Diagnose mathematical readiness

Quantum field theory repeatedly asks you to recognize the mathematical object behind a compact formula: a map rather than an array, a distribution rather than a pointwise function, or an asymptotic statement rather than an exact identity. The three short checks below test that kind of working fluency.

Assess each area separately. Linear and tensor methods and Fourier/distribution/Green-function methods are required background for the Core QFT path; complex and asymptotic methods are helpful background whose importance depends on the route and calculation. There is no total score and no pass/fail verdict about you as a student or researcher.

How to use the check

Try the tasks without consulting a solution, but use your normal scratch tools. A computer algebra system can check arithmetic after you have stated the spaces, conventions, domains, and expected invariant. Handwritten work, typed mathematics, or an oral derivation accompanied by the essential equations are all suitable.

For each of the three areas, choose one result:

• Demonstrated: your derivation contains the stated checks and you can explain why they work.
• Uncertain: your calculation is largely correct, but a convention, hypothesis, or interpretation remains implicit.
• Not yet demonstrated: a key check fails or you cannot yet construct the requested object.

An uncertain or not yet demonstrated result is a useful study pointer. It does not prevent you from reading the site; it tells you where later derivations are likely to demand extra attention.

Linear and tensor methods

Let $V$ be a two-dimensional complex vector space. In a basis $e=(e_1,e_2)$, a vector, covector, Hermitian inner product, and linear map have components $v$, $\alpha$, $G$, and $A$. Change to the basis $e'=eM$ with

$$M= \begin{pmatrix} 1&1\\ 0&2 \end{pmatrix},$$

which is deliberately not unitary.

Task. Derive the primed components of all four objects. Then show directly that the pairing $\alpha(v)$ and the norm $v^\dagger Gv$ are unchanged. Derive the matrix representing the adjoint $A^{\dagger_G}$ defined by

$$\langle u,Av\rangle_G =\langle A^{\dagger_G}u,v\rangle_G,$$

and explain in one sentence why $\operatorname{tr}A$ is independent of the basis. You may choose a convenient positive-definite $G$ and convenient $A$, $v$, and $\alpha$ for the numerical checks, but derive the transformation laws before substituting numbers. These are the finite-dimensional map, duality, inner-product, and adjoint distinctions developed in Axler 2024, Chapters 3, 6, and 7.

Check your reasoning

With column-vector components and $e'=eM$,

$$v'=M^{-1}v, \qquad \alpha'=\alpha M, \qquad G'=M^\dagger GM, \qquad A'=M^{-1}AM.$$

Thus $\alpha'v'=\alpha v$ and

$$v'{}^\dagger G'v'=v^\dagger Gv.$$

Because the inner product is conjugate-linear in its first argument, the defining identity gives

$$A^{\dagger_G}=G^{-1}A^\dagger G,$$

with the identity written equivalently as

$$\langle Av,u\rangle_G =\langle v,A^{\dagger_G}u\rangle_G.$$

In the primed basis the same formula, using $G'$ and $A'$, gives $(A^{\dagger_G})'=M^{-1}A^{\dagger_G}M$. The trace is unchanged because $A'$ is similar to $A$ and $\operatorname{tr}(M^{-1}AM)=\operatorname{tr}A$. The decisive feature is not the displayed formulas alone: each component array must transform according to the abstract object it represents.

Result. Mark demonstrated if you derived the opposing vector/covector laws, obtained the congruence transformation of $G$, and closed both invariant checks. Mark uncertain if the invariants work but you relied on a memorized matrix rule without identifying the spaces or the role of $G$. Mark not yet demonstrated if a scalar changes under the basis change or if transpose and adjoint are treated as interchangeable.

For either weak result, use Linear and tensor methods repair, then repeat the task with a different non-unitary $M$.

Fourier transforms, distributions, and Green functions

Use the site’s Fourier convention:

$$\widetilde f(p)=\int \mathrm dx\,e^{ipx}f(x), \qquad f(x)=\int\frac{\mathrm dp}{2\pi}\,e^{-ipx}\widetilde f(p).$$

Task 1. Derive the momentum-space rule for $\mathrm df/\mathrm dx$ by integration by parts, including the hypothesis that removes the boundary term. Then, for every smooth compactly supported test function $\varphi$, derive the weak derivative of the Heaviside distribution from its pairing with $\varphi$. This pairing definition is the standard way derivatives are extended to distributions Duistermaat and Kolk 2010, Chapters 2–3 and 5.

Task 2. In one-dimensional Euclidean space with $m>0$, find the decaying fundamental solution of

$$\left(-\frac{\mathrm d^2}{\mathrm dx^2}+m^2\right)G(x)=\delta(x).$$

Obtain it by Fourier transformation and verify the source normalization from the jump in $G'$. Finally, say what data distinguish this inverse from another Green function of the same differential expression.

Check your reasoning

Integration by parts gives

$$\widetilde{f'}(p)=-ip\widetilde f(p)$$

when the boundary term vanishes (or, more generally, in the corresponding distributional sense). For the Heaviside distribution $H$,

$$\langle H',\varphi\rangle =-\langle H,\varphi'\rangle =-\int_0^\infty\varphi'(x)\,\mathrm dx =\varphi(0),$$

so $H'=\delta$ as a distribution.

Fourier transformation of the Green equation gives

$$\widetilde G(p)=\frac{1}{p^2+m^2}, \qquad G(x)=\int\frac{\mathrm dp}{2\pi}\, \frac{e^{-ipx}}{p^2+m^2} =\frac{e^{-m\lvert x\rvert}}{2m}.$$

Away from $x=0$ this solves the homogeneous equation. At the source, $G'(0^+)=-1/2$ and $G'(0^-)=+1/2$, so

$$-G'(0^+)+G'(0^-)=1,$$

which supplies the unit delta distribution. Decay at both infinities rules out additions of homogeneous solutions. In Lorentzian problems, retarded, advanced, and Feynman inverses instead differ through support, ordering, state, and pole-prescription data; a shared denominator does not make them the same distribution.

Result. Mark demonstrated if the transform sign and normalization, weak derivative, delta source, and condition selecting the inverse all agree. Mark uncertain if the formulas are right but the test-function meaning or boundary/support data are unstated. Mark not yet demonstrated if the delta is manipulated pointwise, the source check fails, or the inverse is assumed unique without additional data.

For either weak result, use Fourier, distributions, and Green functions repair, then redo the Euclidean source check and compare retarded with advanced support in a Lorentzian example.

Complex and asymptotic methods

This area is recommended rather than required for the Core QFT path. Treat its result independently of the two preceding results.

Task 1. For

$$F(a)=\int_0^\infty e^{-ax}\,\mathrm dx,$$

state where the integral converges, evaluate it there, and state the larger domain on which the result defines an analytic continuation. Explain why the continuation does not make the original integral converge on that larger domain.

Task 2. Evaluate

$$\int_{-\infty}^{\infty}\frac{\mathrm dx}{x^2+1}$$

by closing the contour in the upper half-plane. Record the contour’s orientation, its enclosed poles, and why the large semicircle does not contribute. Say what would change if the real-axis integrand had a pole on the integration path.

Task 3. For fixed real $g\geq0$ and $\lambda\to+\infty$, find the leading term and first correction of

$$I(\lambda,g)=\int_{-\infty}^{\infty} \exp\!\left[-\lambda\left(\frac{x^2}{2} +\frac{gx^4}{4}\right)\right]\mathrm dx.$$

State the saddle, its Hessian, what is held fixed, and the order of the remainder relative to the leading term. Describe one numerical residual test that would probe your remainder claim.

Check your reasoning

The integral for $F$ converges for $\operatorname{Re}a>0$ and equals $1/a$ there. The function $1/a$ continues analytically to $\mathbb C\setminus\{0\}$, but the original positive-real-axis integral still diverges when $\operatorname{Re}a<0$. A continued function and a continued integral representation are different claims.

For the contour task, the upper semicircle is counterclockwise and encloses only the simple pole at $z=i$, whose residue is $1/(2i)$. The residue theorem therefore gives $2\pi i/(2i)=\pi$. On a semicircle of radius $R$, the arc length is $\pi R$ while the integrand is $\mathcal O(R^{-2})$, so the arc contribution vanishes as $R\to\infty$. A pole on the real axis would require an indentation or boundary-value prescription; simply drawing the contour through it would not define the integral.

For $I$, the real contour has a unique minimum at $x=0$ with Hessian $1$. Putting $x=y/\sqrt\lambda$ and expanding at fixed $g$ gives

$$I(\lambda,g) \sim \sqrt{\frac{2\pi}{\lambda}} \left(1-\frac{3g}{4\lambda} +\mathcal O(\lambda^{-2})\right).$$

After the displayed correction, the absolute remainder is $\mathcal O(\lambda^{-5/2})$ at fixed $g$ (equivalently, the relative remainder is $\mathcal O(\lambda^{-2})$). Evaluating the integral at a sequence of large $\lambda$ values and multiplying the absolute residual by $\lambda^{5/2}$ should approach a bounded value in the tested regime. This tests the claimed scaling; it does not prove convergence of the full asymptotic series.

Result. Mark demonstrated if you kept the integral’s convergence domain separate from the continuation domain, justified the contour closure, and stated the saddle expansion with its limit, fixed data, and remainder. Mark uncertain if the calculations are right but a domain, contour, or remainder statement is implicit. Mark not yet demonstrated if analytic continuation is justified only by a formal substitution, a contour crosses a singularity without accounting for it, or the truncated expansion is presented as an exact identity. These checks follow the standard separation of contour hypotheses and fixed-order asymptotics described in Hunter 2004, §§3.5–3.6, pp. 43–47.

For either weak result, use Complex and asymptotic methods repair, then repeat the three elementary tasks with every domain, contour, and limit written explicitly.

Choose your next step

Keep a three-line note rather than combining the results:

Linear and tensor methods: Demonstrated | Uncertain | Not yet demonstrated
Fourier, distributions, and Green functions: Demonstrated | Uncertain | Not yet demonstrated
Complex and asymptotic methods: Demonstrated | Uncertain | Not yet demonstrated

Repair only the area that needs it, then replace that line after re-checking. Return to the page that sent you here, or compare all learning pathways if you have not yet chosen one.

Limitations

These compact tasks sample transferable reasoning; they do not test every mathematical method used in QFT. They do not assess, for example, differential geometry, Lie representation theory, functional analysis, topology, or numerical analysis. A specialist route may name additional preparation, and successful work here does not substitute for checking the hypotheses of a later theorem or approximation.

References

• Sheldon Axler, Linear Algebra Done Right, 4th ed., Springer, 2024, Chapters 3, 6, and 7. Linear maps, duality, inner products, and adjoints.
• J. J. Duistermaat and J. A. C. Kolk, Distributions: Theory and Applications, Birkhäuser, 2010, Chapters 2–3, 5, and 7. Test functions, weak derivatives, support, and Fourier transformation of distributions.
• John K. Hunter, Asymptotic Analysis and Singular Perturbation Theory, PDF, University of California, Davis, 2004, §§3.5–3.6, pp. 43–47. Laplace’s method, Gaussian reduction, and remainder-aware asymptotic reasoning.
• Semyon Dyatlov, Lecture Notes for 18.155: Distributions, Elliptic Regularity, and Applications to PDEs, PDF, Massachusetts Institute of Technology, 2022, §§7.2 and 9.1. Distributional kernels and fundamental solutions.