Renormalisation and scaling degree
6.2. SCALING DEGREE 92 proved 1 in [BF00, Lemma 5.1]
Lemma 6.2.2. Let u∈D0(Rn) and assume degu <∞. Then:
(1) Forα∈Nn, deg(∂αu)≤degu+|α|.
(2) Forα∈Nn, deg(xαu)≤degu− |α|.
(3) Let f ∈ E(Rn) and assume f(α)(0) = 0 for |α| ≤ k−1 , then deg(f u) ≤ degu−k.
(4) Letv ∈D0(Rk) then sd(u⊗v)≤sdu+ sdv.
6.2.1. Extension of distributions. Let us recall the basic ingredients of the construction of extensions of distributions. Essentially, we follow [BF00], but for later purposes, we make systematic use of the following spaces of distributions. Denote by D0({0}) the space of distributions supported at {0}. For r≥0, let D0({0})≤r be the subspace of D0({0}) given by those of maximal degree r,
D0({0})≤r = span {v ∈D0({0}) : degv ≤r}= span
δ(α) ∈D0(Rn) : |α| ≤r . On the other hand, consider the space of all test functions vanishing up to order rat x= 0:
(6.2.1) Dr(Rn)··={ϕ∈D(Rn) : (∂xαϕ)(0) = 0 ∀α∈Nn0,|α| ≤r}.
It will be convenient to generalize this definition in the following way. Let K be a finite dimensional subspace of D0({0}). Set
DK(Rn)··={ϕ∈D(Rn) : hv, ϕi= 0 ∀v ∈ K}.
Clearly,Dr(Rn) equalsDK(Rn) withK =D0({0})≤r. Observe that the scaling degree of a distribution in DK0 (Rn) can be defined in an analogous way to the scaling degree inD0(Rn). We now restate Theorem 5.2 from [BF00] as follows:
Proposition 6.2.3. Let u ∈ D0( ˙Rn) have degree of divergence r ··= degu < ∞.
Then it admits a unique extension ˜u∈Dr0(Rn) with the same degree of divergence r, given by
(6.2.2) h˜u, ϕi ··= lim
ρ→∞hu,(1−ϑρ)ϕi, ϕ∈Dr(Rn)
where ϑρ(x)··=ϑ(2ρx) and ϑ is an arbitrary function in D(Rn) such that ϑ = 1 in a neighbourhood of the origin.
It now remains to find elements of D0(Rn) which correspond to the extension
˜
u ∈ Dr0(Rn). Following the ideas of [BF00], we do so by considering projections2 Wt : D(Rn) → Dr(Rn), and applying their transpose W : Dr0(Rn) → D0(Rn) to
˜
u∈Dr0(Rn).
To this end, let us first state two lemmas which are slight generalizations of results found in [DF04] where they were stated for K =D0({0})≤r.
1Strictly speaking, the third claim is considered there only for k = 0, but the case k ≥1 follows immediately by noting that under the assumptions, such f equals xk1. . . xkng for some g ∈ E(Rn) and then using 2.
2This means that Wt:D(Rn)7→Dr(Rn) is continuous and (Wt)2=Wt.
6.2. SCALING DEGREE 93 Lemma 6.2.4. Let K be a finite dimensional subspace of D0({0}), let {vi}i∈I be a basis of K, and assume {ψi}i∈I is a family ψi ∈D(Rn) s.t. hvi, ψji=δij. Then
(6.2.3) Wtϕ=ϕ−X
i∈I
hvi, ϕiψi
defines a projection Wt:D(Rn)→DK(Rn). Conversely, ifWt :D(Rn)→DK(Rn) is a projection, there is a family {ψi}i∈I with the above properties.
Lemma 6.2.5. Letu∈DK0 (Rn) and letWt:D(Rn)→DK(Rn) be a projection. Then hW u, ϕi=hu, ϕi for all ϕ∈DK(Rn) and degW u= degu.
Taking into account that degδ(α)=|α|, we find the following important result on the existence of extensions with the same degree of divergence.
Corollary 6.2.6. ([BF00]) Let u∈D0( ˙Rn) be a distribution with r··= degu <∞.
Then there is an extension ˙u∈D0(Rn) ofu with deg ˙u= degu. Each such extension can be written as ˙u = Wu, where ˜˜ u is the unique extension of u in Dr0(Rn) and Wt : D(Rn)→ Dr(Rn) is a projection. Moreover, two arbitrary extensions with the above properties differ by an element of D0({0})≤r.
While each extension ofu with the same degree of divergence can be constructed as above by using a projectionWt, it can sometimes be more convenient to use some other operatorVtwhich mapsD(Rn) toDr(Rn) and check whetherVu˜is an extension of u with the correct degree of divergence. In fact, this approach will prove to be more convenient in directly constructing on-shell extensions (cf. Proposition 7.2.7).
95
CHAPTER 7
On-shell extension of distributions
As we outlined in the previous chapter, the renormalisation problem in the Epstein and Glaser approach is reformulated as a problem of extending distributions defined on Rn\ {0} =·· R˙n to distributions on Rn. By construction, any two extensions can differ by a distribution supported in 0, and one way to constrain this ambiguity is to require that the extension should have the same scaling degree as the original distribution. In QFT, the ambiguity is further constrained by imposing physically motivated renormalisation conditions. Such conditions include the requirement that if u ∈ D0( ˙Rn) respects a global symmetry, e.g. it is invariant under the Lorentz group or a global gauge group, then the same should hold for its extension to Rn. Another such condition, which turned out to be essential in renormalisation on both flat and on curved space-time [HW02], is the requirement that if u is homogeneous, its extension should be homogeneous as well, or at least it should behave as much like a homogeneous distribution as possible (a property which can be properly formulated in terms of almost homogeneous distributions).
A problem which at first sight seems to be unrelated to such renormalisation conditions occurs in the construction of on-shell time ordered products involving higher derivatives of quantum fields. Roughly speaking, one wishes to extend toRnan expression inD0( ˙Rn) involving derivatives of the Feynman propagator and Heaviside theta functions such that the extension satisfies the (free) equation of motion. The possibility of finding such an extension can be rephrased using the relation between on-shell time-ordered products (ordinarily used in quantum field theory) and off-shell products, the latter of which have proved to be better suited for a theoretical study of Epstein-Glaser renormalisation [DF03, DF04].
The main idea in the approach proposed in [BW12] is that all these problems can be formulated and solved in a unified framework, by restating them in terms of the existence of extensions which solve a set of (differential) equations. More precisely, we state the following extension problem:
Problem. Let {Qi}ki=1 be a family of differential operators on Rn with smooth coef-ficients, and letu be a distribution in D0( ˙Rn) that satisfies
Qiu= 0 on ˙Rn (i= 1, . . . , k).
Find ˙u ∈ D0(Rn) such that ˙u = u on ˙Rn and Qiu˙ = 0 on Rn (i = 1, . . . , k). If such extensions ˙uexist, we call them on-shell extensions (w.r.t. {Qi}ki=1).
Indeed, invariance of a distribution under the action of a connected Lie group is equivalent to it being a solution of its infinitesimal generators. (Almost) homogeneity
96 is described using (powers of) the operator P
ixi∂i−a. In the construction of on-shell time-ordered products, the differential operator of interest is the Klein-Gordon operator (2+m2). To include discrete symmetries, we will consider a more general class of operators later.
On the mathematical side, the ‘on-shell extension’ problem we consider is closely related to the so-called Bochner’s extension problem, an issue which we explain in the text.
One advantage of our reformulation is the following. The various constructions and prescriptions to implement renormalisation conditions proposed so far in e.g.
[Sch95, Pra99, Gra03, LG03, DF04, HW02], see also [DG12], are each limited to a particular type of symmetry. Therefore, the simultaneous implementation of a number of different conditions requires cumbersome proofs of compatibility. Our framework on the other hand, allows for a compact formulation of e.g. sufficient conditions on the existence of extensions subject to different renormalisation conditions, such as Lorentz invariance considered together with almost homogeneity and (eventually) parity. Moreover, it exhibits a new feature of Epstein-Glaser renormalisation: a renormalisation condition corresponding to a (differential) operatorQcan be imposed by applying a linear map to a generic extension (which is a solution of Q on ˙Rn).
This statement extends to the case of several renormalisation conditions.
Concerning the relation between on-shell and off-shell time-ordered products, we would like to point out that it was given in [DF03, DF04] in terms of a recurrence relation for which an explicit solution was found in [BD08]. In our framework we find a more compact formula, which contrary to that given in [BD08] does not contain unnatural combinatorial factors. Instead, the coefficients which appear in our formula are simply eigenvalues of certain finite-dimensional operators directly related to the Klein-Gordon operator.
This chapter is organized as follows. Section 7.1 contains the main ideas and results. First, in subsection 7.1.1, we introduce the notion of operators of essential order m on D0(Rn). Such operators generalize differential operators and will enable us to treat discrete symmetries. In subsection 7.1.2 we equip the finite dimensional spaces of distributions of given maximal order supported at the origin with a scalar product. The restrictions of operators Q : D0(Rn) → D0(Rn) to these spaces play an important role in subsection 3.2, especially for Theorem 7.2.2. This theorem provides a solution to the extension problem with respect to an operatorQof arbitrary essential order in the sense that it lists different statements which are equivalent to the existence of on-shell extensions, and provides a candidate for such an extension. This candidate can be calculated explicitly and only requires one to find the eigenvalues of a finite dimensional matrix. Special cases, examples and generalizations are then discussed. Of particular interest is Theorem 7.2.8 which states sufficient, and easy-to-check conditions that ensure the existence of on-shell extensions w.r.t. an operator of essential order 0. We then explain how the extension problem with respect to a finite number of operators is solved. Section 8.1 is devoted to the construction of
7.1. OPERATORS OF FINITE ESSENTIAL ORDER 97 on-shell time-ordered products involving higher derivatives of the fields. We clarify how the relation between the on-shell and the off-shell formalism can be formulated and understood in our framework.
7.1. Operators of finite essential order
7.1.1. Essential order. Our aim is to implement symmetries, i.e. we will ask our extensions to satisfy a set of given equations. In order to include discrete sym-metries, we consider more general operators fromD0(Rn) to D0(Rn) rather than just differential operators.
Definition 7.1.1. We say that Q : D0(Rn) → D0(Rn) is an operator of essential order q if
(1) Q is the transpose of a linear operator Qt :E(Rn)→ E(Rn) which continu-ously mapsD( ˙Rn) and D(Rn) to themselves;
(2) q∈N0 is the lowest number such that degQu≤degu+q for allu∈D0(Rn).
Basic examples for such operators are of course differential operators, for which the essential degree was already considered in [Nik07]: a differential operator of order m has essential order smaller or equal m. More precisely, a differential operator Q = P
|α|≤maα(x)∂α has essential order q, where q is the smallest possible non-negative number s.t. (∂βaα)(0) = 0 for |β| ≤ |α| − q −1. In particular, Q has essential order 0 if (∂βaα)(0) = 0 for |β| ≤ |α| −1.
Let us list some basic properties of operators of essential order q.
Lemma 7.1.2. Let Q:D0(Rn)→D0(Rn) be an operator of essential order q. Then (1) Qis continuous in the D0(Rn) topology;
(2) Qmaps D0( ˙Rn) and D0({0}) to themselves.
(3) LetK1 be a linear subspace ofD0({0}). Then Qt mapsDK1(Rn) toDK2(Rn), where
K2 ={v ∈D0({0})| Qv ∈K1}.
In particular,Qt maps D(Rn) to DK(Rn), where K = Ker(Q|D0({0})).
Proof.
(1) By definition of the weak topology.
(2) The first assertion is obvious. For the second one it suffices to notice that for any v ∈ D0({0}) the expression hQv, ϕi only depends on the restriction of ϕto an arbitrary small neighbourhood of 0.
(3) Let ϕ ∈ DK1(Rn), then Qtϕ ∈ DK2(Rn), since for any v ∈ K2, i.e. v ∈ D0({0}) such that Qv ∈K1, we have hv, Qtϕi=hQv, ϕi= 0.
The following two lemmas give examples of operators of essential degree 0, for which we usually reserve the symbol R.
Lemma 7.1.3. Let R be an infinitesimal generator of a Lie group G acting on Rn such that 0 is a fixed point. Then R is an operator of essential degree 0.
7.1. OPERATORS OF FINITE ESSENTIAL ORDER 98
These two cases are of particular importance in our applications. To see this, first recall that a distribution u ∈D0(Rn) is invariant under the induced action of a connected Lie group G acting on Rn if and only if Riu = 0 for all the infinitesimal generators Ri of G. Now, if 0 is a fixed point of the action of G on Rn, then by the first of the above lemmas, the infinitesimal generators are of essential order 0.
Similarly, discrete symmetries entail operators of essential degree 0, as they are of the form discussed in the second lemma. For instance, even distributions are in the kernel of the operatorR+u··=u−u(− ·), and odd ones in that of R−u··=u+u(− ·).
7.1.2. Spaces of distributions supported at the origin. Recall that D0({0})≤r denotes the finite dimensional vector space spanned by derivatives of the delta distribution up to order r. It it will turn out to be very useful to equip it with a scalar product. To this end, for r≥0 define the maps
Sr :D0({0})≤r →E(Rn), Srv ··= X polynomials of degree ≤r. Now set
(v|w)r··=h¯v,Srwi=P
|α|≤r 1
α!h¯v, xαihw, xαi=hw,Srvi,¯ v, w∈D0({0})≤r. where the bar denotes ordinary complex conjugation. Writing elements v, w of D0({0})≤r as linear combinations v = P D0({0})≤r+q. Let us stress that this definition depends on the essential order of Q. The next lemma characterizes the adjoint ofQ|r.
7.2. ON-SHELL EXTENSION — SINGLE OPERATOR CASE 99 Lemma 7.1.5. Let Q be an operator of essential order q. Then the adjoint of Q|r : D0({0})≤r →D0({0})≤r+q is
(7.1.1) (Q|r)∗ =TrQtSr+q :D0({0})≤r+q →D0({0})≤r. Proof. For all v ∈D0({0})≤r+q and w∈D0({0})≤r one has
(v|Qw)r+q =h¯v,Sr+qQwi=hQw,Sr+q¯vi=hw, QtSr+qv¯i
=hTrSrw, QtSr+qvi¯ =hSrw,TrQtSr+qvi¯ = (TrQtSrv|w)r,
so (7.1.1) follows.
We will need for certain results an assumption which guarantees that (Q|r)∗ and (Q|r0)∗ are in a sense compatible for r 6=r0.
Definition 7.1.6. We say that an operator Q of essential order q is essentially ho-mogeneous if Qt maps polynomials of order≤r+q to elements of
(7.1.2) {f ∈C∞(Rn) : f(α)(0) = 0, |α|> r}.
Lemma 7.1.7. Let Qbe essentially homogeneous. Then for all r∈N0 and s ≥r the restriction of the operator (Q|s)∗ to D0({0})≤r+q equals (Q|r)∗.
Proof. We use Lemma 7.1.5 and observe thatSs+qcoincides withSr+qonD0({0})≤r+q
and Ts coincides with Tr on the space (7.1.2). Therefore, TsQtSs+q restricted to
D0({0})≤r+q equals TrQtSr+q.
A sufficient condition for an operatorQ to be essentially homogeneous is thatQt maps polynomials of degree r+q to polynomials of degreer.
Example 2. The operators ∂x, x∂x, x∂x + cosx are essentially homogeneous, but
∂x+m is not unless m= 0.
7.2. On-shell extension — single operator case
Let us specify the problem we already outlined in the introduction, first for the case when only one operator is considered.
Problem. Let Q : D0(Rn) → D0(Rn) be an operator of essential order q. Let u∈D0( ˙Rn) have degree of divergence r··= degu <∞ and assume
Qu= 0 on ˙Rn.
Find ¨u ∈ D0(Rn) such that ¨u = u on ˙Rn and Q¨u = 0 on Rn. If such extension(s) exist, we call themon-shell extensions w.r.t. Q.
The next lemma is an easy observation, which will however play a key role in our approach to the problem of finding on-shell extensions.
Lemma 7.2.1. Let Q : D0(Rn) → D0(Rn) be an operator of essential order q. Let u∈D0( ˙Rn) have r··= degu <∞ and assume
Qu= 0 on ˙Rn.
7.2. ON-SHELL EXTENSION — SINGLE OPERATOR CASE 100