6  Fermions

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6.1 Grassmann Algebra

The Grassmann algebra \(\mathcal{G}_D\) is generated by \(D\) Grassmann variables \(\eta_{A=1,\ldots,D}\) (the generators) together with complex coefficients treated as independent scalars. It consists of all polynomials in \(\eta_A\) with complex coefficients, subject to the fundamental anticommutativity rule \[ \eta_A\eta_B = -\eta_B\eta_A. \tag{$*$} \] In particular, setting \(A=B\) gives \(\eta_A^2 = -\eta_A^2\), so \(\eta_A^2 = 0\).

The first few cases are: \[ \mathcal{G}_1 = \bigl\{a + a_1\eta \;\big|\; a,a_1\in\mathbb{C}\bigr\} \quad\text{(only degree-1 polynomials, since } \eta^2=0\text{)}, \] \[ \mathcal{G}_2 = \bigl\{a + a_1\eta_1 + a_2\eta_2 + a_{12}\eta_1\eta_2 \;\big|\; a,a_1,a_2,a_{12}\in\mathbb{C}\bigr\} \] (note \(\eta_2\eta_1 = -\eta_1\eta_2\), so \(a_{21} = -a_{12}\)). The general element of \(\mathcal{G}_D\) is \[ \vartheta = a + \sum_{n=1}^D \frac{1}{n!} \sum_{A_1,\ldots,A_n=1}^D a_{A_1\cdots A_n}\,\eta_{A_1}\cdots\eta_{A_n}, \] where the coefficients \(a_{A_1\cdots A_n}\) are fully antisymmetric (any symmetric part vanishes by rule \((*)\)).

6.2 Bosons and Fermions

Elements of \(\mathcal{G}_D\) are classified by their behaviour under \(\eta_A \to -\eta_A\):

• Bosonic (even): unchanged, i.e. built from monomials with an even number of Grassmann variables.
• Fermionic (odd): changes sign, i.e. built from monomials with an odd number.

Every element of \(\mathcal{G}_D\) decomposes uniquely into a bosonic plus a fermionic part. If \(b_1,b_2\) are bosonic and \(f_1,f_2\) are fermionic, then \[ b_1 b_2 = b_2 b_1, \qquad bf = fb, \qquad f_1 f_2 = -f_2 f_1. \]

6.3 Functions of Grassmann Variables

Any element of \(\mathcal{G}_D\) can be written as \(a + \vartheta'\) where \(a\in\mathbb{C}\) and \(\vartheta'\) contains at least one power of \(\eta\). Since \((\vartheta')^{D+1} = 0\) (every term in \((\vartheta')^{D+1}\) contains at least \(D+1\) factors of \(\eta_A\), so at least one index must repeat), the Taylor series of any smooth \(f:\mathbb{C}\to\mathbb{C}\) truncates: \[ f(a+\vartheta') \stackrel{\mathrm{def}}{=} \sum_{n=0}^\infty \frac{(\vartheta')^n}{n!}f^{(n)}(a) = \sum_{n=0}^D \frac{(\vartheta')^n}{n!}f^{(n)}(a). \]

Examples. \[ e^\eta = 1+\eta, \] \[ e^{\eta_1+\eta_2} = 1+(\eta_1+\eta_2)+\tfrac{1}{2}(\eta_1+\eta_2)^2 = 1+\eta_1+\eta_2 \quad\text{(check: } (\eta_1+\eta_2)^2 = 2\eta_1\eta_2 + \eta_1^2 + \eta_2^2 = 0\text{)}, \] \[ e^{\eta_A H_{AB}\eta_B} = \sum_{n=0}^D \frac{1}{n!}(\eta_A H_{AB}\eta_B)^n. \]

6.4 Derivatives

The left derivative \(\frac{\partial}{\partial\eta_A}\) is defined by the rules: \[ \frac{\partial}{\partial\eta_A}\frac{\partial}{\partial\eta_B} = -\frac{\partial}{\partial\eta_B}\frac{\partial}{\partial\eta_A}, \qquad \frac{\partial}{\partial\eta_A}a = 0 \;\text{ for } a\in\mathbb{C}, \qquad \frac{\partial}{\partial\eta_A}\eta_B = \delta_{AB}, \] and the graded Leibniz rule: \[ \frac{\partial}{\partial\eta_A}(\theta_1\theta_2) = \left(\frac{\partial}{\partial\eta_A}\theta_1\right)\theta_2 + (-1)^{|\theta_1|}\theta_1\frac{\partial}{\partial\eta_A}\theta_2, \] where \(|\theta_1|=0\) if \(\theta_1\) is bosonic and \(|\theta_1|=1\) if fermionic.

Example. For \(\vartheta = a + a_1\eta_1 + a_2\eta_2 + a_{12}\eta_1\eta_2\): \[ \frac{\partial}{\partial\eta_2}\vartheta = a_2 + a_{12}\frac{\partial}{\partial\eta_2}(\eta_1\eta_2) = a_2 - a_{12}\eta_1. \]

6.5 Berezin Integration

The Berezin integral is defined by the rules (for each generator \(\eta_B\)): \[ d\eta_A\,d\eta_B = -d\eta_B\,d\eta_A, \qquad d\eta_A\,\eta_B = -\eta_B\,d\eta_A, \] \[ \int d\eta_A\,\eta_A = 1, \qquad \int d\eta_A\,1 = 0. \]

Example. For \(\vartheta = a + a_1\eta_1 + a_2\eta_2 + a_{12}\eta_1\eta_2\): \[ \int d\eta_1\,d\eta_2\;\vartheta = a\underbrace{\int d\eta_1}_{=0}\underbrace{\int d\eta_2}_{=0} - a_1\underbrace{\int d\eta_1\,\eta_1}_{=1}\underbrace{\int d\eta_2}_{=0} + a_2\underbrace{\int d\eta_1}_{=0}\underbrace{\int d\eta_2\,\eta_2}_{=1} - a_{12}\underbrace{\int d\eta_1\,\eta_1}_{=1}\underbrace{\int d\eta_2\,\eta_2}_{=1} = -a_{12}. \]

For a general element of \(\mathcal{G}_D\), only the top-degree term survives: \[ \int d\eta_D\cdots d\eta_1\;\vartheta = a_{12\cdots D}, \] where the integral picks out the coefficient of the ordered product \(\eta_1\eta_2\cdots\eta_D\).

6.6 Linear Change of Variables

Under a linear change \(\eta'_A = M_{AB}\eta_B\) with \(M\) an invertible \(D\times D\) complex matrix, \(\eta'_1,\ldots,\eta'_D\) is again a set of generators for \(\mathcal{G}_D\). The integration measure transforms as \[ d\eta_D\cdots d\eta_1 = \det M\; d\eta'_D\cdots d\eta'_1. \] Note the inverse of the Jacobian compared to ordinary integration — this is a characteristic feature of Berezin integration.

6.7 Integration by Parts

Since \(\int d\eta_B\;\eta_B = 1\) and \(\int d\eta_B\;1 = 0\), any element \(\vartheta\) can be split as \(\vartheta = \chi_0 + \eta_B\chi_1\) where \(\chi_0,\chi_1\) contain no \(\eta_B\). Then \(\frac{\partial}{\partial\eta_B}\vartheta = \chi_1\), and \[ \int d\eta_B\;\frac{\partial}{\partial\eta_B}\vartheta = \int d\eta_B\;\chi_1 = 0, \] because \(\chi_1\) contains no \(\eta_B\). Therefore: \[ \int d\eta_B\;\frac{\partial}{\partial\eta_B}\vartheta = 0 \qquad\text{(no sum over }B\text{)}. \] This is the integration-by-parts formula for Grassmann integrals.

6.8 Quark Fields on the Lattice

In QCD, quarks come in \(N_f\) flavours (u, d, s, c, b, t). The quark field \(\psi\) and the antiquark field \(\bar\psi\) carry three indices: \[ \psi_{f\alpha i}(x), \qquad \bar\psi_{f\alpha i}(x), \qquad x\in\Lambda, \] where:

• \(f = 1,\ldots,N_f\) is the flavour index (u, d, s, c, …),
• \(\alpha = 1,2,3,4\) is the spin index (Dirac spinor; gamma matrices act on this),
• \(i = 1,\ldots,N_c\) is the colour index (the gauge group \(SU(N_c)\) acts on this; for QCD, \(N_c=3\)).

In the path-integral formalism, \(\psi\) and \(\bar\psi\) are independent Grassmann variables — there is no algebraic relation between them. The total number of independent Grassmann variables is \[ 2\,(\psi,\bar\psi) \times N_f \times 4\,(\text{spin}) \times N_c \times N^4\,(\text{lattice points}) = 8 N_f N_c N^4. \]

6.9 Fermion Action and Path Integral

The continuum Euclidean fermion action is \[ S_F = \sum_f \int d^4x\;\bar\psi_f(x)\,(\not{D}+m_f)\,\psi_f(x), \qquad \not{D} = \gamma_\mu D_\mu, \] where \(\gamma_\mu\) are the Euclidean gamma matrices satisfying \(\{\gamma_\mu,\gamma_\nu\}=2\delta_{\mu\nu}\). In the chiral basis: \[ \gamma_0 = \begin{pmatrix}0 & -I_2\\-I_2 & 0\end{pmatrix}, \quad \gamma_k = \begin{pmatrix}0 & -i\sigma_k\\ i\sigma_k & 0\end{pmatrix}, \quad \gamma_5 = \gamma_0\gamma_1\gamma_2\gamma_3 = \begin{pmatrix}I_2 & 0\\ 0 & -I_2\end{pmatrix}. \]

The discretized action takes the form \[ S_F = \sum_f\sum_x a^4\;\bar\psi_f(x)\,[\not{D}^f\psi_f](x), \] where \(\not{D}^f\) is a discrete Dirac operator — a \((4N_cN^4)\times(4N_cN^4)\) matrix with spin, colour, and lattice indices: \[ [\not{D}^f\psi_f]_{\alpha i}(x) = \sum_{y,\beta,j} (\not{D}^f)_{\alpha i\,x;\,\beta j\,y}\,\psi_{f\beta j}(y). \] The precise form of \(\not{D}^f\) depends on the discretization scheme and will be specified below.

6.10 Fermion Partition Function

The fermionic path-integral measure is \[ [d\bar\psi\,d\psi] = \prod_{x\in\Lambda}\prod_{\alpha=1}^4\prod_{i=1}^{N_c}\prod_{f=1}^{N_f} d\bar\psi_{f\alpha i}(x)\,d\psi_{f\alpha i}(x). \]

The fermion partition function evaluates to a determinant. Using the change of variable \(\psi' = a^4\not{D}^f\psi\) (with Jacobian \(\det(a^4\not{D}^f)^{-1}\) for ordinary variables, but \(\det(a^4\not{D}^f)\) for Grassmann variables): \[ Z = \int[d\bar\psi\,d\psi]\,e^{-\sum_f\sum_x a^4\bar\psi_f\not{D}^f\psi_f} = \prod_f \det(a^4\not{D}^f). \] The key identity used is \(\int d\bar\eta\,d\eta\,e^{-\bar\eta M\eta} = \det M\) (contrast with the bosonic result \(\propto(\det M)^{-1}\)).

6.11 Fermion Two-Point Function

The fermion propagator (two-point function) is \[ \langle\psi_{f\alpha i}(x)\,\bar\psi_{f'\beta j}(y)\rangle = \delta_{ff'}\,(a^4\not{D}^f)^{-1}_{\alpha i,x;\,\beta j,y} \equiv \delta_{ff'}\,(a^4\not{D}^f)^{-1}(x,y). \]

Proof. Consider the identity (the integral of a total derivative vanishes): \[ 0 = \int[d\bar\psi\,d\psi]\;\frac{\partial}{\partial\bar\psi_{f\alpha i}(x)} \left\{e^{-\sum_{f,z}a^4\bar\psi_f(z)\not{D}^f\psi_f(z)}\,\bar\psi_{f'\beta j}(y)\right\}. \] Evaluating the derivative and dividing by \(Z\): \[ 0 = -\langle(a^4\not{D}^f\psi_f)_{\alpha i}(x)\;\bar\psi_{f'\beta j}(y)\rangle + \delta_{ff'}\delta_{\alpha\beta}\delta_{ij}\delta_{xy}. \] This gives \((a^4\not{D}^f)\langle\psi_f\,\bar\psi_{f'}\rangle = \delta_{ff'}\,I\), i.e. \[ \langle\psi_f\,\bar\psi_{f'}\rangle = \delta_{ff'}\,(a^4\not{D}^f)^{-1}. \]

6.12 Free Fermions and the Doubler Problem

For one flavour of free fermions, the action is \(S_F = \sum_x a^4\,\bar\psi(x)\not{D}\psi(x)\), where \(\not{D}\) is a discretization of \(\not\partial + m\). The three simplest choices all fail:

(a) \(\displaystyle\not{D} = \sum_\mu\gamma_\mu\partial_\mu^f + m\) — breaks charge conjugation \(C\), parity \(P\), and Euclidean time reversal.

(b) \(\displaystyle\not{D} = \sum_\mu\gamma_\mu\partial_\mu^b + m\) — breaks \(C\), \(P\), and Euclidean time reversal.

(c) \(\displaystyle\not{D} = \sum_\mu\gamma_\mu\partial_\mu^s + m\) — produces doublers.

Here \(\partial_\mu^f\), \(\partial_\mu^b\), \(\partial_\mu^s\) denote the forward, backward, and symmetric finite differences: \[ \partial_\mu^s\varphi(x) = \frac{\varphi(x+ae_\mu)-\varphi(x-ae_\mu)}{2a}, \qquad \partial_\mu^s = \frac{\partial_\mu^f+\partial_\mu^b}{2}. \]

6.13 Parity Breaking of Options (a) and (b)

Under parity, fields transform as \(\psi(x_0,\mathbf{x})\to\gamma_0\psi(x_0,-\mathbf{x})\), \(\bar\psi(x_0,\mathbf{x})\to\bar\psi(x_0,-\mathbf{x})\gamma_0\).

In the continuum the parity-transformed action equals the original because \(\gamma_0\gamma_\mu\gamma_0 = \pm\gamma_\mu\) and a change of integration variable \(\mathbf{x}\to-\mathbf{x}\) maps \(\partial_\mu\to\pm\partial_\mu\) consistently. On the lattice with the forward difference, however, \[ \partial_k^f\psi(x_0,\mathbf{x}) \;\xrightarrow{\;P\;}\; \gamma_0\,\partial_k^b\psi(x_0,-\mathbf{x}), \] since the shift \(x\to x+ae_k\) in the forward difference becomes \(x\to x-ae_k\) after \(\mathbf{x}\to-\mathbf{x}\). The parity-transformed action therefore contains \(\gamma_0(\sum_\mu\gamma_\mu\partial_\mu^b\psi + m\psi)\), which differs from the original. Parity is broken. Option (b) fails for the same reason.

6.14 The Doubler Problem for Option (c)

For the symmetric-difference operator, the propagator in momentum space is computed by diagonalising \(\not{D}\) on plane waves \(v_p(x) = a^4 e^{ipx}\): \[ \partial_\mu^s v_p(x) = \frac{e^{iaph_\mu}-e^{-iap_\mu}}{2a}v_p(x) = \frac{i\sin(ap_\mu)}{a}\,v_p(x), \] so \(\not{D}v_p = \bigl\{i\hat{\not{p}} + m\bigr\}v_p\) where \(\hat{p}_\mu \stackrel{\mathrm{def}}{=} \frac{1}{a}\sin(ap_\mu)\).

The propagator is \[ \not{D}^{-1}v_p = \frac{-i\hat{\not{p}}+m}{\hat{p}^2+m^2}\,v_p, \qquad (a^4\not{D})^{-1} = \frac{1}{a^4L^3T}\sum_p \frac{-i\hat{\not{p}}+m}{\hat{p}^2+m^2}\,v_p v_p^\dagger. \]

Finding the poles. In the \(T\to\infty\) limit, the fermion 2-pt function \[ \langle\psi(0)\bar\psi(x)\rangle = \frac{1}{L^3T}\sum_p\frac{-i\hat{\not{p}}+m}{\hat{p}^2+m^2}\,e^{-ipx} \] has poles in the complex \(p_0\) plane wherever \(\hat{p}^2+m^2 = 0\), i.e. \[ \sin^2(ap_0) = -a^2m^2 - \sum_k\sin^2(ap_k), \qquad \sin(ap_0) = \pm i\,\omega(\mathbf{p}), \] with \(\omega(\mathbf{p}) = \sqrt{a^2m^2+\sum_k\sin^2(ap_k)}\). Using \(\sin(iz)=i\sinh(z)\): \[ p_0 = \pm\frac{i}{a}\operatorname{arcsinh}\bigl[\omega(\mathbf{p})\bigr] + \frac{2\pi n}{a}, \qquad n\in\mathbb{Z}. \]

The single-particle energy is \[ E(\mathbf{p}) = \frac{1}{a}\operatorname{arcsinh}\!\sqrt{a^2m^2+\sum_k\sin^2(ap_k)}. \]

Obs 1. \(p_0\) has the wrong periodicity \(\frac{\pi}{a}\) instead of \(\frac{2\pi}{a}\).

Obs 2 (doublers). Given a state with spatial momentum \(\mathbf{p}=(p_1,p_2,p_3)\), there are 7 additional degenerate states obtained by substituting any subset of components \(p_k\to\frac{\pi}{a}-p_k\). All 8 states have the same energy \(E(\mathbf{p})\). The doubler momenta satisfy \(p_k\to\frac{\pi}{a}\neq 0\) as \(a\to 0\), yet their energies remain finite. This gives 16 fermionic species in total (8 per spin degree of freedom), rather than the desired 2.

6.15 Wilson Fermions

The Wilson-Dirac operator adds a second-order derivative term to the symmetric discretization: \[ \not{D} = \sum_\mu\gamma_\mu\partial_\mu^s + m - \frac{ar}{2}\sum_\mu\partial_\mu^b\partial_\mu^f, \] where \(r\) is the dimensionless Wilson parameter (common choice: \(r=1\)). The term \(-\frac{ar}{2}\sum_\mu\partial_\mu^b\partial_\mu^f\) is the discrete Laplacian (as encountered in scalar field theory).

Acting on a plane wave: \[ \not{D}v_p = \Bigl\{i\hat{\not{p}} + m + \tfrac{ar}{2}\hat{p}^2\Bigr\}v_p, \] where \(\hat{p}_\mu = \frac{1}{a}\sin(ap_\mu)\) and \(\hat{p}^2 = \sum_\mu\hat{p}_\mu^2\). Compared to naive fermions, the Wilson term shifts the effective mass: \(m \to m + \frac{ar}{2}\hat{p}^2\).

The propagator is \[ \not{D}^{-1}v_p = \frac{-i\hat{\not{p}}+m+\frac{ar}{2}\hat{p}^2}{\check{p}^2+\bigl(m+\frac{ar}{2}\hat{p}^2\bigr)^2}\,v_p, \] where \(\check{p}_\mu \stackrel{\mathrm{def}}{=} \frac{2}{a}\sin\!\bigl(\frac{ap_\mu}{2}\bigr)\), \(\check{p}^2 = \sum_\mu\check{p}_\mu^2\).

The 2-pt function is \[ \langle\psi(0)\bar\psi(x)\rangle = (a^4\not{D})^{-1}(0,x) = \frac{1}{L^3T}\sum_p\frac{-i\hat{\not{p}}+m+\frac{ar}{2}\hat{p}^2}{\check{p}^2+\bigl(m+\frac{ar}{2}\hat{p}^2\bigr)^2}\,e^{-ipx}. \]

6.16 Wilson Fermions: Pole Analysis and Doubler Decoupling

The single-particle energy for Wilson fermions is \[ E(\mathbf{p}) = \frac{2}{a}\operatorname{arcsinh}\!\left\{\frac{am + \frac{a^2r}{2}\sum_k\hat{p}_k^2}{2\!\left(1+am+\frac{a^2r}{2}\sum_k\hat{p}_k^2\right)^{1/2}}\right\}, \qquad \hat{p}_\mu = \tfrac{2}{a}\sin\tfrac{ap_\mu}{2}. \]

The poles occur at \(p_0 = \pm iE(\mathbf{p}) + \frac{2\pi n}{a}\), \(n\in\mathbb{Z}\), with the correct periodicity \(\frac{2\pi}{a}\).

The 8 states and their continuum limits (\(a\to 0\)):

\(\mathbf{p}\)	\(\hat{\mathbf{p}}\)	\(E(\mathbf{p})\) as \(a\to 0\)
\((0,0,0)\)	\((0,0,0)\)	\(m\) (physical state)
\((\frac{\pi}{a},0,0)\) and 2 permutations	\((\frac{2}{a},0,0)\) and permutations	\(\infty\) (decouple)
\((\frac{\pi}{a},\frac{\pi}{a},0)\) and 2 permutations	\((\frac{2}{a},\frac{2}{a},0)\) and permutations	\(\infty\) (decouple)
\((\frac{\pi}{a},\frac{\pi}{a},\frac{\pi}{a})\)	\((\frac{2}{a},\frac{2}{a},\frac{2}{a})\)	\(\infty\) (decouple)

The doublers acquire infinite energy in the continuum limit and therefore decouple. Only the single physical state with \(\mathbf{p}\to 0\) and \(E\to m\) survives as \(a\to 0\).

In the continuum limit, \(\lim_{a\to 0}E(\mathbf{p}) = \sqrt{m^2+\mathbf{p}^2}\), as required.

6.17 The Nielsen–Ninomiya Theorem

A discretization of the massless free Dirac operator satisfying all four of the following properties does not exist:

(a) Translational invariance: \(\mathcal{D}_{\alpha\beta}(x,y) = \mathcal{D}_{\alpha\beta}(x-y,0)\).

(b) Chiral symmetry and \(\gamma_5\)-hermiticity: \(\{\gamma_5,\mathcal{D}\}=0\) and \(\mathcal{D}^\dagger=\mathcal{D}\).

(c) Locality: \(\mathcal{D}_{\alpha\beta}(z,0)\) decays faster than any inverse power of \(|z|\) for \(|z|\to\infty\).

(d) No doublers: the propagator has only a single pole in each Brillouin zone.

Comments on the assumptions.

• Condition (b) is verified in the continuum: \(\{i\gamma_5,i\not\partial\}=0\) and \((i\not\partial)^\dagger = i\not\partial\). It is equivalent to requiring \(\mathcal{D}\) to be a linear combination of the \(\gamma_\mu\) matrices, \(\mathcal{D}_{\alpha\beta}(z,0) = \sum_\mu(\gamma_\mu)_{\alpha\beta}F_\mu(z)\) with \(F_\mu^*(z)=F_\mu(-z)\).

• Condition (c) is a weak form of locality. The naive Dirac operator satisfies it strictly: \(\mathcal{D}_{\alpha\beta}(z,0)=0\) for \(|z|>2a\).

The theorem explains why every practical lattice fermion formulation must sacrifice at least one of these properties:

Formulation	Property broken
Wilson (\(m=0\))	(b) — chiral symmetry; ultralocal
Staggered	(a) — translational invariance
Ginsparg–Wilson (overlap/domain wall)	(b) — in a modified sense; local in a weaker sense