Representation theory of the Lorentz group (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Representation theory of the Lorentz group" in English language version.

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Dirac suggested the topic of Wigner (1939) as early as 1928 (as acknowledged in Wigner's paper). He also published one of the first papers on explicit infinite-dimensional unitary representations in Dirac (1945) (Langlands 1985), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations (Dalitz & Peierls 1986). Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411. Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proc. R. Soc. A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003, S2CID 202575171 Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822 Dalitz, R. H.; Peierls, Rudolf (1986), "Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984", Biogr. Mem. Fellows R. Soc., 32: 138–185, doi:10.1098/rsbm.1986.0006, S2CID 74547263
Wigner 1939 Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Weyl 1953 Primary source. Weyl, H. (1953), The Classical Groups. Their Invariants and Representations (2nd ed.), Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255{{citation}}: CS1 maint: ignored ISBN errors (link)
Wigner 1939 Primary source. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Wigner 1939, p. 27. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.

archive.org

Weinberg 2002, p. 1 "If it turned out that a system could not be described by a quantum field theory, it would be a sensation; if it turned out it did not obey the rules of quantum mechanics and relativity, it would be a cataclysm." Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equations 5.1.4–5. Weinberg deduces the necessity of creation and annihilation operators from another consideration, the cluster decomposition principle, Weinberg (2002, Chapter 4.) Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7 Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
For instance, there are versions (free field equations, i.e. without interaction terms) of the Klein–Gordon equation, the Dirac equation, the Maxwell equations, the Proca equation, the Rarita–Schwinger equation, and the Einstein field equations that can systematically be deduced by starting from a given representation of the Lorentz group. In general, these are collectively the quantum field theory versions of the Bargmann–Wigner equations.
See Weinberg (2002, Chapter 5), Tung (1985, Section 10.5.2) and references given in these works.
It should be remarked that high spin theories ( $s > 1$ ) encounter difficulties. See Weinberg (2002, Section 5.8), on general $(m, n)$ fields, where this is discussed in some depth, and references therein. High spin particles do without a doubt exist, e.g. nuclei, the known ones are just not elementary.
Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7 Tung, Wu-Ki (1985), Group Theory in Physics (1st ed.), New Jersey·London·Singapore·Hong Kong: World Scientific, ISBN 978-9971966577 Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Combine Weinberg (2002, Equations 5.6.7–8, 5.6.14–15) with Hall (2015, Proposition 4.18) about Lie algebra representations of group tensor product representations. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7 Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
By contrast, there is a trick, also called Weyl's unitarian trick, but unrelated to the unitarian trick of above showing that all finite-dimensional representations are, or can be made, unitary. If $(Π, V)$ is a finite-dimensional representation of a compact Lie group $G$ and if $(\cdot, \cdot)$ is any inner product on $V$ , define a new inner product $(\cdot, \cdot) Π$ by $(x, y) Π = \int G (Π(g) x, Π(g) y) dμ (g)$ , where $μ$ is Haar measure on $G$ . Then $Π$ is unitary with respect to $(\cdot, \cdot) Π$ . See Hall (2015, Theorem 4.28.)

Another consequence is that every compact Lie group has the complete reducibility property, meaning that all its finite-dimensional representations decompose as a direct sum of irreducible representations. Hall (2015, Definition 4.24., Theorem 4.28.)

It is also true that there are no infinite-dimensional irreducible unitary representations of compact Lie groups, stated, but not proved in Greiner & Müller (1994, Section 15.2.).
Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285 Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285 Greiner, W.; Müller, B. (1994), Quantum Mechanics: Symmetries (2nd ed.), Springer, ISBN 978-3540580805
See Simmons (1972, Section 30.) for precise conditions under which two Frobenius method yields two linearly independent solutions. If the exponents do not differ by an integer, this is always the case. Simmons, G. F. (1972), Differential Equations with Applications and historical Notes (T M H ed.), New Dheli: Tata McGra–Hill Publishing Company Ltd, ISBN 978-0-07-099572-7
Weinberg 2002, Section 2.5, Chapter 5. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Greiner & Reinhardt 1996, Chapter 2. Greiner, W.; Reinhardt, J. (1996), Field Quantization, Springer, ISBN 978-3-540-59179-5
Weinberg 2002, Foreword and introduction to chapter 7. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Introduction to chapter 7. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Greiner & Müller (1994, Chapter 1) Greiner, W.; Müller, B. (1994), Quantum Mechanics: Symmetries (2nd ed.), Springer, ISBN 978-3540580805
Greiner & Müller (1994, Chapter 2) Greiner, W.; Müller, B. (1994), Quantum Mechanics: Symmetries (2nd ed.), Springer, ISBN 978-3540580805
Weinberg (2002, Section 3.3) Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg (2002, Section 7.4.) Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equations 5.1.6–7. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equations 5.1.11–12. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Greiner & Reinhardt 1996 Greiner, W.; Reinhardt, J. (1996), Field Quantization, Springer, ISBN 978-3-540-59179-5
Weinberg 2002, Section 2.6 and Chapter 5. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equations 5.6.7–8. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equations 5.6.9–11. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equations 5.6.16–17. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Section 5.6. The equations follow from equations 5.6.7–8 and 5.6.14–15. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Section 2.7 p.88. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Section 2.7. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Chapter 5. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Section 5.6, p. 231. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Section 5.6. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, p. 231. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Sections 2.5, 5.7. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002 This is outlined (very briefly) on page 232, hardly more than a footnote. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equation 5.4.8. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Section 5.4. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, pp. 215–216. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equation 5.4.6. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002 Section 5.4. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Section 5.7, pp. 232–233. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Section 5.7, p. 233. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002 Equation 2.6.5. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002 Equation following 2.6.6. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Section 2.6. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
For a detailed discussion of the spin 0, ⁠1/2⁠ and 1 cases, see Greiner & Reinhardt 1996. Greiner, W.; Reinhardt, J. (1996), Field Quantization, Springer, ISBN 978-3-540-59179-5
Weinberg 2002, Chapter 3. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Simmons 1972, Sections 30, 31. Simmons, G. F. (1972), Differential Equations with Applications and historical Notes (T M H ed.), New Dheli: Tata McGra–Hill Publishing Company Ltd, ISBN 978-0-07-099572-7
Simmons 1972, Sections 30. Simmons, G. F. (1972), Differential Equations with Applications and historical Notes (T M H ed.), New Dheli: Tata McGra–Hill Publishing Company Ltd, ISBN 978-0-07-099572-7
Simmons 1972, Section 31. Simmons, G. F. (1972), Differential Equations with Applications and historical Notes (T M H ed.), New Dheli: Tata McGra–Hill Publishing Company Ltd, ISBN 978-0-07-099572-7
Simmons 1972, Equation 11 in appendix E, chapter 5. Simmons, G. F. (1972), Differential Equations with Applications and historical Notes (T M H ed.), New Dheli: Tata McGra–Hill Publishing Company Ltd, ISBN 978-0-07-099572-7
Weinberg 2002, Equation 2.4.12. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equations 2.4.18–2.4.20. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7
Weinberg 2002, Equations 5.4.19, 5.4.20. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7

arxiv.org

For part of their representation theory, see Bekaert & Boulanger (2006), which is dedicated to representation theory of the Poincare group. These representations are obtained by the method of induced representations or, in physics parlance, the method of the little group, pioneered by Wigner in 1939 for this type of group and put on firm mathematical footing by George Mackey in the fifties. Bekaert, X.; Boulanger, N. (2006), "The unitary representations of the Poincare group in any spacetime dimension", arXiv:hep-th/0611263 Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006).
Bekaert & Boulanger 2006 Bekaert, X.; Boulanger, N. (2006), "The unitary representations of the Poincare group in any spacetime dimension", arXiv:hep-th/0611263 Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006).
Bekaert & Boulanger 2006 p.4. Bekaert, X.; Boulanger, N. (2006), "The unitary representations of the Poincare group in any spacetime dimension", arXiv:hep-th/0611263 Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006).
Gonzalez, P. A.; Vasquez, Y. (2014), "Dirac Quasinormal Modes of New Type Black Holes in New Massive Gravity", Eur. Phys. J. C, 74:2969 (7): 3, arXiv:1404.5371, Bibcode:2014EPJC...74.2969G, doi:10.1140/epjc/s10052-014-2969-1, ISSN 1434-6044, S2CID 118725565
Bekaert & Boulanger 2006, p. 48. Bekaert, X.; Boulanger, N. (2006), "The unitary representations of the Poincare group in any spacetime dimension", arXiv:hep-th/0611263 Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006).

books.google.com

Weyl 1953 Primary source. Weyl, H. (1953), The Classical Groups. Their Invariants and Representations (2nd ed.), Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255{{citation}}: CS1 maint: ignored ISBN errors (link)
Bourbaki 1998, p. 424. Bourbaki, N. (1998), Lie Groups and Lie Algebras: Chapters 1-3, Springer, ISBN 978-3-540-64242-8
Abramowitz & Stegun 1965, Equation 15.6.5. Abramowitz, M.; Stegun, I. A. (1965), Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Dover Books on Mathematics, New York: Dover Publications, ISBN 978-0486612720

doi.org

In 1945 Harish-Chandra came to see Dirac in Cambridge. Harish-Chandra became convinced that theoretical physics was not the field he should be in. He had found an error in a proof by Dirac in his work on the Lorentz group. Dirac said "I am not interested in proofs but only interested in what nature does." Harish-Chandra later wrote "This remark confirmed my growing conviction that I did not have the mysterious sixth sense which one needs in order to succeed in physics and I soon decided to move over to mathematics." Dirac did however suggest the topic of Harish-Chandra's thesis, the classification of the irreducible infinite-dimensional representations of the Lorentz group. See Dalitz & Peierls 1986 Dalitz, R. H.; Peierls, Rudolf (1986), "Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984", Biogr. Mem. Fellows R. Soc., 32: 138–185, doi:10.1098/rsbm.1986.0006, S2CID 74547263
Hall (2015, Section 4.4.)

One says that a group has the complete reducibility property if every representation decomposes as a direct sum of irreducible representations.
Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Dirac suggested the topic of Wigner (1939) as early as 1928 (as acknowledged in Wigner's paper). He also published one of the first papers on explicit infinite-dimensional unitary representations in Dirac (1945) (Langlands 1985), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations (Dalitz & Peierls 1986). Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411. Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proc. R. Soc. A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003, S2CID 202575171 Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822 Dalitz, R. H.; Peierls, Rudolf (1986), "Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984", Biogr. Mem. Fellows R. Soc., 32: 138–185, doi:10.1098/rsbm.1986.0006, S2CID 74547263
Combine Weinberg (2002, Equations 5.6.7–8, 5.6.14–15) with Hall (2015, Proposition 4.18) about Lie algebra representations of group tensor product representations. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7 Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Any discrete normal subgroup of a path connected group $G$ is contained in the center $Z$ of $G$ .

Hall 2015, Exercise 11, chapter 1.
Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
By contrast, there is a trick, also called Weyl's unitarian trick, but unrelated to the unitarian trick of above showing that all finite-dimensional representations are, or can be made, unitary. If $(Π, V)$ is a finite-dimensional representation of a compact Lie group $G$ and if $(\cdot, \cdot)$ is any inner product on $V$ , define a new inner product $(\cdot, \cdot) Π$ by $(x, y) Π = \int G (Π(g) x, Π(g) y) dμ (g)$ , where $μ$ is Haar measure on $G$ . Then $Π$ is unitary with respect to $(\cdot, \cdot) Π$ . See Hall (2015, Theorem 4.28.)

Another consequence is that every compact Lie group has the complete reducibility property, meaning that all its finite-dimensional representations decompose as a direct sum of irreducible representations. Hall (2015, Definition 4.24., Theorem 4.28.)

It is also true that there are no infinite-dimensional irreducible unitary representations of compact Lie groups, stated, but not proved in Greiner & Müller (1994, Section 15.2.).
Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285 Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285 Greiner, W.; Müller, B. (1994), Quantum Mechanics: Symmetries (2nd ed.), Springer, ISBN 978-3540580805
This is one of the conclusions of Cartan's theorem, the theorem of the highest weight.
Hall (2015, Theorems 9.4–5.) Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Section 8.2 The root system is the union of two copies of $A 1$ , where each copy resides in its own dimensions in the embedding vector space. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
"This is as close as one comes to the source of the theory of infinite-dimensional representations of semisimple and reductive groups...", Langlands (1985, p. 204.), referring to an introductory passage in Dirac's 1945 paper. Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822
If finite-dimensionality is demanded, the results is the $(m, n)$ representations, see Tung (1985, Problem 10.8.) If neither is demanded, then a broader classification of all irreducible representations is obtained, including the finite-dimensional and the unitary ones. This approach is taken in Harish-Chandra (1947). Tung, Wu-Ki (1985), Group Theory in Physics (1st ed.), New Jersey·London·Singapore·Hong Kong: World Scientific, ISBN 978-9971966577 Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Bargmann & Wigner 1948 Bargmann, V.; Wigner, E. P. (1948), "Group theoretical discussion of relativistic wave equations", Proc. Natl. Acad. Sci. USA, 34 (5): 211–23, Bibcode:1948PNAS...34..211B, doi:10.1073/pnas.34.5.211, PMC 1079095, PMID 16578292
Dirac 1945 Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proc. R. Soc. A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003, S2CID 202575171
Harish-Chandra 1947 Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Delbourgo, Salam & Strathdee 1967 Delbourgo, R.; Salam, A.; Strathdee, J. (1967), "Harmonic analysis in terms of the homogeneous Lorentz group", Physics Letters B, 25 (3): 230–32, Bibcode:1967PhLB...25..230D, doi:10.1016/0370-2693(67)90050-0
These facts can be found in most introductory mathematics and physics texts. See e.g. Rossmann (2002), Hall (2015) and Tung (1985). Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9 Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285 Tung, Wu-Ki (1985), Group Theory in Physics (1st ed.), New Jersey·London·Singapore·Hong Kong: World Scientific, ISBN 978-9971966577
Hall (2015, Theorem 4.34 and following discussion.) Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Wigner 1939 Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Hall 2015, Appendix D2. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Coleman 1989, p. 30. Coleman, A. J. (1989), "The Greatest Mathematical Paper of All Time", The Mathematical Intelligencer, 11 (3): 29–38, doi:10.1007/BF03025189, ISSN 0343-6993, S2CID 35487310
Coleman 1989, p. 34. Coleman, A. J. (1989), "The Greatest Mathematical Paper of All Time", The Mathematical Intelligencer, 11 (3): 29–38, doi:10.1007/BF03025189, ISSN 0343-6993, S2CID 35487310
Killing 1888 Primary source. Killing, Wilhelm (1888), "Die Zusammensetzung der stetigen/endlichen Transformationsgruppen", Mathematische Annalen (in German), 31 (2 (June)): 252–290, doi:10.1007/bf01211904, S2CID 120501356
Cartan 1913 Primary source. Cartan, Élie (1913), "Les groupes projectifs qui ne laissant invariante aucun multiplicité plane", Bull. Soc. Math. Fr. (in French), 41: 53–96, doi:10.24033/bsmf.916
Brauer & Weyl 1935 Primary source. Brauer, R.; Weyl, H. (1935), "Spinors in n dimensions", Amer. J. Math., 57 (2): 425–449, doi:10.2307/2371218, JSTOR 2371218
Langlands 1985, pp. 203–205 Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822
Harish-Chandra 1947 Primary source. Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Wigner 1939 Primary source. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Bargmann 1947 Primary source. Bargmann, V. (1947), "Irreducible unitary representations of the Lorenz group", Ann. of Math., 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129 (the representation theory of SO(2,1) and SL(2, R); the second part on SO(3; 1) and SL(2, C), described in the introduction, was never published).
Bargmann & Wigner 1948 Primary source. Bargmann, V.; Wigner, E. P. (1948), "Group theoretical discussion of relativistic wave equations", Proc. Natl. Acad. Sci. USA, 34 (5): 211–23, Bibcode:1948PNAS...34..211B, doi:10.1073/pnas.34.5.211, PMC 1079095, PMID 16578292
Dalitz & Peierls 1986 Dalitz, R. H.; Peierls, Rudolf (1986), "Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984", Biogr. Mem. Fellows R. Soc., 32: 138–185, doi:10.1098/rsbm.1986.0006, S2CID 74547263
Dirac 1928 Primary source. Dirac, P. A. M. (1928), "The Quantum Theory of the Electron", Proc. R. Soc. A, 117 (778): 610–624, Bibcode:1928RSPSA.117..610D, doi:10.1098/rspa.1928.0023 (free access)
Hall 2015, Theorem 2.10. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Appendix C.3. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Wigner 1939, p. 27. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Hall 2015, First displayed equations in section 4.6. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Example 4.10. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Equation 4.2. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Equation before 4.5. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Theorems 9.4–5. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Theorem 10.18. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Gonzalez, P. A.; Vasquez, Y. (2014), "Dirac Quasinormal Modes of New Type Black Holes in New Massive Gravity", Eur. Phys. J. C, 74:2969 (7): 3, arXiv:1404.5371, Bibcode:2014EPJC...74.2969G, doi:10.1140/epjc/s10052-014-2969-1, ISSN 1434-6044, S2CID 118725565
Langlands 1985, p. 205. Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822
Langlands 1985, p. 203. Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822
Bargmann 1947 Bargmann, V. (1947), "Irreducible unitary representations of the Lorenz group", Ann. of Math., 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129 (the representation theory of SO(2,1) and SL(2, R); the second part on SO(3; 1) and SL(2, C), described in the introduction, was never published).
Takahashi 1963, p. 343. Takahashi, R. (1963), "Sur les représentations unitaires des groupes de Lorentz généralisés", Bull. Soc. Math. France (in French), 91: 289–433, doi:10.24033/bsmf.1598
Harish-Chandra 1947, Footnote p. 374. Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Harish-Chandra 1947, Equation 8. Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Hall 2015, Proposition C.7. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Appendix C.2. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285

harvard.edu

ui.adsabs.harvard.edu

Dirac suggested the topic of Wigner (1939) as early as 1928 (as acknowledged in Wigner's paper). He also published one of the first papers on explicit infinite-dimensional unitary representations in Dirac (1945) (Langlands 1985), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations (Dalitz & Peierls 1986). Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411. Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proc. R. Soc. A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003, S2CID 202575171 Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822 Dalitz, R. H.; Peierls, Rudolf (1986), "Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984", Biogr. Mem. Fellows R. Soc., 32: 138–185, doi:10.1098/rsbm.1986.0006, S2CID 74547263
If finite-dimensionality is demanded, the results is the $(m, n)$ representations, see Tung (1985, Problem 10.8.) If neither is demanded, then a broader classification of all irreducible representations is obtained, including the finite-dimensional and the unitary ones. This approach is taken in Harish-Chandra (1947). Tung, Wu-Ki (1985), Group Theory in Physics (1st ed.), New Jersey·London·Singapore·Hong Kong: World Scientific, ISBN 978-9971966577 Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Bargmann & Wigner 1948 Bargmann, V.; Wigner, E. P. (1948), "Group theoretical discussion of relativistic wave equations", Proc. Natl. Acad. Sci. USA, 34 (5): 211–23, Bibcode:1948PNAS...34..211B, doi:10.1073/pnas.34.5.211, PMC 1079095, PMID 16578292
Dirac 1945 Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proc. R. Soc. A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003, S2CID 202575171
Harish-Chandra 1947 Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Delbourgo, Salam & Strathdee 1967 Delbourgo, R.; Salam, A.; Strathdee, J. (1967), "Harmonic analysis in terms of the homogeneous Lorentz group", Physics Letters B, 25 (3): 230–32, Bibcode:1967PhLB...25..230D, doi:10.1016/0370-2693(67)90050-0
Wigner 1939 Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Harish-Chandra 1947 Primary source. Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Wigner 1939 Primary source. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Bargmann & Wigner 1948 Primary source. Bargmann, V.; Wigner, E. P. (1948), "Group theoretical discussion of relativistic wave equations", Proc. Natl. Acad. Sci. USA, 34 (5): 211–23, Bibcode:1948PNAS...34..211B, doi:10.1073/pnas.34.5.211, PMC 1079095, PMID 16578292
Dirac 1928 Primary source. Dirac, P. A. M. (1928), "The Quantum Theory of the Electron", Proc. R. Soc. A, 117 (778): 610–624, Bibcode:1928RSPSA.117..610D, doi:10.1098/rspa.1928.0023 (free access)
Wigner 1939, p. 27. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Gonzalez, P. A.; Vasquez, Y. (2014), "Dirac Quasinormal Modes of New Type Black Holes in New Massive Gravity", Eur. Phys. J. C, 74:2969 (7): 3, arXiv:1404.5371, Bibcode:2014EPJC...74.2969G, doi:10.1140/epjc/s10052-014-2969-1, ISSN 1434-6044, S2CID 118725565
Harish-Chandra 1947, Footnote p. 374. Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Harish-Chandra 1947, Equation 8. Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518

jstor.org

Dirac suggested the topic of Wigner (1939) as early as 1928 (as acknowledged in Wigner's paper). He also published one of the first papers on explicit infinite-dimensional unitary representations in Dirac (1945) (Langlands 1985), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations (Dalitz & Peierls 1986). Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411. Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proc. R. Soc. A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003, S2CID 202575171 Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822 Dalitz, R. H.; Peierls, Rudolf (1986), "Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984", Biogr. Mem. Fellows R. Soc., 32: 138–185, doi:10.1098/rsbm.1986.0006, S2CID 74547263
Wigner 1939 Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Brauer & Weyl 1935 Primary source. Brauer, R.; Weyl, H. (1935), "Spinors in n dimensions", Amer. J. Math., 57 (2): 425–449, doi:10.2307/2371218, JSTOR 2371218
Wigner 1939 Primary source. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Bargmann 1947 Primary source. Bargmann, V. (1947), "Irreducible unitary representations of the Lorenz group", Ann. of Math., 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129 (the representation theory of SO(2,1) and SL(2, R); the second part on SO(3; 1) and SL(2, C), described in the introduction, was never published).
Wigner 1939, p. 27. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Bargmann 1947 Bargmann, V. (1947), "Irreducible unitary representations of the Lorenz group", Ann. of Math., 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129 (the representation theory of SO(2,1) and SL(2, R); the second part on SO(3; 1) and SL(2, C), described in the introduction, was never published).

mathnet.ru

Gelfand & Naimark 1947 Gelfand, I. M.; Naimark, M. A. (1947), "Unitary representations of the Lorentz group" (PDF), Izvestiya Akad. Nauk SSSR. Ser. Mat. (in Russian), 11 (5): 411–504, retrieved 2014-12-15(Pdf from Math.net.ru){{citation}}: CS1 maint: postscript (link)

nasonline.org

Green 1998, p=76. Green, J. A. (1998), "Richard Dagobert Brauer" (PDF), Biographical Memoirs, vol. 75, National Academy Press, pp. 70–95, ISBN 978-0309062954
Klauder 1999 Klauder, J. R. (1999), "Valentine Bargmann" (PDF), Biographical Memoirs, vol. 76, National Academy Press, pp. 37–50, ISBN 978-0-309-06434-7
Bargmann was also a mathematician. He worked as Albert Einsteins assistant at the Institute for Advanced Study in Princeton (Klauder (1999)). Klauder, J. R. (1999), "Valentine Bargmann" (PDF), Biographical Memoirs, vol. 76, National Academy Press, pp. 37–50, ISBN 978-0-309-06434-7

nih.gov

ncbi.nlm.nih.gov

Bargmann & Wigner 1948 Bargmann, V.; Wigner, E. P. (1948), "Group theoretical discussion of relativistic wave equations", Proc. Natl. Acad. Sci. USA, 34 (5): 211–23, Bibcode:1948PNAS...34..211B, doi:10.1073/pnas.34.5.211, PMC 1079095, PMID 16578292
Bargmann & Wigner 1948 Primary source. Bargmann, V.; Wigner, E. P. (1948), "Group theoretical discussion of relativistic wave equations", Proc. Natl. Acad. Sci. USA, 34 (5): 211–23, Bibcode:1948PNAS...34..211B, doi:10.1073/pnas.34.5.211, PMC 1079095, PMID 16578292

pubmed.ncbi.nlm.nih.gov

Bargmann & Wigner 1948 Bargmann, V.; Wigner, E. P. (1948), "Group theoretical discussion of relativistic wave equations", Proc. Natl. Acad. Sci. USA, 34 (5): 211–23, Bibcode:1948PNAS...34..211B, doi:10.1073/pnas.34.5.211, PMC 1079095, PMID 16578292
Bargmann & Wigner 1948 Primary source. Bargmann, V.; Wigner, E. P. (1948), "Group theoretical discussion of relativistic wave equations", Proc. Natl. Acad. Sci. USA, 34 (5): 211–23, Bibcode:1948PNAS...34..211B, doi:10.1073/pnas.34.5.211, PMC 1079095, PMID 16578292

semanticscholar.org

api.semanticscholar.org

In 1945 Harish-Chandra came to see Dirac in Cambridge. Harish-Chandra became convinced that theoretical physics was not the field he should be in. He had found an error in a proof by Dirac in his work on the Lorentz group. Dirac said "I am not interested in proofs but only interested in what nature does." Harish-Chandra later wrote "This remark confirmed my growing conviction that I did not have the mysterious sixth sense which one needs in order to succeed in physics and I soon decided to move over to mathematics." Dirac did however suggest the topic of Harish-Chandra's thesis, the classification of the irreducible infinite-dimensional representations of the Lorentz group. See Dalitz & Peierls 1986 Dalitz, R. H.; Peierls, Rudolf (1986), "Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984", Biogr. Mem. Fellows R. Soc., 32: 138–185, doi:10.1098/rsbm.1986.0006, S2CID 74547263
Dirac suggested the topic of Wigner (1939) as early as 1928 (as acknowledged in Wigner's paper). He also published one of the first papers on explicit infinite-dimensional unitary representations in Dirac (1945) (Langlands 1985), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations (Dalitz & Peierls 1986). Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411. Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proc. R. Soc. A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003, S2CID 202575171 Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822 Dalitz, R. H.; Peierls, Rudolf (1986), "Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984", Biogr. Mem. Fellows R. Soc., 32: 138–185, doi:10.1098/rsbm.1986.0006, S2CID 74547263
"This is as close as one comes to the source of the theory of infinite-dimensional representations of semisimple and reductive groups...", Langlands (1985, p. 204.), referring to an introductory passage in Dirac's 1945 paper. Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822
If finite-dimensionality is demanded, the results is the $(m, n)$ representations, see Tung (1985, Problem 10.8.) If neither is demanded, then a broader classification of all irreducible representations is obtained, including the finite-dimensional and the unitary ones. This approach is taken in Harish-Chandra (1947). Tung, Wu-Ki (1985), Group Theory in Physics (1st ed.), New Jersey·London·Singapore·Hong Kong: World Scientific, ISBN 978-9971966577 Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Dirac 1945 Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proc. R. Soc. A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003, S2CID 202575171
Harish-Chandra 1947 Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Wigner 1939 Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Coleman 1989, p. 30. Coleman, A. J. (1989), "The Greatest Mathematical Paper of All Time", The Mathematical Intelligencer, 11 (3): 29–38, doi:10.1007/BF03025189, ISSN 0343-6993, S2CID 35487310
Coleman 1989, p. 34. Coleman, A. J. (1989), "The Greatest Mathematical Paper of All Time", The Mathematical Intelligencer, 11 (3): 29–38, doi:10.1007/BF03025189, ISSN 0343-6993, S2CID 35487310
Killing 1888 Primary source. Killing, Wilhelm (1888), "Die Zusammensetzung der stetigen/endlichen Transformationsgruppen", Mathematische Annalen (in German), 31 (2 (June)): 252–290, doi:10.1007/bf01211904, S2CID 120501356
Langlands 1985, pp. 203–205 Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822
Harish-Chandra 1947 Primary source. Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Wigner 1939 Primary source. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Dalitz & Peierls 1986 Dalitz, R. H.; Peierls, Rudolf (1986), "Paul Adrien Maurice Dirac. 8 August 1902–20 October 1984", Biogr. Mem. Fellows R. Soc., 32: 138–185, doi:10.1098/rsbm.1986.0006, S2CID 74547263
Wigner 1939, p. 27. Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics, 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, JSTOR 1968551, MR 1503456, S2CID 121773411.
Gonzalez, P. A.; Vasquez, Y. (2014), "Dirac Quasinormal Modes of New Type Black Holes in New Massive Gravity", Eur. Phys. J. C, 74:2969 (7): 3, arXiv:1404.5371, Bibcode:2014EPJC...74.2969G, doi:10.1140/epjc/s10052-014-2969-1, ISSN 1434-6044, S2CID 118725565
Langlands 1985, p. 205. Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822
Langlands 1985, p. 203. Langlands, R. P. (1985), "Harish-Chandra", Biogr. Mem. Fellows R. Soc., 31: 198–225, doi:10.1098/rsbm.1985.0008, S2CID 61332822
Harish-Chandra 1947, Footnote p. 374. Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518
Harish-Chandra 1947, Equation 8. Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. R. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047, S2CID 124917518

worldcat.org

search.worldcat.org

Hall (2015, Section 4.4.)

One says that a group has the complete reducibility property if every representation decomposes as a direct sum of irreducible representations.
Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Combine Weinberg (2002, Equations 5.6.7–8, 5.6.14–15) with Hall (2015, Proposition 4.18) about Lie algebra representations of group tensor product representations. Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 978-0-521-55001-7 Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Any discrete normal subgroup of a path connected group $G$ is contained in the center $Z$ of $G$ .

Hall 2015, Exercise 11, chapter 1.
Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
By contrast, there is a trick, also called Weyl's unitarian trick, but unrelated to the unitarian trick of above showing that all finite-dimensional representations are, or can be made, unitary. If $(Π, V)$ is a finite-dimensional representation of a compact Lie group $G$ and if $(\cdot, \cdot)$ is any inner product on $V$ , define a new inner product $(\cdot, \cdot) Π$ by $(x, y) Π = \int G (Π(g) x, Π(g) y) dμ (g)$ , where $μ$ is Haar measure on $G$ . Then $Π$ is unitary with respect to $(\cdot, \cdot) Π$ . See Hall (2015, Theorem 4.28.)

Another consequence is that every compact Lie group has the complete reducibility property, meaning that all its finite-dimensional representations decompose as a direct sum of irreducible representations. Hall (2015, Definition 4.24., Theorem 4.28.)

It is also true that there are no infinite-dimensional irreducible unitary representations of compact Lie groups, stated, but not proved in Greiner & Müller (1994, Section 15.2.).
Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285 Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285 Greiner, W.; Müller, B. (1994), Quantum Mechanics: Symmetries (2nd ed.), Springer, ISBN 978-3540580805
This is one of the conclusions of Cartan's theorem, the theorem of the highest weight.
Hall (2015, Theorems 9.4–5.) Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Section 8.2 The root system is the union of two copies of $A 1$ , where each copy resides in its own dimensions in the embedding vector space. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
These facts can be found in most introductory mathematics and physics texts. See e.g. Rossmann (2002), Hall (2015) and Tung (1985). Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9 Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285 Tung, Wu-Ki (1985), Group Theory in Physics (1st ed.), New Jersey·London·Singapore·Hong Kong: World Scientific, ISBN 978-9971966577
Hall (2015, Theorem 4.34 and following discussion.) Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Appendix D2. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Coleman 1989, p. 30. Coleman, A. J. (1989), "The Greatest Mathematical Paper of All Time", The Mathematical Intelligencer, 11 (3): 29–38, doi:10.1007/BF03025189, ISSN 0343-6993, S2CID 35487310
Coleman 1989, p. 34. Coleman, A. J. (1989), "The Greatest Mathematical Paper of All Time", The Mathematical Intelligencer, 11 (3): 29–38, doi:10.1007/BF03025189, ISSN 0343-6993, S2CID 35487310
Hall 2015, Theorem 2.10. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Appendix C.3. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, First displayed equations in section 4.6. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Example 4.10. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Equation 4.2. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Equation before 4.5. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Theorems 9.4–5. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Theorem 10.18. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Gonzalez, P. A.; Vasquez, Y. (2014), "Dirac Quasinormal Modes of New Type Black Holes in New Massive Gravity", Eur. Phys. J. C, 74:2969 (7): 3, arXiv:1404.5371, Bibcode:2014EPJC...74.2969G, doi:10.1140/epjc/s10052-014-2969-1, ISSN 1434-6044, S2CID 118725565
Hall 2015, Proposition C.7. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
Hall 2015, Appendix C.2. Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285

zenodo.org

Killing 1888 Primary source. Killing, Wilhelm (1888), "Die Zusammensetzung der stetigen/endlichen Transformationsgruppen", Mathematische Annalen (in German), 31 (2 (June)): 252–290, doi:10.1007/bf01211904, S2CID 120501356