-

inter-hosethe

! <pt (;;t)0 (;;t)<p(;;t)-..L 0 (;;t.). .{..(.

and each two-particle operator by the corresponding expressionre-rans-therthe

~II <Pt(~l) <Pt(;;t2)O(~1'~2) <P(~2) <P(~l) \' -..-..+ L O(Jt •• Jt.).. .{. Jz> j

ans-butset.Lea ,I.

+ + + -..where O(Jt) and O(Jtl,Jt2) may contain the derivatives, functions of IL,and spin operators. This procedure leads to the familiar Hamiltonianof the nonre1ativistic quantum electrodynamics. For simplicity, Iwill write it down here only in the case when the positions are notpresent.

heH = J... L CP". - s. t )2 - elh \'"& .B (;;t . )

2m . .{.c. I 2mc. '-:- .{..{. .(.

e2 \' + + -1 +2 +2+-4 L !Jt.-Jt.! +~!(ET+B).

1T i>j.{. j(52)

Hami1-If newunit-dca1ichrve

In order to understand fully the connection between this Hamil-tonian and its field-theoretic counterpart (50), we must clarify theproblem of the electromagnetic mass of the electron. In the lowestorder of perturbation theory in 11m the electromagnetic mass is gen-erated only by the Coulomb interaction.,

The translation formula (53) for the two-particle operatorsassumes the normal ordering of the creation and annihilation oper-ators. The Coulomb energy in the expression (50) is not normally

them toform

(53a)

(53b)

(54)

55

ordered and the reordering introduces an additional term--theCoulomb self-energy:

where

is the electrostatic-field energy of a point charge.

This is the first and the simplest example of the electromag-netic contribution to the electron's rest-mass (or rest-energy).Other terms of this type appear in higher order of perturbationtheory in l/m. All these contributions may be absorbed into themass term in the Hamiltonian by a suitable redefinition of the massparameter (mass renormalization). The observed mass of the elect-ron mob~, therefore, must be identified not with the parameter mintroduced in the original Hamiltonian, but with the final value ofthe coefficient of the mass term, which includes all electromagneticcorrections,

m + om

In higher orders of perturbation theory there appear the elect-romagnetic corrections to the mass of a more complicated characteras compared to those given by the formulas (55) and (56). Theyresult from the renormalization of the mass parameter occurring inother parts of the Hamiltonian. For example, in the second orderof perturbation theory in l/m there will appear the nonrelativistickinetic-energy term multiplied by -om/m , which can be absorbed intothe first-order term by the replacement of l/m by l/mob~'

A similar process of renormalization takes place for the elect-ron's charge. Due to vacuum polarization, the parameter e appear-ing in the initial Hamiltonian cannot be identified with theobserved value of the charge eob~. However, the correction termsappear for the first time in the third-order of perturbation in eand are of no concern to us here.

7. EXTERNAL ELECTROMAGNETIC FIELDS

So far I have treated QED as a theory of a closed system, madeonly of the electrons, the positrons, and the electromagnetic field.However, in most situations there are also other parts of the world,which can not be ignored. In many cases, these additional parts

56

(55)

(56)

(57)

j

1

the may be represented by some given configurations of the electromag-netic field. often called the external electromagnetic field. Avery important example of such a larger system. which can be treatedby the external field method, is the atom.

(55)

(56)

Let us first consider the case of a time-independent externalfield. An arbitrary configuration of the electrostatic and magneto-static fields may be generated by an appropriate distribution ofcharges and magnetic moments (or current loops). Let me denote byPext and mext the corresponding charge density and the magnetismdensity. The total Hamiltonian will now have the form

++Htot = Hep + H&Leld - ! B·mext + Hext

ct romag-argy) •at tonto thethe mass~ elect-seer mvalue ofromagnet.Lc

where Hext describes the energy of the external sources of the field;it does not contain the electron and field degrees of freedom. Ofcourse, the additional charges will also modify Gauss's.

+V'·E= P + Pext

Next. let me introduce the scalar potential Vext and the vectorpotential ~ext generated by the external sources of the field.

(57) - D. Vext = Pextthe elect-aar act.erTheyrring inlid orderltivisticorbed into

+ +V'x V'x Aext = V'xmextThe Hamiltonian (58) will be subject to the following unitary trans-formation:

++U = exp(-i ! E·Aext) • (61)+whose purpose is to shift the vector potential operator by Aext,

the elect-appear-

Ile

11 termsLon in e

-1 + + +2H = U ~oM = Hep[ A + Aext ] + Hnield - ~ ! mext + Hext .(62)

I have written here explicitly the dependence of the electron-positron Ham~ltonian Hep on the vector potential. The field Hamil-tonian contains only the quantizedt and n fields. For the staticdistribution of the external sources the last two terms may bedropped. because they only contribute a constant.

tem, madetic field.the world,L parts

Finally, we have to restrict the Hamiltonian (62) to the gauge-invariant subspace, in order to obtain the Coulomb Hamiltonian.With the use of (59) and (60a). we obtain the well known expression:

(58)

(59)

(60a)

(60b)

57

- -+-+He = He.p[ A + Ae.x:t ] + HMe.f.d +! o Ve.x:t

where I have omitted another constant--the energy of the electro-static interaction of the external charges.

Time-dependent external fields may be introduced by assumingthat the external Hamiltonian generates a nontrivial time evolutionof the external charges and magnetic moments, which leads to acertain time variation of Vex:t and Ae.~' This time variation maybe treated as given a prior1, if the oack reaction on the chargesand magnetic moments, generating the external field, may be neg-lected.

8. RELATIVISTIC CORRECTIONS

The method of unitary transformations is easily extended togive the relativistic corrections to the Hamiltonian (50). Thelowest order relativistic corrections appear in the order 11m2 ofperturbation theory. The generator F of the unitary transformationhas already been calculated to the desired order in Section 5. Allwe have to do now is to collect the terms of the order 11m2 in theexpansion (38). It turns out that such terms are produced only bythe commutator [ F2' HI]' This commutator is slightly more compli-cated than (48),-+becau~e it also contains a contribution from thecommutation of E and A,

_ e. -+-+ -+[ F2, HI ] - 4m2 ff ( [ ~t o'E xt , X o·D ~ ] + h.c. )

where h.c. stands for the hermitian conjugate terms. The integralin the second line will be dropped, since it is off-diagonal and,therefore, it does not contribute to the diagonal part of the Hamil-tonian in the order 11m2• The same is also true about the doublecommutator [ FI, [Fl, HI]]' All such off-diagonal terms occurringin a given order of perturbation theory, in principle, should beeliminated by an appropriate unitary transformation, but after sucha transformation we will obtain a contribution to the diagonal partat best in the next order. With the use of canonical commutationrelations, the first integral in (64) may be rearranged as follows:

= -

-+ e.2 -+-+ -+-+f ~taxt . xo~ - 4m2 Tr([a'D , a·E])

(63)

(64)

(65)

(63)

:0-

lngltionl

may~es,-,

toheof

ationAll

they byompli-the

(64)

gralnd ,Hamil-bleurringbesuch

1 parttionlows:

x )

(65)

The first commutator on the right hand side gives:+7: ++ + -+ -+ -+ -+ -+-+

[ cr·U • cr·E ] = v-E + '<'(0 x E)·cr - '<'(E x O)·cr - 3e. 0(0) • (66)

The analogous result for the second commutator is obtained by re-placing e. by -e..

Skipping some intermediate steps. I will write down the finalexpression for the Hamiltonian in the Coulomb form. To simplifythe notation, I will be rather sloppy with the divergent expression--disregarding them all--and, consistently, I shall also identify mwith the observed mass.

H = mc.2/n.2 + 7: +

1 ( ~t~ + xtx ) - 2m 1 ( ~t (cr.UT)2~ + xt (cr·Or)2X )

1 + -+ + -1 -+ -+ ++87T f!p(lL) !IL-ILI! p(ILI)+~/(q+B2

Le"-2 * -+ -+ -+ -+ + -+ -+- 8m2c. 21 (~t (UTxE-ExDT)·t~-xt (OrxE-ExDr)·cr x

Q_2h2 ++ ~4 1 ~tcrxtm c.

+ "-2 -+-+Xcr~ - 8m2c.2 1 p(lL) p(lL)

-+where E contains both the transverse part and the longitudinal partdetermined by the application of Gauss's law.

;t ~ 1 -+ -+ -1 +1:.= -V-I !IL-ILI! p(lLl)47T

Following the procedure described in the previous Section, wemay easily introduce the external field into the Hamiltonian (67).

Out of the three terms, which desciribe the relativistic corre-tions. the first term is the well-known spin-orbit coupling. Inthe quantum-mechanical notation it has the form:

e. -+ e.-+ +-+mzc! (( P - e A ) x E )·s

from which we may see that it has a classical origin. In the tworemaining terms Planck's constant cannot be absorbed intop ands. They are of a quantum origin and cannot be obtained by thecanonical quantization of the classical theory of charged particles.

(67)

(68)

(69)

59

Our Hamiltonian is still a QED Hamiltonian in the sense thatthe interaction with photons is fully included. We may performanother unitary transformation, which will eliminate all the termslinear in the photon creation and annihilation operators. Thiselimination will lead to new effective interaction terms describ-ing in the instantaneous approximation the result of one-photonexchange.

9. HISTORICAL NOTES

There is a vast literature on many aspects of QED covered inmy lectures and I will only mention here those contributions, whichhad the greatest influence on the subject, or at least on my view ofthe subject.

The version of the relativistic QED presented here is patternedafter two classic papers of the founding fathers of quantum fieldtheory--Heisenberg and Pauli.1 In the second paper they introducedthe concept of the gauge-invariant subspace, on which Gauss's lawis valid.

My unitary transformations, leading to the nonrelativistictheory, are close relatives of the Foldy-Wouthuysen transformationsin the Dirac theory of the relativistic electron.2 There are twodifferences. The F-W transformations apply to the one-particletheory and they are studied in the presence of the external electro-magnetic field only. The field-theoretic form of these transforma-tions resembles the Bogolubov-Holstein-Primakoff transformations ofthe creation and annihilation operators, known from many-bodytheory. 3

The problem of relativistic corrections to the nonrelativisticHamiltonian has a long history, but early derivations were basedeither on classical theory or on the one-particle Dirac theory. 4The first derivation within the framework of QED was given by Itoh.I believe that my derivation is more systematic and that the prob-lems of gauge invariance are handled correctly.

lW. Heisenberg und W. Pauli, Zeitschrift f. Phys. 56, 1 (1929),59, 168 (1930).

2J•D• Bjorken and S.D. Drell, Relativistic Quantum Mechanics,McGraw-Hill, 1964.

3A.L. Fetter and J.D. Walecka, Quantum Theory of Many-ParticleSystems, McGraw-Hill, 1971.

4T• Itoh, Reviews of Mod. Phys. lI, 159 (1965).

60

thatrmtermsisrib-on

d inwhichlTiew of

tternedieldrducedlaw

lcat LonstwoLeLectro-:orma-ms of

risticredv-Itoh 4prob-

)929) ,

dcs ,

ticle

Appendix A

My relativistic metric tensor is:

g~v = diag ( 1,-1,-1,-1 ) (AI)

The Dirac matrices are all used in the Dirac representation,

s = P3 X I + +a = Pl X a (A2)

yO = S + +y = S a (A3)

I use the Heaviside-Lorentz system of electromagnetic units,in which the factor 1/4n appears in Coulomb's law, but not inMaxwell's equations.

The basic tool in the method of unitary transformations usedin these lectures is the following operator identity:

"AA -"AA [ ]"A 2e. Be. =B+"A A,B +21 [A,[A,B]]+ ••• , (A4)

easily proven by differentiating both sides with respect to "A.

In the evaluation of several commutators, the following oper-ator identities were found useful:

[ AB , C ] = A {B , C} - { C , A } B (AS)

[ AB , CD ] = A { B , C } D - C { D , A } B - ACBD + CADB • (A6)

61