HQT: Notation, Glossary and Mathematical Definitions © 2005 Jochem Hauser and Walter Dröscher

In the following some material for reference (no- Appendix 1: Notation and Physi- tation, glossary, and mathematical definitions) as cal Constants well as for review (mathematical definitions) is given. Ã value for the onset of conversion of photons into gravitophotons. This material is constantly updated (August 4, 2005). A denotes the strength of the shielding poten- tial caused by virtual electrons. The notation in Appendix 1 describes the sym- bols used in our publications concerning Heim Compton wave length of the electron Quantum Theory (HQT). h −12 C = =2.43×10 m , ƛC =C /2. me c The glossary of Appendix 2 describes the special terminology used in HQT. c speed of light in vacuum 299,792,458 m/s , 2 ε µ (1/c = 0 0). The glossary of Appendix 3 describes and ex- plains the special mathematical terminology used D diameter of the primeval universe, some in Heim's original work. 10125 m that contains our optical universe.

DO diameter of our optical universe, some The mathematical definitions in Appendix 4 refer 1026 m. to definitions used in modern physics, and are meant to facilitate the reading of our papers. d diameter of the rotating torus.

dT vertical distance between magnetic coil and rotating torus.

-e electron charge -1.602 × 10-19 C.

ez unit vector in z-direction.

Fe electrostatic force between 2 electrons.

Fg gravitational force between 2 electrons.

Fgp gravitophoton force, also termed Heim-Lo- T rentz force, F gp=p e 0 v ×H .

-11 3 -1 -2 G = Gg + Ggp + Gq = 6.67259 × 10 m kg s ,

1 of 14 dius. The distance at which gravitation Gg graviton constant, G ≈G that is Gg de- g changes sign, ρ, is some 46 Mparsec. cribes the gravitational interaction without the postulated gravitophoton and quintes- R+ denotes an upper bound for gravitation and sence interactions. is some type of Hubble radius, but is not the radius of the universe, instead it is the radius 2 Ggp gravitophoton constant, G gp≈1/67 G g . of the optically observable universe. Gravita- tion is zero beyond the two bounds, that is, −18 particles smaller than R- cannot generate Gq quintessence constant, G ≈4×10 G . q g gravitational interactions.

gp g i k metric subtensor for the gravitophoton re classical electron radius in subspace I2 S2 (see glossary for sub- ∪ 2 space description). 1 e −15 re= 2 =3 × 10 m . 40 me c

ph g i k metric subtensor for the photon in sub- 2 2 1 rge ratio of gravitational and electrostatic space I ∪S ∪T (see glossary for subspace forces between two electrons. description). v velocity vector of charges flowing in the mag- h Planck constant 6.626076 × 10-34 Js, netic coil, some 103 m/s in circumferential di- ℏ=h/2. rection. vT bulk velocity vector for rigid rotating ring h metric components for an almost flat space- ik (torus) (see Sections. 3 and 4), some 103 m/s time. in circumferential direction.

3 G ℏ −35 wgp probability amplitude (the square is the ℓ p= =1.6×10 m Planck length. c3 coupling coefficient) for the gravitophoton force (fifth fundamental interaction) -31 me electron mass 9.109390 × 10 kg. 2 2 me −49 wgp=G gp =3.87×10 probability amplitudes m0 mass of proton or neutron 1.672623 × ℏ c 10-27 kg and 1.674929 × 10-27 kg. (or coupling amplitudes) can be distance de- pendent. Nn number of protons or neutrons in the uni- verse. wgpe probability amplitude for emitting a gravitophoton by an electron q electric charge.

wgpe=wgp . R distance from center of coil to location of virtual electron in torus. wgpa probability amplitude for absorption of a gravitophoton by a proton or neutron rN distance from nucleus to virtual electron in torus. m w2 =G m e . gpa gp p ℏc R_ is a lower bound for gravitational struc- tures, comparable to the Schwarzschild ra- wg_q conversion amplitude for the transforma- tion of gravitophotons and gravitons into the

2 of 14 quintessence particle, corresponding to the BPP breakthrough propulsion physics dark energy (rest mass of some 10-33 eV). GP Geomtrization Principle wph probability amplitude (the square is the coupling coefficient for the electromagnetic GR General Relativity force, that is the fine structure constant α) GRP General Relativity Principle

2 2 1 e 1 HQT Heim Quantum Theory w ph= = . 40 ℏ c 137 LQT Loop Quantum Theory wph_qp conversion amplitude for the transforma- tion of photons into gravitophotons. LHS left hand side ls light second wq probability amplitude for the quintessence particle,(sixth fundamental interaction), cor- responding to dark energy (rest mass of ly light year -33 some 10 eV). QED Quantum Electro-Dynamics

Z charge number (number of protons in a nu- RHS right hand side cleus of an atom). SR Special Relativity Z0 impedance of free space, VSL Varying Speed of Light

0 Z 0= ≈376.7. 0 Subscripts α coupling constant for the electromagnetic force or fine structure constant 1/137. e electron

αgp coupling constant for the gravitophoton gp gravitophoton force. gq from gravitons and gravitophotons into quin- γ ratio of probabilities for the electromagnetic tessence and the gravitophoton force ph denoting the photon or electrodynamics 2 w = ph =1.87×1046 . sp space w gp Superscripts -7 2 0 permeability of vacuum 4π × 10 N/m . em electromagnetic t metron area (minimal surface 3Gh/8c3), cur- gp gravitophoton rent value is 6.1510-70 m2. ph photon Φ gravitational potential, Φ=GM/R. T indicates the rotating ring (torus) ω rotation vector.

Abbreviations

3 of 14 Note: Since the discussion is on engineering Appendix 2: Glossary of Physical problems, SI units (Volt, Ampere, Tesla or Weber/m2 ) are used. 1 T = 1 Wb/m2 = 104 G Terms = 104 Oe, where Gauss (applied to B, the aeon Denoting an indefinitely long period of magnetic induction vector) and Oersted (ap- time. The aeonic dimension can be inter- plied to H, magnetic field strength or mag- preted as steering structure governed by the netic intensity vector) are identical. Gauss entelechial dimension toward a dynamically and Oersted are used in the Gaussian sys- stable state. tem of units. In the MKS system, B is meas- ured in Tesla, and H is measured in A/m apeiron the unlimited primeval substance in (1A/m = 4π × 10-3 G). Greek natural philosophy Used to charac- terize the state of existence before the quan- Note: For a conversion from CGS to SI units, tized bang, similar to Penrose's mathematical the electric charge and magnetic field are re- world or world of potentialities. placed as follows: anti-hermetry Coordinates are called anti-her- e e 4 and H 4 H . / 0 0 metric if they do not deviate from Cartesian coordinates, i.e., in a space with anti-her- metric coordinates no physical events can take place.

canonical Conforming to a general rule or ac- ceptable procedure, a canonical form is the simplest form possible (for instance a unit matrix).

condensation For matter to exist, as we are used to conceive it, a distortion from Euclidean metric or condensation, a term introduced by Heim, is a necessary but not a sufficient con- dition.

condensor The Christoffel symbols of the sec- i ond kind k m become the so called con- i densor functions, km that are normalizable. i It can be shown that km the have tensor character. It should be noted that in the ei- genvalue equations for the mass spectrum of i elementary particles, the km are eigenfunc- tions and thus must not be confused with the Christoffel symbols. It should be mentioned that Heim first writes a symbolic eigenvalue equation that he later on using symmetry ar- guments transforms into a mathematically correct eigenvalue equation. The term con- densor is derived from the fact that these functions represent condensations of the spacetime metric. The condenser is an opera-

4 of 14 tor projecting a deformation in the 6, 8, or stance, Newton's law of gravitation is not 12-dimensional metronic lattice into ℝ4, covariant under a →Lorentz transformation. where it appears as an intricate, geometri- cally structured, compressed or “condensed” entelechy (Greek entelécheia, objective, com- lattice configuration. This condensed, struc- pletion) used by Aristotle in his work The tured region is what eventually is interpreted Physics. Aristotle assumed that each phe- as matter constituting an elementary particle. nomenon in nature contained an intrinsic ob- jective, governing the actualization of a condensor flux form-giving cause. The entelechial dimension can be interpreted as a measure of the quality conjunctor of time varying organizational structures (in- verse to entropy, e.g., plant growth) while conversion amplitude Allowing the transmuta- the aeonic dimension is steering these struc- tion of photons into gravitophotons, wph_gp tures toward a dynamically stable state. Any (electromagnetic-gravitational interaction), coordinates outside spacetime can be consid- and the conversion of gravitophotons into ered as steering coordinates. quintessence particles, wgp_q (gravitational- gravitational interaction). differentiable manifold Contains a collection of points, each of which determines a unique coupling constant Value for creation and de- position in Heim space. The smoothness struction of messenger (virtual) particles, → relative to the strong force (whose value is feature is only applicable in the case where set to 1 in relation to the other coupling con- the physical problem considered involves a stants). large number of →metrons. Continuous and differentiable functions are supported. The coupling potential between photon-gravito- differentiable manifold is a topological space photon (Kopplungspotential) As coupling (open sets) and is locally equivalent to a n gp space which is of the same dimension as potential the term g of the metric is de- ℝ i k the corresponding Heim space, i.e., there ex- noted. The reason for using the superscript ists a one-to-one mapping between the open gp is that this part of the photon metric sets of the Heim space and the ℝn. equals the metric for the gravitophoton parti- cle and that a →sieve (conversion) operator energy coordinates exists, which can transform a photon into a gravitophoton by making the second term in epistemology Theory of the nature of knowl- the metric anti-hermetric. In other words, the edge especially with reference to its limits electromagnetic force can be transformed and validity. into a repulsive gravitational like force, and eschatology Concerned with the final events in thus can be used to accelerate a material the history of the universe. body. event cosmogony (Kosmogonie) The creation or ori- gin of the world or universe, a theory of the field activator (Feldkaktivator) flips the spin of origin of the universe (derived from the two a metron, i.e., changes its orientation. Greek words kosmos (harmonious universe) and gonos (offspring)). flucton (Flukton) being movable and compressi- ble. covariant For different inertial frames the laws of physics are varying so as to preserve the flux aggregate mathematical form of these laws. For in-

5 of 14 fundamental kernel (Fundamentalkern) Since ric subtensor is constructed in the subspace 2 2 1 of coordinates I , S and T , denoted as her- the function i m occurs in x m = i m d i ∫ metry form H5. The equation describing the as the kernel in the integral, it is denoted as Heim-Lorentz force has a form similar to the fundamental kernel of the poly-metric. electromagnetic Lorentz force, except, that it exercises a force on a moving body of Galilean spacetime A spacetime in which the → mass m, while the Lorentz force acts upon Galilei transformation is valid moving charged particles only. In other words, there seems to exist a direct coupling Galilei transformation between matter and electromagnetism. In that respect, matter can be considered play- geodesic zero-line process This is a process ing the role of charge in the Heim-Lorentz where the square of the length element in a equation. The force is given by 6- or 8-dimensional →Heim space is zero. T F gp=p e 0 v ×H . Here Λp is a coefficient, gravitational limit(s) There are three distances vT the velocity of a rotating body (insulator) at which the gravitational force is zero. First, of mass m, and H is the magnetic field at any distance smaller than R_, the gravita- strength. It should be noted that the gravito- tional force is 0. Second, photon force is 0, if velocity and magnetic field strength are parallel. gravitophoton (field) Denotes a gravitational like field, represented by the metric sub-ten- Heimian metric gp Heim space A Heim space is a discrete space of sor, gi k , generated by a neutral mass with 6, 8, or 12 dimensions, denoted as 6, 8, 12 a smaller coupling constant than the one for H H H respectively, with three spatial coordinates gravitons, but allowing for the possibility of + signature, while any other coordinate that photons are transformed into gravito- has - signature. In a Heim space an elemen- photons. Gravitophoton particles can be both attractive and repulsive and are always tal surface, termed →metron, exists. If the generated in pairs from the vacuum under surface considered comprises a large number the presence of virtual electrons. The total of metrons, a Heim space can be considered enery extracted from the vacuum is zero, but a →differentiable manifold endowed with a only attractive gravitophotons are absorbed →Heimian metric. by protons or neutrons. The gravitophoton hermetry form (Hermetrieform) The word field represents the fifth fundamental interac- hermetry is an abbreviation of hermeneutics, tion. The gravitophoton field generated by in our case the semantic interpretation of the repulsive gravitophotons, together with the metric. To explain the concept of a hermetry →vacuum particle, can be used to reduce the 6 form, the space H is considered. There are 3 gravitational potential around a spacecraft. coordinate groups in this space, namely 1 2 3 graviton (Graviton) The virtual particle respon- s3= , , forming the physical space sible for gravitational interaction. 3 4 1 ℝ , s2= for space T , and heimline In analogy to →worldline, the path of s =5 ,6 for space S2. The set of all a particle in →Heim space. 1 possible coordinate groups is denoted by S= {s1, s2, s3}. These 3 groups may be com- Heim-Lorentz force Resulting from the newly bined, but, as a general rule (stated here predicted gravitophoton particle that is a 8 without proof, derived, however, by Heim consequence of the →Heim space H . A met- 6 from conservation laws in H , (see p. 193 in

6 of 14 Elementarstrukturen der Materie, Vol I, uniform structure or composition through- Resch Verlag 1996), coordinates 5 and 6 out. must always be curvilinear, and must be pre- sent in all metric combinations. An allowable hyperstructure (Hyperstruktur) Any lattice of combination of coordinate groups is termed a →Heim space that deviates from the iso- hermetry form, responsible for a physical tropic Cartesian lattice, indicating an empty field or interaction particle, and denoted by world, and thus allows for physical events H. H is sometimes annotated with an index, to happen. or sometimes written in the form H=(1, 2 inertial transformation (Trägheitstransforma- . This is a symbolic notation only, and ,...) tion) Such a transformation, fundamentally should not be confused with the notation of an interaction between electromagnetism and an n-tuple. From the above it is clear that the gravitational like gravitophoton field, re- only 4 hermetry forms are possible in H6. It duces the inertial mass of a material object needs a →Heim space H8 to incorporate all using electromagnetic radiation at specific known physical interactions. Hermetry frequencies. As a result of momentum and means that only those coordinates occurring energy conservation in 4-dimensional space- in the hermetry form are curvilinear, all other time, v/c = v'/c', the Lorentz matrix remains coordinates remain Cartesian. In other unchanged. It follows that c < c' and v < v' words, H denotes the subspace in which where v and v' denote the velocities of the physical events can take place, since these test body before and after the inertial trans- coordinates are non-euclidean. This concept formation, and c and c' denote the speeds of is at the heart of Heim's geometrization of all light, respectively. In other words, since c is physical interactions, and serves as the cor- the vacuum speed of light, an inertial trans- respondence principle between geometry formation allows for superluminal speeds. and physics. An inertial transformation is possible only in a 8-dimensional →Heim space, and is in ac- hermeneutics (Hermeneutik) The study of the cordance with the laws of SRT. In an Ein- methodological principles of interpreting the steinian universe that is 4-dimensional and metric tensor and the eigenvalue vector of contains only gravitation, this transformation the subspaces. This semantic interpretation does not exist. of geometrical structure is called hermeneu- tics (from the Greek word to interpret). isotropic The universe is the same in all direc- tions, for instance, as velocity of light trans- hermitian matrix (self adjoint, selbstad- mission is concerned measuring the same jungiert) A square matrix having the prop- values along axes in all directions. erty that each pair of elements in the i-th row and j-th column and in the j-th row and i-th Lorentzian metric column are conjugate complex numbers (i - i). Let A denote a square matrix and A* de- Lorentz transformation This transformation in noting the complex conjugate matrix. A† := spacetime reflects the fundamental fact that (A*)T = A for a hermitian matrix. A hermitian light travels with exactly the same speed c matrix has real eigenvalues. If A is real, the with respect to any inertial frame hermitian requirement is replaced by a re- ' x−vt ' x = , y = y , z '=z , quirement of symmetry, i.e., the transposed 1−v2/c21/2 matrix AT = A . 2 ' t−vx /c t . = 2 2 1/2 homogeneous The universe is everywhere uni- 1−v /c form and isotropic or, in other words, is of

7 of 14 Minkowski spacetime A spacetime in which the considered the beginning of this physical uni- →Lorentz transformation is valid. verse. That condition can be considered as the mathematical initial condition and, when ontology A particular theory about the nature of inserted into Heim's equation for the evolu- being. tion of the universe, does result in the initial diameter of the original universe [1]. Much partial structure (Partialstruktur) For in- later, when the metron size had decreased far 6 stance, in H , the metric tensor that is hermi- enough, did matter come into existence as a tian comprises three non-hermitian metrics purely geometrical phenomenon. from subspaces of H6. These metrics from subspaces are termed partial structure. selector (Selektor) poly-metric The term poly-metric is used with shielding field (Schirmfeld) respect to the composite nature of the metric sieve operator see → transformtion operator tensor in 8D →Heim space. In addition, there 4 → 8→ 4 is the twofold mapping ℝ H ℝ . transformation operator or sieve operator (Sieboperator) The direct translation of probability amplitude With respect to the six Heim's terminology would be sieve-selector. fundamental interactions predicted from the A transformation operator, however, con- →poly-metric of the →Heim space H8, there verts a photon into a gravitophoton by mak- exist six (running) coupling constants. In the ing the coordinate 4 Euclidean. particle picture, the first three describe gravi- tational interactions, namely wg (graviton, at- vacuum particle responsible for the accelera- tractive), wgp (gravitophoton, attractive and tion of the universe, also termed quintes- repulsive), wq (quintessence, repulsive). The sence particle The vacuum particle repre- other three describe the well known interac- sents the sixth fundamental interaction, and tions, namely wph (photons), ww (vector bos- is a repulsive gravitational force whose ons, weak interaction), and ws (gluons, gravitational coupling constant is given by strong interaction). In addition, there are 4.3565×10-18 G. It only interacts with gravi- two →conversion amplitudes predicted that tons and positive (repulsive) gravitophotons, allow the transmutation of photons into but not with real or virtual particles. gravitophotons (electromagnetic-gravita- tional interaction), and the conversion of worldline Path of a particle in spacetime ℝ4. gravitophotons into quintessence particles (gravitational-gravitational interaction). protosimplex prototrope (Prototrope) first in time *protohis- tory* b : beginning : giving rise to *proto- planet quantized bang According to Heim, the uni- verse did not evolve from a hot big bang, but instead, spacetime was discretized from the very beginning, and such no infinitely small or infinitely dense space existed. Instead, when the size of a single metron covered the whole (spherical volume) universe, this was

8 of 14 Appendix 3: Heim's Original Appendix 4: Mathematical Defini- Mathematical Terminology tions

affine transformation is defined by a mapping (→lambda matrix) from coordinates x x': ' x = x a that guarantees the invari- ance of the spacetime interval 2 2 2 x B – x A −c t B−t A =0 between two →worldpoints A and B. It is necessary that the vacuum speed of light as an upper limit remains invariant between any two inertial systems. The →Lorentz transformation satis- fies this requirement.

bijective mapping A mapping f is bijective if f is both →injective and →surjective. The -1 →inverse mapping f is defined by f x0 -1 x0 that is, f associates with each y0∈Y the corresponding x0∈D(f) for which f x0 = y0.

boost parameter

Casimir invariants the scalar product of →Lie group generators is known as Casimir in- variant that commutes with all the genera- 2 tors (e.g., J commutes with J1, J2, and J3) and is therefore invariant under all group transformations. The eigenvalues of the Ca- simir invariants are the conserved quantum numbers of the symmetry group. The groups O(3) and SU(2) have only one invariant while the SU(3) group has two invariants.

Clifford algebra in the Dirac equation four con- stant coefficients occur that are non-com- mutable square matrices. Thus the wave- function ψ(x) is a column matrix. The matrices satisfy the condition { }:= =2 where is the Minkowski space-time metric tensor (diagonal form). The matrices are 4×4 matrices.

cotangent space consider a point P and its →tangent space TPM on a →manifold M. Let

9 of 14 xμ(λ) be a smooth →curve through P on M. compact Lie group a compact Lie group is a The directional derivative of a smooth func- →Lie group whose parameters are defined in tion f on M at P (tangential vector v) with a closed interval, for instance, for U(1) angle respect to the curve xμ(λ) is a differentiable θ varies in the interval [0, 2π]. mapping TPMℝ. The set of all →one- continuous groups contain an infinite number forms ω(v) at P that map TPMℝ forms the of elements. A simple example is the set of dual vector space to TPM, termed the cotan- * * all complex wavefactors of a wavefunction gent space T PM. Bundling together all T PM in quantum mechanics written in the form U at different points on M, gives the cotangent iθ bundle T*M. (θ)=e . The product of two phasefactors is U(θ) U(θ')= U(θ+θ') and the inverse is given curve in a manifold M if is chosen to be the by U-1(θ)=U(-θ). These phase factors from a distance along the curve, a curve para- group called U(1). This group is character- metrized by has the tangent vector ized by a single parameter, the angle θ in the dx/d ,=1,2,... ,n. interval [0, 2π]. The group is differentable since dU = i U dθ and thus the derivative is diffeomorphism A differentiable, bijective map- an element of the U(1). The group of rota- ping f for which both f and f-1 are smooth, tions in three-dimensional space the →O(3) i.e., arbitrarily often differentiable (note: a group and the group of →Lorentz transfor- homomorphism only requires that f and f-1 mations are also continuous groups. are continuous). It should be noted that a diffeomorphic mapping of generalized coor- Lie group the characteristic of a Lie group is dinates q to coordinates q' leaves the equa- that the parameters of a product are analytic tions of motions invariant that are derived functions of each parameter in the product from Lagrange function L. that is, if U(θ) =U(α) U(β) then θ=f(α, β) where f is an analytic function. The analytic differentiable manifold A →manifold can be property guarantees that the group is differ- covered by patches (charts). Different coor- entiable so that an infinitesimal group ele- dinate systems can be set up on any part of a ment dU(θ) can be defined. The →Lorentz manifold. For two overlapping patches, two group is an example of a noncompact group. different coordinate systems can be defined The boosts or transformations from one iner- in the overlap region, xμ and xμ'= xμ'(xμ ). tial frame to another are represented by non- Any function f in the overlap region that is unitary matrices. The →boost parameter μ differentiable with respect to coordinates x η=tanh-1(v/c) is not restricted to a closed in- should also be differentiable with regard to terval. xμ'. This is true if the transformation xμ'(xμ ) between the two coordinate systems is dif- Lorentz group ferentiable. Such a manifold is called a differ- entiable manifold. If there exist n derivatives orthogonal groups O(n) is the group of rota- there is a Cn manifold. For n=∞, the mani- tions in an n-dimensional Euclidean space. fold is C∞. The elements of O(n) are n×n real valued matrices that have n(n-1)/2 independent pa- dual space →cotangent space. rameters. fiber bundle O(3) is the three-dimensional rotation group and leaves the distance x2+y2+z2 invariant. groups: The parameters are the Euler angles α, β, γ

10 of 14 from classical mechanics. The rotation ma- scalar product and a vector product (in form trix is denoted by R(α, β, γ) and can be writ- of the commutation relation). This vector ten as a sequence of three rotations. product is also called Lie product. For in- stance, for the total angular momentum 2 2 2 2 Poincaré group J =J1 +J2 +J3 .

special unitary groups SU(n) has rotation group the rotation of wavefunction det SU(n)=+1 with (n2-1) independent ele- about a direction given by unit vector n is ments. −i n⋅J written as R=e , where J is the special unitary group SU(2) and SU(3) repre- angular momentum operator. For an angle sent two- and three-dimensional matrices dθ, the rotation is given in first order as and are associated with isotopic spin and R=1−idn⋅J . The combined rota- color. A SU(2) rotation leaves the magnitude of the original vector invariant and has tions about the x and y-axis is given by det SU(2)=1. R=1−i d J 11−i d J 2. Revers- The SU(3) group has eight independent ing the order of rotations and forming the difference is written as the commutator →group generators, represented by 3×3 [J1, J2] dθ dϕ. Angular momentum operators →hermitian matrices and denoted as Fi, obey the commutation relation i=1,...,8. The matrices obey special commu- [Ji, Jk]=i εikm Jm. Operators Jm are called the tation rules. group generators. This means that O(3) be- unitary group the elements of the unitary group longs to a non-commutative or non-Abelian group. In contrast, O(2) and U(1) are Abe- U(n) are given by n×n →unitary matrices. lian groups. The determinant det U(n)=1. hermitian matrix group generator: homeomorphism is a →bijective mapping Lie group generator let G be a Lie group and f: AB for which both f and f-1 are continu- operators Fk be its group generator in anal- ous. ogy to the angular momentum operator Jk for the →rotation group. The operator Fk homomorphism of O(3) and SU(2) there is a generates an element of G in the same way homomorphism (many-to-one mapping) be- that Jk generates a rotation, i.e., tween O(3) and SU(2) that is, a real three-di- i F U =e− k k . The number of generators is mensional rotation can be associated with an equal to the number of independent parame- element of SU(2). In other words, a vector ters in G. Thus there are n(n-1)/2 generators (x, y, z) can be defined from the complex for O(n) and n2-1 generators for SU(n). Gen- vector being transformed by SU(2) which is erators of the orthogonal and unitary groups left invariant. It is possible to associate the general matrix R( , , ) with the complex are represented by →hermitian or self-ad- α β γ joint matrices. The commutation rule is parameters of an SU(2) matrix. This means that the 3 3 real valued matrix O(3) can be given by [Fi, Fk]=i cikm Fm. The cikm are called × the structure constants. For O(3) these val- expressed by the 2×2 complex valued matrix ues are either ±1 or 0. The structure of the is SU(2). However, the relation between the different from the group structure. The form two matrices is not one-to-one, hence the the basis of a linear vector space that is denotation homomorphism ( here two possi- known as Lie algebra. There exists both a bilities exist).

11 of 14 injective mapping a mapping is injective if for of elements of B. Therefore for manifold M every x1≠x2∈D(f), x1≠x2 implies f x1≠ f x2 it is required that every point of M belongs that is, different points in the domain have to at least one open set of its basis B and different images. Therefore, the inverse im- there exists a one-to-one correspondence n age is a single point in D(f). with the points of some open set of ℝ . This means that there is a continuous →bijective inverse mapping for an →injective mapping mapping from the open set of M to the open f : D(f) Y the inverse mapping f-1 is defined set in ℝn. When these conditions are met, M as to be the mapping R(f) D(f) such that is called a manifold. Any function f(P) for y0 ∈R(f) is mapped onto that x0∈D(f) for each point PM can be re-expressed as a μ n which f x0= y0. function g(x ) defined on ℝ . A manifold does not possess a metric, hence no scalar lambda matrix given are two coordinate sys- product can be defined on a manifold, →one- tems x and x´ on a manifold. The transforma- form. tion matrix Λ, →spinor, is defined as ´ mapping Given two sets X and Y with A⊂X. A ´ ∂ x . mapping f from A into Y associates with each = ∂ x x∈X a single y∈Y, called the image of x with respect to f, and written as y=f x. The set A Lie algebra →group generator, see Lie group is called the domain, D(f), of f. The range of generator. f, R(f), is the set of all images. The inverse Lorentz transformation this transformation image of element y0∈Y is the set of all (→affine transformation) in spacetime re- x∈D(f) such that f x= y0. flects the fundamental fact that light travels with exactly the same speed c with respect one-form one-forms are the extension of the to any inertial frame (here x and x' denote scalar product of u v of two Euclidean vec- coordinates) tors to a →manifold M. On M no metric is defined, therefore a salar product cannot be ' x−vt ' x , y y , z ' z , formed. A scalar product is considered as a = 2 2 1/2 = = 1−v /c function. For a given vector u, the symbol u• 2 ' t−vx /c t = . acts as a function that assigns to each vector 1−v2/c21/2 v a real number. This mapping is linear. A one-form ω on M is therefore defined as a manifold A manifold is locally equivalent to an linear mapping that is real-valued. Because n μ n-dimensional euclidean space ℝ . That is of the linearity ω(v)=ωμv . Indices of a one- manifold M has the same local topology. M form are in the lower position. A one-form therefore must be a topological space, which field is defined in the same way as a linear means there is a collection of open sets that function on vector fields. An example for a cover it. Second, the structure of these open one-form field is the gradient of a scalar field

sets, within small regions, is equivalent to f, denoted as f. Taking as v the tangent the natural topology of ℝn. ℝn is a Haussdorf vector d/dλ to a curve xμ(λ), one can write space, which means that for any two differ- ∂ f ∂ x ent points there exist non-overlapping neigh- ω v . f = borhoods, and there exists a basis B in form ∂ x ∂ of a collection of open sets such that each subset of ℝn can be represented by a union

12 of 14 partial derivative :=/x . The partial deriva- to M at P. All tangent vectors are in the tan- tives are sometimes also considered as base gential plane through P. For the example of 2 vectors in the corresponding coordinate sys- the sphere TPM is the vetorspace ℝ . tems, see →one-form. tangent bundle The disjoint union of all →tan- rank of tensor field tensors and tensor fields are gent spaces TM :=∪ T P M is itself a classified by their rank, denoted as j . P i manifold of dimension 2n if M has dimension n. The individual tangentspace TPM at point Rank 0 are called scalars and are real 0 P is called a fiber. numbers. A scalar field is a real valued func- tensor field assigns a physical property to every tion f(P) on the manifold. Rank 1 are 0 point on the manifold. called contravariant vectors. They corre- unitary matrix (unitär) let A denote a square spond to a tangential vector on a curve in matrix, and A* denoting the complex conju- differential geometry. In a coordinate system † T -1 a vector can be resolved in its components. gate matrix. If A := (A*) = A , then A is a In general the upper variable denotes the unitary matrix, representing the generaliza- number of contravariant indices and the tion of the concept of orthogonal matrix. If lower one the covariant indices of a tensor. A is real, the unitary requirement is replaced by a requirement of orthogonality, i.e., -1 T spinor the wavefunction of the Dirac function is A = A . The product of two unitary matri- a 4-component column matrix. These com- ces is unitary. ponents will be expressed by a different set of functions and also be rearranged when unitary transformation used in quantum me- transformed from coordinate system x to x. chanics, leaving the modulus squared of the Since the components of are not compo- a complex wavefunction invariant, → unitary nents of a spacetime vector, but represent a group. state in which a particle can exist, they do not transform as vector components, i.e., world point, event, worldline a point (x, t) that they do not follow any tensor transformation specifies both the spatial coordinates and time is called a world point or event. The law The transformation law for the ψ com- β evolution of a signal, represented by a para- ' ponents is given by =S . The metrized curve (x(t), t) is termed worldline. matrix S is determined from the covariant form of the Dirac equation under a Lorentz transformation. For matrix Λ see →lambda matrix. structure constants cijk →Lie group generator. surjective mapping tangent space consider a point P on a manifold M, for instance a point on a sphere embed- ded in ℝ3. The set of all tangent vectors at P of all curves through P forms a vectorspace over ℝ, denoted by TPM as the tangent space

13 of 14 [1] Woan, G., The Cambridge Handbook of Physics Formulas, Cambridge University Press, 2000.,

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