Symmetry-Constrained Quantum Fluctuations in Non-Linear CTP Hydrodynamics

1. Introduction

In the expansive domain of dissipative hydrodynamics, a deliberate and systematic exploration has been initiated with the ambitious goal of deriving a comprehensive set of Navier-Stokes equations. This endeavor extends beyond the conventional scope, incorporating not only the familiar bulk viscosity terms but also delving into the intricate nuances of shear viscosity. Drawing inspiration from a specific dissipative hydrodynamic landscape, our primary objective is to pinpoint the “minimal” dissipative extension. This involves building upon the foundations laid by prior studies to unearth a theoretical framework that yields the all-encompassing Navier-Stokes equations—a universally acknowledged description of fluid behavior, particularly at low energies in the natural world.

1.1. Novelty and Differentiation

While this work builds upon the substantial contributions of researchers such as Glorioso, Liu, Nicolis, and Son, it introduces several key innovations:

• Application of volume-preserving symmetry group to fluctuations: Unlike previous approaches that applied the $\mathrm{SDiff}\left({\mathbb{R}}^{1,3}\right)$ symmetry to the full nonlinear field configuration, we propose applying it solely to the fluctuation component when constructing the effective action. This novel approach allows for a more nuanced treatment of dissipative effects while maintaining essential symmetries.
• Extended theoretical framework: Our formulation goes beyond the standard Navier-Stokes equations by incorporating higher-order nonlinear terms in the effective action. This extension provides a richer description of fluid dynamics, particularly in regimes where conventional theories may break down.
• Quantum considerations: We lay the groundwork for exploring quantum aspects of viscous fluid systems, providing a basic quantization model that opens avenues for future investigations into quantum hydrodynamics.

These innovations, while building on existing research, offer a fresh perspective on dissipative hydrodynamics and pave the way for new insights into fluid behavior across various scales.

The trajectory of our exploration extends beyond theoretical extensions, venturing into the complex and captivating realm of quantum fluctuations. One intriguing avenue of investigation involves the exploration of quantum fluctuations within the context of the effective action on a lattice. This intricate exploration raises pertinent questions about the nature of the non-perturbative vacuum state and introduces the captivating prospect of quantized ideal fluid backreaction contributing significantly to the overall viscosity profile.

A crucial aspect that demands attention in the realm of dissipative hydrodynamics is the application of linear response theory. Understanding how systems respond to external perturbations, especially within the quantum regime, unveils promising avenues for unraveling the underlying dynamics governing fluid behavior. Recent proposals concerning hydrodynamic correlation functions, stemming from an effective action, add a layer of depth to the evolving landscape of dissipative hydrodynamics. These explorations not only enrich our theoretical understanding but also hold the potential for practical applications across diverse fields.

The holographic approach emerges as a valuable and versatile tool in extending our understanding of fluid dynamics. Particularly noteworthy is the derivation of ideal fluid action from holography. The incorporation of holography within the context of the Wilsonian renormalization group in gauge/gravity duality offers novel insights. These approaches provide a unique and powerful perspective, leveraging the intricate interplay between quantum and gravitational theories to enrich our comprehension of fluid dynamics.

At the core of our theoretical framework lies the Schwinger-Keldysh formalism, also known as the closed-time-path (CTP) formalism. This approach serves as a robust foundation for exploring the quantum aspects of dissipative hydrodynamics, providing a comprehensive understanding of the underlying dynamics. Furthermore, the application of kinetic theory within the CTP formalism serves to refine our ability to capture the intricate interplay between quantum effects and fluid behavior, offering a nuanced and comprehensive perspective on dissipation.

The versatility of the CTP formalism is not confined to quantum field theory alone; it finds intriguing applications in classical physics. This extension into classical systems underscores the adaptability of the formalism, enabling the exploration of quantum-inspired dynamics across a broader spectrum of physical phenomena. As we extend the reach of the CTP formalism into classical domains, we uncover new facets of dissipative hydrodynamics that may have previously eluded our understanding.

2. The Schwinger-Keldysh Effective Field Theory

Let us consider a microscopic unitary QFT, such as QED or QCD, in which $\psi$ represents all the fundamental fields in the theory. The generating functional for the correlation functions of fields or composite operators, given an arbitrary initial density matrix of the system, can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill exp\left\{i{W}_{CTP}\left[{\widehat{h}}_{\mu \nu }\right]\right\}& =\int \mathcal{D}\widehat{\psi }{\rho }_i\left[\widehat{\psi }(t_i,\mathit{x})\right]\hfill \\ \hfill & \times exp\left\{i{S}_{CTP}\left[\widehat{\psi }\right]+i\int d^{4}x{\widehat{T}}^{\mu \nu }\left[\widehat{\psi }\right]{\widehat{h}}_{\mu \nu }\right\},\hfill \end{array}\end{array}$$

(1)

where we for concreteness consider the $W_{CTP}$ for the stress-energy tensor $T^{\mu \nu }$. The metric perturbation ${\eta }_{\mu \nu }+h_{\mu \nu }$ of the flat space thus acts as its source. In terms of the operator language, this equation enables us to compute time-dependent expectation values.

The microscopic CTP action has the form

$$\begin{array}{c}\hfill S_{CTP}[{\psi }^{+},{\psi }^{-}]=S_s\left[{\psi }^{+}\right]-S_s^{*}\left[{\psi }^{-}\right],\end{array}$$

(2)

where $S_s$ stands for the unitary single-axis action of $\psi$. $S_{CTP}$ is thus invariant under the CTP symmetry.

3. Hydrodynamics

From the point of view of Wilsonian field theory, hydrodynamics can be formulated as an effective theory of the gapless IR modes that remain in the theory after all the massive modes have been integrated out above some mass gap scale. Such effective theories are most naturally expressed in terms of the gradient expansion of the massless fields.

3.1. Rationale for Applying Symmetry Group to Fluctuations

In our approach, we propose applying the volume-preserving symmetry group ${\mathrm{SDiff}}_{\pi }\left({\mathbb{R}}^{1,3}\right)$ solely to the fluctuation component of the fields when constructing the effective action. This decision is motivated by several key considerations:

• Separation of scales: By applying the symmetry only to fluctuations, we can more effectively separate the long-wavelength, low-energy dynamics (described by the background fields) from the short-wavelength, high-energy fluctuations.
• Preservation of nonlinear effects: This approach allows us to retain important nonlinear effects in the background fields while imposing stringent symmetry constraints on the fluctuations, providing a balance between complexity and tractability.
• Enhanced flexibility: Applying the symmetry only to fluctuations gives us more freedom in constructing the effective action, potentially allowing for the inclusion of terms that would be forbidden if the symmetry were applied to the full fields.
• Consistency with observed phenomena: This approach may better align with experimental observations of fluid behavior, where certain symmetries may be more apparent in small fluctuations than in the overall flow.

3.2. Impact on Degrees of Freedom

Compared to previous well-studied investigations, our approach of applying the ${\mathrm{SDiff}}_{\pi }\left({\mathbb{R}}^{1,3}\right)$ symmetry only to the fluctuation fields has several implications for the physical degrees of freedom in the effective theory:

• Preservation of essential dynamics: By maintaining the symmetry for fluctuations, we ensure that the essential dynamics governed by this symmetry are preserved at the level where they are most relevant.
• Additional flexibility in background fields: Since the symmetry constraints are relaxed for the background fields, we potentially allow for a richer set of background configurations, which could be important for describing complex fluid phenomena.
• Controlled reduction of degrees of freedom: While we impose strong constraints on the fluctuation fields, potentially reducing their degrees of freedom

4. The Dissipative Action and the Navier-Stokes Equations

4.1. Example Lagrangian and Extension Beyond Navier-Stokes

To demonstrate how our approach extends beyond the Navier-Stokes equations, we propose the following Lagrangian:

$$\mathcal{L}={\displaystyle \frac{1}{2}}\left({\partial }_{\mu }{\pi }^i\right)\left({\partial }^{\mu }{\pi }_i\right)-{\displaystyle \frac{\lambda }{4}}\left({\partial }_{\mu }{\pi }^i\right)\left({\partial }^{\mu }{\pi }_i\right)\left({\partial }_{\nu }{\pi }^j\right)\left({\partial }^{\nu }{\pi }_j\right)+\zeta {\left({\partial }_{\mu }{\pi }^{\mu }\right)}^2+\eta {({\partial }_{\mu }{\pi }^{\nu }+{\partial }_{\nu }{\pi }^{\mu })}^2$$

(3)

where ${\pi }^i$ represents the fluctuation fields, $\lambda$ is a nonlinear coupling constant, and $\zeta$ and $\eta$ are the bulk and shear viscosities respectively.

This Lagrangian respects the ${\mathrm{SDiff}}_{\pi }\left({\mathbb{R}}^{1,3}\right)$ symmetry for the fluctuation fields and includes nonlinear terms that go beyond the standard Navier-Stokes formulation.

To illustrate the extension beyond Navier-Stokes, we can derive the equation of motion:

$${\partial }_{\mu }{\partial }^{\mu }{\pi }^i+\lambda \left({\partial }_{\nu }{\pi }^j\right)\left({\partial }^{\nu }{\pi }_j\right){\partial }_{\mu }{\partial }^{\mu }{\pi }^i=\zeta {\partial }^i\left({\partial }_{\mu }{\pi }^{\mu }\right)+\eta {\partial }_{\mu }({\partial }^i{\pi }^{\mu }+{\partial }^{\mu }{\pi }^i)$$

(4)

This equation includes nonlinear terms (the $\lambda$ term) not present in the standard Navier-Stokes equations, demonstrating our extension of the theory.

4.2. Analysis of Kubo Relation and Correlators

To complete the dissipative component of our effective field theory, we now analyze the Kubo relation and associated correlators in the context of our extended theory.

The Kubo formula relates the transport coefficients to retarded Green’s functions of the stress-energy tensor. In our formalism, we can express the shear viscosity $\eta$ as:

$$\eta =-\underset{\omega \to 0}{lim}{\displaystyle \frac{1}{\omega }}\mathrm{Im}{G}_R^{xy,xy}(\omega ,\mathbf{k}=0)$$

(5)

where $G_R^{xy,xy}$ is the retarded Green’s function of the $xy$ component of the stress-energy tensor.

In our extended theory, we can calculate this Green’s function using the path integral formulation:

$$G_R^{xy,xy}(\omega ,\mathbf{k})=-i\int d^{4}x{e}^{i(\omega t-\mathbf{k}·\mathbf{x})}\theta \left(t\right)\langle [T^{xy}(t,\mathbf{x}),T^{xy}(0,\mathbf{0})]\rangle$$

(6)

The presence of the nonlinear terms in our Lagrangian will modify this correlator, potentially leading to frequency-dependent viscosity and other novel transport phenomena not captured by the standard Navier-Stokes theory.

5. Basic Quantization Model

To provide a foundation for the quantized description of our theory, we propose the following basic quantization model:

1. First, we promote the classical fields ${\pi }^i\left(x\right)$ to quantum operators ${\widehat{\pi }}^i\left(x\right)$.

2. We impose equal-time commutation relations:

$$[{\widehat{\pi }}^i(t,\mathbf{x}),{\widehat{\pi }}^j(t,\mathbf{y})]=0$$

(7)

$$[{\widehat{\pi }}^i(t,\mathbf{x}),{\partial }_t{\widehat{\pi }}^j(t,\mathbf{y})]=i\hslash {\delta }^{ij}{\delta }^3(\mathbf{x}-\mathbf{y})$$

(8)

3. The Hamiltonian operator can be constructed from the Lagrangian:

$$\widehat{H}=\int d^{3}x\left[{\displaystyle \frac{1}{2}}{\left({\partial }_t{\widehat{\pi }}^i\right)}^2+{\displaystyle \frac{1}{2}}{(\nabla {\widehat{\pi }}^i)}^2+{\displaystyle \frac{\lambda }{4}}\left({\partial }_{\mu }{\widehat{\pi }}^i\right)\left({\partial }^{\mu }{\widehat{\pi }}_i\right)\left({\partial }_{\nu }{\widehat{\pi }}^j\right)\left({\partial }^{\nu }{\widehat{\pi }}_j\right)\right]$$

(9)

This basic quantization model provides a starting point for exploring quantum effects in our extended hydrodynamic theory. Future work will involve analyzing the quantum corrections and their implications for fluid behavior at microscopic scales.

5.1. Discussion

Delving into the intricate landscape of dissipative hydrodynamics, our quest to unravel the mysteries of fluid behavior takes us into the realm of mathematical structures and theoretical frameworks. Particularly interesting in this exploration is the quantization of the behavior of viscous fluids, prompting a profound investigation into the maximal possible subgroup of $\mathrm{SDiff}\left({\mathbb{R}}^{3,1}\right)$ that could encapsulate their complex dynamics, extending beyond the conventional confines of the Navier-Stokes equations. The theoretical underpinnings of dissipative hydrodynamics, as explored in this article, go beyond the classical understanding of fluid behavior. In our endeavor to comprehend the intricacies of dissipative fluids, it becomes imperative to not only recognize the profound impact of viscosity but also to push the boundaries of our theoretical frameworks. The conventional Navier-Stokes equations, while providing a remarkable description of fluid flow, may fall short in capturing the full spectrum of behaviors exhibited by viscous fluids, especially in scenarios involving non-linear extensions. The mathematical machinery of $\mathrm{SDiff}\left({\mathbb{R}}^{3,1}\right)$, representing the group of volume-preserving diffeomorphisms of four-dimensional spacetime, stands out as a key player in this theoretical exploration.

5.2. Conclusions

In this comprehensive exploration of dissipative hydrodynamics, we have developed a systematic approach rooted in an effective Schwinger-Keldysh field theory of long-range massless modes. This framework has enabled us to derive an extended version of the Navier-Stokes equations, offering a more complete energy-momentum balance equation that captures the intricate dynamics of dissipative fluids.

Our work introduces several key innovations to the field:

• Application of the ${\mathrm{SDiff}}_{\pi }\left({\mathbb{R}}^{1,3}\right)$ symmetry specifically to the fluctuation fields, allowing for a more nuanced treatment of dissipative effects while maintaining essential symmetries in the relevant regime.
• Development of an extended Lagrangian that goes beyond the standard Navier-Stokes formulation, incorporating nonlinear terms that capture more complex fluid behaviors.
• Introduction of a basic quantization model, laying the groundwork for exploring quantum aspects of viscous fluid systems.
• Analysis of the Kubo relation and associated correlators within our extended framework, providing insights into novel transport phenomena not captured by conventional theories.

By maintaining the ${\mathrm{SDiff}}_{\pi }\left({\mathbb{R}}^{1,3}\right)$ symmetry for the fluctuation fields, we have enabled a systematic treatment of dissipative hydrodynamics up to the second order in derivatives. This approach contributes to a more comprehensive understanding of fluid behavior at low energy scales, bridging the gap between microscopic and macroscopic descriptions of fluid dynamics.

The extension beyond the Navier-Stokes equations, demonstrated through our proposed Lagrangian and resulting equations of motion, opens up new avenues for describing complex fluid phenomena, particularly in regimes where conventional theories may break down. This enhanced framework holds promise for applications in diverse fields, from astrophysical plasmas to microfluidics.

Looking ahead, our work suggests several promising directions for future research:

• Further exploration of the quantum aspects of dissipative hydrodynamics, building on our basic quantization model to investigate quantum corrections and their implications for fluid behavior at microscopic scales.
• In-depth analysis of the nonlinear terms in our extended theory, potentially uncovering new physical phenomena in strongly non-equilibrium fluid systems.
• Application of our framework to specific physical systems, such as quark-gluon plasmas or turbulent flows, to test its predictive power and refine the theory based on experimental observations.
• Investigation of the connections between our approach and other advanced techniques in theoretical physics, such as holographic methods and renormalization group approaches.

In conclusion, this work represents a significant step forward in our understanding of dissipative hydrodynamics, providing a more complete and nuanced description of fluid behavior across various scales. By bridging quantum field theory techniques with classical fluid dynamics, we have opened up new possibilities for exploring the rich and complex world of fluid behavior, from the quantum scale to macroscopic phenomena.