PhysicistGermanMax Planck Medal WinnerAcademic

Rudolf Haag

Rudolf Haag, German theoretical physicist, pioneered axiomatic quantum field theory, quantum statistical mechanics, and supersymmetry.

1. Overview

Rudolf Haag was a distinguished German theoretical physicist whose work laid foundational stones for the modern understanding of quantum field theory (QFT) and quantum statistical mechanics. He is widely recognized for his pioneering contributions to the axiomatic and algebraic formulations of quantum field theory, including the development of Haag's theorem and the Haag-Kastler axioms. His research significantly elucidated the formal structure of QFT in terms of the principle of locality and local observables. Beyond QFT, Haag made crucial advancements in quantum statistical mechanics, notably by generalizing the KMS condition for thermal equilibrium states. He also contributed to the classification of supersymmetry through the Haag-Łopuszański-Sohnius theorem and played a pivotal role in establishing the influential journal *Communications in Mathematical Physics*.

2. Biography

Rudolf Haag's life spanned a significant period of scientific and historical change, from his early education and wartime experiences to his impactful academic career and personal life.

Rudolf Haag
Rudolf Haag

2.1. Early Life and Background

Rudolf Haag was born on August 17, 1922, in Tübingen, a historic university town located in the German state of Baden-Württemberg. He came from a cultured middle-class family. His mother was Anna Haag, a notable German writer and politician known for her democratic feminist views and her secret diary during World War II. His father, Albert Haag, was a mathematics teacher at a Gymnasium. After completing his high school education in 1939, shortly before the outbreak of World War II, Haag visited his sister in London. During his stay, he was interned as an enemy alien and spent the war years in a camp for German civilians in Manitoba, Canada. It was during this period that he used his free time, after daily compulsory labor, to diligently study physics and mathematics as an autodidact.

2.2. Education and Academic Path

After World War II, Rudolf Haag returned to Germany and enrolled at the Technical University of Stuttgart in 1946, where he completed his physics degree in 1948. In 1951, he earned his doctorate from the University of Munich under the supervision of Fritz Bopp. His doctoral thesis was titled `Die korrespondenzmässige Methode in der Theorie der Elementarteilchen` (The Correspondence Method in the Theory of Elementary Particles). He continued to work as Bopp's assistant until 1956. In April 1953, Haag joined the CERN theoretical study group in Copenhagen. This group was hosted by the Niels Bohr Institute in Copenhagen because the main laboratory in Geneva was still under construction. After a year, he returned to his assistant position in Munich and completed his German habilitation in 1954 with a thesis titled `On Quantum field theories`. From 1956 to 1957, he collaborated with the renowned physicist Werner Heisenberg at the Max Planck Institute for Physics in Göttingen.

2.3. Academic Career and Appointments

From 1957 to 1959, Rudolf Haag served as a visiting professor at Princeton University in the United States. Following this, he spent a year, from 1959 to 1960, working at the University of Marseille in France. In 1960, he accepted a professorship in Physics at the University of Illinois Urbana-Champaign, where he remained for six years. A significant milestone in his career occurred in 1965 when he, along with Res Jost, co-founded the prestigious journal *Communications in Mathematical Physics*. Haag served as the journal's first editor-in-chief for eight years, until 1973. In 1966, he accepted a professorship in theoretical physics at the University of Hamburg, where he continued his work until his retirement in 1987. After his retirement, Haag dedicated his efforts to developing the concept of the quantum physical event.

2.4. Personal Life

Rudolf Haag developed an early interest in music. He initially began learning the violin but later developed a strong preference for the piano, which he played almost every day. In 1948, Haag married Käthe Fues, who was one of the daughters of the German theoretical physicist Erwin Fues. Together, they had four children: Albert, Friedrich, Elisabeth, and Ulrich. After Käthe's premature death, Haag married his second wife, Barbara Klie. Following his retirement, he and Barbara moved to Schliersee, a tranquil village nestled in the Bavarian mountains.

2.5. Death

Rudolf Haag passed away on January 5, 2016, in Fischhausen-Neuhaus, located in southern Bavaria.

3. Scientific Contributions

Rudolf Haag's scientific contributions are fundamental to modern theoretical physics, particularly in establishing rigorous mathematical frameworks for quantum field theory and quantum statistical mechanics.

3.1. Foundations of Quantum Field Theory

Haag made profound contributions to the theoretical framework of quantum field theory, addressing some of its most challenging conceptual issues.

3.1.1. Haag's Theorem and Scattering Theory

At the beginning of his career, Haag significantly contributed to the foundational concepts of quantum field theory, including Haag's theorem. This theorem demonstrates that the conventional Fock space representation cannot be used to describe interacting relativistic quantum fields with canonical commutation relations, implying that the interaction picture of quantum mechanics does not exist in quantum field theory. This insight necessitated a new approach to describing the scattering processes of particles. In the subsequent years, Haag developed what is now known as Haag-Ruelle scattering theory, which provides a rigorous framework for understanding how particles interact and evolve in quantum field theory.

3.1.2. Axiomatic and Algebraic Quantum Field Theory

During his work on scattering theory, Haag realized that the rigid, pre-assumed relationship between fields and particles was not fundamental. He proposed that the particle interpretation should instead be based on Albert Einstein's principle of locality, which assigns operators to specific regions of spacetime. These insights culminated in the formalization of the Haag-Kastler axioms for local observables of quantum field theories. This framework incorporates elements from the theory of operator algebras and is consequently referred to as algebraic quantum field theory or, from a physical standpoint, as local quantum physics.

This conceptual framework proved highly fruitful for understanding the fundamental properties of any theory in four-dimensional Minkowski space. Without making assumptions about non-observable charge-changing fields, Haag, in collaboration with Sergio Doplicher and John E. Roberts, clarified the possible structure of the superselection sectors of observables in theories with short-range forces. Their analysis, which built upon the Haag-Kastler axioms and the postulate of Haag duality, revealed that sectors can always be composed with one another. Each sector satisfies either para-Bose or para-Fermi statistics, and for every sector, a conjugate sector exists. These findings correspond to the additivity of charges in the particle interpretation, the Bose-Fermi alternative for particle statistics, and the existence of antiparticles. In the specific case of simple sectors, a global gauge group and charge-carrying fields, capable of generating all sectors from the vacuum state, were reconstructed directly from the observables. These results were later generalized for arbitrary sectors in the Doplicher-Roberts duality theorem. The application of these methods to theories in low-dimensional spaces also led to an understanding of the occurrence of braid group statistics and quantum groups.

3.2. Quantum Statistical Mechanics

Rudolf Haag made significant contributions to quantum statistical mechanics. Together with Nicolaas Marinus Hugenholtz and Marinus Winnink, he successfully generalized the Gibbs-von Neumann characterization of thermal equilibrium states using the KMS condition (named after Ryogo Kubo, Paul C. Martin, and Julian Schwinger). This generalization extended the condition to infinite systems in the thermodynamic limit. It was discovered that this condition also plays a prominent role in the theory of von Neumann algebras and led to the development of the Tomita-Takesaki theory. This theory has become a central element in structural analysis and has recently been applied in the construction of concrete quantum field theoretical models. Furthermore, in collaboration with Daniel Kastler and Ewa Trych-Pohlmeyer, Haag succeeded in deriving the KMS condition from the fundamental stability properties of thermal equilibrium states. He also developed a theory of chemical potential in this context, working alongside Huzihiro Araki, Daniel Kastler, and Masamichi Takesaki.

3.3. Quantum Field Theory in Curved Spacetime

The robust framework established by Haag and Kastler for studying quantum field theories in Minkowski space can be extended to theories in curved spacetime. Through his collaborations with Klaus Fredenhagen, Heide Narnhofer, and Ulrich Stein, Haag made important contributions to the understanding of quantum phenomena in these complex environments, including the Unruh effect and Hawking radiation.

3.4. Supersymmetry and Other Theoretical Contributions

While Rudolf Haag held a certain skepticism towards what he considered speculative developments in theoretical physics, such as string theory (arguing it misunderstood the concept of a particle within the conventional QFT framework), he occasionally engaged with such questions. His most notable contribution in this area is the Haag-Łopuszański-Sohnius theorem. This theorem classifies the possible supersymmetries of the S-matrix that are not accounted for by the Coleman-Mandula theorem. The Coleman-Mandula theorem excludes a non-trivial coupling of bosonic inner symmetry groups with geometric symmetries (like the Poincaré group), whereas supersymmetry allows for such a coupling. In his later career, after retirement, Haag also dedicated significant effort to the concept of the quantum physical event.

4. Honors and Awards

Rudolf Haag received numerous honors and awards in recognition of his profound scientific achievements and his foundational role in theoretical physics. In 1970, he was awarded the prestigious Max Planck Medal for his outstanding contributions to theoretical physics. In 1997, he received the Henri Poincaré Prize from the International Association of Mathematical Physics, specifically for his fundamental contributions to quantum field theory as one of the founders of its modern formulation.

Haag was also a member of several esteemed academic institutions. Since 1980, he was a member of the German National Academy of Sciences Leopoldina. In 1981, he became a member of the Göttingen Academy of Sciences and Humanities. He was a corresponding member of the Bavarian Academy of Sciences and Humanities since 1979 and of the Austrian Academy of Sciences since 1987.

5. Publications

Rudolf Haag's influential work is documented in his seminal textbook and numerous scientific papers that have shaped the field of quantum field theory.

5.1. Textbook

  • Haag, Rudolf (1996). Local quantum physics: Fields, particles, algebras. 2nd edition. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-61049-6. [https://doi.org/10.1007/978-3-642-61458-3 DOI: 10.1007/978-3-642-61458-3].

5.2. Selected Scientific Works

  • Haag, Rudolf (1955). "On quantum field theories". Dan. Mat. Fys. Medd.. Vol. 29, No. 12, pp. 1-37. [https://cds.cern.ch/record/212242 (Haag's theorem)].
  • Haag, Rudolf (1958). "Quantum field theories with composite particles and asymptotic conditions". Physical Review. Vol. 112, No. 2, pp. 669-673. [https://doi.org/10.1103/PhysRev.112.669 DOI: 10.1103/PhysRev.112.669]. (Haag-Ruelle scattering theory)
  • Haag, Rudolf; Kastler, Daniel (1964). "An Algebraic approach to quantum field theory". Journal of Mathematical Physics. Vol. 5, No. 7, pp. 848-861. [https://doi.org/10.1063/1.1704187 DOI: 10.1063/1.1704187]. (Haag-Kastler axioms)
  • Doplicher, Sergio; Haag, Rudolf; Roberts, John E. (1971). "Local observables and particle statistics. 1". Communications in Mathematical Physics. Vol. 23, No. 3, pp. 199-230. [https://doi.org/10.1007/BF01877742 DOI: 10.1007/BF01877742]. [https://projecteuclid.org/euclid.cmp/1103857630 (Full text)].
  • Doplicher, Sergio; Haag, Rudolf; Roberts, John E. (1974). "Local observables and particle statistics. 2". Communications in Mathematical Physics. Vol. 35, No. 1, pp. 49-85. [https://doi.org/10.1007/BF01646454 DOI: 10.1007/BF01646454]. [https://projecteuclid.org/euclid.cmp/1103859518 (Full text)]. (Doplicher-Haag-Roberts analysis of the superselection structure)
  • Haag, Rudolf; Hugenholtz, Nico M.; Winnink, Marius (1967). "On the Equilibrium states in quantum statistical mechanics". Communications in Mathematical Physics. Vol. 5, No. 3, pp. 215-236. [https://doi.org/10.1007/BF01646342 DOI: 10.1007/BF01646342]. [https://projecteuclid.org/euclid.cmp/1103840050 (Full text)]. (KMS condition)
  • Haag, Rudolf; Kastler, Daniel; Trych-Pohlmeyer, Ewa B. (1974). "Stability and equilibrium states". Communications in Mathematical Physics. Vol. 38, No. 3, pp. 173-193. [https://doi.org/10.1007/BF01651541 DOI: 10.1007/BF01651541]. [https://projecteuclid.org/euclid.cmp/1103860047 (Full text)]. (Stability and KMS condition)
  • Araki, Huzihiro; Kastler, Daniel; Takesaki, Masamichi; Haag, Rudolf (1977). "Extension of KMS States and Chemical Potential". Communications in Mathematical Physics. Vol. 53, No. 2, pp. 97-134. [https://doi.org/10.1007/BF01609126 DOI: 10.1007/BF01609126]. [https://projecteuclid.org/euclid.cmp/1103900637 (Full text)]. (KMS condition and chemical potential)
  • Haag, Rudolf; Narnhofer, Heide; Stein, Ulrich (1984). "On Quantum Field Theory in Gravitational Background". Communications in Mathematical Physics. Vol. 94, No. 2, pp. 219-238. [https://doi.org/10.1007/BF01209302 DOI: 10.1007/BF01209302]. [https://projecteuclid.org/euclid.cmp/1103941282 (Full text)]. (Unruh effect)
  • Fredenhagen, Klaus; Haag, Rudolf (1990). "On the Derivation of Hawking Radiation Associated With the Formation of a Black Hole". Communications in Mathematical Physics. Vol. 127, No. 2, pp. 273-284. [https://doi.org/10.1007/BF02096757 DOI: 10.1007/BF02096757]. [https://projecteuclid.org/euclid.cmp/1104180137 (Full text)]. (Hawking radiation)
  • Haag, Rudolf; Lopuszanski, Jan T.; Sohnius, Martin (1975). "All possible generators of supersymmetries of the S-matrix". Nuclear Physics B. Vol. 88, No. 2, pp. 257-274. [https://doi.org/10.1016/0550-3213(75)90279-5 DOI: 10.1016/0550-3213(75)90279-5]. (Classification of Supersymmetry)
  • Haag, Rudolf (1990). "Fundamental Irreversibility and the Concept of Events". Communications in Mathematical Physics. Vol. 132, No. 1, pp. 245-252. [https://doi.org/10.1007/BF02278010 DOI: 10.1007/BF02278010]. [https://projecteuclid.org/euclid.cmp/1104201040 (Full text)]. (Concept of Event)

5.3. Other Writings

  • Buchholz, Detlev; Haag, Rudolf (2000). "The Quest for understanding in relativistic quantum physics". Journal of Mathematical Physics. Vol. 41, No. 6, pp. 3674-3697. [https://doi.org/10.1063/1.533324 DOI: 10.1063/1.533324].
  • Haag, Rudolf (2000). "Questions in quantum physics: A Personal view". Mathematical Physics 2000, pp. 87-100. [https://doi.org/10.1142/9781848160224_0005 DOI: 10.1142/9781848160224_0005]. ISBN 978-1-86094-230-3.
  • Haag, Rudolf (2010). "Some people and some problems met in half a century of commitment to mathematical physics". The European Physical Journal H. Vol. 35, No. 3, pp. 263-307. [https://doi.org/10.1140/epjh/e2010-10032-4 DOI: 10.1140/epjh/e2010-10032-4].
  • Haag, Rudolf (2010). "Local algebras. A look back at the early years and at some achievements and missed opportunities". The European Physical Journal H. Vol. 35, No. 3, pp. 255-261. [https://doi.org/10.1140/epjh/e2010-10042-7 DOI: 10.1140/epjh/e2010-10042-7].
  • Haag, Rudolf (2015). "Faces of Quantum Physics". The Message of Quantum Science. Lecture Notes in Physics, Vol. 899, pp. 219-234. Springer, Berlin, Heidelberg. ISBN 978-3-662-46422-9. [https://doi.org/10.1007/978-3-662-46422-9_9 DOI: 10.1007/978-3-662-46422-9_9].
  • Haag, Rudolf (2019). "On quantum theory". International Journal of Quantum Information. Vol. 17, No. 4, pp. 1950037-1-9. [https://doi.org/10.1142/S0219749919500370 DOI: 10.1142/S0219749919500370].

6. Legacy and Influence

Rudolf Haag's legacy in theoretical physics is profound and enduring. His work established a rigorous mathematical foundation for quantum field theory, moving beyond earlier heuristic approaches to define the field's formal structure with unprecedented clarity. The Haag-Kastler axioms and the concept of local quantum physics provided a robust framework that continues to be central to modern research, enabling physicists to analyze quantum phenomena with mathematical precision. His development of Haag's theorem fundamentally reshaped the understanding of the interaction picture in quantum field theory, leading to new approaches like the Haag-Ruelle scattering theory.

Beyond quantum field theory, his generalization of the KMS condition in quantum statistical mechanics was a landmark achievement, connecting thermal equilibrium states to the powerful Tomita-Takesaki theory and influencing the construction of concrete quantum field theoretical models. His contributions to understanding quantum field theory in curved spacetime, including the Unruh effect and Hawking radiation, further demonstrated the versatility and depth of his theoretical insights.

Haag's classification of supersymmetry through the Haag-Łopuszański-Sohnius theorem also remains a crucial result in high-energy physics. As a co-founder and long-time editor-in-chief of *Communications in Mathematical Physics*, he not only contributed groundbreaking research but also fostered a vital platform for the dissemination of mathematical physics, influencing generations of researchers. His insistence on mathematical rigor and conceptual clarity continues to guide the field, making him one of the most influential figures in the history of theoretical physics.