From a proof that 2D Wightman CFT leads to a vertex algebra **[1]**:


Let
$$
Y(a,z):=\frac{1}{(1+z)^{2\Delta\_a}}\Phi\_a\left(i\frac{1-z}{1+z}\right),\quad\text{with}\quad |z|<1.
$$


Here $\Delta\_a\ge 0$ is conformal weight of $\Phi\_a$, and $\Phi\_a$ is the scalar field satisfying the usual Wightman axioms (*Poincare covariance*, *stable vacuum* and *positive spectrum* of momentum, *completeness*, i.e. that polynomials of smeared fields applied on vacuum are dense in the Hilbert space, and *locality* (p. 6 of **[1]**)) and in addition, satisfying conformal covariance
$$
U(q,\Lambda,b)\Phi\_a(x)U(q,\Lambda,b)^{-1}=(1+2x\cdot b+|x|^2 |b|^2)^{-\Delta\_a}\Phi\_a((q,\Lambda,b)\cdot x)
$$
with $(q,\Lambda,b)\mapsto U(q,\Lambda,b)$ being a unitary representation of the *conformal group*, fixing the vacuum vector $|0\rangle$.


On p. 11 of **[1]** the author writes: 



> 
> We expand a chiral field $Y(a,z)$ in a Fourier series:
>  $$
>  Y(a,z)=\sum\_n a\_{(n)}z^{-n-1},\quad\quad(\star)
> $$
>  where $a\_{(n)}\in\mathrm{End}\; \mathcal{D}$. 
> 
> 
> 


Here $\mathcal{D}$ is the span of polynomials in the fields applied on the vacuum and is dense by *completeness* assumption.


I am not certain if this expansion is justified. Especially because it allows the author to prove that we indeed get a vertex algebra and hence we have an operator product expansion of the fields. Moreover, I found in another book **[2]** a similar proof on p. 190 and there the author writes



> 
> The operator product expansion (Axiom 6 on p. 168) of the primary fields allows to understand the fields $\Phi\_a$ as fields
>  $$
> \Phi\_a(z)=\sum a\_{(n)} z^{-n-1} \in \mathrm{End} D [[z,z^{-1}]]
> $$
>  in the sense of vertex algebras.
> 
> 
> 


Here $D$ is some dense domain in Hilbert space which is left invariant by fields, i.e. in **[2]** *completeness* is not assumed.


In other words, the author of **[2]** needs the existance of operator product expansion to write such a Fourier expansion.


My question:


$$ \text{is }(\star)\text{ true?} $$




---


References:


[1] V. Kac. *Vertex Algebras for Beginners*, 1998.


[2] M. Schottenloher. *A Mathematical Introduction to Conformal Field Theory*, 2008.



Fourier series of a Wightman field
Using Marcel Bischoff's comment:
$$
Y(a,z)=\sum\_n a\_{(n)} z^{-n-1} \iff a\_{(m)}=\frac{1}{2\pi i} \oint\limits\_{|z|=1} Y(a,z)z^{m} \,\mathrm{d}z.
$$


Since the author of **[1]** starts with the Schwartz space $\mathscr{S}(\mathbb{R})$, we need to show that $Y(a,z)$ is analytic in $|z|<1$.


By definition,
$$
 Y(a,z)=\frac{1}{(1+z)^{2\Delta\_a}}\Phi\left(i\frac{1-z}{1+z}\right)\quad\text{with}\quad |z|<1.
$$
Clearly,
$$
\frac{1}{(1+z)^{2\Delta\_a}}
$$
is analytic in the domain $|z|<1$ since it is holomorphic. Letting
$$
t:=i\frac{1-z}{1+z}
$$
we note that $|z|<1$ gets mapped to $\mathrm{Im}\, t>0$, so we need to show that $\Phi\_a(t)$ can be analytically continued to $\mathrm{Im}\, t>0$. We now reintroduce the second coordinate, since the *joint* spectrum of momentum operators is positive. From **[1]**
$$
t:=x^0-x^1,\quad \bar{t}:=x^0+x^1\quad\text{and}\quad x:=(x^0,x^1)
$$
with $t,\bar{t}$ the lightcone coordinates.
Since the polynomials in fields applied to the vacuum are dense by *completeness* assumption, it suffices to prove that any
$$
\Phi\_{a\_1}(x\_1)\ldots\Phi\_{a\_n}(x\_n)|0\rangle
$$
can be analytically continued to the domain
$$
\{\mathrm{Im}\, x^0\_1>0\}\times\dots\times\{\mathrm{Im}\, x^0\_n>0\}\text{ such that } \mathrm{Im}\,x^0\_i\ne \mathrm{Im}\,x^0\_j \text{ for } i\ne j.
$$
The fact that coinciding points do not matter follows from locality **[1]**
$$
(z-w)^N[Y(a,z),Y(b,w)]=0 \text{ for } N\gg0.
$$
We now follow Section 2 of **[3]**. Let
$$
 \Psi(x\_1,\ldots,x\_n):=\Phi(x\_1)\ldots\Phi(x\_n)|0\rangle
$$
with $\Phi$ being a Hermitian scalar field. Generalizations to fields of arbitrary spin are given in Section 9 of **[3]** (in 4 dimensions, here 2D). Note that $\Psi$ (smeared with a test function) belongs to the Hilbert space $\mathcal{H}$ by *completeness*. By *positive spectrum*
$$
\Psi(x\_1,\ldots,x\_n)=\int d^2 p\; d^2 q\_1\ldots d^2 q\_{n-1} \tilde{\Psi}(p,q\_1,\ldots,q\_{n-1})e^{i\left(p x\_1+\sum q\_j (x\_{j+1} -x\_j)\right)}.
$$
Here $\tilde{\Psi}$ is non-zero only if $p^0\ge0$ and all $q\_i^0\ge 0$. Thus, $\Psi$ can be analytically continued to a vector-valued analytic function $\Psi\in\mathcal{H}$, i.e. we have
\begin{align}
\Psi(z\_1,\dots,z\_n)\quad\text{of}\quad z\_k=x\_k+i y\_k\quad & \text{defined and holomorphic for}\\
 &y\_1\in V\_+ \text{ and } y\_j-y\_i\in V\_+ \text{ if } j>i\\
 &(V\_+ \text{ is the open forward lightcone}).
\end{align}
In **[4]** it was shown that locality and the edge of the wedge theorem allows to extend the domain of definition and analyticity even further:


*Claim*. $\Psi(z\_1,\ldots,z\_n)$ belongs to $\mathcal{H}$ and is analytic in the $z\_k$ in a connected domain which includes the Euclidean points with $z\_k=(i y^0\_k, x\_k)$ such that $y^0\_k>0$ for all $k$, and $z\_i^0\ne z\_j^0$ if $i\ne j$.


*Proof of Claim*.
 Fix $\pi$ to be any permutation of $(1,\ldots,n)$ and
$$
 \Psi^{\pi} (z\_1,\ldots,z\_n):=\Psi(z\_{\pi(1)},\ldots,z\_{\pi(n)}),\quad z\_k=x\_k+i y\_k.
$$
 By the above, $\Psi^{\pi}$ is well-defined and holomorphic in a domain containing the Euclidean points with $0<y^0\_{\pi(1)}<\ldots<y^0\_{\pi(n)}$. Furthermore, by *locality*
$$
 \Psi^{\pi}(x\_1,\ldots,x\_n)=\Psi(x\_1,\ldots,x\_n)\quad\text{if}\quad x\_k\in\mathbb{R}:(x\_i-x\_j)^2<0 \quad\forall i\ne j,
$$
 i.e. all $\Psi^{\pi}$'s are equal on a real neighborhood. Now the Edge of the Wedge Theorem **[5]** shows that they are analytic continuations of one and the same analytic function. Moreover, the domain of analyticity of this function must contain the domains of analyticity each $\Psi^{\pi}$.
Q.E.D.




---


**Remark.** I think $\Psi(x\_1,\ldots,x\_n)=\Phi(x\_1)\ldots\Phi(x\_n)|0\rangle$ generalizes to arbitrary bosonic fields trivially and for fermionic fields we would pick up some minus signs.




---


References:


[1] V. Kac. *Vertex Algebras for Beginners*, 1998.


[2] M. Schottenloher. *A Mathematical Introduction to Conformal Field Theory*, 2008.


[3] Lüscher, M. and Mack, G. Global conformal invariance in quantum field theory. *Communications in Mathematical Physics*, Springer-Verlag, 1975, 41, 203-234.


[4] Glaser, V. On the equivalence of the Euclidean and Wightman formulation of field theory. *Comm. Math. Phys.*, 1974, 37, 257-272.


[5] Streater, R. F. and Wightman, A. S. *PCT, Spin and Statistics, and All That*, W. A. Benjamin, Inc., 1964.


