Ginsparg-Wilson operators and a no-go theorem

If one uses a general class of Ginsparg-Wilson operators, it is known that CP symmetry is spoiled in chiral gauge theory for a finite lattice spacing and the Majorana fermion is not defined in the presence of chiral symmetric Yukawa couplings. We summarize these properties in the form of a theorem for the general Ginsparg-Wilson relation.Comment: 8 pages, Latex, references updated, version to appear in Phys. Lett.

Quantum and Classical Gauge Symmetries in a Modified Quantization Scheme

The use of the mass term as a gauge fixing term has been studied by Zwanziger, Parrinello and Jona-Lasinio, which is related to the non-linear gauge $A_{\mu}^{2}=\lambda$ of Dirac and Nambu in the large mass limit. We have recently shown that this modified quantization scheme is in fact identical to the conventional {\em local} Faddeev-Popov formula {\em without} taking the large mass limit, if one takes into account the variation of the gauge field along the entire gauge orbit and if the Gribov complications can be ignored. This suggests that the classical massive vector theory, for example, is interpreted in a more flexible manner either as a gauge invariant theory with a gauge fixing term added, or as a conventional massive non-gauge theory. As for massive gauge particles, the Higgs mechanics, where the mass term is gauge invariant, has a more intrinsic meaning. It is suggested to extend the notion of quantum gauge symmetry (BRST symmetry) not only to classical gauge theory but also to a wider class of theories whose gauge symmetry is broken by some extra terms in the classical action. We comment on the implications of this extended notion of quantum gauge symmetry.Comment: 14 pages. Substantially revised and enlarged including the change of the title. To appear in International Journal of Modern Physics

CP breaking in lattice chiral gauge theory

The CP symmetry is not manifestly implemented for the local and doubler-free Ginsparg-Wilson operator in lattice chiral gauge theory. We precisely identify where the effects of this CP breaking appear.Comment: 3 pages, Lattice2002(chiral

On the separability criterion for continuous variable systems

We present an elementary and explicit proof of the separability criterion for continuous variable two-party Gaussian systems. Our proof is based on an elementary formulation of uncertainty relations and an explicit determination of squeezing parameters for which the P-representation condition saturates the $Sp(2,R)\otimes Sp(2,R)$ invariant separability condition. We thus give the explicit formulas of squeezing parameters, which establish the equivalence of the separability condition with the P-representation condition, in terms of the parameters of the standard form of the correlation matrix. Our proof is compared to the past proofs, and it is pointed out that the original proof of the P-representation by Duan, Giedke, Cirac and Zoller(DGCZ) is incomplete. A way to complete their proof is then shown. It is noted that both of the corrected proof of DGCZ and the proof of R. Simon are closely related to our explicit construction despite their quite different appearances.Comment: Some of the issues related to the previous proofs of the separability criterion, which were only briefly touched upon in the original version, are now explained in more detail. 24 page

Chiral Anomaly for a New Class of Lattice Dirac Operators

A new class of lattice Dirac operators which satisfy the index theorem have been recently proposed on the basis of the algebraic relation $\gamma_{5}(\gamma_{5}D) + (\gamma_{5}D)\gamma_{5} = 2a^{2k+1}(\gamma_{5}D)^{2k+2}$. Here $k$ stands for a non-negative integer and $k=0$ corresponds to the ordinary Ginsparg-Wilson relation. We analyze the chiral anomaly and index theorem for all these Dirac operators in an explicit elementary manner. We show that the coefficient of anomaly is independent of a small variation in the parameters $r$ and $m_{0}$, which characterize these Dirac operators, and the correct chiral anomaly is obtained in the (naive) continuum limit $a\to 0$.Comment: 23 pages. Corrected typos and misprints. Made several sentences more precise, and references up-dated. (To appear in Nucl. Phys. B

Does CHSH inequality test the model of local hidden variables?

It is pointed out that the local hidden variables model of Bell and Clauser-Horne-Shimony-Holt (CHSH) gives $||\leq 2\sqrt{2}$ or $||\leq 2$ for the quantum CHSH operator $B={\bf a}\cdot {\bf \sigma}\otimes ({\bf b}+{\bf b}^{\prime})\cdot {\bf \sigma} +{\bf a}^{\prime}\cdot{\bf \sigma}\otimes ({\bf b}-{\bf b}^{\prime})\cdot{\bf \sigma}$ depending on two different ways of evaluation, when it is applied to a $d=4$ system of two spin-1/2 particles. This is due to the failure of linearity, and it shows that the conventional CHSH inequality $||\leq 2$ does not provide a reliable test of the $d=4$ local non-contextual hidden variables model. To achieve $||\leq 2$ uniquely, one needs to impose a linearity requirement on the hidden variables model, which in turn adds a von Neumann-type stricture. It is then shown that the local model is converted to a factored product of two non-contextual $d=2$ hidden variables models. This factored product implies pure separable quantum states and satisfies $||\leq 2$, but no more a proper hidden variables model in $d=4$. The conventional CHSH inequality $||\leq 2$ thus characterizes the pure separable quantum mechanical states but does not test the model of local hidden variables in $d=4$, to be consistent with Gleason's theorem which excludes non-contextual models in $d=4$. This observation is also consistent with an application of the CHSH inequality to quantum cryptography by Ekert, which is based on mixed separable states without referring to hidden variables.Comment: 17 pages. Progress of Theoretical Physics (in press). A typo in the first version was correcte