Constructive Fermionic and Yukawa Models

Fermionic constructive theories replace ordinary probability densities by finite-cutoff Grassmann integrals and control their determinant or Pfaffian expansions uniformly. Anticommutation improves combinatorics, but it does not remove ultraviolet renormalization, infrared singularities, or the need for bosonic stability in Yukawa systems.

Required background. The constructive program and cutoff removal supplies the convergence obligations; cluster expansions and correlation inequalities supplies the connected-volume mechanism.

Helpful background. Gaussian Euclidean fields as measures supplies bosonic reference laws; spin–statistics theorems and failure modes explains what relativistic reconstruction must eventually recover.

Grassmann Gaussian integration and determinants

At finite volume and ultraviolet cutoff, let $C_F$ be a fermion covariance. The normalized Grassmann Gaussian is characterized algebraically by

$$\int \psi_{i_1}\cdots\psi_{i_n}\bar\psi_{j_1}\cdots\bar\psi_{j_n}\,d\mu_{C_F} =\det[C_F(i_a,j_b)]_{a,b=1}^n.$$

Determinant bounds, such as Gram’s inequality when the entries are inner products, replace the factorial sum over Wick pairings by a controlled determinant. A multiscale decomposition of $C_F$ then associates vertices to scales; power counting identifies mass, wave-function, and coupling counterterms; a tree or polymer expansion controls connected kernels uniformly.

For the massive two-dimensional Gross–Neveu model with $N>1$ colors, a regulated action contains

$$\frac{\lambda}{N}\int_\Lambda(\bar\psi\psi)^2d^2x +\delta m_\kappa\int_\Lambda\bar\psi\psi\,d^2x +\delta Z_\kappa\int_\Lambda\bar\psi\,\partial\!\!\!/\,\psi\,d^2x.$$

For sufficiently small renormalized coupling, the ultraviolet and volume cutoffs can be removed from the Schwinger functions. The limits satisfy the fermionic OS axioms and are the Borel sums of their renormalized perturbation series. Gawędzki and Kupiainen construct the model through convergent perturbation expansions; Gawędzki and Kupiainen 1985, pp. 1–30. Summers 2016, §3.5, pp. 23–24 states the cutoff action, small-coupling limit, OS conclusion, and later spectral result.

The determinant mechanism can be checked in the free limit: differentiating the Grassmann generating functional

$$Z[\eta,\bar\eta]=\exp(\bar\eta C_F\eta)$$

gives the signed Wick determinant. If a proposed expansion produces a permanent, the fermionic signs are wrong.

Yukawa models require two kinds of control

For a scalar field $\phi$ coupled by $g\bar\psi\Gamma\psi\phi$, integrating out the fermions gives a regularized determinant $\det(1+gS\Gamma\phi)$. Subtractions in a modified determinant remove its low-order divergent traces, while bosonic Wick ordering and a lower bound control the remaining scalar integral. Alternatively, integrating out the boson gives a nonlocal four-fermion kernel. In either representation the theorem must specify which field was eliminated, the determinant class, counterterms, and norm.

Two-dimensional Yukawa models have constructive OS results under stated mass and coupling conditions; Summers 2016, §3.4, pp. 21–23 gives the renormalized determinant and the cluster-expansion route. This does not mean that every fermion–boson action is stable. A real scalar with no stabilizing self-interaction can acquire an effective potential from the determinant whose large-field behavior defeats integrability.

The worked Gross–Neveu result returns to the Gross–Neveu model and dynamical mass generation. The constructive theorem described above is for the massive model at small renormalized coupling. It does not by itself prove the heuristic mass-generation scenario of the massless model at arbitrary $N$ and coupling.

Adversarial infrared and stability tests

Move the free covariance to a Fermi surface. Scale counting now has extended gapless modes rather than an isolated massive pole, and the same position-space decay and polymer norm fail. A different many-fermion theorem may apply, but the relativistic massive Gross–Neveu estimate does not.

In a Yukawa model, drop the stabilizing bosonic hypothesis and suppose the effective large constant-field action tends to $-c|\Lambda|\phi^4$. The remaining bosonic integral grows like $e^{+c|\Lambda|\phi^4}$ and diverges. A determinant bound alone cannot repair it.

The converse boundary is also strict: convergence of fermionic Schwinger functions need not establish OS positivity, a mass gap, or asymptotic completeness. Those are separate properties; in three-dimensional Gross–Neveu constructions, existence of certain Schwinger functions has historically exceeded the verified OS package.

A useful cross-check compares the two integrations of a Yukawa model. Expanding the modified fermion determinant to second order in the coupling must give the same nonlocal quadratic bosonic term that is obtained by contracting two Yukawa vertices with the free fermion covariance. Conversely, integrating out the boson must yield a four-fermion kernel with the boson covariance and the same sign. Agreement fixes symmetry factors and identifies the mass subtraction; disagreement exposes a convention or Grassmann-sign error before any multiscale estimate is attempted.

Exercises

1. Two-by-two sign. Evaluate $\int\psi_1\psi_2\bar\psi_1\bar\psi_2,d\mu_C$.

Solution

It is $C_{11}C_{22}-C_{12}C_{21}=\det C$. The minus sign records the permutation needed to contract the crossed pairing.

2. Large $N$ normalization. Why is the quartic coupling written $\lambda/N$ when $\bar\psi\psi$ sums over $N$ colors?

Solution

Each color loop contributes a factor $N$. The $1/N$ vertex scaling keeps leading families finite and organizes the large-$N$ expansion; it is not a substitute for the fixed-$N$ constructive bounds.

References

• Gawędzki, Krzysztof, and Antti Kupiainen. “Gross–Neveu Model Through Convergent Perturbation Expansions.” Communications in Mathematical Physics 102 (1985): 1–30. DOI.
• Magnen, Jacques, and Roland Sénéor. “The Wightman Axioms for the Weakly Coupled Yukawa Model in Two Dimensions.” Communications in Mathematical Physics 51 (1976): 297–313. DOI.
• Summers, Stephen J. “A Perspective on Constructive Quantum Field Theory.” arXiv:1203.3991, revised 2016. Open PDF.