#  An unorthodox view on quantum mechanics

Gerard ’t Hooft

(Faculty of Science, Department of Physics
Institute for Theoretical Physics
Princetonplein 5, 3584 CC Utrecht
The Netherlands
e-mail<EMAIL_ADDRESS>
internet: http://www.staff.science.uu.nl/~hooft101/ )

Quantum mechanics was definitely one of the most significant and important
discoveries of 20thcentury science. It all began, I think, in the year 1900
when Max Planck published his paper entitled “On the Theory of the Energy
Distribution Law of the Normal Spectrum” [1]. In it, he describes a simple
observation: if one attaches an entropy to the radiation field as if its total
energy came in packages – now called quanta – then the intensity of the
radiation associated to a certain temperature agrees quite well with the
observations. Planck had described his hypothesis as ‘an act of desperation’.
But it was the only one that worked. Head on. Until that time, the best
attempt at performing such a calculation had resulted in the Rayleigh-Jeans
law, an important result at the time, but this law only worked at the lowest
frequencies of the radiation, whereas it failed bitterly at high frequencies /
smaller wavelengths. The Rayleigh-Jeans law would yield a badly divergent,
hence meaningless, expression for the intensity of the radiation emitted at
high frequencies.

The history of what happened next, is well-known, and has been recounted in
excellent reports [2], which, I think, do not need to be repeated here. It was
realised that all oscillatory motion apparently comes in energy packets, which
seem to behave as particles, and that the converse should also be true: all
particles with definite energies must be associated to waves. This all
culminated in the _theory of quantum mechanics._ The year 1926 provided a new
landmark: Erwin Schrödinger’s equation [3]. The title of my essay was not
intended to cast doubt on this equation; the Schrödinger equation has been and
still is the pivotal equation on which much of physics and all of chemistry is
based, and I am in awe of it just as many researchers before and after me are.

Indeed, in his original paper, Schrödinger went quite far in discussing
Hamilton’s principle, boundary conditions, the hydrogen atom and the
electromagnetic transitions from one energy level to the next. The
‘confusions’ I am referring to arose when more and more researchers began to
dispute the question how the equation is supposed to be interpreted. Why is it
that positions and velocities of particles at one given moment cannot be
calculated, or even defined unambiguously?

We know very well how to use the equation. The properties of atoms, molecules,
elementary particles and the forces between all of these can be derived with
perplexing accuracy using it. The way the equation is used is nothing to
complain about, but what exactly does it say?

The first question one may rightfully ask, and that has been asked by many
researchers and their students, is:

  * _What do these wave functions represent?_ In particular the ones that are _not_ associated to photons (the energy packets of the electromagnetic field, which we think we understand very well). What do those waves stand for that are associated to electrons, or other elementary particles, or even molecules and larger things, including cats, and eventually, physicists? What happens to its wave function when you actually observe a particle?

One extremely useful observation was made by Max Born [4]: The absolute square
of a wave function, at some spot in position space, must simply stand for the
_probability_ to find the particle there. This made a lot of sense, and it was
rightly adopted as a useful recipe for dealing with this equation. But then,
many more questions were asked, many of them very well posed, but the answers
sound too ridiculous to be true, and, as I shall try to elucidate, they _are_
too ridiculous to be true. The truth is much simpler to grasp conceptually,
even if the mathematics needed to back this up can be delicate. This is
astonishing. Almost a full century has passed since the equation was written
down, and fierce discussions have been held, quite a few standpoints were
vigorously defended and equally vigorously attacked. We still do not know what
or whom to believe, but it still goes on, while others get irritated by all
this display of impotence [5]. Why is it that we still do not agree? I think I
know the answers, but almost _everyone_ disagrees with me.

In fact, one possible reply is that one could decide to ignore the question.
Dirac for instance, advised not to ask questions that cannot be answered by
any experiment. Such questions are of secondary importance. We know precisely
how to use Schrödinger’s equation; all that scientists have to do is insert
the masses and coupling parameters of all known particles into the equation,
and calculate. What else can you ask for? Many of my colleagues decided to be
strictly ‘agnostic’ about the interpretation, which is as comfortable a
position to take as what is was for 19thcentury scientists to stay ‘agnostic’
about the existence of atoms.

But yes indeed, there is something else. What _are_ those masses and coupling
strengths? Do particles exist that we have not yet been able to detect? Isn’t
it the scientist’s job to make predictions about things we have not yet been
able to unravel? These are questions that are haunting us physicists. We have
arrived at a splendid theory that accounts for almost anything that could be
observed experimentally. It is called the _Standard Model_ of the subatomic
particles. But this model also tells us that particles and forces may exist
that we could not have detected today. Can we produce any theory that suggests
what one might be able to find, in some distant future? And as of all those
particles and forces that we do know about, is there a theory that _explains_
all their details?

Today’s theories give us little to hang on to. This is why it is so important
to extend our abilities to do experiments as far as we can, Recently,
audacious plans have been unfolded by the European particle physics laboratory
CERN, for building a successor for its highly successful Large Hadron Collider
(LHC). While the LHC has provided for strong evidence supporting the validity
of the Standard Model up to the TeV domain, theoreticians find it more and
more difficult to understand why this model can be all there is to describe
what happens further beyond that scale. There must be more, but our
theoretical reasoning leads to more questions doubting the extent to which
this model can be regarded as ‘natural’ when more of the same particles with
higher energies are included, while the existence of totally new particles
would be denied.

Inspired by what historians of science are telling us about similar situations
in the past history of our field, investigators are hoping for a ‘paradigm
shift’. However, while it is easy to postulate that we ‘are doing something
wrong’, most suggestions for improvement are futile; suggesting that the
Standard Model would be ‘wrong’ is clearly not going to help us. The ‘Future
Circular Collider’ is a much better idea; it will be an accelerator with
circumference of about 100 km, being able to reach a c.m. collision energy of
100 TeV. The importance of such a device is that it will provide a crucial
background forcing theoreticians to keep their feet on the ground: if you have
a theory, it better agree with the newest experimental observations.

Ideally, we should have both experiments _and_ theories. The Standard Model
itself contains more than 20 fundamental parameters, ‘constants of nature’,
that are begging for an explanation. The masses of the fermions (‘matter
particles’) are ascribed to the Brout-Englert-Higgs mechanism, just as the
mass of the Higgs particle itself. Yet the Brout-Englert-Higgs theory does not
tell us why the postulated coupling parameters are such that they generate the
mass spectrum that has been measured, numbers that vary wildly. Other
interaction parameters (those of the gauge fields, due to the ‘force carrying
particles’), can be explained to some extent, though most details of their
origin are also mysterious. What we see is that the relative strengths of the
force couplings hint towards a further unification at extremely high energies,
though we have not seen any verification of predictions arising from this
idea. There is definitely something that we have not understood.

A notorious and brave approach is called _superstring theory_. Superstring
theory has not lead to new predictions – apart from ones that were not
verified by observations at all, such as the prediction of supersymmetry at
LHC energies. The superstring theory approach has been criticised a lot for
this ‘failure’, but it must be emphasised that other approaches achieved far
less. We can note that, although the Standard Model was not explained at all,
string theory does lead to a zoo of particles that are not so unlike the
Standard Model ones, and many of its practitioners interpret this fact as an
encouragement to reinforce their efforts. All I can add is that the more we
keep staring at the Standard Model’s parameters, the more it seems that
natural explanations should be asked for, and the existing theories do not
seem to converge towards answers.

It is here that I suggest to look at the quantum mechanical nature of all
observed phenomena. Superstring theory has not provided for any explanation as
to what quantum mechanics is about. None of today’s cherished theories are
indicating any evidence for conceivable constraints on the quantum mechanical
nature of their equations.

Imagine that further scrutinising what is called quantum mechanics could lead
to new ideas. Perhaps there are things that we don’t see yet but that we can
speculate about. As for quantum mechanics, let me briefly remind the reader
what the developments were that got us entwined into a messy knot of problems.
Back in 1932, John von Neumann gave a “proof” [6] of the statement that
quantum mechanics cannot be explained in terms of “hidden” degrees of freedom,
which would restore some deterministic interpretation of Schrödinger’s wave
function. His proof was later dismissed. Both the proof and its dismissal
showed that the authors understood the mathematical nature of the equation,
but missed the point that the physical world might be much more complex than
just this one abstract equation, in which all sorts of possible complications
were suppressed.

A very important paper was written by Einstein, Podolsky and Rosen [7],
arguing that indeed some essential ingredient was missing in this quantum
theory. In a nut shell: if you consider an atom emitting two photons at once,
quantum mechanics suggests that you can measure the position of one photon and
the momentum of the other photon, but since the two photons are equal
(‘entangled’, to be precise), these measurements together should give you
information for both photons that would be forbidden by the same quantum
theory.

In 1952, D. Bohm took an original idea of L. de Broglie literally: “the wave
function acts as a pilot, telling the particle where to go” [8].

John S. Bell took up the same question [9], but insisted on restoring the
notion of locality and causality in the interpretation of the wave function;
both Bohm’s theory and the EPR argument describe the events while giving up
the notion of locality. Bell tried to rephrase everything in a powerful
theorem, which would become famous. Starting from assumptions that he
considered to be utterly reasonable, he derived: _you can’t build a theory of
hidden variables that explains what Schrödinger’s equation describes, if you
insist that it be local._

A different avenue was the argument that the wave function just displays
_everything_ that might be happening, including all alternatives: an infinite
set of different universes can all be evolving, together forming an even
grander body later called the ‘multiverse’, or ‘omniverse’. This multiverse
would be like a gigantic tree, every branch, every leaf of it being an entire
universe. This ‘many world theory’ was formulated by Hugh Everett [10] and
strongly advocated by Bryce S. DeWitt [11]. This theory would not change the
equations, and basically, it would take for granted that quantum mechanics can
never predict what will happen, beyond the level of statistical statements.
You will always be surprised as to which world you apparently entered into,
while, with different probabilities, all other possible worlds are realised
somewhere else in the multiverse.

I think they are all mistaken. What the Schrödinger equation is describing is
not exactly what is happening; it merely describes the tip of a gigantic
iceberg, in which most processes happen far beneath the water line. All that
needs to be realised is that, like in _any_ scientific theory, our predictions
concerning the outcomes of experiments cannot be expected to be infinitely
accurate. There are margins of error, due to the fact that we have been using
statistics. The statistical assumptions that went _into_ Schrödinger’s
equations are about variables that give shape to space and time, fluctuating
far too fast to be included in such a way that we can control them. We don’t
control them and this is the reason why we discover that our predictions come
with margins of error, like in any other scientific theory. I think that the
internal mathematical nature of Schrödinger’s equation supports this view, but
there is too much that we do not know at present, so that building models for
what “really” happens is still difficult for us.

This idea indeed would be covered by the name “hidden variables”, but the
hidden variables are far more sophisticated than what the earlier researchers
had been thinking of – I return to the consequences of our use of statistics
shortly.

Some of my readers will object that Bell did emphatically include such hidden
variables in his treatment, but Bell made assumptions that need not be valid.
Indeed, what Von Neumann had done, and what Bell did later, was to construct a
so-called ‘no-go theorem’. They both badly wanted to construct a hidden
variable theory, but when they did not succeed, they decided to close the lid
for good, that is, to _prove_ that such a theory cannot be constructed. Time
and again, however, history of science has shown that no-go theorems do not
say much more than that the avenues inspected by their authors do not lead to
a desired result, so that, all that should have been done is modify the
assumptions.

To obtain a more solid understanding of what quantum mechanics is really
telling us, we have to go back a long way into the history of physics as a
science. What happened was that long sequences of discoveries not only
enriched our knowledge tremendously, they also narrowed down our way of
thinking, so much so that some obvious facts are now considered to be
obsolete, just because, in practice, our streamlined vision has become much
more powerful. Take for instance the way we describe a moving object. We use
_real numbers_ to describe positions, velocities, energies and so on. What are
real numbers? They are a natural generalisation of ‘rational numbers’.
Rational numbers are what you get if you divide an integer by an other
integer. _Integers_ are numbers that you get simply by counting, for instance
the number of steps needed to get across your room. Integers can be stored in
a computer, but before doing so we have to realise that there are limits to
how far we can count, in practice. The simplest integers are called ‘bytes’.
They can vary from 0 to 255 but no further than that. Each byte can be
expressed in terms of eight ‘bits’. Bits can take only two values, which you
can call 0 and 1, or alternatively, \+ and –.

Those real numbers used now, are so powerful that we forget that they are man-
made. Every real number that we use to describe our world would require an
infinite amount of computer space to store it, because it contains an infinite
sequence of digits, all of which can be arbitrary. Is this really the correct
way to indicate distances, velocities and so on? It could be that all
positions and all velocities eventually only require integers to specify their
values. This would mean that particles live in a space-time that is not
strictly continuous but could be what is called a _lattice_.

I am not saying that I ‘know’ that space-time is a lattice. I don’t, but I do
know that possibilities such as this would profoundly change the ways one
could think of a ‘particle’, and with that, the way we should formulate the
laws of physics they obey. In practice it is very hard to formulate laws of
physics for lattice-particles that fit well with the tremendously
sophisticated science that we have today. Schrödinger’s beautiful equation
seems to be what you get if you want to describe particles but at the same
time you want to hang on to those, somewhat artificial, real numbers.

There is another aspect of modern science that has become so self-evident that
researchers fail to notice that the deeper physical laws could deviate from
standard dogma. It is called ‘statistics’. Quantum mechanics gives us ways to
estimate very precisely the statistical behaviour of large numbers of
particles, or large numbers of experiments, or large numbers of observers.
When you consider a large reservoir containing molecules or other particles
moving chaotically, then it is only too tempting to say that the whole lot of
them will be fundamentally unpredictable when we ask for all details of the
motion of every one of the particles individually. There will be exceptional
chance events, and they will be fundamentally unpredictable, with or without
quantum mechanics.

To formulate his theorem, John Bell needed to introduce two hypothetical
observers called Alice and Bob. Alice and Bob both had to choose what they
would measure. They had the _free will_ to decide what to choose. But if we
would have an omnipotent theory that prescribes every move with infinite
precision, such a free will does not exist. Nobody today thinks of physical
laws that also control the outcomes of arbitrary, individual, measurements in
a gas consisting of billions of particles, but that _is_ what we should try to
find. Note that I am not pretending that a theory can be found that ‘predicts’
where every single atom or molecule will be going, but I do think that it is
reasonable to suspect that laws of nature exist that would uniquely fix their
behaviour, while our ability to predict will always be limited by our
knowledge, our computation power, and our understanding.

To many of my readers (the ones who may still be with me), what I just said
sounds very much like letters we receive in our daily mail from amateur
physicists. They are amateurs because they usually exhibit a dismal lack of
knowledge and understanding of modern science. Like many of my colleagues, I
quickly discard such letters, but sometimes they are fun to read. More to the
point, by not knowing how our world has been found to hang together, they
could have bounced into some more independent ways of asking questions.
Indeed, when I think of questions concerning quantum mechanics, I know I have
to make giant leaps of logic. Often, these are giant leaps backwards. What I
really want is use kinds of logic that do not make use of conventional number
systems, or other conventional ways to describe our understanding of physical
laws. But then, before irritating my colleagues with my ‘crackpot’ findings, I
search for the proper connections with real science. This is because I do know
what conventional science says about quantum mechanics, particles, and number
systems. And today I think of ways to attach unconventional views to the
beautiful edifice called the Standard Model.

This is the reason why the discussion about quantum mechanics may be an
important discussion after all. We can’t accept a theory that, in the very
end, only predicts statistical properties of the small particles, but leaves
them free, to some extent, to do whatever they like. If you dig deep enough,
you should find some superior scenario of the dynamical laws, and this new
scenario must carry implications for the way we build models.

The Standard Model does not pose any restrictions concerning the quantum
equations on which it is based, but exactly that might be what we need, to
make the next step in particle theory. Today, the known elementary particles
form a bunch of objects with masses ranging from a few milli-electronVolts
(meV) to somewhere around a TeV (tera-electronVolts), a stretch of some 15
orders of magnitude. Particles heavier than a few TeV cannot be observed with
the machines we have today. There is no fundamental limit to the mass of a
particle until we reach the Planck mass, some 16 orders of magnitude beyond
where we are now. It is the meshes of our lattice that forbid elementary
objects beyond that. This is my iceberg: only one part in $10^{16}$ is visible
above the waterline.

The particles hidden from our view today, would undergo interactions and
oscillations that are so much faster as time proceeds, that we may call them
‘fast hidden variables’. Today’s Schrödinger equation has all fast variables
suppressed; we cannot detect such fast phenomena. Now, I claim that the fast,
invisible phenomena could be described by classical laws, but these movements
are fundamentally unobservable, and we can only deal with what we see. In a
way invisible to us now, the things we see are controlled by fast moving
particles, particles that form the bulk of the iceberg. Today’s science gives
us an equation that can be used to evade those particles, but this has come at
a price: we have hit upon unpredictable behaviour that we should not have been
surprised about: these are the whims of the hidden variables. To my taste,
this way to interpret what the quantum mechanical laws are about, sounds much
more reasonable than assumptions concerning ‘pilot waves’ or ‘multiverses’.
But most importantly, _any_ reference to statistical averaging should be
banned from the initial assumptions because these indeed lead to no-go
theorems such as Bell’s. Experimenters do not have the ‘free will’ to do
something that is not controlled by deterministic laws of nature.

When making such claims, one has to come with mathematical models. The models
we have today are imperfect, but encouraging indications exist that the
Schrödinger equation is exactly what one should expect from such hidden
physical processes [12, 13].

I actually found that it may be important to make yet another step backwards:
we have become used to the utility of _complex_ numbers to describe quantum
wave functions. Why? Some people claim it to be one of those quantum
peculiarities, setting quantum mechanics aside from all classical theories,
even though complex numbers have often also proved useful for calculations in
classical topics. What is a complex number? One of the marvels of modern
science is that we learned how to take the square root of negative real
numbers, but then here also, understanding is often replaced by mysticism.

This is when one forgets how complex numbers are defined. Complex numbers are
simply pairs of real numbers, with rules attached on how to add, subtract,
multiply and divide such pairs of numbers. In quantum theories, one is tempted
to forget what this means: the physical state of a system includes one or more
objects that can come in two forms, represented by the two components of the
complex function that represents a wave function. One can also decide to
describe such components in a more direct manner, avoiding the use of complex
quantities, which after all, are also man-made inventions. Such a direct
language may lead to some useful insights. When restricting ourselves to real
wave functions only, one finds that the quantum Hamiltonian is an
antisymmetric (real) matrix. Finding eigenvectors and eigenvalues of matrices
is common practice for easing our calculations, but here, antisymmetry would
force us to split the states into pairs again, leading us straight back to
those useful complex numbers.

The mathematical insights gained through this brief excursion may turn out to
be very useful. Quantum mechanical models with a deeper, deterministic basis
cannot support the use of arbitrary real numbers as input parameters. This
means that constants such as masses and coupling strengths of particles cannot
be assumed to be any real number. In some intricate way, these must be
expressed in terms of integer numbers only. This suggests that one might hit
upon special types of field theories where the input parameters can be
calculated.

## References

  * [1] M. Planck, Verhandl. Dtsch. phys. Ges., 2, 237 (1900), _On the Theory of the Energy Distribution Law of the Normal Spectrum_ , English translation from “The Old Quantum Theory,” ed. by D. ter Haar, Pergamon Press,1967, p. 82.
  * [2] A. Pais, _Inward bound: of matter and forces in the physical world,_ Oxford University Press 1986.
  * [3] E. Schrödinger, _An Undulatory Theory of the Mechanics of Atoms and Molecules_ , Phys. Rev. 28, 1049 – Published 1 December 1926.
  * [4] M. Born, _Zur Quantenmechanik der Stoßvorgänge_ (German) [Toward the quantum mechanics of collision processes], j-z-physik, 37 (1926) 863–867; id., j-z-physik, 38(1026) 803–827.
  * [5] N.G. van Kampen, _Ten theorems about quantum mechanical measurements,_ Physica A 153 (1988) 97
  * [6] J. von Neumann, _Mathematical Foundations of Quantum Mechanics_ (German edition 1932), Princeton: Princeton University Press.
  * [7] A. Einstein, B. Podolsky and N. Rosen, Can Quantum mechanical description of physical reality be considered complete?, Phys. Rev. 47 (1935) 777; M. Jammer, _The conceptual Development of Quantum Mechanics_ , (Mc. Graw-Hill, 1966);
  * [8] D. Bohm, _A Suggested Interpretation of the Quantum Theory in Terms of “Hidden Variables, I”_. Phys. Rev. 85 (1952) 166–179; id., _II_. Phys. Rev. 85 (1952) 180–193.
  * [9] J. S. Bell, _On the Einstein Podolsky Rosen Paradox_ , Physics 1 (1964) 195.
  * [10] H. Everett, _Relative State Formulation of Quantum Mechanics_ , Reviews of Mod. Phys. 29 (1957) 454.
  * [11] B.S. DeWitt, in _The Many-Worlds Interpretation of Quantum Mechanics_ , ed. B.S. DeWitt and N. Graham. Princeton University Press, 1973.
  * [12] G. ’t Hooft, _Deterministic Quantum Mechanics: the Mathematical Equations,_ Subm. to the Article Collection: ”Towards a Local Realist View of the Quantum Phenomenon”, Frontiers Research Topic, ed. A. Casado et al, arxiv:2005.06374.
  * [13] G. ’t Hooft, _Explicit construction of Local Hidden Variables for any quantum theory up to any desired accuracy_ arxiv:2103.04335[quant-ph].


# Amplification of arbitrary frequency chirps of pulsed light on nanosecond
timescales

B.S. Clarke  P.L. Gould  Department of Physics, University of Connecticut,
Storrs, Connecticut 06269, United States<EMAIL_ADDRESS>
###### Abstract

We have developed a system for producing amplified pulses of frequency-chirped
light at 780 nm on nanosecond timescales. The system starts with tunable cw
laser light and employs a pair of fiber-based phase modulators, a
semiconductor optical amplifier, and a tapered amplifier to achieve chirp
rates exceeding 3 GHz/10 ns and peak powers greater than 1 W. Driving the
modulators with an arbitrary waveform generator enables arbitrary chirp
shapes, such as two-frequency linear chirps. We overcome the optical power
limitations of the modulators by duty cycling and avoid unseeded operation of
the tapered amplifier by multiplexing the chirped pulses with “dummy” light
from a separate diode laser.

††preprint: AIP/123-QED

##  I Introduction

Since lasers are ubiquitous in modern science and technology, manipulation of
their output light fields is critically important [1]. Among the properties
which can be controlled are amplitude (or intensity), and phase (or
frequency). The time scales for control cover a wide range, depending on the
application. High-speed modulation is generally done in the frequency domain,
due to electronic speed limitations. On the other hand, modulation at lower
speeds would require very high frequency resolution, and is thus better suited
to control in the time domain. For frequency-domain control with a pulse
shaper, short optical pulses have their broad bandwidth spatially dispersed,
enabling the amplitudes and phases of the various frequency components to be
individually manipulated. These components are then reassembled, yielding a
modified pulse [2, 3]. In the time-domain, control can be accomplished using
acousto-optical modulators (AOMs) [4] and electro-optical modulators (EOMs)
driven by RF waveforms. The former can provide pulses of light with durations
down to 10s of ns and frequency shifts typically in the 100 MHz range. The
latter can modulate either amplitude (intensity modulators) or frequency
(phase modulators). Fiber-based EOMs utilize a waveguide of electro-optical
material (e.g., lithium niobate) and can operate at speeds up to 10s of GHz
with relatively low drive voltages. Modulation in the 10-100 ps range is
difficult to achieve with either frequency-domain or time-domain techniques.

There are several ways one can produce frequency-chirped light in the time
domain. One method is to vary a frequency-controlling parameter of the laser
itself in real time. Direct modulation of the injection current of an
external-cavity diode laser (ECDL) is one such example, though finite system
response limits the modulation speed. In addition, ramping the injection
current leads to undesirable intensity modulation, which can be minimized by
using the frequency-chirped light to injection lock a separate free-running
diode laser (FRDL) [5]. Instead of modulating the laser itself, one can
produce the frequency variation by modulating the light externally, after it
has left the laser [6, 7, 8, 9, 10]. This is the path we have followed here,
using fiber-based electro-optical phase modulators (PMs) driven by an
arbitrary waveform generator (AWG) to produce frequency-chirped light at 780
nm. These chirped pulses are then double-stage amplified by a semiconductor
optical amplifier (SOA) followed by a tapered amplifier (TA). Previous work
along these lines [6] utilized a PM in a multipass fiber loop, followed by
injection locking of a FRDL, to produce pulses of arbitrary frequency-chirped
light. The pulse repetition rate was limited by the fiber loop time, and the
output power was limited to that of the FRDL. Another related scheme [8] used
a combination of fiber-based phase and intensity modulators, followed by a
double-pass tapered amplifier with a fiber delay line. The system described
here has several advantages: it avoids risks of optical damage associated with
a double-pass TA; the repetition rate is not limited by fiber delays; and the
peak output power of the chirped pulses can exceed 1 W.

There are many applications of chirped pulses for which the system described
here should prove useful. In atomic and molecular systems, chirped light can
provide efficient population transfer [11, 12, 13]. In the adiabatic regime,
the transfer efficiency can approach 100% and can be robust with respect to
pulse parameters. Competition with spontaneous emission from excited states,
which typically occurs on the 10s of ns timescale, requires pulses and chirps
of short duration. Specific applications employing chirped-pulse population
transfer include: driving Raman transitions [14, 15, 16]; manipulation of
atomic spin-waves [17, 18]; laser cooling of atoms via sawtooth wave adiabatic
passage [19, 20]; strong optical forces on atoms [21, 22] and molecules [23];
light-induced ultracold collisions [24, 25, 26, 27]; ultracold molecule
formation by photoassociation [28, 29]; and high-speed spectroscopic gas
sensing [30].

##  II Apparatus

###  II.1 Optical Setup

Figure 1: Chirp system optics diagram. Phase modulated Chirp light is
multiplexed with Dummy light, where both are sent through AOM1 and are jointly
coupled to fiber at port FP3, followed by SOA and TA amplification. Additional
information pertaining to the roles of the Chirp, Dummy, AOMs, PMs, SOA, and
TA are described in the text. Symbols: $\lambda/2$, half-wave plate; PBS,
polarizing beam splitter; L, lens; FP, fiber port; POL, polarizer.

The low optical power handling capability of present-day lithium niobate phase
modulators at 780 nm is an unfortunate technical limitation toward realizing
complex frequency-chirp shapes at moderate to high optical powers; however, we
have developed a system to circumvent this problem. As shown in Fig. 1, our
chirp system is comprised of two lasers and two laser amplifiers: a Toptica DL
Pro ECDL (Chirp), a Hitachi HL7852G FRDL (Dummy), an Innolume
SOA-780-20-YY-30dB (SOA), and a Toptica BoosTA Pro (TA). The Chirp laser
provides the light to be frequency-chirped and amplified, where a small
portion of its output is used for frequency locking to an optical cavity. The
rest of its output is directed to a pair of fiber-coupled PMs in series. To
avoid PM optical damage, the time-averaged optical power sent through these
devices is amplitude modulated by AOM2 (80 MHz) as seen in Fig. 1. The phase
modulated (frequency-chirped) light to be amplified is then sent to AOM1 (200
MHz), whose -1st order output is coupled to fiber port FP3. Pulsed -1st order
AOM1 output is then amplified by the fiber-coupled SOA which, after passing
through a fiber isolator and fiber-tap power monitor, seeds the fiber-coupled
TA.

We note that preamplification by the SOA is necessary to take full advantage
of the high power capability of the TA. This is due to the time-averaged power
limitation (5 mW, or 7 dBm) of the PMs in addition to high insertion losses,
2.5 dB and 3.3 dB for PM1 and PM2, respectively. When accounting for
additional losses associated to free-space to fiber and fiber-to-fiber
coupling, the overall loss between FP1 and FP2 is $\sim$11 dB. Ultimately, the
peak FP2 output of the Chirp-PM chain (6 dBm) is far below the moderate seed
power ($>$ 13 dBm) required by the TA to realize a 1 W (30 dBm) output. This
necessitates the use of an intermediate optical amplification stage, such as
an SOA. Since our SOA has a 780 nm low-signal gain of roughly $+$29 dB with an
injection current of 350 mA, FP2’s free-space output is reduced to the
appropriate SOA seed level by controlling the RF amplitude sent to AOM1 in
addition to the combined use of a $\lambda/2$ waveplate and a polarizer, as
depicted in Fig. 1.

Figure 2: Chirp system electronics diagram. A computer (not shown) is used to
program a dual channel arbitrary waveform generator (AWG) with user defined
waveforms for the PMs. All five pulse delay generator (PDG) signals are
synchronized, with the various switches (SW) being active when the PDG TTL
signals are either high (SW1,3,4,6) or low (SW5). See text for additional
details.

To prevent TA thermal damage, it is important to maintain seeding of the TA
when the chirped light is not present. To this end, we multiplex Dummy with
Chirp light using AOM3 (80MHz). The two beams, which are alternated in time,
are aligned such that AOM1 0th order Dummy light is perfectly overlapped with
AOM1 -1st order Chirp light, after which both beams are then fiber coupled at
fiber port FP3. The important consequence of this alignment scheme is that the
light in the fiber is exclusively Dummy light when the AOM1 drive is off, and
primarily Chirp light when the AOM1 drive is on. After amplification by the
SOA and TA, and subsequent beam shaping with free-space optics, the
Dummy/Chirp light is incident on AOM4 (40MHz). The light incident on AOM4 is
unfocused in order to prevent optical damage by the high-power TA output. By
synchronizing the drive for AOM4 with that for AOM1, we can eliminate the
majority of the Dummy light, sending only the chirped light downstream to the
experiment.

In many applications, the presence of far-detuned Dummy light and/or amplified
spontaneous emission (ASE) is harmless; however, residual Dummy/ASE light can
be problematic in some cases. For example, in experiments [29] producing
ultracold Rb2 molecules by photoassociation of ultracold atoms, the driving of
various molecular transitions can lead to photodissociation. If the Dummy’s
wavelength is tuned sufficiently far (e.g., $\sim$8 nm in our case) from the
Chirp’s, such complications due to residual Dummy light and/or broadband ASE
can be circumvented through the use of a narrow-band optical filter.

In order to accurately characterize the light sent to the experiment, a small
portion of the TA’s filtered output is used for diagnostic purposes. The TA
amplified chirped pulses are combined with reference light from a separate
ECDL on a 9.5 GHz amplified photodiode (Thorlabs PDA8GS) and the resulting
heterodyne signal (Fig. 3(b.i)) is sent to an 8 GHz oscilloscope for
averaging. The amplitude profile of the pulse (Fig. 3(b.ii)) and its time-
dependent frequency (Fig. 3(b.iv)) are determined using peak-to-peak analysis.
The spectrograms in Fig. 4(a-c) are obtained using a short-time Fourier
transform (STFT) algorithm [31].

###  II.2 Electronics

The chirp system’s electronics setup is shown in Fig. 2. A computer (not
shown) is used to program an 8 GS/s Euvis AWG 872 with user-defined RF
waveforms for channels A and B. USB digital step attenuators (Telemakus
TEA4000-7) ATT1 and ATT2 are used to precisely set the amplitudes of the
waveforms before amplification. High-gain 6 GHz bandwidth precision RF
amplifiers (Hittite HMC-C075), AMP1 and AMP2, are used to amplify the
waveforms for driving our EOSpace PMs (PM-5K3/5SES-10-PFA-PFA-780-LV-UL), PM1
and PM2 respectively. By attenuating (with ATT3 and ATT4) and 50-ohm
terminating the amplified RF signals for our externally terminated PMs, we are
able to monitor the programmed arbitrary waveform RF output on an 8 GHz
oscilloscope for diagnostic purposes. The four AOMs in Figures 1 and 2 are
driven by voltage-controlled oscillators (VCO) and amplifiers (AMP). The RF
signals are modulated by RF switches (SW) whose timings are controlled by a
BNC 577 pulse delay generator (PDG) or a marker pulse from the AWG. Two AND
gates are used to synchronize the timing of RF and optical signals. The first
AND gate is formed by SW1, which is fed TTL signals from the PDG and AWG, and
is used to trigger two 8 GHz oscilloscopes for RF diagnostics and optical
heterodyne measurement. The second AND gate, which ultimately controls SOA and
TA seeding through AOM1, is formed by SW3 switching the output of AWG
triggered SW2. The AWG Marker 2 signal sent to SW2 is synchronized with the RF
signals that are sent to the PMs.

The multiplexing of Chirp and Dummy light is controlled by AOM2 and AOM3 and
switches SW4 and SW5, where the Chirp light is sent through PM1 and PM2 toward
AOM1, and Dummy light is blocked or set at a reduced level. This multiplexing
of Chirp and Dummy light is crucial for safe and reliable operation of the TA,
where extended absence of seed light could result in irreparable damage to the
device. Note, however, that TAs can be operated un/under-seeded for brief
periods of time without harm, for example, during the $\sim$100 ns intervals
between chirped pulses.

Figure 3:  (a) Chirp system timing diagram with a 100 ms repetition period
where an example chirp window of 10 ms is shown. At $t=0$, Dummy light is set
to a reduced level by AOM3 (a.i) while Chirp light exiting AOM2 is sent to the
PMs (a.ii). The pulses (red vertical lines) within the chirp window (between
0-10 ms and 100-110 ms) in (a.iii) represent AOM1-modulated Chirp light sent
to the SOA and TA. Note that only a small fraction of the many short pulses
(100,000) is shown within the chirp windows depicted in (a.iii). The solid
dark curve in this plot represents amplified Dummy light. The vertical axes
for the Chirp, Dummy, TA, and SOA plots represent the peak powers after AOM2,
AOM3, AOM4, and at the fiber tap respectively. (b) Characteristics of a
positive linear frequency chirp. The amplitude (b.ii) and instantaneous
frequency (b.iv) of the chirp are averages from 50 successive heterodyne (b.i)
measurements, where the Chirp laser is locked 1.5 GHz blue of the reference.
The waveform depicted in (b.iii), described mathematically by Eq 2, represents
the quadratic waveform programmed into both AWG channels for PM1 and PM2.
Except for the frequency plot (b.iv), vertical axis units are arbitrary.

‘

###  II.3 Pulse Timing

A typical pulse sequence for our setup is shown in Fig. 3(a). The repetition
period for the sequence is 100 ms, where the Dummy seeds the SOA/TA for no
less than 90 ms. The maximum chirp duration is restricted to 10 ms (Fig.
3(a.ii)), ensuring that the time-averaged Chirp power sent to PM1 stays below
its 5 mW optical damage threshold. With Dummy light set to a reduced level by
AOM3 at t = 0 ms in Fig. 3(a.i), -1st order AOM1 Chirp light is modulated to
produce pulses of frequency-chirped light (Fig. 3(a.iii)), where the pulse
width is set by AWG Marker 2 and is limited by the rise/fall times of SW2
(MiniCircuits ZASWA-2-50DR+) and AOM1 (Brimrose TEM-200-50-780), which are 5
ns and 10 ns (for a 50 µm beam diameter) respectively. As seen in Fig.
3(a.iii), the SOA/TA combination is always seeded by either Dummy light or
chirped pulses.

The properties of a typical chirped pulse are shown in Figs. 3(b.i-iv). This
30 ns duration positive linear chirp with a repetition period of 100 ns is
repeated 100,000 times during the 10 ms chirp window of Fig. 3(a.iii). As
discussed in Sect. II.D, the time-varying frequency is the time derivative of
the phase, so a linear chirp results from a quadratic voltage applied to the
PMs, which is shown in Fig. 3(b.iii). The AWG programming includes a temporal
offset between channels A and B to compensate for electrical and optical
delays associated with series-connected PM1 and PM2. One single heterodyne
measurement arising from the superposition of the chirped pulse and the cw
reference laser is shown in Fig. 3(b.i). The amplitude vs. time (Fig. 3(b.ii))
and frequency vs. time (Fig. 3(b.iv)) plots are extracted and averaged from 50
heterodyne measurements. We note that apparent frequency fluctuations in
Fig.3(b.iv), apart from the low amplitude regions, are likely due to residual
amplitude modulation (RAM) [32] from imperfect polarization matching in the
polarization-maintaining fibers, where the effect is exacerbated by high RF-PM
drive power.

###  II.4 Phase Modulation

Since the total phase angle of a light field with carrier frequency $f_{0}$
and time-dependent PM-induced phase shift $\phi(t)$ is given by $\Phi(t)=2\pi
f_{0}+\phi(t)$, and since the instantaneous frequency of a light field is
equal to the time derivative of the total phase angle, the time-dependent
frequency of phase modulated light is given by

$\displaystyle
f(t)=\frac{1}{2\pi}\frac{d\Phi(t)}{dt}=f_{0}+\frac{1}{2\pi}\frac{d\phi(t)}{dt}$
(1)

The maximum phase shift that can be realized for any given PM is ultimately
determined by two parameters: the voltage required by the PM to produce a
$\pi$ phase shift $\left(V_{\pi}\right)$ and the maximum RF power handling of
the PM-RF amplifier chain. To help elucidate the following discussion,
consider the voltage waveform $V(t)$ required to produce a positive linear
frequency chirp, such as the one shown in Fig. 3(b.iv):

$\displaystyle\phi(t)=\frac{\pi}{V_{\pi}}V(t)=\frac{\pi}{V_{\pi}}\frac{4V_{p}}{\tau^{2}}\left(t-t_{0}\right)^{2}$
(2)

where the waveform is centered about $t_{0}$, has duration $\tau$, and cycles
between 0 and peak voltage $V_{p}$ twice. Since a chirp’s frequency range
($\Delta f$) is dependent on phase modulation duration ($\tau=\Delta t$), the
capabilities of a PM-RF system are best characterized through the time-
bandwidth product, $\Delta f\Delta t$. For maximum realizable peak voltage
$V_{p\hskip 1.42262ptmax}$, the maximum time-bandwidth product for a linear
frequency chirp is

$\displaystyle\left(\Delta f\Delta t\right)_{max}=4\frac{V_{p\hskip
1.42262ptmax}}{V_{\pi}}$ (3)

With a maximum AWG peak output voltage of 635 mV, a sub-1 GHz Hittite
amplifier gain of +26 dB, 1.5 dB attenuation between the AWG and RF amplifier,
and $V_{\pi}=1.8\hskip 1.42262ptV$, the time-bandwidth product for a linear
chirp produced by one of our PMs is $23.7$. This value is consistent with the
measured frequency range of $\sim 1.6$ GHz for the linear chirp shown in Fig.
3(b), where a quadratic waveform 30 ns in duration with 1.5 dB attenuation was
applied to both PM-RF amplifier chains. Due to the finite frequency response
of the chirp system’s RF components, a linear chirp’s time-bandwidth product
is reduced at high chirp rates. Nevertheless, with 1 dB of RF attenuation, we
have realized a linear chirp of 3 GHz in 10 ns, corresponding to a time-
bandwidth product of 15 for each PM.

##  III Results

###  III.1 Arbitrary frequency chirps

By appropriate programming of the AWG, a variety of chirp shapes can be
realized. Some examples are shown in Fig. 4. These are displayed as
spectrograms [31], which are contour plots of short-time Fourier transforms
(STFTs) of the heterodyne signals. The color coding denotes the relative
intensity of the various frequency components at the indicated time. For all
three examples, the intensity pulse has a FWHM of 22 ns. Fig. 4(a) is a
positive linear chirp spanning 1.6 GHz in 30 ns, similar to that shown in Fig.
3(b). The sum of a slow linear chirp and an arctan chirp is shown in Fig.
4(b). This shape would be particularly useful for driving a Raman transition,
where a slow chirp sweeps adiabatically through the first leg of the
transition, followed by a rapid frequency jump to minimize the time spent in
the excited state, concluding with another slow chirp to adiabatically drive
the final transition. Fig. 4(c) displays a two-frequency chirp in which two
offset frequencies are chirped in parallel. To realize this, a 1.0 GHz
sinusoid is applied to PM1 with the modulation depth, determined by ATT1,
chosen to suppress the carrier. This yields the +1 and -1 order sidebands,
separated by 2.0 GHz. A quadratic voltage is applied to PM2, which provides a
linear chirp to each of the sidebands. Similar to the linear-plus-arctan case,
the two-frequency chirp would prove useful for Raman transitions, but without
requiring the rapid jump, and with the added benefit that the timing of
passing through the two resonances can be adjusted by changing the offset
between the two chirped frequencies. This example highlights the flexibility
offered by using two PMs. The same voltage can be applied to both PMs, as done
for Figs. 4(a) and 4(b), or each PM can be driven independently, as for the
two-frequency chirp in Fig. 4(c).

Figure 4: Short-time Fourier transform (STFT) spectrograms of three arbitrary
frequency-chirps. The carrier frequency for all chirps is 3.0 GHz. (a) Linear,
(b) ArcTan+Linear, and (c) Two-Color Linear.

###  III.2 Multispectral power considerations

When driving transitions with chirped light, the faster the chirp, the greater
is the intensity required to maintain adiabaticity. Therefore, higher peak
powers will allow rapid and robust population transfer. The output power from
the TA is determined by its injection current, I, and the power with which it
is seeded. We typically operate with I = 3.0 A along with Chirp and Dummy
seeded SOA powers of 22 mW and 15 mW, respectively, as shown in Fig. 3(a).
Under these conditions, the peak output power of Chirp and Dummy light during
the chirp window are 1 W and 320 mW, respectively. Since the Chirp pulses are
superimposed on amplified Dummy light, a narrowband filter can be used to
block Dummy light if its wavelength is tuned sufficiently far from the
Chirp’s. With the Dummy tuned 8 nm red of the Chirp’s wavelength, a filter
centered at 780 nm with a 3 nm FWHM (AVR Optics LL01-780-12.5) provides $>$60
dB attenuation of TA amplified Dummy light. This filter also eliminates a
large fraction of the broadband ASE, which would reduce the driving of
unwanted transitions, especially in applications to molecules.

Since our chirp setup employs double-stage optical amplification, the
contribution of ASE to the TA’s output must be considered. To determine how
much of the total output power of the TA can be attributed to ASE, we use a
heated Rb vapor cell to absorb the resonant Chirp light, while allowing
passage of the broadband ($\sim$12 nm FWHM) ASE. With a Rb vapor cell, 5 cm in
length, heated to 65 ∘C, low-intensity light within 400 MHz of the 85Rb (F=3
-> F’=4) transition will be >99% absorbed [33]. Sending attenuated (sub-
saturation intensity) TA output through the cell, with resonance-locked Chirp
light solely seeding the SOA and TA, we measure 3% transmission, meaning that
$<$3% of the TA’s output can be attributed to ASE.

##  IV Conclusion

There are many ways in which the technique described here could be extended
and improved. Different wavelengths could be employed, contingent on the
availability and performance of fiber-based EOMs, SOAs, and TAs or fiber
amplifiers. EOMs capable of handling higher optical powers, such as those
using lithium niobate with MgO doping, would allow a simpler setup. Higher
peak output powers, while sacrificing repetition rate, could be obtained using
additional stages of amplification, e.g., with a multi-pass pulsed Ti-sapphire
amplifier [34]. Shorter pulses could be realized by incorporating a fiber-
based electro-optical intensity modulator in series with the phase modulators
[8]. Higher speed electronics would enable faster chirp rates and larger chirp
ranges. Extended chirp ranges could also be achieved by using the serrodyne
technique [35] to suddenly shift the frequency during the chirp. Recent
developments in optical spectral engineering [36] could potentially yield more
faithful production of desired chirps by compensating for imperfections in
electrical and optical system responses.

In conclusion, we have demonstrated a versatile system for generating high-
power pulses of frequency-chirped light at 780 nm. The use of two independent
phase modulators enables flexibility in the chirps which can be created.
Various chirp shapes have been demonstrated, such as linear, linear plus
arctan, and two-frequency linear. Two stages of amplification, in conjunction
with multiplexed seeding with Dummy light, result in high output power without
exceeding the damage limits of the modulators or the tapered amplifier. These
high-power chirped pulses are well suited to rapidly and robustly drive
transitions in atoms and molecules, and should therefore find use in areas
such as atom optics, ultracold molecule formation, Raman transitions, and
spectroscopic sensing.

###### Acknowledgements.

This work was partially supported by the National Science Foundation, Grant
No. PHY-1506244. We thank D.J. McCarron for useful discussions.

## author declarations

Authors have no conflicts to disclose.

## data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Fregean Flows

Eric Easthope  University of British Columbia, Vancouver, British Columbia,
Canada<EMAIL_ADDRESS>This paper is dedicated to its first adversary Prof.
Dave Gilbert.

###### Abstract.

I humbly introduce a concept I call Fregean flows, a graph theoretic
representation of classical logic, to show how higher-dimensional graph
characteristics might be useful to prove or perhaps at best show the
provability of simple deductive statements typically represented as one-
dimensional strings of characters. I apply these to a very simple proof,
namely proving the equivalence of two definitions for an Abelian group $G$, an
“if-and-only-if” statement, using a re-representation of statements as
vertices and both conjunctions and implications as differently “coloured”
edges. This re-representation of an if-and-only-if is simple but shows
unexpected geometry, and I discuss its possible utility in terms of
provability through ideas of graph topology, similarities of graph contraction
to deductive elimination, and recursion.

###### 2020 Mathematics Subject Classification:

Primary 68V20; Secondary 03F03

††copyright: ©2024: Eric Easthope

> “If the task of philosophy is to break the domination of words over the
> human mind […] then my concept notation, being developed for these purposes,
> can be a useful instrument for philosophers [….] I believe the cause of
> logic has been advanced already by the invention of this concept notation.”
> — Gottlob Frege (preface to Begriffsschrift, 1879; English translation)

##  1\. Motivation

###  1.1. Logic as picture

Gottlob Frege’s visual Begriffsschrift (1879) [2] arguably canonized the
foundations of first-order axiomatic logic and correspondingly the foundations
for mathematical proof by symbolic logical reasoning as well. These early
writings in analytical philosophy were highly visual, recursive, and not so
much adhered to the one-dimensional symbol-sentence style of modern proofs.
Frege was keen to develop purely logical principles of inference that could
represent mathematical proof axiomatically without gaps and unwritten appeals
to “intuition,” and he did so without sentences, higher-dimensionally, with
Boolos describing Frege’s “two-dimensional symbolism [as depicting] the
logical structure of statements visually and vividly.” [1]. It is unfortunate
that Boolos went on to call Frege’s work “nothing so much as wallpaper” [1]
without elaborating on what he indicated he knew about Frege’s interest to
develop arithmetic through geometry, but Boolos’ criticism highlighted an
emergent and rather novel adherence to logic represented as sentence over
logic represented as picture.

###  1.2. The non-linearity of proof

Text, and correspondingly the tasks of reading and writing text, are one-
dimensional. There really is nothing that precludes all written word from
being expressed as an endless, one-dimensional string of characters. And this
how we express threads of logic too, not to mention threads of executed
machine code. Pragmatically we represent text linearly to be maximally useful.
But the comprehension of text does not carry with it such pragmatic
motivations, nor is it necessarily linear; so the comprehension of logic, a
text subset, is not necessarily linear either. There is little about how we
represent and comprehend logic that should demand a fundamental one-
dimensionality like text beyond its expression through language, and language
itself is rarely a systematic and one-to-one parsing of words in sentences to
logical semantics. If language were this systematic the reading and writing
and parsing of any sentence with a characteristic like Russell’s paradox, for
example “I am all of that which I cannot be,” would be catastrophic to
comprehension. Yet despite the paradox of this phrase, we can read it, write
it, parse it, and even derive conclusions, albeit absurd ones. Such sentences
are simple counterexamples to logic and therefore to proofs with logic having
inherently linear comprehension; it is not language that constrains the
dimensionality of logic to text, let alone the dimensionality of proof.

###  1.3. The shadow of classical logic

Perhaps it is out of the crooked squiggles of proofs scrawled over folded and
sometimes crumpled pages that this inherent one-dimensionality also seems less
intrinsic to me. The idea of a “two-dimensional proof” is not that alien:
proofs by cases, for example, can be seen as “paths” of proof satisfied in
conjunction, a view that hearkens to Frege’s own tree-like notations. But
classical logic in its textual symbolic form is a monolith that rules much of
our reasoning and so to even suggest re-representation risks backlash. In kind
and approaching such risks gently this work is not an attempt to thwart or
overturn existing principles, rather to re-cast familiar representations of
mathematics to something non-linear, something novel, but more importantly
something compatible with Frege’s visual conceptions.

##  2\. Concept

###  2.1. Non-linear proof representation

Representing proof in a non-linear sense requires a more general understanding
of what a proof really is. Meta-logic has a lot to say here but the core of
what we need is the concept of proof as sequences of sentences, which on their
own are linear, or linear parallel at most in proofs by cases, combined with
two ideas: that proofs are capable of sharing a namespace of objects within
those sequences of sentences—which is common in bi-conditionals—and that the
ordering of sentences within sequences is a relationship between sentences
unto itself. The overlapping of proofs then makes for non-linear arrangements
through relationships of objects instead of just sentences about objects and
these higher-order relationships can be represented as graphs. The graph
representation extends us a number of desirable concepts: edge directionality,
edge colouring (or labeling if you will) and vertex adjacency, which may not
seem like much for meta-logic’s sake, but it might be enough to represent some
mathematics.

###  2.2. Re-representing AND

Consider the entailment of a sentence C by two preceding statements A, B. If
only A or B is necessary for C to be true, it is common to eliminate one of A
or B. But this sort of elimination is redundant when we only talk about
sufficiency: anywhere we can eliminate a disjunction (A or B) it is just as
well to say that the truth of (A and B) entail C with redundancy in the truth
of one of A or B “through” the non-redundancy of the other. Going further it
just as well to say that the truth of A thru B entails C, or equivalently, the
truth of B thru A entails C; I am highlighting that redundant thru-like
additions such as B to the entailment (A entails C) do not actually change the
truth of the entailment but they introduce intermediate structure that we
might use. Really the entailment A through B entails C represents C through a
causality of the sufficiency of B through the sufficiency of A. If A were
sufficient in place of (A or B) then B is sufficient through A, specifically
sufficiently redundant through the non-redundancy of A, so C is sufficient
through A. With any conjunction (A and B) it is the same, only that we expect
the sufficiency of both A and B without redundancy. But this does not change
the way we can represent it: it is just as well to re-represent (A and B)
entails C as A thru B entails C, or symmetrically B thru A entails C.

###  2.3. e.g. Equivalence of definitions: $G$ is Abelian.

Graphically re-representing how to prove the equivalence of two definitions
for an Abelian group $G$ proceeds as follows:

###  2.4. Proof by contraction

We can speculate from the geometry alone how we might use graph properties to
prove if-and-only-if proofs or at least to assert something about their
provability. If we overlook edge colours then the uncoloured graph is planar
without edge intersections and has a closed circular geometry with no internal
edges, so we can contract the entirety of it to a single vertex. This suggests
the following rule:

> A set of sentences $\\{S\\}$ is provable if-and-only-if its graph
> representation $G=(V,E)$ as vertices $\\{V\\}$ and edges $\\{E\\}$ contracts
> to a single vertex $G_{0}=(\\{v_{0}\\},\varnothing\\}$.

In other words, $\\{S\\}$ is provable if-and-only-if $G$ is homomorphic to
$G_{0}$, or equivalently provable if $G$ is iteratively contractible to
$G_{0}$. Maybe this is useful.

##  3\. Conclusion

I introduced Fregean flows, a graph theoretic representation of classical
logic, and showed that it might be useful to prove or perhaps at best show the
provability of simple deductive statements. I went further to show that an
“if-and-only-if” proof under this representation has the geometry of the
circle and that we can speculate on graph properties like edge contraction to
determine its provability.

## References

  * [1] George Boolos.  Reading the Begriffsschrift.  Mind, 94(375):331–344, 1985.
  * [2] Gottlob Frege.  Begriffsschrift.  In Jean Van Heijenoort, editor, From Frege to Gödel, pages 1–83. Harvard University Press, 1967.