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Appendix C Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Calculation of a Basic Gaussian Integral

Using equation (C.1), evaluate the following Gaussian integral.

\[ \int_{-\infty}^{\infty} dq \; e^{-3q^2} \]
Hint

By comparing \(e^{-3q^2} = e^{-\frac{a}{2}q^2}\), we read off \(a = 6\).

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B-2. Completing the Square in a Gaussian Integral with a Source

Evaluate the following integral using the method of completing the square.

\[ \int_{-\infty}^{\infty} dq \; e^{-2q^2 + 6q} \]
Hint

Rewrite the exponent in the form \(-\frac{a}{2}q^2 + bq\) and apply Eq. (C.3) (with the replacement \(J \to -b\)). First identify \(a = 4\) and \(b = 6\).

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B-3. Gaussian Integral Containing \(q^n\) (Application of Recurrence Relation)

Using the recurrence relation (C.7) repeatedly, evaluate the following integral.

\[ \int_{-\infty}^{\infty} dq \; q^6 \, e^{-\frac{1}{2}q^2} \]
Hint

Setting \(a = 1\), start from \(I_6(1) = \frac{5}{1}\,I_4(1)\), then work downward using \(I_4(1) = 3\,I_2(1)\) and \(I_2(1) = I_0(1) = \sqrt{2\pi}\). You may also use the double factorial \((2m-1)!! = 15\) (for \(m=3\)).

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B-4. Confirming that odd-order Gaussian integrals vanish

Using the symmetry of the integrand, explain why the following integral is zero.

\[ \int_{-\infty}^{\infty} dq \; q^3 \, e^{-\frac{a}{2}q^2} \qquad (\mathrm{Re}(a) > 0) \]
Hint

Perform the variable substitution \(q \to -q\) and show that the integrand is an odd function.

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B-5. Two-Variable Gaussian Integral

For the matrix

\[ A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} \]

use equation (C.8) to evaluate the following integral.

\[ \int_{-\infty}^{\infty} dq_1 \int_{-\infty}^{\infty} dq_2 \; e^{-\frac{1}{2}(q_1, q_2)\,A\begin{pmatrix}q_1\\q_2\end{pmatrix}} \]
Hint

Compute \(\det A = 2 \times 3 - 1 \times 1 = 5\) and substitute \(n = 2\) into equation (C.8).

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B-6. Expansion Using Anticommutation Relations of Grassmann Numbers

For three independent Grassmann numbers \(\eta_1, \eta_2, \eta_3\), simplify the following product.

\[ (\eta_1 + \eta_2)(\eta_2 + \eta_3) \]
Hint

Expand using the distributive law, then apply \(\eta_i^2 = 0\) and \(\eta_i\eta_j = -\eta_j\eta_i\).

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B-7. Basic Calculation of Berezin Integration

Using the definition of Berezin integration (C.16), calculate the following integral.

\[ \int d\eta \; (3 + 5\eta) \]
Hint

Apply \(\int d\eta\;1 = 0\) and \(\int d\eta\;\eta = 1\) to each term.

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B-8. Single-Variable Grassmann Gaussian Integral

For independent Grassmann variables \(\bar{\eta}, \eta\), compute the following by directly verifying equation (C.18):

\[ \int d\bar{\eta}\,d\eta \; e^{-5\bar{\eta}\eta} \]
Hint

Expand \(e^{-5\bar{\eta}\eta} = 1 - 5\bar{\eta}\eta\) (terms of second order and higher vanish since \(\bar{\eta}^2 = \eta^2 = 0\)), then apply the definition of Berezin integration step by step.

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B-9. Sign of Grassmann Differentiation

For two independent Grassmann variables \(\theta, \phi\), compute the following.

\[ \frac{\partial}{\partial\phi}\bigl(\theta\,\phi\,\theta\bigr) \]
Hint

First, simplify \(\theta\,\phi\,\theta\) using the anticommutation relations. Note that \(\theta^2 = 0\).

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B-10. Multivariable Gaussian Integral with Source

Using equation (C.9), compute the following for \(A = \begin{pmatrix}4 & 0\\0 & 4\end{pmatrix}\), \(\mathbf{J} = \begin{pmatrix}2\\0\end{pmatrix}\):

\[ \int d^2 q \; e^{-\frac{1}{2}\mathbf{q}^T A\,\mathbf{q} - \mathbf{J}^T\mathbf{q}} \]
Hint

\(\det A = 16\), \(A^{-1} = \frac{1}{4}\mathbf{1}\). Substitute \(\mathbf{J}^T A^{-1}\mathbf{J} = (2,0)\frac{1}{4}\begin{pmatrix}2\\0\end{pmatrix} = 1\).

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Medium

M-1. Generating Correlation Functions via Gaussian Integrals with Sources

Using the single-variable Gaussian integral with source

\[ Z(J) = \int_{-\infty}^{\infty} dq \; e^{-\frac{a}{2}q^2 + Jq} \]

show the following.

(a) \(\langle q^2 \rangle \equiv \dfrac{1}{Z(0)}\left.\dfrac{\partial^2 Z}{\partial J^2}\right|_{J=0} = \dfrac{1}{a}\)

(b) \(\langle q^4 \rangle \equiv \dfrac{1}{Z(0)}\left.\dfrac{\partial^4 Z}{\partial J^4}\right|_{J=0} = \dfrac{3}{a^2}\)

(c) Verify that the result of (b) is consistent with the combinatorial structure of Wick's theorem (Ch. 8), \(\langle q^4\rangle = 3\langle q^2\rangle^2\), and explain from what pairing combinations the factor "3" arises.

Hint

Expand \(Z(J) = \sqrt{2\pi/a}\;e^{J^2/(2a)}\) as a power series in \(J\), and carry out the derivatives with respect to \(J\). For (c), use the fact that the number of ways to pair 4 factors of \(q\) into 2 pairs is \(4!/(2^2 \cdot 2!) = 3\).

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M-2. Derivation of the Multi-variable Grassmann Gaussian Integral

For \(n\) pairs of independent Grassmann variables \((\bar{\eta}_1, \eta_1), \ldots, (\bar{\eta}_n, \eta_n)\) and an \(n \times n\) matrix \(A\),

\[ \int \prod_{i=1}^n d\bar{\eta}_i\,d\eta_i \; e^{-\sum_{i,j}\bar{\eta}_i A_{ij}\eta_j} = \det A \]

Demonstrate this explicitly for the case \(n = 2\). That is, setting \(A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}\), expand the exponential in powers of Grassmann variables and perform the Berezin integration to obtain \(\det A = ad - bc\).

Hint

Expand \(\bar{\boldsymbol{\eta}}^T A\boldsymbol{\eta} = a\bar{\eta}_1\eta_1 + b\bar{\eta}_1\eta_2 + c\bar{\eta}_2\eta_1 + d\bar{\eta}_2\eta_2\). When expanding \(e^{-X}\), note that only terms in which all four Grassmann variables appear exactly once survive the integration.

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M-3. Source Terms and Inverse Matrix in Grassmann Gaussian Integrals

Derive the integral with source terms \(\bar{\boldsymbol{\xi}}, \boldsymbol{\xi}\) (Grassmann variables)

\[ \int \prod_i d\bar{\eta}_i\,d\eta_i \; e^{-\bar{\boldsymbol{\eta}}^T A\,\boldsymbol{\eta} + \bar{\boldsymbol{\xi}}^T\boldsymbol{\eta} + \bar{\boldsymbol{\eta}}^T\boldsymbol{\xi}} = \det A \; e^{\bar{\boldsymbol{\xi}}^T A^{-1}\boldsymbol{\xi}} \]

by completing the square for Grassmann variables:

\[ \bar{\boldsymbol{\eta}}^T A\,\boldsymbol{\eta} - \bar{\boldsymbol{\xi}}^T\boldsymbol{\eta} - \bar{\boldsymbol{\eta}}^T\boldsymbol{\xi} = (\bar{\boldsymbol{\eta}}^T - \bar{\boldsymbol{\xi}}^T A^{-1})\,A\,(\boldsymbol{\eta} - A^{-1}\boldsymbol{\xi}) - \bar{\boldsymbol{\xi}}^T A^{-1}\boldsymbol{\xi} \]

Also verify that the Jacobian of the change of variables is 1.

Hint

Substitute \(\boldsymbol{\eta}' = \boldsymbol{\eta} - A^{-1}\boldsymbol{\xi}\) and \(\bar{\boldsymbol{\eta}}' = \bar{\boldsymbol{\eta}} - (A^{-1})^T\bar{\boldsymbol{\xi}}\). Verify the transformation rule for the measure under a linear transformation of Grassmann variables \(\eta_i' = \eta_i + c_i\) (where \(c_i\) are Grassmann constants).

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M-4. Gaussian Integral as a Fresnel Integral

Using equation (C.2), evaluate the integral in the limit of a purely imaginary parameter \(a \to -i\alpha\) (\(\alpha > 0\) real):

\[ \int_{-\infty}^{\infty} dq \; e^{\frac{i\alpha}{2}q^2} \]

Show that the result agrees with the Fresnel integral formula

\[ \int_{-\infty}^{\infty} dq \; e^{\frac{i\alpha}{2}q^2} = \sqrt{\frac{2\pi}{\alpha}}\;e^{i\pi/4} \]

Also, discuss how this result relates to the convergence condition of the \(e^{iS}\) weight in the path integral in Minkowski space (Ch. 10).

Hint

From \(-\frac{a}{2}q^2 = \frac{i\alpha}{2}q^2\), we have \(a = -i\alpha\). Write this in polar form as \(a = \alpha e^{-i\pi/2}\) and apply equation (C.2). Note that \(e^{-i(-\pi/2)/2} = e^{i\pi/4}\).

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Advanced

A-1. Ratio of Boson and Fermion Determinants and Supersymmetry

Consider a system in which \(n\) bosonic variables \(q_i\) and \(n\) pairs of Grassmann variables \((\bar{\eta}_i, \eta_i)\) are coupled by the same \(n \times n\) positive-definite matrix \(A\).

\[ Z = \int d^n q \prod_i d\bar{\eta}_i\,d\eta_i \; \exp\!\left[-\frac{1}{2}\mathbf{q}^T A\,\mathbf{q} - \bar{\boldsymbol{\eta}}^T A\,\boldsymbol{\eta}\right] \]

(a) Perform the bosonic and fermionic integrations separately and express \(Z\) in terms of \(\det A\) and \((2\pi)^{n/2}\).

(b) Show that when \(A\) is a constant multiple of the identity matrix, \(A = m^2 \mathbf{1}\), \(Z\) is independent of \(m\).

(c) Discuss how this property—that the bosonic and fermionic determinants cancel—corresponds to the mechanism by which zero-point vacuum energy vanishes in the presence of supersymmetry (SUSY), from the perspective of the one-loop effective potential (the discussion in Ch. 14).

Hint

(a) The bosonic part gives \((2\pi)^{n/2}/(\det A)^{1/2}\), and the fermionic part gives \(\det A\). (b) Substitute \((\det A)^{1/2} = m^n\). (c) The one-loop effective potential takes the form \(V_{\text{1-loop}} \propto \mathrm{STr}\,M^4\ln(M^2/\mu^2)\), and when SUSY holds, the supertrace vanishes.

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A-2. Faddeev–Popov Ghost Representation of Determinants via Grassmann Integration

In the path integral of non-Abelian gauge theory (Ch. 16), the Faddeev–Popov determinant arising from gauge fixing

\[ \det\!\left(\frac{\delta G^a}{\delta\alpha^b}\right) \]

(where \(G^a\) is the gauge-fixing condition and \(\alpha^b\) are the gauge transformation parameters) is to be expressed as an integral over Grassmann variables (ghost fields \(c^a, \bar{c}^a\)). Carry out this procedure through the following steps.

(a) Citing Eq. (C.19), verify that \(\det M = \int \prod_a d\bar{c}^a\,dc^a\;e^{-\bar{c}^a M_{ab}\,c^b}\) holds.

(b) In \(SU(N)\) Yang–Mills theory with the Lorenz gauge \(G^a = \partial_\mu A^{a\mu}\), derive the Faddeev–Popov operator \(M_{ab}(x,y) = -\partial_\mu D^\mu_{ab}\,\delta^4(x-y)\). Here \(D^\mu_{ab} = \delta_{ab}\partial^\mu + g f_{abc}A^{c\mu}\) is the covariant derivative in the adjoint representation.

(c) From the resulting ghost action

\[ S_{\text{ghost}} = \int d^4x \; \bar{c}^a(-\partial_\mu D^\mu_{ab})c^b \]

read off the Feynman rules for the ghost fields (propagator and vertex), and explain, using the properties of Grassmann integration, why the ghosts have the same propagator as a scalar field yet obey Fermi statistics.

Hint

(a) is directly Eq. (C.19). (b) Compute \(\delta G^a / \delta\alpha^b\) from the gauge transformation \(\delta A^a_\mu = D_\mu^{ab}\alpha^b\). (c) The propagator takes the form \(\langle c^a(k)\bar{c}^b(-k)\rangle = \delta^{ab}/k^2\). Connect the fact that ghosts contribute \(\det M\) (in the numerator) in loops with the property (C.19) that Grassmann integration yields the determinant in the numerator.

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