In-In Formalism in Quantum Field Theory

• In-in formalism is a real-time quantum field theory framework that computes expectation values using a doubled field approach and two time contours.
• It employs forward and backward time evolution with an iε prescription to ensure convergence in non-equilibrium and time-dependent scenarios.
• The method is crucial for evaluating non-Gaussian effects and loop corrections in cosmological models, refining predictions for power spectra and bispectra.

The in-in formalism (also known as the Schwinger–Keldysh or closed-time path formalism) is a framework for computing expectation values of quantum operators at finite times in quantum field theory, especially suited for scenarios with time-dependent backgrounds, such as cosmology. Unlike the in-out formalism—used for transition amplitudes between asymptotic states—the in-in formalism is tailored for real-time observables, correlation functions, and non-equilibrium phenomena where typically only a single initial quantum state is specified.

1. Operator and Path Integral Structure

The essential construct of the in-in formalism is the calculation of the expectation value of an operator $W(t)$ at time $t$ given an initial vacuum state, typically the Bunch–Davies vacuum in cosmology. In operator language, this is expressed as: $$\langle W(t) \rangle = \langle \text{in} | U^\dagger(t, -\infty) \, W(t) \, U(t, -\infty) | \text{in} \rangle$$ where $U$ is the time-evolution operator governed by the interaction Hamiltonian $H_\text{int}$. This formalism involves time evolution forward and then backward, resulting in two time contours in complex time space. Selection of the initial vacuum is enforced by an $i\epsilon$ prescription: $t \to t(1 + i\epsilon)$, ensuring convergence and correct boundary conditions (0904.4207).

The path integral representation requires a "doubling" of field variables into $\phi^+$ (forward branch) and $\phi^-$ (backward branch) components, with the generating functional: $$Z[J^+, J^-] = \int D\phi\, D\phi^+\, D\phi^- \, e^{i S[\phi^+, J^+] - i S[\phi^-, J^-]} \Psi_0[\phi^+(-\infty)] \Psi_0^*[\phi^-(-\infty)]$$ and imposes closure conditions at the finite "return" time (Kaya, 2013).

2. Perturbative Expansion and Diagrammatic Rules

Perturbative evaluations in the in-in formalism differ critically from in-out methods. In operator formalism, the time evolution is expanded as nested commutators: $$\langle W(t) \rangle = \sum_n i^n \int_{-\infty}^t dt_n \cdots \int_{-\infty}^{t_2} dt_1 \langle [H_\text{int}(t_1), [H_\text{int}(t_2), \cdots [H_\text{int}(t_n), W(t)] \cdots ]] \rangle$$ With the $i\epsilon$ prescription enforced throughout the contour, explicit regularization of spurious divergences is achieved (0904.4207).

Diagrammatic rules adapt the Feynman diagram vocabulary to the in-in context (1005.3287):

• Draw a horizontal line for the observation time, place external points here.
• Vertices are distributed above (anti-time-ordered, $+ig$) and below (time-ordered, $-ig$) this line.
• Propagators are distinguished as Feynman (below), complex-conjugate Feynman (above), and Wightman (crossing the boundary).
• Symmetry factors and integrations appear as in equilibrium field theory.
• The full answer is obtained by taking $2 \Re[\ldots]$ of the sum.

These prescriptions streamline calculation of real-time correlation functions (e.g., power spectra, bispectra) in time-dependent backgrounds.

3. Treatment of Higher-Order Corrections and Non-Gaussianity

The in-in formalism is central to computations of sub-leading and non-Gaussian effects in cosmological perturbation theory. Tree-level correlators are nearly Gaussian, but higher-order diagrams (e.g., loops and multi-vertex trees) generate non-Gaussian contributions to observables like the bispectrum and trispectrum.

For example, a one-loop correction to tensor (gravitational wave) power spectra from scalar field fluctuations involves two interaction vertices and reads: $$I_\text{loop} \sim \int_{-\infty}^{\tau_*} d\tau_2 \int_{-\infty}^{\tau_2} d\tau_1 \langle H_\text{int}(\tau_1) H_\text{int}(\tau_2) \gamma\gamma \rangle$$ Care is required; naive splitting or "specialization" of momenta prior to performing contour integrals can generate spurious divergences. Inclusion of the vacuum prescription avoids such artifacts and yields finite, physically meaningful results. For instance, the scalar loop correction to the gravitational wave spectrum is: $$\langle \gamma_{ij} \gamma^{ij} \rangle = P_{\gamma\gamma} \left[1 - \frac{35}{4} \pi \epsilon P_{\zeta\zeta} \log k \right]$$ where $P_{\gamma\gamma}$ and $P_{\zeta\zeta}$ denote tensor and curvature power spectra and $\epsilon$ is a slow-roll parameter (0904.4207).

4. Comparison to In-Out Formalism and Quantum Cosmology Implications

The in-in and in-out formulations are fundamentally distinct. While the in-out approach calculates transition amplitudes between prepared in/out vacuum states (appropriate for scattering), the in-in scheme targets real-time expectation values relevant in cosmology, where only the initial (in) state is well-defined. The path integral measure in in-in demands doubled fields and enforces boundary matching at the "return" time, modifying both the diagrammatic rules and renormalization structure (Kaya, 2013).

A counterintuitive implication arises for tunneling and symmetry breaking. The stationary phase (saddle-point) configurations in the in-in path integral, called pseudo-instantons, interpolate between vacua even in infinite volume, in contrast to the suppression found in in-out treatment due to the action scaling with volume. This effect means that vacua are "communicated" even in cosmological contexts, possibly undermining the disjoint Fock space picture and impacting the treatment of symmetry breaking and topological defect formation (Kaya, 2013).

5. Schrödinger Picture Implementation and Consistency

Within the Schrödinger picture, the in-in formalism is realized by evolving the quantum state (density matrix) using a time-ordered exponential of the interaction Hamiltonian in the interaction picture: $$\rho(t) = U_0(t, t_0) \, F(t, t_0) \, \rho(t_0) \, F^{-1}(t, t_0) \, U_0^{-1}(t, t_0)$$ with $F(t, t_0) = T \exp[-i \int_{t_0}^t dt' H_I(t')]$. Observables are computed as expectation values of operators with respect to the time-evolved density matrix.

For primordial cosmological observables, such as the three-point function of inflaton fluctuations: $$\langle \varphi_{k_1} \varphi_{k_2} \varphi_{k_3} \rangle = -i \int_{-\infty}^0 d\eta \, a(\eta) \operatorname{Tr} \big([ \varphi_{k_1}^I \varphi_{k_2}^I \varphi_{k_3}^I, H_\text{int}(\eta) ] \rho_0 \big)$$ where this yields the same form and momentum dependence as in Heisenberg picture in-in calculations. Thus, Schrödinger and Heisenberg approaches to in-in formalism yield consistent results for N-point functions at all orders, reaffirming internal consistency and offering a natural setting for studies of entanglement and decoherence in the early universe (Rostami et al., 2016).

6. Practical Advantages and Computational Strategies

Implementing the in-in formalism within the operator or diagrammatic framework confers several technical advantages:

• Vacuum specification is maintained throughout via contour deformation or $i\epsilon$ prescription, ensuring convergence and proper regularization.
• Multi-vertex diagrams, including loops, are rendered finite when all time integrals are performed before specializing momenta or symmetries; premature simplification leads to spurious divergences.
• Many integrals can be factored into products of an integral and its complex conjugate, simplifying computations.
• Diagrammatic rules allow for modular, intuitive generation of terms, improving transparency and computational tractability—especially in the context of time-dependent backgrounds where traditional in-out (equilibrium) Feynman rules do not apply (1005.3287).

These features are critical for evaluating both tree-level and loop corrections to cosmological power spectra, bispectra, and trispectra, and for producing robust predictions of non-Gaussian statistics relevant to cosmic microwave background analyses.

7. Impact on Cosmological Model Building and Observational Constraints

The in-in formalism underpins modern calculations of non-Gaussianity and higher-order statistics in inflationary cosmology. Systematic and reliable evaluation of loop corrections, higher-point functions, and tensor contributions using the in-in approach refines the comparison between theory and cosmological observations, essential for distinguishing inflationary models and constraining their parameter spaces.

Proper treatment of initial conditions, vacuum selection, and multi-vertex diagrams ensures that predicted corrections (such as $$\delta P_{\gamma\gamma} \sim \epsilon P_{\gamma\gamma} P_{\zeta\zeta} \log k$$) are accurate and free from unphysical divergences, directly informing tests of primordial gravitational waves and the structure of non-Gaussian signatures (0904.4207). Additionally, the recognition that vacua may remain intercommunicated due to pseudo-instantons suggests potential revisions in the understanding of symmetry breaking, with implications for the formation of topological defects and the global structure of the inflationary universe (Kaya, 2013).