Diagnose classical-field and relativity readiness

This untimed diagnostic checks two capabilities used throughout Core QFT: deriving field equations without losing boundary information, and reasoning with Lorentz transformations, causal structure, and representations. Complete either or both parts in any accessible format that makes your assumptions and reasoning visible. The two results remain separate; there is no combined score.

The tasks use QFT.org’s metric and notation conventions. A symbolic or numerical tool may check algebra after you have stated the transformation, field content, allowed variations, and expected invariant.

Variational and classical-field reasoning

Consider a real scalar field on a flat Lorentzian region $\Omega$,

$$S[\phi]=\int_\Omega \mathrm d^d x\, \left[ \frac12\partial_\mu\phi\,\partial^\mu\phi -\frac12m^2\phi^2-\frac{\lambda}{4!}\phi^4 \right].$$

Produce one coherent derivation that does all of the following:

1. Compute $\delta S$ before integrating by parts, then display the bulk and boundary terms separately.
2. Derive the field equation first for compactly supported variations and then for a finite region with one boundary condition you state explicitly.
3. Explain why the surface term vanishes—or identify the boundary term that would have to be added if your chosen data do not make the variational problem well posed.
4. In natural units, determine the engineering dimension of $\phi$ from the kinetic term.

Then examine free Maxwell theory, $\mathcal L=-\tfrac14F_{\mu\nu}F^{\mu\nu}$. Find the momenta conjugate to $A_0$ and $A_i$. Identify the missing velocity in the Legendre transformation and explain why $\pi^0=0$ is a constraint rather than an evolution equation or a boundary condition.

Answer guide

For the scalar field,

$$\begin{aligned} \delta S &=\int_\Omega\mathrm d^d x\, \left[ \partial_\mu\phi\,\partial^\mu\delta\phi -\left(m^2\phi+\frac{\lambda}{3!}\phi^3\right)\delta\phi \right]\\ &=-\int_\Omega\mathrm d^d x\, \left(\Box\phi+m^2\phi+\frac{\lambda}{3!}\phi^3\right)\delta\phi +\int_{\partial\Omega}\mathrm d\Sigma_\mu\, \partial^\mu\phi\,\delta\phi. \end{aligned}$$

Compact support removes the boundary integral. Dirichlet data do so by requiring $\delta\phi|_{\partial\Omega}=0$; suitable Neumann data instead fix the normal derivative and require a compatible boundary functional. Arbitrary interior variations give $\Box\phi+m^2\phi+\lambda\phi^3/3!=0$. Since the action is dimensionless, $[\mathcal L]=d$ and the kinetic term gives $[\phi]=(d-2)/2$.

For Maxwell theory, using $\dot A_i=\partial_0A_i$, $\pi^0=\partial\mathcal L/\partial\dot A_0=0$ and $\pi^i=F^{i0}$ (equivalently the electric field with index/sign placement set by convention). No $\dot A_0$ appears, so the Legendre map cannot be inverted for that velocity. The relation $\pi^0=0$ restricts phase-space data; it is not obtained by specifying spatial boundary data and is not a time-evolution law.

Interpret this part

• Demonstrated: every term in $\delta S$ is accounted for; the allowed variations justify the boundary step; dimensions are consistent; and the Maxwell constraint is separated from dynamics and boundary data.
• Uncertain: the bulk equation and momenta are plausible, but a surface condition, sign, dimensional assumption, or Legendre-map argument remains implicit.
• Not yet demonstrated: the boundary term is discarded without a reason, the field equation does not follow from the stated action, or $\pi^0=0$ is treated as an ordinary equation of motion.

For either of the last two results, use Variational and classical fields, then repeat the scalar derivation with a different boundary choice.

Lorentz, causal, and spin reasoning

Use a proper, orthochronous boost of rapidity $\chi$ along $x^1$,

$$\begin{pmatrix}x'^0\\x'^1\end{pmatrix} = \begin{pmatrix}\cosh\chi&-\sinh\chi\\-\sinh\chi&\cosh\chi\end{pmatrix} \begin{pmatrix}x^0\\x^1\end{pmatrix}.$$

1. Apply it to a general four-vector, one future-directed timelike vector, one null vector, and a future-directed massive momentum $p^\mu=(E,p,0,0)$.
2. Verify the invariant norm and causal type in each case. For the massive momentum, find the rapidity that reaches its rest frame and verify that the transformed energy remains positive.
3. Explain what “proper, orthochronous” protects that norm invariance alone does not.
4. Choose either a Lorentz vector field or a Dirac field. State its field transformation including the shifted spacetime argument, construct a scalar or covariant-vector bilinear, and verify its transformation.
5. Separately describe a one-particle state by its mass shell, momentum orbit, little group, and spin or helicity label. Explain why the finite-dimensional field representation does not by itself determine the particle content.

Answer guide

The boost preserves $(x^0)^2-(x^1)^2$ because $\cosh^2\chi-\sinh^2\chi=1$. It therefore preserves timelike, null, and spacelike type. For $p^\mu=(E,p,0,0)$ with $E^2-p^2=m^2$ and $E>0$, the rest frame satisfies $p'^1=-E\sinh\chi+p\cosh\chi=0$, hence $\tanh\chi=p/E$; then $p'^0=m>0$. Proper transformations preserve orientation, and orthochronous transformations preserve the future time cone. A general Lorentz transformation can preserve the norm while reversing one of these.

For a vector field, one acceptable convention is $A'^\mu(x')=\Lambda^\mu{}_{\nu}A^\nu(x)$, from which $A'_\mu(x')A'^\mu(x')=A_\mu(x)A^\mu(x)$. For a Dirac field, $\psi'(x')=S(\Lambda)\psi(x)$ with $S^{-1}\gamma^\mu S=\Lambda^\mu{}_{\nu}\gamma^\nu$; this makes $\bar\psi\psi$ a scalar. These are transformation laws for local fields. Particle states instead carry unitary representations of the Poincaré group; for a massive particle the little group is $SO(3)$ (or its spin cover), while massless helicity arises from the appropriate massless little-group representation. Field components can create several particle or antiparticle states, and gauge fields contain redundant components, so the labels are not interchangeable.

Interpret this part

• Demonstrated: the norm and future-cone checks close; causal types are correct; the bilinear transforms as claimed; and field and particle representations remain distinct.
• Uncertain: the component algebra is plausible, but the Lorentz-group component, acting space, spin cover, or little-group reasoning is implicit.
• Not yet demonstrated: the interval or mass shell changes, causal type is treated as frame dependent, or a field label is reported directly as the particle’s spin content.

For either of the last two results, use Relativity, Lorentz symmetry, and spin, then repeat the boost and representation tasks with every acting space named.

Decide what to do next

Keep the two capability results separate. “Demonstrated,” “uncertain,” and “not yet demonstrated” describe this work product, never you. If only one part needs review, repair and re-check only that part with changed data.

Core QFT relies on both capabilities. Return to Orientation, conventions, and study contract or to the specialist path that sent you here. You may continue while a result is uncertain, but expect boundary, constraint, causal, or representation steps to need deliberate checking.

References

• Marc Henneaux and Claudio Teitelboim, Quantization of Gauge Systems, Princeton University Press, 1992, publisher page.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, doi:10.1017/CBO9781139644167.