Parallel Universes

Hugh Everett was too wrapped up in his thoughts to be a parent

The Many Worlds of Everett

In April 1959, Hugh Everett III,along with his wife and baby daughter, went to Copenhagen, Denmark, to meet Niels Bohr and his colleagues. He was very excited. He had just completed his PhD thesis, called "Many Worlds Interpratation of Quantum Mechanics" and he was thinking that this was going to change the perspective of the whole quantum physicists. A bit eccentric and self-centered, he even thought he could be the Einstein of his era, to whom he was sending letters at the age of 12.

Niels Bohr talking to Hugh Everett ( 2nd one on the right),at the age 24

Contrary to his expectations, his trip to Copenhagen turned out to be a complete disaster. He was literally mocked by his fellow physicists, to the point that Léon Rosenfeld, one of Bohr's followers, described Everett as being "undescribably stupid and could not understand the simplest things in quantum mechanics". 

But what was Everett claiming that made Bohr and his colleagues so irritated? Let's give some insight first.In the quantum world, an elementary particle, or collection of those particles, can exist in a superposition of two or more possible states at the same time. For instance, an electron can be in a superposition of different locations, velocities and orientations of its spin. Yet anytime scientists try to measure one of these properties, they see a definite result. (just one of the elements of the superposition, not a combination of them).

Many of the founders of the quantum mechanics, namely Bohr, Heisenberg, von Neumann, had agreed on an interpratation of quantum mechanics - Copenhagen Interpratation , to deal with the "observer" problem. According to this view, at the moment of measurement, the wave function describing the superposition of alternatives appears to collapse into one member of the superposition, thereby interrupting the smooth evolution of the wave function and introducing discontinuity. However, there was one man who wasn't buying this: Hugh Everett.

Instead of Schrödinger's wave function, which can be thought of as a list of all the possible configurations of a superposed quantum system, he introduced  a universal wave function,that links observers and objects as parts of a single quantum system.Everett’s radical new idea was to ask, "What if the continuous evolution of a wave function is not interrupted by acts of measurement?". According to him, the universal wave function would contain every probability making up the object's superposition. He thought, all of these alternatives were part of the reality, which are branched and do not influence one another once formed. ( according to a fundamental property of Schrodinger's equation). For instance, during the measurement, although we are observing a definite result, in fact, all the possibilites that could happen, were happening in a different branch. He called these branches "many worlds".

Hugh Everett and his daughter Liz

Everett's family did not end up well. He left physics after completing his PhD, due to the lack of interest to his work and instead worked in defence industry. His daughter Lisa, who were having mental problems, commited suicide in 1996, leaving a note that "she was joining her father in a parallel universe." His wife died of cancer later, and the only surviving member of the Everett's family, Mark Oliver Everett turned out to be a rockstar, the lead singer of the alternative rock band Eels, who prepared a document describing his father and their relationship, called "Parallel Worlds, Parallel Lives".

Everett's great idea , although met scorn at his time, revealed that there are different universes where all of the physically possible events were actually happening. This was also a great cure for paradoxes related with the Double Slit Experiment and Schrödinger's Cat. "Many-Worlds" of Everett are later described by Max Tegmark, who is now a  top-physicist in MIT, as Level-III parallel universes. Apparently, he has 3 more.

The Universes of Tegmark

Max Tegmark is an iconoclast physicist in the Physics Department of MIT and he is the leading supporter of the "multiverse" idea in the physics world today. But why did this whole multiple universes idea suddenly start to sound reasonable to most of the physicists around the world? Well, because the M-theory suggested that our universe was made up of 11 dimensions, and the theory of inflation suggested that our universe was actually infinite.

So according to Tegmark, Level-I universes directly result from the fact that the universe is infinite. If space goes on forever, then there must be other regions like we live in—in fact, an infinite number of them. No matter how unlikely it is to have another planet just like Earth, we know that in an infinite universe it has to happen.

Level-II universes emerge if the fundamental equations of physics, the ones that govern the behavior of the universe after the Big Bang, have in fact more than one solution. So actually at the very moment of Big Bang, other universes with different kinds of physical laws, different solutions to those equations, might have emerged spontaneously . Universes with different realities that we cannot perceive!

Max Tegmark while explaining the birth of the universe

Level-IV universes is a phenomena invented by Max Tegmark himself. Max Tegmark has this crazy idea, called "Mathematical universe hypothesis", that the reality itself is not only described by math, but it is math! Therefore every statement in abstract mathematics in fact describes different realities and physical existences. So these other mathematical universes are made up of "external" realities, which are independent from our owns.

The moral implications of Parallel Universes

So, these all seem very abstract, but what are the implications of this idea? Well, the first implication, that is every physically possible event is actually happening elsewhere could be a comforting or dangerous idea at the same time. It may be comforting because you know that you are living an "ideal life"in another universe, in which you might be a rockstar, the top scorer in Premier League, the Nobel prize-winner, or in which you are never born.

Travis is rocking not only one, but multiple universes- You just can't see them.

It is also a dangerous idea because for instance if you commit armed robbery and shoot the cashier, why should it be a crime, while there are other universes in which it is actually not considered as a crime? Or as an alternative thought, after all, there are millions of parallel universes in which you didn't do it, or in which he shot you! That destroys the whole "universal" morality concept and leaves us with an infinite amount of choice for moral values and standards!

Now, you might think reading this article was totally a waste of time. But I assure you this will be a big hit in another universe.

If you are interested to read more about this subject, see:

"The Many Worlds Interpratation of Quantum Mechanics" Hugh Everett

"The Many Worlds of Hugh Everett", Scientific American

"Parallel Universes", Max Tegmark

"The Mathematical Universe", Max Tegmark

"Parallel Worlds, Parallel Lives", BBC Horizon Documentary

"Parallel Universes", History Channel

Boltzmann & Evolution: An admirer of Darwin

On  May 29, 1886, Boltzmann gave what is now regarded as a very popular lecture at the ‘Festive Session’ of  the Imperial Academy of Sciences in Vienna. When he was asked about his opinions about the new century, he replied as follows:

". . . If you ask me about my innermost conviction whether our century will be called the century of iron or the century of steam or electricity, I answer without hesitation: It will be called the century of the mechanical view of nature, the century of Darwin. . . ."   

Boltzmann was right. Unarguably, Darwin has become one of the key scientists of the century, radically changing our views about the nature, and the meaning of existence and human life.

Boltzmann's views on evolution

Ludwig Boltzmann (1844–1906) was the first scientist who tried to reduce the biological theory of evolution to the thermodynamics and chemistry of the 19th century. For scientists at the end of the last century,  a great challange was that the  second law of thermodynamics seemed to forecast the final disorder, death, and decay of nature, while  on the other hand, Darwin's theory of evolution indicated developing living systems of order with increasing complexity. How was the increase of complexity among living things possible in a sea of disorder and thermal equilibrium?

To understand views of Boltzmann on biology and evolution better, let's go back to his lecture in Vienna and quote him again:

"… The struggle for existence of the living beings is not a fight for basic materials—these materials are available in air, water and soil in sufficient quantities for all organisms– nor for energy, which is plentiful in the form if heat, unfortunately untransformably, in every body. Rather it is a struggle for entropy that becomes available through the flow of energy from the hot Sun to the cold Earth. To make the fullest use of this energy, the plants spread out the immeasurable areas of their leaves and harness the Sun’s energy by a process that is still unexplored, before it sinks down to the temperature level of the Earth, to drive chemical syntheses of which one has no inkling as yet in our laboratories. The products of this chemical kitchen are the object of the struggle ion the animal world. . . ."

Going back to the question about the complexity of life, Boltzmann suggested some explanations which already remind us of modern biochemical concepts of molecular autocatalysis and metabolism. The origin of first primitive living beings like cells was reduced to a selection of molecular building blocks which Boltzmann imagined as a process like Brownian motion. Plants as cellular aggregates are complex systems of order. Thus, in the sense of the second law of thermodynamics, Boltzmann suggests, they are improbable structures which must fight against the spontaneous tendency of increasing entropy in their body with sunlight. According to him, photosynthesis, as he also implied above in his quote, is merely an attempt to compansate the spontaneous increase of entropy in the plants, which makes use of the Earth's energy with relatively low entropy coming directly from the Sun.

Boltzmann on the evolution of nervous systems and brain

 The next step for Boltzmann was extending his views to the evolution of the nervous system and the emergence of memory and consciousness. He claimed that the sensitivity of the earlier primitive organisms to outer impressions led to the development of special nerves and organs of seeing, hearing etc. He believed that  the human brain has been developed with the same perfection as the giraffe's neck or the stork's bill.

Boltzmann on socio-cultural developments

Evolution of morality? Boltzmann believed so.

It may seem quite exaggerated, but Boltzmann even did not hesitate to extend his views about evolution to sociology.  He tried to justify human categories of  space, time and casuality as tools developed by the brain for the survival of the race. He saw "morality" as a "constantly evolving"  weapon for the struggle of life. Later in his life, his "Darwinism" has reached to an incredible point that in 1905, he gave a lecture called  "Explanation of the entropy law and love by the principle of the probability calculus". Well, anyways...

Boltzmann: More Darwinist than anyone else in the history

If you are interested in finding out more about how Darwin effected Boltzmann's views and studies, you can also take a look at his other works, especially "Boltzmann Brain", with which he is trying to explain why the observed entropy is so low.

At the beginning of the 20th century, life still could not be explained by physical and chemical foundations. Earlier on, classical mechanics always considered deterministic and time-reversible patterns in the nature. (A frictionless ball, when you push it, would move forever). But life wasn't working that way. Humans are born, grow and die.-Why? Maybe Boltzmann's statistical interpratations about Darwinian evolution were not enough to explain the origin of life. However, he clearly made his point when he commited suicide in 1909 in Duino,Italy, a person who sees death as a senseless biological and cultural event.

If you are interested to read more about this subject, see:

"Thinking in Complexity", Klaus Mainzer.

"Boltzmann and evolution: Some basic questions of biology seen with atomic glasses",Peter Schuster

"Boltzmann,evolution,atomism,statistics",Peter Schuster
"Dangerous Knowledge", BBC Documentary

Next week: Parallel Universes

Infinitude of prime numbers: Euclid, Euler and the mathematical beauty

Euclid with his students

   

Euclid, as depicted above, used to love teaching and sharing his knowledge with others, and made this teaching process one of his daily routines. One day, a youth who had just begun to learn geometry with Euclid, when he had learnt the first proposition, inquired, "What do I get by learning these things?" So Euclid called a slave and said "Give him three pence, since he must make a gain out of what he learns." 

Euclid's answer probably summarizes why there are still people today, who are really not earning a lot , engaging in pure mathematics and spending their lifetime on a subject which will only be understood by a small portion of the human population on earth. Euclid, who is generally referred as the "Father of Geometry" was probably one of the scientists who enjoyed what he was doing the most. Therefore, it probably comes as no surprise that , he established one of the most elegant proofs in the math history- if not the most elegant one.

Euclid's proof

Are prime numbers infinite? Perhaps one of the most interesting things about the infinitude of prime numbers is that, nobody who is not interested in mathematics very much seemed to have a correct/reasonable answer about it. I usually get answers like "Hmmm,I don't know" from people when I asked about this. We can confidently say that the answer for this question is not intuitive, that is to say, it is not related with our everyday experiences nor observations.

When Euclid asked himself this question, and sought for an answer, he found out that the answer was very simple. A summary can be seen below.

Suppose that p₁=2 < p₂ = 3 < ... < p_r are all of the primes. Let P = p₁p₂...p_r+1 and let p be a prime dividing P; then p can not be any of p₁, p₂, ..., p_r, otherwise p would divide the difference P-p₁p₂...p_r=1, which is impossible. So this prime p is still another prime, and p₁, p₂, ..., p_r would not be all of the primes.

Roughly speaking, as you might have already understood, suppose that the number of prime numbers are finite, and let the last prime number be p_r   .So now if you multiplied all the prime numbers including p_r  and just add 1 to it, you would get another prime number, which contradicts with the first assumption. Hence, prime numbers are infinite.

Not only me, but also lots of mathematicians agree that this is one of the most elegant and beautiful proofs in the math history. The fact that it is so simple and can even be understood by a primary school student makes you want to think : "That's it!".

Euler's proof

Leonhard Euler depicted on a 1983 stamp from the German Democratic Republic. The stamp also includes a diagram of an icosahedron and Euler's famous polyhedral formula 

The underlying idea of Euler's proof is very different than that of Euclid's. In fact,although more complicated,his proof is stronger and makes use of analytical methods. In essence, he proves that the sum of the reciprocals of the primes is infinite.  I can't say I am a big fan of Euler because of his intrigues against Daniel Bernoulli, which he conducted with Daniel's father. But anyways, you know what they say: “Render unto Caesar the things which are Caesar’s”.

 

He used the following equality between the regular harmonic series and the infinite products called "Euler products". In the right hand side, p refers to the prime numbers.Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series. This is a direct result of the fundamental theorem of arithmetic. If you are not convinced, please see: "On the infinitude of prime numbers" by Shailesh A Shirali pg. 9-11.

                                    

Then, Euler took the natural logarithm of both sides and by using the properties of logarithm function he found:

                  

Since

we get e^x > 1 + x and x > ln(1 + x).

So;

Hence diverges. (Because we know the left side diverges as it is equal to the natural logarithm of the sum of the  harmonic series.)But 1/(p_i − 1) < 1/p_i−1 where p_i is the i^th prime. Hence diverges.

This directly implies that , there are infinite number of prime numbers(otherwise the sum would converge).Perhaps none of you could guess a simple divergence test, which you probably learned in Calculus courses, would be the main element of such a beautiful proof.

Mathematical Beauty

Mathematicians usually describe a proof as elegant, when it is simple,unusually succinct, derives a result in a surprising way,is based on new and original insights and can be easily generalized to solve a family of similar problems. Probably one of the main requirements the mathematicians are defending to understand the mathematical beauty is that ,well, "you have to be a mathematician".
 

In his book, "Art of Mathematics", Jerry King draws a figure as depicted above to summarize the  aesthetic view of different people about mathematics.The people in the inner circle (like myself)are people very close to mathematics because they use it day by day, are the ES types (the engineers and scientists). Most engineers and scientists appreciate mathematics solely and only because of its utility. 

The point outside the two concentric circles, point P, represents people who don't care about mathematics. (including most non-science and non-math teachers) These “P people” are unfortunately in the majority.

And according to King, you could only see the beauty of mathematics if you are in the gray zone. That is, if you are a mathematician :) Well a bit arrogant, but who can oppose it ?