Perturbative Algebraic Quantum Field Theory

Abstract

These notes are based on the course given by Klaus Fredenhagen at the Les Houches Winter School in Mathematical Physics (January 29–February 3, 2012) and the course QFT for mathematicians given by Katarzyna Rejzner in Hamburg for the Research Training Group 1670 (February 6–11, 2012). Both courses were meant as an introduction to modern approach to perturbative quantum field theory and are aimed both at mathematicians and physicists.

Appendix—Distributions and Wavefront Sets

We recall same basic notions from the theory of distributions on $\mathbb {R}^n$. Let $\varOmega \subset \mathbb {R}^n$ be an open subset and $\mathcal {E}(\varOmega )\doteq \mathcal {C}^\infty (\varOmega ,\mathbb {R})$ the space of smooth functions on it. We equip this space with a Fréchet topology generated by the family of seminorms:

$$\begin{aligned} p_{K,m}(\varphi )=\sup _{\begin{array}{c} x\in K \\ |\alpha |\le m \end{array}}|\partial ^\alpha \varphi (x)|\,, \end{aligned}$$

(98)

where $\alpha \in \mathbb {N}^N$ is a multiindex and $K\subset \varOmega$ is a compact set. This is just the topology of uniform convergence on compact sets, of all the derivatives.

The space of smooth compactly supported functions $\mathcal {D}(\varOmega )\doteq \mathcal {C}^\infty _c(\varOmega ,\mathbb {R})$ can be equipped with a locally convex topology in a similar way. The fundamental system of seminorms is given by [35]:

$$\begin{aligned} p_{\{m\},\{\epsilon \},a}(\varphi )=\sup _\nu \big (\sup _{\begin{array}{c} |x|\ge \nu , \\ |p|\le m_\nu \end{array}} \big |D^p\varphi ^a(x)\big |/\epsilon _\nu \big )\,, \end{aligned}$$

(99)

where $\{m\}$ is an increasing sequence of positive numbers going to $+\infty$ and $\{\epsilon \}$ is a decreasing one tending to $0$.

The space of distributions is defined to be the dual $\mathcal {D}'(\varOmega )$ of $\mathcal {D}(\varOmega )$ with respect to the topology given by (99). Equivalently, given a linear map $L$ on $\mathcal {D}(\varOmega )$ we can decide if it is a distribution by checking one of the equivalent conditions given in the theorem below [25, 33, 41].

Theorem 3

A linear map $u$ on $\mathcal {E}(\varOmega )$ is a distribution if it satisfies the following equivalent conditions:

• 1. To every compact subset $K$ of $\varOmega$ there exists an integer $m$ and a constant $C>0$ such that for all $\varphi \in \mathcal {D}$ with support contained in $K$ it holds:

  $$ |u(\varphi )|\le C\max _{p\le k}\sup _{x\in \varOmega }|\partial ^p\varphi (x)|\,. $$

  We call $||u ||_{\mathcal {C}^k(\varOmega )}\doteq \max _{p\le k}\sup _{x\in \varOmega }|\partial ^p\varphi (x)|$ the $\mathcal {C}^k$-norm and if the same integer $k$ can be used in all $K$ for a given distribution $u$, then we say that $u$ is of order $k$.

• 2. If a sequence of test functions $\{\varphi _k\}$, as well as all their derivatives converge uniformly to 0 and if all the test functions $\varphi _k$ have their supports contained in a compact subset $K\subset \varOmega$ independent of the index $k$, then $u(\varphi _k)\rightarrow 0$.

An important property of a distribution is its support. If $U' \subset U$ is an open subset then $\mathcal {D}(U')$ is a closed subspace of $\mathcal {D}(U)$ and there is a natural restriction map $\mathcal {D}'(U) \rightarrow \mathcal {D}'(U')$. We denote the restriction of a distribution $u$ to an open subset $U'$ by $u|_{U'}$.

Definition 12

The support $\mathrm {supp}u$ of a distribution $u \in \mathcal {D}'(\varOmega )$ is the smallest closed set $\mathcal {O}$ such that $u|_{\varOmega \setminus \mathcal {O}} = 0$. In other words:

$$ \mathrm {supp}u\doteq \{x\in \varOmega |\, \forall U\,{\mathrm {open neigh. of }}\,x,\, U\subset \varOmega \, \exists \varphi \in \mathcal {D}(\varOmega ), \mathrm {supp}\varphi \subset U,\,{\mathrm {s.t. }}< u,\varphi >\ne 0\}\,. $$

Distributions with compact support can be characterized by means of a following theorem:

Theorem 4

The set of distributions in $\varOmega$ with compact support is identical with the dual $\mathcal {E}'(\varOmega )$ of $\mathcal {E}(\varOmega )$ with respect to the topology given by (98).

Now we discuss the singularity structure of distributions. This is mainly based on [25] and Chap. 4 of [1].

Definition 13

The singular support ${\mathrm {sing\, supp}}\, u$ of $u \in \mathcal {D}'(\varOmega )$ is the smallest closed subset $\mathcal {O}$ such that $u|_{\varOmega \setminus \mathcal {O}} \in \mathcal {E}(\varOmega \setminus$ $\mathcal {O})$.

We recall an important theorem giving the criterion for a compactly distribution to have an empty singular support:

Theorem 5

A distribution $u \in \mathcal {E}'(\varOmega )$ is smooth if and only if for every $N$ there is a constant $C_N$ such that:

$$ |\hat{u}({k} )| \le C_N (1 + |{k} |)^{-N}, $$

where $\hat{u}$ denotes the Fourier transform of $u$.

We can see that a distribution is smooth if its Fourier transform decays fast at infinity. If a distribution has a nonempty singular support we can give a further characterization of its singularity structure by specifying the direction in which it is singular. This is exactly the purpose of the definition of a wave front set.

Definition 14

For a distribution $u \in \mathcal {D}'(\varOmega )$ the wavefront set $\mathrm {WF}(u)$ is the complement in $\varOmega \times \mathbb {R}^n\setminus \{0\}$ of the set of points $(x,{k}) \in \varOmega \times \mathbb {R}^n\setminus \{0\}$ such that there exist

• a function $f \in \mathcal {D}(\varOmega )$ with $f(x)=1$,

• an open conic neighborhood $C$ of ${k}$, with

  $$ \sup _{{k}\in C}(1+|{k}|)^N|\widehat{f \cdot u}({k})|<\infty \qquad \forall N \in \mathbb {N}_0\,. $$

On a manifold $M$ the definition of the Fourier transform depends on the choice of a chart, but the property of strong decay in some direction (characterized now by a point $(x, k)$, $k\ne 0$ of the cotangent bundle $T^*M$) turns out to be independent of this choice. Therefore the wave front set (WF) of a distribution on a manifold $M$ is a well defined closed conical subset of the cotangent bundle (with the zero section removed).

The wave front sets provide a simple criterion for the existence of point-wise products of distributions. Before we give it, we prove a more general result concerning the pullback. Here we follow closely [1, 25]. Let $F:X\rightarrow Y$ be a smooth map between $X\subset \mathbb {R}^m$ and $Y\subset \mathbb {R}^n$. We define the normal set $N_F$ of the map $F$ as:

$$ N_F\doteq \{(F(x),\eta )\in Y\times \mathbb {R}^n| (dF_x)^T(\eta )=0\}\,, $$

where $(dF_x)^T$ is the transposition of the differential of $F$ at x.

Theorem 6

Let $\Gamma$ be a closed cone in $Y\times (\mathbb {R}^n\{0\})$ and $F:X\rightarrow Y$ as above, such that $N_F\cap \Gamma =\varnothing$. Then the pullback of functions $F^*:\mathcal {E}(X)\rightarrow \mathcal {E}(Y)$ has a unique, sequentially continuous extension to a sequentially continuous map $\mathcal {D}'_\Gamma (Y)\rightarrow \mathcal {D}'(X)$, where $\mathcal {D}'_\Gamma (Y)$ denotes the space of distributions with WF sets contained in $\Gamma$.

Proof

Here we give only an idea of the proof. Details can be found in [1, 25]. Firstly, one has to show that the problem can be reduced to a local construction. Let $x\in X$. We assumed that $N_F\cap \Gamma =\varnothing$, so we can choose a compact neighborhood $K$ of $F(x)$ and an open neighborhood $\mathcal {O}$ of $x$ such that $\overline{F(\mathcal {O})} \subset \mathrm int (K)$ and the following condition holds:

$$ \exists \epsilon >0\ \mathrm s.t. \ V\doteq \overline{\bigcup \limits _{x\in \mathcal {O}}\{{k}|(dF_x)^T{k}\}}\,{\mathrm {satisfies}}\, (K\times V)\cap \Gamma =\varnothing \,. $$

Such neighborhoods define a cover of $X$ and we choose its locally finite refinement which we denote by $\{\mathcal {O}_\alpha \}_{\alpha \in A}$, where $A$ is some index set. To this cover we have the associated family of compact sets $K_\lambda \subset Y$ and we choose a partition of unity $\sum \limits _{\alpha \in A}g_\alpha =1$, $\mathrm {supp}g_\alpha \subset \mathcal {O}_\alpha$ and a family $\{f_\alpha \}_{\alpha \in A}$ of functions on $Y$ with $\mathrm {supp}$ $f_\alpha =K_\alpha$ and $f_\alpha \equiv 1$ on $F(\mathrm {supp}g_\alpha )$. Then:

$$ F^*(\varphi ) = \sum \limits _{\alpha \in A} g_\alpha F^*(f_{\alpha }\varphi )\,. $$

This way the problem reduces to finding an extension of $F^*_\alpha \doteq (F\big |_{\mathcal {O}_\alpha })^* : \mathcal {C}^\infty _c(K_\alpha ,\mathbb {R}) \rightarrow \mathcal {C}^\infty (\mathcal {O}_\alpha ,\mathbb {R})$ to a map on $\mathcal {D}'_\Gamma (K_\alpha )$. Note that for $\varphi \in \mathcal {C}_c^\infty (K_\alpha )$, $\mathrm {supp}\chi \subset \mathcal {O}_\alpha$, we can write the pullback as:

$$ \langle F_\alpha ^*(\varphi ),\chi \rangle =\int \varphi (F_\alpha (x))\chi (x) dx=\int \hat{\varphi }(\eta )e^{i\langle F_\alpha (x),\eta \rangle }\chi (x) dxd\eta =\int \hat{\varphi }(\eta )T_\chi (\eta ) d\eta \,, $$

where we denoted $T_\chi (\eta )\doteq \int e^{i\langle F_\alpha (x),\eta \rangle }\chi (x)dx$. We can use this expression to define the pullback for $u\in \mathcal {D}'_\Gamma (K_\alpha )$, by setting:

$$ \langle F_\alpha ^*(u),\chi \rangle \doteq \int \hat{u}(\eta )T_\chi (\eta ) d\eta \,. $$

To show that this integral converges, we can divide it into two parts: integration over $V_\alpha$ and over $\mathbb {R}^n\setminus V_\alpha$, i.e.:

$$ \langle F_\alpha ^*(u),\chi \rangle =\int \limits _{V_\alpha }\hat{u}(\eta )T_\chi (\eta ) d\eta \,+\int \limits _{\mathbb {R}^n\setminus V_\alpha }\hat{u}(\eta )T_\chi (\eta ) d\eta \,. $$

The first integral converges since $K_\alpha \times V_\alpha \cap \Gamma =\varnothing$ and therefore $\hat{u}(\eta )$ decays rapidly on $V_\alpha$, whereas $|T_\chi (\eta )|\le \int |\chi (x)|dx$. The second integral also converges. To prove it, first we note that $\hat{u}(\eta )$ is polynomially bounded i.e. $\hat{\varphi }(\eta )\le C(1+|\eta |)^N$ for some $N$ and appropriately chosen constant $C$. Secondly, we have a following estimate on $T_\chi (\eta )$: for ever $k\in \mathbb {N}$ and a closed conic subset $V\subset \mathbb {R}^n$ such that $(dF_x)^T\eta \ne 0$ for $\eta \in V$, there exists a constant $C_{k,V}$ for which it holds⁸

$$ |T_\chi (\eta )| \le C_{k,V} (1 + |\eta |)^{-k}\,, $$

Since for $\eta \in V_\alpha$ it holds $(dF_x)^T\eta >\epsilon >0$, we can use this estimate to prove the convergence of the second integral.

We already proved that $F^*:\mathcal {D}'_\Gamma (Y)\rightarrow \mathcal {D}'(X)$ exists. Now it remains to show its sequential continuity. This can be easily done, with the use of estimates provided above and the uniform boundedness principle.

Using this theorem we can define the pointwise product of two distributions $t,s$ on an $n$-dimensional manifold $M$ as a pullback by the diagonal map $D:M\rightarrow M\times M$ if the pointwise sum of their wave front sets

$$ \mathrm {WF}(t) + \mathrm {WF}(s) = \{(x, k + k')|(x, k) \in \mathrm {WF}(t), (x, k') \in \mathrm {WF}(s)\}\,, $$

does not intersect the zero section of $\dot{T}^*M$. This is the theorem 8.2.10 of [25]. To see that this is the right criterion, note that the set of normals of the diagonal map $D: x\mapsto (x,x)$ is given by $N_D=\{(x,x,k,-k)|x\in M, k\in T^*M\}$. The product $ts$ is defined by: $ts=D^*(t\otimes s)$ and if one of $t,s$ is compactly supported, then so is $ts$ and we define the contraction by $\langle t,s\rangle \doteq \widehat{ts}(0)$.