In–Out versus In–In Expectation Values

In–out and in–in functionals answer different boundary-value questions. An in–out functional fixes an initial ket and a final bra, so its source derivatives are normalized transition matrix elements. An in–in construction fixes one initial state or density operator, evolves both the ket and the bra, and closes them with a trace, so its derivatives give finite-time expectation values. The distinction is state and boundary data, not a choice of diagram style. Generic in–out matrix elements can be complex; for unitary evolution, a Hermitian observable has a real expectation value in a positive initial state. This page establishes that semantic split and one regulated free-scalar check; the full branch-component grammar comes next.

Required background. The Generating Functional supplies normalized Lorentzian source differentiation with the site’s $+\int J\mathcal O$ convention. Vacua, States, and Representations supplies the distinction among vectors, density operators, and representation-dependent states. Trace formulas below are exact at a finite regulator, or for states normal in the selected representation; an arbitrary algebraic QFT state need not be a trace-class density operator on vacuum Fock space.

Two boundary-value questions

Work first in a regulated closed system on a finite time interval $[t_i,t_f]$. Couple a real source by

$$H_J(t) = H_0(t)-\int\mathrm d^{d-1}\mathbf x\, J(t,\mathbf x)\mathcal O(t,\mathbf x),$$

so the action and path-integral weight contain $+\int J\mathcal O$ and $+i\int J\mathcal O$, respectively. The corresponding evolution operator is

$$U_J(t_f,t_i) = \mathrm T\exp\!\left[ -i\int_{t_i}^{t_f}\!\mathrm dt\,H_J(t) \right].$$

For specified boundary vectors, the in–out amplitude and its normalized source functional are

$$\mathcal A_{fi}[J] = \langle f,t_f|U_J(t_f,t_i)|i,t_i\rangle, \qquad Z_{fi}^{\mathrm{in\text{-}out}}[J] = \frac{\mathcal A_{fi}[J]}{\mathcal A_{fi}[0]}.$$

The normalization requires $\mathcal A_{fi}[0]\neq0$. With $\mathcal O_H$ evolved by $H_0$, source differentiation gives

$$\begin{aligned} (-i)^n \left. \frac{\delta^n Z_{fi}^{\mathrm{in\text{-}out}}} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0} ={}& \frac{ \langle f,t_f|U_0(t_f,t_i) \,\mathrm T\!\left\{ \mathcal O_H(x_1)\cdots\mathcal O_H(x_n) \right\}|i,t_i\rangle }{ \langle f,t_f|U_0(t_f,t_i)|i,t_i\rangle }. \end{aligned}$$

For $W_{fi}^{\mathrm{in\text{-}out}}=-i\log Z_{fi}^{\mathrm{in\text{-}out}}$, one derivative is the corresponding normalized transition matrix element. It is conditioned on both boundaries and need not be real even when $\mathcal O$ is Hermitian. The vacuum specialization supplies the familiar Feynman-ordered in–out correlators; an $S$-matrix still requires the additional asymptotic-state and reduction steps developed in Scattering. Calzetta and Hu display this source sign and the contrast between normalized in–out matrix elements and expectation values in Calzetta and Hu 2008, § 6.2.2, pp. 179–180.

By contrast, an initial-value question starts from one positive, Hermitian, normalized density operator $\rho_i$ and asks for

$$\begin{aligned} \langle\mathcal O(t)\rangle_{\rho_i} &= \operatorname{Tr}\!\left[ \rho_i U_0(t,t_i)^\dagger \mathcal O\,U_0(t,t_i) \right]\\ &= \operatorname{Tr}\!\left[ \mathcal O\,U_0(t,t_i)\rho_i U_0(t,t_i)^\dagger \right]. \end{aligned}$$

A pure initial state is the special case $\rho_i=|\psi_i\rangle\langle\psi_i|$. No final state is postselected: the trace sums the final basis after the ket and bra evolutions have recombined.

Question	In–out	In–in
boundary data	initial ket and final bra	one initial state or density operator
underlying unnormalized object	boundary transition amplitude	trace of the evolved density operator
source derivatives	time-ordered transition matrix elements	expectation values and contour-ordered correlators
natural use	vacuum persistence, amplitudes, Feynman boundary data	finite-time observables and initial-value response
not supplied automatically	probabilities or causal response	thermality, kinetics, or an approximation that preserves unitarity

Closing the evolution on the initial state

The minimal doubled-source functional is

$$Z_{\mathrm{CTP}}[J_+,J_-] = \operatorname{Tr}\!\left[ U_{J_+}(t_f,t_i)\rho_i U_{J_-}(t_f,t_i)^\dagger \right],$$

where $t_f$ lies later than every insertion. The forward and backward sources remain independent until after differentiation. At a finite field regulator, the same statement has the schematic path-integral form

$$\begin{aligned} Z_{\mathrm{CTP}}[J_+,J_-] = \int_{\phi_+(t_f)=\phi_-(t_f)} \!\mathcal D\phi_+\mathcal D\phi_-\, \rho_i[\phi_i^+,\phi_i^-] \exp\!\Bigg\{& iS[\phi_+]-iS[\phi_-]\\ &+i\!\int\!\left(J_+\phi_+-J_-\phi_-\right) \Bigg\}, \end{aligned}$$

with $\rho_i[\phi_i^+,\phi_i^-]=\langle\phi_i^+|\rho_i|\phi_i^-\rangle$. The relative minus signs come from the reversed evolution in $U_{J_-}^\dagger$; they do not signal a negative-norm physical field. Calzetta and Hu derive the branch signs, initial kernel, final sewing, and arbitrary-state trace functional in Calzetta and Hu 2008, §§ 6.3.1–6.3.2, pp. 181–186. Altland and Simons give a compact operator derivation and emphasize that the construction is not intrinsically thermal in Altland and Simons 2023, § 12.2.1, pp. 706–707.

For $\operatorname{Tr}\rho_i=1$ and exact unitary evolution,

$$Z_{\mathrm{CTP}}[J,J] = \operatorname{Tr}\!\left[ U_J\rho_i U_J^\dagger \right] =1, \qquad W_{\mathrm{CTP}}[J,J]=0,$$

where $W_{\mathrm{CTP}}=-i\log Z_{\mathrm{CTP}}$. Nevertheless, differentiating the independent branches first gives

$$\begin{aligned} \langle\mathcal O(x)\rangle_{\rho_i} &= \left. \frac{1}{i} \frac{\delta Z_{\mathrm{CTP}}} {\delta J_+(x)} \right|_{J_+=J_-=0}\\ &= -\left. \frac{1}{i} \frac{\delta Z_{\mathrm{CTP}}} {\delta J_-(x)} \right|_{J_+=J_-=0}, \end{aligned}$$

or $\delta W/\delta J_+=+\langle\mathcal O\rangle$ and $\delta W/\delta J_-=-\langle\mathcal O\rangle$. Setting $J_+=J_-$ before differentiating collapses the functional to $1$ and discards the observable. Closed-Time-Path Grammar develops branch ordering and the four two-point components; the identities here only fix their physical origin and signs. A recent overview begins from the same initial-state rationale and time-folded trace in Haehl and Rangamani 2024, § 1, pp. 3–5 (Open manuscript PDF).

Reality, normalization, and response are separate checks

For Hermitian $\mathcal O$, Hermitian positive $\rho_i$, and unitary $U_0$,

$$\langle\mathcal O(t)\rangle_{\rho_i}^{*} = \operatorname{Tr}\!\left[ \mathcal O\,U_0\rho_i U_0^\dagger \right] = \langle\mathcal O(t)\rangle_{\rho_i}.$$

This reality statement is distinct from diagonal normalization $Z[J,J]=1$. Both are also distinct from causal response. If a source $J$ couples to a Hermitian operator $\mathcal B$, differentiating an in–in expectation gives

$$\left. \frac{\delta\langle\mathcal O(x)\rangle_J} {\delta J(y)} \right|_{J=0} = i\,\theta(x^0-y^0) \langle[\mathcal O(x),\mathcal B(y)]\rangle_{\rho_i}.$$

The plus sign follows from $H_J=H_0-\int J\mathcal B$ and agrees with Retarded, Advanced, and Spectral Correlators. Exact closed evolution therefore supplies a normalized and causal first variation. A truncation or resummation must be checked separately: branch doubling by itself does not guarantee that an approximation preserves reality, unitarity, or causal support. Jordan’s expectation-value construction verifies real and causal effective equations through two-loop order; it is not an unrestricted all-orders statement Jordan 1986, abstract, p. 444.

A free scalar with finite-time initial data

Put the free real scalar in a finite box with a UV mode cutoff. Each real normal mode is a harmonic oscillator,

$$H_{\mathbf k} = \frac12\left( p_{\mathbf k}^{2} +E_{\mathbf k}^{2}q_{\mathbf k}^{2} \right), \qquad E_{\mathbf k}=\sqrt{\mathbf k^2+m^2}.$$

One explicit normalized Gaussian initial density operator is

$$\rho_i = D(\alpha) \left(1-e^{-\eta}\right)e^{-\eta a^\dagger a} D(\alpha)^\dagger, \qquad D(\alpha)=e^{\alpha a^\dagger-\alpha^*a}, \qquad \eta>0.$$

It has

$$\bar q_i = \sqrt{\frac{2}{E_{\mathbf k}}}\, \operatorname{Re}\alpha, \qquad \bar p_i = \sqrt{2E_{\mathbf k}}\, \operatorname{Im}\alpha, \qquad \operatorname{Var}_{\rho_i}(q) = \frac{2n+1}{2E_{\mathbf k}},$$

where $n=(e^\eta-1)^{-1}$. The limit $\eta\to\infty$ is a pure coherent state. Repeating the construction mode by mode with arbitrary $\eta_{\mathbf k}$ does not impose one common temperature.

Now couple a real compactly supported source by $H_j=H_{\mathbf k}-j(t)q_{\mathbf k}$. The Heisenberg equation and the in–in mean obey

$$\left(\partial_t^2+E_{\mathbf k}^{2}\right) \langle q_{\mathbf k}(t)\rangle_{\rho_i,j} =j(t),$$

and the initial-value solution is

$$\begin{aligned} \langle q_{\mathbf k}(t)\rangle_{\rho_i,j} ={}& \bar q_i\cos\!\left(E_{\mathbf k}\Delta t\right) +\frac{\bar p_i}{E_{\mathbf k}} \sin\!\left(E_{\mathbf k}\Delta t\right)\\ &+\int_{t_i}^{t}\!\mathrm dt'\, \frac{\sin\!\left[E_{\mathbf k}(t-t')\right]} {E_{\mathbf k}}\,j(t'), \qquad \Delta t=t-t_i. \end{aligned}$$

This result is real and depends only on initial moments and past source values. It also recovers the retarded kernel $\theta(t-t')\sin[E_{\mathbf k}(t-t')]/E_{\mathbf k}$. A normalized vacuum in–out amplitude does not contain the freely chosen $\bar q_i$, $\bar p_i$, or $n$ unless corresponding boundary data are supplied. The calculation is exact at the regulator; existence of an infinite-volume density operator or a renormalized coincident variance is a separate question.

Restricted computational equivalence does not erase the distinction

Special cases can make results overlap. If the selected final bra is the same stationary vacuum as the initial ket, a normalized in–out time-ordered matrix element is also a vacuum time-ordered expectation value. This does not remove the final-boundary conditioning of the general functional or turn every finite-time initial-value problem into a vacuum amplitude.

There are also restricted computational reformulations. Donath and Pajer show that correlators of nondissipative closed systems can be computed with an in–out prescription under their stated hypotheses, including the absence of infrared divergences; their de Sitter construction uses an auxiliary contracting patch to prepare the future bra, and dissipative systems lie outside the result Donath and Pajer 2024, introduction, pp. 2–7, and § 2, p. 9 (Open PDF). This is a change of calculational representation, not an identification of the initial density operator with a postselected final state. The comparison and its stated limits were checked against literature available through 9 August 2026.

What changes outside the finite closed system

• Infinite volume. A physically valid state may be a positive normalized functional in a representation where no global trace-class $\rho_i$ exists. The finite-regulator trace remains a construction and consistency check, not a theorem that every continuum state has that form.
• Open or reduced dynamics. After tracing an environment, the system evolution need not be unitary. Trace preservation, influence-functional normalization, noise, and dissipation require the developed open-system formalism.
• Initial correlations. Non-Gaussian and correlated states appear as additional initial-boundary data. A Feynman $i0$ prescription does not silently prepare them.
• Gauge theories. The density operator and traced states must obey the physical-state constraints, or the gauge-fixed formulation must carry its auxiliary-sector qualifications.
• Thermal states. A closed time path is not a thermal assumption. KMS periodicity and fluctuation–dissipation relations require extra equilibrium input.
• Approximations. A truncated effective action can violate $Z[J,J]=1$, Hermiticity, or causal response unless its identities are preserved and checked.

Check your understanding

1. Differentiate before aligning sources

Why does substituting $J_+=J_-$ first fail to generate $\langle\mathcal O\rangle$?

Answer

Unitarity makes $Z[J,J]=1$, so the derivative along the diagonal vanishes. The observable is a transverse branch variation: differentiate $J_+$ or $J_-$ independently and only then set the sources equal. Their one-point derivatives have opposite signs.

2. Diagnose a complex one-point result

A Hermitian operator has a complex normalized one-point insertion. Which construction is immediately plausible?

Answer

A generic in–out transition matrix element can be complex because its initial ket and final bra differ. A positive normalized density operator evolved unitarily gives a real one-point expectation for a Hermitian observable; a complex answer there signals a failed hypothesis or approximation.

3. Check the free initial-value limit

Set $j=0$, $\bar p_i=0$, and $t=t_i$ in the mode solution.

Answer

The result is $\langle q_{\mathbf k}(t_i)\rangle=\bar q_i$, as required. Differentiating once gives $\partial_t\langle q_{\mathbf k}\rangle_{t_i}=\bar p_i$, so both pieces of initial data round-trip.

4. Test the claimed in–out reformulation

Which two questions must be answered before using the 2024 computational equivalence?

Answer

At minimum, check that the closed system is nondissipative and that the required correlators are free of the excluded infrared divergences. In de Sitter space one must also account for the auxiliary contracting patch that prepares the bra. Passing those checks licenses the stated calculation, not a universal semantic equivalence.

Where the two constructions lead

• Develop the contour components: Closed-Time-Path Grammar derives branch ordering, normalization identities, and the four basic two-point functions.
• Apply arbitrary initial states: Closed-Time-Path Generating Functionals in Practice develops response, initial-state kernels, KMS conditions, kinetics, and open-system applications.
• Apply causal gravitational expectation values: In-In Effective Actions and Causal Mean Backreaction develops the curved-spacetime and cosmological construction.
• Recheck linear response: Retarded, Advanced, and Spectral Correlators fixes the site’s commutator and source signs.

References

• Altland, Alexander, and Ben Simons. Condensed Matter Field Theory. 3rd ed. Cambridge University Press, 2023. DOI.

• Calzetta, Esteban A., and Bei-Lok B. Hu. Nonequilibrium Quantum Field Theory. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2008. DOI. 2023 open-access reissue.

• Donath, Yaniv, and Enrico Pajer. “The In-Out Formalism for In-In Correlators.” Journal of High Energy Physics 2024, no. 7 (2024): 064. DOI. Open PDF.

• Haehl, Felix M., and Mukund Rangamani. “Records from the S-Matrix Marathon: Schwinger–Keldysh Formalism.” arXiv:2410.10602 (2024); published as “Schwinger–Keldysh Formalism,” in Records from the S-Matrix Marathon, Lecture Notes in Physics 1041, pp. 89–129. Springer, 2025. Book DOI. Open manuscript PDF.

• Jordan, R. D. “Effective Field Equations for Expectation Values.” Physical Review D 33 (1986): 444–454. DOI.