Lecture 10: Weak Sources and Retarded Response

The course has so far described states, correlations, and equilibrium transition weights of systems left alone. Condensed-matter experiments, however, usually intervene: a field, pulse, voltage, or force is applied and a delayed signal is recorded. This lecture supplies the causal map between those two operations. That map becomes the common language for screening, collective modes, spectroscopy, and transport, even though each chooses a different source and readout. Response is consequently not a property of an energy spectrum alone; it belongs to a declared preparation, perturbation, observable, and limiting procedure.

An ESR or NMR experiment supplies a physical protocol for the abstract source. A static magnetic field establishes two spin levels. A weak transverse oscillating field supplies the time-dependent term, and the apparatus detects the transverse magnetization in phase and in quadrature with the drive. For a dilute paramagnetic ion or a nuclear spin sufficiently isolated from other levels, these are the ${\hat\sigma_x}$ source and readout of the two-level anchor up to known magnetic-moment factors.

The idealization already makes falsifiable predictions: no magnetization can respond before the drive, the resonance lies at the level splitting, and the two measured quadratures come from one complex susceptibility. It does not predict a finite material linewidth. Relaxation, dephasing, static disorder, and coupling to other levels must be supplied before a broadened ESR or NMR line can be interpreted quantitatively.

Lecture 9’s directed-transition calculation described what an undriven equilibrium system reveals through its directed transition weights. This lecture changes the protocol: an external agent perturbs the Hamiltonian, and an apparatus asks for the resulting change of a later observable.

We will derive the retarded kernel from two-sided first-order evolution of the driven two-level anchor, then verify causal support, Fourier signs, pole weights, and the static limit against an exact ground-state tilt.

How does first-order unitary dynamics turn a declared source and readout into a causal response kernel, and which sign, pole, and static-limit checks make that kernel physically interpretable?

A source delivers an impulse at ${t_0}$ , yet a proposed susceptibility predicts a nonzero change at ${t<t_0}$ . Before seeing any formula, choose which ingredient must reject that proposal: the equilibrium weights, the transition matrix element, or the time support of the response kernel. The answer will become a calculation rather than an appeal to the word “causal.”

A named susceptibility is never just a property of the unperturbed spectrum; it also depends on the source operator, readout operator, sign convention, preparation, and limiting protocol. Keeping that ledger explicit is what permits formulas from different texts or software packages to be compared safely.

From Lecture 9’s equilibrium-state definition, recover the thermal density operator, the Heisenberg evolution of ${\hat\sigma_x}$ , and the difference between an ordered correlation and a commutator. From quantum mechanics, write the first-order interaction-picture evolution operator and its adjoint. Before continuing, predict which time-ordering statement enforces that a later source cannot change an earlier measurement. No additional statistical-mechanics formalism is assumed.

From a weak source to the retarded commutator

The retarded susceptibility is the derivative of a declared readout with respect to a declared source at zero source strength. It is a causal distribution determined jointly by the equilibrium state, the perturbing operator, and the measured operator; it is neither the raw record nor an equilibrium correlation with a suggestive name. The Kubo kernel is exact as that derivative under the stated regularity assumptions. Replacing a finite drive by its first-order response is controlled only over a specified amplitude and time window; nonlinear response, bath dynamics, and phenomenological broadening are separate corrections [Kub57; KG06].

Source convention and interaction-picture expansion

Keep the same two-level Hilbert space ${\mathcal H_2}$ and ordered energy basis

$\mathcal B_{\mathrm E} :=(|\mathrm g\rangle,|\mathrm e\rangle)$

as Lecture 9. The Pauli operators and the unperturbed Hamiltonian are linear maps ${\hat\sigma_\alpha,\hat H_0:\mathcal H_2\to\mathcal H_2}$ . Their matrices in the declared basis are

${[\hat\sigma_x]}_{\mathcal B_{\mathrm E}} =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad {[\hat\sigma_y]}_{\mathcal B_{\mathrm E}} =\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix},\qquad {[\hat\sigma_z]}_{\mathcal B_{\mathrm E}} =\begin{pmatrix}1&0\\0&-1\end{pmatrix}, \qquad \hat H_0=-\frac{\hbar\Omega}{2}\hat\sigma_z.$

Thus ${|\mathrm g\rangle}$ is the ${+1}$ eigenvector of ${\hat\sigma_z}$ and has energy ${-\hbar\Omega/2}$ . For the core derivation, let the real source amplitude belong to the test-function space ${\mathcal D(\mathbb R;\mathbb R)=C_{\mathrm c}^{\infty}(\mathbb R;\mathbb R)}$ . Ideal impulses, steps, and monochromatic drives will be introduced only as declared distributional or long-window extensions. Declare the protocol before writing a kernel:

$\hat H'(t)=+f_x(t)\hat\sigma_x,$

while the apparatus reads ${\hat\sigma_x}$ . The plus sign is part of the definition of the source function ${f_x(t)}$ . Many texts instead write ${\hat H'=-F_x\hat\sigma_x}$ ; their force is related by ${f_x=-F_x}$ , and their susceptibility has the opposite sign. We retain the Fourier transform ${\mathrm e^{+\mathrm i\omega t}}$ .

Let ${\hat H_0}$ be self-adjoint on a Hilbert space ${\mathcal H}$ , let ${\hat\rho_0}$ be a stationary density operator, and let ${\hat A}$ and ${\hat B}$ be bounded observables with no explicit source dependence. For a real test source ${f\in\mathcal D(\mathbb R;\mathbb R)}$ , introduce a scalar bookkeeping parameter ${\lambda}$ through

$\hat H_\lambda(t):=\hat H_0+\lambda f(t)\hat B.$

Write ${\mathcal E_A[\lambda f](t)}$ for the resulting expectation of the readout at time ${t}$ . The protocol has linear response in the direction ${f}$ if the first Gâteaux variation exists and there is a distributional kernel ${\chi_{AB}(t,t')}$ such that

$\mathcal E_A[\lambda f](t)-\mathcal E_A[0](t) =\lambda\,\langle\chi_{AB}(t,\mathord\cdot),f\rangle +r_t(\lambda;f), \qquad \lim_{\lambda\to0}\frac{r_t(\lambda;f)}{\lambda}=0.$

For a stationary preparation the kernel depends on ${t-t'}$ , and the pairing becomes a convolution. The plus sign in ${\lambda f\hat B}$ , the preparation, source operator, readout, and source space are part of the definition. If ${f}$ has units of the experimental source, then the time-domain kernel has units ${[A]/([f]\,\mathrm{time})}$ , while its Fourier transform has units ${[A]/[f]}$ .

Changing from ${+f_x\hat\sigma_x}$ to ${-F_x\hat\sigma_x}$ does not move a physical pole or reverse a measured tilt. It changes which ordinary source amplitude is called positive and therefore changes the convention-dependent sign assigned to the kernel. Record the coupling sign, response convolution, commutator sign, and Fourier sign as one ledger. A response that changes only one entry is not a consistent convention change.

Predict three properties. A source at ${t'}$ cannot alter a measurement at an earlier ${t}$ . The leading change must be linear in ${f_x}$ . Because an expectation value contains an evolution operator on both sides of the observable, the two first-order corrections should occur in opposite operator orders.

The response function below is not a property of the material with its input left implicit. It is the causal functional derivative of one declared observable expectation with respect to one declared source, evaluated about a specified stationary preparation. Moving a sign or coupling constant between the source Hamiltonian and susceptibility changes a convention but not the predicted observable change. Causality licenses the retarded support and Kramers–Kronig structure; thermal equilibrium is needed only for later fluctuation–dissipation relations.

Before reading Figure 10.1, trace which wire carries the physical source, which path supplies only a phase reference, and whether the two lock-in outputs should be interpreted as two susceptibilities.

Two-sided state correction and the commutator

Figure 10.1. The 3D apparatus (a), enlarged four-terminal device (b), computed synchronous signals (c), and functional drive/reference map (d) connect the response kernel to one weak-source protocol.

The stage and support are representative design choices. The thermometry path shows one representative lead; return wiring and detailed thermal anchoring are omitted.

Panel (c) uses a 17 Hz cosine reference, ${\omega=2\pi\times17\,\mathrm{Hz}}$ , and a response with ${120\,\mu\mathrm V}$ peak amplitude and phase ${-20^\circ}$ , over two cycles. Drive and response are each divided by their own peak; their displayed heights do not encode gain. Here the peak convention is ${X=2\langle V\cos(\omega t)\rangle}$ and ${Y=-2\langle V\sin(\omega t)\rangle}$ , so ${V(t)=X\cos(\omega t)-Y\sin(\omega t)}$ . The selected values give ${X=112.76\,\mu\mathrm V}$ and ${Y=-41.04\,\mu\mathrm V}$ . This declared phase convention, not the sign of an unlabeled sine trace, determines how a calibrated instrument output maps to the response below.

In panel (d), the same oscillator supplies the drive and the dashed phase reference. Differential voltage passes through the preamplifier to the dual-phase lock-in and its ${X,Y}$ record. The generator sets ${f(t)}$ , the sensor estimates the declared observable, and the reference line establishes phase rather than perturbing the sample a second time. After calibration, ${X}$ and ${Y}$ are the two quadratures of one complex response at the drive frequency. They do not define two independent material susceptibilities, and apparatus gain is not part of ${\chi^R_{AB}}$ . The diagram is one implementation of the functional derivative below, not its definition [Sco94].

Historical background

[Dic46, pp. 268–275] — Early switched comparison radiometry with a coherent reference, mixer, filtering, and signed output.

Scale the declared source as ${\hat H'_\lambda(t)=+\lambda f_x(t)\hat\sigma_x}$ . In the interaction picture,

$\hat U_{\mathrm I,\lambda}(t,-\infty) =\hat I-\frac{\mathrm i}{\hbar} \int_{-\infty}^{t}\mathrm dt'\, \hat H'_{\mathrm I,\lambda}(t') +\mathcal O(\lambda^2),$

and

$\hat U_{\mathrm I,\lambda}^{\dagger}(t,-\infty) =\hat I+\frac{\mathrm i}{\hbar} \int_{-\infty}^{t}\mathrm dt'\, \hat H'_{\mathrm I,\lambda}(t') +\mathcal O(\lambda^2).$

Insert both into

$\mathcal E_A[\lambda f_x](t) =\operatorname{Tr}\!\left[ \hat\rho_\beta \hat U_{\mathrm I,\lambda}^{\dagger} \hat A_{\mathrm I}(t) \hat U_{\mathrm I,\lambda}\right].$

The correction from ${\hat U_{\mathrm I}^{\dagger}}$ places ${\hat H'}$ to the left of ${\hat A}$ ; the correction from ${\hat U_{\mathrm I}}$ places it to the right. Their difference gives, for a general source direction ${\hat H'_\lambda(t)=+\lambda f_B(t)\hat B}$ , the first variation ${\delta_f\langle\hat A(t)\rangle :=\left.\partial_\lambda\mathcal E_A[\lambda f_B](t)\right|_{\lambda=0}}$ :

$\delta_f\langle\hat A(t)\rangle =-\frac{\mathrm i}{\hbar} \int_{-\infty}^{t}\mathrm dt'\, f_B(t') {\langle[\hat A(t),\hat B(t')]\rangle}_\beta.$

All three predictions are now explained. Linearity comes from the first-order expansion. The commutator comes from two-sided unitary evolution. Causality is already present in the upper integration limit.

To see the algebra rather than only its result, retain the two first-order terms for ${\hat H'=f_B\hat B}$ :

$\begin{aligned} \delta_f\langle\hat A(t)\rangle &=\frac{\mathrm i}{\hbar} \int_{-\infty}^{t}\mathrm dt'\,f_B(t') {\langle\hat B(t')\hat A(t)\rangle}_\beta\\ &\quad-\frac{\mathrm i}{\hbar} \int_{-\infty}^{t}\mathrm dt'\,f_B(t') {\langle\hat A(t)\hat B(t')\rangle}_\beta. \end{aligned}$

They combine into the displayed commutator with the minus sign appropriate to the plus-source convention. Dropping the correction from either side of the expectation value would replace the commutator by an ordered correlation and would generally violate the independent static-sign check.

Stationarity, ${[\hat\rho_\beta,\hat H_0]=0}$ , makes the expectation depend only on ${t-t'}$ . We may write

$\delta_f\langle\hat A(t)\rangle =\int_{-\infty}^{\infty}\mathrm dt'\, \chi_{AB}^{\mathrm R}(t-t')f_B(t'),$

where the kernel is now defined with all of its conventions visible.

Retarded support and the Fourier convention

Let ${\hat\rho_0}$ be a stationary density operator for ${\hat H_0}$ , and let ${\hat A}$ and ${\hat B}$ be bounded, source-independent observables. For the plus-source convention ${\hat H'_\lambda=+\lambda f_B\hat B}$ , the retarded response function is the distribution on ${\mathbb R}$

$\chi_{AB}^{\mathrm R}(t) =-\frac{\mathrm i}{\hbar}\Theta(t) \operatorname{Tr}\!\left(\hat\rho_0 [\hat A(t),\hat B(0)]\right), \qquad \hat A(t)=\mathrm e^{+\mathrm i\hat H_0t/\hbar} \hat A\mathrm e^{-\mathrm i\hat H_0t/\hbar}.$

It has causal support ${\operatorname{supp}\chi_{AB}^{\mathrm R}\subseteq[0,\infty)}$ . Its Fourier convention is ${\chi^{\mathrm R}(\omega)=\int\mathrm dt\, \mathrm e^{+\mathrm i\omega t}\chi^{\mathrm R}(t)}$ , interpreted as a tempered-distribution transform when necessary. Where the complex-frequency transform exists, it is analytic for ${\operatorname{Im}z>0}$ , and the real-frequency retarded value is the boundary ${z\to\omega+\mathrm i0^+}$ .

This definition is the dynamical commutator contribution for an observable with no explicit source dependence. If the measured observable itself depends on the source, its total first variation may also contain an instantaneous kernel ${K_{AB}^{\mathrm{contact}}(t,t')}$ supported on ${t=t'}$ . Such a contact term must be derived from that model and added separately; it is not hidden inside the commutator above. Unbounded observables require a separately declared common domain.

Dimensions and static-limit checks

Under the hypotheses of Definition 10.1 and Definition 10.2, the two-sided first-order interaction-picture expansion gives

$\mathcal E_A[\lambda f_B](t)-\mathcal E_A[0](t) =\lambda\int_{-\infty}^{\infty}\mathrm dt'\, \chi_{AB}^{\mathrm R}(t-t')f_B(t')+\mathcal O(\lambda^2)$

for bounded finite-system dynamics, with the remainder weakened to the little- ${o(\lambda)}$ condition of the linear-response definition in the general setting. A source-dependent readout contributes the separately derived contact kernel stated in the retarded-response definition.

The Heaviside step ${\Theta(t)}$ is a locally integrable function. In Fourier analysis it is used through its associated distribution. Its value at the single point ${t=0}$ does not change an ordinary integral; its support for ${t\geq0}$ is what makes the response retarded.

An ideal impulse is not an element of ${\mathcal D}$ . Choose ${\varphi\in\mathcal D(\mathbb R)}$ with ${\int\varphi(t)\,\mathrm dt=1}$ and define the ordinary test sources

$f_{B,\varepsilon}(t) =f_0\varepsilon^{-1} \varphi\!\left(\frac{t-t_0}{\varepsilon}\right).$

As ${\varepsilon\to0^+}$ , they converge in ${\mathcal D'(\mathbb R)}$ to the compactly supported distribution ${f_0\delta_{t_0}}$ . Convolution with a delta is defined by translation, so the corresponding distributional extension gives

$\delta_f\langle\hat A(t)\rangle =f_0\chi_{AB}^{\mathrm R}(t-t_0).$

The measured change vanishes for ${t<t_0}$ and reproduces the kernel after the kick. For a general test source, the convolution is a sum of such delayed responses. If ${\chi^{\mathrm R}}$ is tempered and ${f_B\in\mathcal D}$ , the convolution theorem gives, as an equality in ${\mathcal S'(\mathbb R)}$ ,

$\delta_f\langle\hat A(\omega)\rangle =\chi_{AB}^{\mathrm R}(\omega)f_B(\omega),$

where multiplication by the smooth rapidly decreasing transform of ${f_B}$ is well defined. An ideal monochromatic source is read as a long-window limit, not as an undeclared member of the test-source space; it measures the amplitude and phase of the complex susceptibility at its drive frequency.

Apply an ideal impulse ${f_x(t)=F\delta(t)}$ through the plus-source perturbation ${+f_x(t)\hat\sigma_x}$ to the thermal preparation of ${\hat H_0=-(\hbar\Omega/2)\hat\sigma_z}$ , ${\Omega>0}$ . The integrated source ${F}$ has action units. Determine the later transverse expectation to first order in ${\alpha:=F/\hbar}$ , with ${|\alpha|\ll1}$ , and compare it with the exact kick result. The impulse is the short-pulse limit defined above, with free evolution negligible during the pulse.

Evolve both sides of the measured operator; a one-sided change would miss the commutator and its sign. The kick is ${\hat U=\mathrm e^{-\mathrm i\alpha\hat\sigma_x}}$ . Expanding both sides of the readout gives $\hat U^\dagger\hat\sigma_x(t)\hat U =\hat\sigma_x(t)+\mathrm i\alpha\hat\sigma_x\hat\sigma_x(t) -\mathrm i\alpha\hat\sigma_x(t)\hat\sigma_x+O(\alpha^2).$ Reuse the dyad evolution of the transverse-correlation example, ${\hat\sigma_x(t)=\hat\sigma_x\cos\Omega t+\hat\sigma_y\sin\Omega t}$ . Since ${[\hat\sigma_y,\hat\sigma_x]=-2\mathrm i\hat\sigma_z}$ , the thermal trace with ${m:=p_{\mathrm g}-p_{\mathrm e}}$ gives $\delta\langle\hat\sigma_x(t)\rangle =-2\alpha m\,\Theta(t)\sin\Omega t+O(\alpha^2).$ The minus sign comes from ${(-\mathrm i)(-2\mathrm i)=-2}$ . Independently, multiplying the exact kick ${\hat U=\hat I\cos\alpha-\mathrm i\hat\sigma_x\sin\alpha}$ gives ${\langle\hat\sigma_x(t)\rangle=-m\sin(2\alpha)\sin\Omega t}$ after the pulse. Its linear term agrees, with a cubic correction for this readout. The response is zero before the pulse and at its immediate end, since the kick commutes with the instantaneous ${\hat\sigma_x}$ readout. The result is dimensionless and vanishes for equal thermal populations. These checks fix the sign before the next example transforms the same kernel into frequency space. The controlled impulse expansion requires ${|F|/\hbar\ll1}$ .

The two-level susceptibility and its poles

Retain the thermal two-level Hamiltonian and plus-source convention of Example 10.1. Determine the transverse retarded kernel, the isolated first-order response to ${f_x(t)=f_0\Theta(t)}$ , with ${|f_0|/(\hbar\Omega)\ll1}$ , and the kernel’s frequency transform with Fourier factor ${\mathrm e^{+\mathrm i\omega t}}$ . For the transform, also consider the separately prescribed damping factor ${\mathrm e^{-\eta t}}$ , ${\eta>0}$ , and its ${\eta\to0^+}$ limit. The finite damping is a line model, not a request to derive a microscopic relaxation rate.

Reuse the signed impulse check in Example 10.1 to fix the source convention, then calculate the commutator before convolving or Fourier transforming. Heisenberg evolution gives

$\hat\sigma_x(t) =\hat\sigma_x\cos\Omega t +\hat\sigma_y\sin\Omega t.$

The thermal polarization of the two-level equilibrium state is

${\langle\hat\sigma_z\rangle}_\beta =p_{\mathrm g}-p_{\mathrm e} =\tanh\!\left(\frac{\beta\hbar\Omega}{2}\right).$

Using ${[\hat\sigma_y,\hat\sigma_x]=-2\mathrm i\hat\sigma_z}$ , the commutator and its equilibrium expectation are

$\begin{aligned} [\hat\sigma_x(t),\hat\sigma_x] &=\sin(\Omega t)[\hat\sigma_y,\hat\sigma_x] =-2\mathrm i\hat\sigma_z\sin(\Omega t),\\ {\langle[\hat\sigma_x(t),\hat\sigma_x]\rangle}_\beta &=-2\mathrm i \tanh\!\left(\frac{\beta\hbar\Omega}{2}\right) \sin(\Omega t). \end{aligned}$

Multiplying by the plus-source Kubo prefactor ${-\mathrm i\Theta(t)/\hbar}$ gives

$\chi_{xx}^{\mathrm R}(t) =-\frac{2}{\hbar}\Theta(t) \tanh\!\left(\frac{\beta\hbar\Omega}{2}\right) \sin\Omega t.$

The equilibrium spectrum carried ${p_{\mathrm g}}$ and ${p_{\mathrm e}}$ on separate directed lines. The commutator subtracts the opposite operator ordering, so the response carries ${p_{\mathrm g}-p_{\mathrm e}}$ . At high temperature the two directions become equally likely and this net transverse response is suppressed.

Consider a suddenly applied constant source ${f_x(t)=f_0\Theta(t)}$ in the ideal isolated model. Convolving it with the retarded kernel gives, for ${t>0}$ ,

$\delta_f\langle\hat\sigma_x(t)\rangle =-\frac{2f_0}{\hbar\Omega} \tanh\!\left(\frac{\beta\hbar\Omega}{2}\right) \left(1-\cos\Omega t\right).$

The response begins at zero, as it must, then oscillates because an isolated two-level system has no relaxation mechanism. Its time average equals the static isothermal-looking coefficient from the retarded calculation, but the instantaneous state never settles. Adding a bath, switching the source adiabatically, or averaging over a distribution of transition frequencies defines different physical routes to a stationary value. This simple step protocol is why “zero frequency” must always be accompanied by an order of limits and a preparation statement. This is a fixed-time expansion in source strength; higher-order frequency shifts prevent assuming uniform accuracy at arbitrarily long times.

For visualization, introduce a declared damping factor ${\mathrm e^{-\eta t}}$ with ${\eta>0}$ . It is a linewidth model, not the infinitesimal boundary prescription of an exact isolated system. The transform must be calculated before its pole structure is interpreted. With ${M:=\tanh(\beta\hbar\Omega/2)/\hbar}$ ,

$\begin{aligned} \chi_{xx}^{\mathrm R}(\omega) &=-2M\int_0^\infty \mathrm dt\, \mathrm e^{(-\eta+\mathrm i\omega)t}\sin(\Omega t)\\ &=-\frac{M}{\mathrm i} \left[ \frac{\mathrm i}{\omega+\Omega+\mathrm i\eta} -\frac{\mathrm i}{\omega-\Omega+\mathrm i\eta} \right]\\ &=M\left[ \frac{1}{\omega-\Omega+\mathrm i\eta} -\frac{1}{\omega+\Omega+\mathrm i\eta} \right]. \end{aligned}$

The second line uses ${\int_0^\infty\mathrm dt\, \mathrm e^{(-\eta+\mathrm ix)t}=\mathrm i/(x+\mathrm i\eta)}$ . The distributional boundary rule

$\lim_{\eta\to0^+}\frac{1}{x+\mathrm i\eta} =\mathcal P\frac{1}{x}-\mathrm i\pi\delta(x)$

then predicts both the principal-value dispersion and the signed delta peaks of the isolated system. For finite ${\eta}$ , the two components are

$\operatorname{Re}\chi_{xx}^{\mathrm R}(\omega) =M\left[ \frac{\omega-\Omega}{{(\omega-\Omega)}^2+\eta^2} -\frac{\omega+\Omega}{{(\omega+\Omega)}^2+\eta^2} \right],$

$\operatorname{Im}\chi_{xx}^{\mathrm R}(\omega) =M\left[ -\frac{\eta}{{(\omega-\Omega)}^2+\eta^2} +\frac{\eta}{{(\omega+\Omega)}^2+\eta^2} \right].$

The two curves are not independent fits. Their numerators and denominators come from the same causal poles. Near ${+\Omega}$ , the first Lorentzian dominates and the imaginary part is negative; near ${-\Omega}$ , the sign is reversed. As ${\eta\to0^+}$ , these Lorentzians approach delta distributions while the real terms approach principal-value distributions.

For real ${M}$ , the displayed real part is even and the imaginary part is odd, so ${\chi(-\omega)={\chi(\omega)}^*}$ , as required by a real time kernel. Both poles lie at ${\pm\Omega-\mathrm i\eta}$ , and the large-frequency ${1/\omega}$ terms cancel, leaving ${\chi(\omega)\sim2M\Omega/\omega^2}$ . The kernel has units inverse energy after Fourier transformation. Together with the zero initial step response, these are separate symmetry, causality and dimensional checks. Figure 10.2 makes the same calculated pair visible; the static calculation that follows checks its sign by a different route.

Before reading Figure 10.2, predict that the time response is zero for negative time, both poles lie below the real-frequency axis, and the positive-frequency imaginary peak is negative under the declared plus-source convention.

Figure 10.2. Computed causal response of the two-level system. The time-domain kernel vanishes before the source, and the dispersive and absorptive frequency components arise from the same pair of poles.

For ${\beta\hbar\Omega=2}$ and ${\eta/\Omega=0.08}$ , the calculation passes all three predictions. A finite-window numerical principal-value transform reconstructs the real part from the imaginary part to the tolerance recorded with the figure. That test can fail even when two independently drawn curves look plausible.

Mark the two pole coordinates on the frequency panels, then use them to explain why the time panel vanishes for ${t<0}$ and the sign of ${\operatorname{Im}\chi_{xx}^{\mathrm R}(+\Omega)}$ . Finally write one line distinguishing the infinitesimal ${0^+}$ boundary prescription from the declared physical linewidth ${\eta}$ . A pole in the upper half-plane diagnoses a damping-sign error; a correct pole with a reversed peak sign diagnoses a source- or Fourier-convention error. This recovery must occur before adding another formal response relation.

Static and resonant checks in the laboratory

For a sinusoidal transverse field, the component of the magnetization in phase with the field follows the dispersive part of the susceptibility, while the quadrature component follows the absorptive part, with signs fixed by the coupling and Fourier conventions. Under the common magnetic coupling ${\hat H'=-B_1(t)\hat M_x}$ , the cycle-averaged absorbed power is proportional to ${\omega {|B_1|}^2 \operatorname{Im}\chi_{M_xM_x}^{\mathrm R}(\omega)}$ in the usual positive-absorption convention. A peak in that quadrature therefore records dissipation; the sign-changing in-phase curve records the accompanying dispersive response.

The artificial factor ${\mathrm e^{-\eta t}}$ used in the figure can model a Lorentzian line for visualization, but ${\eta}$ has not been derived from the spin environment. A measured linewidth may contain homogeneous relaxation and dephasing as well as inhomogeneous broadening. The infinitesimal ${0^+}$ that chooses a retarded boundary value is not any of those rates.

Two measured resonance channels

Can one experiment show both an absorption-like peak and a dispersive sign change? A room-temperature experiment on the standard EPR reference material DPPH measured the resistance and reactance of a surrounding strip coil. The sample is an insulating pressed-powder bar: the drive current passes through the coil, not through the sample. Use Figure 10.3 to distinguish the magnetic perturbation from the electrical readout.

Figure 10.3. The DPPH system and impedance readout, schematic. The bar retains the reported dimension ratios, but field directions are conceptual rather than a reconstructed mounting orientation. The coil assembly is omitted; its field and readout roles are explicit. [CM20, fig. 1(a) and experimental details, pp. 17312-17313]

At fixed 2-GHz excitation, the static field was swept through resonance. Before reading Figure 10.4, predict which recorded channel will have a peak and which will change sign. Both traces are sample-in minus sample-out impedances, not an absolute susceptibility measurement.

Figure 10.4. Selected digitized measured markers at room temperature and 2 GHz: (a) resistance, (b) reactance. Separate axes retain the original ohm units and baseline scatter. The source’s fitted lines are not reproduced, and no experimental error bars were reported for these markers. [CM20, fig. 1(b), p. 17312]

Adapted from Chaudhuri and Mahendiran’s Figure 1(b) under CC BY-NC 3.0; axes and styling are redrawn, the source fits are omitted, and public use is noncommercial only.

The resistance peak and reactance sign change expose the two-channel structure near resonance. Coil geometry, phase convention and possible channel mixing are part of the conversion to susceptibility; the plotted quantities remain ${R}$ and ${X}$ . This later benchmark is selected for that clear comparison, not as the original discovery of EPR. Nor does it certify our weak-source approximation: the reported panel has no drive-amplitude sweep demonstrating linearity. Such a check would require reducing the drive while testing whether the response divided by drive amplitude and the line shape remain unchanged. Finally, a field sweep at one frequency is not a measured frequency-domain Kramers–Kronig integral.

Before the next calculation, obtain the ground-state tilt from the static Hamiltonian without using the response formula. Compare the two routes only after calculating the static derivative.

Determine the zero-temperature ground-state transverse expectation and its linear slope directly. Add a constant ${f_x}$ directly to the Hamiltonian:

$\hat H_{\mathrm{static}} =-\frac{\hbar\Omega}{2}\hat\sigma_z+f_x\hat\sigma_x.$ Take ${\Omega>0}$ and real ${f_x}$ . Compare the slope at zero source with the isolated, zero-temperature static limit of Example 10.2.

Diagonalize the static Hamiltonian and differentiate its lower eigenvalue; this tests the dynamic response without repeating the time-dependent calculation.

The two eigenvalues are

$E_\pm(f_x) =\pm\frac{1}{2} \sqrt{{(\hbar\Omega)}^2+4f_x^2}, \qquad E_{\mathrm g}(f_x)=E_-(f_x).$

Because the finite-dimensional eigenvalue is nondegenerate near ${f_x=0}$ , the Hellmann–Feynman relation applies:

${\langle\hat\sigma_x\rangle}_{\mathrm g} ={\left\langle \frac{\partial\hat H_{\mathrm{static}}}{\partial f_x} \right\rangle}_{\mathrm g} =\frac{\partial E_{\mathrm g}}{\partial f_x} =-\frac{2f_x} {\sqrt{{(\hbar\Omega)}^2+4f_x^2}} =-\frac{2f_x}{\hbar\Omega}+\mathcal O\!\left({\left(\frac{f_x}{\hbar\Omega}\right)}^3\right).$

Thus the zero-temperature static susceptibility must be negative. Taking ${\omega=0}$ and then ${\eta\to0^+}$ in the dynamic expression gives ${-2/(\hbar\Omega)}$ , in agreement. One route tilts an exact ground state; the other uses time-dependent perturbation theory. Their agreement checks the source sign and Fourier convention.

There is a check using the two-level energy projectors that does not differentiate an eigenvalue. Put ${a=\hbar\Omega/2}$ and ${R=\sqrt{a^2+f_x^2}}$ ; then ${\hat H_{\mathrm{static}}^2=R^2\hat I}$ , so its lower-state projector is ${\hat P_-=(\hat I-\hat H_{\mathrm{static}}/R)/2}$ . Tracing ${\hat P_-\hat\sigma_x}$ gives ${-f_x/R}$ , the same exact tilt. It is odd in ${f_x}$ , bounded in magnitude by one, and its derivative has inverse-energy units. The linear regime requires ${|f_x|\ll a}$ . The dynamic comparison used zero temperature and removed the line regulator; a finite phenomenological ${\eta}$ is not part of this exact equilibrium Hamiltonian. Changing to a longitudinal source changes the preparation question, as the next paragraph explains.

Change the protocol to find the boundary. If ${\hat H'(t)=+f_z(t)\hat\sigma_z}$ and the isolated unperturbed Hamiltonian remains proportional to ${\hat\sigma_z}$ , then ${[\hat\sigma_z(t),\hat\sigma_z(0)]=0}$ . The isolated system has no finite-frequency longitudinal response. This does not mean that its equilibrium population is independent of the level spacing. A bath that slowly re-equilibrates the system after the spacing changes defines a different protocol. Merely setting ${\omega=0}$ cannot erase that difference.

If the source remains ${\hat B=\hat\sigma_x}$ but the readout becomes ${\hat A=\hat\sigma_z}$ , the relevant object is ${\chi_{zx}^{\mathrm R}}$ , not ${\chi_{xx}^{\mathrm R}}$ . Energy differences determine possible resonance locations; matrix elements determine which resonances appear; equilibrium weights determine directional asymmetry; source and readout determine the response being measured. None of these inputs can be inferred from the word “susceptibility” alone.

Transition Weights in Retarded Response

Generalize only what the anchor has exposed. Insert energy eigenstates in the retarded kernel. With ${\omega_{mn}=(E_m-E_n)/\hbar}$ ,

(10.1)

$\chi_{AB}^{\mathrm R}(\omega) =\frac{1}{\hbar} \sum_{n,m} \frac{(p_n-p_m) \langle n|\hat A|m\rangle \langle m|\hat B|n\rangle} {\omega-\omega_{mn}+\mathrm i0^+}.$

This is the Lehmann representation. The symbol ${\mathrm i0^+}$ specifies the retarded boundary value: the poles are approached from the upper half of the complex ${\omega}$ plane and lie below the real axis. It is not a physical decay rate [Leh54].

The numerator can be traced directly to the two operator orderings. The first ordered average contains, with ${A_{nm}:=\langle n|\hat A|m\rangle}$ and ${B_{mn}:=\langle m|\hat B|n\rangle}$ ,

${\langle\hat A(t)\hat B(0)\rangle}_\beta =\sum_{n,m}p_n \mathrm e^{-\mathrm i\omega_{mn}t} A_{nm}B_{mn},$

while relabeling the second ordering gives the same phase and matrix-element product weighted by ${p_m}$ . Their difference is ${p_n-p_m}$ . The remaining time integral is the same boundary-value calculation used in the two-level anchor:

$\lim_{\eta\to0^+} \int_0^\infty\mathrm dt\, \mathrm e^{-\eta t} \mathrm e^{+\mathrm i(\omega-\omega_{mn})t} =\frac{\mathrm i} {\omega-\omega_{mn}+\mathrm i0^+}.$

Multiplication by the kernel’s prefactor ${-\mathrm i/\hbar}$ supplies the positive ${1/\hbar}$ and the retarded denominator in Equation (10.1). Thus the three visible parts of the Lehmann formula have separate origins: preparation probabilities in the numerator, quantum transitions in the matrix elements and ${\omega_{mn}}$ , and causality in ${\mathrm i0^+}$ .

For the two-level anchor, only ${g\leftrightarrow e}$ matrix elements of ${\hat\sigma_x}$ survive. Substituting them into the Lehmann sum reproduces the two denominators already computed. This reverse check matters pedagogically: the general formula has not introduced new physics; it has compressed the same calculation [KG06; MS59].

For ${\hat A=\hat B}$ Hermitian, the boundary identity used above gives

$\begin{aligned} -2\hbar\,\operatorname{Im}\chi_{AA}^{\mathrm R}(\omega) &=2\pi\sum_{n,m}(p_n-p_m) {|A_{nm}|}^2\delta(\omega-\omega_{mn})\\ &=2\pi\sum_{n,m}p_n \left(1-\mathrm e^{-\beta\hbar\omega}\right) {|A_{nm}|}^2\delta(\omega-\omega_{mn}). \end{aligned}$

In the second line, delta support enforces ${p_m/p_n=\mathrm e^{-\beta(E_m-E_n)} =\mathrm e^{-\beta\hbar\omega}}$ . The remaining sum is exactly Lecture 9’s ordered spectrum ${S_{AA}(\omega)}$ . Therefore, under our plus-source convention,

(10.2)

$-2\hbar\,\operatorname{Im}\chi_{AA}^{\mathrm R}(\omega) =\left(1-\mathrm e^{-\beta\hbar\omega}\right) S_{AA}(\omega).$

This fluctuation–dissipation relation is not an unexplained thermal prefactor. It is the population subtraction on each transition. With the conventional perturbation ${-F_A\hat A}$ , the response changes sign and the more familiar positive absorptive convention is recovered [CW51; Kub57].

Return to the isolated transverse two-level system, now with ${\beta\hbar\Omega=\ln3}$ . Check the fluctuation–dissipation factor on both directed lines and determine the sign of work supplied by a weak source. Use ${+f_x(t)\hat\sigma_x}$ with ${f_x(t)=f_a\cos\omega t}$ , ${\omega>0}$ , and the ordered-spectrum convention of Lecture 9. Interpret isolated spectral lines by their integrated weights, not a value at resonance.

Obtain the canonical weights first, compare the thermal factors line by line, then calculate work from the explicit time dependence of the Hamiltonian. The canonical probabilities are ${p_{\mathrm g}=3/4}$ , ${p_{\mathrm e}=1/4}$ , so the two integrated ordered-spectrum weights are ${3/4}$ and ${1/4}$ , while ${M=(p_{\mathrm g}-p_{\mathrm e})/\hbar =1/(2\hbar)}$ . The response therefore has $\operatorname{Im}\chi_{xx}^{\mathrm R}(\omega) =-\frac{\pi}{2\hbar}\delta(\omega-\Omega) +\frac{\pi}{2\hbar}\delta(\omega+\Omega).$ On the positive line, the thermal multiplier is ${1-1/3=2/3}$ , and ${(2/3)(2\pi)(3/4)=\pi}$ . On the negative line it is ${1-3=-2}$ , and ${(-2)(2\pi)(1/4)=-\pi}$ . These are exactly the two coefficients of ${-2\hbar\operatorname{Im}\chi}$ . The test compares distributions through their weights, not values at resonance. Their sum rule also gives ${(3/4)+(1/4)=\langle\hat\sigma_x^2\rangle=1}$ .

For a long weak sinusoidal source ${f_x(t)=f_a\cos\omega t}$ , the readout is ${f_a[\chi'\cos\omega t+\chi''\sin\omega t]}$ . External work follows from ${\partial_t\hat H=\dot f_x\hat\sigma_x}$ ; averaging a cycle gives $\overline P=-\frac{\omega f_a^2}{2}\operatorname{Im}\chi_{xx}^{\mathrm R}(\omega).$ Thus the negative positive-frequency imaginary weight gives nonnegative absorbed work in the plus-source convention. Power has units energy per time. For isolated delta lines this is understood as a weak finite-window spectral pairing, not a finite pointwise power at a delta peak. A linewidth model may smooth the plot, but thermal consistency of that model needs its own check; the exact line calculation alone does not prove it. The next dispersion test uses causality, which does not require these thermal weights.

Causality and Dispersion Relations

Causality also links the real and imaginary parts, but only after the analytic and decay hypotheses are stated.

Historical lineage

[Kro26, pp. 547–557] — Early relation between dispersion and absorption for X-rays.
[Kra27, pp. 545–557] — 1927 paired dispersion relations in the Como proceedings.

Let ${\chi^{\mathrm R}\in\mathcal S'(\mathbb R)}$ be retarded, so its complex-frequency transform is analytic for ${\operatorname{Im}z>0}$ . Suppose that analytic function has boundary values on the real axis and decays fast enough on the upper semicircle that no subtraction is required. Then

$\operatorname{Re}\chi^{\mathrm R}(\omega) =\frac{1}{\pi}\mathcal P \int_{-\infty}^{\infty}\mathrm d\omega'\, \frac{\operatorname{Im}\chi^{\mathrm R}(\omega')} {\omega'-\omega},$

and

$\operatorname{Im}\chi^{\mathrm R}(\omega) =-\frac{1}{\pi}\mathcal P \int_{-\infty}^{\infty}\mathrm d\omega'\, \frac{\operatorname{Re}\chi^{\mathrm R}(\omega')} {\omega'-\omega}.$

If the transform tends to a constant or grows, the corresponding subtracted dispersion relation must be used instead.

where ${\mathcal P}$ denotes the Cauchy principal-value distribution. The computed figure uses this relation as a consistency check. A contour proof and subtracted variants belong in a later mathematical reference, not in the causal spine of Lectures 9–10 [Tol56a].

Test the two-level pole pair from Example 10.2 at ${\omega_* =\Omega/2}$ , with ${M>0}$ , ${\Omega>0}$ . Evaluate one dispersion integral explicitly in the isolated limit ${\eta\to0^+}$ , where the imaginary part is ${-\pi M\delta(\omega'-\Omega)+\pi M\delta(\omega'+\Omega)}$ . Compare it with direct evaluation of the real pole pair. Determine what changes if both denominator prescriptions are replaced by ${-\mathrm i\eta}$ .

First check the analytic premises: for finite ${\eta>0}$ , the original poles lie below the real axis, the real part is even, the imaginary part odd, and the ${1/z^2}$ decay meets the unsubtracted dispersion hypothesis. The evaluation point lies away from either line, so the principal-value pairing reduces to $\frac1\pi\mathcal P\int \frac{\operatorname{Im}\chi(\omega')}{\omega'-\omega_*}\,\mathrm d\omega' =-\frac{M}{\Omega-\Omega/2} +\frac{M}{-\Omega-\Omega/2} =-\frac{8M}{3\Omega}.$ Direct substitution into the real pole pair gives ${M[{(-\Omega/2)}^{-1}-{(3\Omega/2)}^{-1}]=-8M/(3\Omega)}$ , independently checking the dispersion sign and both frequency branches. Its units are inverse energy, since ${M}$ has units inverse action. Dropping the negative frequency pole would fail both this check and the real-kernel symmetry.

Changing both denominator prescriptions to ${-\mathrm i\eta}$ moves the poles into the upper half-plane. The inverse transform then has support at negative time: it is advanced, not retarded. Its factor ${\mathrm e^{+\eta t}}$ decays toward negative infinity; alternatively insisting on positive-time support would require a growing kernel and would lose the assumed real-axis transform. This locates the causal failure before an integral is evaluated. The worked check used an off-resonant, zero-width limit; a finite-width static integral is a distinct calculation.

The larger course role of the resonance experiment is architectural. A weak probe is defined by a preparation, a source operator, a readout operator, and an order of limits. The retarded commutator converts that protocol into a causal kernel; its poles organize resonances, and its real and imaginary parts are not independent fitting functions. Screening, optical conductivity, neutron response, and collective modes will change the operators and material model while retaining this structure.

The wrong-half-plane test in Example 10.5 gives a concrete failure of retarded support. Checking that support and the Fourier convention precedes interpreting any fitted linewidth as a physical decay rate.

Starting only from ${\hat H'(t)=+f_B(t)\hat B}$ , reconstruct the two first-order corrections to ${\langle\hat A(t)\rangle}$ , combine them into the retarded commutator, and state its domain, units, and causal support. Then reverse the source-sign convention and predict which displayed sign changes while pole locations and measurable static agreement remain invariant. Finally distinguish an ideal delta line, an infinitesimal retarded prescription, and a physical linewidth. Those three objects solve different problems and must not be interchanged.

The reconstruction should end with a protocol-invariant statement: reversing the name and sign of the source may reverse the displayed susceptibility, but it cannot alter causal support, transition energies, or agreement with the same exact static experiment. That bounded claim, rather than a memorized formula, is the lecture’s exit criterion.

From a source protocol to a response function

Lectures 9–10 form one chain. Incomplete preparation requires probabilities; the density operator turns them into quantum expectations. Equilibrium supplies normalized energy-dependent probabilities. Those weights appear separately on the two directed peaks of an ordered correlation spectrum and give detailed balance.

A weak source asks a different question. Expanding unitary evolution on both sides of a readout produces a commutator, and the upper time limit produces retarded support. In the anchor system, the same two transitions reappear, but their weights are subtracted. Lehmann and fluctuation–dissipation formulas generalize precisely those already-understood features.

For a new problem, use this order:

state how the system was prepared and whether it is stationary;
name the source operator, source sign, and readout operator;
predict allowed transitions and causal support;
derive the ordered correlation or commutator required by the protocol;
check normalization, temperature limits, static sign, and pole location;
distinguish ideal distributions, boundary prescriptions, and physical linewidths.

A reported susceptibility is incomplete until its preparation, source operator, readout operator, coupling sign, and causal convention are stated. Reassigning a sign or constant between source and response changes the representation but not the predicted observable variation. Retarded support and Kramers–Kronig follow from causality; detailed balance and fluctuation– dissipation require the additional equilibrium state. A pole or peak is then a consequence to interpret, not the operational definition of response.

This source-to-readout map still treats the perturbing field as externally prescribed. It does not determine how the induced density changes the field that acts back on the electrons. Lecture 11 closes that medium-feedback loop in the static limit and thereby turns a bare interaction into a screened one.

Problems

1. Re-derive the response sign

Use ${\hat H'(t)=+f_y(t)\hat\sigma_y}$ and read ${\hat A=\hat\sigma_y}$ .

Derive ${\chi_{yy}^{\mathrm R}(t)}$ from the commutator.
Fourier transform with the convention used in this lecture and locate the poles.
Add a constant ${f_y}$ to the Hamiltonian and compute the zero-temperature ground-state tilt. Show that the static signs agree.
Rewrite the result for a force convention ${\hat H'=-F_y\hat\sigma_y}$ .

2. Transfer: a forbidden line and a new readout

Consider three nondegenerate levels ${E_0<E_1<E_2}$ . A Hermitian operator ${\hat B}$ has nonzero matrix elements only between ${0\leftrightarrow1}$ , while a readout ${\hat A}$ has nonzero matrix elements only between ${1\leftrightarrow2}$ .

Draw all directed transitions in ${S_{BB}(\omega)}$ and label their equilibrium weights.
Verify detailed balance as a distributional relation.
Decide, before calculation, whether ${\chi_{AB}^{\mathrm R}}$ can be nonzero for the declared selection rules.
Change one matrix element minimally so that a cross-response appears, then write its Lehmann numerator and resonance frequencies. Explain which conclusion came from energies, which from weights, and which from selection rules.

3. Causality, damping, and a dispersion check

Consider the real retarded kernel

$\chi^{\mathrm R}(t) =C\,\Theta(t)\mathrm e^{-\gamma t}\sin(\Omega t), \qquad C\in\mathbb R, \quad \gamma,\Omega>0,$

with the Fourier convention of this lecture.

Calculate ${\chi^{\mathrm R}(\omega)}$ and locate all poles.
Show explicitly that the real part is even and the imaginary part is odd on the real-frequency axis.
Calculate the static response and the leading large- ${|\omega|}$ behavior.
Use the zero-frequency Kramers–Kronig relation to connect the static response to ${\operatorname{Im}\chi^{\mathrm R}}$ . Explain which properties would fail if ${\Theta(t)}$ were replaced by a two-sided time dependence with an upper-half-plane pole.

(advanced): Thermal Generators and Probe Interfaces

The response argument is complete without this appendix. The two identities below are retained as a reference-only lookup. They are not exported as a dependency provider unless a later lecture acquires an exact consumer.

Differentiating the normalizers provides

(10.3)

${\langle\hat H\rangle}_\beta =-\frac{\partial}{\partial\beta}\ln Z, \qquad {\langle\hat N\rangle}_{\mathrm{gc}} =\frac{1}{\beta} \frac{\partial}{\partial\mu}\ln\Xi.$

They follow directly by differentiating under the trace when the relevant operators commute. No free-energy formalism is assumed here.

For conductivity and other material responses, the source is often a scalar or vector potential while the readout is a density or current. The microscopic current operator, diamagnetic/contact terms, conserved quantities, boundary conditions, and order of limits must be supplied by the later model [KG06; Kub57]. The general interface is only

(10.4)

$\text{preparation} +\text{source sign/operator} +\text{readout operator} +\text{limiting protocol} \longrightarrow \text{a specified response function}.$

References

[CW51] Callen, H. B., and Welton, T. A. (1951). “Irreversibility and generalized noise.” Physical Review 83 (1), 34–40. https://doi.org/10.1103/PhysRev.83.34.

[CM20] Chaudhuri, U., and Mahendiran, R. (2020). “Detection of l-band electron paramagnetic resonance in the DPPH molecule using impedance measurements.” RSC Advances 10 (29), 17311–17316. https://doi.org/10.1039/D0RA03285A.

[Dic46] Dicke, R. H. (1946). “The measurement of thermal radiation at microwave frequencies.” Review of Scientific Instruments 17 (7), 268–275. https://doi.org/10.1063/1.1770483.

[KG06] Kapusta, J. I., and Gale, C. (2006). “Linear response theory.” in Finite-temperature field theory: Principles and applications, Cambridge University Press, pp. 84–116. https://doi.org/10.1017/CBO9780511535130.007.

[Kel65] Keldysh, L. V. (1965). “Diagram technique for nonequilibrium processes.” Soviet Physics JETP 20 (4), 1018–1026. https://www.jetp.ras.ru/cgi-bin/index/e/20/4/p1018?a=list.

[Kra27] Kramers, H. A. (1927). “La diffusion de la lumière par les atomes.” in Atti del congresso internazionale dei fisici, 10–20 settembre 1927, como–pavia–roma, Nicola Zanichelli, Bologna, pp. 545–557. https://www.lorentz.leidenuniv.nl/IL-publications/sources/Kramers_27.pdf.

[Kro26] Kronig, R. de L. (1926). “On the theory of dispersion of x-rays.” Journal of the Optical Society of America 12 (6), 547–557. https://doi.org/10.1364/JOSA.12.000547.

[Kub57] Kubo, R. (1957). “Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems.” Journal of the Physical Society of Japan 12 (6), 570–586. https://doi.org/10.1143/JPSJ.12.570.

[Leh54] Lehmann, H. (1954). “Über eigenschaften von ausbreitungsfunktionen und renormierungskonstanten quantisierter felder.” Il Nuovo Cimento 11 (4), 342–357. https://doi.org/10.1007/BF02783624.

[MS59] Martin, P. C., and Schwinger, J. (1959). “Theory of many-particle systems. i.” Physical Review 115 (6), 1342–1373. https://doi.org/10.1103/PhysRev.115.1342.

[Mat55] Matsubara, T. (1955). “A new approach to quantum-statistical mechanics.” Progress of Theoretical Physics 14 (4), 351–378. https://doi.org/10.1143/PTP.14.351.

[Sco94] Scofield, J. H. (1994). “Frequency-domain description of a lock-in amplifier.” American Journal of Physics 62 (2), 129–133. https://doi.org/10.1119/1.17629.

[Tol56a] Toll, J. S. (1956). “Causality and the dispersion relation: Logical foundations.” Physical Review 104 (6), 1760–1770. https://doi.org/10.1103/PhysRev.104.1760.

Lecture 10: Weak Sources and Retarded Response

10.1 From a weak source to the retarded commutator

10.1.1 Source convention and interaction-picture expansion

10.1.2 Two-sided state correction and the commutator

Historical background

10.1.3 Retarded support and the Fourier convention

10.1.4 Dimensions and static-limit checks

10.2 The two-level susceptibility and its poles

10.3 Static and resonant checks in the laboratory

10.3.1 Two measured resonance channels

10.4 Transition Weights in Retarded Response

Related imaginary-time formalism

10.5 Causality and Dispersion Relations