Lorentz quantum mechanics

We present a theoretical framework called Lorentz quantum mechanics, where the dynamics of a system is a 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, there exist three types of states, space-like, light-like, and time-like. Fundamental aspects are explored in parallel to the usual quantum mechanics, such as matrix form of a Lorentz transformation, construction of Pauli-like matrices for spinors. We also investigate the adiabatic evolution in this mechanics, as well as the associated Berry curvature and Chern number. Three typical physical systems, where this Lorentz quantum dynamics can arise, are presented. They are one dimensional fermion gas, Bose-Einstein condensate (or superfluid), and one dimensional antiferromagnet.

for a class of generalized PT symmetric Hamiltonian [12], or more precisely, the anti-PT Hamiltonian [13], which can be realized experimentally by making use of refractive indices in optical settings [13,14].

II. 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 σ m,n is given by Equations of this type are usually called Bogoliubov-de Gennes (BdG) equations and are obeyed by bosonic quasiparticles in many different physical system (see Sec. IV). For simplicity, we use the case σ 1,1 to explore the basic structures of the Lorentz quantum mechanics as generalization to σ m,n is straightforward. The BdG equation for spinor (1,1) is Here a(t) and b(t) are the standard Bogoliubov amplitudes, H = H † is a Hermitian matrix, and σ 1,1 = σ z is the familiar Pauli matrix in the z direction, i.e.

A. Complex Lorentz transformation and complex Minkowski space
Suppose the wavefunction's dynamics is governed by the BdG equation (3), then for an arbitrary initial state |ψ(0) = [a(0), b(0)] T , the wavefunction |ψ(t) = [a(t), b(t)] T at times t > 0 can be solved formally from Eq. (3) as Here U(t, 0) is the evolution operator defined by The goal of this section is to show that the operator U(t, 0) defined in Eq. (8) generates a complex Lorentz -instead of a unitary -evolution of |ψ(t) . In particular, defining the interval for a Lorentz spinor we prove below that the interval is conserved under the evolution generated by U(t, 0), i.e.
For above purpose, we first establish the following relation, Expanding σ 1,1 U and (U † ) −1 σ 1,1 in Taylor series, and noting σ 1,1 σ 1,1 = 1, the nth term in the expansions of both σ 1,1 U and (U † ) −1 σ 1,1 are of the form This readily gives from which Eq. (11) ensues. Hence, by virtue of Eq. (11), we obtain and thus Eq. (10). In fact, the normalization of Lorentz-kind Eq. (10) has been extensively demonstrated in non-Hermitian quantum mechanics (see [11] for an example). Why can we refer to the equation (10) as an analogue of Lorentz transformation? Since what we are focusing is the two-mode wavefunction, we can demonstrate this by the two dimensional space-time spanned by (x, t). In special relativity, the interval x 2 − t 2 (in natural units c = 1) for a given inertial frame keeps a constant after the Lorentz boost to any other inertial frame. Here because x and t are both real numbers, x 2 − t 2 = |x| 2 − |t| 2 . The vector (x, t) are called space-like, light-like and time-like as x 2 − t 2 > 0, x 2 − t 2 = 0 and x 2 − t 2 < 0, respectively.
For the current two-mode wavefunction (a, b) T , we can map the first component a as x and the second one b as t. Thus the interval-like quantity |a| 2 − |b| 2 can be accordingly defined. Because a and b may be complex numbers, the notion of modulus is necessary to define the interval. Since we have proven that, during the evolution determined by BdG equation, Eq. (10) holds, we can call this evolution as the Lorentz-like evolution, or complex Lorentz evolution. In analogy with the real Lorentz transformation, we consider that (a, b) T is space-like, light-like and time-like as In((a, b) T ) = |a| 2 − |b| 2 > 0, In((a, b) T ) = |a| 2 − |b| 2 = 0 and In((a, b) T ) = |a| 2 − |b| 2 < 0, respectively. While Eq. (10) formally resembles the conventional Lorentz evolution (transformation) in special relativity, there are delicate differences: (i) in contrast to the conventional Lorentz transformation where only real numbers (spacetime coordinate) are involved, here we are dealing with a complex vector specified by complex numbers, the interval of which requires the notion of modulus (in this sense, we shall refer to the space where these complex vectors reside as the complex Minkowski space); (ii) unlike the real Minkowski space where x(t) must be a space-like (time-like) component, here a freedom is left as we define the space-like axis and time-like one, i.e., we can either define a(b) as the space-like (time-like) component or time-like (space-like) component. Thus whether a wavefunction (a, b) T is space-like or time-like is totally determined by how we define the space-like and time-like component. However, this does not constitute a problem as we can always fix our convention once the definition is determined.
We thus conclude that the evolution generated by U(t, 0) conserves the interval [see Eq. (10)], and therefore, represents a complex Lorentz evolution.

B. Eigen-energies and eigenstates
Although the σ 1,1 H is not Hermitian, under certain conditions, it can admit real eigenvalues -which are relevant for physical processes. We write σ 1,1 H in terms of three basic matrices as (dropping the term involving the identity matrix) where the parameters m i (i = 1, 2, 3) are real. The eigen-energies are the roots of the following equation It is clear that the eigenvalues are real provided the condition is satisfied. In this work, we shall restrict ourselves to this physically relevant regime of real-eigenvalues in the parameter domain specified by (m 1 , m 2 , m 3 ), and we denote the two real eigenvalues as E 1 and E 2 , with the corresponding eigenstates labeled as |1 and |2 , respectively. Two facts are clear from Eq. (16): (i) in the parameter space (m 1 , m 2 , m 3 ), the two eigenstates |1 and |2 exhibit degeneracies on a circular cone (see Fig. 1), which resembles the light-cone in special relativity. This is in marked contrast to a unitary spinor, where the degeneracy occurs only at an isolated point; (ii) unlike a unitary spinor where the constant-energy surfaces are elliptic surfaces, both eigenstates of σ 1,1 H display hyperbolic constant-energy surfaces (see Fig. 2).
We now describe the basic properties of the eigenstates associated with the operator σ 1,1 H. They are solutions to the following eign-equations Keeping in mind that only real eigenvalues are considered, for It can be checked that the two eigenstates of σ 1,1 H can always be specifically expressed as This means that if |1 is space-like then |2 is time-like or vice versa. In the energy representation defined in terms of |1 and |2 , a time-evolved state |ψ(t) = [a(t), b(t)] T [see Eq. (3)] can be written as In transforming |ψ(t) 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 Eq. (20), we find By further assuming a gauge for Lorentz-like normalization, i.e., we obtain from (23) that meaning the interval is conserved for the above representation transformation. The normalization condition |u| 2 −|v| 2 = 1 is different from the eigenstates of a conventional unitary spinor. In fact, if one naively enforce the unitary gauge on Eq. (21), say, |u| 2 + |v| 2 = 1, unphysical consequences would ensue: The time-evolved wavefunction in the original Bogoliubov representation [|ψ = (a, b) T ] could not maintain its ordinary amplitude, such that |a(t)| 2 + |b(t)| 2 = 1 for t > 0, and, in particular, the amplitude in different representation would take different value, e.g., |c 1 | 2 + |c 2 | 2 = |a(t)| 2 + |b(t)| 2 , which can be easily inferred from Eq. (22).

C. Representation transformation and physical meaning of the wavefunction
In the usual quantum mechanics, the change from one representation to another (or from one basis to another) is given by a unitary matrix. As discussed above, the change from the Bogoliubov representation to the energy representation [see Eq. (22) and Eq. (26)] is facilitated by a Lorentz transformation. This motivates us to introduce a complex Lorentz operator L acting on the Lorentz (1, 1)-spinor, defined by where |x| 2 − |y| 2 = 1, with the corresponding inverse Lorentz matrix being Using the identity L † σ 1,1 L = σ 1,1 , it is readily to see that both L and L −1 are Lorentz matrices.
For an arbitrary Lorentz matrix, we have, meaning the interval is preserved. Under a Lorentz transformation, an arbitrary physical operator K transforms as while the corresponding eigenvalues stay unchanged. Note that, since L is no longer a unitary matrix, we have To illustrate the above constructions, consider the transformation from the Bogoliubov to the energy representation as described earlier. In this case, the eigenstates |1 and |2 transform as where the matrix L B is shown in Eq. (27), with x = u * and y = −v, i.e., Because now |x| 2 − |y| 2 = |u| 2 − |v| 2 = 1, L B , as shown in Sec. IIC, must be a Lorentz matrix. Obviously, as we have proven, the interval must be conserved, i.e., |u| 2 − |v| 2 = 1 2 − 0 2 = 1, |v * | 2 − |u * | 2 = 0 2 − 1 2 = −1. In addition, the Bogoliubov operator transforms as, This special Lorentz transformation from the original representation to the energy representation is in fact equivalent to the Bosonic Bogoliubov transformation. This has been studied in the Swanson Hamiltonian [19], where it is found that the energy eigenstates can be constructed from the algebra and states of the harmonic oscillator and transition probabilities governed by the non-Hermitian Swanson Hamiltonian are shown to be manifestly unitary. For a time-dependent Swanson Hamiltonian [20], time-dependent Dyson and quasi-Hermiticity relation is demonstrated clearly.
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 Eq. (22), it is clear that |c 1(2) | 2 , with |c 1 | 2 + |c 2 | 2 = 1, can be interpreted as the probability of finding the spinor in the eigenstate |1(2) , i.e., a wavefunction c 1 |1 + c 2 |2 still describes a probability wave. However, in the Bogoliubov representation, the interpretation of a wavefunction as the probability wave is no longer physically meaningful. For example, consider the eigenstate |1 = (u, v) T , which is usually generated from creating a pair of Bogoliubov quasiparticles in the ground state of the system. Yet, |u| 2 and |v| 2 cannot represent the probabilities in the Bogoliubov basis: the Bogoliubov basis is not a set of orthonormal basis (see Sec. IV for concrete examples), and therefore, instead of |u| 2 + |v| 2 = 1, the convention |u| 2 − |v| 2 = 1 must be taken.

D. Completeness of eigenvectors
or, equivalently, Here, the notation j [for (1 + 1)-mode] is defined by It can be found easily that, ensured by the property of Lorentz matrix L † σ 1,1 L = σ 1,1 , the completeness expression (35) (or (36)) remains in any other representation.

E. 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 τ i (i = 1, 2, 3), is required to fulfill the following conditions: (i) any operator σ 1,1 H, when written in terms of τ i (dropping the term involving identity matrix), i.e., σ 1,1 H = n 1 τ 1 + n 2 τ 2 + n 3 τ 3 , must have real-number components n i ; (ii) the matrices τ i (i = 1, 2, 3) should have the same real eigenvalues, say, ±1, and can transform into each other via Lorentz transformation [see Eq. (30)]. Based on (i) and (ii), we see that the matrices as appeared in Eq. (15) 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 following constructions: It is easy to check that τ i in Eq. (39) satisfy both requirements (i) and (ii). In particular, the transformation between τ 1 and τ 3 is explicitly found to be where L is of the form (27)  i, and that between τ 2 and τ 3 is given by

F. 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). The relations of an operator O and the state |ψ between the two pictures are, where |ψ h keeps constant but O(t) h satisfies the analogue of Heisenberg equation,
In addition, the orthogonal condition for two non-degenerate eigenstates is derived as, j|σ m,n |k = 0, for j = k, Consider a (1, 1)-spinor described by the operator σ 1,1 H(R), which depends on a set of system's parameter R. Suppose the spinor is initially in an eigenstate, say |1 , 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 on the Eq. (18) and using Eq. (19), Here, the last equality is ensured by the real eigenvalues in the considered parameter regimes, together with the condition E 1 = E 2 . We see that the relation (51), except for an additional σ 1,1 , is identical with that in unitary quantum mechanics [21]. This allows us to generalize the familiar adiabatic theorem to the context of Lorentz quantum mechanics: Starting from an initial eigenstate |1(R) (|2(R) ), 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 [21,22], and therefore, here we shall leave out the detailed procedure.)

B. 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 [23], 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 with m = 1, 2. Here, −´E m (R)dt/ denotes the dynamical phase and β the geometric phase. Substituting Eq. (52) into Eq. (3), we find and From Eqs, (53) and (54), we can readily read off the Berry connections as Equations (55) and (56) show that the Berry connection in the Lorentz quantum mechanics is modified from the conventional one, where the Berry connection is given by i m| ∂ ∂R |m . 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 1 = E 2 )? To address these questions, we now calculate the Berry curvature B = ∇ × A. Without loss of generality, we take the eigenvector |1 for concrete calculations.
Our starting point is the identity 1|σ 1,1 |1 = 1. By acting ∇ on both sides, we obtain This indicates that 1|σ 1,1 ∇|1 is purely imaginary (A 1 is real). Hence, B 1 can be evaluated as, where I m represents the imaginary part. In deriving Eq. (58), we have used the completeness relation (35) and the following relation valid for arbitrary scalar µ and vector b. According to Eq. (51), B 1 in Eq. (58) is well defined provided E 1 = E 2 , such that the monopole is expected to be absent in this case. To rigorously establish this, let us calculate the divergence of the Berry curvature, i.e. ∇ · B 1 . Introducing an auxiliary operator, which is Hermitian, F = F † , as ensured by the completeness relation (35), we have In deriving above, we have used Eq. (59). Further noting that i j|F|k = j j|σ 1,1 |∇j j |σ 1,1 |k = j|σ 1,1 ∇|k , the Berry curvature can be expressed in terms of F as Finally, by virtue of ∇ × F in Eq. (61), we find Therefore, as expected, the monopole in the Lorentz quantum mechanics can only appear in the degenerate regime where B 1 diverges, similar as the conventional unitary quantum mechanics.
Next, searching for the monopole, we focus on the degeneracy regime in the parameter space defined by (m 1 , m 2 , m 3 ), which, as shown in Fig. 1, forms a circular cone. There, imagine the path of R = (m 1 , m 2 , m 3 ) realizes a loop in the vicinity of the cone's surface. In this case, the instantaneous eigenstate, say, |1(R) , is expected to vary in a backand-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 R = (m 1 , m 2 , m 3 ) 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 |1 (|2 ). Note that the flux density is proportional to the Berry curvature, which acts as a magnetic field, whose magnitude according to Eq. (64) increases when approaching the cone. Right on the surface of the cone, where θ → π 4 , 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 Fig. 3).
Alternatively, we can write σ 1,1 H in terms of the analogues of Pauli's matrices τ i [see Eq. (38)], which is then mapped onto a vector (n 1 , n 2 , n 3 ) in the parameter space. However, this equivalent kind of decomposition will not contribute anything but modify the slope of Berry curvature θ → θ (tan(θ) = 1/C, while tan(θ ) = 1/(C − √ 2), with C being any constant).

C. Chern number
The Chern number -which reflects the total magnetic charge contained by the monopole on O -can be calculated from Eq. (64) as, with −/+ for the state |1 (|2 ). Hence, the Lorentz spinor not only has distinct distribution of the magnetic flux compared to the unitary spinor, but also possesses unexpectedly the qualitatively different Chern number which is divergent.

IV. 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 Bose-Einstein condensates(BECs) [2]. 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.

A. One dimensional Fermi gas
As the first illustrative example, we investigate the fermion excitations in a one dimensional 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 [24]  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 m3 axis, the two dimensional plot is depicted for clarity.
Here, the operator a † sq (a sq ) creates (annihilates) an excited fermion near the Fermi point (s = R, L) with momentum q (measured with respect to the ground state value). In addition, κ s = 1, −1 for s = R/L,s = L/R, v F labels the fermi velocity, and ρ sq = k a † sk+q a sk is the density operator in the momentum space representation. In writing down Eq. (66), we have taken into account the interactions between two fermions. Specifically, g 2 denotes the strength of interaction between two fermions near opposite Fermi points (i.e. q 2k F ), while g 4 for those close to the same Fermi point (i.e. q 0).
Let |0 denote the state of perfect Fermi sphere (a Fermi line in one dimensional case). A generic state describing density fluctuations near the Fermi points can then be written in terms of a pseduo-spinor as where l is the size of the system. As discussed in Ref. [24], the density operators ρ sq can be effectively treated as bosonic operators within the approximation [ρ sq , ρ s q ] 0[ρ sq , ρ s q ]|0 .
By assuming Eq. (68), it is found that Eq. (67) represents a Lorentz spinor whose dynamics is governed by the BdG equation below i d dt The generator σ 1,1 H of the dynamics in Eq. (69), when written in form of Eq. (15), corresponds to m 1 = g 2 q/(2π), m 2 = 0 and m 3 = v F q + g 4 /(2π)q. Thus, when v F + g4 2π ≥ g2 2π [see Eq. (17)], the σ 1,1 H exhibits real eigenvalues, and has a space-like and a time-like eigenvectors. Due to m 2 = 0, as illustrated in Fig. 3, there is no magnetic flux penetrating a loop in the plane defined by (m 1 , m 3 ). As a result, the Berry phase picked up by the eigenstate, say |1(R) , is always zero when R varies along a loop in the parameter space of (m 1 , m 3 ). According to our theory, it is impossible to implement a geometric force (vector potential or artificial magnetic field) to any fermions in the one dimensional Fermi gas. We must search for other intriguing systems to implement an artificial magnetic field. Below is an example.

V. CONCLUSION
To summarize, we have studied the dynamics of bosonic quasiparticles based on BdG equation for the (1, 1)-spinor. We show that the dynamical behavior of these bosonic quasiparticles is described by Lorentz quantum mechanics, where both time evolution of a quantum state and the representation transformation represent Lorentz transformations in the complex Minkowski space. The basic framework of the Lorentz quantum mechanics for the Lorentz spinor is presented, including construction of basic operators that are analogue of Pauli matrices. Based on it, we have demonstrated the Lorentz counterpart of the Berry phase, Berry connection, and Berry curvatures, etc. Since such Lorentz spinors can be generically found in physical systems hosting bosonic Bogoliubov quasi-particles, we expect that our study allows new insights into the dynamical properties of quasiparticles in diverse systems. In a broader context, the present work provides a new perspective toward the fundamental understanding of quantum evolution, as well as new scenarios for experimentally probing the coherent effect. While our study is primarily based on Bogoliubov equation for the (1, 1)-spinor, we expect the essential features also appear in dynamics described by the Bogoliubov equation of multi-mode, the study of which is of future interest.