Euclidean Preparation and Lorentzian State Dictionaries

A Euclidean path integral that ends on a spatial slice already defines a state wavefunctional on that slice. Gluing embeds this state in a complete complex-time contour and, at saddle level, converts it into Lorentzian classical field and momentum data; the placement of later insertions then selects time-ordered, Wightman, or retarded observables. This page derives that dictionary for a scalar in asymptotically AdS$_{d+1}$, using standard quantization, a cap source supported away from the join, and source-free linearized Lorentzian evolution.

Required background. Timelike Boundary, Causality, and Boundary-Value Problems supplies the admissible AdS boundary data and symplectic-flux condition. The GKPW Generating-Functional Relation supplies the source convention and Euclidean saddle prescription. States, Geometries, and Radial Quantization supplies the state–operator and energy dictionaries.

Helpful background. Initial Density Matrices and Contour Boundary Conditions explains how a general initial density matrix is encoded on a Schwinger–Keldysh contour.

A Euclidean cap defines a wavefunctional

Here a cap is the oriented Euclidean segment $\tau\in(-\infty,0]$ of the complex-time contour, and its join $\Sigma$ is the terminal codimension-one spatial slice at $\tau=0$. A corner, when mentioned in the cited real-time construction, is the lower-dimensional intersection of this join with the radial conformal boundary.

Take Euclidean global AdS with boundary-cylinder time $\tau$,

$$\mathrm ds_E^2=L^2\left( \cosh^2\rho\,\mathrm d\tau^2+\mathrm d\rho^2 +\sinh^2\rho\,\mathrm d\Omega_{d-1}^2\right), \qquad \tau\leq 0,$$

and a scalar with $m^2L^2=\Delta(\Delta-d)$ on the standard branch $\Delta>d/2$. With the GKPW convention $Z_E[J]=\langle\exp(\int J\mathcal O)\rangle$, the boundary source prepares the unnormalized ket

$$\lvert J_-\rangle =\mathcal T_\tau\exp\!\left[ \int_{-\infty}^{0}\mathrm d\tau\,\mathrm d\Omega\, J_-(\tau,\Omega)\mathcal O_E(\tau,\Omega) \right]\lvert 0\rangle .$$

Near the conformal boundary, with defining function $z\sim2e^{-\rho}$,

$$\phi_E =z^{d-\Delta}J_-(\tau,\Omega) +z^\Delta A_-(\tau,\Omega)+\cdots .$$

The bulk cap path integral is not a single number or a unique endpoint field. It is the functional

$$\Psi^{\mathrm{bulk}}_{J_-}[\varphi] = \int_{\substack{\phi_E|_{\tau=0}=\varphi\\ \phi_E\sim z^{d-\Delta}J_-\\ \text{regular at the cap}}} \mathcal D\phi_E\,e^{-S_E[\phi_E]} .$$

For each fixed endpoint configuration $\varphi$, regularity and $J_-$ determine the saddle, when it is unique. The shared endpoint is integrated over—or extremized at leading large $N$—only when the cap is glued to the remaining contour. Confusing these two stages incorrectly makes the source appear to fix $\varphi$ before gluing. The state/wavefunctional construction and the matching data at contour junctions are developed in Skenderis and van Rees 2009, §2.1 and §2.2.1, pp. 7–11 (Open PDF).

A reflected future cap,

$$J_+(\tau,\Omega)=J_-^*(-\tau,\Omega),$$

prepares the adjoint bra. Reflection does not normalize the state: one must still use $\lvert\widehat J\rangle=\lvert J\rangle/\sqrt{\langle J\vert J\rangle}$. Two independent caps instead compute a transition matrix element $\langle\Psi_+\rvert\cdots\lvert\Psi_-\rangle$; they do not by themselves define a Hermitian, positive, unit-trace density matrix.

The roles of the data are worth separating before doing the continuation:

What each part of the preparation contour fixes

Contour part	Data held fixed	What it determines
Past Euclidean cap	boundary source J− and endpoint field φ	ket wavefunctional as a function of φ
Join	one shared endpoint field, integrated or extremized	compatible classical field and momentum data at saddle level
Lorentzian branch	real-time source and later insertions	evolution and contour-ordered observables
Reflected future cap	the reflected complex-conjugate source	the adjoint bra, but not automatic normalization

The join supplies field and momentum data

Use the site’s $(+---)$ convention after continuing $\tau=i t$, equivalently $t=-i\tau$. Direct substitution gives

$$\left.\mathrm ds_E^2\right|_{\tau=i t}=-\mathrm ds_L^2, \qquad \mathrm ds_L^2=L^2\left( \cosh^2\rho\,\mathrm dt^2-\mathrm d\rho^2 -\sinh^2\rho\,\mathrm d\Omega_{d-1}^2\right).$$

Let $h$ be the positive spatial metric on the joining slice and define coordinate momenta along increasing $\tau$ and $t$ by

$$\pi_E=\sqrt h\,N_E^{-1}\partial_\tau\phi_E, \qquad \pi_L=\sqrt h\,N_L^{-1}\partial_t\phi_L .$$

The lower boundary of the Lorentzian action and the upper boundary of the Euclidean action contribute

$$\left.\delta\!\left(iS_L-S_E\right)\right|_\Sigma =-\int_\Sigma \left(i\pi_L+\pi_E\right)\delta\varphi .$$

Stationarity of the joined saddle therefore requires

$$\phi_L(0,\rho,\Omega)=\phi_E(0,\rho,\Omega), \qquad \pi_L(0,\rho,\Omega)=i\pi_E(0,\rho,\Omega),$$

or $\partial_t\phi_L=i\partial_\tau\phi_E$ when the lapses agree. These signs refer to coordinate momenta along increasing coordinates. If both momenta are instead defined with outward normals, the initial Lorentzian normal is past-directed and an extra minus sign appears. The scalar matching conditions follow directly from the joined variational principle; see Skenderis and van Rees 2009, §§3.1.3–3.1.4, eqs. (3.1.10)–(3.1.15), pp. 16–17 (Open PDF).

For source-free Lorentzian evolution, take $J_-$ to vanish in a neighborhood of $\tau=0$. Compact support in $\tau_1<\tau<\tau_2<0$ is a clean choice. The continued field then has no Lorentzian non-normalizable coefficient,

$$\phi_L(z,t,\Omega)=z^\Delta A_L(t,\Omega)+\cdots ,$$

so the cap has prepared normalizable initial data rather than a force that remains switched on. A source that reaches or crosses the join must instead be included in the junction analysis or treated as a Lorentzian boundary source. This support condition and the field-plus-momentum matching are the essential ingredients of the real-time prescription Skenderis and van Rees 2008, “Prescription” and “Example,” printed pp. 2–4, eqs. (1)–(5) (Open PDF).

One oscillator exposes the source-to-state map

Before carrying the normalization into AdS, consider a unit-mass oscillator of frequency $\omega$. Its Euclidean operator is

$$q_E(\tau) =\frac{1}{\sqrt{2\omega}} \left(ae^{-\omega\tau}+a^\dagger e^{\omega\tau}\right).$$

The cap source $J_-$ may be complex. Normal-ordering its Euclidean time-ordered exponential produces a source-dependent scalar factor; the remaining annihilation-operator exponential acts trivially on $\lvert0\rangle$. The state is therefore

$$\lvert J_-\rangle\propto e^{\alpha a^\dagger}\lvert0\rangle, \qquad \alpha=\frac{\mathcal J_\omega}{\sqrt{2\omega}}, \qquad \mathcal J_\omega =\int_{-\infty}^{0}\mathrm d\tau\, e^{\omega\tau}J_-(\tau).$$

After normalization this is the exact Glauber coherent state of the free oscillator. With the reflected bra, its Lorentzian one-point function and initial data are

$$\begin{aligned} q_{\mathrm{cl}}(t) &=\frac{\alpha e^{-i\omega t}+\alpha^*e^{i\omega t}} {\sqrt{2\omega}},\\ q_{\mathrm{cl}}(0)&=\frac{\operatorname{Re}\mathcal J_\omega}{\omega}, & p_{\mathrm{cl}}(0)&=\operatorname{Im}\mathcal J_\omega . \end{aligned}$$

Thus the Euclidean source does not become a Lorentzian function by a blind substitution $\tau\to it$. Its weighted Laplace moment becomes the complex positive-frequency coefficient, and the real and imaginary parts supply the two real pieces of initial data. A real source gives $p_{\mathrm{cl}}(0)=0$; a source phase supplies nonzero initial momentum. In an interacting theory the same construction defines a state, but exact Glauber coherence receives corrections.

Global-AdS modes are weighted Laplace moments

Let the scalar harmonics on $S^{d-1}$ obey

$$\int\mathrm d\Omega\, Y_{\ell\sigma}^*(\Omega)Y_{\ell'\sigma'}(\Omega) =\delta_{\ell\ell'}\delta_{\sigma\sigma'},$$

where $\sigma$ labels the degeneracy at fixed $\ell$. Write the Klein–Gordon-normalized positive-frequency bulk modes and their unit one-particle states as

$$\begin{aligned} U_{n\ell\sigma}(t,\rho,\Omega) &=e^{-i\omega_{n\ell}t} f_{n\ell}(\rho)Y_{\ell\sigma}(\Omega),\qquad \omega_{n\ell}=\Delta+2n+\ell,\\ \lvert n\ell\sigma\rangle &=a_{n\ell\sigma}^\dagger\lvert0\rangle,\qquad n=0,1,\ldots . \end{aligned}$$

Global time and $\omega_{n\ell}$ are dimensionless. Define the normalization rather than hiding it by the one-particle matrix element

$$\langle n\ell\sigma\vert \mathcal O_E(\tau,\Omega)\lvert0\rangle =\kappa_{n\ell}e^{\omega_{n\ell}\tau} Y_{\ell\sigma}^*(\Omega).$$

Rotational symmetry makes $\kappa_{n\ell}$ independent of $\sigma$ once the harmonic phases and one-particle phases are fixed. The complex conjugate on $Y_{\ell\sigma}$ is essential in a complex harmonic basis. At linear order, the source coupling therefore selects

$$J_{\ell\sigma}(\tau) =\int\mathrm d\Omega\, Y_{\ell\sigma}^*(\Omega)J_-(\tau,\Omega),$$

and prepares

$$\begin{aligned} \alpha_{n\ell\sigma} &=\kappa_{n\ell} \int_{-\infty}^{0}\mathrm d\tau\, e^{\omega_{n\ell}\tau}J_{\ell\sigma}(\tau),\\ \phi_L &=\sum_{n,\ell,\sigma} \left( \alpha_{n\ell\sigma}U_{n\ell\sigma} +\alpha_{n\ell\sigma}^*U_{n\ell\sigma}^* \right). \end{aligned}$$

This operational definition includes the scalar-action prefactor, the boundary limit of the radial mode, and the normalization of $\mathcal O$. There is no universal factor $1/(2\omega_{n\ell})$ once the $U_{n\ell\sigma}$ are Klein–Gordon normalized. A source in one angular harmonic $(\ell,\sigma)$ generally excites every radial overtone $n$; angular localization is not full normal-mode isolation. The general mode transform appears in Botta-Cantcheff, Martínez, and Silva 2016, §3.1, eqs. (3.7)–(3.10), pp. 6–7, and the explicit coherent-state identification in §4, especially eqs. (4.18) and (4.32)–(4.35), pp. 13–17 (Open PDF).

Worked preparation: a compact cap pulse

A compact pulse makes the source-free join condition exact. For one oscillator, or for one projected AdS harmonic before multiplication by its AdS normalization, choose the following continuously differentiable test profile:

$$J_-(\tau)= \begin{cases} j\sin^2\!\left[\dfrac{\pi(\tau-\tau_1)}{T}\right], &\tau_1<\tau<\tau_2,\\[6pt] 0,&\text{otherwise}, \end{cases} \qquad T=\tau_2-\tau_1, \qquad k=\frac{2\pi}{T},$$

with $\tau_2<0$. Direct integration gives the reproducible source-to-state map

$$\mathcal J_\omega =j\,\frac{k^2}{2\omega(\omega^2+k^2)} \left(e^{\omega\tau_2}-e^{\omega\tau_1}\right), \qquad \alpha=\frac{\mathcal J_\omega}{\sqrt{2\omega}}.$$

For the concrete choice $\tau_1=-3/\omega$ and $\tau_2=-1/\omega$,

$$\mathcal J_\omega =\frac{j\pi^2}{2\omega(1+\pi^2)} \left(e^{-1}-e^{-3}\right).$$

The coherent-state calculation gives

$$\left(q_{\mathrm{cl}}(0),p_{\mathrm{cl}}(0)\right) =\left( \frac{\operatorname{Re}\mathcal J_\omega}{\omega}, \operatorname{Im}\mathcal J_\omega \right).$$

An independent check comes from solving the Euclidean oscillator with the reflected source: the Green function $e^{-\omega\lvert\tau-\tau'\rvert}/(2\omega)$ gives the same endpoint position, while the joined derivative gives the same momentum. For any source ending at $\tau_2<0$,

$$\left|\mathcal J_\omega\right| \leq e^{\omega\tau_2} \int_{\tau_1}^{\tau_2}\mathrm d\tau\,|J_-(\tau)|,$$

which bounds the raw Laplace moment. A $C^\infty$ bump with the same support may be used when higher source smoothness matters. Marolf et al. derive the linear source-to-initial-data transform in §§2–3, especially eqs. (31)–(38), pp. 11–13 and analyze the tension between sharp localization and perturbative control in §4.1, especially §§4.1.1–4.1.3, pp. 20–26 (Open PDF).

For an AdS mode whose projected source has the same support, the operational normalization above gives

$$|\alpha_{n\ell\sigma}| \leq |\kappa_{n\ell}|e^{\omega_{n\ell}\tau_2} \int_{\tau_1}^{\tau_2}\mathrm d\tau\, |J_{\ell\sigma}(\tau)|.$$

A reproducible truncation at $\omega_{n\ell}\leq\Omega_{\max}$ can monitor both the missed coherent occupation—the Klein–Gordon norm squared of the omitted positive-frequency field—and the missed dimensionless energy:

$$\begin{aligned} N_{\mathrm{tail}}(\Omega_{\max}) &=\sum_{\substack{n,\ell,\sigma\\ \omega_{n\ell}>\Omega_{\max}}} |\alpha_{n\ell\sigma}|^2,\\ H_{t,\mathrm{tail}}(\Omega_{\max}) &=\sum_{\substack{n,\ell,\sigma\\ \omega_{n\ell}>\Omega_{\max}}} \omega_{n\ell}|\alpha_{n\ell\sigma}|^2, \qquad E_{\mathrm{tail}}=\frac{H_{t,\mathrm{tail}}}{L}. \end{aligned}$$

Each sum includes all $n$, $\ell$, and degeneracy labels $\sigma$ above the cutoff. Declare absolute or retained-relative tolerances for both sums, and stop only when both are met and the join residuals $\phi_L-\phi_E$ and $\pi_L-i\pi_E$ vanish to the same numerical accuracy. The physical approximation separately requires a small dimensionless source-induced field amplitude relative to the nonlinear scale and small total bulk energy for negligible backreaction.

Gaussian diagnostic and its endpoint trap

A Gaussian is useful for exposing a common error, but it is not compactly supported. For

$$J_-(\tau)=j\exp\!\left[-\frac{(\tau-\tau_0)^2}{2\sigma^2}\right], \qquad \tau\leq0,$$

the exact half-line moment is

$$\mathcal J_\omega =j\sigma\sqrt{\frac{\pi}{2}}\, \exp\!\left( \omega\tau_0+\frac{\omega^2\sigma^2}{2} \right) \operatorname{erfc}\!\left( \frac{\tau_0+\omega\sigma^2}{\sqrt2\,\sigma} \right).$$

Replacing the half-line by the full Gaussian gives $j\sqrt{2\pi}\sigma\exp(\omega\tau_0+\omega^2\sigma^2/2)$ only when the shifted saddle $\tau_0+\omega\sigma^2$ lies many widths below the join. Merely observing that the unweighted Gaussian is small at $\tau=0$ is not enough. At fixed $\tau_0$ and $\sigma$,

$$\mathcal J_\omega \sim \frac{j}{\omega} \exp\!\left[-\frac{\tau_0^2}{2\sigma^2}\right] \qquad (\omega\to\infty),$$

so the tiny but nonzero endpoint tail eventually dominates. Taper the Gaussian to zero before the join, or restrict it to a declared finite frequency band; otherwise it is not a uniformly ultraviolet-soft, source-free preparation.

Contour branches select the observable

An oriented segment of the complex-time path is a branch. Let $C_1$ run forward in Lorentzian time and $C_2$ return backward; along the full closed contour, every point on $C_2$ is later in contour order than every point on $C_1$. Define $\mathcal G_{ab}=\langle\mathcal T_C\,\mathcal O_a(t)\mathcal O_b(t')\rangle$ without an overall factor of $-i$.

How branch placement selects a real-time two-point function

Placement	Correlator	Meaning
Both insertions on C1	𝒢11 = GF	time ordered
Both insertions on C2	𝒢22 = GF̄	anti-time ordered
First on C1, second on C2	𝒢12 = G<	⟨𝒪(t′)𝒪(t)⟩
First on C2, second on C1	𝒢21 = G>	⟨𝒪(t)𝒪(t′)⟩
Difference of cross-branch responses	GR = +i θ(t − t′)[G> − G<]	causal response in the site’s source convention

For a two-point function with one insertion fixed at the origin, the boundary value $-i\tau=t-i0$ gives $G^>(t)$, not the time-ordered function. The opposite side gives $G^<(t)$, while the Feynman function is piecewise,

$$-i\tau=t-i0\,\operatorname{sgn}(t).$$

With Fourier convention $e^{-ip^0t+i\mathbf p\cdot\mathbf x}$ and signature $(+---)$, a free scalar illustrates the distinction:

$$D_F(p)= \frac{i}{(p^0)^2-\mathbf p^2-m^2+i0}, \qquad \text{retarded poles:}\quad -\frac{1}{(p^0+i0)^2-\mathbf p^2-m^2}.$$

The retarded denominator has both poles below the real $p^0$ axis. A generic CFT correlator uses the corresponding spectral boundary value rather than this simple free pole. Skenderis and van Rees use the opposite Lorentzian source deformation, $S\mapsto S-\int J\mathcal O$, so their $-i\theta$ response translates to the $+i\theta$ convention used here. The branch dictionary and Euclidean boundary values are derived in Skenderis and van Rees 2009, Appendix A.2–A.3, eqs. (A.2.6)–(A.2.10) and (A.3.1)–(A.3.5), pp. 77–80 (Open PDF).

Three adversarial checks

These changes test whether a calculation has retained the state and contour information:

Move a source across the join

Move $j\delta(\tau-\tau_0)$ from the cap to $j\delta(t-t_s)$ on the Lorentzian branch. The cap gives $\alpha=je^{\omega\tau_0}/\sqrt{2\omega}$. With the site convention $S\mapsto S+\int Jq$, the real-time impulse instead gives

$$p(t_s^+)-p(t_s^-)=j$$

and leaves the state before $t_s$ unchanged. This disproves the hypothesis that every source insertion prepares the initial state.

Reverse the continuation side

For the vacuum oscillator, reversing the Euclidean boundary value changes

$$G^>(t)=\frac{e^{-i\omega t}}{2\omega} \quad\longrightarrow\quad G^<(t)=\frac{e^{i\omega t}}{2\omega}.$$

The two answers are complex conjugates rather than identical. This disproves the hypothesis that the $i0$ side is irrelevant.

Drop momentum matching

Keep $\phi_L(0)=\phi_E(0)$ but omit the momentum condition. One may then add $c\sin(\omega t)$ without changing the initial field, while shifting the initial momentum by $c\omega$. This disproves the hypothesis that field continuity alone fixes the state.

The surviving claim is narrower and stronger: a source belongs to the state only when it lies on a preparation segment; an observable is fixed only after its contour branch and boundary value are named; and a second-order Lorentzian problem needs both matched initial data.

Domain, limits, and handoffs

Exact oscillator coherence holds in a free theory and at generalized-free leading order. At this order, the normal-ordered scalar excitation energy associated with dimensionless global time is

$$H_t=\sum_{n,\ell,\sigma} \omega_{n\ell}|\alpha_{n\ell\sigma}|^2, \qquad E_{\mathrm{phys}}=\frac{H_t}{L}.$$

Large $N$ suppresses bulk loops, small source amplitude controls the linearization, and fixed-background evolution additionally requires the following dimensionless ratio to be small:

$$\frac{G_NH_t}{L^{d-1}} =\frac{G_NE_{\mathrm{phys}}}{L^{d-2}} \ll1.$$

These are distinct approximations: large $N$ alone does not remove classical nonlinearities or backreaction. Interactions mix modes and generate noncoherent corrections. A general mixed state requires an explicit initial-density-matrix kernel or an appropriate closed contour, not merely two unrelated caps.

Boundaries and State Preparation develops the underlying path-integral wavefunctional. The prepared normalizable data feed directly into Heavy States, Coherent States, and Semiclassical Geometries. For doubled real-time branches, continue to Schwinger–Keldysh Bulk Geometries and Holographic Initial States and Euclidean Caps.

Common pitfalls

Treating the endpoint field as source-determined. The source and regularity determine a saddle only after the endpoint $\varphi$ has also been specified. Gluing then integrates or extremizes over the shared $\varphi$.

Equating reflection with normalization. A reflected cap supplies the adjoint bra. Divide by the norm when computing normalized expectation values.

Calling an angular harmonic a normal mode. Fixing $(\ell,\sigma)$ still leaves every radial overtone $n$. The factors $\kappa_{n\ell}$ and the different Laplace weights determine their amplitudes.

Using one $i0$ prescription for every correlator. Feynman, Wightman, and retarded functions are different boundary values or branch combinations. Name the branches before continuing.

Exercises

Derive the join sign

For a unit oscillator, let the Euclidean cap end at $\tau=0$ and the Lorentzian segment begin at $t=0$. Using $\delta S_E|_{0}=p_E\delta q$ and $\delta S_L|_{0}=-p_L\delta q$, vary the joined exponent $iS_L-S_E$ and derive the momentum condition. How does the statement change if both momenta are defined with outward normals?

Solution

The shared endpoint value is integrated over, so its coefficient in the saddle variation must vanish:

$$\delta(iS_L-S_E)|_0 =-(ip_L+p_E)\delta q=0.$$

Therefore $p_L=i p_E$ for momenta defined along increasing $t$ and $\tau$. The outward normal at the initial Lorentzian boundary points toward decreasing $t$, so the outward Lorentzian momentum is $-p_L$; in that convention the relation is $p_L^{\mathrm{out}}=-i p_E^{\mathrm{out}}$. The endpoint field itself is continuous because the two path integrals are glued by integrating over one shared $q$.

Derive the compact-pulse transform

For the compact $\sin^2$ pulse above, derive $\mathcal J_\omega$ and evaluate it for $\tau_1=-3/\omega$, $\tau_2=-1/\omega$.

Solution

Put $x=\tau-\tau_1$ and use $\sin^2(\pi x/T)=[1-\cos(kx)]/2$ with $k=2\pi/T$. Then

$$\begin{aligned} \mathcal J_\omega &=je^{\omega\tau_1} \int_0^T\mathrm dx\,e^{\omega x}\sin^2\!\left(\frac{\pi x}{T}\right)\\ &=je^{\omega\tau_1} \frac{e^{\omega T}-1}{2} \left(\frac1\omega-\frac{\omega}{\omega^2+k^2}\right)\\ &=j\,\frac{k^2}{2\omega(\omega^2+k^2)} \left(e^{\omega\tau_2}-e^{\omega\tau_1}\right). \end{aligned}$$

For $T=2/\omega$, $k=\pi\omega$, so

$$\mathcal J_\omega =\frac{j\pi^2}{2\omega(1+\pi^2)} \left(e^{-1}-e^{-3}\right).$$

Diagnose the Gaussian approximation

Complete the square to obtain the exact half-line Gaussian transform. State the condition for replacing it by the full-line integral and find the leading large-$\omega$ behavior.

Solution

The exponent is

$$-\frac{(\tau-\tau_0)^2}{2\sigma^2}+\omega\tau =-\frac{(\tau-\tau_0-\omega\sigma^2)^2}{2\sigma^2} +\omega\tau_0+\frac{\omega^2\sigma^2}{2}.$$

Changing variables at the upper limit $\tau=0$ gives the stated complementary-error-function result. The full-line approximation requires $(\tau_0+\omega\sigma^2)/(\sqrt2\sigma)\ll-1$. For a large positive argument, $\operatorname{erfc}x\sim e^{-x^2}/(\sqrt\pi x)$, which yields

$$\mathcal J_\omega \sim \frac{j\sigma^2}{\tau_0+\omega\sigma^2} \exp\!\left[-\frac{\tau_0^2}{2\sigma^2}\right] \sim \frac{j}{\omega} \exp\!\left[-\frac{\tau_0^2}{2\sigma^2}\right].$$

Recover causal response from contour data

For the vacuum oscillator, use $G^>(t)=e^{-i\omega t}/(2\omega)$ and $G^<(t)=e^{i\omega t}/(2\omega)$ to compute $G_R(t)=+i\theta(t)[G^>(t)-G^<(t)]$. Explain why neither Wightman function alone is retarded.

Solution

Their difference is $-i\sin(\omega t)/\omega$, so with the convention in the question,

$$G_R(t)=+\theta(t)\frac{\sin(\omega t)}{\omega}.$$

The step function makes the response vanish before the perturbation, while the commutator removes the state-dependent symmetric part. Each Wightman function is nonzero for both signs of $t$ and therefore is not by itself a causal response. A source convention $S\mapsto S-\int J\mathcal O$ reverses the overall response sign but not the causal support or pole placement.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Botta-Cantcheff, Marcelo, Pedro J. Martínez, and Guillermo A. Silva. “On Excited States in Real-Time AdS/CFT.” Journal of High Energy Physics 02 (2016): 171. doi:10.1007/JHEP02(2016)171. arXiv:1512.07850 (Open PDF).
• Marolf, Donald, Onkar Parrikar, Charles Rabideau, Ali Izadi Rad, and Mark Van Raamsdonk. “From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories.” Journal of High Energy Physics 06 (2018): 077. doi:10.1007/JHEP06(2018)077. arXiv:1709.10101 (Open PDF).
• Skenderis, Kostas, and Balt C. van Rees. “Real-Time Gauge/Gravity Duality.” Physical Review Letters 101 (2008): 081601. doi:10.1103/PhysRevLett.101.081601. arXiv:0805.0150 (Open PDF).
• Skenderis, Kostas, and Balt C. van Rees. “Real-Time Gauge/Gravity Duality: Prescription, Renormalization and Examples.” Journal of High Energy Physics 05 (2009): 085. doi:10.1088/1126-6708/2009/05/085. arXiv:0812.2909 (Open PDF).