Functional-Integral Representation of Quantum Field Theory

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Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away. Antoine de Saint-Exupery` (1900–1944)

14 Functional-Integral Representation of Quantum Field Theory

In Chapter 7 we have quantized various fields with the help of canonical commutation rules between field variables and their canonical conjugate field momenta. From these and the temporal behavior of the fields determined by the field equations of motion we have derived the Green functions of the theory. These contain all experimentally measurable informations on the quantum field theory. They can all be derived from functional derivatives of certain generating functionals. For a real scalar field this was typically an expectation value

Z[j] 0 T [j] 0 (14.1) ≡h | | i where T [j] was the time-ordered product in (7.827)

4 T [j] Tˆ ei d xj(x)φ(x). (14.2) ≡ R Here the Green functions can all be obtained from functional derivatives of Z[j] of the type (7.841). For complex scalar fields, the corresponding generating functional is given by the expectation value (7.849). Now the Green functions can all be obtained from functional derivatives of the type (7.850). In theoretical physics, Fourier transformations have always played an important role in yielding complementary insights into mathematical structures. Due to the conjugate appearance of fields φ(x) and sources j(x) in expressions like (14.2), this is also true for generating functionals and constitutes a basis for the functional-integral formalism of quantum field theory.

14.1 Functional Fourier Transformations

An important observation is now that instead of calculating these generating functionals as done in Chapter 7 from a formalism of ﬁeld theory, in which φ(x) is a ﬁeld

926 14.1 Functional Fourier Transformations 927 operator, they can also be derived from a functional Fourier transform of another functional Z˜[φ] that depends on a classical ﬁeld φ(x):

D Z[j] φZ˜[φ] ei d xj(x)φ(x). (14.3) ≡ D Z R The symbol φ(x) in this expression is called a functional integral. D The mathematicsR of functional integration is an own discipline that is presented in many textbooks [1, 2]. Functional integrals were first introduced in ordinary quantum mechanics by R.P. Feynman [3], who used them to express physical amplitudes without employing operators. The uncertainty relation that can be expressed by an equal-time commutation relation [x(t),p(t)] = ih¯ between x(t) and the conjugate variable p(t) of quantum mechanics is the consequence of quantum fluctuations of the classical variables x(t) and p(t). In quantum mechanics, the functional integral is merely a path integral of a fluctuation variable x(t). In field theory, there is a fluctuating path for each space point x. Instead of a time-dependent variable x(t) one deals with more general dynamical variables φx(t) = φ(x, t) = φ(x), one for each spacepoint x. The path integrals over all x(t) go over into functional integrals over all fluctuating fields φ(x). Functional integrals may be defined most simply in a discretized approximation. Spacetime is grated into a fine spacetime lattice. For every spacetime coordinate xµ we introduce a discrete lattice point close to it xµ xµ nǫ, n =0, 1, 2, 3, (14.4) → n ≡ ± ± ± where ǫ is a very small lattice spacing. Then we may approximate integrals by sums:

D D D d x j(x)φ(x) ǫ j(xn)φ(xn) ǫ jnφn (14.5) ≈ n ≡ n Z X X where n is to be read as a D-dimensional index (n0, n1,...,nD), one for each component of the spacetime vector xµ. Now we deﬁne φ(x) as the inﬁnite product D of integrals over φn at each point xn: R dφ φ(x)= n . (14.6) D n 2πi/ǫD Z Y Z q Operations with functional integrals are very similar to those with ordinary integrals. For example, the Fourier transform of (14.3) can be inverted, by analogy with ordinary integrals, to obtain:

i dDxj(x)φ(x) Z˜[φ] j(x)Z[j]e− . (14.7) ≡ D Z R There are functional analogs of the Dirac δ-function:

i dDxj(x)φ(x) j(x) e− = δ[φ], (14.8) Z D R D φ(x) ei d xj(x)φ(x) = δ[j], (14.9) D Z R 928 14 Functional-Integral Representation of Quantum Field Theory called δ-functionals. In the lattice approximation corresponding to (14.6), they are deﬁned as inﬁnite products of ordinary δ-functions

D D δ[φ]= 2πi/ǫ δ(φn), δ[j]= 2πi/ǫ δ(jn). (14.10) n n Y q Y q They have the obvious property

φ δ[φ]=1, j δ[j]=1. (14.11) Z D Z D A commonly used notation for the measure (14.6) of functional integrals employs continuously inﬁnite product of integrals which must be imagined as the continuum limit of the lattice product (14.6). In this notation one writes

dφ(x) dj(x) φ(x)= , j(x)= , (14.12) D x √2πi D x √2πi Z Y Z Z Y Z and the associated δ-functionals as

δ[φ]= √2πi δ(φ(x)), δ[j]= √2πi δ(j(x)). (14.13) x x Y Y 14.2 Gaussian Functional Integral

Only very few functional integrals can be solved explicitly. The simplest nontrivial example is the Gaussian integral1

i dDx dDx′ j(x)M(x,x′)j(x′) j(x)e− 2 . (14.14) D Z R In the discretized form, this can be written as

i 2D djn ǫ jnMnmjm e− 2 n,m . (14.15)   n 2πi/ǫD Y Z P  q  We may assume M to be a real symmetric functional matrix, since its antisymmetric part would not contribute to (14.14). Such a matrix may be diagonalized by a rotation

jn j′ = Rnmjm, (14.16) → n which leaves the measure of integration invariant

∂ (j1,...,jn) 1 = det R− =1. (14.17) ∂ (j1′ ,...,jn′ )

1Mathematically speaking, integrals with an imaginary quadratic exponent are more accurately called Fresnel integrals, but ﬁeld theorists do not make this distinction. 14.2 Gaussian Functional Integral 929

In the diagonal form, the multiple integral (14.15) factorizes into a product of Gaus- sian integrals, which are easily calculated:

i 2D ′ ′ djn′ ǫ j Mnj 1 1/2 D e− 2 n n n = = det − ( ǫ M). (14.18) D n 2πi/ǫD n √ ǫ Mn − Y Z P Y − q On the right-hand side we have used the fact that the product of diagonal values Mn is equal to the determinant of M. The ﬁnal result

i 2D djn ǫ jnMnmjm 1/2 D e− 2 n,m = det − ( ǫ M) (14.19)   n 2πi/ǫD − Y Z P  q  is invariant under rotations, so that it holds also without diagonalizing the matrix. This formula can be taken to the continuum limit of inﬁnitely ﬁne gratings ǫ 0. Recall the well-known matrix formula →

det A = elog det A = etr log A, (14.20) and expand tr log A into power series as follows

k ∞ ( ) tr log A = tr log [1 + (A 1)] = tr − (A 1)k . (14.21) − − k − kX=1 The advantage of this expansion is that when approximating the functional matrix A by the discrete matrix ǫDM, the traces of powers of ǫDM remain well-deﬁned objects in the continuum limit ǫ 0: →

D D D tr ǫ M = ǫ Mnn d x M(x, x) TrM, n → ≡ X Z D 2 2D D D 2 tr ǫ M = ǫ MnmMmm d xd x′ M(x, x′)M(x′, x) TrM , (14.22) n,m → ≡ X Z . . .

We therefore rewrite the right-hand side of (14.19) as exp[ (1/2)tr log( ǫDM)], and expand − −

tr log( ǫDM) = tr log 1+ ǫDM 1 − − − k h i ∞ ( ) D D (D) − d x1 d xk M(x1, x2) δ (x1 x2) −−−→ǫ 0 − k ··· − − − ×··· → kX=1 Z h i M(x , x ) δ(D)(x x ) M(x , x ) δ(D)(x x ) . (14.23) × − 2 3 − 2 − 3 − k 1 − k − 1 h i h i The expansion on the right-hand side deﬁnes the trace of the logarithm of the functional matrix M(x, x′), and will be denoted by Tr log( M). This, in turn, − − 930 14 Functional-Integral Representation of Quantum Field Theory

serves to deﬁne the functional determinant of M(x, x′) by generalizing formula (14.20) to functional matrices: −

log det ( M) Tr log( M) Det( M)= e − = e − . (14.24) − Thus we obtain for the functional integral (14.14) the result:

i dDx dDx′ j(x)M(x,x′)j(x′) 1/2 1 Tr log( M) j(x)e− 2 = Det − ( M)= e− 2 − . (14.25) D − Z R This formula can be generalized to complex integration variables by separating the currents into real and imaginary parts, j(x) = [j1(x)+ij2(x)]/√2. Each integral gives the same functional determinant so that

i dDx dDx′ j∗(x)M(x,x′)j(x′) Tr log M j∗(x) j(x)e− = e− . (14.26) D D Z R Here M(x, x′) is an arbitrary Hermitian matrix, and the measure of integration for complex variables j(x) is deﬁned as the product of the measures for real and imaginary parts: j∗(x) j(x) j (x) j (x). D D ≡D 1 D 2 14.3 Functional Formulation for Free Quantum Fields

Having calculated the Gaussian functional integrals (14.25) and (14.26) we are able to perform the functional integrations over the generating functional (14.3) to derive its Fourier transform Z˜[φ]. First we shall do so only for the free-ﬁeld generating functional (7.843), which we shall equip with a subscript 0 to emphasize the free situation: 1 4 4 d y1d y2 j(y1)G0(y1,y2)j(y2) Z0[j]= e− 2 . (14.27) R By writing M(x, x′) as

M(x, x′)= iG (x, x′), (14.28) − 0 the Gaussian functional integral (14.14) becomes

1 D D ′ ′ ′ d x d x j(x)G0(x,x )j(x ) 1/2 j(x)e− 2 = Det − ( iG ). (14.29) D − 0 Z R This result can immediately be extended to calculate the functional Fourier tran- form of the generating functional Z0[j] deﬁned in (14.3). Thus we want to form

1 D D ′ ′ ′ d x d x j(x)G0(x,x )j(x )+i j(x)φ(x) Z˜ [φ]= j(x)e− 2 . (14.30) 0 D Z R R The extra term linear in j(x) does not change the harmonic nature of the exponent. The integral can be reduced to the Gaussian form (14.29) by a simple quadratic completion process. For this manipulation it is useful to omit the spacetime indices, and use an obvious functional vector notation to rewrite (14.30) as

i T 1 T j G0j+ij φ Z˜0[φ]= je− 2 i . (14.31) Z D 14.3 Functional Formulation for Free Quantum Fields 931

The exponent may be completed quadratically as

i T 1 T 1 1 i T 1 j + iG− φ G j + iG− φ + φ iG− φ. (14.32) −2 0 i 0 0 2 0 1 We now replace the variable j + iG0− φ by j′ which, in each of the inﬁnite integrals, amounts only to a trivial shift of the center of integration. Thus

i ′ 1 ′ i T −1 j G0j φ iG φ Z˜0[φ]= j′e− 2 i e 2 0 . (14.33) Z D We can now apply formula (14.26) and ﬁnd

1/2 i dDx dDx′ φ(x)M −1(x,x′)φ(x′) Z˜0[φ] = Det(M) e 2 , (14.34) R Inserting for M(x, x′) the functional matrix (14.28) we see that the exponent contains the free-particle Green function:

1 2 2 (D) M − (x, x′)= iG (x, x′)=( ∂ m )δ (x x′). (14.35) − 0 − − − The determinant is a constant prefactor which does not depend on the ﬁeld φ. We shall abbreviate it as a normalization factor

1/2 1/2 2 2 = Det − ( iG ) = Det − ( ∂ m ). (14.36) N − 0 − − With the help of Eq. (14.20), this can be rewritten as 1 = exp Tr log( ∂2 m2) . (14.37) N − 2 − − Inserting into (14.34) the diﬀerential operator (14.35), and performing a partial integration in spacetime, we can rewrite it as

2 D 1 2 m 2 D i d x [ (∂φ) φ ] i d x 0(φ,∂φ) Z˜ [φ]= e 2 − 2 = e L . (14.38) 0 N N R R Thus the Fourier-transformed generating functional is, up to the normalization factor , just the exponential of the classical free-ﬁeld action under consideration. N By Fourier-transforming Z˜[φ] according to formula (14.3), we recover the initial generating functional Z0[j]. This procedure yields the famous functional integral representation for the free-particle generating functional

D i d x [ 0(φ,∂φ)+j(x)φ(x)] Z [j] = φ(x)e L . (14.39) 0 N D Z R In this representation, the field is no longer an operator but a real variable that contains all quantum information via its field fluctuations. The functional integral over the φ-field is defined in (14.6). By analogy with (14.25), we have for the real field fluctuations the Gaussian formula

i dDx dDx′ φ(x)M −1(x,x′)φ(x′) 1 φ(x)e 2 = (Det M) 2 , (14.40) D Z R 932 14 Functional-Integral Representation of Quantum Field Theory

valid for real symmetric functional matrices M(x, x′). For complex ﬁelds φ = (φ1 + iφ2)/√2, there is a similar functional integral formula

i dDx dDx′ φ∗(x)M −1(x,x′)φ(x′) φ∗(x) φ(x)e = Det M, (14.41) D D Z R in which M(x, x′) is a Hermitian functional matrix. This follows directly from (14.26). Let us use the formula (14.40) to cross-check the proper normalization of (14.39). At zero source, Z0[j] has to be equal to unity [compare (14.27)]. This is ensured if

D 1 1/2 i d x 0(φ,∂φ) − = Det ( iG )= φ(x)e L . (14.42) N − 0 D Z R Indeed, after a spacetime integration by parts, the right-hand side may be written as

D D 1 2 2 2 D −1 ′ ′ i d x 0(φ,∂φ) i d x [(∂φ) m φ ] i d xφ(x)M (x,x )φ(x ) φ e L = φ e 2 − = φ e . (14.43) D D D Z R Z R Z R

Applying now formula (14.39), and using (14.36), we verify that Z0[0] = 1. Using the expression (14.42) for the normalization factor, we can rewrite Eq. (14.38) for the Fourier transform Z˜0[φ] as

D i d x 0(φ,∂φ) ˜ e L Z0[φ]= D . (14.44) R i d x 0(φ,∂φ) φ(x)e L D R This ratio is obviously normalized:R

φ(x)Z˜0[φ]=1. (14.45) Z D Note that the expression (14.44) has precisely the form of the quantum mechanical version (1.491) of the thermodynamical Gibbs distribution (1.489):

1 iEn(t ta)/¯h w Z− (t t )e− b− . (14.46) n ≡ QM b − a There exists a useful formula for harmonically fluctuating fields which are encountered in many physical contexts that can be derived immediately from this. Consider the correlation function of two exponentials of a free field φ(x). Inserting into Eq. (14.59) the special current

D j12(x)= a d x[aφ(x x1) bφ(x x2)], (14.47) Z − − − the partition function (14.27) reads

D 1/2 iaφ(x1) ibφ(x2) i d x [ 0(φ,∂φ)] Z [j ] = Det − ( iG ) φ(x) e e− e L . (14.48) 0 12 − 0 D Z R 14.4 Interactions 933

As such it coincides with the harmonic expectation value

iaφ(x1) ibφ(x2) e e− . (14.49) h i Inserting (14.43) and performing the functional integral in (14.48) yields

1 2 2 iaφ(x1) ibφ(x2) [a G(x1,x1) 2abG(x1,x2)+b G(x2,x2)] e e− = e− 2 − . (14.50) h i In the brackets of the exponent we recognize the expectation values of pairs of ﬁelds

a2 φ(x )φ(x ) 2ab φ(x )φ(x ) + b2 φ(x )φ(x ) , (14.51) h 1 1 i − h 1 2 i h 2 2 i such that we may also write (14.50) as

1 2 iaφ(x1) ibφ(x2) iaφ(x1) ibφ(x2) [aφ(x1) bφ(x2)] e e− = e − = e− 2 h − i. (14.52) h i h i Of course, this result may also be derived by using ﬁeld operators and Wick’s theorem along the lines of Subsection 7.17.1.

14.4 Interactions

Let us now include interactions. We have seen in Eq. (10.24) that the generating functional in the interaction picture [more precisely the functional ZD[j] of Eq. (10.22)] may simply be written as

i dDx int( iδ/δj(x)) Z[j]= e L − Z0[j]. (14.53) R This can immediately be Fourier-transformed to

i dDxj(x) φ(x) i dDx int(φ)( iδ/δj) Z˜[φ]= j(x)e− e L − Z [j] . (14.54) D 0 Z R h R i Removing the second exponential by a partial functional integration, we obtain

i dDx int(φ) i dDxj(x)φ(x) Z˜[φ] = e L j(x)e− Z [j] D 0 R Z R i dDx int(φ,∂φ) i dDx 1 [(∂φ)2 m2φ2] = e L e 2 − (14.55) N R i dDx (φ,∂φ) R e L = D . R i d x 0(φ,∂φ) φ(x)e L D R The functional Fourier transformR of this renders a generalization of (14.39) that includes interactions. In this way we have derived functional integral representation of the interacting theory:

i dDx [ (φ,∂φ)+j(x)φ(x)] Z[j] = φ(x)e L N D Z R i dDx [ (φ,∂φ)+j(x)φ(x)] φ(x)e L = D D . (14.56) R i d x 0(φ,∂φ)(φ) R φ(x)e L D R R 934 14 Functional-Integral Representation of Quantum Field Theory

This representation may be compared with the perturbation theoretic formula of operator quantum ﬁeld theory

i dDx [ int(φ)+j(x)φ(x)] Z[j]= 0 T e L 0 , (14.57) h | | i R where the symbol φ(x) denotes free-field operators, and the vacuum expectation value of products of these fields follow Wick’s theorem. In the functional integral representations (14.54)–(14.56), on the other hand, φ(x) is a classical c-number field. All quantum properties of Z[j] arise from the infinitely many integrals over φ(x), one at each spacetime point x, rather than from field operators. Note that in formula (14.56), the full action appears in the exponent, whereas in (14.57), only the interacting part appears.

In contrast to Z˜0[φ] of Eqs. (14.44) and (14.45), the amplitude for the interacting theory is no longer properly normalized. In fact, we know from the perturbative evaluation of (14.57) that it represents the sum of all vacuum diagrams displayed in (10.81). Since the denominator does not normalize A[φ] anyhow, it is convenient to drop it and work with the numerator only, using the unnormalized

i dDx [ (φ,∂φ)+j(x)φ(x)] Z[j] = φ(x)e L (14.58) D Z R as the generating functional. The normalization in (14.58) has an important advantage over the previous one in (14.56). In the euclidean formulation of the theory to be discussed in Section 14.5, it makes Z[0] equal to the thermodynamic partition function of the system. For free fields, Z[0] is equal to the partition function of a set of harmonic oscil- lators of frequencies ω(k) for all momenta k. This statement can be proved only in the lattice version of the theory. In the continuum limit the statement is nontrivial, since the determinants on the right-hand sides are infinite. However, we shall see in Section 14.7, Eqs. (14.123)–(14.133), that correct finite partition functions are obtained if the infinities are removed by the method of dimensional regularization, that was used in Section 11.5 to remove divergences from Feynman integrals. Even though the operator formula (14.57) and the functional integral formula (14.58) are completely equivalent, there are important advantages of the latter. In some theories it may be difficult to find a canonically quantized set of free fields on which to construct an interaction representation for Z[j] following Eq. (14.57). The photon field is an important example where it was quite hard to interpret the Hilbert space. In particular, we remind the reader of the problem that in the Gupta-Bleuler quantization scheme, the vacuum energy contains the quanta of two unphysical polarization states of the photon. Within the functional approach, this problem can easily be avoided as will be explained in Chapter 17. A second and very important advantage is the possibility of deriving the Feynman rules, without any knowledge of the Hilbert space, directly from the representation (14.58). For this we simply take the non-quadratic piece of the action, which defines 14.4 Interactions 935 the vertices of the perturbation expansion, outside the functional integral as in (14.53), i.e., we rewrite (14.58) as:

D int D i d x ( iδ/δj(x)) i d x [ 0(φ,∂φ)+j(x)φ(x)] Z[j]= e L − φ(x)e L . (14.59) D R Z R Using (14.34), this reads more explicitly

i dDx int( iδ/δj(x)) [ i dDx dDx′ φ(x)iG−1(x,x′)φ(x′)+j(x)φ(x)] Z[j]= e L − φ(x)e 2 0 . (14.60) D R Z R A shift in the ﬁeld variables to

φ′(x)= φ(x)+ dyG0(x, x′)j(x′), (14.61) Z and a quadratic completion lead to

D int 1 D D ′ ′ ′ i d x ( iδ/δj(x)) d x d x j(x)G0(x,x )j(x ) Z[j]= e L − e− 2 R i dDx dDxR′ φ′(x)iG−1(x,x′)φ′(x′) φ′(x)e 2 0 . (14.62) × D Z R The functional integral over the shifted ﬁeld φ′(x) can now be performed with the help of formula (14.25), inserting there M(x, x′) = iG0(x, x′). The result is, recalling (14.36), −

i dDx dDx′ φ′(x)iG−1(x,x′)φ′(x′) 1/2 φ′(x)e 2 0 = Det − ( iG ), (14.63) D − 0 Z R such that we ﬁnd

D int 1 D D ′ ′ ′ 1/2 i d x ( iδ/δj(x)) d x d x j(x)G0(x,x )j(x ) Z[j] = Det − ( iG ) e L − e− 2 . (14.64) − 0 R R Expanding the prefactor in (14.60) in a power series yields all terms of the perturbation expansion (10.29). They correspond to the Wick contractions in Sec- tion 10.3.1, with the associated Feynman diagrams. The free-ﬁeld propagators are the functional inverse of the operators between the ﬁelds in the quadratic part of the Lagrangian. If this is written as

i D D d xd x′ φ(x)D(x, x′)φ(x′), (14.65) 2 Z then

1 G0(x, x′)= iD− (x, x′). (14.66)

This formal advantage of obtaining perturbation expansions from the functional integral representations has far-reaching consequences. We have seen in Chapter 11 that the evaluation of the perturbation series proceeds most conveniently by Wick- rotating all energy integrations to make them run along the imaginary axes in the complex energy plane. In this way one avoids the singularities in the propagators 936 14 Functional-Integral Representation of Quantum Field Theory that would be encountered at the physical particle energies. In Section 10.7, on the other hand, these singularities were shown to be responsible for the fact that particles leave a scattering region and form asymptotic states. Hence, Wick-rotated perturbation expansions describe a theory which does not possess any particles states. In fact, they cannot be described by ﬁeld operators creating particle states in a Hilbert space. Such states can be obtained from a simple modiﬁcation of the above functional integral representation of the generating functional Z[j]. We simply perform the x-space version of the Wick-rotation that was illustrated before on page 495 in Fig. 7.2.

14.5 Euclidean Quantum Field Theory

In Eq. (7.136), we replaced the coordinates xµ (µ = 0, 1, 2, 3) in D = 4 spacetime µ 1 3 4 0 dimensions by the euclidean coordinates xE = (x ,...,x , x = ix ). The same operation may be done to the time x0 in any dimensions. Under− this replacement, the action 2 D D 1 2 m 2 int d x (φ, ∂Eφ) d x (∂φ) φ + (φ) (14.67) A ≡ Z L ≡ Z (" 2 − 2 # L ) goes over into i times the euclidean action 2 D 1 2 m 2 int E = d xE (∂Eφ) + φ + (φ) . (14.68) A Z ("2 2 # L ) The euclidean versions of the Gaussian integral formulas (14.40) and (14.89) are

1 dDx dDx′ φ(x)M(x,x′)φ(x′) 1 φ(x)e− 2 = (Det M)∓ 2 , (14.69) D Z R and

dDx dDx′ φ∗(x)M(x,x′)φ(x′) 1 φ∗(x) φ(x)e− = (Det M)∓ , (14.70) D D Z R for complex ﬁelds φ =(φ1 + iφ2)/√2, where φ∗(x) φ(x) φ1(x) φ2(x). The amplitude (14.55) becomes D D ≡D D

dDx (φ,∂φ) e− E LE wE[φ]= D . (14.71) R d xE E 0(φ,∂φ) φ(x)e− L D The normalized version of this,R R

dDx (φ,∂φ) e− E LE w[φ]= D , (14.72) R d xE E(φ,∂φ) φ(x)e− L D represents the functional versionR of the properR quantum statistical Gibbs distribution corresponding to (14.46) [recall (1.489)]:

En/k T e− B wn = . (14.73) En/kB T n e− P 14.6 Functional Integral Representation for Fermions 937

The functional integral representation for the unnormalized generating functional of all Wick-rotated Green functions corresponding to (14.58) is then

dDx [ (φ,∂ φ) j(x)φ(x)] Z [j] = φ(x)e− E LE E − . (14.74) E D Z R The euclidean action corresponds to an energy of a field configuration. The integrand plays the role of a Boltzmann factor and gives the relative probability for this configuration to occur in a thermodynamic ensemble. We now understand the advantage of working with the unnormalized functional integral: At zero external source, ZE[j] corresponds precisely to the thermodynamic partition function of the system. This will be seen explicitly in the examples in Section 14.7.

14.6 Functional Integral Representation for Fermions

If we want to use the functional technique to also describe the statistical properties of fermions, some modiﬁcations are necessary. Then the ﬁelds must be taken to be anticommuting c-numbers. In mathematics, such objects form a so-called Grassmann algebra G. If ξ,ξ′ are real elements of G, then

θθ′ = θ′θ. (14.75) − A trivial consequence of this condition is that the square of each Grassmann element 2 2 vanishes, i.e., θ =0. If θ = θ1 + iθ2 is a complex element of G, then θ = θ∗θ = 2 2 − 2iθ θ is nonzero, but (θ∗θ) =(θθ) = 0. − 1 2 All properties of operator quantum ﬁeld theory for fermions can be derived from functional integrals if we ﬁnd an appropriate extension of the integral formulas in the previous sections to Grassmann variables. Integrals are linear functionals. For Grassmann variables, these are completely determined from the following basic integration rules, which for real θ are

dθ dθ dθ 0, θ 1, θn 0, n> 1. (14.76) Z √2π ≡ Z √2π ≡ Z √2π ≡

For complex variable θ =(θ1 + iθ2)/√2, these lead to

dθ dθ∗ dθ dθ dθ∗ n 0, θ 1, θ∗ 1, (θ∗θ) δn1 , (14.77) Z √2π ≡ Z √2π ≡ Z √2π ≡ Z √2π √2π ≡ − with the definition dθdθ∗ idθ dθ . ≡ − 1 2 Note that these integration rules make the linear operation of integration in (14.76) coincide with the linear operation of differentiation. A function F (θ) of a real Grassmann variable θ, is determined by only two parameters: the zeroth- and the first-order Taylor coefficients. Indeed, due to the property θ2 = 0, the Taylor 938 14 Functional-Integral Representation of Quantum Field Theory

series has only two terms F (θ) = F + F ′θ, where F = F (0) and F ′ dF (θ)/dθ. 0 0 ≡ But according to (14.76), also the integral gives F ′: dθ F (θ)= F ′. (14.78) Z √2π The coincidence of integration and differentiation has the important consequence that any changes in integration variables will not transform with the Jacobian, but rather with the inverse Jacobian: dθ d(aθ) = a , (14.79) Z √2π Z √2π We shall use this transformation property below in Eq. (14.85). As far as perturbation theory is concerned, it is sufficient to define only Gaussian functional integrals such as (14.40) and (14.89). In the discretized form, we may derive the formula

∞ dθn i 2D 1/2 D exp ǫ θmMmnθn = det (ǫ M). (14.80) D n " √2πiǫ # 2 m,n ! Y Z−∞ X The right-hand side is the inverse of the bosonic result (14.40). In addition, there is an important diﬀerence: only the antisymmetric part of the functional matrix contributes. If the matrix Mmn is Hermitian, complex Grassmann variables are necessary to produce a nonzero Gaussian integral. For complex variables we have

dθndθn∗ 2D D exp iǫ θ∗ Mmnθn = det (ǫ M), (14.81) D D m n " √2πiǫ √2πiǫ # m,n ! Y Z X the right-hand side being again the inverse of the corresponding bosonic result (14.89): We ﬁrst prove the latter formula. After bringing the matrix Mmn to a diagonal form via a unitary transformation, we obtain the product of integrals

dθndθn∗ 2D exp iǫ θ∗ Mnθn . (14.82) D D n n " √2πiǫ √2πiǫ # n ! Y Z X Expanding the exponentials into a power series leaves only the ﬁrst two terms, since 2 (θn∗ θn) = 0, so that the integral reduces to

dθndθn∗ 2D (1 + iǫ θ∗ Mnθn). (14.83) D D n m √2πiǫ √2πiǫ Y Z Each of these integrals is performed via the formulas (14.77), and we obtain the product of eigenvalues Mn, which is the determinant:

Mn = det M. (14.84) m Y 14.6 Functional Integral Representation for Fermions 939

For real fermion ﬁelds, we observe that an arbitrary real antisymmetric matrix Mmn can always be brought to a canonical form C, that is zero except for 2 2 × matrices c = iσ2 along the diagonal, by a real orthogonal transformation T . Thus M = T T CT . The matrix C has a unit determinant so that det T = det 1/2(M). Let θ′ Tmnθn, then the measure of integration in (14.80) changes according to m ≡ (14.79) as follows:

dθn = det T dθn′ . (14.85) n n Y Y Applying now the formulas (14.76), the Grassmannian functional integral (14.80) can be evaluated as follows:

dθn 2D dθn′ exp iǫ θmMmnθn =det T exp θ′ Cmnθ′ D D m n n " √2πiǫ #   n " √2πiǫ # m,n ! Y Z Xk,l Y Z X   = det 1/2(iǫDM). (14.86)

The integrals over θn in one dimension decompose into a product of two-dimensional Grassmannian integrals involving the antisymmetric unit matrix c = iσ2. They have the generic form

dθ2′ n dθ2′ n+1 2D 2D 1+ iǫ θ′ θ′ = ǫ . (14.87) D D 2n 2n+1 n " √2πiǫ # √2πiǫ Y Z There is one such factor for every second lattice site, which changes det T = det 1/2M into det 1/2(ǫDM), thus proving (14.80). See [4]. . In the continuum limit, the result of this discussion can be summarized in an extension of the Gaussian functional integral formulas (14.40) and (14.26) to

i dDx dDx′ φ(x)M(x,x′)φ(x′) 1 φ(x)e 2 = Det ∓ 2 M, (14.88) D Z R where M(x, x′) is real symmetric or antisymmetric for bosons or fermions, respectively, and

i dDx dDx′ φ∗(x)M(x,x′)φ(x′) 1 φ∗(x) φ(x)e = Det ∓ M, (14.89) D D Z R where the matrix is Hermitian. The functional integral formulation of fermions follows now closely that of bosons. For N relativistic real fermion ﬁelds χa, we can obtain an amplitude Z˜[χ] from the Fourier transform

D i d xja(x)χa(x) Z˜[χ]= j(x)Z[j]e− , (14.90) D Z R and ﬁnd

i dDx (χ,∂χ) ˜ e L Z[χ]= D . (14.91) R i d x 0(χ,∂χ) χe L D hR R i 940 14 Functional-Integral Representation of Quantum Field Theory

The functional

i dDx (χ,∂χ) Z[j]= χa e L , (14.92) N a D Y Z R with a normalization factor

D 1 i d x 0(χ,∂χ) − = χa e L , (14.93) N a D Y Z R provides us with a functional integral representation of the generating functional of all fermionic Green functions. An obvious extension of this holds for complex fermion ﬁelds. In the case of a Dirac ﬁeld, for example, where the sources j are commonly denoted by η, we obtain

i dDx [ψ¯(i/∂ m)ψ+ int(ψ)+¯ηψ+ψη¯ ] Z[η, η¯]= ψ ψ∗e − L (14.94) N D D Z R with

1 iψ¯(i/∂ m)ψ 1 − = ψ ψ∗e − = Det iG0− (x, x′) = Det(i/∂ m). (14.95) N Z D D − As in the boson case, we shall from now on work with the unnormalized functional without the factor , N i dDx [ψ¯(i/∂ m)ψ+ int(ψ)+¯ηψ+ψη¯ ] Z[η, η¯]= ψe − L , (14.96) D Z R which again has the advantage that the euclidean version of Z[0, 0] becomes directly the thermodynamic partition function of the system. The functional representations of the generating functionals can of course be continued to a euclidean form, as in Section 14.5, thereby replacing operator quantum physics by statistical physics. The corresponding Gaussian formulas for boson and fermion ﬁelds are the obvious generalization of Eqs. (14.88) and (14.89):

1 dDx dDx′ φ(x)M(x,x′)φ(x′) 1 φ(x)e− 2 = Det ∓ 2 M, (14.97) D Z R and

dDx dDx′ φ∗(x)M(x,x′)φ(x′) 1 φ(x) φ∗(x)e− = Det ∓ M. (14.98) D D Z R Here we have deﬁned the measure of the euclidean functional integration in the same way as before in Eqs. (14.88) and (14.89), except without the factors i under the square roots. The euclidean version of the generating functional (14.94) can be used to obtain all Wick-rotated Green functions from functional derivatives Z[η, η¯]. 14.7 Relation Between Z[j] and the Partition Function 941

As a side result of the above development we can state the following functional integral formulas known under the name Hubbard-Stratonovich transformations:

i d3xdtd3x′dt′ [ϕ(x,t)A(x,t;x′,t′)ϕ(x′,t′)+2j(x,t)ϕ(x,t)δ3 (x x′,t)δ(t t′ )] ϕ(x, t)e 2 − − D Z R i( i Trlog 1 A) i d3xdtd3x′dt′ j(x,t)A−1(x,t;x′,t′)j(x′,t′) = e ± 2 i − 2 , (14.99) n o R i d3xdtd3x′dt′ ψ∗(x,t)A(x,t;x′,t′)ψ(x,t′)+[η∗(x,t)ψ(x)δ3 (x x′)δ(t t′ )+c.c.] ψ∗(x, t) ψ(x, t)e − − D D { } Z R i( iTrlogA) i d3xdtd3x′dtη∗(x,t)A−1(x,t;x′,t′)η(x′,t′) = e ± − . (14.100) R These formulas will be needed repeatedly in the remainder of this text. They are the basis for the reformulation of many interacting quantum ﬁeld theories in terms of collective quantum ﬁelds.

14.7 Relation Between Z[j] and the Partition Function

The introduction of the unnormalized functional integral representation (14.58) for Z[j] was motivated by the fact that, in the euclidean version (14.74), Z[0] is equal to the thermodynamic partition function of the system, except for a trivial overall factor. Let us verify this for a free ﬁeld theory in D = 1 dimension. Then ZE[0] of Eq. (14.74) becomes

β 2 1 2 ω 2 Zω = x exp dτ x˙ (τ)+ x (τ) . (14.101) Z D (− Z0 " 2 2 #) For D = 1, the ﬁelds φ(τ) may be interpreted as paths x(τ) in imaginary time τ = it, and we have changed the notation accordingly. In the exponent, we − recognize the euclidean version

β 2 1 2 ω 2 E = dτ x˙ (τ)+ x (τ) (14.102) A Z0 " 2 2 # of the action of the harmonic oscillator:

2 tb 1 ω = dt x˙ 2(t) x2(t) , (14.103) A Zta "2 − 2 # for tb ta = ihβ¯ = ih/k¯ BT . Thus Zω in expression (14.114) is a quantum- statistical− path− integral for− a harmonic oscillator. The measure of path integration is deﬁned as a product of integrals on a lattice of points τn = nǫ with n =0,...,N +1 on the τ-axis:

N dx x(τ)= n , (14.104) D √2πǫ Z nY=0 Z 942 14 Functional-Integral Representation of Quantum Field Theory

where xn x(τn) and N +1=¯hβ/ǫ. For a ﬁnite τ-intervalhβ ¯ , the paths have to satisfy periodic≡ boundary conditions

x(¯hβ)= x(0), (14.105) as a reﬂection of the quantum-mechanical trace. On the τ-lattice, this implies xN+1 = x0, and the action becomes

N+1 2 N 1 (xn xn 1) 2 2 E = ǫ − − + ω xn . (14.106) A 2 " ǫ2 # nX=1 This can be rewritten as 1 N+1 N = x ( ǫ2 + ǫ2ω2)x , (14.107) AE 2ǫ n − ∇∇ n nX=1 where x denotes the lattice version ofx ¨. It may be represented as an (N +1) (N + 1)-matrix∇∇ ×

2 1 0 ... 0 0 1 − −  1 2 1 ... 0 0 0  2 −. − . ǫ =  . .  . (14.108) − ∇∇      00 0 ... 1 2 1   − −   1 0 0 ... 0 1 2   − −    The Gaussian functional integral can now be evaluated using formula (14.40), and we obtain

2 2 1/2 Z = det [ ǫ + ǫω ]− . (14.109) ω N+1 − ∇∇ The determinant is calculated recursively,2 and yields 1 Z = , (14.110) ω 2 sinh(¯hωβ/˜ 2) whereω ˜ is the auxiliary frequency 2 ωǫ ω˜ arsinh . (14.111) ≡ ǫ 2 In the continuum limit ǫ 0, the frequencyω ˜ goes against ω, and Z becomes → ω 1 Z = . (14.112) ω 2 sinh(¯hωβ/2) This can be expanded as

¯hω/2kB T 3¯hω/2kB T 5¯hω/2kB T Zω = e− + e− + e− + ..., (14.113)

2See the textbook [1] and Section 2.12 of the textbook [2]. 14.7 Relation Between Z[j] and the Partition Function 943 which is the quantum statistical partition function of the harmonic oscillator, as we wanted to prove. The ground state has a nonzero energy, as observed in the operator discussion in Chapter 7.32. For the quantum-mechanical version of the functional integral (14.114)

t 2 b 1 2 ω 2 Zω = x exp i dt x˙ (t) x (t) , (14.114) Z D ( Zta "2 − 2 #) we obtain, with the measure of functional integration analog to (14.104)

N dx x(t)= n , (14.115) D √2πiǫ Z nY=0 Z and the use of the Gaussian integral formula (14.40), the result

2 2 1/2 Z = det [ ǫ ǫω ]− . (14.116) ω N+1 − ∇∇ − Thus we only have to replace ω iω in (14.111)–(14.112). In the continuum limit, → we therefore obtain 1 Z = . (14.117) ω 2 sin(¯hωβ/2) We end this section by mentioning that the path-integral representation of the partition function (14.114) with the integration measure (14.104) can be obtained from a euclidean phase space path integral3

β 2 p(τ) 1 2 ω 2 Zω = x(τ)D exp dτ ipx˙ p x (14.118) Z Z D 2πh¯ (Z0 " − 2 − 2 #) by going to a τ-lattice and integrating out the momentum variables. The momentum integrals give the factors √2πǫ in the denominators of the measure (14.104). In the quantum-mechanical version of (14.118)

tb 2 p(t) 1 2 ω 2 Zω = x(t)D exp i dt px˙ p x , (14.119) Z Z D 2πh¯ ( Zta " − 2 − 2 #) the p-integrals produce the denominators containing the factors √i in the measure (14.104). The measure of functional integration on the τ-lattice

N N+1 ∞ ∞ dpn dxn (14.120) " 2πh¯ # nY=0 Z−∞ nY=1 Z−∞ is the obvious generalization of the classical statistical weight in phase space dp ∞ dx ∞ (14.121) 2πh¯ Z−∞ Z−∞ 3For a detailed discussion of the measure see Chapters 2 and 7 in the textbook [2]. 944 14 Functional-Integral Representation of Quantum Field Theory

to ﬂuctuating paths with many variables xn = x(τn). For completeness, we write down the free energy F = β log Z associated with ω − ω the partition function (14.112):

1 hω¯ 1 β¯hω F = log[2 sinh(¯hωβ/2)] = + log(1 e− ). (14.122) ω β 2 β − At zero temperatures, only the ground-state oscillations contribute. The detailed time-sliced calculation of the partition functions has to be compared with the formal evaluation of the partition function (14.114) according to formula (14.69), which would yield

β 2 1 2 ω 2 Zω = x(τ) exp dτ x˙ (τ)+ x (τ) Z D (− Z0 "2 2 #) 1/2 2 2 = Det − ( ∂ + ω ). (14.123) − τ In this continuum formulation, the right-hand side is at first meaningless. It differs from the results obtained by proper time-sliced calculations (14.109)–(14.112) by a temperature-dependent infinite overall factor. In order to see what this factor is, we note that for a periodic boundary condition (14.105), the eigenvectors of the matrix 2 2 iωmτ 2 2 2 ∂τ + ω are e− with eigenvalues ωm + ω . Since ωm grows with m like m , − 1/2 2 2 the functional determinant Det − ( ∂τ + ω ) is strongly divergent. Indeed, the bosonic lattice result (14.112) can only− be obtained in the limit ǫ 0 after dividing → out of the the lattice result (14.109) an equally divergent ω-independent functional determinant and calculating the ratio

2 2 1/2 1/2 2 2 detN+1[ ǫ ǫω ]− Det − ( ∂τ + ω ) − ∇∇ − −1/2 2 . (14.124) hβ¯ det′ [ ǫ2 ǫω2] 1/2 −−−→ǫ 0 hβ¯ Det ′ ( ∂ ) N+1 − ∇∇ − − → − − τ The prime in the denominator indicates that the zero-frequency ω0 must be omitted to obtain a ﬁnite result. The factor 1/hβ¯ is the regularized contribution of the zero frequency. This follows from a simple integral consideration. An integral

b/2 dx ω2x2/2 e− (14.125) b/2 √2π Z− gives 1/ω only for ﬁnite ω( 1/b). In the limit ω 0, it gives b/√2π. In the path integral, the zero mode is associated≫ with the ﬂuctuations→ of the average of the path x(τ). To indicate this origin of the factorhβ ¯ in the ratio (14.124), we may write (14.124) as 1/2 2 2 Det − ( ∂τ + ω ) 1/−2 , (14.126) Det − ( ∂2) R − τ where the subscript R indicates the above regularization. Explicitly, the ratio (14.124) is calculated from the product of eigenvalues:

1/2 2 2 2 Det − ( ∂τ + ω ) 1 ωm −1/2 2 = 2 2 . (14.127) hβ¯ Det ′− ( ∂τ ) hβω¯ ωm + ω − m>Y0 14.7 Relation Between Z[j] and the Partition Function 945

The product can be found in the tables4: 2 ωm hβω/¯ 2 2 2 = , (14.128) m=0 ωm + ω sinh(¯hβω/2) Y6 so that (14.127) is equal to 1/2 sinh(¯hβω/2), and the properly renormalized partition function (14.123) yields the lattice result (14.112). In a lattice calculation, the 1/2 2 determinant Det R− ( ∂τ ) for β is equal to unity. It is curious to see− that a formal→∞ evaluation of the functional determinant via the analytic regularization procedure of Sections 7.12 and 11.5 is indiﬀerent to this denominator. It produces precisely the same result from the formal expression (14.123) as from the proper lattice calculation. Recalling formula (11.134) and setting D = 1, we ﬁnd

1/2 2 2 1 2 2 Det − ( ∂τ + ω ) = exp Tr log( ∂τ + ω ) − −2 − 1 dω′ 2 2 β hω¯ = exp βh¯ log(ω′ + ω ) = e− 2 . (14.129) (−2 Z 2π ) The exponent gives precisely the free energy at zero temperatures in Eq. (14.122). We now admit ﬁnite temperatures. For this purpose, we have to replace the integral over ω′ by a sum over Matsubara frequencies (2.415), and evaluate

1 2 2 dω′ 2 2 1 dωm 2 2 log(ωm + ω )= log(ω′ + ω )+ log(ωm + ω ). hβ¯ m 2π hβ¯ m − 2π ! X Z X Z (14.130) In contrast to the ﬁrst term whose evaluation required the analytic regularization, the second term is ﬁnite. It can be rewritten as

1 dωm ∞ 2 1 dω′ . (14.131) 2 2 2 − hβ¯ m − 2π ! ω ωm + ω′ X Z Z Recalling the summation formula (2.424), this becomes 2 2 ∞ dω′ coth(¯hβω′/2) ∞ dω′ 2 1 = ′ 2 tanh(¯hβω /2) 2 β¯hω − Zω 2ω′ "( ′ ) − # ± Zω 2ω′ e 1 ∓β¯hω = k T 2 log(1 e− ). (14.132) B ∓ For later applications, we have treated also the case of fermionic Matsubara frequencies (the lower case). Together with the zero-temperature result (14.129), we ﬁnd at any temperature

βFω 1/2 2 2 1 2 2 Zω = e− = Det − ( ∂τ + ω ) = exp Tr log( ∂τ + ω ) − −2 − hω¯ β¯hω = exp β + log(1 e− ) , (14.133) (− " 2 − #) in agreement with (14.122).

4 ∞ 2 2 2 I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.431.2: sinh x = m=1(1 + x /m π ). Q 946 14 Functional-Integral Representation of Quantum Field Theory

14.8 Bosons and Fermions in a Single State

The discussion in the last section cannot be taken over to fermionic variables x(τ). For fermions the action (14.114) vanishes identically, as a consequence of the sym- 2 2 metry of the functional matrix D(τ, τ ′)=( ∂τ + ω )δ(τ τ ′). A fermionic version of the above path integral can only be introduced− within− the canonical formulation (14.119) of the harmonic path integral. With the help of a canonical transformation

a† = 1/2¯hω(ωx ip), a = 1/2¯hω(ωx + ip), (14.134) − q q this may be transformed into the action

tb ω QM = dτ (a∗i∂ta ωa∗a). (14.135) A Zta − A canonical quantization with commutation and anticommutation rules

[â(t), aˆ†(t)] = 1, ∓ [â†(t), aˆ†(t)] = 0, (14.136) ∓ [â(t), aˆ(t)] = 0 ∓ produces a second-quantized Hilbert space of the type discussed in Chapter 2. Since a†(t) and a(t) carry no space variables. They describe Bose and Fermi particles at a single point. The path-integral representation of the quantum-mechanical partition function of this system is

a(t) a∗(t) i ω QM Zω QM D D e A . (14.137) ≡ Z 2π The measure of integration is

a∗ a a a D D ∞ ∞ D 1D 2 , (14.138) 2π ≡ 2π Z Z Z−∞ Z−∞ where a = (a1 + ia2)/√2 and a† = (a1 ia2)/√2 are directly obtained from the canonical measure of path integration in− (14.118). At a euclidean time τ = it, the action of the free nonrelativistic ﬁeld becomes − ¯hβ ω = dτ (a∗∂τ a + ωa∗a), (14.139) A Z0 and the thermodynamic partition function has the path-integral representation

a(τ) a∗(τ) ω Zω D D e−A . (14.140) ≡ Z 2π In this formulation, there is no problem of treating both bosons or fermions at the same time. We simply have to assume the ﬁelds a(τ), a∗(τ) to be periodic or antiperiodic, respectively, in the imaginary-time intervalhβ ¯ : a(¯hβ)= a(0), (14.141) ± 14.9 Free Energy of Free Scalar Fields 947 or in the sliced form a = a . (14.142) N+1 ± 0 Using formula (14.98), the partition function can be written as

¯hβ a∗ a Zω = D D exp dτ (a∗∂τ a + ωa∗a) Z 2π "− Z0 # 1/2 = Det ∓ (∂τ + ω). (14.143) Contact with the previous oscillator calculation is established by observing that in the determinant, the operator ∂τ + ω can be replaced by the conjugate operator ∂ +ω, since all eigenvalues come in complex-conjugate pairs, except for the m =0- − τ value, which is real. Hence the determinant of ∂τ + ω can be substituted everywhere by det (∂ + ω) = det ( ∂ + ω)= det ( ∂2 + ω2). (14.144) τ − τ − τ In the boson case, we thus reobtain the resultq (14.123). In both cases, we may therefore write 1/2 2 2 Z = Det ∓ ( ∂ + ω ). (14.145) ω − τ The right-hand side was evaluated for bosons in Eqs. (14.129)–(14.133). In anticipa- tion of the present result, we have calculated the Matsubara sums up to Eq. (14.132) for bosons and fermions, with the results

βFω 1/2 2 2 1 2 2 Zω = e− = Det ∓ ( ∂τ + ω ) = exp Tr log( ∂τ + ω ) − ∓2 − hω¯ β¯hω = exp β + log(1 e− ) . (14.146) (∓ " 2 ∓ #) This can be written as 1 [sinh(¯hβω/2)]− bosons, Zω = for . (14.147) ( cosh(¯hβω/2) ) ( fermions. ) For bosons, the physical interpretation of this expression was given after Eq. (14.110). The analog sum of Boltzmann factors for fermions is

¯hω/2kB T ¯hω/2kB T Zω = e + e− . (14.148) This shows that the fermionic system at a point has two states, one with no particle and one with a single particle, where the no-particle state has a negative vacuum energy, as observed in the operator discussion in Chapter 2.

14.9 Free Energy of Free Scalar Fields

The results of the last section are easily applied to fluctuating scalar fields. Consider the free-field partition function

dDx 1 [(∂φ)2+m2φ2] Z = φ(x)e− 2 , (14.149) 0 D Z R 948 14 Functional-Integral Representation of Quantum Field Theory which is of the general form (14.74) for j = 0. If the ﬁelds are decomposed into their spatial Fourier components in a ﬁnite box of volume V ,

1 ikx φ(x)= e φk(τ) + c.c. , (14.150) √2V Xk Z h i the partition function becomes

hβ¯ 1 2 2 2 2 2 dt φ˙k(τ) +k φk(τ) +m φk(τ) Z = φk(τ)e− 0 2 | | | | | | . (14.151) 0 D { } Yk Z R For each k, the functional integral is obviously the same as in (14.114), so that

Z0 = Zω(k), (14.152) Yk where ω(k) √k2 + m2, and by (14.112) ≡ 1 Z = . (14.153) ω(k) 2 sinh[¯hω(k)β/2]

For a real field φ(x), the anticommuting alternative cannot be accommodated into the functional integral (14.149). We must first go to the field-theoretic analog of the path integral in phase space

π(x) Z0 = φ(x)D Z Z D 2πh¯ 4 1 2 1 2 2 2 exp d x iπ(x)∂µφ(x) π (x) [(∇φ(x)) + m φ (x)] ,(14.154) × Z − 2 − 2 where π(x) are the canonical ﬁeld momenta (7.1). After the Fourier decomposition (14.150) and a similar one for π(x), we perform again a canonical transformation corresponding to (14.134), 1 1 a† (τ)= [ω(k)φk(τ) iπk(τ)], ak(τ)= [ω(k)φk(τ)+iπk(τ)], (14.155) k √2¯hω − √2¯hω and arrive at the analog of (14.143):

¯hβ ak∗ ak Zω = D D exp dτ (ak∗ ∂τ ak + ωak∗ ak) 2π "− 0 # Yk Z Z 1/2 = Det ∓ (∂τ + ω(k)). (14.156) Yk This can be evaluated as in (14.147) to yield

1 2 sinh[¯hβω(k)/2] − bosons, Zω = { } for . (14.157) ( 2 cosh[¯hβω(k)/2] ) ( fermions. ) Yk 14.10 Interacting Nonrelativistic Fields 949

The associated free energies are

ω(k) β¯hω(k) F0 = β log Z0 = + log 1 e− . (14.158) − ± ( 2 ∓ ) Xk h i For a complex field, the canonical transformation is superfluous. The field in the partition function

dDx 1 (∂φ∗∂φ+m2φ∗φ) Z = φ(x) φ∗(x)e− 2 (14.159) 0 D D Z R can directly be assumed to be of the bosonic or of the fermionic type. A direct application of the Gaussian formula (14.98) leads to

2 2 1 Z = Det [ ∂ + ω (k)]∓ ω − τ Yk 2 2 sinh[¯hβω(k)/2] − bosons, = { } 2 for , (14.160) ( 2 cosh[¯hβω(k)/2] ) ( fermions. ) Yk { } with free energies twice as large as (14.158).

14.10 Interacting Nonrelativistic Fields

The quantization of nonrelativistic particles was amply discussed in Chapter 2 and applied to many-body Bose and Fermi systems in Chapter 3. Her we shall demonstrate that a completely equivalent formulation of the second-quantized nonrelativistic ﬁeld theory is possible with the help of functional integrals. Consider a many-fermion system described by an action

3 0 + int = d xdtψ∗(x, t)[i∂t ǫ( i∇)] ψ(x, t) (14.161) A≡A A Z − − 1 3 3 d xdtd x′dt′ψ∗(x′, t′)ψ∗(x, t)V (x, t; x′t′)ψ(x, t)ψ(x′, t′) −2 Z with a translationally invariant two-body potential

V (x, t; x′, t′)= V (x x′, t t′). (14.162) − − In the systems to be treated in this text we shall be concerned with a potential that is, in addition, instantaneous in time

V (x, t; x′, t′)= δ(t t′)V (x x′). (14.163) − − This property will greatly simplify the discussion. The fundamental ﬁeld ψ(x) may describe bosons or fermions. The complete information on the physical properties of the system resides in the Green functions. 950 14 Functional-Integral Representation of Quantum Field Theory

In the operator Heisenberg picture, these are given by the expectation values of the time-ordered products of the ﬁeld operators

G (x1, t1,..., xn, tn; xn′ , tn′ ,..., x1′ , t1′ ) (14.164)

= 0 Tˆ ψˆ (x , t ) ψˆ (x , t )ψˆ† (x ′ , t ′ ) ψˆ† (x ′ , t ′ ) 0 . h | H 1 1 ··· H n n H n n ··· H 1 1 | i The time-ordering operator Tˆ changes the position of the operators behind it in such a way that earlier times stand to the right of later times. To achieve the final ordering, a number of field transmutations are necessary. If F denotes the number of transmutations of Fermi fields, the final product receives a sign factor ( 1)F . It is convenient to view all Green functions (14.164) as derivatives of the− generating functional

3 Z[η∗, η]= 0 Tˆ exp i d xdt ψˆ† (x, t)η(x, t)+ η∗(x, t)ψˆ (x, t) 0 , (14.165) h | H H | i Z h i namely

G (x1, t1,..., xn, tn; xn′ , tn′ ,..., x1′ , t1′ ) (14.166) n+n′ ′ δ Z[η , η] =( i)n+n ∗ . − δη (x , t ) δη (x , t )δη(x ′ , t ′ ) δη(x ′ , t ′ ) ∗ ∗ 1 1 ∗ n n n n 1 n η=η 0 ··· ··· ≡

Physically, the generating functional (14.165) describes the probability amplitude for the vacuum to remain a vacuum in the presence of external sources η∗(x, t) and η∗(x, t). The calculation of these Green functionals is usually performed in the interaction picture which can be summarized by the operator expression for Z:

3 Z[η∗, η]= N 0 T exp i [ψ†, ψ]+ i d xdt ψ†(x, t)η(x, t) + h.c. 0 . (14.167) h | Aint | i Z h i In the interaction picture, the ﬁelds ψ(x, t) possess free-ﬁeld propagators and the normalization constant N is determined by the condition [which is trivially true for (14.165)]:

Z[0, 0]=1. (14.168)

The standard perturbation theory is obtained by expanding exp i in (14.167) { Aint} in a power series and bringing the resulting expression to normal order via Wick’s expansion technique. The perturbation expansion of (14.167) often serves conveniently to define an interacting theory. Every term can be pictured graphically and has a physical interpretation as a virtual process. Unfortunately, the perturbation series up to a certain order in the coupling constant is unable to describe several important physical phenomena. Examples are the formation of bound states in the vacuum, or the existence and properties of collective excitations in many-body systems. Those require the summation of infinite subsets 14.10 Interacting Nonrelativistic Fields 951 of diagrams to all orders. In many situations it is well-known which subsets have to be taken in order to account approximately for specific effects. What is not so clear is how such lowest approximations can be improved in a systematic manner. The point is that, as soon as a selective summation is performed, the original coupling constant has lost its meaning as an organizer of the expansion and there is need for a new systematics of diagrams. This will be presented in the sequel. As soon as bound states or collective excitations are formed, it is very suggestive to use them as new quantum fields rather than the original fundamental particles ψ. The goal would then be to rewrite the expression (14.167) for Z[η∗, η] in terms of new fields whose unperturbed propagator has the free energy spectrum of the bound states or collective excitations and whose int describes their mutual interactions. In the operator form (14.167), however, suchA changes of fields are hard to conceive.

14.10.1 Functional Formulation In the functional integral approach, the generating functional (14.165) is given by [5]:

Z[η∗, η]= N ψ∗(x, t) ψ(x, t) Z D D 3 exp i [ψ∗, ψ]+ i d xdt [ψ∗(x, t)η(x, t) + c.c.] . (14.169) × A Z It is worth emphasizing that the field ψ(x, t) in the path-integral formulation is a complex number and not an operator. All quantum effects are accounted for by fluctuations; the path integral includes not only the classical field configurations but also all classically forbidden ones, i.e., all those which do not run through the valley of extremal action in the exponent. By analogy with the development in Section 14.4 we take the interactions outside the integral and write the functional integral (14.169) as 1 δ 1 δ Z[η∗, η] = exp i int , Z0[η∗, η], (14.170) ( A " i δη i δη∗ #) where Z0 is the generating functional of the free-field correlation functions, whose functional integral looks like (14.169), but with the action being only the free-particle expression

0[ψ∗, ψ]= dxdt ψ∗(x, t)[i∂t ǫ( i∇)] ψ(x, t), (14.171) A Z − − rather than the full [ψ∗, ψ]= [ψ∗, ψ]+ [ψ∗, ψ] of Eq. (14.161) in the exponent. A A0 Aint The functional integral is of the Gaussian type (14.89) with a matrix

(3) A(x, t; x′, t′) = [i∂ ǫ( i∇)] δ (x x′)δ(t t′). (14.172) t − − − − This matrix is the inverse of the free propagator

1 A(x, t; x′, t′)= iG0− (x, t; x′, t′) (14.173) 952 14 Functional-Integral Representation of Quantum Field Theory where

3 dE d p i[E(t t′) p(x x′)] i G (x, t; x′, t′)= e− − − − . 0 2π (2π)4 E ǫ(p)+ iη Z Z − (14.174)

Inserting this into (14.100), we see that

1 3 3 Z [η∗, η]= N exp i iTr log iG− d xdtd x′dt′η∗(x, t)G (x′, t′)η(x′, t′) . 0 ± 0 − 0 Z (14.175) We now ﬁx N in accordance with the normalization (14.168) to

N = exp [i ( iTr log iG )] (14.176) ± 0 and arrive at

3 3 Z0[η∗, η] = exp d xdtd x′dt′η∗(x, t)G0(x, t; x′, t′)η(x′, t′) . (14.177) − Z This coincides exactly with what would have been obtained from the operator expression (14.167) for Z0[η∗, η] (i.e., without int). Indeed, according to Wick’s theorem [2,A 5, 6], any time ordered product can be expanded as a sum of normal products with all possible contractions taken via Feynman propagators. The formula for an arbitrary functional of free ﬁelds ψ, ψ∗ is

3 3 δ δ T F [ψ∗, ψ] = exp d xdtd x′dt′ G0(x, t; x′, t′) :F [ψ∗, ψ]: . (14.178) Z δψ(x, t) δψ∗(x, t′) Applying this to

0 T F [ψ∗, ψ] 0 = 0 T exp i dxdt(ψ∗η + η∗ψ) 0 (14.179) h | | i h | Z | i one ﬁnds:

Z0[η∗, η] = exp dxdtdx′dt′η∗(x, t)G0(x, t; x′, t′)η(x′, t′) − Z 0 :exp i dxdt(ψ∗η + η∗ψ) : 0 . (14.180) ×h | Z | i The second factor is equal to unity, thus proving the equality of this operatorially deﬁned Z0[η∗, η] with the path-integral expression (14.177). Because of (14.170), this equality holds also for the interacting geerating functional Z[η∗, η].

14.10.2 Grand-Canonical Ensembles at Zero Temperature All these results are easily generalized from vacuum expectation values to thermodynamic averages at ﬁxed temperatures T and chemical potential µ. The change 14.10 Interacting Nonrelativistic Fields 953 at T = 0 is trivial: The single particle energies in the action (14.161) have to be replaced by

ξ( i∇)= ǫ( i∇) µ (14.181) − − − and new boundary conditions have to be imposed upon all Green functions via an appropriate iǫ prescription in G0(x, t; x′, t′) of (14.174) [see [2, 7]]:

3 T =0 dEd p iE(t t′)+ip(x x′) i G (x, t; x′, t′)= e− − − . (14.182) 0 (2π)4 E ξ(p)+ iη sgn ξ(p) Z − Note that, as a consequence of the chemical potential, fermions with ξ < 0 inside the Fermi sea propagate backwards in time. Bosons, on the other hand, have in general ξ > 0 and, hence, always propagate forward in time. In order to simplify the notation we shall often use four-vectors p =(p0, p) and write the measure of integration in (14.182) as

dEd3p d4p 4 = 4 . (14.183) Z (2π) Z (2π) Note that in a solid, the momentum integration is really restricted to a Brillouin zone. If the solid has a ﬁnite volume V , the integral over spacial momenta becomes a sum over momentum vectors, d3p 1 3 = , (14.184) (2π) V p Z X and the Green function (14.182) reads

T =0 dE 1 ip(x x′) i x x′ ′ − − G0( , t; , t ) e 0 . (14.185) ≡ 2π V p p ξ(p)+ iη sgn ξ(p) Z X − The resulting power series expansions for the generating functional at zero- T =0 temperature Z[η∗, η] and nonzero coupling can be written down as before after performing a Wick rotation in the complex energy plane in all energy integrals oc- curring in the expansions of formulas (14.180) and (14.167) in powers of the sources 0 η(x, t) and η∗(x, t). For this, one sets E = p iω and replaces ≡ dE dω ∞ i ∞ . (14.186) 2π → 2π Z−∞ Z−∞ Then the Green function (14.182) becomes

3 T =0 dω d p ω(t t′ )+ip(x x′) 1 G (x, t; x′, t′)= e − − . (14.187) 0 − 2π (2π)3 iω ξ(p) Z − T =0 Note that with formulas (14.170) and (14.177) the generating functional Z[η∗, η] is the grand-canonical partition function in the presence of sources [7]. 954 14 Functional-Integral Representation of Quantum Field Theory

Finally, we have to introduce arbitrary temperatures T . According to the standard rules of quantum field theory (for an elementary introduction see Chapter 2 in Ref. [2]), we must continue all times to imaginary values t = iτ, restrict the imaginary time interval to the inverse temperature5 β 1/T , and impose periodic or antiperiodic boundary conditions upon the fields ψ(≡x, iτ) of bosons and fermions, − respectively [2, 7]: ψ(x, iτ)= ψ(x, i(τ +1/T )). (14.188) − ± − When there is no danger of confusion, we shall drop the factor i in front of the imaginary time in field arguments, for brevity. The same thing− will be done with Green functions. By virtue of (14.177), these boundary conditions wind up in all free Green functions, i.e., they have the property

T T G (x, τ +1/T ; x′, τ ′) G (x, iτ; x′, iτ ′). (14.189) 0 ≡ ± 0 − − This property is enforced automatically by replacing the energy integrations ∞ dω/2π in (14.187) by a summation over the discrete Matsubara frequencies [by −∞ Ranalogy with the momentum sum (14.184), the temporal “volume” being β =1/T ] dω ∞ T , (14.190) 2π → ω Z−∞ Xn which are even or odd multiples of πT

2n bosons ωn = πT for . (14.191) ( 2n +1 ) ( fermions ) The prefactor T of the sum over the discrete Matsubara frequencies accounts for the density of these frequencies yielding the correct T 0-limit. Thus we obtain the following expression for the→ imaginary-time Green function of a free nonrelativistic ﬁeld at ﬁnite temperature (the so-called free thermal Green function)

3 T ′ ′ d p iωn(τ τ )+ip(x x ) 1 x x′ ′ − − − G0( , τ, , τ )= T 3 e . (14.192) − ω (2π) iωn ξ(p) Xn Z − Incorporating the Wick rotation in the sum notation we may write

T = iT = iT , (14.193) p − ω − p X0 Xn X4 where p4 = ip0 = ω. If temperature and volume are both ﬁnite, the Green function is written as−

T ′ ′ T iωn(τ τ )+ip(x x ) 1 G0(x, τ, x′, τ ′)= e− − − . (14.194) − V p p iωn ξ(p) X0 X − 5 We use natural units in this chapter, so that kB =1, ¯h = 1. 14.10 Interacting Nonrelativistic Fields 955

At equal space points and equal imaginary times, the sum can easily be evaluated. One must, however, specify the order in which τ τ ′. Let η denote an inﬁnitesimal → positive number and consider the case τ ′ = τ + η, where the Green function is

3 T d p 1 x x iωnη G0( , τ, , τ + η)= T 3 e . (14.195) − ω (2π) iωn ξ(p) Xn Z − Then the sum is found after converting it into a contour integral

ηz iωnη 1 T e 1 T e = dz z/T . (14.196) ω iωn ξ(p) 2πi C e 1 z ξ Xn − Z ∓ − The upper sign holds for bosons, the lower for fermions. The contour of integration C encircles the imaginary z-axis in a positive sense, thereby enclosing all integer or half-integer valued poles of the integrand at the Matsubara frequencies z = iωm (see Fig. 2.8). The factor eηz ensures that the contour in the left half-plane does not contribute. By deforming the contour C into C′ and contracting C′ to zero we pick up the pole at z = ξ and ﬁnd 1 1 1 iωnη p T e = ξ(p)/T = ξ(p)/T = n(ξ( )). (14.197) ω iωn ξ(p) ∓e 1 ∓e 1 ∓ Xn − ∓ ∓ The phase eηz ensures that the contour in the left half-plane does not contribute. The function on the right is known as the Bose or Fermi distribution function. By subtracting from (14.197) the sum with ξ replaced by ξ, we obtain the important sum formula −

1 1 1 ξ(p) ± T 2 2 = coth . (14.198) ω ωn + ξ (p) 2ξ(p) T Xn

In the opposite limit with τ ′ = τ η, the phase factor in the sum would be iωmη − e− leading to a contour integral

ηz iωmη 1 kBT e− 1 kBT e = dz , (14.199) z/kBT − ω iωm ξ(p) ± 2πi C e− 1 z ξ Xm − Z ∓ − and we would ﬁnd 1 nξ(p). In the operator language,± these limits correspond to the expectation values of free non-relativistic ﬁeld operators

T G (x, τ; x, τ + η) = 0 Tˆ ψˆ (x, τ)ψˆ† (x, τ + η) 0 = 0 ψˆ† (x, τ)ψˆ (x, τ) 0 0 h | H H | i ±h | H H | i T G (x, τ; x, τ η) = 0 Tˆ ψˆ (x, τ)ψˆ† (x, τ η) 0 = 0 ψˆ (x, τ)ψˆ† (x, τ) 0 0 − h | H H − | i h | H H | i = 1 0 ψˆ† (x, τ)ψˆ (x, τ η) 0 . ±h | H H ∓ | i The function n(ξ(p)) is the thermal expectation value of the number operator 956 14 Functional-Integral Representation of Quantum Field Theory

ˆ ˆ ˆ N = ψH† (x, τ)ψH (x, τ). (14.200) In the case of T = 0 ensembles, it is also useful to employ a four-vector notation. The four-vector 6

p (p , p)=(ω, p) (14.201) E ≡ 4 is called the euclidean four-momentum. Correspondingly, we deﬁne the euclidean spacetime coordinate

x ( τ, x). (14.202) E ≡ − The exponential in (14.192) can be written as

p x = ωτ + px. (14.203) E E − Collecting integral and sum in a single four-dimensional summation symbol, we shall write (14.192) as

T T 1 G0(xE x′) exp [ ipE(xE xE′ )] . (14.204) − ≡ −V p − − ip4 ξ(p) XE − It is quite straightforward to derive the general T = 0 Green function from a path-integral formulation analogous to (14.169). For this6 we consider classical ﬁelds ψ(x, τ) with the periodicity or anti-periodicity

ψ(x, τ)= ψ (x, τ +1/T ) . (14.205) ± They can be Fourier-decomposed as

T iωnτ+ipx T ipExE ψ(x, τ)= e− a(ωn, p) e− a(pE) (14.206) V ω p ≡ V p Xn X XE with a sum over even or odd Matsubara frequencies ωn. If a free action is now deﬁned as

1/2T 3 0[ψ∗, ψ]= i dτ d x ψ∗(x, τ)[ ∂τ ξ ( i∇)] ψ(x, τ), (14.207) A − 1/2T − − − Z− Z formula (14.100) renders [1, 8]

1/2T T Tr log A+ dτdτ ′ d3xd3x′η∗(x,τ)A−1(x,τ,x′,τ ′)η(x′,τ ′) −1/2T Z0[η∗, η]= e∓ , (14.208) R R R with the functional matrix

(3) A(x, τ; x′, τ ′) = [∂ + ξ ( i∇)] δ (x x′)δ(τ τ ′). (14.209) τ − − − 14.10 Interacting Nonrelativistic Fields 957

1 Its inverse A− is equal to the propagator (14.192), the Matsubara frequencies arising due to the ﬁnite-τ interval of euclidean space together with the periodic boundary condition (14.205). T Again, interactions are taken care of by multiplying Z0[η∗η] with the factor (14.170). In terms of the ﬁelds ψ(x, τ), the exponent has the form: 1 1/2T int = dτdτ ′ A 2 1/2T Z Z− 3 3 d xd x′ψ∗(x, τ)ψ∗(x′, τ ′)ψ(x′, τ ′)ψ(x, τ)V (x, iτ; x′, iτ ′). (14.210) × Z − − In the case of a potential (14.163) that is instantaneous in time t, the potential of the euclidean formulation becomes instantaneous in τ:

V (x, iτ; x′, iτ ′)= V (x x′) iδ(τ τ ′). (14.211) − − − − In this case can be written in terms of the interaction Hamiltonian as Aint 1/2T int = i dτHint(τ). (14.212) A 1/2T Z− Thus the grand-canonical partition function in the presence of external sources may be calculated from the path integral [8]:

T 1/2T T i + dτ d3x[ψ∗(x,τ)η(x,τ)+c.c.] Z[η∗, η]= ψ∗(x, τ) ψ(x, τ)e A −1/2T , (14.213) Z D D R R where the grand-canonical action is

1/2T T 3 i [ψ∗, ψ]= dτ d xψ∗(x, τ)[∂τ + ξ( i∇)] ψ(x, τ) (14.214) A − 1/2T − Z− Z 1/2T i 3 3 + dτdτ ′ d xd x′ψ∗(x, τ)ψ∗(x′, τ ′)ψ(x, τ ′)ψ(x, τ)V (x, iτ; x, iτ ′). 2 1/2T − − Z− Z The Green functions of the fully interacting theory are obtained from the functional derivatives

G (x1, τ1,..., xn, τn; xn′ , τn′ ,..., x1′ , τ1′ ) (14.215) n+n′ ′ δ Z[η , η] =( i)n+n ∗ . − δη (x , τ ) δη (x , τ )δη(x ′ , τ ′ ) δη(x ′ , τ ′ ) ∗ ∗ 1 1 ∗ n n n n 1 n η=η 0 ··· ··· ≡

Explicitly, the right-hand side represent the functional integrals

T i [ψ∗,ψ] N ψ∗(x, t) ψ(x, t)ψˆ(x1, τ1) ψˆ(xn, τn)ψˆ∗(xn′ , τn′ ) ψˆ∗(x1′ , τ1′ )e A .(14.216) Z D D ··· ··· In the sequel, we shall always assume the normalization factor to be chosen in such a way that Z[0, 0] is normalized to unity. Then the functional integrals (14.216) are obviously the correlation function of the ﬁelds:

ψˆ(x , τ ) ψˆ(x , τ )ψˆ∗(x ′ , τ ′ ) ψˆ∗(x ′ , τ ′ ) . (14.217) h 1 1 ··· n n n n ··· 1 1 i 958 14 Functional-Integral Representation of Quantum Field Theory

In contrast to Section 1.2, the bra and ket symbols denote now a thermal average of the classical fields. If the generating functional of the interacting theory is evaluated in a perturbation expansion using formula (14.170), the periodic boundary conditions for the free Green functions (14.189) will go over to the fully interacting Green functions (14.215). The functional integral expression (14.213) for the generating functional have a great advantage in comparison to the equivalent operator formulation based on (14.180) and (14.170). They share with ordinary integrals the extreme flexibility with respect to changes in the field variables. Summarizing we have seen that the functional (14.213) defines the most general type of theory involving two-body forces. It contains all information on a physical system in the vacuum as well as in thermodynamic ensembles. The vacuum theory is obtained by setting T = 0 and µ = 0, and by continuing the result back from T to physical times. Conversely, the functional (14.169) in the vacuum can be generalized to ensembles in a straight-forward manner by first continuing the times t to imaginary values iτ via a Wick rotation in all energy integrals and then going to periodic functions in− τ. There is a complete correspondence between the real-time generating functional (14.169) and the thermodynamic imaginary-time expression (14.213). For this reason it will be sufficient to exhibit all techniques only in one version, for which we shall choose (14.169). Note, however, that due to the singular nature of the propagators (14.174) in real energy-momentum, the thermodynamic formulation specifies the way how to avoid singularities.

14.11 Interacting Relativistic Fields

Let us see how this formalism works for relativistic boson and fermion systems. Consider a Lagrangian of Klein-Gordon and Dirac particles consisting of a sum ψ, ψ,ϕ¯ = + . (14.218) L L0 Lint As in the case of nonrelativistic fields, all time ordered Green’s functions can be obtained from the derivatives with respect to the external sources of the generating functional ¯ i dx( int+¯ηψ+ψη+jϕ) Z [η, η,¯ j] = const 0 T e L 0 . (14.219) ×h | | i R The fields in the exponent follow free equations of motion and 0 is the free-field vacuum. The constant is conventionally chosen to make Z [0, 0, 0]| i = 1, i. e.

1 ¯ i dx int(ψ,ψ,ϕ) − const = 0 T e L 0 . (14.220) h | | i R This normalization may always be enforced at the very end of any calculation such that Z [η, η,¯ j] is only interesting as far as its functional dependence is concerned. Any constant prefactor is irrelevant. 14.11 Interacting Relativistic Fields 959

It is then straight-forward to show that Z [η, η,¯ j] can alternatively be computed via the Feynman path-integral formula

¯ ¯ i dx[ 0(ψ,ψ,ϕ)+ int+¯ηψ+ψη+jϕ] Z [η, η,¯ j] = const ψ ψ¯ ϕe L L . (14.221) × D D D Z R Here the fields are no more operators but classical functions (with the mental reser- vation that classical Fermi fields are anticommuting objects). Notice that contrary to the operator formula (14.219) the full action appears in the exponent. For simplicity, we demonstrate the equivalence only for one real scalar field ϕ(x). The extension to other fields is immediate [9, 10, 51]. Note that it is sufficient to give the proof for free fields, where

Z [j] = 0 T ei dxj(x)ϕ(x) 0 0 h | | i 1 ✷ 2 R i dx[ ϕ(x)( x µ )ϕ(x)+j(x)ϕ(x)] = const ϕe 2 − − . (14.222) × D Z R For if it holds there, a simple multiplication on both sides of (14.222) by the diﬀer- ential operator

1 δ i dx int( ) e L i δj(x) (14.223) R would extend it to the interacting functionals (14.219) or (14.221). But (14.222) follows directly from Wick’s theorem, according to which any time-ordered product of a free ﬁeld can be expanded into a sum of normal products with all possible time ordered contractions. This statement can be summarized in operator form valid for any functional F [ϕ] of a free ﬁeld ϕ(x):

1 dxdy δ D(x y) δ T F [ϕ]= e 2 δϕ(x) − δϕ(y) : F [ϕ]: , (14.224) R where D(x y) is the free-ﬁeld propagator − 4 i d q iq(x y) i D(x y)= δ(x y)= e− − . (14.225) − ✷ µ2 + iǫ − (2π)4 q2 µ2 + iǫ − x − Z − Applying this to (14.224) gives

1 dxdy δ D(x y) δ i dxj(x)ϕ ˆ(x) Z = e 2 δϕˆ(x) − δϕˆ(y) 0 : e : 0 0 h | | i R1 dxdyj(x)D(x y)j(y) i Rdxj(x)ϕ ˆ(x) = e− 2 − 0 : e : 0 h | | i 1 R dxdyj(x)D(x y)j(y) R = e− 2 − . (14.226) R The last part of the equation follows from the vanishing of all normal products of ϕ(x) between vacuum states. Exactly the same result is obtained by performing the functional integral in (14.222) and by using the functional integral formula (14.99). The matrix A is equal to A(x, y)=( ✷ µ2) δ(x y), and its inverse yields the propagator D(x y): − x − − − 1 1 A− (x, y)= δ(x y)= iD(x y), (14.227) ✷ µ2 + iǫ − − − − x − 960 14 Functional-Integral Representation of Quantum Field Theory yielding again (14.226). The generating functional of a free Dirac ﬁeld theory reads ˆ ˆ¯ Z [η, η¯] = 0 T ei (¯ηψ+ψη)dx 0 0 h | | i ¯ ¯ R i dx[ 0(ψ,ψ)+¯ηψ+ψη] = const ψ ψe¯ L . (14.228) × D D Z R where (x) is the free-ﬁeld Lagrangian L0 (x)= ψ¯(x)(iγµ∂ M) ψ(x)= ψ¯(x)A(x.y)ψ(x), (14.229) L0 µ − By analogy with the bosonic expression (14.226) we obtain for Dirac particles

1 dxdy δ G (x y) δ ˆ ˆ¯ 2 δψ(x) 0 δψ¯(y) i dx(¯ηψ+ψ)η Z0[¯η, η] = e − 0 : e : 0 1 h | ˆ | i R dxdyη¯(x)G0(x y)η(y) i Rdx(¯ηψˆ+ψ¯)η = e− 2 − 0 : e : 0 1 h | | i R dxdyη¯(x)G0(x y)η(y) R = e− 2 − , (14.230) R where G (x y) is the free fermion propagator, related to the functional inverse of 0 − the matrix A(x, y) by

1 1 A− (x, y)= δ(x y)= iG (x y). (14.231) iγµ∂ M + iǫ − − 0 − µ − Note that it is Wick’s expansion which supplies the free part of the Lagrangian when going from the operator form (14.224) to the functional version (14.221).

14.12 Plasma Oscillation

The functional formulation of second-quantized many-body systems allows us to treat eﬃciently various collective phenomena. As a ﬁrst example we shall consider a many-electron system that interacts only via long-range Coulomb forces. The Coulomb forces give rise to collective modes called plasmons. The other extremely important example caused by attractive short-range interactions will be treated in the next chapter.

14.12.1 Plasmon Fields Let us give a ﬁrst application of the functional method by transforming the grand partition function (14.213) to plasmon coordinates. For this, we make use of the Hubbard-Stratonovich transformation (14.99) and observe that a two-body interaction (14.161) in the generating functional (14.213) can always be produced (following Maxwell’s original ideas in electromagnetism) by an auxiliary ﬁeld ϕ(x) as follows: i exp dxdx′ψ∗(x)ψ∗(x′)ψ(x)ψ(x′)V (x, x′) (14.232) −2 Z i 1 = const ϕ dxdx′ ϕ(x)V − (x, x′)ϕ(x′) 2ϕ(x)ψ∗(x)ψ(x)δ(x x′) . × D 2 − − Z Z h i 14.12 Plasma Oscillations 961

To abbreviate the notation, we have used a four-vector notation in which

x (x, t), dx d3xdt, δ(x) δ3(x)δ(t). ≡ ≡ ≡ 1 The symbol V − (x, x′) denotes the functional inverse of the matrix V (x, x′), i.e., the solution of the equation

1 dx′V − (x, x′)V (x′, x′′)= δ(x x′′). (14.233) Z − 1/2 The constant prefactor in (14.232) is [det V ]− . Absorbing this into the always omitted normalization factor N of the functional integral, the grand-canonical partition function Ω = Z becomes

Z[η∗, η]= ψ∗ ψDϕ exp i + i dx (η∗(x)ψ(x)+ ψ∗(x)η(x)) , (14.234) Z D D A Z where the new action is

[ψ∗,ψ,ϕ]= dxdx′ ψ∗(x)[i∂t ξ( i∇) ϕ(x)] δ(x x′)ψ(x′) (14.235) A Z − − − − 1 1 + ϕ(x)V − (x, x′)ϕ(x′) . 2 Note that the eﬀect of using formula (14.99) in the generating functional amounts to the addition of the complete square in ϕ in the exponent:

1 1 dxdx′ ϕ(x) dyV (x, y)ψ∗(y)ψ(y) V − (x, x′) 2 Z − Z ϕ(x′) dy′V (x′,y′)ψ∗(y′)ψ(y′) , (14.236) × − Z together with the additional integration over ϕ. This procedure of going from D (14.161) to (14.235) is probably simpler mnemonically than via the formula (14.99). The fact that the functional Z remains unchanged by this addition is obvious, since 1/2 the integral Dϕ produces only the irrelevant constant [det V ]− . The physical signiﬁcance of the new ﬁeld ϕ(x) is easy to understand: ϕ(x) is directly related to the particle density. At the classical level this is seen immediately by extremizing the action (14.235) with respect to variations δϕ(x):

δ A = ϕ(x) dyV (x, y)ψ∗(y)ψ(y)=0. (14.237) ∂ϕ(x) − Z Quantum mechanically, there will be fluctuations around the field configuration ϕ(x) determined by Eq. (14.237), causing a difference between the Green functions involving the fields ϕ(x) versus those involving the associated composite field operators dyV (x, y)ψ∗(y)ψ(y). Due to the Gaussian nature of the Dϕ integration, the R 962 14 Functional-Integral Representation of Quantum Field Theory difference between the two is quite simple. for example, one can easily see that the propagators of the two fields differ merely by the direct interaction:

T (ϕ(x)ϕ(x′)) (14.238) h i = V (x x′)+ T dyV (x, y)ψ(y) dy′V (x′,y′)ψ∗(y′)ψ(y′) . − Z Z For the proof, the reader is referred to Appendix 14A. Note that for a potential V which is dominantly caused by a single fundamental-particle exchange, the field ϕ(x) coincides with the field of this particle: If, for example, V (x, y) represents the Coulomb interaction e2 V (x, x′)= δ(t t′), (14.239) x x − | − ′| then Eq. (14.237) amounts to 4πe2 ϕ(x, t)= ψ∗(x, t)ψ(x, t), (14.240) − ∇2 revealing the auxiliary field as the electric potential. If the particles ψ(x) have spin indices, the potential will, in this example, be thought of as spin conserving at every vertex, and Eq. (14.237) must be read as spin 4 α contracted: ϕ(x) d yV (x, y)ψ∗ (y)ψ (y). This restriction is initially applied ≡ α only for convenience,R and can easily be dropped later. Nothing in our procedure depends on this particular property of V and ϕ. In fact, V could arise from the exchange of many different fundamental particles and their multiparticle configurations (for example π,ππ,σ,ϕ, etc. in nuclei) so that the spin dependence is the rule rather than the exception. The important point is now that the entire theory can be rewritten as a field theory of only the auxiliary field ϕ(x). For this we integrate out ψ∗ and ψ in Eq. (14.234), and make use of formula (14.100) to obtain

i Z[η∗, η] Ω[η∗, η]= Ne A, (14.241) ≡ where the new action is

1 1 [ϕ] = Tr log iG− + dxdx′η∗(x)G (x, x′)η(x′), (14.242) A ± ϕ 2 ϕ Z with Gϕ being the Green function of the fundamental particles in an external classical ﬁeld ϕ(x):

[i∂ χ( i∇) ϕ(x)] G (x, x′)= iδ(x x′). (14.243) t − − − ϕ − The ﬁeld ϕ(x) is called a plasmon ﬁeld. The new plasmon action can easily be interpreted graphically. For this, one expands Gϕ in powers of ϕ:

Gϕ(x, x′)= G0(x x′) i dx1G0(x x1)ϕ(x1 x′)+ ... (14.244) − − Z − − 14.12 Plasma Oscillations 963

i E ζ(p) ϕ ϕϕ −

Figure 14.1 The last pure-current term of collective action (14.242). The original fundamental particle (straight line) can enter and leave the diagrams only via external currents, emitting an arbitrary number of plasmons (wiggly lines) on its way.

Hence the couplings to the external currents η∗, η in (14.242) amount to radiating one, two, etc. ϕ ﬁelds from every external line of fundamental particles (see Fig. 14.1). A functional expansion of the Tr log expression in powers of ϕ gives

1 1 iTr log(iG− ) = iTr log(iG− ) iTr log(1 + iG ϕ) ± ϕ ± 0 ± 0 1 ∞ n 1 = iTr log(iG− ) iTr ( iG ϕ) . (14.245) ± 0 ∓ − 0 n nX=1 The ﬁrst term leads to an irrelevant multiplicative factor in (14.241). The nth term corresponds to a loop of the original fundamental particle emitting nϕ lines (see Fig. 14.2).

Figure 14.2 Non-polynomial self-interaction terms of plasmons arising from the Tr log in (14.242). The nth term presents a single-loop diagram emitting n plasmons.

Let us now use the action (14.242) to construct a quantum ﬁeld theory of plasmons. For this we may include the quadratic term 1 iTr(G ϕ)2 (14.246) ± 0 2 into the free part of ϕ in (14.242) and treat the remainder perturbatively. The free propagator of the plasmon becomes

0 Tϕ(x)ϕ(x′) 0 (2s + 1)G (x′, x). (14.247) { | | } ≡ 0 This corresponds to an inclusion into the V propagator of all ring graphs (see Fig. 14.3). It is worth pointing out that the propagator in momentum space Gpl(k) contains actually two important physical informations. From the derivation at ﬁxed temperature it appears in the transformed action (14.242) as a function of discrete 964 14 Functional-Integral Representation of Quantum Field Theory

Figure 14.3 Free plasmon propagator containing an inﬁnite sequence of single-loop corrections (“bubblewise summation”).

euclidean frequencies νn = 2πnT only. In this way it serves to set up a time- independent fixed-T description of the system. The calculation (2.16), however, renders it as a function in the whole complex energy plane. It is this function which determines by analytic continuation the time-dependent collective phenomena for real times6. With the propagator (2.16) and the interactions given by (2.13), the original theory of fundamental fields ψ∗, ψ has been transformed into a theory of ϕ fields whose bare propagator accounts for the original potential which has absorbed ringwise an infinite sequence of fundamental loops. This transformation is exact. Nothing in our procedure depends on the statistics of the fundamental particles nor on the shape of the potential. Such properties are important when it comes to solving the theory perturbatively. Only under appropriate physical circumstances will the field ϕ represent important collective excitations with weak residual interactions. Then the new formulation is of great use in understanding the dynamics of the system. As an illustration consider a dilute fermion gas of very low temperature. Then the function ξ( i∇) is ǫ( i∇) µ with 2 − − − ǫ( i∇)= ∇ /2m. − − 14.12.2 Physical Consequences Let the potential be translationally invariant and instantaneous:

V (x, x′)= δ(t t′)V (x x′). (14.248) − − Then the plasmon propagator (2.16) reads in momentum space

1 G (ν, k)= V (k) (14.249) pl 1 V (k)π(ν, k) − where the single electron loop symbolizes the analytic expression7

T 1 1 k π(ν, )=2 2 2 . (14.250) V p iω p /2m + µ i(ω + ν) (p + k) /2m + µ X − − 6See the discussion in Chapter 9 of the last of Ref. [7] and G. Baym and N.D. Mermin, J. Math. Phys. 2, 232 (1961). 7The factor 2 stems from the trace over the electron spin. 14.12 Plasma Oscillations 965

The frequencies ω and ν are odd and even multiples of πT . In order to calculate the sum we introduce a convergence factor eiωη, and rewrite (14.250) as

d3p 1 π(ν, k)=2 (14.251) (2π)3 ξ(p + k) ξ(p) iν Z − − 1 1 T eiωnη . × ω "i(ωn + ν) ξ(p + k) − iωn ξ(p)# Xn − − Using the summation formula (14.197), this becomes

d3p n(p + k) n(p) π(ν, k)=2 − , (14.252) (2π)3 ǫ(p + k) ǫ(p) iν Z − − or, after some rearrangement,

d3p 1 1 π(ν, k)= 2 n(p) + .(14.253) − (2π)3 "ǫ(p + k) ǫ(p) iν ǫ(p k) ǫ(p)+ iν # Z − − − − Let us study this function for real physical frequencies ω = iν where we rewrite it as d3p 1 1 π(ω, k)= 2 n(p) + , (14.254) − (2π)3 " ǫ(p+k) ǫ(p) ω ǫ(p k) ǫ(p)+ ω # Z − − − − which can be brought to the form

k2 d3p 1 π(ω, k)=2 n(p) . (14.255) mω2 (2π)3 (ω p k/m + iη)2 (k2/2M)2 Z − · − For ω >p k/m + k2/2m, the integrand is real and we can expand | | F k2 d3p 2p k p k 2 π(ω, k)=2 n(p) 1+ · +3 · 2 3  mω Z (2π) mω mω !  p k 3 80(p k)4 + m2ω2k4 + · + · + ... . (14.256) mω ! 16m2ω4   Zero Temperature For zero temperature, the chemical potential coincides with the radius of the Fermi sphere µ = pF , and all levels below the Fermi momentum are occupied, to that the Fermi distribuion function is n(p) = Θ(p pF ). Then the integral in (14.256) can be performed trivially using the integral −

d3p N p3 p F 2 3 nT =0( )= = n = 2 , (14.257) Z (2π) V 3π 966 14 Functional-Integral Representation of Quantum Field Theory and we obtain

k2 n 3 p k 2 1 p k 4 1 k4 π(ω, k)= 1+ F + F + + ... . (14.258) ω2 m  5 mω ! 5 mω ! 16 m2ω2    Inserting this into (14.249) we ﬁnd, for long wavelengths, the Green function

1 V (k) n − Gpl(ν, k) V (k) 1 + ... . (14.259) ≈ " − ω2 m # Thus the original propagator is modiﬁed by a factor

4πe2 n ǫ(ω, k)=1 + .... (14.260) − ω2 m The dielectric constant vanishes at the frequency

4πe2 ω = ω = , (14.261) pl s m which is the famous plasma frequency of the electron gas. At this frequency, the plasma propagator (14.249) has a pole on the real-ω axis, implying the existence of an undamped excitation of the system. For an electron gas we insert the Coulomb interaction (14.240), and obtain

1 4πe2 4πe2 − Gpl(ν, k) 1 n + ... . (14.262) ≈ k2 " − mω2 #

Thus the original Coulomb propagator is modiﬁed by a factor

4πe2 ǫ(ω, k)=1 n + ..., (14.263) − mω2 which is simply the dielectric constant. The zero temperature limit can also be calculated exactly starting from the expression (14.256), written in the form

d3p 1 π(ω, k) = 2 Θ(p pF ) +(ω ω) . (14.264) − (2π)3 − "p k + k2/2m ω → − # Z · − Performing the integral yields

mp 1 k mω 2 k2 +2mω 2kp k F 2 2 F π(ω, ) = 2 1 pF + + pF log 2 − − 2π  − 2kpF  − 2 k !  k +2mω +2kpF    + (ω ω).   (14.265) → −  The lowest terms of a Taylor expansion in powers of k agree with (14.258). 14.12 Plasma Oscillations 967

Short-Range Potential

Let us also ﬁnd the real poles of Gpl(ν, k) for a short-rang potential where the 1 singularity at k = 0 is absent. Then a rotationally invariant [V (k)]− has the long- wavelength expansion 1 1 2 [V (k)]− = [V (0)]− + ak + ..., (14.266) 1 as long as [V (0)]− is ﬁnite and positive, i.e., for a well behaved overall repulsive potential satisfying V (0) = d3xV (x) > 0. Then the Green function (14.249) becomes R 1 2 − 2 2 1 2 2 2 n 3 pF k Gpl(ω, k)= ω ω [V (0)]− + aω k k 1+ + ... . (14.267)  − m  5 mω !      This has a pole atω = c0k where  ± n c = V (0) . (14.268) 0 m

This is the spectrum of zero sound with the velocity c0. In the neighbourhood of the positive-energy pole, the propagator has the form k Gpl(k ,k) V (0) | | . (14.269) 0 ≈ × ω c k − 0| | Nonzero Temperature In order to discuss the case of nonzero temperature it is convenient to add and subtract a term n(p + k)n(k) in the numerator of (14.252), and rewrite it as − d3p n(p + k) [1 n(p)] n(p) [1 n(p + k)] π(ν, k)=2 − − − , (14.270) (2π)3 ǫ(p + k) ǫ(p) iν Z − − which can be rearranged to d3p ǫ(p + k) ǫ(p) π(ν, k)= 4 n(p) [1 n(p + k)] − . (14.271) − (2π)3 − [ǫ(p + k) ǫ(p]2 + ν2 Z − In the high-temperature limit the Fermi distribution becomes Boltzmannian, β(p2/2 µ) n(p) e− − , and we evaluate again most easily expression (14.256) as follows: ≈ 3 ∞ d p β(p2/2 µ) σ[ǫ(p+k) ǫ(p) ω] k − − − − − π(ω, )= 2 dσ 3 e +(ω ω) . (14.272) − Z0 Z (2π) → − The right-hand side is equal to 1 3 ∞ d cos θ d p β(p2/2 µ) σ[(pk cos θ/m+k2/2m) ω] dσ e− − − − +(ω ω). (14.273) − 0 1 2 (2π)3 → − Z Z− Z Performing the angular integral yields 2 3 m ∞ dσ d p β(p2/2 µ)+σω 1 − − 2 2 3 e 2 (pk cosh pk sinh pk)+(ω ω). (14.274) − k Z0 2π Z (2π) p − → − 968 14 Functional-Integral Representation of Quantum Field Theory

14.13 Pair Fields

The introduction of a scalar field ϕ(x) was historically the first way, invented by Maxwell, to convert the Coulomb interaction in a theory (14.232) into a local field theory. The resulting plasmon action depends only on the local field ϕ. There exists an alternative way of converting the interaction between four fermions in (14.232) into a new field theory. That is based on introducing a bilocal scalar field which has been very successful to understand the properties of electrons in of superconductors. It is a collective field complementary to the plasmon field. Generically it will be called a pair field. It describes the dominant low-energy collective excitations in systems such as type II superconductors, superfluid 3He, excitonic insulators, etc. The pair field is originally a bilocal field and will be denoted by ∆(x t; x′t′), with two space arguments and two time arguments. It is introduced into the generating functional by rewriting the exponential of the interaction term in (14.232) in the partition function (14.169) as a functional integral i exp dxdx′ψ∗(x)ψ∗(x′)ψ(x′)ψ(x)V (x, x′) = const ∆(x, x′) ∆∗(x, x′) −2 Z ×Z D D i dxdx′ ∆(x,x′) 2 1 ∆∗(x,x′)ψ(x)ψ(x′ ) ψ∗(x)ψ∗(x′)∆(x,x′) e 2 | | V (x,x′) − − . (14.275) × R n o In contrast to the similar-looking plasmon expression (14.232), the inverse 1/V (x, x′) in (14.275) is understood as a numeric division for each x, y, not as a functional inverse. Hence the grand-canonical potential becomes

i [ψ∗,ψ,∆∗,∆]+i dx(ψ∗(x)η(x)+c.c.) Z[η, η∗]= ψ∗ ψ ∆∗ ∆ e A , (14.276) D D D D Z R with the action

[ψ∗, ψ, ∆∗, ∆] = dxdx′ ψ∗(x)[i∂t ξ( i∇)] δ(x x′)ψ(x′) (14.277) A Z { − − − 1 1 1 2 1 ∆∗(x, x′)ψ(x)ψ(x′) ψ∗(x)ψ∗(x′)∆(x, x′)+ ∆(x, x′) , −2 − 2 2| | V (x, x′)) where ξp εp µ is the grand-canonical single particle energy (2.256). This new ≡ − action arises from the original one in (14.169) by adding to it the complete square

i 2 1 dxdx′ ∆(x, x) V (x′, x)ψ(x′)ψ(x) , (14.278) 2 Z | − | V (x, x′) which removes the fourth-order interaction term and gives, upon functional integration over ∆∗ ∆, merely an irrelevant constant factor to the generating func- D D tional. R At the classical level, the ﬁeld ∆(x, x′) is nothing but a convenient abbreviation for the composite ﬁeld V (x, x′)ψ(x)ψ(x′). This follows from the equation of motion obtained by extremizing the new action with respect to δ∆∗(x, x′). This yields δ 1 A = [∆(x, x′) V (x, x′)ψ(x)ψ(x′)] 0. (14.279) δ∆∗(x, x′) 2V (x, x′) − ≡ 14.13 Pair Fields 969

Quantum mechanically, there are Gaussian fluctuations around this solution which are discussed in Appendix 14B. Taking care of the spin components of the Fermi field, we can rewrite the first line in the expression (14.278), which are quadratic in the fundamental fields ψ(x), in a matrix form as 1 f ∗(x)A(x, x′)f(x′) 2 1 [i∂t ξ( i∇)] δ(x x′) ∆(x, x′) = f †(x) − − − − f(x′),(14.280) 2 ∆∗(x, x′) [i∂t + ξ(i∇)] δ(x x′) − ∓ − ! where f(x) denotes the fundamental field doublet

ψ(x) f(x)= (14.281) ψ∗(x) !

T and f † f ∗ , as usual. Here the ﬁeld f ∗(x) is not independent of f(x). Indeed, there is≡ an identity

T 0 1 f †Af = f Af. (14.282) 1 0 !

Therefore, the real-ﬁeld formula (14.99) must be used to evaluate the functional integral for the generating functional

i [∆∗,∆] 1 dx dx′j†(x)G (x,x′)j(x′) Z[η∗, η]= ∆∗ ∆ e A − 2 ∆ , (14.283) D D Z R R where j(x) collects the external source η(x) and its complex conjugate, j(x) ≡ η(x) . Then the collective action (14.278) reads η∗(x) !

i 1 1 2 1 [∆∗, ∆] = Tr log iG− (x, x′) + dxdx′ ∆(x, x′) . (14.284) A ±2 ∆ 2 | | V (x, x ) h i Z ′ 1 The 2 2 matrix G∆ denotes the propagator iA− which satisﬁes the functional equation×

[i∂t ξ( i∇)] δ(x x′′) ∆(x, x′′) dx′′ − − − − G∆(x′′, x′)=iδ(x x′). ∆∗(x, x′′) [i∂t + ξ(i∇)] δ(x x′′) − Z − ∓ − ! (14.285)

G G∆ Writing G∆ as a matrix ˜ , the mean-field equations associated with this G∆† G ! action are precisely the equations used by Gorkov to study the behavior of type II superconductors [12]. With Z[η∗, η] being the full partition function of the system, the fluctuations of the collective field ∆(x, x′) can now be incorporated, at least in principle, thereby yielding corrections to these equations. 970 14 Functional-Integral Representation of Quantum Field Theory

Let us set the sources in the generating functional Z[η∗, η] equal to zero and investigate the behavior of the collective quantum field ∆. In particular, we want to develop Feynman rules for a perturbative treatment of the fluctuations of ∆(x, x′). As a first step we expand the Green function G∆ in powers of ∆ as 0 ∆ 0 ∆ 0 ∆ G∆ = G0 iG0 G0 G0 G0 G0 + ... (14.286) − ∆∗ 0 ! − ∆∗ 0 ! ∆∗ 0 ! with i ∇ δ(x x′) 0  i∂t ξ( i ) −  G0(x, x′)= − − i . (14.287)  0 δ(x x′)   ∓i∂ + ξ(i∇) −   t    We shall see later that this expansion is applicable only close to the critical temperature Tc. Inserting this expansion into (14.283), the source term can be interpreted graphically by the absorption and emission of lines ∆(k) and ∆∗(k), respectively, from virtual zig-zag configurations of the underlying particles ψ(k), ψ∗(k) (see Fig. 14.4)

i ∗ ′ ′ ∆ (ν , q ) ω ξ(p) − i + + . . . (14.286) ν′ ω ξ(q′ p) − − −

i ∆(ν, q) ν ν′+ω ξ(q q′+p) − − −

Figure 14.4 Fundamental particles (fat lines) entering any diagram only via the external currents in the last term of (14.283), absorbing n pairs from the right (the past) and emitting the same number from the left (the future).

The functional submatrices in G0 have the Fourier representation

T i i p0t px ′ − ( − ) G0(x, x ) = 0 e , (14.289) V p p ξp X − T i i p0t px ˜ − ( − ) G0(x, x′) = 0 e , (14.290) ±V p p ξ p X − − − where we have used the notation ξp for the Fourier components ξ(p) of ξ( i∇). − The ﬁrst matrix coincides with the operator Green function

G (x x′)= 0 T ψ(x)ψ†(x′) 0 . (14.291) 0 − h | | i The second one corresponds to

G˜ (x x′) = 0 T ψ†(x)ψ(x′) 0 = 0 T ψ(x′)ψ†(x) 0 0 − h | | i ±h | | i T = G (x′ x) [G (x, x′)] , (14.292) ± 0 − ≡ ± 0 14.13 Pair Fields 971

where T denotes the transposition in the functional sense (i.e., x and x′ are in- terchanged). After a Wick rotation of the energy integration contour, the Fourier components of the Green functions at ﬁxed energy read

1 ip(x x′) G0(x x′,ω) = e − (14.293) − − p iω ξp X − 1 ip(x x′) G˜0(x x′,ω) = e − = G0(x′ x, ω). (14.294) − ∓ p iω ξ p ∓ − − X − − − The Tr log term in Eq. (14.284) can be interpreted graphically just as easily by expanding as in (4.15):

n i 1 i 1 i 0 ∆ 1 Tr log iG∆− = Tr log iG0− Tr iG0 ∆∗ .(14.295) ±2 ±2 ∓ 2 "− ∆∗ 0 ! # n The ﬁrst term only changes the irrelevant normalization N of Z. To the remaining sum only even powers can contribute so that we can rewrite

n n ∞ ( ) i i [∆∗, ∆] = i − Tr δ ∆ ∓ δ ∆∗ A ∓ 2n " i∂t ξ( i∇) ! i∂t + ξ(i∇) ! # nX=1 − − 1 2 1 + dxdx′ ∆(x, x′) 2 Z | | V (x, x′) ∞ 1 2 1 = n[∆∗, ∆] + dxdx′ ∆(x, x′) . (14.296) A 2 | | V (x, x′) nX=1 Z This form of the action allows an immediate quantization of the collective field ∆. The graphical rules are slightly more involved technically than in the plasmon case since the pair field is bilocal. Consider at first the free collective fields which can be obtained from the quadratic part of the action:

i i i 2[∆∗, ∆] = Tr δ ∆ δ ∆∗ . (14.297) A −2 " i∂t ξ( i∇) ! i∂t + ξ(i∇) ! # − − Variation with respect to ∆ displays the equations of motion i i ∆(x, x′)= iV (x, x′) δ ∆ δ . (14.298) " i∂t ξ( i∇) ! i∂t + ξ(i∇) !# − − This equation coincides exactly with the Bethe-Salpeter equation [13] in the ladder approximation. Originally, this was set up for two-body bound-state vertex functions, usually denoted in momentum space by

Γ(p,p′)= dxdx′ exp[i(px + p′x′)]∆(x, x′). (14.299) Z

Thus the free excitations of the field ∆(x, x′) consist of bound pairs of the original fundamental particles. The field ∆(x, x′) will consequently be called pair field. If 972 14 Functional-Integral Representation of Quantum Field Theory

we introduce total and relative momenta q and P = (p p′)/2, then (14.298) can be written as8 − 4 d P ′ i ′ Γ(P q) = i 4 V (P P ) | − (2π) − q /2+ P ′ ξ ′ + iη sgn ξ Z 0 0 − q/2+P i Γ(P ′ q) . (14.300) × | q0/2 P0′ ξq/2 P′ + iη sgn ξ − − − Graphically this formula can be represented as follows: The Bethe-Salpeter wave

Figure 14.5 Free pair ﬁeld following the Bethe-Salpeter equation as pictured in this diagram. function is related to the vertex Γ(P q) by | i Φ(P q) = N | q /2+ P ξq P + iη sgn ξ 0 0 − /2+ i Γ(P q). (14.301) ×q /2+ P ξq P + iη sgn ξ | 0 0 − /2+ It satisﬁes

dP ′ G0 (q/2+ P ) G0 (q/2 P ) Φ(P q)= i 4 V (P,P ′)Φ(P ′ q), (14.302) − | − Z (2π) | thus coinciding, up to a normalization, with the Fourier transform of the two-body state wave functions

ψ(x, t; x′, t′)= 0 T (ψ(x, t)ψ(x′, t′)) B(q) . (14.303) h | | i

If the potential is instantaneous, then (14.298) shows ∆(x, x′) to be factorizable according to

∆(x, x′)= δ(t t′)∆(x, x′; t) (14.304) − so that Γ(P q) becomes independent of P . | 0 8 µ 0 Here q abbreviates the four-vector q = (q , q) with q0 = E. 14.13 Pair Fields 973

Consider now the system at T = 0 in the vacuum. Then µ = 0 and ξp = εp > 0. One can perform the P0 integral in (14.300) with the result

3 d P ′ 1 P P P′ P′ Γ( q)= 4 V ( ) Γ( q). (14.305) | (2π) − q0 εq/2+P′ εq/2 P′ + iη | Z − − − Now the equal-time Bethe-Salpeter wave function

3 3 d Pdq0d q x + x′ x x′ q P x x′ ψ( , ; t) N 7 exp i q0t ( ) ≡ Z (2π) "− − · 2 − · − !# 1 Γ(P q) (14.306) × q0 εq/2+P εq/2 P + iη | − − − satisﬁes

i∂t ǫ( i∇) ǫ( i∇′) ψ(x, x′; t)= V (x x)ψ(x, x′; t), (14.307) − − − − − h i which is simply the Schrödinger equation of the two-body system. Thus, in the instantaneous case, the free collective excitations in ∆(x, x′) are the bound states derived from the Schrödinger equation. In a thermal ensemble, the energies in (14.300) have to be summed over the Matsubara frequencies only. First, we write the Schrödinger equation as

3 d P′ P P P′ P′ P′ Γ( q)= 3 V ( )l( q)Γ( q) (14.308) | − Z (2π) − | | with

l(P q) = i G (q/2+ P ) G˜ (P q/2) (14.309) | − 0 0 − XP0 i i = i . − q0/2+P0 ξq/2+P +iη sgn ξ q0/2 P0 ξq/2 P +iη sgn ξ XP0 − − − − After a Wick rotation and setting q iν, the replacement of the energy integration 0 ≡ by a Matsubara sum leads to

1 1 l(P q) = T | − ω i (ωn + ν/2) ξq/2+P i (ωn ν/2) + ξq/2 P Xn − − − 1 = T ω iν ξq/2+P ξq/2 P Xn − − − 1 1

× "i(ωn + ν/2) ξq/2+P − i(ωn ν/2) + ξq/2 P # − − − nq/2+P + nq/2 P ± − = h i . (14.310) −iν ξq/2+P ξq/2 P − − − 974 14 Functional-Integral Representation of Quantum Field Theory

Here we have used the frequency sum [see (14.197)]

1 1 T = np. (14.311) ξp/T ω iωn ξp ∓e 1 ≡ ∓ Xn − ∓ with np being the occupation number of the state of energy ξp. Using the identity np 1 np, the expression in brackets can be rewritten as N(P, q) where → ∓ − −

N(P q) 1 nq/2+P + nq/2 P | ≡ ± − 1 1 ξq/2+P 1 ξq/2 P = tanh ∓ + tanh ∓ − , (14.312) 2 2T 2T ! so that N(P q) l(P q)= | . (14.313) | −iν ξq/2+P ξ(q/2 P − − − Deﬁning again a Schr¨odinger type wave function for T = 0 as in (14.306), the bound-state problem can be brought to the form (14.305)6 but with a momentum dependent potential V (P P′) N(P′ q). Thus the Bethe-Salpeter equation at any temperature reads − × |

3 d P ′ 1 P P P′ P′ q P′ Γ( q)= 4 V ( )N( ) Γ( q). (14.314) | (2π) − | q0 εq/2+P′ εq/2 P′ + iη | Z − − −

We are now ready to construct the propagator of the pair ﬁeld ∆(x, x′) for T = 0. In many cases, this is most simply done by considering Eq. (14.300) with a potential λV (P,P ′) rather than V , and asking for all eigenvalues λn at ﬁxed q. Let Γn(P q) be a complete set of vertex functions for this q. Then one can write the propagator| as

Γn(P q)Γn∗ (P ′ q) 4 (4) ∆(P q)∆†(P ′ q′)= i | | (2π) δ (q q′) (14.315) | | − λ λ (q) − n n λ=1 X − where a hook denotes, as usual, the Wick contraction of the ﬁelds. Obviously the vertex functions have to be normalized in a speciﬁc way, as discussed in Appendix 14A. n An expansion of (14.315) in powers of [λ/λn(q)] exhibits the propagator of ∆ as a ladder sum of exchanges as shown in Fig. 14.6.

+ + + . . . (14.316)

Figure 14.6 Free pair propagator, amounting to a sum of all ladders of fundamental potential exchanges. This is revealed explicitly by the expansion of (14.315) in powers of (λ/λn(q)). 14.14 Competition of Plasmon and Pair Fields 975

For an instantaneous interaction, either side is independent of P0,P0′. Then the propagator can be shown to coincide directly with the scattering matrix T of the Schr¨odinger equation (14.307) and the associated integral equation in momentum space (14.305) [see Eq. (14A.13)].

1 ∆∆† = iT iV + iV V. (14.317) ≡ E H − Consider now the higher interactions , n 3 of Eq. (14.296). They correspond An ≥ to zig-zag loops shown in Fig. 14.7. These have to be calculated with every possible Γ (P q), Γ∗ (P q) entering or leaving, respectively. Due to the P dependence at n | m |

+ + . . . (14.318)

Figure 14.7 Self-interaction terms of the non-polynomial pair action (14.296) amounting to the calculation of all single zig-zag loop diagrams absorbing and emitting n pair ﬁelds. every vertex, the loop integrals become very involved. A slight simpliﬁcation arises for instantaneous potentials. Then the frequency sums can be performed. Only in the special case of a completely local action, the full P -dependence disappears and the integrals can be calculated. See Section IV.2 in Ref. [4].

14.14 Competition of Plasmon and Pair Fields

The Hubbard-Stratonovich transformation has a well-established place in many- body theory [8, 9, 4]. After it had been successfully applied in 1957 by Bardeen, Cooper, and Schrieﬀer to explain the phenomenon of superconductivity by with the so-called BCS theory [50], Nambu and Jona-Lasinio [14] discovered that the same mechanism which explains the formation of an energy gap and Cooper pairs of electrons in a metal can be used to understand the surprising properties of quark masses in the physics of strongly interacting particles. This aspect of particle physics will be explained in Chapter 26. In many-body theory, the use of the Hubbard-Stratonovich transformation has led to a good understanding of important collective physical phenomena such as plasma oscillations and other charge-density type of waves, for example paramagnons in superﬂuid He3. It has put heuristic calculations such as the Gorkov’s derivation [12] of the Ginzburg-Landau equations [15] on a reliable theoretical ground [4]. In 976 14 Functional-Integral Representation of Quantum Field Theory addition, it is in spirit close to the density functional theory [16] via the Hohenberg- Kohn and Kohn-Sham theorems [17].

In Sections 14.12 and 14.13 we have used the Hubbard-Stratonovich transformation to rewrite the many-body action in two ways. One was theory of a local plasmon field ϕ(x), the other a thery of a bilocal scalar pair field ∆(x, x′). In the theory of collective excitation, either of the two transformations has been helpful to understand either the behavior of electron gases or that of superconductors. In the first case, one is able to deal efficiently with the oscillations of charge distributions in the gas. In the second case one is able to see how the electrons in a superconductor become bound to Cooper pairs, and how the binding gives rise to a frictionless flow of the doubly charged bosonic pairs through the metal. In general, however, there exists a competition between the two mechanisms.

The transformation was cherished by theoreticians since it allows them to re- express a four-particle interaction exactly in terms of a collective ﬁeld variable whose ﬂuctations can in principle be described by higher loop diagrams. The only bitter pill is that any approximative treatment of a many-body system can describe interesting physics only if calculations can be restricted to a few low-order diagrams. If this is not the case, the transformation fails.

The trouble arises in all those many-body systems in which different collective effects compete with similar relevance. Historically, an important example is the fermionic superfluid He3. While BCS superconductivity is described easily via the Hubbard-Stratonovich transformation which turns the four-electron interaction into a field theory of Cooper pairs, the same approach did initially not succeed in a liq- uid of He3-atoms. Due to the strongly repulsive core of an atom, the forces in the attractive p-wave are not sufficient to bind the Cooper pairs. Only by taking into account the existence of another collective field that arises in the competing paramagnon channel was it possible to explain the formation of weakly-bound Cooper pairs [18].

It is important to learn how to deal with this kind of mixture. The answer is found with the help of Variational Perturbation Theory (VPT), which has been discussed in Chapter 3 (see the pages 177 and 216). The key is to abandon the fluctuating collective quantum fields introduced by means of the Hubbard-Stratonovich transformation. Instead one must turn to a variety of collective classical fields. Af- ter a perturbative calculation of the effective action, one obtains a functional that depends on these classical fields. The dependence can be optimized, usually by extremizing their influence upon the effective action. In this way one is able to obtain exponentially fast converging results.

It is the purpose of this section to point out how to circumvent the fatal focussing of the Hubbard-Stratonovich transformation on a speciﬁc channel and to take into account the competition between several competing channels. 14.15 Ambiguity in the Selection of Important Channels 977

14.15 Ambiguity in the Selection of Important Channels

The basic weakness of the Hubbard-Stratonovich transformation lies in the different possibilities of decomposing the fourth order interaction by a quadratic completion with the help of an auxiliary field. The first is based on the introduction of a scalar plasmon field ϕ(x) and the use of the quadratic-completion formula (14.232). The other uses the introduction of a bilocal pair field ∆(x, x′) in combination with the quadratic-completion formula (14.275). The trouble with both approaches is that, when introducing an auxiliary field ϕ(x) or ∆(x, x′) and summing over all fluctuations of one of the fields, the effects of the other is automatically included. At first sight, this may appear as an advantage. Unfortunately, this is an illusion. In either case, even the lowest-order fluctuation effect of the other is extremely hard to calculate. That can be seen most simply in the simplest models of quantum field theory such as the Gross-Neveu model (to be discussed in Chapter 23). There the propagator of the quantum field ∆(x, x′) is a very complicated object. So it is practically impossible to recover the effects OF ϕ(x) from the loop calculations with these propagators. As a consequence, the use of a specific quantum field theory must be abandoned whenever collective effects of different channels are important. To be specific let us assume the fundamental interaction to be of the local form

loc g int = ψα∗ ψβ∗ψβψα = g ψ∗ψ∗ψ ψ , (14.319) ↓ ↑ A 2Zx Zx ↑ ↓ where the subscripts , denote the spin directions of the fermion fields. For brevity, we have absorbed the↑ spacetime↓ arguments x in the spin subscripts and written the symbol x for an integral over spacetime and a sum over the spin indices of the fermionR field. We now introduce auxiliary classical collective fields which are no longer assumed to undergo functional fluctuations, and we replace the exponential in the interacting version of the generating functional (14.276),

∗ loc ∗ i 0[ψ ,ψ]+i +i dx(ψ (x)η(x)+c.c.) Z[η, η∗]= ψ∗ ψ e A Aint , (14.320) D D Z R identically by [34]

loc i i int T new e A = exp ig ψ∗,xψ∗,xψ ,xψ ,x = exp fx xfx exp i int (14.321) ↓ ↑ { Zx ↑ ↓ } −2 Zx M × { A } i new = exp ψβ∆βα∗ ψα + ψα∗ ∆αβψβ∗ + ψβ∗ ρβαψα + ψα∗ ραβψβ exp i int . −2 x × { A } Z Here f(x) is here the doubled spinor ﬁeld (14.281) with spin index:

ψ (x) f(x)= α , (14.322) ψα∗ (x) ! 978 14 Functional-Integral Representation of Quantum Field Theory

T T and fx denotes the transposed fundamental ﬁeld doublet fx = (ψα, ψα∗ ). The new interaction reads

new loc 1 T g int = int + fx xfx = ψα∗ ψβ∗ ψβψα (14.323) A A 2 Zx M Zx 2 1 + ψ ∆∗ ψ + ψ∗ ∆ ψ∗ + ψ∗ ρ ψ . 2 β βα α α αβ β α αβ β We now deﬁne a further free action by the quadratic form

new 1 T 1 ∆,ρ 0 0 fx xfx = fx†Ax,x′ fx′ . (14.324) A ≡A − 2 Zx M 2 ∆,ρ with the functional matrix Ax,x′ being now equal to [i∂ ξ( i∇)]δ +ρ ∆ t − − αβ αβ αβ . (14.325) ∆∗ [i∂ +ξ(i∇)]δ ρ αβ t αβ − αβ! loc The physical properties of the theory associated with the action 0+ int can now be derived as follows: ﬁrst we calculate the generating functional ofA theA new quadratic action new via the functional integral A0 new i new+i dx(ψ∗(x)η(x)+c.c.) Z [η, η∗]= ψ∗ ψ e A0 . (14.326) 0 D D Z R

From its functional derivatives with respect to the sources ηα and ηα† we find the new free propagators G∆ and Gρ. To higher orders, we expand the exponential i new n new n new e Aint in a power series and evaluate all expectation values (i /n!) [ ] with h Aint i0 the help of Wick’s theorem. They are expanded into sums of products of the free N particle propagators G∆ and Gρ. The sum of all diagrams up to a certain order g defines an effective collective action N as a function of the collective classical fields Aeff ∆αβ, ∆βα∗ , ραβ, Obviously, if the expansion is carried to infinite order, the result must be independent of the auxiliary collective fields since they were introduced and removed in (14.322) without changing the theory. However, any calculation can only be carried up to a finite order, and that will depend on these fields. We therefore expect the best approximation to arise from the extremum of the effective action [6, 21, 54]. The lowest-order effective collective action is obtained from the trace of the logarithm of the matrix (14.325):

0 i 1 = Tr log iG− . (14.327) A∆,ρ −2 ∆,ρ h i ∆,ρ 1 The 2 2 matrix G∆,ρ denotes the propagator i[Ax,x′ ]− . To× ﬁrst order in perturbation theory we must calculate the expectation value int of the interaction (14.324). This is done with the help of Wick contractions inhA thei three channels, Hartree, Fock, and Bogoliubov:

ψ∗ψ∗ψ ψ = ψ∗ψ ψ∗ψ ψ∗ψ ψ∗ψ + ψ∗ψ∗ ψ ψ . (14.328) h ↑ ↓ ↓ ↑i h ↑ ↑ih ↓ ↓i−h ↑ ↓ih ↓ ↑i h ↑ ↓ ih ↓ ↑i 14.15 Ambiguity in the Selection of Important Channels 979

For this purpose we now introduce the expectation values

∆˜ ∗ g ψ∗ ψ∗ , ∆˜ g ψ ψ = [∆˜ ∗ ]∗, (14.329) αβ ≡ h α βi βα ≡ h β αi αβ ρ˜ g ψ∗ ψ , ρ˜† [˜ρ ]∗, (14.330) αβ ≡ h α βi αβ ≡ βα and rewrite new as hAint i new 1 ˜ ˜ 1 ˜ ˜ int = (∆∗ ∆ ρ˜ ρ˜ +ρ ˜ ρ˜ ) (∆βα∆βα∗ + ∆αβ∗ ∆βα +2˜ραβραβ). ↓↑ ↑↓ ↓↑ ↑↑ ↓↓ hA i g Zx ↓↑ − − 2g Zx

Due to the locality of ∆˜ αβ, the diagonal matrix elements vanish, and ∆˜ αβ has the ˜ 2 form cαβ∆, where cαβ is i times the Pauli matrix σαβ. In the absence of a magnetic ﬁeld, the expectation valuesρ ˜αβ may have certain symmetries:

ρ˜ ρ˜ = ρ , ρ˜ = ρ 0, (14.331) ↑↑ ≡ ↓↓ ↑↓ ↓↑ ≡ so that (14.331) simpliﬁes to

new 1 ˜ 2 2 ˜ ˜ int = ( ∆ +˜ρ ) (∆∆∗ + ∆∗∆+2˜ρρ) . (14.332) hA i g x | | − Z h i The total ﬁrst-order collective classical action 1 is given by the sum A∆,ρ 1 = 0 + new . (14.333) A∆,ρ A∆,ρ hAint i Now we observe that the functional derivatives of the zeroth-order action 0 are A∆,ρ the free-ﬁeld propagators G∆ and Gρ

δ 0 δ 0 ∆,ρ = [G∆]αβ , ∆,ρ = [Gρ]αβ . (14.334) δ∆αβ A δραβ A Then we can extremize 1 with respect to the collective fields ∆ and ρ, and find A∆,ρ that to this order these fields satisfy the gap-like equations

∆˜ x = g[G∆]x,x, ρ˜x = g[Gρ]x,x. (14.335) If the ﬁelds satisfy (14.335), the extremal action has the value

1 0 1 2 2 ∆,ρ = ∆,ρ ( ∆ + ρ ). (14.336) A A − g Zx | | Note how the theory differs, at this level, from the collective quantum field theory derived via the Hubbard-Stratonovich transformation. If we assume that ρ vanishes 1 identically, the extremum of the one-loop action ∆,ρ gives the same result as of the mean-field collective quantum field action (14.284),A which reads for the present attractive δ-function in (14.319):

1 0 1 2 ∆,0 = ∆,0 ∆ . (14.337) A A − g Zx | | 980 14 Functional-Integral Representation of Quantum Field Theory

1 On the other hand, if we extremize the action ∆,ρ at ∆ = 0, we ﬁnd the extremum from the expression A

1 0 1 2 0,ρ = 0,ρ ρ . (14.338) A A − g Zx The extremum of the ﬁrst-order combined collective classical action (14.336) agrees with the good-old Hartree-Fock-Bogolioubov theory. The essential diﬀerence between this and the new approach arises in two issues:

First, by being able to carry the expansion to higher orders: If the collective • quantum field theory is based on the Hubbard-Stratonovich transformation, the higher-order diagrams must be calculated with the help of the propagators of the collective field such as ∆x∆x′ . These are extremely complicated functions. For this reason, any looph diagrami formed with them is practically impossible to integrate. In contrast to that, the higher-order diagrams in the present theory need to be calulated using only ordinary particle propagators G∆ and Gρ of Eq. (14.334) and the interaction (14.324). Even that becomes, of course, tedious for higher orders in g. At least, there is a simple rule to find the contributions of the quadratic terms 1 f T f in (14.322), given 2 x x Mx x the diagrams without these terms. One calculatesR the diagrams from only the four-particle interaction, and collects the contributions up to order gN in an Ñ Ñ Ñ effective action ∆,ρ. Then one replaces ∆,ρ by ∆ ǫg∆,ρ ǫgρ and re-expands A A A − − everything in powers of g up to the order gN , forming a new series N gi ˜i . i=0 A∆,ρ Finally one sets ǫ equal to 1/g [23] and obtains the desired collectiveP classic action N as an expansion extending (14.336) to: A∆,ρ N N ˜i 2 2 ∆,ρ = ∆,ρ (1/g) ( ∆ + ρ ). (14.339) A A − x | | Xi=0 Z Note that this action must merely be extremized. There are no more quantum fluctuations in the classical collective fields ∆, ρ. Thus, at the extremum, the action (14.339) provides us directly with the desired grand-canonical potential.

The second essential difference with respect to the Hubbard-Stratonovich • transformation approach is the following: It becomes possible to study a rich variety of competing collective fields without the danger of double-counting Feynman diagrams. One simply generalizes the matrix x subtracted from loc new M int to define int in different ways. For instance, we may subtract and add a A Aa a a vector field ψ†σ ψS containing the Pauli matrices σ and study paramagnon fluctuations, thus generalizing the assumption (14.331) and allowing a sponta- neous magnetization in the ground state. Or one may do the same thing with a i a term ψ†σ ψA + c.c. added to the previous term. In this way we derive ∇ ia the Ginzburg-Landau theory of superfluid He3 as in [4, 24]. 14.16 Gauge Fields and Gauge Fixing 981

An important property of the proposed procedure is that it yields good results even in the limit of infinitely strong coupling. It was precisely this property which led to the successful calculation of critical exponents of all φ4-theories in the textbook [21] since critical phenomena arise in the limit in which the unrenormalized coupling constant goes to infinity [57]. This is in contrast to another possibility. For example that of carrying the variational approach to highers order via the so- called higher effective actions [26]. These where discussed in Chapter 13. There one extremizes the Legendre transforms of the generating functionals of bilocal correlation functions, which sums up all two-particle irreducible diagrams. That does not give physically meaningful results [27] in the strong-coupling limit, even for simple quantum-mechanical models, as we have shown in Section 13.12. The mother of this approach is Variational Perturbation Theory (VPT). Its origin was a variational approach developed for quantum mechanics some years ago by Feynman and the author [53]. in the textbook [21] to quantum field theory. It converts divergent perturbation expansions of quantum mechanical systems into exponentially fast converging expansions for any coupling strength [54]. What we have shown in this section is that this powerful theory can easily be transferred to many-body theory, if we identify a variety of relevant collective classical fields, rather than a fluctuating collective quantum field suggested by the Hubbard-Stratonovich Transformation. To lowerst order in the coupling constant this starts out with the standard Hartree-Fock-Bogoliubov approximation, and allows to go to higher oders with arbitrarily high accuracy.

14.16 Gauge Fields and Gauge Fixing

The functional integral formalism developed in the previous sections does not immediately apply to electromagnetism and any other gauge ﬁelds. There are subtleties which we are now going to discuss. These will lead to an explanation of the mistake in the vacuum energy observed in Eq. (7.506) when quantizing the electromagnetic ﬁeld via Gupta-Bleuler formalism. Consider a set of external electromagnetic currents described by the current density jµ(x). Since charge is conserved, these satisfy

µ ∂µj (x)=0. (14.340)

The currents are sources of electromagnetic ﬁelds Fµν determined from the ﬁeld equation (12.49),

νµ µ ∂ν F (x)= j (x), (14.341) if we employ natural units with c = 1. The action reads

4 1 2 µ = d x Fµν (x) j (x)Aµ(x) (14.342) A Z −4 − 1 1 = d4x E2(x) B2(x) ρφ(x) j A(x) . 2 − − − c · Z h i 982 14 Functional-Integral Representation of Quantum Field Theory

The ﬁeld strengths are the four-dimensional curls of the vector potential Aµ(x): F = ∂ A ∂ A , (14.343) µν µ ν − ν µ that satisfy, for single-valued ﬁelds Aµ, the Bianchi identity

ǫµνλκ∂νFλκ =0. (14.344)

The decomposition (14.343) is not unique. If we add to Aµ(x) the gradient of an arbitrary function Λ(x), A (x) AΛ(x)= A (x)+ ∂ Λ(x), (14.345) µ → µ µ µ then Λ does not appear in the field strengths, assuming that it satisfies (∂ ∂ ∂ ∂ )Λ(x)=0, (14.346) µ ν − ν µ i.e., the derivatives in front of Λ(x) commute. In the theory of partial differential equations, this is referred to as the Schwarz integrability condition for the function Λ(x). In general, a function Λ(x) which satisfies (14.346) in a simply-connected domain can be defined uniquely in this domain. Only if Λ(x) fulfills this condition, the transformation (14.345) is called a local gauge transformation. If the domain is multiply connected, there is more than one path along which to continue the function Λ(x) from one spatial point to another and Λ(x) becomes multi-valued. This happens, for example, if (14.346) is nonzero on a closed line in three-dimensional space, in which case the set of paths between two given points decomposes into equivalence classes, depending on how often the closed line is en- circled. Each of these paths allows another continuation of Λ(x). By (14.346), such functions are not allowed in gauge transformations (14.345). Let us now see how we can construct the generating functional of fluctuating free electromagnetic fields in the presence of external currents jµ(x). As in (14.58), we would like to calculate a functional integral, now in the unnormalized version corresponding to (14.58):

3 2 µ µ i d x( F /4 jµA ) Z [j]= A (x)e − µν − . (14.347) 0 D Z R 2 The Fµν-terms in the exponents can be partially integrated and rewritten as

1 4 2 1 4 µ 2 ν d xFµν = d xA (∂ gµν ∂µ∂ν )A . (14.348) −4 Z 2 Z − Hence we can identify the functional matrix D(x, x′) in (14.34) as

2 (4) D (x, x′)=(∂ g ∂ ∂ )δ (x x′). (14.349) µν µν − µ ν − Recalling (14.88) it appears, at ﬁrst, as though the generating functional (14.347) should simply be equal to

1 4 4 ′ ′ ′ 1/2 d xd x jµ(x)G0 µν (x,x )jµ(x ) Z0[j] = (Det G0 µν) e− 2 , (14.350) R 14.16 Gauge Fields and Gauge Fixing 983 where, by analogy with (14.35), we get

1 G0 µν(x, x′)= iDµν− (x, x′). (14.351)

1/2 We have divided out a normalization factor = (Det G )− , assuming that N 0 µν we are dealing with an unnormalized version of Z0[j]. Unfortunately, however, the expression (14.350) is meaningless, since the inverse of the functional matrix (14.349) does not exist. In order to see this explicitly we diagonalize the functional part (i.e., the x, y -part) of G0 µν (x, x′) by considering the Fourier transform

D (q)= q2g + q q . (14.352) µν − µν µ ν For every momentum q, this matrix has obviously an eigenvector with zero eigenvalue, namely qµ. This prevents us from inverting the matrix Dµν(q). Correspond- ingly, when trying to form the inverse determinant of the functional matrix Dµν in (14.350), we encounter an infinite product of infinities, one for every momentum q. The difficulty can be resolved using the fact that the action (14.342) is gauge µ invariant. For Fµν this is trivially true; for the source term j (x)Aµ(x) this is a consequence of current conservation. Indeed, if we change Aµ according to (14.345), the 4 µ source term is changed by d x j (x)∂µΛ(x). With the help of a partial integration, this is equal to d4 x ∂ j (x)(x)Λ(x), and this expression vanishes due to current − µ Rµ conservation (14.340).R Because of this invariance, not all degrees of freedom, which are integrated over in the functional integral (14.347), are associated with a Gaussian integral. The fluctuations corresponding to pure gauge transformations leave the exponent invariant. Since a path integral is a product of infinitely many integrals from minus to plus infinity, the gauge invariance of the integral produces an infinite product of infinite factors. This is precisely the origin of the infinity that occurs in the functional determinant. This infinity must be controlled by restricting the functional integrals to field fluctuations via some specific gauge condition. For example, the restriction is achieved by inserting into the integral gauge-fixing functionals. Several examples have been used:

µ F1[A]= δ[∂µA (x)], (Lorenz gauge) F2[A]= δ[∇ A(x)], (Coulomb gauge) 0 · F3[A]= δ[A (x)], (Hamiltonian gauge) 3 (14.353) F3[A]= δ[A (x)], (axial gauge) F [A] = exp i d4x [∂ Aµ(x)]2 , (generalized Lorenz gauge) 4 − 2α µ i d4xζHζ/2 µ 1/2 F [A]= ζen − R δ[∂ A oζ] Det − H. (’t Hooft gauge) 5 D µ − × R R The first is a δ-functional enforcing the Lorenz gauge at each spacetime point: µ ∂µA (x) = 0. The second enforces the Coulomb gauge, the third corresponds to the axial gauge, and the fourth is a generalized form of the Lorenz gauge used 984 14 Functional-Integral Representation of Quantum Field Theory by Feynman and which also serves to derive the Feynman diagrams of the Gupta- Bleuler quantization formalism, thereby correcting the mistake in the vacuum energy. The fifth, finally, is a generalization of the fourth used by ’t Hooft that arises i d4x ζ2/2α µ by rewriting the fourth as a path integral F [A]= ζe− δ[∂ A ζ] and 4 D µ − generalizing the constant 1/α to a functional matrixR A(x,R x′). If we insert any of these gauge-fixing functionals Fi[A] into the path integral, then gauge-transform the vector potential Aµ(x) àla (14.345), and integrate functionally over all Λ(x), the integral receives a finite contribution from that gauge function Λ(x) which enforces the desired gauge. The result is a gauge-invariant functional of Aµ(x):

Φ[A] = Λ F [AΛ]= Λ F [A + ∂Λ] . (14.354) Z D Z D Explicitly, we ﬁnd for the above cases (14.353), the normalization functionals:

Λ µ 2 Φ1[A] = ΛF1[A ]= Λδ[∂µA (x)+ ∂ Λ], (14.355) Z D Z D Λ 2 Φ2[A] = ΛF2[A ]= Λδ[∇ A(x)+ ∇ Λ], (14.356) Z D Z D · Λ 0 0 Φ3[A] = ΛF3[A ]= δ[∇A (x)+ ∂ Λ], (14.357) Z D Z D Λ i 4 µ 2 2 Φ4[A] = ΛF4[A ]= Λ exp d x [∂µA + ∂ Λ] , (14.358) Z D Z D −2α Z Λ i 4 µ 2 Φ5[A] = ΛF5[A ]= Λ ζ exp d xζHζ δ[∂µA + ∂ Λ ζ] Z D Z D Z D −2 Z − 1/2 2 Det − H Det(∂ ). (14.359) × ×

If we form the ratios Fi[A]/Φi[A], we obtain gauge ﬁxing functionals which all yield unity when integrated over all gauge transformations. If any of these are inserted into the functional integral (14.347), they will all remove the gauge degree of freedom, and lead to a ﬁnite functional integral which is the same for each choice of Fi[A]. Let us calculate the functionals Φi[A] explicitly. For Φ1,2,3[A] we simply observe a trivial identity for δ-functions

1 δ(ax)= a− δ(x). (14.360)

This is proved by multiplying both sides with a smooth function f(x) and integrating over x. The functional generalization of this is

1 δ[ A] = Det − δ[A], (14.361) O O where is an arbitrary differential operator acting on the field Aµ(x). From this we findO immediately the normalization functionals:

1 2 Φ1[A] = Det − (∂ ), (14.362) 1 2 Φ2[A] = Det − (∇ ), (14.363) 1 0 Φ3[A] = Det − (∂ ). (14.364) 14.16 Gauge Fields and Gauge Fixing 985

The fourth functional Φ4[A] is simply a Gaussian functional integral. The addi- µ tive term ∂µA (x) can be removed by a trivial shift of the integration variable µ 2 Λ(x) Λ′(x) = Λ(x) ∂ A (x)/∂ , under which the measure of integration re- → − µ mains invariant, Λ= Λ′. Using formula (14.25) we obtain D D 1 2 Φ4[A] = Det − (∂ /√α). (14.365)

1 The functional determinants Φi− are called Faddeev-Popov determinants [49]. The Faddeev-Popov determinants in the four examples happen to be independent µ of A (x), so that we shall write them as Φi without arguments. This independence is a very useful property. Complications arising for A-dependent functionals Φ[A] will be illustrated below in an example. We now study the consequences of inserting the gauge-ﬁxing factors Fi[A]/Φi into the functional integrands (14.347). For F4[A]/Φ4, the generating functional becomes

4 1 2 1 µ 2 µ 1/2 1/2 4 i d x[ Fµν (∂ Aµ) j Aµ] Z [j] = (Det G ) Det (∂ /α) A (x)e − 4 − α − . (14.366) 0 0 µν D µ Z R The free-ﬁeld action in the exponent can be written in the form

4 1 2 1 µ 2 = d x (∂µAν) + 1 (∂ Aµ) . (14.367) A Z 2 − − α The associated Euler-Lagrange equation is 1 ∂2Aµ 1 ∂µ(∂A)=0, (14.368) − − α which is precisely the ﬁeld equation (7.381) of the covariant quantization scheme. With the additional term in the action, the matrix (14.352) becomes

2 1 qµqν Dµν(q)= q gµν 1 2 . (14.369) − " − − α q # This can be decomposed into projection matrices with respect to the subspaces transverse and longitudinal to the four-vector qµ, q q q q P l (q)= µ ν , P l (q)= g µ ν , (14.370) µν q2 µν µν − q2 as

2 t 1 l Dµν(q)= q Pµν (q)+ Pµν (q) . (14.371) − α It is easy to verify that the matrices are really projections, since they satisfy

l lν t l lν l t lν Pµν P λ = Pµν, Pµν P λ = Pµλ, Pµν P λ =0. (14.372) 986 14 Functional-Integral Representation of Quantum Field Theory

Similar projections appeared before in the three-dimensional subspace [see (4.334) and (4.336)], where the projections were indicated by capital subscripts T , L. Due to the relations (14.372), there is no problem in inverting Dµν (q), and we ﬁnd the free photon propagator in momentum space

1 i t l G (q) = iD− (q)= [P (q)+ αP (q)] 0 µν µν −q2 µν µν

i qµqν = gµν (1 α) . (14.373) −q2 " − − q2 #

For α = 0, this is the propagator Gµν derived in the Gupta-Bleuler canonical ﬁeld quantization in Eq. (7.510), there obtained from canonical quantization rules with a certain ﬁlling of the vacuum with unphysical states. 1 Taking into account the Faddeev-Popov determinant Φ4− of (14.365), we obtain for the generating functional (14.366):

1 4 4 ′ µ ′ ν ′ 1/2 1/2 4 d xd x j (x)G0 µν (x,x )j (x ) Z[j] = [Det ( iG )] Det (∂ /α)e− 2 . (14.374) − 0 µν R µ Let us calculate the functional determinant Det G0 µν . For this we take q in the momentum representation (14.373) along the 0-direction, and see that there are three spacelike eigenvectors of eigenvalues iq2, and one timelike eigenvector of eigenvalue iα/q2. The total determinant is therefore 1 Det( iG )= . (14.375) − 0 µν Det( ∂2)2Det(∂4/α) − The two prefactors of (14.374) together yield a factor 1 [Det ( iG )]1/2Det 1/2(∂4/α) . (14.376) − 0 µν ∝ Det( ∂2) − Recalling the discussion of Eqs. (14.129)–(14.133), we see that the associated free energy is

ω(k) β¯hω(k) F0 = β log Z0 =2 + log 1 e− . (14.377) − ( 2 − ) Xk h i It contains precisely the energy of the two physical transverse photons. The unphysical polarizations have been eliminated by the Faddeev-Popov determinant in (14.374). We now understand why the Gupta-Bleuler formalism failed to get the correct vacuum energy in Eq. (7.506). It has no knowledge of the Faddeev-Popov determinant. Taking into account current conservation, the exponent in (14.374) reduces to

µ ν µ i j (x)G (x, x′)j (x′)= j (x) j (x), (14.378) 0 µν ∂2 µ 14.16 Gauge Fields and Gauge Fixing 987 so that the generating functional becomes

1 4 4 ′ µ 2 ′ 1 2 d xd x j (x)(i/∂ )jµ(x ) Z[j] = const Det − ( ∂ )e− 2 , (14.379) × − R where const is an inﬁnite product of identical constant factors. Let us see what happens in the other gauges (14.353). The results in the Lorenz gauge are immediately obtained by going to the limit α 0 in the previous calcu- → lation. In this limit, the functional Φ4 coincides with Φ1, up to a trivial factor. The Hamiltonian and the axial gauges are quite similar, so we may only discuss one of them. In the Hamiltonian gauge, where the Faddeev-Popov determinant is given by (14.364), the generating functional (14.374) becomes

1 4 4 ′ i ′ j ′ 1/2 d xd x j (x)G0 ij (x,x )j (x ) Z[j] = (Det G0 µν) Det(∂0)e− 2 , (14.380) R where the matrix G0 ij(q) has only spatial entries, and is equal to

1 G0 ij(q)= iDij− (q) (14.381) with 2 Dij(q)= q δij + qiqj (14.382) being the spatial part of the 4 4 -matrix (14.352). With the help of 3 3 -projection matrices × ×

P T (q) = δ q q /q2 (14.383) ij ij − i j L 2 Pij (q) = qiqj/q , (14.384) this can be decomposed as follows:

2 T 0 2 L Dij(q)= q Pij (q)+ q Pij (q). (14.385) The inverse of this is

1 1 T 1 L 1 qiqj Dij− (q)= Pij (q)+ Pij (q)= δij . (14.386) q2 q0 2 q2 − q0 2 !

1 In the exponent of (14.380) we have to evaluate iDij− (q) between two conserved currents, and ﬁnd

i j i 1 j ∗(q)G0 ij(q)j (q)= j∗(q) j(q) q j∗(q) q j(q) . (14.387) −q2 " · − q0 2 · · # Inserting the momentum space version of the local current conservation law µ ∂µj (x) = 0: q j(q)= q0 j0(q), (14.388) · we obtain

i j i µ j ∗(q)G (q)j (q)= j ∗(q)j (q). (14.389) 0 ij q2 µ 988 14 Functional-Integral Representation of Quantum Field Theory

Let us calculate the functional determinant Det G0 µν in this gauge. From 2 (14.386) we see that G0 µν has two eigenvectors of eigenvalue i/q , and one eigenvector of eigenvalue i/q0 2. Hence: − 1 Det G = . (14.390) 0 µν Det( i∂2)2Det(i∂2) − 0 The two prefactors in (14.380) together are therefore proportional to

1 (Det G )1/2Det(∂ ) , (14.391) 0 µν 0 ∝ Det( i∂2) − which is the same as (14.376). Thus the generating functional (14.380) agrees with the previous one in (14.379). Let us also show that the Coulomb gauge leads to the same result. We rewrite the exponent in (14.379) in momentum space as

4 4 µ 1 d q 2 1 2 1 d xj (x) jµ(x)= c ρ(q) ρ(q) c j(q) j(q) . (14.392) ∂2 (2π)4 " q2 − q2 # Z − Z Here we keep explicitly the light velocity c in all formulas since we want to rederive the interaction equivalent to Eq. (12.85) where c is not set equal to unity. Writing 2 2 2 2 2 2 the denominator as q = q0 q and j (q)= jL(q)+ jT (q) with jL(q)= q j(q)/ q and j (q) j (q) = 0, we can− bring (14.392) to the form · | | T · L 4 d q 2 1 1 1 c ρ(q) ρ(q) jL(q) jL(q) jT (q) jT (q) .(14.393) (2π)4 " q2 q2 − q2 q2 − q2 q2 # Z 0 − 0 − 0 − Now we use the current conservation law cq ρ(q)= q j(q) to rewrite (14.393) as 0 · 4 2 d q 2 1 2 q0 1 1 c ρ(q) ρ(q) c ρ(q) ρ(q) jT (q) jT (q) . (14.394) (2π)4 " q2 q2 − q2 q2 q2 − q2 q2 # Z − 0 − 0 − The ﬁrst two terms can be combined to

4 int d q 2 1 L = 4 c ρ(q) 2 ρ(q). (14.395) A − Z (2π) q Undoing the Fourier transformation and multiplying this by e2/c2 we ﬁnd as a ﬁrst term in the interaction (12.84) the Coulomb term which is is the longitudinal part of the interaction (12.83):

1 e2 1 int = d4x E2 (x)= d4x ρ(x) ρ(x) L L ∇2 A −2 Z 2 Z 2 e 3 3 1 = dt d xd x′ ρ(x, t) ρ(x′, t). (14.396) −8π x x Z Z | − ′| 14.17 Nontrivial Gauge and Faddeev-Popov Ghosts 989

The third term in (14.394) involves only the transverse current

4 int d q 1 j∗ j T = 4 T (q) 2 T (q). (14.397) A − Z (2π) q It is the result of the transverse ﬁelds in the Lagrangian (12.54):

int 1 4 2 2 1 T = d x [ET (x) B (x)] + j(x)AT (x) . (14.398) A 4π Z { − c } It is found by integrating out the vector potential. Using Eq. (5.56), the transverse interaction (14.397) can be rewritten as

4 int d q 1 µ 1 2 int int T = 4 jµ∗(q) 2 j (q)+ 2 j0(q) = tot L . (14.399) A Z (2π) " q q | | # A −A The transverse part of the electromagnetic action of a four-dimensional current jµ(q) is the diﬀerence between the total covariant Biot-Savart interaction plus the instantaneous Coulomb interaction.

14.17 Nontrivial Gauge and Faddeev-Popov Ghosts

The Faddeev-Popov determinants in the above examples were all independent of the ﬁelds. As such they were irrelevant for the calculation of any Green function. This, however, is not always true. As an example, consider the following nontrivial gauge-ﬁxing functional (see also [55])

F [A]= δ (∂A)2 + gA2 . (14.400) h i As in Eqs. (14.355)–(14.359), we calculate

Φ[A] = ΛF [AΛ]= Λ δ[∂A + gA2 + ∂2Λ+2gA∂Λ+ g(∂Λ)2]. (14.401) Z D Z D This path integral can trivially be performed by analogy with the ordinary integral 1 dx δ(ax + bx2)= , (14.402) Z a and yields the result

2 µ 1 Φ[A] = Det(∂ +2gAµ∂ )− . (14.403)

The generating functional is therefore

µ 2 µ µ 2 4 1 2 µ Z[j] = A Det(∂ +2gA ∂µ) δ[∂µA + gA ] exp i d x Fµν jµA . Z D Z −4 − (14.404) 990 14 Functional-Integral Representation of Quantum Field Theory

Contrary to the previous gauges, the functional determinant is no longer a trivial overall factor, but it depends now functionally on the field Aµ. It can therefore no longer be brought outside the functional integral. There is a simple way of including its effect within the usual field-theoretic formalism. One introduces an auxiliary Faddeev-Popov ghost field. We may consider the determinant as the result of a fluctuating complex fermion field c with a complex- conjugate c∗, and write

4 ∗ µ ∗ 2 µ i d x(∂c ∂c 2gA c ∂µc) Det(∂ +2gA ∂ )= c∗ ce− − . (14.405) µ D D Z R Note that the Fermi fields are necessary to produce the determinant in the numerator; a Bose field would have put it into the denominator. A complex field is taken to make the determinant appear directly rather than the square-root of it. The ghost fields interact with the photon fields. This interaction is necessary in order to compensate the interactions induced by the constraint δ[∂Aµ + gA2] in the functional integral. It is possible to exhibit the associated cancellations order by order in perturbation theory. For this we have to bring the integrand to a form in which all fields appear in the exponent. This can be achieved for the δ-functional by observing that the same representation (14.404) would be true with any other choice of gauge, say

δ[∂ Aµ(x)+ gA2(x) λ(x)], (14.406) µ − since this would lead to the same Faddeev-Popov ghost term (14.405). Therefore we can average over all possible functions λ(x) with a Gaussian weight and replace (14.406) just as well by

i d4xλ2(x) µ 2 λe− 2 δ[∂ A (x)+ gA (x) λ(x)]. (14.407) D µ − Z R Now the generating functional has the form

4 µ µ i d x( j Aµ) Z[j]= A c∗ c e L− (14.408) D D D Z Z R with a Lagrangian

1 2 1 2 2 µ µ = F ∂A + gA ∂ c∗∂ c +2gA c∗∂ c. (14.409) L −4 µν − 2α − µ µ The photon and ghost propagators are

4 d k ikx i kµkν − Aµ(x)Aν(0) = 4 e 2 gµν (1 α) 2 , (14.410) − Z (2π) k " − − k # 4 d q iqx i ∗ − (14.411) c (x)c(0) = 4 e 2 . − Z (2π) q 14.17 Nontrivial Gauge and Faddeev-Popov Ghosts 991

Contrary to the previous gauges, there are now photon-ghost and photon-photon interaction terms

2 g µ 2 g 2 2 µ ∂ A A A +2gA c∗∂ c (14.412) −α µ − 2α µ µ with the corresponding vertices

(14.413)

(14.414)

. (14.415)

It can be shown that the Faddeev-Popov ghost Lagrangian has the property of canceling all these unphysical contributions order by order in perturbation theory. We leave it as an exercise to show, for example, that there is no contribution of the ghosts to photon-photon scattering up to, say, second order in g, and that self- energy corrections to the photon propagator due to photon and ghost loops cancel exactly. In the context of quantum electrodynamics, there is little sense in using a gauge fixing term (14.400). The present discussion is, however, a useful warm-up exercise to gauge-fixing procedures in nonabelian gauge theories, where the Faddeev-Popov determinant will always be field dependent. Of course, also the previous field-independent Faddeev-Popov determinants can 1 be generated from fermionic ghost fields. The determinant Φ4− in (14.365), for example, can be generated from a complex ghost field c(x) and its complex conjugate c∗ by a functional integral

1 i d4x∂c∗∂c/√α Φ− = c∗ c e . (14.416) 4 D D Z R It should be pointed out that the signs of the kinetic term of these c-field La- grangians are opposite to those of a normal field. If these fields were associated with particles, their anticommutation rules would carry the wrong sign and the states would have a negative norm. Such states are commonly referred to as ghosts, and 992 14 Functional-Integral Representation of Quantum Field Theory

this is the reason for the name of the ﬁelds c,c∗. Note that the determinant cannot be generated by a real fermion ﬁeld via a functional integral

1 ? i d4x ∂2c∂2c/α Φ− = c e− . (14.417) 4 D Z R The reason is that the differential operator ∂4 is a symmetric functional matrix, so that the exponent vanishes after diagonalization by an orthogonal transformation. In the language of ghost fields, the mistake in calculating the vacuum energy in Eq. (7.506), that arose when quantizing the electromagnetic field via the Gupta- Bleuler formalism, can be phrased as follows. When fixing the gauge in the action (7.376) by adding an Lagrangian density (10.86), we must also add a ghost LGF Lagrangian density = ∂c∗∂c/√α. (14.418) Lghost The ghost fields have to be quantized canonically, and the physical states must satisfy, beside the Gupta-Bleuler subsidiary condition in Eq. (7.502), the condition of being a vacuum to the ghost fields:

c ψ =0. (14.419) | “phys”i The Faddeev-Popov formalism is extremely useful oﬀering many other possibilities of ﬁxing a gauge and performing the functional integral for the generating functional over all Aµ-components.

14.18 Functional Formulation of Quantum Electrodynamics

For quantum electrodynamics, the functional integral from which we can derive all time-ordered vacuum expectation values reads [1, 3, 5]:

4 ¯ 4 µ µ i i d x [ψ(x)η(x)+¯η(x)ψ(x)] i d xjµ(x)A (x) Z[j, η, η¯]= ψ ψ¯ A D e A− − , (14.420) D D D D Z R R where is the sum of the free-ﬁeld action (12.86) and the minimal interaction (12.87),A which we shall write here as

4 e 1 µν µ 2 = d x ψ¯ i/∂ /A m ψ F Fµν D∂ Aµ + D /2 . (14.421) A Z − c − − 4 − The Dirac ﬁeld appears only quadratically in the action. It is therefore possible to integrate it out using the Gaussian integral formula (14.95), and we obtain

′ 4 4 ′ ′ ′ 4 µ µ i d x d x η¯(x)G0(x,x )η(x )ψ(x) i d xjµ(x)A (x) Z[j, η, η¯] = Det(i/∂ M) A D e A − − , − D D Z R R (14.422) where ′ is the action A 4 1 µν µ 2 eﬀ = d x F Fµν D∂ Aµ + D /2 + [A]. (14.423) A Z −4 − A 14.18 Functional Formulation of Quantum Electrodynamics 993

The eﬀects of the electron are collected in the eﬀective action

i∆ eff e A = exp [Tr log (i/∂ e/A M) Tr log (i/∂ M)] , (14.424) − − − − where Tr combines the functional trace of the operator and the matrix trace in the 4 O 4 space of Dirac matrices. We have performedO a subtraction of the infinite vacuum× energy caused by the filled-negative energy states. The subtraction is compensated by the determinant in the prefactor of Eq. (14.422).

14.18.1 Decay Rate of Dirac Vacuum in Electromagnetic Fields An important application of the functional formulation (14.422) of quantum electrodynamics was made by Heisenberg and Euler [32, 33]. They observed that in a constant external electric field the vacuum becomes unstable. There exists a finite probability of creating an electron-positron pair. This process has to overcome a large energy barrier 2Mc2, but if the pair is separated sufficiently far, the total energy of the pair can be made arbitrarily low, so that the process will occur with nonzero probability. The rate can be derived from the result (6.254). For a large total time ∆t, the time dependence of an unstable vacuum state will have the form i(E0 iΓ/2)∆t e− − , where E0 is the vacuum energy, Γ the desired decay rate, and ∆t the total time over which the amplitude is calculated. If eff is the effective Lagrangian density causing the decay, and eff the associated action,L we identify A Γ eff = 2 Im A = 2 Im eff . (14.425) V V ∆t L We now make use of the fact that, due to the invariance under charge conjugation, the right-hand side of (14.424) can depend only on M 2. Thus we also have

exp [Tr log (i/∂ e/A M)] = exp [Tr log (i/∂ e/A +M)] − − − 1 = exp [Tr log (i/∂ e/A M)(i/∂ e/A +M)] , (14.426) 2 − − − and may use the product relation (6.107) to calculate (14.424) from half the trace log of the Pauli operator in Eq. (6.241) to ﬁnd the eﬀective action

eﬀ 1 2 e µ ν 2 1 2 2 i∆ = Tr log [i∂ eA(x)] σ νFµ M Tr log ∂ M . A 2 − − 2 − − 2 − − n (14.427)o We now use the integral identity a dτ log = ∞ eiaτ eibτ (14.428) b − 0 τ − Z h i and relation (6.206) to rewrite (14.427) as

eﬀ 1 ∞ dτ iτ(M 2 iǫ) iτ [i∂ eA(x)]2+ e σµν F e iτ∂2 i∆ = e− − tr x e { − 2 µν } e− x . (14.429) A − 2 Z0 τ h | − | i 994 14 Functional-Integral Representation of Quantum Field Theory

Recalling (6.189) and (6.206), the ﬁrst, unsubtracted, term can be re-expressed as

eff 1 ∞ dτ iτHˆ 1 ∞ dτ i∆ 1 = tr x e− x = tr x, τ x 0 . (14.430) A −2 Z0 τ h | | i −2 Z0 τ h | i Inserting (6.254), and subtracting the field-free second term in (14.429), we obtain the contribution to the effective Lagrangian density

eﬀ 1 ∞ dτ eEτ iτ(M 2 iη) − − ∆ = 2 3 1 e . (14.431) L 2(2π) Z0 τ tanh eEτ − The integral over τ is logarithmically divergent at τ = 0. We can separate the divergent term by a further subtraction, splitting

∆ eff = ∆ eff + ∆ eff (14.432) L Ldiv LR into a convergent integral

2 2 2 eﬀ 1 ∞ dτ eEτ e E τ iτ(M 2 iη) − − ∆ R = 2 3 1 e , (14.433) L 2(2π) Z0 τ tanh eEτ − − 3 ! and a divergent one

2 eff e 2 1 ∞ dτ iτ(M 2 iη) − − ∆ div = E 2 e . (14.434) L 2 3(2π) Z0 τ The latter is proportional to the electric part of the original Maxwell Lagrangian density in (4.237). It can therefore be removed by renormalization. We add (14.434) to the Maxwell Lagrangian density, and define a renormalized charge eR by the equation 1 1 1 ∞ dτ iτ(M 2 iη) 1 − − 2 = 2 + 2 e 2 , (14.435) eR e 12π Z0 τ ≡ Z3e to obtain the modified electric Lagrangian density

2 e 2 E = 2 E . (14.436) L 2eR Now we redefine the electric fields by introducing renormalized fields e ER E, (14.437) ≡ eR and identify these with the physical fields. In terms of these, (14.436) takes again the usual Maxwell form 1 = E2 . (14.438) LE 2 R 14.18 Functional Formulation of Quantum Electrodynamics 995

The ﬁnite eﬀective Lagrangian density (14.433) possesses an imaginary part which by Eq. (14.425) determines the decay rate of the vacuum per unit volume

2 2 2 Γ 1 ∞ dτ eEτ e E τ iτ(M 2 iη) − − = Im 2 3 1 e . (14.439) V (2π) Z0 τ tanh eEτ − − 3 ! For comparison we mention that, for a charged boson ﬁeld, the expression (14.424) is replaced by

i∆ eﬀ 2 2 2 2 e A = exp Tr log [i∂ eA(x)] M + Tr log ∂ M . (14.440) − − − − − h n o i Hence the last factor 4 cosh eEτ in (6.254) is simply replaced by 2, and the unsubtracted eﬀective action (14.439) becomes −

eﬀ ∞ dτ i ∞ dτ eEτ iτ(M 2 iη) − − i∆ u = x, τ x 0 = 2 3 e , (14.441) A Z0 τ h | i −4(2π) Z0 τ sinh eEτ implying a twice subtracted eﬀective Lagrangian density

2 2 2 eﬀ 1 ∞ dτ eEτ e E τ iτ(M 2 iη) ∆ R = 2 3 1+ e− − , (14.442) L −4(2π) Z0 τ sinh eEτ − 6 ! and a charge renormalization

1 1 1 ∞ dτ iτ(M 2 iη) 1 − − 2 = 2 2 e 2 . (14.443) eR e − 6π Z0 τ ≡ Z3e The decay rate per unit volume is

2 2 2 ΓKG 1 ∞ dτ eEτ e E τ iτ(M 2 iη) − − = Im 2 3 1+ e . (14.444) V 4(2π) Z0 τ sinh eEτ − 6 ! The integrands in (14.439) and (14.444) are even in τ, so that the integrals for the decay rate can be extended symmetrically to run over the entire τ-axis. After this, the contour of integration can be closed in the lower half-plane and the integral can be evaluated by Cauchy’s residue theorem. To ﬁnd the pole terms we expand the integrand for fermions in Eq. (14.439) as

2 2 eετ ∞ (eετ) ∞ τ nπ 1= 2 2 2 =2 2 2 , τn . (14.445) tanh eετ − n= (eετ) + n π n=1 τ + τn ≡ eε X−∞ X The relevant poles lie at τ = iτ and yield the result − n 2 2 Γ e E ∞ 1 nπM 2/eE = e− . (14.446) V 4π3 n2 nX=1 A technical remark is necessary at this place concerning the integral (14.444). At ﬁrst sight it may appear as if the second subtraction term in the integrand 996 14 Functional-Integral Representation of Quantum Field Theory e2E2τ 2/3 can be omitted [29]. First, it is unnecessary to arrive at a ﬁnite integral in the imaginary part, and second, it seems to contribute only to the real part, since for all even powers α> 2, the integral

∞ dτ α iτ(M 2 iη) 1 − − 3 τ e = 2 α 2 Γ(α 2) (14.447) Z0 τ (iM ) − − is real. The limit at hand α 2, however, is an exception since for α 2, the integral possesses an imaginary→ part due to the divergence at small τ ≈

1 2 π 2 α 2 Γ(α 2) γ log M i + (α 1). (14.448) (iM ) − − ≈ − − − 2 O − The right-hand side of Eq. (14.446) is a polylogarithmic function (2.274), so that we may write 2 2 Γ e E πM 2/eE = ζ (e− ). (14.449) V 4π3 2 For large ﬁelds, this has the so-called Robinson expansion [30]

α ν 1 ∞ 1 k ζ (e− ) = Γ(1 ν)α − + ζ(ν)+ ( α) ζ(ν k). (14.450) ν − k! − − kX=1 This expansion plays an important role in the discussion of Bose-Einstein conden- sation [31]. For ν 2, the Robinson expansion becomes → 2 2 3 α π α α 5 ζ (e− )= +( 1 + log α) α + + (α ). (14.451) 2 6 − − 4 72 O Hence we ﬁnd the strong-ﬁeld expansion

Γ e2E2 π2 πM 2 πM 2 1 πM 2 2 1 πM 2 3 = + 1 + log + + ... . V 4π3  6 "− eE # eE ! − 4 eE ! 72 eE !    (14.452) For bosons, we expand  2 2 eετ ∞ n (eEτ) ∞ n τ nπ 1=2 ( 1) 2 2 2 =2 ( 1) 2 2 , τn . (14.453) sinh eετ − − (eEτ) + n π − τ + τn ≡ eE nX=1 nX=1 Comparison with (14.445) shows that the bosonic result for the decay rate differs from the fermionic (14.446) by an alternating sign, accounting for the different statistics. There is also a factor 1/2, since there is no spin. Thus we find the decay rate per volume 2 2 ΓKG e E 1 ∞ n 1 1 nπM 2/eE = ( 1) − e− . (14.454) V 4π3 2 − n2 nX=1 The sum can again be expressed in terms of the polylogarithmic function (14.450) as follows: k 1 k k 2k ∞ ( 1) − z ∞ z ∞ z 1 ν 2 ζ˜ (z) − = 2 = ζ (z) 2 − ζ (z ). (14.455) ν ≡ kν kν − (2k)ν ν − ν kX=1 kX=1 kX=1 14.18 Functional Formulation of Quantum Electrodynamics 997

α For z = e− 1, the Robinson expansion (14.451) yields ≈ 2 2 3 5 α π α α α 8 ζ˜ (e− )= α log 2 + + + (α ). (14.456) 2 12 − 4 − 24 960 O

˜ α 2 The expansion ζ2(e− ) replaces the curly bracket in (14.452), if we set α = πM /eE.

14.18.2 Constant Electric and Magnetic Background Fields In the presence of both E and B ﬁelds, Eq. (6.253) reads

e e ie (B iE)τ 0 i σµ F ντ − · − . exp ν µ = ie (B+iE)τ (14.457) − 2 0 e− · ! The trace of this can be found by adding the traces of the 2 2 block ma-

e ( iB E)τ eλ1τ eλ2τ × trices e · − ∓ separately. These are equal to e− + e− and their complex

conjugates, respectively, where λ ,λ are the eigenvalues of the matrix ( iB E): 1 2 · − − λ = (E + iB)2 = √E2 B2 +2iEB. (14.458) 1,2 ± ± − q Thus we ﬁnd

e µ ν tr exp i σ νFµ τ = 2(cos eλ1 + cos eλ1∗). (14.459) − 2 The eigenvalues are, of course, Lorentz-invariant quantities. They depend only on the two quadratic Lorentz invariants of the electromagnetic ﬁeld: the scalar S and the pseudoscalar P deﬁned by 1 1 1 S F F µν = E2 B2 , P F F˜µν = EB. (14.460) ≡ −4 µν 2 − ≡ −4 µν In terms of these, Eq. (14.458) reads

λ = √2√S + iP , (14.461) 1,2 ± which can be rewritten as

1/4 λ = √2 S2 + P 2 (cos ϕ/2+ i sin ϕ/2) , (14.462) 1,2 ± where P tan ϕ = , (14.463) S so that S P cos ϕ = , sin ϕ = , (14.464) √S2 + P 2 √S2 + P 2 998 14 Functional-Integral Representation of Quantum Field Theory implying that cos ϕ/2= (1 + cos ϕ)/2 and sin ϕ/2= (1 cos ϕ)/2, or − q q cos ϕ/2 √S2 + P 2 S = ± . (14.465) ( sin ϕ/2 ) q√2(S2 + P 2)1/4 We shall abbreviate the result (14.462) by λ = (ε + iβ) , (14.466) 1,2 ± where ε 1 √S2 + P 2 S = (E2 B2)2 + 4(EB)2 (E2 B2). (14.467) β ≡ ± √ − ± − ( ) q 2rq In terms of ε and β, the invariants S and P in (14.460) become 1 1 1 1 S F F µν = E2 B2 = ε2 β2 , P F F˜µν = EB = εβ, ≡ −4 µν 2 − 2 − ≡ −4 µν (14.468) and the trace (14.459) is simply

e µ ν tr exp i σ νFµ τ = 2cosh(ε + iβ)+2cosh(ε iβ) = 4 cosh ε cos β. (14.469) − 2 − Some special cases will simplify the upcoming formulas: 1. If B = 0, then ε reduces to E , whereas for E = 0, β reduces to B . | | | | 2. If E = 0 and B = 0 are orthogonal to each other, then we have either β = 0 for E6 > B, or ε 6= 0 for B > E. The formulas are then the same as for pure electric or magnetic ﬁelds. 3. If E = 0 and B = 0 are parallel to each other, then ǫ = E = 0, β = B = 0. 6 6 | | 6 | | 6 In all these cases, the calculation of the exponential (14.457) can be done very simply. Take the third case. Due to rotational symmetry, we can assume the ﬁelds to point in the z-direction, B = Bzˆ, E = Eˆz, and the exponential (14.457) has the matrix form

ie σ3(B iE)τ e µ ν e− − 0 exp i σ νFµ τ = ie σ3(B+iE)τ (14.470) − 2 0 e− ! ie(B iE)τ e− − 0 0 0 ie(B iE)τ  0 e − 0 0  = ie(B+iE)τ . 0 0 e− 0    ie(B+iE)τ   0 0 0 e    This has the trace

e µ ν tr exp i σ νFµ τ = 2cosh(E + iB) τ + 2cosh(E iB) τ = 4 cosh Eτ cos Bτ, − 2 − (14.471) 14.18 Functional Formulation of Quantum Electrodynamics 999 in agreement with (14.469). In fact, given an arbitrary constant field configuration B and E, it is always possible to perform a Lorentz transformation to a coordinate frame in which the transformed fields, call them BCF and ECF, are parallel. This frame is called center-of-fields frame. The transformation has the form (4.285) and (4.286) with a velocity of the transformation determined by v/c E B = × . (14.472) 1+( v /c)2 E 2 + B 2 | | | | | | By Lorentz invariance, we see that

E2 B2 = E2 B2 , E B = E B , (14.473) − CF − CF · CF · CF which shows that ECF and BCF in Eq. (14.471) coincide with ε and β in Eq. (14.468). This is| the| reason| why| Eq. (14.471) gives the general result for arbitrary constant fields, if E and B are replaced by E = ε and B = β. | CF| | CF| These considerations permit us to present a simple alternative calculation of a determinantal prefactor that occurred much earlier in Chapter 6, in particular in ν Eqs. (6.214) and (6.225). For a general constant field strength Fµ , the basic matrix that had to be diagonalized was eeF τ in Eq. (6.245). For an electric field pointing iM3eEτ in the z-direction, this has the form e− . In contrast to (6.253), this is a boost matrix with rapidity ζ = eEτ in the defining 4 4 -representation [compare (4.63)]. The fact that the rapidity in the Dirac representation× (6.252) was twice as large, has its origin in the value of the gyromagnetic ratio 2 of the Dirac particle in Eq. (6.119). iMEeτ If the field points in any direction, the obvious generalization is e− . In the presence of a magnetic field, the generators of the rotation group (4.57) enter and eF τ ie(ME+LB)τ (6.245) can be written as e = e− . This is the defining four-dimensional representation of the complex Lorentz transformation, whose chiral Dirac representation was written down in (14.457), apart from the factor 2 multiplying the rapidity and the rotation vectors. As before in the Dirac representation, much labor is saved by working in the center-of-fields frame where electric and magnetic fields are parallel and point in the z-direction. Their lengths have the invariant values ε and β, respectively. The associated transformation eeF τ has then the simple form

cosh εeτ 0 0 sinh εeτ − eF τ i(M3ε+L3β)eτ 0 cos βeτ sin βeτ 0 e = e− =   . (14.474) 0 sin βeτ −cos βeτ 0    sinh εeτ 0 cosh εeτ     −  i(M3ε+L3β)eτ i(M3ε+L3β)eτ From this we ﬁnd a matrix for sin eF τ = [e− e ]/2: − 0 0 0 sinh εeτ − 0 0 sin βeτ 0 sin eF τ =   , (14.475) 0 sin βeτ − 0 0      sinh εeτ 0 1   −  1000 14 Functional-Integral Representation of Quantum Field Theory and from this 0 0 0 εeτ − 0 0 βeτ 0 eF τ =   . (14.476) 0 βeτ − 0 0      εeτ 0 1   −  This leads to the desired prefactor in the amplitude (6.225) 2 1/2 sinh eF τ 2 eεβτ det − = eεβτ . (14.477) eF τ ! sinh ετ sin βeτ Thus we can obtain the imaginary part of the vacuum energy in a constant electromagnetic ﬁeld if we simply replace the term eEτ coth eEτ in the integrand of the rate formula (14.439) as follows: eEτ eετ eβτ . (14.478) tanh eEτ → tanh eετ tan eβτ

14.18.3 Decay Rate in a Constant Electromagnetic Field magnetic field the effective Lagrangian density (14.433) becomes, after the replacement (14.478) 2 2 2 eff 1 ∞ dτ eετ eβτ e (ε β ) iτ(M 2 iη) − − ∆ R = 2 3 1 − e . (14.479) L 2(2π) Z0 τ "tanh eετ tan eβτ − − 3 # The subtracted small-τ divergence lies in the integral [recall the equality ε2 β2 = E2 B2 from Eq. (14.468)] − − 2 eff e 2 2 1 ∞ dτ iτ(M 2 iη) E B − − ∆ div = ( ) 2 e , (14.480) L 2 − 3(2π) Z0 τ which is proportional to the full original Maxwell Lagrangian density in (4.237), and can therefore be absorbed into it by a renormalization of the charge as in (14.435) and of the fields e e BR B, ER E. (14.481) ≡ eR ≡ eR From twice the imaginary part of (14.489) we obtain the decay rate per unit volume extending Eq. (14.439) in an obvious manner. Inserting the expansion (14.445), extending the integral over the entire τ-axis and rotating the contour of integration as we did in evaluating (14.439), we find the generalization of Eq. (14.446) to constant electromagnetic fields 2 2 ΓKG e E ∞ 1 nπβ/ε nπM 2/eE = e− . (14.482) V 4π3 n2 tanh nπβ/ε nX=1 For bosons obeying the Klein-Gordon equation, we obtain, by analogy, the extension of (14.454) to constant electromagnetic fields: 2 2 n 1 Γ e E 1 ∞ ( 1) − nπβ/ε nπM 2/eε = − e− . (14.483) V 4π3 2 n2 sinh nπβ/ε nX=1 14.18 Functional Formulation of Quantum Electrodynamics 1001

14.18.4 Effective Action in a Purely Magnetic Field If there is only a magnetic field, magnetic field the effective Lagrangian density (14.433) becomes, after the replacement (14.478)

2 2 2 eff 1 ∞ dτ eετ eβτ e (ε β ) iτ(M 2 iη) − − ∆ R = 2 3 1 − e . (14.484) L 2(2π) Z0 τ " tanh eετ tan eβτ − − 3 # The subtracted small-τ divergence lies in the integral [recall the equality ε2 β2 = E2 B2 from Eq. (14.468)] − − 2 e 1 dτ 2 eff E2 B2 ∞ iτ(M iη) ∆ div = ( ) 2 e− − , (14.485) L 2 − 3(2π) Z0 τ which is proportional to the full original Maxwell Lagrangian density in (4.237), and can therefore be absorbed into it by a renormalization of the charge as in (14.435) and of the fields e e BR B, ER E. (14.486) ≡ eR ≡ eR From twice the imaginary part of (14.489) we obtain the decay rate per unit volume extending Eq. (14.439) in an obvious manner. Inserting the expansion (14.445), extending the integral over the entire τ-axis and rotating the contour of integration as we did in evaluating (14.439), we find the generalization of Eq. (14.446) to constant electromagnetic fields

2 2 ΓKG e E ∞ 1 nπβ/ε nπM 2/eE = e− . (14.487) V 4π3 n2 tanh nπβ/ε nX=1 For bosons obeying the Klein-Gordon equation, we obtain, by analogy, the extension of (14.454) to constant electromagnetic ﬁelds:

2 2 n 1 Γ e E 1 ∞ ( 1) − nπβ/ε nπM 2/eε = − e− . (14.488) V 4π3 2 n2 sinh nπβ/ε nX=1 14.18.5 Effective Action in a Purely Magnetic Field If there is only a magnetic field, For a constant electromagnetic field the effective Lagrangian density (14.433) becomes, after the replacement (14.478)

2 2 2 eff 1 ∞ dτ eετ eβτ e (ε β ) iτ(M 2 iη) − − ∆ R = 2 3 1 − e . (14.489) L 2(2π) Z0 τ " tanh eετ tan eβτ − − 3 # The subtracted small-τ divergence lies in the integral [recall the equality ε2 β2 = E2 B2 from Eq. (14.468)] − − 2 eff e 2 2 1 ∞ dτ iτ(M 2 iη) E B − − ∆ div = ( ) 2 e , (14.490) L 2 − 3(2π) Z0 τ 1002 14 Functional-Integral Representation of Quantum Field Theory which is proportional to the full original Maxwell Lagrangian density in (4.237), and can therefore be absorbed into it by a renormalization of the charge as in (14.435) and of the fields e e BR B, ER E. (14.491) ≡ eR ≡ eR From twice the imaginary part of (14.489) we obtain the decay rate per unit volume extending Eq. (14.439) in an obvious manner. Inserting the expansion (14.445), extending the integral over the entire τ-axis and rotating the contour of integration as we did in evaluating (14.439), we find the generalization of Eq. (14.446) to constant electromagnetic fields

2 2 ΓKG e E ∞ 1 nπβ/ε nπM 2/eE = e− . (14.492) V 4π3 n2 tanh nπβ/ε nX=1 For bosons obeying the Klein-Gordon equation, we obtain, by analogy, the extension of (14.454) to constant electromagnetic ﬁelds:

2 2 n 1 Γ e E 1 ∞ ( 1) − nπβ/ε nπM 2/eε = − e− . (14.493) V 4π3 2 n2 sinh nπβ/ε nX=1 14.18.6 Eﬀective Action in a Purely Magnetic Field If there is only a magnetic ﬁeld, the integral representation (14.489) reduces to [32, 33]

2 2 2 eﬀ 1 ∞ dτ eBτ e B τ iτ(M 2 iη) − − ∆ R = 2 3 1+ e . (14.494) L 2(2π) Z0 τ tan eBτ − 3 ! The integral still contains a divergence at small τ. The associated divergent integral is precisely a magnetic version of Eq. (14.434). It can be removed by the same type of subtraction as before, with E2 replaced by B2. Going to renormalized quantities as in (14.435) and (14.491) and rotating the contour of integration clockwise to move it away from the poles, a trivial change of the integration variable leads to

2 2 eﬀ e B ∞ ds 1 s sM 2/eB − ∆ R = 2 2 coth s e . (14.495) L −2(2π) Z0 s − s − 3 This is a typical Borel transformation of the expression in parentheses. It implies that its power series expansion leads to coeﬃcients of B2n which grow like (2n)!. The expansion has therefore a vanishing radius of convergence. It is an asymptotic series, and we shall understand later the physical origin of this. For Klein-Gordon particles, the result becomes

2 2 e B ∞ ds 1 1 s sM 2/eB − ∆ R KG = 2 2 + e . (14.496) L (2π) Z0 s sin s − s 6 14.18 Functional Formulation of Quantum Electrodynamics 1003

14.18.7 Heisenberg-Euler Lagrangian By expanding (14.478) in powers of e, we obtain

1 eβτ eετ 1 e2 τ = + ε2 β2 e4 ε4 +5ε2β2 + β4 τ 3 tan eβτ tanh eετ τ 3 3τ − − 45 τ 3 + e6 2ε6 +7ε4β2 7ε2β4 2β6 + .... (14.497) 945 − − Inserting this into (14.489) and performing the integral over τ leads to an expansion in powers of the ﬁelds, whose lowest terms are, with e2 =4πα:

2α2 16πα3 ∆ eff = (E2 B2)2 + 7(EB)2 + 2(E2 B2)3 + 13(E2 B2)(EB)2 . LR 45M 4 − 315M 8 − − n o n (14.498)o In each term we can replace α by α = α(1 + (α)), and the fields by the re- R O normalized fields via (14.491). Then we obtain the same series as in (14.498) but for the physical renormalized quantities, plus higher-order corrections in α for each coefficient, which we ignore in this lowest-order calculation. Each coefficient is exact to leading order in α. To illustrate the form of the higher-order corrections we include, without derivation, the leading correction into the first term, which becomes (see Appendix 14C for details)

2α2 40α 1315α eﬀ E2 B2 2 E B 2 ∆ R = 4 1+ ( ) +7 1+ ( ) + ....(14.499) L 45me 9π − 252π ·

Electrons in a Constant Magnetic Field

For Dirac particles in arbitrary constant ﬁelds we expand

2 2 ∞ (eετ) ∞ τ nπ eετ coth eετ = 2 2 2 = 2 2 , τn,ε , (14.500) n= (eετ) + n π n= τ + τn,ε ≡ eε X−∞ X−∞ 2 2 ∞ (eβτ) ∞ τ mπ eβτ cot eβτ = 2 2 2 = 2 2 , τm,β . (14.501) m= (eβτ) m π m= τ τm,β ≡ eβ X−∞ − X−∞ − The iη accompanying the mass term in the τ-integral (14.439) is equivalent to re- iτ(M 2 iη) iτ(1 iη)M 2 placing e− − by e− − , implying that the integral over all τ has to be performed slightly below the real axis. Equivalently we may shift the τm,β slightly upwards in the complex plane to τm,β +iǫ. This leads to an additional contribution to eff eff eff the action corresponding to a constant electromagnetic field ∆ = ∆ div + ∆ R . It contains a logarithmically divergent part L L L

eﬀ 1 ∞ dτ iτ(1 iη)M 2 ∞ 1 ∞ 1 − − ∆ div = 2 e 2 2 2 , (14.502) L 2(2π) 0 τ τn,ε − τm,β ! Z nX=1 mX=1 1004 14 Functional-Integral Representation of Quantum Field Theory and the ﬁnite part

2 2 1 dτ ∞ τ τ ∞ 1 ∞ 1 ∆ eﬀ = ∞ 1 2τ 2 R 2 3  2 2 2 2 2 2  L 2(2π) Z0 τ n,m= τ + τn,ε τ τm,β − − n=0 τn,ε − m=1 τm,β ! X−∞ − X X iτ(1 iη)M 2   e− − . (14.503) × Performing the sums

k k ∞ 1 eε ∞ 1 eβ k = ζ(k), k = ζ(k), (14.504) τn,ε π τm,β π ! nX=1 mX=1 we see that (14.502) coincides with (14.490), as it should. The remaining sum is ﬁnite:

τM 2 τM 2 eﬀ 1 ∞ 1 ∞ τ e− τ e− ′ ∆ R = 2 2 2 dτ 2 2 2 2 ..., (14.505) L 8π n,m= τn,ε + τm,β Z0 "τ τn,ε +iη − τ + τm,β iη # − X−∞ − − where the dots indicate the subtractions. Now we decompose `ala Sochocki [recall Footnote 9 in Chapter 1]: τ π π = i δ(τ + τ ) i δ(τ τ )+ τ P , (14.506) τ 2 τ 2 + iη 2 n,ε − 2 − n,ε τ 2 τ 2 − n,ε − n,ε where indicates the principal value under the integral. The integrals over the δ-functionsP contribute

2 2 2 2 2 eﬀ i 1 τn,εM e ε β ∞ 1 nπM /eε − − ∆δ = 2 2 e = i 3 2 2 2 2 e . L 8π n>0,m=0 τn,ε + τm,β 8π n>0,m 0 n β + m ε X X≥ 6 (14.507) This leads to a decay rate per volume which agrees with (14.493) if we expand in that expression [recall (14.445) and (14.425)]:

2 2 2 nπβ/ε ∞ (nπβ/ε) n β =1+ 2 2 2 =1+ 2 2 2 2 . (14.508) tanh nπβ/ε m=0 (nπβ/ε) + m π m=0 n β + m ε X6 X6 It remains to do the principal-value integrals. Here we use the formula9

τ ∞ τe− 1 z z J(z) dτ = e− Ei(z)+ e Ei( z) , (14.509) ≡ P 0 τ 2 z2 −2 − Z − h i where

k z ∞ z Ei(z) dt P et = log( z)+ (14.510) ≡ t − kk! Z−∞ kX=1 9I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980, Formulas 3.354.4 and 8.211. 14.18 Functional Formulation of Quantum Electrodynamics 1005 is the exponential integral, γ being the Euler-Mascheroni constant

γ =0.577216 .... (14.511)

The function (14.509) has the small-z expansion

1 z z J(z) = e log(z)+ e− log( z) γ cosh z −2 − − h 2l i 2l+1 ∞ z ∞ z cosh z + sinh z, (14.512) − (2l)(2l)! (2l + 1)(2l + 1)! Xl=1 Xl=1 and behaves for large z like 1 6 120 5040 J(z)= + .... (14.513) −z2 − z4 − z6 − z8 Using these formulas, we ﬁnd from the principal-value integrals in (14.505) [34]:

2 eﬀ e ∞ 1 2 2 ∆ = (β an + ε bn), (14.514) P L −4π4 n2 nX=1 with nπε/β nπm2 nπm2 nπm2 nπm2 an = Ci cos + si sin , (14.515) tanh nπε/β eβ ! eβ eβ ! eβ

1 nπβ/ε nπm2 nπm2 nπm2 nπm2 bn = exp Ei + exp Ei , −2 tanh nπβ/ε eε ! − eε ! − eε ! eε ! (14.516) 1 1 where Ci(z) 2i [Ei(iz) + Ei( iz)] and si(z) 2i [Ei(iz) Ei( iz)] = Si(z) π/2 are the cosine≡ and sine integrals− with the integral≡ representations− − [35]: − dt z dt dt Ci(z) ∞ cos t=γ + log z + (cos t 1) , si(z) ∞ sin t . (14.517) ≡− Zz t Z0 t − ≡− Zz t The prefactors in (14.515) and (14.516) come once more from sums of the type (14.508). For large arguments of J(z), the expansion (14.513) becomes

eff 1 ∞ 1 1 6 120 ∆ = ′ + + + ... . (14.518) P 2 2 2 2 4 4 8 6 12 L 8π n,m= τn,ε + τm,β " τn,ε M τn,ε M τn,ε M # X−∞ Applying the summation formulas (14.504), this is seen to agree with the small-field expansion (14.498). For E = 0, the sum (14.514) contains only the n = 0 -terms. If B is large, the n-sum is dominated by the first term in the expansion (14.512), and yields

eﬀ 1 ∞ 1 2 ∆ = 2 2 2 log(τm,β M ). (14.519) P L 8π τm,β mX=1 1006 14 Functional-Integral Representation of Quantum Field Theory

The leading part of this is

2 2 2 1 ∞ e B M 1 eB ∆ eff = log + ... = e2B2 log , (14.520) P L 4π2 π2m2 eB −24π2 M 2 mX=1 as derived first by Weisskopf [36]. Note that this logarithmic behavior can be understood as the result of a lowest expansion in e2 of an anomalous power behavior of the effective action

1 2 2 ∆ eﬀ = (E2 B2)1+e /12π . (14.521) L 2 − Due to the smallness of e2, this power can be observed only at very large ﬁeld strengths. In self-focussing materials, however, it is visible in present-day laser beams [37].

14.18.8 Alternative Derivation for a Constant Magnetic Field The case of a constant magnetic field can also be treated in a different way [38]. The eigenvalues of the euclidean Klein-Gordon operator of a free scalar field = h¯2∂2 + M 2c2 (14.522) OKG − are 2 2 2 λp = p + M c . (14.523) If a constant magnetic field is present that points in the z-direction, the kinetic 2 2 energy p1 + p2 in the xy-plane changes to Landau energies: 2 2 2 2 p1 + p2 1 p1 + p2 2M =2M hω¯ L n + , n =0, 1, 2,..., (14.524) → × 2M × 2 where ωL = eB/Mc is the Landau frequency. Thus the eigenvalues become

2 he¯ 2 2 λn,p⊥ = p +2 B(n +1/2)+ M c , (14.525) ⊥ c where p is the two-dimensional momentum in the x3 x4 -plane. ⊥ − For a Dirac electron satisfying the Pauli equation (6.110), this changes to

2 he¯ 2 2 λn,p⊥ = p +2 B(n +1/2+ s3)+ M c , s3 = 1/2. (14.526) ⊥ c ± In this expression, the famous g-factor which accounts for the anomalous magnetic moment is approximated by the Dirac value 2. Radiative corrections would insert a factor g/2 in front of σ3, with g =2+ α/π + ... . The contribution of electrons to the eﬀective action in Eq. (14.427) can therefore be written as 2 4 eﬀ 1 d p d p ⊥ i∆ = 2 log λn,p⊥ 4 log λp . (14.527) A 2 n,σ (2πh¯) − (2πh¯) ! X Z Z 14.18 Functional Formulation of Quantum Electrodynamics 1007

These are divergent expressions. In order to deal with them eﬃciently it is useful to introduce the Hurwitz ζ-function employed in number theory [39, 40]:

∞ 1 ζ (s, z)= , Re(s) > 1 , v =0, 1, 2,.... (14.528) H (n + z)s 6 − − nX=0 This Hurwitz ζ-function can be analytically continued in the s-plane to define an analytic function with a single simple pole at s = 1. In quantum field theory, one may extend this definition to an arbitrary eigenvalue spectrum of an operator as O 1 ζ (s)= , (14.529) O (λ/µd)s X where µ is some mass parameter to make the eigenvalues of mass dimension d di- mensionless. For large enough s, this is always convergent. By analytic continuation in s, one can derive a finite value for the functional determinant of : O Tr log Det = e O, Tr log = ζ′ (0). (14.530) O O − O For the Dirac spectrum (14.526), the ζ-function becomes, in natural units,

2 2 2 s eB ∞ d k M + k + eB(2n +1 1) − ⊥ ⊥ ζD(s) = 2 2 ± , (14.531) 2π n=0 (2π) " µ # X X± Z where the prefactor of eB/2π is the degeneracy of the Landau levels which ensures that the sum (eB/2π) n converges in the limit B 0 against the momentum integral dk1dk2/(2π)2 = dk2/4π. Performing the integral→ over the momenta p P ⊥ yields R R

B 2s 1 2 1 s 2 1 s ζ (s) = µ [M +2eBn] − + [M +2eB(n + 1)] − . (14.532) D 8π2 s 1 − n o This can be expressed in terms of the Hurwitz ζ-function (14.528) as

4 2 2 s 2 2 1 s M eB ∞ µ 1 M M − ζD(s) = 2ζH s 1, . (14.533) 4π2 M 2 2eB ! s 1  − 2eB ! − 2eB !  nX=0 −   From this we obtain (eB)2 M 2 M 2 M 2 ζD′ (0) = 2ζH′ 1, log 4π2 (− − 2eB ! − 2eB 2eB ! 1 M 2 2 µ2 + + 1 + log . (14.534)  6 2eB !  " 2eB !#    Choosing µ = M and subtracting the zero-field contribution, which is simply 3M 4/32π2, we find − 2 2 2 2 2 2 eff (eB) M M M 1 1 M ∆ = ζH′ 1, + ζH 1, log + , L 2π2 ( − 2eB ! − 2eB ! 2eB ! −12 4 2eB !   (14.535) 1008 14 Functional-Integral Representation of Quantum Field Theory where we have used the property [41, 39] 1 z z2 ζ ( 1, z)= + . (14.536) H − −12 2 − 2 Contact with the previous result in (14.495) is established with the help of the integral representation of the Hurwitz ζ-function [41]: z t s 1 1 ∞ e− t − ζH (s, z) = dt , Re(s) > 1 , Re(z) > 0. (14.537) Γ(s) 0 1 e t Z − − The integral can be rewritten as 1 s s 1 s s 1 z − z− sz− − 2 − ∞ dt 2z t 1 t ζH (s, z) = + + + e− coth t ,(14.538) s 1 2 12 Γ(s) 0 t1 s − t − 3 − Z − this expression being valid for Re(s) > 2, where the integral converges [42]. From this we evaluate the s-derivative at s =− 1 as follows: 2 − 1 z 1 ∞ dt 2z t 1 t ′ − ζH ( 1, z)= ζH ( 1, z) log z 2 e coth t . (14.539) − 12 − 4 − − − 4 Z0 t − t − 3 Inserting this into (14.535) we recover exactly the previous Eq. (14.495). Let us now derive the strong-field limit. For this we use the following relation between the Hurwitz ζ-function and the Γ-function [41, 39]: z z z ζH′ ( 1, z)= ζ′( 1) log(2π) (1 z)+ log Γ(x)dx . (14.540) − − − 2 − 2 − Z0 This identity follows from an integration of Binet’s integral representation [43] of log Γ(z). Thus we can write

2 2 2 2 2 eﬀ (eB) 1 M 3 M M ∆ = + ζ′( 1) + log(2π) L 2π2 −12 − − 4eB 4 2eB ! − 4eB  2 2 2 2 2 1  M 1 M M M /2eB + + log + dx log Γ(x) . (14.541) −12 4eB − 2 2eB !  2eB ! 0  Z    In the strong-ﬁeld limit, the range of integration in the last term vanishes, so we can use the Taylor expansion [41, 44] of log Γ(x): n ∞ ( 1) log Γ(x)= log x γx + − ζ(n) xn, (14.542) − − n nX=2 where ζ(n) is the usual Riemann ζ-function. This leads to :

2 2 2 2 2 eﬀ (eB) 1 M 3 M M ∆ = + ζ′( 1) + log(2π) L 2π2 −12 − − 4eB 4 2eB ! − 4eB  2 2 2 2 2 2 1 M 1 M M γ M + + log −12 4eB − 2 2eB !  2eB ! − 2 2eB !

2 2  n 2 n+1 M M ∞ ( 1) ζ(n) M + 1 log + − . (14.543) 2eB " − 2eB !# n(n + 1) 2eB !  nX=2   14.18 Functional Formulation of Quantum Electrodynamics 1009

The leading behavior in the strong-ﬁeld limit is (eB)2 2eB ∆ eﬀ = log + ..., (14.544) L 24π2 M 2 in agreement with Weisskopf’s result (14.520).

Charged Scalar Field in a Constant Magnetic Field For a charged scalar ﬁeld obeying the Klein-Gordon equation there is no spin sum and the ζ-function (14.531) becomes

2 2 2 s eB ∞ d k M + k + eB(2n + 1) − ζKG(s) = ⊥ ⊥ 2π (2π)2 " µ2 # nX=0 Z (eB)2 µ2 s 1 1 M 2 = ζH s 1, + . (14.545) 4π2 " 2eB ! (s 1) − 2 2eB !# − Setting again µ = m, and subtracting the zero ﬁeld contribution 3m4/64π2, we obtain

2 2 2 2 eﬀ (eB) 1 M 3 M ∆ KG = ζH′ 1, + + L − 4π2  − 2 2eB ! 4 2eB !  2 2  M 1 M + 1 + log ζH 1, + . (14.546) " 2eB !# − 2 2eB !) In the Bose case, the equivalence with the previous result is shown with the help of the integral representations of the Hurwitz ζ-function [compare (14.538)]

t/2 1 ∞ z t s 1 e− 1 − − ζH (s, 1/2+ z) = e t t dt , Re(s) > 1, Re(z) > Γ(s) Z0 1 e− −2 1 s 1 s −s 1 z − sz− − 2 − ∞ dt 2z t 1 1 t = + e− + .(14.547) s 1 − 24 Γ(s) 0 t1 s sinh t − t 6 − Z − The second expression is valid for Re(s) > 2, where the subtracted integral − converges. We can therefore use it to find the derivative at s = 1 required in Eq. (14.546): − 1 z2 z2 ζ′ ( 1, 1/2+ z) = (1 + log z) + log z H − −24 − 4 2 1 ∞ dt 2z t 1 1 t − 2 e + . (14.548) −4 Z0 t sinh t − t 6 Inserting this into (14.546), we recover exactly the previous result (14.496). In order to find a strong-field expansion of ∆ eff we use the following relation LKG between the Hurwitz ζ-function and the log of the Γ-function [41, 43]:

2 z 1 z log 2 z 1 ζH′ ( 1, 1/2+ z)= ζ′( 1) log 2π + + dz log Γ(x+ 2 ), (14.549) − −2 − − 2 − 24 2 Z0 1010 14 Functional-Integral Representation of Quantum Field Theory where we have used the formula [42, 45, 46]

1 2 5 1 1 3 log Γ(x) dx = log 2 + log π + ζ′( 1) . (14.550) Z0 24 4 8 − 2 − Then we expand

n 1 n 1 ∞ ( 1) − (1 2 ) log Γ x +1/2 = log π (γ + 2 log 2) x + − − ζ(n) xn, (14.551) 2 − n nX=2 and obtain the strong-ﬁeld expansion

2 2 2 2 2 2 eﬀ (eB) 5 M 1 1 M M ∆ KG = + 1 + log L − 4π2 4 2eB ! 24 − 2 2eB !  " 2eB !#    2 1  log 2 M 2 1 M 2 ζ′( 1) log 2 (γ + 2 log 2) −2 − − 24 − 4eB − 2 2eB !

n 1 n 2 n+1 ∞ ( 1) − (1 2 )ζ(n) M + − − , (14.552) n(n + 1) 2eB !  nX=2  with the leading behavior 

(eB)2 2eB ∆ eﬀ = log + .... (14.553) LKG 96π2 M 2 Appendix 14A Propagator of the Bilocal Pair Field

Consider the Bethe-Salpeter equation (14.300) with a potential λV instead of V

Γ= iλV G G Γ. (14A.1) − 0 0 Take this as an eigenvalue problem in λ at ﬁxed energy-momentum q = (q0, q)= (E, q) of the bound states. Let Γ (P q) be all solutions, with eigenvalues λ (q). Then the convenient normalization of n | n Γn is:

4 d P † q q i Γ (P q) G + P G P Γ ′ (P q)= δ ′ . (14A.2) − (2π)4 n | 0 2 0 2 − n | nn Z If all solutions are known, there is a corresponding completeness relation (the sum may comprise an integral over a continuous part of the spectrum) q q i G + P G P Γ (P q)Γ† (P ′ q)=(2π)4δ(4)(P P ′). (14A.3) − 0 2 0 2 − n | n | − n X This completeness relation makes the object given in (14.315) the correct propagator of ∆. In order to see this, write the free ∆-action [∆†∆] as A2 1 1 = ∆† + iG G ∆ (14A.4) A2 2 λV 0 × 0 Appendix 14A Propagator of the Bilocal Pair Field 1011 where we have used λV instead of V . The propagator of ∆ would have to satisfy 1 + iG G ∆∆† = i. (14A.5) λV 0 × 0 Indeed, by performing a short calculation, we can verify that this equation is fulﬁlled by the spectral expansion (14.315). We merely have to use the fact that Γn and λn are eigenfunctions and eigenvalues of Eq. (14A.1), and ﬁnd that that

1 Γ Γ† + iG G iλ n n λV 0 × 0 × − λ λ (q) ( n − n ) 1 X † † ΓnΓ + iG0 G0ΓnΓ = iλ λV n × n − λ λ (q) n n X − λn(q) +1 = iλ − λ ( iG G Γ Γ† ) λ λ (q) − 0 × 0 n n n n X − = i i G G Γ Γ† = i. (14A.6) − 0 × 0 n n i ! X Note that the expansion of the spectral representation of the propagator in powers of λ

Γ Γ† λ k ∆∆† = iλ n n = i Γ Γ† (14A.7) − λ λ (q) λ (q) n n n n n n ! X − Xk X corresponds to the graphical sum over one, two, three, etc. exchanges of the potential λV . For n = 1 this is immediately obvious since (14A.1) implies that λ λ i Γ Γ† = λ (q)V G G Γ Γ† = iλV. (14A.8) λ (q) n n λ (q) n 0 × 0 n n n n n X X For n = 2 one can rewrite, using the orthogonality relation,

2 λ λ † λ † † ′ i ΓnΓn = ΓnΓnG0 G0Γn Γn′ = λV G0 G0λV . (14A.9) λ (q) λ (q) × λ ′ (q) × n n ′ n n X Xnn This displays the exchange of two λV terms with particles propagating in between. The same procedure applies at any order in λ. Thus the propagator has the expansion

∆∆† = iλV iλV G G iλV + .... (14A.10) − 0 × 0

If the potential is instantaneous, the intermediate dP0/2π can be performed replacing R 1 G G i (14A.11) 0 × 0 → E E (P q) − 0 | where q q E (P q)= ξ + P + ξ P 0 | 2 2 − is the free particle energy which may be considered as the eigenvalue of an operator H0. In this case the expansion (14A.10) reads

1 E H ∆∆† = i λV + λV λV + ... = iλV − 0 . (14A.12) E H E H λV − 0 − 0 − 1012 14 Functional-Integral Representation of Quantum Field Theory

We see it related to the resolvent of the complete Hamiltonian as

∆∆† = iλV (RλV +1) (14A.13) where

1 ψ ψ† R = n n (14A.14) ≡ E H λV E E 0 n n − − X − with ψn being the Schr¨odinger amplitudes in standard normalization. We can now easily determine the normalization factor N in the connection between Γn and the Schr¨odinger amplitude ψn. Eq. (29A.3) gives in the instantaneous case

3 d P † 1 Γ (P q) Γ ′ (P q)= δ ′ . (14A.15) (2π)3 n | E H n | nn Z − 0 Inserting ψ from (14.306) renders

3 1 d P † ψ (P q)(E H )ψ ′ (P q)= δ ′ . (14A.16) N 2 (2π) n | − 0 n | nn Z Using ﬁnally the Schr¨odinger equation

(E H )ψ = λV ψ, (14A.17) − 0 we ﬁnd

3 1 d P † ψ (P q) λV ψ ′ (P q)= δ ′ . (14A.18) N 2 (2π)3 n | n | nn Z

For the wave functions ψn (P q) in standard normalization, the integral is equal to the energy diﬀerential | dE λ . dλ For a typical calculation of a resolvent, the reader is referred to Schwinger’s treatment [47] of the Coulomb problem. His result may directly be used for a propagator of electron hole pairs bound to excitons.

Appendix 14B Fluctuations around the Composite Field

Here we show that the quantum mechanical ﬂuctuations around the classical equations of motion (14.237) are quite simple to calculate. The exponent of (14.232) is extremized by the ﬁeld

ϕ(x)= dyV (x, y)ψ†(y)ψ(y). (14B.1) Z Similarly, the extremum of the exponent of (14.278) yields

∆(x, y)= V (x y)ψ(x)ψ(y). (14B.2) − For this let us compare the Green functions of ϕ(x) or ∆(x, y) with those of the composite operators on the right-hand side of Eqs. (14B.1) or (14B.2). The Green functions of ϕ(x) or ∆(x, y) are generated by adding to the ﬁnal actions (14.242) or (14.284) external currents dxϕ(x)I(x) or R Appendix 14B Fluctuations around the Composite Field 1013

1/2 dxdy(∆(y, x)I†(x, y) + h.c.), respectively, and by forming functional derivatives δ/δI. The Green functions of the composite operators, on the other hand, are obtained by adding R dx dyV (x, y)ψ†(y)ψ(y) K(x) (14B.3) Z Z 1 dx dyV (x y)ψ(x)ψ(y)K†(x, y) + h.c. (14B.4) 2 − Z Z to the original actions (14.235) or (14.278), respectively, and by forming functional derivatives δ/δK. It is obvious that the sources K(x) nad K(x, y) can be included in the ﬁnal actions (14.242) and (14.284) by simply replacing

ϕ(x) ϕ′(x)= ϕ(x) dx′K(x′)V (x′, x), (14B.5) → − Z or ∆(x, y) ∆′(x, y) = ∆(x, y) K(x, y). (14B.6) → − If one now shifts the functional integrations to these new translated variables and drops the irrelevant superscript “prime”, the actions can be rewritten as 1 [ϕ] = iTr log(iG−1)+ dxdx′ϕ(x)V −1(x, x′)ϕ(x′)+i dxdx′η†(x)G (x, x′)η(x) A ± ϕ 2 ϕ Z Z 1 + dxϕ(x)[I(x)+ K(x)] + dxdx′K(x)V (x, x′)K(x′), (14B.7) 2 Z Z or i 1 1 [∆] = Tr log iG−1 + dxdx′ ∆(x, x′) 2 A ±2 ∆ 2 | | V (x, x′) Z i 1 + dxdx′j†(x)G (x, x′) 2 ∆ V (x, x′) Z 1 + dxdx′ ∆(y, x) I†(x, y)+ K†(x, y) + h.c. 2 Z 1 + dxdx′ K(x, x′) 2V (x, x′). (14B.8) 2 | | Z In this form the actions display clearly the fact that derivatives with respect to the sources K or I coincide exactly, except for all possible insertions of the direct interaction V . For example, the propagators of the plasmon field ϕ(x) and of the composite operator dyV (x, y)ψ†(y)ψ(y) are related by R δ(2)Z δ(2)Z ϕ(x)ϕ(x′) = = V −1(x, x′) (14B.9) −δI(x)δI(x′) − δK(x)δK(x′) = V −1(x, x′)+ 0 ( dyV (x, y)ψ†(y)ψ(ϕ))( dy′V (x′y′)ψ†(y′)ψ†(y′)ψ(y′)) 0 , h | | i Z Z in agreement with (14.238). Similarly, one finds for the pair fields:

∆(x, x′)∆†(y,y′) = δ(x y)δ(x′ y′)iV (x x′) − − − + 0 (V (x′, x)ψ(x′)ψ(x))(V (y′,y)ψ†(y)ψ†(y′)) 0 . (14B.10) h | | i Note that the latter relation is manifestly displayed in the representation (14A.10) of the propagator ∆. Since

∆∆† = iV G(4)V, (14B.11) 1014 14 Functional-Integral Representation of Quantum Field Theory one has from (14B.10)

0 V (ψψ)(ψ†ψ†V ) 0 = V G(4)V, (14B.12) h | | i which is correct remembering that G(4) is the full four-point Green function. In the equal-time situation relevant for an instantaneous potential, G(4) is replaced by the resolvent R. Appendix 14C Two-Loop Heisenberg-Euler Eﬀective Action

The next correction to the Heisenberg-Euler Lagrangian density (14.489) is [38, 48]

2 ∞ ∞ 4 2 2 ie e β ε 2 ′ ∆(2) eﬀ = dτ dτ ′ e−i(M −iη)(τ+τ ) L −128π4 sin eβτ sin eβτ ′ sinh eετ sinh eετ ′ Z0 Z0 4M 2 [S(τ)S(τ ′)+ P (τ)P (τ ′)] I i I , (14C.1) × 0 − n o where

S(τ) cos eβτ cosh eετ , P (τ) sin eβτ sinh eετ , ≡ ≡ and 1 b (q p) b aq bp I log , I − log − , 0 ≡ b a a ≡ (b a)2 a − ba(b a) − − − a eβ (cot eβτ + cot eβτ ′) , b eε (coth eετ + coth eετ ′) , ≡ ≡ 2e2β2 cosh eε(τ τ ′) 2e2ε2 cos eβ(τ τ ′) p − , q − . (14C.2) ≡ sin eβτ sin(eβτ ′) ≡ sinh eετ sinh eετ ′ This expression contains divergences which require renormalization. First, there is a subtraction of an infinity to make ∆(2) eff vanish for zero fields. Then there are both charge and wave function renormalizations, justL as for the one-loop effective Lagrangian, which involves identifying a divergent term in ∆(2) eff of the form of the zero-loop Maxwell Lagrangian. This is done simply by expanding the integrandL to quadratic order in the fields β and ε. This divergence can be absorbed by redefining the electric charge and the fields as

1/2 −1/2 −1/2 eR = eZ3 , BR = B Z3 , ER = E Z3 (14C.3) where Z3 is some divergent normalization constant, which was given by (14.435) in the previous −1/2 −1/2 result (14.491). The invariants β and ε are renormalized accordingly: βR = βZ3 , εR = εZ3 . Then we re-express ∆(2) eﬀ in terms of the renormalized charges and ﬁelds. Finally, we have to renormalize the mass: L

2 2 2 mR = m0 + δM , (1) 2 (1) eﬀ 2 (1) 2 2 ∂ R (m0) ∆ R (mR) = R (m0)+ δM L 2 . (14C.4) L L ∂m0

2 2 (1) eff The second term in (14C.4) is of the order α , since δM and ∆ R are both of order α. For details of removing the divergencies, see the original papers in Refs.L [48]. The final answer for the renormalized two loop effective Lagrangian is ie2 ∞ τ K (τ) ∆(2) eff = dτ dτ ′ K(τ, τ ′) 0 LR −64π4 − τ ′ Z0 Z0 ie2 ∞ 5 dτK (τ) log(iM 2τ)+ γ (14C.5) −64π4 0 − 6 Z0 Notes and References 1015 where γ 0.577... is Euler’s constant, and the functions K(τ, τ ′) and K (τ) are ≈ 0 2 2 2 ′ (eβ) (eε) K(τ, τ ′) = e−iM (τ+τ ) 4M 2(S(τ) S(τ ′)+ P (τ) P (τ ′))I iI P (τ) P (τ ′) 0 − 1 2i e2(β2 ε2) 5iττ ′ 4M 2 + − 2M 2(ττ ′ 2τ 2 2τ ′2) −ττ ′(τ +τ ′) − τ + τ ′ 3 − − − τ +τ ′ 2 2 2 2 2 ∂ 1 eβτ eετ e (β ε )τ K (τ) = e−iM τ 4M 2 + i 1+ − . (14C.6) 0 ∂τ τ 2 tan eβτ tanh eετ − 3 The fields in this expression can be replaced by the renormalized fields, and everything is finite. The lowest contribution is of the fourth power in the fields and reads

e6 16 263 ∆(2) eﬀ = (β2 ε2)2 + (βε)2 + ..., (14C.7) L 64π4m4 81 − 162 which has been added to the one-loop result in Eq. (14.499). In the limit of strong magnetic ﬁelds it yields

e4β2 eβ ∆(2) eﬀ = log + constant + .... (14C.8) L 128π4 πM 2

Notes and References

[1] R.P. Feynman and A.R. Hibbs, Path Integrals and Quantum Mechanics, McGraw-Hill, New York (1968); [2] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, World Scientific Publishing Co., Singapore 1995, pp. 1–890. [3] R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948); [4] H. Kleinert, Collective Quantum Fields, Fortschr. Physik 26, 565 (1978) (http:// klnrt.de/55). See also (http://klnrt.de/b7/psfiles/sc.pdf). [5] J. Rzewuski, Quantum Field Theory II, Hefner, New York (1968). [6] S. Coleman, Erice Lectures 1974, in Laws of Hadronic Matter, ed. by A. Zichichi, p. 172. [7] See for example: A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover, New York (1975); L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics, Benjamin, New York (1962); A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York (1971). [8] R.L. Stratonovich, Sov. Phys. Dokl. 2, 416 (1958); J. Hubbard, Phys. Rev. Letters 3, 77 (1959); B. Mühlschlegel, J. Math. Phys. 3, 522 (1962); J. Langer, Phys. Rev. 134, A 553 (1964); T.M. Rice, Phys. Rev. 140 A 1889 (1965); J. Math. Phys. 8, 1581 (1967); A.V. Svidzinskij, Teor. Mat. Fiz. 9, 273 (1971); D. Sherrington, J. Phys. C4 401 (1971). [9] The first authors to employ such identities were P.T. Mathews, A. Salam, Nuovo Cimento 12, 563 (1954), 2, 120 (1955). [10] H.E. Stanley, Phase Transitions and Critical Phenomena, Clarendon Press, Oxford, 1971. 1016 14 Functional-Integral Representation of Quantum Field Theory

[11] For the introduction of collective bilocal fields in particle physics and applications see H. Kleinert, On the Hadronization of Quark Theories, Erice Lectures 1976 on Particle Physics, publ. in Understanding the Fundamental Constituents of Matter, Plenum Press 1078, (ed. by A. Zichichi). See also H. Kleinert, Phys. Letters B 62, 429 (1976), B 59, 163 (1975). [12] The mean-field equations associated with the pair fields of the electrons in a metal are precisely the equations used by Gorkov to study the behavior of type II superconductors. See, for example, p. 444 in the third of Refs. [52]. [13] H.A. Bethe and E.E. Salpeter in Encyclopedia of Physics (Handbuch der Physik), Springer, Berlin, 1957, p. 405. [14] Y. Nambu and G. Jona Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961). [15] V.L. Ginzburg and L.D. Landau, Eksp. Teor. Fiz. 20, 1064 (1950). [16] R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford, Ox- ford, 1989; K.U. Gross and R.M. Dreizler, Density Functional Theory, NATO Science Series B, 1995. [17] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). [18] A. Leggett, Rev. Mod. Phys. 47, 331 (1975). ∗ ∗ [19] Note that the hermitian adjoint ∆↑↓ comprises transposition in the spin indices, i.e., ∆↑↓ = ∗ [∆↓↑] . [20] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Finan- cial Markets, World Scientific, Singapore 2004 (http://klnrt.de/b5). [21] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4-Theories, World Scientific, 2001 (klnrt.de/b8). [22] H. Kleinert, Converting Divergent Weak-Coupling into Exponentially Fast Convergent Strong-Coupling Expansions, Lecture presented at the Summer School on ”Approximation and extrapolation of convergent and divergent sequences and series” in Luminy bei Marseille in 2009 (arXiv:1006.2910). [23] The alert reader will recognize her the so-called square-root trick of Chapter 5 in the textbook Ref. [6]. [24] See the www page (http://klnrt.de/b7/psfiles/hel.pdf). [25] H. Kleinert, Phys. Rev. D 57, 2264 (1998); Phys.Rev. D 60, 085001 (1999). (See also klnrt.de/critical). [26] C. De Dominicis, J. Math. Phys. 3, 938 (1962); C. De Dominicis and P.C. Martin, J. Math. Phys. 5, 16, 31 (1964); J.M. Cornwall, R. Jackiw, and E.T. Tomboulis, Phys. Rev. D 10, 2428 (1974); H. Kleinert, Fortschr. Phys. 30, 187 (1982) (klnrt.de/82); Lett. Nuovo Cimento 31, 521 (1981) (klnrt.de/77). [27] H. Kleinert, Annals of Physics 266, 135 (1998) (klnrt.de/255). [28] R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (klnrt.de/159). [29] Such an omission was done in Eq. (4.117) of the textbook [55]. [30] J.E. Robinson, Phys. Rev. 83, 678 (1951). [31] See Chapter 7 in the textbook [6]. Notes and References 1017

[32] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). English translation available at http://klnrt.de/files/heisenbergeuler.pdf. [33] J. Schwinger, Phys. Rev. 84, 664 (1936); 93, 615; 94, 1362 (1954). [34] U.D. Jentschura, H. Gies, S.R. Valluri, D.R. Lamm, E.J. Weniger, Canadian J. Phys. 80, 267 (2002) (hep-th/0107135). [35] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. See Section 5.2. [36] V. Weisskopf, The electrodynamics of the vacuum based on the quantum theory of the electron, Kong. Dans. Vid. Selsk. Math.-Fys. Medd. XIV No. 6 (1936); English translation in: Early Quantum Electrodynamics: A Source Book, A.I. Miller, Cambridge University Press, 1994. [37] G. Mourou, T. Tajima, and S.V. Bulanov, Reviews of Modern Physics, 78, 309, (2006); and references therein. [38] G.V. Dunne, Heisenberg-Euler Effective Actions: 75 years on, Int.J. Mod. Phys. A 27 (2012) 1260004. [39] E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed., Cambridge, 1950, pp. 268–269. [40] http://mathworld.wolfram.com/HurwitzZetaFunction.html. [41] A. Erdelyi (ed.), Higher Transcendental Functions, Vol. I, Kreiger, Florida, 1981. [42] V. Adamchik, Symbolic and Numeric Computation of the Barnes Function, Conference on applications of Computer Algebra, Albuquerque, June 2001; Contributions to the Theory of the Barnes Function, (math.CA/0308086). [43] E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed., Cambridge, 1950. [44] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [45] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1972; Formula 6.441.1. [46] E.W. Barnes, The theory of the G-function, Quart. J. Pure Appl. Math XXXI (1900) 264. [47] J. Schwinger, J. Math. Phys. 5, 1606 (1964). [48] V.I. Ritus, Lagrangian of an intense electromagnetic field and quantum electrodynamics at short distances, Zh. Eksp. Teor. Fiz 69, 1517 (1975) [Sov. Phys. JETP 42, 774 (1975)]; Connection between strong-field quantum electrodynamics with short-distance quantum electrodynamics, Zh. Eksp. Teor. Fiz 73, 807 (1977) [Sov. Phys. JETP 46, 423 (1977)]; The Lagrangian Function of an Intense Electromagnetic Field, in Proc. Lebedev Phys. Inst. Vol. 168, Issues in Intense-field Quantum Electrodynamics, V.L. Ginzburg, ed., (NovaScience Pub., NY 1987); Effective Lagrange function of intense electromagnetic field in QED, (hep- th/9812124). [49] L.D. Faddeev and V.N. Popov Phys. Lett. B 25 29 (1967); See also M. Ornigotti and A. Aiello, (arXiv:1407.7256). [50] J. Bardeen, L. N. Cooper, and J.R. Schrieffer: Phys. Rev. 108, 1175 (1957). See also the little textbook from the russian school: N.N. Bogoliubov, E. A. Tolkachev, and D.V. Shirkov A New Method in the Theory o Su- perconductivity, Consultnts Bureau, New York, 1959. 1018 14 Functional-Integral Representation of Quantum Field Theory

[51] For the introduction of collective bilocal ﬁelds in particle physics and applications see H. Kleinert, On the Hadronization of Quark Theories, Erice Lectures 1976 on Particle Physics, publ. in Understanding the Fundamental Constituents of Matter, Plenum Press 1078, (ed. by A. Zichichi). See also H. Kleinert, Phys. Letters B 62, 429 (1976), B 59, 163 (1975). [52] See for example: A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover, New York (1975); L.P. Kadanoﬀ, G. Baym, Quantum Statistical Mechanics, Benjamin, New York (1962); A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Paricle Systems, McGraw-Hill, New York (1971). [53] R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (klnrt.de/159). [54] H. Kleinert, Converting Divergent Weak-Coupling into Exponentially Fast Convergent Strong-Coupling Expansions, Lecture presented at the Summer School on ”Approximation and extrapolation of convergent and divergent sequences and series” in Luminy bei Marseille in 2009 (arXiv:1006.2910). [55] C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill (1985). See Section 12.2. [56] For more details see (http://klnrt.de/b8/crit.htm). [57] H. Kleinert, Phys. Rev. D 57, 2264 (1998); Phys.Rev. D 60, 085001 (1999). (See also klnrt.de/critical).