Dirac Propagator Research Articles

The Fermionic Entanglement Entropy of the Vacuum State of a Schwarzschild Black Hole Horizon

AbstractWe define and analyze the fermionic entanglement entropy of a Schwarzschild black hole horizon for the regularized vacuum state of an observer at infinity. Using separation of variables and an integral representation of the Dirac propagator, the entanglement entropy is computed to be a prefactor times the number of occupied angular momentum modes on the event horizon.

The Dirac equation across the horizons of the 5D Myers–Perry geometry: separation of variables, radial asymptotic behaviour and Hamiltonian formalism

We analytically extend the 5D Myers–Perry metric through the event and Cauchy horizons by defining Eddington–Finkelstein-type coordinates. Then, we use the orthonormal frame formalism to formulate and perform separation of variables on the massive Dirac equation, and analyse the asymptotic behaviour at the horizons and at infinity of the solutions to the radial ordinary differential equation (ODE) thus obtained. Using the essential self-adjointness result of Finster–Röken and Stone’s formula, we obtain an integral spectral representation of the Dirac propagator for spinors with low masses and suitably bounded frequency spectra in terms of resolvents of the Dirac Hamiltonian, which can in turn be expressed in terms of Green’s functions of the radial ODE.

Singular spin structures and superstrings

There are two main problems in finding the higher genus superstring measure. The first one is that for g ≥ 5 the super moduli space is not projected. Furthermore, the supermeasure is regular for g ≤ 11, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the θ-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szegö kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this holomorphic Fay trisecant identity leads to identities that include points dependent rank 3 quadrics in ℙg−1. Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the θ-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and show, via a new θ-identity, its relation with the Andreotti-Mayer quadric.

Cyclic products of Szegö kernels and spin structure sums. Part I. Hyper-elliptic formulation

The summation over spin structures, which is required to implement the GSO projection in the RNS formulation of superstring theories, often presents a significant impediment to the explicit evaluation of superstring amplitudes. In this paper we discover that, for Riemann surfaces of genus two and even spin structures, a collection of novel identities leads to a dramatic simplification of the spin structure sum. Explicit formulas for an arbitrary number of vertex points are obtained in two steps. First, we show that the spin structure dependence of a cyclic product of Szegö kernels (i.e. Dirac propagators for worldsheet fermions) may be reduced to the spin structure dependence of the four-point function. Of particular importance are certain trilinear relations that we shall define and prove. In a second step, the known expressions for the genus-two even spin structure measure are used to perform the remaining spin structure sums. The dependence of the spin summand on the vertex points is reduced to simple building blocks that can already be identified from the two-point function. The hyper-elliptic formulation of genus-two Riemann surfaces is used to derive these results, and its SL(2, ℂ) covariance is employed to organize the calculations and the structure of the final formulas. The translation of these results into the language of Riemann ϑ-functions, and applications to the evaluation of higher-point string amplitudes, are relegated to subsequent companion papers.

GENERALIZATION OF ELECTRICAL RESISTANCE SCALING TO DIRAC FIELDS ON PERFORATED FRACTALS

I construct perforated, “take-away” fractals that support short-distance power-law scaling with complex exponents for Dirac (spin-1/2) propagators. The construction relies on a fortuitous ansatz for Dirac boundary conditions at the surfaces of spherical voids in a three-dimensional embedding space, and requires that boundary conditions vary from void to void and are distributed statistically. The appropriate distribution can ensure that nonzero charges (but not dipoles, which drive nontrivial power-law scaling) induced in voids cancel out in spatial averaging.

Local Dirac energy decay in the 5D Myers-Perry geometry using an integral spectral representation for the Dirac propagator

We consider the massive Dirac equation in the exterior region of the five-dimensional Myers-Perry black hole. Using the resulting ordinary differential equations (ODEs) obtained from the separation of variables of the Dirac equation, we construct an integral spectral representation for the solution of the Cauchy problem with compactly supported smooth initial data. We then prove that the probability of presence of a Dirac particle to be in any compact region of space decays to zero as , in analogy with the case of the Dirac operator in the Kerr–Newman geometry (Finster et al 2003 Adv. Theor. Math. Phys. 7 25–52).

Majorana propagator on de Sitter space

We study the dynamics of Majorana fermions in an expanding de Sitter space and to that aim, by making use of the method of mode sums, we construct the vacuum Feynman propagator for Majorana fermions in de Sitter space. The Majorana condition implies nontrivial identities for the mode functions, which we carefully implement. We note that, under charge conjugation, the propagator transforms to minus itself, but do not discuss in detail the topological implications of this observation. We construct the propagator for a general complex mass and find that it is identical to the corresponding Dirac propagator, meaning that the coupling of fermions to the classical de Sitter gravitational background does not violate charge symmetry, i.e. it respects the Majorana condition. The complex mass propagator differs from its real mass counterpart in that the usual positive and negative energy projectors acquire a chiral rotation which depends on the phase of the mass term. We use our propagator to calculate the one-loop effective action for Majorana fermions, and find that it differs from that of Dirac fermions by a factor of 1/2, which accounts for the reduced number of degrees of freedom of the Majorana particle. Finally, we derive the Majorana and Dirac propagators for mixing n fermionic flavors and show that the number of CP-violating phases for Dirac fermions exceeds that of the Majorana fermions by {nleft( n-1right) }/{2}. This is vital for understanding how the dynamics of Dirac and Majorana fermions generate different CP-violating effects in the multi-flavor case.

Dressed Dirac propagator from a locally supersymmetric [formula omitted] spinning particle

We study the Dirac propagator dressed by an arbitrary number N of photons by means of a worldline approach, which makes use of a supersymmetric N=1 spinning particle model on the line, coupled to an external Abelian vector field. We obtain a compact off-shell master formula for the tree level scattering amplitudes associated to the dressed Dirac propagator. In particular, unlike in other approaches, we express the particle fermionic degrees of freedom using a coherent state basis, and consider the gauging of the supersymmetry, which ultimately amounts to integrating over a worldline gravitino modulus, other than the usual worldline einbein modulus which corresponds to the Schwinger time integral. The path integral over the gravitino reproduces the numerator of the dressed Dirac propagator.

Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature

The Dirac field, spin 1/2 particles, is investigated in phase space. The Dirac propagator is defined. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature. The energy-momentum tensor is calculated at finite temperature. The Stefan-Boltzmann law is established, and the Casimir effect is calculated for the Dirac field in phase space at zero and finite temperature. A comparative analysis with these results in standard quantum mechanics space is realized.

The Dirac Propagator for One-Dimensional Finite Square Well

The solution of Dirac particles confined in a one-dimensional finite square well potential is solved by using the path-integral formalism for Dirac equation. The propagator of the Dirac equation in case of the bounded Dirac particles is obtained by evaluating an appropriate path integral, directly constructed from the Dirac equation. The limit of integration techniques for evaluating path integral is only valid for the piecewise constant potential. Finally, the Dirac propagator is expressed in terms of standard special functions.

Model of multiple Dirac eikonal scattering of protons by nuclei

The model of multiple Dirac eikonal scattering (MDES) of incident proton by target-nucleus nucleons is developed, in which new expressions for the elastic [Formula: see text]-scattering amplitudes are obtained from the multiple scattering Watson series with employing the eikonal approximation for the Dirac propagators of the free proton motion between successive scattering acts on nucleons. Based on this model, calculations for the complete set of observables of the elastic [Formula: see text] and [Formula: see text]Pb at 800[Formula: see text]MeV have been performed, using proton–nucleon amplitudes determined from the phase analysis and the nucleon densities obtained from describing the target-nucleus structure in the relativistic mean-field approximation. A comparison has been made of the results of these calculations with analogous calculations on the basis of the Glauber multiple diffraction theory.

Strong, random field behavior of the Euclidean Dirac propagator

The one-loop effective action of quantum electrodynamics in four dimensions is shown to be controlled by the Euclidean Dirac propagator $G$ in a background potential. After separating the photon self-energy and photon-photon scattering graphs from the effective action the remainder is known to be the logarithm of an entire function of the electric charge of order 4 under mild regularity assumptions on the potential. This input together with QED's lack of an ultrastable vacuum constrain the strong field behavior of $G$. It is shown that $G$ vanishes in the strong field limit. The relevance of this result to the decoupling of QED from the remainder of the electroweak model for large amplitude variations of the Maxwell field is discussed.

The massive Dirac equation in Kerr geometry: separability in Eddington–Finkelstein-type coordinates and asymptotics

The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr geometry is described in the Newman-Penrose formalism by a regular Carter tetrad and the Dirac spinors and matrices are defined in a chiral Newman-Penrose dyad representation. Applying Chandrasekhar's mode ansatz, the Dirac equation is separated into radial and angular systems of ordinary differential equations. Asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon are derived, and the decay of the associated errors is analyzed. Moreover, specific aspects of the angular eigenfunctions and eigenvalues are discussed. Finally, as an application, the scattering of massive Dirac particles by the gravitational field of a rotating Kerr black hole is studied. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating coordinates and for the construction of an integral spectral representation of the Dirac propagator that yields the dynamics of Dirac particles outside, across, and inside the event horizon, up to the Cauchy horizon.

Quantum Metrology: Towards an alternative definition for the meter

The motivation for this work came from an attempt to give an alternative definition for the meter, the SI unit for measuring length. As a starting point towards this goal we present the underlying theory behind our approach which uses ideas from quantum field theory and non-commutative geometry, in particular the notion of an odd K-cycle which is based on the Dirac operator (and its inverse, the Dirac propagator). Using physics terminology, the key point in our strategy is this: instead of measuring ordinary length in space-time we measure the “algebraic (or spectral) length” in the space of quantum states of some fermion acted upon by the Dirac propagator.

Quantum Field Theory Approach to the Optical Conductivity of Strained and Deformed Graphene

The computation of the optical conductivity of strained and deformed graphene is discussed within the framework of quantum field theory in curved spaces. The analytical solutions of the Dirac equation in an arbitrary static background geometry for one dimensional periodic deformations are computed, together with the corresponding Dirac propagator. Analytical expressions are given for the optical conductivity of strained and deformed graphene associated with both intra and interbrand transitions. The special case of small deformations is discussed and the result compared to the prediction of the tight-binding model.

Gravitational correction to vacuum polarization

We consider the gravitational correction to (electronic) vacuum polarization in the presence of a gravitational background field. The Dirac propagators for the virtual fermions are modified to include the leading gravitational correction (potential term) which corresponds to a coordinate-dependent fermion mass. The mass term is assumed to be uniform over a length scale commensurate with the virtual electron-positron pair. The on-mass shell renormalization condition ensures that the gravitational correction vanishes on the mass shell of the photon, i.e., the speed of light is unaffected by the quantum field theoretical loop correction, in full agreement with the equivalence principle. Nontrivial corrections are obtained for off-shell, virtual photons. We compare our findings to other works on generalized Lorentz transformations and combined quantum-electrodynamic gravitational corrections to the speed of light which have recently appeared in the literature.

Superstatistics approach to path integral for a relativistic particle

Superstatistics permits the calculation of the Feynman propagator of a relativistic particle in a novel way from a superstatistical average over non-relativistic single-particle paths. We illustrate this for the Klein-Gordon particle in the Feshbach-Villars representation, and for the Dirac particle in the Schroedinger-Dirac representation. As a byproduct we recover the worldline representation of Klein-Gordon and Dirac propagators, and discuss the role of the smearing distributions in fixing the reparametrization freedom. The emergent relativity picture that follows from our approach together with a novel representation of the Lorentz group for the Feshbach-Villars particle are also discussed.

Some chirality-related properties of the 4D massive Dirac propagator and determinant in an arbitrary gauge field

For a 4-D massive Dirac field in the background of arbitrary gauge fields, we show that the Dirac propagator and functional determinant are completely determined by knowledge of the corresponding quantities for just one of the chirality sectors of the second-order Dirac operator. This generalizes the related, previously known, statements in (anti-)self-dual background gauge fields. The logarithms of the (renormalized) functional determinants from the two chirality sectors are shown to be different only by a term reflecting the integrated chiral anomaly.

Feynman’s Corner Rule; Quantum Propagation from Special Relativity

Feynman’s sum-over-paths prescription for the Dirac equation in a two dimensional spacetime can be formulated to give an unconventional view of the relationship between quantization and special relativity. By considering a local rule for the maintenance of Lorentz covariance in a discrete space, one is able to see the origin of Feynman’s rule and, taking a continuum limit at the last step, one obtains the Dirac propagator as a manifestation of special relativity, rather than a formal addition to it. In this route to the Dirac equation, the path-dependent phase of wavefunctions, relativistic or not, is a direct manifestations of path-dependent proper time.

Deformation quantization of fermi fields

Deformation quantization for any Grassmann scalar free field is described via the Weyl–Wigner–Moyal formalism. The Stratonovich–Weyl quantizer, the Moyal ★-product and the Wigner functional are obtained by extending the formalism proposed recently in [I. Galaviz, H. García-Compeán, M. Przanowski, F.J. Turrubiates, Weyl–Wigner–Moyal Formalism for Fermi Classical Systems, arXiv:hep-th/0612245] to the fermionic systems of infinite number of degrees of freedom. In particular, this formalism is applied to quantize the Dirac free field. It is observed that the use of suitable oscillator variables facilitates considerably the procedure. The Stratonovich–Weyl quantizer, the Moyal ★-product, the Wigner functional, the normal ordering operator, and finally, the Dirac propagator have been found with the use of these variables.