Gaussian measures and free fields

A free Euclidean field is the infinite-dimensional relative of a Gaussian random vector. This analogy gives a concrete entrance to constructive field theory, but it also reveals why the field usually lives in a space of distributions rather than ordinary functions.

Start in finitely many dimensions

Prerequisites: Gaussian integration and probability measures.
Outcomes: recover moments from a covariance; explain why continuum fields are smeared; distinguish covariance positivity from reflection positivity.

Let $x\in\mathbb R^n$ have a centered Gaussian distribution with positive-definite covariance matrix $C$:

$$d\mu_C(x)= \frac{\exp(-x^{\mathsf T}C^{-1}x/2)} {(2\pi)^{n/2}\sqrt{\det C}}\,d^nx.$$

Its characteristic function is

$$\mathbb E[e^{ij^{\mathsf T}x}] =e^{-j^{\mathsf T}Cj/2}.$$

Differentiating with respect to the components of $j$ generates all moments. In particular $\mathbb E[x_ix_j]=C_{ij}$, while the fourth moment is

$$\mathbb E[x_ix_jx_kx_l] =C_{ij}C_{kl}+C_{ik}C_{jl}+C_{il}C_{jk}.$$

This pairing rule is Wick’s theorem for a Gaussian probability measure.

Replace vectors by smeared fields

For a centered real Gaussian field $\phi$, take real Schwartz test functions $f,g$. The covariance becomes a real bilinear form on these functions:

$$\mathbb E[\phi(f)\phi(g)]=C(f,g).$$

Its characteristic functional is

$$\mathbb E[e^{i\phi(f)}]=e^{-C(f,f)/2}.$$

For a massive scalar field on Euclidean space,

$$C_m(f,g)=\int\frac{d^dp}{(2\pi)^d} \frac{\overline{\widetilde f(p)}\widetilde g(p)}{p^2+m^2}.$$

Under the appropriate continuity and positivity conditions, such a functional defines a Gaussian measure on a distribution space. An infinite-dimensional construction must specify that space and those conditions; the formal notation $D\phi$ does not supply them.

The covariance’s ultraviolet behavior controls sample regularity. Integrating $1/(p^2+m^2)$ at coincident points shows why finite values of $\phi(x)$ cannot generally be assumed. The smeared variables remain well defined.

Three meanings of positivity

It is helpful to distinguish three statements.

1. Covariance positivity: $C(f,f)\geq0$, required for the Gaussian random field.
2. Positive probability measure: nonnegative probabilities for events.
3. Reflection positivity: a further condition involving reflection of Euclidean time, needed in the usual route to a physical Hilbert space.

A general positive Gaussian covariance need not have the third property. For the free massive scalar, reflection positivity can be traced to the positive exponential kernel obtained after Fourier transforming the spatial directions.

Interactions change the question

For the positive-time kernel of one free massive mode, the distinction can be checked directly:

$$\sum_{ij}\overline c_i c_j\frac{e^{-\omega(t_i+t_j)}}{2\omega} =\frac1{2\omega}\left|\sum_j c_je^{-\omega t_j}\right|^2\geq0, \qquad \omega>0,\quad t_j>0.$$

This is a reflected quadratic form, with a sum of positive times rather than their difference. The positivity bridge gives an ordinary positive Gaussian covariance that fails this test. Reconstruction refers to the complete Osterwalder–Schrader framework: the 1973 paper and its 1975 extension and correction, not positivity alone.

At a finite regulator one can try to define an interacting measure by

$$d\mu_V(\phi)=Z^{-1}e^{-V(\phi)}d\mu_C(\phi).$$

If the integral exists, this defines a regulated model. Constructive analysis asks whether renormalized observables converge as the regulator is removed, and whether the resulting theory has the required properties.

A positive mass in a chosen Gaussian covariance does not prove dynamical mass generation in pure Yang–Mills. It describes the Gaussian reference model. Nor does a Gaussian scaling limit automatically give the intended interacting target.

Exercise. Let $X,Y$ be centered jointly Gaussian variables. Show that the connected correlation of $X^2$ and $Y^2$ equals $2\mathbb E[XY]^2$.

Solution. Wick pairing gives $\mathbb E[X^2Y^2]=\mathbb E[X^2]\mathbb E[Y^2]+2\mathbb E[XY]^2$. Subtracting the product of expectations leaves the claimed answer. This also illustrates why a composite operator can probe a different spectral threshold.

Continue

Work through free massive and massless fields for an explicit momentum integral and its infrared restrictions, then use the positivity lab to test the two forms.

Read distributions and reflection positivity. For the free-field physics, see chapter 2 of David Tong’s QFT notes. For constructive field theory’s role in the target problem, see section 6 of Jaffe and Witten.