Researchers have recently discovered that solutions to the Riemann zeta function correspond to the solutions of another function that may make it easier to solve the Riemann hypothesis.
Dorje Brody, a mathematical physicist at Brunel University London, says that “to our knowledge, this is the first time that an explicit—and perhaps surprisingly relatively simple—operator has been identified whose eigenvalues [‘solutions’ in matrix terminology] correspond exactly to the nontrivial zeros of the Riemann zeta function“.
Now what remains to be proven is that all of the eigenvalues are real numbers rather than imaginary ones.
This newly discovered operator has close ties with quantum physics. In 1999, Michael Berry and Jonathan Keating made the conjecture (now called the Berry-Keating conjecture) that if such an operator exists, then it should correspond to a theoretical quantum system with particular properties. However, no one has ever found such a system before now.
In general, mathematicians are optimistic that the eigenvalues are actually real, and there is a strong argument for this based on PT symmetry (a concept from quantum physics which says that you can change the signs of all four components of space-time and the result will look the same as the original).
Ernst Chladni was a German physicist who is sometimes labelled as the ‘father of acoustics‘. His work in this area includes research on vibrating plates and the calculation of the speed of sound for different gases.
One of Chladni’s greatest achievements was his invention of a technique to show the various methods of vibration of a rigid surface, such as a plate, which he detailed in his book Entdeckungen über die Theorie des Klanges (“Discoveries in the Theory of Sound”) in 1787. His technique entailed:
“drawing a bow over a piece of metal whose surface was lightly covered with sand. The plate was bowed until it reached resonance, when the vibration causes the sand to move and concentrate along the nodal lines where the surface is still, outlining the nodal lines.”
The patterns that emerge are beautiful and are now known as Chladni figures, although Chladni was building on experiments and observations by Robert Hooke in 1680 on vibrating glass plates.
Chladni also created a formula that successfully predicted the patterns found on vibrating circular plates.
Chladni’s discovery was extremely important as it inspired many of the acoustic researchers who later extended his work.
Once these patterns were well understood, they began to have many practical uses, for example violin makers use “Chladni figures to provide feedback as they shape the critical front and back plates of the instrument’s resonance box“.
In today’s post I want to take a small diversion into the realm of Physics, in particular Astrophysics, to look at an extremely significant constant: Hubble’s constant.
Edwin Hubble measured the speed of galaxies and their distance from Earth and obtained the following graph:
As the graph is a straight line through the origin it shows that velocity is directly proportional to the distance from Earth. This is known as ‘Hubble’s law’.
Hubble’s Constant can be used to measure astronomical distances, which are too big to be measured by parallax or by using standard candles.
Hubble’s Constanthas a value of 2.3 x 10-18 s^-1or 72 km s^–1 Mpc^–1
[Mpc = megaparsec = 3.26 million light years]
and v = zc (where z is the redshift of the galaxy):
we can find the distance for any distant galaxy, provided we can measure its redshift (z).
We need to assume the straight line remains linear as the redshift becomes bigger and bigger – Hubble’s law holds universally.
This is not true, for example, with Hooke’s law, as it has a limit of proportionality (elastic limit).
Age of the Universe
All distant objects are moving away from us, suggesting that the Universe as a whole is expanding. If we turn back time, then the Universe would contract to a single point. This moment is called the Big Bang.
If we can find the Hubble constant, it will tell us how quickly the Universe is expanding, and from this we can work out how old our universe is.
If the universe was created at a time T ago, for a galaxy that has been moving away from us at a steady rate v for a time T, its distance d from us will now be vT.
Hubble’s Law tells us v = Hd, so v = HvT, which gives us HT = 1. Hence, the age of the universe can be given by:
T = 1/H
This gives an estimate of 14 billion years to 2 significant figures.
However, there are great uncertainties involved with this estimate:
Gravitational forces will mean that the present rate of expansion is less than in the past, so T < 1/H
Although the value for Hubble’s constant has become more accurate since the launch of the Hubble Space Telescope, the current value is only considered accurate to within 5%, so there is an uncertainty to the value for T.
Fate of the Universe?
CLOSED UNIVERSE: since gravity works against expansion, if the density were large enough then the expansion would stop and the universe would collapse in a ‘big crunch’. (Ω > 1)
OPEN UNIVERSE: If the density is small enough, then the expansion would continue forever – steady increase in Hubble’s constant. (Ω < 1)
Hope you enjoyed the post. Let me know what you think of more physics-based posts! M x
Today, to finish off the vectors series, I wanted to briefly discuss three applications of vectors to real life problems.
VECTORS IN MOVIES
Vectors can be extremely useful in computer graphics and computer vision, both used when creating computer generated movies. The characters in these movies are modelled as a surface that is made up of connected polygons – usually triangles. The vertices of these polygons are stored into the computers memory in an order such that the vector produced by these vertices is in the direction that points outwards (using the right hand rule, detailed in my previous post).
In order to establish the lighting of the scene that is being modelled, the ray tracing algorithm is used.
From the camera’s viewpoint, a ray is traced backwards towards the object, and is reflected off it. If the ray reflects off the facet and intersects the light source, it is shaded in a lighter colour, as in real life it would be lit up by this light source. If not, it is shaded with a darker colour.
The act of tracing the ray backwards is done mathematically using vectors. Suppose the equation of the ray is and the equation of the plane defined by a facet with vertices a1, a2 and a3 is , to calculate if and where the ray intersects the facet we need to solve these two expressions.
Furthermore, the rotation of three dimensional objects is done using quaternions (discussed in my blog post on Monday). Let’s say we want to rotate a point A with coordinates (a1, a2, a3), through an angle , about and axis through the origin, given by a vector b = (b1, b2, b3). First, we construct two quaternions:
Then, using the expression A’ = q1Aq2, we can multiply A by these two quaternions to give the point A’ – the position of A after rotation!
Aircraft turbulence is a phenomena that can be explained using Fluid Dynamics, which is a highly mathematical area of science and is of great practical importance. Mathematicians use vector fields to model the speed and direction of a moving fluid through space. A vector field is ‘an assignment of a vector to each point in a subset of space‘. In the case of a velocity field of a moving fluid, a velocity vector is associated to each point in the fluid.
Let’s take the example of the velocity of the air. This may vary from point to point and it may also vary with respect to time. This may be shown using the following notation:
u ( x, t )
where u = velocity, x = position in space and t = time.
Vectors can come in useful when studying the movement of ice, which largely concerns the issue of climate change.
Let’s take a piece of ice with an area of 1m2. The net force on the piece of ice arises from the various stresses acting on the ice, such as air stress (due to wind) and water stress (due to currents), as well as the movement of the Earth, which gives rise to the Coriolis force. Combining this gives us:
Force = Acceleration x Mass = Air Stress + Water Stress + Coriolis Force
This can be written vectorially, as only mass is a scalar quantity:
These vectors encode the direction of the movement of the force acting on the piece of ice, as well as the magnitude.
Calculating the values of water stress, air stress and the Coriolis force using individual equations, we can predict where the ice block will move.
In the example above: the air stress acting on the top of the floe (‘a sheet of floating ice’) is in the direction of the wind; the water drag is opposite to the direction in which the ice floe is moving; the Coriolis force is at right angles to the direction of movement (shown for the Northern hemisphere). The result is steady motion under a triangle of forces.
This week I am going to be doing a series on Vectors, starting off with the history of vectors.
Vectors were born during the start of the 19th century, due to the need to represent complex numbers geometrically. The mathematicians Caspar Wessel (1745-1818), Jean Robert Argand (1768-1822) and Carl Friedrich Gauss (1777-1855) were the first to show complex numbers as being points in a two-dimensional plane, and hence as two-dimensional vectors.
This idea was part of the effort to extend two-dimensional numbers to three dimensions, however at the time no one was able to accomplish this whilst still preserving the basic algebraic properties of real and complex numbers.
In 1827, August Ferdinand Möbius published a book entitled ‘The Barycentric Calculus’, where he introduced line segments which had a direction and where denoted by letters. Basically, these were vectors in all but the name! In the book, he showed how to perform calculations with these line segments – how to add them and multiply them with a real number. However, these accomplishments and their importance were not noticed by the mathematical community.
In 1843, William Hamilton introduced quaternions – a four dimensional system. Hamilton expressed:
“I was walking in to attend and preside along the Royal Canal, an under-current of thought was going on in my mind, which at last gave a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, I could not resist the impulse to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formulae.”
Quaternions were of the form: q = w + ix + jy + kz, where w, x, y and z are real numbers. Hamilton realised that quaternions consisted of two parts – the scalar and the vector part. Hamilton used his ‘fundamental formula’: i2 =j2 = k2 = –ijk = -1, to multiply them together and thus discovered that the product of quaternions was not commutative.
In the 1850s, Peter Guthrie Tait applied quaternions to problems involving electricity and magnetism and other problems in physics.
Around the same time, Hermann Grassman published the book ‘The Calculus of Extension’, which developed a new geometric calculus. He used this to simply large parts of two classical works – Analytical Mechanics by Lagrange and Celestial Mechanics by Laplace. Grassmann expanded the concept of a vector from two or three dimensions to n dimensions, which greatly extended the ideas of space. Furthermore, he anticipated a large amount of modern matrix, linear algebra and vector and tensor analysis. However, his work was largely ignored as it was lacking in explanatory examples and written in a strange style with overcomplicated notation.
The development of the algebra of vectors and vector analysis is largely credited to J. Willard Gibbs. Although British scientists, including Maxwell, relied on Hamilton’s quaternions in order to express the dynamics of physical quantities, Gibbs was the first to note that the product of quaternions always had to be separated into two parts. Hence, calculations with quaternions introduced unnecessary complications and redundancies that could be removed. Therefore, he proposed defining distinct dot and cross products for pairs of vectors and introduced the vector notation we use today.
While working on vector analysis, Gibbs realised that his approach was similar to that of Grassmann and thus sought to publicise Grassmann’s work, stressing that is was more general than Hamilton’s quaternions.
Erwin Schrödinger was an Austrian theoretical physicist, born in 1887, who contributed largely to the wave theory of matter and other fields in quantum mechanics.
Schrödinger’s wide array of interests began at a young age; he not only liked scientific disciplines, but also admired the logic of ancient grammar and the beauty of German poetry. In 1906, he became a student at the University of Vienna, where he studied under Fritz Hasenöhrl, Boltzmann’s successor, and stayed there for 4 years. During these years, Schrödinger mastered eigenvalue problems in the physics of continuous media, thus laying the foundation for his future work.
In 1921, Schrödinger took up a position at the University of Zurich, where he stayed for 6 years. It was here that he enjoyed huge amount of contact and friendship with many of his colleagues such as Hermann Weyl and Peter Debye and it was one of his most fruitful periods. He actively engaged in various subjects in theoretical physics; his papers of that time deal with heats of solids, problems in thermodynamics and atomic spectra, as well as some physiological studies of colour.
His greatest discovery – Schrödinger’s wave equation – was conceived at the end of his time in Zurich (in the first half of 1926), and was published in Annalen der Physik in the paper entitled ‘Quantisierung als Eigenwertproblem’. This wave equation correctly calculated energy values for electrons in an atom. In the paper, Schrödinger gave a derivation of the wave equation for time-independent systems and showed that it gave the correct energy eigenvalues for a hydrogen-like atom. This paper has been celebrated as one of the most important achievements in quantum mechanics in the 20th century.
This result came as a result of his dissatisfaction with the quantum condition in Bohr’s orbit model and his belief that atomic spectra should really be determined by a eigenvalue problem.
“Schrödinger was breaking new ground and did the heroic job of getting the right equation. How you get the right equation, is less important than getting it. He did such a wonderful job of then deriving the hydrogen atom wave function and much more. So did he understand what he had? You bet, he was really right on target.”
– Marlan O. Scully, a physics professor at Texas A&M University
In 1927, he succeeded Max Planck at the Friedrich Wilhelm University in Berlin, where he met Albert Einstein. However, in 1934 Schrödinger decided to leave Germany due to the Nazis’ growing anti-semitism. As a result he moved to the United Kingdom, where he became a Fellow of Magdalen College in the University of Oxford. It was here that Schrödinger learned that he had won the 1933 Nobel Prize in Physics, sharing the award with Paul Dirac. In his Nobel Prize acceptance speech, Schrödinger stated that his mentor, Hasenöhrl, would be accepting the award if he hadn’t died during World War I.
POST NOBEL PRIZE
In 1939, he moved to work at Institute for Advanced Studies in Dublin, Ireland, heading its School for Theoretical Physics. He remained in Dublin until the mid-1950s, returning in 1956 to Vienna, where he continued his career at his alma mater.
In 1944, Schrödinger wrote the book ‘What is Life?‘, which contains a discussion on the negative entropy of living systems, and the concept of a complex molecule with the genetic code for living organisms. According to James Watson‘s memoir ‘The Secret of Life‘, this book gave him the inspiration to research this gene, leading to the discovery of the DNA double helix structure in 1953.
Sadly, in 1961 Schrödinger died in Vienna. To this day, Schrödinger is considered as the father of quantum mechanics. A few of his
During (and after) his lifetime, Schrödinger acquired many honours and awards apart from the Nobel Prize, including:
Max Planck Medal in 1937: awarded for extraordinary achievements in theoretical physics
Elected a Foreign Member of the Royal Society in 1949
The large crater Schrödinger, on the far side of the Moon, is named after him.
The Erwin Schrödinger International Institute for Mathematical Physics was established in Vienna in 1993.
Many of you may recognise the name ‘Schrödinger’ due to Schrödinger’s cat, which is a though experiment devised by Schrödinger in 1935. The video below will help explain it:
Although the realm of Mathematics in Physics is vast, I wanted to start a series where I talk about famous physicists whose work involved a large quantity of mathematics, starting off with Richard Feynman.
Feynman was born in New York City in 1918, and studied at the Massachusetts Institute of Technology (MIT) where he obtained his B.Sc. in 1939 and at Princeton University where he obtained his Ph.D in 1942.
Feynman is most well known for his work in quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercool liquid helium, and particle physics for which he proposed the parton model: “a way to analyse high-energy hadron collisions“. Due to his contributions to quantum electrodynamics, he received, along with Julian Schwinger and Sin-Itiro Tomonaga, the Nobel Prize in Physics in 1965.
He is also well known for diagrams that were used to explain the interaction between the sub-atomic particles, which later became known as ‘Feynman Diagrams’.
Robert R. Wilson, a physicist at Princeton, encouraged Feynman to help develop the atomic bomb during World War II in the Manhattan Project. He was assigned to Hans Bethe’s division, which was theoretical, and impressed Bethe sufficiently to be made group leader. Jointly with Bethe, he developed the Bethe-Feynman formula, which calculated the yield of a fission bomb.
a = internal energy per gram // b = growth rate // c = sphere radius
However, as a junior physicist, Feynman was not central to the project, and thus a large part of his work was administering the computation group of ‘human computers’ in his division. Furthermore, he aided in the establishment of the system for using IBM punched cards for computation.
During his time in Los Alamos – the site of the Manhattan Project – he was sought out by Niels Bohr for one-to-one discussions, due to the fact that he had no inhibitions to say what he thought; most scientists were too much in awe of Bohr to argue with him.
As depicted in his semi-autobiographical book I mentioned above, Feynman became bored in Los Alamos as it was very isolated: “There wasn’t anything to do there“. As a result, Feynman gained a fascination for safecracking. The documents that were generated by bomb work, which contained top secret information, were mostly kept in filing cabinets or combination safes, and there was an assumption of their safety. Feynman set out to prove this to be wrong by becoming an expert safecracker. By reading books by professionals and developing his own methods, he became notorious for his ability to open safes.
“At one instance, just after the close of the war, he had a rare opportunity to put this talent to use, wherein he managed to open a bank of files which contained every document for the construction of the bomb, thus showing conspicuously the edge on which our civilisation sometimes teeters.”
Feynman died of two rare forms of cancer in 1988, aged 69. His last words are noted as:
I’ve tried to condense Richard Feynman’s work into a short blog post, which is truly hard to do as he contributed such a large amount to the scientific community in many different areas. I would highly recommend reading ‘Surely You’re Joking, Mr. Feynman’ to find out more about Feynman’s character and personality, or ‘QED’ to read about his Nobel Prize winning work. Hope you enjoyed this post nonetheless! M x