Initial Density Matrices and Contour Boundary Conditions

An initial density matrix is a boundary interaction at $t_0$. Its quadratic part fixes the Gaussian mean and covariance; higher boundary vertices encode connected non-Gaussian cumulants. These data survive through retarded propagation and memory kernels, so a bulk action alone never specifies a nonequilibrium problem.

Required background. Use the closed-time-path generating functional for branch orientation and source signs.

Helpful background. Initial correlations in Kadanoff–Baym evolution shows how boundary cumulants enter later two-time equations.

Boundary action and density-matrix constraints

In the field basis write

$$\langle\varphi_+|\rho_0|\varphi_-\rangle =\mathcal N e^{iS_0[\varphi_+,\varphi_-]}.$$

$S_0$ is localized at $t_0$ and can be expanded in $r/a$ fields. Hermiticity requires

$$S_0[\varphi_+,\varphi_-]^*=-S_0[\varphi_-,\varphi_+],$$

while trace normalization fixes $\int\mathcal D\varphi\,\langle\varphi|\rho_0|\varphi\rangle=1$. Positivity is stronger: for every wave functional $\Psi$, $\langle\Psi|\rho_0|\Psi\rangle\ge0$. The Schwinger–Keldysh identity $S_0[\varphi,\varphi]=0$ follows from normalization structure but does not by itself prove positivity.

For one oscillator, a general centered Gaussian state is specified by

$$C_{qq}=\langle q^2\rangle, \qquad C_{pp}=\langle p^2\rangle, \qquad C_{qp}=\frac12\langle\{q,p\}\rangle,$$

with

$$C_{qq}C_{pp}-C_{qp}^2\ge\frac14.$$

The inequality is the positivity test. These three numbers give $F(t_0,t_0)$, $\partial_tF|_{t_0}$, and $\partial_t\partial_{t'}F|_{t_0}$; the commutator fixes the corresponding $\rho$ data independently.

Gaussian kernels and non-Gaussian vertices

A convenient quadratic boundary action has the schematic form

$$S_0^{(2)}=\int d^d\mathbf x\,d^d\mathbf y \left[ \varphi_a(\mathbf x)K_R(\mathbf x,\mathbf y)\varphi_r(\mathbf y) +\frac{i}{2}\varphi_a(\mathbf x)K_K(\mathbf x,\mathbf y)\varphi_a(\mathbf y) \right],$$

plus linear terms for the mean. $K_R$ controls phase-space correlations and $K_K$ the statistical width; their relation to $C_{qq},C_{pp},C_{qp}$ depends on the chosen field basis and normalization. There is no pure $r r$ term, consistently with $S_0[\varphi_r,\varphi_a=0]=0$.

Connected initial $n$-point functions require boundary vertices; their role in restoring the interacting equilibrium limit is shown in Garny and Müller 2009, §§ II–IV:

$$S_0^{(n)}=\frac{1}{n!}\int\alpha_n^{a_1\cdots a_n} \varphi_{a_1}\cdots\varphi_{a_n},\qquad n\ge3.$$

Every time argument is $t_0$. A $\mathbb Z_2$-symmetric state has no odd vertices but can have a nonzero connected four-point kernel. Treating such a state as Gaussian changes early-time collision terms even if its two-point function is matched exactly.

The map below organizes the dependencies of a late-time response. Four solid arrows carry the declared state, boundary correlations, sources, and memory kernel into the observable; the dashed arrow marks the separate test required before claiming loss of memory.

The solid arrows identify data on which the response calculation can depend, while the dashed outgoing arrow separates the calculated response from a later memory-loss claim. The initial density matrix fixes normalization, positivity, and boundary kernels, while non-Gaussian cumulants require additional contour vertices. Factorization or Gaussianity removes specific inputs only when imposed and checked; neither implies late-time loss of memory. The diagram is schematic and not to scale.

The sections Boundary action and density-matrix constraints, Gaussian kernels and non-Gaussian vertices, and Preparation protocols and failure tests give the text and equation equivalent of the upper dependencies and their checks.

Checked squeezed-state example

For a pure squeezed oscillator state with squeeze parameter $r$ and angle chosen so $C_{qp}=0$,

$$C_{qq}=\frac{e^{2r}}{2\omega}, \qquad C_{pp}=\frac{\omega e^{-2r}}{2}, \qquad C_{qq}C_{pp}=\frac14.$$

It saturates the uncertainty bound for every $r$. Calling $C_{qq}$ a thermal occupation would be wrong: field and momentum variances correspond to different effective energies. Free evolution rotates the squeezed ellipse and preserves its determinant, so unequal-time $F$ retains the phase information.

By contrast, a thermal oscillator has $C_{qq}=(n_B+1/2)/\omega$, $C_{pp}=\omega(n_B+1/2)$, and determinant $(n_B+1/2)^2>1/4$. The two states can share one equal-time variance but not the full covariance.

Preparation protocols and failure tests

An imaginary-time segment prepares a thermal or Euclidean-correlated state. An adiabatic source prepares a different state and requires a scale separation. A sudden quench is a physical protocol, not a numerical convenience; it injects energy and high-frequency correlations. A finite switch-on must be included in the source history.

Test the state by:

• verifying trace, Hermiticity, and covariance positivity before evolution;
• checking its ultraviolet tail against the renormalized vacuum behavior;
• moving $t_0$ while keeping the actual preparation protocol fixed;
• adding the first omitted connected cumulant; and
• evolving a purported interacting equilibrium state without a quench—any drift is an inconsistency or numerical error.

Exercise

Two Gaussian states have the same $C_{qq}$ but different $C_{pp}$. Can they generate the same future $F(t,t')$ for a free oscillator?

Solution

No. The solution contains $C_{qq}\cos\omega(t-t_0)\cos\omega(t'-t_0)$, $C_{pp}\sin\omega(t-t_0)\sin\omega(t'-t_0)/\omega^2$, and terms proportional to $C_{qp}$. Equal field variance fixes only the first coefficient. The momentum variance and cross covariance are independent initial data subject to the uncertainty inequality.

Continue

Use unitarity and largest-time identities to constrain allowed boundary vertices, then follow their dynamical contribution on initial correlations in Kadanoff–Baym theory.

References

• Chou, K.-C., Su, Z.-B., Hao, B.-L., and Yu, L. (1985). “Equilibrium and Nonequilibrium Formalisms Made Unified.” Physics Reports 118, 1–131. DOI.
• Garny, M., and Müller, M. M. (2009). “Kadanoff–Baym Equations with Non-Gaussian Initial Conditions: The Equilibrium Limit.” Physical Review D 80, 085011. arXiv:0904.3600; DOI.