# Parallel Quasi-concave set optimization:
A new frontier that scales without needing submodularity

Praneeth Vepakomma  Yulia Kempner  Ramesh Raskar

###### Abstract

Classes of set functions along with a choice of ground set are a bedrock to
determine and develop corresponding variants of greedy algorithms to obtain
efficient solutions for combinatorial optimization problems. The class of
approximate constrained submodular optimization has seen huge advances at the
intersection of good computational efficiency, versatility and approximation
guarantees while exact solutions for unconstrained submodular optimization are
NP-hard. What is an alternative to situations when submodularity does not
hold? Can efficient and globally exact solutions be obtained? We introduce one
such new frontier: The class of quasi-concave set functions induced as a dual
class to monotone linkage functions. We provide a parallel algorithm with a
time complexity over $n$ processors of
$\mathcal{O}(n^{2}g)+\mathcal{O}(\log{\log{n}})$ where $n$ is the cardinality
of the ground set and $g$ is the complexity to compute the monotone linkage
function that induces a corresponding quasi-concave set function via a
duality. The complexity reduces to $\mathcal{O}(gn\log(n))$ on $n^{2}$
processors and to $\mathcal{O}(gn)$ on $n^{3}$ processors. Our algorithm
provides a globally optimal solution to a maxi-min problem as opposed to
submodular optimization which is approximate. We show a potential for
widespread applications via an example of diverse feature subset selection
with exact global maxi-min guarantees upon showing that a statistical
dependency measure called distance correlation can be used to induce a quasi-
concave set function.

##  1 Introduction

The rich structure of some set function classes allows for development of
efficient algorithms for combinatorial optimization problem. To be formal, a
set system $(F,\mathcal{Z})$ is a collection $F$ of subsets of a ground set
$\mathcal{Z}$. For example $F$ could be subsets of the power set of
$\mathcal{Z}$ or could be subsets that satisfy the structure of a greedoid
(Korte et al., 2012), semi-lattice (Chajda et al., 2007), independence
systems(Conforti & Laurent, 1989) or an antimatroid(Dietrich, 1989; Kempner &
Levit, 2003; Algaba et al., 2004) and so forth.

Popular set function classes such as submodular functions (Lovász, 1983;
Edmonds, 2003; Nemhauser et al., 1978; Fujishige, 2005; Feige et al., 2011;
Krause & Golovin, 2014; Iyer & Bilmes, 2013) have resulted in a wide array of
powerful algorithms for several tasks across different fields.

Under lack of submodularity, relaxations that characterize approximate
submodularity, (Bian et al., 2017; Bogunovic et al., 2018; Horel & Singer,
2016; Chierichetti et al., 2020; Das & Kempe, 2018) have been introduced to
develop combinatorial algorithms with approximation guarantees. Other set
function classes beyond submodularity include those of subadditive functions,
quasi-submodular functions and the lesser known class of induced quasi-concave
set functions that is relevant to this paper.

This paper introduces a parallel algorithm for optimizing quasi-concave set
functions with global optimality guarantees as opposed to submodular
optimization that provides approximate solutions. Algorithms for optimizing
general quasi-concave set functions do not exist, while a specific sub class
of quasi-concave set functions that can be written in terms of monotone
linkage functions can be optimized to obtain globally optimal solutions. As an
example, we show that certain monotone linkage functions of distance
covariance induce a corresponding quasi-concave set function. We use our
algorithm to find an optimally diverse set of features based on distance
covariance.

###  1.1 Preliminaries

We now list the definition of quasi-concave set functions and state the
induced quasi-concave set function optimization problem which are central to
the focus of this paper.

##  2 Quasi-concave set functions

######  Definition 2.1 (Quasi-Concave Set Function (Mullat, 1976; Kuznecov et
al., 1985; Zaks & Muchnik, 1989; Vepakomma & Kempner, 2019)).

A function $F:\mathcal{F}\mapsto\mathbb{R}$ defined on a set system
$(\mathbf{X},\mathcal{F})$ is quasi-concave if for each
$\mathbf{S,T}\in\mathcal{F}$,

$F(\mathbf{S}\cap\mathbf{T})\geq\min{\\{F(\mathbf{S}),F(\mathbf{T})\\}}$ (1)

Connection: We would like to note its notational similarity to its continuous
counter-part of strictly quasi-concave functions which are those real-valued
functions defined on any convex subset of real-valued vector spaces such that
${\displaystyle f(\lambda
x+(1-\lambda)y)\geq\min{\big{\\{}}f(x),f(y){\big{\\}}}}$ for all
${\displaystyle x\neq y}$ and ${\displaystyle\lambda\in(0,1)}$.

We denote the set $2^{\mathbf{X}}\setminus\left\\{\phi,\mathbf{X}\right\\}$ by
$\mathcal{P}^{-X}$ and we use $i$ indexed subsets like $S_{i}$ to indicate a
singleton (unit cardinality) element of $\mathbf{S}$ labeled by $i$.

######  Definition 2.2 (Monotone Linkage Function (Mullat, 1976)).

A function $\pi(X_{i},\mathbf{Z})$ defined on
$\mathbf{Z}\in\mathcal{P}^{-X},X_{i}\in\mathbf{X}\setminus\mathbf{Z}$ is
called a monotone linkage function if

$\pi(X_{i},\mathbf{S})\geq\pi(X_{i},\mathbf{T}),\mathbf{S}\subseteq\mathbf{T}\in\mathcal{F},\forall
X_{i}\in\mathbf{X}\setminus T$ (2)

We would like to note for the clarity of the reader that $X_{i}$ is an element
while $\mathbf{S},\mathbf{T}$ are sets. Therefore, to make this distinction
clear we denote sets in bold-faced font and elements otherwise.

Monotone linkage functions have been introduced and used for clustering in
(Kempner et al., 1997; Kempner & Muchnik, 2003). A recent work (Seiffarth et
al., 2021) uses these functions to find maximum margin separations in finite
closure systems.

Induced quasi-concave set function optimization This is stated as the problem
of maximizing a quasi-concave set function $M_{\pi}(\mathbf{T})$ over the
modified power set $\mathcal{P}^{-X}$:

$\underset{\mathbf{T}\subset\mathbf{\mathcal{P}^{-X}}}{\mathrm{arg\enskip
max}}\enskip
M_{\pi}(\mathbf{T})=\underset{\mathbf{T}\subset\mathbf{\mathcal{P}^{-X}}}{\mathrm{arg\enskip
max}}\enskip\underset{X_{i}\in\mathbf{X}\setminus\mathbf{T}}{\text{min}}\pi(X_{i},\mathbf{T})$
(3)

where $\pi(X_{i},\mathbf{Z})$ is a monotone linkage function.

##  3 Contributions

  1. 1.

We provide a parallel algorithm to find all the subsets that globally optimize
the induced quasi-concave set function optimization problem in (3).

Type Induced Quasi-concave set function (Parallel: Ours) Induced Quasi-concave
set function Quasi-concave set function (General purpose) Unconstrained
Submodular Robust submodular Unconstrained Quasi submodular Quasi semistrictly
submodular M-/L-convex $SSQM^{\neq}$ under M-convex domain Complexity On $n$
processors, $\mathcal{O}(n^{2}g)+\mathcal{O}(\log{\log{n}})$. For
$n^{2},n^{3}$ processors, check Table 2. $\mathcal{O}(n^{3}g)+\mathcal{O}(n)$
Unknown NP-Hard $\mathcal{O}(nk)$ $\mathcal{O}(n^{2})$
$\mathcal{O}(n^{2}\log{L})+\mathcal{O}(n^{2})$
$\mathcal{O}(n^{4}(\log{L})^{2})$ Solution Globally optimal Globally optimal
Unknown Unknown Approximate Approximate Approximate Approximate

Table 1: We show the computational complexity of our parallel algorithm and
contrast it with that of its non-parallel version (cubic complexity), settings
of submodular optimization and its relaxations. $n$ is the size of the ground
set, $k$ is the cardinality of the returned set =
$\max\left\\{|x(v)-y(v)||x,y\in dom\;f,v\in V\right\\}$ where
$f:Z^{V}\mapsto\mathbb{R}\cup\left\\{+\infty\right\\}$ and $g$ is the
complexity to compute the monotone linkage function.

  2. 2.

The proposed parallel algorithm has a time complexity over $n$ processors of
$\mathcal{O}(n^{2}g)+\mathcal{O}(\log{\log{n}})$ where $n$ is the cardinality
of the ground set and $g$ is the complexity to compute the monotone linkage
function that induces a corresponding quasi-concave set function via a
duality. The complexity reduces to $\mathcal{O}(gn\log(n))$ on $n^{2}$
processors and to $\mathcal{O}(gn)$ on $n^{3}$ processors. The parallel
approach reduces the currently existing cubic computational complexity of the
non parallel version which is $\mathcal{O}(n^{3}g)+\mathcal{O}(n)$.

  3. 3.

As an example, we show that some functions of distance covariance (a measure
of statistical dependence) are quasi-concave set functions. This lets us
optimize them to obtain globally optimal maxi-min solutions for the most
diverse subset of features.

###  3.1 Quasi-concave set function optimization under various set systems

A greedy-type algorithm for finding maximizers of induced quasi-concave set
functions was constructed in (Mullat, 1976; Kuznecov et al., 1985; Zaks &
Muchnik, 1989). Inspired by this work, extensions of these algorithms were
developed for the setting of multipartite graphs in (Vashist, 2006).
Similarly, quasi-concave set functions of distance covariance were derived in
(Vepakomma & Kempner, 2019) and their optimization resulted in a solution for
a diverse feature selection problem with guarantees. Furthermore, quasi-
concave set functions were extended to various set systems including
antimatroids (Levit & Kempner, 2004) and meet-semilattices in (Kempner &
Muchnik, 2008).

| # of
---
processors
| Time
---
Complexity
$n$ (Ours) | $\mathcal{O}(n^{2}g)$
$n^{2}$ (Ours) | $\mathcal{O}(gn\log{n})$
$n^{3}$ (Ours) | $\mathcal{O}(gn)$
Non-parallel | $\mathcal{O}(n^{3}g)+\mathcal{O}(n)$
Table 2: In this table, we show the complexity of our proposed parallel
algorithm with respect to increasing number of processors $n,n^{2}\&n^{3}$.
Here, $n$ is also chosen to be around the order of size of the ground set. We
show that the running times can be drastically reduced from the cubic
complexities in the non-parallel version.

##  4 Related work: Comparing quasi-concave set functions with submodularity

Given the seminal impact of submodular optimization, we would like to compare
the definitions of quasi-concave set functions with submodular functions and
their relaxations. We state some connections inline that we find accordingly.

  1. 1.

Submodular optimization (Fujishige, 2005) Let V be a ground set with
cardinality $|\textbf{V}|=n$, and let $f$ :
$2^{\textbf{V}}\rightarrow\mathbb{R}_{\geq 0}$ be a set function defined on
$\textbf{V}.$ The function $f$ is said to be submodular if for any sets
$\textbf{X}\subseteq\textbf{Y}\subseteq\textbf{V}$ and any element $e\in
V\backslash Y$, it holds that the discrete derivative

$f(\textbf{X}\cup\\{e\\})-f(\textbf{X})\geq
f(\textbf{Y}\cup\\{e\\})-f(\textbf{Y})$

is non-increasing in X. That is, the incremental gain of adding an element to
a subset is $\geq$ (is not smaller) the incremental gain of adding it to a
superset. An equivalent definition is that for every
${\displaystyle\textbf{S,T}\subseteq\textbf{V}}$ we have that

${\displaystyle f(\textbf{S})+f(\textbf{T})\geq
f(\textbf{S}\cup\textbf{T})+f(\textbf{S}\cap\textbf{T})}$ (4)

The problem of maximizing a normalized monotone submodular function subject to
a cardinality constraint has been studied extensively. A celebrated result of
(Nemhauser et al., 1978) shows that a simple greedy algorithm that starts with
an empty set and then iteratively adds elements with highest marginal gains
provides a $(1-1/e)$-approximation.
Connection: Upon defining a linkage function to be equal to a discrete
derivative of a submodular function as

$\pi(e,\textbf{X})=f(\textbf{X}\cup\\{e\\})-f(\textbf{X})$

it can be seen that the derivative of a submodular function is a monotone
linkage function. However, not every monotone linkage function is a derivative
of some submodular function (Muchnik & Shvartser, 1987a, b). Combining
equations (3) and (4), we can say that the functions that are both submodular
and quasi-concave set functions would satisfy
$f(\mathbf{S})+f(\mathbf{T})>=f(\mathbf{S}\cup\mathbf{T})+f(\mathbf{S}\cap\mathbf{T})>=f(\mathbf{S}\cup\mathbf{T})+\min\left\\{f(\mathbf{S}),f(\mathbf{T})\right\\}$.

  2. 2.

Robust submodular optimization Robust versions of submodular optimization
problem were introduced in (Krause et al., 2008; Mirzasoleiman et al., 2017;
Bogunovic et al., 2017; Kazemi et al., 2018; Iyer, 2019; Avdiukhin et al.,
2019; Powers et al., 2016). An earlier variant is of the form introduced in
(Krause et al., 2008) as

$\max_{\textbf{S}\subseteq\textbf{V},|\textbf{S}|\leq
k}\min_{\textbf{Z}\subseteq\textbf{S},|\textbf{Z}|\leq\tau}f(\textbf{S}\backslash\textbf{Z})$

The $\tau$ refers to a robustness parameter, representing the size of the
subset Z that is removed from the selected set S. The goal is to find a set S
such that it is robust upon the worst possible removal of $\tau$ elements,
i.e., after the removal, the objective value should remain as large as
possible. For $\tau=0$, the problem reduces to standard submodular
optimization. The greedy algorithm, which is near-optimal for standard
submodular optimization can perform arbitrarily badly for the robust version
of the problem.
Connection: Note that our statement of induced quasi-concave set function
optimization problem naturally has a robustness component that is similar to
the max-min constraints used in the literature on robust submodular
optimization.

  3. 3.

Quasi submodular and semi-strictly submodular functions (Mei et al., 2015) A
set function $F:2^{N}\mapsto\mathbb{R}$ is quasi-submodular function if
$\forall\textbf{X, Y}\subseteq\textbf{N}$, both of the following conditions
are satisfied

$\begin{array}[]{l}F(\textbf{X}\cap\textbf{Y})\geq F(\textbf{X})\Rightarrow
F(\textbf{Y})\geq F(\textbf{X}\cup\textbf{Y})\\\
F(\textbf{X}\cap\textbf{Y})>F(\textbf{X})\Rightarrow
F(\textbf{Y})>F(\textbf{X}\cup\textbf{Y})\end{array}$

On a similar note, a rich family of semistrictly submodular, discrete Quasi
L-convex and discrete M-convex functions were introduced in (Murota, 1998,
2009).

Figure 1: The proposed parallel algorithm consists of generating a
$\pi$-series at each parallel entity over a copy of the data. The $\pi$-series
at each entity starts with a different $X_{i}$. Each entity then generates a
$\pi$-cluster corresponding to its generated $\pi-series$.The final step
involves picking the best $\pi-cluster$. This is the only step that is not
done in parallel.

##  5 Algorithm and proof of optimality

We now introduce required definitions and corresponding theory to derive the
algorithm. This includes definitions for $\pi$-series and $\pi$-clusters

######  Definition 5.1 ($\pi$-series).

We refer to a series $s_{\pi}=(X_{i_{1}},\ldots,X_{i_{N}})$ as a $\pi$-series
if

$\pi({X}_{i_{k+1}},\bf{\overline{S}_{k}})=\underset{X_{i}\in\mathbf{X}\setminus\mathbf{\overline{S}_{k}}}{\text{min}}\pi(\textnormal{X}_{i},\mathbf{\overline{S}_{k}})$
(5)

for any starting set
$\bf{\overline{S}_{k}}=\\{X_{i_{1}},\ldots,X_{i_{k}}\\},k={1,\ldots,N-1}$.

Therefore, it is a way of greedily populating a series that can start with any
first element $\bf{X}_{i_{1}}$ being the current series, but the subsequent
element to be added to the series, must be the element that minimizes the
element to current series function of
$\pi(\bf{X}_{i_{k+1}},\bf{\overline{S}_{k}})$ where $\bf{X}_{i_{k+1}}$ is the
next element added and $\bf{\overline{S}_{k}}$ is the current series.

######  Definition 5.2 ($\pi$-cluster).

A subset $\bf{S}\in\mathcal{P}^{-\mathbf{X}}$ will be referred to as a
$\pi$-cluster if there exists a $\pi$-series,
$s_{\pi}=(X_{i_{1}},\ldots,X_{i_{N}})$, such that $\bf{S}$ is a maximizer of
$M_{\pi}(\bf{\overline{S}_{k}})$ over all starting sets
$\bf{\overline{S}_{k}}$ of $s_{\pi}$.

######  Theorem 5.1.

(Kempner et al., 1997) If for a $\pi$-series
$s_{\pi}=(X_{i_{1}},X_{i_{2}},\ldots,X_{i_{N}})$, a subset
$\mathbf{S}\subset\mathbf{X}$ contains $X_{i_{1}}$, and if $X_{i_{k+1}}$ is
the first element in $s_{\pi}$ not contained in $\mathbf{S}$ (for some
$k\in\\{1,\ldots,N-1\\}$, then $M_{\pi}(\mathbf{\overline{S}_{k}})\geq
M_{\pi}(\mathbf{S})$

where $\mathbf{\overline{S}_{k}}=\left(X_{i_{1}},\ldots,X_{i_{k}}\right)$. In
particular, if $\mathbf{S}$ is an inclusion-minimal maximizer of $M_{\pi}$
(with regard to $\mathcal{P}^{-\mathbf{X}})$, then
$\mathbf{S}=\mathbf{\overline{S}_{k}}$, that is, $\mathbf{S}$ is a
$\pi$-cluster.

From (Kempner et al., 1997) we have

######  Proposition 5.2.

If $\bf{S_{1}},\bf{S_{2}}\subset\bf{X}$ are overlapping maximizers of a quasi-
concave set function $M_{\pi}(\bf{S})$ over $\mathcal{P}^{-\bf{X}}$, then
$\bf{S_{1}}\cap\bf{S_{2}}$ is also a maximizer of $M_{\pi}(\bf{S})$.

This means that the minimal maximizers of a quasi-convex set function are not
overlapping. Moreover, any nonminimal maximizer can be uniquely partitioned
into a set of the minimal ones.

######  Theorem 5.3.

Each maximizer of a quasi-concave set function on $\mathcal{P}^{-\bf{X}}$ is a
union of its inclusion-minimal maximizers.

###### Proof.

Indeed, if $\bf{S^{\ast}}$ is a maximizer of $M_{\pi}(\bf{S})$ over
$\mathcal{P}^{-\bf{X}}$, then, according to Theorem 5.1, for any
$X_{i}\in\bf{S^{\ast}}$, there exists a minimal maximizer included in
$\bf{S^{\ast}}$ and containing $X_{i}$. ∎

Algorithm 1 Algorithm for induced quasi-convex set function optimization

1:function =DiverseMinimalMaximDCoV($\mathbf{X}$)

2: for all $X_{i}\in\bf{X}$ do

3: Greedily form $\pi$-series $s_{\pi}(x)=(X_{i},X_{i_{2}}\ldots X_{i_{N}})$
starting from $X_{i}$ as its first

4: element.

5: for each $\pi$-series $s_{\pi}(x)$ in step 3 do

6: Find a corresponding smallest starting subset $\bf{T_{x}}$ with
$M_{\pi}(\bf{T_{x}})=\underset{1\leq k\leq
N-1}{\mathrm{max}}\pi(X_{i_{k+1}},\\{X_{i_{1}},\ldots,X_{i_{k}}\\})$

7: end for

8: end for

9: Among the non-coinciding minimal $\pi$-clusters $T_{x}$’s choose those that
maximize
$M_{\pi}(\bf{T_{x}})=\underset{X_{i}\in\mathbf{X}\setminus\mathbf{T_{x}}}{\text{min}}\pi(X_{i},\mathbf{T_{x}})$
all of which are the required minimal maximizers, and we return them as
minimalMax

10:return (minimalMax)

11:end function

######  Theorem 5.4.

The algorithm above finds all the minimal maximizers over
$\mathcal{P}^{-\bf{X}}$.

###### Proof.

From Theorem 5.3 it follows that each element of minimalMax is a maximizer of
$M_{\pi}(\bf{S})$ over $\mathcal{P}^{-\bf{X}}$. Assume that there is a minimal
maximizer $\bf{S}$ that does not belong to minimalMax, and let
$X_{i}\in\bf{S}$. Then, according to Theorem 5.1, there exist $\pi$-series
starting from $X_{i}$ and minimal $\pi$-cluster $T_{x}\subseteq\bf{S}$
containing $X_{i}$ with $M_{\pi}(\bf{T_{x}})\geq M_{\pi}(\bf{S})$. Since
$\bf{S}$ does not belong to minimalMax, and, according to Steps $5$ and $8$ of
the algorithm, $T_{x}$ or some subset of $T_{x}$ belongs to minimalMax, there
is a minimal maximizer strictly included in $\bf{S}$ which contradicts the
minimality of $\bf{S}$. ∎

##  6 Computational complexity

When we have $n$ processors, then we can build each $\pi$-series (in step-3 of
algorithm) in $\mathcal{O}(n^{2}g)$ on one processor (including step 5), and
because we build them in parallel, steps 3-5 take $\mathcal{O}(n^{2}g)$ time.
Finding the maximum in step 8 takes $\mathcal{O}(\log{\log{n}})$ time on $n$
processors, under the CRCW (concurrent-read-concurrent-write) mode (Horowitz &
Sahni, 1978; Horiguchi & Miranker, 1989; Valiant, 1975; Krizanc, 1999). If we
have $n^{2}$ processors, $n$ processors are used to build each $\pi$-series.
To add one element to a series we have to find $\min$ between $n$ elements,
that takes $\mathcal{O}(\log{\log{n}})$ on $n$ processors, so to build each
pi-series takes $g*(\log{1}+\log{2}+\ldots+\log{n})=\mathcal{O}(gn\log{n})$,
and to finish it we have to find $\max$ with $n^{2}$ processors which takes
$\mathcal{O}(1)$ time. This gives us $\mathcal{O}(gnloglogn)$ complexity. If
we have $n^{3}$ processors, then we can use $n^{2}$ processors to build each
$\pi$-series. To add one element to a series we have to find $\min$ between
$n$ elements which takes $\mathcal{O}(1)$ on $n^{2}$ processors. So to build
each $\pi$-series takes $\mathcal{O}(gn)$ time, and to finish we have to find
$\max$ with $n^{3}$ processors, that takes $\mathcal{O}(1)$ time. These are
summarized in Tables 1 and 2.

##  7 Maxi-min Diverse Variable Selection

As an illustrating example, that we derive, we aim to find all the subsets
that maximize the function $M_{\pi}(\mathbf{T})$ which result in the solutions
which are diverse features in the context of statistics/machine learning as
follows

$\underset{\mathbf{T}\subset\mathbf{X}}{\mathrm{arg\enskip max}}\enskip
M_{\pi}(\mathbf{T})=\underset{\mathbf{T}\subset\mathbf{X}}{\mathrm{arg\enskip
max}}\enskip\underset{X_{i}\in\mathbf{X}\setminus\mathbf{T}}{\text{min}}\pi(X_{i},\mathbf{T})$
(6)

For specificity, we use distance covariance upon normalization of the data as
a measure of statistical dependence to model the diversity via
$\pi(\mathbf{X_{i},S})$ as defined in Lemma 8.1.

##  8 Relevant Background on Distance Covariance and Distance Correlation

In this section we introduce some preliminaries about distance correlation and
distance covariance and illustrate a connection between these functions and
quasi-concave set function optimization. Distance Correlation (Székely et al.,
2007) is a measure of nonlinear statistical dependencies between random
vectors of arbitrary dimensions. We describe below distance covariance
$\mathbb{\nu}^{2}(\mathbf{x},\mathbf{y})$ between random variables
$\mathbf{x}\in\mathbb{R}^{d}$ and $\mathbf{y}\in\mathbb{R}^{m}$ with finite
first moments is a non-negative number as

$\mathbb{\nu}^{2}(\mathbf{x},\mathbf{y})=\int_{\mathbb{R}^{d+m}}|f_{\mathbf{x},\mathbf{y}}(t,s)-f_{\mathbf{x}}(t)f_{\mathbf{y}}(s)|^{2}w(t,s)dtds$
(7)

where $w(t,s)$ is a weight function as defined in (Székely et al., 2007),
$f_{\mathbf{x}},f_{\mathbf{y}}$ are characteristic functions of
$\mathbf{x},\mathbf{y}$ and $f_{\mathbf{x},\mathbf{y}}$ is the joint
characteristic function.

The distance covariance is zero if and only if random variables $\mathbf{x}$
and $\mathbf{y}$ are independent. Using the above definition of distance
covariance, we have the following expression for Distance Correlation (Székely
et al., 2007):

The squared Distance Correlation between random variables
$\mathbf{x}\in\mathbb{R}^{d}$ and $\mathbf{y}\in\mathbb{R}^{m}$ with finite
first moments is a nonnegative number is defined as

$\rho^{2}(\mathbf{x},\mathbf{y})=\left\\{\begin{array}[]{cc}\frac{\mathbb{\nu}^{2}(\mathbf{x},\mathbf{y})}{\sqrt{\mathbb{\nu}^{2}(\mathbf{x},\mathbf{x})\mathbb{\nu}^{2}(\mathbf{y},\mathbf{y})}},&\mathbb{\nu}^{2}(\mathbf{x},\mathbf{x})\mathbb{\nu}^{2}(\mathbf{y},\mathbf{y})>0.\\\
0,&\mathbb{\nu}^{2}(\mathbf{x},\mathbf{x})\mathbb{\nu}^{2}(\mathbf{y},\mathbf{y})=0.\end{array}\right.$
(8)

The Distance Correlation defined above has the following interesting
properties.

  1. 1.

${\rho}^{2}(\mathbf{x},\mathbf{y})$ is applicable for arbitrary dimensions $d$
and $m$ of $\mathbf{x}$ and $\mathbf{y}$ respectively.

  2. 2.

${\rho}^{2}(\mathbf{x},\mathbf{y})=0$ if and only if $\mathbf{x}$ and
$\mathbf{y}$ are independent.

  3. 3.

${\rho}^{2}(\mathbf{x},\mathbf{y})$ satisfies the relation
$0\leq\rho^{2}(\mathbf{x},\mathbf{y})\leq 1$.

###  8.1 Sample Distance Covariance and Sample Distance Correlation

We provide the definition of sample version of distance covariance given
samples $\\{(\mathbf{x}_{k},\mathbf{y}_{k})|k=1,2,\ldots,n\\}$ sampled i.i.d.
from joint distribution of random vectors $\mathbf{x}\in\mathbb{R}^{d}$ and
$\mathbf{y}\in\mathbb{R}^{m}$. To do so, we define two squared Euclidean
distance matrices $\mathbf{E}_{\mathbf{X}}$ and $\mathbf{E}_{\mathbf{Y}}$,
where each entry
$[\mathbf{E}_{\mathbf{X}}]_{k,l}=\mbox{}\left\|\mathbf{x}_{k}-\mathbf{x}_{l}\right\|^{2}$
and
$[\mathbf{E}_{\mathbf{Y}}]_{k,l}=\mbox{}\left\|\mathbf{y}_{k}-\mathbf{y}_{l}\right\|^{2}$
with $k,l\in\\{1,2,\ldots,n\\}$. These squared distance matrices are double-
centered by making their row and column sums zero and are denoted as
$\widehat{\mathbf{E}}_{\mathbf{X}},\widehat{\mathbf{Q}}_{\mathbf{X}}$,
respectively. So given a double-centering matrix
$\mathbf{J}=\mathbf{I}-\frac{1}{n}\mathbf{1}\mathbf{1}^{T}$, we have
$\widehat{\mathbf{E}}_{\mathbf{X}}=\mathbf{J}\mathbf{E}_{\mathbf{X}}\mathbf{J}$
and
$\widehat{\mathbf{E}}_{\mathbf{Y}}=\mathbf{J}\mathbf{E}_{\mathbf{Y}}\mathbf{J}$.
The sample distance covariance and sample distance correlation can now be
defined as follows.

######  Definition 8.1.

Sample Distance Covariance (Székely et al., 2007): Given i.i.d samples
$\mathcal{X}\times\mathcal{Y}=\\{(\mathbf{x}_{k},\mathbf{y}_{k})|k=1,2,3,\ldots,n\\}$
and corresponding double centered Euclidean distance matrices
$\widehat{\mathbf{E}}_{\mathbf{X}}$ and $\widehat{\mathbf{E}}_{\mathbf{Y}}$,
the squared sample distance correlation is defined as,

$\hat{\mathbb{\nu}}^{2}(\mathbf{X},\mathbf{Y})=\frac{1}{n^{2}}\sum_{k,l=1}^{n}[\widehat{\mathbf{E}}_{\mathbf{X}}]_{k,l}[\widehat{\mathbf{E}}_{\mathbf{Y}}]_{k,l},$

Using this, sample distance correlation is given by

$\hat{\rho}^{2}(\mathbf{X},\mathbf{Y})=\left\\{\begin{array}[]{cc}\frac{\mathbf{\hat{\nu}}^{2}(\mathbf{X},\mathbf{Y})}{\sqrt{\mathbf{\hat{\nu}}^{2}(\mathbf{X},\mathbf{X})\mathbf{\hat{\nu}}^{2}(\mathbf{Y},\mathbf{Y})}},&\mathbf{\hat{\nu}}^{2}(\mathbf{X},\mathbf{X})\mathbf{\hat{\nu}}^{2}(\mathbf{Y},\mathbf{Y})>0.\\\
0,&\mathbf{\hat{\nu}}^{2}(\mathbf{X},\mathbf{X})\mathbf{\hat{\nu}}^{2}(\mathbf{Y},\mathbf{Y})=0.\end{array}\right.$

Monotonicity of distance covariance under lack of independence: If
$\mathbf{X,Z}\in\mathbb{R}^{p}$ and $\mathbf{Y}\in\mathbb{R}^{q}$ and if
$\mathbf{Z}\rotatebox[origin={c}]{90.0}{$\models$}(\mathbf{X},\mathbf{Y})$
then

$\nu^{2}(\mathbf{X}+\mathbf{Z},\mathbf{Y})\leq\nu^{2}(\mathbf{X},\mathbf{Y})$
(9)

Note that  $\models$ indicates ’statistically independent’ in statistical
literature.

###  8.2 Motivating applications for modeling diversity with quasi-concave set
function optimization

A minor sampling of applications that benefit from the results in this paper
do parallel traditional applications seen in submodular optimization
literature. A few directions are listed below.

  1. 1.

Maximally/minimally correlated marginal selection for private data synthesis
(Zhang et al., 2021).

  2. 2.

Modeling diversity in active learning (Wei et al., 2015), determinantal point
processes (Tschiatschek et al., 2016).

  3. 3.

Diverse sample selection, feature selection and data summarization in machine
learning and statistics. (Prasad et al., 2014; Das et al., 2012)

Figure 2: This illustration refers to the duality between monotone linkage
functions and quasi-concave set functions. Optimization algorithms for general
quqasi-concave set functions do not exist while those that are induced via
monotone linkage functions can be optimized in polynomial time.

###  8.3 A monotone linkage function of distance covariance

######  Lemma 8.1.

The function $\pi(X_{i},\mathbf{S})$ of distance covariance defined on
$X_{i}\notin\mathbf{S}$ as

$\underset{X_{i}\notin\mathbf{S}}{\pi(X_{i},\mathbf{S})}=\sum_{\mathbf{S}_{j}\in\mathbf{S}}-\nu^{2}(X_{i},\mathbf{S}_{j})$
(10)

is a monotone linkage function.

Proof: For $\mathbf{S}\subseteq\mathbf{T}$ we have

$\displaystyle\underset{X_{i}\notin\mathbf{T}}{\pi(X_{i},\mathbf{T})}$
$\displaystyle=\sum_{\mathbf{S}_{j}\in\mathbf{S}}-\nu_{i}^{2}(X_{i},\mathbf{S}_{j})-\sum_{\mathbf{T}_{j}\in\mathbf{T\setminus
S}}\nu_{i}^{2}(X_{i},\mathbf{T}_{j})$ (11)
$\displaystyle\leq\underset{X_{i}\notin\mathbf{T}}{\pi(X_{i},\mathbf{S})}=\sum_{\mathbf{S}_{j}\in\mathbf{S}}-\nu_{i}^{2}(X_{i},\mathbf{S}_{j})$
(12)

We would also like to note that as $\nu(\cdot)$ is a non-negative function the
above inequality does hold true.

By Assertion 1 from (Kempner et al., 1997), we conclude that the function
$M_{\pi}(\mathbf{T})=\underset{X_{i}\in\mathbf{X}\setminus\mathbf{T}}{\text{min}}\pi(X_{i},\mathbf{T})$
is a quasi-concave set function.

######  Theorem 8.2 (Quasi-Concave Distance Covariance Set Function Theorem).

If we have $\mathbf{S}\cap\mathbf{T}\neq\varnothing\text{ and
}\forall\mathbf{S},\mathbf{T},\mathbf{Y}\text{ if
}\nu^{2}(\mathbf{S},\mathbf{T})>0\land\nu^{2}(\mathbf{S},\mathbf{Y})>0\land\nu^{2}(\mathbf{T},\mathbf{Y})>0\text{
then, we have }$

$-\nu^{2}(\mathbf{S}\cap\mathbf{T},\mathbf{Y})\geq
min(-\nu^{2}(\mathbf{S},\mathbf{Y}),-\nu^{2}(\mathbf{T},\mathbf{Y}))$ (13)

###### Proof.

(Vepakomma & Kempner, 2019)

If $\mathbf{S}\cap\mathbf{T}=\mathbf{S}$ then since
$\mathbf{S}\subseteq\mathbf{T}$

the Kosorok’s distance covariance inequality simplifies to give

$-\nu^{2}(\mathbf{S},\mathbf{Y})\geq-\nu^{2}(\mathbf{T},\mathbf{Y})$ (14)

Therefore, we have

$-\nu^{2}(\mathbf{S}\cap\mathbf{T},\mathbf{Y})\geq
min(-\nu^{2}(\mathbf{S},\mathbf{Y}),-\nu^{2}(\mathbf{T},\mathbf{Y}))$

Similarly, if $\mathbf{S}\cap\mathbf{T}=\mathbf{T}$, then since
$\mathbf{T}\subseteq\mathbf{S}$

$-\nu^{2}(\mathbf{T},\mathbf{Y})\geq-\nu^{2}(\mathbf{S},\mathbf{Y})$ (15)

and therefore,

$-\nu^{2}(\mathbf{S}\cap\mathbf{T},\mathbf{Y})\geq
min(-\nu^{2}(\mathbf{S},\mathbf{Y}),-\nu^{2}(\mathbf{T},\mathbf{Y}))$ (16)

In the cases of ${\mathbf{S}\cap\mathbf{T}}\subset{\mathbf{S}}$ and
${\mathbf{S}\cap\mathbf{T}}\subset{\mathbf{T}}$ the Kosorok’s distance
covariance inequality gives

$-\nu^{2}(\mathbf{S}\cap\mathbf{T},\mathbf{Y})>-\nu^{2}(\mathbf{S},\mathbf{Y})$
(17)

and

$-\nu^{2}(\mathbf{S}\cap\mathbf{T},\mathbf{Y})>-\nu^{2}(\mathbf{T},\mathbf{Y})$
(18)

Thus,

$-\nu^{2}(\mathbf{S}\cap\mathbf{T},\mathbf{Y})\geq
min(-\nu^{2}(\mathbf{S},\mathbf{Y}),-\nu^{2}(\mathbf{T},\mathbf{Y}))$ (19)

∎

##  9 Conclusion

We showed that Algorithm 1 gives globally exact solutions that to the induced
quasi-concave set function optimization and is highly parallelizable. This
opens doors to a wide variety of real world applications that we would like to
pursue as part of future work.

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# The angular momentum of electron radiation in a uniform magnetic field

Vladimir Epp<EMAIL_ADDRESS>Ulyana Guselnikova<EMAIL_ADDRESS>Tomsk State Pedagogical University, ul. Kievskaya 60, 634061 Tomsk, Russia

###### Abstract

We study theoretically by means of quantum electrodynamics the vortex
radiation of a relativistic electron in a uniform magnetic field. The exact
expressions for the probability of emission of a photon with a certain angular
momentum are found. The classical asymptotics $\hbar\to 0$ of this probability
does not match the angular momentum flux density calculated by the classical
method using the symmetrized energy-momentum tensor. Although the flux of
angular momentum integrated over the radiation directions is the same in both
cases. We found the angular momentum flux of the radiation field using the
canonical (not symmetrized) energy-momentum tensor and showed that the flux
obtained in this way coincide with the classical limit for the probability of
photon emission.

###### keywords:

vortex radiation, angular momentum, electron , synchrotron radiation, Dirac
equation

###### MSC:

[2010] 78A40, 33E20

## Introduction

Electromagnetic radiation carrying an angular momentum of more than one Planck
constant per photon attracts considerable attention of researchers due to the
possibility of application in various fields of physics: information transfer,
interaction with atoms, high-energy particle collisions and radiation
processes. Vortex light beams have opened up a wide range of applications such
as spatial optical confinement of atoms or microscopic objects and phase
contrast microscopy [1, 2]. The first theoretical and experimental studies of
twisted light or vortex radiation were devoted to laser radiation modified by
an astigmatic optical system or numerically calculated holograms [3, 4]. These
publications stimulated extensive research of the vortex optical beams. A more
extensive bibliography on the history in this area can be found in the reviews
[1, 5].

One of the most promising sources of vortex radiation in the X-ray and the
gamma range of the spectrum are the charged particles moving in a spiral or
vortex beams of charged particles [6, 7]. In particular, spiral motion is
realized in a uniform magnetic field and in a helical undulator. Various
schemes for obtaining a beam of twisted photons in undulators [8, 9, 10] and
free electron lasers [11, 12, 13] have been proposed. Analytical calculations
have shown that a particle moving in a spiral emits photons carrying an
average angular momentum equal to $n\hbar$ with $n$ being the harmonic number
[14, 15, 16]. These theoretical results have been confirmed by a number of
experimental studies of radiation in helical undulators [17, 18, 19].

In the works listed above the angular momentum of the radiation of a particle
moving along a spiral was studied within the framework of classical
electrodynamics. The quantum aspects of this problem were studied in [20, 21]
by use of semiclassical approach.

In the present work, we calculated the probability of radiation of photons
with a high angular momentum using the exact solution of the Dirac equation
for an electron in a uniform magnetic field. This article is a continuation of
our work [15], where the angular momentum of radiation at spiral motion was
studied using classical methods.

The article is organized as follows. The section 1 contains the well-known
solution of the Dirac equation for an electron in a uniform magnetic field.
The main purpose of this section is to remind the wave function of the
electron and to introduce the notation that will be used in what follows. In
the section 2, we calculate the probability of spontaneous transitions with
the emission of a photon in a given direction, carrying an orbital angular
momentum. The classical limit ($\hbar\to 0$) is discussed in section 3.

The result obtained here differs from the angular momentum flux density
calculated earlier by the classical method. Therefore, in section 4, we find
the canonical angular momentum flux density in the Coulomb gauge of the vector
potential by use of the classical method and show that this angular momentum
flux is consistent with the quantum calculation. Finally, in the last section,
we discuss the obtained results.

##  1 Solution to the Dirac equation

Let an electron with charge $e=-e_{0}$ and mass $m$ be in a uniform magnetic
field $\bm{H}$. In a cylindrical coordinate system $r,\varphi,z$ with the $z$
axis directed along the magnetic field, the vector potential can be given as

$\bm{A}=\frac{1}{2}Hr\bm{\hat{\varphi}},$

where $\bm{\hat{\varphi}}$ is the unit vector corresponding to the $\varphi$
coordinate. The solution to the Dirac equation in such a field has the form
[22, 23, 24]

$\displaystyle\Psi_{nl\zeta k_{z}}(\bm{r},t)=\sqrt{\frac{\gamma}{\pi
L}}\exp[i(-Kct+(l-1/2)\varphi+k_{z}z)]\left(\begin{array}[]{r}c_{1}e^{-i\varphi/2}I_{n-1,s}(\rho)\\\
c_{2}e^{i\varphi/2}I_{n,s}(\rho)\\\ c_{3}e^{-i\varphi/2}I_{n-1,s}(\rho)\\\
c_{4}e^{i\varphi/2}I_{n,s}(\rho)\end{array}\right),$ (5)

where $n$ and $l$ are the principal and orbital quantum numbers, respectively,
$n=0,1,2..$, $s$ is the radial quantum number: $s=n-l$. Spin number $\zeta=1$
if the electron spin is directed along the magnetic field and $\zeta=-1$ –
vice versa. The wave function (5) is an eigenfunction of the $z$-component of
the angular momentum operator $\widehat{L}_{z}$ and of the $z$-component of
the linear momentum operator $\widehat{p}_{z}$

$\widehat{L}_{z}\Psi=\left(l-\frac{i}{2}\right)\hbar\Psi,\quad\hat{p}_{z}\Psi=k_{z}\Psi,$

$\hbar$ is the Planck constant. The following notations are used in (5)

$\gamma=\frac{e_{0}H}{2c\hbar},\quad K=\frac{E}{c\hbar},\quad\rho=\gamma
r^{2},$

$c$ is the speed of light, $E$ is the particle energy

$E=c\sqrt{m^{2}c^{2}+\hbar^{2}(k_{z}^{2}+4\gamma n)},$ (6)

$L$ is the formal length restricting motion along the $z$ axis, $I_{ns}(\rho)$
are the Laguerre functions

$I_{ns}(\rho)=\sqrt{\frac{s!}{n!}}e^{-\rho/2}\rho^{(n-s)/2}Q_{s}^{n-s}(\rho),$
(7)

and $Q_{s}^{l}$ are the generalized Laguerre polynomials

$Q_{s}^{l}=\frac{1}{s!}e^{\rho}\rho^{-l}\frac{{\rm d}^{s}}{{\rm
d}\rho^{s}}(\rho^{s+l}e^{-\rho}).$

The $c_{i}$ matrix determines the electron spin polarization

$\begin{pmatrix}c_{1}\\\ c_{2}\\\ c_{3}\\\
c_{4}\end{pmatrix}=\frac{1}{2\sqrt{2}}\begin{pmatrix}B_{+}(A_{+}+A_{-})\\\
B_{-}(A_{-}-A_{+})\\\ B_{+}(A_{+}-A_{-})\\\ B_{-}(A_{+}+A_{-})\end{pmatrix},$

where

$\displaystyle A_{+}=$ $\displaystyle\sqrt{1+k_{z}/K}\,,\quad
A_{-}=\zeta\sqrt{1-k_{z}/K}\,,$ (8) $\displaystyle B_{+}=$
$\displaystyle\sqrt{1+\zeta k_{0}/K_{0}}\,,\quad B_{-}=\zeta\sqrt{1-\zeta
k_{0}/K_{0}}\,,$ (9)

and

$K_{0}=\sqrt{K^{2}-k_{z}^{2}}\,,\quad k_{0}=mc/\hbar.$

##  2 Probability of the transitions $l\to l^{\prime}$

Let us find the probability of emission of a photon with a wave vector
$\bm{k}$ and a $z$-component of angular momentum equal to $L_{z}$. By virtue
of the angular momentum conservation law, this is the probability of
transition from the state with the quantum number $l$ to the state with the
quantum number $l^{\prime}=l-L_{z}/\hbar$. We will specify the photon
polarization by the basis

$\bm{e}_{\pm}=\frac{1}{\sqrt{2}}(\bm{e}_{1}\pm i\bm{e}_{2}),$ (10)

where $\bm{e}_{1}$ and $\bm{e}_{2}$ are mutually orthogonal real vectors
orthogonal to the wave vector $\bm{k}$, so that the vectors listed in the
order $\bm{e}_{1},\bm{e}_{2},\bm{k}$ follow the right-hand rule. The vector
$\bm{e}_{+}$ corresponds to the right-handed polarization, and $\bm{e}_{-}$ to
the left-handed one.

The average value of $z$-component of the photon angular momentum is found by

$\frac{dL_{z}}{{\rm d}t}=\sum_{m^{\prime}}\hbar(l-l^{\prime})w_{\pm},$ (11)

where $m^{\prime}=n^{\prime},l^{\prime},k^{\prime}_{z},\zeta^{\prime}$ are the
quantum numbers of the final state, and $w_{\pm}$ is the probability of
transitions from state $|nlk_{z}\zeta\rangle$ to state
$|n^{\prime}l^{\prime}k^{\prime}_{z}\zeta^{\prime}\rangle$. The transition
probability $w_{\pm}$ was calculated in [23, 24, 25]. Here we briefly repeat
the main steps of the calculations.

$w_{\pm}=\frac{e_{0}^{2}}{2\pi\hbar}\int\frac{{\rm
d}^{3}k}{k}\delta(K-K^{\prime}-k)S_{\pm}.$ (12)

The matrix elements $S_{\pm}$ in the basis (10) take the form

$S_{\pm}=\frac{1}{2}[((\bm{n}\times\langle\bm{\alpha}\rangle^{*})(\bm{n}\times\langle\bm{\alpha}\rangle))\pm((\bm{n}(\langle\bm{\alpha}\rangle\times\langle\bm{\alpha}\rangle^{*}))],$
(13)

The asterisk denotes complex conjugate. The vector $\langle\bm{\alpha}\rangle$
represents the matrix elements of the Dirac matrix $\bm{\alpha}$

$\langle\bm{\alpha}\rangle=\int\psi_{n^{\prime}l^{\prime}k^{\prime}_{z}\zeta}l^{-(\bm{k}\bm{r})}\bm{\alpha}\Psi_{nl\zeta
k_{z}}{\rm d}^{3}x,$ (14)

The vector $\bm{n}=\bm{k}/k$ is the unit vector in the direction of photon
linear momentum. By virtue of axial symmetry, the vector $\bm{n}$ can be
specified as $(n_{x},n_{y},n_{z})=(0,\sin\theta,\cos\theta)$ where $\theta$ is
the angle between the $z$-axis and vector $\bm{n}$. Integration in (12) over
$d^{3}k$ leads to the expression

$w_{\pm}=\frac{e_{0}^{2}}{\hbar}\int S_{\pm}\frac{k\sin\theta
d\theta}{1-\beta_{||}\cos\theta},$ (15)

where

$\beta_{\parallel}=-\frac{\partial K^{\prime}}{\partial
k}=\frac{k_{z}-k\cos\theta}{K-k}.$

In the spherical coordinate system $r,\theta,\varphi$ the matrix elements (13)
are

$\displaystyle S_{\pm}=$
$\displaystyle\frac{1}{2}\left[|\langle\alpha_{1}\rangle|^{2}+|\langle\alpha_{2}\rangle\cos\theta-\langle\alpha_{3}\rangle\sin\theta|^{2}\mp
iS_{\parallel}\right],$ (16) $\displaystyle S_{\parallel}=$
$\displaystyle\bigl{(}\langle\alpha_{1}\rangle^{*}\langle\alpha_{2}\rangle-\langle\alpha_{2}\rangle^{*}\langle\alpha_{1}\rangle\bigr{)}\cos\theta$
$\displaystyle-\bigl{(}\langle\alpha_{1}\rangle^{*}\langle\alpha_{3}\rangle-\langle\alpha_{3}\rangle^{*}\langle\alpha_{1}\rangle\bigr{)}\sin\theta.$

Integration in (14) over the volume gives

$\displaystyle\langle\alpha_{1}\rangle$
$\displaystyle=i[(c^{\prime}_{1}c_{4}+c^{\prime}_{3}c_{2})I_{n,n^{\prime}-1}-(c_{1}c^{\prime}_{4}+c_{3}c^{\prime}_{2})I_{n-1,n^{\prime}}]f,$
$\displaystyle\langle\alpha_{1}\rangle$
$\displaystyle=[(c^{\prime}_{1}c_{4}+c^{\prime}_{3}c_{2})I_{n,n^{\prime}-1}-(c_{1}c^{\prime}_{4}+c_{3}c^{\prime}_{2})I_{n-1,n^{\prime}}]f,$
$\displaystyle\langle\alpha_{3}\rangle$
$\displaystyle=[(c^{\prime}_{1}c_{3}+c_{1}c^{\prime}_{3})I_{n-1,n^{\prime}-1}-(c_{2}c^{\prime}_{4}+c^{\prime}_{2}c_{4})I_{n,n^{\prime}}]f,$
(17)

where the functions $I_{n,n^{\prime}}=I_{n,n^{\prime}}(x)$ are the functions
of the argument $x$

$x=\frac{k^{2}\sin^{2}\theta}{4\gamma},\quad
f=I_{ss^{\prime}}(x)\delta_{k^{\prime}_{z},k_{z}-k\cos\theta}.$ (18)

Finally, substituting (15) into (11), we obtain the rate of radiation of the
$z$-component of angular momentum

$\frac{{\rm d}L_{z}}{{\rm d}t}=e_{0}^{2}\sum_{m^{\prime}}(l-l^{\prime})\int
S_{\pm}\frac{k\sin\theta{\rm d}\theta}{1-\beta_{||}\cos\theta}.$ (19)

This expression differs from the formula for the intensity of radiation [25,
24]

$W=ce_{0}^{2}\sum_{m^{\prime}}\int S_{\pm}\frac{k^{2}\sin\theta{\rm
d}\theta}{1-\beta_{\parallel}\cos\theta}$

by replacing the $z$-component of the photon angular momentum
$\hbar(l-l^{\prime})$ by the photon energy $\hbar\omega$.

The modulus of the wave vector $k$ in (19) is found as a solution to the
system of equations expressing the energy and momentums conservation law

$\displaystyle k_{z}-k_{z}^{\prime}-k\cos\theta=0,$ (20) $\displaystyle
K-K^{\prime}-k=0.$ (21)

This gives

$\displaystyle
k=K\frac{1-\beta_{\parallel}\cos\theta}{\sin^{2}\theta}\left(1-\sqrt{1-\frac{4\gamma(n-n^{\prime})\sin^{2}\theta}{K^{2}(1-\beta_{\parallel}\cos\theta)^{2}}}\,\right).$
(22)

Equations (16) – (19) determine the angular momentum flux carried by photons
during spontaneous transitions to lower energy levels.

##  3 The classic limit

Next, we find the asymptotic expression for the angular momentum flux (19) in
the limit $\hbar\to 0$ and $n,n^{\prime}\gg 1$. In this approximation, the
wave vector (22) is equal to

$\displaystyle
k\approx\frac{(n-n^{\prime})\omega_{0}}{c(1-\beta_{\parallel}\cos\theta)},$
(23)

where $\omega_{0}=eHc/E$ is the frequency of electron motion in a circle. It
can be seen from (6) that the linear momentum component orthogonal to the $z$
axis is equal to $p_{\perp}=2\hbar\displaystyle\sqrt{\gamma n}$. In the
classical limit $p_{\perp}=Ev_{\perp}/c^{2}$, where $v_{\perp}$ is the
velocity part orthogonal to the $z$ axis. Hence, it is convenient to introduce
the notation

$\displaystyle\beta_{\perp}=\frac{v_{\perp}}{c}=\frac{2c\hbar\sqrt{\gamma
n}}{E}$ (24)

Substituting (23) and (24) into the argument of the Laguerre functions (18),
we obtain

$x\approx\frac{(n-n^{\prime})^{2}\beta_{\perp}\sin^{2}\theta}{4n(1-\beta_{\parallel}\cos\theta)}.$
(25)

In the classical limit, the photon energy is much less than the electron
energy, i.e. $n-n^{\prime}\ll n$. Therefore, for $n\to\infty$ and fixed
$n-n^{\prime}$ the variable $x$ decreases as $1/n$. The asymptotic expression
for Laguerre polynomials of the form $Q_{n}^{l}(z/n)$ for large $n$ and fixed
$l$ and $z$ has the form [26, 27]

$Q_{n}^{l}\left(\frac{z}{n}\right)\approx
n^{l}e^{z/2n}z^{-l/2}J_{l}(2\sqrt{z}),$ (26)

where $J_{l}$ are the Bessel functions. Hence,

$\displaystyle I_{n,n^{\prime}}\approx
J_{n-n^{\prime}}(\xi),\quad\xi=\frac{\nu\beta_{\perp}\sin\theta}{1-\beta_{\parallel}\cos\theta}.$
(27)

This asymptotics does not apply to the Laguerre functions
$I_{s,s^{\prime}}(x)$, since in the latter case $x$ tends to zero regardless
of the indices $s$ and $s^{\prime}$. Assuming $s$ and $s^{\prime}$ to be fixed
and letting $x$ tend to zero, we obtain for the Laguerre polynomials

$\displaystyle\lim_{x\to 0}Q_{n}^{l}(x)=\frac{1}{n!}(l+1)(l+2)\dots(l+n).$
(28)

Substituting this into the Laguerre function, we get

$\displaystyle\lim_{x\to 0}I_{s,s^{\prime}}(x)=\delta_{s,s^{\prime}}.$ (29)

Thus, we can the neglect the change in quantum number $s$ in transitions
between states with large numbers $n$. In this case
$n-n^{\prime}=l-l^{\prime}$. This means that the $z$-component of the photon
angular momentum can only be positive (if $z$-axis is directed along the
magnetic field). Substituting the resulting expansions into (16), averaging
the coefficients $c_{i}$ over the initial spin states, and summing up
$c_{i}^{\prime}$ over the final states, we obtain

$\displaystyle
S_{\pm}=\frac{1}{2}\left[\beta_{\perp}J^{\prime}_{\nu}(\xi)\pm\frac{\cos\theta-\beta_{\parallel}}{\sin\theta}J_{\nu}(\xi)\right]^{2}.$
(30)

As a result, the flux of angular momentum in the radiation field (19) takes
the form

$\frac{{\rm d}L_{z\pm}}{{\rm
d}t}=\frac{e_{0}^{2}\omega_{0}}{2c}\sum_{\nu=1}^{\infty}\nu^{2}\int\limits_{0}^{\pi}\left[\beta_{\perp}J^{\prime}_{\nu}(\xi)\pm\frac{\cos\theta-\beta_{\parallel}}{\sin\theta}J_{\nu}(\xi)\right]^{2}\frac{\sin\theta{\rm
d}\theta}{(1-\beta_{\parallel}\cos\theta)^{2}}.$ (31)

This expression was obtained in [20] by use of semiclassical method.

The integrand in Eq. (31) differs from the flux density of angular momentum
obtained by use of classical electrodynamics [15], which reads (for
simplicity, we put $\beta_{\parallel}=0$):

$\frac{{\rm d}L_{z}}{{\rm d}\Omega{\rm
d}t}=\frac{e_{0}^{2}\omega_{0}\beta\sin\theta}{2\pi
c}\sum_{\nu=1}^{\infty}\nu\left\\{\xi\left[{J^{\prime}_{\nu}}^{2}(\xi)+\cot^{2}\theta\,J^{2}_{\nu}(\xi)\right]+J_{\nu}(\xi)J^{\prime}_{\nu}(\xi)\right\\}.$
(32)

The main difference between the Eqs (31) and (32) is that the angular momentum
flux (32) in the direction of the $z$ axis is equal to zero, while the flux
(31) is not. According to standard electrodynamics, the flux of angular
momentum component $(\bm{L}\bm{n})$ in the direction of the vector $\bm{n}$ is
equal to zero [28, 29] The angular momentum flux (32) satisfies this
condition. But the angular momentum flux of radiation in the direction of the
$z$ axis, obtained by quantum theory, is not equal to zero. In particular, if
we put $\theta=0,\pi$ in Eq. (31), we get

$\displaystyle\frac{{\rm d}L_{z\pm}}{{\rm
d}t}=\frac{e_{0}^{2}\omega_{0}\beta_{\perp}}{8\pi
c(1-\beta_{\parallel})^{2}}\left[1\pm\frac{\cos\theta-\beta_{\parallel}}{1-\beta_{\parallel}\cos\theta}\right]_{\theta=0,\pi}.$
(33)

This means that in the direction of the $z$ axis only right-handed photons are
emitted, and only left-handed ones are emitted in the opposite direction.

More detailed information can be obtained from the exact quantum equations
(16) – (19). For $\theta=0,\pi$ the argument of the Laguerre functions (18)
vanishes and $I_{ss^{\prime}}(0)=\delta_{ss^{\prime}}$. Provided that
$n^{\prime}<n$, the matrix elements (17) take the form

$\displaystyle\langle\alpha_{1}\rangle$
$\displaystyle=-i(c_{1}c^{\prime}_{4}+c_{3}c^{\prime}_{2})\delta_{n-1,n^{\prime}}\delta_{s,s^{\prime}}\delta_{k^{\prime}_{z},k_{z}-k\cos\theta},$
(34) $\displaystyle\langle\alpha_{2}\rangle$
$\displaystyle=i\langle\alpha_{1}\rangle,\quad\langle\alpha_{3}\rangle=0.$
(35)

As we can see, the matrix elements are nonzero only for $n^{\prime}=n-1$ and
$s^{\prime}=s$ and, consequently, for $l^{\prime}=l-1$. Thus, in the
$\theta=0,\pi$ directions, photons are emitted only at transitions
$|n,l,\zeta\rangle\to|n-1,l-1,\zeta^{\prime}\rangle$. Decrease the quantum
number $l$ by one means that the angular momentum of these photons is $\hbar$,
and the projection on the direction of radiation is either $\hbar$ at
$\theta=0$, or $-\hbar$ at $\theta=\pi$. Hence, the angular momentum of
photons emitted in the direction of the $z$ axis is represented only by spin.
Their orbital momentum is zero. Which, in fact, is consistent with the
classical representation (31), but not consistent with (32). Note again that
the results of integration over the angles of the Eqs (31) and (32) are the
same. The reasons for this discrepancy are discussed in the next section.

##  4 Canonical angular momentum

The disagreement between the quantum and classical results obtained in the
previous section is due to the fact that the angular momentum density in
classical electrodynamics is defined ambiguously. The straightforward
application of the Noether theorem to electromagnetic field yields the proper
integral conservation laws. In this case, the integrands can be transformed in
such a way that the integral over the entire space does not change. In
particular, the symmetrization of the energy-momentum tensor of the
electromagnetic field is based on this [30, 31].

When calculating the angular momentum flux in our paper [15], we proceeded
from the symmetrized energy-momentum tensor, which leads to the well-known
expression for the angular momentum flux in the wave zone [28, 32]

$\frac{{\rm d}\bm{L}}{{\rm d}t}=\frac{r^{2}}{c}\int(\bm{r}\times\bm{P}){\rm
d}\Omega,$ (36)

where

$\bm{P}=\frac{c}{4\pi}(\bm{E}\times\bm{H})$

is the Poynting vector, and $\bm{E}$ and $\bm{H}$ are the electric and
magnetic fields respectively.

Meanwhile, an immediate consequence of Noether’s theorem is the conservation
of the nonsymmetric canonical energy-momentum tensor [28, 31]

$T^{\mu\nu}=-\frac{1}{4\pi}g^{\mu\alpha}F_{\alpha\rho}\partial^{\nu}A^{\rho}+\frac{1}{16\pi}g^{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}$

The corresponding density of the canonical angular momentum is represented by
the third rank tensor [33]

$M^{\alpha\beta\gamma}=\left(x^{\alpha}T^{\beta\gamma}-x^{\beta}T^{\alpha\gamma}\right)+\frac{1}{4\pi}\left(-A^{\alpha}F^{\beta\gamma}+A^{\beta}F^{\alpha\gamma}\right).$

In this case, the angular momentum density of the electromagnetic field is
given by the component $M^{0\beta\gamma}$ or by the vector
$M_{i}=\varepsilon_{ijk}M^{0jk}$, where $j,k=1,2,3$, and $\varepsilon_{ijk}$
is the unit antisymmetric tensor of the third rank. Accordingly, the angular
momentum flux density in the wave zone is equal to $c\bm{M}$, and the total
flux is determined by the integral

$\frac{{\rm d}\bm{L}}{{\rm d}t}=cr^{2}\int\bm{M}{\rm d}\Omega.$ (37)

The vector $\bm{M}$ in the Coulomb gauge $\bm{\nabla}\bm{A}=0$ can be
represented as the sum [28, 34, 35]

$\bm{M}=\bm{\mathcal{L}}+\bm{\mathcal{S}},$ (38)

where $\bm{\mathcal{L}}$ and $\bm{\mathcal{S}}$ are the “orbital” and “spin”
parts respectively

$\bm{\mathcal{L}}=\frac{1}{4\pi
c}E_{i}(\bm{r}\times\bm{\nabla})A_{i},\quad\bm{\mathcal{S}}=\frac{1}{4\pi
c}(\bm{E}\times\bm{A}).$ (39)

The result of integration over the solid angle in Eqs (36) and (37) is the
same, however the integrands differ significantly. Indeed, let us calculate
the angular momentum flux due to (37), using the expressions for the Fourier
components of the electric field and vector potential obtained in [15]

$\displaystyle A_{\nu r}$ $\displaystyle=\frac{B}{r}J_{\nu}(\xi),\quad
A_{\nu\theta}=\frac{B}{r}\cot\theta J_{\nu}(\xi),$ $\displaystyle
A_{\nu\varphi}$ $\displaystyle=i\frac{B\beta}{r}J^{\prime}_{\nu}(\xi),\quad
B=ee^{i(kr+\nu\varphi-\nu\pi/2)}$ (40) $\displaystyle E_{\nu r}$
$\displaystyle=0,\quad
E_{\nu\theta}=\frac{iB\nu\omega\cot\theta}{rc}J_{\nu}(\xi),\quad
E_{\nu\varphi}=-\frac{B\nu\omega\beta}{rc}J^{\prime}_{\nu}(\xi).$

When using the vector potential, it suffices to restrict ourselves to
quantities of the first order in powers of $1/r$. In the Coulomb gauge one
should put $A_{\nu r}=0$. Let us substitute these values into (39). In order
to find the time-averaged values for the $\nu$-th harmonic, we need to make
the substitution $\bm{E}\to\bm{E}^{*}_{\nu},\,\bm{A}\to\bm{A}_{\nu}$. As
expected, the vector $\bm{\mathcal{L}}_{\nu}$ has only a transverse component
($\bm{\mathcal{L}}_{\nu}\cdot\bm{r}=0$), while the vector
$\bm{\mathcal{S}}_{\nu}$ is directed radially ($\bm{\hat{r}}=\bm{r}/r$):

$\bm{\mathcal{S}}_{\nu}=\frac{e_{0}^{2}\nu\omega_{0}\beta}{2\pi
r^{2}c^{2}}\cot\theta J^{\prime}_{\nu}(\xi)J_{\nu}(\xi)\bm{\hat{r}}.$

The $z$-component of the angular momentum flux is equal to

$\displaystyle{\mathcal{L}}_{\nu z}$
$\displaystyle=\frac{e_{0}^{2}\nu^{2}\omega_{0}}{4\pi
r^{2}c^{2}}\left[\beta^{2}{J^{\prime}}^{2}_{\nu}(\xi)+\cot^{2}\theta
J^{2}_{\nu}(\xi)-\frac{2\beta}{\nu}\frac{\cos^{2}\theta}{\sin\theta}J^{\prime}_{\nu}(\xi)J_{\nu}(\xi)\right],$
(41) $\displaystyle{\mathcal{S}}_{\nu z}$
$\displaystyle=\frac{e_{0}^{2}\nu\omega_{0}\beta}{2\pi
r^{2}c^{2}}\frac{\cos^{2}\theta}{\sin\theta}J^{\prime}_{\nu}(\xi)J_{\nu}(\xi)$
(42)

Substituting these expressions into (37) and (38), we obtain the total flux of
the $z$-component of angular momentum

$\displaystyle\frac{{\rm d}L_{z}}{{\rm
d}t}=\frac{e_{0}^{2}\omega_{0}}{2c}\sum_{\nu=1}^{\infty}\nu^{2}\int\limits_{0}^{\pi}\left[\beta^{2}{J^{\prime}}^{2}_{\nu}(\xi)+\cot^{2}\theta
J^{2}_{\nu}(\xi)\right]\sin\theta{\rm d}\theta$ (43)

in full accordance with Eq. (31) obtained from the probability of photon
emission in the limit $\hbar\to 0,\,\beta_{\parallel}=0$.

In the direction $\theta=0$, the $z$-component of the “orbital” angular
momentum is equal to zero, and in the $z$-component of the “spin” angular
momentum, only the first harmonic remains

$\bm{\mathcal{S}}_{\nu}\Big{|}_{\theta=0}=\frac{e_{0}^{2}\omega_{0}\beta^{2}}{8\pi
r^{2}c^{2}}\delta_{1\nu}\bm{\hat{r}}.$

Wherein, the average angular momentum per one photon is equal to $\hbar$. In
the direction $\theta=\pi/2$, which lies in the plane of the electron orbit,
on the contrary, the “spin” angular momentum ${\mathcal{S}}_{\nu z}$ is equal
to zero, and the “orbital” angular momentum ${\mathcal{L}}_{\nu z}$ takes on a
maximum value. The absence of “spin” angular momentum is due to the fact that
in the classical limit the probability of emission of left-handed photons is
equal to the probability of emission of right-handed ones in this direction.
In this case, the “orbital” angular momentum is associated with the momentum
transfer of the electromagnetic field.

##  5 Conclusion

Using the solution to the Dirac equation for an electron in a uniform magnetic
field, we have found exact expressions for the probability of emission of a
photon with a certain projection of its angular momentum onto the direction of
the magnetic field. We have also found the asymptotic expression for the
angular momentum flux density in radiation field in the limit $\hbar\to 0$.
The expressions obtained in this way do not coincide with the previously found
equations for the angular momentum flux density derived from the symmetrized
energy-momentum tensor of the electromagnetic field. This discrepancy is
explained by the fact that the density of the angular momentum of the field
and, accordingly, the flux of the angular momentum are determined ambiguously.
For comparison, we found the angular momentum flux in radiation using the
canonical (not symmetrized) energy-momentum tensor and showed that the
formulas obtained in this way coincide with the classical limit of equations
for the probability of photon emission. In this case, the flux of angular
momentum integrated over the radiation directions following from the canonical
and symmetrized energy-momentum tensor, as expected, coincide.

The question of the extent to which the canonical or symmetrized energy-
momentum tensor is applicable to real physical measurements is traditionally
the subject of hot discussions [36, 37, 38, 9, 34, 39]. This problem is
closely related to the determination of the angular momentum density of the
field and to the separation of spin and orbital degrees of freedom. Obviously,
the applicability of certain formulas to specific physical measurements is
determined by the specific conditions of the lightmatter interactions.
Currently, various methods for registering the orbital angular momentum of
radiation are being developed [40, 41, 42]. Moreover, there are indications of
the possibility of independent measurement of the spin and orbital angular
momentum [43, 44, 45].

## Acknowledgments

This research was supported by Russian Foundation for Basic Research, project
No. 19-42-700011.

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