Rigidly-rotating quantum thermal states in bounded systems

TL;DR

This paper studies rigidly-rotating thermal states of a massless Klein-Gordon field within a cylinder, analyzing how Robin boundary conditions influence energy density and four-velocity in the Landau frame.

Contribution

It introduces the relationship between Robin boundary condition parameters and physical quantities like energy density and four-velocity in rotating thermal states.

Findings

Robin boundary conditions affect energy density distribution.

The connection between RBC parameters and Landau frame quantities is established.

Rotating thermal states are characterized within bounded cylindrical systems.

Abstract

We consider rigidly-rotating thermal states of a massless Klein-Gordon field enclosed within a cylindrical boundary, where Robin boundary conditions (RBCs) are imposed. The connection between the parameter of the RBCs and the energy density and four-velocity expressed in the Landau frame is revealed.

Equations

$$\square \hat{\Phi }(x)=0.$$

$$f_j=\frac{N_j}{\sqrt{8{\pi }^2{\omega }_j}}{e}^{-i{\omega }_{j}t+i{k}_{j}z+i{m}_j\phi }{J}_{m_j}(q_j\rho ),$$

$$⟨f_j,f_{j^{\prime }}⟩$$

$$q_{m,\ell }R{J}_m^{\prime }(q_{m,\ell }R)+\Psi J_m(q_{m,\ell }R)=0,$$

$$N_{k,m,\ell }=\frac{q_{m,\ell }\sqrt{2}}{|J_m(q_{m,\ell }R)|\sqrt{q_{m,\ell }^2{R}^2+{\Psi }^2-m^2}}.$$

$$\hat{\Phi }(x)=\sum _j\left[f_j(x){\hat{a}}_j+f_j^{\ast }(x){\hat{a}}_j^{†}\right],$$

$$⟨\hat{A}{⟩}_{\beta }=Z^{-1}\mathrm{tr}(\hat{\rho }\hat{A}),\hat{\rho }=\exp \left[-\beta (\hat{H}+\Omega {\hat{L}}_z)\right],$$

$$⟨{\hat{a}}_j^{†}{\hat{a}}_{j^{\prime }}{⟩}_{\beta }=\frac{\delta (j,j^{\prime })}{e^{\beta {\tilde{\omega }}_j}-1},{\tilde{\omega }}_j={\omega }_j-\Omega m_j.$$

$${\hat{T}}_{\mu \nu }=\frac{2}{3}{\nabla }_{(\mu }\hat{\Phi }{\nabla }_{\nu )}\hat{\Phi }-\frac{1}{3}\hat{\Phi }{\nabla }_{(\mu }{\nabla }_{\nu )}\hat{\Phi }-\frac{1}{6}{g}_{\mu \nu }[(\nabla \hat{\Phi }{)}^2+{\mu }^2{\hat{\Phi }}^2],$$

$$T_{\hat{\alpha }\hat{\gamma }}$$

$$F_{\hat{t}\hat{t}}$$

$$F_{\hat{\phi }\hat{t}}$$

$$F_{\hat{\rho }\hat{\rho }}$$

$$F_{\hat{\phi }\hat{\phi }}$$

$$F_{\hat{z}\hat{z}}$$

$$T_{\hat{\alpha }\hat{\gamma }}=\left(\begin{array}{cccc}{T}_{\hat{t}\hat{t}}& 0& T_{\hat{t}\hat{\phi }}& 0\\ 0& T_{\hat{\rho }\hat{\rho }}& 0& 0\\ T_{\hat{t}\hat{\phi }}& 0& T_{\hat{\phi }\hat{\phi }}& 0\\ 0& 0& 0& T_{\hat{z}\hat{z}}\end{array}\right).$$

$$E=\frac{1}{2}\left[T_{\hat{t}\hat{t}}-T_{\hat{\phi }\hat{\phi }}+\sqrt{(T_{\hat{t}\hat{t}}+T_{\hat{\phi }\hat{\phi }}{)}^2-4T_{\hat{t}\hat{\phi }}^2}\right],$$

$$v=-\frac{T_{\hat{t}\hat{\phi }}}{E+T_{\hat{\phi }\hat{\phi }}},\Gamma =\frac{1}{\sqrt{1-v^2}}.$$

$$\frac{v}{1+v^2}=-\frac{T_{\hat{t}\hat{\phi }}}{T_{\hat{t}\hat{t}}+T_{\hat{\phi }\hat{\phi }}}.$$

$$T_{\hat{t}\hat{\phi }}{\rfloor }_R$$

$$(T_{\hat{t}\hat{t}}+T_{\hat{\phi }\hat{\phi }}{)}_R$$

$$\Psi =\frac{3\sum _{m,\ell }{\left(1+\frac{{\Psi }^2-m^2}{q^2{R}^2}\right)}^{-1}{\int }_0^{\infty }\frac{dk}{\omega (e^{\beta \tilde{\omega }}-1)}\left[m\omega R-\frac{v_{\mathrm{b}}({\omega }^2{R}^2+m^2)}{1+v_{\mathrm{b}}^2}\right]}{\frac{v_{\mathrm{b}}}{1+v_{\mathrm{b}}^2}\sum _{m,\ell }{\left(1+\frac{{\Psi }^2-m^2}{q^2{R}^2}\right)}^{-1}{\int }_0^{\infty }\frac{dk}{e^{\beta \tilde{\omega }}-1}}.$$

$$v_{\mathrm{RKT}}=\rho \Omega ,E_{\mathrm{RKT}}=\frac{{\pi }^2{\Gamma }_{\mathrm{RKT}}^4}{30{\beta }^4}.$$

Full text

Keywords: Klein-Gordon field; Finite temperature field theory; Robin boundary conditions; Landau decomposition.

1 Introduction

In quantum field theory, the boundary conditions (b.c.s) are imposed at the level of the field operator $\Phi$, or equivalently, of the quantum modes. The interplay between the b.c. formulation and the ensuing operator expectation values in various states is far from obvious. In this paper, we consider the connection between the choice of b.c.s and the thermal expectation value (t.e.v.) of the stress-energy tensor (SET) operator in rigidly-rotating finite temperature states of the massless Klein-Gordon (KG) field. We show that the free parameter $\Psi$ in the Robin b.c.s (RBCs) can be related to the values of the Landau frame macroscopic four-velocity and energy density on the boundary.

The outline of this paper is as follows. In Sec. 2, the mode solutions of the KG equation inside a cylinder are reviewed. The procedure for constructing t.e.v.s in rigidly-rotating systems is summarised in Sec. 3. The analysis of the SET using the Landau frame decomposition and the connection between the Landau velocity $v$ and $\Psi$ is presented in Sec. 4. Section 5 concludes this paper.

2 Rigidly-rotating thermal expectation values

Let $\hat{\Phi}(x)$ be the field operator for a massless, neutral (real) scalar field which is confined within a cylinder of radius $R$, obeying the KG equation:

[TABLE]

The mode solutions of Eq. (1) can be obtained as follows:[6]

[TABLE]

where $(\rho,\varphi,z)$ are the usual cylindrical coordinates, while $\omega_{j}>0$, $k_{j}$ and $m_{j}$ are the eigenvalues of the Hamiltonian $\hat{H}$, longitudinal momentum $\hat{P}_{z}$ and $z$ component of the angular momentum, $\hat{L}_{z}$. In order to fix the normalisation constant $N_{j}$, we evaluate the KG inner product for $f_{j}$ and $f_{j^{\prime}}$:

[TABLE]

where a standard identity involving integrals of Bessel functions was employed.[9] Orthogonality is ensured when the transverse momenta $q_{j}\rightarrow q_{m,\ell}$ are discretised according to the Robin boundary conditions:[11]

[TABLE]

where $\ell=1,2,\dots$ indexes the non-negative solutions of Eq. (4) for fixed $m$ in ascending order, while $\Psi$ is considered to be a constant, real number. It is easy to see that $\Psi=0$ corresponds to the von Neumann b.c.s [$J^{\prime}_{m}(q_{m,\ell}R)=0$], while the Dirichlet b.c.s [$J_{m}(q_{m,\ell}R)=0$] can be recovered in the limit $\Psi\rightarrow\infty$. Imposing $\langle f_{km\ell},f_{k^{\prime}m^{\prime}\ell^{\prime}}\rangle=\delta(k-k^{\prime})\delta_{m,m^{\prime}}\delta_{\ell,\ell^{\prime}}$ yields:[11]

[TABLE]

The canonical expansion of the field operator with respect to the modes $f_{j}$ is:

[TABLE]

where the one-particle creation ($\hat{a}_{j}^{\dagger}$) and annihilation ($\hat{a}_{j}$) operators obey the standard commutation relation $[\hat{a}_{j},\hat{a}^{\dagger}_{j^{\prime}}]=\delta(j,j^{\prime})$.

3 Rigidly-rotating thermal states

We now consider rigidly-rotating thermal states, corresponding to an inverse temperature $\beta$ and an angular velocity $\Omega$. The thermal expectation value (t.e.v.) of an operator $\hat{A}$ is computed using the density operator $\hat{\rho}$ as follows:

[TABLE]

where $Z={\rm tr}(\hat{\rho})$ is the partition function. It can be shown that:[12]

[TABLE]

Equation (8) is not valid when the co-rotating energy ${\widetilde{\omega}}_{j}<0$, since in this case, the vacuum limit (corresponding to $\beta\rightarrow\infty$) yields a non-vanishing value.[3] Moreover, modes with ${\widetilde{\omega}}<0$ make infinite contributions to rigidly-rotating t.e.v.s.[6] It is noteworthy that finite quantum corrections can still be computed perturbatively.[4] It is reasonable to expect that t.e.v.s should stay finite for all values of $\Omega$ provided that $\Omega R<1$, which requires that $q_{m,\ell}R\geq m$. This property can be ensured only when $\Psi\geq 0$, thus we do not consider negative values of $\Psi$ in this paper.

Starting from the following expressions for the SET operator:[5, 7]

[TABLE]

the t.e.v. of the components of the SET can be obtained using the mode expansion (6) of the field operator. It is convenient to express the results with respect to the tetrad comprised of the vectors $e_{\hat{t}}=\partial_{t}$, $e_{\hat{\rho}}=\partial_{\rho}$, $e_{\hat{\varphi}}=\rho^{-1}\partial_{\varphi}$ and $e_{\hat{z}}=\partial_{z}$. Using the notation $T_{\hat{\alpha}\hat{\gamma}}\equiv\langle:\hat{T}_{\hat{\alpha}\hat{\gamma}}:\rangle_{\beta}$, the following results can be obtained:[1]

[TABLE]

where it is understood that $\omega\equiv\omega_{m,\ell}$ and $q\equiv q_{m,\ell}$, while the Bessel functions and their derivatives take the argument $q_{m,\ell}\rho$. It can be shown that the components of the SET not displayed above vanish for all values of $\rho$, $\beta$ and $\Omega$.

4 Landau decomposition

The matrix structure of the SET given in Eq. (10) can be summarised as follows:

[TABLE]

The energy density $E$ and macroscopic four-velocity $u^{\hat{\alpha}}$ can be obtained in the Landau frame by solving the eigenvalue equation $T^{\hat{\alpha}}{}_{\hat{\gamma}}u^{\hat{\gamma}}=-Eu^{\hat{\alpha}}$.[8, 10] The physically relevant solution for $E$ reads:

[TABLE]

while the Landau velocity $u^{\hat{\alpha}}=\Gamma(1,0,v,0)^{T}$ can be characterised via:

[TABLE]

Further manipulation of the above relations gives:

[TABLE]

When Dirichlet b.c.s are employed, it is easy to see that $v$ vanishes on the boundary. For finite values of $\Psi$, $v$ is in general non-vanishing on the boundary. Let $v_{\rm b}$ denote the value of $v$ on the boundary. We now ask what is the value of $\Psi$ which ensures $v(R)=v_{\rm b}$. Inverting Eq. (14) in order to obtain $\Psi$ as a function of $v_{\rm b}$ does not seem feasible. Instead, an iterative procedure can be established which allows the value of $\Psi$ to be computed numerically. Starting from:

[TABLE]

where the sums over $m$ and $\ell$ run between $-\infty$ and $\infty$ and $1$ and $\infty$, respectively, it can be seen that $\Psi$ can be isolated from the last term of the second equality above:

[TABLE]

Equation (16) is solved iteratively. The value $\Psi^{(n)}$ corresponding to iteration $n$ is obtained by evaluating the right hand side of Eq. (16) after replacing $\Psi$ with the value $\Psi^{(n-1)}$ obtained at iteration $n-1$, while $v_{\rm b}$ is kept fixed at the desired value. Starting from $\Psi^{(0)}=0$ yields the convergence value within a relatively small number of iterations and the process seems to be stable as long as $v_{\rm b}$ can be obtained using $\Psi\geq 0$. To illustrate the procedure, we consider a system with $R=1$ and $\Omega=0.5$. Figure 1(a) shows the variation of $v_{\rm b}$ with $\Psi$ for $\beta\in\{0.25,0.5,1\}$.

It is natural to consider the relation between the Landau energy density $E_{\rm b}$ measured on the boundary and the energy density $E_{{\rm b};{\rm RKT}}$ expected for a rigidly-rotating Bose-Einstein gas, for which[2]

[TABLE]

An iterative scheme for finding $\Psi$ for a prescribed value of $E_{\rm b}$ involves working with quadratic functions with respect to the SET components. The stability and efficiency of such a scheme is questionable. Instead, we employ a bisection algorithm to find the value of $\Psi$ corresponding to $E_{\rm b}$. Typically, the ratio $E_{\rm b}/E_{{\rm b};{\rm RKT}}$ characterising the departure of the quantum state from the expected rigid-rotation profile ranges from $\sim 0.1$ for Dirichlet b.c.s to $\sim 2$ for von Neumann b.c.s. The dependence of $E_{\rm b}/E_{{\rm b};{\rm RKT}}$ on $\Psi$ for $\beta\in\{0.25,0.5,1\}$ is illustrated in Fig. 1(b).

Finally, we examine the profiles of the energy density and velocity when $\beta=0.5$, $\Omega=0.5$ and $R=1$. Fig. 2(a) shows that the RBCs interpolate between the Dirichlet and von Neumann b.c.s. In the former case, the energy density exhibits a strong decreasing trend in the vicinity of the boundary, as also remarked in Ref. [6]. For the von Neumann b.c.s, the energy density is amplified next to the boundary, as compared to the RKT prediction for a rigidly-rotating Bose-Einstein gas. The velocity $v$, shown in Fig. 2(b), shows small variations with respect to $\Psi$.

5 Conclusion

In this paper, a procedure to correlate the parameter $\Psi$ of the RBCs for the massless KG field enclosed within a cylinder and the boundary values of the rigidly-rotating t.e.v. of the SET operator was introduced. The restriction $\Psi\geq 0$ was imposed in order to eliminate modes with negative co-rotating energy $\widetilde{\omega}$ which would otherwise cause t.e.v.s to diverge. Employing the Landau frame decomposition to obtain the macroscopic four-velocity of the state, the velocity on the boundary was shown to take values between [math] (Dirichlet limit) and a maximum value (von Neumann limit), which increases towards the value corresponding to rigid rotation as the temperature is increased. The energy density is strongly quenched compared to the relativistic kinetic theory prediction for a rigidly-rotating Bose-Einstein gas in the vicinity of the boundary when the Dirichlet b.c.s are employed. By contrast, it is amplified in the case of the von Neumann b.c.s.

Acknowledgments

This work was supported by a grant of Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P1-1.1-PD-2016-1423, within PNCDI III.