Magnetism and rotation in relativistic field theory

We investigate the analogy between magnetism and rotation in relativistic theory. In nonrelativistic theory, the exact correspondence between magnetism and rotation is established in the presence of an external trapping potential. Based on this, we analyze relativistic rotation under external trapping potentials. A Landau-like quantization is obtained by considering an energy-dependent potential.


Introduction
In quantum physics, a response to magnetism is one of the most significant subjects. One ubiquitous phenomenon induced by external magnetic backgrounds is the Landau quantization. In condensed matter systems, for example, the quantum Hall effect is essentially described by the Landau levels [1]. In relativistic theory, such as quantum chromodynamics, an external magnetic field enhances a fermion-antifermion condensate and generates a dynamical fermion mass, which is called the magnetic catalysis [2,3]. The magnetic catalysis is also important in condensed matter systems with relativistic dispersion relations, such as graphenes and Dirac semimetals [4].
Rotation has similar effects to magnetism, and so they share many common phenomena: The quantum Hall effect is also induced by rotation instead of an external magnetic field [5]. A quantum vortex is generated by applying a magnetic field to the Bose-Einstein condensate or by rotating it [6,7]. The chiral magnetic effect [8,9] is analogous to the chiral vortical effect [10,11]. Note that, since rotational effects are independent of the electric charge, these rotational phenomena emerge even for neutral particles while magnetic phenomena arise only for charged particles.
Indeed, in nonrelativistic theory, the exact correspondence between magnetism and rotation is known [12,13]. By rotating an atomic gas in a harmonic trap very rapidly, we can experimentally realize this correspondence [14,15,16]. If this correspondence is achieved, the Landau quantization and other magnetic phenomena, such as quantum Hall effect, are expected to be observable in rotating media. On the other hand, the relation between magnetism and rotation is not clear in relativistic theory. Since relativistic theory is more fundamental than nonrelativistic theory, which is derived by taking the nonrelativistic limit of relativistic theory, the nonrelativistic correspondence must be potentially included in relativistic theory. If we can find some correspondence, we can advocate novel phe-nomena in relativistic rotating matters, which are analogous to magnetic phenomena.
In this Letter, we discuss the analogy between magnetism and rotation in relativistic theory. As reviewed in the next section, the exact correspondence in nonrelativistic theory is obtained only after applying an external trapping potential. Based on this, we analyze rotating systems with external trapping potentials in relativistic theory and discuss the quantization of the energy spectrum.

Nonrelativistic theory
In this section, we review the established relation between magnetism and rotation in nonrelativistic theory.
As the simplest example, let us illustrate the correspondence in classical mechanics. A charged particle in a magnetic field receives the Lorentz force On the other hand, a rotating particle feels two apparent forces, i.e., the Coriolis force and the centrifugal force, respectively. The Coriolis force is mathematically equivalent to the Lorentz force under the replacement eB ↔ 2mΩ. Except for the centrifugal force, therefore, we see the correspondence between magnetism and rotation. In other words, the crucial point for the correspondence is how to eliminate the centrifugal force. Also in nonrelativistic quantum mechanics, it can be shown that the centrifugal force is the difference between magnetism and rotation. Suppose a magnetic field and an angular velocity along the z-axis, B = Bẑ and Ω = Ωẑ. The Hamiltonian of a charged particle in a magnetic field is given by with the symmetric gauge A = (−By/2, Bx/2, 0). The energy spectrum is quantized as which is called the Landau quantization. On the other hand, the Hamiltonian for a particle in a rotating frame reads Also here, the correspondence between the kinetic terms in Eqs. (3) and (5) is confirmed. However, the second term in Eq. (5) is missing in Eq. (3). As in classical mechanics, this superfluous term is related to the centrifugal force. Indeed, we obtain the centrifugal force in Eq. (2) from This centrifugal force potential creates the different physical situation from that of magnetism with respect to homogeneity. The presence of the centrifugal force potential gives the inhomogeneity in rotating frames, while systems under external magnetic fields are homogeneous. If we can eliminate the centrifugal force potential by applying the external trapping potential by hand, we can observe the Landau quantized energy spectrum [15] Therefore, when the centrifugal force potential and the external trapping potential cancel out, magnetism and rotation is equivalent with the correspondence eB ↔ 2mΩ.

Relativistic theory
We consider the relativistic scalar field theory in the cylindrical coordinate x µ = (t, r, θ, z). The solutions for the following equations are written as where the z-axis is the direction of B and Ω.
In the presence of a magnetic field, the Klein-Gordon equation is We take eB > 0. Substituting Eq. (10), we obtain The solution of this equation is known as the Landau wave function Φ(r) = r ℓ e −eBr 2 /4 L ℓ n (eBr 2 /2), where L k n (x) is an associated Laguerre polynomial. The corresponding energy eigenvalue, i.e., the Landau energy level, is given by The Landau levels depends only on the radial quantum number n and not on the azimuthal quantum number ℓ, and so the Landau states with different ℓ are energetically degenerate. The normalizability of the wave function (13) restricts the range of ℓ from −n to ∞.
In relativistic theory, rotation is described as spacetime geometry through the corresponding metric tensor. In a curved spacetime with g µν , the Klein-Gordon equation is where V (r) is an external potential. In a rotating frame, the metric tensor is We take Ω > 0. Substituting Eqs. (10) and (16) to Eq. (15), we obtain the equation (17) This equation can be also derived by replacing the energy ε on the flat Minkowski space with ε+ℓΩ. For V (r) = 0, the theory has been investigated in detail about the vacuum state and thermal distribution [17,18,19,20,21]. The solution is [17] with the energy In a rotating frame, there is a singularity at r = r max ≡ 1/Ω (light cylinder) and the frame moves faster than light in r > r max . Particle spectrum is pathological there (see Refs. [19,20] for more detailed discussion). We implicitly assume the region inside the light cylinder and do not discuss this problem.
In order to compare rotation with magnetism, we analyze the energy spectrum of Eq. (17) in three cases of the external potential V (r).
(I) Based on the discussion of the nonrelativistic case, we consider the harmonic trapping potential The solution of Eq. (17) is with the energy spectrum The energy spectrum depends on both of n and ℓ. This is quit different from magnetism. For the nonrelativistic expansion, we explicitly show the speed of light c: m = mc 2 , Ω = Ω/c and p z = p z c. The nonrelativistic energy is then given as Up to the order of 1/c, we reproduce the nonrelativistic Landau quantization (8). Other terms are higher-order relativistic corrections. The relativistic rotation with Ω mc 3 does not give the nonrelativistic Landau quantization because the higher-order correction cannot be negligible.
(III) If we drop the ℓ-dependent potential in Eq. (24), and apply only the trapping potential then the solution is the same as Eq. (27), but the energy spectrum is modified as There are four branches of the dispersion relation. Such a multi-branch structure is explained by the Klein-Gordon equation (17). For V (r) = 0, since the energy appears only in the term (ε+ℓΩ) 2 , there are two branches corresponding to the sign of ε + ℓΩ, as shown in Eq. (19). On the other hand, since Eq. (25) includes not only (ε+ℓΩ) 2 but also ε 2 , there are four branches corresponding the signs of ε + ℓΩ and ε. This is related to whether to regard φ(x) as a particle or an antiparticle. For example, even if φ(x) in the rotating frame is considered as a particle (i.e. ε > 0), there are both cases that φ(x) in the inertial frame is regarded as a particle (i.e. ε + ℓΩ > 0) and as an antiparticle (i.e. ε + ℓΩ < 0) [17]. The ℓ-dependence in Eq. (30) plays a similar role to the spin-dependence of a charged particle under magnetic fields, Thus, large-ℓ modes are tachyonic like the Nielsen-Olesen unstable modes [22]. We note that both the multi-branch dispersion and the instability result from the superfluous term of rotation, i.e. ℓ 2 Ω 2 , as shown in Eqs. (28) and (30).

Summary
We investigated the correspondence between magnetism and rotation in relativistic theory. We discussed the Landau quantization by nonrelativistic rotation and evaluated the relativistic correction to it. Considering the energydependent trapping potential, we found the Landau-like quantization by relativistic rotation.
In this Letter, we discussed the scalar field theory for simplicity. The analysis can be extended to higher spin fields. For example, a rotating spin-1/2 fermion is described by the Dirac equation in a rotating frame. The rotating fermion should have the spin-rotation coupling term, which corresponds to the Zeeman term in magnetism. The spin-rotation coupling can be derived by the Dirac equation, although it depends on the choice of vierbein [23,24]. Starting with relativistic rotation, we can also derive the dispersion relation including the nonrelativistic spinrotation coupling.
If the system has a self-interaction and a chemical potential, the scalar condensate can be nonzero, φ = 0. In the absence of external potentials, the particle spectrum in a rotating frame is trivially obtained by the coordinate transformation from the Minkowski space [19]. The scalar condensate does not change in this transformation. In the presence of external potentials, however, the scalar condensate can change. The same is true for fermion-antifermion condensates. Since rotation couples to all kinds of fields, even to charge-neutral fields, we can universally expect such "rotational catalysis."