On the Hamiltonian and energy operators in a curved spacetime, especially for a Dirac particle
Mayeul Arminjon
备注：
Text of a talk given at the DICE2014 Workshop (Castiglioncello (Livorno), Italy). Submitted to the Proceedings (H.T. Elze et al., eds.). in Seventh International Workshop DICE2014: Spacetime - Matter - Quantum Mechanics, Sep 2014, Castiglioncello (Provincia di Livorno), Italy
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 "Dirac representation" to a "Foldy-Wouthuysen representation"). We also assert that the energy operator ought to be well defined in a given ref-erence frame at a given time, e.g. by comparing the situation for this operator with the main features of the energy for a classical Hamilto-nian particle.
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 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…
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…
Generalizing to relativistic exponential scaling and using the theory of noise from quantum fluctuations, it has been shown that one vacuum (Rindler, Hartle-Hawking, or Gibbons-Hawking for the cases of the uniformly accelerated detector, black hole, and de-Sitter universe, respectively) can be understood as resulting from the scaling of quantum noise in another vacuum. We explore this idea more generally to establish a flat spacetime and curved spacetime analogy. For this purpose, we start by examining noise kernels for free fields in some well-known curved spacetimes, e.g., the spacetime of a…
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 approxim…
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Quasi-classical picture of motion of spin 1/2 massive particle in a curved spacetime is built on base of simple Lagrangian model. The one is constructed due to analogy with Lagrangian of massive vector particle. Equations of motion and spin propagation coincide with Papapetrou equations describing dynamic of classical spinning particle in a curved spacetime.
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Zitterbewegung, as it was originally described by Schrödinger, is an unphysical, non-observable effect. We verify whether the effect can be observed in non-inertial reference frames/curved spacetimes, where the ambiguity in defining particle states results in a mixing of positive and negative frequency modes. We explicitly demonstrate that such a mixing is in fact necessary to obtain the correct classical value for a particle's velocity in a uniformly accelerated reference frame, whereas in cosmological spacetime a particle does indeed exhibit Zitterbewegung.
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The representation of measurements by positive operator valued measures and the description of the most general state transformations by means of completely positive maps are two basic concepts of quantum information theory. These concepts can be trivially extended to field theories in curved spacetime if all the representations of canonical commutation or anticommutation relations are unitarily equivalent. We show that both concepts can be applied even when there is no such unitary equivalence.
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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…
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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…
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The ideal MHD theorem on the conservation of the magnetic connections between plasma elements is generalized to relativistic plasmas in curved spacetime. The connections between plasma elements, which are established by a covariant connection equation, display a particularly complex structure in curved spacetime. Nevertheless, it is shown that these connections can be interpreted in terms of magnetic field lines alone by adopting a $3+1$ foliation of spacetime.
We analyze the classical approximations made in "The general relativistic effects to the magnetic moment in the Earth's gravity", originally published as "Post-Newtonian effects of Dirac particle in curved spacetime - I : magnetic moment in curved spacetime", and work out precisely where in the argument the mistakes are made. We show explicitly that any difference vanishes when properly distinguishing between coordinate and physical distance. In doing this, we illustrate some of the pitfalls in using GR to make predictions.
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In this paper we will investigate how one can create emergent curved spacetimes by locally tuning the coupling constants of condensed matter systems. In the continuum limit we thus obtain continuous effective quantum fields living on curved spacetimes. In particular, using Stingnet condensates we can obtain effective electromagnetism. We will show for example how we obtain quantum electromagnetism (U(1)-Yang-Mills) in a black hole (Schwarzschild) spacetime.
In General Relativity, finding out the geodesics of a given spacetime manifold is an important task because it determines which classical processes are dynamically forbidden. Conserved quantities play an important role in solving geodesic equations of a general spacetime manifold. Furthermore, knowing all possible conserved quantities of a system tells about the hidden symmetries of that system since, conserved quantities are deeply connected with the symmetries of the system, which are very important in their own right. Conserved quantities are also useful to capture certain features of space…
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Earlier we obtained quasi-classical equations of motion of spin 1/2 massless particle in a curved spacetime on base of simple Lagrangian model \cite{al2}. Now we suggest an approach to derive the equations in framework of field theory. Noether theorem formulated in terms of Cartan' formalism of orthonormal frames gives equations for current of spin of the field and tensor of stress-energy. It is shown that under eikonal approximation the above mentioned equations can be reduced to equations for worldline of the particle and equation of spin of the particle along the worldline. This way conform…
In this paper we will investigate how one can create emergent curved spacetimes by locally tuning the coupling constants of condensed matter systems. In the continuum limit we thus obtain continuous effective quantum fields living on curved spacetimes. In particular, using Stingnet condensates we can obtain effective electromagnetism. We will show for example how we obtain quantum electrodynamics in a blackhole (Schwarzschild) spacetime.
We offer a novel derivation of the electromagnetic self-force acting on a charged particle moving in an arbitrary curved spacetime. Our derivation is based on a generalization from flat spacetime to curved spacetime of the extended-body approach of Ori and Rosenthal. In this approach the charged particle is first modeled as a body of finite extension s, the net force acting on the extended body is computed, and the limit s -> 0 is taken at the end of the calculation. Concretely our extended body is a dumbbell that consists of two point charges that are maintained at a constant spacelike separa…
