Quantum Mechanics and Field Theory Diagnostic

Curved-spacetime quantum field theory assumes several distinct capabilities that are easy to blur together: quantum states and operators, oscillator/Fock-space algebra, classical field modes, propagators, perturbative interactions, effective-field-theory power counting, and gauge constraints. This page diagnoses those pieces separately so that one gap does not trigger an indiscriminate restart. It is a transition module, not a quantum-mechanics or QFT course.

Required background. Calculus and Linear Algebra for Relativity supplies complex vector spaces, eigenvalue problems, Fourier transforms, and quadratic forms.

Helpful background. The Wave Equation and Characteristics develops the recommended waves-and-optics capability; Fourier Modes, Dispersion, and Group Velocity supplies the classical mode language used below.

You should leave able to locate a weakness as quantum mechanics, free-field quantization, propagators, interactions/EFT, or gauge constraints, strengthen only that area, and demonstrate the combined capability in a changed problem.

Quantum-mechanics and field-theory diagnostic

1. For a time-independent Hermitian Hamiltonian $H$, show that $U(t)=e^{-iHt}$ preserves the norm. Derive the time derivative of $\langle O\rangle$ for a possibly time-dependent operator $O$.

2. Let $[a,a^\dagger]=1$, $N=a^\dagger a$, and $H=\omega(N+1/2)$. Derive $[N,a^\dagger]$ and the normalized state obtained by applying $a^\dagger$ to $|n\rangle$.

3. A real free scalar in flat spacetime has action $S=-\frac12\int d^4x\,[\partial_\mu\phi\,\partial^\mu\phi+m^2\phi^2]$. Find its equation of motion, canonical momentum, and plane-wave dispersion relation in the site’s $(-,+,+,+)$ convention.

4. Define

   $$D_F(x)=\int\frac{d^4p}{(2\pi)^4} \frac{-i\,e^{ip\cdot x}}{p^2+m^2-i\epsilon}.$$

   Which Green equation does it satisfy, and where lie the positive- and negative-frequency poles in the complex $p^0=\omega$ plane?

5. In four dimensions, determine the mass dimension of $\phi^2\partial_\mu\phi\partial^\mu\phi$. If it appears with coefficient $c/\Lambda^2$, how does its low-energy contribution scale?

6. In Maxwell theory, why are $\pi^0=0$ and Gauss’ law constraints rather than two additional propagating equations, and how many local photon polarizations remain?

Hints and solutions

Hints for 1

Use $U^\dagger=e^{iHt}=U^{-1}$. For the expectation value, differentiate both the bra and ket and include explicit operator dependence.

Solution to 1

Hermiticity gives

$$U^\dagger U=e^{iHt}e^{-iHt}=I,$$

so $\langle\psi(t)|\psi(t)\rangle$ is constant. The Schrödinger equation and its adjoint give

$$\frac{d}{dt}\langle O\rangle =i\langle[H,O]\rangle +\left\langle\frac{\partial O}{\partial t}\right\rangle.$$

Thus an operator with no explicit time dependence has conserved expectation value when it commutes with $H$. Unitarity, Hermiticity, and conservation are related here but are not interchangeable words.

Hints for 2

Use $[AB,C]=A[B,C]+[A,C]B$. Determine the normalization from the norm of $a^\dagger|n\rangle$.

Solution to 2

The commutator is

$$[N,a^\dagger] =a^\dagger[a,a^\dagger] =a^\dagger.$$

Hence $N(a^\dagger|n\rangle)=(n+1)a^\dagger|n\rangle$. Its squared norm is

$$\langle n|aa^\dagger|n\rangle =\langle n|(N+1)|n\rangle=n+1,$$

so

$$|n+1\rangle=\frac{a^\dagger}{\sqrt{n+1}}|n\rangle.$$

Each free-field mode becomes an oscillator only after the mode normalization and canonical commutators are fixed.

Hints for 3

Vary the action and integrate by parts. For $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$, evaluate $\Box=-\partial_t^2+\nabla^2$.

Solution to 3

The variation gives

$$(\Box-m^2)\phi=0.$$

Writing the Lagrangian density as

$$\mathcal L=\frac12\dot\phi^2 -\frac12|\nabla\phi|^2-\frac12m^2\phi^2$$

shows that the canonical momentum is $\pi=\partial\mathcal L/\partial\dot\phi=\dot\phi$. A plane wave obeys

$$\omega^2=|\mathbf k|^2+m^2, \qquad p^2=-m^2.$$

Both signs are tied to the declared metric and time-dependence conventions.

Hints for 4

Act with $\Box-m^2$ under the integral. For the pole locations, solve $-\omega^2+|\mathbf p|^2+m^2-i\epsilon=0$ to first order in $\epsilon$.

Solution to 4

Because $\Box e^{ip\cdot x}=-p^2e^{ip\cdot x}$,

$$(\Box-m^2)D_F(x)=i\,\delta^{(4)}(x)$$

in the stated normalization. If $E_{\mathbf p}=\sqrt{|\mathbf p|^2+m^2}$, the poles approach

$$\omega=+E_{\mathbf p}-i0, \qquad \omega=-E_{\mathbf p}+i0.$$

The positive-frequency pole lies below and the negative-frequency pole above the real axis. This prescription produces time ordering; retarded and advanced Green functions use different pole placements and answer different boundary-value questions.

Hints for 5

In four dimensions $[\phi]=1$ and $[\partial_\mu]=1$. Count dimensions, then evaluate derivatives at a characteristic energy $E$.

Solution to 5

The operator contains four fields and two derivatives, so

$$[\phi^2(\partial\phi)^2]=2+2(1+1)=6.$$

A four-dimensional Lagrangian density has mass dimension four; its coefficient therefore has dimension $-2$ and can be written $c/\Lambda^2$. At characteristic momentum $E\ll\Lambda$, its contact contribution scales as

$$c\frac{E^2}{\Lambda^2}$$

times the appropriate external-state normalization and combinatorial factor. This is a power-counting estimate, not a numerical prediction: symmetries, operator mixing, field redefinitions, and process-dependent coefficients still matter.

Hints for 6

The Lagrangian contains no $\dot A_0$. Vary $A_0$, then count four potential components minus gauge and constraint redundancy.

Solution to 6

Since $\dot A_0$ does not appear, its conjugate momentum vanishes: $\pi^0=0$ is a primary constraint. Varying $A_0$ imposes Gauss’ law,

$$\nabla\cdot\mathbf E=\rho,$$

rather than evolving an independent radiative mode. Gauge equivalence and the constraint remove the longitudinal and scalar redundancies, leaving two local transverse photon polarizations in four-dimensional vacuum. Boundaries, charges, and global gauge sectors can carry additional physical data, but not extra local photon polarizations.

Interpret the result

• Secure: all six ingredients are independently usable, conventions fix every sign, and propagator, interaction, EFT, and gauge statements include their boundary or scale assumptions.
• Uncertain: one algebraic or sign step needs a hint while the physical object is correctly identified. Review that topic and solve a changed example before proceeding.
• Not yet: a classical mode is called a particle without quantization, a propagator prescription is omitted, a nonrenormalizable operator is called meaningless at all energies, or gauge components are all counted as physical states.

Record readiness separately for quantum mechanics, free fields/Fock space, propagators, interactions/EFT, and gauge constraints. A secure result in one category does not cancel a gap in another.

Focused review

Quantum states and operators

Review items 1–2 with Quantum States, Observables, and Measurement. Re-derive one unitary evolution and one oscillator normalization; do not proceed by memorizing ladder-operator arrows without their coefficients.

Free fields and Fock space

The field equation, symplectic normalization, canonical variables, and mode basis come before creation and annihilation operators. Review Classical Field Modes and Canonical Variables and repeat item 3 for a complex scalar or a finite box.

Propagators and boundary conditions

A propagator inverts a differential operator only after a causal, spectral, or Euclidean prescription is chosen. Correlation Functions and Propagators distinguishes Feynman, retarded, and other Green functions. Practice item 4 by drawing the poles and evaluating one elementary contour integral for $t\gt0$ and $t\lt0$.

Interactions and effective theories

Perturbation theory requires a free split, interaction operators, a state, observables, and a small parameter. Effective field theory orders every symmetry-allowed operator by scale rather than discarding higher-dimensional terms. Review Renormalization and Effective Field Theory and power-count a second dimension-six operator.

Gauge constraints

Gauge redundancy changes the state-counting problem: constraints and boundary data must be handled before defining the physical Hilbert space. Gauge Symmetry, Constraints, and Physical States explains how to identify the physical states.

Transfer task: one field, four layers

Put a real scalar field of mass $m$ in a periodic cubic box of side $L$. Add weak interactions

$$\mathcal L_{\rm int} =-\frac{\lambda}{4!}\phi^4 -\frac{c}{4\Lambda^2}\phi^2\partial_\mu\phi\partial^\mu\phi,$$

with characteristic energies well below $\Lambda$. For the free theory, determine the allowed momenta, frequencies, one-particle energies, and momentum-space Feynman denominator. Then compare the parametric size of the two tree-level four-point interactions at energy $E$ and state the regime in which the derivative expansion is controlled.

Hint 1

Periodicity gives $\mathbf k_{\mathbf n}=2\pi\mathbf n/L$. Each real-field mode pair quantizes as oscillators with frequency $\omega_{\mathbf n}$.

Hint 2

The quartic vertex is momentum independent up to its coupling, while two derivatives supply two powers of external momentum. Keep the statement parametric because contractions and channel kinematics affect the exact coefficient.

Solution

The spatial modes and their positive frequencies are

$$\mathbf k_{\mathbf n}=\frac{2\pi}{L}\mathbf n, \qquad \omega_{\mathbf n}=\sqrt{|\mathbf k_{\mathbf n}|^2+m^2}, \qquad \mathbf n\in\mathbb Z^3.$$

After canonical normalization, the free Hamiltonian is a sum of oscillators,

$$H_0=\sum_{\mathbf n}\omega_{\mathbf n} \left(a_{\mathbf n}^\dagger a_{\mathbf n}+\frac12\right),$$

with the usual care that a real field relates opposite classical Fourier coefficients. A one-particle excitation $a_{\mathbf n}^\dagger|0\rangle$ has energy $\omega_{\mathbf n}$. In the large-box momentum description, the Feynman propagator carries the denominator

$$p^2+m^2-i\epsilon,$$

with the pole placement derived in diagnostic 4; the finite box discretizes spatial momentum but not time.

For four external quanta at characteristic energy and momentum $E$, the ordinary quartic contact amplitude scales as $\lambda$, whereas the derivative operator contributes parametrically

$$c\frac{E^2}{\Lambda^2}.$$

The derivative expansion is controlled only when $E/\Lambda\ll1$ and the dimensionless coefficients do not defeat that hierarchy. Perturbation theory additionally requires the relevant combinations of $\lambda$ and $cE^2/\Lambda^2$ to be small. This integrated exercise connects classical modes, Fock excitations, a boundary-conditioned propagator, and EFT power counting without pretending to provide a curved-spacetime vacuum prescription.

Common errors

• Losing the metric-signature change between $p^2=-m^2$ and a mostly-minus textbook formula.
• Quantizing an unnormalized mode expansion and then repairing commutators by hand.
• Treating the $i\epsilon$ prescription as decorative rather than part of the Green function.
• Using an EFT above its cutoff because each individual term is algebraically finite.
• Gauge-fixing before identifying the constraints and boundary degrees of freedom.

Return to your route

After reviewing only the weak topics and completing the changed field problem, return to Quantum Fields in Curved Spacetime or Quantum Gravity.