In physics papers, the massless free boson has a definition involving an action:

$$ S(X) = \frac{1}{8\pi} \int d\sigma^2\, \partial X \overline{\partial X}$$

The random functions $X(z)$ are sampled according to the Gaussian distribution: $e^{-S(X)}$. In this case we can define the correlator to be:

$$ \mathbb{E}[X(z,\overline{z})X(w, \overline{w})] = - \ln |z - w|^2$$

See Ginsparg, applied Conformal Field Theory. I am trying to fill in the details of this derivation... using probability theory, complex analysis or anything else.

Since $X$ is a bosonic field I will just model it as a function... judging from the norm it should live in some hardy space $X \in H^2$ - the complexification of $L^2$ - where we should fill in some Riemann surface with flat structure. I think Ginsparg uses $H^2(\mathbb{R}\times S^1)$.

Then I would like to expand. $\partial X(z) = \sum a_n z_n$ and get

$$ \langle \partial X(z)\partial X(w) \rangle = \sum \mathbb{E}[a_n^2]z \overline{w} = \frac{1}{|z-w|^2}$$

There are holes everywhere in this argument - a real physicists's derivation. How can I make this more rigorous? Especially the intuition about the Gaussian random function in $L^2$ ?

Thanks.

In physics papers, the massless free boson has a definition involving an action:

$$ S(X) = \frac{1}{8\pi} \int d\sigma^2\, \partial X \overline{\partial X}$$

The random functions $X(z)$ are sampled according to the Gaussian distribution: $e^{-S(X)}$. In this case we can define the correlator to be the vacuum expectation value:

$$ \mathbb{E}[X(z,\overline{z})X(w, \overline{w})] \equiv \langle \varnothing | X(z,\overline{z})X(w, \overline{w}) |\varnothing \rangle = - \tfrac{1}{2}\ln |z - w|$$

See Ginsparg, applied Conformal Field Theory. I am trying to fill in the details of this derivation... using probability theory, complex analysis or anything else.

Since $X$ is a bosonic field I will just model it as a function... judging from the norm it should live in some Hardy space $X \in H^2$ - the complexification of $L^2$ - where we should fill in some Riemann surface with flat structure. I think Ginsparg uses $H^2(\mathbb{R}\times S^1)$.

The equations of motion ought to be the Cauchy Riemann equations. The fields in this theory are holomorphic functions. $X(z) = \sum a_n z^n$. Then I would like to expand. $\partial X(z) = \sum n a_n z^n$ and get

$$ \langle \partial X(z)\partial X(w) \rangle = \sum n\mathbb{E}[a_n^2]\,(z \overline{w})^n = \frac{1}{|1-z\overline{w}|^2}$$

There are holes everywhere in this argument - a real physicists's derivation. How can I make this more rigorous? Especially the intuition about the Gaussian random function in $L^2$ ?

Thanks.

In physics papers, the massless free boson has a definition involving an action:

$$ S(X) = \frac{1}{8\pi} \int d\sigma^2\, \partial X \overline{\partial X}$$

The random functions $X(z)$ are sampled according to the Gaussian distribution: $e^{-S(X)}$. In this case we can define the correlator to be:

$$ \mathbb{E}[X(z,\overline{z})X(w, \overline{w})] = - \ln |z - w|^2$$

See Ginsparg, applied Conformal Field Theory. I am trying to fill in the details of this derivation... using probability theory, complex analysis or anything else.

Since $X$ is a bosonic field I will just model it as a function... judging from the norm it should live in some hardy space $X \in H^2$ - the complexification of $L^2$ - where we should fill in some Riemann surface with flat structure. Let's use $H^2(\mathbb{D})$.

Then I would like to expand. $\partial X(z) = \sum a_n z_n$ and get

$$ \langle \partial X(z)\partial X(w) \rangle = \sum \mathbb{E}[a_n^2]z \overline{w} = \frac{1}{|z-w|^2}$$

There are holes everywhere in this argument - a real physicists's derivation. How can I make this more rigorous? Especially the intuition about the Gaussian random function in $L^2$ ?

Thanks.

In physics papers, the massless free boson has a definition involving an action:

$$ S(X) = \frac{1}{8\pi} \int d\sigma^2\, \partial X \overline{\partial X}$$

The random functions $X(z)$ are sampled according to the Gaussian distribution: $e^{-S(X)}$. In this case we can define the correlator to be:

$$ \mathbb{E}[X(z,\overline{z})X(w, \overline{w})] = - \ln |z - w|^2$$

See Ginsparg, applied Conformal Field Theory. I am trying to fill in the details of this derivation... using probability theory, complex analysis or anything else.

Since $X$ is a bosonic field I will just model it as a function... judging from the norm it should live in some hardy space $X \in H^2$ - the complexification of $L^2$ - where we should fill in some Riemann surface with flat structure. I think Ginsparg uses $H^2(\mathbb{R}\times S^1)$.

Then I would like to expand. $\partial X(z) = \sum a_n z_n$ and get

$$ \langle \partial X(z)\partial X(w) \rangle = \sum \mathbb{E}[a_n^2]z \overline{w} = \frac{1}{|z-w|^2}$$

There are holes everywhere in this argument - a real physicists's derivation. How can I make this more rigorous? Especially the intuition about the Gaussian random function in $L^2$ ?

Thanks.