cuprate superconductors $^{4,5,14}$, including the appearance of the dispersing peaks below the high energy and its extinction at the high energy. With increasing doping concentration, the position of the peak along the nodal direction moves towards to the center of the Brillouin zone, while the position of the peak along the antinodal direction is shifted to large momentum region. as, 2 H = t ∑ i ˆ η ( h † i +ˆ η $_{↑}$h$_{i}$$_{↑}$ S + i S − i +ˆ η + h † i +ˆ η $_{↓}$h$_{i}$$_{↓}$ S − i S + i +ˆ $_{η}$) − t ′ i ˆ τ ( h † i +ˆ τ $_{↑}$h$_{i}$$_{↑}$ S + i S − i +ˆ τ + h † i +ˆ τ $_{↓}$h$_{i}$$_{↓}$ S − i S + i +ˆ $_{τ}$) − µ ∑ iσ h † $_{iσ}$h$_{iσ}$ + J$_{eff}$ ∑ i ˆ η S$_{i}$ · S$_{i}$$_{+ˆ}$ $_{η}$, (3) H = − t ∑ i ˆ ησ C † $_{iσ}$C$_{i}$$_{+ˆ}$ ησ + t ′ ∑ i ˆ τ σ C † $_{iσ}$C$_{i}$$_{+ˆ}$ τ σ + µ ∑ iσ C † $_{iσ}$C$_{iσ}$ + J i ˆ η S$_{i}$ · S$_{i}$$_{+ˆ}$ $_{η}$, (2) In cuprate superconductors, the single common feature is the presence of the two-dimensional CuO$_{2}$ plane, and then it is believed that the unconventional physical prop- erties of cuprate superconductors is closely related to the doped CuO$_{2}$ planes $^{18}$. It has been argued that the t - J model on a square lattice, with J$_{eff}$ = (1 − δ ) $^{2}$J , and δ = 〈 h † $_{iσ}$h$_{iσ}$ 〉 = 〈 h † $_{i}$h$_{i}$ 〉 is the hole doping concentration. As a consequence, the kinetic energy term in the t - J model has been transferred as an interaction between charge carriers and spins, which re- flects that even the kinetic energy term in the t - J Hamil- tonian has a strong Coulombic contribution due to the restriction of no double electron occupancy of a given site. For the understanding of the physical properties of cuprate superconductors in the SC state, we have de- veloped a kinetic energy driven SC mechanism $^{17}$, where the interaction between charge carriers and spins arising directly from the kinetic energy term in the t - J model (3) induces a d-wave charge carrier pairing state by exchang- ing spin excitations in the higher power of the hole doping concentration, then the electron Cooper pairs originat- ing from the charge carrier pairing state are due to the charge-spin recombination, and their condensation re- veals the SC ground-state. In particular, this d-wave SC state is controlled by both the SC gap function and the quasiparticle coherent weight, which leads to a fact that the maximal SC transition temperature occurs around the optimal doping, and then decreases in both under- doped and overdoped regimes. Furthermore, it has been shown that this SC state is a conventional BCS-like with the d-wave symmetry $^{25}$, so that the basic BCS formal- ism with a d-wave SC gap function is still valid in quan- titatively reproducing some main low energy features of the SC coherence of quasiparticles, although the pairing mechanism is driven by the kinetic energy by exchanging spin excitations. Within this kinetic energy driven su- perconductivity, we have discussed the low energy elec- tronic structure 21,25 of cuprate superconductors, the dy- namical spin response $^{17,26}$, and the quasiparticle trans- port in the SC state $^{27}$, and qualitatively reproduced some main features of ARPES experiments $^{18}$, inelastic neutron scattering experiments $^{28,29}$, and microwave conductivity measurements 30 on cuprate superconductors. The typi- cal feature of this kinetic energy driven SC mechanism is that the pairing comes out from the kinetic energy by exchanging spin excitations and is not driven by the magnetic superexchange interaction as in the resonant valence bond type theories $^{19}$. In particular, a possible SC theory has been developed for description of super- conductivity in doped hexaborides $^{31}$, where the physical mechanism favoring such a reorientation is the enhanced coherence (and hence lower kinetic energy) of the doped electrons in a ferromagnetic background relative to the paramagnet. Following the previous discussions $^{17,25}$, the charge carrier diagonal and off-diagonal Green’s func- acting on the Hilbert subspace with no double occupied site, i.e., ∑ $_{σ}$C † $_{iσ}$C$_{iσ}$ ≤ 1, captures the essential physics of the doped CuO$_{2}$ plane $^{19}$, where ˆ η = ± ˆ x, ± ˆ y , ˆ τ = ± ˆ x ± ˆ y , C † iσ ( C$_{iσ}$ ) is the electron creation (annihilation) operator, S$_{i}$ = ( S x i , S y $_{i}$, S z i ) are spin operators, and µ is the chem- ical potential. To deal with the constraint of no dou- ble occupancy in analytical calculations, the charge-spin separation (CSS) fermion-spin theory 20 has been devel- oped, where the constrained electron operators are de- coupled as C$_{i}$$_{↑}$ = h † i $_{↑}$S − i and C$_{i}$$_{↓}$ = h † i $_{↓}$S + i , with the spinful fermion operator h$_{iσ}$ = e − i Φ iσ h$_{i}$ describes the charge degree of freedom together with some effects of the spin configuration rearrangements due to the pres- ence of the doped hole itself (charge carrier), while the spin operator S$_{i}$ describes the spin degree of freedom (spin), then the electron local constraint for single oc- cupancy, ∑ $_{σ}$C † $_{iσ}$C$_{iσ}$ = S + i h$_{i}$$_{↑}$h † i $_{↑}$S − i + S − i h$_{i}$$_{↓}$h † i $_{↓}$S + i = h$_{i}$h † $_{i}$( S + i S − i + S − i S + i ) = 1 − h † $_{i}$h$_{i}$ ≤ 1, is satisfied in ana- lytical calculations. In particular, it has been shown 21 that under the decoupling scheme, this CSS fermion- spin representation is a natural representation of the con- strained electron defined in the Hilbert subspace without double electron occupancy. Furthermore, these charge carrier and spin are gauge invariant $^{20,21}$, and in this sense they are real and can be interpreted as physical excitations $^{22}$. This is much different from the usual slave- boson approach $^{23,24}$, where the electron operator is de- composed as the holon and spinon, however, the local constraint for the single occupancy is explicitly replaced by a global constraint in the mean-field level. Due to the constraint, these holon and spinon are also coupled by a strong gauge field $^{23,24}$, allowed by this slave-boson repre- sentation, and therefore these holon and spinon are not gauge invariant. From this point of view, our treatment of constraint for the physical electron may be better than the usual slave-boson approach. In this CSS fermion-spin representation, the t - J Hamiltonian (2) can be expressed