Being a manifestation of quantum phase coherence over macroscopic distances, superfluidity is best described in terms of a complex order parameter, which in the nonrelativistic regime corresponds to a wave function satisfying the nonlinear Schrödinger equation (NLSE). In the relativistic domain this is generalized to the nonlinear Klein–Gordon equation (NLKG)⁴

⁴For the NLSE or the NLKG of the order parameter ψ, the nonlinearity is embodied in a self-interacting term ∝∣ψ∣²ψ, while there could be other interaction terms representing trapping potentials, rotational effects, etc.

(). The order parameter is regarded as the primary state variable of a superfluid, while hydrodynamic quantities, such as the superfluid density and velocity, are derived quantities. Such a view not only gives a more concrete physical picture, but, as shown in [1], also facilitates numerical analysis, especially in regard to quantized vorticity, which is the key ingredient in forming an entangled vortex state—quantum turbulence, a close analogy of the classical turbulence. We shall introduce the normal fluid (and its interaction with the quantized vorticity) in the same framework, and, to put it in historical perspective, start with a brief review of the two-fluid model [2, 3].

Shortly after superfluidity was discovered in liquid ⁴He below the critical temperature of 2.8 K [4], as an apparent absence of viscosity, Tisza [2] suggested that the liquid be modelled as two inter-penetrating liquids, one having ‘super' qualities, and the other behaving in a ‘normal' fashion. Landau [3] made the model more concrete by regarding the superfluid as the ground state of a quantum mechanical many-body system, and the normal fluid as a system of quasiparticle excitations. The mass density ρ and mass current density j are split into superfluid and normal fluid contributions. In the nonrelativistic regime one writeswhere the subscripts s and n refer, respectively, to the super and normal fluids, with the irrotational conditionThe nonrelativistic two-fluid hydrodynamics consists of phenomenological equations based on conservation and thermodynamic laws [3, 5]:where μ is the chemical potential, S is the entropy density, and Π^jk is the energy-momentum tensor:where p is the pressure. The second equation in (3) is the analog of the Euler equation. At absolute zero temperature S vanishes identically, and the equation for j^k becomes the same as the Euler equation, and (3) collapses to the first two equations describing a pure superfluid.

As thermodynamic functions, the quantities S, μ, and p are to be specified in a more detailed model. They can be calculated, for example, if the normal fluid is modelled at low temperatures as a dilute gas of quasiparticles. Hill and Roberts [6] introduced a special pressure term in μ, in order to describe the healing length, the characteristic distance within which the superfluid density decreases to zero at a wall. Geurst [7] has given a general formulation of the two-fluid hydrodynamics in term of an action principle, and a historical review. A relativistic action principle is discussed by Lebedev and Khalatnikov [8].

Even with a built-in healing length, however, the hydrodynamic equations (3) fail to describe one of the signature properties of a superfluid, namely, quantized vorticity. In fact, the irrotational condition (2) rules out vorticity, and to accommodate that one has to add it ‘by hand', by writing something like v_s = ∇α + b, where ∇ × b ≠ 0, and go through another round of phenomenology for b. But all this still does not explain why the vorticity should be quantized, not to mention the impracticality of numerical analysis.

All these difficulties in describing the superfluid are resolved by using a complex order parametera nonrelativistic wave function satisfying the nonlinear Schrödinger equation (NLSE)⁵(⁵The phase representation of the order parameter ${\rm{\Psi }}\left({\boldsymbol{r}},t\right)$ in equation (5) is sometimes referred as the ‘Madelung transformation', which historically led to the hydrodynamic interpretation of quantum mechanics [9]. See the next section for a relativistic version in the framework of the NLKG.) The superfluid velocity is related to the phase of the wave function throughwhere m is the mass scale in the NLSE. The healing length arises automatically, since the superfluid density ρ_s = mF² goes to zero continuously at a boundary. The Hill–Roberts pressure is just the ‘quantum pressure' arising naturally from the NLSE. The quantization of vorticity, namelywhere C is a closed contour in space, follows from the fact that the phase β must be a continuous function [10]. In general ∇ × v_s ≠ 0, even though v_s is a gradient, because $\Psi$ can develop zeros, thus rendering the space non-simply connected, i.e, admitting closed contours that cannot be deformed to zero continuously. Another advantage of the NLSE is that it can be handled numerically very efficiently.

Adopting the NLSE means that we regard the complex wave function $\Psi$ as the fundamental variable, and the hydrodynamic quantities ρ_s, v_s as derived ones. Thus, the first two equations in (3) are replaced by and implied by the NLSE.

To include the normal fluid in the NLSE, we need to introduce four new degrees of freedom associated with ρ_n, v_n. Bogolubov apparently was the first to suggest, in an unpublished note [11], the introduction of gauge-like potentials φ, A via the transformation ∂_t → ∂_t − iφ, ∇ → ∇ − iA. This is done in order to couple the new degrees of freedom to the phase of the wave function, not to impose new gauge symmetry on the system which is, of course, not locally gauge-invariant. Coste [12] shows how one can relate φ, A to ρ_n, v_n through considerations based on Galilean invariance. To obtain the equations of motion for ρ_n, v_n, Coste uses a hybrid variational principle involving $\Psi$, ρ_n, v_n. A modified NLSE is then obtained and the normal fluid enters via an effective potential [12].

This paper is organized in the following manner. After a brief description of the NLKG at absolute zero, we extend it to finite temperature by introducing couplings to the normal fluid, based on Lorentz covariance. We show that the couplings can be expressed in terms of an additional nonlinear potential that has both real and imaginary parts, and discuss the nonrelativistic limit using Coste's approach. We display quantized vorticity by transforming the scalar field theory to its global string dual, and show how the effective potential changes by including more Lorentz invariants. Magnus force and mutual friction are studied and extracted in some simple examples.

In this section, we briefly review our previous formulation of relativistic superfluids at zero temperature using the NLKG (with units ℏ = c = 1) [1, 13, 14]. Consider a complex scalar field φ(x, t), which can be written in the phase representation similar to the NLSE caseIt serves as the order parameter for superfluidity through the dynamics of the phase σ(x, t). The physical superfluid velocity v_s is related to the 4-vectorthroughHere, ω is a frequency given by the time component of v^μ: which ensures ∣v_s∣/c < 1. The Langrangian density is given bywhere g^μν is the metric tensor. The potential V depends only on φ^*φ, and ${V}^{{\prime} }\equiv {\rm{d}}{V}/{\rm{d}}\left({\phi }^{* }\phi \right)$. The actionleads to the NLKGwhere $\square \phi \equiv \tfrac{1}{\sqrt{-g}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }\phi \right)$. Some examples of the NLKG in rotating blackhole backgrounds are given in [14]. In the phase representation, the real and imaginary parts of the NLKG give rise to two hydrodynamic-like equations:The first is the analog of the Euler equation, and the second is the continuity equation $\ {{\rm{\nabla }}}_{\mu }{j}_{0}^{\mu }=0,$ where ∇_μ denotes the covariant derivative, withThis is a number current density, corresponding to the conservation of charge $Q=N-\bar{N}$, where N, $\bar{N}$ are, respectively, the particle and antiparticle number. Unlike the nonrelativistic case, the density ${j}_{0}^{(0)}$ is not positive definite.

If σ were a continuous function, we would have ${\partial }^{\mu }{v}^{\nu }-{\partial }^{\nu }{v}^{\mu }=\left({\partial }^{\mu }{\partial }^{\nu }-{\partial }^{\nu }{\partial }^{\mu }\right)\sigma =0$. But σ is a phase angle, and only continuous modulo 2π. The derivatives ∂^μ, ∂^ν do not commute when operating on σ, and this is the origin of quantized vorticity. Specifically, there is a class of singular solutions in which the modulus F has zeros along a space curve, the vortex line, and ∮_C∇σ · ds = 2πn along any closed circuit C encircling the vortex line, where n is an integer.

Without going into details, we comment on the fact that the first equation in (15) can be rewritten in Euler form as an equation for dv/dt, which contains a ‘quantum pressure'. This pressure naturally vanishes on boundaries where F goes to zero, with a healing length. There is no need for the ‘Hill–Roberts pressure' [6], which is introduced by hand.

In the following, we consider Minkowski spacetime with η^μν = diag (−1, 1, 1, 1). At finite temperatures, new degrees of freedom arise, associated with the normal-fluid velocity field v_n. We represent the new degrees of freedom covariantly by a 4-vector w^μ, and, following Lebedev and Khalatnikov [8], write it in a Clebsch representation of the formwhere α, ξ, χ, are regarded as independent variables, which will be related to physical normal fluid properties. We can construct three Lorentz invariants from v^μ and w^μ:and generalize the zero-temperature Lagrangian density ${{ \mathcal L }}_{0}$ towhere f is a function with derivatives denoted byThe equations of motion are obtained by varying the new action with respect to F, σ, α, ξ, χ: where two current densities j^μ and s^μ are defined asThese are conserved current densities, identified, respectively, with the number current density and the entropy current density. The former reduces to ${j}_{0}^{\mu }$ when f ≡ 0, and the latter is present only at nonzero temperatures. The first two equations of motion in (21) can be combined to give a new NLKG:where W is an effective potential obtained by plugging φ = Fe^iσ into (23) and then comparing with the first two equations of (21)Note that the effective potential W has both real and imaginary parts, indicating that the NLKG for the superfluid is dissipative: there is particle transfer between superfluid and normal fluid. Assuming that α, ξ, χ are continuous functions, we have ∂^μw^ν − ∂^νw^μ = ∂^μξ∂^νχ − ∂^νξ∂^μχ. Multiplying by s_μ yieldsWe will show that the imaginary part of the effective potential W is important in determining the mutual friction coefficients, which describe the coupling between quantized vortices and the normal fluid.

We represent the time and spatial components of current densities j^μ, s^μ asThe first relation defines total number density ρ and current j, with number meaning $N-\bar{N},$ the difference between particle and antiparticle number. The second defines the entropy density s and the normal-fluid velocity v_n. (The definition of ρ agrees with Lebedev and Khalatnikov [8] but differs from our earlier paper [1], in which the density corresponds to $\rho \sqrt{1-{v}_{s}^{2}}$ in the present convention.) The conservation laws can be put in the formThe second equation of (22) can be rewritten asSubstitution into the first equation of (22) giveswhose components are given byWe make the identificationsuch as to giveWe can write, in manifestly covariant form,The energy-momentum tensor of the superfluid is given byUsing (22), we obtain the symmetric formwhere $\tilde{s}\equiv {\rho }_{{\rm{n}}}{F}^{2}{f}_{3}^{{\prime} }$. The spatial components give the stress-energy tensorwhere we have used (10) and (26). The coefficient of ${v}_{{\rm{n}}}^{j}{v}_{{\rm{n}}}^{k}$ should be ωρ_n, as suggested by comparison with (4). Thus, we have the relationUsing equations (37) and (31), we can now express the parameters ${f}_{{\rm{n}}}^{{\prime} }$ in terms of observable normal-fluid properties:where ρ₀ is the density at absolute zero temperature:The NLKG describes the superfluid and its coupling to the normal fluid, whose dynamics requires separate treatment. A general hydrodynamic treatment based on conservation laws and thermodynamics has been given by Lebedev and Khalatnikov [8] and by Carter and Khalatnikov [15]. A general action principle has been discussed by Geurst [7]. A more detailed description of the normal fluid will depend on the specific model. In this respect, a relativistic ideal gas model of the normal fluid may be found in [16], and a treatment based on quantum field theory of the scalar field is given in [17].

Note that the physical meaning of the currents and densities depend on the reference frames and we compare our results with [16]. We consider the following Lorentz scalars:Our choices for v^μ and S^μ are(Here, we changed some notation for the purpose of comparison. We defined before in equations (9)–(11)and we have changed ω to $\tilde{\mu }$, and s to $\tilde{s}$ respectively.) Note that the velocities v_s and v_n are defined in the lab frame. From (40), we obtainWhile μ, s are Lorentz scalars, $\tilde{\mu },\tilde{s}$ are not due to the γ-factors in the above equations. The crossing term yieldsWe define a relative velocity, v_ns, between the superfluid velocity v_s and the normal velocity v_n according to the relativistic velocity-addition formulaand then findPlugging into (44) we obtainwhich is exactly the relative translation speed between the ‘normal' and ‘superfluid' frames used in [16]. One can identify the relativistic generalization of the mass densities of the superfluid and normal fluids, ${\hat{\rho }}_{{\rm{n}}}$ and ${\hat{\rho }}_{{\rm{s}}}$, by considering the decomposition of the stress-energy tensor T^μν. The advantage of finding ${\hat{\rho }}_{n}$ and ${\hat{\rho }}_{{\rm{s}}}$ based on the decomposition of T^μν is that it avoids the use of ‘rest mass', more precisely, the question of which frame should be considered as the rest frame of the fluid element. What is needed is the relative translation velocity between the normal fluid and the superfluid frames [16]. Note thatwhere s_s is the entropy density defined in the superfluid frame and the μ_n is the chemical potential defined in the normal fluid frame. The stress-energy tensor becomesIt is not hard to check that these are consistent with the results in [16]. In fact the explicit correspondences arewhere ${ \mathcal A },{ \mathcal B },{ \mathcal C }$ are quantities defined in [16]

A solution φ of the NLKG contains both positive and negative frequencies. It approaches the nonrelativistic limit when one sign (say, positive) becomes dominant. Formally, we writewhere m is a large mass scale. The nonrelativistic wave function $\Psi$ can be represented in the formwhere ρ is the non-relativistic superfluid density, andis the non-relativistic superfluid velocity. The nonrelativistic phase β is related to the phase σ of the relativistic scalar field φ throughThe wave function $\Psi$ satisfies an NLSE (nonlinear Schrödinger equation) (see [1] for details). To derive it, it is easier to start from the Lagrangian. Let ${{ \mathcal L }}_{0}$ be the nonrelativistic Lagrangian density at absolute zero, which leads to an NLSE with cubic nonlinearity. We show how the normal fluid may be introduced, following Coste [12], but reformulated from our point of view.

Let the superfluid density and current density be denoted, respectively, by ρ and j = ρv_s . The degrees of freedom ρ_n, v_n associated with the normal fluid can be introduced via gauge-like potentials φ, A, through the transformation $\tfrac{\partial }{\partial t}\to \tfrac{\partial }{\partial t}-{\rm{i}}\varphi ,$ ∇ → ∇ − iA [11]. The Lagrangian density at finite temperatures isThis is not locally gauge-invariant (and had better not be, for otherwise the gauge transformation would have no physical effect). The term $\tfrac{1}{2}\rho {A}^{2}$, while crucial for local gauge invariance, is irrelevant here, and will be dropped. Using arguments based on Galillean covariance, Coste [12] writeswhere α is a scalar function, and v_n is the normal-fluid velocity. The equations of motion are then obtained through the action principle. We omit details and just cite the final result. Assuming an original NLSE with quartic nonlinearity, we obtain a modified equationwhere ℏ = m = 1, and all coupling parameters have been scaled to unity. The normal fluid enters via the effective potential U, which vanishes at absolute zero. The real part of U contributes to the phase change of $\Psi$, and thus to superfluid flow. The imaginary part contributes to $\dot{{\rm{\Psi }}}$, rendering ∫d³x∣$\Psi$∣² non-conserved, signifying particle transfer between superfluid and normal fluid. Note that the effective potential U enters the NLSE via Coste's ‘gauge potential' (57), and it will be interesting to investigate how to derive the effective potential U directly from the nonrelativistic limit of the effective potential W in equation (24).

A phenomenological treatment of quantized vorticity in the nonrelativistic domain was pioneered by Schwarz [18], based on the following physical picture (see appendix for notations), which is given in detail in [19, 20]. A vortex configuration is characterized by a space curve called the vortex line, described by the position vector s(ξ, t), where ξ is a parameter that runs along the line. The vortex line may be made up of disjoint closed loops, and curves that terminate on boundaries. The parameter ξ run through all of the components according to some convention. The superfluid density vanishes on the vortex line with a characteristic healing length. We can picture the vortex line as a tube with effective radius a₀ of the order of the healing length. This core size is supposed to be much smaller than any other length in the theory. When we refer a point on the vortex line, we mean some point within the core. Let ${{\boldsymbol{s}}}^{{\prime} }\equiv \partial {\boldsymbol{s}}/\partial \xi $. The triad ${\boldsymbol{s}},{{\boldsymbol{s}}}^{{\prime} },{{\boldsymbol{s}}}^{{\prime\prime} }$ gives a local orthogonal coordinate system. The local radius of curvature is given by $R=\left|{{\boldsymbol{s}}}^{{\prime\prime} }\right|$ ⁻¹.

The superfluid velocity is determined up to a potential flow by the equationwhere κ(r, t) is the vorticity density, which is nonvanishing only on the vortex line:We decompose the superfluid velocity into an irrotational part v₀, and a rotational part b: The velocity field b is like a magnetic field produced by the current density κ, and is given by the Biot–Savart lawwhere s₁ is a particular point on the vortex line.

The velocity of the vortex line at any point is influenced by the shape of the entire vortex line. In a ‘local inducting approximation' (LIA) [19, 20], one considers only the effects from the immediate neighborhood of the point. In this case, the local velocity is the translational velocity of an osculating vortex ring at that point, which is normal to the plane of the vortex ring, and approximately inversely proportional to its radius of curvature R. For a vortex line at absolute zero, this leads to the equationwhere ${\dot{{\boldsymbol{s}}}}_{0}\equiv \partial {{\boldsymbol{s}}}_{0}/\partial t$, $\bar{R}$ is the average radius of curvature, and c₀ is a constant of order unity.

At finite temperatures, there is a normal fluid, which exerts a dissipative force per unit length f_D on the vortex line. It can be fit phenomenologically by the formula [19, 20]where v_ns = v_n − v_s, and $\alpha ,{\alpha }^{{\prime} }$ are temperature-dependent parameters. The vortex line experiences a Magnus force per unit length f_M, when the vortex line velocity ${{\boldsymbol{v}}}_{{\rm{L}}}\left(\xi ,t\right)\equiv \dot{{\boldsymbol{s}}}\left(\xi ,t\right)$ is different from the local superfluid velocity ${{\boldsymbol{v}}}_{\mathrm{sl}}\left(\xi ,t\right)\equiv {{\boldsymbol{v}}}_{{\rm{s}}}\left({\boldsymbol{s}}\left(\xi ,t\right),t\right)$:

The phenomenological equations give physical insight, but for actual computations it is simpler to use the NLKG directly. As shown in [1], complex phenomena such as vortex formation and reconnection can be exhibited in numerical solutions of the NLKG. When quantum vorticity appears, the phase σ of the complex field cannot be smooth everywhere, hence ∇_μv_ν − ∇_νv_μ ≠ 0. From numerical calculations, the phase σ has ambiguity and the modulus F vanishes at the locations of vortices.

To be consistent with the NLKG with effective potential describing the normal-fluid effects, the variational principle should be applied to the Lagrangianwhere f_μ ≡ ∇_μF and ∇σ is generally not a smooth function. Note that one can split the Lagrangian intowhere ${{ \mathcal L }}_{\mathrm{NLKG}}^{0}$ has the same form as in the zero-temperature cases:while ${{ \mathcal L }}^{T}$ includes finite-temperature effects. Following similar notation in [8], we rewritewhere φ is a smooth function whose gradient is curl-free, i.e.,while the vector field b_μ gives the vorticity, described by its ‘field strength' b_μν:Now both the stress-energy tensor and the mass current should include the contribution of the vorticity. For simplicity, let us first consider the zero temperature T = 0 cases without w^μ as in [8]: and the conservation law ${{\rm{\nabla }}}_{\mu }{T}_{\nu }^{\mu }=0$ leads toin comparison with the curl-free or irrotational cases in which b_μν = 0. Note that j^μ also has a vorticity dependence. Therefore after the decomposition (69), besides b_μ the Lagrangian should also contain b_μν explicitly. From the zero-temperature example given in [8], the inclusion of b_μ and b_μν in the Lagrangian is much more complicated than just adding a ‘kinetic' term ∼b^μνb_μν as one may have expected, by analogy to the electromagnetic case. This will be discussed in the next section.

We give a relativistically covariant description of quantized vorticity in Minkowski spacetime with metric diag $\left(-1,1,1,1\right)$. It is easily generalized to curved spacetime. The vortex configuration is specified by a space curve, which sweeps out a world sheet in 4D spacetime. The dynamics of the vortex line is, therefore, that of a relativistic string, which has been widely discussed in the literature [21–29]. We summarize known results from our perspective.

To begin, we use the first equation in (69) to separate rotational flow from irrotational flowwhere φ is a continuous function (whereas σ is only continuous modulo 2π), and b^μ describes vorticity. We define a ‘smooth' order parameter ψ, with the phase φ: The Lagrangian density can be rewritten in terms of ψ and b^μ: This says that we can start with potential flow described by ψ, and introduce vorticity by introducing a ‘gauge field' b^μ. The system is invariant under a local ‘gauge transformation' $\psi \to {\psi }^{{\prime} },$ ${b}_{\mu }\to {b}_{\mu }^{{\prime} }$, withwhere $\alpha \left(x\right)$ is a continuous function; the transformation suggests an emergent ‘gauge symmetry' and is equivalent to a shift χ → χ − α. Note that b has a dual personality: on the one hand, it is like a magnetic field according to (62), and on the other hand, it is like a gauge field in the present context. Note that b_μ is constrained by the vortex quantization condition, which can be represented covariantly asBy means of the Stokes theorem, we can rewrite this aswhere S is a surface bounded by the closed path C , dS^μν is a surface element, and b^μν is the antisymmetric vorticity tensor defined byNow we define the ‘dual' of vorticity b_μν in a similar way as in [21]:In terms of the phase σ, it becomesThe dual vorticity ${\tilde{b}}^{\mu \nu }$ is a distribution, e.g. for a static, straight vortex line lying on the z-axis, the only non-vanishing component isIn general, ${\tilde{b}}^{\mu \nu }$ perform a projection onto the worldsheet swept by the vortex line. Suppose the worldsheet is parametrized by x^μ = x^μ(ζ^a), (a = 0, 1), ${\tilde{b}}^{\mu \nu }$ can be written as [21]:where ${\rm{d}}{\sigma }^{\mu \nu }\equiv {\epsilon }^{{ab}}{x}_{,a}^{\mu }{x}_{,b}^{\nu }{{\rm{d}}}^{2}\zeta $ is the area element of the worldsheet.

These are necessary ingredients connecting the (modified) NLKG to its intrinsic vortex dynamics (or connecting a field theory to a ‘string' theory). From equations (83) and (85), it is clear that the vortex dynamics is determined by the phase σ and the zero of the modulus F.

The vector b_μ can be related to a Kalb–Ramond potential [21]. The decompositionwith the identificationwhere B^μν is the antisymmetric Kalb–Ramond field [23], generalizes the usual Helmholtz decomposition for a 3-vector v, i.e.It is easy to see that in equation (86) the ‘longitudinal' component u_μ = ∂_μφ does not contribute to the vorticity, while the ‘transverse' component b_μ = 1/2 ε_μνλρ∂^νB^λρ is divergenceless for a regular Kalb–Ramond field B^μν. To see why the relation (87) is the relativistic generalization of the three-dimensional analogueone may set the components of the Kalb–Ramond field to be (i, j, k = 1, 2, 3)and then obtainThe relativistic generalization of the three-dimensional vorticity vectoris in a Chern–Simons form [8], whose spatial component contains a termTherefore, the role of the vector potential in v = ∇ × A is played by the Kalb–Ramond field B^μν and the role of the vorticity field ω = ∇ × v is played by the Chern–Simons form K^μ = ε^μνρτb_ν∂_ρb_τ.

Once we have identified b_μ = 1/2 ε_μνλρ∂^νB^λρ, we can use the Kalb–Ramond action [23] as the effective action for a vortex line or a vortex ring. With the Kalb–Ramond field strengththe original nonlinear action for the Klein–Gordon field Φ = Fe^iσ (at T = 0) becomesIntegrating over the massive F modes for a string solution gives the Kalb–Ramond action [21].

Now, we look back at the original Lagrangian and consider its independent variables. We have v² = v^μv_μ, ω² = ω_μνω^μν with ω_μν ≡ ∂_μv_ν − ∂_νv_μ, and h² = h_μh^μ, where a Chern–Simons-type current h^μ is defined as (similar to equation (93))It is easy to see thattherefore, we see that v², ω², h² can be considered as independent variables in the Lagrangian. One can easily write down other Lorentz invariants such aswhere the dual tensor ${\tilde{\omega }}^{\mu \nu }\equiv \tfrac{1}{2}{\epsilon }^{\mu \nu \rho \tau }{\omega }_{\rho \tau }$. However, with the help of two identitiesone can show that what is really important is the ${\omega }_{\mu \nu }{\tilde{\omega }}^{\mu \nu }$ term and the other terms can be reduced to combinations of known terms. Also, note thatsimilar to the gauge theory cases where $F\tilde{F}\sim {\partial }_{\mu }{K}^{\mu }$, i.e. the topological charge term can be written as the divergence of the Chern–Simons current. Therefore, the Lagrangian should only depend on these Lorentz scalars:For the finite temperature cases, w_μ should be included as well. Neglecting classical vorticity, one can writeNote that in the present paper we have not intended to write down an explicit Lagrangian for ${{ \mathcal L }}^{T}$. Instead we aimed to show how the relevant degrees of freedom come from the (modified) NLKG or NLSE. In practice what we propose to solve numerically is the original (modified) NLKG or NLSE, similar to what we have done in [1].

Just like the vorticity is built in the NLKG, forces like the Magnus force, the mutual friction between the quantum vorticity and the normal fluid should be included automatically in the effective potential part of the NLKG. Solving such an NLKG should yield all the effects these forces produce. First let us consider the Magnus force. Any force in vortex dynamics is connected with some velocity by the Magnus relation, connecting the vortex line velocity and the external force per unit length applied to the vortex line (see e.g. [30]):where κ is the circulation vector of magnitude κ, v₀ is the constant velocity that a fluid current passes the vortex line and v_L is the velocity of the vortex line. We use a simple example at zero-temperature to demonstrate the existence of the Magnus force. From the decomposition v_μ = ∂_μφ + b_μ, we takewhich corresponds to a constant backgroundin terms of the Kalb–Ramond field. With proper gauge, the equation of motion of string is [21]Its spatial component givesNote that ${\dot{q}}^{j}$ is the vortex line velocity v_L and ${\boldsymbol{q}}^{\prime} $ is the vortex line tangent. We see that the right-hand side of the above equation can be written as ${{\boldsymbol{v}}}_{L}\times {\boldsymbol{q}}^{\prime} $, which shows the existence of the Magnus force in the superfluid rest frame

Now we consider the mutual friction between quantum vortices and the normal flow. Notice that there is a non-vanishing imaginary part in the effective potential W in equation (24). For simplicity we assume that this imaginary part is a constant ∝γ. At the nonrelativistic limit it reduces to a damped NLSEwhich for small γ becomes approximately [31]where H[ψ, ψ^*] is the Gross–Pitaevskii energy functionalTo connect the order parameter ψ(x, t) to the motion of the vortex line, we follow [32, 33] and consider ψ(x, t) as a functional of the vortex configuration ψ(x, s(ξ, t)). The time-derivative of ψ(x, t), for example, is then related to the vortex line velocity $\dot{{\boldsymbol{s}}}$ and then it has been shown in [33] thatwhere the constant β is defined in equation (63). This allows identifying the mutual friction coefficients [33]

In this paper, we studied a relativistic two-fluid model with quantized vorticity via a modified NLKG. An effective potential is introduced to describe the coupling of the superfluid and the normal fluid. It has been shown that such a formulation incorporates vorticity and the related vortex dynamics, and hence, can facilitate numerical analysis, which is usually quite complicated from a phenomenological point of view (e.g. Schwarz's numerical studies based on vortex filaments [18]). We also considered the connections to other formulations, especially the duality between scalar field and Kalb–Ramond field, and the similarity between quantized vortices and global strings. We propose that just like in the zero-temperature pure superfluid cases (as we have shown numerically in [1]), quantum vorticity and quantum turbulence should be studied using the modified NLKG/NLSE, possibly coupled to other equations depending on the systems or circumstances, in a relativistic or nonrelativistic way.

What is the range of validity of the modified NLKG/NLSE? This is an interesting question, with different answers from different points of view. One can derive the NLKG/NLSE from the quantum N-body wave function, but the validity of this approach is limited to weak interparticle interactions, and in the neighborhood of the ground state of the system, i.e. at low temperatures. The reason is as follows. First, the assumption that the interparticle potential is a delta function can be justified only for weak interactions described through a small S-wave scattering length, which give the equivalent hard-sphere interaction. Secondly, the derivation from the quantum N-body problem corresponds to a mean-field approximation, in which one assumes that effects from excitations from the ground state are small. From this point of view, then, the NLKG/NLSE is a weak-interaction low-temperature approximation.

In the Ginzburg–Landau theory of phase transitions governed by an order parameter, on the other hand, the NLKG/NLSE is a purely phenomenological equation, valid near the transition point of the phase transition. Thus, the order parameter is assumed to be small. One expands the nonlinear potential in powers of the order parameters, and just retains the first few terms From this point of view, the NLKG/NLSE is valid in the neighborhood of the phase transition that creates the order parameter. Which view one adopts would depend on the application.