Use the Principal element in these ways:
As a result, we explicitly obtain the fully gauge invariant partition function, which includes a new type of Wess-Zumino WZ term irrelevant to the gauge symmetry as well as usual WZ action. After their works, we have also quantized other several inter- esting models Sqsm ii by using this BFT formalism.
However, these works  are mainly based on the systematic, but somewhat cumbersome construction of the first class Hamilto- nian as a solution of commuting with the first class constraints.
Here, these BFT fields are obtained from the requirement of commuting with the first class constraints. In fact, we find that the final quantum action in a much simpler form and the final configuration space partition function 2 has the prevailing gauge symmetry contrast to the previous works [7, 11] in conformity with the effectively first class constraint structure of the model.
The organization of the paper is as follows: Sqsm ii section III, according to our recently improved BFT method, we then directly obtain the desired first class Hamiltonian from the canonical Hamiltonian by simply replacing the original fields with the corresponding BFT physical variables defined in the extended phase space and recover the Dirac bracket DB in the original phase space.
In section IV, we obtain the first class quantum Lagrangian, which includes new type of WZ NWZ action which can not be obtained in the usual Faddeev- Popov like path integral framework  as well as the well known WZ terms, and the final theory has the prevailing gauge symmetry contrast to the previous works [7, 11].
This becomes possible by resolving the problem of the non-trivial 5-function in the measure part, which exists in the previous works [7,11]. Section V is devoted to conclusion.
De- pending on the value of a, three different mass spectrums are expected: In addition, these differences are reflected also in the constraint 3 structures: We find that all the constraints are fully second class constraints. In order to carry out simple algebraic manipulations i.
Now, let us introduce new auxiliary fields in order to convert the second class con- straints f2j into first class ones in the extended phase space. Then, the first class constraint algebra of fli imposes the following condition: Therefore, we have all the first class constraints in the extended phase space with only contributing in the series Moreover, Tj may be assumed to be independent of the momenta so that these are considered as FP type gauge conditions .
Before performing the momentum integrations to obtain the desired partition function in the configuration space, it seems appropriate to comment on the strongly involutive Hamiltonian As we have mentioned, all the theories for the extended phase space can be obtained just by replacinging the original variables T with the BFT physical variables T in the theories for the original phase space.
Note that although the Maxwell term appearing in 1 is absent in 33 due to the constraint Q3 in 4the constraints arising from the action 33 are identical to those obtained from 1 and this situation coincides with the previous work .
Now, let us return to the partiton function To obtain the partition function in the configuration space, we perform the momentum integrations by taking the proper order for a simpler calculation without any loss of generality. However, we will show that this problem can be resolved by the applying the Fujiwara's technique of the covariant path integral evaluation [3, 15].FA_ SQSM For Later.
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