Parity Transformation Research Articles

The consequences of parity symmetry for higher-order statistics of cosmic shear and other polar fields

We consider the parity transformation and the consequences of parity invariance on the n-point correlation function of the cosmic shear caused by gravitational lensing by the large-scale structure, or any other polar (or spin-2) field. The decomposition of the shear field into E -a ndB-modes then yields the result that any correlation function which contains an odd number of B-mode shear components vanishes for parity-invariant random shear fields. In particular, this result implies that the expectation value of the third-order cross aperture statistics, D M 3 E , vanishes for parity-invariant shear fields. Therefore, a significant detection of a non-zero value of D M 3 E in a cosmic shear survey does not indicate the presence of B-modes, but of an underestimate of the statistical uncertainty or cosmic variance, or a remaining systematic eect in the shear measurement. We argue that the parity invariance provides a very specific diagnostic of systematic eects in shear data. Our results apply as well to the linear polarization of the cosmic microwave background.

Vacuum, matter and antimatter

We discuss the possibility of producing a new kind of nuclear system by putting a few antibaryons inside ordinary nuclei. The structure of such systems is calculated within the relativistic mean-field model assuming that the nucleon and antinucleon potentials are related by the G—parity transformation. The presence of antinucleons leads to decreasing vector potential and increasing scalar potential for the nucleons. As a result, a strongly bound system of high density is formed. Due to the significant reduction of the available phase space the annihilation probability might be strongly suppressed in such systems.

Can the Wigner function be determined by properties for translation and parity transformation on lattice phase space ?

We show that the Fano operator for one dimensional quantum system is uniquely determined by assuming the reasonable behavior under translation and parity transformation on phase space. Contrarily, for the system with lattice phase space the same procedure does not work.

Chern-Simons theory on the lattice

We present new proposals for the representation of a Chern-Simons term on the lattice. In the first part, such a term is constructed from the fermion determinant, and in the second part directly from the Abelian gauge term. In both cases, the parity transformation is modified on the lattice without affecting its continuum limit.

Autonomous generation of all Wigner functions and marginal probability densities of Landau levels

Generation of Wigner functions of Landau levels and determination of their symmetries and generic properties are achieved in the autonomous framework of deformation quantization. Transformation properties of diagonal Wigner functions under space inversion, time reversal and parity transformations are specified and their invariance under a four-parameter subgroup of symplectic transformations are established. A generating function for all Wigner functions is developed and this has been identified as the phase–space coherent state for Landau levels. Integrated forms of generating function are used in generating explicit expressions of marginal probability densities on all possible two dimensional phase–space coordinate planes. Phase–space realization of unitary similarity and gauge transformations as well as some general implications for the Wigner function theory are presented.

Higher Order Moments of the Cosmic Shear and Other Spin‐2 Fields

We present a method for defining higher order moments of a spin-2 field on the sky using the transformation properties of these statistics under rotation and parity. For the three-point function of the cosmic shear, we show that the eight logically possible combinations of the shear in three points can be divided into two classes; four combinations are even under parity transformations, and four are odd. We compute the expected value of the even-parity ones in the nonlinear regime using the halo model and conclude that on small scales, of the four combinations there is one that is expected to carry most of the signal for triangles close to isosceles. On the other hand, for collapsed triangles, all four combinations are expected to have roughly the same level of signal, although some of the combinations are negative and others positive. We estimate that a survey of a few square degrees area is enough to detect this signal above the noise at arcminute scales.

The three-point correlation function of cosmic shear

The three-point correlation function of cosmic shear, the weak distortion of the images of distant galaxies by the gravitational field of the inhomogeneous matter distribution in the Universe, is studied here. Previous work on three-point statistics of cosmic shear has mainly concentrated on the convergence, or on aperture measures of the shear. However, as has become clear recently for the two-point statistics of cosmic shear, the basic quantity that should be used is the correlation function: first, it is much easier to measure from observational data, since it is immune against complicated geometries of data fields (which contain gaps and holes, e.g. due to masking); second, all other (linear) two-point statistics can be expressed as integrals over the correlation function. The situation is the same for the three-point statistics. However, in contrast to the two-point correlation function, the invariants (with respect to rotations) of the shear three-point correlation function have not been employed yet. Here we consider the transformation properties of the shear three-point correlation function under rotations. We show that there are four complex linear combinations of components of the three-point correlation function, which we shall call “natural components”, since they are multiplied just by a phase factor for arbitrary rotations, but do not mix. In particular, their moduli are invariant under rotations and thus (non-linear) invariants of the three-point correlation function. In terms of these natural components, the invariance of the statistical properties of the shear field under parity transformations are easily obtained. Our results do not apply only to cosmic shear, but also to other quantities with the same mathematical properties – that of a polar. For example, practically every relation derived here applies also to the polarization of the cosmic microwave background radiation.

The Three-Point Correlation Function for Spin-2 Fields

The three-point correlation function (3PCF) of the spin-2 fields, cosmic shear and microwave background polarization, is a statistical measure of non-Gaussian signals. At each vertex of a triangle, the shear field has two independent components. The resulting eight possible 3PCFs were recently investigated by Schneider & Lombardi and Zaldarriaga & Scoccimarro. Using rotation and parity transformations, they posed the question: how many components of the shear 3PCF are nonzero and useful? We address this question using an analytical model and measurements from ray-tracing simulations. We show that all eight 3PCFs are generally nonzero and have comparable amplitude. These eight 3PCFs can be used to improve the signal-to-noise ratio from survey data, and their configuration dependence can be used to separate the contribution from E- and B-modes. This separation provides a new and precise probe of systematic errors. We estimate the signal-to-noise ratio for measuring the shear 3PCF from weak lensing surveys using simulated maps that include the noise due to intrinsic ellipticities. A deep lensing survey with area of the order of 10 deg2 would allow for the detection of the shear 3PCFs; a survey with area exceeding 100 deg2 is needed for accurate measurements.

The variational principle and simple properties of the ground-state wave function

The variational principle is used to show that the ground-state wave function of a one-body Schrödinger equation with a real potential is real, does not change sign, and is nondegenerate. As a consequence, if the Hamiltonian is invariant under rotations and parity transformations, the ground state must have positive parity and zero angular momentum.

Generalized parity transformations in lattice Chern-Simons theory

Regularization modifies the (odd) behaviour of the Abelian Chern-Simons (CS) action under parity. This effect happens for any sensible regularization; in particular, on the lattice. However, as in the chiral symmetry case, there exist generalized parity transformations such that the regularized theory is odd, and the corresponding operator verifies a Ginsparg-Wilson (GW) like relation. We present a derivation of such a relation and of the corresponding symmetry transformations.

THE PIN GROUPS IN PHYSICS: C, P AND T

A simple, but not widely known, mathematical fact concerning the coverings of the full Lorentz group sheds light on parity and time reversal transformations of fermions. Whereas there is, up to an isomorphism, only one Spin group which double covers the orientation preserving Lorentz group, there are two essentially different groups, called Pin groups, which cover the full Lorentz group. Pin(1, 3) is to O(1, 3) what Spin(1, 3) is to SO(1, 3). The existence of two Pin groups offers a classification of fermions based on their properties under space or time reversal finer than the classification based on their properties under orientation preserving Lorentz transformations — provided one can design experiments that distinguish the two types of fermions. Many promising experimental setups give, for one reason or another, identical results for both types of fermions. These negative results are reported here because they are instructive. Two notable positive results show that the existence of two Pin groups is relevant to physics: • In a neutrinoless double beta decay, the neutrino emitted and reabsorbed in the course of the interaction can only be described in terms of Pin(3, 1). • If a space is topologically nontrivial, the vacuum expectation values of Fermi currents defined on this space can be totally different when described in terms of Pin(1, 3) and Pin(3, 1). Possibly more important than the two above predictions, the Pin groups provide a simple framework for the study of fermions; it makes possible clear definitions of intrinsic parities and time reversal; it clarifies colloquial, but literally meaningless, statements. Given the difference between the Pin group and the Spin group it is useful to distinguish their representations, as groups of transformations on "pinors" and 'spinors", respectively. The Pin(1, 3) and Pin(3, 1) fermions are twin-like particles whose behaviors differ only under space or time reversal. A section on Pin groups in arbitrary spacetime dimensions is included.

Ginsparg-Wilson fermions in odd dimensions

The Ginsparg-Wilson relation, if written in a suitable form, can be used as a condition for lattice Dirac operators of massless fermions also in odd dimensions. The fermion action with such a Dirac operator is invariant under a generalized parity transformation, which reduces to the ordinary parity transformation in the (naive) continuum limit. The fermion measure, however, transforms non-trivially under the generalized parity transformation, and hence the parity anomaly arises solely from the fermion measure. The analogy to the lattice construction of chiral gauge theories in even dimensions is clarified by considering a dimensional reduction. We also propose a natural definition of a lattice Chern-Simons term, which is consistent with odd dimensional Ginsparg-Wilson fermions.

Generalized parity transformations in the regularized Chern-Simons theory

We study renormalization effects in the Abelian Chern-Simons (CS) action. These effects can be non-trivial when the gauge field is coupled to dynamical matter, since the regularization of the UV divergences in the model forces the introduction of a parity even piece in the gauge field action. This changes the classical (odd) transformation properties of the pure CS action. This effect, already discussed for the case of a lattice regularization by F. Berruto, M.C. Diamantini and P. Sodano in hep-th/0004203, is also present when the theory is defined in the continuum and, indeed, it is a manifestation of a more general `anomalous' effect, since it happens for every regularization scheme. We explore the physical consequences of this anomaly. We also show that generalized, nonlocal parity transformations can be defined in such a way that the regularized theory is odd, and that those transformations tend to the usual ones when the cutoff is removed. These generalized transformations play a role that is tantamount to the deformed symmetry corresponding to Ginsparg-Wilson fermions [2] (in an even number of spacetime dimensions).

Power Spectra Estimation for Weak Lensing

We develop a method for estimating the shear power spectra from weak-lensing observations and test it on simulated data. Our method describes the shear field in terms of angular power spectra and the cross-correlation of the two shear modes, which differ under parity transformations. Two of the three power spectra can be used to monitor unknown sources of noise in the data. The power spectra are decomposed in a model-independent manner in terms of powers, which are then extracted from the data using a quadratic estimator to find the maximum of the likelihood and its local curvature (for error estimates). We test the method against simulated data from Gaussian realizations and cosmological N-body simulations. In the Gaussian case, the mean band powers and their covariance are well recovered even for irregular (or sparsely) sampled fields. The mild non-Gaussianity of the N-body realizations causes a slight underestimation of the errors that becomes negligible for scales much larger than several arcminutes and does not bias the recovered band powers. These techniques can also be directly applied to cosmic microwave background polarization E and B mode analyses on small fields.

Supersymmetric orientifolds in 6D with D-branes at angles

We study a new class of N=1 supersymmetric orientifolds in six space-time dimensions. The world-sheet parity transformation is combined with a permutation of the internal complex coordinates. In contrast to ordinary orientifolds the twisted sectors contribute to the Klein bottle amplitude leading to new tadpoles to be cancelled by twisted open string sectors. They arise from open strings stretched between D7-branes intersecting at non-trivial angles. We study in detail the Z 3 , Z 4 and Z 6 permutational orientifolds obtaining in all cases anomaly free massless spectra.

Supersymmetric 4D orientifolds of Type IIA with D6-branes at angles

We study a certain class of four-dimensional N=1 supersymmetric orientifolds for which the world-sheet parity transformation is combined with a complex conjugation in the compact directions. We investigate in detail the orientifolds of the Z_3, Z_4, Z_6 and Z_6' toroidal orbifolds finding solutions to the tadpole cancellation conditions for all models. Generically, all the massless spectra turn out to be non-chiral.

Discrete deformations in Type I vacua

We study supersymmetric orientifolds where the world-sheet parity transformation is combined with a conjugation of some compact complex coordinates. We investigate their T-duality relation to standard orientifolds and discuss the origin of continuous and discrete moduli. In contrast to standard orientifolds, the antisymmetric tensor describes a continuous deformation, while the off-diagonal part of the metric is frozen to quantized values and is responsible for the rank reduction of the gauge group. We also give a geometrical interpretation of some recently constructed six-dimensional permutational orientifolds.

Orientifolds of non-supersymmetric, asymmetric orbifolds

We consider certain four-dimensional supersymmetric and non-supersymmetric asymmetric orbifolds with vanishing cosmological constant up to two loops and gauge the world-sheet parity transformation. This leads to new string vacua, in which D p- and D( p - 4)-branes or D p and D(p-4) are identified. Moreover, it is shown that different degrees of supersymmetry can be realized in the bulk and on the brane. We show that for non-supersymmetric models the cosmological constant still vanishes at one-loop order.

A nonunitary version of massless quantum electrodynamics possessing a critical point

Recently, it has been observed that a quantum field theory need not be hermitian to have a real, positive spectrum. What seems to be required is symmetry under combined parity and time-reversal transformations. This idea is extended to massless electrodynamics, in which the photon couples to the axial-vector current with an imaginary coupling constant. The eigenvalue condition necessary for the finiteness of the theory can now be solved; the value for the charge appears to be stable order-by-order. Similarly, the semiclassical Casimir model for the fine-structure constant yields a positive value.

A family of complex potentials with real spectrum

We consider a two-parameter non hermitean quantum-mechanical hamiltonian that is invariant under the combined effects of parity and time reversal transformation. Numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other PT symmetric models, which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.