QED (play)

QED is a play by American playwright Peter Parnell that chronicles significant events in the life of Nobel Prize-winning physicist Richard Feynman. It presents scenes from a fictional day in Feynman's life, less than two years before his death, interweaving many strands from his biography, from the Manhattan project to the "Challenger" disaster inquiry to more personal topics such as the death of Feynman's wife and his own fight with cancer. The play, which grew out of a collaboration between Parnell, actor Alan Alda, and director Gordon Davidson, premiered in 2001. The original production, directed by Davidson and starring Alda as Feynman, was performed first at the Mark Taper Forum in Los Angeles and, from late 2001 to mid-2002, on Broadway.
Plot.
Set in June, 1986, fewer than two years before Feynman's death, in Feynman's office at the California Institute of Technology in Pasadena, the play follows Feynman through a day of his life. As the real Feynman does in his books "What Do You Care What Other People Think?" and "Surely You're Joking, Mr. Feynman!", the stage character talks directly to the audience; we learn from this and from phone calls with off-stage characters that Feynman is to appear that night playing his bongo drums in a student production of the musical "South Pacific", that he is expecting a delegation from the Russian Republic of Tuva, which Feynman is whimsically determined to visit (as detailed in Ralph Leighton's book, "Tuva or Bust!"), and that he is eager to make his views known in the final report of the Rogers Commission charged with the "Challenger" disaster. From phone conversations between Feynman and his doctors, we also learn that Feynman's cancer has returned, and that his doctors are urging him to undergo further surgical procedures, which are not without their own risk. Feynman's conversation with the audience also touches on a number of additional topics well-known to readers of his autobiographical writings: the Manhattan project and safe-cracking, how he learned to draw, his father, as well as musings on physics and, more generally, on the nature of science and knowledge.
In the second act, the play returns to Feynman's study later at night on the same day, after the performance is over. We meet the only other character in addition to the main protagonist: a (fictional) young student by the name of Miriam Field, who has attended one of Feynman's lectures and both witnessed his bongo performance and attended the after-play party. Where Feynman had earlier grown dispirited both by his own condition and by memories of his long-dead wife, Miriam manages to pull him out of his depression. Feynman informs his doctors that he will consent to have surgery, after all; but requests that they awaken him from anesthesia if they determine that he is about to die intraoperatively, because "that would be an interesting experience".
History and productions.
In the mid-1990s, Alan Alda, having read Feynman's autobiographical books, became intent on playing Feynman on stage. He consulted director Gordon Davidson at the Mark Taper Forum in Los Angeles, who suggested that playwright Peter Parnell be the play's author. Over the course of more than six years, the three went through many revisions in their combined effort to bring the many facets of Feynman's character to the stage.
The play premiered on March 25, 2001, at the Mark Taper Forum. Later that year, the production went to Broadway, where it was performed 40 times between November 18, 2001 and June 20, 2002 at the Vivian Beaumont Theater. Six years later, Alda briefly revisited his role in a scenic reading at Columbia University's Miller Theatre as part of the 2008 World Science Festival, once more directed by Davidson.
The Chicago area premiere of "QED" on September 23, 2010, featured Rob Riley as Richard Feynman under the direction of Maureen Payne-Hahner. It was performed at the McCormick School of Engineering and Applied Science of Northwestern University, and was produced by ETOPiA.
It was produced in Athens, Greece in October 2004 with Yorgos Kotanidis as Richard Feynman, directed by Iossif Vardakis. It was a successful run, with a performance attended by Jim Al-Khalili. The production went on to perform the following year at a larger theater. In 2009 it was staged in the virtual reality "Tholos" dome, at the Hellenic Cosmos complex in Athens, once more playing to packed audiences. The show went on a tour of Greece in 2010
The national premiere for Italy of the play was translated and directed by Luca Giberti at the in Genoa (Sala Duse), with Andrea Nicolini playing Feynman.
The play was produced in Berkeley, CA by Indra’s Net Theater in Nov/Dec 2013 with Jeff Garrett playing Feynman and Bruce Coughran directing.
"QED" was produced in Philadelphia, PA by the Lantern Theater in Nov/Dec 2014 with Peter DeLaurier as Richard Feynman and M. Craig Getting directing.


False sunrise

A false sunrise is any of several atmospheric optical phenomena in which the Sun appears to have risen, but is actually still some distance below the horizon. A number of different atmospheric conditions can be responsible for this effect, all of which divert the sunlight in such a way as to allow it to reach the observer's eye, thereby giving the impression that the light comes directly from the Sun itself. The spread of light can sometimes be deceivingly similar to a true sun.
Several atmospheric phenomena that may alternatively be called a "false sunrise" are:
The term "false sunrise" should not be confused with "false dawn", which is a term sometimes used to refer to the zodiacal light.


Paul Nemenyi

Paul Felix Nemenyi (; June 5, 1895 – March 1, 1952) was a Hungarian mathematician and physicist who specialized in continuum mechanics. He was known for using what he called the inverse or semi-inverse approach, which applied vector field analysis, to obtain numerous exact solutions of the nonlinear equations of gas dynamics, many of them representing rotational flows of nonuniform total energy. His work applied geometrical solutions to fluid dynamics. In continuum mechanics, "Nemenyi's theorem" proves that, given any net of isothermal curves, there exists a five parameter family of plane stress systems for which these curves are stress trajectories.
Nemenyi's five constant theory for the determination of stress trajectories in plane elastic systems, was subsequently proven by later mathematicians.
He was the father of the statistician Peter Nemenyi and the probable father of former World Chess Champion Bobby Fischer.
Biography.
Family.
Neményi was born to a wealthy Hungarian-Jewish family on June 5, 1895 in Fiume (Rijeka) in the Kingdom of Hungary. His grandfather was Zsigmond Neményi. His father Dezső Neményi was one of the directors at Rijeka Refinery (now INA d.d.). Nemenyi attended elementary and high school in Fiume (Rijeka). He graduated from high school in Budapest. Nemenyi's uncle was Dr. Ambrus Neményi, born in Pécel, c. 20 km east of Budapest. The family's name was Neumann until 1871 when they magyarized it to Neményi. Paul Nemenyi's aunt was Berta Koppély (Whose parents were Adolf Koppély (1809-1883) and Rózsa von Hatvany-Deutsch). His family's art collection included works by Klimt, Kandinsky and Matisse.
Hungary at the time, was producing a generation of geniuses in the exact sciences, who would be collectively known as Martians, that included Theodore von Kármán (b. 1881), George de Hevesy (b. 1885), Leó Szilárd (b. 1898), Dennis Gabor (b. 1900), Eugene Wigner (b. 1902), John von Neumann (b. 1903), Edward Teller (b. 1908), and Paul Erdős (b. 1913).
Mathematical career.
A child prodigy in mathematics, at the age of 17, Nemenyi won the Hungarian national mathematics competition. Nemenyi obtained his doctorate in mathematics in Berlin in 1922 and was appointed a lecturer in fluid dynamics at the Technical University of Berlin. In the early 1930s, he published a textbook on mathematical mechanics that became required reading in German universities. Stripped of his position when the Nazis came to power, he also had to leave Hungary where anti-Semitic laws had been enacted, and found work for a time in Copenhagen.
He arrived in the US at the outbreak of World War II. He briefly held a number of teaching positions in succession and took part in hydraulic research at the State University of Iowa. In 1941 he was appointed instructor at the University of Colorado (other sources claim Colorado State University), and in 1944 at the State College of Washington.
In Germany, Nemenyi belonged to a Socialist party called the ISK, which believed that truth could be arrived at through neo-Kantian Socratic principles. He was an animal-rights supporter and refused to wear anything made of wool. In 1930, Nemenyi entrusted his 3 year old first son, Peter Nemenyi, to be looked after by the socialist vegetarian community, visiting him once a year.
Theodore von Kármán wrote of Nemenyi: "When he came to this country, he went to scientific meetings in an open shirt without a tie and was very much disappointed as I advised him to dress as anyone else. He told me that he thought this was a country of freedom, and the man is only judged according to his internal values and not his external appearance."
In 1947 Nemenyi was appointed a physicist with the Naval Ordnance Laboratory, White Oak, Maryland. He was head of the Theoretical Mechanics Section at the laboratory and one of the country's principal authorities on elasticity and fluid dynamics. At the US Navy Research Laboratory, Nemenyi became mentor to Jerald Ericksen, where he put Ericksen to work on the study of water bells.
Nemenyi pioneered what he called the inverse or semi-inverse approach, which applied vector field analysis, to obtain numerous exact solutions of the nonlinear equations of gas dynamics, many of them representing rotational flows of nonuniform total energy. In continuum mechanics, "Nemenyi's theorem" proves that, given any net of isothermal curves, there exists a five parameter family of plane stress systems for which these curves are stress trajectories.
In his exposition, "The Main Concepts and Ideas of Fluid Dynamics in their Historical Development", Nemenyi was highly critical of Isaac Newton's inadequate understanding of fluid dynamics. I. Bernard Cohen argues that Nemenyi pays insufficient attention to Newton's empirical experiments. However, Cohen notes that Nemenyi provides the "most thorough and incisive analyses in print of Newton's work on fluids, written by an obvious master of science. For example, Nemenyi is the only author I have encountered who has shown the weakness of Newton's "proof" at the end of Book 2, that vortices contradict the laws of astronomy.
Nemenyi's scientific knowledge extended well beyond the subjects of his researches. He has been described as having "extreme[ly] versatile interests and erudition". Nemenyi's interest and ability encompassed several nonscientific fields. He collected children's art and sometimes lectured upon it. In 1951, he published a critique of the entire Encyclopædia Britannica, and suggested improvements for such diverse sections as psychology and psychoanalysis.
Nemenyi was also deeply interested in the philosophy of mathematics and mathematical education. He translated David Hilbert and Stefan Cohn-Vossen's "Anschauliche Geometrie" into English, giving it the title "Geometry and the Imagination". Clifford Truesdell writes that it was Nemenyi who first taught him "that mechanics was something deep and beautiful, beyond the ken of schools of "applied mathematics" and "applied mechanics"".
Paul Nemenyi died on March 1, 1952, at the age of 56. He was survived officially by one son: Peter Nemenyi, then a student of mathematics at Princeton University.
Supposed fatherhood of Bobby Fischer.
In 2002 Nemenyi was identified as the probable biological father of world chess champion Bobby Fischer, not the man named on Fischer's birth certificate. Additional details on their relationship were reported in 2009.


Representation theory of the Lorentz group

The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group. These have both historical importance in mainstream physics, as well as connections to more speculative present-day theories.
Development.
The full theory of the finite-dimensional representations of the Lie algebra of the Lorentz group is deduced using the general framework of the representation theory of semisimple Lie algebras. The finite-dimensional representations of the connected component formula_1 of the full Lorentz group are obtained by employing the Lie correspondence and the matrix exponential. The full finite-dimensional representation theory of the universal covering group (and also the spin group, a double cover) formula_2 of formula_1 is obtained, and explicitly given in terms of action on a function space in representations of formula_2 and formula_5. The representatives of time reversal and space inversion are given in space inversion and time reversal, completing the finite-dimensional theory for the full Lorentz group. The general properties of the ("m", "n") representations are outlined. Action on function spaces is considered, with the action on spherical harmonics and the Riemann P-functions appearing as examples. The infinite-dimensional case of irreducible unitary representations are realized for the formula_2 principal series and the complementary series. Finally, the Plancherel formula for formula_2 is given, and representations of are classified and realized for Lie algebras.
The development of the representation theory has historically followed the development of the more general theory of representation theory of semisimple groups, largely due to Élie Cartan and Hermann Weyl, but the Lorentz group has also received special attention due to its importance in physics. Notable contributors are physicist E. P. Wigner and mathematician Valentine Bargmann with their Bargmann–Wigner program, one conclusion of which is, roughly, "a classification of all unitary representations of the inhomogeneous Lorentz group amounts to a classification of all possible relativistic wave equations". The classification of the irreducible infinite-dimensional representations of the Lorentz group was established by Paul Dirac's doctoral student in theoretical physics, Harish-Chandra, later turned mathematician, in 1947. The corresponding classification for formula_8 was published independently by Bargmann and Israel Gelfand together with Mark Naimark in the same year.
Applications.
Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in classical field theory, most importantly the electromagnetic field, and of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory and of various objects in string theory and beyond. The representation theory also provides the theoretical ground for the concept of spin. The theory enters into general relativity in the sense that in small enough regions of spacetime, physics is that of special relativity.
The finite-dimensional irreducible non-unitary representations together with the irreducible infinite-dimensional unitary representations of the "inhomogeneous" Lorentz group, the Poincare group, are the representations that have direct physical relevance.
Infinite-dimensional unitary representations of the Lorentz group appear by "restriction" of the irreducible infinite-dimensional unitary representations of the Poincaré group acting on the Hilbert spaces of relativistic quantum mechanics and quantum field theory. But these are also of mathematical interest and of "potential" direct physical relevance in other roles than that of a mere restriction. There were speculative theories, (tensors and spinors have infinite counterparts in the "expansors" of Dirac and the "expinors" of Harish-Chandra) consistent with relativity and quantum mechanics, but they have found no proven physical application. Modern speculative theories potentially have similar ingredients per below.
Classical field theory.
While the electromagnetic field together with the gravitational field are the only classical fields providing accurate descriptions of nature, other types of classical fields are important too. In the approach to quantum field theory (QFT) referred to as second quantization, the starting point is one or more classical fields, where e.g. the wave functions solving the Dirac equation are considered as classical fields "prior" to (second) quantization. While second quantization and the Lagrangian formalism associated with it is not a fundamental aspect of QFT, it is the case that so far all quantum field theories can be approached this way, including the standard model. In these cases, there are classical versions of the field equations following from the Euler–Lagrange equations derived from the Lagrangian using the principle of least action. These field equations must be relativistically invariant, and their solutions (which will qualify as relativistic wave functions according to the definition below) must transform under some representation of the Lorentz group.
The action of the Lorentz group on the space of "field configurations" (a field configuration is the spacetime history of a particular solution, e.g. the electromagnetic field in all of space over all time is "one" field configuration) resembles the action on the Hilbert spaces of quantum mechanics, except that the commutator brackets are replaced by field theoretical Poisson brackets.
Relativistic quantum mechanics.
For the present purposes the following definition is made: A "relativistic wave function" is a set of functions on spacetime which transforms under an arbitrary proper Lorentz transformation as
where is an -dimensional matrix representative of belonging to some direct sum of the representations to be introduced below.
The most useful relativistic quantum mechanics "one-particle" theories (there are no fully consistent such theories) are the Klein–Gordon equation and the Dirac equation in their original setting. They are relativistically invariant and their solutions transform under the Lorentz group as Lorentz scalars () and bispinors respectively (). The electromagnetic field is a relativistic wave function according to this definition, transforming under .
The infinite-dimensional representations may be used in the analysis of scattering.
Quantum field theory.
In quantum field theory, the demand for relativistic invariance enters, among other ways in that the S-matrix necessarily must be Poincaré invariant. This has the implication that there is one or more infinite-dimensional representation of the Lorentz group acting on Fock space. One way to guarantee the existence of such representations is the existence of a Lagrangian description (with modest requirements imposed, see the reference) of the system using the canonical formalism, from which a realization of the generators of the Lorentz group may be deduced.
The transformations of field operators illustrate the complementary role played by the finite-dimensional representations of the Lorentz group and the infinite-dimensional unitary representations of the Poincare group, witnessing the deep unity between mathematics and physics. For illustration, consider the definition an -component field operator: A relativistic field operator is a set of operator valued functions on spacetime which transforms under proper Poincaré transformations according to
Here is the unitary operator representing on the Hilbert space on which is defined and is an -dimensional representation of the Lorentz group. The transformation rule is the second Wightman axiom of quantum field theory.
By considerations of differential constraints that the field operator must be subjected to in order to describe a single particle with definite mass and spin (or helicity), it is deduced that
where are interpreted as creation and annihilation operators respectively. The creation operator transforms according to
and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin of the particle. The connection between the two are the "wave functions", also called "coefficient functions"
that carry "both" the indices operated on by Lorentz transformations and the indices operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection. To exhibit the connection, subject both sides of equation to a Lorentz transformation resulting in for e.g. ,
where is the non-unitary Lorentz group representative of and is a unitary representative of the so-called "Wigner rotation" associated to and that derives from the representation of the Poincaré group, and is the spin of the particle.
All of the above formulas, including the definition of the field operator in terms of creation and annihilation operators, as well as the differential equations satisfied by the field operator for a particle with specified mass, spin and the representation under which it is supposed to transform, and also that of the wave function, can be derived from group theoretical considerations alone once the frameworks of quantum mechanics and special relativity is given.
Speculative theories.
In theories in which spacetime can have more than dimensions, the generalized Lorentz groups of the appropriate dimension take the place of .
The requirement of Lorentz invariance takes on perhaps its most dramatic effect in string theory. "Classical" relativistic strings can be handled in the Lagrangian framework by using the Nambu–Goto action. This results in a relativistically invariant theory in any spacetime dimension. But as it turns out, the theory of open and closed bosonic strings (the simplest string theory) is impossible to quantize in such a way that the Lorentz group is represented on the space of states (a Hilbert space) unless the dimension of spacetime is 26. The corresponding result for superstring theory is again deduced demanding Lorentz invariance, but now with supersymmetry. In these theories the Poincaré algebra is replaced by a supersymmetry algebra which is a -graded Lie algebra extending the Poincaré algebra. The structure of such an algebra is to a large degree fixed by the demands of Lorentz invariance. In particular, the fermionic operators (grade ) belong to a or representation space of the (ordinary) Lorentz Lie algebra. The only possible dimension of spacetime in such theories is 10.
Finite-dimensional representations.
Representation theory of groups in general, and Lie groups in particular, is a very rich subject. The Lorentz group has some properties that makes it "agreeable" and others that make it "not very agreeable" within the context of representation theory; the group is simple and thus semisimple, but is not connected, and none of its components are simply connected. Furthermore, the Lorentz group is not compact.
For finite-dimensional representations, the presence of semisimplicity means that the Lorentz group can be dealt with the same way as other semisimple groups using a well-developed theory. In addition, all representations are built from the irreducible ones, since the Lie algebra possesses the "complete reducibility property". But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, cannot be dealt with in all the aspects as in the simple framework that applies to simply connected, compact groups. Non-compactness implies, for a connected simple Lie group, that no nontrivial finite-dimensional unitary representations exist. Lack of simple connectedness gives rise to spin representations of the group. The non-connectedness means that, for representations of the full Lorentz group, time reversal and space inversion has to be dealt with separately.
History.
The development of the finite-dimensional representation theory of the Lorentz group mostly follows that of the subject in general. Lie theory originated with Sophus Lie in 1873. By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing. In 1913 the theorem of highest weight for representations of simple Lie algebras, the path that will be followed here, was completed by Élie Cartan. Richard Brauer was 1935–38 largely responsible for the development of the Weyl-Brauer matrices describing how spin representations of the Lorentz Lie algebra can be embedded in Clifford algebras. The Lorentz group has also historically received special attention in representation theory, see History of infinite-dimensional unitary representations below, due to its exceptional importance in physics. Mathematicians Hermann Weyl and Harish-Chandra and physicists Eugene Wigner and Valentine Bargmann made substantial contributions both to general representation theory and in particular to the Lorentz group. Physicist Paul Dirac was perhaps the first to manifestly knit everything together in a practical application of major lasting importance with the Dirac equation in 1928.
The Lie algebra.
The irreducible complex linear representations of the complexification, formula_14 of the Lie algebra formula_15 of the Lorentz group are to be found. A convenient basis for formula_15 is given by the three generators of rotations and the three generators of boosts. They are explicitly given in conventions and Lie algebra bases.
The Lie algebra is complexified, and the basis is changed to the components of its two ideals
The components of and separately satisfy the commutation relations of the Lie algebra formula_18 and, moreover, they commute with each other,
where are indices which each take values , and is the three-dimensional Levi-Civita symbol. Let formula_20 and formula_21 denote the complex linear span of and respectively.
One has the isomorphisms
where formula_22 is the complexification of formula_23
The utility of these isomorphisms comes from the fact that all irreducible representations of formula_18, and hence all irreducible complex linear representations of formula_25 are known. The irreducible complex linear representation of formula_22 is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of formula_27
The unitarian trick.
The Lie algebra formula_28 is the Lie algebra of formula_29 It contains the compact subgroup with Lie algebra formula_30 The latter is a compact real form of formula_31 Thus from the first statement of the unitarian trick, representations of are in one-to-one correspondence with holomorphic representations of formula_29
By compactness, Peter–Weyl theorem applies to , and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of are precisely the tensor products of irreducible unitary representations of .
By appeal to simple connectedness, the second statement of the unitarian trick is applied. The objects in the following list are in one-to-one correspondence:
Tensor products of representations appear at the Lie algebra level as either of
where is the identity operator. Here, the latter interpretation, which follows from , is intended. The highest weight representations of formula_22 are indexed by for . (The highest weights are actually , but the notation here is adapted to that of formula_37) The tensor products of two such complex linear factors then form the irreducible complex linear representations of formula_31
Finally, the formula_39-linear representations of the real forms of the far left, formula_15, and the far right, formula_25 in are obtained from the formula_42-linear representations of formula_28 characterized in the previous paragraph.
The ("μ", "ν")-representations of sl(2, C).
The complex linear representations of the complexification of formula_44 obtained via isomorphisms in , stand in one-to-one correspondence with the real linear representations of formula_27 The set of all "real linear" irreducible representations of formula_22 are thus indexed by a pair . The complex linear ones, corresponding precisely to the complexification of the real linear formula_18 representations, are of the form , while the conjugate linear ones are the . All others are real linear only. The linearity properties follow from the canonical injection, the far right in , of formula_22 into its complexification. Representations on the form or are given by "real" matrices (the latter are not irreducible). Explicitly, the real linear -representations of formula_22 are
where formula_51 are the complex linear irreducible representations of formula_22 and formula_53 their complex conjugate representations. (The labeling is usually in the mathematics literature , but half-integers are chosen here to conform with the labeling for the formula_54 Lie algebra.) Here the tensor product is interpreted in the former sense of . These representations are concretely realized below.
The ("m", "n")-representations of so(3; 1).
Via the displayed isomorphisms in and knowledge of the complex linear irreducible representations of formula_28 upon solving for and , all irreducible representations of formula_56 and, by restriction, those of formula_15 are obtained. The representations of formula_15 obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible. Since formula_15 is semisimple, all its representations can be built up as direct sums of the irreducible ones.
Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers and , conventionally written as one of
where is a finite-dimensional vector space. These are, up to a similarity transformation, uniquely given by
where is the -dimensional unit matrix and
are the -dimensional irreducible representations of formula_62 also termed spin matrices or "angular momentum matrices". These are explicitly given as
where denotes the Kronecker delta. In components, with , , the representations are given by
Off-diagonal direct sums.
Since for any irreducible representation for which it is essential to operate over the field of complex numbers, the direct sum of representations and have particular relevance to physics, since it permits to use linear operators over real numbers.
The group.
The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence. The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras. The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted formula_65
If formula_66 for some vector space is a representation, a representation of the connected component of is defined by
This definition applies whether the resulting representation is projective or not.
Surjectiveness of exponential map for SO(3, 1).
From a practical point of view, it is important whether the first formula in can be used for all elements of the "group". It holds for all formula_67, however, in the general case, e.g. for formula_2, not all are in the image of .
But formula_69 "is" surjective. One way to show this is to make use of the isomorphism formula_70 the latter being the Möbius group. It is a quotient of formula_71 (see the linked article). The quotient map is denoted with formula_72 The map formula_73 is onto. Apply with being the differential of at the identity. Then
Since the left hand side is surjective (both and are), the right hand side is surjective and hence formula_75 is surjective. Finally, recycle the argument once more, but now with the known isomorphism between and formula_76 to find that is onto for the connected component of the Lorentz group.
Fundamental group.
The Lorentz group is "doubly connected", i. e. is a group with two equivalence classes of loops as its elements.
To exhibit the fundamental group of , the topology of its covering group formula_2 is considered. By the polar decomposition theorem, any matrix formula_78 may be "uniquely" expressed as
where is unitary with determinant one, hence in , and is Hermitian with trace zero. The trace and determinant conditions imply:
The manifestly continuous one-to-one map is a homeomorphism with continuous inverse given by (the locus of is identified with formula_81)
explicitly exhibiting that formula_2 is simply connected. But formula_84 where formula_85 is the center of formula_2. Identifying and amounts to identifying with , which in turn amounts to identifying antipodal points on formula_87 Thus topologically,
where last factor is not simply connected: Geometrically, it is seen (for visualization purposes, formula_89 may be replaced by formula_90) that a path from to in formula_91 "is" a loop in formula_92 since and are antipodal points, and that it is not contractible to a point. But a path from to , thence to again, a loop in formula_89 and a "double loop" (considering , where formula_94 is the covering map) in formula_92 that "is" contractible to a point (continuously move away from "upstairs" in formula_89 and shrink the path there to the point ). Thus is a group with two equivalence classes of loops as its elements, or put more simply, is "doubly connected".
Projective representations.
Since has two elements, some representations of the Lie algebra will yield projective representations. Once it is known whether a representation is projective, formula applies to all group elements and all representations, including the projective ones — with the understanding that the representative of a group element will depend on which element in the Lie algebra (the in ) is used to represent the group element in the standard representation.
For the Lorentz group, the -representation is projective when is a half-integer. See the section spinors.
For a projective representation of , it holds that
since any loop in traversed twice, due to the double connectedness, is contractible to a point, so that its homotopy class is that of a constant map. It follows that is a double-valued function. It is not possible to consistently chose a sign to obtain a continuous representation of all of , but this is possible "locally" around any point.
The covering group SL(2, C).
Consider formula_5 as a "real" Lie algebra with basis
where the sigmas are the Pauli matrices. From the relations
is obtained
which are exactly on the form of the -dimensional version of the commutation relations for formula_15 (see conventions and Lie algebra bases below). Thus, the map , , extended by linearity is an isomorphism. Since formula_2 is simply connected, it is the universal covering group of .
A geometric view.
Let be a path from to , denote its homotopy class by and let be the set of all such homotopy classes. Define the set
and endow it with the multiplication operation
where formula_101 is the path multiplication of formula_102 and formula_103:
With this multiplication, becomes a group isomorphic to formula_105 the universal covering group of . Since each has two elements, by the above construction, there is a 2:1 covering map . According to covering group theory, the Lie algebras formula_106 and formula_107 of are all isomorphic. The covering map is simply given by .
An algebraic view.
For an algebraic view of the universal covering group, let formula_2 act on the set of all Hermitian matrices formula_109 by the operation
The action on formula_109 is linear. An element of formula_109 may be written in the form
The map is a group homomorphism into formula_112 Thus formula_113 is a 4-dimensional representation of formula_2. Its kernel must in particular take the identity matrix to itself, and therefore . Thus for in the kernel so, by Schur's lemma, is a multiple of the identity, which must be since . The space formula_109 is mapped to Minkowski space , via
The action of on formula_109 preserves determinants. The induced representation of formula_2 on formula_118 via the above isomorphism, given by
preserves the Lorentz inner product since
This means that belongs to the full Lorentz group . By the main theorem of connectedness, since formula_2 is connected, its image under in is connected, and hence is contained in .
It can be shown that the Lie map of formula_121 is a Lie algebra isomorphism: formula_122 The map is also onto.
Thus formula_2, since it is simply connected, is the universal covering group of , isomorphic to the group of above.
Non-surjectiveness of exponential mapping for SL(2, C).
The exponential mapping formula_124 is not onto. The matrix
is in formula_105 but there is no formula_126 such that .
In general, if is an element of a connected Lie group with Lie algebra formula_127 then, by ,

The matrix can be written
Realization of representations of and and their Lie algebras.
The complex linear representations of formula_5 and formula_2 are more straightforward to obtain than the formula_130 representations. They can be (and usually are) written down from scratch. The holomorphic group representations (meaning the corresponding Lie algebra representation is complex linear) are related to the complex linear Lie algebra representations by exponentiation. The real linear representations of formula_5 are exactly the -representations. They can be exponentiated too. The -representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer (but half-integers are used here).
The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of and there is no factor of in the exponential mapping compared to the physics convention used elsewhere. Let the basis of formula_5 be
This choice of basis, and the notation, is standard in the mathematical literature.
Complex linear representations.
The irreducible holomorphic -dimensional representations formula_133 can be realized on the space of homogeneous polynomial of degree in 2 variables formula_134 the elements of which are
The action of formula_2 is given by
The associated formula_5-action is, using and the definition above, for the basis elements of formula_138
With a choice of basis for formula_139, these representations become matrix Lie algebras.
Real linear representations.
The -representations are realized on a space of polynomials formula_140 in formula_141 homogeneous of degree in formula_142 and homogeneous of degree in formula_143 The representations are given by
By employing again it is found that
In particular for the basis elements,
 + \overline{z_2}\frac{\partial}{\partial \overline{z_2}} \\
\phi_{\mu,\nu}(X) &= -z_2\frac{\partial}{\partial z_1} - \overline{z_2}\frac{\partial}{\partial \overline{z_1}} \\
Properties of the ("m", "n") representations.
The representations, defined above via (as restrictions to the real form formula_144) of tensor products of irreducible complex linear representations and of formula_138 are irreducible, and they are the only irreducible representations.
Dimension.
The representations are -dimensional. This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of formula_2 and formula_22. For a Lie general algebra formula_107 the Weyl dimension formula,
applies, where is the set of positive roots, is the highest weight, and is half the sum of the positive roots. The inner product formula_150 is that of the Lie algebra formula_127 invariant under the action of the Weyl group on formula_152 the Cartan subalgebra. The roots (really elements of formula_153 are via this inner product identified with elements of formula_154 For formula_138 the formula reduces to , "where the present notation must be taken into account". The highest weight is . By taking tensor products, the result follows.
Faithfulness.
If a representation of a Lie group is not faithful, then is a nontrivial normal subgroup. There are three relevant cases.
In the case of , the first case is excluded since is semi-simple. The second case (and the first case) is excluded because is simple. For the third case, is isomorphic to the quotient formula_156 But formula_85 is the center of formula_158 It follows that the center of is trivial, and this excludes the third case. The conclusion is that every representation and every projective representation for finite-dimensional vector spaces are faithful.
By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with (abelian) nontrivial non-discrete normal subgroups replaced by (one-dimensional) nontrivial ideals in the Lie algebra, and the center of replaced by the center of formula_159The center of any semisimple Lie algebra is trivial and formula_15 is semi-simple and simple, and hence has no non-trivial ideals.
A related fact is that if the corresponding representation of formula_2 is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding formula_2 representation is not faithful, but is .
Non-unitarity.
The Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary. This is due to the non-compactness of the Lorentz group. In fact, a connected simple non-compact Lie group cannot have "any" nontrivial unitary finite-dimensional representations. There is a topological proof of this. Let , where is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group . Then where is the compact subgroup of consisting of unitary transformations of . The kernel of is a normal subgroup of . Since is simple, is either all of , in which case is trivial, or is trivial, in which case is faithful. In the latter case is a diffeomorphism onto its image, and is a Lie group. This would mean that is an embedded non-compact Lie subgroup of the compact group . This is impossible with the subspace topology on since all "embedded" Lie subgroups of a Lie group are closed If were closed, it would be compact, and then would be compact, contrary to assumption.
In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of and used in the construction are Hermitian. This means that is Hermitian, but is anti-Hermitian. The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm.
Restriction to SO(3).
The representation is, however, unitary when restricted to the rotation subgroup , but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an representation have -invariant subspaces of highest weight (spin) , where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) is -dimensional. So for example, the (, ) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.
Since the angular momentum operator is given by , the highest spin in quantum mechanics of the rotation sub-representation will be and the "usual" rules of addition of angular momenta and the formalism of 3-j symbols, 6-j symbols, etc. applies.
Spinors.
It is the -invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the representation has spin if is half-integral. The simplest are and , the Weyl-spinors of dimension . Then, for example, and are a spin representations of dimensions and respectively. According to the above paragraph, there are subspaces with spin both and in the last two cases, so these representations cannot likely represent a "single" physical particle which must be well-behaved under . It cannot be ruled out in general, however, that representations with multiple subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out "unphysical components", leaving only a single spin.
Construction of pure spin representations for any (under ) from the irreducible representations involves taking tensor products of the Dirac-representation with a non-spin representation, extraction of a suitable subspace, and finally imposing differential constraints.
Dual representations.
The following theorems are applied to examine whether the dual representation of an irreducible representation is isomorphic to the original representation:
Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If is an element of the Weyl group of a semisimple Lie algebra, then . In the case of formula_138 the Weyl group is . It follows that each is isomorphic to its dual formula_166 The root system of formula_167 is shown in the figure to the right. The Weyl group is generated by formula_168 where formula_169 is reflection in the plane orthogonal to as ranges over all roots. Inspection shows that so . Using the fact that if are Lie algebra representations and , then , the conclusion for is
Complex conjugate representations.
If is a representation of a Lie algebra, then formula_171 is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication. In general, every irreducible representation of formula_172 can be written uniquely as , where
with formula_174 holomorphic (complex linear) and formula_175 anti-holomorphic (conjugate linear). For formula_138 since formula_177 is holomorphic, formula_178 is anti-holomorphic. Direct examination of the explicit expressions for formula_179 and formula_180 in equation below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression also allows for identification of formula_174 and formula_175 for formula_183 as
Using the above identities (interpreted as pointwise addition of functions), for yields
where the statement for the group representations follow from = . It follows that the irreducible representations have real matrix representatives if and only if . Reducible representations on the form have real matrices too.
The adjoint representation, the Clifford algebra, and the Dirac spinor representation.
In general representation theory, if is a representation of a Lie algebra formula_127 then there is an associated representation of formula_127 on , also denoted , given by
Likewise, a representation of a group yields a representation on of , still denoted , given by
If and are the standard representations on formula_188 and if the action is restricted to formula_189 then the two above representations are the adjoint representation of the Lie algebra and the adjoint representation of the group respectively. The corresponding representations (some formula_190 or formula_191) always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular.
Applying this to the Lorentz group, if is a projective representation, then direct calculation using shows that the induced representation on is a proper representation, i.e. a representation without phase factors.
In quantum mechanics this means that if or is a representation acting on some Hilbert space , then the corresponding induced representation acts on the set of linear operators on . As an example, the induced representation of the projective spin representation on is the non-projective 4-vector (, ) representation.
For simplicity, consider only the "discrete part" of , that is, given a basis for , the set of constant matrices of various dimension, including possibly infinite dimensions. The induced 4-vector representation of above on this simplified has an invariant 4-dimensional subspace that is spanned by the four gamma matrices. (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, formula_192 whose complexification is formula_193 generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the , a pseudoscalar irrep, also the , but with parity inversion eigenvalue , see the next section below, the already mentioned vector irrep, , a pseudovector irrep, with parity inversion eigenvalue +1 (not −1), and a tensor irrep, . The dimensions add up to . In other words,
where, as is customary, a representation is confused with its representation space.
The spin representation.
The six-dimensional representation space of the tensor -representation inside formula_194 has two roles. The
where formula_195 are the gamma matrices, the sigmas, only of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,
and hence constitute a representation (in addition to spanning a representation space) sitting inside formula_192 the spin representation. For details, see bispinor and Dirac algebra.
The conclusion is that every element of the complexified formula_194 in (i.e. every complex matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on formula_198 making it a space of bispinors.
Reducible representations.
There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained by taking direct sums, tensor products, and quotients of the irreducible representations. Other methods of obtaining representations include the restriction of a representation of a larger group containing the Lorentz group, e.g. formula_199 and the Poincaré group. These representations are in general not irreducible.
The Lorentz group and its Lie algebra have the "complete reducibility property". This means that every representation reduces to a direct sum of irreducible representations. The reducible representations will therefore not be discussed.
Space inversion and time reversal.
The (possibly projective) representation is irreducible as a representation , the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If it can be extended to a representation of all of , the full Lorentz group, including space parity inversion and time reversal. The representations can be extended likewise.
Space parity inversion.
For space parity inversion, the adjoint action of on formula_15 is considered, where is the standard representative of space parity inversion, , given by
It is these properties of and under that motivate the terms "vector" for and pseudovector or "axial vector" for . In a similar way, if is any representation of formula_15 and is its associated group representation, then acts on the representation of by the adjoint action, for formula_202 . If is to be included in , then consistency with requires that
holds, where and are defined as in the first section. This can hold only if and have the same dimensions, i.e. only if . When then can be extended to an irreducible representation of , the orthochronous Lorentz group. The parity reversal representative does not come automatically with the general construction of the representations. It must be specified separately. The matrix (or a multiple of modulus −1 times it) may be used in the representation.
If parity is included with a minus sign (the matrix ) in the representation, it is called a pseudoscalar representation.
Time reversal.
Time reversal , acts similarly on formula_15 by
By explicitly including a representative for , as well as one for , a representation of the full Lorentz group is obtained. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full Poincaré group, four more generators, the , in addition to the and generate the group. These are interpreted as generators of translations. The time-component is the Hamiltonian . The operator satisfies the relation
in analogy to the relations above with formula_15 replaced by the full Poincaré algebra. By just cancelling the 's, the result would imply that for every state with positive energy in a Hilbert space of quantum states with time-reversal invariance, there would be a state with negative energy . Such states do not exist. The operator is therefore chosen antilinear and antiunitary, so that it anticommutes with , resulting in , and its action on Hilbert space likewise becomes antilinear and antiunitary. It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix. This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, is not a "representation".
When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. The Dirac representation, , is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation.
The third discrete symmetry entering in the CPT theorem along with and , charge conjugation symmetry , has nothing directly to do with Lorentz invariance.
Action on function spaces.
If is a vector space of functions of a finite number of variables , then the action on a scalar function formula_205 given by
produces another function . Here is an -dimensional representation, and is a possibly infinite-dimensional representation. A special case of this construction is when is a space of functions defined on the a linear group itself, viewed as a -dimensional manifold embedded in formula_206 (with the dimension of the matrices). This is the setting in which the Peter–Weyl theorem and the Borel–Weil theorem are formulated. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into characters of finite-dimensional representations. The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. formula_158
The following exemplifies action of the Lorentz group and the rotation subgroup on some function spaces.
Euclidean rotations.
The subgroup of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space
where formula_209 are the spherical harmonics. An arbitrary square integrable function one the unit sphere can be expressed as
where the are generalized Fourier coefficients.
The Lorentz group action restricts to that of and is expressed as
where the are obtained from the representatives of odd dimension of the generators of rotation.
The Möbius group.
The identity component of the Lorentz group is isomorphic to the Möbius group . This group can be thought of as conformal mappings of either the complex plane or, via stereographic projection, the Riemann sphere. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere.
In the plane, a Möbius transformation characterized by the complex numbers acts on the plane according to
and can be represented by complex matrices
since multiplication by a nonzero complex scalar does not change . These are elements of formula_2 and are unique up to a sign (since give the same ), hence formula_211
The Riemann P-functions.
The Riemann P-functions, solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as
where the are complex constants. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is
The set of constants in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation. Its "exponents", i. e. solutions of the indicial equation, for expansion around the singular point are and ,corresponding to the two linearly independent solutions, and for expansion around the singular point they are and . Similarly, the exponents for are and for the two solutions.
One has thus
where the condition (sometimes called Riemann's identity)
on the exponents of the solutions of Riemann's differential equation has been used to define .
The first set of constants on the left hand side in , denotes the regular singular points of Riemann's differential equation. The second set, , are the corresponding exponents at for one of the two linearly independent solutions, and, accordingly, are exponents at for the second solution.
Define an action of the Lorentz group on the set of all Riemann P-functions by first setting
where are the entries in
for a Lorentz transformation.
Define
where is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Möbius transformation of the argument is that of shifting the poles to new locations, hence changing the critical points, but there is no change in the exponents of the differential equation the new function satisfies. The new function is expressed as
where
Infinite-dimensional unitary representations.
History.
The Lorentz group and its double cover formula_2 also have infinite dimensional unitary representations, studied independently by , and at the instigation of Paul Dirac. This trail of development begun with where he devised matrices and necessary for description of higher spin (compare Dirac matrices), elaborated upon by , see also , and proposed precursors of the Bargmann-Wigner equations. In he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors. These ideas were incorporated by Harish–Chandra and expanded with expinors as an infinite-dimensional generalization of spinors in his 1947 paper.
The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by and , based on an analogue for formula_2 of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in and .
The theory of spherical functions for the Lorentz group, required for harmonic analysis on the hyperboloid model of 3-dimensional hyperbolic space sitting in Minkowski space is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on formula_215 This theory is discussed in , , and the posthumous text of .
Principal series for SL(2, C).
The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup  of formula_216 Since the one-dimensional representations of correspond to the representations of the diagonal matrices, with non-zero complex entries and , they thus have the form
for an integer, real and with . The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when is replaced by . By definition the representations are realized on sections of line bundles on formula_218 which is isomorphic to the Riemann sphere. When , these representations constitute the so-called spherical principal series.
The restriction of a principal series to the maximal compact subgroup of  can also be realized as an induced representation of  using the identification , where is the maximal torus in  consisting of diagonal matrices with . It is the representation induced from the 1-dimensional representation , and is independent of . By Frobenius reciprocity, on  they decompose as a direct sum of the irreducible representations of  with dimensions with a non-negative integer.
Using the identification between the Riemann sphere minus a point and formula_219 the principal series can be defined directly on formula_220 by the formula
Irreducibility can be checked in a variety of ways:
Complementary series for SL(2, C).
The for , the complementary series is defined on formula_220 for the inner product
with the action given by
The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of , each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of . Irreducibility can be proved by analyzing the action of formula_107 on the algebraic sum of these subspaces or directly without using the Lie algebra.
Plancherel theorem for SL(2, C).
The only irreducible unitary representations of formula_2 are the principal series, the complementary series and the trivial representation.
Since acts as on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided is taken to be even.
To decompose the left regular representation of  on formula_230 only the principal series are required. This immediately yields the decomposition on the subrepresentations formula_231 the left regular representation of the Lorentz group, and formula_232 the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with "k" even and the latter only those with .)
The left and right regular representation and are defined on formula_230 by
Now if is an element of , the operator formula_235 defined by
is Hilbert–Schmidt. Define a Hilbert space  by
where
and formula_239 denotes the Hilbert space of Hilbert–Schmidt operators on formula_240 Then the map  defined on by
extends to a unitary of formula_230 onto .
The map  satisfies the intertwining property
If are in then by unitarity
Thus if formula_245 denotes the convolution of formula_246 and formula_247 and formula_248 then
The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively.
The Plancherel formula extends to all formula_250 By a theorem of Jacques Dixmier and Paul Malliavin, every smooth compactly supported function on formula_251 is a finite sum of convolutions of similar functions, the inversion formula holds for such . It can be extended to much wider classes of functions satisfying mild differentiability conditions.
Classification of representations of SO(3, 1).
The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to "assume" they exist, and to investigate their properties. Thus first assume that an irreducible strongly continuous infinite-dimensional representation on a Hilbert space of is at hand. Since is a subgroup, is a representation of it as well. Each irreducible subrepresentation of is finite-dimensional, and the representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of if is unitary.
The steps are the following:
Step 1.
One suitable choice of basis and labeling is given by
If this were a "finite-dimensional" representation, then would correspond the lowest occurring eigenvalue of in the representation, equal to , and would correspond to the highest occurring eigenvalue, equal to . In the infinite-dimensional case, retains this meaning, but does not. For simplicity, it is assumed that a given occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown that the assumption is possible to avoid (with a slightly more complicated calculation) with the same results.
Step 2.
The next step is to compute the matrix elements of the operators and forming the basis of the Lie algebra of formula_37 The matrix elements of formula_254 and formula_255 (the "complexified" Lie algebra is understood) are known from the representation theory of the rotation group, and are given by
where the labels and have been dropped since they are the same for all basis vectors in the representation.
Due to the commutation relations
the triple is a vector operator and the Wigner–Eckart theorem applies for computation of matrix elements between the states represented by the chosen basis. The matrix elements of
where the superscript signifies that the defined quantities are the components of a spherical tensor operator of rank (which explains the factor as well) and the subscripts are referred to as in formulas below, are given by
Here the first factors on the right hand sides are Clebsch–Gordan coefficients for coupling with to get . The second factors are the "reduced matrix elements". They do not depend on or , but depend on and, of course, . For a complete list of non-vanishing equations, see .
Step 3.
The next step is to demand that the Lie algebra relations hold, i.e. that
This results in a set of equations for which the solutions are
where
Step 4.
The imposition of the requirement of unitarity of the corresponding representation of the "group" restricts the possible values for the arbitrary complex numbers and . Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning
This translates to
leading to
where is the angle of on polar form. For follows formula_266 and formula_267 is chosen by convention. There are two possible cases:
This shows that the representations of above are "all" infinite-dimensional irreducible unitary representations.
Explicit formulas.
Conventions and Lie algebra bases.
The metric of choice is given by , and the physics convention for Lie algebras and the exponential mapping is used. These choices are arbitrary, but once they are made, fixed. One possible choice of basis for the Lie algebra is, in the 4-vector representation, given by:
The commutation relations of the Lie algebra formula_15 are:
In three-dimensional notation, these are
The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol above and in the sequel should be observed.
Weyl spinors and bispinors.
By taking, in turn, and and by setting
in the general expression , and by using the trivial relations and , it follows
These are the left-handed and right-handed Weyl spinor representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) and , whose elements and are called left- and right-handed Weyl spinors respectively. Given
their direct sum as representations is formed,
This is, up to a similarity transformation, the Dirac spinor representation of formula_37 It acts on the 4-component elements of , called bispinors, by matrix multiplication. The representation may be obtained in a more general and basis independent way using Clifford algebras. These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of formula_37 Expressions for the group representations are obtained by exponentiation.
Open problems.
The classification and characterization of the representation theory of the Lorentz group was completed in 1947. But in association with the Bargmann–Wigner programme, there are yet unresolved purely mathematical problems, linked to the infinite-dimensional unitary representations.
The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the (generalized) Lorentz group appears as the little group of the Poincaré group of spacelike vectors in higher spacetime dimension. The corresponding infinite-dimensional unitary representations of the (generalized) Poincaré group are the so-called "tachyonic representations". Tachyons appear in the spectrum of bosonic strings and are associated with instability of the vacuum. Even though tachyons may not be realized in nature, these representations must be mathematically "understood" in order to understand string theory. This is so since tachyon states turn out to appear in superstring theories too in attempts to create realistic models.
One open problem is the completion of the Bargmann–Wigner programme for the isometry group of the de Sitter spacetime . Ideally, the physical components of wave functions would be realized on the hyperboloid of radius embedded in formula_281 and the corresponding covariant wave equations of the infinite-dimensional unitary representation to be known.


Michael Dennis Feit

Michael Dennis Feit (born 1942) is an American physicist at the Lawrence Livermore National Laboratory, California.
Born in Easton, Pennsylvania he studied physics at Lehigh University, graduating B.A in physics in 1964, which was followed by a Ph.D. in 1969 from the Rensselaer Polytechnic Institute. After a brief period as a research associate in the Department of Physics at the University of Illinois he joined the Lawrence Livermore National Laboratory in 1972.
At Livermore his main interests were in the fields of optics and lasers and from 1992 until 1996 he worked as the leader of the optical physics group, afterwards becoming the leader for theoretical optics. Later, from 1997 until 2005, he held the title of leader of the laser damage modeling group at the National Ignition Facility nuclear fusion project.
He was awarded the status of Fellow in the American Physical Society, after they were nominated by their Division of Laser Science in 1988, for "the development and implementation of novel and powerful computational techniques with applications to optical propagation physics and the quantum theory of atoms and molecules and for contributing to the fundamental understanding of complex optical waveguiding devices." In 1992 he was elected a fellow of the Optical Society of America.


Theodore H. Geballe

Theodore Henry Geballe (born 20 January 1920) is an American physicist who is an emeritus professor of applied physics at Stanford University. He is known for his work on the synthesis of novel materials of interest to several areas of physics and many interdisciplinary sciences.
Biography.
Theodore Geballe was born and brought up in a Jewish family in San Francisco, California. His grandfather left the Province of Posen in Prussia to move to the United States in about 1870. He attended the Galileo High School in San Francisco, graduating in 1937. Geballe then travelled across San Francisco Bay to attend college at the University of California, Berkeley. While still an undergraduate student at Berkeley, Geballe worked in William Giauque's lab to accurately measure the specific heat of gold.
In 1941, Geballe was called to active duty as an Army Ordnance Officer during the Second World War. Geballe served in Australia, New Guinea and the Philippines and was responsible for maintaining guns.
After the war, Geballe returned to Berkeley as a graduate student of Giauque, receiving his PhD in chemistry in 1949 - the same year that Giauque won the Nobel Prize in Chemistry. In 1952, he moved to Bell Labs in Murray Hill, New Jersey. While at Bell labs, he worked on studying transport properties in semiconductors at very low temperatures, and also on studying properties of unconventional superconductors. In 1967, Geballe joined Stanford University as a professor in the newly founded Department of Applied Physics as well as the Department of Materials Science and Engineering. Geballe was involved in research in the then-upcoming field of multilayered heterostructure materials.
Geballe served as the head of the Department of Applied Physics at Stanford from 1975 to 1977, and was the director of the Center for Materials Research from 1978 to 1988.
In January 2020, Geballe's 100th birthday was marked by a celebratory conference.
Awards and honours.
Geballe, along with Bernd Matthias won the 1970 Oliver E. Buckley Condensed Matter Prize awarded by the American Physical Society "for experiments that challenged theoretical understanding and opened up the technology of high-field superconductors." Geballe also won the 1991 von Hippel award by the Materials Research Society, and was made a member of the National Academy of Sciences in 1973.
In 1989, he received the Bernd T. Matthias Prize.
In 2000, the new Laboratory for Advanced Materials at Stanford was named "The Theodore H. Geballe Laboratory for Advanced Materials" in his honor.
Personal life.
Geballe was married to Frances Koshland (1921–2019), daughter of Daniel E. Koshland Sr. and granddaughter of Abraham Haas. They have six children: Gordon Theodore Geballe, Alison Frances Geballe, Adam Phillip Geballe, Monica Ruth Geballe, Jennifer Corinne Geballe and Ernest Henry Geballe. He has 16 grandchildren and 10 great grandchildren.


Colin Bruce

Colin Bruce is a British author and physicist. He has written many scientific works, including non-fiction, but he is most well known for his popular science stories. He is an expert in mathematical paradoxes and a lover of mysteries. Living in Oxford, he frequently contributes to university life through giving talks at the Oxford University Physics Society and Oxford University Scientific Society, and attending various events.


Paul Rudolph (physicist)

Paul Rudolph (14 November 1858 – 8 March 1935) was a German physicist who designed the first anastigmatic lens while working for Carl Zeiss. After World War I, he joined the Hugo Meyer optical company, where he designed most of their cine lenses.


Nested neutron spectrometer

The nested neutron spectrometer (NNS) is a tool used for neutron spectroscopy. The NNS is used to measure the energy spectrum of neutrons in a neutron field. This type of detector is used in both research facilities (for characterization of neutron fields) and workplaces, where neutron radiation maybe encountered, for radiation protection purposes. Due to the difficulty associated with the detection of neutrons, the NNS is one of the few pieces of equipment capable of accurately determining the characteristics of a neutron field.
Principles of operation.
The NNS operates under the same principle as a Bonner sphere neutron spectrometer. The sensitive part of the spectrometer is the Helium-3 proportional counter, which detects neutrons through the reaction 3He(n,p)3H. This particular reaction has a significant cross-section only at thermal neutron energies. In order for a higher energy neutrons to be detected their energies must be decreased, or "moderated". To slow down neutrons at higher energies the NNS uses different amounts of high-density polyethylene (HDPE). The He-3 proportional counter is surrounded by varying amounts of HDPE to sample various energy regions of the neutron field. The standard NNS has seven HDPE cylinders which are nested together, similar to Russian nesting dolls, to measure neutrons of higher and higher energies.
Sample neutron spectra.
Shown below are spectra of two standard calibration sources measured by the NNS. The measured spectra (blue) is compared to the ISO standard spectra (red). Both Measurements were made at the NIST low scatter facility in Washington, DC.


VP8 Image Analyzer

The VP8 Image Analyzer is an analog computer produced by Pete Schumacher of Interpretations Systems Incorporated (ISI) in 1972. It has been used by believers to image the Shroud of Turin.


Rutherford gold foil experiment


Jean Becquerel

Jean Becquerel (5 February 1878 – 4 July 1953) was a French physicist, and son of Antoine-Henri Becquerel. He worked on the optical and magnetic properties of crystals, discovering the rotation of the plane of polarisation by a magnetic field. He also published a textbook on relativity. In 1909, he became the fourth in his family to occupy the physics chair at the Muséum National d'Histoire Naturelle, following in the footsteps of his father, his grandfather A. E. Becquerel and his great-grandfather Antoine César Becquerel.
Further reading.


Gerald Guralnik

Gerald Stanford "Gerry" Guralnik (; September 17, 1936 – April 26, 2014) was the Chancellor’s Professor of Physics at Brown University. In 1964 he co-discovered the Higgs mechanism and Higgs boson with C. R. Hagen and Tom Kibble (GHK). As part of "Physical Review Letters" 50th anniversary celebration, the journal recognized this discovery as one of the milestone papers in PRL history. While widely considered to have authored the most complete of the early papers on the Higgs theory, GHK were controversially not included in the 2013 Nobel Prize in Physics.
In 2010, Guralnik was awarded the American Physical Society's J. J. Sakurai Prize for Theoretical Particle Physics for the "elucidation of the properties of spontaneous symmetry breaking in four-dimensional relativistic gauge theory and of the mechanism for the consistent generation of vector boson masses".
Guralnik received his BS degree from the Massachusetts Institute of Technology in 1958 and his PhD degree from Harvard University in 1964. He went to Imperial College London as a postdoctoral fellow supported by the National Science Foundation and then became a postdoctoral fellow at the University of Rochester. In the fall of 1967 Guralnik went to Brown University and frequently visited Imperial College and Los Alamos National Laboratory where he was a staff member from 1985 to 1987. While at Los Alamos, he did extensive work on the development and application of computational methods for lattice QCD.
Guralnik died of a heart attack at age 77 in 2014.


Bethe formula

The Bethe formula or Bethe-Bloch formula describes the mean energy loss per distance travelled of swift charged particles (protons, alpha particles, atomic ions) traversing matter (or alternatively the stopping power of the material). For electrons the energy loss is slightly different due to their small mass (requiring relativistic corrections) and their indistinguishability, and since they suffer much larger losses by Bremsstrahlung, terms must be added to account for this. Fast charged particles moving through matter interact with the electrons of atoms in the material. The interaction excites or ionizes the atoms, leading to an energy loss of the traveling particle.
The non-relativistic version was found by Hans Bethe in 1930; the relativistic version (shown below) was found by him in 1932. The most probable energy loss differs from the mean energy loss and is described by the Landau-Vavilov distribution.
The formula.
For a particle with speed "v", charge "z" (in multiples of the electron charge), and energy "E", traveling a distance "x" into a target of electron number density "n" and mean excitation potential "I", the relativistic version of the formula reads, in SI units:
where "c" is the speed of light and "ε"0 the vacuum permittivity, formula_1, "e" and "me" the electron charge and rest mass respectively.
Here, the electron density of the material can be calculated by
where "ρ" is the density of the material, "Z" its atomic number, "A" its relative atomic mass, "NA" the Avogadro number and "Mu" the Molar mass constant.
In the figure to the right, the small circles are experimental results obtained from measurements of various authors, while the red curve is Bethe's formula. Evidently, Bethe's theory agrees very well with experiment at high energy. The agreement is even better when corrections are applied (see below).
For low energies, i.e., for small velocities of the particle "β" « 1, the Bethe formula reduces to
This can be seen by first replacing "βc" by "v" in eq. (1) and then neglecting "β"2 because of its small size.
At low energy, the energy loss according to the Bethe formula therefore decreases approximately as "v"−2 with increasing energy. It reaches a minimum for approximately "E" = 3"Mc"2, where "M" is the mass of the particle (for protons, this would be about at 3000 MeV). For highly relativistic cases "β" ≈ 1, the energy loss increases again, logarithmically due to the transversal component of the electric field.
The mean excitation potential.
In the Bethe theory, the material is completely described by a single number, the mean excitation potential "I". In 1933 Felix Bloch showed that the mean ionization potential of atoms is approximately given by
where "Z" is the atomic number of the atoms of the material. If this approximation is introduced into formula () above, one obtains an expression which is often called "Bethe-Bloch formula". But since we have now accurate tables of "I" as a function of "Z" (see below), the use of such a table will yield better results than the use of formula ().
The figure shows normalized values of "I", taken from a table. The peaks and valleys in this figure lead to corresponding valleys and peaks in the stopping power. These are called ""Z2"-oscillations" or ""Z2"-structure" (where "Z2" = "Z" means the atomic number of the target).
Corrections to the Bethe formula.
The Bethe formula is only valid for energies high enough so that the charged atomic particle (the ion) does not carry any atomic electrons with it. At smaller energies, when the ion carries electrons, this reduces its charge effectively, and the stopping power is thus reduced. But even if the atom is fully ionized, corrections are necessary.
Bethe found his formula using quantum mechanical perturbation theory. Hence, his result is proportional to the square of the charge "z" of the particle. The description can be improved by considering corrections which correspond to higher powers of "z". These are: the Barkas-Andersen-effect (proportional to "z"3, after Walter H. Barkas and Hans Henrik Andersen), and the Bloch-correction (proportional to "z"4). In addition, one has to take into account that the atomic electrons of the material traversed are not stationary ("shell correction").
The corrections mentioned have been built into the programs PSTAR and ASTAR, for example, by which one can calculate the stopping power for protons and alpha particles. The corrections are large at low energy and become smaller and smaller as energy is increased.
At very high energies, Fermi's density correction has to be added also.


Physics Letters

Physics Letters was a scientific journal published from 1962 to 1966, when it split in two series now published by Elsevier:
"Physics Letters B" is part of the SCOAP3 initiative.


Benjamin Fain

Benjamin Fain (, ) (February 17, 1930 – April 15, 2013) was an Israeli physicist, professor-emeritus, and former refusenik.
Biography.
Fain was born to a Jewish family in Kyiv. His father was a mathematician. He instilled in the child a love for science as well as a strong national sentiment.
Benjamin Fain was named after his grandfather, who was murdered in the Proskurov pogrom. During the Second World War the family was evacuated and changed location several times. After the end of the war the family stayed in Dushanbe, where Fain graduated from school. He became a student in the Moscow Institute of Energetics. During his first year in Moscow he visited synagogue and attempted to learn the Hebrew and Yiddish languages. Fain was strongly impressed by the historical visit of the first Israeli ambassador to USSR, Golda Meir. Fain managed to transfer in 1950 to the Faculty of physics in Gorky University. He graduated there summa cum laude. His instructor was future Nobel prize winner Vitaly Ginzburg.
Academic career.
Fain successfully started his scientific career, and already in 1965 became a professor in his alma mater. He wrote several scientific books translated into English and German. In 1966 he moved to Moscow and started successful work in the Institute of Solid State Physics in Chernogolovka.
Starting from 1972 Fain gradually started to participate in a Zionist movement. He took part in refusenik scientific seminar, and also in Samizdat. He applied for exit visa to Israel in 1974 and became a refusenik. He also became unemployed after dismissal from his work on political grounds.
Sociological research of Soviet Jewry.
In 1976 Fain initiated a sociological research on Soviet Jewry. An attempt to organize an international symposium on the subject was foiled by the KGB, which closely watched all his steps from then on. At the same period Fain gradually started practicing Judaism. After several arrests, searches, interrogations and a hunger strike Fain finally arrived in Israel in 1977.
Fain published his study on Jewish identity of Soviet Jews with the American sociologist Mervin Verbit. Fain and Verbit published their findings in 1984 through the Jerusalem Center for Public Affairs.
In Israel.
He continued to struggle to improve the life of Soviet Jews and also continued his scientific work in Tel Aviv University in the fields of quantum electronics, lasers and condensed matter.
Starting from 1998 his field of interest moved to the philosophy of science and Judaism and the interrelation between them. After retirement Fain wrote his first philosophic book in Hebrew: "Creation Ex Nihilo", where he analyzes the relationship between religion and science. It was published in Hebrew as well as in English and Russian translations. The Russian version of the book also has an autobiographic part to it.
In 2008 Fain completed another book in Hebrew: "Law and Providence". It was published in 2011 in English by Urim Publications. In January 2011 Fain's third book was published by .
Children.
Fain is the father of two sons and one daughter.


Mark Thomson (physicist)

Mark Thomson is a Professor of Experimental Particle Physics at the Cavendish Laboratory at the University of Cambridge and a Fellow at Emmanuel College, Cambridge. In January 2018 he was announced as the new Executive Chair of the Science and Technology Facilities Council.
He works on the Deep Underground Neutrino Experiment, MicroBooNE and the MINOS experiment, amongst others, having previously worked on the OPAL experiment at LEP. He is the author of the 2013 book "Modern particle physics".
Thomson completed a B.A and DPhil in Physics at the University of Oxford. His DPhil was supervised by John H. Cobb and his thesis was entitled "An experimental study of the possible association of deep underground muons with astronomical point sources".


Matvei Petrovich Bronstein

Matvei Petrovich Bronstein (, , Vinnytsia – February 18, 1938) was a Soviet theoretical physicist, a pioneer of quantum gravity, author of works in astrophysics, semiconductors, quantum electrodynamics and cosmology, as well as of a number of books in popular science for children.
He introduced the "cGh" scheme for classifying physical theories. "After the relativistic quantum theory is created, the task will be to develop the next part of our scheme, that is to unify quantum theory (with its constant "h"), special relativity (with constant "c"), and the theory of gravitation (with its "G") into a single theory."
He was married to Lydia Chukovskaya, a writer, prominent human rights activist, and a friend of Andrei Sakharov.
During the Great Purge, in August 1937 Bronstein was arrested. He was convicted by a list trial () in February 1938 and executed the same day in a Leningrad prison. His wife was told that he was sentenced to 10 years of labor camps without the right of correspondence.
Bronstein's books for children "Solar Matter" (), "X Rays" (), "Inventors of Radio" () were republished after his reputation had been rehabilitated in 1957.
The "Bronstein Prize in Loop Quantum Gravity" is offered to post-doctoral scholars in the field, the inaugural winner of which was Eugenio Bianchi in 2013.
Solar Matter.
Samuil Marshak, a renowned children’s writer and outstanding editor, scouted Bronstein to write a popular science book for teens. Bronstein chose to describe spectral analysis, but it took many attempts as well as Marshak's advice to determine that the best plot for the story should be a history of helium.
Solar Matter was first published in book form in April 1936, whereupon Bronstein made an inscription on a copy to L. Chukovskaya, his copy editor, “To my dear Lida, without whom I would have never been able to write this book.” Later, Chukovskaya wrote that the history of helium had become intertwined in her and Bronstein’s life, “The work on the book brought us closer. In fact, the book got us married."


Sameera Moussa

Sameera Moussa () (March 3, 1917 – August 5, 1952) was the first female Egyptian nuclear physicist. Sameera held a doctorate in atomic radiation. She hoped her work would one day lead to affordable medical treatments and the peaceful use of atomic energy. She organized the Atomic Energy for Peace Conference and sponsored a call that set an international conference under the banner "Atoms for Peace." She was the first woman to work at Cairo University.
Youth and college.
Moussa was born in Egypt in Gharbia Governorate in 1917. Her mother died from cancer, and her father was a famous political activist. He moved with his daughter to Cairo and invested his money in a small hotel in the El-Hussein region. At the insistence of her father, Moussa attended Kaser El-Shok primary school, one of the oldest schools in Cairo. After she completed her primary education, she joined the Banat El-Ashraf school, which was built and managed by her father.
Despite the fact that Moussa achieved high grades in her secondary education, and could have pursued a career in engineering, she insisted on joining the Faculty of Sciences at Cairo University. In 1939, Moussa obtained a BSc in radiology with first class honors after researching the effects of X-ray radiation on various materials. Dr. Moustafa Mousharafa, the first dean of the faculty, believed in his student enough to help her become a remarkable lecturer at the faculty. Afterwards, she became the first assistant professor at the same faculty and the first woman to hold a university post, being the first to obtain a PhD in atomic radiation.
Nuclear research.
Moussa believed in Atoms for Peace. She was known to say "My wish is for nuclear treatment of cancer to be as available and as cheap as Aspirin". She worked hard for this purpose and throughout her intensive research, she came up with a historic equation that would help break the atoms of cheap metals such as copper, paving the way for a cheap nuclear bomb.
Moussa organized the Atomic Energy for Peace Conference and sponsored a call for setting an international conference under the banner "Atom for Peace", where many prominent scientists were invited. The conference made a number of recommendations for setting up a committee to protect against nuclear hazards, for which she strongly advocated. Moussa also volunteered to help treat cancer patients at various hospitals especially since her mother went through a fierce battle against this disease.
Visits to the United States.

Omegatetravirus

Omegatetravirus is a genus of viruses, in the family "Alphatetraviridae". Moths and butterflies serve as natural hosts. There are three species in this genus. Infection outcome varies from unapparent to lethal.
Taxonomy.
The following species are assigned to the genus:
Structure.
Viruses in "Omegatetravirus" are non-enveloped, with icosahedral geometries, and T=4 symmetry. The diameter is around 40 nm. Genomes are linear and bipartite, around 2.5kb in length.
Life cycle.
Viral replication is cytoplasmic. Entry into the host cell is achieved by penetration into the host cell. Replication follows the positive stranded RNA virus replication model. Positive stranded RNA virus transcription is the method of transcription. Translation takes place by leaky scanning. Moths and butterflies serve as the natural host. Transmission routes are oral.


Otroeda cafra

Otroeda cafra is a species of moth in the tussock-moth subfamily Lymantriinae. It was first described by Dru Drury in 1782 from Sierra Leone, and is also found in Cameroon, DR Congo, Malawi, and Nigeria.
Description.
Upperside. Antennae deeply pectinated and brown. Head white. Thorax and abdomen yellowish grey. Anterior wings white, edged with a dull blackish, and a large patch of the same at the tips, with two streaks also running from the anterior edges almost across the wings; one near the shoulders, the other in the middle, both narrowing as they go, and ending in points. A little group of spots is also placed at the lower corners. Posterior wings white, but next the body yellow, with a very irregular dark border running along the external edges, on which are several white spots unevenly dispersed, and of various forms.
Underside. Palpi white, tipped with black. Forelegs red underneath, and black above, the ends being black: the other legs grey, the tips black. Breast and abdomen yellow grey. Wings coloured as on the upperside, but fainter. Margins of the wings entire. Wingspan 4 inches (100 mm).


Alicyclobacillus kakegawensis

Alicyclobacillus kakegawensis is a species of Gram positive, strictly aerobic, bacterium. The bacteria are acidophilic and produce endospores. It was first isolated from soil in Kakegawa, Japan. The species was first described in 2007, and the named after the city from which it was first isolated.
The optimum growth temperature for "A. kakegawensis" is 50-55 °C, and can grow in the 40-60 °C range. The optimum pH is 4.0-4.5, and cannot grow at pH 3.0 or pH 6.5.
"A. kakegawensis" was found during a Japanese survey of various beverages and environments, which also discovered 5 other species of "Alicyclobacillus": "A. contaminans", "A. fastidiosus", "A. macrosporangiidus", "A. sacchari", and "A. shizuokensis".


Chaenotheca furfuracea

Chaenotheca furfuracea is a mealy (farinaceous), bright yellow-green leprose pin lichen. It is in the family Coniocybaceae and can be found in European countries like Belgium, Luxembourg, and Switzerland. The species are growing mostly on beeches and oaks, and on tree roots of spruces. They also grow on detritus or sand, and in rare cases on fissures of siliceous rocks. It prefers climates with high humidity and low luminosity.


Boolarra virus

Boolarra virus (BoV) is a member of the family "Nodaviridae". It was named after Boolarra, Victoria where it was first discovered after infecting and killing or severely debilitating a pest larvae. As of now, the only known host for "Boolarra virus" is the "Oncopera intricoides" which a moth that is endemic to Victoria (Australia).
History and discovery.
In the winter of 1977, an outbreak of "Oncopera intricoides" prompted examination of the insect and its habits. While examining the pest, many dead and terminal larvae were found. After looking closely to discover the cause these insects' ailment, researchers were unable to find any traces of bacteria, fungi, or protozoa that could have contributed to the insects' death. After further examination, the researchers found viral particles that were found to be members of the Nodaviridae. After comparing the RNA genome of this virus with that of other members of the Nodaviridae, it was found to be different and therefore a new member of the Nodaviridae. After this discovery, the virus was named after the place where the infected "Oncopera intricoides" were found; Boolarra, Victoria.
Physical data.
"Boolarra virus" typically measures around 30 nano-meters and is approximately 21 percent RNA. "Boolarra virus" is characterized by its polyhedral shape. The virus only shares a significant portion of the same amino acid sequence with the Nodamura virus, which is a similar Nodavirus. Among other Nodoviruses, Boolarra was not very similar concerning amino acid sequences. Still a very rare virus, "Boolarra virus" can only be found in a very small population of moths that is endemic to the region of Victoria, Australia. While only found within this small population, the virus causes severe harm to occur to infected larvae and can even cause death.


Knoellia flava

Knoellia flava is a species of Gram positive, nonmotile, non-sporeforming bacteria. The bacteria are aerobic and mesophilic, and the cells can be irregular rods or coccoid. It was originally isolated from pig manure from Huazhong Agricultural University in Wuhan, China. The species name is derived from Latin "flava" (yellow), referring to the colony color of the species when grown on R2A agar.
The optimum growth temperature for "K. flava" is 28 °C and can grow in the 4-37 °C range. The optimum pH is 7.0, and can grow in pH 5.0-9.0.