Hot Fluids and Nonlinear Quantum Mechanics

Abstract

A hot relativistic fluid is viewed as a collection of quantum objects that represent interacting elementary particles. We present a conceptual framework for deriving nonlinear equations of motion obeyed by these hypothesized objects. A uniform phenomenological prescription, to affect the quantum transition from a corresponding classical system, is invoked to derive the nonlinear Schrödinger, Klein–Gordon, and Pauli–Schrödinger and Feynman-GellMaan equations. It is expected that the emergent hypothetical nonlinear quantum mechanics would advance, in a fundamental way, both the conceptual understanding and computational abilities, particularly, in the field of extremely high energy-density physics.

Appendix A

To accommodate spin (an internal degree of freedom) in a relativistic theory, we must take the following essential steps:

• 1)  Modify the classical energy momentum tensor to include a new, relativistically correct, internal (spin generated) stress term

  $$ \mathcal{T}^{\mu\nu}= g P^{\mu} P^{\nu}+{\Pi}\eta^{\mu\nu}+ g\frac{\hbar^{2}}{8}\partial^{\mu} M_{\alpha\beta}\partial^{\nu} M^{*\alpha\beta}, $$

  (53)

  which will transforms to the “quantum” energy momentum tensor

  $$ \mathcal{T}_{q}^{\mu\nu}= g\left(P^{\mu} P^{\nu}+\frac{\hbar}{2i}\partial^{\mu} \frac{\hbar}{2i}\partial^{\nu}\ln g\right)+{\Pi}\eta^{\mu\nu}+ g\frac{\hbar^{2}}{8}\partial^{\mu} M_{\alpha\beta}\partial^{\nu} M^{*\alpha\beta} $$

  (54)

  when subjected to the prescription (8). The stress is expressed in terms of the fully spin antisymmetric tensor

  $$ M_{\alpha\beta}=\frac{{\Psi}^{\dag}\sigma_{\alpha\beta} {\Psi}}{{\Psi}^{\dag}{\Psi}} , $$

  (55)

  where σ _{α β} =i γ _α γ _β , and γ ^α are the Dirac matrices.

• 2)  In addition to the contribution to the energy momentum tensor, we have to include the effect of spin-electromagnetic field interaction (through the associated magnetic moment). The complete equation of motion

$$ \partial_{\mu} \mathcal{T}_{q}^{\mu\nu}=q g F^{\nu\mu}P_{\mu} + d g M_{\alpha\beta} \partial^{\nu} F^{\alpha\beta}, $$

(56)

then, expands to

$$\begin{array}{@{}rcl@{}} \frac{1}{g}\partial^{\nu}{\Pi}+ P^{\mu}\partial_{\mu} P^{\nu}-\frac{\hbar^{2}}{4}\partial^{\nu}(\frac{1}{2}\partial_{\mu}\zeta\partial^{\mu}\zeta+\Box\zeta)= q F^{\nu\mu}P_{\mu} + κ M_{\alpha\beta} \partial^{\nu} F^{\alpha\beta}\\ -\frac{\hbar^{2}}{8}\left[\partial_{\mu{\zeta}} \partial^{\mu} M_{\alpha\beta}\partial^{\nu} M^{*\alpha\beta}+ \partial_{\mu}(\partial^{\mu} M_{\alpha\beta}\partial^{\nu} M^{*\alpha\beta})\right], \end{array} $$

(57)

where the last three terms on the r.h.s are all spin dependent. Here the constant $κ=-q\hbar /4c$. In spite of the immensely exaggerated complication of (57) vis a vis (11), the passage to the spin Fluidon quantum equation is quite straightforward, though highly tedious. These are the main steps and the final results:

• a)  Following the derivation of the KG Fluidon, we, first find the spin generalized momentum that is a perfect four gradient (vorticity free). Such a combination is found to be

  $$ P^{\mu} + qA^{\mu}-\varsigma\partial^{\mu} \omega=\partial^{\mu} S $$

  (58)

  where ς and ω are functions of spin to be explicitly displayed later. The four curl of the effective momentum P ^μ yields

  $$ \partial^{\mu} P^{\nu}-\partial^{\nu} P^{\mu}=-q F^{\mu\nu}-\hbar{\Omega}^{\mu\nu} $$

  (59)

  where

  $$ {\Omega}^{\mu\nu}=-\partial^{\mu}\varsigma\partial^{\nu} \omega+\partial^{\nu}\varsigma\partial^{\mu} \omega $$

  (60)

  is a kind of spin specific vorticity or curvature, which (with a weight $\hbar$) accentuates the electromagnetic curvature F ^μν (with a weight q).

• b)  then we introduce the following definitions: The wave function [56]

  $$ {\Psi}=\left( \begin{array}{c} \psi \\ -\psi \end{array} \right) $$

  (61)

  is a four component spinor with

  $$ \psi=\sqrt{g} e^{i S/\hbar}\varphi $$

  (62)

  thus the relativistic nature of the spin is represented through a spinor field. Here S is the eikonal defined by (58), g=n/f, as in the KG case, is the thermally modified number density, and

  $$ \varphi=\left( \begin{array}{c} \cos(\theta/2) e^{i\omega/2} \\ i \sin(\theta/2) e^{-i\omega/2} \end{array} \right) $$

  (63)

  is a two-component spinor leading to the normalizations

  $$ \psi^{\dag}\psi=g ,\qquad {\Psi}^{\dag}{\Psi}=2g $$

  (64)

The field 𝜃 is a parametric replacement of ς,

$$ \varsigma=\frac{\hbar}{2}\cos\theta $$

(65)

Using (61)-(65), we identify various intermediate variables in terms of the spinor φ

$$ \varsigma\partial^{\mu}\omega=-i\hbar\varphi^{\dag}\partial^{\mu}\varphi, $$

(66)

$$ M_{\alpha\beta}=\frac{\psi^{\dag}\sigma_{\alpha\beta} \psi}{g}=\varphi^{\dag}\sigma_{\alpha\beta} \varphi , $$

(67)

$$ \frac{1}{8}\partial_{\mu} M_{\alpha\beta}\partial^{\mu} M^{*\alpha\beta}=\partial_{\mu}\varphi^{\dag}\partial^{\mu}\varphi+(\varphi^{\dag}\partial_{\mu}\varphi)(\varphi^{\dag}\partial^{\mu}\varphi) , $$

(68)

and

$$ P^{\mu}=\partial^{\mu} S-qA^{\mu}-i\hbar\varphi^{\dag}\partial^{\mu}\varphi . $$

(69)

With the somewhat elaborate machinery, contained in (61)-(69), (57), and the accompanying continuity equation

$$ 0=\partial_{\mu}(gP^{\mu})= \partial_{\mu}\zeta (\partial^{\mu} S-qA^{\mu}+\varsigma\partial^{\mu} \omega)+\partial_{\mu}(\partial^{\mu} S-qA^{\mu}+\varsigma\partial^{\mu} \omega), $$

(70)

can be manipulated to, finally, derive the nonlinear Feynman-Gell-mann equation

$$ \left[D_{\mu} D^{\mu}+\frac{q\hbar}{2c}\sigma_{\alpha\beta}F^{\alpha\beta} +m^{2}+\lambda \bar{\Pi}({\Psi}^{\dag}{\Psi})\right]{\Psi}=0, $$

(71)

where

$$D^{\mu}=i\hbar\partial^{\mu}-q A^{\mu} $$

is the standard gauge derivative, and where the constant of integration was identified with the fermion mass square m ². The nonlinear term $\lambda \bar {\Pi }({\Psi }^{\dag }{\Psi })$, as before, is fluid specific. The spinor Fluidons, obeying the nonlinear (71), are, then, the quantum objects that embody many body interactions of a thermal fermion assembly. It is totally remarkable that their dynamics is relatively simple- almost like that of an elementary particle.