Using the improved Hamiltonian of lattice gauge field and the truncated eigenvalue equation, we compute the glueball mass (mass gap) and glueball wave function of three-dimensional U(1) lattice gauge field. The result shows a good scaling behavior.

The excited k-boson q -coherent state a(q)(+m) \ z, k, j>(q) is constructed. The m dependence of the antibunching effect is numerically studied. It is shown that the antibunching effect evidently depends on m: For big x ( x = \z\(2)), the excitation can change the region where the antibunching effect appears; For small x, when j = 0, the excited states exhibit strong antibunching effect but the unexcited states exhibit strong bunching effect; For small x, when j not equal 0, both the excited states and the unexcited states exhibit strong antibunching effect.

The squeezing property of the excited odd qs-coherent state a(qs)(+m)\a>(o)(qs) and excited even qs-coherent state a(qs)(+m) \a>(e)(qs) is numerically studied. It is shown that: (1) When the q(qless than or equal to1) or s is far from 1, the state a(qs)(+m)\a>(o)(qs) and state a(qs)(+m)\a>(e)(qs) exhibit strong qs-squeezing, and as r(2) increases the qs-squeezing function Delta(1)(e) (Delta(1)(o)) exhibits the oscillating phenomenon of increasing-amplitude and increasing-period. As m increases and s decreases, the amplitude of Delta(1)(e)(Delta(1)(o)) increases greatly. As s decreases, the period of Delta(1)(e)(Delta(1)(o)) increases but is independent of m. The m can be regard as the third parameter for controling the qs-squeezing. (2) In general, the qs-squeezing function Delta(1)(e) (Delta(1)(o)) is more sensitive to s than to q. That is, it is more resultful to control the qs-squeezing by adjusting the parameter s than by adjusting the parameter q.

The excited odd qs-coherent state and excited even qs-coherent state are constructed. The q , s , and m dependences of the antibunching effect are numerically studied. It is shown that for small r , the excited even qs-coherent state exhibits strong antibunching effect but the even qs-coherent state exhibits strong bunching effect; When the q ( q ≤ 1) is far from 1, as r2 increase, the second-order qs-correlation function exhibits oscillating phenomenon (i.e. alternates between antibunching effect and bunching effect) , whose amplitude and period increase as s and q decrease, but are approximately independent of m; When q→1, the second-order qs-correlation function also exhibits oscillating phenomenon, whose amplitude and period not only increase as 5 decreases but also are dependent on m; In general, the second-order qs -correlation function is more sensitive to s than to q .

Using the improved lattice gauge field Hamiltonian and the truncated eigenvalue equation method, we compute the vacuum wave function and mass gap of three-dimensional SU(2) gauge field theory. Our results show that the improved theory leads to a significant reduction of violation of scaling, that is, using the improved lattice gauge field Hamiltonian, the calculations can be carried out up to a much weaker coupling region than using the unimproved one, with good scaling behavior.

The squeezing property of states generated by the excitation on the SUq(1, 1) even and odd q-coherent states (a(q)(+m) \ alpha>(e)(q) and a(q)(+m) \ alpha>(o)(q)) is numerically studied. It is shown that for small q, the state a(q)(+m) \ alpha>(e)(q) and state a(q)(+m) \ alpha>(o)(q) a can exhibit strong q-squeezing, and as r(2) increases the q-squeezing function Delta(1)(e)(Delta(1)(o)) exhibits a wonderful oscillating phenomenon of increasing-amplitude and increasing-period. As m increases and q decreases, the amplitude of Delta(1)(e)(Delta(1)(o)) increases greatly. As q decreases, the period of Delta(1)(e)(Delta(1)(o)) increases but is independent of m.

In this paper, the excited odd q-coherent state a(q)(+ m) \ alpha > (0)(q) and excited even q-coherent state a(q)(+) (m) \ alpha>(e)(q) are constructed. The q and m dependences of mean photon number and sub-Poissonian character and antibunching effect are numerically studied. It is shown that when the q is far from 1, the second-order q-correlation function exhibits oscillating phenomenon, whose amplitude and period are independent of m; but the mean photon number and the Mandel Q(q) parameter increase greatly as m increases; and the sub-Poissonian character is not equivalent to the antibunching effect unless q-->1.

Using a recently developed Hamiltonian Monte Carlo method, we compute the low-lying energy spectrum and wavefunctions as well as thermodynamical observables in (2+1)-dimensional quantum mechanics, and give an estimate of the statistical errors. Our numerical results are in good agreement with the exact ones.

With our recently proposed effective Hamiltonian via Monte Carlo, we are able to compute low energy physics of quantum systems. The advantage is that we can obtain not only the spectrum of ground and excited states, but also wave functions. The previous work has shown the success of this method in (1+1)-dimensional quantum mechanical systems. In this work we apply it to higher dimensional systems.

Using a recently developed Monte Carlo effective Hamiltonian method,we study the low energy physics of 1 + 1 dimensional quantum mechanical system V(x) = mu(2)x(2) + lambda x(4) (here mu(2) < 0,lambda > 0), which is similar to Higgs potential in the standard model of unified electroweak theory. Good results of the spectra, wavefunctions and thermodynamical observables are obtained. It shows that the new Monte Carlo Hamiltonian method has potential application to systems with many degrees of freedom and lattice gauge theory.

Monte Carlo techniques have been widely employed in statistical physics as well as in quantum theory in the Lagrangian formulation. However, in some areas of application to quantum theories computational progress has been slow. Here we present a recently developed approach: the Monte Carlo Hamiltonian method, designed to overcome the difficulties of the conventional approach.

The Wilson fermion Condensates in 1+1 dimensional Lattice QCD are calculated by using the improved Hamiltonian and the variational method. The results are consistant with the predictions from continuum theory, and are almost independent of the Wilson parameter r. The three-links terms give important contribution to [<(psi)over bar>psi] in the improved Hamiltonian theory.

QCD in two dimensions is investigated using the improved fermionic lattice Hamiltonian proposed by Luo, Chen, Xu, and Jiang. We show that the improved theory leads to a significant reduction of finite lattice spacing errors. The: quark condensate and the mass of the lightest quark and antiquark bound states in the strong coupling phase (different from the 't Hooft phase) are computed. We find agreement between our results and the analytical ones in the continuum. [S0556-2821(99)01711-7].

Although most progress in lattice gauge theory has been achieved by use of the Lagrangian formulation, it is worthwhile to keep the Hamiltonian formulation in mind. The potential usefulness of the Hamiltonian lies in the following areas: Non-perturbative computation of scattering cross sections and decay amplitudes in hadronic systems. Low-lying excited states of the hadronic spectrum and quantum chaos in such a system. Hadron wave functions and hadron structure functions (for small xB and Q ). Finite temperature and finite density in baryonic matter (quark-gluon plasma phase transition, neutron stars and cosmology). Atomic physics: study of spectra and of quantum chaos. Condensed matter: study of spin systems (computation of dynamical structure factors), and high Tc superconductivity models (search for electron pair attraction at very small energy). Here we suggest to construct an effective low energy Hamiltonian, denoted by Heff in the following, via Monte Carlo. We suggest to apply the Monte Carlo method for the numerical computation of matrix elements of the propagator, and second for a stochastic selection of states from a basis of Hilbert states. 2. CONSTRUCTION OF Heff

We study QCD in 2 dimensions using the improved lattice fermionic Hamiltonian proposed by Luo, Chen, Xu and Jiang. The vector mass and the chiral condensate are computed for various SU(Nc) gauge groups. We do observe considerable improvement in comparison with the Wilson quark case.

The vector meson mass in 1 + 1 dimensional lattice QCD is calculated by using the variational method. The predictions are proved: The errors which come from the finite lattice spacing can be greatly reduced evidently by improving the lattice Hamiltonian with Wilson quark.

The truncated eigenvalue equation is derived from the improved U(1) lattice gauge field Hamiltonian. The vacuum wave function in 2 + 1 dimensional U(1) model is computed. The numerical results conform our expectation that the scaling behavior of the vacuum wave function can be greatly improved by, improving the lattice Hamiltonian.

Several improved Hamiltonians for lattice gauge theory with fermions are proposed. A test the Schwinger model and some applications to QCD are presented. The results for the mass spectrum indicate that the use of the improved Hamiltonians gives better results.