Proof of Weinberg’s theorem - Weinberg’s Theorem

5.7 Weinberg’s Theorem

5.7.1 Proof of Weinberg’s theorem

The proof of Weinberg’s theorem is obtained in three steps:

(A) Step of induction: We show that if the theorem is true for dimI ≤k wherek≥1 then it is true for dimI =k+ 1.

(B) Convergence for dimI = 1: We prove that the integral over an arbitrary one-dimensional subspace I is absolutely convergent providedD_I<0.

(C) Covering the one-dimensional subspace I with a finite number subintervals J each of which contributes a certain asymptotic behaviour of the integral.

(D) The sum over the subintervals constructed in (C) is a function of the class An−1 with α_I given by (5.118).

The convergence theorem, i.e., conjecture (a) of the theorem is proven with (A) and (B) which is the simple part of the proof.

Ad (A): Let the theorem be true for any subspaceI ^Rⁿ with dim(I)≤k.

Now letI be a subspace with dimI =k+ 1. Let S₁ and S₂ be arbitrary non-trivial disjoint subspaces of I with I = S₁ ⊕S₂ (there are arbitrary many such splittings of I!). Then necessarily dimS₁+ dimS₂ = dimI =k+ 1 and because S₁ and S₂ are both non-trivial they are both of a dimension less thenk. By the induction hypothesis the theorem holds for these subspaces. Thus we apply it twice making use of Fubini’s theorem:

(a1)f_S₂ converges absolutely ifD_S₂(f)<0 where D_S₂(f) = max

S⁰⁰S2

[α(S⁰⁰) + dimS⁰⁰]. (5.122) (b1) IfD_S₂(f)<0 then f_S₂ ∈A_n−k₂,k₂ = dimS₂, with

α_S₂(S⁰) = max

Λ(S2)S⁰⁰=S⁰[α(S⁰⁰) + dimS⁰⁰−dimS⁰]. (5.123) (a2) Iff_S₂ ∈A_n−k₂ cf. (b1) then f_I converges absolutely if

D_S₁(f_S₂) = max

S⁰S1

[α_S₂(S⁰) + dimS⁰]<0. (5.124) (b2) IffS2 ∈A_n−k₂ and DS1(fS2)<0 then fI ∈A_n−k−1 with

α_I(S) = max

Λ(S1)S⁰[α_S₂(S⁰) + dimS⁰−dimS]. (5.125) (a1) and (a2) together mean thatfI is absolutely convergent if

D⁰_I(f) = max{D_S₂(f), D_S₁(f_S₂)}<0 (5.126)

5.7 · Weinberg’s Theorem

and this can be written as

D⁰_I(f) = max

S⁰⁰

∗[α(S⁰⁰) + dimS⁰⁰] (5.127) where max^∗ indicates that S⁰⁰ has to be a subspace of S₂ or such subspaces of ^Rⁿ that Λ(S2)S⁰⁰ S1. To show that D_I⁰(f) = DI(f) we have to prove that this means that S⁰⁰ is running in fact over all subspaces of I =S₁⊕S₂.

Let S⁰ I but not a subspace of S₂. Then let V~ ∈ S⁰. Because I = S₁⊕S₂ there exist vectors~L₁∈S₁ andL~₂∈S₂ such thatV~ =L~₁+~L₂. So we have Λ(S₂)V~ =L~₁∈S₁ and thus Λ(S₂)S⁰ S₁. Thus since S⁰⁰ in max^∗ runs over all subspaces ofS₂ or such subspaces of^Rⁿ for which Λ(S₂)S⁰⁰S₁ in factS⁰⁰ runs over all subspaces of I and thus

D_I⁰(f) = max

S⁰⁰I[α(S⁰⁰) + dimS⁰⁰] =DI(f) (5.128) Thus we have shown claim (a) of the theorem forf_I.

To show (b) out of the induction hypothesis we have to combine (b1) and (b2) which imme-diately show thatf_I ∈A_n−k−1. Thus we have only to prove (5.118). From (b1) and (b2) we know that

α_I(S) = max

Λ(S1)S⁰=S

dimS⁰−dimS+ max

Λ(S2)S⁰⁰=S⁰[α(S⁰⁰) + dimS⁰⁰−dimS⁰]

= max

Λ(S1)S⁰=S;Λ(S2)S⁰⁰=S⁰[α(S⁰⁰) + dimS⁰⁰−dimS].

(5.129)

To complete the induction step we have just to show that we can combine the both conditions forS⁰⁰ to Λ(I)S⁰⁰=S.

Thus we have to show the equivalence

Λ(S₂)S⁰⁰=S⁰∧Λ(S₁)S⁰ =S

⇔S = Λ(S₁⊕S₂)S⁰⁰ (5.130) LetS = span{L~₁, . . . , ~L_r}thenS = Λ(S₁)S⁰means that in the above maximum we can restrict S⁰ to such subspaces for which existL~⁰₁, . . . , ~L⁰_r ∈S₁such thatS⁰ = span{L~₁+L~⁰₁, . . . , ~L_r+L~⁰_r}. The same argument givesS⁰⁰ = span{L~₁+~L⁰₁+L~⁰⁰₁, . . . , ~L_r+L~⁰_r+L~⁰⁰_r} if Λ(S₂)S⁰⁰ =S⁰ with vectorsL~⁰⁰₁, . . . , ~L⁰⁰_r ∈S₂. But since S₁ and S₂ are disjoint subspaces this is nothing else than the statement thatS⁰⁰ runs over all subspaces with Λ(S1⊕S2)S⁰⁰=S. This proves (A).

Ad (B):Now we like to show (a) for a one-dimensional subspace. Thus we setI = span{~L} and look on

f_~_L(P~) = Z ∞

−∞

f(P~ +y~L). (5.131)

Sincef ∈A_n by definition

f(P~ +Ly)~ ∼=

y→∞O

y^α(I)(lny)^β(^L)^~

(5.132) This integral converges absolutely if α(L) + 1~ < 0 if it exists for any finite integral wrt. y.

Now the only non-trivial subset ofI isI itself because dimI = 1 and thus DI(f) =α(I) + 1 which proves part (a) completely.¹⁵

15The reader who intends just to understand the convergence conjecture (a) for renormalisation theory may stop here because this part is now completely proved. The rest of the section is devoted to the proof of the asymptotic behaviour for a one dimensional integration space.

Ad (C):The rest of the proof is devoted to show that the integral f_I of a functionf ∈A_n over a one-dimensional subspace I = span{L~} is of classA_n−1 providedD_I(f)<0.

To show this we use the form (5.115), i.e. we have to study the asymptotic behaviour of f_L_~(P~) =

Z ∞

−∞

dyf(P~ +y~L) (5.133)

which is the same as that of the functionf_I defined by (5.116).

Let span{L~₁, . . . , ~L_m} ^Rⁿ where theL~_j forj = 1, . . . , mbuild a linearly independent set of vectors which are also linearly independent fromL, and~ W ⊆^Rⁿ compact. Then we have to show that

f_~_L(P~) ∼=

η1,...,ηm→∞O Ym k=1

η_k^α^~^L^(span{^~^L¹^,...,~^L^k^})(lnη_k)^β^L^~⁽^L^~¹^,...,~^L^k⁾

(5.134) with the asymptotic coefficients

α_~_L(S) = max

Λ(I)S⁰=S[α(S⁰) + dimS⁰−dimS]. (5.135) For this purpose we try to split the integration range ^R in (5.133) in regions of definite asymptotic behaviour of the integral. Since we have to find only an upper bound of the integral it is enough to construct a finite coverage which needs not necessarily to be disjoint.

We look on the set of vectors

L~1+u1L, ~~ L2+u2L, . . . , ~~ Lr+ur~L, ~L, ~Lr+1, . . . , ~Lm (5.136) with 0 ≤r≤m and u1, . . . , ur ∈^R.

Now we have to write out what it means thatf ∈A_nfor this set of vectors: There exist num-bersb_l(u₁, . . . , u_r) > (0 ≤l≤r) and M(u₁, . . . , u_r) >0 such that for all η_l > b_l(u₁, . . . , u_r) andC~ ∈W the function fulfils the inequality:

|f[(L~1+ur~L)η1· · ·ηmη0+· · ·+ (~Lr+urL)η~ r· · ·ηmη0+L~r+1ηr+1· · ·ηm+· · ·+L~mηm+C]~ |

≤M(u₁, . . . , u_r) Yr k=1

η_k^α(span{^~^L¹^+u¹^L,...,~^~ ^L^k^+u^k^L})^~ (lnη_k)^β(^L^~¹^+u¹^~^L,...,~^L^k^+u^k^L)^~ ×

×η₀^α(span{^~^L¹^,...,~^L^r^,~^L})ln(η₀)^β(^L^~¹^+u¹^~^L,...,~^L^r^+u^r^L,~^~ ^L)×

×η^α(span{_r+1 ^L^~¹^,...,~^L^r+1^,~^L})(lnη_r+1)^β(^~^L¹^+u¹^L,...,~^~ ^L^r+1^+u^r+1^~^L,~^L)× · · · ×

×η_m^α(span{^~^L¹^,...,~^L^m^,~^L})(lnη_m)^β(^L^~¹^+u¹^~^L,...,~^L^r^+u^r^L,~^~ ^L^r+1^,...,~^L^m^,~^L). (5.137) Now each u ∈ [−b₀, b₀] is contained in a closed interval [u−b⁻¹₁ (u), u+b⁻¹₁ (u)]. Because [−b₀, b₀] is compact in ^R by the Heine-Borel theorem one can find a finite set of points {U_i}i∈J1 (J₁ a finite index set) such that |U₁|< b₀ and 0< λ_i ≤b⁻¹₁ (U_i) such that

[

i∈J1

[U_i−λ_i, U_i+λ_i] = [−b₀, b₀]. (5.138)

5.7 · Weinberg’s Theorem

Next take i∈J₁ and the closed interval [−b₀(U_i), b₀(U_i) which can be covered again due to Heine-Borel with a finite set intervals [U_ij −λ_ij, U_ij +λ_ij] (j ∈ J₂,J₂ finite index set) with 0< λij ≤b⁻¹₂ (Ui, Uij). This construction we can continue and findm finite sets of points

{U_i₁}i1∈J1, {U_i₁_i₂}i1∈J1,i2∈J2, . . . ,{U_i₁_...i_m}i1∈J1,...,im∈Jm (5.139) and numbers

0< λ_i₁_i₂_...i_r ≤b⁻¹_r (i₁, . . . , i_r) withr≤m (5.140) such that

[

ir∈Jr

[U_i₁_...i_r −λ_i₁_...i_r, U_i₁_...i_r +λ_i₁_...i_r]⊆[−b₀(i₁, . . . , i_r−1), b₀(i₁, . . . , i_r−1)]. (5.141) Here we have used the abbreviation

b_l(i₁, . . . , i_r) =b_l(U_i₁, U_i₁_i₂, . . . , U_i₁_i₂_...i_r). (5.142) Now for given η1, . . . , ηm >1 we define intervals J_i^±₁_...i_r(η) with r ≤m which consist of all y which can be written as

y=Ui1η1· · ·ηm+Ui1i2η2· · ·ηm+· · ·+Ui1...imηm+zηr+1. . . ηm (5.143) where z is running over all values with

b0(i1, . . . , ir)≤ |z|=±z≤ηrλi1···ir. (5.144) For r= 0 we defineJ^±(η) to consist of ally with

±y=|y| ≥b₀η₁· · ·η_m. (5.145) Finally we also defineJ_i⁰₁_...i_m to consist of ally that can be written as

y=U_i₁η₁· · ·η_m+U_i₁_i₂η₂· · ·η_m+· · ·+U_i₁_...i_mη_m+z with|z|< b₀(i₁, . . . , i_m). (5.146) Now we show that any y ∈^R is contained in at least one of these intervals J. If y /∈J^±(η) then by (5.145) we know that|y| ≤b0η1· · ·ηm and thus because of (5.138) it exists ani1∈J1

such that

y∈[η₁· · ·η_m(U_i₁ −λ_i₁), η₁· · ·η_m(U_i₁+λ_i₁)]. (5.147) So we can write

y=η1· · ·ηmUi1 +y⁰ with|y⁰| ≤η1· · ·ηmλi1. (5.148) If ±y⁰ ≥ η₂· · ·η_mb₀(i₁) then y ∈ J_i^±₁(η) and we are finished. If on the other hand |y⁰| ≤ η₂· · ·η_mb₀(i₁) by the same line of arguments using our construction below (5.138) we find a y⁰⁰ with

y=η₁· · ·η_mU_i₁ +U_i₁_i₂η₂· · ·η_m+y⁰⁰

| {z }

y⁰

. (5.149)

It may happen that we have to continue this process until the final alternative, which then leads to y∈J_i⁰₁_···i_m(η). Thusy must indeed be in one of the sets J_i^±₁_...i_m(η), J^±(η) or J_i⁰₁_...i_m. Thus we have found an upper boundary for |f_~_L(P~)|given by

|f_L_~(P~)| ≤X

Xm r=0

i1...im

J_i^±_1...ir(η)

dy|f(P~ +~Ly|+ X

i1...im

Ji1···im(η)

dy|f(P~ +y~L)|. (5.150)

Ad (D): The final step is to determine the asymptotic behaviour of each term on the right hand side of (5.150) where P~ is given by

P~ =~L₁η₁· · ·η_m+~L₂η₂· · ·η_m+· · ·+L~_mη_m+C, ~~ C∈W (5.151) where W ⊆^Rⁿ is a compact region.

(i) Lety∈J_i^±₁_...i_r(η)

According to the definition of J_i^±₁_...i_r(η) we can combine (5.151) with (5.143) to P~ +~Ly= (L~₁+U_i₁L)η~ ₁· · ·η_m+ (~L₂+U_i₁_i₂L)η~ ₂· · ·η_m+· · ·+ +(~L_r+U_i₁_...i_rL)η~ _r· · ·η_m+z~Lη_r+1· · ·η_m+L~_r+1η_r+1· · ·η_m+· · ·+~L_mη_m+C~ =

= (L~1+Ui1~L)η1· · · ηr

|z||z|ηr+1· · ·ηm+ (~L2+Ui1i2L)η~ 2· · · ηr

|z||z|ηr+1· · ·ηm+· · ·+ +(~Lr+Ui1...irL)η~ r· · ·ηm± |z|~Lηr+1· · ·ηm+L~r+1ηr+1· · ·ηm+· · ·+L~mηm+C~

(5.152)

Now we define

α(i₁, . . . , i_l) =α(span{L~₁+U_i₁L, ~~ L₁+U_i₁_i₂~L, . . . , ~L_l+U_i₁_...i_l~L}), 1≤l≤m (5.153) and apply (5.137) together with the definition (5.143), (5.144) and (5.140) applied toη₀=|z| with which we find that for

η_l > b_l(i1, . . . , ir) forl6=r (5.154) the following boundary condition is valid:

|f(P~ +~Ly)| ≤M(i₁, . . . , i_r)η₁^α(i¹⁾(lnη₁)^β(i¹⁾· · ·η^α(i_r−1¹^,...,i^r−1⁾(lnη_r−1)^β(i¹^,...,i^r−1⁾×

×η^α(span{_r+1 ^L^~¹^,...,~^L^r+1^,~^L})(lnη_r+1)^β(^~^L¹^,...,~^L^r+1^,~^L)· · ·η^α(span{_m ^L^~¹^,···,~^L^m^,~^L})(lnη_m)^β(^~^L¹^,···,~^L^m^,~^L)×

× η_r

|z|

α(i1,...,ir) ln

η_r

|z|

β(i1,...,ir)

|z|^α(span{^~^L¹^,...,~^L^r^,~^L})(ln|z|)^β(^L^~¹^,...,~^L^r^,~^L). (5.155) Now introducing z according to (5.152) as the integration variable we find by using the boundary condition definingJ_i^±₁_...i_r(η) cf. (5.144) we find

J_i^±

1...ir(η)

dy|f(P~ +y~L)| ≤M(i₁, . . . , i_r)η^α(i₁ ¹⁾(lnη₁)^β(i¹⁾· · ·η^α(i_r−1¹^,...,i^r−1⁾(lnη_r−1)^β(i¹^,...,i^r−1⁾×

×η^α(span{_r+1 ^L^~¹^,...,~^L^r+1^,~^L})(lnη_r+1)^β(^L^~¹^,...,~^L^r+1^,~^L)· · ·η_m^α(span{^~^L¹^,···^,~^L^m^,~^L})(lnη_m)^β(^~^L¹^,···^,~^L^m^,~^L)×

×η_r+1· · ·η_m×

Z ηrλi1···ir

b0(i1,...,ir)

d|z||z|^α(span{^L^~¹^,...,~^L^r^,~^L})(ln|z|)^β(^L^~¹^,...,~^L^r^,~^L)× η_r

|z|

α(i1,...,ir) ln

η_r

|z|

β(i1,...,ir)

. (5.156) Because for our purposes an upper bound of the logarithmic coefficients β(L, . . . , ~~ L_l) is suf-ficient we assume without loss of generality that these are positive integers. Then we use the

5.7 · Weinberg’s Theorem

We give the elementary proof of this lemma at the end of the section. From (5.157) we have with (5.156) Now we look on the infinite intervalsJ^±(η). By definition (5.145) of these intervals we find

where|z| ≤b₀(i₁, . . . , i_m). Now the region

R⁰ ={C~⁰|C~⁰ =Lz~ +C,~ |z| ≤b₀(i₁, . . . , i_m), ~C ∈W} ^Rⁿ (5.164) is compact. Because f ∈ A_n by hypothesis there exist numbers M⁰(i₁, . . . , i_m) > 0 and b⁰_l(i₁, . . . , i_m)>1 such that

|f[(L~1+Ui1L)η~ 1. . . ηm+· · ·+ (L~m+Ui1...imL)η~ m+C~⁰]| ≤

≤M⁰(i₁, . . . , i_m)η₁^α(i¹⁾(lnη₁)^β(i¹⁾· · ·η_m^α(i¹^,...,i^m⁾(lnη_m)^β(i¹^,...,i^m⁾ (5.165) forC~⁰ ∈R⁰ and η_l> b⁰_l(η₁, . . . , η_m). Thus

Ji1...im(η)

dy|f(P~ +~Ly)| ≤2b0(i1, . . . , im)M⁰(i1, . . . , im)×

×η₁^α(i¹⁾(lnη1)^β(i¹⁾· · ·η^α(i_m ¹^,...,i^m⁾(lnηm)^β(i¹^,...,i^m⁾

(5.166)

ifη_l≥b_l(i1, . . . , im) (1≤l≤m).

Now our proof is finished by inspection of (5.158), (5.165) and (5.166) because of the result of the construction in part (C) cf. (5.150). From these estimates of upper bounds forf_L_~(P~) we read off that this function is indeed contained in the classA_n−1and the asymptotic coefficients are given by

α_L_~(span{L~₁, ~L₂, . . . , ~L_r}) = max

i1,...,ir

[α(i₁, . . . , i_r), α(span{~L₁, . . . , ~L_r, ~L) + 1] (5.167) where i₁, . . . ,i_r are running over the index sets defined in (5.153) and it remains to show that this is the same as given by (5.118) for our case of integrating over a one-dimensional subspace.

By definition (5.153) we can rewrite (5.167) by α_~_L(span{L~₁, ~L₂, . . . , ~L_r}) =

= max

u1,...,ur

[α(span{L~₁+u₁L, . . . , ~~ L_r+u_r~L}), α(span{L~₁, . . . ~L_r, ~L}) + 1]. (5.168) Here the u₁, . . . , u_r run over the sets U_i₁, . . . , U_i₁_...i_r defined above cf. (5.139) respectively.

This expression has to be compared with (5.118).

So let S = span{~L₁, . . . , ~L_r} and S⁰ ^Rⁿ such that Λ(span{L~}S⁰ = S. Then of course S⁰ could be either S⊕span{~L}or for each~v∈S there must exist a w~ ∈S⁰ and u∈^Rsuch that

~v=w~ +u~L. In other words in this there are u₁, . . . , u_r∈^R such that

S⁰= span{~L₁+u₁L,~ · · ·, ~L_r+u_rL~}. (5.169) In the first case we have dimS⁰ = dimS+ 1 in the second we have dimS⁰ = dimS since S⁰ in (5.118) runs over all subspaces disjoint with spanL~ and this shows immediately the equivalence of (5.168) with (5.118) for the case of integration over the one-dimensional subspace span{~L} and this finally finishes the proof. Q.E.D.

5.7 · Weinberg’s Theorem

Dans le document Introduction to Relativistic Quantum Field Theory (Page 162-169)