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Scale Transformations of Fields and Correlation Functions H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4-Theories August 28, 2011 ( /home/kleinert/kleinert/books/kleischu/scaleinv.tex) 7 Scale Transformations of Fields and Correlation Functions We now turn to the properties of φ4-field theories which form the mathematical basis of the phenomena observed in second-order phase transitions. These phenomena are a consequence of a nontrivial behavior of fields and correlation functions under scale transformations, to be discussed in this chapter. 7.1 Free Massless Fields Consider first a free massless scalar field theory, with an energy functional in D-dimensions: D 1 2 E0[φ]= d x [∂φ(x)] , (7.1) Z 2 This is invariant under scale transformations, which change the coordinates by a scale factor x → x′ = eαx, (7.2) and transform the fields simultaneously as follows: ′ 0 dφα α φ(x) → φα(x)= e φ(e x). (7.3) 0 From the point of view of representation theory of Lie groups, the prefactor dφ of the parameter α plays the role of a generator of the scale transformations on the field φ. Its value is D d0 = − 1. (7.4) φ 2 Under the scale transformations (7.2) and (7.3), the energy (7.1) is invariant: ′ 1 ′ 0 1 ′ 1 ′ ′ D 2 D 2dφ α α 2 D 2 E0[φα]= d x [∂φα(x)] = d x e [∂φ(e x)] = d x [∂ φ(x )] = E0[φ]. (7.5) Z 2 Z 2 Z 2 0 The number dφ is called the field dimension of the free field φ(x). Its value (7.4) is a direct consequence of the naive dimensional properties of x and φ. In the units used in this text, in which we have set kBT = 1 [recall the convention stated before Eq. (2.6)], the exponential in the partition function is e−E[φ], so that E[φ] is a dimensionless quantity. The coordinates have the dimension of a length. This property is expressed by the equation [x]= L. The field in (7.5) has −d 0 then a naive dimension (also called engineering or technical dimension)[φ]= L φ . To establish contact with the field theories of elementary particle physics, we shall use further natural units in which c =h ¯ = 1. Then the length L is equal to an inverse mass µ−1 (more precisely, L is the Compton wavelength L =¯h/mc associated with the mass m) . It is conventional to specify 91 92 7 Scale Transformations of Fields and Correlation Functions the dimension of every quantity in units of mass µ, rewriting [x] = L and [φ] = L−(D/2−1) as − 0 1 dφ 0 [x]= µ and [φ]= µ . Hereafter, we shall refer to the power dφ of µ as the free-field dimension, rather than the power. This will shorten many statements without danger of confusion. The trivial scale invariance (7.5) of the free-field theory is the reason for a trivial power behavior of the free-field correlation functions. In momentum space, the two-point function reads [recall (4.9)]: 1 G(k)= , (7.6) k2 which becomes in x-space dDk eikx Γ(D/2 − 1) 2D−2 x G( )= D 2 = D/2 D−2 . (7.7) Z (2π) k (4π) |x| It is instructive to rederive this power behavior as a consequence of scale invariance (7.5) of the energy functional. This implies that the correlation function of the transformed fields ′ φα(x) in (7.3) must be the same as those of the initial fields φ(x): ′ ′ hφα(x)φα(y)i = hφ(x)φ(y)i. (7.8) Inserting on the left-hand side the transformed fields from (7.3), we see that 2d 0α e φ hφ(eαx)φ(eαy)i = hφ(x)φ(y)i. (7.9) Translational invariance makes this a function of x − y only, which has then the property 2d 0α e φ G(eα(x − y)) = G(x − y). (7.10) Since G(x − y) is also rotationally invariant, Eq. (7.10) implies the power behavior −2d 0 G(x − y) = const ×|x − y| φ , (7.11) which is precisely the form (7.7). Note that this power behavior is of the general type (1.11) observed in critical phenomena, with a trivial critical exponent η = 0. In order to study the consequences of a continuous symmetry and its violations, it is con- venient to first consider infinitesimal transformations. Then the defining transformations (7.2) and (7.3) read: δx = αx, (7.12) 0 δφ(x) = α(dφ + x∂x)φ(x), (7.13) and the invariance law (7.9) takes the form of a differential equation: 0 0 (dφ + x∂x)hφ(x)φ(y)i +(dφ + y∂y)hφ(x)φ(y)i =0. (7.14) For an n-point function, this reads n 0 (dφ + xi∂xi )hφ(x1) ··· φ(xn)i =0. (7.15) Xi=1 In the notation (2.10) for the correlation functions, this may be written as n (n) 0 (n) xi∂xi G (x1,..., xn)= −ndφ G (x1,..., xn). (7.16) Xi=1 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories 7.2 Free Massive Fields 93 The differential operator on the left-hand side, n Hx ≡ xi∂xi , (7.17) Xi=1 0 measures the degree of homogeneity in the spatial variables xi, which is −ndφ. The power behavior (7.11) shows this directly. − If we multiply Eq. (7.16) by e iΣikixi , and integrate over all spatial coordinates [recall (4.10)], we find the momentum-space equation n (n) 0 (n) ki∂ki G (k1,..., kn)=(ndφ − D) G (k1,..., kn). (7.18) ! Xi=1 The differential operator on the left-hand side, n Hk ≡ ki∂ki , (7.19) Xi=1 measures the degree of homogeneity in the momentum variables xi, which is Hk = −Hx − D = 0 ndφ − D. (n) Turning to the connected one-particle irreducible Green functions Gc (k1,..., kn), and fur- (n) ther to the vertex functions Γ¯ (k1,..., kn), by removing an overall δ-function and multiplying −1 (n) each leg by a factor G (ki), [recall Eq. (4.20)], we find for the vertex functions Γ¯ (k1,..., kn) the differential equation n ¯(n) 0 ¯(n) ki∂ki Γ (k1,..., kn)=(−ndφ + D) Γ (k1,..., kn). (7.20) ! Xi=1 For the two-point vertex function, this implies the homogeneity property Γ¯(2)(k) = const × k2, (7.21) which is, of course, satisfied by the explicit form (7.6) [recall (4.34)]. 7.2 Free Massive Fields We now introduce a mass term to the energy functional, and consider a field φ whose fluctuations are governed by 1 m2 E[φ]= dDx (∂φ)2 + φ2 . (7.22) Z " 2 2 # In the presence of m, the Fourier transform of the two-point function is [recall (4.9)] 1 G(k)= , (7.23) k2 + m2 which becomes, in x-space, D ikx D−2 d k e 1 2 1 D/2−1 x x − x G( )= D 2 2 = D/2 D−2 × D/2−2 |m | KD/2 1(|m |). (7.24) Z (2π) k + m (4π) |x| 2 94 7 Scale Transformations of Fields and Correlation Functions This possesses the scaling form of the general phenomenologically observed type (1.8), with a 2 length scale ξ =1/m. Since m is proportional to the reduced temperature variable t = T/Tc−1, the expression (7.24) has also the general scaling form (1.28), with a critical exponent ν =1/2. In the presence of a mass term, the energy functional (7.22) is no longer invariant under the scale transformations (7.2) and (7.3). Whereas the free part E0[φ] of the energy is invariant as in (7.5), the energy of the mass term of the transformed field φ′(x) is now related to that of the original field by 2 2 2 ′ m ′ 0 m 0− m − D 2 2dφ α D 2 α (2dφ D)α D 2 2α Em[φα] = d x φα (x)= e d x φ (e x)= e d x φ (x)= e Em[φ]. Z 2 Z 2 Z 2 (7.25) 7.3 Interacting Fields 4 Let us now add a φ -interaction to a massless energy functional E0[φ], and consider the energy D 1 2 λ 4 E0,λ[φ]= d x (∂φ) + φ . (7.26) Z "2 4! # This remains invariant under the scale transformations (7.2) and (7.3) only for D = 4. In less than four dimensions, where the critical phenomena in the laboratory take place, the invariance is broken by the φ4-term. The interaction energy behaves under scale transformations (7.2) and (7.3) as follows: ′ λ ′ 0 λ 0− λ − D 4 4dφ α D 4 α (4dφ D)α D 4 (D 4)α Eλ[φα] = d x φα (x)= e d x φ (e x)= e d x φ (x)= e Eλ[φ]. Z 4! Z 4! Z 4! (7.27) This symmetry breakdown may be viewed as a consequence of the fact that, in D =6 4 dimensions, the coupling constant is not dimensionless, but has the naive dimension [λ]= µ4−D. With a new dimensionless coupling constant λˆ ≡ µD−4λ, (7.28) this dimension is made explicit by re-expressing the energy functional (7.26) as 4−D ˆ D 1 2 µ λ 4 E0,λ[φ]= d x (∂φ) + φ (7.29) Z "2 4! # with some mass parameter µ. The theory we want to investigate contains both mass and interaction terms in the energy functional: 1 m2 µ4−Dλˆ E[φ]= dDx (∂φ)2 + φ2 + φ4 .
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