The relativistic acceleration of an electron in a uniform gravitational field is calculated numerically using the generalization of the Dirac equation to curved spacetime. Equivalent results are also obtained analytically using an iterative approach that is not based on the WKB approximation, which has been used extensively in the past. The acceleration is found to be proportional to a factor of $(1-3v^2/c^2)$ using Schwarzschild coordinates, which is consistent with the classical geodesic of a particle. These techniques may be useful in resolving the differences between commonly-used approximations.
We show how to transform a Dirac equation in curved spacetime into a Dirac equation in flat spacetime. In particular, we show that any solution of the free massless Dirac equation in a 1+1 dimensional flat spacetime can be transformed via a local phase transformation into a solution of the corresponding Dirac equation in a curved background, where the spacetime metric is encoded into the phase. In this way, the existing quantum simulators of the Dirac equation can naturally incorporate curved spacetimes. As a first example we use our technique to obtain solutions of the Dirac equation in a par…
We introduce an exact mapping between the Dirac equation in (1+1)-dimensional curved spacetime (DCS) and a multiphoton quantum Rabi model (QRM). A background of a (1+1)-dimensional black hole requires a QRM with one- and two-photon terms that can be implemented in a trapped ion for the quantum simulation of Dirac particles in curved spacetime. We illustrate our proposal with a numerical analysis of the free fall of a Dirac particle into a (1+1)-dimensional black hole, and find that the Zitterbewegung effect, measurable via the oscillatory trajectory of the Dirac particle, persists in the prese…
This paper is motivated by the introduction of a new functional setting of General Relativity (GR) based on the adoption of suitable group non-local point transformations (NLPT). Unlike the customary local point transformatyion usually utilized in GR, these transformations map in each other intrinsically different curved space-times. In this paper the problem is posed of determining the tensor transformation laws holding for the $4-$% acceleration with respect to the group of general NLPT. Basic physical implications are considered. These concern in particular the identification of NLPT-accele…
When quantum fields are coupled to gravitational fields, spontaneous particle creation may occur similarly to when they are coupled to external electromagnetic fields. Gravitational fields can be incorporated as background spacetime if the back-action of matter on it can be neglected, yielding modified Dirac or Klein-Gordon equations for elementary fermions and bosons respectively. The semi-classical description predicts particle creation in many scenarios including expanding universe, near the event horizon of a black hole (the Hawking effect), and an accelerating observer in flat spacetime (…
We find out classical particles, starting from Dirac quantum fields on a curved space-time, by an eikonal approximation and a localization hypothesis for amplitudes. We recover the results by Mathisson-Papapetrou, hence establishing a fundamental correspondence between the coupling of classical and quantum spinning particles with the gravitational field.
An overview is given of recent developments in the field of Dirac equations generalized to curved space-times. An illustrative discussion is provided. We conclude with a variation of Dirac's large-number hypothesis which relates a number of physical quantities on cosmological scales.
Using a generalized Madelung transformation, we derive the hydrodynamic representation of the Dirac equation in arbitrary curved space-times coupled to an electromagnetic field. We obtain Dirac-Euler equations for fermions involving a continuity equation and a first integral of the Bernoulli equation. Comparing between the Dirac and Klein-Gordon equations we obtain the balance equation for fermion particles. We also use the correspondence between fermions and bosons to derive the hydrodynamic representation of the Weyl equation which is a chiral form of the Dirac equation.
We present a recent work on the Dirac equation in a curved spacetime. In addition to the standard equation, two alternative versions are considered, derived from wave mechanics, and based on the tensor representation of the Dirac field. The latter considers the Dirac wave function as a spacetime vector and the set of the Dirac matrices as a third-order tensor. Having the probability current conserved for any solution of the Dirac equation gives an equation to be satisfied by the coefficient fields. A positive definite scalar product is defined and a hermiticity condition for the Dirac Hamilton…
It is shown that the calculation of Dirac operator for the spherical coordinate system with spherical Dirac matrices and using the spin connection formalism is in the contradiction with the definition of standard Dirac operator in the spherical Minkowski coordinate system. It is shown that such contradiction one can avoid by introducing a canonical conjugated covariant derivative for the spinor field. The Dirac equation solution on the Reissner - Nordström background is obtained. The solution describes a bound state of a charged particle.
As opposed to Arminjon statements, in this work we again assert the absence of the non-uniqueness problem of the Dirac theory in a curved and flat spacetime and illustrate this with a number of examples. Dirac Hamiltonians in arbitrary, including time-dependent, gravitational fields uniquely determine physical characteristics of quantum-mechanical systems irrespective of the choice of the tetrad fields. Direct spin-rotation coupling that occurs with a certain choice of tetrads does not manifest itself in final physical characteristics of the systems and therefore does not represent a physicall…
The definition of the Hamiltonian operator H for a general wave equa-tion in a general spacetime is discussed. We recall that H depends on the coordinate system merely through the corresponding reference frame. When the wave equation involves a gauge choice and the gauge change is time-dependent, H as an operator depends on the gauge choice. This dependence extends to the energy operator E, which is the Hermitian part of H. We distinguish between this ambiguity issue of E and the one that occurs due to a mere change of the "represen-tation" (e.g. transforming the Dirac wave function from the "…
Static observers in curved spacetimes may interpret their proper acceleration as the opposite of a local gravitational field (in the Newtonian sense). Based on this interpretation and motivated by the equivalence principle, we are led to investigate congruences of timelike curves in Minkowski spacetime whose acceleration field coincides with the acceleration field of static observers of curved spaces. The congruences give rise to non-inertial frames that are examined. Specifically we find, based on the locality principle, the embedding of simultaneity hypersurfaces adapted to the non-inertial …
We write the Dirac equation in curved 4-dimensional Lorentzian spacetime using concepts from the analysis of partial differential equations as opposed to geometric concepts.
A Lagrangian theory giving rise to a version of the Dirac-Kahler equations on curved backgrounds is considered. The principal pieces are the general fields which have values in the algebra of the Dirac matrices and satisfy a Dirac-type equation. Their components are scalar, pseudo-scalar, vector, axial-vector fields and fields strength which satisfy an irreducible systems of first-order Dirac-Kahler equations having remarkable gauge and duality properties similar to those of the flat case. The vector and axial-vector fields are the physical potentials giving rise to the field strength while th…
These are introductory notes on the study of the Dirac equation in curved spacetime and its relation to hidden symmetries of the dynamics. We present general results on the relation between special spacetime tensors and hidden symmetries, both for the full Dirac equation and for its semi-classical limit, the spinning particle. A concrete application of the general results is provided by the case of rotating higher dimensional black holes with cosmological constant, which we discuss. For these metrics the Dirac equation is separable and the relation between this and hidden symmetries is explain…
The Dirac equation in curved spacetimes is formulated using coordinate-free notation. A Lagrangean density which corresponds to the subject equation is presented. The subject equation is invariant under a local rotation of the coframe. The current is independent of the local orientation of the coframe and it is conserved. The subject equation has an equivalent formulation which uses the Christoffel gamma. The stress-energy tensor is calculated.
Quantum Weak Energy Inequalities (QWEIs) are results which limit the extent to which the smeared renormalised energy density of a quantum field can be negative. On globally hyperbolic spacetimes the massive quantum Dirac field is known to obey a QWEI in terms of a reference state chosen arbitrarily from the class of Hadamard states; however, there exist spacetimes of interest on which state-dependent bounds cannot be evaluated. In this paper we prove the first QWEI for the massive quantum Dirac field on four dimensional globally hyperbolic spacetime in which the bound depends only on the local…
In this paper we propose a definition of quasifree Hadamard states for spinor fields on a curved space-time by specifying the Polarisation Set of the two-point function. We prove that the thermal equilibrium state on an ultrastatic space-time is Hadamard. We then construct an adiabatic vacuum state on a general globally hyperbolic Lorentz manifold using a factorisation of the spinorial Klein-Gordon operator. This state is pure. In what constitutes the main part of the paper, we show that it is also Hadamard. As a side result, we obtain the propagation of singularities of the spinorial Klein-Go…
One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved spacetime. This canonical form is needed to apply the Whitham Lagrangian method. The latter method, unlike the WKB method, places no restricti…
The Dirac wave function in a curved spacetime is usually defined as a quadruplet of scalar fields. It can alternatively be defined as a four-vector field. We describe these two representations in a common geometrical framework and we prove theorems that relate together the different representations and the different choices of connections. In particular, the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the Dirac equation for a vector wave function, in the same spacetime.
We summarize a recent work on the subject title. The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. In this paper, we focus on the standard version of the gravitational Dirac equation, but the non-uniqueness applies also to our alternative versions. We find that the changes which lead to an equivalent operator H, or respectiv…
The quantized Dirac field is known, by a result of Fewster and Verch, to satisfy a Quantum Weak Energy Inequality (QWEI) on its averaged energy density along time-like curves in arbitrary four-dimensional globally hyperbolic spacetimes. However, this result does not provide an explicit form for the bound. By adapting ideas from the earlier work, we give a simplified derivation of a QWEI for the Dirac field leading to an explicit bound. The bound simplifies further in the case of static curves in static spacetimes, and, in particular, coincides with a result of Fewster and Mistry in four-dimens…
The idea of wave mechanics leads naturally to assume the well-known relation $E=\hbar ω$ in the specific form $H=\hbar W$, where $H$ is the classical Hamiltonian of a particle and $W$ is the dispersion relation of the sought-for wave equation. We derive the expression of $H$ in a curved spacetime with an electromagnetic field. Then we derive the Dirac equation from factorizing the polynomial dispersion equation corresponding with $H$. Conversely, summarizing a recent work, we implement the geometrical optics approximation into a canonical form of the Dirac Lagrangian. Euler-Lagrange equations …
In this work, Schrödinger and Dirac equations will be examined in geometries that confine the particles to hypersurfaces. For this purpose, two methods will be considered. The first method is the thin layer method which relies on explicit use of geometrical relations and the squeezing of a certain coordinate of space (or spacetime). The second is Dirac's quantization procedure involving the modification of canonical quantization making use of the geometrical constraints. For the Dirac equation, only the first method will be considered. Lastly, the results of the two methods will be compared an…
We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For ge…
We investigate the spin-1/2 relativistic quantum dynamics in the curved space-time generated by a central massive charged object (black hole). This necessitates a study of the coupling of a Dirac particle to the Reissner-Nordstrom space-time geometry and the simultaneous covariant coupling to the central electrostatic field. The relativistic Dirac Hamiltonian for the Reissner-Nordstrom geometry is derived. A Foldy-Wouthuysen transformation reveals the presence of gravitational, and electro-gravitational spin-orbit coupling terms which generalize the Fokker precession terms found for the Dirac-…
The basic principles of generalization of the group theoretical approach to the relativistic wave equations on curved spaces are examined. The general method of the determination of wave equations from the known symmetry group of a symmetrical curved space is described. The method of obtaining the symmetrical spaces in which invariant wave equations admit the limiting passage to the relativistic wave equations on the flat (not necessarily real) spaces is explained. Starting from the equations for massless particles and from the Dirac equation in the Minkowski space, admissible real symmetrical…
Comment on "Quantum phase transitions of Dirac particles in a magnetized rotating curved background: Interplay of geometry, magnetization, and thermodynamics"
We prove that conformal curved spacetime can be encoded into the initial wave function and that curved propagation can be simulated on a two-dimensional regular lattice with a finite set of homogeneous unitary operators. We generalize recent results shown in [1], where the author transforms flat-spacetime in curved static spacetime via non-unitary rotations. In particular, encoding the metric in the initial quantum state via unitary local rotations, can naturally be useful for quantum simulators due to the efficiency and simplicity of the implementation scheme. We validate our model proving th…
The classical and quantum correlations sharing between modes of the Dirac fields in the noninertial frame are investigated. It is shown that: (i) The classical correlation for the Dirac fields decreases as the acceleration increases, which is different from the result of the scalar field that the classical correlation is independent of the acceleration; (ii) There is no simple dominating relation between the quantum correlation and entanglement for the Dirac fields, which is unlike the scalar case where the quantum correlation is always over and above the entanglement; (iii) As the acceleratio…
By requiring unambiguous symmetric quantization leading to the Dirac equation in a curved space, we obtain a special representation of the spin connections in terms of the Dirac gamma matrices and their space-time derivatives. We also require that squaring the equation give the Klein-Gordon equation in a curved space in its canonical from (without spinor components coupling and with no first order derivatives). These requirements result in matrix operator algebra for the Dirac gamma matrices that involves a universal curvature constant. We obtain exact solutions of the Dirac and Klein-Gordon e…
We perform a qualitative and thermodynamic study of two models when one takes into account adiabatic particle production. In the first one, there is a constant particle production rate, which leads to solutions depicting the current cosmic acceleration but without inflation. The other one has solutions that unify the early and late time acceleration. These solutions converge asymptotically to the thermal equilibrium.
We study the basic quantum mechanics for a fully general set of Dirac matrices in a curved spacetime by extending Pauli's method. We further extend this study to three versions of the Dirac equation: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which are based on the recently proposed linear tensor representations of the Dirac field (TRD). We begin with the current conservation: we show that the latter applies to any solution of the Dirac equation, iff the field of Dirac matrices $γ^μ$ satisfies a specific PDE. This equation is always satisfied for DFW …
We present a semiclassical analysis for Dirac fields on an arbitrary spacetime background and in the presence of a fixed electromagnetic field. Our approach is based on a Wentzel-Kramers-Brillouin approximation, and the results are analyzed at leading and next-to-leading order in the small expansion parameter $\hbar$. Taking into account the spin-orbit coupling between the internal and external degrees of freedom of wave packets, we derive effective ray equations with spin-dependent terms. These equations describe the gravitational spin Hall effect of localized Dirac wave packets. We treat bot…
A general formulation of spinor fields in Riemannian space-time is given without using vierbeins. The space-time dependence of the Dirac matrices required by the anticommutation relation {γ_μ,γ_ν}=2g_{μν} determines the spin connection. The action is invariant under any local spin base transformations in the 32 parameter group Gl(4,c) and not just under local Lorentz transformations. The Dirac equation and the energy-momentum tensor are computed from the action.
Interest on 2 + 1 dimensional electron systems has increased considerably after the realization of novel properties of graphene sheets, in which the behaviour of electrons is effectively described by relativistic equations. Having this fact in mind, the following problem is studied in this work: when a spin 1/2 particle is constrained to move on a curved surface, is it possible to describe this particle without giving reference to the dimensions external to the surface? As a special case of this, a relativistic spin 1/2 particle which is constrained to move on a 2 + 1 dimensional hypersurface …
We study the properties of graphene wormholes in which a short nanotube acts as a bridge between two graphene sheets, where the honeycomb carbon lattice is curved from the presence of 12 heptagonal defects. By taking the nanotube bridge with very small length compared to the radius, we develop an effective theory of Dirac fermions to account for the low-energy electronic properties of the wormholes in the continuum limit, where the frustration induced by the heptagonal defects is mimicked by a line of fictitious gauge flux attached to each of them. We find in particular that, when the effectiv…
We consider a spacetime corresponding to uniform relativistic potential analogus to Newtonian potential as an example of ``minimally curved spacetime''. We also consider a radially symmetric analogue of the Rindler spacetime of uniform proper acceleration relative to infinity.
We study the behaviour of spin-half particles in curved space-time. Since Dirac equation gives the dynamics of spin-half particles, we mainly study the Dirac equation in Schwarzschild, Kerr, Reissner-Nordström geometry. Due to the consideration of existence of black hole in space-time (the curved space-time), particles are influenced and equation will be modified. As a result the solution will be changed from that due to flat space.
We show that a free Dirac quantum field on a globally hyperbolic spacetime has the following structural properties: (a) any two quasifree Hadamard states on the algebra of free Dirac fields are locally quasiequivalent; (b) the split-property holds in the representation of any quasifree Hadamard state; (c) if the underlying spacetime is static, then the nuclearity condition is satisfied, that is, the free energy associated with a finitely extended subsystem (``box'') has a linear dependence on the volume of the box and goes like $\propto T^{s+1}$ for large temperatures $T$, where $s+1$ is the n…
It has been shown that, for all dimensions and signatures, the most general first-order linear symmetry operators for the Dirac equation including interaction with Maxwell field in curved background are given in terms of Killing-Yano (KY) forms. As a general gauge invariant condition it is found that among all KY-forms of the underlying (pseudo) Riemannian manifold, only those which Clifford commute with the Maxwell field take part in the symmetry operator. It is also proved that associated with each KY-form taking part in the symmetry operator, one can define a quadratic function of velocitie…
We first generalise the standard Wigner function to Dirac fermions in curved spacetimes. Secondly, we turn to the Moyal quantisation of systems with constraints. Gravity is used as an example.
The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. The latter equations may be defined when the spacetime is four-dimensional, noncompact, and admits a spinor struc…
It is shown that the main geometrical objects involved in all the symmetries or supersymmetries of the Dirac operators in curved manifolds of arbitrary dimensions are the Killing vectors and the Killing-Yano tensors of any ranks. The general theory of external symmetry transformations associated to the usual isometries is presented, pointing out that these leave the standard Dirac equation invariant providing the correct spin parts of the group generators. Furthermore, one analyzes the new type of symmetries generated by the covariantly constant Killing-Yano tensors that realize certain square…
Quantum fields are well known to violate the weak energy condition of general relativity: the renormalised energy density at any given point is unbounded from below as a function of the quantum state. By contrast, for the scalar and electromagnetic fields it has been shown that weighted averages of the energy density along timelike curves satisfy `quantum weak energy inequalities' (QWEIs) which constitute lower bounds on these quantities. Previously, Dirac QWEIs have been obtained only for massless fields in two-dimensional spacetimes. In this paper we establish QWEIs for the Dirac and Majoran…
Canonical Hamiltonian field theory in curved spacetime is formulated in a manifestly covariant way. Second quantization is achieved invoking a correspondence principle between the Poisson bracket of classical fields and the commutator of the corresponding quantum operators. The Dirac theory is investigated and it is shown that, in contrast to the case of bosonic fields, in curved spacetime, the field momentum does not coincide with the generators of spacetime translations. The reason is traced back to the presence of second class constraints occurring in Dirac theory. Further, it is shown that…
A chiral alternative to the vierbein field in general relativity was considered by 't Hooft in an attempt to facilitate the construction of a quantum theory of gravity. These objects $ f^a{}_{μν}$ behave like the "cube root" of the metric tensor. We try to construct specific representations of these tensors in terms of Dirac $γ$ matrices in Euclidean and Minkowski space, and promote these to curved space through Newman-Penrose formalism. We conjecture that these are new objects with physical significance and are the analog of Killing-Yano tensors. As an application, we try to construct non-Abe…
Higher order symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing-Yano tensors and non-standard supersymmetries is pointed out. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The general results are applied to space-times which appear in modern studies. One presents the infinite dimensional superalgebra of Dirac type operators on the 4-dimensional Euclidean Taub-NUT space that can be s…
On arbitrary spacetimes, we study the characteristic Cauchy problem for Dirac fields on a light-cone. We prove the existence and uniqueness of solutions in the future of the light-cone inside a geodesically convex neighbourhood of the vertex. This is done for data in $L^2$ and we give an explicit definition of the space of data on the light-cone producing a solution in $H^1$. The method is based on energy estimates following L. Hörmander (J.F.A. 1990). The data for the characteristic Cauchy problem are only a half of the field, the other half is recovered from the characteristic data by integr…
