# Motivation for the definition of gaussian states

In algebraic QFT, in the case of a scalar field satisfying the KG equation we can cosntruct the $\ast$-algebra of observables as the free algebra generated by the symbols $\phi(f)$ for $f\in C^\infty_0(M)$ with the relations imposed:

1. Linearity - $\phi(\lambda f + \mu g)=\lambda \phi(f) + \mu \phi(g)$ for all $\lambda,\mu\in \mathbb{C}$ and $f,g\in C^\infty_0(M)$,

2. Equations of Motion - $\phi((\Box+m^2)f)=0$ for all $f\in C^\infty_0(M)$,

3. Hermitian field - $\phi(\bar{f}) = \phi(f)^\ast$

4. Canonical Commutation Relations - $[\phi(f_1),\phi(f_2)]=iE(f_1,f_2)$

where $E$ is the commutator function.

It turns out that any element of the algebra is a linear combination of terms of the form

$$\phi(f_1)\cdots \phi(f_n),$$

so that any state $\omega : \mathscr{A}\to \mathbb{C}$ is determined by its $n$-point functions

$$W(f_1,\dots,f_n)=\omega(\phi(f_1)\cdots \phi(f_n)).$$

In that setting, Wald defines a gaussian state:

Gaussian states (also called "quasi-free states") are defined by the condition that the "connected $n$-point functions" $\omega_n^c(\phi(f_1)\cdots \phi(f_n))$ vanish for all $n > 2$ where $\omega_n^c: \mathscr{A}\times\cdots\times \mathscr{A}\to \mathbb{C}$ is defined by:

$$\omega_n^c(a_1,\dots,a_n)=\dfrac{\partial^n}{\partial t_1\dots \partial t_n}\log \left[\omega(e^{t_1a_1}\cdots e^{t_na_n})\right]\bigg|_{t_i=0}$$

Here the exponentials are to be understood in the sense of a formal series.

He then argues that gaussian states, under the GNS construction, become vacuum states on a Fock space.

Now I can't intuitively understand all of this. What is the motivation for this definition? Why would anyone look to states satisfying this strange condition?

Algebraic QFT is mathematically rigorously formulated. Quite often, the rigorously formulated definitions hide very simple ideas. This is one of these cases. I'll exemplify to you the meaning of this definition for a single Gaussian random variable, the generalizations to a single dimensional harmonic oscillator, then a multiple dimensional harmonic oscillator, then an infinite dimensional harmonic oscillator namely the scalar field become straightforward.

Given a single random variable $Y$ with an unknown distribution $f(y)$, then the conditions that the cumulants: $$c_n = \frac{\partial^n}{\partial t^n}\log(\mathrm{E}(e^{tY}))|_{t=0}$$ ($\mathrm{E}(.)$ denotes the expectation value), vanish for $n>2$ imply that $$\log(\mathrm{E}(e^{tY})) = \mu t+ \frac{1}{2} \sigma^2 t^2$$ Thus, the moment generating function: $$M(t) \equiv\mathrm{E}(e^{tY})= e^{ \mu t+ \frac{1}{2} \sigma^2 t^2}$$ It is clear now why $Y$ is Gaussian: The characteristic function: $M(-it)$ is the Fourier transform of the distribution function $f(y)$, the Fourier transform of a Gaussian is a Gaussian, therefore $f(Y)$ is Gaussian.

How to generalize

1. For an operator algebra generated by one creation operator $a$ and one annihilation operator $a^{\dagger}$ (harmonic oscillator), the condition that the Taylor expansion of: $$\log \omega(e^{\bar{z}a} e^{za^{\dagger}})$$ vanishes beyond the second term implies: $$\omega(e^{\bar{z}a} e^{za^{\dagger}}) = e^{\bar{z} z}$$ (In the above a symmetric distribution is assumed).
2. Similarly, for the multiple dimensional harmonic oscillator" $$\omega(e^{\bar{z}^ia_i} e^{z^ia_i^{\dagger}}) = e^{\bar{z}^i z^i}$$
3. Generalization to a (multiple component) scalar field:

$$\omega(e^{\int\frac{d^3k}{E_k} z_{\mathbf{k}}a^{\dagger}_{\mathbf{k}} + \bar{z}_{\mathbf{k}}a^{\mathbf{k}} }) = e^{\int\frac{d^3k}{E_k} \bar{z}_{\mathbf{k}} z_{\mathbf{k}}}$$

The above is equivalent to the representation in the question using creation and annihilation operators instead of smeared fields.

Why $\omega$ is GNS state

I am not providing a full proof (it is just work), but you can check these facts:

1. The null vectors of this state are exactly the commutation relations: $$a_i a_j – a_j a_i$$ and $$a_i a^{\dagger}_j – a^{\dagger}_j a_i - \delta_{ij}$$
2. A general Gaussian state $\omega$ defined by: (I am writing the case of a multiple dimensional harmonic oscillator)

$$\omega(e^{\bar{z}^ia_i} e^{z^ia_i^{\dagger}}) = e^{\sum_{ij} \Sigma_{ij}\bar{z}^i z^j}$$ Is pure, iff $$Det(\Sigma) =1$$ Thus, the above Gaussian state gives rise to an irreducible GNS representation.