Complex Minkowski Space Research Articles

A Schwarz lemma for weakly Kähler-Finsler manifolds

In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant is necessary constant.

A Physically Meaningful Relativistic Description of the Spin State of an Electron

We introduced a new model to present the states of a two-state quantum system. The space is the complexified Minkowski space. The Lorentz group acts by the linear extension of its action on the four-vectors. We applied this model to represent the spin state of an electron or any relativistic spin 1/2 particle. The spin state of such particle is of the form U+iS, where U is the four-velocity of the particle in the lab frame, and S is the 4D spin in this frame. Under this description, the transition probability between two pure spin states ϱ1 and ϱ2 of particles moving with the same velocity are defined by use of Minkowski dot product as 12<ϱ2|ϱ1>. This transition probability is Lorentz invariant, coincide with the quantum mechanics prediction and thus agree with the experimental results testing quantum mechanics predictions based on Bell’s inequality. For a a particle of mass m and charge q with the spin state ϱ, the total momentum is mcϱ and the electromagnetic momentum is qϱ. This imply that the Landé g factor for such particles must be g=2. We obtain an evolution equation of the spin state in an electromagnetic field which defines correctly the anomalous Zeeman effect and the fine structure splitting.

Quantum measurement of space-time events

The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive-definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four-momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength.

The Isomorphism of Minkowski Space and the Separable Complex Hilbert Space and Its Physical Interpretation

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincare’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”.

Experimental simulation of anti-parity-time symmetric Lorentz dynamics

The recent experimental progress in parity-time (PT) symmetry has attracted great interest. However, compared with PT symmetry, there are only a few reported results on its counterpart, anti-PT symmetry, which would lead to new insights and applications. Experimentally simulating and demonstrating the properties of anti-PT symmetry is of particular interest. Here, we present experimental research for simulating the dynamics of bosonic Bogoliubov quasi-particles with anti-PT symmetry based on single photons generated from a point defect in a gallium nitride film. The dynamical evolution under a non-unitary operator is a continuous complex Lorentz transformation in a complex Minkowski space. The evolved states are located on hyperbolic curves, depending on the values of the new defined inner product, which remain invariant during the evolution. Three types of state, space-like, light-like, and time-like, which are analogous to those predicted by the special relativity, are demonstrated. The results of our work could be helpful to investigate the dynamics of quasi-particles in diverse systems and promote deep understanding of non-Hermitian quantum mechanics in open systems.

The q-linked complex Minkowski space, its real forms and deformed isometry groups

We establish duality between real forms of the quantum deformation of the four-dimensional orthogonal group studied by Fioresi et al. [Quantum Klein space and superspace, preprint (2017), arXiv:1705.01755] and the classification work made by Borowiec et al. [Basic quantizations of [Formula: see text] Euclidean, Lorentz, Kleinian and quaternionic [Formula: see text] symmetries, J. High Energy Phys. 1711 (2017) 187]. Classically, these real forms are the isometry groups of [Formula: see text] equipped with Euclidean, Kleinian or Lorentzian metric. A general deformation, named [Formula: see text]-linked, of each of these spaces is then constructed, together with the coaction of the corresponding isometry group.

Quantum Klein Space and Superspace

We give an algebraic quantization, in the sense of quantum groups, of the complex Minkowski space, and we examine the real forms corresponding to the signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum metrics and providing an explicit presentation of the quantized coordinate algebras. In particular, we focus on the Kleinian signature $(2,2)$. The quantizations of the complex and real spaces come together with a coaction of the quantizations of the respective symmetry groups. We also extend such quantizations to the $\mathcal{N}=1$ supersetting.

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.

Weakly complex Einstein-Finsler vector bundle

Let (E, F) be a complex Finsler vector bundle over a compact Kahler manifold (M, g) with Kahler form Φ. We prove that if (E, F) is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou (1997), then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle (E, F) over a compact Kahler manifold (M, g) is necessarily Φ-semistable and (E, F) = (E1, F1) ⨁ · · · ⨁ (Ek; Fk); where F j := F |E j , and each (E j , F j ) is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a Φ-stable Einstein-Hermitian vector bundle with the same factor c as (E, F).

Light-cones, almost light-cones and almost-complex light-cones

We point out an unusual relationship among a variety of null geodesic congruences; (a) the generators of ordinary light-cones and (b) certain (related) shear-free but twisting congruences in Minkowski Space-time as well as (c) asymptotically shear-free null geodesic congruences that exist in the neighborhood of Penrose's Scri in Einstein or Einstein-Maxwell asymptotically flat-space-times. We refer to these geodesic congruences respectively as: Lignt-Cones (LCs), as Almost-Complex- Light-Cones, (ACLCs), [though they are real they resemble complex light-cones in complex Minkowski space] and finally to a family of congruences in asymptotically flat-spaces as `Almost Light-Cones', (ALC). The two essential points of resemblance among the three families are: (1) they are all either shear-free or asymptotically shear-free and (2) in each family the individual members of the family can be labeled by the points in a real or complex four-dimensional manifold. As an example, the Minkowski space LCs are labeled by the (real) coordinate value of their apex. In the case of (ACLCs) (complex coordinate values), the congruences will have non-vanishing twist whose magnitude is determined by the imaginary part of the complex coordinate values. In studies of gravitational radiation, Bondi-type of null surfaces and their associated Bondi coordinates have been almost exclusively used for calculations. Some surprising relations arise if, instead of the Bondi coordinates, one uses ALCs and their associated coordinate systems in the analysis of the Einstein-Maxwell equations near Scri. More explicitly and surprisingly, the asymptotic Bianchi Identities expressed in the coordinates of the ALCs, turn directly into many of the standard definitions and relations of classical mechanics.

Ambitwistor strings in four dimensions.

We develop ambitwistor string theories for four dimensions to obtain new formulas for tree-level gauge and gravity amplitudes with arbitrary amounts of supersymmetry. Ambitwistor space is the space of complex null geodesics in complexified Minkowski space, and in contrast to earlier ambitwistor strings, we use twistors rather than vectors to represent this space. Although superficially similar to the original twistor string theories of Witten, Berkovits, and Skinner, these theories differ in the assignment of world sheet spins of the fields, rely on both twistor and dual twistor representatives for the vertex operators, and use the ambitwistor procedure for calculating correlation functions. Our models are much more flexible, no longer requiring maximal supersymmetry, and the resulting formulas for amplitudes are simpler, having substantially reduced moduli. These are supported on the solutions to the scattering equations refined according to helicity and can be checked by comparison with corresponding formulas of Witten and of Cachazo and Skinner.

A canonical dilation of the Schrödinger equation

In this paper we shall re-visit the well-known Schrodinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrodinger equation, may be obtained as the expectation of a counting process of object-clock interactions.

Canonical formalism and quantization of a massless spinning bosonic particle in four dimensions

A twistor model of a free massless spinning particle in four-dimensional Minkowski space is studied in terms of space–time and spinor variables. This model is specified by a simple action, referred to here as the gauged Shirafuji action, that consists of twistor variables and gauge fields on the one-dimensional parameter space. We consider the canonical formalism of the model by following the Dirac formulation for constrained Hamiltonian systems. In the subsequent quantization procedure, we obtain a plane-wave solution with momentum spinors. From this solution and coefficient functions, we construct positive-frequency and negative-frequency spinor wave functions defined on complexified Minkowski space. It is shown that the Fourier–Laplace transforms of the coefficient functions lead to the spinor wave functions expressed as the Penrose transforms of the corresponding holomorphic functions on twistor space. We also consider the exponential generating function for the spinor wave functions and derive a novel representation for each of the spinor wave functions.

Extended MacMahon–Schwingerʼs Master Theorem and conformal wavelets in complex Minkowski space

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D 4 = SO ( 4 , 2 ) / ( SO ( 4 ) × SO ( 2 ) ) of the conformal group SO ( 4 , 2 ) (locally isomorphic to SU ( 2 , 2 ) ) in 1 + 3 dimensions. The manifold D 4 can be mapped one-to-one onto the future tube domain C + 4 of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L h 2 ( D 4 , d ν λ ) and L h 2 ( C + 4 , d ν ˜ λ ) of square integrable holomorphic functions with scale dimension λ and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a λ-extension of Schwingerʼs Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of L h 2 ( D 4 , d ν λ ) . SMT is related to MacMahonʼs Master Theorem (MMT) and an extension of both in terms of Louckʼs SU ( N ) solid harmonics is also provided for completeness. Convergence conditions are also studied.

The generalized good cut equation

The properties of null geodesic congruences (NGCs) in Lorentzian manifolds are a topic of considerable importance. More specifically NGCs with the special property of being shear-free or asymptotically shear-free (as either infinity or a horizon is approached) have received a great deal of recent attention for a variety of reasons. Such congruences are most easily studied via solutions to what has been referred to as the ‘good cut equation’ or the ‘generalization good cut equation’. It is the purpose of this paper to study these equations and show their relationship to each other. In particular we show how they all have a four-complex-dimensional manifold (known as -space, or in a special case as complex Minkowski space) as a solution space.

The real meaning of complex Minkowski-space world-lines

In connection with the study of shear-free null geodesics in Minkowski space, we investigate the real geometric effects in real Minkowski space that are induced by and associated with complex world-lines in complex Minkowski space. It was already known, in a formal manner, that complex analytic curves in complex Minkowski space induce shear-free null geodesic congruences. Here we look at the direct geometric connections of the complex line and the real structures. Among other items, we show, in particular, how a complex world-line projects into the real Minkowski space in the form of a real shear-free null geodesic congruence.

The geometry of regular shear-free null geodesic congruences, CR functions and their application to the flat-space Maxwell equations

We describe here what appears to be a new structure, hidden in all asymptotically vanishing Maxwell fields possessing a non-vanishing total charge. Though we are dealing with real Maxwell fields on real Minkowski space, nevertheless, directly from the asymptotic field one can extract a complex analytic worldline defined in complex Minkowski space that gives a unified Lorentz-invariant meaning to both the electric and the magnetic dipole moments. In some sense, the worldline defines a ‘complex center of charge’ around which both electric and magnetic dipole moments vanish. The question of how and where this complex worldline arises is one of the two main subjects of this work. The other subject concerns what is known in the mathematical literature as a CR structure. In GR, CR structures naturally appear in the physical context of shear-free (or asymptotically shear-free) null geodesic congruences in spacetime. In our work, the CR structure is associated with the embedding of Penrose's real three-dimensional null infinity, , as a surface in a complex two-dimensional space, C2. It is this embedding, via a complex function (a CR function), that is our other area of interest. Specifically, we are interested in the ‘decomposition’ of the CR function into its real and imaginary parts and the physical information contained in this decomposition.

Representations of homogeneous quantum Lévy fields

We study homogeneous quantum Lévy processes and fields with independent additive increments over a noncommutative *-monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum Lévy fields on Itô monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set.

Electromagnetic dipole radiation fields, shear-free congruences and complex centre of charge world lines

We show that for asymptotically vanishing Maxwell fields in Minkowski space with non-vanishing total charge, one can find a unique geometric structure, a null direction field, at null infinity. From this structure a unique complex analytic world line in complex Minkowski space can be found and then identified as the complex centre of charge. By ‘sitting’—in an imaginary sense—on this world line both the electric and intrinsic magnetic dipole moments vanish. The intrinsic magnetic dipole moment is (in some sense) obtained from the ‘distance’ the complex world line is from the real space (times the charge). This point of view unifies the asymptotic treatment of the dipole moments. For electromagnetic fields with vanishing magnetic dipole moments the world line is real and defines the real (ordinary centre of charge). We illustrate these ideas with the Lienard–Wiechert Maxwell field. In the conclusion we discuss its generalization to general relativity where the complex centre of charge world line has its analogue in a complex centre of mass allowing a definition of the spin and orbital angular momentum—the analogues of the magnetic and electric dipole moments.

Quantum complex Minkowski space

The complex Minkowski phase space has the physical interpretation of the phase space of the scalar massive conformal particle. The aim of the paper is the construction and investigation of the quantum complex Minkowski space.