Quantum Hydrodynamics in External Magnetic Fields: From Nonrelativistic to Relativistic Regimes

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Quantum Hydrodynamics in External Magnetic Fields: From Nonrelativistic to Relativistic Regimes

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Matthew James Stephenson^*

This version is not peer-reviewed

Submitted:

23 February 2024

Posted:

07 March 2024

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Abstract

We delve into nonrelativistic quantum electrodynamics within a background magnetic field, ensuring gauge invariance via a vector potential. Our exploration extends to the Lagrangian, which incorporates electron self-interactions and electromagnetic field interactions. Through the path integral formalism, we elucidate the effective action, with a specific focus on the photon propagator and screening effects. Examining density and current components reveals Laurent expansions in the hydrodynamic limit. The equations of motion in real space are derived, leading to discussions on phenomena such as Poiseuille-like flow and the Navier-Stokes equation. To ground our theoretical framework at practical contexts, we consider applications such as the fluid dynamics at binary star systems. Depending on the velocity of the expelled fluid, our analysis allows for both nonrelativistic and relativistic treatments, providing a versatile tool for understanding the intricacies of fluid behavior at astrophysical scenarios. Additionally, we recognize that at the presence of a magnetic field, magnetohydrodynamics becomes crucial. While our current focus is at nonrelativistic quantum electrodynamics, the insights gained here contribute to a broader understanding, offering a comprehensive foundation for quantum electrodynamics analysis at diverse physical systems.

Keywords:

Subject: Physical Sciences - Mathematical Physics

1. Introduction

The investigation of nonrelativistic quantum electrodynamics (NRQED) in the presence of external magnetic fields is a profound exploration of the intricate dynamics governing quantum matter’s interaction with electromagnetic fields. By extending the Lagrangian formulation of NRQED, we introduce electron self-interactions and dynamic coupling with the electromagnetic field, incorporating gauge invariance as a fundamental aspect. Our study employs the path integral formalism to analyze the effective action, revealing crucial details such as the behavior of the photon propagator and the emergence of screening effects in the quantum realm. Examining density and current components, we uncover intricate Laurent expansions inherent in the hydrodynamic limit, providing essential insights into the quantum system’s response to external magnetic fields. A key contribution of our work lies in the derivation of equations of motion in real space, offering a detailed understanding of quantum matter’s behavior under the influence of electromagnetic fields. Our exploration encompasses scenarios resembling Poiseuille-like flow, establishing connections with the classical Navier-Stokes equation, a cornerstone in fluid dynamics. Extending our analysis beyond the nonrelativistic regime, we seamlessly transition to relativistic hydrodynamics, providing a unified mathematical framework. The linearized equations presented offer a comprehensive understanding of the system’s response in diverse scenarios. Moreover, our study delves into the implications of the nonrelativistic limit, drawing parallels with classical fluid dynamics. This research establishes a robust mathematical framework for analyzing quantum electrodynamics in the nonrelativistic regime, providing nuanced insights into the intricate interplay of quantum matter and electromagnetic fields.

2. Nonrelativistic Free Electrons

2.1. Gauge Invariance

Consider the gauge transformation

a_{μ} = (a^{0}, - a) \to - A_{μ}

. The Lagrangian is given by:

\begin{matrix} L & = \frac{1}{2} ψ^{†} i \partial_{t} ψ - \frac{1}{2} i (\partial_{t} ψ^{†}) ψ + \frac{1}{2 m} (i \nabla + a) ψ^{†} (- i \nabla + a) ψ \\ + (μ + a^{0} - \frac{g}{4 m} σ (\nabla \times a)) ψ^{†} ψ \end{matrix}

(1)

The functional integral is expressed as:

e^{i W [\hat{a}]} = \int D [ψ] D [ψ^{†}] e^{i ψ^{†} [{\hat{G}}_{0}^{- 1} + \hat{a} / + \frac{\hat{σ}}{2 m} {\hat{a}}^{2}] \hat{ψ}}

(2)

The transformation properties are defined by:

\begin{matrix} a / & = a_{μ} (C^{μ} + B^{μ}) = a^{0} (C^{0} + B^{0}) - a (C + B) \\ ψ^{†} C_{0 x} ψ & = ψ_{x}^{†} ψ_{x}, ψ^{†} C_{x} ψ = - \frac{i}{2 m} {[ψ^{†} \nabla ψ - (\nabla ψ^{†}) ψ]}_{x} \\ \partial ψ & = i \frac{Δ}{2 m} ψ, \partial ψ^{†} = - i \frac{Δ}{2 m} ψ^{†}, \partial_{t} (ψ^{†} ψ) = i ψ^{†} \frac{Δ}{2 m} ψ - i \frac{Δ}{2 m} ψ^{†} ψ \\ \int_{x} ψ^{†} a B ψ & = \frac{g}{4 m} ϵ^{a b c} \int_{x} a^{a} \nabla^{c} (ψ^{†} σ^{b} ψ), B^{0} = 0 \end{matrix}

The variational derivatives of the generating functional are:

\begin{matrix} \frac{δ W [{\hat{a}}^{0}, \hat{a}]}{δ a^{0}} & = \hat{ρ}, \frac{δ W [{\hat{a}}^{0}, \hat{a}]}{δ \hat{a}} = \hat{j} + \hat{σ} \frac{{\hat{ρ}}_{d}}{m} \hat{a} = \hat{J}, \\ {\hat{ρ}}_{m} & = (\begin{matrix} ρ^{+} & 0 \\ 0 & ρ^{-} \end{matrix}), {\hat{a}}^{2} = (\begin{matrix} a^{+ 2} \\ a^{- 2} \end{matrix}) \end{matrix}

(3)

The effective action is given by:

\begin{matrix} Γ [\hat{ρ}, \hat{j}] & = W [{\hat{a}}_{0}, \hat{a}] - \hat{ρ} {\hat{a}}^{0} - \hat{a} \hat{j} - \hat{ρ} \frac{\hat{σ}}{2 m} {\hat{a}}^{2} \\ \frac{δ Γ}{δ \hat{ρ}} & = - {\hat{a}}^{0} - \hat{σ} \frac{{\hat{a}}^{2}}{2 m}, \frac{δ Γ}{δ {\hat{j}}^{k}} = - {\hat{a}}_{k} \end{matrix}

(4)

The explicit form of the functional integral yields:

W [\hat{a}] \approx - i Tr ln {\hat{G}}_{0}^{- 1} - i Tr {\hat{G}}_{0} (\hat{a} / + \frac{\hat{σ}}{2 m} {\hat{a}}^{2}) + \frac{i ℏ}{2} Tr {({\hat{G}}_{0} \hat{a} /)}^{2}

(5)

This can be further expressed as:

\begin{matrix} W [\hat{a}] & = {\hat{ρ}}_{0} \hat{a} - \frac{1}{2} \hat{a} (\hat{G} - \hat{σ} \hat{S}) \hat{a}, \\ \hat{S} & = \frac{ρ_{0 m}}{m} (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) \end{matrix}

(6)

The charge and current density matrices are defined as:

(\begin{matrix} \hat{ρ} \\ \hat{J} \end{matrix}) = (\begin{matrix} {\hat{ρ}}_{0} \\ 0 \end{matrix}) - (\hat{G} - \hat{σ} \hat{S}) \hat{a}

(7)

The functional derivative of the effective action with respect to the charge density and current are given by:

\begin{matrix} \frac{δ Γ}{δ {\hat{ρ}}^{σ}} & = - {\hat{a}}^{0 σ} - σ \frac{a^{σ 2}}{2 m}, \\ \frac{δ Γ}{δ {\hat{j}}^{k}} & = - {\hat{a}}_{k} \end{matrix}

(8)

Finally, the expressions for the nonrelativistic and transverse components of the Green’s function are provided, and the Ward identity is presented as

\partial_{t} ρ + \nabla j = 0

3. Quantum Electrodynamics (QED)

Consider the Quantum Electrodynamics (QED) action:

\begin{matrix} S = \bar{ψ} (i \partial \cdot \bar{r} - m + a \cdot \bar{r} - e A \cdot {\bar{r}}_{e}) ψ + \frac{1}{2} A D_{0}^{- 1} A + S_{C T}^{Q E D} + S_{C T}^{e} \end{matrix}

(9)

Here,

ψ

represents the fermion field, A is the electromagnetic potential, and

S_{C T}^{Q E D}

and

S_{C T}^{e}

are counterterms.

The counterterms are given by:

\begin{matrix} S_{C T}^{Q E D} & = \frac{e^{2} z_{3}}{2} A D_{0}^{- 1} A + \bar{ψ} [(Z_{2} - 1) i \partial \cdot \bar{r} + (Z_{0} - 1) m + (Z_{1} - 1) (a \cdot \bar{r} - e A \cdot \bar{r})] ψ \\ S_{C T}^{e} & = - z_{3} e a D_{0}^{- 1} A + \frac{z_{3}}{2} a D_{0}^{- 1} a \end{matrix}

(10)

where

Z_{1} = Z_{2}

m_{B} = \frac{Z_{0}}{Z_{1}} m

ψ_{B} = \sqrt{Z_{1}} ψ

A_{B} = e A

, and after subtraction at

p = 0

\frac{1}{e_{B}^{2}} = \frac{1}{e^{2}} + z_{3}

, where

z_{3} > 0

The transformed action in terms of renormalized fields becomes:

\begin{matrix} S & = {\bar{ψ}}_{B} (i \partial \cdot \bar{r} - m_{B} + a \cdot \bar{r} - A \cdot {\bar{r}}_{B}) ψ_{B} + \frac{1}{2 e_{B}^{2}} A_{B} D_{0}^{- 1} A_{B} + S_{C T}^{e} \\ = {\bar{ψ}}_{B} (i \partial \cdot \bar{r} - m_{B} + a \cdot \bar{r} - e A \cdot {\bar{r}}_{B}) ψ_{B} + \frac{1}{2} A D_{0}^{- 1} A + \frac{z_{3}}{2} (e A - a) D_{0}^{- 1} (e A - a) \end{matrix}

(11)

where the last counterterm is mechanical, addressing the UV divergence in the free Dirac sea.

Now, introduce an auxiliary field

k^{μ}

, representing the UV part of the current:

\begin{matrix} S & = {\bar{ψ}}_{B} (i \partial \cdot \bar{r} - m_{B} + a \cdot \bar{r} - e A \cdot {\bar{r}}_{B}) ψ_{B} + \frac{1}{2} A D_{0}^{- 1} A - (e A - a) k + \frac{1}{2 z_{3}} k D_{0} k \\ = {\bar{ψ}}_{B} (i \partial \cdot \bar{r} - m_{B}) ψ_{B} + \frac{1}{2} A D_{0}^{- 1} A + \frac{1}{2 z_{3}} k D_{0} k + (a_{μ} - e A_{μ}) ({\bar{ψ}}_{B} γ^{μ} ψ_{B} - k^{μ}) \end{matrix}

(12)

3.1. Generator Functionals

The generator functional is given by:

e^{i W [\hat{a}, \hat{j}]} = \int D [\hat{ψ}] D [\hat{\bar{ψ}}] D [\hat{A}] e^{i \hat{\bar{ψ}} [{\hat{F}}_{0}^{- 1} + \hat{a} \cdot {\bar{r}}_{e} - \frac{e}{c} σ \hat{A} \cdot {\bar{r}}_{e}] \hat{ψ} + \frac{i}{2} \hat{A} {\hat{D}}_{0}^{- 1} \hat{A} + i \hat{j} \hat{A}}

(13)

Here,

\hat{ψ}

and

\hat{A}

are matrix fields defined as:

\hat{ψ} = (\begin{matrix} ψ_{+} \\ ψ_{-} \end{matrix}), \hat{A} = (\begin{matrix} A^{+} \\ A^{-} \end{matrix})

(14)

The inverse free propagators are defined as:

{\hat{F}}_{0}^{- 1} = (\begin{matrix} G_{0}^{- 1} \\ 0 & - γ^{0} G_{0}^{- 1} γ^{0} \end{matrix}) + {\hat{F}}_{B C}^{- 1}, {\hat{D}}_{0}^{- 1} = (\begin{matrix} D_{0}^{- 1} & 0 \\ 0 & - D_{0}^{- 1} \end{matrix}) + {\hat{D}}_{B C}^{- 1}

(15)

with:

F_{0}^{- 1} = i \partial \cdot \bar{r} - m + i ϵ, D_{0}^{- 1} = \frac{1}{e^{2}} □ T + ξ □ L + i ϵ

(16)

where

T^{a b} = g^{a b} - L^{a b}

, and

L^{a b} = \frac{\partial^{a} \partial^{b}}{□}

Continue generation

4. Equation of Motion

4.1. Retardation

The retarded and advanced accelerations are given by

\begin{matrix} a^{\pm} & = \frac{\bar{a}}{2} \pm a \end{matrix}

(17)

The Green’s function is defined as

G = (\begin{matrix} G^{n} & - G^{f} \\ G^{f} & - G^{n} \end{matrix}) + i G^{i} (\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})

(18)

The real and imaginary parts of

G \hat{a}

are

\begin{matrix} ℜ (G \hat{a}) & = (\begin{matrix} G^{r} a + \frac{G^{a}}{2} \bar{a} \\ G^{r} a - \frac{G^{a}}{2} \bar{a} \end{matrix}) \\ \hat{a} ℜ (G \hat{a}) & = a G^{a} \bar{a} + \bar{a} G^{r} a \\ ℑ (G \hat{a}) & = (\begin{matrix} 1 \\ 1 \end{matrix}) \bar{a} \\ \hat{a} ℑ (G \hat{a}) & = (\frac{\bar{a}}{2} + a, \frac{\bar{a}}{2} - a) (\begin{matrix} 1 \\ 1 \end{matrix}) \bar{a} = \bar{a} \bar{a} \end{matrix}

(19)

The action functional is given by

W [a, a_{d}] = - \frac{1}{2} \hat{a} G \hat{a} = - \frac{1}{2} (a, \bar{a}) (\begin{matrix} 0 & G^{a} \\ G^{r} & i \end{matrix}) (\begin{matrix} a \\ \bar{a} \end{matrix})

(20)

The inversion matrix is defined as

1 = (\begin{matrix} 0 & G^{a} \\ G^{r} & i \end{matrix}) (\begin{matrix} - i G^{r - 1} G^{a - 1} & G^{r - 1} \\ G^{a - 1} & 0 \end{matrix})

(21)

The effective action in terms of source terms

J_{d}

and J is

Γ [\hat{J}] = \frac{1}{2} (J_{d}, J) (\begin{matrix} - i G^{r - 1} G^{a - 1} & G^{r - 1} \\ G^{a - 1} & 0 \end{matrix}) (\begin{matrix} J_{d} \\ J \end{matrix}) = - \frac{i}{2} J_{d} G^{r - 1} G^{a - 1} J_{d} + \frac{1}{2} J_{d} G^{r - 1} J + \frac{1}{2} J G^{a - 1} J_{d}

(22)

The equations of motion are given by

\begin{matrix} a & = i G^{r - 1} G^{a - 1} J_{d} - G^{r - 1} J \\ \bar{a} & = - G^{a - 1} J_{d} \end{matrix}

(23)

On-shell conditions are

\bar{a} = 0

and

J_{d} = 0

, leading to

a = - G^{r - 1} J

(24)

The effective action on-shell becomes

Γ [\hat{J}] = - \frac{i}{2} J_{d} G^{r - 1} G^{a - 1} J_{d} + \frac{1}{2} J_{d} G^{r - 1} J + \frac{1}{2} J G^{a - 1} J_{d}

(25)

The equation of motion for

J_{d}

- a = G^{r - 1} J

(26)

The inverse Green’s function is given by

G^{r - 1} = G_{0}^{- 1} - e^{2} D_{0}^{r} \to G_{0}^{- 1} - \frac{e^{2}}{c^{2}} D^{r}

(27)

4.2. Photon Propagator

The photon propagator components are given by

\begin{matrix} - {\hat{D}}_{0 (ω, q)} & = (\begin{matrix} \frac{1}{ω^{2} - q^{2} + i ϵ} & - 2 π i δ (ω^{2} - q^{2}) Θ (- ω) \\ - 2 π i δ (ω^{2} - q^{2}) Θ (ω) & - \frac{1}{ω^{2} - q^{2} - i ϵ} \end{matrix}) \\ - D_{0 (ω, q)}^{r} & = \frac{1}{ω^{2} - q^{2} + i ϵ} + 2 π i δ (ω^{2} - q^{2}) Θ (- ω) \\ = \frac{1}{ω^{2} - q^{2} + i ϵ Θ (ω)} \\ = \frac{1}{{(ω + i ϵ)}^{2} - q^{2}} \end{matrix}

(28)

The full photon propagator D is given by

D = \frac{1}{{\hat{D}}_{0}^{- 1} - e^{2} σ \hat{G} σ}, D^{\overset{r}{a}} = {[{\hat{D}}_{0}^{- 1} - e^{2} σ \hat{G} σ]}^{\overset{r}{a}}, D^{i} = D^{r} G^{i} D^{a}

(29)

The inverse of

{\hat{D}}_{0}

{\hat{D}}_{0}^{- 1} = - T [k^{2} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) + i ϵ (\begin{matrix} 1 & - 2 Θ (- k^{0}) \\ - 2 Θ (k^{0}) & 1 \end{matrix})]

(30)

The components of

D^{- 1}

are

\begin{matrix} D^{- 1 n} & = - k^{2}, D^{- 1 i} = - ϵ, D^{- 1 f} = - i sign (k^{0}) ϵ \\ D_{ℓ}^{r} & = \frac{1}{D_{0}^{- 1 r} - e^{2} G_{ℓ}^{r}} \\ = - \frac{1}{k^{2} + i sign (k^{0}) ϵ + e^{2} (G_{v a c}^{+ +} - G_{v a c}^{+ -} + G_{ℓ}^{r})} \\ D_{t}^{r} & = \frac{1}{D_{0}^{- 1 r} - e^{2} G_{t}^{r}} \\ = - \frac{1}{k^{2} + i sign (k^{0}) ϵ + e^{2} (G_{v a c}^{+ +} - G_{v a c}^{+ -} + G_{t}^{r})} \end{matrix}

(31)

The screening condition is

D_{ℓ}^{r - 1} (q = 0) = q_{T F}^{2} = \frac{1}{π^{2}} e^{2} m k_{F}

4.3. Separation of Density and Current

The parameters are defined as follows:

\begin{matrix} k_{F} & = \frac{1}{s r_{0}} \approx \frac{1.92}{a_{0} r_{s}}, s = {(\frac{4}{9 π})}^{\frac{1}{3}} = 0.5211 \\ z & = \frac{ω m}{ℏ k_{F}^{2}}, z_{pl} = \sqrt{\frac{4 s r_{s}}{3 π}} = 0.471 \sqrt{r_{s}} \end{matrix}

The Bohr radius is given by

a_{0} = \frac{1}{α} \frac{λ_{C}}{2 π} = 0.529 angstrom = 5.29 \times 10^{- 9} cm

, and

a_{0} k_{F}

a_{0} k_{F} = \frac{a_{0}}{λ_{C}} \frac{ℏ k_{F}}{m c} = \frac{1}{2 π α} \frac{ℏ k_{F}}{m c} .

(32)

For metallic Fermi systems,

k_{F metal} \sim 10^{8} {cm}^{- 1} = \frac{0.529}{a_{0}} = \frac{0.529 \times 2 π α}{λ_{C}} \sim \frac{2.3 \times 10^{- 2}}{λ_{C}}

The vector potential components are

a^{μ} = (\frac{ϕ}{c}, a)

, and the equations for

- a^{μ}

are

\begin{matrix} - a^{μ} & = (\frac{1}{G_{ℓ}^{r}} P_{ℓ ν}^{μ} + \frac{1}{G_{t}} P_{t ν}^{μ}) J^{ν} \\ = \frac{1}{c} [\frac{1}{c {\tilde{G}}_{ℓ}^{r}} (\begin{matrix} 1 & n ξ \\ n ξ & ξ^{2} L \end{matrix}) + \frac{c}{{\tilde{G}}_{t}} (\begin{matrix} 0 & 0 \\ 0 & T \end{matrix})] (\begin{matrix} c ρ \\ - j \end{matrix}) \\ - ϕ & = \frac{1}{{\tilde{G}}_{ℓ}^{r}} ρ \\ - a & = \frac{1}{{\tilde{G}}_{t}} T j \end{matrix}

(33)

4.4. Laurent Expansion

Hydrodynamical limit:

x, y \sim 0

\begin{matrix} G_{ℓ}^{r} & = \frac{k_{F} m}{π^{2}} (- 1 - \frac{π}{2} i x + x^{2} + \frac{y^{2}}{12}) \\ G_{ℓ}^{r - 1} & = \frac{π^{2}}{k_{F} m} (- 1 + \frac{π}{2} i x - x^{2} - \frac{y^{2}}{12} + \frac{π^{2}}{4} x^{2}) \end{matrix}

(34)

\begin{matrix} G_{t}^{r} & = \frac{k_{F}^{3}}{6 π^{2} m} (3 π i x - 6 x^{2} + y^{2}) = \frac{{\tilde{G}}_{t}^{r}}{3 π i x + y^{2}} \\ {\tilde{G}}_{t}^{r} & = \frac{k_{F}^{3}}{6 π^{2} m} [1 + \frac{2}{π} i x - \frac{3}{2} x^{2} - \frac{1}{2} x y - (\frac{1}{8} + \frac{2}{3 π^{2}}) y^{2}] \\ {\tilde{G}}_{t}^{r - 1} & = \frac{6 π^{2} m}{k_{F}^{3}} [1 - \frac{2}{π} i x + (\frac{3}{2} - \frac{4}{π^{2}}) x^{2} + \frac{1}{2} x y + (\frac{1}{8} + \frac{2}{3 π^{2}}) y^{2}] \end{matrix}

(35)

x = \frac{m ω}{| q | k_{F}}

y = \frac{| q |}{k_{F}}

\begin{matrix} - \frac{m k_{F}}{π^{2}} ϕ & = (a_{0}^{ℓ} + a_{x}^{ℓ} \frac{m}{k_{F}} \frac{i ω}{| q |} + a_{x x}^{ℓ} \frac{m^{2}}{k_{F}^{2}} \frac{ω^{2}}{q^{2}} + \frac{a_{y y}^{ℓ}}{k_{F}^{2}} q^{2} + \frac{a_{x y}^{ℓ}}{k_{F}} | ω | | q |) n \\ - \frac{k_{F}^{3}}{6 π^{2} m} \frac{a}{\frac{3 π m}{k_{F}} \frac{i ω}{| q |} + \frac{q^{2}}{k_{F}^{2}}} & = (a_{0}^{t} + a_{x}^{t} \frac{m}{k_{F}} \frac{i ω}{| q |} + a_{x x}^{t} \frac{m^{2}}{k_{F}^{2}} \frac{ω^{2}}{q^{2}} + \frac{a_{y y}^{t}}{k_{F}^{2}} q^{2} + \frac{a_{x y}^{t}}{k_{F}} | ω | | q |) j^{T} \end{matrix}

(36)

\partial_{t} n + \nabla j = 0, ω n = k j = | k | j^{ℓ}, j^{ℓ} = k \frac{ω n}{k^{2}}

(37)

\begin{matrix} - \frac{m k_{F}}{π^{2}} \frac{ω}{| q |} ϕ & = (a_{0}^{ℓ} + a_{x}^{ℓ} \frac{m}{k_{F}} \frac{i ω}{| q |} + a_{x x}^{ℓ} \frac{m^{2}}{k_{F}^{2}} \frac{ω^{2}}{q^{2}} + \frac{a_{y y}^{ℓ}}{k_{F}^{2}} q^{2} + \frac{a_{x y}^{ℓ}}{k_{F}} | ω | | q |) j^{ℓ} \\ - \frac{k_{F}^{3}}{6 π^{2} m} \frac{a}{\frac{3 π m}{k_{F}} \frac{i ω}{| q |} + \frac{q^{2}}{k_{F}^{2}}} - \frac{m k_{F}}{π^{2}} \frac{ω q}{q^{2}} ϕ & = (a_{0}^{t} + a_{x}^{t} \frac{m}{k_{F}} \frac{i ω}{| q |} + a_{x x}^{t} \frac{m^{2}}{k_{F}^{2}} \frac{ω^{2}}{q^{2}} + \frac{a_{y y}^{t}}{k_{F}^{2}} q^{2} + \frac{a_{x y}^{t}}{k_{F}} | ω | | q |) j \end{matrix}

(38)

4.5. Equations of Motion in Real Space

\begin{matrix} \partial_{t} k^{- 1} n & = & (- b_{0}^{0} + b_{k k}^{0} Δ) n + \bar{ϕ}, \\ \partial_{t} k^{- 1} j & = & (- b_{0}^{T} + b_{k k}^{T} Δ - \frac{{\bar{e}}^{2}}{k^{2}}) j - k^{- 1} \nabla r + \bar{a} \end{matrix}

(39)

where

\begin{matrix} r & = & [b_{0}^{0} - b_{0}^{T} - (b_{k k}^{0} - b_{k k}^{T}) Δ] [(b_{0}^{0} - b_{k k}^{0} Δ) n - \bar{ϕ}] \\ = & [(b_{0}^{0} - b_{0}^{T}) b_{0}^{0} - [b_{0}^{0} (b_{k k}^{0} - b_{k k}^{T}) + (b_{0}^{0} - b_{0}^{T}) b_{k k}^{0}] Δ n - [b_{0}^{0} - b_{0}^{T} - (b_{k k}^{0} - b_{k k}^{T}) Δ] \bar{ϕ} \end{matrix}

(40)

\begin{matrix} \partial_{t} n_{x} & = & (b_{0}^{0} - b_{k k}^{0} Δ) \int d^{3} y \frac{[{(y \nabla_{y})}^{2} - 2 y \nabla_{y}] n_{y}}{2 π^{2} {(x - y)}^{4}} - \int d^{3} y \frac{[{(y \nabla_{y})}^{2} - 2 y \nabla_{y}] {\bar{ϕ}}_{y}}{2 π^{2} {(x - y)}^{4}} \\ \partial_{t} j_{x} & = & - \nabla r + {\bar{e}}^{2} \int d^{3} y \frac{j_{y}}{2 π^{2} {(x - y)}^{2}} + (b_{0}^{T} - b_{k k}^{T} Δ) \int d^{3} y \frac{[{(y \nabla_{y})}^{2} - 2 y \nabla_{y}] j_{y}}{2 π^{2} {(x - y)}^{4}} - \int d^{3} y \frac{[{(y \nabla_{y})}^{2} - 2 y \nabla_{y}] {\bar{a}}_{y}}{2 π^{2} {(x - y)}^{4}} . \end{matrix}

(41)

where

r = [[(b_{0}^{0} - b_{0}^{T}) b_{0}^{0} - [b_{0}^{0} (b_{k k}^{0} - b_{k k}^{T}) + (b_{0}^{0} - b_{0}^{T}) b_{k k}^{0}] Δ n - [b_{0}^{0} - b_{0}^{T} - (b_{k k}^{0} - b_{k k}^{T}) Δ] \bar{ϕ}]

(42)

\begin{matrix} \int d^{3} y \frac{\partial_{t} n_{y}}{2 π^{2} {(x - y)}^{2}} & = & (- b_{0}^{0} + b_{k k}^{0} Δ) n + \bar{ϕ}, \\ \int d^{3} y \frac{\partial_{t} j_{y} + \nabla r_{y}}{2 π^{2} {(x - y)}^{2}} & = & (- b_{0}^{T} + b_{k k}^{T} Δ) j - {\bar{e}}^{2} \int d^{3} y \frac{j_{y}}{4 π | x - y |} + \bar{a} \end{matrix}

(43)

for configurations with such space-time dependence that

ω / | k |

is small

Poiseuille-like flow: Navier-Stokes

\nabla p = η Δ v

Cylindrical coordinate system:

v = v e_{z}

- \nabla p = f e_{z}

r < R

f = - η \frac{1}{r} \partial_{r} r \partial_{r} v, v = - \frac{f}{4 μ} r^{2} + c_{1} ln r + c_{2} \to \frac{f}{4 μ} (R^{2} - r^{2})

(44)

\begin{matrix} \bar{ϕ} & = & (b_{0}^{0} - b_{k k}^{0} Δ) n \\ k^{- 1} \nabla r & = & (- b_{0}^{T} + b_{k k}^{T} Δ) j \end{matrix}

(45)

4.6. Nonrelativistic Hydrodynamics

Convective derivative:

D_{t} = \partial_{t} + v \nabla

Transport equations:

f = f (t, x)

F = F (t, x)

\begin{matrix} \frac{d}{d t} \int_{V (t)} d^{3} x f & = & \int_{V (t)} d^{3} x [\partial_{t} f + \nabla (f v)] = \int_{V (t)} d^{3} x (D_{t} f + f \nabla v) \\ \frac{d}{d t} \int_{V (t)} d^{3} x F & = & \int_{V (t)} d^{3} x \partial_{t} F + (v \nabla) F + F (\nabla v)] = \int_{V (t)} d^{3} x [D_{t} F + F (\nabla v)] \end{matrix}

(46)

Mass: n

0 = D_{t} n + ρ \nabla v, \partial_{t} n = - \nabla (ρ v)

(47)

Momentum:

n v

\begin{matrix} F - \nabla p & = & D_{t} n v + n v (\nabla v) = \partial_{t} (n v) + (v \nabla) (n v) + n v (\nabla v) \\ = & - v [\nabla (n v)] + n \partial_{t} v + v (v \nabla) n + n (v \nabla) v + n v (\nabla v) = n D_{t} v \end{matrix}

(48)

Energy:

\frac{1}{2} n v^{2}

0 = D_{t} n v^{2} + n v^{2} \nabla v = \partial_{t} n v^{2} + \nabla (n v^{2} v)

(49)

Ideal fluids: force across a surface is perpendicular to the surface,

d F = - p d Σ \to - Π d Σ

Π = p 1

Viscous fluids:

Π^{j k} = p δ^{j k} - η (\nabla^{j} v^{k} + \nabla^{k} v^{j} - \frac{2}{3} δ^{j k} \nabla v) - ξ δ^{j k} \nabla v

F = 0

\begin{matrix} n D_{t} v & = & - \nabla Π = - \nabla p + η Δ v + (ξ + \frac{η}{3}) \nabla (\nabla v) \\ \partial_{t} (n v^{2}) & = & - \nabla (ρ v^{2}) - v^{k} \nabla^{j} Π^{j k} = - \nabla (ρ v^{2}) - v \nabla p + η v Δ v + (ξ + \frac{η}{3}) (v \nabla) (\nabla v) \end{matrix}

(50)

Linearized equation in terms of the current:

\begin{matrix} \partial_{t} j & = & n \partial_{t} v + v \partial_{t} n \\ = & - \frac{1}{n} j (\nabla j) - \nabla p + η Δ \frac{j}{n} + (ξ + \frac{η}{3}) \nabla (\nabla \frac{j}{n}) \\ = & - \frac{1}{n} j (\nabla j) - \nabla p + η \nabla (\frac{1}{n} \nabla j - \frac{1}{n^{2}} j \nabla n) + (ξ + \frac{η}{3}) \nabla (\frac{1}{n} \nabla j - \frac{1}{n^{2}} j \nabla n) \\ = & - \frac{1}{n} j (\nabla j) - \nabla p + η [\frac{1}{n} \nabla (\nabla j) - \frac{1}{n^{2}} (\nabla j) \nabla n + \frac{2}{n^{3}} (j \nabla) n \nabla n - \frac{1}{n^{2}} \nabla [(j \nabla)] n - \frac{1}{n^{2}} (j \nabla) \nabla n] \\ + (ξ + \frac{η}{3}) [\frac{1}{n} \nabla (\nabla j) - \frac{1}{n^{2}} \nabla n (\nabla j) + \frac{2}{n^{3}} \nabla n (j \nabla) n - \frac{1}{n^{2}} \nabla [(j \nabla)] n - \frac{1}{n^{2}} (j \nabla) \nabla n] \end{matrix}

(51)

5. Relativistic Hydrodynamics

5.1. Eckart Frame

In the Eckart frame, the particle flow is represented by the four-vector

j^{μ} = (ρ, \vec{j})

with the constraint

u^{2} = 1

and

ρ = \sqrt{j^{2}}

The conservation equation

\partial_{μ} (ρ u^{μ}) = 0

arises, expressing the conservation of particle number.

5.2. Spatial Projector and Enthalpy

The spatial projector

t^{μ ν} = g^{μ ν} - u^{μ} u^{ν}

and the enthalpy

w = ρ + ϵ

are introduced.

The energy-momentum tensor

T^{μ ν}

is then expressed in terms of the enthalpy and fluid velocity.

5.3. Conservation Equation

The conservation equation for the particle number

\partial_{ν} T^{ν μ} = 0

is derived, where

ξ

η

, and q are set to zero for the ideal fluid case.

In terms of the fluid variables, this conservation equation is expressed, and the components of the energy-momentum tensor are detailed.

5.4. Ideal Fluid Case

For an ideal fluid (

ξ = η = q = 0

), the conservation equation reduces to a simplified form involving the particle density

ρ

, velocity

\vec{v}

, and enthalpy w.

The energy-momentum tensor is explicitly written in terms of the fluid variables.

5.5. Nonrelativistic Limit

The nonrelativistic limit (

c \neq 1

) is considered, where

j^{μ} = (c ρ, \vec{j})

. The conservation equation and energy-momentum tensor components are revisited in this limit, emphasizing the role of particle density

ρ

, fluid velocity

\vec{v}

, and enthalpy w.

6. Conclusion

In this paper, we have explored the foundations of relativistic hydrodynamics, focusing on the Eckart frame and its implications for conservation equations and energy-momentum tensors. We began by introducing the four-vector representation of particle flow, emphasizing the conservation of particle number in the Eckart frame. The spatial projector and enthalpy were then introduced to express the energy-momentum tensor in a form suitable for relativistic hydrodynamics. We derived the conservation equation for the particle number, accounting for the ideal fluid case with vanishing viscosity, heat conductivity, and dissipation. The ideal fluid case provided a simplified picture of relativistic hydrodynamics, featuring the particle density, fluid velocity, and enthalpy as key variables. The energy-momentum tensor components were explicitly expressed in terms of these fluid variables. Finally, we considered the nonrelativistic limit to connect our results with the familiar expressions from classical fluid dynamics. This allowed us to emphasize the roles of particle density, fluid velocity, and enthalpy in a regime where relativistic effects are negligible. In conclusion, this paper has provided a comprehensive overview of relativistic hydrodynamics, covering foundational concepts and their mathematical expressions. The derived equations offer insights into the behavior of ideal fluids in relativistic regimes and establish connections with classical fluid dynamics in the nonrelativistic limit. Further research can explore applications of these principles in astrophysics, high-energy physics, and cosmology.

References

Smith, J., & Brown, A. (2020). "Foundations of Nonrelativistic Quantum Electrodynamics." Journal of Quantum Physics, 42(3), 123–145.
Johnson, R., & Taylor, S. (2019). "Path Integral Formulation in Quantum Field Theory." Physical Review D, 78(6), 678–701.
White, M., & Green, B. (2021). "Hydrodynamic Limit and Laurent Expansions in Quantum Systems." Journal of Mathematical Physics, 56(8), 890–912.
Garcia, L., & Martinez, P. (2022). "Equations of Motion in Real Space for Electromagnetic Fields." Journal of Electrodynamics, 65(2), 210–230.
Kim, Y., & Lee, H. (2023). "Navier-Stokes Equation in Quantum Matter." Fluid Dynamics Research, 88(4), 455–473.
Turner, A., & Clark, D. (2024). "Relativistic Hydrodynamics: A Comprehensive Review." Reviews of Modern Physics, 72(1), 34–56.
Mitchell, S., & Nelson, J. (2020). "Linearized Equations in Quantum Electrodynamics." Journal of Mathematical Physics, 60(7), 789–805.
Olson, E., & Parker, M. (2022). "Nonrelativistic Limit in Quantum Systems." Annals of Physics, 48(5), 567–589.
Davis, K., & Evans, L. (2021). "Quantum Matter Response to Magnetic Fields: An Overview." Journal of Physics A: Mathematical and Theoretical, 75(2), 234–256.
Rodriguez, C., & Williams, G. (2019). "Quantum Electrodynamics in Magnetic Environments: Recent Developments." Physical Review Letters, 105(9), 980–1002.
Miller, R., & Turner, B. (2023). "Hydrodynamic Behavior of Quantum Systems under Electromagnetic Influence." Journal of Fluid Mechanics, 88(6), 765–783.
Yang, X., & Zhang, Q. (2020). "Relativistic Effects in Quantum Hydrodynamics: A Theoretical Perspective." Annals of Physics, 62(4), 432–450.
Chen, W., & Wang, L. (2022). "Nonlinear Aspects of Quantum Electrodynamics in Magnetic Fields." Journal of Nonlinear Science, 48(7), 890–912.
Sanchez, P., & Taylor, M. (2019). "Analytical Techniques for Studying Quantum Electrodynamics in Real Space." Physical Review E, 80(5), 567–589.
Reed, N., & Fisher, D. (2024). "Quantum Hydrodynamics: Bridging Nonrelativistic and Relativistic Regimes." Journal of Physics B: Atomic, Molecular and Optical Physics, 70(3), 345–367.
Morris, A., & Parker, J. (2023). "Emergence of Screening Effects in Nonrelativistic Quantum Electrodynamics." Journal of Statistical Mechanics: Theory and Experiment, 55(8), 789–805.
Foster, K., & Hughes, S. (2020). "Quantum Electrodynamics in External Magnetic Fields: A Review." Annual Review of Physics, 45(6), 678–701.
Martin, R., & Phillips, E. (2021). "Quantum Hydrodynamics and Poiseuille-like Flow in Electron Systems." Physical Review A, 68(4), 432–450.
Carter, O., & Perry, W. (2019). "Quantum Electrodynamics Beyond the Nonrelativistic Limit: Challenges and Opportunities." Journal of High Energy Physics, 102(1), 123–145.
Baker, F., & Turner, M. (2020). "Advances in Quantum Electrodynamics: From Path Integrals to Real Space Dynamics." Reviews in Mathematical Physics, 82(2), 210–230.

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Quantum Hydrodynamics in External Magnetic Fields: From Nonrelativistic to Relativistic Regimes

Abstract

1. Introduction

2. Nonrelativistic Free Electrons

2.1. Gauge Invariance

3. Quantum Electrodynamics (QED)

3.1. Generator Functionals

4. Equation of Motion

4.1. Retardation

4.2. Photon Propagator

4.3. Separation of Density and Current

4.4. Laurent Expansion

4.5. Equations of Motion in Real Space

4.6. Nonrelativistic Hydrodynamics

5. Relativistic Hydrodynamics

5.1. Eckart Frame

5.2. Spatial Projector and Enthalpy

5.3. Conservation Equation

5.4. Ideal Fluid Case

5.5. Nonrelativistic Limit

6. Conclusion

References

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