Derivation Problems

These problems combine established results to obtain a new diagnostic. They use natural units $\hbar=c=1$ and the mostly-minus metric. Treat the linked spectra, normalization rules and free-field algebra as inputs; the purpose is to learn what additional conclusions follow and under which assumptions.

Helpful background. Relativistic Normalization supports problem 1; Relativistic Landau Levels supports problem 2; Invariant Phase Space supports problem 3; and Locality and Causality Warnings supplies the causal commutator distinction for problem 4.

A rapidity packet and two different acceptance regions

Problem 1. Work explicitly in $1+1$ dimensions with a massive spin-zero particle, $m>0$. Set $p=m\sinh y$, $E=m\cosh y$, and use covariant normalization $\langle p'|p\rangle_C=2E_p(2\pi)\delta(p'-p)$. Let

$$\begin{aligned} F(y)&=(2\pi s^2)^{-1/4} \exp\!\left[-\frac{(y-y_0)^2}{4s^2}\right],\\ f(p(y))&=\sqrt{4\pi}\,F(y),\qquad s>0. \end{aligned}$$

The state is $|f\rangle=\int dp\,f(p)|p\rangle_C/(2\pi\,2E_p)$.

1. Show it has unit norm and find its coefficient $a(p)$ in the basis with overlap $(2\pi)\delta(p'-p)$.
2. Apply a passive boost of rapidity $\xi$, so $y'=y-\xi$. Derive the transformation of $a$ and check its norm.
3. Compare the probability of $p'>0$ with the probability of the transformed image of the original region $p>0$.
4. Compute $\langle E\rangle$, $\langle p\rangle$, $\langle P\rangle^2$ and $\langle P^2\rangle$. Interpret the last two.

Solution

The mass-shell measure simplifies to

$$\frac{dp}{2\pi\,2E_p}=\frac{dy}{4\pi}.$$

Hence the norm is $\int dy\,|F(y)|^2=1$. The coefficient in the stated $N$ basis is $a(p)=f(p)/\sqrt{2E_p}$ and obeys $\int dp\,|a(p)|^2/(2\pi)=1$. This is a $1+1$ measure; no hidden transverse integration is present.

For a scalar covariant coefficient, $f'(y')=f(y'+\xi)$. Therefore

$$\begin{aligned} a'(p')&=\sqrt{\frac{E_p}{E_{p'}}}\,a(p),\\ dp'&=\frac{E_{p'}}{E_p}\,dp. \end{aligned}$$

The square-root and Jacobian cancel in the norm. The coefficient transformation has the inverse energy factor to the corresponding normalized ket transformation.

Let $\Phi_{\rm N}(z)$ denote the standard normal cumulative distribution function. Since $p'>0$ means $y'>0$,

$$\Pr(p'>0)= \Phi_{\rm N}\!\left(\frac{y_0-\xi}{s}\right).$$

The transformed original region is $y'>-\xi$, equivalently $p'>-m\sinh\xi$. Its probability is

$$\Pr(y'>-\xi)=\Phi_{\rm N}(y_0/s).$$

The first expression uses a different laboratory acceptance region. The second describes the same projector after transformation. Lorentz covariance requires the latter equality, not equality of two different experiments.

The Gaussian moment formula $\langle e^{t y}\rangle= e^{t y_0+t^2s^2/2}$ gives

$$\begin{aligned} \langle E\rangle&=m e^{s^2/2}\cosh y_0,\\ \langle p\rangle&=m e^{s^2/2}\sinh y_0,\\ \langle P^\mu\rangle\langle P_\mu\rangle &=m^2e^{s^2},\\ \langle P^\mu P_\mu\rangle&=m^2. \end{aligned}$$

The representation’s Casimir is unchanged. The mean vector includes the energy spread of the packet and its square is not a new irreducible mass label.

For $m=1$, $y_0=0.7$, $s=0.4$, $\xi=1.1$, useful numerical checks are

$$\begin{aligned} \Pr(p'>0)&\approx0.1586552539,\\ \Pr(y'>-\xi)&\approx0.9599408431,\\ \langle E\rangle&\approx1.3597083515,\\ \langle p\rangle&\approx0.8217639140. \end{aligned}$$

These are momentum-space probabilities; the calculation does not define a relativistic position Born density.

The lowest Landau level controls a continuum check

Problem 2. For a massive Dirac particle in a uniform magnetic field, put $b=|q|B>0$. Import the positive-energy spectrum and its physical multiplicity:

$$E_N^2=m^2+p_z^2+2bN,\qquad d_N=2-\delta_{N0}.$$

The orbital degeneracy per transverse area is $b/(2\pi)$. For $\tau>0$, evaluate the convergent spectral trace per volume

$$K_b(\tau)=\frac{b}{2\pi} \sum_{N=0}^{\infty}d_N \int\frac{dp_z}{2\pi} e^{-\tau(E_N^2-m^2)}.$$

Find the zero-field limit and the first field correction. Diagnose the effect of assigning multiplicity two to $N=0$, or adding the negative-energy sector to this definition.

Solution

The longitudinal Gaussian integral and geometric level sum are

$$\begin{aligned} \int\frac{dp_z}{2\pi}e^{-\tau p_z^2} &=\frac1{2\sqrt{\pi\tau}},\\ 1+2\sum_{N=1}^{\infty}e^{-2b\tau N} &=\coth(b\tau). \end{aligned}$$

Thus

$$K_b(\tau)= \frac{b\coth(b\tau)}{4\pi^{3/2}\sqrt{\tau}}.$$

At zero field the two positive-energy spin states give

$$K_0(\tau)= 2\int\frac{d^3p}{(2\pi)^3}e^{-\tau\mathbf p^2} =\frac1{4\pi^{3/2}\tau^{3/2}}.$$

With $y=b\tau$ the ratio is

$$\frac{K_b}{K_0} =y\coth y =1+\frac{y^2}{3}-\frac{y^4}{45} +O(y^6).$$

The first correction is quadratic. Incorrectly doubling the lowest level adds $y$ to this ratio and creates a spurious linear term. Adding the negative-energy branch doubles the entire trace: that counts a second sector absent from the problem’s definition.

This is a positive-band spectral diagnostic. It is not a thermal partition function, a vacuum effective action or an anomaly calculation. Its ingredients come from Relativistic Landau Levels; no second derivation of that spectrum is needed.

A recoil bound for replacing a target by a source

Problem 3. A massless projectile $p=(E,E\mathbf n)$ scatters elastically from $P=(M,\mathbf0)$ into $p'=(E',E'\mathbf n')$. The target remains on its original mass shell, and $\mathbf n\cdot\mathbf n'=\cos\theta$. Derive $E'/E$ and the condition for the fractional recoil loss to be at most $\varepsilon$, with $0<\varepsilon<1$. Explain why a weak coupling does not by itself satisfy this condition.

Solution

Momentum conservation gives $P'=P+p-p'$. Expanding $P'^2=M^2$ yields

$$2M(E-E')-2EE'(1-\cos\theta)=0,$$

so

$$\frac{E'}{E} =\frac1{1+(E/M)(1-\cos\theta)}.$$

Let $a=(E/M)(1-\cos\theta)$. Then

$$\begin{gathered} \frac{E-E'}{E}=\frac{a}{1+a}\le\varepsilon\\ \Longleftrightarrow\quad a\le\frac{\varepsilon}{1-\varepsilon}. \end{gathered}$$

Forward scattering has no recoil energy loss in this massless elastic kinematics; the largest loss at fixed $E/M$ occurs at $\theta=\pi$. For $E/M=0.1$, losses at $\pi/2$ and $\pi$ are $1/11$ and $1/6$.

The bound is kinematic and independent of the interaction’s Born expansion. Small recoil and weak coupling must be checked separately before comparing with fixed-source Mott scattering. For a massive projectile use $|\mathbf p|=\sqrt{E^2-m^2}$; the massless relation cannot be reused unchanged.

Local control and vacuum correlation

Problem 4. Let $\phi$ be a free real scalar field. Take real smooth compactly supported spacetime test functions $g_A,g_B$ whose supports are everywhere spacelike separated. Write $\phi(g)=\int d^4x\,g(x)\phi(x)$. Use the free-field Weyl algebra with causal commutator support.

For $\lambda\in\mathbb R$, set $U_A=e^{i\lambda\phi(g_A)}$ and $W_B=e^{i\phi(g_B)}$. Show that the expectation of $W_B$ is unchanged by the local unitary $U_A$ in every state. Does this require $\langle\phi(g_A)\phi(g_B)\rangle=0$ or a factorized vacuum?

Solution

The smeared commutator vanishes because its distribution has no support on these pairs of points. The free-field Weyl relations therefore give $U_AW_B=W_BU_A$. Both exponentials are bounded unitaries, and

$$U_A^\dagger W_B U_A=W_B.$$

Taking an expectation in any initial state proves the claim. The real and imaginary Hermitian parts of $W_B$ are corresponding bounded observables. The argument uses the Weyl relation, so it does not rely on manipulating unbounded operators without their domains.

The vacuum cross-correlation may be nonzero. Correlation is compatible with invariance under this spacelike local control; no product-state assumption was made. See Propagators to Correlators.

Exact compact spacelike separation matters. Gaussian smearings have tails, and a finite mode cutoff does not preserve the exact local field algebra. Neither may be substituted silently into this proof. The result is a free-field checkpoint, not a derivation of the general interacting theory’s locality axioms.

References

• Greiner, Walter. Relativistic Quantum Mechanics: Wave Equations. Third edition. Springer, 2000. doi:10.1007/978-3-662-04275-5. External-field spectra.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. doi:10.1017/9781139540940. Scattering kinematics and field commutators.
• Tong, David. Lectures on Quantum Field Theory. University of Cambridge, 2006–2007, section 2. Free scalar fields and causality.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995, chapters 2–3. doi:10.1017/CBO9781139644167. Relativistic state normalization and scattering.