The Unsolvable Problem in Scientific American of October 2018

This document contains comments about the article A Beacon from the Big Bang by by Toby S. Cubitt, David Perez-Garcia and Michael Wolf In Scientific American of October 2018.
A journey into some of the strangest ideas in modern math and physics
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Half-jokingly, one of asked,"Why don't we prove the undecidability of something people really care about, like the spectral gap?"
At the time we were interested in whether certain problems in physics are "decidable" or "undecidable" - that is, can they ever be solved?
What means solved?
When is a problem solved?
See also: Reflection 1 - What does it mean that we understand physics?

"page 23"

The mathematics of quantum mechanics

But as so often happens, his mathematics turned out to be exactly what was necessary to understand a question that was perplexing physicists at the time.
Accoringly to Wikipedia the relation between quantum mechanics and spectral theory was fortuitos. See :
See also: Reflection 1 - What does it mean that we understand physics?
The frequencies of light emitted by heated materials thus give us a "map" of the gaps between the atom's different energy levels.
Explaining these atomic emissions was one of the problems physicists were wrestling with in the first half of the 20th century.
More or less the same issue as above: What means explain? Explaining a physical phenomena?
The lowest posible energy level of a material is called its ground state.
From a logical point of view there is nothing wrong with this.
The problem is how do you know that physical.
This is the level it will sit in when it has no heat.
Again what does this (i.e. no heat) mean physical.
To get a material into its ground state, scientists must cool it down to extremely low temperatures.
You can now simplify this text as: The ground state is the state of any material at -273.15 degrees C. This does not say anything about the (internal) state, specific energy level, of the material self.
The easiest way is for it to absorb the smallest amount of energy it can, just enough to take it to the next energy level above the ground state - the first excited state.
That seems simple in concept, but in reality different to demonstrate what it really means.
In some materials there is a large gap between the ground state and the first excited state.
This is the case when we study the behaviour of electrons around a nucleus.
In other materials the energy levels extend all the way down to the ground state without any gaps at all.
The question is how can you speak about the energy levels, while there are no gaps in between the levels? i.e. the behaviour is continuous.
Whether a material is "gapped" or "gapless" has profound consequences for its behaviour at low temperatures.
It is easy to assume that different materials can behave different at low temperatures, but IMO it is difficult to prove that this behaviour has anything to do with "gapped" or "gapless", while specific the behaviour of "gapless" is not clear.
It plays a particular significant role in quantum phase transitions.
When the spectral gap disappears - when a material is gapless - the energy needed to reach an excited state becomes zero.
But how do you know that this is an excited state?
This physical process seams rather adhoc.

"page 24"

In fact, thanks to the weird quantum effects that dominate physics at these very low temperatures, the material can temporarily "borrow" this energy from nowhere, go through a phase transition and give the energy back.
Again: The description of this physical proces seems rather adhoc. You need much more physical detail.
Because the spectral gap problem is so fundamental to understanding quantum phases of matter , it crops up all over the place in theoretical physics.
I would write: in experimental physics.
A closely related question even crops up in particle physics: there is very good evidence that the equations describing quarks and their interactions have a "mass gap".
First of all it is very tricky to establish the mass of elementary particles, specific quarks. To calculate the intewractions between these quarks is even more difficult. As such it is very difficult to speak of a "mass gap".
Experimental data from particle colliders such as the LHC support this data, as do massive number-crunching results from supercomputers.
This sentence (the second half) has a large PR content and should be deleted.
To support LHC data you need two models: one with a "mass gap" and one without. The results should indicate that the first should give a better fit as the second.
But proving the idea rigorously from theory seems to be extremely.
You can never prover an idea from a theory, because are 'the same'.
Our proof shows that the general problem is even trickier than we thought.
I can imagine. What you need is a clear description of all what is involved.
The reason comes down to a question called the "Entscheidungsproblem"
I doubt if you can solve a physical problem by means of a mathematical problem.

The Spectral Gap

The authors proved that it will never be possible to determine whether all materials are gapped or gapless
How can you prove this when you first explain that there are materials which are gapped and others which are gapless, independent of what both exactly mean?

Unanswerable questions

Hilbert wanted a rigorous proof.
What means "rigorous proof". You can only (try to) prove something (a mathematical statement) when you use a combination of logical statements which each is considered true.
In 1928 he formulated the Entscheidungsproblem. It asks whether there is a procedure or "algorithm" that can decide whether mathematical statements are true or false.
See my comments about the Entscheidungsproblem in Wikipedia: Entscheidungsproblem Comments

"page 25"

Here is aflavor of Gödel's idea: If someone tells you: "This sentence is a lie," is that person telling the truth or lying?
The first question to ask is: What does it mean: "This sentence is a lie,"?
The answer on the question: "is that person telling the truth or lying?" is impossible to answer, because what the persons tells is garbage.
Even through it appears to be a perfectly reasonable English sentence, there is no way to define if it is true or false.
It is not a 'perfectly reasonable English sentence'
To solve the Entscheidungsproblem, Turing had to pin down precisely what meant to "compute" something.
I think Turing was more involved with the idea "How to compute mechanically"
Turing came up with an idealized, imaginary computer called a Turing machine.
I think it is better to say that he came up with a cenceptual computer called a Turing machine.
The invention of the Turing machine was more important even than the solution to the Entscheidungsproblem.
Compared to the invention of the Turing machine, which is top, the Entscheidungsproblem is nothing.
etc, Turing then went on to prove that there is a simple question about Turing machines that no mathematical procedure can ever decide: Will a Turing machine on a given input ever halt?
I doubt if Turing can either prove or disprove this simple question because the simple question is not clear.
Compared to the invention of the Turing machine, which is top, the halting problem is nothing.
Mathematicians have become accustomed to the fact that any conjecture we are working on could be provable, disprovable or undecible.
One of the first issues is that the conjecture has to be clear and unambiguous.

Where we come in

We wanted to unite the quantum mechanics of the spectral gap, the computer science of undecidability and Hilbert's spectral theory to prove that - like the halting problem - the spectral gap problem was one of the undecidable ones that Gödel and Turing taught us about.
The behaviour of electrons at low temperatures is a physical problem with its own limits and problems. That behaviour has nothing to do with mathematics and computer science.

Burning the midnight Coffee

We attempted to make the next leap by linking the spectral gap problem to quantum computer.
Also here there is no such a link.
The reader is advised to read the text in the article.
But ground states of quantum systems etc. So how can it make a computation?
An atom, in its ground state, cannot make a computation.

"page 26"

In remembrance of tilings past

"page 28"

Exams and deadlines

"page 29"

What it all means

First and most important, they give a rigorous mathematical proof that one of the basic questions of quantum physics cannot be solved in general.
How can you prove that you can solve a physical issue?
By performing an experiment which shows that it can be done.
How can you prove that a physical question cannot be solved?
That is very tricky.
In the travelling salesman problem you can design a graph (which shows the distance between all the towns) and ask the question: which is the shorest route between all the towns.

Reflection 1 - What does it mean that we understand physics?

You could also raise this question slightly different: What does it mean that we understand something? What is involved to understand physics better?
IMO the most important strategy to understand physics lies in the details. The more details we know the better we can understand the inner workings of what we want to understand. That an atom has a nucleus and is surrounded by electrons at individual shells improves our understanding.
The next step is to describe the shells as energy levels. This explain the frequencies of the emitted photons in mathematical language.
However and that is an important the frequencies don't explain the physics of the shells themself.

In the book "Astronomy and Cosmology" by Fred Hoyles, at page 131 we read:

What, one might ask, is the difference between mathematics and physics?
A glance at a few mathematics and physics texts might suggest that the two are really the same. Yes this is not so. The mathematician is concerned entirely with sequential logical statements: If A is true, then B is true. If A is true then K is false. The "pure" mathematician does not seek to establish whether A is really true. This is the physicist's job. What the physicist does is to decide which statements about the world are correct.
It is my interpretation of the article "The Unsolvable Problem" that the authers try to solve this physical problem by using mathematics.

Reflection 2 - Gapped versus Gapless

Considering certain atoms and electrons there is a discrete amount of energy necessary to go from one energy level to another energy level in the form of photons. That means a photon is needed for an electron to jump to a higher level or a photon is released to jump to a lower level. Such materials are called Gapped.

There are different materials consisting also of different levels but the difference between the energy values is tiny. Those materials are called Gapless.

Unfortunate the article does not supply enough physical details to fully understand the physical implications.

Reflection 3 - Cooling down and Heating up

What do you perform by means of an experiment that electrons jump to a higher energy level or to a lower energy level? That is the question.
IMO this is extremely difficult to perform this under controlled conditions.
The article indicates that when material is heated it emits photons. When this is done the electron goes from a higher level to a lower level. The article does not indicate how this is performed.
In the box "The Spectral gap" a certain amount of energy is absorbed in an atom which changes the state of one electron from "Ground state" into "level 1". No details what this amount of energy is.
The book "Astronomy and Cosmology" by Fred Hoyle explains what physical is happening at much more detail. Specific at the pages 136-144 and 397-403. He uses the word radiation to describe an "amount of energy".
This means that radiation is required to bring an electron from its ground state to level 1. The reverse is also assumed: radiation is released when an electron goes from level 1 to its ground state

A much more important question is: what exactly means "heating up" and "cooling down". IMO heating up means that the average speed of all the atoms involved, increase. That means, when we assume that all the atoms are within a closed container, that the outside of this container is heated by means of a Bunsenbrander, which inturn influences the state (average speed) of what is inside. The result in general is that the atoms involved, will collide more, which results in the emission of photons.

The importance of this text that heat (and temperature) is not something directly physical. It is an indication of the average state of movement of a gas or substance.

Reflection 4 - The square in the box problem.

I think this problem comes from Martin Gardner. For details see:
Question 1: Consider a square of 1 by 1. Consider a box (square) of 2 by 2. How many squares can you put in the box? Answer 4. In a box 3 by 3 you can put 9 squares. Simple.
Question 2: What is the smallest size box that you can place 37 squares?

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Created: 26 September 2018

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