We present experimental results of elastic constants as a function of temperature (T ) and magnetic field (H) for a single crystal of PrTi 2 Al 20 for the first time. A clear elastic softening was observed in the temperature dependence of elastic constants, C L =(C 11 +2C 12 +4C 44 )/3 and C T =(C 11 -C 12 +C 44 )/3, being suggestive of a remarkable fluctuation of quadrupolar moments O 20 and O 22 with Γ 3 symmetry. The elastic anomaly associated with a phase transition at 2.0 K exhibits the ferroquadrupolar (FQ) phase behavior under magnetic fields in accordance with the other physical properties such as magnetic susceptibility and specific heat. The softening can be explained well based on the crystalline electric field (CEF) effect model with a Γ 3 ground state. Furthermore, it was found that a coupling constant of the inter-site interaction between relevant quadrupolar moments is positive, indicating the FQ interaction mediated in this system. We argue the FQ scenario and 4 f multiplets of Pr 3+ split by CEF effect in PrTi 2 Al 20
Based on the extended BRST formalism of Batalin, Fradkin and Vilkovisky, we perform a general algebraic analysis of the BRST anomalies in superstring theory of Neveu-Schwarz-Ramond. Consistency conditions on the BRST anomalies are completely solved. The genuine super-Virasoro anomaly is identified with the essentially unique solution to the consistency condition without any reference to a particular gauge for the 2D supergravity fields. In a configuration space where metric and gravitino fields are properly constructed, general form of the super-Weyl anomaly is obtained from the super-Virasoro anomaly as its descendant. We give a novel local action of super-Liouville type, which plays a role of Wess-ZuminoWitten term shifting the super-Virasoro anomaly into the super-Weyl anomaly. These results reveal a hierarchial relationship in the BRST anomalies.
We provide a ERST symmetric version of Yokoyama's Type I gauge on formalism for quantum electrodynamics; the similar theory by Izawa can be considered as a ERST symmetrized Type II theory. With the help of the ERST symmetry, Yokoyama's physical subsidiary conditions are replaced by the Kugo-Ojima type condition. As a result, the formalism becomes applicable even in the background gravitational field. We show how the Hilbert spaces of standard formalism in various gauges are embedded in the single Hilbert space of the present formalism. We also give a path integral derivation of the Lagrangian. § 1. IntroductionIn the standard formalism of canonically quantized gauge theories 1 ),2) we cannot consider the gauge transformation freely. There exists no gauge freedom in the quantum theory, since the theory is defined only after the gauge fixing. Namely, a Hilbert space defined in a particular gauge is quite different from those in other gauges. Thus, if we want to realize the quantum gauge freedom, we need a wider Hilbert space.Yokoyama's gaugeon formalism 3 )-9) provides a wider framework in which we can consider the quantum gauge transformation among a family of Lorentz covariant linear gauges. In this formalism a set of extra fields, so-called gaugeon field, is introduced as the quantum gauge freedom. This theory was first proposed for the quantum electrodynamics 3 )-5) to resolve the problem of gauge parameter renormaliza· tion. )It was also applied later to the Yang-Mills theory.6),g) Thanks to the quantum gauge freedom of this formalism, the gauge parameter independence of the physical S-matrix becomes manifest. )It has also been shown, with the help of certain conjecture, that the wave-function renormalization constant is gauge independent in this formalism. S )The extra gaugeon modes should be removed from the physical Hilbert space since the quantum gauge freedom is unphysical mode. In fact the gaugeon exhibits dipole character and yields negative normed states. To remove these modes Yokoyama imposed the Gupta-Bleuler type subsidiary condition.3 )However, this type of condition does not work well if interaction is present for the gaugeon field. Especially, we cannot use Yokoyama's subsidiary condition in the background gravitational field.In the present paper we improve the subsidiary conditions of Yokoyama's formal-*) Preliminary result was reported at Niigata
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