--- schema_version: 1 id: PHYS-2026-07-23-01 date: 2026-07-23 updated_at: 2026-07-23 title: When spectral broadening is physical summary: From exact real-axis poles to continuum resonances, branch cuts, and the limited regime in which a linewidth becomes a lifetime. language: en entry_kind: daily status: published level: graduate-advanced user_difficulty: unrated domains: - condensed-matter - quantum-theory - mathematical-physics estimated_minutes: 45 --- ## Key claim **The comfortable belief under pressure:** a broadened peak in a spectral function is simply a delta-function energy level with a finite lifetime attached to it. ## Theme **From real-axis poles to resonances and branch cuts: when spectral broadening is physical—and when it is only a regulator.** ## Guiding question The same Lorentzian appears in operator spectral theory, many-body Green functions, and classical damped response. Does it represent the same physical object in all three cases, or are we confusing a mathematical boundary value, an effective decay law, and genuine dissipation? ## Setup Take a discrete state $\ket{d}$ coupled to many environmental states $\ket{k}$: $$ H=\epsilon_d\ket{d}\bra{d} +\sum_k\epsilon_k\ket{k}\bra{k} +\sum_k\left(V_k\ket{d}\bra{k}+\text{h.c.}\right). $$ For a finite system, the projected retarded resolvent has the exact representation $$ G_d^R(z) =\bra{d}(z-H)^{-1}\ket{d} =\sum_n\frac{Z_n}{z-E_n}, $$ and $$ A_d(\omega) =-\frac{1}{\pi}\operatorname{Im}G_d^R(\omega+\ii 0^+) =\sum_n Z_n\delta(\omega-E_n), $$ where $$ Z_n=\left|\braket{d}{n}\right|^2. $$ Eliminating the $k$-states gives $$ G_d^R(z)=\frac{1}{z-\epsilon_d-\Sigma(z)}, \qquad \Sigma(z)=\sum_k\frac{|V_k|^2}{z-\epsilon_k}. $$ After a continuum limit, $$ \Sigma^R(\omega)=\Delta(\omega)-\ii\kappa(\omega), $$ so that $$ A_d(\omega)=\frac{1}{\pi} \frac{\kappa(\omega)} {[\omega-\epsilon_d-\Delta(\omega)]^2+\kappa(\omega)^2}. $$ This is the smallest model in which exact poles, a continuum self-energy, and an apparently decaying quasiparticle can all be separated cleanly. ## Analysis ### Derivation — the resolvent is a spectral measure For every finite-dimensional closed system with self-adjoint $H$, the spectrum is real and $A_d$ is a sum of delta functions. The $\ii 0^+$ does **not** shift the eigenvalues or give them lifetimes. It selects a boundary value of an analytic function: $$ \frac{1}{x+\ii 0^+} =\operatorname{PV}\frac{1}{x}-\ii\pi\delta(x). $$ The imaginary part appears because the real axis contains spectral support, not because an infinitesimal physical damping has been added. More abstractly, $A_d(\omega)\,\dd\omega$ is the spectral measure of $H$ associated with $\ket{d}$. It answers: > How is the vector $\ket{d}$ distributed over the exact energy decomposition of the Hamiltonian? At this stage, it is not a statement about particles, decay, or experimental resolution. ### Scope and assumptions — irreversibility requires a singular limit At finite $N$, $$ \bra{d}\ee^{-\ii Ht}\ket{d} =\sum_n Z_n\ee^{-\ii E_nt} $$ is quasiperiodic. Probability leaves $\ket{d}$, but recurrences remain possible. There is no exact irreversible exponential decay for arbitrarily late times. A smooth linewidth appears only after replacing the discrete bath by a continuum or introducing finite experimental resolution. The dangerous order of operations is $$ N\longrightarrow\infty \quad\text{before probing sufficiently long times}. $$ As the bath level spacing tends to zero, recurrence times diverge and phase information disperses into increasingly many modes. A resonance pole $$ z_\star\simeq E_\star-\ii\kappa_\star $$ can then govern an intermediate-time regime. Even that pole usually does not determine all times: - Very short times are generically quadratic rather than exponential. - Band edges and thresholds generate nonexponential late-time tails. - Bound-state poles outside the continuum can leave a nondecaying component. A “lifetime” is therefore normally an intermediate-scale statement. ### Physical interpretation — classical damping matches only one layer A damped classical mode has susceptibility $$ \chi^R(\omega)\sim\frac{1}{\omega-\omega_0+\ii\kappa}, $$ and its causal time response contains $$ \theta(t)\ee^{-\ii\omega_0t-\kappa t}. $$ The correspondence must be graded carefully. **Mathematical identity:** Fourier transforming a simple lower-half-plane pole gives a causal exponential term. **Controlled physical approximation:** if $\Sigma^R(\omega)$ varies slowly across a narrow resonance, it may be replaced locally by $\Delta_\star-\ii\kappa_\star$. **Suggestive analogy:** the quantum resonance may then be discussed as though it were a classically damped oscillator. The dictionary eventually breaks. A closed quantum system need not possess microscopic friction; its apparent damping can be dephasing into unresolved degrees of freedom. In this model, $A_d$ is a projected spectral measure. In a general many-body problem, a spectral function records addition and removal transition weights. Classical dissipated power is instead tied to $\operatorname{Im}\chi^R$. Relating response to fluctuations requires extra information, such as a state and the fluctuation–dissipation theorem. ## False claim to diagnose > A Lorentzian spectral peak proves that the underlying Hermitian Hamiltonian has a complex eigenvalue $E-\ii\kappa$. Why is this tempting? The Lorentzian is generated by $1/(\omega-E+\ii\kappa)$, whose pole is complex. The failure is that a self-adjoint Hamiltonian has real spectrum. The complex pole belongs to an analytically continued Green function—often on a second Riemann sheet—or to an effective projected non-Hermitian description. It is not an ordinary normalizable eigenstate of the full closed Hamiltonian. What survives is narrower: > An isolated second-sheet pole can accurately govern a resonance's intermediate-time amplitude and line shape when background and threshold contributions vary slowly nearby. ## What follows — and what does not The disagreement is primarily an **order-of-limits problem**, sharpened by a choice of observables. The exact finite-system spectral measure resolves every eigenstate. The effective Green function observes only the $d$-sector after the bath has been eliminated. Taking a continuum limit and discarding bath-resolved phase information can convert reversible dephasing into effective decay. A finite $-\operatorname{Im}\Sigma^R$ is therefore physical when it represents continuum decay channels. But “physical” does not mean “a complex energy eigenvalue of the full Hamiltonian.” The key distinction is: - $\ii 0^+$ selects the causal boundary value. - A finite $-\operatorname{Im}\Sigma^R$ encodes available decay channels after the relevant continuum or resolution limit. ## Exercise Consider a flat continuum with density $\rho$, coupling $V$, and energies $\epsilon\in[-D,D]$: $$ \Sigma^R(\omega)=\rho|V|^2 \int_{-D}^{D}\frac{\dd\epsilon}{\omega-\epsilon+\ii 0^+}. $$ 1. Derive its real and imaginary parts. 2. Take the wide-band regime $|\omega|\ll D$ and obtain the Lorentzian spectral function. 3. Explain why the exact finite-band dynamics cannot be purely exponential for all $t>0$.
Hint 1 Use $$ \frac{1}{x+\ii 0^+} =\operatorname{PV}\frac{1}{x}-\ii\pi\delta(x). $$
Hint 2 Inspect the nonanalytic points at $\omega=\pm D$. Long-time Fourier behavior is controlled by singularities and endpoints, not only by the resonance pole.
**Oral check 1.** What conceptual error is made by replacing $\ii 0^+$ with a finite constant $\ii\eta$ and immediately calling $1/\eta$ a lifetime? **Oral check 2.** Why can exponential decay be extremely accurate experimentally even though it is not exact at arbitrarily short or long times?
Solution The distribution identity gives $$ \Sigma^R(\omega)=\rho|V|^2 \left[ \ln\left|\frac{\omega+D}{\omega-D}\right| -\ii\pi\Theta(D-|\omega|) \right]. $$ Thus, inside the band, $$ \Delta(\omega)=\rho|V|^2 \ln\left|\frac{\omega+D}{\omega-D}\right|, \qquad \kappa=\pi\rho|V|^2. $$ For $|\omega|\ll D$, $$ \ln\left|\frac{\omega+D}{\omega-D}\right| =\frac{2\omega}{D}+O\!\left(\frac{\omega^3}{D^3}\right), $$ so the real part varies slowly and can be absorbed locally into an energy shift and residue renormalization. Then $$ G_d^R(\omega) \simeq\frac{1}{\omega-\epsilon_d^{\mathrm{ren}}+\ii\kappa}, $$ which gives $$ A_d(\omega) \simeq\frac{1}{\pi} \frac{\kappa} {(\omega-\epsilon_d^{\mathrm{ren}})^2+\kappa^2}. $$ The pole contribution to the causal time response is proportional to $\ee^{-\kappa t}$ for $t>0$. The exact self-energy has logarithmic branch points at $\omega=\pm D$. A contour deformation therefore receives both resonance-pole and branch-cut contributions. The branch cut produces nonexponential late-time behavior. Depending on the parameters and band structure, poles outside the continuum can also leave residual nondecaying weight.
## Check your understanding In the present model, is a quasiparticle best defined as an approximate eigenstate, a pole of an analytically continued correlator, or a concentration of spectral weight? Which definition do you regard as most fundamental? Reply with your derivation, or simply say “deeper,” “too easy,” or “too hard.” ## Connections and next step - New pressure point: causal regulators versus physical linewidths. - Main domains: spectral theory, many-body response, and classical damping. - Toy model: a Fano–Anderson discrete level coupled to a flat finite band. - Unresolved: the physical status of second-sheet poles. - Revisit: boundary conditions and effective descriptions in three to four runs.