$\displaystyle\mathchoice{\left[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\bm{Y}_{t^{*}-\ell}}}}}\right]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\bm{Y}_{t^{*}-\ell}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\bm{Y}_{t^{*}-\ell}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\bm{Y}_{t^{*}-\ell}}}}}]}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-\ell+1},\quad\forall\
\ell,\tilde{\ell}\geq 0\ \mbox{and}\ \tilde{\ell}+\ell>k,\ V_{t^{*}}\neq
V_{t^{*}-1}.$ (44)

where $k$ is the Markov order of
$\mathchoice{\left\\{{{{{\bm{Y}_{t}}}}}\right\\}}{\\{{{{{\bm{Y}_{t}}}}}\\}}{\\{{{{{\bm{Y}_{t}}}}}\\}}{\\{{{{{\bm{Y}_{t}}}}}\\}}_{t\in\mathbb{Z}}$,
and only the correlation matrices
$\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell}}}}}\right)}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell}}}}})}$
for $\ell,\tilde{\ell}\geq 0$ and $\tilde{\ell}+\ell<=k$ need to be
parameterized. In this case the dependence between observations in different
regimes is fully modelled. Some advantages of this model formulation compared
with other regime-switching models such as Markov switching vector
autoregressive (MSVAR) models (see Cheng, (2016), Sola and Driffill, (1994),
and Hamilton, (1990)) include the availability of thr stationary joint
distributions of observations within each regime and the possibility to be
extended to models with non-Gaussian margins.

##  Appendix A Parameterization of margin-closed VAR model

Let
$\bm{C}_{i}=\mathchoice{\left({{{{Z_{i,t},\dots,Z_{i,t-k}}}}}\right)}{({{{{Z_{i,t},\dots,Z_{i,t-k}}}}})}{({{{{Z_{i,t},\dots,Z_{i,t-k}}}}})}{({{{{Z_{i,t},\dots,Z_{i,t-k}}}}})}^{\top}$.
To get $R_{\bm{Z}_{t:(t-k)}}$ from $R_{Z_{i,t:(t-k)}}$ for $1\leq i\leq d$ and
$R_{\bm{Z}}$, one can firstly derive

$\displaystyle
R_{\mathchoice{\left\\{{{{{1,\dots,d}}}}\right\\}}{\\{{{{{1,\dots,d}}}}\\}}{\\{{{{{1,\dots,d}}}}\\}}{\\{{{{{1,\dots,d}}}}\\}}}=\operatorname{Corr}\mathchoice{\left({{{{\mathchoice{\left({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}}\right)}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}}}}}\right)}{({{{{\mathchoice{\left({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}}\right)}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}}}}})}{({{{{\mathchoice{\left({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}}\right)}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}}}}})}{({{{{\mathchoice{\left({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}}\right)}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}{({{{{Z_{1,t},\dots,Z_{d,t},\dots,Z_{1,t-k},\dots,Z_{d,t-k}}}}})}}}}})}=\operatorname{Corr}\mathchoice{\left({{{{\mathchoice{\left({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}}\right)}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}^{\top}}}}}\right)}{({{{{\mathchoice{\left({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}}\right)}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}^{\top}}}}})}{({{{{\mathchoice{\left({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}}\right)}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}^{\top}}}}})}{({{{{\mathchoice{\left({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}}\right)}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}{({{{{\bm{C}_{1}^{\top},\dots,\bm{C}_{d}^{\top}}}}})}^{\top}}}}})}.$
(45)

Then $R_{\bm{Z}_{t:(t-k)}}$ can be obtained by reordering rows and columns of
$R_{\mathchoice{\left\\{{{{{1,\dots,d}}}}\right\\}}{\\{{{{{1,\dots,d}}}}\\}}{\\{{{{{1,\dots,d}}}}\\}}{\\{{{{{1,\dots,d}}}}\\}}}$.
The definitions of
$R_{\mathchoice{\left\\{{{{{1,\dots,d}}}}\right\\}}{\\{{{{{1,\dots,d}}}}\\}}{\\{{{{{1,\dots,d}}}}\\}}{\\{{{{{1,\dots,d}}}}\\}}}$
and $R_{Z_{i,t:(t-k)}}$ for $1\leq i\leq d$ lead to

$\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:correlation_matrix_of_Z1_to_Zn}}{e}q:correlation_{m}atrix_{o}f_{Z}1_{t}o_{Z}n}R_{\mathchoice{\left\\{{{{{1,\dots,d}}}}\right\\}}{\\{{{{{1,\dots,d}}}}\\}}{\\{{{{{1,\dots,d}}}}\\}}{\\{{{{{1,\dots,d}}}}\\}}}=\begin{pmatrix}R_{Z_{1,t:(t-k)}}&\operatorname{Corr}\mathchoice{\left({{{{\bm{C}_{1},\bm{C}_{2}}}}}\right)}{({{{{\bm{C}_{1},\bm{C}_{2}}}}})}{({{{{\bm{C}_{1},\bm{C}_{2}}}}})}{({{{{\bm{C}_{1},\bm{C}_{2}}}}})}&\cdots&\operatorname{Corr}\mathchoice{\left({{{{\bm{C}_{1},\bm{C}_{d}}}}}\right)}{({{{{\bm{C}_{1},\bm{C}_{d}}}}})}{({{{{\bm{C}_{1},\bm{C}_{d}}}}})}{({{{{\bm{C}_{1},\bm{C}_{d}}}}})}\\\
\operatorname{Corr}\mathchoice{\left({{{{\bm{C}_{1},\bm{C}_{2}}}}}\right)}{({{{{\bm{C}_{1},\bm{C}_{2}}}}})}{({{{{\bm{C}_{1},\bm{C}_{2}}}}})}{({{{{\bm{C}_{1},\bm{C}_{2}}}}})}^{T}&R_{Z_{2,t:(t-k)}}&\cdots&\operatorname{Corr}\mathchoice{\left({{{{\bm{C}_{2},\bm{C}_{d-1}}}}}\right)}{({{{{\bm{C}_{2},\bm{C}_{d-1}}}}})}{({{{{\bm{C}_{2},\bm{C}_{d-1}}}}})}{({{{{\bm{C}_{2},\bm{C}_{d-1}}}}})}\\\
\vdots&\vdots&\ddots&\vdots\\\
\operatorname{Corr}\mathchoice{\left({{{{\bm{C}_{1},\bm{C}_{d}}}}}\right)}{({{{{\bm{C}_{1},\bm{C}_{d}}}}})}{({{{{\bm{C}_{1},\bm{C}_{d}}}}})}{({{{{\bm{C}_{1},\bm{C}_{d}}}}})}^{T}&\operatorname{Corr}\mathchoice{\left({{{{\bm{C}_{2},\bm{C}_{d-1}}}}}\right)}{({{{{\bm{C}_{2},\bm{C}_{d-1}}}}})}{({{{{\bm{C}_{2},\bm{C}_{d-1}}}}})}{({{{{\bm{C}_{2},\bm{C}_{d-1}}}}})}^{T}&\cdots&R_{Z_{d,t:(t-k)}}\end{pmatrix},$
(46)

where
$\operatorname{Corr}\mathchoice{\left({{{{\bm{C}_{i},\bm{C}_{j}}}}}\right)}{({{{{\bm{C}_{i},\bm{C}_{j}}}}})}{({{{{\bm{C}_{i},\bm{C}_{j}}}}})}{({{{{\bm{C}_{i},\bm{C}_{j}}}}})}$
for $1\leq i<j\leq d$ can be derived by the formulas given below.

For $1\leq p\leq d$, let

$\displaystyle
R_{Z_{p,t:(t-k)}}=\begin{pmatrix}1&\varrho_{p,1}&\cdots&\varrho_{p,k}\\\
\varrho_{p,1}&1&\cdots&\varrho_{p,k-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
\varrho_{p,k}&\varrho_{p,k-1}&\cdots&1\end{pmatrix}$ (47)

and

$H_{p}=\begin{pmatrix}-I_{d_{p}}&\psi_{p,k}&\dots&\psi_{p,2}&\psi_{p,1}&0&0&\cdots&0\\\
0&-I_{d_{p}}&\psi_{p,k}&\dots&\psi_{p,2}&\psi_{p,1}&0&\cdots&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
0&\cdots&0&-I_{d_{p}}&\psi_{p,k}&\cdots&\psi_{p,2}&\psi_{p,1}&0\end{pmatrix},$
(48)

where

$\begin{pmatrix}\psi_{p,1}&\psi_{p,2}&\cdots&\psi_{p,k}\\\
\end{pmatrix}^{\top}=\begin{pmatrix}\varrho_{p,-k}&\varrho_{p,1-k}&\cdots&\varrho_{p,-1}\\\
\end{pmatrix}^{\top}\begin{pmatrix}1&\varrho_{p,1}&\cdots&\varrho_{p,k-1}\\\
\varrho_{p,1}&1&\cdots&\varrho_{p,k-2}\\\ \vdots&\vdots&\ddots&\vdots\\\
\varrho_{p,k-1}&\varrho_{p,k-2}&\cdots&1\end{pmatrix}^{-1},$ (49)

and $\varrho_{p,-m}=\varrho_{p,m}$. Let $H_{p:(k+1)}$ denote the submatrix of
the $(k+1)$-th block column of $H_{p}$ and $H_{p,:-(k+1)}$ denote the
submatrix obtained by removing the $(k+1)$-th block column of $H_{p}$. If
$A=(a_{ij})$ is an $m\times n$ matrix, the Kronecker product $A\otimes B$ is
defined as

$\displaystyle\begin{pmatrix}a_{11}B&\cdots&a_{1n}B\\\ \vdots&\ddots&\vdots\\\
a_{m1}B&\cdots&a_{mn}B\\\ \end{pmatrix}.$ (50)

Let $L$ be the $(2k+1)$-dimensional exchange (or permutation) matrix whose
elements in the anti-diagonal are one and all other elements are zero. Let
$\varrho_{ij,0}=\operatorname{Corr}\mathchoice{\left({{{{Z_{i,t},Z_{j,t}}}}}\right)}{({{{{Z_{i,t},Z_{j,t}}}}})}{({{{{Z_{i,t},Z_{j,t}}}}})}{({{{{Z_{i,t},Z_{j,t}}}}})}$
for $i\neq j$; this is an element of $R_{\bm{Z}}$. Then

$\displaystyle\operatorname{Corr}\mathchoice{\left({{{{\bm{C}_{i},\bm{C}_{j}}}}}\right)}{({{{{\bm{C}_{i},\bm{C}_{j}}}}})}{({{{{\bm{C}_{i},\bm{C}_{j}}}}})}{({{{{\bm{C}_{i},\bm{C}_{j}}}}})}=\begin{pmatrix}\varrho_{ij,0}&\varrho_{ij,1}&\cdots&\varrho_{ij,k-1}\\\
\varrho_{ij,-1}&\varrho_{ij,0}&\cdots&\varrho_{ij,k-2}\\\
\vdots&\vdots&\ddots&\vdots\\\
\varrho_{ij,1-k}&\varrho_{ij,2-k}&\cdots&\varrho_{ij,0}\end{pmatrix},$ (51)

where

$\mathchoice{\left({{{{\varrho_{ij,-k},\dots,\varrho_{ij,-1},\varrho_{ij,1},\dots,\varrho_{ij,k}}}}}\right)}{({{{{\varrho_{ij,-k},\dots,\varrho_{ij,-1},\varrho_{ij,1},\dots,\varrho_{ij,k}}}}})}{({{{{\varrho_{ij,-k},\dots,\varrho_{ij,-1},\varrho_{ij,1},\dots,\varrho_{ij,k}}}}})}{({{{{\varrho_{ij,-k},\dots,\varrho_{ij,-1},\varrho_{ij,1},\dots,\varrho_{ij,k}}}}})}^{\top}=-\varrho_{ij,0}\begin{pmatrix}H_{i,:-(k+1)}\\\
\mathchoice{\left({{{{H_{j}L}}}}\right)}{({{{{H_{j}L}}}})}{({{{{H_{j}L}}}})}{({{{{H_{j}L}}}})}_{:-(k+1)}\end{pmatrix}^{-1}\begin{pmatrix}H_{i,:(k+1)}\\\
H_{j,:(k+1)}\end{pmatrix}.$ (52)

##  Appendix B Proof of Proposition 3.2

##### Part 1 (The process is Markov of order $k$)

Suppose the process changes from regime $g_{b}$ at time $t^{*}-1$ to regime
$g^{\prime}$ at time $t^{*}$. Because $\\{\bm{Y}_{t}\\}$ is a Gaussian
process, $\bm{Y}_{t^{*}}$ is independent of $\bm{Y}_{t^{*}-\ell-1}$ given
$\bm{Y}_{t^{*}-1}$ for any $\ell$ with $\ell\geq 1$, implying

$\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:con_dist1}}{e}q:con_{d}ist1}\mathchoice{\left[{{{{\bm{Y}_{t^{*}}\big{|}\bm{Y}_{t^{*}-1},\dots,\bm{Y}_{1}}}}}\right]}{[{{{{\bm{Y}_{t^{*}}\big{|}\bm{Y}_{t^{*}-1},\dots,\bm{Y}_{1}}}}}]}{[{{{{\bm{Y}_{t^{*}}\big{|}\bm{Y}_{t^{*}-1},\dots,\bm{Y}_{1}}}}}]}{[{{{{\bm{Y}_{t^{*}}\big{|}\bm{Y}_{t^{*}-1},\dots,\bm{Y}_{1}}}}}]}\overset{\mathrm{d}}{=}\mathchoice{\left[{{{{\bm{Y}_{t^{*}}\big{|}\bm{Y}_{t^{*}-1}}}}}\right]}{[{{{{\bm{Y}_{t^{*}}\big{|}\bm{Y}_{t^{*}-1}}}}}]}{[{{{{\bm{Y}_{t^{*}}\big{|}\bm{Y}_{t^{*}-1}}}}}]}{[{{{{\bm{Y}_{t^{*}}\big{|}\bm{Y}_{t^{*}-1}}}}}]}.$
(53)

From the covariance matrix of conditional distributions of multivariate
Gaussian random vectors, Eq. 7 implies

$\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:condition3.1}}{e}q:condition3.1}\begin{split}&\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell-1}\big{|}\bm{Y}_{t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell-1}\big{|}\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell-1}\big{|}\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell-1}\big{|}\bm{Y}_{t^{*}-1}}}}})}=\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-\ell-1}}}}})}\\\
&\hskip
28.45274pt-\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}})}[\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-1}}}}})}]^{-1}\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}=\bm{0},\end{split}$

for any $\ell,\tilde{\ell}\geq 1$. Combining Eq. 6 and Appendix B, by
induction,

$\displaystyle\mathchoice{\left[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\bm{Y}_{t^{*}-\ell-1}}}}}\right]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\bm{Y}_{t^{*}-\ell-1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\bm{Y}_{t^{*}-\ell-1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\bm{Y}_{t^{*}-\ell-1}}}}}]}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-1},$
(54)

i.e.,

$\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:con_dist2}}{e}q:con_{d}ist2}\mathchoice{\left[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{1}}}}}\right]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{1}}}}}]}\overset{\mathrm{d}}{=}\mathchoice{\left[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-1}}}}}\right]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-1}}}}}]}.$
(55)

To prove that process $\\{\bm{Y}_{t}\\}$ is Markov of order $k$ need to show
that the right hand side of Eq. 55 can be reduced to at most $k$ conditioning
variables. Hence, only the situation when $\tilde{\ell}>k$ and
$v_{t^{*}}=v_{t^{*}+1}=\cdots=v_{t^{*}+\tilde{\ell}}=g^{\prime}$ needs to be
considered. Let

$\displaystyle\bm{A}_{t^{*}+\tilde{\ell}}$
$\displaystyle=\mathchoice{\left({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k+1}}}}}\right)}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k+1}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k+1}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k+1}}}}})}^{\top}.$
(56)

Since the process is Markov of order $k-1=\max_{1\leq g\leq G}k_{g}$ within
any regime,

$\displaystyle\mathchoice{\left[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\mathchoice{\left({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}}\right)}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}^{\top}}}}}\right]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\mathchoice{\left({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}}\right)}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}^{\top}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\mathchoice{\left({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}}\right)}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}^{\top}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\bot\mathchoice{\left({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}}\right)}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}{({{{{\bm{Y}^{\top}_{t^{*}+\tilde{\ell}-k},\dots,\bm{Y}^{\top}_{t^{*}}}}}})}^{\top}}}}}]}\Bigm{|}\bm{A}_{t^{*}+\tilde{\ell}}.$
(57)

Moreover, the condition in Eq. 7 implies
$\bm{A}_{t^{*}+\tilde{\ell}}\bot\bm{Y}_{t^{*}-1}$, and leads to

$\displaystyle\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}\big{|}\bm{A}_{t^{*}+\tilde{\ell}}}}}}\right)}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}\big{|}\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}\big{|}\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}\big{|}\bm{A}_{t^{*}+\tilde{\ell}}}}}})}$
(58) $\displaystyle=$
$\displaystyle\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}})}-\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}}\right)}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}[\operatorname{Cov}\mathchoice{\left({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}}\right)}{({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}]^{-1}\operatorname{Cov}\mathchoice{\left({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}}\right)}{({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{Y}_{t^{*}-1}}}}})}$
(59) $\displaystyle=$
$\displaystyle\bm{0}-\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}}\right)}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{Y}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}\operatorname{Cov}^{-1}\mathchoice{\left({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}}\right)}{({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}{({{{{\bm{A}_{t^{*}+\tilde{\ell}},\bm{A}_{t^{*}+\tilde{\ell}}}}}})}\bm{0}=\bm{0}.$
(60)

According to Eq. 55, it implies that if $\tilde{\ell}>k$, then

$\ignorespaces\ignorespaces\mathchoice{\left[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{1}}}}}\right]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{1}}}}}]}\overset{\mathrm{d}}{=}\mathchoice{\left[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-1}}}}}\right]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}-1}}}}}]}\overset{\mathrm{d}}{=}\mathchoice{\left[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}+\tilde{\ell}-k+1}}}}}\right]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}+\tilde{\ell}-k+1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}+\tilde{\ell}-k+1}}}}}]}{[{{{{\bm{Y}_{t^{*}+\tilde{\ell}}\big{|}\bm{Y}_{t^{*}+\tilde{\ell}-1},\dots,\bm{Y}_{t^{*}+\tilde{\ell}-k+1}}}}}]}.$
(61)

Note that Eq. 53, Eq. 55, and LABEL:eq:con_dist3 hold regardless of the number
of regime switches before $t^{*}$.

Since Eq. 5, Eq. 6, and Eq. 7 imply that the above holds for any regime
switching time point,
$\mathchoice{\left\\{{{{{\bm{Y}_{t}}}}}\right\\}}{\\{{{{{\bm{Y}_{t}}}}}\\}}{\\{{{{{\bm{Y}_{t}}}}}\\}}{\\{{{{{\bm{Y}_{t}}}}}\\}}_{t\in\mathbb{Z}}$
is Markov of order $k$.

##### Part 2 (Closure under margins)

Next, we show that Eq. 5, Eq. 6, and Eq. 7 imply that the analogous conditions
to these hold for any sub-process
$\mathchoice{\left\\{{{{{\bm{Y}_{I,t}}}}}\right\\}}{\\{{{{{\bm{Y}_{I,t}}}}}\\}}{\\{{{{{\bm{Y}_{I,t}}}}}\\}}{\\{{{{{\bm{Y}_{I,t}}}}}\\}}_{t\in\mathbb{Z}}$.
Let $P_{I}=\operatorname{diag}(\rho_{i}:i\in I)$. Because $P$ is diagonal, Eq.
5 implies

$\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:uni_condition1}}{e}q:uni_{c}ondition1}\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-1}}}}})}=P_{I}\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-1}}}}})}.$
(62)

Combining Eq. 5 and Eq. 6 leads to

$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:subset_cond}}{e}q:subset_{c}ond}\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-\ell-1}}}}})}=$
$\displaystyle\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-\ell-1}\big{|}\bm{Y}_{t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-\ell-1}\big{|}\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-\ell-1}\big{|}\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-\ell-1}\big{|}\bm{Y}_{t^{*}-1}}}}})}$
(63)
$\displaystyle+\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}},\bm{Y}_{t^{*}-1}}}}})}[\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-1}}}}})}]^{-1}\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}$
(64) $\displaystyle=$ $\displaystyle
PR_{\bm{Y},g}R^{-1}_{\bm{Y},g}\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}\
=P\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{t^{*}-1},\bm{Y}_{t^{*}-\ell-1}}}}})}$
(65)

for any $\ell\geq 1$. Hence,
$\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}})}=P_{I}\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}})}$
for $\ell\geq 1$. Combining with Eq. 62 leads to

$\displaystyle\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}\big{|}\bm{Y}_{I,t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}\big{|}\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}\big{|}\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}\big{|}\bm{Y}_{I,t^{*}-1}}}}})}$
(66) $\displaystyle=$
$\displaystyle\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}})}-\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-1}}}}})}[\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-1}}}}})}]^{-1}\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}})}$
(67) $\displaystyle=$
$\displaystyle\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}},\bm{Y}_{I,t^{*}-\ell-1}}}}})}-P_{I}\operatorname{Cov}\mathchoice{\left({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}}\right)}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}})}{({{{{\bm{Y}_{I,t^{*}-1},\bm{Y}_{I,t^{*}-\ell-1}}}}})}=\bm{0},$
(68)

i.e.,

$\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:uni_condition2}}{e}q:uni_{c}ondition2}\mathchoice{\left[{{{{\bm{Y}_{I,t^{*}}\bot\bm{Y}_{I,t^{*}-\ell-1}}}}}\right]}{[{{{{\bm{Y}_{I,t^{*}}\bot\bm{Y}_{I,t^{*}-\ell-1}}}}}]}{[{{{{\bm{Y}_{I,t^{*}}\bot\bm{Y}_{I,t^{*}-\ell-1}}}}}]}{[{{{{\bm{Y}_{I,t^{*}}\bot\bm{Y}_{I,t^{*}-\ell-1}}}}}]}\big{|}\bm{Y}_{I,t^{*}-1},\quad\forall\
\ell\geq 1.$ (69)

Also, Eq. 7 implies:

$\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:uni_condition3}}{e}q:uni_{c}ondition3}\bm{Y}_{I,t^{*}+\tilde{\ell}}\bot\bm{Y}_{I,t^{*}-\ell},\quad\forall\
\ell,\tilde{\ell}\geq 1.$ (70)

Then, with the same procedure as in Part 1, combining Eq. 62, Eq. 69, and Eq.
70 shows that
$\mathchoice{\left\\{{{{{\bm{Y}_{I,t}}}}}\right\\}}{\\{{{{{\bm{Y}_{I,t}}}}}\\}}{\\{{{{{\bm{Y}_{I,t}}}}}\\}}{\\{{{{{\bm{Y}_{I,t}}}}}\\}}_{t\in\mathbb{Z}}$
is Markov of order $k$. The model is closed under all margins.

##  Appendix C Baum-Welch algorithm

In this section, the Baum-Welch algorithm is given in the notation of the
model in Definition 3.1. All functions depend on $\bm{\theta}$ (the set of all
parameters in the model) as defined in Section 4.

Let $\alpha_{t}$ be functions with $t$ inputs for $1\leq t\leq k$ and $k+1$
inputs for $t\geq k+1$, and they are defined as:

$\begin{split}\alpha_{t}(\omega_{1},\dots,\omega_{t};\bm{\theta})&=f_{\bm{X}_{t:1},V_{t:1}}(\bm{x}_{t},\dots,\bm{x}_{1},\omega_{t},\dots,\omega_{1};\bm{\theta})\quad\text{for
}1\leq t\leq k;\end{split}$

and

$\begin{split}\alpha_{t}(\omega_{1},\dots,\omega_{k+1};\bm{\theta})&=f_{\bm{X}_{t:1},V_{t:(t-k)}}(\bm{x}_{t},\dots,\bm{x}_{1},\omega_{k+1},\dots,\omega_{1};\bm{\theta})\quad\text{for
}t\geq k+1.\end{split}$

It follows that

$\displaystyle\alpha_{1}(\omega_{1};\bm{\theta})$
$\displaystyle=\mathbb{P}(V_{1}=\omega_{1};\bm{\theta})f_{\bm{X}_{1}|V_{1}}(\bm{x}_{1}|\omega_{1};\bm{\theta}),$
(71)

where $\mathbb{P}(V_{1}=\omega_{1};\bm{\theta})$ is the probability that the
initial regime is $\omega_{1}$, $f_{\bm{X}_{1}|V_{1}}$ is the conditional
density of $\bm{X}_{1}$ given $V_{1}$. For $2\leq t\leq k$,

$\begin{split}&\alpha_{t}(\omega_{1},\dots,\omega_{t};\bm{\theta})\\\
=&f_{\bm{X}_{t}|\bm{X}_{(t-1):1},V_{t:1}}(\bm{x}_{t}|\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{t},\dots,\omega_{1};\bm{\theta})\\\
&\times
f_{V_{t}|\bm{X}_{(t-1):1},V_{(t-1):1}}(\omega_{t}|\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{t-1},\dots,\omega_{1};\bm{\theta})\\\
&\times
f_{\bm{X}_{(t-1):1},V_{(t-1):1}}(\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{t-1},\dots,\omega_{1};\bm{\theta})\\\
=&f_{\bm{X}_{t}|\bm{X}_{(t-1):1},V_{t:1}}(\bm{x}_{t}|\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{t},\dots,\omega_{1};\bm{\theta})p(\omega_{t}|\omega_{t-1};\bm{\theta})\alpha_{t-1}(\omega_{1},\dots,\omega_{t-1};\bm{\theta}),\end{split}$

where $p(\omega_{t}|\omega_{t-1};\bm{\theta})$ is the transition probability
from regime $\omega_{t-1}$ to regime $\omega_{t}$, and for $t\geq k+1$,

$\begin{split}&\alpha_{t}(\omega_{1},\dots,\omega_{k+1};\bm{\theta})\\\
=&f_{\bm{X}_{t}|\bm{X}_{(t-1):1},V_{t:(t-k)}}(\bm{x}_{t}|\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{k+1},\dots,\omega_{1};\bm{\theta})\\\
&\times
f_{V_{t}|\bm{X}_{(t-1):1},V_{(t-1):(t-k)}}(\omega_{k+1}|\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{k},\dots,\omega_{1};\bm{\theta})\\\
&\times
f_{\bm{X}_{(t-1):1},V_{(t-1):(t-k)}}(\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{k},\dots,\omega_{1};\bm{\theta})\\\
=&f_{\bm{X}_{t}|\bm{X}_{(t-1):(t-k)},V_{t:(t-k)}}(\bm{x}_{t}|\bm{x}_{t-1},\dots,\bm{x}_{t-k},\omega_{k+1},\dots,\omega_{1};\bm{\theta})\\\
&\times\sum_{g}\mathchoice{\left[{{{{p(\omega_{k+1}|\omega_{k};\bm{\theta})f_{\bm{X}_{(t-1):1},V_{(t-1):(t-k-1)}}(\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{k},\dots,\omega_{1},g|\bm{\theta})}}}}\right]}{[{{{{p(\omega_{k+1}|\omega_{k};\bm{\theta})f_{\bm{X}_{(t-1):1},V_{(t-1):(t-k-1)}}(\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{k},\dots,\omega_{1},g|\bm{\theta})}}}}]}{[{{{{p(\omega_{k+1}|\omega_{k};\bm{\theta})f_{\bm{X}_{(t-1):1},V_{(t-1):(t-k-1)}}(\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{k},\dots,\omega_{1},g|\bm{\theta})}}}}]}{[{{{{p(\omega_{k+1}|\omega_{k};\bm{\theta})f_{\bm{X}_{(t-1):1},V_{(t-1):(t-k-1)}}(\bm{x}_{t-1},\dots,\bm{x}_{1},\omega_{k},\dots,\omega_{1},g|\bm{\theta})}}}}]}\\\
=&f_{\bm{X}_{t}|\bm{X}_{(t-1):(t-k)},V_{t:(t-k)}}(\bm{x}_{t}|\bm{x}_{t-1},\dots,\bm{x}_{t-k},\omega_{k+1},\dots,\omega_{1};\bm{\theta})\times\sum_{g}p(\omega_{k+1}|\omega_{k};\bm{\theta})\alpha_{t-1}(g,\omega_{1},\dots,\omega_{k};\bm{\theta}).\end{split}$

In the backward step, functions $\beta_{t}$ are defined in the similar manner:
for $1\leq t\leq k$,

$\beta_{t}(\omega_{1},\dots,\omega_{t};\bm{\theta})=f_{\bm{X}_{T:(t+1)}|\bm{X}_{t:1},V_{t:1}}(\bm{x}_{T},\dots,\bm{x}_{t+1}|\bm{x}_{t},\dots,\bm{x}_{1},\omega_{t},\dots,\omega_{1};\bm{\theta});$
(72)

for $t\geq k+1$,

$\beta_{t}(\omega_{1},\dots,\omega_{k+1};\bm{\theta})=f_{\bm{X}_{T:(t+1)}|\bm{X}_{t:(t-k)},V_{t:(t-k)}}(\bm{x}_{T},\dots,\bm{x}_{t+1}|\bm{x}_{t},\dots,\bm{x}_{t-k},\omega_{k+1},\dots,\omega_{1};\bm{\theta}).$
(73)

Then

$\beta_{T}(\omega_{1},\dots,\omega_{k+1};\bm{\theta})=1,$ (74)

for $t\geq k+1$,

$\begin{split}&\beta_{t}(\omega_{1},\dots,\omega_{k+1};\bm{\theta})\\\
=&\sum_{g}f_{\bm{X}_{T:(t+1)},V_{t+1}|\bm{X}_{t:(t-k)},V_{t:(t-k)}}(\bm{x}_{T},\dots,\bm{x}_{t+1},g|\bm{x}_{t},\dots,\bm{x}_{t-k},\omega_{k+1},\dots,\omega_{1};\bm{\theta})\\\
=&\sum_{g}\biggl{[}f_{\bm{X}_{T:(t+1)}|\bm{X}_{t:(t-k)},V_{(t+1):(t-k)}}(\bm{x}_{T},\dots,\bm{x}_{t+1}|\bm{x}_{t},\dots,\bm{x}_{t-k},g,\omega_{k+1},\dots,\omega_{1};\bm{\theta})\\\
&\qquad\times
f_{V_{t+1}|\bm{X}_{t:(t-k)},V_{t:(t-k)}}(r|\bm{x}_{t},\dots,\bm{x}_{t-k},\omega_{k+1},\dots,\omega_{1};\bm{\theta})\biggr{]}\\\
=&\sum_{r}\biggl{[}f_{\bm{X}_{T:(t+2)}|\bm{X}_{(t+1):(t-k)},V_{(t+1):(t-k)}}(\bm{x}_{T},\dots,\bm{x}_{t+2}|\bm{x}_{t+1},\dots,\bm{x}_{t-k},g,\omega_{k+1},\dots,\omega_{1};\bm{\theta})\\\
&\qquad\times
f_{\bm{X}_{t+1}|\bm{X}_{t:(t-k)},V_{(t+1):(t-k)}}(\bm{x}_{t+1}|\bm{x}_{t},\dots,\bm{x}_{t-k},g,\omega_{k+1},\dots,\omega_{1};\bm{\theta})p(g|\omega_{k+1};\bm{\theta})\biggl{]}\\\
=&\sum_{g}\biggl{[}\beta_{t+1}(\omega_{2},\dots,\omega_{k+1},g;\bm{\theta})\times
p(g|\omega_{k+1};\bm{\theta})\\\ &\qquad\times
f_{\bm{X}_{t+1}|\bm{X}_{t:(t-k+1)},V_{(t+1):(t-k+1)}}(\bm{x}_{t+1}|\bm{x}_{t},\dots,\bm{x}_{t-k+1},g,\omega_{k+1},\dots,\omega_{2};\bm{\theta})\biggl{]}\end{split}$

and for $t\leq k$,

$\begin{split}&\beta_{t}(\omega_{1},\dots,\omega_{t};\bm{\theta})\\\
=&\sum_{g}f_{\bm{X}_{T:(t+1)},V_{t+1}|\bm{X}_{t:1},V_{t:1}}(\bm{x}_{T},\dots,\bm{x}_{t+1},g|\bm{x}_{t},\dots,\bm{x}_{1},\omega_{t},\dots,\omega_{1};\bm{\theta})\\\
=&\sum_{g}\biggl{[}f_{\bm{X}_{T:(t+1)}|\bm{X}_{t:1},V_{(t+1):1}}(\bm{x}_{T},\dots,\bm{x}_{t+1}|\bm{x}_{t},\dots,\bm{x}_{1},g,\omega_{t},\dots,\omega_{1};\bm{\theta})\\\
&\qquad\times
f_{V_{t+1}|\bm{X}_{t:1},V_{t:1}}(r|\bm{x}_{t},\dots,\bm{x}_{1},\omega_{t},\dots,\omega_{1};\bm{\theta})\biggr{]}\\\
=&\sum_{g}\biggl{[}f_{\bm{X}_{T:(t+2)}|\bm{X}_{(t+1):1},V_{(t+1):1}}(\bm{x}_{T},\dots,\bm{x}_{t+2}|\bm{x}_{t+1},\dots,\bm{x}_{1},g,\omega_{t},\dots,\omega_{1};\bm{\theta})\\\
&\qquad\times
f_{\bm{X}_{t+1}|\bm{X}_{t:1},V_{(t+1):1}}(\bm{x}_{t+1}|\bm{x}_{t},\dots,\bm{x}_{1},g,\omega_{t},\dots,\omega_{1};\bm{\theta})p(g|\omega_{t};\bm{\theta})\biggl{]}\\\
=&\sum_{g}\biggl{[}\beta_{t+1}(\omega_{1},\dots,\omega_{t},g;\bm{\theta})\times
p(g|\omega_{t};\bm{\theta})\times
f_{\bm{X}_{t+1}|\bm{X}_{t:1},V_{(t+1):1}}(\bm{x}_{t+1}|\bm{x}_{t},\dots,\bm{x}_{1},g,\omega_{t},\dots,\omega_{1};\bm{\theta})\biggl{]}.\end{split}$

According to the definition, it follows that

$\displaystyle\alpha_{t}(\omega_{1},\dots,\omega_{t};\bm{\theta})\beta_{t}(\omega_{1},\dots,\omega_{t};\bm{\theta})=f_{\bm{X}_{1:T},V_{1:t}}(\bm{x}_{1},\dots,\bm{x}_{T},\omega_{1},\dots,\omega_{t}|\bm{\theta})$
(75)

for $1\leq t\leq k$, and

$\displaystyle\alpha_{t}(\omega_{1},\dots,\omega_{k+1};\bm{\theta})\beta_{t}(\omega_{1},\dots,\omega_{k+1};\bm{\theta})=f_{\bm{X}_{1:T},V_{(t-k):t}}(\bm{x}_{1},\dots,\bm{x}_{T},\omega_{1},\dots,\omega_{k+1}|\bm{\theta})$
(76)

for $t\geq k+1$. The likelihood of observations is

$f_{\bm{X}_{1:T}}(\bm{x}_{1},\dots,\bm{x}_{T};\bm{\theta})=\sum_{g=1}^{G}\alpha_{1}(g;\bm{\theta})\beta_{1}(g;\bm{\theta}).$
(77)

It leads to that for $1\leq~{}t~{}\leq~{}k$:

$\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{gamma1}}{g}amma1}\begin{split}&\mathbb{P}(V_{1}=\omega_{1},\dots,V_{t}=\omega_{t}|\bm{X}_{1}=\bm{x}_{1},\dots,\bm{X}_{T}=\bm{x}_{T};\bm{\theta})\\\
=&\frac{\alpha_{t}(\omega_{1},\dots,\omega_{t};\bm{\theta})\beta_{t}(\omega_{1},\dots,\omega_{t};\bm{\theta})}{\sum_{g=1}^{G}\alpha_{1}(g;\bm{\theta})\beta_{1}(g;\bm{\theta})},\end{split}$

for $t\geq k+1$:

$\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{gamma2}}{g}amma2}\begin{split}&\mathbb{P}(V_{t-k}=\omega_{1},\dots,V_{t}=\omega_{k+1}|\bm{X}_{1}=\bm{x}_{1},\dots,\bm{X}_{T}=\bm{x}_{T};\bm{\theta})\\\
=&\frac{\alpha_{t}(\omega_{1},\dots,\omega_{k+1};\bm{\theta})\beta_{t}(\omega_{1},\dots,\omega_{k+1};\bm{\theta})}{\sum_{g=1}^{G}\alpha_{1}(g;\bm{\theta})\beta_{1}(g;\bm{\theta})}.\end{split}$

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Spinors, Proper Time and Higher-Spin Fields

N.G. Misuna
_Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut),_

_Am Mühlenberg 1, 14476, Potsdam, Germany_

Tamm Department of Theoretical Physics, Lebedev Physical Institute,
Leninsky prospekt 53, 119991, Moscow, Russia

<EMAIL_ADDRESS>
###### Abstract

We present a Lagrangian formulation for $4d$ integer-spin relativistic fields
in the $5d$ space spanned by two conjugate Weyl spinors and a Lorentz-
invariant proper-time coordinate. We construct a manifestly Poincaré-invariant
free classical action, find a general solution to equations of motion and a
corresponding positive-definite inner product. Our formulation displays a
separation of variables: equations of motion represent ODE in a proper time
only, while spinor coordinates parameterize the Cauchy hypersurface. We also
find momentum eigenstates solutions for massless arbitrary integer-spin fields
and a massive scalar field.

###### Contents

  1. 1 Introduction
  2. 2 $4d$ Poincaré algebra and relativistic fields
  3. 3 Spin-$s$ representation
  4. 4 Free action, e.o.m. and inner product
  5. 5 Momentum eigenstates
    1. 5.1 Scalar field
    2. 5.2 Massless fields
  6. 6 Conclusion

##  1 Introduction

Higher-spin (HS) theories represent an important class of models of
fundamental interactions. Covariant Lagrangian formulations for free higher-
spin fields have been constructed in massive case by Singh and Hagen [1, 2],
and in massless case by Fronsdal and Fang, both in Minkowski [3, 4] and (A)dS
[5, 6] spaces. But it turned out that constructing consistent interactions for
massless HS fields, which problem is of the most interest, gets very involved
in the covariant setup. Therefore the main progress beyond the free level is
due to other approaches.

In particular, cubic HS interactions have been found and studied in detail
within the light-cone framework (see e.g. [7, 8, 9, 10, 11]). However, already
beyond the cubic level the analysis becomes too complicated.

Self-dual HS models are conveniently formulated and analyzed by means of the
methods of twistor theory [12, 13, 14, 15].

The full all-order system of classical e.o.m. of interacting HS gauge fields
has been constructed by Vasiliev [16, 17] in terms of the generating
equations, written in the so-called unfolded form [18, 19, 20] (for a review
of Vasiliev theory see [21, 22]). But extracting HS vertices from Vasiliev
equations represents a very nontrivial task, because one must restrict somehow
the degree of non-locality while solving for auxiliary generating variables,
which problem is currently under the active study (see [23] and references
therein).

More references and a partial review of the recent HS literature can be found
in [24].

Thus, the availability of different implementations of HS fields significantly
enriches our possibilities for constructing and studying HS theories. In this
paper we propose a new realization for the integer-spin representations of the
$4d$ Poincaré group. Instead of dealing with $4d$ Minkowski space, we consider
a $5d$ space spanned by a pair of conjugate spinors and one Lorentz scalar.
This set of coordinates appeared previously in the unfolded formulation of the
$4d$ off-shell fields [25, 26, 27, 28], where they have been playing the role
of the auxiliary fiber coordinates, encoding unfolded descendants of the
space-time fields under consideration. In this paper we use these coordinates
to build a self-contained Lagrangian formulation for $4d$ integer-spin fields
without any reference to a space-time.

To give a preliminary intuitive idea of how such $5d$ space can encode $4d$
fields, let us consider a simple example. An asymptotic one-particle state of
a scalar field is determined by 4-momentum $p^{a}=(E,\overrightarrow{p})$,
which is forced to lie on the mass-shell $p_{a}p^{a}=m^{2}$. Hence, the state
is fixed by three independent parameters: four variables with one constraint.
Alternatively, the same information can be encoded in a Lorentz-scalar
$\pi=\sqrt{E^{2}-\overrightarrow{p}^{2}}$ and a null vector
$n^{a}=(|\overrightarrow{p}|,\overrightarrow{p})$, with the constraint being
$\pi=m$. In its turn, a real null $4d$ vector can be represented in terms of
spinors as
$n^{a}=(\bar{\sigma})^{\dot{\alpha}\beta}\bar{\xi}_{\dot{\alpha}}\xi_{\beta}$.
Thus, a set of 5 variables $\\{\pi,\xi^{\alpha},\bar{\xi}^{\dot{\alpha}}\\}$
(effectively, 4 of them, as the global phase of $\xi$ does not contribute)
determines the 4-momentum, while the mass-shell equation becomes simply
$\pi=m$, putting no restrictions on $\xi$.

In our consideration, however, we make use of a similar $5d$ space as a
substitute not for the momentum $p^{a}$, but rather for the coordinate
$x^{a}$, so that classical e.o.m. become ODE in a scalar coordinate. We find
expressions for Poincaré generators and identify appropriate modules supplied
with a positive-definite inner product.We also construct simple Poincaré-
invariant actions which lead to the appropriate e.o.m. and find their general
solutions. In addition, we find solutions for momentum eigenstates for the
cases of an arbitrary-mass scalar field and of massless arbitrary spin fields.

The paper is organized as follows. In Section 2 we introduce our conventions
for Poincaré generators and give a brief reminder on how covariant quantum
fields are constructed in the standard approach, to be later compared with our
construction. In Section 3 we build a $4d$ integer-spin representation on a
certain $5d$ space. In Section 4 we present a Poincaré-invariant action for a
free field, give a general solution to e.o.m. and propose an inner product for
solutions. In Section 5 we find solutions of e.o.m. corresponding to momentum
eigenstates for a scalar field and massless fields. In Section 6 we sum up our
results.

##  2 $4d$ Poincaré algebra and relativistic fields

Elementary particles are associated with unitary irreducible representations
(UIRs) of the Poincaré group (or an isometry group of the spacetime in
question, more generally) [29].

In the paper we consider $4d$ Poincaré algebra with generators
$P_{\alpha\dot{\alpha}}$, $M_{\alpha\beta}=M_{\beta\alpha}$ and
$\bar{M}_{\dot{\alpha}\dot{\beta}}=\bar{M}_{\dot{\beta}\dot{\alpha}}$, which
correspond to translations, anti-selfdual and selfdual rotations of Minkowski
space, respectively. Here indices belong to two conjugate spinor
representations of the Lorentz algebra $sl(2,\mathbb{C})$. Commutation
relations are

$\displaystyle[M_{\alpha\beta},M_{\gamma\delta}]=\epsilon_{\alpha\gamma}M_{\beta\delta}+\epsilon_{\alpha\delta}M_{\beta\gamma}+\epsilon_{\beta\gamma}M_{\alpha\delta}+\epsilon_{\beta\delta}M_{\alpha\gamma},$
(2.1)
$\displaystyle[\bar{M}_{\dot{\alpha}\dot{\beta}},\bar{M}_{\dot{\gamma}\dot{\delta}}]=\epsilon_{\dot{\alpha}\dot{\gamma}}\bar{M}_{\dot{\beta}\dot{\delta}}+\epsilon_{\dot{\alpha}\dot{\delta}}\bar{M}_{\dot{\beta}\dot{\gamma}}+\epsilon_{\dot{\beta}\dot{\gamma}}\bar{M}_{\dot{\alpha}\dot{\delta}}+\epsilon_{\dot{\beta}\dot{\delta}}\bar{M}_{\dot{\alpha}\dot{\gamma}},$
(2.2) $\displaystyle[M_{\alpha\beta},\bar{M}_{\dot{\gamma}\dot{\delta}}]=0,$
(2.3)
$\displaystyle[M_{\alpha\beta},P_{\gamma\dot{\gamma}}]=\epsilon_{\alpha\gamma}P_{\beta\dot{\gamma}}+\epsilon_{\beta\gamma}P_{\alpha\dot{\gamma}},$
(2.4)
$\displaystyle[\bar{M}_{\dot{\alpha}\dot{\beta}},P_{\gamma\dot{\gamma}}]=\epsilon_{\dot{\alpha}\dot{\gamma}}P_{\gamma\dot{\beta}}+\epsilon_{\dot{\beta}\dot{\gamma}}P_{\gamma\dot{\alpha}},$
(2.5) $\displaystyle[P_{\alpha\dot{\alpha}},P_{\beta\dot{\beta}}]=0,$ (2.6)

where $\epsilon_{\alpha\beta}$ and $\epsilon_{\dot{\alpha}\dot{\beta}}$ are
Lorentz-invariant spinor metrics

$\epsilon_{\alpha\beta}=\epsilon^{\alpha\beta}=\epsilon_{\dot{\alpha}\dot{\beta}}=\epsilon^{\dot{\alpha}\dot{\beta}}=\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right),$ (2.7)

which raise and lower spinor indices according to

$v_{\alpha}=\epsilon_{\beta\alpha}v^{\beta},\quad
v^{\alpha}=\epsilon^{\alpha\beta}v_{\beta},\quad\bar{v}_{\dot{\alpha}}=\epsilon_{\dot{\beta}\dot{\alpha}}\bar{v}^{\dot{\beta}},\quad\bar{v}^{\dot{\alpha}}=\epsilon^{\dot{\alpha}\dot{\beta}}\bar{v}_{\dot{\beta}}.$
(2.8)

UIRs are determined by the values of two Casimir operators: a square of the
momentum, associated with the mass,

$P^{2}=m^{2}$ (2.9)

and, introducing the Pauli–Lubanski pseudovector as

$W_{\alpha\dot{\alpha}}=\frac{1}{2}M_{\alpha\beta}P^{\beta}\text{}_{\dot{\alpha}}-\frac{1}{2}\bar{M}_{\dot{\alpha}\dot{\beta}}P_{\alpha}\text{}^{\dot{\beta}},$
(2.10)

either its square, associated with the spin $s$ when $m^{2}>0$

$W^{2}=-m^{2}s(s+1),$ (2.11)

or the helicity $\lambda$ when $m=0$

$W_{\alpha\dot{\beta}}=\lambda P_{\alpha\dot{\beta}}.$ (2.12)

In (2.9), (2.11) and throughout the paper the square $v^{2}$ of a vector
$v_{\alpha\dot{\beta}}$ is defined as

$v^{2}=\frac{1}{2}v_{\alpha\dot{\beta}}v^{\alpha\dot{\beta}}.$ (2.13)

The standard covariant QFT approach is to implement momentum generators as
coordinate derivatives

$P_{a}=-i\frac{\partial}{\partial x^{a}}$ (2.14)

on the Minkowski space with coordinates $x^{a}$. Then quantum fields look as
$\phi^{I}(x)$, where index $I$ belongs to some finite-dimensional
representation of the Lorentz group (spin), so that rotations are realized as

$M_{a,b}=i(x_{a}\frac{\partial}{\partial x^{b}}-x_{b}\frac{\partial}{\partial
x^{a}})+(S_{a,b})^{I}{}_{J}$ (2.15)

with $S$ being $x$-independent spin generators. In general, however, the
resulting representation of the Poincaré algebra is neither irreducible nor
unitary, and one has to remove undesirable subrepresentations by imposing
additional constraints besides the Klein–Gordon equation (2.9). In order to
represent all of them as following from some Lagrangian equations of motion,
one has to introduce auxiliary fields (for massive fields with $s>1$) and/or
to provide certain gauge symmetry (for massless fields with $s\geq 1$).
Corresponding Lagrangian formulations for arbitrary spin fields have been
constructed by Sing and Hagen for massive fields [1, 2] and by Fronsdal and
Fang for massless fields [3, 4, 5, 6].

##  3 Spin-$s$ representation

In the paper we construct a realization of bosonic UIRs on a $5d$ linear space
spanned by a pair of conjugate commuting $sl(2,\mathbb{C})$ spinors
$Y^{A}=(y^{\alpha},\bar{y}^{\dot{\alpha}})$ and a Lorentz-invariant ’proper
time’ $\tau$. This set of variables $(Y,\tau)$ was previously used in
formulating off-shell unfolded equations for various $4d$ field systems [25,
26, 27, 28]. And spinors $Y$ were initially used in the unfolded Vasiliev
equations [16, 17], where they play the crucial role of the generators of an
associative HS gauge algebra. Here we propose to use $(Y,\tau)$-space instead
of a space-time and build a corresponding Lagrangian formulation for bosonic
fields. All fields are ’scalar’ (i.e. without non-contracted Lorentz indices)
functions $F(Y,\tau)$ on this space.

For the rotation generators we take

$M_{\alpha\beta}=y_{\alpha}\partial_{\beta}+y_{\beta}\partial_{\alpha},$ (3.1)
$\bar{M}_{\dot{\alpha}\dot{\beta}}=\bar{y}_{\dot{\alpha}}\bar{\partial}_{\dot{\beta}}+\bar{y}_{\dot{\beta}}\bar{\partial}_{\dot{\alpha}},$
(3.2)

where $Y$-derivatives are defined as

$\partial_{\alpha}y^{\beta}=\delta_{\alpha}\text{}^{\beta},\quad\bar{\partial}_{\dot{\alpha}}\bar{y}^{\dot{\beta}}=\delta_{\dot{\alpha}}\text{}^{\dot{\beta}}.$
(3.3)

It is easy to check that (3.1)-(3.2) satisfy (2.1)-(2.3). From here it also
directly follows that the proper-time coordinate $\tau$ is Lorentz-invariant
(but not translation-invariant, as we will see). The expressions (3.1)-(3.2)
for rotations operators are universal: we demand that they look the same for
all fields of arbitrary masses and spins, like it is the case for the
translation operator in the standard construction (2.14). The price to pay for
this is that the translation operator now depends on a spin, as we will see.

As $Y$ commute with themselves, they have zero norm

$y_{\alpha}y^{\alpha}=0,\quad\bar{y}_{\dot{\alpha}}\bar{y}^{\dot{\alpha}}=0,$
(3.4)

and the only independent Lorentz-invariant $Y$-combinations one can form are
Euler operators

$N=y^{\alpha}\partial_{\alpha},\quad\bar{N}=\bar{y}^{\dot{\alpha}}\bar{\partial}_{\dot{\alpha}}.$
(3.5)

An appropriate module of a spin-$s$ representation has to contain states with
helicities from $-s$ to $+s$. This can be achieved by considering a set of
functions

$\Phi^{s}(Y,\tau)=\\{\Phi_{\alpha(m),\dot{\alpha}(n)}(\tau)(y^{\alpha})^{m}(\bar{y}^{\dot{\alpha}})^{n},\quad(m+n)\geq
2s,\quad|m-n|\leq 2s\\},$ (3.6)

where we make use of condensed notations for symmetrized indices

$v_{\alpha(m)}=v_{(\alpha_{1}\alpha_{2}...\alpha_{m})},\quad(y^{\alpha})^{m}=y^{\alpha_{1}}y^{\alpha_{2}}...y^{\alpha_{m}}.$
(3.7)

The module (3.6) can be also represented as

$\Phi^{s}(Y,\tau)=\Phi_{A(2s)}(y\bar{y},\tau)(Y^{A})^{2s},$ (3.8)

where $A$ is a Majorana index taking four values $\\{1,2,\dot{1},\dot{2}\\}$.
This form is visually more similar to the standard Minkowski approach, where
an integer spin-$s$ module is a rank-$s$ tensor field $\phi_{a(s)}(x)$. It
should be stressed however, that in (3.8) ’external’ $Y$-s and ’internal’
$y$-s and $\bar{y}$-s are on a completely equal footing, as seen from (3.6).
And $2s$ explicit spinors and indices in (3.8) are highlighted only in order
to show restrictions on the number of $y$ and $\bar{y}$ and play no special
role otherwise.

Now one has to find an expression for the momentum operator
$P_{\alpha\dot{\beta}}$. The most general Ansatz is

$P_{\alpha\dot{\beta}}=a_{N,\bar{N}}\partial_{\alpha}\bar{\partial}_{\dot{\beta}}+b_{N,\bar{N}}y_{\alpha}\bar{y}_{\dot{\beta}}+c_{N,\bar{N}}y_{\alpha}\bar{\partial}_{\dot{\beta}}+\bar{c}_{N,\bar{N}}\partial_{\alpha}\bar{y}_{\dot{\beta}},$
(3.9)

where Lorentz-invariant coefficients $a,b,c,\bar{c}$ are built out of Euler
operators (3.5), as well as of $\tau$ and $\tau$-derivatives. (3.9)
automatically satisfies (2.4) and (2.5), so the only equation to be solved is
(2.6). It can be equivalently reformulated in terms of two conjugate equations

$P_{\alpha\dot{\beta}}P_{\alpha\dot{\gamma}}\epsilon^{\dot{\beta}\dot{\gamma}}=0,$
(3.10) $P_{\beta\dot{\alpha}}P_{\gamma\dot{\alpha}}\epsilon^{\beta\gamma}=0.$
(3.11)

Substituting (3.9), they lead to the following constraints

$\displaystyle(\bar{N}+2)a_{N,\bar{N}}\bar{c}_{N+1,\bar{N}+1}-\bar{N}a_{N+1,\bar{N}-1}\bar{c}_{N,\bar{N}}=0,$
(3.12)
$\displaystyle(\bar{N}+2)b_{N-1,\bar{N}+1}c_{N,\bar{N}}-\bar{N}b_{N,\bar{N}}\bar{c}_{N-1,\bar{N}-1}=0,$
(3.13)
$\displaystyle(\bar{N}+2)a_{N,\bar{N}}b_{N+1,\bar{N}+1}-\bar{N}a_{N-1,\bar{N}-1}b_{N,\bar{N}}+(\bar{N}+2)c_{N,\bar{N}}\bar{c}_{N-1,\bar{N}+1}-\bar{N}\bar{c}_{N,\bar{N}}c_{N+1,\bar{N}-1}=0,$
(3.14)

plus three conjugate equations with $N\leftrightarrow\bar{N}$,
$c\leftrightarrow\bar{c}$ interchanged. In addition, one has to ensure that
the action of (3.9) does not lead outside the module (3.6). This means that
only those solutions are suitable that satisfy

$a_{N,\bar{N}}|_{\varsigma=s-1}=0,\quad
c_{N,\bar{N}}|_{\chi=s+1}=0,\quad\bar{c}_{N,\bar{N}}|_{\chi=-s-1}=0,$ (3.15)

where $\varsigma$ and $\chi$ are important linear combinations of Euler
operators (3.5), which we actively use below,

$\varsigma=\frac{N+\bar{N}}{2},\quad\chi=\frac{N-\bar{N}}{2}.$ (3.16)

Any solution of (3.12)-(3.14) respecting boundary conditions (3.15) defines
some representation of the Poincaré algebra. But many of these representations
are equivalent, and this allows one to put some further constraints.

First, we restrict $\tau$-dependence and provide a ’separation of variables’
$Y$ and $\tau$. Specifically, we require the operator $P^{2}$ to be
$Y$-independent, so that the mass-shell equation (2.9) becomes an ODE in
$\tau$. In addition, we demand $\tau$ to enter (3.9) only through this
$P^{2}$-combination.

Second, we require $P_{\alpha\dot{\beta}}$ to allow for a usual integration by
parts rule

$\int d\tau\int d^{4}Yf(Y,\tau)P_{\alpha\dot{\beta}}g(Y,\tau)=-\int d\tau\int
d^{4}Yg(Y,\tau)P_{\alpha\dot{\beta}}f(Y,\tau).$ (3.17)

To this end one notes that (assuming that one can neglect boundary terms)

$\int d^{4}Y(y^{\alpha}\partial_{\alpha}f(Y))g(Y)=\int
d^{4}Y((\partial_{\alpha}y^{\alpha}-2)f(Y))g(Y)=-\int
d^{4}Yf(Y)(y^{\alpha}\partial_{\alpha}+2)g(Y),$ (3.18)

which allows one to formulate general rules

$\int Nf\cdot g=-\int f\cdot(N+2)g,\quad\int\bar{N}f\cdot g=-\int
f\cdot(\bar{N}+2)g,\quad\int\varsigma f\cdot g=-\int
f\cdot(\varsigma+2)g,\quad\int\chi f\cdot g=-\int f\cdot\chi g.$ (3.19)

These constraints significantly restrict the space of solutions to
(3.10)-(3.11), though still do not fix it unambiguously. We pick up the
following particular solution

$\displaystyle-iP_{\alpha\dot{\beta}}$ $\displaystyle=$
$\displaystyle\frac{(\varsigma-s+1)(\varsigma+s+2)(\varsigma+3/2)}{(N+1)(N+2)(\bar{N}+1)(\bar{N}+2)}\partial_{\alpha}\bar{\partial}_{\dot{\beta}}-\frac{P^{2}}{(\varsigma+1/2)}y_{\alpha}\bar{y}_{\dot{\beta}}+$
(3.20)
$\displaystyle+\frac{1}{(\bar{N}+1)(\bar{N}+2)}[(\chi+s)(\chi-s-1)\Pi^{+}-P^{2}\Pi^{-0}]y_{\alpha}\bar{\partial}_{\dot{\beta}}+$
$\displaystyle+\frac{1}{(N+1)(N+2)}[(\chi-s)(\chi+s+1)\Pi^{-}-P^{2}\Pi^{+0}]\partial_{\alpha}\bar{y}_{\dot{\beta}},$

where projectors $\Pi$ on different $\chi$-components are introduced as

$\displaystyle\Pi^{+}F_{\chi}(Y)=\begin{cases}F_{\chi}(Y),&\chi>0\\\
0,&\chi\leq
0\end{cases};\qquad\Pi^{-}F_{\chi}(Y)=\begin{cases}F_{\chi}(Y),&\chi<0\\\
0,&\chi\geq 0\end{cases};$ (3.21)
$\displaystyle\Pi^{+0}F_{\chi}(Y)=\begin{cases}F_{\chi}(Y),&\chi\geq 0\\\
0,&\chi<0\end{cases};\qquad\Pi^{-0}F_{\chi}(Y)=\begin{cases}F_{\chi}(Y),&\chi\leq
0\\\ 0,&\chi>0\end{cases}.$ (3.22)

Expression (3.20) for $P$ contains manifestly and self-consistently its own
square $P^{2}$, which is $Y$-independent by construction. $P^{2}$ is also
required to be even under integration by parts in order to provide (3.17).

Now for the Pauli–Lubanski pseudovector (2.10) one has

$\displaystyle-iW_{\alpha\dot{\beta}}$ $\displaystyle=$
$\displaystyle-\chi\frac{(\varsigma-s+1)(\varsigma+s+2)(\varsigma+3/2)}{(N+1)(N+2)(\bar{N}+1)(\bar{N}+2)}\partial_{\alpha}\bar{\partial}_{\dot{\beta}}-\chi\frac{P^{2}}{(\varsigma+1/2)}y_{\alpha}\bar{y}_{\dot{\beta}}+$
(3.23)
$\displaystyle+\frac{(\varsigma+1)}{(\bar{N}+1)(\bar{N}+2)}[(\chi+s)(\chi-s-1)\Pi^{+}-P^{2}\Pi^{-0}]y_{\alpha}\bar{\partial}_{\dot{\beta}}-$
$\displaystyle-\frac{(\varsigma+1)}{(N+1)(N+2)}[(\chi-s)(\chi+s+1)\Pi^{-}-P^{2}\Pi^{+0}]\partial_{\alpha}\bar{y}_{\dot{\beta}},$

with its square being

$W^{2}=-P^{2}s(s+1).$ (3.24)

In the case $P^{2}=0$ one finds that $P_{\alpha\dot{\beta}}$ and
$W_{\alpha\dot{\beta}}$ are proportional to each other whenever the module
contains components with $|\chi|=s$ only, in which case

$\displaystyle-iP_{\alpha\dot{\beta}}^{m=0}$ $\displaystyle=$
$\displaystyle\frac{(\varsigma-s+1)(\varsigma+s+2)(\varsigma+3/2)}{(N+1)(N+2)(\bar{N}+1)(\bar{N}+2)}\partial_{\alpha}\bar{\partial}_{\dot{\beta}},\quad
W_{\alpha\dot{\beta}}^{m=0}=-\chi P_{\alpha\dot{\beta}}^{m=0},$ (3.25)

that corresponds to two $\pm s$ helicities (2.12) of the massless field.

Thus, operators (3.1), (3.2) and (3.20) indeed correctly determine a spin-$s$
representation on the module (3.6) after fixing the value of $P^{2}$. In the
massless case $P^{2}=0$ one also has to reduce the module, leaving only $\pm
s$ helicities, which corresponds to setting $|m-n|=2s$ instead of $|m-n|\leq
2s$ in (3.6), or having, instead of (3.8),

$\Phi_{m=0}^{s}(Y,\tau)=\Phi_{\alpha(2s)}(y\bar{y},\tau)(y^{\alpha})^{2s}\oplus\bar{\Phi}_{\dot{\alpha}(2s)}(y\bar{y},\tau)(\bar{y}^{\dot{\alpha}})^{2s}.$
(3.26)

Now, in order to formulate an action principle, one has to realize $P^{2}$ as
a differential operator. As mentioned previously, it must be $\tau$-dependent
only and even under integration by parts, but completely unrestricted
otherwise. This means that in our construction Klein–Gordon equation (2.9) can
be implemented in many different ways. In the next Section we consider one of
the simplest possibilities.

##  4 Free action, e.o.m. and inner product

First we consider the massive case. We take

$P^{2}=-\frac{\partial^{2}}{\partial\tau^{2}}.$ (4.1)

Then a Poincaré-invariant action for a spin-$s$ mass-$m$ field is simply

$S=\frac{1}{2}\sum_{\chi=-s}^{s}\int d^{4}Y\int
d\tau(\dot{\Phi}_{\chi}^{2}-m^{2}\Phi_{\chi}^{2}),$ (4.2)

where the dot means a $\tau$-derivative and $\Phi_{\chi}$ means a subspace of
the spin-$s$ module (3.6) of the definite helicity-$\chi$

$\Phi_{\chi}(Y,\tau)=\Phi_{\alpha(s+\chi),\dot{\beta}(s-\chi)}(y\bar{y},\tau)(y^{\alpha})^{s+\chi}(y^{\dot{\beta}})^{s-\chi}.$
(4.3)

Poincaré-invariance of the action (4.2) is guaranteed by the integration-by-
parts property (3.17), which is obvious for $M$ and $\bar{M}$ (3.1)-(3.2) as
well.

The action (4.2) leads to an e.o.m.

$\ddot{\Phi}_{\chi}+m^{2}\Phi_{\chi}=0.$ (4.4)

Its general solution is

$\Phi_{\chi}(Y,\tau)=e^{-im\tau}f_{\chi}(Y)+e^{im\tau}g_{\chi}(Y),$ (4.5)

where the only requirement to $Y$-functions $f$ and $g$ is to belong to
helicity-$\chi$ subspace. Thus, from the point of view of (4.4), $Y$ are
coordinates on the subspace of Cauchy data, while e.o.m. determines the
evolution in $\tau$-direction.

A Poincaré-invariant inner product for the on-shell states is

$(\text{$\Phi$}_{\chi},\Psi_{\chi^{\prime}})=i\int
d^{4}Y(\bar{\Phi}\dot{\Psi}-\Psi\dot{\bar{\Phi}})\delta_{\chi,\chi^{\prime}}.$
(4.6)

It is $\tau$-independent due to (4.4) and positive-definite for a ’positive-
mass’ subspace of (4.5) with $g=0$. The states with the same $Y$-dependence
but with different mass signs are orthogonal.

The split of the on-shell space into two subspaces, corresponding to
’positive-mass’ $f$ and ’negative-mass’ $g$ contributions in (4.6), is
reminiscent to the split into positive-energy and negative-energy branches in
the standard QFT. However, establishing the rigorous relation between two
these phenomenae requires a separate thorough analysis which we leave for the
future study. Let us note, however, that in our case the split, being
determined by $\tau$-dependence, is manifestly Lorentz-invariant.

Now we move to the massless case. Here using (4.1) potentially leads to
problems: the general solution to (4.4) with $m=0$ is an arbitrary linear
function of $\tau$, so all on-shell states either have zero norm with respect
to (4.6) or are unbounded in $\tau$, which may be unpleasant.

This can be easily fixed by introducing a mass-dimension parameter $\mu$ and
deforming (4.1) to

$P^{2}=-\frac{\partial^{2}}{\partial\tau^{2}}-\mu^{2}.$ (4.7)

Then the zero-mass action becomes

$S=\frac{1}{2}\sum_{\chi=-s}^{s}\int d^{4}Y\int
d\tau(\dot{\Phi}_{\chi}^{2}-\mu^{2}\Phi_{\chi}^{2}),$ (4.8)

and e.o.m. now are

$\ddot{\Phi}_{\chi}+\mu^{2}\Phi_{\chi}=0,$ (4.9)

so the general solution is

$\Phi_{\chi}(Y,\tau)=e^{-i\mu\tau}f_{\chi}(Y)+e^{i\mu\tau}g_{\chi}(Y),$ (4.10)

and one has $\tau$-bounded functions and the split into two branches again.

As said before, in the massless case one also has to reduce the module,
leaving only $|\chi|=s$ components, (3.26). Intermediate components $|\chi|<s$
are necessary to provide off-shell Poincaré invariance of the action (4.8),
but on shell $|\chi|=s$ components decouple into closed subspaces.

It should be stressed that the equation (4.9) describes a massless field,
$m=0$. The parameter $\mu$ does not shift the value of the mass, it only
deforms the functional dependence of $P^{2}$ on $\tau$. In particular, $\mu$
enters directly the expression for the off-shell momentum generator (3.20)
through (4.7). In principle, it can be introduced for the massive fields as
well. Practically, the parameter $\mu$ plays the role of a manifestly
Poincaré-invariant IR-regulator. The possibility of such deformation relies on
the large freedom in choosing the differential realization of the $P^{2}$ and
is specific to the presented construction. In particular, it is unclear how to
locally deform the momentum operator (2.14) of a covariant QFT to have
$P^{2}=-\square+\mu^{2}$.

Let us also give a brief comment on the issue of locality of the constructed
representations. As seen from (3.20), the translations, as opposite to the
rotations (3.1)-(3.2), are realized non-locally: $Y$-differential operators
$N$ and $\bar{N}$ enter (3.20) in a non-polynomial way. But a crucial feature
is that the translations are local in $\tau$, so one cannot e.g. shift the
pole of the propagator by means of Poincaré-transformations. So the evolution
in $\tau$ is completely local, while transformations on the Cauchy
hypersurface with coordinates $Y$ are non-local.

##  5 Momentum eigenstates

Having formulated the classical action and e.o.m., the next natural step is to
look for various partial solutions to them. Of special importance are
solutions that correspond to momentum eigenstates. We restrict ourselves here
to the simplest cases of a scalar field and massless arbitrary spin fields,
for which the momentum operator takes a particularly simple form.

###  5.1 Scalar field

Let us construct momentum eigenstates for the scalar field $s=0$. In this case
the module (3.6) is

$\Phi^{s=0}(Y,\tau)=\Phi(y\bar{y},\tau),$ (5.1)

and the momentum operator (3.20) reduces to

$\displaystyle P_{\alpha\dot{\alpha}}^{s=0}$ $\displaystyle=$
$\displaystyle\frac{i(\varsigma+3/2)}{(\varsigma+1)(\varsigma+2)}\partial_{\alpha}\bar{\partial}_{\dot{\alpha}}+\frac{i}{(\varsigma+1/2)}y_{\alpha}\bar{y}_{\dot{\alpha}}\frac{\partial^{2}}{\partial\tau^{2}}.$
(5.2)

We have to solve an equation

$P_{\alpha\dot{\beta}}\Phi_{p}(Y,\tau)=p_{\alpha\dot{\beta}}\Phi_{p}(Y,\tau)$
(5.3)

with some momentum $p_{\alpha\dot{\beta}}$, $p^{2}=m^{2}$.

A natural Ansatz is

$\Phi_{p}(Y,\tau)=\Phi_{p}(-ip_{\alpha\dot{\alpha}}y^{\alpha}\bar{y}^{\dot{\alpha}})e^{\pm
im\tau},$ (5.4)

where $\tau$-dependence gets fixed by the general solution (4.5) and
$p_{\alpha\dot{\alpha}}y^{\alpha}\bar{y}^{\dot{\alpha}}$ is the only available
Lorentz-invariant combination involving $Y$.

Using that

$\partial_{\alpha}\bar{\partial}_{\dot{\alpha}}f(z_{\beta\dot{\beta}}y^{\beta}\bar{y}^{\dot{\beta}})=z_{\alpha\dot{\alpha}}(\varsigma+1)f^{\prime}-z^{2}y_{\alpha}\bar{y}_{\dot{\alpha}}f^{\prime\prime},$
(5.5)

where the prime means the derivative with respect to the entire argument of
$f$, one can rewrite (5.3) as an ODE with respect to the variable
$u=-ip_{\alpha\dot{\alpha}}y^{\alpha}\bar{y}^{\dot{\alpha}}$

$u\Phi^{\prime\prime}(u)+(\frac{3}{2}-u)\Phi^{\prime}(u)-2\Phi(u)=0.$ (5.6)

This arises from the terms in (5.3), proportional to $p_{\alpha\dot{\beta}}$.
Strictly speaking, there is one more ODE coming from (5.3), which is generated
by terms proportional to $y_{\alpha}\bar{y}_{\dot{\alpha}}$, but it represents
a differential consequence of (5.6).

(5.6) is the Kummer’s equation. Its solution regular at $u=0$ is the confluent
hypergeometric function

$\Phi(u)={}_{1}F_{1}(2;\frac{3}{2};u).$ (5.7)

Thus, momentum-$p_{\alpha\dot{\beta}}$ eigenstate of the scalar field is

$\Phi_{p}(Y,\tau)={}_{1}F_{1}(2;\frac{3}{2};-ipy\bar{y})e^{\pm im\tau}.$ (5.8)

###  5.2 Massless fields

For a massless spin-$s$ field the module is (3.26). It contains two $\pm s$
helicities and for both of them the momentum operator reduces to

$\displaystyle P_{\alpha\dot{\beta}}^{m=0}$ $\displaystyle=$
$\displaystyle\frac{i(\varsigma+3/2)}{(\varsigma+s+1)(\varsigma-s+2)}\partial_{\alpha}\bar{\partial}_{\dot{\beta}}.$
(5.9)

Introducing a polarization vector $\varepsilon_{\alpha\dot{\beta}}$,
orthogonal to the null momentum $p_{\alpha\dot{\beta}}$, $p^{2}=0$,

$\varepsilon_{\alpha\dot{\beta}}p^{\alpha\dot{\beta}}=0,$ (5.10)

we choose following Ansätze for negative and positive helicites

$\Phi_{p,\varepsilon}^{-}(Y,\tau)=(i\varepsilon_{\alpha\dot{\beta}}p_{\alpha}{}^{\dot{\beta}}y^{\alpha}y^{\alpha})^{s}\Psi(-ipy\bar{y})e^{\pm
i\mu\tau},$ (5.11)
$\Phi_{p,\varepsilon}^{+}(Y,\tau)=(i\varepsilon_{\beta\dot{\alpha}}p^{\beta}{}_{\dot{\alpha}}\bar{y}^{\dot{\alpha}}\bar{y}^{\dot{\alpha}})^{s}\Psi(-ipy\bar{y})e^{\pm
i\mu\tau}.$ (5.12)

Here we made use of a $\mu$-deformed realization of $P^{2}$ (4.7). Then for
$\Psi$ one gets, analogously to the scalar field case, the following Kummer’s
equation

$u\Psi^{\prime\prime}(u)+(\frac{3}{2}+s-u)\Psi^{\prime}(u)-2\Psi(u)=0$ (5.13)

whose regular at $u=0$ solution is

$\Psi(u)={}_{1}F_{1}(2;\frac{3}{2}+s;u).$ (5.14)

##  6 Conclusion

In the paper we proposed a new way of implementing bosonic UIR of $4d$
Poincaré group. We presented them as bunches of scalar fields on $5d$ space
with coordinates $\\{Y^{A},\tau\\}$ and found appropriate realizations for
Poincaré generators. These realizations possess some distinguishing features:
the mass operator $P^{2}$ is independent of spinor coordinates $Y$, so that
equations of motion become ODE in a Lorentz-invariant proper time $\tau$ and
follow from a simple manifestly Poincaré-invariant action. Thus, our
construction demonstrates a separation of variables: e.o.m. governs the
evolution in $\tau$, while $Y$ parameterize the space of Cauchy data. The
translation generators are local differential operators in $\tau$, but non-
local in $Y$, hence $\tau$-evolution is local, while translations act non-
locally on the Cauchy hypersurface spanned by $Y$.

The simple form of e.o.m. allowed us to write down their general solutions.
Those contain two branches, corresponding to different sign-dependence on
$\tau$, similarly to positive and negative energy branches in the standard QFT
approach. We found a Poincaré-invariant inner product, which is positive-
definite for one of the branches.

For massless fields we modified the mass operator by introducing an IR-
regulator. This allowed us to have bounded in $\tau$ solutions and the split
into two branches. This modification is manifestly Poincaré-invariant and is
possible due to the large ambiguity in the form of the mass operator, caused
by the separation of variables. Our construction is non-gauge, as we work
directly with helicity-expanded fields: the bunch of scalar fields mentioned
before represents a bunch of helicities of a spin-$s$ representation,
connected by Poincaré transformations. On the zero-mass shell $\pm s$-helicity
components form closed subrepresentations, so ’gauge-fixing’ reduces to direct
putting all intermediate-helicity components to zero.

We also found the momentum-eigenstate solutions for the simplest cases of a
scalar field and massless fields. They have the form of the confluent
hypergeometric functions.

The construction, proposed in the paper, poses many problems for further
research. One of the most urgent is to develop appropriate canonical
structures and to define an analogue of the canonical quantization procedure,
regarding that some necessary elements are already presented (a classical
action, distinguished in a Lorentz-invariant way coordinate $\tau$ that
governs the evolution, two branches of classical solutions etc). Other
interesting directions include considering fermionic and infinite-spin
representations as well as supersymmetric extensions, generalizations to (A)dS
backgrounds and, the most important, introducing interactions. The problem of
interactions, in its turn, immediately rise many questions: can one formulate
a systematic procedure of looking for Poincaré-invariant vertices? what
happens to the separation of $\tau$ and $Y$ variables at the nonlinear level?
how does the $Y$-nonlocality of Poincaré transformations affect the
perturbative analysis? One may hope that answering these questions will
provide us with new powerful formalism for studying higher-spin theories.

## Acknowledgments

The research was supported by the Alexander von Humboldt Foundation.

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