11institutetext: George F R Ellis 22institutetext: Mathematics Department,
University of Cape Town
Rondebosch, Cape Town 7701, South Africa
22email<EMAIL_ADDRESS>33institutetext: The New Institute,
Warburgstrasse 8,
Hamburg 20354, Germany
33email<EMAIL_ADDRESS>
# Quantum physics, digital computers, and life from a holistic perspective

George F R Ellis

###### Abstract

Quantum physics is a linear theory, so it is somewhat puzzling that it can
underlie very complex systems such as digital computers and life. This paper
investigates how this is possible. Physically, such complex systems are
necessarily modular hierarchical structures, with a number of key features.
Firstly, they cannot be described by a single wave function: only local wave
functions can exist, rather than a single wave function for a living cell, a
cat, or a brain. Secondly, the quantum to classical transition is
characterised by contextual wave-function collapse shaped by macroscopic
elements that can be described classically. Thirdly, downward causation occurs
in the physical hierarchy in two key ways: by the downward influence of time
dependent constraints, and by creation, modification, or deletion of lower
level elements. Fourthly, there are also logical modular hierarchical
structures supported by the physical ones, such as algorithms and computer
programs, They are able to support arbitrary logical operations, which can
influence physical outcomes as in computer aided design and 3-d printing.
Finally, complex systems are necessarily open systems, with heat baths playing
a key role in their dynamics and providing local arrows of time that agree
with the cosmological direction of time that is established by the evolution
of the universe.

###### Keywords:

Digital Computers, Life, Hierarchies, Downward Causation, Local Wave
Functions, Values.

##  1 Introduction

A crucial feature of life is the way function, purpose, and agency arise in
biology Ball_2023 ; Mitchell_2023 ; Noble_Noble_2023  despite none of these
attributes occurring at the underlying physics levels Hartwell_1999 ;
Ellis_2016 . In particular these features are not apparent in the quantum
dynamics Landshoff_et_al ; Laughlin_Pines ; Isham_2001  underlying the
functioning of life. Furthermore digital computers are able to carry out
abstract causation Dasgupta_2016 , involving logical branching deployed in
order to attain some desired outcome. This again is based ultimately in the
underlying quantum physics Ellis_Drossel_computers ; but quantum physical
processes do not by themselves involve logical branching such as is key to
digital computation. This paper looks at the way the underlying quantum
physics relates to these issues and enables such real-world emergence to
occur.

A first key feature is that quantum theory, via the Schrödinger equation
Landshoff_et_al ; Laughlin_Pines ; Isham_2001 , is linear in the wave function
$|\Psi\rangle$ Ellis_2012 , but the real universe is highly non-linear;
consequently only local wave functions can exist Ellis_2023 . It is this that
for example allows feedback control loops and homeostasis to exist.

A second key feature is the downward causation that underlies emergence of
complexity in modular hierarchical systems Ellis_2016 . There are two ways
this occurs Ellis_2023a : via time dependent constraints, which is where the
potential term in the Hamiltonian enters; and via creation or deletion of
lower level elements, which is where quantum field theory enters. A crucial
feature of downward causation is multiple realizability of higher levels at
lower levels; and this applies both to physical hierarchies, and the logical
hierarchies they support.

Thirdly, downwards causation from the environment occurs because any complex
system is an open system coupled to heat baths that play a key role in
classical and quantum mechanics Drossel_2014 , and specifically in contextual
wave function collapse CWC . This is how local arrows of time arise out of the
cosmological direction of time associated with the expansion of the universe.

Finally, downward causation associated with mental causation is a product of
the human mind, with choices made crucially shaped by the values of those
making decisions. This is central to the functioning of digital computers
Ellis_Drossel_computers , as is apparent in the way social media influences
the world around us.

###  1.1 Symbol structures and abstract causation

A key point to make at the start is that (Dasgupta_2016 :11,12):

> “As regards digital computers, in its most fundamental, the stuff of
> computers is systems of symbols, forming symbol structures\- that is
> entities that stand for, denote, or represent other entities. Computing is
> symbol processing.”

Then because we are a symbolic species Deacon , essentially the same is true
for the human brain. Inter alia, amongst its capacities is logical thinking:

> As regards the human brain, in its most fundamental, the stuff of thought is
> symbol structures. Thinking is symbol processing.

This is the basis of abstract causation, where symbols cause changes in the
physical world - as happens all the time in society (think traffic lights).

#### Abstract Entities have causal powers

This is clear in digital computers Dasgupta_2016 ; Ellis_Drossel_computers ,
where (1) algorithms, (2) computer programs, and (3) data - all abstract
entities - have causal powers because they alter physical outcomes in a real-
world social context. Digital computers are at their heart symbol processing
systems. As expressed in (Dasgupta_2016 :23),

> Some computational artefacts are entirely abstract: they not only process
> symbol structures, they themselves are symbol structures and are devoid of
> any physicality, though they may be made visible by physical media such as
> marks on paper or computer screens. so physico-chemical laws do not apply to
> them. They neither occupy physical space nor do they consume physical time.

However by the processes detailed in this paper and in Dasgupta_2016 ;
Ellis_Drossel_computers , these abstract entities have causal powers via what
Dasgupta refers to as Liminal Structures where the abstract becomes physical
and can result in images on a screen, marks on paper, 3-d printing of
artefacts, automatic landing of an aircraft, and so on. Similar remarks apply
to the brain.

###  1.2 This paper

The following sections look at, Hierarchies of structure and causation (§2);
Quantum linearity, local wave functions, and wave function collapse (§3);
Quantum Theory and time dependent constraints (§4); Quantum Field Theory:
particle creation and annihilation (§5); Open systems and the arrow of time
(§6); and, Outcomes, How did they get that way, and the key role of values
(§7).

##  2 Hierarchies of structure and causation

All truly complex systems, including digital computers and life, are based in
modular hierarchical structures Ellis_2016 ; Ellis_di_Sia  for good functional
and evolutionary reasons Booch ; Simon_2019 .

#### Orthogonal physical and logical hierarchies

Digital computers involve orthogonal physical and logical hierarchical
structures (Ellis_2016 :§2), together forming computational artefacts
(Dasgupta_2016 :§2) with the first supporting the second. Essentially the same
is true in the case of human beings, because the brain is the basis of
intelligence and understanding Mitchell_2023 ; Donald .

###  2.1 The physical hierarchies for digital computers and for life

The different levels of the emergent hierarchy of physical structure and
function in digital computers Tannen ; Ellis_2016 ; Dasgupta_2016  are shown
on left hand side of Table 2.1 and those for biology Ball_2023 ; Ellis_2016 ;
Noble_2012 ; Campbell  on right hand side. Different kinds of causation emerge
at each level at each of these hierarchies, described by different kinds of
dynamics and associated variables Anderson .

| Digital computers | Human Life
---|---|---
L10 | Internet | Society
L9 | Local Network | The Organism as a whole
L8 | Computer | Physiological structures
L7 | Components | Tissues
L6 | Gates | Cells
L5 | Transistors | Macromolecules, DNA
L4 | Crystalline structures | Molecules
L3 | Atoms | Atoms
L2 | Ions, Carriers | Electrons, Protons
L1 | Particle Physics | Particle Physics

Table 2.1: The emergent hierarchy of physical structure and causation for
digital computers (left) and human life (right) Ellis_2016 .

The higher levels are characterised by larger scales of physical organisation;
furthermore processes come into play that cannot be described in lower level
terms. Thus transistors are either conducting or not; cells are either
dividing or not; animals are either alive or not: and so on, Processes at
level L+1 cannot be characterised in terms of variables defined at level L,
which relate to different kinds of entities.

#### Artificial systems

There are similar hierarchies in the case of artificial systems such as
watches, automobiles, aircraft, and cities Simon_2019 ; Arthur_2010 . In the
case of digital computers, the paper Ellis_Drossel_computers  carefully
relates the higher level operations to the underlying quantum physics level.
The core levels are L5: transistors transistors , (Mellisinos :1.9) and Logic
Gates L6 (Mellisinos :1.9), where logical operations are derived from the
underlying physics.

#### Biology

On the life sciences side, quite different kinds of causation come into play
at levels L6 and upward Ball_2023 ; Campbell , related to biochemical
processes and molecular biology (L6), cellular processes (L7), physiological
processes (L8), mental and psychological processes (L9) Mental_Causation ;
Deacon , and social processes (L10) Berger . Each of these levels is clearly
causally effective; for example mental processes lead to the construction of
aircraft and digital computers Ellis_2016 , which would not otherwise exist
Ellis_2005 . The core level is the cellular level L7 cells , where all the
functions of life first exist. Protein molecules act as computational elements
in living cells Bray . Molecular processes underlie the brain Ellis_Kopel :
membrane potentials alter ion channel shapes and so alter ion locations, and
hence change the molecular Hamiltonian in a time dependent way. Messenger
molecules alter protein conformations Lehn_2004 ; Lehn_2007  in response to
higher level biological needs Hartwell_1999  conveyed by cell signaling
networks Berridge . The link to the underlying quantum physics is laid out in
Karplus_1 ; Karplus_2014 .

###  2.2 The reasons for modularity

In order to obtain genuinely complex behaviour, a modular hierarchy of
emergent levels LN is needed as in Table 2.1, with each level demonstrating
different kinds of behaviour as we see in both digital computers Dasgupta_2016
; Ellis_Drossel_computers  and life Ball_2023 ; Ellis_2016 ; Noble_2012 . Each
level LN will be comprised of networks of modules at the next lower level
LN-1. The reasons this is the necessary kind of structure to attain genuine
complexity are discussed in Booch ; Simon_2019 .

To generate genuine complexity associated with function, one must split a
complex task up into simpler tasks, design units to handle the simpler tasks,
and then knit them together in networks so that when taken together they
handle the complex task. The whole is done in a hierarchical way (sub-modules
of modules do even simpler tasks). At each level, modules are characterised by
(a) Information Hiding, (b) Abstraction, (c) Interface Specification, (d)
Modifiability, and (e) Multiple realizability. This logic applies both to
physical and logical hierarchies.

#### Information Hiding

The module’s internal variables are hidden from outside view, all that is
required is that the module does what it is required to do, as viewed from
outside. The user does not mind what internal variables are chosen and what
logic is used.

#### Abstraction

An abstraction is an external description of what the module does in a
reliable way. This should include a name by which one can refer to the module,
which indicates what it does (it should have a unique identifier).

#### Interface Specification

The interface between each module and the rest of the system must be carefully
specified, as in the case of the ISA: what control variables, parameters, and
data will be sent to the module, and what will be sent out to other modules in
the system from the module.

#### Modifiability

We can change the internal variables and even the internal logic of the
module, as long as the externally viewed functionality, as defined by the
interface and abstraction, is maintained. The system as a whole only cares
that the module does what it is designed to do, not how it does it (although
some implementations are preferred over others if they are either faster, or
use less resources). This leads to multiple realisability in the hierarchy.

#### Multiple realizability

A key feature of downward causation in both physical and abstract hierarchies
is multiple realisation of higher levels at lower levels. Any higher level
structure and dynamics arising out of lower levels can be realised at the
lower levels in numerous ways. Thus there is multiple realizability of every
physical level L and its dynamics in terms of the lower levels L-N in Table
2.1. It follows that the higher level dynamics is not equivalent to lower
level dynamics in any simple way.

In the case of digital computers this occurs for both the physical and logical
hierarchies Abramsky_multiple . Higher level structures can be based on many
different lower level structures in Table 2.1, e.g. using different computer
chips with different instruction sets. Higher level programs can thereby be
based on many different lower level languages (Table 2.2), e.g. using
different operating systems. Even more than that: one can shift the boundary
between hardware and software by different physical implementations
Dasgupta_2016 ; Tannen , so multiple realizability exists in this extended
sense too.

In the case of biology in general this is discussed in
Farnsworth_et_al_multiple , and its crucial role at the molecular level is
powerfully presented in Wagner . In the case of the brain, it is discussed in
Figdor_Multiple ; Ellis_Multiple . Because of this multiple realisability with
huge numbers of lower level elements (cells, molecules, atoms, particles)
involved, the function and agency so patent at higher levels Ball_2023 ;
Mitchell_2023  is indiscernible at these lower levels. That does not mean it
does not exist Noble_Noble_2023 .

###  2.3 Open systems and finiteness

Biological systems are necessarily open systems for thermodynamic reasons
Open1 ; Open2 . They must breathe and obtain food and liquid from the outside
world because living things need energy energy , and discharge solid and
liquid waste into it. Furthermore there is a key issue relating to entropy:
organization is maintained by extracting ’order’ from the environment (Life
:§6)

Life also receives information from the outside (light, sounds, smells) that
influence their behaviour. They are structured so to be able to respond
adaptively to such incoming data, and in turn act on it in various ways,
including sending information into the outside world.

Digital computers are open systems in similar ways. They need to obtain
electrical power from the outside world in order to function, and they
dissipate heat into it. This is related to the physical limits on computation
Landauer2 ; Landauer1 , in particular the fact that all real computers have a
finite memory, so items have to be erased from working memory in order that
the calculation can continue. But such erasure is irreversible, and costs
energy: any logically irreversible transformation of classical information
necessarily involves dissipation of at least $kT\,ln(2)$ of heat per lost bit,
where $k$ is the Boltzmann constant and $T$ the temperature Erasure1 ; Erasure
. Many computer programming languages (e.g. Lisp) have integrated garbage
collection garbage  which carries out the function of freeing up memory which
is not being used - which will generate heat.

Computers also receive information from the outside world via the internet,
which in turn enables them to act on the outside world in multiple ways.

###  2.4 The three kinds of downward causation, and causal closure

Downward causation occurs by three basic mechanisms Ellis_2023a : (a) via time
dependent constraints that shape lower level dynamics, (b) by creating,
altering, or deleting lower level elements, (c) by coupling to the
environment.

#### Non-unitary dynamics

will occur when there is a time dependent potential term $V(r,t):\partial
V(r,t)/\partial t\neq 0$ in the Hamiltonian, representing time-dependent
constraints Ellis_Kopel . The standard uniqueness theorems Arnold ; Arnold1
for Hamiltonian systems with time-independent potentials then don’t hold: the
initial state does not uniquely fix the future evolution. This happens in
engineering and biological systems, where higher level dynamics alters the
context of lower level functioning. Examples are changing voltages $V(t)$
controliing currents in MOSFET transistors in digital computers as a high
level program is executed, the multi-scale dynamics of the heart involving
cardiac sodium channel effects on action potentials and mechano-electric
feedback in which the contraction of the heart influences its electrical
properties Noble_2002 , and action-potential spike chains Fletcher  generated
by voltage gated ion-channels Ellis_Kopel .

#### Creation, alteration, and annihilation of elements

The laws of molecular biology or physics determine specific outcomes in a
given context - provided we know what entities they are operating on. In
molecular biology, gene regulatory networks Karlebach_genes  operating in a
specific context determine what proteins are present in a cell, which in turn
determines what kinds of cells are present at specific locations in a body
through developmental processes Carroll . The lower level elements change all
the time in response to higher level dynamics Ellis_2023a .

In solid state physics contexts, quasi-particles such as phonons are created
by collective oscillations of a crystal structure Simon ; QFT . Ultimately
they are created or destroyed by creation and annihilation operators that are
an essential feature of Quantum Field Theory QFT ; Kittel . The key point then
is when do these operators function? What turns them on and off? I return to
this in §5.

#### Coupling to the environment

As well as imposing time-dependent constraints, which set the context in which
the time evolution of a system takes place, the environment influences a
system also via ongoing exchanges of energy, information, and possibly matter
Ellis_closure ; Ellis_Open . No system is fully isolated from these
influences. This is a further key form of downward causation, whereby arrows
of time get injected into local systems via contact with heat baths which in
turn are in contact with the external environment up to the Universe CWC ;
Ellis_Drossel_time .

#### Computers and users

A key point here is that the two themes of this paper - the way computers
function and the way intelligent life functions - are linked by the fact that
the computer user is part of the environment of the computer, who feeds
information in that then determine its operation Dasgupta_2016 . This is
illustrated in Table 2.2. It is how mental causation (decisions made by the
user) controls the operation of computer hardware, influenced by the society
around Berger , right down to the quantum physics level.

| Society |
---|---|---
|  $\Uparrow$ $\Downarrow$ |
| Computer User |
|  $\Uparrow$ $\Downarrow$ |
| Application software, data |
|  $\Uparrow$ $\Downarrow$ |
| Operating system |
|  $\Uparrow$ $\Downarrow$ |
| Computer Hardware |

Table 2.2 Relation of user to computer The computer is an open system, with
its operation controlled by the user in the context of society.

Knowledge of the internal state of the computer at any time $t_{1}$ will not
determine its state at a later time $t_{2}>t_{1}$ because of this openness.

#### Causal closure

Because of the combination of these upwards and downwards forms of causation,
it is not true that any underlying physical level is by itself causally
closed: all the interacting levels contribute to the final result and affect
the outcome, because of this causal chaining: it would not happen if they were
not all there. That is why every level is causally effective Noble_2012 .
Actually one needs only go down to the level of electrons and protons, which
is the lowest effective physical level, because of the existence of quantum
protectorates Laughlin_Pines : the lower levels (quarks, gluon, and so on) are
irrelevant to outcomes in digital computers and life.

###  2.5 The Logical Hierarchy for digital computers

In the case of digital computers, there are two modular hierarchical
structures: the physical hierarchy discussed above (Table 2.1), and a logical
hierarchy supported and enabled by the structure and function of the physical
hierarchy (Ellis_2016 :§2, Ellis_Drossel_computers ). They can be regarded as
orthogonal to each other, because although the logical hierarchy is realised
by the physical hierarchy, the content of the logical hierarchy is determined
by the software and data loaded, not by the nature of the physical hierarchy.

Level | Language and Logic | Downward Process | Agent
---|---|---|---
$M_{6}$ |  $L_{6}$: Applications Program |  $\Downarrow$ Logic and data | Program Code
$M_{5}$ |  $L_{5}$: High level Language |  $\Downarrow$ Translation: Compiler | Syntax, semantics
$M_{4}$ |  $L_{4}$: Assembly Language |  $\Downarrow$ Translation: Assembler | Syntax, Semantics
$M_{3}$ |  $L_{3}$: Operating system level |  $\Downarrow$ Partial Interpretation | OPerating system
$M_{2}$ |  $L_{2}$: Machine Language |  $\Downarrow$ Interpretation | ISA
$M_{1}$ |  $L_{1}$: Microprograms |  $\Downarrow$ Directly executed | Instruction
$M_{0}$ | Digital Logic Level (binary) | Hardware | Gates

Table 2.3: Logical hierarchy for a digital computer Tannen ;
Ellis_Drossel_computers .

he logical hierarchy (Table 2.3) represents the tower of virtual machines
$M_{I}$ Tannen  each with an associated language $L_{I}$. Multiple
realizability occurs between every pair of logical levels because different
languages can be chosen at lower levels, expressing exactly the same logic
(the algorithm) as do the higher levels, but in a different language. Higher
level data also gets re-expressed in the relevant lower level notation. The
downward arrows are implemented by compilers or interpreters, according to
context,

Here for example $L_{6}$ might be a word processor, internet browser, image
processor, AI program, and so on; $L_{5}$ might a language such as FORTRAN,
$C^{++}$, JAVA, PYTHON, and so on. This logical hierarchy is implemented by
the physical hierarchy (Table 2.1), but not in a 1-1 way. In a particular
context of computer use, every level of the physical hierarchy is at work
representing every level of the logical hierarchy.

A crucial point is that the data and logic represented by the logical
hierarchy is not in any way restricted by the physical hierarchy: the physical
implementation does not in any restrict the logic it represents, provided it
does not use up too much memory or take too much time to process.

> The principle of arbitrary content: digital representations The logical
> hierarchy of a digital computer can represent arbitrary logic and data,
> which is re-expressed in different notations at each of the logical levels,
> until it is expressed in binary notation (0s and 1s) at the lowest level L0.
> This flexibility is based in the possibility of using arbitrary software at
> the higher levels to control flow of electrons in gates.

That is why these hierarchies are called “orthogonal”. This is how abstract
causation works in this context: abstract elements (programs, data) act
downwards to control events at the micro-physical level (Ellis_2016 :§2,
Dasgupta_2016 ), indeed right down to the quantum level
Ellis_Drossel_computers ).

This underlies the extraordinary usefulness of digital computers in the world:
any data whatever, representing pictures, diagrams, videos, scientific
measurements, documents, cooking recipes, engineering variables can be
represented digitally and manipulated logically to calculate probabilities and
possibilities, and even to determine physical outcomes such as manufacturing
processes in automated factories, generating 3-d printed objects of any kind,
and controlling systems such as chemical plant or an aircraft or automobile.

###  2.6 The Symbolic Brain and Abstract thought

A key feature separating human beings from the great apes and all other life
is that we are a symbolic species Deacon  able to represent and analyse the
world in symbolic terms. This occurs at level L9 in Table 2.1. It is enabled
by the physical structure of our brain: an enormously complex neural network
Kandel  with the capacity to learn, analyse, predict, and make choices. Any
book, paper, article, or speech in any language is an abstract modular
hierarchical structure represented in physical form, as are the categories of
objects and actions they express, as set out in dictionaries and
encyclopedias. This logical content is in each case representable in many
different ways both at each level, and at lower levels: the basic principle of
multiple realisability holds again in this context, whereby for example a noun
phrase can act as a noun. This extremely flexible structure has evolved for
good functional Carruthers  and evolutionary Mitchell_2023 ; Sterelny ;
Miyagawa  reasons. It allows us to act on the world with agency Mitchell_2023
, for example creating entities such as digital computers, which would not
otherwise exist.

In parallel with the case of computers just discussed, the data and logic
represented by the functioning of the brain is not restricted by the its
physical structure, apart from memory limitations and processing time.

> The principle of arbitrary content: brain plasticity The physical brain can
> represent, analyse, and make predictions from arbitrary logic and data: any
> thoughts about any topic can occur. These operations of the mind result in,
> and are based in, brain plasticity plasticity : modifications of neural
> network connection weights in the brain enable them, which do not in any way
> restrict what is represented.

This flexibility is what underlies the enormous causal power of abstract
thought and associated mental causation Mental_Causation , which has
transformed the world around us, creating societies, agriculture, money,
economic systems, aircraft, factories, buildings, water supplies, and so on,
as well as transistors, integrated circuits, digital computers, and the
internet.

#### A challenge

We do not know what the equivalent of the logical hierarchy Table 2.2 is in
the case of a brain. Perhaps it includes some manner of representation of
thoughts via coding in adaptive resonant circuits Grossberg ; Grossberg1 ,
chaining down to the structure (logic?) of action potential spike chains
Fletcher  and then to physical implementation via voltage gated ion channels
opening and closing Ellis_Kopel , based in the underlying quantum chemistry
Karplus_2014 . However to clarify this also requires engagement with the
cognitive neuroscience of language Kemmerer .

###  2.7 Branching dynamics at every level

A key feature of both the physical and logical hierarchies is that branching
dynamics occurs at every level, in an integrated way between levels.

#### Logical branching: Computers

In the case of a digital computer, a high level program $L_{6}$ will have a
statement of the logical form (Dasgupta_2016 :36)

$IF\,\,\,\\{\textbf{T1}\\}\,\,\,THEN\,\,DO\,\,\,\\{\textbf{A1}\\}\,\,\,ELSE\,\,\,DO\,\,\,\\{\textbf{A2}\\}$
(1)

where T1 is a logical truth statement (e.g. $\textbf{T1}:=X\geq Y$), while A1
and A2 are two alternative actions. This can represent an arbitrary real world
issue: for example $X$ might be money owing, $Y$ a credit limit, A1 is “refuse
loan” and A2 “grant loan”; or $X$ might be cost of holiday in Rio, $Y$ amount
saved for a holiday, A1 is “stay home” and A2 “go to Rio”. As pointed out in
§2.5, any arbitrary issue whatever can be represented in this way.

#### Physical branching: Computers

Corresponding to the logical branching, physical branching takes place at each
level in Table 2.1:

  * •

At Transistor Level the branching is ON or OFF, depending on the gate voltage
$V(t)$.

  * •

At Gate Level the branching is in terms of truth values of basic logical
operations: AND, NOT, NOR which can then lead to branching in comparators,
adders, decoders, etc. (Mellisinos : pp.37-58;Tannen : pp. 135-164), with
outcomes depending on the data.

  * •

At Computer Level branching occurs via CPU control of the basic instruction
FETCH-EXECUTE cycle (Mellisinos :p.75, Tannen :pp.173–202), with activation of
different instruction memory and data memory locations and output modules via
a bus (Tannen :pp.176–220) controlled by a clock. Branching of electron flows
in the data bus occurs according to the relevant data.

  * •

At Network Level branching is via requests represented by a Hypertext Transfer
Protocol (HTTP), sent between computers identified by their Internet Protocol
address (IP address), as for example sending a request to Google for
information.

#### The relation between them

Logical branching governs physical branching, which is demonstrable by
changing the program loaded. Outcomes alter, although the physical structure
involved (the computer hardware) is identical whatever the logic involved.
This happens via compilers (Dasgupta_2016 :47,compiler ) and interpreters
Tannen  that chain logic down to the machine code level.

The logic of the algorithm Knuth , e.g. Bubblesort, is preserved during the
downward chaining process, as it gets rewritten in different languages LI with
different variables and syntax (Table 2.3). The abstract becomes physical at
the machine level, where digital logic is expressed in a timed sequence of
electron flows that turn transistors ON or OFF Tannen ; Mellisinos .The bits
are translated into electric voltages controlling the transistor logic
Ellis_Drossel_computers ; Mellisinos :

$IF\,\,\,\\{V(t)>V_{threshold}\\}\,\,\,THEN\,\,\\{\textbf{current
flows}\\}\,\,\,ELSE\,\,\,\\{\textbf{not}\\}.$ (2)

This is implemented physically as explained in Table 4.1 in §4.1.

#### Logical and physical branching: the Brain

Essentially the same applies in the brain. The branching logic (1) may apply
at the mental level with regard to some option we consider, or the simpler
case

$IF\,\,\,\textbf{T1}\,\,\,THEN\,\,DO\,\,\,\textbf{A1}$ (3)

may apply, where for example T1 is “I want to learn General Relativity” and A1
is “Learn tensor calculus first” (with the implict logic ‘If NOT A1 then NOT
T1”). This will chain down in ways we do not understand to specific action
potential spike chains in neuronal axons, enabled at the molecular level by
the logic ‘

$IF\,\,\,\\{V(t)>V_{threshold}\\}\,\,\,THEN\,\,\\{allow\,\,ion\,\,flow\\}\,\,\,\,\,\,ELSE\,\,\,\\{not\\}$
(4)

where $V(t)$ is the voltage difference across the axon membrane and voltage
gated ion channels open or close to implement this logic Ellis_Kopel . At an
even lower level this will be implemented by quantum chemistry interactions
that control molecular shape Karplus_2014 .

#### Downward logical and physical branching

In both cases, logical branching at the highest level (algorithms, thoughts)
drive logical branching consistently at all lower levels by the mechanisms
discussed further in Sections 4 and 5, with multiple realizability occurring
at each downward step, until either contextual wavefunction collapse or
particle creation and annihilation operations shape the outcomes at the
quantum level.

##  3 Quantum linearity, local wave functions, and and wave function collapse

Both digital computers and life are enabled at the bottom physical levels by
quantum physics as characterised by the Schrödinger equation. However this is
a linear equation, so it is something of a puzzle as to how this allows
complex structures such as digital computers, cats, human beings, and
societies to come into being Ellis_2023 . None of them is remotely linear:
their dynamics is not unitary.

###  3.1 The Schrödinger equation

The Schrödinger equation Landshoff_et_al ; Laughlin_Pines ; Isham_2001  is

$i\hbar{\frac{\partial}{\partial t}}|\psi(t)\rangle=\hat{H}|\psi(t)\rangle$
(5)

where $\hat{H}$ is the Hamiltonian operator. In position space representation
for a many particle system this is Drossel_Limits

$i\hbar{\frac{\partial}{\partial
t}}|\psi(\vec{r}_{1},...,\vec{r}_{N},t)\rangle=\left[-\frac{\hbar^{2}}{2m}{\frac{\partial^{2}}{\partial
x^{2}}}+V(\vec{r}_{1},...,\vec{r}_{N},t)\right]|\psi(t)|\psi(\vec{r}_{1},...,\vec{r}_{N},t)\rangle$
(6)

This is linear because there is no source term on the right in addition to the
$\hat{H}$ term, and $V(\vec{r}_{1},...,\vec{r}_{N},t)$ is independent of
$|\psi(\vec{r}_{1},...,\vec{r}_{N},t)\rangle$.

###  3.2 Linearity and local wave functions

Because of this linearity, quantum theory for a single wave function
$|\psi(t)\rangle$ cannot describe non-linear systems Ellis_2012 . However,
these exist in the physical world, both in the context of digital computers
and in biology at each emergent level $\textbf{L}\geq\textbf{L4}$ in the
hierarchy of structure (Table 2.1). Consequently, as I now argue, quantum
physics can only hold locally Drossel_Limits ; Drossel_2017 ; Ellis_2023  via
local wavefunctions that hold only in restricted domains.

#### Non-linearities

Where do essential non-linearities occur in digital computers and in biology?
- in five essential cases:

  * •

Hierarchies, as discussed in detail above. An intricate non-linear relation of
functions exists between all the levels in Table 2.1 and Table 2.3.

  * •

Feedback control loops/cybernetic systems such as aircraft autopilots, and
homeostatic systems that are crucial to biology at micro and macro levels

  * •

Networks: these occurs in digital computer integrated circuits; in biology, in
gene regulatory networks, metabolic networks, and neural networks.

  * •

Heatbaths which have a coherence timescale and physical length Drossel_2014 ;
CWC

  * •

Creation and deletion of lower level elements In digital computers, creation
of quasi-particles that shape conduction properties of transistors; in
biology, reading of genes to to produce proteins

None of this is unitary: but they certainly happen. So how do all the non-
unitary dynamics characterised in Table 2.1 arise? The key question is, What
are the limits of quantum mechanics? Drossel_Limits , The answer has to be
that while quantum physics locally underlies all emergent physics and
chemistry everywhere, it does not do so by wave functions with an unrestricted
domain of applicability, such as a single global wave function for the
universe. In fact only local wave functions exist, each valid only in a
restricted domain Ellis_2023 . The proposal is based in the way that in
General Relativity Theory, there are in general no global coordinates systems
covering a spacetime manifold; rather an atlas of local coordinates systems
exist that taken together cover a maximally extended spacetime HE . The same
logic applies in quantum physics.

#### Local wave functions

The idea is that only local wave functions $|\psi\rangle_{\alpha}$ exist that
are each valid only in local domains ${\cal U}_{\alpha}$ where the dynamics
expressed by $|\psi\rangle_{\alpha}$ is linear Ellis_2012 ; Ellis_2023 . The
notation

$|\psi\rangle_{\alpha}=|\psi\rangle_{\alpha}({\cal U}_{\alpha})$ (7)

expresses this limited domain of validity. Any non-linear dynamics such as in
the cases just mentioned will be split up into linear parts each described by
such a local wave function $|\psi\rangle_{\alpha}$; the whole system will be
covered by the union of associated wave function domains
${\cup}_{\alpha}({\cal U}_{\alpha})$. In the overlap ${\cal
U}_{\alpha\beta}:=\\{{\cal U}_{\alpha}\cup{\cal U}_{\beta}\\}$ between any two
of them, they will be related by an invertible transformation

$|\psi(x,t)\rangle_{\alpha}=f_{\alpha\beta}(|\psi(x,t)\rangle_{\beta})$ (8)

and associated Bogoliubov transformation between the relevant Hilbert spaces.
Note that ${\cal U}_{\alpha}$ may be time dependent: ${\cal U}_{\alpha}={\cal
U}_{\alpha}(t)$.

###  3.3 Implications

Consequently there is no single wave function for a living cell in a human
body, a cat, a brain, or a person, for non-linear processes occur in all of
them.

#### Schrödingers’ Cat

In the context of this famous thought experiment Leggett , the “equation”

$|\psi\rangle_{cat}=\alpha|\psi\rangle_{alive}+\beta|\psi\rangle_{dead}$ (9)

is nonsensical, as none of those wave functions exist: the cat’s physiology is
highly non-linear. In the typical Schrödinger cat experiment, the only quantum
entity described by a single wavefunction is the excited atom. All the rest -
the cat, the cage, the poison vial, the hammer, the particle detector are
classical. The cat is never in a superposition of being both alive and dead.

#### The Wave Function of the Universe and the Everett interpretation

Because the Universe contains living cells, cats, and human beings, for each
of which no single wave function exists, there is also no single wave function
$|\psi\rangle_{Universe}$ for the Universe as a whole considered at all
scales, as is routinely claimed. This causes problems for the Everett
interpretation of quantum physics which envisions $|\psi\rangle_{Universe}$
splitting all the time into multiple branches Isham_2001 ; Many_Minds1 . This
proposal relies on existence of a single linearly evolving wave function for
everything, taking for granted that it exists. This is highly implausible, and
at a minimum requires serious justification. Similar problems arise for any
“many minds” interpretation Isham_2001 ; Many_Minds1 ; Many_Minds , whether
related to the Everett interpretation or not. They assume existence of a
single wave function $|\psi\rangle_{brain}$ for each brain, which on the basis
of the above arguments I claim does not exist.

#### An effective Copenhagen Interpretation

Nor is there a single wave function for the apparatus involved in any quantum
physics experiment. Consider for example Alain Aspect’s paper on the Bell test
(Aspect : Figure 1). All is classical: the source, beam splitters, detectors,
coincidence detectors), except a few interacting particles ($\nu_{1}$ and
$\nu_{1}$) described by wave functions. All the rest are conceived of and
described in classical terms. To describe them in quantum terms would make the
analysis impossible: one would have to model each of the elements of the
apparatus quantum mechanically, and they would each have no definite state.
This confirms the claims in CWC  that wave function collapse (the quantum to
classical transition that is necessarily involved in any physics experiment
which produces a classical outcome which can be analysed) is enabled by the
classical context, as discussed in section §3.4.

###  3.4 Collapse of the wave function

A key aspect of downward causation is contextual wavefunction collapse CWC .

Quantum theory has two parts (Penrose_Road :527-533): unitary wavefunction
evolution $U$, plus wave function reduction $R$. The latter leads to definite
physical outcomes, and so is associated with the passage of time. As the very
purpose of the wave function is to determine probabilities of classical
outcomes, quantum theory means nothing physical unless wave function reduction
$R$ occurs Penrose_Road ; Ellis_2012 . However it is often omitted from
discussions of quantum physics.

A simplified description of the process $R$ is as follows: When an event $R$
happens, a wave function $|\Psi\rangle$ that is a superposition of orthonormal
eigenstates $|u_{n}\rangle$ of some operator is projected to a specific
eigenstate $N$ of that operator:

$R:\,\,|\Psi\rangle(t_{0})=\Sigma_{n}c_{n}|u_{n}\rangle\,\rightarrow|\Psi\rangle(t_{1})=\alpha_{N}|u_{N}\rangle.$
(10)

where $\alpha_{N}$ is the eigenvalue for that eigenvector. In reality, this
process is far more complex, as discussed in CWC  (see below). There is
irreducible uncertainty in this irreversible process: the specific outcome $N$
that occurs is not determined uniquely by the initial state
$|\Psi\rangle(t_{0})$ Ghirardi . However the statistics of the outcomes is
reliably determined by the Bohr rule: the probability $p_{N}$ of the specific
outcome $|u_{N}\rangle$ occurring is given by

$p_{N}=|c_{N}|^{2}.$ (11)

The projection process (10) occurs in laboratory experiments such as the
2-slit experiment, and so is often taken as being related only to such
experiments. However it also occurs all the time when physical interactions
take place, such as nucleosynthesis in the early universe (when no observers
were around), when a photon is registered by a CCD, and when a photon hits a
chlorophyll molecule in a leaf and hence releases an electron that starts a
cascade of biochemical reactions during photosynthesis in plants. Thus in many
real-world contexts non-unitary events (10) take place that cannot be
described by (5) Isham_2001 ; Penrose_Road , and are not associated with a
laboratory experiment.

Contextual Wavefunction Collapse (CWC) CWC  proposes that a nonunitary,
stochastic process (10) takes place and obeys the Born rule (11), with the way
this happens being determined by the local physical context. Indeed this is
obviously the case: specific apparatus may measure energy or polarisation, and
outcomes depend on this choice. In the latter case the direction of
polarization measured can be chosen at will, again altering outcomes Susskind
.

I will not repeat in detail the proposal CWC  here, but summarise as follows.
For concreteness that paper focuses on the case of a photodiode, detailing the
processes whereby

  * •

A photon is absorbed and an electron lifted from a bound state to the
conduction band;

  * •

An internal heat bath made of phonons interacts with the electron in the
conduction band and localizes it;

  * •

Interaction with an external heat sink makes the process irreversible and
introduces an arrow of time;

  * •

An electrical field moves the electron from the site of the excitation, with
interactions with other electrons triggering a macroscopic avalanche;

  * •

Detection of the resultant macroscopic current by classical processes results
in movement of a pointer or lighting of a bulb;

  * •

A reset step involves an external heat sink and requires energy; this makes
the process irreversible. The associated direction of time ultimately arises
from the expansion of the universe.

In effect this is a specific form of the Copenhagen interpretation Isham_2001
where the macro apparatus is classical rather than quantum. The reason for
this is limitations of the domain of validity of wave functions $|\Psi\rangle$
Ellis_2012 , as just discussed, and in particular the fact that a heat bath
cannot be described by a many particle wave function Drossel_2017 . Any real
macro apparatus involves heat baths and so is a classical entity, even though
it emerges from a structure that has quantum properties on the microscopic
scale.

An alternative that in many ways has a similar motivation is Relational
Quantum Physics Rovelli . This is an innovative and interesting proposal, but
the Contextual Wavefunction Collapse proposal is tied in more closely to real
physical contexts, such as the operation of photo-diodes or transistors. It
would be interesting to see Relational Quantum Physics applied in specific
contexts.

##  4 Quantum Theory and Time dependent constraints

The first form of downward causation mentioned above (§2.4) is the existence
of higher level time-dependent constraints that shape lower levels outcomes.

The operational question is, Where does $V(\vec{r}_{1},...,\vec{r}_{N},t)$ in
(6) come from? This potential represents downward causation via constraints
representing higher level contexts Noble_2012 ; Juarrero . When quantum
tunneling is involved it may be characterised classically as a potential well.
In the case of digital computers and the quantum chemistry underlying life, it
occurs via the Born-Oppenheimer approximation BO  suitable to the context
(crystals and molecules respectively) and their time dependent emergent
dynamics.

###  4.1 Digital computers and transistors

At a high level time dependent constraints represent voltages resulting from
currents resulting from abstract computation that determine when transistors
are on or off Dasgupta_2016 . The way this works out in the case of digital
computers at the lower levels is set out in (Ellis_Drossel_computers :§4.1,
Table 3), reproduced here as Table 4.1.

| Level | Structure | Outcomes
---|---|---|---
T5 | Gates | Transistor combinations | Boolean Logic
T4 | Transistors | Base, emitter, collector | ON/OFF
|  | Carrier channels |
T3 | Crystal structure | Ions: symmetry breaking | Phonons, band structures
|  | by impurities |
T2 | Electron population | Densities, average flows, | Electron diffusion, current
|  | resistance |
T1 | Individual ions, electrons | Ion bonding; electron velocities, | Electron drift
|  | collisions |

Table 4.1: The lower physical levels in a digital computer
Ellis_Drossel_computers .

The levels are as follows.

Level T5: Gates Transistors are linked via wires and resistors to form the
basic logical gates, perhaps combined to form a single integrated set of
transistors.

Level T4: Transistors are based in semiconductors such as silicon, doped with
donor (n) or acceptor (p) impurities. An applied voltage $V(t)$ on the gate
attracts electrons from the source and thereby opens a conducting channel
between the source and drain, through alterations to the chemical potential
and depletion region.

Level T3: Crystal Structure, Phonons, Electronic Bands The crystal structure
with its particular symmetries Kittel  and degrees of freedom gives rise to
phonons and the electronic band structure. Constraints representing the
periodicity of the crystal lattice underlie the use of Bloch’s Theorem (Kittel
:p.179) Depending on the distance between the bands and their filling, one
obtains the distinction between conductors, insulators, and semi-conductors.

Level T2: Electron Population Electron/carrier flow is due to diffusion:
density gradients leading to depletion regions, and to drift due to an
electric potential $V(t)$, leading to a current. Resistance occurs due to
collisions with impurities, phonons, and other electrons. Electron conduction
is time asymmetric because of interaction with a heat bath. Due to its
dissipative nature, this is not a Hamiltonian process. an emergent description
(a Boltzmann equation) is used: a phenomenological theory for this level.

Level T1:Individual ions and electrons The description at this level is based
on a Hamiltonian for the ions and electrons. The electrons are separated into
conduction band electrons (essentially unbound and so free to move) and
valence band electrons (closely bound to ions and so localised). The
Born–Oppenheimer (adiabatic) approximation BO  is used in the quantum
Hamiltonian with a symmetry-broken ground state: the crystal lattice structure
which leads to phonons existing Kittel . The electron equation is used to
obtain the electronic band structure. Electron-lattice interactions occur via
phonons (Kittel :§2). To model electron-phonon interactions explicitly, a
quantum field theoretical formalism is required based on creation and
annihilation operators for electrons and phonons.

Interlevel effects The steady state situation when the transistor is either
conducting or not depends on whether the applied voltage is above a threshold.
The electric field effects are modelled by adding the gate voltage $V(t)$.
This leads to a potential energy term $H_{V}(t)$ in the Hamiltonian of the
electrons:

$H_{V}(t):=\sum_{i}eV(\textbf{r}_{i}(t))$ (12)

where the Level T4 variable $V(t)$ determines the Level T1 variables
$V(\textbf{r}_{i}(t))$ in a downward way. This leads to a displacement of the
electrons until a new equilibrium is reached where the electrical field
created by the modified charge distribution cancels the electrical field due
to the gate voltage. This alters the band structure and thereby either creates
a conduction channel by changing the depletion zone, or not, according to the
bias voltage applied.

#### The outcome

Outcomes are determined by the time dependent function $V(t)$ determined by
the machine code (Level $M_{1}$ in Table 2.3), and applied in the context of
the specific detailed structure of the transistor concerned. This is all
driven by the programs and data loaded as indicated in Table 2.3.

This is how the logic represented in the abstract structure of the computer
program, expressed in machine code Dasgupta_2016  controls the underlying
physics at the electron level; the electron dynamics is no longer unitary.
Quantum level outcomes are shaped by abstract algorithms and data
Ellis_Drossel_computers . Different programs or data result in different
physical outcomes.

###  4.2 Quantum chemistry and Molecular shape

In living systems, at the molecular level change of shape of molecules in cell
signalling pathways (Berridge :§2) governs the dynamics Lehn_2004 ; Lehn_2007
. However the relevant molecular processes Ball_2023  implemented by metabolic
networks and gene regulatory networks are driven downwards by physiological
needs Noble_2012  and developmental processes Carroll . The underlying quantum
chemistry potential $V(\vec{r}_{1},...,\vec{r}_{N},t)$ in (6) is consequently
time dependent, and hence allows logical branching to arise from the
underlying physics Ellis_Kopel .

Changes in molecular shape are the key active factor in biological function at
the molecular level Ball_2023 ; Lehn_2004 ; Lehn_2007 . However shape is a
classical variable, not a quantum one. A central issue is, how does the
classical concept of shape, key to occurrence of these processes, arise out of
quantum theory? This is discussed in depth in Ramsey_1997  and (Bishop_Ellis
:§5.2). In agreement with the theme of this paper, one can claim that
molecular shape is shaped by the environment Amann_1991 ; Amann_1993 .

The way that change of shape is calculated in terms of quantum theory is
presented by Martin Karplus in Karplus_2014 , again emphasizing (like CWC )
that quantum calculations in complex situations rely on classical concepts to
a large degree. The number of atoms in a macromolecule Lehn_2007  is huge. To
handle their complexity requires multiscale models. Karplus states
Karplus_2014 ,

> “To develop methods to study complex chemical systems, including
> biomolecules, we have to consider the two elements that govern their
> behavior: 1) The potential surface on which the atoms move; and 2) the laws
> of motion that determine the dynamics of the atoms on the potential
> surfaces. …although the laws governing the motions of atoms are quantum
> mechanical, the key realization that made possible the simulation of the
> dynamics of complex systems, including biomolecules, was that a classical
> mechanical description of the atomic motions is adequate in most cases.”

Thus one does not actually solve (6) in this case. A classical background
provides a basis for determining the quantum states, as is generally true when
wavefunction collapse determines quantum outcomesCWC .

How does this relate to biological function? Karplus states further
Karplus_2014 ,

> “First, evolution determines the protein structure, which in many cases,
> though not all, is made up of relatively rigid units that are connected by
> hinges. They allow the units to move with respect to one another. Second,
> there is a signal, usually the binding of a ligand, that changes the
> equilibrium between two structures with the rigid units in different
> positions. …This type of conformational change occurs in many enzymes as an
> essential part of their mechanism. Thus one does not solve the Schrödinger
> equation in this case. A classical background provides a basis for the
> quantum states.

This clearly parallels the situation for digital computers, even though in a
quite different context.

##  5 Quantum Field Theory: Particle Creation and Annihilation

The second key form of downward causation mentioned above (§2.4) is the
creation, modification, or deletion of lower level elements Ellis_2023a . This
is fundamental to both digital computers and biology.

At the quantum level, this is based in the fact that the creation and
annihilation of particles is central to Quantum Field Theory QFT ; Kittel .
These opeartions are implemented by creation and annihilation operators a,
$\textbf{a}^{\dagger}$.

###  5.1 Digital computers

Such creation and annihilation occurs in electron-phonon interactions, as for
example occurs in transistors at Level T1, as just discussed. This is occurs
via creation and annihilation operators Ellis_Drossel_computers ; Transistor-
Create .

###  5.2 Biology

Creation, alteration, and deletion of lower level elements is a key feature of
biology Ball_2023  for example in developmental processes Carroll  controlled
by gene regulatory networks Karlebach_genes ; GRN ; GRN1 ; GRN2  that create
proteins as needed by the higher level physiological context. At the quantum
level this is occurs via creation and annihilation operators as in molecular
structure calculations Molecule-Create ; Molecule-Create1 .

It also happens in photosynthesis photosynthesis ; rhodopsin  where photons
are absorbed and electrons emitted.

###  5.3 When does it happen?

When do creation/annihilation operators act? Standard QFT texts emphasize
existence of these operators, but do not make clear when they change outcomes
QFT . It seems likely that this timing is controlled downwardly by a classical
context, as suggested by these examples.

Hypothesis: The action of creation and annihilation operators in QFT is
controlled in a downward way by the physical context, as occurs generically in
the cases of Contextual Wavefunction Collapse CWC , and specifically in the
quantum physics underlying digital computers Ellis_Drossel_computers  and
macromolecular dynamics Karplus_2014 .

##  6 Open systems and the arrow of time

The third form of downward causation mentioned above (§2.4) is ongoing
interaction with the environment.

Open systems need a heat sink, and this is provided by the dark night sky
Penrose_Road ; Penrose_Fashion  and indeed by the low temperature of the sky
in all directions but that of the Sun during the day. This enables heat baths
to radiate low grade energy into the sky. The existence of a dark night sky is
called Olber’s Paradox, because in static universes the entire sky potentially
would be at the same temperature as the surfaces of stars Bondi . The solution
to this paradox is that the universe is expanding and evolving Harrison ;
Ellis_Sciama . Its thermal history results in the present cosmological
background radiation temperature of $2.73K$ Dod03 ; Peter_Uzan . But because
of the greenhouse effect greenhouse , the sky on Earth is at an effective
temperature of about $15K$ \- which is what life has adapted to.

Heat baths cannot be described by a many particle wave function Drossel_2017
but are a crucial link between an open system (which all digital computers and
living systems are) and the environment. They are the way that the
thermodynamic arrow of time (determined by the cosmological direction of time
EBU ; Ellis_Drossel_time  due to the expansion of the universe Dod03 ;
Peter_Uzan ) underlies the arrows of time that occur in digital computers and
in biology.

##  7 Outcomes, How did they get that way, and the key role of values

This paper has discussed how quantum physics processes are shaped by downward
causation in both the cases of digital computers and biology, with three key
features enabling this. They are, time dependent constraints affecting the
potential term in the Hamiltonian, creation and deletion of electrons via
quantum field theory effects, and interactions with heat baths playing a key
role in wave function collapse and thereby introducing a quantum arrow of time
aligned with the cosmological direction of time.

This section emphasizes, 1. The relation to other aspect of quantum physics
often discussed, 2. What is the proposal? What have we gained? What have we
lost?, 3. How digital computers and humans got to be what they are, and 4. How
the dynamics discussed here regarding digital computers necessarily reflects
the purposes and values of the people that create them.

###  7.1 The usual suspects

This paper has not emphasized some topics common in discussions of quantum
biology (for references, see Ellis_2023 ). They are,

  * •

Quantum tunneling In biology, there are various cases where tunneling matters
tunnel0 ; tunnel1 ; tunnel2 ; tunnel3 . It is also important in digital
computers using tunneling transistors tunneling_transistor . From the
perspective of this paper, the key point is that tunneling is always
calculated relative to some classical structure, which may be modelled in
various ways.

  * •

Superposition will take place within any wave-function domain ${\cal
U}_{\alpha}$, as defined in §3.2. It will be ended by Contextual Wavefunction
Collapse through suitable processes of downward causation (§3.4).

  * •

Entanglement can also take place within any wave-function domain ${\cal
U}_{\alpha}$, however it will normally rapidly be destroyed by decoherence,
for example this will happen in the brain decoherence . It is not important in
the functioning of ordinary digital computers, however it is central to
quantum computing with quantum entanglement as a computational resource Steane
. In this case, the struggle is to maintain entanglement reliably, which in
terms of the present viewpoint is the struggle to maintain wave function
domains ${\cal U}_{\alpha}(t)$ as large as needed, for a substantial time.

  * •

Quantum spin is crucial because it underlies the Pauli exclusion principle,
and hence is the basis of the periodic table of the elements Periodic_table0 ;
Periodic_table ; Periodic_table1  which is key to existence and structure of
digital computers and all biology.

###  7.2 What is the proposal? What have we gained? What have we lost?

If one stands back and asks what the core outcome of this analysis is, it can
be briefly summarised as follows.

What is the proposal?

  1. 1.

Quantum physics applies everywhere locally.

  2. 2.

The crucial feature is that a wave function $|\psi\rangle_{\alpha}$ obeying
the Schrödinger equation (5) will generally only hold in a local domain
$({\cal U}_{\alpha})(t)$ where the dynamics is lunitary: thus
$|\psi\rangle_{\alpha}=|\psi\rangle_{\alpha}({\cal U}_{\alpha})(t)$.

  3. 3.

How big is the domain $({\cal U}_{\alpha})(t)$? The issue is not either
physical scale or energy scale: it is whether the dynamics in this domain is
linear or not.

What have we gained?

  * •

The Copenhagen approach: we can assume a classical context within which
quantum effects occur locally, including effectively classical experimental
apparatus;

  * •

Contextual Wavefunction Collapse: this classical context will shape quantum
outcomes in a downward way;

  * •

Classical molecular shape is a result: the basis of molecular biology,

  * •

A quantum arrow of time, linked to the Direction of Time established by the
expanding universe.

What have we lost?

  * •

Schrödinger’s cat: $|\Psi\rangle_{cat}$ does not exist.

  * •

A single wave function $|\Psi\rangle_{brain}$ for a brain,

hence no many minds interpretation and no “Boltzmann brains”

  * •

A single wave function $|\Psi\rangle_{Universe}$ for the entire Universe at
all scales,

hence no Everett interpretation

###  7.3 How did they get to be that way?

The subjects of this discussion (digital computers, life, human beings) are
enormously complex. How did they come into being?

#### Life, and specifically human beings

Life as we know it came to be what it is by a combination of the processes of
(1) evolution through natural selection Ball_2023 ; Mitchell_2023 ;
Noble_Noble_2023 ; Donald , leading to our genetic inheritance, and (2)
developmental processes  Ball_2023 ; Noble_Noble_2023 ; Carroll  selectively
reading the resultant DNA as needed, which is controlled by gene regulatory
networks GRN ; GRN1 ; GRN2 . This combination leads to adaptation to the
environment: a key example of downward causation Campbell_1 .

This process of natural selection does not, as some claim, act only at the
genetic level L5 Noble_Noble_2023 : on the contrary, it shapes all emergent
levels L5 to L9 in Table 2.1 simultaneously so that they work together as
needed Noble_2012 . Thus it selects proteins at level L5 for specific
biological functions such as enabling vision and transporting oxygen in blood
Wagner , physiological systems at level L8 needed to keep us alive such as the
heart pumping blood Noble_2002  and a brain at Level L8 resulting in organisms
with agency emerging at Level L9 Mitchell_2023 . As noted above (§2.2), higher
level needs can be met in multiple ways at lower levels, and will reach down
to shape configurations of atoms, electrons, and protons at Levels L3 and L2
where quantum effects come into play as discussed above. The specific outcomes
at these lower levels are shaped by this downward chaining resulting from
higher level physiological needs.

#### Digital computers

All technology also comes into being by similar evolutionary processes of
trial and error Arthur_2010 . The design process is a careful analysis of
needs and how to meet them design ; design1 , aiming to lead to a design that
integrates all the emergent levels L4 to L9 in Table 2.1 so that they jointly
lead to the desired operational result through an appropriate computer
architecture (Dasgupta_2016 :§5, Mellisinos :§2, Tannen ). It involves the
specific physical shape of the transistors at level L5 (Mellisinos :§1),
choice of the materials used at level L4 and its doping, the design of
immensely complex integrated circuits expressing the logic of operation in its
design at levels L6 to L8. This shapes the computer at all levels by a process
of intelligent design: it is “the science of the artificial” Simon_2019 .

Corresponding to developmental processes in biology are manufacturing
processes in technology such as those for transistors Mellisinos  and
integrated circuits IC . These again again have gone through a careful process
of adaptive selection: trial and error takes place, discarding those that
don’t work and improving those that do. Furthermore factories to produce chips
and computers must be designed and created and organised, with technicians who
understand the process and know how to handle it.

Both processes are examples of abstract causation (§1.2): these explorations
results in plans for the computer structure at all levels. Those plans are
abstract entities that can be realised in many physical forms (in text, on
paper, in drawings, in computer files for example). They result from the
causal power of human thought, enabled by brains with agency and symbolic
understanding that came into being by biological evolutionary processes
Mitchell_2023 . Obviously digital computers would not exist if this were not
the case. They are what they are because they were designed and manufactured
to be what they are.

#### Assembly Theory

In both cases, what can be done at each time depends on what has already
happened and what components are available: the principles of Assembly Theory
must be fulfilled assembly ; assembly1 . At each moment one takes the next
step from what has already been achieved, in a step by step process. Thus
multicellular organisms can only come into being after single cell organisms
exist; integrated circuits can only come into being after transistors exist;
and so on. Thereby the historical nature of the evolutionary process gets
embodied in the physical product.

###  7.4 The key role of values

The final issue that needs to be emphasized is the key role of values in
determining these outcomes. As stated by Dasgupta (Dasgupta_2016 :31),

> “All artefacts - engineering and computational - have something in common:
> they are the products of human thought, human goals, human needs, human
> desires. Artefacts are purposive: they reflect the goals of their creators.
> … artefacts have entered the world reflecting human needs and goals. It is
> not enough to ask what are the laws and principles governing the structure
> and behaviour of computational artefacts (or for that matter, of pyramids,
> particle accelerators and kitchen knives) if we then ignore the reason for
> their existence.”

Thus in particular, meaning and purpose underlie computation
Ellis_Drossel_computers . The lives of individuals are shaped by their value
systems, which constrain and strongly influence their motivation and actions
Noble_Ellis . Because individuals in turn influence social structures Berger
which have agency Elder_Vass , the functioning of social structures too is
shaped by values to a considerable degree Ellis_Noble  and this has important
real world outcomes Carney .

This then goes on to influence the operation of human creations such as
digital computers, as was emphasized in Ellis_Drossel_computers . Computer
program and associated algorithms have a set of values associated with them
that reflect the purposes and values of the persons writing the program. Their
purpose might for example be to write malware intended to infect someone’s
computer and extract a ransom from them; or it might be to write software used
by an international agency for tracking where floods have occurred and relief
supplies are needed. The very different set of values in these two cases
reflect those of their creators. This is now a key issue because of the way
algorithms used in social media have been designed to maximise profits,
whatever damage is caused.

In the end, the sequence of operations at the underlying quantum physics level
in digital computers are an expression of a specific set of values; just are
the quantum chemistry operations underlying the functioning of our brains.

Acknowledgements: I thank Markus Gabriel and Dean Rickles for useful
discussions, and Barbara Drossel for collaborations leading to papers
Ellis_Drossel_computers , CWC  and Ellis_Drossel_time  that are foundational
to this project.

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# Lie groups as four-dimensional special complex manifolds with Norden metric

Marta Teofilova

###### Abstract

An example of a four-dimensional special complex manifold with Norden metric
of constant holomorphic sectional curvature is constructed via a two-
parametric family of solvable Lie algebras. The curvature properties of the
obtained manifold are studied. Necessary and sufficient conditions for the
manifold to be isotropic Kählerian are given.

2000 Mathematics Subject Classification: 53C15, 53C50.

Keywords: almost complex manifold, Norden metric, Lie group, Lie algebra.

##  1 Preliminaries

Let $(M,J,g)$ be a $2n$-dimensional almost complex manifold with Norden
metric, i.e. $J$ is an almost complex structure and $g$ is a metric on $M$
such that:

(1.1) $J^{2}x=-x,\qquad g(Jx,Jy)=-g(x,y),\qquad x,y\in\mbox{\got X}(M).$

The associated metric $\widetilde{g}$ of $g$ on $M$, given by
$\widetilde{g}(x,y)=g(x,Jy)$, is a Norden metric, too. Both metrics are
necessarily neutral, i.e. of signature $(n,n)$.

If $\nabla$ is the Levi-Civita connection of $g$, the tensor field $F$ of type
$(0,3)$ is defined by

(1.2) $F(x,y,z)=g\left((\nabla_{x}J)y,z\right)$

and has the following symmetries

(1.3) $F(x,y,z)=F(x,z,y)=F(x,Jy,Jz).$

Let $\left\\{e_{i}\right\\}$ ($i=1,2,\ldots,2n$) be an arbitrary basis of
$T_{p}M$ at a point $p$ of $M$. The components of the inverse matrix of $g$
are denoted by $g^{ij}$ with respect to the basis $\left\\{e_{i}\right\\}$.
The Lie 1-forms $\theta$ and $\theta^{\ast}$ associated with $F$ are defined
by, respectively

(1.4) $\theta(x)=g^{ij}F(e_{i},e_{j},x),\qquad\theta^{\ast}=\theta\circ J.$

The Nijenhuis tensor field $N$ for $J$ is given by

(1.5) $N(x,y)=[Jx,Jy]-[x,y]-J[Jx,y]-J[x,Jy].$

It is known [4] that the almost complex structure is complex iff it is
integrable, i.e. $N=0$.

A classification of the almost complex manifolds with Norden metric is
introduced in [2], where eight classes of these manifolds are characterized
according to the properties of $F$. The three basic classes:
$\mathcal{W}_{1}$, $\mathcal{W}_{2}$ of _the special complex manifolds with
Norden metric_ and $\mathcal{W}_{3}$ of _the quasi-Kähler manifolds with
Norden metric_ are given as follows:

(1.6)
$\begin{array}[]{l}\mathcal{W}_{1}:F(x,y,z)=\frac{1}{2n}\left[g(x,y)\theta(z)+g(x,z)\theta(y)\right.\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\qquad\qquad\qquad\qquad\quad\left.+g(x,Jy)\theta(Jz)+g(x,Jz)\theta(Jy)\right];\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\mathcal{W}_{2}:F(x,y,Jz)+F(y,z,Jx)+F(z,x,Jy)=0,\quad\theta=0\hskip
5.05942pt\Leftrightarrow\hskip 5.05942ptN=0,\quad\theta=0;\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \mathcal{W}_{3}:F(x,y,z)+F(y,z,x)+F(z,x,y)=0.\end{array}$

The class $\mathcal{W}_{0}$ of _the Kähler manifolds with Norden metric_ is
defined by $F=0$ and is contained in each of the other classes.

Let $R$ be the curvature tensor of $\nabla$, i.e.
$R(x,y)z=\nabla_{x}\nabla_{y}z-\nabla_{y}\nabla_{x}z-\nabla_{\left[x,y\right]}z$.
The corresponding (0,4)-type tensor is defined by
$R(x,y,z,u)=g\left(R(x,y)z,u\right)$. The Ricci tensor $\rho$ and the scalar
curvatures $\tau$ and $\tau^{\ast}$ are given by:

(1.7)
$\begin{array}[]{c}\rho(y,z)=g^{ij}R(e_{i},y,z,e_{j}),\qquad\tau=g^{ij}\rho(e_{i},e_{j}),\qquad\tau^{\ast}=g^{ij}\rho(e_{i},Je_{j}).\end{array}$

A tensor of type (0,4) is said to be _curvature-like_ if it has the properties
of $R$. Let $S$ be a symmetric (0,2)-tensor. We consider the following
curvature-like tensors:

(1.8) $\begin{array}[]{l}\psi_{1}(S)(x,y,z,u)=g(y,z)S(x,u)-g(x,z)S(y,u)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\phantom{\psi_{1}(S)(x,y,z,u)}+g(x,u)S(y,z)-g(y,u)S(x,z),\vskip 3.0pt plus
1.0pt minus 1.0pt\\\
\pi_{1}=\frac{1}{2}\psi_{1}(g),\quad\pi_{2}(x,y,z,u)=g(y,Jz)g(x,Ju)-g(x,Jz)g(y,Ju).\end{array}$

It is known that on a pseudo-Riemannian manifold $M$ ($\dim M=2n\geq 4$) the
conformal invariant Weyl tensor has the form

(1.9)
$W(R)=R-\frac{1}{2(n-1)}\big{\\{}\psi_{1}(\rho)-\frac{\tau}{2n-1}\pi_{1}\big{\\}}.$

Let $\alpha=\left\\{x,y\right\\}$ be a non-degenerate $2$-plane spanned by the
vectors $x,y\in T_{p}M$, $p\in M$. The sectional curvature of $\alpha$ is
given by

(1.10) $k(\alpha;p)=\frac{R(x,y,y,x)}{\pi_{1}(x,y,y,x)}.$

We consider the following basic sectional curvatures in $T_{p}M$ with respect
to the structures $J$ and $g$: _holomorphic sectional curvatures_ if
$J\alpha=\alpha$ and _totally real sectional curvatures_ if
$J\alpha\perp\alpha$ with respect to $g$.

The square norm of $\nabla J$ is defined by $\left\|\nabla
J\right\|^{2}=g^{ij}g^{kl}g\left((\nabla_{e_{i}}J)e_{k},(\nabla_{e_{j}}J)e_{l}\right)$.
Then, by (1.2) we get

(1.11) $\left\|\nabla J\right\|^{2}=g^{ij}g^{kl}g^{pq}F_{ikp}F_{jlq},$

where $F_{ikp}=F(e_{i},e_{k},e_{p})$.

An almost complex manifold with Norden metric satisfying the condition
$\left\|\nabla J\right\|^{2}=0$ is called an _isotropic Kähler manifold with
Norden metric_ [3].

##  2 Almost complex manifolds with Norden metric of constant holomorphic
sectional curvature

In this section we obtain a relation between the vanishing of the holomorphic
sectional curvature and the vanishing of $\left\|\nabla J\right\|^{2}$ on
$\mathcal{W}_{2}$-manifolds and $\mathcal{W}_{3}$-manifolds with Norden
metric.

In [1] it is proved the following

Theorem A. ([1]) _An almost complex manifold with Norden metric is of
pointwise constant holomorphic sectional curvature if and only if_

(2.1)
$\begin{array}[]{l}3\\{R(x,y,z,u)+R(x,y,Jz,Ju)+R(Jx,Jy,z,u)+R(Jx,Jy,Jz,Ju)\\}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-R(Jy,Jz,x,u)+R(Jx,Jz,y,u)-R(y,z,Jx,Ju)+R(x,z,Jy,Ju)\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ -R(Jx,z,y,Ju)+R(Jy,z,x,Ju)-R(x,Jz,Jy,u)+R(y,Jz,Jx,u)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ =8H\\{\pi_{1}+\pi_{2}\\}\end{array}$

_for some_ $H\in FM$ _and all_ $x,y,z,u\in\mathfrak{X}(M)$. _In this case_
$H(p)$ _is the holomorphic sectional curvature of all holomorphic non-
degenerate 2-planes in_ $T_{p}M$, $p\in M$.

Taking into account (1.7) and (1.8), the total trace of (2.1) implies

(2.2) $H(p)=\frac{1}{4n^{2}}(\tau+\tau^{\ast\ast}),$

where $\tau^{\ast\ast}=g^{il}g^{jk}R(e_{i},e_{j},Je_{k},Je_{l})$.

In [5] we have proved that on a $\mathcal{W}_{2}$-manifold it is valid

(2.3) $\left\|\nabla J\right\|^{2}=2(\tau+\tau^{\ast\ast}),$

and in [3] it is proved that on a $\mathcal{W}_{3}$-manifold

(2.4) $\left\|\nabla J\right\|^{2}=-2(\tau+\tau^{\ast\ast}).$

Then, by Theorem A, (2.2), (2.3) and (2.4) we obtain

######  Theorem 2.1.

Let $(M,J,g)$ be an almost complex manifold with Norden metric of pointwise
constant holomorphic sectional curvature $H(p)$, $p\in M$. Then

(i)

$\left\|\nabla J\right\|^{2}=8n^{2}H(p)$ if $(M,J,g)\in\mathcal{W}_{2}$_;_

(ii)

$\left\|\nabla J\right\|^{2}=-8n^{2}H(p)$ if $(M,J,g)\in\mathcal{W}_{3}$.

Theorem 2.1 implies

######  Corollary 2.2.

Let $(M,J,g)$ be a $\mathcal{W}_{2}$-manifold or $\mathcal{W}_{3}$-manifold of
pointwise constant holomorphic sectional curvature $H(p)$, $p\in M$. Then,
$(M,J,g)$ is isotropic Kählerian iff $H(p)=0$.

In the next section we construct an example of a $\mathcal{W}_{2}$-manifold of
constant holomorphic sectional curvature.

##  3 Lie groups as four-dimensional $\mathcal{W}_{2}$-manifolds

Let g be a real 4-dimensional Lie algebra corresponding to a real connected
Lie group $G$. If $\left\\{X_{1},X_{2},X_{3},X_{4}\right\\}$ is a basis of
left invariant vector fields on $G$ and $[X_{i},X_{j}]=C_{ij}^{k}X_{k}$
($i,j,k=1,2,3,4$) then the structural constants $C_{ij}^{k}$ satisfy the anti-
commutativity condition $C_{ij}^{k}=-C_{ji}^{k}$ and the Jacobi identity
$C_{ij}^{k}C_{ks}^{l}+C_{js}^{k}C_{ki}^{l}+C_{si}^{k}C_{kj}^{l}=0$.

We define an almost complex structure $J$ and a compatible metric $g$ on $G$
by the conditions, respectively:

(3.1) $JX_{1}=X_{3},\quad JX_{2}=X_{4},\quad JX_{3}=-X_{1},\quad
JX_{4}=-X_{2},$ (3.2)
$\begin{array}[]{l}g(X_{1},X_{1})=g(X_{2},X_{2})=-g(X_{3},X_{3})=-g(X_{4},X_{4})=1,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ g(X_{i},X_{j})=0,\quad i\neq j,\quad
i,j=1,2,3,4.\end{array}$

Because of (1.1), (3.1) and (3.2) $g$ is a Norden metric. Thus, $(G,J,g)$ is a
4-dimensional almost complex manifold with Norden metric.

From (3.2) it follows that the well-known Levi-Civita identity for $g$ takes
the form

(3.3)
$2g(\nabla_{X_{i}}X_{j},X_{k})=g([X_{i},X_{j}],X_{k})+g([X_{k},X_{i}],X_{j})+g([X_{k},X_{j}],X_{i}).$

Let us denote $F_{ijk}=F(X_{i},X_{j},X_{k})$. Then, by (1.2) and (3.3) we have

(3.4)
$\begin{array}[]{l}2F_{ijk}=g\big{(}[X_{i},JX_{j}]-J[X_{i},X_{j}],X_{k}\big{)}+g\big{(}J[X_{k},X_{i}]-[JX_{k},X_{i}],X_{j}\big{)}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\qquad\qquad\qquad\qquad+g\big{(}[X_{k},JX_{j}]-[JX_{k},X_{j}],X_{i}\big{)}.\end{array}$

According to (1.6) to construct an example of a $\mathcal{W}_{2}$-manifold we
need to find sufficient conditions for the Nijenhuis tensor $N$ and the Lie
1-form $\theta$ to vanish on $\mathfrak{g}$.

By (1.2), (1.5), (3.2) and (3.4) we compute the essential components
$N_{ij}^{k}$ ($N(X_{i},X_{j})=N_{ij}^{k}X_{k}$) of $N$ and
$\theta_{i}=\theta(X_{i})$ of $\theta$, respectively, as follows:

(3.5)
$\begin{array}[]{l}N_{12}^{1}=C_{34}^{1}-C_{12}^{1}-C_{23}^{3}+C_{14}^{3},\qquad\theta_{1}=2C_{13}^{1}-C_{12}^{4}+C_{14}^{2}+C_{23}^{2}-C_{34}^{4},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
N_{12}^{2}=C_{34}^{2}-C_{12}^{2}-C_{23}^{4}+C_{14}^{4},\qquad\theta_{2}=2C_{24}^{2}+C_{12}^{3}+C_{14}^{1}+C_{23}^{1}+C_{34}^{3},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
N_{12}^{3}=C_{34}^{3}-C_{12}^{3}+C_{23}^{1}-C_{14}^{1},\qquad\theta_{3}=2C_{13}^{3}+C_{12}^{2}+C_{14}^{4}+C_{23}^{4}+C_{34}^{2},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
N_{12}^{4}=C_{34}^{4}-C_{12}^{4}+C_{23}^{2}-C_{14}^{2},\qquad\theta_{4}=2C_{24}^{4}-C_{12}^{1}+C_{14}^{3}+C_{23}^{3}-C_{34}^{1}.\end{array}$

Then, (1.6) and (3.5) imply

######  Theorem 3.1.

Let $(G,J,g)$ be a 4-dimensional almost complex manifold with Norden metric
defined by (3.1) and (3.2). Then, $(G,J,g)$ is a $\mathcal{W}_{2}$-manifold
iff for the Lie algebra $\mathfrak{g}$ of $G$ are valid the conditions _:_

(3.6)
$\begin{array}[]{l}C_{13}^{1}=C_{12}^{4}-C_{23}^{2}=C_{34}^{4}-C_{14}^{2},\qquad
C_{13}^{3}=-\left(C_{12}^{2}+C_{23}^{4}\right)=-\left(C_{14}^{4}+C_{34}^{2}\right),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
C_{24}^{4}=C_{12}^{1}-C_{14}^{3}=C_{34}^{1}-C_{23}^{3},\qquad
C_{24}^{2}=-\left(C_{12}^{3}+C_{14}^{1}\right)=-\left(C_{23}^{1}+C_{34}^{3}\right),\end{array}$

where $C_{ij}^{k}$ ($i,j,k=1,2,3,4$) satisfy the Jacodi identity.

One solution to (3.6) and the Jacobi identity is the 2-parametric family of
solvable Lie algebras $\mathfrak{g}$ given by

(3.7) $\mathfrak{g}:\begin{array}[]{l}[X_{1},X_{2}]=\lambda X_{1}-\lambda
X_{2},\qquad[X_{2},X_{3}]=\mu X_{1}+\lambda X_{4},\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ [X_{1},X_{3}]=\mu X_{2}+\lambda X_{4},\qquad[X_{2},X_{4}]=\mu
X_{1}+\lambda X_{3},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ [X_{1},X_{4}]=\mu
X_{2}+\lambda X_{3},\qquad[X_{3},X_{4}]=-\mu X_{3}+\mu
X_{4},\qquad\lambda,\mu\in\mathbb{R}.\end{array}$

Let us study the curvature properties of the $\mathcal{W}_{2}$-manifold
$(G,J,g)$, where the Lie algebra $\mathfrak{g}$ of $G$ is defined by (3.7).

By (3.2), (3.3) and (3.7) we obtain the components of the Levi-Civita
connection:

(3.8) $\begin{array}[]{ll}\nabla_{X_{1}}X_{2}=\lambda
X_{1}+\mu(X_{3}+X_{4}),&\quad\nabla_{X_{2}}X_{1}=\lambda
X_{2}+\mu(X_{3}+X_{4}),\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\nabla_{X_{3}}X_{4}=-\lambda(X_{1}+X_{2})-\mu
X_{3},&\quad\nabla_{X_{4}}X_{3}=-\lambda(X_{1}+X_{2})-\mu X_{4},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \nabla_{X_{1}}X_{1}=-\lambda
X_{2},\quad\nabla_{X_{2}}X_{2}=-\lambda X_{1},&\quad\nabla_{X_{3}}X_{3}=\mu
X_{4},\quad\nabla_{X_{4}}X_{4}=\mu X_{3},\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \nabla_{X_{1}}X_{3}=\nabla_{X_{1}}X_{4}=\mu
X_{2},&\quad\nabla_{X_{2}}X_{3}=\nabla_{X_{2}}X_{4}=\mu X_{1},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \nabla_{X_{3}}X_{1}=\nabla_{X_{3}}X_{2}=-\lambda
X_{4},&\quad\nabla_{X_{4}}X_{1}=\nabla_{X_{4}}X_{2}=-\lambda
X_{3}.\end{array}$

Taking into account (3.4) and (3.7) we compute the essential non-zero
components of $F$:

(3.9)
$\begin{array}[]{l}F_{114}=-F_{214}=F_{312}=\frac{1}{2}F_{322}=\frac{1}{2}F_{411}=F_{412}=-\lambda,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
F_{112}=\frac{1}{2}F_{122}=\frac{1}{2}F_{211}=F_{212}=-F_{314}=F_{414}=\mu.\end{array}$

The other non-zero components of $F$ are obtained from (1.3).

By (1.11) and (3.9) for the square norm of $\nabla J$ we get

(3.10) $\left\|\nabla J\right\|^{2}=-32(\lambda^{2}-\mu^{2}).$

Further, we obtain the essential non-zero components
$R_{ijks}=R(X_{i},X_{j},X_{k},X_{s})$ of the curvature tensor $R$ as follows:

(3.11)
$\begin{array}[]{l}-\frac{1}{2}R_{1221}=-R_{1341}=-R_{2342}=R_{3123}=\frac{1}{2}R_{3443}=R_{4124}=\lambda^{2}+\mu^{2},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
R_{1331}=R_{1441}=R_{2332}=R_{2442}=-R_{1324}=-R_{1423}=\lambda^{2}-\mu^{2},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ R_{1231}=R_{1241}=R_{2132}=R_{2142}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
=-R_{3143}=-R_{3243}=-R_{4134}=-R_{4234}=2\lambda\mu.\end{array}$

Then, by (1.7) and (3.11) we get the components $\rho_{ij}=\rho(X_{i},X_{j})$
of the Ricci tensor and the values of the scalar curvatures $\tau$ and
$\tau^{\ast}$:

(3.12)
$\begin{array}[]{l}\rho_{11}=\rho_{22}=-4\lambda^{2},\qquad\qquad\quad\rho_{33}=\rho_{44}=-4\mu^{2},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\rho_{12}=\rho_{34}=-2(\lambda^{2}+\mu^{2}),\qquad\rho_{13}=\rho_{14}=\rho_{23}=\rho_{24}=4\lambda\mu,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\tau=-8(\lambda^{2}-\mu^{2}),\qquad\qquad\quad\hskip
3.61371pt\tau^{\ast}=16\lambda\mu.\end{array}$

Let us consider the characteristic 2-planes $\alpha_{ij}$ spanned by the basic
vectors $\\{X_{i},X_{j}\\}$: totally real 2-planes - $\alpha_{12}$,
$\alpha_{14}$, $\alpha_{23}$, $\alpha_{34}$ and holomorphic 2-planes -
$\alpha_{13}$, $\alpha_{24}$. By (1.10) and (3.11) for the sectional
curvatures of the holomorphic 2-planes we obtain

(3.13) $k(\alpha_{13})=k(\alpha_{24})=-(\lambda^{2}-\mu^{2}).$

Then it is valid

######  Theorem 3.2.

The manifold $(G,J,g)$ is of constant holomorphic sectional curvature.

Using (1.9), (3.11) and (3.12) for the essential non-zero components
$W_{ijks}=W(X_{i},X_{j},X_{k},X_{s})$ of the Weyl tensor $W$ we get:

(3.14)
$\begin{array}[]{l}\frac{1}{2}W_{1221}=W_{1331}=W_{1441}=W_{2332}=W_{2442}=\frac{1}{2}W_{3443}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
=-\frac{1}{3}W_{1324}=-\frac{1}{3}W_{1423}=\frac{1}{3}(\lambda^{2}-\mu^{2}).\end{array}$

Finally, by (1.9), (3.10), (3.12), (3.13) and (3.14) we establish the
truthfulness of

######  Theorem 3.3.

The following conditions are equivalent _:_

(i)

$(G,J,g)$ is isotropic Kählerian _;_

(ii)

$|\lambda|=|\mu|$_;_

(iii)

$\tau=0$_;_

(iv)

$(G,J,g)$ is of zero holomorphic sectional curvature _;_

(v)

the Weyl tensor vanishes.

(vi)

$R=\frac{1}{2}\psi_{1}(\rho)$.

## References

  * [1] G. Djelepov, K. Gribachev, _Generalized $B$-manifolds of constant holomorphic sectional curvature_, Plovdiv Univ. Sci. Works – Math. 23(1) (1985), 125–131.
  * [2] G. Ganchev, A. Borisov, _Note on the almost complex manifolds with a Norden metric_ , Compt. Rend. Acad. Bulg. Sci. 39(5) (1986), 31–34.
  * [3] D. Mekerov, M. Manev, _On the geometry of Quasi-Kähler manifolds with Norden metric_ , Nihonkai Math. J. 16(2) (2005), 89–93.
  * [4] A. Newlander, L. Niremberg, _Complex analytic coordinates in almost complex manifolds_ , Ann. Math. 65 (1957), 391–404.
  * [5] M. Teofilova, _Lie groups as four-dimensional conformal Kähler manifolds with Norden metric_ , In:Topics of Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, eds. S. Dimiev and K. Sekigawa, World Sci. Publ., Hackensack, NJ (2007), 319–326.

Faculty of Mathematics and Informatics,
---
University of Plovdiv,
236 Bulgaria Blvd., Plovdiv 4003, Bulgaria.
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