Functional Integral Expression for Density Matrices

• The paper introduces a formal path integral approach to compute density matrices in quantum many-body systems, providing a rigorous framework for representing system trajectories.
• Methodologies such as variational perturbation theory, cumulant expansions, and constrained minimization in nuclear DFT are detailed to capture key spin, parity, and non-Markovian effects.
• The work applies functional integral methods to open quantum systems and phase-space formulations, offering practical computational algorithms and insights into non-equilibrium dynamics.

A functional integral expression for density matrices provides a formal or computational route to represent, calculate, and approximate the density matrix of quantum many-body systems (or subsystems thereof) using functional—typically path—integral techniques. These approaches play a central role in quantum statistical mechanics, equilibrium and non-equilibrium field theory, nuclear density functional theory (DFT), quantum chemistry, and the simulation of open quantum systems.

1. General Structure of Functional Integral Expressions for Density Matrices

The fundamental idea is to express the matrix elements of the density matrix $\rho(x_a, x_b; \beta)$ (where $x_a$ and $x_b$ are coordinates or quantum numbers, and $\beta = 1/(k_B T)$) as a path integral or more generally, a functional integral over trajectories or field configurations: $$(x_a, x_b) = \int_{x(0) = x_a}^{x(\hbar \beta) = x_b} \mathcal{D}x(\tau) \; \exp \left( -\frac{1}{\hbar} \mathcal{A}[x] \right)$$ where $\mathcal{A}[x]$ is the Euclidean action and the measure $\mathcal{D}x$ includes all possible system trajectories with fixed endpoints (Bachmann et al., 2011).

For density matrices in Fock or more abstract Hilbert space bases (as in quantum field theory), the basic structure generalizes to integrals and sums over particle numbers, momenta, or field configurations, often written with suitable operator insertions: $$\rho_{G;H}(K^G|L^H; t) = \sum_{I,J} \int \mathcal{D}\Phi^+ \mathcal{D}\Phi^- \; \mathrm{ker} \; e^{i S^+ - i S^-}$$ where "ker" represents kernel contractions/heavy operator insertions/projectors, and $S^\pm$ are the full forward/backward actions, possibly including environmental couplings and initial conditions (Käding et al., 11 Mar 2025, Käding et al., 2022).

In the context of open quantum systems, the functional integral is used to integrate out (trace over) environmental degrees of freedom, leading to an "influence functional" (after Feynman and Vernon), which modifies the system's effective dynamics (Chen, 2013, Käding et al., 2022).

2. Spin and Radial Density Matrices in Nuclear DFT

Nuclear density functional theory extends functional integral expressions to include spin and angular degrees of freedom. The local spin density matrix (SDM) is defined by: $$\rho_{(\sigma \sigma')}(r) = \operatorname{Tr} \left[ a^\dagger_{r \sigma} a_{r \sigma'} \mathcal{D} \right]$$ where $\mathcal{D}$ is the many-body density operator (Barrett et al., 2010). The SDM can be decomposed into scalar $(S=0)$ and vector $(S=1)$ parts: $$A_{r S M_S} = \sum_{\sigma \sigma'} (-1)^{1/2-\sigma'} \langle \frac{1}{2} \sigma, \frac{1}{2} {-\sigma'} | S M_S \rangle a^\dagger_{r \sigma} a_{r \sigma'}$$ and recoupled into radial and angular components. Only in the presence of parity-violating admixtures (arising from external fields with odd parity) does the vector "hedgehog" density become nonzero.

The functional integral formalism then involves a constrained minimization or search over all many-body density operators $\mathcal{D}$ yielding prescribed radial profiles for both scalar and vector densities: $$F[\bar{\rho}] = \inf_{\mathcal{D} \to \bar{\rho}} \operatorname{Tr}\left[ (H + H_1)\mathcal{D} \right]$$ where $H_1$ is a parity-violating external field. The basic integration variables in the functional integral are these (radial) density profiles, and the Lagrange multipliers or external fields controlling them (Barrett et al., 2010).

3. Variational Path Integral and Smearing Formulas

Systematic computation of density matrices for quantum systems frequently employs variational perturbation theory around a trial action, combined with cumulant expansions: $$(x_a, x_b) = (\text{harmonic prefactor}) \cdot \exp \left( -\frac{1}{\hbar} \langle \mathcal{A}_{\text{int}}[x] \rangle_c + \frac{1}{2\hbar^2} \langle \mathcal{A}_{\text{int}}^2[x] \rangle_c - \cdots \right)$$ where the expectation values are evaluated with respect to a harmonic reference action. The smearing formula allows computation of expectation values of arbitrary nonpolynomial interactions: $$\langle \mathcal{A}_{\text{int}}^n[x] \rangle = \prod_{k=1}^n \int_0^{\hbar\beta} d\tau_k \int_{-\infty}^{+\infty} dz_k V_{\text{int}}(z_k + x_{\min}) \frac{1}{\sqrt{(2\pi)^n \det G}} \exp \left\{ -\frac{1}{2} \sum_{kl} [z_k - x_\text{cl}(\tau_k)] G^{-1}_{kl} [z_l - x_\text{cl}(\tau_l)] \right\}$$ where $G_{kl}(\tau_k, \tau_l)$ is the harmonic Green function. This approach enables complex potentials (e.g., Coulomb) to be handled through effective Gaussian averaging, improving convergence and enabling effective computation across all coupling strengths and temperatures (Bachmann et al., 2011).

4. Functional Integral Approaches for Open Quantum Systems

To analyze open quantum systems, functional integrals provide a rigorous means for constructing reduced equilibrium density matrices (REDM). The influence functional, originally defined in real time by Feynman and Vernon, can under suitable conditions (commuting operators, even kernels) be interpreted as an average over realizations of Gaussian processes: $$\rho_\beta(q'', q') = \frac{1}{Z} \langle \int [Dq] \exp \left\{ -\frac{1}{\hbar} \left[S_\text{sys}^{(E)}[q] + \int_0^{\hbar\beta} d\tau (\mu V[q(\tau)]^2 + V[q(\tau)] \xi(\tau)) \right] \right\} \rangle_\xi$$ Here, $\xi(\tau)$ is a real Gaussian process with covariance determined by the bath spectral function. This stochastic matrix product scheme enables efficient Monte Carlo sampling of REDMs for arbitrary spectral densities and system-bath couplings, properly capturing non-Markovian noise (Chen, 2013).

Stochastic reformulations differ for local or nonlocal baths, requiring either independent noises for local phonon couplings or correlated noises with off-diagonal structure for nonlocal phonon environments. The normalized REDM can be used to compute physical observables and compared with the Boltzmann equilibrium distribution, making this technique broadly applicable (Chen, 2013).

5. Non-Markovianity, Path Integrals, and Dynamical Maps in QFT

Recent developments in quantum field theory leverage path integrals and the Schwinger-Keldysh (SK) formalism combined with thermo field dynamics (TFD) to provide detailed functional integral expressions for density matrix elements of both closed and open systems (Käding et al., 11 Mar 2025, Käding et al., 2022). In TFD-based treatments, one writes the dynamical evolution as a Schrödinger-like equation for the density matrix promoted to a vector in a doubled Hilbert space. The general solution involves a time-ordered exponential: $$\rho^+(t) |1\rangle = \mathcal{T} \exp\left[ i S_I(t) \right] \rho^+(0) |1\rangle$$ Projection into Fock (momentum) bases and suitable path integral representations involving both system and environment fields (with "plus" and "minus" labels for forward/backward contours) yield: $$\rho_{G;H}(K^G|L^H; t) = \sum_{I,J} \int \mathcal{D}\Phi^+ \mathcal{D}\Phi^- \; (\text{kernel}) \; e^{i S^+ - i S^-}$$ After tracing out the environment, the dynamical map for the open system is found to be generally non-divisible (absence of a cocycle property), indicating non-Markovian evolution. The influence action extracted from SK/TFD path integrals quantifies all system-environment correlations and memory effects. In this framework, quantum master equations emerge directly from such functional integral solutions and are notable for not featuring explicit time integrals over the density matrix—contrary to standard Lindblad-like equations (Käding et al., 11 Mar 2025).

6. Special Functional Representations: Moyal, Wigner, and Surfaces

Functional integral representations also underlie phase-space formulations of quantum mechanics, notably the Moyal and Wigner approaches. The density matrix can be mapped to a phase-space function via the Moyal characteristic function

$$M(T, \theta) = \langle e^{i(T \hat{p} + \theta \hat{x})} \rangle$$

or equivalently, the Wigner function, which is simply the density matrix in $(X, P)$-representation. The fundamental evolution of the density matrix in this setting is given by the Moyal equation (quantum Liouville equation in phase space), whose solution can be expressed as a Marinov-type functional integral: $$K_M(\varphi', T; \varphi'', T_0) = \int \mathcal{D}\varphi^a \mathcal{D}\xi^a \exp\left\{ -2i S_M[\varphi, \xi] \right\}$$ where $S_M$ is the "surface" action, parametrizing physical evolution as a sum over random ruled surfaces in phase space rather than single trajectories, thereby encoding full quantum noncommutativity and correlations (Pagani et al., 2022, Hiley, 2014).

7. Theoretical and Practical Implications

Functional integral representations for density matrices provide not only a mathematically rigorous approach to quantum statistical mechanics but also a flexible computational framework. Key implications include:

• Systematic path-integral variational and perturbative approximation schemes are available for nontrivial potentials and coupling strengths (Bachmann et al., 2011).
• Spin, parity, and other symmetries can be encoded through choice of integration variables (density profiles) and constraints in the functional space (Barrett et al., 2010).
• Non-Markovianity and intricate system-bath entanglement can be treated nonperturbatively and numerically using influence functionals, stochastic matrix multiplication, or direct path integrals (SK/TFD) (Chen, 2013, Käding et al., 2022, Käding et al., 11 Mar 2025).
• Connections to Wigner-Moyal and geometric (surface-based) formulations yield both physical insight and suggest new analytic and semiclassical techniques (Pagani et al., 2022).

Functional integral methods thus underpin both formal structural results (e.g., identities for functional derivatives in DFT (Joubert, 2011), mapping to reduced density matrices in geminal theories (Moisset et al., 2022)) and practical algorithms for the evaluation of statistical and dynamical observables in quantum many-body systems.