Lorentz quantum mechanics

We present a theoretical framework for the dynamics of bosonic Bogoliubov quasiparticles. We call it Lorentz quantum mechanics because the dynamics is a continuous complex Lorentz transformation in complex Minkowski space. In contrast, in usual quantum mechanics, the dynamics is the unitary transformation in Hilbert space. In our Lorentz quantum mechanics, three types of state exist: space-like, light-like and time-like. Fundamental aspects are explored in parallel to the usual quantum mechanics, such as a matrix form of a Lorentz transformation, and the construction of Pauli-like matrices for spinors. We also investigate the adiabatic evolution in these mechanics, as well as the associated Berry curvature and Chern number. Three typical physical systems, where bosonic Bogoliubov quasi-particles and their Lorentz quantum dynamics can arise, are presented. They are a one-dimensional fermion gas, Bose–Einstein condensate (or superfluid), and one-dimensional antiferromagnet.


Introduction
Bosonic Bogoliubov quasiparticles arise in many different physical systems [1,2]. They have been studied extensively in condensed matter physics for their static properties, such as dispersion, and in particular, their relation to superfluidity [3][4][5]. Inspired partly by the work in [6], where the dynamics of Bogoliubov quasiparticles in a superfluid with a vortex is studied, we present here a general theoretical framework for such dynamics. As the bosonic Bogoliubov operator is non-Hermitian, we find that the dynamics is a continuous Lorentz transformation of a state in complex Minkowski space. In contrast, the usual quantum dynamics is a continuous unitary transformation of a state in Hilbert space. For this reason, we call the dynamics of bosonic Bogoliubov quasiparticles Lorentz quantum mechanics.
In Lorentz quantum mechanics, we find that the interval of a state is conserved and therefore the complex Minkowski space has three subspaces: space-like, light-like, and time-like, which are invariant during the dynamic evolution. In this work we focus on the 1, 1 ( )-type spinor, the simplest Lorentz spinor, and use this example to explore in which ways the Lorentz quantum mechanics are similar to, and different from, the conventional quantum mechanics. In particular, we construct the matrix representing the Lorentz transformation of complex vectors, and the Lorentz counterpart of the standard Pauli matrices. The Berry phase is also investigated in the context of Lorentz quantum mechanics and it is found to be quite different from the Berry phase in usual quantum mechanics.
In the end, we give three specific physical systems: the spin wave excitations in a one-dimensional (1D) antiferromagnetic system, the phonon excitations on top of a vortex in the Bose-Einstein condensate (BEC), and a 1D fermion gas at low temperatures, where Lorentz quantum mechanics can arise. We use these systems to further illustrate our general results. In particular, with the antiferromagnetic system we point out explicitly how spin-orbit coupling can arise in Lorentz quantum mechanics.
We note that the non-Hermitian Hamiltonian has been extensively studied in the context of PT-symmetric quantum mechanics, where the spectrum (eigenvalue) of non-Hermitian operator is proved to be real [7]. The PT-symmetric structure has found extensive applications in phonon-laser (coupled-resonator) systems, where giant nonlinearity arises in the vicinity of phase transition between PT-symmetric phase and broken-PT phase, resulting in enhanced mechanical sensitivity [8], optical intensity [9], controllable chaos [10] and optomechanically-induced transparency [11], as well as the phonon-rachet effect [12]. The geometric phase of PT-symmetric quantum mechanics [13] and the stability of driving non-Hermitian system has also been studied [14]. The bosonic Bogoliubov operator studied here stands for a class of generalized PT-symmetric Hamiltonian [15], or more precisely, the anti-PT Hamiltonian [16], which can be realized experimentally by making use of refractive indices in optical settings [16,17]. It will be interesting to examine the general theoretical framework of these PT-symmetric quantum mechanics in the future.

Basic structures of Lorentz quantum mechanics
The Lorentz quantum mechanics is described by the following dynamical equation where H H = † is a Hermitian matrix while m n , s is given by Equations of this type are usually called Bogoliubov-de Gennes (BdG) equations and are obeyed by bosonic quasi-particles in many different physical systems (see section 4). For simplicity, we use the case 1,1 s to explore the basic structures of the Lorentz quantum mechanics as generalization to m n , s is straightforward.

Complex Lorentz transformation and complex Minkowski space
Here a(t) and b(t) are the standard bosonic Bogoliubov amplitudes, H H = † is a Hermitian matrix, and For an arbitrary initial state Here t, 0 ( )is the evolution operator defined by t, 0 e . 6 Ht The goal of this section is to show that the operator t, 0 ( )defined in equation (6) generates a complex Lorentzinstead of a unitary-evolution of t y ñ | ( ) . In particular, defining the interval for a Lorentz spinor we prove below that the interval is conserved under the evolution generated by t, 0 For the above purpose, we first establish the following relation from which equation (9) ensues. Hence, by virtue of equation (9), we obtain and thus equation (8). Similarly, we can show that 1   ¹ † and t t y y á ñ ( )| ( ) is not conserved during the dynamical evolution.
It is clear from the above results that the vector space spanned by states t y ñ | ( ) is not a Hilbert space and the evolution operator  is not a unitary transformation. Due to its mathematical similarity to the Lorentz transformation in Minkowski space, we call the vector space spanned by states t y ñ | ( ) complex Minkowski space and any operator satisfying equation (9) Lorentz transformation. For the 1, 1 ( )-spinor, the general matrix form of the Lorentz transformation is , with the corresponding inverse Lorentz matrix being To avoid any confusion, we reiterate that our Lorentz transformation is a mathematical generalization of the Lorentz transformation of special relativity to complex numbers. Physically, they are very different; our Lorentz transformation operates on states of bosonic Bogoliubov quasiparticles which form a complex Minkowski space; the Lorentz transformation in special relativity operates on space-time which is a real Minkowski space.
As the interval defined in equation (8) does not change under Lorentz transformation, the complex Minkowski space where the states of bosonic Bogoliubov quasiparticles reside in has three subspaces up to the normalization constant, which are defined by a b a b In , 0 describe the corresponding anti-particles [5]. We thus conclude that the dynamical evolution generated by t, 0 ( )conserves the interval (see equation (8)), and therefore, is a continuous complex Lorentz transformation in complex Minkowski space.
Before we proceed to explore other properties of this Lorentz quantum dynamics, we take a sidestep to point out that the BdG equation is a special class of PT-symmetric quantum mechanics [15,[18][19][20][21]. The general form of two-mode PT-symmetric Hamiltonian has been written as [15,[18][19][20][21] H cos i sin sin i cos e sin i cos e cos i sin , while the Hermitian Hamiltonian H recovers when 0 m n = = .

Eigen-energies and eigenstates
Although the H  ( ) , and we denote the two real eigenvalues as E 1 and E 2 , with the corresponding eigenstates labeled as 1ñ | and 2ñ | , respectively. Two facts are clear from equation (17) We now describe the basic properties of the eigenstates associated with the operator H This means that if 1ñ | is space-like then 2ñ | is time-like or vice versa. In the energy representation defined in terms of 1ñ (3)) can be written as  (16) forms the surface of a cone. As will be discussed in section 3, the charge (monopole) for the Berry curvature (monopole) is at the tip of the cone rather than distributing over the whole degeneracy cone.
In transforming t y ñ | ( ) from the Bogoliubov representation to the energy representation, the interval of the Lorentz spinor is preserved, i.e. it is a complex Lorentz transformation. To see this, using equation (21) By further assuming a gauge for Lorentz-like normalization, i.e.
meaning the interval is conserved for the above representation transformation. The normalization condition u v 1 could not maintain its ordinary amplitude, such that a t b t 1 2 2

Representation transformation and physical meaning of the wavefunction
In quantum mechanics, the change from one representation to another (or from one basis to another) is given by a unitary matrix. In Lorentz quantum mechanics, the representation transformation is facilitated by Lorentz matrix L in equation (13). Correspondingly, an operator K transforms as Note that, since L is not a unitary matrix, we have 1 ¹ - † L L. As an example, we consider the transformation from the Bogoliubov representation to the energy representation as described earlier. In this case, the eigenstates 1ñ | and 2ñ | transform as Obviously, as we have proven, the interval must be conserved, i.e., u v 1 0 1 In addition, the bosonic Bogoliubov operator transforms as, This special Lorentz transformation from the original representation to the energy representation is just the well-known Bogoliubov transformation for bosons [5,22,23].
In light of the conservation of interval-rather than norm-of the state vector under transformations, a question immediately arises as to whether, or to what extent, the wavefunction in the context of Lorentz quantum mechanics still affords the physical interpretation as the probability wave? Indeed, in the energy representation, see equation (23), it is clear that c 1 | | must be taken.

Analogue of Pauli matrices
In analogy with the conventional spinor that is acted by the basic operators known as Pauli matrices, it is natural to ask, for the Lorentz spinor, if similar matrices can be constructed. Such analogue of the Pauli matrices, denoted by i t (i 1, 2, 3 = ), is required to fulfill the following conditions: (i) any operator H must have real-number components n i ; (ii) the matrices i t (i 1, 2, 3 = ) should have the same real eigenvalues, say, ±1, and can transform into each other via Lorentz transformation (see equation (28)).
Based on (i) and (ii), we see that the matrices as appeared in equation (16) do not represent the analogue of the Pauli matrix for the Lorentz spinor: while they satisfy the requirement (i), the condition (ii) is violated. Instead, we consider the following constructions: It is easy to check that i t in equation (37) satisfy both requirements (i) and (ii). In particular, the transformation between 1 t and 3 t is explicitly found to be

Heisenberg picture
The current Lorentz evolution is in fact defined in the analogue of Schrödinger picture (denoted by subscript s), i.e. any physical operator keeps constant while the wavefunction undergoes Lorentz evolution. In analogy with the conventional spinor, the Lorentz quantum mechanics can also be expressed in the analogue of Heisenberg picture (denoted by subscript h).  s ( ), which depends on a set of system's parameter R. Suppose the spinor is initially in an eigenstate, say 1ñ | , before the parameter R undergoes a sufficiently slow variation, thus driving an adiabatic evolution for the Lorentz spinor. The relevant matrix element capturing the slowly varying time-dependent perturbation can be evaluated as, by acting the gradient operator R  º ¶ ¶ on the equation (19) and using equation (20), Here, the last equality is ensured by the real eigenvalues in the considered parameter regimes, together with the condition E E 1 2 ¹ .
We see that the relation (49), except for an additional 1,1 s , is identical with that in unitary quantum mechanics [24]. This allows us to generalize the familiar adiabatic theorem to the context of Lorentz quantum mechanics; starting from an initial eigenstate R 1 ñ | ( ) ( R 2 ñ | ( ) ), the system will always be constrained in this instantaneous eigenstate so long as R is swept slowly enough in the parameter space. (A rigorous proof would be similar to that in the conventional quantum mechanics [24,25], and therefore, here we shall leave out the detailed procedure.)

Analogue of Berry phase
In conventional quantum mechanics, it is well-known that an eigen-energy state undergoing an adiabatic evolution will pick up a Berry phase [26], when a slowly varying system parameter R realizes a loop in the parameter space. Here we show that in the context of Lorentz quantum mechanics, a Lorentz counterpart of the Berry phase will similarly arise.
The time evolution of an instantaneous eigenstate, which is parametrically dependent on R, can be written as m e e , 5 0 Equations (53) and (54) show that the Berry connection in the Lorentz quantum mechanics is modified from the conventional one, where the Berry connection is given by m m i R á ñ ¶ ¶ | | . Will such modifications give rise to a different monopole structure for the Berry curvature? Or, will the monopole in the Lorentz mechanics still occur at the degeneracy point (where E E . By acting ∇ on both sides, we obtain ( ) , which, as shown in figure 1, forms a circular cone. There, imagine the path of m m m R , , = ( ) realizes a loop in the vicinity of the cone's surface. In this case, the instantaneous eigenstate, say, R 1 ñ | ( ) , is expected to vary in a back-and-forth manner (dropping the overall phases including both the dynamical and Berry phase). This is because the instantaneous eigenstate, apart from an overall phase, is always the same along any straight line emanating from the origin. As a result, the integration of A 1 along this loop vanishes, meaning there is no charge of the Berry curvature on the cone's surface, even though it is in the degeneracy regime. We thus conclude that-just as in the case of unitary spinor-the charge, if exists, can only be distributed on the isolated points, i.e., the original monopole O, in m m m R , , = ( )space. However, different from unitary spinor, the magnetic flux does not uniformly emanate from the monopole O to the parameter space, instead, it emanates only to the region in the cone (more closer to the m 3 axis). In addition, even in this region, the magnetic flux is not uniformly distributed. Specifically, by evaluating the geometric phase along a loop perpendicular to the m 3 axis, we can find the distribution of the magnetic flux density per solid angle as a function of the angle θ from m 3 axis, i.e., with -+ associated with the state 1ñ | ( 2ñ | ). Note that the flux density is proportional to the Berry curvature, which acts as a magnetic field, whose magnitude according to equation (63) increases when approaching the cone. Right on the surface of the cone, where 4 q  p , the magnetic field diverges. Outside the cone, on the other hand, the eigenvalue becomes complex such that the notion of adiabatic evolution and geometric phase become meaningless, i.e. there is no magnetic field emanating outside the cone from the monopole O. Again, due to the aforementioned fact that the instantaneous eigenstate (apart from an overall phase) remains the same along any straight line emanating from the origin, we expect all the magnetic field fluxes to be described by straight lines (see figure 3). Alternatively, we can write H 1,1 s in terms of the analogues of Pauli's matrices i t (see equation (36)), which is then mapped onto a vector n n n , , 1 2 3 ( ) in the parameter space. However, this equivalent kind of decomposition will not contribute anything but modify the slope of Berry curvature q q with C being any constant).

Chern number
The Chern number-which reflects the total magnetic charge contained by the monopole on O-can be calculated from equation (63) as, with -+ for the state 1ñ | ( 2ñ | ). Hence, the Lorentz spinor not only has distinct distribution of the magnetic flux compared to the unitary spinor, both also possesses unexpectedly the qualitatively different Chern number which is divergent. ( ) space as parameterized in equation (16). θ introduced in equation (63) is the angle spanned by m 3 axis and direction of Berry curvature under study. There is no magnetic flux outside of the cone, in the cone the magnetic field becomes stronger as approaching the cone's surface and tends to infinity on the surface. Because the flux density assumes the axial symmetry about the m 3 axis, the two dimensional plot is depicted for clarity.

Physical examples
In previous sections, we have developed and studied the Lorentz quantum mechanics for the simplest Lorentz spinor. Such a Lorentz spinor can arise in physical systems containing bosonic Bogoliubov quasiparticles, for example, in BECs [5]. Specifically, we illustrate our study of Lorentz quantum mechanics by investigating a 1D fermion gas at low temperatures, phonon excitations on top of a vortex in the BEC, and spin wave excitations in a 1D antiferromagnetic system. 4.1. One-dimensional Fermi gas As the first illustrative example, we investigate the fermion excitations in a 1D fermion gas at low temperatures. Since excitations dominantly occur for fermions near the Fermi surface (note at 1D, the Fermi surface shrinks to the left (L) and right (R) Fermi points), the corresponding Hamiltonian can then be written as