# Erlangen Programme at Large 3.1
Hypercomplex Representations of the Heisenberg Group and Mechanics

Vladimir V. Kisil School of Mathematics
University of Leeds
Leeds LS2 9JT
UK<EMAIL_ADDRESS>http://www.maths.leeds.ac.uk/~kisilv/ Dedicated
to the memory of V.I. Arnold

(Date: 27th May 2010)

###### Abstract.

In the spirit of geometric quantisation we consider representations of the
Heisenberg(–Weyl) group induced by hypercomplex characters of its centre. This
allows to gather under the same framework, called p-mechanics, the three
principal cases: quantum mechanics (elliptic character), hyperbolic mechanics
and classical mechanics (parabolic character). In each case we recover the
corresponding dynamic equation as well as rules for addition of probabilities.
Notably, we are able to obtain whole classical mechanics without any kind of
semiclassical limit $\hslash\rightarrow 0$.

###### Key words and phrases:

Heisenberg group, Kirillov’s method of orbits, geometric quantisation, quantum
mechanics, classical mechanics, Planck constant, dual numbers, double numbers,
hypercomplex, jet spaces, hyperbolic mechanics, interference, Segal–Bargmann
representation, Schrödinger representation, dynamics equation, harmonic and
unharmonic oscillator, contextual probability, $\mathcal{PT}$-symmetric
Hamiltonian

###### 2000 Mathematics Subject Classification:

Primary 81R05; Secondary 81R15, 22E27, 22E70, 30G35, 43A65.

On leave from Odessa University.

††copyright: ©:

###### Contents

  1. 1 Introduction
  2. 2 Heisenberg group and $p$-mechanics
    1. 2.1 The Heisenberg group and induced representations
    2. 2.2 Convolutions (observables) on $\mathbb{H}^{n}{}$ and commutator
    3. 2.3 States and Probability
  3. 3 Elliptic characters and Quantum Dynamics
    1. 3.1 Segal–Bargmann and Schrödinger Representations
    2. 3.2 Commutator and the Heisenberg Equation
    3. 3.3 Quantum Probabilities
  4. 4 Hypercomplex Repersentations of the Heisenberg Group
    1. 4.1 Hyperbolic Representations and Addition of Probabilities
      1. 4.1.1 Hyperbolic Representations of the Heisenberg Group
      2. 4.1.2 Hyperbolic Dynamics
      3. 4.1.3 Hyperbolic Probabilities
    2. 4.2 Parabolic (Classical) representations on the phase space
      1. 4.2.1 Classical Non-Commutative Representations
      2. 4.2.2 Hamilton Equation
      3. 4.2.3 Classical probabilities
  5. 5 Discussion

##  1\. Introduction

Complex valued representations of the Heisenberg group (also known as Weyl or
Heisenberg-Weyl group) provide a natural framework for quantum mechanics
[Howe80b, Folland89]. This is the most fundamental example of the Kirillov
orbit method and geometrical quantisation technique [Kirillov99, Kirillov94a].

Following the pattern we consider representations of the Heisenberg group
which are induced by hypercomplex characters of its centre. Besides complex
numbers (which correspond to the elliptic case) there are two other types of
hypercomplex numbers: dual (parabolic) and double (hyperbolic) [Yaglom79]*App.
C [Kisil09c].

To describe dynamics of a physical system we use a universal equation based on
inner derivations of the convolution algebra [Kisil00a] [Kisil02e]. The
complex valued representations produce the standard framework for quantum
mechanics with the Heisenberg dynamical equation [Vourdas06a].

The double number valued representations, with the hyperbolic unit
$\mathrm{j}^{2}=1$, is a natural source of hyperbolic quantum mechanics
developed for a while [Hudson04a, Hudson66a, Khrennikov03a, Khrennikov05a,
Khrennikov08a]. The universal dynamical equation employs hyperbolic commutator
in this case. This can be seen as a Moyal bracket based on the hyperbolic sine
function. The hyperbolic observables act as operators on a Krein space with an
indefinite inner product. Such spaces are employed in study of
$\mathcal{PT}$-symmetric Hamiltonians and hyperbolic unit $\mathrm{j}^{2}=1$
naturally appear in this setup [GuentherKuzhel10a].

The representations with values in dual numbers provide a convenient
description of the classical mechanics. For this we do not take any sort of
semiclassical limit, rather the nilpotency of the parabolic unit
($\varepsilon^{2}=0$) do the task. This removes the vicious necessity to
consider the Planck _constant_ tending to zero. The dynamical equation takes
the Hamiltonian form. We also describe classical non-commutative
representations of the Heisenberg group which acts in the first jet space.

######  Remark 1.1.

It is commonly accepted that the striking difference between quantum and
classical mechanics is non-commutativity of observables in the first case. In
particular the Heisenberg commutation relations, see (2.5), imply the
uncertainty principle, the Heisenberg equation of motion and other quantum
features. However our work shows that quantum mechanics is mainly determined
by the properties of complex numbers. Non-commutative representations of the
Heisenberg group in dual numbers implies the Poisson dynamical equation and
local addition of probabilities in Section 4.2, which are completely
classical.

######  Remark 1.2.

It is worth to note that our technique is different from contraction technique
in the theory of Lie groups [LevyLeblond65a, GromovKuratov05b]. Indeed a
contraction of the Heisenberg group $\mathbb{H}^{n}{}$ is the commutative
Euclidean group $\mathbb{R}^{2n}{}$ which does not recreate neither quantum
nor classical mechanics.

The approach provides not only three different types of dynamics, it also
generates the respective rules for addition of probabilities as well. For
example, the quantum interference is the consequence of the same complex-
valued structure, which directs the Heisenberg equation. The absence of an
interference (a particle behaviour) in the classical mechanics is again the
consequence the nilpotency of the parabolic unit. Double numbers creates the
hyperbolic law of additions of probabilities which were extensively
investigates [Khrennikov03a, Khrennikov05a]. There are still unresolved issues
with positivity of the probabilistic interpretation in the hyperbolic case
[Hudson04a, Hudson66a].

The work clarifies foundations of quantum and classical mechanics. We
recovered from the representation theory the existence of three non-isomorphic
model of mechanics already discussed in [Hudson04a, Hudson66a] from
translation invariant formulation. It also hinted that hyperbolic counterpart
is (at least theoretically) as natural as classical and quantum mechanics are.
The approach provides a framework for description of aggregate system which
have say both quantum and classical components. This can be used to model
quantum computers with classical terminals [Kisil09b].

Remarkably, simultaneously with the work [Hudson66a] group-invariant
axiomatics of geometry lead R.I. Pimenov [Pimenov65a] to description of
$3^{n}$ Cayley–Klein constructions. The connection between group-invariant
geometry and respective mechanics were explored in many works of N.A. Gromov,
see for example [Gromov90a, Gromov90b, GromovKuratov05b]. Those already
highlighted the rôle of three types of hypercomplex units for the realisation
of elliptic, parabolic and hyperbolic geometry and kinematic.

There is a further connection between representations of the Heisenberg group
and hypercomplex numbers. The symplectomorphism of phase space are also
automorphism of the Heisenberg group [Folland89]*§ 1.2. Induced representation
of the symplectic group naturally lead to hypercomplex numbers [Kisil09c].
Hamiltonians, which produce those symplectomorphism, are of interest, for
example, in quantum optic [ATorre10a]. An analysis of those Hamiltonians by
means of creation/annihilation operators recreate hypercomplex coefficients as
well [Kisil11a, Kisil11c].

######  Remark 1.3.

This work is performed within the “Erlangen programme at large” framework
[Kisil06a, Kisil05a], thus it would be suitable to explain the numbering of
various papers. Since the logical order may be different from chronological
one the following numbering scheme is used:

Prefix |  Branch description
---|---
“0” or no prefix |  Mainly geometrical works, within the classical field of Erlangen programme by F. Klein
“1” |  Papers on analytical functions theories and wavelets
“2” |  Papers on operator theory, functional calculi and spectra
“3” |  Papers on mathematical physics

For example, this is the first paper in the mathematical physics area.

##  2\. Heisenberg group and $p$-mechanics

###  2.1. The Heisenberg group and induced representations

Let $(s,x,y)$, where $x$, $y\in\mathbb{R}^{n}{}$ and $s\in\mathbb{R}{}$, be an
element of the Heisenberg group $\mathbb{H}^{n}{}$ [Folland89, Howe80b]. The
group law on $\mathbb{H}^{n}{}$ is given as follows:

(2.1)
$\textstyle(s,x,y)\cdot(s^{\prime},x^{\prime},y^{\prime})=(s+s^{\prime}+\frac{1}{2}\omega(x,y;x^{\prime},y^{\prime}),x+x^{\prime},y+y^{\prime}),$

where the non-commutativity is due to $\omega$—the _symplectic form_ on
$\mathbb{R}^{2n}{}$ [Arnold91]*§ 37:

(2.2) $\omega(x,y;x^{\prime},y^{\prime})=xy^{\prime}-x^{\prime}y.$

The Heisenberg group is non-commutative Lie group with the centre

$Z=\\{(s,0,0)\in\mathbb{H}^{n}{},\ s\in\mathbb{R}{}\\}.$

The left shifts

(2.3) $\Lambda(g):f(g^{\prime})\mapsto f(g^{-1}g^{\prime})$

act as a representation of $\mathbb{H}^{n}{}$ on a certain linear space of
functions. For example, action on $L_{2}{}(\mathbb{H}{},dg)$ with respect to
the Haar measure $dg=ds\,dx\,dy$ is the _left regular_ representation, which
is unitary.

The Lie algebra $\mathfrak{h}^{n}$ of $\mathbb{H}^{n}{}$ is spanned by
left-(right-)invariant vector fields

(2.4) $\textstyle S^{l(r)}=\pm{\partial_{s}},\quad
X_{j}^{l(r)}=\pm\partial_{x_{j}}-\frac{1}{2}y_{j}{\partial_{s}},\quad
Y_{j}^{l(r)}=\pm\partial_{y_{j}}+\frac{1}{2}x_{j}{\partial_{s}}$

on $\mathbb{H}^{n}{}$ with the Heisenberg _commutator relations_

(2.5) $[X_{i}^{l(r)},Y_{j}^{l(r)}]=\delta_{ij}S^{l(r)}$

and all other commutators vanishing. We will omit the supscript $l$ for left-
invariant field sometimes.

We can construct linear representations by induction [Kirillov76]*§ 13 from a
character $\chi$ of the centre $Z$. There are several models for induced
representations, here we prefer the following one, which is presented
stripping off all generalities, cf. [Kirillov76]*§ 13 [MTaylor86]*Ch. 5. Let
$F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ be the space of functions on
$\mathbb{H}^{n}{}$ having the properties:

(2.6) $f(gh)=\chi(h)f(g),\qquad\text{ for all }g\in\mathbb{H}^{n}{},\ h\in Z$

and

(2.7) $\int_{\mathbb{R}^{2n}{}}\left|f(0,x,y)\right|^{2}dx\,dy<\infty.$

Then $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ is invariant under the left shifts and
those shifts restricted to $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ make a
representation ${\rho_{\chi}}$ of $\mathbb{H}^{n}{}$ induced by $\chi$.

If the character $\chi$ is unitary, then the induced representation is unitary
as well. However the representation ${\rho_{\chi}}$ is not necessarily
irreducible. Indeed, left shifts are commuting with the right action of the
group. Thus any subspace of null-solutions of a linear combination
$aS+\sum_{j=1}^{n}(b_{j}X_{j}+c_{j}Y_{j})$ of left-invariant vector fields is
left-invariant and we can restrict ${\rho_{\chi}}$ to this subspace. The left-
invariant differential operators define analytic condition for functions, cf.
[Vourdas06a].

######  Example 2.1.

The function $f_{0}(s,x,y)=e^{\mathrm{i}hs-h(x^{2}+y^{2})/4}$, where
$h=2\pi\hslash$, belongs to $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ for the
character $\chi(s)=e^{\mathrm{i}hs}$. It is also a null solution for all the
operators $X_{j}-\mathrm{i}Y_{j}$. The closed linear span of functions
$f_{g}=\Lambda(g)f_{0}$ is invariant under left shifts and provide a model for
Segal–Bargmann type representation of the Heisenberg group, which will be
considered below.

######  Remark 2.2.

An alternative construction of induced representations is as follow
[Kirillov76]*§ 13.2. Consider a subgroup $H$ of a group $G$. Let a smooth
section $\mathbf{s}:G/H\rightarrow G$ be a left inverse of the natural
projection $\mathbf{p}:G\rightarrow G/H$. Thus any element $g\in G$ can be
uniquely decomposed as $g=\mathbf{s}(\mathbf{p}(g))*\mathbf{r}(g)$ where the
map $\mathbf{r}:G\rightarrow H$ is defined by the previous identity. For a
character $\chi$ of $H$ we can define a _lifting_
$\mathcal{L}_{\chi}:L_{2}{}(G/H)\rightarrow L_{2}^{\chi}{}(G)$ as follows:

(2.8)
$[\mathcal{L}_{\chi}f](g)=\chi(\mathbf{r}(g))f(\mathbf{p}(g))\qquad\text{where
}f(x)\in L_{2}{}(G/H).$

The image space of the lifting $\mathcal{L}_{\chi}$ is invariant under left
shifts. We also define the _pulling_ $\mathcal{P}:L_{2}^{\chi}{}(G)\rightarrow
L_{2}{}(G/H)$, which is a left inverse of the lifting and explicitly cab be
given, for example, by $[\mathcal{P}F](x)=F(\mathbf{s}(x))$. Then the induced
representation on $L_{2}{}(G/H)$ is generated by the formula
${\rho_{\chi}}(g)=\mathcal{P}\circ\Lambda(g)\circ\mathcal{L}$.

###  2.2. Convolutions (observables) on $\mathbb{H}^{n}{}$ and commutator

Using a left invariant measure $dg=ds\,dx\,dy$ on $\mathbb{H}^{n}{}$ we can
define the convolution of two functions:

(2.9) $\displaystyle(k_{1}*k_{2})(g)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k_{1}(g_{1})\,k_{2}(g_{1}^{-1}g)\,dg_{1}.$

This is a non-commutative operation, which is meaningful for functions from
various spaces including $L_{1}{}(\mathbb{H}^{n}{},dg)$, the Schwartz space
$S{}$ and many classes of distributions, which form algebras under
convolutions. Convolutions on $\mathbb{H}^{n}{}$ are used as _observables_ in
$p$-mechanic [Kisil96a, Kisil02e].

A unitary representation ${\rho}$ of $\mathbb{H}^{n}{}$ extends to
$L_{1}{}(\mathbb{H}^{n}{},dg)$ by the formula:

(2.10) ${\rho}(k)=\int_{\mathbb{H}^{n}{}}k(g){\rho}(g)\,dg.$

This is also an algebra homomorphism of convolutions to linear operators.

For a dynamics of observables we need inner _derivations_ $D_{k}$ of the
convolution algebra $L_{1}{}(\mathbb{H}^{n}{})$, which are given by the
_commutator_ :

$\displaystyle D_{k}:f\mapsto[k,f]$ $\displaystyle=$ $\displaystyle k*f-f*k$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k(g_{1})\left(f(g_{1}^{-1}g)-f(gg_{1}^{-1})\right)\,dg_{1},\quad
f,k\in L_{1}{}(\mathbb{H}^{n}{}).$

To describe dynamics of a time-dependent observable $f(t,g)$ we use the
universal equation, cf. [Kisil94d, Kisil96a]:

(2.12) $S\dot{f}=[H,f],$

where $S$ is the left-invariant vector field (2.4) generated by the centre of
$\mathbb{H}^{n}{}$. The presence of operator $S$ fixes the dimensionality of
both sides of the equation (2.12) if the observable $H$ (Hamiltonian) has the
dimensionality of energy [Kisil02e]*Rem 4.1. If we apply a right inverse
$\mathcal{A}$ of $S$ to both sides of the equation (2.12) we obtain the
equivalent equation

(2.13) $\dot{f}=\left\\{\\!\left[H,f\right]\\!\right\\},$

based on the universal bracket
$\left\\{\\!\left[k_{1},k_{2}\right]\\!\right\\}=k_{1}*\mathcal{A}k_{2}-k_{2}*\mathcal{A}k_{1}$
[Kisil02e].

######  Example 2.3 (Harmonic oscillator).

Let $H=\frac{1}{2}(m\omega^{2}q^{2}+\frac{1}{m}p^{2})$ be the Hamiltonian of a
one-dimensional harmonic oscillator, where $\omega$ is a constant frequency
and $m$ is a constant mass. Its _p-mechanisation_ will be the second order
differential operator on $\mathbb{H}^{n}{}$ [BrodlieKisil03a]*§ 5.1:

$\textstyle H=\frac{1}{2}(m\omega^{2}X^{2}+\frac{1}{m}Y^{2}),$

where we dropped sub-indexes of vector fields (2.4) in one dimensional
setting. We can express the commutator as a difference between the left and
the right action of the vector fields:

$\textstyle[H,f]=\frac{1}{2}(m\omega^{2}((X^{r})^{2}-(X^{l})^{2})+\frac{1}{m}((Y^{r})^{2}-(Y^{l})^{2}))f.$

Thus the equation (2.12) becomes [BrodlieKisil03a]*(5.2):

(2.14) $\frac{\partial}{\partial s}\dot{f}=\frac{\partial}{\partial
s}\left(m\omega^{2}y\frac{\partial}{\partial
x}-\frac{1}{m}x\frac{\partial}{\partial y}\right)f.$

Of course, the derivative $\frac{\partial}{\partial s}$ can be dropped from
both sides of the equation and the general solution is found to be:

(2.15) $\textstyle f(t;s,x,y)=f_{0}\left(s,x\cos(\omega t)+m\omega
y\sin(\omega t),-\frac{x}{m\omega}\sin(\omega t)+y\cos(\omega t)\right),$

where $f_{0}(s,x,y)$ is the initial value of an observable on
$\mathbb{H}^{n}{}$.

######  Example 2.4 (Unharmonic oscillator).

We consider unharmonic oscillator with cubic potential, see
[CalzettaVerdaguer06a] and references therein:

(2.16)
$H=\frac{m\omega^{2}}{2}q^{2}+\frac{\lambda}{6}q^{3}+\frac{1}{2m}p^{2}.$

Due to absence of non-commutative products p-mechanisation is straightforward:

$H=\frac{m\omega^{2}}{2}X^{2}+\frac{\lambda}{6}X^{3}+\frac{1}{m}Y^{2}.$

Similarly to the harmonic case the dynamic equation, after cancellation of
$\frac{\partial}{\partial s}$ on both sides, becomes:

(2.17) $\dot{f}=\left(m\omega^{2}y\frac{\partial}{\partial
x}+\frac{\lambda}{6}\left(3y\frac{\partial^{2}}{\partial
x^{2}}+\frac{1}{4}y^{3}\frac{\partial^{2}}{\partial
s^{2}}\right)-\frac{1}{m}x\frac{\partial}{\partial y}\right)f.$

Unfortunately, it cannot be solved analytically as easy as the harmonic case.

###  2.3. States and Probability

Let an observable ${\rho}(k)$ (2.10) is defined by a kernel $k(g)$ on the
Heisenberg group and its representation ${\rho}$ at a Hilbert space
$\mathcal{H}$. A state on the convolution algebra is given by a vector
$v\in\mathcal{H}$. A simple calculation:

$\displaystyle\left\langle{\rho}(k)v,v\right\rangle_{\mathcal{H}}$
$\displaystyle=$
$\displaystyle\left\langle\int_{\mathbb{H}^{n}{}}k(g){\rho}(g)v\,dg,v\right\rangle_{\mathcal{H}}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k(g)\left\langle{\rho}(g)v,v\right\rangle_{\mathcal{H}}dg$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k(g)\overline{\left\langle
v,{\rho}(g)v\right\rangle_{\mathcal{H}}}\,dg$

can be restated as:

$\left\langle{\rho}(k)v,v\right\rangle_{\mathcal{H}}=\left\langle
k,l\right\rangle,\qquad\text{where}\quad l(g)=\left\langle
v,{\rho}(g)v\right\rangle_{\mathcal{H}}.$

Here the left-hand side contains the inner product on $\mathcal{H}$, while the
right-hand side uses a skew-linear pairing between functions on
$\mathbb{H}^{n}{}$ based on the Haar measure integration. In other words we
obtain, cf. [BrodlieKisil03a]*Thm. 3.11:

######  Proposition 2.5.

A state defined by a vector $v\in\mathcal{H}$ coincides with the linear
functional given by the wavelet transform

(2.18) $l(g)=\left\langle v,{\rho}(g)v\right\rangle_{\mathcal{H}}$

of $v$ used as the mother wavelet as well.

The addition of vectors in $\mathcal{H}$ implies the following operation on
states:

(2.19) $\displaystyle\left\langle
v_{1}+v_{2},{\rho}(g)(v_{1}+v_{2})\right\rangle_{\mathcal{H}}$
$\displaystyle=$ $\displaystyle\left\langle
v_{1},{\rho}(g)v_{1}\right\rangle_{\mathcal{H}}+\left\langle
v_{2},{\rho}(g)v_{2}\right\rangle_{\mathcal{H}}$ $\displaystyle{}+\left\langle
v_{1},{\rho}(g)v_{2}\right\rangle_{\mathcal{H}}+\overline{\left\langle
v_{1},{\rho}(g^{-1})v_{2}\right\rangle_{\mathcal{H}}}$

The last expression can be conveniently rewritten for kernels of the
functional as

(2.20) $l_{12}=l_{1}+l_{2}+2A\sqrt{l_{1}l_{2}}$

for some real number $A$. This formula is behind the contextual law of
addition of conditional probabilities [Khrennikov01a] and will be illustrated
below. Its physical interpretation is an interference, say, from two slits.
The mechanism of such interference can be both causal and local, see
[Kisil01c] [KhrenVol01].

##  3\. Elliptic characters and Quantum Dynamics

In this section we consider the representation ${\rho_{h}}$ of
$\mathbb{H}^{n}{}$ induced by the elliptic character
$\chi_{h}(s)=e^{\mathrm{i}hs}$ in complex numbers parametrised by
$h\in\mathbb{R}{}$. We also use the convenient agreement $h=2\pi\hslash$.

###  3.1. Segal–Bargmann and Schrödinger Representations

The realisation of ${\rho_{h}}$ by the left shifts (2.3) on
$L_{2}^{h}{}(\mathbb{H}^{n}{})$ is rarely used in quantum mechanics. Instead
two unitary equivalent forms are more common: the Schrödinger and
Segal–Bargmann representations.

The Segal-Bargmann representation can be obtained from the orbit method of
Kirillov [Kirillov94a]. It allows spatially separate irreducible components of
the left regular representation, each of them is located on the orbit of the
co-adjoint representation, see [Kisil02e]*§ 2.1 [Kirillov94a] for details, we
only present a brief summary here.

We identify $\mathbb{H}^{n}{}$ and its Lie algebra $\mathfrak{h}_{n}$ through
the exponential map [Kirillov76]*§ 6.4. The dual $\mathfrak{h}_{n}^{*}$ of
$\mathfrak{h}_{n}$ is presented by the Euclidean space $\mathbb{R}^{2n+1}{}$
with coordinates $(\hslash,q,p)$. The pairing $\mathfrak{h}_{n}^{*}$ and
$\mathfrak{h}_{n}$ given by

$\left\langle(s,x,y),(\hslash,q,p)\right\rangle=\hslash s+q\cdot x+p\cdot y.$

This pairing defines the Fourier transform $\hat{\
}:L_{2}{}(\mathbb{H}^{n}{})\rightarrow L_{2}{}(\mathfrak{h}_{n}^{*})$ given by
[Kirillov99]*§ 2.3:

(3.1) $\hat{\phi}(F)=\int_{\mathfrak{h}^{n}}\phi(\exp
X)e^{-2\pi\mathrm{i}\left\langle X,F\right\rangle}\,dX\qquad\textrm{ where
}X\in\mathfrak{h}^{n},\ F\in\mathfrak{h}_{n}^{*}.$

For a fixed $\hslash$ the left regular representation (2.3) is mapped by the
Fourier transform to the Segal–Bargmann type representation [Kisil02e]*(2.9)
[deGosson08a]*(1):

(3.2) $\textstyle{\rho_{\hslash}}(s,x,y):f(q,p)\mapsto
e^{-2\pi\mathrm{i}(\hslash
s+qx+py)}f\left(q-\frac{\hslash}{2}y,p+\frac{\hslash}{2}x\right).$

The collection of points $(\hslash,q,p)\in\mathfrak{h}_{n}^{*}$ for a fixed
$\hslash$ is naturally identified with the phase space of the system.

######  Remark 3.1.

It is possible to identify the case of $\hslash=0$ with classical mechanics
[Kisil02e]. Indeed, a substitution of the zero value of $\hslash$ into (3.2)
produces the commutative representation:

(3.3) ${\rho_{0}}(s,x,y):f(q,p)\mapsto
e^{-2\pi\mathrm{i}(qx+py)}f\left(q,p\right).$

It can be decomposed into the direct integral of one-dimensional
representations parametrised by the points $(q,p)$ of the phase space. The
classical mechanics, including the Hamilton equation, can be recovered from
those representations [Kisil02e]. However the condition $\hslash=0$ (as well
as $\hslash\rightarrow 0$) is not completely physical. Commutativity (and
subsequent relative triviality) of those representation is the main reason why
they are oftenly neglected. The commutativity can be outweighed by special
arrangements, e.g. an antiderivative [Kisil02e]*(4.1), but the procedure is
not straightforward, see discussion in [Kisil05c] [AgostiniCapraraCiccotti07a]
[Kisil09a]. A direct approach using dual numbers will be discussed below, cf.
Rem. 4.5.

To recover the Schrödinger representation we use Rem. 2.2, see [Kisil98a]*Ex.
4.1 for details. The subgroup
$H=\\{(s,0,y)\,\mid\,s\in\mathbb{R}{},y\in\mathbb{R}^{n}{}\\}\subset\mathbb{H}^{n}{}$
defines the homogeneous space $X=G/H$, which coincides with $\mathbb{R}^{n}{}$
as a manifold. The natural projection $\mathbf{p}:G\rightarrow X$ is
$\mathbf{p}(s,x,y)=x$ and its left inverse $\mathbf{s}:X\rightarrow G$ can be
as simple as $\mathbf{s}(x)=(0,x,0)$. For the map $\mathbf{r}:G\rightarrow H$,
$\mathbf{r}(s,x,y)=(s-xy/2,0,y)$ we have the decomposition

$(s,x,y)=\mathbf{s}(p(s,x,y))*\mathbf{r}(s,x,y)=(0,x,0)*(s-\textstyle\frac{1}{2}xy,0,y).$

For a character $\chi_{h}(s,0,y)=e^{\mathrm{i}hs}$ of $H$ the lifting
$\mathcal{L}_{\chi}:L_{2}{}(G/H)\rightarrow L_{2}^{\chi}{}(G)$ is as follows:

$[\mathcal{L}_{\chi}f](s,x,y)=\chi_{h}(\mathbf{r}(s,x,y))\,f(\mathbf{p}(s,x,y))=e^{\mathrm{i}h(s-xy/2)}f(x).$

Thus the representation
${\rho_{\chi}}(g)=\mathcal{P}\circ\Lambda(g)\circ\mathcal{L}$ becomes:

(3.4)
$[{\rho_{\chi}}(s^{\prime},x^{\prime},y^{\prime})f](x)=e^{-2\pi\mathrm{i}\hslash(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}\,f(x-x^{\prime}).$

After the Fourier transform $x\mapsto q$ we get the Schrödinger representation
on the configuration space:

(3.5)
$[{\rho_{\chi}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}\,](q)=e^{-2\pi\mathrm{i}\hslash(s^{\prime}+x^{\prime}y^{\prime}/2)-2\pi\mathrm{i}x^{\prime}q}\,\hat{f}(q+\hslash
y^{\prime}).$

Note that this again turns into a commutative representation (multiplication
by an unimodular function) if $\hslash=0$. To get the full set of commutative
representations in this way we need to use the character
$\chi_{(h,p)}(s,0,y)=e^{2\pi\mathrm{i}(\hslash+py)}$ in the above
consideration.

###  3.2. Commutator and the Heisenberg Equation

The property (2.6) of $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ implies that the
restrictions of two operators ${\rho_{\chi}}(k_{1})$ and
${\rho_{\chi}}(k_{2})$ to this space are equal if

$\int_{\mathbb{R}{}}k_{1}(s,x,y)\,\chi(s)\,ds=\int_{\mathbb{R}{}}k_{2}(s,x,y)\,\chi(s)\,ds.$

In other words, for a character $\chi(s)=e^{2\pi\mathrm{i}\hslash s}$ the
operator ${\rho_{\chi}}(k)$ depends only on

$\hat{k}_{s}(\hslash,x,y)=\int_{\mathbb{R}{}}k(s,x,y)\,e^{-2\pi\mathrm{i}\hslash
s}\,ds,$

which is the partial Fourier transform $s\mapsto\hslash$ of $k(s,x,y)$. The
restriction to $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ of the composition formula
for convolutions is [Kisil02e]*(3.5):

(3.6)
$(k^{\prime}*k)\hat{{}_{s}}=\int_{\mathbb{R}^{2n}{}}e^{{\mathrm{i}h}{}(xy^{\prime}-yx^{\prime})/2}\,\hat{k}^{\prime}_{s}(\hslash,x^{\prime},y^{\prime})\,\hat{k}_{s}(\hslash,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$

Under the Schrödinger representation (3.5) the convolution (3.6) defines a
rule for composition of two pseudo-differential operators (PDO) in the Weyl
calculus [Howe80b] [Folland89]*§ 2.3.

Consequently the representation (2.10) of commutator (2.2) depends only on its
partial Fourier transform [Kisil02e]*(3.6):

$\displaystyle[k^{\prime},k]\hat{{}_{s}}$ $\displaystyle=$ $\displaystyle
2\mathrm{i}\int_{\mathbb{R}^{2n}{}}\\!\\!\sin(\textstyle\frac{h}{2}(xy^{\prime}-yx^{\prime}))\,$
$\displaystyle\qquad\times\hat{k}^{\prime}_{s}(\hslash,x^{\prime},y^{\prime})\,\hat{k}_{s}(\hslash,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$

Under the Fourier transform (3.1) this commutator is exactly the Moyal bracket
[Zachos02a] for of $\hat{k}^{\prime}$ and $\hat{k}$.

For observables in the space $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ the action of
$S$ is reduced to multiplication, e.g. for $\chi(s)=e^{\mathrm{i}hs}$ the
action of $S$ is multiplication by $\mathrm{i}h$. Thus the equation (2.12)
reduced to the space $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ becomes the Heisenberg
type equation [Kisil02e]*(4.4):

(3.8) $\dot{f}=\frac{1}{\mathrm{i}h}[H,f]\hat{{}_{s}},$

based on the above bracket (3.2). The Schrödinger representation (3.5)
transforms this equation to the original Heisenberg equation.

######  Example 3.2.

  1. (i)

Under the Fourier transform $(x,y)\mapsto(q,p)$ the p-dynamic equation (2.14)
of the harmonic oscillator becomes:

$\dot{f}=\left(m\omega^{2}q\frac{\partial}{\partial
p}-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$

The same transform creates its solution out of (2.15).

  2. (ii)

Since $\frac{\partial}{\partial s}$ acts on $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$
as multiplication by $\mathrm{i}\hslash$, the quantum representation of
unharmonic dynamics equation (2.17) is:

(3.9) $\dot{f}=\left(m\omega^{2}q\frac{\partial}{\partial
p}+\frac{\lambda}{6}\left(3q^{2}\frac{\partial}{\partial
p}-\frac{\hslash^{2}}{4}\frac{\partial^{3}}{\partial
p^{3}}\right)-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$

This is exactly the equation for the Wigner function obtained in
[CalzettaVerdaguer06a]*(30).

###  3.3. Quantum Probabilities

For the elliptic character $\chi_{h}(s)=e^{\mathrm{i}hs}$ we can use the
Cauchy–Schwartz inequality to demonstrate that the real number $A$ in the
identity (2.20) is between $-1$ and $1$. Thus we can put $A=\cos\alpha$ for
some angle (phase) $\alpha$ to get the formula for counting quantum
probabilities, cf. [Khrennikov03a]*(2):

(3.10) $l_{12}=l_{1}+l_{2}+2\cos\alpha\,\sqrt{l_{1}l_{2}}$

######  Remark 3.3.

It is interesting to note that the both trigonometric functions are employed
in quantum mechanics: sine is in the heart of the Moyal bracket (3.2) and
cosine is responsible for the addition of probabilities (3.10). In the essence
the commutator and probabilities took respectively the odd and even parts of
the elliptic character $e^{\mathrm{i}hs}$.

######  Example 3.4.

Take a vector $v_{(a,b)}\in L_{2}^{h}{}(\mathbb{H}^{n}{})$ defined by a
Gaussian with mean value $(a,b)$ in the phase space for a harmonic oscillator
of the mass $m$ and the frequency $\omega$:

(3.11) $v_{(a,b)}(q,p)=\exp\left(-\frac{2\pi\omega
m}{\hslash}(q-a)^{2}-\frac{2\pi}{\hslash\omega m}(p-b)^{2}\right).$

A direct calculation shows:

$\displaystyle\left\langle
v_{(a,b)},{\rho_{\hslash}}(s,x,y)v_{(a^{\prime},b^{\prime})}\right\rangle=\frac{4}{\hslash}\exp\left(\pi\mathrm{i}\left(2s\hslash+x(a+a^{\prime})+y(b+b^{\prime})\right)\frac{}{}\right.$
$\displaystyle\left.{}-\frac{\pi}{2\hslash\omega m}((\hslash
x+b-b^{\prime})^{2}+(b-b^{\prime})^{2})-\frac{\pi\omega m}{2\hslash}((\hslash
y+a^{\prime}-a)^{2}+(a^{\prime}-a)^{2})\right)$ $\displaystyle=$
$\displaystyle\frac{4}{\hslash}\exp\left(\pi\mathrm{i}\left(2s\hslash+x(a+a^{\prime})+y(b+b^{\prime})\right)\frac{}{}\right.$
$\displaystyle\left.{}-\frac{\pi}{\hslash\omega
m}((b-b^{\prime}+{\textstyle\frac{\hslash
x}{2}})^{2}+({\textstyle\frac{\hslash x}{2}})^{2})-\frac{\pi\omega
m}{\hslash}((a-a^{\prime}-{\textstyle\frac{\hslash
y}{2}})^{2}+({\textstyle\frac{\hslash y}{2}})^{2})\right)$

Thus the kernel $l_{(a,b)}=\left\langle
v_{(a,b)},{\rho_{\hslash}}(s,x,y)v_{(a,b)}\right\rangle$ (2.18) for a state
$v_{(a,b)}$ is:

(3.12) $\displaystyle l_{(a,b)}$ $\displaystyle=$
$\displaystyle\frac{4}{\hslash}\exp\left(2\pi\mathrm{i}(s\hslash+xa+yb)\frac{}{}-\frac{\pi\hslash}{2\omega
m}x^{2}-\frac{\pi\omega m\hslash}{2\hslash}y^{2}\right)$

An observable registering a particle at a point $q=c$ of the configuration
space is $\delta(q-c)$. On the Heisenberg group this observable is given by
the kernel:

(3.13) $X_{c}(s,x,y)=e^{2\pi\mathrm{i}(s\hslash+xc)}\delta(y).$

The measurement of $X_{c}$ on the state (3.11) (through the kernel (3.12))
predictably is:

$\left\langle X_{c},l_{(a,b)}\right\rangle=\sqrt{\frac{2\omega
m}{\hslash}}\exp\left(-\frac{2\pi\omega m}{\hslash}(c-a)^{2}\right).$

######  Example 3.5.

Now take two states $v_{(0,b)}$ and $v_{(0,-b)}$, where for the simplicity we
assume the mean values of coordinates vanish in the both cases. Then the
corresponding kernel (2.19) has the interference terms:

$\displaystyle l_{i}$ $\displaystyle=$ $\displaystyle\left\langle
v_{(0,b)},{\rho_{\hslash}}(s,x,y)v_{(0,-b)}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{4}{\hslash}\exp\left(2\pi\mathrm{i}s\hslash-\frac{\pi}{2\hslash\omega
m}((\hslash x+2b)^{2}+4b^{2})-\frac{\pi\hslash\omega m}{2}y^{2}\right).$

The measurement of $X_{c}$ (3.13) on this term contains the oscillating part:

$\left\langle X_{c},l_{i}\right\rangle=\sqrt{\frac{2\omega
m}{\hslash}}\exp\left(-\frac{2\pi\omega m}{\hslash}c^{2}-\frac{2\pi}{\omega
m\hslash}b^{2}+\frac{4\pi\mathrm{i}}{\hslash}cb\right)$

Therefore on the kernel $l$ corresponding to the state $v_{(0,b)}+v_{(0,-b)}$
the measurement is

$\displaystyle\left\langle X_{c},l\right\rangle$ $\displaystyle=$
$\displaystyle 2\sqrt{\frac{2\omega m}{\hslash}}\exp\left(-\frac{2\pi\omega
m}{\hslash}c^{2}\right)\left(1+\exp\left(-\frac{2\pi}{\omega
m\hslash}b^{2}\right)\cos\left(\frac{4\pi}{\hslash}cb\right)\right).$

(a) (b)

Figure 1. Quantum probabilities: the blue (dashed) graph shows the addition of
probabilities without interaction, the red (solid) graph present the quantum
interference. Left picture shows the Gaussian state (3.11), the right—the
rational state (3.14)

The presence of the cosine term in the last expression can generate an
interference picture. In practise it does not happen for the minimal
uncertainty state (3.11) which we are using here: it rapidly vanishes outside
of the neighbourhood of zero, where oscillations of the cosine occurs, see
Fig. 1(a).

######  Example 3.6.

To see a traditional interference pattern one can use a state which is far
from the minimal uncertainty. For example, we can consider the state:

(3.14) $u_{(a,b)}(q,p)=\frac{\hslash^{2}}{((q-a)^{2}+\hslash/\omega
m)((p-b)^{2}+\hslash\omega m)}.$

To evaluate the observable $X_{c}$ (3.13) on the state $l(g)=\left\langle
u_{1},{\rho_{h}}(g)u_{2}\right\rangle$ (2.18) we use the following formula:

$\left\langle
X_{c},l\right\rangle=\frac{2}{\hslash}\int_{\mathbb{R}^{n}{}}\hat{u}_{1}(q,2(q-c)/\hslash)\,\overline{\hat{u}_{2}(q,2(q-c)/\hslash)}\,dq,$

where $\hat{u}_{i}(q,x)$ denotes the partial Fourier transform $p\mapsto x$ of
$u_{i}(q,p)$. The formula is obtained by swapping order of integrations. The
numerical evaluation of the state obtained by the addition
$u_{(0,b)}+u_{(0,-b)}$ is plotted on Fig. 1(b), the red curve shows the
canonical interference pattern.

##  4\. Hypercomplex Repersentations of the Heisenberg Group

The group of symmetries of classical mechanics—the group preserving the
symplectic form (2.2)—generates automorphisms of the Heisenberg group in a
natural way [Folland89]*§ 1.2. Those common symmetries of quantum and
classical mechanics are behind many important connections, e.g. between
classical “symplectic camel” and the Heisenberg uncertainty relations
[deGossonLuef09a].

The symplectic group of $\mathbb{R}^{2}{}$ is isomorphic to the celebrated
group $SL_{2}{}(\mathbb{R}{})$ [Lang85]. Both groups $\mathbb{H}^{n}{}$ and
$SL_{2}{}(\mathbb{R}{})$ contributes to the symmetries of the paraxial wave
equation [ATorre10a]. There are many other physical links between the
Heisenberg group and $SL_{2}{}(\mathbb{R}{})$, e.g. metaplectic representation
[Folland89]*Ch. 4.

It was demonstrated in [Kisil07a] that dual and double numbers appears very
naturally within the induced representations of the group
$SL_{2}{}(\mathbb{R}{})$. Special relativity [Ulrych05a] and global space-time
model [HerranzSantander02b, Kisil06b] also link the representation theory to
hypercomplex numbers. Physical significance of hypercomplex numbers and
representation theory of Clifford algebras was recently highlighted as well
[BocCatoniCannataNichZamp07] [Ulrych08a] [Plaksa09a] [Ulrych10a]. There is an
explicit similarity between the commutators in the Heisenberg-Weyl Lie algebra
and anticommutators defining Clifford algebra [Kisil93c] [Kisil01d], which can
be unified as a superspace [BieEelbodeSommen09a] [Berezin86]. Thus it would be
an omission to restrict linear representations of $\mathbb{H}^{n}{}$ to
complex numbers only.

###  4.1. Hyperbolic Representations and Addition of Probabilities

Now we turn to double numbers also known as hyperbolic, split-complex, etc.
numbers [Yaglom79]*App. C [Ulrych05a] [KhrennikovSegre07a]. They form a two
dimensional algebra $\mathbb{O}{}$ spanned by $1$ and $\mathrm{j}$ with the
property $\mathrm{j}^{2}=1$. There are zero divisors:

$\mathrm{j}_{\pm}=\textstyle\frac{1}{\sqrt{2}}(1\pm j),\qquad\text{ such that
}\quad\mathrm{j}_{+}\mathrm{j}_{-}=0\quad\text{ and
}\quad\mathrm{j}_{\pm}^{2}=\mathrm{j}_{\pm}.$

Thus double numbers algebraically isomorphic to two copies of $\mathbb{R}{}$
spanned by $\mathrm{j}_{\pm}$. Being algebraically dull double numbers are
nevertheless interesting as a homogeneous space [Kisil05a, Kisil09c] and they
are relevant in physics [Khrennikov05a, Ulrych05a, Ulrych08a]. The combination
of p-mechanical approach with hyperbolic quantum mechanics was already
discussed in [BrodlieKisil03a]*§ 6.

For the hyperbolic character $\chi_{\mathrm{j}h}(s)=e^{\mathrm{j}hs}=\cosh
hs+\mathrm{j}\sinh hs$ of $\mathbb{R}{}$ one can define the hyperbolic
Fourier-type transform:

$\hat{k}(q)=\int_{\mathbb{R}{}}k(x)\,e^{-\mathrm{j}qx}dx.$

It can be understood in the sense of distributions on the space dual to the
set of analytic functions [Khrennikov08a]*§ 3. Hyperbolic Fourier transform
intertwines the derivative $\frac{d}{dx}$ and multiplication by $\mathrm{j}q$
[Khrennikov08a]*Prop. 1.

######  Example 4.1.

For the Gaussian the hyperbolic Fourier transform is the ordinary function
(note the sign difference!):

$\int_{\mathbb{R}{}}e^{-x^{2}/2}e^{-\mathrm{j}qx}dx=\sqrt{2\pi}\,e^{q^{2}/2}.$

However the opposite identity:

$\int_{\mathbb{R}{}}e^{x^{2}/2}e^{-\mathrm{j}qx}dx=\sqrt{2\pi}\,e^{-q^{2}/2}$

is true only in a suitable distributional sense. To this end we may note that
$e^{x^{2}/2}$ and $e^{-q^{2}/2}$ are null solutions to the differential
operators $\frac{d}{dx}-x$ and $\frac{d}{dq}+q$ respectively, which are
intertwined (up to the factor $\mathrm{j}$) by the hyperbolic Fourier
transform. The above differential operators $\frac{d}{dx}-x$ and
$\frac{d}{dq}+q$ are images of the _ladder operators_ in the Lie algebra of
the Heisenberg group. They are intertwining by the Fourier transform, since
this is an automorphism of the Heisenberg group [Howe80a]. A careful study of
ladder operators reveals connections with hypercomplex numbers [Kisil11a,
Kisil11c].

An elegant theory of hyperbolic Fourier transform may be achieved by a
suitable adaptation of [Howe80a], which uses representation theory of the
Heisenberg group.

####  4.1.1. Hyperbolic Representations of the Heisenberg Group

Consider the space $F_{h}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$ of
$\mathbb{O}{}$-valued functions on $\mathbb{H}^{n}{}$ with the property:

(4.1) $f(s+s^{\prime},h,y)=e^{\mathrm{j}hs^{\prime}}f(s,x,y),\qquad\text{ for
all }(s,x,y)\in\mathbb{H}^{n}{},\ s^{\prime}\in\mathbb{R}{},$

and the square integrability condition (2.7). Then the hyperbolic
representation is obtained by the restriction of the left shifts to
$F_{h}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$. To obtain an equivalent
representation on the phase space we take $\mathbb{O}{}$-valued functional of
the Lie algebra $\mathfrak{h}_{n}$:

(4.2)
$\chi^{j}_{(h,q,p)}(s,x,y)=e^{\mathrm{j}(hs+qx+py)}=\cosh(hs+qx+py)+\mathrm{j}\sinh(hs+qx+py).$

The hyperbolic Segal—Bargmann type representation is intertwined with the left
group action by means of the Fourier transform (3.1) with the hyperbolic
functional (4.2). Explicitly this representation is:

(4.3) ${\rho_{\hslash}}(s,x,y):f(q,p)\mapsto\textstyle
e^{-\mathrm{j}(hs+qx+py)}f\left(q-\frac{h}{2}y,p+\frac{h}{2}x\right).$

For a hyperbolic Schrödinger type representation we again use the scheme
described in Rem. 2.2. Similarly to the elliptic case one obtains the formula,
resembling (3.4):

(4.4)
$[{\rho^{\mathrm{j}}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})f](x)=e^{-\mathrm{j}h(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}f(x-x^{\prime}).$

Application of the hyperbolic Fourier transform produces a Schrödinger type
representation on the configuration space, cf. (3.5):

(4.5)
$[{\rho^{\mathrm{j}}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}\,](q)=e^{-\mathrm{j}h(s^{\prime}+x^{\prime}y^{\prime}/2)-\mathrm{j}x^{\prime}q}\,\hat{f}(q+hy^{\prime}).$

The extension of this representation to kernels according to (2.10) generates
hyperbolic pseudodifferential operators introduced in [Khrennikov08a]*(3.4).

####  4.1.2. Hyperbolic Dynamics

Similarly to the elliptic (quantum) case we consider a convolution of two
kernels on $\mathbb{H}^{n}{}$ restricted to
$F_{h}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$. The composition law becomes, cf.
(3.6):

(4.6)
$(k^{\prime}*k)\hat{{}_{s}}=\int_{\mathbb{R}^{2n}{}}e^{{\mathrm{j}h}{}(xy^{\prime}-yx^{\prime})}\,\hat{k}^{\prime}_{s}(h,x^{\prime},y^{\prime})\,\hat{k}_{s}(h,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$

This is close to the calculus of hyperbolic PDO obtained in
[Khrennikov08a]*Thm. 2. Respectively for the commutator of two convolutions we
get, cf. (3.2):

(4.7)
$[k^{\prime},k]\hat{{}_{s}}=\int_{\mathbb{R}^{2n}{}}\\!\\!\sinh(h(xy^{\prime}-yx^{\prime}))\,\hat{k}^{\prime}_{s}(h,x^{\prime},y^{\prime})\,\hat{k}_{s}(h,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$

This the hyperbolic version of the Moyal bracket, cf. [Khrennikov08a]*p. 849,
which generates the corresponding image of the dynamic equation (2.12).

######  Example 4.2.

  1. (i)

For a quadratic Hamiltonian, e.g. harmonic oscillator from Example 2.3, the
hyperbolic equation and respective dynamics is identical to quantum considered
before.

  2. (ii)

Since $\frac{\partial}{\partial s}$ acts on
$F_{2}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$ as multiplication by $\mathrm{j}h$
and $\mathrm{j}^{2}=1$, the hyperbolic image of the unharmonic equation (2.17)
becomes:

$\dot{f}=\left(m\omega^{2}q\frac{\partial}{\partial
p}+\frac{\lambda}{6}\left(3q^{2}\frac{\partial}{\partial
p}+\frac{\hslash^{2}}{4}\frac{\partial^{3}}{\partial
p^{3}}\right)-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$

The difference with quantum mechanical equation (3.9) is in the sign of the
cubic derivative.

####  4.1.3. Hyperbolic Probabilities

(a) (b)

Figure 2. Hyperbolic probabilities: the blue (dashed) graph shows the addition
of probabilities without interaction, the red (solid) graph present the
quantum interference. Left picture shows the Gaussian state (3.11), with the
same distribution as in quantum mechanics, cf. Fig. 1(a). The right picture
shows the rational state (3.14), note the absence of interference oscillations
in comparison with the quantum state on Fig. 1(b).

To calculate probability distribution generated by a hyperbolic state we are
using the general procedure from Section 2.3. The main differences with the
quantum case are as follows:

  1. (i)

The real number $A$ in the expression (2.20) for the addition of probabilities
is bigger than $1$ in absolute value by. Thus it can be associated with the
hyperbolic cosine $\cosh\alpha$, cf. Rem. 3.3, for certain phase
$\alpha\in\mathbb{R}{}$ [Khrennikov08a].

  2. (ii)

The nature of hyperbolic interference on two slits is affected by the fact
that $e^{\mathrm{j}hs}$ is not periodic and the hyperbolic exponent
$e^{\mathrm{j}t}$ and cosine $\cosh t$ do not oscillate. It is worth to notice
that for Gaussian states the hyperbolic interference is exactly the same as
quantum one, cf. Figs. 1(a) and 2(a). This is similar to coincidence of
quantum and hyperbolic dynamics of harmonic oscillator.

The contrast between two types of interference is prominent for the rational
state (3.14), which is far from the minimal uncertainty, see the different
patterns on Figs. 1(b) and 2(b).

###  4.2. Parabolic (Classical) representations on the phase space

After the previous two cases it is natural to link classical mechanics with
dual numbers generated by the parabolic unit $\varepsilon^{2}=0$. Connection
of the parabolic unit $\varepsilon$ with the Galilean group of symmetries of
classical mechanics is around for a while [Yaglom79]*App. C.

However the nilpotency of the parabolic unit $\varepsilon$ make it difficult
if we will work with dual number valued functions only. To overcome this issue
we consider a commutative real algebra $\mathfrak{C}$ spanned by $1$,
$\mathrm{i}$, $\varepsilon$ and $\mathrm{i}\varepsilon$ with identities
$\mathrm{i}^{2}=-1$ and $\varepsilon^{2}=0$. A seminorm on $\mathfrak{C}$ is
defined as follows:

$\left|a+b\mathrm{i}+c\varepsilon+d\mathrm{i}\varepsilon\right|^{2}=a^{2}+b^{2}.$

####  4.2.1. Classical Non-Commutative Representations

We wish to build a representation of the Heisenberg group which will be a
classical analog of the Segal–Bargmann representation (3.2). To this end we
introduce the space $F_{h}^{\varepsilon}{}(\mathbb{H}^{n}{})$ of
$\mathfrak{C}$-valued functions on $\mathbb{H}^{n}{}$ with the property:

(4.8) $f(s+s^{\prime},h,y)=e^{\varepsilon hs^{\prime}}f(s,x,y),\qquad\text{
for all }(s,x,y)\in\mathbb{H}^{n}{},\ s^{\prime}\in\mathbb{R}{},$

and the square integrability condition (2.7). It is invariant under the left
shifts and we restrict the left group action to
$F_{h}^{\varepsilon}{}(\mathbb{H}^{n}{})$.

There is an unimodular $\mathfrak{C}$-valued function on the Heisenberg group
parametrised by a point $(h,q,p)\in\mathbb{R}^{2n+1}{}$:

$E_{(h,q,p)}(s,x,y)=e^{2\pi(\varepsilon
s\hslash+\mathrm{i}xq+\mathrm{i}yp)}=e^{2\pi\mathrm{i}(xq+yp)}(1+\varepsilon
sh).$

This function, if used instead of the ordinary exponent, produces a
modification $\mathcal{F}_{c}$ of the Fourier transform (3.1). The transform
intertwines the left regular representation with the following action on
$\mathfrak{C}$-valued functions on the phase space:

(4.9) ${\rho^{\varepsilon}_{h}}(s,x,y):f(q,p)\mapsto
e^{-2\pi\mathrm{i}(xq+yp)}(f(q,p)+\varepsilon
h(sf(q,p)+\frac{y}{2\pi\mathrm{i}}f^{\prime}_{q}(q,p)-\frac{x}{2\pi\mathrm{i}}f^{\prime}_{p}(q,p))).$

######  Remark 4.3.

Comparing the traditional infinite-dimensional (3.2) and one-dimensional (3.3)
representations of $\mathbb{H}^{n}{}$ we can note that the properties of the
representation (4.9) are a non-trivial mixture of the former:

  1. (i)

The action (4.9) is non-commutative, similarly to the quantum representation
(3.2) and unlike the classical one (3.3). This non-commutativity will produce
the Hamilton equations below in a way very similar to Heisenberg equation, see
Rem. 4.5.

  2. (ii)

The representation (4.9) does not change the support of a function $f$ on the
phase space, similarly to the classical representation (3.3) and unlike the
quantum one (3.2). Such a localised action will be responsible later for an
absence of an interference in classical probabilities.

  3. (iii)

The parabolic representation (4.9) can not be derived from either the elliptic
(3.2) or hyperbolic (4.3) by the plain substitution $h=0$.

We may also write a classical Schrödinger type representation. According to
Rem. 2.2 we get a representation formally very similar to the elliptic (3.4)
and hyperbolic versions (4.4):

$\displaystyle[{\rho^{\varepsilon}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})f](x)$
$\displaystyle=$ $\displaystyle e^{-\varepsilon
h(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}f(x-x^{\prime})$
$\displaystyle=$ $\displaystyle(1-\varepsilon
h(s^{\prime}+xy^{\prime}-\textstyle\frac{1}{2}x^{\prime}y^{\prime}))f(x-x^{\prime}).$

However due to nilpotency of $\varepsilon$ the (complex) Fourier transform
$x\mapsto q$ produces a different formula for parabolic Schrödinger type
representation in the configuration space, cf. (3.5) and (4.5):

(4.11)
$[{\rho^{\varepsilon}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}](q)=e^{2\pi\mathrm{i}x^{\prime}q}\left(\left(1-\varepsilon
h(s^{\prime}-{\textstyle\frac{1}{2}}x^{\prime}y^{\prime})\right)\hat{f}(q)+\frac{\varepsilon
hy^{\prime}}{2\pi\mathrm{i}}\hat{f}^{\prime}(q)\right).$

This representation shares all properties mentioned in Rem. 4.3 as well.

####  4.2.2. Hamilton Equation

The identity $e^{\varepsilon t}-e^{-\varepsilon t}=2\varepsilon t$ can be
interpreted as a parabolic version of the sine function, while the parabolic
cosine is identically equal to one [HerranzOrtegaSantander99a, Kisil07a]. From
this we obtain the parabolic version of the commutator (3.2):

$\displaystyle[k^{\prime},k]\hat{{}_{s}}(\varepsilon h,x,y)$ $\displaystyle=$
$\displaystyle\varepsilon h\int_{\mathbb{R}^{2n}{}}(xy^{\prime}-yx^{\prime})$
$\displaystyle{}\times\,\hat{k}^{\prime}_{s}(\varepsilon
h,x^{\prime},y^{\prime})\,\hat{k}_{s}(\varepsilon
h,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime},$

for the partial parabolic Fourier-type transform $\hat{k}_{s}$ of the kernels.
Thus the parabolic representation of the dynamical equation (2.12) becomes:

(4.12) $\varepsilon h\frac{d\hat{f}_{s}}{dt}(\varepsilon h,x,y;t)=\varepsilon
h\int_{\mathbb{R}^{2n}{}}(xy^{\prime}-yx^{\prime})\,\hat{H}_{s}(\varepsilon
h,x^{\prime},y^{\prime})\,\hat{f}_{s}(\varepsilon
h,x-x^{\prime},y-y^{\prime};t)\,dx^{\prime}dy^{\prime},$

Although there is no possibility to divide by $\varepsilon$ (since it is a
zero divisor) we can obviously eliminate $\varepsilon h$ from the both sides
if the rest of the expressions are real. Moreover this can be done “in
advance” through a kind of the antiderivative operator considered in
[Kisil02e]*(4.1). This will prevent “imaginary parts” of the remaining
expressions (which contain the factor $\varepsilon$) from vanishing.

######  Remark 4.4.

It is noteworthy that the Planck constants completely disappeared from the
dynamical equation. Thus the only prediction about it following from our
construction is $h\neq 0$, which was confirmed by experiments, of course.

Using the duality between the Lie algebra of $\mathbb{H}^{n}{}$ and the phase
space we can find an adjoint equation for observables on the phase space. To
this end we apply the usual Fourier transform $(x,y)\mapsto(q,p)$. It turn to
be the Hamilton equation [Kisil02e]*(4.7). However the transition to phase
space is more a custom rather than a necessity and in many cases we can
efficiently work on the Heisenberg group itself.

######  Remark 4.5.

It is noteworthy, that the non-commutative representation (4.9) allows to
obtain the Hamilton equation directly from the commutator
$[{\rho^{\varepsilon}_{h}}(k_{1}),{\rho^{\varepsilon}_{h}}(k_{2})]$. Indeed
its straightforward evaluation will produce exactly the above expression. On
the contrast such a commutator for the commutative representation (3.3) is
zero and to obtain the Hamilton equation we have to work with an additional
tools, e.g. an anti-derivative [Kisil02e]*(4.1).

######  Example 4.6.

  1. (i)

For the harmonic oscillator in Example 2.3 the equation (4.12) again reduces
to the form (2.14) with the solution given by (2.15). The adjoint equation of
the harmonic oscillator on the phase space is not different from the quantum
written in Example 3.2(i). This is true for any Hamiltonian of at most
quadratic order.

  2. (ii)

For non-quadratic Hamiltonians classical and quantum dynamics are different,
of course. For example, the cubic term of $\partial_{s}$ in the equation
(2.17) will generate the factor $\varepsilon^{3}=0$ and thus vanish. Thus the
equation (4.12) of the unharmonic oscillator on $\mathbb{H}^{n}{}$ becomes:

$\dot{f}=\left(m\omega^{2}y\frac{\partial}{\partial x}+\frac{\lambda
y}{2}\frac{\partial^{2}}{\partial x^{2}}-\frac{1}{m}x\frac{\partial}{\partial
y}\right)f.$

The adjoint equation on the phase space is:

$\dot{f}=\left(\left(m\omega^{2}q+\frac{\lambda}{2}q^{2}\right)\frac{\partial}{\partial
p}-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$

The last equation is the classical Hamilton equation generated by the cubic
potential (2.16). Qualitative analysis of its dynamics can be found in many
textbooks [Arnold91]*§ 4.C, Pic. 12 [PercivalRichards82]*§ 4.4.

######  Remark 4.7.

We have obtained the Poisson bracket from the commutator of convolutions on
$\mathbb{H}^{n}{}$ without any quasiclassical limit $h\rightarrow 0$. This has
a common source with the deduction of main calculus theorems in
[CatoniCannataNichelatti04] based on dual numbers. As explained in
[Kisil05a]*Rem. 6.9 this is due to the similarity between the parabolic unit
$\varepsilon$ and the infinitesimal number used in non-standard analysis
[Devis77]. In other words, we never need to take care about terms of order
$O(h^{2})$ because they will be wiped out by $\varepsilon^{2}=0$.

An alternative derivation of classical dynamics from the Heisenberg group is
given in the recent paper [Low09a].

####  4.2.3. Classical probabilities

It is worth to notice that dual numbers are not only helpful in reproducing
classical Hamiltonian dynamics, they also provide the classic rule for
addition of probabilities. We use the same formula (2.18) to calculate kernels
of the states. The important difference now that the representation (4.9) does
not change the support of functions. Thus if we calculate the correlation term
$\left\langle v_{1},{\rho}(g)v_{2}\right\rangle$ in (2.19), then it will be
zero for every two vectors $v_{1}$ and $v_{2}$ which have disjoint supports in
the phase space. Thus no interference similar to quantum or hyperbolic cases
(Subsection 3.3) is possible.

##  5\. Discussion

In this paper we derive mathematical models for various physical setup from
hypercomplex representations of the Heisenberg group. There are roots for such
hypercomplex characters in the structure of ladder operators associated to
three non-isomorphic quadratic Hamiltonians [Kisil11a, Kisil11c]. Such
hypercomplex representations may be also useful for many other groups as well,
see the example of the $SL_{2}{}(\mathbb{R}{})$ group in [Kisil09c]. Moreover
non-trivial parabolic characters described in [Kisil07a, Kisil09c] are
awaiting a further exploration.

There is a connection of our work with the technique of contractions and
analytic continuations of groups [Gromov90b, Gromov90a], these papers also
highlight the role of hypercomplex numbers of three types. However in our
research we do not modify the group (the Heisenberg group more specifically)
itself, we rather consider its representations in different functional spaces
created by three types of hypercomplex characters. All three cases have a lot
of algebraic similarity and can be written in a unified manner with the help
of parameter, which takes three values, say $u=\mathrm{i}$, $\varepsilon$,
$\mathrm{j}$, with $\mathrm{i}^{2}=-1$, $\varepsilon^{2}=0$,
$\mathrm{j}^{2}=1$. For example, representations (3.4), (4.4) and (4.2.1) can
be unified in:

(5.1)
$[{\rho^{u}_{h}}(s^{\prime},x^{\prime},y^{\prime})f](x)=e^{-uh(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}f(x-x^{\prime}).$

It is noteworthy that this algebraic similarity exists along with the
significant topological and analytic differences between elliptic, parabolic
and hyperbolic cases. An illustration is the distinction of the elliptic (3.5)
and parabolic (4.11) representations in the configuration space, despite of
the fact that both representations are derived from the unified form (5.1).

The parabolic representations (4.2.1) and (4.11) of the Heisenberg group act
in the first order jet spaces. Such spaces have a well established connections
with Lagrangian and Hamiltonian formulations of quantum field theory
[GiachettaMangiarottiSardanashvily97a, Kanatchikov01b, Kisil04a], study of
aggregate quantum-classical systems [Kisil05c, Kisil09a] and spectral theory
of operators [Kisil02a]. Nevertheless the localised non-commutative
representation of $\mathbb{H}^{n}{}$ built in this paper seems to be new and
deserve detailed investigation.

We already seen that it may be useful to consider several hypercomplex units
in the same time. In the case of classical mechanics we combined $\mathrm{i}$
and $\varepsilon$. The algebra generated by $\mathrm{i}$ and $\mathrm{j}$ is
known as (commutative) Segre quaternions. Such commutative algebras with
hypercomplex units and their physical applications attracted attention of many
researchers recently [BocCatoniCannataNichZamp07] [Plaksa09a] [Ulrych05a]
[Ulrych08a].

We may even need to study an algebra which contains all three hypercomplex
units simultaneously. The most straightforward way is to take eight
dimensional commutative algebra with the basis $1$, $\mathrm{i}$,
$\varepsilon$, $\mathrm{j}$, $\mathrm{i}\varepsilon$, $\mathrm{i}\mathrm{j}$,
$\varepsilon\mathrm{j}$, $\mathrm{i}\varepsilon\mathrm{j}$. A reduction of
dimensionality from $8$ to $6$ can be achieved if we replace products
$\varepsilon\mathrm{j}$ and $\mathrm{i}\varepsilon\mathrm{j}$ through the
further identities $\varepsilon\mathrm{j}=\varepsilon$ and
$\mathrm{i}\varepsilon\mathrm{j}=\mathrm{i}\varepsilon$. This do not affect
associativity of the product.

## Acknowledgements

I am grateful to A.Yu. Khrennikov and S. Ulrych for useful discussion on
relation between double numbers and physics. S. Plaksa advised me on various
aspect of commutative hypercomplex algebras. U. Güenther draw my attention to
the connection between $\mathcal{PT}$-symmetric Hamiltonians and Krein spaces.
Prof. N.A. Gromov made several useful suggestions of methodological nature.
Constructive comments of anonymous referees provided further ground for
paper’s improvement.

## References


# Challenges in Understanding the Relationship between Teamwork Quality and
Project Success in Large-Scale Agile Projects

Torgeir Dingsøyr  Norwegian University of Science and
TechnologyTrondheimNorway<EMAIL_ADDRESS>,  Phillip Schneider
Technical University of MunichMunichGermany<EMAIL_ADDRESS>,  Gunnar
Rye Bergersen  University of OsloOsloNorway<EMAIL_ADDRESS>and  Yngve
Lindsjørn  University of OsloOsloNorway<EMAIL_ADDRESS>
(2024)

###### Abstract.

A number of methods for large-scale agile development have recently been
suggested. Much of the advice in agile methods focuses on teamwork. Prior
research has established that teamwork quality influences project success both
for traditional software development teams and agile teams. Further, prior
studies have also suggested that teamwork quality may play out differently in
large projects compared to small. We investigated the relationship between
teamwork quality and project success with a survey of 196 project participants
across 34 teams in four projects, replicating a previous study on single
teams.

The new data do not fit the previously established theoretical model, which
raises several concerns. The observed effect of teamwork quality on project
success operates differently across projects.

We discuss possible reasons, which include disagreements on what characterises
success in large-scale agile development, ”concept drift” of teamwork quality
factors, the possibility that interteam factors might have more influence on
project success than intrateam factors, and finally, that our study design
does not capture all relevant levels and functions.

We conclude with a call for more studies on the quality and frequency of
interaction between teams in addition to internal team factors to further
advance theory and practice within large-scale agile software development.

††conference: 17th International Conference on Cooperative and Human Aspects
in Software Engineering; April 2024; Lisbon, Portugal††journalyear:
2024††copyright: rightsretained††conference: 2024 IEEE/ACM 17th International
Conference on Cooperative and Human Aspects of Software Engineering ; April
14–15, 2024; Lisbon, Portugal††booktitle: 2024 IEEE/ACM 17th International
Conference on Cooperative and Human Aspects of Software Engineering (CHASE
’24), April 14–15, 2024, Lisbon, Portugal††doi: 10.1145/3641822.3641868††isbn:
979-8-4007-0533-5/24/04††submissionid: 13965.6

##  1\. Introduction

Although most advice from fields such as project management and software
engineering is to avoid large software development projects, such projects
often are needed and indeed critical for organisations (dingsoyr2019, ).
Large-scale software development projects are increasingly using agile
methods, and we are starting to see a number of empirical studies focusing on
challenges and success factors for such projects (edison2021, ; uludag2022, ).

Development method is one of many factors which have an influence on project
success (mcleod2011, ). A key characteristic of agile development methods is
that the methods focus on teamwork (baham2021, ). Previous studies have shown
that teamwork quality has a significant impact on software project success,
both for traditional development (Hoegl2001, ) and for agile development teams
(Lindsjorn2016, ), but that teamwork quality affects project success
differently in small and large projects (Lindsjorn2018, ).

We were interested in exploring teamwork in the context of larger projects, in
large-scale agile development, as many organisations put much emphasis on
selecting a method (conboy2019, ), and we know little of how such methods
influence teamwork quality and subsequently project success. Our research
question was, does the effect of teamwork quality on project success differ
between large-scale agile development methods? In the following, we provide
background and then describe the results from this exploratory analysis. We
had access to four projects with data from a total of 34 teams. The main
result was, however, that our data do not fit the previously established
theoretical model. We discuss concerns with the measurement of teamwork
quality and project success. Finally, we conclude this short paper with
recommendations for future studies to explore teamwork in the context of
large-scale agile development.

##  2\. Teamwork quality and multiteam projects

To provide a background, we describe prior work on teamwork quality and
project success, teamwork quality in multiteam projects, and finally, large-
scale agile development methods and teamwork.

###  2.1. Teamwork Quality and Project Success

There is a large body of studies on teamwork in general and a growing number
of studies of teamwork in software development and in the particular context
of agile software development (strode2022, ). Procedures for teamwork are
described in software development methods, for example, on daily meetings in
Scrum and programming in pairs in extreme programming. There are also many
other factors which will affect project success for software teams
(mcleod2011, ). Teamwork is important, however, as it can usually be
influenced at project initiation, for example, in staffing teams with a
facilitator and requiring teams to conduct regular demonstrations and
retrospectives. Such interventions are usually easier than, for example,
changing the staffing in a project.

One of the general studies of teamwork quality with the most impact has been
the survey study on the relationship between teamwork quality and project
success in 145 traditional software teams by Hoegl and Gemuenden (Hoegl2001,
). This study established six teamwork quality (TWQ) constructs, which were
communication, coordination, balance of member contributions, mutual support,
effort, and cohesion. Project success was measured as personal success and
team performance. A modified theoretical model is shown in Figure 1 where
teamwork quality affects team member´s success and team performance.

A replication study by Lindsjørn et al. (Lindsjorn2016, ) with 71 teams found
that the relationship also held in agile software development when team
members and team leaders rated team performance. A negligible effect was found
for product owners. Comparing the effect of teamwork quality on project
success, it was found to be only marginally greater for agile teams than
traditional ones in the original survey, despite the focus on teamwork in
agile software development.

Figure 1. Theoretical model, modified from (Hoegl2001, ) and (Lindsjorn2016,
), used as conceptual model in analysis

A recent mixed method study with quantitative data from two thousand scrum
teams by Verwijs and Russo (verwijs2023, ), finds that scrum team
effectiveness depends on five high-level factors: responsiveness, stakeholder
concern, continuous improvement, team autonomy, and management support.

Table 1. Characteristics of data collection from four projects (CD=Continuous development) Characteristic | LeSS1 | LeSS2 | SAFe1 | SAFe2
---|---|---|---|---
Industry | public sector | defense | finance | utilities
Organisation size | small | large | small | large
Team background | national | national | national | international
Program increment | 11 of 12 | 3 of 4 | 3 in CD | 13 in CD
Agile method | Scrum/XP | Scrum/XP | SAFe 4.6 essential | SAFe 4.6 portfolio
Work style | co-located | partly remote | partly remote | fully remote
Country | Norway | Norway | Germany | Germany
No. of teams | 11 | 10 | 8 | 5
No. of respondents | 78 | 42 | 55 | 21

###  2.2. Teamwork Quality in Multiteam Projects

Projects with multiple teams have been studied in software development and
other domains. A study from the automotive industry found that collaborative
processes such as interteam coordination, project commitment, and teamwork
quality have ”predictive properties in regard to later team performance and
can serve as early warning indicators” (Hoegl2004, ). The study also
identified a positive relationship between interteam coordination, project
commitment, and teamwork quality.

In the context of large-scale agile development, we find several studies
investigating teamwork quality, for example, in modifying the teamwork quality
instrument to the context of agile software development (poth2020, ; poth2021,
).

A multiteam project can be seen as a portfolio of related projects. A study on
how portfolio management practices, such as focusing on a business case
existence, strategic clarity, and operational control, was found in one study
to constrain the relationship between teamwork quality and project success
(bechtel2021, ).

Other teamwork-related factors such as psychological safety, team cohesion,
knowledge sharing, team reflexivity, and team structure are proposed as
influencing team performance and team learning (ahmad2022, ).

###  2.3. Large-scale Agile Development Methods and Teamwork

Large-scale agile development methods provide advice on how to organise
product development with many development teams (dingsoyr2019, ; edison2021,
). Workshops with researchers have put emphasis on topics such as interteam
coordination, interteam communication, and knowledge sharing (bass2019, ).
Empirical studies such as (dingsoyr2018, ; dingsoyr2023, ) and overview
studies (edison2021, ) show a range of extra arenas, work practices, and roles
on top of ones that typically exist in agile development teams. This includes
arenas for interteam coordination and communication such as scrum of scrums or
”product owner coordination meetings”, practices such as ”big room planning”,
and roles such as ”release train engineer” and ”area product owner”. Batra
(batra2020, ) discusses quantitative studies of large-scale development and
states that ”Given that coordination and teamwork are critical success
factors, do we need to create new constructs such as synchronising ability?”

##  3\. Research Method

We conducted an exploratory quantitative multi-case study, collecting data
with a survey as a differentiated replication (Lindsay1993, ) in order to
investigate if the findings from previous studies generalise to teamwork in
large software projects. While Hoegl and Gemuenden (Hoegl2001, ) studied
traditional software teams, and Lindsjorn et al. (Lindsjorn2016, ) agile
teams, we collected data on agile teams in four large-scale agile software
projects. Two Norwegian projects using a method close to Large-Scale Scrum
(LeSS) and two German Scaled Agile Framework (SAFe) projects. LeSS is a
”minimal extension” of Scrum for large-scale product development, while SAFe
is a comprehensive method (dingsoyr2019, ). Our research approach, from the
data collection to the statistical analysis, is detailed in the following
subsections and follows the approach in (Lindsjorn2016, ):

###  3.1. Data Collection and Study Sample

To measure teamwork quality variables in the Norwegian projects, we used the
questionnaire reported in Lindsjørn et al. (Lindsjorn2016, ). A German
translation of the English questionnaire was used for the two German projects.
Depending on the team distribution in each project, the survey was either
conducted at the workplace or as an online survey. The two collected survey
datasets from the Norwegian projects were also used in two previous
publications (Lindsjorn2016, ; Lindsjorn2018, ). The German datasets were
taken from two master theses at the Technical University of Munich (doepp2019,
; styrsky2021, ). We merged the datasets, filtering for a team-based inclusion
criterion. From all questionnaire participants, we only included teams that
came from a large-scale agile project and for which at least one developer, as
well as one product owner, responded.

As can be seen from Table 1, our study sample consists of 196 respondents from
34 teams working in four projects. Because of the team-based inclusion
criterion, there are more LeSS teams (21) than SAFe teams (13) in the merged
dataset. The same holds for the number of respondents. The number of teams per
project ranges from 5 to 11, with a mean of 5.8 respondents per team. In
general, the team sizes between the projects were comparable.

In terms of qualitative differences, the four projects differ in various
characteristics. For instance, all projects came from different industry
sectors: finance, utilities, defense, and the public sector. The projects were
executed by two small and two large organisations. The Norwegian projects have
finished, while the German are continuous development projects. Most of the
projects had teams that were situated in the same country, whereas one project
had an international team. Depending on the specific project, the teamwork
style was co-located, partly remote, or fully remote. The only similarity
between the projects of the same countries was the type of employed agile
development method. While the Norwegian teams used a combination of Scrum and
Extreme Programming close to LeSS, the German teams applied SAFe.

The data and the detailed analyses are available in a replication
package.111https://doi.org/10.5281/zenodo.10571869

###  3.2. Statistical Analysis

We use confirmatory factor and correlation analyses to explore the
interactions between the variables under investigation, as previously detailed
in (Lindsjorn2016, ). Specifically, we calculate the arithmetic mean of the
responses to each question at the team level (our unit of analysis). These
means are then aggregated into the variables of interest. There were
negligible missing data overall (1.5%), but nearly all missing data were from
the product owners (5.1%). For the analysis, we use the lavaan package
(rosseel2012lavaan, ) in R for confirmatory Structural Equation Modeling.

To test the model fit for the specified measurement and structural model, we
report the Root Mean Square Error of Approximation (RMSEA) and its 95%
confidence interval. A close model fit is indicated by RMSEA values below 0.05
and values above 0.10 indicate an unacceptable model fit. As all indices of
model fit have limitations, we also calculated the Comparative Fit Index, the
Tucker-Lewis Index, and the Standardised Root Mean Square Residual; all these
indices support the inferences we make throughout the sections, but we do not
report these. We do, however, report the chi-square statistic ${\chi}^{2}$,
which roughly should be less than two times the degrees of freedom.

##  4\. Results

Table 2. Descriptives and correlations for the investigated variables

| $\overline{X}$ | SD | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Communication‡ (1) | $3.99$ | $0.40$ | $1$ | $0.70$ | $0.69$ | $0.69$ | $0.69$ | $0.25$ | $0.69$ | $0.64$ | $0.31$ | $0.31$ | $0.82$ | $0.26$
Coordination (2) | $3.87$ | $0.48$ | $0.65$ | $1$ | $0.70$ | $0.83$ | $0.69$ | $0.65$ | $0.65$ | $0.64$ | $0.23$ | $0.24$ | $0.77$ | $0.58$
Bal. member contr. (3) | $4.02$ | $0.44$ | $0.70$ | $0.75$ | $1$ | $0.73$ | $0.59$ | $0.65$ | $0.65$ | $0.77$ | $0.64$ | $0.77$ | $0.77$ | $0.49$
Mutual support‡ (4) | $4.26$ | $0.44$ | $0.70$ | $0.80$ | $0.63$ | $1$ | $0.33$ | $0.65$ | $0.55$ | $0.68$ | $0.77$ | $0.68$ | $0.66$ | $0.32$
Effort (5) | $3.94$ | $0.47$ | $0.73$ | $0.72$ | $0.75$ | $0.54$ | $1$ | $0.78$ | $0.31$ | $0.64$ | $0.68$ | $0.64$ | $0.37$ | $0.24$
Cohension (6) | $3.94$ | $0.41$ | $0.84$ | $0.75$ | $0.76$ | $0.75$ | $0.76$ | $1$ | $0.24$ | $0.64$ | $0.64$ | $0.64$ | $0.28$ | $0.69$
Work satisfaction‡ (7) | $4.21$ | $0.52$ | $0.75$ | $0.67$ | $0.59$ | $0.79$ | $0.65$ | $0.83$ | $1$ | $0.54$ | $0.65$ | $0.55$ | $0.91$ | $0.58$
Learning‡ (8) | $4.16$ | $0.47$ | $0.62$ | $0.48$ | $0.48$ | $0.66$ | $0.56$ | $0.73$ | $0.91$ | $1$ | $0.55$ | $0.31$ | $0.62$ | $0.45$
Effectiveness TM (9) | $3.99$ | $0.42$ | $0.61$ | $0.63$ | $0.65$ | $0.58$ | $0.68$ | $0.81$ | $0.68$ | $0.60$ | $1$ | $0.23$ | $0.53$ | $0.50$
Efficiency TM (10) | $3.67$ | $0.52$ | $0.58$ | $0.53$ | $0.58$ | $0.43$ | $0.60$ | $0.67$ | $0.41$ | $0.23$ | $0.71$ | $1$ | $0.34$ | $0.38$
Effectiveness PO (11) | $4.05$ | $0.60$ | $0.16$ | $0.32$ | $0.38$ | $0.13$ | $0.46$ | $0.37$ | $0.31$ | $0.25$ | $0.58$ | $0.46$ | $1$ | $0.67$
Efficiency PO‡ (12) | $3.86$ | $0.84$ | $0.31$ | $0.21$ | $0.32$ | $0.11$ | $0.31$ | $0.46$ | $0.40$ | $0.38$ | $0.51$ | $0.48$ | $0.69$ | $1$

Notes. $\overline{X}$ is the mean and SD the standard deviation. TM is team
member and PO is product owner. N = 34 for all variables. The lower diagonal
is the observed Pearson correlation and the upper diagonal is the transformed
(estimated) correlation matrix using SEM. Correlations above 0.34 are
significant at $p<0.05$, and correlations above 0.44 are significant $p<0.01$.
Variables that are not normally distributed according to the Shapiro-Wilk test
of normality are marked ‡. Variables 1–6 are treated as independent and
variables 7–12 dependent in the analysis.

Table 2 shows the variable descriptives and Pearson correlations for the
merged dataset. Five of the twelve investigated (aggregated) variables have
non-normal distributions. By visual inspection, the lack of normality for
these variables appears to arise from outliers. Moreover, the aggregated
variables associated with teamwork quality (1–6), personal success (7–8)
appear (also by visual inspection) to have approximate linear relations, which
is an assumption in the statistical analyses we report. However, the
aggregated variables associated with team performance as rated by the the
product owners (11–12) do not.

Overall, when inspecting the differences in the patterns of correlations, the
product owners using SAFe appear to be in more agreement (positive
correlations) with the team members than in the LeSS projects.

Table 3 reports model fit for the two different large-scale agile development
methods and the merged dataset. The overall lack of fit of the data to the
theoretical model is clear from the unacceptable RMSEA values and their
confidence intervals. In particular, the SAFe data do not fit the overall
(measurement and structural) model. An approach to improve model fit is to
relax model assumptions by, for example, introducing additional parameters in
the model, allowing for correlated residual (error) variance, or by estimating
only parts of the model, such as the measurement model only. Although we could
improve the model fit slightly with such approaches, the overall model fit was
still clearly unacceptable and consistently worse for the SAFe data. For
example, by estimating the measurement model of teamwork quality only for the
SAFe data, model fit improved (RMSEA = 0.25 [0.11–0.39], ${\chi}^{2}$ (9) =
21.24), but it was still unacceptable.

Table 3. Model fit for the theoretical model in Figure 1 Dataset | n | RMSEA [95% CI] |  ${\chi}^{2}$ (df)
---|---|---|---
LeSS | 21 | 0.15 [0.07–0.22] | 74.86 (51)
SAFe | 13 | 0.42 [0.34–0.49] | 156.43 (51)
Merged | 34 | 0.17 [0.13–0.22] | 102.19 (51)

##  5\. Discussion

This study sought to investigate how and if the effect of teamwork quality on
team performance differs between large-scale agile development methods. We
analysed data from four projects with 5 to 11 development teams, two Norwegian
and two German.

The original study by Hoegl and Gemuenden (Hoegl2001, ) on the relationship
between teamwork quality and project success for innovative projects found a
significant positive relationship between teamwork quality and team
performance as rated by team members, team leaders, and managers.

In the context of agile software development, a replication of the study by
Lindsjørn et al. (Lindsjorn2016, ) found a positive effect of teamwork quality
on team performance as rated by team members and team leaders. A negligible
effect was found for product owners. Comparing the effect of teamwork quality
on project success, it was found to be only marginally greater for agile
teams, despite the focus on teamwork in agile software development.

Further, when comparing the influence of teamwork quality on project success
for small and large agile projects, Lindsjørn et al. (Lindsjorn2018, ) found
that teamwork quality primarily had an impact on product quality in small
projects, while teamwork quality had a positive impact on product quality when
rated by team members but a negative impact when rated by team leaders. The
study further suggests that teamwork quality affects project success
differently in small and large projects.

When analysing data from large-scale agile development projects, what could
explain why our data do not fit the previously established theoretical model?
We discuss some possible explanations in the following, which include aspects
of research design as well as measurements used.

###  5.1. Project Success

There could be larger disagreements as to what characterises success in large-
scale than in small-scale software development. It might be difficult to build
a shared understanding of the quality of the product and if delivery is
according to schedule and budget when the project involves a number of teams
who work on their own parts of the product. Although software projects seldom
are ”megaprojects” (cost of USD one billion or more), large software projects
might suffer some of the same challenges as megaprojects as they are risky due
to long planning horizons and complex interfaces, decision-making involves
multiple stakeholders, and the scope or ambition level can change over time
(Flyvbjerg2014, ).

A strength of the study design of (Hoegl2001, ) compared to, for example,
(verwijs2023, ), is that the dependent variable is measured by someone outside
the team (in our case a product owner). Prior studies have, however, found
that team members rate performance differently than managers, and this is
explained by team members giving high-performance ratings if teams have
healthy internal processes, while managers who have less insight into team
dynamics focus on external factors such as communication with external agents
(cohen1997, ). Still, one would then expect that the SAFe data would fit only
the measurement model of teamwork quality, as this is only evaluated by the
team members; however, these data do not fit the teamwork quality model in
isolation either (RMSEA = 0.25 [0.11–0.39], ${\chi}^{2}{(9)}=21.24$).
Following recommendations by (kahneman2021, ), it could make sense to ask
respondents to rank teams rather than indicating project success on scales. In
particular, in a multiteam context, it will often be one product owner who is
responsible for many teams, and this person should then have a good background
for ranking.

###  5.2. Measuring Teamwork Quality

We have measured teamwork quality with an instrument constructed over 20 years
ago. Clearly, software development practices and processes have changed during
these years to mitigate known problems with teamwork and to help deliver
better software. The recent study by (verwijs2023, ) offers an alternative
theoretical model, indicating that there might be factors other than the
teamwork quality model that better explain project success. The team
effectiveness model by Strode et al. (strode2022, ) is another source of
factors.

Thus, one could argue that the understanding of the ”teamwork quality”
construct may have changed over time. Such a shift over time of what is
measured is sometimes referred to as ”concept drift” (gama2014, ). From a
construct validity perspective (sjoberg2022, ), such a change would involve
the threat of “construct underrepresentation” because key elements of the
teamwork quality that affected team performance is not captured by the
instrument.

One way forward is to use qualitative studies involving teams in an attempt to
establish what they (and their product owners) regard as the main drivers of
team performance in the context of large-scale agile development.

###  5.3. Multiteam Characteristics

Studies of ”teams of teams”, ”multiteam systems” in organisational psychology,
suggest that cross-team (interteam) processes are more important than within-
team (intrateam) processes for the overall performance of the ”multiteam
system” (marks2005, ). Unlike (Hoegl2004, ), our survey instrument did not
include cross-team factors, which could have a larger impact on project
success than what we measure with the teamwork quality instrument at team
level. Such factors have been identified in overview studies on large-scale
agile development, what Edison et al. (edison2021, ) label ”connecting
practices”. Examples of such practices are arenas for interteam coordination,
release planning and architecture, and knowledge sharing and improvement.
There are, of course, also other factors that will influence project success,
such as the skills of developers in important fields (technology and the
domain knowledge of the solution) and how the project is able to engage users
(mcleod2011, ). Also, there may be conflicts between teams or incentive
structures that reduce the motivation for interteam collaboration. The
importance of additional processes for large-scale development is supported by
Rolland et al. (rolland2023, ), who argue that agile practices cannot be
scaled in a ”straightforward linear fashion” such as adding a layer with
”Scrum of Scrums” on top of daily meetings for coordination and communication,
but that additional roles and practices are needed to manage scale.

###  5.4. Multiple Levels

Finally, Batra (batra2020, ) discusses challenges and opportunities in
quantitative studies of large-scale agile development. He identifies
challenges such as operationalising dependent variables and that researchers
”may get overwhelmed” when selecting independent variables as there is a
”plethora of suggested constructs”. A suggestion is to study the phenomenon at
different levels of analysis and three functional perspectives. The levels of
analysis are organisational, team, as well as individual, and the three
functional perspectives are process, governance, and outcome. Our instrument
focused on team level and outcome, combined with a description of process and
governance.

We could also add time as a factor, as for example, project success might be
measured differently when measured during the project or after project
completion.

In line with Batra (batra2020, ), we suggest that future quantitative studies
of large-scale agile development efforts include more analytical levels and
functions. We believe teamwork is a key analytical level, but emphasis should
be made on the quality and frequency of the interaction between teams in
addition to the internal team factors. Literature reviews on large-scale agile
software development (edison2021, ; uludag2022, ) show suggestions for
independent variables that can be measured.

###  5.5. Limitations

The primary limitation of our analysis lies in the limited number of
observations. While we have nearly 200 respondents, the data are grouped into
teams, which are then combined into four large-scale projects representing two
agile methods. This gives a larger uncertainty to conclusions. However, we do
claim that there are concerning issues present, an argument we believe can be
made with less data than if we were to claim that ”the data supports the
theoretical model.”

Another constraint in our study arises from the diverse nature of the
collected data, which varies in at least seven characteristics (see Table 1),
which all might influence the needs for coordination and communication across
teams. This study cannot pinpoint which of the context factors contributes
most significantly to project success. Furthermore, there are likely other
crucial contextual factors, such as team capability factors (see, e.g.,
(boehm1981, )), that are not represented in our dataset.

##  6\. Conclusion and Future Work

One of the main topics in software process research in recent years has been
”large-scale agile development” (dingsoyr2019, ; uludag2022, ). Many methods
are available, and many organisations strive to find a method that fits
(conboy2019, ). Teamwork is a key characteristic of agile software
development. We used a team level study design to focus on multi-team projects
using two different methods for large-scale agile development. We were
interested in exploring if the use of the different methods would impact the
relationship between teamwork quality and project success, which has been
established in prior research both for traditional teams (Hoegl2001, ) and for
agile development teams (Lindsjorn2016, ). However, our data did not fit the
previously established theoretical model, which raised several concerns. We
discussed four possible reasons: Larger disagreements as to what characterises
success in large-scale agile development, the teamwork quality instrument
might not capture all important aspects of teamwork, interteam factors might
have a larger impact on project success than the teamwork quality, and
finally, that our model fails to capture important analytical levels and
functions.

In line with Batra (batra2020, ), we suggest that future quantitative studies
should provide a deeper understanding of the dynamics of large-scale agile
development by exploring more analytical levels and functions. In particular,
we recommend studies to focus on the quality and frequency of interaction
between teams in addition to internal team factors.

### Acknowledgement

We would like to thank the anonymous reviewers for their inspiring comments
and Dr. Ömer Uludağ, Manuel Styrsky, and Maximilian Doepp for their work in
collecting the German data used in this study.

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