Imaginary Effective Actions and Vacuum Instability

An imaginary in–out effective action measures vacuum nonpersistence when in and out vacua, mode normalization, and the contour are defined. A complex determinant obtained from an arbitrary spectral cut or a Euclidean negative mode is not automatically a particle-production probability. The physical interpretation must agree with an independent Bogoliubov or tunneling calculation.

Required background. One-Loop Matter Effective Actions in Curved Space supplies the determinant, Particle Creation in Time-Dependent Backgrounds supplies normalized in/out modes, and In–Out versus In–In Expectation Values fixes the observable.

Helpful background. Matter-Induced Nonlocal Form Factors supplies absorptive branch cuts, and Adiabaticity, Stokes Phenomena, and Production Rates supplies turning-point checks.

Vacuum persistence and mode products

Define the normalized vacuum amplitude by

$$\langle0_{\rm out}|0_{\rm in}\rangle =e^{i\Gamma_{\rm in-out}}.$$

Its modulus gives

$$P_0=|\langle0_{\rm out}|0_{\rm in}\rangle|^2 =e^{-2\operatorname{Im}\Gamma_{\rm in-out}}.$$

For independent bosonic mode pairs,

$$a_{\mathbf k}^{\rm out} =\alpha_{\mathbf k}a_{\mathbf k}^{\rm in} +\beta_{\mathbf k}^*a_{-\mathbf k}^{\rm in\dagger}, \qquad |\alpha_{\mathbf k}|^2-|\beta_{\mathbf k}|^2=1,$$

and $n_{\mathbf k}=|\beta_{\mathbf k}|^2$. The squeezed-vacuum overlap is

$$P_{0,\mathbf k}=\frac1{|\alpha_{\mathbf k}|^2} =\frac1{1+n_{\mathbf k}},$$

so, with each independent pair counted once,

$$2\operatorname{Im}\Gamma_{\rm in-out} =\sum_{\mathbf k}\ln(1+n_{\mathbf k}).$$

For fermions the sign and statistics change: $P_{0,\mathbf k}=1-n_{\mathbf k}$ and $2\operatorname{Im}\Gamma=-\sum_{\mathbf k}\ln(1-n_{\mathbf k})$. Volume factors, degeneracies, and continuum density of states must be stated before calling this a rate. Schwinger’s constant-field calculation is the classic determinant-to-pair-production example Schwinger 1951, pp. 664–679.

First application: an exactly solvable time dependence

Consider the scalar mode equation

$$\ddot v_{\mathbf k}+\omega_{\mathbf k}^2(t)v_{\mathbf k}=0,$$

with

$$\omega_{\mathbf k}^2(t) =\frac{(\omega_{\mathbf k}^{\rm in})^2+(\omega_{\mathbf k}^{\rm out})^2}{2} +\frac{(\omega_{\mathbf k}^{\rm out})^2-(\omega_{\mathbf k}^{\rm in})^2}{2} \tanh\frac t\tau.$$

Normalized hypergeometric in/out solutions give

$$n_{\mathbf k}=|\beta_{\mathbf k}|^2 =\frac{ \sinh^2\!\left[ \frac{\pi\tau}{2} (\omega_{\mathbf k}^{\rm out}-\omega_{\mathbf k}^{\rm in}) \right] }{ \sinh(\pi\tau\omega_{\mathbf k}^{\rm in}) \sinh(\pi\tau\omega_{\mathbf k}^{\rm out}) }.$$

The exact connection formula and its sudden and adiabatic limits are given in Das, Galante, and Myers 2015, § 2. Insert this $n_{\mathbf k}$ into the mode product. It yields

$$2\operatorname{Im}\Gamma_{\rm in-out} =\sum_{\mathbf k} \ln\!\left(1+n_{\mathbf k}\right),$$

which is the independent vacuum-persistence check. When $\omega^{\rm out}=\omega^{\rm in}$, both $n_{\mathbf k}$ and the imaginary part vanish. In the adiabatic limit they are exponentially suppressed. These limits test normalization and mode counting.

The continuum mode sum is a further physical check. One must integrate with the spatial volume, density of states, polarization degeneracy, and the independent-pair convention used in the Bogoliubov transformation. Ultraviolet convergence of $\int d^{d-1}k\,\ln(1+n_{\mathbf k})$ follows only when the background is sufficiently smooth or a physical switching prescription is supplied. A divergent persistence exponent can therefore diagnose an idealized quench rather than a finite production rate; local real counterterms do not remove a genuine positive absorptive part.

Spectral-cut adversary

A Euclidean operator with one negative eigenvalue contributes

$$\frac12\ln(-|\lambda_-|) =\frac12\ln|\lambda_-|\pm\frac{i\pi}{2}$$

depending on the cut. If the background has no defined asymptotic in/out vacua, this phase alone does not provide $P_0$. It may diagnose a nonminimum saddle, a false-vacuum direction, or a contour convention. Change the cut: the determinant phase changes, while normalized Bogoliubov data do not. Only the branch selected by the physical in–out contour and agreeing with $-\ln P_0$ licenses a production interpretation.

Likewise, an imaginary part of an in–out action does not appear as an imaginary force in a causal mean equation. The closed-time-path action reorganizes absorptive data into real dissipation and fluctuations after its two branches and initial density matrix are included.

The structure map shows determinant phases and production data meeting only after the contour and normalization checks.

Spectral cuts define determinant phases; independently normalized in/out modes determine whether that phase has the interpretation $2\operatorname{Im}\Gamma=-\ln P_0$. Schematic; not to scale.

Domain and failure conditions

The persistence formula assumes asymptotically defined in/out vacua and independent normalized modes. Without them, one may still have a spectral instability but not this particle number. Backreaction and real-time expectation values require in–in methods. See Domain and failure conditions.

The failure map distinguishes physical absorption from an arbitrary phase, a negative saddle mode, and a zero-mode singularity. None can be identified solely from the word “imaginary.”

Production requires normalized vacua, mode counting, and agreement between $2\operatorname{Im}\Gamma$ and the independently computed persistence exponent. Schematic; not to scale.

Handoffs

Detailed production diagnostics remain with Particle Creation in Time-Dependent Backgrounds. Causal expectation values continue in In–In Effective Actions and Causal Backreaction.

References

• Das, Sumit R., Damián A. Galante, and Robert C. Myers. “Smooth and Fast versus Instantaneous Quenches in Quantum Field Theory.” Journal of High Energy Physics 2015, 073 (2015). DOI.
• Schwinger, Julian. “On Gauge Invariance and Vacuum Polarization.” Physical Review 82 (1951): 664–679. DOI.