Prerequisite bridges — Geometric QCD

Prerequisite bridges

These short calculations prepare the operations used later. Work the diagnostic before reading its answer. A missed check points to a specific bridge to revisit.

1. Matrix algebra and Gaussian integrals

For finite matrices, $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ follows by exchanging the two summed indices. It does not imply $\mathrm{tr}(AB)=\mathrm{tr}A\mathrm{tr}B$. With $A=B={\sigma }_3$, the left side is $2$ and the right side is zero.

The Pauli matrices satisfy ${\sigma }_i{\sigma }_j={\delta }_{ij}1+i{ϵ}_{ijk}{\sigma }_k$. Therefore $(p\cdot \sigma {)}^2=p^{2}1$ for real $p$, since the antisymmetric epsilon contraction with $p_i{p}_j$ vanishes. This is the finite version of the gamma-algebra step in the Dirac resolvent.

For $a>0$, completing the square gives

$${\int }_{\mathbb{R}}{e}^{-a{x}^2/2+jx}dx=\sqrt{\frac{2\pi }{a}}{e}^{j^2/(2a)}.$$

For real $j$ the shifted Gaussian is immediate. Imaginary $j$ gives the Fourier transform by analytic continuation of this convergent Gaussian identity. Differentiating in $j$ at zero gives $⟨x^2⟩=1/a$.

Diagnostic. Compute $\mathrm{tr}[{\sigma }_1,{\sigma }_2]$ and $\mathrm{tr}({\sigma }_3[{\sigma }_1,{\sigma }_2])$; then find the variance of a normalized weight $e^{-2x^2}$.

Answer. The commutator is $2i{\sigma }_3$, so the traces are zero and $4i$. The Gaussian has $a=4$, hence variance $1/4$.

2. Fourier transforms and distributions

A distribution is defined by its action on test functions. For the delta, $\int \delta (x)\phi (x)dx=\phi (0)$; for its derivative, $\int {\delta }^{\prime }(x)\phi (x)dx=-{\phi }^{\prime }(0)$. These are definitions, not evaluations of an infinite function at a point.

For $\hat{f}(p)=\int e^{ipx}f(x)dx$, integration by parts gives $\widehat{f^{\prime }}=-ip\hat{f}$. A normalized Gaussian of variance $ϵ$ has transform $e^{-ϵp^2/2}$ and tends to the delta against smooth test functions.

In four dimensions a radial integral uses $d^{4}x=2{\pi }^2{r}^{3}dr$. Thus normalization cannot be read off from the one-dimensional graph of a radial kernel. IV.6 uses precisely this measure.

Diagnostic. Normalize $A{e}^{-ar}$ on ${\mathbb{R}}^4$, $a>0$.

Answer. Its integral is $2{\pi }^{2}A(3!)/a^4=12{\pi }^{2}A/a^4$. Unit normalization requires $A=a^4/(12{\pi }^2)$. At $a=2m$ this gives $4m^4/(3{\pi }^2)$.

3. Linear systems and proof by certificate

The equation $Ax=b$ asks whether $b$ lies in the column space of $A$. Row reduction must act on the augmented matrix, since a zero row in $A$ can carry a nonzero right-hand side. A witness $y^{T}A=0$, $y^{T}b\ne 0$ proves inconsistency in one multiplication.

More equations than unknowns does not imply failure. Duplicate equations add no rank. Fewer equations than unknowns does not guarantee success either: a row $0=1$ is impossible at any size.

Diagnostic. Solve $x+y=1$, $2x+2y=2$, and then replace the second right-hand side by $3$.

Answer. Initially $(x,y)=(t,1-t)$ for arbitrary $t$. After the change, subtracting twice the first equation gives $0=1$. The witness is $(-2,1)$.

4. Forms, Hodge star, and complex boundary data

A two-form is $B=\frac{1}{2}{B}_{\mu \nu }d{x}^{\mu }\wedge d{x}^{\nu }$ with antisymmetric components. In oriented Euclidean four-space, $\ast d{x}^{12}=d{x}^{34}$ and $\ast d{x}^{13}=-d{x}^{24}$. The sign is determined by the orientation of the complementary pair.

The bilinear square of a complex vector differs from its Hermitian norm. For $v=(1,i)$, $v\cdot v=0$, but $v^{†}v=2$. This is the distinction in the conformal null constraint.

A real boundary function $C(\theta )=A\cos n\theta$ has holomorphic extension $f(z)=A{z}^n/2$ with $2\mathrm{Re}f=C$. Its real harmonic extension damps the mode by $r^n$. A vector of such harmonic functions need not be a conformal parametrization; III.5 checks this explicitly.

Diagnostic. Is $d{x}^{12}$ self-dual? Does $f(z)=(2z,-iz)/2$ satisfy $f^{\prime }\cdot f^{\prime }=0$?

Answer. No to both. The star of $d{x}^{12}$ is $d{x}^{34}$, and the complex square is $(4-1)/4=3/4$.

5. Grassmann variables and Gaussian integrals

Grassmann generators obey ${\theta }_i{\theta }_j=-{\theta }_j{\theta }_i$, so ${\theta }_i^2=0$. Integration selects the coefficient of the generator: $\int d\theta 1=0$ and $\int d\theta \theta =1$. An orientation of the multiple integration measure fixes all overall signs.

For a single pair, $(\bar{\theta }\theta {)}^2=0$. For $B={\bar{\theta }}_1{\theta }_1+{\bar{\theta }}_2{\theta }_2$, however, the even bilinears commute and $B^2=2{\bar{\theta }}_1{\theta }_1{\bar{\theta }}_2{\theta }_2$ need not vanish. “Grassmann” alone does not imply that every composite squares to zero.

The alternating pairings of a real antisymmetric quadratic form define its Pfaffian. For four variables, $\mathrm{Pf}A=A_{12}{A}_{34}-A_{13}{A}_{24}+A_{14}{A}_{23}$. Squaring it gives the determinant.

Diagnostic. If only $A_{12}=2$ and $A_{34}=3$ are nonzero above the diagonal, find the Pfaffian and determinant.

Answer. They are $6$ and $36$. The matrix consists of two independent antisymmetric blocks.

6. Variations and boundary terms

For a scalar field,

$$\delta {\int }_D\frac{1}{2}|\nabla \rho {|}^2=-{\int }_D\Delta \rho \delta \rho +{\int }_{\partial D}{\partial }_n\rho \delta \rho .$$

The first term determines the bulk equation. The second either vanishes because the boundary values are fixed or combines with a boundary action to determine a natural boundary equation. The two choices describe different variational problems.

Diagnostic. Vary ${\int }_D[\frac{1}{2}|\nabla \rho {|}^2+\sigma e^{2\rho }]+{\int }_{\partial D}m{e}^{\rho }$ with free boundary data.

Answer. The bulk equation is $-\Delta \rho +2\sigma e^{2\rho }=0$ and the boundary equation is ${\partial }_n\rho +m{e}^{\rho }=0$. The coefficients in V.4 differ because its kinetic normalization differs.

7. Stationary phase, exactness, and fit statistics

A Gaussian saddle is exactly integrable. Adding higher powers of the fluctuation generally adds corrections, even when the Hessian is large. Real stationary trajectories and exact quantization are therefore distinct claims.

A fit residual $r_i=(M_i^{\mathrm{model}}-M_i^{\mathrm{obs}})/M_i^{\mathrm{obs}}$ is dimensionless. Its unweighted RMS is $\sqrt{{\sum }_i{r}_i^2/n}$. Experimental error bars enter a different statistic, such as a specified chi-squared. A model error estimate is another distinct input.

Diagnostic. For residuals $(0.1,-0.2)$, calculate the RMS. Is it held-out evidence if both points were used to choose the parameters?

Answer. The RMS is $\sqrt{0.025}\simeq 0.1581$. It is an in-sample diagnostic when both points were fitted.