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$\quad$Using the set of independent Symmetric Tamm-Dancoff (STD) type $q$-deformed quantum oscillators [1] with the standard linear-in-$N$ Hamiltonian, we construct respective gas of $q$-bosons. Such a deformed $q$-Bose gas model admits [2] exact analytic expressions for the $r$-particle momentum correlation function, and corresponding intercepts, of any $r$th order. This is the third example of deformed oscillators' based Bose gas model, along with the $qp$-Bose gas model [3] and the $\mu$-deformed one [4] (based on Jannussis's oscillator), where correlation average $\langle (a^\dagger_{\bf k})^r (a_{\bf k})^r \rangle$ was calculated explicitly. In view of the formal resemblance between the $qp$-bracket and the STD deformation structure function, with $N$-multiplier ignored, we apply the idea of the derivation from [3] for $qp$-bosons, to the current case of STD $q$-bosons.
$\quad$Indeed, using just under-trace algebraic operations, for the STD $q$-bosons in a fixed $\bf k$-mode we find
$\qquad\qquad \langle a^\dagger a\rangle_\mathrm{STD}
= \frac1{(e^{\beta\omega} -\, q)^2}\,
\biggl\{\
\Bigl\langle
\frac{\bigl[a, [a,a^\dagger]_q \bigr]_q\, a^\dagger}{\varphi_\mathrm{STD}(N+1)}
\Bigr\rangle
+ \frac{ [1]^\mathrm{STD}_q!\, e^{\beta\omega} }{\mathrm{Tr}\, e^{-\beta\omega N}}
\biggr\}$
versus $\langle a^\dagger a\rangle_{qp} = \frac{[1]_{qp}!}{e^{\beta\omega}\,-\,q}
\, \bigl\langle[a,a^\dagger]_q\bigr\rangle$ for $qp$-bosons, see [3]. What concerns the quadratic correlations, using the same approach of under-trace transformations we derive
$\qquad \bigl\langle a^{\dagger2}a^2 \bigr\rangle_\mathrm{STD}
=
\frac1{(e^{\beta\omega} -\, q^2)^3 (e^{\beta\omega} -\, 1)^3}\
\biggl\{\
\Bigl\langle \
\sum\limits_{i=0}^4 A_i(q)\; \textrm{g}_q(N\!+\!1\!+\!i)\,\textrm{g}_q(N\!+\!i)
\cdot q^{-2N} \Bigr\rangle$
$\qquad\qquad\qquad\qquad
-\ (e^{\beta\omega}-1)\, \textrm{g}_q(0) P_2\bigl(q;e^{\beta\omega}\bigr)
\biggr\}\
+\
\varphi_\mathrm{STD}(2)! \frac{e^{\beta\omega}-\,1}{(e^{\beta\omega}\,-\,q^2)^3}$
where underlying role belongs to the double commutator as well, through $\textrm{g}_q(N)$-function, $[a,[a,a^\dagger]_q]_q = - \textrm{g}_q({N+1})\, q^{1-N} \cdot a$, and $P_2(q;e^{\beta\omega})$ is some quadratic polynomial in $e^{\beta\omega}$, with coefficients depending on $q$.
- W.S. Chung, A.M. Gavrilik, I.I. Kachurik,
A.P. Rebesh: J. Phys. A: Math. Theor. 47, 305304 (2014). - A.M. Gavrilik, Yu.A. Mishchenko, arXiv preprint,
arXiv:1410.0538 (2014). - L.V. Adamska, A.M. Gavrilik, J. Phys. A: Math. Gen.
37, 4787-4796 (2004). - A.M. Gavrilik, Yu.A. Mishchenko, Phys. Lett. A
376, 2484-2489 (2012).
