Path-Integral Quantization

Derivation. Derives Lemma 106.1. The two terms of \(H\) do not commute, but over a time \(\Delta t\) the error of separating them is second order: by the Baker–Campbell–Hausdorff expansion,

with \(T=p^{2}/2m\). Dropping the correction and inserting the momentum-space resolution of the identity \(\identity=\int\dd p\,\ketbra{p}{p}\) with \(\braket{q}{p}=(2\pi\hbar)^{-1/2}\ee^{\ii pq/\hbar}\),

the \(p\) integral being the Gaussian \(\int\dd p\,\ee^{-ap^{2}/2+bp}=\sqrt{2\pi/a}\,\ee^{b^{2}/2a}\) of Lemma 106.2 below, analytically continued to \(a=\ii\Delta t/(m\hbar)\) on the boundary of its domain of convergence — legitimate because the integrand is the boundary value of a function holomorphic in \(\Re a>0\) and the contour may be rotated by Corollary 8.15. The square root is the principal branch, \(\sqrt{\ii}=\ee^{\ii\pi/4}\).

Finally \(V(q')\) may be replaced by \(V\bigl((q''+q')/2\bigr)\): the two differ by \(\frac{1}{2}(q''-q')V'+O((q''-q')^{2})\), and the prefactor confines \(\abs{q''-q'}\) to order \(\sqrt{\hbar\Delta t/m}\), so the change in the exponent is of order \((\Delta t)^{3/2}\); it cancels against its mirror image on symmetrizing, leaving an error \(O((\Delta t)^{2})\). The midpoint form is not an aesthetic choice, and Section 106.1.3 shows where it becomes compulsory.

The prefactor's dimension is \([\,m/(\hbar\Delta t)\,]^{1/2} =[\mathrm{kg}/\mathrm{J}/\mathrm{s}^{2}]^{1/2} =/\mathrm{m}\), as \(K\) requires.

Derivation. Derives Lemma 106.2. Diagonalize: \(A=O\transpose DO\) with \(O\) orthogonal and \(D=\diag(a_{1},\dots,a_{n})\), \(a_{i}>0\), which is possible by the finite-dimensional spectral theorem for real symmetric matrices. That theorem is quoted here from textbook linear algebra and is not proved anywhere in this book: the subsection it belongs to is Section 12.4.2, and Remark 11.86 records the same import where it is made in the probability chapter. The substitution \(y=Ox\) has unit Jacobian and factorizes the integral into \(n\) copies of the one-dimensional case. Completing the square in one dimension, \(-\frac{1}{2}ay^{2}+\beta y =-\frac{1}{2}a(y-\beta/a)^{2}+\beta^{2}/(2a)\), and \(\int\dd u\,\ee^{-au^{2}/2}=\sqrt{2\pi/a}\) — the Gauss integral, which is Equation (17.42) of Fourier Analysis and Integral Transforms evaluated at zero frequency. Reassembling, \(\prod_{i}a_{i}=\det A\) and \(\sum_{i}\beta_{i}^{2}/a_{i}=b\transpose A^{-1}b\), which is Equation (106.7).

Both sides of Equation (106.7) are holomorphic in the entries of \(A\) on the domain \(\Re A>0\) — the left-hand side because the integral converges absolutely and uniformly there, the right-hand side because \(\det A\neq0\) — so they agree throughout it by the identity theorem (Complex Analysis). For the purely imaginary boundary case, diagonalize \(M\) and rotate each contour: for \(a>0\),

the limit being uniform on compacta by Cauchy's theorem applied to the wedge \(0\leq\arg u\leq\pi/4\), whose arc contribution vanishes by the estimation lemma Lemma 8.9. A negative eigenvalue rotates the other way and contributes \(\ee^{-\ii\pi/4}\); collecting the phases gives the signature.

Derivation. Derives Theorem 106.3. The evolution operator composes exactly, \(U(t'',t')=U(t'',t_{N-1})\cdots U(t_{1},t')\), because \(H\) is time-independent. Insert \(\identity=\int\dd q_{k}\ketbra{q_{k}}{q_{k}}\) between consecutive factors:

which is exact for every \(N\). Now apply Lemma 106.1 to each factor. Each contributes the normalization \((m/2\pi\ii\hbar\Delta t)^{1/2}\) — there are \(N\) of them, one per step, against \(N-1\) integrations — and the phase \(\ee^{\ii\Delta t L_{k}/\hbar}\), and the individual errors \(O((\Delta t)^{2})\) accumulate to \(N\cdot O((\Delta t)^{2}) =O(\Delta t)\), which vanishes with \(N\to\infty\). The exponents add, giving the Riemann sum \(\sum_{k}\Delta t L_{k}\) for the action \(S[q]=\int_{t'}^{t''}\dd t\,L\) of the piecewise-linear path through the sampled points.

Derivation. Derives Proposition 106.4. Write \(q'=q+\eta\). Then

The Gaussian factor suppresses all but \(\eta=O(\sqrt{\hbar \Delta t/m})\), so expanding to the order that survives means keeping \(\eta^{0},\eta^{1},\eta^{2}\) in \(\psi\) and \((\Delta t)^{0}, (\Delta t)^{1}\) elsewhere:

By Equation (106.8) with \(M=m/(\hbar\Delta t)\),

by parity, and differentiating the first with respect to \(m/(\hbar\Delta t)\) gives the second moment

Collecting,

and \((\psi(q,t+\Delta t)-\psi(q,t))/\Delta t\to\pp\psi/\pp t\) gives Equation (106.11) after multiplication by \(\ii\hbar\). Note where the two derivatives came from: the potential from the first-order term of the exponential, the kinetic operator from the second moment of the Gaussian, which is why the dimensionless combination controlling the expansion is \(\hbar\Delta t/m\) and why the paths have the roughness they do.

Derivation. Derives Phenomenon 106.5. The Lagrangian of a particle of charge \(Q\) in an electromagnetic field is \(L=\frac{1}{2}m\dot{\vect{x}}^{2} +Q\,\dot{\vect{x}}\cdot\vect{A}-Q\phi\), whose Euler–Lagrange equation is \(m\ddot{\vect{x}}=Q(\vect{E}+\dot{\vect{x}}\times\vect{B})\), the Lorentz force; that derivation belongs to Generalized Classical Field Theory, with the covariant coupling \(-A_{\mu}J^{\mu}\), and is not repeated here. Along any path \(\gamma\) the action therefore contains the term

which depends on the path but not on its parametrization. For two paths \(\gamma_{1},\gamma_{2}\) with the same endpoints, the difference of the two actions is the circulation around the closed loop \(\gamma_{2}-\gamma_{1}\),

by Stokes's theorem, \(\Sigma\) being any surface spanning the loop. The interference term in Theorem 106.3 therefore acquires the phase

and for the electron, \(Q=-e\), the pattern repeats when \(e\Phi/\hbar=2\pi\), that is when \(\Phi\) advances by \(h/e\). The numerical value in Equation (106.12) is exact in the present SI, both \(h\) and \(e\) being defined constants [Mohr:2025].

The point is the one Aharonov and Bohm made. The surface \(\Sigma\) necessarily passes through the solenoid, where \(\vect{B}\neq0\); the paths do not. In a formulation whose primitive quantities are the fields, the effect is impossible; in a formulation whose primitive quantity is \(\int\vect{A}\cdot\dd\vect{x}\), it is the first thing one would predict. What is measurable is not \(\vect{A}\), which is gauge dependent, but the loop integral \(\oint\vect{A}\cdot\dd\vect{x}\), which is not — a gauge transformation Equation (100.8) changes it by \(\oint\pp_{\mu}\Lambda\,\dd x^{\mu}=0\) for a single-valued \(\Lambda\). The gauge-invariant content of the potential is exactly its holonomy, and Section 106.5.1 is built on that observation.

Derivation, bounded case. Derives Theorem 106.7. Let \(T\) and \(V\) be bounded operators, write \(\Delta t=t/N\), \(X_{N}=\ee^{-\ii(T+V)\Delta t/\hbar}\) and \(Y_{N}=\ee^{-\ii T\Delta t/\hbar}\ee^{-\ii V\Delta t/\hbar}\). Both are contractions, \(\norm{X_{N}}\leq1\) and \(\norm{Y_{N}}\leq1\), and expanding both exponentials gives \(\norm{X_{N}-Y_{N}}\leq K(\Delta t)^{2}\) with \(K=\norm{\comm{T}{V}}/2\hbar^{2}\), since the terms of order \(\Delta t^{0}\) and \(\Delta t^{1}\) agree. The telescoping identity

then bounds \(\norm{X_{N}^{N}-Y_{N}^{N}}\leq N\norm{X_{N}-Y_{N}} \leq Kt^{2}/N\to0\). Unbounded \(T\) and \(V\) — the physical case, since \(p^{2}\) is unbounded — require the strong-topology argument of Trotter [Trotter:1959], whose extension to operators bounded relative to \(T\) is Kato's [Kato:1966]; the mechanism is the same telescoping, applied to the resolvents.

Derivation. Derives Proposition 106.9. Repeat the momentum-space computation Equation (106.6) with \(H\) quadratic in \(\vect{p}-Q\vect{A}\). Writing \(\vect{A}\) at a fixed point \(\vect{x}_{*}\) inside the segment, the \(\vect{p}\) integral is Gaussian with a shifted centre and gives Equation (106.15) with \(\vect{A}(\vect{x}_{*})\) in place of the midpoint value. To fix \(\vect{x}_{*}\), demand that the kernel transform correctly under a gauge transformation \(\vect{A}\to\vect{A}+\nabla\Lambda\), under which the wavefunction acquires the phase \(\ee^{\ii Q\Lambda(\vect{x})/\hbar}\) and hence the kernel must acquire \(\ee^{\ii Q[\Lambda(\vect{x}'')-\Lambda(\vect{x}')]/\hbar}\). The exponent in Equation (106.15) changes by \((\ii Q/\hbar)(\vect{x}''-\vect{x}')\cdot\nabla\Lambda(\vect{x}_{*})\), and

holds only for the midpoint \(\vect{x}_{*}=(\vect{x}''+ \vect{x}')/2\); at any other point the error is quadratic, i.e. of order \(\hbar\Delta t/m\) in the exponent, which survives the limit. Endpoint prescriptions therefore quantize a different Hamiltonian, differing from \(H\) by a term proportional to \(\nabla\cdot\vect{A}\).

Derivation. Derives Theorem 106.10. Away from \(x_{0}\) the integrand oscillates: writing \(\ee^{\ii f/\hbar}=(\hbar/\ii)\nabla f\cdot\nabla \ee^{\ii f/\hbar}/\abs{\nabla f}^{2}\) and integrating by parts \(k\) times produces a factor \(\hbar^{k}\), so those regions contribute beyond all orders. Near \(x_{0}\), substitute \(x=x_{0}+\sqrt{\hbar}\,u\) and expand \(f(x)=f(x_{0})+\frac{\hbar}{2}u\transpose f''(x_{0})u+O(\hbar^{3/2})\). The measure supplies \(\hbar^{n/2}\), the exponent becomes \(\ii f(x_{0})/\hbar+\frac{\ii}{2}u\transpose f''u\), and Equation (106.8) evaluates the remaining integral.

Derivation. Derives Proposition 106.11. The energy eigenfunctions are obtained from \(K\) by \(\int\dd t\,\ee^{\ii Et/\hbar}K(q,t;q',0)\), a Fourier transform of the type of Fourier Analysis and Integral Transforms. Applying Theorem 106.10 in the single variable \(t\), the stationary point is where \(\pp S_{\mathrm{cl}}/\pp t=-E\) by Equation (106.3), i.e. at the classical time of flight at energy \(E\), and the phase there becomes the Legendre transform \(W(q,q'';E)=S_{\mathrm{cl}}+Et=\int p\,\dd q\), Hamilton's characteristic function. The prefactor collects \((\pp^{2}W/\pp q\pp q'')^{1/2}\), which for one degree of freedom is \((\pp p/\pp q'')^{1/2}\propto p(q)^{-1/2}\) after the reciprocity relation. This is Equation (106.18).

Derivation. Derives Proposition 106.13. With \(t=-\ii\tau\), \(\dd t=-\ii\,\dd\tau\) and \(\dd q/\dd t=\ii\,\dd q/\dd\tau\), so \(\frac{m}{2}(\dd q/\dd t)^{2}=-\frac{m}{2}(\dd q/\dd\tau)^{2}\) and

whence \(\ii S/\hbar=\ii\cdot\ii S_{E}/\hbar=-S_{E}/\hbar\).

Derivation. Derives Proposition 106.14. A propagator denominator \(p^{2}-m^{2}c^{2}+\ii\epsilon =(p^{0})^{2}-\vect{p}^{2}-m^{2}c^{2}+\ii\epsilon\) vanishes at

to first order in \(\epsilon\): the positive-energy pole sits just below the positive real axis and the negative-energy pole just above the negative real axis, i.e. in the fourth and second quadrants respectively. The first and third quadrants are therefore free of singularities, and the closed contour consisting of the real axis, the imaginary axis traversed downwards, and the two connecting arcs in those quadrants encloses no pole. By Cauchy's theorem (Theorem 8.12) its integral vanishes; the arcs contribute nothing by the estimation lemma Lemma 8.9 given the assumed fall-off. Rotating \(p^{0}=\ee^{\ii\theta}\abs{p^{0}}\) from \(\theta=0\) to \(\theta=\pi/2\) and writing \(p^{0}=\ii p^{4}\) gives \(\dd p^{0}=\ii\,\dd p^{4}\), which is Equation (106.21). Under it \(p^{2}=(p^{0})^{2}-\vect{p}^{2}\mapsto-(p^{4})^{2}-\vect{p}^{2} =-p_{E}^{2}\), so the denominator becomes \(-(p_{E}^{2}+m^{2}c^{2})\), which never vanishes for real \(p_{E}\): the Euclidean propagator has no poles at all, and no \(\ii\epsilon\) is needed.

Derivation. Derives Theorem 106.16. Repeat the slicing of Theorem 106.3 with \(\ee^{-H\tau/\hbar}\) in place of \(\ee^{-\ii Ht/\hbar}\); every step goes through with \(\ii\varepsilon\mapsto\Delta\tau\), giving

Now the prefactor and the kinetic term together are \(\prod_{k}\rho_{\Delta\tau}(q_{k}-q_{k-1})\) with \(\rho\) as stated, because \(m/(2\hbar\Delta\tau)=1/(4D\Delta\tau)\) with \(D\) of Equation (106.23). That product is precisely the finite-dimensional distribution of Brownian motion of diffusion constant \(D\) sampled at the times \(k\Delta\tau\), and by Kolmogorov's extension theorem it defines a genuine countably additive probability measure on continuous paths — Wiener's measure [Wiener:1923]. What remains in the exponent is \(-\frac{1}{\hbar}\sum\Delta\tau\,V\), the Riemann sum for \(\frac{1}{\hbar}\int V\), and the identification of the limit with the expectation is Kac's theorem [Kac:1949]. Conditioning on the endpoint \(q(\tau)=q''\) produces the bridge and the explicit factor \(\rho_{\tau}(q''-q')\).

Derivation. Derives Proposition 106.17. Insert the resolution of the identity in energy eigenstates into \(\ee^{-H\tau/\hbar}\); each term carries \(\ee^{-E_{n}\tau/\hbar}\) and the smallest exponent dominates. The subleading term is suppressed by \(\ee^{-(E_{1}-E_{0})\tau/\hbar}\).

Derivation. Derives Theorem 106.18. Take Equation (106.22) at \(\tau=\beta\hbar\), set \(q''=q'=q\) and integrate over \(q\): the left-hand side becomes \(\int\dd q\,\bra{q}\ee^{-\beta H}\ket{q}=\tr\ee^{-\beta H}\), and the right-hand side becomes the path integral restricted to closed paths. Fermionic degrees of freedom acquire antiperiodic boundary conditions instead, because the trace of a Grassmann bilinear picks up a sign; that computation is Section 106.3.3.

Derivation. Derives Proposition 106.21. Take \(F=\prod_{i}f_{i}(q(s_{i}))\) with \(0<s_{1}<\cdots<s_{n}\). By Theorem 106.16 the Euclidean expectation of a product of such factors is a time-ordered chain of operators separated by Euclidean evolutions,

because reflecting \(s\mapsto-s\) reverses the order of the factors and conjugates them, while the intervening evolution operators \(\ee^{-H\Delta s/\hbar}\) are positive and self-adjoint. The whole expression is therefore \(\norm{\Psi}^{2}\) with \(\Psi=\hat f_{1}\ee^{-H(s_{2}-s_{1})/\hbar}\cdots\ket{\Omega}\), manifestly non-negative. Linear combinations give Equation (106.30).

Read backwards, this is the content of the axiom: the Euclidean functional integral supplies a bilinear form on functionals of the positive-time half-path, and reflection positivity says exactly that this form is a non-negative inner product. The physical Hilbert space is the completion of that space modulo null vectors, the operator \(\ee^{-H\tau/\hbar}\) is the translation in Euclidean time, and \(H\) is self-adjoint and bounded below because it generates a contraction semigroup. Without Equation (106.30) one obtains a vector space with an indefinite form and no probability interpretation — and no amount of Euclidean invariance repairs it.

Derivation. Derives Theorem 106.25. Differentiating Equation (106.35),

so multiplying by \(\hbar c/\ii\) inserts one factor of \(\phi(x)\) into the integrand. Repeating gives \(n\) insertions; dividing by \(Z[0]\) normalizes. That the insertions come out time-ordered is a property of the slicing: the fields in Equation (106.9) are \(c\)-numbers attached to time slices, and reassembling them into operators reinstates them in the order of their time arguments, whatever order they were written in. This is the technical advantage of the formalism — the object it computes naturally is the object that appears in the Dyson series Equation (100.25) — and also its limitation, since it does not directly give an unordered product.

Derivation. Derives Proposition 106.26. From \(Z=\ee^{\ii W/\hbar}\), \(\delta Z/\delta J=(\ii/\hbar)(\delta W/\delta J)Z\), and combining with Equation (106.36) for \(n=1\) gives \(\avg{\phi}=(\hbar c/\ii)(\ii/\hbar)\delta W/\delta J =c\,\delta W/\delta J\). Differentiating once more,

so that, multiplying by \((\hbar c/\ii)^{2}=-\hbar^{2}c^{2}\),

and the first term is by definition the connected part.

For the general statement, argue combinatorially. Let the connected diagrams (with sources attached) be indexed by \(i\) with values \(C_{i}\). A general diagram is a disjoint union of connected pieces; if it contains \(n_{i}\) copies of the connected diagram \(i\), its symmetry factor includes \(1/n_{i}!\) from the permutations of identical components, because the Wick contractions that produce it differ only by relabelling those components. Summing over all multiplicities independently,

so \(\ln Z\) is the sum of connected diagrams alone. That is the linked-cluster theorem, and it is why the vacuum diagrams — those with no external source — cancel between numerator and denominator in Equation (106.36): they exponentiate into an overall factor \(Z[0]\).

Derivation. Derives Theorem 106.27. Integrate the kinetic term by parts, discarding the surface term:

The exponent of Equation (106.35) is then \(\frac{\ii}{\hbar c}\bigl[\frac{1}{2}\int\phi\mathcal{K}\phi +\int J\phi\bigr]\), a quadratic form in \(\phi\), and Lemma 106.2 applies once the square is completed. Put \(\phi=\phi_{0}+\chi\) with \(\phi_{0}(x)=-\int\dd^{4}y\;G(x-y)J(y)\), so that \(\mathcal{K}\phi_{0}=-J\) by Equation (106.40). Then

the cross term vanishing by construction. The shift \(\phi\mapsto\chi\) is a translation of the integration variable and leaves \(\mathcal{D}\phi\) invariant — this is a property of the sliced definition, each \(\dd\phi_{k}\) being a Lebesgue measure on \(\R\) — so the \(\chi\) integral is the \(J\)-independent constant \(Z_{0}[0]\) and what remains is Equation (106.39).

By Equation (106.37), \(W_{0}[J]=-\frac{1}{2c}\int\!\!\int JGJ\), and Equation (106.38) gives \(\avg{T\phi_{1}\phi_{2}} =(\hbar c^{2}/\ii)(-G/c)=\ii\hbar cG\).

In momentum space, with the convention Equation (100.14), \(\pp_{\mu}\mapsto-\ii p_{\mu}/\hbar\) and \(\Box\mapsto-p^{2}/\hbar^{2}\), so \(\mathcal{K}\mapsto(p^{2}-m^{2}c^{2})/\hbar^{2}\) and \(\widetilde{G}(p)=\hbar^{2}/(p^{2}-m^{2}c^{2})\). The pole on the real axis is resolved by the Feynman prescription \(m^{2}c^{2}\to m^{2}c^{2}-\ii\epsilon\), which is the choice singled out by Proposition 106.14. Multiplying by \(\ii\hbar c\) gives Equation (106.41). The result has exactly the structure of the photon propagator Equation (100.17), the factor \(\hbar^{3}c\) being common to both.

Derivation. Derives Corollary 106.28. Apply Equation (106.36) to Equation (106.39). Each derivative \(\hbar c\,\delta/(\ii\delta J)\) acting on the exponential brings down one factor \(-(\ii/\hbar c)\int G J\) times \(\hbar c/\ii\), i.e. \(-\int GJ\), which is set to zero at \(J=0\) unless a later derivative acts on it. A non-vanishing term therefore requires the derivatives to be used in pairs: one to bring down the bilinear, one to remove the surviving \(J\). The pair \((i,j)\) contributes \((\hbar c/\ii)^{2}\times(-\ii/\hbar c)G=\ii\hbar cG=\Delta_{F}\), and the number of ways of pairing \(2n\) objects is \((2n-1)!!\). An odd number of derivatives always leaves one \(J\) unpaired.

Derivation. Derives Proposition 106.29. Every occurrence of \(\phi(x)\) in \(S_{\mathrm{int}}\) may be replaced by \((\hbar c/\ii)\delta/\delta J(x)\) acting on the integrand of Equation (106.35), by the computation in Theorem 106.25; the resulting operator may then be pulled outside the \(\phi\) integral, leaving \(Z_{0}[J]\). That is Equation (106.43), which is exact. Expanding \(\exp(\ii S_{\mathrm{int}}/\hbar)\) in powers and applying Corollary 106.28 to each term produces sums of products of propagators with four lines meeting at each vertex position, each vertex carrying \((\ii/\hbar)\times(-1/c)\times(g/4!)\) and an integration \(\int\dd^{4}x\); the \(4!\) is cancelled by the number of ways of assigning the four lines to the four fields, except when a symmetry of the diagram leaves some of those assignments equivalent, which is the origin of the symmetry factor. Fourier transforming each propagator by Equation (100.14) turns the position integrals into the momentum-conserving delta functions of Equation (106.44), leaving one free integration per independent loop.

Derivation. Derives Theorem 106.30. By Proposition 106.29 each vertex carries an explicit \(\hbar^{-1}\) (from \(\ii S_{\mathrm{int}}/\hbar\)) and each internal line one factor of \(\hbar\) (the propagator \(\ii\hbar cG\) is linear in \(\hbar\), \(G\) being \(\hbar\)-independent by Equation (106.40) once \(mc/\hbar\) is regarded as the fixed inverse Compton wavelength). A diagram therefore carries \(\hbar^{I-V}\). The second relation of Equation (106.45) is Euler's relation for a connected graph: each internal line carries one momentum, the \(V\) vertices impose \(V\) delta functions of which one is the overall momentum conservation, leaving \(I-(V-1)\) undetermined loop momenta. Hence \(\hbar^{I-V}=\hbar^{L-1}\), and a tree, having \(L=0\), sets the reference power \(\hbar^{-1}\) against which the loops are counted.

Derivation. Derives Proposition 106.31. Suppose the series converged in some disc \(\abs{e^{2}}<r\). Then it would converge for real negative \(e^{2}\), that is for an imaginary charge, and by analyticity the sum would be the physical quantity of that theory. But with \(e^{2}<0\) like charges attract: the vacuum is unstable against the spontaneous creation of a large number of electron–positron pairs, which segregate into two clouds of like charge whose negative potential energy exceeds their rest energy once enough pairs are made. No lowest-energy state exists, so the theory has no ground state, no \(S\)-matrix, and no analytic continuation of the physical quantities to compare with. Hence the radius of convergence is zero.

Dyson's argument is physical, not a theorem: it establishes that the theory at \(e^{2}<0\) is pathological, and infers from this that the function of \(e^{2}\) cannot be analytic at the origin.

Derivation. Derives Proposition 106.33. Apply the two operators in both orders to the eigenstate: \(\hat\psi_{\alpha}(\vect{x})\hat\psi_{\beta}(\vect{y})\ket{\psi} =\psi_{\beta}(\vect{y})\psi_{\alpha}(\vect{x})\ket{\psi}\) — reading the operators from right to left — while the anticommutator gives \(\hat\psi_{\alpha}\hat\psi_{\beta} =-\hat\psi_{\beta}\hat\psi_{\alpha}\), whose eigenvalue is \(-\psi_{\alpha}(\vect{x})\psi_{\beta}(\vect{y})\). Equating the two gives Equation (106.48). Setting \(\alpha=\beta\), \(\vect{x}=\vect{y}\) gives \(\psi^{2}=-\psi^{2}=0\).

Derivation. Derives Theorem 106.35. By linearity the functional is fixed by the two numbers \(I_{0}=\int\dd\theta\,1\) and \(I_{1}=\int\dd\theta\,\theta\). Apply translation invariance to \(f=a+b\theta\):

for every \(b\) and every \(\eta\), which forces \(I_{0}=0\). Normalizing \(I_{1}=1\) then gives Equation (106.49), and since \(\pp(a+b\theta)/\pp\theta=b=aI_{0}+bI_{1}\), integration and differentiation coincide.

Translation invariance is not a technical convenience: it is exactly the property used in Theorem 106.27 to complete the square, and without it there would be no fermionic Gaussian integral and hence no fermionic path integral. It is also the property whose failure under a chiral rotation produces the anomaly of Section 106.7.3 — there the transformation is linear but not a translation, and its Jacobian is the whole story.

Derivation. Derives Corollary 106.36. Only the top monomial survives an \(n\)-fold integration, so the rule is fixed by \(\int\dd^{n}\theta'\,\theta'_{1}\cdots\theta'_{n}=1\). Expanding, \(\theta'_{1}\cdots\theta'_{n} =A_{1j_{1}}\cdots A_{nj_{n}}\theta_{j_{1}}\cdots\theta_{j_{n}} =(\det A)\,\theta_{1}\cdots\theta_{n}\), since reordering the \(\theta\)'s into canonical order produces the sign of the permutation and the sum over permutations with signs is the Leibniz formula for the determinant. To keep the normalization, \(\dd^{n}\theta'\) must carry \((\det A)^{-1}\).

Derivation. Derives Theorem 106.37. Expand the exponential. Because each generator squares to zero, the only term that survives the \(2n\)-fold integration is the one containing each of \(\theta_{1},\dots,\theta_{n}\) and each of \(\bar\theta_{1},\dots,\bar\theta_{n}\) exactly once, namely

the \(n!\) cancelling against the \(n!\) orderings of the identical factors. Reordering each \(\bar\theta_{k}\theta_{\sigma(k)}\) pair into the canonical order \(\theta_{k}\bar\theta_{k}\) produces \(\sgn(\sigma)\) together with \((-1)^{n}\), and the surviving sum is \(\sum_{\sigma}\sgn(\sigma)\prod_{k}A_{k\sigma(k)}=\det A\). The sourced version follows by the shift \(\theta\to\theta+A^{-1}\eta\), \(\bar\theta\to\bar\theta+\bar\eta A^{-1}\), which is legitimate by Theorem 106.35(ii) and has unit Jacobian, being a translation.

Derivation. Derives Proposition 106.38. The exponent is \(\ii S/\hbar\) with \(S=c^{-1}\int\dd^{4}x\,\Lag_{\psi}\), which is a Grassmann bilinear \(-\bar\psi\mathcal{A}\psi\) with \(\mathcal{A}=-(\ii/\hbar c)(\ii\hbar c\gamma^{\mu}D_{\mu}-mc^{2}) =-(\ii/\hbar)(\ii\hbar\gamma^{\mu}D_{\mu}-mc)\) after extracting the common factor \(c\). Apply Equation (106.51) mode by mode: the answer is \(\det\mathcal{A}\), and the field-independent factors \(-\ii/\hbar\) per mode contribute an infinite but \(A\)-independent constant, absorbed into the normalization.

Derivation. Derives Corollary 106.39. By Proposition 106.38, integrating out the fermions contributes \(\det\mathcal{A}\) to \(Z\), hence \(\tr\ln\mathcal{A}\) to \(\ln Z\) and \(-\ii\hbar\tr\ln\mathcal{A}\) to \(W\) by Equation (106.37). Expanding \(\tr\ln(\mathcal{A}_{0}+\delta\mathcal{A}) =\tr\ln\mathcal{A}_{0} +\sum_{k\geq1}\frac{(-1)^{k+1}}{k} \tr\bigl[(\mathcal{A}_{0}^{-1}\delta\mathcal{A})^{k}\bigr]\) produces, at order \(k\), a single trace around a closed chain of \(k\) propagators and \(k\) vertices: a one-loop diagram with \(k\) external legs. For a real boson the same manipulation applied to \((\det)^{-1/2}\) gives \(+\frac{1}{2}\tr\ln\), so the fermion loop differs by the factor \(-2\times\frac{1}{2}=-1\) per loop and by the appearance of a Dirac trace, the boson having no spinor index. The factor \(1/2\) absent in the fermionic case is the symmetry factor of a loop of a real field, which a complex fermion loop does not have because its two orientations are distinct.

Derivation. Derives Proposition 106.42. Vary Equation (106.54) with respect to \(\phi_{c}\), allowing \(J\) to vary with it:

the first two terms cancelling by the definition of \(\phi_{c}\).

Derivation. Derives Theorem 106.43. By definition \(\phi_{c}=c\,\delta W/\delta J\) and, by Equation (106.55), \(J=-c\,\delta\Gamma/\delta\phi_{c}\). The two are inverse maps, so by the chain rule

which is Equation (106.56). Differentiate it with respect to \(J(w)\), using \(\delta/\delta J(w)=c\int\dd^{4}u\;W_{uw}\,\delta/\delta\phi_{c}(u)\):

Contract with \(-c^{2}W_{yb}\) and integrate over \(y\); the left-hand side collapses by Equation (106.56) to \(W_{xbw}\), and the right-hand side becomes \(c^{3}\int W_{xz}W_{by}W_{wu}\Gamma_{zyu}\), which is Equation (106.57) after renaming.

A connected three-point function with a full propagator removed from each external leg cannot be cut into two pieces by severing one internal line, since any such cut would isolate one external leg and the piece so isolated is precisely the propagator already amputated. Hence \(\Gamma_{abd}\) is one-particle irreducible. Differentiating Equation (106.57) once more produces

in which the second group of terms is exactly the set of one-particle-reducible tree graphs built from two cubic vertices joined by one full propagator. The pattern repeats: at each order the new derivative of \(\Gamma\) supplies the irreducible part and the terms generated by differentiating the propagators supply all the reducible trees. Induction on \(n\) gives the general statement.

Derivation. Derives Theorem 106.44. Combining Equations (106.35), (106.37) and (106.54),

using Equation (106.55) to eliminate \(J\). Substitute \(\phi=\phi_{c}+\sqrt{\hbar}\,\chi\) and expand:

At order \(\sqrt{\hbar}\) the term linear in \(\chi\) cancels against the source term, because \(\delta\Gamma/\delta\phi_{c} =\delta S/\delta\phi_{c}+O(\hbar)\); what remains is a Gaussian integral over \(\chi\), evaluated by Equation (106.7) in the form \((\det)^{-1/2}=\exp(-\frac{1}{2}\tr\ln)\). Multiplying by \(\hbar/\ii\) to extract \(\Gamma\) turns \(-\frac{1}{2}\tr\ln\) into \(+\frac{\ii\hbar}{2}\tr\ln\), which is Equation (106.58). The substitution \(\phi=\phi_{c}+\sqrt{\hbar}\chi\) is what makes the \(\hbar\) counting automatic: each additional vertex in the expansion brings \(\hbar^{1/2}\) per field beyond the second, and the Gaussian contractions pair them up.

Derivation. Derives Proposition 106.45. By Equation (106.58) the one-loop term is \(\frac{\ii\hbar}{2}\tr\ln\mathcal{K}_{\phi}\) with \(\mathcal{K}_{\phi}=-(\Box+M^{2}c^{2}/\hbar^{2})\) read off from the second variation of Equation (106.32). For a translation-invariant kernel the trace is \(V_{4}\int\dd^{4}p\,(2\pi\hbar)^{-4}\) of the momentum-space symbol, so

and Equation (106.59) gives \(U_{1}=-c\Gamma_{1}/V_{4}\). Rotate by Proposition 106.14, \(\dd^{4}p=\ii\,\dd^{4}p_{E}\) and \(p^{2}=-p_{E}^{2}\):

up to a field-independent constant. The integral is quartically divergent and is evaluated in \(d=4-2\varepsilon\) dimensions by Lemma 100.28. Writing \(\Lambda^{2}=M^{2}c^{2}\) for the mass-squared parameter, the standard \(d\)-dimensional result is

\(\Gamma\) here denoting the Euler gamma function and not the effective action. Expanding its pole at \(d=4\) by Lemma 100.29 and discarding that pole together with the \(-\gamma_{\mathrm{E}} +\ln4\pi\) which accompanies it — the definition of minimal subtraction, Equation (100.75) — leaves \(\bigl(\Lambda^{2}\bigr)^{2} \bigl[\ln(\Lambda^{2}/\mu^{2})-3/2\bigr]/(32\pi^{2})\). Restoring the \(\hbar^{-4}\) of the measure and the prefactor \(\hbar c/2\) gives Equation (106.60) with \(\mathcal{E}=\mu c\).

Derivation. Derives Proposition 106.47. Insert a complete set of eigenstates of the four-momentum operator between the two fields in \(\bra{0}T\phi(x)\phi(y)\ket{0}\). Poincaré invariance and the spectrum condition (Theorem 105.34) organize them into the vacuum, which contributes only \(\abs{\bra{0}\phi\ket{0}}^{2}\) and is removed by normal-ordering the field; the one-particle states of mass \(m\), whose contribution is fixed by Lorentz invariance to be that of a free field of mass \(m\) times a constant \(Z\); and states whose invariant mass is at least \(2m\), summarized by a density \(\rho(s)\geq0\) supported on \(s\geq(2mc)^{2}\). The positivity of \(\rho\) is the positivity of the Hilbert-space norm, i.e. Equation (106.30) once more, and the sum of all contributions must reproduce the canonical equal-time commutator, which gives \(Z+\int\rho=1\) and hence \(Z\leq1\).

Derivation. Derives Theorem 106.48. Consider the dependence of \(\widetilde{G}_{n}\) on one momentum \(p_{1}\). By Proposition 106.47 applied to the leg carrying \(p_{1}\), the correlator has a simple pole at \(p_{1}^{2}=m^{2}c^{2}\) whose residue is \(\ii\hbar^{3}cZ\) times the matrix element of the remaining fields between the vacuum and the one-particle state \(\ket{p_{1}}\), divided by \(\sqrt{Z}\) — one factor of \(\sqrt{Z}\) being consumed by the overlap \(\bra{0}\phi(0)\ket{p_{1}}\) and the other remaining in the residue. Multiplying by \((p_{1}^{2}-m^{2}c^{2})\) and taking the limit isolates that residue and discards the multiparticle continuum, which has only a branch cut starting at \((2mc)^{2}\) and is regular at the pole. Repeating for each leg replaces every field by an asymptotic one-particle state; what remains between them is by definition the \(S\)-matrix element, which carries the overall momentum-conserving delta function.

The physical statement behind the algebra is that an asymptotic particle is a pole in a correlation function, and nothing else: the interacting field \(\phi\) need not create a one-particle state cleanly, and generically it creates a superposition of one-particle and multiparticle states. The prescription discards the second because it is analytically distinguishable from the first.

Derivation. Derives Corollary 106.49. The derivation above used only two properties of \(\phi\): that it is local, so the spectral decomposition Equation (106.61) applies to it; and that it has non-zero overlap with the one-particle state, so the pole is present. Any \(\mathcal{O}\) with those properties has the same pole at \(p^{2}=m^{2}c^{2}\), differing only in its residue, and the prescription divides that residue out. The \(S\)-matrix element is a property of the states, not of the operator used to reach them.

Derivation. Derives Proposition 106.51. \(S_{\mathrm{YM}}\) is invariant under Equation (106.67) by construction, and the measure \(\mathcal{D}\mathcal{A}\) is invariant because Equation (106.67) is, at each point, a translation of \(\mathcal{A}_{\mu}\) by \(\pp_{\mu}\omega\) plus a rotation in the adjoint index, and both preserve the Lebesgue measure on the fibre. The integral therefore factorizes as \(\bigl(\int_{\mathcal{G}}\mathcal{D}\omega\bigr)\times \bigl(\int_{\mathcal{A}/\mathcal{G}}\ee^{\ii S/\hbar}\bigr)\), the first factor being the divergent volume.

Derivation. Derives Theorem 106.52. For each \(\mathcal{A}\) define \(\Delta[\mathcal{A}]\) by

By hypothesis the delta functional is supported at a single \(\omega_{*}\), and the one-dimensional identity \(\int\dd x\,\delta(f(x))=\abs{f'(x_{*})}^{-1}\) generalizes to \(\Delta[\mathcal{A}]=\abs{\det(\delta G/\delta\omega)}_{\omega_{*}} =\abs{\det\mathcal{M}}\). Also \(\Delta\) is gauge invariant, since \(\mathcal{D}\omega\) is a left- and right-invariant Haar measure on \(\mathcal{G}\) and the defining integral is unchanged by relabelling the orbit representative. Insert \(1=\Delta[\mathcal{A}]\int\mathcal{D}\omega\, \delta[G[\mathcal{A}^{\omega}]-\sigma]\) into the left-hand side of Equation (106.68), exchange the order of integration, and change variables \(\mathcal{A}\to\mathcal{A}^{-\omega}\); the action, the measure and \(\Delta\) are all invariant, so the \(\omega\) integral factorizes out as the group volume and what remains is the right-hand side.

For the covariant condition, \(G^{a}[\mathcal{A}^{\omega}] =\pp^{\mu}\bigl(\mathcal{A}^{a}_{\mu}+(D_{\mu}\omega)^{a}\bigr)\) by Equation (106.67), whose derivative with respect to \(\omega^{b}\) is \(\pp^{\mu}(D_{\mu})^{ab}\), i.e.\ Equation (106.69). The component form of the adjoint covariant derivative follows from \(\ii\comm{\mathcal{A}_{\mu}}{\omega} =\ii\mathcal{A}^{c}_{\mu}\omega^{b}\ii f^{cba}T^{a} =-f^{cba}\mathcal{A}^{c}_{\mu}\omega^{b}T^{a}\), and \(f^{cba}=-f^{bca}=-f^{abc}\) by total antisymmetry and cyclicity, so that \((D_{\mu}\omega)^{a} =\pp_{\mu}\omega^{a}+f^{abc}\mathcal{A}^{c}_{\mu}\omega^{b}\) — the plus sign of Equation (106.69). Equivalently, the adjoint generators are \((T^{a}_{\mathrm{adj}})^{bc}=-\ii f^{abc}\), as they are for \(\SU(3)\) in Proposition 14.55, and \(\delta^{bc}\pp_{\mu}+\ii\mathcal{A}^{a}_{\mu} (T^{a}_{\mathrm{adj}})^{bc} =\delta^{bc}\pp_{\mu}+f^{abc}\mathcal{A}^{a}_{\mu}\) is the same operator with the summed index moved by one cyclic step.

Derivation. Derives Theorem 106.54. The left-hand side of Equation (106.68) does not depend on \(\sigma\), so it may be averaged over \(\sigma\) with any normalized weight; the Gaussian one converts \(\delta[G-\sigma]\) into \(\exp\bigl(\ii S_{\mathrm{gf}}/\hbar\bigr)\) with \(S_{\mathrm{gf}}\) as stated, by Lemma 106.2. The determinant is written as a Grassmann integral by reading Equation (106.51) from right to left with \(A=\mathcal{M}\) — and it is worth noting that this is the one place in physics where the exponent of a determinant being \(+1\) rather than \(-\frac{1}{2}\) decides the statistics of a field: the Jacobian sits in the numerator, so the field that represents it must be the one whose Gaussian integral gives \(\det\) and not \((\det)^{-1/2}\), i.e. an anticommuting one, by Theorem 106.37. The ghosts are spin-zero fields obeying Fermi statistics, which is why they cannot be physical: they violate the spin–statistics connection of Identical Particles.

Written as a single exponent \(\ii S_{\mathrm{tot}}/\hbar\), the ghost term is \(S_{\mathrm{gh}}=\ii\hbar\int\dd^{4}x\, \bar c^{a}(\pp^{\mu}D_{\mu})^{ab}c^{b}\), whose explicit factor of \(\ii\) records that the ghost action is not real. Nothing is wrong: an action is required to be real only when its exponential must be a phase for physical configurations, and the ghosts have no physical configurations. The dimension assignment follows from requiring the exponent in Equation (106.70) to be a pure number: \(\int\dd^{4}x\) supplies \(\mathrm{m}^{4}\), \(\pp^{\mu}D_{\mu}\) supplies \(/\mathrm{m}^{2}\), so \(\bar cc\) must supply \(/\mathrm{m}^{2}\).

Derivation. Derives Proposition 106.55. A massless vector of momentum \(k\) has two physical transverse polarizations, whose sum is \(\sum_{\lambda=1,2}\varepsilon^{\mu}_{(\lambda)} \varepsilon^{*\nu}_{(\lambda)} =-\eta^{\mu\nu}+(k^{\mu}n^{\nu}+n^{\mu}k^{\nu})/(k\cdot n)\) for a lightlike auxiliary \(n\). In a covariant gauge the propagator supplies \(-\eta^{\mu\nu}\), which is the same object plus the longitudinal and timelike pair; those two do not cancel of their own accord when the external current is not conserved, and in a non-abelian theory it is not conserved diagram by diagram, because the gluon carries colour and its own current is only covariantly conserved. The discrepancy is a two-state contribution of positive norm. The ghost loop supplies two scalar states (\(c\) and \(\bar c\)) with the sign \((-1)\) of a closed anticommuting loop, i.e. \(-2\) states, which cancels it identically. The all-orders statement — that the cancellation holds to every order and for every process — is the Kugo–Ojima quartet mechanism of Section 106.4.2, and it is a theorem about the BRST cohomology rather than a diagrammatic accident.

Derivation. Derives Theorem 106.57. On the ghost: \(s^{2}c=-\ii\bigl[(sc)c-c(sc)\bigr] =-\ii\bigl[(-\ii c^{2})c-c(-\ii c^{2})\bigr] =-c^{3}+c^{3}=0\), the sign in the middle being that of \(s\) passing the odd \(c\). Associativity of matrix multiplication has done here exactly the work that the Jacobi identity does in component notation, which is the reason for writing the algebra in matrix form.

On the connection, expand \(s^{2}\mathcal{A}_{\mu}=s\bigl(\pp_{\mu}c +\ii\comm{\mathcal{A}_{\mu}}{c}\bigr)\). Since \(\mathcal{A}\) is even and \(c\) odd, \(s\comm{\mathcal{A}_{\mu}}{c} =\acomm{s\mathcal{A}_{\mu}}{c}+\comm{\mathcal{A}_{\mu}}{sc}\), so

Insert \(sc=-\ii c^{2}\) and use \(\pp_{\mu}(c^{2})=\acomm{\pp_{\mu}c}{c}\) and \(\acomm{\comm{\mathcal{A}_{\mu}}{c}}{c} =\comm{\mathcal{A}_{\mu}}{c^{2}}\):

On \(\bar c\) and \(B\) nilpotency is immediate from \(sB=0\). Since \(s\) is a derivation and vanishes on the square of every generator, \(s^{2}=0\) on any polynomial in the fields.

Derivation. Derives Theorem 106.58. Applying \(s\) to Equation (106.73) and using Equation (106.72),

the sign of the second term coming from \(s\) passing the odd \(\bar c\). The auxiliary field is algebraic and may be integrated out, its equation of motion being \(B^{a}=\pp^{\mu}\mathcal{A}^{a}_{\mu}/\xi\); the first bracket then becomes \((\pp^{\mu}\mathcal{A}^{a}_{\mu})^{2}/2\xi\) and, with the prefactor, is exactly \(S_{\mathrm{gf}}\) of Equation (106.71). The second term is \(+(\hbar/\hat g^{2})\int\dd^{4}x\, \bar c^{a}\pp^{\mu}(D_{\mu}c)^{a}\), which enters the weight \(\exp(\ii S_{\mathrm{tot}}/\hbar)\) as \(\exp\bigl[(\ii/\hat g^{2})\int\bar c\,\pp^{\mu}D_{\mu}c\bigr]\), whereas Equation (106.70) carries \(\exp\bigl[-\int\bar c\,\pp^{\mu}D_{\mu}c\bigr]\). The two differ by the constant factor \(-\ii/\hat g^{2}\) multiplying the ghost bilinear, and are brought into coincidence by rescaling the pair, \(c\mapsto\kappa c\) and \(\bar c\mapsto\kappa\bar c\) with \(\kappa^{2}=\ii\hat g^{2}\). It must be the pair and not one against the other: the ghosts enter only through the product \(\bar cc\), so a rescaling \(c\mapsto\kappa c\), \(\bar c\mapsto\bar c/\kappa\) leaves the ghost term exactly as it was and cannot fix a normalization. The rescaling used here multiplies the Grassmann measure by a field-independent constant, by Corollary 106.36, which cancels between numerator and denominator in every normalized expectation value — equivalently, \(\det(\lambda\mathcal{M})=\lambda^{n}\det\mathcal{M}\) and only ratios of determinants are ever used. Finally \(sS_{\mathrm{YM}}=0\) because \(s\) acts on \(\mathcal{A}\) as a gauge transformation with parameter \(c\), and \(s(s\Psi)=s^{2}\Psi=0\) by Theorem 106.57.

Derivation. Derives Proposition 106.60. From Equation (106.72) the free-field limit of the transformations is \(s\mathcal{A}_{\mu}=\pp_{\mu}c\), \(sc=0\), \(s\bar c=B\), \(sB=0\): the ghost \(c\) is \(Q_{B}\)-closed and maps the longitudinal polarization; the antighost \(\bar c\) maps onto \(B\), which is the timelike combination on shell. So among the four states \(\set{\mathcal{A}_{L},\mathcal{A}_{0},c,\bar c}\), two are \(Q_{B}\)-exact and two are not \(Q_{B}\)-closed. A physical state, being \(Q_{B}\)-closed, has zero inner product with a \(Q_{B}\)-exact one (since \(\braket{\Phi}{Q_{B}\Lambda}=\braket{Q_{B}\Phi}{\Lambda}=0\) by the hermiticity of \(Q_{B}\)), and the states that are not \(Q_{B}\)-closed are excluded from \(\mathcal{H}_{\mathrm{phys}}\) by definition. Hence the quartet decouples entirely, leaving the two transverse polarizations, which is Proposition 106.55 at all orders. The inner product induced on \(\mathcal{H}_{\mathrm{phys}}\) is therefore positive definite and the \(S\)-matrix restricted to it is unitary [Kugo:1979].

Derivation. Derives Theorem 106.61. By Equation (106.74) the entire dependence on the gauge-fixing choice is in \(\Psi\). Let \(\mathcal{O}\) satisfy \(s\mathcal{O}=0\) and vary \(\Psi\to\Psi+\delta\Psi\). Then

the second step because \(s\mathcal{O}=0\) and the last because the expectation value of a BRST variation vanishes: the measure and the action are \(s\)-invariant, so \(\avg{sX}=0\) for every \(X\), by the same argument that gives a Ward identity from any symmetry of the functional integral. In particular the \(\xi\)-dependence of Equation (106.71) cancels out of every physical quantity, which is the checkable statement quoted at Equation (100.18).

Derivation. Derives Theorem 106.62. Change variables in Equation (106.70) by an infinitesimal BRST transformation with a constant anticommuting parameter \(\zeta\): \(\Phi\to\Phi+\zeta\,s\Phi\) for every field \(\Phi\). The action is invariant by Theorem 106.58. The measure is invariant too, and this requires proof rather than assertion: the Jacobian of the transformation is \(\det\bigl(1+\zeta\,\pp(s\Phi)/\pp\Phi\bigr) =1+\zeta\,\tr\bigl(\pp(s\Phi)/\pp\Phi\bigr)\) with a sign for each Grassmann direction, and the bosonic and fermionic traces cancel — the \(\mathcal{A}\)–\(c\) block against the \(c\)–\(c\) block — because \(s\) maps a boson to a ghost and a ghost to a ghost bilinear with matched coefficients. (This cancellation is exactly what fails for a chiral rotation of the fermion measure, and its failure is the anomaly of Section 106.7.3; a gauge anomaly is the statement that it fails here too, and Section 106.7.4 is the condition that it does not.) With action and measure invariant, the integral of a total BRST variation vanishes, which is Equation (106.76).

In the abelian case the ghosts decouple by Corollary 106.53 and Equation (106.72) collapses to \(s\mathcal{A}_{\mu}=\pp_{\mu}c\) with \(c\) a free field, so that a BRST variation is a gauge transformation whose parameter is an anticommuting constant times that free field. Taking \(X=\bar c(y)\,\psi(x_{1})\bar\psi(x_{2})\) then turns Equation (106.76) into the relation between the vertex function and the inverse propagator, \(q_{\mu}\Gamma^{\mu}(p',p)=S^{-1}(p')-S^{-1}(p)\), whose derivation is carried out in full at Theorem 100.48.

Derivation. Derives Corollary 106.63. Add to the action the terms \(\int\dd^{4}x\,\bigl(K^{\mu}_{a}\,s\mathcal{A}^{a}_{\mu} +L_{a}\,sc^{a}\bigr)\). They are BRST invariant by Theorem 106.57, since \(s\) applied to them gives \(K\,s^{2}\mathcal{A}+L\,s^{2}c=0\), so the extended action is still \(s\)-invariant and Equation (106.76) still holds. Take \(X=\Phi\) for each field in turn: the resulting identities say that \(W\) is annihilated by the operator \(\int\dd^{4}x\,\bigl[(\delta W/\delta K^{\mu}_{a}) (\delta/\delta j^{a}_{\mu})+\cdots\bigr]\) acting on the sources. Legendre-transforming in the field variables only — \(K\) and \(L\) are carried along untouched, so that \(\delta\Gamma/\delta K=\delta W/\delta K\) — converts the source derivatives into field derivatives of \(\Gamma\) and produces Equation (106.77). The equation is bilinear in \(\Gamma\) because the composite operators \(s\mathcal{A}\) and \(sc\) are themselves products of two fields, so their sources cannot be eliminated linearly.

Derivation. Derives Proposition 106.64. Under Equation (106.67), \(\pp^{\mu}\mathcal{A}^{\omega}_{\mu} =\pp^{\mu}\mathcal{A}_{\mu}+\pp^{\mu}(D_{\mu}\omega)\), so preserving the condition requires Equation (106.78). In the Euclidean theory the Landau condition \(\pp^{\mu}\mathcal{A}_{\mu}=0\) makes \(-\mathcal{M}\) Hermitian on square-integrable functions, with \(-\mathcal{M}^{ab} =-\delta^{ab}\pp^{2}-f^{abc}\mathcal{A}^{c\mu}\pp_{\mu}\) by Equation (106.69), so its spectrum is real; at \(\mathcal{A}=0\) it is the positive operator \(-\pp^{2}\) on non-constant modes. As the amplitude of \(\mathcal{A}\) is increased along any ray in field space the eigenvalues move continuously, and Gribov exhibited explicit configurations for which the lowest one crosses zero, at which point Equation (106.78) has a solution and the orbit meets the gauge-fixing surface again.

Derivation. Derives Proposition 106.67. Each link carries \(\Omega\) at its start and \(\Omega^{\dagger}\) at its end, so in a product of links forming a path the factors at every interior site cancel, \(\Omega^{\dagger}(m)\Omega(m)=1\). For a closed path the two remaining factors sit at the same site and are removed by the cyclicity of the trace.

Derivation. Derives Theorem 106.68. Multiplying the four exponentials of Equation (106.80) and applying the Baker–Campbell–Hausdorff formula, the leading term of the exponent is the discrete curl of \(\mathcal{A}\), so \(U_{p}=\exp\bigl[\ii a^{2}\mathcal{F}_{\mu\nu}+O(a^{3})\bigr]\) with \(\mathcal{F}_{\mu\nu}=\mathcal{F}^{a}_{\mu\nu}T^{a}\) evaluated at the centre of the plaquette — this is the finite version of Equation (106.64), and the commutator term of Equation (106.64) arises here from the non-commutativity of the four link matrices. Expanding,

with no sum over \(\mu,\nu\), using \(\tr(T^{a}T^{b})=\frac{1}{2}\delta^{ab}\). The sum over plaquettes is a sum over sites and over \(\mu<\nu\), which becomes \(a^{-4}\int\dd^{4}x_{E}\,\frac{1}{2}\sum_{\mu\nu}\); hence \(S_{W}\to(\beta/8N)\int\dd^{4}x_{E}\, \mathcal{F}^{a}_{\mu\nu}\mathcal{F}^{a}_{\mu\nu}\), and matching to Equation (106.82) gives \(\beta=2N/\hat g^{2}\).

Derivation. Derives Proposition 106.71. The two timelike sides of the rectangle are the parallel transporters that carry a static colour source forward in Euclidean time; the two spacelike sides create and annihilate the pair. By Proposition 106.17 the correlator of a creation and an annihilation operator separated by Euclidean time \(\tau\) decays as \(\ee^{-E_{0}\tau/\hbar}\), with \(E_{0}\) the lowest energy in the sector created — here the energy of the static pair, \(V(R)\), measured from the vacuum. Substituting \(A=Rc\tau\) for the enclosed area converts an exponential in the area into an exponential in \(V\tau\) with \(V=\sigma R\). Dimensions: \(\sigma\) carries \(\mathrm{N}\), so \(\sigma A/(\hbar c)\) is dimensionless.

Derivation. Derives Theorem 106.72. Expand \(\ee^{-S_{W}}\) in powers of \(\beta\): for each plaquette, \(\exp\bigl[(\beta/N)\Re\tr U_{p}\bigr] =1+\frac{\beta}{2N}\bigl(\tr U_{p}+\tr U_{p}^{\dagger}\bigr)+O(\beta^{2})\). The integration is over independent Haar measures on each link, and the two identities that are needed are

the first because the integrand transforms in a non-trivial representation and the Haar measure is invariant, the second by Schur's first lemma (Theorem 5.153) applied to the invariant tensor. Every link of the contour must therefore be paired with a link of opposite orientation supplied by some plaquette; the cheapest way to do this is to tile the minimal surface spanned by \(C\) with \(A/a^{2}\) plaquettes, each bringing \(\beta/2N\) from the expansion and each internal link contributing \(1/N\) from the second identity. The single-plaquette case is exact and does the counting transparently:

since \(\int\dd U\,\tr U\tr U^{\dagger} =\sum_{ij}\delta_{ij}\delta_{ij}/N=1\). Iterating over the tiling gives Equation (106.84), and comparing with Equation (106.83) gives \(\sigma\).

That counting used \(\int\dd U\,U_{ij}U_{kl}=0\), i.e.\ \(\int\dd U\,(\tr U)^{2}=0\), which holds only when \(\vect{N}\otimes\vect{N}\) contains no singlet. For \(N\geq3\) it splits into the symmetric and antisymmetric two-index representations and neither is trivial, so the identity holds. For \(\SU(2)\), \(\vect{2}\otimes\vect{2}=\vect{3}\oplus\vect{1}\) does contain the singlet, \(\tr U\) is real, and orthogonality of characters gives \(\int\dd U\,(\tr U)^{2}=1\). The plaquette factor is then \((\beta/2N)(\tr U_{p}+\tr U_{p}^{\dagger})=\tfrac{\beta}{2}\tr U_{p}\) and

which is Equation (106.85): twice the value the generic formula assigns, hence \(\ln(4/\beta)\) in place of \(\ln(8/\beta)\) in the string tension.

Derivation. Derives Proposition 106.74. Fourier transform the difference operator: the shift \(\psi(n\pm\hat\mu)\) becomes \(\ee^{\pm\ii ap_{\mu}/\hbar}\psi\), so the symmetric difference becomes \(\ii\sin(ap_{\mu}/\hbar)/a\), giving Equation (106.87). The propagator is \(\widetilde{D}^{-1}\), and a pole occurs wherever \(\widetilde{D}\) vanishes; \(\sin(ap_{\mu}/\hbar)=0\) has two solutions in the Brillouin zone \(-\pi<ap_{\mu}/\hbar\leq\pi\) for each of the four directions, hence \(2^{4}\) poles, each of which is a physical particle in the continuum limit.

Derivation. Derives Proposition 106.76. The estimator is the sample mean of a random variable whose expectation is \(\avg{\mathcal{O}}\) by construction, so it is unbiased; its variance is \(\mathrm{Var}(\mathcal{O})/M\), and the central limit theorem gives the distribution of the error. Nothing in either statement refers to the dimension. What replaces the dimension in the cost is the autocorrelation of the algorithm that generates the sample, since successive configurations are not independent.

Derivation. Derives Phenomenon 106.77. The procedure is Equation (106.24) applied to a lattice correlator. Choose an interpolating operator \(\mathcal{O}\) with the quantum numbers of the hadron — for the nucleon, a colour-singlet product of three quark fields; Corollary 106.49 guarantees that the choice does not affect the answer. Measure

by Proposition 106.17, and read the mass off the exponential decay rate. Repeat at several lattice spacings, volumes and quark masses, and extrapolate.

The counting of inputs is what makes this a prediction rather than a fit. The Lagrangian of the light sector contains exactly three parameters that are not fixed by symmetry: \(\hat g^{2}\), the degenerate up–down mass, and the strange mass. Measuring \(m_{\pi}\), \(m_{K}\) and one baryon mass fixes all three (the coupling enters only through the overall scale, by dimensional transmutation — The Renormalization Group). Every other hadron mass is then output. The masses to be reproduced include, from the shipped Particle Data Group compilation [Navas:2024], \(m_{p}c^{2}=938.272\,\mathrm{MeV}\), \(m_{n}c^{2}=939.565\,\mathrm{MeV}\), \(m_{\Lambda}c^{2}=1115.683\,\mathrm{MeV}\) and \(m_{\Delta}c^{2}=1232.0(2.0)\,\mathrm{MeV}\), together with the \(\Sigma\), \(\Xi\), \(\Sigma^{*}\), \(\Xi^{*}\) and \(\Omega\); the computed values agree with all of them within the combined statistical and systematic uncertainties, which are at the level of a few percent [Duerr:2008].

Derivation. Derives Theorem 106.80. The Euclidean action is \(S_{E}=(\hbar/4\hat g^{2})\int\dd^{4}x_{E}\, \mathcal{F}^{a}_{\mu\nu}\mathcal{F}^{a}_{\mu\nu}\) by Equation (106.82), and the integrand is a sum of squares, hence non-negative. Consider

using \(\widetilde{\mathcal{F}}\widetilde{\mathcal{F}} =\mathcal{F}\mathcal{F}\), which follows from \(\epsilon_{\mu\nu\rho\sigma}\epsilon_{\mu\nu\alpha\beta} =2(\delta_{\rho\alpha}\delta_{\sigma\beta} -\delta_{\rho\beta}\delta_{\sigma\alpha})\) in four Euclidean dimensions. Hence \(\int\mathcal{F}\mathcal{F}\geq \abs{\int\mathcal{F}\widetilde{\mathcal{F}}}=32\pi^{2}\abs{\nu}\) by Equation (106.91), with equality iff \(\mathcal{F}=\pm\widetilde{\mathcal{F}}\). Multiplying by \(\hbar/4\hat g^{2}\) gives Equation (106.92). A self-dual field automatically solves the Euclidean field equations, since \(D_{\mu}\mathcal{F}_{\mu\nu} =D_{\mu}\widetilde{\mathcal{F}}_{\mu\nu}=0\) by the Bianchi identity; self-duality, a first-order condition, therefore replaces the second-order equations of motion.

Numerically, \(2\pi/0.1180=53.24\) and \(\ee^{-53.24}=8\times10^{-24}\).

Derivation. Derives Proposition 106.81. The integrand of Equation (106.91) is a total divergence, \(\epsilon\tr(\mathcal{F}\mathcal{F})=\pp_{\mu}K_{\mu}\) with the Chern–Simons current \(K_{\mu}=2\epsilon_{\mu\nu\rho\sigma} \tr\bigl(\mathcal{A}_{\nu}\pp_{\rho}\mathcal{A}_{\sigma} +\frac{2\ii}{3}\mathcal{A}_{\nu}\mathcal{A}_{\rho} \mathcal{A}_{\sigma}\bigr)\); this is the transgression form of the second Chern class, and it is Equation (14.155) of Theorem 14.111 specialized to \(r=2\). The volume integral is therefore a surface integral over the sphere at infinity, where \(\mathcal{A}\) is pure gauge; substituting \(\mathcal{A}_{\mu}=-\ii g^{-1}\pp_{\mu}g\) makes the first term of \(K_{\mu}\) combine with the second into Equation (106.94).

That the resulting number is an integer is the degree theorem. Let \(y\) be a regular value of the smooth map \(g:S^{3}\to S^{3}\); its preimage is a finite set of points, and Equation (106.94) evaluates to the sum over those points of the sign of the Jacobian determinant of \(g\), i.e. to the number of times \(g\) covers the target counted with orientation. That count is an integer, and it is unchanged under continuous deformation of \(g\), since the preimages can only be created or destroyed in pairs of opposite sign. The map \(g(x)=\bigl(x_{4}+\ii\,\vect{x}\cdot\vect{\tau}\bigr)/\abs{x}\), with \(\vect{\tau}\) the Pauli matrices, covers \(\SU(2)\cong S^{3}\) exactly once and has \(\nu=1\): it is the identity map, whose Jacobian is everywhere positive.

Derivation. Derives Proposition 106.83. Gauge transformations that tend to the identity at infinity are generated by the Gauss constraint and act trivially on physical states. Those that tend to a non-trivial element — the large transformations, classified by Proposition 106.81 — are symmetries of the theory but are not required to act trivially, exactly as a translation by a lattice vector in a crystal is a symmetry that acts non-trivially on Bloch states. \(T\) is unitary and commutes with \(H\), so \(H\) and \(T\) may be diagonalized together; the eigenvalues of a unitary operator with \(T\ket{n}=\ket{n+1}\) are phases \(\ee^{-\ii\theta}\), and the corresponding eigenvector is Equation (106.95). Because a local operator changes \(n\) by a finite amount, its matrix elements between \(\ket{\theta}\) and \(\ket{\theta'}\) are proportional to \(\delta(\theta-\theta')\): the \(\theta\) sectors are superselected and \(\theta\) is a parameter of the theory, not a state to be chosen.

Derivation. Derives Corollary 106.84. Summing over sectors with the weights of Equation (106.95) inserts \(\ee^{\ii n\theta}\) into the path integral for each topological class \(n\). By Definition 106.79 that class is labelled by \(\nu=\int\dd t\,\dd^{3}x\;q\), so the insertion is \(\exp\bigl(\ii\theta\int\dd t\,\dd^{3}x\,q\bigr) =\exp\bigl(\tfrac{\ii}{\hbar}\cdot\tfrac{1}{c}\int\dd^{4}x\; \theta\hbar q\bigr)\), which is the addition of Equation (106.96) to \(\Lag\). That \(\Lag_{\theta}\) has the dimension of an energy density follows from \([q]=/\mathrm{m}^{3}/\mathrm{s}\) and \([\hbar] =\mathrm{J}\,\mathrm{s}\).

Derivation. Derives Proposition 106.85. The Dirac equation in an external field is \((\ii\hbar\gamma^{\mu}D_{\mu}-mc)\psi=0\), and its conjugate reads \(\ii\hbar\,(D_{\mu}\bar\psi)\gamma^{\mu}+mc\,\bar\psi=0\), the derivative acting on \(\bar\psi\). Then

using \(\acomm{\gamma^{\mu}}{\gamma^{5}}=0\) to move \(\gamma^{5}\) past \(\gamma^{\mu}\) in the second term, which flips the sign of the mass contribution and makes the two add rather than cancel. Multiplying by \(c\) gives Equation (106.98). The gauge field drops out because the covariant derivatives combine into an ordinary one in a neutral bilinear.

Derivation. Derives Lemma 106.86. Expand \(f(\ell+a)-f(\ell)=a_{\mu}\pp^{\mu}f+O(a^{2})\) — the higher terms give integrals of higher divergences of \(f\) and are handled the same way — and apply the divergence theorem to the ball \(\abs{\ell}\leq R\). The surface element grows as \(R^{3}\), so the limit is zero, finite, or infinite according as \(f\) falls faster than, exactly as, or slower than \(R^{-3}\).

Derivation. Derives Proposition 106.88. The integrand of the triangle falls as \(\abs{\ell}^{-3}\) at large loop momentum, since it is a product of three fermion propagators (one power of \(\ell\) each in the denominator, one in the numerator from each \(\gamma\cdot\ell\)) — the superficial degree of divergence is \(D=+1\) by the counting of Theorem 100.17 applied to three external boson lines and a closed fermion loop. By Lemma 106.86 a shift of the routing therefore changes the amplitude by a finite, non-zero, momentum-dependent constant. The three Ward identities — one per vertex — are each linear conditions on that constant, and the constant has fewer free parameters than there are conditions: their sum is fixed independently of the routing and is non-zero. The routing can distribute the failure among the three vertices but cannot remove it. Since the vector current is coupled to the photon and its non-conservation would destroy gauge invariance and unitarity (Section 106.7.4), the failure is assigned entirely to the axial vertex.

The same statement in the language of regulators: Pauli–Villars requires a heavy fermion of mass \(M\), and a Dirac mass term \(M\bar\psi\psi\) is not chirally invariant, so the regulator breaks the axial symmetry by construction and the breaking survives \(M\to\infty\) by Equation (106.98). Dimensional regularization requires \(\gamma^{5}\) in \(d\neq4\) dimensions, where a totally antisymmetric product of four gamma matrices is not available. A lattice regulator is defeated by Theorem 106.75. Every route fails in a different way, which is itself the evidence that the obstruction is physical rather than technical.

Derivation. Derives Proposition 106.90. Apply the commutator of two gauge generators to the single functional \(W\) and use the algebra: \(\comm{G_{a}(x)}{G_{b}(y)}W =\ii f_{abc}\delta^{4}(x-y)G_{c}(x)W\). Writing out the left-hand side as \(G_{a}(x)\bigl(G_{b}(y)W\bigr)-G_{b}(y)\bigl(G_{a}(x)W\bigr)\) and substituting the definition of \(\mathsf{A}\) gives Equation (106.102). The content is that the anomaly is a cocycle of the gauge group: it is not an arbitrary functional but one constrained by the group structure, which is why the possible anomalies in four dimensions form a one-parameter family and the coefficient is the only thing left to compute.

Derivation. Derives Phenomenon 106.91. The anomaly Equation (106.101), applied to the third component of the axial isospin current of the light quarks and combined with the partial conservation of that current, produces an effective interaction between the neutral pion field and two photons,

with \(\widetilde{F}^{\mu\nu} =\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}\) and \(g_{\pi\gamma\gamma}\) of dimension (energy)\(^{-1}\) once the fields are normalized as in Notation 106.23. The colour factor is \(N_{c}\bigl(e_{u}^{2}-e_{d}^{2}\bigr) =N_{c}\bigl(\tfrac{4}{9}-\tfrac{1}{9}\bigr)=\tfrac{N_{c}}{3}\): each quark flavour runs around the triangle once for each colour, weighted by the square of its electric charge, and the two light flavours enter with opposite sign because the current is the third isospin component. That is where the sensitivity to \(N_{c}\) comes from, and it is the only place in this book where a decay rate counts colours directly.

Given Equation (106.106), the width follows from kinematics. The amplitude for \(\pi^{0}(p)\to\gamma(k_{1},\varepsilon_{1})\gamma(k_{2}, \varepsilon_{2})\) is \(\mathcal{M}=g_{\pi\gamma\gamma}\,\epsilon^{\mu\nu\rho\sigma} k_{1\mu}\varepsilon^{*}_{1\nu}k_{2\rho}\varepsilon^{*}_{2\sigma}\). Sum over photon polarizations with \(\sum\varepsilon^{*}_{\mu}\varepsilon_{\nu}\to-\eta_{\mu\nu}\) — legitimate because \(\mathcal{M}\) vanishes when either \(\varepsilon\) is replaced by its momentum. The two metric factors enter with \((-1)^{2}=+1\), and the two Levi-Civita symbols contract by

in the signature \(\diag(+1,-1,-1,-1)\) with \(\epsilon^{0123}=+1\), so that

the first term vanishing because both photons are massless. The coefficient is \(2\), not \(8\): the same number comes out of summing \(\abs{\mathcal{M}}^{2}\) over the two physical transverse polarizations of each photon directly, without the covariant substitution. On the mass shell \(2k_{1}\cdot k_{2}=p^{2}=m_{\pi}^{2}c^{2}\). Measuring every momentum of this kinematic step as an energy, \(k\mapsto ck\), that reads \(k_{1}\cdot k_{2}=(m_{\pi}c^{2})^{2}/2\) and makes \(g_{\pi\gamma\gamma}\) an inverse energy throughout, whence \(\sum\abs{\mathcal{M}}^{2} =\tfrac{1}{2}g_{\pi\gamma\gamma}^{2}(m_{\pi}c^{2})^{4}\). The two-body massless phase-space volume is \(1/8\pi\), the flux factor for a decay at rest is \(1/(2m_{\pi}c^{2})\), and the two photons are identical so the result is halved:

a rate, since \(g_{\pi\gamma\gamma}^{2}(m_{\pi}c^{2})^{3}\) is an energy and \(\hbar\) divides it. The width is \(\hbar\) times that rate, \(\Gamma=g_{\pi\gamma\gamma}^{2}(m_{\pi}c^{2})^{3}/64\pi\), and substituting \(g_{\pi\gamma\gamma}\) from Equation (106.106) gives Equation (106.103). That is the only place \(\hbar\) enters, and it is why the displayed width carries none.

Arithmetic, in \(\mathrm{MeV}\): \(\alpha^{2}=5.32514\times 10^{-5}\); \((m_{\pi}c^{2})^{3}=2.4591\times 10^{6}\,\mathrm{MeV}^{3}\), the cube of \(134.9768\,\mathrm{MeV}\); \(64\pi^{3}=1984.40\); and \(F_{\pi}^{2}=8476.0\,\mathrm{MeV}^{2}\), the square of \(130.2\,\mathrm{MeV}/\sqrt2\). Then

which is Equation (106.104), and \(\Gamma/\hbar=7.79\,\mathrm{eV}/6.582119569\times 10^{-16}\,\mathrm{eV}\,\mathrm{s} =1.18\times 10^{16}\,/\mathrm{s}\). The prediction and the measurement differ by \(0.2\,\mathrm{\%}\) — well inside either uncertainty separately, and \(0.09\) of the two combined in quadrature.

Derivation. Derives Theorem 106.93. Expand the fields in eigenfunctions of the Hermitian Euclidean Dirac operator \(\mathcal{D}=\gamma^{\mu}D_{\mu}\), \(\mathcal{D}\varphi_{n}=\lambda_{n}\varphi_{n}\), with the \(\varphi_{n}\) orthonormal: \(\psi(x)=\sum_{n}a_{n}\varphi_{n}(x)\), \(\bar\psi(x)=\sum_{n}\bar b_{n}\varphi_{n}^{\dagger}(x)\), the coefficients \(a_{n},\bar b_{n}\) being Grassmann. The measure is \(\prod_{n}\dd a_{n}\,\dd\bar b_{n}\).

Under Equation (106.99) the coefficients transform linearly, \(a'_{m}=\sum_{n}C_{mn}a_{n}\) with

and \(\bar b\) transforms with the same matrix, because Equation (106.99) carries the same sign for \(\bar\psi\). By Corollary 106.36 a Grassmann measure transforms with the inverse determinant, so

using \(\det=\exp\tr\ln\) to first order in \(\alpha\), with \(\mathcal{J}(x)=\sum_{n}\varphi_{n}^{\dagger}(x) \gamma^{5}\varphi_{n}(x)\).

The sum \(\mathcal{J}\) is formally \(\tr\gamma^{5}\) times a delta function — zero times infinity — and must be regularized. The regulator that respects gauge invariance is the operator's own spectrum:

where the trace is over Dirac and gauge indices. Insert plane waves, \(\bra{x}\cdots\ket{x}=\int\dd^{4}k\,(2\pi)^{-4}\, \ee^{-\ii k\cdot x}(\cdots)\ee^{\ii k\cdot x}\), and use

by Equation (106.64). Expanding the exponential, the Dirac trace with \(\gamma^{5}\) vanishes unless at least four gamma matrices are present, so the first surviving term is the second order in \(\mathcal{F}\):

Terms with fewer than four gammas vanish on the Dirac trace; terms with more carry additional inverse powers of \(M\) and vanish in the limit. Now \(\int\dd^{4}k\,(2\pi)^{-4}\ee^{-k^{2}/M^{2}}=M^{4}/(16\pi^{2})\) by Lemma 106.2, which cancels the \(M^{-4}\) exactly — so the limit is finite and independent of the regulator scale — and \(\tr(\gamma^{5}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}) =-4\epsilon^{\mu\nu\rho\sigma}\). Collecting the numbers, \(\frac{1}{2}\times(-\frac{1}{4})\times(-4)\times\frac{1}{16\pi^{2}} =\frac{1}{32\pi^{2}}\), so

which is Equation (106.107). Comparison with Equation (106.91) gives \(\int\dd^{4}x\,\mathcal{J}=\nu\), so for constant \(\alpha\) the Jacobian is \(\ee^{-2\ii\alpha\nu}\).

Finally, the anomalous divergence. Invariance of \(Z\) under a change of integration variables requires that the variation of the action cancel the Jacobian. The action's variation under a local rotation produces \(\int\alpha\,\pp_{\mu}j_{5}^{\mu}\) after integrating by parts, so \(\pp_{\mu}j_{5}^{\mu}=2c\,\mathcal{J}\) for a massless field, i.e.\ \(\pp_{\mu}j_{5}^{\mu}=2q(x)\) with the topological charge density \(q\) of Definition 106.79 — not to be confused with the electric charge, which is why the latter is written \(Q\) throughout this chapter. For the abelian case, \(\mathcal{F}_{\mu\nu}=QF_{\mu\nu}/\hbar\) with no trace factor, and \(2c/(32\pi^{2})\times Q^{2}/\hbar^{2} =Q^{2}c/(16\pi^{2}\hbar^{2})\), which is Equation (106.101).

Derivation. Derives Theorem 106.94. Theorem 106.62 derived Equation (106.76) from the invariance of the action and of the measure under BRST. An anomaly is precisely the statement that the measure is not invariant: by Equation (106.107) the Jacobian contributes an extra term to every identity, so \(\avg{sX}=\avg{X\,\mathsf{A}}\neq0\). The chain of consequences then runs backwards through Section 106.4.2: the decoupling of the quartet in Proposition 106.60 used \(\braket{\Phi}{Q_{B}\Lambda}=0\), which requires \(Q_{B}\) to be conserved, and gauge-parameter independence Theorem 106.61 used \(\avg{sX}=0\) directly. Losing them means that a gauge-dependent quantity appears in a physical answer and that states of negative norm contribute to the optical theorem, which is a loss of probability, not a technical inconvenience.

Derivation. Derives Theorem 106.95. The triangle of Proposition 106.88 now carries a group generator at each of its three vertices, and the two orientations of the fermion loop contribute the two orderings, so the group factor is the symmetrized trace \(\tr(T^{a}T^{b}T^{c}+T^{a}T^{c}T^{b}) =\tr(\acomm{T^{b}}{T^{c}}T^{a})\). A right-handed fermion contributes with the opposite sign, because \(\gamma^{5}\) has the opposite eigenvalue and the anomalous piece of the triangle is the one proportional to \(\gamma^{5}\). Summing over all fermions in the theory gives Equation (106.110), and by Proposition 106.90 nothing else can appear.

Derivation. Derives Proposition 106.96. There are four independent conditions, since \(\SU(3)^{3}\) vanishes identically (the quarks appear in a vector-like pair of triplet and antitriplet with respect to colour, once left and right are combined) and \(\SU(2)^{3}\) vanishes because the symmetrized trace of three \(\SU(2)\) generators is zero for any representation, \(\SU(2)\) having only real representations.

(i) \(\SU(3)^{2}\,\U(1)\). Sum \(Y\) over the colour-triplet Weyl fields, weighted by the number of \(\SU(2)\) components:

(ii) \(\SU(2)^{2}\,\U(1)\). Sum \(Y\) over the \(\SU(2)\) doublets, weighted by their colour multiplicity:

which vanishes if and only if \(N_{c}=3\).

(iii) \(\U(1)^{3}\). Sum \(Y^{3}\) over all Weyl fields with their multiplicities \(6,3,3,2,1\):

Carrying \(N_{c}\) instead of \(3\) through the quark multiplicities gives \(N_{c}\bigl[\tfrac{1}{108}-\tfrac{8}{27}+\tfrac{1}{27}\bigr] +\tfrac{3}{4}=\tfrac{3-N_{c}}{4}\), so this condition too holds only for \(N_{c}=3\).

(iv) Mixed gauge–gravitational. The same triangle with two insertions of the energy–momentum tensor requires \(\sum Y=0\):

This one is a statement about the Standard Model in a fixed classical background metric — a one-loop consistency condition on the gauge currents, not a theory of quantum gravity.

In every case the quarks alone and the leptons alone fail; only the complete generation cancels, and only with three colours. This is the theoretical counterpart of the measurement in Section 106.7.2, and the reason Electroweak Unification and the Higgs Boson requires complete generations: a fourth lepton doublet without its quark partners would make the theory inconsistent, so the discovery of a new charged lepton is a prediction that its quarks exist.