Free massive and massless fields

A free field is a reference experiment in which we can solve the spectral question before asking how interactions change it. It distinguishes a mass put into the model from a mass generated by a gauge theory.

Prerequisites: Gaussian measures and Euclidean correlators.
Outcomes: compute a smeared free correlator; locate its spectral threshold; identify infrared and boundary-condition assumptions.

Define the comparison model

Consider a real free scalar on four-dimensional Euclidean space, with three spatial directions, $\hbar=c=1$, and mass $m>0$. This is a scalar Gaussian theory, with no gauge group or gauge constraint. Its momentum covariance is $(p_0^2+\mathbf p^2+m^2)^{-1}$.

Fourier transforming only Euclidean time gives

$$C_m(t,\mathbf p) =\int_{\mathbb R}\frac{dp_0}{2\pi} \frac{e^{ip_0t}}{p_0^2+\omega_{\mathbf p}^2} =\frac{e^{-\omega_{\mathbf p}|t|}}{2\omega_{\mathbf p}}, \qquad \omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.$$

The last integral follows by closing the contour in the half-plane where the exponential decays. The poles at $\pm i\omega_{\mathbf p}$ yield the positive exponential. Free scalar quantization is developed in Tong’s QFT notes, chapter 2; this calculation fixes the conventions used below.

Choose a real Schwartz spatial smearing $f$. For the observable $O_f(t)=\int d^3x\,f(\mathbf x)\phi(t,\mathbf x)$,

$$\langle O_f(t)O_f(0)\rangle =\int\frac{d^3p}{(2\pi)^3} |\widetilde f(\mathbf p)|^2 \frac{e^{-\omega_{\mathbf p}|t|}}{2\omega_{\mathbf p}}.$$

Rapid decrease of $\widetilde f$ controls large momenta. The observable has zero vacuum mean, so this is already connected.

The threshold need not be an isolated eigenvalue

Every energy in this integral is at least $m$. Therefore, for $t\geq0$,

$$0\leq C_f(t)\leq e^{-mt}C_f(0).$$

If $\widetilde f$ does not vanish near $\mathbf p=0$, the spectral measure reaches energies arbitrarily near $m$. On infinite space the momentum eigenvectors are not normalizable, so a threshold at $m$ need not be an isolated Hilbert-space eigenvector. A power multiplying $e^{-mt}$ in a position-space correlator is consistent with a positive threshold.

For the free scalar Fock space, nonvacuum states contain one or more quanta, so their total energy is at least $m$. A nonlinear probe can still miss the one-particle sector. Use the already spatially smeared variable above to define $Y_f(t)=O_f(t)^2-\langle O_f^2\rangle$. Gaussian pairing gives

$$\langle Y_f(t)Y_f(0)\rangle=2C_f(t)^2.$$

When $\widetilde f$ has support arbitrarily near zero momentum, the two-particle spectral support of this correlator begins at $2m$. This example squares one well-defined smeared variable; it does not assume a pointwise Euclidean Wick square exists in four dimensions. A channel threshold and the theory’s gap are different objects.

Setting the mass to zero changes the infrared question

For the massless scalar in four Euclidean dimensions the covariance $1/p^2$ is locally integrable near $p=0$: the radial measure gives $p^3dp/p^2=p\,dp$. Smeared fields can be defined, and one-particle energies $|\mathbf p|$ approach zero. The infinite-volume theory is gapless.

This statement is dimension dependent. In two Euclidean dimensions, the same infrared integral behaves as $dp/p$ and diverges for generic test functions with nonzero Fourier transform at zero. A massless scalar there requires a restricted observable class or a treatment of the additive zero mode. One cannot erase this distinction by declaring all free Gaussian fields identical.

A box supplies another scale

On a spatial interval with Dirichlet boundaries, the mode energies are

$$E_n(L)=\sqrt{m^2+(n\pi/L)^2},\qquad n\geq1.$$

For $m=0$, the first mode is positive at each finite $L$ but tends to zero as $L$ grows. For $m>0$, it tends to $m$. Periodic boundaries instead allow a spatial zero mode. At $m=0$ that mode is not a positive-frequency oscillator; it must be handled separately.

These are exact free-mode statements for the specified boundaries. They are not numerical measurements of a Yang–Mills glueball.

Exercise: a misleading threshold

A free scalar has mass $m=1$ in chosen physical units. A probe couples only to two-particle states with total zero momentum. Its measured long-time decay begins at energy 2. Does the underlying theory have gap 2?

Solution. No. One-particle wave packets have energy arbitrarily near 1, so the nonvacuum spectrum starts there. The selected probe misses that sector. The threshold 2 describes its channel. Changing the probe does not change the Hamiltonian.

What this model cannot transfer

The mass parameter was supplied in the covariance. There is no non-Abelian gauge interaction and no dynamical demonstration of dimensional transmutation. The official Yang–Mills target, §§4–5 cannot be satisfied by substituting this free scalar.

Use this example in the gap-and-decay page, operator overlap, and the finite-volume lab. The model is valuable because every assumption can be inspected.