the above \_k = \_[k]{} - , f(k) = where $k$ is a half-integer and similar formulas hold with $k$ replaced by an integer $n$ for the R-sector.

The interface between the massless and massive theories {#mass_interface}
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Bogolyubov transformation (\[BogR1\]), (\[BogR2\]) is a proper Bogolyubov transformation in the terminology of [@Berezin] which means that it is implemented by a unitary operator $U: {\cal H}_{0}\to {\cal H}_{0}$ so that U |a\^\_[k]{} U\^[\*]{} & = & f(k)(|a\^\_[k]{} - i a\_[k]{}) ,\
U a\^\_[k]{}U\^[\*]{}&=&f(k)(a\_[k]{}\^ + i |a\_[k]{} ) and similarly in the R-sector with the zero modes transforming as U a\_[0]{} U\^[\*]{} = a\_[0]{} , U |a\_[0]{} U\^[\*]{} = [sign]{}(m)|a\_[0]{} .

The Bogolyubov transformation at hand allows one to embed the states from ${\cal H}_{m}$ into ${\cal H}_{0}$ and vice versa. More precisely the first embedding is realised by means of an interface operator D\_[m]{} : [H]{}\_[0]{} \_[m]{} , D\_[m]{} = [O]{} U where |a\^\_[k\_1]{} …a\^\_[k\_n]{}…|0= b\^\_[k\_1]{} …b\^\_[-k\_n]{}…|0\_[NS]{} and similar relations are satisfied in the R-sector with the zero modes transforming as |a\^\_[0]{} …a\^\_[0]{}…|= … …|0\_[R]{} .

The inverse operator D\_[m]{}\^[-1]{} : [H]{}\_[m]{} \_[0]{} , D\_[m]{}\^[-1]{} = [O]{}\^[-1]{} V where $V:{\cal H}_{m} \to {\cal H}_{m}$ is a unitary operator that realises the 