THE LOOP PHASE-FACTOR APPROACH TO GAUGE FIELDS

The gauge field theory is formulated via loop phase factors with a fixed point O as their initial and final point. Let G be the gauge group. When the base space is the Minkowski space E₄, we introduce a set of standard paths Ox （for example, the set of line segments Ox）, where x is arbitrary. The phase factor for the infinitesimal loop Oxx+dxO corresponds to an element in the Lie algebra g and can be expressed as a g-valued differential form k（x, dx） which satisfies the following conditions of consistency （a） k（O, dx）=0, （b） k（x, v）=0, where v is the tangential vector of Ox at x. It is shown that an equivalent class of gauge fields is determined by k（x, dx） or （ad a） k（x, dx） where a is a fixed element of G. Hence if we adopt k（x, dx） for the fundamental physical quantity of a gauge field then a great part of gauge indefiniteness is eliminated. Moreover if the phase factors Φ_xo for standard paths Ox are given then the phase factors for differential arcs x x+dx are easily calculated, and hence a gauge field in the equivalent class is extracted. We call the set of phase factors for standard paths a gauge and k（x, dx） may be interpretated as the gauge potential under a special gauge under which Φ_xo=the unit element of G.The method is useful in considering the equivalence problem and the spacetime symmetry of gauge fields. For example, it is quite easy to determine all spherically symmetric gauge fields if they are free of singularities. By using the method it can also be proved that if two gauge fields have the same gauge and the same field strength then their gauge potentials are equal to each other. Consequently, under a given gauge in the above sense the field strength determines the gauge potential completely.For a general base manifold M_n, every equivalent class of gauge fields over M_n can be defined by loop phase factors also. In this case, M_n is expressed as the sum of a set of neighborhoods each of which is homeomorphic to the Euclidean space. The standard paths are constructed according a certain rule, the phase factors for standard differential loops are also introduced. The transition functions and gauge potentials of a gauge field in the given equivalent class are derived as the phase factors for some finite loops and standard differential loops respectively. Further it is remarkable that a global gauge field is determined completely by the field strength and some discrete loop factors, if the phase factors for the standard paths are gwen.In mathematical terminology principal G-bundle structure as well as a connection in it is determined by the holonomic mapping which maps the set of loops through a fixed point into the group G, provided the mapping is differentible in a certain.The author is very grateful to Prof. Yang Chen Ning for many helpful discussions.     

AN ANALYSIS OF FOUR-ELEMENT MODELS OF LEPTONS AND NON-DIAGONAL NEUTRAL CURRENTS

Due to the possible production of neutral and charge heavy leptons by v_μ, models consisting of four elements per "line", e.g. v_e, e^-, E⁰, E^- v_μ, μ^-, M^0, M^- v_τ, τ^-, T⁰, T^- ……are analysed. Non-diagonal neutral currents cannot be avoided. Nevertheless, a model is proposed within SU_（2）×U_（1） in which the process μ→eγ can still be suppressed by the Bjorken-Lane-Weinberg mechanism; various production cross-sections for heavy leptons are reasonable when compared with experiments. Models corresponding to SU_（3）×U（1） and still higher symmetries are discussed.     

STRONG INTERACTION AND CHIRAL GAUGE FIELD

The concept of e^{iα（x）γ_s} local gauge transformation is introduced in this article. The chiral gauge invariant Lagrangian is given. The relationship between chiral gauge field and strong interaction is also discussed. In a special condition, we obtain the effective Lagrangian for V-A type strong interaction.     

THE CONSTRUCTION OF A TWO-LOOP FINITE SUPERSYMMETRY SU(5) MODEL

By solving the constraint equations of two-loop finiteness another type of two-loop finite SU(5) ground unified model is given in this paper.There is only one pair of light Higgs doublet,and the other Higgs particles are all superheavy (～M_x).The ordinary CKM-mixing can be obtained only in the low energy region.     

THE VERTICAL INTEGRATED MOMENTUM SPECTRUM OF THE MUON WITH MOMENTUM LARGER THAN 4 GeV/c AT 3220m ABOVE SEA-LEVEL

At an altitude of 3220 meters above sea level in Yunnan Province, we measured, by using a G-M counter-magnetic spectrometer, the vertical integratted momentum spectrum of the cosmic muon with momentum larger than 4 GeV/c. The spectrum obtained from this experiment is expressed by a power function in the form of I（>p）=Q （p+4）^-1.86±0.06(cm^-2·sr^-1·s^-1), where p is in GeV/c and Q is a constant. We also measured the charge ratio of muon in the 4—23 GeV/c momentum range. The result is N_μ/N_μ-=1.26±0.11. The error in the momentum measurement with spectrometer has been discussed and calculated.     

A RECURSIVE FORMULA FOR THE CLEBSCH-GORDAN COEFFICIENTS OF SUn GROUP

It is proved that under the Gelfand-Biedenharn phase convention any SU_n singletfactor （SF） belongs to the type of the derivable or underivable,designated as （SU_nSF）d and （SU_r SF） respectively,and （SU_n SF）d=（SU_r SF）,n=r+1,r+2,….Therefore the calculation of any SU_n Clebsch-Gordan coefficients is reduced to thecalculation of a few underivable singlet factor （SU_r SF）.     

FURTHER DISCUSSION ON THE EQUATION AND ITS SOLUTIONS FOR THE BOUND STATES OF MESONS

We continue the investigation of the equation and its solutions for the bound states of mesons discussed previously, emphasizing on large values of v, a parameter characterizing the shape of the potential. The results show that the dependence of the eigenvalues λ and some related physical quantities upon the potential parameters v and E is approximately the same as in our previous work. But the average radii of the bound mesons cease increasing as v increases. For each value of v, there exists a maximum average radius√<R²>_max, and all these √<R²>_max are approximately equal to 12.6 M^-1 （M being the mass of the straton）, which is much smaller than the experimental value of the radius of the π meson. However, if we assume, in addition to the attractive potential, there exists, for the straton-anti-straton superstrong interactions, also a repulsive potential, and, in particular, if we introduce a differencetype potential V_D（r） corresponding to the potential V_v（r） in our previous work, the calculated radii of the bound mesons do approach the experimental value.     

THE SPECTRAL REPRESENTATION OF PROPAGATOR FOR FERMION FIELD AND QUARK CONFINEMENT

The properties of fermion causal Green's function is discussed according to the theory of spectral representation. The definition of effective mass is established from the view point of field theory, and the approximate expression of S_F' (p) as |p²|&#171;M² and the asymptotic formula of S_F' (p) as p²→+∞ is given. In terms of these result the problem of quark confinement is discussed. We find that: (ⅰ) Based on the theory of spectral re presentation the concept of relative confinement is allowed, and if renormalization effect is strong, the quarks can demonstrate definitely lighter effective mass when they are far away from the mass shell; (ⅱ) Based on the theory of spectral representation the concept of absolute confinement is unallowable, even the absolute confinement is taken as the limit of relative confinement. That is if the concept of absolute confinement is introduced, we must discard a part of elementary hypothesis of spectral representation theory.     

ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(I)

In this paper, we analyse the commutation relations of the infintesimal opera-tors of the group SU₃ and find that the eight infinitesimal operators of the groupSU₃ can be written as a scalar operator A, three angular momentum operators (L₁,L_o, L_-1,)and two sets of the irreducible tensor operators of rank 1/2, （T_±1/2,V±1/2）By means of the commutation relations of these operators, all irreducible represen-tations of the group SU₃ can be easily obtained. In this pape, the matrices corresponding to these operators in the irreduciblerepresentation（λμ）, are given; therefore the irreducible representation and its re-presentation space R^λμ are completely defined. Besides, a method for calculatingthe scalar factors of the reduction coefficients and the symmetric relations of thosefactors are also given. As examples, the scalar factors of the reduction coefficientsof （λμ）×（10）, （λμ）×（01）, （λμ）×（20） and （λμ）×（11） are calculated. In the last part of this paper, we define the irreducible tensor operators ofthe group SU₃ and prove the corresponding Wigner-Eckart theory. The method used in the discussion of the group SU₃ be extended to allof the compact simple Lie groups of rank 2 and we shall discuss them in two suc-ceeding papers.     

ON THE PHASE AND THE REPRESENTATION TRANSFORMATIONS OF SUn BARYON AND MESON WAVE FUNCTIONS

The baryon and meson wave functions of SU_n symmetry under the Baird-Bie-denharn phase convention are given explicitly.The phase confusion is clarified andthe matrices of transformations between I-spin,U-spin and V-spin representationsare given for the IR（8）,（27）,（35） of SU₃.     

THE REGGE TRAJECTORY OF MESONS AND THE SU(4) MASS RELATIONS

A relativistic equation of motion of a two particle system under the constraints p_i²-m_i²-U(x²)=0.(i=1,2), where U(x²) is an infinite spherical square well poten-tial, is applied to the SU(4) quark model of mesons. A set of linear mass relations areobtained. Instead of the linear relation between J and M², the Regge trajectory ob-tained is a relation between p_nl² and M², where ζ_nlπ are zeros of the spherical Besselfunctions. The new trajectory seems more satisfactory than the former one.     

THE MULTISYNCHRO-SPHERICAL SOLUTIONS OF THE SU(N)GAUGE THEORY

We study the monopole solutions of the SU(N) gauge group systematically,generalize the current synchro-spherical symmetrical monopole to the multi synchro-spherical symmetrical case,and give the complete classification of the monopole so-lutions.As examples,we have calculated the SU(3) and SU(4) monopoles in detail,and discuss some problems such as angular momenta and topological quantum num-bers.     

A VARIATIONAL METHOD IN LATTICE GAUGE THEORY BASED ON HÖLDER INTEGRAL INEQUALITY

A variational method based on Hölder integral inequality is presented. Taking the parameters in integral inequality as variational parameters, we can calculate the free energy in lattice gauge system. The plaquatte energy of Z₂ 、U(1)、 SU(2) gauge system is found and consistent with that given by Monte Carlo method. But, the higher phase transitions for U(1), SU(2) gauge systems are not found.     

CANONICAL QUANTIZATION OF GAUGE FIELDS(Ⅱ)——SPONTANEOUSLY BROKEN GAUGE FIELDS

In R_ξ gauges a SU（2）spontaneonly broken gauge theory is qnantized within the canonical formalism,and the gauge compensating term is derived.     

SPIN GAINT RESONANCE AND CORE POLARIZATION EFFECTS

The effects of the coupling term between field（Y₀σ₁）_1v and r²（Y₂σ₁）_1v coming fromtensor operator are discussed for the spin gaint dipole resonance.The effects of inclu-ding the field r^λ+1(Y_λ+1σ₁)_λv on the spin gaint multipole resonance and core polariza-tion are considered.     

THE DIFFICULTY OF THE DECAY τ→πντ AND SPIN 3/2 HEAVY LEPTON

We discuss the suppression of the decay of heavy lepton τ into πv_τ,by assuming that τ and/or v_τ carry spin 3/2. For the case of spin 3/2 T and spin 1/2 v_τ, using the weak current ψ_μ^(τ)(a+b_^γ5)Ψ^(v_τ)we find that the πv_τ decay channel relative to the v_τv_ll^-channel is suppressed 12/5 times, while other decay channels are not changed appreciably. △R=σ（e⁺e^-→τ⁺τ^-）/σ（e⁺e^-→μ⁺μ^-） is also discussed.     

A POSSIBLE SU(9) GRAND UNIFIED MODEL

In this letter, we propose a Higgs mechanism which is different from the usualone at the first step of spontaneous symmetry breaking. Using this new Higgs mechanismwe discuss an SU(9) model which can accommodate three generations of light fer-mions and sustains the asymptotic freedom of SU(3)_c.     

MONTE CARLO STUDY OF MANTON'S ACTION FOR FOUR-DIMENSIONAL SU(5) LATTICE GAUGE THEORY

The feature of Manton's lattice action in the SU(5) four-dimensional lattice gauge theory is investigated by means of Monte Carlo simulation. A rather smooth curve of the internal energy is obtained, no discontinuity in the average action perplaquette is found in opposition to the resultes of usually adopted Wilson's action. An evidence of a higher order phase transition with Manton's action is also found.     

THE SU2×SU2 BASIS AND THE PHYSICAL BASES FOR THE STATE VECTORS OF d-BOSON SYSTEMS AND THE TRACELESS BOSON OPERATORS (Ⅰ)

The group chain U₅O₅SU₂×SU₂ used by K. T. Hecht (1965) and by the othersprovides an important representation for expressing the physical basis of d-boson sys-tems. However the methods which have been introduced for this SU₂×SU₂ representa-tion to construct a physical basis is poorer in comparison with those for the otherrepresentations. In view of this we try to find appropriate methods to obtain the SU₂×SU₂ representation wave functions of the existing physical bases constructed byChacon et al. and by Szpikowski et al., In the present paper we analyse the SU₂×SU₂ tensor properties of the,bosonoperators and Vilenkin's traceless boson operators and express succinctly the elementaryvectors of the SU₂×SU₂ basis, the |PP SU₂×SU₂ > vectors, in terms of the tracelessoperators. With the help of this form of the| PP SU₂ SU₂> vectors we derive a simpleformula for obtaining the SU₂×SU₂ -representation wave functions of a physical basisfrom its (n_μ)-representation wave functions. Thus the problem mentioned above is partlysolved. The other parts of the solution of the problem will be found in a coming paper.     

THE ANALYTIC RESULTS FOR U1 AND SU2 AVERAGE PLAQUETTE TO THE THIRD ORDER APPROXIMATION IN THE CUMULANT EXPANSION

In the lattice gauge theory, by using cumulant expansion, we calculate the average plaquette for U₁ and SU₂ group up to the third order approximation in 4 and 5 dimensions analytically. Our results are consistent with MC data. We also discuss the application of cumulant expansion techneque in lattice theory.