The flux phase solutions in the t-J model, the gauge theories with Chern-Simons term, promising new exciting physics, are so far obtained only within slave-boson (or fermion) approaches, where local gauge invariance is not taken into account exactly. We inspect the possibility to do this, mainly, in the example of the t-J model. The analysis of different parametrizations in the functional representation shows that (i) calculations which take into account constraints for the number of on-site-available states are describing quasiparticles in terms of the wrong local statistics, and contain gauge non-invariant objects; (ii) application of the projection technique in the slave-boson(fermion) representation reproduces the correct statistics, and is exactly equivalent to the conventional diagram technique for Hubbard operators. In the language of slave bosons this means that the Fourier transformation leading to Bloch states should be made only after accounting constraints in the measure of functional integral.
Representations for the generating functionals of static correlators of
z-components of spins in the XY and
XX Heisenberg spin chains are obtained in the form of sums of the fermionic functional integrals. The peculiarity of the functional integrals in question is because of the fact that the integration variables depend on the imaginary time ``automorphically''. In other words, the integration variables are multiplied with a certain complex number when the imaginary time is shifted by a period. Therefore, the corresponding boundary conditions at the ends of the imaginary time segment are not of the form corresponding to fermionic, or bosonic, variables taken in the Matsubara representation at nonzero temperature. In fact, one part of sites of the models corresponds to the integration variables which are subjected to the unusual boundary conditions, while the variables on the other sites depend on the imaginary time conventionally, i.e., as fermions (or bosons). Thus a situation, when an ``automorphic'' boundary condition is the same for all sites of a chain spin model, is generalized. The results of the functional integration are obtained in the form of determinants of the matrix operators which are regularized by means of the generalized zeta-function approach. The partition functions of the models and certain correlation functions at nonzero temperature are obtained explicitly thus demonstrating correctness of the functional integral representations proposed.