When spectral broadening is physical

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 $|d⟩$ coupled to many environmental states $|k⟩$:

For a finite system, the projected retarded resolvent has the exact representation

and

where

Eliminating the $k$-states gives

After a continuum limit,

so that

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 $\mathrm{i}{0}^{+}$ does not shift the eigenvalues or give them lifetimes. It selects a boundary value of an analytic function:

The imaginary part appears because the real axis contains spectral support, not because an infinitesimal physical damping has been added.

More abstractly, $A_d(\omega )\mathrm{d}\omega$ is the spectral measure of $H$ associated with $|d⟩$. It answers:

How is the vector $|d⟩$ 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 $N$,

is quasiperiodic. Probability leaves $|d⟩$, 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

As the bath level spacing tends to zero, recurrence times diverge and phase information disperses into increasingly many modes. A resonance pole

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

and its causal time response contains

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 ${\mathrm{\Sigma }}^R(\omega )$ varies slowly across a narrow resonance, it may be replaced locally by ${\mathrm{\Delta }}_{\star }-\mathrm{i}{\kappa }_{\star }$.

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 $\mathrm{Im}{\chi }^R$. 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 $E-\mathrm{i}\kappa$.

Why is this tempting? The Lorentzian is generated by $1/(\omega -E+\mathrm{i}\kappa )$, 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 $-\mathrm{Im}{\mathrm{\Sigma }}^R$ 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:

• $\mathrm{i}{0}^{+}$ selects the causal boundary value.
• A finite $-\mathrm{Im}{\mathrm{\Sigma }}^R$ encodes available decay channels after the relevant continuum or resolution limit.

Exercise

Consider a flat continuum with density $\rho$, coupling $V$, and energies $ϵ\in [-D,D]$:

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

$$\frac{1}{x+\mathrm{i}{0}^{+}}=\mathrm{PV}\frac{1}{x}-\mathrm{i}\pi \delta (x).$$Copy LaTeX

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 $\mathrm{i}{0}^{+}$ with a finite constant $\mathrm{i}\eta$ 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

$${\mathrm{\Sigma }}^R(\omega )=\rho |V{|}^2[\ln \left|\frac{\omega +D}{\omega -D}\right|-\mathrm{i}\pi \mathrm{\Theta }(D-|\omega |)].$$Copy LaTeX

Thus, inside the band,

$$\mathrm{\Delta }(\omega )=\rho |V{|}^2\ln \left|\frac{\omega +D}{\omega -D}\right|,\kappa =\pi \rho |V{|}^2.$$Copy LaTeX

For $|\omega |\ll D$,

$$\ln \left|\frac{\omega +D}{\omega -D}\right|=\frac{2\omega }{D}+O\left(\frac{{\omega }^3}{D^3}\right),$$Copy LaTeX

so the real part varies slowly and can be absorbed locally into an energy shift and residue renormalization. Then

$$G_d^R(\omega )\simeq \frac{1}{\omega -{ϵ}_d^{\mathrm{ren}}+\mathrm{i}\kappa },$$Copy LaTeX

which gives

$$A_d(\omega )\simeq \frac{1}{\pi }\frac{\kappa }{(\omega -{ϵ}_d^{\mathrm{ren}}{)}^2+{\kappa }^2}.$$Copy LaTeX

The pole contribution to the causal time response is proportional to ${\mathrm{e}}^{-\kappa t}$ 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.