Erwin Rudolf Josef Alexander Schrödinger ( SHROH-ding-er; German: [ˈʃʁøːdɪŋɐ] ; 12 August 1887 – 4 January 1961), occasionally spelled as Schroedinger or Schrodinger, was an Austrian-Irish theoretical physicist renowned for his foundational contributions to quantum theory. Notably, he is credited with formulating the Schrödinger equation, which enables the calculation of a system's wave function and its temporal evolution. In 1935, he introduced the concept of "quantum entanglement." Schrödinger was a co-recipient of the 1933 Nobel Prize in Physics, awarded jointly with Paul Dirac, "for the discovery of new productive forms of atomic theory."
Erwin Rudolf Josef Alexander Schrödinger ( SHROH-ding-er; German: [ˈʃʁøːdɪŋɐ] ; 12 August 1887 – 4 January 1961), sometimes written as Schroedinger or Schrodinger, was an Austrian–Irish theoretical physicist who developed fundamental results in quantum theory. In particular, he is recognized for devising the Schrödinger equation, an equation that provides a way to calculate the wave function of a system and how it changes dynamically in time. He coined the term "quantum entanglement" in 1935. Schrödinger shared the 1933 Nobel Prize in Physics with Paul Dirac "for the discovery of new productive forms of atomic theory".
Furthermore, Schrödinger authored numerous publications spanning diverse areas of physics, including statistical mechanics, thermodynamics, dielectric physics, color theory, electrodynamics, general relativity, and cosmology; he also undertook multiple endeavors to formulate a unified field theory. In his seminal work, What Is Life?, Schrödinger explored genetic challenges, examining the phenomenon of life through a physical lens. He also devoted significant attention to the philosophical dimensions of science, ancient and Eastern philosophical traditions, ethics, and religious thought. His writings also encompassed philosophy and theoretical biology. Within popular culture, he is most widely recognized for "Schrödinger's cat," a notable thought experiment. Alongside Dirac, he secured an eighth-place tie in a Physics World survey ranking the most influential physicists throughout history.
In his personal life, Schrödinger maintained a household with both his wife and his mistress, a situation that potentially contributed to his departure from his position at Oxford. Subsequently, he held a position in Graz, Austria, until 1938, when the Nazi annexation prompted his flight. He ultimately secured a long-term appointment in Dublin, Ireland, where he resided until his retirement in 1955. Returning to Vienna in 1956 as an emeritus professor, he passed away from tuberculosis in 1961. Allegations of sexual abuse involving several minors surfaced in 1989.
Early Life and Educational Background
Born on 12 August 1887 in Vienna, Erwin Rudolf Josef Alexander Schrödinger was the sole offspring of Rudolf Schrödinger, a botanist, and Georgine Emilia Brenda Bauer, whose father was a chemistry professor at TU Wien. His mother possessed both Austrian and English heritage. His father adhered to Catholicism, while his mother was Lutheran. Despite identifying as an atheist, Schrödinger exhibited profound interest in Eastern religions and pantheism, incorporating religious symbolism into his scholarly output. He perceived his scientific endeavors as an intellectual pathway toward understanding divinity.
Schrödinger acquired proficiency in English outside formal schooling, attributed to his maternal grandmother's British nationality. Between 1906 and 1910, he pursued studies at the University of Vienna, mentored by Franz S. Exner and Friedrich Hasenöhrl. His doctoral degree was conferred in 1910, under Hasenöhrl's supervision. Additionally, he engaged in experimental research alongside Karl Wilhelm Friedrich "Fritz" Kohlrausch. The subsequent year, he assumed an assistantship under Exner, completing his habilitation (venia legendi) in 1914.
Professional Career
Between 1914 and 1918, Schrödinger served as a commissioned officer in the Austrian fortress artillery, participating in wartime duties across locations such as Gorizia, Duino, Sistiana, Prosecco, and Vienna. In 1920, he commenced an assistantship with Max Wien at the University of Jena, subsequently achieving the rank of ausserordentlicher Professor (associate professor) at the University of Stuttgart in September of that year. The subsequent year, he was appointed ordentlicher Professor (full professor) at the University of Breslau.
Schrödinger relocated to the University of Zurich in 1921. By 1927, he had succeeded Max Planck at the University of Berlin. His strong disapproval of Nazi antisemitism prompted his departure from Germany in 1933. Subsequently, he became a Fellow at Magdalen College, Oxford. Shortly after his arrival, he was awarded the Nobel Prize in Physics jointly with Paul Dirac. However, his tenure at Oxford proved problematic, as his unconventional domestic arrangements, involving shared living quarters with two women, were not socially accepted. In 1934, he delivered lectures at Princeton University and received an offer for a permanent position, which he ultimately declined. This decision may have been influenced by his desire to establish a household with both his wife and his mistress. Although he had prospects for a position at the University of Edinburgh, visa delays intervened, leading him to accept a post at the University of Graz in 1936. He had also previously accepted an offer for a chair position within the Department of Physics at Allahabad University in India.
Amidst these professional uncertainties in 1935, and following extensive correspondence with Albert Einstein, Schrödinger conceptualized the "Schrödinger's cat" thought experiment. Following the Anschluss (German annexation of Austria) in 1938, Schrödinger encountered difficulties in Graz due to his 1933 departure from Germany and his well-known opposition to Nazism. He subsequently issued a statement retracting his opposition, a decision he later regretted, confiding to Einstein: "I wanted to remain free – and could not do so without great duplicity." Nevertheless, this retraction did not fully satisfy the new regime, and the University of Graz consequently dismissed him from his position for "political unreliability." Subjected to harassment and prohibited from leaving the country, he ultimately fled to Italy with his wife. Subsequently, he held visiting appointments at Oxford and Ghent universities.
Dublin
In 1939, Éamon de Valera, Ireland's Taoiseach, extended a personal invitation to Schrödinger to reside in Dublin. The subsequent year, he became the Director of the School of Theoretical Physics at the newly established Dublin Institute for Advanced Studies, a role he maintained until his retirement in 1955. He maintained a modest residence on Kincora Road, Clontarf; commemorative plaques have since been installed at both his Clontarf home and his Merrion Square workplace.
Schrödinger posited that, as an Austrian, he shared a distinctive connection with Ireland; in October 1940, during an interview with a writer from the Irish Press, he discussed the Celtic heritage of Austrians, stating: "I believe there is a deeper connection between us Austrians and the Celts. Names of places in the Austrian Alps are said to be of Celtic origin." He acquired naturalized Irish citizenship in 1948 while simultaneously retaining his Austrian citizenship. He subsequently published approximately fifty additional papers addressing diverse subjects, notably his investigations into unified field theory. In 1943, Schrödinger delivered a series of three significant lectures at Trinity College Dublin, which continue to exert considerable influence within the university. This lecture series initiated annual conferences bearing his name, and several college buildings were subsequently named in his honor.
In 1944, Schrödinger authored What Is Life?, a work that explores negentropy and the notion of a complex molecule containing the genetic code for living organisms. James D. Watson's memoir, DNA, the Secret of Life, indicates that Schrödinger's book inspired Watson's research into the gene, ultimately culminating in the 1953 discovery of the DNA double helix structure. Likewise, Francis Crick, in his autobiographical work What Mad Pursuit, detailed the influence of Schrödinger's hypotheses regarding the molecular storage of genetic information. A manuscript titled "Fragment from an unpublished dialogue of Galileo," dating from this period, was rediscovered at The King's Hospital boarding school in Dublin; it had been composed for the school's 1955 edition of their Blue Coat, coinciding with Schrödinger's final year in Dublin.
Later Life and Demise
In 1956, subsequent to Austria's neutralization in 1955, Schrödinger repatriated to Vienna, assuming a professorship emeritus at the University of Vienna. During a significant address at the World Power Conference, he declined to discuss nuclear power due to his skepticism, opting instead to deliver a philosophical discourse. Within this period, he diverged from the prevailing quantum mechanical interpretation of wave-particle duality, advocating solely for the wave concept, which generated considerable academic debate.
Schrödinger contracted tuberculosis, necessitating multiple stays at a sanatorium in Arosa, Switzerland, during the 1920s. It was within this period that he formulated his seminal wave equation. He succumbed to tuberculosis on January 4, 1961, in Vienna, at the age of 73. Despite not being a Catholic, he was interred in a Catholic cemetery in Alpbach, a decision made after the officiating priest learned of Schrödinger's membership in the Pontifical Academy of Sciences.
Research and Interests
During his early career, Schrödinger conducted experiments across electrical engineering, atmospheric electricity, and atmospheric radioactivity, frequently collaborating with his former mentor, Franz Exner. His studies also encompassed vibrational theory, the theory of Brownian motion, and mathematical statistics. In 1912, upon the invitation of the editors of the Handbook of Electricity and Magnetism, he authored an article entitled Dielectrism. Concurrently, he provided a theoretical estimation of the probable altitudinal distribution of radioactive substances, a crucial factor for elucidating observed atmospheric radioactivity. In August 1913, he performed several experiments in Zeehame, which corroborated both his theoretical predictions and those of Victor Hess. This body of work earned him the Haitinger Prize from the Austrian Academy of Sciences in 1920.
Additional experimental investigations undertaken by the nascent researcher in 1914 included verifying formulas for capillary pressure within gas bubbles and examining the characteristics of soft beta radiation generated by gamma rays impacting a metallic surface. This latter research was conducted in collaboration with his colleague, Fritz Kohlrausch. By 1919, he concluded his experimental work with a study on coherent light, thereafter dedicating his efforts primarily to theoretical investigations.
Quantum Mechanics
New Quantum Theory
During the initial phase of his career, Schrödinger familiarized himself with the tenets of the old quantum theory, a framework established through the contributions of Einstein, Max Planck, Niels Bohr, Arnold Sommerfeld, and other prominent physicists. While this understanding aided his engagement with certain theoretical physics problems, the Austrian scientist remained, at that juncture, disinclined to abandon the conventional methodologies of classical physics.
Schrödinger's initial publications concerning atomic theory and spectral theory commenced in the early 1920s, following his personal interactions with Sommerfeld and Wolfgang Pauli, and his relocation to Germany. By January 1921, Schrödinger completed his inaugural article on this topic, which explored the Bohr–Sommerfeld quantization framework concerning electron interactions and specific characteristics of alkali metal spectra. He was particularly interested in integrating relativistic considerations into quantum theory. In autumn 1922, he geometrically analyzed electron orbits within an atom, employing methodologies pioneered by his colleague Hermann Weyl. This research, demonstrating the association of quantum orbits with specific geometric properties, represented a significant advancement in foreseeing aspects of wave mechanics. Earlier that year, he developed the Schrödinger equation for the relativistic Doppler effect in spectral lines, grounding it in the hypothesis of light quanta and principles of energy and momentum. Attracted by his teacher Exner's concept of the statistical nature of conservation laws, Schrödinger enthusiastically adopted the BKS theory proposed by Bohr, Hans Kramers, and John C. Slater. This theory posited the potential for these laws to be violated in individual atomic processes, such as radiation emission. Despite the Bothe–Geiger coincidence experiment subsequently challenging this premise, the notion of energy as a statistical concept remained a persistent fascination for Schrödinger, which he explored in various reports and publications.
Wave Mechanics
In January 1926, Schrödinger introduced the Schrödinger equation and published his seminal paper, "Quantisierung als Eigenwertproblem" (Quantization as an Eigenvalue Problem), in Annalen der Physik, thereby establishing the foundations of wave mechanics. This initial publication presented a derivation of the wave equation for time-independent systems, demonstrating its capacity to yield accurate energy eigenvalues for hydrogen-like atoms. Widely acclaimed as a pivotal scientific accomplishment of the twentieth century, this work profoundly revolutionized quantum mechanics and significantly impacted physics and chemistry. Subsequently, a second paper, submitted merely four weeks later, addressed the quantum harmonic oscillator, rigid rotor, and diatomic molecule challenges, offering an alternative derivation of the Schrödinger equation. A third installment, released in May, established the conceptual equivalence between Schrödinger's methodology and Werner Heisenberg's matrix mechanics, concurrently detailing the treatment of the Stark effect. The fourth paper in this series elucidated methods for analyzing time-dependent systems, such as those encountered in scattering phenomena. Within this final paper, Schrödinger introduced a complex solution to the wave equation, thereby circumventing the necessity for fourth- and sixth-order differential equations, ultimately simplifying the equation's order to one.
In 1935, Schrödinger advanced the concept of quantum entanglement through a publication that built upon a seminal paper by Einstein, Boris Podolsky, and Nathan Rosen, which had introduced the thought experiment now recognized as the EPR paradox. He characterized this quantum phenomenon as "the one that enforces its entire departure from classical lines of thought," highlighting its fundamental divergence from traditional physics. Despite his foundational contributions, Schrödinger expressed considerable discomfort with the broader implications of quantum theory, often referring to his own work as "wave mechanics" rather than the more encompassing "quantum mechanics." Regarding the probabilistic interpretation of quantum mechanics, he famously stated, "I don't like it, and I'm sorry I ever had anything to do with it." To critique the perspectives of Bohr and Heisenberg on quantum mechanics, Schrödinger devised the renowned thought experiment known as Schrödinger's cat paradox. He reportedly voiced his frustration to students, lamenting that "now the damned Göttingen physicists use my beautiful wave mechanics for calculating their shitty matrix elements.."
Unified Field Theory
Subsequent to his groundbreaking contributions to quantum mechanics, Schrödinger dedicated substantial research to developing a unified field theory. This endeavor aimed to integrate gravity, electromagnetism, and nuclear forces within the foundational structure of general relativity, a pursuit he undertook through extensive correspondence with Albert Einstein. In 1947, he presented a finding, termed "Affine Field Theory," during a lecture at the Royal Irish Academy; however, Einstein deemed this announcement "preliminary," and it ultimately did not culminate in the anticipated unified theory. Following the unsuccessful attempt at unification, Schrödinger discontinued this line of research and shifted his focus to alternative subjects. It is also reported that Schrödinger did not engage in further collaborations with prominent physicists throughout the remainder of his professional life.
Color Perception Research
Schrödinger maintained a profound interest in psychology, specifically focusing on color perception and colorimetry (German: Farbenmetrik). He dedicated several years to investigating these subjects, resulting in the publication of numerous papers within this domain:
- "Theorie der Pigmente von größter Leuchtkraft", Annalen der Physik, (4), 62, (1920), 603–22 (Theory of Pigments with Highest Luminosity)
- "Grundlinien einer Theorie der Farbenmetrik im Tagessehen", Annalen der Physik, (4), 63, (1920), 397–456; 481–520 (Outline of a Theory of Color Measurement for Daylight Vision)
- "Farbenmetrik", Zeitschrift für Physik, 1, (1920), 459–66 (Color Measurement)
- "Über das Verhältnis der Vierfarben- zur Dreifarben-Theorie", Mathematisch-Naturwissenschaftliche Klasse, Akademie der Wissenschaften, Wien, 134, 471 (On the Relationship of Four-Color Theory to Three-Color Theory)
- "Lehre von der strahlenden Energie", Müller-Pouillets Lehrbuch der Physik und Meteorologie, Vol 2, Part 1 (1926) (Thresholds of Color Differences)
Schrödinger's contributions to the psychology of color perception align with the foundational work of Isaac Newton, James Clerk Maxwell, and Hermann von Helmholtz in this field. Several of his relevant papers have been translated into English and are accessible in two key collections: Sources of Colour Science, edited by David L. MacAdam (MIT Press, 1970), and Erwin Schrödinger’s Color Theory, Translated with Modern Commentary, edited by Keith K. Niall (Springer, 2017). ISBN 978-3-319-64619-0 doi:10.1007/978-3-319-64621-3.
Philosophy
Schrödinger possessed a profound interest in philosophical inquiry, drawing significant influence from the writings of Arthur Schopenhauer and Baruch Spinoza. During his 1956 lecture, "Mind and Matter," he articulated the view that "The world extended in space and time is but our representation," directly echoing the opening statement of Schopenhauer's principal philosophical treatise. Schopenhauer's oeuvre also served as Schrödinger's introduction to Indian philosophy, particularly the Upanishads and the Advaita Vedanta interpretation. Schrödinger once pondered a specific philosophical dilemma: "If the world is indeed created by our act of observation, there should be billions of such worlds, one for each of us. How come your world and my world are the same? If something happens in my world, does it happen in your world, too? What causes all these worlds to synchronize with each other?"
Schrödinger posited thatthere is obviously only one alternative, namely the unification of minds or consciousnesses. Their multiplicity is only apparent; in truth, there is only one mind. This perspective aligns with the doctrine of the Upanishads.
In his lectures and publications, Schrödinger explored various philosophical subjects, including consciousness, the mind–body problem, sense perception, free will, and the nature of objective reality. His stance on the interplay between Eastern and Western thought was characterized by caution; he acknowledged the value of Eastern philosophy while also recognizing that some of its concepts diverged from empirical methods in natural philosophy. Certain scholars have proposed that Schrödinger's profound engagement with a non-dualist, Vedântic-like perspective might have provided a foundational framework or subtle inspiration for a significant portion of his work, encompassing his contributions to theoretical physics. Schrödinger conveyed his affinity for the concept of tat tvam asi, asserting that "you can throw yourself flat on the ground, stretched out upon Mother Earth, with the certain conviction that you are one with her and she with you."
Schrödinger articulated that "Consciousness cannot be accounted for in physical terms. For consciousness is absolutely fundamental. It cannot be accounted for in terms of anything else." Furthermore, he foreshadowed the many-worlds interpretation of quantum mechanics. In 1952, he proposed that the distinct terms within a superposition, as they evolve according to the Schrödinger equation, are "not alternatives but all really happen simultaneously." Schrödinger's subsequent works also exhibit characteristics akin to the modal interpretation, a framework developed by Bas van Fraassen. Given Schrödinger's adherence to a form of post-Machian neutral monism, which posits that "matter" and "mind" represent merely different facets or configurations of identical fundamental elements, the conceptualization of the wavefunction as a physical entity and as information became mutually interchangeable.
Personal Life
On April 6, 1920, Schrödinger married Annemarie (Anny) Bertel. Upon his immigration to Ireland in 1938, Schrödinger secured visas for himself, his wife, and Hilde March, who was the spouse of his Austrian colleague Arthur March and with whom Schrödinger had fathered a daughter in 1934. Schrödinger personally corresponded with the Taoiseach, Éamon de Valera, to facilitate March's visa acquisition. By October 1939, the ménage à trois established residence in Dublin. His wife, Anny (born December 3, 1896), passed away on October 3, 1965. Notably, one of Schrödinger's grandchildren, Terry Rudolph, has pursued a career as a quantum physicist and is currently a faculty member at Imperial College London.
Sexual Abuse Allegations
Around 1926, at the age of 39, Schrödinger served as a tutor for Itha "Ithi" Junger, then 14 years old. Walter Moore's 1989 biography of Schrödinger details that these lessons "included 'a fair amount of petting and cuddling'" and that Schrödinger "had fallen in love with his pupil." Moore further indicates that "not long after her seventeenth birthday, they became lovers." This relationship persisted, and in 1932, at the age of 20, Junger became pregnant. Moore documented that "Erwin tried to persuade her to have the child; he said he would take care of it, but he did not offer to divorce [his wife] Anny ... in desperation, Ithi arranged for an abortion."
Moore characterized Schrödinger as exhibiting a "Lolita complex," citing an entry from Schrödinger's diary where he articulated that "men of strong, genuine intellectuality are immensely attracted only by women who, forming the very beginning of the intellectual series, are as nearly connected to the preferred springs of nature as they." A 2021 article in the Irish Times summarized this inclination as a "predilection for teenage girls," further condemning Schrödinger as "a serial abuser whose behaviour fitted the profile of a paedophile in the widely understood sense of that term." Schrödinger's grandson and daughter expressed dissatisfaction with Moore's accusations, leading to a cessation of family contact with Moore following the biography's publication.
In his book Helgoland, Carlo Rovelli observes that Schrödinger "always kept a number of relationships going at once – and made no secret of his fascination with preadolescent girls." Rovelli also states that while in Ireland, Schrödinger fathered children with two women, whom a Der Standard article identified as a 26-year-old and a married political activist of unspecified age. Moore's book elaborated on both these incidents, using the pseudonym Kate Nolan for the first woman and naming the second as Sheila May, although neither was a student. The biography also recounted an instance in Ireland where Schrödinger became "infatuated" with a twelve-year-old girl, Barbara MacEntee. He ceased his advances after receiving a "serious word" from an unnamed individual, subsequently "listing her among the unrequited loves of his life." This particular episode from Moore's book was highlighted by the Irish Times article and other publications.
Moore asserted that Schrödinger's perspective on women was "that of a male supremacist," yet he reportedly disapproved of the "official misogyny" prevalent at Oxford, which socially marginalized women. Helge Kragh, in his review of Moore's biography, commented that "the conquest of women, especially very young women, was the salt of life for this sincere romantic and male chauvinist." In January 2022, the physics department at Trinity College Dublin announced its recommendation to rename a lecture theatre, which had borne Schrödinger's name since the 1990s, due to his documented history of sexual abuse. Concurrently, a portrait of the scientist would be removed, and the renaming of an eponymous lecture series would be considered.
Recognition
Awards
Memberships
Chivalric Orders
Commemorations
The philosophical dilemmas presented by Schrödinger's cat continue to be debated and represent his most enduring contribution to popular science, while Schrödinger's equation stands as his most significant legacy at a more technical level. Schrödinger is recognized as one of several figures credited as "the father of quantum mechanics." The Schrödinger crater, located on the Moon's far side, is named in his honor. Furthermore, the Erwin Schrödinger International Institute for Mathematical Physics was established in Vienna in 1992.
Schrödinger's likeness was the central design element of the 1983–97 Austrian 1000-schilling banknote, which was the second-highest denomination. A building at the University of Limerick in Ireland bears his name, as do the Erwin Schrödinger Zentrum in Adlershof, Berlin, and Route Schrödinger at CERN, Prévessin, France. His 126th birthday anniversary in 2013 was commemorated with a Google Doodle.
Publications
- Science and the Human Temperament, Allen & Unwin (1935), translated and introduced by James Murphy, with a foreword by Ernest Rutherford.
- Nature and the Greeks and Science and Humanism, Cambridge University Press (1996) ISBN 978-0-521-57550-8.
- The Interpretation of Quantum Mechanics, Ox Bow Press (1995) ISBN 978-1-881987-09-3.
- Statistical Thermodynamics, Dover Publications (1989) ISBN 978-0-486-66101-8.
- Collected Papers. Friedr. Vieweg & Sohn, 1984. ISBN 978-3-7001-0573-2.
- My View of the World. Ox Bow Press, 1983. ISBN 978-0-918024-30-5.
- Expanding Universes. Cambridge University Press, 1956.
- Space-Time Structure. Cambridge University Press, 1950. ISBN 978-0-521-31520-3.
- What Is Life?. Macmillan, 1944.
- What Is Life? & Mind and Matter. Cambridge University Press, 1974. ISBN 978-0-521-09397-2.
A list of Erwin Schrödinger's publications was compiled by Auguste Dick, Gabriele Kerber, Wolfgang Kerber, and Karl von Meyenn.
Sources
- Moore, Walter J[ohn]. (1989). Schrödinger – Life and Thought. Cambridge University Press. ISBN 978-0-521-43767-7. Retrieved 23 January 2024.Moore, Walter J[ohn]. (1992). Schrödinger – Life and Thought. Cambridge University Press. ISBN 978-0-521-43767-7. Retrieved 7 November 2011.Moore, Walter J[ohn]. (1994). A Life of Erwin Schrödinger (Canto ed.). Cambridge University Press. Bibcode:1994les..book.....M. ISBN 978-0-521-46934-0.Gribbin, John. (2012). Erwin Schrödinger and the Quantum Revolution. Transworld. ISBN 978-1-4464-6571-4. Retrieved 11 February 2017.Erwin Schrödinger and other notable figures are depicted on Austrian banknotes.
- Erwin Schrödinger and others on Austrian banknotes
- A 1927 Solvay video features an opening shot of Schrödinger.
- "Biography from the Austrian Central Library for Physics" (in English)
- An Encyclopædia Britannica article discusses Erwin Schrödinger.
- Erwin Schrödinger's Nobel Lecture, delivered on 12 December 1933, was titled The Fundamental Idea of Wave Mechanics.
- Vallabhan, C. P. Girija. "Indian Influences on Quantum Dynamics." [ed. Schrödinger's interest in Vedanta].
- The Schrödinger Medal is awarded by the World Association of Theoretically Oriented Chemists (WATOC).
- The Nobel Banquet speech, delivered in German, was titled The Discovery of New Productive Forms of Atomic Theory.
- An annotated bibliography is available for Erwin Schrödinger.
- A critical interdisciplinary review (in Italian) examines Schrödinger's "What Is Life?".
- Newspaper clippings concerning Erwin Schrödinger are preserved.
- Schrödinger in Oxford by Sir David C. Clary. World Scientific, 2022.