Undergrad Field Operator for Edge States

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The discussion focuses on confusion regarding the derivation of the field operator for Bogoliubov quasiparticles in edge states, specifically in equation (1.15). The user has a clear understanding of the preceding equation (1.14), which addresses the edge state of the first-quantized Hamiltonian. However, the transition to the field operator in (1.15) is unclear, prompting a request for clarification. Additional references to equations (1.49) and (1.50) are provided, indicating a shift from k-space to a generalized real-space formulation. The conversation highlights the complexities involved in understanding edge state physics and the associated mathematical formulations.
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I'm currently looking at the following set of notes and am confused at equation (1.15) where they discuss the Bogoliubov quasiparticle for the edge states. I understand up to equation (1.14), where they have solved for the edge state of the first-quantized Hamiltonian. What I don't understand is how they derived the field operator in (1.15).
 
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In case people are curious, see the following eqns (1.49) and (1.50) in these notes and references therein. It is a generalized real-space version of what is usually used in k-space.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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