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by KangSoo Lee and Uwe R. Fischer
I apologize for not finishing this paper. I just open this in the internet. But this document is flawed as it is originally written in LaTeX and not translated to the docuK format perfectly yet. This paper is published in Int. J. Mod. Phys. B 84, 1550021 (2014) and arXiv>condmat>arXiv:1301.2199.
##[.nosecN] Abstract
We introduce a scheme to describe the evolution of an interacting system of bosons, for which the field operator expansion is truncated after a finite number of modes, in a rigorously controlled manner. Using McLachlan's principle of least error , we find a set of equations for the manybody state. As a particular benefit, and in distinction to previously proposed approaches , our approach allows for the dynamical increase of the number of orbitals during the temporal evolution, because we can rigorously monitor the error made by increasing the truncation dimension. The additional orbitals, determined by the condition of least error of the truncated evolution relative to the exact one, are obtained from an initial trial state by a method we call steepest constrained descent.
## PH
 20220706: Update (images, bold, italic and so on.)
 20141217: To the docuK.
 20140623: First posting?
For fermions, the commutator is replaced by the anticommutator \{\ \ , \ \ \} ,
:
\begin{align*}
\{a^{\,}_i, a^\dagger_j\} &\equiv a^{\,}_i a^\dagger_j + a^\dagger_j a^{\,}_i = \delta_{i j}, \\
\{ a^\dagger_i, a^\dagger_j \} &= \{ a^{\,}_i, a^{\,}_j \} = 0.
\end{align*}
Therefore, exchanging disjoint (i.e. i \ne j ) operators in a product of creation or annihilation operators will reverse the sign in fermion systems, but not in boson systems Wiki :: Creation and annihilation operators :: Creation and annihilation operators in quantum field theories
and Wiki :: Second quantization
and Wiki :: Quantum field theory. But I am not currently understanding the reason why the composite of even fermions satisfy bosonic commutation relation
\begin{align*}
[ a^{\,}_i, a^\dagger_j ] &= a^{\,}_i a^\dagger_j  a^\dagger_j a^{\,}_i = \delta_{i j}, \\
[ a^\dagger_i, a^\dagger_j ] &= [ a^{\,}_i, a^{\,}_j ] = 0.
\end{align*}
I thought the composition of any particles in a set state \alpha , the creation and annihilation operators can be expressed like
\begin{align*}
z^\dagger_\alpha = \Big( a^\dagger_{1_\alpha} a^\dagger_{2_\alpha} b^\dagger_{3_\alpha} \cdots c^\dagger_{N_\alpha} \Big) \\
z^{\,}_\alpha = \Big( a^{\,}_{1_\alpha} a^{\,}_{2_\alpha} b^{\,}_{3_\alpha} \cdots c^{\,}_{N_\alpha} \Big) ,
\end{align*}
where a^\dagger means creation of a particle, b^\dagger means creation of b particle, and so on.
Since these a , b , \cdots c particles are fermions, I thought that the commutation relation of composite particle z breaks down.
\begin{align*}
[ z^{\,}_\alpha, z^\dagger_\beta ] &= z^{\,}_\alpha z^\dagger_\beta  z^\dagger_\beta z^{\,}_\alpha = \delta_{\alpha \beta}, \\
[ z^\dagger_\alpha, z^\dagger_\beta ] &= [ z^{\,}_\alpha, z^{\,}_\beta ] = 0.
\end{align*}
\begin{align*}
[ z^{\,}_\alpha, z^\dagger_\beta ] &= \Big( a^{\,}_{1_\alpha} a^{\,}_{2_\alpha} b^{\,}_{3_\alpha} \cdots c^{\,}_{N_\alpha} \Big) \Big( a^\dagger_{1_\beta} a^\dagger_{2_\beta} b^\dagger_{3_\beta} \cdots c^\dagger_{N_\beta} \Big)  \Big( a^\dagger_{1_\beta} a^\dagger_{2_\beta} b^\dagger_{3_\beta} \cdots c^\dagger_{N_\beta} \Big) \Big( a^{\,}_{1_\alpha} a^{\,}_{2_\alpha} b^{\,}_{3_\alpha} \cdots c^{\,}_{N_\alpha} \Big) = \delta_{\alpha \beta}, \\
[ z^\dagger_\alpha, z^\dagger_\beta ] &= [ z^{\,}_\alpha, z^{\,}_\beta ] = 0.
\end{align*}
And I thought any more than double creation of the same state \alpha of z kills any manybody Fock state.
z^\dagger_\alpha z^\dagger_\alpha  \text{any} \rangle = 0 ,
since the individual fermions are created twice on the same state.
So treating a composite particle, which is composed of even fundamental fermion particles, as a boson seems to be wrong for me.
The extension to a true manybody physics, incorporating quantum correlations beyond meanfield, requires, however, vast computational resources when both the number of particles and the interaction between those particles increases. Therefore, a simplification of the problem by truncating the field operator expansion to a finite number of modes (or, as an equivalent term, singleparticle orbitals) has been commonly utilized to obtain results relevant to the prediction of experiments in trapped bosonic quantum gases.
The most extreme truncation, the semiclassical form of meanfield theory, retaining just one orbital, gives the wellknown GrossPitaevskiand Wiki :: Second quantization
and Wiki :: Quantum field theory. But I am not currently understanding the reason why the composite of even fermions satisfy bosonic commutation relation
 Fermionic Algebra and Fock Space.pdf, The University of Texas at Austin

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 A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. Lett. 100, 130401 (2008). %Formation and dynamics of manyboson fragmented states in onedimensional attractive ultracold gases
 P. Bader and U. R. Fischer, Phys. Rev. Lett. 103, 060402 (2009). %Fragmented ManyBody Ground States for Scalar Bosons in a Single Trap
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 We illustrate this statement as follows. The reason why we can use
df = \frac{\partial f}{\partial z}dz + \frac{\partial f}{\partial z^*}dz^* is not becausedz anddz^* are linearly independent differentials, but becausedz anddz^* are differentials, that is infinitesimal quantities;dz^* is in fact 100\% dependent ondz sincedz^* is simply the complex conjugate ofdz . Therefore we cannot simply argue that becausedz anddz^* are independent variations, the partial derivatives\frac{\partial f}{\partial z} and\frac{\partial f}{\partial z^*} must be zero for the stationarity off . There is a rather widespread misunderstanding of this mathematical fact found in the literature.  When the variables and constraints are given in complex form, the method of Lagrange multipliers is expressed by
\frac{\partial f} {\partial z_k^*} = \sum_l \bigg[ \lambda_l \frac{\partial g_l} {\partial z_k^*} + \lambda_l^* \frac{\partial g_l^*} {\partial z_k^*} \bigg] . Thoughg_l (z_k, z_k^*)=c andg_l^* (z_k, z_k^*)=c^* are the same constraint, we have to add the complex conjugate of it in the Lagrange equation. Be careful to note that(\frac{\partial g_l}{\partial z_k^*})^* is not equal to\frac{\partial g_l^*}{\partial z_k^*} . The above equation gives the condition for the stationarity of the realvalued functionf . \vec{m}_k^l indicates the shorthanded notation of\hat{a}_{l}^{\dagger} \hat{a}_{k}  \vec{m} \rangle = \sqrt{(m_{l}+1)m_{k}}  \vec{m}_k^l \rangle which means that, from the given configuration\vec{m} , one particle is removed fromk orbital and one particle is added tol orbital and the state \vec{m}_k^l \rangle is properly normalized. A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. A. 73, 063626 (2006). %General variational manybody theory with complete selfconsistency for trapped bosonic systems.
 J. J. McKeown, D. Meegan, and D. Sprevak, An Introduction to Unconstrained Optimization, IOP Publishing, 1990.
 The steepest descent direction in terms of real variables is given by the gradient:
d\vec{x} =  \vec{\nabla} f . On the other hand, the steepest descent direction in terms of complex variables is given by the differential with respect to the complex conjugate of the variables:dz_k =  \frac{\partial f}{\partial z_k^*} . Then the steepest constrained descent is given by\frac{d z_k} {d \tau} = \Delta(\tau) \bigg[ \frac{\partial f} {\partial z_k^*}  \sum_l \Big[\lambda_l \frac{\partial g_l} {\partial z_k^*} + \lambda_l^* \frac{\partial g_l^*} {\partial z_k^*} \Big] \bigg] where\Delta(\tau) can be any arbitrary positive definite function.  A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. Lett. 99, 030402 (2007). %Role of Excited States in the Splitting of a Trapped Interacting BoseEinstein Condensate by a TimeDependent Barrier
 kipid's blog  Method of Lagrange multipliers
 Second quantization of manybody Quantum mechanics
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