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Andrius Kulikauskas

  • m a t h 4 w i s d o m - g m a i l
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  • My work is in the Public Domain for all to share freely.

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  • 读物 书 影片 维基百科

Introduction E9F5FC

Questions FFFFC0

Software

Bott periodicity, Hamiltonians, Modeling the self

Write out the Lie algebra decomposition for the Lie algebra embeddings.

Lie algebra decomposition

  • Relate the perspective's symmetry with the equations for {$C, T, S$}.

Note that {$\mathbb{H}_{ij}=H^{(0)}\partial_{ij}+H^{(1)}_{ij}$}. I think {$H^{(0)}$} is just a constant, a scalar multiple of the identity. When {$i\neq j$} then we can simply deal with {$H^{(1)}$}. But with the diagonal elements we have to be careful because with regard to them we may have {$H^{(1)T}\neq H^{(1)*}$} when {$H^{(0)}_{jj}=a+bi$} and {$H^{(1)}_{jj}=c-bi$} for they can compensate each other's imaginary diagonal elements.

Note that Stone-Chiu-Roy relate {$J_k$} with the second quantization operators {$\mathcal{T}, \mathcal{C}$}

Hypothesis: Perspective

My hypothesis: A perspective {$J_k$} is an operator which establishes a symmetry of the form {$J_kH^*=HJ_k$}.

  • {$\mathbf{U_TH^*U_T^\dagger=H}$} symmetry on {$H^*$} accessible
  • {$\mathbf{U_CH^TU_C^\dagger=-H}$} symmetry on {$H^T$} inaccessible
  • {$\mathbf{U_SHU_S^{-1}=-H}$} symmetry on {$H$} defined

In Stone-Chiu-Roy we have

  • linear {$SH=-HS$}
  • antilinear {$TH^*T^{-1}=H$}
  • antilinear {$CH^*C^{-1}=-H$}

Thus time reversal establishes a symmetry of the kind we want. And it commutes with the Hamiltonian by considering the newest {$P_{k+1}$} which anticommutes with {$m\in\frak{m}_k$} for {$H=J_1m$}. But the operator may not always exist in the desired space or square as required.

Hypothesis: Shift in perspective

A shift in perspective may be the consequence of the reduction of the space to its subspace. Then operators with off-diagonal blocks may be thrown out. And operators may be expressed in terms of each other, {$J_2J_1=J_3$}, on the smaller subspace but not on the larger one.

Note that in this smaller subspace {$J_2J_1=J_3$} implies {$J_3J_2=J_2J_1J_2=J_1$} and {$J_1J_3=J_3J_2J_3=J_2$}.

Note that {$S=J_2J_3$} anticommutes with {$H_3$} and commutes with {$J_1=i$}. What does this mean for the shift in perspective?

A shift in perspective may perhaps be understood as multiplying by {$J_1=i$}, yielding a dependent perspective, as with {$J_3=J_2J_1$}.

Twofold complex Bott periodicity

Lie algebra decomposition {$\frak{g}_i=\frak{h}_i+\frak{m}_i$}{$\frak{u}$}{$(M)\oplus\frak{u}$}{$(M)=\frak{u}$}{$(M)+\frak{m}$}{$\frak{u}$}{$(2M)=\frak{u}$}{$(M)\oplus\frak{u}$}{$(M)+\frak{m}$}
Operator yielding decomposition{$iJ_1$}{$iJ_2$}
{$2\times 2$} blocks of Lie subalgebra {$\frak{h}_i$}{$\begin{pmatrix} z & \\ & z \\ \end{pmatrix}$}
{$z\in\mathbb{C}$}		{$\begin{pmatrix} ? & ? \\ ? & ? \\ \end{pmatrix}$}
{$2\times 2$} blocks of subspace {$\frak{m}_i$}{$\begin{pmatrix} z & \\ & -z \\ \end{pmatrix}$}
{$z\in\mathbb{C}$}		{$\begin{pmatrix} ? & ? \\ ? & ? \\ \end{pmatrix}$}
Hamiltonian {$H=im$}{$\mathbf{H=H^\dagger}$}{$\begin{pmatrix} & \mathbf{h}_{pq} \\ \mathbf{h}_{pq}^\dagger & \\ \end{pmatrix}$}
Quantum symmetries {$S^2=1$}

Eightfold real Bott periodicity

{$\frak{o}$}{$(2M)=\frak{u}$}{$(M)+\frak{m}$}{$\frak{u}$}{$(2M)=\frak{sp}$}{$(M)+\frak{m}$}{$\frak{sp}$}{$(2M)=\frak{sp}$}{$(M)\oplus\frak{sp}$}{$(M)+\frak{m}$}{$\frak{sp}$}{$(M)\oplus\frak{sp}$}{$(M)=\frak{sp}$}{$(M)+\frak{m}$}
{$J_1$}{$J_2$}{$J_3$}{$J_4$}
{$\begin{pmatrix} a & -b \\ b & a \\ \end{pmatrix}$}
{$a+bi\in\mathbb{C}$}		{$\begin{pmatrix} X & Y \\ -\overline{Y} & \overline{X} \\ \end{pmatrix}$}
{$X+Yj\in\mathbb{H}$}		{$\begin{pmatrix} q_1 & \\ & q_2 \\ \end{pmatrix}$}
{$q_1,q_2\in\mathbb{H}$}		{$\begin{pmatrix} q & \\ & q \\ \end{pmatrix}$}
{$q\in\mathbb{H}$}
{$\begin{pmatrix} a & b \\ b & -a \\ \end{pmatrix}$}
{$a,b\in\mathbb{R}$}		{$\begin{pmatrix} X & Y \\ \overline{Y} & -\overline{X} \\ \end{pmatrix}$}
{$X,Y\in\mathbb{C}$}		{$\begin{pmatrix} & q_1 \\ q_2 & \\ \end{pmatrix}$}
{$q_1,q_2\in\mathbb{H}$}		{$\begin{pmatrix} q & \\ & -q \\ \end{pmatrix}$}
{$q\in\mathbb{H}$}
{$\begin{pmatrix} & \mathbf{h}_{MM} \\ -\mathbf{h}_{MM}^* & \\ \end{pmatrix}$}
{$m=\mathbf{h}_{MM}=-\mathbf{h}_{MM}^*$}		{$\begin{pmatrix} \mathbf{h}_{aa} & \mathbf{h}_{ab} \\ -\mathbf{h}_{ab}^* & \mathbf{h}_{aa}^* \end{pmatrix}$}{$\begin{pmatrix} & \mathbf{h}_{rr} & \mathbf{h}_{rs} \\ & -\mathbf{h}_{rs}^* & \mathbf{h}_{rr}^* \\ \textrm{h.c} & & \\ \end{pmatrix}$}{$\begin{pmatrix} \mathbf{h}_{aa} & \mathbf{h}_{ab} \\ \mathbf{h}_{ab}^* & -\mathbf{h}_{aa}^* \end{pmatrix}$}
{$C^2=+1, T^2=-1, S^2=1$}{$T^2=-1$}{$T^2=-1,C^2=-1,S^2=1$}{$C^2=-1$}
{$\frak{sp}$}{$(M)=\frak{u}$}{$(M)+\frak{m}$}{$\frak{u}$}{$(M)=\frak{o}$}{$(M)+\frak{m}$}{$\frak{o}$}{$(2M)=\frak{o}$}{$(M)\oplus\frak{o}$}{$(M)+\frak{m}$}{$\frak{o}$}{$(M)\oplus\frak{o}$}{$(M)=\frak{o}$}{$(M)+\frak{m}$}
{$J_5$}{$J_6$}{$J_7$}{$J_8$}
{$\begin{pmatrix} z & \\ & z \\ \end{pmatrix}$}
{$z\in\mathbb{C}\subset\mathbb{H}$}		{$\begin{pmatrix} a & \\ & a \\ \end{pmatrix}$}
{$a\in\mathbb{R}\subset\mathbb{H}$}		{$\begin{pmatrix} a_1 & \\ & a_2 \\ \end{pmatrix}$}
{$a_1,a_2\in \mathbb{R}\subset M_8(\mathbb{R})$}		{$\begin{pmatrix} a & \\ & a \\ \end{pmatrix}$}
{$a\in\mathbb{R}\subset M_{8}(\mathbb{R})$}
{$\begin{pmatrix} zj & \\ & zj \\ \end{pmatrix}$}
{$z=a+bi\in\mathbb{C}\subset\mathbb{H}$}
{$j\in\mathbb{H}$}		{$\begin{pmatrix} bi & \\ & bi \\ \end{pmatrix}$}
{$b\in\mathbb{R}\subset\mathbb{H}, i\in\mathbb{H}$}		{$\begin{pmatrix} & a_1 \\ a_2 & \\ \end{pmatrix}$}
{$a_1,a_2\in \mathbb{R}\subset M_8(\mathbb{R})$}		{$\begin{pmatrix} a & \\ & -a \\ \end{pmatrix}$}
{$a\in\mathbb{R}\subset M_{8}(\mathbb{R})$}
{$\begin{pmatrix} & \mathbf{h}_{MM} \\ \mathbf{h}_{MM}^* & \\ \end{pmatrix}$}
{$\mathbf{h}_{MM}=\mathbf{h}_{MM}^T$}		{$\mathbf{H=H^*}$}{$\begin{pmatrix} & \mathbf{h}_{pq} \\ \mathbf{h}_{pq}^T & \\ \end{pmatrix}$}
{$\mathbf{h}_{pq}=\mathbf{h}_{pq}^*$}		{$\mathbf{H=-H^*=\begin{pmatrix} & -m \\ m & \\ \end{pmatrix}}$}
{$C^2=-1, T^2=+1, S^2=1$}{$T^2=+1$}{$C^2=+1, T^2=+1, S^2=1$}{$C^2=+1$}

Analyzing the Hamiltonians

Lie algebra decomposition
{$\frak{g}_i=\frak{h}_i+\frak{m}_i$}		{$2\times 2$} blocks of
Lie subalgebra {$\frak{h}_i$}		{$2\times 2$} blocks of
subspace {$\frak{m}_i$}		Hamiltonian {$H$}simplified Hamiltonian {$\{\mathbf{Q}(k)\}$}remarks
{$\frak{u}$}{$(M)\oplus\frak{u}$}{$(M)=\frak{u}$}{$(M)+\frak{u}$}{$(M)$}   {$G_{m,m+n}(C)$} 
{$\frak{u}$}{$(2M)=\frak{m}$}{$+\frak{u}$}{$(M)\oplus\frak{u}$}{$(M)$}   {$\begin{pmatrix} \mathbf{0}_n & \mathbf{q} \\ \mathbf{q}^\dagger & \mathbf{0}_m \\ \end{pmatrix}$}
{$ \mathbf{q}\in U(m)$}		 
Lie algebra decomposition
{$\frak{g}_i=\frak{h}_i+\frak{m}_i$}		{$2\times 2$} blocks of
Lie subalgebra {$\frak{h}_i$}		{$2\times 2$} blocks of
subspace {$\frak{m}_i$}		Hamiltonian
{$H$}		simplified Hamiltonian {$\{\mathbf{Q}(k)\}$}remarks
{$\frak{o}$}{$(M)\oplus\frak{o}$}{$(M)=\frak{m}$}{$+\frak{o}$}{$(M)$}{$\begin{pmatrix} a & \\ & a \\ \end{pmatrix}$}
{$a\in\mathbb{R}$}		{$\begin{pmatrix} a & \\ & -a \\ \end{pmatrix}$}
{$a\in\mathbb{R}$}		{$\mathbf{H=-H^*=\begin{pmatrix} & -m \\ m & \\ \end{pmatrix}}$}{$G_{m,2m}(\mathbb{C})$}
{$\tau_x\mathbf{Q}(k)^*\tau_x =-\mathbf{Q}(-k)$}		 
{$\frak{o}$}{$(2M)=\frak{m}$}{$+\frak{u}$}{$(M)$}{$\begin{pmatrix} a & -b \\ b & a \\ \end{pmatrix}$}
{$a+bi\in\mathbb{C}$}		{$\begin{pmatrix} a & b \\ b & -a \\ \end{pmatrix}$}
{$a,b\in\mathbb{R}$}		{$\begin{pmatrix} & \mathbf{h}_{MM} \\ -\mathbf{h}_{MM}^* & \\ \end{pmatrix}$}
{$m=\mathbf{h}_{MM}=-\mathbf{h}_{MM}^*=-\mathbf{h}_{MM}^T$}		{$\begin{pmatrix} \mathbf{0}_n & \mathbf{q} \\ \mathbf{q}^\dagger & \mathbf{0}_m \\ \end{pmatrix}$}
{$ \mathbf{q}\in U(2m), \mathbf{q}(k)^T=-\mathbf{q}(-k)$}		{$J_1=S$}
{$J_1H_1^*=H_1J_1$}
absolute frame
{$\frak{u}$}{$(2M)=\frak{m}$}{$+\frak{sp}$}{$(M)$}{$\begin{pmatrix} X & Y \\ -\overline{Y} & \overline{X} \\ \end{pmatrix}$}
{$X+Yj\in\mathbb{H}$}		{$\begin{pmatrix} X & Y \\ \overline{Y} & -\overline{X} \\ \end{pmatrix}$}
{$X,Y\in\mathbb{C}$}		{$\begin{pmatrix} \mathbf{h}_{aa} & \mathbf{h}_{ab} \\ -\mathbf{h}_{ab}^* & \mathbf{h}_{aa}^* \end{pmatrix}$}{$G_{2m,2m+2n}(\mathbb{C})$}
{$J\mathbf{Q}(k)^*=\mathbf{Q}(-k)J$}		{$J_2=T$}
{$J_2H_2^*=H_2J_2$}
{$\frak{sp}$}{$(2M)=\frak{m}$}{$+\frak{sp}$}{$(M)\oplus\frak{sp}$}{$(M)$}{$\begin{pmatrix} q_1 & \\ & q_2 \\ \end{pmatrix}$}
{$q_1,q_2\in\mathbb{H}$}		{$\begin{pmatrix} & q_1 \\ q_2 & \\ \end{pmatrix}$}
{$q_1,q_2\in\mathbb{H}$}		{$\begin{pmatrix} & \mathbf{h}_{rr} & \mathbf{h}_{rs} \\ & -\mathbf{h}_{rs}^* & \mathbf{h}_{rr}^* \\ \textrm{h.c} & & \\ \end{pmatrix}$}{$\begin{pmatrix} \mathbf{0}_n & \mathbf{q} \\ \mathbf{q}^\dagger & \mathbf{0}_m \\ \end{pmatrix}$}
{$ \mathbf{q}\in U(2m), J\mathbf{q}(k)^*=\mathbf{q}(-k)J$}		{$J_3=T$}
{$J_3H_3^*=H_3J_3$}
{$\frak{sp}$}{$(M)\oplus\frak{sp}$}{$(M)=\frak{m}$}{$+\frak{sp}$}{$(M)$}{$\begin{pmatrix} q & \\ & q \\ \end{pmatrix}$}
{$q\in\mathbb{H}$}		{$\begin{pmatrix} q & \\ & -q \\ \end{pmatrix}$}
{$q\in\mathbb{H}$}		{$\begin{pmatrix} \mathbf{h}_{aa} & \mathbf{h}_{ab} \\ \mathbf{h}_{ab}^* & -\mathbf{h}_{aa}^* \end{pmatrix}$}{$G_{m,2m}(\mathbb{C})$}
{$\tau_y\mathbf{Q}^*(k)\tau_y =-\mathbf{Q}(-k)$}		 
{$\frak{sp}$}{$(M)=\frak{m}$}{$+\frak{u}($}{$M)$}{$\begin{pmatrix} z & \\ & z \\ \end{pmatrix}$}
{$z\in\mathbb{C}\subset\mathbb{H}$}		{$\begin{pmatrix} zj & \\ & zj \\ \end{pmatrix}$}
{$z=a+bi\in\mathbb{C}\subset\mathbb{H}$}
{$j\in\mathbb{H}$}		{$\begin{pmatrix} & \mathbf{h}_{MM} \\ \mathbf{h}_{MM}^* & \\ \end{pmatrix}$}
{$\mathbf{h}_{MM}=\mathbf{h}_{MM}^T$}		{$\begin{pmatrix} \mathbf{0}_n & \mathbf{q} \\ \mathbf{q}^\dagger & \mathbf{0}_m \\ \end{pmatrix}$}
{$ \mathbf{q}\in U(m), \mathbf{q}(k)^T=\mathbf{q}(-k)$}		 
{$\frak{u}$}{$(M)=\frak{m}$}{$+\frak{o}$}{$(M)$}{$\begin{pmatrix} a & \\ & a \\ \end{pmatrix}$}
{$a\in\mathbb{R}\subset\mathbb{H}$}		{$\begin{pmatrix} bi & \\ & bi \\ \end{pmatrix}$}
{$b\in\mathbb{R}\subset\mathbb{H}, i\in\mathbb{H}$}		{$\mathbf{H=H^*}$}{$G_{m,m+n}(\mathbb{C})$}
{$\mathbf{Q}(k)^*=\mathbf{Q}(-k)$}		 
{$\frak{o}$}{$(2M)=\frak{m}$}{$+\frak{o}$}{$(M)\oplus\frak{o}$}{$(M)$}{$\begin{pmatrix} a_1 & \\ & a_2 \\ \end{pmatrix}$}
{$a_1,a_2\in \mathbb{R}\subset M_8(\mathbb{R})$}		{$\begin{pmatrix} & a_1 \\ a_2 & \\ \end{pmatrix}$}
{$a_1,a_2\in \mathbb{R}\subset M_8(\mathbb{R})$}		{$\begin{pmatrix} & \mathbf{h}_{pq} \\ \mathbf{h}_{pq}^T & \\ \end{pmatrix}$}
{$\mathbf{h}_{pq}=\mathbf{h}_{pq}^*$}		{$\begin{pmatrix} \mathbf{0}_n & \mathbf{q} \\ \mathbf{q}^\dagger & \mathbf{0}_m \\ \end{pmatrix}$}
{$ \mathbf{q}\in U(m), \mathbf{q}(k)^*=\mathbf{q}(-k)$}		 
{$\frak{o}$}{$(M)\oplus\frak{o}$}{$(M)=\frak{m}$}{$+\frak{o}$}{$(M)$}{$\begin{pmatrix} a & \\ & a \\ \end{pmatrix}$}
{$a\in\mathbb{R}\subset M_{8}(\mathbb{R})$}		{$\begin{pmatrix} a & \\ & -a \\ \end{pmatrix}$}
{$a\in\mathbb{R}\subset M_{8}(\mathbb{R})$}		{$\mathbf{H=-H^*}$}  
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This page was last changed on May 11, 2025, at 08:28 AM