arXiv subjects

Kazuo Fujikawa

Publications and source records attributed to Kazuo Fujikawa.

At least 127 records · Page 7Linked to original sources

arXiv1999

Note on the Gauge Fixing in Gauge Theory

In the absence of Gribov complications, the modified gauge fixing in gauge theory $ \int{\cal D}A_μ\{\exp[-S_{YM}(A_μ)-\int f(A_μ)dx] /\int{\cal D}g\exp[-\int f(A_μ^{g})dx]\}$ for example, $f(A_μ)=(1/2)(A_μ)^{2}$, is identical to the conventional Faddeev-Popov formula $\int{\cal D}A_μ\{δ(D^μ\frac{δf(A_ν)}{δA_μ})/\int {\cal D}gδ(D^μ\frac{δf(A_ν^{g})} {δA_μ^{g}})\}\exp[-S_{YM}(A_μ)]$ if one takes into account the variation of the gauge field along the entire gauge orbit. Despite of its quite different appearance,the modified formula defines a local and BRST invariant theory and thus ensures unitarity at least in perturbation theory. In the presence of Gribov complications, as is expected in non-perturbative Yang-Mills theory, the modified formula is equivalent to the conventional formula but not identical to it:Both of the definitions give rise to non-local theory in general and thus the unitarity is not obvious. Implications of the present analysis on the lattice regularization are briefly discussed.

hep-th↗

arXiv1999

Implications of $Trγ_{5}=0$ in Lattice Gauge Theory

We analyze the implications of the relation $Trγ_{5}=0$, which is customarily assumed in practical lattice calculations. On the basis of the finite dimensional representations of the Ginsparg-Wilson algebra, it is shown that this relation reflects the species doubling in lattice theory; topological excitations associated with species doublers, which have eigenvalue $2/a$, contribute to $Trγ_{5}$ without any suppression. In this sense, the relation $Trγ_{5}=0$ is valid only when we allow the presence of unphysical states in the Hilbert space; this statement is also valid in the Pauli-Villars regularization. If one eliminates the contributions of the unphysical states, the trace $Trγ_{5}$ is replaced by $TrΓ_{5}\equiv Tr γ_{5}(1 -{1/2}aD)$ which gives rise to the Pontryagin index, to be consistent with the continuum analysis.

hep-lat↗

arXiv1999

Relation $Tr γ_{5}= 0$ and the index theorem in lattice gauge theory

The relation $Tr γ_{5}= 0$ implies the contribution to the trace from unphysical (would-be) species doublers in lattice gauge theory. This statement is also true for the Pauli-Villars regularization in continuum theory. If one insists on $Tr γ_{5}= 0$, one thus inevitably includes unphysical states in the Hilbert space. If one truncates the trace to the contribution from physical species only, one obtains $\tilde{T}r γ_{5} = n_{+} - n_{-}$ which is equal to the Pontryagin index. A smooth continuum limit of $\tilde{T}r γ_{5} = Tr γ_{5}(1-(a/2)D) = n_{+} - n_{-}$ for the Dirac operator $D$ satisfying the Ginsparg-Wilson relation leads to the natural treatment of chiral anomaly in continuum path integral. In contrast, the continuum limit of $Tr γ_{5}= 0$ is not defined consistently. It is shown that the non-decoupling of heavy fermions in the anomaly calculation is crucial to understand the consistency of the customary lattice calculation of anomaly where $Tr γ_{5}= 0$ is used. We also comment on a closely related phenomenon in the analysis of the photon phase operator where the notion of index and the modification of index by a finite cut-off play a crucial role.

hep-lat↗

arXiv1999

A continuum limit of the chiral Jacobian in lattice gauge theory

We study the implications of the index theorem and chiral Jacobian in lattice gauge theory, which have been formulated by Hasenfratz, Laliena and Niedermayer and by Lüscher, on the continuum formulation of the chiral Jacobian and anomaly. We take a continuum limit of the lattice Jacobian factor without referring to perturbative expansion and recover the result of continuum theory by using only the general properties of the lattice Dirac operator. This procedure is based on a set of well-defined rules and thus provides an alternative approach to the conventional analysis of the chiral Jacobian and related anomaly in continuum theory. By using an explicit form of the lattice Dirac operator introduced by Neuberger, which satisfies the Ginsparg-Wilson relation, we illustrate our calculation in some detail. We also briefly comment on the index theorem with a finite cut-off from the present viewpoint.

hep-th↗

arXiv1998

Comment on anomaly matching in N=1 supersymmetric QCD

An attempt is made at a systematic approach to anomaly matching problem in non-Abelian electric-magnetic duality in N=1 supersymmetric QCD. A strategy we employ is somewhat analogous to anomaly analyses in grand unified models where the anomaly cancellation becomes more transparent if one embeds SU(5) multiplets into a multiplet of (anomaly-free) SO(10). A complication arises in the treatment of $U^{AF}_{R}(1)^{3}$ matching where $U^{AF}_{R}(1)$ is anomaly-free $R$ symmetry. It is noted that a relatively systematic analysis of the anomaly matching is possible if one considers the formal breaking sequence of color gauge symmetry: $SU(N_{f})_{c}\to SU(N_{c})_{c}\times SU(\tilde{N}_{c})_{c}$ with $N_{f}= N_{c}+ \tilde{N}_{c}$, where $N_{f}$ stands for the number of massless quarks.

hep-th↗

arXiv1998

Fluctuation-dissipation theorem and quantum tunneling with dissipation at finite temperature

A reformulation of the fluctuation-dissipation theorem of Callen and Welton is presented in such a manner that the basic idea of Feynman-Vernon and Caldeira -Leggett of using an infinite number of oscillators to simulate the dissipative medium is realized manifestly without actually introducing oscillators. If one assumes the existence of a well defined dissipative coefficient $R(ω)$ which little depends on the temperature in the energy region we are interested in, the spontanous and induced emissions as well as induced absorption of these effective oscillators with correct Bose distribution automatically appears. Combined with a dispersion relation, we reproduce the tunneling formula in the presence of dissipation at finite temperature without referring to an explicit model Lagrangian. The fluctuation-dissipation theorem of Callen-Welton is also generalized to the fermionic dissipation (or fluctuation) which allows a transparent physical interpretation in terms of second quantized fermionic oscillators. This fermionic version of fluctuation-dissipation theorem may become relevant in the analyses of, for example, fermion radiation from a black hole and also supersymmetry at the early universe.

hep-th↗

arXiv1998

Fluctuation-dissipation theorem and quantum tunneling with dissipation

We suggest to take the fluctuation-dissipation theorem of Callen and Welton as a basis to study quantum dissipative phenomena (such as macroscopic quantum tunneling) in a manner analogous to the Nambu-Goldstone theorem for spontaneous symmetry breakdown. It is shown that the essential physical contents of the Caldeira-Leggett model such as the suppression of quantum coherence by Ohmic dissipation are derived from general principles only, namely, the fluctuation-dissipation theorem and unitarity and causality (i.e., dispersion relations), without referring to an explicit form of the Lagrangian. An interesting connection between quantum tunneling with Ohmic dissipation and the Anderson's orthogonality theorem is also noted.

hep-th↗

arXiv1998

BRST gauge fixing and the algebra of global supersymmetry

A global supersymmetry (SUSY) in supersymmetric gauge theory is generally broken by gauge fixing. A prescription to extract physical information from such SUSY algebra broken by gauge fixing is analyzed in path integral framework. If $δ_{SUSY}δ_{BRST}Ψ= δ_{BRST}δ_{SUSY}Ψ$ for a gauge fixing ``fermion'' $Ψ$, the SUSY charge density is written as a sum of the piece which is naively expected without gauge fixing and a BRST exact piece. If $δ_{SUSY}δ_{SUSY}δ_{BRST}Ψ= δ_{BRST}δ_{SUSY}δ_{SUSY}Ψ$, the equal-time anti-commutator of SUSY charge is written as a sum of a physical piece and a BRST exact piece. We illustrate these properties for N=1 and N=2 supersymmetric Yang-Mills theories and for a D=10 massive superparticle (or ``D-particle'') where the $κ$-symmetry provides extra complications.

hep-th↗

arXiv1997

Duality in Landau-Zener-Stueckelberg potential curve crossing

It is pointed out that there exists an interesting strong and weak duality in the Landau-Zener-Stueckelberg potential curve crossing. A reliable perturbation theory can thus be formulated in the both limits of weak and strong interactions. It is shown that main characteristics of the potential crossing phenomena such as the Landau-Zener formula including its numerical coefficient are well-described by simple (time-independent) perturbation theory without referring to Stokes phenomena. A kink-like topological object appears in the ``magnetic'' picture, which is responsible for the absence of the coupling constant in the prefactor of the Landau-Zener formula. It is also shown that quantum coherence in a double well potential is generally suppressed by the effect of potential curve crossing, which is analogous to the effect of Ohmic dissipation on quantum coherence.

hep-th↗

arXiv1997

SO(9,1) invariant matrix formulation of supermembrane

An $SO(9,1)$ invariant formulation of an 11-dimensional supermembrane is presented by combining an $SO(10,1)$ invariant treatment of reparametrization symmetry with an $SO(9,1)$ invariant $θ_{R} = 0$ gauge of $κ$-symmetry. The Lagrangian thus defined consists of polynomials in dynamical variables (up to quartic terms in $X^μ$ and up to the eighth power in $θ$), and reparametrization BRST symmetry is manifest. The area preserving diffeomorphism is consistently incorporated and the area preserving gauge symmetry is made explicit. The $SO(9,1)$ invariant theory contains terms which cannot be induced by a naive dimensional reduction of higher dimensional supersymmetric Yang-Mills theory. The $SO(9,1)$ invariant Hamiltonian and the generator of area preserving diffeomorphism together with the supercharge are matrix regularized by applying the standard procedure. As an application of the present formulation, we evaluate the possible central charges in superalgebra both in path integral and in canonical (Dirac) formalism, and we find only the two-form charge $[X^μ, X^ν]$.

hep-th↗

arXiv1997

Duality in potential curve crossing: Application to quantum coherence

A field dependent $su(2)$ gauge transformation connects between the adiabatic and diabatic pictures in the (Landau-Zener-Stueckelberg) potential curve crossing problem. It is pointed out that weak and strong potential curve crossing interactions are interchanged under this transformation, and thus realizing a naive strong and weak duality. A reliable perturbation theory should thus be formulated in the both limits of weak and strong interactions. In fact, main characteristics of the potential crossing phenomena such as the Landau-Zener formula including its numerical coefficient are well-described by simple (time-independent) perturbation theory without referring to Stokes phenomena. We also show that quantum coherence in a double well potential is generally suppressed by the effect of potential curve crossing, which is analogous to the effect of Ohmic dissipation on quantum coherence.

hep-th↗

arXiv1997

An extended q-deformed su(2) algebra and the Bloch electron problem

It is shown that an extended q-deformed $su(2)$ algebra with an extra (``Schwinger '') term can describe Bloch electrons in a uniform magnetic field with an additional periodic potential. This is a generalization of the analysis of Bloch electrons by Wiegmann and Zabrodin. By using a representation theory of this q-deformed algebra, we obtain functional Bethe ansatz equations whose solutions should be functions of finite degree. It is also shown that the zero energy solution is expressed in terms of an Askey Wilson polynomial.

hep-th↗

arXiv1997

On a Lorentz covariant matrix regularization of membrane theories

A Lorentz covariant matrix regularization of membrane thories is studied.It is shown that the action for a bosonic membrane can be defined by matrix regularization in a Lorentz covariant manner. The generator of area preserving diffeomorphism can also be consistently defined by matrix regularization, and we can make the area preserving gauge symmetry manifest. However, the reparametrization BRST charge explicitly depends on a specific basis set introduced to define the matrix regularization. We also briefly comment on an extension of the present formulation to a supermembrane.

hep-th↗

arXiv1997

A Schwinger term in q-deformed su(2) algebra

An extra term generally appears in the q-deformed $su(2)$ algebra for the deformation parameter $q = \exp{ 2 πiθ}$, if one combines the Biedenharn-Macfarlane construction of q-deformed $su(2)$, which is a generalization of Schwinger's construction of conventional $su(2)$, with the representation of the q-deformed oscillator algebra which is manifestly free of negative norm. This extra term introduced by the requirement of positive norm is analogous to the Schwinger term in current algebra. Implications of this extra term on the Bloch electron problem analyzed by Wiegmann and Zabrodin are briefly discussed.

hep-th↗

arXiv1996

Path integral of the hydrogen atom, the Jacobi's principle of least action and one-dimensional quantum gravity

A path integral evaluation of the Green's function for the hydrogen atom initiated by Duru and Kleinert is studied by recognizing it as a special case of the general treatment of the separable Hamiltonian of Liouville-type. The basic dynamical principle involved is identified as the Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the appearance of a gauge freedom is naturally understood. The separation of variables in operator formalism corresponds to a choice of gauge in path integral, and the Green's function is shown to be gauge independent if the operator ordering is properly taken into account. Unlike the conventional Feynman path integral,which deals with a space-time picture of particle motion, the path integral on the basis of the Jacobi's principle sums over orbits in space. We illustrate these properties by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates, and thus avoiding the use of the Kustaanheimo-Stiefel transformation. In the present formulation , the Hamiltonian for Stark effect is converted to the one for anharmonic oscillators with an unstable quartic coupling. We also study the hydrogen atom path integral from a view point of one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. A simple BRST analysis of the problem with an evaluation of Weyl anomaly is presented .

hep-th↗

arXiv1996

A toy model of quantum gravity with a quantized cosmological constant

The path integral for the hydrogen atom, for example, is formulated as a one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. The (renormalized) cosmological constant, which corresponds to the energy eigenvalue, is thus quantized in this model. Possible implications of the quantized cosmological constant are briefly discussed.

hep-th↗

arXiv1996

Exact path integral of the hydrogen atom and the Jacobi's principle of least action

The general treatment of a separable Hamiltonian of Liouville-type is well-known in operator formalism. A path integral counterpart is formulated if one starts with the Jacobi's principle of least action, and a path integral evaluation of the Green's function for the hydrogen atom by Duru and Kleinert is recognized as a special case. The Jacobi's principle of least action for given energy is reparametrization invariant, and the separation of variables in operator formalism corresponds to a choice of gauge in path integral. The Green's function is shown to be gauge independent,if the operator ordering is properly taken into account. These properties are illustrated by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates.

hep-th↗

arXiv1996

Physical Implications of a Vector-like Extension of the Standard Model

Physical implications of a vector-like extension of the standard model for heavier quarks and leptons with $SU(2)\times U(1)$ gauge symmetry and only one Higgs doublet are discussed. This scheme incorporates infinitely many fermions, and the chiral structure is realized by a non-vanishing analytic index of mass matrices. The model is perturbatively well controllable, and the physical Higgs particle generally mediates leptonic as well as quark flavor-changing processes at a rate below the present experimental limit. This model illustrates a general feature of models where some of the fermion masses are of non-Higgs origin.

hep-ph↗