Notations and Definitions

Appendix A Notations and Deﬁnitions

A.1 Four-Component Vector Notations

A four-dimensional contravariant vector is deﬁned as1

x D x D .x0;x1;x2;x3/ D .x0; x/ D .ct; x/; (A.1) where D0; 1; 2; 3 and x is the three-dimensional coordinate vector xD.x1;x2;x3/ .x; y; z/. The four-dimensional differential is

d 4x D cdt d3x and d3x D dx dy dz:

A corresponding covariant vector is deﬁned as

x D .x0;x1;x2;x3/ D gx D .x0; x/ D .ct; x/; (A.2) which implies that

0 i x D x0 x Dxi .i D 1; 2; 3/; (A.3) g is a metric tensor, which can raise the so-called Lorentz indices of the vector. Similarly, an analogous tensor can lower the indices

x D g x: (A.4)

These relations hold generally for four-dimensional vectors. There are various possible choices of the metric tensors, but we shall use the following 0 1 10 0 0 B C B 0 10 0C g D g D @ A : (A.5) 0010 00 01

1 In all appendices, we display complete formulas with all fundamental constants. As before, we use the Einstein summation rule with summation over repeated indices.

I. Lindgren, Relativistic Many-Body Theory: A New Field-Theoretical 273 Approach, Springer Series on Atomic, Optical, and Plasma Physics 63, DOI 10.1007/978-1-4419-8309-1, c Springer Science+Business Media, LLC 2011 274 A Notations and Deﬁnitions

The four-dimensional scalar product is deﬁned as the product of a contravariant and a covariant vector:

0 ab D a b D a b0 a b; (A.6) where a b is the three-dimensional scalar product

a b D axbx C ay by C azbz:

The covariant gradient operator is deﬁned as the gradient with respect to a contravariant coordinate vector: @ 1 @ @ D D ; r (A.7) @x c @t and the contravariant gradient operator analogously @ 1 @ @ D D ; r ; (A.8) @x c @t r is the three-dimensional gradient operator

@ @ @ r D eO x C eO y C eO ; @x @y @z z where .eO x; eO y; eO z/ are unit vectors in the coordinate directions. The four-dimensional divergence is deﬁned as

0 1 @A @A D C rA DrA; (A.9) c @t where rA is the three-dimensional divergence

@Ax @Ay @A rA D C C z : @x @y @z

The d’Alembertian operator is deﬁned as

2 1 @ 2 2 D @ @ D r Dr ; (A.10) c2 @t2 where @2 @2 @2 r 2 D D C C @x2 @y 2 @z2 is the Laplacian operator. A.2 Vector Spaces 275 A.2 Vector Spaces

A.2.1 Notations

X;Y;:::are sets with elements x;y;:::. x 2 X means that x is an element in the set X. N is the set of nonnegative integers. R is the set of real numbers. C is the set of complex numbers. Rn is the set of real n-dimensional vectors. Cn is the set of complex n-dimensional vectors. A X means that A is a subset of X. A [ B is the union of A and B. A \ B is the intersection of A and B. A Dfx 2 X W P g means that A is the set of all elements x in X that satisfy the condition P . f W X ! Y represents a function or operator, which means that f maps uniquely the elements of X onto elements of Y . A functional is a unique mapping f W X ! R .C/ of a function space on the space of real (complex) numbers. The set of arguments x 2 A for which the function f W A ! B is deﬁned is the domain, and the set of results y 2 B which can be produced is the range. .a; b/ is the open interval fx 2 R W a

A.2.2 Basic Deﬁnitions

A real (complex) vector space or function space X is an infinite set of elements, x, referred to as points or vectors,whichisclosed under addition, x C y D z 2 X,and under multiplication by a real (complex) number c, cx D y 2 X. The continuous functions f.x/ on the interval x 2 Œa; b form a vector space, also with some boundary conditions, like f.a/ D f.b/D 0. A subset of X is a subspace of X if it fulfills the criteria for a vector space. A norm of a vector space X is a function p W X ! Œ0; 1 with the properties (1) p.x/ Djjp.x/ (2) p.x C y/ p.x/ C p.y/ for all real ( 2 R)andall x;y 2 X (3) that p.x/ D 0 always implies x D 0 The norm is written as p.x/ Djjxjj.Wethenhave jjxjj D jjjjxjj , jjx C yjj jjxjj C jjyjj ,and jjxjj D 0 ) x D 0. If the last condition is not fulfilled, it is a seminorm. 276 A Notations and Definitions

A vector space with a norm for all its elements is a normed space, denoted by .X; jjjj/. The continuous functions, f.x/, on theR interval Œa; b form a normed b 2 1=2 space by defining a norm, for instance, jjf jj D Œ a dt jf.t/j . By means of the CauchyÐSchwartz inequality, it can be shown that this satisfies the criteria for a norm [4, p. 93]. If f is a function f W A ! Y and A X,thenf is defined in the neighborhood of x0 2 X,ifthereisan>0such that the entire sphere fx 2 X Wjjx x0jj <g belongs to A [4, p. 309]. A function/operator f W X ! Y is bounded, if there exists a number C such that jjfxjj sup D C<1: 0¤x2X jjxjj Then C Djjf jj is the norm of f . Thus, jjfxjj jjf jj jjxjj . A function f is continuous at the point x0 2 X,ifforevery ı>0there exists an >0such that for every member of the set x Wjjx x0jj <we have jjfx fx0jj ı [4, p. 139]. This can also be expressed so that f is continuous at the point x0, if and only if fx ! fx0 whenever xn ! x0, fxng being a sequence in X, meaning that fxn converges to fx0,ifx converges to x0 [5, p. 70]. A linear function/operator is continuous if and only if it is bounded [4, p. 197, 213], [2, p. 22]. A functional f W X ! R is convex if

f.txC .t 1/y/ tf .x/ C .t 1/f .y/; for all x;y 2 X and t 2 .0; 1/. A subset A X is open,ifforevery x 2 A there exists an >0such that the entire ball Br .x/ Dfy 2 Xjjjy xjj <g belongs to A,i.e.,Br .x/ A [1, p. 363], [4,p.98],[5, p. 57]. A sequence fxng ,wheren is an integer (n 2 N), is an inﬁnite numbered list of elements in a set or a space. A subsequence is a sequence, which is a part of a sequence. A sequence fxn 2 Ag is (strongly) convergent toward x 2 A, if and only if for every >0there exists an N such that jjxn xjj < for all n>N[4, p. 95, 348]. A sequence is called a Cauchy sequence if and only if for every >0there exists an N such that jjxn xmjj <for all m; n > N. If a sequence fxng is convergent, then it follows that for n; m > N

jjxm xnjj D jj.xn x/ C .x xm/jj jjxn xjj C jjxm xjj <2; which means that a convergent sequence is always a Cauchy sequence. The opposite is not necessarily true, since the point of convergence need not be an element of X [3, p. 44]. A.3 Special Functions 277

A subset A of a normed space is termed compact, if every inﬁnite sequence of elements in A has a subsequence, which converges to an element in A. The closed interval [0,1] is an example of a compact set, while the open interval (0,1) is non- compact,sincethesequence1,1/2,1/3...andallofitssubsequencesconvergeto 0, which lies outside the set [5, p. 149]. This sequence satisﬁes the Cauchy convergence criteria but not the (strong) convergence criteria. A dual space or adjoint space of a vector space X, denoted as X , is the space of all functions on X. An inner or scalar product in a vector space X is a function h; i W X X ! R with the properties (1)

hx;1y1 C 2y2iD1hx;y1iC2hx;y2i ; hx;yiDhy; xi; for all x;y;y1;y2 2 X and all j 2 R ,and(2) hx;xiD0 only if x D 0.

A.2.3 Special Spaces

A.2.3.1 Banach Space

A Banach space is a normed space in which every Cauchy sequence converges to a point in the space.

A.2.3.2 Hilbert Space p A Banach space with the norm jjxjj D C hx;xi is called a Hilbert space [1, p. 364].

A.2.3.3 Fock Space

A Fock space is a Hilbert space, where the number of particles is variable or un- known.

A.3 Special Functions

A.3.1 Dirac Delta Function

We consider the integral Z L=2 dx eikx: (A.11) L=2 278 A Notations and Deﬁnitions

Assuming periodic boundary conditions,eiLx=2 D eiLx=2, limits the possible k values to k D kn D 2n=L.Then

Z L=2 1 iknx dx e D ıkn;0 D ı.kn;0/; (A.12) L L=2 where ın;m is the Kronecker delta factor 1 if m D n ın;m D : (A.13) 0 if m ¤ n

If we let L !1, then we have to add a “damping factor”ejxj,where is a small positive number, to make the integral meaningful, Z 1 ikx jxj 2 dx e e D 2 2 D 2 .k/: (A.14) 1 k C

In the limit ! 0,wehave Z 1 1 ikx jxj lim .k/ D lim dx e e D ı.k/; (A.15) 0 0 ! 2 ! 1 which can be regarded as a deﬁnition of the Dirac delta function, ı.k/. Formally, we write this relation as Z 1 dx eikx D ı.k/: (A.16) 1 2

The function also has the following properties

lim .x/ D ıx;0; Z !0 1 dx .x a/.x b/ D C.a b/: (A.17) 1 In three dimensions, (A.12) goes over into Z 1 3 iknx 3 d x e D ı .kn;0/D ı.knx;0/ı.kny ;0/ı.knz;0/: (A.18) V V

In the limit where the integration is extended over the entire three-dimensional space, we have in analogy with (A.16) Z d3x eikx D ı3.k/: (A.19) .2/3 A.3 Special Functions 279 A.3.2 Integrals over Functions

We consider the integral Z Z 1 1 dxı.x a/ f .x/ D lim dx .x a/ f .x/ 0 1 ! 1Z 1 1 2 D lim dx f.x/: (A.20) 0 2 2 2 ! 1 .x a/ C

The integral can be evaluated using residue calculus and leads to Z 1 dxı.x a/ f .x/ D f.a/ (A.21a) 1 provided the function f.x/has no poles. In three dimensions, we have similarly Z 3 d xı.x x0/f.x/ D f.x0/ (A.21b) integrated over all space. The relations above are often taken as the deﬁnition of the Dirac delta function, but the procedure applied here is more rigorous. Next, we consider the integral over two functions Z Z 1 2 2 dx .x a/ .x b/ D 2 dx 2 2 2 2 .2/ Z .x a/ C .x b/ C 1 1 1 D dx 42i x a i x a C i 2 .x b C i/.x b i/ 1 1 1 D 2i b a i. C / b a C i. C / 1 2. C / D (A.22) 2 .a b/2 C . C /2 after integrating the ﬁrst term over the negative and the second term over the positive half plane. Thus, Z

dx .x a/ .x b/ D C.a b/ (A.23) and we see that here the widths of the functions are added. 280 A Notations and Deﬁnitions

Now, we consider some integrals with the functions in combination with electron and photon propagators that are frequently used in the main text. First, we consider the integral with one function and an electron propagator (4.10) Z Z 1 d! 1 2 d! ."a !/ D 2 2 ! "j C i 2 ! "j C i ."a !/ C Z d! 1 2 D : 2 ! "j C i ."a ! C i/."a ! i/

The pole of the propagator yields the contribution ."a "j /, which vanishes in the limit ! 0,if"a 6D "j . Nevertheless, we shall see that this pole has a significant effect on the result. Integrating above over the positive half plane, with the single pole "a Ci, yields 1 "a "j C i C i and integrating over the negative half plane, with the two poles "j i; "a i, yields 2i 1 C ."a "j C i C i/."a "j i C i/ "a "j i C i 1 D ; "a "j C i C i which is identical to the previous result. We observe here that the pole of the propagator, which has a vanishing contribution in the limit ! 0, has the effect of reversing the sign of the i term. The parameter originates from the adiabatic damping and is small but finite, while the parameter is infinitely small and only determines the position of the pole of the propagator. Therefore, if they appear together, the term dominates, and the term can be omitted. This yields Z 1 1 d! ."a !/ D (A.24) ! "j C i "a "j C i noting that the parameter of the propagator is replaced by the damping parameter . Second, we consider the integral with the photon propagator (4.31) Z Z 1 1 d! 1 1 d! ."a !/ D !2 2 C i 2 2 ! C i ! C i 2 ."a ! C i/."a ! i/ A.3 Special Functions 281 1 1 1 D 2 "a C i C i "a i C i 1 D 2 2 (A.25) "a . i i/ or, neglecting the term, Z 1 1 d! 2 2 ."a !/ D 2 2 (A.26) ! C i "a C i noting that 0. Finally, we consider the integrals of two functions and the propagators. With the electron propagator, we have Z 1 d! ."a !/ ."b !/ ! "j C i Z 1 1 1 1 D 2 d! .2i/ ! "j C i "a ! i "a ! C i 1 1 : "b ! i "b ! C i

Here, three of the combinations with poles on both sides of the real axis contribute, which yields 1 1 1 C 2i ."b "j C i/."a "b 2i/ ."a "j C i/."b "a 2i/ 1 : ."a "j C i/."b "j C i/

The last two terms become 1 1 1 1 "a "j C i "b "a 2i "b "j C i ."b "a 2i/."b "j C i/ neglecting an imaginary term in the numerator. This leads to Z 1 1 d! ."a !/ ."b !/ 2 ."a "b/: ! "j C i "a "j C i

(A.27) 282 A Notations and Deﬁnitions

Similarly, we ﬁnd for the photon propagator Z 1 1 d! 2 2 ."a !/ ."b !/ 2 2 ."a "b/: ! C i "a C i (A.28)

Formally, we can obtain the integral with propagators by replacing the function by the corresponding Dirac delta function, noting that we then have to replace the imaginary parameter in the denominator by the damping factor .

A.3.3 The Heaviside Step Function

The Heaviside step function is deﬁned as

.t/ D 1t0 >t D 0t0

The step function can also be given the integral representation Z 1 d! ei!t .t/ D lim i (A.30) 0 ! 1 2 ! C i from which we obtain the derivative of the step function Z d.t/ 1 d! ! D lim ei!t D ı.t/ (A.31) 0 dt ! 1 2 ! C i where ı.t/ is the Dirac delta function.

References

1. Blanchard, P., Br¬uning, E.: Variational Methods in Mathematical Physics. A Uniﬁed Approach. Springer-Verlag, Berlin (1992) 2. Debnath, L., Mikusi«nski, P.: Introduction to Hilbert Spaces with Applications., second edn. Academic Press, New York (1999) 3. Delves, L.M., Welsh, J.: Numerical Solution of Integral Equations. Clarendon Press, Oxford (1974) 4. Griffel, D.H.: Applied Functional Analysis. Wiley, N.Y. (1981) 5. Taylor, A.E.: Introduction to Functional Analysis. Wiley and Sons (1957) Appendix B Second Quantization

B.1 Deﬁnitions

(See, for instance [2, Chap. 5], [1, Chap. 11].) In second quantization Ð also known as the number representation Ð a state is represented by a vector (see Appendix C.1) jn1;n2; i, where the numbers represent the number of particles in the particular basis state (which for fermions can be equal only to one or zero). Second quantization is based on annihilation/creation operators cj /cj ,which annihilate and create, respectively, a single particle. If we denote byj0i the vacuum state with no particle, then cj j0i Djj i (B.1) represents a single-particle state. In the coordinate representation (C.19), this corresponds to the wave function j .x/ Dhxjj i (B.2) satisfying the single-electron Schr¬odinger or Dirac equation. Obviously, we have

cj j0i D 0: (B.3)

For fermions, the operators satisfy the anticommutation relations

fci ;cj gDci cj C cj ci D 0; fci ;cj gDci cj C cj ci D 0; fci ;cj gDci cj C cj ci D ıij ; (B.4)

where ıij is the Kronecker delta factor (A.13). It then follows that

ci cj j0i Dcj ci j0i ; (B.5)

283 284 B Second Quantization

which means that ci ci j0i represents an antisymmetric two-particle state, which we denote in the following way1

ci cj j0i Djfi;jgi : (B.6)

The antisymmetric form is required for fermions by the quantum-mechanical rules. A corresponding bra state is

h0jcl ck D hfk; lgj (B.7)

and it then follows that the states are orthonormal. In the coordinate representation, the state above becomes

1 hx1x2jfi;jgi D p i .x1/j .x2/ j .x1/i .x2/ : (B.8) 2

Generalizing this to a general many-particle system leads to an antisymmetric product, known as the Slater determinant,

1 hx1; x2; xN jcacb cN j0i D p Detfa; b; N g NŠ ˇ ˇ ˇ ˇ ˇ 1.x1/1.x2/ 1.xN / ˇ ˇ ˇ 1 ˇ 2.x1/2.x2/ 2.xN / ˇ D p ˇ ˇ : (B.9) ˇ ˇ NŠ ˇ ˇ N .x1/N .x2/ N .xN /

For an N -particle system, we deﬁne one- and two-particle operators by

Xn F D fn; (B.10) nD1 Xn G D gmn; (B.11) m

respectively, where the fn and the gmn operators are identical, differing only in the particles they operate on. In second quantization these operators can be expressed (see, for instance [1, Sect. 11.1])2

1 We shall follow the convention of letting the notation ji;ji denote a straight product function ji;jiDi .x1/j .x2/, whilejfi;jgi represents an antisymmetric function. 2 Occasionally, we use a “hat” on the operators to emphasize their second-quantized form. We also use the Einstein summation rule with summation over all indices that appear twice. Note the order between the annihilation operators. B.1 Deﬁnitions 285

FO D ci hijf jj i cj ; 1 GO D c c hij jgjkli cl ck ; (B.12) 2 i j etc. (note order between the operators in the two-particle case). Here, Z 3 hijf jj i D d x1 i .x1/f j .x1/; ZZ 3 3 hij jgjkli D d x1 d x2 i .x1/j .x2/gk.x1/l .x2/: (B.13)

We can check the formulas above by evaluating ˝ ˇ ˇ ˛ ˇ1 ˇ hfcdgjGOjfabgi D fcdg c c hij jgjkli cl ck fabg 2 i j ˝ ˇ ˇ ˛ 1 ˇ ˇ D 0 cd cc c c hij jgjkli cl ck cac 0 : (B.14) 2 i j b Normal-ordering the operators yields

cl ck cacbj0i D ık;aıl;b ıl;aık;b and similarly h0jcd cc ci cj D ıi;dıj;c ıj;cıi;d: Then we have hfcdgjGOjfabgi D hcdjgjabi hdcjgjabi ; which agrees with the results using determinantal wave functions (see, for instance [1, Eq. 5.19]) 1 ˝ ˇ hfcdgjGOjfabgi D cd dcˇGO jab bai : (B.15) 2 We deﬁne the electron-ﬁeld operators in the Schrodinger¨ representation (3.1) by

O x x O x x S. / D cj j . /I S . / D cj j . /: (B.16) Then, the second-quantized one-body operator can be expressed as Z Z O 3x 3x O O F D d ci i .x/ f cj j .x/ D d S .x/ f S.x/ (B.17) and similarly ZZ 1 3 3 GO D d x1d x2 O .x1/ .x2/g O .x2/ O .x1/: (B.18) 2 S S S S 286 B Second Quantization

The nonrelativistic Hamiltonian for an N -electron system (2.11) consists of a single-particle and a two-particle operator XN 2 XN „ 2 H1 D rn C v .xn/ D h1.n/; 2m ext nD1 nD1 N N X e2 X H2 D D h2.m; n/; (B.19) m

B.2 Heisenberg and Interaction Pictures

In an alternative to the Schr¬odinger picture, the Heisenberg picture (HP), the states are time independent and the time dependence is transferred to the operators,

iHt=O „ iHt=O „ iHt=O „ jHi DjS.t D 0/i D e jS.t/i I OOH D e OOS e : (B.21)

In perturbation theory, the Hamiltonian is normally partitioned into a zeroth- order Hamiltonian H0 and a perturbation V (2.48), which using second-quantized notations becomes HO D HO0 C V:O (B.22) We can then deﬁne an intermediate picture, known as the interaction picture (IP), where the operators and state vectors are related to those in the Schr¬odinger picture by iHO 0t=„ iHO 0t=„ iHO 0t=„ jI.t/i D e jS.t/i I OOI.t/ D e OOS e : (B.23) The relation between the Heisenberg and the interaction pictures is3

iHt=O „ iHO0t=„ iHt=O „ iHO0t=„ iHO0t=„ iHt=O „ jHi D e e jI.t/i I OOH.t/ D e e OOI e e : (B.24) Using the relation (3.9), we then have

jHi D U.0;t/jI.t/i I OOH.t/ D U.0;t/OOI U.t;0/: (B.25)

O O O 3 iHt=„ iH0t=„ iVt=„ Note that HO and HO0 generally do not commute, so that in general e e ¤ e . References 287

The state vector of time-independent perturbation theory corresponds in all pictures considered here to the time-dependent state vectors with t D 0,

ji DjHi DjS.0i/ DjI.0/i : (B.26)

In the Heisenberg picture (B.21), the electron-ﬁeld operators (B.16) become

O iHt=O „ O x iHt=O „ O iHt=O „ O x iHt=O „ H.x/ D e S. / e I H.x/ D e S . / e (B.27) andintheinteraction picture (IP) (B.23)

iHO0t=„ iHO 0t=„ iHO 0t=„ iHO 0t=„ O I.x/ D e O S.x/ e D e cj j .x/ e i"j t=„ D cj j .x/ e D cj j .x/ O x i"j t=„ I .x/ D cj j . / e D cj j .x/; (B.28) where j .x/ is an eigenfunction of H0. We also introduce the time-dependent creation/annihilations operators in the IP by

i"j t=„ i"j t=„ cj .t/ D cj e I cj .t=„/ D cj e ; (B.29) which gives

O x O x I.x/ D cj .t/ j . / I I .x/ D cj .t/ j . /: (B.30)

From the deﬁnition (B.23), we have h i h i @ @ iHO0t=„ iHO0t=„ OO .t/ D e OO e D i H0; OO .t/ : (B.31) @t I @t S I

References

1. Lindgren, I., Morrison, J.: Atomic Many-Body Theory. Second edition, Springer-Verlag, Berlin (1986, reprinted 2009) 2. Schweber, S.S.: An Introduction to Relativistic Quantum Field Theory. Harper and Row, N.Y. (1961) Appendix C Representations of States and Operators

C.1 Vector Representation of States

A state of a system can be represented by the wave function or Schrodinger¨ function .x/,wherex stands for all (space) coordinates. If we have a complete basis set available in the same Hilbert space (see Appendix A.2), fj .x/g, then we can expand the function as .x/ D aj j .x/ (C.1) with summation over j according to the Einstein summation rule. If the basis set is orthonormal, implying that the scalar or inner product satisﬁes the relation Z hijj i D dxi j .x/ D ıi;j; (C.2) then the expansion coefﬁcients are given by the scalar product Z aj D dxj .x/ Dhjji : (C.3)

These numbers form a vector, which is the vector representation of the state or the state vector, 0 1 h1ji B C B 2 C ˇ ˛ B h j i C ˇ B C D B C : (C.4) @ A hNji Note that this is just a set of numbers Ð no coordinates are involved. N is here the number of basis states, which may be ﬁnite or inﬁnite. [The basis set need not be numerable and can form a continuum in which case the sum over the states is replaced by an integral.] The basis states are represented by unit vectors jj i

289 290 C Representations of States and Operators 0 1 0 1 1 0 B C B C B 0 C B 1 C ˇ ˛ B C ˇ ˛ B C ˇ B C ˇ B C 1 D B 0 C 2 D B 0 C etc: (C.5) @ A @ A

The basis vectors are time independent, and for time-dependent states the time dependence is contained in the coefﬁcients

j.t/i D aj .t/jj i ; (C.6)

ji is a ket vector, and for each ket vector there is a corresponding bra vector ˝ ˇ ˇ D a1 ;a2 ; ; (C.7) where the asterisk represents complex conjugate. It follows from (C.1)that

aj D hjj i: (C.8)

The scalar product of two general vectors with expansion coefﬁcients aj and bj , respectively, becomes hj˚iDaj bj (C.9) with the basis vectors being orthonormal. This is identical to the scalar product of the corresponding vector representations 0 1 b1 B C B b2 C hj˚iD a1 ;a2 ; @ A : (C.10)

The ket vector (C.4) can be expanded as

ji Djj ihj ji: (C.11)

But this holds for any vector in the Hilbert space, and therefore we have the formal relation in that space jj ihj j I ; (C.12) where I is the identity operator. This is known as the resolution of the identity.Us- ing the expression for the coefﬁcients, the scalar product (C.9) can also be expressed as hj˚iDhjj i hj j˚i; (C.13) which becomes obvious, considering the expression for the identity operator. C.2 Matrix Representation of Operators 291 C.2 Matrix Representation of Operators

The operators we are dealing with have the property that when acting on a function in our Hilbert space, they generate another (or the same) function in that space,

OO .x/ D ˚.x/ (C.14) or with vector notations OOji Dj˚i : (C.15) Expanding the vectors on the left-hand side according to the above yields

jiihijOOjj i hjji Dj˚i : (C.16)

Obviously, we have the identity

OO jiihijOOjj i hj: (C.17)

The numbers hijOOjj i are matrix elements Z hijOOjj i D dxi .x/ OO j .x/ (C.18) and they form the matrix representation of the operator 0 1 h1jOOj1ih1jOOj2i OO ) @ h2jOOj1ih2jOOj2i A :

Standard matrix multiplication rules are used in operations with vector and matrix representations, for instance, 0 1 0 1 h1ji h1j˚i 0 1 B C B C 1 OO 1 1 OO 2 B 2 C B 2 ˚ C h j j ih j j i B h j i C B h j i C OO @ OO OO A B C B C ji Dj˚i ) h2j j1ih2j j2i B C D B C ; @ A @ A hNji hNj˚i where hkj˚i D hkjOOjj i hjj˚i summed over the index j . 292 C Representations of States and Operators C.3 Coordinate Representations

C.3.1 Representation of Vectors

The coordinate representation of the ket vectorji (C.4) is denoted as hxji,and this is identical to the corresponding state or (Schr¬odinger) wave function

hxji.x/I hjxi .x/: (C.19)

This can be regarded as a generalization of the expansion for the expansion coef- ﬁcients (C.1), where the space coordinates correspond to a continuous set of basis functions. The basis functions j .x/ have the coordinate representation hxjj i, and the coordinate representation (C.1) becomes

hxji D aj j .x/ D aj hxjj i : (C.20)

The scalar product between the functions .x/ and ˚.x/ is Z hj˚iD dx.x/ ˚.x/; (C.21) which we can express as Z hj˚iD dx hjxihxj˚i : (C.22)

We shall assume that an integration is always understood, when Dirac notations of the kind above are used, i.e.,

hj˚iDhjxi hxj˚i (C.23) in analogy with the summation rule for discrete basis sets. This leads to the formal identity jxihxj I; (C.24) which is consistent with the corresponding relation (C.12) with a numerable basis set.

C.3.2 Closure Property

From the expansion (C.1), we have Z 0 0 0 .x/ D dx j .x /.x/j .x/: (C.25) C.3 Coordinate Representations 293

This can be compared with the integration over the Dirac delta function Z .x/ D dx0 ı.x x0/.x0/; (C.26) which leads to the relation known as the closure property

0 0 j .x /j .x/ D ı.x x / (C.27)

(with summation over j ). In Dirac notations, this becomes

hxjj ihj jx0iDı.x x0/ or ˛ hxjIjx0 D ı.x x0/; (C.28) which implies that the delta function is the coordinate representation of the identity operator (C.12). Note that there is no integration over the space coordinates here.

C.3.3 Representation of Operators

The coordinate representation of an operator is expressed in analogy with that of a state vector ˛ OO ) hxjOOjx0 D OO .x; x0/: (C.29) which is a function of x and x0. An operator OO acting on a state vectorji is represented by Z ˛ hxOOji ) hxjOOjx0 hx0ji D dx0 OO .x; x0/.x0/; (C.30) which is a function of x. Appendix D Dirac Equation and the Momentum Representation

D.1 Dirac Equation

D.1.1 Free Particles

The standard quantum-mechanical operator representation @ E ! EO D i„ I p !Op Di„rI x !Ox D x; (D.1) @t where E; p; x represent the energy, momentum, and coordinate vectors, and E;O pO ; xO the corresponding quantum-mechanical operators, was previously used to obtain the nonrelativistic Schr¬odinger equation (2.9). If we apply the same procedure to the relativistic energy relation

2 2 2 2 4 E D c p C me c ; (D.2) where c is the velocity of light in vacuum and me the mass of the electron, this would lead to 2 @ .x/ 2 „2 D c2pO C m2c4 .x/; (D.3) @t2 e which is the Schrodinger¨ relativistic wave equation. It is also known as the Klein–Gordon equation. In covariant notations (see Appendix, Sect. A.1), it can be expressed as 2 2 2 „ C me c .x/ D 0: (D.4) In contrast to the nonrelativistic Schr¬odinger equation (2.9), the KleinÐGordon equation is nonlinear, and therefore the superposition principle of the solutions cannot be applied. In order to obtain a linear equation that is consistent with the energy relation (D.2) and the quantum-mechanical substitutions (D.1), Dirac proposed the form for a free electron

@ .x/ i„ D c˛ pO C ˇm c2 .x/; (D.5) @t e

295 296 D Dirac Equation and the Momentum Representation where ˛ and ˇ are constants (but not necessarily pure numbers). This equation is the famous Dirac equation for a relativistic particle in free space. The equivalence with the equation (D.3) requires

˛ p 2 ˛ p 2 2p2 2 4 .c O C ˇmec /.c O C ˇmec / c O C me c ; which leads to

2 2 2 2 ˛x D ˛y D ˛z D ˇ D 1; ˛x ˛y C ˛y˛x D 0.cyclic/; ˛ˇ C ˇ˛ D 0; (D.6) where “cyclic” implies that the relation holds for x ! y ! z ! x. The solution proposed by Dirac is the so-called Dirac matrices 0 10 ˛ D I ˇ D ; (D.7) 0 0 1 where D .x;y ;z/ are the Pauli spin matrices 01 0 i 10 x D I y D I D : (D.8) 10 i 0 z 0 1

The Dirac matrices anticommute

˛ˇ C ˇ˛ D 0: (D.9)

With the covariant four-dimensional momentum vector (A.2) p D .p0; p/, and the corresponding vector operator i„ @ pO D .pO0; Op/ D ; i„r ; (D.10) c @t the Dirac equation (D.5) becomes 2 cpO0 c˛ pO ˇmec .x/ D 0 (D.11) or with ˛ D .1; ˛/ (D.12) and ˛ pO DOp0 ˛ pO we obtain the covariant form of the Dirac Hamiltonian for a free particle

2 HD Dc˛ pO C ˇmec (D.13) D.1 Dirac Equation 297 and the corresponding Dirac equation 2 c˛ pO ˇmec .x/ D 0: (D.14)

With the Dirac gamma matrices

D ˇ˛; (D.15) this can also be expressed as (ˇ2 D 1) O pO mec .x/ D 6 p mec .x/ D 0; (D.16) where 6 pO is the “p-slash” operator

6 pO D pO D ˇ˛ pO D ˇ.pO0 ˛ pO /: (D.17)

It should be observed that in the covariant notation pO0 is normally disconnected from the energy, i.e., q p2 2 2 pO0 ¤ O C me c : (D.18) This is known as off the mass shell. When the equality above holds, it is referred to as on the mass shell, which can also be expressed as

2 2 p2 2 2 O pO DOp0 O D me c or 6 p D mec: (D.19)

In separating the wave function into space and time parts,

i"p t=„ .x/ D p.x/ e ; (D.20) the time-independent part of the Dirac equation (D.5) becomes

Ofree x x hD .pO/p. / D "p p. /; (D.21) where Ofree ˛ 2 hD D c pO C ˇmec (D.22) is the free-electron Dirac Hamiltonian. The Dirac equation can also be expressed as "p ˇ ˇ˛ pO m c p.x/ D 0: (D.23) c e Here, p.x/ is a four-component wave function, which can be represented by

1 ip x p.x/ D p ur .p/ e I pO p.x/ D p p.x/: (D.24) V 298 D Dirac Equation and the Momentum Representation

(Note the difference between the momentum vector p and the momentum operator ip x pO). e represents a plane wave,andur .p/ is a four-component vector function of the momentum p. For each p, there are four independent solutions (r D 1; 2; 3; 4). The parameter p in our notations p and "p represents p and r or, more explicitly,

p.x/ D p;r .x/ I "p D "p;r :

With the wave function (12.1), the Dirac equation (D.23) leads to the following equation for the ur .p/ functions "p ˇ ˇ˛ pO m c ur .p/ D 0 c e or "p=c mec pO ur .p/ D 0; (D.25) pO "p=c mec where each element is a 2 2 matrix. This equation has two solutions for each momentum vector p: "p=c C mec p uC.p/ D NC I u.p/ D N (D.26) p "p=c C mec corresponding to positive (r D 1; 2) and negative (r D 3; 4) eigenvalues, respectively. Deﬁning the momentum component p0 Ð to be distinguished from the corresponding operator component pO0 (D.10)Ðby q 2 2 2 j"pjDEp D cp0 I p0 D p C me c (D.27) gives p0 C mec p uC.p/ D NC I u.p/ D N : (D.28) p p0 C mec

The corresponding eigenfunctions (12.1) are

1 ip x iEp t=„ 1 ip x iEp t=„ pC .x/ D p uC.p/ e e p .x/ D p u.p/ e e V V (D.29) including the time dependence according to (D.20)and(D.27). The vectors p u.p/ D uC.p/ and v.p/ D u.p/ D N (D.30) p0 C mec D.1 Dirac Equation 299 satisfy the equations

.6p mec/u.p/ D 0 and .6p C mec/v.p/ D 0; (D.31) where p0 isgivenby(D.27). Note that the negative energy solution corresponds here to the momentum p for the electron (or Cp for the hole/positron).

Normalization

Several different schemes for the normalization of the u matrices have been used (see, for instance, Mandl and Shaw [1, Chap. 4]). Here, we shall use

ur0 .p/ ur .p/ D ır0;r ; (D.32) which leads to 2 p0 C mec u .p/ DjN j .p0 C m c; p/ C C e p 2 2 2 2 DjNCj .p0 C mec/ C . p/ DjNCj 2p0 .p0 C mec/ (D.33)

2 2 2 2 2 using . p/ D p D p0 me c .Thisgives

1 NC D p (D.34) 2p0 .p0 C mec/ and the same for N. With the normalization above, we have 2 p0 C mec u .p/ u .p/ DjN j .p0 C m c; p/ C C C p e 1 p0 C m c p p0 C ˛ p C ˇmec D e D (D.35) 2p0 p p0 mec 2p0 and similarly

p0 .˛ p C ˇmec/ u.p/ u.p/ D ; (D.36) 2p0 which gives

uC.p/ uC.p/ C u.p/ u.p/ D I: (D.37) 300 D Dirac Equation and the Momentum Representation D.1.2 Dirac Equation in an Electromagnetic Field

Classically, the interaction of an electron with electromagnetic ﬁelds is given by the “minimal substitution”(E.15), which in covariant notations can be expressed as1

p !Op C eA (D.38) with the four-dimensional potential being .x/ A.x/ D ; A.x/ : (D.39) c

This implies that the Dirac Hamiltonian (D.13) becomes

2 HD Dc˛ .pO C eA/ C ˇmec (D.40) and that the interaction with the ﬁelds is given by the term

Hint Dec˛ A: (D.41)

D.2 Momentum Representation

D.2.1 Representation of States

Above in Sect. C.3 we have considered the coordinate representation of a state vector, a.x/ Dhxjai. An alternative is the momentum representation, where the state vector is expanded in momentum eigenfunctions. A statejai is then represented by a.pr/ D hprjai, which are the expansion coefﬁcients of the state in momentum eigenfunctions hxjaiDhxjpri hprjai (D.42) with summations over p and r. The expansion coefﬁcients become Z r Z 3 1 3 ipx hprjaiD d x hprjxi hxjaiD d x e u .p/a.x/: (D.43) V r In the limit of continuous momenta, the sum over p is replaced by an integral and V replaced by .2/3.

1 e In many text books, the minimal substitution is expressed as p !Op C c A, because a mixed unit system, like the cgs system, is used. In the SI system Ð or any other consistent unit system Ð the substitution has the form given in the text. The correctness of this expression can be checked by means of dimensional analysis (see Appendix K). D.2 Momentum Representation 301

Note that the momentum representation is distinct from the Fourier transform. The latter is deﬁned as r Z 1 3 ipx hpjaiDur .p/ hprjaiD d x e a.x/ Z V 3=2 3 ipx ! .2/ d x e a.x/ (D.44) using the identity (D.37). In analogy with (C.23), we have

hajbiDhajp;ri hp;rjbi; (D.45) which yields jp;rihp;rj I (D.46) with implicit summation/integration over p and summation over r.

D.2.2 Representation of Operators

Coordinate representation of an operator OO : O.x2; x1/ D hx2jOOjx1i. Momentum representation of an operator OO : O.p2r2; p1r1/ D hp2r2jOOjp1r1i. Transformation between the representations ZZ 3 3 hp2r2jOOjp1r1i D d x2 d x1 hp2r2jx2i hx2jOOjx1ihx1jp1r1i: (D.47)

The corresponding Fourier transform is according to (D.44)

OO ur2 .p2/ hp2r2j jp1r1i ur1 .p1/: (D.48)

Any operator with a complete set of eigenstates can be expanded as

OO Djj i "j hj j where OOjj i D "j jj i : (D.49)

This gives the coordinate and momentum representations

hx2jOOjx1i D hx2jj i "j hj jx1i; (D.50a)

hp2r2jOOjp1r1i D hp2;r2jj i "j hj jp1r1i: (D.50b) 302 D Dirac Equation and the Momentum Representation D.2.3 Closure Property for Momentum Functions

In three dimensions, we have the closure property (C.27)

0 3 0 j .x/j .x / D ı .x x / (D.51) and for a continuous set of momentum eigenfunctions this becomes Z 3 x x0 3 x x0 d p pr . /pr . / D ı . / (D.52) with summation over r. This can also be expressed as

hxjpri hprjxiDı3.x x0/ (D.53) also with integration over p. From the closure property (D.51), we have

0 0 3 0 j .p;r/j .p ;r / D ır;r 0 ı .p p /; (D.54) which leads to 0 0 3 0 hp;rjj i hj jp ;r iDır;r 0 ı .p p /: (D.55)

D.3 Relations for the Alpha and Gamma Matrices

From the definition of the alpha matrices and the definitions in Appendix A,wefind the following useful relations:

2 ˛ ˛ D 1 ˛ D2; ˛ ˛˛ D ˛˛ ˛ D2˛; ˛ ˇ˛ D ˇ ˛ˇ˛ D 4ˇ; ˛ˇ˛ D ˇ C ˛ˇ˛ D2ˇ; ˛ A˛6 D ˛ ˇ˛ A ˛ D 4 A;6 (D.56) where 6A is deﬁned in (D.17). The gamma matrices satisfy the following anticommutation rule:

C D 2g; A6 B6 C B6 A6 D 2AB: (D.57) This leads to

D2; A6 D2 A;6 Reference 303

D 4; D2; A6 D2 A;6 0 0 0 0 D D 1; 0 D 0e ; A6 0 D 0eA; 0 0 D e ; 0 A6 0 D eA; 0 0 D e e ; 0 A6 6B0 D eAB;e 0 ˇ 0 D e ˇe e ; 0 A6 6B 6C0 D eABeC;e (D.58)

e 0 i 0 where A D A0 Ai D A0 C A With the number of dimensions being equal to 4 ,tobeusedindimensional regularization (see Chap. 12), the relations become

D 4 ; D.2 / ; A6 D.2 / A;6 D 4g ; A6 B6 D 4AB A6 B;6 ˇ ˇ ˇ D2 C ; i i D 3 ; i i D.2 / e ; i e i A6 D.2 / A6 A; i i D 4g e e ; i ee i A6 B6 D 4AB AB A6 B;6 ˇ i ˇ ˇ ˇ i D2 e e e C ; i eee i A6 6B 6C D2 C6 B6 A6 ABC C A6 B6 C:6 (D.59)

Reference

1. Mandl, F., Shaw, G.: Quantum Field Theory. John Wiley and Sons, New York (1986) Appendix E Lagrangian Field Theory

Concerning notations, see Appendix A.

E.1 Classical Mechanics

In classical mechanics, the Lagrangian function for a system is deﬁned as

L D T V; (E.1) where T is the kinetic energy and V the potential energy of the system. Generally, @qi this depends on the coordinates qi , the corresponding velocities qPi D @t ,and possible explicitly on time (see, for instance [2, Sect. 23])

L.tI q1;q2 Iq P1; qP2 /: (E.2)

The action is deﬁned as Z

I D dtL.tI q1;q2 Iq P1; qP2 /: (E.3)

The principle of least action implies that

ıI.q1;q2 Iq P1; qP2 / D 0; (E.4) whichleadstotheLagrange equations d @L @L D 0: (E.5) dt @qPi @qi

The Hamilton function can be deﬁned as

H D pi qPi L; (E.6)

305 306 E Lagrangian Field Theory where pi is the canonically conjugate momentum to the coordinate qi

@L pi D : (E.7) @qPi

It then follows that @H @qi DPqi D : (E.8a) @pi @t Furthermore, from the deﬁnitions above and the Lagrange equations, we have

@H DPpi : (E.8b) @qi

These are Hamilton’s canonical equations of motion. We consider a general function of time and the coordinates and canonical momenta f.tI pi ;qi /. Then the total derivative with respect to time becomes

df @f @f @qi @f @pi @f @f @H @f @H D C C D C : (E.9) dt @t @qi @t @pi @t @t @qi @pi @pi @qi

With the Poisson bracket of two functions A and B,deﬁnedby

@A @B @B @A fA; BgD ; (E.10) @qi @pi @qi @pi the derivative can be expressed as

df @f D Cff; H g: (E.11) dt @t For a single-particle system in one dimension (x), the kinetic energy is T D p2=2m,wherem is the mass of the particle, which yields

p2 mv2 L D V D V; 2m 2

2 where v DPx is the velocity of the particle. Furthermore, pi qPi D pxP D p =m D mv2, yielding with (E.6)

p2 mv2 H D C V D C V; 2m 2 E.1 Classical Mechanics 307 which is the classical energy expression. The canonically conjugate momentum (E.7)isthen @L @L p D D D mv; @xP @v which is the classical momentum.

E.1.1 Electron in External Field

The Lagrangian for an electron (charge e) in an external ﬁeld, A D ..x/=c; A/, is [1, p. 25] 1 L.x; xP / D mxP 2 eA xP C e.x/; (E.12) 2 where the last two terms represent (the negative of) a velocity-dependent potential. The conjugate momentum corresponding to the variable x is then according to (E.7)

pi ! p D mxP eA: (E.13)

Using the relation (E.6), we get the corresponding Hamilton function

1 1 2 H D p xP L D mxP 2 e.x/ D .p C eA/ e.x/: (E.14) 2 2m We see that the interaction with the ﬁelds (;A) is obtained by means of the substitutions H ! H e.x/ p ! p C eA (E.15) known as the minimal substitutions. The corresponding equations of motion can be obtained from either the La- grange’s or Hamilton’s equation of motion. We then have d @L d ! .mxP eA/ dt @qPi dt and @L !er.A xP / C er.x/: @qi The same equations are obtained from the Hamilton’s equations of motion (E.8b). The total time derivative can in analogy with (E.9) be expressed as

d @ dx @ @ D C CD CPx r dt @t dt @x @t 308 E Lagrangian Field Theory giving d @A .mxP eA/ D mxR e e.xP r/A: dt @t From the identity

xP .r A/ D r.A xP / .xP r/A; we then obtain the equations of the motion

@A mxR D er.x/ C e e xP .r A/ De.E C v B/ (E.16) @t with v DPx being the velocity of the electron. This is the classical equations of motion for an electron of charge e in an electromagnetic field. The right-hand side is the so-called Lorentz force on an electron in a combined electric and magnetic field. This verifies the Lagrangian (E.12).

E.2 Classical Field Theory

In classical ﬁeld theory, we consider a Lagrangian density of the type

L D L.r ;@r /; (E.17) where r D r .x/ represents different ﬁelds and

@r @r D : (E.18) @x The requirement that the action integral Z 4 I D d x L.r ;@r / (E.19) be stationary over a certain volume leads to the Euler–Lagrange equations

@L @L @ D 0: (E.20) @r @.@r /

The ﬁeld conjugate to r .x/ is

@L r .x/ D ; (E.21) @Pr E.3 Dirac Equation in Lagrangian Formalism 309 where the “dot” represents the time derivative. The Lagrangian function is deﬁned as Z 3 L.t/ D d x L.r ;@r /: (E.22)

The Hamiltonian density is deﬁned as

H.x/ D r .x/Pr .x/ L.r ;@r /: (E.23)

In quantized Lagrangian field theory, the fields are replaced by operators, satisfying the Heisenberg commutation rules at equal times [1, Eq. 2.31] h i 0 3 0 O.x/; O .x / D i„ ı; ı .x x / (E.24) with the remaining commutations vanishing. In our applications, the quantized field will normally be the electron field in the interaction picture (B.28) or the electromagnetic field (G.2).

E.3 Dirac Equation in Lagrangian Formalism

From the Dirac equation for a free electron (D.14), we can deduce the corresponding Lagrangian density 2 L.x/ D O .x/ i„c˛ @ ˇmec O .x/: (E.25)

Using the relation (B.17), the space integral over this density yields the corresponding operator Z 3 2 2 L D d x L.x/ D i„c˛ @ ˇmec D c˛ p ˇmec (E.26)

(with pO D i„ @) and the corresponding Hamilton operator (E.6)

2 H DL Dc˛ p C ˇmec ; (E.27) since the ﬁelds are time independent. This leads to the Dirac equation for a free electron (D.14). We can also apply the EulerÐLagrange equations (E.20) on the Lagrangian (E.25), which leads to @L @ D @ O .x/ i„ c˛ ; @.@ O /

@L 2 D O .x/ ˇmec ; @ O 310 E Lagrangian Field Theory and 2 @i„ c˛ O .x/ C ˇmec O .x/ D 0 with the hermitian adjoint 2 i„ c˛ @ C ˇmec O .x/ D 0; (E.28) which is consistent with the Dirac equation for the free electron. In the presence of an electromagnetic ﬁeld, we make the minimal substitution (D.38) ie p ! p C eA.x/ or @ ! @ A.x/; (E.29) „ whichleadstotheLagrangian density in the presence of an electromagnetic ﬁeld 2 L.x/ D O .x/ c˛ p ˇmec C ec˛ A.x/ O .x/: (E.30)

This gives the the corresponding Hamiltonian density 2 H.x/ D O .x/ c˛ p C ˇmec ec˛ A.x/ O .x/; (E.31) where the last term represents the interaction density

Hint.x/ D O .x/ ec˛ A.x/ O .x/: (E.32)

The corresponding Hamilton operator can then be expressed as Z 3 2 HO D d x1 O .x1/ c˛ p C ˇmec ec˛ A.x/ O .x1/: (E.33)

References

1. Mandl, F., Shaw, G.: Quantum Field Theory. John Wiley and Sons, New York (1986) 2. Schiff, L.I.: Quantum Mechanics. McGraw-Hill Book Co, N.Y. (1955) Appendix F Semiclassical Theory of Radiation

F.1 Classical Electrodynamics

F.1.1 Maxwell’s Equations in Covariant Form

The Maxwell equations in vector form are1

rE D =0; (F.1a)

1 @E r B D C 0 j ; (F.1b) c2 @t rB D 0; (F.1c)

@B r E C D 0; (F.1d) @t where is the electric charge density and j the electric current density. Equation (F.1c)gives B D r A; (F.2) where A is the vector potential.From(F.1d), it follows that the electric ﬁeld is of the form @A E D r; (F.3) @t where is the scalar potential.The(F.1a)and(F.1b) give together with (F.3)and (F.2)

2 @ 0 r rA D =0 D c0 j @t 2 2 1 @ A 1 @ r A r rA C D0 j (F.4) c2 @2t c2 @t

1 As in the previous Appendices, the formulas are here given in a complete form and valid in any consistent unit system, like the SI system (see Appendix K).

311 312 F Semiclassical Theory of Radiation using the vector identity r .r A/ D r.rA/ r 2A. Here, j 0 D c (with 2 00 D c )isthescalar or “time-like” part of the four-dimensional current density j D j D .c; j /; (F.5) where the vector part is the three-dimensional current density j . Similarly, the four- dimensional vector potential

A D .=c; A/A D .=c; A/ (F.6) has the scalar part =c and the vector part A. With the d’Alambertian operator (A.10), these equations can be expressed as2

@ 0 .rA/ D c0 j ; (F.7) @t A Cr.rA/ D 0 j ; (F.8) which leads to Maxwell’s equations in covariant form

A r.rA/ D 0 j (F.9) or @ @ A @ .@A / D 0 j : (F.10)

F.1.1.1 Electromagnetic-Field Lagrangian

We introduce the ﬁeld tensor [2, Eq. 5.5]

F D @A @A: (F.11)

Then we ﬁnd for instance

01 1 0 0 1 @=c @Ax F D @ A @ A D D Ex; @x @ct

@Ax @Ay F 12 D @2A1 @1A2 D D B ; @y @x z etc., leading to the matrix 0 1 0Ex=c Ey=c Ey=c B C B Ex=c 0 Bx Bz C F D @ A : (F.12) Ey=c Bz 0Bx Ez Bx Bx 0

2 Concerning covariant notations, see Appendix A.1. F.1 Classical Electrodynamics 313

The Maxwell equations (F.10) can now be expressed as [2, Eq. 5.2]

@F D 0 j (F.13) using the identity @@ A @ @A :

With r D A, the EulerÐLagrange equations (E.20) become

@L @L @ D 0: (F.14) @A @.@ A/

Using the ﬁeld tensor (F.11) and the form of the metric tensor (A.5), we have F F D @ A @A .@ A @ A / D @ A @A .g @ g A g @ g A /: (F.15)

Here, and are running indices that are summed over, and we can replace them with 0 and 0, respectively. The derivative with respect to ﬁxed and then gives

@ 00 0 0 0 0 F00 F D F F C F00 g g F00 g g D 4F : @.@A/ (F.16) We then ﬁnd that with the Lagrangian

1 L D FF j A; (F.17) 40 the EulerÐLagrange equations (F.14) lead to the Maxwell equations (F.13). With the same Lagrangian, the conjugate ﬁelds (E.21)are

@L 1 @L .x/ D D 0 ; (F.18) @AP c @.@ A/

0 @ 1 @ 0 where the dot represents the time derivative and @ D @x0 D c @t D @ .Usingthe relation (F.16), this yields 1 .x/ D F 0.x/: (F.19) 0c The Hamiltonian is given in terms of the Lagrangian by [2, 5.31] Z 3 H D d xsNf .x/AP.x/ Lg; (F.20) where N fg represents normal order [1, Chap. 11] (see Sect. 2.2). 314 F Semiclassical Theory of Radiation

F.1.1.2 Lorenz Condition

The Lorenz condition is3

1 @ rA D @A D rA C D 0 (F.21) c2 @t and with this condition the Maxwell equations get the simple form

A D 0 j: (F.22)

Then also the electromagnetic ﬁelds have particularly simple form, given in (G.2).

F.1.1.3 Continuity Equation

Operating on Maxwell’s equations (F.9) with r yields:

r .A/ rr.rA/ D 0rj:

Since Dr2 and r commute, this leads to the continuity equation

rj D @j D 0: (F.23)

F.1.1.4 Gauge Invariance

A general gauge transformation is represented by

A ) A Cr; (F.24) where is an arbitrary scalar.

Inserted into the Maxwell equations (F.9), this yields:

.r/ r.rr/ D .r/ r./ D 0; which shows that the Maxwell equations are gauge invariant.

3 This condition is named after the Danish physicist Ludvig Lorenz, not to be confused with the more well-known Dutch physicist Hendrik Lorentz. F.1 Classical Electrodynamics 315 F.1.2 Coulomb Gauge

F.1.2.1 Transverse and Longitudinal Field Components

The vector part of the electromagnetic ﬁeld can be separated into transverse (divergence-free) and longitudinal (rotation-free) components

A D A? C Ak I rA? D 0 I r Ak D 0: (F.25)

The electric field can be similarly separated @A @A E D E C E I E D ? I E D k r; ? k ? @t k @t while the magnetic field has only transverse components due to the relation (F.2). The separated field equations (F.4) then become

2 @ r C rA D=0; (F.26a) @t k

2 2 1 @ Ak 1 @ r A r rA C D0 j ; (F.26b) k c2 @2t k c2 @t k

2 2 1 @ r A D0j : (F.26c) c2 @2t ? T

The longitudinal and the scalar or “time-like” components (Ak;)representthe instantaneous Coulomb interaction and the transverse components (A?)represent retardation of this interaction and all magnetic interactions, as well as the electromagnetic radiation ﬁeld (see Sect. F.2.4). The energy of the electromagnetic ﬁeld is given by Z ˇ ˇ ˇ ˇ 1 3 1 ˇ ˇ2 ˇ ˇ2 Erad D d x B C 0 E 2 0 Z Z ˇ ˇ ˇ ˇ ˇ ˇ 1 3 1 ˇ ˇ2 ˇ ˇ2 1 3 ˇ ˇ2 D d x B C 0 E ? C d x 0 E k : (F.27) 2 0 2

The last term represents the energy of the instantaneous Coulomb field, which is normally already included in the hamiltonian of the system. The first term represents the radiation energy. Semiclassically, only the transverse part of the field is quantized, while the longitudinal part is treated classically [3, Chaps. 2 and 3]. It should be noted that the separation into transverse and longitudinal components is not Lorentz covariant and therefore, strictly speaking, not physically justified, when relativity is taken into ac- count. It can be argued, though, that the separation (as made in the Coulomb gauge) should ultimately lead to the same result as a covariant gauge, when treated properly. 316 F Semiclassical Theory of Radiation

In a fully covariant treatment also the longitudinal component is quantized. The ﬁeld is then represented by virtual photons with four directions of polarizations.A real photon can only have transverse polarizations. The Coulomb gauge is deﬁned by the condition

rA.x/ D 0: (F.28)

Using the Fourier transform Z A.x/ D d4kA.k/eikx; (F.29) this condition leads to

i Z @A 4 i ikx D d kA .k/.i/ki e D 0 @xi or A.k/ k D 0: (F.30) This is also known as the transversally condition and implies that there is no longitudinal component of A. Maxwell’s equations then reduce to

2 r D=0: (F.31)

This has the solution Z 1 .x0/ .x/ D d3x0 ; (F.32) 40 jx x0j which is the instantaneous Coulomb interaction. In free space, the scalar potential can be eliminated by a gauge transformation. Then the Lorenz condition (F.21) is automatically fulﬁlled in the Coulomb gauge. The ﬁeld equation (F.4) then becomes

1 @2A r 2A D 0: (F.33) c2 @2t The relativistic interaction with an atomic electron (D.41) is then in the Coulomb gauge given by Hint D ec ˛ A? (F.34) and in second quantization (see Appendix B) X HOint D cj hijec ˛ A?jj i cj ; (F.35) ij F.2 Quantized Radiation Field 317 where c;c represent creation/annihilation operators for electrons. In the interaction picture, this becomes X i."i "j /t=„ HOint;I.t/ D ci hijec ˛ A?jj i cj e : (F.36) ij

F.2 Quantized Radiation Field

F.2.1 Transverse Radiation Field

Classically the transverse components of the radiation ﬁeld can be represented by the vector potential [3, Eq. 2.14]

X X2 h i i.kx!t/ i.kx!t/ A.x;t/D ckp "p e C ckp "p e ; (F.37) k pD1

where k is the wave vector, ! D cjkj the frequency, and ckp =ckp represents the amplitude of the wave with a certain k vector and a certain polarization "p.The energy of this radiation can be shown to be equal to [3, p.22] X X 2 2 Erad D 20 ! ckp ckp D 0 ! ckp ckp C ckr ckp : (F.38) kp kp

By making the substitution s s

„ „ ckp ! akp and ckp ! akp; 20 !V 20 !V

where akp;=akp are photon creation/annihilation operators, the radiation energy goes over into the hamiltonian of a collection of harmonic oscillators X 1 Hharm:osc D „!.akp akp C akp akp/: 2 kp

Therefore, we can motivate that the quantized transverse radiation ﬁeld can be represented by the operator [3, Eq. 2.60] s X X2 h i „ i.kx!t/ i.kx!t/ A?.x;t/D akp "p e C akp "p e : (F.39) 20!V k pD1 318 F Semiclassical Theory of Radiation F.2.2 Breit Interaction

The exchange of a single transverse photon between two electrons is illustrated by the time-ordered diagram (left) in Fig. F.1, where one photon is emitted at the time t1 and absorbed at a later time t2. The second-order evolution operator for this process, using the interaction picture, is given by (see Sect. 3.1)

2 Z 0 Z 0 .2/ i .t1Ct2/ U .0; 1/ D dt2 HOint;I.t2/ dt1 HOint;I.t1/ e ; (F.40) „ 1 1 where is the parameter for the adiabatic damping of the perturbation. The interaction Hamiltonians are in the Coulomb gauge given by (F.36) with the vector potential (F.39) s X X2 ˝ ˇ ˇ ˛ „ ˇ ikx ˇ HOint;I.t1/ D cr r akp ec ˛ "kp e a 20!1V 1 k1 p1D1

it1."a"p „!1/=„ case ; 2 X „ X ˝ ˇ ˇ ˛ O ˇ ˛ " ikx ˇ Hint;I.t2/ D cs r akp ec kp e 2 a 20!2V k2 p2D1

it2."b "s C„!2/=„ cb e ; (F.41) which leads to the evolution operator X 2 2 .2/ e c U .0; 1/ Dcr cacs cb p 0 1 2 k 2„ V ! ! X ˝ ˇ ˇ ˛ ˇ ikx ikx ˇ rs akp ˛ "p e akp ˛ "p e ab I; 2 1 p1p2 (F.42) where I is the time integral. The contraction between the creation and annihilation operators (G.10) yields .! D !1 D !2/

s 2 r s r B12 6 2 1 b 1 a b a

Fig. F.1 Diagrammatic representation of the exchange of a single, transverse photon between two electrons (left). This is equivalent to a potential (Breit) interaction (right) F.2 Quantized Radiation Field 319 X ˝ ˇ ˇ ˛ ˇ ikx ikx ˇ rs akp ˛ "p e akp ˛ "p e ab 2 1 p1p2 X2 ˝ ˇ ˇ ikr12 D rs .˛ "p/2 .˛ "p/1 e .r12 D x1 x2/: (F.43) pD1

The time integral in (F.42)is Z Z 0 t2 it2."b "sC„!Ci/=„ it1."a"r „!Ci/=„ I D dt2 e dt1 e 1 1 1 D (F.44) .cq C cq0 C 2i/.cq ! C i/

0 with cq D ."a "r /=„ and cq D ."b "s/=„. The result of the opposite time ordering t1 >t2 is obtained by the exchange 1 $ 2.r12 $r12/, a $ b,andr $ s, and the total evolution operator, including both time-orderings, can be expressed as

2 2 X X2 ˝ ˇ ˇ ˛ .2/ e c ˇ ˇ U .0; 1/ D cr cacs cb rs .˛ "p/1.˛ "p/2 M ab 2„0!V k pD1 (F.45) with eikr12 eikr12 MD C : .cq C cq0 C 2i/.cq ! C i/ .cq C cq0 C 2i/.cq0 ! C i/ (F.46)

This can be compared with the evolution operator corresponding to a potential interaction B12 between the electrons, as illustrated in the right diagram of Fig. F.1,

Z 0 .2/ i it."aC"b "r "s Ci/=„ U .0; 1/ D cr cacs cb hrsjB12jabi dt e „ 1 cr cacs cb hrsjB12jabi D : (F.47) „ cq C cq0 C i

Identiﬁcation then leads to e2c2 X eikr12 eikr12 12 ˛ "p 1 ˛ "p 2 B D . / . / C 0 : (F.48) 20!V kp cq ! C i cq ! C i

We assume now that energy is conserved by the interaction, i.e.,

0 "a "r D "s "b or q C q D 0: (F.49) 320 F Semiclassical Theory of Radiation

It is found that the sign of the imaginary part of the exponent is immaterial (see Appendix J.2), and the equivalent interaction then becomes

2 X i kr12 e ˛ " ˛ " e B12 D . p/1. p/2 2 2 (F.50) 0 V kp q k C i with ! D ck. The "p vectors are orthogonal unit vectors, which leads to [3, Eq. 4.312] X3 .˛ "p/1.˛ "p/2 D ˛1 ˛2: (F.51) pD1

This gives X2 .˛ "p/1.˛ "p/2 D ˛1 ˛2 .˛1 kO /.˛2 kO / (F.52) pD1 assuming "3 D kO to be the unit vector in the k direction. The interaction (F.50)then becomes in the limit of continuous momenta (Appendix D) Z h i 2 3k i kr12 e d ˛ ˛ ˛ kO ˛ kO e B12 D 3 1 2 . 1 /. 2 / 2 2 : (F.53) 0 .2/ q k C i

With the Fourier transforms in Appendix J, this yields the retarded Breit interaction " # e2 eijqjr12 eijqjr12 1 Ret ˛ ˛ ˛ r ˛ r B12 D 1 2 . 1 1/. 2 2/ 2 : (F.54) 40 r12 q r12

Setting q D 0, we obtain the instantaneous Breit interaction (real part)

2 ˛ ˛ Inst e 1 2 1 B12 D C .˛1 r1/.˛2 r2/r12 40 r12 2 or using

˛1 ˛2 .˛1 r12/.˛1 r12/ ˛ r ˛ r . 1 1/. 2 2/r12 D C 2 ; r12 r12 we arrive at 2 ˛ r ˛ r Inst e 1 . 1 12/. 1 12/ B12 D ˛1 ˛2 C ; (F.55) 40 r12 2 2r12 which is the standard form of the instantaneous Breit interaction. F.2 Quantized Radiation Field 321 F.2.3 Transverse Photon Propagator

s r r 2 1 s r s - 6 + 6 = 12 1 b a 2 a b a b

Fig. F.2 The two time-orderings of a single-photon exchange can be represented by a single Feynman diagram

We shall now consider both time-orderings of the interaction represented in Fig. F.2 simultaneously. The evolution operator can then be expressed as

2 Z 0 Z 0 h i .2/ i .jt1jCjt2j/ U .0; 1/ D dt2 dt1 T HOint;I.t2/ HOint;I.t1/ e ; „ 1 1 (F.56) where 8 h i < HOint;I.t2/ HOint;I.t1/t2 >t1 O O T Hint;I.t2/ Hint;I.t1/ D : : (F.57) HOint;I.t1/ HOint;I.t2/t1 >t2 In the Coulomb gauge, the interaction is given by (F.36) and the vector potential is given by (F.39). The evolution operator for the combined interactions will then be Z Z e2c2 0 0 .2/ ˛ A ˛ A U .0; 1/ Dcr cacs cb 2 dt2 dt1 TŒ. ?/1 . ?/2 „ 1 1 eit1."a"r Ci/=„ eit2."b "s Ci/=„: (F.58)

Here X „ TŒ.˛ A?/1 .˛ A?/2 D .˛ "p/1 .˛ "p/2 kp 2!0V 8 i.k1x1!t1/ i.k2x2!t2/ < e e t2 >t1 : i.k2x2!t2/ i.k1x1!t1/ e e t1 >t2 or with r12 D x1 x2 and t12 D t1 t2

2 „ X 1 X ei.kr12!t12/ TŒ.˛ A?/1 .˛ A?/2 D .˛ "p/1 .˛ "p/2 ; 0 V 2! pD1 k (F.59) 322 F Semiclassical Theory of Radiation where the upper sign is valid for t2 >t1. This yields

2 2 Z 0 Z 0 X2 .2/ e c ˛ " ˛ " U .0; 1/ Dcr cacs cb 2 dt2 dt1 . p/1 . p/2 0„ 1 1 pD1 1 X ei.kr12!t12/ eicq t12 e.t1Ct2/ (F.60) V k 2! using the energy conservation (F.49). The boxed part of the equation above is essentially the photon propagator (4.23) Z 1 X ei.kr12!t12/ d3k ei.kr12!t12/ DF.1; 2/ D ) 3 : (F.61) V k 2! .2/ 2!

This can be represented by a complex integral Z Z d3k dz eizt12 D .1; 2/ D i eikr12 ; (F.62) F .2/3 2 z2 !2 C i where is a small, positive quantity. As before, the sign of the exponent ik r12 is immaterial. The integrand has poles at z D˙.! i/, assuming ! to be positive. 1 i!t12 ikr12 For t2 >t1, integration over the negative half plane yields 2! e e and for 1 i!t12 ikr12 t1 >t2 integration over the positive half plane yields 2! e e ,whichis identical to (F.61). The evolution operator (F.60) can then be expressed as

2 2 Z 0 Z 0 .2/ e c U .0; 1/ Dcr cacs cb dt2 dt1 0„ 1 1 X2 icq t12=„ .t1Ct2/„ .˛ "p/1 .˛ "p/2 DF.1; 2/ e e : pD1 (F.63)

F.2.4 Comparison with the Covariant Treatment

It is illuminating to compare the quantization of the transverse photons with the fully covariant treatment, to be discussed in the next chapter. Then we simply have to replace the sum in (F.42) by the corresponding covariant expression X2 X3 .akp ˛ "p/1 .akp ˛ "p/2 ) .akp ˛ "p /1 .akp ˛ "p /2: p1p2D1 p1p2D0 (F.64) F.2 Quantized Radiation Field 323

The commutation relation (G.10) yields X3 .akp ˛ "p /1 .akp ˛ "p/2 D ˛1 ˛2 1: (F.65) p1p2D0

We then ﬁnd that the equivalent potential interaction (F.50) under energy conservation is replaced by Z 2 3k i kr12 e d ˛ ˛ e V12 D 3 .1 1 2/ 2 2 (F.66) 0 .2/ q k C i and with the Fourier transform given in Appendix J.2

2 e ijqjr12 V12 D .1 ˛1 ˛2/ e : (F.67) 40 r12

We shall now compare this with the exchange of transverse photons, treated above. We then make the decomposition 8 < 1 .˛1 kO /.˛2 kO / ˛ ˛ 1 1 2 D : : (F.68) ˛1 ˛2 C .˛1 kO /.˛2 kO /

The last part, which represents the exchange of transverse photons, is identical to (F.52), which led to the Breit interaction. The ﬁrst part, which represents the exchange of longitudinal and scalar photons, corresponds to the interaction Z h i e2 3k i kr12 d ˛ kO ˛ kO e VC D 3 1 . 1 /. 2 / 2 2 : (F.69) 0 .2/ q k C i

This Fourier transform is evaluated in Appendix J.3 and yields Z Z e2 d3k q2 ei kr12 e2 d3k ei kr12 VC D 3 1 2 2 2 D 3 2 (F.70) 0 .2/ k q k C i 0 .2/ k i provided that the orbitals are generated in a local potential. Using the transform in Appendix J.2, this becomes

e2 VCoul D : (F.71) 40 r12 Thus, we see that the exchange of longitudinal and scalar photons corresponds to the instantaneous Coulomb interaction, while the exchange of the transverse photons corresponds to the Breit interaction. Note that this is true only if the orbitals are generated in a local potential. 324 F Semiclassical Theory of Radiation

If instead of the separation (F.68), we would separate the photons into the scalar part .p D 0/ and the vector part .p D 1; 2; 3/, 8 < 1 ˛ ˛ 1 1 2 D : ; (F.72) ˛1 ˛2 then the result would be

2 Ret e ijqjr12 VCoul D e ; 4õ0 r12 e2 Ret ˛ ˛ ijqjr12 VGaunt D 1 2 e ; (F.73) 40 r12 which represents the retarded Coulomb and the retarded magnetic (Gaunt) interaction.Thisimpliesthatthe longitudinal photon represents the retardation of the Coulomb interaction, which is included in the Breit interaction (F.54). If we would set q D 0, then we would from (F.73)retrievetheinstantaneous Coulomb interaction (F.71)and

e2 ˛1 ˛2; (F.74) 40 which is known as the Gaunt interaction. The Breit interaction will then turn into the instantaneous interaction (F.55). This will still have some effect of the retardation of the Coulomb interaction, although it is instantaneous. We shall see later that the interactions (F.73) correspond to the interactions in the Feynman gauge (4.56), while the instantaneous Coulomb and Breit incinerations correspond to the Coulomb gauge.

References

1. Lindgren, I., Morrison, J.: Atomic Many-Body Theory. Second edition, Springer-Verlag, Berlin (1986, reprinted 2009) 2. Mandl, F., Shaw, G.: Quantum Field Theory. John Wiley and Sons, New York (1986) 3. Sakurai, J.J.: Advanced Quantum Mechanics. Addison-Wesley Publ. Co., Reading, Mass. (1967) Appendix G Covariant Theory of Quantum ElectroDynamics

G.1 Covariant Quantization: GuptaÐBleuler Formalism

With the Lorenz condition (F.21) @A D 0, the Maxwell equations have a particularly simple form (F.22)

A D 0j: (G.1)

In this case, the covariant electromagnetic radiation ﬁeld can be expressed in analogy with the semiclassical expression (F.39) and represented by the four-component vector potential [4, Eq. 5.16] s X h i C „ ikx ikx A.x/ D A .x/ C A.x/ D "r akr e C akr e : 0 2! V kr (G.2)

However, different equivalent choices can be made, as further discussed in Sect. G.2. Here, we use the covariant notations

k D k D .k0; k/ I k0 D !=c DjkjI kx D !t k x defined in Appendix A.1,andr D .0;1;2;3/represents the four polarization states. Normally, the polarization vector for r D 3 is defined to be along the k vector Ð longitudinal component Ðandforr D 1; 2 to be perpendicular Ð transverse components. The component r D 0 is referred to as the time-like or scalar component (see Sect. F.1.2). The electromagnetic fields components are Heisenberg operators and should satisfy the canonical commutation (quantization) rules (E.24) at equal times 0 3 0 A.t; x/; .t; x / D i„ı; ı .x x /; (G.3) where .x/ is the conjugate field (E.21). With the Lagrangian (F.17)thefield0 vanishes according to the relation (F.19), which is inconsistent with the quantization

325 326 G Covariant Theory of Quantum ElectroDynamics

2 rule (G.3). In order to remedy the situation, we add a term 20 .@ A / to the Lagrangian (F.17), where is an arbitrary constant [3, Eq. 1-49, Eq. 3-98]

1 2 L D FF .@ A / j A: (G.4) 40 20

We can rewrite the extra term as

.@g A /.@A /: 20

Then the conjugate ﬁeld (F.19) becomes [3, Eq. 3-100]

@L 1 0 0 .x/ D 0 D F g @A (G.5) @.@ A/ c0 0 and 0 ¤ 0 for ¤ 0. The extra term in the EulerÐLagrange equations (F.14) leads to

@ @ .@g A /.@A / D @g .@ A / 20 @.@ A/ 0

D @g .@ A / D @ .@ A /: 0 The Maxwell equations (F.10) then take the modiﬁed form [3, Eq. 3-99]

@@ A .1 / @ .@A / D 0j : (G.6)

Setting D 1, we retrieve the same simple form of Maxwell’s equations as with the Lorenz condition (G.1)Ðwithout introducing this condition explicitly.Thisis usually referred to as the Feynman gauge. The Lagrangian (G.4) is incompatible with the Lorenz condition, and to resolve the dilemma this condition is replaced by its expectation value

hj@A ji D 0; (G.7) which is known as the GuptaÐBleuler proposal [4, 5.35]. In the Feynman gauge, the commutation relations (G.3) become [4, 5.23] 0 2 3 0 A.t; x/; AP .t; x / D ic 0„g ı; ı .x x /: (G.8)

To satisfy this relation, we can assume that the polarization vectors fulﬁll the or- thogonality/completeness relations [4, Eq. 5.18,19]

"r "r0 D grr0 ; X grr"r "r D g; (G.9) r G.2 Gauge Transformation 327 and the photon creation and absorption operators the commutation relation [4, Eq. 5.28] h i 0 0 akr ;ak0r0 Dık;k grr : (G.10)

Considering that the g-matric (A.5) used is diagonal, this leads to X h i X h i 0 0 0 "r akr ;"r ak0r0 D "r "r akr ;ak0r0 Dg;ık;k (G.11) rr0 rr0 and then it follows that the field operators (G.2) satisfy the commutation relation (G.8). With the Lagrangian (G.4) and the conjugate fields (G.5), the Hamiltonian of the free field (F.20) becomes in the Feynman gauge ( D 1)[4, Eq. 5.32] X HRad D „!grr akr akr : (G.12) k;r

G.2 Gauge Transformation

G.2.1 General

The previous treatment is valid in the Feynman gauge, where the Maxwell equations have the form (G.1), and we shall here investigate how the results will appear in other gauges. The interaction between an electron and the electromagnetic ﬁeld is given by the Hamiltonian interaction density (D.41)

Hint D jA ; (G.13) where j is the current density. The Maxwell equations are invariant for a gauge transformation (F.24) A ) A Cr, which transforms this interaction to @ Hint D jA ) A C j: @x

Integration over space leads after partial integration to Z Z 3 @ 3 @j d x j D d x D 0: @x @x 328 G Covariant Theory of Quantum ElectroDynamics

Since is arbitrary, it follows that

@j D ı j Drj D 0; @x which is the continuity equation (F.23). In analogy with (F.30), the corresponding relation in the k space is j.k/ k D 0: (G.14) The single-photon exchange is represented by the interaction (4.44)

2 2 I.x2;x1/ D e c ˛1 ˛2 DF.x2;x1/; (G.15)

which corresponds to the interaction density j DF j . In view of the relation (G.14), it follows that the transformation

DF.k/ ) DF.k/ C kf.k/ C kf.k/; where f.k/ and f.k/ are arbitrary functions of k, will leave the interaction un- changed.

G.2.2 Covariant Gauges

In a covariant gauge, the components of the electro-magnetic ﬁeld are expressed in a covariant way. We shall consider three gauges of this kind.

G.2.2.1 Feynman Gauge

The photon propagator in the Feynman gauge is given by the expression (4.28)

g 1 DF.k/ D 2 : (G.16) c0 k C i

G.2.2.2 Landau Gauge

With 1 k f.k/ D 2 2 ; c0 .k C i/ the propagator (G.16) becomes 1 1 kk DF.k/ D 2 g 2 : (G.17) c0 k C i k C i G.2 Gauge Transformation 329

This leads to k DF D 0, which is consistent with the Lorenz condition (F.21)

rA D @ A D 0:

G.2.2.3 FriedÐYennie Gauge

With 1 k f.k/ D .1 / 2 2 ; 2c0 .k C i/ the propagator (G.16) becomes 1 1 kk DF .k/ D 2 g .1 / 2 : (G.18) c0 k C i k C i

With D 1, this yields the Feynman gauge, and with D 0 we retrieve the Lan- dau gauge. The value D 3 yields the FriedÐYennie gauge [2], which has some improved properties, compared to the Feynman gauge, in the infrared region.

G.2.3 Noncovariant Gauge

We consider only one example of a noncovariant gauge, the Coulomb gauge, which is of vital importance in treating the combined QED-correlation problem. Here, the Coulomb interaction is treated differently from the transverse part.

G.2.3.1 Coulomb Gauge

With

1 1 k0 1 1 ki f0 D 2 2 I fi D 2 2 .i D 1; 2; 3/; 2c0 k C i k 2c0 k C i k the propagator (G.16) can be expressed as ! 1 1 k2 0 DF00.k/ D 1 ki kj ; (G.19) c0 0 ıi;j k2Ci k2 where the ﬁrst row/column corresponds to the component D 0 and the second row/column to D 1; 2; 3. i This leads to k DFij D 0, which is consistent with the Coulomb condition (F.30)

i rA D @ Ai D 0.iD 1; 2; 3/: 330 G Covariant Theory of Quantum ElectroDynamics

The formulas above can be generalized to be used in dimensional regularization (see Sect. 12.4), where the number of dimensions is noninteger (mainly from Adikins [1], see also t’Hooft and Veltman [6]). Following the book by Peskin and Schroeder [5], we can by means of Wick rotation evaluate the integral

Z D Z D d l 1 m d l 1 D 2 m D i.1/ D 2 m .2/ .l / .2/ .lE C / Z Z 1 D1 m d˝D 0 lE D i.1/ 4 dlE 2 m : .2/ 0 .lE C /

0 0 We have here made the replacements l D ilE and l D lE and rotated the inte- o gration contour of lE 90 , which with the positions of the poles should give the D same result. The integration over d lE is separated into an integration over the 0 D-dimensional sphere ˝D and the linear integration over the component lE .This corresponds in three dimensions to the integration over the two-dimensional angular coordinates and the radial coordinate (see below).

Z dDk 1 i.1/n .n D=2/ 1 D ; (G.20) .2/D .k2 C s C i/n .4/D=2 .n/ snD=2 Z k d4k D 0; (G.21) .k2 C s C i/n Z dDk kk i.1/n .n D=2 1/ 1 D : (G.22) .2/D .k2 C s C i/n .4/D=2 .n/ snD=21

G.2.3.2 Covariant Gauge

Compared to Adkins [1] (A1a), (A3), and (A5a):

2 p !qI M !sI ! ! D=2I ˛ ! nI ! nI Q D p !qI A ! gI

! w D q2 s; Z dDk 1 i.1/n 1 .n D=2/ D ; (G.23) .2/D .k2 C 2kq C s C i/n .4/D=2 .n/ wnD=2

Z dDk k i.1/n 1 .n D=2/ D q ; (G.24) .2/D .k2 C 2kq C s C i/n .4/D=2 .n/ wnD=2 G.2 Gauge Transformation 331 Z dDk kk .2/D .k2 C 2kq C s C i/n i.1/n 1 .n D=2/ g .n 1 D=2/ D qq : (G.25) .4/D=2 .n/ wnD=2 2 wn1D=2

G.2.3.3 Noncovariant Gauge

Compared to Adkins [1] (A1b), (A4), and (A5b):

p !qI M 2 !sI ! ! D=2I ˛ ! nI ˇ ! 1I ! n C 1I k2 !k2I Q D py !qyI 1 y A ! g C ı;0ı;0 I .AQ/ !q y ı0 .1 y/q0; y

2 2 2 2 ! w D q y C .1 y/yq0 sy C .1 y/ 2 2 2 2 Dq y C yq0 sy C .1 y/; Z dDk 1 1 .2/D .k2 C 2kq C s C i/n k2 2 n Z 1 i.1/ 1 n11=2 .nC 1 D=2/ D D=2 dyy n 1 D=2 ; (G.26) .4/ .nC 1/ 0 w C Z dDk k 1 .2/D .k2 C 2kq C s C i/n k2 2 Z n 1 i.1/ 1 n11=2 .nC1D=2/ D D=2 dyy q y C ı;0 q0.1 y/ n 1 D=2 ; .4/ .n/ 0 w C (G.27)

0 0 6k !6qy q0.1 y/ D q y q0;

Z dDk kk 1 .2/D .k2 C 2kq C s C i/n k2 2 Z n 1 ˚ i.1/ 1 n11=2 D D=2 dyy q y C ı;0 q0.1 y/ .4/ .n/ 0

.nC 1 D=2/ Œq y C ı;0 q0.1 y/g n 1 D=2 w C ˚ 1 .n D=2/ g C ı;0ı;0.1 y/=y (G.28) 2 wnD=2 332 G Covariant Theory of Quantum ElectroDynamics 0 1 6k ! q y q0 in ﬁrst part and 6k 6k ! C .1 y/=y in second. 2 Z dDk ki kkj 1 .2/D .k2 C 2kq C s C i/n k2 2 Z n 1 ˚ i.1/ 1 n11=2 i j 3 i j 2 D D=2 dyy q q q y C q q q ı0.1 y/y .4/ .n/ 0 ˚ .nC 1 D=2/ 1 i j j i ji n 1 D=2 C y g q C g q C g q w C 2 ij .n D=2/ ı0 g q0.1 y/ (G.29) wnD=2

G.3 Gamma Function

The Gamma function can be deﬁned by means of Euler’s integral Z 1 .z/ D dttz1et : (G.30) 1

For integral values, we have the relation

.n/D .n 1/Š (G.31) and generally .z/ D .z 1/ .z 1/: (G.32) The Gamma function can also be expressed by means of

Y1 1 z =n D ze Ez 1 C ez ; (G.33) .z/ n nD1 where E is Euler’s constant, E D 0:5772 : : : The Gamma function is singular, when z is zero or equal to a negative integer. Close to zero, the function is equal to

1 ./D C O./; (G.34) E which follows directly from the expansion above. We shall now derive the corresponding expression close to negative integers. G.3 Gamma Function 333 G.3.1 z D1

Y1 1 .1 / 1 C .1 /=n D.1 C / e E C 1 e C (G.35) .1 / n nD1 Q The ﬁrst few factors of the product are (to orders linear in )

e1C D e1.1 C /; 1 C 1 1 e.1C/=2 D .1 / e1=2.1 C =2/; 2 2 1 C 2 1 e.1C/=3 D .1 =2/ e1=3.1 C =3/; 3 3 1 C 3 1 e.1C/=4 D .1 =3/ e1=4.1 C =4/; 4 4 which in the limit becomes e E 1 C Œ1 1=2 1=.2 3/ 1=.3 4/ e E using the expansion

1 C 1=2 C 1=3 C 1=4 CC1=M ) ln M C E: (G.36)

This gives 1 .1 / D C 1 C O./: (G.37) E This can also be obtained from (G.32).

G.3.2 z D2

Y1 1 .2 / 2 C .2 /=n D.2 C / e E C 1 e C : (G.38) .2 / n nD1 Q The ﬁrst few factors of the product are (to orders linear in )

.1 C / e2C D.1 C / e2.1 C /;

=2 e.2C/=2 D=2 e1.1 C =2/; 334 G Covariant Theory of Quantum ElectroDynamics 2 C 1 1 e.2C/=3 D .1 / e2=3.1 C =3/; 3 3 2 C 2 1 e.2C/=4 D .1 =2/ e2=4.1 C =4/; 4 4 2 C 3 1 e.2C/=5 D .1 =3/ e2=5.1 C =5/; 5 5 which in the limit becomes

2 2 e E .1 C Œ5=2 2=.1 3/ 2=.2 4/ 2=.3 5/ / e E .1 C /:

This gives 1 1 .2 / D C 1 1=2 C O./ : (G.39) 2 E This is consistent with the formula

.2z/ D .2/1=222z1=2.z/ .z C 1=2/: (G.40)

The step-down formula (G.32) yields 1 1 .3 / D C 1 1=2 1=3 ; (G.41) 2 3 E which can be generalized to

" # N .1/N 1 1 X 1 .N / D C : (G.42) NŠ E n nD1

References

1. Adkins, G.: One-loop renormalization of Coulomb-gauge QED. Phys.Rev.D27, 1814Ð20 (1983) 2. Fried, H.M., Yennie, D.M.: New techniques in the Lamb shift calculation. Phys. Rev. 112, 1391Ð1404 (1958) 3. Itzykson, C., Zuber, J.B.: Quantum Field Theory. McGraw-Hill (1980) 4. Mandl, F., Shaw, G.: Quantum Field Theory. John Wiley and Sons, New York (1986) 5. Peskin, M.E., Schroeder, D.V.: An introduction to Quantun Field Theory. Addison-Wesley Publ. Co., Reading, Mass. (1995) 6. t’Hooft, G., Veltman, M.: Regularization and renormalization of gauge ﬁelds. Nucl. Phys. B 44, 189Ð213 (1972) Appendix H Feynman Diagrams and Feynman Amplitude

In this appendix, we shall summarize the rules for evaluatiig Feynman diagrams of the different schemes, discussed in this book. These rules are based on the rules formulated by Feynman for the so-called Feynman amplitude, a concept we shall also use here.

H.1 Feynman Diagrams

H.1.1 S-Matrix

The Feynman diagrams for the S-matrix have an outgoing orbital line for each electron-ﬁeld creation operator

O 66 .x/ and an incoming orbital line for each electron-ﬁeld absorption operator

O 66 .x/ and a vertex diagram

A iec˛ A for each interaction point.

335 336 H Feynman Diagrams and Feynman Amplitude

This leads to an electron propagator for each contracted pair of electron-ﬁeld operators:

2 R O O d! x x ! 66 .x1/ .x2/ D i SF.x2;x1/ D i 2 SF.!I 2; 1/ 1 and a photon propagator for each contracted pair of photon-ﬁeld operators: R -z dz x x 1, ,2 A.x1/A.x2/ D i DF.x2;x1/ D i 2 DF.zI 2; 1/.

Thus, there is a photon interaction (4.45), including the vertices,

R R -z dz x x 2 2 x x ; 1 ; 2 2 .i/I.zI 2; 1/ D dz .i/e c ˛1 ˛2 DF.zI 2; 1/ for each photon exchange, and a corresponding diagram

2 iV Di e 1 2 C 40r12 for each Coulomb interaction, VC, between the electrons, and a potential diagram

iV for each energy potential, V (Fig. 4.6).

H.1.2 Green’s Function

The Feynman diagrams of the Green’s function are identical to those of the S-matrix with the exception that all outgoing and incoming lines represent electron orbitals.

H.1.3 Covariant Evolution Operator

The Feynman diagrams of the Covariant evolution operator are identical to those of the Green’s function with the exception that there are creation/absorption operator lines attached to all outgoing/incoming orbital lines. H.2 Feynman Amplitude 337 H.2 Feynman Amplitude

The Feynman amplitude, M, contains An electron propagator (4.10) Z d! i S .!I x2; x1/ 2 F

for each internal orbital line. A photon interaction Z Z dz dz 2 2 .i/I.zI x2; x1/ D .i/e c ˛ ˛2 D .zI x2; x1/ 2 2 1 F

for each photon exchange, including the vertices. At each vertex space integrations and a time integral 2 .arg/,wherethe argument is equal to incoming minus outgoing energy parameters. A factor of 1 and a trace symbol for each closed orbital loop. The integration over the energy parameters leads to a factor of i for each “nontrivial.” (The integral is considered to be “trivial,” when it contains a function from the time integration.) The S-matrix is related to the Feynman amplitude by (4.110)

S D 2 ı.Ein Eout/ M; (H.1) which gives the energy contribution (4.111)

M E D ıEin;Eout hji ji: (H.2)

The Green’s function is in the equal-time approximation related to the Feynman amplitude, M, by (5.25)

0 0 it.E0H0/ 0 0 it0.E0H0/ G.x; x I x0;x0/ D e M.x; x I x0; x0/ e (H.3) and analogously for the covariant-evolution operator (6.10)

U.t;1/jabi D eit.E0H0/ jrsi hrsjiMjabi eit0.E0H0/: (H.4)

In this formalism, the contribution to the effective interaction is

hjVeffjiDhjMji: (H.5) 338 H Feynman Diagrams and Feynman Amplitude

Illustrations

6 6 6 6 iSF 6 6iSF iSF 6 6iSF - iSF 6 6iSF - i Vsp - i Vsp i Vsp 6 6 iSF 6 6iSF 6 6 6 6 A:i B:1 C:i

6 6 6 6 6 6 6 6 iI Pair iI Pair iSF 6 6iSF iSF 6 6iSF iSF 6 6iSF - - - i Vsp i Vsp i Vsp iSF 6 6iSF iSF 6 6iSF iSF 6 6iSF iI Pair iI Pair iI Pair 6 6 6 6 6 6 D:1 E:i F:1

Diagram A is a ﬁrst-order S-matrix (Sect. 4.4), and the Feynman amplitude is

M DiVsp.E0/:

Diagram B is a ﬁrst-order covariant evolution-operator diagram with the unper- turbed state as input and with outgoing electron propagators (Sect. 6.2). Here, there are three energy parameters and two subsidiary conditions. This leaves one nontrivial integration, giving a factor of i. This gives a factor of i2.i/2 D 1 and

M D .E0/Vsp.E0/:

Diagram C is a ﬁrst-order covariant evolution-operator diagram with incoming and outgoing electron propagators (Sect. 8.1). Here, there are ﬁve parameters and three conditions, leaving two nontrivial integrations. This gives the factor i4.i/3 D iand(8.9)

M D .E0/ iVsp.E0/.E0/:

Diagram D is a ﬁrst-order covariant evolution-operator diagram with incoming pair function (Sect. 6.2.1). This gives i4.i/4 D 1

Pair Pair M D I .E0/Vsp.E0/ .E0/I : H.2 Feynman Amplitude 339

Diagram E is an S-matrix diagram with incoming and outgoing pair functions (Sect. 6.2.1). This gives i4.i/5 Diand

Pair Pair M DiI .E0/Vsp.E0/ .E0/I :

Diagram F is a ﬁrst-order covariant evolution-operator diagram with incoming and outgoing pair functions (Sect. 6.2.1). Here, there are seven parameters and four subsidiary conditions, yielding i6.i/6 D 1 and

Pair Pair M D .E0/I .E0/Vsp.E0/ .E0/I : Appendix I Evaluation Rules for Time-Ordered Diagrams

In nonrelativistic (MBPT) formalism, all interaction times are restricted to the interval .t; 1/, and the Goldstone diagrams are used for the graphical representation. In the relativistic (QED) formalism, on the other hand, times are allowed in the entire interval .1; 1/, and then Feynman diagrams, which contain all possible time orderings, are the relevant ones to use. For computational as well as illustrative purpose, it is sometimes useful also in the relativistic case to work with time-ordered diagrams. It should be observed, though, that time-ordered Feynman diagrams are distinct from Goldstone diagrams, as we shall demonstrate here.1 When only particles states (above the Fermi level) are involved, time runs in the positive direction, and the time-evolution operator can be expressed as (3.12) Z Z Z t t t1 2 U.t;1/ D 1 i dt1 V.t1/ C .i/ dt1 V.t1/ dt2 V.t2/ C ; 1 1 1 (I.1) where V.t/ is the perturbation in the interaction picture (3.16) Z 3 V.t/ D d x O .x/ec˛ A.x/ O .x/: (I.2)

Core states and negative energy states are regarded as hole states below the Fermi level with time running in the negative direction. Then, the corresponding time integration should be performed in the negative direction.

1 The treatment here is partly based on that in [2, Appdices C and D].

341 342 I Evaluation Rules for Time-Ordered Diagrams I.1 Single-Photon Exchange

t r s -z 2 1 a b E We consider ﬁrst the time-ordered diagram for single-photon exchange with only particle states involved. The time restrictions are here

t>t2 >t1 > 1; which corresponds to the evolution operator (I.1) Z Z t t2 2 it2d2 it1d1 .i/ dt2 V.t2/ e dt1 V.t1/ e : (I.3) 1 1

The contraction of the radiation-ﬁeld operators gives rise to a photon propagator (4.18)

A.x2/A.x1/ D i DF.x2;x1/ and this leads to the interaction (4.46)

Z Z 2 x1 x2 2 2 dz iz.t2t1/ 2c k dkf.kI ; / I.x2;x1/ D e c ˛ ˛2 D .x2;x1/ D e : 1 F 2 z2 c2k2 C i

The time dependence at vertex 1 then becomes eit1d1 ,where

d1 D "a "r z C i:

This parameter is referred to as the vertex value and given by the incoming minus the outgoing orbital energies/energy parameters at the vertex. Similarly, we deﬁne

d2 D "b "s C z C i d12 D d1 C d2 D E "r "s (I.4) with E D "a C "b. This leads to the time integrals Z Z t t2 itd12 2 it2d2 it1d1 e 1 .i/ dt2 e dt1 e D : (I.5) 1 1 d12 d1 I.2 Two-Photon Exchange 343

Together with the opposite time ordering, t>t1 >t2 > 1, the denominators become 1 1 1 1 1 1 C D C : (I.6) d12 d1 d2 E "r "s "a "r z C i "b "s C z C i

This leads to the Feynman amplitude (6.11) Z dz 1 1 1 Msp D i C 2 E "r "s "a "r z C i "b "s C z C i Z 2c2k dkf.k/ (I.7) z2 c2k2 C i or 1 hrsjMspjabi D hrsjVsp.E/jabi (I.8) E "r "s with Z 1 1 Vsp.E/ D c dkf.k/ C : (I.9) "a "r ck C i "b "s ck C i

If the interaction is instantaneous, then the time integral becomes Z t itd12 it.E"r "s / it12d12 e e i dt12 e D D ; (I.10) 1 d12 E "r "s which for t D 0 is the standard MBPT result.

I.2 Two-Photon Exchange

Next, we consider the diagrams in Fig. I.1. We extend the deﬁnitions of the vertex values:

0 0 d1 D "a "t z I d2 D "b "u C z I d3 D "t "r z d4 D "u "s C z ;

0 0 d12 D d1 C d2 D E "t "u I d13 D "a "r z z I d24 D "b "s C z C z ;

0 0 d123 D E "r "u z I d124 D E "t "s C z I d1234 D E "r "s; i.e., given by the incoming minus the outgoing energies of the vertex. There is a damping term ˙i for integration going to 1. 344 I Evaluation Rules for Time-Ordered Diagrams

r s r s r s t r s t t z 4 t r s - 0 r s r r s -z 4 1 -z 1 2 3 u t 4 t u z u t -z 2 3 u 3 1 1 t -z0 4 a b a 2 a 2 b b a 3 b

E D "a C "b E E E

Fig. I.1 Time-ordered Feynman diagrams for the two-photon ladder with only particle states (left) and with one and two intermediate hole states (right)

I.2.1 No Virtual Pair

We consider now the ﬁrst diagram above, where only particle states are involved. We assume that it is reducible, implying that the two photons do not overlap in time. Then the time ordering is

t>t4 >t3 >t2 >t1 > 1:

This leads to the time integrations Z Z Z Z t t4 t3 t2 4 it4d4 it3d3 it2d2 it1d1 .i/ dt4 e dt3 e dt2 e dt1 e 1 1 1 1 eitd1234 D : (I.11) d1234 d123 d12 d1

Changing the order between t1 and t2 and between t3 and t4 leads to the denominators 1 1 1 1 1 1 C C : (I.12) d1234 d123 d124 d12 d1 d2

Here, all integrations are being performed upward, which implies that all denominators are evaluated from below. If the interaction 1Ð2 is instantaneous, then the integrations become Z Z Z t t4 t3 itd1234 3 it4d4 it3d3 it12d12 e .i/ dt4 e dt3 e dt12 e D 1 1 1 d1234 d123 d12 (I.13) I.2 Two-Photon Exchange 345 and together with the other time ordering eitd1234 1 1 1 C : (I.14) d1234 d123 d124 d12 If both interactions are instantaneous, we have Z Z t t34 itd1234 2 it34d34 it12d12 e .i/ dt34 e dt12 e D (I.15) 1 1 d1234 d12 consistent with the MBPT result [1, Sect. 12.2].

I.2.2 Single Hole

Next, we consider the two-photon exchange with a single hole, represented by the second diagram above. We still assume that the diagram is reducible, implying that the two photons do not overlap in time. The time ordering is now

t>t4 >t3 >t2 > 1 and 1 >t1 >t4; but the order between t1 and t is not given. If this is considered as a Goldstone diagram, all times (including t1) are restricted to tn

Note that the last integration is being performed in the negative direction, due to the core hole. This is illustrated in Fig. I.2 (left). Considered as a Feynman diagram, the time t1 can run to C1, which leads to Z Z Z Z t4 t t4 t3 it1d1 it4d4 it3d3 it2d2 dt1 e dt4 e dt3 e dt2 e 1 1 1 1 eitd1234 D : (I.17) d1234 d1 d23 d2

Here, the last integration is still performed in the negative direction, this time from C1 to t4, and this leads to a result different from the previous one. In the Goldstone case all denominators are evaluated from below, while in the Feynman case one of them is evaluated from above (see Fig. I.2, right). For diagrams diagonal in energy we have d1234 D 0, and hence d1 Dd234, which implies that in this case the two results are identical. 346 I Evaluation Rules for Time-Ordered Diagrams

t t r s r s d1234 d1234 ? ? 1 -z d234 1 -z d1 6 t t 4 d23 ? 4 d23 ? ? 3 d2 3 d2 ? ? a 2 a 2 b b

Goldstone Feynman

Fig. I.2 Time-ordered Goldstone and Feynman diagrams, respectively, for two-photon exchange with one virtual pair. In the latter case, one denominator (at vertex 1) is evaluated from above

Let us next consider the third diagram in Fig. I.1, where the time ordering is

t>t4 >t1 >t3 >t2 > 1:

Here, all times are limited from above in the Goldstone as well as the Feynman interpretation, and this leads in both cases to Z Z Z Z t 1 t1 t3 it4d4 it1d1 it3d3 it2d2 dt4 e dt1 e dt3 e dt2 e 1 t4 1 1 eitd1234 D : (I.18) d1234 d123 d23 d2

I.2.3 Double Holes

The last diagram in Fig. I.1, also reproduced in Fig. I.3, represents double virtual pair. Considered as a Goldstone diagram, the time ordering is

t>t2 >t1 >t4 >t3 > 1; which yields Z Z Z Z t t2 t1 t4 it2d2 it1d1 it4d4 it3d3 dt2 e dt1 e dt4 e dt3 e 1 1 1 1 eitd1234 D : (I.19) d1234 d134 d34 d3 I.3 General Evaluation Rules 347

t t r s d1234 r s d1234 ? ? 2 2 d134 d2 6 z z u ? u 1 d34 1 d12 6

t -z0 4 d3 ? t -z0 4 d3 ? ? a 3 b a 3 b

Goldstone Feynman

Fig. I.3 Time-ordered Goldstone and Feynman diagrams, respectively, for two-photon exchange with two virtual pairs. In the latter case, two denominators (at vertices 1 and 2) are evaluated from above

This is illustrated in Fig. I.3 (left). Considered as a Feynman diagram, we have instead 1 >t2 >t1, which leads to Z Z Z Z t t4 t4 t1 it4d4 it3d3 it1d1 it2d2 dt4 e dt3 e dt1 e dt2 e 1 1 1 1 eitd1234 D ; (I.20) d1234 d3 d12 d2 where two integrations are performed in the negative direction.

I.3 General Evaluation Rules

We can now formulate evaluation rules for the two types of diagrams considered here. For (nonrelativistic) Goldstone diagrams, the rules are equivalent to the standard Goldstone rules [1, Sect. 12.2] There is a matrix element for each interaction. For each vertex, there is a denominator equal to the vertex sum (sum of vertex values): incoming minus outgoing orbital energies and zCi for crossing photon line (leading to ck after integration) below a line immediately above the vertex. For particle/hole lines, the integration is performed in the positive/negative direction. 348 I Evaluation Rules for Time-Ordered Diagrams

For the relativistic Feynman diagrams, the same rules hold, with the exception that For a vertex where time can run to C1, the denominator should be evaluated from above with the denominator equal to the vertex sum above a line immediately below the vertex (with z i for crossing photon line, leading to Cck).

References

1. Lindgren, I., Morrison, J.: Atomic Many-Body Theory. Second edition, Springer-Verlag, Berlin (1986, reprinted 2009) 2. Lindgren, I., Salomonson, S., As«û en, B.: The covariant-evolution-operator method in bound- state QED. Physics Reports 389, 161Ð261 (2004) Appendix J Some Integrals

J.1 Feynman Integrals

In this section, we shall derive some integrals, which simplify many QED calcula- tions considerably (see the books of Mandl and Shaw [1, Chap. 10] and Sakurai [2, Appendix D]), and we shall start by deriving some formulas due to Feynman. We start with the identity Z 1 1 b dt D 2 : (J.1) ab b a a t

With the substitution t D b C .a b/x, this becomes Z Z 1 1 dx 1 dx D 2 D 2 : (J.2) ab 0 Œb C .a b/x 0 Œa C .b a/x

Differentiation with respect to a yields Z 1 1 xdx 2 D 2 3 : (J.3) a b 0 Œb C .a b/x

Similarly, we have Z Z 1 1 x 1 D 2 dx dy 3 abc 0 0 Œa C .b a/x C .c b/y Z Z d 1x 1 D 2 x dy 3 : (J.4) 0 0 Œa C .b a/x C .c a/y

Next we consider the integral Z Z Z 1 4 1 k 2 k dk0 d k 2 3 D 4 j j dj j 2 3 : .k C s C i/ 1 .k C s C i/

349 350 J Some Integrals

The second integral can be evaluated by starting with Z 1 dk0 i D p 2 2 2 1 k0 jkj C s C i jkj s evaluated by residue calculus, and differentiating twice with respect to s. The integral then becomes Z Z 1 3i2 jkj2djkj i2 d4k D D : (J.5) .k2 C s C i/3 2 .jkj2 C s/5=2 2s

The second integral can be evaluated from the identity

x2 1 s D .x2 C s/5=2 .x2 C s/3=2 .x2 C s/5=2 and differentiating the integral Z p dx p D ln x C x2 C s x2 C s yielding Z x2 1 D : .x2 C s/5=2 3s For symmetry reason, we ﬁnd Z k d4k D 0: (J.6) .k2 C s C i/3

Differentiating this relation with respect to k leads to Z Z kk g 1 i2g d4k D d4k D (J.7) .k2 C s C i/4 3 .k2 C s C i/3 6s using the relation (A.4). By making the replacements

k ) k C q qnd s ) s q2; the integrals (J.5)and(J.6) lead to Z 1 i2 d4k D ; (J.8) .k2 C 2kq C s C i/3 2.s q2/ Z Z k q i2q d4k D d4k D : .k2 C 2kq C s C i/3 .k2 C 2kq C s C i/3 2.s q2/ (J.9) R d 3k eikr12 J.2 Evaluation of the Integral .2/3 q2k2Ci 351

Differentiating the last relation with respect to q leads to Z kk i2 g 2qq d4k D C ‹‹: (J.10) .k2 C 2kq C s C i/4 12 s q2 .s q2/2

Differentiating the relation (J.8) with respect to s yields Z 1 i2 d4k D ; (J.11) .k2 C 2kq C s C i/4 6.s q2/2 which can be generalized to arbitrary integer powers 3 Z 1 .n 3/Š 1 d4k D i2 : (J.12) .k2 C 2kq C s C i/n .n 1/Š .s q2/n2 This can also be extended to nonintegral powers Z 1 .n 2/ 1 d4k D i2 (J.13) .k2 C 2kq C s C i/n .n/ .s q2/n2 and similarly Z k .n 2/ q d4k Di2 ; (J.14) .k2 C 2kq C s C i/n .n/ .s q2/n2 Z kk .n 3/ .2n 3/ qq: g d4k D i2 C ; .k2 C 2kq C s C i/n 2 .n/ .s q2/n2 .s q2/n3 (J.15)

Z d 3k eikr12 J.2 Evaluation of the Integral .2/3 q2 k2 C i

Using spherical coordinates k D .;;/, . Djkj/, we have with d3k D 2 d sin d d˚ and r12 Djx1 x2j Z Z Z 3k ik.x1x2/ 1 2 d e 2 d ir12 cos 3 2 2 D .2/ 2 2 d sin e .2/ q k C i 0 q C i 0 Z i 1 d eir12 eir12 D 2 2 2 4 r12 0 q C i Z i 1 d eir12 eir12 D 2 2 2 ; (J.16) 8 r12 1 q C i 352 J Some Integrals where we have in the last step used the fact that the integrand is an even function of . The poles appear at D˙q.1Ci=2q/.eir12 is integrated over the positive ir12 ˙iqr12 and e over the negative half-plane, which yields e =.4 r12/ with the upper sign for q>0. The same result is obtained if we change the sign of the exponent in the numerator of the original integrand. Thus, we have the result

Z Z 3 ˙ik.x1x2/ 1 ijqjr12 d k e 1 2 d sin.r12/ e 3 2 2 D 2 2 2 D : (J.17) .2/ q k C i 4 r12 0 q C i 4 r12

The imaginary part of the integrand, which is an odd function, does not contribute to the integral.

J.3 EvaluationZ of the Integral 3 ikr12 d k O O e ˛1 k ˛2 k .2/3 q2 k2 C i

The integral appearing in the derivation of the Breit interaction (F.53)is Z d3k eikr12 I2 D .˛1 k/.˛2 k/ .2/3 q2 k2 C i Z d3k eikr12 D.˛1 r1/.˛2 r2/ : (J.18) .2/3 k2.q2 k2 C i/

Using (J.16), we then have Z 1 d eir12 eir12 i ˛ r ˛ r I2 D 2 . 1 1/. 2 2/ 2 2 8 r12 1 .q C i/ Z 1 1 ˛ r ˛ r 2 d sin.kr12/ D 2 . 1 1/. 2 2/ 2 2 2 : (J.19) 4 r12 0 .q C i/

The poles appear at D 0 and D˙.q C i=2q/. The pole at D 0 can be treated with half the pole value in each half plane. For q>0, the result becomes

1 eiqr121 2 4 r12 q and for q>0the same result with q in the exponent. The ﬁnal result then becomes R d 3k O O eikr12 J.3 Evaluation of the Integral .2/3 ˛1 k ˛2 k q2k2Ci 353 Z 3k ikr12 ijqjr121 d ˛ k ˛ k e 1 ˛ r ˛ r e 3 . 1 /. 2 / 2 D . 1 1/. 2 2/ 2 .2/ q2 k C i 4r12 q

1 ˛ r ˛ r D 2 . 1 1/. 2 2/ 4 r12 Z 1 2 d sin.r12/ 2 2 2 : (J.20) 0 .q C i/

Assuming that our basis functions are eigenfunctions of the Dirac hamiltonian hOD, we can process this integral further. Then the commutator with an arbitrary function of the space coordinates is h i hOD;f.x/ D c ˛ pO f.x/ C ŒU; f .x/ : (J.21)

The last term vanishes if the potential U is a local function, yielding h i hOD;f.x/ D c ˛ pO f.x/ Dic ˛ rf.x/: (J.22)

In particular h i ikx ikx ikx hOD; e Dic ˛ re D c ˛ kO e : (J.23)

We then ﬁnd that ikx 1 ikx ikx .˛ r/1.˛ r/2 e D hD; e hD; e (J.24) c2 1 2 with the matrix element

ikx 2 ikx hrsj .˛ r/1.˛ r/2 e jabi D q e (J.25) using the notation in (F.49). The integral (J.18) then becomes Z Z d3k eikr12 d3k q2 eikr12 I2 D .˛1 kO /.˛2 kO / D .2/3 q2 k2 C i .2/3 k2 q2 k2 C i (J.26) provided that the orbitals are generated by a hamiltonian with a local potential. 354 J Some Integrals References

1. Mandl, F., Shaw, G.: Quantum Field Theory. John Wiley and Sons, New York (1986) 2. Sakurai, J.J.: Advanced Quantum Mechanics. Addison-Wesley Publ. Co., Reading, Mass. (1967) Appendix K Unit Systems and Dimensional Analysis

K.1 Unit Systems

K.1.1 SI System

The standard unit system internationally agreed upon is the SI system or System Internationale.1 The basis units in this system are given in the following table:

Quantity SI unit Symbol Length Meter m Mass Kilogram kg Time Second s Electric current Ampere A Thermodynamic temperature Kelvin K Amount of substance Mole mol Luminous intensity Candela cd

For the deﬁnition of these units, the reader is referred to the NIST WEB page (see footnote). From the basis units Ð particularly the ﬁrst four Ð the units for most other physical quantities can be derived.

K.1.2 Relativistic or “Natural” Unit System

In scientific literature, some simplified unit system is frequently used for conve- nience. In relativistic field theory the relativistic unit system is mostly used, where the first four units of the SI system are replaced by

1 For further details, see The NIST Reference on Constants, Units, and Uncertainty (http://physics.nist.gov/cuu/Units/index.html).

355 356 K Unit Systems and Dimensional Analysis

Quantity Relativistic unit Symbol Dimension Mass Rest mass of the electron m kg Velocity Light velocity in vacuum c ms1 Action Planck’s constant divided by 2 „ kg m2s1 2 4 1 3 Dielectricity Dielectricity constant of vacuum 0 A s kg m

In the table, the dimension of the relativistic units in SI units are also shown. From these four units, all units that depend only on the four SI units kg, s, m, A can be derived. For instance, energy that has the dimension kg m2m2 has the relativistic 2 unit mec , which is the rest energy of the electron (511 keV). The unit for length is „ D =2 0; 386 1012 m; mec where is Compton wavelength and the unit for time is 2c= 7; 77 104 s).

K.1.3 Hartree Atomic Unit System

In atomic physics, the Hartree atomic unit system is frequently used, based on the following four units

Quantity Atomic unit Symbol Dimension Mass Rest mass of the electron m kg Electric charge Absolute charge of the electron e As Action Planck’s constant divided by 2 „ kg m2 s1 2 4 1 3 Dielectricity Dielectricity constant of vacuum times 40 A s kg m 4

Here, the unit for energy becomes

me4 1H D 2 3 ; .40/ „ which is known as the Hartree unit and equals twice the ionization energy of the hydrogen atom in its ground state (27.2 eV). The atomic unit for length is

2 40„ a0 D me2 known as the Bohr radius or the radius of the ﬁrst electron orbit of the Bohr hydrogen model (0:529 1010 m). The atomic unit of velocity is ˛c,where

e2 ˛ D (K.1) 40c„ K.2 Dimensional Analysis 357 is the dimensionless ﬁne-structure constant (1/137,036). Many units in these two systems are related by the ﬁne-structure constant. For instance, the relativistic length unit is ˛a0.

K.1.4 cgs Unit Systems

In older scientiﬁc literature, a unit system, known as the cgs system, was frequently used. This is based on the following three units:

Quantity cgs Unit Symbol Length Centimeter cm Mass Gram g Time Second s

In addition to the three units, it is necessary to deﬁne a fourth unit to be able to derive most of the physical units. Here, two conventions are used. In the electrostatic version (ecgs), the proportionality constant of Coulombs law, 40, is set equal to unity, and in the magnetic version (mcgs) the corresponding magnetic constant, 0=4, equals unity. Since these constants have dimension, the systems cannot be used for dimensional analysis (see below). The most frequently used unit system of cgs type is the so-called Gaussian unit system, where electric units are measured in ecgs and magnetic ones in mcgs. This implies that certain formulas will look differently in this system, compared to a system with consistent units. For instance, the Bohr magneton, which in any consistent unit system will have the expression e„ B D 2m will in the mixed Gaussian system have the expression e„ B D 2mc which does not have the correct dimension. Obviously, such a unit system can easily lead to misunderstandings and should be avoided.

K.2 Dimensional Analysis

It is often useful to check physical formulas by means of dimensional analysis, which, of course, requires that a consistent unit system, like the SI system, is being 358 K Unit Systems and Dimensional Analysis used. Below we list a number of physical quantities and their dimension, expressed in SI units, which could be helpful in performing such an analysis. In most parts of the book, we have set „D1, which simpliﬁes the formulas. This also simpliﬁes the dimensional analysis, and in the last column below we have (after the sign )) listed the dimensions in that case.

kg m 1 Œforce D N D ) ; s2 ms kg m2 1 Œenergy D J D Nm D ) ; s2 s kg m2 Œaction; „ D Js D ) 1; s J kg m2 1 Œelectric potential D V D D ) ; As As3 As2 kg m 1 Œelectric ﬁeld; E D V=m D ) ; As3 Ams2

2 kg 1 Œmagnetic ﬁeld; B D Vs=m D ) ; As2 Am2s kg m 1 Œvector potential; A D Vs=m D ) ; As2 Ams kg m 1 Œmomentum;pD ) ; s m As Œcharge density;D ; m3 A Œcurrent density; j D ; m2

2 kg m 1 Œ0 D N=A D ) ; A2s2 A2ms 2 4 2 3 2 A s A s Œ0 D Œ1=0c D ) : kg m3 m

Fourier transforms Z dz iz.t2t1/ D .x1;x2/ D D .zI x1; x2/ e ; F 2 F ŒA.!; x/ D sŒA.x/; ŒA.!; k/ D sm3ŒA.x/; ŒA.k/ D m4ŒA.x/: K.2 Dimensional Analysis 359

Photon propagator

1 ŒD .x; x/ D ; F A2m2s2 s Œ0D .x; x/ D ; F m3

Œ0DF.k/ D sm; s Œ0D .k0; x/ D ; F m2 1 Œ0D .t; x/ D ; F m2 s2 Œ0D .z; x/ D z D ck0; F m3 2 Œ0DF.z; k/ D s ;

2 2 1 Œe c D .z; x/ D I.z; x/ D ; F s e2 m D : 0 s

Electron propagator

SOF.x; x/ ) s; 1 S .x; x/ ) ; F m3 s S .z; x/ ) ; F m3

SF.z; k/ ) s;

SF.k/ ) m:

S-matrix

S.x;x/O ) 1; S.z; x/ ) s; S.z; k/ ) m4: 360 K Unit Systems and Dimensional Analysis

Self energy

1 ˙.O z/ ) ; s 1 ˙.z; x/ ) ; sm3 1 ˙.z; k/ ) ; s m ˙.k/ ) : s2

Vertex .z; k/ ) 1; m .p; p0/ ) : s Abbreviations

CCA Coupled-cluster approach CEO Covariant evolution operator GML Gell-MannÐLow relation HP Heisenberg picture IP Interaction picture LDE Linked-diagram expansion MBPT Many-body perturbation theory MSC Model-space contribution NVPA No-virtual-pair approximation PWR Partial-wave regularization QED Quantum electrodynamics SCF Self-consistent ﬁeld SI International unit system SP Schr¬odinger picture

361 Index

A covariant evolution operator, 3, 119 Adiabatic damping, 51 covariant vector, 273 all-order method, 30 creation operator, 283 annihilation operator, 283 cut-off procedure, 248 antisymmetry, 22

B D Banach space, 277 d’Alembertian operator, 274 BetheÐSalpeter equation, 3, 119, 193, 199, 225 de Broglie’s relations, 13 effective potential form, 205 density operator, 120 BetheÐSalpeterÐBloch equation, 6, 156, 173, difference ratio, 139 193, 199, 206 dimensional analysis, 355 Bloch equation dimensional regularization, 163 for Green’s operator, 146 Dirac delta function, 277 generalized, 20, 23 Dirac equation, 1, 295 bra vector, 290 Dirac matrices, 296 Breit interaction, 38, 73, 318 Dirac sea, 38 BrillouinÐWigner expansion, 153 Dirac-Coulomb Hamiltonian, 37 BrownÐRavenhall effect, 2, 38, 178, 194, 216, discretization technique, 42 219 dot product, 135 Dyson equation, 108, 201, 243

C Cauchy sequence, 276 closure property, 292 E complex rotation, 37 effective Hamiltonian, 5, 20 conﬁguration, 22 intermediate, 37 conjugate momentum, 15, 306 effective interaction, 23, 31 connectivity, 37 Einstein summation rule, 289 continuity equation, 314 electron ﬁeld operators, 285 contraction, 18 electron propagator, 61, 98 contravariant vector, 273 equal-time approximation, 96, 120, 124, 173, coordinate representation, 63, 292 205, 208 counterterms, 138 Euler-Lagrange equations, 308 coupled-cluster approach, 30 evolution operator, 47 normal-ordered, 34 exponential Ansatz, 33 coupled-cluster-QED expansion, 196 normal-ordered, 34 covariance, 1 external-potential approach, 4, 208

363 364 Index

F intermediate normalization, 21 Feynman amplitude, 88, 96, 124, 174, 335 intruder state, 23, 36 Feynman diagram, 28, 60, 87, 125, 335 irreducible diagram, 40, 127 fine structure, 165 first quantization, 13 Fock space, 133, 277 K photonic, 6, 51, 132, 211 ket vector, 290 fold, 126 Klein-Gordon equation, 295 folded diagram, 23 Kronecker delta factor, 278 Fourier transform, 63 functional, 275 Furry picture, 21, 27 L Furry’s theorem, 86 Lagrange equations, 305 Lagrangian function, 309 Lamb shift, 2, 79, 84, 157 G Laplacian operator, 274 g-factor, 164 Lehmann representation, 97 gamma function, 332 linked diagram, 28, 107 gauge LippmannÐSchwinger equation, 205 Coulomb, 59, 68, 76, 81, 315, 324, 329 Lorentz covariance, 1, 39, 60, 94, 119 covariant, 59, 65, 122, 125, 328 Lorentz force, 308 Feynman, 59, 65, 66, 81, 324, 326, 328 Lorentz transformation, 1 Fried-Yennie, 329 Lorenz condition, 314 Landau, 328 non-covariant, 329 gauge invariance, 314 M gauge transformation, 327 many-body Dirac Hamiltonian, 133 Gaunt interaction, 213, 217 matrix elements, 291 Gell-MannÐLow theorem, 51 matrix representation, 291 relativistic, 131 Maxwell’s equations, 311, 312 Goldstone diagram, 25 metric tensor, 273 Goldstone rules, 24, 28, 125 minimal substitutions, 307 Green’s function, 3, 91 model space, 20 projected, 113 complete, 22 Green’s operator, 119, 134 extended, 23 Grotsch term, 164 model state, 20 Gupta-Bleuler formalism, 325 model-space contribution, 29, 56, 126, 138, 142 momentum representation, 300 H MSC, 56, 126 Hamiltonian density, 309 multi-photon exchange, 127 HartreeÐFock model, 26 Heaviside step function, 282 Heisenberg picture, 202, 286 N Heisenberg representation, 92 no-virtual-pair approximation, 2, 37 helium fine structure, 225 non-radiative effects, 39 Hilbert space, 277 norm, 275 hole state, 119 normal order, 18 Hylleraas function, 5, 163

P I pair correlation, 30 instantaneous approximation, 4 parent state, 52, 131 interaction picture, 47 partitioning, 21 Index 365

Pauli spin matrices, 296 photon, 87, 246 perturbation proper, 108 BrillouinÐWigner, 111 sequence, 276 RayleighÐSchr¬odinger, 1, 24 set, 275 photon propagator, 65 size consistency, 30 Poisson bracket, 15, 306 size extensive, 207 polarization tensor, 87 size extensivity, 30, 207 principal-value integration, 64 Slater determinant, 22, 284 spin-orbital, 22 spline, 41 Q state-speciﬁc approach, 37 QED effects, 39, 59, 79 state-universality, 36 QED potential, 181 subset, 275 quantization condition, 15 Sucher energy formula, 61 quasi-degeneracy, 23, 56 quasi-potential approximation, 4 quasi-singularity, 56 T target states, 20 time-ordering, 49 Wick, 18 R trace, 85 radiative effects, 39 transverse-photon, 77 reducible diagram, 40, 127 two-times Green’s function, 3, 111 reference-state contribution, 78, 142 regularization, 79, 237 BrownÐLangerÐSchaefer, 252 U dimensional, 257 Uehling potential, 86, 128 partial-wave, 255 union, 275 PauliÐWillars, 248 unit system, 355 renormalization, 79, 237 cgs, 357 charge, 244 Hartree, 356 mass, 242 mixed, 300 resolution of the identity, 290 natural, 355 resolvent, 24 relativistic, 59, 355 reduced, 24 SI , 355 unlinked diagram, 28

S S-matrix, 3, 60, 157 V scalar potential, 311 vacuum polarization, 2, 39, 84, 87 scalar product, 274 valence universality, 21, 36 scalar retardation, 213, 217 vector potential, 311 scattering matrix, 60 vector space, 275 Schr¬odinger equation, 15 vertex correction, 39, 82, 186, 240, 246 Schr¬odinger picture, 47 Schwinger correction, 164 second quantization, 16, 283 W self energy Ward identity, 83, 242 electron, 39 wave operator, 20 self energy, 108 Wick’s theorem, 19 electron, 79, 157, 186, 239, 244 Wickmann-Kroll potential, 86