{
"schema_version": 1,
"id": "PHYS-2026-07-23-01",
"canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-23-01/",
"source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-23-01.md",
"metadata": {
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"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",
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"content_markdown": "\n## Key claim\n\n**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.\n\n## Theme\n\n**From real-axis poles to resonances and branch cuts: when spectral broadening is physical—and when it is only a regulator.**\n\n## Guiding question\n\nThe same Lorentzian appears in operator spectral theory, many-body Green functions, and classical damped response.\n\nDoes 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?\n\n## Setup\n\nTake a discrete state $\\ket{d}$ coupled to many environmental states $\\ket{k}$:\n\n$$\nH=\\epsilon_d\\ket{d}\\bra{d}\n+\\sum_k\\epsilon_k\\ket{k}\\bra{k}\n+\\sum_k\\left(V_k\\ket{d}\\bra{k}+\\text{h.c.}\\right).\n$$\n\nFor a finite system, the projected retarded resolvent has the exact representation\n\n$$\nG_d^R(z)\n=\\bra{d}(z-H)^{-1}\\ket{d}\n=\\sum_n\\frac{Z_n}{z-E_n},\n$$\n\nand\n\n$$\nA_d(\\omega)\n=-\\frac{1}{\\pi}\\operatorname{Im}G_d^R(\\omega+\\ii 0^+)\n=\\sum_n Z_n\\delta(\\omega-E_n),\n$$\n\nwhere\n\n$$\nZ_n=\\left|\\braket{d}{n}\\right|^2.\n$$\n\nEliminating the $k$-states gives\n\n$$\nG_d^R(z)=\\frac{1}{z-\\epsilon_d-\\Sigma(z)},\n\\qquad\n\\Sigma(z)=\\sum_k\\frac{|V_k|^2}{z-\\epsilon_k}.\n$$\n\nAfter a continuum limit,\n\n$$\n\\Sigma^R(\\omega)=\\Delta(\\omega)-\\ii\\kappa(\\omega),\n$$\n\nso that\n\n$$\nA_d(\\omega)=\\frac{1}{\\pi}\n\\frac{\\kappa(\\omega)}\n{[\\omega-\\epsilon_d-\\Delta(\\omega)]^2+\\kappa(\\omega)^2}.\n$$\n\nThis is the smallest model in which exact poles, a continuum self-energy, and an apparently decaying quasiparticle can all be separated cleanly.\n\n## Analysis\n\n### Derivation — the resolvent is a spectral measure\n\nFor every finite-dimensional closed system with self-adjoint $H$, the spectrum is real and $A_d$ is a sum of delta functions.\n\nThe $\\ii 0^+$ does **not** shift the eigenvalues or give them lifetimes. It selects a boundary value of an analytic function:\n\n$$\n\\frac{1}{x+\\ii 0^+}\n=\\operatorname{PV}\\frac{1}{x}-\\ii\\pi\\delta(x).\n$$\n\nThe imaginary part appears because the real axis contains spectral support, not because an infinitesimal physical damping has been added.\n\nMore abstractly, $A_d(\\omega)\\,\\dd\\omega$ is the spectral measure of $H$ associated with $\\ket{d}$. It answers:\n\n> How is the vector $\\ket{d}$ distributed over the exact energy decomposition of the Hamiltonian?\n\nAt this stage, it is not a statement about particles, decay, or experimental resolution.\n\n### Scope and assumptions — irreversibility requires a singular limit\n\nAt finite $N$,\n\n$$\n\\bra{d}\\ee^{-\\ii Ht}\\ket{d}\n=\\sum_n Z_n\\ee^{-\\ii E_nt}\n$$\n\nis quasiperiodic. Probability leaves $\\ket{d}$, but recurrences remain possible. There is no exact irreversible exponential decay for arbitrarily late times.\n\nA smooth linewidth appears only after replacing the discrete bath by a continuum or introducing finite experimental resolution. The dangerous order of operations is\n\n$$\nN\\longrightarrow\\infty\n\\quad\\text{before probing sufficiently long times}.\n$$\n\nAs the bath level spacing tends to zero, recurrence times diverge and phase information disperses into increasingly many modes. A resonance pole\n\n$$\nz_\\star\\simeq E_\\star-\\ii\\kappa_\\star\n$$\n\ncan then govern an intermediate-time regime.\n\nEven that pole usually does not determine all times:\n\n- Very short times are generically quadratic rather than exponential.\n- Band edges and thresholds generate nonexponential late-time tails.\n- Bound-state poles outside the continuum can leave a nondecaying component.\n\nA “lifetime” is therefore normally an intermediate-scale statement.\n\n### Physical interpretation — classical damping matches only one layer\n\nA damped classical mode has susceptibility\n\n$$\n\\chi^R(\\omega)\\sim\\frac{1}{\\omega-\\omega_0+\\ii\\kappa},\n$$\n\nand its causal time response contains\n\n$$\n\\theta(t)\\ee^{-\\ii\\omega_0t-\\kappa t}.\n$$\n\nThe correspondence must be graded carefully.\n\n**Mathematical identity:** Fourier transforming a simple lower-half-plane pole gives a causal exponential term.\n\n**Controlled physical approximation:** if $\\Sigma^R(\\omega)$ varies slowly across a narrow resonance, it may be replaced locally by $\\Delta_\\star-\\ii\\kappa_\\star$.\n\n**Suggestive analogy:** the quantum resonance may then be discussed as though it were a classically damped oscillator.\n\nThe dictionary eventually breaks. A closed quantum system need not possess microscopic friction; its apparent damping can be dephasing into unresolved degrees of freedom.\n\nIn this model, $A_d$ is a projected spectral measure. In a general many-body problem, a spectral function records addition and removal transition weights.\n\nClassical 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.\n\n## False claim to diagnose\n\n> A Lorentzian spectral peak proves that the underlying Hermitian Hamiltonian has a complex eigenvalue $E-\\ii\\kappa$.\n\nWhy is this tempting? The Lorentzian is generated by $1/(\\omega-E+\\ii\\kappa)$, whose pole is complex.\n\nThe 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.\n\nIt is not an ordinary normalizable eigenstate of the full closed Hamiltonian.\n\nWhat survives is narrower:\n\n> 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.\n\n## What follows — and what does not\n\nThe disagreement is primarily an **order-of-limits problem**, sharpened by a choice of observables.\n\nThe exact finite-system spectral measure resolves every eigenstate. The effective Green function observes only the $d$-sector after the bath has been eliminated.\n\nTaking a continuum limit and discarding bath-resolved phase information can convert reversible dephasing into effective decay.\n\nA 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.”\n\nThe key distinction is:\n\n- $\\ii 0^+$ selects the causal boundary value.\n- A finite $-\\operatorname{Im}\\Sigma^R$ encodes available decay channels after the relevant continuum or resolution limit.\n\n## Exercise\n\nConsider a flat continuum with density $\\rho$, coupling $V$, and energies $\\epsilon\\in[-D,D]$:\n\n$$\n\\Sigma^R(\\omega)=\\rho|V|^2\n\\int_{-D}^{D}\\frac{\\dd\\epsilon}{\\omega-\\epsilon+\\ii 0^+}.\n$$\n\n1. Derive its real and imaginary parts.\n2. Take the wide-band regime $|\\omega|\\ll D$ and obtain the Lorentzian spectral function.\n3. Explain why the exact finite-band dynamics cannot be purely exponential for all $t>0$.\n\n\nHint 1
\n\nUse\n\n$$\n\\frac{1}{x+\\ii 0^+}\n=\\operatorname{PV}\\frac{1}{x}-\\ii\\pi\\delta(x).\n$$\n\n \n\n\nHint 2
\n\nInspect the nonanalytic points at $\\omega=\\pm D$. Long-time Fourier behavior is controlled by singularities and endpoints, not only by the resonance pole.\n\n \n\n**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?\n\n**Oral check 2.** Why can exponential decay be extremely accurate experimentally even though it is not exact at arbitrarily short or long times?\n\n\nSolution
\n\nThe distribution identity gives\n\n$$\n\\Sigma^R(\\omega)=\\rho|V|^2\n\\left[\n\\ln\\left|\\frac{\\omega+D}{\\omega-D}\\right|\n-\\ii\\pi\\Theta(D-|\\omega|)\n\\right].\n$$\n\nThus, inside the band,\n\n$$\n\\Delta(\\omega)=\\rho|V|^2\n\\ln\\left|\\frac{\\omega+D}{\\omega-D}\\right|,\n\\qquad\n\\kappa=\\pi\\rho|V|^2.\n$$\n\nFor $|\\omega|\\ll D$,\n\n$$\n\\ln\\left|\\frac{\\omega+D}{\\omega-D}\\right|\n=\\frac{2\\omega}{D}+O\\!\\left(\\frac{\\omega^3}{D^3}\\right),\n$$\n\nso the real part varies slowly and can be absorbed locally into an energy shift and residue renormalization. Then\n\n$$\nG_d^R(\\omega)\n\\simeq\\frac{1}{\\omega-\\epsilon_d^{\\mathrm{ren}}+\\ii\\kappa},\n$$\n\nwhich gives\n\n$$\nA_d(\\omega)\n\\simeq\\frac{1}{\\pi}\n\\frac{\\kappa}\n{(\\omega-\\epsilon_d^{\\mathrm{ren}})^2+\\kappa^2}.\n$$\n\nThe pole contribution to the causal time response is proportional to $\\ee^{-\\kappa t}$ for $t>0$.\n\nThe exact self-energy has logarithmic branch points at $\\omega=\\pm D$. A contour deformation therefore receives both resonance-pole and branch-cut contributions.\n\nThe branch cut produces nonexponential late-time behavior. Depending on the parameters and band structure, poles outside the continuum can also leave residual nondecaying weight.\n\n \n\n## Check your understanding\n\nIn 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?\n\nWhich definition do you regard as most fundamental? Reply with your derivation, or simply say “deeper,” “too easy,” or “too hard.”\n\n## Connections and next step\n\n- New pressure point: causal regulators versus physical linewidths.\n- Main domains: spectral theory, many-body response, and classical damping.\n- Toy model: a Fano–Anderson discrete level coupled to a flat finite band.\n- Unresolved: the physical status of second-sheet poles.\n- Revisit: boundary conditions and effective descriptions in three to four runs.\n",
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"tex": "A_d(\\omega)\n\\simeq\\frac{1}{\\pi}\n\\frac{\\kappa}\n{(\\omega-\\epsilon_d^{\\mathrm{ren}})^2+\\kappa^2}.",
"line": 276,
"sha256": "25a1d3706121c6a96ca00f542aef46a86310c4cd439f9b496d011664e234013f"
},
{
"index": 53,
"display": false,
"tex": "\\ee^{-\\kappa t}",
"line": 283,
"sha256": "9ecc0b041160b3baa62cfc7fb4a2b1a52172aff290c3fc6c783af004932fb4d3"
},
{
"index": 54,
"display": false,
"tex": "t>0",
"line": 283,
"sha256": "953c0794968549f7f8580d93fc444696c0aec372099eb60533896c15fecb81a3"
},
{
"index": 55,
"display": false,
"tex": "\\omega=\\pm D",
"line": 285,
"sha256": "1231b3db94ffe7d6f0d7f7096f93f3a377697ff0cc01d24835cf74a67a96711d"
}
]
}