Matrix elements of unitary group generators in many-fermion correlation problem. I. tensorial approaches

You should be warned that acquaintance with only one of the approaches will deprive you of techniques and understanding reflected by the other approaches ...

(Serge Lang, Algebraic Number Theory, Addison Wesley, Reading, MA, 1970, p. 176.)

Abstract

The objective of this series of papers is to survey important techniques for the evaluation of matrix elements (MEs) of unitary group generators and their products in the electronic Gel’fand–Tsetlin basis of two-column irreps of U(n) that are essential to the unitary group approach (UGA) to the many-electron correlation problem as handled by configuration interaction and coupled cluster approaches. Attention is also paid to the MEs of one-body spin-dependent operators and of their relationship to a standard spin-independent UGA formalism. The principal goal is to outline basic principles, concepts, and ideas without getting buried in technical details and thus to help an interested reader to follow the detailed developments in the original literature. In this first instalment we focus on tensorial techniques, particularly those designed specifically for UGA purposes, which exploit the spin-adapted tensorial analogues of the standard creation and annihilation operators of the ubiquitous second-quantization formalism. Subsequent instalments will address techniques based on the graphical methods of spin-algebras and on the Green–Gould polynomial formalism. In the “Appendix A” we then provide a succinct historical outline of the origins of the Lie group and Lie algebra concepts.

Appendices

Appendix A: historical notes

This appendix is intended as a brief (and thus very incomplete) historical outline of an exciting 19th century development of the mathematical theory of symmetry as pioneered by Sophus Marius LIE (1842–1899) and Wilhelm Karl Joseph KILLING (1847–1923) (for a more detailed, excellent, and accessible account see Ref. [104]). These advances provided a fertile ground for the 20th century progress in many fields of mathematics, pioneered primarily by Elie Joseph CARTAN (1869–1951), and Wilhelm Karl Joseph KILLING (1847–1923), as well as by Hermann Klaus Hugo WEYL (1885–1955), Eugene Paul WIGNER (1902–1995), and Wolfgang PAULI (1900–1958), whose work turned out to be of a great significance for quantum theory and relativity [105].

For a start, let us first point out that the history of the mathematical theory of symmetry is very young. It is truly remarkable that in spite of the omnipresence of symmetry all around us from times immemorial, in nature, arts, and, indeed, geometry and mathematics (for example, the so-called Platonic solids were known and admired for millennia), the very concept of a “group"— and of the basic group theory—originated from a rather esoteric and abstract studies of algebraic structures (which we would nowadays call field extensions, constituting the basis of Galois theory). Indeed, almost all present day textbooks introduce the concept of a group by relying on the symmetry properties of simple geometric figures in ${\mathbb {R}}^2$ or objects in ${\mathbb {R}}^3$, leading to the well-known point groups.

Thus, it was Evariste GALOIS (1811–1832), “révolutionnaire et géomètre", who is generally credited with the creation of a modern group theory and who was the first mathematician to use this term. During his short and turbulent life (he failed twice the entrance exams to École Polytechnique and was expelled from École Normale—the second best school in France at that time—for his revolutionary activities as a republican) he developed a new theory of algebraic equations, introducing the concept of a group in the process (specifically for the groups of permutations), which enabled him to prove the impossibility of expressing the roots of higher than quartic equations in terms of radicals.

As is generally known, Galois died prematurely in a duel (defending the “honour" of an “infâme coquette"), yet prior to this tragic event managed to dispatch four “mèmoirs" to various members of the “Academie des Sciences". The one he sent to Siméon Denis POISSON (1781–1840) was returned to him as “incomprehensible", while those sent to the baron Jean-Baptiste Joseph FOURIER (1768–1830) and (two) to baron Angustin Louis CAUCHY (1789–1857) remained unanswered. Since Cauchy himself made seminal contributions to group theory in about 1845 (although he never used the term “group"), he was sometimes suspected of deliberately hiding Galois’ manuscripts. However, it seems more likely that he never read these mèmoirs, since he was himself mixed in political upheavals of the time (downfall of Bourbons to whom Cauchy was loyal) and in fact left France for several years to Italy. It was only three years after Cauchy’s death when Marie Ennemond Camille JORDAN (1838–1922), one of the leading mathematicians of the time, was appointed to edit Cauchy’s collective works that he found a sealed envelope with Galois’ manuscripts. Fortunately, Jordan was himself very much interested in these problems and thanks to the advancements in mathematics that occurred in the meantime was able to appreciate Galois’ pioneering work. He tried to find all of Galois’ work, further elaborated and developed it (he was the first to study group representations, quotient groups, and infinite groups, particularly the so-called classical groups), and popularized this work in his “Traitè des substitutions et des equations algèbriques ", published in 1870. The influence of this work on the subsequent development of the subject would be hard to overestimate.

We would be amiss not to mention, however briefly, other important contributions to this field. One of the great 19th century mathematicians, whose work and destiny parallels that of Galois in many respects, was a young Norwegian Niels Henrik ABEL (1802–1829). His “mèmoire" on general properties of transcendental functions, which he presented to the French Academy of Sciences, was also “lost" by Cauchy and “forgotten" by LEGENDRE (Adrien-Marie, 1752–1833) to whom it was entrusted by Fourier. Abel’s work was no more appreciated in Germany at the time, except by an amateur mathematician and a prominent rich entrepreneur August Leopold CRELLE (1780–1855), who founded the famous “Crelle Journal ", properly called the “Journal für reine und angewandte Mathematik " [106]. In the very first issue of this journal (from 1826) appeared an extensive mèmoire (in French) by Abel on “Démonstration de l’impossibilité de la résolution algébrique des équations générales qui passent le quatrième degré ". Although this work precedes that of Galois, it lacks the generality of Galois’ theory (essentially, in modern terminology, he treated the cases when the Galois group is commutative or cyclic; hence the present day term for “abelian groups"). Coming from an impoverished millieu and being shy and very modest, Abel received little credit during his short lifetime. Only in 1828 was he elected a professor in Berlin, but unfortunately did not live to receive the official notice of his appointment (he was in Christiania—present day Oslo—at the time, visiting his fiancé before Christmas), dying of tuberculosis at the age of 27.

The concept of abstract groups was perhaps first developed in the early 1850s by Arthur CAYLEY (1821–1895), the father of matrix algebra. In fact, at this time the only known groups were still the permutation (or symmetric) groups, yet Cayley realized that matrices and quaternions may also be endowed with a group structure. After many year’s hiatus, he resumed his studies of group theory in the later 1870s. The crucial turning point for the study of finite groups came at the turn of the century (1897) with the publication of the monograph on “Theory of groups of finite order " by William BURNSIDE (1852–1927).

By the time when Jordan published his “Traitè des substitutions " (understand “treatise on permutations"), which systematized Galoi’s ideas and developed the basic group theory, two young mathematicians, the Norwegian Sophus LIE and the seven years younger German, Christian Felix KLEIN (1849–1925) went to Paris to meet him and to study with him. Lie came from the family of a Lutheran pastor near Bergen and initially wished to follow in his father’s footsteps. Later he contemplated a military career, but had to give up the idea due to his poor eyesight. He eventually entered the University of Christiania (which later became Kristiania and in 1925 Oslo) to study mathematics and natural sciences. Although his teachers were highly qualified [e.g., he attended lectures by Peter Ludwig Mejdell SYLOW (1832–1918) on Abel’s and Galois’ work], he showed neither exceptional ability nor liking for the subject. He simultaneously taught private pupils while trying to decide on the course of his academic career and studied some astronomy and mechanics, and even botany and zoology. The turning point came in 1868 when he read the works of Jean Victor PONCELET (1788–1867) and Julius PLÜCKER (1801–1868). Poncelet was a student of the founder of “gèometrie descriptive", Gaspard MONGE (1746–1816). As an officer in Napoleon’s army, Poncelet was taken a prisoner during the 1812 Russian campaign, and during his two year incarceration near Saratov on the Volga river he lectured to his fellow officers developing the field of projective geometry. (Could this happen in today’s World?) Plücker—who preferred a more algebraic approach to analytical geometry in contrast to a purely geometric approach of Poncelet—was in fact a teacher of Klein at the University of Bonn. Since he was more closely associated with the French and British schools rather than the German ones, his revolutionary ideas that broke completely with Cartesian view of coordinates as line segments, were opposed by his compatriots (especially by Steiner and Jacobi). Since he was an extremely talented individual who was just as well at home in abstract mathematics as in experimental physics, the discouragement he received from his fellow-geometers eventually made him turn from geometry to physics (he is the founder of spectral analysis and with his student Hittorf the discoverer of the cathode rays). Although he tried to make a physicist out of Klein, the latter was nonetheless more impressed by his “line geometry" (cf. Plücker coordinates), just as his friend Sophus Lie was [107].

The first paper by Lie on line geometry was accepted by Crelle’s Journal in 1869 and its impact enabled him to secure himself a scholarship to travel. He went to Prussia and visited both Göttingen and Berlin, two key centers of mathematics at that time. He was not enticed by Weierstrass’s mathematics that dominated the Berlin school at that period of time and attended instead Kummer’s seminar on the geometry of ray systems, which is intimately related with line geometry [Ernst Eduard Kummer (1810–1893)]. Here he met and befriended Klein who came to Berlin from Bonn for his postdoctoral studies.

Both Klein and Lie were experts in Plückerian line geometry and started close friendship and intense collaboration. As we already noted above, in the Summer of 1870, they decided to go to Paris to study with Jordan and Gaston DARBOUX (1824–1917) [108], the latter an expert in differential geometry who focussed on local properties of smooth curves and surfaces, thus combining the differential geometry with the theory of differential equations: the topic which played the key role in Lie’s later work. Unfortunately, their studies with Jordan and Darboux were soon interrupted by the 1871 Franco-Prussian War. Klein, who was a great patriot, immediately returned to Germany to join the Prussian army. However, his military adventure was cut short when he contracted typhus and by the time he recuperated the war was over.

Lie, who was always a great nature fan and very homesick for his native Norway, decided to use the opportunity to make a big hike in the Alps rather than to be alone in Paris. But, alas, this was not a good time for hiking in the border regions of the country at war. Moreover, Lie’s French was rather limited and his physical Nordic appearance (according to his contemporaries, he was an embodiment of a true Viking: very tall, blond and blue-eyed) made him suspicious and eventually imprisoned in Fontainebleau. This suspicion was further enhanced by his aloofness when he was immersed in his thoughts only to intermittently and hastily jot down his ideas in his notebook. Since the guards could not make sense of his mathematical formulas and Norwegian notes, they took it for a secret code, thus further enhancing their misgivings. Fortunately, Darboux soon learned of his arrest and used his contacts and influence to have him freed. Undaunted, Lie resumed his hiking.

Nonetheless, the results he obtained while in the Fontainebleau prison and hiking brought Lie high regard and recognition [109]. After spending some time with Klein in Göttingen and a year in Lund, Sweden, he was offered a professorship at his alma mater in the present day Oslo, where he worked for the subsequent 14 years.

Klein’s and Lie’s friendship and collaboration continued after the Franco-Prussian War and lasted for many years in spite of their very different personalities and approach to mathematics. While Lie was a very sensitive man, generally of ill-tempered nature, who often suffered from homesickness and a lack of recognition—a condition that later developed into neurasthenia, requiring hospitalization— Klein, on the other hand, was a very outgoing and multitalented man, an excellent expositor and organizer as well as a researcher. Klein’s development was certainly very much influenced by Plücker, who tried to steer him towards physics, as already noted above. Although he did not succeed, he certainly influenced Klein’s approach to mathematics and science in general.

After the Franco-Prussian War, Klein joined for awhile his mentor, Rudolph Friedrich Alfred CLEBSCH (1833–1872) in Göttingen and continued his collaboration with Lie. In 1872, after Clebsch passed away, he was appointed a full professor in the recently established Mathematics Department of the University of Erlangen. Apparently Lie even accompanied him to Erlangen, both intensely discussing their future work in geometry. These ideas were summarized by Klein in his inaugural lecture and became known as The Erlangen Program [110]. Nonetheless, their interests started to bifurcate at this stage, with Klein devoting himself to discrete groups, while Lie pursued continuous groups.

Being very successful with his “Erlanger Programm", Klein received “a call" to the prestigious Technical University (Technische Hochschule) in Munich in 1875, and five years later (in 1880) he joined the Department of Geometry at the University of Leipzig. Here he developed the theory of automorphic functions, “combining Galois and Rieman", as he himself put it (i.e., implanting group theoretical formalism into the Riemanian geometry). In 1886 Klein had finally managed to return to Göttingen, where following a sudden demise of Clebsch in 1872 (who died unexpectedly of diphtheria at the age of 39) the Berlin establishment, so much resented by Klein (and vice versa, see above), dominated the scene thanks to Weierstrass’s very able student Karl Herman Amandus SCHWARZ (1843–1921). Moreover, he managed to arrange the appointment of Lie, who was rather isolated and thus professionally rather unhappy in Christiania [even though Klein sent him there an excellent collaborator Friedrich ENGEL (1861–1941), who co-authored several books with Lie and later published Lie’s collected works], to succeed him in Leipzig (much to the annoyance of both Schwarz and Weierstrass).

Klein also seriously suffered from exhaustion—both due to overwork and various rivalries, especially with Jules Henri POINCARÉ (1854–1912), and the Berlin establishment, and suffered even a nervous breakdown while in Leipzig. Although he never fully regained his youthful vigor, he was extremely influential and productive (he edited “Mathematische Annalen " after Clebsch’s extinction and made it into a leading journal worldwide, he undertook an enormous task of publishing Gauss’s collective works in 1898, directed work on the “Enzyklopädie der Mathematischen Wissenschaften ", etc.), and by the end of the nineteenth century, the Göttingen school was firmly associated with his and David HILBERT’s (1862–1943) names. Their talents greatly complemented one another, Klein being a great organizer and administrator while Hilbert directed the scientific activities. Klein even organized (in 1898) and headed the International Commission on Mathematical Education, trying very hard “to eliminate the ‘China wall’ separating different mathematical disciplines", was elected a corresponding member of the Berlin Academy of Sciences in 1913, received the title of a “Geheimrat ", and represented the University of Göttingen in the Upper Chamber of the Prussian Parliament. Yet, in spite of these official functions and his patriotism, he detested chauvinism of some of his compatriots (for example, he refused to sign Wilhelm Ostwald’s infamous anti-British and anti-French manifesto at the start of WWI), and established an undelible humanistic tradition in Göttingen, which survived him for a long time. It was primarily for this heritage that the University of Göttingen was decimated by Nazi’s when they came to power.

Lie’s succession of Klein in Leipzig—which lasted for 12 years— turned out to be very productive researchwise, though not very happy in personal terms. He had excellent students during this time [Engel, with whom he collaborated already in Christiania, Georg SCHEFFERS (1866–1945), an outstanding pedagogue and differential geometer, Friedrich SCHUR (1856–1932) [not to be confused with Issaï Schur of Schur’s Lemma], topologist Felix HAUSDORFF (1868–1942), and others.

Unfortunately, an unceasing overwork, feelings of alienation, and the resulting depression led to an ill-fated mishap that poisoned— at least temporarily—his long-time friendship with Klein. In the Introduction to the third volume of his “Theorie der Transformationsgruppen " (1893 edition, co-authored by Engel), Lie gives Klein credit for his work on discrete groups (while leaving no doubt about his own contribution to the “Erlanger Programm"), but makes it clear that the work on “continuous groups defined by DEs" is entirely his own, as is the important concept of differential invariants, of which there is no trace in Klein’s Program. He writes: “Klein made no contribution to this concept, which is the basis of the general invariant theory, and only thanks to me he found out that every DE determines a group of differential invariants..." [111].

To add insult to injury, he continues to state that he was induced to this discussion by the statements of Klein’s students and friends who falsely present his and Klein’s contribution to the subject and concludes: “I am no pupil of Klein, nor is the opposite the case, although it might be closer to the truth" [112]. He states that he highly appreciates Klein’s many talents and the help he provided to him, yet in the same sentence he chastises him for his inability to make a clear distinction between the derivation and the proof, and between the idea of a concept and its exploitation [113].

Fortunately, Klein was aware of Lie’s medical problems and was magnanimous enough not to reciprocate in this unfortunate squabble, in spite of being genuinely hurt. Lie, when he recuperated from his depression, himself realized his mistake. It seems that both Klein and Lie were sufficiently high-minded to forget the incident, and Lie was again welcome in Klein’s house as before.

Lie’s affection for his homeland made him to return to Christiania in 1898 (in fact, he never resigned his chair in Christiania and was on a 12-year leave of absence while in Leipzig!). Unfortunately, by this time, his health severely deteriorated and he died of pernicious anemia in February of 1899.

In 1897, shortly before his passing away, Lie was awarded, as the very first recipient, Lobachevsky’s prize and medal of the Physical-Mathematical Society of the Imperial Kazan University. In fact, it was Felix Klein who nominated him and wrote the detailed reivew of his work. This prize became very prestigious in the subsequent years, the next three recipients being Wilhelm Karl Joseph KILLING (1847–1923), David Hilbert and Felix Klein himself. Later on, Poincaré, Weyl, Cartan, Georges DE RHAM (1903–1990), and Heinz HOPF (1894–1971) were similarly honoured.

The subject of Lie groups provided a very fertile ground for the subsequent development in the 20th century mathematics, even though the focus has shifted towards the structure, classification, and representation theory of Lie groups and corresponding Lie algebras, pioneered by Cartan, Killing, Weyl, Wigner, Pauli and others.

Appendix B: Tables

For reader’s convenience we present here the relevant part of Tables 1 and  2 of Ref. [88] (Tables 1 and 2).

Table 1 Relevant scaled isoscalar factors

Table 2 Nonvanishing raising-generator RMEs (cf., Eq. (51))