Influential Mathematicians: Sophus Lie
Hello quantum enthusiasts! For those who recently joined the community, this is our pre-course newsletter series aimed at highlighting mathematicians that have directly or indirectly contributed to the mathematical formalism of quantum mechanics. It is our humble attempt of taking any fear of mathematics out of your mind by showing that mathematicians are ordinary humans too!
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Going back to the topic of influential mathematicians series, a LinkedIn connection recently asked if I was covering more French mathematicians because of my French connection! The answer is no, it’s just French mathematicians happen to have made significant contributions to the branches of mathematics relevant to the abstract formalism of quantum mechanics.
Anyway, today I'll cover Sophus Lie, a Norwegian mathematician!:)
Who was Sophus Lie?
The School of Mathematics and Statistics University of St Andrew has a pretty comprehensive bio on Sophus Lie that I've summarised for you below.
Sophus Lie's father was Johann Herman Lie, a Lutheran minister. His parents had six children and Sophus was the youngest of the six. Sophus first attended school in the town of Moss, which is a port in south-eastern Norway, on the eastern side of the Oslo Fjord. In 1857 he entered Nissen's Private Latin School in Christiania (the city which became Kristiania, then Oslo in 1925) . While at this school he decided to take up a military career, but his eyesight was not sufficiently good so he gave up the idea and entered University of Christiania.
At university Lie studied a broad science course. There was certainly some mathematics in this course, and Lie attended lectures by Ludwig Sylow in 1862. Although not on the permanent staff, Sylow taught a course, substituting for Broch, in which he explained Abel's and Galois' work on algebraic equations. Lie also attended lectures by Carl Bjerknes on mathematics, so he certainly had teachers of considerable quality, yet he graduated in 1865 without having shown any great ability for the subject, or any great liking for it.There followed a period when Lie could not decide what subject to pursue and he taught pupils while trying to make his decision. The one thing he knew he wanted was an academic career and he thought for a while that astronomy might be the right topic. He learnt some mechanics, wondered whether botany or zoology or physics might be the right subjects and in general became rather confused. However, there are signs that from 1866 he began to read more and more mathematics and the library records in the University of Christiania show clearly that his interests were steadily turning in that direction.
It was during the year 1867 that Lie had his first brilliant new mathematical idea. It came to him in the middle of the night and, filled with excitement, he rushed to see his friend Ernst Motzfeldt, woke him up and shouted:‘‘I have found it, it is quite simple!’’
This was not the end of Lie's problems of course (far from it for Lie would always have problems), but at least in his own mind he now knew the career he wanted and it would be fair to say that from that moment on Lie became a mathematician. The type of mathematics that Lie would study became more clearly defined during 1868 when he avidly read papers on geometry by Plücker and Poncelet.
Lie wrote a short mathematical paper in 1869, which he published at his own expense, based on the inspiration which had struck him in 1867. He wrote up a more detailed exposition, but the world of mathematics was too cautious to quickly accept Lie's revolutionary notions. The Academy of Science in Christiania was reluctant to publish his work, and at this stage Lie began to despair that he would become accepted in the mathematical world. His friend Motzfeldt did a superb job of encouraging Lie to press on with his mathematical ideas and the breakthrough came later in 1869 when Crelle's Journal accepted his paper. He sent letters to two Prussian mathematicians, Reye and Clebsch, still attempting to gain recognition for his ideas. The paper in Crelle's Journal, however, proved vital for, on the strength of the paper, Lie was awarded a scholarship to travel and meet the leading mathematicians.
Setting off near the end of the year 1869, Lie went to Prussia and visited Göttingen and then Berlin. In Berlin he met Kronecker, Kummer and Weierstrass. Lie was not attracted to the style of Weierstrass's mathematics which dominated Berlin. His interests fitted more closely with Kummer, and Lie lectured on his own results in Kummer's seminar and was able to correct some errors that Kummer had made in his work on line congruences of degree 3. Most important to Lie, however, was the fact that in Berlin he met Felix Klein. It was easy to see that these two would instantly find common ground in mathematics since Klein had been a student of Plücker, and Lie, although he never met Plücker, always said that he felt like Plücker's student.
In the spring of 1870 Lie and Klein were together again in Paris. There they met Darboux, Chasles and Camille Jordan. Jordan seems to have succeeded in a way that Sylow did not, for Jordan made Lie realise how important group theory was for the study of geometry. Lie started to develop ideas which would later appear in his work on transformation groups. He began to discuss with Klein these new ideas on groups and geometry and he would collaborate later with Klein in publishing several papers. This joint work had as one of its outcomes Klein's characterisation of geometry in his Erlangen Program of 1872 as properties invariant under a group action. While in Paris Lie discovered contact transformations. These transformations allowed a 1-1 correspondence between lines and spheres in such a way that tangent spheres correspond to intersecting lines.
While Lie and Klein thought deeply about mathematics in Paris, the political situation between France and Prussia was deteriorating. The popularity of Napoleon III, the French emperor, was declining in France and he thought a war with Prussia might change his political fortunes since his advisers having told him that the French Army could defeat Prussia. Bismarck, the Prussian chancellor, saw a war with France as an opportunity to unite the South German states. With both sides feeling that a war was to their advantage, the Franco-Prussian War became inevitable. On 14 July, Bismarck sent a telegram which infuriated the French government and on the 19 July France declared war on Prussia. For Klein, a Prussian citizen who happened to be in Paris when war was declared, there was only one possibility: he had to return quickly to Berlin.
However, Lie was a Norwegian and he was finding mathematical discussions in Paris very stimulating. He decided to remain but became anxious as the German offensive met with only an ineffective French reply. In August, the German army trapped part of the French army in Metz and Lie decided it was time for him to leave and he planned to hike to Italy. He reached Fontainebleau but there he was arrested as a German spy, his mathematics notes being assumed to be top secret coded messages. Only after the intervention of Darboux was Lie released from prison. The French army had surrendered on 1 September, and on 19 September the German army began to blockade Paris. Lie fled again to Italy, then from there he made his way back to Christiania via Germany so that he could meet and discuss mathematics with Klein.
In 1871 Lie became an assistant at Christiania, having obtained a scholarship, and he also taught at Nissen's Private Latin School in Christiania where he had been a pupil himself. He submitted a dissertation On a class of geometric transformations (written in Norwegian) for his doctorate which was duly awarded in July 1872. The dissertation contained ideas from his first results published in Crelle's Journal and also the work on contact transformations, a special case of these transformations being a transformation which maps a line into a sphere, which he had discovered while in Paris.
It was clear that Lie was a remarkable mathematician and the University of Christiania reacted in a very positive way, creating a chair for him in 1872. The famous Norwegian mathematician Abel had died more than 40 years before this (some 14 years before Lie was born) but, despite Abel's short career, his complete works had not been published at that time. It was natural that Norwegian mathematicians would undertake the task, and between 1873 and 1881 Sylow and Lie prepared an edition of Abel's complete works. Lie, however, always claimed that most of the work was done by Sylow. Another event which took place within two years of Lie being appointed to his chair was his marriage. He married Anna Birch and they would have three children, one daughter and two sons.
Lie had started examining partial differential equations, hoping that he could find a theory which was analogous to the Galois theory of equations. He examined his contact transformations considering how they affected a process due to Jacobi of generating further solutions of differential equations from a given one. This led to combining the transformations in a way that Lie called an infinitesimal group, but which is not a group with our definition, rather what is today called a Lie algebra. It was during the winter of 1873-74 that Lie began to develop systematically what became his theory of continuous transformation groups, later called Lie groups leaving behind his original intention of examining partial differential equations. Later Killing was to examine the Lie algebras associated with Lie groups. He did this quite independently of Lie (and not it would appear in a manner which Lie found satisfactory), and it was Cartan who completed the classification of semisimple Lie algebras in 1900.
Although Lie was producing highly innovative mathematics, he became increasingly sad at the lack of recognition he was receiving in the mathematical world. One reason was undoubtedly his isolation in Christiania, but a second reason was that his papers were not easily understood, partly through his style of writing and partly because his geometrical intuition greatly exceeded that of other mathematicians. Klein, realising the problems, had the excellent idea of sending Friedrich Engel to Christiania to help Lie.
Engel had received his doctorate from Leipzig in 1883 having studied under Adolph Mayer writing a thesis on contact transformations. Klein recognised that he was the right man to assist Lie and, at Klein's suggestion, Engel went to work with Lie in Christiania starting in 1884. He worked with Lie for nine months leaving in 1885. Engel then was appointed to Leipzig and, when Klein left the chair at Leipzig in 1886, Lie was appointed to succeed him. The collaboration between Engel and Lie continued for nine years culminating with their joint major publication Theorie der Transformationsgruppen in three volumes between 1888 and 1893. This was Lie's major work on continuous groups of transformations. In Leipzig, life for Lie was rather different from that in Christiania. He was now in the mainstream of mathematics and students came from many countries to study under him.
Quantum Formalism Influence
Although Lie did not directly contribute to the actual quantum formalism, Lie Groups and Lie Algebras named after him are very important to the Hilbert space formalism of quantum mechanics. For example, their unitary representations in the context of symmetry generated by operators ‘commuting’ with the energy operator i.e. Hamiltonian.
That’s it for today! I’ll end with some reading materials related to Sophus Lie:
By the way, Lie Groups & Lie Algebras will be covered in the advanced module of the course ;)
Author: Bambordé Baldé, co-founder at Zaiku Group. Please feel free to connect with us on social media below!;)