Conceptual Problems

These six problems ask what a calculation does and does not establish. Give a verdict, one supporting equation and the assumption that makes the verdict valid. The stated formulas are inputs; the task is to interpret them before attempting a longer derivation. Use natural units $\hbar=c=1$ throughout.

Helpful background. Relativistic Normalization supports the packet problem; Gauge Covariance supports the two gauge questions; Propagators to Correlators and Pair Creation support the field interpretation; Invariant Phase Space supplies the recoil kinematics. Each problem can be attempted independently after its corresponding background.

Mass labels and wave-packet averages

Problem 1. In $1+1$ dimensions a massive scalar one-particle representation has $P^\mu P_\mu=m^2I$. A normalized packet has a Gaussian rapidity distribution of mean $y_0$ and variance $s^2>0$, with $p=m\sinh y$ and $E=m\cosh y$. Its mean momentum is

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

A reader calls $\sqrt{\langle E\rangle^2-\langle p\rangle^2}$ the packet’s “renormalized particle mass.” Is this a changed Casimir eigenvalue? What does the excess mean energy at $y_0=0$ represent?

Solution

No. The nonlinear square of the mean vector is $\langle P^\mu\rangle\langle P_\mu\rangle =m^2e^{s^2}$, while $\langle P^\mu P_\mu\rangle=m^2$. Averaging and taking a quadratic function are different operations. Every momentum component remains on the same mass shell.

At $y_0=0$ the momenta average to zero, but their positive kinetic energies do not: $\langle E\rangle=m e^{s^2/2}>m$. No interaction, self-energy correction or superposition of different irreducible mass labels has been introduced. The squared mean vector is a valid Lorentz scalar of this prepared distribution; it is not the representation’s mass Casimir. See Massive and Massless Representations.

Gauge-dependent energy signs

Problem 2. A free positive-frequency mode of a particle with $q\ne0$ has energy $E>0$. Apply the gauge function $\chi(t)=-V_0t/q$. In the accepted convention,

$$\mathbf A'=0,\qquad q\Phi'=V_0,\qquad \psi'=e^{-iV_0t}\psi.$$

Choose $V_0<-E$. Has the mode become an antiparticle because its new canonical frequency is negative? Can this change create pairs?

Solution

The canonical energy label becomes $E'=E+V_0$, but its kinetic combination is $E'-q\Phi'=E>0$. The background has zero electric and magnetic fields and is gauge equivalent to the original free problem. The mode and the sector definition must be transformed together.

The sign of a gauge-dependent canonical energy alone is not a particle classification. This pure gauge transformation supplies no physical pair-producing field. It changes the energy origin and phase description, not the occupation of physical in/out states. See Gauge Covariance and Antiparticle Decoupling.

A conserved norm can hide a derivative error

Problem 3. A periodic spectral code represents a Dirac eigenstate after multiplication by $e^{i\chi(x)}$, where the charge is absorbed into the dimensionless phase $\chi$. Its sampled norm is one to roundoff at every grid size. On the coarsest grid the full eigen-equation residual has norm $5.6\times10^{-2}$, with energy and mass scales of order one. The author accepts the calculation because the gauge phase has unit modulus. Is that acceptance justified? Name two further tests.

Solution

Pointwise phase multiplication preserves the sampled norm even if the derivative is poorly resolved. A finite Fourier truncation does not retain every frequency generated by the phase. Thus norm preservation does not verify the differential equation or the coupling sign.

First refine the spatial grid and check $\|H_A\psi_A-E\psi_A\|$ against the known continuum solution. Second use a deliberately wrong coupling sign as a control: it should fail even when the sampled norm remains one. Distinguish discretization error from roundoff saturation. The notebook diagnostics use independent residuals and limits for this reason.

A switched vacuum detector

Problem 4. An inertial two-level detector begins in its ground state, with positive gap $\Omega$. A prescribed time-dependent coupling to a free scalar vacuum is specified in the detector’s rest frame. Take real smooth compact switching $\chi(t)$, a real smooth rapidly decreasing spatial profile $f(\mathbf x)$, and a detector transition matrix element of one. At leading perturbative order its excitation probability is

$$P_{\rm exc}^{(2)} =\lambda^2\int d\Pi_p\, |\widetilde f(\mathbf p)|^2 |\widetilde\chi(\Omega+E_p)|^2.$$

Here $d\Pi_p=d^3p/[(2\pi)^3 2E_p]$ and

$$\begin{aligned} \widetilde\chi(\omega)&=\int dt\,\chi(t)e^{i\omega t},\\ \widetilde f(\mathbf p)&=\int d^3x\,f(\mathbf x)e^{-i\mathbf p\cdot\mathbf x}. \end{aligned}$$

A finite switching can make the result nonzero. Does a click demonstrate a particle already present in the vacuum? Where can the excitation energy come from, and what differs in the stationary positive-gap excitation rate?

Solution

The outcome is a prediction of the specified coupling and switching, not a pre-existing particle inventory. At this order the excitation term can create both a detector excitation and a field quantum. The external switching apparatus can supply their energy.

For an eternally stationary inertial coupling, energy conservation would require $\Omega+E_p=0$, which has no solution at positive gap and positive field energy. The vacuum excitation rate therefore vanishes. Finite switching transients are a different observable; stretching a plateau while holding its switching edges fixed need not erase their total contribution.

Use a Wightman response with the specified detector model, not a Feynman kernel selected merely by its name. See Propagators to Correlators and Fewster, Juárez-Aubry and Louko (2016) for the role of switching.

Coherent pair mixing can be reversed

Problem 5. Initially in the incoming vacuum, a single particle–antiparticle channel of a complex scalar is driven by a neutral external mass-parameter pulse. Its frequency jumps from $\omega_0$ to $\omega_1$, stays there for time $T$, and returns to $\omega_0$, with both frequencies positive. The pulse preserves global charge. Exact matching gives

$$N_{\mathbf p} =\frac{(\omega_1^2-\omega_0^2)^2} {4\omega_0^2\omega_1^2} \sin^2(\omega_1T).$$

Each separate interface has nontrivial Bogoliubov mixing when $\omega_1\ne\omega_0$. Why can the final pair occupation vanish at $\omega_1T=\pi$? Does this contradict positive pair probabilities? Can the two interfaces be assigned independent positive rates?

Solution

The two transformations combine as amplitudes with an intervening phase. At this duration their pair-producing contributions cancel. The final occupation is the nonnegative squared modulus of their sum.

Adding two interface occupations would discard the coherent relative phase. An independent-rate description needs an additional dephasing, averaging or appropriate long-time argument. Reversal of one mode does not mean one common $T$ reverses every momentum mode, since $\omega_1$ depends on momentum. This is a scalar-parameter pump, not an electromagnetic pulse. The mode and statistical definitions are those of Pair Creation.

A relativistic projectile and a heavy target

Problem 6. A massless projectile of energy $E$ scatters elastically from a target of mass $M$ initially at rest. At projectile scattering angle $\theta$,

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

A calculation uses a fixed source because the projectile is ultrarelativistic and the Born coupling is weak. Are those assumptions enough to neglect recoil? Evaluate the fractional energy loss at $E/M=0.1$, for $\theta=\pi/2$ and $\pi$. State the condition for a recoil loss of at most $2\%$.

Solution

Put $a=(E/M)(1-\cos\theta)$. The loss is $(E-E')/E=a/(1+a)$, equal to $1/11$ at $\pi/2$ and $1/6$ at $\pi$ for the stated ratio. Neither is small at the requested accuracy. Requiring $a/(1+a)\le0.02$ gives $a\le1/49$.

Small projectile rest mass, small Born coupling and small target recoil are three independent conditions. The last is angle dependent. A dynamical amplitude includes the target state and four-momentum conservation; a prescribed static source uses a different state and delta normalization. See Invariant Phase Space and Mott Scattering.

References

• Fewster, Christopher J., Benito A. Juárez-Aubry, and Jorma Louko. “Waiting for Unruh.” Classical and Quantum Gravity 33, 165003 (2016). doi:10.1088/0264-9381/33/16/165003. Switching and detector response; only the inertial response distinction is used here.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. doi:10.1017/9781139540940. State normalization, scattering and field observables.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995. doi:10.1017/CBO9781139644167. Particle representations and the distinction between amplitudes and probabilities.